Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1211–1218. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1251–1258. Original Russian Text Copyright c 2003 by Gangrsky, Zhemenik, Maslova, Mishinsky, Penionzhkevich, Sz¨oll¨os. NUCLEI Experiment
Independent Yields of Kr and Xe Isotopes in the Photofission of Heavy Nuclei
Yu.P.Gangrsky*, V.I.Zhemenik,N.Yu.Maslova, G. V. Mishinsky, Yu. E. Penionzhkevich, and O. Szoll¨ os¨ Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received July 15, 2002
Abstract—The yields of Kr (A =87–93)andXe(A = 138–143) primary fission fragments produced in 232Th, 238U,and244Pu photofission upon the scission of a target nucleus and neutron emission were measured in an experiment with bremsstrahlung from electrons accelerated to 25 MeV by a microtron, and the results of these measurements are presented. The experimental procedure used involved the transportation of fragments that escaped from the target by a gas flow through a capillary and the condensation of Kr and Xe inert gases in a cryostat at liquid-nitrogen temperature. The fragments of all other elements were retained with a filter at the capillary inlet. The isotopes of Kr and Xe were identified by the γ spectra of their daughter products. The mass-number distributions of the independent yields of Kr and Xe isotopes are obtained and compared with similar data on fission induced by thermal and fast neutrons; the shifts of the fragment charges with respect to the undistorted charge distribution are determined. Prospects for using photofission fragments in studying the structure of highly neutron-rich nuclei are discussed. c 2003 MAIK “Nauka/Interperiodica”.
INTRODUCTION At the same time, a wider range of experimental data and new theoretical approaches to describing The distributions of product fragments with re- them are required for obtaining deeper insight into spect to their mass numbers A and charge numbers Z the dynamics of the fission process. Measurement of are among the main properties of the nuclear-fission the isotopic and isobaric distributions of fragments process. The shape of these distributions is controlled originating from reactions induced by γ rays is one by the dynamics of the fission process from the saddle of the promising lines of investigation in this region to the scission point. This intricate process depends of nuclear fission since such reactions possess some on a number of factors, including the relief of the special features: energy surface, the configurations of nuclear shapes γ at the instant of scission, and the nuclear viscosity (1) The interaction of radiation with nuclei is of collective motion. Therefore, measurements of the purely electromagnetic, and its properties are well isotopic and isobaric distributions of fragments (that known; therefore, an adequate calculation of this in- is, those in A at given Z and those in Z at given A, teraction can be performed without resort to addi- respectively) provide important data on the dynamics tional model concepts. of the fission process. The distributions of fragments (2) In the energy range 10–16 MeV, the interac- produced after fissile-nucleus scission and the emis- tion of photons with nuclei is of resonance origin (this sion of neutrons (primary fragments) are the most is the region of a giant dipole resonance); the energy of informative of them. this resonance corresponds to the frequency of proton oscillations with respect to neutrons in a nucleus. However, measurement of such distributions in- volves some difficulties. Each of the techniques used (3) The absence of the binding energy and of the has its limitations and can be applied efficiently only Coulomb barrier allows one to obtain fissile nuclei of to a certain range of fission fragments. Therefore, iso- any (even extremely low) excitation energy immedi- topic and isobaric distributions of primary fragments ately after photon absorption. have received adequate study only for the fission of the (4) Over a wide range of γ-ray energies, the Th, U, and Pu isotopes that is induced by low-energy angular-momentum transfer to the irradiated nucleus neutrons and for the spontaneous fission of 252Cf. undergoes virtually no change—it is as low as 1 in Data on these distributions are compiled in [1, 2]. the dipole absorption of photons. These features peculiar to reactions induced by γ *e-mail: [email protected] rays allow one to obtain new information about the
1063-7788/03/6607-1211$24.00 c 2003 MAIK “Nauka/Interperiodica” 1212 GANGRSKY et al.
Ar 238U,and244Pu fission induced by γ rays of energy corresponding to a giant dipole resonance and to compare these yields with similar data obtained in neutron-induced fission. The measurements of the Chamber 238 – cumulative yields of these isotopes in Uphoto fis- e γ sion are reported in [9]; preliminary results on the independent yields of Xe fragments in 232Th and 238U fission are given in [10]. Capillary Pump EXPERIMENTAL PROCEDURE The extraction of Kr and Xe isotopes from the bulk of photofission fragments is based on the use of their special chemical properties, which differ significantly Cryostat from the properties of other elements and which re- Fig. 1. Layout of the experimental setup. veal themselves in a number of phenomena, includ- ing the adsorption of their atoms on a filter and the walls of the capillary through which the fragments are nuclear-fission process. For example, experiments transported from the irradiated target to radioactive- that studied the photofission of nuclei furnished a radiation detectors. The point is that Kr and Xe inert gases are efficiently adsorbed only at liquid-nitrogen wide set of data on the structure of the fission barrier ◦ and the shape of the potential surface [3]. The above temperature (−195.8 C), while all other fragments features of γ-ray interaction with nuclei are expected are adsorbed even at room temperature. It is precisely to manifest themselves in the process of fission- this property that is employed in our experimental fragment formation as well. This can be illustrated setup, which comprises a reaction chamber, a cryo- by the results presented in [4], according to which the stat, and Teflon capillaries for a gas flow. The layout γ-ray-energy dependence of the yield of fragments of the setup is displayed in Fig. 1, and a more detailed (115Cd, 117Cd) from symmetric fission has a rather description of it is given elsewhere [11]. unusual character, with a maximum at 6 MeV and a The irradiated targets were placed in the reaction kink at 5.3 MeV. chamber, which had the shape of a cylinder 30 mm in However, data on the isotopic and especially height, its inner diameter being 40 mm. The cylinder isobaric distributions of photofission fragments (first walls had a thickness of 1.5 mm. The inlet and out- of all, on their independent yields) are considerably let holes for a buffer gas were located symmetrically scarcer than data on reactions induced by neutrons opposite each other along a diameter of the reaction and charged particles. We can mention only the chamber. One or two targets placed at the end faces of results of the experiments performed in Ghent (Bel- the cylinder were used in the experiments. The prop- gium) for a limited set of nuclei [5–8]. erties of the targets are presented in Table 1 (these are The objective of this study is to measure the in- the chemical composition, admixtures, dimensions, dependent yields of fragments (these are the isotopes layer thickness, and the substrates). of Kr and Xe inert gases) produced directly in 232Th, An inert gas was supplied from a gas-container to the reaction chamber through a polyethylene capillary 4 mm in diameter. The gas pressure in the chamber Table 1. Properties of the targets used was adjusted by a container valve and measured with a manometer at the chamber inlet. Chemically pure Target 232Th 238U 244Pu He, Ar, and N2 at a pressure of 1 to 2 atm were used as buffer gases. The choice of inert gas and pressure Material Metal U3O8 PuO2 in the chamber was determined by the conditions of Content 100 99.85 96 the experiment (the efficiency of fragment collection, (%) the rate of their transportation, the levels of the γ- radiation background caused by the activation of the Admixtures (%) – 0.15 (235U) 4(240Pu + 242Pu) inert gas owing to microtron bremsstrahlung). Thickness 25 3 0.3 2 The fission fragments were transported, together (mg/cm ) with the buffer gas, from the reaction chamber to the Dimensions (mm) ∅15 ∅20 and 30 10 × 10 cryostat through a Teflon capillary of length 10 m and inner diameter 2 mm by means of evacuation Substrate – Al Ti with a pump. A fibrous filter placed at the capillary
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 INDEPENDENT YIELDS OF Kr AND Xe 1213
A = 91 93 140 142 Se 0.37 s ↓ Br 0.64 s Br 0.1 s I 0.86 s I 0.2 s ↓ ↓ ↓ ↓ Kr 8.6 s Kr 1.3 s Xe 14 s Xe 1.2 s ↓ ↓ ↓ ↓ Rb 58 s Rb 5.8 s Cs 64 s Cs 1.8 s ↓ ↓ ↓ ↓ Sr 9.5 h Sr 7.4 min Ba 13 d Ba 11 min ↓ ↓ ↓ ↓ Y 5.8 d Y 10.5 h La 40 h La 92 min ↓ ↓ ↓ ↓ Zr stable Zr 1.5 × 106 y Ce stable Ce stable
Fig. 2. Examples of the chains of β decays of the fission fragments that include Kr and Xe isotopes. The isotopes for which we measured γ spectraaregiveninboldfacetype. inlet efficiently adsorbed all fission fragments, with the cryostat. Therefore, only Kr and Xe fragments that exception of Kr and Xe. were produced directly at the instant of the scission A spirally twisted copper tube placed in a Dewar of a fissile nucleus reached the cryostat, and their vessel containing liquid nitrogen was used as a cryo- yields can be considered to be independent. Frag- stat. The total length of the tube was 3 m, its inner ments of other elements could appear in the cryostat diameter was 2 mm, and its walls had a thickness only after the β decay of Kr and Xe isotopes, and of 0.5 mm. This design of the cryostat enabled us to their yields must also correspond to the independent maintain the temperature at a level close to −200◦C yields of the inert gases. It was the yields of precisely and, hence, to ensure the condensation of Kr and Xe these nuclides that were measured in the experiment, (their temperatures of condensation are −157◦Cand because they had longer half-lives and more easily −112◦C, respectively). In this way, it was possible to observed γ lines. Short-lived fragments (forerunners separate Kr and Xe isotopes efficiently and quickly of Kr and Xe) that decayed in the reaction chamber from the bulk of fission fragments, which remained on had low independent yields and did not distort the the filter and the capillary walls. A comparison of the Kr and Xe yields. Figure 2 shows examples of β- spectra of γ radiation from fragments in the target, at decay chains involving the Kr and Xe isotopes; the the filter, and in the cryostat revealed that more than nuclides that were used to determine the yields in 90% of fragments not associated with the β decay question are given in boldface type. The properties chains involving Kr and Xe were retained by the filter of the radioactive decay of these nuclides (half-lives and that more than 50% of Kr or Xe fragments were T1/2,energiesEγ , intensities Iγ of measured γ lines) stopped in the cryostat. are listed in Table 2 [12]. The present approach to measuring independent yields enabled us to improve The time it takes to transport Kr and Xe fragments the accuracy of measurements and to observe frag- from the reaction chamber to the cryostat was de- −3 termined in dedicated tests that involved measuring ments whose yields are as low as 10 of the number the pressure drop in the chamber for the case where of fission events. the valve was closed while the pump was operating. This time, which comprised the time of fragment dif- EXPERIMENTAL RESULTS fusion to the outlet hole and the time of transportation through the capillary, ranged between 0.5 and 1.0 s, The experiments measuring the independent its specific value being dependent on the type of gas yields of Kr and Xe fragments were performed in and its pressure [11]. It was the time period that a bremsstrahlung-photon beam from the MT-25 determined the lower limit on the half-lives of the microtron of the Flerov Laboratory for Nuclear Re- fragments under study (about 0.3 s). If the half-lives actions at the Joint Institute for Nuclear Research of the fragments that were the forerunners of Kr and (JINR, Dubna). The description of this microtron and Xe were greater than the time of their diffusion to its basic properties were presented in [13]. The energy the outlet chamber hole (0.1–0.2 s), they were also of accelerated electrons was 25 MeV (which deter- adsorbed on the filter, making no contribution to the mined the endpoint energy of the bremsstrahlung- yield of the Kr and Xe fragments transported to the photon spectrum); the beam current was 15 µA. A
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1214 GANGRSKY et al.
Table 2. Properties of the radioactive decay of the nuclides explored in our experiments
Primary Daughter T T E ,keV I ,% fragment 1/2 isotope 1/2 γ γ 89Kr 3.2 min 89Rb 15 min 1032 58 1248 43 91Kr 9s 91Sr 9.5 h 750 23.6 1024 33.4 92Kr 2s 92Sr 2.7 h 1384 90.1 92Y 934 13.9 93Kr 1s 93Y 10.5 h 267 7.32 138Xe 14 min 138Cs 32 min 1436 76.3 2218 15.2 139Xe 40 s 139Ba 83 min 166 23.6 140Xe 14 s 140Ba 12.8 d 537 24.4 140La 40 h 1596 95.4 141Xe 1.8 s 141Ba 18 min 190 46 141La 4h 1355 1.64 142Xe 1.2 s 142La 93 min 641 47.4 2398 13.3 143Xe 0.8 s 143Ce 33 h 293 42.8
water-cooled tungsten disk 4 mm thick was used as radiation; Iγ is the relative intensity of a γ line in the a converter. Behind the converter, there was a 20- decay of the measured nuclide; α is the coefficient of mm-thick aluminum shield preventing electrons from the internal conversion of γ radiation; f(t) is the time entering the reaction chamber. The electron beam had factor that takes into account the decay of Xe and Kr the shape of an ellipse whose horizontal and vertical nuclei in the course of their transportation to the cryo- axes were 7 and 6 mm, respectively. stat and the accumulation and decay of the measured Typical periods of target irradiation were 30 min. nuclides; N is the number of decaying nuclei in the After a time interval of 5 min, the copper spiral was layer from which fission fragments escape; and η is removed from the Dewar vessel and brought to the the intensity of γ rays in the bremsstrahlung spec- γ -radiation spectrometer located in a room that was trum that induce the nuclear-fission process being protected from microtron radiation; the β decay of Kr considered. This method was used to measure, for the and Xe isotopes occurred within this period. first time, the independent yields of four Kr isotopes The areas of the γ lines associated with the nu- (A =89–93) and six Xe isotopes (A = 138–143)in clides of the daughter products of the β decay of the Kr 232 238 244 and Xe isotopes were obtained from the γ-radiation Th, U,and Pu photofission. spectra measured with this detector. These γ-line Since the Kr and Xe isotopes being studied were areas and the yields of the corresponding isotopes are obtained within a single exposure and since the pro- related by the equation cedure of subsequent measurements was identical for S (1 + α) Nf (t) Y (A)= , (1) all of them, the relative yields of these isotopes can tIγε1ε2ε3η be obtained from the measured γ-line areas by using where S is the γ-line area after background subtrac- the aforementioned parameters Iγ , α, ε3,andf(t) without recourse to the parameters ε and ε , which tion; t is the time of measurements; ε1, ε2,andε3 1 2 are the efficiencies of, respectively, the transportation are not known to a sufficient precision. The yields of Kr and Xe fragments along the capillary, their ad- obtained by this method and normalized with respect sorption in the cryostat, and the detection of their γ to the 91Kr and 139Xe isotopes are given in Table 3.
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 INDEPENDENT YIELDS OF Kr AND Xe 1215
Table 3. Independent yields of Kr and Xe fragments
232Th(γ,f) 238U(γ,f) 244Pu(γ,f) Fragment Yrel Yfr Yrel Yfr Yrel Yfr 89Kr 0.15(2) 0.10(2) 0.29(1) 0.18(2) 91Kr 1.00 0.62(5) 1.00 0.60(5) 92Kr 0.95(1) 0.59(5) 0.80(2) 0.46(4) 93Kr 0.25(1) 0.15(2) 0.25(2) 0.15(2)
137Xe 0.27(3) 0.16(2)∗ 138Xe 0.85(1) 0.53(4) 0.65(3) 0.38(4) 0.84(4) 0.47(4) 139Xe 1.00 0.63(5) 1.00 0.59(5) 1.00 0.56(5) 140Xe 0.85(7) 0.53(5) 0.94(7) 0.56(5) 1.03(8) 0.57(5) 141Xe 0.19(1) 0.12(1) 0.53(3) 0.31(3) 0.73(4) 0.41(4) 142Xe 0.09(1) 0.06(1) 0.26(2) 0.16(2) 0.46(4) 0.26(3) 143Xe 0.08(2) 0.05(1) 0.24(4) 0.14(2) Note: The value labeled with an asterisk was borrowed from [6].
The absolute values of the independent yields of DISCUSSION OF THE RESULTS 91Kr and 139Xe isotopes can be deduced by employ- ing the known data on their yields in the reactions The curves in Fig. 3 have a typical shape charac- induced by γ rays [6–8, 14] and by fission-spectrum terizedbyamaximumataspecificvalueofA and a and 14.7-MeV neutrons [2] (the isotopic distributions smooth decrease to the right and to the left of it. Such are similar for these reactions, and the yields are close inmagnitude[15]).Onthebasisofananalysisof these data, the independent 91Kr and 139Xe yields Y, (fission events)–1 divided by the sums of the yields of all A =91and 139 –1 10 Kr Xe isobars (fractional yields), respectively, were assigned values that are given in Table 3, and these values were used to determine the fractional independent yields for the rest of the measured Kr and Xe isotopes with 232Th 232Th allowance for the variation in the total yields of frag- 10–2 ments with a specific mass number and the emission of delayed neutrons from the fragments (these results are also presented in Table 3). The above yields di- vided by the total number of fission events are shown in Fig. 3 versus the mass number. 10–3 238U 238U ×10 × Owing to the fact that the bremsstrahlung spec- 10 trum is continuous, these yields are associated with a particular range of excitation energies. This range can 244Pu × be obtained from the experimentally measured exci- 10–4 100 tation function for the reaction 238U(γ,f) [16] (the 90 92 94 138 140 142 144 excitation functions for the remaining fissile nuclei A were assumed to be similar to that) and from the bremsstrahlung spectrum calculated for the specific Fig. 3. Mass-number distributions of the independent yields of Kr and Xe isotopes in the photofission of 232Th, conditions of the present experiment [17]. The mean 238U,and244Pu nuclei. The circles represent experi- excitation energy appears to be 12.5 MeV, and the mental data, while the dashed curves correspond to their half-width of the distribution is about 5 MeV. approximation by Gaussian curves.
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1216 GANGRSKY et al.
Table 4. Parameters of the isotopic distribution of Kr and Xe fragments
N − Z Kr Xe References Reaction N A¯ σ A¯ σ 232Th(γ,f) 0.366 91.3(2) 1.1(1) 138.9(2) 1.2(1) This study 238U(γ,f) 0.370 91.1(2) 1.3(1) 139.4(2) 1.5(1) This study 138.9(3) [5] 244Pu(γ,f) 0.373 139.7(2) 1.8(2) This study 235U(γ,f) 0.357 89.4(3) 1.3(1) 137.4(4) 1.4(1) [5] 233U(n, f) 0.352 89.3(1) 1.5(1) 137.8(1) 1.5(1) [1] 235U(n, f) 0.364 90.1(1) 1.5(1) 138.4(1) 1.6(1) [1] 238U(n, f) 0.379 91.5(1) 1.6(1) 139.5(1) 1.8(2) [1]
Table 5. Shifts of charges of Kr and Xe fragments with respect to the undistorted charge distribution
Z A¯ ν ∆Z Reaction Z0/A0 Kr Xe Kr Xe Kr Xe Kr Xe 232Th(γ,f) 0.388 36 54 91.3(2) 138.9(2) 1.0 0.8 −0.19(8) +0.20(8) 238U(γ,f) 0.387 36 54 91.1(2) 139.4(2) 1.0 0.8 −0.36(9) +0.25(8) 244Pu(γ,f) 0.385 54 139.7(2) 0.8 +0.13(6) curves can be described by a Gaussian distribution, (ii) The standard deviation of the distribution shows a sizable growth with increasing charge num- A − A¯ 2 Y (A)=K exp − , (2) ber of the fissile nucleus (by way of example, we 2σ2 indicate that, in going over from 232Th to 244Pu, it in- creases by a factor of 1.5). This implies a considerable where K is the normalization factor, A¯ is the mean distinction between the yields of the most neutron- mass number, and σ is the standard deviation. rich fragments. For example, the yield of 143Xe in The experimental values of the yields were ap- 244Pu photofission is an order of magnitude higher proximated by such distributions, which are shown in than that for 232Th, and this difference grows fast with Fig. 3, the fitted values of their parameters being given increasing number of neutrons in the fragment. ¯ 238 in Table 4 (the A value for Uis in agreement with (iii) Deviation of the experimental values of the that obtained in [7]). The corresponding parameters yields from the Gaussian distribution, as well as their 235 233 for Uand U fission induced by thermal neutrons odd–even differences, is small and does not go beyond and for 238U fission induced by 14.7-MeV neutrons the uncertainties of the measurement. [2] are also presented for the sake of comparison. A In 232Th photofission, the Kr and Xe fragments comparison of these parameters leads to the following are conjugate—that is, they are produced in a single conclusions: fission event. In comparing the sum of their mean (i) The mean mass number for Kr and Xe frag- mass numbers [A¯(Xe) = 138.9, A¯(Kr)=91.3]with ments exhibits a slow growth with increasing Z and the mass number of the fissile nucleus (A¯ = 232), A of the fissile nucleus and is close to the A¯ value one can assess the number of neutrons emitted from in 238U fission induced by 14.7-MeV neutrons, but these fragments. It appears to be ν =1.8(2). If it is it is noticeably greater than that in 235Uand 233U considered that the fractional yields of 91Kr and 139Xe fission induced by thermal neutrons. This suggests are, respectively, 0.62(5) and 0.63(5) (see Table 3), the growth of A¯ with increasing neutron excess in the this value is in satisfactory agreement with the known fissile nucleus [this excess can be characterized by the values of the number of fission neutrons from the frag- ratio (N − Z)/N , whose values are quoted in Table 4]. ments with a specific mass number and their depen-
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 INDEPENDENT YIELDS OF Kr AND Xe 1217 dence on the excitation energy [15, 18]. It follows from fission. It is obvious that photon-absorption-induced these data that the ratio of the numbers of neutrons dipole vibrations of the electric charge of nuclei are from a light and the complementary heavy fragment is damped completely by the instant of fragment for- 1.3 in the fragment-mass-number range under study mation. Probably, some other of the aforementioned and that it varies only slightly in the excitation-energy features of γ-radiation interaction with nuclei can range being considered. This corresponds to the value affect fragment formation at lower excitation energies of ν =1.0 for Kr isotopes and to the value of ν =0.8 (in the vicinity of the fission barrier or below it). for Xe isotopes. The results of this study give sufficient grounds By using the above values of ν, we can assess to conclude that photofission of nuclei is an effi- the shift of the fragment charge with respect to the cient means for obtaining the most neutron-rich nu- undistorted charge distribution, which corresponds to clides. A high yield of such nuclides is caused by the ratio Z0/A0 of the charge number of the fissile a low excitation energy of nascent fission fragments nucleus to its mass number. The expression for the (and, consequently, by a small number of neutrons charge shift in the fragment of charge number Z has emitted from them), by a high penetration capacity the form of bremsstrahlung radiation (this makes it possible Z to employ large amounts of fissile substance), and ∆Z = 0 A¯ + ν − Z. (3) A by a rather large standard deviation of the isotopic 0 distribution of fission fragments. Therefore, photofis- The ∆Z values obtained by this method for the mean sion provides a promising method both for explor- mass numbers of Kr and Xe fragments in 232Th, 238U, ing the nuclear structure of fission fragments and and 244Pu photofission are presented in Table 5. They for obtaining intense beams of accelerated neutron- appear to be close to the corresponding ∆Z values in rich radioactive nuclei by using these fragments. In- the neutron-induced fission of heavy nuclei [1, 19]. vestigation of reactions induced by such nuclei is of A wide variety of models have been used to de- great interest since they permit obtaining radically scribe the isotopic and isobaric distributions of fis- new information about the properties of nuclei, which sion fragments. In recent years, the diffusion model is inaccessible with other methods. A large num- [20, 21] has become widely accepted. Within this ber of projects devoted to the acceleration of fission model, the breakup of a fissile nucleus is a quasiequi- fragments have been developed in recent years or librium process with respect to the charge mode, so are being executed at present. These include DRIBs that the statistical approach can be used to describe project, which is being carried out at the Laboratory the charge distribution. In this case, the collective for Nuclear Reactions at JINR. In DRIBs, fission motion of nucleons, which manifests itself as dipole fragments are obtained by irradiating a thick uranium isovector vibrations, will be the main mechanism of target with bremsstrahlung from the microtron and formation of the isobaric charge distribution of frag- are accelerated at the 4-m isochronous cyclotron [23]. ments. However, the single-particle motion of nu- cleons can also make a nonvanishing contribution. ACKNOWLEDGMENTS The characteristic time of both kinds of motion (about 10−22 s) is much shorter than the time it takes for We are grateful to Yu.Ts. Oganessian, M.G. Itkis, the nucleus to descend from the saddle to the scission and S.N. Dmitriev for support of our studies; to point (about 3 × 10−20 s). The criterion of potential- Ya. Kliman and V.V. Pashkevich for stimulating energy minimum is used to determine the fragment discussions; and to A.G. Belov and G.V. Buklanov charge. The deformation dependence of the potential for assistance in performing the experiments. energy was calculated on the basis of the liquid-drop This work was supported by the Russian Founda- model with allowance for shell corrections. For the tion for Basic Research (project nos. 01-02-97038, fission of nuclei in the Th–Pu region, the calculations 01-02-16455) and INTAS (grant no. 2000-00463). of the isobaric charge distributions within this model showed that the shift of the charge with respect to the undistorted charge distribution is 0.2–0.4 [22]. These REFERENCES values of ∆Z are consistent with the experimental 1. A. C. Wahl, At. Data Nucl. Data Tables 39, 1 (1988). data obtained in our study of photofission, as well as 2. T. R. England and B. E. Rides, Preprint La-Ur-94- with those from other studies of the fission process 3106, LANL (Los Alamos, 1994). induced by various particles. 3. Yu. M. Tsipenyuk, Yu. B. Ostapenko, G.N.Smirenkin,andA.S.Soldatov,Usp.Fiz. The results of our experiments reveal that the iso- Nauk 144, 3 (1984) [Sov. Phys. Usp. 27, 649 (1984)]. topic distributions of fragments originating from the 4. B. V. Kurchatov, V. I. Novgorodotseva, V. A. 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PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1219–1228. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1259–1268. Original Russian Text Copyright c 2003 by Kadmensky, Rodionova. NUCLEI Theory
Angular Distributions, Relative Orbital Angular Momenta, and Spins of Fragments Originating from the Binary Fission of Polarized Nuclei S.G.KadmenskyandL.V.Rodionova Voronezh State University, Universitetskaya pl. 1, Voronezh, 394693 Russia Received June 17, 2002; in final form, November 5, 2002
Abstract—The angular distributions of fragments originating from the binary fission of odd and odd–odd nuclei capable of undergoing spontaneous fission that are polarized by a strong magnetic field at ultralow temperatures and from the low-energy photofission of even–even nuclei that is induced by dipole and quadrupole photons are investigated. It is shown that the deviations of these angular distributions from those that are obtained on the basis of the A. Bohr formula make it possible to estimate the maximum relative orbital angular momentum of fission fragments, this estimate providing important information about the relative orientation of the fragment spins. The angular distributions of fragments originating from subthreshold fission are analyzed for the case of the 238U nucleus. A comparison of the resulting angular distributions with their experimental counterparts leads to the conclusion that the maximum relative orbital angular momentum of binary-fission fragments exceeds 20, the fragment spins having predominantly a parallel orientation. The possibility is considered for performing an experiment aimed at measuring the angular distributions of fragments of the spontaneous fission of polarized nuclei in order to determine both the spins of such nuclei and the maximum values of the relative orbital angular momenta of fission fragments. c 2003 MAIK “Nauka/Interperiodica”.
1. INTRODUCTION the fragments in question emit prompt photons. Upon photon emission, the fragment spins assume values The problem of determining the relative orbital of J that correspond to the states of final fragments. angular momenta l and spins J (i =1,2)offission if i We denote by J and J the total spins of, respec- fragments is one of the key problems in the physics in iγ tively, prompt neutrons and prompt photons emitted of binary nuclear fission. The conservation of the total from the ith fragment. From the spin-conservation angular momentum J of a fissile nucleus in the fission law, it then follows that J = J + J + J .Experi- process leads to the vector relation i in iγ if ments devoted to measuring the multipole orders and J = l + F; F = J1 + J2. (1) multiplicities of prompt photons emitted by fission fragments revealed [2] that the mean values J¯iγ of It was shown in [1] that, because of the validity of the total spins J carried away by photons can be the adiabatic approximation in the asymptotic region iγ J¯ ≈ 6 8 of a fissile system, where primary fission fragments estimated (in units) at iγ – . are already formed, their deformation parameters hav- In the case of predominantly antiparallel orien- ing nonequilibrium values in this region, rotational tation of fragment spins, which follows from the bands characterized by rather wide sets of values of mechanism proposed by Mikhailov and Quentin [3], the fragment spins J1 and J2 are excited in these who considered the orientation pumping of fragment fragments. In the course of a further evolution, pri- spins, the mean value of the total spin of primary mary fission fragments go over to excited states that fragments, F¯, is close to zero. Therefore, the mean have equilibrium values of the deformation parame- value of the relative orbital angular momentum of ters, but which retain the initial values of the spins fragments is ¯l ≈ J, so that the values of ¯l are small for J1 and J2. At the next stage, these states of the typical values of the spin J of fissile nuclei undergoing fragments undergo transitions, via the emission of, spontaneous or low-energy induced fission (these first, prompt neutrons and, then, prompt photons, to values do not exceed a few units, being much less predominantly the ground states of final fragments than the mean spins J¯1 and J¯2). that are recorded in experiments. Upon the emission This conclusion is at odds with another impor- of prompt neutrons, the spins of fission fragments tant result that was obtained, within the quantum- 0 0 change values, from Ji to Ji , Ji being the initial val- mechanical theory of binary nuclear fission [1], from ues of the fragment spins for the stage within which an analysis of the angular distributions of fragments
1063-7788/03/6607-1219$24.00 c 2003 MAIK “Nauka/Interperiodica” 1220 KADMENSKY, RODIONOVA originating from the fission of polarized nuclei. These theory. Concurrently, the concept of angular distribu- distributions depend on the solid angle Ω=(θ,ϕ) tions of fission fragments in the internal coordinate determining, in the laboratory frame, the direction of frame of a fissile nucleus was also introduced in [1]. theradiusvectorR of the relative motion of fission The objective of the present study is to analyze, − fragments, R = R1 R2,whereR1 and R2 are the within the approaches proposed in [1], the angu- coordinates of the centers of mass of, respectively, the lar distributions of fragments originating from the first and the second fragment. The A. Bohr formula low-energy photofission of even–even nuclei into two (see [4]) was successfully used in describing these fragments that is induced by dipole and quadrupole angular distributions for the binary fission of polarized photons (the angular distributions for this case have nuclei [4] and in describing the coefficient of P -even already been investigated experimentally) and the an- and P -odd asymmetry in the angular distributions of gular distributions for the as-yet-unexplored case of fission fragments [5–8]. It was shown in [1] that, for the binary fission of J>1/2 odd and odd–odd nuclei this formula to be approximately valid, it is neces- capable of undergoing spontaneous fission that are sary that fission fragments have coherently coupled polarized by a strong magnetic field at ultralow tem- relative orbital angular momenta l lying in a rather peratures, these analyses being aimed at deducing wide range 0 ≤ l ≤ lm; that is, the maximum possible information about the values of the relative orbital value lm of the relative orbital angular momentum angular momenta and spins of fission fragments. must be quite large. Therefore, the mean value ¯l is also large; if ¯l is much greater than the spin J of a fissile nucleus, formula (1) then leads to the relation 2. ANGULAR DISTRIBUTIONS ¯≈ ¯ | | ¯ ¯ OF FRAGMENTS ORIGINATING l F = J1 + J2 > J1, J2, whence it follows that the FROM THE BINARY FISSION ¯ ¯ spins J1 and J2 are predominantly parallel to each OF POLARIZED NUCLEI other, but this contradicts the results presented in [3]. Since the nuclear-fission process induced by spe- 0 The upper limit lm on lm can be deduced from cific particle species (n, α, γ, ...) proceeds through the the law of conservation of the fissile-nucleus spin formation of a compound nucleus [4] having a spin 0 [see Eq. (1)]: lm = J1m + J2m + J. In order to assess J and a spin projection M onto the z axis in the the maximum values Jim of the fragment spins, we laboratory frame (this axis is chosen to be aligned will make make use of the relation Jim =(Jin)m + with the incident-beam axis), the differential cross section for the emission of fragments from the binary (Jiγ )m +(Jif )m, which follows from the law of con- servation of the fragment spin. If we assume that the fission of nuclei can be represented in the form values of the maximum possible spin carried away by dσf (Ω) σf (θ) ≡ (2) photons [(Jiγ)m]differ by a few units from the mean dΩ value J¯ obtained in [2] for this spin, consider that J J iγ Γ (JK) the maximum value of the total spin carried away = σ (JM) f T J (θ), Γ(J) MK by neutrons [(Jin)m] is about a few units, and use J M=−J K=0 values of about a few units for the fissile-nucleus spin σ (JM) J and for the maximum spins (J ) of final fission where is the cross section for the formation if m of a compound nucleus, Γ (JK) is the partial-wave fragments, we can derive an estimate of the upper f 0 0 ≈ decay width of this nucleus with respect to the chan- limit lm on lm: lm 25. It follows that the possible nel characterized by a specific value of the projection values of the relative orbital angular momenta l of K of the spin J of the axisymmetric compound nu- fission fragments are constrained by the condition ≤ cleus onto the z axis of the internal coordinate frame, l lm < 25. and Γ(J) is the total decay width of the compound With the aid of the methods proposed in [9, 10], nucleus. a quantum-mechanical theory of the nuclear-fission In the K channel, the angular distribution process was developed in [1] by introducing the con- J TMK (θ) of fission fragments that is normalized to cepts of fission phases and the amplitudes of partial- unity is usually calculated on the basis of the A. Bohr wave fission widths and by relying on the adiabatic formula [4]; that is, approximation in describing the asymptotic region of 2J +1 a fissile system, the idea of A. Bohr [4] that transition T J (θ)= (3) MK 8π fission states formed at the saddle point of the defor- mation potential of a fissile nucleus play a crucial role × J 2 J 2 DMK (ω) + DM,−K (ω) , in the fission process being taken into account in that β=θ
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J where DMK (ω) is a generalized spherical harmonic Such a situation can be simulated mathematically [1] that depends on the Euler angles α, β, γ ≡ ω char- by describing the distribution of θ by a formula of the acterizing the orientation of the axes of the internal type in (4) under the condition that the quantity lm in coordinate frame with respect to the axes of the labo- it takes finite, albeit rather large, values. In that case, ratory frame. In applying this formula, it is considered nonzero values of the angle θ lie within cones whose 2 that DJ (ω) is independent of the angles α and axes correspond to the values of θ =0and θ = π and MK ≈ γ. Formula (3) is based on the qualitative physical as- whose opening angle at a cone apex is ∆θ 1/lm. sumption [4] according to which fission fragments are The investigation performed in [1] by using the emitted only along or against the direction of the sym- adiabatic approximation in the asymptotic region of metry axis (z axis) of the fissile nucleus being con- a fissile system showed that the generalization of the sidered. This means that, in the internal coordinate A. Bohr formula (3) to this case can be represented as frame, the angular distribution of fission fragments ¯J 2J +1 ∓ TMK (θ)= (5) has a delta-function-like character of the form δ(ξ 16π2 1),whereξ =cosθ , θ being the angle between the 2 2 × dω DJ (ω) + DJ (ω) F 2 (θ ), vector R and the symmetry axis (z axis) of the fissile MK M,−K lm nucleus. Taking into account the azimuthal symmetry where the function F 2 (θ) is a smeared delta function of the angular distribution of fission fragments, going lm over from the argument θ to the argument ξ in the of the form in (4) with an amplitude; that is, spherical harmonic Yl0(θ ), redenoting the resulting Flm (θ ) (6) Y (ξ) function by the symbol l0 as before,√ and using l m the completeness of the set of functions 2πYl0(ξ ) l = b(lm) Yl0(ξ )Yl0(1) 1+ππ1π2(−1) . ξ −1 ≤ ξ ≤ 1 in the space of ( ), we can recast the l=0 aforementioned delta functions into the form In expression (6), the factor [1 + ππ π (−1)l],where lm 1 2 δ(ξ ∓ 1) = 2πY (ξ )Y (±1) (4) π, π1,andπ2 are the parities of, respectively, the l0 l0 parent nucleus, the first fission fragment, and the sec- l=0 ond fission fragment, takes into account the parity- lm conservation law, while the quantity b(l ) is deter- ± m = (2l +1)πYl0(ξ )Pl( 1), mined from the condition requiring that the angular l=0 distribution in (5) be normalized to unity. As a result, where P (ξ) is a Legendre polynomial [P (+1) = 1, we obtain l l − l l 1/2 Pl(−1) = (−1) ] and the quantity lm is considered in m 2 (2l +1) l the limit l →∞. Formula (4) reflects the quantum- b(lm)= 1+ππ1π2(−1) . m 4π mechanical uncertainty relation between the operator l=0 of the square of the orbital angular momentum of a (7) particle, l2,andthesquareoftheangleθ, which char- Formula (5) reduces to formula (3) for lm →∞. acterizes the direction of the radius vector R of this But at finite and rather small values of lm,thean- particle. From this relation, it follows that the angle gular distribution of fission fragments in (5) differs θ can be specified exactly only if the orbital angular significantly from the angular distribution of fission momentum l oftheparticleisabsolutelyuncertain fragments in (3). ≤ ≤∞ (0 l ). With an eye to applying the above formulas to Since only finite values of the spins J1 and J2 and a more general case, we will analyze the angular of the relative orbital angular momentum l of fission distributions of fission fragments in (3) and (5) at fragments may appear in the nuclear-fission process, arbitrary values of J, M,andK.Byemployingthe formula (3) is only approximate. At the same time, definition of a complex-conjugate generalized spheri- an analysis of the angular distributions of fragments cal harmonic [4], ∗ − originating from the fission of polarized nuclei [4] and DJ (ω)=(−1)M K DJ (ω) , (8) an analysis of the coefficients of P -even and P -odd MK −M,−K asymmetries in the angular distributions of fission and the multiplication theorem for D functions, fragments [5–8] on the basis of expression (3) re- DJ1 (ω) DJ2 (ω) (9) veal that this expression reflects actual properties of M1K1 M2K2 the nuclear-fission process. This means that angular J= J1+J2 JM1+M2 JK1+K2 J distributions of fission fragments do not vanish only = C C DM +M ,K +K (ω), J1J2M1M2 J1J2K1K2 1 2 1 2 at values of the angle θ that are close to 0 and π. J=|J1−J2|
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1222 KADMENSKY, RODIONOVA we can recast the A. Bohr formula (3) for the angular allowance for the orthogonality property of D func- distribution of fission fragments into the form tions [4], we obtain T J (θ)= B0 P (ξ), (10) | ˜J |2 | |2 MK JMKL L dω AM,±K (ω,Ω) = b(lm) (17) L lm lm where × jM−m j±K jM−m CJlM−mCJl±K0CJlM−m 0 L B = 1+(−1) l=0 l=0 jm JMKL (11) 2 2J +1 L0 L0 M+K j±K 8π 2l +1 2l +1 ∗ × C − C − (−1) × C ± Y (Ω) 8π JJM M JJK K Jl K0 2j +1 4π 4π l m l l and L takesonlyevenvaluesintherangefrom0 × Ylm(Ω)[1 + ππ1π2(−1) ][1 + ππ1π2(−1) ]. to 2J. We note that, by virtue of the properties of Clebsch–Gordan coefficients, the quantity B0 is By using the multiplication theorem for spherical JMKL harmonics [4], equal to 1/4π at L =0. m ∗ (−1) Let us now reduce the normalized (to unity) an- Ylm (Ω) Ylm (Ω) = gular distribution of fission fragments in (5) to the 4π × L0 L0 form of an expansion in Legendre polynomials that is (2l +1)(2l +1)Cll00Cllm−mPL (ξ), analogous to the expansion in (10). We introduce the L notation we can now recast expression (17) into the form J ≡ J AM,±K ω,Ω DM,±K (ω) Flm (θ ) (12) dω|A˜J (ω,Ω) |2 = |b(l )|2 (18) and represent the angular distribution (5) as M,±K m 2J +1 lm lm ¯J jM−m j±K jM−m j±K TMK (Ω) = (13) × C C C C 16π2 Jl M−m Jl ±K0 JlM−m Jl±K0 l=0 l=0 jmL × J 2 J 2 dω AM,K ω,Ω + AM,−K ω,Ω . L0 L0 (2l +1)(2l +1) × C C − P (ξ) ll 00 ll m m L 2(2j +1) Making use of the Wigner transformation [4] for go- l l ing over from the internal coordinate frame to the × [1 + ππ1π2(−1) ][1 + ππ1π2(−1) ]. laboratory frame, This formula can be significantly simplified by per- l∗ Ylk(Ω )= Dmk(ω)Ylm(Ω), (14) forming summation over the projection m of the rel- m ative orbital angular momentum l with the aid of the formulathatmakesitpossibletoevaluatesumsofthe we can recast the function F (θ) (6) into the form lm products of Clebsch–Gordan coefficients [11], Flm (θ )=b(lm) (15) cγ eε dδ Cabαβ Cdbδβ Cafαϕ l m αβδ × l∗ − l Dm0(ω)Ylm(Ω)Yl0(1)[1 + ππ1π2( 1) ]. =(−1)b+c+d+f (2c +1)(2d +1) l=0 m Taking into account relation (15) and the represen- abc × eε tation of a complex-conjugate D function in the Ccfγϕ , form (8), we can rewrite expression (12) as efd J ˜J AM,±K ω,Ω = AM,±K (ω,Ω) (16) abc where isaWigner6j coefficient, and the lm j= J+l jM−m j±K efd = b(lm) CJlM−mCJl±K0 l=0 m j=|J−l| property of symmetry of Clebsch–Gordan coefficients under the permutation of angular momenta, × j − m DM−m,±K(ω)( 1) Ylm(Ω)Yl0(1) JM J1−M1 2J +1 J2M2 l C =(−1) C − × [1 + ππ π (−1) ]. J1J2M1M2 J1JM1 M 1 2 2J2 +1 2J +1 Substituting expression (16) into (13) and perform- − J2+M2 J1M1 =( 1) CJ J−M M . ing integration with respect to the Euler angles with 2J1 +1 2 2
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Table 1. Values of the coefficients αJK, βJK,andγJK for the A. Bohr angular distribution (10) and for the angular distribution (20) in the problem of the low-energy photofission of nuclei
lm α10 α11 α20 α21 α22 β10 β11 β20 β21 β22 γ20 γ21 γ22
∞ 0 0.75 0 1.25 0 0.75 −0.375 0 −0.625 0.625 0.9375 −0.625 0.15625
30 0.024 0.74 0.060 1.200 0.030 0.714 −0.357 −0.022 −0.580 0.592 0.854 −0.569 0.142
20 0.04 0.73 0.087 1.163 0.044 0.697 −0.349 −0.033 −0.560 0.576 0.815 −0.543 0.136
19 0.04 0.73 0.091 1.159 0.046 0.695 −0.348 −0.034 −0.557 0.574 0.809 −0.539 0.135
10 0.07 0.72 0.162 1.088 0.082 0.652 −0.326 −0.061 −0.503 0.533 0.709 −0.473 0.118
9 0.07 0.72 0.177 1.072 0.089 0.643 −0.322 −0.066 −0.492 0.525 0.688 −0.459 0.115
5 0.12 0.69 0.283 0.965 0.139 0.577 −0.289 −0.105 −0.411 0.463 0.538 −0.358 0.090
4 0.14 0.68 0.332 0.915 0.169 0.545 −0.272 −0.122 −0.372 0.434 0.468 −0.312 0.078 Expression (18) then assumes the form 2L +1 j+J × (2l +1) (−1) 2J +1 | ˜J |2 | |2 dω AM,±K (ω,Ω) = b(lm) (19) l l × [1 + ππ1π2(−1) ][1 + ππ1π2(−1) ]. lm lm At L =0, the quantity B is equal to 1/4π. × j±K j±K L0 JM JMKL CJl±K0CJl±K0Cll00CJLM0 l=0 l =0 jL 3. ANGULAR DISTRIBUTIONS ljJ × (2l +1)(2l +1) OF FRAGMENTS ORIGINATING FROM THE LOW-ENERGY PHOTOFISSION JLl 2 OF NUCLEI 2L +1 j+J+L Let us consider the case of the low-energy photo- × (−1) PL(ξ) 2J +1 fission of an even–even target nucleus in the ground l l state, its total angular momentum being I =0.Inor- × [1 + ππ1π2(−1) ][1 + ππ1π2(−1) ]. der to describe the differential photofission cross sec- Since l and l have the same parity, it follows from tion as a function of the angle θ between the directions the properties of the Clebsch–Gordan coefficient of fission-fragment emission, which are defined with L0 respect to the axis of the incident-photon beam, one C that L can assume only the even values of L = ll 00 can employ formula (2) at M = ±1, since the photon 0, 2,...,2J. Upon the substitution of (19) into (13), ±1 ¯J helicity is , and set the value of the compound- the angular distribution TMK (Ω) of fission frag- nucleus spin to J =1 in the case of dipole pho- ments, which is normalized to unity, can be reduced toabsorption and to J =2in the case of quadrupole to a form that is analogous to (10). Specifically, we photoabsorption. have Since the angular distribution of fission fragments ¯J TMK(θ)= BJMKLPL(ξ), (20) that is specified by Eqs. (3) and (10) and the angular L distribution that is specified by Eqs. (5) and (20) are invariant under the substitution of −M for M, where formula (2) can be represented in the form lm lm 2 2 2J +1 2 jK σf (θ)=a0 + b0 sin θ + c0 sin (2θ) , (22) B = |b(l )| C (21) JMKL 16π2 m Jl K0 l=0 l=0 jL where the coefficients a , b ,andc are defined as 0 0 0 a = α P (JK); b = β P (JK); jK L0 JM ljJ 0 JK 0 JK × C C C (2l +1) JlK0 ll 00 JLM0 JK JK JLl (23)
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Table 2. Values of the coefficients G20, a,andb for the A. Bohr angular distribution (10) and the angular distribution (20) in the problem of the low-energy fission of nuclei at G22 =0and G22 =1
G22 =0 G22 =1 lm
a G20 a G20
∞ 0 0.73 ± 0.04 0 1.17 ± 0.07
30 0.096 ± 0.002 0.82 ± 0.04 0.098 ± 0.003 1.35 ± 0.08
20 0.15 ± 0.006 0.87 ± 0.05 0.15 ± 0.005 1.43 ± 0.08
10 0.29 ± 0.013 1.06 ± 0.06 0.29 ± 0.014 1.79 ± 0.11
Experiment [14] 0.09 ± 0.024 0.09 ± 0.024 c0 = γJKP (JK), rather small values of lm,thecoefficients αJK, βJK, JK and γJK change significantly upon going over from the A. Bohr formula (10) to formula (20). with +1 Our further analysis of low-energy photofission Γ (JK) 238 P (JK)= f σ (JM), (24) will be performed for the fission of U nuclei that is Γ(J) M=−1 induced by bremsstrahlung from 5.2-MeV electrons immediately below the fission threshold. We will in- while the coefficients αJK, βJK,andγJK are super- vestigate the quantity positions of the coefficients BJ1KL for L =0, 1, 2. σf (θ) It can be seen from Table 1 that, in the case of W (θ)= ◦ , (25) σf (90 )
W(θ) which characterizes the asymmetry in the angular distribution of fission fragments with respect to the 1.6 photon-beam axis. From Eqs. (25) and (22), it follows that the asymmetry W (θ) is given by expression (22), where the coefficients a0, b0,andc0 must be replaced a b 1.2 by the coefficients a = 0 , b = 0 ,andc = a0 + b0 a0 + b0 c 0 , which satisfy the relation a + b =1. 0.8 a0 + b0 The experimental values of the coefficients a and c are aexpt =0.04 ± 0.04 and cexpt =1.02 ± 0.07 [12], 0.4 aexpt =0.10 ± 0.035 and cexpt =0.91 ± 0.08 [13], aexpt =0.09 ± 0.024 and cexpt =0.907 ± 0.045 [14], expt +0.06 expt ± and a =0.08−0.03 and c =0.97 0.10 [15], 0306090 whence we can see that the values cexpt change θ , deg only slightly from measurements of one experimental group to measurements of another group, but that Fig. 1. Asymmetry W (θ) in the angular distribution expt 238 a of fragments originating from Uphotofission induced the values change more sharply, remaining, expt expt by bremsstrahlung photons whose endpoint energy is however, much smaller than b and c .Ascan 5.2 MeV,this asymmetry being defined with respect to the be seen from Table 1, the coefficients αJK are positive photon-beam axis. The curves depict theoretical results for all positive values of lm; therefore, the smallness of for various values of the maximum relative orbital angular the coefficient a mayonlybeduetothesmallnessof momentum of fission fragments: (solid curve) l = ∞ m P (11) P (21) (A. Bohr formula), (dashed curve) lm =20, and (solid the quantities and appearing as factors curve labeled with a diamond) lm =10.Opencircles in front of the coefficients α11 and α21, which are large represent experimental data reported in [14]. in magnitude. Retaining, in (23), only the K =0and
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 ANGULAR DISTRIBUTIONS, RELATIVE ORBITAL ANGULAR MOMENTA 1225 K =2channels, we can then find for the coefficients displayed along with the analogous asymmetry ob- a, b,andc that tained experimentally in [14], the A. Bohr formula (10) ∞ α + α G + α G corresponding to lm = is unable to reproduce a = 10 20 20 22 22 ; (26) the experimental asymmetry in the vicinity of the d angle θ =0. β10 + β20G20 + β22G22 b = ; If we compare the interval 20
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– J θ TJJ( ) 0.16 J = 3/2
J = 1
0.08
0 60 120 180 θ, deg
¯J Fig. 2. Angular distributions TJJ (θ) (J = 3/2, 1) of fragments originating from the fission of polarized nuclei for various values of the maximum relative orbital angular momentum of fission fragments: (solid curve) lm = ∞ (A. Bohr formula), (dashed curve) lm =20, (solid curve labeled with a diamond) lm =10, and (solid curve labeled with a circle) lm =5.
– J θ TJJ( ) 0.3
J = 3
0.2 J = 2
0.1
0 60 120 180 θ, deg
¯J Fig. 3. Angular distributions TJJ (θ) of fragments originating from the fission of polarized nuclei for various values of the maximum relative orbital angular momentum of fission fragments: (solid curve) lm = ∞ (A. Bohr formula), (dashed curve) lm =20, (solid curve labeled with a diamond) lm =10, and (solid curve labeled with a circle) lm =5.
T J (θ) (10) are both symmetric with respect to an of undergoing protonic decay [9], the latter having JJ ◦ angle of θ =90◦ and take maximum values at θ =0◦ maxima at an angle of θ =90 . The deviations of ◦ (180 ). With increasing spin J of the parent nucleus, the distributions in (20) from the A. Bohr distribu- T J ◦ T J ◦ T¯J ◦ T¯J ◦ tions in (10) are the most pronounced at angles of the ratios JJ (0 ) JJ (90 ) and JJ (0 ) JJ (90 ) ◦ ◦ ◦ grow rather fast, which makes it possible to use the θ =0 (180 ) and 90 , growing in magnitude with angular distributions in (10) and (20) to determine increasing spin J of the parent nucleus and with the value of the spin J. We note that the angular decreasing maximum value lm of the relative orbital distributions in (20) and (10) for polarized nuclei are angular momentum of fission fragments. At rather ¯J qualitatively different from the angular distributions of high values of lm, the angular distributions TJJ (θ) protons emitted by polarized odd–odd nuclei capable of fission fragments in (20) change only slightly upon
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Table 3. Deviation from the A. Bohr angular distribution in the problem of the spontaneous fission of polarized nuclei
T¯(0◦) − T (0◦) T¯(90◦) − T (90◦) T (90◦) ◦ , % ◦ , % ◦ , % lm T (0 ) T (90 ) T (0 )
J =1 J =3/2 J =2 J =3 J =1 J =3/2 J =2 J =3 J =1 J =3/2 J =2 J =3
20 −2.33 −3.49 −4.36 −5.72 2.34 6.96 14.83 48.62 0.5 0.25 0.125 0.03
19 −2.45 −3.66 −4.57 −6.00 2.45 7.31 15.56 51.00 0.5 0.25 0.125 0.03
10 −4.35 −6.52 −8.15 −10.69 4.32 13.04 27.70 90.92 0.5 0.25 0.125 0.03
9 −4.77 −7.14 −8.90 −11.70 4.78 14.27 30.36 99.70 0.5 0.25 0.125 0.03
5 −7.70 −11.54 −14.41 −18.85 7.69 23.07 49.10 161 0.5 0.25 0.125 0.03
4 −9.10 −13.63 −17.02 −22.24 9.10 27.27 58.02 191 0.5 0.25 0.125 0.03 going over from even values of l to their odd neighbors interaction via strongly nonspherical Coulomb and (that is, those that differ from their even counterparts nuclear potentials, ensures the emergence of such ◦ by unity), as is illustrated in Table 3 at angles of 0 high values of lm and, hence, of the total fission- and 90◦. Therefore, the angular distributions in (20), fragment spin F . as well as the A. Bohr angular distributions in (10), It would be of great interest to perform experi- are virtually independent of the parities of the parent ments devoted to studying angular distributions of nucleus and fission fragments. It can be seen from fission fragments originating from the fission of odd Table 3 that the relative deviations of the angular dis- nuclei capable of undergoing spontaneous fission that ◦ ◦ tributions in (20) from those in (10) at θ =0 (180 ) are polarized in a strong magnetic field at ultralow ◦ and 90 grow with decreasing lm, reaching 20% for temperatures. We note that similar experiments that ◦ ◦ ◦ θ =0 (180 ) and 191% for θ =90 at lm =4 and studied the angular distributions of alpha particles J =3. The emergence of such large deviations of emitted by nuclei undergoing alpha decay that were the angular distributions in (20) from the angular polarized in strong magnetic fields at ultralow tem- distributions in (10) at low values of lm indicates that peratures were successfully performed in [16, 17]. experiments aimed at studying the angular distribu- Because of small branching fractions of the fission tions of fragments originating from the spontaneous process with respect to alpha decay or ε capture for fission of polarized nuclei would make it possible to known odd and odd–odd spontaneously fissile nuclei, determine the lower limit on lm. it is necessary to overcome serious difficulties in order to record angular distributions for such nuclei with a high statistical accuracy. Nonetheless, such experi- 5. CONCLUSION ments are quite promising from the point of view of From the analysis of angular distributions of fis- their theoretical treatment, since, in contrast to what occurs in the case of induced fission, one transition sion fragments that was performed here for the case fission state characterized by only one value of K where parent nuclei are polarized, it was deduced that manifests itself in such experiments. the maximum relative orbital angular momentum of fission fragments produced under such conditions is It also seems important to continue investigating the angular distributions of photofission fragments rather high, lm > 20. It follows that the deviations of the angular distributions of fission fragments from for various nuclei and various endpoint energies of incident photons. that predicted by the A. Bohr formula (3) must be quite modest, so that this formula appears to be ap- In addition, it would be of interest to obtain an plicable in many cases. experimental corroboration of the conclusion drawn here that the spins of fragments originating from the At the same time, the fact that values obtained for binary fission of nuclei are predominantly parallel. the maximum relative orbital angular momentum lm of fission fragments exceed 20 indicates that it is nec- essary to consider a specific physical mechanism that, ACKNOWLEDGMENTS at the stage of the scission of a fissile nucleus into We are grateful to V.E. Bunakov, W.I. Furman, and fission fragments and the stage of their subsequent A.A. Barabanov for stimulating discussions.
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1228 KADMENSKY, RODIONOVA This work was supported by INTAS (grant no. 99- 9. S. G. Kadmensky and A. A. Sanzogni, Phys. Rev. C 00229) and by the foundation Universities of Russia 62, 054601 (2000). (project UR-01.01.011). 10. S. G. Kadmensky, in Proceedings of the IX Inter- national Seminar on Interaction of Neutrons with Nuclei, Dubna, 2001, p. 128. REFERENCES 11.D.A.Varshalovich,A.N.Moskalev,andV.K.Kher- 1. S. G. Kadmensky, Yad. Fiz. 65, 1424 (2002) [Phys. sonskiı˘ , Quantum Theory of Angular Momentum At. Nucl. 65, 1390 (2002)]. (Nauka, Leningrad, 1975). 2. Yu. N. Kopach et al., Phys. Rev. Lett. 82, 303 (1999). 12. A. S. Soldatov, G. N. Smirenkin, et al., Phys. Lett. 3. I. N. Mikhailov and P. Quentin, Phys. Lett. B 462,7 14, 217 (1965). (1999). 13. N.S.Rabotnov,G.N.Smirenkin,et al.,Yad.Fiz.11, 4. A. Bohr and B. Mottelson, Nuclear Structure (Ben- 508 (1970) [Sov. J. Nucl. Phys. 11, 285 (1970)]. jamin, New York; Amsterdam, 1969, 1975; Mir, 14. A. V. Ignatyuk, N. S. Rabotnov, et al.,Zh.Eksp.´ Moscow, 1971, 1977), Vols. 1, 2. Teor. Fiz. 61, 1284 (1971) [Sov. Phys. JETP 61, 684 ´ 5. A.K.Petukhov,G.V.Petrov,et al.,Pis’maZh.Eksp. (1971)]. 30 30 Teor. Fiz. , 324 (1979) [JETP Lett. , 439 (1979)]. 15. L. J. Lindgren, A. Alm, and A. Sandell, Nucl. Phys. A 6. O. P. Sushkov and V. V. Flambaum, Usp. Fiz. Nauk 298, 43 (1978). 136, 3 (1982) [Sov. Phys. Usp. 25, 1 (1982)]. 16. J. Wouters et al., Phys. Rev. Lett. 56, 1901 (1986). 7. V.E. Bunakov and L. B. Pikel’ner, Fiz. Elem.´ Chastits At. Yadra 29, 337 (1997). 17. P. Schuurmans et al., Phys. Rev. Lett. 77, 4420 8. A. Barabanov and W. Furman, Z. Phys. A 357, 441 (1995). (1997). Translated by A. Isaakyan
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1229–1238. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1269–1278. Original Russian Text Copyright c 2003 by Ishkhanov, Orlin. NUCLEI Theory
Semimicroscopic Description of the Gross Structure of a Giant Dipole Resonance in Light Nonmagic Nuclei B. S. Ishkhanovand V. N. Orlin Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received April 15, 2002; in final form, August 12, 2002
Abstract—A simple model is formulated that makes it possible to describe the configuration and defor- mation splittings of a giant dipole resonance in light nonmagic nuclei. The gross structure of the cross sections for photoabsorption on 12C, 24Mg, and 28Si nuclei is described on the basis of this model. c 2003 MAIK “Nauka/Interperiodica”.
1. INTRODUCTION was first considered in 1964 by Neudatchin and A giant dipole resonance in light nonmagic nuclei Shevchenko [8] and was then investigated in detail has a number of special features: (i) a very large by the authors of [9–11]. Nonetheless, the problem of theoretically describing the gross structure of a width (up to 15 MeV in 23Na [1]), (ii) a manifest giant resonance in light nonmagic nuclei has yet to intermediate structure that varies from one nucleus be solved conclusively. to another strongly, and (iii) a rather complicated gross structure. In order to explain these features, it In the present study, we make an attempt at de- is sufficient to recall that, in contrast to what we have scribing the gross structure of a giant dipole reso- in heavy nuclei, shell effects in light nuclei are much nance within the semimicroscopic model of vibrations stronger than collective effects; as a result, the os- that was proposed in [12]. For the sake of simplicity, cillator strengths of single-particle dipole transitions we take into account only the deformation and the are not concentrated predominantly in one or two (if configuration splitting of a giant dipole resonance, the nucleus being considered is deformed) collective which are closely interrelated in light nonmagic nuclei states. (see, for example, [13]). For N = Z nuclei, the isospin In the present study, we consider a number of splitting of a giant dipole resonance can addition- problems connected with describing the gross struc- ally be taken into account by using recipes proposed ture of a giant dipole resonance in light nonmagic in [2–5]. nuclei. This structure is formed, first, by the defor- mation splitting of a giant dipole resonance; second, 2. CONFIGURATION SPLITTING OF by its configuration splitting, which is due to the fact SINGLE-PARTICLE E1 TRANSITIONS IN that, in light nuclei, the energies of single-particle E1 LIGHT NUCLEI transitions between principal (oscillator) shells (from a filled inner shell to the valence shell and from the As was indicated above, principal (oscillator) valence shell to an unfilled outer shell) strongly differ shells in light nuclei are separated by unequal energy from one another; and, finally, by the isospin splitting gaps. This is illustrated in Fig. 1, which displays of a giant dipole resonance. data on the energies of the knockout of s-, p-and Each of the above types of giant-resonance split- d-wave protons from the first three oscillator shells ting has received quite an adequate study in the corresponding to the excitation of N =0,1,and2 literature. For example, a theoretical analysis of the oscillator quanta [9]. spectrum of various isospin E1 modes was performed There is a wide scatter of the experimental val- as far back as 1965–1968 in [2–5], where simple ues of the energies EN l in the figure. This is due theoretical estimates were obtained for the isospin not only to errors in the relevant measurements but splitting of a giant dipole resonance. Ten years earlier, also to the effect of spin–orbit splitting of single- Danos [6] and Okamoto [7] successfully explained particle levels (for l =0 ) and (in unfilled shells) the the deformation splitting of a giant dipole resonance effect of nucleon–nucleon correlations on these lev- within a hydrodynamic model. The configuration els. It can be seen from the figure that, for nuclei splitting of a giant resonance in light nonmagic nuclei heavier than oxygen, the spacing between the “0s”
1063-7788/03/6607-1229$24.00 c 2003 MAIK “Nauka/Interperiodica” 1230 ISHKHANOV, ORLIN
Energy, MeV
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s 60 s s s s 0s s s
p s p p s p 40 p s p 11 s p p s s s p 1p p p 10 ? s p 7 p d 7 s 8 p p 2d 11 s p p d 20 s 6 s ? ? d p d p d d ? 10 p d 11 p p ? d s s ? d s s d s s p p s s s s s s s s 2s 7 p d s s s 7 11 p s f s d s f p s d f 10 p s 6 p 0 He Li Be B C N O F NaMg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe CoNi Zn Atomic number
Fig. 1. Energies of the knockout of s-, p-, and d-wave protons from the first three oscillator shells [9]. and “1p” curves remains virtually unchanged as the independent of the number of nucleons occurring in charge number Z is increased. For nuclei lighter than this shell and in higher lying shells. oxygen—that is, in the region where the 1p shell is not Analyzing experimental data on the energies E1p, closed—these curves somewhat approach each other, E2d,andE2s and considering that the 2s-wave pro- which may be attributed to the effect of residual forces tons shift the centroid of the 2d2s shell only slightly, (for example, pairing), which shift the energy E1p we can arrive at the conclusion, in a similar way, that upward. We are interested here in the mean spacing the mean energy of single-particle transitions from between the 0s and 1p single-particle oscillator shells. the filled 1p shell to the 2d2s shell is also independent This quantity is not expected to depend on proton of the number of protons occurring in states that lie higher than the 1p shell. pairing. In estimating the difference E0s − E1p over 4 → 16 In all probability, this property is common to all the He O segment, where the filling of the closed shells. It implies a significant spatial separa- 1p shell occurs, it is therefore necessary to take into tion of nucleons occurring in different principal shells account only data on odd-Z nuclei (see Fig. 1). After of a nucleus. Indeed, it is clear that, if nucleons be- − this correction, the quantity E0s E1p appears to be longing to outer shells are situated by and large far- approximately constant for any values of Z greater ther from the center of the nucleus being consid- than that for helium. This means that, for a first ered than nucleons belonging to inner shells, then approximation, the mean energy of single-particle the energy of a nucleon transition from a filled inner transitions from the filled 0s shell to the 1p shell is shell to the next higher lying shell will be determined
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 SEMIMICROSCOPIC DESCRIPTION OF THE GROSS STRUCTURE 1231 predominantly by the mean field generated by “inter- of the nucleus, the greater the configuration splitting. nal” (core) nucleons. A spatial separation of shells In light nonmagic nuclei, there can occur a situation is directly corroborated by fluctuations of the radial where a major part of nucleons reside in the valence dependence of the nuclear density ρ(r) (see, for ex- shell, while, in heavy nuclei, the greater part of the ample, [14]). nuclear mass is always concentrated in filled inner Let us now estimate the mean spacings between shells. It therefore comes as no surprise that the con- the 0s, 1p,and2d2s shells. This can be done by figuration splitting of a giant dipole resonance man- two methods: either with the aid of experimental data ifests itself primarily in light nonmagic nuclei. (This presented in Fig. 1 or with the aid of the formula [15] statement can be illustrated by considering the exam- ple of the 12Cand 159Tb nuclei, for which the above (0) ≈ −1/3 ωcore 41Acore MeV, (1) (0) (0) ratio of the mean transition energies, ω2 / ω1 , (0) takes values of about 1.44 and 1.06, respectively.) where ωcore is the frequency parameter of the os- cillator potential simulating the mean field that is generated by Acore core nucleons (Acore =4for the 3. DESCRIPTION OF THE MODEL USED → 0s 1p single-particle transition, and Acore =16for In this section, we will first present fundamentals → the 1p 2d2s transition). of the semimicroscopic model of dipole vibrations that Using the approximating curves in Fig. 1, we find, was proposed in [12] and then show how it can be in the first case, that the 1p shell is separated by generalized to take into account the configuration and approximately 22.5 MeV from the 0s shell and by the deformation splitting of a giant dipole resonance. approximately 18.5 MeV from the 2d2s shell. In the second case, we instead obtain 25.8 and 16.3 MeV, respectively. It can easily be seen that the discrep- 3.1. Semimicroscopic Model of Dipole Vibrations ancy between the two types of estimates does not Basic properties of a nuclear dipole resonance can exceed the uncertainty in the approximating curves be explained in terms of the interplay of single-particle “0s” and “1p.” Indeed, perfect agreement between the nucleon excitations and the isovector dipole field that estimates can be attained if we draw the first curve is generated by these excitations and which, for vibra- somewhat higher and the second curve somewhat tions along the x axis, is given by lower (there are sufficient grounds for doing this). A | | + We now have all the required data at our dis- F = (2tzx)i = α 2tzx β aα aβ + h.c., (3) posal for estimating the configuration splitting of i=1 α>β single-particle dipole transitions. In a nonmagic + nucleus, two types of single-particle E1 transitions where aλ and aλ are the operators of, respectively, are possible: type-1 transitions proceed from the creation and absorption of a nucleon in a single- valence shell of the nucleus to an empty outer particle state |λ and tz = ±1/2 is the isospin variable shell, while type-2 transitions proceed from the last for a nucleon. filled inner shell to the valence shell. The nuclear The application of the dipole operator F to the shell mean field in which valence nucleons move can be ground state of a nucleus (physical vacuum |0 )gen- approximately described in terms of the oscillator erates a superposition of particle–hole (1p1h) states; potential corresponding to the oscillator-quantum that is, (0) ≈ −1/3 energy of ω 41A MeV, where A is the | | | + | mass number of the nucleus. Therefore, the mean F 0 = α 2tzx β aα aβ 0 . (4) (0) ≈ (0) α>β energy of type-1 transitions is ω1 ω .On the other hand, the mean energy of type-2 transi- In the case of heavy and medium-mass spheri- tions can be calculated by formula (1), which yields cal nuclei, the state F |0 is approximately an eigen- (0) (0) −1/3 state of the single-particle Hamiltonian H ,thecor- ω ≈ ωcore ≈ 41Acore MeV. 0 2 responding excitation energy being Thus, the ratio of the mean energies for two types − of single-particle E1 transitions in nonmagic nuclei ω(0) ≈ 41A 1/3 MeV, (5) can be approximated by the expression since the energy scatter of single-particle E1tran- ω(0) A 1/3 sitions is much less than ω(0). The excitation F |0 2 ≈ . (2) (0) is formed by a great number of single-particle exci- ω Acore 1 tations. In view of this, the operator F can be repre- From this formula, it can be seen that the greater sented in the form ∗ the contribution of valence nucleons to the total mass F = f (0)c(0)+ + f (0) c(0), (6)
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1232 ISHKHANOV, ORLIN where c(0)+ and c(0) are the operators of creation This estimate of the energy of dipole vibrations is in and annihilation of a vibrational quantum of energy good agreement with the experimental values of the ω(0), these operators approximately satisfying com- giant-dipole-resonance energy for heavy nuclei. mutation rules for bosons, and f (0) is the probability In the representation of normal vibrations (eigen- amplitude for the production of such a quantum by the states), the dipole-moment operator F has the form ∗ dipole field F . F = fc+ + f c, (15) The amplitude f (0) can be estimated with the aid where f is the probability amplitude for the excitation of the classical sum rule of normal vibrations. This amplitude is related to the 2 (0) ω(0)|f (0)|2 = A, (7) amplitude f by the equation 2M ω|f|2 = ω(0)|f (0)|2, (16) or it can be calculated directly by using the relation | (0)|2 | + | | | | |2 from which it follows that the canonical transforma- f = 0 F F 0 = α 2tz x β , (8) tion (11) has no effect on the oscillator strength of αβ dipole transitions. where M is the nucleon mass and where summation is performed over filled (β) and free (α) single-particle 3.2. Inclusion of the Configuration Splitting levels. of a Giant Dipole Resonance Single-particle dipole vibrations cannot be con- sidered as normal vibrations (eigenstates) of a nu- Ifthereisaconfiguration splitting of E1transi- cleon system about the equilibrium position, since tions, the dipole operator F must be broken down into they effectively interact with one another via the exci- two terms, tation of the isovector single-particle potential. This F = F1 + F2, (17) circumstance can be taken into account by supple- where the first term menting the single-particle Hamiltonian H0 with that κ 2 (1) | | + for dipole–dipole forces ,F /2, whereupon the vi- F1 = α 2tzx β aα aβ + h.c. (18) brational Hamiltonian assumes the form α<β 1 H = ω(0)c(0)+c(0) + κF 2. (9) exhausts single-particle transitions between the va- 2 lence and outer shells, while the second term By using the semiempirical Weizsacker¨ mass for- (2) | | + mula, the coupling constant for dipole–dipole inter- F2 = α 2tzx β aα aβ + h.c. (19) action can be expressed [12] in terms of the symmetry α<β V potential as takes into account transitions from a filled inner shell V to the valence shell and the transitions inverse to κ = , (10) 4Ax2 these. Following the same line of reasoning as in Sub- where x2 ≈R2/5 ≈ (1.2A1/3)2/5 fm2 is the mean- section 3.1, we introduce the operators of creation x (0)+ (0)+ (0) (0) square value of the coordinate of intranuclear nu- (c , c ) and annihilation (c , c )ofsingle- cleons. 1 2 1 2 particle quanta whose energies are (in MeV) With the aid of the linear canonical transformation (0) ≈ −1/3 (0) ≈ −1/3 c+ = Xc(0)+ − Yc(0) (X2 − Y 2 =1), (11) ω1 41A and ω2 41Acore (20) the Hamiltonian in (9) can be reduced to the diagonal (see Section 2); further, we employ the approximation form (0) (0)+ (0)∗ (0) Fi = fi ci + fi ci (i =1, 2), (21) H = ωc+c + const, (12) where the probability amplitudes for the excitation of where single-particle vibrations are given by ω = ( ω(0))2 +2κ ω(0)|f (0)|2 (13) | (0)|2 (i) | | | |2 fi = α 2tz x β (i =1, 2). (22) is the energy of a normal mode of dipole vibrations. α<β From (5), (7), (10), and (13), we find at V ≈ The next step consists in representing the vibra- 130 MeV that tional nuclear Hamiltonian in the form −1/3 ω ≈ 80A MeV. (14) H = H1 + H2 + H12, (23)
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 SEMIMICROSCOPIC DESCRIPTION OF THE GROSS STRUCTURE 1233 where of the calculations, which reproduces the calcula- (0) (0)+ (0) 1 tions described in Subsection 3.1, the Hamiltonian in H = ω c c + κ F 2 (24) Eq. (23) reduces to the form 1 1 1 1 2 1 1 2 is the Hamiltonian that describes normal vibrations H = ω c+c + κ F F + const, (29) for type-1 dipole transitions, i i i 12 1 2 i=1 (0) (0)+ (0) 1κ 2 + H2 = ω2 c2 c2 + 2F2 (25) where ci and ci are the operators of, respectively, cre- 2 ation and annihilation of a quantum of eigenvibrations generated by the Hamiltonian H ; is the analogous Hamiltonian for type-2 dipole tran- i sitions, and (0) 2 κ (0)| (0)|2 ωi = ( ωi ) +2 i ωi fi (30) H12 = κ12F1F2 (26) is the energy of such a quantum; is the operator that takes into account the interaction + ∗ of dipole transitions of the two types. Fi = fici + fi ci (31) In order to take into account the possible distinc- is operator Fi in the representation of the eigenstates tions for three types of 1p1h 1p1h interactions— of the Hamiltonian Hi;andfi is the probability ampli- +| 1 1, 2 2,and1 2—we have introduced three tude for the excitation of vibrations of the ci 0 type, different coupling constants for dipole–dipole forces this amplitude being related to the amplitude f (0) by κ κ κ i ( 1, 2,and 12). This can be justified by the follow- the equation ing considerations. According to the estimate in (10), | |2 (0)| (0)|2 the coupling constant for dipole–dipole forces is pro- ωi fi = ωi fi . (32) −5/3 portional to A . This mass dependence is obvi- Let us estimate the ratio of the energies ω and κ 2 ously appropriate for the constant 1 and, to some ω . We assume that the fraction q of the valence κ κ 1 extent, for the constant 12, but the constant 2 is shell is not occupied and that its fraction (1 − q) is −5/3 most likely to be proportional to Acore ,sinceouter accordingly filled. Using the dipole sum rule, we then (valence) nucleons have but a slight effect on the obtain oscillator excitations of nucleons belonging to a filled 2 ω(0)|f (0)|2 ≈ qA , (33) inner shell (see the analysis of data on quasielastic 2 2 2M core nucleon knockout in Section 2). Moreover, the in- 2 (0)| (0)|2 ≈ − teraction within the group of type-1 excitations is ω1 f1 (A qAcore). expected to be additionally suppressed in relation to 2M the interaction within the group of type-2 excita- By using these estimates and relations (20), (27), tions, since the former occurs at the nuclear periphery, and (30), we obtain where the mean density of nuclear matter is lower. 1/3 K Further, it can also be assumed that the intergroup ω2 ≈ A 1+0.0247q 2 − K . interaction is weaker than the intragroup interaction ω1 Acore 1+0.0247(1 qAcore/A) 1 because the overlap integral is small for type-1 and (34) type-2 particle–hole states. The configuration splitting of a giant dipole res- Taking the aforesaid into account, we represent onance does not play a significant role either in the the coupling constants for dipole–dipole interaction case where the valence shell is nearly empty (q → 1) in the form or in the case where it is nearly filled (q → 0). But in κ K −5/3 κ K −5/3 other cases, it can be seen from relations (28) that the 1 = 1A , 2 = 2Acore , (27) second factor on the right-hand side of (34) is close to −5/3 κ12 = K12A , unity. Therefore, we assume that (0) K K K ω ω A 1/3 where the factors 1, 2,and 12 satisfy the relations 2 ≈ 2 ≈ . (35) (0) ω1 ω Acore K12 K1 K2 ≈ 100 MeV. (28) 1 In the representation specified by Eqs. (23)–(26), We diagonalize the Hamiltonian in (23) in two the coupling constants κ1, κ2,andκ12 play the role steps: first, we find the eigenstates of the Hamilto- of free parameters of the model. Upon recasting the nians H1 and H2, whereupon we take into account Hamiltonian H into the form (29), it becomes possi- the interaction of these states. After the first step ble to choose parameters in an alternative way: ω1,
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1234 ISHKHANOV, ORLIN
5/3 ω2,andK12 = κ12A . This parametrization has a 3.3. Generalization to the Case of Deformed Nuclei number of advantages: on one hand, the number of In a spherical nucleus, normal dipole vibrations in independent variable parameters decreases by virtue three mutually orthogonal directions are degenerate of the condition in (35); on the other hand, the region in energy. In view of this, we have only considered where phenomenological dipole–dipole forces are ap- above vibrations along the x axis. In a deformed plied, which are only used to describe a relatively weak spheroidal nucleus, this degeneracy is partly removed. interaction of different configuration modes, shrinks. In that case, it is necessary to distinguish between The Hamiltonian in (29) can be diagonalized by vibrations along the symmetry axis of the nucleus means of the linear canonical transformation (z axis) and vibrations along an orthogonal direction 2 (say, along the x or the y axis). Each of these types + + − cˆi = (Xijcj Yijcj), (36) of vibrations is described in precisely the same way j=1 as vibrations along the x axis of a spherical nucleus (see Subsection 3.2). It is only necessary to employ, where the expansion coefficients X and Y satisfy ij ij in Eqs. (18), (19), and (22), the single-particle states the orthogonality conditions |α and |β of the deformed single-particle potential 2 and to take into account the effect of deformation ∗ − ∗ (XijXkj YijYkj)=δik, (37) on the oscillator frequencies (20) of single-particle j=1 (0) excitations—that is, to multiply the quantities ω1 2 (0) and ω by the factor 1 − 4δ/3 (by δ , we mean (X Y − Y X )=0. 2 ij kj ij kj here the deformation parameter that was introduced j=1 in [15]) for longitudinal vibrations (vibrations along The eigenenergies of the Hamiltonian H are given the z axis) or by the factor 1+2δ/3 for transverse by vibrations (vibrations along the x and y axes). 2 2 2 2 2 2 − 2 2 2 Relation (35) can now be recast into the form ω1 + ω2 ∓ ( ω1 ω2) ωˆi = (38) || ⊥ 1/3 2 4 ω2 ≈ ω2 ≈ A || ⊥ , (42) ω Acore 1/2 1/2 ω1 1 κ2 | |2 | |2 +4 12 ω1 f1 ω2 f2 , || || where ω1 and ω2 are the energies of, respectively, type-1 and type-2 intermediate normal vibrations for ≤ where i =1, 2 and ωˆ1 ωˆ2. the longitudinal mode of a giant dipole resonance, The dipole operator F reduces to the form ⊥ ⊥ while ω1 and ω2 are the analogous quantities for 2 its transverse mode [compare with the energies ω1 ˆ + ˆ∗ F = (ficˆi + fi cˆi), (39) and ω2 of the Hamiltonian in (29)]. i=1 The equalities in (42) make it possible to reduce ˆ the number of parameters varied in describing the where fi are the probability amplitudes for the excita- structure of a giant resonance in deformed nuclei to +| tion of the normal vibrations cˆi 0 . These amplitudes three; for these, one can use, for example, the energies can be found from the equation || ⊥ K ω1 and ω1 and the constant 12. | ˆ|2 | |2 2 2 − 2 2 ωˆi fi =[ ω1 f1 ( ωˆi ω2) (40) | |2 2 2 − 2 2 + ω2 f2 ( ωˆi ω1) 4. APPLICATION TO LIGHT NONMAGIC κ | |2 | |2 2 2 − 2 2 − 2 2 NUCLEI +4 ω1 f1 ω2 f2 ]/(2 ωˆi ω1 ω2) (i =1, 2). The model considered above was used to describe the gross structure of a giant dipole resonance in Finally, the relations the 12C, 24Mg, and 28Si nuclei. These are three self- 2ωˆ2 − 2ω2 conjugate nonmagic nuclei, for which the effects of |X |2 −|Y |2 = i 2 , (41) both the configuration and the deformation splitting i1 i1 2 2 − 2 2 − 2 2 2 ωˆi ω1 ω2 of a giant dipole resonance are of importance. 2ωˆ2 − 2ω2 We employed the following computational scheme. |X |2 −|Y |2 = i 1 i2 i2 2 2ωˆ2 − 2ω2 − 2ω2 First, the energies and oscillator strengths of normal i 1 2 excitations for longitudinal and transverse dipole vi- specify the contribution of type-1 and type-2 config- brations were calculated on the basis of the formalism +| urations to the dipole states cˆi 0 (i =1, 2). developed in Section 3. After that, the resulting dipole
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 SEMIMICROSCOPIC DESCRIPTION OF THE GROSS STRUCTURE 1235 resonances, whose number is equal to four with to reproduce, in the calculations, the positions of the || allowance for the degeneracy of vibrations along the x centroids E and E⊥ of dipole states for the longitu- and y axes, were approximated by Lorentzian curves. dip dip dinal and transverse modes of vibrations. After that, a Below, we expound on the details of our computa- different value was chosen for the parameter K12,and tional procedure. the procedure was repeated. || ⊥ The energies Edip and Edip were calculated with 4.1. Choice of Single-Particle Potential the aid of the relations In order to calculate the single-particle states |α , ⊥ || 2Edip + Edip we employed the Nilsson spheroidal potential [16], Edip = , (46) setting its parameters to 3 ⊥ E 2 − 4 dip z 1+2δ /3 ω(0) =41A 1/3 1 − δ, (43) = = . z 3 || x2 1 − 4δ/3 Edip − 2 ω(0) = ω(0) =41A 1/3 1+ δ, The giant-resonance energy E ,whichappears x y 3 dip in these relations, was estimated by the formula [18] 2 δ = δ/(1 + δ), 2 2 3 − 1+π η E ≈ 86A 1/3 [MeV], where δ is the parameter of the quadrupole deforma- dip 1+10π2η2/3+7π4η4/3 tion of a nucleus, (47) 3 Q δ = 0 . (44) where the quantity η was defined in (45). 2 4 Z r The best agreement with experimental data for 2 K ≈ Here, Q0 is the internal quadrupole moment and r the nuclei considered here was obtained at 12 is the mean-square radius of the nuclear-charge dis- 25 MeV. As might have been expected, this value, tribution. which characterizes the strength of interaction of dif- ferent dipole configurations, is rather small in relation For the nuclei considered here, the parameter δ to a value of about 100 MeV, which follows from the was calculated theoretically by means of the proce- estimate in (10). In order to assess the sensitivity of dure described in [17]. This procedure leads to over- calculations to the choice of value for the constant estimated values of the parameter δ for light nuclei, 28 K12, we have performed calculations for the Si nu- since it disregards the effect of nuclear-surface dif- cleus, employing various values of this constant. The 2 2 2 fuseness on x , y ,and z . The value corrected results are given in Fig. 2. with allowance for this effect can be obtained by the || ⊥ formula The intermediate energies ω1 and ω1 do not have an independent physical meaning. Nonetheless, 1+2π2η2 + 2 π4η4 15 it is interesting to note that, in varying the relevant δcorr ≈ δ , (45) 1+ 10 π2η2 + 7 π4η4 model parameters, they are grouped around values 3 3 that follow from formulas of the type in (30) if, in where η ≡ a/R0,witha ≈ 0.55 fm and R0 ≈ estimating the coupling constant for dipole–dipole 1.07A1/3 fm being, respectively, the diffuseness pa- forces [see Eq. (10)], use is made of a symmetry rameter of the nuclear surface and the distance potential V that faithfully reproduces the position between the center of the nucleus and the point at of the giant dipole resonance [compare the results which its density decreases by a factor of 2. in (14) and (47)]. The root-mean-square distinctions For the 12C, 24Mg, and 28Si nuclei, the eventual between the two types of estimates of the energies || ⊥ values of the parameter δ proved to be 0.13, 0.28, and ω1 and ω1 are less than 1 MeV. This confirms that 0.15, respectively. the approximations made in the model are physically justified. 4.2. Varying Model Parameters 4.3. Approximation of Resonance Widths || We varied the following three parameters: ω1 , ⊥ K Our calculations revealed (see Figs. 3–5) that ω1 ,and 12 (see Subsection 3.3). First, the value K type-1 and type-2 dipole configurations are weakly of the parameter 12 was fixed, whereupon the en- mixed in light nonmagic nuclei. On the other hand, || ⊥ ergies ω1 and ω1 were varied in such a way as the damping of vibrations in light nuclei is governed
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1236 ISHKHANOV, ORLIN
σ, mb σ, mb
20 (a) 50
10 25
0 0 10 15 20 25 30 E, MeV 20 (b)
Fig. 2. Results of our calculations for the 28Si nucleus at various values of the parameter K12. Points correspond to the experimental values of the photoabsorption cross 10 section from [19], while the dashed, the solid, and the dotted curve represent the results of the calculations with K12 =40, 25, and 10 MeV,respectively.
