Nuclear-Physics Aspects of Controlled Thermonuclear Fusion: Analysis of Promising Fuels and Gamma-Ray Diagnostics of Hot Plasma V

REVIEW

Nuclear-Physics Aspects of Controlled Thermonuclear Fusion: Analysis of Promising Fuels and Gamma-Ray Diagnostics of Hot Plasma V. T. Voronchev1) and V. I. Kukulin Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received February 15, 2000

Abstract—A brief survey of nuclear-physics aspects of the problems of controlled thermonuclear fusion is given. Attention is paid primarily to choosing and analyzing an optimal composition of a nuclear fuel, reliably extrapolating the cross sections for nuclear reactions to the region of low energies, and exploring gamma-ray methods (as a matter of fact, very promising methods indeed) for diagnostics of hot plasmas (three aspects that are often thought to be the most important ones). In particular, a comparative nuclear-physics analysis of hydrogen, DT, and DD thermonuclear fuels and of their alternatives in the form of D3He, D6Li, DT6Li, H6Li, H11B, and H9Be is performed. Their advantages and disadvantages are highlighted; a spin-polarized fuel is considered; and the current status of nuclear data on the processes of interest is analyzed. A procedure for determining cross sections for nuclear reactions in the deep-subbarrier region is discussed. By considering the example of low-energy D + 6Li interactions, it is shown that, at ion temperatures below 100 keV, the inclusion of nuclear- structure factors leads to an additional enhancement of the rate parameters 〈σv〉 for the (d, pt) and (d, nτ) channels by 10Ð40%. The possibility of using nuclear reactions that lead to photon emission as a means for determining the ion temperature of a thermonuclear plasma is discussed. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION reactions for extracting required information about the dynamics of hot plasmas. In our opinion, attention Among important applications of nuclear physics, given to these important problems in review articles on that which is dealing with the problem of controlled controlled thermonuclear fusion and in original investi- thermonuclear fusion stands out in many respects. One gations into the allied range of problems is insufficient. of the main problems to be solved here is to evolve and It is shown in the present study, however, that such implement a large-scale thermonuclear reactor that issues are of nonnegligible importance for further would represent an economical source of energy and advancements in the problem of controlled thermonu- which would be safer than fission reactors. Searches for clear fusion and that they may even become crucial for an optimum composition of a nuclear fuel have so far solving some problems. We hope that the present sur- been one of the main lines in such investigations. Both vey, which is pioneering in these realms, will fill, at one- and multicomponent mixtures of light elements least partly, the gap between a vast body of currently have been considered. Despite many years of efforts in available information about the structure of light nuclei these realms, preference has not yet been given to a and specific investigations into the problem of con- unique fuel cycle. Of factors that are of prime impor- trolled thermonuclear fusion. tance for this, we would like to mention knowledge of the properties of light isotopes and the possibilities for Bearing in mind the technological potential of the their production, understanding of mechanisms that first experimental facilities of the tokamac type for govern nuclear reactions between light nuclei, and pre- heating and confining hot plasmas, researchers had cise information about cross sections for such pro- focused, for a long time, on two types of hydrogen ther- cesses. monuclear fuel, deuterium (DD) and deuteriumÐtritium (DT) fuels [1]. It should be recalled that large cross sec- The present survey, which is based in part on previ- tions for the process T(d, n)4He at low energies (E ≤ ous studies of the present authors, is devoted to nuclear- keV) and a considerable energy release in the reac- physics aspects of controlled thermonuclear fusion— 100 namely, to the role of nuclear-structure factors in reaction, Q = 17.6 MeV, seemed to give sufficient ground to tions between light nuclei, to radiative-capture pro- hope for experimentally implementing the ignition of cesses of the A(B, γ)C type in the deep-subbarrier the reaction as early as the 1970s at plasma tempera- region of energies, and to the possibilities of using such tures of a few keV, which were thought to be quite realistic at that time. These were the reasons why, despite 1) Institute of Crystallography, Russian Academy of Sciences, Le- serious drawbacks of a DT mixture—a low efficiency ninskiœ pr. 59, Moscow, 117333 Russia. of the transformation of nuclear energy into electric

2052 VORONCHEV, KUKULIN energy because of the generation of an extremely fast particles, both charged particles and neutrons, on intense neutron radiation in the reaction being dis- the ions of the fuel. The latter process proceeds at small cussed and a need for producing and employing radio- values of the impact parameter, but it involves high active tritium, which is extremely hazardous—it was momentum transfers. In the literature, the above elas- justifiably thought that there was no serious alternatives tic-scattering processes are referred to as CES (Cou- to DT fuels. lomb elastic scattering) and NES (nuclear elastic scat- However, radical improvements of tokamacs, the tering) processes. Either leads to the heating of the advent of a new fusion reactor within the ITER interna- plasma and to an increase in its reactivity. Ion scattering tional project, and a vigorous development of different is a dominant mechanism of heating, but the role of reactor schemes (for example, open-type devices and a neutron scattering is nonnegligible in severely compressed laser targets characterized by large values of configuration featuring a so-called inverted magnetic ρ ρ field), as well as the emergence of the concept of iner- the parameter R, where and R are, respectively, the tial thermonuclear fusion, created preconditions for density and the radius of the target. considering initial plasma heating to a few tens of keV The second class of secondary phenomena accom- followed by the working range of burning temperatures panying the burning of a thermonuclear fuel includes reaching a few hundred keV. This gave impetus to those that lead to nuclear transformations. These are, in studying alternative, non-hydrogen, fuels like D3He, particular, catalytic nuclear reactions between fuel ions H6Li, D6Li, and H11B, which require higher ignition and active isotopes produced in the plasma. Catalytic temperatures (in relation to D + T fusion), but which processes proceed on both thermalized and nonther- possess a number of important ecological and econom- malized nuclei. In the latter case, the processes being ical advantages. For example, burning in the H + 6Li discussed are in-flight reactions—that is, some fast par- and H + 11B systems generates neutron fluxes and haz- ticles enter into nuclear reactions prior to undergoing ardous radionuclides only in negligible amounts. At the thermalization. It is necessary to take such phenomena same time, such fuels are not expensive, since they con- into account because particles produced in a plasma sist of stable light-element isotopes abundant on the have, on average, the mean energy of a few MeV, at Earth. which the cross sections for nuclear reactions are much larger than the corresponding cross sections at thermal At present, however, physical processes in the energies (10Ð50 keV). The set of in-flight processes is majority of non-hydrogen fuels have received much not exhausted by reactions on the products of nuclear less study than analogous processes in DT or DD fusion—reactions between fast fuel ions accelerated in cycles. Even the first investigations revealed that a the- elastic collisions are also of importance. The above oretical analysis of some alternative cycles is very suprathermal fusion reactions produce charged parti- involved because burning in hot plasmas is a highly cles and neutrons of energies as great as a few tens of ramified process involving a few tens of exothermic MeV [2]. Among other secondary processes of nonneg- reactions proceeding simultaneously with commensu- ligible importance, mention should be made of neu- rate probabilities. This type of process dynamics is tron-induced reactions leading to the production of markedly different from the character of the burning of active isotopes. The role of second-generation pro- a DT fuel, where the channel T d n 4 dominates over ( , ) He cesses is greater in non-hydrogen fuels, where a large all possible reactions. number of active-isotope species are produced owing to In studying thermonuclear fuels that show consider- the occurrence of a wide variety of nuclear reactions able promise, it is necessary to take into account two (see, for example, [3]). types of processes. Of these, the first is associated with Inputs necessary for a realistic analysis of the kinet- first-generation reactions representing direct channels ics of processes proceeding in thermonuclear-fusion of the burning of a nuclear fuel in reactions between reactors must include reliable data on the cross sections light elements originally present in the reactor region. for many reactions in a broad range of energies. Above As a rule, such processes provide a major part of the all, this concerns reactions involving light nuclei and energy released in the fusion process. There are also, proceeding in the low-energy region E < 500 keV, however, secondary phenomena that accompany the which is of prime importance for controlled thermonu- burning of any fuel and which play a very important clear fusion. An additional incentive to study the rele- role in some cases. We will break up secondary pro- vant cross sections at a high-precision level comes from cesses into two classes that differ in the physics under- nuclear astrophysics, where many fundamental prob- lying the phenomena that occur. lems cannot be solved without this. At present, experi- In the first class, we include processes that do not mentalists have accumulated a vast body of relevant lead directly to nuclear transformations, but which data. Compilations of measured cross sections and appear to be a basic mechanism of fuel self-heating. computed reaction-rate parameters are presented in the Above all, this is the elastic Coulomb scattering of fast well-known reference literature [4Ð8]. For example, the charged particles, products of nuclear reactions, on DATLIB database [8] contains 270 data files for plasma ions and electrons. In addition to Coulomb scat- 77 channels of nuclear reactions involving isotopes of tering, there can occur the nuclear elastic scattering of light elements from hydrogen to boron. References to

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NUCLEAR-PHYSICS ASPECTS OF CONTROLLED THERMONUCLEAR FUSION 2053 other databases can be found in [8, 9]. We would also examples, we demonstrate there, among other things, like to mention the Russian handbook [10] on nuclear how the special features of the nuclear structure of 6Li processes and cross sections, which covers a very broad and 7, 9Be affect the behavior of the cross sections for range of energies and which additionally presents a low-energy thermonuclear reactions. In Section 4, we spline approximation of experimental data. A precision consider the possibility of determining the ion temper- R-matrix parametrization of the cross sections for low- ature of a fuel by measuring the fluxes of reactor pho- energy D + T, D + D, and D + 3He interactions, as well tons. The basic results of our consideration are summa- as of their Maxwell reactivities, is given in [11]. rized in the Conclusion (Section 5). In some cases, however, available experimental data are insufficient for a detailed analysis of fuel burning 2. THERMONUCLEAR FUELS and of the possibility for monitoring this burning. Here, According to the classification of McNally [14], we only note the following. First, low-energy radiative- thermonuclear fuels can be broken down by convention capture reactions like T(t, γ)6He, 3He(τ, γ)6Be, 6Li(d, γ 8 6 τ γ 9 7 γ 9 7 γ 10 into three groups: classical (DT), promising (DD, ) Be, Li( , ) B, Li(d, ) Be, and Li(t, ) Be, which D3He, D6Li), and exotic (3He3He, H6Li, H11B, H9Be) can be used to determine the ion temperature of a fuels. We will consider them precisely in this order. plasma or to monitor its dynamics, have not yet received adequate study. Moreover, there are no data on some of such processes whatsoever. Second, neither 2.1. DT Fuels reliable measurements of some reactions involving A DT mixture of hydrogen isotopes is the most tra- light radionuclides and proceeding at subbarrier ener- ditional form of fuel for nuclear reactors. The following gies nor relevant theoretical investigations have been processes are usually considered in studying the deute- performed so far. For example, the interesting catalytic riumÐtritium cycle: reaction 7Be(d, p)24He in a D6Li fuel—this reaction, which is accompanied by a large heat release of Q ~ (i) the primary exothermic reactions 17 MeV, produces only charged particles—has been D + T 4He + n + 17.59 MeV, (1) studied insufficiently. Third, the situation is not abso- D + D T + p + 4.03 MeV, (2) lutely clear in what is concerned with cross sections for low-energy reactions on polarized nuclei. This is espe- D + D 3He + n + 3.27 MeV, (3) cially important since the use of spin-polarized fuels T + T 4He + 2n + 11.33 MeV; (4) can increase the energy released by a thermonuclear plasma and suppress simultaneously the generation of (ii) the main secondary exothermic reaction neutron fluxes. By way of example, we indicate that, D + 3He 4He + p + 18.34 MeV; (5) for the D + T resonance process, the estimate of the nuclear-spin-polarization effect is known [12], but that, (iii) the tritium production in a lithium blanket, for the D + D direct reaction, the situation is much more n + 6Li T + 4He + 4.78 MeV; (6) complicated [13]. For the majority of non-hydrogen (iv) the neutron-breeding processes thermonuclear fuels, no detailed investigations of polarization processes have been undertaken thus far. n + 9Be 2n + 24He – 1.57 MeV, (7) One of the objectives of the present study is to ana- n + D 2n + p – 2.23 MeV. (8) lyze various thermonuclear cycles based on the use of Among all promising reactions, the D + T process hydrogen, helium, lithium, beryllium, and boron iso- provides the highest energy release and is characterized topes. We pay attention primarily to nuclear-physics by the largest value of the reaction-rate parameter aspects of the issue—in particular, to the nuclear-struc- (reactivity) r = 〈σv〉, a quantity obtained by averaging ture effect on the cross sections for fusion reactions and the product of the nuclear cross section σ(E) and the on the rates of these reactions. relative velocity v of the reacting particles over their The ensuing exposition is organized as follows. In Maxwell velocity distribution f(v). That the reactivity Section 2, we consider DT, DD, D3He, D6Li, DT6Li, is so high in this case is due to a manifestly resonance 6 11 9 character of the D + T reaction in the region of low sub- H Li, H B, and H Be fuel mixtures and analyze their π advantages and disadvantages. The important case of barrier energies. It proceeds through the J T = [(3/2)+, spin-polarized fuels is also analyzed. It should be 1/2] level of the compound nucleus 5He at the excita- emphasized that difficulties encountered in simulating tion energy of E* = 16.76 MeV; this corresponds to an the burning of some alternative fuels are associated incident-deuteron kinetic energy of Ed ~ 100 keV [15]. with uncertainties in the predicted cross sections for In [16], it was shown that this “thermonuclear” reso- nuclear reactions in the deep-subbarrier energy region, nance is a typical three-body near-threshold resonance where there are no experimental data. For this reason, of the t + n + p structure and that the coupling of the we further discuss (in Section 3) extrapolation methods input and output channels (td αn) is due both to for determining cross sections in the region of low noncentral (tensor) and to central forces. Additional (thermonuclear) energies. By studying some important gain in the reactivity can be obtained owing to a spin

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2054 VORONCHEV, KUKULIN polarization of the fuel [12, 17]. In the presence of an very serious technological problems of protecting external magnetic field B, the cross section for the D + structural reactor materials from formidable neutron T interaction can be represented in the form fluxes. Moreover, the energy of neutrons escaping from the reactor can be utilized with an efficiency not  2 1  1 2  σ = a + --b + --c σ + --b + --c σ , (9) exceeding 40%—in other words, immense energy  3 3  3/2 3 3  1/2 fluxes will circulate in the system at a typical thermonuclear-reactor power of 5 × 103 GW if energy convert- where a = d+t+ + d–t–, b = d0, and c = d+t– + d–t+, d+(t+), ers are included in the reactor circuit. But this is not the d–(t–), and d0 being the fraction of deuterons (tritons) whole story: tritium is radioactive; it must be produced oriented, respectively, parallelly, antiparallelly, and artificially; and the extraction of tritium from a lithium orthogonally to the field B. Since the reaction being blanket presents a nontrivial problem. To summarize, considered proceeds through the J = 3/2 channel at low the presence of a source of high-energy and high-den- energies, a parallel orientation of the D and T spins sity neutron radiation, together with the use of a increases the cross-section value by 50% in the reso- strongly radioactive material, imposes severe radiation- nance region [12]. However, the eventual estimate of safety requirements on the implementation of a DT the gain in the energy release is complicated by various reactor. depolarization effects associated with binary collisions, fluctuations, nonuniformities of magnetic fields, and other similar factors. In the literature, there are both 2.2. DD Fuels optimistic and pessimistic estimates of the depolariza- Primary D + D interactions have two almost tion rate (see, for example, [12, 17Ð21]). equiprobable exothermic channels (2) and (3). They In contrast to the D + T process (1), the D + D pro- lead to the production of the active isotopes T and 3He cesses (2) and (3) are not of a resonance character. in the plasma. This in turn initiates the important sec- These reaction channels have approximately identical ondary catalytic processes (1) and (5). Although the probabilities and are both characterized by smaller burning of a DD fuel is also accompanied by the gener- energy-release values and considerably reduced (in ation of neutron fluxes, they are less intense than those relation to the D + T process) cross sections at low ener- in the D + T process. The DD cycle does not require gies. As a result, the above D + D processes make a rel- specially producing tritium; hence, the application of atively small contribution to the energy released in the this cycle makes it possible to avoid using a lithium burning of DT fuels, and they are often referred to as a blanket and involved tritium technologies associated satellite component. There exists, however, a third pos- with this. It has already been indicated, however, that, sible channel of D + D interactions, D(d, γ)4He; this at low energies, the D + D cross sections are much channel involves the production of high-energy pho- smaller than the D + T cross sections and that triggering tons and can have diagnostic applications—in particu- the D + D process requires much higher plasma temper- lar, it can be used to determine the ion temperature of atures. By way of example, we indicate that the ideal the fuel [22]. The T + T process (4), which is the last in threshold temperature becomes as high as about 40 keV the list of primary processes, does not make a signifi- in this case [23]. cant contribution to the reaction energy release either and appears to be a by-process. At actual temperatures, its rate is two orders of magnitude less than the rate of 2.3. D3He Fuels reaction (1). By convention, such fuels are categorized as neu- A DT mixture possesses important advantages. tron-free fuels. In such a heliumÐhydrogen mixture, the First, the nτ value satisfying the Lawson criterion is D + 3He process (5), the D + D processes (2) and (3), much less for D + T interactions than for other kinds of and the reaction thermonuclear fuels. Second, the D + T fusion reaction is triggered at relatively low temperatures. The ideal 3He + 3He 4He + 2p + 12.86 MeV (10) threshold temperature determined as the temperature at appear to be primary reactions. Predominantly, useful which the energy release is equal to the energy loss by energy is released in the D + 3He channel (5). This reac- bremsstrahlung in the case of a complete particle con- tion, which has the greatest Q value of 18.34 MeV, finement is about 4 keV [23]; an optimum temperature leads to the production of only charged particles, a cir- of burning is estimated at 15 keV in [23] and at 20 keV cumstance that improves conditions for fuel self-heat- in [1]. Finally, the specific-power release is at least two ing. Of course, the neutron-free process (10), which is orders of magnitude greater than similar energy charac- accompanied by a considerable energy release per teristics of other mixtures. fusion event, is also very appealing. The use of 3He Nonetheless, the DT cycle has some serious draw- fuels as such seems tempting, but it is very difficult to backs. First of all, the burning process is accompanied implement reaction (10) in practice because of a high by intense neutron fluxes. High-energy neutrons carry threshold for its ignition. At an operating ion tempera- about 80% of the energy released in the D + T fusion ture of the plasma about 100 keV, its rates are approxi- process (about 14 MeV per reaction event). This entails mately two orders of magnitude less than the rates of

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NUCLEAR-PHYSICS ASPECTS OF CONTROLLED THERMONUCLEAR FUSION 2055 the D + 3He reaction; as a consequence, the contribu- value corresponds to a 4.5-fold suppression of the D + tion of reaction (10) to the burning of D3He fuels is neg- D process. That the neutron channels in a polarized ligibly small. D3He fuel can be suppressed so strongly gave incentive 3 The group of secondary processes in the D3He cycle to developing a conceptual design of a D He fusion includes the catalytic reactions (1) and (4) (D + T and reactor on the basis of a linear tandem device [33, 34]. T + T, respectively), as well as two reactions involving Apart from a high ignition temperature, D3He fuels fast protons, have yet another serious drawback: 3He is a very rare isotope on the Earth; moreover, there are presently no D + p 2p + n – 2.23 MeV, (11) efficient sources for its production. However, the possi- T + p 3He + n – 0.76 MeV. (12) bility of delivering 3He from the Moon’s surface, which can contain, according to estimates based on studying Owing to dominance of the D + 3He channel (5), the Moon rocks, about 106 t of this isotope [35], is dis- fuel being discussed can be thought to be neutron-free cussed in the literature. by convention [24]. Here, the primary neutron channel (3) of D + D interactions and the D + T catalytic reaction (1) appear to be the sources of neutron contamina- 2.4. D6Li and DT6Li Fuels tion in the D3He cycle, the contribution of the latter The presence of the 6 often being dominant [25]. The fraction of the neutron Li isotope in a hydrogen plasma complicates dramatically an analysis of ther- component in the energy release is estimated at about 6 3% [26]. This is a serious advantage of reactors employ- monuclear fusion. Nuclear reactions in a D Li mixture are multistep branched processes, a number of isotopes ing D3He mixtures, and the relevant possibility attracts of light elements being produced in them with com- much attention at present.2) A detailed survey of the cur- mensurate probabilities. In the first generation alone, rent status of the problem can be found in [29]. the possible processes include the two D + D channels Some gain in the energy release can be achieved by (2) and (3) and the seven exothermic D + 6Li reactions burning a spin-polarized D3He fuel. It is difficult, however, to obtain an accurate estimate of the increase in D + 6Li 4He + 4He + 22.37 MeV, (13) τ the reactivity because, near the d + threshold, there D + 6Li 7Li + p + 5.03 MeV, (14) are two (not one as in the case of d + t) excited states of the intermediate nucleus 5Li at E* ~ 16.66 MeV [JπT = D + 6Li 7Li* + p + 4.55 MeV, (15) + ± + (3/2) , 1/2] and E* ~ 18 1 MeV [(1/2) , 1/2] [15] that 6 7 contribute to the S = 1/2 and 3/2 reaction channels (both D + Li Be + n + 3.38 MeV, (16) channels of the reaction being considered). The cross- D + 6Li 7Be* + n + 2.95 MeV, (17) section-enhancement values adopted in the literature fall within the range 44Ð49%. Another important objec- D + 6Li 4He + T + p + 2.56 MeV, (18) tive pursued in using nuclear spin polarization is to sup- 6 4 3 n MeV press selectively the D + D reactions and neutron yields D + Li He + He + + 1.80 . (19) associated with them [12, 30], but the degree of this We do not present here other possible reactions in the suppression is still debated. Previously, it was shown in 6Li + 6Li system, which are characterized by very small [31] that, for the low-energy D + D reactions, the con- cross sections and which do not therefore make a 5 tribution of the S2 input channel is small in relation to noticeable contribution against the background of reac- 1 the contribution of the S0 channel, and this made it pos- tions (13)Ð(19). Among secondary processes, whose sible to consider the possibility of suppressing this number exceeds 80, we only mention the strong reac- reaction in the case of parallel deuteron spins. Later on, tions (1) and (5) between D and T and between D and however, a pessimistic estimate of this suppression was 3He, respectively, and the high-Q catalytic processes obtained upon taking into account the D-wave state in 7 4 3He [32], because an additional allowed contribution of D + Be 2 He + p + 16.77 MeV, (20) central forces arises in this case. After a number of D + 7Li 24He + n + 15.12 MeV. (21) studies on the subject, the highly reliable analysis of Zhang et al. [13], who considered the D + D reactions In studying prospects for a D6Li fuel, it is necessary in a polarized deuterium plasma, nonetheless revealed to take into account its special features [3, 36]. η σ σ that the ratio = 1, 1/ 0 (ratio of the cross sections for (A) Branched character of burning. The reaction reactions involving spin-polarized and unpolarized cross sections at low energies are small in relation to nuclei) for deuteron energies between 30 and 90 keV the D + T cross sections, but the number of processes decreases monotonically from 0.86 to 0.22. The last proceeding in the D6Li cycle is as large as a few tens. For this reason, the total contribution of a large number 2)Yamagiwa [27] proposed using an 18O admixture in a D3He reac- 7 18 of channels—of these, some (for example, D + Be and tor in order to produce the positron-unstable radionuclide F in 7 the reaction 18O(p, n)18F. Along with 11C, 13N, 15O, and 19Ne, D + Li) have energy releases commensurate with the this element can be employed in positron tomography and find energy release in the D + T processes—may prove to be diagnostic applications in cancer radiotherapy [28]. significant.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2056 VORONCHEV, KUKULIN (B) Relatively large yields of fast and slow the handbook of Abramovich et al. [10], are known to charged particles. Slow charged particles can effi- a somewhat poorer precision, which is nonetheless ciently heat a fuel via elastic Coulomb collisions. As to quite sufficient for simulating the ignition and burning fast charged particles, their presence enhances the role processes. We would also like to mention the interest- of other secondary phenomena like NES processes and ing study of Cherubini et al. [39], who used quasielastic in-flight reactions. By way of example, we indicate the data on the reaction 6Li(6Li, 2α)4He in the region of following catalytic chain [37]: above-barrier energies to extract the 6Li(d, α)4He exci- the inelastic-scattering process tation function at very low relative energies. This indirect method of analysis is based on the so-called Trojan 4He (fast) + 6Li 6Li*(2.18 MeV) + 4He (slow), Horse model proposed in [40]. However, there also the decay process remain unresolved questions. For example, the contribution to the reaction being discussed from 8Be highly 6Li*(2.18 MeV) 4He (slow) + D (fast), (22) excited states near the d + 6Li threshold at about 22 to 23 MeV has yet to be clarified conclusively. In particu- and the reaction lar, it was shown in [41] that the D + 6Li processes can D (fast) + 6Li 24He (fast) + 22.37 MeV. be described satisfactorily without resort to a resonance mechanism. On the contrary, the contribution of the This chain results in the doubling of the number of fast excited state 8Be(2+) at about 22.28 MeV was found to alpha particles (they appear to be some kind of a cata- be large in the experimental study of Czersky et al. lyst for the process) and releases a large amount of [42], who measured the cross sections for the reaction energy. 6 α 4 Li(d, ) He in the energy range Ed = 50Ð180 keV. (C) Monotonic energy dependence of the reac- Thus, the highly exothermic reaction (13) receives a tion cross sections. As was indicated above, the main considerable contribution from a resonance mecha- reaction of thermonuclear fusion (D + T) is of a reso- nism. In this connection, it should be recalled that, nance character. Accordingly, its cross sections are despite many years of strenuous efforts, there is pres- large in the region of subbarrier energies, but they begin ently no reliable general method for extrapolating the to decrease upon achieving a maximum at Ed ~100 keV. cross sections for low-energy reactions to those near- On the contrary, the cross sections for the D + D and threshold energies that are of immediate importance for D + 6Li processes at low energies are much smaller than calculating the plasma reactivity 〈σv〉. In Section 3, we the D + T cross section, but they increase monotoni- will discuss some extrapolation procedures and allied cally (and quite sharply for D + 6Li) as the deuteron problems. Among other things, it will be shown that energy increases up to Ed ~ 1 MeV. This type of behav- nuclear-structure factors play an important role in ior is peculiar to many secondary reactions of the D6Li extrapolating cross sections to the deep-subbarrier cycle as well. As a result, there can arise an additional region. positive feedback in the mechanism of plasma self- For some second-generation nuclear reactions in the heating. D6Li cycle, the cross sections required for studying the (D) Presence of the fluxes of neutrons having role of these reactions in the heating of the fuel and moderate energies. It should be recalled that D6Li is their contribution to the energy release have not yet not a neutron-free fuel, but that its burning is accompa- been determined to a sufficient precision. In some nied primarily by the generation of neutrons with ener- cases, there are only fragmentary experimental data or gies between 1 and 3 MeV (in contrast to 14-MeV neu- only theoretical estimates. For example, neither exper- trons coming from the D + T process), so that a major imental nor theoretical studies have been performed part of the energy release is associated with charged thus far for the interesting second-generation reaction particles. Neutron radiation makes but a small contribu- 7Be(d, p)24He + 16.77 MeV, which has an energy tion to the total energy release, but it leads to the addi- release virtually identical to that from one event of the tional production of tritium in the blanket reaction D + T process, but which produces, in contrast to the 6Li(n, t)4He, which proceeds in this case directly in the latter, only charged particles. The two-step process region of burning. We also note that neutrons from the D + 6Li processes can be used in mixed reactors of the D + 7Be 8Be*(2+) + p + 0.05 MeV, (23) breeder type [38]. 8Be*(2+) 24He + 16.72 MeV (24) (E) Uncertainties in the cross-section values. For the first-generation D + D and D + 6Li processes, the appears to be the most probable mechanism of the reac- situation around nuclear data on low-energy cross section being discussed, since there is no Coulomb barrier tions is quite clear. The D + D processes were inten- for neutron transfer in reaction (23) and since the 8Be sively investigated earlier—for example, reliable data nucleus features the well-known Jπ = 2+ 16.63-MeV on their cross sections in the region of thermonuclear level [15], whose population in the above process cor- energies and on the rate parameters 〈σv〉 were pub- responds to a nearly vanishing energy release. The first lished in [11]. The D + 6Li cross sections, which can be step leads to the formation of a relatively long-lived found, for example, in the DATLIB database [8] or in 8Be* state, which then decays into two fast alpha parti-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NUCLEAR-PHYSICS ASPECTS OF CONTROLLED THERMONUCLEAR FUSION 2057 cles. Since Q ≈ 0 in process (23), the reaction cross sec- (F) Effect of nuclear-spin polarization. At low tion is given by energies, the reaction 6Li(d, α)4He (13), which is characterized by the highest energy release among the first- ϕ ( 2) Coul q dq generation processes, proceeds predominantly through σ ∼ + 8 ∫------2------, (25) the S = 2 channel, provided that the 2 level in Be at Q + q E* ≈ 22.28 makes a dominant contribution (see above). If this process is implemented with polarized particles where the function ϕ q2 is proportional to the pen- Coul( ) whose spins are parallel, the energy release must there- etrability of the Coulomb barrier. It follows that, for fore increase by a factor close to 2, all other conditions Q 0, the reaction cross section can sharply grow 3) being the same. slightly above the threshold [43] and that its values at 6 moderately small energy values must considerably In analyzing the prospects of D Li fuels, it is impos- exceed simple barrier estimates. sible to take into account completely all the aforementioned features. However, a simplified model incorpo- The above exemplifies problems encountered in rating all first-generation reactions and some secondary these realms when one studies only one of more than 80 processes that was developed in [3, 36] permitted draw- second-generation reactions in the D6Li cycle. A non- ing some important conclusions on the role of 6Li in a conventional procedure for directly determining the plasma consisting of hydrogen isotopes. In the first of cross sections for a broad range of nuclear reactions at those studies, the kinetics of 13 D + D, D + 6Li, D + T, low energies not yet explored by that time was pro- D + 3He, D + 7Be, and D + 7Li nuclear processes was posed in [44] on the basis of a laser compression of a investigated under the conditions of magnetic plasma thermonuclear target. In contrast to conditions preva- confinement. There, the mechanism of self-heating was lent in conventional nuclear-physics experiments, high described within the CES process; that is, the plasma matter densities, exceeding typical solid-state densities temperature increased owing to the elastic Coulomb by many orders of magnitude, are achieved in a laser scattering of fast particles on fuel ions. compression of a target. Despite very small cross-section values for energies E ≤ 10 keV and despite a com- On the basis of the results obtained in [3], one can draw two important conclusions. First, a 10% admix- paratively low hydrodynamic efficiency of targets, we 6 can therefore expect that the yields of nuclear prod- ture of Li to a deuterium fuel can increase the plasma reactivity. This means that an additional energy release ucts—in particular, neutrons and photons—will be suf- 6 ficient for purposes of reliable detection. Within this owing to the presence of Li exceeds the enhancement approach, one determines not the absolute value of the of radiative losses because of a contamination of a cross section for the reaction of interest at a specific hydrogen plasma by an admixture with charge number energy value, but its relative yield with respect to some ZLi = 3. At the same time, the threshold ignition temper- reference reaction [like D(d, n)3He] whose cross sec- ature increases insignificantly, amounting to Tthr ~ tion is well known. The absolute yield of products from 50 keV—the corresponding temperature for a pure DD thermonuclear reactions is then found as the convolu- fuel is about 40 keV [23]. The second conclusion con- tion of the required cross section with the velocity dis- cerns a dominant mechanism of energy generation. It tribution of plasma ions, which is not known very well. turned out that it is of manifest catalytic character and is due to the production of the active elements T, 3He, However, the parameters of this distribution can be cor- 7 rected on the basis of a series of measurements of the and Be. Their fusion proceeds so vigorously that, yields of particles from a reference reaction thoroughly within some 10 to 14 s from the commencement of burning, the concentration of these nuclei exceeds the studied in advance. The yields from both the reference 6 reaction and the reaction under study change in current value of the Li concentration. Characteristic curves describing the dynamics of fuel burnup and the response to variations in the conditions under which the 6 laser target is compressed. If one has sufficient statistics production of active isotopes in a D + Li plasma are of particle yields in the two reactions at his disposal and presented in Fig. 1. if one knows the effective plasma temperature in the For yet another example of the use of 6Li nuclei in target upon each shot of the laser gun used, it is possible fusion reactors, we can indicate a relatively simple to establish the behavior of the required cross section in means for the ignition of a DD fuel at initial tempera- ≤ 6 the deep-subbarrier region [44]. By way of example, tures of T0 10 keV by injecting a T Li pellet into the we indicate that, according to the estimates presented in reactor core [36]. This composition of a solid-state [44], the cross sections for the D + 6Li processes can be igniter is recommended for the following two reasons. determined at extremely low energies of E = 1Ð10 keV First, it initiates a self-sustaining process of burning, by applying the proposed procedure, provided that the with an energy release in the three-component DD + T6Li mixture being not less than that in the two-compo- relative yield of fast particles from these processes has 4) been measured. nent DD + T mixture. Second, the use of lithium trit- ide (that is, tritium-substituted lithium hydride) makes 3)In the absence of a Coulomb barrier—that is, for neutrons—the reaction cross section must grow indefinitely in this case from the 4)This notation for the plasma composition emphasizes that a T 6Li threshold. or a T admixture is introduced in a deuterium plasma.

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Concentration, 1/cm3 D + D, and D + 6Li processes of suprathermal fusion on 1016 deuterons and tritons accelerated in neutron scattering. For the case of the volume fuel-ignition mechanism, D typical results are quoted in Table 1. The initial conditions were characterized by the value of ρR = 10 g/cm2, 6 Li the equal ion and electron temperatures of Ti = Te = 1014 1.5 keV, the degree of compression (on the scale of the ρ ρ 6 3 solid-state density) of / 0 = 5000, and the Li content He of 5% in the DT mixture. Table 1 shows that the D + T T 7Be process, the main thermonuclear-fusion reaction, is 1012 dominant here. This result might have been expected 0 10 20 from the outset, but it is of interest to compare the rela- Time, s tive contributions of other channels. It can be seen that the energy release is 30% greater in the D + 6Li pro- Fig. 1. Concentrations of a fuel and of some active isotopes 6 cesses than in the D + D processes. The data quoted produced in a D + Li plasma as functions of time [3]. here cannot be considered to be conclusive, since some important issues associated with the optimization of the unnecessary a separation of tritium from a lithium blan- nonhomogeneous structure of the target and with the ket and its subsequent transportation to the zone of regime of its compression have yet to be clarified. In burning. addition, it is of interest to study the kinetics of nuclear reactions at higher initial temperatures as well. A diagnostic application of the reactions 6Li(d, p)7Li* (478 keV) and 6Li(d, n)7Be* (431 keV) furnishes an additional motivation for introducing a 6Li 2.5. H6Li Fuels admixture in a deuterium-containing plasma [45]. By The above special features of the D + 6Li nuclear detecting monochromatic photons produced in these 6 reactions, one can attempt to measure the ion tempera- processes are inherent in part in H Li fuels. As in the case of D6Li, the burning of the fuel is highly branched ture Ti (see Section 4) or to control the dynamics of burning. here, and the number of relevant reactions is as high as a few tens [49]. However, the spectrum of the first-gen- However, Kernbichler and Heindler [46] assessed 6 eration reactions is much narrower here than that for more pessimistically the use of Li as a possible com- D + 6Li. If we discard the suppressed 6Li + 6Li channel ponent of a thermonuclear fuel and showed that the 6 γ 7 6 and the weak radiative-capture process Li(p, ) Be, energy released in a D Li mixture decreases monotoni- there remains, in a H6Li mixture, only one exothermic cally as its concentration is increased. In order to draw reaction a definitive conclusion on the prospects for the use of a D6Li fuel, we need more detailed models taking into H + 6Li 4He + 3He + 4.02 MeV, (26) account many second-generation processes; it is also which is pure in the sense that it produces neither neu- necessary to consider various reactor designs. Such an trons nor radionuclides. The second-generation exo- analysis is hindered, however, by formidable computa- thermic reactions tional difficulties and by the absence of some nuclear data on cross sections. 3He + 6Li 24He + H + 16.88 MeV, (27) A new attempt at refining the model for laser ther- 3He + 6Li 7Be + D + 0.11 MeV (28) monuclear fusion was made by Nakao et al. [47], who additionally studied the role of suprathermal nuclear do not lead to neutron production either; the first of reactions in a DT + 6Li target. A numerical simulation these regenerates protons. The processes in (26) and was performed there on the basis of coupled transport (27) can be combined into the chain and hydrodynamic processes [48]. This simulation took 6 4 MeV into account all first-generation reactions in the DT + H + 2 Li H + 3 He + 20.90 . (29) 6Li system; charged-particle and neutron scattering; the In relation to the direct reaction 6Li(6Li, α)24He, this deuteron-breakup reaction D(n, 2n)p; and the D + T, chain provides an easier means for burning 6Li with the

Table 1. Energy release in a DT + 6Li laser target Relative contribution of channels, % Initial internal Energy release, MJ thermal suprathermal energy, MJ D + T D + D D + 6Li D + T D + D D + 6Li 0.634 476.2 90.1 0.8 1.2 7.6 0.2 0.1

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NUCLEAR-PHYSICS ASPECTS OF CONTROLLED THERMONUCLEAR FUSION 2059 generation of three fast alpha particles. This process implementing the H + 11B process and that magnetic was simulated, for example, in [28]. confinement proves to be inefficient because of large Despite the obvious attractiveness of H6Li fuels radiative losses. According to [58], the volume ignition ρ 2 (owing to the absence of strong neutron fluxes and their is achievable at R = 16 g/cm under certain conditions. cheapness), it has proven to be impossible at present to For ignition, Martinez-Val et al. [59] propose using par- accomplish its ignition.5) An analysis of the cycle in ticle beams under the conditions of the inertial com- (29) with allowance for the 6Li + 6Li thermonuclear pression of a target with the aim of generating a ther- reactions does not lead to ignition even in the case of mal-detonation wave. A new type of a thermonuclear β 2 reactor, CBFR (colliding-beam fusion reactor, which is large values of ~ nT/B (where n and T are the plasma 11 density and temperature, respectively, while B is the appropriate for the use of a H B fuel), was proposed by external-magnetic-field induction), in which case Rostoker et al. [52], who argue that a CBFR is conve- losses by cyclotron radiation are minimal [51]. The nient for the burning of a spin-polarized fuel, since the more recent investigation of Kernbichler and Heindler depolarization rate is insignificant on the time scale of nuclear fusion or diffusion [60]. It was shown that the [46] also demonstrated that, even at optimal operating 11 temperatures in the range 300Ð650 keV, the energy use of a polarized H B mixture ensures a 60% characteristics of H6Li mixtures are below the thresh- enhancement of the cross section [60]. old values by about one order of magnitude. Nonethe- 6 less, H + Li plasmas may still be of interest for differ- 2.7. H9Be Fuels ent settings—for example, in the case of collisions between plasma bunches [52]. The H + 9Be process has two approximately equiprobable exothermic channels of the production of slow charged particles, 2.6. H11B Fuels H + 9Be 4He + 6Li + 2.13 MeV, (32) As in the preceding case, the almost complete 9 4 absence of neutron generation and cheapness are H + Be 2 He + D + 0.65 MeV, (33) important advantages of boronÐhydrogen mixtures. In which are able to transfer a major part of their energy to the first generation, the two possible exothermic reac- plasma ions. Although the total energy release is as tions are small as some 2.8 MeV, the low-energy behavior of the reaction is determined by the presence of a strong res- H + 11B 34He + 8.68 MeV, (30) onance at an incident-proton energy of Ep ~ 330 keV H + 11B γ + 12C + 15.96 MeV, (31) [61], which corresponds to the excited state 10B(1–) at E* ~ 6.88 MeV [15]. This conclusion was confirmed in the latter being severely suppressed. In the energy the earlier experimental study of Sierk and Tombrello region of our prime interest, the branching ratio for [62] as well. In this region, the reaction cross sections –5 these channels is about 10 [53]. The dominant reac- are extremely large; the product σv exceeds that for the tion (30) yields alpha particles of moderate energies, D + 3He processes, falling short of only the correspond- Eα ~ 2.9 MeV; this favors the heating of the ion compo- ing quantity for the D + T process [63]. It was also indi- nent with an efficiency of 80Ð90% [54]. Over a broad cated in [61] that the additional reason for extremely energy range, the reaction in question is of a resonance large low-energy cross sections may be concealed in character associated with a few excited states of the 12C the structural features of the 9Be nucleus as well. This nucleus. Of these, the lowest is responsible for the low- nucleus has a neutron weakly bound to the 8Be core, its energy peak in the cross section at Ep ~ 163 keV. The separation energy being as low as some 1.7 MeV [15, reactivity of secondary processes in H11B fuels is very 64]. The 9Be density profiles found in [65] indicate that low. The weak reactions 11B(p, n)11C and 11B(α, n)14N the neutron wave function extends over a large dis- may prove to be neutron sources, the neutron contribu- tance, so that the direct reaction mechanism is quite tion to the energy release being estimated at about 0.1% possible. This is confirmed by the latest experimental [55]. A H11B fuel was proposed in [56] and was inves- study of the reaction on polarized protons at Ep ~ 80– tigated, for example, in [53Ð57]. The reactivity attains 300 keV in [66], where it was shown that the direct a maximum only at high temperatures (about 300 keV); mechanism provides a good description of the cross together with large losses by radiation because of a sections for the (p, d) channel (33) at very low ener- large boron charge of Z = 5, this complicates fulfillment gies. of the energy-balance conditions. The general conclu- It is of paramount importance that all products of the sion from the aforementioned investigations is that it is 9 necessary to use laser thermonuclear facilities for H + Be processes (32) and (33) cause the second-generation exothermic processes [63] 5)It is interesting to note that, under astrophysical conditions, the D + 9Be 4 reactions with total Q ~ 21 MeV, (34) 6 α 3 reaction Li(p, ) He plays an important role. Together with the 6Li + 9Be 8 reactions with Ò Q ≈ 2–15 MeV, (35) radiative-capture process 4He(d, γ)6Li, it is responsible for the abundance of the 6Li isotope in the Universe [50]. 4He + 9Be n + 12C + 5.70 MeV, (36)

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VD6Li, keV illustration, we indicate that, in the D + D system, the 1200 p-wave contribution is of importance up to 15 keV, so that a conventional use of an s-wave extrapolation (discard of the p-wave contribution) leads to large errors. 800 2 Within the standard extrapolation, the penetrability factor is usually calculated for a purely Coulomb shape of the potential barrier in the semiclassical approxima- 400 tion, which leads to the Gamow formula [67] 1 π2 2 ∼  4 e Z1Z2 PGamow expÐ------ , (39) 0 10 20 hv R, fm where v is the relative velocity of particles with charge numbers Z and Z . Fig. 2. (1) Realistic ([70]) and (2) purely Coulomb potential 1 2 barrier in the D + 6Li system. For the Coulomb barrier, the Within the more precise quantum-mechanical channel radius was taken to be a = 4.3 fm. model, we have (see, for example, [68])

kR 6 3 4 ∼ Li + H He + He + 4.02 MeV, (37) Pquant ------, (40) F 2 + G 2 and that their role in plasma heating may prove to be R = a significant. However, no detailed investigations of 9 where F and G are, respectively, the regular and the H Be fuels have been performed thus far. Moreover, a irregular Coulomb wave function, k is the wave num- comprehensive simulation of its burning has not yet ber, and a is the channel radius. The extrapolation pro- been performed, and the role of secondary processes cedure developed in [69, 70] is more correct. There, the has not been clarified. It is clear, however, that the res- 9 quantum-mechanical-tunneling factor P(E) is deter- onance H + Be process can at least utilize moderated mined for a realistic shape of the potential barrier protons very efficiently in the energy region below (rather than for an ideal Coulomb shape) that contains 1 MeV [63]. information about the inner structure of particles involved in the reactions being considered. This is espe- 3. EXTRAPOLATION OF CROSS SECTIONS cially important in describing the interaction of nuclei FOR NUCLEAR REACTIONS TO THE REGION having a pronounced cluster structure. The realistic OF LOW ENERGIES AND ROLE barrier is constructed with allowance for the peripheral OF NUCLEAR-STRUCTURE FACTORS attractive component of nuclear interaction in the system, whereby the shape of the barrier is strongly modi- In order to extrapolate the cross sections for nuclear fied, its height being reduced.6) This can clearly be seen reactions to the region of low (deep-subbarrier) ener- in Fig. 2, which displays the Coulomb and the realistic gies, where there are usually no experimental data, the potential barrier for the D + 6Li system [70]. The latter cross section σ(E) is represented as the product of a fac- was computed on the basis of the double-folding model tor that changes slowly with energy and a factor that with reliable three-body wave functions of the 6Li changes fast. Specifically, we set nucleus, Ψ6 (αn p ), that provide a good description Li 1 1 S(E) σ(E) = ------P(E), (38) of its structure and of many processes in which this E nucleus participates [72Ð74] and with the realistic deu- Ψ where the structural factor S(E) is weakly dependent on teron wave functions D(n2 p2) obtained by using the energy or is taken to be a constant, a dominant energy Reid soft-core nucleonÐnucleon (NN) potential [75]. In our approach, both the nuclear and the Coulomb pair dependence being absorbed primarily in the potential- α barrier penetrability factor P(E). potentials (Vij and Uij, respectively) of the N and NN It is clear, however, that, if the potential-barrier interactions are weighted with the internal wave functions of the nuclei involved in the reactions being con- shape or a method for computing the penetrability P(E) 6 is chosen inappropriately, the relevant error will trans- sidered. In the D + Li system, the binary interaction is late into the factor S. Thereby, the factor S(E) will represented as acquire an additional unphysical energy dependence V 6 (R) = 〈Ψ6 (αn p )Ψ (n p )|V D Li Li 1 1 D 2 2 nucl adversely affecting the extrapolation of cross sections (41‡) |Ψ (α )Ψ ( )〉 to the deep-subbarrier region. We recall in this connec- + UCoul 6 n1 p1 D n2 p2 , tion that, since direct measurements at low energies of Li E < 300 keV usually involve sizable errors, any uncon- 6)In passing, we would like to mention the independent investiga- trollable energy dependence in S(E) will inevitably tion of Rowley and Merchant [71], who analyzed the effect of the impair the accuracy of the extrapolation. By way of shape of the barrier on its penetrability in astrophysical reactions.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NUCLEAR-PHYSICS ASPECTS OF CONTROLLED THERMONUCLEAR FUSION 2061 where ê(Ö) 100 V = V α + V α + V + V + V + V , nucl n2 p2 n1n2 n1 p2 p1n2 p1 p2 ...... (41b) êpres ...... UCoul = Uα p + U p p ...... 2 1 2 ...... and R is the distance between the D and 6Li centers of ...... ê mass. It is important to note that the Ψ6 wave func- ...... semicl Li –3 tions in (41a) were found by solving the three-body 10 ...... problem with the same αN and NN potentials (41b) as ...... those used in the folding model. Thus, the shape of the ...... potential barrier in the D + 6Li system was calculated in a fully self-consistent way without resort to adjustable ...... parameters (ab initio calculation); hence, it can be thought to be quite reliable. 10–6 For the realistic barrier in (41), the tunneling factors P(E) are depicted in Fig. 3. The Psemicl curve corresponds to the penetrability factor found in the semiclassical approximation [76],

 R   2  10–9 ( ) ∼ 2 µ( ( ) ) 0 100 200 300 Psemicl E expÐ-- ∫ 2 V 6 R Ð E dR, (42)  ប D Li  Ö, keV  R1  Fig. 3. Precise and semiclassical penetrability [P (E) and where µ is the reduced mass of the nuclei, E is their pres Psemicl(E), respectively] of a realistic potential barrier in the c.m. energy, and R1, 2 are the coordinates of the classical 6 turning points [they are determined from the condition D + Li system.

V 6 (R ) = V 6 (R ) = E]. The P curve corre- D Li 1 D Li 2 pres êpres(Ö)/êsemicl(Ö) sponds to a precise calculation where the tunneling penetrability factor is defined as the ratio of the flux 2 densities for the transmitted and reflected waves. It can be seen that these two approaches lead to different 1 energy dependences for the tunneling factor P(E), as is additionally illustrated in Fig. 4, which displays the ratio of the penetrability factors Ppres(E) and Psemicl(E). 0 100 200 300 This comes as no surprise: the semiclassical expression Ö, keV (42) is highly accurate only in the case of smooth Fig. 4. Ratio P /P for a realistic potential barrier in potentials (that is, potentials slowly varying with dis- 6 pres semicl tance), but the interaction potential in (41) does not the D + Li system as a function of energy. possess this property—it decreases fast to the left of the maximum (see Fig. 2), not ensuring the required accu- The equilibrium rate parameters 〈σv〉 calculated for the racy. Thus, we have revealed that there arise sizable (d, nτ) and (d, pt) channels of D + 6Li interaction, which errors when precise quantum-mechanical penetrability are responsible for the production of 3He and tritium, of the actual potential barrier is replaced by that in the ≡ are quoted in Table 2. In the ion-temperature range Ti semiclassical approximation. kT = 1Ð100 keV, these results exceed those presented in The above barrier corrections proved to be of impor- [77] by about 10Ð40%, the difference between the rate tance in describing the reaction between loosely bound parameters found from the two calculations becoming nuclei D and 6Li, which are characterized by a high greater with decreasing ion temperature. degree of clustering. We will demonstrate this by considering the example of the reaction rate in a D + 6Li In the above methods for extrapolating relevant plasma under the condition of thermal equilibrium at cross sections, the slowly changing structural factor temperature kT. In this case, the velocity distribution of S(E) is parametrized analytically—for example, in the particles has a Maxwellian form, and the reaction-rate form of a Padé approximant. In many cases, this parameter 〈σv〉 is given by approach unfortunately does not ensure the required ∞ accuracy and reliability of extrapolation, since the 1/2 errors in extracting the factor S from experimental data 〈σ 〉  8  ( )Ð3/2 σ( ) ÐE/kT v = µ----π-- kT ∫E E e dE. (43) often grow so sharply with decreasing energy that the 0 confidence interval ∆S for a determination of S(0) in an

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Table 2. Rate parameters 〈σv〉 of the reactions the a + b channel. Although this approach had been 6Li(d, nτ)4He and 6Li(d, pt)4He for the case of a Maxwell ve- known for quite a long time, its specific realization on locity distribution of ions the basis of a new method for constructing the multichannel potential being discussed was proposed only in 〈σv〉 3 Ion temperature Rate parameter , cm /s recent years (see [78Ð80]). The potential in question is Ti, keV (d, nτ) (d, pt) based on a direct iterative solution to the inverse scattering problem. This solution proceeds directly from 1 2.03 (Ð31) 2.46 (Ð31) experimental data on cross sections, as well as on vector and tensor analyzing powers. Although this method 2 1.29 (Ð27) 1.59 (Ð27) has hitherto been harnessed only in elastic-scattering 3 8.67 (Ð26) 1.08 (Ð25) problems featuring channel coupling, it can obviously 4 1.20 (Ð24) 1.52 (Ð24) be applied to the general problem of predicting near- threshold cross sections for rearrangement reactions. 5 7.71 (Ð24) 9.83 (Ð24) 6 3.12 (Ð23) 4.03 (Ð23) 4. TEMPERATURE DIAGNOSTICS 7 9.49 (Ð23) 1.23 (Ð22) OF A THERMONUCLEAR PLASMA 8 2.35 (Ð22) 3.14 (Ð22) BY THE GAMMA-RAY METHOD 9 5.11 (Ð22) 6.68 (Ð22) A determination of the ion temperature and of its profiles and time evolution is an important problem of 10 9.77 (Ð22) 1.27 (Ð21) diagnostics of a thermonuclear plasma. At present, a 20 3.82 (Ð20) 5.43 (Ð20) great number of diagnostic procedures that primarily employ data from atomic physics have been proposed 30 2.03 (Ð19) 3.22 (Ð19) for measuring the parameters of a thermonuclear 40 5.54 (Ð19) 9.70 (Ð19) plasma. By and large, these procedures furnish only 50 1.11 (Ð18) 2.05 (Ð18) indirect information about the ion plasma component, but direct nuclear physics methods that make it possible 60 1.87 (Ð18) 3.63 (Ð18) to monitor straightforwardly the dynamics of ions may 70 2.79 (Ð18) 5.65 (Ð18) prove to be of paramount importance here. Of particular value are nuclear-physics methods relying on reac- 80 3.91 (Ð18) 8.01 (Ð18) tions between charged particles. Such reactions may 90 5.14 (Ð18) 1.08 (Ð17) proceed in a fuel initially, but they can also be activated in a dedicated way by introducing diagnosing admix- 100 6.54 (Ð18) 1.37 (Ð17) tures in a plasma. Note: In the second and in the third column, the numbers in front Let us consider the possibility of determining the of parentheses and the numbers presented parenthetically stand, respectively, for the mantissa and the integral (nega- ion temperature of a fuel, Ti, by using processes that tive) power of ten in the floating-point representation of produce high-energy photons freely escaping from the numbers—for example, 2.03 (Ð31) denotes 2.03 × 10Ð31. reactor core both in the case of a magnetic plasma confinement and in the case of inertial confinement. We restrict ourselves to considering DT and D3He fuels, extrapolation to the threshold becomes commensurate which have received so far the most detailed study, and with the S(0) value itself. choose, for activating (diagnosing) admixtures, 3He and 6 isotopes for DT mixtures and T and 6 for In those frequent cases where there are no pro- Li Li 3 mixtures. Presented below are the radiative-cap- nounced compound resonance states near the threshold D He ture reactions proceeding in the systems under consid- (that is, where the existing near-threshold resonances eration and involving the emission of photons of ener- are of a so-called potential character—this is so in sys- 3 3 gies Eγ in excess of 10 MeV [processes (15) and (17) tems like D + D and He + He), it is possible to develop could also be included in this list]: an alternative, very promising approach to extrapolating cross sections, which is especially efficient for reac- T(d, γ)5He + 16.70, 3He(t, γ)6Li + 15.79, tions featuring spin-polarized particles. It consists in using well-known data at not very low energies of E ~ D(d, γ)4He + 23.85, T(t, γ)6He + 12.31, 0.5–5 MeV to construct a reliable multichannel interac- 3 5 6 8 tion potential with allowance for important reaction He(d, γ ) Li + 16.39, Li(d, γ ) Be + 22.28, (44) channels a + b c + d (i = 0, 1, …, n). In contrast to 3 6 6 9 i i He(τ, γ ) Be + 11.49, Li(τ, γ ) B + 16.60, the scattering amplitude, this potential is in general a very smooth function of E, and the threshold energy is 6Li(d, p)7Li*[0.478] ( 7Li + γ + 0.478) + 4.55, not a peculiar point for it. Therefore this potential can be used to predict cross sections near the threshold for 6Li(d, n)7Be*[0.431] ( 7Be + γ + 0.431) + 2.95

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(the energies released in these reactions are given in slightly in subbarrier region [82], so that the question of MeV). Of the reactions listed here, T(d, γ), D(d, γ), and whether this distinction is sufficient for attaining the 3He(d, γ) have received adequate experimental study; required degree of accuracy in determining the temper- the same can be said about the last two processes 6Li(d, ature is still open. Anyway, a direct measurement of pγ) and 6Li(d, nγ) [22, 81Ð83]. For the first time, the use photon yields from nuclear reactions proceeding in a of these reactions for a hot-plasma thermometer was thermonuclear reactor would nevertheless furnish proposed in [45, 84Ð86]. unique information about the most important properties of plasmas. The idea of determining the ion temperature Ti is based on the fact that the yield of photons from A + B interactions, Yγ(AB), is proportional to the concentra- 5. CONCLUSION tion of reaction products, n , and to the γ-channel A(B) In the present review article, we have considered reactivity 〈σv〉 , which depends greatly on T . By ABγ i some important nuclear-physics facets of the problem choosing an appropriate combination of a few reactions of controlled thermonuclear fusion. Attention has been from (44), it is possible to find the required tempera- given primarily to (i) choosing an optimum thermonu- ture, irrespective of plasma-density values [45]. By clear cycle for future thermonuclear power engineer- way of example, we indicate that, for a DT mixture, a ing, (ii) analyzing the problem of a reliable extrapola- simultaneous detection of photon yields in the reactions tion of measured cross sections to the deep-subbarrier T(d, γ) and D(d, γ) leads to energy region, and (iii) discussing gamma-ray diagnos- 〈σ 〉 tics of a hot plasma on the basis of radiative-capture Yγ (TD) 2n v TD T γ ( ) reactions. ------(------)- = ------〈---σ------〉------= f Ti , (45) Yγ DD nD v DDγ We have shown that special features of nuclear structure—for example, pronounced clustering in light where the ratio of the concentrations nT and nD can be nuclei—often play an important role in the low-energy considered as a constant to a high precision. It follows behavior of relevant cross sections for nuclear reactions that the ratio of the yields of photons from the above and can also affect the choice of strategies for investi- two reactions is a known function of the ion tempera- gations in these realms. At the same time, these factors ture Ti. Therefore, a simultaneous independent mea- have been virtually disregarded not only in monographs surement of the two photon fluxes provides a means for and review articles devoted to the problem of controlled determining T . The monitoring of three photon fluxes i γ γ γ thermonuclear fusion but also in a large number of orig- from the reactions T(d, ), D(d, ), and T(t, ) could be inal specific studies. Fortunately, interest in these issues very useful, but there are no nuclear data on the cross has quickened somewhat in recent years, and a series of sections for the last process. studies, which are surveyed in our article, have been The smallness of cross sections is a drawback com- performed along these lines. mon to all radiative-capture reactions, which are gov- We have conducted a comparative nuclear-physics erned by electromagnetic interaction. For example, the analysis of the following eight promising thermonu- ratio of the branching fractions for the radiative and the clear fuels based on isotopes of light elements: DT, DD, × –5 main channel at low energies is about 5 10 for D + D3He, D6Li, DT6Li, H6Li, H11B, and H9Be. Of these, 3 –7 T and D + He interactions and about 10 for D + D DT, DD, and D3He mixtures have received the most interactions [22]. Therefore, the problem of experimen- detailed study. Among all cycles considered above, a tally detecting photons in such reactions is nontrivial. DT fuel possesses the lowest ignition temperature (T < However, Lerche et al. [87] reported that they recorded 10 keV). However, practical uses of this fuel are 16.7-MeV gamma radiation from D + T interactions at restricted by many disadvantages like the presence of the Nova laser facility and used it in studying the intense neutron fluxes carrying about 80% of the dynamics of burning. energy released in D + T interactions, a relatively low The last two processes in (44), which yield mono- reactor efficiency, and the fact that one has to deal with chromatic photons of energy Eγ about 500 keV, do not radioactive tritium. In what is concerned with radiation have this drawback. These reactions are governed by safety and with the efficiency of conversion of nuclear strong interaction, and the relevant low-energy cross energy into electric energy, a D3He fuel is preferable sections are relatively large. Their photon yield as esti- because, here, neutrons carry only about 3% of the total mated for a facility implementing a magnetic plasma energy release. Unfortunately, the ignition of this fuel confinement [45] is quite sizable even at a 1% concen- requires higher plasma temperatures of a few tens of tration of 6Li. That the reactions 6Li(d, nγ) and 6Li(d, pγ) keV; in addition, there are practical difficulties in represent different output channels of the same process obtaining the 3He isotope, one of the fuel components. is also of importance, because any current fluctuation Additional gain in the energy release can be achieved of the plasma density or some other parameter of the through the use of spin-polarized fuels, where the cross plasma under study would not affect the accuracy in sections for the two reactions increase by about 50%. determining Ti. However, the energy dependences of However, it is difficult to estimate this gain definitively, the cross sections σ(E) for these channels differ only since the result depends on the role of depolarization

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2064 VORONCHEV, KUKULIN effects. Another objective pursued by the use of spin- the ion temperature, irrespective of the initial plasma oriented nuclei is that there are hopes for suppressing densities and their current fluctuations. the neutron flux associated with the D + D by-process. The degree of this suppression, which depends on the cross sections for low-energy D + D interactions ACKNOWLEDGMENTS involving polarized deuterons, is still hotly debated; We are grateful to many physicists, including sometimes, the estimates presented in the literature are V.I. Kogan, V.N. Pomerantsev, G.V. Sklizkov, V.A. Chu- at odds with one another. yanov, and H. Nakashima, with whom we discussed, over various periods of time, the range of problems A high ignition temperature is the main (and some- considered in the present review article. times the only) drawback of non-hydrogen fuels. If, Special thanks are due to V.M. Krasnopolsky, however, operating temperatures of a few hundred keV Y. Nakao, and K. Kudo, who were our collaborators in are achieved, the use of alternative fuels can prove to be many original studies. advantageous from the ecological and economical points of view. For example, the cheap and “pure” fuel H11B, for which the adopted estimate of the neutron REFERENCES contribution to the total energy release is as low as 1. L. A. Artsimovich, Controlled Thermonuclear Reac- 0.1%, is very appealing in this respect. Unfortunately, tions, Ed. by A. Kolb and R. S. Pease (Fizmatgiz, Mos- the kinetics of nuclear processes has been studied much cow, 1961; Gordon and Breach, New York, 1964). more poorly in alternative fuels than in hydrogen fuels. 2. Y. Nakao, T. Honda, Y. Honda, et al., Fusion Technol. 20, Virtually no investigations have been performed for 824 (1991). 9 σ 9 H Be fuels, although v values for H + Be interactions 3. V. T. Voronchev, V. I. Kukulin, and V. M. 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Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2067Ð2072. From Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2163Ð2168. Original English Text Copyright © 2000 by Collins, Davanloo, Iosif, Dussart, Hicks, Karamian, Ur, Popescu, Kirischuk, Pouvesle, McDaniel, Crist.

