Independent Yields of Kr and Хе Fragments in the Photofission Of

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Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1417–1420. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1475–1478. Original Russian Text Copyright c 2005 by Gangrsky, Zhemenik, Mishinsky, Penionzhkevich. NUCLEI Experiment

Independent Yields of Kr and Хе Fragments in the Photoﬁssion of 237Np and 243Am Odd Nuclei

Yu.P.Gangrsky*, V. I. Zhemenik, G. V. Mishinsky, and Yu. E. Penionzhkevich Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received August 5, 2004

Abstract—Results are presented that were obtained by measuring the independent yields of Kr (A = 89−93)andХе(A = 135−142) appearing as fragments in the photofission of 237Np and 243Am odd nuclei. The respective experiments were performed in a beam of bremsstrahlung photons from electrons accelerated to an energy of 25 MeV at the microtron of the Laboratory of Nuclear Reactions at the Joint Institute for Nuclear Research (JINR, Dubna). Use was made of the procedure involving the transportation of fragments emitted from the target by a gas flow along a capillary and the condensation of inert gases in a cryostat at liquid-nitrogen temperature. The identification of Kr and Хе appearing as fragments was performed by the gamma spectra of their daughter products. The mass-number distributions of the independent yields of Kr and Хе isotopes were obtained, along with those for the complementary fragments (Y and La in the fission of 237Np and Nb and Pr in the fission of 243Am). c 2005 Pleiades Publishing, Inc.

INTRODUCTION Хе inert gases—originating from the photofission of 237 243 Measurement of fragment yields and of their de- Np and Am odd nuclei and is a continuation 232 238 pendences on various features of fissile nuclei and of similar experiments performed with Th, U, fragments formed (nucleonic composition, excitation and 244Pu even–even nuclei and reported in [3, 4]. energy, angular momentum) is one of the lines of in- An analysis of the measured fragment-mass-number vestigation into the mechanism of the nuclear-fission dependences of the yields with allowance for known process. Measurement of the yields of primary frag- similar data on even nuclei would make it possible to ments (independent yields)—that is, those fragments explore the effect of an odd particle in a fissile nucleus that were formed upon the scission of a fissile nucleus on the formation of fragments. There are virtually and neutron emission, but which have not yet under- no data on the independent yields of fragments in gone beta decay—is of particular interest. Such mea- the photofission of nuclei featuring an odd number of surements furnish important information about the protons or neutrons (we can only indicate the study formation of the nucleonic composition of fragments reported in [5]). Moreover, the use of known data on in the process of their transition from the saddle to the the number of prompt fission neutrons would make scission point. it possible to deduce additional information about the However, data on this process are obviously in- yields of the complementary fragments as well—Nb sufficient and refer predominantly to the neutron- and Pr in the photofission of 243Am and Y and La in induced fission of U and Pu nuclei and to the spon- the fission of 237Np. taneous fission of 252Cf [1, 2]. It is of interest to extend the range of such investigations by including in them fission reactions caused by other bombarding EXPERIMENTAL PROCEDURE particles—for example, photons. Special features of photofission reactions (a fixed angular momentum In the present experiments, we employed the same that is introduced in a fissile nucleus by a photon and procedure as in the studies of our group that were the absence of Coulomb energy and binding energy) devoted to the photofission of even–even nuclei and make it possible to obtain new information about the which were reported in [3, 4]. Within this procedure, effect of external conditions on the formation of the one implements an efficient separation of Kr and Хе nucleonic composition of fragments. isotopes from other fission fragments by using the fact The present study is devoted to measuring the that the chemical properties of the former are strongly independent yields of fragments—isotopes of Kr and different from the chemical properties of the latter. Krypton and xenon nuclei originating as fission frag- *e-mail: [email protected] ments from a target irradiated with bremsstrahlung

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Table 1. Independent yields of Kr and Хе fragments after the scission of a fissile nucleus and neutron emission, but which did not have time to undergo beta 237Np (γ, f) 243Am (γ, f) decay. The measurements were performed in a room Frag- that was protected from microtron radiation—that is, ment −1 −1 Yrel, % Yabs, fiss. Yrel, % Yabs, fiss. under conditions of a low gamma-ray and neutron 89Kr 65(8) 0.0019(2) 81(3) 0.019(2) background. 91Kr 100 0.025(3)∗ 100 0.023(2)∗ For targets subjected to irradiation, we used 50-µg/cm2-thick layers made from Np and Am 92 Kr 44(7) 0.011(1) 42(2) 0.0095(9) oxides and deposited onto an aluminum substrate 93Kr 18(4) 0.0045(5) 20 µm thick. At these layer and substrate thick- 135 nesses, half of the fragments formed in a target were Xe 61(3) 0.023(3) 42(5) 0.015(2) emitted from it and were moderated in the gas. 137 ∗ ∗ Xe 100 0.038(4) 100 0.035(4) In order to measure the gamma spectra in ques- 138Xe 80(10) 0.030(3) 76(8) 0.027(3) tion,weemployedanHpGedetectorofvolume 100 cm3 and resolution 2.1 keV for the 1332-keV 139 31(4) 0.012(1) 35(3) 0.012(1) Xe gamma line of 60Со. The resulting spectra were saved 140Xe 8.0(9) 0.0030(3) 9.8(6) 0.0034(3) in the memory of a PC for a subsequent analysis with 141Xe 3.0(3) 0.0011(1) 6.0(6) 0.0021(2) theaidoftheAKTIVcode[7].Theyieldsofidentified Kr and Хе fragments were determined from the areas 142Xe 2.0(3) 0.00076(8) 3.4(7) 0.0012(1) of their gamma lines (or the gamma lines of their ∗ Estimated yield values. daughter products) in the spectra with allowance for the detection efficiency, the intensity per decay event, the irradiation time, the time of transportation along photons were moderated in a gas and were trans- the capillary, and the time of the measurements. ported by its flow along a Teflon capillary to a cryostat, The experiments were performed at an accelerated- where they were condensed on the walls of a 1-m- electron energy of 25 MeV,the corresponding average long copper pipe coiled into a spiral and placed in excitation energy of a fissile nucleus being 13 MeV (it a Dewar flask that was filled with liquid nitrogen. was determined from the shape of the bremsstrahlung All other fragments stopped in the gas were blocked spectrum [8] and the excitation function for the by a filter at the inlet of the capillary. Thus, only Kr photofission of 237Np and 243Am nuclei under the and Хе isotopes, whose half-lives were longer than assumption that this function is similar to the well- 0.2 s (the time of fragment transportation along the known dependence that was determined in [9] for capillary was about 0.5 s), and the products of their 238U). beta decay reached the copper pipe. For the carrier gas, we employed pure helium at a pressure of 2.5 atm in the chamber. This gas was characterized by the RESULTS OF THE MEASUREMENTS highest velocity of fragment transportation and was not activated by neutrons or bremsstrahlung from the By analyzing the measured gamma spectra, we microtron used. were able to identify four krypton isotopes and seven xenon isotopes. The ratios of the independent yields In identifying Kr and Хе isotopes and in determin- 91 137 ing their yields, we relied on the spectra of gamma of these isotopes to the yields of Kr and Хе, radiation that they emitted (for data on these spectra, respectively, are given in Table 1. In order to deter- see [6]). The identification of short-lived Kr and Хе mine these yields per fission event, we measured the isotopes (in the case of half-lives shorter than 1 min) cumulative yields of fragments characterized by the was based on the gamma spectra of their daughter mass numbers of A =91and A = 137 and estimated the fractions of Kr and Хе (fractional yields) in these products (Rb and Sr for Kr and Ва and La for Хе). isobars. The cumulative yields were obtained from These daughter products, which were formed directly 91 in the scission of a fissile nucleus, were blocked by the intensities of the gamma lines of the Sr and 137 the filter at the inlet of the capillary. Therefore, they Cs isotopes, which are at the end of the beta-decay did not contribute to the measured yields of Kr and chains for these A values. The results proved to be Хе fragments. Also, the filter blocked Se, Br, Te, and 4.2(2)% and 6.3(3)% of the number of fission events. I fragments, whose beta decay could produce Kr and The fractional yields of 91Kr and 137Хеfragments Хе isotopes being studied. Thus, one can see that, in were estimated by using the systematics of yields in order to measure gamma spectra, we separated only the neutron-induced fission of U and Pu nuclei [1] those Kr and Хе fragments that emerged immediately and were found to be 0.60(6) for 237Np and 0.55(6)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 INDEPENDENT YIELDS OF Kr AND Хе FRAGMENTS 1419

243 for Am. These values make it possible to deter- Y, fiss.–1 Xe mine the independent yields per fission event for 91Kr Kr and 137Хе and, from them, for all other Kr and Хе fragments. The yield values obtained in this way are 10–2 also given in Table 1, while their dependences on the fragment mass number are displayed in the figure. These dependences (isotope distributions of fragments) are usually approximated by the Gaussian 10–3 distribution (A − A¯)2 Y (A)=Kexp − , (1) 2σ2 ×10 ×10 10–4 where A¯ and σ are, respectively, the mean mass number and the variance of the distribution, while K is 88 90 92 134~ ~ 136 138 140 142 a normalization factor. From the figure, it can be A seen that, in just the same way as in the photofission of even–even nuclei, the measured distributions are Mass-number (isotope) distributions of the independent well described by expression (1). However, an excess yields of Kr and Хе fragments in the reactions (closed of the measured yield values above their calculated circles) 237Np(γ, f)and(opencircles)243Am(γ, f). counterparts is observed for neutron-rich xenon frag- The points represent experimental data, while the dashed 141 142 curves correspond to the calculation by formula (1) whose ments ( Хеand Хе). This may be associated with parameters are set to the values in Table 2. the nuclear structure of these fragments.

The fitted values of the parameters of isotope dis- 237 243 tributions are given in Table 2. A comparison with the energies of the Np and Am nuclei undergoing known distributions for even–even nuclei shows that, fission, one can evaluate ν. The result is ν =4for 237 243 both for 237Np and for 243Am, the values of A¯ are well Np and ν =46for Am. These values of ν make ¯ below those for 232Th, 238U, and 244Pu [3], but that it possible to determine the mean mass numbers A of they are close to the value of A¯ for 235U fission induced Y, Nb, La, and Pr fragments. The results are given in by thermal neutrons [2]. This relationship between Table 2. the A¯ values reflects their dependence on the neutron The data obtained in the present study for the 237 243 excess in a fissile nucleus. The ratio N/A is 1.587 for photofission of Np and Am nuclei supplement 238U (similar values were found previously for 232Th the systematics of independent yields of fission frag- and 244Pu), 1.549 for 237Np, and 1.558 for 243Am. The ments. They give sufficient grounds to conclude that an odd particle in a fissile nucleus does not change σ 237 243 values of for Np and Am correspond to the significantly the isotope distribution of fragments in systematics of variances of isotope distributions in the relation to what was observed for even–even nuclei neutron-induced fission and photofission of nuclei. and that the distributions themselves are close to From the results obtained here for the isotope dis- those that were obtained in the fast-neutron-induced tributions of Kr and Хе, one can derive the analogous fission of the corresponding nuclei. distributions for the complementary fragments. For Kr fragments, these are La isotopes in 237Np fission Table 2. and Pr isotopes in 243Am fission. For Хе fragments, Parameters of the isotope distributions of photofission fragments these are, respectively, Y and Nb isotopes. For fragments characterized by identical independent yields, 237Np(γ, f) 243Am(γ, f) we have Element A¯ σ A¯ σ Al + Ah + ν = A0, (2) Kr 903(2) 1.2(1) 90.1(3) 1.4(2) where Al, Ah,andA0 are the mass numbers of, respectively, a light fragment, a heavy fragment, and the Y 96.3(3) 1.3(1) fissile nucleus being considered, while ν is the total Nb 101.5(2) 1.5(1) number of neutrons emitted from the two fragments. By employing the known numbers of neutrons Xe 136.7(2) 1.3(1) 136.9(1) 1.5(1) emitted in the neutron-induced fission of nuclei in La 142.7(3) 1.3(1) the reactions 236Np(n, f)and242Am(n, f) [10, 11] and taking into account corrections for the excitation Pr 148.3(5) 1.4(2)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1420 GANGRSKY et al. ACKNOWLEDGMENTS 5. D. De Frenne, H. Thierens, B. Proot, et al.,Phys. Rev. C 26, 1356 (1982). We are grateful to Yu.Ts. Oganessian and M.G. Itkis for their support of this study, A.G. Belov 6. E. Broune and R. B. Firenstone, Table of Radioac- tive Isotopes (Wiley, New York, 1986). for his assistance in irradiating the samples at the microtron, and G.V. Buklanov for manufacturing the 7. V. E. Zlokazov, Comput. Phys. Commun. 28,27 targets. (1982). 8. Ph. G. Kondev, A. P. Tonchev, Kh. G. Khristov, and V.E.Zhuchko,Nucl.Instrum.MethodsPhys.Res.B REFERENCES 71, 126 (1992). 1. A. C. Wahl, At. Data Nucl. Data Tables 39, 1 (1988). 9. J. T. Caldwell, E. L. Dowly, B. L. Berman, et al.,Phys. 2. T. R. England and B. E. Rides, Preprint No. La-Ur- Rev. C 21, 1215 (1980). 94-3106, LANL (Los Alamos National Laboratory, 10. F.Manero and V.A. Konshin, At. Energy Rev. 10, 657 Los Alamos, 1994). (1972). 3. Yu. P. Gangrsky, V. I. Zhemenik, N. Yu. Maslova, 11.Yu.P.Gangrsky,B.Dalkhsuren,andB.N.Markov, et al.,Yad.Fiz.66, 1251 (2003) [Phys. At. Nucl. 66, Nuclear-Fission Fragments (Energoatomizdat,´ 1211 (2003)]. Moscow, 1986) [in Russian]. 4.Yu.P.Gangrsky,V.I.Zhemenik,S.G.Zemlyanoı˘ , et al., Izv. Akad. Nauk, Ser. Fiz. 67, 1475 (2003). Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1421–1432. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1479–1490. Original Russian Text Copyright c 2005 by Kadmensky, Rodionova. NUCLEI Theory

Subthreshold Photoﬁssion of Even–Even Nuclei S. G. Kadmensky* andL.V.Rodionova Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia Received February 14, 2005

Abstract—Within quantum-mechanical fission theory, the angular distributions of fragments originating from the subthreshold photofission of the even–even nuclei 232Th, 234U, 236U, 238U, 238Pu, 240Pu, and 242Pu are analyzed for photon energies below 7 MeV. Special features of various fission channels are assessed under the assumption that the fission barrier has a two-humped shape. It is shown that the maximum value of the relative orbital angular momentum Lm of fission fragments can be found upon taking into account deviations from the predictions of A. Bohr’s formula for the angular distributions of fission fragments. The result is Lm ≈ 30.Theexistenceofan“isomeric shelf” for the angular distributions of fragments from 236Uand238Uphotofission in the low-energy region is confirmed. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION of the qualitative physical assumption that fission fragments are emitted along or against the direc- The discovery [1] of a sizable anisotropy in the tion of the fissile-nucleus-symmetry axis. Within the angular distributions of fragments originating from 232 238 quantum-mechanical theory of binary nuclear fis- the subthreshold photofission of Thand Unu- sion [9], it is shown that this assumption is at odds clei served as a basis for introducing the concept with the quantum-mechanical uncertainty relation of A. Bohr’s transition fission states [2, 3] (fission between the orbital angular momentum and the par- channels) that are formed at the saddle points of the ticle emission angle. Therefore, A. Bohr’s formula deformation potential and which are associated with can only be valid approximately under the condition “cold” intrinsic states of a fissile nucleus. The analysis that, in the asymptotic region of a fissile nucleus, of the angular distributions of photofission fragments where primary fission fragments have already been that was performed in the review article of Ostapenko formed, there arise rather high orbital angular mo- et al. [4] on the basis of a vast body of experimental menta of these fragments, L Lm. A virtually uni- information and under the assumption [5] that the versal mechanism of pumping of the orbital angular deformation potential of a fissile nucleus has a two- momenta and spins of fission fragments was justified humped shape confirmed that transition fission states in [10]. This mechanism is due to a strong non- associated both with the inner and with the outer sphericity of the fragment-interaction potential in the fission barrier play a decisive role in the formation of vicinity of the scission point of a fissile nucleus and these distributions. leads to a value of Lm ≈ 30 for the maximum rela- For the angular distributions of fragments from tive orbital angular momentum of fission fragments. 236 238 Uand Uphotofission in the low-energy region, So large a value of Lm is consistent with the limit- an interesting phenomenon was discovered in [6, 7] ing estimate obtained in [10] for the maximum value that was associated with the emergence of an “iso- of the total spin of primary fragments, F = J1 + J2, meric shelf,” which was determined by the radiative on the basis of experimental data on the multiplic- population of shape-isomer states [8] in the second ity of prompt neutrons and on the multiplicity and well of the deformation potential of a fissile nucleus. multipolarity of prompt photons emitted by primary fragments, the condition that the maximum spins of We note that the results obtained previously from || an analysis of angular distributions of photofission these fragments are parallel, (J1)max (J2)max,being fragments—for example, in [4]—were based on the satisfied. use of A. Bohr’s [2, 3] formula for the angular dis- On the basis of quantum-mechanical fission the- tributions of fragments originating from the fission ory [9], we analyzed in [11] the angular distribu- channel where an axisymmetric fissile nucleus has tion of fragments from 238Uphotofission induced by aspinJ whose projection onto the symmetry axis bremsstrahlung photons of endpoint energy Eγ = of this nucleus is K. This formula is a realization 5.2 MeV, at which the experimental parameters of this distribution were determined in [12] to a fairly *e-mail: [email protected] high degree of precision. For the maximum relative

1063-7788/05/6809-1421$26.00 c 2005 Pleiades Publishing, Inc. 1422 KADMENSKY, RODIONOVA orbital angular momentum in question, we found the М1 photons. At the same time, one can disregard + range 25 PHOTOFISSION OF NUCLEI first through the stage of formation of a compound We will perform a further analysis for the sub- nucleus in the first well and which, thereupon, is not threshold photofission of axisymmetric deformed associated with the mechanism of delayed fission (it is even–even actinide nuclei, assuming that the fission induced by the radiative population of shape-isomer barrier has a two-humped shape [4]. At a rather states [8] in the second well [6, 7]) can be represented low endpoint energy of the photons (Eγ < 7 MeV), in the form [3] the cross sections for photofission that leads to dσ (θ) σ (θ) ≡ γf = P JKT J (θ), (1) the formation of compound-nucleus states in the γf dΩ MK first well of the deformation potential decrease with J,K0 increasing photon multipolarity J according to the ΓJK long-wavelength law; that is, JK f EJ P = (2 − δK,0)σ , (2) EJ ≈ 2(J−1) E1 MJ ≈ 2J E1 ΓJ γ σγ (kγ RA) σγ ; σγ (kγ RA) σγ , where the factor (2 − δK,0) takes into account a dou- where kγ is the photon wave vector and RA is the 2 ble degeneracy of K>0 states, the photofission cross radius of the nucleus involved, the quantity (kγ RA) EJ section σγ (Eγ ) involves only E1andE2photons, taking values in the interval 0.03–0.05 for actinide ≡ nuclei [4]. Therefore, one can disregard the contribu- and the solid angle Ω (θ,ϕ) determines the directions to the photofission cross sections from electric tion of fission-fragment emission in the laboratory J photons EJ for J>2 and from magnetic photons frame. In (2), Γ is the total decay width of the com- MJ for J>1. Subthreshold photofission via the ab- pound state in the first well with allowance for all π + JK sorption of M1 photons is determined by J =1 decay channels, including radiative ones, and Γf is transition fission states at the first and the second the partial fission width of this compound state, the fission barrier, where J is the spin of these states and latter being determined by the projection K of the spin π is their parity. For axisymmetric even–even nuclei, J onto the symmetry axis of a fissile nucleus in the the projection K of the spin J onto the nuclear- external region of this nucleus, where K is already an symmetry axis in Jπ =1+ low-lying collective states integral of the motion. It is assumed that this region is K =1. In the region of actinide nuclei, the JπK = coincides with the region where there occurs the tran- 1+1 collective state in the first well has a high ex- sition from the outer saddle point of the deformation citation energy (about 1.5 MeV) with respect to the potential through the scission point to the space of ground state, whose quantum numbers are JπK = already formed primary fission fragments, the fissile 0+0. Therefore, one can expect that 1+1 transition nucleus retaining its axisymmetric shape through- JK fission states formed at the first and the second fission out this transition. The quantity Γf depends in a barrier will also have high excitation energies with complicated way on the properties of transition fission respect to 0+0 and 2+0 transition fission states and, states at the first and the second fission barrier, the in the case of subthreshold fission, very low penetra- properties of collective deformation modes of motion tion factors in relation to the penetration factors for of the nucleus, and the character of the coupling of 2+0 transition fission states initiated in photofission these modes to the compound states of the nucleus in by quadrupole electric photons. For the subthreshold the first and the second well. Here, it is assumed that, photofission of even–even nuclei, one can therefore in the compound states of the first well, all projections eliminate effects associated with the absorption of of the spin J onto the symmetry axis of the nucleus

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1423 undergo a uniform statistical mixing owing to the where the quantities a , b ,andc are defined as 0 0 0 dynamical enhancement of the Coriolis interaction JK JK effect [13]. A similar idea is also often used [3, 4] for a0 = αJKP ,b0 = βJKP , (7) compound states in the second well, but it has yet to JK JK JK be rigorously justified. c0 = γJKP . The normalized (to unity) angular distribution JK T JK(θ) of the fragments of photofission in the JK The values of the coefficients αJK, βJK,andγJK for exit fission channel is independent of the spin projec- →∞ ± the cases of Lm (A. Bohr’s limit), Lm =30,and tion M = 1 and is usually calculated by A. Bohr’s L =20are given in Table 1. formula [3] m In analyzing the experimental angular distribu- JK 2J +1 | J |2 | J |2 tions of photofission fragments, one usually deter- T (θ)= [ D1K (ω) + D1−K (ω) ]β=θ, ◦ 8π mines the asymmetry Wγf(θ) ≡ σγf(θ)/σγf(90 ) of (3) these distributions with respect to the angle θ =90◦. where DJ (ω) is a generalized spherical harmonic This asymmetry can be represented by formula (6) MK upon the replacement of the quantities a , b ,andc that depends on the Euler angles ω ≡ (α, β, γ) deter- 0 0 0 by the quantities mining the orientation of the symmetry axes of the a b c fissile nucleus with respect to the axes in the lab- a = 0 ,b= 0 ,c= 0 , (8) oratory frame. Within quantum-mechanical fission a0 + b0 a0 + b0 a0 + b0 JK theory [9–11], the quantity T (θ) takesadifferent respectively; obviously, we have a + b =1. form, JK 2J +1 | J |2 T (θ)= dω[ D1K (ω) (4) 3. ANALYSIS OF THE ANGULAR 16π2 DISTRIBUTIONS OF FRAGMENTS | J |2 ORIGINATING FROM THE SUBTHRESHOLD + D1−K (ω) ]F (Lm,θ). PHOTOFISSION OF NUCLEI Here, F (L ,θ) is the normalized (to unity) angular m Let us investigate the photofission of 232Th, 234U, distribution of photofission fragments in the intrinsic 236 238 238 240 242 coordinate system of the fissile nucleus, U, U, Pu, Pu, and Pu even–even nuclei that is induced by photons of energy in the range F (Lm,θ)=b(Lm) (5) E ≤ 7 MeV. We select only those results of the ex- γ 2 perimental studies reported in [12, 14–18] for which Lm − L 1+ππ1π2( 1) relative root-mean-square deviations of the measured × YL0(ξ )YL0(1) , 2 quantities a and c from a and c are less than 0.5. L=0 For the ensuing analysis, we employ the quanti- −1 JK JK 10 Lm − L 2 ties G = P /P , which possess the following (2L +1) 1+ππ1π2( 1) JK 10 11 11 10 b(Lm)= , properties: G > 0, G =1, G =Γ /Γ ,and 4π 2 f f L=0 G22

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1424 KADMENSKY, RODIONOVA

Table 1. Values of the coeﬃcients αJK, βJK,andγJK for Lm →∞, Lm =30,andLm =20

Lm α10 α11 α20 α22 β10 β11 β20 β22 γ20 γ22 ∞ 0 0.75 0 0 0.75 −0.375 0 0.625 0.9375 0.15625 30 0.024 0.74 0.060 0.030 0.714 −0.357 −0.022 0.592 0.854 0.142 20 0.04 0.73 0.087 0.044 0.697 −0.349 −0.033 0.576 0.815 0.136

11 20 − c(α11 + β11)(α22 − a(α22 + β22))], For all cases studied here, the G and G values calculated with allowance for the experimental d = c[(α11 + β11)α20 − α11(α20 + β20)] (11) errors in determining a and c appear to be positive definite in A. Bohr’s limit (Lm →∞). As to finite − a(α11 + β11)γ20 + α11γ20. values of Lm, the situation for them proves to be By substituting, into these formulas, the experimental more complicated, as might have been expected. One values of the quantities a and c, one can calculate G11 can single out cases where the values of G11 and 20 22 20 and G versus the quantities Lm and G ,whichare G for Lm =30and Lm =20provetobepositive considered as parameters. Since the quantities G11 definite and rather close to the values of G11 and 20 and G20 are positive by definition, a comparison of the G , respectively, in A. Bohr’s limit. These cases calculated values of G11 and G20 with the conditions arise at rather large values of experimentally deter- G11 > 0 and G20 > 0 enables one to determine the mined a and c and correspond to the situation where A. Bohr’s formula (3) for the angular distributions of possible minimal values of Lm. fragments is highly accurate. Table 2 gives the G11 It was indicated above that, by using experimental and G20 values calculated for cases where A. Bohr’s data on the multiplicity and multipolarity of prompt formula (3) is no longer correct, so that there arise photons emitted by primary fission fragments, the significant distinctions between the absolute values maximum value of the relative orbital angular mo- G11 G20 L L mentum of fission fragments was estimated in [10] of and for m =30and m =20and their ≈ counterparts in A. Bohr’s limit. From Table 2, one as Lm 30. In order to test this estimate, we will 11 20 can see that, at Lm =30, the values of G and G consider the cases of Lm =30, Lm =20,andLm → ∞ (A. Bohr’s limit) in our ensuing calculations. provetobepositivedefinite in the majority of cases studied here, but the current experimental accuracy in 22 20 The ratio G /G is equal to the fission-width determining a and c is insufficient for the root-mean- 22 20 11 ratio Γf /Γf , which is much less than unity if the square deviations of all G and, in some cases (for axisymmetric shape the nucleus is conserved in the example, for 232Th), of G20 from their mean values to process of subthreshold fission. This is explained by be much less than unity. At the same time, it turns out thefactthatthe2+2 transition fission state, which that, in eight of the 78 cases presented in Table 2 (at 232 is associated with the γ vibrations of a nucleus, has energies in the range 5.9 ≤ Eγ ≤ 6.4 MeV in Th, a rather high excitation energy (0.7 MeV [19]) with at energies in the range 5.75 ≤ E ≤ 5.97 in 234U, + γ respect to the 2 0 transition fission state. Therefore, 236 22 22 and at an energy of Eγ =5.75 MeV in U), positive the quantity G was varied in the interval 0 ≤ G ≤ 11 20 values of G can only be obtained by shifting the ex- 0.2G , where the upper boundary was chosen to perimental values of a toward greater values by more 22 exceed, by a good margin, the possible values of G . than two standard deviations in determining a.We For all cases investigated here, these variations of the note in passing that, in some cases, the most recent quantity G22 lead to changes of not more than 25% measurements of a lead to positive definite values in the absolute values of G11 and G20 calculated by of G11, in contrast to measurements of a that were formulas (9)–(11). Therefore, one can disregard, with performed previously. By way of example, we indicate a high degree of precision, the effect of G22 on G11 that such a situation arises for 232Thphotofission 20 22 and G and employ a value of zero for G . This at Eγ =6.4 MeV, in which case the results of the conclusion, along with the above conclusion that the more recent measurements of a that were performed effect of the 2+1 fission channel is negligible, actually in [15] differ significantly from those presented earlier validates the procedure extensively used earlier [4] in [12]. Therefore, the above situation where there that describes the angular distributions of fragments appear negative values of G11 is likely to stem from from the subthreshold photofission of nuclei without an insufficient experimental accuracy in determining taking into account the 2+1 and 2+2 fission channels. a in the region of its smallest values (a≤0.014

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1425

11 20 Table 2. Values of G and G for Lm →∞, Lm =30,andLm =20

L =30 L =20 L = ∞ Eγ , c a m m m MeV G11 G20 G11 G20 G11 G20 238Pu 5.25 [14] 1.412 ± 0.139 0.408 ± 0.103 0.368 ± 0.169 1.540 ± 0.049 0.284 ± 0.173 1.603 ± 0.050 0.523 ± 0.163 1.416 ± 0.048 5.5 [14] 1.513 ± 0.112 0.330 ± 0.063 0.240 ± 0.096 1.571 ± 0.056 0.153 ± 0.099 1.635 ± 0.057 0.399 ± 0.090 1.448 ± 0.053 5.75 [14] 0.654 ± 0.055 0.414 ± 0.037 0.431 ± 0.062 0.711 ± 0.043 0.381 ± 0.065 0.738 ± 0.044 0.523 ± 0.059 0.659 ± 0.04 6.0 [14] 0.370 ± 0.018 0.526 ± 0.011 0.644 ± 0.022 0.434 ± 0.018 0.607 ± 0.023 0.450 ± 0.018 0.714 ± 0.020 0.402 ± 0.017 240Pu 5.45 [14] 1.147 ± 0.07 0.102 ± 0.044 −0.009 ± 0.052 1.038 ± 0.039 −0.075 ± 0.054 1.077 ± 0.04 0.109 ± 0.049 0.966 ± 0.037 5.7 [14] 0.710 ± 0.052 0.222 ± 0.034 0.160 ± 0.046 0.685 ± 0.037 0.11 ± 0.047 0.711 ± 0.038 0.251 ± 0.043 0.638 ± 0.035 5.95 [14] 0.331 ± 0.013 0.533 ± 0.010 0.660 ± 0.020 0.390 ± 0.013 0.625 ± 0.02 0.405 ± 0.013 0.727 ± 0.019 0.361 ± 0.012 242Pu 5.0 [14] 3.702 ± 0.424 0.532 ± 0.308 0.485 ± 0.593 4.646 ± 0.530 0.275 ± 0.583 4.892 ± 0.570 0.850 ± 0.598 4.119 ± 0.402 5.25 [14] 0.965 ± 0.032 0.448 ± 0.053 0.461 ± 0.089 1.078 ± 0.003 0.396 ± 0.090 1.122 ± 0.003 0.580 ± 0.088 0.995 ± 0.001 5.35 [14] 1.018 ± 0.069 0.418 ± 0.046 0.408 ± 0.078 1.114 ± 0.042 0.341 ± 0.080 1.159 ± 0.044 0.531 ± 0.074 1.028 ± 0.040 5.5 [14] 0.734 ± 0.034 0.310 ± 0.022 0.272 ± 0.033 0.748 ± 0.025 0.219 ± 0.034 0.776 ± 0.026 0.367 ± 0.031 0.695 ± 0.023 5.75 [14] 0.422 ± 0.012 0.488 ± 0.008 0.572 ± 0.015 0.482 ± 0.011 0.532 ± 0.015 0.500 ± 0.011 0.646 ± 0.014 0.447 ± 0.010 232Th 5.90 [14] 0.084 ± 0.014 0.010 ± 0.005 −0.028 ± 0.006 0.072 ± 0.012 −0.054 ± 0.006 0.074 ± 0.012 0.01 ± 0.005 0.068 ± 0.011 5.95 [14] 0.074 ± 0.010 0.014 ± 0.004 −0.024 ± 0.005 0.063 ± 0.008 −0.049 ± 0.005 0.066 ± 0.009 0.014 ± 0.004 0.06 ± 0.008 6.2 [14] 0.079 ± 0.010 0.012 ± 0.003 −0.026 ± 0.004 0.068 ± 0.008 −0.051 ± 0.004 0.07 ± 0.009 0.012 ± 0.003 0.064 ± 0.008 6.4 [12] 0.028 ± 0.006 0.021 ± 0.003 −0.014 ± 0.003 0.024 ± 0.005 −0.037 ± 0.004 0.025 ± 0.005 0.021 ± 0.003 0.023 ± 0.005 6.4 [15] 0.060 ± 0.040 0.060 ± 0.030 0.025 ± 0.034 0.052 ± 0.034 0 ± 0.036 0.054 ± 0.035 0.062 ± 0.032 0.049 ± 0.032 6.4 [15] 0.028 ± 0.006 0.044 ± 0.002 0.01 ± 0.002 0.024 ± 0.005 −0.014 ± 0.003 0.025 ± 0.005 0.045 ± 0.002 0.023 ± 0.005 6.5 [14] 0.022 ± 0.014 0.022 ± 0.005 −0.012 ± 0.006 0.019 ± 0.012 −0.035 ± 0.006 0.02 ± 0.012 0.022 ± 0.005 0.018 ± 0.011 6.7 [14] 0.009 ± 0.008 0.023 ± 0.002 −0.01 ± 0.003 0.008 ± 0.007 −0.033 ± 0.003 0.008 ± 0.007 0.023 ± 0.002 0.007 ± 0.006 6.9 [14] 0.020 ± 0.022 0.032 ± 0.007 −0.002 ± 0.009 0.017 ± 0.019 −0.025 ± 0.009 0.018 ± 0.02 0.033 ± 0.007 0.016 ± 0.018 7.0 [14] 0.038 ± 0.012 0.036 ± 0.004 0.001 ± 0.005 0.033 ± 0.01 −0.023 ± 0.005 0.034 ± 0.011 0.037 ± 0.004 0.031 ± 0.01 234U 5.6 [18] 0.195 ± 0.018 0.032 ± 0.008 −0.014 ± 0.009 0.169 ± 0.015 −0.044 ± 0.01 0.175 ± 0.015 0.033 ± 0.008 0.158 ± 0.014 5.75 [16] 0.152 ± 0.012 0.025 ± 0.004 −0.018 ± 0.005 0.131 ± 0.01 −0.046 ± 0.005 0.136 ± 0.01 0.025 ± 0.004 0.123 ± 0.009 5.97 [16] 0.158 ± 0.012 0.028 ± 0.004 −0.016 ± 0.005 0.136 ± 0.01 −0.044 ± 0.005 0.141 ± 0.01 0.028 ± 0.004 0.128 ± 0.009 6.05 [16] 0.110 ± 0.014 0.041 ± 0.005 0.001 ± 0.006 0.096 ± 0.012 −0.025 ± 0.007 0.099 ± 0.012 0.042 ± 0.005 0.09 ± 0.011 6.17 [16] 0.158 ± 0.012 0.050 ± 0.004 0.007 ± 0.005 0.138 ± 0.01 −0.021 ± 0.005 0.143 ± 0.011 0.051 ± 0.004 0.13 ± 0.01 6.2 [16] 0.183 ± 0.017 0.086 ± 0.007 0.043 ± 0.009 0.163 ± 0.015 0.014 ± 0.009 0.169 ± 0.015 0.09 ± 0.008 0.153 ± 0.014 6.3 [16] 0.089 ± 0.018 0.084 ± 0.008 0.047 ± 0.01 0.079 ± 0.016 0.022 ± 0.011 0.082 ± 0.016 0.088 ± 0.009 0.074 ± 0.015 6.35 [16] 0.083 ± 0.015 0.133 ± 0.006 0.102 ± 0.008 0.076 ± 0.013 0.076 ± 0.008 0.078 ± 0.014 0.142 ± 0.007 0.071 ± 0.013 6.4 [16] 0.020 ± 0.016 0.133 ± 0.008 0.106 ± 0.01 0.018 ± 0.015 0.083 ± 0.011 0.019 ± 0.015 0.143 ± 0.009 0.017 ± 0.014 6.45 [16] 0.087 ± 0.013 0.163 ± 0.005 0.136 ± 0.007 0.081 ± 0.012 0.11 ± 0.007 0.084 ± 0.012 0.177 ± 0.006 0.076 ± 0.011 6.5 [16] 0.046 ± 0.019 0.157 ± 0.009 0.132 ± 0.012 0.042 ± 0.017 0.108 ± 0.013 0.044 ± 0.018 0.17 ± 0.011 0.04 ± 0.016 6.6 [16] 0.044 ± 0.018 0.185 ± 0.009 0.165 ± 0.012 0.041 ± 0.017 0.141 ± 0.013 0.043 ± 0.017 0.204 ± 0.011 0.039 ± 0.016 6.7 [16] 0.042 ± 0.018 0.238 ± 0.009 0.231 ± 0.013 0.041 ± 0.017 0.207 ± 0.014 0.042 ± 0.018 0.27 ± 0.012 0.038 ± 0.016

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1426 KADMENSKY, RODIONOVA

Table 2. (Contd.)

Lm =30 Lm =20 Lm = ∞ Eγ , c a MeV G11 G20 G11 G20 G11 G20 236U 4.3 [15] 1.71 ± 0.35 0.72 ± 0.25 1.064 ± 0.665 2.369 ± 0.004 0.957 ± 0.677 2.481 ± 0.012 1.249 ± 0.625 2.133 ± 0.021 5.0 [15] 0.250 ± 0.15 0.200 ± 0.10 0.175 ± 0.133 0.23 ± 0.129 0.143 ± 0.139 0.238 ± 0.134 0.229 ± 0.124 0.215 ± 0.121 5.1 [15] 0.187 ± 0.048 0.04 ± 0.02 −0.005 ± 0.024 0.162 ± 0.040 −0.034 ± 0.0254 0.167 ± 0.042 0.041 ± 0.021 0.152 ± 0.028 5.25 [15] 0.12 ± 0.04 0.06 ± 0.03 0.020 ± 0.034 0.105 ± 0.034 −0.006 ± 0.036 0.105 ± 0.035 0.062 ± 0.032 0.099 ± 0.032 5.5 [15] 0.479 ± 0.016 0.048 ± 0.006 −0.018 ± 0.007 0.419 ± 0.013 −0.058 ± 0.008 0.434 ± 0.013 0.049 ± 0.006 0.393 ± 0.012 5.5 [15] 0.500 ± 0.13 0.150 ± 0.07 0.094 ± 0.089 0.459 ± 0.103 0.052 ± 0.094 0.475 ± 0.106 0.165 ± 0.082 0.429 ± 0.096 5.6 [15] 0.503 ± 0.028 0.040 ± 0.012 −0.027 ± 0.014 0.438 ± 0.022 −0.069 ± 0.015 0.454 ± 0.023 0.041 ± 0.013 0.411 ± 0.020 5.75 [15] 0.452 ± 0.015 0.027 ± 0.005 −0.037 ± 0.006 0.391 ± 0.012 −0.076 ± 0.007 0.405 ± 0.012 0.027 ± 0.005 0.367 ± 0.011 6.0 [15] 0.166 ± 0.013 0.042 ± 0.005 −0.002 ± 0.006 0.144 ± 0.011 −0.03 ± 0.006 0.149 ± 0.011 0.043 ± 0.005 0.136 ± 0.01 6.25 [15] 0.110 ± 0.013 0.069 ± 0.005 0.03 ± 0.006 0.097 ± 0.011 0.004 ± 0.007 0.1 ± 0.012 0.071 ± 0.005 0.091 ± 0.011 6.5 [15] 0.080 ± 0.014 0.112 ± 0.006 0.078 ± 0.008 0.072 ± 0.012 0.053 ± 0.008 0.075 ± 0.013 0.119 ± 0.007 0.068 ± 0.012 6.75 [15] 0.051 ± 0.015 0.199 ± 0.008 0.181 ± 0.011 0.048 ± 0.014 0.157 ± 0.011 0.05 ± 0.014 0.221 ± 0.01 0.045 ± 0.013 238U 4.85 [12] 1.380 ± 0.134 0.231 ± 0.079 0.124 ± 0.107 1.346 ± 0.069 0.045 ± 0.11 1.399 ± 0.072 0.266 ± 0.101 1.245 ± 0.066 4.93 [12] 1.350 ± 0.088 0.171 ± 0.053 0.052 ± 0.067 1.272 ± 0.045 −0.023 ± 0.069 1.322 ± 0.047 0.189 ± 0.063 1.18 ± 0.043 5.0 [12] 1.139 ± 0.031 0.135 ± 0.020 0.026 ± 0.024 1.051 ± 0.017 −0.04 ± 0.025 1.091 ± 0.018 0.145 ± 0.023 0.977 ± 0.016 5.13 [12] 1.026 ± 0.054 0.162 ± 0.033 0.065 ± 0.042 0.96 ± 0.033 0.003 ± 0.043 0.996 ± 0.034 0.177 ± 0.039 0.893 ± 0.031 5.2 [14] 0.910 ± 0.080 0.100 ± 0.035 0.006 ± 0.043 0.821 ± 0.057 −0.051 ± 0.046 0.851 ± 0.059 0.106 ± 0.039 0.765 ± 0.053 5.2 [12] 0.907 ± 0.045 0.090 ± 0.024 −0.005 ± 0.029 0.814 ± 0.03 −0.062 ± 0.03 0.844 ± 0.031 0.095 ± 0.026 0.759 ± 0.028 5.4 [12] 0.306 ± 0.026 0.050 ± 0.012 −0.004 ± 0.014 0.267 ± 0.021 −0.037 ± 0.015 0.277 ± 0.022 0.051 ± 0.013 0.251 ± 0.02 5.45 [14] 0.155 ± 0.021 0.038 ± 0.009 −0.005 ± 0.011 0.134 ± 0.018 −0.033 ± 0.011 0.139 ± 0.018 0.039 ± 0.009 0.126 ± 0.017 5.5 [14] 0.039 ± 0.014 0.078 ± 0.005 0.045 ± 0.006 0.035 ± 0.012 0.021 ± 0.007 0.036 ± 0.013 0.081 ± 0.005 0.032 ± 0.012 5.5 [12] 0.161 ± 0.007 0.040 ± 0.004 −0.004 ± 0.005 0.14 ± 0.006 −0.032 ± 0.005 0.145 ± 0.006 0.041 ± 0.004 0.131 ± 0.005 5.5 [18] 0.172 ± 0.033 0.049 ± 0.013 0.005 ± 0.016 0.15 ± 0.028 −0.024 ± 0.017 0.155 ± 0.029 0.050 ± 0.014 0.141 ± 0.026 5.6 [12] 0.074 ± 0.012 0.054 ± 0.005 0.017 ± 0.006 0.065 ± 0.01 −0.008 ± 0.006 0.067 ± 0.011 0.056 ± 0.005 0.061 ± 0.01 5.65 [14] 0.040 ± 0.010 0.034 ± 0.005 −0.001 ± 0.006 0.035 ± 0.009 −0.025 ± 0.006 0.036 ± 0.009 0.035 ± 0.005 0.033 ± 0.008 5.65 [12] 0.085 ± 0.011 0.034 ± 0.006 −0.004 ± 0.007 0.074 ± 0.009 −0.03 ± 0.007 0.076 ± 0.01 0.035 ± 0.006 0.069 ± 0.009 5.7 [12] 0.041 ± 0.009 0.054 ± 0.004 0.019 ± 0.005 0.036 ± 0.008 −0.005 ± 0.005 0.037 ± 0.008 0.056 ± 0.004 0.034 ± 0.007 5.75 [18] 0.043 ± 0.014 0.041 ± 0.005 0.006 ± 0.006 0.037 ± 0.012 −0.018 ± 0.007 0.039 ± 0.012 0.042 ± 0.005 0.035 ± 0.011 5.8 [12] 0.052 ± 0.014 0.068 ± 0.007 0.033 ± 0.008 0.046 ± 0.012 0.009 ± 0.009 0.047 ± 0.013 0.07 ± 0.008 0.043 ± 0.011 5.9 [12] 0.058 ± 0.007 0.054 ± 0.004 0.018 ± 0.005 0.051 ± 0.006 −0.006 ± 0.005 0.052 ± 0.006 0.056 ± 0.004 0.048 ± 0.006 5.97 [18] 0.128 ± 0.013 0.079 ± 0.005 0.039 ± 0.006 0.113 ± 0.011 0.012 ± 0.007 0.117 ± 0.012 0.082 ± 0.005 0.107 ± 0.011 6.0 [12] 0.048 ± 0.012 0.099 ± 0.006 0.066 ± 0.007 0.043 ± 0.011 0.042 ± 0.008 0.044 ± 0.011 0.104 ± 0.007 0.04 ± 0.01 6.05 [18] 0.125 ± 0.016 0.087 ± 0.007 0.048 ± 0.009 0.111 ± 0.014 0.021 ± 0.009 0.115 ± 0.014 0.091 ± 0.008 0.105 ± 0.013 6.17 [18] 0.123 ± 0.012 0.090 ± 0.005 0.051 ± 0.006 0.11 ± 0.01 0.024 ± 0.007 0.114 ± 0.011 0.094 ± 0.005 0.103 ± 0.010 6.2 [12] 0.034 ± 0.038 0.128 ± 0.019 0.1 ± 0.024 0.031 ± 0.034 0.076 ± 0.026 0.032 ± 0.035 0.137 ± 0.022 0.029 ± 0.032 6.2 [18] 0.138 ± 0.016 0.086 ± 0.007 0.046 ± 0.009 0.123 ± 0.014 0.019 ± 0.009 0.127 ± 0.014 0.090 ± 0.008 0.115 ± 0.013 6.3 [18] 0.038 ± 0.018 0.110 ± 0.008 0.079 ± 0.01 0.034 ± 0.016 0.056 ± 0.011 0.035 ± 0.017 0.116 ± 0.009 0.032 ± 0.015 6.35 [18] 0.032 ± 0.013 0.103 ± 0.005 0.072 ± 0.006 0.029 ± 0.012 0.049 ± 0.007 0.03 ± 0.012 0.109 ± 0.006 0.066 ± 0.009 6.4 [14] 0.034 ± 0.008 0.127 ± 0.004 0.098 ± 0.005 0.031 ± 0.007 0.075 ± 0.005 0.032 ± 0.007 0.136 ± 0.005 0.029 ± 0.007 6.4 [18] 0.038 ± 0.020 0.114 ± 0.009 0.084 ± 0.011 0.034 ± 0.018 0.06 ± 0.012 0.035 ± 0.019 0.121 ± 0.010 0.032 ± 0.014 6.45 [18] 0.078 ± 0.011 0.111 ± 0.004 0.077 ± 0.005 0.07 ± 0.01 0.052 ± 0.006 0.073 ± 0.01 0.118 ± 0.004 0.066 ± 0.009 6.5 [18] 0.052 ± 0.018 0.123 ± 0.009 0.093 ± 0.011 0.047 ± 0.016 0.068 ± 0.012 0.049 ± 0.017 0.131 ± 0.010 0.044 ± 0.015 6.6 [18] 0.043 ± 0.018 0.126 ± 0.008 0.097 ± 0.01 0.039 ± 0.016 0.073 ± 0.011 0.04 ± 0.017 0.135 ± 0.009 0.037 ± 0.015

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1427

G11 G20 102 1 (‡) 10 (b)

101 100

100

10–1

10–2 10–2

4 5 6 7 4 5 6 7 Eγ, MeV

11 20 236 11 Fig. 1. Energy dependence of (a) G and (b) G for U (open circles) in the limit Lm →∞and of G (closed circles) at 11 Lm =30.Inthisfigure and those that follow, the values of G for Lm →∞were multiplied by a factor of 10. for 232Th, a≤0.028 for 234U, and a≤0.027 for For all cases presented in Table 2, the values of 236 11 U) and may serve as a basis for performing more G for Lm =30prove to be significantly smaller than accurate investigations. From an analysis of our re- the values of G11 that were found in A. Bohr’s limit. sults for G11 that are presented in Table 2 for the This circumstance, which is illustrated in Figs. 1–7, nuclei and the energies Eγ investigated here, one can results in the need for revisiting some parameters of draw the conclusion that the maximum possible value the model of doorway states [20, 21], which was used of L =30is by and large compatible with the entire in [4] to calculate cross sections and angular distribu- m tions of fragments from the subthreshold photofission body of available experimental data. This conclusion →∞ is in accord with the results presented in [10, 11]. of nuclei in A. Bohr’s approximation (Lm ). As to the quantities G20, one can see from Ta- The situation changes radically at Lm =20.From ble 2 that their values at Lm =30are virtually co- Table 2, it can be seen that, for 25 of the 78 cases incident with those in A. Bohr’s limit. In Figs. 1–7, ≤ ≤ studied here (at energies in the range 5.9 Eγ we therefore present the values of G20 for A. Bohr’s 232 7 MeV in Th, at energies in the range 5.75 ≤ Eγ ≤ limit, which tend to increase with decreasing energy 6.17 MeV in 234U, at energies in the range 5.5 ≤ Eγ and, in some cases, reach a plateau region at 236 E Eγ ≤ 6 MeV in U, and at energies in the range extremely low values of the energy γ , in accordance 238 11 with the expected properties of the fission widths Γ10 5.2 ≤ Eγ ≤ 5.65 MeV in U), positive values of G f 20 can be obtained only upon shifting the values found and Γf in the subbarrier region [4]. Here, one can for a experimentally toward greater values by more see resonance structures in the behavior of G20 in than two standard deviations in the definition of a. 236U, 238U, and 232Th nuclei. The origin of these It follows that, for the experimental-data array being structures was discussed in [4] and is associated with considered, the value of Lm =20can be realized only the appearance of resonance states of the deformation with a probability that is negligible in relation to the fission mode in the second well of the deformation probability of realization of the value of Lm =30. potential.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1428 KADMENSKY, RODIONOVA

G11 G20 2 10 (‡) 101 (b)

101 100

100

10–1

10–2 10–2

4 5 6 7 5 6 7 Eγ, MeV

11 20 238 Fig. 2. Energy dependence of (a) G and (b) G for U (open circles and triangles) in the limit Lm →∞and (closed circles and triangles) Lm =30. Presented here are experimental data from (open and closed circles) [12, 14] and (open and closed triangles) [15, 17, 18].

G11 G20 101 100 (‡) (b)

100

10–1

10–2 10–2

5 6 7 5 6 7 Eγ, MeV

Fig. 3. As in Fig. 1, but for 234U.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1429

G11 G20 0 100 10 (‡) (b)

10–1 10–1

10–2 10–2

5 6 7 5 6 7 Eγ, MeV

Fig. 4. As in Fig. 1, but for 232Th.

G11 G20 102 1 (a) 10 (b)

101

100

10–1 10–1

5 6 7 5 6 7 Eγ, MeV

Fig. 5. As in Fig. 1, but for 238Pu.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1430 KADMENSKY, RODIONOVA

11 G 20 2 G 10 (a) 101 (b)

101

100 100

10–1 10–1

10–2

5 6 7 5 6 7 Eγ, MeV

Fig. 6. As in Fig. 1, but for 240Pu.

G11 G20 102 1 (a) 10 (b)

101 100

100 10–1

5 6 7 5 6 7 Eγ, MeV

Fig. 7. As in Fig. 1, but for 242Pu.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SUBTHRESHOLD PHOTOFISSION OF EVEN–EVEN NUCLEI 1431 4. “ISOMERIC SHELF” IN THE ANGULAR were employed to deduce information about the val- DISTRIBUTIONS OF FRAGMENTS ues of G11 and G20. At the same time, the experi- FROM THE SUBTHRESHOLD mental values of a that were used in our calculations PHOTOFISSION OF NUCLEI actually correspond to the quantities a¯ introduced above, which take into account the isomeric-shelf- The “isomeric shelf” phenomenon discovered ex- is 11 perimentally in analyzing the total cross sections for induced correction a0 . As a result, the quantity G 236 238 the subthreshold photofission of 238U [6, 7] and 236U for the Uand Unuclei(seeFigs.1,2)first [7,15] nuclei consists in the following: as one goes decreases with decreasing Eγ and attains a minimum, 11 over from the photon endpoint energies in the re- whereupon there occurs a sharp increase in G in gion E >Eis to those in the range E ≤ Eis,the the region of the lowest energies Eγ . This behavior γ γ 11 rate of the decrease in the photofission cross section of G differs from the behavior expected in the case with decreasing Eγ abruptly becomes much lower. where the prompt-fission mechanism is dominant [4] 11 Concurrently, the isotropic component of the angular and where G first decreases with decreasing Eγ distribution of photofission fragments grows in the and then reaches the region of constant values. It is is is interesting to note that, as can be seen from Figs. 1 region Eγ ≤ E , becoming dominant at Eγ ≈ E . 0 and 2, the regions where the isomeric shelf manifests An explanation of this phenomenon was proposed 236 238 in [22, 4] on the basis of introducing the mechanism itself for Uand U prove to be close in A. Bohr’s →∞ of delayed fission. Finding its way to the second well limit (Lm )andatLm =30.FromFigs.3– 234 232 238 240 in the process of fission, a fissile nucleus can undergo 7, one can see that, for U, Th, Pu, Pu, fission either via fission without any change in energy and 242Pu nuclei, the behavior of the quantity G11 and without delay (prompt fission) or via radiative in all regions of the energy Eγ that were studied decay accompanied by the population of a shape- here has a character that corresponds to the prompt- isomer state and followed by fission from this state— fission mechanism and which does not feature the that is, with a delay characterized by the lifetime of isomeric-shelf effect. This observation is in line with the nucleus in the second well (delayed fission). Since the conclusions drawn in [4] from an analysis of the the spin of a shape isomer is zero for an even–even subthreshold photofission of 234Uand232Thnucleiin nucleus, the angular distribution of fragments from the region of extremely low values of the energy Eγ . delayed fission is isotropic. Therefore, the quantity a0, which appears in the definition (6) of the angular 5. CONCLUSION distribution of photofission fragments and which has Within quantum-mechanical fission theory, we the form (7), is replaced, upon taking into account have analyzed the angular distributions of fragments is is delayed fission, by a¯0 =(a0 + a0 ),wherea0 grows originating from the subthreshold photofission of is seven even–even nuclei. The results of this analysis with decreasing Eγ from negligible values (a0 a0) is is is have confirmed the conclusion drawn in [10, 11] for Eγ >E to a0 b0 a0 at Eγ = E0 .Inorder that the maximum value Lm of the relative orbital to observe the isomeric-shelf effect in the angular angular momentum L of fission fragments cannot distributions of nuclear-photofission fragments, one differ significantly from a limiting value of Lm ≈ 30, usually studies [4] the experimental values of the ratio which was obtained in [10, 11] from an analysis of the b0/a¯0, which can be represented in the form b0/a¯0 = multiplicity of prompt neutrons and the multiplicity −1 ¯b/a¯ =(1− a¯)/a¯ =(¯a) − 1 and which can be ex- and multipolarity of prompt photons emitted by pri- − pressed in terms of (¯a) 1 [the quantities a¯ and ¯b are mary fission fragments. This result can be considered given by formulas (8), where a¯0 must be substituted as an additional substantiation of the mechanism of −1 −1 for a0]. For prompt fission, (¯a) = a first increases pumping of high spins and orbital angular momenta monotonically with decreasing Eγ and then reaches of primary fission fragments that was proposed in [10] values that remain constant to the smallest observ- and which was associated with a strong nonsphericity able energies Eγ [4]. But if one takes into account of the potential of interaction of these fragments. delayed fission associated with the population of a shape isomer, the behavior of (¯a)−1 in response to a ACKNOWLEDGMENTS decrease in Eγ is the following: upon reaching a max- We are grateful to V.M.Vakhtel’, G.A. Petrov, and is V.E. Bunakov for stimulating discussions. imum at Eγ ≈ E , it begins decreasing and attains a ≈ is This work was supported by the Russian Foun- minimum in the region around Eγ E0 . This effect dation for Basic Research (project no. 03-02-17469), was studied in detail by Ostapenko et al. [4]. the Dynasty Foundation for support of noncommer- In the present study, formulas (1), (2), (7), and (8), cial programs, and the International Center for Fun- which are based on the prompt-fission mechanism, damental Physics in Moscow.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1432 KADMENSKY, RODIONOVA REFERENCES 13. S. G. Kadmenskiı˘ ,V.P.Markushev,andV.I.Furman, 1. E. J. Winhold, P.T. Demos, and I. Halpern, Phys. Rev. Yad. Fiz. 35, 300 (1982) [Sov. J. Nucl. Phys. 35, 166 87, 1139 (1952). (1982)]. 2. A. Bohr, in Proceedings of the International Con- 14.N.S.Rabotnov,G.N.Smirenkin,A.S.Soldatov, ference on the Peaceful Uses of Atomic Energy, et al.,Yad.Fiz.11, 508 (1970) [Sov. J. Nucl. Phys. New York, 1956, Vol. 2, p. 175. 11, 285 (1970)]. 3. A. Bohr and B. Mottelson, Nuclear Structure (Ben- jamin, New York, 1969, 1974; Moscow, 1971, 1977), 15.V.E.Zhuchko,Yu.B.Ostapenko,G.N.Smirenkin, Vols. 1, 2. et al.,Yad.Fiz.30, 634 (1979) [Sov. J. Nucl. Phys. 4.Yu.B.Ostapenko,G.N.Smirenkin,A.S.Soldatov, 30, 326 (1979)]. and Yu. M. Tsipenyuk, Fiz. Elem.´ Chastits At. Yadra 16. L. J. Lindgren, A. S. Soldatov, and Yu. M. Tsipenyuk, 12, 1364 (1981) [Sov. J. Part. Nucl. 12, 545 (1981)]. Yad. Fiz. 32, 335 (1980) [Sov. J. Nucl. Phys. 32, 173 5. S. BjørnholmandV.M.Strutinsky,Nucl.Phys.A (1980)]. 136, 1 (1969). 6.C.D.Bowman,I.G.Schroder,C.E.Dick,and¨ 17. V.E.Rudnikov,G.N.Smirenkin,A.S.Soldatov,and H. E. Jackson, Phys. Rev. C 12, 863 (1975). Sh. Yukhas, Yad. Fiz. 48, 646 (1988) [Sov. J. Nucl. 7. V. E. Zhuchko, A. V. Ignatyuk, Yu. B. Ostapenko, Phys. 48, 414 (1988)]. et al.,Pis’maZh.Eksp.´ Teor. Fiz. 22, 255 (1975) 18. P. P.Ganich, V. E.Rudnikov, D.I. Sikora, et al.,Yad. [JETP Lett. 22, 118 (1975)]. Fiz. 52, 36 (1990) [Sov. J. Nucl. Phys. 52, 23 (1990)]. 8. S. M. Polikanov, Shape Isomerism of Nuclei (Atom- 19.R.Kuiken,N.J.Pattenden,andH.Postma,Nucl. izdat, Moscow, 1977) [in Russian]. Phys. A 190, 401 (1972). 9. S. G. Kadmensky, Yad. Fiz. 65, 1424 (2002) [Phys. At. Nucl. 65, 1390 (2002)]. 20. P. D. Goldstone and P. Paul, Phys. Rev. C 18, 1733 10. S. G. Kadmensky, Yad. Fiz. 67, 167 (2004) [Phys. At. (1978). Nucl. 67, 170 (2004)]. 21. U. Goerlach, D. Habs, M. Just, et al.,Z.Phys.A287, 11. S. G. Kadmensky and L. V. Rodionova, Yad. Fiz. 66, 171 (1978). 1259 (2003) [Phys. At. Nucl. 66, 1219 (2003)]. 12. A. V. Ignatyuk, N. S. Rabotnov, et al.,Zh.Eksp.´ 22. C. D. Bowman, Phys. Rev. C 12, 856 (1975). Teor. Fiz. 61, 1284 (1971) [Sov. Phys. JETP 34, 684 (1971)]. Translatedby A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1433–1442. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1491–1500. Original Russian Text Copyright c 2005 by Kadmensky, Rodionova. NUCLEI Theory

Angular Distributions of Fragments Originating from the Spontaneous Fission of Oriented Nuclei and Problem of the Conservation of the Spin Projection onto the Symmetry Axis of a Fissile Nucleus

S. G. Kadmensky* andL.V.Rodionova Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia Received April 19, 2004; in ﬁnal form, September 7, 2004

Abstract—The concept of transition fission states, which was successfullyused to describe the angular distributions of fragments for the spontaneous and low-energyinduced fission of axisymmetric nuclei, proves to be correct if the spin projection onto the symmetry axis of a fissile nucleus is an integral of the motion for the external region from the descent of the fissile nucleus from the external fission barrier to the scission point. Upon heating a fissile nucleus in this region to temperatures of T ≈ 1 MeV (this is predicted bymanytheoretical models of the fission process), the Coriolis interaction uniformlymixes the possible projections of the fissile-nucleus spin for the case of low spin values, this leading to the loss of memory about transition fission states in the asymptotic region where the angular distributions of fragments are formed. Within quantum-mechanical fission theory, which takes into account deviations from A. Bohr’s formula, the angular distributions of fragments are calculated for spontaneously fissile nuclei aligned byan external magnetic field at ultralow temperatures, and it is shown that an analysis of experimental angular distributions of fragments would make it possible to solve the problem of spin-projection conservation for fissile nuclei in the external region. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION estimate of lm obtained with allowance for the mechanism of pumping of high spins and relative orbital On the basis of the observation that there is a angular momenta of fragments (this mechanism was sizable anisotropyin the angular distributions of frag- proposed in [12]). ments originating from the subthreshold photofis- The angular distributions of fragments originating sion of axisymmetric deformed even–even nuclei [1], 236 234 A. Bohr [2, 3] introduced the concept of transition from the fission of Uand U nuclei formed upon fission states that are formed at the saddle points of the capture of resonance and low-energyneutrons of energyin the range 0.4

1063-7788/05/6809-1433$26.00 c 2005 Pleiades Publishing, Inc. 1434 KADMENSKY, RODIONOVA between the fission amplitudes for different compound a rather high statistical accuracyof experiments that states of the fissile nucleus in the first well of the studythe angular distributions in question. deformation potential, which have different energies The objective of the present studyis to explore the and, in general, different spins, manifests itself in problem of conservation of the spin projection K for cross sections and in the angular distribution of frag- a fissile axisymmetric nucleus in the fission process ments for nuclear fission induced byresonance and and to analyze, within quantum-mechanical fission low-energyneutrons. Such an interference does not theory[9 –12], the potential of experiments that would contribute to cross sections and angular distributions studythe angular distributions of fragments for the of fragments for photofission, since the character- spontaneous fission of oriented nuclei with the aim of istic energyinterval of averaging photo fission cross solving this problem. sections is verywide ( ≥100 keV) in the experiments reported in [8]. The veryfact that there are distinctions between 2. PROBLEM OF CONSERVATION the results of descriptions of photofission, on one OF THE FISSILE-NUCLEUS-SPIN PROJECTION IN THE FISSION PROCESS hand, and fission induced byresonance and low- energyneutrons, on the other hand, demonstrates Let us consider the case where the shape of the typical difficulties that arise in extracting reliable in- deformed fissile nucleus in the first well of its deforma- formation about fission physics in analyzing the fis- tion potential possesses axial symmetry. If we elim- sion of nuclei from highlyexcited compound states of inate the effect of weak interactions, the spin J and a fissile nucleus that possess a complicated structure. the parity π of fissile-nucleus states will be integrals In this connection, it would be of paramount im- of the motion at all stages of the fission process. As to portance to implement the project [16] of studying the projection K of the fissile-nucleus spin onto the the angular distributions of fragments for the spon- symmetry axis of the fissile nucleus, it can change in taneous fission of nuclei aligned byan external mag- the fission process because of the effect of two main netic field at ultralow temperatures. For anisotropyin factors. The first factor is associated with the possibil- these angular distributions to be observed, it is neces- itythat, in the fission process, the nucleus undergoing sarythat the ground-state spin J of the fissile nucleus fission maydevelop triaxial deformations speci fied by γ satisfythe condition J>1/2. This means that the a nonzero parameter , which lead to the mixing of states characterized bydi fferent values of K. This nuclei being studied must be odd or odd–odd, which possibilitywas discussed in [17]. The second factor creates difficulties for planned experiments since the is associated with the effect of Coriolis interaction, branching fractions of spontaneous fission for odd which, as was shown in [18, 19], undergoes a dynam- and especiallyodd –odd nuclei are small, falling well ical enhancement in the case of rather high excitation below the analogous branching fractions for even– energies E∗ of the fissile nucleus and thermalization even spontaneously fissile nuclei. of this energyover a large number of multiquasipar- Investigation of the spontaneous fission of axisym- ticle states (compound states) and leads to a uniform metric deformed nuclei is advantageous in that, for statistical mixing of the projections K of the spins J the ground states of these nuclei, the projection K of of these states. the spin J always takes the value K = J. It follows For actinide nuclei, it is generallyaccepted that, as that, within the concept of transition fission states [2, collective deformation coordinates of the nucleus un- 3], spontaneous fission is determined exclusivelyby dergo changes in the process of its fission, the nuclear the onlydeep-subbarrier transition fission state of shape retains axial symmetry. Therefore, we will dis- π π quantum numbers J K = J J. This circumstance regard the effect of the first factor on the conservation simplifies substantiallythe analysisof the angular of the spin projection K in the fission process. As to distributions of fragments for the spontaneous fission the second factor, which is associated with Coriolis of oriented nuclei in relation to the analysis of the interaction, its inclusion seems mandatory. angular distributions of fragments for fission from In calculating fission widths and the angular dis- compound states of fissile nuclei (see above). tributions of fragments for induced nuclear fission, On the basis of quantum-mechanical fission the- use is usuallymade [3, 11] of the concept that, at ory[9], we demonstrated in [10] the deviations of the initial stage of the fission process—this stage the angular distributions of fragments for the spon- involves the formation of highlyexcited compound taneous fission of nuclei oriented to the maximum states of a fissile nucleus at the first well of the de- possible degree byan external magnetic field from formation potential—the projections K of the spins the predictions of A. Bohr’s formula, this making it J of these compound states are distributed uniformly possible to determine the maximum value lm of rela- at rather low values of J. As a rule, this concept is tive orbital angular momenta of fission fragments at also applied [3, 11] to the compound states of the

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1435 fissile nucleus in the second well of the deformation mixing of all values of the projections K of the fissile- potential. However, there is presentlyno consistent nucleus spin J at rather low J and to the loss of validation of this concept for the case of practical memoryabout the spin projection in the transition interest where the excitation energies of compound fission states of the nucleus. states in the second well are much lower than the excitation energies of the analogous states in the first This conclusion contradicts the fact that there well. appear sizable anisotropies in the angular distribu- In describing spontaneous and low-energyin- tions of fragments for the subthreshold fission of nu- duced fission of nuclei, use is extensivelymade of the clei [8, 10, 11] and for the fission of aligned nuclei concept of transition fission states [2, 3], which are that is induced bylow-energyand resonance neu- formed at the saddle points of the deformation poten- trons [12, 13]; it is also at odds with the observation tial of a fissile nucleus and which are characterized of P -odd [28] and P -even [29] asymmetries in the bya spin J,aparityπ, and a spin projection K onto angular distributions of fragments originating from the symmetry axis of the fissile nucleus. For the states nuclear fission induced bypolarized cold neutrons, as being considered, the spin projection K takesasingle well as with the discoveryof signi ficant fluctuations value for the reason that, since the internal state of in the fission widths of neutron resonances [30]. By the fissile nucleus is cold, Coriolis mixing in K does employing the formula obtained for the angular dis- not playanysigni ficant role at the saddle points. tributions of fragments originating from subthreshold For the concept of transition fission states to be nuclear fission [3, 8, 10, 11] and fragments originating operative and for these states to be crucial for the for- from the fission of aligned nuclei that is induced by mation of fission widths and the asymptotic behavior low-energyand resonance neutrons [13, 14], as well of the fissile-nucleus wave function in the region of al- astheformulasforthecoefficients of P -odd and readyformed primary fission fragments, the following P -even asymmetries in the angular distributions of condition of paramount importance must be satisfied: fragments in nuclear fission induced bycold polarized the spin projections K associated with transition fis- neutrons [31–33], it can be proven rigorouslythat the sion states at the outer saddle point of the deformation anisotropies and the coefficients of P -odd and P -even potential must remain integrals of the motion over asymmetries in the angular distributions of fission the entire external region of the fissile system from fragments vanish in the case of a uniform distribution the descent of the nucleus from this saddle point to of all projections K of the spins J of fissile-nucleus the point of its scission into primary fission fragments states in the vicinityof the scission point. and a subsequent divergence of these fragments. This means that the fissile nucleus must remain quite cold Upon considering that, at nuclear eхcitation ener- throughout this region in order that the Coriolis in- gies in the range E∗ < 2 MeV,the Coriolis interaction teraction not come into playand not mix, with equal effect on the mixing of the projections K of the spins J probabilities, all possible projections K of the fissile- of excited states of an A ≈ 250 axisymmetric nucleus nucleus spin. is weak at rather low values of J [18], the use of The majorityof modern fission models [20–27], formula (1) makes it possible to obtain a qualitative which describe successfullythe mass, charge, and estimate of maximum possible fissile-nucleus tem- energydistributions of fragments originating from the peratures Tm at which the projection K of the fissile- spontaneous and induced fission of various nuclei, in- nucleus spin J can still be considered as an integral volve the concept that, because of dissipation effects, of the motion, Tm ≤ 0.25 MeV. This estimate con- a considerable part of the difference of the potential tradicts the result obtained within the fission models energies of a fissile nucleus at the outer saddle point that were proposed in [20–27] and in which, as was and at the point of its scission into fragments goes indicated above, the temperature T of a fissile nucleus into heating the fissile nucleus, whose state near the in the vicinityof its scission point is about 1 MeV. It scission point is characterized bya temperature of is of importance to develop new fission models that T ≈ 1 MeV.According to [3], the relation between the ∗ would be able to describe basic features of the fission nuclear excitation energy E and the nuclear temper- process, such as the mass, charge, and energydistri- ature T at rather high T is butions of fission fragments, and which, at the same ∗ E = aT 2, (1) time, would be compatible with the experimentally corroborated concept of transition fission states [2, 3]. where a =(A/8) MeV−1 [3]. For an A ≈ 250 fissile nucleus, this yields the estimate E∗ ≈ 30 MeV.It was An investigation of the angular distributions of indicated above that, upon the thermalization of so fragments originating from the spontaneous fission high an excitation energy, Coriolis interaction in an of aligned nuclei mayfurnish additional arguments in axisymmetric nucleus leads to [18, 19] a complete favor of the above concept.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1436 KADMENSKY, RODIONOVA 3. ANGULAR DISTRIBUTIONS fragments in the bodyframe of the fissile nucleus. OF FRAGMENTS FOR THE FISSION Specifically, we have OF ALIGNED NUCLEI lm √ Within quantum-mechanical fission theory[9 – F (lm,θ)=b(lm) Yl0(θ ) 2l +1Rl , (6) 12], the normalized (to unity) angular distribution l=0 dP Jπ(θ)/dΩ of fragments originating from the spon- −1/2 taneous fission of axisymmetric nuclei aligned by an lm 2 external magnetic field at ultralow temperatures can b(lm)= (2l +1)Rl , (7) be represented in the form l=0 Jπ JπK JπK l dP (θ) (2 − δ )Γ dP (θ) where Rl =1+ππ1π2(−1) , lm is the maximum = K,0 , (2) dΩ ΓJπ dΩ value of the relative orbital angular momentum l of K≥0 fission fragments, and Yl0(θ ) is a spherical harmonic. JπK As follows from specific calculations, the function where dP (θ)/dΩ is the angular distribution of fragments for fission from the JπK state of the fissile F (lm,θ) at high values of lm is virtuallyindependent nucleus, of the parity π of the fissile nucleus and the parities π1 and π2 of fission fragments—that is, on the parity JπK dP (θ) J Jπ of the relative orbital angular momentum l of fission = aM TMK(θ); (3) →∞ dΩ fragments. For lm , the angular distribution M 2 F (lm,θ) reduces to the sum of delta functions of Ω ≡ (θ,ϕ) is the solid angle that determines the di- the form rection of light-fission-fragment emission in the lab- 1 [δ(ξ − 1) + δ(ξ +1)], oratoryframe; ΓJπK is the partial fission width for 4π the transition of the parent nucleus to the asymptotic where ξ =cosθ , so that the angular distribution (4) state of spin J,parityπ, and spin projection K (M) assumes the form predicted byA. Bohr’s formula [2, 3], onto the symmetry axis of the nucleus (z axis in the laboratoryframe); and ˜Jπ 2J +1 J TMK(θ)= BMK(ω) , (8) Jπ JπK 8π β=θ Γ = (2 − δK,0)Γ K>0 which is usuallyused [3, 8, 13, 14] in calculating the is the total fission width of the parent nucleus, the fac- angular distributions of fission fragments. tor (2 − δK,0) taking into account double degeneracy The mechanism of pumping of relative orbital an- of K>0 states of an axisymmetric nucleus. In (3), gular momenta and spins of fission fragments due to aJ is a parameter that characterizes the relative pop- a stronglynonspherical character of the potential of M interaction between primary fission fragments in the ulation of the JπM sublevels of the fissile nucleus, J J Jπ vicinityof the point where the fissile nucleus under- M=−J aM =1,andTMK(θ) is the normalized (to goes scission into fragments was proposed in [12]. unity) angular distribution of fragments for fission This mechanism leads to a value of lm ≈ 30 for the from the JπMK states of the fissile nucleus. For this maximum relative orbital angular momentum of fis- angular distribution, we have [9, 10] sion fragments. This value of lm was confirmed in [10, 2J +1 11], where the angular distributions of fragments were T Jπ (θ)= dωBJ (ω)F 2(l ,θ), (4) MK 16π2 MK m analyzed for the subthreshold photofission of a set of even–even nuclei and will therefore be used below. BJ (ω)=|DJ (ω)|2 + |DJ (ω)|2, (5) Byusing the multiplication theorem for spheri- MK MK M−K cal harmonics and the formula for transforming the where θ is the angle between the direction of light- spherical harmonic Yl0(θ ) from the bodyframe to fission-fragment emission and the symmetry axis the laboratoryframe, we can represent the angular J F 2(l ,θ) of the fissile nucleus and DMK(ω) is a generalized distribution m in the form spherical harmonic, which depends on the Euler 2 2 [b(lm)] (2l + 1)(2l +1) angles ω ≡ (α, β, γ) determining the orientation of F (lm,θ)= √ (9) 4π (2L +1) the symmetry axes of the fissile nucleus with respect Lk ll ∗ to the axes of the laboratoryframe. The function ×| L0 |2 L Cll00 YLk(θ)Dk0 (ω)RlRl , F (lm,θ) in (4) coincides with the amplitude of the L0 normalized (to unity) angular distribution of fission where Cll00 is a Clebsch–Gordan coefficient.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1437 In turn, the use of the multiplication theorem for can recast the angular distribution dP JπK(θ)/dΩ (3) generalized spherical harmonics makes it possible to into the form J JπK represent the quantity BMK(ω) in (5) as dP (θ) 1 2J +1 = + (15) J JL L dΩ 4π 4π BMK(ω)=2 BMKD00(ω), (10) × J JL L=0,2,4... aM BMKDL(lm)PL(θ). M L=2,4,... − BJL =(−1)M K CL0 CL0 (11) MK JJM−M JJK−K For a further analysis of the angular distributions (15), =(−1)2J−M−K CL0 CL0 . J JJM−M JJK−K it is convenient to employthe quantity fL that is Performing integration in (4) with respect to ω and referred to as the nuclear-orientation parameter and which is given by[34] taking into account the explicit expressions (10) BJ (ω) F 2(l ,θ) −1/2 2 and (9) for MK and m , respectively, we J (2J + L +1)! (L!) −L J Jπ fL = J GL, can recast the angular distribution TMK(θ) into the (2L + 1)(2J − L)! (2L)! form (16) 2J Jπ 2J +1 JL where TMK(θ)= DL(lm)BMKPL(θ), 4π J J − J−M L0 L=0,2,4... GL = aM ( 1) CJJM−M (17) (12) M where PL(θ) is a Legendre polynomial and the coeffi- is the Fano statistical tensor. If a fissile nucleus is cient DL(lm) is given bythe expression not oriented because of an equiprobable population of all JπM sublevels of this nucleus at different values of − DL(lm) (13) J 1 M, in which case we have aM =(2J +1) ,theFano lm statistical tensor GJ and the nuclear-orientation 2 (2l + 1)(2l +1) L0 2 L =[b(lm)] RlRl (C ) , J J 2L +1 ll 00 parameter fL become, respectively, GL =(2J + ll −1/2 J 1) δL,0 and fL = δL,0 byvirtue of the following whence it follows that DL(lm) ≤ 1 and D0(lm)=1 properties of Clebsch–Gordan coefficients: for anyvalues of lm. L0 00 C − C − (18) If, for the generalized spherical harmonic DL (ω), JJM M JJ MM 00 M which appears in the definition of BJ (ω),one MK L0 J−M 1 L | = C − (−1) √ = δL,0. employs the relation D00(ω) β=θ = PL(θ), then the JJM M 2J +1 ˜Jπ M angular distribution TMK(θ) (8) in A. Bohr’s limit (lm →∞) can be represented in the form In this case, all of the L =0 terms vanish in the angu- 2J lar distribution (15), which becomes purelyisotropic, 2J +1 ˜Jπ JL is TMK(θ)= BMKPL(θ). (14) dP JπK(θ) 1 4π = . L=0,2,4... dΩ 4π Formula (14) can also be obtained from (12) upon With the aid of formulas (11), (12), (16), and (17), considering that, in A. Bohr’s limit, the quantity the angular distribution (15) can in general be rep- l D l →∞ DL( m) in (13) reduces to L( m )=1for any resented in the form values of L, whence it can be seen that, at finite values JπK JπK is of lm, the angular distribution in (12) differs from dP (θ) dP (θ) = W JπK(θ), (19) A. Bohr’s angular distribution in (14) onlybecause dΩ dΩ of the deviation of the coefficients DL(lm) from unity. where At lm =30,thecoefficients DL(lm) are independent of the parity π of the fissile nucleus or the parities JπK J W (θ)=1+ DL(lm)f AL(J, K)PL(θ) π and π of fission fragments, taking the values L 1 2 L=2,4,... of D2(30) = 0.95, D4(30) = 0.91, D6(30) = 0.87, (20) D8(30) = 0.82, ..., which decrease with increasing L. is the reduced angular distribution of fission frag- Employing formula (12) and considering that, at ments with JL −1 − J−K L0 L =0, the quantity BMK is equal to (2J +1) ,we AL(J, K)=( 1) CJJK−K (21)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1438 KADMENSKY, RODIONOVA

J Orientation parameters fL and coeﬃcients AL(J, K) for J =7/2 and J =9/2 and various values of K versus L

J =7/2 J =9/2

L A (J, K) A (J, K) J L J L fL fL K =7/2 K =5/2 K =3/2 K =1/2 K =9/2 K =7/2 K =5/2 K =3/2 K =1/2

2 0.57 4.08 0.58 −1.75 −2.92 0.59 4.6 1.53 −0.76 −2.30 −3.07

4 0.08 7.96 −14.78 −3.41 −10.23 0.11 10.75 −13.14 −10.16 1.79 10.75

6 0.003 10.21 −51.06 91.91 −51.06 0.008 17.30 −63.43 57.67 34.60 −46.13

8 0.0002 20.85 −145.96 417.04 −583.86 291.92

(2J +1)JL(2L)! (2L + 1)(2J − L)! 1/2 unityif the distribution of all projections K of the × . (L!)2 (2J + L +1)! fissile-nucleus spin J is equiprobable in the asymptotic region, in which case the partial fission widths For a complete alignment of a spontaneously fissile ΓJπK in (24) are independent of K. This result is nucleus in an external magnetic field at ultralow tem- associated with the propertyof Clebsch –Gordan co- J J peratures, in which case aM becomes aM = δM,±J , efficients that arises from (18) upon the substitution depending on the sign of the gyromagnetic ratio for a of the index K for the index M and which leads to J fissile nucleus, the Fano tensor GL (17) assumes the the vanishing of all L =0terms in (24). This means form that the discoveryof a sizable anisotropyin experi- Jπ 2 1/2 mental reduced angular distributions W (θ) would J L0 (2L + 1)(2J!) GL = CJJJ−J = , indicate that, in the external region introduced above, (2J − L)!(2J + L +1)! the fissile nucleus being studied is not heated to sig- while the nuclear-orientation parameter reduces nificant temperatures. It would also be of interest to to [34] obtain a piece of evidence in support of an exact spin- projection conservation for a spontaneously fissile nu- 2 (L!) − (2J)! f J = J L . (22) cleus in the external region, in which case the sum L (2L)! (2J − L)! over K in (24) would be dominated bythe K = J term. The total angular distribution dP Jπ(θ)/dΩ of fragments for the spontaneous fission of oriented nuclei is obtained bysubstituting (19) and (20) 4. ANGULAR DISTRIBUTIONS into (2), OF FRAGMENTS FOR THE SPONTANEOUS FISSION OF COMPLETELY ORIENTED dP Jπ(θ) dP JπK(θ) is NUCLEI 255Es AND 257Fm = W Jπ(θ), (23) dΩ dΩ A further analysis will be performed by consid- Jπ ering spontaneous fission from the ground states of where the reduced angular distribution W (θ) is 255 257 JπK the Es and Fm odd nuclei that are completely expressed in terms of the distribution W (θ) (20) oriented in an external magnetic field at ultralow tem- as peratures and which were chosen for investigation in JπK (2 − δK,0)Γ the project formulated in [16]. Although there have W Jπ(θ)= W JπK(θ). (24) Jπ so far been no reliable measurements of the spins of ≥ Γ K 0 these nuclei, one can conclude from the systematics One can then see that, in just the same wayas in the of spins of neighboring odd isotopes of Es and Fm case of angular distributions of fragments for the sub- that the spins J of the 255Es and 257Fm nuclei are 7/2 threshold fission of nuclei [8, 10, 11] and fragments and 9/2, respectively. Since the angular distributions originating from the fission of aligned nuclei that is of fragments in the spontaneous fission of aligned induced bylow-energyand resonance neutrons [13, nuclei are sensitive to the spins J of these nuclei, one 14], the reduced angular distribution W Jπ(θ) of fis- can obtain direct information about the spins J from sion fragments becomes fullyisotropic and equal to an analysis of these distributions, thereby confirming

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1439

π WJ K(θ) 1.6 L = 0, 2 255Es J = 7/2, K = 3/2 L = 0, 2, 4, 6 1.2

0.8

0.4

0 30 60 90 120 150 180 L = 0, 2, 4 θ, deg –0.4

Fig. 1. Reduced angular distributions W JπK(θ) (20)inthecaseof255Es for various L values included in their calculation.

π WJ K(θ)

257Fm 1.5 J = 9/2, K = 5/2

1.0 L = 0, 2, 4

L = 0, 2 L = 0, 2, 4, 6 0.5 L = 0, 2, 4, 6, 8

0 30 60 90 120 150 180 θ, deg

–0.5

Fig. 2. As in Fig. 1, but for 257Fm.

(or disproving) the spin values proposed above for the in (20). The calculations reveal that, in describing nuclei in question. angular distributions of fragments for fission from the From the table, which presents the values calcu- ground state of a J = K fissile nucleus, one can dis- lated byformulas (22) and (21) for the coe fficients regard the contributions of the L =6terms for 255Es J 257 fL and AL(J, K) at J =7/2 and J =9/2,onecan and the contributions of the L =8terms for Fm. J In describing the angular distributions W JπK(θ) for see that the quantities fL decrease with increasing K2 terms in the sum over L and at J =9/2 and K =5/2.FromFig.1,onecan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1440 KADMENSKY, RODIONOVA

π WJ K(θ)

255Es 4 J = 7/2 K = 7/2

K = 1/2 2

K = 5/2

1 K = 3/2

0 30 60 90 120 150 180 θ, deg

Fig. 3. Reduced angular distributions W JπK(θ) (20) in the case of 255Es for various values of K. The solid and dashed curves represent the results of the calculations at, respectively,lm =30and lm = ∞ (A. Bohr’s formula).

π WJ K(θ) 257Fm J = 9/2 5 K = 9/2

K = 1/2 2 K = 3/2 K = 7/2

1 K = 5/2

0 30 60 90 120 150 180 θ, deg

Fig. 4. As in Fig. 3, but for 257Fm.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ANGULAR DISTRIBUTIONS OF FRAGMENTS 1441 see that, at J =7/2 and K =3/2, the shape of the ACKNOWLEDGMENTS angular distribution changes markedlyin the vicinity ◦ WearegratefultoV.E.Bunakov,G.A.Petrov,and of the angle θ =0 upon taking into account the L = W.I. Furman for stimulating discussions. 6 term. Figure 2 shows that, at J =9/2 and K =5/2, the inclusion of the L =8term is necessarysince, This work was supported bythe program “Uni- versities of Russia” (grant no. 02.01.011), the Dy- otherwise, the distribution W JπK(θ) takes negative ◦ ◦ nastyfoundation for support of noncommercial pro- values in the vicinities of the angles 0 and 180 . grams, and the International Center for Fundamental As can be seen from Figs. 3 and 4, the shape of Physics in Moscow. the reduced angular distributions W JπK(θ) for J = 7/2 and J =9/2 changes stronglywith K—from the REFERENCES distributions at low K, which have a maximum at θ =90◦, to the distributions at K = J,whichhavea 1. E. J. Winhold, P.T. Demos, and I. Halpern, Phys. Rev. minimum at θ =90◦. At the same time, a compar- 87, 1139 (1952). ison of the data in Figs. 3 and 4 shows that, in the 2. A. Bohr, in Proceedings of the International Con- JπK ference on the Peaceful Uses of Atomic Energy case of K = J, the distributions W (θ) change (New York, 1956), Vol. 2. substantiallyupon going over from J =7/2 to J = 3. A. Bohr and B. Mottelson, Nuclear Structure,Vol.2: 9/2. Therefore, a comparison of experimental reduced Nuclear Deformations (Benjamin, New York, 1975; angular distributions with the distributions W Jπ(θ) Mir, Moscow, 1977). in (24) would make it possible to determine not only 4. V. M. Strutinskiı˘ ,Yad.Fiz.3, 614 (1966) [Sov. J. the spin J of a fissile nucleus but also the spin pro- Nucl. Phys. 3, 449 (1966)]; V. M. Strutinsky, Nucl. jections K that manifest themselves in the external Phys. A 95, 420 (1967). 5. J. E. Lynn, Report AERE-R5891 (1968). region of the fissile system. 6. H. Weigmann, Z. Phys. 214, 7 (1968). Upon attaining a rather high statistical accuracy 7. C. D. Bowman, Phys. Rev. C 12, 856 (1975). in measuring reduced angular distributions W Jπ(θ), 8.Yu.B.Ostapenko,G.N.Smirenkin,A.S.Soldatov, ´ one can hope to reveal deviations of W Jπ(θ) from and Yu. M. Tsipenyuk, Fiz. Elem. Chastits At. Yadra the predictions of A. Bohr’s formula (24) and to find 12, 1364 (1981) [Sov. J. Part. Nucl. 12, 545 (1981)]. 9. S. G. Kadmensky, Yad. Fiz. 65, 1424 (2002) [Phys. the true value of lm. Indeed, the angular distributions JπK At. Nucl. 65, 1390 (2002)]. W (θ) calculated byformula (20) at lm =30for ◦ 10. S. G. Kadmenskyand L. V. Rodionova, Yad. Fiz. 66, angles around θ =90 differ from the angular distri- 1259 (2003) [Phys. At. Nucl. 66, 1219 (2003)]. butions calculated byA. Bohr formula (14) byfactors 11. S. G. Kadmenskyand L. V. Rodionova, Yad. Fiz. (in of 1.57 and 2.66 for, respectively, J =7/2 and J = press). 9/2 (see Figs. 3, 4). 12. S. G. Kadmensky, Yad. Fiz. 67, 167 (2004) [Phys. At. Nucl. 67, 170 (2004)]. 13. N. J. Pattenden and H. Postma, Nucl. Phys. A 167, 5. CONCLUSION 225 (1971). 14.R.Kuiken,N.J.Pattenden,andH.Postma,Nucl. Our analysis of the problem of conservation of Phys. A 190, 401 (1972). the projection K of the spin J of an axisymmetric 15. W. I. Furman, in Proceedings of FG/OM Spring fissile nucleus at various stages of the fission process Session, Geel, Belgium, 1999, p. 248. has revealed that, because of the Coriolis interaction 16. G. M. Gurevich et al.,inProceedings of the 53rd In- effect, onlyif the fissile nucleus being considered re- ternational Conference on Nuclear Spectroscopy, St. Petersburg, Russia, 2003, p. 36. mains cold up to the point of its scission into fission 17. L. Wilets, Theories of Nuclear Fission (Clarendon, fragments do there appear noticeable anisotropies in Oxford, 1964; Atomizdat, Moscow, 1967). the angular distributions of fragments for the sponta- 18. S. G. Kadmenskiı˘ ,V.P.Markushev,andV.I.Furman, neous and low-energyinduced fission of nuclei. The Yad. Fiz. 35, 300 (1982) [Sov. J. Nucl. Phys. 35, 166 planned experiments to studythe angular distribu- (1982)]. tions of fragments originating from the spontaneous 19. S. G. Kadmenskiı˘ , V. P.Markushev, Yu. P.Popov, and fission of oriented nuclei mayfurnish additional argu- V. I. Furman, Yad. Fiz. 39, 7 (1984) [Sov. J. Nucl. ments in favor of the ideas developed above. Phys. 39, 4 (1984)]. 20. J. R. Nix, Nucl. Phys. A 130, 241 (1969). It is of importance to emphasize that the con- 21. A. V. Ignatyuk, Yad. Fiz. 9, 357 (1969) [Sov. J. Nucl. clusions drawn from the present analysis cannot be Phys. 9, 208 (1969)]. reconciled with existing fission models, which lead to 22. M. Brack et al.,Rev.Mod.Phys.44, 320 (1972). significant temperatures in the prescission configura- 23. Y. Bonch and Z. Fraenkel, Phys. Rev. C 10, 893 tion of a fissile nucleus. (1974).