0 by different mechanisms for vibrations of different types: these are predominantly the emission of an 10 20 30 40 excited particle from a nucleus for type-1 vibrations E, åeV and the spreading of a collective mode over a large number of noncollective nuclear states interacting Fig. 3. Structure of the giant dipole resonance in the 12C with this mode for type-2 vibrations. It follows that, nucleus according to the calculations (a)withand(b) depending on the type of dominant configurations, the without allowance for the configuration splitting of the +| giant dipole resonance. Points represent the experimental normal excitation cˆi 0 (i =1–4) of a light nucleus values of the photoabsorption cross section from [20]. The ↑ may be assigned either the emission width Γ ( ωˆi) or description of the curves and histograms presented in this ↓ figure and in Figs. 4 and 5 below is given in the main body the spreading width Γ ( ωˆi). of the text. For 16 A 240 nuclei, the total width of a giant dipole resonance can be approximated by the for- mula [18] Each of these figures consists of two panels (a, b). In the panels carrying the label a, the experimental 0.0293 1+(π2/3)η2 Γ(E) ≈ 1 − 3η E2, (48) values of the photoabsorption cross section (points) 1+π2η2 1+π2η2 are contrasted against the results of the calcula- tion based on the model described above (curves and where, by η,weimplythesamequantityasin histograms). The panels labeled with b display the Eqs. (45) and (47). results of the calculation that was performed under As is well known, the approximate relation Γ ≈ the assumption that a giant dipole resonance is split Γ↑ +Γ↓ holds. We introduce the notation ν ≡ Γ↑/Γ. only into deformation modes whose energies are de- We then find that Γ↑(E) ≈ νΓ(E) and Γ↓(E) ≈ (1 − termined by relations (46) and (47). The thick solid ν)Γ(E),whereΓ(E) is given by expression (48). curves represent the theoretical values of the pho- In the present study, the dipole-resonance widths toabsorption cross section. In each case, two thin were estimated by using the values of ν =0.6,0.4, solid curves show the contributions to this cross sec- tion that come from, respectively, longitudinal and and 0.3 for 12C, 24Mg, and 28Si, respectively. In se- transverse dipole vibrations. Two dashed curves rep- lecting these values, we have taken into account the resent the configuration splitting of the giant dipole experimental trend toward a decrease in the contribu- resonance. The histograms depict the distributions of tion to Γ(E) from the emission width with increasing the oscillator strengths of dipole states (in arbitrary mass number A. units), the narrow and broad rectangles correspond- ing to, respectively, longitudinal and transverse dipole 5. DISCUSSION OF THE RESULTS modes. The open (closed) part of a rectangle repre- The basic results of our calculations are given in sents the contribution to a given state from type-1 Figs. 3–5. (type-2) configurations.
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 SEMIMICROSCOPIC DESCRIPTION OF THE GROSS STRUCTURE 1237
σ, mb σ, mb (a) 40 (a) 50
20 25
0 0 40 (b) ( ) 50 b
20 25
0 0 15 20 25 30 10 15 20 25 30 E, åeV E, åeV
Fig. 4. Structure of the giant dipole resonance in the Fig. 5. Structure of the giant dipole resonance in the 28Si 24Mg nucleus according to the calculations (a)withand nucleus according to the calculations (a)withand(b) (b) without allowance for the configuration splitting of the without allowance for the configuration splitting of the giant dipole resonance. Points represent the experimental giant dipole resonance. Points represent the experimental values of the photoabsorption cross section from [21]. values of the photoabsorption cross section from [19].
Figures 3–5 clearly demonstrate that the configu- from an inner filled shell [10]. The present calculation ration splitting of a giant dipole resonance plays a very basically confirms this point of view, but with one important qualification: the main contribution to the important role in the formation of its gross structure high-energy region of a giant dipole resonance is in light nonmagic nuclei. Indeed, experimental data formed by a collective mode that emerges from single- foranyofthenucleiconsideredherecannotbeade- particle excitations under the effect of residual forces quately reproduced without taking this phenomenon rather than by individual type-2 1p1h configurations. into account (compare panels a and b). The effect of Indeed, the residual interaction shifts this state (see the configuration splitting of a giant dipole resonance extreme right rectangles in Figs. 3–5) by aproxi- is especially pronounced in nuclei characterized by a 12 28 mately 8 MeV upward with respect to the energy small deformation ( C, Si). However, it can easily position of type-2 single-particle transitions. be singled out in the structure of the cross section for From the histograms presented in Figs. 3–5, it photoabsorption on the strongly deformed nucleus of 24 can be seen that type-1 and type-2 dipole configura- Mg as well (see Fig. 4). tions interact with each other rather weakly. This im- At the same time, the formation of the gross struc- plies that normal vibrational modes are formed owing ture of a giant dipole resonance is also greatly af- primarily to interactions of configurations belonging fected by the deformation splitting of a giant dipole to the same type. resonance (even in weakly deformed nuclei). This be- Finally, it should be noted that the main strength comes obvious in comparing the rectangles of the his- of dipole transitions from an inner shell to the valence tograms with the special features of the experimental shell (that is, of type-2 transitions) is associated with photoabsorption cross sections. the transverse mode of dipole vibrations. In the literature, the high-energy tails of the cross sections for photoabsorption on nonmagic 1p-and 6. CONCLUSIONS 2d2s-shell nuclei (see the notation in Section 2) are The model developed in the present study has been usually associated with single-particle excitations used to clarify the global features of the photoabsorp-
PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1238 ISHKHANOV, ORLIN tion cross section in the mass range 10 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1239–1241. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1279–1281. Original Russian Text Copyright c 2003 by Isakov. NUCLEI Theory Charge-Exchange (p, n) Reaction on 48Ca as a Means for Determining the Isospin Structure of the Mean Nuclear Spin–Orbit Field V. I. Isakov* Institute of Nuclear Physics, Russian Academy of Sciences, Gatchina, 188300 Russia Received April 3, 2002 Abstract—The isotopic structure found previously in spin–orbit splitting by studying the spectra of heavy nuclei close to doubly magic ones is tested in polarization effects that arise in charge-exchange (p, n) reactions between the isobaric states of A =48nuclei. c 2003 MAIK “Nauka/Interperiodica”. On the basis of an analysis of available exper- in an isotopic-invariant form (Lane potential) that imental data for nuclei close to the doubly magic is appropriate for describing the processes that are 208 132 nuclides of Pb and Sn, it was shown in [1], by both diagonal and nondiagonal in t3 [single-particle invoking various theoretical approaches, that neutron spectra and elastic scattering in the first case and spin–orbit splitting in N>Znuclei is stronger, for (p, n) reactions leading to the excitation of isoanalog states in the second case]: identical orbitals, than the corresponding splitting for protons.It was found in [1], among other things, that Tˆ · ˆt various theoretical models lead to a stronger splitting Uˆ = V0 1 − 2β f(r) (2) for neutrons than for protons in the case of the 1d and A 48 1p orbitals in Ca, where experimental data on the ˆ · ˆ T t 1 df ˆ energies of the corresponding single-particle levels + Vls 1 − 2βls l · ˆs. are incomplete (in particular, because of their strong A r dr fragmentation).Also, it was revealed in [1] that, in terms of a phenomenological nuclear mean-field po- The spin–orbit term in the potential leads to polar- tential, the experimental values of “true” energies as ization phenomena in scattering.From (2), it can be determined for single-particle levels in nuclei with seen that, whereas polarization in elastic scattering allowance for configuration mixing (averaged over is determined by a combination of parameters in the the values of single-particle spectroscopic factors— form − see [2]), including spin–orbit splitting, can be well − N Z ≈ described by the potential Vls 1 βls t3 Vls, A 1 N − Z Uˆ (r, σ,τˆ )=V 1+ β τ f(r) (1) similar effects in charge-exchange reactions involv- 3 0 2 A 3 ing the excitation of isoanalog states depend on βlsVls − and are therefore controlled by the isovector param- 1 N Z 1 df ˆ· 1+τ3 + Vls 1+ βls τ3 l ˆs + UCoul, eter β of the mean spin–orbit field, because the 2 A r dr 2 ls parameter V is well known.Thus, all of the conclu- − ls r − R 1 sions drawn in [1] on β and based on the description f(r)= 1+exp ,R= r A1/3, ls a 0 of nuclear spectra can be verified by using the data on quasielastic (p, n) scattering.The corresponding − where V0 = 51.5 MeV; r0 =1.27 fm; Vls = information about polarization effects observed for 2 ≈ ∼− 33.2 MeV fm ; a 0.6 fm; β =1.39; βls 0.6;and nuclei in the vicinity of 48Ca can be found in [4], where − τ3 = 1 for neutrons, and τ3 =+1for protons. the reaction 48Ca(p, n)48Sc leading to the excitation − Introducing the quantities t3 = τ3/2 and T3 = of the 0+ isoanalog state at 6.67 MeV was investi- (N − Z)/2 and making, as in [3], the substitution gated with polarized protons, but the theoretical anal- ˆ ˆ T3 · t3 → T · ˆt,whereT and ˆt are the isospin vector ysis was performed there in terms of a microscopic operators for the core and a nucleon, respectively, approach to describing the nuclear structure and in we obtain the nuclear component of the potential (1) terms of nucleon–nucleon amplitudes for describing the scattering process (distorted-wave impulse ap- *e-mail: [email protected] proximation).Below, we perform our analysis within 1063-7788/03/6607-1239$24.00 c 2003 MAIK “Nauka/Interperiodica” 1240 ISAKOV A case of a R, this leads to WS ≈−(R/3)WV .For 1.0 the absorptive term, we therefore use the expression 1 R d 5 iWV α − (1 − α) f(r), (3) 3 dr 0.5 3 ≤ ≤ 4 where 0 α 1.This results in polarization e ffects that are independent of α for small scattering an- gles, but which are strongly dependent on α for high 0 momentum transfers.Thus, the scattering process, along with polarization phenomena, is described here in terms of the optical potential in the form (2), where –0.5 Vo →−51.5(1 − 0.0058E) (4) 2 R d − i3.3 · (1 + 0.03E) α − (1 − α) , 3 dr –1.0 0102030identical energy dependences being assumed for the θ Ò.m., deg isoscalar and isovector components of the central nuclear potential. Fig. 1. Experimental data on the analyzing power from The results of the present calculations for the an- [4] (points) and results of various calculations (curves). alyzing power A in the (p, n) reaction occurring on Curve 1 corresponds to the microscopic calculation in the 48Ca and leading to the excitation of an isoanalog distorted-wave impulse approximation from [4].Curves 2–5 were calculated in this study with (2) α =1(vol- state, V . 2 β − . − ume absorption), ls =332 MeV fm ,and ls = 0 6; dσ↑↑/dω dσ↑↓/dω | |≤ (3) α =0 (surface absorption), V =33.2 MeV fm2, Atheor = , A 1, (5) ls dσ↑↑/dω + dσ↑↓/dω and βls = −0.6;(4) α =0, βls = −0.6, and an energy- dependent parameter Vls;and(5) α =0.5,allotherpa- are displayed in the figure, along with experimental rameters being set to the same values as those for curves data and the results of the microscopic calculations 2 and 3. from [4].The quantities σ↑↑ and σ↑↓ are the reaction cross sections for the cases where the proton polar- the Lane model, relying on the mean-field parameters ization vector ε is, respectively, parallel and antipar- determined in [1]; in addition, we employ the Born allel to the vector [ki × kf ].It can be seen that, in approximation to describe the scattering process.In the case of surface absorption (α =0), our calcula- describing (p, n) reactions, Gosset et al. [5] also con- tions in which we use the corresponding spin–orbit sidered polarization effects in the Born approxima- parameters from [1] agree well with experimental data tion. up to large values of the scattering angle.At the same It is well known that, in the Born approximation, time, the description of the analyzing power becomes there are no polarization effects caused by a spin– unsatisfactory upon introducing, in the spin–orbit orbit potential [6, 7] if use is made of a real-valued parameter Vls, an energy dependence similar to that central potential.Therefore, an imaginary part (ab- for the central potential. ∗ sorption) must be introduced in the optical potential For the reaction 48Ca(p, n) 48Sc (IAS) on unpo- in order to describe such effects.The energy depen- larized protons, where the abbreviation IAS stands dence must also be taken into account in the real for an isoanalog state, our calculations with α =0 and imaginary parts of the potential, since the inci- yield a value of about 7.7 mb/sr for the differential dent energy was quite high (E = 134 MeV) in [4]. cross section at zero angle, this cross section being − In [8, 9], V0 was parametrized as follows: V0 = V0 (1 a very slowly increasing function of the parameter − 0.0058E),whereV0 = 52 MeV, which is very close α.The cross section decreases sharply with increas- to the value of −51.5 MeV obtained in [1].In this ing scattering angle and exhibits some structure at ≈ ◦ case, the form proposed in [8] for the corresponding Θc.m. 20 .The above value can be compared with absorptive term in the optical potential is iWV f(r), the experimental result obtained in [10] for the differ- where WV (MeV) = −3.3(1 + 0.03E).Surface ab- ential cross section at zero angle (about 7 mb/sr) and sorption is usually taken into account via the term with the theoretical result obtained in [10] on the basis iWS(df / d r ).For low momentum transfers (small an- of microscopic calculations (about 7.5 mb/sr). gles), the two versions of absorption must provide an The following conclusions can be drawn from the identical description of the scattering process.In the above results: PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 CHARGE-EXCHANGE (p, n) REACTION 1241 (i) Experimental data on the isotopic structure also considerably affect the predictions for nucleon of spin–orbit splitting in nuclei are compatible with drip lines in nuclei characterized by an extremely high data on polarization effects in quasielastic (p, n) neutron excess or an extremely high neutron deficit. scattering.The mean- field parameters determined in [1], which made it possible to reproduce proton ACKNOWLEDGMENTS and neutron spin–orbit splitting in nuclei near 132Sn 208 and Pb—in particular, the parameter βls—also I am grateful to B.Fogelberg, H.Mach, V.E.Bu- faithfully reproduce experimental data for quasielastic nakov, and K.A. Mezilev for numerous stimulating (p, n) scattering on 48Ca. discussions on the questions considered in this (ii) A good description of the analyzing power study—in particular, those concerning the problem at high energies of incident protons with the val- of spin–orbit splitting in nuclei. ues taken for spin–orbit parameters from low-energy This work was supported by the Russian Founda- spectroscopy is compatible with the assumption that tion for Basic Research (project no.00-15-96610). these parameters of the optical model are weakly de- pendent on energy . REFERENCES (iii) A satisfactory description of the cross section 1. V.I.Isakov, K.I.Erokhina, H.Mach, et al.,Eur.Phys. for the relevant (p, n) reaction involving the excitation J.A 14, 29 (2002); nucl-th/0202044. of an isoanalog state indicates that the energy de- 2. M.Baranger, Nucl.Phys.A 149, 225 (1970). pendence of the isovector terms in the central nuclear 3. A.M.Lane, Nucl.Phys. 35, 676 (1962). potential was parametrized correctly. 4. B.D.Anderson, T.Chittrakarn, A.R.Baldwin, et al., (iv) The present results unambiguously corrobo- Phys.Rev.C 34, 422 (1986). rate the presence of a significant surface component 5. J.Gosset, B.Mayer, and J.L.Escudie, Phys.Rev.C [(1 − α) 0.5] in the imaginary part of the optical 14, 878 (1976). potential. 6.L.D.Landau and E.M.Lifshitz, Quantum Me- chanics: Non-Relativistic Theory (Nauka, Moscow, In conclusion, we note that, although spin–orbit 1989, 4th ed.; Pergamon, Oxford, 1977, 3rd ed.). interaction makes a relatively small contribution to 7.A.S.Davydov, Theory of the Nucleus (Fizmatgiz, the total mean-field potential for nuclear single- Moscow, 1958). particle levels, it is of paramount importance for 8.A. G. Sitenko, Theory of Nuclear Reactions the formation of the nuclear shell structure.For (Energoatomizdat,´ Moscow, 1983). example, the density of single-particle levels is high in 9. A.Bohr and B.Mottelson, Nuclear Structure, Vol.1: superheavy nuclei; therefore, their small shifts due to Single-Particle Motion (Benjamin, New York, 1969; refining the isospin dependence of the spin–orbit field Mir, Moscow, 1971). can be decisive in determining shell corrections that 10. B.D.Anderson, T.Chittrakarn, A.R.Baldwin, et al., control the stability of such nuclei.A correct inclusion Phys.Rev.C 31, 1147 (1985). of the isotopic dependence of spin–orbit splitting can Translated by V. Bukhanov PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1242–1251. From Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1282–1291. Original English Text Copyright c 2003 by Karnaukhov, Avdeyev, Duginova, Petrov, Rodionov, Oeschler, Budzanowski, Karcz, Janicki, Bochkarev, Kuzmin, Chulkov, Norbeck, Botvina. NUCLEI Theory Thermal Multifragmentation of Hot Nuclei and Liquid–Fog Phase Transition* V.A.Karnaukhov 1)**,S.P.Avdeyev1),E.V.Duginova1),L.A.Petrov†1), V.K.Rodionov 1),H.Oeschler2)***,A.Budzanowski3),W.Karcz3),M.Janicki3), O.V.Bochkarev 4),E.A.Kuzmin4), L.V.Chulkov 4),E.Norbeck5),andA.S.Botvina6) Received April 30, 2002; in final form, October 4, 2002 Abstract—Multiple emission of intermediate-mass fragments (IMF) in the collisions of protons (up to 8.1 GeV), 4He (4 and 14.6 GeV), and 12C (22.4 GeV) on Au has been studied with the 4π setup FASA. In all the cases, thermal multifragmentation of the hot and diluted target spectator takes place. The fragment multiplicity and charge distributions are well described by the combined model including the modified intranuclear cascade followed by the statistical multibody decay of the hot system. IMF– IMF-correlation study supports this picture, giving a very short time scale of the process (≤70 fm/c). This decay process can be interpreted as the first-order nuclear “liquid–fog” phase transition inside the spinodal region. The evolution of the mechanism of thermal multifragmentation with increasing projectile mass was investigated. The onset of the radial collective flow was observed for heavier projectiles. The analysis reveals information on the fragment space distribution inside the breakup volume: heavier IMFs are formed predominantly in the interior of the fragmenting nucleus possibly due to the density gradient. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION the nuclear liquid–gas phase transition. One of the first nuclear models, suggested by N. Bohr, C. von Study of the decay properties of the hot nuclei is Weizsacker,¨ and Ya.I. Frenkel 65 years ago, is the liquid-drop model, which has been successfully used one of the most challenging topics of modern nu- up to now. The liquid–gas phase transition in nuclear clear physics. The excitation energy of the hot nuclei matter was predicted much later [1–3] on the basis of (500–700 MeV) is comparable with the total binding the similarity between van der Waals and nucleon– energy. They disintegrate via a new multibody decay nucleon interactions. In both cases, the attraction mode—thermal multifragmentation. This process is between particles is replaced by repulsion at a small characterized by the copious emission of intermediate interaction range. As a result, the equations of state mass fragments that are heavier than alpha particles are similar for such different systems. This is well ≤ but lighter than fission fragments (IMF,2 1063-7788/03/6607-1242$24.00 c 2003 MAIK “Nauka/Interperiodica” THERMAL MULTIFRAGMENTATION OF HOT NUCLEI 1243 nucleus expands due to thermal pressure and enters P the metastable region. Due to density fluctuations, a Skyrme homogeneous system converts into the mixed-phase Van der Waals state, consisting of droplets (IMFs) surrounded by 2 nuclear gas (nucleons and light composite particles). In fact, the final state of this transition is a nuclear T = 1.5 Gas fog [3]. 1.25 The neutrons fly away with energies correspond- 1 1.0 ing to the system temperature (5–7 MeV), while the Liquid 0.75 charged particles are additionally accelerated in the 0.5 Coulomb field of the system. Disintegration time is 0 determined by the time scale of the thermodynamic fluctuations and is expected to be very short. This is a scenario of nuclear multifragmentation as the spinodal decomposition, considered in a number of –1 theoretical and experimental papers (see, for example, [4–10] and review papers [11, 12]). This picture was 01234 proved experimentally with a significant contribution V from the FASA collaboration; a short review of the results obtained is presented below. Fig.1. Comparison of the equation of state for a van der Waals gas and for a nuclear system interacting through a As for the critical temperature for the liquid–gas Skyrme force (relative units are used). phase transition Tc (at which the surface tension van- ishes), its value is not known definitely . There are many theoretical calculations of Tc for finite nuclei. In excitation energy is almost entirely thermal. Light rel- [1, 2], the calculation is done using a Skyrme effective ativistic projectiles therefore provide a unique possi- interaction and thermal Hartree–Fock theory. The bility of investigating “thermal multifragmentation,” values of Tc were found to be 18.1 MeV [1] and in which was realized in the FASA project. the range 8.1–20.5 MeV [2] depending on the chosen To study multifragmentation with the beams of Skyrme interaction parameters. There are not yet the Dubna synchrophasotron, a 4π setup FASA was reliable experimental data for Tc, despite the claims created [14]. The device consists of two main parts: (i) Five dE–E telescopes (at θ =24◦, 68◦, 87◦, made in a number of papers. The latest of them is [13], ◦ ◦ where it is stated that Tc =6.7 ± 0.2 MeV. We return 112 ,and156 with the beam direction), which serve to the discussion of this point in Section 6. as triggers for the readout of the system allowing the measurement of the fragment charge and energy dis- tributions. Ionization chambers and Si(Au) detectors 2. HOW TO PRODUCE are used as dE and E counters, respectively. AND STUDY HOT NUCLEI (ii) A fragment multiplicity detector (FMD) in- cluding 64 CsI(Tl) counters (with a scintillator thick- An effective way to produce hot nuclei is collisions 2 of heavy ions with energies up to hundreds of MeV per ness averaging 35 mg/cm ), which cover 89% of 4π. nucleon. Around a dozen sophisticated experimental TheFMDgivesthenumberofIMFsintheeventand devices were created to study nuclear multifragmen- their angular distribution. tation with heavy-ion beams. But in this case, heat- A self-supporting Au target 1.0 mg/cm2 thick is ing of nuclei is accompanied by compression, strong located at the center of the FASA vacuum chamber rotation, and shape distortion, which may essentially (∼1 m in diameter). influence the decay properties of hot nuclei. The following beams were used: protons at en- 4 Investigation of dynamic effects caused by excita- ergies of 2.16, 3.6, and 8.1 GeV [15]; He at 4 and 12 tion of collective (or “mechanical”) degrees of free- 14.6 GeV; and C at 22.4 GeV [16]. The mean beam dom is interesting in itself, but there is a great prob- intensity was around 7 × 108 p/spill for protons and lem of disentangling all these effects to get informa- helium and 1 × 108 p/spill for carbon projectiles (spill tion on the thermodynamic properties of a hot nuclear length 300 ms, spill period 10 s). system. One gains simplicity and the picture becomes In recent years, FASA has been significantly up- clearer when light relativistic projectiles (first of all, graded, and a new counter array consisting of 25 protons, antiprotons, pions) are used. In contrast to dE–E telescopes was developed. At present, the total heavy-ion collisions, fragments are emitted only by number of detectors in FASA is 129, supplied with the source—the slowly moving target spectator. Its 193 electronic channels. PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 1244 KARNAUKHOV et al. β⊥ 0.16 (a) (b) 0.12 0.08 0.04 0 –0.1 0 0.1 –0.1 0 0.1 β|| Fig.2. Transverse vs. longitudinal velocity plot for carbon isotopes produced in 4He (14.6 GeV) (a)and12C (22.4 GeV) (b) collisions with Au target. Circles are drawn through points of equal invariant cross section corresponding to isotropic emission in the moving source frame. dN/dM, arb. units dN/dM, arb. units (a) (b) 101 p(8.1 GeV), × 25 101 p(8.1 GeV), × 25 α(14.6 GeV), × 5 α(14.6 GeV), × 5 C(22.4 GeV) C(22.4 GeV) 100 100 10–1 10–1 10–2 10–2 10–3 10–3 08246 2468 10 FMD-multiplicity (IMF) Primary IMF multiplicity Fig.3. (a) Associated IMF-multiplicity distributions and fits with Fermi functions (folded with the experimental filter) for p + Au collisions at 8.1 GeV, 4He + Au at 14.6 GeV, and 12C + Au at 22.4 GeV. (b) Symbols represent directly reconstructed primary IMFdistributions; histograms are the Fermidistributions used to fit the data in the left part. The curves are calculated with the combined model INC + Expansion + SMM. 3. EVIDENCE FOR THERMALIZATION of (0.01–0.02) c. The IMFangular distribution in the OFTHE TARGET SPECTATOR AT BREAKUP laboratory system exhibits a forward peak caused by Let us consider a very important point of thermal- the source motion. ization of the system at breakup. To check whether this state is close to thermal equilibrium, plots were composed for the fragment yields in terms of the Another finding in favor of the thermal equilib- longitudinal versus transverse velocity components. rium is that the fragment kinetic energy spectra look They look similar for all collisions investigated. Fig- like Maxwellian ones with the maximum around the ure2showssuchplotsfor4He + Au and 12C + Au Coulomb barrier followed by an exponential tail. All interactions [16]. The symbols correspond to the con- these observations can be considered as good moti- stant invariant cross sections taken for emitted car- vation for using statistical approaches to describe the bon fragments. The lines connecting the experimental data. It was done in our studies rather successfully. points form circles demonstrating isotropic emission in the frame of the moving source. This indicates that Some details of the model used are given in the next the fragment emission proceeds from a thermalized section, but here for illustration we present in Fig. 3 state. The center positions of circles determine the the fragment multiplicity distributions for different source velocities, which are found to be in the range collisions in comparison with the model calculations. PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 THERMAL MULTIFRAGMENTATION OF HOT NUCLEI 1245 σ 4. DENSITY OFTHE SYSTEM AT BREAKUP d /dvrel, arb. units What is the size of a fragmenting target spectator? 800 Is it true that a very hot nucleus expands due to v1 v2 v1 v2 the thermal pressure to get into the phase instability 600 (spinodal) region? To answer this question, we mea- sured [17] the distribution of the relative velocities for coincident fragments at large correlation angles. 400 The fragment kinetic energy is determined in the main by the acceleration in the Coulomb field of the fragmenting nucleus. Therefore, the fragment velocity 200 is sensitive to the configuration of the system at the moment of breakup. In the upper part of Fig. 4, two variants of fragment emission are shown: evaporation 0 from the surface of the nucleus with normal density 2 3 4 5 6 (right) and the volume decay of the expanded system vrel, cm/ns (left). The measured distribution is shifted to lower ve- Fig.4. Distribution of relative velocities for the coincident fragments from 4He + Au collisions (at 14.6 GeV) mea- locities relative to the calculated one for the surface ◦ ◦ emission; this observation is in favor of the volume sured at correlation angles of 150 –180 .Thevertical fragment emission. After quantitative analysis of the line shows the expected maximum position for fragment evaporation from the nucleus surface. The data are shifted data by means of a combined model (see below), it to lower velocities, corresponding to the volume disinte- was concluded that the fragment emission occurs gration of the expanded system (see text). from the expanded system with a mean density that is 3 to 4 times smaller than normal [18, 19]. The same conclusion is drawn from considerations of the i.e., the values for the residual (after INC) masses and fragment kinetic energy spectra [15]. Thus, one can their excitation energies are scaled on an event-by- say that thermal multifragmentation is indeed the event basis (for details, see [15, 16]). The final stage spinodal decomposition process. This conclusion is of the combined model INC + Expansion + SMM supported by measuring the time scale of fragment is the multibody Coulomb trajectory calculations for emission, which is very fast (see the next section). all charged particles in the exit channel (again on an Now, let us consider the combined model. The event-by-event basis). As a result, the fragment ener- reaction mechanism for light relativistic projectiles gies and momenta are obtained and can be compared is usually divided into two steps. The first one is a with the experimental data. fast energy deposition stage, during which energetic light particles are emitted and the nuclear remnant is excited. The fast stage is usually described by 5. THERMAL the intranuclear cascade model (INC). We use the MULTIFRAGMENTATION—A NEW DECAY version of the INC from [20] to get the distribu- MODE OFHOT NUCLEI tion of the nuclear remnants in charge, mass, and The time scale of IMFemission is a crucial char- excitation energy. The second stage is described by acteristic for understanding the mechanism of this the statistical multifragmentation model (SMM) [21]. decay process: whether it is a “slow” successive and Within the SMM, the probability of different decay independent evaporation of IMFs or a new (multi- channels of the excited remnant is proportional to body) decay mode with “simultaneous” ejection of their statistical weight. The breakup volume deter- the fragments governed by the total accessible phase mining the Coulomb energy of the system is taken to space. “Simultaneous” means that all the fragments be Vb =(1+k)A/ρ0,whereA is the mass number are liberated during a time which is smaller than −21 of the decaying nucleus, ρ0 is the normal nuclear the characteristic Coulomb time τc ≈ 10 s [22], density, and k is the model parameter. Thus, ther- which is the mean time of fragment acceleration in the mal expansion before the breakup is assumed. The Coulomb field of the system. In that case, emission of breakup density is ρb = ρ0/(1 + k).Itisfoundthat the fragments is not independent; they interact with this traditional approach fails to describe the observed each other via the Coulomb forces during the acceler- IMFmultiplicities, whose mean values saturate at ation. Thus, measurement of the IMFemission time 2.2 ± 0.2. The expansion stage is inserted between τem (the mean time separation between two consecu- the two parts of the calculation. In fact, the excitation tive fragment emissions) is a direct way to answer the energies and the residual masses are finely tuned to question as to the nature of the multifragmentation get agreement with the measured IMFmultiplicities; phenomenon. PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 1246 KARNAUKHOV et al. / Ω, arb. units (θ ) dY d FCoul Cf rel 2 10 1.0 0.8 101 0.6 0.4 100 0.2 0 50 100 150 θ , deg τ rel 1.0 c Multibody decay Sequential emission 0.8 1 2 3 τ 10 10 10 em, fm/c 0.6 Fig.5. Distribution of relative angles between coincident IMFs for 4He (14.6 GeV) + Au collisions. The solid curve 0.4 is calculated for the simultaneous emission of fragments; the dashed curve corresponds to the sequential, indepen- 0.2 dent evaporation. 50 100 150 θ rel, deg There are two procedures to measure the emission Fig.6. Comparison of the measured correlation functions time: analysis of the IMF–IMFcorrelation function (for p + Au at 8.1 GeV) with the calculated ones for with respect to the relative velocity and the relative different mean decay times of the fragmenting system: angle. We used the second method. Figure 5 shows solid, dashed, dotted, and dash-dotted curves for τem = the IMF–IMFrelative angle correlation for the frag- 0, 50, 100, and 200 fm/c. Calculations are made with mentation target spectator in 4He (14.6 GeV) + Au combined model INC + Expansion + SMM assuming two values of breakup volumes: 8V0 (upper panel) and collisions [18]. The correlation function exhibits a 4V0 (lower panel). minimum at θrel =0arising from the Coulomb re- pulsion between the coincident fragments. The mag- nitude of this effect drastically depends on the time nucleus, which is around 70 fm/c according to model scale of emission, since the longer the time distance estimation [15, 24]. between the fragments, the larger their space sep- aration and the weaker the Coulomb repulsion. The 6. EVOLUTION OFTHE REACTION multibody Coulomb trajectory calculations fittheda- MECHANISM WITH INCREASING ta on the assumption that the mean emission time is PROJECTILE MASS −22 τem ≤ 70 fm/c (2.3 × 10 s). This value is signifi- cantly smaller than the characteristic Coulomb time It is shown in a number of papers that the mul- tifragment emission in the central collisions of very τc. The trivial mechanism of IMFemission (indepen- dent evaporation) is definitely excluded. heavy ions is not described by the statistical models. Initial compression of the system is tremendous and A similar result is obtained in our recent paper [23], the collective part of the excitation energy is so large devoted to the time scale measurement for multifrag- that the partition of the system into fragments is likely ment emission in p + Au collisions at 8.1 GeV. The to be a very fast dynamic process [25]. In that case, model dependence of the results was carefully inves- the fragment kinetic energy is largely determined by tigated. Figure 6 shows the experimental correlation the decompressional collective flow. It is interesting to function and the calculated ones for two values of the follow the evolution of the multifragmentation mech- breakup volumes: Vb =4V0 and Vb =8V0. The mean anism (as the projectile mass increases) from pure emission time is found to be τem =50± 10 fm/c.One thermal to that influenced by the dynamic effects. should notice that this value is in fact the mean time of fragment formation under breakup conditions. The total duration time of the reaction is larger. It includes 6.1. Fragment Charge Distribution the thermalization time (10–20 fm/c) and the mean We performed a comparative study of multifrag- expansion time before the disintegration of the hot mentation induced in a gold target by relativistic PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 THERMAL MULTIFRAGMENTATION OF HOT NUCLEI 1247 Yield, arb. units Yield, arb. units τ 7 2.6 106 10 106 2.2 105 105 1.8 104 2 10 40 4 10 Eproj, GeV 103 103 102 102 1 10 101 246810 Z 246810 Z ◦ Fig.7. Fragment charge distributions obtained at θ =87 for p + Au at 8.1 GeV (•), 4He + Au at 4 GeV (), 4He + Au at 14.6 GeV (), and 12C + Au at 22.4 GeV (). The lines (left side) are calculated by INC + Expansion + SMM (normalized at Z =3). The power law fits are shown in the right panel with τ parameters given in the inset as a function of the beam energy. Thelastpointintheinsetisfor12C + Au collisions at 44 GeV (from a preliminary experiment). protons and helium and carbon ions [16, 26]. It was excitation energy (E∗) with the minimal value around already demonstrated that, in all cases, one dealt 2.0 at E∗/A =3–5 MeV [31]. with disintegration of a thermally equilibrated system In [32], the value of the critical temperature is (Fig. 2), and IMF multiplicity distributions were well estimated from the data on the fission probabilities. In reproduced by the statistical model (Fig. 3). It was this paper, the temperature dependence of the liquid- also found that charge distributions of fragments were drop fission barrier is calculated as in [1]. The crit- similar for all the collisions studied, and they are ical temperature Tc (at which the surface tension very well described by the combined model INC + vanishes) is taken as a parameter. It is found that Expansion + SMM ( Fig. 7). the barrier height is very sensitive to the ratio T/Tc. The general trend of the IMFcharge distribu- Experimental data [33] and calculations are compared tions is also well reproduced by a power law Y (Z) ∼ 188 − for highly excited Os. It is concluded that Tc is defi- Z τ . In earlier papers on multifragmentation [3, 27– nitely higher than 10 MeV.The results of recent paper 29], such a power-law dependence for the fragment [13] are in conflict with that conclusion. The fragment charge yield was interpreted as an indication of the mass distributions obtained by the ISIS collaboration proximity of the decaying state to the critical point were analyzed in the framework of the Fisher model for the liquid–gas phase transition in nuclear matter. with the Coulomb energy taken into account. The ex- This was stimulated by the application of the classical tracted critical temperature Tc =6.7 ± 0.2 MeV. We Fisher drop model [30], which predicted a pure power believe that this result should be treated with caution, law droplet-size distribution with τ =2–3 at the crit- keeping in mind the shortcomings of the Fisher model ical point. According to this model, the τ parameter in application to the hot nuclear system [34].7) has a minimal value at the critical temperature. So, in the spirit of the Fisher model, the data in the inset of Fig. 7 should be considered as an indication of the 6.2. Fragment Kinetic Energy Spectra “critical behavior” of the system at beam energies of 5–10 GeV. But this is not the case. The power law is The fragment kinetic energy spectra change with well explained at temperatures far below the critical increasing projectile mass. The spectral shapes show point. As is seen in Fig. 7, the pure thermodynamical an increase in the number of high-energy fragments SMM predicts that the IMFcharge distribution is 7) close to a power law at freeze-out temperatures of 5– Note added in proof: In our recent article [Karnaukhov et al., Phys.Rev.C67, 011601(R) (2003)], the critical temperature 6 MeV, while the critical temperature is assumed to was found to be Tc =20± 3 MeV.This was done by analyz- be Tc =18MeV. The statistical model also predicts ing the charge distribution of fragments within the statistical the parabolic dependence of the exponent τ on the multifragmentation model. PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 1248 KARNAUKHOV et al. the fragment charge. The figure reveals enhance- ment in the kinetic energies for the light fragments 6 (Z<10) emitted in 4He (14.6 GeV) + Au and 12C(22.4 GeV) + Au collisions as compared to the , MeV p(8.1 GeV) + Au case. The calculated values (curves) + IMF 4 are obtained with the combined model INC expan- A / + 〉 sion SMM. The measured energies are close to E 〈 the calculated ones for p + Au collisions in the range of fragment charges between 4 and 9. However, the 2 experimental values for heavier projectiles exceed the theoretical ones, which are similar for all three cases. 3 What is the cause of that? 12 C (22.4 GeV) The kinetic energy of fragments is determined by , MeV four terms: thermal motion, Coulomb repulsion, ro- IMF 2 tation, and collective expansion energies, E = Eth + A )/ ECoul + Erot + Eflow. The Coulomb term is signifi- cantly larger than the thermal one, as was shown in no flow 〉 [18, 23]. The contribution of the collective rotational E 〈 1 energy is negligible even for C + Au collisions [16, – 〉 26]. We suggest that the observed energy enhance- E 〈 ( ment is caused by the expansion flow in the system, 0 which is assumed to be radial as the velocity plot 0.08 12 0.96 (Fig. 2) does not show any significant deviation from C (22.4 GeV) circular symmetry. Note that the contribution of the collective flow for p(8.1 GeV) + Au collisions is in- 0.06 0.72 conspicuous. As was estimated in [15], the mean flow c / sys 〉 R velocity for that case is less than 0.02c. We believe / 〉 flow Z that the observed flow for heavier projectiles is driven v 0.04 0.48 R 〈 〈 by the thermal pressure, which is expected to be larger than that for the proton beam. 0.02 0.24 An estimate of the fragment flow energies may be obtained as a difference between the measured 0 46810 12 14 IMFenergies and those calculated without taking Z into account any flow in the system. This difference for 12C + Au collisions is shown in Fig. 8 (middle panel). Fig.8. Upper part: Mean kinetic energies of fragments ◦ In an attempt to describe the data, we modified the per nucleon measured at θ =87 for p(8.1 GeV) ( , SMM code in the INC + Expansion + SMM concept 4 dashed curve), He(14.6 GeV) ( , dotted curve), and by including a radial velocity boost for each particle 12C(22.4 GeV) (•, solid curve) collisions with Au. The curves are calculated within the INC + Expansion + at freeze-out. In other words, the radial expansion SMM approach assuming no flow. Middle part: Flow velocity was superimposed on the thermal motion in energy per nucleon (dots) obtained as a difference of the the calculation of the multibody Coulomb trajectories. measured fragment kinetic energies and the values calcu- Self-similar radial expansion is assumed when the lo- lated under the assumption of no flow in the system. The cal flow velocity is linearly dependent on the distance dashed curve represents a calculation assuming a linear 0 of the particle from the center of mass. The expansion radial profile for the expansion velocity with vflow =0.1c. Lower part: Experimentally deduced mean flow veloci- velocity of particle Z located at radius RZ is given by ties (dots) for 12C + Au collisions as a function of the the following expression: fragment charge (left scale) and the mean relative radial (Z)=v0 /R , coordinates of fragments (right scale), obtained under the vflow flowRZ sys (1) assumption of a linear radial profile for the expansion ve- 0 locity.The dashed curve shows the mean radial coordinate where vflow is the radial velocity on the surface of according to SMM. the system. Note that, in this case, the density dis- tribution changes in dynamic evolution in a self- similar way, being a function of the scaled radius for heavier projectiles. This observation is sum- RZ /Rsys.The use of the linear profile of the radial marized in Fig. 8 (upper panel), which shows the velocity is motivated by the hydrodynamic model cal- mean kinetic energies per nucleon as a function of culations for an expanding hot nuclear system (see, PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 THERMAL MULTIFRAGMENTATION OF HOT NUCLEI 1249 2 d N/dEdΩ, arb. units EMF* /AMF, MeV 103 Ω 6 p dEd 2 α / 10 N 2 12C 2 d 10 101 4 50 100 150 200 E, MeV 101 2 20 40 60 80 100 120 E, MeV 0 5 10 15 20 Fig.9. Energy distribution of carbon fragments (at θ = ◦ E , GeV 87 )from12C + Au collisions. Solid curves are calculated proj assuming radial flow with the velocity on the surface equal Fig.10. Mean excitation energy of the fragmenting nu- to 0.1c. Dashed curves are calculated assuming no flow. ∗ cleus per nucleon EMF/AMF as a function of the beam energy: the closed points refer to the thermal part; the flow 0 energy added is shown as open symbols and gray area. for example, [35]). The value of vflow was adjusted to describe the mean kinetic energy measured for the carbon fragment. only goes down slightly with increasing fragment Figure 9 shows the comparison of the measured charge. This trend of the calculations is to be ex- and calculated energy spectra (for 12C + Au colli- 0 pected. The mean fragment flow energy is propor- sions) assuming vflow =0.1c. The agreement is good. 2 tional to RZ . This value varies only slightly with The calculation without a flow deviates strongly. the fragment charge in the SMM code due to the There is a longstanding problem of a qualitative dif- assumed equal probability for fragments of a given ference between the chemical or thermal equilibrium charge to be formed at any point of the available temperature and the “kinetic” or so called “slope breakup volume. This assumption is a consequence temperature.” A recent discussion of that point can of the model simplification that considers the system be found in [36]. The mean equilibrium temperature as a uniform one with ρ(r)=const for r ≤ R .The obtained in our calculations is 6.2 MeV. At the same sys time, the slope temperature found from the spectrum data in Fig. 8 indicate that this is not the case. In fact, the dense interior of the expanded nucleus may favor shape is Ts =14.5 MeV for the “no-flow” case (see dashed curve in inset). This is the mutual result of the appearance of larger IMFs if fragments are formed the thermal motion, Coulomb repulsion during the via the density fluctuations. This observation is also volume disintegration, and the secondary decay of the in accordance with the analysis of the mean IMF excited fragments. Introducing a rather modest radial energies performed in [15] for proton-induced frag- flow results in an increase in the slope temperature up mentation. It is also seen in Fig. 8 (top) that, for p + to Ts =24MeV. Au collisions, the measured energies are below the Let us return to Fig. 8 (middle). The model- theoretical curve for fragments heavier than Ne. This calculated flow energy is given as the difference of the may be explained by the preferential location of the 0 heavier fragments in the interior region of the freeze- calculated fragment energies obtained for vflow =0.1c 0 out volume, where the Coulomb field is reduced. and vflow =0. The data deviate significantly from the calculated values for Li and Be. This may be caused The experimentally deduced mean flow velocities in part by the contribution of particle emission during of IMFs for 12C + Au collisions are presented in Fig. 8 the early stage of expansion from a hotter and denser (lower panel). The values for Li and Be are considered system. This explanation is supported by the fact as upper limits because of the possible contribution that the extra energy of Li fragments with respect of the preequilibrium emission. The corresponding to the calculated value is clearly seen in Fig. 8 (top) values of RZ /Rsys, obtained under the assumption even for the proton-induced fragmentation, where no of the linear radial profile for the expansion velocity, significant flow is expected. can be read on the right-hand scale of the figure. As to fragments heavier than carbon, the calcu- The dashed curve shows the mean radial coordinates lated curve in Fig. 8 (middle) is above the data and of the fragments according to the SMM code. The PHYSICS OFATOMIC NUCLEI Vol. 66 No. 7 2003 1250 KARNAUKHOV et al. calculated values of RZ /Rsys decrease only slightly observed for heavier projectiles. It is believed to be with Z in contrast to the data. driven by the thermal pressure. The mean total flow The total expansion energy can be estimated by energy at the moment of breakup is estimated to be integrating the nucleon flow energy over the available around 115 MeV for both 4He (14.6 GeV) and 12C volume at freeze-out. For a uniform system, one gets (22.4 GeV) beams, while the mean thermal excitation is around 400 MeV. Etot =(3/10)Am (v0 )2(1 − r /R )5, (2) flow N flow 0 sys The flow energy of fragments decreases as their where mN and r0 are the nucleon mass and radius. charge increases. The analysis of the data reveals 12 tot ∼ For C + Au collisions, it gives Eflow = 115 MeV, interesting information on the fragment space dis- corresponding to the flow velocity on the surface equal tribution inside the breakup volume: heavier IMFs to 0.1c. are formed predominantly in the interior of the frag- Similar results are obtained for 4He(14.6 GeV) + menting nucleus possibly due to the density gradient. Au collisions. The excitation energies of the frag- This conclusion is in contrast to the predictions of the menting systems studied are largely thermal ones; SMM. therefore, we deal with thermal multifragmentation. It This study of multifragmentation using a range of is reflected in Fig. 10, where the mean total excitation projectiles demonstrates a transition from pure “ther- ∗ p energy per nucleon EMF/AMF is shown as a function mal decay” (for + Au collisions) to disintegration of the incident energy. The closed symbols corre- “decorated” by the onset of a collective flow (for heav- spond to the thermal part of the excitation energy ier projectiles). Nevertheless, the decay mechanism obtained via analysis of the data on fragment mul- should be considered as thermal multifragmentation. tiplicity and charge distributions with the combined The partition of the system is governed by the nuclear model of the process. Open symbols include the flow heating, and IMFcharge distributions in all the cases energy. Thermal energies for these cases are 4 times considered are well described by the statistical model larger than collective ones. The onset of the collective neglecting any flow. flow driven by the thermal pressure takes place at the excitation energy around 4 MeV/nucleon, which is in good agreement with the results of [37]. The ACKNOWLEDGMENTS mean fragmenting masses are equal to 158, 103, We are grateful to Profs. A. Hrynkiewicz, A.N. Sis- and 86 for proton (8.1 GeV), 4He (14.6 GeV), and sakian, S.T. Belyaev, A.I. Malakhov, and N.A. Rus- 12C(22.4 GeV) collisions with Au, respectively. Note sakovich for support and to the staff of the JINR that selection of the events with the IMFmultiplicity synchrophasotron for running the accelerator. M ≥ 2 (for the correlation measurements) results in The research was supported in part by the Russian an increase in the mean excitation energy by 0.5– Foundation for Basic Research (project no. 03-02- 0.7 MeV/nucleon [23]. 17263), by a grant from the Polish Plenipotentiary in JINR, by grant NATO PST.CLG.976861, by grant 7. CONCLUSION no. 1P03 12615 from the Polish State Committee for Scientific Research, by contract no. 06DA453 with In this work, we study the mechanism of mul- the Bundesministerium fur¨ Forschung und Technolo- tifragment emission in collisions of relativistic pro- gie, and by the US National Science Foundation. tons, 4He, and 12C with an Au target. 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Glushkov* Institute for Cosmophysical Research and Aeronomy, Siberian Division, Russian Academy of Sciences, pr. Lenina 31, Yakutsk, 677891 Russia Received May 13, 2002 Abstract—Results are presented that were obtained from an analysis of arrival directions for cosmic rays 16.9–17.2 of energy in the range E0 ≈ 10 eV that were recorded by the Yakutsk array between 1974 and 2001 at zenith angles of θ ≤ 45◦. It is shown that a considerable part of them form clusters that have small-scale cellular structure. In all probability, these showers are generated by neutral particles of an extragalactic origin. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION PCR that generate neutral particles. Recently, this More than 40 years ago, a group from Moscow hypothesis was additionally confirmed in [13, 14] at ≈ × 17 State University (MSU) discovered a knee in the energies of E0 (1.3–4) 10 eV. Several new spectra of the extensive air showers (EAS) at an experimental results supporting this opinion are given 15 below. energy of E0 ≈ 3 × 10 eV [1]. Revealing its nature would greatly contribute to solving the problem of the origin of cosmic rays having extremely high energies, 2. FEATURES UNDER INVESTIGATION up to about 1020 eV. Many experiments have so far In the present article, we consider showers of en- been performed to study this phenomenon, but there ≈ × 17 is still no an unambiguous understanding of it. ergy in the range E0 (0.8–1.6) 10 eV that were detected by the Yakutsk EAS array from 1974 to 2001 It is widely believed that, at energies in the range at zenith angles of θ ≤ 45◦. We select only those × 15 ≤ 17 3 10 1063-7788/03/6607-1252$24.00 c 2003 MAIK “Nauka/Interperiodica” 17 SMALL-SCALE ANISOTROPY AND COMPOSITION OF E0 ≈ 10 eV COSMIC RAYS 1253 90 N 400 60 60 (a) Nexpt 30 30 200 Nran 0180 240 300 360 –30 –30 0 nσ (b) –60 –60 2 –90 0 Fig. 1. Individual arrival directions of 14 318 showers 17 ◦ with E0 ≈ (0.8–1.6) × 10 eV and θ ≤ 45 in galactic coordinates. –2 –60 –30 0 30 60 90 Equal areas in this chart correspond to sky regions bG, deg of equal area. The coordinates of the North Pole of ◦ Fig. 2. The distribution of 14 318 showers with E ≈ the Galaxy in right ascension are α1950 = 192.3 and 0 ◦ (0.8–1.6) × 1017 eV and θ ≤ 45◦ with respect to arrival δ1950 =27.4 . directions in terms of the galactic latitude of their arrival: In Fig. 1, one can see a small-scale anisotropy and (a)observed(Nexpt) and expected (Nran) distributions numerous compact groups and chains. Let us con- for an isotropic flux and (b) deviation of the number of sider some special features of these data. The distri- observed events√ from the number of expected ones, nσ = − bution of showers presented in Fig. 1 with respect to (Nexpt Nran)/ Nran. The shaded area corresponds to a arrival directions is displayed in Fig. 2 in terms of the 3.3σ dip. galactic latitude of their arrival (with a step of ∆bG = ◦ 1.5 ). Figure 2a shows the observed (Nexpt)andex- (=3× 6) samples by using the same method as in pected (Nran) distributions, while Fig. 2b presents the [10–14]. For each shower, we determined all “neigh- deviation of the number of observed events from the bors” spaced in arrival direction by an angular dis- number of√ expected ones in units of the standard devi- tance of d ≤ 3◦. If a circle contained n ≥ 3 showers, ation σ = Nran; nσ =(Nexpt − Nran)/σ. The values their coordinates were averaged and these averaged Nran were determined from a Monte Carlo simulation. coordinates were used in the following as new points, At first glance, the experimental data in Fig. 2 are referred to as nodes. quite consistent with an isotropic PCR flux. How- For each of the three energy ranges individually, ever, we would like to call the attention of the reader we further analyzed nodes in any of the six samples for to the shaded area in the latitude range from 4.5◦ intersections of multiplicity not less than two (under to 0◦, where there is a pronounced dip of |527 − the condition that their centers are within an angular √ ≤ ◦ 608|/ 608 ≈ 3.3σ. A similar dip, which is, however, distance of d 3 ). If such nodes were found, the more statistically significant (≈ 9.2σ), was obtained arrival-direction coordinates of all showers belong- ≈ × 17 ing to them were averaged, and this new large node at energies of E0 (1.3–4) 10 eV in [13, 14]. (referred to as a cluster) was used in the ensuing In the present article, this dip is explained by the analysis. absorption of extragalactic PCR. This issue will be discussed in what follows. In order to get a clearer idea of the structure of 3. SMALL-SCALE ANISOTROPY OF PCR thedatainFig.1,webrokedownthewholesample The superimposed chart of the positions of nodes of 14 318 showers into three energy ranges: E0 = for six samples of showers of energy in the range 16.9–17.0 17.0–17.1 17.1–17.2 17.0–17.1 10 , 10 ,and10 eV. In each of E0 =10 eV is shown in Fig. 3 in terms these, we took six independent samples that con- of supergalactic coordinates (the coordinates of the tained approximately 800 events each, but which dif- North Pole of the Supergalaxy are α = 286.2◦ and fered from one another only in that their axes traversed δ =14.1◦). The longitude of the Supergalaxy was different annular regions within the central circle of reckoned in the counterclockwise direction from the the array, all other conditions being the same. The vector pointing to the anticenter. For the purpose presence of local shower groups over the celestial of convenience, the equatorial coordinates are also sphere was checked individually for each of the 18 shown in this chart. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1254 GLUSHKOV 90 90 60 60 60 60 α = 270 α = 270 δ 30 = 30 30 30 30 δ δ = –30 δ δ = –30 α = 0 = 0 α = 0 = 0 0 60 180 240 300 360 0 60 180 240 300 360 δ = –60 δ α = 180 α = 180 = –60 –30 –30 –30 δ = 30 –30 α = 90 α = 90 –60 –60 –60 –60 –90 –90 Fig. 3. Superimposed chart of the positions of nodes Fig. 4. Superimposed chart of the positions of clusters for six independent samples of 2439showers with E0 = containing two or more nodes from three groups of show- 17.0–17.1 ◦ 16.9–17.0 17.0–17.1 17.1–17.2 10 eV and θ ≤ 45 in supergalactic coordinates. ers with E0 =10 , 10 ,and10 eV in terms of the supergalactic coordinates. The rectangle singles out the fragment presented in Fig. 5. One can see that this sample of 2439showers (in 17.0–17.1 all, there are 4861 showers in the E0 =10 eV type [19] or even something like a giant quasicrystal group) forms chains, which in turn form a cellular [20], with the bulk of matter being concentrated in its structure. A similar pattern is observed at other val- nodes. ues of the PCR energy. The positions of the clusters having different energies coincide in the majority of One of the hypotheses concerning the formation of the cellular structure of the Universe was widely cases. This can be seen from Fig. 4, which displays developed within the adiabatic theory of Zeldovich the chart of clusters from three groups corresponding 16.9–17.0 17.0–17.1 17.1–17.2 [21], where some stages of concentration of mat- to E0 =10 , 10 ,and10 eV. ter are represented in the evolution chain of the ex- This chart is a superposition of three original charts panding Universe as bright surfaces–bright curves– (one of these is shown in Fig. 3) obtained by the bright drops sequence. same methods. Here, the phenomenon in question becomes more pronounced, but the general structure One cannot rule out the possibility that the nodes of the distribution of clusters over the sky area under and clusters found above have some bearing on consideration remains unchanged. the cellular structure of the Universe. We will now estimate a typical size of the voids in Figs. 3–5; A fragment of Fig. 4 is shown in Fig. 5. Points this is of interest since such voids may additionally correspond to clusters where the number of showers, characterize the small-scale anisotropy of ultrahigh- n, in one of these three groups in energy is not less energy PCR. For this purpose, we analyze the dis- than 10; crosses represent n ≥ 10 clusters common tributions of angles between the nodes by using the to any two groups of these three; and closed circles E =1017.0–17.1 eV data in Fig. 3 as an example. ≥ 0 correspond to n 10 clusters common to all three Let us consider any two (of six independent) samples groups in energy E0. It can clearly be seen that the containing equal numbers of events (370) and find majority of clusters condense into chains separated ◦ ◦ the minimal angular distances dmin between the by voids of characteristic angular dimension 5 –10 . nodes from sample 1 with respect to the nodes from According to a Monte Carlo simulation, the probabil- sample 2. To improve the statistical accuracy, we − ity of such a result is less than 10 5. use all 15 pairs of such independent combinations. Observations reveal that the Universe contains The eventual distribution (Jexpt) is constructed as large black holes of dimension 100–130 Mpc sepa- the sum of these distributions. In order to find the rated by relatively thin (20–30 Mpc) layers, and the expected distribution for an isotropic flux (Jran), we theory confirms this. About 60 to 80% of galaxies are perform a such procedure once more (beginning from concentrated in these layers. These galaxies have a the selection of nodes), as was done in deriving the tendency to gather in extended and flattened super- data presented in Fig. 3. The random directions are clusters. The Supergalaxy (of diameter 50 to 60 Mpc) evaluated on the basis of real showers by replacing considered here is one of such superclusters. In all their measured right ascension by that determined from a Monte Carlo simulation in the angular range probability, many of the superclusters are adjacent to ◦ ◦ one another, forming a unique cellular structure of the between 0 and 360 . − Universe [15–18]. There is the opinion that it may The angular√ correlation functions kσ =(Jexpt be a structure of the three-dimensional-chessboard Jran)/σ(σ = Jran) for 2220 (= 370 × 6)eventscon- PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 17 SMALL-SCALE ANISOTROPY AND COMPOSITION OF E0 ≈ 10 eV COSMIC RAYS 1255 bSG, deg α = 270 45 30 α = 0 15 0 α = 180 –15 α 90 = 90 120 150 lSG, deg Fig. 5. Fragment singled out by the rectangle in Fig. 4 and (points) original data from Fig. 4 and (+ and •) clusters common to, respectively, any two and all three groups in energy (lSG is the longitude of shower arrival from the Supergalactic disk). σ sidered above are shown in Fig. 6. The closed circles (Jexpt – Jran)/ in Fig. 6a correspond to showers from the nodes in 8 Fig. 3, while those in Fig. 6b correspond to other (a) showers not belonging to these nodes. It can be seen that, in Fig. 6a, there is a deficit of events (of about ◦ 4 three standard deviations) for dmin ≤ 2.5 ,butthat, ◦ ◦ for dmin ≈ 5 –10 , there is a stable and statistically significant excess of events at a level of (4–5.5)σ. 0 The correlation function kσ in Fig. 6b exhibits an absolutely different behavior. At first glance, this result is very strange. The –4 presence of a dip in Fig. 6a instead of a peak at dmin ≤ 2.5◦ is especially surprising. However, a further sim- 2 (b) ulation revealed that such a pattern is quite possible 0 if the data have a structure ordered in some specific way rather than a random one. For a rough model, we –2 considered a rectangular network in equatorial coor- ◦ dinates that has a cell size of 5 . For the directions 0 5 10 15 20 of initial events, we took the quantities obtained by dmin, deg smearing the celestial coordinates of the nodes of this network according to a normal law with a standard Fig. 6. Angular correlation functions for the E0 = deviation of 2.5◦. If we treat these simulated events 1017.0–17.1 eV sample. Closed circles represent (a)show- [by using the same method as for the data in Figs. 3 ers in nodes (see Fig. 3) and (b) showers outside these nodes, while open circles represent nodes from the rect- and 6 (closed circles)], we arrive at the distribution ◦ angular network of size 5 in equatorial coordinates. represented by open circles in Fig. 6a. One can see that this distribution is similar to the actual distribution even in many of its details. If we reduce the smearing of the network nodes to 0.5◦–1◦, ◦ ◦ ◦ then distinct peaks at angles of dmin ≤ 2 , 5 ,and10 Fig. 6 are observed for other PCR energies considered will appear. Distributions similar to those presented in in the present study. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1256 GLUSHKOV σ 90 (a) (Jexpt – Jran)/ 60 60 (a) 30 30 4 α α δ = 0 δ = –30 = 270 δ = 60 = 90 0 60120 180 240 300 360 δ = 30 δ = –60 –30 –30 α = 0 0 –60 –60 –90 90 –4 (b) (b) 60 60 2 30 30 0 α α = 90 δ = 0 δ = –30 = 270 δ = 60 0 60120 180 240 300 360 δ = 30 δ = –60 –2 –30 –30 α = 0 0 5 10 15 20 d , deg –60 –60 min –90 Fig. 8. Angular correlation between the nodes of E0 = 1016.9–17.2 eV PCR and quasars from the samples in Fig. 7. Chart of the distributions in galactic coordinates: Fig. 7: (a)showersinthenodesand(b)showersoutside (a) 74 quasars with redshifts of z ≤ 0.3 [22] and (b)nodes the nodes. 16.9–17.2 ◦ of PCR with E0 =10 eV and θ ≤ 45 from the data of Yakutsk EAS array. Figure 8a exhibits a statistically significant peak ◦ (5.5σ) at dmin ≤ 1 , but there is no peak in Fig. 8b. 4. QUASARS AS A POSSIBLE SOURCES This suggests that primary particles forming nodes OF PCR and clusters might have been generated by quasars. The peak in Fig. 8a at d ≈ 5◦ can be related to the The above results give grounds to assume that min analogous peak in Fig. 6b; that is, it can be considered the ultrahigh-energy cosmic rays forming nodes are as an additional piece of evidence for the ordered in all probability of an extragalactic origin. It was character of the large-scale cellular structure of the shown in [8, 13, 14] that quasars might be one of Universe. their sources, since there is a similarity in the spatial distribution of PCR and quasars over the celestial sphere. We consider the correlation between them. 5. ESTIMATES OF THE PCR COMPOSITION In terms of the galactic coordinates, Fig. 7a shows Most likely, primary particles entering into the the chart of 74 quasars with redshifts of z ≤ 0.3 from ◦ clusters are neutral; otherwise, their motion in the the catalog [22] with declinations of δ ≥ 30 .Since, galactic magnetic fields would destroy any correlation in the catalog [22], there are virtually no data on the in direction with their production sources, and they | |≤ ◦ equatorial region of the Galaxy ( bG 30 ) because could not reveal the structure that we see in Figs. 1 of strong light absorption, all quasars and PCR nodes and 3–6. (Fig. 7b) from this area of the sky were eliminated Let us return once again to the distribution in from the ensuing analysis. Fig. 2. Figure 9displays this distribution for 7104 In Fig. 8, we present the angular correlation showers entering into the clusters (Fig. 9a)andfor functions kσ =(Jexpt − Jran)/σ, which are similar to the other 7214 showers, which do not belong to the those in Fig. 6. The measured distributions (Jexpt) clusters (Fig. 9b). There are substantial distinctions and the distributions expected for random variables between these samples. First, the measured distri- bution (histogram) in Fig. 9a differs substantially, (Jran) were evaluated as the sum of 18 (=3× 6) original distributions of minimal angular distances in terms of the χ2 criterion, from that which is ex- dmin between the nodes (from 123 to 206) of the above pected for random events (smooth curve). Indeed, one groups of showers and 74 quasars. Figures 8a and 8b has χ2 = 161 for k =80degrees of freedom, which display the results for, respectively, showers from the corresponds to a probability of P<10−5 for a ran- nodes and showers not belonging to the nodes. dom result. Second, there is a statistically significant PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 17 SMALL-SCALE ANISOTROPY AND COMPOSITION OF E0 ≈ 10 eV COSMIC RAYS 1257 N 200 Nexpt 7104 (a) 100 N 4.8σ ran 0 7214 (b) 100 0 –60 –30 0 30 60 90 bG, deg Fig. 9. Distributions similar to those in Fig. 2 for (a) showers in the clusters and (b) other showers not entering into the clusters. The numbers of showers are indicated in the corresponding panels. ◦ ◦ ≈ ≈ deficit of events in the√ range from 4.5 to 0 (shaded where N0 350, N 259 (see Fig. 9), and l is area): |259 − 350|/ 350 ≈ 4.8σ. On the other hand, the average thickness (in g/cm−2)ofthediskofthe the distribution in Fig. 9b has an absolutely different Galaxy in our observation sector. From (4), we then shape featuring no dip of the above type. It leads to the obtain value of χ2 =93at k =90and suggests an isotropic λA ≈ 3.3 l . flux of cosmic rays. The result presented in Fig. 9can be considered as In order to find the quantity l , we assume that a further piece of evidence that the PCR fraction in- neutral hydrogen plays the most important role in the side the clusters is of an extragalactic origin. It seems attenuation of PCR, because the dust contribution that the Galaxy only absorbs this radiation, and it is very small (about 1%) [23]. It can be seen from does this more strongly in the disk. The mean shift of Fig. 1 that, within the disk of the Galaxy, PCR arrives ◦ ◦ ◦ the dip from the plane of the Galaxy is about −2.1 .In from the sector ∆lG ≈ 60 –180 . Over this sector, all probability, this shift is due to the fact that the Sun the path r from an observer to the outer boundary of is not strictly situated in the symmetry plane of the the Galaxy changes from about 10 to 5 kpc ( r ≈ disk, but it is slightly shifted to the Northern Hemi- 8 kpc). According to data from [23], the average hy- ◦ sphere with respect to the plane of the Galaxy. If this drogen concentration in the latitude band |bG|≤10 is so, the PCR flux coming from southern latitudes at decreases from ρ ≈ 0.5 cm−3 (at the distance of R = small angles with respect to disk must undergo much 10 kpc from the center of the Galaxy) to ρ ≈ 0.1 cm−3 stronger absorption in relation to the analogous flux − at R =15kpc ( ρ ≈0.25 cm 3). On this basis, we coming at the same angles from northern latitudes. obtain This assumption is consistent with the distribution −2 −2 of neutral hydrogen in the disk of the Galaxy, where, l ≈ r ρ mp ≈ 10 gcm , according to data from [23], there is a −1.4◦ shift of × −24 the plane of the highest hydrogen concentration in where mp =1.67 10 g is the proton mass. Fi- relation to the commonly used position. nally, we arrive at Let us assume that the dip in Fig. 9a is associated −2 −2 λA ≈ 3.3 × 10 gcm . with a relatively stronger absorption of extragalactic PCR within the disk of the Galaxy in relation to that The value derived for the path of mysterious A at higher latitudes. In this case, we can roughly esti- particles appeared to be about 1000 times shorter mate the absorption range of unknown extragalactic than that for the nuclear interactions of E ∼ 1017 eV A 0 particles (for the sake of brevity, we refer to them as protons. Nevertheless, these particles can arrive with- particles) on the basis of the relation out losses almost from the horizon of the observed N = N0exp(− l /λA), (4) Universe. Considering that the age of the expanding PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1258 GLUSHKOV Universe is t ≈ 13 × 109 years and that the average The results presented in this article and in [24] −30 −3 matter density is ρ0 ≈ 10 gcm , we obtain could be explained if one assumes that PCR consists ≈ × −2 −2 of two components. One is likely to contain charged l0 = ρ0ct 1.23 10 gcm , particles, which are strongly mixed by the magnetic where c is the speed of light. Using Eq. (4), we find field of the Galaxy, while the other, which produces clusters, most probably features neutral particles of that the attenuation factor is exp(− l0 /λA) ≈ 0.7. extragalactic origin. In the Earth’s atmosphere, A particles begin to in- teract at an altitude hA above sea level. This quantity can be found from the barometric equation 6. CONCLUSION λ = p exp(−h /h ), (5) The data presented above indicate that some A 0 A 0 17 fraction of cosmic rays of energy E0 ≈ 10 eV are −2 where p0 = 1020 gcm is the air pressure at sea grouped into numerous nodes and clusters within ◦ level and h0 =6.85 km. It follows that hA ≈ 71 km. solid angles with d ≤ 3 . They form a cellular struc- These particles initiate the development of EAS ture (see Fig. 3–5) having a characteristic angular much earlier than particles belonging to the ordinary scale of about 5◦–10◦ (Fig. 6). This structure has a composition of cosmic rays. If we take, by way of correlation with quasars characterized by a redshift of 17 ≤ example, protons of energy E0 ∼ 10 eV, their mean z 0.3 (Fig. 8) and a negative correlation with the path in the Earth’s atmosphere to the first event of nu- disk of the Galaxy (see Fig. 9). It can be conjectured clear interaction is about 50 g cm−2. The analogous that the nodes and clusters are an indication of some path of iron nuclei is shorter by a factor of about 2 to 3. extragalactic pointlike sources of PCR and that these sources are associated with the large-scale structure It is likely that, after the first interaction, these of the Universe. mysterious A particles disappear, giving way, in the Primary particles forming the clusters are most development of EAS, to the ordinary cascade of likely neutral. Otherwise, they could not correlate in secondary particles. Otherwise, showers initiated by direction with their sources because of the motion these mysterious particles would be substantially dif- in the magnetic fields of the Galaxy and exhibit the ferent from ordinary ones and could easily be detected. cellular structure of the matter distribution in the Because of so short a range of A particles before the Universe. first nuclear interaction, showers initiated by them In all probability, these particles have a very short must lead to a very fast development of EAS, with range with respect to nuclear interaction (λA ≈ 3.3 × a higher maximum of a cascade curve in relation to −2 −2 that for primary protons. In view of this, showers from 10 gcm ) and initiate an earlier development A particles could be misidentified as showers from of EAS in relation to what occurs in the case of iron nuclei, especially as the majority of the methods the usual composition of PCR. Their fraction in the ≈ 17 for determining the composition of PCR are indirect: whole flux of cosmic rays at E0 10 eV is about ≈ they are based on a comparison of the observed 7104/14 318 0.5. This is severalfold greater than × features of EAS with the results of calculations that that in the adjacent energy range E0 =(1.3–4) rely on model assumptions concerning the develop- 1017 eV [13, 14]. ment of EAS for one presumed composition of PCR So fast an increase in the fraction of neutral parti- 17 or another. Therefore, one cannot rule out here the cles at E0 ≈ 10 eV could be erroneously attributed situation where the appearance of any new primary to an increase in the content of heavy particles in particles having a very short range is interpreted in PCR. We cannot rule out the possibility that such terms of a noticeable increase in the fraction of heavy hypothetical unknown particles manifest themselves nuclei. 