NUCLEI Experiment

Study of the Gamma Emission from the 31-yr Isomer of 178Hf Induced by X-Ray Irradiation* C. B. Collins1), F. Davanloo1), M. C. Iosif1), R. Dussart1), J. M. Hicks1), S. A. Karamian**, C. A. Ur2), I. I. Popescu2), V. I. Kirischuk3), J. M. Pouvesle4), P. McDaniel5), and C. E. Crist6) Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received September 20, 1999; in ﬁnal form, January 14, 2000

Abstract—A sample containing 6.3 × 1014 nuclei of the 16+ isomer of 178Hf having a half-life of 31 yr and an excitation energy of 2.446 MeV was irradiated with x-ray pulses from a device operated at 15 mA to produce bremsstrahlung with an endpoint energy of 90 keV. The gamma spectra of the isomeric target were taken with a Ge detector. The intensity of the 325.5-keV (6+ 4+) transition in the ground-state band of 178Hf was found to increase by about 2%. Such an enhanced decay of the 178Hf isomer is consistent with an integrated cross section value of 3 × 10–23 cm2 keV if resonance absorption occurs within energy ranges corresponding to the maxima of the x-ray ﬂux, either near 20 keV or at the energies of the characteristic emission lines of W. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION excite some fraction of a high-K isomeric population to the K-mixing level. Then, the decay to the ground state The four- and ﬁve-quasiparticle isomers of Lu, Hf, γ and Ta are of interest because they have relatively long via one or more cascades could subsequently release lifetimes for states at excitation energies of 2 to 3 MeV. the total energy of the isomer plus that of the absorbed They are referred to as K isomers because spontaneous trigger photon. The types of K-mixing states needed in such schemes to induce the decay of nuclear isomers radiative decay is hindered by structural changes for- 180 174 bidden by K-quantum numbers. In this mass region, have been reported [2] in Ta and described in Hf nuclei are deformed and the projection of the total and other isomers [3]. angular momentum onto the symmetry axis contributes In 1999, the use of soft x-ray irradiation to enhance this quantum number K, which can change by no more the decay of the 178Hf isomer was reported in [4, 5]. In than the multipolarity of the electromagnetic transition. that study, the continuous x-ray spectrum was most Decays from a high-K isomer to the rotational states of intense in the range between 20 and 60 keV. While the a low-K band are forbidden; therefore, a relatively long energy of the particular component causing the transi- lifetime is inevitable. The most interesting example tion that initiated the process was not determined, the may be the 31-yr, four-quasiparticle 178Hf isomer hav- data were analyzed under the assumption that the ing an excitation energy of 2.446 MeV. energy lies near the 40-keV mean value of the spectral Proposals to trigger the energy release of a nuclear distribution. isomer by exciting it to some higher level associated with freely radiating states have been known for over a The two-step process leading to the excitation of the decade [1]. To be efﬁcient, such schemes should be intermediate level and following the decay to the applied at an energy near that of the K-mixing state of ground state is assumed here to be the same as earlier for 180mTa [2]. The integrated cross section of σΓ ≈ the isomer. It was proposed in [1] to use the resonant − absorption of x rays from a bremsstrahlung source to 10 25 cm2 keV measured for 180mTa corresponds to the presence of an activation level of width Γ ≈ 0.5 eV. The same 178m * This article was submitted by the authors in English. 2 σΓ ≈ Ð21 2 1) Center for Quantum Electronics, University of Texas at Dallas, width assumed for Hf leads to 10 cm keV Richardson, USA. when the cross section in the BreitÐWigner resonance 2) H. Hulubei National Institute of Physics and Nuclear Engineer- is used. This is in accord with the experimental results ing and IGE Foundation, Bucharest, Romania. 180m 3) reported in [4, 5], but the rescaling from Ta to Institute for Nuclear Research, National Academy of Sciences of 178m Ukraine, pr. Nauki 47, UA-252028 Kiev, Ukraine. 2Hf looks straightforward. The level schemes of the 4) GREMI, CNRSÐUniversite d’Orleans, France. two nuclei are very different, and so are the energy and 5) Air Force Research Laboratory, DEPA, Kirtland Air Force Base, Albuquerque, New Mexico, USA. the wavelength of the incident radiation: nearly 3 MeV 180 178 6) Sandia National Laboratory, Albuquerque, New Mexico, USA. for Ta [2] and about 40 keV for Hf [4, 5]. The inte- ** e-mail: [email protected] grated cross section (ICS) for photon resonance absorp-

2068 COLLINS et al.

4-mm Pb shield 2. EXPERIMENTAL DETAILS Cone of irradiation 3-mm Cu The experimental work was performed at the Center 41.6 cm +2-mm Pb for Quantum Electronics, University of Texas at Dallas. Pb shield The irradiating bremsstrahlung beam was provided by an x-ray unit operated at 15 mA and an endpoint energy up to 90 keV. The device was operated in a way that ensured a duty cycle for the irradiation of about 0.6%. The irradiated sample consisted of a sealed plastic 178m target containing 6.3 × 1014 2Hf isomeric nuclei x-ray 133Ba position placed in a 1-cm diameter well. The main radioactive head Proximate position contaminants in the sample were the 172Hf nuclide and its daughters at a level of activity comparable to the intensity of the 31-yr spontaneous decay of the 178Hf isomer. The sample was placed at 5.5-cm distance from the emission point of the x-rays, and the only absorp- Fig. 1. Scheme of the experimental arrangement showing tion was due to the glass of the x-ray tube and the 2-mm the geometric placements and dimensions of the components. plastic sealing of the x-ray device. The absorption in the sealing of the sample was negligible. Figure 1 shows the experimental arrangement. The tion has been recently discussed in [6], and the possibil- HP coaxial Ge detector was placed at 41.6-cm distance ity of modifying the recommended equations was con- from the sample being irradiated. The detector had the cluded. More experimental information is necessary for efficiency of 10% relative to the standard NaI for 60Co calibrating ICS at photon energies in the region Eγ < lines. A shielding built of 3-mm Cu and 2-mm Pb was 100 keV. used in order to prevent detection of scattered x-rays in the Ge detector. Thus, the pulse rate of scattered x-rays On the other hand, the systematics of nuclear transi- allowed to get into the detector was measured to be tion strengths do not predict the widths of levels as about 1500 counts/s. The count rate produced by the large as those known from the 180mTa experiment [2] radioactive target in the absence of the x-rays was about 4000 counts/s. Such an experimental arrangement 178m and obtained for 2Hf [4, 5]. The value of Γ = 0.5 eV resulted in a low value of the total dead time of the corresponds to a short lifetime (≈10Ð15 s), while a typi- acquisition tract of about 9% during the irradiation. The energy and efficiency calibration of the Ge detector was cal lifetime of nuclear states at excitation energies of 2 done using standard calibration sources, 60Co, 133Ba, to 3 MeV is two orders of magnitude greater. Accord- and 137Cs. We estimated the maximum absolute detec- ingly, σΓ ≈ 3 × 10Ð27 cm2 keV could be predicted for tion efficiency to be of about 1.5 × 10Ð4 at a γ-ray 180mTa by using the value of B(E1) = 0.01 Weisskopf energy near 300 keV in the described geometry with units recommended in systematics. This is lower than absorbers. the experimental value by one order of magnitude. The Data acquisition was enabled only when a signal corresponding disagreement is even more impressive in was presented from a pÐiÐn diode which monitored the the 178Hf case, mainly because of the deficit of levels x-ray beam. The signal from the diode was processed in with appropriate quantum numbers for the decay of the order to produce a 4-ms gate centered on the burst of the x-ray flux. We recorded γ-ray energies up to 2 MeV high-spin intermediate state excited after the absorp- with amplification set to give 0.25 keV/channel, allow- 178m tion of a photon by the 2Hf isomer. As a conse- ing for a good analysis of the possible admixtures in the quence, the properties of intermediate K-mixing states lines of interest. Spectra were stored during each three hours and gain matched using internal γ lines before appear as an extraordinary phenomenon that naturally adding them in order to compensate a weak variation stimulates more focused studies. In particular, more (within 0.05%) of the electronics gain. 178m details on the enhancement of the decay of the 2Hf Special attention was devoted to the measurement of isomer are necessary in order to clarify the mechanism the incident x-ray flux. For this purpose, the x-ray of the x-ray induced deexcitation of the isomer. By device was aligned to direct the flux to the Ge detector placed at 8.2-m distance. The direct x-ray flux could focusing upon a confirmation of the work previously reach the active area of the detector only through a reported [4, 5], while extending it to include a study of 1-mm-diameter hole placed near the detector. High a fragment of a cascade not present in the spontaneous count rate still required the application of absorbers for decay of the isomer, the present study is aimed at meet- the measurement of the direct radiation spectrum. The ing these requirements. final spectra could be reconstructed from a few mea-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 STUDY OF THE GAMMA EMISSION 2069 surements with different absorbers. They are shown in Spectral density, photons/keV cm2 s Fig. 2 for two conditions, with and without a 2.7-mm equivalent thickness of Al available for covering the 1011 output window. The spectrum taken with 1.5-mm Cu absorber is given in the inset. It was one of the compo- 12 nent measurements from which the composite spec- 10 1010 trum was assembled. In previous work [4, 5], the 178m 2Hf sample was irradiated with the presence of the 1 Al absorber between x-ray tube and the sample, while 109 in the experiment reported here it had been removed. One can see from curve 1 in Fig. 2 that there is a high intensity, above 4 × 1011 photons cmÐ2 keVÐ1 sÐ1, at low 108 energies near 20 keV; drastic decrease of the intensity 1011 with increasing Ex values; and a clear manifestation of 40 60 80 the characteristic K-X lines of W (material of the con- verter in the x-ray device). All these special features of the incident radiation are important for conclusively 2 determining the integrated cross section, as discussed below.

3. RESULTS 1010 Acquired with the Ge detector, spectra of the induced emission of γ radiation generally resembled those obtained in the earlier work [4, 5]. However, in the present experiment, normalization of the spectra taken with and without x-ray irradiation was facilitated by the deliberate inclusion of lines from the 133Ba source placed near the Hf target, but not irradiated. Those fiducial lines were within about 30 keV of the 109 325.5-keV (6+ 4+) component of the ground-state band (GSB) and therefore reduced any effects of a drift or a nonlinear energy dependence of the efficiency. An empty target holder, the “blank,” of a mass and construction similar to the one carrying the isomeric nuclei 10 30 50 70 90 was available for use. Comparative measurements Ex, keV showed that over 95% of the elastic and inelastic scattering of the irradiating beam arose from the mass of Fig. 2. Photon spectral flux expected at the inbeam position of the isomeric target. These experimental data for a 90-keV the holder, not from its contents. endpoint were measured by using a Ge detector from input Three geometric arrangements were important dur- attenuated with a pin hole and placed at 8.2-m distance from ing the collection of data: (1) “inbeam,” when the iso- the x-ray tube. The radiation spectra correspond to this exper- meric target was centered in the cone of irradiation as iment (curve 1) and the measurements in [4, 5] (curve 2), where the Al absorber was used. The inset shows the raw shown in Fig. 1; (2) “outbeam,” when the target was data taken with a 1.5-mm Cu absorber. placed out of the beam of x-rays at the position denoted as “proximate” in Fig. 1 and the “blank” target holder replaced it in the cone of irradiation; and (3) “baseline,” were as nearly equal as could be arranged. The counts when the isomeric target was placed in the position of in the areas under the relevant peaks are summarized in the cone of irradiation, but the x-ray source was turned Tables 1 and 2. In this case, more counts were used for off. During analysis, both inbeam and outbeam spectra a better accuracy of the baseline spectrum accumulated. were scaled to the baseline spectrum, so that the areas γ of the fiducial line in each spectra were the same. It was mentioned above that the efficiency of the - ray detection had a maximum near 300 keV, decreasing Figure 3 shows the results of 16 h of acquisition to lower energies due to a presence of absorbers and to time, during which there were 340 s of actual counting higher energies due to an intrinsic efficiency of the Ge time enabled by the gate coincident with the detection detector. Accordingly, in this series of measurements, of x-rays. From top to bottom are shown the inbeam, we concentrated to detect the enhancement of the outbeam, and baseline spectra: when the spectra were 325.5-keV line providing the best resolution and rea- subtracted, the data were acquired so that the total num- 178m bers of photons collected in the 356.0-keV line of 133Ba sonably good statistics. Some other lines of the 2Hf

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2070 COLLINS et al.

Counts 200 (a)

100

–100

200 (b)

100

–100

2000 302.9 372 + 373 377.5 383.9 323.9 296.8

0 300 320 340 360 380 Eγ, keV

178m Fig. 3. Spectra of the 2Hf from the irradiated target: (a) target inbeam minus scaled baseline, (b) target outbeam minus scaled baseline, and (c) baseline spectrum. The region shown could be dependably normalized by comparing the areas under the 356.0-keV line of 133Ba and included the 325.5-keV (6+ 4+) component of the GSB of 178Hf. decay could be also enhanced (for instance 213.4 and line 210.3 keV was known as a member of the Kπ = 6+ 426.6 keV) under irradiation, but a statistical accuracy band in 178Hf, but it was not previously seen in sponta- for them was lower than that for the 325.5-keV line. neous decay of the isomer [7]. Other transitions by this

In addition, the difficulties in establishing precise 178m2 fiducial lines exist in other regions of the spectra; dif- band were observed neither for the Hf decay pre- ferences in areas under peaks could not be attained with viously nor in present measurement. the same high level of confidence as accomplished for According to [7], very weak line of 172Lu should be the 325.5-keV line. Previous reports [4, 5] suggested detected at Eγ = 210.28 keV, and it was really seen in the that not all components of the spontaneous decay cas- baseline spectrum. However, its intensity, being in cade were enhanced by the x-rays. If several transitions agreement with [7], was fourfold lower than that in the feeding the GSB in spontaneous decay are not enhanced, spectrum taken “inbeam.” Thus, the 210.6-keV line there naturally arises the question of what channels are shown in Fig. 4 cannot be explained by a contribution involved in the induced decay of the isomer. from 172Lu. An analysis of the spectra suggested several “new” components, some of them were obviously seen in the spectra stored during 1998 series of experiments, but 4. DISCUSSIONS AND CONCLUSION the statistics was not enough to discuss them in papers The first reports [4, 5] of the great cross section for [4, 5]. Figure 4 shows the best of such lines observed in 178m2 the experiment reported here for Eγ = 210.6 keV. The the decay of the Hf isomer induced by low-energy

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 STUDY OF THE GAMMA EMISSION 2071

Table 1. Comparison of the gamma emission from the target irradiated at the inbeam position with the baseline obtained without irradiation

With irradiation Baseline Eγ, keV Nucleus Excess, counts area area norm. area* 296.8 178Hf 1692 (68) 14925 (201) 1685 (27) 7 (73) 302.9 133Ba 6503 (103) 58052 (304) 6554 (64) Ð51 (121) 323.9 172Lu 1403 (71) 12891 (222) 1455 (28) Ð52 (76) 325.5 178Hf 17035 (146) 147952 (451) 16703 (148) 332 (208)** 356.0 133Ba 22617 (180) 200333 (482) Normalizing line 372 + 373 172Lu 2439 (84) 21860 (250) 2468 (90) Ð29 (123) 377.5 172Lu 3098 (75) 27445 (225) 3098 (36) 0 (83) 383.8 133Ba 3225 (76) 28160 (261) 3179 (40) 46 (86) * Normalized to the 356-keV line of 133Ba. Statistical inaccuracy of normalization is included. ** Estimation of the effect: (2.0 ± 1.2)%.

Table 2. Comparison of gamma emission acquired from the target in the outbeam geometry with the baseline obtained without irradiation (normalization is identical to that in Table 1) Baseline Eγ, keV Nucleus Outbeam area Excess, counts area norm. area 296.8 178Hf 785 (52) 20826 (262) 832 (14) Ð47 (54) 302.9 133Ba 3021 (70) 75754 (352) 3028 (36) Ð7 (79) 323.9 172Lu 722 (49) 17737 (250) 709 (13) 13 (51) 325.5 178Hf 7961 (103) 199062 (517) 7957 (88) 4 (135) 356.0 133Ba 10408 (110) 260372 (551) Normalizing line 372 + 373 172Lu 1258 (57) 29652 (284) 1185 (55) 73 (79) 377.5 172Lu 1428 (51) 36429 (260) 1456 (19) Ð28 (54) 383.8 133Ba 1509 (52) 36928 (261) 1476 (19) 33 (55)

178m Table 3. Integrated cross section σΓ calculated from a 2% enhancement of the 325.5-keV 2Hf line under the assumption of different activation-energy values

Ex, keV 20 30 40 50 Kα1 (W)* 60 Kβ1 (W)* 70 σΓ, cm2 keV 3.2 × 10Ð23 5.4 × 10Ð23 1.6 × 10Ð22 3.4 × 10Ð22 1.6 × 10Ð23 3.9 × 10Ð22 3.3 × 10Ð23 1.1 × 10Ð21 Flux with no filter σΓ, cm2 keV 3.5 × 10Ð22 1.2 × 10Ð22 2.4 × 10Ð22** 4.5 × 10Ð22 2.0 × 10Ð23 4.7 × 10Ð22 4.0 × 10Ð23 1.3 × 10Ð21 Flux with 2.7-mm Al filter * Under the assumption that the natural width of the emission line is 50 eV. ** For a comparison with [5]. x-rays seems to be unexpected because it was not pre- [4, 5]. Table 1 shows that with the same type of small dicted by any model. The problem has not been x-ray generator traditionally used in dental medicine resolved by the results of the experiment reported here. enhancements of the order of 2% can be induced in the However, some illumination of the unexpected nature rate of spontaneous decay of the Hf isomer. As can be of the phenomenon has been realized, it is still instructive to examine the details. seen from Table 2, there is no comparable value of a spurious enhancement found when the empty target Of prime importance is the fact that general phe- 178m nomenology has been reproduced in accordance with holder was irradiated while the 2Hf sample was

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Counts in the x-ray spectrum with resolution of 0.9 keV. The 5000 natural width is much smaller, and, in reality, the flux is respectively higher in the characteristic X-ray peaks. For the case of isomeric 178Hf, we have confirmed 30000 4600 210.6 that the irradiation by photons with the energy of the order of 20Ð60 keV can induce the prompt release of Hf the 2.446 MeV stored by the isomer into freely radiat- 213.4 ing states. This is an energy gain of about 60. 4200 Hf 20000 216.7 Further research is needed to provide greater precision to the measurements of the transition energy to the K-mixing level and to clarify properties of the cascade 3800 Lu 10000 203.4 feeding the GSB. Such data will then facilitate a better understanding of these first evidences of the triggering of induced gamma emission from the 31-yr isomer of 3400 178 0 Hf with very low energy photons through large inte- 202 206 210 214 218 grated cross sections, σΓ ≥ 2 × 10Ð23 cm2 keV. Eγ, keV

Fig. 4. Spectra obtained under the different conditions. From ACKNOWLEDGMENTS top to bottom: inbeam spectrum, baseline spectrum normalized to the previous one by the peak counts in the 213.4 line We are grateful to the US Air Force Office of Scien- (left scale for both), and full range baseline spectrum (right tific Research’s European Office of Aerospace scale). The FWHM values for the marked peaks are 1.10, Research and Development (EOARD) (contract 0.91, 1.06, and 1.04 keV from left to right, as given by the fit. no. F61708-98-W0027) and the US Air Force Office of Scientific Research (AFOSR contract no. F49620-99- 1-0082) for support of this experiment; we also grate- placed in the outbeam geometry. The excess counts in fully acknowledge the supportive efforts of our partic- the 325.5-keV line are essentially zero, as can be seen ular colleagues, students, and staff members, who as well in Fig. 3. assisted so generously in the collection and analysis of The yield of triggering events would equal the prod- the data. uct of the number of isomeric atoms in the target, the spectral flux density from Fig. 2 at the appropriate energy, and the unknown integrated cross section, σΓ, REFERENCES for the branch of the excitation of a K-mixing level that 1. C. B. Collins et al., J. Appl. Phys. 53, 4645 (1982). ends in a state other than that of the initial isomer. Since 2. C. B. Collins, C. D. Eberhard, J. W. Glesener, and each quantity is known, except for the integrated cross J. A. Anderson, Phys. Rev. C 37, 2267 (1988). section for the “triggering branch,” the latter one can be obtained if the transition energy is estimated. Possible 3. P. M. Walker et al., Phys. Lett. B 408, 42 (1997). values of σΓ are summarized in Table 3. The specific 4. C. B. Collins et al., Phys. Rev. Lett. 82, 695 (1999). value of σΓ is strongly dependent on the position of an 5. C. B. Collins et al., Laser Phys. 9, 8 (1999). intermediate level because of the strong variation of the 6. S. A. Karamian, Invited Talk at LASERS’99 Conference, x-ray flux with energy. Assuming the resonance band Quebec, Canada, 1999. lies near the maxima of flux, one can derive σΓ ≈ (2Ð3) × 7. NNDC Online Data Service, Brookhaven National Lab- 10Ð23 cm2 keV. The emission lines of W were detected oratory, http://www.nndc.bnl.gov

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NUCLEI Theory

Investigating the Hydrogen Atom in Crossed External Fields by the Method of Moments* V. A. Gani1), ** and V. M. Weinberg*** Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received February 9, 2000; in ﬁnal form, April 26, 2000

Abstract—Recurrence relations of perturbation theory for the hydrogen ground state are obtained. With the aid of these relations, polarizabilities in constant mutually perpendicular electric and magnetic ﬁelds are computed up to the 80th order of perturbation theory. The high-order asymptotic expression is compared with its semiclassical estimate. For the case of an arbitrary relative orientation of external ﬁelds, a general sixth-order formula is given. The energy and the width of the ground state are obtained by means of a perturbation-series summation. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION present study, the method of moments permits computing sufficiently high orders of this expansion. The hydrogen atom in constant uniform electric and The high-order asymptotic behavior can be obtained magnetic fields still remains the subject of theoretical by using the imaginary-time method [13Ð15]. This investigations. For example, the authors of [1] devel- asymptotic behavior is determined by the contribution oped a recurrent nonperturbative method for construct- of an extreme subbarrier classical trajectory to the ing a convergent double series representing the exact atom-ionization probability [16, 17]. A pair of extreme wave function of the hydrogen atom in a magnetic field. paths replaces this trajectory at some value of the ratio For a more extensive discussion of the problem, the of external fields, γ = Ᏼ/Ᏹ. The γ dependence of high- reader is referred to the review article of Lisitsa [2]. order terms in a perturbation series reflects this change The well-known technical problem of the impossi- of extreme trajectory and should be especially sharp for bility of separating the variables only stimulates the mutually perpendicular external fields. This is the case application of new investigation methods, including that we study here. perturbative ones. The method of moments, which was first used to treat perturbatively an anharmonic oscilla- 2. RECURRENT EVALUATION tor [3], does not require separating relevant variables. OF A PERTURBATION SERIES The recent application of this method to the Zeeman effect problem [4] allowed a check of the behavior of Let us consider the ground state of the hydrogen the high-order asymptotic expression for a perturbation atom placed in mutually perpendicular electric and series. Independently of [3], the method of moments magnetic fields. These fields are assumed to be constant was developed by Fernandez and Castro [5] in a form and uniform. We restrict ourselves to the nonrelativistic similar to that used in [4]. It was then applied to the approximation and neglect the electron spin. From the hydrogen atom in parallel electric and magnetic fields outset, we attempt to simplify numerical computations [6]; later on, the Zeeman effect problem was considered and to achieve a sufficiently high order of perturbation for four sequences of hydrogen-atom states [7]. theory. For this purpose, we treat γ as a fixed parameter, replacing the double expansion in external fields by the It seems even more important to apply it to the single-variable series Ᏹ ∞ ∞ hydrogen atom in crossed electric ( ) and magnetic k ⊥ 2j ψ ==∑Ᏹ |〉k , E ∑E Ᏹ , (1) Ᏼ 2j ( ) fields because only initial terms of power expan- k=0 j=0 | 〉 sion in terms of Ᏹ and Ᏼ have hitherto been consid- where the wave-function corrections k and hyperpo- ⊥ γ ered for this case [8Ð12]. As will be shown in the larizabilities Ek depend on . We also introduce circu- lar coordinates * This article was submitted by the authors in English. 1) Moscow State Engineering Physics Institute (Technical Univer- x±=xiy±; sity), Kashirskoe sh. 31, Moscow, 115409 Russia. ** e-mail: [email protected] after that, all relations given below will have real coef- *** e-mail: [email protected] ficients. In terms of these coordinates, the Hamiltonian

2074 GANI, WEINBERG

| | ln( Ek /k!) where 100 k = 80 k ≡ 1[ k Ð 1 k Ð 1 γ (α β) k Ð 1] Rσαβ -- Pσ + 1, α + 1, β + Pσ + 1, α, β + 1 + Ð Pσαβ 80 2 [ ] 60 2 k/2 γ k Ð 2 ⊥ k Ð 2 j 60 + ----Pσ , α , β Ð ∑ E Pσαβ . 8 + 2 + 1 + 1 2 j 40 j = 1 40 The right-hand side of relation (5) and hyperpolariz- 20 ⊥ 20 ability Ek depend only on the moments of preceding orders. As was usually done in the method of moments 0 [3], the following orthogonality condition is accepted: 0 2 4 6 8 10 〈 | 〉 δ k δ γ 0 k = 0, k P0, 0, 0 = 0, k. (6) An expression for hyperpolarizability arises from (5) at γ | ⊥ | σ α β Fig. 1. Functions fk( ) = ln( Ek /k!) resulting from the = = = 0 and even k: recurrently computed hyperpolarizabilities. 2 ⊥ 1 k Ð 1 k Ð 1 γ k Ð 2 E = --(P , , + P , , ) + ----P , , . (7) k 2 1 1 0 1 0 1 8 2 1 1 of the problem in question has the form The closed system of recurrence relations (5)Ð(7) 2 1 1 enables us to calculate, at least in principle, an arbitrary Hˆ = Hˆ + ᏱHˆ + Ᏹ Hˆ , Hˆ = Ð --∆ Ð --, 0 1 2 0 2 r high order of the perturbation theory. The computation starts from γ 1 γ  ∂ ∂  ˆ -- ˆ --( ) -- (σ ) α H1 = x + Lz = x+ + xÐ + x+∂ Ð xÐ∂  , (2) 0 + 2 ! ! δ 2 2 2 x+ xÐ Pσαβ = ----σ------α------αβ. 2 + 1 Ð (2α + 1)!! 2 2 γ 2 2 γ Hˆ 2 = ----(x + y ) = ----x x . The sequence of manipulations is similar to that used in 8 8 + Ð [4] to compute Zeeman shift of a nondegenerate state. k We use atomic units. The electric and magnetic fields At every order k, only moments Pσαβ from the sector Ᏹ 2 5 ប4 × are measured in the units of at = m e / = 5.142 σ ≥ α + β Ð 1, α ≥ 0, β ≥ 0 are necessary. They are eval- 9 Ᏼ 2 3 ប3 × 9 10 V/cm and at = m e c/ = 2.35 10 G, respec- uated by successively increasing of σ, α, and β values tively. The wave-function correction of order k satisfies with the use of relation (5). the differential equation We have obtained hyperpolarizabilities in mutually perpendicular fields up to the 80th order of perturbation ( ˆ )| 〉 ˆ | 〉 ˆ | 〉 H0 Ð E0 k = Ð H1 k Ð 1 Ð H2 k Ð 2 theory. This order is large enough to compare the [k/2] dependence of these coefficients on γ (see Fig. 1), with ⊥ (3) + ∑ E |k Ð 2 j〉. the predictions following from semiclassical consider- 2 j ations. One can see from Fig. 1 that the function f (γ) ≡ j = 1 k | ⊥ | ln( Ek /k!) has two features. It has a minimum at In just the same way as in other problems to which the γ ≈ method of moments was used [3, 4], it is easy to trans- 3.4 and a sequence of singular points to the right of ⊥ γ form Eq. (3) into the algebraic relation between the this point. Note that the function Ek ( ) changes its sign γ moments of order k: at every singular point of fk( ).

k σ Ð α Ð β α β Pσαβ = 〈0|r x x |k〉. (4) + Ð 3. HIGH-ORDER ASYMPTOTIC BEHAVIOR α β 〈 | σ – α – β Multiplying (3) from the left by 0 r x+ xÐ and It is well known [16] that the dispersion equation relates the asymptotic expression of high-order coefﬁ- considering that the Hamiltonian is Hermitian, we ⊥ obtain the recurrence relation cients Ek to the ionization probability of the atom, i.e., to the penetrability of the potential barrier. This relation 1(σ α β)(σ α β ) k is a consequence of the fact that the energy eigenvalue -- Ð Ð + + + 1 Pσ Ð 2, αβ 2 Ᏹ2 Γ Ᏹ2 (5) E = E0( ) Ð (i/2) ( ) has essential singularity at αβ k σ k k 2 2 + 2 Pσ Ð 2, α Ð 1, β Ð 1 Ð Pσ Ð 1, αβ = Rσαβ , Ᏹ = 0 and a cut along Ᏹ > 0 semiaxis. (Similarly,

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 INVESTIGATING THE HYDROGEN ATOM 2075

2 E(Ᏼ ) has essential singularity at Ᏼ = 0 and a cut along a(γ) 2 1.8 Ᏼ < 0.) To evaluate the ionization probability Γ, the imaginary-time method was previously developed [13Ð15]. 1.6 ⊥ The leading term of the asymptotic expression E˜ of k 1.4 ⊥ ∞ the Ek coefficients at k is determined by the classical subbarrier path with extremal value of the 1.2 abbreviated action. Time takes complex values during this subbarrier motion. There are two kinds of complex classical trajectories. As in the Stark effect case, the 1.0 0 1 2 3 4 5 6 ionization may be induced by the electric field in the γ case of the stabilizing effect of the magnetic field. The path of this kind produces the asymptotic expression Fig. 2. Parameter a(γ) of the perturbation-series asymptotic ⊥ behavior. The solid line follows from the semiclassical esti- ˜ k γ Ӷ γ ӷ Ek (γ ) ∼ k!a (γ ), k is even. (8) mate at 1 [see (13)]; the same estimate for 1 is shown by dashed lines [see (14) and (15)]. Values obtained This asymptotic expression is applicable at a moderate numerically are denoted by asterisks. γ γ magnetic field for below some critical value c. γ According to [18], c = 3. 54 for mutually perpendicular where external fields. It is possible to penetrate through the Ᏼ2 3 barrier also at < 0 as in the Zeeman effect problem. a(γ ) = ------. (12) Subbarrier trajectories of this kind are responsible for 2g(γ ) ⊥ ˜ γ γ γ the form of Ek ( ) in the opposite case, > c. This The last equality is valid also in the region γ > γ , where γ γ γ c γ change of asymptotic expression explains the origin of g( ) and a( ) are complex functions. At < c, the the left minimum in Fig. 1. resulting approximate expressions for a(γ) are To obtain an estimate for the function a(γ) entering ⊥ 3 1 2 71 4  ˜ a(γ ) = -- 1 Ð -----γ Ð ------γ + … , γ Ӷ 1; (13) into Ek , we apply the results of [18, 19] and represent 2 30 2100  here some necessary expressions for the special case of mutually perpendicular external fields. For more gen- γ 3 (γ ) Ӎ 4 γ ӷ a ------2------, 1. (14) Ᏹ Ᏼ 2 2 Ð γ Ð 1 eral considerations related to arbitrary and (γ Ð 1) (1 Ð 2e ) mutual orientation, see [18]. γ γ The time τ of subbarrier motion satisfies the equa- In the region > c, another representation is application [19] ble: τ2 (τ τ )2 γ 2 γ 2 8π2  1 Ð coth Ð 1 = , (9) a(γ ) = -- 1 Ð ---- + ------+ 3 ---- + … . (15) π γ 2  3  γ 4 τ π τ which has a set of solutions n = in + n' . The minimum value of the imaginary part of the subbarrier On the other hand, in the limit of large k, the following τ γ γ action is provided by 0 for < c and by a pair of solu- simple relation appropriate for numerical evaluation τ γ γ γ γ tions ±1 for > c. In the region < c, the energy half- holds: width is (γ ) d Ek 2 B(γ ) 2g(γ ) ln a = ----- ln------. (16) Γ(Ᏹ ) = ------exp Ð------, dk k! Ᏹ 3Ᏹ (10) Evaluating a(γ) above γ , we used smoothed function τ c (γ ) 3 (γ 2 τ2 γ 2) ⊥ γ g = ------3 Ð Ð . Ek ( ) excluding the nodes vicinities. Figure 2 shows 2γ the function a(γ) obtained numerically in such a man- The dispersion relation in Ᏹ2 then leads to ner as compared to expressions (13)Ð(15). γ γ ∞ Now we turn to the region > c. Two solutions of ⊥ τ τ ˜ 1 Γ(z)dz 2 j Eq. (9), 1 and Ð1, lead to complex conjugate values of E2 j = Ð------∼ (2 j)!a , (11) γ τ π∫ j + 1 g( ). Substituting approximate 1 value into second 2 z 0 expression (10), we obtain the phase of the function

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2076 GANI, WEINBERG a(γ) in the form can be easily obtained from hyperpolarizabilities ⊥ π E (γ). Thus, in the sixth order, taking into account that arg(a) = Ðarg(g) = Ð -- + α(γ ), k 2 Stark and Zeeman coefficients are fixed, it is enough to γ 2 choose four different values and to solve a set of four 2 π + 2 5 α(γ ) = -- Ð ------+ O(1/γ ). linear equations. As a result we obtain the representa- γ 3 3γ tion Finally, the sign-alternating asymptotic behavior arises: ⊥ 1  γ 2 953869γ 4 ⊥ E6 = Ð------2512779 Ð 521353 + ------˜ 2 j + 1 512 27 E2 j = 2 B(γ ) (2 j)! a 3 (20)  π  5581 6 ⊥ 2 j × cos (2 j + 1) Ð -- + α(γ ) + β(γ ) Ð ------γ ≡ ∑ γ , (Ᏼ/Ᏹ) .  2  (17) 9  6 Ð 2 j 2 j j = 0 j 2 j + 1 ∼ (Ð1) (2 j)! a sin[(2 j + 1)α(γ ) + β(γ )], (The last identity introduces notation of [12].) Using j ӷ 1. the linear relation between expansions (1) and (18) and the known magneto-electric susceptibilities in parallel β γ Here, ( ) = arg(B) is the phase of the preexponential fields [20], it is easy to obtain the following term of factor. Its relative contribution to the total phase series (18): decreases as 1/j. 6 4 2 When the perturbation theory order 2j is fixed and γ (6) 2512779 254955 E = Ð ------Ᏹ + ------Ᏹ Ᏼ increases, expression (17) changes its sign at every 512 512 point where the argument of the sinus turns to zero. 133199 2 2 49195 2 4 This could explain the singular points in Fig. 1 in terms + ------Ᏹ [Ᏼ × Ᏹ] Ð ------Ᏹ Ᏼ (21) of the asymptotic expression. But rather lengthy calcu- 256 1536 lations are required to establish a detailed quantitative 255557 2 2 5581 6 relationship between asymptotic expression (17) and Ð ------Ᏼ [Ᏼ × Ᏹ] + ------Ᏼ . ⊥ 6912 4608 exact E2 j coefficients including node vicinities. A simple approximate expression for α(γ) is not enough for Some next terms of series (18) can be obtained by the this aim. same procedure. Expressions (20) and (21) are convenient to check term by term the sixth-order correction. As follows from [12], 4. CROSSED EXTERNAL FIELDS OF ARBITRARY MUTUAL ORIENTATION ⊥[12] 1610197 ⊥[12] 2417015 γ = Ð ------and γ = ------, (22) For the general case of the ground-state-energy 24 27648 42 1536 expansion in powers of crossed external fields, the term of the fourth power was known for a long time [9]: while the results of our computation are ∞ 2 2 1 (2 j) (2) 9 1 γ ⊥ 953869 γ ⊥ 521353 E = Ð -- + ∑ E , E = Ð --Ᏹ + --Ᏼ , (18) 24 = Ð ------and 42 = ------. (23) 2 4 4 13824 512 j = 1 4 2 2 All other coefficients from [12] coincide with our (4) 3555 159 E = Ð ------Ᏹ + ------Ᏹ Ᏼ respective results. We carried out an additional inde- 64 32 (19) pendent calculation by means of the method from [9] 10 2 53 4 and obtained + -----[Ᏼ × Ᏹ] Ð ------Ᏼ . 3 192 ⊥[9] 953869 γ = Ð ------. (24) The value of E(4) is confirmed for mutually perpen- 24 13824 dicular fields in [12] and for parallel fields in [12, 17, ⊥ 20]. The E coefficient computed by means of recur- Note that [9] contains complete correction of the sixth 4 power in external fields for the case of parallel fields rence relations (5)Ð(7) exactly agrees with (19). How- and only a part of it for the case of perpendicular fields. ever, there is a numerical difference between our coef- ⊥ These “celebrated” sixth-order terms result as a by- ficient E6 and the corresponding quantity from [12]. product of fourth-order calculations in [9]. The agree- ⊥ Therefore, the sixth order of the perturbation theory ment between high-order hyperpolarizabilities E and was analyzed in detail. k ˜ ⊥ The magneto-electric susceptibilities, i.e., coeffi- their asymptotic Ek confirms once again the correct- cients of the double series in powers of external fields, ness of recurrence relations (5)Ð(7).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 INVESTIGATING THE HYDROGEN ATOM 2077

2 E0 Ᏹ = 0 is approximately reproduced. In practice, suc- –0.48 2 cessions of approximants [L, L, L](Ᏹ ) up to L = 13 Ᏼ = 0.2 were computed. As an example of their convergence, –0.52 we note that the obtained energy eigenvalues have five exact decimal digits at Ᏼ = 0.2, Ᏹ = 0.1 and three exact decimal digits at Ᏼ = 0.2, Ᏹ = 0.5. The resultant –0.56 Ᏼ = 0.1 ground-state energy and width are presented in Figs. 3 and 4. The magnetic field stabilizes the level diminish- –0.60 ing its binding energy and width. We give now some technical remarks. The energy convergence is worsened in the vicinity of the branch- –0.64 Ᏹ Ᏼ 0 0.2 0.4 0.6 0.8 ing point, at ~ 0.01Ð0.07 for from 0.05 to 0.2, Ᏹ respectively. At large electric fields, convergence is observed up to Ᏹ ~ 1. It is rather obvious that the preci- Fig. 3. Real part of the lowest energy eigenvalue of hydro- sion of the final result may be increased and the range gen atom placed in mutually perpendicular external fields. of fields for which the convergence takes place may be extended by increasing the order of employed approxi- Γ/2 mants. Besides, it appears practically optimal if the maximal order of the approximants used is almost 10–2 ⊥ equal to the number of exact decimal digits of Ek coefficients. 10–4 Ᏼ = 0.05 Ᏼ = 0.2 6. CONCLUSION 10–6 The problem considered above demonstrates once –8 again the high efficiency and convenience of the 10 Ᏼ = 0.1 method of moments. The resulting recurrence relations have allowed advance up to the 80th order of perturba- 10–10 tion theory. The high-order asymptotic behavior was 0.04 0.08 0.12 0.16 also analyzed. Basic parameters of this asymptotic Ᏹ expression are in good agreement with those previously obtained in the semiclassical approximation with the Fig. 4. Half-widths of the lowest hydrogen state in mutually use of the imaginary-time method. perpendicular external fields. In the problem under consideration, the method of moments can also be applied to excited states as well 5. ENERGY AND WIDTH CALCULATION just as it was previously used for the Zeeman effect case [4]. The most substantial points of this application are Summation of the obtained perturbation series was the following. The moments performed with the use of HermiteÐPadé approximants, k 〈˜| σ Ð α Ð β α β| 〉 just as it was previously done for the case of Stark Pσαβ = 0 r x+ xÐ k effect [21]. Let us recall that this method is based on m/2 (26) employing the relation |˜〉 imφ Ð r/n x+ Ðr/n with 0 = e = ---- e xÐ A (x)E2(x) + B (x)E(x) + C (x) L M N (25) allow consideration of an arbitrary hydrogen state. Note L + M + N + 2 = O(x ), that bra 〈0˜ | in this definition is the nodeless exponential Ᏹ2 Ᏹ2 factor of the unperturbed wave function, rather than the where x = ; E( ) is series (1); and AL(x), BM(x), and unperturbed wave function. The degeneracy of each CN(x) are polynomials of the powers L, M, and N, excited state should be taken into account from the very respectively. Normalization AL(0) = 1 is accepted. First, 0 beginning, i.e., at the stage of Pσαβ computation. In the Eq. (25) is considered as an identity in x, and the resulting system of linear equations is solved for the second approximation, a constraint excluding ambiguity of energy correction is introduced. This is enough to unknown coefficients of polynomials AL, BM, and CN. Then, Eq. (25) is solved as a quadratic equation for the uniquely determine the mixing parameters inherited by 2 the moments from the unperturbed wave function. sum of series E(Ᏹ ). This way, the branching of exact 2 It is well known that very strong magnetic fields are function E(Ᏹ ) at the point of its essential singularity created by some astrophysical objects. Pulsars possess

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2078 GANI, WEINBERG a magnetic field up to at least Ᏼ ~ 103Ᏼ . In addition, 5. F. M. Fernández and E. A. Castro, Int. J. Quantum Chem. Ᏼ at Ᏼ magnetic white dwarfs with ~ (0.2Ð0.5) at are 26, 497 (1984). observed [22, 23]. An atom moving fast in vicinity of 6. F. M. Fernández and E. A. Castro, Int. J. Quantum Chem. such an object is subjected in its rest frame to intense 28, 603 (1985). mutually perpendicular electric and magnetic fields. 7. F. M. Fernández and J. A. Morález, Phys. Rev. A 46, 318 Quasistatic perpendicular fields are also present in the (1992). radio waves emitted by a pulsar. These fields may be 8. Yu. N. Demkov, B. S. Monozon, and V. N. Ostrovsky, strong enough near the surface of the radiating star. The Zh. Éksp. Teor. Fiz. 57, 1431 (1969) [Sov. Phys. JETP possibility of the investigations of the neutron-star 30, 775 (1969)]. atmospheres in the ultraviolet and x-ray ranges of their 9. P. Lambin, J. C. van Hay, and E. Kartheuser, Am. J. Phys. absorption spectra is discussed in [23]. 46, 1144 (1978). For astrophysical applications, the properties of 10. E. A. Solov’ev, Zh. Éksp. Teor. Fiz. 85, 109 (1983) [Sov. hydrogen in crossed external fields were successfully Phys. JETP 58, 63 (1983)]. computed in an adiabatic approach with a Landau level 11. A. V. Turbiner, Zh. Éksp. Teor. Fiz. 84, 1329 (1983) [Sov. as the initial approximation [24]. However, avoided Phys. JETP 57, 770 (1983)]; Zh. Éksp. Teor. Fiz. 95, crossings of hydrogen levels give no way to use the adi- 1152 (1989) [Sov. Phys. JETP 68, 664 (1989)]. Ᏼ Ᏼ abatic approximation below ~ 100 at. The perturba- 12. N. L. Manakov, S. I. Marmo, and V. D. Ovsyannikov, tive approach described above seems to be more appro- Zh. Éksp. Teor. Fiz. 91, 404 (1986) [Sov. Phys. JETP 64, priate for moderately strong external fields. 236 (1986)]. 13. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Éksp. Teor. Fiz. 50, 1393 (1966) [Sov. Phys. JETP ACKNOWLEDGMENTS 23, 924 (1966)]; Zh. Éksp. Teor. Fiz. 51, 309 (1966) We would like to express our deep gratitude to [Sov. Phys. JETP 24, 207 (1967)]. V.S. Popov and A.E. Kudryavtsev for valuable com- 14. A. M. Perelomov and V. S. Popov, Zh. Éksp. Teor. Fiz. ments. We are grateful to N.L. Manakov, S.I. Marmo, 52, 514 (1967) [Sov. Phys. JETP 25, 336 (1967)]. and V.D. Ovsyannikov for correspondence and discus- 15. V. S. Popov, V. P. Kuznetsov, and A. M. Perelomov, sion and to F.M. Fernandez, who drew our attention to Zh. Éksp. Teor. Fiz. 53, 331 (1967) [Sov. Phys. JETP 26, the studies reported in [5Ð7]. Discussion with V.D. Mur 222 (1968)]. is also gratefully acknowledged. 16. C. M. Bender and T. T. Wu, Phys. Rev. D 7, 1620 (1973). This work was supported in part by the Russian 17. V. S. Popov and A. V. Sergeev, Pis’ma Zh. Éksp. Teor. Foundation for Basic Research (project no. 98-02- Fiz. 63, 398 (1996) [JETP Lett. 63, 417 (1996)]. 17007). The work of V. A. Gani was supported by the 18. V. S. Popov and A. V. Sergeev, Zh. Éksp. Teor. Fiz. 113, INTAS (grant no. 96-0457) within the research pro- 2047 (1998) [JETP 86, 1122 (1998)]. gram of the International Center for Fundamental Phys- 19. V. S. Popov, B. M. Karnakov, and V. D. Mur, Phys. Lett. ics in Moscow. A 229, 306 (1997). 20. B. R. Johnson, K. F. Scheibner, and D. Farrelly, Phys. Rev. Lett. 51, 2280 (1983). REFERENCES 21. V. M. Weinberg, V. D. Mur, V. S. Popov, and A. V. Ser- 1. Yu. P. Kravchenko, M. A. Liberman, and B. Johansson, geev, Zh. Éksp. Teor. Fiz. 93, 450 (1987) [Sov. Phys. Phys. Rev. Lett. 77, 619 (1996); Phys. Rev. A 54, 287 JETP 66, 258 (1987)]. (1996). 22. P. Fassbinder and W. Schweizer, Astron. Astrophys. 314, 2. V. S. Lisitsa, Usp. Fiz. Nauk 153, 379 (1987) [Sov. Phys. 700 (1996); Phys. Rev. A 53, 2135 (1996). Usp. 30, 927 (1987)]. 23. G. G. Pavlov et al., in The Lives of the Neutron Stars, Ed. 3. J. P. Ader, Phys. Lett. A 97A, 178 (1983). by M. Alpar, Ü. Kizilo°glu, and J. van Paradijus (Kluwer, 4. V. M. Weinberg, V. A. Gani, and A. E. Kudryavtsev, Dordrecht, 1995), p. 71. Zh. Éksp. Teor. Fiz. 113, 550 (1998) [JETP 86, 305 24. A. Y. Potekhin, J. Phys. B 27, 1073 (1994); 31, 49 (1998)]. (1998).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NUCLEI Theory

Energies of Mass-Asymmetric Coulomb Systems T. K. Rebane and A. V. Filinsky1) Institute of Physics (Petrodvorets Branch), St. Petersburg State University, ul. Ul’yanovskaya 1, Petrodvorets, 198904 Russia Received May 19, 1999; in ﬁnal form, October 29, 1999

Abstract—It is found that particle-mass-symmetric and particle-mass-asymmetric Coulomb systems are adiabatically similar. Expressions are proposed for the mass dependence of upper and lower bounds on the energies of asymmetric systems, and an expression approximating these energies is given. The energies of the families of mesic molecules that are adiabatically similar to the mesic molecules dpµ, tpµ, and tdµ are investigated with the aid of these expressions. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION operators in (1) and (2) satisfy the equalities ε() ()(),,() In [1], precise variational calculations were per- 0 = E 2m1m2/ m1 + m2 2m1m2/m1+m2m3, formed for a large number of three-particle Coulomb ε() (),, (3) systems symmetric in the masses of likely charged par- 1=Em1 m2 m3 . ticles, and analytic formulas that describe accurately The reversal of the sign of the parameter λ in (2) corre- the energies of these systems were constructed on this sponds to the interchange of the numbers of particles 1 basis. In the present study, we consider relations and 2. Since this does not change the energy of the sys- between the energies of mass-asymmetric three-parti- tem, ε(λ) is an even function of λ, cle systems, which have yet to receive adequate study. ελ()Ð= ελ(), (4) The system of three particle having the unit charges so that its derivative vanishes at the origin: ± +− of q1 = q2 = 1 and q3 = 1 and masses m1, m2, and m3 ε'0()= 0. (5) is described by the Hamiltonian Since the operator in (2) depends on λ linearly, the sec- (),, ()∆[]∆ ∆ H r1 r2 r3 =Ð1/2 1/m1++2/m2 3/m3 ond derivative of its lowest eigenvalue ε(λ) is nonposi- (1) tive (see Appendix), +1/r12 Ð1/1/r13 Ð r23. ε''()λ ≤ 0. (6) For physical quantities, we use here the system of ប | | From (5) and (6), it follows that ε'(λ) ≤ 0 for λ ≥ 0. atomic units ( = me = e = 1). The likely charged par- ≥ Hence, the function ε(λ) decreases monotonically in ticles 1 and 2 are numbered in such a way that m1 m2. The eigenvalue spectrum of the operator in (1) begins the interval 0 ≤ λ ≤ 1, from E(m1, m2, m3), the ground-state energy of the sys- ε()0 ≥≥ ελ() ε()1. (7) tem with a ﬁxed center of mass. Substituting the values of the function ε(λ) at the points λ = 1 and 0 from (3) into (7), we arrive at 2. UPPER BOUND ON ENERGY (),, Em1 m2 m3 (8) An upper bound on the energy E(m1, m2, m3) of a sys- ≤()(),,() tem asymmetric in the masses of the likely charged par- E2m1m2/m1+m22m1m2/m1+m2m3. ticles 1 and 2 can be expressed in terms of the energy of This inequality expresses an upper bound on the energy its symmetric analog. Let us consider the operator of an asymmetric system involving likely charged par- H()r ,,r r ; λ =Ð()∆1/4 [()+∆()1/m + 1/m ticles of unequal masses in terms of the energy of the 1 2 3 1 2 1 2 symmetric system in which the masses of these parti- λ∆()∆ ()]∆ +1Ð 2 1/m1 Ð 1/m2 Ð1/3/2m3 + r12 (2) cles coincide. Ð1/r Ð 1/r , 13 23 3. HAMILTONIAN IN TERMS which depends on the parameter λ. We denote by ε(λ) OF JACOBI COORDINATES its lower eigenvalue. A comparison with (1) shows that the operator in (2) describes the system of particles We introduce the scaled Jacobi coordinates λ []()() with masses m1, m2, and m3 at = 1 and the system of s=m1m2/m1+m2r2Ðr1, two particles 1 and 2 with identical masses equal to λ []()1/2 2m1m2/(m1 + m2) at = 0. Hence, the eigenvalues of the t= m1m2m3/m1++m2m3 (9) × []()() 1) Admiral Makarov State Marine Academy, St. Petersburg, Russia. r3 Ðm1r1 + m2r2 /m1+m2

1063-7788/00/6312-2079 $20.00 © 2000 MAIK “Nauka/Interperiodica” 2080 REBANE, FILINSKY and rewrite the Hamiltonian (1) of the system of parti- radius vector t, for a particle of charge Ð1 that moves in cles with a fixed center of mass in terms of s and t as the field of two fixed Coulomb centers having the equal ( , ) [ ( )] ( , ) charges of q = 2x(m1, m2, m3). These centers are located H s t = m1m2/ m1 + m2 h s t , (10) at the points where the denominators of the fractions in where the operator h(s, t) is given by the bracketed expressions on the left-hand side of (17) t ( , ) ≡ ∆ ∆ ( , , ) vanish—that is, at the points whose coordinates are h s t Ð s/2 Ð t/2 + 1/ s Ð 2x m1 m2 m3 t = Ð[x(m , m , m ) + y(m , m , m )]s, × [1/ t + [x(m , m , m ) + y(m , m , m )]s (11) 1 1 2 3 1 2 3 1 2 3 1 2 3 [ ( , , ) ( , , )] [ ( , , ) ( , , )] ] t2 = x m1 m2 m3 Ð y m1 m2 m3 s. + 1/ t Ð x m1 m2 m3 Ð y m1 m2 m3 s They are separated by the distance R = |t – t | = 2sx(m , and where we have introduced the following two com- 2 1 1 m2, m3). The eigenvalue W of the particle energy binations of the particle masses: depends on the charges q of the centers and on the dis- ( , , ) x m1 m2 m3 tance R between them; therefore, this eigenvalue is a (12) function of the quantities s (the absolute value of the [ ( )2 ( )]1/2 = m3 m1 + m2 /4m1m2 m1 + m2 + m3 , vector s) and x. At the same time, it is independent of y since a change in y implies parallel translation of the ( , , ) y m1 m2 m3 two centers at a fixed distance between them, in which (13) case the particle energy W remains unaffected. [ ( )2 ( )]1/2 = m3 m1 Ð m2 /4m1m2 m1 + m2 + m3 . Since W is independent of y, the operator of the ∆ | | We will refer to h(s, t) as the operator of the reduced reduced energy of the slow subsystem [– s/2 + 1/ s + energy of the system. Its eigenvalues η depend on the W] and its eigenfunction χ do not depend on y either. particle masses through the quantities x and y and are The function χ, together with the adiabatic reduced- η related to the eigenvalues of the Hamiltonian for the energy levels ad of the total system, is determined from system with a fixed center of mass by the equation the equation E(m , m , m ) [Ð ∆ /2 + 1/ s + W(s; x)]χ(s; x) = η (x)χ(s; x). 1 2 3 (14) s ad (18) [ ( )]η( ( , , ), ( , , )) = m1m2/ m1 + m2 x m1 m2 m3 y m1 m2 m3 . Since y does not appear in (18), the reduced-energy adiabatic levels depend on the particle masses only 4. ADIABATIC APPROXIMATION through the combination x(m1, m2, m3). Taking into account (14), we find that the adiabatic energy levels of In [2], it was shown that the lowest energy value Ead the system are given by of a quantum-mechanical system in the adiabatic ( , , ) approximation coincides with its ground-state energy: Ead m1 m2 m3 (19) ( , , ) ≤ ( , , ) [ ( )]η ( ( , , )) Ead m1 m2 m3 E m1 m2 m3 . (15) = m1m2/ m1 + m2 ad x m1 m2 m3 .