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24. B. D. Wilkins et al., Phys. Rev. C 14, 1832 (1976). 31. O. P. Sushkov and V. V. Flambaum, Usp. Fiz. Nauk 25. V.A. Rubchenya and S. G. Yavshits, Yad. Fiz. 40, 649 136, 3 (1982) [Sov. Phys. Usp. 25, 1 (1982)]. (1984) [Sov. J. Nucl. Phys. 40, 416 (1984)]. 32. S. G. Kadmensky, Yad. Fiz. 66, 1739 (2003) [Phys. 26. W. Greiner, J. P. Park, and W. Schield, Nuclear Molecules (World Sci., Singapore, 1995). At. Nucl. 66, 1691 (2003)]. 27. T. M. Shneideman et al., Phys. Rev. C 65, 064302 33. S. G. Kadmensky, Yad. Fiz. 67, 258 (2004) [Phys. At. (2002). Nucl. 67, 241 (2004)]. 28. G. V. Danilyan et al.,Pis’maZh.Eksp.´ Teor. Fiz. 26, 68 (1977) [JETP Lett. 26, 186 (1977)]. 34. Alpha-, Beta-, andGamma-ray Spectroscopy ,Ed. byK. Siegbahn (North-Holland, Amsterdam, 1968; 29. A.K.Petukhov,G.A.Petrov,et al.,Pis’maZh.Eksp.´ Teor. Fiz. 30, 324 (1979) [JETP Lett. 30, 439 (1979)]. Atomizdat, Moscow, 1969), Vol. 3. 30. I. Halpern, Nuclear Fission (Fizmatgiz, Moscow, 1962). Translatedby A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1443–1452. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1501–1510. Original Russian Text Copyright c 2005 by Kuklin, Adamian, Antonenko. NUCLEI Theory

Spectroscopic Factors and Barrier Penetrabilities in Cluster Radioactivity

S. N. Kuklin1), 2), G.G.Adamian1), 3)*, and N. V. Antonenko1) Received December 2, 2004

Abstract—The cold cluster decay model is presented in the framework of a dinuclear system concept. Spectroscopic factors are extracted from barrier penetrabilities and measured half-lives. The deformation of the light cluster and residual nucleus is shown to affect the nucleus–nucleus potential and decay characteristics. Half-lives are predicted for neutron-deficient actinides and intermediate-mass nuclei. The connection between spontaneous fission and cluster radioactivity is discussed. c 2005 Pleiades Publish- ing, Inc.

1. INTRODUCTION description of cluster decay is similar to the classical description of the nuclear fission process. Models Spontaneousbinarynucleardecayispossibleif based on this description are called adiabatic or fission this process is exothermic, i.e., occurs with positive models [4–7]. At the same time, one can assume that Q>0 energy release . This condition is satisfied for a light cluster is formed on the surface of the parent many decay modes of numerous nuclei in the second nucleus. The decay of such a system consisting of half of the Periodic Table. Only α decay (decay super- two daughter nuclei does not require a significant symmetric in product masses) and/or spontaneous adjustment of the system to the final decay channel. fission (symmetric or asymmetric decay) is primarily In this nonadiabatic approach [1, 2, 8, 9], α decay is realized. The yields of carbon, fluorine, oxygen, neon, well described. Thus, in adiabatic models, the shape magnesium, and silicon isotopes are measured only 208 of a fissioning nucleus is assumed to change con- for the case where one of daughter nuclei is Pb tinuously and smoothly with time, and the process is or nuclei close to it in narrow ranges 90 ≤ Z ≤ 93 ≤ ≤ described dynamically beginning with the parent nu- and 206 A 212 of the charge and mass numbers, cleus and ending with the prescission configuration. α respectively. Light clusters, as well as the particle, In nonadiabatic models, the probability of the almost are in the ground state, and the heavy residual nucleus instantaneous formation of a two-cluster system and is in the ground state or in one of low-lying states 223 → 14 209 the penetrability of the potential barrier are calculated (e.g., the fine structure of the Ra C + Pb separately. decay [1, 2]). This phenomenon is called cluster radioactivity and has many common features with α The model discussed in this work is based on decay and cold fission in which the decay fragments the assumption that the ground nuclear state can are almost unexcited [1, 2]. Experiments point to be represented as a superposition of cluster states the decisive role of structure effects such as the fine and a mononucleus [10]. Cluster states are described structure and exclusion of transitions from odd parent as dinuclear systems (DNSs) [11]. A DNS decays nuclei [1–3]. if nuclei overcome a potential barrier preventing the The process of cold nuclear decay can be repre- instantaneous decay of the DNS. There are many sented in the form of the collective flow of nucleons methods for calculating a nucleus–nucleus poten- of the initial nucleus to the daughter nuclei, when tial (e.g., the proximity potential, folding potential, nuclei have time to occupy levels with the minimum double-folding potential, optical potential, etc.) [1, 2, energy. This process is accompanied with the consid- 12]. In addition, different models involve different pa- erable rearrangement of the parent nucleus. Such a rameters for calculating the interaction potential. The form of the potential is determined primarily by the 1) Joint Institute for Nuclear Research, Dubna, Moscow spectroscopic factor and barrier penetrability. We use oblast, 141980 Russia. a nucleus–nucleus potential based on the double- 2)Mari State University, Ioshkar Ola, Russia. 3)Institute of Nuclear Physics, Academy of Sciences of folding procedure. The aim of this work is to extract Uzbekistan, Tashkent, 702132 Uzbekistan. the spectroscopic factors of clusters from the available *e-mail: [email protected] experimental half-lives and calculated penetrabilities

1063-7788/05/6809-1443$26.00 c 2005 Pleiades Publishing, Inc. 1444 KUKLIN et al.

U, MeV resonances in the classically allowed surface of the potential energy for a particular parent nucleus. In 130 the case under consideration, this frequency is related to the frequency of zero modes in the variable η near 110 |η| =1. The allowed region arises because the potential energy of the α configuration is comparable or 90 even lower (particularly for neutron-deficient nuclei) | | R0 R1 than the potential energy of the mononucleus ( η = 70 1) [10]. The angular frequency has a specificvaluefor 10 14 18 R, fm aspecific parent nucleus, but we take the mean value ω =1.2 28 206 0 MeV [10] in calculations for all decays Fig. 1. Interaction potential between the Mg and Hg under consideration. nuclei in the decay of 234U. Calculations were performed for two values of the quadrupole nuclear deformation: The probability of tunneling through a potential (solid curve) βx = βf =0and (dashed curve) βx =0.49, barrier in R, i.e., its penetrability P ,iscalculatedby βf =0.05. The horizontal straight line shows the abso- the formula lute value of Q for the given decay. P =1/(1 + exp G), (2) where of potential barriers and to predict the half-lives of R1 neutron-deficient nuclei. 2 − G = 2m[U(R) Q]dR. 2. MODEL R0 The R0 and R1 values are given in Fig. 1 for the Cluster decay can be described as the evolution 28 206 of the system along the mass-asymmetry coordinates interaction potential between the Mg and Hg nuclei, m = m A A /(A + A ) is the reduced mass η =(Af − Ax)/(Af + Ax) (Af and Ax are the mass 0 x f f x numbers of the heavy and light clusters, respectively) of the system (m0 is the nucleon mass), and Q is the ≈ and along the distance R between the centers of decay energy. The contact configuration at R = R0 mass of clusters. With low probability, a strongly Rx(1 + 5/(4π)βx)+Rf (1 + 5/(4π)βf )+0.5 fm asymmetric DNS exists in the ground nuclear state. 1/3 (Rx,f = r0x,f A are the radii of the light and heavy Such a DNS decays due to tunneling through the x,f clusters, and β are the parameters of quadrupole barrier of the nucleus–nucleus potential. In the first x,f approximation, the process can be divided into two nuclear deformation) corresponds to the position of stages: the formation of the DNS can be treated as the local minimum of the nucleus–nucleus poten- evolution along η, and the decay of the DNS, as tial U. It is assumed that the DNS is axisymmetric, evolution along R. The probability S of the formation because the potential interaction energy for all other of the DNS can be found using the ground state orientations of the DNS nuclei is higher. We note that of the Schrodinger¨ equation in the mass asymmetry only quadrupole nuclear deformation is taken into variable. The probability P of tunneling in R can be account. considered in the WKB approximation. The potential U(R) can be represented as the sum In order to calculate the characteristics of cluster of three terms [12]: decay, we use formulas similar to the formulas of U(R)=UCoul(R)+UN (R)+UL(R), (3) α-decay theory [1]. The half-life of a nucleus with 2 respect to cluster decay is given by the expression where UCoul(R), UN (R),andUL(R)= L(L + 1)/(2(R)) are the Coulomb, nuclear, and centrifu- tx = /Γx, gal potentials, respectively, where (R) is the orbital where moment of inertia. Calculation of UN (R) is most ω difficult. Let us take it in the form of the double- Γ = 0 SP folding potential: x π is the decay width. The half-life can be expressed in UN (R)= ρx(rx)ρf (R − rf )F (rf − rx)drxdrf . terms of tx as π ln 2 The nucleon–nucleon forces T = t ln 2 = . (1) 1/2 x F (r − r ) ω0SP f x In the expression, the energy ω0 related to the angu- ρ0(rx) ρ0(rx) = C0 Fin + Fex 1 − δ(rf − rx), lar frequency is the distance between the neighboring ρ00 ρ00

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1445

Table 1. Calculated penetrabilities P and spectroscopic factors S, as well as measured half-lives T1/2; quadrupole- deformation parameters βx and βf of the cluster and residual nucleus, respectively [15]; and energy release Q in decay (the calculations were performed with the parameters a0x =0.55 fm, a0f =0.56 fm, r0x =1.15 fm, and r0f =1.16 fm)

A → Ax + Af βx βf Q,MeV T1/2,s P S 221Fr → 14C + 207Tl −0.36 0.05 31.29 2.9 × 1014 8.2 × 10−30 5.2 × 10−7 221Ra → 14C + 207Pb −0.36 0.03 32.29 1.5 × 1013 8.9 × 10−28 9.1 × 10−8 222Ra → 14C + 208Pb −0.36 0.00 33.05 1.7 × 1011 1.9 × 10−26 3.8 × 10−7 223Ra → 14C + 209Pb −0.36 0.02 31.85 7.6 × 1015 6.0 × 10−29 2.9 × 10−9 224Ra → 14C + 210Pb −0.36 0.02 30.54 7.3 × 1015 3.2 × 10−31 5.3 × 10−7 226Ra → 14C + 212Pb −0.36 0.02 28.21 1.7 × 1021 1.0 × 10−36 6.8 × 10−7 225Ac → 14C + 211Bi −0.36 0.02 30.48 1.5 × 1017 9.3 × 10−33 8.6 × 10−9 228Th → 20O + 208Pb 0.26 0.00 44.72 5.4 × 1020 7.7 × 10−29 2.9 × 10−14 230Th → 24Ne + 206Hg 0.45 0.05 57.78 4.1 × 1024 3.4 × 10−26 5.8 × 10−21 231Pa → 23F + 208Pb 0.45 0.00 51.87 9.5 × 1025 3.9 × 10−27 3.3 × 10−21 231Pa → 24Ne + 207Tl 0.45 0.05 60.24 7.9 × 1022 8.0 × 10−23 1.9 × 10−22 232U → 24Ne + 208Pb 0.45 0.00 62.31 2.5 × 1020 2.3 × 10−22 2.1 × 10−20 233U → 24Ne + 209Pb 0.45 0.02 60.50 6.9 × 1024 1.5 × 10−24 1.1 × 10−22 233U → 25Ne + 208Pb 0.50 0.00 60.75 6.9 × 1024 3.9 × 10−24 4.4 × 10−23 234U → 24Ne + 210Pb 0.45 0.02 58.84 8.5 × 1025 8.3 × 10−27 1.7 × 10−21 234U → 26Ne + 208Pb 0.50 0.00 59.47 8.5 × 1025 6.5 × 10−26 2.1 × 10−22 234U → 28Mg + 206Hg 0.49 0.05 74.13 5.4 × 1025 6.3 × 10−22 3.5 × 10−26 235U → 24Ne + 211Pb 0.45 0.02 57.36 2.8 × 1027 3.6 × 10−29 1.2 × 10−20 235U → 25Ne + 210Pb 0.50 0.02 57.83 2.8 × 1027 6.2 × 10−28 6.5 × 10−22 235U → 26Ne + 209Pb 0.50 0.02 58.11 2.8 × 1027 1.1 × 10−27 1.5 × 10−21 236U → 30Mg + 206Hg 0.43 0.05 72.51 3.8 × 1027 7.7 × 10−25 4.0 × 10−25 236Pu → 28Mg + 208Pb 0.49 0.00 79.67 3.5 × 1021 8.6 × 10−19 4.2 × 10−25 238Pu → 28Mg + 210Pb 0.59 0.02 75.93 4.7 × 1025 2.0 × 10−21 1.3 × 10−26 238Pu → 30Mg + 208Pb 0.43 0.00 77.03 4.7 × 1025 9.9 × 10−23 2.5 × 10−24 238Pu → 32Si + 206Hg 0.22 0.05 91.21 1.9 × 1025 6.3 × 10−23 9.5 × 10−25 242Cm → 34Si + 208Pb 0.18 0.00 96.53 1.4 × 1023 1.5 × 10−21 1.5 × 10−23

A − 2Z A − 2Z −3 x x f f where ρ00 =0.17 fm and a0x,f are the nuclear dif- Fin,ex = ζin,ex + ζin,ex , Ax Af fusivity parameters. depend on the nuclear density, because ρ0(rx)= − ρx(rx)+ρf (R rf ). The constants ζin =0.09, ζex = The Coulomb potential UCoul is calculated by the − 3 2.59, ζin =0.42, ζex =0.54,andC0 = 300 MeV fm formula [14] are calculated by ﬁtting the measured nuclear charac- 2 2 teristics [13]. The spatial axisymmetric nucleon den- e ZxZf 3 e ZxZf sity has the form of the Woods–Saxon distribution: UCoul(R)= + 3 R 5 R ρ00 2 ρ (r)= , × R βiY20(θi) x,f 1+exp(|r − R |/a ) 0i x,f 0x,f i=x,f

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1446 KUKLIN et al.

Table 2. Calculated penetrabilities P and spectroscopic factors S for spherical clusters and residual nuclei, as well as measured half-lives T1/2 and energy release Q in decay (the calculations were performed with the parameters a0x =0.55 fm, a0f =0.56 fm, r0x =1.15 fm, and r0f =1.16 fm)

A → Ax + Af Q,MeV T1/2,s P S 221Fr → 14C + 207Tl 31.29 2.9 × 1014 1.8 × 10−27 2.3 × 10−9 221Ra → 14C + 207Pb 32.29 1.5 × 1013 1.6 × 10−26 5.0 × 10−9 222Ra → 14C + 208Pb 33.05 1.7 × 1011 6.6 × 10−25 1.1 × 10−8 223Ra → 14C + 209Pb 31.85 7.6 × 1015 1.9 × 10−27 8.4 × 10−11 224Ra → 14C + 210Pb 30.54 7.3 × 1015 1.2 × 10−29 1.4 × 10−8 226Ra → 14C + 212Pb 28.21 1.7 × 1021 1.2 × 10−34 5.8 × 10−9 225Ac → 14C + 211Bi 30.48 1.5 × 1017 5.8 × 10−33 1.4 × 10−8 228Th → 20O + 208Pb 44.72 5.4 × 1020 1.5 × 10−31 1.5 × 10−11 230Th → 24Ne + 206Hg 57.78 4.1 × 1024 6.0 × 10−32 4.8 × 10−15 231Pa → 23F + 208Pb 51.87 9.5 × 1025 2.1 × 10−32 6.0 × 10−16 231Pa → 24Ne + 207Tl 60.24 7.9 × 1022 1.1 × 10−29 1.4 × 10−15 232U → 24Ne + 208Pb 62.31 2.5 × 1020 7.7 × 10−28 6.3 × 10−15 233U → 24Ne + 209Pb 60.50 6.9 × 1024 1.7 × 10−30 1.0 × 10−16 233U → 25Ne + 208Pb 60.75 6.9 × 1024 2.6 × 10−30 6.7 × 10−17 234U → 24Ne + 210Pb 58.84 8.5 × 1025 9.9 × 10−33 1.4 × 10−15 234U → 26Ne + 208Pb 59.47 8.5 × 1025 2.3 × 10−32 6.0 × 10−16 234U → 28Mg + 206Hg 74.13 5.4 × 1025 6.2 × 10−30 3.5 × 10−18 235U → 24Ne + 211Pb 57.36 2.8 × 1027 3.6 × 10−35 1.2 × 10−14 235U → 25Ne + 210Pb 57.83 2.8 × 1027 9.4 × 10−35 4.3 × 10−15 235U → 26Ne + 209Pb 58.11 2.8 × 1027 5.5 × 10−34 3.2 × 10−15 236U → 30Mg + 206Hg 72.51 3.8 × 1027 2.2 × 10−32 1.4 × 10−17 236Pu → 28Mg + 208Pb 79.67 3.5 × 1021 1.1 × 10−25 3.3 × 10−18 238Pu → 28Mg + 210Pb 75.93 4.7 × 1025 3.8 × 10−30 6.9 × 10−18 238Pu → 30Mg + 208Pb 77.03 4.7 × 1025 4.2 × 10−29 6.0 × 10−18 238Pu → 32Si + 206Hg 91.21 1.9 × 1025 1.2 × 10−27 5.0 × 10−20 242Cm → 34Si + 208Pb 96.53 1.4 × 1023 1.5 × 10−24 1.5 × 10−20

2 12 e ZxZf 3. RESULTS OF THE CALCULATIONS + [R β Y (θ )]2. 35 R3 0i i 20 i i=x,f Figure 1 shows the nucleus–nucleus potential for the DNS 28Mg + 206Hg at two diﬀerent values of In terms of the measured half-lives T1/2 and cal- nuclear deformation. The inclusion of quadrupole de- culated penetrabilities P , the probabilities S of the formation changes the position and height of the formation of various clusters (spectroscopic factors) barrier, as well as the depth of the local potential are expressed as minimum corresponding to the DNS at the contact π ln 2 point; i.e., R0 and the area under the curve U(R) − Q S = . (4) change. As a result, penetrability P increases for βx > ω0T1/2P 0 and decreases for βx < 0 by one to eight orders Numerical results are given in the next section. of magnitude as compared to the values calculated

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1447 for spherical nuclei (Tables 1 and 2). Penetrability 96 for identical clusters emitted from neighboring nuclei (a) always increases with charge asymmetry. 92 , MeV 88 If an additional degree of freedom—the angle θi between the emission direction of one of the nu- 84 clei (i = x, f) and the axial symmetry axis—is introduced, the nucleus–nucleus potential is a function PU 30 (b) of this collective variable. A change in the nucleus– nucleus potential leads to changes in the barrier pen- –log 26 etrability P and the potential energy Uc = Bx + Bf + U (B and B are the binding energies of the nuclei) 22 x f 0 π/6 π/3 π/2 of the contact configuration. As θi for deformed nuclei θ x increases, Р decreases and Uc increases. These dependences become sharper for large deformations and Fig. 2. (а) Contact-configuration potential U and (b)the large products ZxZf .Asanexample,P and U for the sign-reversed logarithm of the barrier penetrability P for 208 30 208 30 DNS Pb + Mg are shown in Fig. 2 as functions the dinuclear system Pb + Mg vs. the angle θx of the deviation of the light-nucleus axis of symmetry from the of θx.Bydefinition, the state with minimum potential energy is most probable for the contact system (it axis passing through the centers of masses of the nuclei. has the largest spectroscopic factor). In this case, it corresponds to the cluster configuration with θi =0. and by seven and eight orders of magnitude for 24Ne The S value for θi = π/6 is approximately 1/30 of the and 28Mg, respectively. value for θ =0. The barrier penetrability is maximal i According to Tables 1 and 2, the S value for 28Mg for θ =0. Therefore, the probability SP of cluster i is 17 orders of magnitude smaller than the value decay is also maximal, and the main contribution to for 14C. The spectroscopic factors for carbon ra- averaging of SP over θi comes from the vicinity of 222,224,226 θi ≈ 0. dioactivity of the even–even isotopes Ra as Analysis of the decays of odd nuclei requires calculated including deformation are close to the respective values calculated disregarding deformation. the inclusion of DNS states with nonzero orbital 223 angular momenta L. These states are effectively The S value in the odd isotope Ra is two orders of magnitude smaller than the values for the neighbor- taken into account by the centrifugal potential UL in nucleus–nucleus potential (3). Possible L values ing even–even isotopes. This relation indicates that are determined by the conservation laws for parity structure effects are noticeable for a nuclear decay and angular momentum. In particular, for the decay mechanism and expresses the known suppression of − the emission of an even particle from an odd nucleus 235U(7/2 ) → 26Ne(0+)+209Pb(9/2+), L can be as compared to the neighboring even–even nuclei. equalto1,3,5,and7.Figure3showsthedepen- This suppression explains strong deviation of the dences of the barrier penetrability on the possible widths measured for the decays 221Ra → 14C + 207Pb orbital angular momenta for the decays 235U → 26Ne, and 235U → 24,25,26Ne + 209,210,211Pb from known 223 → 14 229 → 21 θ Ra C, and Th Oat =0in compar- dependences. ison with P at L =0. It is seen that, as the orbital angular momentum increases, the penetrability de- The spectroscopic factors presented in Table 1 234 28 206 creases slightly due to an increase in the moment of (except for S for the decay U → Mg + Hg) inertia of the system. In the decay considered above, are one to four orders of magnitude smaller than the L does not exceed 10. All calculations are performed values for the phenomenological potential described with the inclusion of the angular momentum. in [16]. Our S values for 14C are two orders of magni- Tables 1 and 2 present spectroscopic factors S tude larger than the shell-model results [1], and our S 24 28 30 calculated by Eq. (4) using experimental data with, values for Ne, Mg, and Mg are two, five, and one respectively, the inclusion and exclusion of the static order of magnitude lower, respectively, than the cor- quadrupole deformation of the light cluster and resid- responding shell-model results [1]. The spectroscopic ual nucleus. The deformation parameters of the nu- factors obtained for 24Ne and 28Mg in [5] are three and clei in the ground state are taken from experimen- four orders of magnitude larger than the respective tal systematics [15], where only the absolute values spectroscopic factors in Table 1. The spectroscopic of the quadrupole deformation parameters were pre- factors obtained for 14С in [5] agree well with our sented. It is remarkable that, as the deformation of the calculations. light cluster is varied, S for a given channel changes Figures 4 and 5 show the spectroscopic factor strongly by more than one order of magnitude for 14C S extracted from experimental data as a function of

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1448 KUKLIN et al.

–logP 31

0246L

Fig. 3. Plot of −log P vs. L for the decays () 235U → 26Ne, () 223Ra → 14C, and ( ) 229Th → 21O.

–logSx

15 20 25 30 Ax

Fig. 4. Plot of −log Sx,whereSx is the spectroscopic factor calculated using the measured half-life, vs. the mass Ax of the light cluster. Quadrupole deformations of clusters and the residual nuclei () 208Pb, () 206Hg, and () 207Tl were taken into account in the calculations. The same symbols are connected by straight segments for clearness.

the cluster mass number Ax. Points connected by necessary in order to corroborate the half-life that has line segments correspond to decays with the same been already obtained [17]. heavy daughter nuclei indicated in the ﬁgures. It is Figure 6 shows the half-life as a function of barrier seen that the spectroscopic factor decreases expo- penetrability for “carbon,”“neon,” and “magnesium” nentially on average as the cluster mass increases. radioactivity. Points for each type of radioactivity are When deformation is taken into account, decays ac- located along the lines with approximately the same companied by the emission of 32,34Si and the de- slope. This property in principle follows from Eq. (1) cay 236U → 206Hg + 30Mg do not exhibit this de- if the S values for the identical clusters are approxi- pendence. This tendency is likely associated with the mately equal to each other. This tendency is characteristic of α decay. Analysis of the resulting S values fact that these cluster conﬁgurations are beyond the (Tables 1 and 2) shows that the spectroscopic factors Z ≈ Businaro–Gallone critical mass asymmetry ( x for the identical clusters emitted from neighboring 8–12), after which the potential of the DNS de- even–even nuclei coincide with each other within creases with decreasing mass asymmetry, and the one order of magnitude. This conclusion is corrobo- simple two-stage representation of cluster decay be- rated by microscopic calculations [1]. This fact makes comes doubtful. Moreover, new experimental data are it possible to predict T1/2 values for certain decays

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1449

–logSx 20

15 20 25 30 Ax

Fig. 5. The same as in Fig. 4, but for spherical residual nuclei and clusters.

(Tables 3 and 4). We use the following decays as In [21, 22], the possibility of fission through a the reference decays: 222Ra → 14C + 208Pb, 228Th → quasimolecular configuration, i.e., through the mass 20O + 208Pb, 232U → 24Ne + 208Pb, 234U → 28Mg + asymmetry coordinate, was discussed. A fissioning 206Hg, and 238Pu → 28Mg + 210Pb. The reliability nucleus sequentially passes through a number of of our semiempirical predictions for even–even ac- stages corresponding to different DNSs. Let the tinides is entirely determined by the accuracy of the spontaneous fission (s.f.) also occur through the measured widths of decays for neighboring reference cluster mechanism; i.e., fluctuations of the nuclear shape in the collective coordinate of mass asymmetry nuclei. The S value for the decay 232Pa → 25Ne + 207 give rise to the formation of the initial DNS with Tl is calculated as the product of the spectroscopic which the system begins to evolve in the energetically 232 24 208 factor of the decay U → Ne + Pb and the allowed region of the potential surface. Thus, the − structure suppression factor F =10 1 for the easy transition from cluster decay to spontaneous fission cluster transition [1]. Predictions obtained for T1/2 occurs when the Q value for the initial DNS is equal including (Table 3) and disregarding (Table 4) nuclear to the potential energy U(R0) at the contact point deformation differ only slightly for the same decays. of the nuclei. Since the system inevitably decays (Ps.f. =1) in the process of evolution in the energet- The spectroscopic factors S (Table 5) for decays ically allowed region (Q becomes much higher than in the “tin-radioactivity” region are simply obtained the deformed Coulomb barrier U(R ≈ R +1fm)) as the products of our spectroscopic factors for the b 0 indicated reference “lead-radioactive” nuclei and the ratios of the spectroscopic factors for the tin and log(T1/2[s]) lead regions as calculated in the microscopic theory [1]. The results give hope to detecting new de- 28Mg 24 cays with the yield of various clusters: 112,114,116Ba → 25 Ne 12C + 100,102,104Sn, 118Ce → 12C + 106Te, 118Ce → 16 102 124 → 20 104 124 → O + Sn, Sm Ne + Te, and Sm 20 14C 24Mg + 100Sn. For this region of nuclei, only the up- × 4 per limit T1/2 > 1.2 10 s is experimentally obtained 15 for the decay 114Ba → 12C + 102Sn [20]. Comparison shows that our predictions (Tables 3 and 5) strongly 10 differ from those made in [1, 2] in the tin radioactivity 20 25 30 35 region. The difference arises due primarily to the dif- –logP ference in the barrier penetrabilities. Our predictions of T1/2 (Table 5) for decays with the emission of the Fig. 6. Logarithm of the measured half-life T1/2 vs. 12C, 16O, and 20Ne nuclei from the tin radioactivity −log P ,whereP is the calculated barrier penetrability for 14 24 28 region nearly coincide with those made in [5]. the light clusters ( ) С, ( ) Ne, and () Mg.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1450 KUKLIN et al.

Table 3. Estimates of T1/2 for certain possible decays [here, S is the spectroscopic factor taken from the same decay of the nearest reference even nucleus (see Table 1 and the main text), P is the penetrability of the nucleus–nucleus potential calculated for a given dinuclear system, and the parameters of the clusters and residual nuclei are the same as in Table 1]

A → Ax + Af Q,MeV S P T1/2,s 224Th → 14C + 210Po 32.9 3.8 × 10−7 5.1 × 10−29 2.8 × 1014 226Th → 14C + 212Po 30.5 3.8 × 10−7 1.5 × 10−33 9.5 × 1018 229Th → 21O + 208Pb 43.18 2.9 × 10−15 1.3 × 10−31 1.4 × 1022 232Pa → 25Ne + 207Tl 58.95 2.1 × 10−21 1.1 × 10−23 2.4 × 1020 230U → 24Ne + 206Pb 61.40 2.1 × 10−20 3.7 × 10−23 7.2 × 1021 236U → 24Ne + 212Pb 59.9 2.1 × 10−20 4.2 × 10−25 6.9 × 1023 234Pu → 24Ne + 210Po 62.3 2.1 × 10−20 1.1 × 10−24 2.5 × 1023 234Pu → 28Mg + 206Pb 79.2 1.3 × 10−26 2.3 × 10−18 1.9 × 1023 237Pu → 29Mg + 208Pb 77.51 1.3 × 10−27 5.7 × 10−21 7.5 × 1026

Table 4. Estimates of T1/2 for certain possible decays under the assumption that the clusters and residual nuclei are spherical [here, S is the spectroscopic factor taken from the same decay of the nearest reference even nucleus (see Table 1 and the main text), P is the penetrability of the nucleus–nucleus potential calculated for a given dinuclear system, and the parameters of the clusters and residual nuclei are the same as in Table 1; the data from Table 2 are used]

A → Ax + Af Q,MeV S P T1/2,s 224Th → 14C + 210Po 32.9 1.1 × 10−8 5.7 × 10−27 9.6 × 1013 226Th → 14C + 212Po 30.5 1.1 × 10−8 1.6 × 10−31 3.4 × 1018 230U → 24Ne + 206Pb 61.4 6.3 × 10−15 2.6 × 10−29 3.5 × 1022 236U → 24Ne + 212Pb 59.9 6.3 × 10−15 6.2 × 10−31 1.5 × 1024 234Pu → 24Ne + 210Po 62.3 6.3 × 10−15 3.0 × 10−30 3.1 × 1024 234Pu → 28Mg + 206Pb 79.2 6.9 × 10−18 1.9 × 10−26 4.3 × 1022

s.f. and the measured half-lives T1/2 of actinides in mononucleus as Z increases. Comparison of Ss.f. and spontaneous fission are known, the spectroscopic S(Ax) provides the assumption that the initial DNS is an asymmetric DNS, where the light nucleus is factors Ss.f. of the initial DNS can be estimated by Eq. (4) as from the nuclear region Ca–Fe. The upper limit of this region is determined by the fact that the light nuclei π ln 2 S = . 66Cr, 66Mn, and 66Fe are observed in experiments on s.f. s.f. (5) ω0T1/2 spontaneous fission [23]. However, the observation of light clusters is limited because their yields are s.f. Then, T1/2/T1/2(Ax)=S(Ax)P (Ax)/Ss.f.. Corre- low due to the low penetrabilities of the nucleus– s.f. nucleus potential barrier. With further improvement spondingly, T ≤ T1/2(Ax) if Ss.f. ≥ S(Ax)P (Ax). 1/2 of experimental technique, the observation of such As is known from experiments, spontaneous fission rare events will apparently become possible. competes with cluster decay for nuclei heavier than 232U. Table 6 presents the weights of the initial DNS in the wave functions of the ground state of 4. CONCLUSIONS actinides. As is seen, Ss.f. increases with increasing charge number Z of the fissioning nucleus. This The above mechanism of cluster decay is associ- dependence is likely due to the fact that the absolute ated with dynamic oscillations of the decaying nu- value of the potential energy of the DNS at the cleus in the mass-asymmetry coordinate. The pro- Businaro–Gallone point decreases with respect to the cesses of the formation of a particular DNS and its

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 SPECTROSCOPIC FACTORS 1451

∗ 222 Table 5. Estimates of the half-lives T1/2 for tin radioactivity (the following reference decays are used: Ra → 14 208 228 20 208 230 24 206 234 28 206 ˜ C + Pb, Th → O + Pb, Th → Ne + Hg, and U → Mg + Hg; T1/2 values are the half-lives obtained with the spectroscopic factors taken from [1]; βx, βf ,andQ are taken from [18, 19]; and the calculations are performed with the parameters a0x = a0f =0.52 fm, r0x =1.15 fm, and r0f =1.16 fm)

˜ A → Ax + Af βx βf Q,MeV S P T1/2,s T1/2,s 112Ba → 12C + 100Sn 0.00 0.00 22.09 2.4 × 10−6 3.3 × 10−19 7.2 × 103 5.9 × 105 112Ba → 12C + 100Sn 0.00 0.00 21.46 2.4 × 10−6 2.0 × 10−20 1.2 × 105 9.8 × 106 114Ba → 12C + 102Sn 0.00 0.00 19.00 2.4 × 10−6 1.3 × 10−25 1.8 × 1010 1.5 × 1012 116Ba → 12C + 104Sn 0.00 0.02 17.30 2.4 × 10−6 1.3 × 10−29 1.8 × 1014 1.5 × 1016 118Ba → 12C + 106Sn 0.00 0.03 15.48 2.4 × 10−6 6.7 × 10−35 3.5 × 1019 2.5 × 1021 118Ce → 12C + 106Te 0.00 0.12 18.00 2.4 × 10−6 3.6 × 10−29 6.6 × 1013 5.4 × 1015 124Sm → 12C + 112Ba 0.00 0.21 15.95 2.4 × 10−6 2.2 × 10−39 1.1 × 1024 9.0 × 1025 126Sm → 12C + 114Ba 0.00 0.24 16.01 2.4 × 10−6 9.3 × 10−39 2.5 × 1023 2.1 × 1025 114Ba → 16O + 98Cd 0.01 0.01 26.46 3.9 × 10−10 1.5 × 10−27 9.6 × 1015 1.2 × 1016 116Ba → 16O + 100Cd 0.01 0.02 24.11 3.9 × 10−10 1.9 × 10−32 7.7 × 1020 9.6 × 1020 118Ce → 16O + 102Sn 0.01 0.00 29.44 3.9 × 10−10 1.6 × 10−24 8.8 × 1012 1.1 × 1013 122Nd → 16O + 106Te 0.01 0.12 27.54 3.9 × 10−10 3.2 × 10−29 4.5 × 1017 5.6 × 1017 124Sm → 16O + 108Xe 0.01 0.15 27.44 3.9 × 10−10 9.1 × 10−32 1.5 × 1020 2.0 × 1020 126Sm → 16O + 110Xe 0.01 0.17 26.75 3.9 × 10−10 6.7 × 10−33 2.1 × 1021 2.7 × 1021 118Ce → 20Ne + 98Cd 0.36 0.01 34.54 4.1 × 10−14 5.8 × 10−26 2.4 × 1019 2.0 × 1017 122Nd → 20Ne + 102Sn 0.36 0.00 36.54 4.1 × 10−14 1.8 × 10−26 7.7 × 1018 6.4 × 1016 124Sm → 20Ne + 104Te 0.36 0.05 38.65 4.1 × 10−14 3.7 × 10−25 3.7 × 1017 3.1 × 1015 126Sm → 20Ne + 106Te 0.36 0.12 35.35 4.1 × 10−14 7.0 × 10−30 2.0 × 1022 1.6 × 1020 122Nd → 24Mg + 98Cd 0.41 0.01 46.23 3.0 × 10−19 1.1 × 10−22 1.6 × 1020 3.8 × 1015 124Sm → 24Mg + 100Sn 0.41 0.00 51.97 3.0 × 10−19 3.4 × 10−18 5.6 × 1015 1.3 × 1011 126Sm → 24Mg + 102Sn 0.41 0.00 48.94 3.0 × 10−19 5.5 × 10−22 3.4 × 1019 7.8 × 1014

∗ The calculation of S is explained in the main text.

Table 6. Weights of the initial dinuclear systems in the wave function of the ground state of actinides

A A A Z Ss.f. Z Ss.f. Z Ss.f. 232U 4.8 × 10−43 234U 2.4 × 10−45 236U 1.5 × 10−45 236Pu 7.6 × 10−39 238Pu 9.5 × 10−40 240Pu 2.4 × 10−40 242Pu 5.6 × 10−40 244Pu 5.8 × 10−40 246Cm 2.1 × 10−36

decay are considered separately. The inclusion of the Ax increases. When nuclear deformations are taken positive static quadrupole nuclear deformation no- into account, the spectroscopic factors S for Ax > 28 ticeably reduces the nucleus–nucleus potential and, diﬀer from each other much less than the spectro- correspondingly, increases the barrier penetrability by scopic factors for lighter clusters, and the exponential one to eight orders of magnitude. For spherical nuclei, decrease in S is violated. Final conclusions require the spectroscopic factor S decreases exponentially as new experiments on the detection of Si nuclei.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1452 KUKLIN et al.

The half-lives T1/2 predicted with allowance for 7. D. N. Basu, Phys. Rev. C 66, 027601 (2003). the deformations of the light cluster and residual nu- 8. S. G. Kadmenskiı˘ et al., Izv. Akad. Nauk SSSR, Ser. cleus are nearly identical to those for spherical nuclei. Fiz. 50, 1786 (1986); 57 (1), 12 (1993). The half-lives have been predicted for various clus- 9. R. Blendowske, T. Fliessbach, and H. Walliser, Nucl. Phys. A 464, 75 (1987). ters in the tin-and lead-radioactivity regions. The 10. T. M. Shneidman et al., Phys. Lett. B 526, 322 connection between spontaneous ﬁssion and cluster (2002); Phys. Rev. C 67, 014313 (2003). radioactivity in this model is discussed. The weights 1 1 . V. V. Vo l k o v, DeepInelastic Nuclear Reactions of the initial DNS in the wave function of the ground (Energoizdat,´ Moscow, 1982) [in Russian]. states of certain actinides have been estimated. 12. G. G. Adamian et al.,Int.J.Mod.Phys.E5, 191 (1996). ACKNOWLEDGMENTS 13. A. B. Migdal, Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Nauka, Moscow, We are grateful to Yu.M. Tchuvil’sky and 1982; Interscience, New York, 1967). S.P. Tret’yakova for stimulating discussions and 14. C. Y. Wong, Phys. Rev. Lett. 31, 766 (1973). valuable remarks. This work was supported in part 15. S. Raman et al., At. Data Nucl. Data Tables 78,1 by the Russian Foundation for Basic Research and (2001). Deutsche Forschungsgemeinschaft. 16. P. R. Christensen and A. Winter, Phys. Lett. B 61B, 19 (1976). REFERENCES 17. A. A. Ogloblin et al., Phys. Rev. C 61, 034301 (2000). 1. Yu. M. Tchuvil’sky, Cluster Radioactivity (Mosk. 18. P. Moeller and J. J. Nix, At. Data Nucl. Data Tables Gos. Univ., Moscow, 1997) [in Russian]. 39, 213 (1988); Preprint LA-UR-86-3983, LANL (Los Alamos National Library, Los Alamos, 1986). 2. Yu.S.Zamyatinet al.,Fiz.Elem.´ Chastits At. Yadra 19. J. K. Tuli, Nuclear Wallet Cards (BNL, New York, 21, 537 (1990) [Sov. J. Part. Nucl. 21, 231 (1990)]. 2000). 3.S.P.TretyakovaandA.A.Ogloblin,inProceedings of the International Symposium on Nuclei Clus- 20. A. Guglielmetti et al., Phys. Rev. C 56, R2912 (1997). ters, Ed. by R. Jolos and W. Scheid (EP Systema, 21. V. A. Shigin, Yad. Fiz. 3, 756 (1966) [Phys. At. Nucl. Debrecen, 2003), p. 207. 3, 555 (1966)]; 14, 695 (1971) [14, 391 (1971)]. 4. G. A. Pik-Pichak, Yad. Fiz. 44, 1421 (1986) [Sov. J. 2 2 . V. V. Vo l k o v, Ya d . F i z . 62, 1159 (1999) [Phys. At. Nucl. Phys. 44, 923 (1986)]. Nucl. 62, 1086 (1999)]. 5. D. U. Poenaru et al., At. Data Nucl. Data Tables 34, 23. D. U. Poenaru, Nuclear Decay Modes (IOP Publ., 423 (1986). Bristol, 1996). 6. R.K.Gupta,S.Singh,andR.K.Puri,J.Phys.G18, 1533 (1992). Translated by R. Tyapaev

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1453–1486. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1511–1544. Original English Text Copyright c 2005 by Pomerantsev, Kukulin, Voronchev, Faessler. NUCLEI Theory

Dibaryon Model for Nuclear Force and the Properties of the 3N System*

V. N. Pomerantsev, V. I. Kukulin **, V.T.Voronchev,andA.Faessler1) Institute of Nuclear Physics, Moscow State University, Moscow, Russia Received March 21, 2005

Abstract—The dibaryon model for NN interaction, which implies the formation of an intermediate six- quarkbag dressed by a σ field, is applied to the 3N system, where it results in a new three-body force of scalar nature between the six-quarkbag and a third nucleon. A new multicomponent formalism is developed to describe three-body systems with nonstatic pairwise interactions and nonnucleonic degrees of freedom. Precise variational calculations of 3N bound states are carried out in the dressed-bag model including the new scalar three-body force. The unified coupling constants and form factors for 2N-and 3N-force operators are used in the present approach, in sharp contrast to conventional meson-exchange models. It is shown that this three-body force gives at least half the 3N total binding energy, while the weight of nonnucleonic components in the 3Hand3He wave functions can exceed 10%.Thenewforce model provides a very good description of 3N bound states with a reasonable magnitude of the σNN coupling constant. A new Coulomb 3N force between the third nucleon and dibaryon is found to be very important for a correct description of the Coulomb energy and rms charge radius in 3He. In view of the new results for Coulomb displacement energy obtained here for A =3nuclei, an explanation for the long-term Nolen–Schiffer paradox in nuclear physics is suggested. The role of the charge-symmetry-breaking effects in the nuclear force is discussed. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION: CURRENT PROBLEMS models based on the Yukawa concept. Among all IN A CONSISTENT DESCRIPTION OF 2N such puzzles, we mention here only the most re- AND 3N SYSTEMS WITH TRADITIONAL markable ones, such as the A puzzle in N + d FORCE MODELS y and N + 3He scattering [1, 2] and disagreements A few historical remarks should be made first. on the minima of differential cross sections (Sagara The current rather high activity in few-body physics puzzle) at E ∼ 150−200 MeV and polarization data started at the beginning of the 1960s, after mathe- for N + d [3], N + d, N + d [4], and N + 3He scat- matical formulation of the Faddeev equations for the tering. The strongest discrepancy between current three-body problem. The aim was claimed to estab- theories and respective experiments has been found lish unambiguously off-shell properties of the two- in studies of the short-range NN correlations in the body t matrix, which cannot be derived from two-body 3He(e, epp) [5], 4He(γ,pp) [6], and 3He(e, eNN) [7] scattering data only. It was hoped during that time processes. In addition to these particular problems, that just accurately solving the 3N scattering prob- there are more fundamental problems in the current lem would enable one to establish strong constraints theory of nuclear forces, e.g., strong discrepancies for the off-shell properties of the two-nucleon t ma- between the πNN, πN∆,andρNN form factors trix. However, more than forty years has passed since used both in one-boson-exchange (OBE) models then, but we are still unable to formulate such a two- for the description of elastic and inelastic scattering nucleon t matrix that can explain fully quantitatively and in the consistent parametrization of 2N and 3N the properties of even 3N systems. forces [8–11]. Many of these difficulties are attributed to a rather poor knowledge of the short-range behav- Moreover, since that time, many puzzles in few- ior of nuclear forces. This behavior was traditionally nucleon scattering experiments have been revealed associated with vector ω-meson exchange. However, which could not be explained by the current force the characteristic range of this ω exchange (for mω ∗ 780 MeV) is equal to about λω 0.2−0.3 fm; i.e., it This article was submitted by the authors in English. is deeply inside the internucleon overlap region. 1)Institute for Theoretical Physics, University of Tubingen,¨ Germany. In fact, since Yukawa, the NN interaction is ex- **e-mail: [email protected] plained by a t-channel exchange of mesons between

1063-7788/05/6809-1453$26.00 c 2005 Pleiades Publishing, Inc. 1454 POMERANTSEV et al. two nucleons. The very successful Bonn, Nijmegen, the whole short-range interaction (in the S wave).2) Argonne, and other modern NN potentials prove Assuming a reasonable qq-interaction model, many the success of this approach. But the short- and authors (see, e.g., [20–23]) have suggested that intermediate-range region in these potentials is more this mixture will result in both strong short-range parametrized than parameter-free microscopically repulsion (associated mainly with the s6 component) described. and intermediate-range attraction (associated mainly Besides the evident difficulties with the description with the above mixed-symmetry s4p2 component). for short-range nuclear force, there are also quite se- However, recent studies [22, 23] for NN scattering rious problems with a consistent description of basic on the basis of the newly developed Goldstone bo- intermediate-range attraction between nucleons. In son exchange qq interaction have resulted in purely the traditional OBE models, this attraction is de- repulsive NN contributions from both s6[6] and scribed as a t-channel σ exchange with artificially s4p2[42] 6q enhanced σNN vertices. However, accurate modern components. There is no need to say calculations of the intermediate-range NN interac- that any quark-motivated model for the NN force tion [12, 13] within the 2π-exchange model with ππ with π exchange between quarks inevitably leads to s-wave interaction have revealed that this t-channel the well-established Yukawa π-exchange interaction mechanism cannot give a strong intermediate-range between nucleons at long distances. attraction in the NN sector, which is necessary for Trying to solve the above problems (and to under- binding of a deuteron and fitting the NN phase shifts. stand more deeply the mechanism for the intermediate- This conclusion has also been corroborated by recent and short-range NN interaction), the Moscow– independent calculations [14]. Thus, the t-channel Tubingen¨ group suggested replacing the conven- mechanism of the σ-meson exchange should be re- tional Yukawa meson-exchange (t-channel) mech- placed by some other alternative mechanism that anism (at intermediate and short ranges) with the should result in the strong intermediate-range attrac- contribution of an s-channel mechanism describing tion required by even the existence of nuclei. the formation of a dressed 6q bag in the intermediate When analyzing the deep reasons for all these state such as |s6 + σ or |s6 +2π [8, 24]. It has been failures, we must lookat a most general element shown that, due to the change in the symmetry of which is common for all the numerous NN-force the 6q state in the transition from the NN channel to models tested in few-nucleon calculations for the last the intermediate dressed-bag state, a strong scalar 40 years. This common element is just the Yukawa σ field arises around the symmetric 6q bag. This concept for the strong interaction of nucleons in nu- intensive σ field squeezes the bag and increases its clei. Hence, if, after more than 40 years of devel- density [25]. The high quarkdensity in the symmetric opment, we are still unable to explain quantitatively 6q bag enhances the meson field fluctuations around and consistently even the basic properties of 3N and the bag and thereby partially restores the chiral 4N systems at low energies and relatively simple → symmetry [26]. Therefore, the masses of constituent processes like pp ppγ, this concept, which is a quarks and σ mesons decrease [8]. As a result of this cornerstone of all building of nuclear physics, should phase transition, the dressed-bag mass decreases be analyzed critically, especially in the regions where considerably (i.e., a large gain in energy arises), which applicability of this concept looks rather questionable. manifests itself as a strong effective attraction in the Since the quarkpicture and QCD have been de- NN channel at intermediate distances. The contri- veloped, the “nucleon–nucleon force community” is bution of the s-channel mechanism would generally more and more convinced that, at short ranges, the be much larger due to resonance-like enhancement.3) quarkdegrees of freedom must play an important role. One of the possible mechanisms for short-range In our previous works [8, 24], on the basis of NN interaction is the formation of the six-quark (6q) the above arguments, we proposed a new dibaryon bag (dibaryon) in the s channel. Qualitatively, many model for the NN interaction [referred to further as would agree with this statement. But to obtain a the “dressed-bag model” (DBM)], which provided a quantitative description of the nucleon–nucleon and quite good description of both NN phase shifts up the few-nucleon experimental data with this approach 2) with the same quality as the commonly used Bonn, We will denote the NN partial waves by capital letters (S,P,...), while the partial waves in all other cases will be Nijmegen, Argonne, and other equivalent potentials denoted by lower-case letters. is a quite different problem. 3)In the theory of nuclear reactions, the t-channel mechanism Within the 6q dynamics, it has long been can be associated with the direct nuclear reaction, where known [15–19] that the mixing of the completely only a few degrees of freedom are important, while the s- 6 channel mechanism can be associated with resonance-like symmetric s [6] component with the mixed-symmetry (or compound-nucleus-like) nuclear reactions with much s4p2[42] component can determine the structure of larger cross sections at low energies.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1455 to 1 GeV and the deuteron structure. The developed smooth matching between the nucleonic (at low mo- model includes the conventional t-channel contri- mentum transferred) and quarkcurrents (at very high butions (Yukawa π and 2π exchanges) at long and momentum transferred) and, at the same time, results intermediate distances and the s-channel contribu- in correct counting rules at high momentum trans- tions due to the formation of intermediate dressed- ferred. bag states at short distances. The most important distinction of such an approach from conventional So, it should be very important to test the above models for nuclear forces is the explicit appearance dibaryon concept of nuclear force in concise and con- of a nonnucleonic component in the total wave sistent 3N calculations and to compare the predic- function of the system, which necessarily implies the tions of the new model with the results of the conven- presence of new three-body forces (3BF) of several tional meson-exchange models. kinds in the 3N system. These new 3BF differ from conventionally used models for three-body forces. Thus, the aim of this workis to makea com- One important aspect of the novel 3BF should be prehensive study of the properties of the 3N system emphasized here. In conventional OBE models, the with 2N and 3N forces given by the DBM. However, main contribution to NN attraction is due to the the DBM introduces explicitly nonnucleonic (quark– t-channel σ exchange. However, the 3BF models meson) channels. Therefore, it is necessary to in- suggested until now (such as Urbana–Illinois or troduce a self-consistent multichannel few-body for- Tucson–Melbourne) are mainly based on the 2π malism for the study of the 3N system with DBM exchange with intermediate ∆-isobar production, interaction. We develop in this worksuch a general and the σ exchange either is not taken into account formalism, based on the approach which was sug- at all or is of little importance in these models. In gested in the 1980s by Merkuriev’s group [29–31] contrast, the σ exchange in our approach dominates for the boundary-condition-type model for pairwise in both 2N and 3N forces. In fact, in our approach, interactions. This general formalism leads immedi- just the unified strong σ field couples both two- and ately to a replacement of all two-body forces related three (and more)-nucleon systems; i.e., the general to the dibaryon mechanism by the respective three pattern of the nuclear interaction appears to be more (and many)-body forces, leaving a two-body charac- consistent. ter only for long-range Yukawa π and 2π exchanges, Our recent considerations have revealed that this which are of little importance for the nuclear binding. dibaryon mode is extremely useful in the explanation Another straightforward sequence of the formalism of very numerous facts and experimental results in developed here is a strong energy dependence of these nuclear physics, in general. We note here only a few many-body forces. In this work, we study all these of them. aspects in detail by means of applications to the (1) The presence of a dibaryon degree of freedom 3N-system properties. The preliminary version of this (DDF) can result in a very natural explanation of workis published in [32]. cumulative effects (e.g., the production of cumulative particles in high-energy collisions [27]). This paper is organized as follows. In Section 2, we present a new general multichannel formalism for (2) DDF leads to automatic enhancement of near- description of a two- and three-body system of par- threshold cross sections for one- and two-meson production in pp, pd, etc., collisions, which is required by ticles having inner degrees of freedom. In Section 3, many modern experiments (e.g., the so-called ABC we give a brief description of the DBM for the NN puzzle [28]). This is due to an effective enhancement system. In Section 4, we treat the 3N system with of meson–dibaryon coupling as compared to meson– DBM interactions, including a new 3BF.In Section 5, nucleon coupling. some details of our variational method are discussed, including calculation of the matrix elements for new (3) The incorporation of DDF makes it possi- Coulomb 3BF. The results of our calculations for ble (without the artificial enhancement of meson– ground states of 3Hand3He are given in Section 6, nucleon form factors) to share the large momentum of while in Section 7 we discuss the role of the new an incident probe (e.g., high-energy photon) among 3BF and present a new explanation for the Coulomb other nucleons in the target nucleus. displacement energy in 3He within our interaction (4) The DDF produces in a very natural way new model. A comprehensive discussion of the most im- short-range currents required by almost all experi- portant results found in the workis given in Section 8. ments associated with high momentum and energy In the Conclusion, we summarize the main results of transfers. the work. In the Appendix, we give the formulas for (5) Presence of the dressed-6q-bag components the matrix elements of all DBM interactions taken in in nuclear wave functions leads automatically to a the Gaussian symmetrized variational basis.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1456 POMERANTSEV et al. 2. THE GENERAL MULTICOMPONENT has straightforwardly been extended to three-body FORMALISM FOR 2N problem. In particular, it has been shown for the AND 3N SYSTEMS WITH COUPLED above two models that elimination of the internal INTERNAL AND EXTERNAL CHANNELS channels leads to the following recipe for embedding the energy-dependent pair interactions into the three- In the 1980s, the former Leningrad group body problem: the replacement of the pair energy by (Merkuriev, Kurepin, Motovilov, Makarov, Pavlov, et al.) constructed and substantiated with mathe- the difference between the three-body energy and ε → E − t matical rigor a model of strong interaction with the kinetic energy of the third particle: α α [31, coupling of external and internal channels [29–31, 33]. It has also been proved that the resulting Faddeev 33]. This model was a particular realization of a gen- equations for the external channel belong with such eral approach to interaction of particles having inner energy-dependent potentials to the Fredholm class degrees of freedom. The basic physical hypothesis is and are equivalent to the four-channel Schrodinger¨ that an energy-dependent interaction appears as a re- equation. sult of the internal structure of interacting particles.4) Our aim here is to extend our new NN force A general scheme proposed by S.P. Merkuriev et al. model—DBM—by using the above Merkuriev et al. was based on the assumption of existence of two in- approach to the 3N system. There are external dependent channels: an external one, which describes (nucleon–nucleon) and internal (quark–meson) chan- the motion of particles considered as elementary nels in our model, and coupling between them is bodies, i.e., neglecting their inner structure, and an determined within a microscopic quark–meson ap- internal one, which describes the dynamics of inner proach. In this section, we present a general mul- degrees of freedom. These channels can have quite ticomponent formalism for description of systems different physical and mathematical nature and their of two and three particles having internal structure, dynamics are governed by independent Hamiltonians. without assuming any specific form for coupling The main issues here were how to define the coupling between the external and internal channels. between external and internal channels and how to derive the corresponding dynamical equations (of Schrodinger¨ or Faddeev type) for particle motion in 2.1. Two-Body System the external channel. In [29–31], this coupling was postulated via We assume that the total dynamics in the two- boundary conditions on some hypersurface. Thus, body system is governed by a self-conjugated Hamil- such an approach is well applicable to hybrid models tonian h acting in the orthogonal sum of spaces: for NN interaction, which were rather popular in H = Hex ⊕Hin, the 1980s, e.g., the quarkcompound bag (QCB) model suggested by Simonov [34]. As for the 3N where Hex is the external Hilbert space of states de- system, the formalism of incorporation of the internal scribing motion of particles neglecting their internal channels (6q bags) was proposed for the first time structure and Hin is the internal Hilbert space cor- also within the QCB model [35]. The general scheme responding to internal degrees of freedom. Thus, the by Merkuriev et al. has allowed one to substantiate total state of the system Ψ ∈Hcanbewrittenasa this formalism. two-component column: In QCB-like models, the coupling between the   NN ex ex external ( ) and the inner (bag) channels was Ψ ∈H  given just on some hypersurface, similarly to the Ψ= . well-known R-matrix approach in nuclear physics. Ψin ∈Hin Later on, such a general approach was applied to the two-channel Hamiltonian model, where the internal The two spaces, Hex and Hin, can have quite a differ- Hamiltonian had a purely discrete spectrum and the ent nature, e.g., in the case of the NN system, Ψex only restriction imposed on the operators coupling depends on the relative coordinate (or momentum) of the external and internal channels was their bound- two nucleons and their spins, while Ψin can depend ness [33]. The above general multichannel scheme on quarkand meson variables. The two independent Hamiltonians are defined in each of these spaces: hex 4) From a more general point of view, the explicit energy depen- acts in Hex and hin acts in Hin. Here, hex includes dence of interaction reflects its nonlocality in time, while this the kinetic energy of relative motion and some part of time nonlocality, in turn, is a result of some excluded degrees ex of freedom. So, the explicit energy dependence is signaling two-body interaction v : some inner hidden (e.g., quark) degrees of freedom in NN ex ex interaction. h = t + v .

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1457 For the NN system, vex includes the peripheral part (ijk) = (123), (231), (312) of the meson-exchange potential and Coulomb inter- in action between nucleons (if they are protons). Cou- where hjk is the two-body internal Hamiltonian for pling between external and internal channels is deter- the pair (jk), I is the unity operator and ti is the mined formally by some transition operators: hex,in = kinetic energy of the third particle (i) in respect to the (hin,ex)∗. Further, one can write down the total Hamil- center of mass of the pair (jk). (Here and below we tonian h as a matrix operator, use capital letters for three-body quantities and small   letters for two-body ones.) ex ex,in  h h  The external three-body Hamiltonian acts in the h = , (1) Hex hin,ex hin external space 3 and includes the total kinetic energy T and the sum of external two-body interactions, not specifying so far the coupling operators (if opera- which were incorporated into the external two-body ex in ex,in Hamiltonians: tors h and h are self-adjoint and h is bounded, then the Hamiltonian h is the self-adjoint operator in Hex = T + vex. H). 3 ij i

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1458 POMERANTSEV et al. effective Schrodinger¨ equation for the external three- i.e., each term Wβγ is, generally speaking, a three- ex bodywavefunctionΨ3 , body force. In spite of the three-body character of eff ex ex such effective potentials, the corresponding Faddeev H3 (E)Ψ3 = EΨ3 , (9) equations have the Fredholm property and are equiv- with an effective (pseudo-)Hamiltonian alent to the four-channel Schrodinger¨ equation (it has been proved for a model with a discrete internal eff ex ex,in in in,ex H3 (E)=H3 + Hi Gi (E)Hi , (10) spectrum [33]). i in 2.3. A New Three-Body Force in the Three-Body where the resolvent of internal Hamiltonian Gi is a in System with External and Internal Channels convolution of the two-body internal resolvent gjk of pair (jk) and the free motion resolvent for the third In each internal channel one can introduce a new particle (i): interaction between the third particle and the pair as a whole. This leads to replacement of the operator for in − in −1 Gi =(E Hi ) (11) kinetic energy of the third particle ti by some (single- ∞ particle) Hamiltonian hi, 1 in 0 in = g (z)g (E − z)dz = g (E − t ) ⊗ I . ti ⇒ hi = ti + vi, (13) 2πi jk i jk i i −∞ in in Eq. (5) for Hi ,viz., in in ⊗ I I ⊗ Thus, the effective Hamiltonian in the external Hi = hjk i + jk hi. (14) three-body channel takes the form The physical meaning of such interactions will be eff { ex − } H = T + vjk + wjk(E ti) ; (12) discussed below, and here we treat only the formal jk aspects of their introduction. As the internal Hamilto- nian (14) is still a direct sum of the two-body internal i.e., the total effective interaction in the external chan- Hamiltonian and the Hamiltonian corresponding to nel of the three-body system is a sum of the two-body relative motion of the third nucleon, then its resolvent ex external potentials vjk and the two-body effective in- can be expressed as a convolution of two subresol- teractions with replacement of pair energy εi with the vents: difference between the total three-body energy and ∞ operator for the relative-motion kinetic energy of the in − in −1 1 in − Gi =(E Hi ) = gjk(z)gi(E z)dz, third particle: ε → E − t . 2πi i i −∞ Just this recipe for inclusion of energy-dependent (15) pair interactions in the three-body problem is widely −1 used in Faddeev calculations. This recipe has been where gi(ε)=(ε − hi) . Now, of course, the effec- rigorously proved in the works of Merkuriev et al. for tive interaction in the external channel is not reduced a two-channel model without a continuous spectrum to a sum of pairwise effective interactions with re- → − in the internal channel [33] and, in particular, for placement εi E ti. Nevertheless, this interaction the boundary-condition model [31]. We see, however, includes three terms Wjk and is still suitable for Fad- that this result is a direct consequence of the above deev reduction. But now there are no pure pairwise ex two assumptions and by no means is related to usage forces (except vjk)intheeffective Hamiltonian for the of any specific interaction model.5) external three-body channel. The resulting form of the effective three-body Moreover, if even the external interaction vi is dis- Hamiltonian (12) is suitable for the Faddeev reduc- regarded at all, each term wjk in the effective Hamil- tion. However, it should be emphasized that each tonian (12) includes a dependence on the kinetic energy of the third particle, i.e., can be considered, gen- term Wβγ in the effective Hamiltonian (12) includes a dependence on the kinetic energy of the third particle, erally speaking, as a three-body force. This dependence on the third-particle momentum reduces the 5)In the literature, however, there were also discussions of strength of the effective interaction between the other the alternative variants for embedding energy-dependent two particles due to a specific energy dependence of pairwise force into the three-body system [36, 37]. These the coupling constants (see below). Therefore, one schemes assume that the effective total energy ε12 of the can say that there are no pure two-body forces in two-body subsystem in the three-body system is obtained from the total three-body energy E in the following way: the three-body system in such an approach, with ε = E − t −v −v ,wherev is the two-body the exception of that part of the interaction which 12 12 13 23 ij ex interaction between particles i and j and an averaging is is included in v (for the NN system, it is just the assumed with the exact 3N wave function. peripheral part of meson exchange).

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3. DRESSED-BAG MODEL FOR NN σ FORCES N N Here, we give a brief description of the two- component DBM for the NN interaction. A detailed AB6q description has been presented in our previous pa- N N pers [8, 24]. (The effective-field theory description for the dibaryon model of nuclear force has also been Fig. 1. Effective NN interaction induced by the produc- developed recently [38].) The main assumptions of tion of an intermediate dressed bag. the DBM are following: (i) Interacting nucleons form at small and inter- The effective interaction w(E) resulting from the mediate distances (rNN ∼ 1 fm) a compound state— the dibaryon or six-quarkbag dressed with π, σ,and coupling of the external NN channel to the interme- ρ fields. diate dressed-bag state is illustrated by the graph in Fig. 1. (ii) The coupling of the external NN channel with this state gives the basic attractive force between To derive the effective interaction w for the NN nucleons at intermediate and small distances, the σ- channel in such an approximation, knowledge of the dressed bag giving the main contribution. full internal Hamiltonian hin of the dressed bag, as Thus, nucleon–nucleon system can be found in well as the full transition operator hin,ex,isnotnec- two different phase states (channels): the NN phase essary. We need only to know how the transition and the dressed-6q-bag phase. In the NN (external) operator acts on those dressed-bag states which are channel, the system is described as two nucleons in- included in the resolvent (16): hin,ex|α, k. The cal- teracting via OBE; in the internal 6q + σ channel, the culation of this quantity within a microscopic quark– system is treated as a 6q bag surrounded by a strong meson model results in a sum of factorized terms [24]: scalar–isoscalar σ field (a “dressed” bag).6) The ex- ex,in| JM | JM J ternal two-nucleon Hamiltonian includes the periph- h α , k = ϕL BL(k), (19) eral part of one-pion- and two-pion-exchange (OPE L and TPE, respectively) interaction and Coulomb in- where ϕJM ∈Hex is the NN transition form factor teraction: L and BJ (k) is the vertex function dependent on the σ- ex { OPE TPE} Coul L h = t + v + v (with soft cutoff) + v . meson momentum. In the simplest version of the DBM, we used a Here, we should elucidate our notation in re- pole approximation for the dressed-bag (internal) respect to the quantum numbers of angular momenta. gin In general, the 6q-state index α includes all the solvent : ≡ quantum numbers of the dressed bag, i.e., α |α, kα, k|d3k { } gin(E)= , (16) J, M, S, T, Lb,Lσ ,whereLb, S, T , J,andM are E − E (k) α α the orbital angular momentum of the 6q bag, its spin, isospin, total angular momentum, and its projection | where α is the 6q partofthewavefunctionforthe on the z axis, respectively, and Lσ is the orbital dressed bag and |k represents the plane wave of the angular momentum of the σ meson. However, in σ-meson propagation. Here, Eα(k) is the total energy the present version of the DBM, the s-wave state of the dressed bag: of the 6q bag with the s6 configuration only is taken Eα(k)=mα + εσ(k), (17) into account, so that Lb =0, J = S, and thus the isospin of the bag is uniquely determined by its spin. where The states of the dressed bag with Lσ =0 should lie 2 2 εσ(k)=k /(2mα)+ωσ(k) mσ + k /(2m ¯ σ), higher than those with Lσ =0. For this reason, the (18) former states are not included in the present version mσmα of the model. Therefore, the state index α is specified m¯ σ = , here by the total angular momentum of the bag J and mσ + mα (if necessary) by its z projection M: α ⇒{J(M)}. ω (k)= m2 + k2 is the relativistic energy of the σ σ σ Thus, the effective interaction in the NN channel meson, and mσ and mα are the masses of the σ meson ≡ ex,in in in,ex and 6q bag, respectively. w(E) h g (E)h canbewrittenasasumof separable terms in each partial wave: 6)A full description of the NN interaction at energies E ∼ J 1 GeV still requires other fields in the bag, such as 2π, ρ, w(E)= wLL (r, r ,E), (20) and ω. JLL

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Table 1. Deuteron properties in the DBM and other current NN models

2 −1/2 Model Ed,MeV PD,% rm,fm Qd,fm µd,n.m. AS,fm η(D/S) RSC 2.22461 6.47 1.957 0.2796 0.8429 0.8776 0.0262 Moscow 99 2.22452 5.52 1.966 0.2722 0.8483 0.8844 0.0255 Bonn 2001 2.224575 4.85 1.966 0.270 0.8521 0.8846 0.0256

DBM(I) P6q =3.66% 2.22454 5.22 1.9715 0.2754 0.8548 0.8864 0.02588

DBM(II) P6q =2.5% 2.22459 5.31 1.970 0.2768 0.8538 0.8866 0.0263 Experiment 2.224575 – 1.971 0.2859 0.8574 0.8846 0.0263a) a) An average value of the asymptotic mixing parameter η over the results of a few of the most accurate experiments is presented here (see [40–43]). with As one can see from the comparison between ∗ J J JM J JM Eqs. (22) and (24), the integral I in Eq. (24) is wLL (r, r )= ϕL (r)λLL (E)ϕL (r ). (21) LL M equal to the energy derivative (with opposite sign) of J J the coupling constant λLL (E): The energy-dependent coupling constants λ (E) LL J appearing in Eq. (21) are directly calculated from the J dλLL (E) I = − , loop diagram shown in Fig. 1; i.e., they are expressed LL dE in terms of the loop integral of the product of two and thus we get an interesting result: transition vertices B and the convolution of two prop- dλ(E) agators for the meson and quarkbag with respect to ||Ψin||2 ∼− ; the momentum k: dE ∞ J J ∗ i.e., the weight of the internal 6qN state is propor- J BL(k)BL (k) λLL (E)= dk . (22) tional to the energy derivative of the coupling con- E − Eα(k) 0 stant of effective NN interaction. In other words, the stronger the energy dependence of the interaction in J The vertex form factors BL(k) and the potential the NN channel, the larger the weight of the channel JM ∈Hex form factors ϕL have been calculated in the corresponding to nonnucleonic degrees of freedom. microscopic quark–meson model [8, 24]. This result is in full agreement with the following When the NN-channel wave function Ψin is well-known hypothesis: energy dependence of inter- obtained by solving the Schrodinger¨ equation with action is a sequence of underlying inner structure of the effective Hamiltonian heff(E), the internal (6qN) interacting particles. component of the wave function is found from Eq. (4): The total wave function of the bound state Ψ must be normalized to unity. Assuming that the external J k ex in | JM BL( ) JM| ex (nucleonic) part of the wave function Ψ found from ΨJM(E)= α ϕL Ψ (E) , E − Eα(k) the effective Schrodinger¨ equation has the standard L normalization ||Ψex|| =1,onefinds that the weight of (23) the internal part, i.e., the dressed-bag component, is equal to where the underlined part can be interpreted as the mesonic part of the dressed-bag wave function. ||Ψin||2 P = . (25) The weight of the internal dressed-bag component in 1+||Ψin||2 of the bound-state wave function (with given value J) in Thus, the NN interaction in the DBM approach is is proportional to the norm of ΨJM: a sum of peripheral terms (vOPE and vTPE) represent- in 2 JM 2 JM ex ex JM || || || || | | ΨJM = α ϕL Ψ Ψ ϕL (24) ing OPE and TPE with soft cutoff parameter ΛπNN LL and an effective interaction w(E) [see Eqs. (20), (21)], J J ∗ B (k)B (k) which is expressed (in a single-pole approximation) × L L dk . (E − E (k))2 as a one-term separable potential with the energy- α dependent coupling constants (22). The potential IJ JM LL form factors ϕL (r) are taken as the conventional

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| | 7) W (E) harmonic oscillator wave functions 2S and 2D . 3 1 3 Therefore, the total NN potential in the DBM model σ ex can be represented as v1 (OPE + TPE) 6q OPE TPE Coul vNN = v + v + v (26) 2 2 + w(E)+λΓ, 3BF W1 where Γ=|ϕ0ϕ0| is the projector onto |0S harmonic oscillator function and the constant λ should 1 1 be taken to be sufficiently large. The model described above gives a very good de- Fig. 2. Different interactions in the 3N system for one 1 3 3 of three possible combinations (1+23) of three nucle- scription for singlet S0 and triplet S1− D1 phase ex ons: the peripheral two-nucleon interaction v1 is due to shifts and mixing parameter ε1 in the energy region OPE + TPE; the effective two-body interaction W1(E) from zero up to 1 GeV [24]. The deuteron observables is induced by the production of the dressed 6q bag and 3BF obtained in this model without any additional or free meson-exchange 3BF W1 . parameter are presented in Table 1 in comparison with some other NN models and experimental values. The quality of agreement with experimental data for the and m¯ = mN mα/(mN + mα) is a reduced mass of NN phase shifts and deuteron static properties found the nucleon and 6q bag. In the pole approximation, with the presented force model, in general, is higher this effective interaction reduces to a sum of two- than those for the modern NN potential model such body separable potentials with the coupling constants as Bonn, Argonne, etc., especially for the asymp- depending on the total three-body energy E and the totic mixing parameter η and the deuteron quadrupole third-particle momentum qi: moment. The weight of the internal (dressed-bag) Wi(pi, p , qi, q ; E)=δ(qi − q ) (29) component in the deuteron is varied from 2.5 to 3.6% i i i q2 in different versions of the model [8, 24]. × JiMi Ji − i JiMi ϕ (pi)λ E ϕ (pi). Li LiLi Li 2¯m JiMiLiLi 4. THREE-NUCLEON SYSTEM WITH DBM INTERACTION When using such an effective interaction, one must also include an additional 3BF due to the For description of the three-body system with meson-exchange interaction between the dressed DBM interaction, the momentum representation is bag and the third nucleon (see the next subsection). more appropriate. We will employ the same notation The pattern of different interactions arising in the 3N for functions in both the coordinate and momentum representations. The following notation for coordi- system in such a way is illustrated in Fig. 2. nates and momenta are employed: ri (pi) is relative In the single-pole approximation, the internal coordinate (momentum) of pair (jk), while ρi (qi)is (dressed-bag) components of the total wave function the Jacobi coordinate (momentum) of the ith particle, are expressed in terms of the nucleonic component ex and k is usually the momentum of the σ meson. Ψ (pi, qi) as Ji JiMi B (k)χ (qi) Ψin(k, q ; E)= |αJiMi Li Li , 4.1. Effective Interaction Due to Pairwise NN i i − − 2 E Eα qi /(2m) Forces JiMiLi (30) One obtains an effective Hamiltonian for the external 3N channel according to a general recipe for JiMi where χ (qi) are the overlap integrals of the ex- transition from a two- to three-particle system: Li ternal 3N component and the potential form factors eff { ex } JiMi H = T + vi + Wi(E) , (27) ϕ : Li i χJiMi (q )= ϕJiMi (p )Ψex(p , q )dk . (31) where each of three effective potentials takes the form Li i Li i i i i W (E)=δ(q − q )w (E − q2/(2m ¯ )), i i i i i (28) These overlap functions depend on the momentum 7)It was first suggested [39] long ago and then confirmed in (or coordinate), spin, and isospin of the third nucleon. detailed 6q microscopic calculations [15] that the 6q wave For brevity, the spin–isospin parts of the overlap function in the NN channel corresponds just to 2Ω excited functions and corresponding quantum numbers are 6q-bag components |s4p2[42]LST , while the ground state omitted unless they are needed. In Eqs. (29)–(31) and |s6[6] describes the wave function in the bag channel. below, we keep the index i in the quantum numbers

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(a) (b) (c) N N N N N N

π σ σ σ N N N N N N

AB6q AB6q AB6q N σ N N σ N N N

Fig. 3. The graphs corresponding to three new types of three-body force.

Li and Ji in order to distinguish the orbital and total Using Eq. (33), the norm of the 6qN component angular momenta attributed to the 2N form factors can eventually be rewritten as from the respective angular momenta J and L of the whole 3N system. ||Ψin||2 = ||αJi || χJiMi (q ) (34) i Li i JiMi LiLi It should be noted that the angular part of the J M i i d J J M function χL (qi) in Eq. (31) is not equal to YLiMi (ˆq). × − i − 2 i i i λL L (E qi /(2m)) χL (qi)dqi. This part also includes other angular orbital momenta dE i i i due to coupling of the angular momenta and spins of Due to explicit presence of the meson variables in the dressed bag and those for the third nucleon. In our approach, it is generally impossible to deﬁne the the next section, we consider the spin-angular and wave function describing the relative motion of the isospin parts of the overlap functions χJiMi (q ) in N Li i third nucleon ψ(q) in the 6qN channel. However, more detail. in by integrating Ψi (k, q) with respect to the meson momentum k, one can obtain an average momentum The norm of each 6qN component for the 3N distribution of the third nucleon in the 6qN channel bound state is determined by the sum of the integrals: (i.e., that weighted with the σ-meson momentum distribution). On the basis of Eq. (33), we can attribute ||Ψin||2 = ||αJiMi || χJiMi (q ) (32) the meaning of the third nucleon wave function in the i Li i 6qN channel to the quantity JiMi LiLi   JiMi ψ˜ (qi) (35)   Li    Ji Ji  BL (k)B (k) 1/2 i Li JiMi × d J J M 2 dk χL (qi)dqi. − i − 2 i i  2  i = λ (E qi /(2m)) χL (qi).  q  LiLi i  E − E − i  dE α 2m With this “quasi wave function,” one can calculate The internal loop integral with respect to k in Eq. (32) the mean value of any operator depending on the (in braces) can be replaced by the energy derivative momentum (or coordinate) of the third nucleon. We J note that the derivative −dλ/dE is always positive. of λL:

Ji Ji B (k)B (k) Li L i dk (33) 4.2. Three-Body Forces in the DBM − − 2 2 E Eα qi /(2m) In this study, we employ the eﬀective interac- d q2 − Ji − i tion (29) and take into account the interaction be- = λL L E . dE i i 2m tween the dressed bag and the third nucleon as an additional 3BF. We consider here three types of 3BF: Thus, the weight of the 6qN component in the 3N one-meson exchange (π and σ) between the dressed system is determined by the same energy dependence bag and the third nucleon (see Figs. 3a and 3b) J of the coupling constants λLL (ε) as the contribution and the exchange by two σ mesons where the third- of the 6q component in the NN system but at a shifted nucleon propagator breaks the σ loop of the two-body energy. force—2σ process (Fig. 3c).

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All these forces can be represented in the effective q3 q0 q3' Hamiltonian for the external 3N channel as some integral operators with factorized kernels: q0 – q3 q0 – q3' σσ 3BF p2 p2' W(i) (pi, pi, qi, qi; E) (36) –q0 JM 3BF JJ J M = ϕL (pi) WLL (qi, qi; E)ϕL (pi). 6q bag JMJM LL p1 p1' Therefore, matrix elements for 3BF include only the overlap functions, and thus the contribution of 3BF is Fig. 4. The graph illustrating the three-body scalar force proportional to the weight of the internal 6qN compo- due to two-sigma exchange (2σ process). nent in the total 3N wave function. To our knowledge, the first calculation of the 3BF contribution induced by OPE between the 6q bag and the third nucleon 2 was done by Fasano and Lee [44] in the hybrid QCB the values for constant λ(E − q /(2m)), so that the model using perturbation theory. They used the model vertex functions B(k) can be excluded from formulas where the weight of the 6q component in a deuteron for OME 3BF matrix elements. The details of cal- was about 1.7%, and thus they obtained a very small culations for such matrix elements are given in the value of −0.041 MeV for the 3BF OPE contribution Appendix. to the 3N binding energy. Our results for the OPE 3BF agree with the results obtained by Fasano and 4.2.2. 2σ process. The 2σ process (TSE) shown Lee (see Table 2 in Section 7), because the OPE in Fig. 4 also contributes significantly to 3BF. This contribution to 3BF is proportional to the weight of 3N interaction seems less important than the OSE the 6q component, and in our case, it should be at force, because this interaction imposes a specific least twice their calculation. However, we found that a kinematic restriction on the 3N configuration.8) much larger contribution comes from scalar σ-meson exchanges: one-sigma exchange (OSE) and two- The operator of the TSE interaction includes sigma exchange (TSE). We emphasize that, due to explicitly the vertex functions for the transitions (proposed) restoration of chiral symmetry in our ap- (NN ⇐⇒ 6q + σ), so that these vertices cannot be proach, the σ-meson mass becomes about 400 MeV, excluded similarly to the case of OME. Therefore, and thus the effective radius of the σ-exchange inter- we have to choose some form for these functions. action is not as small as that in conventional OBE It is natural to require that these vertices be the models. Therefore, we cannot use perturbation theory same as those assumed in the two-body DBM; i.e., anymore to estimate the 3BF contribution but have they can be normalized by means of the coupling to do the full calculation including 3BF in the total constants λ(E), which, in turn, are chosen in the three-body Hamiltonian. 2N sector to accurately describe NN phase shifts 4.2.1. One-meson exchange between the and deuteron properties [see below Eq. (41) for vertex dressed bag and third nucleon. For the one- normalization]. We use the Gaussian form factor for meson-exchange (OME) term, the three-body inter- these vertices: 3BF JiJi action W takes the form LiLi J J OME i i −b2k2 WL L (qi, qi; E) (37) e i i BJ (k)=BJL , (39) J L 0 B i (k) 2ωσ(k) = dk Li − − 2 E Eα qi /(2m) where k is the meson momentum and the parameter Ji b is taken from the microscopic quark model [24]: BL (k) × V OME(q , q ) i . i i − − 2 2 5 2 E Eα qi /(2m) b = b ,b=0.5 fm. (40) 24 0 0 Therefore, the matrix element for OME can be expressed in terms of the internal “bag” compo- 8)It follows from the intuitive picture of this interaction that this in nents Ψi : force can be large only if the momentum of the third nucleon is almost opposite to the momentum of the emitted σ meson. ex| | ex in| OME| in Ψ OME Ψ =3Ψi V Ψi . (38) Thus, a specific 3N kinematic configuration is required when two nucleons approach close to each other to form a bag, The integral with respect to the σ-meson momentum while the third nucleon has a specific space localization and k (37) can be shown to be reduced to a difference of momentum.