17 for E0 < 10 eV as well. Further investigations are According to data from the Yakutsk EAS ar- required for clarifying this point. ≈ ray [24], the global flux of PCR of energy E0 Showers that do not enter into the composition of 1017 eV does not exhibit any statistically significant the nodes and clusters are distributed almost isotrop- anisotropy. From these data, it follows that the result ically over the celestial sphere. This PCR fraction is obtained for the amplitude of the first harmonic in likely to consist of charged particles (protons and right ascension by the conventional method of har- nuclei of various chemical elements) propagating monic analysis is (0.45 ± 0.55)%. However, special through the Galaxy via diffusion. One of the intense attention should be given to the phase of the first sources occurs in the vicinity of the center of the ◦ ◦ harmonic, ϕ1 = 192 ± 70 . Albeit involving a large Galaxy [25–27]. It is very difficult to observe other uncertainty associated with the isotropy of PCR, it sources of charged particles at present, because their provides an indication of the center of the Super- positions are strongly “smeared” by the magnetic field galaxy. of the Galaxy. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 17 SMALL-SCALE ANISOTROPY AND COMPOSITION OF E0 ≈ 10 eV COSMIC RAYS 1259 ACKNOWLEDGMENTS 14. A. V.Glushkov and M. I. Pravdin, Pis’ma Astron. Zh. 28, 341 (2002) [Astron. Lett. 28, 296 (2002)]. The financial support extended by the Ministry for Science of the Russian Federation to the Yakutsk 15. B. A. Vorontsov-Vel’yaminov, I. D. Karachentsev, multipurpose EAS array (grant no. 01-30) is grate- E. Turner, et al.,inThe Large-Scale Structure of the fully acknowledged. This work was also supported by Universe, Ed. by M. Longeir and J. Einasto (Reidel, the Russian Foundation for Basic Research (project Dordrecht, 1978; Mir, Moscow, 1981). no. 00-15-96787). 16. M. Seldner, B. 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Astrophys. 268, 726 63, 1 (1987). (1993). 23. I. Oort, in Handbuch der Physik. Bd. 53. Astro- 6. A. V. Glushkov, Pis’ma Zh. Eksp.´ Teor. Fiz. 48, 513 physikIV: Sternesysteme (West Berlin, 1959; In- (1988) [JETP Lett. 48, 555 (1988)]. ostr. Lit., Moscow, 1962). 7. A. V.Glushkov, Priroda 5, 111 (1989). 24. M. I. Pravdin, A. A. Ivanov, A. D. Krasil’nikov, et al., 8. A. V. Glushkov, Pis’ma Zh. Eksp.´ Teor. Fiz. 73, 355 Zh. Eksp.´ Teor. Fiz. 119, 881 (2001) [JETP 92, 766 (2001) [JETP Lett. 73, 313 (2001)]. (2001)]. 9. A.V.GlushkovandI.E.Sleptsov,Izv.Akad.Nauk, 25. N. Hayashida, M. Nagano, D. Nishikawa, et al.,As- Ser. Fiz. 65, 437 (2001). tropart. Phys. 10, 303 (1999). 10. A. V.Glushkov and M. I. Pravdin, Zh. Eksp.´ Teor. Fiz. 119, 1029(2001) [JETP 92, 887 (2001)]. 26. M. Teshima, M. Shikawa, M. Fukushima, et al., 11. A. V.Glushkov and M. I. Pravdin, Pis’ma Astron. Zh. in Proceedings of the 27th ICRC, Hamburg, 2001, 27, 577 (2001) [Astron. Lett. 27, 493 (2001)]. p. 337. 12.A.V.GlushkovandM.I.Pravdin,Yad.Fiz.66, 886 27. J. A. Bellido, B. W. Clay, R. B. Dawson, and (2003) [Phys. At. Nucl. 66, 854 (2003)]. M. Johnston-Hollitt, astro-ph/0009039. 13. A. V. Glushkov, Pis’ma Zh. Eksp.´ Teor. Fiz. 75,3 (2002) [JETP Lett. 75, 1 (2002)]. Translated by S. Slabospitsky PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1260–1268. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1300–1308. Original Russian Text Copyright c 2003 by Povarov, Smirnov. ELEMENTARY PARTICLES AND FIELDS Theory Contributions of Scalar Leptoquarks to the Cross Sections for the Production of Quark–AntiquarkPairs in Electron –Positron Annihilation A. V. Povarov* andA.D.Smirnov** Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000 Russia Received May 13, 2002; in final form, November 5, 2002 Abstract—The contributions of scalar-leptoquark doublets to the cross sections σQQ˜ for the production of quark–antiquark pairs in electron–positron annihilation are calculated within the minimal model based on the four-color symmetry of quarks and leptons. These contributions are analyzed versus the scalar- leptoquark masses and the mixing parameters of the model at colliding-particle energies in the range 250–1000 GeV. It is shown that the contributions in question are of greatest importance for processes leading to t-quark production. In particular, it is found that, with allowance for the contribution of the scalar leptoquark of charge 5/3 and mass in the range 250–500 GeV, the cross section σtt˜ calculated at ∼ (SM) a mixing-parameter value of kt 1 may be severalfold larger than the corresponding cross section σtt˜ within the Standard Model. The possibility of setting constraints on the scalar-leptoquark masses and on the mixing parameters by measuring such contributions at future electron–positron colliders is indicated. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION 1 TeV or below). We note that, because of their Higgs origin, the doublets of scalar leptoquarks and scalar Searches for possible versions of new physics be- gluons interact with fermions in a particular way: yond the Standard Model form one of the lines of their fermionic coupling constants are proportional to investigations into elementary-particle physics. Such the ratios of fermion masses to the Standard Model versions include that which is based on a four-color expectation value. It follows that, within the approach symmetry of quarks and leptons [1] that belongs to the being considered, the characteristic values of these Pati–Salam type. This symmetry predicts vector lep- constants for specific processes are known (apart toquarks of mass about the scale of breakdown of this from mixing parameters), which makes it possible symmetry; it also admits the existence of scalar lepto- to obtain quantitative estimates for the contribution quarks [2–4] and of some other particles. If mass gen- of these particles to processes involving quarks and eration and the splitting of quark and lepton masses leptons. are due to the Higgs mechanism [5–7], the four-color Within the minimal model involving the four-color symmetry of quarks and leptons predicts, in addition symmetry of quarks and leptons [5–7], we calcu- to vector leptoquarks, the existence of the doublets late, in the present study, the contributions from the of scalar leptoquarks and the doublets of scalar glu- doublets of scalar leptoquarks to the production of ons. In the approach being considered, these par- quark–antiquark pairs in electron–positron annihi- ticles are responsible for the splitting of quark and − lation, e+e → Q Q˜ ; also, we give a quantita- lepton masses via the Higgs mechanism and appear ia ja to be some kind of partner of the standard Higgs tive analysis of these contributions, contrasting them doublet. Estimates of the masses of these particles— against the corresponding Standard Model cross sec- tions at energies of about 1 TeV and below. in particular, estimates obtained from the parameters S, T ,andU of radiative corrections [8–10]—reveal that, in contrast to vector leptoquarks, these parti- 2. SCALAR-LEPTOQUARK INTERACTION cles may be relatively light (of mass about 1 TeV or WITH CHARGED LEPTONS below) and, hence, lead to various manifestations of IN THE MINIMAL MODEL INVOLVING four-color symmetry at relatively low energies (about QUARK–LEPTON SYMMETRY *e-mail: [email protected] We use here the minimal model involving quark– ** e-mail: [email protected] lepton symmetry [5, 6]. It is based on the SUV (4) × 1063-7788/03/6607-1260$24.00 c 2003 MAIK “Nauka/Interperiodica” CONTRIBUTIONS OF SCALAR LEPTOPQUARKS 1261 × SUL(2) × UR(1) group and is constructed by mini- handed fields. With respect to the SUL(2) UR(1) mally extending the Standard Model in such a way group, left-handed fermions are doublets of zero hy- as to include the four-color symmetry of quarks percharge, while right-handed fermions are singlets and leptons. In this model, the left-handed (L)and of hypercharge YR = ±1 for up (a =1) and down L,R L,R (a =2) fermions, respectively. right-handed (R)quarks,Qiaα = ψiaα ,andleptons, L,R L,R lia = ψia4 —here, a =1, 2 are the SUL(2) indices, In the gauge sector, this model predicts the exis- while α =1, 2, 3 are the SUc(3) color indices—form tence of a charged color triplet of vector leptoquarks L,R V ± and of an extra neutral Z boson. the fundamental quartets ψiaA , A =1, 2, 3, 4,ofthe α SUV (4) group in each generation of number i = In general, the scalar sector of the model contains 1, 2, 3 .... With allowance for the possible mixing (1) (2) (3) (4) L,R four multiplets Φ , Φ , Φ ,andΦ transform- of fermions, the basis quark and lepton fields Q iaα ing according to the (4,1,1), (1,2,1), (15,2,1), and L,R L,R × × and l ia , which form the basis quartets ψ iaA of (15,1,0) representations of the SUV (4) SUL(2) the SUV (4) group, can in general be represented as UR(1) group and having vacuum expectation values (1) superpositions of the corresponding mass eigenstates η1, η2, η3,andη4, respectively. The multiplets Φ and L,R L,R (4) (1) (4) Qiaα and lia of physical quarks and gluons; that is, Φ involve the scalar leptoquarks S and S of α α L,R L,R L,R electric charge 2/3, which are singlets of the SUL(2) Q iaα = AQ Qjaα, (1) a ij (3) j group, while the multiplet Φ contains two color SUL(2) doublets of scalar leptoquarks, L,R L,R L,R l ia = Al lja , a ij (±) j (±) S1α Saα = . (2) L,R L,R (±) where A and A are unitary matrices that diag- S2α Qa la (±) onalize the matrices of quarks and leptons, respec- The up components of these doublets, S1α ,have tively. electric charges of 5/3 and 1/3, while their down Transformations of the basis fields in (1) under the (±) ± components S2α have electric charges of 2/3,re- SU (4) group have a vectorial character; that is, V spectively. From an analysis of the scalar sector of L,R → L,R L,R the model, it follows (see, for example, [9]) that, in ψiaA ψ iaA = UABψ iaB , ∗ (+) (−) (1) B general, the scalar leptoquarks S2α , S2α , Sα ,and (4) where U is the 4 × 4 matrix of gauge SUV (4) trans- Sα of electric charge 2/3 are superpositions of the formations, which is common to the left- and right- Goldstone mode * − (+) ( ) 2 η1 (1) 2 S2 +S2 (4) η1 2 2 2 S0 = − S + η3 √ + η4S + (η3 + η4 ) (3) 2 3 2 4 3 and three physical scalar leptoquarks S , S ,andS , (+) *(−) 1 2 3 to the coefficients of the fields S , S , S(1),and which are orthogonal to it; that is, 2 2 S(4) in Eq. (3), while its remaining elements must 3 3 (+) (+) *(−) (−) only satisfy constraints imposed by the unitarity of S2 = Ck Sk, S2 = Ck Sk, (4) this matrix. k=0 k=0 In the unitary gauge, two triplets of the up lep- (+) (−) 3 3 toquarks S1α and S1α of electric charge 5/3 and (1) (1) (4) (4) S = Ck Sk,S = Ck Sk, 1/3 and, in the general case of arbitrary mixing, k=0 k=0 three scalar leptoquarks Skα (k =1, 2, 3)ofelectric (±) (1) (4) charge 2/3 appear as physical fields upon the elimi- where Ck , Ck ,andCk are complex numbers × nation of Goldstone modes. forming a 4 4 unitary matrix that describes the In a general form, the interactions of scalar dou- mixing of the scalar leptoquarks of electric charge (±) (1) (4) blets with fermions in the minimal model involv- 2/3. The elements C0 , C0 ,andC0 of this matrix, ing quark–lepton symmetry are presented in [7]. which correspond to the Goldstone mode, are equal In order to perform calculations for processes like PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1262 POVAROV, SMIRNOV L,R L,R (a) mixing matrices A and A in Eqs. (1); C = Qa la Q e(p1) Qja(–q2) (AL )+AL is the Cabibbo–Kobayashi–Maskawa Q1 Q2 γ, Z matrix, which is known to be due to the distinction between the mixing matrices AL and AL in (1) for Q1 Q2 – – e(–p2) Q (–q ) up and down left-handed quarks; C =(AL )+AL is ia 1 l l1 l2 (b)(c) the matrix that is analogous to the preceding one in – – the lepton sector and which is due to the possible e(p ) u (q ) e(p ) d (q ) 1 i 1 1 i 1 distinction between the mixing matrices AL and AL l1 l2 S(+) (p – q ) S (p – q ) in (1) for up and down left-handed leptons; β is the 1 1 1 k 1 1 angle of mixing of two color-singlet scalar doublets – – available in the model, their expectation values being e(–p2) uj(–q2) e(–p2) dj(–q2) 2 2 η√2 and η3 (tan β = η3/η2); and η = η2 + η3 = −1/2 Fig. 1. Diagrams describing the production of quark– ( 2GF) ≈ 246 GeV is the Standard Model ex- antiquark pairs in electron–positron annihilation with al- pectation value. We note that, in general, there are five + − ˜ lowance for scalar leptoquarks, e e → QiaQja. L,R nonidentity matrices, Ka , a =1, 2,andCl, and this leads to [6] an additional mixing of quarks and leptons + − e e → QiaQ˜ja, we need the interactions of the in their interaction with vector leptoquarks and to the scalar-leptoquark doublets (2) with charged leptons additional mixing of leptons in their interaction with lj2 =(e, µ, τ) and with up [Qi1 ≡ ui =(u, c, t)]and W bosons. down [Qi2 ≡ di =(d, s, b)]quarks(i =1, 2, 3 are the indices of generations). These interactions can be represented as 3. CROSS SECTIONS FOR THE PROCESS + − e e → QiaQ˜ja OF THE PRODUCTION ¯ S P (+) L (+) = Qi1α (g )ij +(g )ijγ5 lj2S1α + h.c., OF QUARK–ANTIQUARK PAIRS Q1l2S1 (5) WITH ALLOWANCE FOR SCALAR LEPTOQUARKS ¯ S P LQ2l2Sk = Qi2α (gk )ij +(gk )ijγ5 lj2Skα + h.c. The production of a quark–antiquark pair in (6) electron–positron annihilation with allowance for the scalar leptoquarks (2) and (4) is described by the dia- (hereafter, a tilde symbol above a fermion denotes its grams in Fig. 1. The diagram in Fig. 1a corresponds antiparticle, while an overbar denotes the Dirac con- to the contribution of a photon and a Z boson in the jugation of its wave function). In the minimal model Standard Model to the process under consideration featuring quark–lepton symmetry, the matrices that (the standard Higgs boson may also contribute within are formed by the scalar and pseudoscalar coupling the Standard Model, but we disregard this contri- constants and which are involved in Eqs. (5) and (6) bution here since the constant of its coupling to the are given by electron is small, m /η ∼ 10−6), while the diagrams e in Figs. 1b and 1c describe the contributions to the (gS,P ) = 3/2 ∓ (C M KR) (7) ij Q 2 2 ij production of up and down quarks from the scalar + (+) L leptoquarks S1 and Sk (k =1, 2, 3), respectively. +(M1K1 Cl)ij /(2η sin β), S,P ∓ R (+) + − → (gk )ij = 3/2 (M2K2 )ijCk (8) 3.1. Cross Section for the Process e e uiu˜j + − √ + − L ( ) The cross section for the process e e → uiu˜j of − (M2K )ijC /( 2η sin β), 2 k theproductionofupquarksisdescribedbythesumof + the diagrams in Fig. 1a and 1b and can be represented − L R where Ma = MQa Ka MlaKa ,with(MQa )ij = in the form m δ and (M ) = m δ being the diagonal − Qia ij la ij lia ij σ(e+e → u u˜ ) ≡ σ (9) mass matrices of the up (a =1) and down (a = i j uiu˜j L,R L,R + L,R 2)quarksandleptons;Ka =(A ) A are (+) (+) Qa la (SM) XnS1 S1 = δij σ + σ + σ , unitary mixing matrices that are due to the pos- uiu˜i uiu˜i uiu˜j sible distinctions between the quarks and lepton n=1,2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CONTRIBUTIONS OF SCALAR LEPTOPQUARKS 1263 (SM) γ γ where σ is the cross section for this process in mγ =0, v = Qf ,anda =0. The functions uiu˜i fa a fa (+) XnS1 f±(s, m, m3) appearing in (12) are given by the Standard Model; σu u˜ is the contribution that i i 2 is due to the interference between the contributions of m3 w+(s, m, m3,m3) f+(s, m, m3)= ln , (15) the Xn boson, Xn =(γ,Z), n =1, 2,andthescalar s w−(s, m, m3,m3) S(+) leptoquark S(+);andσ 1 is the scalar-leptoquark 1 uiu˜j 2 − 2 − (+) 2m 2m3 s v(s, m3,m3) (S ) contribution associated with the diagram in f−(s, m, m3)= (16) 1 s 2 Fig. 1b. 2 − 2 2 (SM) (m m3) w+(s, m, m3,m3) In the Standard Model, the cross section σu u˜ + 2 ln , i i s w−(s, m, m3,m3) has the form 2 2 where (SM) 4πα 2mui σ = v(s, mui ,mu˜i ) 1+ (10) ± uiu˜i s s w±(s, m, m3,m4)= sv(s, m3,m4) + m2 + m2 − s − 2m2. s2 3 4 × Q2 + (vZ )2 +(aZ )2 (vZ )2 ui − 2 2 e e ui Similarly, the calculation of the contributions (s mZ) S(+) σ 1 yields 2Q s s2 uiu˜j − ui vZ vZ + v2(s, m ,m ) − 2 e ui ui u˜i − 2 2 (1) (1) ∗ s m (s m ) (+) Ncg (g ) Z Z S1 i j σ = f (s, m (+) ,m ,m ), (17) uiu˜j 5/3 ui u˜j 16πs S1 × (vZ )2 +(aZ )2 (aZ )2 , e e ui where 2 − 2 − 2 2m m3 m4 where f5/3(s, m, m3,m4)= (18) s 2(m2 + m2) (m2 − m2)2 − 3 4 3 4 w+(s, m, m3,m4) v(s, m3,m4)= 1 + 2 ; × ln s s w−(s, m, m ,m ) (11) 3 4 4 2 2 − 2 2 − 2 2m +2m3m4 (2m3 +2m4 s)m − 2 + (τ3)aa 4Qfa s (τ3)aa 4 2 2 − 2 2 − 2 Z W Z m + m3m4 (m3 + m4 s)m vfa = ,afa = 4sWcW 4sWcW × v(s, m3,m4). are the vector and axial-vector fermionic coupling Formulas (9), (10), (12), and (17) determine the cross constants of the Z boson in the Standard Model; + − → s =sinθ and c =cosθ , θ being the Wein- section for the process e e uiu˜j of the produc- W W W W W tion of up quarks in electron–positron annihilation berg angle; τ3 is the third Pauli matrix; and Qfa is the fermion electric charge expressed in units of the with allowance for the contributions of the scalar lep- (+) proton charge. toquark S1 . Upon taking into account Eq. (7), the X S(+) constants that describe electron coupling to quarks of For the interference term σ n 1 , our calculations uiu˜i the ith generation and which appear in (13) take the yield form (+) 3 XnS1 Ncα (m) σ = − g (12) S,P ∓ uiu˜i 2(s − m2 ) i (g )i1 = 3/2 (CQ)ii (19) Xn ± m=1,2 i =1 (m)Xn × V ± f±(s, m (+) ,m ), R L ui S ui × m (K ) − (K ) m 1 di 2 i 1 2 i 1 e where + L (1) S 2 P 2 R − ≡| | | | + (K1 )ii mν mui (K1 )ii (Cl)i 1 /(2η sin β), gi (g )i1 + (g )i1 , (13) i g(2) ≡ (gS ) (gP )∗ +(gP ) (gS )∗ , i i1 i1 i1 i1 where me is the electron mass, while mui , mdi ,and mνi are the masses of, respectively, the up quark, the V (1)Xn = vXn vXn ± aXn aXn , (14) down quark, and the neutrino of the ith generation. Qia± Qia e Qia e In the particular case where there is no additional (2)Xn Xn Xn Xn Xn V ± = ±a v + v a , L R Qia Qia e Qia e mixing in the fermion sector—that is, K = K = PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1264 POVAROV, SMIRNOV C = I—the coupling constants in (19) can be sim- 3 l (SM) XnSk S = δij σ + σ + σ ˜ . plified to become did˜i did˜i didj n=1,2 k=1 S,P (g )i1 = 3/2[∓(CQ)i1(md − me) (20) − + δi1(mνe mu)] /(2η sin β). It should be noted that, in the limit of mass- The Standard Model cross section σ(SM) and the less quarks, the functions in (11) and (15) reduce to did˜i v(s, 0, 0) = 1 and f+(s, m, 0) = 0, respectively, while XnSk interference terms σ ˜ appearing in (23) can be the functions in (16) and (18) take a much simpler didi form, obtained from formulas (10) and (12) by means of (+) (m) 2 x → → → f−(s, m, 0) ≡ f (x)=x − 1/2+x ln , (21) the substitutions ui di, S1 Sk,andgi 1 1+x (g(m)) . For the contribution σS from the diagram i kk d d˜ x 1+2x i j f (s, m, 0, 0) ≡ f (x)=2x ln + , in Fig. 1c, the calculations yield 5/3 2 1+x 1+x (22) 3 Nc (1) (1) ∗ 2 S where x = m /s. σ ˜ = (gi )kl(gj )kl (24) didj 16πs k,l=1 × + − f2/3(s, mSk ,mSl ; mdi ,md˜ ), 3.2. Cross Section for the Process e e → did˜j j + − → ˜ The process e e didj of the production of where down quarks with allowance for scalar leptoquarks is ∗ ∗ described by the sum of the diagrams in Figs. 1a and (g(1)) =(gS) (gS ) +(gP ) (gP ) , (25) 1c. Similarly to (9), the cross section for this process i kl k i1 l i1 k i1 l i1 can be represented in the form (2) S P ∗ P S ∗ (gi )kl =(gk )i1(gl )i1 +(gk )i1(gl )i1, − σ(e+e → d d˜ ) ≡ σ (23) i j did˜j m4 + m4 +2m2m2 − 2(m2 + m2)(m2 + m2) f (s, m ,m ; m ,m )= 1 2 3 4 3 4 1 2 (26) 2/3 1 2 3 4 2s(m2 − m2) 1 2 2 2 − 2 − 2 w+(s, m1,m3,m4)w−(s, m2,m3,m4) m1 + m2 m3 m4 × ln + v(s, m3,m4)+ w−(s, m1,m3,m4)w+(s, m2,m3,m4) 2s (w (s, m ,m ,m ) − m2 + m2)2 − (m2 − m2)2 × ln + 2 3 4 1 2 1 2 . − 2 2 2 − 2 − 2 2 (w−(s, m2,m3,m4) m1 + m2) (m1 m2) √ With allowance for formula (8), the constants de- × ∓ (+) − (−) Ck Ck /( 2η sin β). scribing scalar-leptoquark (Sk) coupling to the elec- tron and the down quark and appearing in (25) as- sume the form At equal masses of the scalar leptoquarks, the S,P function in (26) coincides with that in (18), (gk )i1 = 3/2 (27) f2/3(s, m, m; m3,m4)=f5/3(s, m, m3,m4), while, × ∓ R − L (+) (mdi (K2 )i1 me(K2 )i1)Ck in the limit of massless quarks, it takes the form √ − f (s, m ,m ;0, 0) ≡ f (x ,x ) (29) − L − R ( ) 2/3 Sk Sl 3 k l (mdi (K2 )i1 me(K2 )i1)Ck /( 2η sin β). xl xl In the particular case where there is no additional = 1+xl ln xl − xk 1+xl mixing in the fermion sector—that is, KL = KR = x x I—the nonvanishing (at i =1) constant appearing in − k 1+x ln k , x − x k 1+x (27) has the form l k k S,P − where x = m2 /s. At equal masses of the scalar (gk )11 = 3/2(md me) (28) k Sk PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CONTRIBUTIONS OF SCALAR LEPTOPQUARKS 1265 σ ~ σ ~ tt , pb tt, pb 4 4 3 3 3 1 2 2 4 2 1 1 2 1 3 0 0 400 600 800 1000 400 600 800 1000 s, GeV s, GeV + − → ˜ + − Fig. 2. Cross section√ for the process e e tt as a Fig. 3. Cross section for√ the process e e → tt˜ as function of the energy s at the scalar-leptoquark mass afunctionoftheenergy s at the mixing-parameter of m (+) = 300 GeV for the mixing-parameter values of value of kt =1for the scalar-leptoquark-mass values of S1 m (+) = (1) 250, (2) 500, (3) 700, and (4) 1000 GeV. kt = (1) 0 (Standard Model), (2)0.5,and(3)1.0. S1 The dotted curve represents the corresponding result in the Standard Model. leptoquarks, the function in (29) coincides with the function f2(xk) appearing in (22). suppressed by the additional element (CQ)i3 of the 4. ANALYSIS Cabibbo–Kobayashi–Maskawa matrix], we find that OF THE SCALAR-LEPTOQUARK the coupling constants for processes involving c and t quarks have the form CONTRIBUTIONS TO THE CROSS + − → ˜ SECTIONS FOR e e QiaQja PROCESSES 1 3 m (gS,P ) ≈− ui k , (30) The constants in (19) and (27), which describe i1 2 2 η ui scalar-leptoquark coupling to the quarks and the electron and which lead to the processes being where, by kui [i =2, 3,ui =(c, t)] we have denoted considered, are determined by the ratios of the quark, the effective mixing parameter for the c or the t quark, electron, and neutrino masses to the Standard Model | L |2 K1 Cl i1 vacuum expectation value; the Cabibbo–Kobayashi– kui = 2 . (31) L sin β Maskawa matrix CQ; arbitrary unitary matrices Ka , R From (30), we obtain the coupling constants Ka ,andCl describing additional fermion mixing; and the mixing angle β in the scalar sector. We note S,P −2 (g )21 ≈−0.3 × 10 kc, (32) that, in the absence of additional fermion mixing S,P [see formulas (20) and (28)], the above coupling (g )31 ≈−0.43 kt constants prove to be on the order of the ratio of for the c and the t quark, respectively. One can see the masses of the first-generation quarks to the that, in the case of identical assumptions on the mix- Standard Model vacuum expectation value—that is, −5 on the order of mu/η ∼ md/η ∼ 10 ; in this case, σ ~ the contributions of scalar leptoquarks to the process tc , pb being studied are negligible. We note that the latest 3×10–4 data on neutrino oscillations seem to suggest [11] that the lepton-mixing matrix C is nondiagonal— l 2×10–4 that is, an additional fermion mixing is possible, in particular, in the lepton sector. In the presence of 1 × –4 an arbitrary fermion mixing, the coupling constants 1 10 2 in (19) and (27) will be determined predominantly by 3 the masses of the heavy t, c,andb quarks and by the 0 mixing parameters. 200 400 600 800 1000 s, GeV 4.1. Cross Sections for the Processes − + → ˜ + − e e tt, tc,˜ cc˜ of the Production of Heavy Up Fig. 4. Cross section for√ the process e e → tc˜ as a Quarks function of the energy s at kckt =1 for the scalar- leptoquark-mass values of m (+) = (1) 250, (2) 500, and Retaining, in (19), the contributions of the c and S1 t quarks [the possible contribution of the b quark is (3) 1000 GeV. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1266 POVAROV, SMIRNOV √ (S,P ) ing parameters, the coupling constants (g )31 for over the entire range of s, and this leads to a sig- scalar-leptoquark interaction with the electron and (SM) nificant excess of σ ˜ over σ (by a factor greater the t quark appear to be dominant. tt √ tt˜ than five at k =1.0 and s ∼ 600 GeV). Taking into account expression (30) and using t relations (9), (10), (12), and (17), we obtain the cross In Fig. 3,√ the cross section σtt˜ as a function of ˜ the energy s is displayed at kt =1 for m (+) = section for the production of a tt pair in the form S1 (+) (+) 250–1000 GeV. From this figure, it can be seen that, (SM) XnS1 S1 σtt˜ = σ + σ + σ , (33) if the scalar-leptoquark mass takes values in the tt˜ tt˜ tt˜ ∼ n range m (+) 250–500 GeV, the cross section σtt˜ S1 (at kt =1) can be severalfold larger than the cross (+) − 2 XnS1 3Nc mt α (SM) σ = kt (34) section σ , the characteristic value of the cross tt˜ 8 η (s − m2 ) tt˜ Xn section σtt˜ being a few picobarns. We note that mea- × (m)Xn Vt± f±(s, m (+) ,mt), surement of the cross section σtt˜ at electron–positron S1 ± m=1,2 colliders of the TESLA type [13] could efficiently set constraints on the scalar-leptoquark mass and on the (+) S1 9Nc mt 4 2 1 mixing parameter of the model. σ ˜ = ( ) kt f5/3(s, m (+) ,mt,mt), tt 256π η s S1 Using Eqs. (9) and (17) and taking into account (35) the coupling constants in (30), we can obtain the cross section for the process e+e− → tc˜ in the form (m)Xn where the constants Vt± for the t quark are given 2 + − 9Nc mc by Eqs. (14), while the functions f±(s, m (+) ,m ) → ≡ S t σ(e e tc˜) σtc˜ = 1 256π η and f (s, m (+) ,m ,m ) are defined by Eqs. (15), 5/3 S t t 2 1 mt 1 (16), and (18). × kckt f5/3(s, m (+) ,mt,mc). η s S1 The cross sections given√ by Eqs. (33)–(35) as a function of the energy s are determined by two √ In Fig. 4, the cross section σtc˜ as a function of parameters, the scalar-leptoquark mass m (+) and S s is shown at kckt =1for m (+) = 250–1000 GeV. 1 S1 the mixing parameter kt. According to the estima- As can be seen from this figure, the cross section tions of the scalar-leptoquark masses on the basis of in question takes values of about 10−4 pb,whichis currently available data on the parameters S, T ,and much smaller, in particular, than the currently avail- U of radiative corrections [12], the scalar leptoquarks able upper limit from LEP II data [14]: σtc˜ < 0.55 pb. can be relatively light [10] (lighter than 1 TeV), We recall that this process cannot proceed in the a lower experimental limit on the mass of scalar Standard Model. leptoquarks from data on their direct production being An analysis reveals that the cross section for the about 250 GeV [12]. From Eqs. (31) and (32), it − process e+e → cc˜ differs only slightly from the pre- can be seen that, without disturbing the smallness of dictions of the Standard Model. By way of example, the perturbation-theory parameter (gS,P )2 /4π,the 31 we indicate that, at k ∼ 1 and m (+) = 250 GeV,the c S mixing parameter kt can vary from zero to values of 1 −6 about unity or even somewhat greater. In the ensuing scalar-leptoquark√ contribution is about 10 pb in analysis, we restrict ourselves to values of the scalar- the range s =180–500 GeV. We also note that the leptoquark masses in the range 250–1000 GeV scalar-leptoquark contribution to processes leading and values of the mixing parameter kt in the range to the production of tu˜, cu˜,anduu˜ pairs involving 0.0–1.0. a light u quark is negligible (10−8 pb) because of Figure 2 shows√ the cross section σtt˜ as a func- the smallness of the constants of scalar-leptoquark tion of the energy s at m (+) = 300 GeV for kt = coupling to the electron and the u quark. S1 0.0–1.0.Forkt ≤ 0.5, the destructive-interference term is operative,√ which, in particular, leads, at en- 4.2. Cross Section for the Production of a b˜b Pair ≤ − ergies of s 500 GeV, to a reduction of the cross in the Process e+e → b˜b section below that in the Standard Model, σ(SM);at √ tt˜ Disregarding the electron mass against the mas- energies of s ≥ 500 GeV,however, the cross section ses of down quarks and using Eq. (27), we approxi- σ already exceeds σ(SM) considerably. At k > 0.5, mately obtain tt˜ tt˜ t the purely leptoquark contribution σS is dominant S,P ≈ tt˜ (gk )i1 3/2mdi PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CONTRIBUTIONS OF SCALAR LEPTOPQUARKS 1267 − √ × ∓ (KR) C(+) − (KL) C( ) /( 2η sin β). Thus, our analysis has revealed that the contri- 2 i1 k 2 i1 k bution of scalar leptoquarks to the cross sections for the processes considered here can be significant. In The coupling constants in (25), which appear in the particular, this is so for the cross sections σtt˜ and σtc˜ cross section for the production of down quarks, can for processes involving the production of a t quark— be represented in the form as can be seen from Figs. 2 and 3, the cross section σtt˜ 3 m 2 for the production of a tt˜pair due to the contribution (g(m)) ≈ di (k(m)) , i kl di kl (+) 2 η of the scalar leptoquark S1 can be severalfold as (SM) where large as the corresponding cross section σ in the tt˜ * Standard Model. Measurement of such cross sec- (k(m)) = δ |(KR) |2C(+)C(+) (36) tions will become possible at facilities of the TESLA di kl m 2 i1 k l type [13], and this would yield stringent constraints − * − on the scalar-leptoquark masses and mixing param- | L |2 ( ) ( ) 2 + (K2 )i1 Ck Cl / sin β eters. is the effective mixing parameter for the production of down quarks di with δm = ±1 at m =1, 2. 5. CONCLUSION Additionally√ √ disregarding the b-quark mass On the basis of the minimal model featuring the against s ( s>>mb), we can represent the cross four-color symmetry of quarks and leptons, we have section (23) for the production of a b˜b pair in the form calculated the contribution of the scalar-leptoquark doublets to the cross section σQ Q˜ for processes + − → ˜ ≡ (SM) XS S ia ja σ(e e bb) σ ˜ = σ + σ ˜ + σ ˜, (37) + − bb b˜b bb bb of the e e → QiaQ˜ja type. We have analyzed these where cross sections versus the scalar-leptoquark masses 2 and fermion-mixing parameters. This analysis has XS −3Ncα mb been performed for energies of a colliding electron– σ ˜ = (38) √ bb 4 η positron pair in the range s = 250–1000 GeV. 3 (m)Xn It has been shown that, owing to special features V − × b (k(m)) f (x ), of the interaction of the scalar-leptoquark doublets (s − m2 ) b kk 1 k n=1,2 m=1,2 k=1 Xn with fermions (because of the Higgs origin of these doublets, the respective coupling constants are pro- 3 9N m 4 1 portional to the ratios of the fermion masses to the σS = c b |(k(1)) |2f (x ,x). Standard Model expectation value), these contribu- b˜b 64π η s b kl 3 k l k,l=1 tions are the most significant for processes leading to (39) the production of a heavy t quark. By way of exam- ple, we indicate that, at the scalar-leptoquark mass (m)Xn ∼ Here, V − is given by (14); the functions f1(xk) in the range m (+) 250–500 GeV and a mixing- b S1 ∼ and f3(xk,xl) are specified by Eqs. (21) and (29), parameter value of kt 1.0, the cross section σ√tt˜ for respectively; and x = m2 /s. A numerical analysis ˜ k Sk the production of a tt pair in the energy range s = reveals that the contributions (38) and (39) of scalar 400–1000 GeV may be severalfold larger than the leptoquarks to the cross section (37) are small. To (SM) corresponding Standard Model cross section σ ˜ . | (m) |∼ tt illustrate this statement, we set (kb )kl 1 and Measurement of the cross section σtt˜ at electron– ∼ mSk 500 GeV by√ way of example and find from (37) positron colliders of the TESLA type could set limits ∼ that σb˜b 4 pb at s = 180 GeV in this case, the on the scalar-leptoquark mass and on the mixing contribution of scalar leptoquarks, which is given by parameter. −5 Eqs. (38) and (39), being about 10 pb. This is much For the cross section σtc˜, which vanishes at the smaller than known experimental errors, for exam- tree level in the Standard Model, we have obtained −4 ± ± the estimate σ ∼ 10 pb at m (+) ∼ 250–500 GeV ple, in the experimental value of σb˜b =4.6 0.6 tc˜ S 0.3 1 pb,√ which was obtained by√ the OPAL Collabora- and kckt ∼ 1.0. tion at s = 183 GeV[15].At s = 250–1000 GeV, It has been shown that the scalar-leptoquark | (m) |∼ ∼ (kb )kl 1,andmSk 500 GeV, the contribu- contributions to the cross sections σcc˜ and σb˜b are tion of scalar leptoquarks, which is given Eqs. (38) small—in the mass and mixing-parameter region and (39), remains small (about 10−5–10−6 pb). considered here, they are below 10−5 pb. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1268 POVAROV, SMIRNOV Our results—in particular, estimates of the cross 5. A. D. Smirnov, Phys. Lett. B 346, 297 (1995). section σtt˜—may be of interest for projects of the 6. A. D. Smirnov, Yad. Fiz. 58, 2252 (1995) [Phys. At. TESLA type that are being presently discussed in the Nucl. 58, 2137 (1995)]. literature. 7. A. V. Povarov and A. D. Smirnov, Yad. Fiz. 64,78 (2001) [Phys. At. Nucl. 64, 74 (2001)]. 8. A. D. Smirnov, Phys. Lett. B 431, 119 (1998). ACKNOWLEDGMENTS 9. A. D. Smirnov, Yad. Fiz. 64, 367 (2001) [Phys. At. This work was supported in part by the program Nucl. 64, 318 (2001)]. Universities of Russia—Basic Research (project 10. A. D. Smirnov, Phys. Lett. B 531, 237 (2002); hep- no. 02.01.25) and by the Russian Foundation for ph/0202229. Basic Research (project nos. 00-02-17883, 01-02- 11. M. Fukugita and M. Tanimoto, Phys. Lett. B 515,30 06221). (2001). 12. Particle Data Group (D. E. Groom et al.), Eur. Phys. REFERENCES J. C 15, 1 (2000), and partial update for the edition of 1. J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974). 2002 (http://pdg.lbl.gov). 2. W.Buchmuller,¨ R. Ruckl,¨ and D. Wyler, Phys. Lett. B 13. R-D. Heuer et al., DESY 2001-011; hep- 191, 442 (1987). ph/0106315. 3. J. L. Hewett and T. G. Rizzo, Phys. Rev. D 36, 3367 14. DELPHI Collab., Phys. Lett. B 446, 62 (1999). (1987). 15. OPAL Collab., Eur. Phys. J. C 6, 1 (1999). 4. J. L. Hewett and T. G. Rizzo, Phys. Rev. D 56, 5709 (1997). Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1269–1281. From Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1309–1321. Original English Text Copyright c 2003 by Belozerova, Frick, Henner. ELEMENTARY PARTICLES AND FIELDS Theory Wavelet Analysis of Data in Particle Physics: Vector Mesons in e+e− Annihilation* T. S. Belozerova, P. G. Frick, and V. K. Henner** Perm State University and Institute of Continuous Media Mechanics, Perm, Russia Received February 28, 2002; in final form, November 26, 2002 Abstract—The advantages that wavelet analysis (WA) provides for resolving the structures in experimen- tal data are demonstrated. Due to good scaling properties of the wavelets, one can consider data with various resolutions, which allows the resonances to be separated from the background and from each other. The WA is much less sensitive to noise than any other analysis and allows the role of statistical errors to be substantially reduced. The WA is applied to the e+e− annihilation into hadron states with quantum numbers of ρ and ω mesons, and to p-wave ππ scattering. Distinguishing the resonance structures from an experimental noise and the background allows us to make more reliable conclusions about the ρ and ω states. The WA yields a useful set of starting conditions for analysis of ω states with the multiresonance Breit–Wigner method preserving unitarity in the case of overlapping resonances. We also apply the WA for the ratio Re+e− . c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION ωππ channels are described, as well as p-wave ππ scattering. We consider the use of wavelet transforms to re- The two isovector ρ(1450) and ρ(1700) states duce the presence of statistical noise in experimental were found many years ago. There are also some data and argue that the subsequent smoothed forms indications of the state with mass about 2.1–2.2 GeV. allow the extraction of physical parameters which Another possible ρ state has a mass in the range would otherwise be obscured. The method is com- 1.1–1.3 GeV. The situation is probably similar to the monly used in image and signal processing and is isoscalar ω states. potentially quite interesting in the context of experi- Parameters of all these resonances are not well mental particle physics. established, which makes the nature of these states The data on e+e− annihilation in different hadron not clear. They are considered as two-quark states— states in the energy range from ρ-meson mass to radial and orbital excitations of the ρ and ω mesons— about 2 GeV indicate a behavior related to several or as a mixture of two-quark, four quark, and hybrid resonances associated with states with quantum states. The situation is really complicated and the numbers of ρ and ω mesons. The properties of these poor accuracy of experimental data opens a possibil- states and even their number are not well defined. ity of drawing very different conclusions about these Major difficulties in understanding the situation with states. Therefore, an appropriate analysis of data, them are large statistical errors in the data and which allows one to extract all the important struc- overlapping of these states decaying into several final tures that can be of different scales, is useful before channels. Correspondingly, the major goal of this any model based on some physical assumptions is paper is to resolve structures in the existing data applied. relevant to this study and then to find parameters of Section 2 features a brief discussion of the wave- the excitations related to these structures using an lets, their properties, and applicability to the purposes appropriate method. Such a method has to preserve of resolving the resonance-like structures observed unitarity even when the resonances r and r with the in high-energy and nuclear physics. In Section 3, i j e+e− same quantum numbers overlap, i.e., |E − E |∼ the wavelet analysis (WA) of annihilation in ri rj different final hadron channels and of p-wave ππ scat- Γ +Γ . ri rj tering is presented. Then, in Section 4, these wavelet- The experimental data on e+e− annihilation in ππ, analyzed data are used as an input to study ω states π+π−π+π−, π+π−π0π0, ωπ0, ηπ+π−, π+π−π0,and with the Breit–Wigner method that preserves uni- tarity for overlapping multichannel resonances. The ∗ This article was submitted by the authors in English. application of the WA to “smear” the ratio Re+e− is **e-mail: [email protected] presented in the Conclusion. 1063-7788/03/6607-1269$24.00 c 2003 MAIK “Nauka/Interperiodica” 1270 BELOZEROVA et al. 2. WAVELETS AND STRUCTURE The continuous WT of a function f(t) represent- RECOGNITION ing the data is defined as +∞ Our main goal is to detect essential structures in − − ∗ t − t w(a, t)=C 1/2a 1/2 ψ f(t )dt , (2) data sets in a wide range of scales using the most ψ a suitable tool for such purposes. A convenient method −∞ for such analysis is a wavelet transformation (WT), where which is known to be an efficient multiscale technique +∞ (for example, see the books of Holschneider [1] or | −1|| ˆ |2 Torresani [2]). It is often said that the WA works like Cψ = ω ψ(ω) dω (3) a “microscope” discriminating different scales—both −∞ the characteristic scales and the positions of any local is a constant defined through the Fourier transforma- structures are obtained independently of the general tion of ψ(t): structure of the data. +∞ − Before giving a very brief introduction to wavelets, ψˆ(ω)= ψ(t)e iωtdt. (4) we would like to say that we are not the only ones −∞ performing a type of “optimization” of the data in ele- mentary particle physics. Let us point out an analogy Similar to most of the references, we call an argu- between this work and the works on optimization of ment t as “time,” even though in our actual problem the ratio Re+e− in QCD [3–5]. The data on Re+e− it is an energy variable. The decomposition (2) is have a wide range of structures and large statistical performed by convolution of the function f(t) with errors, making a direct comparison with QCD im- a biparametric family of self-similar functions gen- possible. However, a meaningful comparison can be erated by dilatation and translation of the analyzing function ψ(t): done if some kind of smearing procedure is used. The smeared ratio t − b s ψa,b(t)=ψ , (5) max a ∆ Re+e− (s ) R¯ + − (s, ∆) = ds (1) e e π (s − s)2 +∆2 where a scale parameter a characterizes the dilatation 0 and b characterizes the translation in time or space. Wavelet function ψ(t) isasortof“window function” s = E2 ∆ was introduced [3] ( ,and is a smearing with a nonconstant window width: high-frequency R + − parameter) to even out any rapid variations in e e . wavelets are narrow (due to the factor 1/a), while The difference between the described procedure low-frequency wavelets are much broader. Very im- and the analysis needed to obtain resonance parame- portant for WA is the choice of the proper analyzing ters is that, contrary to the smearing, we need to re- wavelet, which depends on the goal of the study. The solve the structures and, what is most important, we function ψ(t) should be well localized in both time and want to detect structures corresponding to different Fourier spaces and must obey the admissibility con- +∞ scales (global and local). The WA seems to be a very dition, −∞ ψ(t)dt =0. This condition implies that ψ good tool for that. We will return in the conclusion is an oscillatory (but with limited support) function to the discussion of the ratio Re+e− to point out that and, if the integrals (3) and (4) converge, provides the the WA provides a very reasonable and simple way to existence of the inverse transformation: +∞ +∞ smear this ratio. −1/2 t − t dt da Most of the applications of WA to physics are f(t)=C ψ w(a, t ) . (6) ψ a a5/2 related to spectral analysis and turbulence, where −∞ 0 scaling is its inherent feature. Over the past few years, WA has been getting more popular in different areas The WT is usually used for two kinds of problems: of physics and many applications have been described time-frequency analysis and time-scale analysis. In in reviews [6–8]. In high-energy nucleus–nucleus in- the first case, one studies, for example, some quasi- teractions, it was successfully applied for studying the periodic signals/data with time-dependent spectral angular distributions of secondary particles [9, 10]— properties. For these purposes, the wavelets with more references related to the processes of multiple good spectral resolution (having enough oscillations) production can be found in [7, 8]. In particle physics, are required. The complex Morlet wavelet is com- the WA was recently used for evaluations of a back- monly used for such purposes: − 2 ground in the π−p → ηπ0 reaction [11]. ψ(t)=e t /2eiω0t, (7) PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1271 where the proper frequency ω0 defines its spectral nonperiodic noisy data is not suitable for our goals: resolution. The admissibility condition is satisfied ex- when the number of harmonics is small enough to actly for the imaginary part of (7); for its real part, eliminate the noise, the restored signal does not re- ∞ 2 2 + −t /2 −ω0 /2 −∞ e cos(ω0t)dt = e and, if we take ω0 produce many important features of the original data. to be large enough, we can consider this average to be It means that further theoretical analysis based on the Fourier-reconstructed data (for instance, to obtain zero. A conventional choice is ω0 =2π—in this case, a signal with the period T gives a stripe on a wavelet the resonance masses and partial widths) would give plane w(a, t) just at the level a = T −1. wrong results, no matter what physical model is used for this analysis. For another kind of problem, the goal is to rec- ognize the local features of data and to find the The situation is quite different when the WA is ap- parameters of dominating structures (location and plied. A very helpful representation of the WT, which scale/width). For these purposes, wavelets with a allows one to see the features of the signal, gives good localization in physical space with a small the “time–frequency” plane. This is a multiresolution number of oscillations are used. One of the most spectrogram that shows the frequency (scale) con- popular wavelets of this type is called “Mexican Hat” tents of the signal as a function of time. Each pixel (MH), on the spectrogram represents w(a, t) for a certain a − 2 (scale) and t (time). Figure 1 (top) shows the wavelet ψ(t)=(1− t2)e t /2. (8) transformation w(a, t) performed by the MH. The We end this short introduction to wavelets with sequences of dark and light spots indicate variations two examples similar to real data in physics of res- at corresponding scales. The location of a spot on the onances. They will give the reasons why the WA is vertical axis (scale axis, a) corresponds to the width so suitable for this situation and help to understand of the extremum. The intensity of dark spots shows the figures that will be presented when we analyze real the amplitudes of maxima. The highest maximum at data. t =10corresponds to the darkest spot that is located The first example is the model signal composed of at the smallest value of a ≈ 0.2 corresponding to this several Gaussians with different widths and intensi- narrowest maximum. The spot at t ≈ 1 is located at ties: larger a ≈ 0.35 and is not so dark [lower and wider − − 2 − − 2 − − 2 f(t)=e 5(t 1) +2e (t 6) /20 +3e 10(t 10) (9) maximum in (9)]. The dark region at the top of the − − 2 − − 2 figure demonstrates that, in the large-scale region, + e (t 14) /10 + e (t 18) . the whole signal is nothing but a wide maximum. Then it was discretized with a 0.1 step on the inter- The reconstructed signal is stable under small per- val 0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1272 BELOZEROVA et al. a 2.8 101 2.1 1.4 100 0.7 0 –0.7 10–1 –1.4 0 5 10 15 20 t n = 5 a = 0.37 7 0.29 10 0.22 20 0.14 100 0.08 0 10 20 0 10 20 t t Fig. 1. (Top) Wavelet plane. (Bottom) Model signal based on Eq. (9) and its Fourier reconstruction for different n (n is the number of harmonics) (left side); wavelet reconstruction with MH wavelet for different values of the boundary scale (right side). at small numbers n—both these cases correspond to of them substantially overlap. The signal was dis- damped noise. cretized with a step of 0.1 on the interval 0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1273 the WA of the signal (10) + noise are similar to those a 2.8 obtained for the signal based on (9). Rigorously, the 2.1 continuous Gaussian-shape MH wavelet may not be 101 completely adequate for the problem of searching for 1.4 Breit–Wigner resonances, but for practical purposes, as our examples demonstrate, it works well for both 0.7 Gaussian- and Lorentz-like signals. It is important 100 0 that the contributions of different bands to WT are reasonably separated along the vertical axis. That –0.7 is why the reconstruction is stable under small 10–1 –1.4 variations of anoise, which enables one to distinguish 0 5 10 15 20 between the informative low-frequency bands and t contributions of higher frequencies usually generated by noise. Here, to choose a boundary scale, anoise, that sep- arates noiselike high-frequency components of data, a = 0.30 we hold to a pragmatic line of reasoning: the best option of anoise is the smallest one which will smooth out any rapid variations in data and enable us to re- 0.22 produce stable results for low frequencies (resonances area). A similar pragmatic strategy was applied in [3, 4] to choose the parameter ∆ that had to be large 0.10 ¯ enough for a comparison of the smeared Re+e− with QCD models, but not too large in order to keep some Analyzed fine details of the data. signal The reconstructed data are obtained using the in- verse transformation (6) 0 5 10 15 20 t a t max max − dt da 1/2 Fig. 2. (Top) Wavelet plane. (Bottom) Model signal based fr(t)= f + Cψ ψa,t(t )w(a, t ) . a5/2 on Eq. (10) and its reconstructions for different boundary a t noise min scale a and different continuations below tmin: f(t<0) = (11) 0 (solid curve); f(t<0) = f(0) (dashed curve). The f must be added to the reconstructed signal to restore the mean value of the original signal (recall continuations outside of a few half-width intervals that the mean value of the WT is zero because an from the Gaussian center. We directly checked for average value of any wavelet is zero). Formula (11) in several processes studied below that the WA proce- → →∞ →−∞ → the limit anoise 0, amax , tmin , tmax dure is very stable from this point of view—the WA is +∞ is equivalent to the exact relation (6), but in not sensitive to the behavior of the analyzed function practice a limited number of experimental points on beyond the region of reconstruction (region of the the restricted energy interval leads to a limited do- actual data). main in the integral (11). To fill in the gaps between experimental points, we use a linear interpolation. In practice, different interpolations lead to a small dif- 3. WAVELET ANALYSIS OF DATA ference in the restored signal (which produces a very The experimental data relevant to ρ and ω states small difference on a lower part of the wavelet plane in e+e− annihilation and ππ scattering are available corresponding to noise). To show that, in the case of from a variety of sources and are shown in Figs. 3– rather rare coverage on the energy scale in the data in 10. The wavelet transformation images obtained with Fig. 8 (below), we use two interpolations: linear and the MH wavelet are shown on the upper panels of quadratic spline. Figs. 4–10. The WT localizes the structures in a One of the big advantages of the WA is that the fashion that allows us to estimate the masses of the sensitivity of the restored signal to any physically resonances and their widths. Thus, the wavelet im- reasonable continuation of the data outside the in- ages themselves provide us with some useful infor- terval (tmin,tmax), where the data are known, is very mation before the reconstruction of the data and their low. Simply speaking, the Gaussian-like shape of physical analysis is made. The straight horizontal wavelets makes the integration insensitive to any lines correspond to the boundary scale anoise which PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1274 BELOZEROVA et al. δ, deg a 0 200 (a) 10 12 150 9 6 100 –1 10 3 50 0 0 –3 0.8 1.2 1.6 2.0 10–2 –6 η 0.4 0.8 1.2 1.6 2.0 2.4 M(ππ), GeV 1.2 (b) σ, nb 1.0 3 0.8 10 0.6 102 0.4 0.2 1 1.0 1.2 1.4 1.6 1.8 10 M(ππ), GeV 0 Fig. 3. Experimental data for the phase shift (a)and 10 inelasticity (b)ofthep-wave ππ scattering [12, 13] and their reconstruction. 10–1 0.4 0.8 1.2 1.6 2.0 2.4 cuts off the small scale structures. The curves in the M(ππ), GeV figures are the reconstructed data obtained using the − inverse transformation (11). Fig. 4. Cross section of e+e → ππ.(Top)Wavelet plane. (Bottom) Experimental data [14–16] and recon- The cutoff value anoise in the figures below cor- struction. responds to the smallest structure included in the reconstructed data. The width of such a structure can be estimated with the width of the MH wavelet at tribution (large spot near 0.8 GeV), there are maxima ≈ its half-height, Γ 1.5a. It is reasonable to assume around 1.03, 1.4, 1.65–1.85, and 2.2 GeV (ln σ was that a resonance structure to be reliably resolved used for WA because of a logarithmic scale for σ in should include at least three experimental points. In Fig. 4). The first one is the least stable to variation of the data we analyze, the distance between the experi- value of a . mental points is 0.01 GeV or larger, and the smallest noise + − → + + − − structures we are looking for have Γmin not less than For e e π π π π (Fig. 5), the most obvi- 0.03 GeV. This value 0.03 GeV corresponds to the ous maxima are located at about 1.45 and 1.7 GeV value anoise =0.02 GeV,which we use in this section. with widths about 0.1 and 0.05 GeV, respectively. At The data on ππ scattering do not contain much in- greater values of a, these two maxima appear as one formation about ρ mesons. The dips at 1.6–1.7 GeV wide maximum around 1.55 GeV and width about 0.2 GeV.The situation we are describing recalls a for- in Fig. 3 are clearly seen; that is why in order to save space we do not present the wavelet planes. mer ρ (1600) that was later resolved into two states. The reconstruction of ππ-scattering data is stable For energy above the 1.7-GeV maximum, several under variation of the boundary scale (the curves are small maxima unstable to anoise variation are seen. . − − plotted for anoise =002 GeV), which means that the For e+e → π+π π0π0 with subtracted contri- experimental noise can be very reliably separated— bution from ωπ0 (Fig. 6), two obvious maxima are here and elsewhere, this can be considered as one of seen around 1.55 and 1.7 GeV with widths about the criteria of quality of the experiment. 0.08 and 0.1 GeV. Two other maxima are seen to Figure 4 shows that, on the wavelet plane of occur around 1.3 and 2.0 GeV. All these structures e+e− → ππ, along with the obvious ρ-meson con- are stable for variation of the cutoff scale: a reduction PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1275 a a 0 10 24 100 12 18 9 12 6 –1 –1 10 6 10 3 0 0 –6 –3 10–2 –12 10–2 –6 1.0 1.2 1.4 1.6 1.8 2.0 1.1 1.3 1.5 1.7 1.9 2.1 M(π+π+π–π–), GeV M(π+π–π0π0), GeV σ, nb σ, nb 30 15 24 12 18 9 12 6 6 3 0 0 1.0 1.2 1.4 1.6 1.8 2.0 1.1 1.3 1.5 1.7 1.9 2.1 M(π+π+π–π–), GeV M(π+π–π0π0), GeV Fig. 5. Cross section of e+e− → 2π+2π−. (Top) Wavelet Fig. 6. Cross section of e+e− → π+π−2π0 with sub- plane. (Bottom) Experimental data [17, 18] and recon- tracted contribution from ωπ0. (Top) Wavelet plane. struction. (Bottom) Experimental data [17, 18] and two reconstruc- tions for different cutoff values anoise. of anoise by a factor of almost two does not change the locations of the peaks and their shapes. the wavelet plane, we present the plot corresponding to a linear interpolation). + − → 0 For e e ωπ (Fig. 7), a wide structure is lo- cated at about 1.25 GeV. A decrease in the cutoff Thus, the WA indicates the ρ states with masses scale causes splitting of that major maximum—two 1.05–1.25, 1.4, 1.6–1.85, and 2.0–2.2 GeV and maxima occur at 1.05 and 1.45 GeV. Three more widths of about 0.1 GeV, which makes sense to be smaller structures can be recognized between these included in further analysis that should be based two. There are also two structures at about 1.85 and on some physical models. The first and the last 2.0 GeV. The scale values corresponding to these states are sensitive to high-frequency noise, which structures are close to reliable values of anoise.Itcan makes them questionable. A discussion above on the be observed that, except for the 1.25- and 1.45-GeV stability of those states under noise contribution can states, no dominating scales can be attributed to help evaluate the reliability of the results of further other states. analyses. Note that the 1.01–1.25 GeV state has a long and complicated history; one of the last pieces of e+e− → ηπ+π− The data (Fig. 8) are rather poor evidence was presented by the LASS group [25], but and the only visible structure is a wide peak around 1) 1.55 GeV. Here, to fill in rather big gaps between it still needs confirmation. the experimental points, we use linear and quadratic 1)The likely existence of ρ (1200) with total width of about spline interpolations. They lead to a small difference in 0.2 GeV, the main decay mode into the ωπ0 channel and the restored signal and to a tiny difference on a lower smaller fractions into 2π and 4π states, was recently pre- part of the wavelet plane corresponding to noise (on sented by Crystal Barrel [26]. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1276 BELOZEROVA et al. a a 100 16 100 4 12 3 8 2 –1 –1 10 4 10 1 0 0 –4 –1 10–2 –8 10–2 –2 0.9 1.1 1.3 1.5 1.7 1.9 2.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 M(ωπ0), GeV M(ηπ+π–), GeV σ, nb σ, nb 7 16 6 12 5 4 8 3 4 2 0 1 0.91.1 1.3 1.5 1.7 1.9 2.1 M(ωπ0), GeV 0 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 ηπ+π– Fig. 7. Cross section of e+e− → ωπ0. (Top) Wavelet M( ), GeV plane. (Bottom) Experimental data from [17, 19] and re- − − construction. Fig. 8. Cross section of e+e → ηπ+π . (Top) Wavelet plane. (Bottom) Experimental data [20, 21] and two re- constructions for different interpolations between exper- For the process e+e− → π+π−π0 (Fig. 9), a wide imental points: solid curve for linear interpolation and ω structure dominates at about 1.25 GeV. It decays dashed curve for quadratic spline interpolation. into three “horns” without characteristic scales as the cutoff scale decreases. There is also a relatively stable the fact that even data with good statistics do not state at about 1.65 GeV with a rather well resolved directly provide masses and widths of the resonances. characteristic scale position a. Due to overlapping, the observed positions of maxima The wavelet plane in Fig. 10 for e+e− → ωππ can differ substantially from the physical resonance indicates a wide ω state at about 1.68 GeV and a masses, and partial and total widths can differ signifi- weaker state at around 1.8 GeV—both of these are cantly from preliminary estimations. The WA in this rather well resolved in a. case yields a useful set of starting conditions for a As we noted above, the argument in favor of de- more accurate analysis. For the sake of clarity in the veloping a further more comprehensive analysis is following section, and because the main goal of this paper is the WA, we do such an analysis for the ω states only (it involves a smaller number of states and Table 1. Parameters of two ω mesons (in GeV) channels than those related to the ρ states). Meson Mass Width 4. BREIT–WIGNER ANALYSIS OF ω STATES ± ± ω1 1.15 0.033 0.197 0.051 USING THE WAVELET-ANALYZED DATA ω 1.69 ± 0.014 0.297 ± 0.028 The interference of the resonances with the same 2 decay channels is the key aspect of any analysis and PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1277 Table 2. Branching ratios of two ω mesons (in %) and phases of ω2 (relative to ω1,indeg) Channel B B ϕ ω1 ω2 ω2 e+e− (1.7 ± 0.4) × 10−5 (1.4 ± 0.6) × 10−4 0.16 ± 1.9 ρπ 86.2 ± 53.6 13.9 ± 11.1 0.07 ± 13.3 ωππ 13.8 ± 53.6 86.1 ± 11.1 0.14 ± 2.1 interpretation. In the Breit–Wigner (BW) approach, set. Whether or not these phases are included, such a this interference is often taken into account by rel- sum of BWterms is depleted of unitarity, which is the ative phases in BWterms that are treated as free basic point in the BWdescription. parameters, the most often just 0 or π. The results of There are several unitarity-preserving approaches, analysis critically depend on the choice of the phase such as an often used K-matrix method. Contrary to a a 100 4 100 4 3 3 2 2 –1 10 1 10–1 1 0 0 –1 –1 10–2 –2 10–2 –2 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.4 1.6 1.8 2.0 2.2 M(ωππ), GeV M(π+π–π0), GeV σ, nb σ, nb 5 3 4 2 3 2 1 1 0 0 1.4 1.6 1.8 2.0 2.2 1.01.2 1.4 1.6 1.8 2.0 2.2 M(ωππ), GeV M(π+π–π0), GeV Fig. 10. Cross section of e+e− → ωππ. (Top) Wavelet plane. (Bottom) Experimental data [23, 24] (in the figure, Fig. 9. Cross section of e+e− → π+π−π0. (Top) Wavelet the cross section from [23, 24] is multiplied by the factor plane. (Bottom) Experimental data [22, 23] and recon- 1.5 to include the nonobserved ωπ0π0 state) and recon- struction (solid curve); fits with two ω (dashed curve) and struction (solid curve); fits with two ω (dashed curve) and three ω (dotted curve). three ω (dotted curve). PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1278 BELOZEROVA et al. Re+e– 8 6 4 2 024 6 E, GeV Fig. 11. Ratio Re+e− (E). Experimental points are from [29]. Dashed and dotted curves correspond to two different treatments of data reconstruction with anoise =0.25 GeV for energy below 1.5 GeV (see the text); solid curve represents the theoretical approach [5] Dash-dotted curve represents the result of wavelet reconstruction for anoise =0.6 GeV. this approach, the BWmethod directly provides the preserves unitarity (note that the number of free pa- physical parameters of the resonances, their masses, rameters is smaller than or equal to that in a “naive” widths, and branching fractions. BWor K-matrix approach). As is often used for the vector states, here we use the approximation of Let us briefly formulate the unitarity-preserving energy-independent widths. BWmethod (details and connection to di fferent meth- + − ods are described in [27, 28]). Let the scattering am- Index i =1corresponds to the initial state e e plitudes be written as (or to the virtual photon), index j =2, 3 corresponds to the final states π+π−π0 and ωππ (which are the N mrΓrgrigrj two main channels of the ω -state decay [29]), and in- fij(s)= 2 (12) s − m + imrΓr dex r enumerates the ω resonances. Here, we use da- r=1 r ta, some of which are rather recent [22–24], that lead N to ω states with masses rather different from those mrΓr|gri||grj| = ei(ϕri+ϕrj) , cited most often (at about 1.45 and 1.6 GeV) [29]. s − m2 + im Γ r=1 r r r The ω masses reported from the SND detector [24] are 1.25, 1.40, and 1.77 GeV. In the previous work of where the phases ϕ are not free independent param- ri the same team [22], the PDG [29] resonance ω (1420) eters and should be determined in such a way that was not observed and the ω(1200) state was reported as a replacement for ω(1420). Interestingly, the 1.2- Table 3. Parameters of three ω mesons (in GeV) GeV state is clearly observed on the wavelet plane in Fig. 9, and the important aspect is that ω(1420) is rather sensitive to the anoise value. Meson Mass Width A comparison of the reconstructed data and ex- pression (12) is seen in Figs. 9 and 10. For two ± ± ω1 1.204 0.014 0.250 0.048 ω , there are eight free parameters; for three ω , the number of free parameters is 12. The three-ω ± ± fit better describes the region 1.4–1.5 GeV for the ω2 1.550 0.019 0.212 0.077 cross section of e+e− → π+π−π0, but the overall ± ± description for the two processes is about the same ω3 1.700 0.021 0.300 0.039 2 for both cases with χ /nD ≈ 1.3. Starting from the PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1279 Table 4. Branching ratios of three ω mesons (in %) and phases of ω2 and ω3 (relative to ω1,indeg) Channel B B ϕ B ϕ ω1 ω2 ω2 ω3 ω3 e+e− (1.9 ± 1) × 10−5 (1.1 ± 0.5) × 10−5 −24.4 ± 9.3 (1.3 ± 1) × 10−4 −16.1 ± 2.5 ρπ 83.2 ± 35.1 81.6 ± 11.8 −34.5 ± 5.9 15.7 ± 10.5 −10.8 ± 6.4 ωππ 16.8 ± 35.1 18.4 ± 11.8 −28.4 ± 10.9 84.3 ± 10.5 −21.5 ± 6.6 WA masses (and widths), we allow big deviations works related to the ρ and ω states with the masses of about 150 MeV from these positions in either close to all the ones listed above. The parameters of direction. The masses, widths, branching ratios, these states are known with a large uncertainty due to and relative phases of ω states are presented in the poor accuracy, troubles in analyzing overlapping Tables 1–4. states, and conflicting experimental data. The e+e− annihilation into hadrons gives the most direct obser- vation of the vector states. Other processes (such as τ 5. CONCLUSION decays) also give very important information, but with Numerous applications of wavelets to data analy- substantially greater uncertainty in the parameters of sis in different fields of mathematics and physics have resonances. proved themselves to be a powerful tool for studying The WA shows that some experimental data are the fractal signals and data. This technique can be statistically inadequate in the sense that they do not successfully applied to some problems of nuclear and allow the noise contribution to be separated. The high-energy physics where wavelet analysis will work method gives the criteria for distinguishing between as a tool for studying the energy scaling of the pro- “stable” and “unstable” data—the latter do not re- cess. produce the same essential structures when a con- When studying the data containing the informa- tribution of the experimental noise changes slightly. tion on ρ and ω states, we performed the WA of Technically, it means that the structure (resonance) is questionable if it is sensitive to the noise-cutoff value. the data in order to single out their contribution. Due to good scaling properties of the wavelets, one can Interestingly, this criterion supports ρ at about 1.45 consider experimental data with various resolutions, and 1.65–1.70 GeV, in accord with the assessment which allows us to separate the resonances from in the current PDG [29] compilation. (We cannot directly compare the parameters of ω states that we noise and from each other. Such a local analysis (and the corresponding reconstruction) is very significant obtained with those in [29], because the data on ω we when it is necessary to distinguish between several analyze [22, 24] are not included in [29] review yet.) The only conclusion that we can draw with help of resonances in experimental data with large errors, as in the case of ρ and ω mesons. The WA is much the WA about other states (likeρ (1200)) is that their less sensitive to the noise than any other analysis existence is consistent with the experimental data, and allows us to reduce the role of statistical errors but improvement of the cross-section measurement substantially. accuracy is needed. In the case of one isolated resonance (or several Let us note one more thing related to fluctuations. resonances that are very distant from each other), the While small-scale structures located at the bottom wavelet image of data gives practically direct infor- of wavelet plane correspond to experimental noise, mation about the mass and total width of each reso- the structures in the intermediate part of the wavelet nance. When resonances overlap, their physical pa- plane can be of a different nature. Dark spots (re- rameters are found by applying the BWapproach that lated to the large amplitude of the wavelet coeffi- preserves unitarity of the amplitudes for such cases. cient w(a, E)), resolved in frequency, occur where the Fitting the data with suppressed noise gives more structure of the signal is similar to the wavelet shape reliable results for resonance parameters. The WA of and size at this location (a, E). Thus, such spots are thedataweuseindicatestheρ states with masses most likely associated with resonances. There might of 1.05–1.25, 1.4, 1.6–1.85, and 2.0–2.2 GeV and also be dark “horns” (usually less intense than in the ω states at about 1.2, 1.6, and maybe at 1.4 and the previous case) reaching high frequencies from 1.8 GeV.The Review of Particle Physics [29] contains the area of low or intermediate frequencies, and not the references to the experimental and theoretical having a relatively well resolved maximum intensity PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1280 BELOZEROVA et al. (looking like a kneecap). Such horns should also be data from [29] up to 60 GeV) contribute a negligible checked for association with resonances (using some amount below 8 GeV in the restored data. As seen physical models), but most likely they are related to in Fig. 11, our WA-smeared ratio Re+e− corresponds noise fluctuations of large amplitude (when one or very well to the theoretical prediction for the QCD- a few experimental points substantially jump up or smeared ratio. down with respect to their neighbors). To conclude this paper, let us return to the ra- ACKNOWLEDGMENTS tio Re+e− to demonstrate another application of the wavelet technique to high-energy physics. To even We thank V.G. Solovyev for helping with the com- out any rapid variations in Re+e− , we remove high- puting part of this study. We also thank M.N. Acha- frequency noise with WT, which provides the smear- sov for providing us with the recent e+e− data. One ing alternative to the procedure (1). The wavelet ap- of the authors (V.H.)thanks D.V.Shirkov for attract- proach corresponds remarkably to this goal because ing attention to the smearing of the ratio R,thus of its multiscale nature. inspiring our application of the WA to that ratio, and The restored data in Fig. 11 keep all the main S.B. Gerasimov for very useful discussions. features of Re+e− with statistical errors and threshold The research described in this publication was singularities damped, which makes a direct compari- made possible in part by award no. PE-009-0 of son with the corresponding QCD-smeared quantity the US Civilian Research and Development Founda- possible. The restored data are stable under rather tion for the Independent States of the Former Soviet large variations of the cutoff scale, which reflects the Union (CRDF). quality of the Re+e− data. We restore the Re+e− on the same interval as in [3, 4], from about 1.4 to 7.5 GeV. 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Above 1.5 GeV, for a big cutoff value anoise,itjust gives some small vertical shift for the restored curve— 11. V. L. Korotkikh and L. I. Sarycheva, in Proceedings of the 9th International Conference on Hadron the dotted curve on Fig. 11 is obtained by including Spectroscopy, “Hadron 2001”, Protvino, 2001, all experimental points above the 2mπ threshold, and p. 615. the dashed curve is obtained when the data from 12. B. Hyams et al.,Nucl.Phys.B64, 134 (1973). the threshold are continued to the first experimental 13. S. D. Protopopescu et al., Phys. Rev. D 7, 1279 point in the figure at about 1.4 GeV using a linear (1973). approximation (thus excluding the ρ peak). When the 14. A. Quenzer et al., Phys. Lett. B 76B, 512 (1978). cutoff value increases, the difference between these 15. L. M. Barkov et al.,Nucl.Phys.B256, 365 (1985). two curves becomes very small—corresponding re- 16. D. Bisello et al., Preprint LAL 85-15 (Orsay, 1985). constructed data for the value anoise =0.6 GeV are 17. L. Stanco, in Proceedings of the International Con- indicated by the dashed-dotted curve. The data from ference on Hadron Spectroscopy, College Park, an energy interval well beyond 8 GeV (we use the Maryland, 1991, p. 84. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 WAVELET ANALYSIS OF DATA IN PARTICLE PHYSICS 1281 18. R. R. Akhmetshin et al., Phys. Lett. B 446, 392 26. B. Pick, in Proceedings of the 9th International (1999). Conference on Hadron Spectroscopy, “Hadron 19. M. N. Achasov et al., Phys. Lett. B 486, 29 (2000). 2001”, Protvino, 2001, p. 683. 20. V. P.Druzhin et al., Phys. Lett. B 174, 115 (1986). 21. B. Delcourt et al., Phys. Lett. B 113B, 93 (1982). 27. V. K. Henner and T. S. Belozerova, Fiz. Elem.´ 22. M. N. Achasov et al., Phys. Lett. B 462, 365 (1999). Chastits At. Yadra 29, 148 (1998) [Phys. Part. Nucl. 23. A. Antonelly et al.,Z.Phys.C56, 15 (1992). 29, 63 (1998)]. 24. M. N. Achasov et al.,inProceedings of the 9th In- ternational Conference on Hadron Spectroscopy, 28. V. K. Henner and T. S. Belozerova, Yad. Fiz. 60, 2180 “Hadron 2001”, Protvino, 2001, p. 30. (1997) [Phys. At. Nucl. 60, 1998 (1997)]. 25. D. Aston et al., Preprint SLAC-PUB 5657 (1991); Preprint SLAC-PUB 5606 (1994). 29. D. E. Groom et al.,Eur.Phys.J.C15, 1 (2000). PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1282–1288. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1322–1328. Original Russian Text Copyright c 2003 by Mokeev, Ripani, Anghinolfi, Battaglieri, De Vita, Golovach, Ishkhanov, Markov, Osipenko, Ricco, Sapunenko, Taiuti, Fedotov. ELEMENTARY PARTICLES AND FIELDS Theory Helicity Components of the Cross Section for Double Charged-Pion Production by Real Photons on Protons V. I. Mokeev1),M.Ripani2), M. Anghinolfi2), M. Battaglieri2),R.DeVita2), E. N. Golovach1),B.S.Ishkhanov1), N.S.Markov3), M.V.Osipenko1), G. Ricco2), V.V.Sapunenko2),M.Taiuti2),andG.V.Fedotov3) Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received October 9, 2002 Abstract—The helicity components σ1/2 and σ3/2 of the cross section for double charged-pion production by real photons on a nucleon are calculated within a phenomenological approach developed previously. A high sensitivity of the σ1/2–σ3/2 asymmetry to the contribution of nucleon resonances having strongly different electromagnetic helicity amplitudes A1/2 and A3/2 is demonstrated. This feature is of importance for seeking “missing” baryon states. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION its behavior obeys the 1/Q2 law. This behavior follows Experiments with polarized electron (photon) from the calculations within perturbative QCD in the beams and polarized proton targets made it possible region Q2 > 5 GeV2, but experimental data are in ac- to determine the cross sections σ1/2 and σ3/2 corre- cord with the asymptotic behavior of photon–proton sponding to the total proton and photon helicities of interaction at lower values of Q2 down to 1 GeV2. 1/2 and 3/2 in the initial states both for the inclusive The integral IGDH grows fast from the photon channel and for various exclusive channels [1–6]. point to Q2 ≈ 1–2 GeV2, and its absolute value de- Interest in studying the helicity components σ1/2 and creases approximately by an order of magnitude at 2 2 σ3/2 is motivated by the following factors. Q =1.0 GeV . Under the assumption of photon On the basis of the most general theoretical prin- interaction with proton partons, the difference σ1/2 − ciples, Gerasimov [7] and Drell and Hern [8] (GDH in σ3/2 is determined by the difference of the probabilities the following) predicted the value of the integral of finding a parton with spin orientation along and dν against the photon-spin direction. Thus, the integral IGDH = (σ − σ ) , (1) 1/2 3/2 ν IGDH is related to the contribution of the parton spin to the total proton spin. According to the analyses where σ1/2 and σ3/2 are the total photoabsorption performed in [9, 10], the contribution of the parton cross sections for the case in which the total photon– spin to the total nucleon spin in the asymptotic region proton helicity is 1/2 and 3/2, respectively, while ν is (Q2 > 1.0 GeV2) does not exceed 30%; at the photon the photon energy. It follows from [7, 8] that the GDH Q2 =0 2 − point ( GeV ), the main contribution to the integral must take the value of 204 µb at the photon nucleon spin comes from constituent quarks. point. The experimental results reported in [1, 2] were 2 compared with the predictions made in [7, 8]. The Thus, the variation in the quantity Q over the 2 definition of IGDH can be extended to the case of interval from 0 to 1.0 GeV leads to a significant 2 virtual-photon absorption, IGDH = IGDH(Q ),where variation in the helicity amplitudes of photon–proton Q2 is the sign-reversed square of the virtual-photon interaction and in the contribution of the quarkspin 4-momentum. The investigation of the Q2 depen- to the total proton spin. In order to understand mech- dence of I in [4] revealed that, for Q2 > 1.0 GeV2, anisms behind such strong changes in the spin struc- GDH ture of the photon–proton interaction, it is necessary 1)Institute of Nuclear Physics, Moscow State University, to analyze contributions of various exclusive channels Vorob’evy gory, Moscow, 119899 Russia. to the cross-section difference σ1/2 − σ3/2. 