Going over to the adiabatic approximation, where A systems of particles with masses m1, m2, and m3 t s we consider the coordinates and as, respectively, a and a system of particles with masses m' , m' , and m' fast and a slow one, we can represent an approximate 1 2 3 eigenfunction of the reduced-energy operator (11) in are referred to as adiabatically similar systems if they the form are characterized by the same value of x (12); that is, ( , , ) ( , , ) Ψ(t; s, x, y) = ψ(t; s, x, y)χ(s; x), (16) x m1 m2 m3 = x m1' m2' m3' , (20) where ψ and χ are the wave functions for, respectively, whence it follows that the fast and the slow subsystem. The wave function ψ ( )2 ( ) for the fast subsystem and the eigenvalue W of its adia- m3 m1 + m2 /4m1m2 m1 + m2 + m3 batic energy are determined by the equation 2 (21) = m' (m' + m' ) /4m' m' (m' + m' + m' ). {Ð ∆ /2 Ð 2x(m , m , m ) 3 1 2 1 2 1 2 3 t 1 2 3 It can be seen from (18) that the ratio of the adia- × [1/ t + [x(m , m , m ) + y(m , m , m )]s batic energies of such systems is equal to the ratio of the 1 2 3 1 2 3 (17) [ ( , , ) ( , , )] ] } reduced masses of likely charged particles, + 1/ t Ð x m1 m2 m3 Ð y m1 m2 m3 s ( ' , ' , ' ) ( , , ) × ψ( , , ) ( )ψ( , , ) Ead m1 m2 m3 : Ead m1 m2 m3 t; s x y = W s; x t; s x y . (22) This wave function is dependent on the coordinate t = [m' m' /(m' + m' )] : [m m /(m + m )]. playing the role of a dynamical variable and parametri- 1 2 1 2 1 2 1 2 cally on the coordinate s and the particle-mass combi- This means that, for the system formed by particles nations x and y. If the vector s is ﬁxed, Eq. (17) assumes with masses m1, m2, and m3, Ead(m1, m2, m3) determines the form of the Schrödinger equation, in terms of the the adiabatic value of the energy and, hence, a lower

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Energies of asymmetric mesic molecules as estimated in terms of the energies of systems that are adiabatically similar to them and which represent, in each case, a symmetric mesic molecule involving nuclei of identical masses and a mesic molecule involving one fixed nucleus (see main body of the text) versus precise results (all energy values are given in atomic units) Energies of mesic molecules Mesic mole- Asymmetry Energies of symmetric Energies of mesic mole- under consideration cules under parameter u mesic molecules cules with a fixed nucleus consideration estimate on the precise results basis of Eq. (28) from [4, 5] dpµ 0.1112615 Ð105.7065 Ð108.2807 Ð105.99 Ð106.01 tpµ 0.2402140 Ð106.9404 Ð109.0532 Ð107.47 Ð107.49 tdµ 0.0396933 Ð111.3272 Ð112.2274 Ð111.36 Ð111.36 Note: Precise energy values from [4, 5] were rounded off to five significant decimal places. bound on the precise energies [see the inequality in quantity x takes the same value for all members of a fam- (15)] for the entire family of adiabatically similar sys- ily, which differ by y values varying in the range 0 ≤ y < tems, those that consist of particles whose masses m' , x. The masses of the first two particles of family mem- 1 bers are given by m2' , and m3' satisfy Eq. (21). ( , ) ( 2 2) [ ( )] m1 x y = 1 + y Ð x m3/ 2x x Ð y , (24) ( , ) ( 2 2) [ ( )] 5. APPROXIMATE EXPRESSION m2 x y = 1 + y Ð x m3/ 2x x + y . FOR THE ENERGIES OF ADIABATICALLY SIMILAR SYSTEMS At y = 0, we have a symmetric mesic molecule featuring nuclei of identical masses, m1(x, 0) = m2(x, 0) = (1 – Extending equality (22) from the adiabatic energies 2 2 x )m3/2x ; at y = x, the system reduces to a molecule Ead to the precise energies E of adiabatically similar where an infinitely heavy (immobile) first particle gen- systems, we arrive at the approximate relation erates the field for the remaining two (mobile) particles, ( ' , ' , ' ) ( , , ) the second particle having the mass m2(x, x) = (1 – E m1 m2 m3 : E m1 m2 m3 2 2 (23) x )m3/4x and the charge identical to that of the first particle and a muon, which has an opposite charge. Ӎ [m' m' /(m' + m' )] : [m m /(m + m )]. 1 2 1 2 1 2 1 2 Upper bounds on the energies of the family mem- The form of the operator in (11) indicates that, for the bers were calculated according to (8). The required systems under consideration, the measure of inaccu- energies of symmetric systems with identical masses of racy of this relation is controlled by the difference of the first two particles, M = 2m1(x, y)m2(x, y)/(m1(x, y) + y(m1' , m2' , m3' ) and y(m1, m2, m3). Under the condition m2(x, y)), were calculated by the formula [1] y(m' , m' , m' ) = y(m , m , m ), the approximate rela- ( , , ) 1 2 3 1 2 3 E M M m3 tion (23) becomes exact. 12 (25) [ ( )] [ ( )]k/2 = Mm3/ 2M + m3 ∑ Ck m3/ 2M + m3 , 6. ENERGIES OF THREE FAMILIES k = 0 OF ADIABATICALLY SIMILAR MESIC MOLECULES where C0 = –1.20526924, C1 = 0.641781090, C2 = 0.285160388, C3 = –0.177735530, C4 = –0.259757745, Let us apply relations (8), (22), and (23) to three C5 = 2.38013126, C6 = –21.1917686, C7 = 113.523948, families of systems that are adiabatically similar to the C = –376.094845, C = 777.364202, C = µ µ µ −8 9 10 mesic molecules dp , tp , and td . 978.197128, C11 = 687.774491, and C12 = Using the values of the muon, proton, deuteron, and −207.711422. triton masses (mµ = 206.76826, mp = 1836.1527, md = For the members of the above families of mesic 3670.4830, mt = 5496.9216), we find that, for these molecules, the adiabatic energy values (which coincide molecules, x and y take the values with lower bounds on the exact energies) were calcu- x(dpµ) = 0.2017599, x(tpµ) = 0.1911185, lated by formulas (15) and (22) by using the adiabatic energies Ead(M, M, m) computed in [3] for the corre- x(tdµ) = 0.1515551, y(dpµ) = 0.06729878, sponding symmetric systems, the latter being interpo- y(tpµ) = 0.09540891, y(tdµ) = 0.03019459. lated in the form The mesic molecules under consideration belong to 7 ( , , ) [ ( )]1 + k/2 three different families of adiabatically similar systems Ead M M m3 = ∑ Dk m/ 2M + m3 , (26) featuring the same third particle (muon, m3 = mµ). The k = 0

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E, a.u. of the members of a family of adiabatically similar (a) –104 mesic molecules, expression (15) yields the adiabatic lower bound 1 ( , , ) ≥ ( , , ) E m1 m2 mµ Ead m1 m2 mµ 4 –108 7 (28) [ ( )] [ ( )]1 + k/2 2 = 2m1m2/ m1 + m2 ∑ Dk mµ/ 2M + mµ , k = 0 dpµ –112 3 where m1 and m2 depend on x and y according to (24), while M is given by (27). In addition to deriving the above upper and lower (b) bounds on the energies of the systems considered here, –105 we have also calculated the corresponding approxi- 1 mated values on the basis of (23). The results of our calculations for the families of 4 systems that are adiabatically similar to the mesic mol- µ µ µ –110 2 ecules dp , tp , and td are presented in Figs. 1‡, 1b, and 1c, respectively, along with the variational values obtained for the corresponding energies by using a tpµ 3 broad basis of Laguerre functions of the perimetric coordinates of the relevant particles. These results are –115 in perfect agreement with those from [4, 5] for the mesic molecules dpµ, tpµ, and tdµ. (c) –111 1 From Fig. 1, we can see that the adiabatic lower bound on the energy as given by (15) lies 1.5 to 2.5 a.u. 4 below the precise value. The approximation in (23), which relies on the exact value of the symmetric-mole- –113 cule energy and which takes into account the property 2 of adiabatic similarity, yields a more accurate lower bound on the energy. tdµ Exact energy values for the members of the families –115 3 of adiabatically similar mesic molecules lie between the lower bound in (23) and the upper bound in (8). 0 4 8 u Within five decimal places, the energy values calculated precisely for the mesic molecules dpµ, tpµ, and Fig. 1. Ground-state energies of the families of mesic mole- tdµ agree with the arithmetic mean of the above cules that are adiabatically similar to the (a) dpµ, (b) tpµ, bounds. and (c) tdµ molecules versus the parameter u = (m – 2 2 1 As can be seen from the figure, the precise energy m2) /(m1 + m2) of the mass asymmetry of likely charged particles. Vertical dashed lines in Figs. 1a, 1b, and 1c corre- values for the members of the families of adiabatically spond to the mesic molecules dpµ, tpµ, and tdµ, respec- similar systems depend almost linearly on the parame- tively. Solid curves represent (1) the upper bounds accord- ter u = (y/x)2 = (m – m )2/(m + m )2 (which character- ing to Eq. (8), (2) the estimates according to Eq. (23), (3) the 1 2 1 2 adiabatic lower bounds according to Eq. (15), and (4) the izes the mass asymmetry of likely charged particles), results of a variational calculation. and so do the estimates in (8), (15), and (23). This enables us to approximate the energies of adiabatically similar systems by the expression where D = –1.205278, D = 0.643267, D = –0.499109, 0 1 2 ( ( , ), ( , ), ) D3 = 2.085603, D4 = –27.07942, D5 = 172.0880, D6 = E m1 x y m2 x y m3 Ð543.8975, and D = 674.7041. 7 Ӎ (1 Ð u)E(m (x, 0), m (x, 0), m ) (29) Taking into account relations (24), we find that the 1 2 3 ( ( , ), ( , ), ) mass of particles 1 and 2 of the symmetric mesic mol- + uE m1 x x m2 x x m3 , ecule belonging to the family of adiabatically similar systems that is characterized by a given value of x is where the dependence of the particle masses on the equal to parameters x and y is given by (24). By using the expression for the parameters x, y, and u in terms of the ( 2) 2 M = 1 Ð x m3/2x . (27) particle masses m1, m2, and m3, we can recast the result presented in (29) in such a way as to obtain an approx- Setting m1' = m2' = M and m3' = m3 = mµ in (22) and imate formula that relates the energy of an asymmetric taking into account (26), we find that, for the energies mesic molecule involving particles of arbitrary masses

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 ENERGIES OF MASS-ASYMMETRIC COULOMB SYSTEMS 2083 to the energies of two systems that are adiabatically We assume that, at λ and δλ values being consid- similar to the system being considered and which rep- ered, the ground state of the system at energy ε(λ + δλ) resent a mesic molecule symmetric in the masses of the is bound, so that the wave function Ψ(λ + δλ) is normal- constituent nuclei and a mesic molecule featuring one ized to unity. We expand the lowest eigenvalue ε(λ + infinitely heavy nucleus, δλ) and the corresponding wave function Ψ(λ + δλ) in ( , , ) Ӎ [ ( , , ) a series in powers of the perturbation parameter, E m1 m2 m3 4m1m2E M M m3 (30) ε(λ + δλ) ( )2 (∞, , ) ] ( )2 + m1 Ð m2 E m m3 / m1 + m2 , ( ) ( ) ( ) (A.3) = ε 0 (λ) + ε 1 (λ)δλ + ε 2 (λ)(δλ)2 + …, 2 2 where M = 2m1m2/(m1 + m2) – (m1 – m2) m3/2(m1 + m2) 2 and m = m1m2(m1 + m2 + m3)/(m1 + m2) . Ψ(λ + δλ) (A.4) This expression was used to estimate the energies of Ψ(0)(λ) Ψ(1)(λ)δλ Ψ(2)(λ)(δλ)2 … asymmetric mesic molecules. The values that are = + + + . obtained in this way and which are displayed in the Substituting expansions (A.3) and (A.4) into the table demonstrate that the approximation in (30) is equation quite accurate. H(λ + δλ)Ψ(λ + δλ) = ε(λ + δλ)Ψ(λ + δλ) (A.5) 6. CONCLUSION of the relevant eigenvalue problem and equating the coefficient at each power of the parameter δλ to zero, The results presented in this study give sufficient we arrive at an infinite set of perturbation-theory equa- ground to believe that the property of adiabatic similar- tions. The first three of these are given by ity can be employed in calculating the energies of Cou- ( ) ( ) lomb mesic molecules that are asymmetric in the [H(λ) Ð ε 0 (λ)]Ψ 0 (λ) = 0, (A.6) masses of likely charged particles. That the energy of ( ) ( ) the members of families of adiabatically similar sys- [H(λ) Ð ε 0 (λ)]Ψ 1 (λ) tems is a nearly linear function of the mass-asymmetry ( ) ( ) (A.7) parameter has enabled us to construct a highly accurate + [W Ð ε 1 (λ)]Ψ 0 (λ) = 0, formula that expresses the energy of an asymmetric ( ) ( ) system in terms of the energy of a symmetric mesic [H(λ) Ð ε 0 (λ)]Ψ 2 (λ) (A.8) molecule and the energy of a mesic molecule involving (1) (1) (2) (0) one infinitely heavy nucleus. + [W Ð ε (λ)]Ψ (λ) Ð ε (λ)Ψ (λ) = 0. Multiplying Eq. (A.7) by Ψ(1)(λ) from the left and ACKNOWLEDGMENTS integrating the resulting product with respect to dynamical variables, we obtain This work was supported by the Russian Foundation ( ) ( ) ( ) for Basic Research (project no. 97-02-16126). 〈Ψ 1 (λ)|H(λ) Ð ε 0 (λ)|Ψ 1 (λ)〉 ( ) ( ) ( ) (A.9) + 〈Ψ 1 (λ)|W Ð ε 1 (λ)|Ψ 0 (λ)〉 = 0. APPENDIX The inequality in (6) follows from basic equations Since ε(0)(λ) is the lowest eigenvalue of the operator of perturbation theory. We now consider a quantum- H(λ), the operator [H(λ) – ε(0)(λ)] is positive definite. mechanical system whose Hamiltonian depends lin- Therefore, the first term in (A.9) is nonnegative: it rep- early on a parameter λ; that is, resents the expectation value of the positive-definite Ψ(1) λ (λ) λ operator for the wave function ( ). Hence, the sec- H = L + W, (A.1) ond term on the left-hand side of (A.9) is nonpositive, where the operator L is independent of λ. ( ) ( ) ( ) 〈Ψ 1 (λ)|W Ð ε 1 (λ)|Ψ 0 (λ)〉 ≤ 0. (A.10) We will rely on perturbation theory, associating the operator in (A.1) with the Hamiltonian of the unper- Further, we multiply Eq. (A.9) by Ψ(0)(λ) from the turbed system [H(0) = H(λ)] and taking the operator left and integrate the resulting product with respect to H(λ + δλ) = H(λ) + δλW (A.2) dynamical variables. By virtue of Eq. (A.6) and the fact that the operator H(λ) is self-conjugate, the contribu- for the total Hamiltonian of the perturbed system. tion of the term involving the second-order correction Here, we treat W and δλ as the perturbation operator function Ψ(2)(λ) vanishes. Considering that the unper- and the parameter of perturbation, respectively. For the turbed wave function is normalized by the condition sake of simplicity, we will omit below the dependence 〈Ψ(0)(λ)|Ψ(0)(λ)〉 = 1, we find that the second-order cor- of the operators and of the wave functions on dynami- rection to the energy is given by cal variables, retaining only their dependence on the (2) (1) (1) (0) parameters λ and δλ (if any). ε (λ) = 〈Ψ (λ)|W Ð ε (λ)|Ψ (λ)〉. (A.11)

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From the last two formulas, it follows that the sec- 2. V. F. Brattsev, Dokl. Akad. Nauk SSSR 160, 570 (1965) ond-order correction to the ground-state energy is non- [Sov. Phys. Dokl. 10, 44 (1965)]. positive, ε(2)(λ) ≤ 0. Taking into account the expansion 3. D. S. Bystrov and T. K. Rebane, Khim. Fiz. 5, 451 in (A.3) for the energy, we can say that the second (1986). derivative of the ground-state energy with respect to the 4. S. I. Vinitskiœ and L. I. Ponomarev, Fiz. Élem. Chastits parameter that enters linearly into the Hamiltonian is At. Yadra 13, 1336 (1982) [Sov. J. Part. Nucl. 13, 557 indeed nonpositive, ε''(λ) ≤ 0. (1982)]. 5. I. V. Puzynin and S. I. Vinitsky, Muon Catal. Fusion 3, REFERENCES 307 (1988). 1. T. K. Rebane and A. V. Filinsky, Yad. Fiz. 60, 1985 (1997) [Phys. At. Nucl. 60, 1816 (1997)]. Translated by O. Chernavskaya

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Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2085Ð2090. Translated from Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2181Ð2186. Original Russian Text Copyright © 2000 by Mebonia, Abusini, Saralidze, Sulakadze, Skhirtladze.

NUCLEI Theory

Approach to Studying Three-Body Processes J. V. Mebonia, M. A. Abusini, P. J. Saralidze, K. I. Sulakadze, and G. É. Skhirtladze Tbilisi State University, Tbilisi, Georgia Received August 19, 1999; in ﬁnal form, April 11, 2000

Abstract—An approach to studying three-body reactions that takes consistently into account the single-collision mechanism is discussed. Speciﬁc calculations are performed for elastic and quasielastic nucleon scattering by a deuteron. The ability of the proposed simple approach to account for a wide range of experimental data suggests that it can be applied to more complicated nuclear reactions. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In the CTBIA, it is stated that, under certain conditions, the Faddeev equations can be solved in the T- Investigation of three-body processes in nonrelativ- matrix formalism by retaining only first-order terms. istic scattering theory furnishes important information However, such terms would correspond to a single col- about fundamental problems in nuclear physics. It is lision proper only if the incident particle (say, particle 1) sufficient to mention that many serious difficulties does not interact simultaneously with the two particles inherent in general many-body problems arise even at forming the bound state (particles 2 and 3). These unde- the three-body level. We mean here the problem of sirable interactions can be eliminated by cutting off the deriving and solving correct equations. For this reason, Fourier transform of the radial wave function ϕ(r) for not only did the advent of the set of Faddeev integral the bound state: G(q) G(q, R), where equations [1] give impetus to a vigorous development ∞ of the theory of three-body reactions proper, but it also 2 sin()qr GqR(), = 2/π r drϕ()r ------. (1) inspired hopes for evolving a consistent approach to ∫ qr complicated processes. R The cutoff radius R must be greater than the de Broglie However, a straightforward application of the Fad- wavelength λ associated with the motion of particle 1 deev equations to three-body problems involves techni- with respect to the (2, 3) system. It can be expressed in cal difficulties. For this reason, theorists usually resort terms of the absolute value of the relevant momentum to various approximate schemes. In particular, unitary as (see Fig. 1) schemes were proposed in [2Ð4]. However, these w schemes did not become popular because they do not RC=/w, (2) simplify calculations substantially. The method of where C is a constant that ensures fulfillment of the straightforwardly summing the truncated WatsonÐFad- requirement λ ≤ R. The amplitude for three-body scat- deev iteration series [5] also proved to be inefficient. tering within the CTBIA then assumes the form Two formally different unitary approaches to study- ˆ Φ Φ Mfi =A f ∑tj i , (3) ing three-body reactions—the cutoff three-body j =23, impulse approximation (CTBIA) [6] and the unitarized where tj is the two-body scattering matrix for particles three-body impulse approximation (UTBIA) [7]— l and n ( j = 123, 231, 312); Φ and Φ are the asymp- were proposed by one of the present authors (J.V. Me- ln i f bonia). Either approach is based on consistently taking totic functions associated with the initial and the final into account the single-collision mechanism, but the state, respectively; and Aˆ is the operator of antisymme- specific implementations of this were different. A trization with respect to identical particles. method for unitarizing the relevant amplitude on the basis of approximately solving the Faddeev equations in the K-matrix formalism was devised within the 2 UTBIA. 1 w Provided that the stringent constraints of the eikonal R approximation are met and that the particles constitut- ing the target nucleus are frozen, the differential elastic- scattering cross section calculated within the UTBIA 3 coincides with the well-known GlauberÐSitenko formula [8, 9]. Fig. 1.

2086 MEBONIA et al.

dσ/dΩ, mb/sr These potentials leading to the same results for the on- energy-shell amplitudes of nucleonÐnucleon scattering may yield, however, different results for the off-energy- 160 1600 shell amplitudes. The off-energy-shell partial-wave amplitudes are required for solving the Faddeev equations at various energies. Owing to this, a comparison of such solutions obtained with a sufficiently high precision with experimental data may be helpful in choosing between various NN potentials. In this respect, valu- 80 800 able information comes not only from the differential cross sections but also from the so-called polarization asymmetry. There is yet another important possibility associated with studying Nd scattering. Such studies make it pos- 0 sible to test various approximate methods for solving 0 80 160 three-body problems in order to extend them to more θ c.m., deg complicated cases. This served as motivation for applying the CTBIA, the simplest unitary scheme, to elastic Fig. 2. Differential cross section for elastic Nd scattering as Nd scattering. θ a function of the scattering angle c.m. in the c.m. frame at Formula (3) implies that the differential cross sec- the laboratory incident-neutron energy of En = 14.1 MeV. Presented in the figure are the results of the calculations per- tion for elastic Nd scattering in the c.m. frame can be formed (solid curve and left scale) with and (dashed curve represented in the form and right scale) without a cutoff. The experimental data (left scale) were taken from [19]. 2 dσ 42m 2 ------= (2π) ------∑ M , (4) dΩ 27 spins This brings up the question of why the cutoff according to Eq. (1) leads to unitarization of the ampli- where tude for single scattering. As early as 1973, Nakamura ˆ Ψ ( ) ( , ε) Ψ ( ) [10] showed that, in the expansion of the amplitude for M = A ∑ ∫dq d* p t j x h; d p 0 , (5) three-body scattering in two-particle partial-wave j = 2, 3 amplitudes, three-particle unitarity is violated by par- k k tial-wave amplitudes associated with low orbital angu- p = Ð-- Ð q, p = Ð -----0 Ð q, 2 0 2 lar momenta. For this reason, it was proposed to intro- (6) duce a cutoff in orbital space. In the semiclassical q q 3 2 2 x = k + -- , h = k Ð -- , ε = ------(k Ð q ) Ð Q, approximation, the orbital angular momentum, the lin- 2 0 2 4m ear momentum, and the radius vector are related as L ~ Ψ Rk. Therefore, a cutoff in orbital space at a given value d is the total deuteron wave function, m is the nucleon of energy must be equivalent to a cutoff in coordinate mass, Q is the deuteron binding energy, and k0 (k) is the space. For this reason, the CTBIA can be considered as momentum of the incident nucleon in the initial (final) a qualitative alternative to the UTBIA, because the lat- state. Summation in formula (4) is performed over the ter has a firmer theoretical ground than the former. Nev- nucleon- and deuteron-spin projections prior to and ertheless, the CTBIA proved to be an efficient scheme after a collision event; explicitly, these spin projections, for treating various three-body processes [11Ð13]. as well as the functions G(p ) and G(p , R), appear after Ψ 0 0 Later on, the equivalence of the two approaches was expanding d and tj in partial waves. proven in [14] for the example of nucleonÐdeuteron We use the system of units in which ប = c = 1. In our collisions. calculation, the two-nucleon off-energy-shell T matrix The objective of the present study is to test the and the radial part of the deuteron wave function were potential of the CTBIA by extending the analysis of constructed for the nonlocal separable Mongan poten- three-body reactions to the cases of the elastic and tial [18]. quasielastic nucleonÐdeuteron scattering processes We have calculated the differential cross section for d(N, N)d and d(N, 2N)N. elastic Nd scattering as a function of the scattering θ angle c.m. The results of this calculation are displayed in Figs. 2 and 3, along with relevant experimental data 2. ELASTIC-SCATTERING PROCESS d(N, N)d at energies 14.1 and 22.7 MeV in the laboratory frame. The advent of the Faddeev equations initiated inten- The solid curve and the left scale show the results sive investigations into the physics of elastic Nd scatter- obtained within the CTBIA, whereas the dashed curve ing [15Ð17]. The problem can be solved in a closed and the right scale correspond to similar calculations form for any realistic nucleonÐnucleon (NN) potential. without a cutoff. Experimental data (left scale) were

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 APPROACH TO STUDYING THREE-BODY PROCESSES 2087 taken from [19, 20]. In the calculations, we took into dσ/dΩ, mb/sr 1 1 1 3 3 3 3 3 3 account the S0, P1, D2, S1 + D1, P0, P1, P2 + F2, 3 and D2 two-nucleon states. However, the eventual results are dominated by the contribution of zero par- 96 960 tial-wave amplitudes, since we restrict our consideration to sufficiently low energies. As can easily be seen, the results that the cutoff-free impulse approximation, which is usually associated with the single-collision mechanism, yields for the differential cross section differ sizably from experimental 48 480 data both in shape and in magnitude. On the other hand, even the unitarization procedure as simple as the application of cutoff to the bound-state wave function improves considerably the agreement between the theoretical results and the experimental 0 data. Nevertheless, some qualitative discrepancies still 0 80 160 θ remain. This might have been expected because we use c.m., deg an approximate method for solving the problem. It is quite natural that rigorous calculations on the basis of Fig. 3. As in Fig. 2, but for En = 22.7 MeV. The experimental the Faddeev equations adequately describe elastic Nd data (left scale) were taken from [20]. scattering. We denote by k the laboratory momentum of the 3. QUASIELASTIC-SCATTERING 0 incident nucleon and by k1, k2, and k3 the momenta of REACTION d(N, 2N)N the scattered nucleons. Usually, experiments of this All that was said in the preceding section about elas- type are performed in coplanar geometry, and the quan- tic Nd scattering remains in force for the quasielastic- tity subjected to investigation is the differential cross scattering reaction d(N, 2N)N. Moreover, the latter reac- section as a function of the scattering angles of two tion offers additional possibilities for studying a three- final-state nucleons and the energy of one of these. The nucleon system in the region of the continuous spec- remaining kinematical quantities are determined from trum. This allows one to obtain new information about the law of energyÐmomentum conservation. the properties of the two-nucleon off-energy-shell Within the CTBIA, the differential cross section for amplitudes. In what follows, we show that, within the the reaction d(N, 2N)N has the form CTBIA, the matrix element for the quasielastic-scattering reaction d(N, 2N)N can be expressed directly in d3σ ------terms of the half-off-energy-shell amplitudes for dΩ dΩ dE nucleonÐnucleon scattering, a circumstance of para- 1 2 1 mount importance indeed, which renders the analysis 2 (7) 2 ∑ M of NN interactions clearer than the analysis of elastic 8 4 3k k = --π m ---1------2 ------spins------, Nd scattering, because, in the latter case, one deals with 3 k 2k Ð k cos(θ ) + k cos(θ + θ ) integrals involving the off-energy-shell NN amplitudes 0 2 0 2 1 1 2 [see Eq. (5)] and not the amplitudes themselves. where Finally, the reaction d(N, 2N)N represents the simplest process from a wide class of quasielastic-scattering ˆ ( , ε )Ψ ( ) M = A ∑ t j x j h j; j d p j , (8) reactions. The majority of such reactions involve rather j = 2, 3 complex fragments, and the relevant matrix elements are expressed in terms of two-fragment off-energy-shell 1( ) 1( ) x j = -- kn Ð kl ; h j = k0 Ð -- kn + kl ; amplitudes, which are usually determined from experi- 2 2 (9) mental data on free fragment scattering. Such a determi- ε ξ 2 nation is possible only if the difference between the off- j = j /4m; p j = kn + kl Ð k0. and on-energy-shell amplitudes is disregarded or if two- fragment phenomenological potentials are used. It is con- The rest of the notation and the computational proce- ceivable that, at moderately low energies, a microscopic dure are identical to those in the preceding section. description of the interaction between the fragments as Measurements are usually performed in such a way composite objects consisting of nucleons is essential. But that the scattering angles are fixed, so that the differen- prior to proceeding to study these reactions at the micro- tial cross section is determined as a function of the scopic level, it is reasonable to test the method to be used energy E1 of one of the recorded nucleons. In recent by applying it to simpler processes like the reaction years, however, measurements often determine the dif- d(N, 2N)N. ferential cross section as a function of the so-called arc

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2088 MEBONIA et al.

3σ Ω Ω –2 –1 3σ Ω Ω –2 –1 d /d 1d 2dE1, mb sr MeV d /d 1d 2dE1, mb sr MeV 101 (a) 12 100

10–1 6 10–2

0 5 11 17 14 32 50 E1, MeV θ θ (b) Fig. 5. As in Fig. 4‡, but for 1 = 43.57°, 2 = –43.57°, and 16 E0 = 85 MeV. The experimental data were taken from [27].

3σ Ω Ω –2 –1 d /d 1d 2dE1, mb sr MeV

0 8 10

10–1 0 5 11 17 E1, MeV

10–2 Fig. 4. Differential cross sections for the reactions (a) d(p, 2p)n and (b) d(p, pn)p as functions of the energy E of a θ 1 θ recorded proton at scattering angles of 1 = 42.5° and 2 = –42.5° and the incident-proton energy of E0 = 30 MeV. 25 45 65 85 Solid curves represent the results of the calculations that E , MeV were performed (thick curves) with and (thin curves) with- 1 out a cutoff and which take into account all phase shifts in θ θ each case. Dash-dotted and dotted curves correspond to Fig. 6. As in Fig. 4‡, but for 1 = 52°, 2 = –40° and E0 = similar calculations that were performed, respectively, with 156 MeV. The experimental data were taken from [28]. and without a cutoff and which take into account only the ᏿-wave phase shifts in each case. The experimental data were taken from (‡) [25] and (b) [26]. 3 d σ display the differential cross section ----Ω------Ω------as a d 1d 2dS length S [21Ð24], which is related to the energies of the function of S. Solid curves represent the results of the ﬁnal-state particles by the equation calculations that take into account all phase shifts (thick curves) with and (thin curves) without a cutoff, while 2 2 dash-dotted and dotted curves correspond to analogous dS = dE1 + dE2 (10) calculations taking into account only the ᏿-wave and which is required to satisfy the condition S = 0 at phases shifts. Experimental data were taken from [22, ≠ E2 = 0 and E1 0. 25Ð28]. It can easily be seen that the introduction of the cutoff improves the agreement between the theoretical A preliminary CTBIA analysis of the differential result and experimental data on the differential cross cross section for the reaction d(N, 2N)N as a function of section both in magnitude and in shape. With increas- the energy E1 was performed in [11Ð14]. Here, we pur- ing energy, the role of the cutoff becomes less pro- sue the analysis of this dependence (see Figs. 4Ð6) fur- nounced, which primarily concerns the differential ther, considering quite a wide range of incident- cross section. This must have been expected because, as nucleon energies (E0 = 10Ð160 MeV). Figures 7 and 8 the energy is increased, the wavelengths of the collid-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 APPROACH TO STUDYING THREE-BODY PROCESSES 2089

3σ Ω Ω –2 –1 3σ Ω Ω , –2 –1 d /d 1d 2dS, mb sr MeV d /d 1d 2dS mb sr MeV

12 1.2

6 0.6

0 0 5 15 25 15 35 55 75 S, MeV S, MeV Fig. 7. Differential cross section for the reaction d(p, 2p)n θ θ as a function of the arc length S at 1 = 52°, 2 = –63°, and θ θ E0 = 19 MeV. The notation for the curves is identical to that Fig. 8. As in Fig. 4a, but at 1 = 35.2°, 2 = –35.2°, and E0 = in Fig. 4‡. The experimental data were taken from [22]. 65 MeV. The experimental data were taken from [22]. ing particles decrease, and so is, according to Eqs. (1) here, the parameter C changed within 12% (C = 1.20 ± and (2), the contribution associated with the discarded 0.15). part of the bound-state wave function. It is interesting The above results reveal that the single-collision to note that, at sufﬁciently low energies, the cross sec- mechanism, when consistently taken into account, has tions for the reactions d(p, 2p)n and d(p, pn)p differ not yet exhausted its potential, which can be of use in substantially even under identical kinematical condi- studying complicated nuclear reactions. That the tions (see Figs. 4a, 4b). This can be explained as fol- CTBIA, a simple unitary method, is capable of lows. The leading contribution to the differential cross accounting for the experimental results considered here section comes from the amplitude describing the inter- demonstrates the importance of respecting basic physi- action between the recorded particles. At low energies, cal principles in constructing approximate methods. 1 the protonÐneutron pair can be either in the S0 or in the 3 S1 state, whereas the two protons can be only in the REFERENCES ﬁrst of these two state—the second is forbidden by the Pauli exclusion principle. As a consequence, the maxi- 1. L. D. Faddeev, Zh. Éksp. Teor. Fiz. 39, 1459 (1960) [Sov. mum of the differential cross section for the reaction Phys. JETP 12, 1014 (1960)]. d(p, pn)p is approximately twice as large as the maxi- 2. R. T. Cahill, Nucl. Phys. A 194, 599 (1972). mum of the cross section for the reaction d(p, 2p)n. 3. K. L. Kowalski, Phys. Rev. D 5, 39 (1972). This difference is reproduced by the theoretical calcu- 4. T. Sasakawa, Nucl. Phys. A 203, 496 (1973). lations only upon introducing the cutoff—that is, if the 5. J. M. Wallac, Phys. Rev. C 7, 10 (1973). single-collision mechanism is consistently taken into 6. J. V. Mebonia, Phys. Lett. A 48A, 196 (1974). consideration. With increasing energy, the contribution 7. J. V. Mebonia and T. I. Kvarackheliya, Phys. Lett. B 90B, of other states increases, so that the difference between 17 (1980). the protonÐproton and the protonÐneutron amplitudes 8. R. J. Glauber, Lect. Theor. Phys. 1, 315 (1959). gradually vanishes. Of particular interest is the depen- 9. A. G. Sitenko, Ukr. Fiz. Zh. (Russ. Ed.) 4, 152 (1959). dence of the differential cross section for the reaction 10. H. Nakamura, Nucl. Phys. A 208, 207 (1973). d(p, 2p)n on the arc length S (Figs. 7, 8). What is worthy 11. T. I. Kvarackhelia and J. V. Mebonia, J. Phys. G 10, 1677 of note here above all is that both the magnitude of the (1984). cross section and the shape of the corresponding curve 12. O. L. Bartaya and J. V. Meboniya, Yad. Fiz. 33, 987 greatly depend on the incident-proton energy. For this (1981) [Sov. J. Nucl. Phys. 33, 521 (1981)]. reason, we eagerly expect new experimental results in this region at various energies and scattering angles. 13. V. Sh. Jikia, T. I. Kvaratskhelia, and J. V. Mebonia, Yad. Fiz. 58, 30 (1995) [Phys. At. Nucl. 58, 27 (1995)]. Finally, we note that some degree of arbitrariness in 14. T. I. Kvaratskhelia and J. V. Mebonia, Soobshch. Akad. choosing the cutoff parameter C ≥ 1 was used to nor- Nauk Gruz. SSR 136, 53 (1989). malize the theoretical plots to experimental data. It 15. M. Sawada et al., Phys. Rev. C 27, 1932 (1983). turned out nonetheless that, in all cases considered 16. P. Doleschall, Nucl. Phys. A 220, 491 (1974).

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17. W. Klüge, R. Schlüfter, and W. Ebenhoh, Nucl. Phys. A 24. J. Zejma et al., Phys. Rev. C 55, 42 (1997). 228, 29 (1974). 25. J. L. Durand, Phys. Rev. C 6, 393 (1972). 18. Th. Mongan, Phys. Rev. 178, 1957 (1968). 26. J. L. Durand, Nucl. Phys. A 224, 77 (1974). 19. G. Rauprich et al., Few-Body Syst. 5, 67 (1988). 27. V. K. Cheng, Nucl. Phys. A 225, 397 (1974). 20. H. Witala and W. Glöckle, Nucl. Phys. A 528, 48 (1991). 21. M. Karus et al., Phys. Rev. C 31, 1112 (1985). 28. F. Takeutchi, Nucl. Phys. A 152, 434 (1970). 22. J. Strate et al., Nucl. Phys. A 501, 51 (1989). 23. G. Rauprich et al., Nucl. Phys. A 535, 313 (1991). Translated by R. Rogalyov

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NUCLEI Theory

Formation of Two Pions on Nuclei in A(g, ppN)B Reactions I. V. Glavanakov Institute of Nuclear Physics, Tomsk Polytechnic Institute, Tomsk, Russia Received July 28, 1999; in ﬁnal form, February 15, 2000

Abstract—A model is proposed for double pion photoproduction on nuclei that is accompanied by nucleon emission. Simple models that faithfully reproduce single-particle differential cross sections are used to describe photon interactions with intranuclear nucleons. The calculated cross sections for pion photoproduction on 12C nuclei are compared with inclusive pion spectra measured in the second resonance region of photon energies. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Single quasifree pion photoproduction has been studied for more than 20 years. As a result, a vast body In recent years, much attention has been given to of experimental data has been accumulated over this studying modifications to the properties of hadrons period of time, and theoretical models have been devel- placed in a nuclear medium. The problem in question is oped [8Ð12] that explain a major part of experimental of interest since the internal structure of hadrons and results. As to investigation of double pion photoproduc- nonnucleonic degrees of freedom of the nucleus can tion, the second process under discussion, it is still in its play an important role in this phenomenon of nuclear infancy. There is only one experimental study, that physics [1, 2]. The question of modifications to the which is reported in [13], where the inclusive spectrum properties of nucleons, mesons, and nucleon reso- of negative pions was measured in the energy region of nances in a nucleus arises in studying (e, e') reactions interest. Only in recent years has double pion photopro- [3], pion-production processes [4], vector-meson-production on nuclei attracted the attention of theorists duction processes [5, 6], and other similar processes. [14Ð16]: the inclusive total cross sections for pion pho- Results obtained by measuring the total cross sections toproduction in A(γ, ππ)X reactions were studied in [14, for photoabsorption on a proton and on C and Pb nuclei 15], while the coherent production of pion pairs on [7] exemplify data whose analysis leads to the assump- nuclei was explored in [16]. Experimental data are tion that the properties of hadrons suffer changes in a available neither for total inclusive cross sections nor nucleus. In particular, virtually no evidence for the for coherent double pion photoproduction. Double pion excitation of N(1520)D13 and N(1680)F15 resonances photoproduction on complex nuclei that is accompa- could be seen in the energy dependences of the cross nied by nucleon emission has been investigated neither sections for photoabsorption on nuclei. There are at theoretically nor experimentally. least three possible reasons for this behavior of the Information about some isotopic channels of single cross section: (i) There exists a mechanism suppressing and double quasifree pion photoproduction on nuclei in the photoexcitation of the above resonances in nuclei the photon-energy region of interest will be obtained in (modification of γN interaction in a nuclear medium). the near future from an experiment at the Tomsk syn- (ii) The masses and widths of resonances excited in a chrotron, where the yields of π0p and π±p pairs originat- nuclear medium differ significantly from those in a vac- ing from interactions between a beam of bremsstrahl- uum. (iii) The dynamics of photonÐnucleus interaction ung photons having an endpoint energy of 900 MeV is such that it smooths the energy dependence of the and H, Li, C, and Al nuclei are being measured as func- cross sections. tions of the secondary-proton energy and of the azimuthal angle of pion emission. It is obvious that a detailed analysis of individual reactions leading to photon absorption is the most effi- For double pion photoproduction on nuclei that is cient way toward solving this problem, but it is quite accompanied by nucleon emission, it is desirable to laborious. It can easily be verified that, at an energy of develop a model that could be used, together with a about 700 MeV, which corresponds to a cross-section similar model for the production of single pions, to ana- maximum associated with the photoexcitation of the lyze experimental data in the second resonance region, N(1520) resonance, two processes—single and double and it is precisely the objective that is pursued in the pion photoproduction—nearly saturate the photoab- present study. sorption cross section. It can be expected that photonÐ The model proposed here to describe double pion nucleus interaction will be dominated by the processes production on nuclei is based on the impulse approxi- of single and double quasifree pion photoproduction. mation, where the amplitude of photon interaction with

2092 GLAVANAKOV a nucleon bound in a nucleus is replaced by the ampli- (c) γp π0π0p, (d) γn π+π–n, (1) tude of photon interaction with a free nucleon. Within this approach, which takes no account of intermediate- (e) γn π–π0p, (f) γn π0π0n. isobar interaction with a nucleus, it is possible to explain a significant part of experimental data on the The most vigorous experimental investigations into cross sections for single pion production in the reaction the photoproduction of two pions on a proton were con- 12C(γ, π–p)11C in the ∆(1232) region [10Ð12]. Two man- ducted in the late 1960s by using bubble chambers. The ifestations of a nucleus are usually considered in its cross sections for pion production on neutrons were interaction with an isobar: (i) Because of the Pauli measured in the 1970s. A comprehensive list of refer- exclusion principle, intranuclear nucleons restrict the ences to experimental studies performed by that time phase space of the nucleon formed in isobar decay, can be found in [18]. In recent years, the advent of high- whereby the isobar width is reduced. (ii) The scattering current electron accelerators made it possible to con- of an isobar on intranuclear nucleons opens new chan- tinue investigating double pion photoproduction on a nels of its decay, whereby the isobar width is increased. proton by new methods ensuring a 4π coverage of mul- According to [17], these two mechanisms governing tiparticle events. Some of the results obtained in Mainz the changes in the isobar width compensate each other at a facility of this type (DAPHNE) are quoted in [18]. near the ∆(1232) pole. This is one of the possible rea- Measured total cross sections constitute the bulk of sons why quasifree pion production in the ∆(1232) information about double pion photoproduction. Pres- region is satisfactorily described within the impulse ently, the cross section for reaction (1a) have been mea- approximation. That the degree to which the isobar sured to a high statistical precision. The cross sections width changes depends on the density of nuclear matter for reactions (1b) and (1c) are known to a somewhat may be another reason for this. Because of the presence poorer precision of (2Ð4)%. The accuracy achieved in of two strongly interacting particles in the final state of measuring the cross sections for reactions (1d) and (1e) A(γ, πN)B reactions, the cross sections for these reac- occurring on a neutron and leading to the formation of tions are strongly suppressed by final-state interaction. charged particles in the final state is about 10%. There It follows that pions are more efficiently produced at are no experimental data on reaction (1f). the nuclear periphery, where the modifying properties Concurrently, the theoretical model of the process of nuclei are weaker. In the second resonance region, evolved, becoming ever more complicated, which is constraints associated with the Pauli exclusion princi- reflected in the growth of the number of Feynman dia- ple are weaker for isobars of higher mass, N(1520) and grams taken into account in the calculations: from 5 in N(1680); as a result, their widths prove to be somewhat γ ππ [19] to 67 in [20]. The model that was proposed in [20] greater than in a vacuum [17]. In the case of A( , N)B for reaction (1a) and which includes the intermediate reactions, however, pion production is superficial to a baryonic states N, ∆(1232), N*(1440), and N*(1520) much greater extent, which validated the disregard of and the ρ meson as a two-pion intermediate resonance the interaction of intermediate isobars with the partici- was extended in [18] to other isotopic channels of the pant nucleus in the conceptual framework of the model photoproduction of two pions. (impulse approximation). The ensuing exposition is organized as follows. Three A calculation of cross sections for particleÐnucleus different approximations of the amplitude for the ele- interactions in the impulse approximation involves mentary process γN ππN that describe satisfactorily considerable technical difficulties associated with the single-particle pion and proton spectra and the experi- presence of multidimensional integrals. Such problems arise even in the analysis of single pion production in mental azimuthal dependence of the charged-pion yield γ π as determined in Tomsk are considered in Section 2. In reactions like A( , N)B. An increase in the number of Section 3, the A(γ, ππN)B amplitude is represented as the final-state particles can render these problems next to sum of terms corresponding to quasifree, exchange, and insurmountable. For this reason, the model used to cal- quasielastic pion-production mechanisms. The last two culate the amplitudes for the reactions in (1) must be as mechanisms are dominant at low nucleon momenta. The simple as possible. Three comparatively simple model problem of kinematically defining one-nucleon ampli- versions satisfactorily reproducing available experi- tudes is considered. Numerical results presented in Sec- mental data will be considered here. Each of these is tion 4 for pion-photoproduction cross sections are com- based on experimental results for the total cross sec- pared with experimental data. tions. Let us represent the differential cross section for the reaction γN ππN in the form 2. PHOTOPRODUCTION OF TWO PIONS ON NUCLEONS σ ( π)Ð5δ4( ) d = 2 Pγ + PN Ð Pπ Ð Pπ Ð PN The production of two pions in photon interactions i 1 2 with nucleons occurs in six reactions dpπ dpπ dp (2) × 1 1 2 1 2 N ------M fi ------, + – + 0 j 4Eγ E 2Eπ 2Eπ 2E (a) γp π π p, (b) γp π π n, Ni 1 2 N

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 FORMATION OF TWO PIONS ON NUCLEI 2093 where Pγ = (Eγ, pγ), PN = (EN , pN ), and PN = (EN, pN) Number of events i i i 30 are the 4-momenta of the incident photon, the initial- θ = 38°–44° state nucleon, and the final-state nucleon, respectively; p Pπ = (Eπ , pπ ) and Pπ = (Eπ , pπ ) are the 4- 1 1 1 2 2 2 20 momenta of two final-state pions; j is the particle flux; 2 and M fi is the quantity obtained by averaging the squared modulus of the transition matrix element over 10 the polarization states of the photon and of the initial- state nucleon and by summing the result over the polarization states of the final-state nucleon. 0 + – θ From an analysis of γp π π p events recorded p = 18°–24° by a bubble chamber, it was deduced [21] that the distribution of these events with respect to the recoil-pro- 20 ton momentum pN is satisfactorily described by the phase-space distribution normalized to the total cross σ section tot. Therefore, the squared modulus of the 10 matrix element (2) can be set to σ 2 5 tot M = (2π) × 4Eγ E j------, (3) fi Ni 0 Stot 200 400 600 800 p , MeV/Ò π2 p 2 m0 Ð mN where Stot = ------∫ pN* pπ**dmππ is the total phase m 2mπ Fig. 1. Recoil-proton-momentum (p ) distribution of p(γ, 0 + – p π π )p events at Eγ = 600Ð700 MeV in two ranges of the space of the reaction in question; mN is the nucleon recoil-proton emission angle (see main body of the text). mass; mπ is the pion mass; mππ and m0 are, respectively, The experimental distributions were obtained on the basis of γ data reported in [21]. the two-pion and the N invariant mass; pN* is the 2 –1 nucleon momentum in the γN c.m. frame; and pπ** is d σ/dpπdΩπ, nb (MeV/Ò sr) the pion momentum in the two-pion c.m. frame. 80 (a) (b) (c) In the approximation specified by Eq. (3), the differential cross section with respect to the recoil-nucleon 60 momentum and the direction of emission in the laboratory frame is given by 40 2σ 2 **σ d π pN pπ tot 20 ------Ω------= ------. (4) d pNd N 2EN mππ Stot In Fig. 1, the recoil-proton-momentum distribution 0 200 400 0 200 400 0 200 400 of γp π+π–p events that was averaged over photon pπ, MeV/Ò energies in the range 600Ð700 MeV and which was obtained on the basis of data presented in [21] is dis- Fig. 2. Differential cross sections for charged-pion produc- played for proton emission angles in the ranges 18°– tion in the reactions (a) γp π+ + charged particles, (b) 24° and 38°–44°. The dashed curve in Fig. 1 represents γp π– + all the remaining particles, and (c) γp π+ the distribution of events that was calculated by averag- + neutral particles as functions of the pion momentum pπ ing the differential cross section in (4) in relevant kine- (Eγ = 730 MeV, θπ = 40°). Points represent experimental matical regions and by performing normalization to the data from (᭹) [22] and (᭺) [23] as quoted in [22]. total number of events recorded in the chosen photon- energy interval. As can be seen, the distribution of tron of Tokyo University [22] at the polar pion emission recoil protons is satisfactorily described in the approx- θ imation that is specified by Eq. (3) and which does not angle of π = 40°. The formation of positively charged rely on any assumptions on the dynamics of the reac- pions that contributed to the experimental cross section γp π+π–p. tions in Fig. 2‡ was accompanied by the emergence of For charged-pion photoproduction on a proton, charged particles. At the same time, no charged parti- Fig. 2 shows the differential cross section measured in cles were found in recording pions that contributed to a beam of 730-MeV tagged photons from the synchro- the data in Fig. 2c. It can therefore be conjectured that

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2094 GLAVANAKOV π π (a) (b) (c) the momentum of the pion 2 in the 1N c.m. frame, and γ π π π π π 1 2 2 2 ( ) γ 1 2 γ 1 2 π m1 = mN + mπ + 2 EN Eπ + pN pπ , 2 2 2 = + ∆ ∆ ∆ 2 2 2 ( ) m2 = mN + mπ + 2 EN Eπ Ð pN pπ . Ni N Ni N Ni N 2 2 π π For the scattered and the decay pion ( 1 and 2 in Fig. 3. Diagrams representing the isobaric mechanism of Fig. 3, respectively), the momentum distributions cal- double pion photoproduction on a proton. culated in the approximation specified by Eq. (5) are represented by, respectively, the solid and the dotted the data in Figs. 2‡ and 2b correspond, respectively, to curves in Fig. 2. It can be seen that, within the experi- positively and to negatively charged pions from the mental errors, the differential cross sections (6) and (7) reaction γp π+π–p and that the data in Fig. 2c cor- satisfactorily reproduce experimental data. If, however, respond to positive pions from the reaction γp we focus on the high-momentum slope of the cross sec- π+π0 tion, where the assumption that the recorded events are n. The dashed curves in Fig. 2 represent the cross associated with double pion production is more justi- sections computed in the approximation specified by fied, we can conclude that the differential cross section Eq. (3). It can be seen that, in contrast to the proton for the formation of positively charged pions in the spectra, the pion spectra are not described by the phase- γ π+π– space distributions. reaction p p complies better with the decay- pion cross section (7) and that the cross section for neg- According to [19, 20], the production of two pions atively charged pions rather agrees with the scattered- is dominated by the isobaric-reaction mechanism illus- pion cross section. The calculated cross sections for trated by the diagram in Fig. 3a. Taking this circum- negatively charged pions are in better agreement with stance into account, we represent the expression for experimental data from [23]. The recoil-proton- M 2 in the form momentum distribution of γp π+π–p events that fi was calculated in the approximation specified by σ 2 5 B(m) tot Eq. (5) and which is depicted by the solid curves in M = (2π) ⋅ 4Eγ E j------, (5) fi Ni 2m S' Fig. 1 differs only slightly from the results of the calcu- where m is the invariant mass of the π N system; lations in the approximation specified by Eq. (3). 2 The exclusive differential cross sections exhibit the 1 Γ/2 B(m) = ------is the BreitÐWigner func- highest sensitivity to reaction dynamics. Unfortunately, π 2 2 (m∆ Ð m) + Γ /4 there are presently no experimental data on the exclu- tion describing the mass distribution of the delta isobar; sive cross sections for double pion photoproduction on π2 ( ) a nucleon, so that the models used cannot be subjected 2 m0 Ð mπ B m S' = ------pπ* pπ**------dm is the convolution of to a detailed test on this basis. Information about the ∫mπ + m 1 2 m0 N 2m azimuthal angular correlation of protons and charged the phase space of the cascade in Fig. 3‡ with the mass pions in the reaction γp π+π–p was obtained from distribution of the isobar; m∆ and Γ are the ∆(1232) an experiment that studied the production of πp pairs on nuclei at the Tomsk synchrotron. For the reaction in mass and width, respectively; pπ* is the momentum of 1 question, this experiment measured, as a function of the the first pion in the γN c.m. frame; and pπ** is the azimuthal charged-pion emission angle φπ, the differen- 2 3 π tial yield d Y/dE dΩ dΩπ, which is related to the cross momentum of the second pion in the 2N c.m. frame (for p p further details of the notation used here, see Fig. 3‡). section In the approximation specified by Eq. (5), the 3σ d ( π)Ð51 1 2 dependence of the cross sections on the momenta of the ------Ω------Ω----- = 2 ------M fi π π dE pd pd π j 4Eγ m p pions 1 and 2 is given by 3 (8) 2 2 p pπ pπ pπ** ( )σ p d σ 1 2 B m tot × ------, ------= π------, (6) ( ) 2 ⋅ ( ) d pπ dΩπ 2Eπ m 2m S' 8 E0 Ð E p pπ Ð Eπpπ p0 Ð p p 1 1 1 where E0 and p0 are, respectively, the energy and the 2 2 pπ γ d σ 2 1 momentum of the N system, by the equation ------= ------d pπ dΩπ 2Eπ 4m' pπ' 3 2 2 2 2 d Y (7) ------dE dΩ dΩπ σ p p × m1 Ð m∆ m2 Ð m∆ tot (9) arctan------ Ð arctan------ ------, 3 3 Γ/2 Γ/2 S'  d σ + d σ Ð  ( ) π π  = ∫dEγ f Eγ ------Ω------Ω------+ ------Ω------Ω------. where m' is the invariant mass of the π N system, pπ' is dE pd pd π dE pd pd π 1 2

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 FORMATION OF TWO PIONS ON NUCLEI 2095

3 2 –1 In expression (9), f(Eγ) is the bremsstrahlung spectrum d σ/dE dΩ dΩπ, µb (MeV sr ) normalized by the condition p p (a) π 1 Eγ max 75

f (Eγ )Eγ dEγ = Eγ , ∫ max 50 0 where Eγmax is the maximal photon energy equal to 25 900 MeV. In the experiment, the azimuthal angle φπ was varied in the interval 0°–40° with a step of 10°. The azimuthal angle of proton emission was kept at a con- 0 π φ (b) π stant value equal to . At zero value of π, the momenta 50 2 of all particles participating in the reaction lie in the same plane. The polar angles of pion emission and of proton emission were chosen to be 61° and 41°, respec- 25 tively. A polyethylene target was used in the experiment. The effect from hydrogen was determined from the difference of the yields from the reactions occurring on a polyethylene and on a carbon target. The differen- 0 (c) tial cross section (8) for the scattered pions, that for the 100 π + π decay pions, and their sum are displayed in Figs. 4‡, 4b, 1 2 and 4c, respectively, as functions of the azimuthal pion emission angle at the proton kinetic energy of T = p 50 140 MeV and the incident-proton energy of Eγ = 900 MeV. The cross sections calculated in the approximation speciﬁed by Eq. (3) and in the approximation speciﬁed by Eq. (5) are depicted by the dotted and by 0 the dashed curves, respectively. The solid curves in 3 Ω Ω µ 2 –1 Fig. 4 represent the results of the calculations based on d Y/dEpd pd π, b (MeV sr ) the isobaric model proposed in [24]. This model takes 150 ( ) π+ π– into account the contribution of two dominant diagrams d + in Fig. 3: the contact diagram in Fig. 3b and the one- 100 pion-exchange diagram in Fig. 3c. Within this model, the squared modulus of the transition matrix element is given by 50 σ 2 5 1 + tot M = (2π) ⋅ 4Eγ E j--∑tr(mλmλ )------, (10) 0 fi Ni 4 S'' λ 0 40 80 φπ, deg where Fig. 4. Differential cross section for the reaction γ p mλ + – π π p as a function of the azimuthal pion emission angle φπ Eπ 2pπ eλ at the proton kinetic energy of Tp = 140 MeV and the incident-  2  + 1 S pπ Ð ------p∆ S eλ + ------(pπ Ð pγ ) photon energy of Eγ = 900 MeV for (‡) scattered, (b) decay,  2  2 2 1 m∆ (Pπ Ð Pγ ) Ð mπ and (c) all product pions; (d) reaction yield as a function of the 1 φ ± = ------Γ------. azimuthal pion emission angle π at Tp = 160 20 MeV. E∆ Ð E∆s + i /2 Points represent experimental data obtained in Tomsk.

Here, λ and eλ are the photon polarization index and the photon polarization vector, respectively; E∆ and E∆s are It can be seen that the aforementioned three approx- the delta-isobar energies in the intermediate state and 2 on the mass shell, respectively; p∆ is the delta-isobar imations for M fi lead to markedly different depen- momentum; S is the operator connecting the spin-3/2 dences of the cross sections on the azimuthal angle. The and spin-1/2 states [25]; the isobar mass m∆ and width cross sections for scattered and decay pions in the Γ were chosen in accordance with [26]; and the quantity S'' in (10) was determined in just the same way as approximation speciﬁed by Eq. (5) differ considerably from those in the approximation speciﬁed by Eq. (10). the quantities Stot and S' in (3) and (5), respectively— that is, by introducing a normalization of the cross sec- However, the azimuthal dependence of the sum of the σ tion in (2) to the total cross section tot measured exper- cross sections for two pions is weakly dependent on the imentally. model used. The results of the calculations for the azi-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2096 GLAVANAKOV Ψ muthal dependences of the differential reaction yields and the wave function β(x2, …, xA) of the residual (9) (see Fig. 4d, where these results are presented along nucleus B in the state β and the multiparticle wave func- with experimental data obtained in Tomsk for protons tion of nucleus A as a linear combination of the prod- ± of kinetic energy Tp = 160 20 MeV) show a similar ucts of the wave functions of the system formed by A Ð degree of distinctions. Satisfactorily reproducing the 1 nucleons and one more nucleon, we can recast the shape of the experimental azimuthal dependence, the expression for the matrix element TIA into the form [29] calculated yield falls short of the experimental data in absolute value. T IA = T QF + T Ex, (12) where 3. A(γ, ππN)B CROSS SECTION The differential cross section for the production of A 〈 | | 〉〈 , β|α〉 π π T QF = ------∑ n tππNγ n' n' , two pions, 1 and 2, on a nucleus with the emission of (2π)3 a nucleon N, n' γ + A B + N + π + π , 1 2 A T = ------∑〈β|tππ( )γ |β'〉〈n, β'|α〉. can be represented in the form Ex ( π)3 A Ð 1 2 β' dσ = 2πδ(Eγ + M Ð Eπ Ð Eπ Ð E Ð E ) T 1 2 N r Here, the ∑ sign denotes summation over spin and dpπ dpπ dp dp × 2 1 2 N r T ------3 ------3 ------3 ------3, isospin states and integration with respect to the (2π) (2π) (2π) (2π) momentum of the relevant particle; where T is the matrix of the transition from the initial iq ⋅ r state involving a photon and a nucleus to the ﬁnal state 〈n|tππ γ |n'〉 = dxϕ*(x)e tππ γ ϕ (x) π π N ∫ n N n' comprising two pions ( 1 and 2), a nucleon, and a residual nucleus; (pr, Er) is the 4-momentum of the is the amplitude of pion production on a nucleon; residual nucleus; and MT is the mass of the target nucleus. 〈β| |β 〉 … Ψ*( , …, ) The transition matrix T can be written as [27] tππ(A Ð 1)γ ' = ∫dx1 dxA Ð 1 β x1 xA Ð 1

Ð1 A Ð 1 ( )( η ) iq ⋅ r ( ) T = V ππAγ + V ππA + V NB E + i Ð H V ππAγ , (11) × j j Ψ ( , …, ) ∑ e tππNγ β' x1 xA Ð 1 where H is the Hamiltonian of the system, E is an j = 1 eigenvalue of H, Vππ γ is the photonÐnucleus interac- A is the amplitude of pion production on a nucleus tion resulting in the production of two pions, VππA is the interaction of the pions with the residual nuclear sys- formed by A Ð 1 nucleons; and tem, and VNB is the interaction of the nucleon involved 〈 , β |α〉 with the set of nucleons forming the nucleus B. n ' In the impulse approximation, the matrix element of … ϕ ( )Ψ ( , …, )Ψ ( , …, ) = ∫dx1 dxA n* x1 β*' x2 xA α x1 xA the interaction VππAγ has the form [28]

… Ψ*( , …, ) is an overlap integral that characterizes the probability T IA = ∫dx1 dxA f x1 xA of the virtual-decay process A (A – 1) + 1. A ⋅ ( ) Two pole diagrams—that which features a nucleon × iq r j j Ψ ( , …, ) ∑ e tππNγ i x1 xA , in a virtual state (Fig. 5‡) and that which features an j = 1 (A Ð 1)-nucleon nucleus in a virtual state (Fig. 5b)— correspond to two terms in (12) [10]. These terms rep- where x is the complete set of variables (spatial, spin, j resent, respectively, the amplitude of quasifree pion and isospin ones) of the jth nucleon; the integral sign photoproduction and the exchange amplitude. denotes integration with respect to spatial variables and summation over spin and isospin variables; tππNγ is the Let us consider the second term in the transition operator of pion photoproduction on a free nucleon; q = matrix as given by (11). For a ﬁrst approximation, the mechanism of A(γ, ππN)B reactions is determined by pγ – pπ – pπ ; Ψ = Ψα(x , …, x ) is the antisymmetric 1 2 i 1 A α Ψ the dynamics of the nuclear system involved. In wave function of nucleus A in the state ; and f is the expression (11), we therefore disregard the interaction wave function of the system consisting of the residual VππA . Under the assumption that the Green’s function nucleus B and the knock-on nucleon. (E + iη – H)–1 is diagonal in the intermediate states, the Representing Ψ as the antisymmetrized product of matrix element for the simplest intermediate state— ϕf the wave function n(x1) of a free nucleon in the state n that is, a state where there is only one virtual particle,

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 FORMATION OF TWO PIONS ON NUCLEI 2097 an excited nucleus A* in our case—can be represented (a) (b) (c) as [30] γ π γ π π γ N N B A* 1 〈n, β|V |α*〉〈α*|tππ γ |α〉 π T = ------N--B------A------, N' B' π QE 3∑ η ( π) EN + EB Ð Eα* + i 2 α* A B A N A π B 〈α | |α〉 where * tππAγ is the amplitude of partial pion photoproduction accompanied by the transition of the tar- Fig. 5. Diagrams representing (‡) the quasifree, (b) the α exchange, and (c) the quasielastic mechanism of pair pion get nucleus to the * state, Eα* is the energy of the photoproduction on a nucleus. nucleus in this state, and the ∑ sign denotes summation over discrete states of the nucleus and integration Ψα(r) is the single-particle wave function of a bound with respect to its momentum. nucleon in the state α; and pˆ and pˆ are the nucleon- This expression is represented by the diagram in i f Fig. 5c; following the terminology adopted in [27], we momentum operators in, respectively, the initial and the will refer to the corresponding pion-production mecha- ﬁnal state (these operators appear in the amplitude nism as a quasielastic mechanism. tππNγ). Thus, the transition matrix T taken in the approxi- Substituting the bound-nucleon wave functions in mation described above appears to be the sum of three the form of expansions in the eigenfunctions of the terms, momentum operator into (14), we obtain

T = TQF + T Ex + T QE, (13) ( ) ( , )Ψ ( ) QF tπ π N γ p N Ðpr α Ðpr , which correspond to the quasifree, the exchange, and  A Ð 2  ( ) Ψ* the quasielastic mechanism of pion production in Ex ∫d p β ' q ------+ p  A(γ, ππN)B relations. A Ð 1 (15) In the resonance energy region, where the cross sec- ×  A Ð 2 pr , pN  Ψ ( ) tion shows sharp variations, it is of paramount impor- tππNγ q------+ p + ------Ð ------+ p β p ,  A Ð 1  tance to give a correct kinematical deﬁnition of the A Ð 1 A Ð 1 pion-photoproduction amplitude—to a great extent, what must be done for this reduces to taking correctly  A Ð 1  (QE) d p Ψ α* q ------+ p tπ π γ ( q + p , p ) Ψ α(p), into account the Fermi motion of intranuclear nucleons ∫ '  A  N and to considering that these nucleons are off the mass (16) shell. We will proceed as follows. In the terms of the amplitude in the laboratory frame that correspond to where Ψα(p) is bound-nucleon wave function in the the above three reaction mechanisms, the structures momentum representation, while tππNγ(pf, pi) is the determining the kinematics of the amplitude for pion amplitude for pion photoproduction on a nucleon (pi photoproduction on a nucleon in the plane-wave and pf are, respectively, the initial-state and the ﬁnal- approximation are given by state momentum).

⋅ In expressions (15) and (16) for the exchange and iq rA Ð 1 (QF) d r φ * e tππ γ (pˆ , pˆ )Ψα(r ), ∫ A Ð 1 pN N f i A Ð 1 the quasielastic mechanism, respectively, the one- nucleon amplitude is averaged over nucleon momenta with a weight appearing to be the transition density of (Ex) d r Ψβ*(r ) ∫ A Ð 2 ' A Ð 2 the momentum distribution of intranuclear nucleons. Commonly, use is made here of the factorization p iq ⋅ r ×  r  A Ð 2 ( , ) approximation: the one-nucleon amplitude is factored expÐi------rA Ð 2 e tππNγ pˆ f pˆ i A Ð 1 outside the integral sign at the momentum value corre- (14) sponding to the maximal transition density. In the oscil-  pN  × exp Ði------r Ψβ(r ), lator model, the transition density  A Ð 1 A Ð 2 A Ð 2 Ψ ( )Ψ ( ) α' p' + p α p

(QE) d r Ψ α* ( r ) ∫ A Ð 1 ' A Ð 1 of the momentum distribution—generally, it depends ⋅ on the initial and on the ﬁnal state of the nucleon—is  pN + pr  iq rA Ð 1 × exp Ði------r e tππ γ (pˆ , pˆ )Ψα(r ), 1  A A Ð 1 N f i A Ð 1 maximal at p = –-- p' for the majority of the transitions. 2 where rA – 1 and rA – 2 are the coordinates of the nucle- In this case, which occurs quite frequently, the labora- ons with respect to the centers of mass of the (A Ð 1)- tory momenta of the initial- and ﬁnal-state nucleons for nucleon and the (A Ð 2)-nucleon system, respectively; the exchange and the quasielastic pion-production

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2098 GLAVANAKOV

σ γ ππ d /dpp, arb. units reaction on a nucleus, A( , N)B. In the Ni rest frame, 101 0 the photon energy Eγ is given by

2 2 QF mππ Ð m Ex 0 N Ni Eγ = ------, 0 2m 10 Ni

QE where m is the free-nucleon mass and mππ = [(Eπ + Ni N 1 2 2 1/2 Eπ + E ) – (pπ + pπ + p ) ] is the invariant mass 2 N 1 2 N 10–1 π π of the 1 2N system. It is of importance that the active nucleon is taken to be off the mass shell. From (17) and 0 400 800 from the conservation of energy at the vertex of the vir- pp, MeV/Ò tual decay A (A – 1) + Ni, it follows that 4 γ π+π– 2 2 2 < Fig. 6. Differential ross section for the reaction He( , Ei = MA Ð Er, mi = Ei Ð pr mN . p)3H as a function of the proton momentum at the incident- i photon energy of Eγ = 700 MeV. The curves labeled with the In this case, the energy and the momentum are con- QF, Ex, and QE symbols represent results obtained by γ ππ assuming the quasifree, the exchange, and the quasielastic served in the Ni( , )N vertex as well. That the one- reaction mechanism, respectively. nucleon amplitude was kinematically deﬁned in this way made it possible to describe satisfactorily the energy dependence of the cross section for quasifree mechanism in the factorization approximation and for single negative-pion photoproduction in the ∆(1232) the quasifree mechanism are given by region [31].