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JL Then, the vertex constants B0 should be found from Hamiltonian, we used an iterational procedure in re- the equation spect to the total energy E for solving this equation: JL JL −2b2k2 tot (n−1) ex(n) (n) ex(n) 1 B0 B0 e J H (E )Ψ = E Ψ . dk = λ (E), 3 − − · LL (2π) (E mα εσ(k)) 2ωσ(k) Such iterations can be shown to converge if the en- (41) ergy derivative of effective interaction is negative (for J our case, this condition is valid always). For our cal- where λLL (E) are the coupling constants employed in the construction of the DBM in the 2N sector and culations, 5–7 iterations usually provide the accuracy are fixed by NN phase shifts. For the σNN vertices, of five decimal digits for the 3N binding energy. 2 2 we take also the Gaussian form factor: g e−α k Construction of a 3N variational basis. Here, σNN we give the form of the basis functions used in this with α2 = b2/6. 0 workand the corresponding notation for the quantum Then, the box diagram in Fig. 4 can be expressed numbers. The wave function of the external 3N chan- in terms of the integral over the momentum q0 of the nel, Ψex, can be written in the antisymmetrized basis third nucleon in the intermediate state: as a sum of three terms: TSE JJ 2 JL JL W (q, q ; E)=δJJ g B B (42) ex (1) (2) (3) LL σNN 0 0 Ψ =Ψex +Ψex +Ψex , (45) 1 exp[−(α2 + b2)(q − q)2] × dq 0 where the label (i =1, 2, 3) enumerates one of three (2π)3 0 m2 +(q − q)2 σ 0 possible sets of the Jacobi coordinates (ri, ρi).Every 1 exp[−(α2 + b2)(q − q)2] term in Eq. (45) takes the form × 0 . − − 2 2 − 2 E mα q0/(2m) mσ +(q0 q ) (i) γ (i) Ψex = CnΦγn. (46) Thus, the matrix element for the total contribution of γ n TSE eventually takes the form (i) The basis functions Φγn are constructed from Gaus- TSE =3 χJiMi (q) (43) sian functions and corresponding spin-angular and Li isospin factors: JiMi,Li,Li λ l TSE JiJi JiMi (i) γ i i {− 2 − 2} × W (q, q ; E)χ (q )dqdq . Φγn = Nn ri ρi exp αγnri βγnρi (47) LiLi Li ×F(i) T (i) After the partial wave decomposition, these six- γ (ˆri, ρî) γ , dimensional integrals can be reduced to two-dimen- F(i) T (i) sional integrals, which are computed numerically by where the spin-angular γ (ˆri, ρî) and isospin γ means of the appropriate Gaussian quadratures. components of the basis functions are given in the Appendix and the composite label γ ≡ γ(i)= We should emphasize here that both the two- { } nucleon force induced by the DBM and two parts of λiliLSjkStjk represents the respective set of quan- the 3BF contribution in our approach, i.e., OSE and tum numbers for the basis functions (47): λi is the TSE, are all taken with unified coupling constants orbital angular momentum of the (jk)pair;li is the and unified form factors in Eqs. (37), (39)–(41), in orbital angular momentum of the third nucleon (i) a sharp contrast to the traditional meson-exchange relative to the center of mass for the (jk)pair;L is the models (see also Section 8). total orbital angular momentum of the 3N system; Sjk and tjk are the spin and isospin of the (jk)pair, respectively; and S is the total spin of the system. We 5. VARIATIONAL CALCULATIONS OF 3N omit here the total angular momentum J =1/2 and SYSTEM WITH DBM INTERACTION its z projection M,aswellasthetotalisospinofthe The effective Schrodinger¨ equation for the external system T =1/2 and its projection Tz (in this work, 3N part of the total wave function Htot(E)Ψex(E)= we neglect the very small contribution of the T =3/2 EΨex(E) with Hamiltonian component). The nonlinear parameters of the basis functions 3 α and β are chosen on the Chebyshev grid, Htot(E)=T + {vex + W (E)+W 3BF(E)} γn γn i i i which provides the completeness of the basis and i=1 (44) fast convergence of variational calculations [46]. As was demonstrated earlier [47], this few-body Gaus- has been solved by variational method using an an- sian basis is very flexible and can represent quite tisymmetrized Gaussian basis [45]. Because of the complicated few-body correlations. Therefore, it leads explicit energy dependence of the three-body total to accurate eigenvalues and eigenfunctions. The

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1465 formulas for the matrix elements of the Hamiltonian li and J are the orbital and total angular momenta (for local NN interactions) on the antisymmetrized of the third (ith) nucleon, respectively; and T 1 is 2 tzi Gaussian basis are given in paper [45]. The matrix isospinor corresponding to the third nucleon. In the elements of DBM interactions on this basis are given present calculation for the ground states of 3Hand in the Appendix. 3He (with J =1/2), we have considered the two low- Wave function in the internal 6qN channel. est even partial wave components (S and D)in3N Having the 3N component Ψ found in the above 3N wave functions only. Therefore, l can take only two variational calculation, one can construct the inner i (i) values: 0 or 2. Moreover, the total angular momentum 6qN-channel wave function Ψin ,whichdependson of the third nucleon J is uniquely determined by value the coordinate (or momentum) of the third nucleon of li: J =1/2 at li =0and J =3/2 at λi =2.So, and the σ-meson momentum and includes the bag actually, there is no summation over J in Eq. (49). wave function [see Eq. (30)]. By integrating the mod- It is easy to see that the three form factors ϕJi used ulus squared of this function with respect to the me- Li 0 1 1 son momentum and inner variables of the bag, one in the present work( ϕ0, ϕ0,andϕ2) determine five ra- obtains the density distribution of the third nucleon JiLi dial components of the overlap function Φ J (qi) and relative to the 6q bag in the 6qN channel. This density li five respective components of the quasi wave function can be used to calculate further all observables whose operators depend on the variables of the nucleons and for the 6qN channel. To specify these components, it the bag. However, it is much more convenient and is sufficient to give three quantum numbers, e.g., Sd, l ,andL , and we will use the notation Ψin (q ) for easier to deal with the quasi wave function of the i i Sdli,Li i third nucleon in the 6qN channel, which has been these radial components: introduced by Eq. (35). in 1 To calculate matrix elements of the 3BF Coulomb Ψ : Ji = Sd =0,td =1,Li =0,li =0, J = , 00,0 2 and OPE forces, one needs the spin–isospin part of 6qN components of the total wave function. Here, we in 1 Ψ : Ji = Sd =1,td =0,Li =0,li =0, J = , give them explicitly. The potential form factors ϕJiMi 10,0 2 Li JiMi (i) now include the spin–isospin part Y (pî)T with in 3 LiSd td Ψ : J = S =1,t =0,L =0,l =2, J = , 12,0 i d d i i 2 quantum numbers corresponding to the dressed bag: J M t t 1 i i d dz Ji YJiMi T (i) in ϕ = φ (pi) (pî) ; (48) Ψ : Ji = S =1,t =0,Li =2,li =0, J = , LiSd Li LiSd td 10,2 d d 2 (i) T = |tjtk : tdtd . td z in J 3 Ψ12,2 : Ji = Sd =1,td =0,Li =2,li =2, = . The full set of the quantum numbers labeling the 2 L form factors includes the total (Ji) and orbital ( i) Finally, we give a formula for the total quasi wave angular momenta, related to the vertex form factor, function in internal channel (i), separating out explic- and also the spin and isospin numbers S , t ,andt , d d dz itly its spin-angular and isospin parts, which include related to the dressed bag. However, since the present the spin–isospin part of the bag wave function: version of the DBM involves the bag states with zero   orbital angular momentum, we have Sd = Ji, while   the bag spin and isospin are supplementary to each Ψin = Ψin (q ) (50) i  Sdli,Li i  other: td + Sd =1. Hence, we will omit the quantum l S L i d i numbers Sd and td where they are unnecessary. 1 1 J M J M × l (J )S : JM t : TT . The total overlap function χ i i (i)=ϕ i i |Ψ i 2 d d 2 z Li Li 3N can be written (with its spin–isospin part), e.g., as The explicit dependence of this function on the isospin JiMi projection T is important for calculation of the χ (qi) (49) z Li Coulomb matrix elements and rms charge radius. J L = Φ i i (q )J mJ J M |JM liJ i i i The interaction matrix elements include the over- J li lap integrals of the potential form factors with the J mJ 1 (1) (2) (3) ×Y (qˆ ) t t t |TT T 1 . basis functions Φγ,n =Φγ,n +Φγ,n +Φγ,n,whereall l 1 i d dz 2 zi z tz i 2 2 i five above components of the overlap function enter Here, J and M are the total angular momentum of the matrix elements independently (certainly, some the 3N system and its z projection; T and Tz are the of the matrix elements can vanish). The explicit for- total isospin of the 3N system and its z projection; mulasfortheaboveoverlapfunctionsanddetailed

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1466 POMERANTSEV et al. formulas for the matrix elements of all DBM inter- 6. COULOMB EFFECTS IN 3He actions are given in the Appendix. When calculating both the normalization of the internal components In this section, we will demonstrate that the DBM and observables, the 6qN components distinguished approach leads to some new features related to the 3 only by their radial parts can be summed. Thus, only Coulomb effects in nuclei, and, in particular, in He. three different components of the 6qN quasi wave First of all, the additional Coulomb force arises be- 6q function remain: one S-wave singlet (Sd =0), cause the bag and rest nucleon can have electric in ≡ in charges. We have found that this new Coulomb three- Ψ00 Ψ00,0, body force is responsible for a significant part of the 3 and two triplet ones (Sd =1): total He Coulomb energy (this three-body Coulomb 3N Ψin =Ψin +Ψin , Ψin =Ψin +Ψin . (51) force has been missed fully in previous calcula- 10 10,0 10,2 12 12,0 12,2 tions within hybrid 6qN models [48]). The total weight of each of the three 6qN components The second feature of the interaction model used is equal to here is the absence of the local NN short-range (i) || in ||2 || in ||2 || in ||2 repulsive core. The role of this core is played by the Pin = Ψ00 + Ψ10 + Ψ12 , (52) condition of orthogonality to the confined 6q states i =1, 2, 3. forbidden in the external NN channel. This orthog- Now, let us introduce the relative weights of the indi- onality requirement imposed on the relative-motion vidual 6qN components: NN wave function is responsible for the appearance ||Ψin ||2 ||Ψin ||2 of some inner nodes and respective short-range loops P in = 00 ,Pin = 10 , S0 (i) S1 (i) (53) in this wave function. These short-range nodes and Pin Pin loops lead to numerous effects and general conse- ||Ψin ||2 quences for the nuclear structure. One of these con- P in = 12 . D (i) sequences is a rather strong overestimation of the Pin Coulomb contribution when using the interaction be- After renormalization of the full four-component tween pointlike nucleons. Thus, it is necessary to take wave function, the total weight of all internal compo- into account the finite radius of the nucleon charge nents is equal to distribution.9) 3P (i) Finally, in order to obtain an accurate Coulomb P = in 3 3 in (i) (54) displacement energy ∆E = E ( H) − E ( He), 1+3P C B B in one should take into consideration the effects as- (here, we assume that the 3N component of the to- sociated with the small mass difference between tal wave function, Ψex, obtained from the variational the proton and neutron. It is well known [49] that calculation, is normalized to unity), while the total theabovemassdifference makes a rather small 3 weight of the 3N component Ψ3N is equal to contribution to the difference between the He and 3 1 H binding energies. Therefore, it was usually taken P = =1− P . ex (i) in (55) into account in a perturbation approach. However, 1+3Pin since the average kinetic energy in our case is twice It is also interesting to find the total weight of the kinetic energy in conventional force models, this the D wave with allowance for nonnucleonic compo- correction is also expected to be much larger in our nents: case. Hence, we present here the estimation for such ex − in a correction term without using perturbation theory. PD = PD (1 Pin)+PD Pin. (56) Numerical values of all above probabilities for internal and external components are given below in 6.1. “Smeared” Coulomb Interaction Table 2. The total weight of all 6qN components P6qN ≡ Pin in the 3N system turns out to be rather The Gaussian charge distribution ρ(r) that corre- large and approaches or even exceeds 10%.Further- sponds to the rms charge radius rch and is normalized more, taking into account the short-range character to the total charge z, 4π ρr2dr = z, can be written of these components and the much harder nucleon as momentum distribution (closely associated with the α 3/2 − 2 − 2 ρ(r)=z e αr ,α1 = r2 . (57) first property) for these components, as well as very π 3 ch strong scalar three-body interaction in the internal 6qN channels, one can conclude that these nonnu- 9)We recall at this point that accounting for the finite radii cleonic components are extremely important for the of nucleons in the conventional approaches leads to fully properties of nuclear systems. negligible corrections to the Coulomb energy.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1467 The Coulomb potential for the interaction between is evident that the matrix element of the operator (60) such a charge distribution ρ(r) and a pointlike can be expressed in terms of the integrals of the prod- JiMi charged particle has the well-known form uct of the overlap functions χ (qi) of NN form Li drρ(r) z √ factors and three-body basis functions. The method V (R)= = erf(R α). |R − r| R for calculation of such Coulomb integrals is given in the Appendix. We have derived here a similar formula for the Coulomb interaction between two Gaussian distributions with different widths α1 and α2 and rms radii 7. RESULTS OF CALCULATIONS rch and rch , respectively: 1 2 √ Here, we present the results of the 3N bound-state z1z2 V (R; α1,α2)= erf(R α˜), (58) calculations based on two variants of the DBM. R (I) In the first version developed in [24], the α1α2 −1 2 2 2 dressed-bag propagator includes three loops (two α˜ = , or α˜ = (rch1 + rch2 ). α1 + α2 3 loops are with pions and one loop is with the σ In our calculations, we used the following charge radii meson). Two of them are of the type shown in Fig. 2 for the nucleon and dibaryon: of [24], in which each loop was calculated within the 3 10) P0 model for quark–meson interaction. The third (rch)p =0.87 fm, (rch)6q =0.6 fm. loop consists of two such vertices and a convolution These values lead to the “smeared” Coulomb interac- of the σ-meson and 6q-bag propagators [24]. tions in the NN and 6qN channels: (II) In the second version, we replaced the two 2 √ above pionic loops with the effective Gaussian form Coul e −1/2 → VNN (r)= erf(r αNN),αNN =1.005 fm, factor B(k), which describes the direct NN 6q + r σ transition, i.e., the direct transition from the NN (59) channel to the dressed-dibaryon channel. 2 e √ − V Coul(ρ)= erf(ρ α ),α1/2 =0.863 fm. Both versions have been fitted to the NN phase in ρ in in shifts in low partial waves up to an energy of 1 GeV with almost the same quality. Therefore, they can be considered on equal footing. However, version II has 6.2. Matrix Elements of the Three-Body Coulomb one important advantage. Here, the energy depen- Force dence arising from the convolution of the two prop- The Coulomb interaction between the charged agators involved into the loop, i.e., the propagators bag and the third nucleon in the 6qN channel is of the σ meson and bare dibaryon, describes (with no determined by the three-particle operator with the further correction) just the energy dependence of the separable kernel [see Eq. (36)]: effective strength of the NN potential λ(II)(E), which Coul (i) is thereby taken directly from the above loop integral. W (pi, pi; qi, qi) (60) In contrast, in the first version of the model, two (i) JiMi 1+τ3 additional qqππσ loops give a rather singular three- = ϕ (pi) Li 2 dimensional integral for λ(I)(E), where the energy J M L L i i i i dependence at higher energies should be corrected by × ˆ Coul Ji JiMi (1 + tdz ) W (qi, qi; E)ϕ (pi), a linear term. LiLi Li (i) where (1 + τ3 )/2 is the operator of the ith nucleon 7.1. Bound-State Energies of 3H and 3He ˆ charge and 1+tdz is the operator of the bag charge. It and Individual Contributions to Them

10)This value is simply the rms charge radius of the 6q bag The main difference between the results for both with the parameters given in [24]. The neutral σ field of the versions is that the energy dependence of λ(E) for bag changes this value only slightly. The evident difference the second version is much weaker than that for the between the charge radii of the nucleon and dibaryon can be first one. In addition, this energy dependence leads to well understood as follows: the charge radius of the 3q core some decrease in the contribution of the 6qN compo- r3q 0.5−0.55 of the nucleon is usually taken as ch fm, while nent to all 3N observables and thus to the respective remaining 0.3 fm is assumed to come from the charge distri- increase of the two-body force contribution as com- bution of the π+ cloud surrounding the 3q core in the proton. In contrast, the meson cloud of the dibaryon in our approach pared to the three-body force one. Table 2 presents is mainly due to the neutral scalar–isoscalar σ meson, so the calculation results for the two above versions that the dibaryon charge distribution is characterized by the for the following characteristics: the weights of the charge radius of the bare 6q core only. internal 6qN channels and D wave in the total 3N

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Table 2. Results of the 3N calculations with two- and three-body forces for two variants of the DBM

Contributions to H,MeV Model E,MeV PD, % PS , % P6qN (Pin), % T T + V (2N) V (3N) 3H DBM(I) g =9.577a) –8.482 6.87 0.67 10.99 112.8 –1.33 –7.15 DBM(II) g =8.673a) –8.481 7.08 0.68 7.39 112.4 –3.79 –4.69 AV18 + UIXb) –8.48 9.3 1.05 – 51.4 –7.27 –1.19 3He DBM(I) –7.772 6.85 0.74 10.80 110.2 –0.90 –6.88 DBM(II) –7.789 7.06 0.75 7.26 109.9 –3.28 –4.51 AV18 + UIX b) –7.76 9.25 1.24 – 50.6 –6.54 –1.17 a) These values of the σNN coupling constant in 3H calculations have been chosen to reproduce the exact binding energy of the 3H nucleus. The calculations for 3He have been carried out without any free parameters. b)The values are taken from [50].

function, as well as the weight of the mixed-symmetry where ρi is Jacobi coordinate in the set (i)andN{ p } is n S component (only for the 3N channel); the average the number of protons (neutrons). Due to the property individual contributions from the kinetic energy T , (2N) 3 ± (i) two-body interactions V plus the kinetic energy ex 1 τ3 ex ex 3 ˆ ex Ψ Ψ = Ψ ± T3 Ψ T , and three-body force (V (3N))duetoOSEandTSE 2 2 i=1 to the total Hamiltonian expectation. 3 The percentages of the D wave and the inter- = ± Tˆ3 = N{ p }, nal components given in Table 2 were obtained with 2 n incorporation of the three internal components; i.e., the above densities are normalized to unity, provided these values correspond to the normalization of the that the external wave function Ψex is also normalized total (four-component) wave function of the system to unity: to unity. ex 2 To compare the predictions of the new model with ρ{ p }(r)r dr =1. the respective results for the conventional NN po- n tential models, Table 2 also presents the results of ex| (i)| ex The matrix element Ψ τ3 Ψ is proportional to recent calculations with the Argonne potential AV18 the z projection of the total isospin T ; therefore, and Urbanna–Illinois 3BF UIX [50]. 3 the nucleon densities can be separated into isoscalar (matter) density and isovector parts: 7.2. The Densities, rms Radii and Charge 3 3 3 δ r − ρ1 Distributions in H and He ρ (r)=ρ = Ψex 2 Ψex , s m 2 (62) First, we give deﬁnitions of the nucleon and charge r distributions in the multichannel system. 3 The external 3N channel. The proton (ρp)and 3 δ(r − ρ1) ρ (r)= Ψex 2 τ (1) Ψex . (63) neutron (ρn) densities in this channel are deﬁned in v 2T r2 3 the standard way [51]: 3 ex ρ{ p }(r) (61) Both latter densities are also normalized to unity. n Then the nucleon densities can be expressed in terms 1 3 δ(r − 3 ρ ) 1 ± τ (i) of isoscalar and isovector densities as ex 2 i 3 ex = Ψ 2 Ψ N{ p } r 2 ex 3 ex 3 1 n i=1 ρp ( He)=ρn ( H)= (3ρs + ρv), (64) 3 (1) 4 3 δ(r − ρ1) 1 ± τ ex 2 3 ex ex 3 ex 3 1 = Ψ 2 Ψ , ρ ( H)=ρ ( He)= (3ρs − ρv). N{ p } r 2 p n n 2

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1469 The rms radii of the corresponding distributions are These average numbers of nucleons in 6qN chan- equal to nel can be expressed through relative probabilities of the 6qN components with a definite value of isospin t, 2ex ex 4 r {s,v,p,n} = ρ{s,v,p,n}(r)r dr. (65) which in our case are equal to P in ≡ P in = P in + P in,Pin ≡ P in = P in , The rms charge radius in the 3N sector is also defined 0 t=0 S1 D 1 t=1 S0 conventionally: (70) 2 ex 2ex 2 Nn 2 where P in , P in,andP in are determined by Eq. (53). rch = r p + Rp + Rn, (66) S1 D S0 Np in in Hence, Pt=0 + Pt=1 =1. Then one can write the av- 2 2 2 − 2 erage numbers of nucleons as where Rp =0.7569 fm and Rn = 0.1161 fm are 2 the squared charge radii of the proton and neutron, N in(3H)=N in(3He)= P in , (71) respectively. p n 3 t=1 1 The various types of one-particle densities (isoscalar, N in(3He)=N in(3H)=P in + P in . isovector, proton, neutron) in the external 3N channel p n t=0 3 t=1 for the 3Hand3He ground states calculated in The nucleon densities (61) can be expressed by a DBM(I) are shown in Fig. 5. similar formula through components of the internal Below, we present also the two-proton density for wave function with a definite value of isospin t. 3He,whichisdefined usually as [52] The total densities of nucleon distributions. The total nucleon densities (normalized to unity) for δ(r − r ) 1+τ (2) 1+τ (3) ρ (r)=6 Ψex 1 3 3 Ψex . the whole 3N system, with allowance for both the 3N pp r2 2 2 and 6qN components, can now be defined as − ex in in (67) (1 Pin)ρ{ p }N{ p } + Pinρ{ p }N{ p } n n n n p 2 ρ{ } = , (72) This density is normalized to 2: ρ (r)r dr =2.(As n N{ p } pp n there is only a single nucleon in the 6qN channel, we in in do not attach the index “ex” to this quantity.) The where Pin =3P1 /(1 + 3P1 ) is the total weight of all two-neutron density ρ (r) for 3Hisdefined similarly three internal channels [Eq. (54)] and the denomina- nn tor (i) → − (i) [with replacement 1+τ3 1 τ3 in Eq. (67)]. In in N{ p } =(1− Pin)N{ p } + PinN{ p }

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ρ, fm–3 1.2

ρ 3 s[ H] 1.0 ρ 3 v[ H] ρ 3 p[ H] 0.8 ρ 3 n[ H] ρ 3 s[ He] ρ 3 0.6 v[ He] ρ 3 p[ He] ρ 3 0.4 n[ He]

0.2

0 0.5 1.0 1.5 2.0 2.5 3.0 r, fm

3 3 Fig. 5. The isoscalar ρs,isovectorρv,protonρp, and neutron ρn densities in the external 3N channel for the Hand He systems obtained with DBM (I).

is equal to the average number of protons (neutrons) where α = md/(md + mN ), mN is a nucleon mass, in the whole multicomponent system. The densities and md is the mass of the bag (dibaryon). Then the of the total proton and neutron distributions and also total matter density (normalized to unity) is equal to external- and internal-channel distributions for 3He (1 − P )ρex · 3m + P ρin (m + m ) calculated for DBM(I) are presented in Fig. 7. r2ρ (r)= in m N in m N d . m m One can also deﬁne a (normalized) density of the (75) matter (or mass) distribution in the 6qN channel as 1 The rms radius of any distribution normalized to unity r2ρin (r)= (74) m (m + m )P is deﬁned by Eq. (65). The denominator in Eq. (75) N d in determines the average mass of the whole system × in| − − − | in Ψ1 δ(r αρ1)mN + δ(r (1 α)ρ1)md Ψ1 , taking into account nonnucleonic channels: − m =(1 Pin)3mN + Pin(mN + md) (76) 2ρ –1 r (r), fm =3mN + Pin(md − 2mN ) > 3mN . 0.8 In Table 3 we present some characteristics of isospin structure for wave functions in the 6qN 0.6 channel: the relative probabilities for the components in in with t =0and t =1(i.e., Pt=0 and Pt=1), average 0.4 numbers of protons and neutrons in all three 6qN components (3N{ p }), and also the average number n of nucleons N and the average mass m (divided 0.2 by 3mN value) in the whole four-component 3N system. It should be noted that the average number of nucleons in our multicomponent model, N{ p },is n 0 2 4 6 8 10 always less than the numbers of nucleons in the 3N r, fm channel just due to the existence of the nonnucleonic

3 components. For example, for DBM(I), the average Fig. 6. The two-proton density in He (solid curve) and 3 the two-neutron density in 3H (dashed curve) calculated number of protons in H is approximately equal to the 3 3 with DBM (I) in comparison with the two-proton density average number of neutrons in He, viz., Np( H) ≈ for the Bonn potential [52] (triangles). 3 Nn( He), and is equal to 0.92, while the average

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ρ, fm–3 1.2

ρex p 1.0 ρin p ρtot p ρex 0.8 n ρin n ρtot n 0.6

0.4

0.2

0 0.5 1.0 1.5 2.0 2.5 3.0 r, fm

Fig. 7. The external ρex, internal ρin, and total ρtot densities of proton and neutron distributions in 3He found with DBM (I). number of neutrons in 3H is approximately equal In our DBM, N is equal to 2.78 and 2.85 for ver- to the average number of protons in 3He, viz., sions I and II, respectively. 3 3 Nn( H) ≈Np( He),andisequalto1.86.Hence, the average number of nucleons found with the total The charge distributions. The charge distribu- multicomponent 3Hand3He functions is also always tion for the pointlike particles in the 6qN channel less than 3: can be written as the charge density of a system N = Np + Nn =3− 2Pin < 3. consisting of a pointlike nucleon and a pointlike bag:

1 1+τ (1) r2ρin (r)= Ψin δ(r − αρ ) 3 + δ(r − (1 − α)ρ )(1 + tˆ ) Ψin , (77) ch–point Z 1 1 2 1 3 1

ˆ where 1+t3 is the operator of the bag charge. The The last term in Eq. (78) includes the projectors Γt,t3 total charge radius in 6qN channel includes the rms onto the 6q-bag isospin state with deﬁnite values of 2in radius of this pointlike distribution r ch–point,the isospin t and its projection t3 and is equal to (for the 3 3 nucleon charge radius (Rp or Rn), and the charge bag Hand He states with total isospin T =1/2) radius Rd, which depends on the bag isospin t and its ∆(r2 )in (79) ch bag projection t3: 1 2 in 1 2 1 = Rd(0, 0)Pt=0 + Rd(1, 0) r2in = r2in + N inR2 + N inR2 (78) Z 3 ch ch–point Z p p n n 2 2 2 2 − in + Rd(1, 1)δT , 1 + Rd(1, 1)δT ,− 1 Pt=1 . 3 3 2 3 3 2 in 2 in + Ψ1 Γt,t3 Rd(t, t3) Ψ1 . 2 t,t3 The characteristic charge radius of the 6q bag Rd(t, t3)

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Table 4. The total rms radii (in fm) for the proton (rp), neutron (rn), matter (rm), and charge (rch) distributions in the DBM approach and their separate values for external and internal channels

3H 3He Model

rp rn rm rch rp rn rm rch

DBM(I) 3N 1.625 1.770 1.723 1.779 1.805 1.648 1.754 1.989

6qN 1.608 1.823 1.142 1.188 1.854 1.618 1.159 1.412

Total 1.625 1.773 1.663 1.724 1.807 1.647 1.694 1.935

DBM(II) 3N 1.613 1.761 1.713 1.769 1.795 1.636 1.744 1.980

6qN 1.573 1.797 1.124 1.171 1.829 1.583 1.141 1.396

Total 1.613 1.762 1.672 1.732 1.796 1.635 1.703 1.944

AV18 + UIX a) 1.59 1.73 1.76 1.61

Experiment 1.60b) 1.755 1.77b) 1.95 a)Taken from [53]. b)These “experimental” values are taken from [53]. They have been obtained by subtraction of the characteristic proton and neutron charge radii squared (0.743 and –0.116 fm2, respectively) from the experimental values of the charge radii squared. has, in general, different values in different isospin of our model (viz., DBM(I) and DBM(II)) give quite states (which is related to the different multiquarkdy- similar values for all the radii. The most interesting namics in the channels with different isospin values), point here is the importance of 6qN-component but we assume in this workthat their di fference can contributions. In fact, the contribution of the 6qN 3 be ignored, viz., channel shifts all the radii, i.e., rch and rp in Hand 2 2 2 2 2 3 Rd(0, 0) = Rd(1, 0) = Rd(1, 1) = Rd = b0, (80) He, predicted with pure nucleonic components in 2 − our approach, much closer to the respective experi- b0 =0.6 fm, and Rd(1, 1) = 0 mental values. For example, the value rch =1.822 fm (which corresponds to an nn bag). calculated for 3H with only the nucleonic part of With the above assumptions, the 6q-bag contribution the wave function is substantially larger than the respective experimental value 1.755 fm. However, to the 3Hand3He charge radius is reduced to an admixture of a rather compact 6qN component 2 3 R 2+2T3 (r =1.22 fm) immediately shifts the Hcharge ∆(r2 )in = d P in + P in . (81) ch ch bag Z 0 3 1 radius down to a value of 1.766 fm, which is very close to its experimental value. The rms charge radius of whole multicomponent system is defined as Thus, the dibaryon–nucleon component works in a right way also in this aspect. It is interesting to r2 =(1− P )r2ex + P r2ex. ch in ch in in note that, in general, the predictions of our two-phase model are quite close to those of the conventional In Table 4, we give the rms radii for all the above pure nucleonic AV18 + UIX model. This means that distributions in 3Hand3He found in the impulse our multichannel model is effectively similar to a con- approximation, as well as the respective experimen- ventional purely nucleonic model (at least for many tal values and results obtained for AV18(2N)+ static characteristics). However, this similarity will UIX(3N) forces. To demonstrate the separate contri- surely hold only for the characteristics that are sen- butions of the 3N and 6qN channels to these observ- sitive mainly to low-momentum transfers, while the ables, we also present the values calculated separately properties and processes involving high-momentum with only nucleonic and 6qN parts of the total wave transfers will be treated in two alternative approaches function. It is seen from Table 4 that both versions in completely different ways.

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7.3. Coulomb Displacement Energy Table 5. Contribution of various terms (in keV) of the 3 3 and Charge-Symmetry-Breaking Effects Coulomb interaction to the H− He mass difference ∆EC The problem of accurate description of Coulomb effects in 3He in the current 3N approach of the Contribution DBM(I) DBM(II) AV18 + UIX Faddeev or variational type has attracted much attention for the last three decades (see, e.g., [49, 54] Point Coulomb 3N only 598 630 677 and the references therein to the earlier works). It is Point Coulomb 3N +6qN 840 782 – 3 interesting that the Coulomb puzzle in He, being Smeared Coulomb 547 579 648 related to the long-range interactions, is treated in a 3N only different manner in our and conventional approaches. Smeared Coulomb 710 692 – The ∆EC problem dates backto the first ac- 3N +6qN curate 3N calculations performed on the basis of np mass difference 46 45 14 the Faddeev equations with realistic NN interactions in the mid-1970s [55]. These pioneering cal- Nuclear CSB (see Table 6) 0 0 +65 culations first exhibited a hardly removable difference Magnetic moments 17 17 17 of about 120 keV between the theoretical prediction and spin–orbita) th for ∆EC 640 keV and the respective experimental Total 773 754 754 ∆Eexp 760 value C keV. In the subsequent 30 years, a)Here we use the value for this correction from [49]. numerous accurate 3N calculations have been performed over the world using many approaches, but this puzzle is still generally unsolved. The most plau- pointlike particles within our approach. Hence, we sible quantitative explanation (but yet not free of se- must take into account the finite radii of charge rious questions) for the puzzle has recently been sug- distributions in the proton and 6q bag. Otherwise, gested by Nogga et al. [49]. They have observed that all Coulomb energies will be overestimated. 1 the difference in the singlet S0 scattering lengths of (ii) Another important effect following from our pp (nuclear part) and nn systems [originating from calculations is a quite significant contribution of the the effects of charge-symmetry breaking (CSB)] can internal 6qN component to ∆EC. In fact, just this increase the energy difference between the 3Hand interaction, which is completely missing in conven- 3 He binding energies and thus contribute to ∆EC. tional nuclear force models, makes the main contri- Our results obtained in this workwith the DBM bution (163 and 113 keV for versions I and II, respectively) to filling the gap in ∆E between conventional give an alternative explanation of the ∆EC puzzle and C other Coulomb effects in 3He without any free pa- 3N calculations and experiment if the CSB effects are of little significance in ∆EC. rameter. The Coulomb displacement energies ∆EC, together with the individual contributions to the ∆EC The large magnitude of this three-body Coulomb- value, are presented in Table 5. force contribution in our models can be explained We emphasize three important points where our by two factors: first, a rather short average distance results differ from those for conventional models. ρ21/2 between the 6q bag and the third nucleon (i) First, we found a serious difference between (which enhances the Coulomb interaction in the 6qN the conventional and our approaches in the short- channels) and, second, a relatively large weight of the range behavior of wave functions even in the nu- 6qN components, where the 6q bag has the charge cleonic channel. Conventional 3N wave functions +1 (i.e., it is formed from an np pair). This specific are strongly suppressed along all three interparticle Coulomb repulsion in the 6qN channel should also coordinates rij due to the short-range local repulsive appear in all other nuclei, where the total weight of core, while our wave functions (in the 3N channel) such components is about 10% or higher. There- have stationary nodes and short-range loops along fore, it should strongly contribute to the Coulomb all rij and the third Jacobi coordinates ρk.Sucha displacement energies over the entire periodic table node along the ρ coordinate is seen also in the 6qN and could somehow explain the long-term Nolen– relative-motion wave function. This very peculiar Schiffer paradox [56] in this way. short-range behavior of our wave functions leads to (iii) The third specificeffect that has been found a strong enhancement of the high-momentum com- in this study and contributes to the quantitative exponents of nuclear wave functions, which is required planation of ∆EC is a strong increase in the average by various modern experiments. On the other hand, kinetic energy T of the system. This increase in T these short-range radial loops lead to significant has already been discovered in the first early 3N cal- errors when using the Coulomb interaction between culations with the Moscow NN potential model [57]

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Table 6. Contribution of charge-symmetry-breaking ef- The first value has been extracted from the previous 3 3 − fects to the H− He mass difference ∆EC analysis of experiments d(π ,γ)nn [58] (see also [59] and references therein) and is used in all current ∆EC,keV NN potential models, while the second value in (82) ann,fm DBM(I) DBM(II) has been derived from numerous three-body breakup experiments n + d → nnp done for the last three −16.3 −18 −39 decades. In recent years, such breakup experiments − 18.9 +45 +26 are usually treated in the complete Faddeev formalism, which includes most accurately both two-body (2) and results in a similar nodal wave function behavior and 3BF [60]. Thus, this ann valueisconsideredtobe along all interparticle coordinates but without any quite reliable. However, the quantitative explanation nonnucleonic component. for the ∆EC value in conventional force models uses The increase in T leads to the proportional in- just the first value of ann as an essential point of all the crease in the np mass-difference correction to ∆EC. construction. At the same time, the use of the second Since the average kinetic energy in our case is twice value ann(= −16.3 fm) (which is not less reliable the kinetic energy in conventional force models, this than the first one) invalidates completely the above correction is expected to be also much larger in our explanation! case. Hence, we evaluate such a correction term in Therefore, in order to understand the situation the following way (without using perturbation the- more deeply and to determine the degree of sensitivity ory). In the conventional isospin formalism, one can of our prediction for ∆E to variation in ann,wealso 3 3 C assume that the Hand He nuclei consist of equal- made 3N calculations with two possible values of ann mass nucleons, from Eq. (82). These calculations have been carried m + m out with the effective values of the singlet-channel m = p n , 2 coupling constant corresponding to the VNqN part of the NN force: so that mp = m +∆m/2,mn = m − ∆m/2,where ∆m = m − m . The simplest way to include the eff 1 1 2 p n λ ( S0)= λpp + λnp, (83) correction due to the mass difference ∆m is to as- He 3 3 sume that all particles in 3H have the average mass 1 2 2m + m 1 λeff(1S )= λ + λ . (84) m¯ = n p = m − ∆m, H 0 3 nn 3 np H 3 6 3 In the above calculations, we employ the value λnp = while in He they have the different average mass 328.9 MeV,which provides an accurate description of 2mp + mn 1 the 1S np phase shifts and the experimental value m¯ He = = m + ∆m. 0 3 6 of the np scattering length anp = −23.74 fm [24]. In spite of the smallness of the parameter ∆m/m, Here, for the pp channel, we use the value λpp = perturbation theory in respect to this parameter does 325.523 MeV fitted to the well-known experimental − not work. So we used the average mass m¯ H in cal- magnitude app = 8.72 fm, and for the nn channel, 3 3 culation of Handm¯ He in calculation of He. The two λnn values corresponding to two available alter- contribution of this np mass difference to the ∆EC native values of the nn scattering length (82) have value is given in the fifth row of Table 5. As is seen been tested. The calculation results are presented in from the table, this correction is not very small in our Table 6. case and contributes to ∆EC quite significantly. As is seen in Tables 5 and 6, the DBM (version I) Many other effects attributed to increasing the can precisely reproduce the Coulomb displacement average kinetic energy of the system will arise in our energy ∆EC with the lower (in modulus) value ann = approach, e.g., numerous effects associated with the −16.3 fm, while this model overestimates ∆EC by enhanced Fermi motion of nucleons in nuclei. 54 keV (=45+9keV) with the larger (in modulus) Charge-symmetry-breaking effects in DBM. value ann = −18.9 fm. Thus, the DBM approach, As was noted above, the best explanation for the ∆EC in contrast to the conventional force models, prefers value in the frameworkof conventional force models the lower (in modulus) possible value −16.3 fm of published to date [49] is based on the introduction the nn scattering length, which has been extracted of some CSB effect, i.e., the difference between nn from very numerous 3N breakup experiments n + and pp strong interactions. At present, two alternative d → nnp [60]. values of the nn scattering length are assumed: Now, let us brieflydiscussthemagnitudeofCSB (1) − (2) − ann = 18.7 fm and ann = 16.3 fm. (82) effects in our model. The measure of CSB effects at

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1475 low energies is used to consider the difference be- “pairwise” NN forces (including their q2 dependence tween ann and so-called “pure nuclear” pp scattering on the momenta of the third nucleon) give only about N length app that is found from pp scattering data when half the total 3N binding energy, leaving the second the Coulomb potential is disregarded. The model de- half for the 3BF contribution. Therefore, the follow- pendence of the latter quantity was actively discussed ing question is decisively important: Can the 3BF in the 1970s–1980s [61–63]. However, the majority (inevitably arising in our approach) give the large of modern NN potentials fitted to the experimen- missing contribution to the 3N binding energy? Use- tal value a = −8.72 fm results in the value aN = fulness of the developed model for the description pp pp of nuclear systems depends directly on the answer −17.3 fm when the Coulomb interaction is discarded. It is just the value that is adopted now as an “em- to this important question. It is appropriate here to pirical” value of the pp scattering length [64]. Thus, recall that, in the conventional 3BF models, such as Urbana–Illinois or Tucson–Melbourne, the contri- the difference between this value and a is usually nn bution of 3BF to the total 3N binding energy does not considered as the measure of CSB effects. However, our model (also fitted to the same experimental value exceed 1 MeV; i.e., this contribution can be considered as some correction (∼15%), although it is sig- a = −8.72 fm) gives a quite surprising result: pp nificant for the precise description of the 3N system. N − app(DBM)= 16.57 fm, (85) (iii) Fortunately, the contribution of 3BF induced which differs significantly from the above conven- by OSE and TSE enables one to fill this 4.3-MeV gap tional value (by 0.8 fm) due to the explicit energy between the two-body force contribution and exper- dependence of the NN force in our approach. imental value. In fact, including both OSE and TSE contributions to 3BF, taken with the same coupling aN − a Thus, if the difference pp nn is still taken as constants and form factors as in the driving NN- the measure of CSB effects, the smallness of this force model, together with a quite reasonable value difference obtained in our model attests to a small for the σNN coupling constant, gσNN =8−10,one magnitude of the CSB effects, which is remarkably obtains a 3N binding energy that is very close to smaller than the values derived from conventional the experimental value (see rows 1, 2 and 4, 5 in OBE models for the NN force. Table 2). Thus, the presented force model leads to a very reasonable binding energy for the 3N system, 8. DISCUSSION however, with the much larger (as compared to the The 3N results presented in the previous section traditional 3N force model) contribution of 3BF. In differ significantly from the results found with any fact, the unification of the basic 2N-and3N-force conventional model for 2N and 3N forces (based parameters provides strong support for the whole on Yukawa’s meson-exchange mechanism) and also force model suggested here and is in a sharp contrast from the results obtained in the frameworkof hybrid with all traditional force model based on the t-channel models [65], which include the two-component repre- exchange mechanism. We remind the reader that the N N sentation of the NN wave function Ψ=Ψ +Ψ . 2 and 3 forces in conventional approaches (where NN 6q the latter is induced by an intermediate ∆-isobar It is convenient to discuss these differences in the following order. production) are taken with different cutoff parameters 2 values ΛπNN and ΛπN∆ in the 2N and 3N sectors in (i) We found that the q dependence of pair NN order to explain the basic features of 3N nuclei and forces on the momentum of the third particle in the N + d scattering! Thus, in the traditional approach, 3N system is more pronounced in our case than in one has some serious inconsistency in parameter val- other hybrid models [34, 48, 65, 66]: the 3N bind- ues for the 2N and 3N sectors. ing energy decreases by about 1.7 MeV, from 5.83 (iv) The contributions of the pairwise and different to 4.14 MeV, when one takes into account the q2 three-body forces to the total 3N binding energy for dependence [32]. From a more general point of view, 3 it means that, in our approach, pairwise NN in- H are given in the sixth and seventh columns of teractions (except Yukawa OPE and TPE terms), Table 2. From the results presented in this table, one being “embedded” into a many-body system, lose can conclude that just the total 3BF contribution their two-particle character and become substantially to the 3N binding energy dominates and, in fact, 3 3 many-body forces (i.e., depending on the momenta of determines the whole structure of the Hand He 11) other particles of the system). ground states. Moreover, comparing the entries q2 (ii) Due to such a strong dependence (of “re- 11)It should be noted here that the relative contribution of the pulsive” character), the 3N system calculated includ- pairwise effective force W (E) to the 3N binding energy ing only the pairwise forces turns out to be strongly decreases noticeably when including 3BF (due to strength- underbound (E = −4.14 MeV). In other words, the ening of the q2 dependence arising from the pairwise forces).

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1476 POMERANTSEV et al. in the first and second rows and sixth and seventh (v) Dependence of the two-body coupling con- columns of the table, one can see a “nonlinear” ef- stants λ(ε) upon the average momentum of the other fect of self-strengthening for the 3BF contribution. nucleon in the 3N system [see, e.g., Eq. (29)] can In fact, the comparison of the results presented in be interpreted generally as a density dependence of these rows of the table (see the fourth and the seventh the resulting many-body force in a many-nucleon columns) shows clearly that the 3BF contribution to system. It is easy to show that the appearance of the the total 3N binding energy is largely determined by energy-dependent pairwise potentials of the above- the weight Pin of the 6qN component in the total mentioned type leads unavoidably to a repulsive 3N wave function. So, when the weight of the 6qN many-body force. In other words, the effects of the component increases, the 3BF contribution, which is two-body interactions of this type can be reinter- related directly only to this component of the total preted in terms of the conventional static interaction wave function, increases accordingly. However, the as an additional contribution of the effective repulsive enhancement of the pure attractive 3BF contribu- density-dependent many-body force. For example, tion squeezes the 3N system and thus reduces its if we remove the q2 dependence from the coupling rms radius, i.e., the mean distance between nucleons, constant λ(E − q2/(2m)) of our two-body force (this which, in turn, again increases the weight of the q2 dependence leads to a weakening of the two- 6qN component. In other words, some chain pro- body force in a many-nucleon system, when q2 is cess, which strengthens the attraction in the system, rising), then the neglected q2 dependence must be arises. This process is balanced both by the weak- compensated by an additional repulsive density- ening of the effective pairwise interaction due to the 2 dependent effective three-body force. Thus, we can q dependence and by the repulsive effect of the or- replace this energy-dependent two-body interaction thogonalizing pseudopotentials included in each pair by an effective static two-body potential (as is usually interaction. done) plus a repulsive density-dependent 3BF. There are two other important stabilizing factors On the other hand, it is well known from Skirme weakening the strong three-body attraction in the 3N model calculations of nuclei that just similar repulsive system. The first factor is the generation of the short- phenomenological density-dependent 3BF should be range repulsive vector ω field, where all three nucleons added to conventional 2N and 3N forces to guarantee are close to each other [25]. Since the ω meson is the saturation properties of heavy nuclei. Thus, in this heavy, this field is located in the deep overlap region respect, the present force model is also in qualitative of all three nucleons. In the present study, we omitted agreement with the phenomenological picture of nu- the three-body contribution of this repulsive ω field. clear interactions. This repulsive contribution will keep the whole system from further collapse due to the strong attractive 9. CONCLUSION 3N force induced by the scalar field. In this paper, we have developed a general for- The second factor weakening slightly the effective malism for the multicomponent description of the 3N attraction is associated with the conservation of three-body system with particles having inner de- thenumberofscalarmesonsgeneratedinthe2N- grees of freedom. We have applied our new approach and 3N-interaction process. The problem is that TSE to studying the 3N system with 2N and 3N inter- giving the 3BF contribution (see Fig. 4) arises due actions based on the dressed dibaryon intermediate to the breakof the σ-meson loop, which induces the state and σ-field generation. It has been shown that main 2N force. In other words, the σ meson generated the DBM applied to the 3N system inevitably re- in the transition of pair nucleons from the NN phase sults in new three-body scalar and also new (three- state to the 6q state is absorbed either in the 6q body) Coulomb forces due to the (strong + Coulomb) bag with closing of the loop or by the third nucleon, interaction between the dressed dibaryon and the resulting in the 3BF contribution. Thus, the appear- third nucleon. These forces play a crucial role in the ance of such a 3BF should weaken the attraction structure of few-nucleon systems and very likely in between nucleons in the pair. We carefully estimated the whole nuclear dynamics. Our accurate variational the effect of the meson-number conservation for the calculations have demonstrated that the new 3BF TSE contribution on the total 3N binding energy. giveshalfofthe3N binding energy, whereas the 3BF Its magnitude turned out to be rather moderate on contribution in the traditional NN-force approaches the absolute energy scale (about 0.3–0.4 MeV), but gives about 15% of the total binding energy. Thus, the quite noticeable within the whole TSE contribution. suggested approach to the 2N and 3N interactions However, when the total nucleon density increases can lead to significant revision of relative contribu- (and the relative TSE contribution also increases), tions of two- and many-body forces in all nuclear the effect is enhanced. systems.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1477 The developed model gives the precise value for data corresponding to large-momentum transfers the Coulomb displacement energy ∆EC of the A =3 (∼1 GeV/c), many types of meson-exchange and system. Two basic sources of this contribution, which isobar currents are often introduced into theoretical differ from conventional force models, should be indi- frameworks. However, these currents are often un- cated here: related to the underlying force model. Hence, it is the three-body Coulomb force between the dressed rather difficult to checkthe self-consistency of such bag and the charged third nucleon; and calculations, e.g., the validity of gauge invariance, etc. the quite significant correction to the kinetic en- Numerous modern experiments could corroborate ergy of the system due to the np mass difference and these results. In particular, according to the recent high average kinetic energy. 3He(e, e pp) experiments [5] and their theoretical interpretation on the basis of fully realistic 3N calcu- It should be emphasized that, contrary to other studies based on conventional force models (using the lations, the cross sections for the 3He(e, e pp) pro- 2N and 3N forces generated via the meson-exchange cess are underestimated by about five times with a mechanism), this explanation does not require any fully realistic 3N model and incorporation of final- noticeable CSB effect, although our model is still state interaction and meson-exchange currents. This compatible with such effects. However, these CSB important conclusion has been further confirmed in effects do not contribute remarkably to ∆EC in our recent experiments at the Jefferson Laboratory when approach. the incident-electron-beam energy was increased up It is crucially important that the DBM leads to Ee =2.2 and 4.4 GeV [7]. The data of the two dif- to significant nonnucleonic components in the 3N ferent experiments give clear evidence of very strong 3 wave function (8–11%), while this component in the short-range NN correlation in the He ground state. deuteron is about 3%, which results in a reformulation This correlation still cannot be explained within the of many basic effects in few-nucleon systems and traditional pattern for the 3N system. other nuclei as well. It is probable that the weight of In addition, our approach has recently been par- such nonnucleonic components in heavy nuclei can tially supported [67] from the other side by consid- be even higher with an increase in the mass number ering a model for 2π production in pp collisions at and nuclear density. Ep = 750 and 900 MeV. The authors have found that Thereisaveryspecific new interplay between two- almost all particle energy and angular correlations − and three-body forces: the stronger the three-body (e.g., π+π , pp, πpp, etc.) can be explained quanti- force, the smaller the attractive contribution of the tatively by assuming that π+π− production occurs two-body force to the nuclear binding energy! This through the generation of an intermediate light σ gives a very important stabilization in nuclei and nu- meson with the mass mσ 380 MeV. These values clear matter. In this way, a very natural density depen- generally agree with the parameters adopted in our dence of nuclear interactions appears from the begin- NN model [8, 24] and drastically disagree with the ning. Thus, the general properties of the 3N system, values assumed in OBE and other potential models. where forces differ so much from any conventional A very interesting general implication of the re- force model, would appear also to be much different sults presented here is their evident interrelation to from the predictions of any conventional model and, the famous Walecka hydrodynamic model for nu- hence, from experiment. clei [68]. It is well known that the Walecka model Therefore, it was very surprising for us to find that describes nuclei and nuclear matter in terms of the the static characteristics of the 3N system in our case scalar σ and vector ω fields, where the σ field gives turned out to be very close to the predictions of the the attractive contribution, while the vector ω field modern force model (such as AV18 + UIX) and thus balances this attraction by short-range repulsion. It to experiment. This gives us a good test of the self- is very important that both basic fields appear (in consistency and accuracy of the new force model. the model) as explicit degrees of freedom (together However, predictions of the present 2N-and3N- with relativistic nucleons), in contrast to conventional force model in other aspects will strongly deviate from meson-exchange models for nuclear forces, where those for conventional models. First, these are the mesons appear as carriers of forces rather than as properties determined by the high-momentum com- explicit field degrees of freedom. Our approach does ponents of nuclear wave functions. The point is that include the σ-meson (and potentially the ω-meson) the system described by our multicomponent wave degrees of freedom in an explicit form, similarly to the functions including the dibaryon components ex- Walecka model. Moreover, since the average kinetic plicitly can easily absorb high-momentum transfers, energy of the 3N system in our model is high (it which can hardly be absorbed by the system consist- is much higher than that in the conventional OBE ing of nucleons only. Therefore, to fit the experimental approach), nucleon motion is closer to the relativistic

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1478 POMERANTSEV et al. case, and thus the similarity with the Walecka model Appendix gets even closer. There is also an additional strong argument in OVERLAP FUNCTIONS BETWEEN 3N favor of a tight interrelation between our and the SYMMETRIZED BASIS AND NN FORM above Walecka-type nuclear model. Very recently, we FACTORS AND MATRIX ELEMENTS have formulated [38] the dibaryon model for NN in- OF DBM INTERACTIONS FOR A 3N teraction in terms of relativistic effective field theory SYSTEM with the intermediate dibaryon being represented as a 1. The Construction of Basis Functions color quantum string with color quarkclusters at its The total wave function in the 3N channel with the π σ ρ ω ends. This theory includes , , ,and mesons as angular momentum (J, M) and the isospin (T,T ) N∆ z a dressing of the dibaryon together with the and is written as (below we omit the quantum numbers ∆∆ loops. Thus, the 3N scalar force introduced in the JMTT ): present work “by hand” can be derived in the field- z (JMTTz) (1) (2) (3) theory Lagrange language within the effective field Ψex =Ψex +Ψex +Ψex , (A.1) theory. Moreover, in the mean-field approximation, this effective field theory approach, being applied to (i) γ (i) Ψex = CnΦγn (i =1, 2, 3), (A.2) nuclei, should result in the Walecka–Serot relativistic γn model with the dominating collective σ field, which couples the nucleons in a nucleus together. where (i) γ λ l Thus, the alternative description given here by the Φγn(ri, ρi)=Nn ri ρi (A.3) new force model looks to be more self-consistent 2 2 JMTTz (i) × exp(−αnr − βnρ )F (ˆri, ρî)T . and straightforward than the conventional OBE- i i γ tjk type models. One aspect of this new picture is We use the following notation: ri (pi) is the relative evident—the present model being applied to any coordinate (momentum) of the pair (jk), while ρi electromagnetic process on nuclei leads automat- (qi) is the Jacobi coordinate (momentum) of the ith ically to a consistent picture of the process as a particle relative to the center of mass for the pair (jk), whole: single-nucleon currents at low-momentum (i, j, k)=(1, 2, 3) or their cyclic permutations. Here, transfers, meson-exchange currents (including new the composite label γ = {λlLSjkStjk} represents the meson currents) at intermediate-momentum trans- set of quantum numbers for the basis functions: the fers, and quarkcounting rules at very high mo- angular momenta λ and l correspond to the Jacobi mentum transfers, because the model wave function coordinates ri and ρi, respectively; Sjk(tjk) is the includes explicitly multinucleon, meson-exchange, spin (isospin) of the two-body subsystem (jk);and and multiquarkcomponents. L(S) is the total orbital momentum (spin) of the system. The normalizing coefficient in (A.3) is From all the above-mentioned arguments, one can conclude that the dibaryon concept of nuclear force λ+3/2 l+3/2 γ λ+l+3 2αn βn advocated in the workresults in a new picture for Nn =2 . (A.4) nuclear structure and dynamics. π(2λ + 1)!!(2l +1)!! The spin-angular F and isospin T (i) parts of γ tjk ACKNOWLEDGMENTS the basis function are defined by a standard vector- coupling scheme: F JMTTz |{ }{ } We are grateful to Dr. I. Obukhovsky for discus- γ = λili : L sjsk(Sjk)si : S : JM , sions and help in calculation of some matrix elements, and also to many of our colleagues at Tubingen¨ Uni- (A.5) versity for continuous encouragement in the course T (i) = |t t (t )t : TT , (A.6) of this work, and also to the staff of Institut fur¨ tjk j k jk i z Theoretische Physikder Universit at¨ Tubingen,¨ where most of the calculations were performed. where si(= 1/2) and ti(= 1/2) are the spin and isospin of the ith nucleon. This workwas supported in part by der Deutsche Now we define the symmetrized basis functions as Forschungsgemeinschaft (grant no. 436 RUS sym (i) 113/790) and the Russian Foundation for Basic Φγn = Φγn, (A.7) Research (grant nos. 05-02-17404, 05-02-04000). i=1,2,3

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1479 so that the total wave function in an external (3N) namely, one “diagonal” (I(i)i) and two nondiagonal channel takes the form ones: γ sym If,γn(ρ )=ϕ (r )|Φ(j)(r , ρ ) = T (i)|T (j) Ψex = CnΦγn . (A.8) (i)j i f i γn j j f tik γ,n (A.14) 1 × Df N γ rλi exp(− η2 r2)rλρl 2. Nucleon–Nucleon Form Factors m n i 2 m i j j m NN × − 2 − 2 F |F 3 The form factors in the separable DBM in- exp( αnrj βnρj ) f (ˆri) γ (ˆrj, ρˆj) d ri. J M t t teraction and in the projectors ϕ i i d dz (r ), corre- λiSd i sym Due to symmetry of the basis functions Φγn ,three sponding to the orbital momentum λi,spinSd,the f,γn overlap integrals I (ρi) (i =1, 2, 3) are identical, total angular momentum (Ji,Mi),andisospin(tdtdz ) (i) of the subsystem (jk) (Ji = λi + Sd), have the form so that further we present formulas for the case of i =2. For example, JiMitdtdz ϕ (ri) ≡ ϕf (ri) (A.9) λiSd γ f,γn f,γn f,γn χf (2) = Cn I + I + I . (A.15) f λ 1 2 2 (i) (2)2 (2)1 (2)3 = D r i exp(− η r )F (ˆr )T , γn m i 2 m i f i f m (i) Diagonal overlap integrals I(2)2 : where f,γn ε l ≡{ } I(2)2 (ρ2)= G22ρ2 (A.16) f λi,Sd,Ji,Mi,td,tdz , (A.10) J m (i) F | T | J t t f = λiSd : JiMi , f = tjtk : tdtdz , × − 2 Y JiJM X d dz exp( βnρ2) l (ρˆ2)δtdt31 2 , f ≡{ J } and Dm and ηm are linear and nonlinear parameters, ε γ,f, ,m , respectively, of the Gaussian expansion. (In this Ap- where pendix, we have altered the notation for the quantum Gε = δ δ (−1)λ+l+L (A.17) numbers of the NN form factor as compared with the 22 λλi SdS31   main text of the paper: we have replaced L → λ for   i i  l 1 J  the orbital momentum and also included the isospin J  2  × f γ (2λ +1)!! π[L][S][Ji][ ] quantum numbers td, tdz .) In the single-pole approx- DmNn λS J , λ+2 λ+3/2  31 i imation, the DBM includes only one form factor for 2 αnm   each set f,sothatindexf determines the form factor LS J uniquely. In the present version of the DBM, we use only 0S, 2S,and2D oscillator functions as the form [X] ≡ 2X +1, (A.18) factors. So, we need to expand in Gaussians the 2S 1 function only. α = α + η2 , (A.19) nm n 2 m J J Y JiJM J | Y mJ 3. Overlap Integrals l (ρˆ2)= mJ JiMi JM l 1/2 (ρˆ2), (A.20) The total overlap function (49) JiMitdtdz ≡ | χ (i) χf (i)= ϕf (i) Ψex (A.11) tdtd 1 λiSd X z = t t t TT |t t . (A.21) 2 d dz 2z z 2 2z γ | sym 2 = Cn ϕf (i) Φγn γn (ii) Nondiagonal overlap integrals: and also matrix elements (m.e.) of any DBM inter- f,γn − Sd+S23 ε˜ t I(2)1 (ρ2)=( 1) G21ρ2 (A.22) action include the overlap integrals between the form J sym g,t,m factors ϕf and symmetrized basis functions Φγn : 2 J JiJM tdtdz × exp(−ωnmρ )Y (ρˆ2)τ21(td,t23)X , f,γn | sym 2 g 2 I(i) (ρi)= ϕf (ri) Φγn . (A.12) If,γn(ρ )=(−1)λi+λ Gε˜ ρt (A.23) This overlap integral consists of three terms: (2)3 2 21 2 J g,t,m f,γn f,γn f,γn fγn J t t I(i) = I(i)i + I(i)j + I(i)k , (A.13) × − 2 Y JiJM X d dz exp( ωnmρ2) g (ρˆ2)τ23(td,t12) 2 ,

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1480 POMERANTSEV et al. where where ε˜ ≡{γ,n,f,J ,g,t,m}, (A.24) 1 3 3 1 µ = α + β ,ν= α + β , (A.28) n 4 n 4 n n 4 n 4 n ε˜ √ G21 (A.25) 3 − J σn = (αn βn). = G21(γ,n,f, ,g,m,ξ)δt,λi +λ+l−L1−L3−L4 , 2 ξ

ξ ≡{L1,L2,L3,L4,j4,g1,g4}. The coeﬃcients A in (A.26) are related to rotation In Eq. (A.25), the summation is carried out over all of the basis functions from one Jacobi set to the other one: intermediate quantum numbers incorporated into the LL1L2 λ+l L1 λ−L1 L2 composite index ξ. Note that the overlap functions A (Rˆ )=(−1) (R11) (R12) (R21) I I λlj1j2 (2)1 and (2)3 are distinguished by a phase factor and (A.29) isospin functions only.    

The algebraic coeﬃcients G21 in (A.25) are equal − L1 L2 J1 λ − L1 l − L2 J2 × l L2     to (R22)

Ji+g1+L+1/2−J 000 000 G21(γ,n,f,J ,g,m,ξ)=(−1) (A.26) [λ]![l]![λ][l][L1][L2][λ − L1][l − L2][j1][j2] f γ λiL10 LL3L4 × × D N A (Pnm)A (Qnm) m n λi0L1(λi−L1) λlL1j4 − − [L1]![L2]![λ L1]![l L2]!   [g1][g4] [λi][L][S][S23][Sd][Ji][λi − l][j4][J ] ×  −  L1+L3+L4+3 L1 λ L1 λ (2µ ) 2 nm × L2 l − L2 l . L1 + L3 + L4 +3   × Γ (λi − L1)0j40|g0   2 j1 j2 L       1 1  − S23 (λi L1) L1 λi 2 2 × The rotation matrices P and Q in (A.26)   1  nm nm S0 jg1 SSd have the form    2        σn J (λi − L1) g4 J J J 1 −  × i 2µ 1 Pnm =  nm  , (A.30)    −  j4 g (λi L1) g1 g4 01 2    1   j g   √   4 2 4 × 1 3 L S g . − −   1 d 1 √2 2    Qnm =   Pnm. (A.31) LSJ 3 −1 2 2 In formulas (A.22), (A.23), (A.26) the following notation is used: 2 The overlaps between the basic isospin functions 1 2 − σn µnm = µn + ηm,ωnm = νn , (A.27) τik are equal to 2 4µnm

  δt t for i = j,  jk jk    (i) (j) 1 1  τij(t ,t ) ≡T |T =  t  jk ik t tik  jk − tik jk  2 2 ( 1) for (ij) = (13), (21), (32),  (2tjk + 1)(2tik +1)  1  − tjk   Tt ( 1) for (ij) = (12), (23), (31). 2 ik (A.32)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1481 f γn 4. Conversion to Momentum Representation where Mfγn (i2j) are the corresponding basis m.e.: One of the advantages of the Gaussian basis is the M f γ n (i2j)=If ,γ n |O |If,γn. (A.39) fact that Gaussian functions have the same form in fγn (2)i (2) (2)j both the coordinate and momentum representations. Any scalar–isoscalar operator O(q − q) which So expressions for the overlap integrals given below does not depend on spin and isospin variables (e.g., in the coordinate representation can be directly used the DBM two-body force, the projector, the 3BF due for the calculation of m.e. of DBM interaction op- to σ exchange, the norm of nonnucleonic component) erators in the momentum representation. We use a can be expanded in spherical harmonics as “symmetrized” momentum representation: − ∗ d3x O(q q)= OL(q , q)YLM (qˆ )YLM (qˆ). f(p)= f(x)eip·x . 3/2 (A.33) LM (2π) (A.40) Therefore, due to properties of the Gaussian func- In this case, the spin-angular and isospin parts of the tions, the (normalized) basis functions Φ(i)(p , q ) in overlaps give γn i i the momentum representation have the same form J ∗ J Y JiJM | |Y JiJM (A.3): g (qˆ ) YLM (qˆ )YLM (qˆ) g (qˆ2) M (i) Φγn(pi, qi) (A.34) (A.41) ˜ γ λ l − 2 − ˜ 2 F JMTTz T (i) = δJ J δgLδgL, = Nn pi qi exp( αñpi βnqi ) γ (pî, qî) , tjk t t t t where X d dz |X d dz = δ . (A.42) tdtd 1 1 t α˜ = , β˜ = . (A.35) dz n 4α n 4β n n Therefore, nine m.e.’s for such an operator M(i2j) ≡ Moreover, as all the NN form factors (A.9) are the Mi2j (i, j =1, 2, 3) can be reduced to radial integrals sums of Gaussians, the form of the overlap integrals of four types (here, we omit the indices fγn,fγn for (A.16)–(A.29) remains invariable when passing from brevity): the coordinate to momentum representation if one M222 = δJ J δllδt t R222, replaces in these formulas i i 13 13 → → ˜ → − Sd+S23 αn αñ,βn βn,ηm η˜m =1/ηm. (A.36) M122 =( 1) τ12(t23,td)R122,

Below, we use the symbols with a tilde for designation λ +λ M =(−1) i τ (t ,t )R , of the corresponding quantities in the momentum 322 32 12 d 122 representation, e.g., α˜nm =˜αn +1/(2˜ηm),etc. Sd+S23 M221 =(−1) τ21(t23,td)R221,

M =(−1)λi+λτ (t ,t )R , 5. Matrix Elements for DBM Operators 223 23 12 d 221 − S23+S23 All quantities related to the nonnucleonic channels M121 =( 1) τ12(t23,td)τ21(t23,td)R121, in the DBM can be expressed in the momentum − λ+λ +λi+λi representation as the sums of integral operators with M323 =( 1) τ32(t12,td)τ23(t12,td)R121, factorized kernels [see Eq. (36)]: − Sd+S23+λi+λ M123 =( 1) τ12(t23,td)τ23(t12,td)R121, DBM f f O(i) = ϕf (pi)O (q i, qi; E)ϕf (pi). (A.37) S +S +λ +λ M =(−1) d 23 i τ (t ,t )τ (t ,t )R . Therefore, the m.e. of such an operator is equal 321 32 12 d 21 23 d 121 to the sum of the m.e.’s for one-particle opera- Here, tors Of f (q , q ; E) between the overlap functions i i R121 (A.43) χf (qi): ε˜ ε˜ tt = G G δJJ δgg R (˜ωnm , ω˜nm; O), | DBM| f | | f 21 21 g M2 = Ψex O(2) Ψex = χ(2) O(2) χ(2) JJgg,mmtt ff (A.38) R (A.44) 122 γ f γn γ ε˜ ε t l ˜ = Cn Cn Mfγn (i2j), = G21G22δJJ δg lRl (˜ωn m , βn; O), γnγn i,j=1,2,3 JJg,mmt

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1482 POMERANTSEV et al. R (A.45) therefore, 221 ε ε˜ l t ˜ δ(q − q ) = G22G21δJJ δgl Rl (βn , ω˜nm; O), Ji Ji − 2 Wλ λ = 2 λλ λ (E q /(2m ˜ )). JJg,mmt i i L q i i (A.54) R (A.46) 222 In the present version of the DBM we employed a ε ε ll ˜ ˜ = G22G22δJJ Rl (βn , βn; O), rational approximation for the energy dependence of JJ,mm J the coupling constant λLL (E) [24]:

tt J J E0 + aE R (˜ω , ω˜; O) (A.47) λ (E)=λ (0) , (A.55) L LL LL E − E ∞∞ 0 2 2 = (q)t +2qt+2e−ω˜ (q ) e−ωq˜ O (q,q)dqdq. where the parameters E0 and a can be taken to be L the same for all λ. We found that this simple rational 0 0 form can reproduce quite accurately the exact energy J dependence of the coupling constants λ (E) cal- Below, we give the explicit formulas for the radial LL tt culated from the loop diagram in Fig. 1. Therefore, in integrals RL for all speciﬁc terms of the DBM inter- the 3N system, the corresponding coupling constants action. Ji λλ λ take the form Projector. The total projector onto the state i i Ji Ji − 2 Ji ϕλ ,S has the form λλ λ (E q /(2m ˜ )) = λλ λ (0) (A.56) i d  i i i i  ¯ Γf ≡ ΓJi (A.48) λi,Sd    E0 1  JiMitdtdz JiMitdtdz × −a +(a +1) . = |ϕ δ(q − q )ϕ |.  − 2  λi,Sd λi,Sd E0 E q Mi,td 1+ z 2˜m(E0 − E) After expanding the δ function into partial waves, The ﬁrst term in Eq. (A.56) leads to the expression for ∗ 2 the radial m.e. like (A.51). For calculating the second δ(q − q )= Y (q )YLM (q)δ(q − q )/q , LM 1/(1 + q2) LM term, we expand the function into a sum of Gaussians: (A.49) 1 2 2 the corresponding operator O in Eq. (A.47) is re- = BM exp(−θMq /q ), (A.57) L 1+q2/q2 0 duced to 0 M ¯ δ(q − q ) 2 − f where q0 =2˜m(E0 E) > 0(for E

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1483 For the energy dependence such as in Eq. (A.55), the Three-body force due to OSE. The exchange derivative takes the form operator for scalar mesons does not include any spin– d 1 isospin variables. Therefore, Eq. (A.62) can be sim- − λ(E)=λ(0) . (A.60) dE (E − E)2 plified and, in view of the energy dependence given in 0 Eq. (A.55), reduces to the form Therefore, OSE JiJi Ji W (q , q ; E)=δ λ λ λ i i J Ji λ λ (0) (A.65) ¯ ¯ E (1 + a) i i i i i t t N f f Ji 0 RL (˜ω , ω˜; )=λλ λ (0) 2 (A.61) − 2 i i (E − E0) 1 gσNN × E0(1 + a) q2 − 2 2 t +t+3 E − E − i (qi qi) + mσ Γ( 2 ) 0 2m × BM BM . t +t+3 2 1 M M 2(˜ω +˜ω +(θM + θM)/q0) 2 × . q2 E − E − i Three-body force due to OME. When calculat- 0 2m ing the m.e.’s for 3BF due to OME [Eq. (37)], viz., Using the expansion of the OME interaction into J J OME i i partial waves, Wλ λ (qi, qi; E) (A.62) i i 1 ∗ QL(z) Ji = Y (q ) YLM (q), Bλ (k ) − 2 2 LM i OME (qi qi) + mσ 2qq = dk 2 V (qi, qi) LM q (A.66) E − E − i α 2m where mσ is the mass of the σ meson, QL(z) is a BJi (k) × λi Legendre function of the second kind, and 2 , qi 2 2 2 E − E − q + q + mσ α 2m z = , 2qq we use a similar trickas in the calculation of the norm for the 6qN component. It enables us to exclude one gets the following radial integral for OSE m.e.: J ¯ ¯ the vertex functions B i (k) from the formulas for the tt OSE f f λi RL (˜ω , ω˜; W ) (A.67) m.e.’s. By replacing the product of propagators in the E (1 + a) integral [over the meson momentum k in Eq. (A.62)] 2 Ji 0 = gσNN λλ λ (0) i i (E − E )2 with their difference, one obtains the following ex- 0 pression free of the vertex functions: × BM BMR(t , ω˜ + θ ,t,ω˜ + θ; L, mσ), Ji Ji MM Bλ (k)Bλ (k) i i dk 2 2 where q q E − ε(k) − E − ε(k) − R(t ,ω ,t,ω; L, m) (A.68) 2m 2m 2 Q (z) 2 (A.63) = (q)t +2e−ω q L qt+2e−ωq dqdq. 2qq q2 q2 Ji − − Ji − λλ λ E λλ λ E i i 2m i i 2m We calculate integrals like those in Eq. (A.68) in = the following way. In the integral (A.68), t ≥ L and 2 − 2 q q t ≥ L, and it can be shown that t − L and t − L are Ji even numbers. Introducing the auxiliary indices k and =∆λλ λ (q ,q). i i k,sothat This quantity is the finite-difference analog of the t = L +2k, t = L +2k, derivative of λ with respect to q2, which, in the case of the energy dependence (A.55), takes the form the integral (A.68) can be written as ∆λ(q ,q) (A.64) R(L +2k ,ω ,L+2k, ω; L, m) (A.69) k k 1 1 ∂ ∂ = λ(0)E0(1 + a) . − − R − 2 2 = (L, ω ,L,ω; L, m). E q /(2m) E − q /(2m) ∂ω ∂ω Thus, the m.e.’s for OME can be found without ex- The last integral with t = t = L is easily calculated in plicit usage of the vertex functions BJi (k). λi the coordinate representation. Using the well-known

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1484 POMERANTSEV et al. formulas and Z1 cannot be used for large values of x.So,at 2 2 2 ∞ large x, we use the following asymptotic series: q +q +m −mρ QL( 2qq ) e = j (q ρ) j (qρ)ρ2dρ Yuk 2qq L ρ L J2L (b, m) (A.78) ∞ 0 2b i (2L +2i − 1)!!(L + i)! (A.70) = − , m2 i! and i=0 ∞ L Yuk 2 −βq2 L+2 ρ −ρ2/(4β) J2L+1(b, m) (A.79) e q jL(qρ)dq = e , π (2β)L+3/2 ∞ i 2b (2L +2i + 1)!!(L + i)! 0 = − . (A.71) m2 i! i=0 we get the result Three-body force due to OPE. For OPE, we R(L, ω ,L,ω; L, m) (A.72) take the interaction operator in the standard form 1 −1 −1 = JYuk((ω + ω )/4,m), g2 (4ωω)L+3/2 L (i) − πNN (i) · VOPE = 2 (σ p) (A.80) (2mN ) where 1 ∞ × · (i) · −mρ 2 2 (Sd p)(τ Td), e − 2 p + m JYuk(b, m)= ρ2L+2 e bρ dρ. (A.73) π − L ρ p = q q , 0 where σ(i) and τ (i) are the spin and isospin operators This integral of the Yukawa potential is calculated by of the third (ith) nucleon, whereas Sd and Td are the recursions: operators of the total spin and isospin of the 6q bag, Yuk 1 m respectively. We found that the contribution of OPE J (b, m)= ZL(x),x= √ , (A.74) L mL+1 2 b is so small that it is suﬃcient to include only S waves in its evaluation. In this case, only the central part of where √ the OPE interaction remains: x2 Z0(x)= πxe (1 − erf(x)), (A.75) 2 OPE 2 mπ 1 (i) · Vc = gπNN 2 (σ Sd) (A.81) 2 (4mN ) 3 Z1(x)=2x (1 − Z0(x)), (A.76) × (i) · 1 2 (τ Td) 2 2 . ZL(x)=2x ((k − 2)ZL−2(x) − ZL−1(x)) . (A.77) p + mπ These recursions (especially for large L) for functions The spin–isospin m.e. is nonzero only for a singlet– Z(x) and also expressions (A.75) and (A.76) for Z0 triplet transition:

1 (i) (i) Sd =0,Td =1 (σ · Sd)(τ · Td) Sd =1,Td =0 =4/9. (A.82) 3

Then, the m.e. of the OPE contribution for S waves from each other only by a constant. Therefore, using takes the form (for Sd =0, Sd =1or Sd =1, Sd =0) Eq. (A.63), one can exclude these functions from the formula for the m.e. f¯ f¯ R00(˜ω , ω˜; OPEW ) (A.83) 0 Three-body Coulomb force. The m.e. of the 4 E (1 + a) = f 2 λ0 (0)λ1 (0) 0 operator for the three-body Coulomb force (60) (for 9 πNN 00 00 (E − E )2 0 pointlike charges) can be expressed in terms of integrals over the overlap functions χ (q): × B BMR(0, ω˜ + θ , 0, ω˜ + θ;0,m ). f M π M M Coul 2 Ji Mi2j( W )=e λ (0)(1 + a) (A.84) LiLi 0 1 Here,wetakethevertexfunctionsB0 and B0 diﬀering JiMiLiLi

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DIBARYON MODEL FOR NUCLEAR FORCE 1485 (2) 1+τ The latter integral is evaluated analytically in the form If ,γ n (q) 3 (1 + tˆ )If,γn(q) (2)i dz (2)j of a ﬁnite sum: 2 × dqdq , 1 1 2 Coul 2 JL (b, a)= L+1 (A.91) q 2 q 2 a E − E0 − (q − q ) E − E0 − 2m 2m L k! (2L − 2k +1)!! × Ck , L k+1 L−k+1/2 where (1 + τ (2))/2 is the operator of the nucleon b b 3 k=0 2L−k +1 charge (nucleon with number 2). The isospin part of a a this m.e. is equal to where Ck are the binomial coeﬃcients. τ Coul(t ) (A.85) L d 1+τ (2) X tdtdz 3 X td REFERENCES = (1 + tdz ) 2 1. Proceedings of the XVIII European Conference on td z  Few-Body Problems in Physics, Bled, Slovenia,  1 for td =0, 2002; Few-Body Syst. Suppl. 14 (2003). = 1 2. W. Tornow, H. Witala, and A. Kievsky, Phys. Rev. C for td =1. 57, 555 (1998); D. Hueber, Few-Body Syst. Suppl. 3 10, 305 (1999); L. D. Knutson, Nucl. Phys. A 631,9c Thus, for calculation of 3BF Coulomb m.e., one can (1998). apply formulas Eqs. (A.69)–(A.79) for the isoscalar 3. Y. Koike and S. Ishikawa, Nucl. Phys. A 631, 683c operator with this additional isospin factor (A.85): (1998). ¯ ¯ 4. H. O. Meyer, Talk Presented at Symposium on Cur- tt Coul f f RL (˜ω , ω˜; W ) (A.86) rent Topics in the Field of Light Nuclei, Cracow, Poland, 1999, p. 858 (unpublished). Coul 2 Ji E0(1 + a) = δ τ (t )e λ (0) 5. D. L. Groep et al., Phys. Rev. C 63, 014005 (2001). t td d λ λ d i i (E − E )2 0 6. D. P.Watts et al., Phys. Rev. C 62, 014616 (2000). Coul 7. L. Weinstein, Few-Body Syst. Suppl. 14, 316 (2003). × B BMR (t , ω˜ + θ ,t,ω˜ + θ; L). M 8.V.I.Kukulin,I.T.Obukhovsky,V.N.Pomerantsev, M M and A. Faessler, Yad. Fiz. 64, 1748 (2001) [Phys. At. Here, the Coulomb integral RCoul(t, ω,t,ω; L) for Nucl. 64, 1667 (2001)]; J. Phys. G 27, 1851 (2001). the pointlike charges is obtained from the Yukawa 9. B. F. Gibson, Lect. Notes Phys. 260, 511 (1986). integral R (A.68) by setting m =0: 10. B. F. Gibson et al., Few-Body Syst. 3, 143 (1988). 11. P. Sauer, Talk Presented at Symposium on Cur- rent Topics in the Field of Light Nuclei, Cracow, RCoul(t ,ω ,t,ω; L)=R(t ,ω,t,ω; L, 0). (A.87) Poland, 1999, p. 503 (unpublished). 12. N. Kaiser, R. Brockmann, and W. Weise, Nucl. Phys. Hence, these Coulomb integrals are reduced by dif- A 625, 758 (1997); N. Kaiser, S. Gerstendorfer,¨ and ferentiating [see Eq. (A.69)] to the integrals W. We i s e , N u c l . P h y s . A 637, 395 (1998). 1 13. E. Oset et al.,Prog.Theor.Phys.103, 351 (2000). RCoul(L, ω ,L,ω; L)= (A.88) (4ωω)L+3/2 14. M. M. Kaskulov and H. Clement, Phys. Rev. C 70, 70 ∞ 014002 (2004); , 057001 (2004). 2 1 ρ − 15.A.M.Kusainov,V.G.Neudatchin,and × ρ2L+2 exp − (ω−1 + ω 1) dρ I. T. Obukhovsky, Phys. Rev. C 44, 2343 (1991). ρ 4 16. F. Myhrer and J. Wroldsen, Rev. Mod. Phys. 60, 629 0 (1988). 1 − ≡ JCoul((ω−1 + ω 1)/4), 17. A. Faessler and F. Fernandez, Phys. Lett. B 124B, (4ωω)L+3/2 L 145 (1983); A. Faessler, F.Fernandez, G. Lubeck,¨ and K. Shimizu, Nucl. Phys. A 402, 555 (1983). Γ(L +1) 18. V. I. Kukulin and V. N. Pomerantsev, Prog. Theor. JCoul(b)= . (A.89) L 2bL+1 Phys. 88, 159 (1992). 19. Y. Yamauchi and M. Wakamatsu, Nucl. Phys. A 457, Now, we can replace the Coulomb potential 1/ρ 621 (1986). between the pointlike charges in the integrand in 20. M. Oka and K. Yazaki, Nucl. Phys. A 402, 477 (1983); Eq. (A.88) with the corresponding Coulomb potential E. M. Henley, L. S. Kisslinger, and G. A. Miller, Phys. between the “smeared” charges: Rev. C 28, 1277 (1983); K. Brauer,E.M.Henley,and¨ ∞ √ G. A. Miller, Phys. Rev. C 34, 1779 (1986). erf(ρ a) − 2 21. T. Fujita et al.,Prog.Theor.Phys.100, 931 (1998). JCoul(b, a)= ρ2L+2 e bρ dρ. (A.90) L ρ 22. Fl. Stancu, S. Pepin, and L. Ya. Glozman, Phys. Rev. 0 C 56, 2779 (1997).

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PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1487–1496. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1545–1554. Original English Text Copyright c 2005 by Isakov, Mach, Fogelberg, Erokhina, Aas, Hagebø. NUCLEI Theory

Structure of States and Transition Rates in the Even–Even N = 82 Nucleus 136Xe* V. I. Isakov1)**,H.Mach2),B.Fogelberg2), K.I.Erokhina3), A.J.Aas4),andE.Hagebø4) Received May 24, 2004; in ﬁnal form, January 17, 2005

Abstract—The properties of the 136Xe nucleus are theoretically investigated by using two diﬀerent approaches: the two-quasiparticle RPA method and the shell-model calculation. The investigated characteristics include both the energy levels and the electromagnetic properties of 136Xe. Comprehensive comparison with the experiment that includes all the currently available experimental data is performed. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION 132Sn were studied. Here, the experimental single- particle energies (except that for the 3s1/2 state) were The singly magic nuclei provide a detailed testing used and the two-body matrix elements of [8] were of model predictions for residual interaction between employed as startingones in the fittingprocedure. At like nucleons and checkingof di fferent theoretical the same time, an extended shell-model calculation usingrealistic G-matrix interaction was performed approaches for their description. Such nuclei with Z =52−62 N =82 136 for nuclei with even , in [12]. A N =82, includingthe Xe, have been interesting systematic study of the N =82isotones from 132Sn from a theoretical viewpoint for a longtime and there to 146Gd based on the RPA method [13] was reported exist several calculations for them. The shell model by us earlier [14]. was applied to these nuclides by Wildenthal in [1], However, the whole bulk of experimental evi- while in [2–5] their properties were considered using 136 the TD approximation accountingfor the pairingcor- dence [15–26] obtained by now for Xe, including the new data on electromagnetic properties, stim- relations and particle number projection. The nature ulates us to make a new theoretical review of the of isomerism in the even N =82isotones was cleared situation in this nuclei. We begin the interpretation of up by Heyde et al. in [6]. Paper [7] treated their the 136Xe properties also in the framework of the RPA properties in the framework of the generalized senior- method. The only difference with [14] is a new set of ity scheme. Wildenthal [8] calculated the spectra of 133 154 values for single-particle energies that is summarized even and odd N =82isotones from Sb to Hf in our work [27] and for residual interaction that in the framework of the multiparticle shell model that was also recently used by us [28] to interpret the used a restricted basis and 160 values of fitted pair odd–odd N =83 nucleus 136I. Startingfrom this matrix elements representingthe e ffective interac- approach, we continue to the shell-model calculation. tion. A similar procedure, initiated due to the updated The intercomparison between two calculations is and extended experimental information on the single- presented at the end of this paper. proton nucleus 133Sb [9] and the two-proton nucleus 134Te [10], but with the full model space of the proton − 2. BASIC RELATIONS DEFINING 50 82 shell, was also demonstrated by Blomqvist THE SPECTRA OF LEVELS in [11], where nuclei with up to six protons added to AND TRANSITION PROBABILITIES IN RPA ∗This article was submitted by the authors in English. Assumingthe presence of correlations in the true 1)PetersburgNuclear Physics Institute, Russian Academy of ground state |0 of an even–even nucleus, we define Sciences, Gatchina, 188350 Russia. Q+ 2)Department of Radiation Sciences, University of Uppsala, the creation operator n,JM of the one-phonon ex- Sweden. | + | cited state ωnJM = Qn,JM 0 as 3)Ioffe Physicotechnical Institute, Russian Academy of Sci- ences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 + n,J + + n,J Q = X [ξ ξ ]JM − Y [ξcξd]JM, Russia. n,JM jajb a b jcjd ≥ ≥ 4)Department of Chemistry, University of Oslo, Norway. a b c d **E-mail: [email protected] (1)

1063-7788/05/6809-1487$26.00 c 2005 Pleiades Publishing, Inc. 1488 ISAKOV et al. where elements of the eﬀective interaction ϑˆ in the particle– + + 1 JM + + particle and particle–hole channels with a given spin [ξ ξ ]JM = C ξ ξ , a b jamajbmb jama jbmb (see [29, 30]). 1+δj j a b mamb (2) From the explicit form of the system (7), there follows the normalization condition on the X and Y 1 amplitudes, which looks like [ξcξd]JM = (3) 1+δ jcjd JM n,J m,J n,J m,J × C ξ − ξ − ϕ ϕ , X X − Y Y = δmn, (10) jcmcjdmd jc mc jd md c d jajb jajb jcjd jcjd ≥ ≥ mcmd a b c d Xn,J = ω ; JM|[ξ+ξ+]|0, (4) jajb n a b which in terms of the RPA bosons corresponds to the n,J | | condition Yj j = ωn; JM [ξcξd] 0 . (5) c d + 0˜|Qn,JM Q |0˜ = δmn. (11) Here, ξ+ are the quasiparticle operators deﬁned by m,JM the equations In [2, 3, 14], which described systems with valence + + − 2 2 aα = u|α|ξα v|α|ϕαξ−α and u|α| + v|α| =1, (6) protons above the Z =50, N =82shells, the inter-

α+jα−mα action of the followingform was used: where ϕα =(−1) is the phase of the particle–hole transformation (we use below the des- ˆ ˆ ˆ − 2 ϑ = V0(Ps + tPt)exp( βr12), (12) ignations u|α| = uα and v|α| = vα, omittingthe mag- −2 netic quantum numbers that enter only in ϕα), and where V0 = −33.2 MeV, t =0.2, β =0.325 fm , |− α is a state with the opposite sign of magnetic and Pˆs and Pˆt are singlet and triplet spin projectors. quantum number with respect to |α. Here, we involve the basis that includes not only pro- The set of RPA equations that define the ampli- tons, but also neutrons, which gives rise to neutron tudes “X” and “Y ” of the states |ωn,JM and the particle–hole excitations and admixtures. So, now eigenvalues ω has the followingform: we use a more general interaction, introduced by us n   in [28] for description of odd–odd nuclei in the vicinity [(E − ω)I + A] B X of A = 132, namely, =0. −B −[(E + ω)I + A] Y ϑˆ = V + Vσσˆ1 · σˆ2 + VT Sˆ12 (13) (7) −r2 /r2 + Vτ + Vτσσˆ1 · σˆ2 + VτTSˆ12 τˆ1 · τˆ2 e 12 0 In Eq. (7), E = Eab = Eja + Ejb , Icd,ab = δjajc δjbjd , Ej is a quasiparticle energy, while the matrix elements of the submatrices A and B in the case of even–even with r12 = |r1 − r2| and Sˆ12 beinga tensor operator. nuclei look as follows: Pair Coulomb interaction between protons was also ≡ J taken into consideration. The parameters used in the Acd,ab Aj j ,j j =(ujc ujd uja ujb (8) c d a b calculation were the following: V = −16.65, V = |ˆ| σ + vjc vjd vja vjb )a jcjd; J ϑ jajb; J a 2.33, VT = −3.00, Vτ =3.35, Vτσ =4.33, VτT = 3.00 (all in MeV), and r0 =1.75 fm. With these +(ujc vj uja vj + vjc uj vja uj )a d b d b parameters and for a system of only like nucleons, × ¯ |ˆ| ¯ − − ja+jb+J jcjd; J ϑ jajb; J a ( 1) (vjc ujd uja vjb interaction (13) identically transforms into that given ¯ |ˆ| ¯ by Eq. (12). We also mention here that the interaction + ujc vj vja uj )a jcjd; J ϑ jbja; J a, d b in (13) was used by us not only for calculation of ≡ J the particle–particle and the particle–hole matrix Bcd,ab Bj j ,j j =(ujc uj vja vj (9) c d a b d b elements that enter the RPA equations (7)–(9), but |ˆ| + vjc vjd uja ujb )a jcjd; J ϑ jajb; J a also for definingthe pairingcharacteristics ( u, v − 136 (ujc vjd vja ujb + vjc ujd uja vjb )a coefficients and the Ej values) in Xe. Our basis included 11 proton and 5 neutron orbitals that are ×j j¯ ; J|ϑˆ|j j¯ ; J +(−1)ja+jb+J (v u v u c d a b a jc jd ja jb closest to the proton and neutron Fermi levels. ¯ |ˆ| ¯ + ujc vjd uja vjb )a jcjd; J ϑ jbja; J a. Corresponding single-particle energies were either borrowed from the experiment [9, 31] or generated |ˆ| In Eqs. (8) and (9), a jcjd; J ϑ jajb; J a and by the suitable phenomenological potential [27, 29, ˆ ajcj¯d; J|ϑ|jaj¯b; Ja are the antisymmetrical matrix 30], if the experimental values were absent.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 STRUCTURE OF STATES AND TRANSITION RATES 1489 Consideringbelow the electromagnetic transi- 3. SHELL-MODEL CALCULATIONS tions that are described by the operator Mˆ (λµ), The RPA method considered by us before has Mˆ (λµ)= i|mˆ (λµ)|ka+a , (14) stronglimitations, since it reproduces only states of i k the “two-quasiparticle” nature [see Eq. (1)] with a i,k certain class of correlations, includingthe ground- we must distinguish between two different cases, i.e., state ones, taken into account. Indeed, in the energy the phonon–phonon (between the two excited states) interval that is of interest to us, one can see some and the phonon–ground-state transitions. In the last other excitations, which are out of the RPA approx- case, the transition matrix element reads as imation. For example, the total number of 2+ states 136 0˜||Mˆ (λ)||ω ,J =(−1)λδ δ (15) in Xe with excitation energies less than 3.5 MeV is n Jλ πnπλ at least six, while the RPA gives only four levels. The + n,J excess of 4 states with energies less than 3 MeV is × X (uj vj ± vj uj ) jajb a b a b also seen, as compared to the RPA predictions. The ≥ ja jb extra excitations offer the levels that, in the spirit of (−1)lb the multiparticle shell model, are nothingbut states × ja||mˆ (λ)||jb 1+δ with the total seniority values s,fromdifferent pos- jajb sible orbitals, more than two (while the RPA states − Y n,J (v u ± u v ) correspond to shell-model states with s =2). jajb ja jb ja jb ≥ 136 ja jb Below, the structure of Xe is considered in a multiparticle shell-model approach, formally in the (−1)lb × ja||mˆ (λ)||jb , framework of the diagonal approximation, but with 1+δjajb the values of interaction matrix elements obtained from the empirical data, which implicitly include the where the upper signs refer to T -even (Eλ)tran- bulk of correlations. sitions, while the lower ones refer to T -odd (Mλ) Beginning from the lowest configuration transitions. 4 {(π1g7/2) },wenotethattheRPAschemegivesus In Eq. (15), δ are the Kronecker symbols, while πn 2 + + only {(π1g7/2) , I =0} (ground state), I =2 , 4 , and πλ are the parities of the state |ωn and of the + { 4 transition operator Mˆ (λ). and 6 levels, which are equivalent to the 7/2 ; s = } { 4 } { 4 } At the same time, the phonon–phonon matrix el- 0, I =0 , 7/2 ; s =2,I =2 , 7/2 ; s =2,I =4 , { 4 } ement in the case of λ =0 has the form and 7/2 ; s =2,I =6 shell-model states. How- ever, the configuration {7/24} gives rise [34] to ω ,J||Mˆ (λ)||ω ,J =[(2J + 1)(2J +1)]1/2 n m additional seniority-four levels, namely, {7/24; s = (16) } { 4 } { 4 } 4,I =2 , 7/2 ; s =4,I =4 , 7/2 ; s =4,I =5 , Xm,J Xn,J ± Y m,J Y n,J {7/24; s =4,I =8} jajb jcjd jajb jcjd and , which have no analogin the × RPA scheme. ≥ ≥ (1 + δjajb )(1 + δjcjd ) { n ja jb,jc jd Consider the splittingof the con figuration j ; s, α, I},wheren is the number of particles, I is spin, s is × ∓ δjbjd W [λjcJjb; jaJ ](ujc uja vjc vja ) seniority, and α is some additional quantum number

(when necessary). Usingthe formalism of fractional × || || − − jc+jd+J jc mˆ (λ) ja ( 1) δjbjc parentage expansions, one may easily represent the × ∓ || || diagonal matrix element of the interaction ϑˆ in the W [λjdJjb; jaJ ](ujd uja vjd vja ) jd mˆ (λ) ja form − − ja+jb+J ( 1) δjajd W [λjcJja; jbJ ] n × (u u ∓ v v )j ||mˆ (λ)||j n ˆ n jc jb jc jb c b j sα; I ϑ(i, k) j sα; I (17) − ja+jb+J+jc+jd+J i

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3 Table 1. Energy levels and E2-transition rates, B(E2),in Table 2. Energies of the |(π1g7/2) Js,π2d5/2; I states the diagonal {7/24} shell-model calculations for 136Xe; all in 136Xe; only the levels with J =7/2, s =1have their the nondiagonal matrix elements for the M1 operator are analogs in the RPA calculations (note that the L =3inter- equal to zero (note that the calculated value of the ground- action matrix element of the {π1g7/2π2d5/2} configuration 136 state bindingenergyof Xe is equal to B = 1141.84 MeV, does not enter the energies of the I =10and I =0states) 136 which may be compared to B( Xe)exp = 1141.88 MeV) J s I E,MeV J s I E,MeV → E, B(E2; Iisi I s Ii si If sf 2 4 7/2 1 1 2.553 9/2 3 2 3.804 MeV If sf ), e fm 0 0 g.s. 2 2 0 0 215 2 2.634 3 3.738 2 2 1.279 4 4 6 2 278 3 2.408 4 3.698 4 2 1.577 4 4 4 2 229 4 2.385 5 3.617 6 2 1.691 4 4 2 2 238 5 2.421 6 3.400 4 4 1.976 2 4 4 2 106 6 2.192 7 3.392 2 4 2.204 2 4 2 2 218 3/2 3 1 3.587 11/2 3 3 3.689 5 4 2.296 5 4 6 2 164 2 3.337 4 3.855 8 4 2.664 5 4 4 2 206 3 3.373 5 3.597 8 4 6 2 154 4 3.318 6 3.434 7 3.462 interaction. Instead, we define their values from the 8 3.233 experiment by employingthe ansatz based on the 5/2 3 0 3.483 15/2 3 5 4.193 Koopmans theorem [36] and with account of residual interaction between the valence particles: 1 3.083 6 4.018 2 B 133 −B 132 2 3.181 7 4.027 VJ0 ((π1g7/2) )=2 ( Sb) ( Sn) (18) −B 134 J0 134 2 3 3.304 8 3.940 ( Te)+Eexc( Te, (π1g7/2) ), 4 3.106 9 3.591 where B are ground-state binding energies. Then the excitation energy of the |(7/2)4sα; I state in 136Xe is 5 2.996 10 3.500 given by the equation EI (136Xe; sα)=B(136Xe)+3B(132Sn) (19) exc correspondingto the eeff(E2) = 1.85|e| (see [10]) and 133 − 4B( Sb)+MI (7/2,n =4,s,α) the Woods–Saxon wave functions, while || || (no additional quantum number α is really present in π1g7/2 mˆ (M1) π1g7/2 =4.70µN , the configuration {7/24}). correspondingto the experimental [32] value of the Let us consider now the transition probabilities. If ground-state magnetic moment of 133Sb. |i≡|7/24s ; I and |f≡|7/24s ; I ,then i i f f The theoretical energy levels and transition rates ||Mˆ || f (λ) i =4 (2Ii + 1)(2If +1) (20) arisingfrom this approach are shown in Table 1. Note × 3 |} 4 that, amongthe nondiagonal E2 matrix elements only 7/2 s0J0, 7/2; Ii 7/2 siIi those with ∆s =2are different from zero. s0J0 × 3 |} 4 Now let us pass on to the higher lying configu- 7/2 s0J0, 7/2; If 7/2 sf If { 3 }≡{ 3 } ration j1 ,j2 (π1g7/2) ,π2d5/2 . Schematically, × W [If J0λ7/2; 7/2Ii]π1g7/2||mˆ (λ)||π1g7/2, its wave function may be represented as |J1,j2; Ia, where ...|} ... are the fractional parentage coeffi- where index a means antisymmetrization and cients. In practical calculations we used the following |J = |j3s ,α ; J (21) values of reduced matrix elements enteringthe right- 1 1 1 1 1 a hand side of (20): 2 |} 3 | 2 = j1 J0,j1; J1 j1 s1α1J1 j1 J0,j1; J1 . 2 J π1g7/2||mˆ (E2)||π1g7/2 = −40.20|e| fm , 0 even

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 STRUCTURE OF STATES AND TRANSITION RATES 1491 The diagonal matrix element of interaction for the =2B(133Sb) −B(132Sn) −B(134Te) 3 conﬁguration {j J ,j ; I} looks like 2d 1 1 2 + EL (134Te,π1g π2d ) − E 5/2 (133Sb), exc 7/2 5/2 exc 4 3 ˆ 3 j1 s1α1J1,j2; I ϑ(i, k) j1 s1α1J1,j2; I (22) where Eexc are the correspondingexcitation energies. i

= MJ (j1,n=3,s1,α1)+3(2J1 +1) 1 At the present time, the excitation energies for all 2 × (2L +1)W [j1j2J0I; LJ1] 134 members of the {π1g7/2,π2d5/2} multiplet in Te, J0 even; L except the L =3level, are known [33]. For this level, × 2 |} 3 2 |ˆ| j1 J0,j1; J1 j1 s1α1J1 a j1j2L ϑ j1j2L a. we used the value of 2.7 MeV,obtained by us from the proper interpolation based on the parabolic rule [37, j j L|ϑˆ|j j L The values of a 1 2 1 2 a are deﬁned in our 38]. This quantity is a cause of some uncertainty in approach from the formula our prediction of energies, the values of which are ˆ aπ1g7/2π2d5/2L|ϑ|π1g7/2π2d5/2La (23) deﬁned from the formula

2d EI (136Xe,s J )=B(136Xe)+3B(132Sn) − 4B(133Sb)+E 5/2 (133Sb) (24) exc 1 1 exc 4 3 ˆ 3 + (π1g7/2) s1J1,π2d5/2; I ϑ(i, k) (π1g7/2) s1J1,π2d5/2; I i

and are listed in Table 2. In addition to the single-particle transition matrix Let us consider now the transitions from the state elements given earlier, we now need some new ones: 3 4 |i = |j siαiJi,j2; Ii to the |f = |j sf αf If (in our 2 1 1 π1g7/2||mˆ (E2)||π2d5/2 =8.19|e| fm , case j = π1g and j = π2d ). The transition 1 7/2 2 5/2 || || − | | 2 matrix element looks like π2d5/2 mˆ (E2) π2d5/2 = 33.82 e fm , || || − f||Mˆ (λ)||i (25) π1g7/2 mˆ (M1) π2d5/2 = 0.135µN , and π2d5/2||mˆ (M1)||π2d5/2 =5.45µN . = (2Ii + 1)(2If +1)W [If Jiλj2; j1Ii] All these quantities were also calculated by usingthe × 3 |} 4 || || j1 siαiJi,j1; If j1 sf αf If j1 mˆ (λ) j2 . effective charges and effective M1 operators, defined in [10] and [33]. At last, we show the formula for transitions within 3 the configuration {j ,j2}, which has the form 1 4. INTERCOMPARISON j3s α J ,j ; I ||Mˆ (λ)||j3s α J ,j ; I (26) BETWEEN THE TWO MODELS 1 f f f 2 f 1 i i i 2 i Comingto the consistency between both theo- = (2Ii + 1)(2If +1) retical models, we again mention that, among the shell-model states, there are ones, namely, with total × || || 3W [Jf j2λIi; If Ji] j1 mˆ (λ) j1 seniority values (from different orbitals) equal to two, which have their analogs in the RPA scheme. How- × 2 |} 3 j1 J0,j1; Ji j1 siαiJi ever, there also exist states which appear only in one J0 approach or the other. The comparison of theoretical energies obtained in the framework of the mentioned ×j2J ,j ; J |}j3s α J 1 0 1 f 1 f f f methods with the results of experiment is demon- × (2J + 1)(2J +1)W [J J λj ; j J ] strated in Table 3. Parallel with our data, we show i f f 0 1 1 i here the intercomparison with experimental results + W [If Jf λj2; j2Ii]j2||mˆ (λ)||j2 from [17, 22, 26]. As one can easily see, the total number of states in such combined method increases, × δ(JiJf )δ(sisf )δ(αiαf ) . as compared to only RPA or empirical limited shell- model approaches, and one can conciliate it with the

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Table 3. Comparison of experimental and theoretical energy levels in 136Xe (experimental states and their interpretation in the framework of approaches considered in this work are exhibited below; in the case of RPA calculations the leading components of eigenvectors are speciﬁed; all the energies are given in MeV)

π π Ii ; E(Ii )exp Phonon description Empirical shell-model description + { 4 } 01 ;g.s. Vacuum; g.s. (g7/2) , I =0, s =0 ;g.s. + 02 ; 2.581 Pairingvibration; 1.872 + { } { 3 } 11 ; 2.634 g7/2d5/2 ; 2.509 (g7/2) , J =7/2, s =1, d5/2; I =1 ; 2.553 + { 3 } (12 ); 3.212 (g7/2) , J =5/2, s =3, d5/2; I =1 ; 3.083 + { } { 4 } 21 ; 1.313 g7/2g7/2 ; 1.301 (g7/2) , s =2, I =2 ; 1.279 + { 4 } 22 ; 2.289 (g7/2) , s =4, I =2 ; 2.204 + { } { 3 } 23 ; 2.414 g7/2d5/2 ; 2.315 (g7/2) , J =7/2, s =1, d5/2; I =2 ; 2.634 + { } 24 ; 2.849 d5/2d5/2 ; 2.528 + { } 25 ; 2.869 g7/2d3/2 ; 3.189 3 {(g7/2) , J =5/2, s =3, d5/2; I =2}; 3.181 3 {(g7/2) , J =3/2, s =3, d5/2; I =2}; 3.337 + { } { 3 } 31 ; 2.560 g7/2d5/2 ; 2.384 (g7/2) , J =7/2, s =1, d5/2; I =3 ; 2.408 + { } (32 ); 3.873 g7/2d3/2 ; 3.703 3 {(g7/2) , J =5/2, s =3, d5/2; I =3}; 3.304 + { } { 4 } 41 ; 1.694 g7/2g7/2 ; 1.672 (g7/2) , s =2, I =4 ; 1.577 + { 4 } 42 ; 2.126 (g7/2) , s =4, I =4 ; 1.976 + { } { 3 } 43 ; 2.465 g7/2d5/2 ; 2.260 (g7/2) , J =7/2, s =1, d5/2; I =4 ; 2.385 + { 4 } 51 ; 2.444 (g7/2) , s =4, I =5 ; 2.296 + { } { 3 } 52 ; 2.608 g7/2d5/2 ; 2.353 (g7/2) , J =7/2, s =1, d5/2; I =5 ; 2.421 3 {(g7/2) , J =5/2, s =3, d5/2; I =5}; 2.996 + { } { 4 } 61 ; 1.892 g7/2g7/2 ; 1.782 (g7/2) , s =2, I =6 ; 1.691 + { } { 3 } 62 ; 2.261 g7/2d5/2 ; 2.090 (g7/2) , J =7/2, s =1, d5/2; I =6 ; 2.192 + { 4 } 81 ; 2.867 (g7/2) , s =4, I =8 ; 2.664 + { 3 } 82 ; 3.229 (g7/2) , J =11/2, s =3, d5/2; I =8 ; 3.233 + { 3 } 101 ; 3.484 (g7/2) , J =15/2, s =3, d5/2; I =10 ; 3.500 − 31 ; 3.275 Collective; 3.848

+ → + experiment. Attention should also be drawn to the the 42 21 E2-transition probability is found fact that the allied (RPA and shell-model) states have to be small, about 0.21 W.u. (experimental value rather similar energies. ≥ + → + 1.7 W.u.), while the M142 41 transition is + One can see from Table 3 that the second 2 and rather strong, with B(M1) of about 0.025 W.u. Then 4+ levels really offer not the second RPA states, but the ratio γ(4+ → 4+)/γ(4+ → 2+) becomes equal to the four-quasiparticle {(πg )4,s=4}, i.e., two- 2 1 2 1 7/2 ∼20, which also strongly contradicts the experiment, phonon configurations. This prescription is fortified while the pattern presented in Table 4 gives the by examiningthe electromagnetic properties of these + levels, shown amongthe others in Table 4. Re- better agreement. The situation with the 22 level is + ally, if one considers the phonon 2.260-MeV state more complicated. Here, if one takes as a 22 level { } + + with the structure g7/2d5/2 as a 42 level, then the phonon 2.315-MeV state and as 23 level also

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Table 4. Comparison of experimental and theoretical transition probabilities in 136Xe (the quantities γ represent both experimental and theoretical branchingratios for the γ decays from the initial to diﬀerent ﬁnal states; they are normalized to 100 for the strongest decay seen in the experiment (or given by calculations), for each initial state)

π → π a) b) Ii If XL B(XL)exp,W.u. B(RPA), W.u. B(sh. mod.), W.u. γ(exp.) γ(RPA) γ(sh. mod.) + → + 21 01 E2 8.5(35) 14.5 5.2 100 100 100 + → + 41 21 E2 1.29(2) 0.34 0 100 100 100 + → + 61 41 E2 0.0132(5) 0.13 0 100 100 100 + → + ≥ × −3 42 41 M1 2.8 10 0 25(2) 4.1 E2 ≥9.2 5.5 + → + ≥ 42 21 E2 1.7 5.7 100(8) 100 + → + × −3 × −1 × −5 62 61 M1 2.7(5) 10 0.75 10 7.08 10 100(6) 100 100 E2 121(24) 0.29 2.55 × 10−2 + → + × −1 × −3 62 41 E2 0.34(7) 0.6 10 0.67 10 2.4(2) 0.14 1.6 + → + ≥ × −4 22 21 M1 2.4 10 0 25.8(14) 25 E2 ≥0.16 5.25 + → + ≥ × −3 c) 22 01 E2 8.8 10 0.28 100(5) 100 + → + ≥ × −3 51 62 M1 6.5 10 0 11(1) 0 E2 ≥121 0 + → + ≥ × −3 51 42 M1 1.1 10 0 10(1) 0 E2 ≥6.9 0 + → + ≥ × −4 51 61 M1 2.9 10 0 14(1) 17 E2 ≥0.6 3.95 + → + ≥ × −4 51 41 M1 9.3 10 0 100(8) 100 E2 ≥1.0 4.96 + → + × −1 × −3 23 21 M1 0.16 10 0.25 10 5.3(5) 10.9 6.3 E2 0.133 1.14 × 10−3 + → + 23 01 E2 1.53 0.040 100(5) 100 100 + → + × −3 × −2 d) 43 42 M1 1.9(8) 10 0.21 10 22(2) 79 E2 10.1(46) 0.24 × 10−3 + → + × −4 × −1 × −3 43 41 M1 7.2(32) 10 0.25 10 0.182 10 100(5) 100 100 E2 0.76(34) 0.88 × 10−1 0.024 + → + × −3 × −2 e) 31 22 M1 1.24(3) 10 1.11 10 11.4(15) 12 E2 11.0(54) 3.58 × 10−3 + → + × −4 × −2 d) 31 42 M1 9.6(46) 10 0.61 10 35(2) 27 E2 3.2(15) 0.50 × 10−2 + → + × −4 × −3 × −3 31 41 M1 1.1(5) 10 0.56 10 2.85 10 30.1(19) 49 100 E2 8.7(41) × 10−2 1.53 × 10−2 1.68 × 10−3 + → + × −4 × −3 × −5 31 21 M1 1.1(5) 10 3.72 10 5.75 10 100(4) 100 12 E2 4.7(22) × 10−2 1.43 × 10−2 2.29 × 10−2

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Table 4. (Contd).