2) Istituto Nazionale di Fisica Nucleare, Sezione di Genova, 2 Genova, Italy. A detailed description of the Q and W (W is the 3)Faculty of Physics, Moscow State University, Vorob’evy total energy in the c.m. frame) dependences of the gory, Moscow, 119899 Russia. cross sections σ1/2 and σ3/2 and of the GDH integral 1063-7788/03/6607-1282$24.00 c 2003 MAIK “Nauka/Interperiodica” HELICITY COMPONENTS OF THE CROSS SECTION 1283 in total-photoabsorption and single-pion-production γγπ+ π– reactions was given in [11]. At W>1.6 GeV,double- pion-production reactions begin to contribute to the π– = π+ ∆++ cross section for the total-photoabsorption reaction p p' p on a proton significantly; at W>1.9 GeV, their con- ' tribution becomes dominant. Therefore, it is inter- p esting to investigate the helicity components of the π+ cross section for double pion production by photons on a proton. In [12–14], a model was developed for π– γγπ + ρ describing double pion production on a proton by virtual and real photons at W<4.0 GeV and Q2, + ++ C(W, Q2) ∆0 π – −t<5.0 GeV2,wheret is the square of the difference p p p' of the initial- and final-proton 4-momenta. In this kinematical region, the model proposed in [12–14] p' describes well the entire body of available data on the + − cross sections for double charged-pion production by Fig. 1. Mechanisms of the production of π π pairs by real and virtual pions. This approach was used to photons on a proton. analyze the first data of the CLAS Collaboration on thedoubleproductionofchargedpionsbyphotonson nucleon resonances in photon–proton interaction in a proton in the energy region corresponding to the ex- the input channel (Figs. 2a,2f) and nonresonance citation of nucleon resonances (E-93-006 experiment mechanisms (Figs. 2b–2e and 2g). at JLAB; spokespersons V.D. Burkert and M. Ri- pani) [15, 16]. In the present study, we calculate the In calculating the resonance components in the helicity components σ1/2 and σ3/2 of the cross section amplitudes for the quasi-two-particle reactions for double charged-pion production by real photons at (Fig. 2), we tookinto account all well-known nu- W<2.0 GeV, relying on the approach developed in cleon resonances having masses below 2.0 GeV and [12–14]. We determine the model parameters from a significant widths with respect to decays through the fit to all available data on the cross sections for double π∆ and ρp channels [12, 13]. charged-pion production by photons in the energy The resonance amplitudes were calculated in the region of nucleon-resonance excitation. Breit–Wigner approximation. The details of the cal- culations can be found in [12]. The electromagnetic amplitudes corresponding to the γpR vertex were cal- 2. DESCRIPTION OF THE CROSS culated on the basis of data on the helicity amplitudes SECTIONS FOR DOUBLE CHARGED-PION A and A from [17]. Here, a nucleon resonance is PRODUCTION ON PROTONS AND THEIR 1/2 3/2 HELICITY COMPONENTS denoted by R. For these amplitudes, one can also use the results obtained within any quarkmodel. Quasi-two-particle mechanisms involving the production and subsequent decay of ∆ and ρ res- Having calculated the cross section for double onances in the intermediate state (see Fig. 1) are charged-pion production within the model proposed known to make the main contribution to the reaction in [12–14] with the model amplitudes A1/2 and A3/2 of double charged-pion production on a proton target. estimated on the basis of a quarkmodel, we can The amplitude of each mechanism in Fig. 1 was compare the results with experimental data and, in calculated in the Breit–Wigner approximation as the this way, confirm that the model description of the product of the relevant two-particle amplitude, the electromagnetic form factors for nucleon resonances amplitudes for the decay of unstable intermediate is adequate. On the other hand, we can treat the particles into stable final-state particles, and the electromagnetic form factors as free parameters of corresponding Breit–Wigner propagators [13, 14]. the model and extract their values from a fitofthe We describe all other processes that contribute to calculated cross sections to experimental data. We the double-pion-production reaction in the phase- determined the amplitudes of the strong nucleon- space approximation; that is, their total amplitude resonance decays corresponding to the Rpρ (Rπ∆) is approximated by a quantity C(W, Q2) that is vertex from the results obtained in [18] by analyzing → independent of the final-state kinematical variables the cross sections for πN ππN reactions. and which is a free model parameter to be determined We described nonresonance processes in the from an analysis of experimental data. The amplitudes quasi-two-particle reaction γp → ρp in the diffraction for the quasi-two-particle mechanisms in Fig. 2 are approximation [13, 14]. The common factor in the described by a superposition of the excitations of nonresonance amplitude is a free parameter that is PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1284 MOKEEV et al. (‡) γπ γ π =∑ p N*, ∆* ∆ p ∆ (b) (c) γπ γ π + + π p ∆ p ∆ (d) (e) γ ∆ γπ + + ∆ ; p N ∆ p π (f)(g) γρ γργρ ∑ bt =+p N*, ∆* p ~ Ae p p p p Fig. 2. Diagrams describing the quasi-two-particle reactions γp → π∆ [resonance mechanisms (a) and Born terms: (b) contact term, (c)on-flight pion, (d)nucleonterm,and(e)on-flight delta] and γp → ρp [(f) resonance mechanisms and (g) diffractive production of a ρ meson]. independent of W and which must be determined where Rπ(t) is the Regge propagator proposed in [19] from experimental data. for the pion trajectory, t is the Mandelstam vari- able corresponding to the amplitude in Fig. 2c,and We described the nonresonance amplitude for mπ is the pion mass. The multiplication of the one- the channel γp → π∆ by the set of gauge-invariant pion-exchange amplitude (Fig. 2c) by the factor in Born amplitudes corresponding to the diagrams in (2) corresponds to the substitution of the Reggeized Figs. 2b–2e.Off-shell pion formation in the mech- propagator Rπ(t) for the one-pion propagator in this anism represented by the diagram in Fig. 2c was amplitude. Since the gauge-invariant sum of Born taken into account by introducing the πN∆ vertex amplitudes involving one-pion exchange is multiplied function determined from data on NN scattering by a common factor, the gauge invariance of the sum and the pion electromagnetic form factor [12]. The is preserved upon the substitution of the Reggeized product of the strong and electromagnetic vertices propagator Rπ(t) for the one-pion-exchange propa- for the diagram involving the exchange of a ∆ isobar gator. (Fig. 2e), as well as the t and Q2 dependence of The use of the Reggeized gauge-invariant Born the contact term (Fig. 2b), was determined from the amplitudes made it possible to obtain a satisfactory condition requiring that the sum of Born terms be description of experimental data from [20] on the gauge-invariant [12]. The possibility of the exchange angular distributions of π− mesons in the quasi- of various mesons with pion quantum numbers via the two-particle channel γp → π∆ (see Fig. 3). There- mechanism in Fig. 2c—this possibility is of particular by, problems concerning the description of the π−- importance at W>1.7 GeV—is effectively described meson angular distribution in the reaction γp → π∆, by means of the substitution of the pion Regge tra- which were discussed in [12], have been successfully jectory for the one-pion propagator, as was proposed solved; at the present stage, the Born terms in the in [19]. To restore the gauge invariance of Born channel γp → π∆ are gauge-invariant in our model. terms, we follow the procedure used in [19]. All Born Coupling to open inelastic channels in the initial amplitudes corresponding to one-pion exchange are and final states is of importance in describing the multiplied by the common factor nonresonance Born amplitudes for the reaction γp → π∆. In [12], a dedicated approach was developed ac- − 2 t mπ Rπ(t), (2) cording to which this coupling is effectively taken into PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 HELICITY COMPONENTS OF THE CROSS SECTION 1285 dσ/dΩ, µb/sr 10 * * W =1.51 GeV * W =1.57 GeV * * * * * 5 * * ** * * * * * * * * * * 0 * 10 * * 1.63 GeV * 1.71 GeV * * 5 * * * * * * * * * * * * * * * * 0 * 10 * * 1.82 GeV * 1.92 GeV 5 * * * * * * * *** * * **** * * 0 100 2000 100 200 θπ, deg Fig. 3. Angular distribution of π− mesons in the reaction γp → π−∆++ according to our calculations with Reggeized Born terms (see main body of the text), along with experimental data from [20]. The angle between the γ and π− momenta in the c.m. frame is plotted along the abscissa. account as the absorption of projectile particles in the The element dτ of the final-state three-particle initial state and the absorption of emitted particles in phase space is the final state. The absorption factors are determined 1 1 from data on πN scattering. dτ = dS + dS + − dΩdα, (5) 32W 2 (2π)5 π p π π The helicity differential-cross-section compo- where S + and S + − are the squares of the invariant nents dσ1/2 and dσ3/2 are defined as π p π π masses of, respectively, the π+p and the π+π− system 1 1 dσ = 4πα (3) in the final state; Ω is the solid angle of proton (or 1/2(3/2) 2 4K M − L N π -meson) emission; and α is the angle between the 2 plane spanned by the photon and proton (or photon × ππλp |T | λγ λp dτ, and π−-meson) 3-momenta and the plane spanned λγ λpλ p by the π+-andπ−-meson (or π+-meson and proton) | | λγ + λp =1/2(3/2), 3-momenta. where ππλp |T | λγλp are the helicity amplitudes for In calculating the helicity components dσ1/2(3/2), the reaction of double charged-pion production at the terms characterized by the total helicities λγ + the incident-photon helicity of λγ , the target-proton λp are included in the sum in (3) according to the helicity of λp,andthefinal-state-proton helicity of lower line in (3) and the conventions in [21]. The λp . sum dσ1/2 + dσ3/2 is related to the total unpolarized cross section as follows: The quantity KL is the equivalent-photon wave vector dσ1/2 + dσ3/2 =2dσ. (6) W 2 − M 2 K = N , (4) The amplitudes parameterized in the phase-space L 2M N approximation, C(W, Q2), are shared between the he- where MN is the nucleon mass and α =1/137 is the licity components of 1/2 and 3/2 under the assump- fine-structure constant. tion of their equal contributions to all helicity states. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1286 MOKEEV et al. σ, µb σ, µb γp → π+π–p 80 γp → π+π–p 80 (‡) 60 σ 0.5 3/2 60 40 σ 20 0.5 1/2 γp → π–∆++ 40 0 γp → ρp Asymmetry of resonance processes 20 0 (b) –20 0 Asymmetry of nonresonance 1.2 1.4 1.6 1.8 2.0 processes W, GeV –40 σ σ Total asymmetry, 1/2 – 3/2 Fig. 4. Experimental data from [20] on the W depen- –60 dence of the integrated cross sections for the reaction − 1.2 1.4 1.6 1.8 2.0 γp → π+π p (•) and contributions of the quasi-two- , GeV − W particle channels γp → π ∆++ ()andγp → ρp (), along with the results of the calculations based on the Fig. 5. (a) Total cross section (solid curve) and its helicity model described in the main body of the text. components for the reaction γp → π+π−p along with experimental data from [20]; (b) σ1/2–σ3/2 asymmetry. Upon including, in the model developed in [12– 3. RESULTS OF THE CALCULATIONS 14], the calculations of the polarization components FOR THE HELICITY COMPONENTS dσ1/2(3/2) of the cross sections, the values of the OF THE CROSS SECTIONS FOR DOUBLE electromagnetic form factors for nucleon resonances CHARGED-PION PRODUCTION BY PHOTONS can be extracted from a simultaneous fittounpo- larized differential cross sections dσ and their he- Within the approach described above, we have licity components dσ1/2(3/2). Such an extension of calculated the W dependences of the helicity cross- the range of fitted data may significantly improve the section components σ1/2 and σ3/2 for the exclusive accuracy with which one extracts the electromagnetic channel of double charged-pion production by pho- form factors for nucleon resonances. In addition, a tons. The electromagnetic form factors A1/2 and A3/2 calculation of the helicity components dσ1/2(3/2) with for nucleon resonances were taken in accordance with the amplitudes A and A from various quark data presented by the Particle Data Group [17]. The 1/2 3/2 amplitudes of strong resonance decays were extracted models and a comparison of the results obtained in from [18] by using the method described in [12]. The this way with experimental data extend considerably following parameters of the model were taken to be the possibilities for validating the description of the free: the amplitude of the three-particle phase space, structure of nucleon resonances as objects formed by C(W, Q2);theeffective coupling constant for the π interacting quarks and gluons. Regge trajectory; and the multiplicative factor of the diffractive nonresonance amplitude for ρ-meson pro- duction. The free parameters were determined from a The approach developed in the present study can fit to data of the ABBHM Collaboration [20] on the also provide information about the contributions of W dependences of the integrated cross sections for resonance and nonresonance mechanisms and vari- double charged-pion photoproduction. ous two-quasi-particle channels to the cross-section The integrated cross sections that were calculated components dσ . This information is of great 1/2(3/2) for double charged-pion photoproduction on the basis value for obtaining deeper insight into the spin struc- of our model with the parameter values corresponding ture of nucleons and into the dynamics of processes to the best fit are displayed in Fig. 4 along with experi- that are responsible for the evolution of the helicity mental data from [20]. The calculations are seen to re- structure of photon–proton interaction. produce these data well. In order to test our model, we PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 HELICITY COMPONENTS OF THE CROSS SECTION 1287 calculated the integrated cross sections for the quasi- of the cross section can aid in determining the rela- two-particle reactions γp → π−∆++ and γp → ρp, tive phase of resonance and nonresonance amplitudes employing the parameters determined from a fitto from experimental data, and this is of importance the total cross section (solid curve in Fig. 4). The W because it is difficult to do this theoretically. dependences of the integrated cross sections for these reactions (dashed and dash-dotted curves in Fig. 4) are contrasted against experimental data from [20]. 4. CONCLUSION It can be seen that these theoretical results faithfully On the basis of the model developed in [12–14] reproduce the data from [20] on the W dependences for describing double charged-pion photoproduction of the integrated cross sections for these quasi-two- on a proton, we have predicted here the W depen- particle channels. dence of the helicity cross-section components σ1/2 The calculated helicity components σ1/2 and σ3/2 and σ3/2 and of their difference. The results exhibit and their difference for the integrated cross sections a rather high sensitivity of the difference σ1/2 − σ3/2 are shown in Fig. 5 versus W . The lower panel in to the excitation of resonance states characterized Fig. 5 displays the contributions of the resonance by significantly different values of the electromag- mechanisms (dashed curve) and nonresonance pro- netic form factors A and A . The excitation of − 1/2 3/2 cesses (dotted curve) to the difference σ1/2 σ3/2. such resonances leads to the formation of resonance A feature peculiar to the difference σ1/2 − σ3/2 structures in the W dependence of the difference of is the formation of a dip at W =1.50 GeV. Such the helicity cross-section components σ1/2 and σ3/2. a structure was observed in the MAMI experimen- Thus, measurement of the helicity components of the tal data [3] on the difference σ1/2 − σ3/2 in double cross section for double charged-pion photoproduc- charged-pion production. The predicted structure tion will play an important role in the investigation is in qualitative agreement with the data from [3]. of nucleon resonances having significantly different The W dependence for nonresonance processes is values of A1/2 and A3/2—in particular, in searches for smooth, while the W dependence for the excita- “missing” baryon states that possess such properties. tion of nucleon resonances features a structure at Our approach makes it possible to extract the W =1.51 GeV (Fig. 5b). The structure in the W electromagnetic form factors for nucleon resonances − dependence of the difference σ1/2 σ3/2 results from from a global fittodataonthedifferential cross sec- the contribution of the D13(1520) state, which is tions for double charged-pion photoproduction and characterized by significantly different values of the their helicity components. The development of the electromagnetic helicity amplitudes A1/2 and A3/2 model from [12–14] in the course of the present in- (−0.024 and 0.166 GeV−1/2, respectively). So great vestigation, along with the extension of the range of a distinction between these helicity amplitudes leads data subjected to analysis, significantly enhances its to strongly different values of the helicity components potential for determining the electromagnetic form of the resonance part of the cross section; this in turn factors for nucleon resonances and improves the re- leads to a dip in the W dependence of the difference liability of the results obtained in this way. of the helicity total-cross-section components σ1/2 and σ at a W value close to the mass of the 3/2 REFERENCES D13(1520) resonance. Thus, measurement of the helicity components of the cross section for double 1. J. Ahrens et al., Phys. Rev. Lett. 84, 5950 (2000). pion photoproduction is sensitive to the existence 2. P. Pedroni, in Proceedings of the Symposium on the Gerasimov–Drell–Hern Sum Rule and the of resonances characterized by significantly different Nucleon Spin Structure in the Resonance Region, values of the electromagnetic form factors A1/2 and Germany, 2000, Ed. by D. Drechsel and L. Tiator, A3/2 and can be used as an effective tool in searches p. 99. for such states. In particular, measurements of the 3. M. Lang, in Proceedings of the Symposium on the helicity components of the cross section will play an Gerasimov–Drell–Hern Sum Rule and the Nu- important role in seeking “missing” baryon states cleon Spin Structure in the Resonance Region, having significantly different electromagnetic form Germany, 2000, Ed. by D. Drechsel and L. Tiator, p. 125. factors. 4. K. Ackerstaff et al., Phys. Lett. B 444, 531 (1998). From the theoretical results presented in Fig. 5b, 5. R. C. 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Kobrinsky PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1289–1296. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1329–1336. Original Russian Text Copyright c 2003 by Braguta, Likhoded, Chalov. ELEMENTARY PARTICLES AND FIELDS Theory Transverse Polarization of the Lepton in the Decay Process B → Dlνl due to Electromagnetic Final-State Interaction V.V.Braguta,A.A.Likhoded,andA.E.Chalov Institute for High Energy Physics, Protvino, Moscow oblast, 142281 Russia Received May 31, 2002; in final form, October 18, 2002 0 − + Abstract—The emergence of transverse polarization of the lepton in the decay processes B → D l νl + 0 + and B → D¯ l νl for l = τ, µ is studied on the basis of the Standard Model in the leading approximation of heavy-quark effective theory. It is shown that a nonzero transverse polarization appears owing to electromagnetic final-state interactions at the one-loop level. Diagrams involving D and D∗ mesons in the intermediate state and making a nonzero contribution to the transverse polarization of the outgoing lepton are considered. If only these mesons are taken into account in evaluating the mean values of the 0 − + + 0 + −3 τ-lepton polarization in the decays B → D τ ντ and B → D¯ τ ντ , the results are 2.60 × 10 and −1.59 × 10−3, respectively. The corresponding values of the transverse muon polarization averaged over the Dalitz plot are 2.97 × 10−4 and −6.79 × 10−4. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION amounts to 4.76 × 10−4 [2]. The transverse polariza- tion of the muon in the decay K+ → µ+νπ0 is as Although the Standard Model has correctly pre- small as about 10−6 in the Standard Model [4, 5]; dicted the results of many experiments, the mecha- therefore, it is advisable to seek effects of new physics nism of CP violation has yet to be established conclu- in this decay. The current experimental value obtained sively. In the Standard Model, CP violationisdueto in the KEK-E246 experiment [6] for the transverse the complex-valuedness of the Cabibbo–Kobayashi– polarization of the muon in this process is Maskawa matrix; however, there are many models − that offer different mechanisms of CP violation. The PT = 0.0042 + 0.0049(stat.)+0.0009(syst.). (1) Weinberg model involving three doublets of Higgs bosons [1] provides an example of a model where CP This result does not give sufficient grounds to state violation occurs in the Higgs sector. Investigation that the transverse polarization of the muon in the of CP-odd phenomena may contribute to obtaining process being discussed is due to new physics. How- deeper insight into the mechanism of CP violation ever, the accuracy of the experiments will be improved and, hence, to clarifying some basic questions of in the near future, and this looks quite promising for elementary-particle physics. studying CP violation. + → As an example of physical quantities sensitive to The T -odd correlation in the decay K CP violation, we can consider the transverse polar- π0µ+νγ (it is defined in terms of the distribution ization of leptons and T -odd correlations in weak of the decay width with respect to the T -odd kine- decays. The transverse polarization of the muon in the matical variable pπ · [pµ × q]) is also small within decays K+ → µ+νπ0 and K+ → µ+νγ has been the the Standard Model [7]. The reason is identical to subject of many theoretical and experimental investi- that in the case of the transverse polarization of the gations. In some extensions of the Standard Model, it outgoing lepton in Kl2γ decays. It is expected that appears even at the tree level [2, 3]. In the Standard this T -odd correlation will be measured in the OKA Model, the transverse polarization of an outgoing experiment [8] on the basis of some 7 × 105 events. lepton is small because it vanishes in the tree approx- The transverse polarization of the outgoing lepton imation. A nonvanishing (CP-conserving) contribu- can be measured not only in the decays of K mesons, tion to this quantity is due to final-state interaction. but also in the decays of B mesons. The latter de- The transverse polarization of the muon in the decay cays are particularly sensitive to the CP-violating K+ → µ+νγ arises at the one-loop level and varies interactions of Higgs particles with fermions. Obvi- in the range (0.0–1.1) × 10−3 over the Dalitz plot. Its ously, the transverse lepton polarization induced by average over the kinematical domain Eγ ≥ 20 MeV the interactions of Higgs particles in the decay K → 1063-7788/03/6607-1289$24.00 c 2003 MAIK “Nauka/Interperiodica” 1290 BRAGUTA et al. µ ¯ µ µ ¯ µ πµν is (mbmτ )/(msmµ) ∼ 800 times less than the where the quantities V = bγ c and A = bγ γ5c are ∗ analogous polarization in the decay B → D(D )τντ . associated with B-meson decays. The form factor µ In [9–11], effects associated with the emergence D(k)|A |B(p) is equal to zero because an axial of a CP-violating component in the transverse polar- vector cannot be generated by the two momentum ization of the lepton in the decay B → D(D∗)lν were vectors that we have at our disposal. The definition of the Levi-Civita tensor appearing in the expression considered within various extensions of the Standard ∗ Model. If CP violation occurs in the Higgs sector, for the transition current D (k, )|V µ|B(p) is such the transverse polarization of the τ lepton can take that e0123 =1. values in the region PT < 1 [9, 10]; the corresponding We calculate the transverse polarization in the constraint in the leptoquark model is PT < 0.26 [11]. leading order of HQET—that is, under the assump- Thus, the magnitude of transverse polarization is tion that mb,mc →∞. In this approximation, all form expected to be quite large in some extensions of the factors can be expressed in terms of the Isgur–Wise Standard Model. In order to perform an exhaustive function ξ(ω) [12, 13]; accordingly, expressions (2) investigation of this issue, it is necessary, in our can be recast into the form opinion, to begin by computing, within the Stan- µ ξ(ω) µ µ dard Model, the transverse polarization of outgoing D(k)|V |B(p) = √ (mDp + mBk ), (3) leptons in the above decay processes. The present mDmB study is devoted to analyzing the CP-conserving D(k)|Aµ|B(p) =0, component of the transverse polarization of the lepton ∗ ξ(ω) ∗ 0 → − + + → ¯ 0 + D (k, )|V µ|B(p) = −i√ eµναβ k p , in the decays B D l νl and B D l νl, m m ν α β where l = τ,µ. D B ξ(ω) For the sake of simplicity, our calculations are D∗(k, )|Aµ|B(p) = √ performed on the basis of heavy-quark effective theory mDmB (HQET) in the leading order of the expansion in pow- ∗µ ∗ µ × ((mBmD + pk) − ( p)k ), ers of 1/mQ. We show that the transverse polarization does not vanish only if the phases of the form factors where ω =(pk)/(mDmB). that appear in the expression for the amplitude are In addition to the matrix elements given above, we different. It is electromagnetic final-state interaction will need the matrix elements of the vector current ∗ that gives rise to a nonvanishing phase even in the describing the D–D and D –D transitions. Within one-loop approximation, and this in turn leads to a HQET, the form factors that appear in the expressions nonzero value of the transverse polarization. Also, we for these matrix elements are expressed in terms of the take into account the contribution of only the (D, D∗) Isgur–Wise function as doublet in our calculations. D(p )|cγ¯ µc|D(k) = ξ(ω )(p µ + kµ), (4) The ensuing exposition is organized as follows. Expressions describing the form factors for the pro- | µ | ∗ − ξ(ω ) µναβ D(p ) cγ¯ c D (k, ) = i e νpαkβ, cesses under study are given in Section 2. The pro- mD cedure for computing the transverse polarization in where ω =(kp )/(mDmB). In formulas (3) and (4), question is outlined in Section 3. The results of nu- we disregard the difference of the D-andD∗-meson merical computations are presented and discussed in masses because, in the leading order of HQET, this Section 4. difference does not contribute to the quantities in question. 2. FORM FACTORS FOR THE PROCESS For the function ξ, we use the standard parametri- UNDER STUDY zation In calculating the transverse polarization of the ξ(ω)=1− ρ2(ω − 1). (5) outgoing lepton (that is, the polarization component In the numerical computations, we employ the value orthogonal to the decay plane), we will need some 2 form factors; the most general expressions for them of ρ =0.94, which was obtained in [14] on the basis have the form of the potential quark model and which is in agree- ment with experimental data from [13]. The ultimate | µ| µ µ µ − µ D(k) V B(p) = f+(p + k )+f−(p k ), (2) result cannot change significantly upon replacing it D(k)|Aµ|B(p) =0, by another value because the kinematical domain of ∗ ∗ the decay is rather narrow. The parameter ω for the D (k, )|V µ|B(p) = −iveµναβ k p , ν α β decay B → Dτν ω varies from 1 to (m2 + m2 − ∗ µ ∗ µ ∗ µ τ B D D (k, )|A |B(p) = a1( ) + a2( p)p 2 mτ )/(2mB mD)=1.43, whereas that for the decay ∗ µ + a3( p)k , B → Dµνµ varies from 1 to 1.59. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 TRANSVERSE POLARIZATION OF THE LEPTON 1291 In the calculations, we also use the matrix ele- It should be noted that expression (11) offers the µ ∗ µ µ ments D|Jem|D and D|Jem|D ,whereJem is the most general form of the amplitude for the process electromagnetic current, which can be represented as under study for the case where only a left-handed the sum of the “heavy” and “light” components, neutrino is involved. Substituting the matrix ele- µ µ µ ments (3) of the transition currents into (10), we Jem = J + J . (6) h l obtain expressions for C1 and C2 in the leading order The matrix elements of the “heavy” component of of HQET, the electromagnetic current can be expressed in terms ξ(ω) √ of the transition amplitudes (4) as C1 = (mD + mB), (12) mBmD | µ| − µ µ D(p ) Jh D(k) = qcξ(ω )(p + k ), (7) ξ(ω) C2 = √ (mBml). ∗ ξ(ω ) m m D(p )|Jµ|D (k, ) = iq eµναβ p k , B D h c m ν α β D In the B-meson rest frame, the partial width with where qc is the c-quark charge. The matrix elements respect to the decay B → Dl+ν has the form µ l of Jl can be represented in the form | |2 M 4 µ − − − | | 1 2 µ µ dΓ= (2π) δ(p pD pl pνl ) (13) D(p ) Jl D(k) = qlfl (q )(p + k ), (8) 2mB D(p )|Jµ|D∗(k, ) = iq βf2(q2)eµναβ p k , d3p d3p d3p l l l ν α β × D l νl 3 3 3 , (2π) · 2ED (2π) · 2El (2π) · 2Eν where ql is the charge of the light quark in the meson l 1 2 2 2 and fl (q ) and fl (q ) are the form factors describ- where summation is performed over the spin states of ing the q2 dependence of the matrix elements, these the muon and photon involved. form factors being normalized in such a way that We now introduce a unit vector directed along (1,2) the spin of the lepton in its rest frame, s, and unit fl (0) = 1. The constant β obtained in [15] is equal − vectors along the longitudinal, normal, and transverse to 1.9 GeV 1.Theq2 dependence of the form factors i 2 polarizations of the lepton, ei (i = L, N, T ). In terms fl (q ),i=1, 2, that is used in our computations was of these vectors, the expression for the square of the obtained under the assumption that the contribution matrix element for a transition to a state characterized of the ω and ρ resonances is dominant [16]. Disre- by a specific polarization of the lepton has the form ω ρ garding the difference of the -and -meson masses, 2 i |M| = ρ0[1 + (PLeL + PN eN + PT eT ) · s], (14) we reduce the expressions for fl to the form 1 where ρ0 is the probability density averaged over spin f i = f (q2)= ,i=1, 2, (9) l l q2 states in the Dalitz plot. The unit vectors ei are ex- − pressed in terms of the momenta of final particles as 1 2 mρ pl pl × (pD × pl) where m is the ρ-meson mass. eL = , eN = , (15) ρ |pl| |pl × (pD × pl)| pD × pl eT = . 3. TRANSVERSE POLARIZATION |pD × pl| OF THE LEPTON For this definition of the vectors e , P , P ,andP ∗ + i T L N The amplitude of the decays B → D(D )l νl has stand for, respectively, the transverse, longitudinal, the form and normal components of the lepton polarization. √GF ∗ | µ − µ| The probability density ρ0 is given by M = Vcb D V A B u¯(pν)(1 + γ5)γµv(pl), 2 2 | |2 | |2 | |2 ρ0 = GF Vcb (4 C1 (ppν)(ppl)+2C2 (pνpl) (10) (16) where G is the Fermi constant and V ∗ is the rele- − | |2 2 − ∗ F cb 2 C1 (pν pl)mB 4mlRe(C2C1 )(ppν)). vant element of the Cabibbo–Kobayashi–Maskawa matrix. The transition currents D|V µ − Aµ|B are The expression for the transverse polarization can given in the preceding section. For the ensuing calcu- then be represented in the form ρ lations, it is convenient to parametrize the amplitude P = T , (17) + T for the process B → Dl νl as follows: ρ0 G ∗ where ρ is given by M = √F V u¯(p )(1 + γ )(С pˆ + C )v(p ). (11) T cb ν 5 1 2 l 2 | |2 ∗ | × | 2 ρT =4GF Vcb mBIm(C1C2 ) pD pl . (18) PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1292 BRAGUTA et al. τ ν which is equal to 500 or 160 MeV for a lepton or l a muon, respectively. These values indicate that the difference of the D-andD∗-meson masses must be B l taken into account in performing integration over the phase space. γ D(D*) It is obvious that the quantity under study re- D ceives contributions from the excited states of the D meson as well. We assume that such contributions Fig. 1. Feynman diagrams describing the contribution have only a small effect on the results because the to the transverse polarization of the muon in the decay 0 − Isgur–Wise function describing a transition to an B → D lνl in the one-loop approximation of the Stan- dard Model. excited state of the D meson is smaller than the function defined in (5). For example, the Isgur–Wise function for the B-meson transition to the (0+, 1+) From formula (18), it can be seen that the trans- doublet, τ (ω), was evaluated in [18] by using QCD verse polarization of the muon is nonzero only if the 1/2 sum rules. The value of τ1/2(1) obtained there is phases of the form factors C1 and C2 are different. In the tree approximation of the Standard Model, the 0.24. It is obvious that the light-quark contribution transverse polarization of the muon vanishes because is proportional to τ1/2(1), whereas the heavy-quark the above form factors are real-valued there. A non- contribution is proportional to the square of τ1/2(1). vanishing transverse polarization results from final- Moreover, the diagrams involving a heavy D meson in state interaction. The relevant phase difference can the intermediate state make a nonvanishing contribu- be calculated on the basis of the unitarity condition, tion to the transverse polarization only in the domain 0 → 2 2 as was done in [17] for the decay K πµν.The bounded by the condition (pl + pD) ≥ (mD + ml) . diagrams contributing to the transverse polarization For the D∗ meson, this condition leads to only an ofthemuonareshowninFig.1.Here,wetakeinto insignificant decrease in the physical domain (mD∗ − account the contribution to the transverse polariza- ∼ + + ∗ mD 0.15 GeV), whereas, for the (0 , 1 ) doublet, tion of the muon only from D and D mesons. The it reduces the physical domain substantially, because contribution of the diagrams in Fig. 1 has the form the mass difference in this case ranges up to about G ∗ α 0.5 GeV. Therefore, the contribution to the average ImM = √F V (19) 2 cb 2π polarization in the latter case is considerably smaller than that in the former case. dρ × u¯(p )(1 + γ )γ (kˆ − m )γ v(p ) The details of the procedure for computing the k2 ν 5 σ l l λ l γ average polarization are presented in the Appen- × D|Jλ |n n|V σ − Aσ|B , dices. The integrals used in the computations by em formula (19) are listed in Appendix 1, while the n=D,D∗ ultimate result is given in Appendix 2. where kl is the lepton momentum, kD is the mo- mentum of the D (or D∗) meson in the intermediate 2 − 2 4. RESULTS AND DISCUSSION state, kγ =(kD pD) is the square of the momen- tum transfer, and dρ is an element of the two-body Before proceeding to discuss the results, we intro- phase space. The expressions for the matrix elements duce the notation of the transition currents D|Jλ |n and n|V σ − mB mB em E = x, E = y. (21) Aσ|B are given in Section 2. D 2 l 2 It should be noted that the D-meson contribution All the results will be formulated in terms of this in (19) involves an infrared divergence, but it does not notation. contribute to the polarization. The soft-photon con- tribution factorizes completely; therefore, it does not Three-dimensional plots over the kinematical give rise to a nonvanishing phase, which is necessary domain (x, y) for the transverse polarization of the 0 → − 0 → − for the emergence of a nonzero transverse polariza- lepton in the reactions B D τντ ,B D µνµ, + 0 + 0 tion. For this reason, we do not take the divergent B → D¯ τντ ,andB → D¯ µνµ are given in Figs. 2, terms into consideration [17]. The D∗-meson contri- 3, 4, and 5, respectively. The corresponding isolines bution in (19) does not involve an infrared divergence are depicted in Figs. 6–9. The average values of the because, in this case, the lower bound on the momen- transverse polarization are as follows: tum transfer is given by the expression −3 0 − PT =2.60 × 10 ,B→ D τντ , − 2 2 − 2 ml ( kγ)min =(mD∗ mD) , (20) × −4 0 → − ml + mD PT =2.97 10 ,B D µνµ, PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 TRANSVERSE POLARIZATION OF THE LEPTON 1293 PT PT 0.006 0 0.004 –0.002 0.8 0.002 0.9 0 –0.004 0.6 0.8 0.8 0.8 0.4 y y 0.9 0.9 0.2 x 0.7 x 1.0 1.0 1.1 Fig. 2. Transverse polarization of the τ lepton in the decay 0 − Fig. 5. Transverse polarization of the muon in the decay B → D τντ on a three-dimensional plot. + 0 B → D¯ µνµ on a three-dimensional plot. y × 3 1.0 êí 10 –PT 0 0.9 –0.001 0.8 5.0 3.0 2.0 1.0 0.8 –0.002 0.6 0.8 0.4 y 0.9 0.7 0.2 x 1.0 1.1 0.7 0.8 0.9 1.0 x Fig. 3. Transverse polarization of the muon in the decay Fig. 6. Isolines for the transverse polarization of the τ − 0 − 0 → B → D µνµ on a three-dimensional plot. lepton in the decay B D τντ . y 1.0 × 3 êí 10 0.8 – 1.0 0.5 0.2 0.1 PT 0.6 0.004 0.002 0.4 0 0.9 0.2 0.8 0.8 y 0.9 x 0.7 0.7 0.8 0.9 1.0 1.1 x 1.0 Fig. 7. Isolines for the transverse polarization of the muon 0 − Fig. 4. Transverse polarization of the τ lepton in the decay in the decay B → D µνµ. + 0 B → D¯ τντ on a three-dimensional plot. ization of the lepton can be represented as the sum − × −3 + → ¯ 0 PT = 1.59 10 ,B D τντ , of the “heavy” and the “light” component. The form −4 + 0 PT = −6.79 × 10 ,B→ D¯ µνµ. factors for photon–heavy-quark and photon–light- quark interactions depend on the scale parameter that As was mentioned above, the transverse polar- varies from about mρ to about mD over the range PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1294 BRAGUTA et al. y y × 3 1.0 – êí 10 1.0 × 3 – êí 10 0.8 0.9 4.0 1.7 1.0 0.5 0.3 3.0 0.6 0.8 2.0 0.4 1.0 0.7 0.2 0.7 0.8 0.9 1.0 x 0.7 0.8 0.9 1.0 1.1 x Fig. 8. Isolines for the transverse polarization of the τ + 0 Fig. 9. Isolines for the transverse polarization of the muon lepton in the decay B → D¯ τντ . + 0 in the decay B → D¯ µνµ. of variation of the momentum transfers kγ . For this reason, the contribution of the diagrams involving a µ µ − (PpD) µ l = pD P , neutral D¯ 0 meson to the transverse polarization of (P 2) + → ¯ 0 the lepton in the decay B D lν is evanescently 1 2 2 − 2 2 2 PkD = (P + mD∗ ml ), small at low momentum transfers (kγ mρ)and 2 is nonzero over the bulk of the kinematical domain, 2 2 2 − (PpD) 2 2 l = mD , where kγ mρ. However, the contributions of the P 2 light and heavy quarks partly cancel each other owing 2 2 2 (PkD) ¯ 0 l = m ∗ − , to the fact that the D meson is neutral. As a con- 1 D P 2 sequence, the contribution of the diagram involving a D∗ meson in the intermediate state is greater than 2 2 2 − (PkD)(PpD) η = mD + mD∗ 2 . the analogous contribution involving a D meson. The P 2 distinction between these contributions accounts for In this notation, the integrals under consideration can the fact that the sign of the average polarization in be written as + 0 the decay B → D¯ lνl is opposite to that in the decay 0 − dρ 2 B → D lν . J1 = fl(k ) l k2 γ In conclusion, we note that the average values of γ the transverse polarization in the Standard Model are π 1 −η2 − 2 l2l2 considerably smaller than the corresponding average = √ ln 1 4 −l2P 2 −η2 +2 l2l2 values in various extensions of this model [9–11]. 1 The results obtained in [9–11] give sufficient grounds −η2 − 2 l2l2 + m2 to conclude that B-meson decays may provide an − ln 1 ρ , − 2 2 2 2 efficient tool for seeking effects of new physics. η +2 l1l + mρ 2 ACKNOWLEDGMENTS 2 π mρ J2 = dρf (k )=− √ l γ 4 − 2 2 We are grateful to V.V. Kiselev and A.K. Likhoded lP for stimulating discussions and enlightening com- −η2 − 2 l2l2 + m2 ments. × ln 1 ρ , − 2 2 2 2 This work was supported in part by the Russian η +2 l1l + mρ Foundation for Basic Research (project nos. 99-02- 16558, 00-15-96645), the Ministry of the Russian − 2 2 l1P Federation for Higher Education (grant no. E00-3.3- J3 = dρ = π 2 , 62), and CRDF (grant no. MO-011-0). P α kD 2 α α APPENDIX 1 fl(k )dρ = a1P + b1l , k2 γ To perform integrations in formula (19), we use the γ following notation: PkD µ µ µ a1 = J1, P = pl + pD, P 2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 TRANSVERSE POLARIZATION OF THE LEPTON 1295 2 η 1 1 2 b = J − J . c3 = (η b2 − B2), 1 2l2 1 2l2 2 2l2P 2 PkD α 2 α α d3 = 2 2 d2, kDfl(kγ )dρ = A1P + B1l , P l 1 2 − − 2 e3 = 2 2 (η d2 D2) 2 b3, PkD 2(l ) l A1 = J2, P 2 PkD f3 = (b2 − 2a2). η2 m2 (P 2)2 B = J − ρ (J − J ). 1 2l2 2 2l2 2 3 α β γ γ β α α β α β kDkDkDdρ = A3(gαβ P + gαγ P + gβγP ) kDkD 2 αβ P P fl(k )dρ = a2g + b2 k2 γ P 2 γ β α γ + B3(gαβ l + gαγ l + gβγl ) lαlβ α β γ β α γ γ α β α β β α + C3(l P P + l P P + l P P ) + c2(l P + l P )+d2 2 , l α β γ β γ α γ α β + D3(l l P + l l P + l l P ) 2 2 l η B1 α β γ α β γ a = 1 J − b + , + E3l l l + F3P P P , 2 2 1 4 1 4 (Pk )2 D − PkD b2 = J1 a2, A = A , P 2 3 P 2 2 PkD 1 1 c2 = b1, B = (η2A − m2A + m2l2J ), P 2 3 2 2 ρ 2 ρ 1 3 2l 3 η2 B 2 − 1 − 1 2 2 (PkD) d2 = b1 a2. C = η B − m B − J 2 2 3 2l2P 2 2 ρ 2 P 2 3 α β 2 α β 2 αβ P P l1 k k f (k )dρ = A g + B + J3 , D D l γ 2 2 P 2 3 α β α β β α l l PkD + C (l P + l P )+D , D3 = D2, 2 2 l2 P 2l2 1 2 2 2 2 2 2 − − l η m B1 E3 = 2 2 (η D2 mρD2) 2 B3, A = 1 J − B + ρ , 2(l ) l 2 2 2 4 1 4 PkD (Pk )2 F3 = (B2 − 2A2). B = D J − A , (P 2)2 2 P 2 2 2 PkD C2 = B1, P 2 APPENDIX 2 2 2 η m B1 D = B − ρ − A . 2 2 1 2 2 Let us first consider the contribution of the D meson to the transverse polarization of the lepton. We ∗ α β γ represent the expression for the quantity Im(C1C ), kDkDkD γ β α 2 dρ = a3(gαβ P + gαγ P + gβγP ) which appears in formula (18), as the sum of the k2 γ “heavy” and the “light” component, γ β α + b3(gαβ l + gαγ l + gβγl ) ∗ α ml α β γ β α γ γ α β Im(C1C2 )= ξ(ω) + c3(l P P + l P P + l P P ) 2π mBmD α β γ β γ α γ α β ρ2 + d3(l l P + l l P + l l P ) × 2 − ql (1 + ρ )Cl Cl α β γ α β γ mBmD + e3l l l + f3P P P , 2 − 2 − ρ Pk qc (1 + ρ )Ch Ch , a = D a , mBmD 3 P 2 2 1 where ξ(ω) is the Isgur–Wise function; ρ2 is the slope b = (η2a − A ), 3 2l2 2 2 of this function; and ql and qc are the charges of the PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1296 BRAGUTA et al. light and the c quark, respectively. The coefficients Cl REFERENCES and Cl appearing in the last formula are given by 1. S. Weinberg, Phys. Rev. Lett. 37, 657 (1976). Pp 2. V. Braguta, A. Likhoded, and A. Chalov, hep- C =2b D (m2 m2 − M 2P 2), l 1 P 2 B l ph/0105111. 3. G. Belanger and C. Q. Geng, Phys. Rev. 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B 232, 113 D-meson mass, whereupon the integral J1 becomes divergent; however, this integral does not contribute (1989); 237, 527 (1990). to the transverse polarization because the transverse 13. M. Neubert, Phys. Rep. 245, 259 (1994). polarization does not involve divergences. For this 1 4 . V. V. K i s e l e v, M o d . P h y s . L e t t . 10, 1049 (1995). reason, it can be safely set to zero in all formulas. In 15. R. Casalbuoni et al., hep-ph/9605342. order to obtain the coefficients Ch and Ch,wemust 16. P.Colangelo et al., Phys. Lett. B 316, 555 (1993). 2 expand the coefficients Cl and Cl in powers of 1/mρ 17. L. D. Okun’ and I. B. Khriplovich, Yad. Fiz. 6, 821 2 2 2 (1967) [Sov. J. Nucl. Phys. 6, 598 (1967)]. up to linear terms and set 1/mρ to ρ /(2mD). 18. P.Colangelo, G. Nardulli, and N. Paver, Phys. Lett. B We do not present here the resulting formulas for 293, 207 (1992). the D∗-meson contribution to the transverse polar- ization since they are rather cumbersome. Translated by R. Rogalyov PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1297–1309. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1337–1349. Original Russian Text Copyright c 2003 by Mikheev, Narynskaya. ELEMENTARY PARTICLES AND FIELDS Theory Energy and Momentum Losses of a Magnetized Plasma via Familon Emission N. V. Mikheev* and E. N. Narynskaya Yaroslavl State University, Sovetskaya ul. 14, Yaroslavl, 150000 Russia Received February 19, 2002; in final form, December 5, 2002 Abstract—Familon emission from a dense magnetized plasma in the processes e− → e−φ and e− → µ−φ is investigated.The contribution of these processes to the energy losses of a supernova remnant is calculated.It is shown that, at a late stage of the cooling of a supernova remnant, the energy loss of a plasma via familon emission may become commensurate with the loss via neutrino emission.It is found that, because of asymmetry of familon emission in the process e− → µ−φ, there arises a force acting on the plasma. c 2003 MAIK “Nauka/Interperiodica”. 1.INTRODUCTION where cij is a model-dependent dimensionless factor on the order of unity; F is the scale of horizontal- In the past decades, there has been permanent symmetry breaking; Φ and Ψi are, respectively, the interest in studying electroweak-interaction models familon and the fermion wave function; the subscripts that involve an extended Higgs sector.Special at- i and j indicate fermion flavors; and g2 + g2 =1. tention has been given to light and strictly massless a v In the particular case of interaction diagonal in particles playing the role of pseudo-Goldstone and flavors (i = j), the conservation of the vector current Goldstone bosons, including axions, arions, familons, makes it possible to recast the Lagrangian in (1) into and Majorons [1].The possible existence of such par- the form ticles has many important implications both for the ci α theoretical aspects of particle physics and for astro- L = (barΨiγαγ5Ψi)∂ Φ. (2) physical applications. F At the present time, the Standard Model due to The existing astrophysical estimates are close to Weinberg, Salam, and Glashow is the most conve- those that are based on laboratory experiments, both 9 nient theory that describes the physics of electroweak yielding F>(1.1–3.1) × 10 GeV [6]. interactions.Despite its impressive successes in Since the familon is a particle whose interaction describing experimental data, some problems have with matter is weak, it may penetrate deep into a sub- remained, however, unsolved within this model. stance, whence it follows that various processes in- These include those concerning the existence of a volving its emission may provide an additional mech- few fermion generations and the mass difference anism through which stars and other astrophysical between particles belonging different generations. objects lose energy.Moreover, the possible asym- These problems could possibly be solved by intro- metry of familon emission may generate a reactive ducing an additional horizontal symmetry between force that in turn could clarify the problem of pulsar fermion generations [2, 3].A spontaneous breakdown velocities. of this symmetry may lead to the observed hierarchy In studying quantum processes under astrophys- in the fermion-mass spectrum, a Goldstone boson, ical conditions, it is necessary to take into account referred to, in this case, as a familon, concurrently the effect of an active external medium, and not arising in the theory (see, for example, [4]). only may this be a plasma, but also a strong mag- netic field is capable of playing such a role.By In general, familon coupling to fermions is of a strong, one usually means a magnetic field whose nondiagonal character and can be described in terms strength considerably exceeds the critical value of of the Lagrangian [5] B = m2/e =4.41 × 1013 G.1) According to the c e e L = ij (Ψ¯ γ (g + g γ )Ψ )∂αΦ+h.c., (1) concepts adopted at present, fields of strength in F i α v a 5 j 1)Here, use is made of a natural system of units where c = = *e-mail: [email protected] 1,ande>0 is an elementary charge. 1063-7788/03/6607-1297$24.00 c 2003 MAIK “Nauka/Interperiodica” 1298 MIKHEEV, NARYNSKAYA the range B ∼ 1015–1017 G could be generated in Considering that the energy of an ultrarelativistic astrophysical cataclysms, such a supernova explosion electron in a magnetic field in the Nth level is or the merger of neutron stars [7]. E p2 +2eBN, An external magnetic field playing, along with a z plasma, the role of an active medium has a pro- where pz is the projection of the electron momentum nounced effect of the properties of particles and on onto the magnetic-field direction, we obtain their interactions.Owing to a change in the disper- E2 µ2 sion relation, channels forbidden in a vacuum may Nmax ∼ 1. open in this case—for example, the splitting of a 2eB 2eB photon into two photons, γ → γγ [8], and photon Under the physical conditions specified by (4), − decay into an electron–positron pair, γ → e e+ [9], in which case a large number of Landau levels is become possible.Further, we would like to emphasize excited, investigation of quantum processes effec- that, in the presence of an external magnetic field, tively reduces to calculations in a constant crossed the interaction specified by Eq.(1) may lead to the field (B ⊥ E, |B| = |E|), the technique of such cal- synchrotron radiation of a familon, e− → e−φ,and culations being well known [11, 12].This is be- the “decay” of an electron, e− → µ−φ [10]. cause an arbitrary relatively weak and slowly varying It is important to note that an external magnetic external electromagnetic field is close, in the rest field also induces a new effective familon–photon in- frame of an ultarelativistic electron, to a crossed field.As a matter of fact, the result in the leading teraction of the form approximation then depends on only one dynamical ˜αβ 2 1/2 Lφγ = gφγF (∂βAα)Φ, (3) invariant, [e (pFFp)] ,wherepµ is the particle 4- momentum [here and below, it is assumed that the where Aµ is the 4-potential of a quantized electro- magnetic field, F is the strength tensor of the ex- tensor indices of 4-vectors and tensors appearing αβ within parentheses are consecutively contracted— ternal magnetic field, F˜αβ =(1/2)εαβρσ Fρσ is its dual for example: (pFFp)=pαFαβ Fβγpγ].We would like counterpart, and gφγ is the effective coupling constant to note that calculations in a crossed field possess a for familon–photon interaction in the external mag- high degree of generality and are of interest in and netic field.Familon photon transitions generate a of themselves, since they appear to be a relativistic new channel of synchrotron familon radiation from limit of the corresponding calculation in an arbitrary γ∗ an electron through an intermediate plasmon, e− → relatively weak electromagnetic field. e−φ. Under the conditions specified in (4), which are In the present study, we examine familon radiation considered here, plasma electrons are ultrarelativistic from a dense magnetized plasma through the pro- particles.In the ensuing calculations, we will there- − − − − fore disregard the electron mass in all cases where this cesses e → e φ and e → µ φ, which are forbid- does not lead to confusion. den in a vacuum by the law of energy–momentum conservation, but which become possible in an ex- For the processes being considered, the differential ternal magnetic field owing to a nontrivial particle probability per unit time is given by 2 kinematics. |S| dW = dn (1 − f − )dn , (5) We consider the physical situation in which the T f e φ typical energy of plasma electrons is a dominant pa- where S is the S-matrix element for the relevant rameter of the problem; that is, process; T is the total interaction time; and dnf and 2 2 2 µ ,T eB me. (4) dnφ are, respectively, the number of states of a final fermion in a crossed field2) and the number of familon Such conditions could be realized, for example, states, in the core of a supernova, where, according to 2d3p d3k present-day concepts, the plasma chemical poten- dnf = 3 V, dnφ = 3 V. (6) tial is µ ∼ 500me, while the plasma temperature is (2π) (2π) T ∼ 60 me [1].In this case, even very strong magnetic 17 Here, V = LxLyLz is a normalization volume; k is fields, those of strength B ∼ 10 G, satisfy the condition in (4); therefore, they can be considered the familon-momentum vector; p is the vector of the as relatively weak.In this case, plasma electrons 2)It will be shown below that, in a relatively weak magnetic occupy a large number of Landau levels.In order to field, in which case electrons occupy a large number of Lan- demonstrate that this is indeed so, we will estimate dau levels, the number of states is approximately described the maximum number of populated Landau levels. by the same formula. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 ENERGY AND MOMENTUM LOSSES 1299 momentum that fixes the state of the final fermion in (a)(b) a crossed field; and f − is the final-fermion distribu- e φ(q) φ(q) tion with allowance for the presence of a plasma, the invariant form of this distribution being γ* 1 fe− = − , e((p u) µ)/T +1 e–(p) e–(p') e–(p) e–(p') where uα is the medium-velocity 4-vector, which, in α Fig. 1. Diagrams describing familon emission by an elec- the plasma rest frame has the form u =(1, 0). tron in a magnetized plasma. It is worthy of note that it is the mean energy and momentum losses of a magnetized plasma via Considering that integration with respect to p familon emission rather than the probabilities of the y determines the degeneracy multiplicity s of an energy processes that are of greatest practical interest for eigenstate according to the relation possible astrophysical applications.It is convenient to L dp eBL L begin by calculating the mean energy and momentum s =2 y y = x y losses of one plasma electron, which are determined 2π π by the 4-vector and going over from summation over N to integration dE dp with respect to the transverse momentum p⊥,we Qµ = −E( , ), (7) dt dt obtain the total number of primary-electron states in the form where E and p are, respectively, the energy of this 2d3p electron and its 3-momentum vector.The zeroth dn − V. (11) 0 e (2π)3 component Q is associated with the mean energy N lost by one plasma electron per unit time, while the spatial components Q are associated with the mo- Thus, we can see that, in a relatively weak field, mentum loss of the electron per unit time.Therefore, the number of electron states is given by the same the spatial components determine the force acting on formula as in a crossed field [see (6)].With allowance the plasma electron from the emitted familon. for Eq.(11), the de finition in (9) can be recast into the form Thecomponentsof4-vector(7),whichspecifies 2d3p (Q0, Q) the losses, can be calculated by the formula (˙ε, F)= f − . (12) (2π)3 e E µ qµEdW, Q = (8) In the present study, we further calculate the en- ergy and momentum losses of a magnetized plasma where qµ =(ω,k) is the familon 4-momentum. via familon emission.The ensuing exposition is or- In order to estimate the volume density of energy ganized as follows.In Section 2, we examine two lost by a plasma per unit time via familon emission channels of the synchrotron radiation of familons from electrons in a magnetized plasma.In Section 3, we (familon luminosity), ε˙,andthevolumedensityof − → − force acting on the plasma from the emitted familon, consider the “decay” process e µ φ for the case F, it is necessary to sum the energy and momentum where both electrons and muons occupy a great num- losses of one plasma electron over all states of primary ber of Landau levels.In Section 4, we present nu- electrons; that is, merical estimates of the results that we obtained. In particular, we estimate the contributions of the (Q0, Q) processes in question to the energy loss of the core of (˙ε, F)= dn − f − , (9) e e VE a supernova and the asymmetry of familon emission N in the process e− → µ−φ. where fe− is the distribution of primary electrons. In the plasma rest frame, where the electromag- 2.SYNCHROTRON RADIATION netic field reduces to a purely magnetic field [B = OF A FAMILON BY PLASMA ELECTRONS µ (0, 0,B) in the gauge A =(0, 0,Bx,0)], the number 2.1. Process e− → e−φ in a Model Involving a Direct of electron states is Familon–Fermion Coupling dpydpz dne− = LyLz (10) In this section, we examine familon emission by (2π)2 plasma electrons due to a direct familon–fermion PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1300 MIKHEEV, NARYNSKAYA coupling.This proceeds via the process e− → e−φ to the process e− → e−φ induced by a direct familon– (see Fig.1 a).The S-matrix element for this process electron coupling.We have can be obtained from the Lagrangian in (2) by directly 2 2 cφceme eB 2/3 13/3 substituting solutions to the Dirac equation into it; − − ε˙(e →e φ) 4 2 ( ) T , (17) that is, π F µ − where √ ce S(e−→e−φ) = (13) 14 3 1/6 3 1 13 F 2ωV cφ = ( ) Γ ( )ζ( ) 3.38. 81 4 3 3 × d4xψ¯ (p,x)qµγ γ ψ (p, x)eiqx, e µ 5 e We note that our result in (17) is valid under the conditions in (4), which are more lenient than the where ψe stands for solutions to the Dirac equation α condition B Be, which is necessary for the validity in a constant crossed field [12]; q =(ω,k) is the of the result in [13]. α α familon 4-momentum; and p =(p0, p) and p = (p , p) 0 are constant 4-vectors that characterize the − − states of, respectively, the primary and the final elec- 2.2. Process e → e φ in a Model without Direct tron in a crossed field and which become coincident Familon–Fermion Coupling with the particle momentum upon switching the field. In an external magnetic field, not only may a By means of the procedure for performing calcula- familon interact with an electron at the tree level, tions in a crossed field, the result for the probability (5) but it may also feature an induced interaction via of the process e− → e−φ can be expressed in terms an intermediate photon.As a result, the channel of familon emission from plasma electrons through a of a single integral with respect to the dynamical ∗ − γ − parameter χq;thatis, virtual plasmon as an intermediate state, e → e φ 2 4 (see diagram in Fig.1 b),opensupinanactive ceme − − external medium (field + plasma). W(e →e φ) = 2 2 (14) 2π F p0 The amplitude for this process can be obtained χ from the Lagrangian dχq χq 4/3 × ( ) (−Φ (z))(1 − f − ), 1/3 e ˜αβ ¯ µ (χ − χq) χ L = gφγF (∂βAα)Φ + e(ΨeA γµΨe), (18) 0 g where φγ is the effective familon–photon coupling where fe− is the distribution of the final electron and constant in a magnetized plasma.In general, this Φ(z) is the Airy function defined as coupling involves a contribution induced by the ex- ∞ ternal magnetic field and a contribution caused by x3 the presence of an electron–positron plasma.Direct Φ(z)= dx cos(zx + ), (15) 3 calculations (see Appendix) reveal that, under the 0 physical conditions specified in (4), which correspond dΦ(z) tothecoreofasupernova,theplasmacontribution Φ (z)= . g dz to φγ is much smaller than the field contribution and can be discarded.In this case, the coupling constant We have also used the following notation: gφγ proves to be independent of the familon (photon) χ e2(qFFq) momentum and has a simple form; that is, z =( q )2/3,χ2 = , (16) χ(χ − χ ) q m6 2αce q e gφγ = , (19) 2 πF 2 e (pFFp) 2 χ = 6 . where α = e /4π is the fine-structure constant. me Although the process of familon emission through It should be noted that, apart from a change in the a virtual plasmon arises in a higher order of pertur- notation, expression (14) reproduces the result pre- bation theory, its amplitude may appear to be on the sented in Eq.(6) of [13] for the probability of the syn- same order of magnitude as the tree amplitude, espe- chrotron radiation of axions in a weak magnetic field cially in the case of a longitudinal plasmon.This is be- (that is, a field whose induction satisfies the condition cause the familon and longitudinal-plasmon disper- B Be).In order to prove this statement, it is nec- sion curves intersect at some familon energy ω = ω0 essary to make the substitution gae → 2mece/F in (see Fig.2), with the result that the process of familon Eq.(6) of [13].In view of this, we immediately present emission by plasma electrons has a resonance char- the eventual expression for the familon luminosity due acter.Concurrently, it turns out that, in the case PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 ENERGY AND MOMENTUM LOSSES 1301 of ultrarelativistic plasma, the contribution from a ω2 transverse plasmon is negligible. By using the Lagrangian in (18) and taking into ω2 0 ω2 ω2 account relation (19), we can represent the S-matrix = L(k) element for the process under study in the form 2αece L µ S ∗ = √ (qFG˜ ) J , (20) −γ→ − µ (e e φ) πF 2ωV 2 q = 0 4 ¯ iqx L where Jµ = d xψe(p ,x)γµψe(p, x)e and Gαβ ω2 2 0 0 k is the longitudinal-plasmon propagator, the rest of the notation being identical to that in Eq.(13).In Fig. 2. Dispersion curves for (solid line) a longitudi- 2 2 a relatively weak external electromagnetic field, the nal plasmon, ω = ωL(k), and (dashed line) a familon, longitudinal-plasmon propagator can be represented ω2 = k2. in the form l l GL α β , (21) Here, E and p are, respectively, the energy and the αβ q2 − ΠL momentum of plasma electrons (positrons) and fe+ is the distribution of plasma positrons, which is ob- q2 (uq) l = (u − q ), tained from the distribution f − of electrons by means α 2 − 2 α 2 α e (uq) q q of the substitution µ →−µ. where q is the virtual-plasmon 4-momentum, while In order to find the resonance position, which µ is determined by the point of intersection of the l ΠL α and are, respectively, the eigenvector and the longitudinal-plasmon and familon dispersion curves, eigenvalue of the polarization operator that corre- ω ω0, it is necessary to find a simultaneous solu- spond to the longitudinal plasmon. tion to the dispersion equations for the longitudinal The main contribution to the probability of the plasmon, ∗ − γ − process e → e φ comes from the vicinity of the res- k2 = F (c), (25) onance, where q2 =0; throughout the ensuing calcu- lations, we will therefore set q2 =0, unless this leads and the familon, 2 2 to confusion.Also, that part of lα which is propor- k = ω . (26) tional to qα does not contribute to the amplitude by virtue of gauge invariance and the asymmetry of the As a result, we obtain the familon energy at which the dispersion curves intersect: tensor F˜αβ .Without loss of generality, the propagator 2 in (21) can therefore be represented in the form ω0 = F (1). (27) u u q2 GL α β . (22) At the point c =1, the function F (c) can be re- αβ (uq)2 q2 − ΠL duced to the form −2α L F (1) (28) In general, Π is a complex-valued function, π L L L ∞ Π = ReΠ + iImΠ . (23) E + p d × dEE E ln − 2p (fe− + fe+ ). L E − p dE In a weak field, the real part of Π is virtually m coincident with its value in a nonmagnetized plasma e and can be represented in the form [14] It should be emphasized that formula (28) is univer- sal—it is valid both for a cold (µ T ) and for a hot F (c) |k| L 2 (µ T ) plasma.Upon the expansion of the function ReΠ = q 2 ,c= , (24) k ω F (c)/k2 in a power series in the vicinity of the reso- where nance, the expression for the real part of ΠL can be ∞ recast into the form 4α dpp E E + pc F (c) ln − p ω2 π E c E − pc L 2 ReΠ q 2 . (29) 0 ω0 E2p(1 − c2) − − As to the imaginary part, it can undergo a signifi- 2 2 2 (fe + fe+ ). E − c p cant change even under the effect of a relatively weak PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1302 MIKHEEV, NARYNSKAYA field, since a magnetic field opens new channels of sufficient to know the value of the function Φ1(z) at photon decay and new channels of photon absorption the origin. by plasma electrons and positrons, γ ↔ e−e+ and ± ± It is convenient to perform a further integration in e ↔ e γ, these channels being forbidden in a non- the rest frame of the plasma, where the electromag- magnetized plasma.According to the unitarity con- netic field reduces to a purely magnetic one and where dition, the imaginary part of ΠL and the total width the dynamical parameters take the form with respect to longitudinal-plasmon “annihilation,” L eBE eBω Γ (ω), are related by the equation χ = 3 sin θ, χq = 3 sin θ . (34) me me ZLImΠL = −ωΓL(ω), (30) Here, θ is the angle between the familon 3-momen- where ZL is the renormalization factor for the plas- tum k and the magnetic-field direction and the effec- mon wave function. tive angle θ is defined by the relation p⊥ With allowance for relations (29) and (30), the sin θ = . (35) longitudinal-plasmon propagator (22) in the vicinity p2 + p2 of the resonance can be represented in the form ⊥ z u u GL α β , (31) An analysis reveals that, in a relatively weak mag- αβ 2 − 2 2 (ω0 ω + iγω0) netic field (eB µω), we have L ω0Γ (ω0) γ = . sin θ sin θ , (36) q2ZL which means that, under the conditions being con- γ∗ Substituting the explicit form (31) of the propa- sidered, the kinematics of the process e− → e−φ is gator into expression (20) for the relevant S-matrix collinear. element, we obtain In expression (33), we further go over to the rest ˜ 2αece (qFu)(uJ) frame of the plasma with allowance for (34) and (36) S ∗ = √ . (32) −γ→ − 2 − 2 2 (e e φ) πF 2ω0V (ω ω0 + iγω0) and substitute the result into definition (8).In this γ∗ way, we obtain, for the process e− → e−φ,thezeroth From expression (32), it can clearly be seen that, component of the loss 4-vector in the form at an emitted-familon energy close to ω0, the process 2 2 3 2 of familon emission by plasma electrons has a distinct 0 ceα ω0(eB) 2 Q γ∗ cos θ (37) resonance character. (e−→e−φ) 3 2 ∗ 3π F − γ − The probability of the process e → e φ is calcu- E (E − ω)dω lated directly by formula (5) with the aid of expres- × (1 − f − ), 2 − 2 2 2 4 e sion (32) for the S-matrix element.After straight- (ω ω0) + γ ω0 forward, albeit rather cumbersome, calculations, the 0 result can be represented in the form where it has been considered that Φ1(z) Φ1(0) = 2 3 2 π/3. 4ceα ω0 W ∗ −γ − 3 2 (33) (e →e φ) π F p0 The resonance factor in expression (37) can be χ approximated by a delta function, 2 dχq (χ − χq)(qFu˜ ) × Φ1(z)(1 − f − ), 1 π 2 2 − 2 2 2 4 e δ(ω − ω0), (38) χq [(ω ω0) + γ ω0] 2 − 2 2 2 4 3 0 (ω ω0) + γ ω0 2ω0γ where and this makes it possible to perform integration with ∞ respect to the familon energy quite easily. Φ1(z)= Φ(x)dx, In order to calculate the dimensionless factor γ,it is necessary to know the total longitudinal-plasmon- z annihilation width ΓL, which is defined as the differ- with the argument z being given by (16).Under the ence of the widths with respect to the absorption and conditions in (4), which are considered in the present the creation of a longitudinal plasmon in an electron– study, the dynamical parameter χ for electrons is positron plasma and which can be represented in the much greater than unity.The main contribution to the form [15] relevant integral with respect to the variable χ then q L L − L − −ω/T L comes from the region where z 1.In this case, it is Γ =Γabs Γcr =(1 e )Γabs. (39) PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 ENERGY AND MOMENTUM LOSSES 1303 An analysis reveals that the main contribution to ΓL comes from the process e−γL → e−,wherea abs µ–(p') longitudinal plasmon is absorbed by plasma elec- φ(q) trons.The S-matrix element for this process can be obtained from the electromagnetic-interaction La- – grangian by directly substituting into it solutions to e (p) the Dirac equation in a crossed field.In the lowest order of perturbation theory, the result has the form Fig. 3. Diagram describing familon emission in the pro- 4 µ − → − S = ie d xψ¯e(p ,x)A γµψe(p, x). (40) cess e µ φ in an external field. The plasmon-absorption width is determined in a From the calculations, it follows that, under the standard way; that is, conditions considered here, the effect of the interfer- | |2 3 3 ence between the contributions from the above two L S d p d p Γ = Vf − V (1 − f − ). channels of the synchrotron radiation of familons is abs T (2π)3 e (2π)3 e s,s nonexistent. (41) − → − Omitting the details of integration over the phase 3.TRANSITION e µ φ spaces of the particles involved, we only present the In this section, we consider the electron-decay result obtained by evaluating expression (41): process e− → µ−φ as a possible additional source of 2α µ2 q2ZL the familon emissivity of a magnetized plasma (see ΓL . (42) Fig.3).This process was previously studied by Averin abs 3 − −ω/T 3 ω 1 e et al. [10] in the limit of strong magnetic fields whose 2 − 2 With allowance for (39) and (42), the dimensionless induction satisfies the condition eB µ mµ,in parameter γ reduces to the form which case muons are produced only in the first Lan- 2α µ2 dau level.Among other things, those authors showed that, under the conditions that they considered, the γ = 2 . (43) 3 ω0 familon emissivity is competitive with the neutrino Substituting (37) into (12) and performing integra- emissivity caused by the magnetic bremsstahlung of tion with respect to the primary-electron momentum neutrino pairs from electrons.It should be noted, and with respect to the familon energy with allowance however, that the magnetic bremsstrahlung of neu- for Eq.(38), we obtain the following expression for the trinos does not saturate the actual emissivity of a ∗ − γ − supernova remnant or a nascent neutron star.As a familon luminosity due to the process e → e φ: matter of fact, the actual emissivity of these astro- 2 2 3 physical objects is determined by modified URCA αce(eB) ω0 ε˙ γ∗ . (44) processes [16]. (e−→e−φ) 12π4F 2 eω0/T − ( 1) Here, we consider the limit of relatively weak mag- netic fields, which corresponds to a situation that is We note that the result in (44) is valid not only in a inverse to that in [10] and which could occur, for ex- degenerate plasma but also in the general case.Only ample, in the core of a supernova, where µ2 − m2 the energy ω0 at which the familon and longitudinal- µ plasmon dispersion curves intersect depends on spe- eB and where muons, as well as electrons, therefore cific physical conditions.In the case of a degenerate populate a large number of Landau levels. ultrarelativistic plasma, we find from (27) with al- The general form of the S-matrix element for the − − lowance for (28) that process e → µ φ can be obtained from the La- grangian in (1) by directly substituting into it solu- 2 4α 2 2µ − ω0 µ (ln 1). tions to the Dirac equation in a crossed field; that is, π me − √ cµe In the case of a hot plasma, ω2 is given by S(e−→µ−φ) = 0 2ωV F 2 4πα 2 4T 1 − ζ (2) × 4 ¯ µ iqx ω0 T (ln + γE + ), d xψµ(p ,x)q γµ(gv + gaγ5)ψe(p, x)e , 3 me 2 ζ(2) α where γE is the Euler constant (γE 0.577)and where q =(ω,k) is the familon 4-momentum, while − α α ζ (2)/ζ(2) 0.570. p =(p0, p) and p =(p0, p ) are constant vectors PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1304 MIKHEEV, NARYNSKAYA that characterize the states of, respectively, the elec- In a relatively weak external electromagnetic field, tron and the muon in a crossed field. the dynamical parameter for electrons satisfies the The probability of the process is determined ac- condition χ 1, while, for the argument of the Airy cording to Eq.(5), where a muon appears as the function, we have z 1.Performing integration with final fermion.In the relevant calculations, we will respect to the variable χq with allowance for the disregard the electron mass and the statistical factor known asymptotic behavior of the functions Φ(z) of muons, assuming that the concentration of ther- and Φ1(z) at large values of the argument [12], we malizedmuonsislowinthecoreofasupernova.The eventually obtain results of the calculations can be reduced to the form c2 m4 χ √ 2 4 √µe µ − 3/χ cµemµ W(e−→µ−φ) = e . (49) − − 2 W(e →µ φ) = 3 2 (45) 36 3πF p0 8π F p0 +∞ χ We would like to emphasize that the exponential χ2dχ × q q 2 2 2 2 2 smallness of the probability in (49) is characteristic dτ 2 [Φ (y)r (τ + χ /χq) rχ (χ − χq) of all processes that are forbidden in a vacuum, but −∞ 0 which are allowed in a relatively weak field.However, 2 the calculations reveal that, despite an exponential +Φ (y) − 4gagvΦ(y)Φ (y)rτ], suppression, the decay probability W(e−→µ−φ) proves where the Airy function Φ(y) was defined in (15), its to be on the same order of magnitude as the probabil- argument being ities of the two channels of the synchrotron radiation χ of a familon from plasma electrons.This is because of y = r2 τ 2 + , χ the fact that, by virtue of (4), the last two probabili- q ties also involve a small parameter that is associated and where we have also introduced the following con- either with the fine-structure constant α (33) or with ventional notation [12]: the electron mass me (14) and which is the smallest parameter in the problem being considered. e(pFq˜ ) χ 1/3 τ = ,r= q . (46) According to our analysis, the main contribu- 4 − χqmµ 2χ(χ χq) tion to the integral with respect to the variable τ We recall that expression (45) was obtained in the in expression (45) comes from the region around ultrarelativistic limit, where the electron mass can be τ 2 ∼ 1/r3 1; in the rest frame of the plasma and at disregarded. relatively weak fields (eB ωmµ), this is equivalent In contrast to what we had in (16), the dynamical to the collinearity condition.In order to make sure of parameters χ and χq are now naturally defined in this, it is sufficient to employ the explicit expressions for τ and r in the rest frame.We then have terms of the muon mass; that is, 2 2 1/2 e (qFFq) e (pFFp) eB 1/2 χ2 = ,χ2 = . cos θ − cos θ ∼ sin θ(sin θ ) 1. q 6 6 ωm mµ mµ µ (50) The last term in the bracketed expression on the right-hand side of (45) describes the interference be- This makes it possible to go over, in expres- tween the vector and the axial-vector coupling con- sion (47), to the rest frame of the plasma prior to stant entering into the Lagrangian in (1); although performing integration with respect to the variable this term does not contribute to the total probability χq and to recast this expression into the form of the decay process in question, it leads to the asym- 2 4 − − cµemµ metry of familon emission in the process e → µ φ. W −→ − = (51) (e µ φ) 16π2F 2E2 Upon integration with respect to the variable τ,we E obtain 2ωΦ(z) χ × (Φ (z) − )dω, 2 4 1 − cµemµ dχq (E ω)z − − 0 W(e →µ φ) = 2 2 (47) 16π F p0 χ 0 where the argument z oftheAiryfunctionbecomes 2χqΦ (z) 6 1/3 × Φ (z) − , E mµ 1 (χ − χ )z z = . q ω(E − ω)2 (eB sin θ)2 1/3 χ Substituting expression (51) into the definition in (12) z = 2 > 0. (48) χq(χ − χq) with allowance for (8) and performing integration PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 ENERGY AND MOMENTUM LOSSES 1305 with respect to the familon energy and with respect to ε. 3 the primary-electron momenta, we obtain the familon , erg/(cm s) e− → µ−φ luminosity due to the process .The result 1026 of these calculations can be expressed in terms of a 24 single integral as 10 √ 22 2 4 3 10 cµe 2mµµ I(y) ε˙ −→ − , (52) (e µ φ) 216π5/2F 2 y3/2 10 20 30 T, MeV ∞ e−y/xdx √ m3 I(y)= x7/2 ,y= 3 µ , Fig. 4. Familon luminosity of a degenerate plasma eµ(x−1)/T +1 eBµ (µ T ) as a function of temperature: (solid curve) con- 0 tribution of the synchrotron radiation of a familon through ∗ − γ − where the variable x is related to the primary-electron a virtual plasmon (e → e φ), (dashed curve) contribu- energy by the equation x = E/µ. tion of the synchrotron radiation of a familon due to direct coupling (e− → e−φ), and (dash-dotted curve) contribu- The integral I(y) in (52) can easily be calculated tion of the transition e− → µ−φ. in two limiting cases. First, there is the case of a cold plasma (T → 0), where the distribution of primary electrons can under such conditions, the contributions from the two be approximated by a Heaviside step function, f processes in question are commensurate. Θ(µ − E).This leads to √ A comparison of the familon luminosity with the 2 4 3 −y known neutrino luminosity under the same condi- cµe 2mµµ e 34 3 ε˙(e−→µ−φ) , (53) ε˙ ∼ 10 / 216π5/2F 2 y5/2 tions, ν erg (cm s) [1], reveals that, at the initial stage of cooling, when the plasma temperature Second, there is the case of a relatively “warm” is T ∼ 35 MeV,the processes being considered do not relativistic plasma: T>µ/y,butT µ; in this case, have a significant effect on the dynamics of cooling. where the main contribution comes from electrons of It is interesting to note, however, that, at a later energy E>µ, the result is stage of cooling, when the plasma temperature de- 2 4 3 creases to values of about a few MeV units, the cµemµµ T 5/2 −y(2−1/t)/t ε˙ −→ − 2y( ) e , − → − (e µ φ) 216π2F 2 µ contribution of the process e µ φ with respect to (54) the contribution of the synchrotron-radiation process e− → e−φ becomes dominant.In this case, the en- yT ergy loss of the plasma by familon radiation is given t = . µ by × 25 erg Since t>1, the density of losses, ε˙, increases signif- ε˙(e−→µ−φ) 3.4 10 cm3 s icantly in this case. and can be commensurate with the loss by neutrino radiation at the same stage, ε˙ ∼ 1026 erg/(cm3 s) [1]. 