(QF) p = Ð p , p = p , i r f N 4. NUMERICAL RESULTS 1 1 A Ð 2 ( ) Ex p i = Ð p N ------Ð -- q ------, From the numerical results displayed in Fig. 6, we A Ð 1 2 A Ð 1 deduce qualitative information about the kinematical 1 A Ð 2 1 (17) regions where the reaction mechanisms corresponding p = --q------+ p ------, f 2 A Ð 1 r A Ð 1 to the three diagrams in Fig. 5 are operative. The contributions of the three amplitudes in (13) to the cross 1 A Ð 1 1 A + 1 4 γ π+π– 3 (QE) p = Ð--q------, p = --q------. section for the reaction He( , p) H are shown in i 2 A f 2 A Fig. 6 as functions of the proton momentum. The calculation that yielded these results was performed in the It is well known that the factorization approxima- plane-wave approximation on the basis of expression tion is quite satisfactory if the amplitude of the reaction (3) for the amplitude of the reaction p(γ, π+π–)p. The occurring on a nucleon changes slowly in the region contribution from the two lowest excited states of the being considered. Otherwise, the exact expressions 4He nucleus at 20.1 and 21.1 MeV (their quantum num- (15) and (16) must be used to calculate the amplitudes bers are 0+0 and 0–0, respectively) were taken into for the exchange and for the quasielastic mechanism. account in calculating the amplitude for the quasielastic reaction mechanism. It was assumed that the conﬁgura- The averaging of the amplitude for pion production tions of these 0+0 and 0–0 states are (1s)–1(2s) and on a free nucleon in (15) and (16) over the states of (1s)−1(1p). It can be seen that the quasifree pion-photo- intranuclear nucleons smooths the energy dependence production mechanism is dominant for proton of the cross section for the reaction occurring on a momenta exceeding the characteristic intranuclear- nucleus. In view of this, the region dominated by the nucleon momentum of about 200 MeV/c. The quasifree mechanism of pion photoproduction is more exchange- and the quasielastic-mechanism contribu- appropriate for experimentally studying in-medium tion overlap in Fig. 6, but they are separated to a con- changes in the properties of resonances because, there, siderable extent in the dependence of the cross section the kinematics of ﬁnal-state particles furnishes the vast- on the invariant mass of the p3H system; at the minimal est amount of information for determining the one- excitation energy, these mechanisms can also be sepa- nucleon amplitude. rated by studying the differential cross section with respect to the angle of divergence of the proton and the In order to specify kinematically the one-nucleon 3H nucleus. amplitude for the quasifree reaction mechanism, quan- Figure 7 shows the inclusive spectra of negative tities that describe the state of the particles participating pions that were formed in the interactions of photons γ ππ in reactions of the Ni( , )N type are assigned the val- having energies in the intervals 515Ð595, 595Ð675, and ues of the analogous quantities for the corresponding 675Ð755 MeV with 12C nuclei and which were emitted

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 FORMATION OF TWO PIONS ON NUCLEI 2099

2 –1 at an angle of 41° with respect to the photon-beam axis d σ/dpπdΩπ, µb (MeV/Ò sr) [13]. The relevant data come from the experiment that 0.8 was performed at the synchrotron of Tokyo University. Eγ = 515–595 MeV Eγ = 595–675 MeV Eγ = 675–755 MeV The experimental results are presented in the form of the dependence on the difference ∆p of the momentum p0 that follows from the kinematics of single pion pro- 0.4 duction on a free nucleon and the momentum of the recorded pion. It is rather difficult to analyze these data since negatively charged pions can be formed in many π ππ π ππ π ππ processes at these energies. However, the contribution 0 of some single-pion-production processes where the –200 0 200 400 0 200 400 0 200 400 final nucleus occurs in a bound state can be disregarded p0 – pπ, MeV/Ò because, under the kinematical conditions being considered, the minimal absolute value of the momentum Fig. 7. Differential cross section for the reaction 12C(γ, π–)X | | transfer, pγ – pπ , exceeds 300 MeV/c. For the same as a function of the difference of the momentum p0 that fol- reason, it is legitimate to disregard the exchange and lows from the kinematics of single pion production on a free the quasielastic mechanism of single pion photopro- nucleon and the recorded-pion momentum pπ (θπ = 41°). duction accompanied by nucleon emission. Experimental data were borrowed from [13]. In the quasifree approximation, negative pions can be formed via single pion photoproduction in the reac- substantially improved. It is worth noting that the filling tion of the cross section minimum is more pronounced in the region where the cross sections for single and dou- Ð 12C(γ, π p)11C (18) ble photoproduction overlap. One of the possible reasons behind underestimating the experimental cross sec- and via three double-pion-photoproduction processes tions in this region of the pion spectra may be associated 12 Ð + 11 12 Ð + 11 with the contribution of the coherent photoproduction of C(γ, π π p) B, C(γ, π π n) C, (19) two pions on a carbon nucleus in the reaction 12C(γ, π– 12 Ð 0 11 π+ 12 C(γ, π π p) C. ) C [16]. In the same interval of pion momenta, it is natural to expect manifestations of the exchange and of In Fig. 7, the curves labeled with the symbol π rep- the quasielastic mechanism of the reactions in (19). resent the cross section calculated for reaction (18) in the quasifree approximation with distorted waves, In analyzing experimental results for the photoab- while the curves labeled with the symbol ππ corre- sorption cross section, it is of interest to compare the spond to the sum of the cross sections calculated for total cross sections for single and double pion photo- three double-pion-photoproduction reactions (19) production on a proton with the total cross sections (integrated over the total phase space) for quasifree sin- within the approximation specified by Eq. (5). The 12 final-state interaction was taken into account in the gle and double pion photoproduction on a C nucleus. eikonal approximation. The potential that was pro- This comparison makes it possible to assess a minimal posed in [32] and which describes satisfactorily the degree of the changes that the energy dependence of the effect of proton interaction with a residual nucleus in cross section for the reaction occurring on a nucleus the single-pion-production process at energies in the may suffer within the impulse approximation. It can be ∆(1232) region [33] was taken here for the nucleonÐ expected that the contributions from other photoab- nucleus optical potential. The pion wave function was sorption channels will provide a poorer description of distorted by the optical potential used in [34]. The the cross section for photonÐnucleon interaction. states of intranuclear nucleons were described by oscil- In Fig. 8‡, the dashed and the dash-dotted curve rep- lator wave functions characterized by the oscillator γ π0 12 resent the total cross sections for the reactions p( , )p parameter equal to the charge radius of the C nucleus. and p(γ, π+)n, respectively, as functions of the total c.m. The high-momentum section of the pion spectrum energy s1/2. These cross sections were obtained by inte- (small negative values of ∆p) is satisfactorily described grating the experimental differential cross sections by the contribution from the single quasifree photopro- dσ/dΩ* represented in the form of an expansion in duction of negatively charged pions. The contribution θ* of double quasifree pion photoproduction faithfully powers of cos π . The expansion coefficients were reproduces the rate of the cross-section growth with ∆p taken from [35]. The dotted curve corresponds to the in the region ∆p > 100 MeV/c. At the minimal photon sum of the total cross sections for the double-pion-pho- energy, the absolute value of the cross section for the toproduction reactions p(γ, π+π–)p, p(γ, π+π0)n, and reactions in (19) around the maximum is nearly one- p(γ, π0π0)p [18]. In Fig. 8‡, the solid curve represents half as large as the experimental inclusive cross section. the sum of the total cross sections for single and double As the photon energy is increased, the agreement pion photoproduction on a proton as a function of between the computed and measured cross sections is energy. The same notation for the curves is used in

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2100 GLAVANAKOV

σ µ 1/2 tot, b for double pion production—at s ~ 1500 MeV, this 600 cross section reaches a maximum value; and the reso- (a) nance contribution of N(1520) to the cross section for all processes. Effects resulting from the fact that intra- 400 nuclear nucleons are in a bound state are vividly exem- π0 plified by observables of the reactions p(γ, π0)p and 12C(γ, π0p)11B: the cross-section maximum associated 200 ππ with the excitation of the ∆(1232) isobar is shifted to π+ the region of higher energies with increasing width of 0 the peak. Qualitatively similar changes in the cross section are observed in the N(1520) region. 750 (b) Final-state interaction has a pronounced effect on 500 the energy dependence of the cross section. At a reso- π0 nance photon energy, the mean energy of product pions does not have a resonance value with respect to the ππ 250 π+ interaction with nucleons at rest. This leads to a shift of the maximum of the cross-section suppression to the 0 region of higher photon energies. It follows that not 1200 1400 1600 only does the final-state interaction suppress the cross s1/2, MeV section in the ∆(1232) region by a factor greater than 3, but it also deforms considerably the energy dependence Fig. 8. (a) Cross sections for the reactions p(γ, π0)p and p(γ, of the cross section. The latter effect is especially pro- π+)n and sum of the cross sections for double pion photo- nounced when one compares the cross sections for the production in the reactions p(γ, π+π–)p, p(γ, π+π0)n, and p(γ, production of positively charged pions in the reactions p γ π+ n and 12 γ π+n 11 . In the N region, the π0π0)p and (b) cross section for single pion photoproduction ( , ) C( , ) B (1520) final-state interaction changes the relationship between in the reactions 12C(γ, π0p)11B and 12C(γ, π+n)11B and sum of the cross sections for double pion photoproduction in the the cross sections for single and double pion photopro- reactions 12C(γ, π+π–p)11B, 12C(γ, π+π0n)11B, and 12C(γ, duction. Because of the presence of two pions, the sec- π0π0p)11B as functions of the total c.m. energy s1/2. ond process is more sensitive to nuclear-medium effects than the first one. The suppression of the cross section for double pion production because of pion Fig. 8b to depict the cross sections for single pion pho- interaction with the residual nucleus is enhanced when toproduction in the reactions we approach the region where both pions reach the resonance energy with a high probability, in which case 12C(γ, π0 p)11B, 12C(γ, π+n)11B (20) the cross-section maximum in the N(1520) region becomes more pronounced. and the sum of the cross sections for double pion photoproduction in the reactions By comparing the sum of the total cross sections for single and double pion production on a free proton with 12C(γ, π+πÐ p)11B, 12C(γ, π+π0n)11B, that for the analogous processes on a nucleus, we can (21) conclude that, although the energy dependence of the 0 0 12C(γ, π π p)11B. cross section for the reaction occurring on a nucleus basically reproduces typical features of the cross sec- These cross sections were calculated in the quasifree tion for the analogous reactions on a proton, the energy approximation with distorted waves by using Eq. (5). In dependence of the former has a less pronounced struc- the quasifree approximation, reactions (20) and (21) ture. The cross-section minimum between the two res- proceed via photon interaction with the protons of the 12 onance regions is filled sizably, but not to an extent suf- C nucleus. The final-state interaction was taken into ficient for explaining the absence of a maximum in the account in the eikonal approximation (the relevant energy dependence of the photoabsorption cross sec- details were described above in connection with inter- tion at the position of the N(1520) resonance. It should preting the negative-pion spectra displayed in Fig. 7). be borne in mind, however, that, with allowance for the The data in Figs. 8‡ and 8b are presented in the same isotopically symmetric reactions form as the measured cross sections in [7]: the same photon energy in the laboratory frame for the reaction 12C(γ, π0n)11C, 12C(γ, πÐ p)11C; on a proton and for the reaction on a nucleus corresponds to a specific value of s1/2. 12C(γ, π+πÐn)11C, 12C(γ, πÐπ0 p)11C, In the energy range being considered, the energy 12 (γ, π0π0 )11 dependence of the cross section for pion photoproduc- C n C, tion on a free proton is governed by three factors: the cross sections displayed in Fig. 8b saturate only 30Ð ∆(1232) excitation; a sharp growth of the cross section 40% of the photoabsorption cross section and that the

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 FORMATION OF TWO PIONS ON NUCLEI 2101 effect of N(1520) excitation is somewhat smaller in production on nuclei, the energy dependence of the photonÐneutron interactions [18, 36]. total cross sections for some isotopic channels through 12 γ π 11 12 γ ππ 11 A result that is qualitatively similar (to that which the reactions C( , N) B and C( , N) B proceed owing to prompt photon interactions with the described above) in what is concerned with the mani- 12 festations of the cross-section maximum at the position protons of the C nucleus has been calculated in the of the N(1520) resonance was obtained in the semiclas- quasifree approximation. Together with the cross sec- sical transport model used in [17] to explain the cross tions for isotopically symmetric channels for pion pro- section for photoabsorption on 40Ca. According to the duction, the cross sections for the above processes sat- estimates presented there, the isobar width increases in urate 30 to 40% of the photoabsorption cross section. a nucleus by 20Ð40 MeV, which is one order of magni- These results have been compared with the cross sec- tude less than what is required for explaining the pho- tion for single and double pion production on a free toabsorption cross sections [37]. proton. In addition to trivial broadening of resonance peaks, which is due to the Fermi motion of the nucleons, and a shift to the region of higher energies, a pro- 5. CONCLUSION nounced effect of the final-state interaction on the γ ππ energy dependence of the cross section has been Reactions of the A( , N)B type, which represent revealed. The final-state interaction deforms noticeably one of the main channels of photon absorption by a the energy dependence of the cross section in the nucleus in the second resonance region of energies, ∆(1232) region and changes the relationship between have been analyzed. The reaction amplitude has been the cross sections for single and for double pion pro- represented as the sum of the amplitude for quasifree duction at the position of the N(1520) resonance. The pion production, the exchange amplitude, and the cross-section minimum between two resonance regions quasielastic amplitude. In the quasielastic process, the is partly filled. The majority of the factors listed above production of a pion pair is accompanied by the excita- smooths the energy dependence of the cross sections tion of the nucleus, which further decays via the emis- for above-type reactions on a nucleus. Nonetheless, this sion of a nucleon. The quasifree mechanism of pion pro- provides no way to explain the absence of the maxi- duction is dominant in the kinematical region where the mum in the cross section for photoabsorption on 12C in proton momentum exceeds a value of about 200 MeV/c. the N(1520) region. For the N(γ, ππ)N reactions, three models have been considered here. These are (i) the phase-space model, which satisfactorily reproduces the results of bubble- ACKNOWLEDGMENTS chamber measurements for recoil-proton-momentum This work was supported by the Russian Foundation distributions of events of the reaction p(γ, π+π–)p; for Basic Research (project no. 97-02-17765). (ii) the phase-space model embedded in the isobaric model—this approach makes it possible to describe the spectra of charged pions from the reactions p(γ, π+π–)p REFERENCES γ π+π0 and p( , )n induced by a beam of tagged photons 1. P. J. Mulders, Phys. Rep. 185, 83 (1990). from the synchrotron of Tokyo University; and (iii) the 2. N. C. Mukhopadhyay and V. Vento, RPI Internal Report isobaric model that takes into account the contribution RPI-97, No. 117 (1997); nucl-th/9712073. of the contact diagram and the contribution of the pion- exchange diagram. For the shape of the azimuthal 3. R. M. Sealock, K. L. Giovanetti, S. T. Thornton, et al., dependence of the yield from the reaction p(γ, π+π–)p, Phys. Rev. Lett. 62, 1350 (1989). all these models provide results that are in satisfactory 4. D. Pelte, nucl-ex/9902006. agreement with data obtained at the Tomsk synchro- 5. Ye. S. Golubeva, L. A. Kondratyuk, and W. Cassing, tron. Nucl. Phys. A 625, 832 (1997). Within the model that takes into account single and 6. S. H. Lee, nucl-th/9904007. double quasifree pion photoproduction, the spectra of 7. A. Deppman, N. Bianchi, E. De Sanctis, et al., nucl- negatively charged pions from photon interactions with th/9809085. a 12C nucleus have been calculated in the second reso- 8. A. I. Shebeko, Yad. Fiz. 14, 1191 (1971) [Sov. J. Nucl. nance region of energies. The results of the calculations Phys. 14, 664 (1971)]. satisfactorily reproduce the high-momentum contribu- 9. J. M. Laget, Nucl. Phys. A 194, 81 (1972). tions to the experimental inclusive pion spectrum, which 10. I. V. Glavanakov, Yad. Fiz. 49, 91 (1989) [Sov. J. Nucl. are due to single pion production; the low-momentum Phys. 49, 58 (1989)]. spectrum is described at energies Eγ > 600 MeV. At low 11. C. Bennhold, X. Li, and L. E. Wright, Phys. Rev. C 48, momenta, the calculated spectrum associated with dou- 816 (1993). ble pion photoproduction falls slightly short of the 12. J. I. Johansson and H. S. Sherif, Nucl. Phys. A 575, 477 experimental spectrum. (1994). In order to clarify the nuclear-medium effect on the 13. I. Arai, H. Fujii, S. Homma, et al., J. Phys. Soc. Jpn. 45, energy dependence of the cross section for pion photo- 1 (1978).

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14. J. A. Gómez Tejedor, M. J. Vicente-Vacas, and E. Oset, 27. M. L. Goldberger and K. M. Watson, Collision Theory Nucl. Phys. A 588, 819 (1995). (Wiley, New York, 1964; Mir, Moscow, 1967). 15. M. Effenberger, A. Hombach, S. Teis, and U. Mosel, 28. M. Lax and H. Feshbach, Phys. Rev. 81, 189 (1951). Nucl. Phys. A 614, 501 (1997). 29. I. V. Glavanakov, Vopr. At. Nauki Tekh., Ser. Obshch. 16. S. S. Kamalov and E. Oset, Nucl. Phys. A 625, 873 Yad. Fiz. 1 (34), 94 (1986). (1997). 30. I. V. Glavanakov, Yad. Fiz. 50, 1231 (1989) [Sov. J. Nucl. 17. M. Effenberger and U. Mosel, nucl-th/9707010. Phys. 50, 767 (1989)]. 18. J. A. Gómez Tejedor and E. Oset, hep-ph/9506209. 31. I. V. Glavanakov, Yad. Fiz. 35, 875 (1982) [Sov. J. Nucl. 19. L. Lüke and P. Söding, Springer Tracts Mod. Phys. 59, Phys. 35, 509 (1982)]. 39 (1971). 32. C. J. Batty, Nucl. Phys. 23, 562 (1961). 20. J. A. Gómez Tejedor and E. Oset, Nucl. Phys. A 571, 667 33. I. V. Glavanakov, Yad. Fiz. 31, 342 (1980) [Sov. J. Nucl. (1994). Phys. 31, 181 (1980)]. 21. P. Brinckmann and W. Mohr, Bonn-PI-1-111 (1970). 34. Y. Futami and J. Suzumura, Prog. Theor. Phys. 66, 534 22. K. Maruyama, H. Fujii, S. Homma, et al., J. Phys. Soc. (1981). Jpn. 46, 1403 (1979). 35. H. Genzel, P. Joos, and W. Pfeil, Photoproduction of Ele- 23. S. Fukui, Y. Inagaki, S. Iwata, et al., Nucl. Phys. B 81, mentary Particles (Springer-Verlag, New York, 1973). 378 (1974). 36. F. X. Lee, C. Bennhold, S. S. Kamalov, and L. E. Wright, 24. J. M. Laget, Phys. Rev. Lett. 41, 89 (1978). nucl-th/9806024. 25. H. Sugawara and F. von Hippel, Phys. Rev. 172, 1764 37. L. A. Kondratyuk and Ye. S. Golubeva, Yad. Fiz. 61, 951 (1968). (1998) [Phys. At. Nucl. 61, 865 (1998)]. 26. I. Blomqvist and J. M. Laget, Nucl. Phys. A 280, 405 (1977). Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NUCLEI Theory

Reactions p(d, p)d and p(d, p)pn as a Tool for Studying the Short-Range Internal Structure of the Deuteron* V. P. Ladygin** Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received August 5, 1999; in ﬁnal form, January 18, 2000

Abstract—In recent years, the deuteron structure at short distances has often been treated from the point of view of nonnucleonic degrees of freedom. In this study, measurements of T-odd polarization observables by using a tensorially polarized deuteron beam and a polarized proton target or a proton polarimeter are proposed as a means for seeking quark conﬁgurations inside the deuteron. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tion with ~4% of the |6q〉 probability at 55° of the relative phase between |6q〉 and the S component from the Recent experimental results concerning the struc- RSC potential [9]. ture of the deuteron led to the speculations that manifestations of the quarkÐgluon degrees of freedom are Recent measurements of the tensor analyzing power ° present even at relatively large distances between T20 for deuteron inclusive breakup at 0 that were per- nucleons. Measurements of the cross section for the formed in Saclay [7] and in Dubna [14Ð16] at different inclusive deuteron-breakup reaction A(d, p)X on carbon energies and for different targets showed the strong with the proton emitted at zero angle [1] showed a rel- deviation from the IA predictions at k ≥ 0.2 GeV/c. The atively broad shoulder at internal nucleon momenta of behavior of the coefficient of polarization transfer from κ k ~ 0.35 GeV/c in the deuteron defined in the light-cone a vectorially polarized deuteron to a proton ( 0) [17Ð dynamics [2Ð4]. This enhancement was observed later 19] also disagrees with the calculations using conven- at different initial energies and for different A values of tional deuteron wave functions at k ≥ 0.2 GeV/c. On the κ the target [5Ð7]. This shoulder could not be reproduced other hand, both T20 and 0 data demonstrate some fea- by calculations within relativistic impulse approxima- tures of the IA like a weak dependence on the target tion (IA) by using the standard deuteron wave functions mass number A and an approximate energy indepen- [8Ð10] or by including rescattering corrections [11]. dence. A consideration of the mechanisms additional to The theoretical study of Kobushkin and Vizireva led to the IA one [20, 21] cannot explain the experimental the possibility that there exists a 6q admixture in the data. deuteron wave function [12]. This 6q amplitude, arising The most intriguing feature of experimental data is from the S configurations of six quarks, must be added that the tensor analyzing power T20 in deuteron inclu- to the S component of the standard deuteron wave func- sive breakup and backward elastic deuteronÐproton χ tion with a relative phase . A fit to the experimental scattering show, at high internal proton momenta, the data [5] gave the probability of the 6q configuration same negative value of about Ð(0.3Ð0.4) [15, 16, 22], χ ° ° about 4% and relative angle of ~ 82 and ~61 for which is incompatible with the predictions using any the Paris [8] and the Reid soft-core (RSC) [9] NN reasonable nucleonÐnucleon potential. Various potential, respectively. An admixture of the 6q state of attempts were undertaken to explain the T20 data by tak- about 3.4% was introduced in [13] as well to describe ing into account nonnucleonic degrees of freedom in the tail of the momentum spectrum for the reaction the deuteron. An asymptotic negative limit on T 12 20 was C(d, p)X [5]. obtained within the QCD-motivated approach pre- One of the important features of this hybrid wave sented in [23] and based on the reduced-nuclear-ampli- function is that an additional 6q admixture masks the tude method [24]. The results of calculations [20] with node of the NN S wave, and this is drastically reflected the hybrid deuteron wave function [25] made it possible to describe satisfactorily the T data up to k ~ in the behavior of polarization observables. For 20 κ instance, data obtained at Saclay for the tensor analyz- 1 GeV/c [15]. Recently, the data on T20 and 0 in the 12C(d, p)X reaction at 0° were reasonably reproduced ing power T20 and the cross section in inclusive deuteron breakup at zero angle and at an initial energy of within a model that incorporates multiple scattering 2.1 GeV [7] were explained by the hybrid wave func- and the Pauli exclusion principle at the quark level [26]. By additionally taking into account the exchanges of * This article was submitted by the author in English. negative-parity nucleon resonances, Azhgirey and ** e-mail: [email protected] Yudin [27] were able to improve the agreement

2104 LADYGIN between the results of the calculations and experimen- where u(p) and w(p) are S and D components of the tal data on T20 in backward elastic dp scattering. The standard deuteron wave function based on the NN iχ iχ tensor analyzing power Ayy obtained for deuteron inclu- potentials, and v0(p)e and v2(p)e are the complex 6q sive breakup up to proton transverse momenta of admixtures to the S and D components of the standard 600 MeV/c [28] also disagrees with the calculations deuteron wave function, respectively. within the hard-scattering model [29] using conventional deuteron wave functions. However, the sign of Using parity conservation, time reversal invariance, 1) and the Pauli exclusion principle, we can write the Ayy at high proton momenta at zero angle [14Ð16], as well as at an angle of about 90° in the deuteron rest matrix of NN elastic scattering in terms of five indepen- frame [28], is identical to that predicted by the QCD- dent complex amplitudes [32] (when isospin invariance motivated approach used in [23]. is assumed): These special features of experimental data and rel- 1 M(k', k) = --((a + b) + (a Ð b)(s ⋅ n) ⋅ (s ⋅ n) atively successful attempts at describing them by con- 2 1 2 sidering nonnucleonic degrees of freedom stimulate ( )( ⋅ ) ⋅ ( ⋅ ) (4) possible efforts to measure additional polarization + c + d s1 m s2 m observables that are crucial to quark degrees of free- + (c Ð d)(s ⋅ l) ⋅ (s ⋅ l) + e((s + s ) ⋅ n) ), dom in the deuteron. 1 2 1 2 In [30], it was proposed to study the deuteron struc- where a, b, c, d, and e are the scattering amplitudes; s × 1 ture at short distances by using a polarized proton target and s2 are the Pauli 2 2 matrices; k and k' are the unit and a proton polarimeter. Here, investigation of T-odd vectors in the directions of the incident and scattered polarization observables [31] in deuteron exclusive particles, respectively; and center-of-mass basis vec- breakup in collinear geometry and backward elastic dp tors n, l, and m are defined as scattering is considered as a means for identifying the exotic 6q configurations inside the deuteron. k' × k k' + k k' Ð k n = ------, l = ------, m = ------. (5) k' × k k' + k k' Ð k 2. MATRIX ELEMENTS OF THE REACTIONS dp ppn AND dp pd However, at a zero angle there are only three independent amplitudes, and the matrix element (4) can be In this section, we analyze the polarization effects in written as [32] two processes: deuteron breakup in the strictly collinear geometry, d + p p(0°) + p(180°) + n, and ᏹ( ) 1( ( ⋅ ) ( ⋅ )( ⋅ )) backward elastic deuteronÐproton scattering, d + p 0 = -- A + B s1 s2 + C s1 k s2 k , (6) p + d, using the hybrid deuteron wave function with the 2 complex 6q admixture. where amplitudes A, B, and C are related to the ampli- This function can be presented in the momentum tudes defined in [32] as follows: space in the following form: α A = a(0) + b(0), B = c(0) + d(0), Φ ( ) i 1 ψ +  ( )( ⋅ ) d p = ------p U p s x -- (7) 2 4π C = Ð2d(0). (1) W( p)  β+ We consider deuteron breakup reaction in the spe------( ( ˆ ⋅ )( ⋅ ˆ ) ( ⋅ )) σ ψ Ð 3 p x s p Ð s x  y n , cial kinematics, i.e., with the emission of the spectator 2 αβ ψ ψ proton at zero angle, while the neutron interacts with where p and n are the proton and neutron spinors, the proton target and the products of this interaction go respectively; x is the deuteron polarization vector along the axis of the reaction. defined in a standard manner: Using both expressions (1) and (6), the matrix ele- 1 ( , , ) 1 ( , , ) x1 = Ð------1 i 0 , xÐ1 = ------1 Ði 0 , ment of deuteron breakup process in collinear geome- 2 2 (2) try can be written as [30] x = (0, 0, 1); 0 ᏹ i 1 ψ +ψ +  ( )( ⋅ ) = ------1 2 U p s x p is the relative protonÐneutron momentum inside the 2 2 4π deuteron; and pˆ = p/|p| is the unit vector in the p direc- ( ) tion. Here, S and D components are defined as W p ( ( ⋅ )( ⋅ ) ( ⋅ )) σ (8) Ð ------3 pˆ x s pˆ Ð s x  y ( ) ( ) ( ) iχ 2 U p = u p + v 0 p e , (3) iχ × ( ( ⋅ ) ( ⋅ )( ⋅ ))ψ ψ ( ) ( ) ( ) A + B s1 s2 + C s1 k s2 k 1* 2. W p = w p + v 2 p e , The matrix element of backward elastic dp scatter- 1) At zero emission angle, we have Ayy = ÐT20/ 2 . ing within the framework of one-nucleon exchange has

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 REACTIONS p(d, p)d AND p(d, p)pn 2105 the following form: 104

1 + ᏹ = ------ψ U*(p)(s ⋅ x*) -- 8π f  f f 2 1 φ 10 ( ) W* p ( ( ⋅ *)( ⋅ ) ( ⋅ *)) Ð ------3 pˆ x f s f pˆ Ð s f x f  2 (9) ( ) 10–2 ×  ( )( ⋅ ) W p ( ( ⋅ )( ⋅ ) U p si xi Ð ------3 pˆ xi si pˆ 2 1 ( ⋅ ) ) ψ --Ð si xi  i. 20

T 0

3. SIX-QUARK CONFIGURATIONS –1 In this section, we consider different models considering the quark (or baryonÐbaryon) degrees of freedom inside the deuteron. 1 In the hybrid model of the deuteron wave function [12], the 6q amplitude arising from the s6 conﬁgura- tions of six quarks must be added to the S component of 0 the standard deuteron wave function according to the κ 0 expression

χ U(k) = 1 Ð β2u(k) + βv (k)ei , (10) –1 0 0 0.4 0.8 where the parameter β and the phase χ represent the k, GeV/c value of the 6q admixture in the deuteron and the degree of nonorthogonality between np and 6q compo- Fig. 1. Momentum density φ2(k) [5], tensor analyzing power κ nents of the deuteron wave function, respectively. T20 [16], and polarization transfer coefficient 0 [17] (open squares), [18] (full triangles), and [19] (full circles and The 6q admixture has the following form: squares) versus internal momentum k in deuteron inclusive breakup with the emission of proton at 0°. Solid, dashed, 6 3/4 2 2/3 2   2  Ðk /3ω and dotted curves correspond to calculations with hybrid v (k) = I 10 ⋅ 2 ------e . (11) 0   3πω wave function [12] using the RSC [9], Paris [8], and Bonn 1 + 2 (C) [10] deuteron wave functions, respectively. Here, factor I ≈ 0.332 is the overlap factor of color spinÐisospin wave functions, ω defines the root-mean- 2 ω In [20, 25], the nonnucleonic degrees of freedom square radius of the 6q configuration r = 5/4 , and k is (NN*, NNπ, and higher components of the Fock space) internal momentum of a nucleon in the deuteron were taken into account in the following way: defined in the light-cone dynamics [2Ð4]. β χ Φ2(α, ) ( β2)φ 2 (α, ) ( α( α)) The parameters of the 6q admixture r, , and were kt = 1 Ð NN kt / 2 1 Ð obtained in [33] from the fit of the experimental data on (12) φ2 β2 (α, ) the momentum density of the nucleon in deuteron (k) + Gd kt , [5], tensor analyzing power T [16], and polarization- κ 20 where Φ2(α, k ) is the distribution of constituents in the transfer coefficient 0 [17Ð19] for inclusive deuteron φ tα breakup with the emission of the proton at a zero angle deuteron; NN( , kt) is the relativized standard deuteron wave function; G (α, k ) is the distribution of NN*, using standard deuteron wave functions [8Ð10]. The π d t results of the fit are given in Table 1 and shown in Fig. 1 NN , …, or 6q component in the deuteron. Parameter β2 by the solid, dashed, and dotted curves for the RSC [9], gives the probability of this nonnucleonic compo- Paris [8], and Bonn (version C) [10] deuteron wave functions, respectively. One can see the satisfactory Table 1. Parameters of the 6q admixture in the hybrid mo- description of the experimental data. The probability of del [12] for various standard deuteron wave functions [8Ð10] the 6q admixture is found to be 3Ð4%. The relative phase is ~40° for the RSC [9], ~47° for the Paris [8], and DWF β2, % χ, deg r, fm ° 55 for the Bonn (C) deuteron wave function [10]. The ± ± ± radius is r ~ 0.6 fm for all used deuteron wave functions. [8] 3.42 0.09 47.2 0.6 0.578 0.009 The parameters are comparable with the results obtained [9] 4.07 ± 0.10 40.1 ± 0.6 0.590 ± 0.009 in [7] using the RSC deuteron wave function [9]. [10] 2.79 ± 0.09 55.1 ± 0.7 0.595 ± 0.010

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2106 LADYGIN

104 tudinal momentum fraction α are deﬁned as [2Ð4] m 2 + k 2 k2 = ------p------t---- Ð m 2 , 4α(1 Ð α) p 2 1 φ 10 2 2 (14) k|| + k|| + m α = ------p-. 2 2 2 k|| + m p 10–2 Here, k|| is the longitudinal momentum in inﬁnite momentum frame and mp is the nucleon mass. α 1 The expression for nonnucleonic component Gd( , kt) is written as [25]

20 2 Ðbkt

T 0 (α, ) ( π) (α) Gd kt = b / 2 G1 e (15) with –1 (α) Γ( ) (Γ( )Γ( )) G1 = A2 + B2 + 2 / A2 + 1 B2 + 1 A B (16) × α 2(1 Ð α) 2, 1 where Γ(…) denotes the Γ function. The parameter b is chosen to be 5 GeV/c. We assume that nonnucleonic component (15), (16) has the relative phase χ with the

0 S wave of the standard deuteron wave function [20]. κ 0 The results of the fit of the experimental data [5, 16Ð 19] are given in Table 2 and shown in Fig. 2 by the solid, dashed, and dotted curves for RSC [9], Paris [8], –1 0 0.4 0.8 and Bonn (C) [10] deuteron wave functions, respec- k, GeV/c tively. The probability of the nonnucleonic component is found to be also ~3%. The relative phase χ between ° ° Fig. 2. Momentum density φ2(k) [5], tensor analyzing power NN and nonnucleonic components is 40 Ð60 . The κ T20 [16], and polarization transfer coefficient 0 [17Ð19] parameters A2 and B2 are found to be approximately the versus internal momentum k in deuteron inclusive breakup same for Paris [8], RSC [9], and Bonn (C) [10] deu- with the emission of proton at 0°. Solid, dashed, and dotted teron wave functions. Note that all the used NN deu- curves correspond to calculations with the wave function teron wave functions provide satisfactory agreement adopted in [20, 25] using the RSC [9], Paris [8], and Bonn (C) [10] deuteron wave functions, respectively. The notation with the existing data; however, using the RSC deu- for the points is identical to that in Fig. 1. teron wave function gives a better description of the κ polarization transfer coefficient 0. nent. The relativistic form of the deuteron wave func- φ α tion NN( , kt) can be written according to [2Ð4] as 4. T-ODD POLARIZATION EFFECTS Let us define the general spin observable of the third 2 2 1/4  m + k  order in terms of Pauli 2 × 2 spin matrices σ for protons φ (α, k ) = ------p------t---- φ(k), (13) NN t 4α(1 Ð α) and a set of spin operators Sλ for the spin-one particle for both reactions as [34] where φ(k) is the standard deuteron wave function (for p d + p tr(ᏹσα Sλ ᏹ σβ ) instance, [8Ð10]) and internal momentum k and longi- Cα, λ, β, 0 = ------+------, (17) tr(ᏹᏹ ) Table 2. Parameters of the 6q admixture [20, 25] for various where indices α and λ refer to the initial proton and standard deuteron wave functions [8Ð10] deuteron polarization, and index β refers to the final proton, respectively. DWF β , % A B χ, deg 2 2 2 We use a right-hand coordinate system, defined in [8] 2.96 ± 0.18 10.0* 20.23 ± 0.42 47.0 ± 0.6 accordance with Madison convention [35]. This system [9] 3.70 ± 0.43 10.0 ± 0.9 19.87 ± 2.72 40.2 ± 0.6 is specified by a set of three orthogonal vectors L, N, ± ± ± and S, where L is the unit vector along the momenta of [10] 2.67 0.19 10.0* 19.46 0.48 54.6 0.7 the incident particle, N is taken to be orthogonal to L, * This parameter is fixed. and S = N × L.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 REACTIONS p(d, p)d AND p(d, p)pn 2107

C0, SL, N, 0 CN, SL, 0, 0 0 (a) 0.2 (a)

–0.5 0 –1.0 –0.2 0 (b) (b) –0.5 0 –1.0 –0.2 0 (c) (c) –0.5 0 –1.0 –0.2 0 0.4 0.8 0 0.2 0.4 0.6 k, GeV/c k, GeV/c Fig. 3. Tensor-vector polarization transfer coefﬁcient C0, SL, N, 0 in deuteron exclusive breakup in the collinear Fig. 4. Tensor-vector spin correlation parameter CN, SL, 0, 0 geometry and backward elastic dp scattering using 6q in deuteron exclusive breakup in the collinear geometry at admixture adopted in [12] (solid curves) and [20, 25] 2.1 GeV of the deuteron initial energy using 6q admixture (dashed curves). The curves are obtained with the use of the adopted in [12] (solid curves) and [20, 25] (dashed curves). (a) Paris [8], (b) RSC [9], and (c) Bonn (C) [10] deuteron The curves are obtained with the use of the (a) Paris [8], (b) wave functions. RSC [9], and (c) Bonn (C) [10] deuteron wave functions.

In this paper, we consider T-odd polarization k ~ 600 MeV/c. The dependence on the used NN deu- observables, namely, tensor-vector spin correlations teron wave function occurs only at high k of about CN, SL, 0, 0 due to tensor polarization of the beam and 900 MeV/c. Therefore, the observation of a large neg- polarization of the initial proton and polarization trans- ative value of C0, SL, N, 0 could indicate that quark fer coefficient C0, SL, N, 0 from tensor polarized deuteron degrees of freedom play quite an important role in the to proton in the dp pd and dp p(0°) + p(180°) + deuteron at large k. n reactions. Note that such observables must be zero in Spin correlation parameter CN, SL, 0, 0 due to tensor the framework of one-nucleon exchange using standard polarization of the beam and polarization of the initial deuteron wave functions; however, they do not vanish with the existing 6q admixture in the deuteron wave function. CN, SL, 0, 0 Using the formulas for the matrix elements of the 0.2 (a) p(d, p)pn and p(d, p)d reactions (8) and (9), respectively, one can obtain the expression for the polariza- 0 tion transfer coefficient C0, SL, N, 0: χ 3 wv 0 sin –0.2 C , , , = ------. (18) 0 SL N 0 2 2 2 χ (b) 2u + w + v 0 + 2uv 0 cos 0 One can see that C0, SL, N, 0 does not depend on the initial energy and is defined only by the interference between the D wave of the standard deuteron wave function and –0.2 6q admixture. The results of the calculations with the (c) use of Paris [8], RSC [9], and Bonn (C) [10] deuteron 0 wave functions are presented in Figs. 3a, 3b, and 3c for two different models of the 6q admixture: [12] (solid –0.2 curves) and [20, 25] (dashed curves). These two types 0 0.2 0.4 0.6 of hybrid deuteron wave functions give quite similar k, GeV/c behavior of the C0, SL, N, 0 up to k ~ 800 MeV/c; however, they differ at higher momenta. Both models pre- Fig. 5. As in Fig. 4, but for the collinear geometry at dict the smooth variation of the C0, SL, N, 0 of about Ð1 at 1.25 GeV.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2108 LADYGIN ° CN, SL, 0, 0 where Aoonn(0 ) is spin correlation of elastic neutronÐ 0 (a) proton scattering at a zero angle for vertically polarized particles (see notation used in [32, 30]). Therefore, the ° ° –0.4 behavior of CN, SL, 0, 0 in the dp p(0 ) + p(180 ) + n reaction is deﬁned by both the deuteron wave function –0.8 and the np elementary amplitude, which is energy dependent. The calculations of C for the deu- 0 N, SL, 0, 0 (b) teron initial energy of 2.1 and 1.25 GeV using the results of phase-shift analysis performed in [36] are –0.4 shown in Figs. 4 and 5, respectively. One can see that C is positive at 2.1 GeV up to k ~ 550 MeV/c –0.8 N, SL, 0, 0 and negative at 1.25 GeV at k ~ 300Ð400 MeV/c. The 0 (c) difference between the two models of 6q admixture shown by the solid [12] and dashed [20, 25] curves for –0.4 (a) Paris [8], (b) RSC [9], and (c) Bonn (C) [10] deuteron wave functions is not dramatic at both energies. –0.8 0 0.4 0.8 Spin correlation parameter CN, SL, 0, 0 in backward k, GeV/c elastic deuteronÐproton scattering is given by

CN, SL, 0, 0 Fig. 6. Tensor-vector spin correlation parameter CN, SL, 0, 0 in backward elastic deuteronÐproton scattering using 6 q 2 2 2 (20) 1 wv sinχ((u + v cosχ Ð 2w) + v sin χ) admixture adopted in [12] (solid curves) and [20, 25] = ------0------0------0------. (dashed lines). The curves are obtained with the use of the 2 ( 2 2 2 χ)2 (a) Paris [8], (b) RSC [9], and (c) Bonn (C) [10] deuteron u + w + v 0 + 2uv 0 cos wave functions. The behavior of this observable for different types of 6q admixture in the deuteron wave functions is shown in C0, SL, N, 0 Fig. 6 for (a) Paris [8], (b) RSC [9], and (c) Bonn (C) 0 (a) [10] deuteron wave functions by the solid [12] and dashed [20, 25] curves. One can see that the spin correlation CN, SL, 0, 0 has a small negative value at low k, then –0.5 it approaches a minimum of Ð(0.7Ð0.8) at k ~ 400 MeV/c, and afterwards it goes smoothly to a zero for both mod- –1.0 els of 6q component. However, the use of deuteron wave functions with the 6q admixture adopted in [20,

CN, SL, 0, 0 25] gives systematically more negative value of spin (b) correlation at internal momenta ≥300 MeV/c. The use 0 of different NN potentials [8Ð10] (see Figs. 6a, 6b, and 6c, respectively) gives slightly different behavior of –0.4 CN, SL, 0, 0 for both models of 6q admixture. Neverthe- less, one can conclude that the measurements of spin correlation C in backward elastic dp scattering –0.8 N, SL, 0, 0 0 0.4 0.8 can help to distinguish between these two models. k, GeV/c Note that nonorthogonality in the deuteron wave function results in the T-invariance violation, which Fig. 7. (a) Tensor-vector polarization transfer coefficient contradicts the experiment. However, NN and 6q com- C0, SL, N, 0 in deuteron exclusive breakup in the collinear ponents can be orthogonalized following the procedure geometry and backward elastic dp scattering and (b) tensor- described in [37]. Such a procedure only slightly vector spin correlation parameter CN, SL, 0, 0 in backward elastic deuteronÐproton scattering using results of [39] and changes the probability of 6q admixture [37], but does Paris deuteron wave function [8]. not affect the behavior of the considered observables. For instance, the probability of 6q component changes from 2.96 to 3.31% and from 3.42 to 4.17% for the proton for the dp p(0°) + p(180°) + n process can models [12] and [20, 25], respectively, in the case of the be written as use of the Paris deuteron wave function [8]. The six-quark wave function of the deuteron has CN, SL, 0, 0 been calculated recently not only from s6, but also from 4 2 3 wv sinχ (19) s p configurations [38]. Such configurations are = ------0------A (0°), orthogonal to the usual S and D waves in the deuteron. 2 2 2 χ oonn 2u + w + v 0 + 2uv 0 cos Tensor analyzing power T20 and polarization transfer

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 REACTIONS p(d, p)d AND p(d, p)pn 2109

CN, SL, 0, 0 higher than 600 MeV/c could be a rather clear indication of the exotic 6q conﬁgurations. 0.4 (a)

5. CONCLUSION 0 We have considered T-odd observables in deuteron exclusive breakup and backward elastic dp scattering, –0.4 namely, tensor-vector polarization transfer coefficient (b) C0, SL, N, 0 and tensor-vector spin correlation CN, SL, 0, 0. These observables, which are associated with the tensor polarization of the deuteron and the polarization of the 0 proton, are sensitive to quark degrees of freedom in the deuteron and to their spin structure. The calculations give a sizable effect at large internal momenta, which –0.4 could be measured with the existing experimental tech- 0 0.4 niques. k, GeV/c Measurement of these observables could be performed at COSY at the Zero Degree Facility (ANKE) Fig. 8. Tensor-vector spin correlation parameter CN, SL, 0, 0 in deuteron exclusive breakup in the collinear geometry at using an internal polarized target with the detection of (a) 2.1 GeV and (b) at 1.25 GeV of the deuteron initial two charged particles in case of deuteron breakup and energy results of [39] and Paris deuteron wave function [8]. with the detection of the fast proton in case of backward elastic dp scattering. κ Such experiments could be also performed at the coefficient 0 in deuteron inclusive breakup at zero proton emission angle have been qualitatively reproduced Laboratory for High Energies of the Joint Institute for at large internal momenta using the results of these cal- Nuclear Research. The rotation of the primary deuteron culations [39]. The probability of the D wave originated spin could be provided by the magnetic field of the from s4p2 configurations was found to be about 0.5% (a beam line upstream of the target or by the special spin- small part of the total D wave probability ~6%). The rotating magnet. results on the polarization transfer C0, SL, N, 0 and spin correlation CN, SL, 0, 0 in backward elastic dp scattering ACKNOWLEDGMENTS using the Paris deuteron wave function [8] and the 6q projection on NN component from [39] are given in I am indebted to Dr. N. B. Ladygina for critically Figs. 7a and 7b, respectively. The behavior of this revising this manuscript and for permanent help and observable differs significantly from the results shown support. I am also grateful to Prof. A.P. Kobushkin for in Figs. 3 and 6. This deviation is due to presence of D helpful comments and discussions. I thank I.M. Sitnik wave in the six-quark wave function. The results on for estimating the bending angle of the magnetic sys- tensor-vector spin correlation C in the reaction tem upstream of the polarized target installed now at N, SL, 0, 0 the Laboratory for High Energies of JINR. dp p(0°) + p(180°) + n at the initial deuteron energy of 2.1 and 1.25 GeV are shown in Figs. 8a and 8b, respectively. The behavior is qualitatively the same as REFERENCES shown in Figs. 4 and 5; however, the value of C SL, N, 0, 0 1. V. G. Ableev et al., Nucl. Phys. A 393, 491 (1983); Erra- at 2.1 GeV and k ~ 300 MeV/c is twice as much as that tum: 411, 541 (1983); Pis’ma Zh. Éksp. Teor. Fiz. 37, in the case of absence of D wave. 196 (1983) [JETP Lett. 37, 233 (1983)]. One of the interesting features of QCD consists in 2. P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949). the possible existence of resonances in the dibaryon 3. S. Weinberg, Phys. Rev. 150, 1313 (1966). system corresponding to six-quark states with domi- 4. L. L. Frankfurt and M. I. Strikman, Phys. Rep. 76, 215 nantly hidden color, i.e., the states orthogonal to the (1981). usual np states. The rich structure in the behavior of the 5. V. G. Ableev et al., Pis’ma Zh. Éksp. Teor. Fiz. 45, 467 tensor analyzing power T20 in backward elastic dp scattering [22, 40] can be an indication of such dibaryon (1987) [JETP Lett. 45, 596 (1987)]; JINR Rapid Com- resonances [41]. mun., No. 1[52]-92, 10 (1992). 6. L. Anderson et al., Phys. Rev. C 28, 1224 (1983). Of course, the mechanisms additional to one- 7. C. F. Perdrisat et al., Phys. Rev. Lett. 59, 2840 (1987); nucleon exchange can contribute to CN, SL, 0, 0 and V. Punjabi et al., Phys. Rev. C 39, 608 (1989). C0, SL, N, 0. However, the calculations taking into account such mechanisms [20] show that their contribution is 8. M. Lacombe et al., Phys. Lett. B 101B, 131 (1981). small at large internal momenta. Thus, the observation 9. R. V. Reid et al., Ann. Phys. (N.Y.) 50, 411 (1968). of large values of CN, SL, 0, 0 and C0, SL, N, 0 at momenta 10. R. Machleidt et al., Phys. Rep. 149, 1 (1987).

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11. L. Bertocchi and A. Tekou, Nuovo Cimento A 21, 223 28. S. V. Afanasiev et al., Phys. Lett. B 434, 21 (1998); (1974); L. Bertocchi and D. Treleani, Nuovo Cimento A V. P. Ladygin et al., Few-Body Syst., Suppl. 10, 451 36, 1 (1976). (1999); L. S. Azhgirey et al., Preprint No. R1-98-199, 12. A. P. Kobushkin and L. Vizireva, J. Phys. G 8, 893 OIYaF (Joint Institute for Nuclear Research, Dubna, (1982). 1998); Yad. Fiz. 62, 1796 (1999) [Phys. At. Nucl. 62, 13. M. A. Ignatenko and G. I. Lykasov, Yad. Fiz. 46, 1080 1673 (1999)]. (1987) [Sov. J. Nucl. Phys. 46, 625 (1987)]. 29. L. S. Azhgirey and N. P. Yudin, Yad. Fiz. 57, 160 (1994) 14. V. G. Ableev et al., Pis’ma Zh. Éksp. Teor. Fiz. 47, 558 [Phys. At. Nucl. 57, 151 (1994)]. (1988) [JETP Lett. 47, 649 (1988)]; JINR Rapid Com- 30. V. P. Ladygin, Yad. Fiz. 60, 1371 (1997) [Phys. At. Nucl. mun. 4[43]-90, 5 (1990). 60, 1238 (1997)]. 15. T. Aono et al., Phys. Rev. Lett. 74, 4997 (1995). 31. G. G. Ohlsen, Rep. Prog. Phys. 35, 717 (1972). 16. L. S. Azhgirey et al., Phys. Lett. B 387, 37 (1996). 32. J. Bystricky, F. Lehar, and P. Winternitz, J. Phys. (Paris) 17. N. E. Cheung et al., Phys. Lett. B 284, 210 (1992). 39, 1 (1978). 18. A. A. Nomoﬁlov et al., Phys. Lett. B 325, 327 (1994). 33. V. P. Ladygin, PhD Thesis (Dubna, 1998). 19. B. Kuehn et al., Phys. Lett. B 334, 298 (1994); L. S. Azh- 34. V. Ghazikhanian et al., Phys. Rev. C 43, 1532 (1991). girey et al., JINR Rapid Commun., No. 3[77]-96, 23 (1996). 35. Proceedings of the Third International Symposium on Polarization Phenomena in Nuclear Reactions, Ed. by 20. G. I. Lykasov and M. G. Dolidze, Z. Phys. A 336, 339 H. H. Barschall and W. Haeberli (Madison, 1970). (1990); G. I. Lykasov, Fiz. Élem. Chastits At. Yadra 24, 140 (1993) [Phys. Part. Nucl. 24, 59 (1993)]. 36. R. A. Arndt et al., Phys. Rev. C 50, 2731 (1994); Code 21. C. F. Perdrisat and V. Punjabi, Phys. Rev. C 42, 1899 SAID-95. (1990). 37. V. V. Burov et al., Preprint No. R2-81-621, OIYaF (Joint 22. L. S. Azhgirey et al., Phys. Lett. B 391, 22 (1997); Yad. Institute for Nuclear Research, Dubna, 1981). Fiz. 61, 494 (1998) [Phys. At. Nucl. 61, 432 (1998)]. 38. L. Ya. Glozman, V. G. Neudatchin, and I. T. Obukhovsky, 23. A. P. Kobushkin, J. Phys. G 19, 1993 (1993). Phys. Rev. C 48, 389 (1993); L. Ya. Glozman and 24. S. J. Brodsky and J. R. Hiller, Phys. Rev. C 28, 475 E. I. Kuchina, Phys. Rev. C 49, 1149 (1994). (1983). 39. V. S. Gorovoj and I. T. Obukhovsky, in Proceedings of 25. A. V. Efremov et al., Yad. Fiz. 47, 1364 (1988) [Sov. J. the XII International Symposium on High Energy Phys- Nucl. Phys. 47, 868 (1988)]; Z. Phys. A 338, 247 (1991). ics Problems, Dubna, September 1994, Ed. by A. M. Bal- 26. A. P. Kobushkin, Phys. Lett. B 421, 53 (1998). din and V. V. Burov (1997), JINR E1.2-97-79, Vol. 2, p. 189. 27. L. S. Azhgirey and N. P. Yudin, Preprint No. R1-99-157, OIYaF (Joint Institute for Nuclear Research, Dubna, 40. V. Punjabi et al., Phys. Lett. B 350, 178 (1995). 1999); Yad. Fiz. 63, 2280 (2000) [Phys. At. Nucl. 63, 41. S. V. Afanasiev et al., JINR Rapid Commun., No. 4[84]- 2184 (2000)]. 97, 5 (1997).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2111Ð2114. Translated from Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2207Ð2210. Original Russian Text Copyright © 2000 by Akhobadze, Grigalashvili, Chikovani, Ioramashvili, Khizanishvili, Mailian, Nicoladze, Shalamberidze.

ELEMENTARY PARTICLES AND FIELDS Theory

Inclusive Features of Neutral Strange Particles in pA Interactions at 40 GeV/c K. G. Akhobadze, T. S. Grigalashvili, L. D. Chikovani, E. Sh. Ioramashvili, L. A. Khizanishvili, E. S. Mailian, M. I. Nicoladze, and L. V. Shalamberidze Institute of Physics, Georgian Academy of Sciences, ul. Tamarashvili 6, GE-380077 Tbilisi, Georgia Received June 21, 1999

Abstract—Mean inclusive features of neutral strange particles produced in antiproton interactions with D, Li, C, S, Cu, and Pb nuclei at an incident momentum of 40 GeV/c are found as functions of the target mass number. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION distribution of K0 mesons. The leading-particle effect Λ Inclusive characteristics of neutral strange particles was observed for hyperons in the nucleus-fragmenta- produced in antiproton interactions with nuclei at inter- tion region. mediate energies have not yet received adequate study. Such investigations remain topical at present. Our group has already analyzed the yields of neutral strange 2. DESCRIPTION OF THE EXPERIMENT AND PROCEDURE FOR DATA PROCESSING particles (K 0, Λ, Λ ) [1] and the associated multiplici- ties of charged particles [2] produced in antiprotonÐ An experiment that employed a relativistic ioniza- nucleus collisions at a momentum of 40 GeV/c. The tion streamer chamber (RISK) was performed at the present article reports on an investigation of inclusive Serpukhov accelerator U-70 with the aim of studying features of neutral strange particles in antiproton colli- hadronÐhadron and hadronÐnucleus interactions under sions with D, Li, C, S, Cu, and Pb nuclei. A global anal- the conditions of 4π coverage. A 5-m streamer chamber ysis of the aforementioned issues would make it possi- that was placed in a magnetic field of strength 1.5 T and ble to obtain deeper insights into the dynamics of had- which made it possible to measure charged-particle ronÐnucleus interactions. momenta to within 10% served as a detector. An unsep- In studying antiprotonÐproton collisions at a projec- arated beam that contained π–, K–, and p particles in tile momentum of 32 GeV/c, Bogolyubsky et al. [3] the proportion 98 : 1.7 : 0.3 traversed D, Li, C, S, Cu, compared the momentum characteristics of antiprotonÐ and Pb nuclear targets of thickness less than 1% of the proton and protonÐproton collisions at the same energy. nuclear range that were arranged at a distance of 30 cm It was shown that K 0 mesons produced in the annihila- from one another to minimize systematic errors. Eleven S targets featuring seven nuclear combinations—D, Li, tion channel have broader longitudinal-momentum dis- C, S, Cu, CsI, and Pb—were installed in the chamber of tributions than those in the nonannihilation channel. On × × 3 average, the kaon transverse momenta are somewhat fiducial volume 470 93 80 cm . In fact, we analyzed higher in antiprotonÐproton than in protonÐproton six pure nuclear species, discarding data on CsI interactions. because of its complex composition. The targets were of nonidentical thicknesses, which were reduced in In antiprotonÐproton collisions at c.m. energies of accordance with the growth of the cross section for the 200 and 900 GeV, the UA5 experiment [4] determined 〈 〉 production of secondary particles with increasing mass the mean transverse momenta p⊥ of neutral strange number of the nucleus. Table 1 presents the parameters particles in the rapidity region |y| ≤ 2 for Λ hyperons and of the targets in the same order as they were positioned | | ≤ 0 in the chamber. in the rapidity region y 2.5 for KS mesons. The mean values 〈p⊥〉Λ appeared to be greater than 〈p⊥〉 0 by a fac- Film data were viewed on a scanning table with a K tor of 1.5 at 200 GeV and by a factor of about 1.2 at magnification of 1 : 4 : 5 with respect to the actual 900 GeV. chamber dimensions. In scanning, we recorded all secondary two-prong stars (V 0 particles) whose total For pXe interactions at 200 GeV/c, the NA5 exper- momentum was directed toward the interaction vertex. iment [5] analyzed the rapidity and transverse-momen- These events were further considered as candidates for tum distributions of K 0 mesons and Λ hyperons. A shift strange particles. After double viewing, the detection toward low rapidity values was found in the rapidity efficiency was 99%.

2112 AKHOBADZE et al.

Table 1. Parameters of targets π hypotheses. The distributions in p⊥ , the transverse Number Target Thickness, mm Labs, % Lrad, % momenta of decay products in the rest frame with 1 D 198 Ð Ð respect to the direction of the neutral-particle momen- 2 Pb 0.867 0.315 15.480 tum in the laboratory frame, were used to separate Λ 0 Λ 0 3 S 7.123 0.890 6.720 ambiguously identified /KS and /KS particles. 4 Pb 0.612 0.222 10.930 0 Unseparated particles were associated with KS mesons 5 Li 16.500 0.840 1.060 π 6 Pb 0.819 0.298 14.630 if p⊥ ≥ 105 MeV/c including the errors in the measure- 7 CsI 1.666 0.283 8.820 ment of this momentum. Only unambiguously identi- 8 Cu 1.416 0.641 9.880 fied neutral strange particles were analyzed to deter- 9 C 7.040 0.939 2.570 mine their physical properties. More details on the 10 CsI 3.315 0.563 16.980 selection and identification of the relevant events can be 11 Cu 0.978 0.443 6.820 found in [1].

3. ANALYSIS OF EXPERIMENTAL DATA The vertex coordinates and the V 0-particle tracks were measured and reconstructed within the chamber In this article, the inclusive features of neutral volume. The momentum vectors and the polar and azi- strange particles as functions of the target mass number muthal emission angles for products originating from are presented for the reactions 0 V -particle decays were calculated on the basis of the pA K 0 + X, (1) results of the reconstruction. pA Λ + X, (2) The candidate events were then kinematically pro- Λ cessed. This included calculating the coplanarity, the pA + X. (3) mass, the total momentum, the transverse momentum, Figures 1a and 1b show the mean emission angles the energy, the emission angle with respect to the pri- 〈θ〉 for strange particles from reactions (1)Ð(3) with mary-particle momentum, and the decay range for V0 respect to the primary-particle momentum and their particles. Each V 0 event characterized by a deviation decay ranges 〈R〉 as functions of the target mass number from coplanarity in excess of 6° at a root-mean-square A. It is obvious that 〈θ〉 is independent of A for Λ error of about 2° was excluded from a further analysis. hyperons and increases in nearly the same way for K 0 Such events were either two-prong inelastic interac- mesons and Λ hyperons. For deuteron targets, the mean tions of neutral particles (NI) or three-body decays. emission angle for Λ hyperons is smaller than those for After that, the V 0 events were identified by means of a K0 and Λ by a factor of about 2.5, and this difference 0 Λ least squares kinematical fit to four hypotheses (KS , , increases with increasing atomic number. For D, Li, C, 0 Λ γ, Λ ) at three degrees of freedom (3C fit). Each two- and S targets, the K and decay ranges differ, but they prong event was taken to be unambiguously identified, become indistinguishable within the measurement χ2 errors in the case of heavy target nuclei (Cu and Pb). provided that did not exceed 11 for one of the With increasing A, the mean decay range is constant for Λ hyperons and decreases in nearly the same way for K0 〈θ〉, deg 〈R〉, cm mesons and Λ hyperons. 30 (a) 100 (b) As might have been expected, the mean decay range 80 〈R〉 of Λ antihyperons is sizably greater than the those 20 0 Λ 60 for K mesons and hyperons (by factors of about 2 βγ τ and 3, respectively). Since the decay range L0 = c 0 40 10 depends not only on the lifetime, but also on the parti- 20 cle momentum, Λ hyperons that have β values in 0 0 excess of 0.98 decay at distances of R > 0.7 m from the 0 1 2 0 1 2 10 10 10 A 10 10 10 A interaction vertex, and this explains the loss of Λ

0 hyperons having momenta above 20Ð25 GeV/c. Fig. 1. (a) Mean emission angles 〈θ〉 for K mesons, Λ Lambda antihyperons that escape detection for this rea- hyperons, and Λ hyperons and (b) their mean decay ranges son originate predominantly from the targets in the sec- versus the target mass number A. In both panels, data for K0, ond half of the chamber in the downstream direction (C Λ, and Λ are represented by crosses, open triangles, and and Cu), and that is why their mean momenta are open circles, respectively. biased for the production processes on these targets.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 INCLUSIVE FEATURES OF NEUTRAL STRANGE PARTICLES 2113

〈 〉 p , GeV/Ò 〈p⊥〉, GeV/Ò 10 (a) 0.7 (b) 8 0.6 6 0.5 4 0.4 2

0 0.3 100 101 102 A 100 101 102 A

0 Fig. 2. (a) Mean momenta 〈 p〉 and (b) mean transverse momenta 〈 p⊥〉 of (crosses) K mesons, (open triangle) Λ hyperons, and (open circles) Λ hyperons versus the target mass number A.