π → π a) b) Ii If XL B(XL)exp,W.u. B(RPA), W.u. B(sh. mod.), W.u. γ(exp.) γ(RPA) γ(sh. mod.) + → + ≥ × −3 52 51 M1 9.2 10 0 18.0(21) 0 E2 ≥232 0 + → + ≥ × −3 52 62 M1 5.1 10 0.16 0.107 87(4) 100 100 E2 ≥28.7 0.93 0.984 + → + ≥ × −3 × −3 d) 52 42 M1 1.1 10 3.66 10 53(3) 11 E2 ≥3.3 1.52 × 10−2 + → + ≥ × −4 × −3 × −4 52 61 M1 1.8 10 3.5 10 4.18 10 28.3(17) 19 4 E2 ≥0.24 9.4 × 10−3 1.85 × 10−2 + → + ≥ × −4 × −3 × −3 d) 52 41 M1 3.5 10 5.4 10 3.66 10 100(5) 63 62 E2 ≥0.26 7.32 × 10−2 1.52 × 10−2 + → + × −3 11 23 M1 2.2(8) 10 0.58 0.35 3.5(2) 11 47 E2 29(10) 1.26 3.18 + → + × −3 × −2 e) 11 22 M1 1.9(7) 10 1.64 10 11.4(6) 9 E2 9.8(35) 0.15 + → + × −4 × −2 × −4 11 21 M1 2.9(10) 10 2.41 10 1.46 10 100(5) 100 8 E2 0.10(4) 1.34 × 10−2 3.99 × 10−2 + → + × −6 × −3 × −4 11 01 M1 9.8(35) 10 1.35 10 4.23 10 26.7(12) 44 100 + → + 81 61 E2 3.7 + → + 82 62 E2 4.3 + → + 101 82 E2 3.2 − → + 31 01 E3 16.9 15 a) For transitions where the mixed M1/E2 mode is allowed but the mixingparameter δ is not deﬁned, the listed rates are given separately for the cases when the pure M1 or E2 mode is assumed. b) Weighted average of branchings obtained by us [26] and the values adopted in [17]. c) + + Obtained as a result of the 23% admixture of 21 to 22 . d) + + + Takinginto account the 15 % admixture of 43 to 42 and 41 in the amplitude. e) + + By virtue of the 22% admixture of 23 to 22 .

+ → + the phonon, but 2.528-MeV state, then both the 22 01 transition. Small configuration mixing for transition probabilities and branchingratios can be other levels is also needed for better description of the conciliated with the experiment. However, in this branchingratios. 4 + 136 case, there is no room for the {(g7/2) s =4I =2} At present, two 5 levels of Xe with exci- two-phonon state, which really is the next one, after tation energies less than 3 MeV are found in the + experiment [26]. At the same time, the calculations the 21 level (see Table 3). As in the harmonic picture, show the existence of three 5+ states in this in- there is no transition between the two-phonon and terval of energies. Among them, the shell-model + → + 4 ground states; the B(E2; 22 01 ) value in the {(g7/2) ,s=4,I =5} state that has the lowest en- diagonal shell-model approximation is equal to zero. ergy (see Table 3) corresponds to the experimental + In order to reproduce the experimental branching 51 (2.444-MeV) level. Meanwhile, the identifica- + + from the 22 level, we must include small configuration tion of the structure of higher 5 levels is not so + mixing(see footnote to Table 4), which “opens” the synonymous. We assume that the 52 (2.608-MeV)

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3 state has the shell-model structure {(g7/2) ,J = ACKNOWLEDGMENTS } 7/2,s =1,d5/2; I =5 , its RPA analogbeingof the This work was supported in part by the Research { } + g7/2d5/2 type. In this case, the half-life of the 52 Council of Norway, the Royal Swedish Academy of state proves to be about ∼3 ps as compared to the Sciences, and the Russian Foundation for Basic Re- search (grant no. RSGSS-1124.2003.2). A.J.A. and experimental value of T1/2 ≤ 27 ps [26], while the V.I.I. would also like to thank the OSIRIS group at alternative attribution of the structure {(g )3,J = 7/2 Studsvik, Sweden, for their generous hospitality. 5/2,s =3,d5/2; I =5} for this level leads to T1/2 ∼ 300 ps, which contradicts the experiment. REFERENCES At the present time, there exist several measurements concerningmagnetic properties of the 1. B. H. Wildenthal, Phys. Rev. Lett. 22, 1118 (1969). 136 + 2. M. Waroquier and K. Heyde, Nucl. Phys. A 164, 113 Xe states. In [39], the magnitude of g(41 )= (1971). 0.80(15) was measured, while in [40] and [41] the 3. K. Heyde and M. Waroquier, Nucl. Phys. A 167, 545 + + (1971). values of g(21 ) turned out to be g(21 )=0.77(5) and g(2+)=1.20(25), respectively. These numbers may 4. M. Waroquier and K. Heyde, Z. Phys. A 268,11 1 (1974). be compared with the results of our RPA calculations: 5. G. Wenes, K. Heyde, M. Waroquier, and P. Van + + g(41 )=0.81 and g(21 )=0.86. We again mention Isacker,Phys.Rev.C26, 1692 (1982). here that, in calculations, we used magnetic parame- 6. K. Heyde, M. Waroquier, and G. Vanden Berghe, ters close to that from our work [33]. Unfortunately, at Phys. Lett. B 35B, 211 (1971). the present time, there is no experimental evidence on 7. O. Sholten and H. Kruse, Phys. Lett. B 125B, 113 the values of quadrupole moments of states in 136Xe, (1983). 8. B. H. Wildenthal, in Proceedings of the 3rd Interna- which are extremely dependent on the shell-model tional Spring Seminar on Nuclear Physics, Ischia, structure and the occupancy of single-particle levels. Italy, 1990, Ed. by A. Covello (World Sci., Singapore, Amongthe positive-parity states, we mark out 1991), p. 35. the high-spin levels, where configuration mixing is 9. M. Sanchez-Vega, B. Fogelberg, H. Mach, et al., small. Calculated empirical shell-model energies of Phys. Rev. Lett. 80, 5504 (1998). these levels are very close to the experimental ones. 10. J. P. Omtvedt, H. Mach, B. Fogelberg, et al.,Phys. Rev. Lett. 75, 3090 (1995). At the same time, the theory predicts rather strong E2 11. J. Blomqvist, Acta Phys. Pol. B 30, 697 (1999). transitions between these levels (see Table 4), which 12. A. Holt, T. Engeland, E. Osnes, et al.,Nucl.Phys.A is also qualitatively confirmed by the experiment. 618, 107 (1997). 13. M. Baranger, Phys. Rev. 120, 957 (1960). 14. K. I. Erokhina, V. I. Isakov, B. Fogelberg, and 5. SUMMARY AND CONCLUSIONS H. Mach, Preprint No. 2136, PIYaF (PetersburgNu- clear Physics Institute, Gatchina, 1996). This work has essentially expanded our under- 15. H. Mach and B. Fogelberg, Phys. Scr., T 56, 270 standingof the structure of the N =82 nucleus (1995). 136Xe. Joint theoretical calculation of the energies 16. T. W.Burrows, Nucl. Data Sheets 52, 273 (1987). of levels and their decay properties performed in 17. J. K. Tuli, Nucl. Data Sheets 71, 1 (1994). the framework of two different models enabled us 18. W. R. Western, J. C. Hill, W. L. Talbert, and 15 to make a synonymous definition of the structure W. C. Schick, Phys. Rev. C , 1822 (1977). 19. P.A.Moore,P.J.Riley,C.M.Jones, et al., Phys. Rev. for the majority of excited states in this nucleus. C 1, 1 (1970). In our calculations, we used the minimal amount 20. E. Achterberg, F. C. Iglesias, A. E. Jech, et al.,Phys. of free parameters, which we borrowed either from Rev. C 5, 1759 (1972). our own previous investigations (as in the RPA 21. S. Sen, P. J. Riley, and T. Udagawa, Phys. Rev. C 6, approach) or from the empirical data on other nuclei 2201 (1972). (as in the shell-model calculations). In particular, the 22. P.J.Daly,P.Bhattacharyya,C.T.Zhang,et al.,Phys. parameters of interaction that we used in the RPA Rev. C 59, 3066 (1999). scheme agree with those used by us in other works, 23. P.F. Mantica, B. E. Zimmermann, W.B. Walters, and as well as with the parameters used by other authors K. Heyde, Phys. Rev. C 43, 1696 (1991). for description of the N =82nuclei. Our combined 24. K. Kawade, G. Battistuzzi, H. Lawin, et al.,Z.Phys. 136 A 298, 187 (1980). theoretical approach for description of Xe enabled 25. J. A. Morman, W. C. Schick, and W. L. Talbert, Phys. us to describe most of the experimental information Rev. C 11, 913 (1975). on this nucleus and may be successfully applied to 26. A. J. Aas, PhD Thesis (University of Oslo, Oslo, other nuclei with one filled shell. 1999).

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PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1497–1509. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1555–1567. Original English Text Copyright c 2005 by Balbutsev, Schuck. NUCLEI Theory

The Nuclear Scissors Mode by Two Approaches (Wigner Function Moments versus RPA)*

E. B. Balbutsev** and P. Schuck1) Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received January 11, 2005

Abstract—Two complementary methods to describe the collective motion, the RPA and the method of Wigner function moments, are compared using a simple model as an example—a harmonic oscillator with quadrupole–quadrupole residual interaction. It is shown that they give identical formulas for eigenfrequencies and transition probabilities of all collective excitations of the model, includingthe scissors mode, which is a subject of our special attention. The normalization factor of the “synthetic” scissors state and its overlap with physical states are calculated analytically. The orthogonality of the spurious state to all physical states is proved rigorously. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION Lebovitz [4]. Instead of writingthe equations of motion for microscopic amplitudes of particle–hole exci- The full analysis of the scissors mode in the tations (as in RPA), one writes the dynamical equa- framework of a solvable model [harmonic oscillator tions for various multipole phase-space moments of with quadrupole–quadrupole residual interaction a nucleus. This allows one to achieve better physical (HO + QQ)] was given in [1]. Many obscure points in interpretation of the studied phenomenon without go- the understandingof the mode’s nature were clari fied: inginto its detailed microscopic structure. The WFM for example, its coexistence with the isovector giant method was successfully applied to study isoscalar quadrupole resonance (IVGQR), the decisive role and isovector giant multipole resonances and low- of the Fermi surface deformation in its creation, lyingcollective modes of rotatingand nonrotating and so on. nuclei with various realistic forces [5]. The results of calculations were always very close to similar re- The method of Wigner function moments (WFM) sults obtained with the help of the RPA. In principle, was applied to derive analytic expressions for cur- this should be expected, because the basis of both rents of both coexistingmodes (this was done for methods is the same: time-dependent Hartree–Fock the first time), their excitation energies, and mag- (TDHF) theory and a small-amplitude approxima- netic and electric transition probabilities. Unexpect- tion. On the other hand, it is evident that they are edly, our formulas for energies turned out to be identi- not equivalent, because one deals with equations of cal to those derived by Hamamoto and Nazarewicz [2] motion for different objects. The detailed analysis of in the framework of the RPA. This fact generated the the interplay of the two methods turns out to be useful natural motivation for this work: to check the relation also from a “practical” point of view: firstly, it allows between formulas for transition probabilities derived one to obtain additional insight into the nature of the by the two methods. The obvious development of this scissors mode; secondly, we find new exact mathe- investigation is the systematic comparison of the two matical results for the considered model. approaches with the aim to establish the connection between them. The HO + QQ model is a very convenient provingground for this kind of research, 2. THE WFM METHOD because all results can be obtained analytically. There is no need to describe the merits and demerits of the A detailed description of the method of WFM can RPA—they are known very well [3]. It is necessary, be found in [1, 5, 6]. Here, we recall brieflyonlyits however, to say a few words about the WFM. Its idea main points. The basis of the method is the TDHF τ is based on the virial theorems of Chandrasekhar and equation for the one-body density matrix: i ∂ρˆ /∂t = Hˆ τ , ρˆτ , where Hˆ τ is the one-body self-consistent ∗This article was submitted by the authors in English. 1)Institut de Physique Nucleaire, Orsay, France. Hamiltonian dependingimplicitly on the density ma- ** τ τ e-mail: [email protected] trix ρ (r1, r2,t)=r1|ρˆ (t)|r2 and τ is an isotopic

1063-7788/05/6809-1497$26.00 c 2005 Pleiades Publishing, Inc. 1498 BALBUTSEV, SCHUCK index. It is convenient to modify this equation, intro- with d{p, r}≡2(2π)−3 d3p d3r. ducingthe Wignertransform of the density matrix [3], 2 τ Integration of Eq. (1) with the weights rλµ, known as the Wigner function f (r, p,t): 2 (rp)λµ ≡{r ⊗ p}λµ,andp yields the followingset ∂fτ 2 λµ = sin (∇H ·∇f −∇H ·∇f ) Hτ f τ , of equations [1]: ∂t 2 r p p r W d 2 (1) Rτ − Lτ =0,λ=0, 2; (4) dt λµ m λµ where the upper index on the nabla operator stands d 1 Lτ − P τ + mω2Rτ for the function on which this operator acts and HW dt λµ m λµ λµ is the Wigner transform of the Hamiltonian H. √ 2 It is shown in [5, 6] that, by integrating Eq. (1) − 2 30 2j +1 11j (Zτ Rτ ) =0, { } 2λ1 2 j λµ over the phase space p, r with the weights j=0 x x ...x p ...p p ,wherek runs from 0 i1 i2 ik ik+1 in−1 in λ =0, 1, 2; to n, one can obtain a closed ﬁnite set of dynamical d equations for Cartesian tensors of the rank n.Taking P τ +2mω2Lτ linear combinations of these equations, one is able to dt λµ λµ represent them through various multipole moments, √ 2 which play the roles of collective variables of the − 11j τ τ 4 30 2j +1 2λ1 (Z2 Lj )λµ =0, problem. Here, we consider the case n =2. j=0 λ =0, 2, 2.1. Model Hamiltonian, Equations of Motion where 11j is the Wigner 6j symbol, r2 ≡{r ⊗ The microscopic Hamiltonian of the model is 2λ1 λµ } λµ r λµ = σ,ν C1σ,1ν rσrν is a tensor product [7], and A p2 1 H = i + mω2r2 (2) rν are cyclic variables 2m 2 i √ i=1 r = −(x + ix )/ 2,r= x , +1 1 2 √ 0 3 2 Z N µ r−1 =(x1 − ix2)/ 2. +¯κ (−1) q2µ(ri)q2−µ(rj) µ=−2 i j Further, the followingnotation is introduced:  2 Z τ 2 τ 1 µ R (t)= d{p, r}r f (r, p,t), + κ (−1) q (r )q − (r ) λµ λµ 2  2µ i 2 µ j µ=−2 i= j  τ { } τ Lλµ(t)= d p, r (rp)λµf (r, p,t), N 

+ q2µ(ri)q2−µ(rj) , i= j P τ (t)= d{p, r}p2 f τ (r, p,t). λµ λµ where the quadrupole operator q = 16π/5r2Y , √ 2µ 2µ q = 6r2 κ¯ is the strength constant of the neutron–proton In terms√ of cyclic variables 2µ 2µ; therefore,√ τ τ τ − τ interaction, κ is the strength constant of the neutron– Q2µ = 6R2µ.Bydefinition R00 = Q00/ 3 with neutron and proton–proton interactions, and N and τ τ 2 Q00 = N r beingthe mean square radius. The Z are the numbers of neutrons and protons, respec- τ tensor L1ν is connected with the angular momentum tively. The mean field potential for protons (or neu- τ √i τ τ trons) is by the followingrelations: L10 = I3 ,L1±1 = 2 2 1 τ 1 2 2 µ τ (Iτ ∓ iIτ ). V (r,t)= mω r + (−1) Z (t)q − (r), 2 1 2 2µ 2 µ 2 µ=−2 We rewrite Eqs. (4) in terms of the isoscalar and (3) n p ¯ n − isovector variables Rλµ = Rλµ + Rλµ, Rλµ = Rλµ n n p p p n p where Z2µ = κQ2µ +¯κQ2µ, Z2µ = κQ2µ +¯κQ2µ, Rλµ (and so on) with the isoscalar κ0 =(κ +¯κ)/2 τ − and the quadrupole moments Q2µ(t) are defined as and isovector κ1 =(κ κ¯)/2 strength constants. It is no problem to solve these equations numerically. τ { } τ However, we want to simplify the situation as much Q2µ(t)= d p, r q2µ(r)f (r, p,t) as possible to get the results in an analytic form that

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 THE NUCLEAR SCISSORS MODE 1499 gives us a maximum of insight into the nature of the For κ0, we take the self-consistent value κ0 = 2 2 2 modes. −mω¯ /(4Q00),whereω¯ = ω /(1 + 2δ/3) (see Ap- (i) The problem is considered in a small-amplitude pendix A) with the standard definition of the deforma- approximation. Writingall variables as a sum of their 4 tion parameter Q20 = Q00 3 δ.Then equilibrium value plus a small deviation, Ω2[Ω2 − 2¯ω2(1 + δ/3)] = 0. (8) eq R eq P Rλµ(t)=Rλµ + λµ(t),Pλµ(t)=Pλµ + λµ(t), The nontrivial solution of this equation gives the fre- eq L Lλµ(t)=Lλµ + λµ(t), quency of the µ =1branch of the isoscalar GQR R¯ (t)=R¯eq + R¯ (t), P¯ (t)=P¯eq + P¯ (t), 2 2 2 λµ λµ λµ λµ λµ λµ Ω =Ωis =2¯ω (1 + δ/3). (9) ¯ ¯eq L¯ Lλµ(t)=Lλµ + λµ(t), Takinginto account relation (A.4) from Appendix A, R P L we find that this result coincides with that of [8]. The we linearize the equations of motion in λµ, λµ, λµ trivial solution Ω=Ω =0 is characteristic of the R¯ P¯ L¯ 0 and λµ, λµ, λµ. nonvibrational mode, correspondingto the obvious (ii) We study nonrotatingnuclei, i.e., nuclei with integral of motion L11 = const responsible for the eq ¯eq rotational degree of freedom. This is usually called the L1ν = L1ν =0. eq “spurious” mode. (iii) Only axially symmetric nuclei with R2±2 = eq ¯eq ¯eq R2±1 = R2±2 = R2±1 =0are considered. (iv) Finally, we take 2.3. Isovector Eigenfrequencies R¯eq = R¯eq =0. (5) The information about the scissors mode is con- 20 00 tained in the set of isovector equations with µ =1: This means that equilibrium deformation and mean ˙ R¯21 − 2L¯21/m =0, (10) square radius of neutrons are supposed to be equal to that of protons. ˙ 2 eq eq L¯21 − P¯21/m + mω + κQ +4κ1Q R¯21 =0, Due to approximation (5), the equations for isosca- 20 00 P¯˙ 2 eq L¯ − eq L¯ lar and isovector systems are decoupled. Further, 21 +2[mω + κ0Q20] 21 6κ0Q20 11 =0, due to the axial symmetry, the angular momentum L¯˙ +3¯κQeq R¯ =0. projection is a good quantum number. As a result, 11 20 21 every set of equations splits into five independent Imposingthe time evolution via e−iΩt, one trans- ± ± subsets with quantum numbers µ =0, 1, 2.The forms (10) into a set of algebraic equations. Again, detailed derivation of formulas for eigenfrequencies the eigenfrequencies are found from the characteristic and transition probabilities together with all nec- equation, which reads essary explanations is given in [1]. Here, we state only the final results required for comparison with the 4 2 2 8 eq 2 eq Ω − Ω 4ω + κ1Q + (κ1 +2κ0)Q respective results obtained in the framework of the m 00 m 20 RPA. (11)

36 eq 2 + (κ0 − κ1)κ0(Q ) =0. 2.2. Isoscalar Eigenfrequencies m2 20 Let us analyze the isoscalar set of equations with Supposing, as usual, the isovector constant κ1 to be µ =1: proportional to the isoscalar one, κ1 = ακ0,andtak- ingthe self-consistent value for κ0,weﬁnally obtain R˙ 21 − 2L21/m =0, (6) Ω4 − 2Ω2ω¯2(2 − α)(1 + δ/3) + 4¯ω4(1 − α)δ2 =0. L˙ −P 2 eq eq R 21 21/m + mω +2κ0(Q20 +2Q00) 21 =0, (12) P˙ 2 eq L 21 +2[mω + κ0Q20] 21 =0, The solutions to this equation are L˙ 2 2 11 =0. Ω± =¯ω (2 − α)(1 + δ/3) (13) − Imposingthe time evolution via e iΩt for all variables, ± ω¯4(2 − α)2(1 + δ/3)2 − 4¯ω4(1 − α)δ2. one transforms (6) into a set of algebraic equations. The eigenfrequencies are found from its characteristic The solution Ω+ gives the frequency Ωiv of the µ =1 equation, which reads branch of the isovector GQR (IVGQR). The solution Ω− gives the frequency Ωsc of the scissors mode. 2 2 − 2 − 6κ0 eq 4 eq Ω Ω 4ω Q20 + Q00 =0. (7) We adjust α from the fact that the IVGQR is m 3 experimentally known to lie practically at twice the

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1500 BALBUTSEV, SCHUCK ˆ 2 energy of the isoscalar GQR. In our model, the ex- B(E2)iv =2|iv|O21|0| (21) perimental situation is satisﬁed by α = −2.Then   e2 5 (1 + δ/3)Ω2 − 2(¯ωδ)2 = Q iv , 2 00 2 − 2 δ δ 3 m 8π Ωiv(Ωiv Ωsc) Ω2 =4¯ω2 1+ + 1+ − δ2  , (14) iv 3 3 4 B(E2) =2| |Oˆ |0|2   is is 21 (22) 2 δ δ 2 3 e 5 2 − 2 3 Ω2 =4¯ω2 1+ − 1+ − δ2 . = Q00[(1 + δ/3)Ωis 2(¯ωδ) ]/[Ωis] . sc 3 3 4 m 8π Usingrelations (18) in formulas (20) and (21), we obtain the expression for B(E2), valid for all three 2.4. Linear Response and Transition Probabilities excitations: | | ˆ | |2 A direct way of calculatingthe reduced transition B(E2)ν =2 ν O21 0 (23) probabilities is provided by the theory of linear re- e2 5 (1 + δ/3)Ω2 − 2(¯ωδ)2 = Q ν . sponse of a system to a weak external ﬁeld 00 2 − 2 − m 16π Ων[Ων ω¯ (2 α)(1 + δ/3)] ˆ ˆ − ˆ† O(t)=Oexp( iΩt)+O exp(iΩt). The isoscalar value (22) is obtained by assuming For magnetic excitations, α =1.

Oˆ = Oˆ1µ = −i∇(rY1µ) · [r ×∇]µN , (15) 3. RPA e µN = ; Standard RPA equations in the notation of [3] are 2mc {[δijδmn(,m − ,i)+¯vmjin] Xnj +¯vmnij Ynj} B(M1) =2|sc|Oˆ |0|2 (16) sc 11 n,j − 2 2 − 2 1 α mω¯ 2 Ωsc 2(1 + δ/3)¯ω 2 (24) = Q00δ µ , 4π Ω (Ω2 − Ω2 ) N = ΩX , sc sc iv mi ˆ 2 {v¯ijmnXnj +[δijδmn(,m − ,i)+¯vinmj] Ynj} B(M1)iv =2|iv|O11|0| (17) n,j − 2 2 − 2 1 α mω¯ 2 Ωiv 2(1 + δ/3)¯ω 2 = Q δ µ . = −ΩYmi. 00 2 − 2 N 4π Ωiv(Ω Ωsc) iv Accordingto the schematic model (2), the matrix These two formulas can be joined into one expression element of the residual interaction is by the simple transformation of the denominators. τ∗ τ Indeed, we have from (13) v¯mjin = κττ DimDjn

±(Ω2 − Ω2 )=±(Ω2 − Ω2 ) (18) with D = q = 16π/5r2Y and κ = κ = κ, iv sc + − 21 21 nn pp κ =¯κ. This interaction distinguishes between pro- = ±2 ω¯4(2 − α)2(1 + δ/3)2 − 4¯ω4(1 − α)δ2 np tons and neutrons, so we have to introduce the 2 2 =2Ω± − 2¯ω (2 − α)(1 + δ/3) isospin projection indices τ,τ into the set of RPA 2 − − 2 2 equations (24): =2Ω± (2 α)(ωx + ωz ). ∗ τ − τ τ τ τ τ Usingthese relations in formulas (16) and (17), we (,m ,i )Xmi + κττ DimDjnXnj (25) obtain the expression for B(M1), valid for both exci- n,j,τ ∗ tations: τ τ τ τ + κττ DimDnjYnj = ΩXmi, − 2 2 1 α mω¯ n,j,τ B(M1)ν =2|ν|Oˆ11|0| = (19) 8π τ∗ τ τ τ τ τ κ D D X +(, − , )Y Ω2 − 2(1 + δ/3)¯ω2 ττ mi jn nj m i mi × Q δ2 ν µ2 . n,j,τ 00 2 − 2 − N Ων[Ων ω¯ (2 α)(1 + δ/3)] ∗ τ τ τ − τ + κττ DmiDnjYnj = ΩYmi. 2 For electric excitations, Oˆ = Oˆ2µ = er Y2µ, n,j,τ ˆ 2 B(E2)sc =2|sc|O21|0| (20) Its solution is ∗ ∗ 2 2 − 2 Dτ Dτ e 5 (1 + δ/3)Ωsc 2(¯ωδ) Xτ = im Kτ ,Yτ = − mi Kτ (26) = Q00 , mi − τ mi τ 2 − 2 Ω ,mi Ω+,mi m 8π Ωsc(Ωsc Ωiv)

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τ τ − τ τ τ with ,mi = ,m ,i and K = τ κττ C . different major shells with ∆N =2. All correspond- τ τ − The constant C is defined as C = ing transition energies are degenerate too: ,m ,i = τ τ τ τ ≡ n,j(DjnXnj + DnjYnj). Usinghere the above- (ωx + ωz) ,2. Therefore, the secular equation can τ τ be rewritten as written expressions for Xnj and Ynj,onederivesthe useful relation 1 ,0D0 ,2D2 = + . (35) τ τ τ τ τ 2κ E2 − ,2 E2 − ,2 C =2S K =2S κττ C , (27) 1 0 2 τ 2 The sums D0 = |Dmi| and D2 = where the followingnotation is introduced: mi(∆N=0) 2 ,τ |Dmi| can be calculated analytically (see Sτ = |Dτ |2 mi (28) mi E2 − (,τ )2 mi(∆N=2) mi mi Appendix B): with E = Ω. Let us write relation (27) in detail: Q Q D = 00 , ,D= 00 , . (36) Cn − 2Sn(κCn +¯κCp)=0, (29) 0 mω¯2 0 2 mω¯2 2 p p n p C − 2S (¯κC + κC )=0. Let us transform the secular equation (35) in the The condition for existence of a nontrivial solution to polynomial this set of equations leads to the secular equation 4 − 2 2 2 E E [(,0 + ,2)+2κ1(,0D0 + ,2D2)] n p n p 2 (1 − 2S κ)(1 − 2S κ) − 4S S κ¯ =0. (30) 2 2 +[,0,2 +2κ1,0,2(,0D2 + ,2D0)] = 0. Makingobvious linear combinations of the two equations in (29), we write them in terms of isoscalar and Usinghere expressions (36) for D0 and D2 and the isovector variables C = Cn + Cp, C¯ = Cn − Cp: self-consistent value of the strength constant (A.3), we find C − 2(Sn + Sp)κ C − 2(Sn − Sp)κ C¯ =0, 0 1 (31) 4 − 2 − 2 2 − 2 2 n p n p E E (1 α/2)(,0 + ,2)+(1 α),0,2 =0, C¯ − 2(S − S )κ0C − 2(S + S )κ1C¯ =0. or Approximation (5) allows us to decouple equations 4 − 2 − 2 − 4 for isoscalar and isovector variables. Really, in this Ω Ω (2 α)ω+ +(1 α)ω− =0, (37) n p ≡ case, S = S S/2; hence, we obtain two secular 2 2 2 4 2 − equations: where the notation ω+ = ωx + ωz and ω− =(ωx ω2)2 is introduced. This result coincides with that 1 − 2Sκ =0 or 1 − Sκ = Sκ¯ (32) z 0 of [2]. By a trivial rearrangement of the terms in (37), in the isoscalar case and one obtains the useful relation 1 − 2Sκ =0 or 1 − Sκ = −Sκ¯ (33) 2 2 − 2 − 2 2 − 4 1 Ω (Ω ω+)=(1 α)(Ω ω+ ω−). (38) in the isovector case, the difference between them Substitutingexpressions (A.3) from Appendix A for beingin the strengthconstants only. Havingin mind ω2 and ω2 into (37), we reproduce formula (12) for the the relation κ1 = ακ0,wecometotheconclusionthat x z it is sufficient to analyze the isovector case only—the isovector case: results for isoscalar case are obtained by assuming Ω4 − 2Ω2ω¯2(2 − α)(1 + δ/3) + 4¯ω4(1 − α)δ2 =0. α =1. Takinghere α =1, we reproduce formula (8) for the isoscalar case: 3.1. Eigenfrequencies 4 − 2 2 The detailed expression for the isovector secular Ω 2Ω ω¯ (1 + δ/3) = 0. equation is 1 , = |D |2 mi . (34) 3.2. B(E2) Factors 2κ mi E2 − ,2 1 mi mi The operator D hasonlytwotypesofnonzeromatrix Accordingto [3], the transition probability for a ˆ A ˆ elements Dmi in the deformed oscillator basis. Matrix one-body operator F = i=1 fi is calculated with the elements of the first type couple the states of the same help of the formula major shell. All correspondingtransition energiesare | ˆτ | ˆτ τ,ν ˆτ τ,ν degenerate: ,m − ,i = (ωx − ωz) ≡ ,0.Matrixele- 0 F ν = (fimXmi + fmiYmi ). (39) ments of the second type couple the states of the mi

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2 To calculate quadrupole excitations, one has to take it with the secular equation (37) (multiplied by ω+), 5 which obviously does not change its value, we find fˆp = er2Y =˜eDp with e˜ = e . The expres- 2µ 16π after elementary combinations sions for Xτ and Y τ are given by formulas (26). 4 2 − 2 4 2 4 mi mi Denom =Ωνω+ 2Ων ω− + ω+ω− (45) Combiningthese results, we have 2 4 − 2 − 2 − 4 + ω+[Ων Ων(2 α)ω+ +(1 α)ω−] ,p 0|Dp|ν =2˜eKp |Dp |2 mi (40) = ω2 Ω2[2Ω2 − (2 − α)ω2 ] ν mi E2 − (,p )2 + ν ν + mi ν mi − 4 2 − − 2 ω−[2Ων (2 α)ω+] =2˜eKpSp =˜eCp. ν ν ν 2 2 − 4 2 − − 2 =(Ωνω+ ω−)[2Ων (2 α)ω+]. The constant Cp is determined by the normalization ν This result allows us to write the final expression condition 2 2 2 − 4 τ,ν∗ τ,ν τ,ν∗ τ,ν 5 e Ωνω+ ω− δν,ν = (X X − Y Y ), B(E2) = Q , mi mi mi mi ν 2 00 2 − − 2 mi,τ 16π mω¯ Ων[2Ων (2 α)ω+] (46) which gives which coincides with (23) (we recall that ω2 = | p |2 p + 1 Dmi ,mi 2 4 2 4 = E (41) 2¯ω (1 + δ/3) and ω− =4δ ω¯ ). By simple transfor- p 2 ν p 2 2 − p 2 2 (Cν ) (Sν) [Eν (,mi) ] mations, we can reduce this formula to the result of mi (Cn)2 |Dn |2 ,n Hamamoto and Nazarewicz [2] (consideringthat they + ν mi mi . published it without the constant factor (Cp)2 (Sn)2 [E2 − (,n )2]2 ν ν ν mi 2 5 e 0 The ratio Cn/Cp is determined by any of Eqs. (29): Q00). 32π mω0 Cn 1 − 2Spκ 2Snκ¯ = = . (42) Cp 2Spκ¯ 1 − 2Snκ 3.3. B(M1) Factors Formula (41) is considerably simplified by the In accordance with formulas (39), (26), and (15), Sp = Sn ,p = ,n the magnetic transition matrix element is approximation (5), when , mi mi,and p n Dmi = Dmi. Applyingthe second parts of formu- (Oˆp ) Dp* (Oˆp ) Dp* | ˆp | p 11 im im − 11 mi mi las (32) and (33), we can easily find that, in this 0 O11 ν = Kν p p . n p E − , E + , case, C /C = ±1. As a result, the final expression mi ν mi ν mi for B(E2) is (47) p 2 B(E2)ν =2|0|D |ν| (43) As is shown in Appendix B, the matrix element p p −1 (O11)im is proportional to Dim [formula (B.15), 2 2 2 ,mi Appendix B]. So, expression (47) is reduced to =2˜e 16Eν κ |Dmi| . 1 (E2 − ,2 )2 mi ν mi | ˆp | − p e˜√ 2 − 2 p 0 O11 ν = Kν (ωx ωz ) (48) With the help of formulas (36), this expression can be 2c 5 transformed into p p* p p* DimDim DmiDmi 2 × − 5 e Q00 p − p p p ,im(Eν ,mi) ,mi(Eν + ,mi) B(E2)ν = 2 2 (44) mi 8π mω¯ α Eν e˜ |Dp |2 2 2 −1 p √ 2 − 2 p mi , , = Kν (ωx ωz ) Eν . × 0 + 2 c 5 ,p [E2 − (,p )2] 2 − 2 2 2 − 2 2 mi mi ν mi (Eν ,0) (Eν ,2) 5 e2Q (E2 − ,2)2(E2 − ,2)2 With the help of approximation (5) and expres- = 00 ν 0 ν 2 sions (36) for D and D ,wefind 2 2 2 − 2 2 2 2 − 2 2 2 0 2 8π mω¯ α Eν (Eν , ) , +(E , ) , 2 0 0 2 Cp e˜ 2 2 2 − 4 2 | ˆp | ν √ 2 − 2 5 e Q00 (Ωνω+ ω−) 0 O11 ν = p (ωx ωz ) (49) = 2 4 2 2 4 2 4 . 2Sν c 5 16π mω¯ Ω Ω ω − 2Ω ω− + ω ω− ν ν + ν + Q E E × 00 ν + ν At a glance, this expression has nothing in common 2mω¯2 E2 − ,2 E2 − ,2 with (23). Nevertheless, it can be shown that they ν 0 ν 2 e˜ Q Ω (Ω2 − ω2 ) are identical. To this end, we analyze carefully the = −2κ Cp √ (ω2 − ω2) 00 ν ν + 1 ν x z 2 2 2 − 4 denominator of the last expression in (44). Summing c 5 mω¯ α(Ων ω+ ω−)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 THE NUCLEAR SCISSORS MODE 1503 p − Cν √e˜ 2 − 2 1 α step, it is necessary to show that the secular equa- = (ωx ωz ) . 2 c 5 Ων tion (30) has the solution E =0. We need the expression for Sτ (E =0)≡ Sτ (0). In accordance with (28), Relation (38) and the self-consistent value of the we have κ = ακ strength constant 1 0 were used at the last τ step. For the magnetic transition probability, we have τ ,0D0 ,2D2 S (E)= 2 + 2 , p 2 E2 − , E2 − , B(M1)ν =2|0|Oˆ |ν| (50) 0 2 11 τ p 2 2 − 2 4 − 2 τ D0 D2 (Cν ) e˜ 4 (1 α) ω− (1 α) S (0) = − + . =2 2 ω− 2 = 2 2 B(E2). ,0 ,2 4 5c Ων 20c Ων τ τ This relation between B(M1) and B(E2) was also The expressions for D0 and D2 are easily extracted 2 from formulas (B.10) and (B.11) (see Appendix B): found (to the factor 1/(20c )) by Hamamoto and τ Nazarewicz [2]. Substitutingexpression (46) for 1+ 4 δ 1 − 2 δ τ τ 3 − 3 B(E2) into (50), we reproduce [with the help of D0 = Q00 , (51) m ωx ωz relation (38)] formula (19). 4 − 2 τ τ τ 1+ 3 δ 1 3 δ D2 = Q00 + . 3.4. “Synthetic” Scissors and Spurious State m ωx ωz The nature of collective excitations calculated by the method of WFM is ascertained quite easily by So, we find analyzingthe roles of collective variables describing 1+ 4 δ τ − τ 3 1 1 the phenomenon. The solution of this problem in the S (0) = Q00 + (52) RPA approach is not so obvious. That is why the m ωx ,2 ,0 nature of the low-lyingstates has often been estab- τ 1 − 2 δ 1 1 lished by consideringoverlaps of these states with the + 3 − “pure scissors state” [10, 11] or “synthetic state” [2], ωz ,2 ,0 produced by the action of the scissors operator 2 τ τ τ 4δ Q00 1 3Q20 N −1 n2 p − p2 n = − = − , Sx = ( Ix Ix Ix Ix) τ τ 2 − 2 τ m ,2,0 m (ωx ωz ) on the ground state: where, in accordance with (B.12) from Appendix B, | | Syn = Sx 0 , 6 (ω2 − ω2)p = − (κQp +¯κQn ), (53) N beingthe normalization factor. Due to axial sym- x z m 20 20 τ τ metry,onecanusetheIy component instead of Ix , 6 (ω2 − ω2)n = − (κQn +¯κQp ), or any linear combination of them, for example, the x z m 20 20 τ variable L11, which is much more convenient for us. n τ 2 and the deformation of neutrons δ is allowed to be The terms Ix are introduced to ensure the orthog- different from the deformation of protons δp. Finally, onality of the synthetic scissors to the spurious state we get |Sp =(In + Ip)|0. However, we do not need these terms, because the collective states |ν of our model Qp 2Sp(0) = 20 , are already orthogonal to |Sp (see below); hence, p n κQ20 +¯κQ20 the overlaps Syn|ν will be free from any admixtures κQ¯ n of |Sp. So, we use the followingde finitions of the − p 20 1 2S (0)κ = p n , synthetic and spurious states: κQ20 +¯κQ20 n | N −1 p − n | n Q20 Syn = γ (L11 L11) 0 2S (0) = , κQn +¯κQp N −1 ˆp − ˆn | 20 20 = (O O11) 0 , 11 κQ¯ p | ˆp ˆn | 1 − 2Sn(0)κ = 20 . Sp =(O11 + O11) 0 , n p κQ20 +¯κQ20 where It is easy to see that, substitutingthese expressions e 3 γ = −i . into (30), we obtain the identity; therefore, the secular 2mc 2π equation has the zero solution. Let us demonstrate the orthogonality of the spu- At the second step, it is necessary to calculate rious state to all the rest states |ν.Asthefirst the overlap Sp|ν. Summing(47) with an analogous

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1504 BALBUTSEV, SCHUCK expression for neutrons, we get and Q2 ≡ Q20. The expression in the last curly brackets obviously coincides with the secular equa- | e˜√ τ 2 − 2 τ Sp ν = Eν Kν (ωx ωz ) (54) tion (30), which proves the orthogonality of the c 5 τ spurious state to all physical states of the considered | τ |2 model. Thus, we can conclude that, strictly speaking, Dmi e˜√ × = Eν this is not a spurious state, but one of the exact ,τ (E2 − ,2 )τ c 5 mi mi ν mi eigenstates of the model corresponding to the integral | τ |2 τ of motion In + Ip. In other words [3]: “In fact, these τ 2 2 τ Dmi ,mi × K (ω − ω ) . excitations are not really spurious, but they represent ν x z (,2 )τ (E2 − ,2 )τ τ mi mi ν mi adifferent type of motion, which has to be treated Applyingthe algebraic identity separately.” The same conclusion was made by Lo Iudice [12], who solved this problem approximately 1 1 1 1 = + with the help of several assumptions (a small defor- ,2(E2 − ,2) E2 ,2 E2 − ,2 mation limit, for example). and rememberingthe de finition (28) of Sτ ,wecan The problem of the “spurious” state beingsolved, | rewrite (54) as the calculation of the overlaps Syn ν becomes | ˆn Sp|ν (55) trivial. Really, we have shown above that 0 O11 + Oˆp |ν =0. This means that 0|Oˆn |ν = −0|Oˆp |ν; e˜ 11 11 11 = √ Kτ (ω2 − ω2)τ (Sτ − Sτ (0)) | N −1 | ˆp − ˆn | N −1 × ν x z then Syn ν = 0 O11 O11 ν =2 c 5Eν τ 0|Oˆp |ν and p 11 e˜ Kν − = √ (ω2 − ω2)p(Sp − Sp(0)) U 2 ≡|Syn|ν|2 =2N 2B(M1) . E x z ν c 5 ν n The nontrivial part of the problem is the calculation 2 − 2 n n − n Kν N +(ωx ωz ) (S S (0)) p . of the normalization factor . It is important not to Kν forget about the time dependence of the synthetic In accordance with (27) and (42), state, which should be determined by the external field: Kn 1 − 2Spκ ν = . (56) | N −1 ˆp − ˆn −iΩt p n Syn(t) = [(O O11)exp Kν 2S κ¯ 11 ˆp − ˆn † iΩt | 2 − 2 τ τ +(O11 O11) exp ] 0 . Notingnow [see formula (52)] that (ωx ωz ) S (0) = − 3 τ Then we have Q20 and takinginto account relations (53), we m N 2 = Syn(t)|Syn(t) (58) find | ˆp − ˆn † ˆp − ˆn | =20 (O11 O11) (O11 O11) 0 Sp|ν = β [(κQp +¯κQn)2Sp − Qp] (57) 2 2 2 | ˆp − ˆn †| | ˆp − ˆn | =2 0 (O11 O11) ph ph (O11 O11) 0 1 − 2Spκ ph +[(κQn +¯κQp)2Sn − Qn] 2 2 2 2Snκ¯ | | ˆp − ˆn | |2 | | ˆτ | |2 =2 ph (O11 O11) 0 =2 ph O11 0 . p − p p n ph τ,ph = β [(2S κ 1)Q2 +2S κQ¯ 2] With the help of relation (B.15) from Appendix B, we − p n n n p 1 2S κ find +[(2S κ − 1)Q +2S κQ¯ ] τ 2 2 2Snκ¯ 2 | | 2 | |2 N 2 2 e 4 ph r Y21 0 − p = ω− 2 (59) p n n n 1 2S κ 5 2c , = β 2S κQ¯ +(2S κ − 1)Q τ,ph ph 2 2 2Snκ¯ 2 τ n 1 e 4 τ D0 D2 Q2 n p n = (ω−) + . = β 2S κ¯2S κ¯ − (1 − 2S κ) 8π 2c ,2 ,2 2Snκ¯ τ 0 2 τ τ τ τ × − p Expressions for D0 , D2 , ωx,andωz are given by for- (1 2S κ) =0, mulas (51) and in Appendix (B.12). To get a definite number, it is necessary to make some assumption where concerningthe relation between neutron and proton p − 3 e˜√ Kν equilibrium characteristics. As usual, we apply the β = n p m c 5 Eν approximation (5); namely, we suppose Q00 = Q00

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n p and Q20 = Q20. It is easy to check that, in this case, It is possible to extract from the histogram of [2] τ + formulas for ωx,z are reduced to the ones for the the value of the overlap of calculated low-lying 1 isoscalar case, namely, (A.3) from Appendix A, and excitations with the synthetic scissors state: U 2 ≡ τ τ + 2 2 D0 = D0/2 and D2 = D2/2,whereD0 and D2 are |Syn|1 | ≈ 0.4. The result of our calculation U = given by (36). So, we get 0.43 agrees with it very well. So the natural conclu- 2 sion of this section is that the correct treatment of ω4 e Q N 2 − 00 1 1 pair correlations is more important for a reasonable = 2 + (60) 8π 2c mω¯ ,0 ,2 description of the scissors mode than the thorough δ mω choice of an interaction. = µ2 x Q . N 2π 00 The estimation of the overlap for 156Gd with δ =0.27 4. CONCLUSION N 2 2 2 gives =34.72µN and U =0.53, which is two The properties of collective excitations (the scis- times larger than the result of [10] obtained in QRPA sors mode, isovector and isoscalar giant quadrupole calculations with the Skyrme forces. The disagree- resonances) of the harmonic oscillator Hamiltonian ment can naturally be attributed to the difference in with the quadrupole–quadrupole residual interaction forces and especially to the lack of pair correlations in (HO + QQ) were studied by two methods: WFM our approach (see the next section, nevertheless). In and RPA. We have found that both methods give the 1 3 same analytic expressions for energies and transi- a small-deformation limit, U 2 = ≈ 0.6. 2 2 tion probabilities of all considered excitations. Does it 3.4.1. Superdeformation. A certain drawback of mean that WFM and RPA are identical approaches? our approach is that, so far, we have not included Certainly not. First of all, we have the experience of the superfluidity in our description. Nevertheless, our previous WFM calculations [5] with realistic forces, formulas (14), (19) can be successfully used for the which show that, for example, we reproduce only cen- description of superdeformed nuclei, where the pair- troids of giant resonances, whereas RPA describes ingis very weak [2, 9]. For example, applyingthem their fine structure. Secondly, we suppose that one 152 can find such nuclear characteristics that will be de- to the superdeformed nucleus Dy (δ 0.6, ω0 = scribed differently by the two approaches even in this 41/A1/3 MeV), we get simple model. Thirdly, to establish completely (and 2 Eiv =20.8 MeV,B(M1)iv =15.9µN finally) the relation between the two approaches, it is necessary to analyze the equations of motion for for the isovector GQR and multipole moments from the point of view of RPA. 2 Esc =4.7 MeV,B(M1)sc =20.0µN This will be done in a subsequent publication. for the scissors mode. There are not so many results There is no sense in speakingabout advantagesor of other calculations to compare with. As a matter disadvantages of one of the two discussed methods— of fact, there are only two papers consideringthis they are complementary. Of course, RPA gives com- problem. plete, exhaustive information concerningthe microscopic (particle–hole) structure of collective excita- The phenomenological TRM model [9] predicts tions. However, sometimes, considerable additional 2 Eiv 26 MeV,B(M1)iv 26µN , efforts are required to understand their physical nature. On the contrary, the WFM method gives in- E 6.1 MeV,B(M1) 22µ2 . sc sc N formation only on the physical nature of excitations The only existingmicroscopic calculation [2] in the and does not touch their microscopic structure. Our framework of QRPA with separable forces gives results serve as a very good illustration of this situation. What do we really know about the scissors mode E 28 ,B(M1) 37µ2 , iv MeV iv N and IVGQR from each method? RPA says that the − 2 Esc 5 6 MeV,B(M1)1+ 23µN . scissors mode is mostly created by ∆N =0particle– hole excitations with a small admixture of ∆N =2 Here, B(M1)1+ denotes the total M1 orbital strength, particle–hole excitations and vice versa for IVGQR. π + carried by the calculated K =1 QRPA excitations And this is all! One cannot even suspect the key role modes in the energy region below 20 MeV. of the relative angular momentum in the creation of It is easy to see that, in the case of IVGQR, one the scissors mode. On the other hand, the WFM can speak, at least, about qualitative agreement. Our method says that the scissors mode appears due to results for Esc and B(M1)sc are in good agreement oscillations of the relative angular momentum with a with that of the phenomenological model and with Esc small admixture of the quadrupole moment oscilla- and B(M1)1+ of Hamamoto and Nazarewicz [2]. tions and vice versa for IVGQR. Further, it informs

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1506 BALBUTSEV, SCHUCK us about the extremely important role of the Fermi where Σ0 and ω0 are defined in the spherical case, we surface deformation in the formation of the scissors get mode. Q ω2 − ω2 2σδ 4 Two new mathematical results are obtained for the 20 =2 x z = = δ. 2 2 − HO + QQ model. We have proved exactly, without Q00 ωx +2ωz 1 σδ 3 any approximations, the orthogonality of the “spu- Solvingthe last equation with respect to σ,wefind rious” state to all physical states. In this sense, we 2 have generalized the result of Lo Iudice [12], derived σ = . (A.2) in a small-deformation approximation. The analytic 3+2δ expressions are derived for the normalization factor of Therefore, the oscillator frequencies and the strength the synthetic scissors state and overlaps of this state constant can be written as with eigenstates of the model. 2 2 2 4 ωx = ωy =¯ω 1+ δ , Appendix A 3 2 mω¯2 ω2 =¯ω2 1 − δ ,κ= − (A.3) It is known that the deformed harmonic oscillator z 3 0 4Q Hamiltonian can be obtained in a Hartree approxi- 00 mation “by makingthe assumption that the isoscalar 2 2 2 with ω¯ = ω /(1 + 3 δ). The condition for volume part of the QQ force builds the one-body container conservation ω ω ω = const = ω3 makes ωδ-de- well” [13]. In our case, it is obtained quite easily by x y z 0 pendent: summingthe expressions for V p and V n [formula (3)]: 1+ 2 δ 1 p n ω2 = ω2 3 . V (r,t)= (V (r,t)+V (r,t)) (A.1) 0 4 2/3 − 2 1/3 2 (1 + 3 δ) (1 3 δ) 2 So, the final expressions for oscillator frequencies are 1 2 2 µ = mω r + κ0 (−1) Q2µ(t)q2−µ(r). 2 4 1/3 µ=−2 1+ δ ω2 = ω2 = ω2 3 , x y 0 − 2 In the state of equilibrium (i.e., in the absence of an 1 3 δ external field), Q2±1 = Q2±2 =0. Usingthe de fini- 4 2 2 2/3 tion [14] Q20 = Q00 δ and the formula q20 =2z − 1 − δ 3 ω2 = ω2 3 . x2 − y2, we obtain the potential of the anisotropic z 0 1+ 4 δ harmonic oscillator 3 m It is easy to see that they correspond to the case, V (r)= [ω2(x2 + y2)+ω2z2] 2 x z where the deformed density ρ(r) is obtained from the with oscillator frequencies spherical density ρ0(r) by the scale transformation [8] − 2 2 2 2 2 − (x, y, z) → (xeα/2,yeα/2,ze α) ωx = ωy = ω (1 + σδ),ωz = ω (1 2σδ), with where 1/3 8Q00 4 3α − σ = −κ . α 1+ 3 δ 3 e 1 0 2 e = ,δ= , (A.4) 3mω − 2 3α 1 3 δ 2 e +2 The definition of the deformation parameter δ must be reproduced by the harmonic oscillator wave func- which conserves the volume and does not destroy the tions, which allows one to fix the value of σ.Wehave self-consistency, because the density and potential are transformed in the same way. Σx Σy Σz Q00 = + + , It is necessary to note that Q00 also depends on δ: m ωx ωy ωz Σx Σy Σz Σz Σx Q00 = + + Q20 =2 − , m ωx ωy ωz m ωz ωx A 2 1 where Σ = (n +1/2) and n is the oscillator = Σ0ω0 + x i=1 x i x m ω2 ω2 quantum number. Usingthe self-consistency condi- x z tion [14] 1 = Q0 , 00 4 1/3 − 2 2/3 Σxωx =Σyωy =Σzωz =Σ0ω0, (1 + 3 δ) (1 3 δ)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 THE NUCLEAR SCISSORS MODE 1507 0 3 2 1/3 where Q00 = A 5 R , R = r0A . As a result, the ﬁnal 2 5 express the zx component of r Y21 = D = expression for the strength constant becomes 16π 1/3 15 2 − 2 2 − z(x + iy) as mω0 1 3 δ mω0 −α 8π κ0 = − = − e , 4Q0 1+ 4 δ 4Q0 00 3 00 zx = P0 + P2. which coincides with the respective result of [8]. Hence, we have A 2 Appendix B ,0 0 zsxsmi (B.3) mi(∆N=0) s=1 A 2 D = |D |2 To calculate the sums 0 mi(∆N=0) mi = ,0 0 P0(s)mi and D = |D |2,weemploythesum- mi s=1 2 mi(∆N=2) mi A A rule techniques of Suzuki and Rowe [8]. The well- 1 known harmonic oscillator relations = 0 P0(s), H, P0(s) 0 , 2 √ √ s=1 s=1 − xψnx = ( nxψnx 1 + nx +1ψnx+1), where ,0 = (ωx − ωz). The above commutator is 2mωx (B.1) easily evaluated for the Hamiltonian with the potential (A.1), as √ √ A A − m ωx − pˆxψnx = i ( nxψnx−1 nx +1ψnx+1) 0 P0(s), H, P0(s) 0 (B.4) 2 s=1 s=1 allow us to write | A 2| | A 2| 0 s=1 zs 0 − 0 s=1 xs 0 xzψ ψ (B.2) = ,0 . nx nz 2m ωx ωz √ = √ ( n n ψ − ψ − 2m ω ω x z nx 1 nz 1 Takinginto account the axial symmetry and usingthe x z deﬁnitions + (nx +1)(nz +1)ψnx+1ψnz+1 A 2 2 Q00 = 0 (2xs + zs )0 , + (nx +1)nzψnx+1ψnz−1 s=1 − + nx(nz +1)ψnx 1ψnz+1), A 2 − 2 Q20 =2 0 (zs xs)0 , pˆ pˆ x z ψ ψ s=1 m2ω ω nx nz 4 x z Q = Q δ, √ 20 00 3 = − √ ( n n ψ − ψ − 2m ω ω x z nx 1 nz 1 x z we transform this expression to + (n +1)(n +1)ψ ψ A A x z nx+1 nz+1 − 0 P0(s), H, P0(s) 0 (B.5) (nx +1)nzψnx+1ψnz−1 s=1 s=1 − 4 2 nx(nz +1)ψnx−1ψnz+1). − 1+ 3 δ 1 3 δ = ,0Q00 − . These formulas demonstrate in an obvious way that 6m ωx ωz the operators With the help of the self-consistent expressions for ωx 1 1 ω P = zx + pˆ pˆ and and z (A.3), one comes to the followingresult: 0 2 x z 2 m ωxωz A A 1 − 1 0 P0(s), H, P0(s) 0 (B.6) P2 = zx 2 pˆxpˆz 2 m ωxωz s=1 s=1 Q ,2 2 ω ω 2 contribute only to the excitation of the ∆N =0 00 0 0 0 − 0 = 2 = Q00 . and ∆N =2 states, respectively. Following[8], we 6m ω¯ 6m ωz ωx

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1508 BALBUTSEV, SCHUCK 4 By usingthe fact that the matrix elements for the (ωp)2 = ω2 1+ (κQp +¯κQn ) , 2 z mω2 20 20 zy component of r Y21 are identical to those for the zx component, because of axial symmetry, we finally n 2 2 − 2 n p obtain (ωx) = ω 1 (κQ20 +¯κQ20) , mω2 A 2 2 2 2 4 p ,0 0 rs Y21mi (B.7) n n (ωz ) = ω 1+ 2 (κQ20 +¯κQ20) . mi(∆N=0) s=1 mω 1/3 The above-written formulas can also be used to 5 Q 5 Q0 ,2 1+ 4 δ = 00 ,2 = 00 0 3 . calculate the analogous sums for various compo- 2 0 2 − 2 16π mω¯ 16π m ω0 1 3 δ nents of the angular momentum. Indeed, by definition, Iˆ = ypˆ − zpˆ and Iˆ = zpˆ − xpˆ . In accor- By calculatinga double commutator for the P oper- 1 z y 2 x z 2 dance with (B.1), we have ator, we find ω A 2 xpˆ ψ ψ = −i z (B.13) 2 z nx nz 2 ω ,2 0 rs Y21mi (B.8) √ x mi(∆N=2) s=1 × n n ψ − ψ − . x z nx 1 nz 1 4 1/3 0 2 1+ δ − (nx +1)(nz +1)ψn +1ψn +1 5 Q00 2 5 Q00 ,2 3 x z = 2 ,2 = 2 2 , 16π mω¯ 16π m ω 1 − δ + (n +1)n ψ ψ − 0 3 x z nx+1 nz 1 − where ,2 = (ωx + ωz). nx(nz +1)ψnx−1ψnz+1 . We need also the sums Dτ and Dτ calculated sep- 0 2 Therefore, arately for neutron and proton systems with the mean ω ω fields V n and V p, respectively. The necessary for- Iˆ ψ ψ = i z − x (B.14) 2 nx nz 2 ω ω mulas are easily derivable from the already obtained √ x z results. There are no reasons to require the fulfillment × nxnzψnx−1ψnz−1 of the self-consistency conditions for neutrons and − (n +1)(n +1)ψ ψ protons separately, so one has to use formula (B.5). x z nx+1 nz+1 The trivial change of notation gives ωz ωx + i + Z Z 2 ωx ωz p 0 P0(s), H , P0(s) 0 (B.9) × − s=1 s=1 (nx +1)nzψnx+1ψnz 1 4 p − 2 p p p 1+ δ 1 δ − 3 − 3 nx(nz +1)ψnx−1ψnz+1 . = ,0Q00 p p , 6m ωx ωz Havingformulas (B.2) and (B.14), one derives the fol- lowingexpressions for matrix elements couplingthe Z 2 p 2 ground state with ∆N =2and ∆N =0excitations: ,0 0 rs Y21mi (B.10) s=1 n +1,n +1|Iˆ |0 mi(∆N=0) x z 2 1+ 4 δp 1 − 2 δp 2 − 2 5 p p 3 − 3 (ωx ωz ) (nx +1)(nz +1) = ,0Q00 p p , = i , 16π m ωx ωz 2 ωx + ωz ωxωz Z 2 nx +1,nz − 1|Iˆ2|0 p 2 , 0 r Y21mi (B.11) 2 s 2 − 2 s=1 (ωx ωz ) (nx +1)nz mi(∆N=2) = i , 2 ωx − ωz ωxωz 1+ 4 δp 1 − 2 δp 5 p p 3 3 = ,2Q00 p + p . 16π m ωx ωz (nx +1)(nz +1) nx +1,nz +1|xz|0 = , The nontrivial information is contained in oscillator 2m ωxωz frequencies of the mean fields V p and V n [formula (3)]: 2 (nx +1)nz p 2 2 − p n n +1,n − 1|xz|0 = . (ωx) = ω 1 (κQ20 +¯κQ20) , (B.12) x z mω2 2m ωxωz

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 THE NUCLEAR SCISSORS MODE 1509 It is easy to see that REFERENCES n +1,n +1|Iˆ |0 1. E. B. Balbutsev and P. Schuck, Nucl. Phys. A 720, x z 2 293 (2003); 728, 471 (2003). 2 − 2 (ωx ωz ) 2. I. Hamamoto and W. Nazarewicz, Phys. Lett. B 297, = im nx +1,nz +1|xz|0, ωx + ωz 25 (1992). 3. P. Ringand P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980). nx +1,nz − 1|Iˆ2|0 4. S. Chandrasekhar, Ellipsoidal Figures of Equilib- 2 − 2 (ωx ωz ) rium (Yale Univ. Press, New Haven, 1969; Mir, = im nx +1,nz − 1|xz|0. ωx − ωz Moscow, 1973). ´ Due to the degeneracy of the model, all particle–hole 5. E. B. Balbutsev, Fiz. Elem. Chastits At. Yadra 22, 333 (1991) [Sov. J. Part. Nucl. 22, 159 (1991)]. excitations with ∆N =2have the same energy ,2 and all particle–hole excitations with ∆N =0have the 6. E. B. Balbutsev and P. Schuck, Nucl. Phys. A 652, 221 (1999). energy ,0. This fact allows one to join the last two formulas into one general expression 7.D.A.Varshalovich,A.N.Moskalev,andV.K.Kher- sonskiı˘ , Quantum Theory of Angular Momentum 2 − 2 |ˆ | (ωx ωz ) | | (Nauka, Leningrad, 1975; World Sci., Singapore, ph I2 0 = i m ph xz 0 . 1988). ,ph 8.T.SuzukiandD.J.Rowe,Nucl.Phys.A289, 461 Takinginto account the axial symmetry, we can write (1977). the analogous formula for Iˆ1: 9. N. Lo Iudice, Riv. Nuovo Cimento 23 (9) (2000). 10. R. R. Hilton et al.,inProceedings of the 1st In- 2 − 2 (ωx ωz ) ternational SpringSeminar on Nuclear Physics ph|Iˆ1|0 = −im ph|yz|0. ,ph “Microscopic Approaches to Nuclear Structure,” Sorrento, 1986, Ed. by A. Covello (Editrice Compos- ˆ The operator O1±1 is proportional (15) to the angular itori, Bologna, 1986), p. 357. ie 3 11. A. E. L. Dieperink and E. Moya de Guerra, Phys. Lett. momentum: Oˆ ± = − (Iˆ ∓ iIˆ ).There- B 189, 267 (1987). 1 1 4mc 2π 2 1 12. N. Lo Iudice, Nucl. Phys. A 605, 61 (1996). fore, we can write 13. R. R. Hilton, Ann. Phys. (N.Y.) 214, 258 (1992). ph|Oˆ1±1|0 14. A. Bohr and B. Mottelson, Nuclear Structure,Vol.2: 2 − 2 Nuclear Deformations (Benjamin, New York, 1975; e (ωx ωz ) 2 = − √ ph|r Y2±1|0. (B.15) Mir, Moscow, 1977). 2c 5 ,ph

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1510–1524. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1568–1582. Original Russian Text Copyright c 2005 by Sukhovoj, Khitrov, Chol. NUCLEI Theory

Experimental Arguments in Favor of Reﬁning Model Ideas of the Cascade Gamma Decay of a Compound State of a Complex Nucleus

A. M. Sukhovoj*, V.A.Khitrov**,andLiChol*** Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received July 14, 2004; in ﬁnal form, November 23, 2004

Abstract—An analysis of the entire body of data on the intensities of two-stepgamma cascades studied in thermal-neutron capture for more than 50 nuclei from the range 27 ≤ A ≤ 199 suggests that such processes should be described in terms of model concepts that are much more involved than those currently adopted by experimentalists. According to the results of this analysis, models of relevant parameters, such as the density of excited levels and radiative strength functions for dipole gamma transitions, should take into account more explicitly the coexistence and interaction of quasiparticle and phonon interactions. A direct inclusion of the idea that a second-order phase transition occurs and aﬀects not only level densities but also radiative strength functions for dipole transitions may prove to be necessary. These conclusions concern primarily the excitation-energy region below a value of about 0.5Bn. c 2005 Pleiades Publish- ing, Inc.

1. INTRODUCTION One extracts two basic parameters from experimental data and then uses them to calculate nuclear- In analyzing experimental data—for example, on physics constants [1]. These parameters are the cascade gamma decay of a neutron resonance— − (i) the level density ρ = D 1 at a given excitation one usually uses some model ideas of the process energy Eexc for specific values of the quantum num- being studied. If there exist alternative models of the bers characterizing the levels being considered; process, experimentalists give preference, as a rule, to the simplest models. As the volume of experimental (ii) the radiative strength functions data increases and, what is more important, as one 3 2/3 k =Γλi/(E A Dλ), (1) goes over to a previously inaccessible region of the γ parameters of the phenomenon under study, there which determine the intensities of gamma transitions arises, however, the need for invoking more involved that deexcite the excited levels λ. concepts. The ideas used to describe the level density range A large volume of experimental data on the prop- between the simplest model [2] of a noninteracting erties of excited states from about 1 or 3 MeV to Fermi gas and the rather complicated generalized nearly Bn for more than 50 nuclei from the mass model of a superfluid nucleus [3]. Within the latter, range 27 ≤ A ≤ 199 was accumulated in experiments one directly employs theoretical concepts of the phase transition in a nucleus from a superfluid to a normal performed in Dubna, Riga, and Reˇ zˇ and has been state. However, there is no direct experimental in- comprehensively analyzed by now at the Laboratory formation about this transition in a nucleus; in view of Neutron Physics at the Joint Institute for Nuclear of this, Rastopchin et al. [3] fixed its energy on the Research (JINR, Dubna). This region of excitation basis of available information about the transition of energies (especially for deformed nuclei) has been an electron gas to a superfluid state. studied for the first time in such minute detail. The entire body of resulting information gives sufficient In order to specify the shape of the radiative grounds to assume that the concepts underlying the strength function for an E1 transition having an en- existing models of cascade gamma decays call for a ergy Eγ and a mean width Γλi and connecting λ and i ≤ radical improvement. states in the excitation-energy range 0 Eexc

1063-7788/05/6809-1510$26.00 c 2005 Pleiades Publishing, Inc. EXPERIMENTAL ARGUMENTS 1511 resonances for primary transitions of the cascades or level density multiplied either by the photon-emission from the density of intermediate levels for secondary probability or by the model-dependent factor of transitions of the cascades, which was obtained in nuclear-surface penetrability to an evaporated nu- one way or another. cleon (nucleons). For this reason, the reliability of ρ For M1 transitions, models relying on the as- and k values determined from the respective spectra sumption that there exists a giant magnetic dipole [cases (a) and (b)] is adversely affected by a strong and resonance were developed in addition to the existing irremovable correlation between the density of excited idea that k(M1) is constant. That models of radiative levels and the probability of emission of reaction strength functions are less diverse than level-density products. models is due to paucity of reliable experimental data In the case of λ → i → f two-stepcascades, the on k in the range of transition energies Eγ less than presence of the second quantum is responsible for a Bn by several megaelectronvolts. different form of the functional dependence of their The relationshipbetween E1andM1 transition intensity on the energy of the primary transition of a ≈ widths for Eγ Bn was determined experimentally cascade: for almost all stable target nuclei and can readily be I = (Γ /Γ m )n (Γ /Γ m ). (2) fixed in the calculations of the parameters of a cascade γγ λi λi λi λi if if if gamma decay, irrespective of the model assumptions J,π used. The partial and total cascade-transition widths Γ,as Present-day theoretical ideas that make it possible well as the number m(n) of levels excited in respective to calculate ρ and k, in principle, are more realistic. transitions in various energy ranges (the bin width However, they are hardly appropriate for analyzing that is optimal for calculations is ∆E ≈ 50 keV), experimental data. Despite this, an analysis of their are unambiguously determined by the functions ρ conceptual framework reveals the main flaw in level- and k under the condition that gamma transitions of density models used to treat experimental data and given multipolarities are taken into account in ex- to calculate nuclear-physics constants. It consists pression (2). This is precisely the circumstance that either in completely ignoring [2] or in inadequately makes it possible to reveal, in the structure of ex- taking into account [3] the coexistence and interac- cited states of complex nuclei, special features that tion of quasiparticle and phonon excitations in nuclei. are inaccessible in experiments employing different In any case, a solution to this problem cannot be methods. obtained by purely theoretical methods. As before, the Naturally, the reliability and accuracy of the con- problem of deriving the most reliable data on ρ and k cepts that can be deduced from an analysis of Iγγ for by methods that invoke model-dependent concepts to the properties of the excited states of the nucleus be- the minimum possible degree is the most important ing studied become higher as the volume of additional for experimentalists. information included in such an analysis is enlarged. For example, one can employ known values of the total radiative width of a decaying compound state, 2. POTENTIAL OF THE EXPERIMENT The possibility of obtaining completely model- Γλ = ( Γλi mλi), (3) independent information about ρ and k upto Bn by i currently available methods of nuclear spectroscopy this being necessary for determining the absolute has yet to be realized for nuclei characterized by a value of k (but not the shape of its dependence on Eγ ). high level density. The only available way to solve In addition, a minimum uncertainty in the sought this problem is to select, for the corresponding pa- value of ρ is ensured by taking into account the mean rameters, values that would provide the best fittothe spacing Dλ between neutron resonances, which is following observables: determined spectroscopically by the time-of-flight (a) evaporation spectra of nucleons originating method, and the known number of low-lying levels. from nuclear reactions; The use of spectroscopic data from the estimated (b) various spectra of gamma rays emitted in these schemes of the gamma decay of the nuclei being reactions; studied serves the same goal. Accordingly, the region (c) the intensities [4] of two-stepcascades trig- of excitations under study is broken down in all cal- gered by slow-neutron capture that connect a neu- culations into a “discrete” and a “continuous” part. tron resonance with low-lying levels of the nucleus The parameters ρ and k, which are discussed below, being considered. are characterized by minimum fluctuations and are In the first two cases, the amplitude of the ex- naturally determined only in the continuous part of perimentally measured spectra is determined by the excitation energies. The lower boundary of this part,

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1512 SUKHOVOJ et al.

60 118 Edis, is 1.5, 2.9, and 1.1 MeV in Co, Sn, and ρ and k but also if use is made of their values obtained 187W, respectively. In order to simplify the figures according to [4]. below, the calculated quantities are everywhere rep- This situation manifests itself in [5] to a lesser ex- resented by lines of the same type. Since fluctuations tent for the 156Gd and 168Er nuclei. In these deformed of the calculated quantities are maximal in the dis- nuclei, the ρ and k values determined according to [4] crete part of excited levels, the corresponding data are reproduce fairly well the experimentally determined represented by broken lines. population of low-lying levels only up to an excitation energy of about 3.5 MeV. The population of levels at higher energies cannot be estimated because there 3. EXPERIMENTAL DATA are no data from other experiments on the intensities ON THE PARAMETERS OF THE CASCADE i of gamma rays emitted in the radiative capture of GAMMA DECAY OF A COMPLEX NUCLEUS thermal neutrons in the required energy intervals. The accuracy to which a calculation reproduces However, the calculation relying on the data the entire body of experimental data (it should not from [4] reproduces the total spectra of gamma be lower than the accuracy of the experiments being rays from the radiative capture of thermal and fast considered) is a natural characteristic of correctness neutrons much more poorly in the region A ∼ 160 [5] of our ideas of processes occurring in a complex nu- than in the region A>180 (however, the results cleus. Unfortunately, it has not been possible so far in the former region are better than their counter- to attain this degree of agreement between the the- parts produced by models traditionally used for this ory and experiment since the volume of information purpose). As a matter of fact, this means that the accessible to an experimental determination has been ideas of cascade gamma decays are inconsistent with insufficient. experimental data, at least for deformed even–even Only four functionals can be used both to assess nuclei. It is natural to expect that this inconsistency the intervals of probable values of ρ and k and to test will manifest itself directly in the cascade population the process of a cascade gamma decay. These are of levels in nuclei of this type only at excitation (i) the total radiative width (as well as the cross energies in excess of about 4 MeV. sections for neutron interaction with nuclei); Naturally, the ρ value determined according to [4] (ii) the intensity of cascades in a given interval of should correspond to an independent estimate of ρ energies of their intermediate and final levels; from the procedure [6] involving an approximation of the distribution of random deviations of the inten- (iii) the population [5] of low-lying levels by cas- sities of cascades in narrow intervals of energies of cades involving various numbers of transitions (in- their intermediate levels from a mean value. However, cluding the total spectra of gamma radiation); the level density determined according to [4] at low (iv) the expected [6] number of intermediate levels excitation energies sometimes proves to be less in a of individual cascades whose intensities are below a number of nuclei than that obtained in [6] for zero threshold value. threshold of the detection of the intensities of cas- Inasmuch as the first two functionals were already cades resolved in energy (in the form of two peaks) used in [4] to determine the quantities considered that feature intermediate levels at energies below 3 or here, the values obtained for ρ and k should ensure 4MeV. an accurate calculation of the population of low-lying Of course, we cannot rule out the possibility that levels that was extracted in [5] from a comparison of the existing disagreement between the measured the intensities of cascades featuring specific interme- functionals of cascade gamma decays and their coun- diate levels with the intensities determined indepen- terparts calculated within conventional models is due dently for the primary and secondary gamma transi- to systematic errors in the experiment itself or in tions of these cascades, as well as from a comparison known nuclear-physics constants used in analyzing of the total spectra of gamma radiation accompanying respective experimental data, but they may also be neutron capture and various nuclear reactions. Un- due to inadequate ideas of the properties of the fortunately, data obtained according to [4] are such process being considered. that the required degree of agreement between the experiment and the calculation could not be attained. 3.1. Estimating the Possible Role of Systematic For example, the total populations of 118Sn [7] and Errors in Determining ρ and k 183,187W [8] levels in the ranges between about 3 and 5 MeV and between about 1 and 3 MeV, respectively, The main qualitative conclusion that can be drawn disagree sharply with the results of the calculations from an analysis of the results obtained according not only if use is made of model-dependent values of to [4] is that it is necessary to take into account, in

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 EXPERIMENTAL ARGUMENTS 1513 greater detail than in conventional models, the co- spectra of radiative thermal-neutron capture. The set existence and interaction of a superfluid and a nor- of relevant data [10] on neutron capture in samples mal phase of matter. An alternative idea assumes of all stable elements of natural isotopic composition the presence of a so-called pygmy resonance in the was obtained in experiments performed more than radiative strength function [9]. 30 years ago. Nevertheless, data obtained recently by The line of the most appropriate modification of our measuring the same spectra in Budapest [11] with ideas concerning the properties of nuclei below the modern γ spectrometers did not show any evidence neutron binding energy cannot be chosen correctly in support of the statement that, in data presented without specifying the sources of systematic errors in [10], there was a systematic error that led to a regu- in the sought quantities ρ and k within various pro- lar exaggeration of the gamma-transition intensity— cedures for obtaining relevant experimental data and for example, a severalfold exaggeration. without estimating these errors. The root-mean-square discrepancy between the As a matter of fact, the results obtained by apply- data from [10] and [11] was estimated in [12] at 20% ing the procedure employed in [4] may involve a sig- for all of the observed gamma transitions. nificant error and, accordingly, may distort our ideas Therefore, a variation of Iγγ within a factor of 1.2 of the properties of nuclei in the energy range from 2 to 1.3 with respect to the experimental intensity value or 3 MeV to Bn for only three reasons. These are both toward smaller and toward greater values gives (i) the presence of a regular systematic error that a reasonable idea of the effect of a first-type error would result in a severalfold exaggeration of the ex- on the ρ and k values determined according to [4]. perimental value of the cascade intensity for almost In the presence of a greater discrepancy between the all of the 50 nuclei being studied; absolute intensities of primary cascade transitions as (ii) the existence of two or more groups of excited determined in [10] and in a modern experiment, the states such that they all have the same spin–parity Jπ probable systematic error in the cascade intensities (but different structures of wave functions) and that increases. Accordingly, the tested cascade intensity the mean intensities of two-step cascades populating must be increased (or decreased) to a greater extent them differ by more than one order of magnitude in if the intensities of maximum-energy gamma transi- any interval of energies of intermediate levels that is tions in [11] are higher (or lower) than the analogous narrow against the scale of Bn; quantities used previously for the normalization of (iii) a difference in the forms of the energy depen- Iγγ. The result of such an estimation is given in [13]. dence of radiative strength functions for primary and The basic conclusion of the analysis performed in [13] secondary cascade transitions of the same multipo- is that the discrepancy between the ρ values obtained larity and energy (this difference should be so pro- according to [4] and those predicted by the models nounced as to ensure that the calculated value of the of a noninteracting Fermi gas cannot be explained ratio Γif /Γi for transitions to Ef < 1 MeV levels was by the systematic error in the experimental values on average severalfold overestimated for all cascades of Iγγ for its maximum possible values. By way of observed experimentally). example, we indicate that, of 14 nuclei for which a The effect of any of the above errors on the param- direct comparison of the spectra of gamma rays from eters to be determined was estimated by means of a radiative thermal-neutron capture is possible, the Iγγ standard formula for the transfer of errors. It becomes values found previously were underestimated by a factor of about 2 in relation to those obtained with very difficult to estimate the resulting probable errors 114 124 in the parameters ρ and k since the set of Eqs. (2) the aid of data from [11] for Cd and Te. For and (3) is strongly correlated and essentially nonlin- 182Ta and 192Ir, they are overestimated by a factor ear. It should be noted that, although the number of of about 1.5, the mean intensity in the high-energy unknown quantities exceeds the number of equations section of the spectra of radiative thermal-neutron in this set, the form of the functional relations between capture being identical for the remaining 10 nuclei. ρ and k, on one hand, and the measured intensities of If one assumes that the data presented in [11] involve the cascades and the total width, on the other hand, a systematic error much less than that in the earlier constrain the range of their variation quite efficiently data compared with them [4], then the amplitude of for any excitation energies and any photon energies. fluctuations of the difference between the ρ and k (There is no such constraints for the spectra of evapo- values determined according to the procedure em- rated nucleons and primary gamma transitions in any ployed in [4] and the model concepts existing for them nuclear reactions.) will increase somewhat for the former two nuclei and (1) All of the intensities obtained thus far for two- decrease insignificantly for the latter two. stepcascades were normalized to known intensities (2) Experiments of all types known to date have of the strongest primary gamma transitions from the one special feature in common—they are plagued by

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1514 SUKHOVOJ et al. an irremovable ambiguity in ρ and k values deter- threshold of their detection, the relative error in determined from them. The uncertainties in the sought mining the dependence Iγγ = F (E1) may increase. parameters ρ and k depend greatly on the volume of There are presently no specific data that would sug- information that one invokes and on the procedure gest the presence or absence of this effect. But if used to analyze it. the mean intensity of cascades does indeed depend By way of example, we indicate that, in the ex- greatly on the structure of their intermediate level, perimentally measured distributions of the intensities then the prevalent ideas of the process being studied of cascades whose total energy is equal to a few [which are reflected in Eqs. (2) and (3)] would require megaelectronvolts, the order in which the two gamma arefinement more pronounced than that which is transitions occur cannot be determined by means of associated with taking into account the previously experimental equipment exclusively. As a result, such unobserved [4] stepwise structure in the level density. 2 spectra can be faithfully reproduced (χ /f < 1)by (3) The question of whether the level density does using a very wide variety of the functional depen- indeed have a stepwise structure and whether it is due dences ρ = F (Eexc) and k = φ(Eγ ). Moreover, the to [3, 15] the rupture of some Cooper pairs of nucleons maximum value of either parameter may severalfold is directly related to the problem of assessing the de- exceed its minimal value. This uncertainty can be gree to which radiative strength functions depend on reduced only by establishing the order in which the the structure of levels excited by a gamma transition. transitions follow each other in a cascade. For this, it is necessary to isolate that section in the experimental Indirect pieces of evidence that R = k(Eγ , spectra which is equal to the sum of the intensities Eλ)/k(Eγ ,Ei) = const for transitions that have the of all possible cascades where the primary-transition same multipolarity and energy, but which occur in energy lies in a preset interval. At the present time, the deexcitation of levels having different excitation this problem can be solved [14] by means of the energies Eλ and Ei,werefirst obtained in [5] in an nuclear-spectroscopy technique and by using data attempt at describing the total spectra of gamma accumulated in relevant experiments. radiation in the capture of thermal and fast neutrons. In the experimental spectrum of two-step cas- It was found that, for the accuracy of the reproduction cades such that E1 + E2 = const, one usually ob- of these spectra in the calculations to be improved, serves (in the form of pairs of resolved peaks) not the radiative strength functions for secondary gamma less than 90 to 95% of the intensity of cascades transitions must feature a weaker dependence on the going to Ei ≤ 0.5Bn levels at a nearly zero level of emitted-photon energy than their counterparts for primary transitions. background. The dependence Iγγ = F (E1) obtained according to the method used in [14] involves a sys- The most detailed, albeit still indirect, data on the tematic error that is caused by a nonzero threshold for behavior of a nucleus in the region of the stepwise individual-cascade detection. It decreases asymptot- structure and on the expected change in the radia- ically with increasing efficiency of germanium detec- tive strength functions come from an analysis of the tors used to record coincidences. total and cascade populations of levels at excitation A method for estimating the error in determin- energies upto 3 or 4 MeV in heavy nuclei ( 181Hf and ing [14] the shape of the functional dependence Iγγ = 183,184,185,187W) and at somewhat higher energies in F (E1) was described in [6]. It consists in extrapo- nuclei of lower mass, such as 60Co and 118Sn. lating the cumulative sums of random values of cas- The intensity of an arbitrary cascade, i , is related cade intensities to zero value of the experimental- γγ sensitivity threshold. For a present-day experiment, to the intensities of its primary and secondary transitions, i1 and i2, respectively, by the equation the threshold cascade-intensity value below which such an extrapolation should be performed does not iγγ = i1i2/ i2. (4) exceed, as a rule, approximately 10−4 events per decay. By applying this equation to the entire body of ex- Therefore, the relative error in the dominant part perimental data on the aforementioned intensities, we of the dependences Iγγ = F (E1) obtained according can determine (see Fig. 1) the total population P = to [14] is in fact smaller than the error in absolute cas- i2 of about 100 or more intermediate levels of the cade intensities if random deviations of the cascade cascades in any nucleus among those that have been intensities from their mean values have a variance studied to date by the method of summation of the whose magnitude does not exceed that which was amplitudes of coinciding pulses. In the presence of used in [6]. maximally comprehensive data on i1 and i2, this can If this is not so and if the majority of the E1 < be done for the excitation energies of intermediate 0.5Bn cascades have intensities below the sensitivity cascade levels upto 3 or 4 MeV or even for higher

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P, % 102 187W 101 100 10–1 10–2 1234 102 118Sn 101

100

10–1

10–2 123456 102 60Co 101

100

10–1

10–2 12 34 Eexc, MeV

Fig. 1. Total population P of intermediate levels of two-stepcascades in 187W, 118Sn, and 60Co nuclei: (points) experimental data; (thin lines) results of the calculation with the ρ and k values obtained according to the procedure used in [4]; (dashed lines) results of the calculation based on the models proposed in [2, 17]; and (thick lines) results of the calculation employing the level density from [4], the corresponding strength functions being modified according to expressions (5) and (6). energies, but, if there are no such data, the popula- therefore, it is reasonable to compare experimental tions in question can be determined upto much lower and calculated results for P − i1 values summed over energies [5]. a moderately narrow interval of excitation energies The difference of P and the intensity i of the and consider these sums as a lower bound for each 1 of the intervals. As an example, we present below the primary transition to each of these levels is equal results of such a comparison for the compound nuclei to the sum of their populations by two-step, three- 60 118 187 step, etc., cascades. It can be calculated by using Co, Sn, and W (Fig. 2). Among 17 nuclei ac- various assumptions on the density of levels excited cessible to the data-analysis method described below, in thermal-neutron capture and on radiative strength they were chosen because the population in them was functions for cascade gamma transitions. This may be determined for the maximum number of intermediate levels of the cascades, because their masses differ to done, for example, within the existing models of these the maximum possible degree, and because they rep- parameters or possible hypotheses concerning them resent the three possible combinations of parities of (including the k and ρ values obtained according the numbers of neutrons and protons in a compound to [4]). nucleus. At the present time, the populations of all (without The degree of the discrepancy between the calcu- exception) intermediate levels of two-step cascades lated total population P (see Fig. 1) and its precise can hardly be determined even at moderate excita- value is determined both by the incompleteness of tion energies (because of the presence of the detec- data on the intensities of cascades and transitions and tion threshold for the intensities of the cascades, iγγ, by the possible strong effect of the structure of the and for the respective gamma transitions, i1 and i2); wave function for an excited level on the probability

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P – i1, % Iγγ, % 102 6 187W 187W 101 4

100 2

10–1 0 1234142635 102 4 118Sn 118Sn 3 101 2 100 1 0 10–1 1 23456 1537 102 25 60Co 60 20 101 Co 15 100 10 5 10–1 0 10–2 –5 12341537 , MeV Eexc E1, MeV

Fig. 3. Sums of the intensities of two-stepcascades in Fig. 2. Total cascade population of intermediate levels of 187 118 60 187 118 60 W, Sn, and Co nuclei over intervals of width cascades in W, Sn, and Co nuclei. The notation 0.5 MeV versus the primary-cascade-transition energy. for the curves is identical to that in Fig. 1. The dashed curve represents the results of the calculation according to expression (2) for the set of models from [2] and [17]. of its cascade population. The need for taking this possibility into account follows from the fact that the cascade population of levels at excitation energies stepcascades, which deexcite levels from a wider above 1 to 3 MeV changes strongly from one level to spin window than two-step cascades. Therefore, it another. This is suggested by very strong fluctuations seems incorrect to relate, on the basis of the data in of the populations of neighboring levels. Fig. 1, the difference in the population of neighboring In the isotopes being considered, primary dipole intermediate levels of the cascades to the difference in transitions of two-stepcascades excite J =1/2, 3/2 their spins exclusively [as is usually done, for example, in analyzing data on (n, nγ) reactions]. levels in 187W, J =0–2 levels in 118Sn, and J =2–5 levels in 60Co. Since the entire body of data on two- The number of available versions of the functional stepcascades can be described only with allowance dependences of strength functions and level densities for electric and magnetic dipole transitions, a further on, respectively, the photon energy and the excitation analysis is performed under the assumption that they energy of a level is quite large. However, general excite levels of both parities. The fractions of captured regularities of the change in the population of levels in neutrons in one or another spin channel for cobalt and response to a change in their excitation energy can be tin were taken into account in all calculations on the revealed by using only three versions of calculations: basis of data from [16]. (i) The level density is predicted by any version The cascade population of any level is determined of the model of a noninteracting Fermi gas; the E1 not only by the total intensity of all two-stepcas- radiative strength function is specified by known ex- cades but also by the intensity of three- and four- trapolations of a giant electric dipole resonance to the

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–9 –3 region below Bn;andk(M1) = const, k(M1)/k(E1) k(E1) + k(M1), 10 MeV being normalized to the experimental value. 102 187 (ii) One employs ρ and k values [4] that faithfully W reproduce the intensities of two-step cascades versus 101 the energy of their primary transition (Fig. 3).