4.ASTROPHYSICAL APPLICATIONS ν The asymmetry of familon emission is yet another In this section, we present quantitative estimates interesting feature of radiation processes.This asym- of our results, choosing, for a scale of horizontal- metry can be defined as symmetry breaking, the upper bound on the relevant constraints, F 3.1 × 109 GeV. F3 A = 0, 0, , (55) The volume density of the energy loss of the ε˙ plasma by familon radiation can be represented as the sum of the contributions from two processes: where it is considered that the asymmetry does not vanish only in the magnetic-field direction (which is ε˙φ =˙ε(e−→e−φ) +˙ε(e−→µ−φ). coincident with the direction of the z axis) and where F We further estimate the energy loss per unit 3 is the force acting on the plasma from the familon volume of the plasma by familon radiation under along the magnetic field.This force can be calculated the conditions prevalent in the core of a supernova by formula (12). (µ = 250 MeV, T =35MeV), setting the magnetic Direct calculations reveal that only the transi- − − field to B =1017 G.The results of these numerical tion e → µ φ contributes to the asymmetry given calculations are given in Fig.4, which shows that, by (55), which proved to be independent of either the PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1306 MIKHEEV, NARYNSKAYA γ∗ V, km/s e− → e−φ has been shown to be of a resonance 120 character at a certain value of the emitted-familon 80 energy. 40 As a possible astrophysical application, the vol- umedensityoftheenergylossofaplasmabyfamilon 2 6 10 radiation has been calculated under the conditions B, 1016 G of the core of a supernova.The estimates obtained in this way indicate that, under realistic conditions Fig. 5. Velocity of a supernova remnant with a mass on prevalent in the core of a supernova at the initial stage the order of the Sun’s mass versus the magnetic-field of cooling, the contributions to the energy loss of the induction. plasma from the two processes being considered are on the same order of magnitude, but that they are chemical potential of the plasma or its temperature insufficient for significantly affecting the dynamics of and which has the simple form supernova cooling. At the same time, the contribution to the energy g g eV − − a v loss from the transition e → µ φ at a later stage A 2 . (56) 3 mµ of cooling considerably exceeds the contribution from the synchrotron radiation of a familon and becomes We note that the emergence of the asymmetry commensurate with the energy loss by neutrino ra- in (56) is due to the interference between the vector diation under the same conditions.It follows that, at and the axial-vector coupling constant entering into the stage of cooling, the role of processes involving the Lagrangian in (1). familons may become significant against the back- Owing to this asymmetry, the total momentum ground of neutrino processes when the plasma tem- carried away by a familon is nonzero, which in turn perature decreases to a value of T ∼ 1 MeV. leads to a “push” velocity V of a supernova remnant. It has also been found that, in the case of familon This velocity can be estimated as emission by plasma electrons in the process e− → εA µ−φ, there occurs an interesting interference effect V = , (57) M that leads to the asymmetry of the familon momentum with respect to the direction of the magnetic field of where ε = dtε˙ is the total energy loss of the plasma a supernova remnant.In magnetic fields of induction − → − by familon radiation in the process e µ φ and M B ∼ 1017 G, this generates a push velocity of the is the mass of the supernova remnant. supernova remnant on the order of 100 km/s. Figure 5 shows a numerical estimate of the ve- Our results may be of use in performing a detailed locity of a supernova remnant versus the magnetic- investigation into, for example, astrophysical phe- field induction.It can be seen that, even in strong nomena that accompany the creation of a magnetized ∼ 17 magnetic fields (B 10 G), the velocity of a super- neutron star (pulsar) upon supernova explosion and nova remnant with a mass on the order of the Sun’s in describing the dynamics of its cooling. mass does not exceed 130 km/s.Thus, we conclude that, while being of interest in and of itself, the effect of the asymmetry of familon emission in the process ACKNOWLEDGMENTS − → − e µ φ unfortunately cannot clarify the problem This work was supported in part by the Min- of pulsar velocities. istry for Higher Education of the Russian Federation (project no.Е02-11.0-48)and by the Russian Foun- 5.CONCLUSION dation for Basic Research (project nos.01-02-17334, 03-02-06215). We have considered familon emission from a magnetized plasma in the electron-“decay” process e− → µ−φ and in the process of synchrotron radi- APPENDIX ation of a familon by plasma electrons, e− → e−φ. In general, the familon → photon conversion in a New transitions of the familon photon type, which magnetized plasma is affected both by the magnetic generate an additional channel of the synchrotron field and by the plasma itself. radiation of a familon by an electron through a virtual The one-loop external-field-induced contribution γ∗ → plasmon as an intermediate state, e− → e−φ,become to the amplitude for the familon photon transition, field possible in an external magnetic field.In the case Mφ→γ , is described by the diagram in Fig.6, where where a virtual plasmon is longitudinal, the process double lines denote that the effect of the external field PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 ENERGY AND MOMENTUM LOSSES 1307 f φ(q) γ(q') φ(q) γ(q') γ φ(q) (q) + xy xy x y e–(p) e–(p) e–(p) e–(p) f Fig. 7. Contribution of a magnetized plasma to the Fig. 6. Diagram describing the familon → photon tran- familon → photon transition. sition in an external magnetic field through a fermionic loop. 4 2/3 η = , − 2 has been taken into account in the propagators for χf (1 t ) virtual charged fermions f. 2 where f(η) is the Hardy–Stokes function and χf = The plasma contribution M plasma comes from 2 6 φ→γ ef (qFFq)/mf is the dynamical parameter of the rel- Compton-like processes of familon-to-photon tran- evant virtual fermion f. sitions occurring on plasma electrons and positrons The asymptotic behavior of the integral J in (A.2) (see Fig.7). at small and large values of the dynamical parameter field The expression for the amplitude Mφ→γ can be de- χf is as follows: rived, for example, from the results presented in [17], J 1+O(χ2 ),χ 1, (A.3) where the one-loop contributions to the generalized f f → ¯ → −2/3 amplitude for the transition j ff j in an exter- J O(χf ),χf 1. nal electromagnetic field were calculated for arbitrary combinations of the scalar, pseudoscalar, vector, and Considering that the familon–photon interaction pseudovector coupling of the currents j and j to is free from the Adler anomaly, we can recast expres- fermions.Performing, in Eq.(3.9)of [17], the substi- sion (A.1) into a form from which it is clearly seen → tutions jAµ → (icf /F )qµ and jVµ →−ef εµ and set- that the amplitude for the familon photon transition ting q2 =0, one can obtain the transition amplitude does not involve a term that is linear in the field; that in the form is, ˜ ∗ ˜ ∗ field i(qFε ) 2 field i(qFε ) 2 M → = c e J, (A.1) M → = cf e [J − 1]. (A.4) φ γ 2π2F f f φ γ 2π2F f f f where ef is the charge of a virtual fermion in the By virtue of the asymptotic expressions in (A.3), loop; qµ is the familon (photon) 4-momentum; εµ only relatively light fermions, those for which χf 1, is the photon polarization 4-vector; F˜αβ is the dual contribute to the transition amplitude (A.4). In view of counterpart of the strength tensor of the external the fermion-mass hierarchy, a virtual electron makes electromagnetic field; and J is a field form factor that, a dominant contribution under the conditions being in general, depends both on the field and on the 4- considered: momentum q . 2iαc ∗ µ M field e (ε Fq˜ ). (A.5) Under physical conditions corresponding to a φ→γ πF magnetized plasma in the core of a supernova (B Comparing expression (A.5) with the Lagrangian 18 µ 10 G, 250 MeV), the limit of a relatively weak in (3), we find that the external-electromagnetic- external magnetic field is realized, in which case many field-induced contribution to the effective familon– Landau levels are excited [see (4)].The form factor J photon coupling constant gφγ can be represented in then proves to be dependent only on the combination the form (qFFq) and has the form field 2αce 1 g = . (A.6) φγ πF J(χ )= ηf(η)dt, (A.2) f We note that, in contrast to axion–photon interac- 0 field tion, the effective coupling gφγ features no vacuum ∞ term, because, as was indicated above, the familon 1 f(η)=i dz exp −i ηz + z3 , does not involve the vacuum anomalies Φ(GG˜) and 3 ˜ 0 Φ(F F ) (G and F are, respectively, the gluonic and PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1308 MIKHEEV, NARYNSKAYA electromagnetic field tensors), so that its interaction curring on plasma electrons can be obtained from the with photons becomes possible only in an external Lagrangian electromagnetic field. − 2imece α Let us now consider the plasma contribution to L = (Ψ¯ eγ5Ψe)Φ + e(Ψ¯ eγαΨe)A (A.7) the amplitude for the familon → photon transition F corresponding to the diagrams in Fig.7. and, with allowance for coherent scattering on all The S-matrix element for the process φ → γ oc- plasma electrons, is given by − 2iem c S(e ) = e e d4xd4y[Φ(q,x)S(y,x)γ ψ (p, x)ψ¯ (p, y)γ Aα(q ,y) (A.8) F 5 e e α ¯ β +Φ(q,y)ψe(p, y)γ5S(y,x)γβA (q ,x)ψe(p, x)]dne− fe− , where Aµ is the 4-potential of a quantized electro- Under the conditions of a degenerate ultrarela- magnetic field, Φ is the familon wave function, ψe tivistic plasma, the dynamical parameter χq is much stands for solutions to the Dirac equation, S(y,x) is greater than unity, in which case the integral with the electron propagator in an external magnetic field respect to the variable v can easily be calculated.As (see, for example, [18]), dne− is an element of the a result, the coupling constant can be represented in phase space of a plasma electron, and f − is its distri- the form e bution.The S-matrix element for the familon → pho- 1/3 4 plasma αce3 1 1 (e+) g = Γ . (A.11) ton transition occurring on positrons, S , is ob- φγ π2F 3 2/3 tained from expression (A.8) by means of the substi- χq tution p →−p in the solutions to the Dirac equation Comparing expression (A.11) with the field con- and the substitution µ →−µ in the distributions of tribution given by (A.6), one can see that, under the primary electron. the physical conditions being considered, the plasma Upon integration with respect to the coordinates contribution to the coupling constant gφγ is relatively x and y and over the phase space of electrons and small and can be discarded. positrons, the four-dimensional delta function δ4(q − q) can be singled out in the relevant S-matrix ele- ment.Owing to this, the standard invariant amplitude REFERENCES for the familon → photon transition in a relatively 1.G.G.Ra ffelt, Stars as Laboratories for Fundamen- weak magnetic field can be written in the form tal Physics (Univ.of Chicago Press, Chicago, 1996). 2. Dzh.L.Chkareuli, Pis’ma Zh. 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Isaakyan PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1310–1318. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1350–1358. Original Russian Text Copyright c 2003 by Agababyan, Ammosov, Atayan, Grigoryan, Gulkanyan, Ivanilov, Karamyan, Korotkov. ELEMENTARY PARTICLES AND FIELDS Theory Effect of Nuclear Matter on the Production of Hadrons in Deep-Inelastic Neutrino Scattering N. M. Agababyan1), 2), V. V. Ammosov, M. R. Atayan2), N. G. Grigoryan2), G. R. Gulkanyan2),A.A.Ivanilov*, Zh. K. Karamyan2),andV.A.Korotkov Institute for High-Energy Physics, Protvino, Moscow oblast, 142284 Russia Received June 28, 2002; in final form, October 29, 2002 Abstract—The inclusive spectra of hadrons are measured with the aid of the SKAT propane–freon bubble chamber irradiated with a beam of 3- to 30-GeV neutrinos from the Serpukhov accelerator. The resulting data indicate that the intranuclear absorption of leading quark-fragmentation products is enhanced as the energy transfer to the quark involved decreases or as the quark-energy fraction z acquired by the product hadron increases. An analysis of the data on the basis of the color-string model reveals that the cross section for the intranuclear absorption of positively charged hadrons that are characterized by z values in the range between 0.7 and 0.9 is close to the inelastic cross section for pion–nucleon interaction. c 2003MAIK “Nauka/Interperiodica”. INTRODUCTION emission [10, 11], it follows that lh depends on the quark-energy fraction transferred to the hadron, z = Deep-inelastic lepton–nucleon scattering is ac- Eh/ν, this dependence being such that, with decreas- companied by the formation of a color string between ing z, the length of formation of the most energetic the knock-on quark and the nucleon residue. In scat- fragmentation products decreases, which entails an tering on a peripheral intranuclear nucleon, the frag- increase in the probability of their absorption in a mentation of such a string is analogous to its frag- nuclear medium. mentation in the case of a hydrogen or a deuterium target. In the case of scattering on a nonperiph- Acomparison of the inclusive spectra of hadrons eral nucleon, the fragmentation process is affected to that are produced on nuclei and the analogous spectra some extent by a nuclear medium. for the production process on a deuteron [12–16] corroborates the predicted weakening of nuclear- According to some theoretical predictions (see, for absorption effects with increasing ν and their en- example, [1–5]), the mean spacetime gap l required h hancement for z → 1. These data were obtained m for the formation of a hadron having a mass h and at comparatively high energies (up to 400 GeV); E an energy h is determined by the Lorentz factor at the same time, data at intermediate energies l ∼ E /m ( h h h). In a number of other models, it is (ν<10 GeV) are rather scanty. For the sake of com- assumed that lh is proportional to the Lorentz factor ∼ ∗ pleteness, it would be of interest to perform detailed of the parent quark lh ν/mq,whereν is the en- investigations in this energy region as well, where the ergy transfer to the quark involved, the virtual-quark ∗ nuclear-medium effect on the fragmentation process mass mq being dependent on the momentum transfer is expected to be more pronounced. squared Q2 [6] or being determined by the kinematics Of special interest is a comparison of the spectra of hadron “emission” from the quark [7]. of hadrons from different samples of events of lepton From a more detailed treatment of the hadron- scattering on the same nucleus—namely, samples formation process on the basis of the Lund fragmen- that feature no explicit indication of secondary nuclear tation model [8], as well as on the basis of models interactions and samples where there are such indi- that take into account quark moderation either owing cations (in the following, samples of quasinucleonic to the tension of a color string [9] or owing to gluon and cascade events, respectively). In this comparison, manifestations of nuclear effects are expected to be 1)Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia. more distinct in cascade events. Such a comparison 2)Yerevan Physics Institute, ul. Brat’ev Alikhanian 2, Yerevan, was performed for νNe interactions at comparatively 375036 Armenia. high values of the hadron-system energy W (up to *e-mail: [email protected] about 25 GeV) [17]. 1063-7788/03/6607-1310$24.00 c 2003 MAIK “Nauka/Interperiodica” EFFECT OF NUCLEAR MATTER 1311 The objective of the present study is to explore assigned a weight that took into account losses of nuclear-absorption effects in neutron–nucleus inter- events. The mean weight of the sample used in the actions at intermediate energies, ν =2-15 GeV and present study was 1.43. W =2–5 GeV. The ensuing exposition is organized For the quantity ν, the eventual value that takes as follows. The experimental procedure used is de- into account undetected neutrons and photons was scribed in Section 1. An account of our methods for determined on the basis of the measured value of νvis selecting interactions of the cascade and quasinucle- by using the relation ν = a + bνvis,wherethecoeffi- onic types is given in Section 2. In Section 3, the cients a =0.15 ± 0.24 GeV and b =1.07 ± 0.05 were spectra of hadrons from cascade events are compared found by means of the procedure applied in [19]. Close with those for quasinucleonic events. Aquantitative values of a and b were obtained from a Monte Car- estimate of intranuclear-interaction effects is pre- lo simulation of neutrino interactions in the cham- sented in Section 4. In Section 5, data obtained in ber [20]. our experiment are compared with the predictions of For a further analysis, we selected 2223 events the color-string model. The basic results of this study where 3 1. EXPERIMENTAL PROCEDURE 2. SELECTION OF QUASINUCLEONIC, Our experiment employed the SKAT bubble CASCADE, AND DEUTERON EVENTS chamber [18] irradiared with a wideband neutrino The selection of quasinucleonic and cascade beam from the Serpukhov accelerator at a primary- events according to the procedure described in de- proton energy of 70 GeV. The chamber was filled tail in [21] was performed by using a number of with a mixture containing 87% (in volume) propane topological and kinematical criteria. The subsample (C3H8)and13% freon (CF3Br), the percentage of BN of quasinucleon interactions included events nuclei in this mixture being H : C : F : Br =67.9: exhibiting no indication of a secondary interaction 26.8:4.0:1.3. The density of admixtures was in the target nucleus: the total charge of secondary 3 0.55 g/cm ; the radiation length was X0 =50cm; hadrons was required to be q =+1(for the subsample and the nuclear-interaction range was 149 cm. The Bn of interactions with a neutron) or q =+2 (for 3 total volume of the chamber was 6.5 m ,itseffective the subsample Bp of interactions with a proton), volume used being 1.73 m3. Auniform magnetic field while the number of recorded baryons (these included of strength 20 kG was maintained in the chamber. identified protons and Λ hyperons, along with neu- We selected charged-current-interaction events trons that suffered a secondary interaction in the at negative-muon momenta satisfying the condition chamber) was forbidden to exceed unity, baryons flying in the backward directions being required to pµ > 0.5 GeV/c. Amuon was identi fied as a neg- ative particle that possessed the highest transverse be absent among them. Moreover, a constraint from above was imposed on the effective target mass Mt, momentum among particles that did not suffer a sec- 2 ondary interaction in the chamber. Other negative M t < 1.2 GeV/c , this mass being defined as Mt = − − i particles were assumed to be π mesons. Protons (Ei p||), where summation is performed over of momentum below 0.6 GeV/с and some protons the energies Ei of secondary particles and over the i of momentum in the range 0.6 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1312 AGABABYAN et al. Mean features of deep-inelastic neutrino scattering for the total sample of events and for the subsamples Bp, Bn,andBS 2 2 2 2 Sample Eν ,GeV ν ,GeV W ,GeV Q ,GeV x Total 10.8 ± 0.1 6.5 ± 0.1 9.5 ± 0.1 3.6 ± 0.1 0.30 ± 0.001 Bp 11.0 ± 0.3 6.6 ± 0.2 9.8 ± 0.3 3.5 ± 0.1 0.29 ± 0.001 Bn 10.9 ± 0.2 6.5 ± 0.2 9.3 ± 0.3 3.9 ± 0.1 0.33 ± 0.001 BS 10.7 ± 0.2 6.5 ± 0.1 9.5 ± 0.2 3.5 ± 0.1 0.29 ± 0.001 the mean multiplicities of positively and negatively that the subsamples Bp and Bn may contain only charged hadrons in the subsamples Bp and Bn an insignificant admixture of events where there are in satisfactory agreement with data from [23] occurred a secondary intranuclear interaction. on νp and νn interactions, this agreement being Some averaged kinematical features of the total observed both at negative and at positive values of the sample and of the subsamples of events are listed in Feynman variable. In addition, the inclusive spectra the table. of hadrons were compared with available data for a hydrogen (deuterium) target in the region 2 ≤ W ≤ 5 GeV. Here, we obtained satisfactory agreement 3. COMPARISON OF THE INCLUSIVE between the inclusive spectra of π− mesons in νp SPECTRAOF HADRONS interactions [24] and those in the subsample B IN THE SUBSAMPLES OF CASCADE p AND QUASINUCLEONIC EVENTS and between the inclusive spectra of hadrons in νD ∗ interactions [25] and those in the combined subsam- The ratio Ry∗ (BS/BN ) of the pseudorapidity (y ) ple BD of quasideuteron events. The subsample of distributions of charged particles in the c.m. frame of quasideuteron events contained the subsample Bn the hadronic system in the subsamples BS and BN and 60% of the subsample Bp (this fraction effectively is shown in Fig. 1 for two intervals of the invariant corresponds to νD interactions), the remaining 40% energy, 2 sion being especially pronounced at low W —for the Ry*(BS/BN) ∗ fastest particles (y > 1.8),wehaveRy∗ (BS/BN )= 0.44 ± 0.13, which is compatible with the results pre- sented in [17], where a value of 0.6 ± 0.1 at W = 2–7 GeV was obtained for the ratio of the yields of y∗ > 2 100 charged hadrons in the cascade and quas- inucleonic subsamples of νNe interactions. We also note that, with increasing W ,effects associated with the absorption of fast quark-fragmentation products become weaker, disappearing almost completely, ac- cording to data from [17], in the region W>7 GeV. 10–1 –2 –1 0 1 2 For the subsamples BS and BN , the distributions y* ρ(z)=(1/Ntot)dN/dz with respect to the variable ∗ z are displayed in Fig. 2 separately for positively Fig. 1. Ratio of the pseudorapidity (y ) distributions charged hadrons (identified protons are not included of charged hadrons in the subsamples BS and BN for − two regions of the invariant energy W : (closed circles) in these distributions) and for π mesons. It can 2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 EFFECT OF NUCLEAR MATTER 1313 ρ (z) Rz(BS/BN) 101 2.0 1.0 0.8 100 0.6 2 6 10 ν, GeV Fig. 3. Ratio of the yields of charged hadrons in the subsamples BS and BN as a function of ν for various 10–1 regions of the variable z: (closed circles) z<0.2,(open circles)0.2 will be a monotonically growing function of the kine- 0 0.2 0.4 0.6 0.8 1.0 η ∼ l z matical variable h. Otherwise, the theoretical prediction that there is a linear relation between η Fig. 2. Distributions of (circles) positively charged and lh would be at odds with experimental data on hadrons and (triangles) π− mesons with respect to the the leptonic production of hadrons. The data in Fig. 4 variable z in the subsamples (open symbols) BS and make it possible to test various theoretical predictions (closed symbols) BN . qualitatively. From the data in Fig. 4a,itfollowsthat,incon- trast to the input assumption that the length lh of losses in a nuclear medium, the spectra of hadrons formation of a leading hadron is linearly related to in the subsample B are shifted toward small values S Ech/mπ, the degree of suppression of its yield and, of z. According to the data in Fig. 3, which illus- hence, the path that it travels in a nuclear medium trates the ν dependence of the ratio Rz(BS/BN )= after formation undergo virtually no changes over ρch (z)/ρch (z) of the analogous charged-hadron a broad region of these variables. For the variable BS BN ∗ distributions integrated over three ranges of the ν/mq, which is the Lorentz factor of the quark whose variable z—z<0.2, 0.2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1314 AGABABYAN et al. + ch ch Rη(BS/BN) Rz (BA/BD) Rη (BS/BN) 1.0 1.0 1.0 (‡) (b) 0.8 0.8 0.9 0.6 0.8 0.6 0 20 40 0 5 10 15 η 1 2 + 10 10 + Rη(B /B ) Rη(BS/BN) S N A (c) 1.0 (d) 1.0 ch Fig. 6. Ratio Rz (BA/BD) for z>0.2 charged hadrons 0.8 0.8 at the mean energy of ν =11.5 GeV versus the mass 0.6 number A of the target nucleus: (closed circles) data from 0.6 our present study and (open circles) data from [13, 15, 16]. 0.4 0.4 The description of the straight line is given in the main 0 2 4 0 2 4 body of the text. η, GeV Fig. 4. Ratios of the yields of (a)chargedhadronsand range 2 <ν<15 GeV, which was studied in our (b, c, d) positively charged hadrons in the subsamples experiment described here. BS and BN versus the kinematical variables defined in the main body of the text and denoted generically by η: − −1/2 ∗ (a) η = Ech/mπ,(b) η = ν[z(1 z)] /pT ,(c) η = 4. RATIO OF THE INCLUSIVE SPECTRA −2 2 ν(1 − z),and(d) η = νz[−1+ln(z )/(1 − z )]. OF HADRONS IN THE SUBSAMPLES OF NUCLEAR AND QUASIDEUTERON EVENTS R (B /B ) z A D In order to obtain quantitative estimates and to 1.4 perform a comparison with the results of other experi- mental studies, it is necessary to represent data in the 1.0 form of the ratio of the inclusive spectra in question for neutrino interactions with nuclei of the propane– 0.6 freon mixture to those for neutrino interactions with deuterons. The subsample BA of nuclear interactions 0 0.3 0.6 0.9 can be obtained by eliminating the contribution that z comes from events of interaction with hydrogen and which, in our experiment, is about 40% of the number Fig. 5. Ratio of the yields of (closed circles) of positively − of events in the subsample B . Thus, the subsample charged hadrons and (open circles) π mesons in the p B can be symbolically represented as B = B + subsamples BA and BD versus the variable z. A A S Bn +0.6Bp = BS + BD. The ratio of the inclusive spectra in the subsam- z → 1, a similar dependence was obtained within the ples BA and BD, for example, with respect to the −2 Lund string model [8]: lh ∼ νz[−1+ln(z )/(1 − variable z is given by 2 z )]. In either case, lh decreases monotonically with Rz(BA/BD)=ρA(z)/ρD(z) increasing z (over the region z>0.3 in the second N tot +0.6N tot case); therefore, the suppression of the hadron yield = n p becomes more pronounced. N tot + N tot +0.6N tot S n p + ∆N h(z) The dependence of the ratio Rη (BS/BN ) on × S 1+ h h , these variables in Figs. 4c and 4d is in qualitative ∆Nn (z)+0.6∆Np (z) agreement with this prediction. As was shown in [16], tot tot tot theoretical predictions based on the dependence where NS , Nn ,andNp are the total numbers of weighted events in the subsamples B , B ,andB , lh ∼ ν(1 − z) agree satisfactorily with data on the S n p h h h electroproduction of z>0.2 hadrons on a nitrogen respectively, while ∆NS (z), ∆Nn (z),and∆Np (z) nucleus in the energy range 7 <ν<24 GeV, which, are the numbers of hadrons having a given value of on average, lies noticeably higher than the energy z in the corresponding subsamples. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 EFFECT OF NUCLEAR MATTER 1315 ch ch Rz (BA/BD) Rz (BA/BD) 1.1 1.0 0.9 0.8 0.7 5 10 15 20 ν, GeV 0.6 0.2 0.4 0.6 0.8 1.0 z Fig. 8. Yields of z>0.5 charged hadrons in the subsam- ples BA and BD (closed circles) and in the interactions Fig. 7. Ratio of the yields of charged hadrons in the of electrons with nitrogen nuclei and deuterons [16] (open circles) versus ν. subsamples BA and BD (closed circles; 2 <ν<15 GeV) and in the interactions of electrons with nitrogen nuclei and deuterons [16] (open circles; 7 <ν<24 GeV) ver- sus z. 15 GeV, along with data obtained at higher energies of 7 <ν<24 GeV for nitrogen nuclei (A =14) [16]. Although stronger intranuclear-absorption effects The ratio Rz(BA/BD) asafunctionofz is shown ch are expected here, values that the ratio Rz (BA/BD) in Fig. 5 individually for positive hadrons (in which takes in the region 0.2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1316 AGABABYAN et al. loss per unit length. In the approximation where the ν(1 − z), which is a combination of our kinematical emission of gluons from the quark is disregarded, variables; averaging over this quantity was performed we have κ ≈ 1 GeV/fm [26, 27], while, upon the by using its experimental distribution in the subsam- A inclusion of this emission, the effective value of κ ple BD. The mean values Rz calculated in this way 2 2 is given by [10] κ =8αs(Q )Q /9π ≈ 1.8 GeV/fm were averaged over target nuclei, and the resulting 2 2 at Q =3.6 (GeV/c) and αs =0.35 [28]. If the theoretical value Rz was compared with the experi- production of a leading qq¯ pair (predominantly, a ud¯ mentally measured ratio Rz(BA/BD). pair in the present case) proceeds through the rupture The calculations were performed at two values of of the string at l = lh, then its energy—that is, the the parameter κ, 1.0 and 1.8 GeV/fm. The parameter energy of the leading final hadron (predominantly, a σh was fitted in such a way as to obtain the best de- + π meson in our case)—is Eh ≈ νq = ν − κlh,where scription of data in the region of large z, z =0.7–0.9. lh = ν(1 − z)/κ. The region z<0.7 was not included in the fitting Immediately after the formation of a qq¯ pair, its procedure because a partial compensation of the sup- mean transverse dimension may be smaller than the pression of the hadron yield due to the secondary in- pion radius, in which case the color quark charge is teractions of more energetic particles was disregarded partly screened; as a result, the interaction of this in the calculations. qq¯ +6 pair in a nucleus is characterized by an effective Our fit resulted in the value of σh =19−4.5 mb cross section that is smaller than the cross section +4.6 in ≈ at κ =1GeV/fm and the value of σh =15.5−3.8 mb for inelastic pion–nucleon interaction, σπN 20 mb. From theoretical predictions obtained in [10, 11], it at κ =1.8 GeV/fm. In Fig. 9, the curves calcu- follows that, with increasing ν,theeffective cross lated at these parameter values are contrasted against experimental data. For 0.7 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 EFFECT OF NUCLEAR MATTER 1317 + Rz (BA/BD) 1.2 (‡) (b) (c) 〈ν〉 = 6.5 GeV 〈ν〉 = 3.8 GeV 〈ν〉 = 8.7 GeV 1.0 0.8 0.6 0.4 0.6 0.8 0.4 0.6 0.8 0.4 0.6 0.8 z + Fig. 9. Comparison of data on the ratio Rz (BA/BD) for positively charged hadrons (points) along with the predictions of the color-string model (the solid and dashed curves calculated, respectively, at σh =19mb and κ =1GeV/fm and at σh =15.5 mb and κ =1.8 GeV/fm) for (a) 2 <ν<15 GeV,(b) 2 <ν<5 GeV,and (c) 5 <ν<15 GeV. is commensurate with internucleon distances in a the prehadronic qq¯ state acquires the properties of nucleus. the nascent pion within a spacetime interval that is commensurate with internucleonic distances in a nucleus. We have also obtained a piece of evidence CONCLUSION in favor of the statement that gluon emission from a New experimental data on the neutrino-induced knock-on quark does not play a significant role in the production of hadrons on nuclei have been obtained evolution of the color string at moderately low values in the intermediate-energy region specified by the of Q2 ≈3.6 (GeV/с)2 and ν ≈6.5 GeV, which inequalities 2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1318 AGABABYAN et al. 1 8 . V. V. Am m o s o v et al.,Fiz.Elem.´ Chastits At. Yadra 24. P.Allen et al.,Nucl.Phys.B214, 369 (1983). 23, 648 (1992) [Sov. J. Part. Nucl. 23, 283 (1992)]. 25. D. Allasia et al.,Z.Phys.C24, 119 (1984). 19. A. E.´ Asratyan et al.,Yad.Fiz.41, 1193 (1985) [Sov. 26. A. Casher, H. Neuberger, and S. Nussinov, Phys. Rev. J. Nucl. Phys. 41, 763 (1985)]. D 20, 179 (1979). 20. N. Agababyan et al., Preprint No. 1539 (Yerevan. 27. E. G. Gurvich, Phys. Lett. B 87B, 386 (1979). Phys. Inst., Yerevan, 1999). 28. Review of Particle Physics, Eur. Phys. J. C 3,87 21. N. Agababyan et al., Preprint No. 1578 (Yerevan. (1998). Phys. Inst., Yerevan, 2002). 29. H. De Vries, C. W. De Jager, and C. De Vries, At. 22. D. Zieminska et al., Phys. Rev. D 27, 47 (1983). Data Nucl. Data Tables 36, 495 (1987). 23. J. Brunner et al.,Z.Phys.C45, 361 (1989). Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1319–1327. Translatedfrom Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1359–1367. Original Russian Text Copyright c 2003 by Babenko, Petrov. ELEMENTARY PARTICLES AND FIELDS Theory Correlation between the Properties of the Deuteron and the Low-Energy Triplet Parameters of Neutron–Proton Scattering V. A. Babenko and N. M. Petrov* Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Metrologicheskaya ul. 14b, Kiev, 03143 Ukraine Received June 4, 2002; in final form, October 9, 2002 Abstract—The correlation between the asymptotic normalization constant for the deuteron, AS,andthe neutron–proton scatteringlengthfor the triplet case, at, is investigated. It is found that 99.7% of the asymptotic constant AS is determined by the scatteringlength at. It is shown that the linear correlation −2 between the quantities AS and 1/at provides a good test of correctness of various models of nucleon– nucleon interaction. It is revealed that, for the normalization constant AS and for the root-mean-square deuteron radius rd, the results obtained with the experimental value recommended at present for the triplet scatteringlength at are exaggerated with respect to their experimental counterparts. By using the latest experimental data obtained for phase shifts by the group headed by Arndt, it proved to be possible to derive, for the low-energy parameters of scattering (at, rt, Pt) and for the properties of the deuteron (AS , rd), results that comply well with experimental data. c 2003 MAIK “Nauka/Interperiodica”. 1. Basic features of the deuteron—such as the both theoretically and experimentally (see [4, 7, 15– bindingenergy εd; the electric quadrupole moment 17]). The majority of the theoretical estimates of this Q; the root-mean-square radius rd; the asymptotic quantity are in perfect agreement with its experimen- normalization constants for the S and the D wave, tal value of η =0.0272 [15]. At the same time, the AS and AD; and the correspondingmixingparame- values of some features of the deuteron, such as the ter η = AD/AS —play a significant role in construct- asymptotic normalization constant AS and the root- ingrealistic models of nucleon –nucleon interaction mean-square radius rd, were the subject of contro- and are important physical characteristics of nuclear versy. For example, the value obtained for AS directly forces. Of equally great importance are low-energy from experimental data by analyzingelastic proton – parameters of neutron–proton scatteringin the triplet deuteron scattering[18], state. These include the scatteringlength at;theef- −1/2 AS =0.8781 fm , (3) fective range rt; and the shape parameters v2, v3, v4, ... appearingin the e ffective-range expansion is at odds with the theoretical estimates derived for this quantity for many realistic potentials [19–26], as 1 1 2 4 6 well as with those found from an analysis of phase k cot δt(k)=− + rtk + v2k + v3k (1) at 2 shifts [17, 27] and on the basis of the effective- 8 range expansion [4, 8]. Values of the asymptotic + v4k + ..., constant AS that are discussed in the literature where δt(k) is the triplet neutron–proton phase shift vary within a rather broad range—from 0.7592 to 3 −1/2 proper correspondingto the S1 state. For this reason, 0.9863 fm [28, 29]. much attention has been given to these quantities In a number of studies, the result reported by −1/2 both in theoretical and in experimental studies [1– Ericson in [5], AS =0.8802 fm , is used for an 17]. At the present time, the experimental value of “experimental” value. It should be noted, however, the deuteron bindingenergy εd is known to a high that this value was obtained from an analysis of the precision [9]: linear relationship between the asymptotic constant A and the root-mean-square radius r rather than ε =2.22458900 MeV. (2) S d d from experimental data directly. This linear correla- The value of the asymptotic constant for S–D mix- tion between AS and rd was established empirically ing, η, was also determined to a fairly high precision, for various models of nucleon–nucleon interaction [5, 30]. The procedure that Ericson used to obtain the * −1/2 e-mail: [email protected] above value of AS =0.8802 fm on the basis of 1063-7788/03/6607-1319$24.00 c 2003 MAIK “Nauka/Interperiodica” 1320 BABENKO, PETROV this relationship involved averaging the values of the Accordingto Eq. (4), the followingvalues of the scat- root-mean-square radius rd that were found exper- teringlength at correspond to the values of the radius imentally in [10, 11]. In view of this, the use of Er- rd in (5): icson’s result for an experimental value is not quite a =5.4050 fm (B), (6a) correct. The value that is presented in (3) and which t was derived in [18] on the basis of a direct method at =5.3863 fm (S), (6b) for determiningthis normalization constant from an at =5.3713 fm (K). (6c) analysis of data on elastic proton–deuteron scattering is more justified. On the other hand, the values of the triplet scat- teringlengththat were calculated for many realistic Other values sometimes used for the experi- nucleon–nucleon potentials [19–21, 24, 25] are close mental asymptotic normalization constant include to the experimental value [32] −1/2 AS =0.8846, 0.8848, and 0.8883 fm (see [4], [30, 31], and [8], respectively). In just the same way as at =5.424 fm, (7) Ericson’s result, they can hardly be treated, however, which is recommended at present, but which, in ac- as correct experimental values, since they were found cordance with Eq. (4), leads to a root-mean-square in the effective-range approximation with allowance radius exaggerated in relation to the experimental for some corrections associated with the form of inter- values in (5a)–(5c); that is, action; moreover, the low-energy triplet parameters r =1.9711 fm. (8) of neutron–proton scatteringthat were employed in d doingthis were not determined from experimental Thus, we can see that, frequently, values obtained and data unambiguously. By way of example, we indicate used in various studies for the features of the deuteron that, in [1–3, 12–14, 32], values between 5.377 fm [1] and for the triplet low-energy parameters of neutron– and 5.424 fm [32] are given for experimental values of proton scatteringare contradictory and deviate from the triplet scatteringlength at, but the asymptotic experimental results. normalization constant AS greatly depends on at. 2. In accordance with [8], the asymptotic normal- As will be shown below, 99.7% of the normalization ization constant AS for the deuteron can be repre- constant AS is determined by the triplet scattering sented in the form a length t. 2 2α AS = , (9) In the following, the value in (3) from [18] will 1 − αρd be used for the experimental value of the normaliza- tion constant A . It is close to the value of A = where α is the deuteron wave number defined by the S S 2 2 ≡ − − 0.8771 fm−1/2, which corresponds to the vertex- relation εd = α /mN and ρd ρ( εd, εd) is the constant value of G2 =0.427 fm for the d → n + p deuteron effective radius correspondingto S-wave d interaction. The definition and properties of the ra- vertex function and which was obtained earlier in [33]. dius ρ and of the function ρ(E ,E ) are discussed As we have already indicated, the authors of [18, 33] d 1 2 in detail elsewhere [1]. The quantity ρ appears in employed a direct method for determiningthe con- d the expansion of the function k cot δ (k) at the point stants A and G2 that relies on extrapolatingthe ex- t S d k2 = −α2—that is, at the energy value equal to the perimental cross sections for elastic proton–deuteron deuteron bindingenergy. The expansion at the point scattering to the exchange-singularity point. k2 = −α2 is similar to the expansion in (1), which In just the same way as the constant AS, the root- is performed at the origin, involving the ordinary ef- r mean-square radius d of the deuteron is determined, fective range r ≡ ρ(0, 0) of scatteringtheory —that a t to a great extent, by the triplet scattering length t. is, the effective range at zero energy. It can easily be r A linear correlation between the quantities d and found that the quantities ρ and r are expanded in a was established empirically in [6, 30]. To a high d t t α2 precision, this relationship can be approximated as powers of the parameter as 2 4 6 follows: ρd = ρm − 2v2α +4v3α − 6v4α + ..., (10) − 2 4 6 rd =0.4at 0.1985 (fm). (4) rt = ρm +2v2α − 2v3α +2v4α − ..., (11) At present, the followingexperimental values are used where ρm ≡ ρ(0, −εd) is the so-called mixed effective in the literature for the root-mean-square radius of range [1], for which the following relation holds: the deuteron: 2 − 1 rd =1.9635 fm [10] (B), (5a) ρm = 1 . (12) α αat rd =1.9560 fm [11] (S), (5b) The shape parameters vn in expansions (1), (10), and rd =1.950 fm [30] (K). (5c) (11) are dimensional quantities. Instead of them, one PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CORRELATION BETWEEN THE PROPERTIES OF THE DEUTERON 1321 often introduces the dimensionless shape parameters the scatteringlength at and the effective range rt— Pt, Qt, ... related to the parameters vn by the equa- are expressed in terms of bound-state parameters tions (deuteron radius Rd and dimensionless normalization − 3 5 constant Cd)as v2 = Ptrt ,v3 = Qtrt ,.... (13) 2Rd From the expansions in (10) and (11), it follows that at = −2 , (24) the quantities rt, ρd,andρm are related as 1+Cd − 4 6 − 2 ρd + rt =2ρm +2v3α − 4v4α + ..., (14) rt = Rd(1 Cd ). (25) 2 6 ρd +2rt =3ρm +2v2α − 2v4α + .... (15) With the aid of parameters that characterize the bound state of the neutron–proton system (deute- The asymptotic normalization constant AS for the ron), the behavior of the phase shift at low energies deuteron is directly expressed in terms of the residue can be predicted in the effective-range approxima- of the S matrix S(k) at the pole k = iα corresponding tion (19) with allowance for expressions (24) and (25). to a bound state of the two-nucleon system; that is, 3. As was indicated above, a great number of stud- 2 ies have been devoted to exploringand calculatingthe AS = i Res S(k). (16) k=iα asymptotic normalization constant AS. The values of the constant AS (and of the quantities εd and at) Alongwith the constant AS, other physical quanti- ties, such as the nuclear vertex constant G and the for some realistic potentials [19–25, 27, 34–36] are d quotedinTable1,alongwiththevaluesofA that dimensionless asymptotic normalization constant Cd, S are frequently used in the literature [28, 29]. These were found from the analysis of phase shifts in [17, 27] and on the basis of the effective-range expansion two quantities are directly related to the constant AS by the equations in [4, 8]. Also given in the same table are the values of the constant A that were calculated in the present 2 2 2 S Gd = πň AS, (17) study by formulas (18) and (21), which correspond to C2 = A2 /2α, (18) the approximation where there is no dependence on d S the form of interaction. In addition, Table 1 presents where ň =2ňN ,withňN ≡ /mN c beingthe Comp- the absolute (∆) and the relative (δ) error in the ton wavelength of the nucleon. calculation of AS in this approximation. In the approximation where there is no dependence It can be seen from Table 1 that, for the majority on the form of interaction (v2 = v3 = ...=0), of the models, the relative error in the AS value cal- culated in the effective-range approximation does not 1 1 2 k cot δt(k)=− + rtk , (19) exceed 0.3%, while the absolute error is not greater a 2 t than 0.003 fm−1/2, as a rule. This is not so only the quantities rt, ρd,andρm are equal to each other, for some early models of the Bonn potential (lines as follows from Eqs. (10) and (11): 15–17 in Table 1), in which case the relative error rt = ρd = ρm. (20) in the approximate value of the constant AS is 2 to 3%. For the same potentials, the relative error in the In this approximation, the use of relations (12) and approximate values of the root-mean-square radius (20) makes it possible to recast Eq. (9) into the more rd of the deuteron that were obtained by formula (4) convenient form is also overly large. For example, the relative error in R −2 − d rd is 2.52% for the HM-2 potential and 1.49% for the Cd = 1+2 , (21) at Bonn F potential. At the same time, this error is small where the quantity for more correct models of the Bonn potential, Bonn R and Bonn Q (0.081 and 0.137%, respectively). For the ≡ Rd 1/α, (22) Paris potential, the error in question is 0.035%. which characterizes the spatial dimensions of the Thus, we can see from Table 1 that the values deuteron, is referred to as the deuteron radius [2]. By of the normalization asymptotic constant AS for usingthe value in (2) for the deuteron bindingenergy, realistic potentials are strongly correlated with the one can easily obtain the followingnumerical value for values of the triplet neutron–proton scatteringlength the deuteron radius: at.ThesameisalsotrueforthevaluesofAS that were found from the analysis of phase shifts Rd =4.317688 fm. (23) (lines 18, 19) and on the basis of the effective- In the approximation where there is no dependence range expansion in [4] (line 20). As to the value of −1/2 on the form of interaction, the low-energy param- AS =0.8883 fm (line 21), which was calculated eters of neutron–proton scattering—we mean here in [8] on the basis of the effective-range expansion, PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1322 BABENKO, PETROV Table 1. Asymptotic normalization constant AS for the deuteron within various models of nucleon–nucleon interaction −1/2 AS ,fm −1/2 no. References εd,MeV at,fm ∆,fm δ,% effective-range precise value approximation 1 RHC [34] 2.22464 5.397 0.88034 0.87867 0.00167 0.216 2 RSC [34] 2.2246 5.390 0.87758 0.87711 0.00047 0.054 3 Paris [19] 2.2249 5.427 0.8869 0.88528 0.00162 0.183 4 Moscow [35] 2.2246 5.413 0.8814 0.88211 0.00071 0.080 5 Nijm I [21, 27] 2.224575 5.418 0.8841 0.88272 0.00138 0.156 6 Nijm II [21, 27] 2.224575 5.420 0.8845 0.88316 0.00134 0.152 7 Reid 93 [21, 27] 2.224575 5.422 0.8853 0.88360 0.00170 0.193 8 Argonne v18 [25] 2.224575 5.419 0.8850 0.88341 0.00159 0.180 9 GK-4 [36] 2.226 5.364 0.87462 0.87201 0.00261 0.299 10 GK-8 [36] 2.226 5.413 0.88434 0.88262 0.00172 0.194 11 GK-7 [36] 2.226 5.477 0.89776 0.89678 0.00098 0.109 12 HM-1 [22, 23] 2.224 5.50 0.901 0.90119 0.00019 0.021 13 Bonn R [20] 2.2246 5.423 0.8860 0.88430 0.00170 0.193 14 Bonn Q [20] 2.2246 5.424 0.8862 0.88452 0.00168 0.190 15 Bonn F [20] 2.2246 5.427 0.9046 0.88517 0.01942 2.195 16 HM-2 [22, 23] 2.2246 5.45 0.919 0.89024 0.02876 3.230 17 M [24] 2.22469 5.424 0.9043 0.8845 0.01975 2.233 18 SCSS [17] 2.224575 5.4193 0.8838 0.88300 0.00079 0.089 19 STS [27] 2.224575 5.4194 0.8845 0.88303 0.00147 0.166 20 ER-C [4] 2.224575 5.424 0.8846 0.88451 0.00009 0.011 21 KMMA [8] 2.224644 5.412 0.8883 0.88191 0.00639 0.725 it is strongly overestimated in relation to the value taken to have approximately the same value in all of −1/2 of AS =0.88191 fm , which was found in the the calculations, one can conclude on the basis of the present study in the approximation where there is results in Table 1 that 99.7% of the asymptotic nor- no dependence on the form of interaction. In that malization constant AS is determined by the triplet case, the relative error in the approximate value scatteringlength at. The inverse also holds: knowing was 0.725%. An incorrect choice of the shape pa- the values of the constant AS, one can determine the triplet scatteringlength a to a high degree of rameter Pt in [8] is the reason for this discrepancy— t namely, the followingdata from [37] on the low- precision. energy parameters of scattering were used in [8] Takingthe aforesaid into consideration, we will to calculate the constant AS: at =5.412 fm and investigate the asymptotic constant AS asafunction rt =1.733 fm; however, the value of Pt = −0.0188, of the triplet scatteringlength at.Itisconvenient chosen there for the shape parameter, corresponded to perform this investigation for the dimensionless −2 to the Paris potential, for which the scatteringlength quantity Cd , which, in the approximation where and the effective range take values (at =5.427 fm, there is no dependence on the form of interaction, is rt =1.766 fm) that considerably exceed those that a linear function of the quantity Rd/at [see Eq. (21)]. were employed in [8]. −2 The Rd/at dependence of Cd isshowninthefig- Consideringthat the deuteron bindingenergyhas ure. The straight line represents the results of the been determined to a high degree of precision and is calculation based on the approximate formula (21), PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CORRELATION BETWEEN THE PROPERTIES OF THE DEUTERON 1323 –2 C d 9 0.60 2 4 18 1 20 5 10 6, 19, 8 3 7 1413 21 0.58 12 11 15 17 0.56 16 0.54 0.785 0.790 0.795 0.800 0.805 Rd/at Correlation between the dimensionless asymptotic normalization constant for the deuteron, Cd, and the triplet scattering length at. Points represent values obtained within the various models of nucleon–nucleon interaction in Table 1 (the numerals in the figure correspond to the numbers of the lines in this table), while the straight line was calculated by the approximate formula (21). while the points in the figure correspond to εd, at,and which is well above the experimental value of this AS values computed by various authors and quoted quantity in (3). As was indicated above, the exper- in Table 1. As can be seen from the figure, there imental scattering-length value in (7) also leads to −2 is a linear relationship between the quantities Cd the exaggerated value in (8) for the root-mean-square and Rd/at. Points correspondingto their values lie radius rd of the deuteron. in a close proximity of the straight line specified by Thus, it can be concluded from the above anal- Eq. (21). As was mentioned above, this is not so only ysis that the currently recommended experimental for points correspondingto early models of the Bonn value of the triplet neutron–proton scatteringlength potential and the point correspondingto the values in (7) does not comply with the experimental values of the quantities in question from [8]. Points that of the asymptotic normalization constant A for −2 S represent values of Cd and Rd/at for more correct the deuteron and its root-mean-square radius rd versions of the Bonn potential model (Bonn R, Bonn in (3) and (5a)–(5c), respectively. Therefore, it is of Q) lie near the straight line specified by Eq. (21) paramount importance to determine, for the features (points 13, 14). of the deuteron and for the parameters of low-energy Thus, it follows from Table 1 and from the figure neutron–proton scatteringin the triplet case, such that the asymptotic normalization constant AS and values that would be consistent with one another, the triplet scatteringlength at are well correlated on one hand, and which would be compatible with quantities, so that any of these can be determined to a experimental data on the other hand. high degree of precision if the other is known. For the experimental value presented in (3) for the constant 4. Fixingthe features εd, AS,andrd of the deuteron, we will study the behavior of the S-wave AS, the correspondingvalue of the scatteringlength phase shift at low energies in the approximation that at can easily be determined in the effective-range approximation by formulas (18) and (21). The result takes into account the shape parameter Pt in the is effective-range expansion; that is, a =5.395 . t fm (26) − 1 1 2 − 3 4 k cot δt(k)= + rtk Ptrt k . (28) At the same time, the currently recommended exper- at 2 imental value of the triplet scatteringlengthin (7) For the scatteringlength at, use is made here of the leads, in the effective-range approximation, to the values in (6a)–(6c), which correspond to the values asymptotic-normalization-constant value of the root-mean-square radius rd of the deuteron −1/2 AS =0.88451 fm , (27) in (5a)–(5c), while, in accordance with (10), (13), PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1324 BABENKO, PETROV Table 2. Parameters of effective-range theory that were calculated on the basis of the experimental values of εd, AS,and rd, as well as those found from the analysis of the experimental phase shifts for neutron–proton scattering Version at,fm rt,fm Pt ρm,fm ρd,fm B 5.4050 1.75047 −0.02312 1.73716 1.72385 S 5.3863 1.70257 0.02010 1.71321 1.72385 K 5.3713 1.66391 0.06064 1.69388 1.72385 ER 5.395 1.72385 0 1.72385 1.72385 PWA 5.4030 1.7495 −0.02592 1.73461 1.7197 Table 3. Triplet phase shift calculated for neutron–proton scatteringin the shape-parameter approximation (28) with the parameter values from Table 2 as a function of the laboratory energy Elab Phase shift δt,deg Elab,MeV experimental data [38] PWA B S K ER 0.1 169.32 169.315 169.311 169.349 169.380 169.331 0.5 156.63 156.645 156.637 156.728 156.802 156.686 1.0 147.83 147.823 147.812 147.954 148.068 147.889 2.0 136.56 136.548 136.536 136.770 136.958 136.664 5.0 118.23 118.236 118.228 118.750 119.168 118.516 10.0 102.55 102.598 102.612 103.672 104.521 103.200 20.0 85.84 86.049 86.127 88.385 90.211 87.379 30.0 75.61 76.043 76.195 79.702 82.592 78.131 40.0 68.11 68.822 69.046 73.796 77.806 71.652 45.0 64.96 65.843 66.103 71.462 76.050 69.032 50.0 62.11 63.171 63.466 96.424 74.602 66.709 100.0 42.34 45.705 46.265 57.612 69.487 52.131 and (14), the effective range rt and the shape param- latest partial-wave analysis (PWA) reported by Arndt eter Pt are given by et al. [38] for the case of nucleon–nucleon scattering, r =2ρ − ρ , (29) which, as can be seen from this table, are in very good t m d agreement with the corresponding values for version ρd − ρm Pt = , (30) B, where the features of the deuteron were set to 2r3α2 −1/2 t εd =2.22458900 MeV [9], AS =0.8781 fm [18], and r =1.9635 fm [10]. where the effective radius ρd of the deuteron and the d mixed effective range ρm are determined from Eqs. (9) The quantity δρ defined as the difference of the and (12), respectively. The parameters at, rt, Pt, ρm, effective radius ρd of the deuteron and the mixed and ρd calculated in this approximation on the basis effective range ρm, of the experimental values of the deuteron bindingen- δρ = ρd − ρm, (31) ergy εd in (2), the asymptotic normalization constant AS in (3), and the root-mean-square radius rd of the is often discussed in the literature. Accordingto the deuteron in (5a)–(5c) are given in Table 2 (versions estimates obtained by Noyes in [3] on the basis of dis- B, S, K), alongwith the values calculated for these persion relations, this difference arises owingto one- parameters in the effective-range approximation (ER) pion exchange and is positive, its magnitude being on the basis of the experimental values of εd and AS, 0.016 fm. For many potential models, the difference as well as values found in the present study from the δρ is also positive; as was established in [5, 6, 31], it PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 CORRELATION BETWEEN THE PROPERTIES OF THE DEUTERON 1325 Table 4. Difference ∆ of the experimental value of the phase shift and its theoretical value calculated in the shape- parameter approximation (28) with the parameter values from Table 2 ∆,deg Elab,MeV PWA B S K ER 0.1 0.005 0.009 −0.029 −0.060 −0.011 0.5 −0.015 −0.007 −0.098 −0.172 −0.056 1.0 0.007 0.018 −0.124 −0.238 −0.059 2.0 0.012 0.024 −0.210 −0.398 −0.104 5.0 −0.006 0.002 −0.520 −0.938 −0.286 10.0 −0.048 −0.062 −1.122 −1.971 −0.650 20.0 −0.209 −0.287 −2.545 −4.371 −1.539 30.0 −0.433 −0.585 −4.092 −6.982 −2.521 40.0 −0.712 −0.936 −5.686 −9.696 −3.542 45.0 −0.883 −1.143 −6.502 −11.090 −4.072 50.0 −1.061 −1.356 −7.314 −12.492 −4.599 100.0 −3.365 −3.925 −15.272 −27.147 −9.791 correlates well with the triplet scatteringlength at.In PWA versions. These sets provide a nearly precision our case, this difference is given by description (with a relative error of about 0.005%)of experimental phase shifts up to an energy value of δρ =2P r3α2; t t (32) 5 MeV. Thus, the accuracy of existingexperimental for the parameter values used in the S and K versions data that is available at present is quite sufficient for from Table 2, it is positive, takingthe values of 0.011 removingambiguities that arise in determiningthe and 0.030 fm, respectively. For the cases of B and scatteringlength at and the effective range rt [1] and PWA in Table 2, the difference δρ is negative, its val- which are associated with the form of the potential. In ues being −0.013 and −0.015 fm. For this reason, the view of this, the problem of deducingthe scattering problem of the sign and magnitude of the difference of length, the effective range, and parameters of higher the effective radius ρd of the deuteron and the mixed order (shape parameters) in expansion (1) directly effective range ρm calls for a further investigation. from experimental data is pressing. It should be noted For the low-energy parameters given in Table 2, that the B and PWA sets describe well, in contrast to other parameter sets, experimental phase shifts up to we have calculated the triplet phase shift δ (k).The t an energy value of 50 MeV. For the B and PWA sets, results are displayed in Table 3, alongwith the latest the absolute error is about 0.5◦ at E =30MeV and experimental data of Arndt et al. [38] on the triplet ◦ lab about 1 at Elab =50MeV. From Table 4, it can be phase shift. Table 4 presents the energy dependence seen that, for other parameter sets, the absolute error of the difference is much greater. − ∆=δexpt δtheor (33) Thus, neutron–proton scatteringin the triplet of the experimental value of the phase shift and its state can be described rather well within the B set up theoretical counterparts calculated by formula (28) to an energy value of 50 MeV,the experimental values and quoted in Table 3. As can be seen from Tables 3 used in this description for the features of the deuteron and 4, all sets of low-energy parameters a , r ,and beingthat in (2) for the bindingenergy, that in (3) t t for the asymptotic normalization constant, and that Pt from Table 2 describe well experimental data up to an energy value of 5 MeV (the absolute error being in (5a) for root-mean-square radius. On the other less than 1◦). Nonetheless, the distinction between hand, parameters that characterize the neutron– the sets of low-energy parameters in describing ex- proton bound state (deuteron) can be determined perimental data becomes noticeable at an energy as from experimental data on neutron–proton scatter- low as 1 MeV, and we can see that preference should ing. By using the values be given to the parameter sets employed in the B and at =5.4030 fm,rt =1.7495 fm, (34) PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1326 BABENKO, PETROV Pt = −0.0259, Even at an energy of 10 MeV, the absolute error exceeds 1◦ in this case. which we found here for the low-energy parameters of The values found for the low-energy parameters scatteringfrom an analysis of the latest data on phase of scatteringwith the aid of the latest experimen- shifts [38], we will now determine the asymptotic nor- tal results reported by Arndt et al. [38] for the malization constant AS for the deuteron and its root- phase shifts are a =5.4030 fm, r =1.7495 fm, and r t t mean-square radius d. In accordance with Eqs. (4), P = −0.0259. They are in good agreement with the (9), (10), (12), and (13), we obtain t parameter values of at =5.4050 fm, rt =1.7505 fm, −1/2 and P = −0.0231, which were obtained on the basis AS =0.8774 fm , (35) t of the experimental values of the features of the rd =1.9627 fm. (36) deuteron. Both sets of these parameters make it As might have been expected, these results are in possible to describe well, in the shape-parameter very good agreement with the experimental values of approximation, the experimental phase shift up to an −1/2 energy of 50 MeV, this indicating that the parameter AS =0.8781 fm [18] and rd =1.9635 fm [10]. Qt and parameters of higher order in the effective- 5. To summarize, we will formulate our basic re- range expansion are small. sults and conclusions. We have investigated the cor- On the basis of the parameter values of at = relation between the asymptotic normalization con- 5.4030 fm, rt =1.7495 fm, and Pt = −0.0259, which stant for the deuteron, AS, and the triplet neutron– correspond to the experimental phase shifts ob- proton scatteringlength at. It has been established tained by Arndt et al. [38], we have found the that 99.7% of the asymptotic constant AS is deter- asymptotic normalization constant for the deuteron, mined by the triplet scatteringlength a . It has been −1/2 t AS =0.8774 fm , and its root-mean-square ra- shown that, in the effective-range approximation, the dius, r =1.9627 fm, these results beingin ex- 2 d linear correlation between the quantities 2α/AS and cellent agreement with the experimental values of R /a −1/2 d t provides a good test of correctness of various AS =0.8781 fm [18] and rd =1.9635 fm [10]. potential models and methods that are used in study- In summary, we arrive at the basic conclusion that ingnucleon –nucleon interaction. the latest experimental results of Arndt et al. [38] for It has been found that evaluatingthe asymptotic the phase shift comply very well with the experimental normalization constant for the deuteron and its root- values of parameters that characterize the deuteron— mean-square radius with the currently recommended specifically, with the bindingenergyin (2), the asymp- triplet-scattering-length value of at =5.424 fm [32] totic normalization constant determined in [18] and −1/2 leads to results, AS 0.8845 fm and rd = given in (3), and the root-mean-square radius of the 1.9711 fm, that are exaggerated in relation to the deuteron as obtained in [10] and presented in (5а). correspondingexperimental values in (3) and (5a) – (5c). REFERENCES By usingthe experimental values of εd = 1. L. Hulthen´ and M. Sugawara, Handb. Phys. 39,1 −1/2 2.22458900 MeV [9], AS =0.8781 fm [18], and (1957). 2. R. 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Stoks, and R. Schiavilla, Phys. (Laval, Quebec, 1975), p. 235. Rev. C 51, 38 (1995). 38. R. A. Arndt, W. J. Briscoe, R. L. Workman, and 26. J. W. Humberstone and J. B. G. Wallace, Nucl. Phys. I. I. Strakovsky, Partial-Wave Analysis Facility SAID, A 141, 362 (1970). URL http://gwdac.phys.gwu.edu. 27. J. J. de Swart, C. P.F. Terheggen, and V. G. J. Stoks, 3rd International Symposium “Dubna Deuteron Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1328–1334. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1368–1374. Original Russian Text Copyright c 2003 by Davidovsky, Struminsky. ELEMENTARY PARTICLES AND FIELDS Theory Nucleon Structure Functions, Resonance Form Factors, and Duality V.V.Davidovsky and B.V.Struminsky †1) Institute for Nuclear Research, National Academy of Sciences of Ukraine, pr. Nauki 47, Kiev, 03680 Ukraine Received April 3, 2002 Abstract—The behavior of nucleon structure functions in the resonance region is explored. For form factors that describe resonance production, expressions are obtained that are dependent on the photon virtuality Q2, which have a correct threshold behavior, and which take into account available experimental data on resonance decay. Resonance contributions to nucleon structure functions are calculated. The resulting expressions are used to investigate quark–hadron duality in electron–nucleon scattering by taking the example of the structure function F2. c 2003MAIK “Nauka/Interperiodica”. 1. INTRODUCTION number of hadrons. Despite the distinction of mecha- nisms responsible for the interaction in the resonance To obtain deeper insights into the structure of and in the deep-inelastic region, it turns out that hadrons and their interactions in terms of quarkand relevant amplitudes (or structure functions) behave gluon degrees of freedom is the most important the- similarly in these regions, and it is possible to describe oretical problem in nuclear and elementary-particle them in terms of only quarkor only hadronic degrees physics. of freedom, irrespective of the kinematical region. The equivalence of quarkand hadronic description Quarkand gluon degrees of freedom provide a is referred to as quark–hadron duality. convenient basis for constructing QCD as a gauge- invariant theory of strong interaction. Hadronic de- Formally, quark–hadron duality is exact; however, grees of freedom form a different basis. Since no the fact that, in practice, we must truncate the ex- physical observable can be dependent on the choice of pansion of any Fockstate leads to di fferent mani- basis, the description of processes in terms of quark– festations of this duality under different kinematical conditions and in different reactions. gluon degrees of freedom is expected to be equivalent to the corresponding description in terms of hadronic We recall that the historically first concept of dual- degrees of freedom. ity related resonances at low energies to Regge poles at high energies. Finite-energy sum rules that were The choice of degrees of freedom for describing independently introduced in [1–3] provided an exact hadronic processes depends on specific kinemat- mathematical formulation of the duality concept. ical conditions. By way of example, we indicate The Veneziano amplitude was the first example that, in lepton–nucleon scattering, the resonance- where hadron–Reggeon duality was fulfilled. A dual production region is usually described in terms of amplitude featuring Mandelstam analyticity [4, 5], hadronic degrees of freedom, while the region of deep- inelastic scattering is naturally described by using quark–gluon degrees of freedom. In the first region, e– a virtual photon that the lepton and the nucleon involved in lepton–nucleon scattering exchange in- teracts with the nucleon as a discrete unit, changing e– the orientation of the spins of relevant quarks and, hence, their angular momentum, whereby resonance γ production occurs. In the second region, a virtual * photon undergoes scattering immediately on the quarks involved, causing the production of a large R †Deceased. N 1)Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Metrologicheskaya ul. 14b, Fig.1. Production of nucleon resonances in the lowest Kiev, 03143 Ukraine. order in the electromagnetic-coupling constant. 1063-7788/03/6607-1328$24.00 c 2003 MAIK “Nauka/Interperiodica” NUCLEON STRUCTURE FUNCTIONS 1329 where one can introduce nonlinear complex Regge Experimental data obtained recently in the Jeffer- trajectories with correct thresholds, provides yet an- son Laboratory [8, 9] provide a strong motivation for other example of this kind. performing such investigations. A manifestation of quark–hadron duality was first noticed by Bloom and Gilman [6] in analyzing data on nucleon structure functions in the region of nucleon- 2. FORM FACTORS resonance excitation from experiments that studied In the lowest order in the electromagnetic coupling electron–nucleon scattering. It turned out that, on constant, the process of lepton–hadron scattering average, the resonance curve for the structure func- proceeds via the exchange of a virtual photon having 2 tion F2(x ,Q ) reproduces the scaling curve. In other a virtuality Q2 = −q2 (q is the photon 4-momentum) words, the area under the resonance curve is equal and an energy (in the nucleon rest frame) ν =(pq)/m to the area under the scaling curve. This is the so- (p is the nucleon 4-momentum, and m is the nucleon called global duality. It was mentioned that, to some mass). In the case under consideration, Q2 also plays specific accuracy, a local duality holds. It means that the role of the square of the momentum transfer from all the aforesaid holds for averaging not only within the lepton to the nucleon involved. the entire resonance region but also in the vicinity of an individual resonance. The mechanism of virtual-photon interaction with The duality provided additional possibilities for a nucleon depends on the momentum transfer to the 2 studying the nucleon structure on the basis of data on nucleon, Q , and on the photon energy. At high values the properties of nucleon resonances. For example, of Q2 and ν, there occur the noncoherent scattering corrections to the scaling behavior of the structure of the virtual photon by the quarks of the nucleon function F2—such corrections were calculated within and the production of a large number of hadrons. At QCD and have been measured over the past decade— Q2 ∼ 1 GeV2 and moderate values of the energy ν, can be extracted from resonance data [7]. the excitation of internal spin states of the nucleon Presently, it is of interest to describe duality occurs, which leads to resonance production (see theoretically—in particular, to construct nucleon Fig. 1). structure functions in the resonance region (threshold Let us introduce basic conventions and define the energies and low values of the photon virtuality). quantities that will be used below. We denote by p, q, For this, it is necessary to know the dependence of and P , respectively, the nucleon, the photon, and the the form factors for resonance-production processes resonance 4-momenta. In the resonance rest frame, γ∗N → R on the photon virtuality Q2. they are given by M 2 + m2 + Q2 (M 2 − m2 − Q2)2 +4M 2Q2 p = , 0, 0, − , (1) 2M 2M M 2 − m2 − Q2 (M 2 − m2 − Q2)2 +4M 2Q2 q = , 0, 0, , (2) 2M 2M P =(M, 0, 0, 0), (3) 2 2 2 2 where p = m , P = M ,andM is the resonance nance, the nucleon, and the photon, respectively, λγ mass. taking the values of −1, 0, +1,andJ(0) is the current The γ∗N → R vertex of virtual-photon absorption operator. by a nucleon is described by three independent form The nucleon structure functions can be expressed 2 factors G±,0(Q ) (or by two form factors if the res- in terms of form factors (4) [10] as 2 onance is equal to 1/2, in which case G−(Q ) ≡ 0), 2 2 2 2 F1(x, Q )=m δ(W − M ) (5) which, apart from a factor, coincide (in the resonance × | 2 |2 | 2 |2 rest frame) with the helicity amplitudes for the transi- G+(Q ) + G−(Q ) , tion γ∗N → R, these amplituces having the form 2 1 ν 2 2 − 2 − | | 1+ 2 F2(x, Q )=mνδ(W M ) (6) Gλγ = R, λR = λN λγ J(0) N,λN , (4) Q 2m 2 2 2 2 2 2 × |G (Q )| +2|G (Q )| + |G−(Q )| , where λR, λN ,andλγ are the helicities of the reso- + 0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1330 DAVIDOVSKY, STRUMINSKY Q2 dependence of the form factors of known resonances 1+ g (x, Q2)=m2δ(W 2 − M 2) (7) ν2 1 has not yet received adequate experimental study: the tables given in [11] only display their values at Q2 =0. 2 2 2 2 ∗ → × |G+(Q )| −|G−(Q )| There are two types of γ N R transitions in parity: normal transitions, √ − − 1/2+ → 3/2 , 5/2+, 7/2 ,..., (11) − J−1/2 Q 2 ∗ 2 2 +( 1) η G0(Q )G+(Q ) , ν and anomalous transitions, + → − + − 2 1/2 1/2 , 3/2 , 5/2 ,.... (12) Q 2 2 2 2 1+ g2(x, Q )=−m δ(W − M ) (8) 2 ν2 For the form factors G±,0(Q ) corresponding to the above transitions, we know (i) their threshold be- 2 2 2 2 havior for |q|→0 [12], (ii) their asymptotic behavior × |G+(Q )| −|G−(Q )| at high Q2, and (iii) their values at Q2 =0[11]. √ It was shown in [12] that, if a γ∗N → R transition − − J−1/2 ν 2 ∗ 2 2 ( 1) η G0(Q )G+(Q ) , is normal in parity [see (11)], the form factors for the Q production of a spin-J resonance have the following threshold behavior: where J and η are the resonance spin and the 2 ∼| |J−3/2 resonance parity, respectively; W 2 =(p + q)2 is the G±(Q ) q , (13) square of the total energy of the photon–nucleon 2 q0 J−1/2 system in the c.m. frame; and x is the Bjorken G0(Q ) ∼ |q| . (14) |q| variable. We recall that the structure functions F1 and F2 are related to the quantities W1 and W2 by the For transitions that are anomalous in parity [see (12)], 2 2 2 equations F1(x, Q )=mW1(ν, Q ) and F2(x, Q )= we have 2 2 J−1/2 νW2(ν, Q ). G±(Q ) ∼|q| , (15) Formulas (5)–(8) control the contribution of one infinitely narrow resonance to the nucleon structure 2 q0 J+1/2 G0(Q ) ∼ |q| . (16) functions. For a resonance of width Γ, it is neces- |q| sary to replace, in these formulas, the delta function + → + δ(W 2 − M 2) by Transitions of the 1/2 1/2 type stand out as a particular case. Specifically, they are controlled only 1 MΓ . (9) by two form factors, G+ and G0—the form factor G− π (W 2 − M 2)2 + M 2Γ2 corresponds to the helicity of a spin-3/2 resonance and, therefore, vanishes for a spin-1/2 resonance. For In principle, this expression is not the only approx- + → + imation of the resonance shape. However, a specific 1/2 1/2 , their threshold behavior is as follows: 2 form of the resonance contribution is immaterial at G+(Q ) ∼|q|, (17) this stage. It should be noted, however, that expres- sion (9) originates from the resonance propagator. 2 q0 2 G0(Q ) ∼ |q| . (18) The idea of the approach used in the present study |q| is to take into account the contributions of all reso- The form factors for 1/2+ → 1/2− transitions are nances for which there are data in the literature [11]. R R given by expressions (15) and (16) at J =1/2;that If we denote by F1,2 and g1,2, the contributions of the is, resonance R to, respectively, the spin-independent 2 ∼ and the spin-dependent structure function, the reso- G+(Q ) const, (19) nance contributions to the structure functions can be q represented as sums G (Q2) ∼ 0 |q|. (20) 0 |q| F = F R ; g = gR . (10) 1,2 1,2 1,2 1,2 2 R R The high-Q behavior of the form factors is deter- mined by quark-counting rules [13, 14], according to In order to calculate the resonance contribution to which − − the structure functions, it is necessary to construct G (Q2) ∼ Q 3,G(Q2) ∼ Q 4, (21) the form factors for resonance production as functions + 0 2 ∼ −5 of the photon virtuality Q2.WenotethattheQ2 G−(Q ) Q . PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 NUCLEON STRUCTURE FUNCTIONS 1331 Thus, the expressions representing the form fac- sections for virtual-photon absorption, R(Q2)= 2 2 tors and possessing all of the aforementioned proper- σL(Q )/σT (Q ), since the following relation holds: ties can be written as 2 G (Q2) 2 2 2 | |2 0 G±(Q ) = G±(0) (22) 2 R R(Q )= 2 2 . (29) 2J−3 (|G (Q2)| + |G−(Q2)| ) |q| Q2 Q2 m± + × 0 0 , R | | 2 2 2 2 q Q=0 Q + Q 0 Q + Q0 By the way, expression (29), together with expres- 2 2a 2 sions (22)–(25), makes it possible to determine the 2 Q q G (Q2) = C2 0 (23) behavior of R, and this may be helpful in analyzing 0 2 2 | |2 Q + Q 0 q experimental data. 2J−1 2 |q| Q2 Q2 m0 The values of the form factors at Q =0and the × 0 0 helicity photoproduction amplitudes A and A |q| 2 2 Q2 + Q2 1/2 3/2 Q=0 Q + Q 0 0 presented in [11] are related by the equation [10] in the case of normal transitions and as 2 − 2 | | −1 M m | | 2 2 2 G+,−(0) = e A1/2,3/2 , (30) G±(Q ) = |G±(0)| (24) m 2J−1 |q| Q2 Q2 m± where e = 4π/137 is the electron charge. We note × 0 0 , 2 | | 2 2 2 2 that, at Q =0, the longitudinal form factor vanishes, q Q=0 Q + Q 0 Q + Q0 G0(0) = 0. 