Since measurements of short tracks after the ( p , π+) 4. DISCUSSION OF THE RESULTS decay resulted in signiﬁcant errors (in order that ∆p/p ~ In [1, 2], our group analyzed data on the yields of 10%, the length of such “energetic” tracks must exceed neutral strange particles and on the associated multi- 1.5 m), these tracks were excluded from statistics. plicities of charged particles in events leading to the According to our estimate, their number was about production of neutral strange particles in pA interac- 1.5% of all observed Λ hyperons. tions at 40 GeV/c. The results of those analyses suggest the development of an intranuclear cascade in reaction Figures 2‡ and 2b display, respectively, the mean (2) and the suppression of such cascades in reactions momenta 〈p〉 and the mean transverse momenta 〈p⊥〉 of (1) and (3). The present analysis conﬁrms these conclu- strange particles as functions of the mass number A. As sions. was mentioned above, the mean momenta of Λ hyper- A possible explanation of the decrease in the mean momentum 〈p〉 of K 0 mesons with increasing target ons produced on C and Cu nuclei are somewhat under- 0 estimated. The mean momentum of K 0 mesons mass number is that K mesons, which are products of decreases with increasing target mass number, while a primary interaction or annihilation, have time to the mean momentum of Λ hyperons remains constant. undergo rescattering in the nucleus. It is well known that Λ hyperons are predominantly A slight growth of the transverse momenta 〈p⊥〉 with produced in cascade processes, but the secondary inter- increasing target mass number is observed for almost actions of annihilation kaons with intranuclear nucle- all neutral strange particles. ons is an additional source of Λ hyperons in antiprotonÐ nucleus collisions. Our analysis of the experimental The rapidity distributions are known to indicate the data reveals that the mean momentum 〈p〉 of Λ hyper- kinematical regions where the particles under study are produced. Figure 3 shows the mean laboratory rapidities 〈y〉 as functions of the target mass number. It is 〈y〉 obvious that, in the interactions on deuterons, K0 2.5 mesons and Λ hyperons are produced in the central rapidity region (the projectile-fragmentation region 2.0

〈y〉 0 = 2.27 ± 0.07 for kaons and 〈y〉 = 2.24 ± 0.10 for K Λ lambda antihyperons), while Λ hyperons are produced 1.5 in the target-fragmentation region (〈y〉Λ = 1.39 ± 0.05). 〈 〉 0 With increasing mass number, y values for K mesons 1.0 are shifted to the left, while those for Λ remain constant. 0.5 100 101 102 A All the inclusive features studied above were ﬁtted α in terms of the power-law function aA . Table 2 shows Fig. 3. Mean laboratory rapidities 〈 y〉 of (crosses) K 0 the ﬁtted parameter values and the corresponding val- mesons, (open triangle) Λ hyperons, and (open circles) Λ ues of χ2/NDF. hyperons versus the target mass number A.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2114 AKHOBADZE et al.

Table 2. Fitted values of the parameters in the power-law α while, for Λ hyperons, the mean rapidities are always form aA used to parametrize the inclusive features of neutral strange particles in the projectile-fragmentation region and remain invariable with increasing target mass number. Particle α a χ2/NDF 〈p〉 5. CONCLUSIONS K0 Ð0.07 ± 0.01 4.3 ± 0.4 2.9 To summarize the results obtained from our investi- Λ Ð0.035 ± 0.007 2.8 ± 0.2 0.5 gation of the mean inclusive features of neutral strange 0 Λ Ð0.03 ± 0.01 8.2 ± 0.7 0.6 particles (K mesons and Λ and Λ hyperons) produced 〈 〉 in antiprotonÐnucleus interactions at an incident p⊥ momentum of 40 GeV/c, we list here the changes that K0 0.027 ± 0.006 0.48 ± 0.02 1.1 these features have been found to undergo in response Λ 0.027 ± 0.004 0.40 ± 0.01 0.2 to the increase in the mass number of the target nucleus. 0 〈θ〉 Λ 0.044 ± 0.003 0.48 ± 0.01 0.1 (i) For K mesons, the mean emission angle increases, the mean decay range 〈R〉 decreases, the 〈y〉 mean momentum 〈p〉 decreases, the mean transverse 0 K Ð0.051 ± 0.007 2.3 ± 0.1 3.8 momentum 〈p⊥〉 increases, and the mean rapidity 〈y〉 is Λ Ð0.034 ± 0.009 1.3 ± 0.1 3.6 shifted from the projectile-fragmentation region to the target-fragmentation region. Λ ± ± Ð0.001 0.003 2.20 0.05 0.3 (ii) For Λ hyperons, the mean emission angle 〈θ〉 〈θ〉 increases, the mean decay range 〈R〉 and the mean K0 0.12 ± 0.01 12.0 ± 1.3 3.5 momentum 〈p〉 both remain virtually constant, the 〈 〉 Λ 0.112 ± 0.008 11.9 ± 0.8 0.6 mean transverse momentum p⊥ increases, and the mean rapidity 〈y〉 is always in the target-fragmentation Λ 0.02 ± 0.02 6.0 ± 1.0 3.8 region. 〈R〉 (iii) For Λ hyperons, the mean emission angle 〈θ〉 K0 Ð0.069 ± 0.008 56.0 ± 4.0 1.6 remains unchanged, the mean decay range 〈R〉 and the Λ Ð0.001 ± 0.010 36.0 ± 4.0 1.0 mean momentum 〈p〉 decrease, the mean transverse momentum 〈p⊥〉 increases, and the mean rapidity 〈y〉 is Λ ± ± Ð0.07 0.01 92.0 9.0 2.0 always in the projectile-fragmentation region. ons assumes a near-threshold value (about 2.5 GeV/c) ACKNOWLEDGMENTS 〈 〉 owing to these two processes. As a consequence, p is We are grateful to our collaborators in the RISC virtually independent of A, in contrast to what occurs in experiment, who participated in all its stages. π–A interactions, where the mean momentum 〈p〉 of Λ hyperons decreased from 4.5 to 2.3 GeV/c [6], because This work was supported by the Georgian Academy there occurred only cascade processes. of Sciences (project no. 2.11). Because of a high production threshold, Λ antihyperons originate only from primary interactions with REFERENCES nuclei—they can be generated neither in cascade pro- 1. T. Grigalashvili et al., Z. Phys. C (in press). cesses nor in the interactions of annihilation kaons 2. K. G. Akhobadze et al., Yad. Fiz. 63, 904 (2000) [Phys. within the target nucleus; as a result, the mean momen- At. Nucl. 63, 834 (2000)]. tum of Λ hyperons is virtually independent of the tar- 3. M. Yu. Bogolyubskiœ et al., Yad. Fiz. 50, 683 (1989) get mass number. [Sov. J. Nucl. Phys. 50, 424 (1989)]. 4. R. E. Ansorge et al., Nucl. Phys. B 328, 36 (1989). All the aforesaid is conﬁrmed by the results of the analysis of the mean rapidities 〈y〉 versus the target 5. I. Derado et al., Z. Phys. C 50, 31 (1991). mass number A for neutral strange particles. For K 0 6. K. G. Akhobadze et al., Yad. Fiz. 61, 245 (1998) [Phys. mesons, 〈y〉 is shifted from the projectile-fragmentation At. Nucl. 61, 196 (1998)]. region to the target-fragmentation region. For Λ hyperons, 〈y〉 is always in the target-fragmentation region, Translated by E. Kozlovskiœ

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2115Ð2122. Translated from Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2211Ð2218. Original Russian Text Copyright © 2000 by Arakelyan, G. Grigoryan, R. Grigoryan.

ELEMENTARY PARTICLES AND FIELDS Theory

LienardÐWiechert Potentials and Synchrotron Radiation from a Relativistic Spinning Particle in Pseudoclassical Theory S. A. Arakelyan, G. V. Grigoryan*, and R. P. Grigoryan** Yerevan Physics Institute, ul. Brat’ev Alikhanian 2, AM-375036 Yerevan, Armenia Received March 29, 1999; in ﬁnal form, December 3, 1999

Abstract—For a relativistic spinning particle with an anomalous magnetic moment, LienardÐWiechert potentials are constructed within the pseudoclassical approach. Some specific cases of the motion of a spinning particle are considered on the basis of general expressions obtained in this study for the LienardÐWiechert potentials. In particular, the intensity of the synchrotron radiation from a transversely polarized particle moving along a circle at a constant speed is investigated as a function of the particle spin. In the specific case of particles having no anomalous magnetic moment and moving in an external uniform magnetic field, the resulting expressions coincide with familiar formulas from the quantum theory of radiation. The spin dependence of the polarization of synchrotron radiation is investigated. © 2000 MAIK “Nauka/Interperiodica”.

1. It is well known that theories of pointlike particles This corresponds to the inclusion of the spin in the where particle spins are described even at the classical semiclassical approximation. In the case where the par- level can be constructed in terms of variables represent- ticle being considered moves in a uniform external ing elements of a Grassmann algebra [1, 2]. Such theo- magnetic field, the formulas obtained here in this ries, referred to as pseudoclassical theories, exemplify approximation (with allowance for the external-mag- constrained gauge systems. A detailed account of meth- netic-field dependence of the effective particle mass) ods for quantizing systems of this type can be found, for particles having no anomalous magnetic moment for example, in [3Ð5]. That the spin in external fields coincide with the analogous formulas derived previ- can be consistently described even within classical the- ously either in the semiclassical theory of radiation [10] ory by invoking Grassmann variables opens the possi- or in the corresponding quantum theory [11]. We also bility of studying spin effects in some physical pro- investigate the spin dependence of synchrotron-radia- cesses on the basis of classical equations of motion. It tion polarization. should be noted that the spin can also be introduced in It should be emphasized that a derivation of expres- classical theories without resort to Grassmann vari- sions describing the spin dependence of observables of ables, but all such theories do not lead, upon quantiza- the radiation from a relativistic particle is much more tion, to the conventional Dirac equation (see the review straightforward within the approach being discussed article of Frydryszak [6] and references therein). than within QED. In the present study, the pseudoclassical approach is The ensuing exposition is organized as follows. In used to construct LienardÐWiechert potentials for a rel- Section 2, an action functional is formulated within ativistic spinning charged particle having an anomalous pseudoclassical theory that describes the interaction of magnetic moment (AMM) and interacting with an elec- a relativistic spinning particle having an anomalous tromagnetic field (a consistent pseudoclassical theory magnetic moment with an electromagnetic field. We for the interaction of a spinning particle having an present there relevant equations of motion, introduce anomalous magnetic moment with an electromagnetic subsidiary gauge-fixing conditions, and derive general field was developed in [7–9]). Some specific cases of expressions for LienardÐWiechert potentials. In Sec- the motion of such a particle are considered here on the tion 3, we consider the case of a spinning particle at rest basis of general expressions that we obtain for the with a precessing spin. In Section 4, we obtain formulas LienardÐWiechert potentials. In particular, we derive for the strengths of the electromagnetic field of the syn- general expressions that describe the spin dependence chrotron radiation from a transversely polarized relativ- of the intensity and polarization of the synchrotron istic spinning particle moving along a circle at a con- radiation from a transversely polarized particle moving stant speed and investigate the spin dependences of the along a circle at a constant speed. Our analysis is per- intensity and the polarization of this radiation. The formed in the second-order approximation in Grass- motion of such a particle in a uniform external mag- mann variables (first-order approximation in the spin). netic field is investigated in detail.

* e-mail: [email protected] 2. Let us consider the action functional within a the- ** e-mail: [email protected] ory describing the interaction of a relativistic spinning

2116 ARAKELYAN et al. particle having an anomalous magnetic moment with We fix one of the gauge degrees of freedom by an electromagnetic field in D = 4 space. We have [7Ð9] imposing the constraint µ ( µ)2 µ ξ 1 τ xú 2 (ξ ξú ξ ξú ) xú µ = 0. (6) S = --∫d ------+ em Ð i µ Ð 5 5 2 e By virtue of relation (4), it is equivalent to the condition µ µ ν µ ν ξ ξ ξ ξ ξ 5 = 0. By using this condition and considering that, Ð 2gxú Aµ Ð igMeFµν + 4iGFµν xú 5 ξ (1) from relation (6), it follows that the quartic terms in µ ξ µ vanish, we find from Eqs. (3) and (5) that the auxiliary  µ xú µ ν  2 µ ν 2 χ Ð iχ ------+ mξ + iGFµνξ ξ ξ Ð eG (Fµνξ ξ ) fields e and are given by  e 5 5 xú2 1 4 ( ) µν( ) e = Ð ------, Ð --∫ d zFµν z F z , 2 µ ν 4 m Ð igMFµνξ ξ µ µ (7) ξ µ ν where x are the particle coordinates; are Grassmann µξν ξ ξ 4GFµν xú 4GFµν xú  iG µ ν variables describing spin degrees of freedom; 5, e, and χ = ------= ------1 Ð ----- Fµνξ ξ . χ χ ξ µ ν   are auxiliary fields, e ( and 5) being an even element m + iGFµνξ ξ m m (odd elements) of the Grassmann algebra; Aµ is the 4- (The sign of e is chosen in such a way that the energy is potential of the electromagnetic field; Fµν = ∂µAν – positive in the nonrelativistic limit.) ∂νAµ; g is the charge; G is the anomalous magnetic moment; M = 1 + 2Gm/g is the total magnetic moment Substituting expression (7) for e and the condition ξ of the particle in units of the Bohr magneton; overdots 5 = 0 into (2), we obtain the equation for the field Aµ in denote differentiation with respect to the parameter τ the form along the particle trajectory; and the derivatives with (∂ ∂µ λ ∂λ∂ µ)( ) λ( ) respect to Grassmann variables are left-hand. µ A Ð µ A y = j y , (8) λ The action functional in Eq. (1) possesses three where j (y) is given by gauge symmetries: reparametrization and supergauge λ( ) τ λδ( (τ) ) symmetries [8] and the gauge U(1) symmetry of elec- j y = g∫d xú x Ð y tromagnetic interactions. These symmetries correspond ∂ (9) igM τδ( (τ) ) 2ξµξλ to the presence of three primary first-class constraints + ------µ-∫d x Ð y xú . in the Hamiltonian formulation of the theory. The rele- m ∂y vant degrees of freedom will be fixed below. This equation can be further simplified as follows. In order to derive expressions for the LienardÐ We fix the remaining two gauges in the theory. For this, Wiechert potentials, we write down explicitly the equa- we choose the gauge condition for the field Aµ in the µ tions of motion for the electromagnetic field Aµ and for form ∂µA = 0 and take the parameter τ for the proper ξ χ the fields e, 5, and : time, whereby we fix, respectively, the U(1) gauge free- δ µλ λ S ∂ ( ) τ δ( (τ) ) dom and the reparametrization freedom. In this case, ------= µF y Ð g∫d xú x Ð y 2 δAλ(y) xú (τ) = 1. We note that Eq. (8) involves only second- ξ ξ τ( µξλ λξµ)ξ ∂ δ( (τ) ) order terms in µ. Since second-order terms in will be + 2iG∫d xú Ð xú 5 µ x Ð y proportional to ប upon quantization, our consideration τ ξµξλ∂ δ( (τ) ) (2) is equivalent to taking into account spin in the semiclas- Ð igM∫d e µ x Ð y sical approximation. With allowance for the above, Eq. (8) takes the form 2 τ ξσξρξµξλ∂ δ( (τ) ) Ð 2G ∫d eFσρ µ x Ð y µ µ A = j , (10) µ λ τχξ ξ ξ ∂ δ( (τ) ) µ µ + G∫d 5 µ x Ð y = 0, where ᮀ = ∂ ∂µ, and j is given by relation (9) in which 2 2 we have taken into account the condition xú = 1. It is δS xú 2 µ ν ----- = Ð ---- + m Ð igMFµνξ ξ µ δe 2 convenient to represent the current vector j as the sum e of two terms, (3) ξ µ µ µ 2( ξµξν)2 χ µ xú j (y) = g∫dτxú δ(x(τ) Ð y) Ð G Fµν + i ------2--- = 0, e ∂ (11) τδ( (τ) ) νµ(τ) µ + ν∫d x Ð y p , δ ξ µ ν ∂ 1 S µ xú ξ ξ ξ ξ y ------= ------+ m 5 + iGFµν 5 = 0, (4) i δχ e where δ µ ν µ ν 1 S ξú ξ χ ξ ξ µν igMξµξν µν νµ --δ----ξ---- = 2 5 Ð 4GFµν xú + (m + iGFµν ) = 0. p = ------, p = Ðp . (12) i 5 (5) m

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The first term in Eq. (11) corresponds to the current of polarization vector, the proportionality factor being a charged particle having no dipole moment, while the equal to the total magnetic moment of the particle. second term represents the contribution to the current By using the identity (see, for example, [12]) from the dipole moment of a charged spinning particle. α β β α λ σ It can easily be seen that, by virtue of relation (6), pµν = Ð (δµ δν Ð δµ δν )qα xúβ Ð εµνλσm xú (19) the particle electric dipole moment, which is related to ν and taking into account Eqs. (13) and (17), we obtain qµ by the equation qµ = –pµν xú (see, for example, [12]), the equality vanishes, gM λ σ pµν = Ð------εµνλσW xú , igM 2 (20) qµ = Ð------ξµ(xúξ) = 0, (13) m m as it must in the case of a pointlike particle. We will which will be needed below. now show that, from the pseudoclassical expression A solution to Eq. (10) with the current represented (12) for pµν, it follows that the particle magnetic in the form (11) was found in [12] in terms of retarded moment is expected to be aligned with the spin vector. fields. This solution can be written as For this, we consider that the particle-magnetic- µ µ µ 1 gxú moment pseudovector mµ is related to the dipole- A (y) = A (y) = ------µν ret π ρ τ τ moment tensor p by the equation [12] 4 = r (21) ν λσ 1  µν µν µν µν  mµ = Ð--εµνλσ xú p . (14) 1 1 1 Ð --------( pú kν Ð p (xúúk)kν) + -----( p kν Ð p xúν) , 2 4π ρ ρ2  τ τ By using Eq. (1), we further express the particle 4- = r ˜ momentum ᏼµ = (Ᏹ , 3) as µν g 2 [µ ν ] [µ ν ] 2 [µ ν ] F (y) = ------(k xúú Ð (xúúk)k xú ) + -----k xú ret π ρ 2 ∂ ν χξ 4 ρ τ = τ ᏼ L xúµ ξ ξ µ r µ = ------+ gAµ = ---- + 2iGFµν 5 Ð i------. (15) ∂ µ e 2e [µν] [µν] [µν] (22) xú 1 2P 2Q 2R  + ------------1------+ ------1------+ ------1------ , Using the gauge specified by Eq. (6), taking into 4π ρ 2 3 4  ρ ρ τ τ account (7), and disregarding the quartic terms in ξ , we = r 2 µ µ µ µ ν find from (15) (at xú = 1) that where R ≡ y – x (τ), R Rµ = 0, ρ = xú Rν, and kµ = µ µ ν λ µ Rµ ρ; all the quantities in (21) and (22) are taken at the ᏼ 2iG ᏼ ξ ξ / xú = Ð /meff Ð ------Fνλ , τ τ 2 time instant r specified by the equation t0 = t0( r) = t – m 1) (16) R(t0). The explicit expressions for the quantities igM µνξ ξ appearing in (22) are meff = m Ð ------F µ ν. 2m µν µ µ 2 µ µ α ν P = { púú α Ð 3(xúúk) pú α + 3(xúúk) p α Ð (úxúúk) p α}k k , For the case being considered, the analog of the relation 1 between the particle 4-momentum and the particle (23) 2 2 µν µν µ α ν µ (α ν ) µν mass has the form ᏼµ – m = 0. The signs on the Q = pú + 3 pú αk k Ð 4 pú αk xú Ð (xúúk) p eff 1 (24) right-hand side of the first equation in (16) are governed ( ) µ α ν ( ) µ (α ν ) µ (α ν ) by the definition of the generalized momentum conju- Ð 6 xúúk p αk k + 6 xúúk p αk xú Ð 2 p αk xúú , gate to the particle coordinate. µν µν µ α ν µ (α ν ) µ α ν R1 = p + 3 p αk k Ð 6 p αk xú + 2 p α xú xú . Substituting (12) and (16) into (14), we obtain (25) ᏼν In Eqs. (21) and (22), the bracketed expressions cor- i gMε ξλξσ gMWµ g mµ = ------µνλσ------= ------= M------aµ, (17) respond to the contributions of a charged particle to Aµ 2 m m m m 2m and Fµν without allowing for its spin, whereas the where Wµ is the pseudoclassical analog of the PauliÐ braced expressions represent the contribution of the Lubanski vector and is given by (see [9]) dipole moment of this particle. i ν λ σ Substituting the explicit expressions for the quanti- Wµ = --εµνλσᏼ ξ ξ , (18) µν µ 2 ties p and x and for their derivatives into (21)Ð(25), we obtain, within pseudoclassical theory, the LienardÐ while aµ/2 is a relativistic generalization of the pseudoclassical spin vector (upon a canonical quantiza- 1)Hereafter, brackets […] in the subscript denote a complete anti- tion of the theory, the vector aµ goes over to the polar- symmetrization—for example, A[αβγ] = 1/3!{Aαβγ + Aβγα + ization 4-vector of the spinning particle being consid- Aγαβ – Aβαγ – Aαγβ – Aγβα}; parentheses (…) denote a complete ered). From (17), it follows that the vector of the mag- symmetrization—for example, A(αβγ) = 1/3!{Aαβγ + Aβγα + Aγαβ + netic moment of a particle is proportional to its Aβαγ + Aαγβ + Aγβα}.

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Wiechert potentials in the general form and the corre- By using (29) and (32) and the equality p0i = 0, we sponding strength tensor for the electromagnetic field find from (26) that the strengths of the electric and mag- created by a charged spinning particle. netic fields created by an immobile charged particle 3. In order to clarify the correspondence between with a precessing spin are given by the pseudoclassical expressions obtained here for the g gM LienardÐWiechert potentials and the well-known for- E = F = ------n + ------{ω ([w × S] ⋅ n) i i0 2 i π i mulas for the strengths of the electric and magnetic 4πR 4 mR (33) fields of a charged particle having a magnetic moment, gM we consider, by way of example, the case of a particle Ð (w ⋅ n)[w × S] } + ------{ω (S ⋅ n) Ð (w ⋅ n)S }, i π 2 i i i , µ δµ 4 mR at rest (xú = 0 xú = 0 ) with a precessing spin. From Eqs. (22)–(25), we can easily find that the 1ε gM { ( ⋅ ) } Bi = Ð-- iklFkl = ------3 n S ni Ð Si strength tensor of the electromagnetic field created by a 2 4πmR3 particle at rest is given by gM {[ × ] ([ × ] ⋅ ) ( ⋅ )( ⋅ )ω µν g µ ν 1 µβ ν Ð ----π------w n i w S n + w n n S i F (y) = ------n δ + ------púú nβn 4 mR ret 2 0 π 4πR 4 R (34) ( ⋅ )2 } gM { [ × ] Ð w n Si + ------2 w S i 1 ( µν µβ ν µβ δ ν) 4πmR2 + ------2 pú + 3 pú nβn Ð 2 pú nβ 0 4πR (26) [ × ] ( ⋅ ) ( ⋅ )[ × ] } Ð 3 w n i n S + 3 w n S n i . 1 ( µν µβ ν µβ δ ν) + ------3 p + 3 p nβn Ð 3 p nβ 0 From these formulas, we can easily obtain expres- π τ τ 4 R = r sions for the strengths of the electric and magnetic fields of an immobile particle having a fixed magnetic- Ð [µ ν]τ τ , = r moment vector. The results are µ µ 2 2 2 g where n = R /R = (1, n) and R = R = R1 + R2 + R3 0 Ei = ------ni, (for retarded fields, we have R = y – x (τ) > 0). For- 4πR2 0 0 0 (35) mula (26) was obtained by considering that, in the rest gM 0i { ( ⋅ ) } frame, p = 0, which immediately follows from (20). Bi = ------3 3 n S ni Ð Si . From (20) at v = 0, we can also obtain 4πmR

gM k 0 gM Expressions (33)Ð(35) coincide with the corresponding p = Ð------ε W xú = Ð------ε W (27) ij 2 ijk0 2 ijk k expressions for the ﬁelds generated by a particle having m m the charge g and the total magnetic-moment vector m = ε ε ε0ijk ε ( ijk = – 0ijk = , 123 = 1). gM ------S (see, for example, [14, 15]). Considering that, in the particle rest frame, we have [13] m

Wi = mSi, (28) 4. In this section, we obtain expressions for the where S is the particle spin, we arrive at strengths of the electromagnetic ﬁeld of the synchro- i tron radiation generated by a relativistic spinning parti- gM p = Ð------ε S . (29) cle moving along a circle at a constant speed, as well as ij m ijk k for the power and polarizations of this radiation. This is done for the case of a transverse particle polarization Accordingly, pú and púú are given by ij ij (the vector of the particle spin is orthogonal to the plane of its rotation). The problem is of interest for testing the gMε ú gMε úú púij = Ð------ijkSk, púúij = Ð------ijkSk. (30) possibility of determining particle polarization in m m cyclic accelerators on the basis of measured observ- Let us now assume that the particle spin precesses ables of synchrotron radiation. about a certain axis at a constant angular velocity w; that is, It is well known that the radiation ﬁeld is controlled by those terms in expression (22) that are proportional ú [ × ] ε ω to 1/ρ; that is, Si = w S i = ikl kSl. (31) µν ( ) Substituting (31) into (30), we obtain Frad y gM [µν] (36) ------(ω ω ) g 2 [µ ν ] [µ ν ] 1 2P1 púij = Ð iS j Ð jSi , = ------(k xúú Ð (xúúk)k xú ) + ------, m πρ π ρ τ τ (32) 4 4 = r gM {ω [ × ] ω [ × ] } µν púúij = Ð------i w S j Ð j w S i . m where P1 is given by (23).

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Let us now transform expression (36) with allow- v ance for the relations θ n µ xú = γ (1, v), ρ = γ (R Ð (v ⋅ R)) = γR(1 Ð (v ⋅ n)), a ϕ γ = 1/ 1 Ð v2, µ xúú = (γ 4(v ⋅ a), γ 2a + γ 4v(v ⋅ a)), (37) B(S)

α R R R α Fig. 1. Reference frame for specifying angles in considering  ,  angular features of radiation. k = -ρ-- -ρ-- = -ρ--n ,

µ úxúú = (aú0, γ 3b + 3γ 5(v ⋅ a)a + aú0v), aligned with the external magnetic field; see Fig. 1)— this is precisely the case that is considered here—we aú0 = γ 5[(v, b) + a2] + 4γ 7(v ⋅ a)2, have where v ≡ (vi) = dxi/dt is the particle 3-velocity, a = (v ⋅ a) = (S ⋅ v) = (S ⋅ a) = (S ⋅ b) = 0, dv/dt, b = da/dt, and dt/dτ = (1 – v2)–1/2 = γ. (42) ú γ ω2 Upon substituting expressions (37) into the first S = 0, ú = 0, b = Ð v, term in (36), we arrive at the well-known formula for where ω is the angular velocity of particle rotation. ch the strength Ei of the electric field of the radiation Taking into account these equalities, we recast expres- from a charged spinless particle: sion (41) into the form

ch g gMγ ε E = ------pij = Ð------ijkSk. (43) i 3 m 4πR(1 Ð (v ⋅ n)) (38) × [(n Ð v )(a ⋅ n) Ð a (1 Ð (v ⋅ n))]. Using (40) and (43) and taking into account (42), we i i i obtain µν In order to transform the spin component P1 of the gM 2 pú = Ð------γ ε S a , µν µν µν µ 0i m ijk j k tensor Frad [ P1 contains p , which is related to W by Eq. (20)], we make use of the equation that relates Wµ gMγ 3ε gMω2γ 3ε (44) ≡ púú0i = Ð------ijkS jbk = ------ijkS jv k, and the vector S (Si) of the particle spin in the rest m m frame and which has the form [9, 13] púij = púúij = 0. D D( D ⋅ ) Wµ (3 ⋅ S) 3i 3 S  , µν ------= ------Si + ------ m m m(Ᏹ + m) With the aid of the explicit expression (23) for P1 , (39) the contributions that the dipole moment of a particle v (v ⋅ S) γ ( ⋅ ), γ 2 i  makes to the strengths Erad and Brad of the electromag- =  v S Si + ------ , γ + 1 netic ﬁeld of the radiation generated by this particle can D D be represented as where 3µ = –ᏼµ = (Ᏹ, 3 ) stands for the physical particle 4-momentum and where the second equality has dip 1 j µ E = Ð------(T k k Ð T µk k ) been written in the approximation linear in S. i 4πρ 0 j i i 0 Using Eq. (20) and taking into account Eq. (39), we 2 R arrive at = Ð ------(T (δ Ð n n ) + T n ), (45) πρ3 0 j ij i j ij j gM j k gM 4 p = Ð------ε W xú = Ðγ ------ε S v , (40) 0i 2 0ijk m ijk j k rad ch dip rad [ × rad] m Ei = Ei + Ei , B = n E , gMε µ ν where pij = Ð------2-- ijµνW xú m 2 (41) Tµν = púúµν Ð 3(xúúk) púµν + 3(xúúk) pµν Ð (úxúúk) pµν. (46) gM γ 2 = Ð ------ε γ S Ð ------(v ⋅ S)v . m ijk k γ + 1 k From (37) and (42), we obtain µ µ In the case of a transversely polarized particle mov- xúú = (0, γ 2ai), úxúú = (0, γ 3bi) = (0, Ðγ 3ω2v i). ing along a circle at a constant speed (the particle spin is (47)

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2120 ARAKELYAN et al. Taking into account (40), (43), (44), and (47), we find where θ and ϕ are angles that specify the direction of from (46) that the vector n in the coordinate frame determined by the vectors v, a, and B(S) (see Fig. 1). 3(a ⋅ n)a gMγ 3ε k Integrating expression (54) over the solid angle, we T0i = Ð------ijkS j ------m 1 Ð (n ⋅ v) find that the radiation power is given by (48) 2 2 ω 2 2 dI g 2a M SB 3v (a ⋅ n) ω v ------= ------1 Ð ------. (55) + ------k------Ð ------k------, dt 4π 2 2 m 2 2 0 3(1 Ð v ) 1 Ð v (1 Ð (n ⋅ v)) 1 Ð (n ⋅ v) In order to compare this expression with the analogous 2 2 gM 3 3(a ⋅ n) ω (n ⋅ v) expression obtained in the quantum theory of radiation, T = Ð------γ ε S ------Ð ------. (49) ij ijk k 2 ( ⋅ ) we consider that, for the case of a uniform external m (1 Ð (n ⋅ v)) 1 Ð n v magnetic field, the effective particle mass, which is In the case of the motion of a transversely polarized par- given by (16), takes the simple form ticle along a circle at a constant speed, we also have, in i gM 1 addition to the equalities in (42), the obvious relations m = m Ð ------F ξ ξ = m Ð --F p . (56) eff 2 m ij i j 2 ij ij S ε S v = [S × v] = Ð----B-a , The right-hand equality in (56) is written with allow- ijk j k i ω i (50) ance for Eq. (12). Substituting the expression for pij ε [ × ] ω ijkS jak = S a i = SB v i, from (43) into (56), we find that, in the first order in the spin, meff is given by where SB is the spin-vector projection onto the direction gM of the external magnetic field B. Taking these relations m = m Ð γ ------B S . (57) into account, we can recast expression (48) into the eff m i i form ᏼ D ω With the aid of (16), the expression for µ in the case gMγ 3 ai T0i = Ð------of above-type motion in an external magnetic field can m 1 Ð (n ⋅ v) be reduced to the form

(51) D 2 2 2 2 2 (ᏼ ) Ᏹ γ γ v ( ⋅ ) 3(a ⋅ n)ωv 3(a ⋅ n) a µ = meff , = meff + 2G B S , + ------i Ð ------i----- S . (58) 1 Ð (n ⋅ v) 2 B D γ γ 2( ⋅ ) ω(1 Ð (n ⋅ v)) 3 = meffv + 2G B S v. Substituting (51) and (49) into (45), we obtain By means of expression (57) and the expression for Ᏹ from (58), we find in the first order in the spin (for dip gM E = ------γ ӷ 1) that i π ( ( ⋅ ))3 4m R 1 Ð n v n n Ᏹ  g Ᏹ  γ = --- 1 + n------BS  2 2 ω    2 B  3(a ⋅ n) ω (n ⋅ v)  a j m m ×  ------Ð ------[n × S] + ------m  2 ( ⋅ ) i ( ⋅ ) (59)  ( ( ⋅ )) 1 Ð n v 1 Ð n v n 2 1 Ð n v Ᏹ n Ᏹ (52) = --- 1 + ------S ω . m mm B 3(a ⋅ n)ωv 3(a ⋅ n)2a  + ------j Ð ------j----- S (δ Ð n n ) . 1 Ð (n ⋅ v) ω( ( ⋅ ))2 B ij i j In deriving the last equality, we have made use of 1 Ð n v  the relation gB = mωγ. Further taking into account (59), Using relations (38) and (52) and the equality we can recast (55) into the form 2 2 4 2 SB dI g 2a Ᏹ 4 Ð M Ᏹ [n × S] = ------((v ⋅ n)a Ð (a ⋅ n)v ), (53) ------ω --- i 2 i i ------= π   1 + SB   . (60) ωv dt0 4 3 m m m which is obtained with allowance for (42) and (50), we 1 ζ ζ ± can represent the energy of the synchrotron radiation Considering that S = -- , where = 1 is the spin com- Ω 2 emitted by a particle into a solid-angle element d per ponent along the magnetic field, we find that, in the unit of “particle time” (at the radiation instant) t0 as absence of an anomalous magnetic moment (M = 1), 2 2 formula (60) (in the first approximation in the spin) dI  g  a 2 ------= ------(1 Ð v cosθ) -- coincides with the analogous formulas obtained in the Ω  π 5 dt0d 4 (1 Ð v cosθ) (54) semiclassical ([10]) or in the quantum ([11]) theory of radiation. Thus, formula (60) appears to be a generali- ( 2) 2 θ 2 ϕ Mω 2 θ 2 ϕ Ð 1 Ð v sin cos Ð 2---- SB sin sin , zation of the formula for the synchrotron-radiation m intensity in the first order in the spin to the case where

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 LIENARDÐWIECHERT POTENTIALS AND SYNCHROTRON RADIATION 2121 the particle in question possesses an anomalous mag- B netic moment. n As in [16], we will investigate the polarization properties of synchrotron radiation by decomposing the β electric-field strength Erad into two components aligned v with two mutually orthogonal unit vectors e = a/a and [n × e]: ψ rad [ × ] a E = E1e + E2 n e . (61) This choice of unit vectors is reasonable when use is Fig. 2. Reference frame for specifying angles in describing made (with the aim of describing radiation from fast radiation from fast particles. particles) of the reference frame in Fig. 2 for specifying the relevant angles. This reference frame is convenient  ω because the dominant contribution comes from small 2gaβψ SB angles β and ψ of about 1/γ, since the angle between the E2 = Ð------1 Ð ------π (µ 2 ψ2)3 m vectors n and v is about 1/γ. As was indicated in [16], R 0 + the representation of Erad in the form (61) is disadvan- (66) (µ2 ψ2) ( )( µ2 ψ2)  tageous in that the right-hand side of Eq. (61) is not 2 0 Ð 5 4 M Ð 1 2 0 Ð × ------+ ------ . orthogonal to the vector n, but this uncertainty is about (µ 2 ψ2)2 (µ 2 ψ2)2  0 + 0 + τ τ 1/γ, so that the decomposition in (61) can be used to = r compute leading terms (to a precision of 1/γ). From µ 2 ψ2 Eqs. (38) and (52), we find, according to the decompo- At 0 – = 0, it follows from (65) and (66) that sition in (61), that E and E are given by 1 2 ω 2 ga SB 2Mβ ga E = ------1 Ð ------ , (67) E = ------1 π µ 6 m  µ 2  1 2 2 3 4 R 0 0 τ τ πR(µ + ψ ) = r (62) ( ) ω 2 2 2 gaβψ  3 Ð M SB  2 2 M 4β (µ Ð 5ψ ) E = Ð------1 + ------. (68) × (ψ µ ) ---- ω------2 6  2  Ð + SB 2 , 4πRµ mµ τ τ m (µ2 ψ2) 0 0 = r + τ = τ r From the above formulas, it can be seen that, in the 2ga (a, v) plane (β = 0), there is radiation (proportional to E = Ð------2 2 2 3 the spin) in the direction ψ = m/Ᏹ, in contrast to what πR(µ + ψ ) we have in the case of a spinless particle. (63) M 4βψ(2µ2 Ð ψ2) × βψ Ð ----S ω------, m B 2 2 2 ACKNOWLEDGMENTS (µ + ψ ) τ τ = r We are grateful to I.V. Tyutin and A. Airapetyan for where µ2 = γ–2 + β2. enlightening discussions on the problems considered here. Taking into account Eq. (59), we represent the expression for µ2 in the form This work was supported in part by grants nos. 96- 538 and 93-1038 from INTAS and the joint grant 2 2 2 m 2 2 2 no. 95-0829 from INTAS and the Russian Foundation µ = β + --- Ð --- S ω = µ Ð --- S ω, Ᏹ m B 0 m B for Basic Research. (64) 2 2 2 m µ β --- REFERENCES 0 = + Ᏹ . 1. F. A. Berezin and M. S. Marinov, Ann. Phys. (N.Y.) 104, Further substituting the expression for µ2 into (62) 336 (1977). and (63), we find in the first order in the spin that 2. R. Casalbuoni, Nuovo Cimento A 33, 115 (1976). 3. P. A. M. Dirac, Lectures on Quantum Mechanics  (Yeshiva Univ., New York, 1964). ga (ψ2 µ 2) E1 = ------ Ð 0 -- 4. D. V. Gitman and I. V. Tyutin, Canonical Quantization of π (µ 2 ψ2)3 R 0 + Constrained Fields (Nauka, Moscow, 1986). (65) 5. M. Henneaux and C. Teitelboim, Quantization of Gauge 2 2 2 2 2 4 2ψ Ð µ Mβ (µ Ð 5ψ )  Systems (Princeton Univ. Press, Princeton, 1992). + --- S ω ------0- + ------0------ , m B µ 2 ψ2 2 2 2 6. A. Frydryszak, hep-th/9601020. 0 + (µ + ψ ) τ τ 0 = r 7. A. A. Zheltukhin, Teor. Mat. Fiz. 65, 151 (1985).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2122 ARAKELYAN et al.

8. D. Gitman and A. Saa, Class. Quantum Grav. 10, 1447 14. V. V. Batygin and I. N. Toptygin, Problems in Electrody- (1993). namics (Nauka, Moscow, 1970; Academic, London, 9. G. V. Grigoryan and R. P. Grigoryan, Yad. Fiz. 58, 1146 1964). (1995) [Phys. At. Nucl. 58, 1072 (1995)]. 15. V. A. Bordovitsyn, N. N. Byzov, G. K. Razina, and 10. I. M. Ternov and V. A. Bordovitsyn, Vestn. Mosk. Univ., V. Ya. Épp, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 5, 12 Ser. 3: Fiz., Astron. 24, 69 (1983). (1978). 11. I. M. Ternov, V. G. Bagrov, and R. A. Rzaev, Zh. Éksp. 16. V. N. Baœer, V. M. Katkov, and V. S. Fadin, Radiation Teor. Fiz. 46, 374 (1964) [Sov. Phys. JETP 19, 255 from Relativistic Electrons (Atomizdat, Moscow, 1973); (1964)]; Synchrotron Radiation, Ed. by A. A. Sokolov V. N. Baier, V. M. Katkov, and V. M. Strakhovenko, Elec- and I. M. Ternov (Nauka, Moscow, 1966). tromagnetic Processes at High Energies in Oriented Sin- 12. J. R. Ellis, J. Math. Phys. 7, 1185 (1965). gle Crystals (World Sci., Singapore, 1998). 13. G. V. Grigoryan and R. P. Grigoryan, Zh. Éksp. Teor. Fiz. 103, 3 (1993) [JETP 76, 1 (1993)]. Translated by A. Isaakyan

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

ELEMENTARY PARTICLES AND FIELDS Theory

Semileptonic Decays of the Bc Meson* V. V. Kiselev, A. K. Likhoded, and A. I. Onishchenko1) Institute for High Energy Physics, Protvino, Moscow oblast, 142284 Russia Received September 1, 1999; in ﬁnal form, February 1, 2000

ψ η Abstract—The semileptonic decays of the Bc meson into heavy quarkonia J/ ( c) and a pair of leptons are investigated on the basis of three-point sum rules of QCD and nonrelativistic QCD (NRQCD). Calculations of analytic expressions for the spectral densities of QCD and NRQCD correlation functions with allowance for Coulomb-like α /v terms are presented. At recoil momenta close to zero, generalized relations due to the spin s ψην symmetry of NRQCD are derived for the Bc J/(c)ll form factors, with l denoting one of the leptons (e, µ, or τ). This allows one to express all NRQCD form factors in terms of a single universal quantity, an analog of the IsgurÐWise function at the maximal lepton-pair invariant mass. The gluon-condensate corrections to three-point functions are calculated both in full QCD in the Borel transform scheme and in NRQCD in the moment scheme. This enlarges the parametric-stability region of the sum-rule method, thereby rendering the ψην results of the approach more reliable. Numerical estimates of the widths for the transitions Bc J/(c)ll are presented. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION heavy-quark mass. Thus, we see that the production of σ σ Ð3 In [1], the CDF collaboration reported on the first Bc is relatively suppressed, (Bc)/ (bb ) ~ 10 , experimental observation of the Bc meson, a heavy because of an additional heavy-quark pair in final quarkonium with a mixed heavy flavor [1]. This meson states. Basic special features of production mechanisms is similar to the families of the charmonium cc and the appear owing to higher orders of QCD even in the leading approximation: the fragmentation regime at high bottomonium bb in what is concerned with its spectro- transverse momenta much greater than the quark scopic properties: two heavy quarks move nonrelativis- masses and a strong role of nonfragmentation contribu- tically, since the confinement scale, determining the tions at pT ~ mQ, which admit a precise perturbative cal- presence of light degrees of freedom (sea of gluons and culation; a negligible contribution of the octet mecha- quarks), is suppressed with respect to the heavy-quark nism [4] because there is no enhancement due to a masses m , and Coulomb-like exchanges result in trans- α Q lower order in s. The predictions for the cross sections α 2 fers about s mQ , which is again much less than the and distributions of Bc in various interactions are dis- heavy-quark mass. Therefore, the nonrelativistic picture cussed in [5], where we see good agreement with the of binding the quarks leads to the well-known arrange- CDF measurements [1]. ment of system levels, which is very similar to those in In contrast to cc and bb states decaying owing to the families mentioned above. The calculations of the the annihilation into light quarks and gluons, the Bc bc mass spectrum were reviewed in [2, 3]. We expect meson is a long-lived particle, since it decays owing to M = 6.25± 0.03 GeV, weak interaction. The lifetime and various modes of Bc decays were analyzed within (a) potential models [6], (b) whereas the measurement yielded technique of operator-product expansion in the effective theory of nonrelativistic QCD (NRQCD) [7] by consid- M exp = 6.40± 0.39± 0.13 GeV. Bc ering the series in both a small relative velocity v of heavy quarks inside the meson and the inverse heavy- There is a significant difference in the mechanisms quark mass [8], and (c) QCD sum rules [9, 10] applied to that govern the production of the heavy quarkonia cc , the three-point correlation functions [11Ð13]. The results bb, and Bc. To bind the b and c quarks, one has to pro- of potential models and NRQCD are in agreement with duce four heavy quarks in flavor-conserving interac- each other. Thus, we expect that the total lifetime is 2) tions, which allows one to use perturbative QCD, τ = 0.55± 0.15 ps, since the virtualities are determined by the scale of the Bc which agrees, within the errors, with the experimental * This article was submitted by the authors in English. value given by CDF [1]: 1) Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia. exp +0.18 2) τ = 0.46 ± 0.03 ps. We do not consider the production in weak interactions. Bc Ð0.16

2124 KISELEV et al. µ τ Further, a consideration of exclusive Bc decays within where l denotes one of the leptons e, , or . This pro- QCD sum rules showed that the role of Coulomb cor- cedure is similar to that of two-point sum rules, and rections to the bare-quark-loop results could be very information from the latter on the coupling of mesons important to reach the agreement with the other to their currents is required in order to extract the values approaches mentioned [13]. This requires evaluating of the form factors. Thus, in our work, we will use the α s /v corrections in NRQCD, which possesses spin meson couplings, defined by the equations symmetry providing some relations between the exclu- 2 sive form factors. For semileptonic decays, such a rela- 〈 | γ | ( )〉 f PMP tion was derived in [14]. Note that the CDF collabora- 0 q1i 5q2 P p = ------(1) m1 + m2 tion observed 20 B + J/ψe+(µ+)ν events, so that a c and consistent calculation of semileptonic decay modes is of interest. For theoretical reviews on B -meson phys- 〈 | γ | ( , ⑀)〉 ⑀ c 0 q1 µq2 V p = i µMV f V , (2) ics, the reader is referred to [15]. In this study, we perform a detailed analysis of semi- where P and V represent the scalar and vector mesons with desired flavor quantum numbers, respectively, and leptonic Bc decays within QCD and NRQCD sum rules. We recalculate the double spectral densities available m1 and m2 are the quark masses. Now we would like to previously in [11] in full QCD for the massless leptons describe the method used. and add the analytic expressions for the form factors necessary in evaluation of decays to massive leptons 2.1. Description of the Method and P-wave levels of quarkonium with different quark masses. We analyze the NRQCD sum rules for the As we have already said, for the calculation of had- three-point correlation functions for the first time. We ronic matrix elements relevant to the semileptonic Bc derive generalized relations between the NRQCD form decays into the pseudoscalar and vector mesons in the factors, which extends the consideration in [14], framework of QCD, we explore the QCD sum-rule because we explore a soft limit of recoil momentum method. The hadronic matrix elements for the transi- close to zero, wherein the velocities of initial and final + ψ η +ν tion B J/ ( c)l l can be written down as fol- heavy quarkonia v1, 2 are not equal to each other, when c their product tends to zero, in contrast to the hard limit lows: v v 1 = 2. The spin symmetry relations between the form 〈η ( )| | ( )〉 ( ) factors are conserved after taking into account the Cou- c p2 V µ Bc p1 = f + p1 + p2 µ + f Ðqµ, (3) α lomb s /v corrections, which can be written down in covariant form. We investigate numerical estimates in 1 ν α β --〈J/ψ( p )|V µ|B ( p )〉 = iF ⑀µναβ⑀* ( p + p ) q ,(4) the sum-rule schemes of spectral-density moments and i 2 c 1 V 1 2 Borel transform and show an important role of the Cou- lomb corrections. Next, we perform the calculation of 1〈 ψ( )| | ( )〉 A⑀ -- J/ p2 Aµ Bc p1 = F0 µ* gluon condensate contribution to the three-point sum i (5) rules of both full QCD and NRQCD for the case of A(⑀ ⋅ )( ) A(⑀ ⋅ ) three massive quarks, for the first time. + F+ * p1 p1 + p2 µ + FÐ * p1 qµ, The QCD sum rules of three-point correlation func- ⑀µ ⑀µ where qµ = (p1 Ð p2)µ, = (p2) is the polarization vec- tions are considered in Section 2, where the spectral ψ densities are calculated in the bare quark-loop approx- tor of the J/ meson, and Vµ and Aµ are the flavor- imation and with allowance for the Coulomb correc- changing vector and axial electroweak currents. The tions, and the gluon-condensate term in the Borel trans- A A 2 form factors f±, FV, F0 , and F± are functions of q form scheme is presented. Section 3 is devoted to the only. It should be noted that by virtue of transversality NRQCD sum rules for recoil close to zero. The spin of the lepton current lµ = lγµ(1 + γ )ν in the limit m symmetry relations are derived, and the gluon conden- 5 l l +ν sate is taken into account in the scheme of moments. 0, the probabilities of semileptonic decays into e e and µ+ν A The numerical results are summarized in Section 4. µ are independent of fÐ and FÐ . Thus, in calculation Appendices A and B contain technical details of evalu- of these particular decay modes of Bc meson, these ation of decay widths for the massive leptons and gluon form factors can be consistently neglected [6, 11, 13]. condensate in full QCD, respectively. However, since the calculation of both semileptonic decay modes, including e, µ, or τ, and some hadronic 2. THREE-POINT QCD SUM RULES decays, stands among the goals of this paper, we will present the results for the complete set of form factors In this article, we will use the approach of three- given in (3)Ð(5). point QCD sum rules [9, 10] to study form factors and Following the standard procedure for the evaluation + ψ η +ν decay rates for the transitions Bc J/ ( c)l l, of form factors in the framework of QCD sum rules, we

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SEMILEPTONIC DECAYS 2125 consider the three-point functions The leading QCD term is a triangle quark loop dia-

( ⋅ ⋅ ) gram, for which we can write down the double disper- Π ( , , 2) 2 i p2 x Ð p1 y sion representation at q2 ≤ 0: µ p1 p2 q = i ∫dxdye (6) ρpert( , , 2) × 〈0|T{q (x)γ q (x), V µ(0), b(y)γ c(y)}|0〉, pert 2 2 2 1 s s Q 1 5 2 5 Π ( p , p , q ) = Ð------i------1------2------ds ds i 1 2 ( π)2∫( 2)( 2) 1 2 , ( ⋅ ⋅ ) 2 s1 Ð p1 s2 Ð p2 Π V A( , , 2) 2 i p2 x Ð p1 y (12) µν p1 p2 q = i ∫dxdye (7) + subtractions, × 〈 | { ( )γ ( ), V, A( ), ( )γ ( )}| 〉 0 T q1 x µq2 x Jµ 0 b y 5c y 0 , where Q2 = Ðq2 ≥ 0. The integration region in (12) is determined by the condition γ γ where q1 (x) 5q2(x) and q1 (x) µq2(x) are interpolating η ( 2)( 2 2 ) currents for states with the quantum numbers of c and 2s1s2 + s1 + s2 Ð q mb Ð mc Ð s1 , Ð1 < ------< 1 (13) ψ V A λ1/2( , , 2)λ1/2( 2, , 2) J/ , respectively, and Jµ are the currents Vµ and Aµ s1 s2 q mc s1 mb of relevance to the various cases. and The Lorentz structures in the correlation functions can be written down as λ( , , ) ( )2 x1 x2 x3 = x1 + x2 Ð x3 Ð 4x1 x2. Π Π ( ) Π µ = + p1 + p2 µ + Ðqµ, (8) ρ pert 2 The calculation of spectral densities i (s1, s2, Q ) and Π V Π ⑀ α β µν = i V µναβ p2 p1 , (9) gluon-condensate contribution to (11) will be considered in the next sections. Now let us proceed with the Π A Π A Π A Π A physical part of three-point sum rules. The connection µν = 0 gµν + 1 p2, µ p1, ν + 2 p1, µ p1, ν (10) to hadrons in the framework of QCD sum rules is A A obtained by matching the resulting QCD expressions of + Π p , µ p , ν + Π p , µ p , ν. 3 2 2 4 1 2 current correlation functions with the spectral represen- A A tation, derived from the double dispersion relation at The form factors f±, FV, F0 , and F± will be determined, q2 ≤ 0: A respectively, from the amplitudes Π±, Π , Π , and V 0 ρphys( , , 2) 2 2 2 1 i s1 s2 Q A 1 A A Π ( , , ) Π Π ± Π i p1 p2 q = Ð------∫------ds1ds2 ± = -- ( 1 2 ). In (8)Ð(10) the scalar amplitudes ( π)2 ( 2)( 2) 2 2 s1 Ð p1 s2 Ð p2 Π Π (14) i are the functions of kinematical invariants, i.e., i = + subtractions. Π ( p 2 , p 2 , q2). i 1 2 Assuming that the dispersion relation (14) is well con- To calculate the QCD expression for the three-point vergent, the physical spectral functions are generally correlation functions, we employ the operator product saturated by the lowest hadronic states plus a contin- expansion (OPE) for the T-ordered product of currents uum starting at some effective thresholds s th and s th : in (6) and (7). The vacuum correlations of heavy quarks 1 2 are related to their contribution to the gluon operators. ρ phys( , , 2) ρ res( , , 2) i s1 s2 Q = i s1 s2 Q For example, for the 〈QQ 〉 and 〈QGQ 〉 condensates, (15) θ( th)θ( th)ρ cont( , , 2) the heavy-quark expansion gives + s1 Ð s1 s2 Ð s2 i s1 s2 Q , α α 1 s 2 1 s 3 where 〈QQ〉 = Ð ------〈G 〉 Ð ------〈G 〉 + …, 12m π 3 π Q 360mQ ρ res( , , 2) 〈 | γ (γ ) | ψ(η )〉 i s1 s2 Q = 0 c µ 5 c J/ c m α α 2 2 2 〈 〉 Q ( 2 ) s 〈 2〉 1 s 〈 3〉 … × 〈J/ψ(η )|F (Q )|B 〉〈B |bγ c |0〉(2π ) δ( s Ð M ) QGQ = ------log mQ --π--- G Ð ------π--- G + . c i c c 5 1 1 2 12mQ (16) × δ(s Ð M 2) + higher state contributions, Then, in the lowest order for the energy dimension of 2 2 operators, the only nonperturbative correction comes and M1, 2 denote the masses of quarkonia in the initial from the gluon condensate: and ﬁnal states. The continuum of higher states is modeled by the perturbative absorptive part of Π , i.e., by Π ( 2, 2, 2) Π pert( 2, 2, 2) ρ i i p1 p2 q = i p1 p2 q i. Then, the expressions for the form factors Fi can be derived by equating the representations for the three- 2 α (11) G 2 2 2 s 2 Π + Π ( p , p , q ) -----G . point functions i in (11) and (14), which means the i 1 2 π formulation of sum rules.

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2.2. Calculating the Spectral Densities ρ A( , , 2) 0 s1 s2 Q In this section, we present the analytic expressions  to one-loop approximation for the perturbative spectral 3 ( ) 2 1( ∆ 2 ∆ 2 ∆ ∆ ) = ------ m1 Ð m3 m3 + -- s1 2 + s2 1 Ð u 1 2 functions. We have recalculated their values, already 1/2 k available in the literature [11]. Among new results there k A A A ∆ ∆ (20) are the expressions for ρ , ρ , and ρ±' , where ρ±' are  2   2  – Ð Ð m m Ð -----1 Ð m m Ð -----2 spectral functions, which come from the double disper- 2 3 2  1 3 2  A 1 A A sion representation of Π±' = -- (Π ± Π ). These  2 3 4 2 1(∆ ∆ ) + m3 m3 Ð -- 1 + 2 Ð u + m1m2 , spectral densities are not required for the purposes of 2  this paper, but they will be useful for calculation of form factors for the transition of the B meson into a A 2 c ρ (s , s , Q ) scalar meson.3) The procedure of evaluating the spec- + 1 2 tral functions involves the standard use of Cutkosky  rules [16]. There is, however, one subtle point in using 3 [ ∆ ∆ ∆ ∆ ∆ 2] = ------m1 2s2 1 Ð u 1 + 4 1 2 + 2 2 these rules. At Q2 > 0, there is no problem in applying 2k3/2 ρ 2 the Cutkosky rules in order to determine i(s1, s2, Q ) and the limits of integration with respect to s and s . At 2[ ] [ ∆ ∆ ] 1 2 + m1m3 4s2 Ð 2u + m2 2s1 2 Ð u 1 Q2 < 0, which is the physical region, non-Landau-type [ ( ∆ ∆ ) ( ∆ ∆ ) singularities appear [17, 18], which makes the determi- Ð m3 2 3s2 1 + s1 2 Ð u 3 2 + 1 + k (21) nation of spectral functions quite complicated. In our ∆ ∆ ∆ 2 2( ) ] + 4 1 2 + 2 2 + m3 4s2 Ð 2u case, we restrict the region of integration in s1 and s2 by s th and s th , so that at moderate values of Q2 the non- 6 1 2 + --(m Ð m )[4s s ∆ ∆ Ð u(2s ∆ ∆ Landau singularities do not contribute to the values of k 1 3 1 2 1 2 2 1 2 spectral functions. For spectral densities ρ (s , s , Q2), i 1 2  we have the expressions ∆ 2 ∆ 2 ) ( ∆ 2 ∆ 2) ] --+ s1 2 + s2 1 + 2s2 s1 2 + s2 1 ,   ρ ( , , 2) 3 k(∆ ∆ ) + s1 s2 Q = -------- 1 + 2 3/22 2k ρ A( , , 2) 3 { ( 2 ) Ð s1 s2 Q = Ð------kum3 2m1m3 Ð 2m3 + u [ ( ) ( )] 2k5/2 Ð k m3 m3 Ð m1 + m3 m3 Ð m2 (17) [ ( ∆ ∆ ) (∆ ∆ )] ( ) 2∆ 2 ∆ [( ) Ð 2 s2 1 + s1 2 Ð u 1 + 2 + 12 m1 Ð m3 s2 1 + k 2 m1 + m3 u

2 2 u  ( ) ] ∆ ( )( ) × m Ð -- + m m Ð m m Ð m m , Ð 2s1 m2 Ð m3 + 2 2 k + 3us1 m1 Ð m3 3 2 1 2 2 3 1 3 (22)  ∆ [ ( ) ∆ ( 2)( )] + 1 ku m2 Ð m3 + 2 2 k Ð 3u m1 Ð m3  ρ ( , , 2) 3 k(∆ ∆ ) 2 2 Ð s1 s2 Q = Ð-------- 1 Ð 2 + 2s (m Ð m )[2km Ð k∆ + 3u∆ Ð 6u∆ ∆ ] 2k3/22 2 1 3 3 1 1 1 2 ( ∆ 2( )) } [ ( ) ( )] Ð 2s1s2 km3 Ð 3 2 m1 Ð m3 , Ð k m3 m3 Ð m1 Ð m3 m3 Ð m2 (18) [ ( ∆ ∆ ) (∆ ∆ )] A 2 3 2 + 2 s2 1 Ð s1 2 + u 1 Ð 2 ρ ( , , ) { ( )[( )∆ +' s1 s2 Q = Ð------Ð2 m1 Ð m3 k Ð 3us2 1 2k5/2 2  × u 2 2 2 2 m3 Ð -- + m1m2 Ð m2m3 Ð m1m3 , ∆ ] ( )( ∆ ) 2  + 6s1 2 + ku m1 Ð m3 2m3 + 2 + ku m3 ∆ [ ( ) + 1 ku 2m1 Ð m2 Ð 3m3 2 3 ρ (s , s , Q ) = ------{(2s ∆ Ð u∆ )(m Ð m ) V 1 2 3/2 1 2 1 3 2 ( )( ∆ 2∆ ) ] k (19) Ð 2 m1 Ð m3 ks2 Ð k 2 + 3u 2 (23) + (2s ∆ Ð u∆ )(m Ð m ) + m k }, 2 1 2 3 1 3 [( )( 2 ∆ ∆ ∆ 2) Ð 2s1 m1 Ð m3 2km3 Ð 6u 1 2 Ð 3u 2 3) 〈 | γ | 〉 2 2 The meson at the P-wave level, for which 0 q1 µq2 P(p) = ( ∆ ∆ ) + 2s2 km3 + 3m1 1 Ð 3m3 1 ifPpµ, where P(p) denotes the scalar P-wave meson under consid- ≠ ∆ ( ) ] } eration, and m1 m2. + k 2 2m1 Ð m2 Ð 3m3 ,

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SEMILEPTONIC DECAYS 2127

ρ A( , , 2) 3 { ( )[( )∆ 2 q Ð' s1 s2 Q = ------2 m1 Ð m3 k + 3us2 1 2k5/2 2∆ 2 ] ( )( 2 ∆ ) 2 + 6s1 2 + ku m1 Ð m3 2m3 + 2 + ku m3 ∆ [ ( ) p1 + k m1 m2 p2 + k + 1 ku Ð 2m1 Ð m2 + m3 ( )( ∆ 2∆ ) ] Ð 2 m1 Ð m3 ks2 Ð k 2 + 3u 2 (24) m3 [( )( 2 ∆ ∆ ∆ 2) p p + 2s1 m1 Ð m3 2km3 Ð 6u 1 2 + 3u 2 1 k 2 Ð 2s (km Ð 3m ∆ 2 + 3m ∆ 2) 2 3 1 1 3 1 Fig. 1. The triangle diagram giving the leading perturbative ∆ ( ) ] } term in the OPE expansion of the three-point function. + k 2 2m1 + m2 Ð m3 . Here, k = (s + s + Q2)2 Ð 4s s , u = s + s + Q2, ∆ = 1 2 1 2 1 2 1 q 2 2 ∆ 2 2 s1 Ð m1 + m3 , and 2 = s2 Ð m2 + m3 ; m1, m2, and m3 are the masses of quark ﬂavors relevant to the various decays (see prescriptions in Fig. 1). α π We neglect hard O( s / ) corrections to the triangle diagrams, as they are not available yet. Nevertheless, we expect that their contributions are quite small (~10%); bearing in mind the accuracy of QCD sum rules, we therefore conclude that the correction will not drastically change our results.

In expressions (12), the integration with respect to s1 k1 . . . kn l1 . . . lk and s2 is performed in the near-threshold region, where α E E instead of s, the expansion should be done in the 1 p1 . . . pn q1 . . . qk 2 α p parameters ( s/v13(23)), with v13(23) meaning the relative velocities of quarks in bc and cc systems. For the Fig. 2. The ladder diagram of the Coulomb-like quark inter- heavy quarkonia, where the quark velocities are small, action. these corrections take an essential role (as in the case α for two-point sum rules [19, 20]). The s/v corrections, caused by the Coulomb-like interaction of quarks, are tion is given by related to the ladder diagrams shown in Fig. 2. It is well 4 2 4π 2 4π known that, numerically, the Born value of the spectral V˜ (k) = Ð--α (k )------, α (k ) = ------, 3 s 2 s ( 2 Λ2) density is doubled or tripled upon taking into account k b0ln k / these corrections in two-point sum rules [21, 22]. Now, let us comment on the effect of these correc- 2 1 31 5  b = 11 Ð --n , Λ = Λ exp ------Ð --n , tions in the case of three-point sum rules [13]. Con- 0 3 f MS b  6 9 f Π 0 sider, for example, the three-point function µ(p1, p2, q) 2 2 2 with n being the number of ﬂavors, while the fermionic at q = qmax, where qmax is the maximum invariant mass f η ν propagators, corresponding to either a particle or an of the lepton pair in the decay Bc cl l. Introduce ≡ ≡ antiparticle, have the following forms: the notation p1 (mb + mc + E1, 0) and p2 (2mc + E2, 2 2 ( γ ) Ӷ i 1 + 0 /2 0). At s1 = M1 and s2 = M2 , we have E1 (mb + mc) ( ) SF k + pi = ------2------, and E Ӷ 2m . In this kinematics, the quark velocities 0 k 2 c E + k Ð ------+ i0 are small, and, thus, the diagram in Fig. 2 may be con- i 2m sidered in the nonrelativistic approximation. We will use the Coulomb gauge, in which the ladder diagrams ( γ ) ( ) Ði 1 Ð 0 /2 with the Coulomb-like gluon exchange are dominant. SF p = ------2------. 0 k The gluon propagator then has the form Ð k Ð ------+ i0 µν µ ν 2m D = iδ 0δ 0/k2. (25) The notation concerning Fig. 2 is given by In this approximation, the nonrelativistic potential of ≡ ≡ heavy-quark interaction in the momentum representa- ki = pi + 1 Ð pi, li = qi Ð qi Ð 1, pn + 1 q0 p.