(iii) One chooses the set of values for the level 100 density and the strength function in such a way as to reproduce simultaneously precise values of Iγγ = − 10–1 F (E1) and Γγ and to reproduce the values of P i1 13524 to the highest possible precision. In order to solve 3 this problem exactly, it would be necessary to specify 10 118Sn the form of the dependence of k(E1) and k(M1) not 102 only on the photon energy but also on the excitation 1 energy of a nucleus for electric and magnetic transi- 10 tions individually. (Naturally, the state of the art in the 100 experiment presently gives little hope to obtain this –1 information in the near future.) A possible approxi- 10 mate solution is to multiply the function k for the first 10–2 photon of a cascade—it depends only on the energy 1357 E —by some correcting function h = f(E ,E ) for γ γ exc 102 60Co all subsequent photons of the cascades. Through this choice of the form of the dependence 101 of radiative strength functions on the energy U of a 0 level excited by a gamma transition, U = Eexc − Eγ , 10 one can also effectively introduce, in the iteration –1 process proposed in [4], additional experimental in- 10 formation about the distinctions between the forms 1357 of the energy dependence of strength functions for E1, MeV primary and secondary gamma transitions, concurrently retaining the efficiency of this iteration process and a relatively high rate of convergence to values of Fig. 4. Sums of radiative strength functions that corre- χ2/f < 1 for the calculated cascade intensity. spond to primary electric and magnetic dipole transitions of cascades and which make it possible to reproduce An implementation of version (iii) is possible in faithfully their intensity with allowance for the probable an iterative regime: for the k values obtained accord- distinction between their values and the values of the ing to [4], one selects a functional dependence that strength functions for secondary transitions in the cases of (closed circles) h given by Eqs. (5) and (6) and (open changes the values of the strength functions for sec- circles) h = const. The upper and lower dashed curves ondary transitions with respect to the strength func- represent the predictions of the models proposed in [17] tion from [4] is such a way as to reproduce the values and [18], respectively; k(M1) = const. The thick curve of P − i1 to the highest possible degree of precision. corresponds to the function h for gamma transitions to For this, it is quite sufficient to multiply the strength the levels at Ei. functions for secondary gamma transitions to levels below some boundary excitation energy by a function with some parameters α, U1, U2,andUc.Thecon- h that involves a few very narrow peaks. It is very dition (Uc − U1) =( U2 − Uc) ensures, if necessary, convenient to specify the dependence of the shapes the asymmetry of peaks and a somewhat closer re- of these peaks on the nuclear excitation energy U by production of the cascade population of levels at the analogy with the heat capacity of a uniform system at tails of the peaks than, for example, in the case of a the point Uc of a second-order phase transition as Lorentzian curve. h =1+α(ln|Uc − U1|−ln |Uc − U|) (5) With decreasing excitation energy U, the parameter α must increase (for example, linearly), in the best in the case of UUc results found for the correcting functions are then

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Nlev 104 187W

102

100

1 2 3 4 5 6 118 104 Sn

102

100

1 3 5 7 9 60Co 102

101

100

10–1 1 3 5 7 Eexc, MeV

Fig. 5. Total number of all intermediate levels of two-stepcascades for the cases of (closed circles with error bars) h given by Eqs. (5) and (6) and (open circles) h = const. The thin and thick curves represent, respectively, the predictions of the model proposed in [2] and the best correcting function h, while the triangles correspond to the observed number of intermediate levels in the allowed cascades. The width of the summation interval is 100 keV. included in the analysis according to [4] in order to data within the approximations used gives sufficient determine ρ and k values faithfully reproducing the grounds to expect that the cascade-gamma-decay cascade intensities. The values of the latter are given parameters found according to [4] reflect basic spe- in Fig. 3, while the corresponding strength functions cial features of actual level densities and strength and level densities are displayed in Figs. 4 and 5, re- functions, so that these features must remain un- spectively. After that, this cycle can be repeated—not changed as the total uncertainty in the determination more than once if use is made of the hypothesis that of the parameters in question is reduced further. By the distortions of the values of k(E1) and k(M1) grow and large, the impossibility of explaining, in terms linearly with decreasing energy of decaying levels and of known systematic errors, the discrepancy between a few times for the version where α = const. In order the experimental functionals of cascade gamma decay to minimize the number of adjustable parameters, it and their counterparts calculated according to data was assumed that the correcting functions in Figs. 4 from [4] provides the main argument in support of the and 5 are identical for electric and magnetic gamma need for invoking more involved model ideas of the transitions. properties of heavy nuclei (first of all, deformed nuclei) at excitation energies below 5 to 8 MeV than those The growth of the parameter α with decreasing currently used by experimentalists. energy of a decaying nucleus reduces the derivative dk/dEγ , this being in perfect agreement with 4. OBSERVED PATTERN OF THE CASCADE the conclusions drawn in [5] from a comparison of GAMMA DECAY OF A NEUTRON the total experimental and calculated spectra for the RESONANCE IN A COMPLEX NUCLEUS radiative capture of thermal and fast neutrons. The The largest volume of the most detailed infor- possibility of consistently describing all experimental mation about the properties of heavy nuclei in their

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 EXPERIMENTAL ARGUMENTS 1519 excitation-energy range 1–2 decay scheme are shown below Edis in spectroscopic experimental data (see Fig. 6); for a in Fig. 1. Above it, only its spin-averaged value is first approximation, the parameter of equidistance is given to avoid encumbering the figure. For P values proportional to the number of boson pairs in unfilled summed over a narrow interval of excitation energies, nucleon shells]. all three versions of the calculation of populations (see The equidistant character of the spectrum may be Subsection 3.1) yield results differing insignificantly. an indication that the structures of the wave functions However, quite a sizable discrepancy between the for intermediate levels of the cascades feature large calculated and experimental values of P − i1 for the components corresponding to the existence of one, level density [2] is a very strong argument against two, or more phonons—that is, this fact may suggest employing the ideas of the model of a noninteracting that excitations of the boson type have a significant Fermi gas in attempts at constructing a precise effect on cascade gamma decay at energies of cascade description of the cascade gamma decay of this odd– levels at least below a value of about 0.5Bn. odd nucleus. A sizable increase in the strength functions for Yet, it should be noted that quite a significant dis- secondary gamma transitions to levels in the region tinction between the best values of k and their coun- of the stepwise structure also counts in favor of local terparts predicted by the models proposed in [17, 18]

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–1 4 Iγγ, keV per 10 decays 8 187W 6

0500 1000 1500 2000 2500 3000 6 118Sn 4

0 20004000 6000 40 60Co 30

20 10

0 0 1000 2000 3000 4000 Eexc, keV

Fig. 6. Possible “bands” of nearly equidistant intermediate levels of the most intense cascades and their multiplets (intensities of the strongest cascades are smoothed by using a Gaussian curve of mean width σ =50keV).

(this is manifested to the same degree in other odd– argument in support of the above assumption that odd nuclei as well [4]) indicates that the model con- unpaired nucleons have a pronounced eﬀect on the cepts used are sharply at odds with experimental re- mean probability of photon emission below Bn and sults for nuclei of this type and that this discrepancy the statement that the conditions under which the is probably of a general character. calculations are performed are inadequate to the phenomenon being studied. 4.2. 118Sn Even–Even Nucleus

The populations of intermediate levels of the cas- 187 cades were determined on the basis of available data 4.3. WEven–Odd Nucleus on the spectra of gamma rays in the even–even compound nuclei 74Ge, 118Sn, and 184W upto energies Available data on the intensities of individual two- 114 124 150 156,158 stepcascades and on the spectraof gamma rays orig- of Eexc > 4 MeV and Cd, Te, Sm, Gd, 168 196 inating from radiative thermal-neutron capture made Er, and Pt upto Eexc ≈ 3 MeV.In the first three of these nuclei, the function h increases greatly in it possible to obtain a vast body of information about the populations of levels in 181Hf and 183,185,187Wand the region Eexc ≥ 4 MeV. Only in this way can one reduce significantly the discrepancy between the cal- a smaller amount of information for 165Dy and 175Yb. − culated and experimental distributions of P − i1.At For all of these nuclei, the cascade population P i1 the same time, the distributions P = f(Eexc) feature could not be reproduced without taking into account asignificant noncompensated excess of the experi- local peaks in the function h, but the value found in mental population of levels above the results of all this case for α is not as great as that for even–even model calculations. This discrepancy is an additional compound nuclei.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 EXPERIMENTAL ARGUMENTS 1521 5. UNEXPLAINED EFFECTS IN CASCADE all, these data rule out the possibility of reproducing GAMMA DECAY its parameters to a precision of present-day experi- The entire body of data and parameters concerning ments on the basis of employing, for the level density, gamma-decay processes that comes from the inves- model concepts like those in the model of a noninter- tigation of two-stepcascades has not yet furnished acting Fermi gas or some other model concepts that a completely consistent pattern of such processes. lead to ρ values close to that in [2]. This conclusion [4] For example, the distinction between the level density is in line with the results of our present analysis (see Fig. 5); moreover, it also follows directly and inde- required for reproducing experimental Iγγ values that was obtained according to the procedure proposed pendently from the maximum discrepancy between in [4] or according to its modification where R = the experimental and calculated total populations of relevant levels (see Fig. 1). k(Eγ ,Eλ)/k(Eγ ,Ei) = const and the level density estimated according to the procedure used in [6] has Yet another conclusion of considerable importance yet to be explained. from our analysis of experimental data is that it is Of course, the number of intermediate levels of necessary to take directly into account, within models cascades that is extrapolated to zero value of their of the level density and radiative widths, large-scale detection threshold in the form of two peaks resolved effects associated with the influence of the nuclear in energy may correspond to experimental data only structure on these parameters up to an excitation under conditions that would ensure the correctness energy of about 0.5Bn or higher. This conclusion of the application of the Porter–Thomas distribu- follows from the impossibility of faithfully reproducing tion [19] to describing [6] the distribution of random the sums of cascade populations of levels (see Fig. 2) values of the intensities of primary transitions in the with the aid of the ρ and k values deduced from the region of their smallest values. This assumption may analysis performed in [4]. The degree to which the be erroneous, for example, in the case where the details of the nuclear structure should be taken into distribution of terms for the primary-transition am- account in model concepts will of course depend on plitude differs from a normal distribution owing to an the degree of discrepancies between the calculated excess of terms having the same sign. functionals of cascade gamma decay and their coun- An alternative possibility consists in that primary terparts determined experimentally. or secondary gamma transitions (or both of them) of The entire body of experimental data accumulated the same multipolarity and nearly the same energy thus far gives sufficient grounds to put forth the hy- fluctuate with respect to different mean widths. As to pothesis that model concepts of the process being their markedly different values, they are determined by studied should be refined by taking into account more aspecific structure of the excited (decaying) interme- directly and precisely the interplay of the superfluid diate level of the cascades. In this case, it is natural to and normal phases in a nucleus. However, the follow- assume that a nucleus features excited states forming ing questions would remain open in all probability: two systems different in properties and that these (a) Is the present-day experimental level of un- systems go over to each other in response to a change derstanding of cascade gamma decay sufficient for in the nuclear excitation energy. Among intermediate creating new models of level densities and radiative levels of the cascades, those whose wave functions in- strength functions? volve dominant components of the multiquasiparticle or the phonon type may be candidates for this role. (b) Is it possible to obtain all experimental data that are required for fully parametrizing such models? There are no obstacles of fundamental character for taking into account this possibility in expres- The following points may form the basis of new sions (2) and (3), but it is impossible to verify this model concepts: assumption without additional experimental informa- (i) The transition of a nucleus from a superfluid to a tion. The possibilities of such a verification are esti- normal phase with increasing excitation energy of the mated in Section 7. nucleus occurs well below the neutron binding energy for its values within some range. 6. POSSIBLE FORM OF A MORE INVOLVED (ii) The probability of excitation of levels whose MODEL OF CASCADE GAMMA DECAY wave functions involve both quasiparticle and vibra- AND METHODS FOR AN EXPERIMENTAL tional components depends on the relationship be- DETERMINATION OF ITS PARAMETERS tween their contributions to the normalization (in All experimental data that we have obtained to particular, on the position of the excited level with re- date for the cascade gamma decay of compound nu- spect to the energy of the presumed phase transition). clear states are indicative of a strong dependence of Since the spins of neutron resonances are low, this process on the structure of excited levels. First of one can skip, for the majority of deformed nuclei, the

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1522 SUKHOVOJ et al. question of a possible distinction between the proba- cannot reproduce the intensities Iγγ for all nuclei be- bilities of excitation and decay of levels of rotational ing studied, most likely because of an approximately bands, on one hand, and the analogous values for twofold overestimation of the energy of the phase bandheads, on the other hand, paying no attention, transition from a superfluid to a normal state. It is also at this stage, to the effect of rotational excitations on necessary to revisit its ideas of the thermodynamic cascade gamma decay, at least in slow-neutron cap- parameters of a nucleus below the point of the phase ture by even–even targets. It is very unlikely that at- transition for a lower phase-transition energy. tempts at taking into account the first point would run The possibilities for going over to more involved into difficulties of a fundamental character. It can be model concepts of radiative strength functions for incorporated by various methods, including extremely performing a more precise analysis of experimental simple ones. This is illustrated by the attempt made data are severely restricted by the absence of reliable in [20] to create some phenomenological version of data on the probability of photon emission in a wide the level-density model. By using the concept of the interval of photon energies and a wide interval of heat capacity of nuclear matter and by applying, to excitation energies of decaying levels. For primary nuclei, the ideas of the form of its variation with en- transitions of the cascades, the values obtained in [4] ergy in the region of a second-order phase transition for k(E1) + k(M1) have a minimum possible, albeit that were inferred from the study of superfluidity in actually unknown, discrepancy with the sought value. liquid helium, it was shown in [20] that the use of this, Data presented in [5] give every reason to assume extremely simple, modification of models belonging to that, as the excitation energy of a nucleus decreases, the type in [2] makes it possible to reproduce stepwise the sum of the E1andM1 strength functions features structures in the level density. a weaker dependence on the photon energy than for Unfortunately, it is then necessary to solve the primary transitions of neutron-resonance decay. This following problems: comment is probably quite correct for strongly de- (a) a determination of the phenomenological deformed nuclei, but is may be insufficiently accurate, say, for spherical and transition nuclei from the re- pendence of the nuclear heat capacity for a mixture ∼ ∼ of a normal and superfluid phase in the presence of gions around A 100 and around A 190, respec- unpaired nucleons; tively. Of course, questions like that of the number of (b) the inclusion of the expected dependence of broken Cooper pairs at an excitation energy of 3 to mean partial widths with respect to gamma transi- 4 MeV for a deformed rare-earth nucleus and that of tions on the energies of a decaying and an excited level the effect of the phonon components of a decaying that are characterized by wave functions of different level at an excitation energy above the region of the structures. presumed phase transition remain unresolved experi- That an increase in the sums of radiative strength mentally. functions and a decrease in the level density (the latter The procedures developed thus far for studying is necessary for the dependence of the intensity of a excited nuclear states by means of two-stepcas- cascade on the energy of its intermediate level to be cades do not have restrictions of a fundamental ori- reproduced in the calculations) occur synchronously gin [4, 6, 13] on the accuracy in determining their may suggest the need for taking into account the parameters nearly over the entire region of excitations relationshipbetween these two nuclear parameters. of intermediate and final levels, at least in the cap- In particular, the set of available data [4] on level ture of thermal neutrons and neutrons of “filtered” densities and radiative strength functions for cascade beams from stationary nuclear reactors. In attempts gamma transitions gives sufficient grounds to assume at deriving more precise ideas of the properties of that these parameters may depend on the parity of the complex nuclei below the neutron binding energy and number of nucleons in a compound nucleus (number at developing new phenomenological models of the of excited quasiparticles). This is one of the possible nucleus, the existing experimental procedures may (in principle) explanations for the significant discrep- prove to be insufficient, however, primarily because ancy between the absolute values of radiative strength of the absence of detailed experimental data on the functions for even–even and odd–odd nuclei that was intensities of two-stepcascades to Ef > 1 MeV final observed in [4]. levels. Yet another problem that was not solved in [20] consists in the need for directly including the density of vibrational levels in the modified version of the 7. POSSIBILITIES OF EXPERIMENTAL Fermi gas model. This problem requires a dedicated TESTS OF NEW IDEAS analysis within the generalized model of a superfluid A direct observation of enhanced secondary gamma nucleus, since, within the existing version, the model transitions to levels in the region of “stepwise” struc-

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 EXPERIMENTAL ARGUMENTS 1523 tures is the most pressing experimental problem of The required result can be attained with recording the near future. systems formed by at least three high-purity germa- In order to prove directly a local enhancement of nium detectors or by only two such spectrometers radiative strength functions for gamma transitions to in the case where the Compton background is sup- levels at an excitation energy of about 3 MeV in a pressed. Experimental data reported by Bondarenko deformed heavy even–odd nucleus, it is necessary to et al. [21] make it possible to obtain a preliminary determine the reduced relative probability of gamma estimate of the required degree of suppression of the transitions from higher lying levels in the interval Compton background: the suppression factor must of Eγ from a few hundred kiloelectrovolts to a few not be less than 10 for each detector. In one version or megaelectronvolts. This problem cannot be solved by another, the efficiency of the detectors used must be means of classical nuclear spectroscopy by using any commensurate with the efficiency of modern crystal types of existing detectors and procedures for deter- ball detectors. mining the energies of excited levels and their decay modes. 8. CONCLUSION The only realistic possibility to solve it is to measure experimentally the distribution of the intensities The results of experiments performed thus far in ˇ of two-stepcascades to all possible final levels up Dubna, Riga, and Rezˇ to study two-stepcascades to an excitation energy of about 3 MeV or higher in in thermal-neutron capture and the results of their a deformed heavy even–odd nucleus. For even–even analysis carried out at the Laboratory of Neutron and lighter spherical nuclei, the energy of these levels Physics at JINR have revealed that various func- must somewhat exceed 4 to 5 MeV. In accumulating tionals of the cascade gamma decay of nuclei that useful statistics of a few thousand events for each have a high level density cannot be reproduced, to a spectrum where E1 + E2 = Bn − Ef = const, some precision of present-day experiments, by a calculation fraction of secondary gamma transitions will be re- that employs only the existing ideas of the factors solved in energy in these spectra in the form of pairs determining this process. The set of available data and of isolated peaks. By using presently elaborate proce- their interpretation indicates that it is necessary to dures, one will be able to determine, from these data, elaborate on the model concepts of this process that the intensities of individual secondary gamma tran- underlie an analysis of any experiment in this region of sitions to Ef ≤ 3–4 MeV levels, thereby obtaining a low-energy nuclear physics. The most probable line set of values for the reduced probabilities of gamma of such a modification is to take into account more transitions that makes it possible to assess directly precisely, in new models of ρ and k, the coexistence the shape of the energy dependence of strength func- of quasiparticle and phonon excitations in a nucleus tions in the excitation-energy region of interest to us. and the effect of their interplay on the level density and on the probability of gamma decay, including a The special features of the energy dependence direct use of the idea that there exists a second-order of the observed population of levels gives sufficient phase transition in a nucleus between a superfluid and k grounds to expect a maximum increase in values for a normal state. secondary transitions of the cascades in even–even nuclei and a less pronounced increase in even–odd nuclei. REFERENCES The only reason why such spectra have not been 1. Handbook for Calculation of Nuclear Reaction recorded [21] so far is that, in the spectra of the sums Data, IAEA-TECDOC-1034 (Vienna, 1998). of amplitudes of coinciding pulses, the corresponding 2. W. Dilg et al.,Nucl.Phys.A217, 269 (1973). peaks of ever decreasing area appear against an ever 3. E.M.Rastopchin,M.I.Svirin,andG.N.Smirenkin, increasing Compton background. In principle, the Yad. Fiz. 52, 1258 (1990) [Sov. J. Nucl. Phys. 52, 799 problem in question can be tackled by means of a pro- (1990)]. ´ cedure where cases of a simultaneous absorption of 4. E. V. Vasilieva, A. M. Sukhovoj, and V. A. Khitrov, the total energy of three photons sequentially emitted Yad. Fiz. 64, 195 (2001) [Phys. At. Nucl. 64, to the ground and low-lying states of a nucleus are 153 (2001)]; V. A. Khitrov and A. M. Sukhovoj, INDC(CCP)-435, IAEA (Vienna, 2002), p. 21; nucl- isolated in the form of peaks in the spectrum of sums ex/0110017. of amplitudes of three coinciding peaks. Since the en- 5. A. M. Sukhovoj, V. A. Khitrov, and E. P. Grigor’ev, ergies of secondary transitions of two-stepcascades INDC(CCP)-432, IAEA (Vienna, 2002), p. 115. are known (and can be found in the same experiment), 6. A. M. Sukhovoj, and V. A. Khitrov, Yad. Fiz. 62,24 it proves to be possible to determine, from the cases (1999) [Phys. At. Nucl. 62, 19 (1999)]. of detection of the total energy of three sequentially 7. J. Honzatko,V.A.Khitrov,C.Panteleev,´ et al., emitted photons, the distributions of the intensities of Preprint No. E3-2003-88, JINR (Joint Inst. Nucl. two-stepcascades to Ef > 1 MeV final levels. Res., Dubna, 2003); nucl-ex/0305020.

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8. V. A. Bondarenko, J. Honzatko,´ V. A. Khitrov, et al., 16. S. F. Mughabghab, M. Davadeenom, and in Proceedings of the XI International Seminar on N. E. Holden, Neutron Cross Sections (Academic, Interaction of Neutrons with Nuclei, Dubna, Rus- New York, 1984), Vol. 1, Part A. sia, 2003 (JINR, Dubna, 2003), E3-2004-9, p. 73. 17. P. Axel, Phys. Rev. 126, 671 (1962); J. M. Blatt and 9. M. Igashira et al.,Nucl.Phys.A457, 301 (1986). V.F.Weisskopf, Theoretical Nuclear Physics (Wiley, 10.M.A.Lone,R.A.Leavitt,andD.A.Harrison,Nucl. New York, 1952; Inostrannaya Literatura, Moscow, Data Tables 26, 511 (1981). 1954). 11. http://www-nds.iaea.org/pgaa/egaf.html 18. S. G. Kadmenskiı˘ ,V.P.Markushev,andV.I.Furman, 12. G. L. Molnar et al., Appl. Radiat. Isot. 53, 527 (2000). Yad. Fiz. 37, 277 (1983) [Sov. J. Nucl. Phys. 37, 165 13. V. A. Khitrov, Li Chol, and A. M. Sukhovoj, in Pro- (1983)]. ceedings of the XI International Seminar on In- 19.C.F.PorterandR.G.Thomas,Phys.Rev.104, 483 teraction of Neutrons with Nuclei, Dubna, 2003 (1956). (JINR, Dubna, 2003), E3-2003-9, p. 98. 20. A. M. Sukhovoj and V. A. Khitrov, Izv. Ross. Akad. 14. S. T. Boneva et al.,Z.Phys.A338, 319 (1991); Nucl. Nauk, Ser. Fiz. 61, 2068 (1997). Phys. A 589, 293 (1995). 21. V. A. Bondarenko, J. Honzatko,´ V. A. Khitrov, et al., 15. A. V. Ignatyuk and Yu. V. Sokolov, Yad. Fiz. 19, 1229 Fizika B (Zagreb) 11, 201 (2002). (1974) [Sov. J. Nucl. Phys. 19, 628 (1974)]. Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1525–1539. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1583–1598. Original Russian Text Copyright c 2005 by Ryabov, Adeev. NUCLEI Theory

Static and Statistical Properties of Hot Rotating Nuclei in a Macroscopic Temperature-Dependent Finite-Range Model

Е. G. Ryabov* and G. D. Adeev** Omsk State University, pr. Mira 55A, Omsk, 644077 Russia Received May 13, 2004; in ﬁnal form, October 29, 2004

Abstract—A macroscopic temperature-dependent model that takes into account nuclear forces of finite range is used to calculate the static and statistical properties of hot rotating compound nuclei. The level- density parameter is approximated by an expression of the leptodermous type. The resulting expansion coefficients are in good agreement with their counterparts proposed previously by A.V. Ignatyuk and his colleagues. The effect of taking simultaneously into account the temperature of a nucleus and its angular momentum on the quantities under study, such as the heights and positions of fission barriers and the effective moments of inertia of nuclei at the barrier, is considered, and the importance of doing this is 2 demonstrated. The fissility parameter (Z /A)crit and the position of the Businaro–Gallone point are studied versus temperature. It is found that, with increasing temperature, both parameters are shifted to the region of lighter nuclei. It is shown that the inclusion of temperature leads to qualitatively the same effects as the inclusion of the angular momentum of a nucleus, but, quantitatively, thermal excitation leads to smaller effects than rotational excitation. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION relativistic mean-field theory [13, 14]. However, such calculations are complicated and cumbersome even In recent decades, there has appeared the pos- at the present-day level of development of computa- sibility of experimentally studying the fission of nu- tional techniques. By using a simpler macroscopic– clei formed in heavy-ion reactions. This has given microscopic approachproposed by Strutinsky [15], impetus to theoretical investigations into processes Diebel et al. [1] calculated fission barriers and crit- involving the formation and decay of nuclei charac- ical angular momenta for excited nuclear systems. terized by high angular momenta and relatively high However, an approach that is based on the liquid- excitation energies [1–12]. In this case, a theoretical drop model of the nucleus is the simplest and the consideration includes a quantitative estimate of the most successfully used in fission dynamics. Despite effect of excitation energy and angular momentum its more than half-century history, the liquid-drop on the evolution of the shapes of a fissile compound model, albeit having undergone a number of modifica- nucleus. This information makes it possible to de- tions and improvements, remained essentially macro- scribe the fission of rotating heavy nuclei and the scopic. For many years, the fact that the temperature competition between fusion and fission in heavy-ion of a compound nuclear system was not taken into reactions. In particular, the theoretically predicted account in this model has been a significant draw- probability of the formation of relatively stable super- back of the model. The absence of vast experimental heavy nuclei depends primarily on excitation energy data generally used to determine the parameters of and angular momentum. The answer to the question the liquid-drop model (nuclear binding energies in of whether the nucleus formed will exist, prior to the ground state and fission barriers, some fusion undergoing fission, long enoughto be recorded exper- barriers, an equivalent nuclear radius, as well as the imentally is determined precisely by these quantities. diffuseness of the charge distribution) for hot nuclei In theoretical calculations, the most comprehen- was an excuse for this situation to some extent. In sive and precise information about the effect of the determining the coefficients used in the model, one temperature and rotation of a nucleus on its prop- has had to rely, in this case, on the results of mi- erties can of course be obtained within microscopic croscopic calculations performed, for example, within approaches, such as the Hartree–Fock method [9], the extended temperature-dependent Thomas–Fermi the extended Thomas–Fermi method [3, 4, 7, 12], or method [3]. It is the approach that was used in [2, 4, 7, 10, 16]. The rotating-liquid-drop model that takes *e-mail: [email protected] into account nuclear forces of finite range and the **e-mail: [email protected] diffuseness of the nuclear density [17, 18] is widely

1063-7788/05/6809-1525$26.00 c 2005 Pleiades Publishing, Inc. 1526 RYABOV, ADEEV used bothin dynamical and in statistical approaches 2. TEMPERATURE-DEPENDENT to nuclear fission. It was generalized to the case of FINITE-RANGE LIQUID-DROP MODEL hot nuclei by Krappe in [16]. Within the macroscopic approach (the very applicability of this approach is determined by nuclear excitation and rotation), this Hot rotating compound nuclei produced in heavy- finally makes it possible to take into account ade- ion reactions form a thermodynamic system. Natu- quately and self-consistently the effect of boththe rally, various parameters of this system and its sta- angular momentum and the temperature of a com- bility conditions must be determined by some of its pound system on the statistical properties of nuclei, thermodynamic potentials, either the free energy [16] on one hand, and on fission dynamics, on the other or entropy [20]. The choice of one thermodynamic po- hand. It should be emphasized that the parameters of tential or another is a matter of taste and convenience this model were determined by Krappe on the basis of in performing calculations since they are related by calculations within the microscopic Thomas–Fermi − method (a detailed description of this procedure can the well-known equation F (q,T)=E(q,T) TS. be found in [16]). In the earlier studies reported in [1–5, 7, 9, 10] and For a set of collective coordinates q describing mentioned above, temperature effects and the rotation the nuclear shape, we choose geometric parame- of a nucleus were not taken simultaneously into ac- ters of the nuclear shape—that is, q = {c, h, α}. count in determining static and statistical properties In doing this, we employ a modified version of the of nuclei (fission barriers Bf ; level-density parameters well-known {c, h, α} parametrization from [21]. The in the ground state and at the barrier, an and af , parametrization used here in the equation deter- respectively; effective moments of inertia, Jeff ;and mining the profile function differsfromtheoriginal critical angular momenta L ). The importance of crit {c, h, α} parametrization in that the former involves a taking these effects into account cannot be overestimated, but this does not reduce the value of the new mass-asymmetry parameter α that is related to previous investigations. α by the scale transformation [22] Some preliminary results of an analysis of the 3 properties of the temperature-dependent finite-range α = αc . (1) liquid-drop model were described by our group in [19]. Within the parametrization being considered, the The present article is aimed at more detailed calcula- shape of the nuclear surface in terms of cylindrical tions and discussions. coordinates is given by the equation [21, 23]

  − c 2 c2 − z2 A c2 + Bz2 + α z/c2 ,B≥ 0, 2 s ρs(z)= (2)  −2 2 2 2 2 2 c c − z Asc + α z/c exp(Bcz ),B<0, where z is the coordinate along the symmetry axis and ρs is the value of the coordinate ρ at the nuclear surface. The quantities As and B are expressed in terms of the nuclear-shape parameters (c, h) as B =2h +(c − 1)/2;

 c−3 − B/5,B≥ 0, As = 4 B (3) − √ √ ,B<0. 3 exp(Bc3)+(1+1/(2Bc3)) −πBc3erf( −Bc3)

In Eqs. (1)–(3), c is the elongation parameter (2c is symmetric withrespect to the z =0plane correspond the elongation of a nucleus), h is a parameter that to the case of α =0. A formula for the free energy was proposed in [16] speciﬁes the neck thickness at a given elongation, and onthebasisoftheﬁnite-range liquid-drop model. The α is the mass-asymmetry parameter. Shapes that are free energy of a nucleus as a function of the mass

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Table 1. Coefficients in Eq. (4) [values at zero temperature (first row) and temperature coefficient xi (second row)]

r0 a ad av kv as ks

ai(0) 1.16 0.68 0.7 16.0 1.911 21.13 2.3

3 −2 10 xi [MeV ] −0.736 −7.37 −7.37 −3.22 5.61 4.81 −14.79 number A = N + Z, the neutron excess per nucleon The temperature dependence of the seven coeffi- [I =(N − Z)/A], temperature, and the collective co- cients appearing in Eq. (4), av, as, kv, ks, r0, a,and ordinates q characterizing the nuclear shape has the ad, is parametrized in the form [16] form [16] 2 ai(T )=ai(T = 0)(1 − xiT ), (7) 2 F (A, Z, q,T,L)=−av(1 − kvI )A (4) ≤ 2 2/3 which can be considered as adequate for T 4 MeV + as(1 − ksI )Bn(q)A [3]. Information about the temperature coefficients xi Z2 was obtained from self-consistent microscopic calcu- + c A0 + a B (q) 0 C A1/3 C lations on the basis of the extended Thomas–Fermi method with the SkM∗ interaction for the effective in- 5 3 2/3 Z4/3 2L(L +1) − a + , teraction between nucleons [3, 4]. In [16], the results c 4 2π A1/3 2J(q) of these calculations for the thermodynamic Gibbs potential were used to derive formula (4). The values where av, as,andaC are the parameters of the vol- of 14 parameters that were obtained in [16] and which ume, surface, and Coulomb energies in the finite- are used in the present study are given in Table 1. range liquid-drop model at zero temperature, while kv and ks are, respectively, the volume and the surface Considering a nucleus within the Fermi gas parameter of the symmetry energy. The dependence model, using thermodynamic relations, and knowing on the deformation in Eq. (4) is determined by the the free energy, one can obtain both the entropy and functionals Bn(q), BC(q),andJ(q): the level-density parameter: 1 B (q)= (5) ∂F(q,T) S(q,T) n 2 4 2 2/3 S(q,T)=− ,a(q,T)= . 8π a r0A ∂T 2T V |r − r | exp (−|r − r |/a) (8) × 2 − drdr , a |r − r|/a V V Within the Fermi gas model, we also have the relation − 2 15 Eint(q,T)=E(q,T) E(q,T =0)=a(q)T , B (q)= C 2 5 5/3 (6) (9) 32π r0A |r − r| where Eint(q,T) and E(q,T) are, respectively, the × 1 − 1+ internal energy of the system and its total excita- 2ad V V tion energy, while a(q) is the level-density parameter. From formulas (8) and (9), it follows, among other | − | × − r r drdr things, that the temperature dependence of the free exp . ad |r − r | energy has the form

One can easily see that the functionals in (5) and F (q,T)=V (q) − a(q)T 2, (10) (6) are in fact the nuclear-energy and the Coulomb energy functional within the finite-range liquid-drop where V (q) is the potential energy of the nucleus model[18].Thelasttermin(4)istherotationalen- at T =0 (V (q)=F (q,T =0)). The microscopic ergy involving the deformation-dependent rigid-body calculations performed in [3] within the extended moment of inertia of a nucleus withallowance for the temperature-dependent Thomas–Fermi method re- diffuseness of the nuclear density (perfect analog of vealed that Eq. (10) for the free energy F provides the rotational energy in the finite-range liquid-drop quite an accurate approximation in the region T ≤ model [18]). 4 MeV.

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Table 2. Coefficients av and as obtained in the present nuclear excitations. We note that the level-density study by using a one-, a two-, and a three-dimensional parameter a(q) calculated on the basis of the liquid- grid and in [24, 27] drop-model version used here weakly depends on temperature. The same result was obtained in [16] for a ,MeV−1 a ,MeV−1 spherical nuclei. Because of this, we further assume v s that the level-density parameter in the finite-range one-dimensional 0.0603 0.1186 liquid-drop model is temperature-independent. We approximated the deformation dependence of two-dimensional 0.0600 0.1210 the level-density parameter in the finite-range liquid- drop model by the widely used expression (11). In three-dimensional 0.0598 0.1218 order to obtain the approximation coefficients av and a , we used 70 nuclei along the beta-stability line [24] 0.073 0.095 s from Z =47to Z = 116. The equation determining the beta-stable nuclei has the form [29] [27] 0.0685 0.274 A 0.4A Z = 1 − . (12) 2 A + 200 3. RESULTS OF THE CALCULATIONS AND DISCUSSIONS In constructing the approximation in question, we 3.1. Asymptotic Level-Density Parameter employed the least squares method and took into account all nuclear shapes, with the exception of The level-density parameter a isoneofthemost scission shapes whose neck radius was smaller than important features of an excited nucleus considered 0.3R0 (R0 is the radius of the original spherical nu- within the Fermi gas model [24, 25]. By using the cleus). The results of the approximation are given in thermodynamic relations (8) and (9) and formula (4), Table2.Thecoefficients av and as were estimated we can determine the level-density parameter within for the one- (h = α =0), two- (α =0), and three- the liquid-drop model that takes into account nuclear dimensional cases. It can easily be seen that the forces of finite range and nuclear excitations. In this coefficients are not sensitive to the dimensionality of case, an implicit dependence of a on the nuclear de- the grid used in the approximation; therefore, all of formation is specified by the dependence of the free the results discussed below were obtained with the energy F appearing in Eqs. (8) and (9) on the collec- coefficients calculated witha three-dimensional grid. tive coordinates. A variation of the approximate coefficients av and At the same time, the dependence of the level- as along the beta-stability line is shown in Fig. 2a.It density parameter on the deformation of a nucleus is can be seen that the value of av is in fact constant, often represented by a leptodermous-type expansion while the coefficient as decreases by 20% as Z varies of the form [24–27] from 47 to 116. It can be concluded from Figs. 2b and a(q)=a A + a A2/3B (q). (11) 2c that our approximation is quite correct within the v s s range of nuclear deformations under consideration. In this equation, A is the mass number of the fissile The coefficient as specifies the dependence of the nucleus and the dimensionless factor Bs(q) specifies level-density parameter on deformation; hence, it is the area of the deformed-nucleus surface in units of of importance for statistical and dynamical models of thesurfaceofaspherehavingthesamevolume— nuclear fission. The value obtained for this coefficient the surface-energy functional in the model of a liquid in the approximation proposed here is close to that −1 drop having a sharp surface [28]. Among a number of from [24], as =0.095 MeV ,anddiffers by a factor −1 sets of the parameters av and as in the dependence of more than two from the value of as =0.274 MeV given by (11), use is most frequently made of that from [27]. The second coefficient, av, takes approx- which was proposed by Ignatyuk et al. [24] (av = imately the same value in all of the approximations −1 −1 0.073 MeV and as =0.095 MeV )andthatwhich considered here. was recommended by Toke¨ and Swiatecki [27] (av = For the ground states of the nuclei lying along −1 0.0685 MeV and as =4av). the beta-stability line, Fig. 3 displays the level- For two values of the nuclear temperature, Fig. 1 density parameter as a function of the mass number. shows the level-density parameters versus deforma- From this figure, one can see that the results of tion (elongation) that were calculated by formula (11) the calculations with Ignatyuk’s coefficients [24] withthecoe fficients borrowed from [24] and [27] and and the results of the calculations performed within by formulas (4) and (8) within the liquid-drop model the temperature-dependent liquid-drop model are in taking into account nuclear forces of finite range and better agreement with the values obtained on the

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 STATIC AND STATISTICAL PROPERTIES 1529

a, MeV–1 28

0.8 1.2 1.6 2.0 c

Fig. 1. Level-density parameter as a function of deformation for the 224Thnucleus: results for theliquid-drop model taking into account nuclear forces of finite range and nuclear excitations at (solid curve) T =0.001 MeV and (dash-dotted curve) T =4MeV,forthecaseof(dashedcurve)Ignatyuk’scoefficients and (dotted curve) Toke¨ –Swiatecki coefficients, and for the case of (thin straight line) a(q)=A/10.

–1 as, av, MeV 0.14 as (a) 0.10

av 0.06 50 70 90 110 Z a, MeV–1 (b) 25 306116

10.0 (c) 109Ag 9.6

9.2 0.8 1.2 1.6 2.0 c

Fig. 2. (a)Coefficients av and as as obtained here from an approximation along the beta-stability line; (b, c) level-density parameter as a function of deformation for the element 306116 and for 109Ag within the finite-range liquid-drop model (solid curves) and its approximation (dashed curves) (for these nuclei, the accuracy of the approximation is the lowest). basis of microscopic approaches than with those dependent finite-range liquid-drop model were deter- that were calculated by employing the set of Toke¨ – minedin[16]onthebasisofanapproximationofthe Swiatecki parameters [27]. In turn, the curve for the results obtained within the extended Thomas–Fermi level-density parameters that was calculated with method by using effective Skyrme interaction forces Ignatyuk’s coefficients is virtually coincident with ∗ the results obtained within relativistic mean-field SkM , the curves corresponding to them in Fig. 3 theory [14]. Since the parameters of the temperature- coincide.

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a, MeV–1 superdeformed nuclei—but also, in the case of the 24 decay of an equilibrium compound nucleus, the competition between the processes of fission and particle 20 emission. It should be noted that the angular distribu- 16 tion of fission fragments also depends on the effective moment of inertia of a fissile nucleus at the barrier 12 (Jeff), this moment of inertia being determined by the saddle-point configuration of the nucleus. 8 The effect of nuclear rotation on the fission process 4 is well known [18, 30]: the fission barrier decreases, 40 80 120 160 200 disappearing at some value of the angular momentum A (Lcrit); the ground state of the nucleus with respect to the main fission coordinate (c, which is the elonga- Fig. 3. Level-density parameter for some nuclei along tion parameter of the nucleus) is shifted toward the the beta-stability line in the ground state: (dash- scission point, while the saddle point is, on the con- dotted curve) results within the finite-range liquid-drop trary, shifted toward the ground state, these effects model [16], (solid curve) results obtained withIgnatyuk’s becoming more pronounced withincreasing angular coefficients [24], and (dashed curve) results obtained withT oke¨ –Swiatecki’s coefficients [27]. The boxes rep- momentum of the rotating nucleus. resent the level-density parameter as a function of the However, the inclusion of temperature effects is no mass number for beta-stable nuclei according to calcu- less important [2–5, 9]. It should be noted that only lations within the extended Thomas–Fermi method with effective-interaction Skyrme forces SkM∗ [4], while the a few studies have been devoted to a macroscopic circles correspond to self-consistent calculations within treatment of the temperature dependence of fission relativistic mean-field theory [14]. barriers (among those that were mentioned above, there are [2, 6, 7, 31]). If, in a theoretical description, we want to approachthepattern observed in Bf, MeV heavy-ion reactions, we must take self-consistently 50 into account bothfactors (temperature and angular T = 0 momentum), considering a hot rotating nucleus [1, 7, 40 10, 11] within a unified model. T = 2 In analyzing the temperature or the angular- T = 3 30 momentum dependence of static and statistical prop- T = 4 erties of nuclei, we will assume here and below that a factor whose magnitude and variation are not 20 discussed is constant and is equal to zero. For hot beta-stable nuclei, Fig. 4 presents the 10 dependence of the fission barriers on the mass number. Here, the triangles represent experimental data 0 from [12] that were corrected with allowance for shell 50 100 150 200 250 300 effects. As was indicated in [2], the liquid-drop model Ä is inapplicable in the region of medium-mass and light nuclei, this being by and large confirmed by Fig. 4. Fission barriers in hot beta-stable nuclei at var- overestimated values of theoretically predicted barri- ious temperatures T and L =0(solid curves, the temperature being indicated in MeV) and at T =2MeV and ers in the region of light nuclei (see Fig. 4). Qual- L =40 (dashed curve). The triangles represent experi- itative agreement between the temperature depen- mental data reported in [12] corrected withallowance for dence of the barrier height (a decrease in Bf with shell effects. increasing temperature) and data obtained in [2, 10] is worthy of note, albeit the barriers calculated in the 3.2. Effect of the Temperature of a Nucleus and Its present study are systematically below those data. Angular Momentum on the Barrier Height The same conclusions can be drawn from a comparison of our results for the temperature dependence For nuclei formed in heavy-ion reactions, fission of barriers in heavy and superheavy nuclei with the barriers (Bf ) play an important role in their fusion– results reported in [1], where use was made of Struti- fission dynamics. The height and position of the bar- nsky’s macroscopic–microscopic approach. At the rier determine not only the character of a nuclear same time, good qualitative agreement is obtained process—fast fission, quasifission, or the existence of for barrier heights if a comparison is performed with

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 STATIC AND STATISTICAL PROPERTIES 1531 the results of calculations based on the Thomas– Bf, MeV Fermi method [7, 12], although an interaction dif- 4 ∗ (a) fering from the SkM interaction was used for an 240 effective nucleon–nucleon potential. As might have Pu been expected, the results of our calculations at zero 3 L = 0 temperature for barriers agree withthebarriers in the finite-range liquid-drop model employing Sierk’s 20 parameters [18] (in Fig. 4, the respective curves are 2 indistinguishable). The dashed curve, which represents the barrier heights for the same beta-stable 1 40 nuclei of temperature 2 MeV whose rotation is characterized by an angular momentum of L =40, il- lustrates vividly the need for simultaneously taking 0 into account the rotation of a nucleus and its thermal af/an excitation. The difference in the barrier heights as calculated with and without allowance for the rotation 1.04 (b) 220Fr of a hot nucleus becomes ever more pronounced with decreasing mass number of nuclei. 1.03 L = 0 The effect of simultaneously taking into account 20 the thermal excitation of a nucleus and its rotation 40 as a discrete unit is illustrated in Fig. 5a,wherethe 1.02 50 effect that both factors considered in the present study (a decrease in the barrier height with increasing nu- 1.01 clear temperature and angular momentum of nuclear 240 rotation) exert on the barrier is shown for the Pu 1.00 nucleus. It is worthmentioning thatan increase in 012 3 4 the angular momentum produces a greater effect, the T, MeV temperature effect becoming less pronounced with 240 increasing L. For a comparison of our results with Fig. 5. (a) Fission barrier for Pu as a function of data of Garcias et al. [7], who considered the effect the temperature and angular momentum of a compound 240 nucleus: (solid curves) results of a theoretical calcula- of the same two factors on the fission barrier in Pu tion by formula (4) and (dashed curves) approximation within the Thomas–Fermi method, to be complete, by expression (13); (b)ratioaf /an as a function of the 220 we approximated the dependence of Bf on T and L temperature and angular momentum of the Fr nucleus by the expression (dashed curve in Fig. 5a) according to calculations by formulas (4), (8), and (9) for the saddle-point configuration and for the ground state of −3 2 −6 3 Bf =3.889 − (1.544 × 10 L − 5.71 × 10 L ) the nucleus. (13) −1 −4 2 2 − (5.03 × 10 − 0.63 × 10 L )T we can conclude that temperature effects are only +0.068T 3 [MeV], slightly correlated with nuclear-rotation effects (the term proportional to L2T 2 is only a slight correction). which is similar to expression (2) from [7]. The coefficients in the approximation are very close to In contrast to the quadratic dependence in [7], the values obtained in [7], naturally with the exception we obtained a cubic temperature dependence of the of that term in (13) which is cubic in temperature. barrier height (it descends slowly as the temperature The fact that there is no such term in relation (2) approaches the values of T = Tc, at which the bar- from [7] may be due to the following: although the rier disappears). A cubic temperature dependence is authors of [7] relied on the Thomas–Fermi method observed for barriers in all of the nuclei considered in realized withan e ffective nucleon–nucleon interac- the present study and is a general feature peculiar to tion of the SkM∗ type, they calculated the effective the finite-range model of a hot liquid drop [16]. Bar- Weizsacker¨ coefficient (β/36) in the expansion in riers for symmetric binary fission that were calculated powers of 2 withthevalue of β =1.6 rather than in [6], where diffuseness was simulated in terms of the withthevalue of β =1, which is usually used. A “proximity” energy [32] and where the temperature detailed discussion of the effect of this factor is be- dependence was constructed on the basis of data yond the scope of the present study. We note that from [5], also have a cubic temperature dependence. a cubic character of the temperature dependence of Following the same line of reasoning as Garcias the barrier height was also observed in [10], where et al. [7] and employing the approximate relation (13), use was made of the generalized liquid-drop model

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1532 RYABOV, ADEEV due to Myers and Swiatecki [28] and the “proximity” 3.3. Effect of Temperature and Angular Momentum energy (for details, see [10] and references therein). on the Saddle-Point Configuration of a Fissile Although the fission barriers calculated in [10] are Nucleus and on Features Depending on It systematically higher than those that were obtained in the present study, their temperature and angular- The saddle-point configuration of a fissile nucleus momentum dependences are similar: withincreasing is the shape that this nucleus assumes at the saddle angular momentum, the temperature effect becomes point. The saddle point determines the position of the weaker. fission barrier on the quasisurface that specifies the collective-coordinate dependence of the free energy The critical angular momentum Lcrit characterof the nucleus being considered and is a point of izes the fission stability of a nucleus withrespect to conditional equilibrium (a maximum withrespect to rotation [30]. Strictly speaking, this is the angular- the main fission coordinate c and a minimum with momentum value at which the fission barrier in a respect to other coordinates). Accordingly, the barrier compound nucleus vanishes. Since this feature is height is definedasthedifference of the free energy determined by the barrier height, which in turn de- at the saddle point and the minimum value of the creases withincreasing nuclear temperature, as we free energy. In turn, the point of minimum specifies have seen above, Lcrit must decrease as the nucleus the coordinates of the ground state of the nucleus. being considered is heated. The critical angular mo- In the case of the {c, h, α } parametrization, which mentum as a function of the mass number A for beta- stable nuclei is displayed in Fig. 6a for various values is used here, the position of the saddle point in the of the temperature. It seems quite natural that, at space of collective coordinates is determined by three c h α T =0, the critical angular momentum determined in quantities, sd, sd,and sd. From the outset, we took zero value for the coordinate α , since it is the present study is consistent with Lcrit calculated by sd Sierk [18] on the basis of the finite-range liquid-drop well known that, within the model of a rotating liquid model. Since Lcrit only decreases upon taking into drop [18, 33], asymmetric shapes of a fissile nucleus account nuclear temperature, there is no agreement, are less favorable energetically for nuclei lying on the over the whole temperature range, with the critical- right of the Businaro–Gallone point (medium-mass angular-momentum values calculated for superheavy and heavy nuclei), so that, for the saddle point, the profile of the free-energy surface along the coordinate elements within the macroscopic–microscopic approachproposed in [1] (see Table 3 in [1]). We ob- α hastheshapeofaparabolawithaminimumat tained Lcrit values close to zero even at zero temper- α =0. For light nuclei (which lie on the left of the ature, while, at T =1.5 MeV, Lcrit [1] is about a few Businaro–Gallone point), the profile in question has, tens of units (yet, these values were obtained with on the contrary, the shape of an inverted parabola with allowance for shell effects). a maximum at α =0. In this case, searches for a minimum in the mass-asymmetry coordinate involve Our estimates of L also fall systematically short crit the problem of grid finiteness in this coordinate. In of data reported in [10], where, with increasing mass order to circumvent this problem, we made the follow- number, the critical angular momentum decreases ing “simplification” in our calculations: we set to zero in the region of heavy nuclei faster than its coun- 190 the mass-asymmetry coordinate at the saddle point, terpart calculated here. For the Hg nucleus, we α , for all nuclei studied here. This simplification is found agreement to within 1 for L withthere- sd crit consistent with the experimentally observed fact that sults of Myers and Swiatecki [12], who employed the symmetric fission is dominant for hot nuclei [33]. Thomas–Fermi formalism to calculate the barriers. A comparison withdata from [7] reveals that,for On the basis of our calculations, we can draw 240Pu, the values that we obtained for the critical the following conclusion: similarly to the case of a angular momentum are systematically greater—for decrease in the barrier height, the position of the example, 62 versus 50 at zero temperature and 40 barrier is most strongly affected by temperature and versus 32 at T =2MeV. However, this fact is not angular momentum in the region of heavy nuclei; on unexpected, highlighting the validity of our statement the contrary, the lighter the nucleus, the weaker the that it is necessary to consider simultaneously the effect of these two factors. By way of example, we effect of the temperature of a nucleus and the effect indicate that, at any temperature considered here (not of its rotation on the static properties of a compound higher than 4 MeV), the saddle-point configuration nuclear system. It is the difference in the character of of nuclei lighter than 122Te is virtually independent of the temperature dependence of the barriers (we ob- either a thermal or a rotational excitation. If, however, tained here a cubic dependence, which descends with thetemperaturerangeisboundedfromabovebythe increasing angular momentum, while it is quadratic value of 2 MeV,this effect is observed for nuclei lighter 156 in [7]) that is responsible for the difference in Lcrit. than Gd. A similar pattern emerges in the case of

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Lcrit 90 (a)

60 T = 0 T = 1 T = 2 45 T = 3 30

T = 4 15 20 60 100 140 180 220 sd Q2

80 (b) T = 0, L = 40 = 2, = 40 70 T L 60 50 40

30 T = 0 T = 1 20 T = 2 10 T = 3 = 4 0 T 60 100 140 180 220 260 A

Fig. 6. (a) Critical angular momentum Lcrit (in units) as a function of the mass number A for beta-stable nuclei at various sd values of the temperature T (in MeV); (b) quadrupole moment in the saddle-point configuration of a nucleus, Q2 (in barns), as a function of the mass number A for beta-stable nuclei at various values of the temperature. varying the angular momentum at a constant temper- It is impossible to compare directly the results of ature. our calculations for saddle-point configurations with The above is illustrated in Fig. 6b,wherethetem- experimental data. However, this can be done indi- perature dependence of the quadrupole moment in rectly via a comparison of nuclear-shape-dependent saddle-point configurations is shown for beta-stable quantities at the barrier (such as the ratio of the level- nuclei. In order to compare the effect of rotation of density parameters at the barrier and in the ground a nucleus on the saddle-point configuration and the state, af /an,andtheeffective moment of inertia at the effect of its heating, curves drawn through boxes and barrier, Jeff). circles are also displayed in Fig. 6b. They correspond The importance of the ratio a /a in a statistical to the saddle-point configurations of nuclei at the f n angular momentum of L =40 and the temperatures simulation of fission reactions induced by heavy ions of T =0and 2 MeV (solid and dashed curves, re- is well known [34, 35]. Governing neutron emission, spectively). One can see that the heating of a non- as a matter of fact, this parameter determines, along rotating nucleus to a temperature of 2 MeV and the with the barrier height, the fissility of a compound rotation of a cold nucleus at the angular momentum nucleus. The difference in af and an isamacro- of 40 lead to approximately identical saddle-point scopic effect, which is described within the liquid- configurations of fissile beta-stable nuclei. Similar drop model, the origin of this effect being related in the conclusions can also be drawn for the T =3, L =0 theory to the effect of the nuclear surface and shape and T =2, L =40 cases. We note that the pattern on the level density [24, 27]. Although the ratio af /an observed here is in qualitative agreement with the is close to unity, even a change of a few percent in it results obtained in [2, 7]. changes the ratio of the mean neutronic and fission

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af/an being considered) at an angular momentum fixed at L =20 (this choice was motivated by the fact that, 1.08 in the reactions being studied, the fission process is induced by light charged particles, so that the formation of a compound nucleus that would carry a 1.06 high angular momentum is improbable). From Fig. 7, one can see that the results obtained within the above two theoretical approaches (dashed and dash-dotted 1.04 curves) agree quite well, but that the two curves deviate markedly from the empirical dependence. According to [35], the scatter of the results of different 1.02 170 180 190 200 210 A theoretical analyses [24, 27] for the ratio af /an,as well as the scatter of its experimental values, may be

Fig. 7. Ratio af /an for preactinide nuclei along the beta- as large as 15%. A large arbitrariness in describing stability line: (solid curve) averaged empirical estimates the level density of excited nuclei in the studies of from [37], (dashed curve) results based on relation (11) different authors may be one of the reasons for this withIgnatyuk’s coe fficients [24], and (dash-dotted curve) uncertainty. By way of illustration, we indicate that, results of our calculation (see main body of the text for the angular-momentum and temperature values at which we in [38], the ratio af /an changes in response to going performed this calculation). over from one phenomenological systematics of level densities to another in analyzing the ratio of the fission and neutronic width. decay widths by a value that grows with increasing The barrier heights and the ratios af /an deter- excitation energy [36]. mined by Krappe within the rotating-liquid-drop The aforementioned effect of the change in the model generalized to the case of hot nuclei have not position of the saddle point in the space of collective yet been used in statistical calculations. However, variables in response to a change in the temperature the study of Newton et al. [39], who calculated the and angular momentum naturally leads to a varia- mean prescission neutron multiplicity for 181Ta + 19F tion in the ratio af /an. Moreover, the conclusions and 159Tb + 19F reactions on the basis of the statis- on the character of the dependence of the barrier tical model, is worthy of note. In order to calculate heights on T and L fully apply to the ratio af /an: the temperature-dependent barrier heights, use was it decreases bothin response to an increase in the made of data from [7, 8], while the parameter af /an angular momentum and in response to an increase in was adjusted to fitthefission cross sections calcu- the temperature (see, for example, Fig. 5b). Although lated for the above two reaction types to their experi- this decrease is as small as about a few percent, it can mental counterparts. Also, the af /an value at which substantially affect, as has already been mentioned, the agreement with experimental data on neutron the results of statistical calculations for reactions multiplicities is good was estimated in [39], but use involving the fission of compound nuclei. We note was made there of the barriers that were calculated by that the lighter the nucleus involved, the weaker the the Thomas–Fermi method [3, 4] and which are ac- temperature dependence of af /an; on the contrary, it tually equivalent to the barriers determined within the becomes more important to take into account nuclear model considered here. The value of af /an =0.97, rotation. which was obtained by using the barriers from [7, 8] On the basis of an analysis of the fissility of com- proves to be less than its counterpart in the case of the pound preactinide nuclei bombarded withcharged barriers from [3, 4] (af /an =1.00). So small a value, particles, Ignatyuk et al. [37] estimated the averaged which is less than unity, is at odds with some results empirical ratio af /an (the results are shown by the obtained in a number of studies [24, 35–37], where solid curve in Fig. 7). The dashed curve in this figure a statistical description of level densities in nuclei led represents the ratio af /an calculated for preactinide to af /an > 1. In view of this, the self-consistency of nuclei on the basis of relation (11) with Ignatyuk’s the calculations employing the barriers from [3, 4] and coefficients [24] without explicitly taking into account resulting in a greater value of af /an is less disputable. the temperature and angular-momentum depen- The problem of consistency of the model concepts dences. The dash-dotted curve corresponds to our and arbitrariness in choosing the ratio af /an can be calculation for the same nucleus with allowance for fully solved by employing barriers and level-density- the excitation energy (which specifies the nuclear parameter ratios (af /an) determined within a unified temperature according to the relation T = Eint/a, model—for example, that which was considered in where Eint is the internal energy of the nucleus the present study.

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J0/Jeff 2.0

1.6

1.2

0.8

0.4

0 30 34 38 42 46 Z2/A

−1 Fig. 8. Inverse effective moments of inertia Jeff for nuclei along the beta-stability line: (solid curves) results of Krappe’s calculation [16] within the temperature-dependent macroscopic finite-range liquid-drop model [at the temperature values of 0, 1, 2, 3, and 4 MeV (from up down), the angular momentum being set to zero], (dotted curve) results of the calculation on −1 the basis of the rotating-liquid-drop model due to Cohen, Plasil, and Swiatecki [30], and (points) Jeff values obtained from an analysis of experimental data on the angular anisotropy of fission fragments: (open triangles) experimental data on heavy-ion reactions [40] and (closed triangles) results of an experimental analysis from [41].

The effective moment of inertia Jeff at the saddle positionsofthesaddlepointscoincideinthesemodels point is yet another feature that is determined by the at zero temperature, while the moments of inertia position of the barrier. The inverse of Jeff is given by are calculated in the rigid-body approximation. From 1 1 1 Fig. 8, one can see that the experimental data being = − , (14) considered are in better agreement withtheresults Jeff J|| J⊥ 2 −1 of our calculations, while, for Z /A < 36,theJeff where J|| and J⊥ are the moments of inertia of the values obtained from the analysis of experimental nucleus under consideration withrespect to thesym- data lie considerably higher than the curves that we metry axis and the axis orthogonal to it (nuclear- obtained and comply better with the Cohen–Plasil– rotation axis), correspondingly. As a rule, the effective Swiatecki model [30]. The existence of experimental moment of inertia is measured in units of J0, which is points below the solid curve, which corresponds to the moment of inertia of a sphere of the same volume. zero temperature (T =0) is worthy of special note: The experimental value of the effective moment if one considers that, within the model used here, an of inertia can be obtained from an analysis of the increase in the angular momentum of a nucleus at −1 anisotropy of the angular distribution of fission frag- fixed temperature also leads to a decrease in Jeff ,the ments. This was precisely the method that was experimental values are reproduced at specificvalues −1 of T and L, this counting in favor of the model that used calculate the Jeff values represented by closed and open triangles in Fig. 8. This figure displays we apply, which takes into account the temperature dependence of relevant parameters. the fissility-parameter dependence of Jeff for nuclei along the beta-stability line. The effective moments The analysis of experimental data that was per- of inertia calculated here on the basis of the Krappe formed in [40, 41] and which is illustrated in Fig. 8 did model [16] for various values of the temperature are not take into account the emission of prescission neu- shown by the solid curves in the figure. The dotted trons. This shortcoming was remedied in [42], where curve, which represents the results of the calculations the data are given with allowance for prefission neu- based on the model of a liquid drop with a sharp trons. However, these corrected data do not change surface [30], lies markedly higher. The curve calcu- the conclusions that we have drawn above. Anyway, lated withtheSierk coe fficients [18] coincides with overall agreement between a theoretical description the solid curve, which corresponds to the temperature and experimental data could not be achieved. The value of T =0. This comes as no surprise since the model used here is more realistic than the idealized

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1536 RYABOV, ADEEV Cohen–Plasil–Swiatecki model of a liquid drop with 3.4. Stiffness withRespect to a Mass-Asymmetry a sharp surface [30] and takes into account the tem- Variation in the Nuclear Shape within the Model perature dependence of the liquid-drop parameters. of Hot Rotating Nuclei As was indicated in [43], the quantitative disagree- In the fission of Z2/A < 32 nuclei whose rotation ment may be due bothto an approximate characterof is not very fast (low values of L), the “dynamics of the Halpern–Strutinsky statistical theory of angular- motion” of a nucleus from the saddle to the scission distribution formation [44] and to the uncertainty in point does not play any significant role [46]. In this some parameters used in analyzing experimental data case, the static properties of the potential-energy sur- (for example, a variation in the level-density parame- face of a deformed nucleus play the most important ter in [41]). role in a theoretical analysis of the fission process. In describing the potential energy within the liquid-drop By calculating the effective moment of inertia, one model, the stiffness (stability) of nuclei withrespect can determine, for the ratio (Z2/A),thecriticalvalue to mass-asymmetric deformations (shape variations), ∂2V/∂η2 [46], is one of the main parameters. (Z2/A) at which a nucleus loses stability with re- crit η spect to fission, this being a parameter of paramount For the mass-asymmetry coordinate ,wefol- lowed Strutinsky’s definition [45] importance for nuclear physics. For nuclei character- 2 V − V ized by this value of (Z /A), the saddle-point defor- η =2 l r , (15) mation coincides with that in the ground state and Vl + Vr →∞ Jeff , the ratio J0/Jeff tending to zero. Within the where Vl and Vr are the volumes of those parts of 2 model considered here, (Z /A)crit developsatemper- the asymmetric body representing the nucleus being ature and an angular-momentum dependence, as all considered that are, respectively, on the left and on of the other static parameters that we explored in this the right of the midpoint of the neck. It was indicated above that, in our calculations, we used the parameter study do. This circumstance becomes obvious if one recalls that the parameter (Z2/A) is determined α for the coordinate describing the mass asymmetry crit of the nucleus; therefore, we calculated the stiffness in terms of the ratio of the surface and Coulomb by the formula constants of the liquid-drop model, as and aC, which depend explicitly on temperature in the model being ∂2F ∂2F ∂α 2 = . (16) considered; as to the angular momentum, its effect ∂η2 ∂α 2 ∂η is due to the shift of the saddle point, at which we − We calculated the second factor in this formula by calculate J 1. An excitation of a nucleus (bothof a eff employing Eq. (15) and the expressions chosen for thermal and a rotational character) leads to a shift the volumes Vl and Vr of the left- and right-hand 2 of (Z /A)crit toward the region of lighter nuclei, the parts in accordance withtheparametrization being effect of the two factors being approximately identical. considered. By way of example, we indicate that, at zero temper- On the basis of the liquid-drop model involving ature, the change in temperature from 0 to 4 MeV temperature-dependent parameters, we calculated 2 leads to the reduction of (Z /A)crit from 44.5 to 40.5; here stiffness for hot nuclei, employing the free energy at zero temperature, approximately the same values F instead of the potential energy V . This enables us correspond to a variation in the angular momentum to assess the effect of temperature on the magnitude from L =0to L =60. The value that we obtained of stiffness for the case of mass-asymmetric defor- at T =0is in good agreement withthepredictions mations and on the critical points of the stiffness of other realistic liquid-drop models [18, 45]. The curve. The results are presented in Fig. 9. In this figure, the experimental data are given only for the experimental estimates obtained in [41] for (Z2/A) crit region Z2/A < 31,since,forZ2/A > 32 nuclei, the J on the basis of determining eff from an analysis saddle and scission points differ significantly, as was of the angular anisotropies of fission fragments are indicated in [46, 47], while the experimental values different for different sets of nuclei and for different of the stiffness for them lie higher, corresponding to values of the nuclear-level-density parameter that some point that is intermediate between the saddle 2 were used in that analysis. The estimate (Z /A)crit = and scission points. This range of nuclei will not be 44.3–44.9 [41], whichisin good agreement withthe considered in the present analysis. results of our calculations and which is markedly less In [46–48], it was indicated that, in the interval 2 2 than the value of (Z /A)crit 50, predicted by the Z /A ∼ 20–30,thefinite-range liquid-drop model model of a liquid drop with a sharp surface, seems the involving Sierk’s coefficients provides the best de- most plausible. scription of experimental data. In those studies, the

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∂2F/∂η2 16

12 T = 0

T = 1 8 T = 2 T = 3

T = 4 4

0 18 22 26 30 34 38 42 Z2/A

Fig. 9. Stiﬀness of ﬁssile nuclei at the saddle point, ∂2F/∂η2 (in MeV), for nuclei along the beta-stability line at various values of the temperature T (in MeV). Boxes represent the results obtained by analyzing experimental data from [48].

2 2 position of the Businaro–Gallone point, where a x (x =(Z /A)/(Z /A)crit). We will arrive at the same nucleus completely loses stability against a mass- conclusion if, in calculating the fissility parameter x, asymmetric deformation (the stiffness vanishes), we take into account the temperature dependence of 2 ± 2 was determined to be (Z /A)BG =22.0 0.6.It (Z /A)crit described in the preceding section. How- should be noted that the theoretical curves calculated ever, it is worth noting that, within the model being in [46–48] were obtained withtheset of coe fficients 2 considered, the decrease in (Z /A)crit withincreasing from [17] rather than with the set of constants pro- 2 temperature is faster than the decrease in (Z /A)BG, posed by Sierk in [18]. Our results at zero temperature since, as the temperature increases from T =2 to agree with the results obtained on the basis of the 4 MeV, the position of the Businaro–Gallone point Sierk model with the constants from [18]. The value x 2 on the fissility axis, BG, changes from 0.47 to 0.49. of (Z /A)BG =21.1, which was found in the present But in [31], the situation was qualitatively different: study, depends greatly on the set of constants used. an increase in the temperature there led to a sizable 2 For example, (Z /A)BG =21.6 if use is made of decrease in xBG. the parameter set from [17]. This is less than the A second critical point on the curve of the stiff- estimate 22 ± 0.6 from [46–48], but the difference ness withrespect to mass-asymmetric nuclear-shape is likely to be due to employing different nuclear- variations is the point of maximum; at T =0,itoc- shape parametrizations in the respective calculations Z2/A ∼ 30 (generalized Cassini ovals in [49] versus the {c, h, α } curs at (see Fig. 9). As was found in [47, 48], this point of inflection on the curve of the stiffness parametrization in the present study). For a more 2 detailed discussion of all uncertainties associated plotted against Z /A corresponds to a change in the 2 2 withtheanalysis of experimental data and withthe sign of the derivative dσM /dL —this means that, calculation of the stiffness on the basis of these data withincreasing angular momentum, thevariance of in [46–48], the interested reader is referred to [47]. the mass distribution decreases for nuclei characterized by Z2/A values on the left of this point and From Fig. 9, one can see that an increase in the Z2/A temperature leads to a shift of the Businaro–Gallone increases for nuclei characterized by values on point toward the region of lighter nuclei, with the the right of it. Figure 9 clearly shows that, as the temperature is elevated from 0 to 4 MeV, the position result that, at T =4MeV, (Z2/A) =19.8. Similar BG 2/A ∼ behavior was obtained in [31] within the model of a of the maximum is shifted from Z 30 to 27. rotating liquid drop withallowance for “proximity” The effect of the angular momentum on the stiff- forces [32]. It was also found there that, up to the ness withrespect to mass-asymmetric nuclear-shape temperature of 2 MeV, the position of the Businaro– variations was studied in detail by Rusanov et al. [46]. Gallone point remains unchanged on the fissility axis Their conclusions that the Businaro–Gallone point

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1538 RYABOV, ADEEV and the point of maximum are shifted toward the re- version of the liquid-drop model, embraces nuclei gion of light nuclei, this corresponding to an effective heated to such an extent that single-particle pairing increase in the masses of nuclei, also apply to the and shell effects can be disregarded. For superheavy results obtained within the model being considered. nuclei, shell corrections play a significant role; there- This comes as no surprise since the potential energy fore, the liquid-drop-model version due to Krappe of a rotating drop was calculated in [46] on the basis of is inapplicable to such nuclei. The model that we the Sierk model [18], which is equivalent to the zero- have considered here does not reproduce the results temperature Krappe model, which was used here. obtained in [1]: for superheavy elements, there are Analyzing and comparing the results obtained in [46] no stable states characterized by a relatively high and in the present study, we would like to emphasize temperature and high angular momenta. that the rotation of a nucleus has a substantially The results that we have obtained here for the stronger effect than temperature on the position of effective moments of inertia at the saddle point are the Businaro–Gallone point. This was also observed in good agreement withexperimental estimates only in [47, 48] in analyzing experimentally determined for Z2/A > 36 nuclei. For Z2/A ∼ 33–34 nuclei, the stiffnesses. model of a liquid drop witha sharp surface leads to 2 better agreement. However, the parameter (Z /A)crit 4. SUMMARY AND OUTLOOK extracted from information about Jeff within the model of a liquid drop witha sharp surface appears to In calculating static and statistical properties 2 ∼ of hot nuclei, we have applied the temperature- be (Z /A)crit 50, which is markedly larger than dependent macroscopic model that takes into ac- experimental estimates, 44.3–44.9, which, in turn, count nuclear forces of finite range. are well reproduced by the temperature-dependent finite-range liquid-drop model that was generalized The main coefficients of the model that were by Krappe and which has been used in the present obtained by Krappe [16] on the basis of approxi- study. mating the results of calculations within the microscopic Thomas–Fermi method feature an ex- In calculating the stiffness of nuclei, we have fo- plicit temperature dependence. Within a unified ap- cused our attention primarily on temperature effects. proach, we have therefore been able to calculate Withincreasing temperature, theBusinaro –Gallone self-consistently both fission barriers and level- point is shifted toward the region of light nuclei on the density parameters. The importance of this self- Z2/A axis, this corresponding to the results reported consistency was previously highlighted in [19]. By in [31]. As in [31], the increase in the temperature relying on the temperature-dependent finite-range from 0 to 2 MeV does not change the position of the liquid-drop model, we have approximated the level- Businaro–Gallone point on the x axis, but, in con- density parameter by a leptodermous-type equation. trast to the results of Haddad and Royer [31], a further The values that we have obtained for the coefficients, increase in the temperature leads, in our case, to an −1 −1 2 2 av =0.0598 MeV and as =0.1218 MeV ,agree increase in xBG =(Z /A)BG/(Z /A)crit, this being 2 well withthecoe fficients obtained in [24]. These due to a faster decrease in the parameter (Z /A)crit coefficient values can be of use bothin a statistical withincreasing temperature. Thiscircumstance con- and in a dynamical simulation of the fission process firms once again the importance of consistently ap- onthebasisofthefinite-range liquid-drop model. plying nuclear models in theoretical calculations. By way of example, we have calculated barrier Summarizing the aforesaid, we emphasize that, heights in nuclei and have determined their saddle- even within the best (at least at the present time) point configurations and ratios af /an. From these ex- version of the liquid-drop model, one can hardly re- amples, it is obvious that, in theoretically describing produce experimental data over the whole range of heavy-ion fusion–fission reactions (where the afore- mass numbers. In view of this, our present con- mentioned parameters play a crucial role), it is nec- clusions concerning the simultaneous effect of an essary to take into account the effect of boththe increase in the temperature and angular momen- temperature and the rotation of a nascent compound tum of a fissile nucleus are only qualitative. In or- nuclear system. In the majority of cases, the effect of der to perform a more detailed quantitative com- the angular momentum proves to be more sizable, but parison withexperimental data, one needs dynami- it weakens withincreasing temperature. cal calculations that would employ the temperature- As matter of fact, the liquid-drop-model version dependent macroscopic model featuring finite-range used here, which involves the coefficients that were nuclear forces that has been studied here. That the determined on the basis of microscopic calculations, application of this model in our future dynamical remains macroscopic. Its applicability range, as well calculations would be quite straightforward explains as the applicability range of any other macroscopic our attention to it and partly justifies the qualitative

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 STATIC AND STATISTICAL PROPERTIES 1539 character of the results that we obtained for the static 27. J. Toke¨ and W. J. Swiatecki, Nucl. Phys. A 372, 141 and statistical features of ﬁssile nuclei. (1981). 28. W. D. Myers and W. J. Swiatecki, Ark. Fys. 36, 343 ACKNOWLEDGMENTS (1967). We are grateful to A.V. Karpov and P.N. Nad- 29. A. E. S. Green, Nuclear Physics (McGraw-Hill, New tochy for stimulating discussions and generous as- York, 1985), p. 185. sistance in writing some of the computational pro- 30. S. Cohen, F. Plasil, and W. J. Swiatecki, Ann. Phys. grams used. We are also indebted to A.Ya. Rusanov (N.Y.) 82, 557 (1974). for enlightening comments and for carefully reading 31. F. Haddad and G. Royer, J. Phys. G 21, 1357 (1995). the manuscript. 32.J.Blocki,J.Randrup,W.J.Swiatecki,and C. F. Tsang, Ann. Phys. (N.Y.) 105, 427 (1977). REFERENCES 33. M. G. Itkis, V. N. Okolovich, A. Ya. Rusanov, and 1. M. Diebel, K. Albrecht, and R. Hasse, Nucl. Phys. A G. N. Smirenkin, Fiz. Elem.´ Chastits At. Yadra 19, 355, 66 (1981). 701 (1988) [Sov. J. Part. Nucl. 19, 301 (1988)]. 2. M. Pi, X. Vinas,˜ and M. Barranco, Phys. Rev. C 26 ´ 21 (2), 733 (1982). 34. J. O. Newton, Fiz. Elem. Chastits At. Yadra , 821 (1990) [Sov. J. Part. Nucl. 21, 349 (1990)]. 3. M. Brack, C. Guet, and H. B. Hakansson,˚ Phys. Rep. 123, 275 (1985). 35. E. M. Rastopchin, Yu. B. Ostapenko, M. I. Svirin, and 4. C. Guet, E. Strumberger, and M. Brack, Phys. Lett. G. N. Smirenkin, Yad. Fiz. 49, 24 (1989) [Sov. J. Nucl. B 205, 427 (1988). Phys. 49, 15 (1989)]. 5. H.R.Jaqaman,Phys.Rev.C40, 1677 (1989). 36. G. A. Pik-Pichak, Physics of Nuclear Fission 6. G. X. Dai, J. B. Natowitz, R. Wada, et al.,Nucl.Phys. (Gosatomizdat, Moscow, 1962) [in Russian]. A 568, 601 (1994). 37. A. V. Ignatyuk, G. N. Smirenkin, M. G. Itkis, et al., 7. F.Garcias, M. Barranco, H. S. Wio, et al.,Phys.Rev. Fiz. Elem.´ Chastits At. Yadra 16, 709 (1985) [Sov. J. C 40, 1522 (1989). Part. Nucl. 16, 307 (1985)]. 8. F. Garcias, M. Barranco, J. Nemeth, et al.,Nucl. 38. A. S. Iljinov, M. V. Mebel, et al.,Nucl.Phys.A543, Phys. A 495, 169 (1989). 517 (1992). 9. J. Bartel and P. Quentin, Phys. Lett. B 152B,29 (1985). 39. J.O.Newton,D.G.Popescu,andJ.R.Leigh,Phys. 10. G. Royer and J. Mignen, J. Phys. G 18, 1781 (1992). Rev. C 42, 1772 (1990). 11.J.Bartel,K.Mahboub,J.Richert,andK.Pomorski, 40.S.A.Karamyan,I.V.Kuznetsov,Yu.A.Muzychka, Z. Phys. A 354, 59 (1996). et al.,Yad.Fiz.6, 494 (1966) [Sov. J. Nucl. Phys. 6, 12. W. D. Myers and W. J. Swiatecki, Nucl. Phys. A 601, 360 (1967)]. 141 (1996). 41. R. F. Reising, G. L. Bate, and J. R. Huizenga, Phys. 13. P.Ring, Prog. Part. Nucl. Phys. 37, 193 (1996). Rev. 141 (3), 1161 (1966). 14. B. Nerlo-Pomorska, K. Pomorski, J. Bartel, and 42. W. U. SchroderandJ.R.Huizenga,Nucl.Phys.A¨ K. Dietrich, Phys. Rev. C 66, 051302 (2002). 502, 473c (1989). 15. B. M. Strutinsky, Nucl. Phys. A 95, 420 (1967); 122, 43. Yu. Ts. Oganessian and Yu. A. Lazarev, in Treatise on 1 (1968). Heavy Ion Science,Ed.byD.A.Bromley(Plenum, 16. H.J.Krappe,Phys.Rev.C59, 2640 (1999). New York, 1985), Vol. 4, p. 1. 17. H.J.Krappe,J.R.Nix,andA.J.Sierk,Phys.Rev.C 20, 992 (1979). 44. I. Halpern and V. M. Strutinsky, in Proceedings of 18. A. J. Sierk, Phys. Rev. C 33, 2039 (1986). the Second United Nations International Con- 19. A. V. Karpov, P. N. Nadtochy, E. G. Ryabov, and ference on the Peaceful Uses of Atomic Energy, G. D. Adeev, J. Phys. G 29, 2365 (2003). Geneva, Switzerland, 1957 (United Nations, Gene- 20. P. Frobrich¨ and I. I. Gontchar, Phys. Rep. 292, 131 va, 1958), p. 408. (1998). 45. V.M. Strutinskiı˘ ,Zh.Eksp.´ Teor. Fiz. 45, 1891 (1963) 21. M. Brack, J. Damgaard, A. S. Jensen, et al.,Rev. [Sov. Phys. JETP 18, 1298 (1963)]. Mod. Phys. 44, 320 (1972). 46.A.Ya.Rusanov,V.V.Pashkevich,andM.G.Itkis, 22.A.Mamdouh,J.M.Pearson,M.Rayet,andF.Ton- Yad. Fiz. 62, 595 (1999) [Phys. At. Nucl. 62, 547 deur, Nucl. Phys. A 644, 389 (1998). (1999)]. 2 3 . A . V. K a r p o v, P. N . N a d t o c hy, D . V. Va n i n , a n d 47. M. G. Itkis and A. Ya. Rusanov, Fiz. Elem.´ Chastits 63 G. D. Adeev, Phys. Rev. C , 054610 (2001). At. Yadra 29, 389 (1998) [Phys. Part. Nucl. 29, 160 24.A.V.Ignatyuk,M.G.Itkis,V.N.Okolovich,et al., (1998)]. Yad. Fiz. 21, 1185 (1975) [Sov. J. Nucl. Phys. 21, 616 (1975)]. 48. M. G. Itkis, Yu. A. Muzychka, Yu. Ts. Oganessian, 25. A. V. Ignatyuk, Statistical Properties of Excited et al.,Yad.Fiz.58, 2140 (1995) [Phys. At. Nucl. 58, 2026 (1995)]. Nuclei (Energoatomizdat,´ Moscow, 1983). 26. R. Balian and C. Bloch, Ann. Phys. (N.Y.) 60, 401 4 9 . V. V. Pa s hk e v i c h, N u c l . P hy s . A 169, 275 (1971). (1970). Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1540–1547. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1599–1606. Original Russian Text Copyright c 2005 by Bogdanov, Shablya, Vokal, Kosmach, Plyuschev. NUCLEI Theory

Interaction of Gold Nuclei with Photoemulsion Nuclei at Energies in the Range 100–1200 MeV per Nucleon and Cascade–Evaporation Model

S. D. Bogdanov1), E. Ya. Shablya1)*,S.Vokal2),V.F.Kosmach1),andV.A.Plyuschev3) Received September 9, 2004; in ﬁnal form, March 10, 2005

Abstract—The interaction of gold nuclei with photoemulsion nuclei at energies in the range 100– 1200 MeV per nucleon was studied experimentally. A consistent comparison of the experimental data obtained in this way with the results of the calculations based on the cascade–evaporation model is performed. c 2005 Pleiades Publishing, Inc.

INTRODUCTION 197Au nuclei accelerated at the Bevalac accelerator (Berkeley, USA) to an energy of 1147.2 MeV per nu- On the basis of the nuclear-photoemulsion method, cleon. In the irradiation, the mean ﬂuence ranged be- the propagation of heavy (gold) nuclei through a ho- tween 500 and 2000 particlesper 1 cm 2. The detailsof mogeneousmedium isstudiedhere under conditions the development and primary treatment of the results of a complete experiment in the energy range 100– obtained in the chamber are given elsewhere [6, 7]. 1200 MeV per nucleon. Also, the features of the interactionsbetween 197Au nuclei and photoemulsion By using the method of fast and slow scanning, nuclei are calculated within the cascade–evaporation 1122 trackswere traced from their entrance into the model [1], thisincluding an analysisof the multiplici- emulsion to their stop or interaction. As a result, 197 tiesof secondaryparticlesversusthe basicproperties 585 interactionsof Au nuclei over the energy range of the process. Further, our experimental data are 0–1147 MeV per nucleon were found over a length compared with the results of the calculations. of 30 m. Among these, we processed 332 inelas- 197 A similar investigation of the applicability of the tic interactionsof Au nuclei whose mean energy cascade–evaporation model to describing the inter- ranged between 100 and 1147 MeV per nucleon. The actionsof lighter nuclei (Ne, Ar, Fe) with photoemul- procedure employed here to process interactions was sion nuclei in the same energy range was previously described in [8, 9]. In processing data on projectile performed in [2–5]. disintegration, we were able to determine the charges of all projectile fragmentsand their polar and azimuthal emission angles. EXPERIMENT AND CALCULATIONS For a sample comprising 108 inelastic interactionsof 197Au nuclei whose mean energy fell within In order to obtain quantitative results, we em- ± ployed a photoemulsion chamber formed by 34 lay- the range 741.2 19.8 MeV per nucleon, we mea- ers of BR-2 photoemulsion having a standard com- sured the production angles, ionization losses, and position (3.148 × 1022 H, 1.412 × 1022 C, 0.396 × free paths of all charged secondaries (the sample included 4444 particles). 1022 N, 0.956 × 1022 O, 0.004 × 1022 S, 0.002 × 1022 I, 1.031 × 1022 Br, and 1.036 × 1022 Ag nuclei per In addition, we processed 173 nonrelativistic in- 197 1cm3). The dimensions of each layer were close to teractionsof Au nuclei where the atomic number 10 × 10 × 0.05 cm3. The chamber wasirradiated with of a fragment in excess of 20 was determined by the residual free path of the fragment [8], measurements 1)St. Petersburg State Polytechnical University, Politekh- for other charged particlesnot being performed. nicheskaya ul. 29, St. Petersburg, 195251 Russia. The calculated statistical sample was obtained by 2) Joint Institute for Nuclear Research, Dubna, Moscow generating the interactionsbetween 197Au nuclei of ˇ oblast, 141980 Russia; University of P.J. Safarik,´ Jesenna5,´ energy 700 MeV per nucleon and individual emulsion SK-041 54 Kosice,ˇ Slovak Republic. 1 12 14 16 80 107 3)Khlopin Radium Institute, Vtoroı˘ Murinskiı˘ pr. 28, St. Pe- nuclei ( H, C, N, O, Br, Ag) on the ba- tersburg, 194021 Russia. sis of the Monte Carlo method and by subsequently *e-mail: [email protected] performing summation with weights corresponding

1063-7788/05/6809-1540$26.00 c 2005 PleiadesPublishing,Inc. INTERACTION OF GOLD NUCLEI WITH PHOTOEMULSION NUCLEI 1541 to the calculated cross sections and the composi- 〈L〉, cm tion of the emulsion. In doing this, we took into account meson-production processes, a Lorentz con- 11 traction, and the effect of the Pauli exclusion principle at the fast (first) stage and the effect exerted by the change in the nuclear-matter density in the course of cascade propagation. The deexcitation of nuclear 9 residues after the completion of the fast stage (the second stage, which is slow) is described on the basis of the statistical model. The total statistical sam- 7 ple obtained within the cascade–evaporation model comprised 1000 events. 5 INELASTIC-INTERACTION FREE PATHS OF 197Au NUCLEI 3 On the basis of our experimental measurements, 0 200 400 600 800 1000 inelastic-interaction free paths of 197Au nuclei in E, MeV/nucleon BR-2 nuclear photoemulsion were determined in the energy ranges100 –400, 400–900, 900–1200, and Fig. 1. Inelastic-interaction mean free paths L of 197Au 100–1200 MeV per nucleon. The results obtained in nuclei in emulsion versus the projectile energy E:(closed this way are given in Fig. 1. Also shown in this figure boxes) our present experimental data, (open circles) experimental data from [10], (solid curve) results of the cal- are experimental results reported in [10], where Ilford- culation based on the model proposed in [11], and (dash- G5 emulsions and 197Au nuclei were used for targets dotted line) results of the calculation within the model and projectiles, respectively. considered in [12]. Among a great variety of modelsfor calculating cross sections for inelastic nucleus–nucleusinterac- MULTIPLICITIES OF SECONDARY tions, we choose the Karol model [11] and the Bradt– PARTICLES Petersmodel [12], according to which we calculated the cross sections for the interaction of photoemul- For a further analysis of our data, all secondaries sion nuclei with 197Au nuclei and the corresponding were partitioned into the following groups: inelastic-interaction free paths for BR-2 photoemul- (i) the group of (g + s) particlesde fined as first- sion in the energy range between 100 and 1200 MeV stage particles, which comprise singly charged target per nucleon. In thiscalculation, we took into account fragmentsof energy E ≥ 26 MeV per nucleon, prod- the change in the energy of the nuclei asthey traverse uct mesons, and singly charged projectile fragments emulsion layers and the corresponding change in the whose transverse momentum exceeds 222.6 MeV/c mean cross section for nucleon–nucleon interactions. (thiscorrespondsto E ≥ 26 MeV per nucleon in the From the experimental data and from the results of transverse direction); the calculations(seeFig. 1), it followsthat, although (ii) the group of b particlesde fined astarget frag- the cross sections for nucleon–nucleon interactions mentsof energy in the range E ≤ 26 MeV per nu- exhibit a pronounced energy dependence in the en- cleon (their free path isshorterthan 3 mm), which are ergy range being considered, the inelastic-interaction emitted predominantly at the second reaction stage; 197 free pathsof Au nuclei in a nuclear photoemulsion (iii) the group of (s + g + b ) particleshaving are rather weakly dependent on the projectile energy. energiesin the region E ≥ 26 MeV per nucleon and The experimental data obtained here and in [10] prove transverse momenta below 222.6 MeV/с per nucleon, to be quite close to each other if one takes into account the distinctions between the composition and thisgroup including singly ( s ), doubly (g ), and mul- ≥ densities of BR-2 and Ilford-G5 emulsions. The cal- tiply (b , Z 3) charged projectile fragments. culationslead to larger crosssectionsfor the interac- In each interaction event, we determined the mul- tions of gold nuclei and, consequently, to shorter free tiplicitiesof particlesbelonging to the above groups, paths. The observed discrepancies between the cal- the total number N of charged secondaries, and their culated and measured results may be due to missing total charge Zint.ch. (in electron-charge units). This few-prong eventsin the experimentsor to employ- made it possible to obtain particle-number distribu- ing inaccurate valuesof adjustableparametersin the tionsin disintegration eventsinduced by gold ions above models or to both of these factors. of energy 0.741 GeV per nucleon and correlations

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1542 BOGDANOV et al.