2 2a 2 2 2 2 Q q0 Substituting into (10) relations (5)–(8) written for G0(Q ) = C (25) Q2 + Q2 |q|2 each specific resonance, taking into account transi- 0 tion parities, and using the corresponding expressions 2J+1 | | 2 2 m0 for the form factors, we find that, in the resonance × q Q 0 Q0 | | 2 2 2 2 region, the structure functions have the form q Q=0 Q + Q 0 Q + Q0 m2 F (x, Q2)= (31) in the case of anomalous transitions, where 1 π R (M 2 − m2 − Q2)2 +4M 2Q2 |q| = , (26) × MΓ 2M (m2 + Q2(1/x − 1) − M 2)2 + M 2Γ2 2 − 2 M m 2 n+ |q|Q=0 = , |q| Q 2M × 0 |G (0)|2 | | 2 2 + q Q=0 Q + Q 0 m+ =3, m− =5,andm0 =4. 2 m+ 2 m− 1/2+ → 1/2+ Q0 2 Q0 The form factors for transitions can × + |G−(0)| , be written in the form Q2 + Q2 Q2 + Q2 0 0 2 2 | |2 G+(Q ) = G+(0) (27) 2m2x 1 2 2 | | 2 2 m+ F2(x, Q )= 2 2 2 (32) q Q 0 Q 1+4m x /Q π × 0 , R | | 2 2 2 2 q Q=0 Q + Q 0 Q + Q0 × MΓ (m2 + Q2(1/x − 1) − M 2)2 + M 2Γ2 2 2a 2 2 2 2 Q q0 2 n+ G0(Q ) = C (28) |q| Q Q2 + Q2 |q|2 × 0 |G (0)|2 0 |q| 2 2 + 4 Q=0 Q + Q 0 | | 2 2 m0 q Q 0 Q0 2 m+ 2 m− × . Q0 2 Q0 |q| Q2 + Q2 Q2 + Q2 × + |G−(0)| Q=0 0 0 Q2 + Q2 Q2 + Q2 0 0 2a 2 n0 2 2 Q2 q2 |q| Q In expressions (22)–(25), the quantities Q0, Q 0, +2C2 0 0 2 2 2 | |2 | | 2 2 Q ,anda are free parameters that can be deter- Q + Q 0 q q Q=0 Q + Q 0 0 mined from a fit to experimental data. The coefficient 2 m0 C can be found if there are experimental data on × Q0 2 2 , the ratio of the longitudinal and transverse cross Q + Q0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1332 DAVIDOVSKY, STRUMINSKY 1 m2 production of mesons and other hadrons forms a g (x, Q2)= (33) 1 1+4m2x2/Q2 π nonresonance background, which must also be taken R into account. MΓ × The contribution of the nonresonance background (m2 + Q2(1/x − 1) − M 2)2 + M 2Γ2 can be parametrized as [9] 2 n+ |q| Q 2 − × 0 |G (0)|2 nr 2 Q 1 x | | 2 2 + F2 (x, Q )= (35) q Q=0 Q + Q 4π2α 1+4m2x2/Q2 0 2 m+ 2 m− N Q0 2 Q0 × −|G−(0)| 2 2 n−1/2 2 2 2 2 × (1 + R(x, Q )) Cn(Q )[W − Wth] , Q + Q0 Q + Q0 √ n=1 2 a J−1/2 2 2mx Q +(−1) η C |G+(0)| where N =3, Cn areadjustablecoefficients, and W Q Q2 + Q2 thr 0 is the threshold energy. (n +n )/2 | | | | 2 0 + × q0 q Q 0 | | | | 2 2 q q Q=0 Q + Q 0 3. DUALITY Q2 (m0+m+)/2 Thus, we have obtained expressions that describe × 0 , Q2 + Q2 the nucleon structure functions in the resonance re- 0 gion and which depend on the Bjorken variable and the photon virtuality. 1 m2 g (x, Q2)=− (34) Following [6], we will consider how expressions 2 1+4m2x2/Q2 π R (31)–(34) for the structure functions—these expres- MΓ sions involve resonance terms, which are strongly × 2 (m2 + Q2(1/x − 1) − M 2)2 + M 2Γ2 dependent on Q owing to the form factors—reduce, 2 n+ in the high-Q limit, to scaling expressions, which are | | 2 2 q Q 0 2 weakly dependent on Q . × |G+(0)| | | 2 2 q Q=0 Q + Q 0 Let us go over to the variable x = Q2/(Q2 + W 2), 2 m+ 2 m− which is related to x by the equation Q0 2 Q0 × −| − | 2 2 G (0) 2 2 2 Q + Q0 Q + Q0 xQ x = 2 2 . (36) Q Q2 a Q + xm − (−1)J−1/2η √ C |G (0)| 2 2 + 2mx Q + Q 0 Let us consider the case of infinitely narrow → | | | | 2 (n0+n+)/2 resonances—that is, Γ 0. In this case, the struc- × q0 q Q 0 ture function will be different from zero only for those | | | | 2 2 2 2 q q Q=0 Q + Q 0 values of x at which W = M —that is, for x = 2 2 2 2 (m0+m+)/2 Q /(M + Q ). It can be seen that, with increasing Q × 0 Q2, the resonance is shifted to a region around x ∼ 1. 2 2 , Q + Q0 As is suggested by experimental data, the resonance then drifts along the scaling curve, whose form can n =2J − 3 n =2J − 1 where + and 0 for normal tran- be obtained from the expressions for the structure n =2J − 1 n =2J +1 sitions, + and 0 for anomalous functions in the resonance region by means of the transitions, and summation is performed over reso- 2 M 2x/ − x nances. We tookinto account the contributions of the substitution Q = (1 ). following resonances: N(1440), N(1520), N(1535), In the case of finite-width resonances, the equality N(1650), N(1675), N(1680), N(1700), N(1710), W 2 = M 2 is approximate, which leads to corrections N(1720), N(1990), ∆(1232), ∆(1550), ∆(1600), to the scaling behavior that are proportional to the ∆(1620), ∆(1700), ∆(1900), ∆(1905), ∆(1910), ratio Γ/M . ∆(1920), ∆(1930),and∆(1950). In this study, we will consider only the structure The resulting formulas determine the resonance function F2. In what is concerned with the structure contribution to the nucleon structure functions. It functions g1, it should be noted that, in contrast to F2, is obvious that the production of resonances in it is sensitive to the longitudinal form factor and to the electron–nucleon scattering is not the only pro- relative values of G+ and G−—in (33), they appear in cess contributing to the structure functions. The the form of the difference. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 NUCLEON STRUCTURE FUNCTIONS 1333 ξ 2 F2( , Q ) 0.4 Q2 = 0.2 GeV2 Q2 = 0.45 GeV2 Q2 = 0.85 GeV2 0.3 Q2 = 1.4 GeV2 Q2 = 2.4 GeV2 Q2 = 3.3 GeV2 0.2 Scaling curve 0.1 0 0.2 0.4 0.6 0.8 1.0 ξ 2 Fig.2. Resonance contribution to the structure function F2(ξ,Q ) in the resonance region. Letusgoovertothevariablex in (32). At high in [8] and which was obtained from a fit to available Q2, we then obtain data on deep-inelastic scattering. 2 It can be seen that a dominant contribution to the 2m x F (x )= (37) structure function comes from the ∆(1232) isobar. 2 πMΓ R The resonance background is relatively small in this 2 m+ − m+ region, in contrast to what occurs in regions that × | |2 Q0 1 x correspond to more massive resonances. It can also G+(0) 2 M x be seen that, in response to variations in Q2,the 2 m− − m− ∆(1232) peakexactly follows the scaling curve, in 2 Q0 1 x + |G−(0)| accord with the duality concept. M 2 x In order to perform a comparison with experimen- m0 m Q2 1 − x 0 tal data, it is necessary to parametrize the contribu- +2C2 0 . M 2 x tion of the nonresonance background. At this stage, however, this problem does not have an unambiguous From (37), it follows that, in the limit x → 1,the solution. structure function under study behaves as F2(x ) ∼ The inclusion of the nonresonance background (1 − x )m+ . must lead to the growth of the structure function in the region of higher lying resonances to such an In the resonance region, use is often made of the extent that it would follow the scaling curve (as Q2 is Nachtman variable ξ =2x/(1 + 1+4m2x2/Q2) varied) over a wide region of ξ. (the variable x = Q2/2mν is not convenient in an- alyzing structure functions in the resonance region, since all resonances are closely spaced in the region 4. CONCLUSION x ∼ 1). Considering that 0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1334 DAVIDOVSKY, STRUMINSKY processes in terms of only hadronic or only quark REFERENCES degrees of freedom. 1. K. Igi and S. Matsuda, Phys. Rev. Lett. 18, 625 In practice, this implies that investigation of (1967). nucleon resonances—for example, in terms of form 2. A. Logunov, L. Soloviev, and A. Tavkhelidze, Phys. factors—would provide additional information about Lett. B 24B, 181 (1967). the properties of nucleons in the deep-inelastic re- 3. R. Dolen, D. Horn, and C. Schmid, Phys. Rev. 166, gion. 1768 (1968). Expressions that were obtained in the present 4.A.I.Bugrij,L.L.Jenkovszky,andN.A.Kobylinsky, study for the form factors describing the production Nucl. Phys. B 35, 120 (1971). of nucleon resonances and which take into account a 5. L. L. Jenkovszky, V. K. Magas, and E. Predazzi, Eur. correct threshold behavior and experimental data on Phys. J. A 12, 361 (2001). resonance decays can form a basis for studying the 6.E.D.BloomandF.J.Gilman,Phys.Rev.D4, 2901 (1971). properties of nucleons in terms of hadronic degrees of 7. C. E. Carlson and N. C. Mukhopadhyay, Phys. Rev. freedom. Lett. 74, 1288 (1995). Expressions obtained here for the structure func- 8. I. Niculescu, C. S. Armstrong, J. Arrington, et al., tions make it possible to demonstrate qualitatively Phys. Rev. Lett. 85, 1186 (2000). the manifestation of duality by using the structure 9. I. Niculescu, PhD Thesis (Hampton Univ., 1999). function F2 as an example. 10. C. E. Carlson and N. C. Mukhopadhyay, Phys. Rev. D In the future, we plan to study the structure func- 58, 094029 (1998). tion g1, which receives a significant contribution from 11. Particle Data Group, Phys. Lett. B 204, 379 (1988). the longitudinal form factor. A numerical test of the 12. J. D. Bjorken and J. D. Walecka, Ann. Phys. (N.Y.) model upon the appearance of new experimental data 38, 35 (1966). would also be of interest. 13. S. J. Brodsky and G. Farrar, Phys. Rev. Lett. 31, 1153 (1973). 14.V.A.Matveev,R.M.Muradian,andA.N.Tavkhe- ACKNOWLEDGMENTS lidze, Lett. Nuovo Cimento 7, 719 (1973). We are grateful to L.L. Jenkovszky for discussions. Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1335–1341. From Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1375–1381. Original English Text Copyright c 2003 by Andreev. ELEMENTARY PARTICLES AND FIELDS Theory Fermion Production and Correlations due to Time Variation of Effective Mass* I. V. Andreev** Lebedev Institute of Physics, Russian Academy of Sciences, Leninski ˘ı pr. 53, Moscow, 117924 Russia Received May 22, 2002; in final form, August 20, 2002 Abstract—The fermion production arising due to time variation of effective mass has been considered. The diagonal polarization states have been found to be the definite helicity states. The strength of the production process and specificfermion–antifermion correlations have been calculated. The production of the fermion–antifermion pairs and the relative two-particle correlations appeared to be large for a sharp and significant change in the mass depending also on fermion occupancy in the initial state. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION anticommutation relations † − It is well known that, in quantum chromodynam- [bν (k1),bµ(k2)]+ = δµν δ(k1 k2), (2) ics, the chiral invariance is spontaneously broken at [d (k ),d† (k )] = δ δ(k − k ). low energies and the fermion masses (say, nucleon ν 1 µ 2 + µν 1 2 or constituent quark mass) arise essentially due to Bispinors uν(k) and vµ(k) are orthonormal, chiral-symmetry breaking. In the course of phase † † transition at high fermion density or at high temper- uν(k)uµ(k)=vν(k)vµ(k)=δµν , (3) ature, the approximate chiral invariance is restored u (k) v (−k) and the light (u, d, s) quark masses become small and bispinors ν and µ having opposite mo- (ensuring small explicit violation of the symmetry). In menta are orthogonal, † − † − practice, the change in the fermion masses is treated uν (k)vµ( k)=vν(k)uµ( k)=0. (4) (semi)classically (for example, see [1]). However, it appears that, if the change in the mass is fast enough, For bispinors uν (k) and vµ(k) related to the Dirac equation then the quantum fermion field changes in a specific way producing correlated fermion–antifermion pairs. ∂ iγ0 − γnkn − m ψ(k,t)=0, (5) Similar effects were considered earlier for bosons ∂t (mesons [2–5] and photons [6]) with application to heavy-ion collisions [6, 7]. Below, we consider the n =1, 2, 3, fermions in a spatially homogeneous large volume. we take the standard representation Application of this effect to heavy-ion collisions will 1 1 s be given elsewhere. u (k)= U (k)= (m + γk) ν , (6) ν N ν N 0 Decomposition of the free fermion field is taken in the form 1 1 − 0 vν (k)= Vν(k)= (m γk) 3 N N sν d k · ψ(x)= eik x (1) (2π)3/2 with two-component unit spinors (s1)ρ = δ1ρ, 2 (s2)ρ = δ2ρ and normalization factor × −iEt − † − iEt uν (k)bν (k)e + vν( k)dν ( k)e , N ≡ N(E,m)=(2E(E + m))1/2 . (7) ν=1 where E =(k2 + m2)1/2; m is the fermion mass; and † † 2. STEPWISE VARIATION bν , bν and dν, dν are annihilation and creation oper- OF THE FERMION MASS ators of particles and antiparticles, obeying standard Let the mass m in (5) depend on time, m = m(t). ∗This article was submitted by the author in English. We consider in this section the stepwise variation of **e-mail: [email protected] the mass. This simple case reveals the main features 1063-7788/03/6607-1335$24.00 c 2003 MAIK “Nauka/Interperiodica” 1336 ANDREEV of the phenomenon. We suggest that at time t =0the It appears useful to introduce the normalized linear † † mass changes instantly from mi to mf . According combinations of bν, ˜bν and dν, d˜ν operators as follows: to Dirac Eq. (5), the time derivative of the field ψ 1 has a jump discontinuity at t =0. Therefore, the field ´bν(k)= (bν(k)+˜bν (k)), (14) ψ(k,t) is continuous at the point t =0, 2(1 + n3) − → ´† − 1 † − ˜† − ψi(k, δt)=ψf (k, +δt),δt 0. (8) dν( k)= (dν( k)+dν ( k)), 2(1 + n3) In view of continuity of the field ψ,wehave so that, in accordance with (12) and (14), − † − ui,ν(k)bi,ν (k)+vi,ν( k)di,ν ( k) (9) ´bµ(k)=wµν (n)bν (k), (15) − † − = uf,ν(k)bf,ν(k)+vf,ν( k)df,ν( k) ´† − † − dµ( k)=wµν (n)dν ( k) † (sum over ν). Multiplying both sides of (9) by u (k) f,µ with † and then by v (−k) and using (3) and (4), we get the 3 1 2 f,µ 1 1+n n − in transformation of the annihilation and creation oper- w = (16) 2(1 + n3) −n1 − in2 1+n3 ators expressing final-state operators through initial- state operators: cos(θ/2) sin(θ/2)e−iφ ˜† = , bf,ν(k)=αν(k)bi,ν (k)+βν (k)d (−k), (10) i,ν − sin(θ/2)eiφ cos(θ/2) † d (−k)=−βν(k)˜bi,ν (k)+αν(k)di,ν (−k) f,ν where θ and φ are spherical angles of the momentum with coefficients k. The rotation matrix w is the SU(2) matrix, and 2 the operators ´b (k), d´ (k) obey correct commutation (Ef + mf )(Ei + mi)+k ν ν αν(k)= , (11) relations; satisfy the orthogonality conditions (3) and N N i f (4); and (contrary to input operators b (k), d (k))un- − − ν ν k(Ef + mf Ei mi) dergo a simple evolution transformation, which does βν(k)=∓ NiNf not mix polarizations of the new operators ´bν, d´ν, ´ ´ ´† − for ν =1, 2, respectively, where Ni and Nf are nor- bf,ν(k)=αν(k)bi,ν (k)+βν (k)di,ν ( k), (17) malization factors of the spinors u ,v in the initial ν ν ´† − − ´ ´† − and final states given by (7). In (10), the combinations df,ν( k)= βν (k)bi,ν(k)+αν (k)di,ν ( k), b b of the polarization states of 1, 2 operators arise: for every ν. As can be seen from (11), the coefficients 3 1 2 k k ˜b1(k)=n b1(k)+(n − in )b2(k), (12) αν ( ) and βν( ) in (17) satisfy the condition 1 2 3 | |2 | |2 ˜b2(k)=−(n + in )b1(k)+n b2(k) αν (k) + βν (k) =1, (18) † so that the time evolution is described by the unitary (and also the same combinations d˜ (−k) of the ν Bogolyubov transformation [8] as it must be to pre- † − i i dν ( k) operators), where n = k /k are projections serve the anticommutation relations of the creation of the unit vector n directed along the momentum and annihilation operators in the course of the time k. The transformation (12) is the unitary SU(2) evolution. † transformation, and the operators ˜bν(k),˜bν (k) (as Therefore, it is useful to rewrite the fermion field † decomposition (1) in terms of the new operators well as d˜ν (k), d˜ν (k)) satisfy canonical commutation ´ ´† relations. Their time evolution is given by (10) and it bν (k), dν(−k) and corresponding bispinors u´ν (k), reads v´ν (−k) so that ˜ ˜ † − ´ bf,ν(k)=αν(k)bi,ν (k)+βν (k)di,ν ( k), (13) uν(k)bν (k)=´uν (k)bν (k), (19) ˜† − − ˜† − v (−k)d† (−k)=´v (−k)d´† (−k) df,ν( k)= βν (k)bi,ν(k)+αν (k)di,ν ( k). ν ν ν ν The closed transformations of the time evolution con- (sums over ν) with new bispinors † † ˜ − ˜ − −1 ∗ nect the pairs bν(k), dν( k) and bν (k), dν( k) for u´µ(k)=wνµ uν (k)=wµν uν (k), (20) ν every , but they mix the initial polarizations, as fol- − −1 − ∗ − lows from (10), (12), and (13). v´µ( k)=wνµ vν( k)=wµν vν( k), PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 FERMION PRODUCTION AND CORRELATIONS 1337 where w∗ is the matrix with complex-conjugated el- As one can see from (26), the evolution parameter ements [that is, it differsfrom(16)bythesignofthe r(k) is equal to zero at k =0; it is maximal at azimuthal angle φ in (16)]. In explicit form, 2 2 mi mf k2 = k2 = m m + ; (27) m i f (m − m )2 1 (E + m)ξν i f u´ν(k)=√ , (21) N ±kξ and it falls slowly at large k, ν |mi − mf | ∓ r(k) ≈ . (28) 1 kξν 2k v´ν (−k)=√ ,ν=1, 2, N (E + m)ξν At the point of the maximum, the parameter r de- pends only on the ratio mf /mi and it is not small if with spinors the mass ratio is small (or large), say, mi mf .In this case, −iφ cos(θ/2) − sin(θ/2)e ξ1 = ,ξ2 = tan rmax = |β/α|max ≈ 1 − 2 mf /mi (29) sin(θ/2)eiφ cos(θ/2) 2 ≈ (22) for k mimf . If one takes the fermion masses m and m to be dependent on direction of the momentum k.Itcanbe i f the constituent quark mass (∼350 MeV) and the seen that the spinors ξ (and thus the bispinors u´ (k), ν ν current quark mass (∼5 MeV), then tan r ≈ 0.76 −k max v´ν ( ))havedefinite helicities, at k ≈ 42 MeV. 1 1 In reality, the change in the fermion mass has finite σiniξ = ± ξ ,ν=1, 2 (23) 2 ν 2 ν time duration τ and the stepwise approximation is i valid only if the momentum is much less than the (σ are the Pauli matrices), which are conserved in the inverse time duration, kτ 1.Theeffect of finite time course of time evolution in view of (17). duration will be discussed in the next section. Therefore, the Bogolyubov transformation (17) connects creation and annihilation operators that 3. SMOOTH TRANSITION have opposite directions of momentum k and equal Let us consider the smooth variation of the fermion helicities (opposite spin projections). Let us note mass m(t). In this case, the coefficients α(k) and that, for a stepwise transition, the Bogolyubov co- β(k) of the Bogolyubov transformation can be ex- efficients (11) are real-valued. The coefficient βν(k) pressed through solutions of Dirac Eq. (5) [9]. To changes its sign if we change the sign of the momen- solve this equation, it is helpful to use its squared tum (k →−k), or change ν(1 ↔ 2), or interchange form, representing the Dirac ψ function in (5) in the ↔ the initial and final states (f i). form d In general, the coefficients αν(k) and βν (k) of the ψ(k,t)= iγ0 − γnkn + m(t) χ(k,t). (30) SU(2) transformation (17) can be represented in the dt form Then Eq. (5) takes the form iϕα iϕβ α(k)=cosr(k)e ,β(k)=sinr(k)e , (24) d2 dm − iγ0 + k2 + m2(t) χ(k,t)=0, (31) where dt2 dt − r(k)=tan 1 |β/α| (25) which splits into two complex-conjugated equations for two upper and two lower components of χ.The is the main evolution parameter and the phases ϕα bispinor χ canbewrittenintheform and ϕβ do not play an important role and will not be sν 0 considered here. For a stepwise transition, the phases χν(k,t)= ϕ(k,t)+ ϕ(k,t), (32) are absent. 0 sν Using (11), we get ν =1, 2, 1 k2 + m m where sν are unit two-component spinors [see (6)]. α2(k)= + i f , (26) As a result, we have the second-order equation for 2 2EiEf scalar function ϕ(k,t), 2 2 1 − k + mimf d2 dm β (k)= . − i + k2 + m2(t) ϕ(k,t)=0, (33) 2 2EiEf dt2 dt PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1338 ANDREEV which is nothing but an oscillator equation with Bogolyubov coefficient. The choice of another unit complex-valued variable frequency (energy). spinor in (40) gives a similar result. The momentum Let us first consider the stepwise variation of the k in (40) may have arbitrary direction. If we take the mass to establish the correspondence of the above k direction along the third (spin-quantization) axis, approach with results of Section 2. In the region of then bispinors u1 in (40) have definite helicity h =1/2 constant mass, Eq. (33) is satisfied by arbitrary su- that is conserved in the course of transition. Note that perposition of exponential functions exp(±iEt).Let the sign of the coefficient βν is not determined by the us have in the initial state the function ϕ(k,t) in the above procedure because the momentum k in βν(k) 2 − 2 form of a single wave: [see (11)] appears as the square root of Ef mf − in (39). Considering below the smooth variation of the ϕ(t)=e iEit,t<0. (34) mass, we will use the squared Dirac equation to get For t>0, the general solution of (33) has the form the explicit expressions for Bogolyubov coefficients α and β . So we will keep in mind the sign assign- + −iEf t − iEf t ν ν ϕ(t)=c e + c e ,t>0. (35) ment arising in (11) for stepwise transition. Choosing one of the upper spinors sν (say, s1) in (32) Considering the smooth variation of the mass and substituting (32), (34), and (35) into (31), we get m(t), we use the reference model with the corresponding (not normalized) expressions for m + m m − m the Dirac field ψ(k,t) at t<0 and at t>0. Then the m(t)= f i + f i tanh(2t/τ), (41) continuity condition (8) reads 2 2 + which has an exact solution and contains the impor- U1(k,mi,Ei)=c U1(k,mf ,Ef ) (36) 1 tant parameter τ giving the characteristic time of the − − + c1 U1(k,mf , Ef ), mass variation. where bispinors U1 are not normalized [see (6)], and Substituting (41) into (33), we get the equation − we find the coefficients c+ and c : d2 1 1 + k2 + e + f tanh(2t/τ) (42) E + E − m + m dt2 c+ = f i f i , (37) 1 2E f + g tanh2(2t/τ) ϕ(t)=0 − Ef − Ei + mf − mi c1 = . 2Ef with Note that bispinors on the right hand side of (36) are (m + m )2 i(m − m ) e = f i − f i , (43) orthogonal. Using normalized bispinors uν instead of 4 τ thosein(36),weget m2 − m2 f = f i , u1(k,mi,Ei)=α1(k)u1(k,mf ,Ef ) (38) 2 − 2 + β1(k)u1(k,mf , Ef ) (m − m ) i(m − m ) g = f i + f i . with 4 τ N(E ,m ) Up to numerical coefficients e, f,andg, (42) co- α (k)= f f c+, (39) 1 N(E ,m ) 1 incides with the equation for charged particles in i i 2 the electric field Eel =1/cosh (2t/τ). The solution N(−Ef ,mf ) − β1(k)= c1 , of (42) can be expressed through hypergeometric N(Ei,mi) function F (see [9]): − where N(E,m) is given by (7). The coefficients α1(k) ϕ(z)=(−z) iµ(1 − z)iλF (α, β, γ|z), (44) and β (k) coincide with Bogolyubov coefficients (11). 1 4t/τ Therefore, the solution of the classical Dirac equation z = −e , can describe the evolution of the normalized fermion with parameters of the hypergeometric function given field [9], by −iEit ψ(k,t)=u1(k,mi,Ei)e ,t<0, (40) α = −iµ + iν + iλ, (45) −iEf t ψ(k,t)=α1(k)u1(k,mf ,Ef )e β = −iµ − iν + iλ, γ =1− 2iµ. iEf t + β1(k)u1(k,mf , −Ef )e ,t>0, The parameters µ, ν, and λ are expressed through showing the appearance of the negative-energy initial parameters of the problem as follows: wave (creation of antiparticles) with the correct µ = ±Eiτ/4,ν= ±Ef τ/4, (46) PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 FERMION PRODUCTION AND CORRELATIONS 1339 { ↔ } λ =(mf − mi)/4. + α β . We choose the upper signs of µ and ν in (46). Then, in the initial state (t →−∞, z → 0), we get from (44) Then, one gets the final state (t →∞, z−1 → 0) of the the incoming wave of the form (34): form (35): − ϕ(t)=e iEit,t→−∞. (47) − − To find the asymptotic behavior of ϕ(t) at t →∞,one ϕ(t)=c+(τ)e iEf t + c (τ)eiEf t, (49) can use the relation [10] t →∞, F (α, β, γ|z) (48) Γ(γ)Γ(β − α) − = F (α, α +1− γ,α +1− β|z 1) with Γ(β)Γ(γ − α) ± Γ(∓iE τ/2)Γ(1 − iE τ/2) c (τ)= f i . (50) Γ(iτ(−Ei ∓ Ef + mf − mi)/4) Γ (1 + iτ(−Ei ∓ Ef − mf + mi)/4) The coefficients α and β of the Bogolyubov transformation (17) are given by (39), where we can use the normalization factors (7) because these factors are related to asymptotic states and so they do not depend on the sharpness of the transition. Using Γ-function properties π π Γ(z)Γ(1 − z)= , |Γ(ix)|2 = , (51) sin(πz) sinh(πx) we get from (39) and (50) after some transformations 2 |α(k,τ)| sinh (πτ(±Ef + Ei − mf + mi)/4) sinh (πτ(Ef ± Ei ± mf ∓ mi)/4) 2 = , (52) |β(k,τ)| sinh (πτEf /2) sinh (πτEi/2) where upper signs correspond to α and lower signs maximum of β(k,τ) and r(k,τ) is reduced for finite τ, correspond to β. The Bogolyubov coefficients in (52) 2 1 satisfy the crucial unitarity condition (18). In the limit |βmax(k,τ)| ≈ − h(πmiτ/2)mf /mi, (56) 2 of stepwise transition, they correspond to (11), as one and shifted to smaller momentum, can check by straightforward calculation. As can be seen from (52), the coefficient β(k, τ) → km ≈ mimf /h(πmiτ/2), (57) 0 at k → 0 and it falls for large k: due to the h factor. At large k τ −1, the evolution sinh (πτ|mi − m |/4) | |∼ − |β(k,τ)|≈r(k,τ) ≈ f , (53) effect is exponentially small, β exp( πkτ/2) for sinh (πτk/2) smooth transition. k mi,mf . In the region m k m ,wherethecoefficient 4. CREATION OF FERMIONS AND THEIR f i CORRELATIONS β(k) and the evolution parameter r(k) are maximal for stepwise transition, we get Using Bogolyubov transformation (17), one can find the final-state fermion momentum distributions 2 1 1 mf k and the final-state fermion correlations. We confine |β(k,τ)| ≈ − + h(πmiτ/2) (54) 2 2 k mi ourselves to the symmetric case where the fermions with opposite momenta are produced in an equivalent with way (say, central collisions of identical nuclei). For 1 simplicity, the Bogolyubov coefficients will be taken h(x)= (1 + x coth x),h(x) ≥ 1, (55) | | 2 to be real-valued and k = k dependent. We also consider the simple model—fast simultaneous tran- where mi mf and kτ 1 were taken in ac- sition of a large homogeneous system at rest. The cordance with typical parameters: mi ∼ 350 MeV, movement of the system can be taken into account mf ∼ 5 MeV, and k ∼ 40 MeV in Section 2 and by shifting each moving element to its rest frame and with typical time duration τ ∼ 1 fm. As a result, the integrating over proper times and spacetime rapidities PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1340 ANDREEV of the elements of the system as was done for photon for the final state. Therefore, in zero-momentum production [6] in heavy-ion collisions. states that are initially completely occupied (both The final-state momentum distribution of the with fermions and antifermions), the occupation fermions (single-particle inclusive cross sections) is number decreases, contrary to initially empty states, where the occupation number increases. For thermal given by (below, we use notation bν , dν for operators having definite helicity ν = ±1/2) equilibrium, n(k) is the Fermi distribution, depending on temperature and chemical potential, and both 1 dσν † effects are present. Nf,ν(k)= = b (k)bf,ν(k) , (58) σ d3k f,ν The transition effect is better seen in two-particle where brackets mean averaging over initial state. Us- correlations. Two-particle inclusive cross sections are ing (17), we get given by 2 ++ 2 † 1 d σµν Nf,ν(k)=|α(k)| b (k)bi,ν(k) (59) i,ν 3 3 (65) σ d k1d k2 2 † 2 V † † −|β(k)| d (−k)di,ν (−k) + |β(k)| k k k k i,ν (2π)3 = bf,ν( 2)bf,µ( 1)bf,µ( 1)bf,ν( 2) † † and a similar expression for antifermions (with inter- = bf,ν(k2)bf,ν(k2) bf,µ(k1)bf,µ(k1) ↔ † † change b d). The last term on the right-hand side − δ b (k )b (k ) b (k )b (k ) of (59) reflects the result of the vacuum rearrange- µν f,ν 2 f,µ 1 f,µ 1 f,ν 2 ment (the initial vacuum of fermions having mass mi for two fermions with helicities µ, ν (correlation of is not the ground state of bf , df operators). The factor identical fermions) and V/(2π)3 replaces δ3(0) in the real case of large but 2 +− 1 d σµν finite volume V (see [2]). 3 3 (66) σ d k1d k2 It is convenient to introduce the level population † † = b (k )d (k )d (k )b (k ) function nν(k): f,ν 2 f,µ 1 f,µ 1 f,ν 2 † † V = bf,ν(k2)bf,ν(k2) df,µ(k1)df,µ(k1) Nν(k)= 3 nν(k). (60) (2π) † † + δµν bf,ν(k2)df,µ(k1) df,µ(k1)bf,ν(k2) Then, (59) and the corresponding equation for an- tiparticles take the simple form (independently for for an fermion–antifermion pair. The last term on every helicity) the right-hand side of (66) is essential only in the presence of the time evolution of the fermion field. 2 2 nf,ν(k)=|α(k)| ni,ν(k)+|β(k)| (1 − n¯i,ν(−k)) , Here, for simplicity, we do not take into account the (61) dynamical correlations arising due to Coulomb and strong interactions of the fermions. 2 2 n¯f,ν(k)=|α(k)| n¯i,ν(k)+|β(k)| (1 − ni,ν(−k)) , Considering correlations in finite volume V ,we (62) can use the modified creation and annihilation op- erators [2, 7] that are nonzero only in the volume V where the notation n¯ is used for antifermions. In gen- and satisfy modified anticommutation relations of the eral, one has to consider simultaneously the fermions form and antifermions with momenta ±k to get the full † V − picture of the particle creation. For example, if there bµ(k1),bν (k2) = δµν 3 G(k1 k2), (67) are no particles in the initial state, then we have + (2π) |β(k)|2 particles of each kind in the final state. If there G(0) = 1, is one particle having momentum k and helicity ν in where G(k) is the normalized form factor of the the initial state, then we have fermion source (normalized Fourier transform of the nf,ν(k)=1, n¯f,ν(−k)=0, (63) source density). The corresponding correlator of the modified creation and annihilation operators has the 2 and |β| particles in the rest of the states. If there form is a fermion–antifermion (singlet) pair having zero † bµ(k1),bν (k2) (68) momentum in the initial state, that is, ni,ν(k)=1, V n¯i,ν(−k)=1,thenweget ≈ − δµν 3 n((k1 + k2)/2)G(k1 k2). 2 (2π) nf,ν(k)=¯nf,ν(−k)=|α(k)| , (64) The right-hand side of (67) and (68) represents the − | |2 nf,ν( k)=¯nf,ν(k)= β(k) smeared δ function having a width of the order of the PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 FERMION PRODUCTION AND CORRELATIONS 1341 2 inverse size of the source, which is much less than the ×|α(k)β(k)(1 − ni,ν(k) − n¯i,ν(k))G(k1 + k2)| , width of the momentum distribution n(k). where smooth functions α(k) and β(k) can be evalu- Using Bogolyubov transformation (17), substitut- ≈± ing (68) for arising fermion correlators of initial-state ated at any of momenta k1, k2 k and we suggest ↔− operators, and taking into account the small width of that the process is k k symmetric. Taking the the form factor G(k), one can express the final-state sum over helicities ν,wefinally obtain the relative correlators through Bogolyubov coefficients α, β and (divided by the product of single-particle distribu- the form factor G: tions) correlation functions that are measured in ex- † † periment: bf,ν(k2)bf,ν(k1) bf,ν(k1)bf,ν(k2) (69) 1 2 ++ 2 V 2 C (k1, k2)=1− |G(k1 − k2)| , (71) = |α(k)|2n (k)+|β(k)|2(1 − n¯ (k)) 2 (2π)6 i,ν i,ν 2 ×|G(k − k )| , − 1 1 2 C+ (k , k )=1+ R2(k)|G(k + k )|2, (72) 1 2 2 1 2 2 † † V b (k )d (k ) d (k )b (k ) = (70) with f,ν 2 f,ν 1 f,ν 1 f,ν 2 (2π)6 | − − |2 2 α(k)β(k)(1 ni(k) n¯i(k)) R (k)= 2 2 2 2 , (73) [|α(k)| ni(k)+|β(k)| (1 − n¯i(k))] [|α(k)| n¯i(k)+|β(k)| (1 − ni(k))] where we suggest that the helicities ±1/2 are equally ACKNOWLEDGMENTS represented in the initial state. Equation (71) represents the identical-particle The work was supported by the Russian Founda- correlation in its simplest (no-interaction) form. The tion for Basic Research (project no. 00-02-16101). minus sign is a distinctive feature of the fermions and the factor 1/2 arises due to the sum over polarizations REFERENCES (only two of four final polarization states contain 1. L. P. Csernai, I. N. Mishustin, and A. Mocsy, in identical particles). Proceedings of the 4th Rio de Janeiro Interna- Equation (72) describes the time evolution effect tional Workshop on Relativistic Aspects of Nu- that depends on evolution parameter r (or β, α). clear Physics (World Sci., Singapore, 1995), p. 137. For the vacuum initial state, the relative correlation 2. I. V. Andreev and R. M. Weiner, Phys. Lett. B 373, function is big, being equal to α2/2β2 > 1/2.Itis 159 (1996). especially big for small β, but in this case the pair 3. M. Asakawa and T. Csorgo, Heavy Ion Phys. 4, 233 production itself is weak [see (61), (62)]. In the case (1996). of cold dense matter, the correlation function is big for 4. I. V.Andreev, Mod. Phys. Lett. A 14, 459 (1999). empty (high-momentum) initial states. In the case of 5. M. Asakawa, T. Csorgo, and M. Gyulassy, Phys. Rev. hot quark–gluon plasma in the initial state, one has to Lett. 83, 4013 (1999). take into account the presence of an effective thermal quark mass that is much bigger than the original 6. I. V. Andreev, hep-ph/0105208; Yad. Fiz. 65, 1961 (2002) [Phys. At. Nucl. 65, 1908 (2002)]. quark mass (mi ∼ gT for light quarks, where g is the QCD coupling constant and T is the temperature). 7. I. V. Andreev, Yad. Fiz. 65, 2088 (2002) [Phys. At. Nucl. 63, 1988 (2000)]. 8. N. N. Bogoliubov, Izv. Akad. Nauk SSSR, Ser. Fiz. 5. CONCLUSION 11, 77 (1947). Calculation of the evolution effect for fermions 9. A. I. Nikishov, Tr. Fiz. Inst. Akad. Nauk SSSR 111, shows that opposite-side fermion–antifermion corre- 152 (1979). lations can be large. They can serve as a sign of chiral 10. L. D. Landau and E. M. Lifshits, Quantum Mechan- phase transition in quantum chromodynamics. ics (Nauka, Moscow, 1974), p. 746. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1342–1348. From Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1382–1388. Original English Text Copyright c 2003 by Badalian. ELEMENTARY PARTICLES AND FIELDS Theory 3 * Is the a0(1450) a Candidate for the Lowest qq P 0 State? A. M. Badalian Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received December 23, 2002 3 Abstract—For a0(1450),consideredastheqq 1 P0 state, “experimental” tensor splitting, cexp = −150 ± 40 MeV, appears to be in contradiction with the conventional theory of fine structure.There is no such 3 discrepancy if a0(980) belongs to the qq 1 PJ multiplet.The hadronic shift of a0(980) is shown to be greatly dependent on the value of strong coupling in spin-dependent interaction. c 2003 MAIK “Nau- ka/Interperiodica”. 1.TWO POSSIBLE CANDIDATES to the mass of the b1(1235) meson (see the dis- 3 FOR THE GROUND P0 STATE cussion in [7]).In the other case, if a0(980) is the 3 At present, a few low-lying P -wave light mesons, 1 P0 state, FS splittings and Mcog appear to be in in both isovector and isoscalar channels, were exper- agreement with the theoretical picture, and the value imentally observed [1, 2].However, theoretical iden- of the hadronic shift is strongly correlated with the tification of these states faces serious difficulties, first strong coupling in spin-dependent interaction.For 3 3 of all, because the P0 member of the lowest PJ mul- large αs(one-loop)=0.53, the hadronic shift appears tiplet has still not been identified in an unambiguous to be equal to zero and the masses of the a0, a1, a2 way.Here, we shall discuss only a simpler isovector mesons just coincide with their experimental values. multiplet, where mixing with other channels can be For αs(two-loop)=0.43, the hadronic shift about neglected. 100 MeV is obtained. As is well known, a0(980) couples strongly to ηπ and KK channel, and therefore this state was 2.FINE-STRUCTURE SPLITTINGS suggested to be interpreted as a multiquark meson [3] 3 or KK molecule [4].Then, under such assumptions, For any n PJ multiplet, the FS parameters— 3 spin–orbit (SO) splitting a(nP ) and tensor split- the qq 1 P0 state is to be identified with the a0(1450) meson [5].There also exist many arguments in favor ting c(nP )—can be expressed through the masses 3 MJ (J =0, 1, 2) of the members of the multiplet in of considering the a0(980) as the ground qq P0 state, the standard way [8]: which, however, has a large KK component in its wave function [6]. 5 1 1 a = M2 − M1 − M0, (1) We shall discuss here both possibilities, perform- 12 4 6 3 ing a fine structure (FS) analysis of the lowest qq PJ 5 5 5 multiplet.It is evident that an identi fication of the c = M1 − M2 − M0, (2) members of this multiplet should be in accord with 6 18 9 values and sign of FS parameters.We shall show where the experimental values of M1 and M2 for the here that, in two cases depending on whether a0(980) a1 and a2 mesons are well known [1], or a (1450) is the qq 13P state, a drastic differ- 0 0 M1 = 1230 ± 40 MeV, (3) ence in value and even sign of tensor splitting takes M2 = 1318 ± 0.6 MeV, place.In particular, if a0(1450) is identified as the 3 3 qq 1 P0 state, then the large magnitude and negative while the 1 P0 state will be identified either with sign of tensor splitting appears to be in contradiction a0(1450) with the mass M0A [1], with conventional QCD theory of the FS.Also, in ± 3 case A: M0A = 1452 8 MeV, (4) this case, the center of gravity of the 1 PJ multiplet Mcog has a large value and large shift with respect or with a0(980) having the mass M0B [1], ∗ ± This article was submitted by the author in English. case B: M0B = 984.8 1.4 MeV. (5) 1063-7788/03/6607-1342$24.00 c 2003 MAIK “Nauka/Interperiodica” IS THE a0(1450) A CANDIDATE 1343 Then, from Eqs.(1) –(4) in case A, the following a light meson is calculated here by using the spinless “experimental” values of SO and tensor splittings can Salpeter equation, be extracted: (1) 2¯αs −3 aP (nP )=a = r nP , (11) aA(exp.)=−0.3 ± 11.6 MeV, (6) P 2 mn 2 4 α¯ c (exp.)=−148 ± 38 MeV; (7) (1) (1) s −3 A cP (nP )=cP = aP = 2 r nP . 3 3 mn i.e, SO splitting is compatible with zero or even a small negative value, while tensor splitting (7) is al- These expressions contain the constituent mass mn ways negative and has large magnitude: of a light quark, which was supposed to be fixed and just the same for all states with different quan- −186 ≤ c (exp.) ≤−110 MeV. (8) ∼ A tum numbers nL; mn = 300–350 MeV is usually taken [11]; the strong coupling αs is denoted as α¯s Note that large experimental error in cA(exp.) (7) since the renormalization scale in αs is still not de- comes from the error in the a1(1260) mass (3). fined and also to distinguish between αst in the static 3 In case B, if a0(980) is the ground qq P0 state, potential and αs in spin-dependent interaction.Here, the situation is complicated by strong coupling of ...nP denotes the m.e. over the wave function of the the qq channel with other hadronic channels, ηπ and nP state. KK, and for our analysis it is convenient to introduce More detailed analysis of spin-dependent po- ˜ 3 the mass M0 of the 1 P0 state in the one-channel tentials for light mesons, both perturbative and approximation.Then, nonperturbative (NP), was done in [12], where the ˜ Feynman–Schwinger–Fock (FSF) representation of M0B(exp.) = 985 MeV = M0 − ∆had, (9) the light-meson Green’s function was used.Keep- where the hadronic shift ∆had is an unknown value, ing only bilocal correlators of the fields, the spin- while the mass M˜0 can be calculated in different dependent potentials were expressed through these theoretical approaches, e.g., in the paper [9] of God- correlators (see Appendix) which define perturbative frey and Isgur, M˜0(GI) = 1090 MeV, which corre- SO and tensor splittings sponds to ∆had(GI) = 105 MeV.In the QCD string (1) 2¯αs −3 approach, our calculations give close number for the a (nL)= r nL, (12) P µ2(nL) choice of αs(FS)=0.42. (1) 2 (1) It is of interest to note that, if a (980) is the qq 13P c (nL)= a (nL), 0 0 P 3 P state, then both SO and tensor splittings are positive and the situation looks like that in charmonium.For and NP contributions are the χ (1P ) mesons a (cc) and c (cc) are both σ − c exp exp a (nL)=− r 1 , (13) positive with tensor splitting being 13% larger than NP 2µ2(nL) nL the SO one [10]: cNP(nL) is compatible with zero. aexp(cc)=34.6 ± 0.2 MeV, (10) It is important to stress that, in the FSF represen- ± cexp(cc)=39.1 0.6 MeV. tation, the expansion in inverse quark masses as in heavy quarkonia is not used, and the constituent mass µ(nL) in Eqs.(12), (13) is de fined by the 3.THEORY OF THE P -WAVE FINE average over the kinetic energy term of the string STRUCTURE Hamiltonian [13, 14].This “constituent,” or dynam- Spin-dependent interaction in light nn mesons for ical, mass µ(nL) for a light quark with current mass the L-wave multiplets with L =0 can be considered m =0appears to be as a perturbation since FS splittings are experimen- 2 1 tally small in relation to the meson masses.It is also µ(nL)= p nL = σrnL (14) assumed that the short-range one-gluon exchange 2 (OGE) gives the dominant contribution to perturba- for linear potential and can be expressed only through P tive potentials, in particular, to SO potential VSO(r) string tension σ and the universal number. P and tensor potential VT (r) [11].As a result, the ma- In contrast to the constant mass mn in Eqs.(11), P P trix elements (m.e.) aP = VSO and cP = VT in the mass µ(nL) depends on the quantum numbers n light mesons are defined as the first-order terms in and L of a given state and increases with growing n αs.Note that the wave function for a given state of and L. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1344 BADALIAN In light mesons, which have large radii linear static Thus, the NP contribution is present only in SO potential σr dominates for all states with the excep- splitting, so that total SO splitting is tion of the 1S and 1P states, where the Coulomb term 2¯α − turns out to be important [14]. a = a + a = s r 3 (22) P NP µ2(1P ) 1P The choice of the string tension σ and coupling σ −1 α in the static potential can be fixed from the slope − r 1P . st 2µ2(1P ) and the intercept of leading Regge L trajectory which is defined for spin-averaged masses Mcog(L) for the Due to the negative Thomas precession term, a can- multiplets 1L.As shown in [14], the experimental cellation of perturbative and NP terms in a(total) values of Mcog(L) put strong restrictions on σ and takes place and, in principle, a(total) could be small 2 also on strong coupling αst: σ =0.185 ± 0.005 GeV , or a negative number. αst 0.40.Here, we present our calculations for the In contrast to that, NP tensor splitting is small and linear σr potential with σ =0.18 GeV2 and also for positive, so that total tensor splitting appears to be the linear plus Coulomb potential with always positive, 2 4¯αs − αst =0.42,σ=0.18 GeV , (15) c = c = r 3 . (23) P 3µ2(1P ) 1P and take into account that the experimental Regge 2 L trajectory [14] gives Note that, in the χc mesons, the correction of order α¯s to cP was found to be also positive and small [10]. Mcog(1P ) = 1260 ± 10 MeV. (16) The linear potential with σ =0.18 GeV2 fits very well ≥ 4.REMARKS ABOUT STRONG all orbital excitations with L 2, and for this poten- COUPLING α tial, the m.e. can be taken from [14], s −1 In heavy quarkonia, strong coupling αs(µren) at µ0(1P ) = 448 MeV, r 1P =0.236 GeV, (17) the renormalization scale µren can explicitly be ex- −3 3 r 1P =0.0264 GeV , tracted from experimental values of FS splittings due to a rather simple, renorm-invariant relation between and the calculated constituent mass for the 1S state, αs(µren) and the combination η =(3/2)c − a [10, 15]. µ(1S) = 335 MeV, (18) In light mesons, one-loop perturbative corrections 2 (αs order) are yet to be calculated and OGE con- coincides with the conventional value mn ≈ tributions (12) with a fitting α¯s are assumed to be 300–350 MeV, usually used in potential models.For dominant. the 1P state, the calculated m.e. for the Cornell potential are However, at present, we know some useful fea- tures of αs: −1 µ(1P ) = 486 MeV, r 1P =0.260 MeV, (19) (i) The strong coupling freezes at large distances, 3 3 and therefore α¯ in OGE terms have to be less than, r 1P =0.0394 GeV ; s or equal to, the critical value αcr [17, 9]. −3 i.e., in this case, r 1P turns out to be 50% larger (ii) The critical value αcr was calculated in back- than for the linear potential. ground field theory [17], and the value obtained for Nonperturbative contributions to SO and tensor αcr(r) appears to be in good agreement with lattice potentials can be correctly defined in a bilocal ap- measurement of the static potential in the quenched proximation (see Appendix) when the Thomas term (0) approximation [18].For QCD constant ΛQCD = dominates in NP SO splitting, 385 ± 30 MeV, defined in lattice calculations [19], the σ − a (nP )=− r 1 , (20) calculated αcr [17] is NP 2µ2(nP ) nP αcr(one-loop)=0.59; (24) while NP tensor splitting c (1P ) for light mesons +0.05 NP αcr(two-loop)=0.43−0.04 (nf =0), (as well as for heavy mesons) is compatible with zero, where in the quenched approximation the number of 0 ≤ cNP(1P ) < 5 MeV, (21) flavors nf =0. and therefore can be neglected in our later analysis. (iii) The characteristic size of FS interaction in the This result follows from the fact, established in lattice P waves, RFS,canbedefined as QCD, that vacuum correlator D ,whichdefines c , 1 NP −3 −1/3 ∼ is small [15, 16]. RFS =[r 1P ] = 0.60 fm. (25) PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 IS THE a0(1450) A CANDIDATE 1345 3 From the study of αs(r) in coordinate space [17], it 6. a0(980) IS THE qq 1 P0 STATE was observed that, at distances r ∼ 0.6 fm, strong 3 Here, the mass M˜0( P0) will be calculated in the coupling αs(r) is already close to the critical value one-channel approximation, (24), being only about 10% less.Therefore, one can ˜ 3 − − expect that α¯s in OGE splittings (22), (23) has to be M0 = Mcog(1 PJ ) 2a c, (33) equal to 3 with Mcog(1 PJ ) = 1260 ± 10 MeV from the exper- α¯s(RFS)=0.41 ± 0.02. (26) imental Regge trajectory.We give below FS split- tings a(1P ) and c(1P ) for two values of strong cou- pling (24). (iv) The size RFS (25) coincides remarkably with 1Pcc 2Pbb For αs =0.43, when radiative corrections of order theradiusofthe and also of the state, 2 −3 which both are about 0.60 fm.For them, the strong αs were supposed to be small, r 1P = 3 coupling extracted from experiment is 0.0394 GeV , one obtains (aP = 143 MeV, aNP = −99 MeV) α (cc, 1P )=0.38 ± 0.03(exp.); (27) exp a =44MeV,c=96MeV (34) i.e., this number is very close to that in Eq. (26). To and, from the definition (33), check the sensitivity of FS splitting to the choice of M˜ (α =0.43) = 1076 ± 10 MeV, (35) α¯s, here in our calculations we shall take 0 s and the corresponding hadronic shift of the a0 meson, α¯s(two-loop)=0.43, (28) ∼ according to the definition (9), is α¯s(one-loop)=0.53 = 0.9αcr. ∆had(αs =0.43) = 91 ± 10 MeV. (36) If radiative corrections are not suppressed, then it 5. a (1450) IS THE qq 13P STATE would be more consistent to take in OGE terms the 0 0 strong coupling in the one-loop approximation [17], ∼ We start with the NP contribution to SO splitting α¯s(one-loop) = 0.9αcr(one-loop)=0.53. (37) 2 aNP,andforσ =0.18 GeV (µ(1P ) = 448 MeV), Then, FS splittings appear to be in good agreement with the experimental numbers, aNP = −106 MeV. (29) a(theory)=78MeV,c(theory) = 118 MeV, It can be shown that aNP is weakly dependent on the (38) choice of parameters of static interaction varying in ± the range 99–106 MeV. aexp(∆had =0)=77.5 10 MeV, cexp(∆had = 0) = 112 ± 34 MeV, From Eq.(6), aA(exp.) is compatible with zero, and from Eq.(22), this condition can be reached only which correspond to the hadronic shift equal to zero, and if aP = |aNP|.Therefore, we have M˜0 = 986 ± 10 MeV (39) aP = 106 MeV and α¯s(fit)=0.40. (30) coincides with the mass of a0(985).Thus, the hadro- Note that this α¯s is in agreement with expected nic shift turns out to be very sensitive to the choice value (26).Correspondingly, from (23) in theory of strong coupling and, for αs =0.53, is compatible tensor splitting cP =(2/3)aP is positive, with zero, ∼ ± cP = c =71MeV. (31) ∆had(¯αs =0.53) = 3 10 MeV. (40) This positive sign of c(1P ) obtained within a conven- It is essential that, in [9], as well as in Eq.(37), a tional theoretical approach appears to be in contra- relatively large value of one-loop coupling was used. It assumes that α2 corrections to FS splitting are diction with the “experimental” number (7): cA(exp.)= s −148 ± 38 MeV. not small, being about 20–30%.To understand which choice of α¯s is preferable, one needs to study many The second discrepancy is that spin-averaged other P -wave multiplets.In any case, one can con- mass considered in case A, clude that the predicted value of the hadronic shift is correlated with the choice of strong coupling in SO Mcog = 1303 MeV, (32) and tensor potentials. is larger than the experimental value (16) following For FS splittings (38), the masses of the a1 and from the linear Regge L trajectory for spin-averaged a2 mesons are equal to 1242 and 1325 MeV, i.e., very masses [14]. close to their experimental values. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1346 BADALIAN 7.HYPERFINE SHIFT AS A TOOL would lie above the linear Regge trajectory for spin- TO IDENTIFY THE MEMBERS averaged masses. 3 OF THE PJ MULTIPLET (iii) The hyperfine shift of b1(1235) with respect to Hyperfine (HF) splitting of the P -wave multiplet Mcog would be two times larger than the predicted which comes from spin–spin interaction, ∆HF(1P ), number in [7]. is defined as There are no such discrepancies if a0(980) is the ∆ (1P )=M (1PJ ) − M(1P1), (41) 3 3 HF cog qq 1 P0 state.In this case, the mass M˜0( P0) can where M is not sensitive to the taken value of M˜ be calculated in the one-channel approximation, and cog 0 ˜ 3 − since it enters M with small weight equal to 1/9. from the difference M0( P0) M0(a0(980)) = ∆had, cog the hadronic shift is found to be very sensitive to the In (41), M(1P ) is the mass of b (1235). 1 1 chosen value of strong coupling α¯ . 3 s Then, if the a0(1450) belongs to the lowest qq PJ If radiative corrections of order α2 are small, as multiplet, Mcog = 1303 MeV and ∼ s in charmonium, and α¯ = 0.40 is used, then ∆ ± s had ∆HF(1P ) =73 3(exp.) MeV, case A. (42) is large, ∆had = 100 ± 10 MeV.If for α¯s the value α¯ (one-loop)=0.53 is taken, the hadronic shift ap- a (980) qq 3P s Identifying 0 as the ground 0 state, one pears to be equal to zero. obtains Mcog = 1252 MeV and ± ± ∆HF(1P ) =28 6(theory) 3(exp.) MeV, case B, (43) ACKNOWLEDGMENTS i.e. substantially smaller. One can compare these I am grateful to Yu.A. Simonov for fruitful discus- numbers with theoretical predictions from [7], where sions.This work has been supported by the Russian the perturbative contribution to the HF shift was Foundation for Basic Research, project no.00-02- shown to be small and negative, while the NP term 17836, and INTAS, grant no.00-00110. is positive and not small, ± ∆HF(theory)=30 10 MeV. (44) APPENDIX In ∆ (theory) (44), the theoretical error comes from HF Spin-Dependent Potentials in Light Mesons uncertainty in our knowledge of gluonic correlation length Tg in lattice calculations: Tg =0.2 fm in In [12, 20, 21], all NP spin-dependent potentials in quenched QCD and Tg =0.3 fm in full QCD [16]. light mesons, VˆLS = L · SVLS, tensor potential VˆT = From comparison of (44) and (43), one can see Sˆ V Vˆ S · S V that theoretical number (44) appears to be in good 12 T , and HF potential HF = 1 2 HF, were ob- tained being expressed through bilocal vacuum corre- agreement with experimental number (43) if a0(980) 3 lation functions (v.c.f.) D(x) and D1(x): is the qq 1 P0 and this statement does not depend on ∞ r a value of hadronic shift ∆had. NP − 1 − 4λ On the contrary, if a0(1450) is considered to be the VLS (r)= 2 dν dλ 1 (A.1) 3 µ0r r qq 1 P0 state, then “experimental” value (42) is two 0 0 times larger than ∆HF(theory). ∞ 3 × 2 2 NP 2 2 D( λ + ν )+ 2 dνD1 ( r + ν ), 2µ0 8.CONCLUSIONS 0 Experimental data on FS splittings in light mesons ∞ appear to be a useful tool to identify the members of 2r2 ∂ V NP(r)=− dν DNP( r2 + ν2), (A.2) the 3P multiplet.Our study of the FS has shown T 2 2 1 J 3µ0 ∂r that a0(1450) cannot be a candidate for the ground 0 3 P0 state for several reasons: ∞ (i) The “experimental” value of tensor splitting NP 2 2 2 turns out to be negative (with large magnitude), VHF (r)= 2 dν D( r + ν ) (A.3) µ0 c = −148 ± 38) MeV, in contradiction with the 0 (exp.) conventional theory. 2r2 DNP( r2 + ν2) + DNP( r2 + ν2)+ 1 (ii) Also for such interpretation, spin-averaged 1 3 ∂r2 mass Mcog = 1303 MeV would be too large and PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 IS THE a0(1450) A CANDIDATE 1347 (µ0 is the average kinetic energy for a given state). and correspondingly the integral Here, v.c.f. D and D1 are defined through the gauge- T invariant bilocal vacuum correlators over fields dt J¯ = σ(2)F (A.11) strengths Fµν : q 2¯µ(t) µν µν g2 0 Fµν (x)φ(x, y)Fλσ (y)φ(y,x) (A.4) Nc is defined for the antiquark.In the bilocal approxima- 1 tion after averaging, the exponents (A.8) will contain =(δ δ − δ δ )D(x − y)+ µλ νσ µσ νλ 2 the bilocal correlators, or the cumulants.To obtain spin-dependent potentials, the important approxima- × − − ∂µ [(hλδνσ hσδνλ)+permutation]D1(x y) , tion is that the spin factors (A.8) are considered as where hµ = xµ − yµ and the standard factor a perturbation and therefore µ(t) and µ¯(t) in (A.10), x (A.11) can be changed by corresponding values µ0 and µ¯0 (µ0 =¯µ0) calculated for the unperturbed φ(x, y)=P exp Aµ(z)dzµ (A.5) Hamiltonian HR which is defined for a meson with y a spinless quark and antiquark.It is inportant that, provides the gauge invariance of the correlators (A.4). to derive spin-dependent potentials in light mesons It was shown in [22] that, in the bilocal approxi- in the FSF representation, the expansion in inverse mation, there is no perturbative contribution to v.c.f. powers of the quark mass was not used. D(x), while the correlator D1 contains both pertur- The derivation of the meson relativistic Hamilto- nian H and the definition of the constituent mass bative and NP contributions: R P NP 2 2 D1 = D1 + D1 (A.6) µ0 = p + m nL, (A.12) with are discussed in detail in [12–13]. 16 α P s The v.c.f. D and D were calculated in lattice D1 (x)= 4 . (A.7) 1 3π x NP QCD [16], where it was shown that D1 is small We give some comments on how the average energy, when compared to D(x) and even compatible with 2 or the constituent mass, µ0, appears in the denomi- zero in full QCD.Therefore, in the potentials (A.1) – nators in Eqs.(A.1) –(A.3). To derive the expressions NP (A.3), the terms containing D1 can be omitted, in (A.1)–(A.3), the meson Green’s function GM (x, y) is particular, the NP contribution to tensor splitting, studied in the FSF representation (which is gauge- c = V . invariant), where spin terms enter GM (x, y) through NP T is compatible with zero (A.13) the exponential factors The perturbative contribution to SO and tensor po- s s¯ tentials that are defined by v.c.f. (A.7) just corre- (1) − (2) sponds to OGE terms (12) in Section 3. exp g dτσµν Fµν exp g dτσ¯ µν Fµν . 0 0 Lattice measurements have also shown that v.c.f. (A.8) D can be parametrized with good accuracy as an exponential at distances x 0.2 fm, i.e., (i) Here, σµν =(γ γ − γ γ )/4, i =1, 2 refer to the µ ν ν µ x quarks (antiquarks), Fµν is the field strength, and s D(x)=d exp − . (A.14) (¯s) is the proper time of the quark (antiquark).The Tg proper time τ (¯τ) plays the role of ordering parameter Then, from (A.1) and (A.9), along the quark (antiquark) trajectory z(τ)(¯zµ(τ)). To obtain the Hamiltonian and potentials from the r/T g meson Green’s function, it is necessary to go over σ V NP(r)=− dttK (t) (A.15) t LS 2 1 from the proper time to the actual time of the quark, µ0rπ thus defining the new quanity µ(t): 0 dt dt 4σ 2T 1 r 2µ(t)= , 2¯µ(t¯)= . (A.9) g − + 2 2 K2 , dτ dτ¯ πµ0 r Tg Tg Then, in (A.8), the integrals can be rewritten as s T where the string tension, ∞ ∞ (i) dt (1) Jq = dτσ Fµν = σ Fµν (z(t)), µν 2µ(t) µν σ =2 dν dλD ( λ2 + ν2), (A.16) 0 0 (A.10) 0 0 PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1348 BADALIAN for the exponential form of D(x) is 7. A.M.Badalian and B.L.G.Bakker, Phys.Rev.D 64, 114010 (2001); hep-ph/0105156. σ = πdT 2. g (A.17) 8. J.Pantaleone, S.-H.H.Tye, and Y.J.Ng, Phys.Rev. 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N.A.T ornqvist¨ and M.Ross, Phys.Rev.Lett. 76, 23. Yu.A.Simonov, Phys.Rep. 320, 265 (1999); Pis’ma 1575 (1996); S.Narison, Nucl.Phys.B (Proc.Suppl.) Zh. Eksp.Teor.Fiz.´ 69, 471 (1999) [JETP Lett. 69, 96, 244 (2001). 505 (1999)]; hep-ph/9902233. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1349–1356. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1389–1397. Original Russian Text Copyright c 2003 by Zinovjev, Molodtsov. ELEMENTARY PARTICLES AND FIELDS Theory Diquark Condensate and Quark Interaction with an Instanton Liquid G.M.Zinovjev1) and S. V. Molodtsov Institute of Experimental and Theoretical Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received May 23, 2002 Abstract—The interaction of light quarks with an instanton liquid is considered at nonzero density of quark/baryon matter in a phase where the diquark condensate is nonzero. It is shown that the inclusion of the relevant perturbation of the instanton liquid leads to an increase in the quark chemical potential µc. This in turn induces a considerable growth of the threshold quark-matter density at which one expects the emergence of color superconductivity. c 2003 MAIK “Nauka/Interperiodica”. The possible existence of strongly interacting occur as the result of the corresponding phase tran- matter in the phase of quark–gluon plasma has sitions whose critical temperatures are virtually coin- remained one of the most fundamental and intriguing cident. This observation shows that, most likely, the predictions of QCD nearly since the time of its two phenomena in question are closely related, which advent. At a phenomenological level, this phase is may heuristically play an important role in obtaining treated as a state where quarks and gluons at finite deeper insight into the confinement mechanism. values of temperature (T ) and (or) the quark/baryon The behavior of strongly interacting matter at fi- chemical potential (µ) can propagate rather freely nite quark/baryon densities has received much less over significant (naturally, with respect to the hadron study. From the theoretical point of view, the reason scale) distances. Even the simplest qualitative esti- for this is that QCD runs into known serious difficul- mates of physical features of this phase appeared to ties in introducing fermions in the lattice model and be so plausible that this made it possible to begin especially in performing a generalization to nonzero experimental searches for quark–gluon plasma in quark/baryon densities. As to a phenomenological collisions of relativistic heavy ions. Although the first aspect, available experimental data proved to be of investigations yielded, in a sense, quite promising a low information value for developing the theory results, a new generation of experiments is required along these lines. It is obvious that, both at present for drawing compelling conclusions. This in turn and in the future, an increase in the energy of col- urgently calls for a serious quantitative analysis of liding heavy ions would result in the production of the QCD phase diagram. strongly interacting matter that occurs ever closer to the temperature axis of the µT plane. In view of It seems that lattice QCD would make it possible this circumstance, a phenomenological development to solve this problem on the basis of the first prin- of investigations along the µ axis would predomi- ciples of the theory; however, a solution within this nantly rely on astrophysical observations of compact framework has so far been obtained (at a level that is stars. In recent years, it has been revealed that the rather far from reliability) mainly in the vicinity of the effect of diquark pairing in the 3¯ color channel ow- temperature axis of the µT plane. Nevertheless, the ing to instanton-induced interaction may prove to lattice approximation revealed that, in all probability, be significantly stronger than attraction that is in- the phase structure of QCD is quite rich [1]. In partic- duced by one-gluon exchange and which was con- ular, we firmly know at present that, if the temperature sidered previously and may lead, in particular, at high falls below some critical value, the chiral symme- quark/baryon densities, to the phase of color super- try of the original Lagrangian is broken, whereupon conductivity; in general, the pattern that arises in quarks are bound by confinement forces. Moreover, the µT plane upon taking this into account is by no lattice calculations indicate that these phenomena means trivial [2]. 1)Bogolyubov Institute for Theoretical Physics, National The present article reports on a continuation of Academy of Sciences of Ukraine, Metrologicheskaya ul. 14b, our previous study [3], where, on the basis of the 03143 Kiev, Ukraine. method proposed in [4], we examined, in the phase of 1063-7788/03/6607-1349$24.00 c 2003 MAIK “Nauka/Interperiodica” 1350 ZINOVJEV, MOLODTSOV †L L †L L broken chiral symmetry, the effect of quarks on the in- V =2λ(ψ1 L1ψ1 )(ψ2 L2ψ2 ) stanton liquid at nonzero chemical potentials. These †R R †R R effects are rather weak and can manifest themselves +2λ(ψ1 R1ψ1 )(ψ2 R2ψ2 ), in the second order of the expansion in the effective T R L where ψf =(ψf ,ψf ), f =1, 2, are the quark fields coupling constant. Nevertheless, we found that, at L,R a qualitative level, allowance for quark interaction composed from spinors of specific helicity, ψf = with the instanton liquid can lead to an increase in 1 ± γ5 P±ψ with P± = ; n is the instanton-liquid the threshold quark-matter density at which one can f 2 expect the emergence of color superconductivity. It density; V4 is the 4-volume of the system; µν = is well known that, if there is no perturbation of the (0,µ);andN is a renormalization factor, which, for instanton liquid, this density appears to be paradoxi- the sake of definiteness, we set here to unity, but −3 cally low (nq ∼ 0.062 fm ), so that the diquark phase which can in principle be considered as a free pa- should be formed even in ordinary nuclei (nnucl rameter of the model. For models where the packing- 0.45 fm−3—the quark-matter density is taken to be fraction parameter nρ¯4 is fixed, this factor is im- three times greater than the nuclear-matter density) material, while, if this parameter is variable, there [5, 6]. In this study, we will obtain a quantitative esti- appears a weak logarithmic dependence on N.The mate for the critical value of the quark/baryon chem- factors of 2 in expression (1) appear upon going over to dimensionless variables. The term V representing ical potential µc. For this purpose, we will analyze the corresponding set of Gor’kov equations for a color the four-fermion quark interaction can be directly superconductor. In doing this, we rely on an “exact” expressed in terms of the helicity components; for example, we have Lagrangian for four-quark interaction generated by (anti)instantons without resorting to extensively used †L L dpf dqf †L (ψ Lf ψ )= ψ (pf ) simplifications of the original Lagrangian that are f f π8 fαf if associated with isolating the channel where attraction β j Lβ j × L f f (p ,q ; µ)ψ f f (q ). is the strongest. We believe that this is dictated by the αf if f f f f very formulation of the problem, where the instanton- For the right-handed fields, it is necessary to make liquid-perturbation effects being discussed may prove β j the substitution L → R.ThekernelsL f f are de- to be of the same order of smallness as effects induced αf if by terms that are usually omitted. fined in terms of zero modes (solutions to the Dirac We recall that, in the instanton-liquid model [7], equation with the chemical potential µ) by means of Z the generating functional is represented as the the functions hi, i =1,...,4, which are given by product of a gluon and a quark component; that is, π { − − − h4(k4,k; µ)= (k µ ik4)[(2k4 + iµ)f1 Z = Zg ·Zψ. 4k − − − The first factor provides information about the gluon + i(k µ ik4)f2 ] Z + − + } condensate, while the fermion factor ψ serves for +(k + µ + ik4)[(2k4 + iµ)f1 i(k + µ + ik4)f2 ] , describing quarks in an instanton medium [5, 7]. Hereafter, we everywhere use the notation borrowed πk − h (k ,k; µ)= i (2k − µ)(k − µ − ik )f from [3], where we introduced dimensionless variables i 4 4k2 4 1 (dictated by the form of the interquark-interaction +(2k + µ)(k + µ + ik )f + kernel)—for example, we make the substitutions µ → 4 1 → µρ/¯ 2 and kiρ/¯ 2 ki, i =1,...,4,forthechemical − − − potential and for the momenta, respectively, with ρ¯ + 2(k µ)(k µ ik4) being the mean size of pseudoparticles. With the aid 1 of an auxiliary integration with respect to the parame- − (µ + ik )[k2 +(k − µ)2] f − Z k 4 4 2 ter λ, the quark determinant ψ can be reduced to the form [here, we consider the SU(3) groupfor quarks of + 2(k + µ)(k + µ + ik ) two flavors, Nf =2] 4 nρ¯4 Z † − 1 2 2 + ψ dλ Dψ Dψ exp nV4 ln 1 + (µ + ik4)[k4 +(k + µ) ] f2 ,i=1, 2, 3. λN k 2 | | dk † Here, k = k if we consider the spatial compo- × exp ψ (k) · 2(−kˆ − iµˆ)ψ (k)+V , π4 f f nents of the 4-vector kν and f=1 ± ± ± ± ± I (z )K (z ) − I (z )K (z ) (1) f = 1 0 0 1 , 1 z± PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 DIQUARK CONDENSATE AND QUARK INTERACTION 1351 I (z±)K (z±) ρ¯ ± 1 1 ± 2 ± 2 which are of interest in the case of a diquark conden- f2 = 2 ,z = k4 +(k µ) , z± 2 sate, the use of the expression that follows from (1) for the effective action makes it possible to obtain a where Ii and Ki(i =0, 1) are modified Bessel func- set of Gor’kov equations that is similar to the set of tions. In passing, we introduce the scalar function equations obtained in [6]. The result is h(k ,k; µ) associated with the three-dimensional 4 + −1 L − R LR T − ki G0 (p) F (p) Σ (p)G ( p)=0, (2) component: hi(k4,k; µ)=h(k4,k; µ) , i =1, 2, 3 − k G+(p) 1 GLR(p) − ΣR(p)F +RT(p)=1, (we will not present the arguments of the functions 0 − −1 R L RL T hi explicitly unless this leads to confusion). We then G (p) F (p) − Σ (p)G (−p)=0, 0 have − −1 RL L +LT † G (p) G (p) − Σ (p)F (p)=1, Lβj(p, q; µ)=Sik(p; µ) klU αU σ σnS+ (q; −µ). 0 αi l β nj − G±(p) 1 = −2(p + iµ)± − + + where 0 refers to a free Here, S(p; µ)=(p + iµ) h (p; µ) and S (p; −µ)= R R T ∗ Green’s function, Σ (p)=∆ T (p) T (−p), h−(p; −µ)(p + iµ)+, with the conjugate function be- ΣL(p)=∆LT +(p) T +T (−p),and∆L,R is a gap. ∗ The form of the Σ matrices is determined by the ing given by (p; −µ)=h (p; µ), while is an anti- hµ µ structure of the kernels of the equations that arise symmetric matrix, 12 = − 21 =1. In these formulas, ± upon averaging over color orientations (it should p and similar symbols are used for 4-vectors asso- be recalled that we consider an ensemble that is ± ± ± ciated with the matrices σν ,whereσν =( iσσσ,1), σ stochastic in color). The set of Eqs. (2) can be closed ± ± being the 3-vector of the Pauli matrices, p = pνσ , by supplementing it with the gap equation ν while U is the matrix of rotations in color space. For R 2λ dq + R +T − the right-handed components, we have the analogous ∆ = 4 [T (q)F (q)T ( q) Nc(Nc − 1) π relations − T +(−q)F RT (q)T +T (q)], †R R dpf dqf †R (ψ Rf ψ )= ψ (pf ) f f π8 fαf if L 2λ dq L T βf jf Rβf jf ∆ = [T (q)F (q)T (−q) × R (p ,q ; µ)ψ (q ) − 4 αf if f f f f Nc(Nc 1) π − − LT T with the kernel T ( q)F (q)T (q)]. βj ik kl α †σ σn + − It can easily be seen that the right-hand sides of Rαi (p, q; µ)=T (p; µ) Ul U Tnj(q; µ), β these equations involve the difference of a matrix and where T (p; µ)=(p + iµ)+h−(p; µ) and T +(p; −µ)= its transposed counterpart; therefore, the right-hand ∗ sides are automatically proportional to . + − − h (p; µ)(p + iµ) . Since the vector function h(p) A similar set of equations is valid for the conjugate is associated only with the vector p,thecomponents matrices; that is, of the matrices (p + iµ)± and h∓(p; µ) can be inter- +LT − −1 RLT +R changed. As a result, it can easily be shown that the F (p) G (p) − G (−p)Σ (p)=0, (3) 0 following identities hold: RL − −1 − R +R G (p) G0 (p) F (p)Σ (p)=1, + + T (p; µ)=S (p; −µ),T(p; −µ)=S(p; µ). −1 F +RT (p) G+(p) − GLRT (−p)Σ+L(p)=0, 0 Further, we will omit the dependence of the matrices −1 GLR(p) G+(p) − F L(p)Σ+L(p)=1. T and T + on µ since the chemical potential appears 0 in the matrix T only with a plus sign and in the matrix The relevant closing gapequation has the form T + only with a minus sign. In addition, we present +R 2λ dq T − +RT two useful identities ∆ = 4 [T ( q)F (q)T (q) Nc(Nc − 1) π σ T T (p)σ = T +(p),σT +T (p)σ = T (p). 2 2 2 2 − T T (q)F +R(q)T (−q)], where T T is the transposed matrix. 2λ For the expectation values ∆+L = Nc(Nc − 1) L,R L,R 4 L,R ψ1αi (p)ψ2βj (q) = 12 αβπ δ(p + q)Fij (p), dq × [T +T (−q)F +LT (q)T +(q) π4 L †R 4 LR ψ (p)ψ (q) = δ δ π δ(p − q)G (p), +T +L + fαi gβj fg αβ ij − T (q)F (q)T (−q)], PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1352 ZINOVJEV, MOLODTSOV where Σ+R(p)=∆+RT +T (−p) T +(p), Σ+L(p)= or ∆+LT T (−p) T (p),and∆+L,+R are the correspond- − LR + [1 H(p)]G (p)=G0 (p), ing gaps. We note that, in this study, we will re- + R − − +L strict ourselves to the case of a diquark condensate; where H(p)=G0 (p)C (p)G0 ( p)C (p). however, it is well known that, in the vicinity of the The form of the matrices H(p) is such that the sum ∼ phase transition at µc 300 MeV,there exists a small − intermediate region where the mixed phase of non- H(p)+H( p)=α(p) vanishing chiral and diquark condensates occurs [6], and product realizing the regime of a transition to the state of color H(p)H(−p)=β(p) superconductivity. In order to describe this phase, we must extend the set of equations by supplementing it are proportional to identity matrices [by definition, with the additional expectation values we see that α(−p)=α(p); similarly, the relation L,R †L,R 4 − LL,RR β(−p)=β(p) also holds]. Denoting the functions ψfαi (p)ψgβj (q) = δfgδαβ π δ(p q)Gij (p). hν (p) of sign-reversed argument by gν = hν (−p),we From the sets of Eqs. (2) and (3), we obtain can write the functions α(p) and β(p) in the compact form GLR(p)=G+(p)+G+(p)ΣR(p)F +RT(p), 0 0 α(p)=4∆R∆+L F +RT(p)=GLR T (−p)Σ+L(p)G+(p). 0 × [−4A(p)(hg)+2(p2 + µ2)(h2)(g2)], +L Let us introduce auxiliary matrices C (p)= where ∆+LT +(−p)T (p) and CR(p)=∆RT (p)T +(−p). A(p)=(p2 + µ2)(hg) − 2iµp(g h − h g). With the aid of them, the Σ matrices can be recast 4 4 into the form In these expressions, we have used a natural definition +L +L R R 4 Σ (p)= C (p), Σ (p)=C (p) . of the scalar product in the form (hg)= i=1 higi and similar definitions for the squares of these func- By using identities that relate the T matrices, we can tions, (h2) and (g2). In the last term appearing in the show that the following equalities hold: expression for A(p), we have employed the aforemen- C+LT(−p) T = C+L(p), tioned notation for the scalar function associated with the three-dimensional component h.Forthefunction CRT(−p) T = CR(p). β, we can in turn obtain R 2 +L 2 2 2 2 2 2 2 2 For the Green’s function, we have β(p)=16 ∆ ∆ p + µ (h ) (g ) . LR + + R G (p)=G0 (p)+G0 (p)C (p) Since the functions α and β are associated with × GLR T (−p) C+L(p)G+(p). identity matrices, the solution for the Green’s func- 0 tion can be represented in the form C By using the properties of matrices and considering G+(p)[1 − H(−p)] that the free Green’s functions satisfy the identities GLR(p)= 0 . 1 − α(p)+β(p) ±T ∓ σ2G (p)σ2 = G (p), 0 0 In this case, the gapequation can be rewritten as we obtain a closed equation for determining the func- 2λ dp α(p) − 2β(p) tion GLR: ∆L = . N (N − 1) π4 ∆+L(1 − α(p)+β(p)) LR − T − − − − c c G ( p) = G0 ( p)+G0 ( p) (4) × C+L(p)GLR(p)CR(p)G−(−p). 0 We are interested in a solution of the form ∆R = +L L +R The fact that the vector function h(p) is associated ∆ =∆ =∆ =∆ for λ<0. This solution is with the vector p leads to a useful property—all of dictated by the symmetry properties of the kernel of ± the matrices G , C+L,andCR commute with one the four-quark interaction. We recall that, in the ker- 0 nel,thefactorforeachsortofquarksistraditionally another. Seeking a solution for the Green’s function LR written with the factor of i [7, 8]. This explains the G in the form of an iterative series, one can see that choice of the sign of λ. It should be noted, however, it also commutes with these matrices. We can now R easily derive a final equation for the Green’s function that, in principle, there exists an alternative—∆ = +L − L − +R in the form ∆ = ∆ = ∆ for λ>0—if we redеfine the LR + 2 LR kernels. An analysis shows that the denominator on G (p)=[1+H(p)]G0 (p)+H (p)G (p) the right-hand side of Eq. (4) is always positive; PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 DIQUARK CONDENSATE AND QUARK INTERACTION 1353 λ 0.006 0.004 Fig. 1. Diagram of the tadpole approximation (see main body of the text): (solid lines) fermion fields and (dashed line) scalar field. 