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0 0 x Integration with respect to p , p0, and q by means of Provided = 0, only the terms with l = 0 are retained in i i the sum. For the spectral density we then have residues yields the expression ρ ( , , 2 ) Ψ C( )Ψ C( ) Π ( , , ) µ E1 E2 qmax = Ð2Ncgµ0 1 0 2 0 µ E1 E2 q = 2Ncgµ0 C C dp (30) ∞ n × Ψ˜ (p)Ψ˜ (p)------, dp ∫ 1E1 2E2 3 × ------i------˜ (( )2) (2π) ∑ ∏ 2 V pi + 1 Ð pi 3 p  n = 1 i = 1 (2π) -----i---- Ð E Ð i0 Ψ C  µ 1  where i are the Coulomb wave functions for the bc 2 1 or cc systems. An analogous expression can also be 1 1 × ------(26) derived in the Born approximation: p 2 p 2 ------Ð E Ð i0 ------Ð E Ð i0 ρ B( , , 2 ) µ 1 µ 2 µ E1 E2 qmax 2 1 2 2 f f (31) ∞ k f f ˜ ˜ dp dq = Ð 2N gµ Ψ (0)Ψ (0) Ψ1E (p)Ψ2E (p)------. × j ˜ (( )2) dp c 0 1 2 ∫ 1 2 3 ∑∏------V q j Ð q j Ð 1 ------, (2π) q 2 (2π)3 k = 1 j = 1 ( π)3 j  2 ----µ----- Ð E2 Ð i0 Ψ f 2 2 Here, i stands for the function of free quark motion. Since the continuous-spectrum Coulomb functions V˜ ((p Ð p )2) ≡ V˜ ((p Ð p )2), have the same normalization as the free states, we n + 1 n n obtain the approximation V˜ ((q Ð q )2) = V˜ ((q Ð p)2), 1 0 1 Ψ C(0)Ψ C(0) ρ ( , , 2 ) ≈ ρ B( , , 2 )-----1------2------µ µ µ E1 E2 qmax µ E1 E2 qmax f f where Nc denotes the number of colors, and 1 and 2 Ψ ( )Ψ ( ) 1 0 2 0 (32) are the reduced masses of the bc and cc systems, ≡ ρB( , , 2 ) respectively. This three-point function may be µ E1 E2 qmax C, expressed in terms of the Green’s functions for the relative motion of heavy quarks in the bc and cc systems 4πα  4πα  Ð1 C = ------s 1 Ð exp Ð------s (i) 3v  3v  in the Coulomb ﬁeld, GE (x, y):  13 13 (33) 1/2   4πα  4πα  Ð1  ∞  n  × ------s 1 Ð exp Ð------s  , (i) dpk   G (x, y) =  ------ 3v 23 3v 23  E ∑ ∏∫ 2  3 p  n = 1 k = 1 (2π) -----k---- Ð E Ð i0   µ i  (27) where v13 and v23 are relative velocities in the bc and 2 i n Ð 1 cc systems, respectively. For them, we have the fol- ⋅ ⋅ × ˜ (( )2) ip1 x Ð ipn y lowing expressions: ∏V pk Ð pk + 1 e . k = 1 4m m v = 1 Ð ------1------3------, (34) 13 2 ( )2 Comparing expressions (26) and (27), we ﬁnd p1 Ð m1 Ð m3

2 Πµ(E , E , q ) 4m m 1 2 max v = 1 Ð ------2------3------. (35) (28) 23 2 ( )2 (1) (2) dp p2 Ð m2 Ð m3 = 2N gµ G (x = 0, p)G (p, y = 0)------c 0∫ E1 E2 3 (2π) Equation (32) is exact for the identical quarkonia in the (1) (2) 3 initial and ﬁnal states of transition under consideration. = 2N gµ G (0, z)G (z, 0)d z. c 0∫ E1 E2 However, if the reduced masses are different, then the overlapping of Coulomb functions can deviate from For the Green’s function, we use the representation unity, which breaks the exact validity of (32). From a pessimistic viewpoint, this relation can serve as an esti- ∞  Ψ ( )Ψ* ( ) mate of the upper bound on the form factor at zero ( , ) nlm x nlm y GE x y = ∑ ∑ ------recoil. In reality, this boundary is practically saturated,  Enl Ð E Ð i0 l, m n = l + 1 which means that in sum rules at low momenta inside (29) the quarkonia, i.e., in the region of physical resonances, Ψ (x)Ψ* (y) the most essential effect comes from the normalization dk klm klm  + ∫(------π----)------. factor C, determined by the Coulomb function at the 2 k Ð E Ð i0  origin. The latter renormalizes the coupling constant at

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SEMILEPTONIC DECAYS 2129 the quarkÐmeson vertex from the bare value to the 0 0 “dressed” one. After that, the motion of heavy quarks in k2 the triangle loop is very close to that of free quarks. k1 k2 In accordance with (8) for the Lorentz decomposi- p1 + k p1 + k ρ 2 tion of µ(p1, p2, q ), we have p2 + k 2 2 2 ρµ( p , p , q ) = ( p + p ) ρ (q ) + qµρ (q ). (36) p p p p 1 2 1 2 µ + Ð 1 k 2 1 k 2 As we have seen, the nonrelativistic expression of k ρ 2 1 µ(E1, E2, qmax ) is proportional to the vector gµ0, which 0 0 allows us to isolate the evident combination of form k2 factors f±. The relations between the form factors p1 + k + k1 k appearing in NRQCD at a recoil momentum close to p + k – k 1 p2 + k zero will be considered below. Here, we stress only that 2 2 we have 2 B 2 ρ ( ) ρ ( ) p1 p2 p p + qmax = + qmax C, (37) k 1 k 2 where the factor C has been speciﬁed in (33). k1 k2 ψ ν In the case of Bc J/ l l transition, one can eas- 0 0 ρ A 2 k k ily obtain an analogous result for 0 (q ) (note that the 1 2 A ρ A k form factor F0 ~ 0 gives the dominant contribution 2 p2 + k p1 + k p2 + k to the width of this decay [14]). In the nonrelativistic approximation, we have , ρ A( 2 ) ρ A B( 2 ) p1 p2 p1 p2 0 qmax = 0 qmax C. (38) k k

To conclude this section, note that the derivation of for- k1 mulas (37) and (38) is purely formal, since the spectral 2 Fig. 3. The gluon-condensate contribution to three-point densities are not specified at qmax (one can easily show QCD sum rules. The directions of p1, k1, and k2 momenta ρ B 2 are incoming, and that of p is outgoing. i (q ) to be singular in this point). Therefore, the 2 resultant relations are valid only for q2 approaching 2 Γ q . Unfortunately, this derivation does not give the q2 show up as incomplete functions in the resulting max expression and the absence of them in the final answer dependence of the factor C. But we suppose that C does leads authors of [18] to the conclusion that such contri- not crucially affect the pole behavior of the form fac- butions are actually absent in the processes they consid- tors. Therefore, the resultant widths of transitions can ered. be treated as the saturated upper bounds in the QCD sum rules. The gluon condensate contribution to three-point sum rules is given by diagrams depicted in Fig. 3. For calculations, we have used the Fock–Schwinger fixed- 2.3. Gluon-Condensate Contribution point gauge [20, 23]: In this subsection, we will discuss the calculation of µ a x Aµ (x) = 0, (39) Borel transformed Wilson coefficient of the gluon-con- a densate operator for the three-point sum rules with where Aµ (a = {1, 2, …, 8}) is the gluon field. arbitrary masses. The technique used is the same as in In the evaluation of the diagrams in Fig. 3, we [18] with some modifications to simplify the resulting 4) expression. As was noted in [18], this method does not encounter integrals of the type allow for the subtraction of continuum contributions, d4k Iµ µ …µ (a, b, c) = ------which, however, only change our results a little, as the 1 2 n ∫ 4 total contribution of the gluon condensate to the three- (2π) ≤ (40) point sum rule is small by itself ( 10%), and, thus, its kµ kµ …kµ continuum portion is small too. The form of the × ------1------2------n------. [ 2 2]a[( )2 2]b[( )2 2]c obtained expression does not permit us to use the same k Ð m3 p1 + k Ð m1 p2 + k Ð m2 argument as in [18] to argue over the absence of those contributions at all. Their argument was based on an 4)Since the diagrams under consideration do not have UV diver- expectation that the typical continuum contribution can gences, there is no need for a dimensional regularization.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2130 KISELEV et al. Continuing to Euclidean spacetime and employing the expression Schwinger representation for propagators ∞ ( , ) ( 2 2 )a b ∞ U0 a b = ∫dy y + Mbc + Mcc y 2 2 1 1 a Ð 1 Ðα( p + m ) ------= ------dαα e , (41) 0 (47) 2 2 a Γ(a)∫ [ p + m ] B 0 × exp Ð -----Ð--1 Ð B Ð B y , y 0 1 we ﬁnd the following expression for the scalar integral (n = 0): where

1 2 4 2 4 a + b + c B = ------(m M + m M (Ð1) i Ð1 2 2 2 bc 1 cc I (a, b, c) = ------M M 0 Γ(a)Γ(b)Γ(c) cc bc 2 2 2 2 2 ∞ ∞ ∞ ( ) ) + Mcc Mbc m1 + m2 Ð Q , × α β γαa Ð 1βb Ð 1γ c Ð 1 ∫∫∫d d d (42) 1 ( 2 ( 2 2) 2 ( 2 2)) (48) B0 = ------Mcc m1 + m3 + Mbc m2 + m3 , 0 0 0 2 2 Mbc Mcc 4 α( 2 2) β( 2 ⋅ 2) γ ( 2 ⋅ 2) d k Ð k + m3 Ð s1 + k + 2 p1 k + m1 Ð s2 + k + 2 p2 k + m2 2 × ------e . m ∫ 4 ------3------(2π) B1 = 2 2 . Mbc Mcc This representation proves to be very convenient for One can then express the results of calculation for any applying the Borel transformation with diagram in Fig. 3 through Iˆ0 (a, b, c), Iˆµ (a, b, c), and

2 2 Ðα p 2 Iˆµν (a, b, c) and their derivatives with respect to the Bˆ 2(M )e = δ(1 Ð αM ). (43) p Borel parameters, using the partial fractioning of the integrand expression together with the following rela- We then have tion:

2 2 2 m 2 n a + b + c ˆ ( ) ˆ ( )[ ] [ ] ( , , ) ( ) 2 Ð a Ð c B 2 Mbc B 2 Mcc p1 p2 Iµ µ …µ a b c ˆ ( , , ) Ð1 i ( 2 ) p1 p2 1 2 n I0 a b c = ------2 Mcc Γ(a)Γ(b)Γ(c) ⋅ 16π (44) m n [ 2 ]m[ 2 ]n d d = Mbc Mcc ------(49) × ( 2 )2 Ð a Ð b ( , ) ( 2 )m ( 2 )n Mbc U0 a + b + c Ð 4 1 Ð c Ð b , d Mbc d Mcc 2 m 2 n × [ ] [ ] ˆµ µ …µ ( , , ) a + b + c + 1 Mbc Mcc I 1 2 n a b c . (Ð1) i  p1µ p2µ  Iˆµ(a, b, c) = ------+ ------Γ( )Γ( )Γ( ) ⋅ π2 2 2  The obtained expression can be further written down in a b c 16 Mbc Mcc (45) terms of three quantities Iˆ0 (1, 1, 1), Iˆµ (1, 1, 1), and × ( 2 )3 Ð a Ð c( 2 )3 Ð a Ð b ( , ) Mcc Mbc U0 a + b + c Ð 5 1 Ð c Ð b , Iˆµν (1, 1, 1) and their derivatives with respect to the Borel parameters and quark masses by means of a + b + c (Ð1) i  p1µ p2µ  a Ð 1 Iˆµν(a, b, c) = ------+ ------1 d Γ( )Γ( )Γ( ) ⋅ π2 2 2  Iˆµ µ …µ (a, b, c) = ------a b c 16 Mbc Mcc 1 2 n Γ(a)Γ(b)Γ(c) ( 2)a Ð 1 d m3  p ν p ν  2 4 Ð a Ð c 2 4 Ð a Ð b (50) × -----1---- + -----2---- (M ) (M ) b Ð 1 c Ð 1  2 2  cc bc d d M M × ------Iˆµ µ …µ (1, 1, 1). bc cc (46) ( 2)b Ð 1 ( 2)c Ð 1 1 2 n × ( , ) d m1 d m2 U0 a + b + c Ð 6 1 Ð c Ð b a + b + c + 1 However, contrary to the case discussed in [18], in gµν (Ð1) i 2 3 Ð a Ð c 2 3 Ð a Ð b ------( ) ( ) such calculations values of parameters a and b will + ------2 Mcc Mbc 2 Γ(a)Γ(b)Γ(c) ⋅ 16π arise for which the U0(a, b) function has no analytic expression (it is connected to nonzero m mass in our × ( , ) 3 U0 a + b + c Ð 6 2 Ð c Ð b , case). The analytic approximations for U0(a, b) at these values of parameters lead to very cumbersome expres- 2 2 sions. The search for the most compact form of the ﬁnal where Mbc and Mcc are the Borel parameters in the s1 answer leads to the conclusion that the best decision in and s channels, respectively. Here, we have intro- 2 this case is to express the result in terms of the U0(a, b) duced the U0(a, b) function, which is given by the function at different values of its parameters. For this

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SEMILEPTONIC DECAYS 2131 purpose, we have used the following transformation f1(0) properties of U0(a, b): 0.17 ( , ) dU0 a b ( , ) ------= aU0 a Ð 1 b dM 2 bc 0.15  2 2 2  m2 m1 Mcc ( , ) Ð ------Ð ------ U0 a b Ð 1 (51)  2 4  Mcc Mbc 0.13 m 2 + m 2 m 2 + -----1------3-U (a, b) + ------3------U (a, b + 1), 4 0 4 2 0 5 15 25 35 Mbc Mbc Mcc 2 2 Mcc, GeV dU (a, b) ------0------= aU (a Ð 1, b) 2 0 Fig. 4. The gluon-condensate contribution to the f1(0) form dMcc 2 factor in the Borel transformed sum rules at fixed Mbc = 2 2 2   2 m1 m2 Mbc ( , ) 70 GeV . The dashed curve represents the bare-quark-loop Ð ------Ð ------ U0 a b Ð 1 (52)  2 4  results, and the solid curve is the form factor including the Mbc Mcc α s 2 Ð2 4 gluon-condensate term at --π---G = 10 GeV . m 2 + m 2 m 2 -----2------3- ( , ) ------3------( , ) + 4 U0 a b + 2 4 U0 a b + 1 . Mcc Mbc Mcc to the leading order leads to various relations between the different form factors. Solving these relations In Appendix B, we have presented an analytic expres- results in the introduction of a universal form factor (an sion, obtained in this way, for the Wilson coefficient of 2 2 the gluon-condensate operator, contributing to the Π = analog of the IsgurÐWise function) at q qmax . Π Π 1 + + – amplitude. One can see that, even in this form, In this subsection, we consider the limit the obtained results are very cumbersome. Thus, we µ µ have realized the gluon-condensate corrections as C++ v ≠ v , codes, where the functions U (a, b) are evaluated 1 2 (53) 0 ⋅ numerically. Analytic approximations that can be made w = v 1 v 2 1, for the U0(a, b) functions are discussed in Appendix B. µ µ 2 In Fig. 4, we have shown the effect of gluon conden- where v 1, 2 = p1, 2 / p1, 2 are the four-velocities of sate on the f1(0) form factor in the Borel transformed heavy quarkonia in the initial and final states. The study three-point sum rules. of region (53) is reasonable enough, because in the rest We can draw the conclusion that the calculation of µ 2 gluon-condensate term in full QCD sum rules allows frame of the Bc meson ( p1 = ( p1 , 0), the four-veloc- one to enlarge the stability region in the parameter | | µ 2 space for the form factors, which indicates the reliabil- ities differ only by a small value p2 ( p2 = ( p2 , p2)), ity of the sum-rule technique. whereas their scalar product w deviates from unity only | | due to a term proportional to the square of | p2 : w = 2 2 3. THREE-POINT NRQCD SUM RULES p 1 p 1 + -----2----- ~ 1 + ------2----- . Thus, in the linear approxima- The formulation of sum rules in NRQCD follows 2 2 2 p2 p2 the same lines as in QCD; the only difference is the tion at |p | 0, relations (53) are valid and take place. Lagrangian, describing strong interactions of heavy 2 quarks. Here, we would like to note that (53) generalizes the investigation of [14], where the case of v1 = v2 was considered. This condition severely restricts the relations 3.1. Symmetry of Form Factors in NRQCD of spin symmetry for the form factors, and, as a conse- and One-Loop Approximation quence, it provides a single connection between the At the recoil momentum close to zero, the heavy form factors. quarks in both the initial and final states have small rel- As can be seen in Fig. 1, since the antiquark line ative velocities, so that the dynamics of heavy quarks is with the mass m3 is common to the heavy quarkonia, essentially nonrelativistic. This allows us to use the the four-velocity of an antiquark can be written down as NRQCD approximation in the study of mesonic form a linear combination of four-velocities v1 and v2: factors. As in the case of heavy quark effective theory µ µ µ (HQET), the expansion in the small relative velocities v˜ 3 = av 1 + bv 2 . (54)

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In the leading order of NRQCD for the kinematical 〈 | γ γ µ | 〉 PQ Q Q1 5 Q3 V Q Q invariants, determining the spin structure of quark 1 3 2 3 (62) propagators in the limit w 1, we have the following µ µ µ ( ⋅ ⑀ ⋅ v (⑀ ⋅ v ) ⋅ v (⑀ ⋅ v )) expressions: = c⑀ + c1 1 1 + c2 2 1 h, where 2 ( )2 ᏹ 2 p1 m1 + m3 = 1 , c⑀ = Ð2, 2 ( )2 ᏹ 2 p2 m2 + m3 = 2 , m (3m + m ) c = Ð-----3------1------3---, ∆ ᏹ 1 1 2m3 1, 4m1m2 (55) ∆ ᏹ 2 2m3 2, 1 ( 2) c2 = ------4m1m2 + m1m3 + 2m2m3 + m3 , ᏹ ᏹ ⋅ 4m1m2 u 2 1 2 w, (63) 1 2 2 2 2 2 2 ( ) λ( p , p , q ) 4ᏹ ᏹ (w Ð 1). cV = Ð------2m1m2 + m1m3 + m2m3 , 1 2 1 2 2m1m2 In this kinematics, it is an easy task to show that in (54) P m3 m3 1 c1 = 1 + ------Ð ------, a = b = Ð-- , i.e., 2m1 2m2 2 P m3 m3 µ 1 µ c2 = 1 Ð ------+ ------. v˜ = Ð--(v + v ) . (56) 2m1 2m2 3 2 1 2 For the form factors in NRQCD, we then have the fol- Applying momentum conservation at the vertices in lowing symmetry relations: Fig. 1, we derive the following formulas for the four- ( P ⋅ ᏹ P ⋅ ᏹ ) ( P ⋅ ᏹ P ⋅ ᏹ ) velocities of quarks with the masses m1 and m2: f + c1 2 Ð c2 1 Ð f Ð c1 2 + c2 1 = 0, A µ µ m3 µ ⋅ ⋅ ᏹ ᏹ v˜ = v + ------(v Ð v ) , (57) F0 cV Ð 2c⑀ FV 1 2 = 0, 1 1 2m 1 2 1 A( ) ⋅ ᏹ ( A(ᏹ ᏹ ) F0 c1 + c2 Ð c⑀ 1 F+ 1 + 2 (64) µ µ m µ v˜ = v + ------3--(v Ð v ) , (58) A(ᏹ ᏹ ) ) 2 2 2 1 + FÐ 1 Ð 2 = 0, 2m2 A P ⋅ ᏹ ( ) 2 2 F0 c1 + c⑀ 1 f + + f Ð = 0. and in the limit w 1, we have v˜ 1 = v˜ 2 = 1, as it should be. Thus, we can claim that, in the approximation of After these definitions have been made, it is NRQCD, the form factors of weak currents responsible straightforward to write down the transition form factor for the transitions between two heavy quarkonium Γ Γ states are given in terms of the single form factor, say, for the current Jµ = Q1 µQ2 with the spin structure µ A A γ γ γ F . An exception is observed for the form factors F = { µ, 5 µ} 0 + and F A , since a definite value is only taken by their lin- 1 µ Ð 〈H |Jµ|H 〉 = tr Γµ--(1 + v˜ γ µ) A A Q Q3 Q Q3 1 ᏹ ᏹ ᏹ ᏹ 1 2 2 ear combination F+ ( 1 + 2) + FÐ ( 1 Ð 2). This (59) fact has a simple physical explanation. Indeed, the × Γ 1( νγ )Γ 1( λγ ) ( , , ) ⑀µ 1-- 1 + v˜ 3 ν 2-- 1 + v˜ 2 λ h m1 m2 m3 , polarization of vector quarkonium has two compo- 2 2 ⑀ µ ⑀ µ nents: the longitudinal term L and transverse one T where Γ determines the spin state in the heavy quarko- µ 1 (i.e., (⑀ á v ) = 0). The quantity ⑀ can be decomposed Γ T 1 L nium Q1Q3 (in our case it is pseudoscalar, so that 1 = µ µ γ Γ in terms of v and v : 5), and 2 determines the spin wave function of 1 2 µ quarkonium in the final state: Γ = {γ , ⑀ γµ} for the µ µ µ 2 2 2 5 ⑀ = α⑀ + β⑀ , α + β = 1, (65) pseudoscalar and vector states, respectively (H = P, V). L T The quantity h is a universal function at w 1, inde- where pendent of the quarkonium spin state. So, for the form ⑀ µ 1 ( µ µ) 1 ( µ µ) factors discussed in our paper we have L = ------Ð2s2 p1 + u p2 ------Ðv 1 + wv 2 , 2 µ µ µ s2k w Ð 1 〈 | γ | 〉 ( P ⋅ P ⋅ ) P Q1 Q3 P = c1 v 1 + c2 v 2 h, (60) Q1Q3 Q2Q3 (66) 2 s2 1 µ µναβ α (⑀ ⋅ ) (⑀ ⋅ v ) 〈 | γ | 〉 ⋅ ⑀ ⑀ ⋅ = Ð------L p1 Ð------1 . P Q1 Q3 V = icV νv 1αv 2β h, (61) 2 Q1Q3 Q2Q3 k w Ð 1

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, From (66), one can see that the decomposition of polar- correlation function Π A NR , we have the expression ization vector ⑀ into the longitudinal and transverse 0 parts in NRQCD is singular in the limit w 1: A, NR 1 dω˜ dω˜ A, NR Π = Ð------1------2------ρ 0 2∫(ω˜ ω )(ω˜ ω ) 0 µ 1 µ µ µ (2π) 1 Ð 1 2 Ð 2 ⑀ = Ð ------(Ð v + wv )(⑀ ⋅ v ) + β ⋅ ⑀ . (67) 2 1 2 1 T k w Ð 1 th 2 (70) 3 k dk = ----- ∫ ------, A π2  k2   k2  It means that the introduction into the form factor F0 0 ω Ð ------ω Ð ------ 1 2m   2 2m  ∆ A 2 δ 13 23 of an additional term F0 = (w Ð 1) h, which vanishes at w 1 and, thus, is not under control in NRQCD, where kth denotes the resonance region boundary. In the ω A method of moments in NRQCD sum rules, we set 1 = leads to a ﬁnite correction for the form factors F+ and ω Ð(m1 + m3) + q1 and 2 = Ð(m2 + m3) + q2, so that in the A limit q 0 we have FÐ . This correction is canceled in the special linear 1, 2 combination of form factors presented in (64). n + m 1 1 d A, NR A, NR ------Π Π [ , ] n m 0 = 0 n m In the case of v1 = v2, we reproduce the single rela- n!m! dq1 dq2 A tion between form factors F0 and f±, as it was obtained k th 2 (71) earlier in [14]. 3 k dk = -----2 ∫ ------2------n---+---1------2------m----+---1-. Thus, we have obtained the generalized relations π ᏹ k  ᏹ k  0  1 + ------  2 + ------ due to the spin symmetry of NRQCD Lagrangian for 2m13 2m23 the case v ≠ v in the limit where the invariant mass of 1 2 In the hadronic part of NRQCD sum rules in the limit lepton pair takes its maximum value, i.e., at a recoil | | momentum close to zero. p2 0, we model the resonance contribution using the following representation: In the one-loop approximation for the three-point Q1Q3 2 Q2Q3 NRQCD sum rules, i.e., in the calculation of bare- , f M , f M , Π A res ------i------1---i------j------2----j A quark loop, the symmetry relations (64) already take 0 = ∑ 2 2 F0, ij (m + m )M , M , ρ NR i, j 1 3 1 i 2 j place for the double spectral densities j in the limit (72) |p | 0. We have checked that the spectral densities  2  l 2  m 2 q1 q2 of the full QCD in the NRQCD limit w 1 satisfy × ∑------ ------ .  2   2  the symmetry relations (64). l, m M1, i M2, j It is easily seen that in this approximation 2 2 Saturating the p1 and p2 channels by ground states of A , 6m m m ρ A NR 1 2 3 mesons under consideration for the F0, 1S → 1S form fac- 0 = Ð------(------)-. (68) p2 m1 + m3 tor, we have Π A, NR[ , ]( ) When integrating over the resonance region, we must A 0 n m m1 + m3 n m F0, 1S → 1S = ------M1, 1S M2, 1S . (73) take into account that Q1Q3 2 Q2Q3 2 f 1S M1, 1S f 1S M2, 1S ω Ð ω m ⁄ m m ------2------1------13------23------2 ≤ 1, (69) In the next subsection, we will proceed with the gluon- ω p2 2 1m13 condensate contribution to NRQCD sum rules. |p | so we see that, in the limit 2 0, the integration 3.2. Contribution of the Gluon Condensate region tends to a single point. Here, p = (m + m + ω , 1 1 3 1 Following a general formalism for the calculation of ω mim j gluon-condensate contribution in the FockÐSchwinger 0), p2 = (m2 + m3 + 2, p2), and mij = ------is the mi + m j gauge [20, 23], we have considered diagrams depicted reduced mass of system Q Q . in Fig. 3. In our calculations, we have used the NRQCD i j approximation and analyzed the limit where the invari- 2 ant mass of the lepton pair takes its maximum value. ω k Here, we would like to note that the spin structure, in After the substitutions of variables 1 = ------and 2m13 the leading order of relative velocity of heavy quarks, 2 does not change, in comparison with the bare-loop  ω k  m2 | | x = - 2 Ð ------ ------in the limit p2 0 for the result for the nonrelativistic quarks. Thus, we can con- 2m23 p2 k clude that in this approximation the symmetries of

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2134 KISELEV et al. NRQCD Lagrangian lead to a universal Wilson coefﬁ- After two differentiations of the nonrelativistic propa- cient for the gluon-condensate operator. As a conse- gators quence, relations (64) remain valid. 1 1 Below, we perform calculations for the form factor ------= ------, (80) ( , ) 2 A 2 2 2 P p mQ p F ( p , p , q2) in the limit q2 q . The contribu- pv + ------0 1 2 max 2m tion of gluon condensate to the corresponding correla- Q tion function is given by the expression two types of contributions to the gluon-condensate cor- 2 α π rection appear. The ﬁrst is equal to ∆ G s 2 [ ] F0 = -----Gµν ----- 3R0 Ð R2 , (74) π 48 αβ α a a ⋅ 〈α 2 〉 3 v µv ν sGµα Gνβ g sGµν -----, (81) where 12

2 and it leads to the term with R0. The second expression 1 k dkdk0 R0 = Ð------∫------Rg, π2 P (k, k )P (k, k , ω )P (k, k , ω ) α 2 ( ⋅ )2 i 3 0 1 0 1 2 0 2 α a a β 〈α 2 〉 k Ð v k (75) v µv ν sGµα Gνβ k k sGµν ------4 12 1 k dkdk0 (82) R2 = Ð------∫------Rk. 2 π2 P (k, k )P (k, k , ω )P (k, k , ω ) 2 k i 3 0 1 0 1 2 0 2 = Ð 〈α Gµν〉 ------s 12 In these expressions, the inverse propagators are leads to the term with R2, which is of the same order in k2 the relative velocity of heavy quarks as R , but it is sup- P (k, k , ω ) = ω + k Ð ------, 0 1 0 1 1 0 2m pressed numerically in the region of moderate numbers 1 for the momenta of spectral densities, where the non- 2 ( , , ω ) ω k perturbative contributions (condensates) of higher P2 k k0 2 = 2 + k0 Ð ------, (76) dimension operators are not essential. 2m2 2 It is easy to show that the contributions of R0 and R2 ( , ) k can be obtained by the differentiation of two basic inte- P3 k k0 = Ð k0 Ð ------. 2m3 grals: ∞ The functions R and R are symmetric under the per- 2 g k 1 k dkdk 2m ⋅ 2m mutation of indices 1 and 2 and are given by 0 ------13------23--- E0 = Ð----2---∫------= , π i P1P2P3 k13 + k23 1 1 1 1 1 1 1 1 0 R = Ð ------+ ------Ð ------+ ------, ∞ g 2 2 4 m1 P3 P1 m2 P3 P1 P1 P2 1 k dkdk0 (83) E2 = Ð----2---∫------π i P1P2P3 3 1 3 1 3 1 3 1 0 Rk = ------+ ------+ ------+ ------m 2 P 4 m 2 P 4 m 2 P 3P m2 P 3P ⋅ 1 1 2 2 1 1 3 2 2 3 2m13 2m23( 2 2 ) (77) = Ð ------k13 + k13k23 + k23 , 1 1 1 1 1 1 1 1 k13 + k23 + ------+ ------Ð ------Ð ------m m 2 2 m m 2 2 2 4 m m 3 1 3 P1 P3 2 3 P2 P3 m3 P3 1 3 P1P3 ω ω where k13 = Ð2m13 1 and k23 = Ð2m23 2 . For R0 1 1 1 1 1 1 and R2, we then have Ð ------Ð ------Ð ------. m m 3 m m 2 m m 2 2 2 3 P2P3 1 2 P1P2P3 1 2 P1 P2 ˆ ⋅ R0 = Rg E0, Let us note that, in the calculation of the diagrams in ˆ ⋅ Fig. 3, we have used the following vertex for the inter- R2 = Rk E2, action of a heavy quark with a gluon: ˆ ˆ where operators Rg and Rk can be obtained from Rg v µ Lint = Ðgshv v µ A hv , (78) and Rk in (77) after the substitutions

a n n a λ 1 (Ð1) ∂ where Aµ = Aµ ----- and its Fourier transform in FockÐ ------, n n! ∂ω n 2 P1 1 Schwinger gauge has the form m m ∂ 1 (Ð1) ∂ a ( ) i a ( ) ( π)4δ( ) ------, Aµ kg = Ð--Gµν 0 ∂ 2 kg . (79) m m! ∂ω m 2 kg P2 2

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l l A 1 (Ð1) ∂ ∂ F0 (0), GeV ------+ ------. l l! ∂ω ∂ω P3 1 2 1.5 As a result, we have 1.0  1 ∂3 1 ∂3 1  ∂3 ∂3  0.5 R = ------+ ------+ ------------+ ------ 0 3 3 3 2 6m1 ∂ω 6m2 ∂ω 2m1 ∂ω ∂ω ∂ω  14  1 2 1 1 2 18 (84) 10 3 3 14 n 1  ∂ ∂  2m 2m nbc cc + ------------+ ------ ------13------23---, 6 3 2 10 2m2 ∂ω ∂ω ∂ω  k13 + k23 2 2 1  Fig. 5. The QCD sum-rule results in the moment scheme for 4 4  3 ∂ 3 ∂ A R = ------+ ------the F0 (0) form factor in the bare approximation. 2 2 ∂ω 4 2 ∂ω 4 4!m1 1 4!m2 2

4 4 4 4 2 3  ∂ ∂  1  ∂ ∂  and expand ∆F G in a series in {q , q } at the point {0, + ------------+ ------ + ------------+ ------ 0 1 2 2 ∂ω 4 ∂ω 3∂ω  2 ∂ω 4 ∂ω 3∂ω  2 3!m1 1 1 2 3!m2 2 2 1 ∆ G 0}, which allows us to determine F0 [n, m] =

n + m 2  4 4 4  1 1 d G 1 ∂ ∂ ∂ ------∆F . Further analysis has been per- + ------------+ 2------+ ------ n m 0 4 3 2 2 n!m! 4m1m3 ∂ω ∂ω ∂ω ∂ω ∂ω  dq1 dq2 1 1 2 1 2 formed numerically with the help of MATHEMATICA. 1  ∂4 ∂4 ∂4  1  ∂4 To conclude, we have calculated the contribution of + ------------+ 2------+ ------ Ð ------------4 3 2 2 2 4 gluon condensate to the three-point sum rules for heavy 4m2m3 ∂ω ∂ω ∂ω ∂ω ∂ω  ∂ω 2 2 1 1 2 4!m3 1 quarkonia in the leading order of relative velocity of α heavy quarks and in the ﬁrst order of s. Due to the ∂4 ∂4 ∂4 ∂4  symmetry of NRQCD in the limit where the invariant + 4------+ 6------+ 4------+ ------ (85) mass of the lepton pair takes its maximum value, i.e., at ∂ω 3∂ω ∂ω 2∂ω 2 ∂ω ∂ω 3 ∂ω 4 1 2 1 2 1 2 2 a recoil momentum close to zero, the Wilson coefﬁcient A for the form factor F0 is universal in the sense that it 1  ∂4 ∂4 ∂4 ∂4  ------------+ 3------+ 3------+ ------ determines the contributions of gluon condensate to Ð 4 3 2 2 3 3!m1m3 ∂ω ∂ω ∂ω ∂ω ∂ω ∂ω ∂ω  other form factors, in accordance with relations (64). 1 1 2 1 2 1 2

1  ∂4 ∂4 ∂4 ∂4  ------------+ 3------+ 3------+ ------ 4. NUMERICAL RESULTS Ð 4 3 2 2 3 3!m2m3 ∂ω ∂ω ∂ω ∂ω ∂ω ∂ω ∂ω  ON THE FORM FACTORS 2 2 1 1 2 2 1 First, we evaluate the form factors in the scheme of 1  ∂4 ∂4 ∂4  spectral density moments. This scheme is not strongly ------------+ 2------+ ------ Ð 3 2 2 3 sensitive to the values of threshold energies determin- 2m1m2 ∂ω ∂ω ∂ω ∂ω ∂ω ∂ω  1 2 1 2 1 2 ing the region of resonance contribution. In the QCD sum rule calculations, we set (in GeV)  4  1 ∂  2m 2m  2 2 ------------ 13 23 ( ) k (bc) = 1.5, Ð 2 2 Ð------ k13 + k13k23 + k23 . th 4m1m2∂ω ∂ω  m13 + m23 1 2 k (cc) = 1.2, th (86) Equations (74), (84), and (85) represent the most com- m = 4.6, pact analytic expression for the contribution of gluon b A mc = 1.4, condensate to the form factor F0 , whereas performing the differentiations leads to very cumbersome expres- where kth is the momentum of quark motion in the rest sions. frame of quarkonium. The chosen values of threshold In the moment scheme of sum rules, we suppose momenta correspond to the minimal energy of heavy- ω ( ) meson pairs in speciﬁed channels. 1 = Ð m1 + m3 + q1, The typical behavior of form factors in the moment ω ( ) 2 = Ð m2 + m3 + q2 scheme of QCD sum rules is presented in Fig. 5.

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+ 2 Table 1. Form factors for Bc decay into heavy quarkonia at q = 0 in the bare-quark-loop approximation with allowance for the Coulomb correction

Ð1 A A Ð1 A Ð1 Approximation f+ fÐ FV, GeV F0 , GeV F+ , GeV FÐ , GeV Bare 0.10 Ð0.057 0.016 0.90 Ð0.011 0.018 Coulomb 0.66 Ð0.36 0.11 5.9 Ð0.074 0.12

The evaluation of the Coulomb corrections strongly zero recoil is close to zero for the states with the differ- α 6) depends on the appropriate set of s for the quarkonia ent quantum numbers. Thus, we can easily modify the under consideration. The corresponding scale of gluon relation (73) due to the second transition and justify the virtuality is determined by quite a low value close to the A A value of F , → to reach the stability of F , → at average momentum transfer in the system. So, the 0 2S 2S 0 1S 1S α low values of moment numbers. We find expected s is about 0.5. To decrease the uncertainty, A A ≈ we consider the contribution of the Coulomb rescatter- F0, 2S → 2S /F0, 1S → 1S 3.7 and present the behavior of ing in the two-point sum rules giving the leptonic con- A stants of heavy quarkonia. These sum rules are quite the form factor F0 at zero recoil in Fig. 8. sensitive to the value of the strong-coupling constant, As can be seen in these figures, the gluon contribu- as the perturbative contribution depends on it linearly. tion, while varying the moment number in the region of ≈ The observed value for the charmonium, fJ/ψ 410 MeV, bound cc states, plays an important role, because it α C can be obtained in this technique at s (cc ) = 0.6. The allows us to extend the stability region for the form fac- value f = 385 MeV, as it is predicted in QCD sum tor F A up to three times (from n < 5 to n < 15) and thus Bc 0 α C the reliability of sum-rule predictions. Let us also note rules [22], gives s (bc ) = 0.45. We present the results that, in the scheme of saturation for the hadronic part of of the Coulomb enhancement for the form factors in sum rules by the ground states in both variables s and Table 1. 1 s2, the allowance for the gluon condensate leads to the The result after the introduction of the Coulomb A correction is shown in Fig. 6. Such large corrections to 20% reduction for the value of form factor F0 . the form factors should not lead to confusion, as they The analysis of the dependence on the moment resulted from the fair account of the Coulomb corrections both for the bare-quark-loop diagram and for the number nbc in the region of bound states bc at fixed ncc meson-coupling constant. shows that the contribution of gluon condensate in this case does not affect the character of this dependence. Working in the same scheme for the case of This may be explained by the fact that the Coulomb NRQCD sum rules, we have plotted in Fig. 7 the value corrections to the Wilson coefficient of gluon operator A of F0, 1S → 1S at fixed ncc = 4 with the following values 2 7) Gµν play an essential role. The summation of α /v of the parameters:5) s terms for bc may give a sizable effect, contrary to the k = 1.3 GeV, th situation with the cc system, where the relative veloc- mb = 4.6 GeV, ity of heavy quarks is not too small.

mc = 1.4 GeV. In the scheme of the Borel transformation for QCD sum rules, we find a strong dependence on the thresh- α s 2 Ð2 4 olds of continuum contribution. We think that this -----Gµν = 1.7 × 10 GeV . π dependence reflects the influence of contributions coming from the excited states. So, the choice of kth values Further, in (72) we can take into account the dominant in the same region as in the scheme of spectral-density subleading term, which is the contribution by the moments results in form factors which are approxi- 2S 2S transition. In this case, one could expect that mately 50% greater than the predictions in the the form factor is not suppressed in comparison with the contribution by the 1S 2S transition, since in 6)The corresponding estimates were performed in [24], where the the potential picture the latter decay has to be neglected 1S 2S transition is suppressed with respect to 1S 1S as because the overlapping between the wave functions at 1/25. 7)At present, there is an analytic calculation of initial six moments 2 for the Wilson coefficient of gluon condensate in the second order 5) G α The evaluation of the ∆F [n , n ] dependence in a broad of s [26]. In this region, the influence of gluon condensate on the 0 cc bc sum-rule results is negligibly small for the heavy quarkonia, con- range of [ncc, nbc] takes too much calculation time, so we restrict taining the b quark, which does not allow one to draw definite ourselves by showing the results in Figs. 7 and 8 for the fixed n = 4. α cc conclusions on the role of such s corrections.

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A A F0 (0), GeV F0 (0), GeV 4 7.5 3 2.5 2 18 14 14 10 n nbc cc 1 6 10

Fig. 6. The QCD sum-rule results in the moment scheme for 0 10 20 nbc A the F (0) form factor with allowance for the Coulomb cor- 0 A rections. Fig. 7. The NRQCD sum-rule results for the F0 (0) form factor. The solid curve represents the bare-quark-loop con- moments scheme, where the higher excitations numer- tribution, the short dashed curve is the result obtained by ically are not essential. In this case, we can explore the taking into account the gluon-condensate term R0 only, and the long dashed curve is the form factor including the full ideology of finite-energy sum rules [25], wherein the expression for the gluon condensate. choice of interval for the quarkÐhadron duality, expressed by means of sum rules, allows one to isolate A the contribution of basic states only. So, if we set F0 (0), GeV 4 k (bc) = 1.2 GeV, th (87) ( ) 3 kth cc = 0.9 GeV, then the region of the lowest bound states is taken into account in both channels of initial and final states, and 2 the Borel transform scheme leads to results that are very close to those of the moment scheme. The dependence of calculated values on the Borel parameters is 1 presented in Figs. 9 and 10, in the bare and Coulomb approximations, respectively. 10 20 As for the dependence of form factors on q2, the con- 0 nbc sideration of bare-quark-loop term shows that, say, for A Fig. 8. The NRQCD sum-rule results for the F (0) form A 2 0 F0 (q ), it can be approximated by the pole function factor. The contribution of 2S 2S transition has been A taken into account. The notation is the same as in Fig. 7. F (0) A( 2) ------0------F0 q = 2 2 , (88) 1 Ð q /Mpole ≈ with Mpole 4.5 GeV. The latter is in good agreement with the value given in [12]. However, we believe that α the pole mass can change after the inclusion of s cor- A rections.8) From the naive meson dominance model, we F0 (0), GeV ≈ expect that Mpole 6.3Ð6.5 GeV. We have calculated the total widths of semileptonic 0.95 decays in the region of Mpole = 4.5Ð6.5 GeV, which result in the 30% variation of predictions for the modes 0.90 with the massless leptons and more sizable dependence 30 for the modes with the τ lepton (see Table 2). 0.85 To compare with other estimates, we calculate the 20 20 +ν 2 2 width of b ce e transition as the sum of decays 40 Mcc, GeV 9) into the pseudoscalar and vector states and find 2 2 10 Mbc, GeV 60 8)In HQET, the slope of IsgurÐWise function acquires a valuable α correction due to the s term. Fig. 9. The Borel transformed sum-rule results for the 9) For normalization of the calculated branching ratios, we used the A F (0) form factor in the bare approximation of QCD. total Bc width obtained in the framework of the OPE approach [8]. 0

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[ + η ] ± Br Bc c + light hadrons = (4.0 0.5)%. (90)

Neglecting the decays b ccs , which are sup- FA(0), GeV 0 pressed by both the small phase space and the negative Pauli interference of decay product with the charmed 5.6 quark in the initial state [8], we evaluate the branching fraction of beauty decays in the total width of Bc as 5.4 30 [ + ] ± 5.2 Br Bc cc + X = (23 5)%, 20 20 which is in agreement with the estimates in other 2 2 40 Mcc, GeV approaches [6, 8], where this value is equal to 25%. 10 2 2 Mbc, GeV 60 5. CONCLUSIONS Fig. 10. The Borel transformed sum-rule results for the A F (0) form factor with allowance for the Coulomb correc- We have calculated the semileptonic decays of Bc 0 meson in the framework of sum rules in QCD and tions. NRQCD. We have extended the previous evaluations in QCD to the case of massive leptons: the complete set of double spectral densities in the bare-quark-loop + +ν ≈ ± Br(Bc cce e ) (3.4 0.6)%, which is in good approximation have been presented. The analysis in the agreement with the value obtained in potential models sum-rule schemes of density moments and Borel trans- [6] and in OPE calculations [8], where the following form has been performed, and consistent results have + +ν ≈ been obtained. estimate was obtained: Br(Bc cce e ) 3.8%. As for our estimates for exclusive decay channels, they We have taken into account the gluon-condensate also agree with the results obtained previously in the contribution for the form factors of semileptonic transi- framework of potential models [6]. tions between the heavy quarkonia in the Borel transform sum rules of QCD, wherein the analytic expres- In the results presented, we have supposed the quark | | sions for the case of three nonzero masses of quarks mixing matrix element Vbc = 0.040. have been presented. As for the hadronic decays, in the approach of factorization [27], we assume that the width of transition We have considered the soft limit on the form factors in NRQCD at a recoil momentum close to zero, + ψ η Bc J/ ( c) + light hadrons can be calculated with which has allowed one to derive the generalized rela- the same form factors after the introduction of QCD tions due to the spin symmetry of the effective corrections, which can be easily written down as the Lagrangian. The relations have shown good agreement 2 α with the numerical estimates in full QCD, which means factor H = Nc a1 . The factor a1 represents the hard s that the corrections in both the relative velocities of corrections to the four-fermion weak interaction. heavy quarks inside the quarkonia and the inverse ≈ Numerically, we set a1 = 1.2, which yields H 4.3. So, heavy-quark masses are small within the accuracy of we ﬁnd the method. Next, we have presented the analytic results on the gluon-condensate term in the NRQCD [ + ψ ] ± Br Bc J/ + light hadrons = (11 2)%, (89) sum rules within the moments scheme. In both the QCD and NRQCD sum rules, the

+ account of the gluon condensate has allowed one to Table 2. Width with respect to Bc decay into heavy quarko- enforce the reliability of predictions, since the region of nia and leptonic pair and branching fractions calculated at physical stability for the form factors evaluated was τ B = 0.55 ps signiﬁcantly expanded in comparison with the leading- c order calculations of bare-quark-loop contribution. Γ Ð15 Mode , 10 GeV Br, % Next, we have investigated the role played by the η +ν ± ± α ce e 11 1 0.9 0.1 Coulomb s /v corrections for the semileptonic transi- η τ+ν ± ± tions between the heavy quarkonia. We have shown c τ 3.3 0.9 0.27 0.07 ψ +ν ± ± that, as in the case of two-point sum rules, the three- J/ e e 28 5 2.5 0.5 point spectral densities are enhanced due to the Cou- + J/ψτ ντ 7 ± 2 0.60 ± 0.15 lomb renormalization of quarkÐmeson vertices.

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The complete analysis shows that the numerical d3p estimates of various branching fractions, 2δ4( ) ˜ Wαβ = ∫------p1 Ð p2 Ð q Wαβ 2E2 (Ä.3) [ + ψ +ν ] ± Br Bc J/ l l = (2.5 0.5)%, 2 2 2 θ( )δ( ⋅ ) ˜ αβ = E2 M1 Ð 2M1 q0 + q Ð M2 W p2 = p1 Ð q, Br[B + cc + X] = (23 ± 5)%, c where agree with the results obtained in the framework of ˜ ( ( )( ) ( ) ) potential models and OPE in NRQCD. More detailed Wαβ = f + t p1 + p2 α + f Ð t qα results are presented in tables. (Ä.4) × ( f (t)( p + p )β + f (t)qβ) Thus, we draw the conclusion that at present the the- + 1 2 Ð oretical predictions on the semileptonic decays of Bc for the pseudoscalar particle in the final state, and meson give consistent and reliable results. ˜ ( 4 A ( ) A Wαβ = Ð iF0 gαe + iF+ p1e p1 + p2 α + iFÐ p1eqα ACKNOWLEDGMENTS ⑀ i j )( A A ( ) Ð FV αeij(p1 + p2) q iF0 + iF+ p1k p1 + p2 β (Ä.5) We are grateful to Prof. A. Wagner and the members of the DESY Theory Group for their kind hospitality   A m n e k during the visit to DESY, where this article was written, + iF p qβ + F ⑀β ( p + p ) q ) ∑ ⑀ ⑀*  Ð 1k V kmn 1 2   and to Prof. S.S. Gershtein for discussions. We also polarizations thank Prof. A. Ali for reading this manuscript and for enlightening comments. for the vector meson in the final state with the polariza- ⑀ This work was supported in part by the Russian tion µ, where Foundation for Basic Research (RFBR), project nos. e k p p 99-02-16558 and 00-15-96575. The work of A.I. Oni- ⑀e⑀*k = ----2------2- Ð gek, ∑ 2 (Ä.6) shchenko was supported in part by grants from the M2 International Center for Fundamental Physics in Mos- polarizations ψ η cow and the International Science Foundation and by a and M2 is the mass of final-state meson (J/ or c). joint grant (no. 95I1300) from INTAS and RFBR. The integral over the leptonic phase space in (A.1) is given by

APPENDIX A τ αβ 1 pl 〈 αβ〉 ∫d l L = ------L , (Ä.7) 4π 2 Semileptonic Decay Width q In this Appendix, we present the derivation of exclusive semileptonic widths for the B -meson decays into with ψ η c the J/ and c mesons with allowance for lepton αβ 1 αβ 〈 L 〉 = ------dΩ L masses. 4π∫ l Γ The exclusive semileptonic width SL for the decay (Ä.8) ψ η ν µ τ  α β αβ αβ  Bc J/ ( c)l l , where l = e, , or , can be written 2 ( λ )( 2) 3λ 2 = -- 1 + 1 q q Ð g q + -- 2g q , down in the form [28] 3 2  2 2 1 GF V bc 4 αβ 2 2 Γ = ------d q dτ L Wαβ, (Ä.1) where λ ≡ λ Ð 2λ and λ ≡ λ Ð λ . Introducing the SL 3 M ∫ ∫ l 1 + Ð 2 + Ð (2π) Bc ≡ 2 2 dimensionless kinematical variable t q /mb and inte- 4 π| | 2 τ | | Ω π2 2 grating with respect to q , the semileptonic width takes where d q = 2 q dq dq0, d l = pl d l /(16 q ) is the 0 Ω the following form: leptonic-pair phase space, d l is the solid angle of 2 2 2 | | 2 Φ 1 G V m charged lepton l, pl = q l/2 is its momentum in the Γ = ------F------b---c------b- Φ ≡ SL 3 2 dilepton center-of-mass system, and l π 64 M1 2 2 2 2 (Ä.9) 1 Ð 2λ + λ , with λ± ≡ (m ± mν )/q . The tensors tmax + Ð l l αβ × Φ ( ) 〈 αβ〉 ˜ L and Wαβ in (A.1) are given by ∫ dt l t q L Wαβ,

tmin αβ 1 α β L = -- ∑ (l O ν )(ν O l) where 4 l l spins (Ä.2) 1 2 2 2 2 2 2 α β β α αβ αβγδ ( ) q = ------M1 + mb t Ð M2 Ð 4mb M1 t. = 2[ p pν + p pν Ð g ( p ⋅ pν ) + ie p γ pν δ], l l l l l l l l 2M1

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2 m 2 Π 〈 G 〉 (Ð1, Ð2) ( , ) (Ð1, Ð1) ( , ) -----l- 1 = C1 U0 Ð1 Ð2 + C1 U0 Ð1 Ð1 In (A.9), the limits of integration are given by tmin = 2 mb 0 Ð1 (0, i) ( , ) (1, i) ( , ) 1 + ∑ C1 U0 0 i + ∑ C1 U0 1 i and t = ------(M Ð M )2. (B.1) max 2 1 2 i = Ð4 i = Ð5 mb 0 0 ( , ) ( , ) αβ 2 i ( , ) 3 i ( , ) Calculation of 〈L 〉W˜ αβ yields the expressions + ∑ C1 U0 2 i + ∑ C1 U0 3 i , i = Ð6 i = Ð6 〈 αβ〉 ˜ L Wαβ where 1 2 2 4 2 2 2 = --{3t f (t)λ m + 6t f (t) f (t)(M Ð M )λ m 2 2 3 Ð 2 b Ð + 1 2 2 b (Ð1, Ð2) Mbc + Mcc C1 = Ð------2------, (B.2) 2 [ 2( λ λ ) 4 ( 2 2)( λ 12Mbc + f +(t) t 2 1 Ð 3 2 + 2 mb Ð 2t M1 + M2 2 1 (Ð1, Ð1) 1 λ ) 2 ( 2 2)2(λ ) ] } C = Ð------, (B.3) Ð 3 2 + 2 mb + 2 M1 Ð M2 1 + 1 1 2 12Mbc for the pseudoscalar meson in the ﬁnal state, and (0, Ð4) C1 αβ 1 2 4 2 〈 〉 ˜ αβ { [ λ (( λ ) L W = ------2 t ( 1 + 1)mb Ð 2t 1 + 1 M1 1 10 2 2 2 8 2 12M2 ------( ( ) = 6 4 Mbc 6m2 + m3 + 8m2 Mbc Mcc 48Mbc Mcc (B.4) ( λ ) 2 2λ ) 2 ( 2 2)2( Ð 5 1 + 1 M2 + 9M2 2 mb + M1 Ð M2 1 ( 2 2 2) 6 4 ( 2 2 + Ð 5m1 + 5m2 + m3 Mbc Mcc + Ð 7m1 + 3m2 2 + λ ) ](F A) Ð 2[t2m 4 Ð 2t(M 2 + M 2)m 2 ( ) ) 4 6 2 2 8 2 10 ) 1 0 b 1 2 b + 2m1 m2 Ð m3 Mbc Mcc Ð 4m1 Mbc Mcc Ð 2m1 Mcc ,

2 ( 2 2) ][ A( ( λ ) 2 λ 2 (0, Ð3) 1 2 2 8 + M1 Ð M2 F+ 2 1 + 1 tmb Ð 3t 2mb ------(( ) C1 = 6 4 m1 Ð 2m1m3 Ð 2m3 Mcc 48Mbc Mcc Ð 2(1 + λ )M 2 + 2(1 + λ )M 2 ) Ð 3tF Am 2λ ]F A 1 1 1 2 Ð b 2 0 ( 2 2 ) 8 ( 2 2 2 + 6m2 + 4Mcc Mbc Ð 2m1 + 2m2 + 6m2m3 + 2m3 ( 2 4 ( 2 2) 2 ( 2 2)2) + t mb Ð 2t M1 + M2 mb + M1 Ð M2 ( ) 2 2 ) 6 2 + 2m1 Ð 5m2 + m3 Ð 11Mcc Ð 2Q Mbc Mcc (B.5) × [( 2 4 2λ 4 2λ 4 ( λ ) 2 2 ( 2 2 ( ) 2 2t mb + 2t 1mb Ð 3t 2mb Ð 4 1 + 1 tM1 mb + m1 Ð m2 + 2m1 m2 Ð 5m3 Ð 2m2m3 + 2m3 ( λ ) 2 2 2λ 2 2λ 2 2 2 ) 2 6 ( 2 2 Ð 4 1 + 1 tM2 mb + 6tM1 2mb + 6tM2 2mb + 5Mcc + Q Mbc Mcc + Ð 7m1 + 4m1m2 Ð 2m2 2 2 2 4 4 4 4 2 2 ) ) ( λ ) ( λ ) ( λ ) ) Ð 10m1m3 Ð 8m2m3 + 2m3 + 12Mcc + 2Q Mbc Mcc , + 2 1 + 1 M1 + 2 1 + 1 M2 Ð 4 1 + 1 M1 M2 (0, Ð2) 1 2 2 6 A 2 A A 2 2 2 2 A 2 (( ) × ( ) λ ( ) ( ( ) C1 = ------3m1 Ð 2m1m3 Ð 2m3 Mcc F+ + 6tFÐ F+ 2mb M1 Ð M2 + tmb 3t FÐ 6 4 48Mbc Mcc 2 2 2 2 2 2 2 2 6 2 2 × λ ( λ )( ) λ ( ) ) ] } ( ) ( 2mb + 16 1 + 1 F V M2 Ð 24 2M2 FV + m2 Ð 5m3 + 9Mcc Mbc + Ð 4m1 Ð 7m2 for the vector meson in the ﬁnal state. ( ) 2 2 + 2m1 6m2 Ð 5m3 Ð 8m2m3 Ð 13m3 + 16Mcc (B.6) 2 ) 4 2 ( 2 ( ) APPENDIX B + 4Q Mbc Mcc Ð 5m1 Ð 4m1 m2 Ð 4m3 ( 2 2 2 2) 2 4 ) Gluon-Condensate Contribution to QCD Sum Rules + 2 m2 + 2m2m3 + m3 Ð Mcc Ð Q Mbc Mcc , ( , ) In this appendix, we illustrate the kind of expres- C 0 Ð1 sions that arise for the gluon-condensate contribution to 1 i the form factors F (t) in the framework of Borel trans- 1 ( 2 4 ( 2 2 ) 4 = ------Ð 2m3 Mcc + Ð 6m3 + 5Mcc Mbc formed three-point sum rules in the case of the f1(t) = 6 4 48Mbc Mcc (B.7) f+(t) + fÐ(t) form factor. ( 2 2 Following the algorithm described in Subsection + Ð m1 + 2m1m2 Ð m2 + 4m2m3 〈 2〉 Π G 2 2 2 ) 2 2 ) 2.3, for 1 , we have the expression Ð 16m3 + Mcc + Q Mbc Mcc ,