Table 1. Multiplicity (in number of particlesper interaction event) of charged secondariesproduced in the interactions between 197Au nuclei of mean energy 0.741 GeV per nucleon and photoemulsion nuclei

Interaction N N N N Z type g+s b s +g +b int.ch. 197Au + Em 21.71 ± 0.45 3.24 ± 0.17 16.24 ± 0.39 41.19 ± 0.62 91.97 ± 0.92 (19.06) (3.94) (13.04) (36.03) (95.08) 197Au + H 0.82 ± 0.27 0.09 ± 0.09 6.27 ± 0.76 7.18 ± 0.81 80.00 ± 2.69 (1.34) (0.02) (4.14) (5.49) (80.02) 197Au + CNO 13.39 ± 0.57 0.49 ± 0.11 20.51 ± 0.71 34.39 ± 0.92 82.51 ± 1.42 (10.74) (1.75) (15.26) (27.75) (87.56) 197Au + AgBr 31.91 ± 0.76 5.88 ± 0.32 15.07 ± 0.52 52.86 ± 0.97 101.25 ± 1.34 (35.52) (7.85) (16.33) (59.70) (118.40) between the numbersof di fferent-type particlesin an experimental counterparts, describing the increase in interaction event. the particle multiplicity and in Zint.ch. with increas- Table 1 givesthe mean multiplicitiesof secondary ing mass of colliding particles. particles, Ng+s, Nb,andNs+g+b ,andthe A comparison of our experimental data with the mean multiplicity of all charged secondaries, N,in results reported in [2, 3] revealed that the mean mul- the interactions of nuclei having various masses and tiplicity of b particlesisvirtually independent of the energies,aswell asthe mean value of the total charge projectile mass in the mass range 20–197 amu. For in an interaction event, Zint.ch..Wenotethatthe example, the increase in the projectile mass by a fac- statistical errors in the values calculated on the basis tor of 10—from that of 20Ne to that of 197Au—leads of the cascade–evaporation model (these values are to only a 15% increase in the multiplicity of b particles given in parentheses) are approximately three times in the interaction with heavy emulsion nuclei, the less than the experimental errors; therefore, they are multiplicity of b particlesin the interactionswith light not presented here. emulsion nuclei even decreasing. From Table 1 and from the data obtained previ- In order to clarify the possible reasons behind this ously for the interactions of lighter nuclei (Ne, Ar, behavior of the multiplicity of b particles, we plotted Fe) [2–5], it follows that an increase in the mass the mean multiplicity of b particlesasa function of of colliding particleshasthe greateste ffect on the the multiplicity of (g + s) particlesin disintegration number of particlesproduced at the first stage of eventsinduced by the interaction between 197Au nu- the process. For interactions of energy below 1 GeV clei of energy 0.741 GeV per nucleon and photoemul- per nucleon, the mean multiplicity of (g + s)parti- sion nuclei (see Fig. 2). Figure 2 also displays the cles is proportional to the product of the masses of results of the calculations based on the cascade– interacting nuclei that is raised to a power close to evaporation model that were performed for 197Au nu- 0.7 0.7, Ng+s∼(AprAtar) . Also, the total interaction clei of energy 0.7 GeV per nucleon and the results 20 charge Zint.ch. increases with increasing dimensions presented in [2, 3] for the interactions of Ne and 40 of interacting nuclei. We note that Zint.ch. doesnot Ar nuclei whose energies were, respectively, 0.28 include the target-nucleuscharge, which wasnot and0.27GeVpernucleon. detected in our experiment. The masses of interacting From the data in Fig. 2, it followsthat, up to a nuclei have a weaker effect on the mean multiplicity of value of N ≈ 40, N growsalmostlinearly with all charged secondaries, N,thanonN .Forin- g+s b g+s N in the interactions of gold nuclei. The emission teractionsof energy below 1 GeV per nucleon, N is g+s 0.5 of each of the (g + s) particles(a particle that es- approximately proportional to (AprAtar) . caped from the nucleus under consideration at the fast As can be seen from Table 1, the results obtained stage) leads, on average, to an additional emission of by the Monte Carlo method for the mean multiplici- 0.2 to 0.3 of a b particle. In the region Ng+s > 40,a tiesof ( g + s), b,and(s + g + b) particlesand for the decrease in the mean multiplicity of b particleswith mean multiplicity of all charged secondaries, as well increasing multiplicity of (g + s) particlesisobserved asfor the total interaction charge, are closeto their in disintegration events involving heavy target nuclei.

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〈 〉 Nb 16

0 0 40 80 120 Ng + s

Fig. 2. Experimental correlationsbetween the mean multiplicity of b particlesand the multiplicity of fast( g + s)particles in disintegration events induced by the interactions of (closed boxes) 197Au nuclei of energy 0.741 GeV per nucleon, (open circles) 20Ne nuclei of energy 0.28 GeV per nucleon, and (open triangles) 40Ar nuclei of energy 0.27 GeV per nucleon with photoemulsion nuclei. The histogram represents the respective correlation calculated on the basis of the cascade–evaporation model for 197Au nuclei of energy 0.7 GeV per nucleon.

Experimentswith light projectilesof 20Ne and 40Ar light nuclei decreasesasone goesover from neon to exhibited an increase in Nb with increasing Ng+s; gold projectilescan alsobe explained by the finiteness however, the respective correlation dependences did effect. Upon undergoing a collision with gold nucle- not show (see Fig. 2) a decrease in Nb as Ng+s ons, the bulk of the nucleons of light nuclei go over to growsfurther sincethere were no interactionschar- theenergyregionabove26MeVpernucleon. acterized by Ng+s > 40. We also note that, in the The histogram in Fig. 2 represents the results of interaction of 660-MeV protonswith heavy emul- the calculations based on the cascade–evaporation sion nuclei, the number of emitted (g + s) particles model. One can easily see that this model provides doesnot exceed three, the mean multiplicity being a fairly good description of the complicated experi- 1.03 particlesper interaction event [12]. mental dependence Nb (Ng+s), including the exper- The simplest way to explain the observed form of imentally observed effect of finiteness of a heavy target nucleus. the Nb(Ng+s) correlation, which characterizesthe interplay of the slow and the fast reaction stage, is to take into account effects that are associated with TARGET-NUCLEUS FRAGMENTATION the “finiteness” of a heavy target nucleus. If the heavy For each target-nucleusfragment ( b particle), we target nucleusof 107Ag loses 35 to 47 protons at the determined here the polar (θ) and azimuthal (ψ)emis- first reaction stage, there will be no sizable target- sion angles and its free path in the photoemulsion. nucleus residue, with the result that the multiplicity of Under the assumption that all b particlesare protons, b particles will not be large. Previously, the observed we found angular, energy, and momentum properties effect of finitenessof a heavy target nucleuswasnot of each b particle. The mean valuesof the multiplic- found in the projectile-mass range between 2 and ity ( Nb ), polar emission angle, ratio of the num- 56 amu, since the required number of nucleons could ber of particlesemitted into the forward hemisphere not be knocked out of a heavy photoemulsion nucleus to the number of particlesemitted into the back- because of small projectile dimensions. In our opin- ward hemisphere (forward/backward), and longitudi- ion, the experimentally observed fact that the mean nal and transverse momenta (P|| and P⊥,respec- multiplicity of b particlesin interactionsinvolving tively) are given in Table 2 for the interactionsof gold

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Table 2. Propertiesof target fragmentsof energy in the range E ≤ 26 MeV per nucleon that are emitted at the second reaction stage

Target nucleus Nb,part./int. θ,deg Forward/backward E,MeV/nucleon P||,MeV/c Р⊥,MeV/c Au, E =0.741 GeV/nucleon Em 3.24 ± 0.17 77.81 ± 1.91 1.76 ± 0.09 9.23 ± 0.37 26.26 ± 3.92 94.96 ± 2.62 (3.94) (80.03) (1.59) (9.72) (19.6) (94.29) C, N, O 0.49 ± 0.11 64.63 ± 5.14 5.67 ± 0.58 5.25 ± 1.15 36.76 ± 9.31 71.92 ± 9.15 (1.76) (84.30) (1.38) (7.24) (13.67) (79.02) Ag, Br 5.88 ± 0.32 78.71 ± 2.00 1.65 ± 0.09 9.47 ± 0.38 25.43 ± 4.12 96.36 ± 2.71 (7.85) (79.32) (1.62) (10.14) (20.54) (96.93)

Au, Eint ≤ 0.873 GeV/nucleon Ag, Br 5.53 ± 0.43 75.85 ± 0.23 1.77 ± 0.13 9.92 ± 0.04 30.31 ± 0.47 96.54 ± 0.30

Au, Eint > 0.873 GeV/nucleon Ag, Br 6.27 ± 0.49 81.62 ± 0.21 1.55 ± 0.13 9.01 ± 0.04 20.46 ± 0.43 96.17 ± 0.30 nuclei at an average energy of 741 MeV per nucleon. emission angle decreases by 20%. Thise ffect waspre- The resultsof the calculationswithin the cascade – viously observed in [2–5], where it wasexplained by evaporation model are presented in parentheses. the contribution of first-stage particles to the energy From Table 2, one can see that a change in the region E<26 MeV per nucleon, thiscontribution target mass by nearly a factor of 7 (in going over increasing as the projectile energy decreases. from С, N, and O to Ag and Br) leadsto an increase The measurements of the free paths and of the in the multiplicity of b particlesby almosta factor azimuthal and polar emission angles in our experi- of 12, the energy by a factor of 1.8, and the trans- ment made it possible to determine the 3-momentum verse momentum by a factor of 1.3. Concurrently, the componentsfor each of the b particles. For our anal- forward orientation becomeslesspronounced —the ysis, we choose Pz(P||),whichisthemomentum forward/backward ratio decreases by almost a factor component along the direction of projectile motion, of 3.5, P|| decreases by a factor of 1.5, and the polar and Px(P⊥ cos ψ), which isthe projection of the mo- emission angle increases by a factor of 1.2. mentum onto an axisin the azimuthal plane that is The results of the calculations within the cascade– orthogonal to the direction of projectile motion. evaporation model are in satisfactory agreement with In Fig. 3, we present the results obtained for the experimentally determined propertiesof b parti- the distributions of b particles with respect to these clesin interactionswith heavy photoemulsionnu- 3-momentum componentsfrom our measurements clei, the discrepancies being within 5 to 10%. The (Figs. 3a,3c) and from the calculationswithin the calculationsdescribe qualitatively the dependencesof cascade–evaporation model (Figs. 3b,3d). Both the multiplicity, energy, and transverse momentum of the experimental and the measured distributions are low-energy protonson the target-nucleusmass. close to Gaussian distributions, this giving sufficient In order to analyze the effect of the projectile ve- grounds to consider a system that emits particles locity on the propertiesof b particles, the entire body isotropically and which has a characteristic temper- of data on interactionswith heavy photoemulsionnu- atureandmovesataspecificvelocity.Theeffective clei waspartitioned into two groupswhere the in- temperature of the system emitting particles can be estimated by assuming that the distribution of singly teraction energieswere Eint 873 MeV per nucleon charged particleswith respectto each 3-momentum and Eint > 873 MeV per nucleon (samples containing identical numbersof b particles). The properties of b component is described by a Gaussian distribution 0.5 particles belonging to these samples are also given in characterized by the parameter σ =(2/π) P⊥, Table 2. One can see that, with decreasing projectile the variance being related to the temperature by 2 velocity, the forward orientation of b particlesbe- the equation T0 = σ /m. Both for the experimental comesmore pronounced, the longitudinal momentum eventsand for their counterpartscalculated within the component increasing by a factor of 1.5. The mean cascade–evaporation model, the estimates performed

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N N 50 500 40 (a) 400 (b) 30 300 20 200 10 100 0 0 –250 –150 –50 50 150 250 –250 –150 –50 50 150 250 Pz, MeV/Ò per nucleon N N 50 500 40 (c) 400 (d) 30 300 20 200

10 100 0 0 –250 –150 –50 50 150 250 –250 –150 –50 50 150 250 Px , MeV/Ò per nucleon

197 Fig. 3. Distributions of b particleswith respectto two 3-momentum components( Pz, Px) in the interactionsof Au nuclei with photoemulsion nuclei at an average energy of 0.741 GeV per nucleon. The histograms represent experimental data in Figs. 3a and 3c and the results of the calculations within the cascade–evaporation model in Figs. 3b and 3d.Thedashed curves correspond to approximations by a Gaussian distribution. for the interaction with heavy nuclei yield a value the experimental data. By way of example, we indicate of about 6 MeV for the effective particle-emission that, in the first region, the adequate calculated data temperature and a value of β|| =0.025 of the speed for our statistical sample (1000 events) at a projectile of light for the velocity in question. energy of about 700 MeV per nucleon should have been only 180 fragmentsinsteadof 590 actually obtained in the calculation—that is, the discrepancy in PROJECTILE-NUCLEUS FRAGMENTATION thisregion of fragment atomic numbersisasgreat as a factor of 3. In the region Z =3−20, the experimen- For the interactionsof gold nuclei with pho- tal data differ from the resultsof the calculationsby a toemulsion nuclei at an average energy of 741 MeV factor of 18. per nucleon, Table 3 givesthe mean multiplicities The data that we obtained show that the model of doubly and multiply charged fragments( N g applied here isunable to take adequately into account and Nb , respectively), the mean multiplicities of the multifragmentation and fission of excited heavy multiply charged fragmentsin variouscharge inter- ≥ nuclear residues. As a result, the calculations in- vals(columns4 –6), and the mean charge of Z 3 volve virtually no interactions leading to the emission fragments( Zfr ). The respective values obtained of two or more heavy fragments. The mean mul- from the calculationswithin the cascade –evaporation tiplicity of heavy fragmentsof the incident nucleus model are presented in parentheses. 197Au was1.7 to 2.7 particlesper interaction event in From an analysis of the data in this table on the the experiment (see Table 3), whereas its calculated multiplicity of b particles, one can easily see that there value amounted to unity. An increase in the proba- are two regionsboth in the calculated data and in bility of the multifragmentation and fission of excited the experimental data at E ≈ 700 MeV per nucleon: projectile fragmentswithin the model would lead to these are the region of fragment charges close to the an increase in the multiplicity of fragments in the projectile charge (Z =61−79), where the calculated range Z =3−20 and would increase, by and large, valuesexceed considerablytheir experimental coun- the multiplicity of Z ≥ 3 fragmentsin the interactions terparts, and the fragment-charge region Z =3−20, of nuclei, and this would make it possible to describe where the calculation predictsfragment yieldsbelow the experimental data more adequately.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1546 BOGDANOV et al.

Table 3. Multiplicity of multiply charged projectile fragmentsformed in the interaction of 197Au nuclei with photoemulsion nuclei at an average energy of 0.741 GeV per nucleon

Target Ng , Nb , Intervalsin Z Zfr,el.ch. nucleus part./int. part./int. 3–20 21–60 61–79

Em 4.76 ± 0.21 2.52 ± 0.15 1.74 ± 0.13 0.62 ± 0.08 0.16 ± 0.04 18.79 ± 0.26

(1.97) (1.00) (0.08) (0.35) (0.57) (58.10)

H 2.27 ± 0.46 1.73 ± 0.40 0.27 ± 0.16 1.09 ± 0.31 0.36 ± 0.18 41.53 ± 1.48

(0.91) (1.00) (0.00) (0.03) (0.97) (74.70)

C, N, O 5.83 ± 0.38 3.37 ± 0.29 2.68 ± 0.26 0.63 ± 0.12 0.05 ± 0.03 13.38 ± 0.31

(2.52) (1.00) (0.06) (0.39) (0.56) (57.66)

Ag, Br 4.46 ± 0.28 2.05 ± 0.19 1.34 ± 0.15 0.52 ± 0.10 0.21 ± 0.06 21.51 ± 0.43

(2.14) (1.00) (0.14) (0.51) (0.35) (49.04)

In order to separate fission events from the whole effective temperature of about 6 MeV and movesat array of processed events of 197Au disintegration at avelocityofβ|| =0.025 of the speed of light. an energy of 0.741 GeV per nucleon, we selected In studying projectile fragments, we have found four events that involved at least two fast projectile that the main discrepancy between the calculated ≥ fragmentscarrying Zfr 20 elementary charge units and experimental data consists in that the calcula- each. Thus, the ratio of the cross section for the for- tion underestimates significantly the probability of the mation of two massive fragments to the total inelastic disintegration of a fast nucleus into several Z ≥ 3 cross section was 0.037 ± 0.010, which iscloseto the fragments, with the result that the charge spectra of 197 probability of the binary fission of Au nuclei that is 197Au fragments appear to be distorted. Otherwise, induced by high-energy protons(3 –5% [12]). the results of the calculations within the cascade– evaporation model describe qualitatively and quantitatively (to within 10 to 15%) our experimental data, CONCLUSION including correlation dependencesand the angular, New experimental and calculated data have been energy, and momentum featuresof charged secon- obtained for inelastic-interaction free paths of 197Au dariesversusthe target massandparticle type. nuclei in photoemulsion and for the multiplicities of various charged secondaries produced in the inelastic ACKNOWLEDGMENTS interaction of gold nuclei with photoemulsion nuclei We are grateful to the personnel of the Bevatron at energiesin the range 100 –1147 MeV per nucleon, in Berkeley and the staff of the Laboratory of High aswell asfor the angular, energy, and momentum Energies at the Joint Institute for Nuclear Research properties of these particles. The fragmentation of (JINR, Dubna), who prepared photoemulsion data target and projectile nuclei hasbeen considered.The and placed them at our disposal. propertiesof the interaction have been analyzed versus the energy of interacting nuclei. For particles emitted at the slow stage, we have REFERENCES discovered the effect of finiteness of heavy photoemul- 1. S. D. Bogdanov, S. S. Bogdanov, V. F. Kosmach, sion nuclei in their interaction with gold ions. This et al., Izv. Akad. Nauk, Ser. Fiz. 60, 132 (1996). 2. V. E. Dudkin, E. E. Kovalev, N. A. Nefedov, et al., effect was not observed previously in the projectile- Nucl. Phys. A 568, 906 (1994). mass range between 2 and 56 amu. The behavior 3. S. D. Bogdanov, V. F. Kosmach, V. M. Molchanov, of particles originating at the second stage of the and V. A. Plyushchev, Izv. Akad. Nauk, Ser. Fiz. 63, process from target nuclei can be described quite 427 (1999). accurately within the model of a thermalized source 4. V.A.Bakaev,S.D.Bogdanov,S.S.Bogdanov,et al., that emitsparticlesisotropically and which hasan Izv. Akad. Nauk, Ser. Fiz. 64, 2144 (2000).

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 INTERACTION OF GOLD NUCLEI WITH PHOTOEMULSION NUCLEI 1547

5. S.D.Bogdanov,S.S.Bogdanov,V.E.Dudkin,et al., 9. V.A.Bakaev,S.D.Bogdanov,S.S.Bogdanov,et al., Nucl. TracksRadiat. Meas. 25, 111 (1995). Poverkhnost, No. 4, 45 (2004). 6. S. D. Bogdanov, S. S. Bogdanov, E. E. Zhurkin, and 10. C. J. Waddington and P. S. Freier, Phys. Rev. C 31, V. F. K o sm a c h , Z h . Eksp.´ Teor. Fiz. 115, 404 (1999) 888 (1985). [JETP 88, 220 (1999)]. 11. P.J. Karol, Phys. Rev. C 11, 1203 (1975). 7. V.A. Bakaev, S. D. Bogdanov, S. S. Bogdanov, et al., 12.V.S.BarashenkovandV.D.Toneev,Interactions Surf. Inv. 13, 681 (1998). of High-Energy Particles and Nuclei with Nuclei 8. V.A. Bakaev, S. D. Bogdanov, S. S. Bogdanov, et al., (Atomizdat, Moscow, 1972) [in Russian]. Poverkhnost, No. 4, 41 (2003). Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1548–1550. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1607–1610. Original Russian Text Copyright c 2005 by Bazarov, Glagolev, Gulamov, Lutpullaev, Olimov, Khamidov, A. Yuldashev, B. Yuldashev. ELEMENTARY PARTICLES AND FIELDS Experiment

Some Special Features of the Momentum Spectrum of Protons Produced in 16Оp Collisions at a Momentum of 3.25 GeV/с per Nucleon

E.Kh.Bazarov,V.V.Glagolev1),K.G.Gulamov,S.L.Lutpullaev, K. Olimov*, Kh. Sh. Khamidov2), А.А.Yuldashev,andB.S.Yuldashev2) Institute for Physics and Technology, Fizika–Solntse Research and Production Association, Uzbek Academy of Sciences, ul. Timiryazeva 2b, Tashkent, 700084 Republic of Uzbekistan Received August 10, 2004; in ﬁnal form, December 2, 2004

Abstract—The momentum features of protons originating from 16Оp collisions at a momentum of 3.25 GeV/с per nucleon are analyzed.It is shown that the degree of excitation of the fragmenting nucleus aﬀects predominantly the shape of the momentum spectrum of protons emitted into the backward hemisphere in the rest frame of the projectile nucleus and partly the shape of the spectra of protons emitted into the forward hemisphere and formed via the mechanisms of Fermi breakup and evaporation. c 2005 Pleiades Publishing, Inc.

It is well known that the formation of light frag- of either the energy or the mass number of the frag- 1 2 3 3 ments like Н1, H1, H1,and Не2 is a feature pecu- menting nucleus. liar to the nuclear-fragmentation process [1].In par- The present study, which is a continuation of the ticular, it was proven experimentally that the bulk analysis reported in [2], is devoted to a more detailed of protons are formed at the initial, fast, stage of exploration of correlations between the properties of high-energy hadron–nucleus and nucleus–nucleus the momentum spectrum of protons and the degree of interactions.At the same time, the cross section for excitation of the oxygen nucleus. nucleon production appeared to be commensurate The experimental data subjected to the present with the inelastic nuclear-reaction cross section.In analysis were obtained by exposing the 1-m hydro- view of this, more information about the dynam- gen bubble chamber of the Laboratory for High En- ics of the nuclear-fragmentation process can be de- ergies at the Joint Institute for Nuclear Research duced from an analysis of the production of light (JINR, Dubna) to a beam of 16О nuclei accelerated fragments than from an analysis of the production at the Dubna synchrophasotron to a momentum of of heavy fragments.Since it is quite straightforward 3.25 GeV/c per nucleon.This data sample consisted to identify secondary protons and to measure their 16 kinematical features, they have been explored fairly of 11 098 measured Оp events.The methodological well.Nevertheless, correlations in the production of aspects of our experiment are described in [3–5]. ≥ protons and multiply charged fragments in relativistic The total charge of multiply charged (zf 2)frag- hadron–nucleus collisions have not yet received ade- ments ( zf ) can be considered as a measure of the quate study. degree of excitation of the fragmenting oxygen nu- In [2], it was shown that the shape of the momen- cleus.In this connection, it is of interest to explore the eﬀect of the degree of excitation of a fragmenting nu- 16 p tumspectrumofprotonsproducedin О collisions cleus on the momentum features of protons.We will at a momentum of 3.25 GeV/с per nucleon depends now consider the dependence of the mean momentum on the degree of excitation of the fragmenting oxygen of protons emitted into the forward and backward nucleus.It was also shown there that the mecha- hemispheres (forward and backward protons, respec- nism of the production of relatively fast protons (p> tively) in the rest frame of the oxygen nucleus on 0.45 GeV/c) traveling in the forward hemisphere in the total charge of multiply charged fragments.This the rest frame of the projectile nucleus is independent dependence is shown in Fig.1.From Fig.1 a,one can see that, within the statistical errors, the mean 1) Joint Institute for Nuclear Research, Dubna, Moscow momentum of forward protons is everywhere (with oblast, 141980 Russia. z =6 2)Institute of Nuclear Physics, Uzbek Academy of Sciences, the exception of the point at f ) independent pos.Ulughbek, Tashkent, 702132 Republic of Uzbekistan. of zf —that is, of the degree of excitation of the *E-mail: [email protected] fragmenting nucleus.The solid curve represents the

1063-7788/05/6809-1548$26.00 c 2005 Pleiades Publishing, Inc. SOME SPECIAL FEATURES OF THE MOMENTUM SPECTRUM 1549 result obtained (without allowance for the experimen- 〈p〉, MeV/Ò tal point at zf =6) by approximating the experi- 500 mental data by a function of the form (a) p = α. (1) 450 It can be seen that this approximation describes fairly ...... well the fact that the mean momentum of forward 400 protons is independent of the total charge of multiply charged fragments.The fitted value of the parameter α proved to be 435.9 ± 4.0 MeV/c at χ2 =0.74 per 350 five degrees of freedom, this corresponding to a 98% 300 confidence level.We also note that, within the statis- (b) tical errors, the standard deviations for the momen- 250 tum spectra of forward protons remain unchanged for these groups of events ( = 273.7 ± 2.9 MeV/c).If, in approximating the experimental data by expres- 200 sion (1), one includes the point at zf =6 (the ± dotted curve in Fig.1 a), then α = 425.0 3.0 MeV/c 150 at χ2 =14.96 per six degrees of freedom, this cor- 0 2 4 6 responding to a 2% confidence level.Thus, one can ∑z f conclude that the deviation of the experimental point at zf =6from p is significant in relation to other Fig. 1. Mean total momenta of (a)forwardand(b)back- values of the total charge of multiply charged frag- ward protons in the rest frame of the oxygen nucleus ments. versus the total charge, zf ,ofmultiplychargedfrag- ments. The observed decrease of about10% in the mean momentum of forward protons at zf =6in relation to p at other values of zf can be explained the momentum distributions decrease almost linearly; by a relatively large contribution of the evaporation that is, the momentum spectra in question become mechanism to the spectrum of forward protons in narrower. these events.This can be qualitatively interpreted as Here, it is also of interest to study the dependence follows.By virtue of the isotopic invariance of strong of the shapes of the forward- and backward-proton interaction, the probabilities of pp and pn interactions momentum spectra on the degree of excitation of the are identical since the projectile nucleus contains oxygen nucleus.Figure 2 shows the ratios R of the equal numbers of protons and neutrons.Further, the forward- and backward-proton momentum spectra processes of multiple intranuclear rescattering can be for various groups of events to the total spectra of, disregarded since events in which the total charge is respectively, forward and backward protons.The Ro- 16 6 or 7 are due to peripheral Оp collisions.In the man numerals there indicate the numbers of groups case where z =7, the observed proton is, with f corresponding to different values of zf .Because of identical probabilities, a product of either a direct scarce statistics of zf =0and zf =3events, knockout or evaporation.For the group of zf =6 they were combined with events characterized by events, the final state features two protons; of these, close total charges of multiply charged fragments— one is due to evaporation, while the other may be, with that is, with zf =2and zf =4events, respec- a probability of 0.5, a knock-on proton, as in the case tively.Thus, groups I, II, III, IV, and V contain, of zf =7.This is precisely the circumstance that − − respectively, zf =0 2, zf =3 4, zf =5, leads to an increased fraction of evaporated protons zf =6,and zf =7events. in the zf =6group and, accordingly, to a decrease in the mean momentum in this group. As follows from Fig.2 a,theratiosR display a From Fig.1 b, one can see that, with increasing significant momentum dependence only in the region total charge of multiply charged fragments, the mean p<0.25 GeV/c, which corresponds to the spectrum momentum of backward protons decreases mono- of evaporated protons.One can see that the depen- tonically, which is likely to be due to an increase in dence of R on p is linear; however, the slope of the the fraction of evaporated protons.It should be noted respective straight lines changes from negative values that the contribution of cascade protons to the spec- for groups I–III to positive values for groups IV and trum of backward protons can be disregarded.With V, r e flecting the dynamics of changes in the relative increasing total charge, the standard deviations for yields of protons of different origin versus momentum.

PHYSICS OF ATOMIC NUCLEI Vol.68 No.9 2005 1550 BAZAROV et al.

R, arb. units group III, R remains virtually unchanged, within the (a) statistical errors, over the entire momentum range; for groups IV and V, R decreases monotonically with 100 V increasing momentum. From the experimental data considered above, it IV follows that the degree of excitation of the fragmenting nucleus affects differently the shapes of the mo- III mentum spectra of forward and backward protons. A substantial effect of the degree of excitation of II the fragmenting nucleus on the shape of the mo-

–1 mentum spectrum of backward protons can be ex- 10 I plained by the origin of these protons.Protons emitted into the backward hemisphere are predominantly produced via evaporation and Fermi breakup, these 0 0.2 0.4 0.6 0.8 1.0 1.2 mechanisms being directly associated with the degree of excitation of the fragmenting nucleus, as is 101 V observed experimentally.In addition to these mech- (b) anisms, intranuclear-cascade processes make a significant contribution to the production of forward pro- IV tons.It is interesting to note that, although there is an interrelation between the degree of excitation of the fragmenting nucleus and the multiplicity of the intranuclear cascade, the former has no effect on the III shape of the momentum spectrum of protons of this II origin. 100 Thus, we can conclude that the degree of excitation of the fragmenting nucleus has a substantial I effect only on the shape of the momentum spectrum of backward protons and affects partly the shape of 0 0.1 0.2 0.3 0.4 0.5 the spectra of forward protons that do not originate p, GeV/Ò from a cascade.The observed decrease in the mean momentum of forward protons at zf =6in relation Fig. 2. Ratios R of (a) forward- and (b) backward-proton to other charge groups is due to an increased fraction momentum spectra in the rest frame of the oxygen nu- of evaporated protons. cleus for various charge groups of events to the total momentum spectra of, respectively, forward and backward protons. REFERENCES 1. V.I.Zagrebaev and Yu.E.Penionzhkevich, Fiz. Elem.´ In the region p>0.25 GeV/c, where there are vir- Chastits At.Yadra 24, 295 (1993) [Phys.Part.Nucl. tually no evaporated protons, R is independent of p 24, 125 (1993)]. within the statistical errors.Thus, one can conclude 2. E.Kh.Bazarov,´ V.V.Glagolev, K.G.Gulamov, et al., that the degree of excitation of the projectile nucleus Yad.Fiz. 67, 736 (2004) [Phys.At.Nucl. 67, 708 changes the shape of the spectrum of forward protons (2004)]. that do not originate from a cascade. 3. V.V.Glagolev, K.G.Gulamov, M.Yu. Kratenko, et al., From Fig.2 a, one can also see that the fraction of Pis’ma Zh. Eksp.Teor.Fiz.´ 58, 497 (1993) [JETP Lett. evaporated protons is maximal in group IV—that is, 58, 497 (1993)]. in the case where zf =6.The bulk of events in this 4. V.V.Glagolev, K.G.Gulamov, M.Yu. Kratenko, et al., group involve the formation of one sextuply charged Pis’ma Zh. Eksp.Teor.Fiz.´ 59, 316 (1994) [JETP Lett. fragment and three doubly charged fragments. 59, 336 (1994)]. The ratio R for backward protons displays an 5. V.V.Glagolev, K.G.Gulamov, M.Yu. Kratenko, et al., interesting momentum dependence (Fig.2 b).For Yad.Fiz. 58, 2005 (1995) [Phys.At.Nucl. 58, 1896 groups I and II, R increases slowly with increasing (1995)]. momentum over the entire region of the spectrum; for Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol.68 No.9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1551–1553. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1611–1613. Original Russian Text Copyright c 2005 by Bazarov. ELEMENTARY PARTICLES AND FIELDS Experiment

On the Energy Spectrum of Protons Produced in 16Оp Collisions at a Momentum of 3.25 GeV/с per Nucleon E. Kh. Bazarov Institute for Physics and Technology, Fizika–Solntse Research and Production Association, Uzbek Academy of Sciences, ul. Timiryazeva 2b, Tashkent, 700084 Republic of Uzbekistan Received August 10, 2004; in ﬁnal form, December 2, 2004

Abstract—New experimental data concerning the mechanisms of the production of protons originating as fragments from oxygen-nucleus interactions in a hydrogen bubble chamber at high energies are presented. It is shown that anomalies observed in the energyspectrum of protons at kinetic energies in the range T =70−90 MeV are associated with the absorption of slow pions bya quasideuteron nucleon pair. c 2005 Pleiades Publishing, Inc.

According to present-daytheoretical ideas, the the observed structure is the result of pion or meson- production of the lightest nuclear fragments, nu- resonance absorption bystronglybound few-nucleon cleons (protons and neutrons), can proceed at all systems. stages of hadron–nucleus interaction at high ener- The present studyis devoted to exploring the gies, which is an extremelyintricate process. Such energyspectrum of protons originating as oxygen- nucleons mayemerge at the stage of an intranuclear nucleus fragments from 16Оp interactions at a mo- cascade or from the decayof excited multinucleon mentum 3.25 GeV/с per nucleon. It is a continuation fragments; also, theymaybe products of either of the series of investigations into the mechanisms evaporation or an explosive decay(Fermi breakup) of relativistic-hydrogen-nucleus fragmentation with of a thermalized residual nucleus. The interaction of primaryparticles with intranuclear systems,where the aid of the 1-m liquid-hydrogen bubble chamber nucleons occur at veryshort distances ( ≤1 fm), at the Joint Institute for Nuclear Research (JINR, Dubna). It should be noted that the use of a liquid mayin principle lead to the formation of so-called bubble chamber in a magnetic field proved to be cumulative nucleons—that is, nucleons whose momenta have values forbidden bythe usual kinematics an efficient method for studying many features of the fragmentation process under conditions of 4π of particle scattering on free intranuclear nucleons geometry. The methodological issues concerning (with allowance for their Fermi motion). In particular, the identification of particles and fragments were relativelyenergetic nucleons appearing as fragments described in detail elsewhere [4–7]. mayoriginate from the absorption of slow product pions byquasideuteron nucleon pairs in a nucleus via The experimental data presented below are based the reaction on a statistical sample of about 15 000 measured 16Оp events. In analyzing the energy spectra of the π + d → N + N. (1) protons, a clear mass separation of fragments was If the energyof the pion involved is rather high, the ensured byconsidering onlythose events in which final state mayinvolve isobars. Such reactions, if they the measured length of tracks of fast singlycharged indeed occur, mayresult in the formation of cumula- particles in the fiducial volume of the chamber ex- tive nucleons. ceeded 35 cm. In the case of this selection, the mean error in momentum measurements was less than 3% Previously, a structure in the momentum spec- for 1Нand2Нand5% for 3Н. The following mo- trum of protons emitted in the backward hemisphere mentum intervals were introduced in order to per- in the laboratoryframe was discovered in [1 –3] in pNe form a mass identification of these fragments: p = interactions at 300 GeV/c and in π−C collisions at 4.75−7.75 GeV/с for 2Нandp>7.8 GeV/c for 3Н. 4and40GeV/c—more specifically, the differential All singlycharged positive particles in the momen- cross section as a function of the proton momentum tum range 2.0 ≤ p ≤ 4.75 GeV/c were classified as was found there to deviate from a monotonic behavior protons, since the production of positivelycharged in the region p ≈ 0.3−0.5 GeV/c. It was shown that pions is kinematicallyimpossible in the momentum

1063-7788/05/6809-1551$26.00 c 2005 Pleiades Publishing, Inc. 1552 BAZAROV

Number of protons the kinetic-energydistribution of protons appearing as fragments (in the range 40 2 GeV/c. It should be noted that all mul- in a nucleus (π + d → N + p) is stronglysup- tiplycharged fragments of the projectile nucleus were pressed. Indeed, one can see from Fig. 1b that, in unambiguouslyidenti fied bythe charge value. This events involving deuteron production, there are no made it possible to studythe fragmentation process anomalies in the range T ≈ 70−90 MeV. In passing, in individual topological channels. In describing and we emphasize that the presence or absence of three- discussing the results obtained here, all energyfea- nucleon fragments (3Hand3He nuclei), which could tures of protons are presented below in the antilabo- be formed upon the fusion of a deuteron with another ratoryframe —that is, in the rest frame of the oxygen nucleon from a cascade [7], has virtuallyno e ffect on nucleus. the result. As was shown in [7], the distribution of the invari- The absence of a sizable peak in topological chan- ant structure function f(T ) with respect to the proton nels where the total charge of z ≥ 2 fragments is kinetic energyhas a sharplydescending character f as the energyincreases in the region T ≤ 10 MeV, less than five can be explained qualitativelyas follows. but, at higher energies, the slope of the spectrum Since a greater number of intranuclear collisions and decreases; in the region T>50 MeV, the shape of a disintegration of the primarynucleus to a greater the spectrum approaches that of an exponential. In degree occur in those channels, protons formed in the region T ≈ 60−90 MeV, the inclusive kinetic- reaction (1) lose their initial kinematical parameters, energyspectrum of protons exhibits a deviation from the energyspectrum of these protons being strongly monotononicityin its descending part. This structure smeared. manifests itself most clearlyin topological channels Thus, we can conclude that the structure observed where the total charge of multiplycharged fragments in the energyspectrum of protons in the range T ≈ falls within the range 5 ≤ zf ≤ 7.Fortwogroups 70−90 MeV is associated with the decayof a two- of events belonging to these channels, Fig. 1 displays nucleon system upon the absorption of a slow pion.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ON THE ENERGY SPECTRUM OF PROTONS 1553 ACKNOWLEDGMENTS 4. V.V.Glagolev, K. G. Gulamov, M. Yu. Kratenko, et al., Yad. Fiz. 58, 2005 (1995) [Phys. At. Nucl. 58, 1896 IamgratefultoV.V.Glagolev,K.G.Gulamov, (1995)]. K. Olimov, B. S. Yuldashev, and A. A. Yuldashev for helpful advice and a discussion on the results obtained 5. V.V.Glagolev, K. G. Gulamov, M. Yu. Kratenko, et al., in this study. Pis’ma Zh. Eksp.´ Teor. Fiz. 59, 316 (1994) [JETP Lett. 59, 336 (1994)]. REFERENCES 6. K. G. Gulamov, V. D. Lipin, S. L. Lutpullaev, et al., 1. S. O. Edgorov, S. L. Lutpullaev, A. A. Yuldashev, and Yad. Fiz. 59, 1042 (1996) [Phys. At. Nucl. 59, 997 B. S. Yuldashev, Yad. Fiz. 55, 2962 (1992) [Sov. J. (1996)]. Nucl. Phys. 55, 1656 (1992)]. 7. V.V.Glagolev,K.G.Gulamov,V.D.Lipin,et al.,Yad. 2. V.M. Asaturyan et al.,Nucl.Phys.A496, 770 (1989). Fiz. 62, 1472 (1999) [Phys. At. Nucl. 62, 1388 (1999)]. 3. D. Armutliı˘ ski, A. P. Gasparyan, V. G. Grishin, et al., Yad. Fiz. 46, 1712 (1987) [Sov. J. Nucl. Phys. 46, 1023 (1987)]. Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1554–1572. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1614–1632. Original English Text Copyright c 2005 by A. Anisovich, V. Anisovich, Markov, Nikonov, Sarantsev. ELEMENTARY PARTICLES AND FIELDS Theory

Decay φ(1020) → γf0(980):Analysis in the Nonrelativistic Quark Model Approach* A. V. Anisovich, V. V. Anisovich, V. N. Markov, V. A. Nikonov, andA. V. Sarantsev Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina, 188350 Russia Received May 18, 2004; in ﬁnal form, January 17, 2005

Abstract—We demonstrate the possibility of a good description of the processes φ(1020) → γππ and φ(1020) → γf0(980) within the framework of the nonrelativistic quark model assuming f0(980) to be a dominantly quark–antiquark system. Diﬀerent mechanisms of the radiative decay, that is, the emission of a photon by the constituent quark (additive quark model) and charge-exchange current, are considered. We also discuss the status of the threshold theorem applied to the studied reactions, namely, the behavior of → → the decay amplitude at Mππ mφ and mf0 mφ. In conclusion, the arguments favoring the qq¯ origin of f0(980) are listed. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION poles lying on two different sheets of the complex- M plane, at M = 1020 − 40i MeV and M = 960 − The K-matrix analysis of meson spectra [1–3] and 200i MeV. It should be emphasized that these two meson systematics [4, 5] point determinately to the poles are important for the description of f0(980). quark–antiquark origin of f0(980). However, there Therefore, we use the method below as follows: we exist hypotheses where f0(980) is interpreted as a calculate the radiative transition to a stable bare ¯ bare ± four-quark state [6], KK molecule [7], or vacuum f0 state (this is f0 (700 100)); its parameters scalar [8]. The radiative and weak decays involving were obtained in the K-matrix analysis [1]. In this f0(980) may be a decisive tool for understanding the way, we find out the description of the process → bare ± nature of f0(980). φ(1020) γf0 (700 100), and furthermore, we In the present paper, the reaction φ(1020) → switch on the hadronic decays and determine the transition φ(1020) → γππ; just the residue in the γf0(980) is considered in terms of a nonrelativistic f (980) pole of this amplitude is the radiative transition quark model assuming 0 to be dominantly → the qq¯ state. The nonrelativistic quark model is a amplitude φ(1020) γf0(980).Hence,weobtain → good approach for the description of the lowest qq¯ a successful description of data for φ(1020) γππ → states of pseudoscalar and vector nonets, so one may and φ(1020) γf0(980) within the assumption that hope that the lowest scalar qq¯ states are described f0(980) is dominated by the quark–antiquark state. with reasonable accuracy as well. The choice of The conclusion about the nature of f0(980) cannot a nonrelativistic approach for the analysis of the be based on the study of one reaction only but should reaction φ(1020) → γf0(980) was motivated by the be motivated by the whole aggregate of data. In the fact that, in its framework, we can take account of not article, we also list the other processes, which provide only the additive-quark-model processes (emission us with arguments in favor of the dominant qq¯ struc- of a photon by a constituent quark) but also those ture of f0(980). beyond it within the use of the dipole formula (photon Section 2 is introductive: here, we consider a sim- emission by the charge-exchange current gives such ple model for the description of composite vector an example). The dipole formula for the radiative (V ) and scalar (S) particles, the composite parti- transition of a vector state to a scalar one, V → γS, cles consisting of one-flavor quark, charge-exchange was applied before for the calculation of reactions with currents being absent. In such a model, the decay heavy quarks (see [9, 10] and references therein). Still, transition V → γS is completely determined by the a straightforward application of the dipole formula to additive-quark-model process: a photon is emitted the reaction φ(1020) → γf0(980) is hardly possible only by one or another constituent quark. Two alter- → since the f0(980) surely cannot be represented as a native representations of the V γS decay ampli- stable particle: this resonance is characterized by two tude are given, namely, the standard additive-quark- model formula and that of a photon dipole emission; ∗ This article was submitted by the authors in English. in the latter, the factor ω = mV − mS is written in

1063-7788/05/6809-1554$26.00 c 2005 Pleiades Publishing, Inc. DECAY φ(1020) → γf0(980) 1555 explicit form. The comparison of these two represen- and M = 960 − 200i MeV, and these two poles are tations helps us to formulate the problem of appli- important for the description of f0(980). The essential cation of the threshold theorem [11] to the reaction role of the second pole is seen by considering the ππ V → γS. Using a simple example with exponential spectrum in φ(1020) → γππ (Section 6): the visible wave functions, we demonstrate that the ω3 factor width of the pick in the ππ spectrum is of the order of occurs in the partial decay width when the transition 150 MeV, and the spectrum decreases slowly with a → V γS is considered in terms of the additive quark further decrease in Mππ. model. The threshold theorem has a straightforward for- Another problem to be discussed is the choice of mulation for the stable V and S states, but it is not method for the consideration of radiative decay ampli- the case for resonances, which are the main objects tude. One can work within two alternative represen- → of our present study. That is why we intend to refor- tations for the Aφ γf0 amplitude. One representation mulate the threshold theorem as the requirement of uses the additive quark model complemented with amplitude analyticity—this is given in Section 3 on the contribution from the charge-exchange current the basis of [12]. Working with nonstable particles, processes. Whence the additive-quark-model ampli- when V and S are resonances, the V → γS amplitude tude can be calculated rather definitely, at least for should be determined as a residue of a more general the lowest qq¯ states, the charge-exchange current amplitude, with stable particles in the initial and final processes are vaguely determined. states. For example, the φ(1020) → γf (980) ampli- 0 The other way to deal with the transition amplitude should be definedasaresidueofthee+e− → tude consists in using the dipole emission formulas, γππ amplitude in the poles corresponding to reso- + − where the amplitude is defined by the mean transi- nances φ(1020) and f0(980) (the φ pole in the e e − tion radius and factor (mφ mf0 ). For the lowest channel and the f0 pole in the ππ channel). The only residue in the pole defines the universal amplitude qq¯ states, we have a good estimate of the radius. which does not depend on the considered reaction. But there is a problem of the determination of the − factor (mφ mf0 ), because the f0(980) is an unstable In Section 4, we discuss the reaction φ → γf0 for the case when the f is a multicomponent system and particle characterized by two poles. The pole M = 0 960 − 200i MeV is disposed on the same sheet of the f0 and φ are stable states with respect to hadronic decays. The analysis of meson spectra (e.g., see the complex-M planeasthepoleofφ meson, and the | − | ∼ latest K-matrix analyses [1, 5]) definitely tells us that distance mφ mf0 is 200 MeV, while the pole the f mesons are the mixture of the quarkonium M = 1020 − 40i MeV is located on another sheet, 0 √ ∼ (nn¯ =(uu¯ + dd¯)/ 2 and ss¯) and gluonium compo- and the distance from the φ meson is 70 MeV, which is also not small on the hadronic scale. The nents. Such a multichannel structure of f0 states reveals itself in the existence of the t-channel charge- problem is what the mass difference factor means in the case of complex masses and which pole should be exchange currents. Therefore, the transition φ → γf0 goes via two mechanisms: the photon emission by used to characterize this mass difference. constituent quarks (additive-quark-model process) To succeed in the description of the decay and charge-exchange current. We write down the for- φ(1020) → γf0(980), we use the results of the mulas for the amplitudes initiated by these two mech- K-matrix analysis of the IJPC =00++ wave [1]. anisms. The equality to zero of the whole amplitude ∼ Thefactisthat,ontheonehand,theK-matrix at small mass difference of φ and f0, Aφ→γf0 ω at → analysis allows us to get the experimentally based ω 0, resulted from the cancellation of contributions information on masses and full widths of resonances of these two mechanisms. We also give a dipole rep- together with the pole residues needed for the decay resentation of the transition amplitude, where A → φ γf0 couplings and partial widths. On the other hand, the is determined through the mean transition radius and knowledge of the K-matrix amplitude enables us to mass difference (m − m ). φ f0 trace the evolution of states by switching on/off the Section 5 is devoted to the reaction φ(1020) → decay channels. In such a way, one may obtain the γf0(980). First, we discuss whether it is possible characteristics of the bare states, which are prede- to treat f0(980) as a stable particle. Our answer cessors of real resonances. With such characteristics, is “no”; in fact, the f0(980) is an unstable parti- one can perform a reverse procedure: to retrace the cle characterized by two poles located near the KK¯ transformation of the amplitude written in terms of threshold. As was stressed above, strong transitions bare states to the amplitude corresponding to the f0(980) → ππ, KK¯ reveal themselves in the two am- transition to a real resonance. Just this procedure has plitude poles, which are located on different sheets been applied in Section 5 for the calculation of the of the complex-M plane, at M = 1020 − 40i MeV decay amplitude φ(1020) → γf0(980).

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1556 ANISOVICH et al. Therefore, within the framework of the nonrela- when the charge-current-exchange forces are absent tivistic quark model, we have calculated the reac- and the V → γS amplitude is given by the additive- → bare bare tion φ(1020) γf0 (n),wheref0 (n) are bare quark-model contribution. states found in [1]. Furthermore, with the K-matrix technique, we have taken account of the decays 2.1. Wave Functions for Vector and Scalar f bare(n) → ππ,KK 0 , thus having calculated the reac- Composite Particles tion φ(1020) → γππ and the amplitude of φ(1020) → γf0(980) (the pole residue in the ππ channel). In The qq¯ wave functions of vector and scalar parti- this way, we see that the main contribution is given cles are defined as follows: → bare ± 2 by the transition φ(1020) γf0 (700 100).The ΨVµ(k)=σµψV (k ), (1) bare ± characteristics of f0 (700 100) are fixed by the Ψ (k)=(σ · k)ψ (k2), K-matrix analysis [1]: this is a qq¯ state close to the S S flavor octet and it is just the predecessor of f0(980). where, by using Pauli matrices, the spin factors are In the framework of this approach, we succeed in singled out. The blocks dependent on the relative the description of data for the reactions φ(1020) → momentum squared are related to the vertices in the γf (980) (Section 5) and φ(1020) → γππ (Sec- following way: 0 √ tion 6). 2 2 m GV (k ) Letusnotethatsuchamethod,theuseofbare ψV (k )= 2 , (2) states for the calculation of meson spectra, has been 2 k + mεV 2 applied before for the study of weak hadronic decays 2 √1 GS(k ) + → + + − ψS(k )= 2 . D π π π [13] and description of the ππ spec- 2 m k + mεV tra in photon–photon collisions γγ → ππ [14]. The question of what the accuracy of the addi- Here, m is the quark mass, and ε is the composite- system binding energy: εV =2m − mV and εS = tive quark model is in the description of the reac- − → bare ± → 2m mS,wheremV and mS are the masses of tions φ(1020) γf0 (700 100) and φ(1020) bound states. The normalization condition for the γf0(980) is discussed in Section 7. We compare wave functions reads the results of the calculation of the φ(1020) → d3k γfbare(700 ± 100) reaction by using the dipole for- Sp Ψ+(k)Ψ (k) (3) 0 (2π)3 2 S S mula with that of the additive quark model. It is seen that, within error bars given by the K-matrix d3k = ψ2 (k2)Sp [(σ · k)(σ · k)] = 1, analysis [1], the results coincide. Still, one should (2π)3 S 2 emphasize that the dipole-calculation accuracy is low, which is due to a large error in the determination 3 d k + of the bare-state masses. The coincidence of results Sp Ψ (k)ΨVµ (k) (2π)3 2 Vµ in the dipole and additive-quark-model formulas 3 should point to a small contribution of processes d k 2 2 = ψ (k )Sp [σ σ ]=δ . which violate additivity, such as photon emission (2π)3 V 2 µ µ µµ by the charge-exchange current: this smallness is natural, provided the hadrons are characterized by two sizes, namely, the hadron radius (Rh ∼ Rconf) 2.2. Amplitude within the Additive Quark Model and constituent-quark radius (rq) under the condition 2 2 Whenaphotonisemittedbyaquarkorantiquark, rq Rh (see [15] and references therein). the V → γS process is described by the triangle di- The performed analysis demonstrates that the agram (see Fig. 1a) that is actually the contribution → reaction φ(1020) γf0(980) does not produce any from the additive quark model. Relativistic consider- difficulty with the interpretation of f0(980) as a qq¯ ation of the triangle diagram is presented in [16, 17], state. Still, to draw a conclusion about the content of while the discussion of the nonrelativistic approxima- f(980), we list in Section 8 the arguments in favor of tion is given in [12, 18] (recall that, in [18], the corre- the qq¯origin of f0(980). sponding wave functions were determined in another 2 2 2 −1 way, namely, ψV (k )=GV (k )(4k +4mεV ) and → 2 2 2 −1 2. THE PROCESS V γS ψS(k )=GS(k )(4k +4mεS ) ). WITHIN THE NONRELATIVISTIC ADDITIVE In terms of wave functions (1), the triangle- QUARK MODEL diagram contribution reads Here, in the framework of the nonrelativistic quark → (V ) (γ) V →γS (V ) (γ) V →γS model, we consider the transition V γS in the case "µ "α Aµα = eZV →γS"µ "α Fµα , (4)

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(a)(b)(c) q γγ

k1 k1' ' – + ppk2 1– 0+ 1 0

Fig. 1. Transitions V → γS in the additive quark model. 3 → d k + Let us consider the case when the quark–quark F V γS = Sp Ψ (k)4k Ψ (k) . µα (2π)3 2 S α Vµ interaction is rather simple, say, it depends on the relative interquark distance with the potential U(r). (V ) (γ) Here, "µ and "α are polarization vectors for V For vector and scalar composite systems, we also (V ) (γ) and γ: "µ pVµ =0and "α qα =0. The charge factor use an additional simplifying assumption: vector and scalar mesons consist of quarks of the same flavor ZV →γS being different for different reactions is specified below (see also [16, 17]). The expression for the (qq¯). Then we have the Hamiltonian transition amplitude (4) can be simplified after the ∇2 H = − + U(r) (11) substitution in the integrand m · → 2 2 ⊥⊥ and can rewrite (10) as Sp2[σµ(σ k)]kα k gµα , (5) 3 − − ∇ ⊥⊥ 2im(Hrα rαH)=4( i α). (12) where gµα is the metric tensor in the space orthogonal to total momentum of the vector particle p and After substituting the commutator in (9), the transi- V tion form factor for the reaction V → γS reads photon q. The substitution (5) results in V →γS ⊥⊥ V →γS 3 + Aµα = egµα AV →γS, (6) Fµα = d rSp2 ΨS (r)rαΨVµ(r) (13) where × 2im(εV − εS). ∞ 2 dk 2 2 2 3 Here, we have used the fact that (H + εV )ΨV =0 AV →γS = ZV →γS ψS(k )ψV (k ) k . (7) π 3π and (H + εS)ΨS =0. 0 The factor (εV − εS) on the right-hand side of (13) V →γS − The amplitudes Aµα and AV →γS were used in is a manifestation of the threshold theorem: at εV − → V →γS [16, 17] for the decay amplitude φ(1020) → γf0(980) εS = mS mV 0, the form factor Fµα goes to within the relativistic treatment of the quark transi- zero. Actually, in the additive quark model, the ampli- tions. tude of the V → γS transition, being determined by However, for our purpose, it would be suitable not the process of Fig. 1a, cannot be zero if V and S are V →γS basic states with radial quantum number n =1:in to deal with Eq. (7) but use the form factor Fµα 2 2 of Eq. (4) rewritten in the coordinate representation. this case, the wave functions ψV (k ) and ψS(k ) do One has not change sign, and the right-hand side of (7) does not equal zero. In order to clarify this point, let us con- 3 ik·r ΨVµ(k)= d re ΨVµ(r), (8) sider as an example the exponential approximation for 2 2 the wave functions ψV (k ) and ψS(k ). 3 ik·r ΨS(k)= d re ΨS(r). 2.3. Basic Vector and Scalar qq¯ States: An Example V →γS Then the form factor Fµα can be represented as of the Exponential Approach to Wave Functions follows: We parametrize the ground-state wave functions V →γS 3 + · of scalar and vector particles as follows: Fµα = d r Sp2 ΨS (r) 4kαΨVµ(r) , (9) 2 ΨVµ (r)=σµψV (r ), (14) where kα is the operator: kα = −i∇α. This oper- 1 r2 ator can be written as the commutator of rα and ψ (r2)= exp − , 2 V 3/4 −∇ /m = T (kinetic energy): 5/4 3/4 4bV 2 π bV 2im(Tr − r T )=4(−i∇ ). (10) 2 α α α ΨS(r)=(σ · r)ψS (r ),

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1558 ANISOVICH et al. i r2 JP =0+ ψ (r2)= √ exp − . photon and then turn into the scalar state. S 5/4 25/4π3/4b 3 4bS This amplitude has as a subprocess a bound-state S transition. Namely, the blocks for the rescattering of The wave functions with n =1 have no nodes; constituents in Fig. 1b contain the poles related to numerical factors take account of the normalization bound states (see Fig. 1c), and the residues in these conditions poles determine the bound-state transition amplitude 3 + (triangle diagram shown as intermediate block in d rSp2 ΨS (r)ΨS(r) =1, (15) Fig. 1c). With the notation for invariant mass squares in the 3 + d rSp2 ΨVµ(r)ΨVµ (r) = δµµ . initial and final states of Fig. 1b as follows: 2 2 PV = sV ,PS = sS, (19) With exponential wave functions, the matrix element for V → γS given by the additive-quark-model we can write the spin structures for this more general diagram [Eq. (9)] is equal to transition V → γS. The standard representation of → this amplitude is "(V )"(γ)F V γS(additive) (16) µ α µα V →γS 2 → Aµα (sV ,sS,q 0) (20) 7/2 3/4 5/4 (V ) (γ) 2 bV bS =(" " ) √ . 2qµPVα 5/2 − 3 (bV + bS) = gµα AV →γS(sV ,sS, 0). sV − sS V →γS The formula for Fµα written in the framework Here, we stress that the amplitude AV →γS describes of dipole emission [Eq. (13)] reads the emission of a real photon, q2 =0. In (20), it was → − "(V )"(γ)F V γS(dipole) (17) taken into account that (PV q)=(sV sS)/2.The µ α µα requirement of analyticity, i.e., the absence of a pole 7/4 5/4 27/2 b b at sV = sS, leads to the condition =("(V )"(γ)) √ V S m(m − m ). 5/2 V S 3 (bV + bS) AV →γS(sV ,sS, 0) → 0, (21) → In the case under consideration [one-flavor quarks sV sS with the Hamiltonian given by Eq. (11)], Eqs. (16) and which is the threshold theorem for the transition am- V →γS V →γS → (17) coincide, Fµα (additive) = Fµα (dipole); plitude V γS. therefore, It should be now emphasized that the form of the − m(m − m )=b 1, (18) spin factor in Eq. (20) is not unique. Alternatively, V S V one can write the spin factor as the metric tensor − ⊥⊥ which means that the factor (εS εV )ontheright- gµα working in the space orthogonal to PV and q, hand side of (13) relates to the difference between the ⊥⊥ ⊥⊥ i.e., PVµgµα =0and gµα qα =0[see Eq. (5)]. This V and S levels and is defined by bV only. In this way, V →γS metric tensor reads the form factor Fµα goestozeroonlywhenbV (or ⊥⊥ 4s b )tendstoinfinity. g (0) = g + V q q (22) S µα µα (s − s )2 µ α The considered example does not mean that the V S threshold theorem for the reaction V → γS does not 2 − (PVµqα + qµPVα), work; this tells us only that we should interpret and sV − sS use it carefully. In the next section, we discuss how and we have used it in Eq. (6). The uncertainty in to formulate the threshold theorem based on the re- the choice of spin factor is due to the fact that the quirement of amplitude analyticity, thus getting more difference information on its applicability. ⊥⊥ − − 2qµPVα gµα (0) gµα =4Lµα(0), (23) sV − sS 3. ANALYTICITY OF THE AMPLITUDE AND THE THRESHOLD THEOREM where s 1 The threshold theorem can be formulated as the V − Lµα(0) = 2 qµqα PVµqα, requirement of analyticity of the amplitude. To clarify (sV − sS) 2(sV − sS) this statement, we consider here not only the tran- (24) sition of the bound states but also a more general is the nilpotent operator [12]: process shown in Fig. 1b, where the interacting constituents being in the vector JP =1− state emit a Lµα(0)Lµα(0) = 0. (25)

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The addition of the nilpotent operator L (0) to spin γ µα + factor of the transition amplitude V → γS does not e γ* A → (s ,s , 0) change the expression V γS V S (see [12] for φ more detail). Here, in discussing the analytical struc- f0 e– ture of the amplitude, it is convenient to work with the π operator (20), since it is the least cumbersome. Consider now the reaction V → γS (V and S be- π ing quark–antiquark bound states), say, of the type of → → − − φ γf0 or φ γa0. Because of the conﬁnement, the Fig. 2. Process e+e → γππ:residuesinthee+e and | quarks are not the particles which form the in and ππ channels determine the φ → γf0 amplitude. | out states; therefore, the amplitudes like Aφ→γf0 are to be deﬁned as the amplitude residue for the process with the scattering of the stable particles, for example, special discussion. We shall come back to this point for e+e− → γπ+π− (see Fig. 2): below, and now let us investigate how the requirement (28) is realized in quantum mechanics when φ and f0 e+e−→γπ+π− − 2qµPVα are stable particles. Aµα (sV ,sS, 0) = gµα sV − sS (26) 4. QUANTUM MECHANICS 2 2 A → (m ,m , 0) φ γf0 φ f0 CONSIDERATION OF THE REACTION × G + −→ g → + − e e φ 2 2 f0 π π φ → γf WITH φ AND f BEING STABLE (sV − m )(sS − m ) 0 0 φ f0 PARTICLES

+ B(sV ,sS, 0) . In Section 2, we have considered the model for the reaction V → γS when V and S are formed by quarks of the same flavor (one-channel model for V and S). 2 2 We see that A(m ,m , 0), up to the factors G + −→ φ f0 e e φ The one-channel approach for φ(1020) (the dom- + − and gf0→π π , is the residue in the amplitude poles inance of ss¯ component) looks acceptable, though s = m2 and s = m2 : just this value supplies us for f0 mesons it is definitely not so: scalar–isoscalar V φ S f0 with the transition amplitude for the reactions with states are multicomponent ones. bound states φ → γf . If we deal with stable com- The existence of several components in the f0 0 → posite particles, in other words, if φ and f0 can be in- mesons changes the situation with the φ γf0 de- cluded in the set of fields |in and out|, the transition cays. First, the mixing of different components may amplitude φ → γf0 can be written in a form similar result in close values of masses of the low-lying vector to (20): and scalar mesons. Second, Eqs. (9) and (13) for the → φ → γf0 decay turn out to be nonequivalent because Aφ γf0 (m2 ,m2 , 0) (27) µα φ f0 of the photon emission by the t-channel exchange currents. 2qµpα 2 2 = gµα − Aφ→γf m ,m , 0 , Here, we consider in detail a simple model for φ m2 − m2 0 φ f0 φ f0 and f0:theφ meson is treated as an ss¯ system, with no admixture of the nonstrange quarkonium, nn¯ = where we have substituted PV → p.For √ ¯ 2 2 (uu¯ + dd)/ 2, nor gluonium (gg), while the f0 meson Aφ→γf m ,m , 0 , the threshold theorem is ful- 0 φ f0 is a mixture of ss¯ and gg. filled: This model can be considered as a guide for the 2 2 2 2 A → (m ,m , 0) ∼ m − m ; (28) → φ γf0 φ f0 m2 →m2 φ f0 study of the reaction φ(1020) γf0(980). Indeed, φ f0 the φ(1020) is an almost pure ss¯ state; the admix- this means that the threshold theorem of Eq. (28) ture of the nn¯ component in φ(1020) is small, ≤ 5%, reveals itself as a requirement of analyticity of the and it can be neglected in a rough estimate of the → amplitude φ γf0 determined by Eq. (27). φ(1020) → γf0(980) decay. Let us emphasize again that formula (27) has been The resonance f0(980) is a multicomponent state. φ f written for the and 0 mesons assuming them to Analysis of the IJPC =00++ wave in the K-matrix be stable; i.e., they can be treated as the states which fit to the data for meson spectra ππ, KK¯ , ηη, ηη, belong to the sets |in and out|. However, by consid- ππππ gives the following constraints for the ss¯, nn¯, ering the process φ → γf , we deal with resonances, 0 gg f (980) not stable particles, and whether this assumption is and components in 0 [1, 5]: valid for resonances is a question which deserves 50% Wss¯[f0(980)] < 100%, (29)

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(a)(b)(c) The Schrodinger¨ equation for the two-component states, ss¯ and gg, reads ss k2 m + Uss¯→ss¯(r) Uss¯→gg(r) (31) – – + k2 s s Uss¯→gg(r) + Ugg→gg(r)  mg   Fig. 3. Examples of diagrams which contribute to the Ψ (r) Ψ (r) ×  ss¯   ss¯  potentials Uss¯→ss¯(r), Ugg→gg(r),andUss¯→gg(r). = E . Ψgg(r) Ψgg(r)

0 Wnn¯[f0(980)] < 50%, Furthermore, we denote the Hamiltonian on the left- hand side of (31) as H . 0 W [f (980)] < 25%. 0 gg 0 We set the gg component in φ to zero. It means Also, the f0(980) may contain a long-range KK¯ that the potential Uss¯→gg(r) satisfies the following component, on the level of 10–20%. constraints: + + The restrictions (29) permit the variant when the 0 ss¯|Uss¯→gg(r)|0 gg =0, (32) probability for the nn¯ component is small and f (980) − − 0 1 ss¯|U → (r)|1 gg =0. is a mixture of ss¯ and gg only. Bearing this variant in ss¯ gg mind, we consider such a two-component model for φ These constraints do not look surprising for mesons and f0 supposing these particles are stable in respect in the region 1.0–1.5 GeV because the scalar glueball to hadronic decays. is located just in this mass region, while the vector ∼ It is not difficult to generalize our consideration for one has considerably higher mass, 2.5 GeV [19]. the three-component f0 state (nn¯, ss¯,andgg); the The t-exchange diagrams shown in Figs. 3a,3b, corresponding formulas are given in this section too. and 3c are an example of interaction leading to the potentials Uss¯→ss¯(r), Ugg→gg(r),andUss¯→gg(r).The potential Uss¯→gg(r) contains the t-channel charge 4.1. Two-Component Model (ss¯, gg)forf0 and φ exchange. 4.1.1. Dipole emission of the photon in φ → γf0 Now let us discuss the model where f0 has two 0 components only: strange quarkonium (ss¯ in the P decay. The Hamiltonian for the interaction of an wave) and gluonium (gg in the S wave). The spin electromagnetic field with two-component composite structure of the ss¯ wave function is written in Section systems (quarkonium and gluonium components) is 2: it contains the factor (σ · r) in the coordinate repre- presented in the Appendix. → sentation. For the gg system, we have δab or, in terms For the transition V γS, keeping the terms of polarization vectors, the convolution ((g)(g)). proportional to the charge e, we have the following 1 2 operator for the dipole emission: Here, we consider a simple interaction when the potential does not depend on spin variables—in this ˆ 2kα irαUss¯→gg(r) case, one may forget about the vector structure of dα = . (33) − + gg working as if the gluon component consists of irαUss¯→gg(r)0 spinless particles. As concerns φ,itisconsidered asapuress¯ state in the S wave, with the wave- The transition form factor is given by a formula similar function spin factor ∼ σ (see Section 2). So, the to Eq. (9) for the one-channel case; it reads µ wave functions of f0 and φ mesons are written as → φ γf0 3 ˆ + ˆ ˆ follows: Fµα = d rSp Ψ (r)2dαΨφµ(r) . (34)     2 f0

Ψf (ss¯)(r) (σ · r)ψf (ss¯)(r) Drawing explicitly the two-component wave functi- Ψˆ (r)= 0  =  0  , f0 ons, one can rewrite Eq. (34) as follows: Ψf0(gg)(r) ψf0(gg)(r) → (30) F φ γf0 = d3rSp Ψ+ (r)4k Ψ (r)     µα 2 f0(ss¯) α φ(ss¯)µ Ψφ(ss¯)µ(r) σµψφ(ss¯)(r) (35) Ψˆ φµ(r)= = . 3 + Ψφ(gg)µ(r) 0 + d rSp Ψ (r) 2 f0(gg) The normalization condition is given by Eq. (15) with × − → ˆ → ˆ ( irαUgg→ss¯(r)) Ψφ(ss¯)µ(r) . the obvious replacement ΨS Ψf0 and ΨVµ Ψφµ.

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The ﬁrst term on the right-hand side of (35), with Re M, MeV the operator 4kα, is responsible for the interac- 850 900 950 1000 1050 – tion of a photon with constituent quark that is the KK additive-quark-model contribution, while the term –50 1020 – 40i (−irαUgg→ss¯(r)) describes interaction of the photon with the charge ﬂowing through the t channel— –100 2nd sheet this term describes the photon interaction with the fermion-exchange current. –150 Let us return to Eq. (34) and rewrite it in a form similar to (13). One can see that –200 960 – 200i ˆ − ˆ ˆ 3rd sheet im H0rˆα rˆαH0 = dα, (36) Im M, MeV where Hˆ0 is the Hamiltonian for composite systems Fig. 4. Complex-M plane and location of the poles corre- ¯ written on the left-hand side of (31), and the operator sponding to f0(980); the cut related to the KK threshold rˆ is determined as is shown as a broken line. α   rα 0 rˆ =   . (37) α = d3rSp (σ · r)ψ (r)r σ ψ (r) 00 2 f0(nn¯) α µ φ(nn¯) × 2im(ε − ε ). Substituting Eq. (36) into (34), we have φ f0

φ→γf0 3 The partial width reads Fµα = d r (38) m2 − m2 1 φ f0 × σ · r → Sp2 ( )ψf0(ss¯)(r)rασµψφ(ss¯)(r) mφΓφ γf0 = α 2 (42) 6 mφ × 2im(εφ − εf ). 0 2 × A → + A → , This formula for the dipole emission of a photon is φ γf0(ss¯) φ γf0(nn¯) similar to that of (13) for the one-channel model. with A → deﬁned by Eqs. (4) and (6). The charge → φ γf0 4.1.2. Partial width of the decay φ γf0.The factors, which were separated in Eq. (4), are equal to partial width of the decay φ → f0 in the case when φ is apuress¯state is determined by the following formula: (ss¯) 2 (nn¯) 1 Z = − ,Z = ; (43) φ→γf0 φ→γf0 m2 − m2 3 3 1 φ f0 2 m Γ → = α A → , (39) φ φ γf0 6 m2 φ γf0(ss¯) they include the combinatorics factor 2 related to φ two diagrams with photon emission by a quark and where α =1/137 and the Aφ→γf0(ss¯) amplitude is antiquark (see [16, 17] for more detail). determined by Eq. (6) (here, it is speciﬁed that we deal with ss¯ quarks in the intermediate state).

5. DECAY φ(1020) → γf0(980) 4.2. Three-Component Model (ss¯, nn¯, gg) for f0 and φ The vector meson φ(1020) has a rather small de- The above formula can be easily generalized for the cay width, Γφ(1020) 4.5 MeV; from this point of case when f0 is the three-component system (ss¯, nn¯, view, there is no doubt that treating φ(1020) asasta- gg)andφ is the two-component one (ss¯, nn¯), while ble particle is reasonable. As to f0(980),thepictureis gg is supposed to be negligibly small. We have two not so determinate. In the PDG compilation [20], the transition form factors: ≤ ≤ f0(980) width is given in the interval 40 Γf0(980) → F φ γf0(ss¯) (40) 100 MeV, and the width uncertainty is related not µα to the data inaccuracy (experimental data are rather 3 · = d rSp2 (σ r)ψf0(ss¯)(r)rασµψφ(ss¯)(r) good) but to a vague deﬁnition of the width. × − 2im(εφ εf0 ) The mass and width of the resonance are determined by the pole position in the complex-mass and plane, M = m − iΓ/2—just this magnitude is a uni- φ→γf0(nn¯) Fµα (41) versal characteristic of the resonance.

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5.1. The f0(980): Position of Poles Figure 7 demonstrates the evolution of coupling constants at the onset of the decay channels: follow- The definition of the f (980) width is aggravated 0 ing [21], relative changes of the coupling constants bytheKK¯ threshold singularity that leads to the are shown for f (980) after switching on/off the decay existence of two poles, not one. According to the 0 channels. K-matrix analyses [1, 5], there are two poles in the Let us bring attention to a rapid increase in the IJPC =00++ wave at s ∼ 1.0 GeV2, coupling constant f0 → KK¯ on the evolution curve M I 1020 − 40i GeV, (44) bare ∼ f0 (700)–f0(980) in the region x 0.8–1.0,where M II 960 − 200i GeV, γ(x =1.0) − γ(x =0.8) 0.2 (see Fig. 7). Actually, this increase allows one to estimate a possible admix- which are located on different complex-M sheets re- ture of the long-range KK¯ component in the f (980): ¯ 0 lated to the KK threshold (see Fig. 4). By switching it cannot be greater than 20%. off the decay f0(980) → KK¯ , both poles begin to move toward one another, and they coincide com- → pletely after switching off the KK¯ channel. Usually, 5.3. Calculation of the φ(1020) γf0(980) Decay when one discusses the f0(980), the resonance is Amplitude characterized by the closest pole, M I. However, when The above discussion of the location of the am- we are interested in how far from each other the plitude poles of f0(980),aswellasthemovementof φ(1020) and f0(980) are, one should not forget about poles by switching off the decay channels, tells us the second pole. definitely that the smallness of the amplitude of the Keeping in mind the existence of two poles, φ(1020) → γf0(980) decay due to a visible proxim- one should accept that φ(1020) and f0(980) are ity of masses of vector and scalar particles is rather considerably “separated” from each other, and the questionable. As to f0(980),itspoles“dived” into f0(980) resonance can hardly be represented as a the complex plane, on the average by ∼ 100 MeV stable particle—we return to this point once more (40 MeV for one pole and 200 MeV for another). But in Section 6 in discussing the ππ spectrum in when we intend to represent f0(980) as a stable level, φ(1020) → γππ. one should bear in mind that the mass of the stable level is below the mass of φ(1020) by ∼ 300 MeV (this value is given by the K-matrix analysis). In both 5.2. Switching off Decay Channels: Bare States cases, we deal with shifts in the mass scale of the in K-Matrix Analysis of the IJPC = 00 ++ Wave order of the pion mass, which is hardly small on the K hadronic scale. Asignificant trait of the -matrix analysis is that ++ it also gives us, along with the characteristics of real The K-matrix amplitude of the 00 wave re- resonances, the positions of levels before the onset constructed in [1] gives us the possibility to trace of the decay channels, i.e., it determines the bare the evolution of the transition form factor φ(1020) → bare ± states. In addition, the K-matrix analysis allows one γf0 (700 100) during the transformation of the bare ± to observe the transform of bare states into real reso- bare state f0 (700 100) into the f0(980) resonances. The onset of the decay channels is regulated nance. Using diagrammatic language, one can say by the parameter x,andthevaluex =0corresponds (bare) that the evolution of the form factor F → is due to to the bare state (amplitude pole on the ReM axis) φ γf0 and the value x =1stands for the resonance observed the processes shown in Fig. 8: the φ meson goes into bare bare experimentally. In Fig. 5, one can see such a trans- f0 (n), with the emission of a photon; then f0 (n) ++ bare → ¯ form of the 00 -amplitude poles by switching off the decays into mesons f0 (n) hh = ππ, KK, ηη, ¯ decays f0 → ππ,KK, ηη, ηη ,ππππ. It is seen that, ηη , ππππ. The decay yields may rescatter, thus com- after switching off the decay channels, the f0(980) ing to final states. turns into a stable state, approximately 300 MeV Theresidueoftheamplitudepoleφ(1020) → lower: γππ gives us the transition amplitude φ(1020) → −→ bare ± f0(980) f0 (700 100). (45) γf0(980).So,intheK-matrix representation, the amplitude of the reaction φ(1020) → γππ (Fig. 8) The transform of bare states into real resonances reads can be illustrated by Fig. 6 for the levels in the po- (bare) tential well: bare states are the levels in a well with F → bare φ(1020) γf0 (n) bare an impenetrable wall (Fig. 6a); at the onset of the A → (s)= g (n) φ(1020) γππ M 2 − s a decay channels (underbarrier transitions, Fig. 6b), a,n n the stable levels transform into real resonances. (46)

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Im M, MeV 0

f (980) 0 f0(1500)

f0(1300) f0(1750)

–200

–400

f0(1200–1600)

–600 0 1000 2000 Re M, MeV

Fig. 5. Complex-M plane: trajectories of poles corresponding to the states f0(980), f0(1300), f0(1500), f0(1750), f0(1200–1600) within a uniform onset of the decay channels. The value x =0.5 is marked by a square point.

(a) (b) V(r) Bare states V(r) Real states Continuous 2460 2390 hadron 2330 spectrum 3 – 2140 4 P0qq 2105 2070 3 – 2005 3 P0qq 1820 33P qq– 1750 1750 0 23P qq– 1600 0 Glueball 1500 1400±200 3 – 1280 2 P0qq 1300 1250 3 – 1 P0qq 980

3 – 720 1 P0qq r r

Fig. 6. The f0 levels in the potential well depending on the onset of the decay channels: bare states (a) and real resonances (b). − 1 (1 − iρˆ(s)Kˆ (s)) 1 takes account of the rescatterings × . 1 − iρˆ(s)Kˆ (s) of the formed mesons. Here, ρˆ(s) is the diagonal ma- a,ππ bare trix of phase spaces for hadronic states (for example, Here, Mn is the mass of the bare state, and ga (n) 2 bare → for the ππ system, it reads ρππ(s)= (s − 4m )/s), is the coupling for the transition f0 (n) a,where π a = ππ, KK¯ , ηη, ηη , ππππ. The matrix element and the K-matrix elements Kab(s) contain the poles

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γ II a A may be represented as the sum of φ(1020)→γf0 (980) 1.0 contributions from diﬀerent bare states: f (980) 0 AI (49) – φ(1020)→γf0 (980) 0.8 KK (I) (bare) = ζn [f0(980)]F → bare , φ(1020) γf0 (n) n 0.6 ππ II A → φ(1020) γf0 (980) (II) (bare) 0.4 = ζn [f0(980)]F → bare . φ(1020) γf0 (n) n

0.2 To calculate the constants ζn[f0(mR)],weusethe K-matrix solution for the 00++-waveamplitudede- noted in [1] as II-2. In this solution, there are five bare 0 bare 0 0.2 0.4 0.6 0.8 1.0 states f0 (n) in the mass interval 290–1950 MeV: 3 x four of them are members of the qq¯ nonets (1 P0qq¯ 3 and 2 P0qq¯)andthefifth state is the glueball. Namely: Fig. 7. The evolution of normalized coupling constants 2 3 bare ± bare ± γa = ga/ b gb at the onset of the decay channels for 1 P0qq¯ : f0 (700 100),f0 (1220 30), (50) f0(980). 3 bare ± bare ± 2 P0qq¯ : f0 (1230 40),f0 (1800 40), bare ± glueball : f0 (1580 50). corresponding to the bare states: For the first pole of the f0(980) resonance located at gbare(n)gbare(n) − K (s)= a b + f (s). (47) M[f0(980)] = 1020 40i MeV, the renormalization ab M 2 − s ab constants are as follows: n n (I) ◦ ζ [f0(980)] = 0.62 exp(−144 i), (51) The function fab(s) is analytical in the right-hand 700 half-plane of the complex-s plane, at Res>0 (see [1] (I) − ◦ ζ1220[f0(980)] = 0.37 exp( 41 i), for more detail). (I) ◦ ζ1230[f0(980)] = 0.19 exp(1 i), Near the pole corresponding to the f0 resonance − (I) − ◦ (resonance poles are contained in the factor (1 ζ1800[f0(980)] = 0.02 exp( 112 i), −1 iρˆ(s)Kˆ (s)) φ(1020) → γππ (I) ◦ ), the amplitude is ζ [f (980)] = 0.02 exp(5 i). written as follows: 1580 0 A → (s) (48) These constants are complex-valued. One should pay φ(1020) γππ attention to the fact that the phases of constants Aφ(1020)→γf0 (980) (I) (I) g → + smooth terms, ζ [f0(980)] and ζ [f0(980)] have a relative shift M 2 − s f0(980) ππ 700 ◦ 1220 f0(980) close to 90 . This means that the contributions from f bare(700 ± 100) and f bare(1220 ± 30) (which are where M is the complex-valued resonance 0 0 f0(980) members of the basic 13P qq¯nonet) do not interfere in → I − 0 mass: Mf0(980) M 1020 40i MeV for the first practice in the calculation of probability for the decay → II − φ(1020) → γf (980) pole, and Mf0(980) M 960 200i MeV for the 0 . second one. The transition amplitude Aφ(1020)→γf0(980) Actually, one may neglect the bare states bare bare bare is different for different poles; the gf0(980)→ππ cou- f0 (1230), f0 (1800), f0 (1580) in the calcula- plings are different as well. tion of the φ(1020) → γf0(980) reaction because the We see that the radiative transition φ(1020) → form factors for the production of radial excited states are noticeably suppressed (see [17]): γf (980) is determined by two amplitudes, 0 (bare) (bare) I F 3 F 3 . A I ≡ A , φ(1020)→γf0 (2 P0qq¯) φ(1020)→γf0 (1 P0qq¯) φ(1020)→γf0 (M ) φ(1020)→γf0(980) (I) II In addition, the coefficients ζ1230[f0(980)] A II ≡ A , φ(1020)→γf0(M ) φ(1020)→γf0 (980) (I) ζ1800[f0(980)] are also comparatively small [see (51)]. and just these amplitudes are the subject of our The second pole located on the third sheet, interest. The amplitudes AI and − φ(1020)→γf0(980) M[f0(980)] = 960 200i MeV, has renormalizing

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γ π h

F φ bare (1020) f0 h π

Fig. 8. Diagram for the transition φ(1020) → γππ in the K-matrix representation, Eq. (46). constants as follows: the closest one to the real axis (1020 − 40i MeV), one (II) ◦ has ζ700[f0(980)] = 1.00 exp(6 i), (52) AI (55) (II) ◦ φ(1020)→γf0 (980) ζ1220[f0(980)] = 0.33 exp(113 i), (0.58 ± 0.04)F (bare) , (II) ◦ φ(1020)→γfbare (700) ζ1230[f0(980)] = 0.32 exp(148 i), 0 (II) ◦ and for the distant one (960 − 200i MeV), ζ1800[f0(980)] = 0.08 exp(4 i), ◦ II (II) Aφ(1020)→γf (980) (56) ζ1580[f0(980)] = 0.04 exp(98 i). 0 ± (bare) φ(1020) → (0.92 0.06)F → bare . Here, as before, the transitions φ(1020) γf0 (700) bare bare bare γf0 (1230), γf0 (1580), γf0 (1800) are negli- II We see that, in practice, the A → ampli- gibly small. φ(1020) γf0(980) tude does not change its value during the evolution φ(1020) nn¯ In , the admixture of the component from bare state to resonance, while the decrease in is small. In the estimates given below, we assume I A → is significant. φ(1020) to be a pure ss¯ state. The bare states φ(1020) γf0 (980) bare bare f0 (700) and f0 (1220) aremixturesofthenn¯ and ss¯ components, 5.4. Comparison to Data nn¯ cos ϕ + ss¯sin ϕ, Comparing the above-written formulas to experimental data, we have parametrized the wave func- and, according to [1], the mixing angles are as follows: tions of the qq¯ states in the simplest, exponent-type, ◦ ◦ φ(1020) ϕ f bare(700) = −70 ± 10 , (53) form (see Section 2.3). For , we accept its 0 mean radius square to be close to the pion radius, 2 2 R Rπ: both states are members of the same ϕ f bare(1220) =20◦ ± 10◦. φ(1020) 0 36-plet. This value of the mean radius square for −2 Because of that, the transition amplitude for φ(1020) → φ(1020) fixes its wave function by bφ =10GeV . bare γf0(980) reads For f0 (700), we change the value bf0 in the N interval Aφ(1020)→γf (980) (54) − 0 5 ≤ b(bare) ≤ 15 GeV 2, f0 (N) bare ζ [f0(980)] sin ϕ f (700) 2 700 0 which corresponds to the interval (0.5–1.5)Rπ of the (bare) (bare) × F mean radius square of f0 (700). φ(1020)→γfbare (700) 0 Using the branching ratios [22, 23] (N) bare BR[φ(1020) → γf0(980)] (57) + ζ1220[f0(980)] sin ϕ f0 (1220) ± +1.3 × −4 =(3.5 0.3−0.5) 10 , × F (bare) . φ(1020)→γfbare (1220) 0 BR[φ(1020) → γf0(980)] (N) (N) =(2.90 ± 0.21 ± 1.54) × 10−4, Here, ζ700 [f0(980)] and ζ1220[f0(980)] (N = I, II) are given by formulas (51), (52). One can see that numer- and the definition of the radiative decay width, ically the factor ζ [f (980)] sin ϕ × f bare(1220) is 1220 0 0 m2 − m2 1 φ f0 2 small, and we may neglect the second term on the → | → | mφΓφ γf0 = α 2 Aφ γf0 , right-hand side of (54). Then, for the pole which is 6 mφ

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| φ → γ | 0 0 4 A( (1020) f0(980)) , GeV BR (φ(1020) → π π γ) × 10 8 0.3 (a)

6 (calc) 0.2 Adipole

0.1 A(exp) 2

0 5 7 9 111315 0 –2 bf , GeV 8 0 (b) Fig. 9. Amplitudes for the decay φ(1020) → γf0(980): the calculated amplitude A(calc) vs. the experimental one dipole 6 A(exp). we have the following experimental value for the decay amplitude: 4 A(exp) =0.115 ± 0.040 GeV. (58) φ(1020)→γf0(980) 2 Here, α =1/137, mφ =1.02 GeV, and mf0 = 0.975 GeV (the mass reported in [22, 23] for the measured γf (980) signal) and Γ [φ(1020)] = 4.26 ± 0 tot 0 0.05 MeV [20]. The right-hand side of (58) should 0.4 0.6 0.8 1.0 be compared with Aφ(1020)→γf0(980) calculated with Mππ, GeV Eqs. (17), (38), and (55): Fig. 10. The ππ spectrum of the reaction φ(1020) → γππ (calc) calculated with the Flatteformula(´ a) and Eq. (68) (b). A → (dipole) (59) φ(1020)γf0 (980) bare (ss¯) lated amplitude (59) is in a perfect agreement with da- (0.58 ± 0.04) Wqq¯[f (700)]Z → 0 φ γf0 bare ta when Mf0 750–800 MeV,which is just inside 7/4 5/4 27/2 b b theerrorbarsgivenbytheK-matrix analysis [1]. × √ φ f0 m [m − (0.7 ± 0.1) ] . 5/2 s φ GeV 3 (bφ + bf ) 0 6. PION–PION SPECTRUM IN φ(1020) → γππ Recall that, in (59), the factor (0.58 ± 0.04) takes The f0(980) resonance is seen in the reaction into account the change of the transition amplitude φ(1020) → γππ as a peak at the edge of the ππ caused by the final-state hadron interaction, Eq. (55). spectrum. So, it is rather enlightening to calculate The probability to find the quark–antiquark com- the ππ spectrum to be sure that its description agrees bare ponent in the bare state f0 (700) is denoted as both with the quark-model calculation of the form bare Wqq¯[f0 (700)]: one can guess that it is of the order factor Fφ(1020)→γf0 (980) and with the threshold the- of 80–90%, or even more. The mass of the strange orem (cross section tending to zero as ω3 at ω → 0, constituent quark is equal to ms 0.5 GeV.The wave where ω = mφ − Mππ). bare functions of φ(1020) and f0 (700) are parametrized The partial cross section of the decay φ(1020) → −2 0 0 as exponents: we fix bφ =10GeV (which gives for γπ π is given by the following formula: the mean radius of φ(1020) a value of the order of the dΓφ(1020)→γπ0 π0 1 pion radius, R R )andvaryb in the interval 5– = Γ → (60) φ π f0 dM 3 φ(1020) γf0(980) 15 GeV−2. ππ 2 − 2 m Mππ 2M The comparison of the data (58) to the calculated × φ ππ ρ m2 − m2 π ππ amplitude is shown in Fig. 9. We see that the calcu- φ f0

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DECAY φ(1020) → γf0(980) 1567 g 2 form × π + B(M 2 ) . 2 2 2 2 ππ 2 2 2 M − M − ig ρππ − ig ρ ¯ B(M )=C 1+a(M − m ) (64) 0 ππ π K KK ππ ππ φ 2 − 2 The factor 1/3 in front of the right-hand side of (60) is mφ Mππ 0 0 × exp − , associated with the π π channel: Γφ(1020)→γπ0π0 = µ2 (1/3)Γφ(1020)→γππ. Here, for the description of the f (980),weusetheFlatte´ formula [24] with the and the parameter C is ﬁxed by the constraint 0 phase-space factors g π + C =0. 1 2 − 2 − 2 − 2 ρ = M 2 − 4m2 , (61) M0 Mππ igπρππ igK ρKK¯ M =m ππ M ππ π ππ φ 0 (65) 1 2 − 2 ρKK¯ = Mππ 4mK . 0 0 M0 Fitting to the π π spectrum [22] (see Fig. 10a), we have the following values for other parameters: At M 2 < 4m2 , one should replace M 2 − 4m2 → ππ K ππ K 1 = −0.2 GeV2,µ=0.388 GeV. (66) 2 − 2 a i 4mK Mππ. In line with [22, 23, 25], we use the

Flatte´ formula with the parameters For Γφ(1020)→γf0 (980) entering (60), we have used A → =0.13 GeV, which satisﬁes both g2 =0.12 GeV2,g2 =0.27 GeV2, (62) φ(1020) γf0 (980) π K (58) and (59). M0 =0.975 GeV. The Flatte´ formula gives us a rather rough de- The threshold theorem requires scription of the ππ amplitude around the f0(980) resonance. A more precise description may be obtained gπ 2 2 2 2 (63) by using in addition the nonzero transition length M − M − ig ρππ − ig ρ ¯ → ¯ 0 ππ π K KK for ππ KK [21]. For this case, we have formulas analogous to Eq. (60), after replacing the resonance 2 ∼ − + B(Mππ) Mππ mφ, factor gπ Mππ→mφ 2 2 (67) M − M 2 − ig2 ρ − ig ρ ¯ which gives a constraint for the background term 0 ππ π ππ K KK 2 2 B(Mππ).ThetermB(Mππ) is parametrized in the by the following one:

g + iρ ¯ g f π KK K . (68) 2 − 2 − 2 − 2 2 2 − 2 M0 Mππ igπρππ iρKK¯ [gK + iρππ(2gπgK f + f (M0 Mππ))]

The parameters found in [21] are equal to Let us emphasize that the visible width of the f (980) ππ g =0.386 GeV,g=0.447 GeV, (69) 0 signal in the spectrum is comparatively π K large, ∼ 150 MeV, which is related to an essential M0 =0.975 GeV,f=0.516. contribution of the second pole at 960 − 200i MeV.

The transition length aππ→KK¯ is determined by the parameter f as follows: aππ→KK¯ =2f/M0. The description of the π0π0 spectra [22] with- 7. THE ADDITIVE QUARK MODEL, in the resonance formulas (68) is demonstrated in DOES IT WORK? Fig. 10b. In this ﬁt, we have the following parameters 2 for B(Mππ): Let us point to the two aspects of this question. One is the problem of the applicability of the addi- a =0,µ=0.507 GeV. (70) tive quark model to the production of the resonance In this variant of the ﬁtting to spectra, we also used f0(980); the other is the production of the bare state bare ± Aφ(1020)→γf0 (980) =0.13 GeV. f0 (700 100).

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1568 ANISOVICH et al.