0.002 0 400 800 1200 hence, it is guaranteed that this equation has a so- µ lution at sufficiently large λ. , åeV The state of quark matter at a specificvalueof Fig. 2. Saddle-point value λ (to the one-loop approxima- the chemical potential is determined by the saddle tion, it is proportional to the free energy of the system) point of the functional in (1). We consider this func- as a function of the chemical potential µ at Nc =3and tional, retaining the first nonvanishing contribution Nf =2. The solid curves were obtained for the phase and restricting ourselves to the right (figure-of-eight) where chiral symmetry is broken (the lower curve was calculated with allowance for perturbations of the instan- diagram (see, for example, [5]), whose contribution I ton liquid). The dashed curves correspond to the phase of has the form color superconductivity (left curve) for the case where the dp α(p) − 2β(p) tadpole contribution is disregarded and (right curve) for I =2(N − 1) . c 4 − the case where the interaction of the instanton liquid is π 1 α(p)+β(p) included. In our case, this contribution to the generating func- ∼ W tional (Zψ e ) reduces to few hundred MeV, M 2 = ν/κ,whereκ is the kinetic W = −nρ¯4 ln λ + λ Y , Y I. coefficient that is calculated within the semiclassi- cal approach and ν =(b − 4)/2 with b =(11Nc − In the simplest case of a constant instanton-liquid 2Nf )/3, Nc and Nf being the numbers of quark colors density, the saddle-point equation can be written in and flavors, respectively. For this kinetic coefficient, the form our estimations yield κ ∼ cβ [9], where β =8π2/g2 is 4 nρ¯ − λ Y =0. the instanton action and c ∼ 1.5–6, the specificvalue Comparing this equation with the gap equation in (4), of the factor c being dependent on the ansatz adopted one can notice an interesting property: the saddle- for the saturating configuration. point equation leads to the condition according to Eventually, we find that, in addition to the four- which the gapas a function of µ is constant. This leg diagrams [see the term V in (1)], there appear property provides a useful test for numerical calcula- diagrams featuring a scalar field connected through a tions. vertex where differentiation with respect to ρ occurs It was shown in [4, 9] that the reaction of quarks and where it is necessary to take into account the to the instanton liquid can be described within per- variation of the functions describing zero modes, turbation theory in terms of small variations of the ∂h instanton-liquid parameters, δn and δρ, in the vicinity h → h + i δρ, i =1,...,4. (5) of their equilibrium values n and ρ¯. The variation i i ∂ρ of the parameters is included in the theory if use is made of deformable field configurations (crumpled Owing to the presence of a condensate (in our case, instantons)—in our case, instantons of dimension ρ this is a diquark condensate), we can significantly as a function of x and z: ρ → ρ(x, z)—and if the simplify the Lagrangian under consideration if we variation of zero modes in the quark determinant is retain only the main contributions, which stem from taken into account in the interaction vertices upon the tadpole-type diagrams. The leading contribution the substitution ρ¯ → ρ¯ + δρ. It appears that, for long- actually comes from the term V and from the term wave perturbations, such as π mesons, the deforma- associated with two four-leg diagrams connected by tion field describes scalar colorless excitations of the the scalar field whose propagator is 1/M 2,oneof instanton liquid with mass gap M on the order of a these diagrams being closed into a figure-of-eight PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1354 ZINOVJEV, MOLODTSOV –3 –3 nq, fm nq, fm 3 12 2 8 1 4 0 200 400 0 200 400 600 800 µ, åeV µ, åeV Fig. 3. Quark density determined from the intersection of Fig. 4. Quark density determined from the intersection of the upper curve in Fig. 2 (for the chiral condensate) with the lower curve in Fig. 2 (for the chiral condensate, the the curve representing the diquark condensate (without diquark condensate being taken without allowing for the including perturbations of the instanton liquid) in the tadpole contribution) and the curve representing the di- same figure. The solid and the dashed curve show the quark condensate (without allowing for the perturbations result for the stable and metastable phases, respectively. of the instanton liquid) in the same figure. The solid and the dashed curve depict the results for, respectively, the stable and the metastable phase. (see Fig. 1); because of a condensate, it is considered with fully paired lines. external line, the scalar-field propagator, Analyzing the modified Lagrangian, we can eas- ily see that the form of the set of Gor’kov equa- 2λ 1 1 J = 4 2 I. tions remains unchanged except that, instead of Σ, Nc(Nc − 1) nρ¯ κ 4M we must everywhere use the corrected expression Σ+δΣ, where the correction δΣ is constructed from It should be emphasized that, with allowance for the the modified functions (5). Further, one can see that explicit form of M, the dependence on the kinetic term the result for the Green’s function also remains un- κ is effectively canceled, so that its exact value is changed. By Σ in final expressions, one should imply immaterial in the approximation being considered. By differentiating the figure-of-eight diagram, we obtain the corrected functions. In particular, the form of the gapequation does not change since, in the kernel, ∂I dp 1 − β(p) =2(N − 1) δα(p) there arises once again the familiar combination of a ∂ρ c π4 (1 − α(p)+β(p))2 matrix and its transposed counterpart, this combina- α(p) − 2 tion being constructed from the matrix functions T + δβ(p) . and T + differentiated with respect to ρ (and consid- (1 − α(p)+β(p))2 ered, for example, in terms of finite differences) and We must also take into account the modification of being associated with an identity matrix. Finally, it the generating functional. It should be recalled that turns out that, in the gapequation, we must make the this functional is calculated to the first nonvanishing substitutions contribution, → α(p) α(p)+δα(p), nρ¯4 W = −nρ¯4 ln − 1 + λ Y . β(p) → β(p)+δβ(p), λ where With allowance for a variation of the instanton-liquid density, the saddle point is now determined from the ∂α δα(p)=J , equation ∂ρ nρ¯4 nρ¯4 − λ(nρ¯4) ln − λ Y =0. ∂β |λ| δβ(p)=J , ∂ρ As to the instanton-liquid density, it is given by [4] with J describing the contribution of a diagram be- ν nρ¯4 = (6) longing to the figure-of-eight type and having, for the 2βξ2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 DIQUARK CONDENSATE AND QUARK INTERACTION 1355 1/2 n /Λ4 2 δI IL ν δρ Γ(ν +1/2) + + √ , 2βξ2 βξ2 2 νΓ(ν) 2.0 where the prime denotes differentiation with respect 27 N 1.5 to λ, the constant ξ2 = c π2 characterizes 2 − 4 Nc 1 interaction in the stochastic ensemble of pseudopar- ticles, and the mean size of pseudoparticles is defined 1.0 2N as ρ¯Λ=exp − c . 2ν − 1 The derivative of the figure-of-eight diagram with 0.5 respect to λ is given by the expression 0 400 800 1200 µ, åeV δI ∆ =4(N − 1) δρ c ∆ Fig. 5. Instanton-liquid density at Nc =3and Nf =2 for two regimes. The solid curve and two dashed curves dp 1+α − 6β + αβ + β2 represent the results of the calculations with, respec- × δα tively, the chiral and the diquark condensate. The lower π4 (1 − α + β)3 dashed curve was obtained by estimating the instanton- −4+7α +4β − αβ − α2 liquid density without self-consistently taking into ac- + δβ . count quark interaction with the instanton liquid (one (1 − α + β)3 finds the diquark condensate and then determines the instanton-liquid density on the basis of it), while the Finally, the derivative ∆ is determined from the rela- upper curve is discussed in the main body of the text. tion N (N − 1) 2∆ dp 2β − 2β2 +2αβ − α2 c c = . of the “exact” Lagrangian would be premature. We 2λ2 ∆3 π4 (1 − α + β)2 recall that the value of µc 300 MeV is obtained if use is made of an approximate Lagrangian where the The calculations were performed to the one-loop channel characterized by the strongest attraction is approximation, where the saddle-point parameter λ is proportional to the free energy of the system. In selected. The point is that the instanton-liquid pa- Fig. 2, this parameter is represented by the dashed rameters obtained within different approaches differ curves (the left curve was obtained without taking somewhat from one another, but optimal agreement into account quark interaction with the instanton with phenomenological results known from analy- liquid). In the same figure, the solid curves represent ses of the QCD vacuum can be reached by adjust- similar results, but for the phase where chiral sym- ing the model parameters—for example, by changing metry is broken (see [3]). It should be noted that the the scale Λ. In this sense, the shift of the phase- transition point is probably within the accuracy that saddle-point parameter λ1 from [3] and that which is used in the present study are related by the equation the instanton-liquid model can ensure. λnρ¯4 2 − The situation becomes more certain if we take λ1 = . The intersection of the curves 2(Nc − 1)Nc into account the disturbance of the instanton liquid determines the point of the transition to the supercon- by quarks. In this case, the phase-transition point ducting phase. It was mentioned above that, in fact, is shifted noticeably toward greater values of µc (the there exists a narrow intermediate zone of a mixed point at which the lower solid curve intersects the phase where a nonvanishing chiral and a nonvanish- left-hand dashed curve). The quark-matter density ing diquark condensate coexist, so that the transition for the former version (“exact” Lagrangian) and for is not very sharp. But refinements following from this the version discussed immediately above versus the are not very important for our consideration. First of chemical potential is given in Figs. 3 and 4, whence it all, we note that the phase-transition point obtained can be seen that the threshold densities at which the without allowance for the tadpole contribution (the emergence of the phase involving color superconduc- point at which the upper solid curve intersects the tivity is expected may significantly exceed the den- left-hand dashed curve) lies farther than the value of sity of normal nuclear matter. In all probability, these µc 300 MeV, which was found previously in [5, 6]. results suggest that, in principle, additional weak However, the conclusion that the phase-transition is (against the background of the instanton interaction) shifted toward greater values of µ owingtotheuse interactions of light quarks can significantly change PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1356 ZINOVJEV, MOLODTSOV the estimate of the threshold nuclear-matter density alternating, which, as can be seen from Fig. 5, leads at which diquark condensation occurs. to a sharpchange in the character of the chemical- But if we also take into account the interaction potential dependence of the instanton-liquid density, of quarks in the phase involving a nonvanishing di- causing a significant reduction of the gluon conden- quark condensate with the instanton liquid (right- sate. Yet, this effect was previously indicated in [10]. hand dashed curve), the intersection point is shifted to unjustifiably large values of µ—that is, to the region where the simple instanton-liquid model can hardly ACKNOWLEDGMENTS be used to describe the QCD vacuum. The restric- This work was supported in part by CERN– tion from the side of large values of the chemical INTAS (grant no. 2000-349) and NATO (grant potential is due to the fact that, at some value of µ, no. 2000-PST.CLG 977482). We thank the Faberge´ interquark distances become so small that Coulomb Foundation for providing excellent conditions for (perturbative) fields become commensurate with in- work. S.V. Molodtsov is grateful to Prof. M. Namiki stanton fields, with the result that a superposition of and the HUUJUKAI Fund for financial support and (anti)instantons ceases to be an appropriate configu- to Prof. P. Giubellino for interesting discussions and ration that saturates the relevant path integral. From support. our results, we can therefore draw only the conserva- tive conclusion that the curves describing the chiral and diquark condensates move apart upon taking REFERENCES into account the perturbation of the instanton liquid 1. F. Karsch, hep-ph/0103314. by quarks. The chiral curve moves downward, while 2. K. Rajagopal, Nucl. Phys. A 651, 150c (1999); M. Al- thediquarkcurvedriftstowardlargervaluesofµ, ford, hep-ph/0110150. simultaneously displacing the phase-transition point. 3. S. V. Molodtsov and G. M. Zinovjev, Nucl. Phys. B In order to obtain more accurate numerical data, it (Proc. Suppl.) 106, 471 (2002). is necessary to improve the instanton-liquid model in 4. G. M. Zinovjev, S. V. Molodtsov, and A. M. Snigirev, such a way that the contribution of perturbative fields Yad. Fiz. 65, 961 (2002) [Phys. At. Nucl. 65, 929 could be taken into account more reliably. (2002)]; hep-ph/0004212. Figure 5 shows the instanton-liquid density as a 5.G.W.CarterandD.I.Diakonov,Phys.Rev.D60, function of µ with allowance for quark interaction 016004 (1999). with the instanton liquid in the phase where chiral 6. R. Rapp, T. Schafer,¨ E. V.Shuryak, and M. Velkovsky, Ann. Phys. (N.Y.) 280, 35 (2000). symmetry is broken (solid curve) and in the phase of 7. D. I. Diakonov and V. Yu. Petrov, Nucl. Phys. B color superconductivity (dashed curves). The lower 245, 259 (1984); in Hadronic Matter under Ex- curve gives an estimate obtained for the instanton- treme Conditions, Ed. by V.Shelest and G. Zinovjev liquid density without self-consistently taking into (Nauk. Dumka, Kiev, 1986), p. 192; D. I. Diakonov, account quark interaction with the instanton liquid V.Yu.Petrov,andP.V.Pobylitsa,Phys.Lett.B226, (one first finds the diquark condensate and then de- 471 (1989); C. G. Callan, R. Dashen, and D. J. Gross, termines the instanton-liquid density on the basis of Phys.Rev.D17, 2717 (1978); E.-M. 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In the first case, 2000), p. 307; G. M. Zinov’ev, S. V. Molodtsov, and the corresponding analog of the tadpole contribution A. M. Snigirev, Yad. Fiz. 63, 975 (2000) [Phys. At. δI/(δρ) in (6) is strictly positive; therefore, the inclu- Nucl. 63, 903 (2000)]. sion of instanton-liquid interaction with quarks in this 10. N. O. Agasyan, D. Ebert, and E.-M. Ilgenfritz, Nucl. phase may only lead to an increase in the instanton- Phys. A 637, 135 (1998); N. O. Agasyan, B. O. Ker- liquid density (gluon condensate). However, this in- bikov, and V. I. Shevchenko, Phys. Rep. 320, 131 crease appears to be insignificant, merely reflecting (1999); D. Ebert, V. V. Khudyakov, K. G. Klimenko, the sensitivity of the model to variations in the dy- and V. Ch. Zhukovsky, hep-ph/0106110; D. Ebert, namical quark mass [3]. In the phase of a nonvan- V. V. K h u d y a k o v, H . To k i , et al., hep-ph/0108185. ishing diquark condensate, the tadpole contribution is Translated by A. Isaakyan PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1357–1374. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1398–1415. Original Russian Text Copyright c 2003 by Danilov. ELEMENTARY PARTICLES AND FIELDS Theory Application of the Unitarity Conditions to CalculatingInteraction Amplitudes for Ramond States in SuperstringTheory G. S. Danilov Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina, 188350 Russia Received July 9, 2002 Abstract—The unitarity conditions for superstring amplitudes of boson interaction are used to calculate the Green’s function for Ramond states—that is, for 10-spinor states and Ramond bosons.It is shown that, from the unitarity conditions, it follows, among other things, that local quantities specifying the sought amplitudes satisfy some integral relations.The amplitude for the transition of two massless Neveu – Schwarz bosons into a system of two massless Ramond states is obtained in an arbitrary order in the number of loops.For this amplitude, the aforementioned integral relations are veri fied in the tree approximation. c 2003 MAIK “Nauka/Interperiodica”. 1.INTRODUCTION the Schottky groups.Such a parametrization is espe- cially convenient for investigating the unitarity con- Within Ramond–Neveu–Schwarz superstring ditions since it arises in a natural way in matching [8] theory [1], it is rather difficult to calculate the interac- tree amplitudes. tion amplitudes for Ramond states (ten-dimensional spinors and Ramond bosons) [1–3].In order to Although only the contributions from the Neveu– calculate such amplitudes for an arbitrary number of Schwarz sector were known explicitly within this loops, it is proposed in this study to use the unitarity supercovariant approach for a long time [7–9], all conditions for the amplitudes of the interaction of contributions to boson-emission amplitudes for an Neveu–Schwarz boson states.These amplitudes arbitrary number of loops, including the contribution were obtained in [4–6] for a closed oriented string to from the Ramond sector, have been calculated by now any order of the coupling constant.In this study, we [4–6].The required amplitudes are given in the form derive the amplitude for the transition of two massless of finite-dimensional integrals of explicit functions bosons from the gravitational multiplet to the system with respect to the parameters of the super Schottky groups [7, 8, 16, 17] and the coordinates of interac- of two massless Ramond states.Here, local quantities | that determine the amplitude in question for an tion vertices in the complex (1 1) supermanifold [18]. arbitrary number of loops (n − 1) are expressed in The relevant integrand is the sum over the superspin terms of local quantities that determine the n-loop structures defined in terms of the super Schottky amplitude for the scattering of massless bosons in the groups.For all spin structures, including the Ramond region where one of the handles degenerates.In the sector, the super Schottky groups were obtained in [4, future, we hope to generalize the proposed method for 5, 17].A supergroup transformation depends on three calculating more general amplitudes. Riemann and two Grassmann (odd) complex-valued parameters [18, 19].At nonzero odd parameters, this The problem of calculating multiloop superstring transformation mixes boson and fermions, whence it amplitudes was discussed by a number of authors [7– follows that a supermanifold of genus n>1 is not 14].In the scheme proposed in [10], supersymmetry is split in accordance with [20].Therefore, superspin lost, since multiloop superstring amplitudes depend structures differ from ordinary spin structures [21], on the choice of basis for gravitino zero modes [11, where boson fields are single-valued on Riemann 12].Yet, Hoker and Phong [15] recently proposed surfaces and where fermion fields can acquire, upon a procedure for eliminating this dependence.Any- rounds about noncontractible cycles, only a sign fac- way, the approaches considered in [10, 13] require an tor.Functions that characterize a transition to the intricate modular parametrization, and this compli- split description of supermanifolds are singular [22], cates their application to specific calculations.There and a superstring is not invariant under this trans- is no such drawback in the supercovariant calculation formation.It is for this reason that supersymmetry is [4–9], which employs a parametrization in terms of lost in the approach [10] that implies the split descrip- supergroups that are superconformal extensions of tion of supermanifolds.A more detailed comparison 1063-7788/03/6607-1357$24.00 c 2003 MAIK “Nauka/Interperiodica” 1358 DANILOV of the supercovariant approach with the approaches this study, we obtain such relations for the aforemen- proposed in [10, 15] is planned to be given in a forth- tioned four-leg diagram, which involves two massless coming publication. bosons and two massless Ramond particles.So far, The contribution to the amplitude of each super- we have verified these relations only for this four- spin structure is given by the product of an integra- leg diagram in the tree approximation.The relations tion measure and the vacuum expectation value of that arise for loop amplitudes require a dedicated in- the product of vertices.The integration measures for vestigation.It should be noted that only some gen- even superspin structures1) were calculated explicitly eral properties of integration measures and of vac- by solving equations that were obtained in [4] on uum correlation functions upon the degeneration of the basis of the condition requiring that multiloop handles are of importance in deriving the unitarity amplitudes not depend on the choice of gauge of conditions for boson amplitudes—the details of the the zweibein and gravitino fields.For odd superspin expressions obtained in [4–6] are immaterial. structures, the partition functions were calculated by This article is organized as follows.In Section 2, factorizing appropriate even structures [6].The vac- we present an original expression for the amplitudes uum expectation values of the product of vertices are describing boson emission and some formulas that expressed in terms of the vacuum correlation func- will be used below.In Section 3, we derive the two- tions for matter superfields; in turn, these correla- particle unitarity condition for the amplitude being tion functions are expressed in terms of holomorphic considered.In Section 4, we discuss the unitarity Green’s functions and quantities related to them (pe- condition for a boson loop.In Section 5, we discuss riod matrix and holomorphic superscalar functions the amplitude describing the transition of two bosons having periods).For a handle of the Ramond type, to two Ramond particles in an arbitrary order in the a nonidentity transformation corresponds to a round coupling constant.In Appendix А, we present, for the about an A cycle.At nonzero Grassmann moduli, this sake of convenience, explicit expressions for integra- transformation is not split, and neither is the trans- tion measures and for vacuum correlation functions. formation corresponding to a round about a B cycle. Some details of the calculations are given in Appen- In this case, holomorphic Green’s functions cannot dix B. be represented in the form of a series of the Poincare´ type.A method for calculating such Green’s functions 2.EXPRESSIONS FOR AMPLITUDES and quantities related to them was proposed in [23, 24].All of the functions being discussed, including As was indicated in the Introduction, the super- those in the Ramond sector, were calculated in [4–6]. string n-loop amplitude can be expressed in terms of It will be shown in Section 4 below that bo- an integral of the sum over superspin structures de- son amplitudes satisfy the unitarity conditions upon fined on a complex (1|1) supermanifold [18] in terms cutting boson loops.The unitarity condition upon of super Schottky groups [4, 5, 17] of genus n.This cutting fermion loops can be used to calculate the supergroup is specified by the set of transformations interaction amplitudes for states of the Ramond type. Γa,s(l1s) and Γb,s(l2s) corresponding to rounds about For this purpose, the discontinuity in energy (more As and Bs cycles on a Riemann surface of genus precisely, in the corresponding bilinear 10-invariant) n,wheres =1, 2,...,n.In general, each transfor- in the Neveu–Schwarz boson-interaction amplitude mation depends on three even and two odd (Grass- must be represented in the form of a trace associ- mann) parameters.Each pair including Γa,s(l1s) and ated with a fermion loop.Since Ramond states are Γb,s(l2s) is a superconformal extension of the Schot- described by Majorana–Weyl spinors [1], there are tky transformations Γ(0)(l ) and Γ(0)(l ), which are in the interaction amplitudes for Ramond states, no a,s 1s b,s 2s 10-spinor structures featuring an even number of given by (below, z is a local coordinate on the super- manifold and ϑ is its superpartner) Dirac 10-matrices sandwiched between spinor states. Therefore, the number of 10-spinor structures that − 2l2s (0) → asz + bs →−( 1) ϑ can be involved in the amplitude is in general less Γb,s (l2s)= z ,ϑ , csz + ds csz + ds than the number of conditions that can be locally (1) satisfied by representing the aforementioned discon- (0) { → → − 2l1s } tinuity in the energy variable as the trace associated Γa,s(l1s)= z z, ϑ ( 1) ϑ , with a fermion loop.Thus, the unitarity conditions require fulfillment of specific integral relations for local where l1s and l2s are the characteristics of the theta quantities determining the amplitude in question.In function that are attributed to a particular handle s. They can only take values of 0 and 1/2—more specif- 1)A superspin structure is even (odd) if the spin structure ically, l1s =0 for a boson loop and l1s =1/2 for a corresponding to it is even (odd). fermion loop.In the last case, l2s =0and l2s =1/2 PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1359 define a loop of, respectively, even and odd spin struc- zs being related to z by transformation (4).Under − { → a} { → ture.Further, we have asds bscs =1with the transformations Γa,s = t ts and Γb,s = t − − b} ˆ us ksvs ksus vs ts , the supersymmetric p tensors Fp(t) change as as = √ ,ds = √ , (2) ks(us − vs) ks(us − vs) ˆ a ˆ(s) p ˆ b ˆ p Fp(ts )=Fp (t)Q (t), Fp(ts)=Fp(t)Q (t), − Γa,s Γb,s 1 ks (6) cs = − , ks(us vs) (s) where the function Fˆp (t) is obtained from Fˆp(t) by where ks is the Schottky dilatation parameter and us − means of a round about the circle C( ) or C(+).By and vs are fixed points of the transformation.With- s s Q (t) and Q (t), we denote those factors that the out loss of generality, we can assume that |ks| < 1 Γa,s Γb,s and | arg ks|≤π.Expressions (2) can be supersym- spinorial derivative D(t) acquires upon applying the metrized in different ways, but in this case, the 1/2- transformations Γa,s(l1s) and Γb,s(l2s), respectively. form space will not in general have a basis [25].There For an arbitrary superconformal transformation Γ= will then arise difficulties in constructing vacuum {t → tΓ}, the factor QΓ(t) is defined as follows: correlation functions for scalar superfields, with the D(tΓ)=QΓ(t)D(t),D(t)=θ∂z + ∂θ. (7) result that such a supersymmetrization will be inap- | propriate for describing superstrings.In order to con- Thus, (3 2) complex parameters (ks,us,vs) and struct [4, 26] the required extension of formulas (1), (µs,νs) correspond to each handle.The n-loop super- it is necessary to consider that, at n =1, superstring- string amplitude is given by an integral with respect interaction amplitudes are obtained by means of sum- to these parameters and their complex conjugate mation over ordinary spin structures.In calculating counterparts and with respect to the coordinates | the pole contribution to the amplitude, in which case tj =(zj ϑj) of interaction vertices on the relevant a given handle is separated from the others, the odd supermanifold.Any (3|2) complex variables {N0} are parameters of this handle therefore cease to be mod- then fixed owing to SL(2) symmetry, and this leads 2 uli.For each number s, all odd modular parameters in to emergence of an additional factor |H({N0})| in Γa,s(l1s) and Γb,s(l2s) must then reduce to zero by the the integrand.This factor was calculated in [4].In the same transformation Γ˜s.Therefore, we have following, we fix two local variables u and v, together with their Grassmann partners, which are chosen to Γ (l )=Γ˜−1Γ(0)(l )Γ˜ , a,s 1s s a,s 1s s (3) be zero, and the third local variable z4.In this case, ˜−1 (0) ˜ we have Γb,s(l2s)=Γs Γb,s (l2s)Γs, 2 2 |H({N0})| = |(z4 − u)(z4 − v)| . (8) (0) (0) where Γa,s(l1s) and Γb,s (l2s) are transformations of Intheamplitudefortheinteractionofm bosons, each term of the sum over superspin structures is the prod- the type in (1), while Γ˜s is the (z → zs,ϑ→ ϑs) (n) { } transformation, which depends on the odd parameters uct of the integration measure ZL,L ( q,q ) and the µs and νs: vacuum expectation value ˜ { (n) (j) Γs = z = zs + ϑsεs(zs), (4) Fm ({tj, tj}, {pj ,' }, {q,q}; L, L ) of the product of all interaction vertices (from j =1to j = m)with ϑ = ϑs(1 + 1/2εs(zs)ε (zs)) + εs(zs)}, s 2) the coordinates tj =(zj|ϑj) on the supermanifold. where We denote by {q} the set of parameters of the super µs(z − vs) − νs(z − us) (j) Schottky groups and by pj and ' the momentum of εs = ∂zεs(z) and εs(z)= . us − vs the jth boson and its polarization tensor, respectively. It follows from (1) and (4) that, under the transfor- By L and L , we denote the superspin structures mations in (3), the points (us|µs) and (vs|νs) on the of, respectively, holomorphic and antiholomorphic complex (1|1) supermanifold remain unchanged.At fields.According to (1) and (3), superspin structures l =1/2, the transformation Γ (l =1/2) differs are specified, in just the same way as ordinary spin 1s a,s 1s structures [21], by the characteristics l and l of the from the identity transformation, but Γ2 =1.This 1s 2s a,s theta function that are attributed to a given handle s. means that superfields develop a square-root cut on (n) { (j)} the complex plane of z.Each transformation in (3) Thus, the n-loop amplitude Am ( pj,' ) for the − interaction of m bosons has the form converts the circle C(+) into the circle C( ),where s s g2n+2 − (n) { (j)} ( ) { | | } Am ( pj,' )= (9) Cs = z : cszs + ds =1 , (5) 2nn! (+) { |− | } Cs = z : cszs + as =1 , 2)Hereafter, an overbar denotes complex conjugation. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1360 DANILOV (n) × | { } |2 { } and, accordingly, κj =2ηjζ D(tj).In these formu- H( N0 ) ZL,L ( q,q¯ ) (j) L,L las, ζ(j) and ζ(j) describe the polarization of the jth (n) (j) (j) × F ({tj, tj}, {pj ,' }, {q,q¯}; L, L )(dqdqdtd¯ t¯) , m boson, so that ζ(j)M ζ(j)N = 'MN.Allofthemomenta where (dqdqdtd¯ t¯) stands for the product of the dif- are considered to be entering the diagram. ferentials of all coordinates of interaction vertices and A vacuum correlation function is expressed in modular parameters, with the exception of those (3|2) terms of the holomorphic Green’s function that are fixed owing to SL(2) symmetry.Further, g (n) { } RL (t, t ; q ) and the superscalar functions is the coupling constant and the rest of the notation (n) was given above.Since we will discuss the four-leg Js (t; {q}; L),wheres =1,...,n;thatis, diagram featuring massless bosons, only even super- (n) ˆ { } { } spin structures (including those that involve an even 4XL,L (t, t; t , t ; q,q¯ )=RL (t, t ; q ) (12) number of fermion loops of odd spin structures) are (n) (n) + R (t, t ; {q})+I (t, t; t , t ; {q,q¯}), present in expression (9).The factor of 1/2n takes L LL into account symmetry under the permutation of the (n) { } (n) { } fixed points Us =(us|µs) and Vs =(vs|νs) for a given ILL (t, t; t , t ; q,q¯ )=[Js (t; q ; L) (13) handle, while the factor of 1/n! reflects symmetry (n) { } (n) { } −1 under the permutation of handles.For any boson vari- + Js (t; q ; L )][ΩL,L ( q,q )]sr able x,wedefine dxdx = d(Rex)d(Imx)/(4π).For × [J(n)(t ; {q}; L)+J(n)(t; {q}; L)], any Grassmann variable η,wedefine dηη =1.The r r string tension will be taken to be 1/π.Our normal- (n) where the matrix Ω ({q,q}) is expressed in terms ization conditions correspond to the normalization L,L (n) { } conditions in [1].The vertex describing the emission of the period matrix ω ( q ,L) as of a massless boson (in our normalization) character- (n) { } (n) { } − (n) { } ΩL,L ( q,q )=2πi[ω ( q ,L) ω ( q ,L)]. ized by a 10-momentum p = {pM } and polarization { } { } (14) 10-vectors ζ = ζM and ζ = ζM for, respectively, left- and right-handed fields (M =0,...,9)hasthe Because of boson–fermion mixing, the period matrix form [2] ω(n)({q},L) depends [4, 8] on the superspin structure V (t, t; p; ζ) (10) L.The Green’s function is normalized by the condi- tion =4[ζD(t)X(t, t)][ζD(t)X(t, t)] exp[ipX(t, t)], R(n)(t, t; {q})=ln(z − z − ϑϑ)+R˜(n)(t, t; {q}), where, for the two 10-vectors a and b,thequan- L L M (15) tity ab = aM b denotes their scalar product, the mostly plus metric being predominantly used here. where R˜(n)(t, t; {q}) is not singular at z = z.As N L By X (t, t¯), we imply the scalar superfield of a string. (n) usual, the function R (t, t ; {q}) at coinciding ar- Further, pζ = pζ =0 and p2 =0; the spinorial L ˜(n) { } derivative D(t) was defined in (7).In this case, guments t = t is taken to be RL (t, t; q ).The cor- (n) relation function in (12) at coinciding arguments is the quantity F ({t , t }, {p ,'(j)}, {q,q¯}; L, L ) ≡ m j j j calculated in the same way.When the coordinates (n) (j) Fm ({tj, tj}, {pj ,' }) appearing in (9) is expressed → b → a change as t ts and t ts under the transforma- [1] in terms of the vacuum correlation function tions of the super Schottky group that correspond XˆL,L (tj, tj; tl, tl; {q}) ≡ Xˆ(j, l) for scalar superfields to the rounds about, respectively, the Bs and the in the form of an integral with respect to auxiliary As cycle, the holomorphic Green’s functions being { } Grassmann parameters ηj, ηj associated with each discussed satisfy the relations boson; that is, (n) b { } (n) { } (n) { } RL (ts,t; q )=RL (t, t ; q )+Js (t ; q ; L), (n) { } { (j)} (16) Fm ( tj, tj , pj ,' )= (dηdη¯) (11) (n) a { } (n)(s) { } RL (ts ,t; q )=RL (t, t ; q ), 1 × exp − (ˆκ + ip )(ˆκ + ip )Xˆ(j, l) , (n)(s) 2 j j l l where R (t, t ; {q}) is the function obtained from j,l L R(n)(t, t; {q}) by means of a round about the Schot- where the operator κˆ is defined as the sum κˆ = L j j tky circle (5) corresponding to the As cycle [it should κj + κj of the two operators; here, κj =2ηjζ(j)D(tj) be noted that, according to formulas (1) and (3), the PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1361 Green’s function for a Ramond handle has a square- form of an integral where known functions of genus 1 root branch point].Further, we have appear as integrands (Appendix A). (n) b { } (n) { } { } The integration measures in (9) are calculated Jr (ts; q ; L)=Jr (t; q ; L)+2πiωsr( q ,L), (17) from the equations derived in [4, 9] from the require- ment that the relevant amplitude be independent of (n) a { } (n)(s) { } Jr (ts; q ; L)=Jr (t; q ; L)+2πiδrs, the choice of the zweibein and the field of the two- dimensional gravitino.Owing to the separation of (n)(s) (n) where Jr is obtained by means of a round of Jr holomorphic and antiholomorphic fields [29], the in- about the As cycle [from (3), it follows that, because of tegration measure in (9) has the form [4] (n) boson–fermion mixing, Jr at nonzero Grassmann (n) 5n (n) −5 Z ({q,q})=(8π) [det Ω ({q,q})] (19) moduli has a branch point associated with a round L,L L,L (n) (n) about the Schottky circle for a Ramond handle].The × Z ({q})Z ({q}), problem of constructing the required functions will L L (n) be solved if we construct functions that satisfy the where the matrix Ω ({q,q}) is given by (14) and conditions in (16) and (17).In the Neveu –Schwarz L,L (n) { } sector and for a bosonic string [7–9, 27, 28], these ZL ( q ) is a holomorphic function of the variables functions can be specified in the form of Poincare-´ {q} that can be written as type series.In particular, the relevant expressions Z(n)({q})=Z˜(n)({q},L) for a bosonic string are given in [4] (Appendix B). L (20) In the Neveu–Schwarz sector, the expressions for n (1) × Z (ks; l1s,l2s) a superstring can be obtained from the expressions − . (3 2l1s)/2 − − for a bosonic string by substituting the superinterval s=1 ks (us vs µsνs) 8(t1,t2)=(z1 − z2 − ϑ1ϑ2) for the interval (z1 − z2). Here, l1s,l2s takes the values of 0 or 1/2, as was (n) ≡ (n) { } (n) In this case [RL (t, t ) RL (t, t ; q )], we have indicated above; the function Z˜ ({q},L) is given in (1) − Appendix А; and, for Z (k; l1),wehave (n) (ϑ ϑΓ)D(t )ϑΓ D(t )R (t, t )= , (18) L 8(t, t ) (1) − 2l1+2l2 2l1 Γ Γ Z (k; l1,l2)=( 1) 16 (21) ∞ − (r) − 2l2 p (2l1 1)/2 8 8(t, U ) × [1 + ( 1) k k ] J(n)(t; {q}; L)= ln Γ , . r (r) [1 − kp]8 p=1 Γ 8(t, VΓ ) 2πiωsr({q},L)=δrs ln ks The domain of integration in (9) will be discussed in a future publication.We only note now that the 8(U (s),U(r))8(V (s),V(r)) + ln Γ Γ , integral with respect to the vertex coordinates in (9) (s) (r) (s) (r) is calculated over the fundamental domain [4, 5] of Γ 8(U ,VΓ )8(V ,UΓ ) the super Schottky group—for this domain, one can (s) (s) where, as before, U =(us|µs) and V =(vs|νs) choose the domain that is external with respect to the and tΓ =(zΓ,ϑΓ) is the result of applying the trans- circles specified in (5).The domain of integration with formation Γ to t =(z|ϑ).In the first formula in (18), respect to the modular parameters is predominantly summation is performed over all group products Γ of determined by modular symmetry.Here, the dilata- | |→ the transformations Γb,s(l2s) of the super Schottky tion parameters kj 1 can be eliminated from the group [see (3)].In the second formula, the sum does region being considered.There then appear unitarity not include those Γ that contain powers of the trans- discontinuities of the amplitudes [1] if some or all dilatation parameters tend to zero, k → 0. formation Γb,r(l2r) on the right.In the third formula, j those Γ that involve powers of the transformation Γb,s(l2s) on the left are additionally excluded from the 3.TWO-PARTICLE UNITARITY CONDITION sum.At s = r, the sum does not involve the term corresponding to the identity transformation either. Two-particle discontinuities arise upon integra- In the Ramond sector, the Green’s functions can- tion over the region where the only one dilatation not be written in the form of Poincare-type´ series at parameter tends to zero, k → 0.In this case, it is nonzero Grassmann moduli, since, according to (3), convenient to fix in (9) a local coordinate of one of the the transformation corresponding to a round about interaction vertices—for example, z4—and both lim- the circles in (5) is not split in this case.In any case, iting points U =(u|µ) and V =(v|ν) of that trans- they can nevertheless be specified in the form of series formation in (3) for which k → 0.We set µ = ν =0. 2 where each term of the series can be represented in the The factor |H({N0})| in (9) is then given by (8). PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1362 DANILOV For the sake of definiteness, we will calculate the tree and the (n − 1)-loop amplitude.It is determined discontinuities in the invariant quantity s = −(p1 + byaconfiguration where the limiting points of the re- 2 2 p2) = −(p3 + p4) .They emerge upon integration maining basis transformations of the Schottky group over the region where z1 and z2 tend to u or to v, are not close to u.We discuss only massless states. remaining beyond the Schottky circles.The region In the limit k → 0, the singularity of the integra- where z2 tends to v makes the same contribution to tion measures in (19) emerges owing to the factor → theintegralastheregionwherez2 u.Therefore, (3−2l1 )/2 → 1/k in (20).The leading singularity of the we assume that z2 u and then double the result 3/2 obtained in this way.In addition, we must multiply 1/k form for a handle of the Neveu–Schwarz type the result by n since each of the n handles can play (in this case, l1 =0; see Section 2) cancels in the sum the role of the degenerate handle being considered. of two terms corresponding to l2 =0and l2 =1√/2. If z2 → u,thenz1 → u or z1 → v.The second case Indeed, expression (9) is an even function of k. reduces to the first by the corresponding Schottky Therefore, the leading singularity appears to be of the 2 2 transformation of the coordinate z1,whereuponz1 1/|k| type, the coefficient of 1/|k| being logarithmi- appears to be within a circle containing the point u. cally dependent on |k| through the period matrix (see Therefore, we assume that z1 → u and z2 → u,but Appendix А).Terms of higher order in powers of k do z1 can be either beyond or within the Schottky circle, not contribute to the sought discontinuity, which is while z2 lies beyond this Schottky circle.According caused by massless particles.To the required accu- to (2) and (5), we then have √ racy, we represent the contribution W of the region |z − u|≥ k|u − v|. (22) under discussion to the integral in (9) as an integral 2 of the sum of expressions factorized in the initial and For the sake of simplicity, we will calculate that part of final states with respect to the momentum p˜1 flowing the discontinuity which is given by the product of the along the loop in question; that is, 10 2d p˜1 W = Aˆ ({p, '} , p˜ ; λ, λ )Aˆ − (λ, λ ;˜p , {p, '} ), (23) p˜2p˜2 (0) (i) 1 (n 1) 1 (f) 1 2 λ,λ where p˜1 +˜p2 = P = p1 + p2 = −(p3 + p4) and the each term Wˆ in the integrand in (9) is written to the summation is taken over the polarizations λ, λ of required accuracy in the form of a Gaussian integral intermediate states of holomorphic (λ) and antiholo- with respect to the momentum flowing in the loop: morphic (λ) fields.In each term, the first factor de- ˆ 10 ˜ ˜ 2 ˜ pends on the momenta and polarizations of parti- W = d p˜1 exp[G0 + G1p˜1 + B1p˜1] (24) cles in the initial [{p, '} =(p ,p ,'(1),'(2))]and (i) 1 2 × exp[G p˜2 + G p˜ p˜ + G p˜2 + B p˜ the intermediate state, while the second factor de- 1 1 12 1 2 2 2 1 1 { } { } { } { } pends on the parameters of the intermediate and fi- + B2p˜2 + G0]OL˜ ( p , ζ , p˜1, t , q ) (3) (4) nal [{p, '} =(p3,p4,' ,' )] states.These fac- × { } { } { } { } (f) OL˜ ( p , ζ , p˜1, t , q ) tors were calculated at p˜2 =˜p2 =0.The sought uni- − 1 2 × Zˆ(n 1)({q,q¯})|(z − u)(z − v)|2. tary discontinuity, which emerges from the vanish- L,˜ L˜ 4 4 2 2 ing of the denominator (˜p1p˜2) can easily be calcu- Below we demonstrate that, upon performing the lated.From a comparison with the unitarity condi- above integrations in (9), expression (24) reduces tion, it follows that, under the normalization con- to the sought form (23).In (24), all 10-vectors ˆ { } dition adopted in this study, A(0)( p, ' (i), p˜1; λ, λ ) are assumed to be Euclidean.The quantities in ˆ the integrand depend on the type of the degenerate and A(n−1)(λ, λ ;˜p1, {p, '}(f)) are, respectively, the ˜ ˜ tree and the (n − 1)-loop amplitude.In this case, loop being considered.The coe fficients G0, G1,and ˜ the corresponding unitarity condition is satisfied for B1 depend on the initial state, as well as on y = a boson loop and the sought interaction amplitudes ln |k| and on (u, u¯).The coe fficients in the exponent for Ramond states are calculated from the unitarity of the second exponential function depend on the condition for a fermion loop.In order to derive for- final state and on modular parameters (with the mula (23) itself, the vacuum expectation value in (9) exception of the dilatation parameters k =0).They is represented in the form of the integral (11) with also depend on the spin structures (L,˜ L˜) formed respect to (dηdη¯), whereupon, in the limit k → 0, by all handles, apart from the degenerate one.The PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1363 2 expression |(z4 − u)(z4 − v)| in (24) is the factor Concurrently, the tree amplitude arises in the form of 2 | ∞| |H({N0})| in (9) as specified by formula (8).The an integral where the coordinates (0 0) and ( 0) of sets of the momenta, polarizations, and coordinates the particles in the intermediate state (with momenta of the vertices describing boson emission in the p˜1 and p˜2, respectively) and the local coordinate z2 are initial and final states are denoted by {p}, {ζ}, fixed.In the integral for the (n − 1)-loop amplitude, { } { ¯} { } { } { } { } the coordinates (v|0) and (u|0) of the particles having ζ ,and t, t .The factors OL˜ ( p , ζ , p˜1, t , q ) the momenta p˜1 and p˜2, respectively, and the local and O ({p}, {ζ }, p˜ , {t}, {q}) emerged from the L˜ 1 coordinate z are fixed. expansion of holomorphic and antiholomorphic func- 4 tions in (19) and (11) in the small parameters k, 4.UNITARITY FOR A BOSON LOOP (z1 − u),and(z2 − u).Owing to the nonholomorphic factor in (19) and the last term in (12), the factor First, we discuss the unitarity condition for a { } { } { } { } {¯} k → 0 OL˜ ( p , ζ , p˜1, t , q ) at n>1 depends on t , boson loop.In the case of ,wethenhave { } (n) { } ≡ q¯ ,andL as well, while the factor J1 (t; q ; L) J1(t).In addition, the elements { } { } { } { } { } { } (n) { } ≡ (n) { } ≡ OL˜ ( p , ζ , p˜1, t , q ) depends on t , q¯ ,and ω ( q ,L)11 ω11 and ω ( q ,L)1l ω1l of the L.These factors can depend polynomially on p˜1. period matrix can be expressed in terms of the holo- − Each of the factors is the sum of expressions that morphic Green’s function R(n 1)(t, t; {q}) ≡ R(t, t) are factorized with respect to the initial and the L˜ (n−1) (n−1) { } ˜ ≡ final state.The factor Zˆ ({q,q¯}) is caused by and the scalar functions J1 (t; q ; L) Jl(t) on L,˜ L˜ the supermanifold of genus (n − 1) formed by the the partition function.The explicit form of all of the remaining handles as quantities being discussed will be obtained in the sections that follow.In deriving expression (24), we 2πiω11 =lnk + R(U, U)+R(V,V ) (26) use the following formulas for the determinant of the − R(U, V ) − R(V,U), n-dimensional matrix Ω and for the quadratic form 2πiω = J (U) − J (V ), −1 1l l l JΩ J of n quantities Js (where s =1,...,n): J1(t)=R(t, U) − R(t, V ), −1 −1 2 JΩ J =[J1(1 − Ω1j Ω˜ )Jl ] (25) 1 j1l1 1 where U =(u|0) and V =(v|0).Formula (26) di- × − ˜ −1 −1 ˜ −1 → [Ω11 Ω1jΩjl Ωl1] + JjΩjl Jl, rectly follows from (18) for k 0.We examine a loop − that is bosonic both for holomorphic and for anti- { } − ˜ 1 det Ω = det Ωjl [Ω11 Ω1j1 Ω Ωl 1]. j1l1 1 holomorphic fields—that is, l1 = l1 =0.By taking Here, we imply summation over dummy indices, these into account (26), we then find that the coefficients running through the values from 2 to n.In addition, determining the exponential functions in (24) are 4 we have used the notation Ω=˜ {Ωjl}.The coe fficients − ˆ ¯ ¯ of JpJs (including p, s =1) in (25) are nothing but B1 = i (ˆκj + ipj)X(tj, tj; V,V ), (27) −1 j=3 the corresponding elements Ωps expressed in terms of the elements of the matrix Ω.The second formula 4 in (25) follows from the equality ln det Ω = tr ln Ω if B2 = −i (ˆκj + ipj)Xˆ(tj, t¯j; U, U¯), − we calculate tr ln Ω in terms of Ω , Ω , Ω˜ 1 ,and j=3 11 1j1 j1l1 4 Ωl11. 1 G = − (ˆκ + ip )(ˆκ + ip )Xˆ(t , t¯ ; t , t¯ ), If, instead of z1, we introduce the variable z = 0 2 j j l l 3 3 4 4 (z1 − u)/(z2 − u), the entire dependence on z2 and y j,l=3 in (24) is factored out in the form G12 = Xˆ(U, U¯; V,V¯ ), |z − u|−2 exp[x(˜p2 − p˜2)/4+yp˜2/4],wherex = 2 2 1 1 ˆ ¯ ¯ | − | ln |z2 − u|.Here, −∞ PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1364 DANILOV the correlation function (12) on the supermanifold of For n-loop amplitudes, the function − (n−1) genus (n 1) formed by all handles, with the excep- Zˆ ({q,q¯}) in (24) is equal to the partition func- L,˜ L˜ tion of the degenerate handle being considered.The − ˜ tion on the supermanifold of genus (n 1) that is quantity G0, as well as each term in the expression formed by all handles, with the exception of the ˜ for B1, is a correlation function defined on a surface of degenerate being under considered.For an arbitrary genus 0 and calculated for the corresponding points. n, the right-hand part of the integral in (24) can The nonexponential factors in (24) are independent of be written, in just the same way as for n =1,in p˜1 and p˜2.In calculating them in the expansion in k, the form of the sum over physical polarizations of | − |∼ one must retain, in view of the estimates z1 u bosons in the intermediate state; upon integration 1/4 |z2 − u|∼ |k| and ϑ1 ∼ ϑ2 ∼|k| , terms that are with respect to y, t1, t2, t3 = t,andϑ4,therethen not less than unity.In order to verify the validity of arises expression (23).By applying a similar method, expressions (27), we must substitute them into (24) we can perform a complete verification of the unitarity and calculate the resulting integral.In performing a of boson loops, at least for massless states (such a comparison with the amplitude in (9), use is made of verification will be given elsewhere). formulas (25), where the subscript “1” is associated with the degenerate handle. 5.RAMOND INTERMEDIATE STATES By way of example, we will consider a one-loop Let us now consider the two-particle unitarity amplitude (n =1).The vacuum correlation function condition for a loop involving Ramond states in the → Xˆ(t, t¯; t , t¯) in (27) is then equal to (ln |z1 − z2 − holomorphic sector (l1 =1/2).For k 0, the holo- − ϑ ϑ |)/2, while the factor Zˆ(n 1)({q,q¯}) in (24) morphic Green’s function then remains singular at 1 2 L,˜ L˜ the points u and v.Thus, this function, along with reduces to unity.We note in addition that the product the other quantities in (12), differs from the corre- { } { } { } { } { } { } { } { } − OL˜ ( p , ζ , p˜1, t , q )OL˜ ( p , ζ , p˜1, t , q ) sponding functions of genus (n 1).For example, the holomorphic Green’s function R(0)(t1,t2) on a is written in the form O(b)({ζ})O(b)({ζ }), the holo- morphic function O ({ζ}) here being the sum over sphere as obtained upon going over to the above limit (b) in the corresponding function of genus 1 (both at ˜ ˜ the polarization 10-vectors ζ1 and ζ2 of bosons whose l2 =0and at 2l2 =1)hastheform momenta are p˜1 and p˜2;thatis, − − ϑϑ 2 R(0)(t, t )=ln(z z ) − (29) ˜ ˜ ˜ (κj + ipj)ϑj 2(z z ) O(b) = −ζ1ζ2 + ζ1 (28) 2(z − u) ˜ ˜ j=1 j (z − u)(z − v) (z − v)(z − u) ζ1,ζ2 × + . (z − v)(z − u) (z − u)(z − v) 2 1 ζ˜ ζ˜ × ζ˜ (κ + ip )ϑ − 1 2 2 2 l l l u − v Formula (29) immediately follows from expressions l=1 (A.2) and (A.7) for the Green’s functions (see Ap- 4 (κ + ip )ϑ 4 (κ + ip )ϑ 2 pendix А).Green’s functions of higher genus also + ζ˜ r r r ζ˜ s s s − , 1 2(z − v) 2 2(z − u) u − v have a singularity of this type, as follows from (A.2) r=3 r s=3 s for the case where all Grassmann moduli are equal where κ is the holomorphic part of the operator κˆ to zero.Of course, this singularity occurs in the case of nonzero Grassmann moduli as well (see Appendix [see the comments immediately after formula (11)]. → In formula (28), summation is performed over all ten B).It follows that, if z u and if z lies at a fi- u independent polarizations of each boson.The term nite distance from the point , the Green’s function (n) { } ≡ −2/(u − v) emerged because of the appearance of RL (t, t ; q ) R(t, t ) has the form 8/(u − v) instead of 10/(u − v) from the expansion of ϑφˆ(t ) (30) expression (21) in powers of k.This term is the contri- R(t, t )=R (t, t )+R (U, t ) − , (0) (r) − bution from the loop of Faddeev–Popov ghosts, and 2 (z u) it cancels the contribution of unphysical polarizations where R (t, t) is given by formula (29), the function t t t = t (0) upon integration with respect to 1, 2,and 3 (the → ˆ proof of this natural cancellation of unphysical polar- R(r)(U, t ) is regular for z u,andφ(t ) is a factor in izations will be given elsewhere).Upon integration the singular part.The function φˆ(t) is not singular with respect to y, there then arises expression (23), for z → u since the Green’s function is symmetric where summation is performed only over physical (apart from the emergence of the term ±πi)underthe polarizations. permutation of arguments.We recall that U =(u|0). PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1365 In the limit z → v, the Green’s function has a similar in the limit z → u and z → u.Here, the function form.The singularities being discussed are associated ϕ(t) is not singular at z = u.In addition, the quantity with the fermion part of the Green’s function, but, ϕ(U) is proportional to the Grassmann modular pa- in the presence of Grassmann modular parameters rameters that correspond to nondegenerate handles. corresponding to nondegenerate handles, they also If l1 =1/2, the two terms in (34) that are singular arise in its bosonic part and, hence, in the scalar func- in the limit z → u must be supplemented with the (n) tions Js (t; {q}; L).The explicit form of the functions terms that are complex conjugate to them.We note ˆ R(r)(U, t ) and φ(t ) in (30) can be obtained (see Ap- that, because of the presence of the Grassmann pa- pendix B) from formulas presented in Appendix А, but rameters, the function ϕ(t) depends not only on t but this is immaterial for the ensuing analysis.If z,z → also on t¯.In this case, the singularity being studied u,then does indeed appear in scalar functions and, hence, in the last term on the right-hand side of (12).The R(t, t )=R (t, t )+R (U, U) (0) (rr) (31) validity of these expressions for the coefficients in (24) ϑφˆ(U) ϑφˆ(U) is verified by a direct calculation of the integral in (24) − − , 2 (z − u) 2 (z − u) upon substituting in it expressions (27) and (32) and expressions that are given below for nonexponential where R(rr)(U, U) is that part of R(r)(U, t ) which is factors. → ˆ − regular for z u.The quantity φ(U) is proportional In particular, the function Zˆ(n 1)({q,q¯}) in (24) to the Grassmann modular parameters correspond- L,˜ L˜ ing to nondegenerate handles.If one argument of has the form (19) where Z(n)({q}) must be re- L the function R(t, t ) tends to U =(u|µ =0) or to (n) placed by the expression kZ ({q})/16 at k =03) V =(v|ν =0)(or both of its arguments tend to these L (n) { } quantities), the Green’s function has no singularity and where the period matrix ω ( q ,L) in for- of the above type since ϑ =0in this case.Therefore, (n) { } mula (14) for ΩL,L ( q,q ) is replaced by the period relations (26) are valid in the case of a fermion loop (n−1) { } ˜ as well (see Appendix B).In accordance with for- matrix ω˜ ( q , L) whose elements are equal (n) { } mula (29) in the limit z,z → u and formula (12), the to the elements [ω ( q ,L)]jl at j, l =2,...,n of the period matrix at k =0.As in formula (25), quantity G˜0 in (24) is given by the subscripts j, l =2,...,n number nondegenerate 1 ˜ − | − | handles.If, in addition, l1 =1/2, a similar replace- G0 = (ˆκ1 + ip1)(ˆκ2 + ip2) ln z1 z2 (32) (n) 2 ment must be made for the quantities Z ({q}) and L (n) (n) − − ω ({q},L).If l =0, then the quantities Z ({q}) − ϑ1ϑ2 z1 u z2 u 1 L + . (n) { } 8(z1 − z2) z2 − u z1 − u and ω ( q ,L) are replaced by the corresponding values calculated on the supermanifold of genus In the case of a fermion loop and antiholomorphic (n − 1) that is formed by all handles, with the excep- fields (that is, l1 =1/2), the bracketed expression tion of the degenerate handle being studied.We note in (32) also involves the term that is complex con- that all of the functions being discussed can be explic- jugate to the last term in this bracketed expres- itly obtained (see Appendix B) from formulas given sion.The remaining coe fficients in the exponents in Appendix А.Finally, we note that, for the fermion appearing in (24) are given by formulas (27), where loop under study, the factor O˜ ({p}, {ζ}, p˜1, {t}, {q}) L the correlation function Xˆ(t, t¯; t , t¯) is defined as in (24) has the form the limit of the vacuum correlation function (12) { } { } { } { } for k → 0.According to the formulas obtained for OL˜ ( p , ζ , p˜1, t , q ) (35) the Green’s functions, we can write the correlation 2 2ηjζ(j) + ipjϑj function Xˆ(t, t¯; t, t¯) in (27) as =exp √ Ψˆ , 8 z − u j=1 j ˆ ˆ ¯ ϑϕ(t ) X(t, t¯; t , t¯)=X(r)(U, U; t , t¯) − (33) 2 (z − u) where the Grassmann quantities ηj are identical to ˆ for z → u and as those in (11) and the 10-vector Ψ depends on the ˆ ˆ ¯ ¯ X(t, t¯; t , t¯)=Xrr(U, U; U, U) (34) 3)In this expression, the factor of 1/16 is due to the fact that, ϑϕ(U) ϑϕ(U) at 2l1 =1, the factor of 16 in (21) at 2l1 =1arises from the − − trace of the products of Dirac 10-matrices in the fermion loop 2 (z − u) 2 (z − u) [see the text after formula (38) below]. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1366 DANILOV ˜ parameters of the final state and on p˜1;thatis, where P = p1 + p2 =˜p1 +˜p2, p˜ = (p1 − p˜2)/2, 4 1 ˆ Φ= [ϕ(t )+ϕ(t ) − ϕ(U) − ϕ(V )], (40) Ψ= [2ηj ζ(j)D(tj)+ipjϑj]ϕ(tj) (36) 2 3 4 j=3 Ψ=2η ζ D(t )ϕ(t )+2η ζ D(t )ϕ(t ) (41) + ip˜1ϕ(V )+ip˜2ϕ(U). 3 (3) 3 3 4 (4) 4 4 1 Here, the spinorial derivative D(t) and the function + i(p3 − p4) [ϕ(t3) − ϕ(t4)]. ϕ(t) were defined in (7) and (33), respectively.In just 2 the same way as ϕ(U), the quantity ϕ(V ) is propor- The function ϕ(t) was defined in (33).In this case, tional to the Grassmann modular parameters corre- expression (35) takes the form sponding to nondegenerate handles.At l =1/2,the 1 O ({p}, {ζ}, p˜ , {t}, {q}) (42) { } { } { } { } L˜ 1 expression for the factor OL˜ ( p , ζ , p˜1, t , q ) 2 4ηrζ + i(p1 − p2)(ϑ1 − ϑ2) in (24) has a similar form.At l1 =0, this factor is =exp (r) √ Ψ calculated by a method similar to that used for a boson 16 zr − u loop.As has already been indicated, the expressions r=1 obtained above are verified according to the scheme 2 2iη (Pζ )+(p p )(ϑ + ϑ ) − j (j) √ 1 2 1 2 where their substitution into (24) is followed by a Φ 8 zj − u calculation of the emerging integral with respect to j=1 p˜1 and a comparison of the result with expression (9). 2 ϑ1 + ϑ2 The sought n-loop amplitude A˜(n) for the transition + i √ (P Ψ) 16 z − u to Ramond states has the form j=1 j ˜(n) ¯ ˜(n) ˜(n) 2 A = ψ(˜p2)(T + T )ψ(˜p1), (37) 4ηs(ζ p˜)+i(p1 − p2)˜p(ϑ1 − ϑ2) 3 1 + (s) √ where ψ(p) is a Majorana—Weyl 10-spinor satisfying 16 zs − u s=1 the condition Γ11ψ(p)=ψ(p) and the Dirac equation M M (Γ pM )ψ(p)=0.Here, Γ is a Dirac matrix, and × [ϕ(V ) − ϕ(U)] . Γ11 is the product of all ten matrices.The quantity ˜(n) ˜(n) T1 in (37) involves one Dirac matrix, while T3 The tree amplitude can be calculated by considering contains the antisymmetrized product of three matri- { } { } { } { } the case of n =1, where there are no Grassmann ces.Therefore, OL˜ ( p , ζ , p˜1, t , q ) must have modular parameters.Representing, in this case, the the form { } { } { } { } factor OL˜ ( p , ζ , p˜1, t , q ) appearing in (24) in O˜ ({p}, {ζ}, p˜1, {t}, {q}) (38) theform(38),wearriveat L (0) (0) (n−1) (n−1) √ = tr[(T + T )(Γ˜p )(T + T )(Γ˜p )] (0) 1 3 1 1 3 1 2 4 2T = − Γ[(ζ ζ ) (p − p ) (43) 1 3 4 2 3 4 + ..., η ϑ η ϑ where the ellipsis stands for the contribution that − (ζ p )ζ +(ζ p )ζ ] 3 3 4 4 vanishes in (9) upon taking the relevant integrals and 4 3 3 3 4 4 2(z − z ) 3 4 performing summation over spin structures.In (38), − − × v u v u T (n) involves one Dirac matrix, while T (n) contains + 1 3 (z3 − u)(z4 − v) (z3 − v)(z4 − u) the antisymmetrization product of three Dirac ma- [(ζ (˜p − p˜ ))ζ +(ζ (˜p − p˜ ))ζ ](v − u)2 trices.Since the 10-spinors in (37) obey the Weyl + 3 3 4 4 4 3 4 3 , condition, the trace of the identity matrix is equal 4(z3 − u)(z3 − v)(z4 − u)(z4 − v) √ to 16 in (38). (0) { } { } { } { } 4 2T3 The expression for OL˜ ( p , ζ , p˜1, t , q ) in [(Γζ )(Γζ )(Γ(p − p ))] η ϑ η ϑ (v − u)2 (35) is a fourth-order polynomial in the exponent, = 3 4 3 4 a 3 3 4 4 , which is the sum of the products of Grassmann 4(z3 − u)(z3 − v)(z4 − u)(z4 − v) variables.Odd powers of this polynomial vanish upon integration with respect to Grassmann variables. where [...]a symbolizes an antisymmetrized (with a Even powers must be represented in the form (38), factor of 1/6) expression.In (43), we have discarded whereby one defines the integrand in the expressions terms that vanish upon integration with respect to − Grassmann variables.We recall that v and u are the A˜(0) A˜(n 1) for the amplitudes and .Concurrently, it coordinates of vertices describing the emission of a is convenient to recast (36) into the form fermion with a momentum p˜1 and p˜2, respectively. Ψ=Ψˆ − iP Φ+ip˜[ϕ(V ) − ϕ(U)], (39) For the amplitude involving bosons whose momenta PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1367 are p1 and p2, the corresponding expressions arise For an arbitrary n (including n =0), the structure →∞ (n−1) as the limit u of formulas (43), whereupon T in (38) is obtained owing to the terms that → 3 one must make the change of variable v u and are proportional to ∼Ψ Ψ Ψ Φ.These terms → → M1 M2 M3 the substitutions 3 1 and 4 2 for the indices. arise upon expanding the exponential function in (42) The successive substitutions of formula (43) into in a series as the result of the multiplication of the expression on the right-hand side of (38) and the third power of the sum over r in (42) by the of the resulting expression into (24) lead (apart from term proportional to Φ.Here, Ψ is the component normalization) to the amplitude obtained in [2].In de- M of the 10-vector Ψ.These ΦΨ Ψ Ψ terms riving (43), we integrated by parts those terms in (24) M1 M2 M3 are also proportional to (˜p1p˜2).They correspond that are proportional to η1η2p1p2/[(z1 − u)(z1 − z2)] to the (˜p1p˜2) terms in (38), which originate from and η1η2p1p2/[(z2 − u)(z1 − z2)].The first and the (0) (n−1) (n−1) second term were integrated with respect to z2 and tr[T3 (Γ˜p1)T3 (Γ˜p2)].Here, the quantity T3 is calculated unambiguously and is given by z1, respectively.By using expression (32) for G˜0, we eventually find that the first and the second − i − − − T (n 1) = (ΓΨ)(ΓΨ)(ΓΨ)Φ. (47) term reduce to η1η2p2p˜1/[(z1 u)(z2 u)] and 3 48 η1η2p1p˜1/[(z1 − u)(z2 − u)], respectively.In some cases, the product of the last factor in the first term Since the products of the Dirac matrices are anti- (n−1) (0) on the right-hand side of the first equation in (43) and symmetrized in T3 and T3 , there are no terms 1/(z3 − z4) canbewrittenas (0) (n−1) in tr[T3 (Γ˜p1)T3 (Γ˜p2)] that are proportional to 2(v − u) the scalar products of those vectors that are both (44) (0) (n−1) (z4 − u)(z4 − v)(z3 − z4) contained in T3 or T3 .For the same reason, the − − (v u) trace being studied does not involve terms in which (z3 − u)(z4 − u)(z4 − v) eachofthevectorsp˜1 and p˜2 in (38) is scalarly mul- − tiplied by the vectors that are both included either in (v u) − − , T (0) or in T (n 1).Only those terms in the expansion (z3 − v)(z4 − v)(z4 − u) 3 3 of the exponential function in (42) that involve the or as the expression obtained from (44) by means of square of the sum over r can generate nonzero terms the substitutions z z and v u.When expres- (n−1) 3 4 in the trace being studied.If the quantity T is sion (44) was multiplied by η η p p ,thefirst term in 3 3 4 3 4 given by (47), the part of the trace that is quadratic in (44) was integrated by parts with respect to z3.As a result, the contribution of this term proved to be the components of the 10-vector (˜p1 +˜p2) stems from proportional to the expression the term where the square of the above sum over r is multiplied by the product of the sum proportional 2η η (v − u) p p˜ p p˜ to P Ψ and the sum proportional to Φ.The remaining − 3 4 3 1 + 3 2 . (45) (z4 − u)(z4 − v) z3 − v z3 − u terms of this trace are quadratic in the components of the 10-vector (˜p1 − p˜2).They cannot be obtained by From (29), it follows that, in the case of n =1, which expanding the exponential function in (42) since the is being considered at present, the function ϕ(t) remaining terms of this expansion do not involve the in (42) has the form quantities proportional to ΨΨΨΦ.For the required u − v contribution to the unitarity condition for the boson ϕ(t)=−ϑ . (46) amplitude to arise in spite of this, it is necessary that (z − u)(z − v) 1 i Ψ Ψ 1 − [ϕ(V ) − ϕ(U)] P 2Φ+i(P Ψ) − Ψ Ψ (Ψ˜p)Φ =0, (48) M N 16 4 M N where ... denotes integration of the expression spect to variables that determine the amplitude obtained by multiplying that in (48) [for details, see A˜(n−1).Relation (48) follows from a comparison of formula (50) above and the text after it] by the factors the corresponding contributions in the expansion remaining in (24) [all of the quantities there are fac- of the exponential function (42) with the quantity − − tors, with the exception of O˜ ({p}, {ζ}, p˜1, {t}, {q})] (0) (n 1) (0) (n 1) L tr[T3 (Γ˜p1)T3 (Γ˜p2)],where,forT3 and T3 , and by summing the result over the superspin struc- we use the expressions obtained above [see formu- tures (L,˜ L˜).The integral is calculated with re- las (43) and (47)].Three independent conditions for PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1368 DANILOV (n−1) the coefficients of ζ ζ , ζ p˜N (ζ p3),and the n-loop amplitude (9).In addition, Zˆ ({q,q¯}) (3)M (4)N (3)M (4) L,˜ L˜ ζ(4)M p˜N (ζ(3)p4),where,asbefore,p˜ =(˜p1 − p˜2)/2, (n) is calculated in terms of Z ({q,q¯}), as was ex- follow from formula (48) if we take into account the L,L (n−1) plained after formula (34).It should be noted that identity of bosons.In order to calculate T1 in (38), (n−1) − Zˆ ({q,q¯}) depends, in particular, on the coordi- (0) (n 1) L,˜ L˜ we consider the term tr[T (Γ˜p1)T (Γ˜p2)].Iso- 3 1 nates v and u of the vertices describing the emission lating this term from the terms appearing in the − of Ramond states.Further, F˜(n 1) is calculated in expansion of the exponential function in (42), we L,˜ L˜ obtain ˆ(n−1) terms of the function F (η1,η2, η¯1, η¯2), which is − i L,˜ L˜ T (n 1) = −i(ΓΨ)Φ + (ΓΨ)(P Ψ)[ϕ(V ) − ϕ(U)]Φ, 1 8 given by the expression obtained by integrating the (49) integrand in formula (11) with respect to (η3, η¯3) and (η4, η¯4).In this expression, the vacuum correlation where the notation is identical to that in (42).With functions (12) on a supermanifold of genus n must allowance for (49) and (48), the remaining terms in Xˆ(t, t¯; t, t¯) the expansion of the exponential function in (42) that be replaced by the functions ,whichare the corresponding vacuum correlation functions (12) are linear in Ψ yield all the remaining terms in (38), → with the exception of the term proportional to the of genus n that are calculated for k 0 [see the (0) (n−1) explanation in the text after formula (27); see also product of tr[(Γ˜p)T1 ] and tr[(Γ˜p)T1 ],wherep˜ = formula (33) and Appendix B].In the resulting ex- (˜p1 − p˜2)/2 as before.This term cannot be obtained pression, we must additionally make the substitu- from the unit term in the expansion of expression (35) tions (z1|ϑ1) → (v|0), (z2|ϑ2) → (u|0), p1 → p˜1,and because it does not involve Ψ.For the required con- p2 → p˜2.If states characterized by the momenta p˜1 tribution to the unitarity condition for the boson am- and p˜2 are Ramond bosons, then we have plitude to arise in spite of this, it is necessary that ¯ (n−1) (n−1) √ O˜ = ψ(˜p2)(T + T )ψ(˜p1), (52) 1 − L 3 1 16+ tr (ΓT (n 1))(Γ˜p) + 2[ϕ(V ) (50) 2 1 where ψ(p) is a Majorana–Weyl 10-spinor and other quantities are given by formulas (47) and − 2 O ϕ(U)][P Φ+(ΨP )] =0, (49).The quantity L˜ is calculated in a similar − way.Here, the function F˜(n 1) in (51) is equal to where, as in (48), ... denotes integration of the L,˜ L˜ (n−1) expression obtained by multiplying that in (48) by all Fˆ (η ,η , η¯ , η¯ ) at η = η =0 and η¯ =¯η = L,˜ L˜ 1 2 1 2 1 2 1 2 factors following from (24) and summing the result 0;thatis, ˜ ˜ over the superspin structures (L, L ).The rest of the − − F˜(n 1) = Fˆ(n 1)(0, 0, 0, 0). (53) notation is identical to that in (42). L,˜ L˜ L,˜ L˜ (n−1) Thus, the (n − 1)-loop amplitude A˜ de- − We recall that the function Fˆ(n 1)(η ,η , η¯ , η¯ ) is scribing the transition of two massless Ramond L,˜ L˜ 1 2 1 2 states having the momenta p˜1 and p˜2 to two mass- defined in the text given between formulas (51) and less bosons whose momenta and polarizations are (52).If the particles that we consider and which have (p3,ζ(3)) and (p4,ζ(4)) is given by the momenta p˜1 and p˜2 are Ramond fermions, either 2n the holomorphic or the antiholomorphic part of the (n−1) 2g 2 above two-particle state is described by a bosonic A = |(z4 − u)(z4 − v)| (51) 2n(n − 1)! wave function.If it is the holomorphic part of the wave − − O (n 1) (n 1) function that corresponds to it, then L˜ =1and × Zˆ ({q,q¯})F˜ O O (dqdqdtd¯ t¯) , L,˜ L˜ L,˜ L˜ L˜ L˜ − − L,˜ L˜ F˜(n 1) = dη dη Fˆ(n 1)(η ,η , 0, 0). (54) L,˜ L˜ 1 2 L,˜ L˜ 1 2 where g is the coupling constant, as in (9), and sum- O mation is performed over superspin structures on a In addition, the quantity L˜ is calculated according supermanifold of genus (n − 1).We note that, in to (52).If the antiholomorphic part of the Ramond contrast to the boson amplitudes for massless-boson state under discussion corresponds to bosons, the scattering, the sum in question involves the contri- factors in question are calculated in a similar way. bution of odd superspin structures that is induced In (37), the coordinates of the interaction vertices by the contribution to the unitarity condition from corresponding to Ramond particles are fixed along the l2 =1/2 fermion loop.The quantities in (51) are with the local coordinate of the vertex describing the expressed in terms of the corresponding quantities for emission of one of the bosons.Here, the Grassmann PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1369 partners of the local coordinates of the vertices cor- where Rb(z,z ) and Rf (z,z ; L) are the Green’s func- responding to the Ramond particles are set to zero. tions for boson and fermion fields, respectively.For Since the integrand in (51) possesses SL(2) symme- even spin structures, the fermion Green’s functions try, any (3|2) variables could be fixed in the integral can be written, for example, as [4] in (51); however, we will not discuss this subject in 1 the present article. R (z,z ; L)=exp [R (z,z)+R (z ,z )] (A.2) f 2 b b One can verify that, if the conditions in (48) and (0) (50) are satisfied, the amplitude in (51) vanishes in the Θ[l ,l ](J|ω ) − R (z,z ) 1 2 , b (0) case of longitudinal-boson emission, as is prescribed Θ[l1,l2](0|ω ) by gauge invariance (we plan to discuss this issue in detail elsewhere).The aforementioned conditions (48) where Rb(z,z) for z = z is defined as the limit and (50) and the analogous conditions arising in the Rb(z,z ) − ln(z − z ) for z → z , Θ is a theta func- case of l =1/2 can be obtained by replacing, in (51), tion, and J stands for the set of scalar functions 1 O O (Js(z) − Js(z ))/2πi that are its arguments.Here, the factor L˜ (or the factor L˜ or both these factors if the quantities R (z,z), J (z),andω(0) are given by l1 = l1 =1/2) by the expressions within the angular b s brackets ... in (48) and (50) (or by their complex formulas (18) at zero values of the parameters µ and conjugate counterparts).At present, we have proven ν.For a surface of arbitrary genus n, in particular, we the validity of the relations being discussed only for have −1 −2 the tree amplitudes in (51).In this case, the first ∂z Rb(z,z )= [z − g(z )] (cz + d) , (A.3) factor under the sum sign in (51) is equal to unity; g the remaining quantities are calculated in terms of correlation functions on a sphere and in terms of ex- where summation is performed over all elements of pressions (32) and (46).In doing this, it is convenient the Schottky group.Formula (A.2)can also be recast to consider the limit u →∞or v →∞; this is suffi- into the form of Poincare´ series, but we do not dwell cient since the expressions being discussed possess on this point here.At nonzero Grassmann moduli, L(2) symmetry.In this case (for u →∞), expres- the Green’s functions in the Ramond sector cannot be directly expressed in terms of theta functions or sions that involve η3η4p3p4/[(z4 − v)(z3 − z4)] are integrated by parts in just the same way as was done written in the form of Poincare-type´ series (see In- for the expressions considered after formula (43).This troduction and Section 2).In any case, they can nev- verification can also be performed for arbitrary u and ertheless be specified in the form of series where each v.In this case, it is convenient to use formula (44) and term of a series is represented as the integral of an integration by parts leading to (45).The problem of expression constructed from functions of genus 1 that are associated with each of the handles.Each such verifying the relations in question for loop amplitudes (1) requires a further investigation. function Rs (t, t) was calculated for the parameters (ks,us,vs) and (µs,νs) of a given handle and its spin structure ls =(l1s,l2s).By virtue of relations (3), this ACKNOWLEDGMENTS function has the form This work was supported by the US Civilian (1) Rs (t, t ) (A.4) Research and Development Foundation (CRDF) for (1) − (1) the Independent States of the Former Soviet Union = R(b)s(zs,zs) ϑsϑsR(f)s(zs,zs; l1s,l2s) (grant no.RP1-2108) and by the Russian Foundation + ε ϑ Υ (∞,z )+ε ϑ Υ (z , ∞), for Basic Research (project nos.00-02-16691, 00- s s s s s s s s 15-96610). − (1) Υs(z,z )=(z z )Rf (z,z ; l1s,l2s), where z and ϑ are given by (4) and R(1) (z,z) and APPENDIX A s s (b)s (1) Here, we present expressions for holomorphic R(f)s(z,z ; l1s,l2s) are the relevant boson and fermion Green’s functions and quantities related to them functions.The terms proportional to Υs were added in (in particular for the Ramond sector), as well as (1) order to ensure a decrease in Ks (t, t ) for z →∞or for the holomorphic function Z˜(n)({q},L) in (20) [it (1) z →∞.Here, the function K (t, t ) has the form determines the integration measure (19)].At zero s (1) (1) Grassmann parameters, the sought holomorphic Ks (t, t )=D(t )Rs (t, t ). (A.5) Green’s functions R (t, t; L) have the form (0) For the case of n =1, the boson function is given by R (t, t ; L)=R (z,z ) − ϑϑ R (z,z ; L), (A.1) (1) (0) b f formula (A.3). The scalar function Js (t) of genus 1 PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1370 DANILOV is given by + [(1 − Kˆ (1))−1Kˆ (1)] (t, t )dt R˜(1)(t ,t) − rs 1 1 r 1 (1) zs us r,s Js (t)=ln , (A.6) Cs zs − vs 1 −1 − Φ (t; L; {q})Vˆ Φ (t ; L; {q}), where, for zs, we employ expressions (4).For even 2 m mm m spin structures, the fermion function m,m (1) R (z,z ; l1s,l2s) has the form (A.2), where L = (1) −1 (1) (f)s where [(1 − Kˆ ) Kˆ ]rs(t, t1)dt1 is the kernel of (l1s,l2s), while, for odd spin structures, we use the the operator function − ˆ (1) −1 ˆ (1) ˆ (1) ˆ (1) ˆ (1) (1) (1 K ) K = K + K K + ..., R (z,z ;1/2, 1/2) (A.7) (A.12) (f)s (1) (1) (1) { | } and the sum over m, m is taken only over odd spin ∂z Θ[1/2, 1/2](J(0)s ωs ) ∂z J(0)s(z ) = , structures of genus 1.Here, Vˆmm are defined only for (1) | (1) (1) Θ[1/2, 1/2](J(0)s ωs ) ∂zJ(0)s(z) those m, m that number these odd spin structures.In this case, we have (1) where J(0)s(z) is given by (A.6) at zs = z.In this case, Φ (t; L; {q})=ϕ (t) m m (A.13) we have − ˆ (1) −1 ˆ (1) (1) (1) (1) − + [(1 K ) K ]pm(t, t )dt ϕm(t ), Rs (tb,t)=Rs (t, t )+Js (t ) ϕs(t)ϕs(t ), p (A.8) Cm where t corresponds to a round about the B cycle; 4) b s where ϕm(t) is given by (A.9) and (1) the scalar function Js (t ) has the form (A.6); and the 1 spinor zero mode ϕ (t) is Vˆ = − D(t)ϕ (t)dt (A.14) s mm 2 m 1/2 p= m ϑ (u − v ) Cp ϕ (t)= s s s + ε (u − v )1/2; s 1/2 s s s [(zs − us)(zs − vs)] (1) −1 (1) × [(1 − Kˆ ) Kˆ ] (t, t )dt ϕ (t ) (A.9) pm m Cm and (zs|ϑs) is defined according to (4).The last term ϕ (t) 1 in (A.9) ensures a decrease of s in the limit − (1 − δmm ) D(t)ϕm(t)dtϕm (t), z →∞. 2 In order to construct the required Green’s func- Cm ˆ (1) tion, we define (see [4]) the matrix operator K = with δmm being the Kronecker delta.In the Neveu – (1) {Kˆsr},whereKˆsr is an integral operator equal to Schwarz sector, where the function defined in (A.5) zero for s = r.For s = r, the kernel of the operator is has no cuts, the integrals in (A.11) can easily be (1) (1) calculated.In this case, we can obtain a series for K˜s (t, t )dt .Here, the function K˜s (t, t ) is related R(n)(t, t; {q}) by using the Poincare´ series (18) at to R˜(1)(t, t) via (A.5); that is, L s n =1for the Green’s functions.In general, the proof ϑ − ϑ of (A.11) reduces to verifying that expression (A.11) K(1)(t, t )= + K˜ (1)(t, t ), (A.10) s z − z s satisfies the conditions in (16).In order to test rela- K˜ (1)(t, t)=D(t)R˜(1)(t, t). tions (16) for the transformations associated with the s s rth handle, we rewrite (A.11) as Following [4], we define the kernel along with the R(n)(t, t ; {q})=R(1)(t, t ) (A.15) differential dt = dz dϑ /2πi.The operator under dis- L r cussion integrates the above function multiplied by (1) ˆ (1) + Kr (t, t1)dt1R(t1,t), K˜s (t, t ) with respect to t along the contour Cr, s= r which circumvents the limiting points of the Schot- Cs tky supertransformation associated with the rth han- (1) (1) dle.This contour also circumvents the cuts that are where Rr (t, t ) and Kr (t, t1) are the total Green’s present for handles of the Ramond type.The required functions of genus 1 with allowance for the singular expression for the Green’s function has the form terms in (15) and (A.10). Calculating the contribution to (A.15) from the pole term in (A.10), we do indeed (n) { } − − ˜(1) RL (t, t ; q )=ln(z z ϑϑ )+ Rs (t, t ) 4) ˆ s The matrix Vmm differs slightly from the corresponding (A.11) matrix in [4]. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1371 arrive at (A.11). Simultaneously, we calculate the × [(1 − Kˆ (1))−1Kˆ (1)] (t, t)dtJ(1)(t) pr s quantity Rˆ(t1,t; {q}; s).For a handle of an odd spin Cs structure, we obtain [4] 1 (r) −1 (s) − Φ (L; {q})Vˆ Φ (L; {q}), ˆ { } 2 m mm m ϕr(t )+ ϕr(t1)dt1R(t1,t; q ; s)=0. m,m s= r Cs (A.16) where the quantities appearing in this expression are given by (A.18) and (A.6). For expression (A.15), relations (16) for the transfor- mations associated with the rth handle obviously hold As was indicated in Section 2, the integration [upon taking into account relations (A.7) and (A.16) measures in (9) are calculated on the basis of the for the handle of odd spin structure].Simultaneously, equations derived in [4, 9] by requiring that the am- plitude be independent of the choice of the zweibein J(n)(t; {q}; L) the scalar function r corresponding to and the field of the two-dimensional gravitino.For the r the th handle is determined in the form integration measures in question, we thereby obtain J(n)(t; {q}; L)=J(1)(t) (A.17) ˜(n) { } r r formulas (19)–(21), where the function Z ( q ,L) in (20) can be written in the form (1) ˜(1) + D(t1)Jr (t1)dt1Rs (t1,t) (n) (1) s= r ln Z˜ ({q},L)=−5tr ln(I − Kˆ ) (A.20) Cs ˆ − ˆ(1) − ˆ (1) +5lndetV + tr ln(I G ) ln det U, + D(t1)Jr (t1)dt1 p= r s (1) Cp where I is the identity operator and Kˆ is the same operator as in (A.11). The matrix Vˆ is defined × [(1 − Kˆ (1))−1Kˆ (1)] (t ,t )dt R˜(1)(t ,t) ps 1 2 2 s 2 in (A.14). The matrix operator Gˆ(1) is calculated in Cs (1) terms of the Green’s function Gs (t, t ) of genus 1 1 − − (r) { } ˆ 1 { } for ghosts (see below) in just the same way as the Φm (L; q )Vmm Φm (t; L; q ), 2 (1) (1) m,m quantity Kˆ is calculated in terms of Ks (t, t ). (1) (1) (1) Thus, we obtain Gˆ = {Gˆsr }, where the integral where Js (t) is given by (A.6) and (1) operator Gˆsr vanishes at s = r.For s = r,itskernel (r) { } − (1) (1) Φm (L; q )=(1 δmr) (A.18) Gˆ is equal to the regular part G˜ (t, t ; s)dt of the sr ls × (1) (1) D(t1)Jr (t1)dt1ϕm(t1) ghost correlation function Gs (t, t );thatis, Cm − (1) ϑ ϑ − ˜(1) Gs (t, t )= G (t, t ; s). (A.21) (1) z − z ls + D(t1)Jr (t1)dt1 p= r As in the case of the matrix in (A.14), the matrix Cp elements Uˆ are defined only for those m, m that × − ˆ (1) −1 ˆ (1) mn [(1 K ) K ]pm(t1,t2)dt2ϕm(t2). number odd spin structures of genus 1.They are (1) Cm calculated in terms of the (3/2) zero modes χm (t) (1) The period matrix is calculated [4] on the basis of and the (−1/2) zero modes φm (t) of genus 1.As a equalities (17) for the functions in (A.17). The ma- result, we obtain (n) { } trix element ωrs ( q ; L) of the period matrix then 1 (1) Uˆ = − χ (t)dt (A.22) assumes the form mm 2 m p= m 2πiω(n)({q}; L)=δ ln k +(1− δ ) (A.19) Cp rs rs r rs − (1) × − ˆ 1 ˆ × (1) (1) [(1 G) G]pm (t, t )dt φm (t ) D(t)Jr (t)dtJs (t) Cm Cs 1 (1) − − (1) (1) (1 δmm ) χm (t)dtφ (t). + D(t)Jr (t)dt 2 m p C Cp m PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1372 DANILOV √ − Here, the zero modes are given by (1) 1 √ k = Gf (z,z ; l1,l2)+ pµ(z)χµ(z ; l1,l2) (u − v )2 k χ(1)(t)=− m m , √ m − − 2 3/2 − − [(zm um)(zm vm)Qm(t)] (1 k)pν (z)χν (z ; l1,l2), (A.23) where (cz + d) corresponds to the transformation ϑ Q2 (t) (z − u )(z − v ) g(z) being considered; p (z) and p (z) have the form φ(1)(t)= m m m m m m , µ ν m (u − v ) − − m m 2(z v) −2(z u) pµ(z)= − ,pν(z)= − ; (A.29) where variables (zm|ϑm) are defined in (4) at s = m, u v u v while the quantity Qm defined in (7) corresponds to and z -dependent functions are the aforementioned transformation (4) at s = m.As − (1) (u v)W1(z; l1,l2)+W2(z; l1,l2) G (t, t) χµ(z; l1,l2)=− , for scalar fields, the function s is expressed z − u z − v (1) 2( )( ) in terms of the boson function G(b)s(z,z ) and the (A.30) (1) W (z; l ,l ) fermion function G(f)s(z,z ; l1s,l2s) as [4] − 2 1 2 χν(z; l1,l2)= − − . 2(z u)(z v) G(1)(t, t )=Q2(t)[G(1) (z ,z )ϑ (A.24) s s (b)s s s s For an odd spin structure, where there are no Green’s (1) − ˜ ∞ −3 functions possessing the property specified by + ϑsG (zs,zs; l1s,l2s) εsΥs( ,zs)]Qs (t ), (f)s Eq.(A.28)because of the (−1/2) zero mode, we ˜ − (1) Υs(z,z )=(z z )G(f)s(z,z ; l1s,l2s), define the Green’s function by the formula (1) (1) where z and ϑ are definedin(4)andtheterms G (z,z ;1/2, 1/2) = G (z,z ) (A.31) s s f (σ=1) ˜ (1) proportional to Υs ensure a decrease in Gl (t, t ) for z − u s − 2 − 1 χ (z ;1/2, 1/2), z →∞ or z →∞.The boson part of the Green’s z − v ν functions in (A.24) has the form [4] − 1 1 χν(z;1/2, 1/2) = (1) − 2 (z − u)(z − v) G(b) (z,z )= n 4 , (z − g (z ))(cnz + dn) n 1 (A.25) × , (c z + d )2 n n n where we omit the subscript s for the sake of brevity. (1) Summation is performed here over the group prod- where G(σ)(z,z ) has the form ucts of the corresponding Schottky transformation l1σ (1) (1) (z − u)(z − v) g(z).The fermion Green’s function G (z,z ; l1,l2) G (z,z )= f (σ) − − (A.32) for even spin structures (we again omit the subscript (z v)(z u) − − s for the sake of brevity) has the form l1σn − (2l2 1)n × k ( 1) n 3 . (1) (z − g (z ))(cnz + dn) Gf (z,z ; l1,l2) (A.26) n (z − u)(z − v) Here, summation is performed over all powers of the = R(1)(z,z; l ,l ) (z − u)(z − v) f 1 2 transformation g(z).We also note that, for even spin structures, we have (z − v)W1(z ; l1,l2)+W2(z ; l1,l2) − (1) (1) , (z − u)(z − v) Gf (z,z ; l1,l2)=G(σ=1)(z,z ) (A.33) where the last term is calculated in terms of the +[pµ(z)χµ(z ; l1,l2) − pν(z)χν (z ; l1,l2)] (1) asymptotic expression for Rf (z,z ; l1,l2) in the limit 1 z − u z →∞: − [pµ(z)(3χµ(z ; l1,l2)+χν(z ; l1,l2)) 2 z − v W (z; l ,l ) W (z; l ,l ) R(1)(z,z; l ,l ) → 1 1 2 + 2 1 2 . − p (z)(χ (z ; l ,l ) − χ (z ; l ,l ))], f 1 2 z − u (z − u)2 ν µ 1 2 ν 1 2 (A.27) where the notation is specified by formulas (A.29) and (A.30). The distinction between formula (A.20) and (1) It can easily be verified that Gf (z,z ; l1,l2) de- the corresponding expression in [4] is due to the fact →∞ →∞ (1) creases for z for z .In addition, we have that,in[4],thefunctionG(σ=1)(z,z ) is used instead (1) (1) (cz + d)Gf (g(z),z ; l1,l2) (A.28) of Gf (z,z ; l1,l2).In order to obtain the formula PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 APPLICATION OF THE UNITARITY CONDITIONS 1373 given in [4] from formula (A.20), we must first use can prove formula (31) and derive formulas for scalar expressions (A.33) and (A.31) to calculate Gˆ(1) in functions.Taking into account (12), we thereby arrive (A.20) and then invoke the formula at expressions (33) and (34).Following the same method and using formulas (19)–(21) and (A.20), we tr ln[A + A1A2] (A.34) − can calculate the function Zˆ(n 1)({q,q¯}) in (24). −1 L,˜ L˜ = tr ln A + tr ln[1 + A A1A2] P −1 = tr ln A +(−1) tr ln[1 + A2A A1], REFERENCES which is valid for any of the operators A1, A2,andA, det A =0 P =1 A 1.M.Green, J.Schwartz, and E.Vitten, The Theory of provided that .Here, if both 1 and Superstrings (Cambridge Univ.Press, Cambridge, A2 obey Fermi statistics; otherwise, P =0. 1987, 1988; Mir, Moscow, 1990), Vols.1, 2. 2. D.Friedan, E.Martinec, and S.Shenker, Nucl.Phys. B 271, 93 (1986). APPENDIX B 3. B.E.W.Nilsson and A.K.Tollst en,´ Phys.Lett.B 240, In order to obtain the first relation in (26), we 96 (1990). 4. G.S.Danilov, Phys.Rev.D 51, 4359 (1995); Erra- calculate (A.19) for ω11 in the limit k → 0 by sub- stituting there formula (A.6) for the scalar function tum: 52, 6201 (1995). (1) 5. G.S.Danilov, Nucl.Phys.B 463, 443 (1996). J1 (t) at µ = ν =0.The contour of integration with 6. G.S.Danilov, Yad.Fiz. 59, 1837 (1996) [Phys.At. respect to z circumvents nondegenerate handles, but Nucl. 59, 1774 (1996)]. it can be transformed into a contour circumventing 7. P.Di Vecchia, K.Hornfeck, M.Frau, et al.,Phys. the points z = u and z = v.Thus, the integral with Lett.B 211, 301 (1988). respect to z is given by the sum of the residues at 8. 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G.Moore and A.Morozov, Nucl.Phys.B 306, 387 gular at z = u and z = v and using expression (A.11), (1988); A.Yu.Morozov, Teor.Mat.Fiz. 81, 24, 293 we can obtain the required relation.In order to de- (1989). rive the second relation in (26), we again substitute 13. N.Berkovits, Nucl.Phys.B 395, 77 (1993). expression (A.6) for J(1)(t) into formula (A.19) for 14.S.Mandelstam, Phys.Lett.B 277, 82 (1992). 1 15. E.D’Hoker and D.H.Phong, hep-th/0110247; hep- ω1l and calculate the integral with respect to t in the same way as in deriving the first relation in (26).In th/0110283. the limit k → 0, the integral in (A.19) with respect to 16. E.Martinec, Nucl.Phys.B 281, 157 (1986). ´ t is equal to the sum of the residues of the integrand 17.G.S.Danilov, Pis’ma Zh. Eksp.Teor.Fiz. 58, 790 at the points z = u and z = v.Finally, the relation (1993) [JETP Lett. 58, 736 (1993)]. 18. M.A.Baranov and A.S.Shvartz, Pis’ma Zh. Eksp.´ for J1 in (26) can be obtained by substituting ex- (1) Teor.Fiz. 42, 340 (1985) [JETP Lett. 42, 419 (1985)]; pression (A.6) for J1 (t) into the integral in (A.17) D.Friedan, in Proceedings of the Santa Bar- (n) for Jr (t; {q}; L) and by subsequently calculating bara Workshop on Unified String Theories, Ed. the integral in the limit k → 0.The contour of in- by D.Gross and M.Green (World Sci.,Singapore, 1986). tegration with respect to t1 in this integral circum- vents nondegenerate handles.In just the same way 19. D.V.Volkov, A.A.Zheltukhin, and A.I.Pashnev, Yad.Fiz. 27, 243 (1978) [Sov.J.Nucl.Phys. 27, 131 as in the preceding cases, the integral is equal to the (1978)]. sum of the residues at the points z = u and z = v. 20. L.Crane and J.M.Rabin, Commun.Math.Phys. With allowance for formula (A.11), there arises the 113, 601 (1988); J.D.Cohn, Nucl.Phys.B 306, 239 required relation.In order to obtain formula (30), we (1988). use expression (A.11). For z → u, the regular part 21. 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Isaakyan PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1375–1383. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1416–1425. Original Russian Text Copyright c 2003 by Kaloshin, Radzhabov. ELEMENTARY PARTICLES AND FIELDS Theory Unitary Scalar–Vector Mixing in the ξ Gauge A.E.Kaloshin * and A.E.Radzhabov Institute of Applied Physics, Irkutsk State University, bul’v. Gagarina 20, Irkutsk, 664003 Russia Received February 14, 2002; in final form, August 16, 2002 Abstract—The effect of the unitary mixing of scalar and vector fields is considered in the ξ gauge.For this effect to emerge, it is necessary that the vector current not be conserved; in the ξ gauge, there arise additional complications because of the presence of an unphysical scalar field.Solutions to the Dyson – Schwinger equations are obtained, and the renormalization of complete propagators is investigated.The use of the Ward identity, which relates a few different Green’s functions, is a key point in performing this renormalization.It is shown that the dependence on the gauge parameter ξ disappears in the renormalized matrix element. c 2003 MAIK “Nauka/Interperiodica”. 1.INTRODUCTION ξ gauge, pursuing our analysis to the stage of deriv- The mixing of scalar and vector fields (S–V mix- ing a renormalized matrix element and investigating ing) arises at a loop level if there exists an off-diagonal the dependence on ξ.We consider both the case of loop that relates a scalar and a vector propagator to boson loops and the case of fermion loops because each other.For this, it is necessary that the relevant either case has its own special features.In particular, vector current not be conserved. our consideration is applicable to electroweak models 2) A similar effect was noticed long ago [1, 2] in involving an extended Higgs sector. studying the Standard Model in the ξ gauge, where In the ξ gauge, the Lagrangian is supplemented there arose the mixing of gauge-boson fields and an with the gauge-fixing term unphysical field—that is, the so-called Higgs ghost, 1 µ 2 2 2 1) L = − (∂ A ) . (1) whose propagator has a pole at the point p = ξM . gf 2ξ µ However, a physical scalar field may also partici- pate in mixing.By way of example, we indicate that, In taking into account loop contributions, there arises in [3], this effect was examined in the π–a1 system the mixing of three bare propagators (see Fig.1): the and that, in [4], the S–V mixing between gauge scalar-particle propagator, the vector-field propaga- bosons and Higgs particles was explored in various tor in the ξ gauge, and the ghost propagator. extended electroweak models.However, no attention It is convenient to break down the vector propa- was given in [4] either to the renormalization problem, gator into the transverse and the longitudinal compo- which is nontrivial in the case being considered, or nent; that is, to the problem of gauge dependence.We note that, 1 ξ in the ξ gauge, S–V mixing leads to an interesting µν µν µν π22 = T + L , (2) effect that was discovered in [5]: upon taking into p2 − M 2 ξM2 − p2 account Ward identities, the dressing procedure leads pµpν pµpν T µν = −gµν + ,Lµν = . to a transition to a different type of singularity for p2 p2 some propagators—namely, a simple pole of a bare propagator at the point p2 = ξM2 transforms into a We recall that only the longitudinal vector-propa- second-order pole of the corresponding total propa- gator component not associated with the spin of J = gator.In view of this, there naturally arises the ques- 1 can be mixed with a scalar field.For this e ffect to tion of whether the Standard Model is renormalizable emerge, it is necessary that the vector current with in principle in this gauge [5]. 2) In the present study, we examine the unitary mix- Loop transitions between massive gauge fields and Higgs scalars may arise only if the theory being considered features ingofaphysicalscalarfield and a vector field in the charged or pseudoscalar Higgs particles.It should also be recalled that, upon a spontaneous breakdown of symmetry, *e-mail: [email protected] currents remain conserved, but that currents with which vec- 1)Different terms are used for this field.In what follows, we will tor fields interact are not Noether currents; therefore, there refer to it merely as a ghost, nourishing the hope that this will is no contradiction between spontaneous symmetry breaking not lead to a terminological confusion. and the occurrence of scalar–vector transitions. 1063-7788/03/6607-1375$24.00 c 2003 MAIK “Nauka/Interperiodica” 1376 KALOSHIN, RADZHABOV π 1 11 = ------p2 Ð µ2 µν µν µ ν ()1 Ð ξ µνπ 1 Ðg + p p ------22 = ------2 2 p2 Ð M2 p Ð ξM π 1 33 = ------p2 Ð ξM2 Fig.1. Scalar-particle, vector-particle, and ghost propagators. which this vector field interacts not be conserved. contributions.In these equations, we will break down In the longitudinal component of the vector-particle each quantity carrying two indices into a transverse propagator, there appeared an unphysical pole at the and a longitudinal component; that is, 2 2 point p = ξM , but it was noticed long ago (see, for µν Π = T µν ΠT (p2)+Lµν ΠL (p2), (4) example, [6]) that the sign of this pole is opposite with 22 22 22 µν µν T 2 µν L 2 respect to the scalar-meson contribution.In order π22 = T π22(p )+L π22(p ), to cancel this pole at the tree level, it is therefore µν J = T µν JT (p2)+Lµν JL (p2), sufficient to introduce an unphysical scalar field of 22 22 22 mass ξM2.As a result, there arises, at a loop level, the where mixing of the aforementioned three propagators; in pµpν pµpν T µν = −gµν + ,Lµν = . addition, there appear total off-diagonal propagators, p2 p2 which were absent at the tree level.Upon dressing, there therefore arise the problem of renormalizing In the quantities carrying one index, we go over to coupled propagators and the problem of the gauge- scalar functions according to the relations parameter (in)dependence of the relevant S matrix. µ µ 2 µ µ 2 Π12(p)=p Π12(p ), Π21(p)=p Π21(p ), (5) µ µ 2 µ µ 2 J12(p)=p J12(p ),J21(p)=p J21(p ), 2.SET OF DYSON –SCHWINGER EQUATION µ µ 2 µ µ 2 Π23(p)=p Π23(p ), Π32(p)=p Π32(p ), In the case of mixing, propagators and loops Jµ (p)=pµJ (p2),Jµ (p)=pµJ (p2). become matrices; as a result, the set of Dyson– 23 23 32 32 Schwinger equations takes the form3) The equation for the transverse component is sep- − − µ µ − arated from the set of equations in question.As a Π11 = π11 Π11J11π11 Π12J21π11 Π13J31π11, (3) result, it assumes the same form as in the absence of scalar–vector mixing; that is, µ ν νµ γ γν νµ ν νµ Π = −Π11J π − Π J π − Π13J π , 12 12 22 12 22 22 32 22 ΠT = πT − ΠT JT πT . (6) − − µ µ − 22 22 22 22 22 Π13 = Π11J13π33 Π12J23π33 Π13J33π33, µ µ µγ γ µ A solution to this equation can be written in the form Π = −Π J11π11 − Π J π11 − Π J31π11, 21 21 22 21 23 1 Πµν = πµν − Πµ Jγ πγν − Πµγ Jγρπρν − Πµ Jγ πγν, ΠT = . (7) 22 22 21 12 22 22 22 22 23 32 22 22 p2 − M 2 + JT µ − µ − µγ γ − µ 22 Π23 = Π21J13π33 Π22 J23π33 Π23J33π33, µ µ As to the longitudinal components, the following Π31 = −Π31J11π11 − Π J π11 − Π33J31π11, 32 21 ξ 4) µ − ν νµ − γ γν νµ − ν νµ set of equations arises for them in the gauge: Π32 = Π31J12π22 Π32J22 π22 Π33J32π22 , Π11 = π11 − Π11J11π11 Π = π − Π J π − Πµ Jµ π − Π J π , 33 33 31 13 33 32 23 33 33 33 33 2 −p Π12J21π11 − Π13J31π11, (8) where πij are bare propagators, Πij are total prop- Π12 = −Π11J12π22 − Π12J22π22 − Π13J32π22, agators, and Jij are single-particle-irreducible loop Π21 = −Π21J11π11 − Π22J21π11 − Π23J31π11, 3) In relation to the analysis of the unitary gauge in [3], we Π = π − p2Π J π redefined here the off-diagonal scalar–vector propagators 22 22 21 12 22 → → 2 as iΠ12 Π12 and iΠ21 Π21 in order to obtain a more −Π22J22π22 − p Π23J32π22, symmetric form of the equations in question.In addition, we 2 do not assume that off-diadonal transitions possess some Π13 = −Π11J13π33 − p Π12J23π33 − Π13J33π33, specific symmetry or asymmetry properties—the form of − − 2 − symmetry relations is dictated by the form of interaction, Π31 = Π31J11π11 p Π32J21π11 Π33J31π11, so that these relations may be different in different cases. We also note that a change in the form of the equations 4)Below, we use longitudinal components, suppressing the corresponds to redefining the loops involved. superscript L, unless otherwise stated. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 UNITARY SCALAR–VECTOR MIXING 1377 − −1 Π23 = −Π21J13π33 − Π22J23π33 − Π23J33π33, (π33 + J33)sJ21J12 + sJ12J23J31 + sJ32J21J13. − − − Π32 = Π31J12π22 Π32J22π22 Π33J32π22, We note that the transverse and the longitudinal com- 2 Π33 = π33 − Π31J13π33 − p Π32J23π33 ponent of the vector-particle propagator are not com- pletely independent.For the relevant matrix element −Π J π . 33 33 33 to be free from a 1/p2 pole, fulfillment of the condition T L A solution to this set of equations has the form J22(0) + J22(0) = 0 is necessary. 1 Π = (π−1 + J )(π−1 + J ) − sJ J , 11 D 22 22 33 33 23 32 3.BOSON LOOPS: π–a SYSTEM 1 1 − Π = − J (π 1 + J ) − J J , , (9) We will examine the model identical to that which 12 D 12 33 33 13 32 was studied in the unitary gauge in [3]—specifically, 1 − − 1 − we consider the π–a1 system dressed by a πσ inter- Π21 = J21(π33 + J33) J23J31 , D mediate state.5) 1 Π = (π−1 + J )(π−1 + J ) − J J , The Feynman rules for this model are illustrated in 22 D 11 11 33 33 31 13 Fig.2. − 1 −1 − Π13 = J13(π22 + J22) sJ12J23 , For unphysical poles to be canceled in the matrix D element at the tree level, it is necessary that 1 − −1 − g Π31 = J31(π22 + J22) sJ32J21 , aπσ D gGπσ = . (11) M 1 Π = − J (π−1 + J ) − J J , 23 D 23 11 11 21 13 We now present results obtained by calculating 1 − the relevant self-energy parts at the one-loop level. Π = − J (π 1 + J ) − J J , 32 D 32 11 11 31 12 We note that, upon specifying the Feynman rules, loops must be determined consistently with Dyson– 1 −1 −1 − Π33 = (π11 + J11)(π22 + J22) sJ21J12 , Schwinger equations.With respect to the unitary D gauge, there appear new contributions that relate the where s = p2 and ghost propagator to other particles.Loops are calcu- − − − lated by means of a procedure in which the application D(s)=(π 1 + J )(π 1 + J )(π 1 + J ) (10) 11 11 22 22 33 33 of a unitary cut is followed by the restoration of an − −1 − −1 (π11 + J11)sJ32J23 (π22 + J22)J31J13 integral over the discontinuity; that is, d4l 1 J (p2)=−ig2 , (12) 11 σππ (2π)4 (l2 − µ2)((l − p)2 − m2) d4l (2l − p)µ Jµ (p)=− g g , 12 a1πσ σππ (2π)4 (l2 − µ2)((l − p)2 − m2) g g d4l (p · (2l − p)) J (p2)=−i ππσ a1πσ , 13 M (2π)4 (l2 − µ2)((l − p)2 − m2) d4l (2l − p)µ(2l − p)ν Jµν (p)=−ig2 , 22 a1πσ (2π)4 (l2 − µ2)((l − p)2 − m2) g2 d4l (2l − p)µ(p · (2l − p)) Jµ (p)= a1πσ , 23 M (2π)4 (l2 − µ2)((l − p)2 − m2) g2 d4l (p · (2l − p))(p · (2l − p)) J (p2)=−i a1πσ , 33 M 2 (2π)4 (l2 − µ2)((l − p)2 − m2) where m = mσ and µ = mπ. J31 = J13. The Feynman rules lead to the following symmetry In just the same way as in the case of the unitary relations for off-diagonal loops: µ µ 5) J = −J , (13) We recall that π–a1 transitions at the tree level arise in chiral 21 12 models [7, 8]; therefore, their emergence at a loop level is µ − µ J32 = J23, quite natural. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1378 KALOSHIN, RADZHABOV k k igππσ igππσ p q q p k µ k µ –gAπσ(k–q) gAπσ(k–q) p q q p k µ µ k µ µ ig πσ(k–q) p ig πσ(k–q) p q H q H p p Fig.2. Vertices for the π–a1 model. gauge in [3], all loops are expressed in terms of one (ii) The total pion propagator Π11 must have a pole function H(p2), apart from subtraction polynomi- with a residue equal to unity, in just the way as the als to be determined in the renormalization process. bare pion propagator π11 does have such a pole.This Specifically, we have means the sum of all loop insertions into the external pion line is equal to zero. 1 ds λ(s, m2,µ2) 1/2 H(p2)= , (14) These requirements lead to conditions on loops at − 2 2 π s(s p ) s the point s = µ2—that is, on subtraction polynomi- als: J (s)=g2[P + sH(s)], (15) 11 1 11 2 2 J11(µ )=J11(µ )=0, (17) J12(s)=−ig1g2[P12 + H(s)], 2 2 g g J12(µ )=0,J13(µ )=0. J (s)= 1 2 [P + sH(s)], 13 M 13 2 Renormalization of ξ. Since the mass of a vector 2 g2 particle is renormalized in the transverse part of the J22(s)=g2[P22 + H(s)],J23(s)=i [P23 + H(s)], M vector-particle propagator, the quantity M can be g2 treated as a mass that has already been renormalized. J (s)= 2 [P + sH(s)], 33 M 2 33 Therefore, the renormalization of the unphysical pole at the point s = ξM2 is the renormalization of the where Pij are polynomials in s with real-valued coef- gauge parameter ξ. ficients.We have√ also introduced the following√ nota- 2 − 2 We will try to follow a line of reasoning similar tion: g1 = gσππ/ 16π, g2 =(µ m )ga1πσ/ 16π, and λ(a, b, c)=(a − b − c)2 − 4bc. to that in the case of the pion pole and formulate → the following requirements on the renormalization in For total propagators, the πσ πσ matrix ele- 6) ment has the form question: 1 g g (i) The function D(s) must possess a second- MJ=0 = −g2Π − 2ig g Π − 2 1 2 Π 16π 1 11 1 2 12 M 13 order pole at the point s = ξM2 for any values of the 2 2 2 coupling constants. g2 g2 g2 +2i Π23 − Π22 − Π33. (16) M s M 2 (ii) At this point, the total propagators Π22 and Π33 must have a simple pole. Renormalization of the pion pole. We will per- form renormalization by means of on-shell subtrac- By using the explicit form of the relevant solutions, tions.Needless to say, the procedure is somewhat we find that this leads to the conditions 2 2 2 modified with allowance for the mixing of the prop- J22(ξM )=J33(ξM )=J12(ξM ) (18) agators.The most economical formulation of the re- 2 2 quirements on pion-pole renormalization is as fol- = J13(ξM )=J23(ξM )=0. lows: 6) (i) The function D(s) must possess a simple zero These requirements are minimal, implying that the charac- 2 ter of the singularity remains unchanged upon dressing the at the point s = µ for any values of the coupling bare propagators.Since a ghost appears only in the form constants g1 and g2, which are assumed to be inde- of propagators—and not in the form of external legs—we pendent. impose no conditions on the residues of the propagators. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 UNITARY SCALAR–VECTOR MIXING 1379 It can easily be seen that, among total propaga- With allowance for (18), the polynomial in the loop L tors, only Π23,apartfromΠ22 and Π33,canhave J22 must have the form a pole at the point s = ξM2.In order to trace the s s unphysical pole in the matrix element, it is therefore P = E 1 − − H(ξM2), (22) sufficient to consider only these contributions: 22 ξM2 ξM2 2 1 Mˆ J=0 −g2 2 where E is a constant that is determined in the trans- = 2 Π22(p ) (19) 16π p verse part of the loop J22.We can now write renor- 2 2 malized loops that satisfy the condition in (20): g2 2 g2 2 − Π33(p )+2i Π23(p ). M 2 M s J = g2 E 1 − (23) Substituting solutions to the Dyson–Schwinger 22 2 ξM2 equations and using conditions (18), we find that, s for an unphysical pole to be absent from the matrix − H(ξM2)+H(s) , element, it is necessary that the function Y (s), ξM2 2 g2 Y (s)=M J33(s)+sJ22(s)+2iMsJ23(s), (20) J = i 2 H(ξM2) − H(s) , 23 M have a second-order zero at the point ξM2.This g2 imposes constraints on the subtraction polynomials 2 J33 = 2 since, from (15), it can be seen that the loop function M H(s) cancels in (20). s × −ξM2H(ξM2)− EξM2 1− + sH(s) . We recall that the absence of a 1/p2 pole relates ξM2 JT to JL : 22 22 The remaining loops, which are not involved in the T L J22(0) + J22(0) = 0. (21) function Y (s) (20), are given by J = g2 −sH(µ2) − µ2H (µ2)(s − µ2)+sH(s) , (24) 11 1 ξM2H(µ2) − µ2H(ξM2) H(ξM2) − H(µ2) J = −ig g + s + H(s) , 12 1 2 µ2 − ξM2 µ2 − ξM2 g g H(ξM2) − H(µ2) ξM2H(ξM2) − µ2H(µ2) J = 1 2 µ2ξM2 − − sH(s) . 13 M µ2 − ξM2 µ2 − ξM2 Having determined the subtraction polynomials in 1 2 1 2 1 2 2 − (∂A) + (∂µϕ) − ξM ϕ loops, we can now calculate the matrix element (16). 2ξ 2 2 Upon the substitution of total propagators, we arrive 1 + A Jµ + ∂ ϕJ µ, at a cumbersome expression in which there remains µ M µ a dependence on the gauge parameter ξ.This means µ that an unphysical pole cannot be renormalized by a where ϕ is the ghost field and J isavectorcurrent method similar to that applied to the physical pole, so whose explicit form is immaterial here.We have pre- that it is necessary to seek some other ways. sented here only terms involving the vector field and the ghost field.The equations of motion have the form Ward identity. TheuseofaWardidentitythat relates a few different total propagators is a key point 1 (∂ ∂α + M 2)A − 1 − ∂ (∂A)=−J , (26) in renormalizing ξ.This identity was derived in [5] by α µ ξ µ µ means of the Bechi–Rouet–Stora–Tyutin (BRST) transformation in a somewhat simpler situation where α 2 1 there is no a physical scalar.To test this derivation, we (∂α∂ + ξM )ϕ = − (∂J). (27) M will obtain the Ward identity by a different method. The Feynman rules presented above correspond to From these two equations, it follows that 7) α 2 the Lagrangian (∂α∂ + ξM )((∂A) − ξMϕ)=0, (28) 1 2 1 2 µ L = − (∂µAν − ∂ν Aµ) + M AµA (25) which means that the combination (∂A) − ξMϕ, 4 2 which appeared in (28), is a noninteracting field.If 7)We do not explicitly write isotopic indices here because they one examines the vacuum expectation value of the are trivial in the model being considered. time-ordered product of these fields, such a Green’s PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1380 KALOSHIN, RADZHABOV m1 m1 ig1I ig1I p m2 m2 p m m µ 1 γµ γ5 1 µ γµ γ5 ig2 (1 + ) ig2 (1 + ) p m2 m2 p m m µ 1 γµ γ5 µ 1 µ γµ γ5 µ ig2 (1 + )(–ip ) ig2 (1 + )(+ip ) ppm2 m2 Fig.3. Vertices for the W ±–H ± model. − 2 2 2 2 function must not change upon switching on the 2isMJ12J13 M J13 + s J12 =0, interaction.Calculating it for the respective bare −J J2 + sJ J2 +2sJ J J =0. quantities, we find that the result vanishes: 22 13 33 12 12 13 23 0|T {((∂A(x)) − ξMϕ(x))((∂A(y)) (29) Solving these equations, we arrive at simple relations between the loops: − ξMϕ(y))}|0 =0. s J = J , (33) We note that, in order to go over in this expression 33 M 2 22 to the propagators, it is necessary to differentiate i s the time-ordered product of vector fields accurately. J23 = J22,J13 = i J12. M M The point is that this operation generates additional terms that are proportional to single-time commuta- We note that, in (32), there appeared the same func- tors of interacting fields.It is well known, however, tion Y (s) (20), which ensures the absence of unphys- that single-time commutation relations for interact- ical poles in the matrix element. ing fields commute with those for the corresponding Let us now calculate the total propagators by free fields (see, for example, [9]).Upon performing the using relations (32), which follow from the Ward required calculations, we arrive at identity.It immediately turns out that, in the function ∂ ∂ | { }| D(s), the dependence on the gauge parameter factors 0 T Aµ(x)Aν (y) 0 (30) 9) ∂xµ ∂yν out: = 0|T {∂A(x)∂A(y)}|0 −iξδ4(x − y). (s − ξM2)2 D(s)=− Dˆ(s), (34) ξM2 As a result, we find that, in the momentum represen- tation, the required Ward identity in terms of the total where there appeared the function 8) propagators assumes the form −1 2 2 Dˆ(s)=(π + J11)(M + J22)+s(J12) , (35) L − 2 2 11 sΠ22(s) 2isξMΠ23(s)+ξ M Π33(s)+ξ =0. (31) which plays the role of the determinant in the unitary gauge [3]. Substituting the explicit form (9) of the total prop- The total propagators assume a rather simple agators into the Ward identity (31) and equating to form; that is, zero the coefficients of the identical powers of the 2 coupling constants g1 and g2, we obtain the following M + J22 Π11 = , (36) set of relations: Dˆ 2 2 M J33 + sJ22 +2isMJ23 =0, (32) ξM J12 MsJ12 Π12 = Π13 = −i , 2 2 2 J22J33 + s(J23) =0, (s − ξM )Dˆ (s − ξM )Dˆ 8)We note that, in the literature, there is no consensus on the 9)Upon dressing the propagator, the pole lying above the exact form of this relation.By way of example, we indicate threshold usually goes to the complex plane.In the presence that, in [10], it is written without the term ξ, although the oftheWardidentity,however,thepoleremainshereonthe corollaries from it coincide with our Eqs.(33). real axis. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 UNITARY SCALAR–VECTOR MIXING 1381 −1 2 ξM (π11 + J11)J22 + s(J12) Π23 = i , m1 m1 (s − ξM2)2Dˆ γ5 γ5 m ig1 m ig1 pp2 2 Π22 −1 2 2 2 2 (π +J11)(M (s−ξM )+sJ22)+s (J12) = −ξ 11 , Fig.4. Vertices for Z0–pseudoscalar Higgs boson model. (s−ξM2)2Dˆ Π 33 The expressions for the loops are given by −1 − 2 − − 2 2 (π11 + J11)((s ξM ) ξJ22) ξs(J12) 4 = M . 2 − 2 d l (s − ξM2)2Dˆ J11(p )= ig1 4 (39) (2π) Substituting the total propagators into the matrix × 1 1 → tr , element πσ πσ (16), we obtain ˆl − pˆ − m2 ˆl − m1 1 M 2 + J MJ=0 = −g2 22 (37) d4l 16π 1 ˆ Jµ (p)=−ig g D 12 1 2 (2π)4 −1 J12 − 2 π11 + J11 +2ig1g2 g2 . 1 µ 5 1 Dˆ Dˆ × tr γ (1 + γ ) , ˆl − pˆ − m2 ˆl − m1 The dependence on the gauge parameter ξ disap- peared, and this expression coincides with the matrix g g d4l J (p2)= 1 2 element in the unitary gauge [3] if one imposes no 13 M (2π)4 conditions on the loop J11, J12,andJ22 at the point 2 1 1 s = ξM . × tr pˆ(1 + γ5) , ˆ− − ˆ− Finally, we note that, if the terms in the matrix l pˆ m2 l m1 element (16) are arranged as 4 µν − 2 d l 1 i J22 (p)= ig2 MJ=0 = −g2Π − 2ig g Π − Π (2π)4 16π 1 11 1 2 12 M 13 µ 5 1 ν 5 1 1 1 2i × tr γ (1 + γ ) γ (1 + γ ) , − 2 − ˆl − pˆ − m ˆl − m g2 Π22 + 2 Π33 Π23 , (38) 2 1 s M M 2 4 one can easily notice that, not only is the entire sum µ g2 d l J23(p)= 4 independent of the gauge parameter ξ, but also each M (2π) of the above three terms individually possesses this µ 5 1 5 1 property. × tr γ (1 + γ ) pˆ(1 + γ ) , ˆl − pˆ − m2 ˆl − m1 2 4 4.FERMION LOOPS: MIXING OF GAUGE − g2 d l J33(p)= 2 4 BOSONS AND HIGGS PARTICLES M (2π) IN EXTENDED MODELS 1 1 × tr pˆ(1 + γ5) pˆ(1 + γ5) . We recall that the unitary mixing of gauge bosons ˆl − pˆ − m2 ˆl − m1 with Higgs particles is possible only in extended elec- troweak models since either pseudoscalar or charged The symmetry properties differ somewhat from those Higgs particles are required for this.We will not spec- of boson loops; that is, µ µ ify the choice of model—it is sufficient to fix the form J21(p)=J12(p), (40) of vertices.For the sake of simplicity, we assume that J (p)=−J (p),Jµ (p)=−Jµ (p). only one physical scalar participates in mixing. 31 13 32 23 W ±–H± mixing. The interaction vertices are In the case of fermion loops, all longitudinal loops 2 displayed in Fig.3. 10) are expressed in terms of two functions H1(p ) and 2 H2(p ), apart from subtraction polynomials.Speci fi- 10)The longitudinal component of the W ± field is mixed with cally, we have the field of the charged Higgs particle.As will be seen from 2 − Eq.(41), which is given below, o ff-diagonal transitions van- 2 1 (m1 + m2) s H1(p )= 2 ish for the neutral Higgs field (m1 = m2). π s(s − p ) PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 1382 KALOSHIN, RADZHABOV 1/2 λ(s, m2,m2) − − i 5 · × 1 2 ds, g1g2 Π12 Π13 v(q2)γ u(q1) u(k1) s2 M p2 (m − m )2 − s(m2 + m2) × 5 − i H (p2)= 1 2 1 2 pˆ(1 + γ )v(k2) g1g2 Π21 + Π31 v(q2) 2 π s2(s − p2) M 5 5 2 2 1/2 × pˆ(1 + γ )u(q1) · u(k1)γ v(k2) λ(s, m1,m2) × ds, 1 1 i i s2 − g2 Π + Π − Π + Π v(q ) 2 s 22 M 2 33 M 23 M 32 2 2 J11 = f1 [P11 + sH1(s)] ,J12 = f1f2 [P12 + H1(s)] , × pˆ(1 + γ5)u(q ) · u(k )ˆp(1 + γ5)v(k ). (41) 1 1 2 In relation to the case of the scalar Higgs bosons, f1f2 2 J13 = i [P13 + sH1(s)] ,J22 = fˆ [P22 + H2(s)] , some of the loops changed: M 2 4 fˆ2 fˆ2 2 2 d l 2 2 J11(p )=−ig (44) J23 = i [P23 + H2(s)] ,J33 = [P33 + sH2(s)] , 1 4 M M 2 (2π) 1 1 where Pij are polynomials in s with real-valued coeffi- × tr γ5 γ5 , ˆ ˆ cients.We√ have also used the following√ notation:√ f1 = l − pˆ − m2 l − m1 g1/ 8π, f2 =(m1 − m2)g2/ 8π, fˆ2 = g2/ 4π. 4 Examining relations (33), which follow from the µ − d l J12(p)= ig1g2 4 Ward identity, we can easily see that the Ward identity (2π) imposes constraints only on the subtraction polyno- 1 1 mials, the loop integrals H and H identically sat- × tr γ5 γµ(1 + γ5) , 1 2 ˆ− − ˆ− isfying Eqs.(33).Upon imposing the conditions in l pˆ m2 l m1 Eqs.(33), we arrive at a very simple dependence of 4 the propagators on ξ in the form (36).It only remains 2 g1g2 d l J13(p )= 4 to trace the ξ dependence in the matrix element. M (2π) → The f1(q1)f 2(q2) f1(k1)f 2(k2) matrix element × 5 1 5 1 has the form tr γ pˆ(1 + γ ) , ˆl − pˆ − m2 ˆl − m1 MJ=0 − 2 · = g1Π11 v(q2)u(q1) u(k1)v(k2) (42) µ µ 2 − 2 i J21(p)=J12(p),J31(p )= J13(p ). (45) − g g Π − Π v(q )u(q ) · u(k ) 1 2 12 M 13 2 1 1 In relation to the mixing of charged fields, the only change is that the function H (p2) took a different × 5 − i 1 pˆ(1 + γ )v(k2) g1g2 Π21 + Π31 v(q2) form; that is, M − 2 − × 5 · 2 1 (m1 m2) s pˆ(1 + γ )u(q1) u(k1)v(k2) H (p )= 1 π s(s − p2) 2 1 1 i i − g Π + Π − Π + Π v(q ) 1/2 2 22 2 33 23 32 2 λ(s, m2,m2) s M M M × 1 2 ds, 5 5 s2 × pˆ(1 + γ )u(q1) · u(k1)ˆp(1 + γ )v(k2). This expression can be simplified by using the equa- 2 J11 = f1 [P11 + sH1(s)] ,J12 = f1f2 [P12 + H1(s)] , tions of motion for spinors; however, it can be seen f f even at this stage that different spinor matrix ele- J = i 1 2 [P + sH (s)] , 13 M 13 1 ments appear with factors that are independent of ξ √ [see Eq.(38)].Thus, the dependence on the gauge where f2 =(m1 + m2)g2/ 8π. parameter disappeared. The renormalized matrix element for the transition Mixing in the system formed by the pseu- f1(q1)f (q2) → f1(k1)f (k2) assumes the form doscalar Higgs boson and the Z0 gauge boson. 2 2 2 The vertex of Higgs boson interaction with fermions MJ=0 − 2 (M + J22) = g1 v(q2) (46) is shown in Fig.4. Dˆ → The form of the f1(q1)f 2(q2) f1(k1)f 2(k2) ma- 5 5 J12 × γ u(q1) · u(k1)γ v(k2)+2g1g2 trix element changed slightly to become Dˆ MJ=0 − 2 5 · 5 × 5 · 5 = g1Π11v(q2)γ u(q1) u(k1)γ v(k2) (43) (v(q2)γ u(q1) u(k1)ˆp(1 + γ )v(k2) PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 UNITARY SCALAR–VECTOR MIXING 1383 5 5 + v(q2)ˆp(1 + γ )u(q1) · u(k1)γ v(k2)) fulfillment of relations (33) for the loops.If these rela- −1 tions are satisfied, the position of second-order poles − 2 (π11 + J11) 5 g2 v(q2)ˆp(1 + γ ) is ultraviolet stable [see Eqs.(36)].It can therefore be Dˆ conjectured that, in the course of the calculations per- 5 × u(q1) · u(k1)ˆp(1 + γ )v(k2). formed in [5], loops did not satisfy the Ward identity for some reason or another, although this identity was There are no other changes—in particular, the used there in a general form. renormalized matrix element remains independent of ξ. After our study, there arises the impression that the use of the ξ gauge is not very convenient in extended Higgs models.Nonetheless, this gauge (or 5.DISCUSSION particular cases of ξ) is extensively used in studying electroweak models.In particular, one can test the Thus,wehaveexamined,atthelooplevel,theef- ξ fect of scalar–vector mixing in the ξ gauge and found correctness of calculations by varying and tracing that, if one employs a Ward identity, the renormalized the respective variation of the matrix element. matrix element becomes independent of the gauge As to physical implications of unitary scalar– parameter.A complete disappearance of the depen- vector mixing in electroweak models, this issue dence on ξ is quite surprising, since we have studied deserves a dedicated investigation. only that part of Dyson–Schwinger equations which have bearing on the dressing of propagators.There REFERENCES are therefore reasons to conjecture that relations (33) 1. L.Baulieu and R.Coquereaux, Ann.Phys.(N.Y.) 140, are of a more general character. 163 (1982). A change in the character of the singularity upon 2.K.Aoki et al., Prog.Theor.Phys.Suppl. 73, 1 (1982). dressing is an ineresting phenomenon, which was 3. A.E.Kaloshin, Yad.Fiz. 60, 1306 (1997) [Phys.At. previously found in [5] for a simpler situation.Speci fi- Nucl. 60, 1179 (1997)]. cally, a simple pole 1/(p2 − ξM2) in bare propagators 4. M.C.Peyranere, Int.J.Mod.Phys.A 14, 429 (1999). transforms, upon dressing, into the second-order pole 5. Hung Cheng and S.P.Li,hep-ph/0107103. of total propagators.There is always such a possibility 6.J.C.Taylor, Gauge Theories of Weak Interac- in the mixing of two bare propagators corresponding tion (Cambridge Univ.Press, Cambridge, 1976; Mir, to equal masses, but it is realized only if loops satisfy Moscow, 1978). some specific relations that are dictated by the Ward 7. S.Gasiorowicz and D.A.Ge ffen, Rev.Mod.Phys. 41, identity. 531 (1969). Hung Cheng and S.P. Li [5] calculated bosonic 8.M.K.Volkov, Fiz. Elem.Chastits´ At.Yadra 17, 433 loop contributions and found that the position of the (1986) [Sov.J.Part.Nucl. 17, 186 (1986)]. second-order unphysical pole is ultraviolet-divergent 9. N.N.Bogolyubov and D.V.Shirkov, Introduction to with allowance for the Ward identity.In this connec- the Theory of Quantized Fields (Nauka, Moscow, tion, it was argued that, in the ξ gauge, the Standard 1976). Model is unrenormalizable in principle.11) As was 10. G.Passarino, Nucl.Phys.B 574, 451 (2000). shown above, however, the Ward identity leads to Translated by A. Isaakyan 11)The authors of [5] studied the mixing of the longitudinal part of W and a ghost in the Standard Model.This is a particular case of our consideration in the absence of a physical scalar. PHYSICS OF ATOMIC NUCLEI Vol.66 No.7 2003 Physics of Atomic Nuclei, Vol. 66, No. 7, 2003, pp. 1384–1394. Translated from Yadernaya Fizika, Vol. 66, No. 7, 2003, pp. 1426–1436. Original Russian Text Copyright c 2003 by Lantsman, Pervushin. ELEMENTARY PARTICLES AND FIELDS Theory Monopole Vacuum in Non-Abelian Theories L.D.Lantsman 1)* and V.N.Pervushin ** Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received May 10, 2002; in final form, October 17, 2002 Abstract—It is shown that, in the theory of interacting Yang–Mills fields and a Higgs field, there is a topological degeneracy of Bogomol’nyi–Prasad–Sommerfield (BPS) monopoles and that there arises, in this case, a chromoelectric monopole characterized by a new topological variable that describes transitions between topological states of the monopole in Minkowski space (in just the same way as an instanton describes such transitions in Euclidean space). The limit of an infinitely large mass of the Higgs field at a finite density of the Bogomol’nyi–Prasad–Sommerfield monopole is considered as a model of the stable vacuum in pure Yang–Mills theory. It is shown that, in QCD, such a monopole vacuum may lead to a growing potential, a topological confinement, and an additional mass of the η0 meson. The relationship between the result obtained here for the generating functional of perturbation theory and the Faddeev– Popov integral is discussed. c 2003 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION AND FORMULATION and to construct, in order to describe processes in OF THE PROBLEM terms of probability amplitudes normalized per time The problem of choosing, in a non-Abelian theory, and spatial-volume units, a generating functional for a physical vacuum and variables that adequately re- the S matrix in the class of functions such that the energy density is finite [5–7]. In QED, such functions flect the topological properties of the manifold of initial 1+m data for non-Abelian fields [1–3] in Minkowski space have an O(1/r ) behavior at spatial infinity. They is still considered to be one of the most important are referred to as monopoles if m =0and as multi- problems in these realms. There are reasons to believe poles if m>0. that solving this problem will contribute to obtaining, Solving differential equations in theoretical phy- within QCD, deeper insights into the nature of con- sics presumes specifying initial data. Such initial data finement, hadronization, and a spontaneous break- are measured by a set of physical instruments with down of scale invariance in the infrared region. which one associates a reference frame. In the present The present study is devoted to employing mono- study, we will consider reference frames that are de- (0) pole solutions [4] to the equations of a non-Abelian termined by the timelike unit vector lµ =(1, 0, 0, 0) theory to construct a model of a topologically in- (1) and various Lorentz transformations of it, lµ . variant vacuum of Yang–Mills theory in Minkowski There are two types of groups of transformations of space. The respective Lagrangian of the theory has the differential equations of a gauge theory. These are the form relativistic transformations, which change initial data 1 L = − Ga Gµν , (1) (that is, reference frame), and gauge transformations, 4 µν a ˆu ˆ −1 where Aµ(t, x):=u(t, x) Aµ + ∂µ u (t, x), (4) Ga = ∂ Aa − ∂ Aa + gε Ab Ac . (2) τ a µν µ ν ν µ abc µ ν Aˆ = g Aa , µ 2i µ From the mathematical point of view, the main prob- lem in the theory of gauge fields is to find general which are associated with the gauge of physical fields solutions to the equations of the theory, and which do not affect the readings of an instrument. ab µν ab ab c The set of Eqs. (3) is referred to as a relativistically Dµ Gb =0 (Dµ = δ ∂µ + gεacbAµ), (3) covariant set of equations if the total manifold of its solutions for each specific reference frame coincides 1)Permanent address: Wissenschaftliche Gesellschaft, bei with its counterpart for any other reference frame Judische Gemeinde zu Rostock, Wilhelm-Kulz¨ Platz, 18055 (see [7–12]). Rostock, Germany. *e-mail: [email protected] In each reference frame, the set of all equations is ** ab µi e-mail: [email protected] broken down into the equations of motion Dµ Gb = 1063-7788/03/6607-1384$24.00 c 2003 MAIK “Nauka/Interperiodica” MONOPOLE VACUUM 1385 0 (i =1, 2, 3)—to solve these equations, it is nec- must also belong to this class of monopole functions essary to measure initial data—and the constraint and have an O(1/r1+m), m>0 asymptotic behavior. ab µ0 equations Dµ Gb =0, which relate initial data for Thus, our objective is to quantize non-Abelian the spatial components of the fields involved to initial fields in the class of monopole functions that involves data for their time components. The time component a topological degeneracy. Such a quantization pre- of a field is singled out since it has no canonical sumes the choice of Dirac’s variables in which this momenta. In view of this, Dirac [13] and, after him, degeneracy occurs. The first Hamiltonian quantiza- other authors of the first classic studies devoted to tion of non-Abelian gauge theories in terms of Dirac’s quantizing gauge theories (see [14, 15]) eliminated variables without allowing for their topological de- the time component by a gauge transformation. In our generacy was due to Schwinger [10], who proved the case, such a transformation, relativistic covariance of the transverse gauge at the t level of commutation relations for the generators of ˆD ¯ˆ ¯ ˆ the algebra of the Poincare´ groupthat were con- Ak = v(x)T exp dtA0(t, x) Ak + ∂k (5) structed in the above class of functions. This Hamil- t0 tonian quantization of non-Abelian fields was repro- − t 1 duced by Faddeev [16], who employed the method of a path integral Z (0) explicitly dependent on a reference × ¯ˆ ¯ l v(x)T exp dtA0(t, x) frame. In [17], it was shown that the relativistic trans- t0 formation at the level of fundamental operator quan- tization according to Schwinger [10] corresponds to (here, the symbol Т denotes time ordering of the (0) → (1) matrices under the exponential sign), specifies a non- the relativistic transformation l l of the time Abelian analog of Dirac’s variables, apart from arbi- axis on which the path integral Zl(0) depends. The trary time-independent matrices v(x), which are con- dependence of this integral on a reference frame is 2) sidered as initial data at the instant t0 for solving the called an implicit relativistic covariance. −1 equation U(Aˆ0 + ∂0)U =0. At the level of Dirac’s That the path integral Zl(0) is independent of a ref- variables, Lorentz transformations of original fields erence frame for on-shell amplitudes of elementary- become nonlinear, while the groupof gauge trans- particle scattering was first discovered by Feyn- formations reduces to a groupof time-independent man [21] and was proven by Faddeev [16] as a transformations that specify the degeneracy of ini- validation of the heuristic Faddeev–Popov path inte- tial data for physical fields (including the classical gral [22]. This integral was proposed as a generating vacuum A0 = Ai =0,whichisdefined as the zero- functional of unitary perturbation theory for any energy state). By gauge fixing, one means, in this gauges, including those that are independent of a case, presetting initial data in perturbation theory as reference frame. Schwinger noticed that gauges that the transversality condition [9–12]. are independent of a reference frame may be physically In a non-Abelian theory, the set of time-indepen- inadequate to fundamental operator quantization; dent gauge transformations is a set of three-dimen- that is, they may distort the spectrum of the original 3) sional paths in the space of the SUc(2) groupthat are system. broken down into topological manifolds characterized In the present study, we verify Schwinger’s state- by integers (exponents of mapping): ment in a non-Abelian theory, answering the question concerning the spectrum of a theory quantized in the N − 1 3 ijk [n]= 2 d x (6) class of monopole functions that involves a topolog- 24π ical degeneracy of initial data and the question con- − − − (n) (n) 1 (n) (n) 1 (n) (n) 1 × tr v ∂iv v ∂jv v ∂kv = n. cerning the relationshipbetween fundamental quan- tization and the heuristic Faddeev–Popov integral in The exponent of mapping indicates how many times a gauge that is independent of a reference frame. the three-dimensional path v(x) goes about SUс(2) The ensuing exposition is organized as follows. as the coordinate xi runs over the entire three- Section 2 is devoted to describing the topological dimensional space where this coordinate is specified. degeneracy of known monopole solutions and to con- The condition in (6) means that the entire set of sidering zero modes of the constraint equation. In three-dimensional paths has the homotopic group Section 3, we examine the limiting transition to pure − (n) (n) 1 π(3)(SUc(2)) = Z and that all fields v ∂iv are 2)The choice of the time axis for such an integral is discussed defined in the class of functions for which the integral elsewhere [18–20]. in (6) is finite. This is the class of monopole func- 3)“We reject all Lorentz gauge formulations as unsuited to the D tions [1, 2]. Naturally, the fields Ai (t, x) themselves role of providing the fundamental operator quantization” [10]. PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1386 LANTSMAN, PERVUSHIN Yang–Mills theory having a monopole vacuum. In in the class of functions such that the energy density Section 4, we analyze the U(1) problem. In the Con- is finite. clusion, we discuss the connection with the Faddeev– In addition to the trivial vacuum in (9), there are Popov integral. monopole solutions in the system described by the Lagrangian density in (7). These are solutions having an O(1/r) type of behavior at spatial infinity: 2. TOPOLOGICAL DEGENERACY OF m BOGOMOL’NYI–PRASAD–SOMMERFIELD r →∞; ϕa − na √ = O(1/r),Aa = O(1/r). MONOPOLE λ i (10) Let us consider the well-known example of in- teracting Yang–Mills fields and a scalar Higgs field The Bogomol’nyi–Prasad–Sommerfield (BPS) so- for the case where there is a spontaneous breakdown lution [4] of symmetry. This situation is described by the La- xa 1 1 grangian density [4] ϕa = f BPS(r),fBPS(r)= − , gr 0 0 tanh(r/ ) r L −1 a µν (11) = Gµν Ga (7) 4 2 2 xk 1 a µ − λ m − 2 a ≡ aBPS BPS + (Dµϕ )(D ϕa) ϕ , Ai (t, x) Φi (x)=εiak 2 f1 (r), (12) 2 4 λ gr where BPS − r f1 = 1 , a a − a b c sinh(r/ ) Gµν = ∂µAν ∂νAµ + gεabcAµAν which was obtained in the limit is the strength tensor of our non-Abelian field and 1 gm ϕa, a =1, 2, 3,isascalarfield forming a triplet of λ → 0,m→ 0; ≡ √ =0 , (13) the adjoint representation of the SU(2) group. The λ potential energy depends on the square of the vector is a monopole solution satisfying the Yang–Mills ϕa, while the covariant derivative of the field has the equations, having a finite minimum energy, and vi- form olating SU(2) symmetry. In the limit specified in (13), a a b c the Bogomol’nyi–Prasad–Sommerfield solution has Dµϕ = ∂µϕ + gεabcAµϕ . (8) a finite minimum energy that is proportional to The Lagrangian density (7) possesses a manifest gm 4π gauge invariance under transformations of the SU(2) d3x[BaBa]=4π √ = , (14) group. i i g2 λ g2 The classical vacuum is defined as the asymptotic where Ba is the magnetic induction, solution i | |→∞ → g r = x ,EminE, Ba(ΦcBPS)=ε ∂ ΦaBPS + εabcΦbBPSΦcBPS . i k ijk j k 2 j k providing the minimum of the field energy E. 2 ≥ ≥ That this energy value corresponds to a minimum is For m 0 and λ 0, the vacuum loses the ensured by the requirement that the magnetic field in SU (2) symmetry of the Lagrangian; that is, question be of a potential character; that is, | |→∞ a → a √m r = x ,ϕn , Ba(ΦcBPS)=Dab(ΦcBPS)ϕ , (15) λ i k i k b with na being an arbitrary unit vector (|n| =1)in where the covariant derivative is specified by Eq. (8). isotopic space. Choosing a specific vacuum reduces It is precisely this condition (which ensures, as was to choosing a specific direction of the vector n,and indicated immediately above, the potential character this violates the symmetry of the SU(2) group. This of the magnetic field)—it is referred to as the Bo- phenomenon is referred to as a spontaneous break- gomol’nyi equation—that will play an important role down of symmetry. in our construction of the stable vacuum of a non- Abelian theory (see below). In the Standard Model, a non-Abelian vector field We will now show that the equation of potentiality develops a mass in precisely this way. Usually, quan- means a topological degeneracy of fields under the tum field theory is then constructed as a perturbation time-independent gauge transformations theory over this vacuum, −1 m Aˆ(n)(t , x)=v(n)(x) Aˆ(0)(t , x)+∂ v(n)(x) . ϕa = na √ , (9) i 0 i 0 i λ (16) PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 MONOPOLE VACUUM 1387 Dynamical fields can be represented in the form of solution in the form of the Bogomol’nyi–Prasad– the sum of the Bogomol’nyi–Prasad–Sommerfield Sommerfield monopole (11): BPS ¯(0) a a monopole Φi (x) and perturbations Ai : τ x 1 Φˆ 0 = −iπ − . (25) ˆ(0) ˆ BPS ¯ˆ(0) r tanh(r/ ) r Ai (t, x)=Φi (x)+Ai (t, x). (17) Thus, we have shown that the Bogomol’nyi–Prasad– Perturbations are considered as weak multipole Sommerfield monopole and transverse gauge physi- fields [23]: cal fields have Gribov’s replicas in the form of topo- 1 logical degeneracy (22). A¯ (t, x)| = O (l>1). (18) i asymp r1+l It should be recalled that a topological degeneracy is associated primarily with a classical vacuum of In the lowest order of perturbation theory, the equa- zero energy, where this degeneracy is characterized tion for the time component, by the Pontryagin index or by the Chern–Simons 2 ac c ac c D (A) A0 =[Di (A)∂0Ai ], (19) functional [1] (which we consider in a finite spacetime of volume V within the time interval tin PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1388 LANTSMAN, PERVUSHIN not commute with the Hamiltonian Hˆ .Oneofthe Apart from the factor N˙ (t), which plays the role of a simplest ways to remove all of these flaws, including new variable (that is, a zero mode), the topological- a the nonnormalizability of the wave function degeneracy phase Φ0(x) is the required solution Z to the homogeneous equation in the lowest order of Ψins[A]=exp{iPX X[A]} (30) perturbation theory: and the unphysical values of the energy and the mo- Zˆ ˙ ˆ mentum PX of the topological motion, (t, x)=N(t)Φ0(x). (39) 8π2 The solution to the homogeneous equation describes H =0,P= ±i , (31) X g2 an electric monopole, a Z ab BPS Zb ˙ ab BPS b consists in separating the topological motion from the Gi0( )=Di (Φ ) = N(t)Di (Φ )Φ0, (40) local variables via the introduction of an independent topological degree of freedom N(t) by means of a which cannot be completely eliminated from the ac- gauge transformation [25, 26]: tion functional of the theory or from the Pontryagin index by going over to Dirac’s variables with the aid of ˆ(N) ˆ ˆ(0) Ai =exp[N(t)Φ0(x)][Ai (32) the gauge transformation (32). As was shown in [27], the Pontryagin index + ∂i]exp[−N(t)Φˆ 0(x)]. tout By means of a direct calculation, it can be proven [27] g2 ν[A]= dt d3xGa G˜aµν (41) that, for the vacuum of Bogomol’nyi–Prasad–Som- 16π2 µν (n) merfield monopoles Φ , this degree of freedom is tin V i − completely separated from the local degrees of free- = N(tout) N(tin) dom that are specified in the class of multipole func- depends only on the difference of the final and initial tions: values of the topological degree of freedom. X[A(N)]=X[A(0)]+N(t). (33) i i In the lowest order of perturtion theory in the In this case, the instanton wave function (30) ac- coupling constant, the action functional of the the- quires a new independent degree of freedom, ory being considered contains, in addition to the Bogomol’nyi–Prasad–Sommerfield monopole, an Ψ [AN ]=exp{iP X[AN ]} (34) ins N electric monopole and describes the dynamics of the (0) new topological variable N(t) in the form of a free =exp iPN (X[A ]+N) , i rotor; that is, and describes the topological motion of this degree tout of freedom at physical values of the momentum PN . BPS 3 1 WZ [N,Φ ]= dt d x (42) An independent topological motion arises as the in- 2 evitable consequence of a general solution to the tin V µ0 equation DµG =0for the time component of the × b Z 2 − b bBPS 2 field [25, 26]. This equation has the form [Gi0( )] [Bi (Φk )] [D (A)]ab [D (A)]bc Ac = Dac(A)∂ Ac, (35) k k 0 i 0 i 1 4π = dt IN˙ 2 − , with the initial data being those that correspond to the 2 g2 Bogomol’nyi–Prasad–Sommerfield monopole: where c BPS ∂0Ai =0,Ai(t, x)=Φi (x). (36) 4π I = d3x(Dac(Φ )Φc)2 = (2π)2 (43) According to the theory of differential equations, a i k 0 g2 general solution to a nonhomogeneous equation can V be represented as the sum is√ the angular momentum of the rotor and = a Za ˜a A0 = + A0, (37) λ/gm is the size of the Bogomol’nyi–Prasad– Sommerfield monopole in terms of the canonical ˜a where A0 is a particular solution to the nonhomoge- ˙ a momentum PN = NI: neous equation being considered and Z isasolution 2 2 to the corresponding homogeneous equation 2π 2 g H = PN +1 . (44) (D2(A))abZb =0. (38) g2 8π2 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 MONOPOLE VACUUM 1389 Upon introducing new Dirac’s variables with the not a stable state. As a rule, uniform ([31]) or singular aid of the transformation in (32), the topological de- fields including instantons ([1]), the equation for generacy of all fields reduces, for such Dirac’s vari- which involves delta-function-like singularities in ables, to the degeneracy of only one topological vari- Euclidean space, are used for nonzero vacuum fields. able N(t) with respect to a change in this variable If one explains physical effectsbyauniform(orby by integers (N → N + n, n = ±1, ±2, ...).Thewave an instanton) vacuum, it is also necessary to explain function for the topological motion in Minkowski the emergence of capacitors at spatial infinity that space has the form of the free-rotor wave function generate uniform fields (or the origin of sources of delta-function-like singularities). Ψ [N]=exp{iPN N} , (45) mon Bogomol’nyi–Prasad–Sommerfield monopoles with the momentum spectrum being determined from provide a unique possibility for introducing, in the iθ the condition Ψmon[N +1]=e Ψmon[N]. The result class of regular functions associated with topologi- is cally nontrivial gauge transformations, vacuum fields P = NI˙ =2πk + θ, (46) in such a way that the equations of Yang–Mills theory N [Eqs. (1) in our case] do not developany additional where k is the number of a Brillouin zone and θ sources. is the angle that specifies the spectrum of physical In order to introduce such a monopole vacuum, values of the Hamiltonian in (44). This Hamilto- we include the interaction of gauge fields with a nian has zero eigenvalue (H =0) for the unphysical Higgs field in space of finite volume V = d3x. ± 2 2 momentum values of PN = i8π /g , at which the The scalar-field condensate forms a Bogomol’nyi– instanton wave function (34) coincides with the wave Prasad–Sommerfield monopole characterized by a function (45) for the monopole vacuum under the as- finite mass of the scalar field: sumption that the topological degree of freedom is de- 1 m g2 termined by a functional of local variables exclusively. = √ ≡ V B2 . (48) Thus, the basic distinction between the monopole λ 4π vacuum and the instanton vacuum is that, in the first If we go over to the infinite-volume limit V →∞ case, there arises an independent Goldstone mode under the condition that B2 is finite, the scalar associated with the spontaneous breakdown of sym- field acquires an infinitely large mass and disappears metry of physical states under the transformations of from the spectrum of physical excitations, while the π(3)(SU(2)) = Z homotopy group. the regular solution representing a Bogomol’nyi– Equation (46) and condition (15), which ensures Prasad–Sommerfield monopole smoothly transforms the potential character of the magnetic field, deter- into a Wu–Yang monopole [33]; this means that the mine the spectrum of the electric-field strength: equations of the theory do not develop, at the origin of coordinates, a singularity inherent in the Wu– b ˙ (0) b θ b (0) Gi0 = N[Di(Φ )A0] = αs + k Bi (Φ ). Yang monopole and that the energy density does not 2π go to infinity. In this limit, the finite energy density (47) of the Bogomol’nyi–Prasad–Sommerfield monopole has the form Expression (47) is an analog of the spectrum of the electric-field strength, G = e(θ/(2π)+k),intwo- 10 3 a 2 ≡ 2 dimensional electrodynamics [28–30] characterized d x[Bi ] V B , (49) by the same topology of degeneracy of initial data: π(1)(U(1)) = π(3)(SU(2)) = Z. where B2 is the quantity that one has for the order The vanishing of the topological momentum does not parameter of the physical vacuum of the gauge field imply that the degeneracy of physical states disap- upon the the elimination of the scalar field. pears. Physical implications of such a degeneraсy will We recall that the Wu–Yang monopole [33] is be considered in the next section. an exact solution to the classical equations of pure Yang–Mills theory, Eqs. (3), everywhere, with the exception of a small vicinity of the origin of coordi- 3. YANG–MILLS THEORY FEATURING nates. It is precisely in this small region around the TOPOLOGICAL DEGENERACY origin of coordinates that the Wu–Yang monopole is OF PHYSICAL STATES regularized by the scalar-field mass, and this region It is well known that perturbation theory con- disappears in the infinite-volume limit V →∞[see structed for non-Abelian fields by analogy with Eq. (49)]. It should be noted here that, in quantum QCD [10, 16] is infrared-unstable [31, 32]. A con- field theory, a transition to the limit V →∞is per- ventional vacuum of perturbation theory, А =0,is formed upon calculating physical observables, such PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1390 LANTSMAN, PERVUSHIN as scattering cross sections and decay probabilities, Reproducing the calculations performed in [16] (see that are normalized per unit time and per unit vol- also [17]), one can obtain, as the generating func- ume. Therefore, all special features of the above the- tional for such a perturbation theory, a Feynman path ory involving a Bogomol’nyi–Prasad–Sommerfield integral in a reference frame with a specified time axis, monopole, which include a topological degeneracy of lµ =(1, 0, 0, 0);thatis, initial data and an electric monopole, survive at any 3 finite value of the volume. Z∗[l, J∗]= dN(t) [d2A∗cd2E∗c] On the other hand, there are, in the Yang–Mills t theory specified by the Lagrangian density (1), direct c=1 indications that the scale symmetry of the vacuum is (54) broken by solutions belonging to the type of a Wu– ∗ ∗ ∗ ∗ ∗ × exp{iW [N,A ,E ]+i d4xJ A }, Yang monopole [33]. In particular, the topological classification of classical solutions to pure Yang– ∗ ∗ Mills theory specifies the class of solutions that pos- where A are Dirac’s variables (32); E are their canonically conjugate momenta; J∗ are their sources; sess zero topological index (n =0 ), ∗ ∗ ∗ and W [N,A ,E ] is the original action functional (0) δX[A] taken on the manifold spanned by solutions of the X[A =Φ ]=0, c =0, (50) δAi A=Φ(0) constraint equation and which have the form δW ⇒ cd d a k =0 Di (A)G0i =0 (55) τ x δA0 Φˆ (0) = −i ε f(r), (51) i 2 iak r2 d for the non-Abelian electric-field strength G0i rep- where there is only one unknown function, f(r).An resented in the form of the sum of the transverse momentum E∗ and the longitudinal component: equation for this function can be obtained by substi- tuting expression (51) into the classical Eq. (3): d ∗d db b cd N(WY) ∗d G0i = Ei + Di (Φ)σ Di (Φ )Ei =0 . d2f f(f 2 − 1) Dab(Φ(0))Gkj(Φ(0))=0⇒ + =0. (56) k i b i dr2 r2 (52) If one assumes that, in perturbation theory, indepen- ∗d dN(WY) ¯∗d In the region r =0 , there exist the following three dent Dirac’s variables Ai =Φi + Ai given solutions to this equation: by (32) satisfy the gauge conditions PT WY ± cd N(WY) ∗d f1 =0,f1 = 1 . (53) Di (Φ )Ai =0, (57) PT The first, trivial, solution f1 =0corresponds to or- which were considered above, then the constraint dinary unstable perturbation theory involving Eq. (55) reduces to an equation for the function σb; “asymptotic freedom” [31, 32]. The two nontrivial that is, solutions f WY = ±1 represent Wu–Yang monopoles, cd ∗ db N(WY) b c 1 Di (A )Di (Φ )σ = j0, (58) which, in the model being considered, emerge from Bogomol’nyi–Prasad–Sommerfield monopoles in where the quantity on the right-hand side is the cur- the infinite-volume model without their singularities, rent of independent non-Abelian variables, ∗ ∗ along with the Goldstone model accompanying the ja = gεabc[A b − ΦbN(WY)]E c. (59) breakdown of scale invariance. 0 i i i Thus, the monopole vacuum characterized by a One can solve Eq. (58), which involves a zero mode topological degeneracy of all physical states has the (described above), by means of perturbation theory, following features distinguishing it from the topo- employing a Green’s function of the Coulomb type. In logically degenerate instanton vacuum: Minkowski the lowest order of perturbation theory, this Green’s space; a topological Goldstone mode associated with function Cbc(x, y) in the field of the usual Wu–Yang scale-symmetry breaking that generates a nonzero monopole ΦWY is determined by the equation order parameter B2=0 ; and a clear physical ori- [D2(ΦWY)]ab(x)Cbc(x, y)=δacδ3(x − y). (60) gin of scale-symmetry breaking, which is due to the condensate of a scalar Higgs field and which survives A solution to this equation specifies, in the Hamil- upon the elimination of the scalar field. tonian, an instantaneous interaction of non-Abelian Physical implications of the theory being consid- currents, ered, which involves a monopole vacuum, are con- 1 − d3xd3yjb(x)Cbc(x, y)jc(y), (61) trolled by the generating functional for unitary per- 2 0 0 turbation theory in the covariant Coulomb gauge. V0 PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 MONOPOLE VACUUM 1391 as an analog of the Coulomb interaction of the cur- Growing potentials do not remove poles of Green’s rents in QED. A solution to Eq. (60) in the presence functions in perturbation theory for amplitudes of pro- of a Wu–Yang monopole, where cesses not involving color degrees of freedom. Pertur- 2 WY ab ab bation theory is formulated in terms of fields that are [D (Φ )] (x)=δ ∆ characterized by zero topological quantum numbers nanb + δab n n and which can be called partons: − +2 a ∂ − b ∂ , 2 b a ˆ∗ | (0) ˆ(0) −1 r r r A (N A )=UN [A + ∂]UN . na(x)=xa/r, r = |x|, was obtained in [27] by means By virtue of gauge invariance, the phase factors of ab of an expansion of C in terms of a complete set of topological degeneracy, UN =exp{N(t)Φˆ 0(x)},dis- orthogonal vectors; that is, appear. However, these factors survive at the sources of physical fields in the generating functional (54). Cab(x, y)=[na(x)nb(y)V (z) 0 A theory featuring a topological degeneracy of initial a b data, where the sources of physical fields involve the + e (x)e (y)V1(z)],z= |x − y|, α α Gribov factors α=1,2 − ˆ (n) ¯ˆ(0) (n) 1 tr[Jiv Ai v ], where V0(z) and V1(z) are potentials. Substituting this expansion into Eq. (60), one can derive an equa- differs from a theory that is free from degeneracy ˆ ¯ˆ(0) tion for the potentials. The result is and which involves the sources tr[JiAi ].Inatheory d2 2 d n featuring degeneracy of initial data, it is necessary − to average amplitudes over degeneracy parameters. 2 Vn + Vn 2 Vn =0,n=0, 1. dz z dz z Such averaging may lead to the disappearance of a Solving this equation, we obtain the potentials number of physical states. n n In [26, 37], it was shown that amplitudes for the | − | | − |l1 | − |l2 Vn( x y )=dn x y + cn x y ,n=0, 1, production of physical color particles may vanish (62) because of the destructive interference between the n n phase factors of topological degeneracy. In this case, where dn and cn are constants, while l1 and l2 are the n 2 n the probability-conservation law for the S-matrix roots of the equation (l ) + l = n;thatis, elements i|S = I + iT |j in the form √ √ 1+ 1+4n −1+ 1+4n | | | ∗| | | ln = − ,ln = . (63) i T f f T j =2Im i T j 1 2 2 2 √ f 0 − − 0 At n =0√,wehavel1 = (1 + 1)/2= 1 and l2 = is saturated exclusively by the production of color- (−1+ 1)/2=0, so that there arises the Coulomb singlet states (hadrons) f = h. By virtue of the potential probability-conservation law, the sum over all hadro- nic channels becomes equal to the doubled imaginary −1 V0(|x − y|)=−(1/4π)|x − y| + c0; (64) part of the color-singlet amplitude (2Imi|T |j).In √ turn, the dependence on the factors of topological 1 − ≈− 1 at n =1√ , l1 = (1 + 5)/2 1.618 and l2 = degeneracy disappears completely in the color-singlet (−1+ 5)/2 ≈ 0.618,inwhichcaseonearrivesat amplitude. Owing to gauge invariance, the Hamil- a growing potential for the golden-section equation tonian of the theory, H[A(n)]=H[A(0)], depends (l1)2 + l1 =1: only on the fields of the zero topological sector, (0) −1.618 0.618 A , which play the role of Feynman’s partons. In V1(|x − y|)=−d1|x − y| + c1|x − y| . (65) the high-energy parton region, where the imaginary part of the color-singlet amplitude, Imi|T |j,can As was shown in [34–36], the instantaneous inter- be calculated on the basis of perturbation theory, action of color currents through a growing potential quark–hadron duality, which is used to measure rearranges perturbation-theory series and leads to the directly parton quantum numbers coinciding with the constituent mass of the gluon field in Feynman dia- quantum numbers of physical color particles, arises grams; this changes the asymptotic-freedom formula from the probability-conservation law. at low momentum transfers, so that the coupling 2 2 constant αQCD(q ∼ 0) becomes finite. The growing 4. ESTIMATING THE QUANTITY B potentials of the instantaneous interactions of color WITHIN QCD currents [34–36] also lead to a spontaneous break- Let us estimate the quantity B2 within QCD. down of chiral invariance for quarks. In the monopole vacuum of QCD, the antisymmetric PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 1392 LANTSMAN, PERVUSHIN Gell-Mann matrices λ , λ ,andλ play the role of the C2 N 2 α2 B2 2 5 7 m 2 = η = f QCD . (68) matrices τ , τ ,andτ . The simultaneous interaction η 2 2 1 2 3 IQCDV F 2π of color quark currents through a growing potential π leads to the spectrum of mesons—in particular, to This result makes it possible to assess the strength the pseudoscalar η0 meson. Its anomalous interaction of the vacuum chromomagnetic field in QCD(3+1), with gluons is described in terms of the Veneziano 2 2 2 2 2 2 4 B αQCD =2π Fπ mη /Nf =0.06 GeV ,by effective action [38] 2 ∼ ∼ using the estimate αQCD(q 0) 0.24 [34, 39]. 1 Upon the calculation, we can remove infrared reg- W = dt η˙2 − M 2η 2 V + C η X˙ [A(N)] eff 2 0 0 0 0 0 ularization by going over to the limit V →∞. (66) in the rest frame of this meson.√ Here, V is the volume CONCLUSION of space, C0 =(Nf /Fπ) 2/π is the coupling con- The monopole-vacuum model considered here stant for the anomalous interaction of the meson with demonstrates that the quantization of a non-Abelian (N) the topological functional X[A ]=X[A]+N, Fπ theory featuring a topological degeneracy of the initial is the weak-pion-decay constant, and Nf =3is the data for all physical states in a specific reference number of flavors. A calculation of a similar action frame describes a destructive interference of degen- functional for QCD and for QED(3+1),wherethe eracy phase factors, which leads to quark–hadron topological functional describes the decay of para- duality; a Goldstone mode that is associated with positronium into two photons, is presented in [12]. a spontaneous breakdown of scale invariance and In all probability, expression (66) for the effective which leads to an extra mass of the η0 meson; and anomalous interaction of a pseudoscalar state with a growing potential that controls the interaction of gauge fields is common to all gauge theories. For currents and which is thought to be responsible electrodynamics in two-dimensional spacetime, one for a spontaneous breakdown of chiral invariance. can obtain the same effective action [29, 30], where it These hidden features of non-Abelian fields manifest leads to the mass of the Schwinger bound state. themselves upon switching on and off gauge-field interaction with a Higgs field, which acquires an In QCD , the extra mass of a bound pseu- (3+1) infinitely large mass in the the infinite-volume limit. doscalar state, There arises the question of the extent to which such 2 2 mη = C0 /IQCDV, a fantastic possibility may be realized in nature. The generating functional found in the form of a can be determined, upon adding the action functional Feynman path integral with respect to Dirac’s vari- that is specified by Eqs. (42) and (43) and which ables can be recast [16] into the form of a Faddeev– controls the topological dynamics of zero modes, Popov integral [22] by means of the change of vari- ˙ 2 ables 1 3 2 N IQCD WQCD = dt d xG0i = dt ˆ∗ | D ˆ D −1 2 2 Ai (N A)=UN U [A] Ai + ∂i (UN U [A]) , V 2π 2 1 where I = , QCD 2 D 1 WY k αQCD V B U [A]=exp D (Φ )Aˆ (69) D2(ΦWY) k to the anomalous Veneziano action, by diagonalizing is Dirac’s “dressing” of non-Abelian fields. Upon this the total Lagrangian change of variables, the Feynman integral reduces to ˙ 2 N IQCD the Faddeev–Popov integral L = + C0η0N˙ 2 3 Z∗[l, J∗]= dN(t) d4Acδ(f(A)) ˙ 2 2 (N + C0η0/IQCD) IQCD − C0 2 t c=1 = η0 . (70) 2 2IQCD 4 ∗ ∗ × detMFP exp {iW [A]+i d xJ A (N|A)} In QED(1+1), the analogous formula describes the mass of the Schwinger state [29, 30], whereas, in QCD , we obtain the extra mass of the η meson: in an arbitrary gauge of the physical variables, f(A)= (3+1) 0 0; here, the Faddeev–Popov determinant is deter- 1 2 − 2 2 2 mined in terms of the linear response of this gauge to a Leff = [˙η0 η0(t)(m0 + mη)]V, (67) Ω −Ω 2 gauge transformation, f(e (A + ∂)e )=MFPΩ+ PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 7 2003 MONOPOLE VACUUM 1393 O(Ω2), while W is the original action functional in 5. N. N. Bogolyubov and D. V.Shipkov, Introduction to the theory being considered. At the same time, there the Theory of Quantized Fields (GITTL, Moscow, remains the Dirac gauge of the sources in (69). As 1957). a relic of the fundamental quantization, the Dirac 6. A. I. Akhiezer and V.B. Berestetskiı˘ , Quantum Elec- phase factors in the integral in (70) “remember” the trodynamics (Fizmatgiz, Moscow, 1959). entire body of information about the reference frame; 7. S. Schweber, An Introduction to Relativistic Quan- monopoles; the growing potential of instantaneous tum Field Theory (New York, 1961; Inostr. Lit., Moscow, 1963). interaction; and other initial data, including their 8. J. Schwinger, Phys. Rev. 74, 1439 (1948). topological degeneracy and confinement. As was pre- 9. B. Zumino, J. Math. Phys. (N. Y.) 1, 1 (1960). dicted by Schwinger [10], all these effects disappear, 10. J. Schwinger, Phys. Rev. 127, 324 (1962). leaving no trace, if these Dirac factors are removed 11. I. V. Polubarinov, Preprint No. P2-2421 (Joint Inst. ∗ | → by means of the substitution A (N A) A [16], for Nucl. Res., Dubna, 1965); Fiz. Elem.´ Chastits At. which is made with the only purpose of removing the Yadra 34 (2003) (in press). dependence of the path integral on a reference frame 12. V.N. Pervushin, Dirac Variables in Gauge Theories, and initial data. On getting rid of this dependence, Lecture Notes in DAAD Summerschool on Dense we obtain, instead of hadronization and confinement Matter in Particle Physics and Astrophysics, JINR, in Dirac’s quantization, only the amplitudes for Dubna, Russia, 2001; hep-th/0109218; Fiz. Elem.´ the scattering of free partons in the “relativistic” Chastits At. Yadra 34 (2003) (in press). Faddeev–Popov integral, which do not exist as physi- 13. P.A. M. Dirac, Proc. R. Soc. London, Ser. A 114, 243 cal observables in Dirac’s scheme of quantization that (1927); Can. J. Phys. 33, 650 (1955). is dependent on initial data. 14. W. Heisenberg and W. Pauli, Z. Phys. 56, 1 (1929); The same metamorphosis occurs in QED as well: 59, 166 (1930). going over from the Dirac gauge of sources to the 15. E. Fermi, Rev. Mod. Phys. 4, 87 (1932). Lorentz gauge in order to remove the dependence on a 16. L. D. Faddeev, Teor. Mat. Fiz. 1, 3 (1969). 17. 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