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2( 2 2 ) (1, Ð2) 1 2 2 2 6 (0, 0) m3 Mbc + Mcc C = ------(m (3m Ð 2m m + 6m )M ------1 8 6 3 1 1 3 3 cc C1 = 6 4 , (B.8) 48Mbc Mcc 48Mbc Mcc ( 2 ( 2 2 ) 2 (1, Ð5) 1 2 4 2 4 + 14m2m3Mcc + 16m3 Ð 13Mcc Mcc ------( ) C1 = Ð 8 6 m2 Mbc Ð m1 Mcc 2 2 2 6 2 2 3 48Mbc Mcc ( ) ) ( + m2 4m3 + Mcc Mbc + Ð 5m2 m3 Ð 2m2m3 2 2 8 2 6 2 × (( ) 4 2 2 2 2 2 4 5m2 + m3 Mbc Ð m3 Mbc Mcc (B.9) Ð 2m3 + 4m2 Mcc + 8m2m3Mcc + 32m3 Mcc Ð 8Mcc ( 2 2) 4 4 ( ) 2 6 Ð 4m1 + 3m2 Mbc Mcc + 2m1 m2 Ð m3 Mbc Mcc 2( 2 2 ) ( 2 3 + m1 Ð 3m3 + 4Mcc + 2m1 3m2m3 Ð 2m3 (B.12) + 2m 2M 8 ), 1 cc 2 2 ) 2 2 2 2 ) 4 2 Ð 2m2Mcc + 6m3Mcc + 3m3 Q Ð 4Mcc Q Mbc Mcc (1, Ð4) 1 2 2 2 10 ------( ( ) 3 2 2 2 C1 = 8 6 m1 3m1 + 2m1m3 + 6m3 Mcc ( ( ) Ð 2m1 m3 + m1 Ð 4m2m3 + 4m3 + Mcc 48Mbc Mcc ( 4 2 2 2 2 ) 10 ( 4 3 2( 2 2 2 2) + Ð 2m2 Ð 7m2 Mcc + m3 Mcc Mbc + 8m2 + 4m2 m3 + m3 m2 + 2m2m3 + 4m3 Ð 7Mcc Ð Q 2 2 2 2 2 4 2 2 3 2( 2 2) ( 3 + 2m m (m Ð m m + 3m Ð 2M Ð Q ) )M M ), + 3m2 m3 + 2m2m3 + m1 4m2 + m3 Ð 2m1 4m2 1 3 2 2 3 3 cc bc cc 2 2 3 2 2 2 2 Ð 3m m + m m Ð m ) Ð 11m M Ð 3m M (1, Ð1) 1 ( 3 4 ( 4 2 3 2 3 3 2 cc 3 cc C1 = Ð------2m1m3 Mcc + 4m3 48M 8 M 6 Ð 4m 2Q2 Ð m 2Q2 )M 8 M 2 Ð (3m 4 + 6m 2m 2 bc cc 2 3 bc cc 1 2 3 (B.13) Ð 17m 2M 2 + M 4 )M 4 + 2m 2(m 2 Ð 2m m 3( ) ( 2 2 3 cc cc bc 3 1 1 2 + m1 Ð 6m2 + 4m3 + 2m1 m2 m3 + 2m2Mcc (B.10) + m 2 + 2m m + 4m 2 Ð 6M 2 Ð Q2 ) ), 2 ) 2( 2 2 2 2 1 3 3 cc Ð 2m3Mcc + m1 10m2 + 2m2m3 + 2m3 Ð 8Mcc ( 2 2 )( 2 4 2 4 )3 2 ) ) 4 6 ( 2 2( 2 2 ) (2, Ð6) Mbc Ð Mcc m2 Mbc Ð m1 Mcc Ð 3Q Mbc Mcc + 4m1m2 m3 + m1 9m2 + 16Mcc C1 = ------, (B.14) 96M 10 M 8 2( 2 2 2 2)) bc cc + m2 2m2 + 4m2Mcc Ð 3m2m3 Ð 4m3 Ð Mcc Ð 2Q (2, Ð5) 1 2 4 2 4 6 4 3 2 2 ( ) × ( ( ) C1 = Ð------m2 Mbc Ð m1 Mcc Mbc Mcc Ð m1 6m1 + 4m1 m3 + 2m3 Ð m2 + m3 8 6 96Mbc Mcc ( 2 2 2 2 ) ) 2 8 ) + m 1 m 2 + 2 m 2 m 3 Ð 4m3 + 5Mcc Ð Q Mbc Mcc , × (( 2 2) 6 2 2 4 2( 2 m2 + 2m3 Mbc + 7m1 Mbc Mcc + m2 m1 (1, Ð3) 1 3 2 3 8 ( ( ) 2 2 C1 = ------2m3 m1 + 5m1 m3 + 2m3 Mcc 8 6 Ð 2m1m2 + m2 + 2m1m3 + 2m2m3 + 3Mcc (B.15) 48Mbc Mcc 2 4 2 2 2 ( 4 2( 2 2 ) 2 2 2 Ð Q )M Ð m (m Ð 2m m + m + 2m m + m3 + m2 9m3 Ð 4Mcc + 8m2m3Mcc Ð 3m3 Mcc bc 1 1 1 2 2 1 3 2 2 4 4 8 4 2 2 4 2 2 ) ) ) ( + 2m2m3 + 7Mcc Ð Q Mcc , Ð 11Mcc Mbc + 2m2 + 10m2 m3 + m3 + 7m2 Mcc 2 2 2 4 ( ) (2, Ð4) 1 4 2 10 3 2 + 18m2m3Mcc + 2m3 Mcc Ð 20Mcc Ð 4m1 m2 Ð m3 ------( ( C1 = 10 8 Ð 3m1 m3 Mcc + m1 6m2Mbc 96Mbc Mcc × (m 2 + 2M 2 ) + m 2(2m 2 + 5M 2 ) Ð 2m 2Q2 2 cc 1 2 cc 2 2 2 2 2 ) 2 8 Ð 6m3Mbc + m1m3 Ð 2m1Mcc Ð 4m3Mcc Mbc Mcc 2 2 ) 6 2 ( 2 2 4 Ð 5Mcc Q Mbc Mcc + Ð 7m2 m3 Ð 4m3 (B.11) ( 3 2 3 2 2 4 2 4 2 2 2 4 2( Ð 12m2 m3Mcc + 4m2m3 Mcc Ð 12m2 Mcc Ð 4m3 Mcc + 4m2 Mcc + 10m2m3Mcc Ð 7Mcc + 2m1 2m2m3 + m 4(3m 2 + 4M 2 ) )M 10 + 2m (Ð3m 3M 2 2 2 ) ( 3 3 2 2 2 3 cc bc 1 2 bc Ð m3 + 5Mcc + m1 6m2 Ð 6m3 Ð 2m2 m3 + 4m2m3 2 4 2 2 2 2 2 2 2 2 2 2 6 + 4m m m M Ð 8m M + m (3m M Ð m m + 4m M +10m M ) + 2m Q Ð 4M Q )M M 1 2 3 cc 1 cc 2 3 bc 1 3 2 cc 3 cc 3 cc bc cc (B.16) ( 4 3( ) 2 2 ( 2 2 2 ) ) 6 4 2( ( + m1 Ð 2m1 m2 Ð 4m3 + 10m3 Mcc + 2m1m3 m2 + 3m1Mcc + 2m3Mcc Mbc Mcc + 2m1 Ð4m2 m1 2 2 2 ) 2( 2 ) 2 2( 2 2 ) ( 2 + m2m3 + m3 Ð Mcc Ð Q + m1 m2 Ð 2m3 Mcc + m2 m3 + 3Mcc + 3m1 + 6m1m3 2 2 ) 8 ) 2 2 ) 2 ) 4 6 2( 2( 2 2 ) + 2m2m3 + 7Mcc Ð Q Mcc , + 2Mcc Ð 3Q Mcc Mbc Mcc + m2 m2 m3 Ð 4Mcc

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( ) 2 ( 2 2 2 ) 2 ( 2 2 + 4m2 m1 Ð 4m3 Mcc + 2 Ð 2m1 Ð 4m1m3 + 3Mcc + 6m3Mcc Ð 2Mcc Ð 3m1 m3 Ð 6m1m3 2 ) 2 ) 8 2 ) 2 2 2 ) ) ) + 2Q Mcc Mbc Mcc , + 2m1Mcc + 20m3Mcc + 3m3Q ,

(2, Ð3) 1 2 2 C = ------(m (9m M Ð m m (2, Ð1) 1 2 2 2 1 8 6 1 3 bc 1 3 ------( ( C1 = 8 6 m3 Mbc 4m2m3 + 15m3 48Mbc Mcc 96Mbc Mcc (B.19) 2 2 2 2 4 ( ) ) 2 2 2 2 + m2 Ð 9Mbc Ð 4m1m3 + 4m3 + 4m1Mcc Mbc Mcc ) ( ( ) Ð 16Mcc + m3 Ð m1 m3 + 2m1 m2m3 Ð m3 + 2Mcc ( 3 2 2 4 2 ( 2 2 2) ) ) Ð 2m2 m3 + 9m2 m3 + m3 + 10m2m3Mcc + m3 Ð m2 Ð 2m2m3 + 13Mcc + Q , 4 ) 6 ( 4 2 3 3 2 2 2 Ð 10Mcc Mbc + m2 m3 + 2m2 m3 Ð 3m2 m3 Mcc 4( 2 ( 2 2 ) 2 2 ) (2, 0) m3 Mbc m3 Ð 4Mcc + m3 Mcc 2 4 4 6 2 2 2 ------( C1 = Ð 10 8 , (B.20) Ð 3m2 Mcc Ð 14m2m3Mcc + 9Mcc + m1 m2 m3 96Mbc Mcc 2 2 4 ) ( 3 2 2 3 + 2m3 Mcc Ð 3Mcc + 2m1 Ð m2 m3 + m2 m3 (B.17) 4 2 2 4 3 ( , ) (M m Ð m M ) C 3 Ð6 = ------b---c------2------1------c--c-----, (B.21) + m 2m M 2 Ð 2m m 2M 2 + m M 4 Ð 3m M 4 ) 1 10 8 2 3 cc 2 3 cc 2 cc 3 cc 96Mbc Mcc 2 2 2 2 2 2 4 2 ) 4 Ð m2 m3 Q Ð 2m3 Mcc Q + 3Mcc Q Mbc (3, Ð5) 1 ( 4 2 2 4 ) C1 = ------Mbc m2 Ð m1 Mcc ( 3 2 ( ) 2 2 2( 2 8 8 + m1 m1 m3 + 3 m2 Ð m3 m3 Mbc + m1 3m2Mbc 96Mbc Mcc (B.22) 2 2 3 2 ) 2( 2 × ( 4 2( 2 ) 2 4 ( 2 )) Ð 3m3Mbc Ð 2m2m3 + 2m3 Ð 2m3Mcc + m1m3 m2 Mbc m2 2m2m3 + Mcc Ð m1 Mcc 2m2m3 + 7Mcc , 2 2 4 + 2m m Ð 3M Ð Q ) )M ), (3, Ð4) 1 4 2 8 8 2 2 2 2 3 cc cc ------( ( C1 = Ð 10 8 Ð 3m1 m3 Mcc + Mbc m2 m2 m3 (2, Ð2) 1 2 4 6 96Mbc Mcc C = ------(3m m M 1 10 8 1 3 cc (B.23) 4 ) 4 2 4 ( 2 2 96Mbc Mcc Ð 8m2m3Mcc + 2Mcc + 2Mbc m1 Mcc m2 m3 2 2 4 ( 2 2 2 2 4 ) ) + m1m3 Mbc Mcc 6m2Mbc Ð 6m3Mbc + m1m3 + 6m2m3Mcc + 3Mcc ,

2 2 6 2 4 2 Ð 2m M + 4m M ) + M (3m m + 8m m (m (3, Ð3) 1 2 2 4 2 1 cc 3 cc bc 2 3 2 3 3 ------( ( ) C1 = 8 8 m1 m3 Mcc Ð 2m2m3 + Mcc 2 ) 2 4 ( 2 2 ) ) 48M M Ð Mcc Mcc + 4Mcc Ð 11m3 + 2Mcc (B.18) bc cc (B.24) 4 2 ( ( ) 2 2( 3 4 ( 3 3 2 2 2 4 6 ) ) + Mbc m3Mcc Ð 8m2 m1 Ð 2m3 m3Mcc + m2 m3 + Mbc Ð 2m2 m3 + 2m2 m3 Mcc + 6m2m3Mcc Ð 3Mcc ,

2 2 2 4 4 2 2 2 4 ( , ) m (3m m M + M (m m + 12m m M Ð 18M )) 3 Ð2 -----3------1------3------c--c------b--c------2------3------2------3------c--c------c--c------C1 = Ð 10 8 , 96Mbc Mcc (B.25)

4 2 2 ≥ ( ) for a 0. Here, Kn[z] is the modiﬁed Bessel function of (3, Ð1) m3 2m2M Ð 3m2 Ð 9Mcc C = ------, (B.26) the second order. In the case of U0(Ð1, Ð2) and U0(Ð1, 1 8 8 − 96Mbc Mcc 1), we have failed to obtain exact analytic expressions, so we present their analytic approximations: 6 ( , ) m 3 0 ------3------C1 = 10 8 . (B.27) exp[ÐB ]  B 96M M U (Ð1, Ð2) = ------0---2exp Ð -----Ð--1 B v 2 bc cc 0 3 B 1 2v BÐ1 B1 1 For functions U0(a, b), we have the expressions [ 1/2 3/2] 2 [ 1/2 3/2] 1 + a + b + 2exp ÐBÐ1 B1 BÐ1 B1v + exp ÐBÐ1 B1 U (a, b) = ∑ 2C a exp[ÐB ] 0 n Ð b Ð 1 0 × B 3/2 B 1/2(2 + v B )v Ð 2v 2 B B 2 n = 1 + b Ð1 1 1 Ð1 1 n/2 a + b + 1 Ð n B B × ( 2 2 ) BÐ1 [ ] ×  Ð1 Γ , Ð1 BÐ1 Γ( , 1/2 3/2) Mbc + Mcc ------ KÐn 2 BÐ1 B1 exp ------0 ------+ ------ + 0 BÐ1 B1  B1 v v B1

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Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2145Ð2156. From Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2241Ð2252. Original English Text Copyright © 2000 by Kulikov, Narodetskiœ. ELEMENTARY PARTICLES AND FIELDS Theory

Light-Front Quark Model and Analysis of the Spectrum of Photons from Inclusive B Xsg Decays* P. Yu. Kulikov and I. M. Narodetskiœ Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received January 21, 2000

Γ γ Abstract—The electroweak-decay width (B Xs) is investigated in a light-front (LF) constituent quark Γ γ model. A new partonlike formula is derived that establishes a simple relation between (B Xs) and the b sγ decay width. A treatment of the b quark as an on-mass-shell particle and the inclusion of effects that arise from the transverse motion of the b quark in the B meson are basic features of this approach. Adopting different b-quark LF distribution functions, both phenomenological ones and those that are derived from constituent quark models, and neglecting perturbative corrections, we compute the photon energy spectra and the moments of the shape function. It is shown that the LF approach can be matched completely with a heavy-quark expansion (HQE), provided that the constituent b-quark mass is redefined in a way similar to that used in HQE to define the pole mass of the b quark. In this way, the correction to first order in 1/mb can be eliminated from the total width in agreement with the general statement of HQE. We also show that the photon energy spectra calculated in the LF approach agree well with those obtained in the model of Altarelli et al., provided that the same distribution function is used as an input in both cases. Despite the simplicity of the model, our results are in fairly good agreement both with HQE predictions and with available experimental data. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION are under consideration at SLAC (BaBar) and KEK (Belle) and the upgraded B factory at CESR (CLEO III In the Standard Model (SM), rare B-meson decays collaboration) will drastically change the experimental are induced by one-loop W-boson- or up-quark- situation concerning rare B decays in the near future. exchange diagrams. In particular, the flavor-changing γ With an expected high luminosity, radiative B decay neutral-current decay b s may proceed via the will no longer be rare events. An experimental error in electroweak penguin diagram [1] where the quark the inclusive mode b sγ below 10% is possible. emits and reabsorbs a W boson, thus changing the fla- This is also a motivation for increasing the accuracy of vor twice, and a photon is radiated from either the W theoretical predictions. boson or one of the charged-quark lines. Penguin decays became increasingly appreciated in recent The existence of loop decays was first confirmed years. They give insight into the SM because these experimentally by an observation of an electromag- decays can be used to determine various elements of netic penguin in the exclusive mode B K*γ by the CabibboÐKobayashiÐMaskawa (CKM) matrix that the CLEO II collaboration [4]. The branching ratio γ enter into the SM Lagrangian. In particular, the penguin found for the decay B Xs by the CLEO collabo- γ γ | | transitions b s and b d are sensitive to Vts ration in 1994 is [5] and |V |, respectively, which will be very difficult to td ()γ measure in top decay. In addition, penguin decays may Br BXs (1) be quite sensitive to new physics [2]. Ð4 = ()2.32± 0.57() stat. ±0.35() syst. × 10 , At present, b-quark decays are under investigation with high-statistics samples from the following three and the very recent preliminary updated data from colliders: the CESR at Cornell, which produces BB CLEO yield [6] pairs at the ϒ(4S) resonance just above the e+eÐ BB ()γ Br BXs threshold; the LEP collider at CERN, which produces (2) ()± ()±()× Ð4 bb pairs in Z0 decays; and the Tevatron at FNAL, = 3.15 0.80 stat. 0.72 syst. 10 . which produces bb pairs in pp collisions. For the The ALEPH measurement of the corresponding review of various aspects of these machines, the reader branching ratio for B hadrons (mesons and baryons) is referred to [3]. Presently, the B-meson factories that produced in Z0 decays yields [7] γ ± ± × Ð4 * This article was submitted by the authors in English. Br(B Xs) = (3.11 0.80 0.72) 10 . (3)

2146 KULIKOV, NARODETSKIŒ

Note that the measurements by CLEO and ALEPH 2 2 2 were quite different and should not be expected to give mass mf(p ) = MB Ð p + msp , where msp is the mass identical results. The experimental results presented of the spectator quark, and p is the maximally above were obtained by measuring the high-energy sec- max Φ 2 tion of the photon spectrum and by extrapolating it to allowed value of p. Distribution (p ) is usually assumed the total rate. This requires the theoretical understand- to have the Gaussian form determined by a nonperturba- ing of the photon spectrum. tive parameter, pF, Processes of the B X γ type are expected to be  2 s 2 4 p dominated by the two-body decays such as B Φ( p ) = ------expÐ----- , (6) π 3  2 K*(892)γ and by final states with soft-gluon emission pF pF that kinematically resemble two-body decays. In this case, the photon energy in the B-meson rest frame has with the wave-function normalization a spectrum that peaks approximately at half of the B- ∞ meson mass. In contrast, charged-current-initiated radi- Φ( p2) p2dp = 1. ative b decays, such as b cW*γ and b uW*γ ∫ (where W* is a virtual W boson), produce photons with 0 a spectrum resembling that from bremsstrahlung and, Based on calculation of [13] for the photon energy hence, of much lower energy. The experimental results spectrum and adopting the Fermi motion model of [12], for the inclusive decay rate were obtained by measuring CLEO used the value R = 85Ð94% [8] to determine the high-energy section of the photon spectrum, γ Br(B Xs ) from the measured partial branching

max ratio. Eγ In contrast to the exclusive decays, the inclusive ( γ )( min) dBr Br B Xs Eγ = ∫ ------dEγ , (4) decay modes are theoretically simpler, because no spe- dEγ min cific model is needed to describe the final hadronic Eγ states. Assuming the quarkÐhadron duality and the fact min that the b quark is heavy as compared to the character- with the present experimental cut of Eγ ~ 2.1 GeV by Λ istic low-energy scale QCD of the strong interactions, CLEO collaboration [8] and the extrapolation to the the inclusive B X γ decay rate was traditionally cal- total rate. To determine the total branching ratio for s γ culated in a systematic QCD-based expansion [14] in B Xs from such a measurement, one must know powers of Λ/m : γ b theoretically the fraction R of the B Xs events with min  1  ( γ ) ( γ ) ᏻ ------photon energies above Eγ . Therefore, the study of the Br B Xs = Br b s +  2 . (7) photon spectrum from the weak radiative decay B mb γ Xs is important for understanding the possible accu- This formula is known under the name of HQE. The racy that can be attained in the prediction of the total free quark decay width including the QCD perturbative rate in the presence of an experimental cut on the pho- corrections from the real and virtual gluons emerges as ton energy Eγ. Gross features of the spectrum, such as the leading term in this expansion. The nonperturvative the average photon energy, may be used to measure the Λ2 2 1) fundamental heavy-quark-expansion (HQE) parame- corrections are suppressed by the factor /mb . The ters determining the quark pole mass and the kinetic fact that there are no first-order corrections is based on energy [9, 10]. a particular definition of mb [16] that can be regarded as a nonperturbative generalization of the pole mass. The experimental results given above have been solely based on the ACM model [11, 12]. This model In the free-quark approximation neglecting the radi- treats the heavy hadron as a bound state of the heavy ative corrections, the spectrum of photons is a mono- quark and a spectator, with a certain momentum distri- chromatic line: bution Φ(p2). The photon energy spectrum from the Γ( γ ) d b s Γ( γ )δ mb decay of the B meson at rest is then given by ------= b s Eγ Ð ------ . (8) dEγ 2 dΓ(B X γ ) ------s------For energy Eγ not too close to its maximal value, the dEγ photon spectrum dΓ(b sγ)/dEγ has a perturbative

pmax (5) expansion in the strong interaction coupling constant Γ( γ ) α α 2Φ( 2)d b s ( , ( 2)) s, and the first ( s independent) term of the expansion = ∫ dp p p ------Eγ m f p . dEγ 0 1)In addition, the nonperturbative contributions from cc intermedi- Γ γ 2 Here, d (b s )/dEγ is the photon energy spectrum ate states were found which scale like 1/mc and enlarge the from the in-ﬂight decay of the b quark with ﬂoating branching ratio by ≈3% [15].

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 LIGHT-FRONT QUARK MODEL AND ANALYSIS OF THE SPECTRUM OF PHOTONS 2147 is given by (8). This expansion is known at the order of energy and the invariant mass distributions of the had- α α 2β β rons recoiling against the photon. We will compare the s [10]. The s 0 contributions, where 0 = 11 Ð 2nf /3 Γ β photon spectra d /dEγ calculated in the LF and ACM is the one-loop function, have been recently com- approaches and will show that the discrepancy between puted [17]. However, near the end point, the energy the two is very small. released for a light quark system is not of the order The paper is organized as follows. In Section 2, we ᏻ Λ Λ (mb) but of the order of = MB Ð mb ~ QCD. Thus, collect the main facts concerning the calculation of the 2 γ the expansion parameter is no longer 1/m , but rather Br(b s ) in QCD. In Section 3, we derive the par- b tonic formula for the photon energy spectra from B 1/m Λ ~ ᏻ (1 GeV), and the operator product expan- γ b Xs . In Section 4, we consider the ansätze for the distri- sion breaks down. The divergent series in the effective 2 theory corresponding to the nonperturbative effect due bution function f(x, p⊥ ). In Section 5, we discuss the to the soft interactions of the b quark with the light con- choice of the b-quark mass in our approach. The stituents have to be resumed. This so-called “Fermi numerical results for the photon energy spectra and the motion” can be included in the heavy-quark expansion moments are presented in Section 6, where we also by summing an infinite set of leading-twist contribu- present the comparison of our results with those tions into a shape function F(x) [18], where a scaling obtained in the ACM model. In Section 7, we discuss variable is defined as the invariant mass spectra and the calculation of Br(B K*γ) in our approach. Finally, Section 8 con- 2Eγ Ð m x = ------b , tains our conclusions. Λ (9) with Λ = M Ð m + ᏻ(1/m ). 2. PERTURBATIVE TREATMENT B b b OF Γ(b sγ) IN QCD This is quite similar to what happens for the structure In this section, we just briefly recall the main points function in deep-inelastic scattering in the region, γ where the Bjorken variable x 1. A model-inde- of the calculation of the b s decay in QCD (for B general discussion see [28]). pendent determination of the shape function in the µ γ effective theory is not possible at present; however, it At high energy scale ~ MW, the b s quark decay may be possible to solve this problem by using lattice is governed by an electroweak penguin diagram. To obtain µ QCD [19]. An ansatz for the shape function constrained an effective low-energy theory relevant for scales ~ mb, by the information on its few first moments has been heavy degrees of freedom related to W and Z0 bosons and recently used in [20] including the next-to-leading t quark must be integrated out to obtain an effective cou- (NLO) perturbative QCD corrections. pling for the pointlike interactions of initial and final par- In this paper, we consider the light-front (LF) ticles. After the integration, the effective Lagrangian for approach to calculation for the photon energy spectra b s penguin decays takes the form γ from inclusive B Xs decay. This approach was 10 originally suggested for the inclusive semileptonic ᏸ 4GF (µ) (µ) eff = ------VtbV t*s ∑Ci Oi , (10) transitions [21–24] and has been recently refined in 2 [25]. The corresponding ansatz from [25] reduces to a 1 specific choice of the input LF distribution function where Oi are currentÐcurrent (i = 1, 2), gluonic penguin 2 2 (i = 3, …, 6), and electroweak penguin (i = 7, …, 10) |ψ(ξ, p⊥ )| , which represents the probability that the b operators. In (10), GF is the Fermi constant and Vij are quark carries a LF fraction ξ and a transverse momen- the elements of the CKM matrix. The effects of the 2 2 tum squared p⊥ = |p⊥| . As a result, a new partonlike heavy degrees of freedom are hidden in the effective formula for the width of inclusive semileptonic b gauge coupling constants, running quark masses, and the Wilson coefficients C describing the effective c, u decays has been derived [25], which is similar to i strength of the local operators O in (10). The coeffi- the formula obtained by Bjorken et al. [26] in the case i cients C at µ ~ M serve in leading order as initial con- of infinitely heavy b and c quarks. i W ditions to the renormalization group evolution from µ Some of the dynamical features of this model get µ W down to b. An explicit form of the operators Oi in (10) obscured by the integration over the lepton energy. can be found, e.g., in [28]. The operators relevant for They are better manifested in the spectrum of the pho- leading approximation are the electroweak penguin tons in the radiative X γ transitions. In this paper, B s e we extend the work of [25] to compute the nonpertur- (µ) σ ( γ ) O7γ = ------2mb sα µν 1 + 5 bαFµν (11) bative corrections to the photon spectrum and the inclu- 32π γ 2) sive rate B Xs . We attempt to take into account and the gluonic penguin the B-meson wave function effects on the photon gs (µ) σ ( γ ) a a O8G = ------mb sα µν 1 + 5 T αβ bβGµν , (12) 2)A preview of this work can be found in [27]. 32π2

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(0)eff where e and gs are the electromagnetic and strong cou- µ coefficient” C7γ ( b) [30] a pling constants, and Fµν and Gµν denote the electro- (µ ) (0)eff(µ ) η16/25 (0)( ) magnetic and the gluonic field strength tensor, respec- c7γ b C7γ b = C7γ MW tively. The contribution of other operators O3, …, O6 8 (18) 8 14/25 16/25 (0) (0) ai can be neglected to an excellent approximation. Wilson + --(η Ð η )C (M ) + C (M )∑h η coefficients of these operators originate from operator 3 8G W 2 W i mixing only and are numerically small. 1 (0) η α µ α µ Technically, the calculations are performed at a with C2 (MW) = 1 and = ( W)/ ( b). The “magic µ ᏻ high-energy scale W ~ (MW) at which the full theory numbers” hi and ai are given, e.g., in Table 23 from is matched with the effective five-quark theory and then [28]. The scale µ enters into (18) only through η. µ µ b is evolved to a low-energy scale ~ b using renormal- (0) µ For m = 170 GeV, C (µ ) = Ð0.096. Then, using ization group equations. At W ~ MW, the contribution t 8G W γ of the matrix element of O7γ to the b s decay width α ( ) 3) α (µ ) s MZ has the form s b = ------α------(------)------(19) β s MZ MZ Γ = Γ(b sγ ) 1 Ð 0------π------ln--µ----- 0 2 b α 2 (13) µ α GF 2 2 2 3 with b = 5 GeV and s(MZ) = 0.118, one obtains = ------V V c γ (µ )m (µ ) m . 4 tb ts 7 W b W b (0)eff µ (0) µ (0) µ 32π C7γ ( b) = 0.695C7γ ( W) + 0.085C8G ( W) Ð (0) µ 3 0.158C2 ( W) = Ð0.3. In the absence of QCD correc- The factor mb in (13) originates from the twoÐbody η (0)eff µ (0) µ phase space. In HQE, mb is usually associated with the tions, we would have = 1 and C7γ ( b) = C7γ ( W). µ b-quark pole mass mb, pole. For b ~ mb, one has Therefore, the leading QCD corrections cause the large QCD enhancement of the B X γ rate [31]. α (µ ) s 2(µ ) 2  8 s b  The predictions for Br(B X γ) are usually mb b = mb, pole 1 Ð ------ . (14) s 3 π obtained by normalizing the result to that for the semileptonic rate, thereby eliminating the uncertainties due This difference does not enter the leading approxima- 5 (0) to the CKM matrix elements and the factor mb . In the tion. The basic functions c γ (M ) = C γ (x ) and 7 W 7 t leading approximation, one obtains (0) 2 2 c8G(MW) = C8G (xt), xt =mt /MW were calculated by Γ( γ ) Γ( γ ) B Xs ≈ b s many authors, in particular, by Inami and Lim [29]. Γ----(------ν-----)- -Γ---(------ν-----)- = Rquark, (20) Their explicit form is B Xc e b c e

3 2 where (0)( ) 2 3xt Ð 2 8xt + 5xt Ð 7xt C7γ MW = xt ------ln xt + ------, (15) Rquark 4(x Ð 1)4 24(1 Ð x )3 t t 2 α ( ) 2 δ δ (21) 2 3 2 V t*s V tb 6 QED 0 eff  SL RAD = ------C γ (µ ) 1 Ð ------+ ------, (0) Ð3xt Ð xt + 5xt + 2xt 2 π (ρ) 7 b  2 2  C (M ) = ------ln x + ------. (16) V g m m 8G W ( )4 t ( )3 cb b b 4 x Ð 1 8 xt Ð 1 g(ρ) = 1 Ð 8ρ + 8ρ2 Ð ρ4 Ð 12ρ2lnρ is the phase space (0) (0) factor in the semileptonic b ceν decay, and ρ = For mt = 170 GeV, C7γ = Ð 0.193, and C8G = Ð0.096. 2 2 Expressions (15) and (16) do not include QCD cor- mc, pole /mb, pole . The last factor in (21) parametrizes the µ rection. The evolution of (10) down to b mixes the nonperturbative corrections in HQE to the semileptonic operators and radiative decay rates [9] ( ρ)4 δ 1λ 3 6 1 Ð  λ (µ ) (µ , µ ) ( ) SL = -- 1 + -- Ð ------ 2, Ci b = ∑U b W C j MW . (17) 2 2 f (ρ) (22) j λ λ δ 1 9 2 Γ γ RAD = ----- Ð ------, In particular, (b s ) in the leading order is again 2 2 given by (13) in which c γ(µ ) is replaced by “effective 7 b λ λ where 1 and 2 parametrize the matrix elements of the 3)The strange quark mass will be neglected throughout this paper; it kinetic and chromomagnetic operators, respectively, 2 2 which appear in the effective Lagrangian of HQE at the only enters the ﬁnal results quadratically as ms /mb . λ order of 1/mb. The value of 2 is known from B*ÐB

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λ 2 λ splitting, 2 = 0.12 GeV . The value of 1 is controver- In our phenomenological consideration, we associate sial, fortunately it cancels on the right-hand side of mb with the constituent b-quark mass (see below). The (22). The two nonperturbative corrections in (22) both ξ 2 are of a few percent and tend to cancel each other. normalization of f( , p⊥ ) in (25) is as follows: Therefore, the total nonperturbative correction in (22) 1 ∞ is much smaller than the accuracy of the perturbative π ξ 2 (ξ, 2 ) ∫d ∫dp⊥ f p⊥ = 1, (27) calculation of Rquark. The final result of the leading- order analysis is [30] 0 0 where the factor 1/ξ comes from the normalization of Br(B X γ ) = (2.8 ± 0.8) × 10Ð4, s (23) the B bd vertex [33]. The phase space factor dτ is µ given by where uncertainty reflects mainly the scale ambiguity b. α τ δ[( )2] The NLO corrections include the s corrections to d = pb Ð q Eγ dEγ . (28) (0)eff α C7γ [32] and the s corrections originating from one- We choose the z axis as being parallel to the 3-vector q, 〈 γ| | 〉 〈 γ| | 〉 ± τ loop matrix elements s O7γ b and s O8G b and two- so that q+ = 2Eγ, q– = 0, where q± = q0 qz, then d takes 〈 γ| | 〉 loop matrix elements s Oi b of the remaining opera- the form tors (see Eq. (21) of [32]). A part of NLO corrections  2 2 originates from the bremsstrahlung corrections. For an τ δ 2 p⊥ + mb d = mb Ð ------ Eγ dEγ . (29) updated review of NLO calculation, see [20]. All the  p +  theoretical predictions with allowance of the NLO b QCD corrections in the SM fall in the range 2 In the first approximation, we neglect p⊥ in the argu- ( γ ) ( ± ) × Ð4 δ Br B Xs = 3.3 0.3 10 ; (24) ment of the function. Then, in terms of the scaling variable i.e., the LO result is shifted by about 20%, and the y = 2Eγ /M , (30) present theoretical uncertainty is reduced to about 10%. B This theoretical result is in good agreement with Γ γ the photon spectrum d /dy in B Xs takes the sim- updated experimental data (2) and (3). ple form Γ( γ ) 1 d B Xs ( ) 3. THE PARTON FORMULAS -Γ------= RLF y , (31) 0 dy As a preliminary, we briefly discuss a derivation of where γ the inclusive photon energy spectrum for the B Xs decays in the context of the LF approach of [25]. Simi- ( ) 1 ˜ ( ) RLF y = ----y f y , (32) lar to the ACM model, the LF quark model treats the x0 beauty meson as consisting of the heavy b quark plus a and spectator quark. Both quarks have fixed masses, mb and ∞ msp, though. This is at variance with the ACM model that has been introduced in order to avoid the notion of ˜ (ξ) π 2 (ξ, 2 ) f = ∫dp⊥ f p⊥ . (33) the heavy quark mass at all. 0 γ To calculate the B Xs decay width, we use the Equivalently, one can write the spectrum in the stan- approach of [25] which assumes that the sum over all dard QCD form [34] possible strange final states X can be modeled by the s Γ decay width of an on-shell b quark into an on-shell c 1 d 2 ( ) Γ------= ---F x , (34) quark weighted with the b-quark distribution function 0 dE Λ ξ 2 f( , p⊥ ). Going through the intermediate steps (for the where details, see [25]), we obtain the partial decay width of γ 2Eγ Ð m y Ð x the inclusive B Xs decay in the form x = ------b = ------0-, (35) Λ 1 Ð x0 2 d p⊥dξ Γ Γ (ξ, 2 ) τ and x is defined by (26). Therefore, the specific choice d = 4x0 0∫------ξ------f p⊥ d , (25) 0 adopted for ˜f (ξ) corresponds to the following particu- where lar form of the QCD shape function ( ) (Λ ) ˜ ( ) x0 = mb/MB. (26) F x = /MB x0 y f y . (36)

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2150 KULIKOV, NARODETSKIŒ Note that if one uses the infinite momentum frame pre- using the ACM model. In what follows, we will adopt ξ 2 ξ ξ2 2 both a phenomenological LF wave function and the LF scription [21, 22] pb = PB, i.e., mb ( ) = MB , then Γ ∝ functions corresponding to the various equal time (ET) one easily derives from (25) and (29) that d quark-model wave functions. As to the phenomenolog- 3 ˜ Eγ f (y)dEγ [35]. ical ansatz, we use a model first proposed in [36] and also employed in [22, 24] to take into account the We take into account now the transverse motion of bound state effects in B-meson decays. It is written in 2 the b quark, i.e., the term p⊥ in (29), to find a little the Lorentz-invariant form more complicated expression for RLF(y) [25]  λ   λ ε  ψ( p ) = ᏺexp Ð--v v = ᏺexp Ð------p-- , (40) 1  b sp   2 2msp 2 R (y) = 2m π f (ξ, p⊥* )dξ, (37) LF b ∫ where vb and vsp are the 4-velocities of the b quark and y the spectator quark, respectively, and ε = p 2 + m 2 where the integration limits follow from the condition p sp is the energy of the spectator. We use the normalization 2 p⊥* ≥ 0, with condition ∞ 2 2 2 2 2 ψ ( p ) p⊥* = p⊥* (ξ, E) = m (ξ/y Ð 1). (38) b ∫ p dp------ε------= 1; (41) 2 p In this case, the shape of the spectrum is obtained by 0 2 in this case, direct integration of the distribution function f(ξ, p⊥ ). We will see below that the difference between R (y) 2 2λ LF ᏺ = ------, (42) given by (32) and (37) is very small. m 2 K (λ) In the free-quark approximation, sp 1 λ where K1( ) is the Macdonald function. The function (ξ, 2 ) δ(ξ ξ )δ( 2 ) Φ 2 ψ2 | | ε f p⊥ = Ð 0 p⊥ , (39) (p ) = ( p )/2 p represents a momentum distribution Γ γ of the spectator quark in the rest frame of B meson. We the total inclusive width (B Xs ) of the B meson Γ γ go over from ET to LF momenta by leaving the trans- is the same as the radiative width (b s ), and the verse momenta unchanged and letting spectrum of photons is monochromatic [see (8)]. Due to the heavy-quark motion, the δ function in (8) is trans-  2 2 1 + Ð 1 + p ⊥ + m formed into a finite width peak. This effect is solely p = --( p Ð p ) = -- p Ð ----i------i- (43) iz 2 i i 2 i +  responsible for the filling in of the windows between pi mb/2 and the kinematical boundary in the B-meson 4) for both the b quark (i = b) and the spectator quark (i = decay, Emax = MB/2. Expressions in (32) and (37) ξ sp). The longitudinal LF momentum fractions i are exhibit a pronounced peak, which is rather asymmetric. defined as It is a gratifying feature of the LF model, since it is in + + qualitative agreement with both findings in QCD and ξ psp ξ pb sp = ------, b = ------, (44) experimental data. The perturbative corrections arising P + P + from gluon bremsstrahlung and one-loop effects [13] B B also lead to a nontrivial photon spectrum at the partonic with p + + p + = P + . In the rest frame of B meson, level. Since our primary object here is to discuss nonper- b sp B + turbative effects due to the Fermi motion, we will implic- PB = MB. Then, for the distribution function itly ignore perturbative gluon emission throughout our 2 2 analysis. In this case, the parton matrix element squared 2 1 ψ (ξ, p⊥ ) f (ξ, p⊥ ) = ------(45) is a constant and can be taken out of the integral in (25). 8π 1 Ð ξ ξ ξ ( = b), normalized according to (27), one obtains 4. ANSÄTZE FOR THE LF HEAVY-QUARK 2 DISTRIBUTION 2 ᏺ f (ξ, p⊥ ) = ------Since we do not have an explicit representation for 8π(1 Ð ξ) the B-meson Fock expansion in QCD, we shall proceed (46)  ξ  2  λ 1 Ð ξ 0 p⊥ by making an ansatz for the momentum representation of × exp Ð--------+ ------1 + ------ , the wave function. This is a model-dependent enterprise, ξ ξ 2 2 0 1 Ð  m  but it has close equivalence in studies of B X γ by sp s ξ where 0 = msp/MB. Function (46) is sharply peaked at 4) 2 2 ≈ 2 ξ ξ The true endpoint is actually located at [ MB Ð (mK + mπ) ]/2MB p⊥ = 0, = 0. In what follows, we will refer to the LF ≈ 2.60 GeV, i.e., slightly below MB/2 2.64 GeV. wave function of (46) as the case A.

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Ð3/2 α Ð2 Table 1. Values of the parameters ci (in units of GeV ) and i (in units of GeV ) in (48) for the analytic parametrization of the B-meson wave function corresponding to AL1 and DSR potentials (the wave function is normalized by the condition ∞ ψ2 2 ∫0 (p)p dp = 1) α α α α Model 1 2 3 4 c1 c2 c3 c4 AL1 4.979 2.184 0.638 0.097 4.845 1.839 0.128 0.097 DSR 5.294 2.022 0.758 0.233 1.448 2.303 0.650 0.067

A priori, there is no relation between the ET mined in the previous section. In the case A, we take 2 λ momentum distribution Φ(p ) of a constituent quark mb = 4.8 GeV, msp = 0.3 GeV, and = 2 as reference val- 2 ues. These values are motivated by a study of the b model and LF wave function ψ(x, p⊥ ). However, the c decays [24]. For the models B to D, we use the con- mapping between the variables described above constituent quark masses listed in Table 2. verts a normalized solution of the ET equation of motion into a normalized solution of the different-look- The choice of mb in our approach deserves some ing LF equation [37]. Because the ET function depends comments. The numerical calculations using the con- on the relative momentum, it is more convenient to use stituent b-quark masses show large deviations of the Γ γ Γ ∝ 3 ≈ the quarkÐantiquark rest frame instead of the B-meson (B Xs ) from the free decay rate 0 mb ( 10% rest frame. Recall that these two frames are different in for the cases B to D, see Table 2). This points to the the LF formalism. As a result, one obtains the LF wave appearance of the linear 1/mb corrections to the free function in the form quark limit. The reason is that the constituent quark ψ(ξ, 2 ) (∂ ∂ξ)Φ( 2 2(ξ, 2 )) models usually employ the b-quark masses that are p⊥ = pz/ p⊥ + pz p⊥ . (47) 300Ð400 MeV higher than the pole b-quark mass. This The explicit form of this function is given by formulas fact seems to be a subtlety in applying the constituent (10) of [24]. It is wave functions made kinematically quark model to calculate the nonperturbative correc- γ relativistic in this manner that were used in the recent tions to the B Xs inclusive rate. To overcome the calculation of the Bc lifetime [38]. We calculate the uncertainties induced by the mass of the constituent b photon energy spectra using the three representative LF quark, we use a simple phenomenological recipe that wave functions corresponding to the nonrelativistic considerably improves the situation. Notice that, as in ISGW2 [39], AL1 [40], and relativized DSR [41] con- the ACM model [34], 1/mb corrections can be absorbed stituent quark models. The ISGW2 equal-time function into the deﬁnition of the b-quark mass. We introduce Φ 2 corresponds to the Gaussian distribution (p ) of (6) δ m˜ b = mb + mb (49) conventionally employed in the ACM model with pF = 0.43 GeV. For AL1 and DSR models, we use simple by imposing the condition y (m˜ b ) = x0, where analytic parametrizations of the ET wave functions [42] 1 ψ( p) = c exp(Ðα p2). (48) ( ) ∑ i i y = ∫yRLF y dy. (50) α 0 The parameters ci and i are listed in Table 1. The main difference between the ET wave functions of these models lies in the behavior at a high value of the internal Table 2. Values of the constituent quark mass mb and msp momentum (for further discussion see [38]). We believe (in GeV) for the models A to D [also indicated are the values that the spread of results obtained for these distribution 〈 〉 ACM of m˜ b as defined by the HQET condition x = 0); mb are functions is a fair representation of model dependence 2 resulting from the inclusion of Fermi motion. the average values of the floating mass mf(p )] Model A B C D 5. CHOICE OF m b mb 4.80 5.20 5.227 5.074 Having speciﬁed the nonperturbative aspects of our msp 0.30 0.33 0.315 0.221 calculations, we proceed to present numerical results γ ˜ a) 4.76 4.72 4.76 4.68 for the photon spectrum in the B Xs decay. In this mb section, we analyze the sensitivity of the calculated m˜ b) 4.73 4.68 4.73 4.60 spectra and the energy moments to the choice of the B- b ACM meson radial functions. We will calculate the photon mb 4.73 4.68 4.73 4.60 energy spectra using both expressions (32) and (37) and a) b) 2 Note: Cases and correspond to RLF(y) given by (32) and (37), using the four distribution functions f(ξ, p⊥ ) deter- respectively.

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Table 3. Integrated photon spectra RLF and RACM calculated This condition coincides with that used in HQE to with the use of (32) (case a)) and (37) (case b)) for the wave- define the pole mass of the b quark. As a result, the cor- function models A to D [the integrated spectra R were ACM rection to first order in 1/mb will be eliminated from the calculated by using Eq. (55)] total width in agreement with the general statement of 5) Model A B C D HQE. To illustrate our arguments, we consider the exactly a) RLF(mb) 0.9 0.9 0.9 0.9 solvable case of the photon spectrum of (32) with the b) distribution function given by (46). In this case, we RLF(mb) 0.973 0.900 0.905 0.899 have ( )a) RLF m˜ b 0.995 0.994 0.995 0.99 2 λ ξ 1 Ð ξ ( ˜ )b) ˜f (ξ) = ------exp Ð------0---- + ------, (51) RLF mb 0.987 0.986 0.989 0.974 λξ (λ)  ξ ξ  0K1 2 1 Ð 0 ACM RACM(mb ) 1.008 1.005 1.008 1.003 1 ( ) and simple analytic expressions for RLF = ∫RLF y dy Table 4. Energy moments yn , n = 1, 2, 3, and HQE mo- 0 and y have the form ments 〈x2〉 and 〈x3〉 for cases A to D [the LF photon energy spectra were calculated by using Eq. (32); the scaling vari- 1 ( ξ κ ) ables x and y are defined in Eqs. (9) and (30), respectively] RFL = ---- 1 Ð 0 2 , x0 Model A B C D (52) 1 2 y = ----(1 Ð 2ξ κ + ξ κ ), x 0 2 0 3 y (mb) 0.889 0.822 0.831 0.823 0 ( ) where κ = K (λ)/K (λ). The 1/M correction to R can y m˜ b 0.897 0.888 0.897 0.877 n n 1 B LF be included into the definition of m˜ b . Indeed, neglect- 2( ) 0.806 0.744 0.758 0.740 y mb ξ 2 ≈ ξ κ ing 0 and letting m˜ b (1 Ð 0 2)MB, one obtains y2(m˜ ) 0.813 0.797 0.812 0.783 ᏻ 2 b RLF =1 + (1/MB ). The B-meson mass can be elimi- 3 0.733 0.676 0.693 0.670 nated in favor of the b-quark mass, so we have the y (mb) desired result in the form 3 ( ˜ ) 0.739 0.718 0.737 0.705 2 y mb ᏻ( ) RLF = 1 + 1/mb . (53) 〈x2〉 0.390 0.313 0.316 0.493 For the models B to D, we have no analytic results. 3 〈Ðx 〉 0.377 0.155 0.195 0.415 However, the numerical effect of introducing m˜ b is the same (see Table 3). We have calculated numerically the

values of m˜ b in different models using both (32) and n Table 5. Energy moments y , n = 1, 2, 3, and HQE mo- (37) for R (y). Although m˜ depends on the assumed 〈 2〉 〈 3〉 LF b ments x and x for cases A to D [the LF photon energy shape of distribution, this dependence is marginal: the spectra were calculated by using Eq. (37); the scaling variables x and y are defined in Eqs. (9) and (30), respectively] uncertainty on m˜ b is between 4.6 and 4.7 GeV depend- ξ 2 Model A B C D ing on the choice of f( , p⊥ ) (see Table 2). These values are consistent with the b-quark pole mass mb, pole = y (mb) 0.873 0.806 0.818 0.795 4.8 ± 0.15 GeV [43]. ( ) y m˜ b 0.885 0.873 0.885 0.851 6. PHOTON ENERGY SPECTRA 2( ) 0.787 0.726 0.742 0.709 y mb AND THE ENERGY MOMENTS y2(m˜ ) 0.798 0.778 0.796 0.751 We study ﬁrst the photon spectra using (32) and (37). b Our results for the integrated photon spectra and energy y3 (m ) 0.712 0.655 0.676 0.636 moments are presented in Fig. 1 and in Tables 3Ð5. The b different curves in Fig. 1 correspond to the models A to 3( ) D. For each case, we show separately the spectra calcu- y m˜ b 0.722 0.695 0.718 0.669 lated from (32) and (37) by using the corresponding 〈x2〉 0.378 0.295 0.302 0.456 5) 3 In this way, the bound-state effects are largely compensated by 〈Ðx 〉 0.348 0.130 0.176 0.348 the shift in the b-quark mass.

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Γ Γ –1 Γ Γ –1 values of m˜ b and the photon energy spectra calculated (1/ 0)d /dE, GeV (1/ 0)d /dE, GeV in the ACM model (see below). The influence of the 3 various choices of the distribution function and the b- A B quark mass can be seen from Table 3, where we show 3 both RLF (mb), RLF(m˜ b ) and analogous quantities RACM 2 for the ACM model. In Tables 4 and 5, we show the 2 average photon energy y normalized to MB /2, and the 2 3 1 moments y and y . We present the results obtained 1 using both (32) (Table 4) and (37) (Table 5). The replacement mb m˜ b modifies the predictions for 0 0 the total rate and the moments y and y2 by about 8Ð9% for the models BÐD, while it changes the corresponding C D quantities by less than 1.5% for the model A. Our final 3 results for the integrated photon spectra RLF(m˜ b) agree 2 well with the corresponding OPE prediction [9] ROPE = δ δ λ 2 1 + RAD, where RAD has been defined in (22). For 1 = − ± 2 λ ± 2 0.3 0.2 GeV and 2 = 0.12 0.02 GeV , ROPE changes 1 ± the free quark result by a few percent, ROPE = 0.975 1 0.005. Note also that the sensitivity of RLF to the functional form chosen for the distribution function is small. This behavior agrees again with that based on the global 0 0 quarkÐhadron duality that ensures insensitivity of even 1.8 2.2 2.6 1.8 2.2 2.6 E, GeV partially integrated quantities to the bound state effects. Note that the effect of the transverse momentum is con- Fig. 1. Theoretical predictions for the photon energy spec- siderably smaller than the model-dependent uncertainty. trum using the LF quark models described in the text. Each n plot shows the spectra calculated by using the b-quark The dependence of the energy moments y on mb is masses m˜ b from (32) (solid lines) and (37) (dashed lines). rather weak, in contrast to that of the QCD moments of The dash-dotted lines show the photon energy spectra calcu- the shape function lated in the ACM model. The curves A to D show the results for the corresponding models. 1 〈 xn〉 = ∫ xnF(x)dx, (54) f Λ Ð2MB/ where mb is the floating b-quark mass and 2 2 which are very sensitive to the difference between MB 2 1m  p (ε) = ------sp-- Ð ε , ε = M (1 Ð y). (56) and mb. In particular, change of the b-quark mass from 0 4 ε  B m to m˜ modifies 〈x2〉 and 〈x3〉 drastically. Note that the b b We have used the momentum distributions Φ(p2) of the 〈 2〉 〈 3〉 resulting values of x and x are still considerably spectator quark for the models A to D. In all cases, we model dependent. Our predictions for 〈x2〉 are relatively have found that the spectra calculated in ACM and LF small, although in agreement (for the model D) with the parton models are almost identical. This is not surpris- result of [20] and compatible with the results obtained ing because we have checked numerically that the 〈 2〉 ≈ from the QCD sum rules; x 0.5. This means that the quark masses m˜ defined using the LF models practi- LF ansatz can be made consistent with the QCD b description, provided the spectator quark is relativistic. cally coincide with the values of the floating b-quark f Φ 2 This conclusion agrees with the similar conclusion mass mb averaged over the distribution (p ) (see obtained in the ACM model in [34]. Table 2). The integrated energy spectra RACM for the In order to compare our results with those derived in models A to D are presented in Table 3. They coincide the ACM model, we have calculated the inclusive with RLF within a percent accuracy. The energy γ B Xs photon spectra in a simplified ACM model moments for the ACM model are shown in Table 6. [34] assuming the monochromatic distribution (8) for Although in this paper we do not consider perturba- the free b quark. In this case, one easily obtains tive corrections, it is instructive to compare theoretical ∞ Γ predictions for the Doppler shifted spectrum dB/dElab d ACM ∼ Γ 2Φ( 2)( f ( 2))3 in the laboratory frame with the CLEO data. To per------0 ∫ dp p mb p , (55) dEγ form a fit to the data, we rebin the boosted photon spec- 2(ε) p0 tra in the same energy intervals as used by CLEO and,

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2154 KULIKOV, NARODETSKIŒ for each choice of the distribution function, adjust the decays can be studied. The invariant mass MH of the overall normalization to provide the best fit to the data. hadronic final state is related to the scaling variable y by The results are reported in Table 7, and the best fits are 2 2 χ2 Ӷ MH = MB (1 Ð y). Therefore, the theoretical results for displayed in Fig. 2. All fits have /ndof 1, indicating the present accuracy of experiment. Averaging over the the photon spectrum can be translated into predictions models, we obtain for the hadronic mass spectrum. The Xs mass distribution from the reconstruction analysis of CLEO inclu- ( γ ) Br B Xs sive data shows a clear K* peak followed by a dip (57) which is expected since the next excited-kaon reso- = (2.5 ± 0.5(exp.) ± 0.3(model)) × 10Ð4, nance is K1(1270). A broad enhancement at and above where the last error originates from the model depen- K1(1270) is observed; this is also expected since many dence. This result is consistent with both the updated resonance states exist in this region. The present exper- CLEO value (2), and the recent reanalysis [20] imental statistics are insufficient to identify any of the γ ± +0.37 × –4 individual resonances beyond K*(892). Combining the Br(B Xs ) = (2.62 0.60(exp.)Ð0.30 (theor.)) 10 of exclusive and inclusive measurements, CLEO deter- the GLEO data. mines Br(B K*(892)) 7. INVARIANT-MASS SPECTRUM RK*(892) = ------(------γ----)------Br B Xs (58) In addition to the photon energy spectrum, the = (18.1 ± 6.8)%. invariant hadronic mass distribution in radiative B Figure 3 shows the invariant mass distribution of the –4 –1 –4 –1 dB/dElab, 10 GeV dB/dElab, 10 GeV hadrons recoiling against the photon for the models AÐ 10 D. Our predictions for hadronic mass spectra must be A 8 B understood in the sense of quarkÐhadron duality. The 2 ∝ 2 8 true hadronic mass spectrum for low MH MK* has a resonance structure that looks rather different from our 6 predictions. A realistic model for the hadronic mass 6

4 Γ Γ –1 Γ Γ –1 4 (1/ 0)d /dMH, GeV (1/ 0)d /dMH, GeV A B 2 2 0.8 0.8 0.6 0 0 0.6 0.4 10 8 0.4 C D 0.2 8 0.2 6 0 0 6 4 0.8 C D 4 0.6 0.6 2 2 0.4 0.4 0 0 0.2 2.0 2.4 2.8 2.0 2.4 2.8 0.2 Elab, GeV

Fig. 2. Theoretical predictions for the photon energy spectrum in the laboratory frame for the different distribution 0 1 2 3 0 1 2 3 M , GeV ξ 2 H functions f( , p⊥ ) and the corresponding m˜ b. The notation is the same as in Fig. 1. The experimental data are the CLEO Fig. 3. Comparison of the theoretical predictions for the results [46]. Both the left-hand plot and the right-hand plot invariant hadronic mass spectrum for the LF and ACM mod- show the results of the best ﬁt reported in Table 7. els. The notation is identical to that in Fig. 1.

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Table 6. Energy moments yn , n = 1, 2, 3, and HQE moments (and model-dependent) correction of order 1/mb, but at the expense of introducing the shift in the constituent 〈x2〉 and 〈x3〉 for cases A to D [the photon energy spectra were calculated by using Eq. (55); the scaling variables x and y are quark mass which substantially compensates the bound defined in Eqs. (9) and (30), respectively] state effects. A summary of our results presented in Tables 3Ð5 shows a fairly good agreement with both the Model A B C D QCD results and the available experimental data. We 0.903 0.894 0.901 0.883 have also found that the photon energy spectra calcu- y lated in our LF partonlike approach agree well with the y2 0.817 0.800 0.813 0.786 spectra obtained in the ACM model, provided the same ET distribution function Φ(p2) is used as input in both 3 0.743 0.718 0.737 0.706 y cases. Finally, we note that it would be interesting to 〈x2〉 0.355 0.308 0.317 0.489 check whether the effective values of the b-quark mass 3 〈Ðx 〉 0.289 0.141 0.206 0.427 m˜ b are approximately the same for different channels (b c vs. b u or b s) and for different beauty Table 7. Branching ratios Br(B X γ) obtained from a fit hadrons. This work is in progress, and the results will s be reported elsewhere. to the CLEO data and partial fractions RK*(892)

Model A B C D ACKNOWLEDGMENTS Γ Γ × 4 0/ B 10 2.54 2.59 2.50 2.91 This work was supported in part by the INTAS a) RK*(892) 0.192 0.158 0.174 0.218 (grant no. 96-155). b) RK*(892) 0.168 0.131 0.149 0.181 a) b) Note: Cases and correspond to RLF(y) given by Eqs. (32) and (37), respectively. REFERENCES 1. J. Ellis et al., Nucl. Phys. B 131, 285 (1977); T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); Erra- spectrum consists of a single peak located at the mass tum: 65, 1772 (1981); B. A. Campbell and P. J. O’Don- K*(892) followed by a continuum (above a threshold nell, Phys. Rev. D 25, 1989 (1982); N. G. Deshpande and G. Eilam, Phys. Rev. D 26, 2463 (1982). value Mth) which is given by the inclusive spectrum and is dual to a large number of overlapping resonances. In 2. A. Masiero, in Proceedings of 8th Symposium on Heavy Flavour, Southampton, UK, 1999. Table 7, we show the ratios RK*(892) obtained by the integration of the inclusive spectrum in the range M ≤ M . 3. K. Lingel, T. Skwarnicki, and J. G. Smith, Annu. Rev. H th Nucl. Part. Sci. 48, 253 (1998). The result crucially depends on the choice of Mth; we use a value of M = 1.15 GeV adopted in [20]. Averag- 4. CLEO Collab. (R. Ammar, S. Ball, P. Barnes, et al.), th Phys. Rev. Lett. 71, 674 (1993). ing over the different model predictions, we obtain +0.24 5. CLEO Collab. (M. S. Alam, I. J. Kim, Z. Ling, et al.), RK*(892) = 0.157Ð0.44 . This result agrees with both exper- Phys. Rev. Lett. 74, 2885 (1995). imental data and theoretical estimates based on QCD 6. T. Skwarnicki, Talk presented at XXIX International sum rules [44] and lattice QCD calculation [45]. Conference on High Energy Physics, Vancouver, BC, Canada, 1998, No. 702. 7. ALEPH Collab. (R. Barate et al.), Phys. Lett. B 429, 169 8. CONCLUSION (1998). 8. CLEO Collab. (S. Ahmed et al.), hep-ex/9908022. We have derived a new parton formula that establishes a simple relation between the electroweak decay 9. A. F. Falk, M. Luke, and M. Savage, Phys. Rev. D 49, Γ γ 3367 (1994). rate (B Xs ) and the rate of free b-quark decay. The main features of our approach are the treatment of 10. A. Kapustin and Z. Ligeti, Phys. Lett. B 355, 318 (1995). the b quark as an on-mass-shell particle and the inclu- 11. A. Ali and E. Pietarinen, Nucl. Phys. B 154, 512 (1979). sion of the effects arising from the b-quark transverse 12. G. Altarelli, N. Cabibbo, G. Corbo, et al., Nucl. Phys. B motion in the B meson. Formulas (32) and (37) present 202, 512 (1982). our main result. Using various b-quark distribution 13. A. Ali and C. Greub, Z. Phys. C 49, 431 (1991); 60, 433 functions, we have calculated the photon energy spec- (1993); Phys. Lett. B 259, 182 (1991); 287, 191 (1992); tra and the corrections to the free decay rate. We have 361, 146 (1995); N. Pott, Phys. Rev. D 54, 938 (1996). shown that the decay width has no corrections linear to 14. J. Chay, H. Georgy, and B. Grinstein, Phys. Lett. B 247, 1/mb only if it is expressed not in terms of the constitu- 399 (1990); I. I. Bigi, N. G. Uraltsev, and A. I. Vainsh- ent quark mass, but in terms of a mass m˜ which is tein, Phys. Lett. B 293, 430 (1992); Erratum: 297, 477 b (1993); I. I. Bigi, M. Shifman, N. G. Uraltsev, and deﬁned in the way similar to that used in the HQE to A. I. Vainshtein, Phys. Rev. Lett. 71, 496 (1993); A. V. Ma- deﬁne mb, pole. In this way, one avoids an otherwise large nohar and M. B. Wise, Phys. Rev. D 49, 1310 (1994).

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PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

ELEMENTARY PARTICLES AND FIELDS Theory

Nonforward Color-Octet Kernel of the BalitskyÐFadinÐKuraevÐLipatov Equation* V. S. Fadin** and D. A. Gorbachev*** Budker Institute of Nuclear Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrentieva 11, Novosibirsk, 630090 Russia, and Novosibirsk State University, Novosibirsk, Russia Received February 11, 2000

Abstract—The contribution to the kernel of the nonforward BalitskyÐFadinÐKuraevÐLipatov equation from two-gluon production is calculated for the case of the antisymmetric color-octet state of two Reggeized gluons in the t channel. The one-gluon contribution to the kernel in the one-loop approximation is also obtained by using the one-loop expression for the effective vertex of the one-gluon production in ReggeonÐReggeon collisions. An explicit form of the total kernel is presented. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Both goals—a stringent test of gluon Reggeization and an examination of the calculations—can be solved The most common basis for describing processes at 2 2 simultaneously by the check of the bootstrap equations small values of x = Q /s (Q is a typical virtuality and [13, 14] appearing as the requirement of the compati- s is the c. m. s. energy) within perturbative QCD is bility of gluon Reggeization with s-channel unitarity. In the BalitskyÐFadinÐKuraevÐLipatov (BFKL) equation fact, the BFKL equation is the equation for the Green’s [1], originally derived in the leading-logarithm approx- function of two Reggeized gluons. In the color-singlet imation (LLA), which means resummation of all terms state, these Reggeized gluons create a Pomeron. Self- α n α 2 π of the type [ slns] [ s = g /(4 ) is the QCD coupling consistency requires that, in the antisymmetric color- constant]. The calculation of radiative corrections to octet state, the two Reggeized gluons reproduce a the kernel of this equation took many years of hard Reggeized gluon itself (bootstrap condition). The work [2Ð7]. Recently, the kernel was obtained in the above statements are valid in the NLA, as well as in the next-to-leading approximation (NLA) [8] for the case LLA. Along with the stringent test of gluon Reggeiza- of forward scattering—that is, for zero momentum tion, the check of the bootstrap equations provides a transfer (t = 0) and the vacuum quantum numbers in the global test of the calculations of the NLA kernel, t channel. In the MS renormalization scheme with a because these equations contain almost all quantities reasonable scale setting, corrections appear to be large. appearing in the calculations. At present, this problem is being widely discussed in In the BFKL approach, amplitudes of high-energy the literature [9]. In this situation, it is very important to processes are expressed in terms of the aforementioned be sure that both the basic hypotheses used and the cal- Green’s function and impact factors of scattered parti- culations performed in deriving the equation are cor- cles, which are defined by ReggeonÐparticle scattering rect. amplitudes. The nonforward impact factors for gluon It should be recalled that the derivation of the BFKL [15] and quark [16] scattering were recently calculated equation (in the NLA, as well as in the LLA) is based on in the NLA, and fulfillment of the bootstrap conditions one of the remarkable properties of QCD—gluon for them was demonstrated [15Ð17] for both helicity- Reggeization [10], which was proved in the LLA [1, 11]. conserving and helicity-nonconserving parts, in an In the NLA, this property was only checked in the first arbitrary spacetime dimension D = 4 + 2⑀. three orders of perturbation theory [6]. Since gluon Reggeization forms a basis for deriving the BFKL equa- The quark contribution to the nonforward BFKL tion, it is clear that more powerful tests are necessary. kernel was also calculated [18], and fulfillment of the bootstrap conditions for the kernel in the part associ- As for the calculations of radiative corrections to the ated with this contribution was explicitly demonstrated kernel, they are very complicated, and only some of them in the NLA [18, 19], in an arbitrary spacetime dimen- has been independently performed [7] or checked [12]. sion as well. Only one (but the most complicated) boot- Therefore, the calculations must be carefully verified. strap condition remains unchecked—for the gluon part * This article was submitted by the authors in English. of the kernel. In this study, we calculate the gluon con- ** e-mail: [email protected] tribution to the nonforward color-octet kernel of the *** e-mail: [email protected] BFKL equation.