→ | φ → γ | φ γf0 A( (1020) f0(980)) , GeV The use of Fµα (additive), Eq. (16), for the calculation of A(calc) results in agreement φ(1020)→γf0(980) 0.14 A(additive) with experimental data. Thus, we have (calc) A → (additive) (72) φ(1020) γf0(980) A(exp) (0.58 ± 0.04) W [f bare(700)]Z(ss¯) qq¯ 0 φ→γf0 0.08 3/4 5/4 27/2 b b × √ φ f0 . 5/2 3 (bφ + bf0 ) (calc) In Fig. 11, one can see A → (additive) 0.02 φ(1020) γf0 (980) versus A(exp) : there is good agreement φ(1020)→γf0(980) 579111315with data. b , GeV–2 f0 The existence of two characteristic sizes in a hadron, namely, hadronic radius and that of the Fig. 11. The additive quark-model-amplitude (72) for constituent quark, may be the reason why the contri- (exp) φ(1020) → γf0(980) vs. A → . bution of the charge-exchange current is small in the φ(1020) γf0 (980) → bare reaction φ(1020) γf0 (700). The relatively small radius of the constituent quark assumes that charge- → bare ± ss¯ → gg → nn¯ 7.1. Process φ(1020) γf0 (700 100) exchange interaction is a short-range one, which causes a smallness of the second term on The additive quark model describes well the pro- the right-hand side of (35). duction of the bare state f bare(700 ± 100),provided Thehadronicsizeisdefined by the confinement 0 ∼ its mass is in the region 750–800 MeV. To see it, radius Rh Rconf, which is of the order of 1 fm for → light hadrons. The constituent-quark size r is much consider Eq. (38) for F φ γf0 (dipole), or Eq. (17), q µα smaller; it is defined, as one may believe, by the where the exponential representation of the quark relatively large mass of the soft gluon (experimental wave functions is used. Formula (17) takes into ac- data [29] and lattice calculations [30] give us m ∼ count both the additive-quark-model processes and g 700 1000 r2/R2 ∼ 0.1 0.2 photon emission by the charge-exchange current, – MeV). So, we get q h – ;the while Eq. (16) gives us the triangle-diagram contri- same value follows from the analysis of soft hadron bution within the additive quark model. The contri- collisions (see [15, 31] and references therein). bution of the charge-exchange current is small when → − 1 7.2. Process φ(1020) γf0(980) ms[mφ Mf bare ] . (71) 0 bφ 2 2 The two sizes, rq and Rh, being accepted, the −2 additive-quark-model contribution dominates the re- At ms =0.5 GeV and bφ =10 GeV ,theequa- action φ(1020) → γf0(980) too, thus allowing direct lity (71) is almost fulfilled, when Mf bare 0.8 GeV. 0 use of the triangle diagram of Fig. 1a for the calcula- Such a magnitude is allowed by the K-matrix fit[1], tion of this process. Such calculations were performed which gives M bare =0.7 ± 0.1 GeV. f0 in [17], revealing reasonable agreement with data. However, let us emphasize that the error bars Once again it should be emphasized that the triangle diagram contribution does not have a particular ±0.1 GeV are rather large in the difference m − φ smallness related to a deceptive proximity of φ(1020) Mf bare : with the lower possible limit Mf bare =0.6 GeV, 0 0 and f0(980). In addition, as was explained above, the we face a two-times disagreement in Eq. (71). Still, poles associated with these resonances are separated one may hardly hope that the K-matrix analysis of from each other in the complex-M plane by nonsmall the 00++ wave would provide us with a tighter re- distances on the hadronic scale. striction for the mass of this bare state, since a large In the literature, there exist rather opposite state- uncertainty in the definition of M bare is not related f0 ments about the possibility to describe the reaction to the data accuracy but to the problem of the light φ(1020) → γf0(980) within the framework of the hy- σ-meson existence (see the discussions in [5, 26–28] pothesis of the qq¯ nature of f0(980). Using the QCD and references therein). sum-rule technique, the authors of [32] evaluated

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DECAY φ(1020) → γf0(980) 1569

M2, GeV2 (a) (b) 6 2450 2330 5 2240 2260 2175 2105 2120 2100 4 2005 1975 1975 1800 1820 1800 3 1750 1750 1650 1500 1535 2 1415 (1/20+) 1220 bare 1300 K0 K0 f (00++) 1230 0 bare 1220 f0 1 980 980 960 ++ ++ bare f (00 ) a0(10 ) f bare 0 0 700 a0

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 n

Fig. 12. Linear trajectories on the (n, M 2) plane for bare states (a) and scalar resonances (b).

the rate of the decay φ(1020) → γf0(980),withfair 8. CONCLUSIONS agreement with data, supposing a sizeable ss¯ com- f ponent in the f (980). Correct determination of the origin of 0(980) is 0 a key for understanding the status of the light σ and The results of the calculation performed in [33] classification of heavier mesons f (1300), f (1500), in the framework of the additive quark model do not 0 0 f0(1750), and the broad state f0(1200–1600). agree with data on the reaction φ(1020) → γf0(980). This calculation, though similar to those of [16, 17], We have shown that experimental data on the led to a different result, so it would be instructive reaction φ(1020) → γf0(980) do not contradict the to compare model parameters used in these two ap- suggestion about the dominance of the quark–anti- proaches. quark component in the f0(980). However, as was In [33] as well as in [16, 17], the exponential emphasized in the Introduction, the final conclusion parametrization of the wave function was used; about the origin of f0(980) should be made on the however, the slopes bφ and bf0 in [33] were consid- basis of the whole availability of arguments, so let us erably smaller (constituent quark masses are smaller enumerate them briefly. −2 too). In [33], bss¯ =2.9 GeV and buu¯ = bdd¯ = (1) There are data on the hadronic decays of the −2 3.7 GeV (mu = md = 220 MeV, ms = 450 MeV), lowest mesons, and the most reliable information on −2 while in [16, 17] bφ 10 GeV and bφ ∼ bf (mu = scalar resonances is given by the K-matrix analy- 0 sis. Summing up, one can state that, according to md = 350 MeV, ms = 500 MeV). In addition, in [33], the scheme of mixing of f0 states was used that was the K-matrix analysis [1], the lowest states f0(980), suggested in [33, 35], where the transitions f bare → f0(1300), a0(980), K0(1430) are the descendants of 0 3 real mesons were not accounted for. Still, as was bare mesons, which have created the 1 P0 multiplet. emphasized above (Section 5.2), just the transitions Because of that, all the decays, namely, bare → ¯ f0 ππ,KK,ηη,ππππ afford the final disposition f0(980) → ππ,KK,¯ (73) of poles in the complex plane, for they are responsible f → ππ,KK,¯ ηη, for the resonance mass shift of the order of 100 MeV 0(1300) (see Fig. 5). a0(980) → KK,πη,¯ In our opinion, the failure of the qq¯ model demon- K0(1430) → Kπ, strated in [33] can attest only to the fact that not any model enables the description of radiative decays. The are described by two parameters only in the leading qq¯ model should be based on the whole set of exper- terms of the 1/Nc expansion, which are the universal imental data but not on the reproduction of several coupling constant and mixing angle for the nn¯ and ss¯ levels of the lowest states. components in the scalar–isoscalar sector.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1570 ANISOVICH et al. + → + (2) Another argument is the systematics of scalar reaction Ds π f0(980) can be reliably calculated 2 states on linear trajectories on the (n, M ) plane. under the assumption that f0(980) is close to the qq¯ For scalar mesons, the trajectories are shown in flavor octet. + → + Fig. 12a—one can see that f0(980) lies comfortably The study of Ds π f0(980) decay in [44–46] on a linear trajectory, together with the other scalars. also led to the conclusion about the ss¯ nature of Such trajectories are formed not only for the scalar f0(980). sector but also for all aggregate of data (see [4, 5]), (5) Concerning radiative decays with the forma- and all the trajectories are characterized by a universal tion of f (980), we see that the transition φ(1020) → slope. 0 γf0(980) can be well described within the approach Similar qq¯ trajectories exist in the (J, M 2) plane of the additive quark model, with the dominant qq¯ too, and f0(980) belongs to one of them. component in the f0(980). Another radiative decay, (3) The alternative to the qq¯ system may be f0(980) → γγ, the partial width of which was mea- the four-quark qqq¯ q¯ or molecular KK¯ structu- sured [47], can also be treated in terms of the qq¯ re [6, 7, 36, 37]. Such a nature of f0(980) would structure of f0(980) [16, 48]. The values of partial mean that f0(980) was a loosely bound system, but widths in both decays support the conclusion made the experiment tells us that this is not so. The point is in [1] that the flavor content of f0(980) is close to the that octet one. (i) f0(980) is easily produced at large momenta ACKNOWLEDGMENTS − → transferred to the nucleon in the reaction π p We thank Ya.I. Azimov, L.G. Dakhno, S.S. Gersh- f0(980)n [38, 39], tein, Yu.S. Kalashnikova, A.K. Likhoded, M.A. Mat- 0 (ii) f0(980) is produced in Z -boson decays [40], veev, V.A. Markov, D.I. Melikhov, A.V. Sarantsev, and W.B. von Schlippe for helpful and stimulating (iii) f0(980) is produced in central pp collisions at high energies [41]. discussions of the problems involved. This work is supported by the Russian Foundation Were the f0(980) a loosely bound system, these processes would be suppressed. for Basic Research, project no. 04-02-17091. (4) The f0(980) resonance is produced in hadronic + → + decays of the Ds meson, Ds π f0(980),witha Appendix probability comparable with that for the transition D+ → π+φ(1020) [42]. These two reactions are due s DIPOLE EMISSION OF PHOTON to the weak decay of the c quark, c → π+s;asto f0(980), it is formed, like φ(1020),bythess¯ system To describe the interaction of a composite system in the transition ss¯ → f0(980). The calculation of this with an electromagnetic field, we consider the full process [43] shows us that the f0(980) yield in the Hamiltonian

2 2 k1 k2 + + U → (r − r ) U → (r − r ) 2m 2m ss¯ ss¯ 1 2 ss¯ gg 1 2 Hˆ (0) = 2 2 . (A.1) − k1 k2 − Uss¯→gg(r1 r2) + + Ugg→gg(r1 r2) 2mg 2mg

Here, the coordinates (ra) and momenta (ka = The electromagnetic interaction is included by sub- −i∇a) of the constituents are related to the charac- stituting in (8) as follows: teristics of the relative movement, entering (31), as 2 → − 2 follows: k1 (k1 e1A(r1)) , (A.3) 2 → − 2 1 1 k2 (k2 e2A(r2)) , r = r + R, r = − r + R, (A.2) 1 2 2 2 1 −1 k1 = k + P, k2 = k + P. 2 2 Uss¯→gg(r1 − r2) → Uss¯→gg(r1 − r2)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 DECAY φ(1020) → γf0(980) 1571   r1 r2 and matrix χˆ reads ×   exp ie1 drαAα(r )+ie2 drαAα(r ) , −∞ −∞ exp[ie χ(r ) − ie χ(r )] 0 χˆ = s 1 s 2 . (A.6) with e1 = −e2 = es. After that, we obtain the gauge- 01 invariant Hamiltonian Hˆ (A): Hˆ (A)=ˆχ+Hˆ (A + ∇χ)ˆχ, (A.4) For the transition φ → γf0, keeping the terms pro- ∇ where A + χ means the substitution portional to the s-quark charge, es, we have the fol- A(ra) → A(ra)+∇χ(ra), (A.5) lowing operator for the dipole emission:

− − ˆ − ˆ 2(k1α k2α) i(r1α r2α)Uss¯→gg(r1 r2) dα = . (A.7) −i(r1α − r2α)Uˆss¯→gg(r1 − r2)0

There exist other mechanisms of photon emis- 12. V. V. Anisovich and M. A. Matveev, Yad. Fiz. 67, 634 sion which, being beyond the additive quark model, (2004); hep-ph/0303119. lead us to the dipole formula for V → γS transition; 13. S. Malvezzi, Talk Presented at “Hadron-03,” an example is given by (L · S) interaction in the Aschaﬀensburg, Germany, August–September, quark–antiquark component [9, 10, 49]. The short- 2003. range (L · S) interaction in the qq¯ systems was 14. A. V. Anisovich and V. V. Anisovich, Phys. Lett. B 467, 289 (1999). discussed [50, 51] as a source of the nonet split- · 15. V. V. Anisovich, M. N. Kobrinsky, J. Nyiri, and ting. Actually, the pointlike (L S) interaction gives Yu. M. Shabelski, Quark Model and High-Energy (v/c) corrections to the nonrelativistic approach. In Collisions, 2nd ed. (World Sci., Singapore, 2004). the relativistic quark model approaches based on the 16. A. V. Anisovich, V. V. Anisovich, and V. A. Nikonov, Bethe–Salpeter equation, the gluon-exchange forces Eur.Phys.J.A12, 103 (2001). result in similar nonet splitting as for the (L · S) in- 17. A. V. Anisovich, V. V. Anisovich, V. N. Markov, and teraction (see, for example, [52]). V. A. Nikonov, Yad. Fiz. 65, 523 (2002) [Phys. At. Nucl. 65, 497 (2002)]. 18. A. V. Anisovich, V. V. Anisovich, and V. A. Nikonov, REFERENCES hep-ph/0305216. 1. V. V. Anisovich and A. V. Sarantsev, Eur. Phys. J. A 19. G. S. Bali et al., Phys. Lett. B 309, 378 (1993); 16, 229 (2003). J. Sexton, A. Vaccarino, and D. Weingarten, Phys. 2. V. V.Anisovich, A. A. Kondashov, Yu. D. Prokoshkin, Rev. Lett. 75, 4563 (1995); C. J. Morningstar and et al.,Yad.Fiz.63, 1489 (2000) [Phys. At. Nucl. 63, M. Peardon, Phys. Rev. D 56, 4043 (1997). 1410 (2000)]. 20. PDG (D. E. Groom et al.),Eur.Phys.J.C15,1 3. V. V. Anisovich and A. V. Sarantsev, Phys. Lett. B (2000). 382, 429 (1996); V. V. Anisovich, Yu. D. Prokoshkin, 21. V. V. Anisovich, V. A. Nikonov, and A. V. Sarantsev, and A. V. Sarantsev, Phys. Lett. B 389, 388 (1996). Yad. Fiz. 65, 1583 (2002) [Phys. At. Nucl. 65, 1545 4. A. V.Anisovich, V.V.Anisovich, and A. V. Sarantsev, (2002)]; 66, 772 (2003) [66, 741 (2003)]. Phys.Rev.D62, 051502 (2000); V.V.Anisovich, hep- 22. M. N. Achasov et al., Phys. Lett. B 485, 349 (2000). ph/0310165. 23. R. R. Akhmetshin et al., Phys. Lett. B 462, 380 5 . V. V. A n i s o v i c h , U s p . F i z . N a u k 174, 49 (2004) [Phys. (1999). Usp. 47, 45 (2004)]; hep-ph/0208123, v3. 24. S. M. Flatte,´ Phys. Lett. B 63B, 224 (1976). 6. R. L. Jaﬀe, Phys. Rev. D 15, 267 (1977). 25. A. V. Sarantsev, private communication. 7. J. Weinshtein and N. Isgur, Phys. Rev. D 41, 2236 26. S. F. Tuan, hep-ph/0303248. (1990). 27. E. van Beveren et al., Mod. Phys. Lett. A 17, 1673 8. F. E. Close et al., Phys. Lett. B 319, 291 (1993). (2002). 9. N. Byers and R. McClary, Phys. Rev. D 28, 1692 28. W. Ochs and P. Minkowski, Nucl. Phys. B (Proc. (1983). Suppl.) 121, 123 (2003). 10. A. LeYaouanc, L. Oliver, O. Pene, and J. C. Raynal, 29. G. Parisi and R. Petronzio, Phys. Lett. B 94B,51 Z. Phys. C 40, 77 (1988). (1980); M. Consoli and J. H. Field, Phys. Rev. D 49, 11. A. J. F. Siegert, Phys. Rev. 52, 787 (1937). 1293 (1994).

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PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1573–1586. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1633–1646. Original English Text Copyright c 2005 by A. Anisovich, V. Anisovich, Markov, Matveev, Sarantsev. ELEMENTARY PARTICLES AND FIELDS Theory

Quark–Antiquark Composite Systems: the Bethe–Salpeter Equation in the Spectral-Integration Technique intheCaseofDiﬀerent Quark Masses* A. V. Anisovich, V. V. Anisovich**, V.N.Markov,M.A.Matveev,andA.V.Sarantsev Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina, 188350 Russia Received May 24, 2004; in ﬁnal form, December 22, 2004

Abstract—The Bethe–Salpeter equations for quark–antiquarkcomposite systems with di fferent quark masses, such as qs¯ (with q = u,d), qQ¯,andsQ¯ (with Q = c, b), are written in terms of spectral integrals. For mesons characterized by the mass M,spinJ, and radial quantum number n, the equations are written for the (n, M 2) trajectories with fixed J. The mixing between states with different quarkspin S and angular momentum L is also discussed. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION In this paper, we present final formulas for the (n, M 2) The relativistic description of quark–antiquark trajectories; the details of the calculations states is a necessary step for meson systematics may be found in [7]. It should be immediately empha- and the search for exotic states. The standard way sized that the case of different masses requires more to take account of relativistic effects is to use the cumbersome calculations. In particular, the mixing of Bethe–Salpeter equation [1]. Different versions of the states with J = L, S =0and J = L, S =1should be Bethe–Salpeter equations applied to the description accounted for (here, L and S are the orbital momen- of quark–antiquarksystems may be found in [2 –6]. In tum and spin of quarks, respectively). the present paper, we develop the approach suggested Note that, for systems with equal quarkmasses, in [7] for the Bethe–Salpeter equation written for we have already obtained numerical results for a set 2 quark–antiquarksystems in terms of spectral inte- of the (n, M ) trajectories such as a0,a1,a2,π,ρ,and grals. In [7], the systems of quarks with equal masses b1. So, we hope that we have elaborated a rather have been considered, such as uu,¯ ud,¯ dd¯,andss¯.In efficient technique allowing us to find realistic wave this paper, the systems with unequal masses, like qs¯ functions for the quark–antiquarksystems. Certain (q = u, d)andqQ¯ (Q = c, b), are treated. aspects of numerical solutions of the Bethe–Salpeter A detailed presentation of the spectral-integration equations in terms of the spectral integrals are dis- method as well as the emphasis on its advantages was cussed in [8]. given in [7], so we need not repeat ourselves. Let us The paper is organized as follows. In Section 2, only stress a particular feature of the method: it is we define the quantities entering the Bethe–Salpeter rather easy to control the quark–gluonium content of equation: for equal masses, they were introduced the composite system that is rather important for the in [7]; now, we expand the definition for unequal search for exotics. Another advantage consists in the masses. In Section 3, the equations for the (n, M 2) easy treatment of the systems with high spins. trajectories are considered for two different cases, Similarly to [7], we present here the equations for for J = L, S =0 and J = L, S =1 states. The the group of states lying on the (n, M 2) trajectory. technicalities related to the trace calculations of loop Such trajectories are linear, and they are suitable for diagrams as well as trace factor convolutions are the reconstruction of interaction between quarks at considered in Appendices A and B. large distances. We hope, by investigating high-spin quark–antiquarkstates, to obtain decisive information on the structure of forces in the region of r ∼ 2. QUARK–ANTIQUARK COMPOSITE Rconf. SYSTEMS

∗This article was submitted by the authors in English. In the spectral integral technique, the Bethe– ** e-mail: [email protected] Salpeter equation for the wave function of the q1q¯2

1063-7788/05/6809-1573$26.00 c 2005 Pleiades Publishing, Inc. 1574 ANISOVICH et al. system with the total momentum J, angular momen- + g⊥ g⊥ k⊥ k⊥ ...k⊥ + ... + ... , tum L = |J − S|,andquark–antiquarkspin S can be µ1µ2 µ3µ5 µ4 µ6 µJ schematically written as − (J−1) 2J 1 2 (S,L,J) Z (k⊥)= s − M Ψ (k⊥) (1) µ1...µJ ,α 2 (n)µ1...µJ L d3k J ⊥ ˆ (J−1) ⊥ 2 = Φ(s )V s, s , (k⊥k⊥) (k1 + m1) × X (k⊥)g − (2π)3 µ1...µi−1µi+1...µJ µiα 2J − 1 i=1 (S,L,J) × Ψ (k⊥)(−kˆ + m2), J (n) µ1...µJ 2 ⊥ (J−1) × g X (k⊥) . 2 2 2 µiµj µ1...µi−1µi+1...µj−1µj+1...µJ α where the quarks are mass-on-shell: k1 = k1 = m1 i,j=1 2 2 2 i

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1575 +(iσ ⊗ iσ )V s, s , (k⊥k ) 3.1. The Equation for the S = 0,1, J = L State µν µν T ⊥ +(iγ γ ⊗ iγ γ )V s, s , (k⊥k ) There are two equations for the two states with µ 5 µ 5 A ⊥ S =0, 1 and J = L. Their wave functions are denoted +(γ5 ⊗ γ5)VP s, s , (k⊥k⊥) . (0,J,J) (1,J,J) as CiΨ(n)µ ...µ (k⊥)+DiΨ(n)µ ...µ (k⊥),withi = (S,L,J) 1 J 1 J LetusmultiplyEq.(1)bytheoperatorQµ1...µJ (k⊥) 1, 2. These wave functions are orthogonal to each and convolute over the spin–momentum indices: other. Normalization and orthogonality conditions give three constraints for four mixing parameters Ci 2 (S,L,J) and D . s − M tr Ψ (k⊥)(k1 + m1) (12) i (n)µ1...µJ Each wave function obeys two equations: 2 (J) (s − M )X (k⊥) (16) (S,L,J) µ1...µJ × Q (k⊥)(−k + m ) µ1...µJ 2 2 × (J) (0,J,J) 2 tr iγ5CiXµ ...µ (k⊥)ψn (k⊥) 1 J (S,L,J) = tr F (k + m )Q (k⊥)(−k + m ) ⊥ (J) (1,J,J) 2 c 1 1 µ1...µJ 2 2 + γ D ε P Z (k⊥)ψ (k ) α i αν1ν2ν3 ν1 ν2µ1...µJ ,ν3 n ⊥ c 3 × − (J) d k⊥ (k1 + m1)iγ5( k2 + m2) = Xµ ...µ (k⊥) × Φ(s )V s, s , (k⊥k⊥) 1 J (2π)3 c × tr F (k + m )iγ (−k + m ) × − (S,L,J) c 1 1 5 2 2 tr (k1 + m1)Fc( k2 + m2)Ψ (k⊥) . c (n)µ1...µJ 3 d k⊥ We have four states with the q1q¯2 spins S =0and × ⊥ 3 Φ(s )Vc s, s , (k k⊥) S =1: (2π) (i) S =0; L = J; × (J) (0,J,J) 2 tr iγ5CiXµ ...µ (k⊥)ψn (k⊥) (ii) S =1; L = J +1,J,J − 1, 1 J ⊥ (J) (1,J,J) 2 which are mixed and form two final states. The wave + γα Diεα ν1ν2ν3 Pν Zν µ ...µ ,ν (k⊥)ψn (k⊥) functions read 1 2 1 J 3 for S =0, 1, J = L, × − (k1 + m1)Fc( k2 + m2) (Si,J,J) (0,J,J) Ψ(n)µ ...µ (k⊥)=CiΨ(n)µ ...µ (k⊥) (13) 1 J 1 J and (1,J,J) 2 (J) + DiΨ (k⊥), s − M εβν ν ν Pν Z (k⊥) (17) (n)µ1...µJ 1 2 3 1 ν2µ1...µJ ,ν3 × (J) (0,J,J) 2 where Ci and Di are the mixing coefficients with i = tr iγ5CiXµ ...µ (k⊥)ψn (k⊥) 1, 2; 1 J ⊥ (J) (1,J,J) 2 for S =1,L= J ± 1,J, ⊥ + γα Diεαν1ν2ν3 Pν1 Zν2µ1...µJ ,ν3 (k )ψn (k⊥) (1,(J±1)j ,J) ⊥ Ψ(n)µ ...µ (k⊥) (14) × − 1 J (k1 + m1)γβ ( k2 + m2) (1,J−1,J) (1,J+1,J) = AjΨ (k⊥)+BjΨ (k⊥), (J) (n)µ1...µJ (n)µ1...µJ = εβν ν ν Pν Z (k⊥) 1 2 3 1 ν2µ1...µJ ,ν3 where Aj and Bj are the mixing coefficients with j = × ⊥ − 1, 2. tr Fc(k1 + m1)γβ ( k2 + m2) These wave functions are normalized: c 3 3 d k⊥ d k⊥ × Φ(s )V s, s , (k⊥k ) Φ(s)(−1) (15) 3 c ⊥ (2π)3 (2π) × (J) (0,J,J) 2 (S ,L ,J ) tr iγ5CiXµ ...µ (k⊥)ψn (k⊥) × j (S,Lj ,J) 1 J tr Ψ(n)µ ...µ (k⊥)(k1 + m1)Ψ(n)µ ...µ (k⊥) 1 J 1 J ⊥ (J) (1,J,J) 2 + γ D ε P Z (k )ψ (k ) α i α ν1ν2ν3 ν1 ν2µ1...µJ ,ν3 ⊥ n ⊥ J × − × (−k2 + m2) =(−1) δS SδL L δJ J δn n. (k1 + m1)Fc( k2 + m2) . j j Now consider the left-hand side of Eq. (16). Using the traces written in Appendix A and convolution of 3. EQUATIONS FOR (n, M 2)TRAJECTORIES operators from Appendix B, we have In this section, we write the trajectories for J = (J) X (k⊥) iγ (k + m )iγ (−k + m ) L, S =0, 1 and J = L ± 1,S =1states. µ1...µJ tr 5 1 1 5 2 2

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1576 ANISOVICH et al. (J) 2 2J × ⊥ − − − Xµ1...µJ (k )= 2(s ∆ )α(J)k⊥ , = tr Fc(k1 + m1)iγ5( k2 + m2) c=T,A (J) X (k⊥) (18) ⊥ µ1...µJ × − tr γα (k1 + m1)Fc( k2 + m2) × ⊥ − tr γα (k1 + m1)iγ5( k2 + m2) × Vc s, s , (k⊥k⊥) . (J) × ε P Z (k⊥)=0. αν1ν2ν3 ν1 ν2µ1...µJ ,ν3 In Appendix A, the trace calculations are presented, Here and below, we use the following notation: and the values Ac (s, s , (k⊥k⊥)), Cc (s, s , (k⊥k⊥)) − δ = m2 m1,σ = m2 + m1. Also, the left-hand side are given. So, after the summation, Ac, Bc, Cc are of (17) contains two convolutions: written as follows: (J) εβν1ν2ν3 Pν1 Zν µ ...µ ,ν (k⊥) (19) A s, s , (k⊥k⊥) (23) 2 1 J 3 × ⊥ − tr iγ5(k1 + m1)γβ ( k2 + m2) = Ac s, s , (k⊥k⊥) Vc s, s , (k⊥k⊥) c=T,A,P (J) × X (k⊥)=0, µ1...µJ √ = −16(k⊥k⊥) 2 ss VT s, s , (k⊥k⊥) (J) ε P Z (k⊥) βν1ν2ν3 ν1 ν2µ1...µJ ,ν3 ⊥ ⊥ 2 2 2 × − +∆ VA s, s , (k⊥k⊥) − 4(s − ∆ )(s − ∆ ) tr γα (k1 + m1)γβ ( k2 + m2) (J) × εαν ν ν Pν Z (k⊥) σσ 1 2 3 1 ν2µ1...µJ ,ν3 × √ ⊥ ⊥ VA s, s , (k k⊥) + VP s, s , (k k⊥) , J(2J +3)2 ss = −2s(s − ∆2) α(J)k2J . 3 ⊥ (J +1) Bβα s, s , (k⊥k⊥) (24) The right-hand side of Eq. (17) is calculated in two steps. First, we summarize over c: = (B ) s, s , (k⊥k⊥) c β α c=T,A,V,S A s, s , (k⊥k⊥) (20) × Vc s, s , (k⊥k⊥) = Ac s, s , (k⊥k⊥) Vc s, s , (k⊥k⊥) ⊥ 2 2 c=T,A,P =4g (s − ∆ )(s − ∆ ) β α = tr Fc(k1 + m1)iγ5(−k2 + m2) × VV s, s , (k⊥k⊥) c=T,A,P 2 × − √σ tr iγ5(k1 + m1)Fc( k2 + m2) +2 VT s, s , (k⊥k⊥) ss × √ Vc s, s , (k⊥k⊥) , +4 k2 k 2z ss V s, s , (k⊥k ) ⊥ ⊥ A ⊥ Bβα s, s , (k⊥k⊥) (21) 2 +2∆ VT s, s , (k⊥k⊥) = (Bc)βα s, s , (k⊥k⊥) ⊥ ⊥ 2 +16k k σ V s, s , (k⊥k⊥) c=T,A,V,S β α S × V s, s , (k⊥k ) 2 2 c ⊥ σ ∆ + √ VV s, s , (k⊥k⊥) ⊥ − ss = tr Fc(k1 + m1)γβ ( k2 + m2) c=T,A,V,S 2 2 +4 k⊥ k⊥zVV s, s , (k⊥k⊥) ⊥ × γ k m F −k m √ tr α ( 1 + 1) c( 2 + 2) ⊥ ⊥ − 16k k ss V s, s , (k⊥k⊥) β α A × Vc s, s , (k⊥k⊥) , 2 +2∆ VT s, s , (k⊥k⊥) and ⊥ ⊥ − 2 Cα s, s , (k⊥k⊥) (22) +16kβ kα s ∆ VV s, s , (k⊥k⊥) c ⊥ ⊥ − 2 = Cα s, s , (k⊥k⊥) Vc s, s , (k⊥k⊥) +16kβ kα s ∆ VV s, s , (k⊥k⊥) , c=T,A

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1577 and 2 × P (z) − JPJ+1(z) J +1 J Cα s, s , (k⊥k⊥) (25) c × s ∆VA s, s , (k⊥k⊥) = C s, s , (k⊥k⊥) Vc s, s , (k⊥k⊥) α √ c=T,A +2∆ ss VT s, s , (k⊥k⊥) . =8 2∆ε V s, s , (k⊥k ) α kk P A ⊥ + σε V s, s , (k⊥k ) For the right-hand side of (17), we have α Pk P A ⊥ (J) +4∆ε V s, s , (k⊥k ) εβν ν ν Pν Z (k⊥) (28) α Pkk T ⊥ 1 2 3 1 ν2µ1...µJ ,ν3 × (J) +2σεα PkP VT s, s , (k⊥k⊥) . Cβ s, s , (k⊥k⊥) Xµ ...µ (k⊥) 1 J J+1 ≡ Here we used a shorthand notation: εα kk P 2J +3 2 2 =16 α(J) k⊥ k⊥ kβkµPνεαβµν. Second, the convolution of operators J +1 is performed by using equations of Appendix B and 2 recurrent formulas for the Legendre polynomials × P (z) − JP (z) J +1 J J+1 J +1 J zP (z)= P (z)+ P − (z), × J 2J +1 J+1 2J +1 J 1 s∆VA s, s , (k⊥k⊥) √ which allows us to represent the Bethe–Salpeter +2∆ ss VT s, s , (k⊥k⊥) equation in terms of the Legendre polynomials. As a result, we get and (J) (J) (J) ε P Z (k⊥) Xµ ...µ (k⊥)A s, s , (k⊥k⊥) Xµ ...µ (k⊥) (26) β ν1ν2ν3 ν1 ν2µ1...µJ ,ν3 (29) 1 J 1 J J × Bβα s, s , (k⊥k⊥) εαν ν ν − 2 2 1 2 3 = 4α(J) k⊥ k⊥ (J) × P Z (k⊥) ν1 ν2µ1...µJ ,ν3 J √ √ 2 J +1 2 2 J(2J +3) × 4 k⊥ k⊥ 2 ss VT s, s , (k⊥k⊥) = −4 ss α(J) k2 k 2 2J +1 (J +1)3 ⊥ ⊥ √ 2 J 2 2 +∆ VA s, s , (k⊥k⊥) PJ+1(z) × 4 k k ss V s, s , (k⊥k⊥) 2J +1 ⊥ ⊥ A − 2 − 2 +(s ∆ )(s ∆ ) 2 +2∆ VT s, s , (k⊥k⊥) PJ+1(z) 2 σ × √ VA s, s , (k⊥k⊥) ss − 2 − 2 +(s ∆ )(s ∆ ) VV s, s , (k⊥k⊥) + VP s, s , (k⊥k⊥) 2 σ +2√ VT s, s , (k⊥k⊥) PJ (z) ss J 2 2 × PJ (z)+4 k⊥ k⊥ √ 2J +1 J +1 2 2 +4 k k ss V s, s , (k⊥k⊥) √ 2J +1 ⊥ ⊥ A × 2 ss VT s, s , (k⊥k⊥) 2 +2∆ VT s, s , (k⊥k⊥) PJ−1(z) . 2 +∆ VA s, s , (k⊥k⊥) PJ−1(z) Expanding the interaction blockin a series of Leg- and endre polynomials, (J) X (k⊥)C s, s , (k⊥k ) (J) µ1...µJ α ⊥ (27) Vc s, s , (k⊥k⊥) = Vc s, s PJ (z) (30) (J) J × ε P Z (k ) α ν1ν2ν3 ν1 ν2µ1...µJ ,ν3 ⊥ J J+1 (J) 2J +3 = V s, s α(J) − k2 k 2 P (z), =16 α(J) k2 k2 c ⊥ ⊥ J J +1 ⊥ ⊥ J

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1578 ANISOVICH et al. ∞ and integrating over angle variables on the right-hand ds 2J +3 2 2 J+1 side by taking account of the standard normalization = − ρ(s ) · 8 k⊥(−k⊥) π J +1 dz 2 1 2 (m1+m2) condition − P (z)=1/(2J +1),wehaveﬁnally 1 2 J × (0,J,J) 2 − − 2 2 (0,J,J) ψn (s )Dj (2J 4a 1) (s − M ) (s − ∆ )ψ (s)Ci (31) J +1 n a ∞ − 2(a+1) ds · − 2 J (0,J,J) × ξ(J − 2a − 1) − k2 k2 = ρ(s ) 2( k⊥) ψn (s )Cj ⊥ ⊥ π 2 (m1+m2) × (J−2a−1) s∆VA s, s √ J +1 2 2 − − × − 4ξ(J +1) k⊥k⊥ (J 2a 1) 2J +1 +2∆ ss VT s, s √ (J+1) 2 (J+1) − J+1 × 2 ss V s, s +∆ V s, s Jξ(J +1) s∆VA s, s T A 2 2 √ + ξ(J)(s − ∆ )(s − ∆ ) J+1 +2∆ ss VT s, s 2 σ (J) (J) × √ V s, s + V s, s ∞ ss A P √ 2 ds J(2J +3) 2 J + ρ(s ) · 2 ss (−k⊥) J π (J +1)3 − − 2 4ξ(J 1) (m1+m2) 2J +1 √ J (J−1) (J−1) × (1,J,J) − 2 2 × 2 ψn (s )Dj 4ξ(J +1) k⊥k⊥ 2 ss VT s, s +∆ VA s, s 2J +1 √ ∞ × (J+1) 2 (J+1) ss VA s, s +2∆ VT s, s ds 2J +3 2 2 J+1 − ρ(s ) · 8 k⊥(−k⊥) π J +1 + ξ(J)(s − ∆2)(s − ∆2) 2 (m1+m2) 2 σ (J) (J) 2 × 2√ V s, s + V s, s × ψ(1,J,J)(s)D (2J − 4a − 1) ss T V n j J +1 a J +1 −2(a+1) − 4ξ(J − 1) 2J +1 × ξ(J − 2a − 1) − k2 k2 ⊥ ⊥ √ − − × ssV (J 1) s, s +2∆2V (J 1) s, s . × (J−2a−1) A T s ∆VA s, s √ (J−2a−1) The normalization and orthogonality conditions look +2∆ ssV s, s T as follows: ∞ (J+1) − Jξ(J +1) s ∆V s, s 2 A ds 2 (0,J,J) 2 · ρ(s) Ci ψn (k⊥) 2α(J) (34) π √ 2 (J+1) (m2+m1) +2∆ ss V s, s , T 2 × − 2 J − 2 2 (1,J,J) 2 ( k⊥) s ∆ + Di ψn (k⊥) where J(2J +3)2 × 2α(J)(−k2 )J s s − ∆2 =1, − − ⊥ 3 PJ (z)= (2J 4a 1)PJ−2a−1(z). (32) (J +1) a i =1, 2, and Equation (17) reads ∞ ds 2 2 (0,J,J) 2 · J(2J +3) ρ(s) C1C2 ψn (k⊥) 2α(J) (s − M 2) s(s − ∆2) ψ(1,J,J)(s)D π 3 n i 2 (J +1) (m2+m1) (33) (35)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1579 2 (J+1) (1,J+1,J) 2 × − 2 J − 2 (1,J,J) 2 + B X (k⊥)ψ (k⊥) ( k⊥) s ∆ + D1D2 ψn (k⊥) j µ1...µJ α n 2 (J−1) 2 J 2 J(2J +3) = Zµ ...µ ,β (k⊥) × 2α(J)(−k⊥) s s − ∆ =0. 1 J (J +1)3 × ⊥ − tr Fc(k1 + m1)γβ ( k2 + m2) As the result of our calculations, we observe c the mixing of singlet (J = L, S =0) and triplet 3 d k⊥ × ⊥ (J = L, S =1) states and that is proportional to 3 Vc s, s , (k k⊥) the quarkmass di ﬀerence. This is a consequence of (2π) C-parity breaking for the quark–antiquarksystem × ⊥ − tr γα (k1 + m1)Fc( k2 + m2) with nonequal quarkmasses. The best knownexam- ples of such a system is K and B mesons. × (J−1) (1,J−1,J) 2 AjZµ ...µ ,α (k⊥)ψn (k⊥) The mixing between S =1, J = L and S =0, J = 1 J L states gives rise to a system of equations where, (J+1) (1,J+1,J) 2 + BjX (k⊥)ψn (k⊥) . for states with total spin J, we need to know all µ1...µJ α lower projections of the potential on the Legendre First, consider Eq. (36). On the left-hand side of (36), polynomials, not only the J +1, J, J − 1 ones. one has two convolutions: (J+1) ⊥ X (k⊥)tr γ (k + m ) (38) µ1...µJ β α 1 1 3.2. The Equations for the S = 1, J = L ± 1 States × ⊥ − (J+1) We have two equations for the two states with γβ ( k2 + m2) Xµ ...µ α(k⊥) S =1and J = L ± 1. Their wave functions are de- 1 J − (1,J 1,J) (1,J+1,J) 2J +1 noted as AjΨ(n)µ ...µ (k⊥)+BjΨ(n)µ ...µ (k⊥),with 2(J+1) 2 2 1 J 1 J =2α(J)k⊥ (s − ∆ )+4k⊥ , j =1, 2. These wave functions are orthogonal. The J +1 normalization and orthogonality conditions give three (J+1) ⊥ ⊥ constraints for four mixing parameters Aj and Bj. X (k⊥)tr γ (k + m )γ (−k + m ) µ1...µJ β α 1 1 β 2 2 Each wave function obeys two equations: (J−1) 2(J+1) × Z (k⊥)=8α(J)k⊥ . 2 (J+1) µ1...µJ ,α s − M X (k⊥) (36) µ1...µJ β Also, the left-hand side of (37) contains two convolu- × ⊥ ⊥ − tions: tr γα (k1 + m1)γβ ( k2 + m2) (J−1) ⊥ ⊥ Z (k⊥)tr γ (k1 + m1)γ (−k2 + m2) × (J−1) (1,J−1,J) 2 µ1...µJ ,β α β AjZµ ...µ ,α(k⊥)ψn (k⊥) 1 J (39) (J+1) (1,J+1,J) 2 (J+1) 2(J+1) + BjXµ ...µ α(k⊥)ψn (k⊥) × X (k⊥)=8α(J)k , 1 J µ1...µJ α ⊥ (J+1) = X (k⊥) − µ1...µJ β (J 1) ⊥ Zµ ...µ ,β(k⊥)tr γα (k1 + m1) × ⊥ − 1 J tr Fc(k1 + m1)γβ ( k2 + m2) c ⊥ (J−1) × γ (−k2 + m2) Z (k⊥) 3 β µ1...µJ ,α d k⊥ × ⊥ 3 Vc s, s , (k k⊥) (2π) 2(J−1) 2J +1 2 2 =2α(J)k⊥ (s − ∆ )+4k⊥ . × ⊥ − J tr γα (k1 + m1)Fc( k2 + m2) The right-hand sides of Eqs. (36) and (37) are (J−1) (1,J−1,J) 2 × AjZ (k⊥)ψ (k⊥) determined by the convolutions of the trace factor µ1...µJ ,α n Bβα (s, s , (k⊥k⊥)) [see Eqs. (24)] with angular- (J+1) (1,J+1,J) 2 + B X (k⊥)ψ (k⊥) momentum wave functions; the corresponding for- j µ1...µJ α n mulas may be found in Appendix B. Taking them into and account, one has for the right-hand side of (36) − 2 (J 1) (J+1) (J+1) s − M Z (k⊥) (37) µ1...µ ,β X (k⊥)Bβ α s, s , (k⊥k⊥) X (k⊥) J µ1...µJ β µ1...µJ α × ⊥ ⊥ − (40) tr γα (k1 + m1)γβ ( k2 + m2) J+1 (J−1) (1,J−1,J) 2 2 2 × ⊥ =4α(J) k⊥ k⊥ AjZµ1...µJ ,α(k )ψn (k⊥)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1580 ANISOVICH et al. 2J +1 2 2 2 J +1 2 × (s − ∆ )(s − ∆ ) × s − ∆ +4 k⊥ J +1 2J +1 2 2 σ × k⊥VV (s, s , (k⊥k⊥))PJ+1(z) × ⊥ √ ⊥ VV (s, s , (k k⊥)) + 2 VT (s, s , (k k⊥)) ss 2 2 2 σ√∆ − 2 2 + σ VS(s, s , (k⊥k⊥)) + VV (s, s , (k⊥k⊥)) +4(s ∆ )k⊥VV (s, s , (k⊥k⊥)) ss 2 2 +4(s − ∆ )k⊥VV (s, s , (k⊥k⊥)) √ 2 − ss VA(s, s , (k⊥k⊥)) − 2∆ VT (s, s , (k⊥k⊥)) J +1 2 2 +16 k⊥k⊥V (s, s , (k⊥k⊥)) 2J +1 V 2 2 2 J 2 × k k P (z)+ s − ∆ +4 k⊥ ⊥ ⊥ J 2J +1 2 × PJ+1(z)+4 σ VS(s, s , (k⊥k⊥)) 2 × k⊥VV (s, s , (k⊥k⊥))PJ−1(z) , 2 2 σ ∆ + √ VV (s, s , (k⊥k⊥)) ss and J √ (J−1) Z (k⊥)Bβα s, s , (k⊥k⊥) (43) + ss V (s, s , (k⊥k⊥)) µ1...µJ ,β J +1 A J−1 (J−1) 2 2 × Z (k⊥)=4α(J) k⊥ k⊥ 2 2 2 µ1...µJ ,α +2∆ VT (s, s , (k⊥k⊥)) k⊥ k⊥PJ (z) J +1 2 2 × 16 k⊥k⊥VV (s, s , (k⊥k⊥))PJ+1(z) J 2 2 2J +1 +16 k⊥k VV (s, s , (k⊥k⊥))PJ−1(z) 2J +1 2 +4 σ VS(s, s , (k⊥k⊥)) and (J+1) 2 2 X (k⊥)Bβα s, s , (k⊥k⊥) (41) σ ∆ µ1...µJ β √ + VV (s, s , (k⊥k⊥)) (J−1) 2 ss × Z (k⊥)=16α(J)k⊥ µ1...µJ ,α √ J +1 J−1 + ss VA(s, s , (k⊥k⊥)) 2 2 2 J +1 2 J × k⊥ k⊥ s − ∆ +4 k⊥ 2J +1 2 2 2 +2∆ VT (s, s , (k⊥k⊥)) k⊥ k⊥PJ (z) 2 × k V (s, s , (k⊥k ))P (z) ⊥ V ⊥ J+1 2 2 2J +1 2 σ ∆ − 2 − 2 + σ VS(s, s , (k⊥k⊥)) + √ VV (s, s , (k⊥k⊥)) + (s ∆ )(s ∆ ) ss J √ 2 − ss VA(s, s , (k⊥k⊥)) σ × VV (s, s , (k⊥k⊥)) + 2√ VT (s, s , (k⊥k⊥)) ss − 2 2∆ VT (s, s , (k⊥k⊥)) 2 2 +4(s − ∆ )k⊥VV (s, s , (k⊥k⊥)) 2 2 +4(s − ∆ )k⊥V (s, s , (k⊥k⊥)) 2 2 2 J 2 V × k k P (z)+ s − ∆ +4 k⊥ ⊥ ⊥ J 2J +1 J 2 2 +16 k⊥k⊥VV (s, s , (k⊥k⊥)) PJ−1(z) . 2 2J +1 × k⊥VV (s, s , (k⊥k⊥))PJ−1(z) . On the right-hand sides of Eqs. (36) and (37), we expand the interaction blockin a series of Legen- For the right-hand side of (37), dre polynomials and integrate over angle variables: (J−1) 1 dz/2 Z (k⊥)B s, s , (k⊥k⊥) (42) −1 . As a result, Eq. (36) reads µ1...µJ ,β β α − J 1 − (J+1) 2 2 2 (s − M 2) 4ψ(1,J 1,J)(s)A × X (k⊥)=16α(J)k⊥ k k n j (44) µ1...µJ α ⊥ ⊥

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1581 2J +1 J +1 4 4 (J+1) − 2 2 (1,J+1,J) × 16ξ(J +1) k⊥k⊥V (s, s ) + (s ∆ )+4k⊥ ψn (s)Bj 2J +1 V J +1 ∞ 2 2 2 (J) − 4ξ(J)k⊥k⊥ σ V (s, s ) ds 2 J−1 S = ρ(s ) · 8(−k⊥) π 2 2 2 σ ∆ (J) J +1 (m1+m2) √ + VV (s, s )+ ss J (1,J−1,J) 4 × ψ (s )Aj ξ(J +1)k⊥ √ n × (J) 2 (J) ss VA (s, s )+2∆ VT (s, s ) 2 J +1 2 (J+1) × s − ∆ +4 k⊥ V (s, s ) 2J +1 V 2J +1 2 2 + ξ(J − 1) (s − ∆ )(s − ∆ ) 2 2 J 2 2 (J) σ ∆ (J) − ξ(J)k⊥ σ V (s, s )+ √ V (s, s ) 2 S V (J−1) σ (J−1) ss × V (s, s )+2√ V (s, s ) √ V ss T (J) 2 (J) − ss V (s, s ) − 2∆ V (s, s ) + ξ(J − 1) − A T − 2 2 (J 1) +4(s ∆ )k⊥VV (s, s ) − 2 J 2 (J−1) − 2 2 (J 1) × s − ∆ +4 k⊥ V (s, s ) +4(s ∆ )k⊥VV (s, s ) 2J +1 V ∞ J 2 2 (J−1) +16 k⊥k⊥VV (s, s ) ds · − 2 J+1 (1,J+1,J) 2 2J +1 + ρ(s ) 2( k⊥) ψn (k⊥) ∞ π 2 (m1+m2) ds 2 J+1 (1,J+1,J) + ρ(s ) · 8(−k⊥) ψ (s )B π n j 2 2J +1 2 2 (m1+m2) × Bj ξ(J +1) (s − ∆ )(s − ∆ ) J +1 4 2 J +1 2 × ξ(J +1)k⊥ s − ∆ +4 k⊥ σ2 2J +1 × V (J+1)(s, s)+2√ V (J+1)(s, s) V ss T × V (J+1)(s, s) − ξ(J)k2 V ⊥ (J+1) 2 2 − 2 2 (J) σ ∆ (J) +4(s ∆ )k⊥VV (s, s ) × 2 √ σ VS (s, s )+ VV (s, s ) (J+1) ss +4(s − ∆2)k 2V (s, s ) ⊥ V √ − ssV (J)(s, s) − 2∆2V (J)(s, s) J +1 2 2 (J+1) A T +16 k⊥k⊥V (s, s ) 2J +1 V 2 J 2 (J−1) + ξ(J − 1) s − ∆ +4 k⊥ V (s, s ) . σ2∆2 2J +1 V − 4ξ(J) σ2V (J)(s, s)+ √ V (J)(s, s) S ss V The normalization and orthogonality conditions are √ ∞ J (J) (J) 2 2 + ss VA (s, s )+2∆ VT (s, s ) ds 2 (1,J−1,J) 2 J +1 ρ(s) A ψ (k⊥) (46) π j n 2 (m2+m1) J (J−1) +16ξ(J − 1) V (s, s ) . − 2J +1 V × 2α(J)(−k2 )(J 1) ⊥ 2J +1 2 2 The second equation, (37), reads × (s − ∆ )+4k⊥ J 2 2J +1 2 2 (s − M ) (s − ∆ )+4k⊥ (45) (1,J−1,J) 2 (1,J+1,J) 2 +2AjBjψ (k⊥)ψ (k⊥)8α(J) J n n 2 − × − 2 (J+1) 2 (1,J+1,J) 2 × (1,J 1,J) 4 (1,J+1,J) ( k⊥) + Bj ψn (k⊥) ψn (s)Aj +4k⊥ψn (s)Bj 2 (J+1) ∞ × 2α(J)(−k⊥) ds · − 2 J−1 (1,J−1,J) = ρ(s ) 2( k⊥) ψn (s )Aj 2J +1 2 2 π × (s − ∆ )+4k⊥ =1 2 J +1 (m1+m2)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1582 ANISOVICH et al. for j =1, 2,and Appendix A ∞ 2 ds (1,J−1,J) 2 TRACES FOR LOOP DIAGRAMS ρ(s) A1A2 ψn (k⊥) (47) π Here, we present the traces used in the calcula- (m +m )2 2 1 tion of loop diagrams. Recall that, in the spectral- 2 (J−1) 2J +1 2 2 integral representation, there is no energy conser- × 2α(J)(−k⊥) (s − ∆ )+4k⊥ J vation, s = s,whereP 2 = s, P 2 = s,butallcon- (1,J−1,J) 2 stituents are mass-on-shell: +(A1B2 + A2B1)ψn (k⊥) 2 2 2 2 2 2 2 2 k1 = m1,k2 = m2,k1 = m1,k2 = m2. (1,J+1,J) 2 2 (J+1) × ψ (k⊥) · 8α(J)(−k⊥) n We have used the notation for the quarkmomenta 2 (1,J+1,J) 2 2 (J+1) + B B ψ (k ) · 2α(J)(−k ) 1 1 1 2 n ⊥ ⊥ k = (k − k ) ,k= (k − k ) , (A.1) ν 2 1 2 ν ν 2 1 2 ν 2J +1 2 2 ⊥ ⊥ ⊥ ⊥ × (s − ∆ )+4k⊥ =0. k = k g ,k= k g , J +1 µ ν νµ µ ν νµ m2 − m2 m2 − m2 k = 1 2 P + k⊥,k = 1 2 P + k⊥, Let us emphasize again: all the above equations are µ 2s µ µ µ 2s µ µ written for J>0. and for the quarkmasses

∆=m2 − m1,σ= m2 + m1. (A.2) We workwith the following de finition of the matrices: 4. CONCLUSION 1 γ = −iγ0γ1γ2γ3,σ= [γ γ ] . 5 µν 2 µ ν We have presented the Bethe–Salpeter equations for the quark–antiquarksystems when the quarkand TRACES FOR THE S =0STATES antiquarkhave di fferent masses. The main difference For the S =0 states, we have the following from the equal-mass case is that there is the mixture nonzero traces: of states J = L, S =0and J = L, S =1and that is ˆ −ˆ proportional to the quark-mass difference. The mix- TP = tr iγ5(k1 + m1)γ5( k2 + m2) (A.3) ing between S =1, J = L and S =0, J = L states 2 =2i(s − (m2 − m1) ), gives rise to a strongly correlated system of equations. ˆ −ˆ In the equation for states with total spin J, we need TA = tr iγ5(k1 + m1)iγµγ5( k2 + m2) to know all lower projections of the potential on the = −2 2k (m − m )+P (m + m ) , Legendre polynomials, not only the J +1, J, J − 1 µ 2 1 µ 2 1 ones. The numerical study of these equations is now ˆ −ˆ TT = tr iγ5(k1 + m1)iσµν ( k2 + m2) in progress. − = 4iεµναβ Pαkβ and ˆ ˆ ACKNOWLEDGMENTS TP = tr iγ5(−k2 + m2)γ5(k1 + m1) (A.4) =2i(s − (m − m )2), 2 1 We are grateful to L.G. Dakhno, M.N. Kobrinsky, TA = tr iγ5(−kˆ2 + m2)iγµγ5(kˆ1 + m1) Yu.S. Kalashnikova, B.Ch. Metsch, V.A. Nikonov, H.R. Petry, and V.V. Vereshagin for stimulating and =2[2k (m − m )+P (m + m )] , µ 2 1 µ 2 1 useful discussions. TT = tr iγ5(−kˆ2 + m2)iσµν (kˆ1 + m1)

The paper was supported by the Russian Foun- =4iεµναβ Pαkβ. dation for Basic Research, project no. 01-02-17861. The convolutions of the traces A =(T T ), A = One of us (V.N.M.) was supported in part by INTAS P P P A (T T ), A =(T T ) are equal to call 2000, project 587, RFBR no. 01-02-17152, and A A T T T 2 2 the Dynasty Foundation. AP = −4(s − ∆ )(s − ∆ ), (A.5)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1583 2 σ − 2 2 2 − √ − 2 − 2 ⊥ 2 2 AA = 16∆ k⊥k⊥z 4 (s ∆ )(s ∆ ), (B ) =4 g (s − ∆ )(s − ∆ ) ss V β α β α √ 2 2 AT = −32 ss k⊥k⊥z. ⊥ ⊥ 2 2 ⊥ ⊥ − 2 +16kβ kα k⊥k⊥z +4kβ kα (s ∆ ) 2 2 TRACES FOR THE S =1STATES ⊥ ⊥ 2 ⊥ ⊥ σ ∆ +4k k (s − ∆ )+4k k √ , β α β α ss For the S =1states, the traces are equal to √ ⊥ ⊥ ⊥ ⊥ ˆ −ˆ − − 2 2 TS = tr γα (k1 + m1)( k2 + m2) (A.6) (BA)β α = 16 ss kβ kα gβα k⊥k⊥z , ⊥ =4kα (m2 + m1), 2 ⊥ ⊥ ⊥ 2 2 (B ) = −8 4∆ k k − g k k z ⊥ ˆ −ˆ T β α β α β α ⊥ ⊥ TV = tr γα (k1 + m1)γµ( k2 + m2) ⊥ 2 ⊥ 2 =2 g (s − (m2 − m1) )+4k k , ⊥ σ 2 2 α µ α µ − g √ (s − ∆ )(s − ∆ ) . β α ss ⊥ ˆ ⊥ −ˆ TA = tr γα (k1 + m1)iγµ γ5( k2 + m2) =4εαµαβ kαPβ, Appendix B ⊥ ˆ −ˆ TT = tr γα (k1 + m1)iσµν ( k2 + m2) CONVOLUTIONS OF TRACE FACTORS − ⊥ − ⊥ =2i 2(m2 m1) gανkµ gαµkν Here, we present the convolutions of the angular- momentum factors. Let z be (k⊥k ) ⊥ ⊥ ⊥ +(m1 + m2) g P − g P z = . (A.9) α ν µ α µ ν 2 2 k⊥ k⊥ and The convolutions for the S =1states read ⊥ −ˆ ˆ TS = tr γβ ( k2 + m2)(k1 + m1) (A.7) (J) (J) Xµ µ ...µ (k⊥)Xµ µ ...µ (k⊥) (A.10) 1 2 J 1 2 J ⊥ J =4kβ (m1 + m2), 2 2 = α(J) k⊥ k⊥ PJ (z). ⊥ −ˆ ˆ TV = tr γβ ( k2 + m2)γµ(k1 + m1) Analogous convolutions for S =1states are writ- ⊥ − − 2 ⊥ ten as follows: =2 gµβ (s (m2 m1) )+4kβ kµ , (J+1) (J+1) X (k⊥)X (k⊥) (A.11) µ1µ2...µJ β µ1µ2...µJ α ⊥ −ˆ ˆ TA = tr γβ ( k2 + m2)iγµγ5(k1 + m1) 2 α(J) J k⊥ = k2 k2 A (z)k⊥k⊥ − ⊥ ⊥ PJ,J +1 β α = 4εβ µαβ kαPβ, J +1 2 k⊥ ⊥ −ˆ ˆ TT = tr γβ ( k2 + m2)iσµν (k1 + m1) 2 k⊥ ⊥ ⊥ + BPJ,J +1 (z)kβ kα − ⊥ − ⊥ 2 =2i 2(m2 m1)(gβµkν gβνkµ) k⊥ ⊥ ⊥ ⊥ ⊥ + C (z)k k + D (z)k k PJ,J +1 β α PJ,J +1 β α ⊥ − ⊥ +(m2 + m1) gβµPν gβνPµ . 2 2 ⊥ + k⊥ k⊥ EPJ,J +1 (z)gβα , The corresponding convolution BS =(TcTc) reads ⊥ ⊥ 2 (J+1) (J−1) (B ) =16k k σ , X (k⊥)Z (k⊥) (A.12) S β α β α (A.8) µ1µ2...µJ β µ1µ2...µJ ,α

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1584 ANISOVICH et al. J (J) α(J) 1 ε P Z (k⊥) (A.14) − 2 2 βν1ν2ν3 ν1 ν2µ1...µJ ,ν3 = 2 k⊥ k⊥ J k⊥ (J) × εαλ λ λ P Z (k⊥) 1 2 3 λ1 λ2µ1...µJ ,λ3 2 2 k k J−1 ⊥ ⊥ ⊥ ⊥ 2 × (2J +3) 2 2 APJ,J +1 (z)kβ kα + = α(J) k⊥ k⊥ (PP ) 2 2 (J +1)3 k⊥ k⊥ ⊥ ⊥ × − 2 BPJ,J +1(z) (2J +1)AJ (z) kβ kα × − 2 2 − k⊥ k⊥ (z 1)DPJ,J +1 (z) − ⊥ ⊥ +(CPJ,J +1 (z) (2J +1)BJ (z))kβ kα + D (z)k⊥k⊥ ⊥ − PJ,J +1 β α + zEPJ,J +1 (z) gβα DPJ,J +1 (z)   ⊥ 2 2 2 2 + k⊥ k⊥ EPJ,J +1 (z)gβα , k⊥ k⊥ ×  ⊥ ⊥ ⊥ ⊥ − ⊥ ⊥ kβ kα + kβ kα zkβ kα 2 2 k⊥ k⊥ (J−1) (J−1) ⊥ Zµ µ ...µ ,β(k )Zµ1µ2...µ ,α(k⊥) (A.13) 1 2 J J J−2 ⊥ ⊥ J +1 + zDP (z)+EP (z) k k , = α(J) k2 k2 J,J +1 J,J +1 β α J2 ⊥ ⊥ and k2 ⊥ ⊥ ⊥ (J) (J) × − ⊥ APJ,J +1 (z) (2J +1)AJ (z) kβ kα Xµ1...µJ (k )εαν1ν2ν3 Pν1 Zν2µ1...µJ ,ν3 (k⊥) (A.15) 2 k⊥ 2 k − ⊥ ⊥ ⊥ J 1 − 2J +3 + BPJ,J +1 (z) (2J +1)AJ (z) kβ kα 2 2 2 = α(J) k⊥ k⊥ APJ,J +1 (z)εαP kk . k⊥ J +1 (2J +1)2 + C (z)+ P (z) Here, PJ,J +1 J +1 J APJ,J +1 (z)=BPJ,J +1(z) (A.16) − ⊥ ⊥ ⊥ ⊥ 2(2J +1)BJ (z) k kα + DPJ,J +1 (z)k kα 2 β β 2zPJ (z)+ Jz − (J +2) PJ+1(z) = − , (1 − z2)2 2 2 ⊥ + k⊥ k⊥ EPJ,J +1 (z)gβα ,

2 2 (1 − J)z +(J +1) PJ (z)+ (2J +1)z − (2J +3) zPJ+1(z) C (z)= , PJ,J +1 (1 − z2)2

2 − − (J+1) (J+1) (J +2)z J PJ (z) 2zPJ+1(z) kβX (k⊥)X (k⊥)k D (z)= , µ1µ2...µJ β µ1µ2...µJ α α PJ,J +1 − 2 2 (1 z ) √ J+2 zP (z) − P (z) = α(J) k2 k2 P (z), E (z)= J J+1 . ⊥ J PJ,J +1 (1 − z2) (J+1) (J+1) k X (k⊥)X (k⊥)k (J+1) × β µ1µ2...µJ β µ1µ2...µJ α α For the factors KβXµ µ ...µ β(k⊥) 1 2 J J+1 (J+1) 2 2 2 Xµ1µ2...µJ α(k⊥)Kα,whereK = k, k , we need more = k⊥α(J) k⊥ k⊥ PJ+1(z), complicated convolutions, namely: (J+1) (J+1) (J+1) (J+1) ⊥ k X (k⊥)X (k⊥)k kβXµ µ ...µ β(k )Xµ1µ2...µ α(k⊥)kα β µ1µ2...µJ β µ1µ2...µJ α α 1 2 J J J+1 J+2 2 2 2 2 2 = k⊥α(J) k⊥ k⊥ PJ+1(z), = α(J) k⊥ k⊥

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK–ANTIQUARK COMPOSITE SYSTEMS 1585 2J +1 J J−1 × − 2J +1 zPJ+1(z) PJ (z) , = α(J) k2 k 2 P − (z); J +1 J +1 J ⊥ ⊥ J 1 ⊥ (J+1) (J+1) g X (k⊥)X (k⊥) (J) βα µ1µ2...µJ β µ1µ2...µJ α and for the factors K ε P Z (k⊥) × β βν1ν2ν3 ν1 ν2µ1...µJ ,ν3 2J +1 J+1 (J) 2 2 εαλ1λ2λ3 Pλ Z (k⊥)Kα: = α(J) k⊥ k⊥ PJ+1(z); 1 λ2µ1...µJ ,λ3 J +1 (J) kβεβν1ν2ν3 Pν1 Zν µ ...µ ,ν (k⊥) (J+1) 2 1 J 3 as well as for the factors KβX (k⊥) × (J) µ1µ2...µJ β × − εαλ1λ2λ3 Pλ1 Zλ µ ...µ ,λ (k⊥)kα =0, Z(J 1) (k )K : 2 1 J 3 µ1µ2...µJ ,α ⊥ α (J) kβεβν ν ν Pν Z (k⊥) (J+1) (J−1) 1 2 3 1 ν2µ1...µJ ,ν3 k X (k⊥)Z (k⊥)k β µ1µ2...µJ β µ1µ2...µJ ,α α (J) × εαλ λ λ P Z (k⊥)k =0, J−1 1 2 3 λ1 λ2µ1...µJ ,λ3 α 4 2 2 (J) = k⊥α(J) k⊥ k⊥ PJ−1(z), k ε P Z (k⊥) β βν1ν2ν3 ν1 ν2µ1...µJ ,ν3 (J) (J+1) (J−1) × ε P Z (k⊥)k =0, k X (k⊥)Z (k⊥)k αλ1λ2λ3 λ1 λ2µ1...µJ ,λ3 α β µ1µ2...µJ β µ1µ2...µJ ,α α J (J) kβεβν1ν2ν3 Pν1 Zν µ ...µ ,ν (k⊥) 2 2 2 2 1 J 3 = k⊥α(J) k⊥ k⊥ PJ (z), (J) × ε P Z (k⊥)k αλ1λ2λ3 λ1 λ2µ1...µJ ,λ3 α (J+1) (J−1) ⊥ 2 J+1 kβXµ µ ...µ β(k )Zµ1µ2...µ ,α(k⊥)kα (2J +3) 1 2 J J = α(J) k2 k2 J+1 (J +1)3 ⊥ ⊥ 2 2 = α(J) k⊥ k⊥ PJ+1(z), × (PP )[zPJ (z) − PJ+1(z)] , (J+1) (J−1) ⊥ (J) ⊥ g εβν ν ν Pν Z (k⊥) kβXµ µ ...µ β(k )Zµ1µ2...µ ,α(k⊥)kα βα 1 2 3 1 ν2µ1...µJ ,ν3 1 2 J J J (J) × εαλ λ λ P Z (k⊥) 2 2 2 1 2 3 λ1 λ2µ1...µJ ,λ3 = k⊥α(J) k⊥ k⊥ PJ (z), J(2J +3)2 J − 2 2 ⊥ (J+1) (J−1) = 3 α(J) k⊥ k⊥ (PP )PJ (z). g X (k⊥)Z (k⊥)=0; (J +1) βα µ1µ2...µJ β µ1µ2...µJ ,α (J−1) and for the factors K Z (k⊥) × Considering the Bethe–Salpeter equation with β µ1µ2...µJ ,β (J−1) unequal masses, one may face the problem of the Zµ1µ2...µJ ,α(k⊥)Kα: definition of the momentum transfer squared t in the (J−1) (J−1) interaction block(this means the relation between kβZ (k⊥)Z (k⊥)kα µ1µ2...µJ ,β µ1µ2...µJ ,α z, t) when retardation effects are accounted for. We J−1 thinkthat the most convenient form, symmetrical 2 2 2 = k⊥α(J) k⊥ k⊥ PJ−1(z), under the permutation of the indices 1 ⇐⇒ 2,was suggested in [10]: (J−1) (J−1) ⊥ kβZµ µ ...µ ,β(k )Zµ1µ2...µ ,α(k⊥)kα − − 1 2 J J t =(k1 k1)µ(k2 k2)µ. (A.17) J 2 2 The other variants of accounting for retardation ef- = α(J) k⊥ k⊥ PJ (z), fects may be found in [11, 12]; see also discussion (J−1) (J−1) in [10]. ⊥ kβZµ µ ...µ ,β(k )Zµ1µ2...µ ,α(k⊥)kα 1 2 J J J−1 2 2 2 = k⊥α(J) k⊥ k⊥ PJ−1(z), REFERENCES 1. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (J−1) (J−1) ⊥ (1951); E. Salpeter, Phys. Rev. 91, 994 (1953). kβZµ µ ...µ ,β(k )Zµ1µ2...µ ,α(k⊥)kα 1 2 J J 2. D. Merten, R. Ricken, M. Koll, et al.,Eur.Phys.J.A J 2 2 13, 477 (2002). = α(J) k⊥ k⊥ 3. R. Ricken, M. Koll, D. Merten, et al.,Eur.Phys.J.A 9, 221 (2000); 9, 79 (2000). 2J +1 J +1 4. M. Harada, M. Kurachi, and K. Yamawaki, Phys. Rev. × zP − (z) − P (z) , J J 1 J J D 68, 076001 (2003). 5. A. D. Lahiff and I. R. Afnan, Phys. Rev. C 66, 044001 ⊥ (J−1) (J−1) g Z (k⊥)Z (k⊥) βα µ1µ2...µJ ,β µ1µ2...µJ ,α (2002).

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6.R.Alkofer,P.Watson,andH.Weigel,Phys.Rev.D 9. A. V.Anisovich, V.V.Anisovich, V.N. Markov, et al., 65, 094026 (2002). J. Phys. G 28, 15 (2002). 7. A. V.Anisovich, V.V.Anisovich, V.N. Markov, et al., 10. H. Hershbach, Phys. Rev. C 50, 2562 (1994); Phys. Yad. Fiz. 67, 794 (2004) [Phys. At. Nucl. 67, 773 Rev. A 46, 3657 (1992). (2004)]; hep-ph/0208150. 8. V. N. Markov, in Proceedings of the XXXIII PNPI 11. F.Gross and J. Milana, Phys. Rev. D 43, 2401 (1991). Winter School “The Bethe–Salpeter Equation for 12. K. M. Maung, D. E. Kahana, and J. W.Norbury, Can. the Meson Quark–Antiquark States,” 2003. J. Phys. 70, 86 (1992).

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1587–1598. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1647–1658. Original Russian Text Copyright c 2005 by Saleev, Vasin. ELEMENTARY PARTICLES AND FIELDS Theory

Electroproduction of D∗± Mesons at High Energies

V. A. Saleev* and D. V. Vasin** Samara State University, ul. Akademika Pavlova 1, Samara, 443011 Russia Received February 4, 2004; in ﬁnal form, August 18, 2004

Abstract—A comparative analysis of the predictions of the collinear parton model and the kT -factorization approach is performed for the case of the D∗-meson electroproduction at the HERA ep collider. It is shown that, owing to eﬀectively taking into account, in noncollinear distributions, next-order corrections in the strong coupling constant αs,thekT -factorization approach increases, in contrast to the predictions of the collinear parton model, the absolute value of the cross sections for charmed-meson electroproduction by approximately a factor of 1.5 to 2. As a result, the agreement with experimental data is improved. This is not so only for the pseudorapidity spectrum, whose shape diﬀers considerably from the experimental one and depends greatly on the choice of parametrization of the noncollinear gluon distribution within the proton. c 2005 Pleiades Publishing, Inc.

1. INTRODUCTION At some values of kinematical parameters, our present results for the D∗-meson-production spec- The cross sections for photo- and electroproduc- tra differ by a factor of 1.5 to 2 from the results of tion of D∗ mesons have been calculated many times our previous study [2], where an embarrassing error within the kT -factorization approach (see, for ex- was made at the stage of numerical calculations. An ample, the discussion on the results of the Small-x improved version of the study reported in [2] was Collaboration in [1]). However, there are still a large presented in the form of a preprint [3]. number of disputable aspects in the kT -factorization approach, the ambiguities in their interpretation substantially affecting the results. Previous calculations 2. FORMALISM of D∗-meson spectra ensured fairly good agreement OF THE kT -FACTORIZATION APPROACH with experimental data [1] via optimizing the choice of At the present time, there exist two approaches parameters of the approach (which include the quark to describing high-energy processes that lead to masses and the renormalization scale) and the choice the production of hadrons containing heavy quarks. of parametrization for the noncollinear gluon distribu- These are the collinear parton model and the kT - tion in the proton. The objective of the present study factorization approach, which is also known as the is to perform a unified calculation of the spectra for theory of semihard processes. Either approach is ∗ D -meson photo- and electroproduction in order to based on the hypothesis that the effects of long- and assess the uncertainty that is inherent in the approach short-distance physics factorize in hard processes. because it is necessary to choose the parametrization In the collinear parton model, it is assumed that for the noncollinear gluon distribution in the pro- the cross section for the hadronic process being ton. Moreover, analytic expressions that describe the considered—σ(ep → D∗X, s) in the present case— squares of the absolute values of amplitudes for par- and the cross section for the corresponding partonic ∗ tonic subprocesses within the kT -factorization ap- subprocess, σˆ(eg → D X, sˆ), are related by the proach and which are convenient for numerical cal- equation culations are given for the first time in this article, this σPM(ep → D∗X, s) being of paramount importance for a comparison with (1) the results obtained by other authors. Our results are 2 → ∗ compared below with available data on various spec- = dxG(x, µ )ˆσ(eg D X, sˆ), tra (with respect to pT , η, W , xBj,andQ), whereas the comparison in the literature is usually performed only where sˆ = xs, s is the square of the total energy of the participant electron and proton in their c.m. for the pT and η spectra. frame, G(x, µ2) is the collinear gluon distribution in *e-mail: [email protected] the proton, k = xpN is the gluon 4-momentum, x is **e-mail: [email protected] the proton-momentum fraction carried by the gluon,

1063-7788/05/6809-1587$26.00 c 2005 Pleiades Publishing, Inc. 1588 SALEEV, VASIN and µ2 is the characteristic scale of the hard scat- from the study of Blumlein [9], the JS parametrization tering process. Within perturbative QCD, the evo- from the study of Jung and Salam [10], and the KMR lution of the gluon distribution G(x, µ2) is described parametrization from the study of Kimber, Martin, by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi and Ryskin [11]. evolution equation [4]. For the sake of comparison, Figs. 1 and 2 display Within the kT -factorization approach, the par- unintegrated gluon distributions in the proton versus 2 2 tonic and hadronic cross sections are related by the x at fixed kT (Fig. 1) and versus kT at fixed x (Fig.√ 2). equation One can see that, in the x region around x ∼ m / s c and the transverse-momentum region around k2 ≈ KT ∗ dx 2 T σ (ep → D X, s)= dk (2) 2 x T 50 GeV (and these are the regions in which we are interested), the parametrizations in question differ dϕ × Φ(x, k2 ,µ2)ˆσ(eg∗ → D∗X, sˆ), substantially. 2π T where σˆ(eg∗ → D∗X, sˆ) is the cross section for ∗ 3. FRAGMENTATION FUNCTION D -meson production in electron scattering on a ∗ Reggeized gluon, k = xp + k is the gluon 4- The production of D mesons, which consist of a N T ¯ momentum, kT =(0, kT , 0) is the transverse gluon heavy c quark and a light u¯ (d) antiquark, is described 4-momentum, k2 = k2 = −k2 is the virtuality of phenomenologically on the basis of the fragmentation T T model [2, 12–14] by introducing a universal fragmen- the primary gluon, pN = EN (1, 0, 0, 1) is the 4- 2 2 tation function Dc→D∗ (z,µ ) for c-quark fragmenta- momentum of the primary proton, sˆ = xs − k , ϕ is ∗ T tion into a D meson or on the basis of the fusion the azimuthal angle in the xy plane between the vec- model [15, 16]. tor k and the fixed x axis (p ∗ ∈ xz,thez axis being T D At high energies, the cross sections for c-quark ep Φ(x, k2 ,µ2) ∗ aligned with the -reaction axis), and T and D -meson production in the fragmentation is the noncollinear distribution of Reggeized glu- model are related by the equation ons in the proton. The QCD evolution of the non- ∗ 2 2 dσ(ep → D X) (5) collinear gluon distribution Φ(x, kT ,µ ) is described by the Balitsky–Fadin–Kuraev–Lipatov [5] or the 2 = D → ∗ (z,µ )dz · dσ(ep → cX), Ciafaloni–Catani–Fiorani–Marchesini [6] evolution c D equation. It is usually assumed that the function 2 describing the noncollinear distribution of gluons where Dc→D∗ (z,µ ) is the above fragmentation func- in the proton and satisfying the Balitsky–Fadin– tion, µ2 is the characteristic scale of the hard scatter- Kuraev–Lipatov evolution equation [5] can be related ing process, and z is the fragmentation parameter. to the collinear gluon distribution G(x, µ2) in the The probability of c-quark fragmentation into a ∗± proton as follows: D meson is given by µ2 1 2 2 2 2 2 ωc→D∗± = Dc→D∗± (z,µ )dz. (6) xG(x, µ )= Φ(x, kT ,µ )dkT . (3) 0 0 According to data of the OPALCollaboration [17], we The need for reconstructing the gauge invariance have of amplitudes for hard scattering processes involv- ± ing Reggeized gluons leads to effective Feynman ωc→D∗± =0.222 0.014. (7) rules [7], where the polarization vector of primary off- In the present study, we use the Peterson phe- shell gluons is represented in the form nomenological scaling fragmentation function [18] µ 2 µ kT z(1 − z) ε (k )= . (4) ∗ T | | Dc→D (z)=N . (8) kT [(1 − z)2 + z]2 At the stage of numerical calculations, use is made In the limit of massless quarks and hadrons, the of the GRV LO parametrization [8] of the collinear fragmentation parameter z relates the c-quark and 2 gluon distribution G(x, µ ) in the proton. In the D∗-meson 4-momenta as follows: case of calculations within the k -factorization ap- T p ∗ = zp . (9) proach, we employed the following parametrizations D c for the unintegrated (noncollinear) gluon distribution Experimental data on D∗±-meson production at the 2 2 ≥ Φ(x, kT ,µ ) in the proton: the JB parametrization HERA ep collider correspond to the region p mc,

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Φ 2 µ2 (x, kT, ) 101

JB JS KMR KMR JS 10–1 KMR JB

10–3 JS JB 2 2 2 2 2 2 kT = 10 GeV kT = 30 GeV kT = 50 GeV

10–5 10–4 10–3 10–2 10–1 10–4 10–3 10–2 10–1 10–4 10–3 10–2 10–1 x

2 2 2 2 Fig. 1. Unintegrated gluon distribution Φ(x, kT ,µ ) in the proton versus x at kT =10, 30,and50 GeV for µ =10GeV.

Φ 2 µ2 (x, kT, ) 102 JB 100 JS KMR KMR

10–2 JB JS JB 10–4 JS KMR x = 0.001 x = 0.01 x = 0.1 10–6 10–1 100 101 102 10–1 100 101 102 10–1 100 101 102 2 2 kT, GeV

2 2 2 Fig. 2. Unintegrated gluon distribution Φ(x, kT ,µ ) in the proton versus kT at x =0.001, 0.01,and0.1 for µ =10GeV. so that the use of the massless approximation is not the fragmentation process. For the sake of definite- quite correct; therefore, we employ a different defini- ness, we set the c-quark mass to mc =1.5 GeV in tion of the fragmentation parameter z, our numerical calculations. E ∗ + |p ∗ | z = D D , (10) Alternative definitions of the fragmentation pa- Ec + |pc| rameter z are also possible, for example, assuming that pD∗ = zpc. (12) θD∗ = θc, (11) As was shown in [20], the definitions in (9), (10), and ∗ ∗ where θD∗ and θc are, respectively, the D -meson and (12) lead to virtually identical spectra of D mesons c-quark emission angles in the laboratory frame. This over a wide range of kinematical variables. assumption is justified by the fact that the c-quark mass is approximately equal to the D∗±-meson mass, Adefinition of z close to that in (10) was used ≈ → ∗ mc mD∗± (mD∗± =2.010 GeV [19]); therefore, the in [21–24] to extract the c D fragmentation func- c quark does not change the direction of its motion in tion in the processes of electro- and photoproduction

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1590 SALEEV, VASIN 2 2 on protons; namely, where Q is the photon virtuality; xBj = Q /(ys) is 2 2 ED∗ + |pD∗ | the Bjorken variable; y =(W + Q )/s; W is the en- z = , (13) ergy of photon–proton interaction in the с.m. frame; 2Ehad ϕc is the polar angle between the directions of pcT and where E is the total energy of the hadron jet. ∈ had the fixed axis x (pe xz); pe is the 4-momentum of For the free parameter of the Peterson fragmen- the final electron; η is the pseudorapidity, tation function [18], a fit to experimental data on ∗± − θ D -meson production in e+e annihilation gives η = − ln tan ; (16) the value of =0.036 [25] in the model that takes 2 into account the c-quark mass and the value of = 2 2 2 2 µ = m ∗ + p ∗ + Q is the characteristic scale of 0.116 [12, 26] in the model of massless c quarks. We D D T the hard interaction; and, for other quantities, we use use the value the notation =0.06, (14) Q2 +2|p |Qβ + syb x = cT 1 , (17) which is in accord with the analysis in [21]. s|y − a|

β1 =cosϕc, (18) 4. DEEP-INELASTIC SCATTERING E − p a = c cz , (19) 4.1. Cross Section for the Process 2E → ∗ e e + p e + D + X E + p b = c cz . (20) By using relations (1), (2), and (5) and the defi- 2EN nition of the fragmentation parameter z in the form In the kT -factorization approach, we have specified by Eqs. (10) and (11), one can rewrite the quadruple-differential production cross section in the dσKT(ep → eD∗X) 2 2 (21) parton model as dηD∗ d|pD∗T |dQ dW ∗ dσPM(ep → eD X) dϕ Φ(x, k2 ,µ2) ∗ 2 2 T 2 2 (15) = dzDc→D (z,µ ) dkT dηD∗ d|pD∗T |dQ dW 2π x 2 2 | || | | ∗ → |2 = dzD → ∗ (z,µ )G(x, µ ) 1 pc pcT 1 M(eg ecc¯) c D × dϕc , 4 ∗ | − | 2 (2π) 2ED s y a 8xs | || | | → |2 × 1 pc pcT 1 M(eg ecc¯) dϕc 4 2 , where (2π) 2ED∗ s|y − a| 8xs

|k |2 + Q2 +2|k ||q |β − 2|p ||k |β +2|p ||q |β + s(yb + av) x = T T T 2 cT T 3 cT T 1 , (22) s|y − a|

v = −Q2/s, (23) where E is the energy of the ﬁnal electron and ϕ is e e the azimuthal angle of ﬁnal-electron emission. |qT | = Q 1 − y, (24)

β2 =cosϕ, (25) 2 In the high-energy region (xBj 1, Q s), the β3 =cos(ϕ − ϕc). (26) electroproduction cross section can be represented as the convolution of the photoproduction cross sec- To go over from one set of diﬀerential cross sections tion σˆ(γ∗g∗ → cc¯) and the equivalent-virtual-photon to another, one can employ the formulas 2 spectrum fγ∗/e(y,Q );thatis, 3 2 2 2 d pe dQ dW dϕe dydQ dϕe KTQT → ∗ = = (27) dσ (ep eD X) 2Ee 4s 4 2 2 (28) dηD∗ d|pD∗T |dQ dW 2 ydxBjdQ dϕe 2 2 = , 2 dϕ 2 Φ(x, kT ,µ ) 4x = dzDc→D∗ (z,µ ) dk Bj 2π T x

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ELECTROPRODUCTION OF D∗± MESONS 1591 | || | 2 2 1 pc pcT a 6 − 4 2 − × dϕcfγ∗/e(y,Q ) M12 = 2Q 4mc(Q 2sx) (37) 2 ∗ 4˜ (2π) 2ED Q tu˜ 1 |M(γ∗g∗ → ecc¯)|2 +2Q4(2bs + tˆ− sx)+Q2 2s2(2b2 − 2bx + x2) × , s|y − a| 2xys2 + tˆ(2bs − uˆ − sx) − uˆ(2bs − uˆ − 3sx) where qT =(0, qT , 0) is the transverse momentum of the virtual photon, q2 = −Q2; q = yp + q is the 4- + s(−btˆ2 + buˆ2 − 2bstxˆ + tˆ2x +2bsuxˆ +3tûxˆ T e T momentum of the virtual photon; p = E (1, 0, 0, −1) e e ˆ 2 − 2 4 − 2 ˆ− − is the 4-momentum of the initial electron; ϕ is the +2stx ) mc 2Q Q (8bs + t 5û 6sx) c p polar angle between the directions of cT and the fixed + s 2b(4s(x − b) − 3(tˆ− uˆ)) + 9txˆ ∈ ∗ 2 x axis (q xz); and fγ /e(y,Q ) is the equivalent- virtual-photon spectrum, which can be represented in +3ûx +2sx2 , the form [27] − 2 2 4 2 α 1+(1 y) 2mey a − 4 2 − fγ∗/e(y,Q )= − . (29) M22 = 4 2 uˆ Q + Q ( sˆ + sx) (38) 2π yQ2 Q4 Q u˜ In this approximation, the polarization tensor of vir- + s(btˆ+ buˆ +2bsx − txˆ − 2ûx − 2sx2) tual photons is given by 4 − 2 4 2 µν µ ν 2 + mcs(6b 7x)+mc 3Q + Q (4bs +ˆs Le (q)=qT qT /Q . (30) − 4û − 5sx)+s(4b2s − btˆ− 7buˆ − 10bsx + txˆ 4.2. Amplitudes of Parton Subprocesses +9ûx +6sx2) . Let us introduce the notation ˆ t˜= tˆ− m2, (31) Here, sˆ, t,anduˆ are the standard Mandelstam vari- c ables for the process γ∗ + g → c +¯c. − 2 u˜ =û mc (32) For the electroproduction of c quarks on a Reggei- and set zed gluon via the process 2 ∗ fkk = |kT | ,fkp = β3|kT ||pcT |, (33) e + g → e + c +¯c, (39) | || | fkq = β2 kT qT , the result analogously obtained by averaging the 2| |2 | || | square of the absolute value of the amplitude over the fpp = β3 pcT ,fpq = β2β3 pcT qT , 2| |2 electron polarizations and over the color states of the fqq = β2 qT . initial gluon is For the electroproduction of a pair of c quarks on a |M(eg∗ → ecc¯)|2 (40) Yang–Mills gluon via the process =64α2α e2π3(M b +2M b + M b ), e + g → e + c +¯c, (34) s c 11 12 22 where the result obtained upon averaging the absolute value 2s of the amplitude over the electron and gluon polariza- M b = v(Q2 − 2m2 +2bs + t˜) (41) 11 4˜2 c tions and over the color states of the initial gluon has Q t the form 2 − 2b s (fkk − 4fkp − 4fkq +4fpp |M(eg → ecc¯)|2 (35) 2 2 3 a a a +8fpq +4fqq) − bt˜(fkk − 8fkp − 4fkq +8fpp =32α αsec π (M11 +2M12 + M22), 2 − ˜ ˜ 2 where + fpq +4fqq + Q +ˆs) xt(t + Q +2bs) , 4 M a = − m4 4Q2 + s(6b + x) (36) 11 Q4t˜2 c s M b = − 4bs(b − x − v)(4f − 4f (42) 12 4˜ pp kp + stˆ (Q2 + tˆ)x + b(tˆ+û +2sx) Q tu˜ + f +4f − 2f ) − 4bu˜(f +2f − f ) − 2 4 2 kk pq kq qq pq kq mc 2Q + Q s(4b + x) − 2xt˜(4fpp − 4fkp + fkk +2fpq)+(û − tˆ) + s(4b2s +7btˆ+ buˆ +2bsx +2txˆ ) , 2 × b(ˆs + Q )+˜u +3vfkk + v 2sx 4fpp

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1592 SALEEV, VASIN − − 2 − 2 2 4fkp + fkk +4fpq 4mc fkk + fqq 4fpp × β2 − (ˆs − 3fkk − Q )Qβ2 +2|kT |Q β3 − 7fpq +4û fpp + fqq − 4fkp − 2fkq +4f Q Q +2|k |β − 2|p | (f +3Q2 2 pp T 2 cT kk − 4ˆs 2fkp +2fpq + fkq − 2Q 4fpp +8fpq − | | 2 sˆ) kT β2 +2fkkQ +2(˜u +2fkk)Qβ2 + fkk − 4tˆ fpp +3fpq +2fkk fkk − 8fkp − 2|p |(4|k |Q +(3f +3Q2 − sˆ)β 2 cT T kk 2 − 4fkq +4fpp +2fqq + x Q + t˜ +4|kT |Qβ2)β3 +8fpp(Q + |kT |β2) β1 × 2sx + t˜+˜u +2s b(tˆ− uˆ)+vu˜ | |2 | | −| | +4pcT 2 pcT β3 kT +2 m2(û + tˆ) − m4 − tûˆ , c c × −| |− | | 2 kT 2Qβ2 +2pcT β3 β1 , b 2s M = − 2s(x − b)(x − b − v) (43) c 2 2 22 4 2 M = − f (˜u + Q ) − (tˆ+ f )û (48) Q u˜ 22 u˜2 kk kk − Q2v (f − 4f +4f )+4ûv(−f + f +4(|k |Qβ + f β2)(˜u + f ) − m2(ˆs − m2 kk kp pp kp pp T 2 kk 2 kk c c 2 + fpq)+˜u(x − b)(fkk − 8fkp +8fpp +4fpq) + Q ) − 4|pcT | Q(˜u +4fkk)β2 2 − − +2mcv(fkk 2fkp +2fpp 2fpq) 2 +2|kT |(˜u +2fkk)β2 + |kT |Q β3 +˜u(Q2 +ˆs)(b − v)+xu˜ u˜ − sˆ +2s(x − b + v) . 2 +4fpp Q +4|kT |Qβ2 +4fkkβ2 Upon averaging over the color states of the initial 2 gluon, the square of the absolute value of the ampli- | |2 | |− | | 2 +4pcT kT 2 pcT β3 β1 tude for the photoproduction of c quarks by a virtual photon on a Reggeized gluon via the process − 4|pcT |β1 −|kT | +2|pcT |β3 −|kT |Q ∗ ∗ γ + g → c +¯c (44) − (˜u +2f )β +2|p |Qβ +2|k ||p | kk 2 cT 3 T cT takes the form × cos(2ϕ − ϕc)+2|kT ||pcT |β1 . |M(γ∗g∗ → cc¯)|2 (45) 2 2 c c c We have tested analytically that, in the limit |k |→0 =8ααsec π (M11 +2M12 + M22), T and upon averaging over the azimuthal angle ϕ,the where amplitudes for the processes involving a Reggeized c − 2 ˜ 2 initial gluon reduce exactly to the amplitudes for the M11 = (fkk +4fpp + t)Q (46) t˜2 process involving a real gluon in the initial state. By − − 2 2 − | | | | way of example, we indicate that, for the processes (fkk mc +ˆs)mc 4 pcT Q kT Q in (34) and (39), ˜ 2 ˜ 2 | | 2 2 2π +(t +2Q )β2 β3 +4(t + Q )( kT Qβ2 + Q β2 ) dϕ |M(eg → ecc¯)|2 = lim |M(eg∗ → ecc¯)|2, | |→ − (Q2 +û)tˆ− 4|p | 4f Qβ +(t˜+4Q2)|k | kT 0 2π cT pp 1 T 0 (49) × β +2(t˜+2Q2)Qβ2 − 2|p |(2|k |Q 2 2 cT T ϕ x ˜ 2 | |2 where is the angle between the fixed axis and the +(t +4Q )β2)β3 + Qfkk +4pcT k vector T . × 2 2 | | 2 β1 fkk +2Q +4kT Qβ2 +2Q (cos 2ϕ) 4.3. Results of the Calculations − 4fkp − 4|pcT | Qβ1 − Q cos(2ϕ − ϕc) In this section, we compare theoretical predictions + |pcT |β3 , made within various approaches and experimental data obtained at the HERA collider for the spectra ∗± c 2 2 2 of D mesons in ep interactions at the energies of M = − −t˜u˜ +2(˜u + fkk)(t˜+ Q )β (47) 12 t˜u˜ 2 EN = 820 GeV and Ee =27.5 GeV. The experimental data of the ZEUS Collaboration [23] are presented 2 − | | − | | ˜ 2 | | ∗± + fkkQ 2ˆs kT Qβ2 2 pcT 2(t +2Q ) kT in the form of various spectra of D mesons versus

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2 2 2 σ the variable y = Q /(xBjs)=(Q + W )/s in the d /dpT, nb/GeV interval 0.02

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4 0.08

3 0.06 JB GRV JS JS , nb η , nb/GeV d 2 / 0.04

σ KMR dW d /

σ KMR d

1 0.02 JB

(a) GRV (b) 0 0 –20 2 40 140 240 η W, GeV 101 JB 6×100 JB GRV GRV KMR , nb , nb 2 Bj Q JS x KMR 100 log 6×10–1 log d d JS / / σ σ d d

Fig. 4. Diﬀerential cross section for the deep-inelastic electroproduction of D∗ mesons as a function of various variables.

The Williams–Weizs acker¨ spectrum of quasireal diﬀerential production cross section within the parton photons can be derived by performing integration of model and within the kT -factorization approach. (29) with respect to Q2 [28]. The result is In the parton model, we have

− 2 2 PM ∗ α 1+(1 y) Qmax dσ (ep → eD X) fγ/e(y)= log 2 (51) (52) 2π y Q ∗ ∗ min dηD dpD T 2 1 1 dz 2 2 − → ∗ +2mey 2 2 . = dyfγ/e(y) Dc D (z,µ )G(x, µ ) Qmin Qmax z | || | | → |2 By using the relation between the cross sections at × 2 pc pcT M(γg cc¯) 2 2 , the hadron level and at the quark level as deter- Ec(W − 2EN (Ec − pcz)) 16πxW mined by Eq. (5), the relation between the hadron 2 2 process and the parton subprocess as determined where ymin = Wmin/s and ymax = Wmax/s are, corre- by Eq. (1) in the parton model and by Eq. (2) in spondingly the lower and the upper limit of integration the kT -factorization approach, and the deﬁnition of with respect to y; W is the energy of photon–proton 2 2 2 the fragmentation parameter z in the form speciﬁed interaction in the с.m. frame; and µ = mD∗ + pD∗T by Eqs. (10) and (11), one can obtain the double- is the characteristic scale of hard interaction.

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Within the kT -factorization approach, the double- + 4p2 k2 − k4 +3(−2m2 + s)2 tˆ2 +2tˆ4 diﬀerential cross section in question takes the form T T T c KT → ∗ dσ (ep eD X) − 2 − ˆ3 | | | | − 6 (53) 4(2mc sˆ)t +4pT kT cos ϕ 2mc dηD∗ dpD∗T dz − 2 − − ˆ ˆ 2 − 4 2 ˆ ∗ 2 (kT sˆ 2t)t(mc u˜)+mc(kT +3ˆs +6t) = dyfγ/e(y) Dc→D (z,µ ) z + m2 3k4 +ˆs2 − 6ˆstˆ+ k2 (4ˆs − 2tˆ) − 6tˆ2 dϕ Φ(x, k2 ,µ2) c T T × dk2 T 2π T x | | 4 2 2 ˆ 2 − +cos(2ϕ) pT mc kT + kT t(mc u˜) 2|p ||p | |M(γg∗ → cc¯)|2 × c cT , 2 2 − 2 4 2 2 ˆ Ec(W − 2EN (Ec − pcz)) 16πxW mc 2kT +ˆs + kT (3ˆs +2t) . where q = ype is the 4-momentum of the real photon; 2 q =0;andk = xpN + kT is the 4-momentum of the If we use the polarization vector of the Reggeized 2 2 −| |2 gluonintheform Reggeized gluon, k = kT = kT . µ µ xpN ε (x, kT )=− , (58) 5.2. Amplitudes of Parton Subprocesses |kT | For the process the amplitude squared appears to be [30, 31] γ + g → c +¯c, (54) |M(γg∗ → cc¯)|2 (59) the result of averaging the square of the absolute 2 2 value of the amplitude over the photon and gluon 2 2 2 2 α1 + α2 polarizations and over the color states of the initial =16π ecαsα(ˆs + kT ) (tˆ− m2)(ˆu − m2) gluon is well known and can be represented in the c c form [29] 2m2 α α 2 − c 1 − 2 , 2 − 2 ˆ− 2 |M(γg → cc¯)|2 (55) kT uˆ mc t mc

2 2 2 2 2 − mc mc where =16π ecαsα 4 + t˜ u˜ 2 2 mc + pcT 2 2 α = , (60) m m u˜ t˜ 1 2 − ˆ − 4 c + c + + . mc t ˜ ˜ t u˜ t u˜ m2 + p2 α = c cT¯ ; (61) For the analogous process involving a Reggeized 2 2 − mc uˆ gluon, ∗ γ + g → c +¯c, (56) pcT and pcT¯ are the transverse momenta of the c and c¯quarks, respectively; and kT = pcT + pcT¯ is the the averaging of the square of the absolute value of the transverse momentum of the Reggeized gluon. respective amplitude over the photon polarization and over the color states of the initial gluon was performed Formula (59) can be reduced to the form (57), in [2]. The result can be written in the form whereby the gauge-invariance of the amplitude for the −16π2e2α α process involving a Reggeized gluon is conﬁrmed. |M(γg∗ → cc¯)|2 = c s (57) t˜2u˜2 In the limit |kT |→0, the matrix element involving × m2 2m6 − (8p2 + m2 +3k2 )k4 a Reggeized gluon reduces to the matrix element c c T c T T involving a real gluon, as this occurred in the case of +4m2p2 k2 − (4m4 +5k4 +12p2 k2 )ˆs electroproduction. Speciﬁcally, we have c T T c T T T 2π +(3m2 − 4p2 − 3k2 )ˆs2 − sˆ3 dϕ c T T |M(γg → cc¯)|2 = lim |M(γg∗ → cc¯)|2, |kT |→0 2π 4 − 2 2 2 2 4 0 + 4mc (3ˆs 2mc )+(ˆs +4pT + kT +2mc )kT (62)

2 2 − 2 2 − 2 2 ˆ where ϕ is the angle between the ﬁxed axis x and the +4pT kT (ˆs 2mc )+(s + kT 6mc)ˆs t vector kT .

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σ σ η d /dpT, nb/GeV d /d , nb 12 101 JB

JB 8 KMR 100 KMR JS 4

JS GRV GRV 0 10–1 JB 2

KMR 10–2 JS 1 246 8 10 12 pT, GeV GRV

Fig. 5. Differential cross section for the photoproduction 0 of D∗ mesons as a function of the transverse momentum 0.6 pD∗T . KMR JB 0.4 5.3. Results of the Calculations JS In this section, we compare the theoretical predictions obtained within various approaches with exper- 0.2 imental data on the spectra of D∗±-meson photopro- GRV duction in ep interactions at the HERA collider for the energies of E = 820 GeV and E =27.5 GeV. 0 N e –1.5 –1.0 –0.5 0 0.5 1.0 1.5 Experimental data of the H1 [21] and ZEUS [22] η Collaborations are represented in the form of various ∗± spectra of D mesons—specifically, the spectrum Fig. 6. Differential cross section for D∗-meson photo- with respect to W in the range 130 2, 4, 6 GeV (from top to bottom)] and for W 4-momentum in the range Q < 1 GeV , the spec- from the interval 130

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 ELECTROPRODUCTION OF D∗± MESONS 1597 uncertainties that are associated with the choice of dσ/dW, nb/GeV various parametrizations for the noncollinear gluon 0.20 distribution in the proton. JB 0.15 JS KMR 6. CONCLUSIONS 0.10 In the present study, we have shown that only for a GRV specific choice of parametrization for the noncollinear 0.05 gluon distribution in the proton do calculations with parton amplitudes in the leading order in the strong 0 coupling constant within the kT -factorization approach describe experimental data on D∗±-meson 0.03 electro- and photoproduction at the HERA collider JB JS energies better than calculations within the collinear KMR parton model (see also [3, 32]). 0.02 ∗ Our results for the pT and η spectra of D mesons in photoproduction processes are in good agreement 0.01 with the results obtained in [13], where the JB GRV parametrization [9] was used for the noncollinear gluon distribution. 0 0.008 The main sources of uncertainties in theoreti- JB cal calculations are the following: (i) the existing JS arbitrariness in the choice of c-quark mass (1.3– 0.006 KMR 1.8 GeV), a variation in the c-quark mass leading to a 30% change in the result (this is in accord with 0.004 the results obtained in [14]); (ii) the uncertainties in the choice of fragmentation function and of the 0.002 GRV fragmentation parameter, they being about 15%; (iii) the errors in measuring the parameter in the 0 fragmentation function for the transformation of a c 50 100 150 200 250 ∗± quark into a D meson, which are about 25% of the W, GeV result; (iv) the distinctions (by a factor of 1.5 to 2.5) Fig. 7. Differential cross section for the photoproduction between the predictions obtained by using different ∗ parametrizations of the unintegrated gluon distribu- of D mesons as a function of the total energy W of the photon–proton interaction in the c.m. frame for various tion in the proton; and (v) the choice of renormaliza- ∗ cuts on the transverse momentum of D mesons [p ∗ > 2 D T tion scale µ in the running strong coupling constant, 2, 4, 6 GeV (from top to bottom)] and for |ηD∗ | < 1.5. this leading to a change in the total cross section by a factor of 1.5 to 2. By way of example, we indicate 2 that, in [13, 14, 33], the argument of the running We note that the choice of µ with allowance 2 2 coupling constant was set to the Reggeized gluon for kT in the quark–gluon interaction vertex, µ = 2 2 2 2 2 2 m ∗ + p + Q + k , introduces virtually no changes virtuality µ = kT . Concurrently, the value of αs is D T T 2 2 in the results because of a low power of α in the fixed in the region of low k (αs(k )=αs,max 0.5), s T T eg∗ → ecc¯ or integration with respect to k2 is performed over square of the absolute value of the matrix T element, a weak logarithmic dependence in α (µ2) the region k2 ≥ Q2 2 GeV2. In either case, one s T 0 at high µ2, and a relatively small contribution from has to invoke additional assumptions and introduce 2 new phenomenological parameters. Moreover, such a the region of high kT due to the decrease in the 2 choice for processes involving a few Reggeized gluons unintegrated distributions with increasing kT . or a few quark–gluon interaction vertices some of which have no bearing on a Reggeized gluon may ACKNOWLEDGMENTS lead to a violation of the gauge-invariance condition We are grateful to H. Jung for information about 2 for the amplitudes if different values of µ are chosen parametrizations of unintegrated gluon distributions for different vertices. In the present calculations, we and to А. Likhoded, О. Teryaev, and А. Kotikov 2 2 2 2 2 → have set µ = mD∗ + pT + Q ,whereQ 0 in the for enlightening discussions on problems of low-x case of photoproduction. physics.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1598 SALEEV, VASIN The work of D.V. Vasin was supported by the 15. A. V. Berezhnoy, V. V. Kiselev, and A. K. Likhoded, Dynasty foundation for support of noncommercial hep-ph/9901333; Yad. Fiz. 63, 1682 (2000) [Phys. At. programs and by the International Center for Funda- Nucl. 63, 1595 (2000)]. mental Physics in Moscow. 16. M. Kramer, Prog. Part. Nucl. Phys. 47, 141 (2001); E.BraatenandT.C.Yuan,Phys.Rev.D52, 6627 (1995). REFERENCES 17. OPAL Collab. (R. Akers et al.), Z. Phys. C 67,27 1. Small-x Collab. (B. Andersson et al.),Eur.Phys.J.C (1995). 25, 77 (2002). 18. C. Peterson, D. Schlatter, I. Schmitt, and P. M. Zer- 2.V.A.SaleevandD.V.Vasin,Phys.Lett.B548, 161 was, Phys. Rev. D 27, 105 (1983). (2002). 19. Rev. Part. Phys. (Meson Particle Listing), Phys. Rev. 3. V. A. Saleev and D. V. Vasin, hep-ph/0209220 v2. D 54, 468 (1996). 4. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781 (1972) [Sov. J. Nucl. Phys. 15, 438 (1972)]; 20. K. Anikeev et al., Fermilab-Pub-01/197 (2001). Yu.L.Dokshitser,Zh.Eksp.´ Teor. Fiz. 73, 1216 21. H1 Collab. (C. Adolf et al.), Nucl. Phys. B 545,21 (1977) [Sov. Phys. JETP 46, 641 (1977)]; G. Altarelli (1999). and G. Parisi, Nucl. Phys. B 126, 298 (1977). 22. ZEUS Collab. (J. Breitweg et al.), Eur. Phys. J. C 6, 5. E.A.Kuraev,L.N.Lipatov,andV.S.Fadin,Zh.Eksp.´ 67 (1999). Teor. Fiz. 71, 840 (1976) [Sov. Phys. JETP 44, 443 23. ZEUS Collab. (J. Breitweg et al.),Eur.Phys.J.C12, (1976)]; I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. 35 (2000). Phys. 28, 822 (1978). 24. ZEUS Collab. (J. Breitweg et al.), Phys. Lett. B 481, 6. M. Ciafaloni, Nucl. Phys. B 296, 49 (1988); S. Catani, 213 (2000). F. Fiorani, and G. Marchesini, Phys. Lett. B 234, 339 25. P.Nason and C. Oleari, Phys. Lett. B 447, 327 (1999). (1990); G. Marchesini, Nucl. Phys. B 445, 49 (1995). 26. J. Binnewies, B. A. Kniehl, and G. Kramer, Phys. Rev. 7. J. C. Collins and R. K. Ellis, Nucl. Phys. B 360,3 D 58, 014014 (1998). (1991). 27. C. F. Weizsacker,¨ Z. Phys. 88, 612 (1934); 8. M. Gluck,¨ E. Reya, and A. Vogt, Z. Phys. C 67, 433 E. J. Williams, Phys. Rev. 45, 729 (1934). (1995). 9. J. Blumlein, DESY 95-121 (1995). 28. S. Frixione et al., Phys. Lett. B 319, 339 (1993). 10. H. Jung and G. P. Salam, Eur. Phys. J. C 19, 351 29. J. Babcock, D. Sivers, and S. Wolfram, Phys. Rev. D (2001). 18, 162 (1978). 11. M.A.Kimber,A.D.Martin,andM.G.Ryskin,Phys. 30. S. Catani, M. Ciafaloni, and F. Hautmann, Nucl. Rev. D 63, 114027 (2001). Phys. B 366, 135 (1991). 12. B.A.Kniehl,G.Kramer,andM.Spira,Z.Phys.C76, 31. V. A. Saleev and N. P. Zotov, Mod. Phys. Lett. A 11, 689 (1997). 25 (1996). 13. S. P. Baranov, H. Jung, and N. P. Zotov, hep- 32. V.A. Saleev and D. V.Vasin, Phys. Rev. D 68, 114013 ph/9910210. (2003). 14. S. P.Baranov, H. Jung, L. Jonsson, et al.,Eur.Phys. 33. A. V. Lipatov, V. A. Saleev, and N. P. Zotov, Mod. J. C 24, 425 (2002); S. P. Baranov, H. Jung, and Phys. Lett. A 15, 1727 (2000). N. P. Zotov, Nucl. Phys. B (Proc. Suppl.) 99, 192 (2001). Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1599–1615. From Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1659–1675. Original English Text Copyright c 2005 by Bogdan, Fadin. ELEMENTARY PARTICLES AND FIELDS Theory

Quark Regge Trajectory in Two Loops from Unitarity Relations*

A. V. Bogdan** and V. S. Fadin*** Budker Institute for Nuclear Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia Received April 15, 2004; in ﬁnal form, February 28, 2005

Abstract—The two-loop quark Regge trajectory is obtained at arbitrary spacetime dimension D using the s-channel unitarity conditions. Although explicit calculations are performed for massless quarks, the method used allows one to ﬁnd the trajectory for massive quarks as well. At D → 4, the trajectory turns into one derived earlier from the high-energy limit of the two-loop amplitude for the quark–gluon scattering. The comparison of two expressions obtained by quite diﬀerent methods serves as a strict cross check of many intermediate results used in the calculations, and their agreement gives strong evidence of accuracy of these results. c 2005 Pleiades Publishing, Inc.

2 1. INTRODUCTION with the property jP (mP )=sP (mP and sP are respective mass and spin values), called Regge trajec- Perturbative QCD is widely used for the descrip- tory, describe the motion of poles of the correspond- tion of semihard [1] as well as hard processes [2]. But ing t-channel partial waves in the complex angular whereas the theory of the latter is clear and plain, a lot momentum plane. In this respect, QCD sharply dif- of problems remain unsolved for the former processes. fers from QED, where only amplitudes with electron t The applicability of perturbation theory, improved by exchange in the channel [7], but not with photon the renormalization group, to a hard process with a one [8], acquire the Regge factors. large typical virtuality Q2 is justified by the small- The Reggeization phenomenon is extremely im- 2 ness of the strong coupling constant αs(Q ).Inthe portant at high energy. The gluon Reggeization is semihard case, however, smallness of the ratio x of especially significant, since gluon exchanges in the the typical virtuality Q2 to the squared c.m.s. energy t channel provide nondecreasing cross sections at s requires resummation of the terms strengthened large s. In particular, the gluon Reggeization consti- by powers of ln(1/x). In the scattering channel, this tutes the basis of the famous BFKL approach [5, 9] to problem is related to the theoretical description of the theoretical description of high-energy processes high-energy amplitudes at fixed (not growing with s) in QCD. Initially, the BFKL approach was formulated momentum transfer t. It turns out, that the Gribov– in the leading logarithmic approximation (LLA). Now Regge theory of complex angular momenta, which it is developed in the next-to-leading order (NLO) (for was developed much before the appearance of QCD, references, see, for instance, [10]) since LLA is not is eminently suitable for description of the QCD am- sufficiently reliable, especially because this approxi- plitudes, due to the remarkable property of QCD— mation does not fix scales either longitudinal (for s)or Reggeization of its elementary particles, gluons and transverse (for running coupling αs) momenta. This quarks [3–6]. The Reggeization means, in particular, development extensively uses gluon Reggeization, that, with account of radiative corrections in the high- which has been proved in LLA [11], but in the next- energy limit, the s dependence of QCD amplitudes to-leading approximation (NLA) still remains a hy- with gluon (G)orquark(Q) quantum numbers in pothesis, although it has successfully passed through a set of stringent tests on self-consistency (see, for the t channel is given by Regge factors (s)jP (t),with instance, [12] and references therein). Accordingly, P = G or P = Q, accordingly. The functions j (t) P the NLO gluon Regge trajectory and Reggeized ∗This article was submitted by the authors in English. gluon vertices are calculated; moreover, a way for the **e-mail: [email protected] proof of gluon Reggeization in the NLA is outlined ***e-mail: [email protected] (see, e.g., [10] and references therein).

1063-7788/05/6809-1599$26.00 c 2005 Pleiades Publishing, Inc. 1600 BOGDAN, FADIN

G Q' give a strict test for quark Reggeization. A proof of the Reggeization is also possible in this way. The calculation of the two-loop corrections to the q quark trajectory is performed explicitly for massless quarks; but the method used here allows one to do it for massive quarks as well, since all necessary one- QG' loop Reggeon vertices for the massive case are known now. Fig. 1. Schematic representation of the backward quark– gluon scattering process G + Q → Q + G. The triple The paper is organized as follows. In the next line denotes an intermediate t-channel state with mo- section, all necessary notation is introduced and the − − mentum q = pQ pG = pQ pG. method of calculation is discussed. Section 3 is devoted to the calculation of the two-particle contribution to the s-channel discontinuity of the quark– Along with the Pomeron, which appears in the gluon scattering amplitude. The contribution of the BFKL approach as a compound state of two Reggeized three-particle intermediate state is calculated in Sec- gluons, the hadron phenomenology requires Reggeons, tion 4. The final expressions for the discontinuity and which can be constructed in QCD as colorless states the two-loop corrections to the quark trajectory are of Reggeized quarks and antiquarks. It demands presented and discussed in Section 5. For conve- further development of Reggeized quark theory, which nience, the integrals encountered in Sections 3 and 4 remains in a worse state than Reggeized gluon theory, are listed in Appendices A and B, respectively. Details although noticeable progress was achieved in recent of the calculation of a new nontrivial integral arising years. In particular, multiparticle Reggeon vertices in present calculations are given in Appendix C. required in the NLA were found [13] and the NLO corrections to the LLA vertices were calculated [14, 15] assuming quark Reggeization in NLA. Note that 2. NOTATION AND METHOD the Reggeization hypothesis is extremely powerful; OF CALCULATION but in the quark case, it actually was not proved even Let us consider the backward quark–gluon scat- in the LLA, where merely its self-consistency was tering process (see Fig. 1) in the limit of large (tend- shown, in all orders of αs but only in a particular ingtoinfinity) c.m.s. energy and fixed momentum 2 2 case of elastic quark–gluon scattering [6]. Recently, transfer t ≡ q =(pQ − pG ) . the hypothesis was tested at the NLO in order of 2 We use the Sudakov decomposition of momenta αs in [16], where its compatibility with the two-loop amplitudeforthequark–gluon scattering was exhib- p = βp1 + αp2 + p⊥, ited and the NLO correction to the quark trajectory 2 2 2 was found in the limit of the spacetime dimension D p1 = p2 =0, (p1 + p2) = s, (1) D 2 2 tending to the physical value =4. sαβ = p − p⊥, In this paper, we investigate the quark Reggeiza- supposing that the momenta p ,p and p ,p are tion in the NLO by the method based on the s- G Q Q G close to the light-cone momenta p1 and p2, respec- channel unitarity and the analyticity of scattering tively, that is, amplitudes, which was developed for analysis of pro- ∼ ∼ ∼ cesses with gluon exchanges [4, 5] and was already βG βQ αQ αG 1, (2) successfully applied to processes with fermion ex- |t| β β α α ∼ , changes [6]. The two-loop quark trajectory at arbi- Q G G Q s D trary spacetime dimension is obtained as a particular result of the investigation. At D → 4,thetra- and all transverse momenta are limited, so that q jectory goes into one derived in [16]. This agreement q⊥. We do not suppose that p1 and p2 are contained testifies to the accuracy of many intermediate results in the initial momentum plane, in order to main- used in both derivations. In the method used here, tain symmetry between cross channels and to make the trajectory is obtained from the requirement of the more evident substitutions for transitions between channels. For a gluon having momentum k (k ) compatibility of the Reggeized form of the amplitudes a b with predominant component along p (p ), we use with the s-channel unitarity at the two-loop level. 1 2 physical polarization vectors in the light-cone gauge A possible generalization of this requirement to all e(k )p =0(e(k )p =0), so that orders of perturbation theory should give the “boot- a 2 b 1 strap” conditions on the Reggeized quark vertices (e(ka)ka)⊥ e(ka)=e(ka)⊥ − p2, (3) and the trajectory in QCD. Verification of them will kap2

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⊥ (e(kb)kb) where ΓQG and ΓGQ are effective vertices for in- e(kb)=e(kb)⊥ − p1, kbp1 teraction of particles (quarks and gluons) with the Reggeized quark; we call them PPR vertices, and where (ab)⊥ means (a⊥b⊥). Furthermore, since with we call δ (ˆq) the quark trajectory. Strictly speaking, our choice of gauges gluon polarization vectors are T for massive quarks, there are two trajectories, in ac- expressed in terms of their transverse components, cordance with two possible parity states for an off- from this point forward, we will use only these com- mass-shell quark. These trajectories are determined ponents, without explicit indications, so that every- by eigenvalues of δT (ˆq). We perform actual calcula- where below e means e⊥. The same we will do for the momentum transfer q. tions for the massless case, when δT (ˆq) depends in fact not on qˆ,butonqˆ2 = t,andwriteitasδ (t). The large-s and fixed-t limit of scattering ampli- T Note, however, that the PPR vertices Γ and Γ t Q G G Q tudes is related to quantum numbers in the chan- are known now for massive quarks [15], so that all nel. For the gauge group SU(Nc),thet-channel consideration presented below can be transferred to color state of the process depicted in Fig. 1 contains the massive case in a straightforward way. three irreducible representations of the color group ⊕ ¯ ⊕ We demonstrate that the Reggeized form (4) is (for QCD with Nc =3,itis3 6 15). Therefore, compatible with the s-channel unitarity and obtain it is natural to decompose the quark–gluon scatter- the NLO contribution to the trajectory δT .Morepre- ing amplitude into three parts, in accordance with cisely, we calculate, using the unitarity relation, both the representations [6, 16]. At the same time, in the logarithmic and nonlogarithmic terms in the two- Gribov–Regge theory, each part must be decom- loop s-channel discontinuity of the backward quark– posed into two pieces according to the new quantum gluon scattering amplitude with positive signature number—signature—which is introduced in the the- and prove that only color triplet t-channel states con- ory. Besides quantum numbers, commonly used for tribute to the discontinuity. It means that only the particle classification, a Reggeon has definite signa- (+) ture, positive or negative, which is actually a parity of color triplet part A3 of the amplitude survives at the t-channel partial waves with respect to the sub- NLO as well as at LO. We compare the calculated stitution cos θt →−cos θt (which turns into s → u = discontinuity with the discontinuity of the Reggeized −s in the limit of large s and fixed t). Consequently, quark contribution (4). The logarithmic terms of con- (±) fronted discontinuities turn out to be equal. The non- A ,χ , ¯ there are six terms χ =3 6,and15 in full de- logarithmic terms in the discontinuity of (4) are ex- composition [6, 16] of the quark–gluon scattering pressed through the one-loop corrections to the PPR amplitude. As is known [6], in the LLA only one of the vertices, which are known, and the two-loop cor- (+) positive signature amplitudes, namely A3 ,doessur- rection to the trajectory, which makes it possible to vive; and at the same time, it has the Reggeized form. obtain the last correction. This is not so for the negative signature amplitudes. For massless quarks, the PPR vertices entering The Bethe–Salpeter-type equation obtained for them in (4) have the form [14] in the LLA [6] does not have a simple Regge-type G ΓQG = −gu¯(pQ )t (5) solution (in fact, no solution has been found at all). (eGq)ˆq Note that they are not actually leading in each order of × eˆG(1 + δe(t)) + δq(t) , perturbation theory, because leading logarithms can- q2 cel in them as the result of antisymmetry with respect ∗ ∗ (e q)ˆq to the exchange s → u = −s. Below, we consider only − G ΓG Q = g eˆG (1 + δe(t)) + δq(t) the amplitudes of positive signature. As we will see, q2 intheNLA,aswellasintheLLA,onlyamplitudes G × t u(pQ). with a color triplet in the t channel survive among them. For quark–gluon scattering, the contribution Here, tG and tG are the color group generators in of the Reggeized quark can be represented as [here the fundamental representation; we omit color wave and below, we write symbols of initial (final) particles functions for gluons and assume that they are in- as lower (upper) indices of scattering amplitudes and cluded in u(pQ), u¯(pQ ) for quarks. The one-loop cor- place first the particles with momenta close to p1] rections δe(t), δq(t) canbewrittenas

QG 1 1 (1) (1) (1) (1) R δe(t)=ω (t)δe ,δq(t)=ω (t)δq , (6) GQ =ΓQ G (4) mQ − qˆ2 where − δT (ˆq) δT (ˆq) s s C 1 3(1 − ) × + ΓGQ, (1) F − −t −t δe = (7) 2Nc 2(1 + 2 )

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1602 BOGDAN, FADIN 1 right-hand side of (10) with ∆s instead of ∆R, + ψ(1) + ψ(1 − ) − 2ψ(1 + ) + − , 2 2(1 + 2 ) we see that ∆s has the same helicity structure as ∆R (11) and that their logarithmic terms coincide; 1 moreover, the nonlogarithmic terms at the helicity δ(1) = 1+ , q 2 ˆ 2(1 + 2 ) Nc nonconserving structure E also coincide. After that, the requirement of equality of the nonlogarithmic ψ(x)=Γ(x)/Γ(x) is the logarithmic derivative of the ˆ (1) terms at the helicity conserving structure Q gives us Euler gamma function, and ω (t) is the one-loop (2) gluon Regge trajectory: δT . − 2N g2 µ2 The calculation of ∆s is the main content of this ω(1)(t)=− c s Γ , (8) paper. It is determined from the s-channel unitarity (4π)2+ −t relation: 2 Γ (1 + )Γ(1 − ) Q G Γ = . A(+) ( two-loop) (13) Γ(1 + 2 ) GQ s 2 We use conventional dimensional regularization; the 2 (1) iπg 1 ω (t) G G spacetime dimension D =4+2 and gs = gµ is the = u¯(pQ )t t ∆su(pQ) t 2 Nc dimensionless bare coupling constant. To avoid uncertainties, let us note that the vertices (+) n n ∗ = iP dΦ A A , ΓGQ¯ and ΓQG¯ are obtained from (5) by the sub- n GQ Q G ∗ → → → n stitutions u¯(pQ ) v¯(pQ¯ ),eG eG and u(pQ) ∗ → v(pQ¯),eG eG , respectively. where Φn is the n-particle phase-space element and (+) Representing the quark trajectory as P is the positive signature projector. The summa- 2 tion is performed over two-and three-particle inter- (1) (1) ω (t) (1) 1 ω (t) (2) mediate states; accordingly, we represent the discon- δT (t)= δT + δT , (9) tinuity as the sum of two contributions Nc 2 Nc (2) (3) (1) ∆s =∆s +∆s . (14) where δT = CF [6], and the two-loop s-channel discontinuity of the Reggeized quark contribution (4) as The projection on the positive signature means the half-sum of the s-channel discontinuities for the di- RQ G → ˜ → ¯ GQ (two-loop) (10) rect (GQ Q G ) and crossed (GG Q Q )pro- s cesses. More precisely, if one represents the disconti- 2 2 (1) Q G |M|GQ iπg 1 ω (t) G G nuity of the direct processes as ,where = u¯(pQ )t t ∆Ru(pQ), |GQ isaspinandcolorquark–gluon wave function, t 2 Nc and the discontinuity for the process GG˜ → QQ¯ we have from (4), (5), (9), and (6) ¯ | | ˜ with pG˜ = pQ,pQ¯ = pG as Q Q Mc GG , then the (2) QG|(M + ∆R = δ Qˆ +2CF (11) projection on the positive signature is T M )/2|GQ. Note that, calculating ∆ , we always s c s × (1) Qˆ (1)Eˆ use the fact that the rightmost pˆ and leftmost pˆ in 2Ncδe + CF ln − +2CF Ncδq , 2 1 t M and Mc give negligible contributions because of where the Dirac equation. Qˆ ∗ Eˆ ∗ ∗ =êGqêˆG , =êG(eG q)+êG (eGq), (12) (1) (1) 3. TWO-PARTICLE CONTRIBUTION and δe , δq are given by (7). TO THE DISCONTINUITY Below we calculate the discontinuity In the direct channel, a two-particle intermediate (+) Q G A (two-loop) of the backward quark– state can be solely a quark–gluon one. Since only GQ s gluon scattering amplitude with positive signature limited transverse momenta of intermediate particles from the s-channel unitarity condition. We show that are important in the unitarity relation (we will see it RQ G directly; actually, it is a consequence of the renor- it has the same color structure as GQ (two-loop) s malizability), non-negligible contributions are given (10); i.e., in this approximation, only the color triplet by two nonoverlapping kinematical regions. In one state survives in the positive signature. Writing the of them, the intermediate gluon momentum is close calculated discontinuity in the same form as the to pG (see Fig. 2a), and in another, it is close to pQ

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1603

(a) (b)

G G1 Q' G Q1 Q'

q1 q2 q2 q1

QGQ1 ' QGG1 '

Fig. 2. Schematic representation of the two-particle contribution to the s-channel discontinuity of the backward quark–gluon scattering amplitude. The double lines represent Reggeized quark and gluon, and the blobs, PPR vertices.

r r (Fig. 2b). In both cases, the amplitudes on the right- where ΓAA and ΓBB are Reggeized gluon vertices, hand side of the unitarity relation (13) are in Regge- which can be found in [12]. In the direct channel, type kinematics. we need quark–gluon scattering amplitudes. For As we need to calculate the two-loop contribution the process G + Q → G1 + Q1 (see the left part of to the discontinuity, one of them has to be taken in Fig. 2a), we obtain the Born approximation and another one in the one- 2g2s pˆ loop approximation. An important point is that since AG1Q1 = T r u¯(p )tr 1 u(p ) GQ 2 G1G Q1 Q (17) Born amplitudes are real, only real parts of one-loop q1 s amplitudes are essential for the calculation of the discontinuity. Therefore, required amplitudes are deter- × − ∗ (1) 2 (eG eG) 1+ω (q1) mined by Reggeized quark and gluon contributions. 1 ∗ Q G n Moreover, we can use An instead of A ,as Q G − s × δ(1+) + δ(1 ) + δ(1) +ln imaginary parts of the amplitudes are not important. G G Q − 2 q1 The amplitudes with t-channel quarks can be ∗ obtained by evident substitutions from (4). Using (eG q1)(eGq1) (1−) → + ω(1)(q2)(D − 2) 1 δ , Eqs. (5), (6), for the process G + Q Q + G ,we 1 q2 G have with required accuracy 1 2 where T r are color generators in the adjoint repre- QG g ∗ G1G G G AGQ = u¯(pQ )t t eˆGqêˆG (15) − ⊥ −q2 sentation; q1 =(pG1 pG) ; δQ represents the one- loop corrections to the quark–quark–Reggeon ver- × (1) 2 (1) CF s tex, 1+ω (q ) 2δe + ln 2 Nc −q 1 1 δ(1) = + ψ(1 − )+ψ(1) − 2ψ(1 + ) (18) (1) 2 (1) ∗ ∗ Q + ω (q )δ [ê (e q)+(e q)ê ] u(p ), 2 q G G G G Q 2+ 1 2 + − 1+ where q = pQ − pG . 2 2(1 + 2 )(3 + 2 ) 2Nc (1 + 2 ) In the following, in the amplitudes on the right- n 1+ hand side of (13), we denote Reggeized gluon and − f , quark momenta q1 and q2, respectively, q1 + q2 = Nc (1 + 2 )(3 + 2 ) q, and Reggeized gluon color index r.Sinceq1 q1⊥,q2 q2⊥, everywhere below we omit the sign ⊥ (1+) nf being the number of quark flavors; and δG at q1,2,sothatqi means qi⊥,i=1, 2. (1−) δ Amplitudes for processes AB → A B with gluon and G represent helicity-conserving and helicity- exchanges with our accuracy are written as violating corrections to the gluon–gluon–Reggeon vertex, AB 2s r (1) 2 s r A AB = 2 ΓA A 1+ω (q1)ln 2 ΓB B, q −q (1+) 1 2 1 1 δ = + ψ(1 − )+ψ(1) (19) (16) G 2

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1604 BOGDAN, FADIN 2 9(1 + ) +2 Note that, in order to obtain AQ G (see the right − 2ψ(1 + ) − Q1G1 2(1 + )(1 + 2 )(3 + 2 ) part of Fig. 2b) from (17), one has to change pˆ1 to 3 2 pˆ besides evident substitution of symbols. nf (1 + ) + 2 + 2 , Nc (1 + ) (1 + 2 )(3 + 2 ) In addition to the presented amplitudes, in the (1−) crossed channel, we need gluon–gluon and quark– δG = antiquark forward scattering amplitudes (see Fig. 3). 2(1 + )(1 + 2 )(3 + 2 ) n × −1+ f . The ﬁrst (see the left part of Fig. 3a) is written as Nc(1 + )

2 − G1G˜1 2g s r r ∗ ∗ (1) 2 (1+) (1 ) s A = TG GT ˜ ˜ (eG eG)(e ˜ e ˜) 1+ω (q1) 2δ +2δ +ln (20) GG˜ q2 1 G1G 1 G1 G G G −q2 1 1 ∗ ∗ ∗ ∗ (eG eG)(e ˜ q1)(eG˜ q1)+(eG q1)(eGq1)(e ˜ eG˜) − (1) 2 − 1 G1 1 G1 (1−) ω (q1)(D 2) 2 δG , q1

and the second (see the right part of Fig. 3b) means that contributions of other color states cancel 2 in the sum of the direct and cross channels. The QQ¯ 2g s pˆ2 − r case when both upper vertices are Born ones can be A ¯ = u¯(pQ )t u(pQ1 )¯v(pQ¯ ) (21) Q1Q1 q2 s 1 1 considered quite analogously. One can come to the same conclusion seeing that sum of contributions of × r pˆ1 (1) 2 (1) s t v(p ¯ ) 1+ω (q1) 2δ +ln . Fig. 2a and Fig. 3b,aswellasFig.2b and Fig. 3a, s Q Q −q2 1 is proportional to tGtr.Itisnotdifficult to understand Before starting with the calculation, let us show that the cancellation of color states different from a triplet is not restricted by the considered diagrams that only a color triplet state survives in the dis- or by the two-loop approximation, but is a general continuity, due to cancellation of contributions of all property of the NLA, as well as the LLA. other color states in the direct and crossed channels. In fact, this cancellation has the same nature as in Since we have shown that only a color triplet the leading order [6]. The point is that, if a one-loop survives in the t channel, we can write contribution is taken for one of the PPR vertices in ∗ (+) P1P2 P1P2 Figs. 2 and 3, then all other vertices must be taken P3 AGQ AQG (23) at NLO in the Born approximation. Therefore, either P1P2 both lower or both upper vertices are Born ones. Let 4 (1) G G M(2) us consider the first case. Since the upper parts of the = g ω (t)su¯(pQ )t t u(pQ). diagrams Fig. 2a and Fig. 3a are equal, contributions With the amplitudes listed above, calculation of to M + Mc from the lower lines enter as the sum M(2) is straightforward. Dividing it into two pieces, pˆ (2) (2) γµ tG uλ(p )¯uλ(p )tr 1 (22) M(2) = M + M , one of which contains one- ⊥ Q1 Q1 s Q G λ loop corrections for a quark channel and another for a ∗ ˜ µ gluon channel, we obtain − λ G1 λ r µ r G eˆG˜ t eG˜ TG˜ G = γ⊥t t . 1 1 1 2 − λ q C M(2) = F (24) Q q2 q2q2 Here, we have omitted terms with leftmost pˆ1 because 2 1 2 of the reason explained above and have taken the (1) CF s ∗ same Lorentz and color indices of the gluons G and × 2δ + ln eˆ qˆ eˆ e − 2 G 2 G ˜ Nc q2 G as we do not write their wave functions. It is easy to see that the lower lines of Fig. 2b and Fig. 3b ∗ ∗ r G (1) give in sum the same result. Since t t projects + δq [êG(eG q2)+(eGq2)êG ] the t-channel quark–gluon state on a color triplet, it

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1605

(a) (b)

G G1 Q' GQQ1 '

q1 q2 q2 q1

~ ~ – ~ – – G G1 Q' G Q1 Q'

Fig. 3. Schematic representation of the two-particle contribution to the cross-channel discontinuity of the backward quark– gluon scattering amplitude. and 4. THREE-PARTICLE CONTRIBUTION 2 − TO THE DISCONTINUITY M(2) q 1 − 1 (1) G = 2 2 2 δQ (25) q q q Nc 1 2 1 Let us denote intermediate particles in the unitarity condition (13) Pj and their momenta kj,j = − s + N δ(1+) + δ(1 ) + C ln 1–3. Just as before, only limited transverse momenta c G G F − 2 q1 are important. As for longitudinal momenta, let us ∗ Nc D − 2 (1−) set (without loss of generality) α3 1; that is, for × eˆ qˆ eˆ − δ → G 2 G 2 q2 G the direct process G + Q Q + G , a particle P3 1 is produced in the fragmentation region of the initial ˜ × ∗ ∗ quark (note that, for the crossed process G + G → [êGqˆ2qˆ1(eG q1)+(eGq1)ˆq1qˆ2eˆG ] . Q + Q¯, it is the region of G˜ fragmentation). Then β1 + β2 1; i.e., at least one of the particles Pi,i= Integration over the phase-space element dΦ2 = D−2 D−2 1, 2, is produced in the gluon fragmentation region. d q2/(2s(2π) ) is quite simple. The integrals Let it be P1;thenβ1 ∼ 1,butβ2 can vary from β2 ∼ 1 are well convergent at large |q2|, so that the in- (which means P2 as well as P1 is produced in the tegration region can be expanded to infinity. For ∼| 2 | gluon fragmentation region) to β2 k2⊥ /s (which convenience of the reader, we present necessary ∼ formulas in Appendix A. As a result, we obtain for means α2 1, i.e., P2 is in the quark fragmenta- the two-particle contribution to ∆ tion region). Of course, the same with substitution s ↔ 2 1− 1 2 can be said for the case when in the gluon 2N (−q ) − ∆(2) = c dD 2q M(2) fragmentation region particle P2 is produced. Note s (D−2)/2 2 (26) Γπ 1 β |k2 |/s that region i i⊥ is usually called multi- N 2 − Regge or central region for a particle Pi. But this =2N X 2C δ(1) + c δ(1+) − δ(1 ) c Γ F e 2 G 1 − G region does not require separate consideration, because amplitudes for production of Pi in any of the 2 fragmentation regions are applicable in it. Actually, − 1 (1) CF s δQ + ln +Ψ1 natural bounds for domains of applicability of these 2Nc Nc −t amplitudes are αi 1 for the gluon and βi 1 for CF s 1 Qˆ the quark fragmentation regions. Therefore, it is suffi- + ln − +Ψ1 + 2 t 2 cient to consider two regions: 1 ≥ β ≥ |k2 |/s and i i⊥ (1 + ) − + X C δ(1) + N δ(1 ) Eˆ, 1 ≥ α ≥ |k2 |/s. For brevity, we will say that, in Γ F q c 1 − G i i⊥ the first (second) case, there are two particles in the where gluon (quark) fragmentation region. Moreover, the Γ(1 − 2 )Γ2(1 + 2 ) ↔ X = , (27) symmetry with respect to α β in the definition of Γ Γ(1 + )Γ(1 + 3 )Γ2(1 − ) theregionspermitsonetoconsideronlyoneofthem. → and Indeed, as regards the inverse reaction (Q + G G + Q Ψ = ψ(1 − 2 )+ψ(1 + 3 ) (28) ), the names of the regions must be changed; 1 therefore their contribution to the discontinuity is re- − − − ψ(1 + 2 ) ψ(1 ). lated by the substitutions Q ↔ Q ,G↔ G and com-

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1606 BOGDAN, FADIN plex conjugation. We will consider the gluon frag- phase-space element dΦ3 in the gluon fragmentation mentation region. region looks like In this region, amplitudes on the right-hand side D−2 D−2 dβ1dβ2 d k1d k2 of the unitarity relation (13) can be written, in accor- dΦ3 = δ(1 − β1 − β2) , (34) 4sβ β (2π)2D−3 dance with our agreement about notation, as 1 2 2s with the help of (29)–(31), we obtain the contribution AP1P2P3 = Γr Γr (29) AB 2 {P1P2}A P3B of this region to ∆s (13) in the form q1 2(−q2)1−2 dβ dβ ∆(3)G = − 1 2 and s (D−2) 2 (35) − π Γ β1β2 P P P qˆ2 A 1 2 3 =Γ{ } Γ D−2 D−2 AB P1P2 A 2 P3B (30) d k1d k2 ∗ q2 × F qˆ eˆ δ(1 − β − β ) q2q2 G 2 G 1 2 for gluon and quark exchanges, respectively, with the  1 2    r 2 2 same vertices ΓP B and ΓP3B as for the elastic ampli- |k | |k | 3 × θ β − 1⊥  θ β − 2⊥  , tudes, but taken now in the Born approximation only. 1 s 2 s Therefore, as well as in the two-particle contribution, only the t-channel color triplet survives in positive where q and q are the momenta of t-channel gluon signature since 1 2 and quark, respectively, q1 + q2 = q. As was already (+) r ∗ pointed out, the total three-particle contribution can P Γ (ΓP G ) (31) P3Q 3 bewrittenthenas P3 (3) (3)G ¯ (3)G → 2 ∆s =∆s + ∆s (G G ), (36) (+) r ∗ g r G ∗ = −P ΓP Q Γ = − t t eˆ u(pQ). 3 P3G 2 G ∆=¯ γ0∆†γ0 s P where . Note that logarithms of appear 3 (3)G r in ∆s from integration over βi of those terms in FG The vertices Γ{ } and Γ{P P }A can be found P1P2 A 1 2 which do not go to zero at βi → 0. It is always possi- in [12] and [13], respectively. Actually, they can be ble to rewrite FG as a sum of terms which go to zero easily calculated since are given by Born amplitudes either at β =0or at β =0.Forthefirst (second) of → 1 2 for processes A + R P1 + P2,whereR is either a | 2 | | 2 | gluon (for Γr )withmomentumq , color index them, the limitation β1 > k1⊥ /s (β2 > k2⊥ /s) {P1P2}A 1 can be taken away; the change of variables k1⊥ = r, and polarization vector p2/s (p1/s)foraparticle A with predominant momentum components along β1l1⊥ (k2⊥ = β2l2⊥) often turns out to be useful, and we meet integrals p1 (p2)oraquark(forΓ{P1P2}A)withmomentumq2 and omitted quark wave function. An important point 1 dβi δ is that the corresponding light-cone gauge [see (3)] (1 − βi) (37) must be taken not only for real gluons, but for virtual √ βi | 2 | ones as well. Note that the Hermitian property of Born ki⊥ /s amplitudes gives the relations 1 s − ∗ = ln 2 + ψ(1) ψ(δ +1), r r 2 |k ⊥| Γ =Γ , (32) i {P1P2}Q Q {P1P2} † where δ is proportional to . 0 Γ{P1P2}Q =ΓQ {P1P2}γ , | 2 | Calculation of integrals without ln ki⊥ does not which we use in the following. present a problem. The list of basis integrals is presented in Appendix B. On the contrary, two of the Let us denote 2 integrals with ln |k ⊥| cannot be expressed in terms r r r i Γ Γ { } − Γ{ } Γ t {P1P2}G Q P1P2 P1P2 G Q {P1P2} of elementary functions at arbitrary . Besides the P1P2 integral I0 [see (46)] already encountered in the cal- (33) culations of the two-loop gluon Regge trajectory, here 4 G we meet a new nontrivial integral I [see (47)] below, = g u¯(p )t F . 1 Q G which is considered in Appendix C. The particles P1 and P2 can be two gluons (G1G2), In the following, we will use Eqs. (33), (35), (36) quark and gluon (Q1G2), and quark and antiquark and the notation Q Q¯ ( 1 2). Evidently, only the first (second) term on the k =(k − β p )⊥,k =(k − β p )⊥, (38) left-hand side of (33) contributes in the first and third iG i i G iQ i i Q − − cases (in the second case). Using the fact that the k12 = k21 =(β2k1 β1k2)⊥.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1607

It is easy to understand that kiA (i =1, 2)isthe One should pay attention that all terms in (42) corre- transverse part of ki with respect to the pA,p2 plane, spond to planar diagrams. It is an important property multiplied by βA. of a color triplet state in the t channel which greatly Since the integration in (35) is symmetric with simpliﬁes calculations. respect to exchange k1 ↔ k2, we will systematically The convolutions entering in (42) give omit in FG contributions antisymmetric in relation to µν 2 { this exchange, without further reminding. γµν (p)γ12 = 2 2 eˆG[2β2(k1Q p) (43) k1Q p

− ˆ 4.1. Fragmentation into Two Gluons + β1(1 2β2)k1Q pˆ]+4β1β2(eGk1Q )ˆp − − ˆ } The vertex for production of the gluons G1,G2 + β1β2[β1(D 2) 4](eGp)k1Q ; with color indices i ,i by the initial gluon G can be 1 2 −4 written as γ (p)γµν = µν [12] 2 2 (44) k12p r 2 G r µν Γ{ } = g T T γ (k1G) (39) ×{ − − ˆ G1G2 G i1i2 β1β2(2 β1β2(D 2))(eGp)k12 − 2β1β2(eGk12)ˆp − (1 − 2β1β2)(k12p)ˆeG}. µν ∗ ∗ − γ (k12) e e +(1↔ 2) , (3)G 1µ 2ν We obtain the two-gluon contribution in ∆s by substituting (42) into (35) and performing integra- where tion. Note that, if we integrate over all phase space 2 in (35), we have to take into account equivalence of γµν (p)= β β gµν (e p) − β eµ pν − β pµeν . p2 1 2 ⊥ G 1 G 2 G produced gluons by the factor 1/2!. With account of the quark fragmentation region according to (36), The vertex for a quark exchange can be taken in [13]. we obtain, denoting the contributions related to the Using Sudakov parametrization and omitting terms µν QQ terms γµν (p)γN in FG (42) by ∆N∗p, with rightmost pˆ2,wehaveforit 2 2 Γ { } = −g u¯(p ) (40) Nc 2 3 Q G1G2 Q ∆ ∗ = 2 − +Ψ (45) 12 k1G 4 1+2 2 × i1 i2 µν − µν ↔ t t γ12 γ[12] e1µe2ν +(1 2) , s 5 1 Qˆ 2 Eˆ where +ln + XΓ + (I0 + I1) + ; −t 2 2 1+2 µν 1 µ µ ˆ − ν γ12 = 2 β1k1Q γ⊥ 2k1Q γ⊥, (41) k1Q N 2 c ∆12∗k12 = XΓ µν 2 µν ˆ − µ ν − µ ν 4 γ[12] = 2 β1β2g⊥ k12 β1γ⊥k12 β2k12γ⊥ . k12 × 3 − 3 − s − Q−ˆ 2 Eˆ 2 ln − Ψ ; Note that the lower indices of γµν vertices in (40) 1+2 t 1+2 are determined by sequences of color matrices in ∆ ∗ =0; corresponding group factors, and square brackets in [12] k12 µν γ emphasize its antisymmetry with respect to the 2 [12] Nc s 5 ∆ ∗ = − X 2 ln +Ψ+ permutation 1 ↔ 2, as well as its relation to the color [12] k1G 4 Γ −t 2 factor [ti1 ti2 ]. 4 1 As was pointed out already, only the ﬁrst term on − + Qˆ the left-hand side of (33) contributes in the case of 1+2 (1 − )(1 + 2 )(3 + 2 ) two-gluon production. Putting there the vertices (39) and (40) and performing summation over spin and 4 2 − Eˆ , color of intermediate gluons, after simple color alge- (3 + 2 )(1 − )(1 + 2 ) bra, we obtain for the two-gluon contribution to F G where N 2 F GG = − c (γ (k ) − γ (k )) (42) 2(−q2)1−2 G 4 µν 1G µν 12 I = − 0 2 (D−2) (46) Γ π − − µν µν dD 2k dD 2k q2 q2 × γ − γ +(1↔ 2) . × 1 2 ln , 12 [12] 2 − 2 2 − 2 − 2 k1⊥(k1 q)⊥k2⊥(k2 q)⊥ (k1 k2)⊥

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1608 BOGDAN, FADIN 2 − 2 1−2 − ( q ) Using vertices (50) and (52), we obtain for the quark– I1 = − (47) Γ2π(D 2) gluon contribution to FG after summation over spin D−2 D−2 2 2 and color d k1d k2(k1 − k2)⊥ q × ln . QG − 2 − 2 2 − 2 − 2 FG = β1 Lµ(kij ) (54) k1⊥(k1 q)⊥k2⊥(k2 q)⊥ (k1 k2)⊥ ij Here and below, we use the notation × ij µ ij µ ij µ CG2γG2 + C2Gγ2G + C[G2]γ[G2] , Ψ2 =2[ψ(1) − ψ(1 + 2 )], (48) where ij takes on values 1Q, 2Q, 12 and Ψ=ψ(1 − 2 )+ψ(1 + 3 ) 1 1 1 C1Q = ,C2Q = 1+ , (55) − ψ(1 + ) − ψ(1) + Ψ2. G2 G2 2 4 4 Nc GG 1 The contribution ∆s of fragmentation into two glu- C12 = ; (3) G2 4N 2 ons to ∆s is c GG 1Q 1 2Q 1 ∗ ∗ C = ,C = , ∆s =∆12 k1G +∆12 k12 +∆[12]∗k1G . (49) 2G 2G 2 4 4Nc 12 −1 − 1 C2G = 1 2 ; 4.2. Fragmentation into Quark and Gluon 4 Nc 1 1 C1Q =0,C2Q = − ,C12 = − . Let us denote particles produced in the gluon frag- [G2] [G2] 4 [G2] 4 mentation region Q1 and G2 and their momenta k1 2Q µ and k2, respectively. The vertex Γ{Q1G1}G can be ob- Note that the term CG2 Lµ(k2Q )γG2 here corre- tained by crossing and appropriate substitutions from sponds to a nonplanar diagram and leads to a ΓQ {G1G2} (40). We will represent it as complicated integral. Fortunately, it is canceled by the respective contribution from quark–antiquark 2 Γ{ } = −g u¯(k ) (50) Q1G2 G 1 production, as we will see in the next subsection. µ ∗ For the products Lµ(p)γmn, we have after some × G i2 ν − ν t t γG2 γ[G2] e2ν simpliﬁcations ∗ µ 1 i2 G µ µ Lµ(p)γ = {2β1β2(D − 2) (56) + t t γ2G + γ[G2] e2µ , G2 2 2 k1Gp

ν ν ν µν µν × (eGk1G)ˆp − β2(D − 6)ˆpkˆ1GeˆG − 2ˆeGkˆ1Gpˆ}; where γG2,γ[G2],andγ2G, are obtained from γ12 ,γ[12], νµ and γ21 , respectively, by the substitutions µ 1 Lµ(p)γ2G = 2 2 1 β2 k1G k12p β1 → ,β2 →− ,k1Q →− , (51) β1 β1 β1 ×{β2(β2(D − 2) − 4)ˆpkˆ12 +4(k12p)}eˆG; k2G k12 k12 → ,k2Q → β1 β1 µ 2β2 β2 − ˆ Lµ(p)γ[G2] = 2 2 1 eˆGpˆk2G µ r k p β1 and convolution with e .ThevertexΓ can be 2G G Q {Q1G2} +(β (D − 2) − 4)(e k )ˆp found in [12] and represented as 2 G 2G r 2 pˆ2 2 Γ = g u¯(p ) (52) ˆ Q {Q1G2} Q + (k2Gp)ˆeG +4(eGp)k2G . s β2 i2 r ×{t t (Lµ(k2Q )+Lµ(k1Q )) (3)G µ The quark–gluon contribution to ∆ is given r i2 − } s + t t (Lµ(k12) Lµ(k1Q )) e2 u(k1), by (35) with (54) instead of FG.Denotingby∆mn·ij ij where the contributions proportional Cmn in (54), after µ µ integration, we obtain for them with account of the β2pγˆ ⊥ − 2p Lµ(p)= . (53) quark fragmentation region according to (36) p2 1 3 − 2 s In the case of quark–gluon production, only the sec- ∆G2·1Q = 2 − Ψ2 − ln (57) ond term on the left-hand side of (33) contributes. 4 1+2 −t

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1609 5 1 2 where − X − (I + I ) Qˆ + Eˆ ; 2 Γ 2 0 1 1+2 −pêˆ +2β (pe ) L(p)= G 1 G . (60) p2 1 ∆ · = X G2 12 2 Γ The vertex Γ { ¯ } can be obtained from [13]; 4Nc Q Q1Q2 s 1 3 − 2 2 a direct calculation does not encounter difficulties × −2 ln +Ψ+ − Qˆ + Eˆ ; either. We have −t 1+2 1+2 2 c σ c Γ { ¯ } = g v¯(k )t γ u(k )¯u(p )t A (61) 1 Q Q1Q2 2 1 Q σ ∆2G·1Q = XΓ 4 − c σ · c s 3 3 u¯(pQ )t γ u(k1) v¯(k2)t BσδQ1Q , × 2 ln +Ψ+ − Qˆ; −t 2 2(1 + 2 ) where δ shows that the last contribution exists Q1Q 1 3 − 2 1 only when an intermediate quark has the same flavor · − Qˆ ∆2G 2Q = 2 XΓ 2 ; as the initial one, and the values Aσ and Bσ can be 4Nc 2(1 + 2 ) written as − σ ∆2G·12 =0; σ β1β2 σ 2p2 A = γ⊥ − (kˆ + kˆ )⊥ ; (62) k2 1 2 s 12 1 3 2 σ − − β1 2p ∆[G2]·2Q = 2 + +Ψ2 σ σ ˆ − 2 4 1+2 B = 2 γ⊥ +(k1 pˆQ )⊥ . k1Q sβ2 s XΓ 1 2 By substituting the vertices (59) and (61) into (33), +ln − − I1 Q−ˆ Eˆ ; −t 2 1+2 after summation over spin and color states of intermediate particles, we obtain for the quark–antiquark contribution to F 1 s G ∆ · = XΓ 2 ln +Ψ [G2] 12 4 −t pˆ F QQ = Cijtr kˆ 2 L(k )kˆ γ Aσ (63) G A 1 s ij 2 σ 3 3 2 ij + − Qˆ + Eˆ . 1+2 1+2 pˆ − Cijγ kˆ 2 L(k )kˆ Bσ , B σ 1 s ij 2 The contribution ∆G2·2Q is not presented here since it is canceled with the respective contribution from where ij takes on values 1G, 2G, 12 and quark–antiquark production, as was pointed out. The (3) Nc Nc Nc quark–gluon contribution to ∆s is given by C1G = n ,C2G = − n ,C12 = − n ; A 8 f A 8 f A 4 f QG ∆ = ∆ · ,ij=1Q , 2Q , 12; (64) s mn ij (58) ij,mn 1 1 1 1 C1G = ,C2G = 1+ ,C12 = . mn = G2, 2G, [G2]. B 2 B 2 B 4Nc 4 Nc 4 Taking into account, as usual, external Dirac spinors, 4.3. Fragmentation into Quark–Antiquark Pair we obtain As well as in the two-gluon case, only the first pˆ 2β β tr kˆ 2 L(p)kˆ γ Aσ = 1 2 (65) term on the left-hand side of (33) exists. We denote 1 2 σ 2 2 s p k12 the momenta of particles Q and Q¯ in the gluon 1 2 × − − ˆ fragmentation region k1 and k2, respectively. The (k12p)êG (eGk12)ˆp +(1 4β1β2)(eGp)k12 ; vertex Γr can be obtained by crossing and {Q1Q¯2}G r corresponding substitutions from Γ { } (52): pˆ2 σ Q Q1G2 γ kˆ L(p)kˆ B σ 1 s 2 r 2 pˆ2 G r Γ = g u¯(k ) t t L(k ) (59) −β1 {Q1Q¯2}G 1 1G { − ˆ s = 2 2 2β1β2(D 2)(eGp)k1Q k1Q p − L(k ) + trtG (L(k )+L(k )) v(k ), ˆ ˆ 12 12 2G 2 − β2(D − 6)k1Q pêˆG − 2êGpˆk1Q }.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1610 BOGDAN, FADIN (3)G The quark–antiquark contribution to ∆s is given where ∆B·2G = −∆G2·2Q , as was shown above. Be- by (35) with (63) instead of FG. As has been men- cause of cancellation of these contributions in tioned above, in F QQ we also have a term correspond- (3) GG GQ QQ G ∆s =∆s +∆s +∆s , (69) ing to a nonplanar diagram. In (63), it appears with 2G 2G 2Q ∆B·2G is not present in (67), nor ∆G2·2Q in (57). the coeﬃcient CB .NotethatCB = CG2 [see (55)]. Moreover, comparing the ﬁrst equations in (56) and the second in (65), one can see that the substitution 5. TWO-LOOP CORRECTION k1Q →−k2Q ,k2G →−k1G (66) TO THE QUARK TRAJECTORY

σ µ The total discontinuity ∆ is given by the sum of turns γ kˆ (ˆp /s)L(k )kˆ B into β L (k )γ s σ 1 2 2G 2 1 µ 2Q G2 the contributions of two-and three-particle interme- with opposite sign. Note that, for the quark–antiquark diate states in the unitarity relation. The two-particle contribution, q2 in (35) is equal to pQ⊥ − k1⊥ − ∆(2) k2⊥ = β2q − k1Q − k2G. Upon the substitution (66), contribution s is given explicitly by (26). All nec- (3) it turns into β2q + k2Q + k1G = k1⊥ + k2⊥ − pG⊥, essary contributions to ∆s (69) are calculated in the which is just the t-channel quark momentum for preceding section. They are given by (49) and (45) for the quark–gluon contribution. An important point fragmentation into two gluons, by (58) and (57) for is that the theta functions in (35) can be omitted for fragmentation into a quark and gluon, and by (68), the contributions of the nonplanar diagrams due to (67) for fragmentation into a quark and antiquark. convergence of integrals, after which the substitu- Now we can compare the calculated discontinuity tion (66) does not influence the integration region. with the form (11) required by the quark Reggeiza- Therefore, these contributions cancel each other. tion. First of all, we note coincidence of the terms with lns. Actually, this coincidence must be expected, Performing integration and denoting by ∆A·ij and 1j since it is required by the quark Reggeization in the ∆B·ij the contributions to (63) proportional to CA LLA, which was already checked on this level. Much 1j more important is that the calculated discontinuity and CB , respectively, we obtain with account of the quark fragmentation region according to (36) has the same helicity structure as ∆R (11), with the same coefficientatthestructureEˆ. It is a serious Nc ∆ · = n X (67) argument in favor of validity of the Reggeization hy- A 1G 8 f Γ pothesis in the NLA. Then, comparing coefficients at −2 1 the structure Qˆ, we obtain × 1 − Qˆ 1+2 (1 − 2)(3 + 2 ) (2) XΓ(1 + ) δ = CF − nf (70) 4 2 T (1 + 2 )(3 + 2 ) − Eˆ ; (1 − 2)(1 + 2 )(3 + 2 ) 1 + N I + I − X [ψ(1 + ) − ψ(1 + 2 )] c 2 0 1 Γ ∆A·2G =∆A·1G;∆A·12 =0; 7X X (11 + 7 ) + Γ + Γ 1 3 − 2 2 2(1 + 2 )(3 + 2 ) ∆ · = 2 − +Ψ B 1G 4N 2 1+2 2 c 1 +2CF − I1 + ψ(1) − ψ(1 − ) s − XΓ − 1 Q−ˆ 2 Eˆ 2 +ln− I1 ; t 2 1+2 +(2− X )[ψ(1 + ) − ψ(1 + 2 )] Γ 1+X (1 − X )(3 − ) 1 − Γ − Γ , ∆B·12 = XΓ 2(1 + 2 ) 4 3 − 2 s 1 2 × − − − − Q−ˆ Eˆ where X is given by (27), and I and I are defined 2 ln − Ψ . Γ 0 1 1+2 t 1+2 in (46) and (47). (3) The two-loop correction to the quark Regge tra- Therefore, the quark–antiquark contribution to ∆s canbewrittenas jectory jQ =1/2+δT (t) is determined by Eqs. (9), QQ (70) at arbitrary spacetime dimension D =4+2 . ∆ =2∆ · +∆ · +∆ · +∆ · , (68) s A 1G B 1G B 12 B 2G Unfortunately, at arbitrary D, the integrals I0 and I1

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1611 cannot be written in terms of elementary functions. In a particular case of elastic quark–gluon scattering, the limit → 0, we have for them (see Appendix C) although in all orders of αs. In the NLA, it is tested 2 1 in the same process only in order of αs.Apossible I = +15ψ(2)(1) 2 + O( 3), (71) 0 way of stricter testing and, in principle, a complete proof is examination of “bootstrap” conditions on 4 (2) 2 3 I1 = − +6ψ (1) + O( ), the Reggeized quark vertices and the trajectory in QCD. These conditions appear from comparison of where ψ(2) means the second derivative of the ψ func- the Reggeized form of discontinuities of amplitudes tion. Using this result and the proportion ψ(2)(1) = with the discontinuities calculated by using the s- −2ζ3,whereζn is the Riemann zeta function, from channel unitarity. Eq. (70), we obtain for the two-loop correction up to We have used dimensional regularization for both terms nonvanishing at → 0 infrared and ultraviolet divergences and the bare coupling constant g = gsµ , so that, besides infrared (2) 2 2 404 poles in , our result contains the ultraviolet poles. δ = CF β0 − K + Nc − 2ζ3 (72) T 27 To remove them, it is suﬃcient to express the bare coupling through the renormalized one. In the MS 56 − n +(C − N )16ζ , renormalization scheme, f 27 F c 3 2 − − gµΓ(1 ) g = gµµ 1+β0 , (73) where (4π)2+ 11 2 67 5 β = N − n ,K= − ζ N − n , 0 6 c 3 f 18 2 c 9 f where gµ is the renormalized coupling at the normalization point µ. in agreement with the result of [16].

ACKNOWLEDGMENTS 6. SUMMARY † In this paper, we have checked compatibility of the A.V.B. is grateful to M.I. Kotsky for helpful dis- quark Reggeization hypothesis with the s-channel cussions. V.S.F. thanks the Alexander von Hum- unitarity by explicit two-loop calculations and have boldt Foundation for the research award, the Uni- versitat¨ Hamburg and DESY, the Dipartimento di found in the case of massless quarks the two-loop Fisica dell’Universita` della Calabria, and the Istituto correction to the quark trajectory at arbitrary space- Nazionale di Fisica Nucleare—gruppo collegato di D =4+2 time dimension .The expansion of the Cosenza for their warm hospitality while part of this correction gives the result obtained in [16]. We have work was done. This work is supported in part by s calculated the two-loop -channel discontinuity of INTAS (grant no. INTAS-00-00366) and in part by the backward quark–gluon scattering amplitude with the Russian Foundation for Basic Research (project positive signature keeping nonlogarithmic terms and no. 04-02-16685-a). have proved that only a color triplet part of the amplitude survives at NLO as well as at LO. It was shown that the calculated discontinuity has a form re- Appendix A quired by the Reggeization hypothesis. The two-loop correction to the trajectory has been obtained from For convenience of the reader, we list here the comparison of the calculated discontinuity with the (2) Reggeized form. In the case of massive quarks, the integrals encountered in the calculation of ∆s (note trajectory can be found by the same method, since all that, everywhere below, we use Euclidean transverse momenta and omit the transeverseness sign): necessary quantities for such calculation are known. The cancellation of contributions of color states dD−2kki Ji = (A.1) diﬀerent from triplet in positive signature is not re- 1 (k2)1−(k − q)2 stricted by the two-loop approximation, but is a gen- π(D−2)/2 eral property of the NLA. Therefore, in this approxi- = qi Γ · X , mation, as well as in the LLA, real parts of amplitudes (q2)1−2 Γ with positive signature are completely determined by Reggeized quark contributions. dD−2kki s Ji = ln (A.2) Note that testing of the quark Reggeization per- 2 (k2)1−(k − q)2 k2 formed up to now is rather limited. Even in the LLA, self-consistency of the hypothesis was shown only in † Deceased.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1612 BOGDAN, FADIN i s eˆGkˆ(kˆ − qˆ)ˆp s = J ln +Ψ1 , × ln 1 −t 2 − 2 2 − 2 − 2 k (k q) p (p q) (k p) 1 5 s dD−2kqi 3 =ê qˆ (I + I )+X +ln , Ji = = Ji, (A.3) G 2 0 1 Γ 2 −t 3 (k2)1−(k − q)2 2 1 (D−2) (D−2) D−2 i dk dp d kq s K4 = (B.5) Ji = ln (A.4) Λ(Γ )2 4 (k2)1−(k − q)2 k2 eˆG(kˆ − qˆ)kˆpˆ s 3 i 1 i × ln = J + J , 2 − 2 2 − 2 − 2 2 2 4 1 k (k q) p (p q) (k p) 1 XΓ s D−2 l m i =ê qˆ − I − +ln , µ d kk k q G 2 1 −t J5 = − (A.5) (k2)2 (k − q)2 (D−2) (D−2) k l kl dk dp 1 q q δ i = (1 − 2 )+ J . K5 = 2 (B.6) (1 − ) q2 4 1 Λ(Γ) ˆ Recall that × (eGk)kpˆ k2(k − p)2p2(p − q)2 Γ2(1 + )Γ(1 − ) Γ = , (A.6) 1 X Γ(1 + 2 ) = − eˆ qˆ +(e q)(1 + ) Γ , G 2 G (1 + 2 ) Γ(1 − 2 )Γ2(1 + 2 ) XΓ = , Γ(1 + )Γ(1 + 3 )Γ2(1 − ) dk(D−2)dp(D−2) K6 = 2 (B.7) Λ(Γ) Ψ1 = ψ(1 + 3 )+ψ(1 − 2 ) ˆ − − − ψ(1 + 2 ) − ψ(1 − ). × (eGk)(k pˆ)(ˆp qˆ) 2 − 2 2 − 2 k (k p) p (p q) 1 X Appendix B = eˆ qˆ +(e q) 2 Γ , G 4 G (1 − )(1 + 2 ) Apart from I0 (46) and I1 (47), considered in the (D−2) (D−2) next appendix, integrals required for calculation of dk dp K7 = 2 (B.8) (3) Λ(Γ) ∆s can be transformed to the ones listed below. 2 Using (A.6), (48), and eˆGkpˆ × =êGqXˆ Γ, π(D−2) k2(k − p)2p2(p − q)2 Λ= − , (B.1) 2(q2)1 2 − − dk(D 2)dp(D 2) K = (B.9) we have 8 2 Λ(Γ) (D−2) (D−2) dk dp ˆ 2 K = eˆGkp s 1 2 (B.2) × ln Λ(Γ) 2 − 2 2 − 2 − 2 k (k p) p (p q) (k p) eˆ (kq)ˆp × G =ê q,ˆ s 1 2 − 2 2 − 2 G =ê qˆ ln +Ψ− Ψ + X , k (k q) p (p q) G −t 2 Γ (D−2) (D−2) dk dp dk(D−2)dp(D−2) K2 = (B.3) K = Λ(Γ )2 9 2 (B.10) Λ(Γ) (e k)(kˆ − qˆ)ˆp eˆ (kp)(ˆp − qˆ) 1 × G × G = − eˆ qXˆ , k2(k − q)2p2(p − q)2 k2(k − p)2p2(p − q)2 2 G Γ

1 − − = −eˆGqˆ +(eGq) , dk(D 2)dp(D 2) 2(1 + 2 ) 1+2 K = (B.11) 10 Λ(Γ )2 (D−2) (D−2) − dk dp × eˆG(kp)(ˆp qˆ) s K3 = 2 (B.4) 2 2 2 2 ln 2 Λ(Γ) k (k − p) p (p − q) (k − p)

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 QUARK REGGE TRAJECTORY IN TWO LOOPS 1613 1 s 3 Representing the integral as = − eˆGqˆ ln +Ψ− Ψ2 + XΓ. 2 −t 2 d I1 = I(ν) , (C.3) dν Appendix C ν=0 D−2 D−2 d k1d k2 I(ν)= 2 At arbitrary D =4 , the integrals I0 (46) and Λ(Γ) I1 (47) can be expressed only through infinite series. (q2)ν × , They belong to the class of integrals which were 2 − 2 2 − 2 − 2 ν−1 studied particularly in [17]. The first of them has k1(k1 q) k2(k2 q) [(k1 k2) ] already appeared in the calculation of the two-loop we have from Eqs. (1), (4), (9) of [17] correction to the gluon Regge trajectory [18], where − its limit at → 0 was found: 2 2 ν 1(ν − ) I(ν)= (C.4) D−2 D−2 2 d k d k 2 − 1 (Γ) I = 1 2 (C.1) 0 Λ(Γ )2 × G (1, 1 − + ν)G (1,ν) 2 2 q2 × × ν(ν − 1) lim S( − 1,b,2 − ν, ν − 1 − ) , 2 − 2 2 − 2 → k1(k1 q) k2(k2 q) b 0 2 × q 1 (2) 2 O 3 where ln 2 = +15ψ (1) + ( ). (k1 − k2) G2(α1,α2)=G1(α1)G1(α2) (C.5) (2) Here, Γ and Λ are given in (A.6) and (B.1); ψ × G1(2 + 2 − α1 − α2), Γ(1 + − α) means the second derivative of ψ. We have obtained G (α)= . the limit of the second integral: 1 Γ(α) D−2 D−2 d k1d k2 The function S(a, b, c, d)isdefined by Eqs. (17), (16), I1 = 2 (C.2) Λ(Γ) and (10) of [17]: (k − k )2 × 1 2 π cot(πc) − 1 2 − 2 2 − 2 S(a, b, c, d)= (C.6) k1(k1 q) k2(k2 q) H(a, b, c, d) c q2 4 × − (2) 2 O 3 − b + c − − ln 2 = +6ψ (1) + ( ). F (a + c, b, c, b + d), (k1 − k2) bc Below, some details of the calculation are given. where

Γ(1 + a)Γ(1 + b)Γ(1 + c)Γ(1 + d)Γ(1 + a + b + c + d) H(a, b, c, d)= , (C.7) Γ(1 + a + c)Γ(1 + a + d)Γ(1 + b + c)Γ(1 + b + d)

and F (a, b, c, d) is expressed through the generalized where hypergeometric function 3F2: S1(ν)=π cot[π(2 − ν)] (C.10)   Γ(3 − ν)Γ(1 + ν) ν(ν − 1) − − × −  a, b, 1  − − , F (a, b, c, d)=3F2 ;1 − 1. (C.8) Γ( )Γ(2 1) 2 ν 1+c, 1+d 3 − 1 − ν S2(ν)=ν(ν − 1) (C.11) The limit b → 0 in (C.4) can be easily taken. We can ( − 1)(ν − 2 ) also use for this purpose Eq. (12) of [17] and obtain × F (1 − , 2 − ν, ν − 2 ,ν − 2). with our values of parameters With this notation, we have − − − − − ν(ν 1) lim S( 1,b,2 ν, ν 1 ) (C.9) d b→0 I = [S (ν)(S (ν)+S (ν))] , (C.12) 1 dν 0 1 2 = S1(ν)+S2(ν), ν=0

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1614 BOGDAN, FADIN 1 1 where d 2 1−2 −1− J2 = dzdxz ln(1 − x) (1 − zx) . dx S0(ν)= 2 (C.13) (Γ) 0 0 2 Γ2( )Γ(2 − ν)Γ(1 − 2 + ν)Γ(1 − ν + ) (C.20) × . 2 − 1 Γ(3 − ν)Γ(1 + ν)Γ(1 + 2 − ν) The integral J1 can be easily found: We need to know S (0) and S(0), i =0–2.Fori = 1 Γ(1 − )Γ(2 − 2 ) i i J = + (C.21) 0, 1, they are easily obtained from (C.13), (C.10): 1 4(1 − )2 Γ(2 − 3 ) 3 (ψ(2 − 3 ) − ψ(2 − 2 )) S0(0) = − XΓ, (C.14) × . 2 1 1 − S0(0) = S0(0) +Ψ Ψ2 ; For the integral J2,replacingd/dx by (z/x)d/dz in 6 the representation (C.20) and integrating over z by (2 − 1) Γ(1 + 3 ) parts, we obtain in the limit → 0 S1(0) = 2π cot[2π ] ; 3 Γ(1 + )Γ(1 + 2 ) − 1 (1) J +1+ψ (1) (C.22) S (0) = S (0) 2 2 1 1 1 2π 1 ψ(2)(1) × ψ(1) − ψ(1 + 3 )+ + + , + 5 − ψ(1)(1) + . 3 sin(4π ) 2 2 where XΓ is defined in (A.6), and Ψ and Ψ2 are defined Using in (C.12) the results (C.21), (C.22), (C.14) and in (48). To find S2(0) and S2(0),weusetheintegral representation (C.17), (C.18), we get the final result (C.2). Γ(1 + d)Γ(1 + c) F (a, b, c, d)=−1+ Γ(d)Γ(−b)Γ(1 + c + b) (C.15) REFERENCES 1 1 1. L. V. Gribov, E. M. Levin, and M. G. Ryskin, Phys. − − − Rep. 100, 1 (1983). × dx(1 − x)d 1 dzz b 1(1 − z)c+b(1 − zx)a, 0 0 2. G. Altarelli, Phys. Rep. 81, 1 (1982). which follows from the standard representation for 3. M. T. Grisaru, H. J. Schnitzer, and H.-S. Tsao, Phys. hypergeometric functions. Performing integration Rev. Lett. 30, 811 (1973); Phys. Rev. D 8, 4498 over x by parts three times, we obtain (1973). S (ν)=(3 − 1 − ν) (C.16) 2 4. L. N. Lipatov, Yad. Fiz. 23, 642 (1976) [Sov. J. Nucl. Phys. 23, 338 (1976)]. × ν − + − 1 2 + ν 2 2 + ν 5. V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, Phys. 1 1 Lett. B 60B, 50 (1975); L. N. Lipatov, E. A. Kuraev, − d − − and V. S. Fadin, Zh. Eksp.´ Teor. Fiz. 71, 840 (1976) + dxdzz1 2+ν (1 − x)ν (1 − zx) 1 , dx [Sov. Phys. JETP 44, 443 (1976)]. 0 0 ´ so that 6. V. S. Fadin and V. E. Sherman, Pis’ma Zh. Eksp. − − Teor. Fiz. 23, 599 (1976) [JETP Lett. 23, 548 (1976)]; − Γ(1 )Γ(1 2 ) ´ S2(0) = (1 2 ) − , (C.17) Zh. Eksp. Teor. Fiz. 72, 1640 (1977) [Sov. Phys. Γ(1 3 ) JETP 45, 861 (1977)]. S2(0) − 7. M. Gell-Mann, M. L. Goldberger, F. E. Low, et al., S2(0) = − +(3 1) 1 3 Phys. Rev. 133, B145 (1964); B. M. McCoy and T. T. Wu, Phys. Rev. D 13, 369 (1976). × 1 − 2 + (J1 + J2) , (C.18) 1 − 2 4(1 − ) 8. S. Mandelstam, Phys. Rev. 137, B949 (1965). where 9.E.A.Kuraev,L.N.Lipatov,andV.S.Fadin,Zh. 1 1 ´ − d − − Eksp. Teor. Fiz. 72, 377 (1977) [Sov. Phys. JETP 45, J = dxdzz1 2 ln z (1 − zx) 1 , (C.19) 1 dx 199 (1977)]; Ya. Ya. Balitskii and L. N. Lipatov, Sov. 0 0 J. Nucl. Phys. 28, 822 (1978).

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PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1616–1617. Translated from Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1676–1677. Original Russian Text Copyright c 2005 by the Editorial Board. CURRENT EVENTS

70th Birthday of Yuriı˘ Fedorovich Smirnov

2002 and is a member of the Editorial Council of the journal Physics of Atomic Nuclei. The first important contribution of Smirnov to science dates back to his postgraduate years and concerns the theory of nucleon clustering in light nuclei on the basis of the multiparticle shell model. Together with two colleagues, he introduced coefficients describing the transformation of the wave function for a system of a few nucleons in the oscillator shell model from single-particle to cluster Jacobi coordinates and proposed a method for calculating these coefficients, which are presently referred to as Talmi–Moshinsky– Smirnov coefficients. By using this result, Smirnov showed (1959–1961) that by no means does the fact that reduced widths for cluster (for example, alpha- particle) emission from p-shell nuclei are close to Wigner’s single-particle limit favor the alpha-particle model of nuclei; moreover, he proved that such values are quite compatible with the multiparticle shell model. This theory made it possible to describe successfully and to predict a wide variety of experimental data. In 1966–1968, Smirnov and his colleagues proposed the method of quasielastic electron knockout by a fast electron [(e, 2e) reactions] at energies of about 10 to 50 keV from atoms, molecules, and solid bodies. This was an extremely successful example of extending experience gained in nuclear physics On May 26, 2005, Professor Yuriı˘ Fedorovich [(p, 2p) reactions at energies of about 500 MeV] to Smirmov, leading researcher at the Institute of Nu- adifferent region. An experimental implementation clear Physics at Moscow State University, member of this method in several countries made it possible of the Mexican Academy of Sciences, celebrated his to “feel” (via measuring respective momentum dis- 70th birthday. Yu. Smirnov is a brilliant representative tributions), for first time, various individual electron of Soviet and Russian science. The main focus of his orbitals and to test, at this level, which is the most scientific interests lies in the realms of theoretical nu- adequate, the Hartree–Fock method, aswellastoex- clear and atomic physics and mathematical (group- plore, by means of (γ,2e) and (e, 3e) experiments per- theory) methods in theoretical physics. He is the formed in the double-and triple-coincidence modes author of about 300 scientific publications, including (such experiments were also proposed by Smirnov ten well-known monographs. The citation index of his and his colleagues), electron–electron correlations studies exceeds 1500. He supervised the work of 16 in various multielectron systems. Many experimental candidates of science in physics and mathematics and and theoretical studies have been conducted in this taught many different lecture courses at the Physics field as well. Department of Moscow State University. Smirnov Finally, Smirnov and his colleagues obtained an was a recipient of the K.D. Sinelnikov Prize of the outstanding result in the realms of group-theory Ukrainian Academy of Sciences in 1982 and the methods in quantum physics. They developed a Lomonosov Prize of Moscow State University in universal method of projection operators (currently

1063-7788/05/6809-1616$26.00 c 2005 Pleiades Publishing, Inc. 70th BIRTHDAY OF YURII˘ FEDOROVICH SMIRNOV 1617 known as Asherova–Smirnov–Tolstoy projection scientific problems with colleagues and disciples. It operators) for all semisimple Lie groups. This method is a pleasure for us to wish Smirnov good health, makes it possible to construct efficiently wave func- prosperity to him and his relatives, and many years tions for quantum systems possessing preset symme- of creative activity. try properties. A complete and consistent mathematical theory of the SU(3) spin (including respective R. M. Asherova, V. V. Balashov, Clebsch–Gordan and Racah coefficients) was con- L. D. Blokhintsev, G. Ya. Korenman, structed for the first time by this method. Smirnov and V. V. Kukulin, D. E. Lansko ˘ı , his colleagues also applied the method in question V. I. Man’ko, V. G. Neudatchin, to quantum Lie algebras, which form a very rapidly I. T. Obukhovsky, I. M. Pavlichenkov, developing field of investigations. V. N. Pomerantsev, V. P. Popov, At seventy, Smirnov continues his studies vigor- V. N. Tolstoy, A. M. Shirokov, ously. He writes articles (in particular, in Physics of and Yu. M. Tchuvil’sky Atomic Nuclei) and reviews and discusses modern

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 Physics of Atomic Nuclei, Vol. 68, No. 9, 2005, pp. 1618–1620. Translatedfrom Yadernaya Fizika, Vol. 68, No. 9, 2005, pp. 1678–1680. Original Russian Text Copyright c 2005 by the Editorial Board. FUTURE PUBLICATIONS

Production of Ωscb Baryons in Photon–Photon Collisions S. P.Baranov and V. L. Slad

Methods for calculating the total and diﬀerential cross sections for the production of Ωscb baryons in photon–photon collisions and the results that these methods produce are described.

Charge Asymmetry in the Photoproduction of Charmed Mesons A. V. Berezhnoy and A. K. Likhoded Within the perturbative-recombination model, the charge asymmetries in the D∗+ − D∗−, D∗0 − D∗0,and + − − Ds Ds yields are estimated under the kinematical conditions of the COMPASS experiment. Corrections + that arise owing to the mass of a light quark in a charmed meson are taken into account. The yield of Ds − mesons is predicted to be large in relation to the yield of Ds mesons.

Gauge-Invariant Spatial Shift in Calculating a Color Axial Anomaly I. T. Dyatlov A nonsinglet axial anomaly is calculated for the case where the interaction of fermions with a non-Abelian gauge field is regularized by means of a coordinate shift according to Schwinger. The methods used makes it possible to obtain a covariant expression for the anomaly directly from the effective action for the field. The anomaly in question was calculated many times by numerous different methods, but the application of the shift method to this problem (from the study of W. Bardeen in 1969) led to the result only through a number of intermediate stages and additional subtractions of specially chosen polynomials of fields.

SolvingFaddeev –Merkuriev Equations within the J -Matrix Approach: Application to Coulomb Problems S. A. Zaitsev, V. A. Knyr, and Yu. V. Popov Within the J-matrixmethod for solving Faddeev –Merkuriev three-particle differential equations, a version is proposed that makes it possible to take into account the full spectrum of the two-particle Coulomb subsystem. Laguerre functions are used as a basis in which the total wave function for the problem being considered is expanded. In order to test the efficiency of the proposed method, the differential cross section for a single ionization of a helium atom is calculated for the case where the emerging He+ ion remains in an excited state. The result is in satisfactory agreement with experimental data both in magnitude and in the shape of the respective curve.

Neutrino Corona of a Protoneutron Star and Analysis of Its Convective Instability V. S. Imshennik and I. Yu. Litvinova A numerical solution to the problem of the structure of the neutrino corona of a protoneutron star that is formed upon an star-iron-core collapse, which is peculiar to all massive stars at the end of their thermonuclear evolution, is given. The structure of a neutrino corona, which is semitransparent to neutrino radiation from the spherical envelope between the neutrinosphere and the front of the accretion shock wave, is determined by a set of ordinary nonlinear equations of spherically symmetric neutrino hydrodynamics with allowance for a full set of beta processes in a Boltzmann gas of free nucleons and an ultrarelativistic Fermi–Dirac electron–positron gas that form neutrino-corona matter. The problem of consistently taking into account nonequilibrium neutrino- absorption and neutrino-emission processes and the problem of formulating boundary conditions for a neutrino corona were the main problems in constructing the numerical solution in question, which was obtained by means of a dedicated algorithm. The problem at hand features a number of parameters: the protoneutron- star mass, М0; the rate of accretion of the outer layers of the collapsing star being considered, М˙ 0;the

1063-7788/05/6809-1618$26.00 c 2005 Pleiades Publishing, Inc. FUTURE PUBLICATIONS 1619 effective temperature of the neutrinosphere and the effective neutrino chemical potential there, Тν eff and ψν eff, respectively; and, finally, the total neutrino emissivity of the neutrinosphere, Lνν¯. Two of these parameters, М0 and Lνν¯, are varied within broad intervals in accordance with the hydrodynamic theory of a collapse. On one hand, the numerical solutions constructed in the present study give an idea of the physical conditions in the immediate vicinity of a protoneutron star in the course of its continuing gravitational collapse; on the other hand, they make it possible to obtain exhaustive information about its convective instability, which is the most important property of a so-called soundless collapse—that is, a collapse not accompanied by an explosion on the supernova scale. The increment of development of a convective instability is obtained at a linear stage, this giving sufficient grounds to introduce the hypothesis that the instability in question plays a key role in the origin of observed gamma-ray bursts. More precisely, these bursts may result from the development of the instability at the subsequent nonlinear stage, which has yet to be studied theoretically—in particular, on the basis of non-one-dimensional numerical models of neutrino hydrodynamics.

Nucleus–Nucleus Scattering in the High-Energy Approximation and Optical Potential in the FoldingModel

V. K. Lukyanov, E. V. Zemlyanaya, and K. V. Lukyanov

A microscopic complexfolding-model potential that reproduces the scattering amplitude of Glauber – Sitenko theory in its optical limit is obtained. The real and imaginary parts of this potential are dependent on energy and are determined by known data on nuclear-density distributions and on the nucleon–nucleon scattering amplitude. For the real part, use is also made of a folding-model potential obtained with effective nucleon–nucleon forces and with allowance for nucleon exchange. Three forms of semimicroscopic optical potentials where the contributions of the inputs—that is, the real and the imaginary folding-model potential— are controlled by adjusting two parameters are constructed on this basis. The efficiency of these microscopic and semimicroscopic potentials is tested by means of a comparison with the experimental differential cross sections for the elastic scattering of heavy ions 16OonnucleiatanenergyofE ∼ 100 MeV per nucleon.

Noncompact Quantum Algebra uq(2, 1): Intermediate Discrete Series of Unitary Irreducible Representations

Yu. F. Smirnov, Yu. I. Kharitonov†,andR.M.Asherova

Unitary irreducible representations of the uq(2, 1) quantum algebra that belong to an intermediate discrete series are considered. A q-analog of the Mikelson–Zhelobenko algebra is developed. Use is made of the U basis that corresponds to the uq(2, 1) ⊃ uq(2) reduction. Explicit expressions are obtained for the matrix elements of generators in this basis. An operator is found that projects an arbitrary vector onto the extremal vector for a representation from this series.

Nontraditional Role of Interparticle Forces in Quantum Mechanics

A. I. Steshenko

A quantum-mechanical model for systems of interacting bodies that takes into account noncommutativity of the operators of coordinates and momenta of different particles and correlation equalities for the uncertainties of these quantities is considered. Here, the noncommutativity of these operators is due to the effect of interparticle forces and arises as a natural generalization of the usual commutation relation between the coordinate and momentum operators for a single particle. The efficiency of the model is proven by calculations for systems known in atomic and nuclear physics.

† Deceased.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005 1620 FUTURE PUBLICATIONS A Study of the Nuclear-Medium Influence on the Production of Neutral Strange Particles in Deep-Inelastic Neutrino Scattering N. M. Agababyan, V. V. Ammosov, M. Atayan, N. Grigoryan, H. Gulkanyan, A. A. Ivanilov, Zh. Karamyan, andV.A.Korotkov The influence of nuclear effects on the production of neutral strange particles (V 0) is investigated using data obtained with the SKAT propane–Freon bubble chamber irradiated in a neutrino beam (with Eν =3−30 GeV) 0 ± at the Serpukhov accelerator. The mean multiplicity of V particles in nuclear interactions, nV 0 A =0.092 ± 0.010, is found to exceed significantly that in “quasideuteron” interactions, nV 0 D =0.063 0.013.Theratio ± − ± ρV 0 = nV 0 A/ nV 0 D =1.46 0.23 is larger than that for π mesons, ρπ− =1.10 0.03.Itisshownthat the multiplicity gain of V 0 particles can be explained by intranuclear interactions of product pions.

Mass and Decays of Brout–Englert–Higgs Scalar with Extra Generations J.-M. Frere,` A. N. Rozanov, and M. I. Vysotsky The upper bound on the mass of the Brout–Englert–Higgs scalar boson arising from radiative corrections is not stable when the Standard Model is extended to include nondecoupling particles. In particular, additional generations of fermions allow for a heavier scalar. We investigate how the branching ratios for scalar-boson decays are aﬀected by the opening of new channels.

Antineutrino Background from Spent-Fuel Storage in Sensitive Searches for θ13 at Reactors V.I.Kopeikin,L.A.Mikaelyan,andV.V.Sinev Sensitive searches for antineutrino oscillations in the atmospheric-mass-parameter region, much discussed in recent years, are based on an accurate comparison of the inverse-beta-decay positron spectra measured in two (or more) detectors, far and near, positioned, for example, at about 1000 and 100 m from the reactor(s) used. We show that antineutrinos emitted from the stored irradiated fuel can diﬀerently distort the soft part of the positron spectra measured in the far and near detectors, thereby mimicking (or hiding) an oscillation signal.

Coulomb Deexcitation and Nonresonant Charge Exchange of Muonic Hydrogen in Mixtures of Hydrogen Isotopes A. V. Kravtsov and A. I. Mikhailov The process of inelastic collisions of excited muonic hydrogen is considered within the asymptotic theory of nonadiabatic transitions. The Coulomb deexcitation and charge-transfer rates are calculated in the energy range 0.001–100 eV with allowance for the electron-screening eﬀect for excited states in the range n =3−10 for various isotopic conﬁgurations.

Experimental Study of Direct Photon Emission in the Decay K− → π−π0γ with the ISTRA+ Detector V. A. Uvarov, S. A. Akimenko, V. N. Bolotov, G. I. Britvich, K. V. Datsko, V. A. Duk, A. P.Filin, A. V. Inyakin, V. A. Khmelnikov, A. S. Konstantinov, V. F. Konstantinov, I. Y. Korolkov, S. V. Laptev, V. M. Leontiev, V. P. Novikov, V. F. Obraztsov, V. A. Polyakov, A. Yu. Polyarush, V. I. Romanovsky, V. M. Ronjin, V. I. Shelikhov, N. E. Smirnov, O. G. Tchikilev, and O. P.Yushchenko The branching ratio in the charged-pion kinetic energy region between 55 and 90 MeV for direct photon emission in the decay K− → π−π0γ was measured by using in-flight decays detected with the ISTRA+ setup operating in a 25-GeV/c negative secondary beam of the U-70 PS. The value of Br(DE) =[0.37 ± 0.39(stat.) ± 0.10(syst.)] × 10−5, which was obtained from an analysis of 930 completely reconstructed events, is consistent with the average value of two stopped-kaon experiments, but it differs by 2.5 standard deviations from the average value of three in-flight-kaon experiments. The result is also compared with recent theoretical predictions.

PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 9 2005