2158 FADIN, GORBACHEV The significance of the nonforward octet kernel is dimension D = 4 + 2⑀ is taken different from 4 to regu- not exhausted by the check of the bootstrap condition. larize the infrared divergences. We use the normaliza- The kernel of the nonforward BFKL equation for an tion adopted in [13]. arbitrary color state in the t channel is expressed in The nonforward kernel, as well as the forward one, terms of the gluon Regge trajectory and the part related is given by the sum of the “virtual” part, defined by the to real particle production in ReggeonÐReggeon scat- gluon trajectory j(t) = 1 + ω(t), and the “real” part tering (“real” part, for brevity). The trajectory is known ᏷ (᏾) [4] and enters into the kernel in the universal (not r , related to the real particle production in the depending on a color state) way [13]. The “real” part ReggeonÐReggeon collisions includes contributions from one-gluon, two-gluon, and quarkÐantiquark pair production. The last two contri- ᏷(᏾)( , ) [ω( 2) ω( 2)] q1 q2; q = Ðq1 + Ðq1' butions can be separated (for an arbitrary color state in (2) the t channel) in two pieces, one of which being deter- (᏾) × 2 2δ(D Ð 2)( ) ᏷ ( , ) mined by the color-octet state. Therefore, the color- q1 q1' q1 Ð q2 + r q1 q2; q . octet piece enters (with some group coefficient) into As is seen from (2), the gluon trajectory enters the kernels for other color states—in particular, for the ᏾ color-singlet state (Pomeron channel). In the Pomeron equation in the universal (independent from ) way. In channel, the nonforward BFKL equation can be used the one-loop approximation (LLA), the trajectory is directly to describe experimental data. Evidently, the purely gluonic: applicability range of this equation is much wider than 2 D Ð 2 (1) g Nt d q the forward-case one. ω (t) = ------1, (3) ( π)D Ð 1∫ 2 2 The ensuing exposition is organized as follows. In 2 2 q1 q1' Section 2, we present a general form of the gluon contribution to the kernel and an explicit form of the gluon where t = q2 = Ðq2, and N is the number of colors (N = piece of the gluon trajectory. In Section 3, we derive the 3 in QCD). In the NLA, the trajectory was calculated in gluon part of the contribution to the kernel from one- [4]. Since the quark contribution to the nonforward ker- gluon production. In Section 4, we consider two-gluon nel was already considered [18], we present here the production in collision of two Reggeized gluons. The two-loop gluon contribution: contribution of this process to the kernel and the result 2 for the total gluon contribution to the kernel are pre- ω (2)( ) g t G t = ------sented in Sections 5 and 6, respectively. (2π)D Ð 1 (D Ð 2) (4) × d q1[ ( , ) ( , )] 2. DEFINITIONS AND BASIC EQUATIONS ∫------FG q1 q Ð 2FG q1 q1 , 2 '2 In the BFKL approach, the high-energy-scattering q1 q1 amplitudes are expressed in terms of the impact factors Φ of the scattering particles and of the Green’s function where

G for the scattering of Reggeized gluons [13]. Consid- (D Ð 2) ering the Green’s function, we can take, without any g2 N2q2 d q  q2  F (q , q) = Ð------2---- ln------ loss of generality, masses of the colliding particles with G 1 D Ð 1∫ 2 2 2 4(2π) q (q Ð q) (q Ð q )  2 2 2 2 1 2 momenta pA and pB equal to zero: pA = pB = 0 (pA + p )2 = 2(p p ) = s. As usual, in an analysis of high- ψ( ) ψ D ψD  ψ( ) B A B Ð 2 D Ð 3 Ð 3 Ð ---- + 2 ---- Ð 2 + 1 energy processes, we use the Sudakov decomposition 2 2 for particle momenta. The Mellin transform of the (5) Green’s function with the initial momenta of the 2 D Ð 2 Ӎ β + ------+ ------; Reggeized gluons in the s channel q1 pA + q1⊥ and (D Ð 3)(D Ð 4) ( )( ) − Ӎ α Ӎ 4 D Ð 1 D Ð 3 q2 pB Ð q2⊥ and the momentum transfer q q⊥ obeys the equation [13] Γ'(x) ψ(x) = ------, Γ(x) ω (᏾)( , , ) 2 2δ(D Ð 2)( ) Gω q1 q2 q = q1 q1' q1 Ð q2 and Γ(x) is the Euler gamma function. In Eqs. (3)Ð(5) D Ð 2 (1) d r᏷(᏾)( , ) (᏾)( , ) and below, g is the bare coupling constant related to the + ∫-----2------2--- q1 r; q Gω r q2; q , r r' renormalized coupling gµ in the MS scheme by the relation where ᏾ denotes the representation of the color group in the t channel. The transverse momenta are spacelike 2 µÐ⑀ 11 2n f gµ and we use the bold font for them. Here and below, we g = gµ 1 + ----- Ð ------ ----- , (6) use for brevity v ' ≡ v Ð q for any v. The spacetime 3 3 N 2⑀

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NONFORWARD COLOR-OCTET KERNEL 2159 where nf is the number of the quark ﬂavors, The remarkable properties of the kernel are (᏾) (᏾) 2 ᏷ ( , ) ᏷ ( , ) 2 gµ NΓ(1 Ð ⑀) r 0 q2; q = r q1 0; q gµ = ------⑀------. (7) (11) ( π)2 + ᏷ (᏾)( , ) ᏷ (᏾)( , ) 4 = r q q2; q = r q1 q; q = 0 Let us stress that in this paper we will systematically and use the perturbative expansion in terms of the bare cou- (᏾) (᏾) pling g. ᏷ (q , q ; q) = ᏷ (q' , q' ; Ðq) r 1 2 r 1 2 (12) The “real” part for the nonforward case in the LLA (᏾) ᏷ ( , ) is [1, 21] = r Ðq2 Ðq1; Ðq . The properties in (11) mean that the kernel turns into 2  2 2 2 2  (᏾)B g c q q' + q q' 2 zero at zero transverse momenta of the Reggeons and ᏷ (q , q ; q) = ------R---------1------2------2------1-- Ð q  , (8) r 1 2 ( π)D Ð 1 ( )2  follow from the gauge invariance; expressions (11) are 2 q1 Ð q2 the consequences of the symmetry of the imaginary part of the ReggeonÐReggeon scattering amplitude (see where the superscript B means the LLA (Born) approx- below (13)). imation and the color group coefﬁcients cR for the singlet (R = 1) and octet (R = 8) cases are Using the operator ᏼˆ ᏾ for the projection of the two-gluon color states in the t channel on the irreduc- N ible representation ᏾ of the color group, we can repre- c1 = N, c8 = ----. (9) 2 sent the imaginary part of the ReggeonÐReggeon scattering amplitude entering (10) in the form In the NLA, the “real” part of the kernel can be presented as [13] (᏾) 〈 ' |ᏼˆ | ' 〉 Ꮽ ( , ) c1c1 ᏾ c2c2 (᏾) Im RR q1 q2; q = ------᏷ ( , ) 2n᏾ r q1 q2; q (13) ∞ × γ { f } ( , )(γ { f } ( , ))* ρ ∑ c c q1 q2 q1' q2' d f . dsRR (᏾) ∫ 1 2 c' c' Ꮽ ( , )θ( ) { } 1 2 = ∫------Im RR q1 q2; q sΛ Ð sRR f (2π)D 0 Here, n᏾ is the number of independent states in the rep- D Ð 2 (10) resentation ᏾, the sum {f} is performed over all states 1 d r᏷ (᏾)B( , )᏷ (᏾)B( , ) Ð --∫-----2------2--- r q1 r; q r r q2; q f which are produced in the ReggeonÐReggeon colli- 2 r r' sions and over all discrete quantum numbers of these { f }  2  states. γ (q , q ) is the effective vertex for the pro- sΛ c1c2 1 2 × ln------ . ρ ( )2( )2 duction of the state f, and d f is the phase space element r Ð q1 r Ð q2 for this state,

(᏾) 1 D (D)  Here, Ꮽ (q , q ; q) is the scattering amplitude of the dρ = ----(2π) δ q Ð q Ð ∑ k RR 1 2 f n!  1 2 i β i ∈ f Reggeons with the initial momenta q1 = pA + q1⊥ and α (14) Ðq2 = pB Ð q2⊥ at the momentum transfer q = q⊥ and D Ð 1 ᏾ × d ki the representation of the color group in the t channel, ∏------D----Ð---1------, 2 αβ 2 ∈ (2π) ⋅ 2⑀ and sRR = (q1 Ð q2) = s Ð (q1 Ð q2) is the squared i f i invariant mass of the Reggeons. The sRR-channel imag- where n is the number of identical particles in the state Ꮽ (᏾) f. In the LLA, only the one-gluon production does con- inary part Im RR (q1, q2; q) is expressed in terms of the effective vertices for the production of particles in tribute in (13), and Eq. (10) gives for the kernel its LLA the ReggeonÐReggeon collisions [13]. The second term value (8); in the NLA, the contributing states include on the right-hand side of Eq. (10) serves for the subtrac- also the two-gluon and the quarkÐantiquark states. The tion of the contribution of the large s region in the first normalization of the corresponding vertices is defined RR in [13]. term, in order to avoid a double counting of this region ᏾ in the BFKL equation. The intermediate parameter sΛ The most interesting representations are the color singlet (Pomeron channel) and the antisymmetric color in (10) must be taken tending to infinity. At large sRR, only the contribution of the two-gluon production does octet (gluon channel). We have for the singlet case survive in the first integral, so that the dependence of sΛ δ δ ' c c' 〈 ' |ᏼˆ | ' 〉 c1c1 2 2 disappears in (10) due to the factorization property of c1c1 1 c2c2 = ------2------, n1 = 1, (15) the two-gluon production vertex [13]. N Ð 1

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2160 FADIN, GORBACHEV and for the octet case 2 2 2 2 ( ) q q q  G 11 1 2  1 k f c c' c f c c' c f 3 = ------ln ----- + ----, 〈c c' |ᏼˆ |c c' 〉 = ------1---1------2----2--, n = N2 Ð 1, (16) 3 (q 2 Ð q 2) q 2 6 1 1 8 2 2 N 8 1 2 2 ζ where fabc are the structure constants of the color group. where (n) is a Riemann zeta function. Note that the one-loop contribution to the RRG vertex is not known for arbitrary ⑀. Therefore, Eqs. (19), contrary to all the 3. THE ONE-GLUON CONTRIBUTION preceding formulas, are valid only in the limit ⑀ 0. TO THE KERNEL The only term of these equations which remains unex- The gluon contribution to the ReggeonÐReggeonÐ panded in ⑀ is (k2)⑀. For this term, the expansion is not gluon (RRG) vertex was calculated in [3]. Recall that performed because the RRG vertex is singular at k2 = 0, the complicated analytic structure [20, 3] of the vertex and in subsequent integrations of its contribution to the is irrelevant in the NLA, where only the real parts of the kernel the region ⑀|ln(k2)| ~ 1 does contribute. In (19), production amplitudes do contribute (only these parts all terms are retained, which gives, after these integra- interfere with the LLA amplitudes, which are real). tions, contributions nonvanishing in the limit ⑀ 0. Recall also that, in the NLA, the vertex depends on the energy scale sR used in the Regge factors. In (10) and The vertex (17) is explicitly invariant under the 2 (13), it was assumed that sR = k , where k is the pro- gauge transformation duced gluon momentum. Neglecting the imaginary µ µ µ part, we have for the gluon contribution to the RRG e (k) e (k) + k χ, (20) vertex with this choice of s [21] R so that we can use the relation

G d  µ γ ( , ) ( ) ( , ) (λ) (λ) c c q1 q2 = gT c c eµ* k C q2 q1 -- ( ) ( ) 1 2 1 2  ∑eµ* k eν k = Ðgµν. (21) λ 2 Γ( ⑀) × 2g N 1 Ð ( (G) (G)) Substituting (17) into (13) and using (21) for the sum 1 + ------⑀------f 1 + f 2 (4π)2 + over polarizations and the relations (17) 2 δ δ d ( d ) ( 2 )  pA pB  g NΓ(1 Ð ⑀) ' ' T c c T ' ' * = N N Ð 1 , + ------Ð ------c1c1 c2c2 1 2 c1 c2 ( ) ( ) 2 + ⑀ k pA k pB µ (4π) (22) f f T d (T d )* = N2(N2 Ð 1)/2 c c'c c c' c c1c2 c' c'  1 1 2 2 1 2 × [ (G) ( 2 2 2) (G)] -- f 3 Ð 2k Ð q1 Ð q2 f 2 .  for the sum over color indices, with the help of Eqs. (18), (19), and (14)Ð(16), we obtain from (10) the gluon part Here, d is the color index of the product gluon, eµ(k) is of the contribution to the kernel from the one-gluon d production in ReggeonÐReggeon collisions: its polarization vector, T = Ði f are the matrix c1c2 dc1c2 elements of the color-group generator in the adjoint  2  2 2 2 2  representation, k = q Ð q , G(᏾) g c᏾  q q' + q' q 2 1 2 ᏷ ( , ) ------ 1 2 1 2  RRG q1 q2; q = D Ð 1 ------2------Ð q  2 (2π)  k  ( , ) ( ) 2q1  C q2 q1 = Ð q1⊥ Ð q2⊥ + q1 Ð q1⊥ 1 Ð ------  k2  (18)  2   1 g NΓ(1 Ð ⑀) 2 ⑀ 2 2 2 × ( ) π ⑀ζ( )   --- + ------⑀----- Ð k ---- Ð + 4 3  ( ) 2q2  ( π)2 +  ⑀2  + q2 Ð q2⊥ 1 Ð ------ , 2 2 4  k2   2  2  '2 '2 2 2  2  2 2 q1 g NΓ(1 Ð ⑀) q1 Ð q2 (G) (q + q ) q 2 q -----  ------ 11 1 2 1 1 1 Ð ln 2 + 2 + ⑀ ------2------2-- 2 f 1 = ------ln----- Ð -- ln -----    ( π)  6 ( 2 2)  2 2  2 q2 6 4 q1 Ð q2 q1 Ð q2 q2 q2 2   k 2 2 2 2 ⑀ 1 Ð ------(q + q + 4q' ⋅ q' Ð 2q ) (23) Ð (k ) ---- Ð 3ζ(2) + 2⑀ζ(3) , 2 2 2 1 2 1 2 2 ( ) ⑀  q1 Ð q2 (19) 2 2 2 2 2 2 2 2  2 2q q q  2q q (G) k 2 2 q1 q2 q1 × 1 2 ----1- 2 2 1 2 2 f = ------q + q Ð 2------ln----- , ----2------2 ln 2 Ð q1 Ð q2 + 11 ----2------2 2 2 2 2 1 2 2 2 2   ( ) ( )   q1 Ð q2 q2 q1 Ð q2 3 q1 Ð q2 q1 Ð q2 q2

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NONFORWARD COLOR-OCTET KERNEL 2161

 we can present the polarization vectors as 2 2 2 2 2 2  2  q q' Ð q' q q + q 2 q  ( ( )) + ----1------2------1------2- Ð ----1------2-q ln----1- Ð 2q' ⋅ q'   ( ) ( ) k1e⊥ k1 2 2 2  2 1 2  e k1 = e⊥ k1 Ð ------p1, k q1 Ð q2 q2  k1 p1  (30) ( ( )) k2e⊥ k2 e(k ) = e⊥(k ) Ð ------p , 2 2 2 k p 2 g c᏾ { } 2 2 + ------qi qi' . D Ð 1 α α (2π) 1 2 and their convolution with the tensor A (k1, k2) as G(᏾) α α ᏷ 1 2 The symmetry properties (12) of RRG (q , q ; q) eα* (k )eα* (k )A (k , k ) 1 2 1 1 2 2 1 2 are evident from (23). The properties in (11) are not so (31) α α 1 2 evident, but can be easily checked. ≡ 4e⊥*α (k )e⊥*α (k )c (k , k ). 1 1 2 2 1 2 µν The tensor c (k1, k2) in the transverse space was 4. THE TWO-GLUON PRODUCTION defined in [6]. It can be presented in the form µ ν Let us consider the production of two gluons with µν (q Ð k )⊥ (q Ð k )⊥ c (k , k ) = ------1------1------1------1------momenta k1 and k2 in collisions of two Reggeons with 1 2 ˜t1 momenta q1 and Ðq2. We will use the Sudakov parametrization for the product-gluon momenta: ( ) µ β ν α µ ( ) ν q1 Ð k1 ⊥     q1 Ð k1 ⊥ Ð ------k Ð -----1k + k Ð -----2k ------β α 2  1 β 2  2 α 1 2 ki = i pA + i pB + ki⊥, k 2 ⊥ 1 ⊥ k α β 2 2 , µ ν µ ν s i i = Ðki⊥ = ki , i = 1 2, k ⊥ k ⊥ t k ⊥ k ⊥ t (24) + ---1------1------2---- + ---2------2------1---- (32) β β β α α α 2 α β 2 αβ 1 + 2 = , 1 + 2 = , k s 1 k s 2 µ ν µ ν k ⊥ + k ⊥ = q ⊥ Ð q ⊥, 1 2 1 2 k1⊥ k2⊥  ˜t1  k2⊥ k1⊥ 1 µν ˜t1 Ð ------1 + ------+ ------Ð --g⊥ 1 + ---- 2  α β  2  2 and the notation k s 1 2 k 2 k α β α β α β α β k = k + k = q Ð q , s 1 2 s 2 1 s 1 2 2t1 1t2 1 2 1 2 (25) + ------Ð ------+ ------Ð ------Ð ------, ˜ 2 2 2 2  2 t1 k k αk βk so that sRR = k . For the effective vertex of the two-gluon production in the ReggeonÐReggeon collision, we have where the notation 2 2 t = q ⊥ , i = 1, 2; ˜t = (q Ð k ) (33) γ GG ( , ) 2 ( ) ( ) i i 1 1 1 c c q1 q2 = g eα* k1 eα* k2 1 2 1 2 (26) µν d d α α d d α α is used, and g⊥ is the metric tensor in the transverse × [T 1 T 2 A 1 2(k , k ) + T 2 T 1 A 2 1(k , k )], c1 j jc2 1 2 c1 j jc2 2 1 plane, µ ν µ ν µν µν where di are the color indices of the produced gluons, pA pB + pB pA µ g⊥ = g Ð ------(------)------. (34) and e (ki) are their polarization vectors. The tensor pA pB α α 1 2 Since different gauges are used for each of the pro- A (k1, k2) obtained in [2] satisfies the transversality conditions duced gluons, to obtain the contribution of the second term in (26) one has to change not only k1 k2 in the α α 1 2 contribution of the first term, but also the gauges k Aα α (k , k ) = k Aα α (k , k ) = 0. (27) 1 1 2 1 2 2 1 2 1 2 e(k)p1 = 0 e(k)p2 = 0, so that α α Due to these conditions, the two terms in (26) are sep- ( ) ( ) 2 1( , ) eα* k1 eα* k2 A k2 k1 arately invariant with respect to independent gauge 1 2 (35) α β α β transformations of the gluon polarization vectors 1 1 2 2 = 4e⊥*α (k )e⊥*α (k )Ω (k )Ω (k )cβ β (k , k ), 1 1 2 2 1 2 2 1 2 1 α( ) α( ) αχ e ki e ki + ki i, (28) where so that we can use different gauges for each of the pro- α β Ωαβ( ) αβ k⊥ k⊥ duced gluons and for each of the terms. Choosing k = g⊥ Ð 2------2----. (36) k⊥ ( ) α ( ) α eα k1 k1 = eα k1 p1 = 0, In order to calculate the two-gluon contribution to (29) the imaginary part of the ReggeonÐReggeon scattering ( ) α ( ) α eα k2 k2 = eα k2 p2 = 0, amplitude (13) entering Eq. (10) for the kernel, we need

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2162 FADIN, GORBACHEV to sum over gluon polarizations and color indices. The has the same form for both choices: first sum can be obtained using the relation D Ð 2 dk2 dx d k α(λ) β(λ) αβ ------dρ = ------1--- ( ( ))* ( ) π GG ( ) (D Ð 1) ∑ e⊥ k e⊥ k = Ðg⊥ , (37) 2 4x 1 Ð x (2π) λ D Ð 2 and the second, for the most interesting singlet and dy d k (44) = ------2--- , octet representations, with the help of ( ) (D Ð 1) 4y 1 Ð y (2π) δ δ d1 d2 d1 d2 2( 2 ) T c i T ic T T = N N Ð 1 , ≥ ≥ ≥ ≥ c1c1' c2c2' 1 2 c1' j jc2' 1 x 0, 1 y 0. 2 2 Note that d1 d2 d2 d1 N (N Ð 1) δ δ T T T T = ------, c c' c c' c1i ic2 c' j jc' ' ' 1 1 2 2 1 2 2 k1 + k2 = k = q1 Ð q2 = q1 Ð q2 (45) (38) 3 2 is fixed. d1 d2 d1 d2 N (N Ð 1) f f T T T T = ------, The important symmetry properties of the convolu- c c' c c c' c c1i ic2 c' j jc' 1 1 2 2 1 2 4 tions d1 d2 d2 d1 α α f f T T T T = 0. ( , ) 1 2( , ) ( , ) c c' c c c' c c1i ic2 c' j jc' f k k = c k k cα' α k k (46) 1 1 2 2 1 2 a 1 2 1 2 1 2 1 2 Using these formulas, we obtain and ( , ) f b k1 k2 〈c c' |ᏼˆ ᏾|c c' 〉 -----1-----1------2-----2---∑γ GG (q , q )(γ GG (q' , q' ))* (47) c1c2 1 2 c' c' 1 2 α β α β 2n᏾ 1 2 1 1 2 2 GG = cα α (k , k )cβ' β (k , k )Ω (k )Ω (k ) 1 2 1 2 2 1 2 1 1 2 α α 4 2[( 1 2( , ) ( , ) entering (39) are their invariance with respect to the = 8g N aRc k1 k2 cα' α k1 k2 1 2 (39) “left–right” transformation α β α β α β α β 1 1 2 2 k1 k2, 1 2 , 2 1 , + b cα α (k , k )cβ' β (k , k )Ω (k )Ω (k ) ) R 1 2 1 2 2 1 2 1 1 2 (48) ( ) ] q1 Ð q 2, q1' Ð q2' . + k1 k2 , This invariance follows from the transformation law for µν where cα' α (k , k ) is obtained from cα α (k , k ) by the 1 2 1 2 1 2 1 2 the tensor c (k1, k2). It is easily seen from (32) that this νµ tensor reduces c (k1, k2) under transformation (48). In substitution qi qi' and the coefficients aR and bR for terms of the variables ki, x, and y, this transformation is the singlet (R = 1) and octet (R = 8) representations are represented as

1 1 k k , x y, q Ð q , q' Ð q' . a0 = 1, b0 = --; a8 = --, b8 = 0. (40) 1 2 1 2 1 2 2 4 (49) Evidently the term (k k ) in (39) gives the same One can check by a direct inspection that the tensor 1 2 µν contribution to the kernel as the preceding terms, so c (k1, k2) (32) is equal to zero when the transverse that in the following only these terms will be consid- momentum of one the Reggeons is zero, q1 = 0 or q2 = 0. ered and their contribution to the kernel will be dou- This guarantees another important property of the func- bled. tions fa(k1, k2) and fb(k1, k2)—they vanish at zero trans- To perform in Eqs. (10) and (13) the integration over verse momenta of the Reggeons [compare with (11)]. longitudinal components of the produced gluon Let us adopt the ﬁrst choice (x, k1) of variables for the ≡ β β integration in Eqs. (10) and (13). The two-gluon contri- momenta, we will use the variable x 1/ , so that β bution to the kernel is then represented in the form i , 2 2 D Ð 2 x1 = x, x2 = 1 Ð x, xi = ----, i = 1 2. (41) β ᏷ (᏾) ( , ) 4g N d k1 RRGG q1 q2; q = ------D----Ð---1∫------(--D----Ð--1--) The alternative choice is y, with (2π) (2π) α 1 2 i , dxθ(sΛ Ð k ) y2 = y, y1 = 1 Ð y, yi = -α---, i = 1 2. (42) × ------[a f (k , k ) + b f (k , k )] ∫ x(1 Ð x) R a 1 2 R b 1 2 The variables x and y are connected by the relations 0 (50) D Ð 2 2 2 1 d r (᏾)B (᏾)B xk2 yk1 Ð ------᏷ (q , r; q)᏷ (r, q ; q) y = ------, x = ------, (43) 2∫ 2 2 r 1 r 2 2 ( ) 2 2 ( ) 2 r r' xk2 + 1 Ð x k1 yk1 + 1 Ð y k2 2 which are inverse to each other. Recall that the vector  sΛ  × ln------ , sign is used for the transverse components. The integra- ( )2( )2 tion measure in (10) and (13) with allowance for (14) r Ð q1 r Ð q2

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NONFORWARD COLOR-OCTET KERNEL 2163 where the group coefficients aR and bR are defined in This result and Eqs. (30), (31), and (18) give us the rela- (40) and the functions fa(k1, k2) and fb(k1, k2) in (46) and tion (47), respectively. The functions must be expressed in *( ) *( ) µν( , ) eµ k1 eν k2 A k1 k2 x = 1 terms of x and k1. It can be done by using (41) and the relations ( ) µ( , ) ( ) ν( , ) (54) eµ* k1 C q1 Ð k1 q1 eν* k2 C q2 q1 Ð k1 2 = Ð ------2------, 2 ((1 Ð x)k Ð x k ) (q Ð k ) k = ------1------2-----, 1 1 x(1 Ð x) which means that in the multi-Regge limit, the vertex 1 2 2 for the two-gluon production is expressed in terms of ˜t = Ð--((1 Ð x)k + x(k Ð q ) ), (51) 1 x 1 1 1 the one-gluon production vertices. For the function fa(k1, k2) (46), we obtain from (53), using (8), 2 k 2 k + k = k, t = q 2, α = -----i-, i = 1, 2.  2  1 2 i i i β 2g cR ( , ) 1 s i ------ f a k1 k2 = ------( π)D Ð 1 x = 1 ( )2( )2 2 q1 Ð k1 q1' Ð k1 To understand the behavior of the functions fa(k1, k2) (᏾)B (᏾)B and fb(k1, k2) in the integration region of (50), it is con- × ᏷ ( , )᏷ ( , ) r q1 q1 Ð k1; q r q1 Ð k1 q2; q . venient to express the tensor cµν(k , k ) in (32) in terms 1 2 Therefore, the subtraction term in (50) can be written as of x and k1. After this, it is not difficult to show that, for D Ð 2 any x in the interval [0, 1], the tensor decreases in pro- 1 d r᏷ (᏾)B( , )᏷ (᏾)B( , ) Ð------r q r; q r r q ; q 2 2∫ 2 2 1 2 portion to 1/k1 , so that the integration over k1 is well r r' convergent in the ultraviolet region. As for the x behav- 2 2 2  sΛ   2g c  ior at fixed k , it is easy to see that in the limit x 0 ×   1 R  1 ln ------2------2- = Ð------D----Ð---1 α α ( ) ( )  2( π)  (55) 1 2 r Ð q1 r Ð q2 2 the tensor c (k1, k2) tends to zero, whereas at x 2 1 the tensor has a finite value. It means that the function 1 Ð k2 /sΛ D Ð 2 dx fb(k1, k2) [see (47)] turns to zero both at x = 0 and x = 1, × d k ------f (k , k ) . ∫ 1 ∫ x(1 Ð x) a 1 2 x = 1 so that performing the integration of the term with fb(k1, 2 k1 /sΛ k2) in (50) we can ignore the restrictions on the integra- θ 2 2 2 tion region imposed by (sΛ Ð k ). Recall that the Taking into account that cR = N aR [compare (9) and parameter sΛ must be taken tending to infinity, there- (40)], we obtain that the two-gluon contribution to the kernel can be presented as fore, due to the convergence of the integral over k1 in the ultraviolet region, the restrictions have the form 4 2 1 (᏾) 4g N dx ᏷ (q , q ; q) = ------2 2 RRGG 1 2 D Ð 1∫ ( ) k k (2π) x 1 Ð x 1 Ð ----2- ≥ x ≥ ----1-. (52) 0 sΛ sΛ 2 + 2⑀ × d k1 { [ ( , ) ( ( , ) )] ∫------aR f a k1 k2 Ð x f a k1 k2 x = 1 From the discussion above, it is clear that the restriction (2π)D Ð 1 from below does not play any role, but the upper limit (56) 4 2 is important for the integration of f (k , k ) in (50). 2g N a 1 2 + b f (k , k ) } + ------R b 1 2 ( π)D Ð 1 The limit x 1 corresponds to the multi-Regge 2 limit of large relative rapidities of the produced gluons, 2 + 2⑀  2 d k1 k1 so that the value of fa(k1, k2) at this limit is related to the × ------a f (k , k ) ln---- . ∫( π)D Ð 1 R a 1 2 x = 1  2 ᏷ (᏾)B 2 k2 LLA kernel r . Indeed, by using (41) and (51), we find from (32) that Recall that, from general arguments, the kernel must be symmetric [see (12)] with respect to the substitutions µ 2 ( ) qi qi' , i = 1, 2, and q1 Ðq2, q1' Ð q2' µν , Ð1 q1 Ð k1 c (k1 k2) = ------q1 Ð k1 + ------k1 (note that at both of them q changes its sign). The sym- x = 1 ( )2 2 q1 Ð k1 k1 ⊥ metries of the two-gluon contribution to the kernel can ν (53) be explicitly demonstrated using representation (50). (q Ð k )2 First of all, it is easy to show with the help of expression × ------1------1----- q1 Ð k1 Ð 2 k2 . (8) for the Born kernel that the subtraction term is sym- k2 ⊥ metric under these substitutions. After this, the symme-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2164 FADIN, GORBACHEV try of the total contribution under the ﬁrst transforma- 2 ( ⋅ )( ⋅ ) q (L ⋅ k) q1' L q1 L tion follows from the evident invariance of the convolu- + (1 Ð 2x)(1 Ð x)----1------+ ------Σ  2 tions in (46) and (47) under this transformation. The k symmetry under the second transformation follows from the invariance of these convolutions as well as the ( ) 2 ( ) 2 µ ν  x 1 Ð x q1' x 1 Ð x q1 µ ν k⊥ q1⊥ ------------⊥ ⊥  integration measure [see (44)] with respect to the “left– + 2 2 k k Ð ------2----- right” transformation given by (48) and (49). Turning to k Σ  2k Σ k  (56), we see that the contribution of f (k , k ) is symmet- b 1 2 2 2 2 2 2 ric with respect to both transformations. As for the 2 ⊥ x q1' q2 xq1' q2 × (2(1 + ⑀)ΛµΛν + L gµν ) + ------+ ------terms with fa(k1, k2), the last of them gives the contribu- 2Σ 2 2 4k1 4(1 Ð x)k1 k tion antisymmetric under the substitution q1 Ðq2, 2 2 2 q1' Ðq2' . Therefore, this term can be omitted x(1 Ð x) q1' q1 1 + ⑀  + ------Ð (3 + 2⑀)x(1 Ð x) together with antisymmetric contributions from the  Σ  2  2  remaining terms. So, the total contribution of all three 2 terms with f (k , k ) can be obtained by integration of ' a 1 2 x(1 Ð x)q1 [( )(( ⑀)( ( ⋅ ) 2) δ Ð ------1 Ð x 1 + 2 k1 q1 Ð xq1 the ﬁrst term with respect to x from zero to 1 Ð at arbi- 2Σ trary small δ with subsequent omission of terms pro- 2k portional to lnδ and terms antisymmetric under the sub- 2 2 ⑀ ( 2 2) ) ( ⑀) 2 2 ] xq1 q ' ' Ð x k Ð q2 Ð 1 + k2 + 2q2 + ------stitution q1 Ðq2, q1 Ð q2 . ˜ 2 4t1k1 2 (q' Ð 2(k ⋅ q' )) ( ⑀)( )2 5. THE TWO-GLUON CONTRIBUTION ( ⋅ )( ) 1 1 1 1 + 1 Ð x Ð q1 q 1 Ð x ------+ ------TO THE OCTET KERNEL ˜t1˜t1' 4 Up to now, our results could be used for any color (59) ᏾ ( 2 ( ⋅ ))( 2 ( ⋅ )) representation . Starting from this point, we will con- q1' Ð 2 k1 q1' q1 Ð 2 k1 q1 sider only the case of the gluon channel (antisymmetric × ------˜ ˜ octet representation). In this case, only the function t1t1' fa(k1, k2) does contribute to the kernel. From the discussion at the end of the preceding section, it follows that 1 (( ⋅ )( ⋅ ' ) ( ' ⋅ )( ⋅ )) + ----2----- 2 q1 q2 k1 q1 Ð q1 q2 k1 q1 - the two-gluon contribution to the octet kernel can be k ˜t1 presented as 2 2 2( ⋅ ) 2( ⋅ ) 4 2 + xq2' q1 Ð xk q1 q + q1 q1' q ᏷ (8) ( , ) g N ᏿ˆ RRGG q1 q2; q = ------D----Ð---1 2 ( π) 2 (1 Ð x)k ⑀ 2 ' 2 1 1 + ( ) Ð q1 q2 ------Σ------+ ------1 Ð x 1 2 + 2⑀ (57) 2 dx d k f (k , k ) × ------1- ----a------1------2---, ∫( ) ∫ D Ð 1  2 2 1 Ð x + ( π) x × ( ( ⋅ ))( ( ) ( ⋅ )) 2  q1 Ð 2 k1 q1 q1' 1 Ð x Ð 2 q1' L 0  ˆ where ᏿ denotes the operator of symmetrization with 2 2 2  2 2 2 2 2 2( 2)2 k1 k2 q1' (q ) xq1 q k xq1' q2 q q ' ' + ------ + ------+ ------Ð ------respect to the substitution 1 Ð 2, q1 Ð q2 Σ ˜ ˜' 2˜ 2 2Σ˜  8t1t1 4k t1k1 4k t1 and (1 Ð x)+ means the subtraction 2 1 1 2( 2)2 2( 2( ' ⋅ ) ' ( ⋅ )) xq2' q1 xq2 q1 q1 k1 Ð q1 q1 k1 dx ( ) ≡ dx [ ( ) ( )] Ð ------+ ------∫-(------)--- f x ∫(------)- f x Ð f 1 . (58) 2˜ 2 ( ) 2˜ 2 1 Ð x + 1 Ð x 4k t1k1 2 1 Ð x k t1k1 0 0 2( ⋅ ) According to (46), the function fa(k1, k2) is determined q2 q1' q  µν ------ { '} by the convolution of the tensor c (k , k ) given by (32) Ð 2 + qi qi , 1 2 2(1 Ð x)k ˜t  ' 1 with the tensor cµν (k1, k2) given by the same formula ≡ where ˜t1' is obtained from ˜t1 (51) by the substitution with the substitution qi qi' qi Ð q. We obtain (details of the calculation are given in Appendix A) q1 q1' 2 Λ = ((1 Ð x)k Ð xk ) , 1 + ⑀ x (1 Ð x) 2 1 2 ⊥ f (k , k ) = ------q' q ⋅ L - (60) a 1 2 2 Σ 1  1 2 2  k Σ = L + x(1 Ð x)k .

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NONFORWARD COLOR-OCTET KERNEL 2165

Unfortunately, integral (57) cannot be expressed in 2  2 x(q2(k ⋅ q' ) Ð q' (k ⋅ q )) terms of elementary functions (and dilogarithms) at × xq2 1 1 1 1 1 1 ( ⋅ ) ⑀ ------Ð q q1' arbitrary . Therefore, we present the result (see details  2˜ 2 of the integration in Appendix B) in a “combined” 2L t1 k1 form, leaving untouched the terms in fa(k1, k2) which 2 4 4 2 2 2 2 x2(1 Ð x) q' q q q' Ð q q k  cannot be integrated in elementary functions: Ð ------1------2- + ----1------2------1------ 4 2 2 2 Σ 2 ( ) Γ( ⑀) Γ ( ⑀) 4L ˜t k  ᏷ 8 ( , ) 4g N 1 Ð 1 + 1 1 (62) RRGG q1 q2; q = ------⑀------( π)Dπ1 + Γ(1 + 2⑀) 2 2 ⑀ Ð 1 4 Γ (⑀) (k ) 2 2 2 2 2 2 Ð ------(q q' + q q' Ð q k ) 2 ⑀ Ð 1 Γ(2⑀) 2⑀ 1 2 2 1 (k ) 2 2 2 2 1 × ------(q' q + q' q ) -- + ψ(1) + ψ(1 + ⑀) ⑀ 1 2 2 1 ⑀ 2 2 2 + 2⑀  4 (q ) 1 d l ------Ð ----1---+---⑀------∫------11 + 7⑀  q2k2 4 π Γ(1 Ð ⑀) (q Ð l)2(q' Ð l)2 Ð 2ψ(1 + 2⑀) Ð ------Ð ------1 1 2(1 + 2⑀)(3 + 2⑀) ⑀ 2( )2 2 ⑀ + 1 × l l Ð k  (q )  1 ln------ . + ------Ð -- + ψ(1) Ð ψ(1 Ð ⑀) + 2ψ(1 + ⑀) q4 4⑀  ⑀ ⑀ The ﬁrst of the symmetries in (12) of the two-gluon ψ( ⑀) 11 + 7  Ð 2 1 + 2 Ð ----(------⑀---)--(------⑀---)- contribution is explicit in (61); the second is also easily 2 1 + 2 3 + 2 seen. The properties in (11) are not so evident. It takes 2 ⑀ 2 ⑀ some work to demonstrate their existence. In particular, 11 + 7⑀ 2 2(q ) Ð (q ) Ð ------q q ------1------2----- one has to calculate the function I(q , q ; q) at q = 0 4⑀(1 + 2⑀)(3 + 2⑀) 1 2 2 2 1 2 2 q1 Ð q2 and q2' = 0. It is not too easy, but possible (for details, ⑀ ⑀ 2 ( 2) + 1 ( 2) + 1 2 q q1 Ð q2 q  1 see Appendix C), so that we have checked the fulﬁll- + ------+ ------Ð ψ(1 + ⑀) ⑀(1 + 2⑀) 2 2 4⑀2⑀ ment of (11) for the two-gluon contribution at arbi- q1 Ð q2 trary ⑀.  2 ⑀ 2 ⑀ 1 In the limit ⑀ 0, we obtain + ψ(1 + 2⑀) ((q ) + (q ) ) + ------ 1 2 ⑀ ⑀ ⑀ 4 2 8 (1 + 2 )(3 + 2 ) (61) ( ) Γ( ⑀) ᏷ 8 ( , ) g N 1 Ð RRGG q1 q2; q = ------D------1---+---⑀---- ⑀ ⑀ (4π) π × (2(1 + ⑀)q2q 2((q 2) Ð (q 2) ) Ð ⑀(q 2 + q 2) - 1 2 1 2 1 2  × ( 2)⑀ Ð 1( 2 2 2 2 2 2) 1 11 67  k q1' q2 + q2' q1 Ð q k ---- Ð ----- + ----- 2 ⑀2 ⑀ ⑀ ⑀   6 18 × (( 2) + 1 ( 2) + 1) ) k ( 2 2 '2 q1 Ð q2 ------3 q1 + q2 + 2q1 (q 2 Ð q 2) 1 2  202 11  Ð 4ζ(2) + ⑀Ð ------+ 9ζ(3) + ------ζ(2) 2 2  27  2 q' Ð q'  6 + 2q' Ð 2k2 Ð 2q2 ) Ð ------1------2----- 2 2 2 2 ( )  2 2 q1 Ð q2   2 11 2 q1 q2 + q -- -- + lnq ln------ Ð 2ζ(2) 2 ⑀ + 1 2 ⑀ + 1 ⑀  4 (q ) Ð (q ) 2 2 2  q  1 2 (⑀( ' ' 2) + ------2------2------q1 + q2 Ð k (q Ð q ) 2 2 2 2 2 1 2 11 q q  (q + q ) q  + -----ln----1------2- + ------1------2----ln----1- (63) Γ( ⑀)  6   2 2 ( 2 2)  2 ⑀ 2 ) 1 + 2 ( , ) k q q1 Ð q2 q2 -----Ð 2(1 + )q + ------I q1 q2; q  4Γ2(1 + ⑀)   2  2 ⋅ 1 2 q1 1 2 q2 q1 q2 11 2 2 4 2 2 + -- ln ----- + -- ln ----- + ------Ð -----(q + q ) 4g N Γ (1 + ⑀) 2 2 1 2 + ------Γ(1 Ð ⑀)------4 q  4 q  3 6 D 1 + ⑀ Γ ⑀ (4π) π (1 + 2 )  2 1 k 2 2 2 × { , }  ( ( ' ⋅ ' ) ) q1 q1' q2 q 2' , Ð - - 11 Ð ------2 q1 + q2 + 4 q1 q2 Ð 2q 6 (q 2 Ð q 2) where 1 2 1 2 + 2⑀ 2 2   2 2  2  dx d k q1' Ð q2' 2q1 q2 q1 2 2 I(q , q ; q) = 4᏿ˆ ------1------+ ------ ------ln----- Ð q Ð q  1 2 ∫ ( ) ∫ 1 + ⑀ 2 2 2 2 2 1 2 x 1 Ð x + π Γ(1 Ð ⑀) (q Ð q ) (q Ð q ) q   0 1 2 1 2 2

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2166 FADIN, GORBACHEV 6. THE NONFORWARD OCTET BFKL KERNEL  g4 N2Γ(1 Ð ⑀) ( , )  { } The general form of the kernel (for arbitrary repre- + I q1 q2; q + ------D------1---+---⑀---- qi q i' .  (4π) π sentation ᏾ of the color group in the t channel) is given In this limit, the function I(q , q ; q) takes the form by Eq. (2). The “virtual” part is universal (does not 1 2 ᏾ 1 depend on ) and is determined by the gluon Regge 1 dx trajectory, which is given by Eqs. (3)Ð(5). (Recall that ( , ) ------I q1 q2; q = ∫ 2 in this paper we consider pure gluodynamics.) The 2 (q (1 Ð x) + q x) 0 1 2 quark part of the kernel was considered in [18]. The  2( ) 2  “real” part, related to real particle production in × q1 1 Ð x + q2 x 2( 2 2 2) 2 2 ReggeonÐReggeon collisions, in the NLA is given by ln------ q k Ð q1 Ð q2 + 2q1 q2 -  k2 x(1 Ð x)  the one-gluon and two-gluon contributions considered in Section 3 and Section 5, respectively. Since the radi- 2 2 2 2 2 2 q q' Ð q q' ative corrections to the effective vertex of the one-gluon 2 ' 2 ' 1 2 2 1 ( 2 2) ⑀ Ð q1 q2 Ð q2 q1 + ------2------q1 Ð q2 production are known only in the limit 0, the total k “real” part of the kernel can be obtained only in this (64) limit. It is given by the sum of (23) and (63). After pow- 2  2 2 2 2 2 q 1 2 q q q' Ð q q' erful cancellations (in particular, between the terms + -----4ζ(2) Ð -- ln ----1- Ð ----1------2------2------1--- 2  2  2 2 with singularities 1/⑀2 and all terms with (q 2 Ð q 2 ) in q2 4k 1 2 denominators), we obtain  2  2 2 2  2 2 2 2 q1 q1 q2 q 1 2 q1 q2 q1' q2' × ----- ------ ----- ------ 2 2 2 2 2 ln 2 ln 4 Ð -- + lnq  ln 8 ( )  ' '      4 ⑀   ᏷ G 8 , g N q1 q2 + q1 q2 2 q2 k q r (q1 q2; q) = ------------Ð q  2(2π)D Ð 1 k2   2   2  1 2 q1 1 2 q2 + -- ln ------ + -- ln ------ . ⑀ 2  '2 2  '2 1 g2 NΓ(1 Ð ⑀)(k2)  11 67 q1 q2 × -- + ------Ð ----- + ----- Ð ζ(2) 2 + ⑀  ⑀ The integral in (64) can be presented in another form: 2 (4π) 6 18

1 2 2 dx q (1 Ð x) + q x 2 ----------1------2----  202 11   g NΓ(1 Ð ⑀) ∫ 2 ln 2 + ⑀ Ð ------+ 7ζ(3) + -----ζ(2) + ------(q (1 Ð x) + q x)  k x(1 Ð x)   6   2 + ⑀ 0 1 2 27 (4π) ∞ dz 1   2 2  2  '2 = ∫------(65) 2 11 q1 q2 1 q1 q1 2 2 × q ----- ln------ + -- ln----- ln------ z + k ( 2 2 ) 2 2 2 2 2 2 0 q1 + q2 + z Ð 4q1 q2  6 q k  4 q   q 

 2  2 q 2 + q 2 + z + (q 2 + q 2 + z) Ð 4q 2q 2  2  '   2  ×  1 2 1 2 1 2 1 q2 q2 1 2 q1 ln------ . + -- ln----- ln------ + -- ln -----  2 2 2 2 2 2 2 4  2  2  4  2  q + q + z Ð (q + q + z) Ð 4q q  q q q2 1 2 1 2 1 2 (68) 2 2 2 2 It is also possible to express the integral in (64) in terms q 2q' + q 2q' q2 q 2q' Ð q 2q' of dilogarithms, but this expression is not too conve- Ð ----1------2------2------1--ln2 ----1- + ----1------2------2------1--- 2  2 2 nient: 2k q2 k 1 2 2  ( )  2 2 2 dx q1 1 Ð x + q2 x      ------ln------ q1 11 1 q1 q2 1 2 2 2 2 ∫ 2 2 × ln----- ----- Ð -- ln------ + -- q (k Ð q Ð q ) ( ( ) )  ( )  2 4 1 2 q1 1 Ð x + q2 x k x 1 Ð x    6 4   2 0 q2 k

2 ρsinφ 2 2 2 2 ρ 2 2 q q' Ð q q' = Ð ------φ-- ln arctan-(------ρ------φ----)- (66) 2 2 2 2 1 2 2 1 ( 2 2) q1 q2 sin 1 Ð cos + 2q1 q2 Ð q1 q2' Ð q2 q1' + ------q1 Ð q2 k2

--+ Im (L(ρexpiφ)) , 1 2 2 dx q (1 Ð x) + q x  × ------ln----1------2----  φ ∫ 2 2 where is the angle between q1 and q2, (q (1 Ð x) + q x)  k x(1 Ð x)   0 1 2 1  q q  dt 2 ρ = min -----1--, -----2-- , L(z) = ---- ln(1 Ð t). (67) g N { }   ∫ + ------qi qi' . q2 q1 t ( π)D Ð 1 0 2 2

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⑀2 µ ν µ After cancellation of the terms ~1/ , the leading Λ  2  2 µν 1 q1⊥ ⊥ xk ν singularity of the kernel is 1/⑀. It turns again into ~1/⑀ l = ---- Ð ------+ k Ð ------k  q ⊥ 2 2 1 Ð x  2 1 1 after subsequent integrations of the kernel because of k k1 ⊥ the singular behavior of the kernel at k2 = 0. The addi- (Ä.3) ν tional singularity arises from the region of small k2, 2x µ q ⋅ k  µν ------1------1 ( ⋅ ) ( ⋅ ) ⑀| 2| Ð 2 k1⊥  k2 Ð q1 k k1 Ð q1 L g⊥ , where lnk ~ 1. Therefore, we have not expanded in k 1 Ð x ⊥ ⑀ the term (k2)⑀. The terms ~⑀ are taken into account in 1 the coefficient of the expression divergent at k2 = 0 in µ ν µ ν µν xk ⊥ (q Ð k ) (q Ð k )⊥ (q Ð k )⊥ order to conserve all nonvanishing contributions in the l = Ð ------1------1------1-----⊥- + ------1------1------1------1------3 2 ˜ limit ⑀ 0 after the integrations. k t1 1 (Ä.4) µν The symmetries (12) of the kernel are easily seen. ( ) g⊥ 1 Ð x ( 2 ⋅ ) The first of them is explicit in (68). To notice the sec- + ------q1 Ð 2k1 q1 . 2 ˜t1 ond, it is sufficient to change x (1 Ð x) in the integral in (68). Analogous decomposition is made for the tensor cµ' ν (k , k ) obtained from cµν(k , k ) by the substitution In order to check that the kernel (68) vanishes at 1 2 1 2 zero transverse momenta of the Reggeons (11), one qi qi' , i = 1, 2, and we denote as ln' µν (n = 1, 2, 3) needs to know the behavior of the integral in (68). A the tensors lnµν after this substitution. suitable representation for this purpose is given in (65). µν From this representation, one can see that singularities The calculation of the products ln ln' 'µν is signifi- of the integral at zero transverse momenta of the cantly simpler than the calculation of the whole convo- Reggeons are no more than logarithmic. Verifying lution fa(k1, k2) (46), though still rather tedious. The Eq. (11) therefore involves no more problems. results are In conclusion, let us note that in [17] the octet kernel 2 2 2  ⋅ µν x q1' q1 2k1 k2 1 Ð x 1  was obtained using as a basis the bootstrap relation and l l' µν = ------Ðk ------+ ------Ð ------1 1 4  2Σ xΣ 2 a specific ansatz to solve it. Our results disagree with k  k1 x(1 Ð x)k1 the results obtained in [17]. To see the disagreement, it is sufficient to observe that the kernel obtained in [17] (Ä.5) 1 + ⑀ 1 Ð x 2 2L ⋅ k 2  + ------k 1 Ð 2x + ------, is expressed in terms of elementary functions. We con- Σ  2  clude that the ansatz used in [17] is not correct. 2 k  2 µν xq'  ⋅ ' 1 ⑀  2L k l2 l1µν = ------Ð2(1 + )(1 Ð x)1 Ð 2x + ------ ACKNOWLEDGMENTS 2k4  k2

This work was supported by the Russian Foundation ( ⋅ ) 2  2 q1 L k 2 xk 1 Ð x for Basic Research. × ------Ð k (q ⋅ k )------Ð ------(Ä.6) Σ 1 1  2Σ Σ k1

APPENDIX A 2 1 Ð 3x + x 2 1 Ð 2x 2   Ð ------ + k (q ⋅ k) ------Ð ----- , ( ) 2  1  Σ 2 In this appendix, we present the details of the calcu- 1 Ð x k1 k1  lation of the convolution fa(k1, k2) (46). 2 2 µν µν xq1' 2 k 2 It is suitable to represent the tensor c (k , k ) in the l l' µν = ------(1 + ⑀)(1 Ð x) ------(q 1 2 3 1 2 Σ˜ 1 form 2k  t1 µν µν µν µν  2L ⋅ k 1 c (k , k ) = l + l + l , (Ä.1) Ð 2k ⋅ q ) 1 Ð 2x + ------Ð ------1 2 1 2 3 1 1  2  ( ) 2˜ k 2 1 Ð x k1 t1 where × [ 2( 2 ( )( 2 2)) ( ) 2( 2 q1 q2 + 1 Ð x q1 Ð k Ð 2 1 Ð x k q1 2 µ ν µ (( ) ) ν µν xq k ⊥ k ⊥ k1⊥ 1 Ð x k1 Ð k2 ⊥ l = Ð------1- ---2------2--- + ------q 2 1 2 Σ ( ) 2 Ð k ⋅ q ) ] Ð ------2--- [q2 Ð 4(1 Ð x)(q 2 Ð 2k ⋅ q )] k 1 Ð x k1 1 1 Σ˜ 2 1 1 1 (Ä.2) 2 t1 µν 2 2 2 2 2 2 (Ä.7) g⊥ (1 Ð x) Σ k1 4(1 Ð x) q Ð q xk (2q ⋅ k Ð k ) -----+ ------------Ð ----- , Ð ------1------2- + ------1------2------2 Σ 1 Ð x x 2x(1 Ð x)˜t 2Σ 1 2k1

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2 2 2 1 2 1 Ð 2x 2 2 × [( ⋅ ' )( ( ) ) ( ' ⋅ ) ] + --- (1 Ð x)(q Ð k ⋅ q ) + ------(q Ð k ) q1 q1 q1 + 2x Ð 1 k Ð 2x q1 k q1 Σ 1 1 1 2 2 1 {( ⋅ )[ 2( ) ( )2 2] + ------q1 q1' q1 3 Ð 4x Ð 1 Ð 2x k 1 2 ( )˜ 2 + ------(Ð2(1 Ð 3x + x )(q ⋅ k ) 4 1 Ð x t1k ( ) 2 1 1 2 1 Ð x k1 2( ⋅ ) ⋅ ( ( )) ( ⋅ )( ⋅ ) } Ð q1 q1' k 2 1 + 4x 1 Ð x + 4 q1 q1' q1 k  2 2 2 x [ 2( ⋅ ) ( ⋅ )( 2 2)] -+ (1 Ð x)k + xq Ð 2q ) , + ------2q1 q1' L + q1 q1' q1 + k 2 1 ( ) 2 2  4 1 Ð x k1 k

⋅  2 ( ⋅ ' ) 2 µν x(1 Ð x) q1 q1' 1 k q1 q1 k l l' µν = ------------+ ----- Ð x(1 Ð x)------+ [(k ⋅ q )(q' ⋅ k) 2 2 2 2 ( )2 2 2 2 1 1 1 k 1 Ð x k1 2k k1 2  2 xq  (q ⋅ k )(q' ⋅ L) (q ⋅ L)(q' ⋅ L) Ð (q' ⋅ k )(q ⋅ k) ]------+ ------1------ . 1 1 1 ( ⑀) 1 1 1 1 1  2˜ ( ) 2˜ 2 Ð ------+ 1 + ------k t1 1 Ð x k1 t1k (1 Ð x)k 2 k4 1 (Ä.8) With the aid of Eqs. (A.1)Ð(A.10) and the equations { } + q1 q 1' , obtained from them by the substitution qi qi' , we come to (59). ⋅ 2( ⋅ ' ) 2 µν 2q1 q1' q q1 q1 (1 + ⑀)(1 Ð x) l l' µν = x ------Ð ------+ ------3 3 2 ˜ ˜' 4 2k1 4t1t1 APPENDIX B 2 ( 2 ⋅ ) In this appendix, we present the details of the calcu- (q Ð 2k ⋅ q ) q1' Ð 2k1 q1' x(1 Ð x) × ------1------1------1------Ð ------lation of the integrals in (57) with fa(k1, k2) given by ˜t ˜' 2 (59). First, let us recall the notation used: 1 t1 2k1 k = k + k = q Ð q , 2 2 1 2 1 2 (q' Ð 2(k ⋅ q' ))(k ⋅ q ) q' Ð 2k ⋅ q' × ------1------1------1------1------1--- Ð ----1------1------1- (Ä.9) L = (1 Ð x)k Ð xk , q' = q Ð q, i = 1, 2; ˜t1' 2˜t1˜t1' 1 2 i i ( ⋅ ) 2 (B.1) 2x k1 q1 ((1 Ð x)k Ð x k ) × (1 Ð 2x)(q ⋅ q) + ------q ⋅ (q Ð k ) 2 1 2 Σ 2 2 1 2 1 1 k = ------(------)------, = L + x(1 Ð x)k , k1 x 1 Ð x { , } 1 2 2 + q1 q 1 ' q Ð q , ˜t = Ð--((1 Ð x)k + x(k Ð q ) ), 1 x 1 1 1 ' ⋅ µν ( ⑀)( )( 2 ⋅ )q1 L l3 l2' µν = Ð 1 + 1 Ð x q1 Ð 2k1 q1 ------˜ ˜ 2 and t1' is obtained from t1 by the substitution q1 q1' . k ˜t1 It is easy to see that ⋅ q1' k1 2 + x(1 Ð x)------(q Ð 2k ⋅ q ) 1 ⑀ 2˜ 1 1 1 2 + 2  ⑀ 2 2k1 t1 dx d k1 1 + x (1 Ð x) J0 = ∫------∫------⑀------------x x(1 Ð x) π1 + Γ( ⑀) 2 Σ ------[( ⋅ )( ' ⋅ ) (( ' ⋅ )( ⋅ ) 1 Ð  k + 2 2 k1 q1 q1 L Ð q1 k1 q1 k 0 k k1 2 2 q (L ⋅ k) 2( ⋅ ) ( ⋅ ) × q' q ⋅ L + (1 Ð 2x)(1 Ð x)----1------x q1 q1' x q1 q1' 1  1 Σ  Ð (q ⋅ k )(q' ⋅ k) ) ] Ð ------Ð ------1 1 1 2 ( ) 2 2k1 2 1 Ð x k ( ' ⋅ )( ⋅ ) ( ) 2 ( ⋅ ) q1 L q1 L x 1 Ð x q1' x q1 q1' 2 + ------+ ------(B.2) ( ⋅ ) 2 2 Ð ------q1 Ð q1 k1 Σ ˜ 2 k k 2˜t1k1 2 µ ν 2  ( )  q (q' ⋅ k ) Ð (k ⋅ q )(q' ⋅ q ) x 1 Ð x q1 µ ν k⊥ q1⊥ 1 1 1 1 1 1 1 × ------k⊥ k⊥ Ð ------ + ------2 2 2  2k Σ k  k ˜t1 2 2 2 (Ä.10) x q1 + k 2 xq1 2 ⊥  × 2 + ------Ð k + ------× (2(1 + ⑀)ΛµΛν + L gµν )  = 0. 2 1 Ð x  ( )˜ 2 2 2k1 4 1 Ð x t1k1 k 

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The zero appears as a result of the integration with ⑀ ( ) '2 respect to k (or, equivalently, with respect to L). The × 1 + ( ⑀) ( ) x 1 Ð x q1 1 ------Ð 3 + 2 x 1 Ð x  Ð ------ﬁrst two terms here give zero due to parity, the third van- 2 2k2Σ (B.5) ishes due to the dimensional regularization, and the × [( )(( ⑀)( ( ⋅ ) 2) ⑀ ( 2 2)) remaining terms vanish due to the isotropy of the trans- 1 Ð x 1 + 2 k1 q1 Ð xq1 Ð x k Ð q2 verse space leading to the replacement 2(1 + ⑀)ΛµΛν ⊥ 2 2 2  2  −L gµν after the angular integration. 2 2 Γ (1 + ⑀) q1' q2 2 ⑀ -Ð (1 + ⑀)k + 2q ]  = ------(k ) ψ(1) 2 2 Γ( ⑀) 2 The calculation of the contributions of terms with  1 + 2 2⑀k  the denominators Σ, Σ2, k2Σ, and k2k2 is straightforward. We obtain, using usual Feynman parametrization 11 + 7⑀  if necessary, + ψ(⑀) Ð 2ψ(2⑀) Ð ------, 2(1 + 2⑀)(3 + 2⑀)  2 + 2⑀ d k1 1 Ð1 2 ⑀ IΣ = ------= -----[x(1 Ð x)k ] , ∫ 1 + ⑀ Σ ⑀ 1 2 + 2⑀ 2 2 π Γ(1 Ð ⑀) dx d k xq q ------1 ------1------J2 = ∫ ∫----1---+---⑀------2 ⑀ x(1 Ð x) π Γ( ⑀) ˜ 2 + 2 + 1 Ð 4t1k1 d k1 1 [ ( ) 2]⑀ Ð 1 0 (B.6) IΣ2 = ∫------⑀------= x 1 Ð x k , 2 π1 + Γ( ⑀)Σ2 Γ (⑀) ⑀ 1 Ð ( 2) 2[ψ(⑀) ψ( ⑀)] (B.3) = Ð ------q1 q Ð 2 . 2 + 2⑀ 4Γ(2⑀) d k1 1 1 2 ⑀ Ð 1 I 2 = ------= --[x(1 Ð x)k ] , Λ Σ ∫π1 + ⑀Γ( ⑀) 2Σ ⑀ 1 Ð L The integrals with ˜t1˜t1' in denominators can be cal-

2 + 2⑀ 2 culated with the help of the trick used in [18]. Let us d k1 1 Γ (⑀) 2 2 ⑀ Ð 1 consider in (59) the first such term. It can be repre- 2 2 ------[ ] IΛ = ∫----1---+---⑀------2 2 = x k . k1 π Γ( ⑀) Γ(2⑀) sented as 1 Ð L k1 The subsequent integration of these terms with respect 2 q' Ð 2(k ⋅ q' ) to x can be done without difficulties. ( ⋅ )( ) 1 1 1 Ð q1 q 1 Ð x ------The integrals of the terms with the denominators ˜t1˜t1' 2Σ 2˜ k 2 (B.7) k1 and k1 t1 with respect to 1 cannot be expressed in 1 k ⑀ = (q ⋅ q)(1 Ð x) --- + ------1---- . terms of elementary functions at arbitrary . Neverthe- 1 ˜t ˜ ˜' less, these terms do not create problems. For such 1 xt1t1 terms, it is convenient to make integration with respect The first term on the right-hand side can be integrated to k1, with respect to k1 and then with respect to x. For the sec- 2 + 2⑀ 1 ond, it seems more convenient to begin with the inte- d k1 1 2 ⑀ Ð 1 dz 2 ( ) ------gration with respect to x in (57), getting I Σ = ∫----1---+---⑀------2----- = k ∫ 1 Ð ⑀ , k1 π Γ(1 Ð ⑀)k Σ (xz(1 Ð xz)) 1 0 1 2 2 + 2⑀ dx ( ⋅ )( ) k1 d k1 1 ∫------q1 q 1 Ð x ------2 ------x(1 Ð x) ˜ ˜' I = ∫----1---+---⑀------2 (B.4) xt1t1 k1 t1 π Γ(1 Ð ⑀)k ˜t 0 1 1 (B.8) 1 ( )2 ⑀ Ð 1 q ⋅ q q' Ð k ( 2) dz 1 1 1 = Ðx q1 ∫------⑀ , = ------ln------2 . (xz(1 Ð xz))1 Ð (q' Ð k )2 Ð (q Ð k )2 (q Ð k ) 0 1 1 1 1 1 1 then to introduce the variable y = xz instead of z and to change the order of the integrations with respect to x With the help of the representation and y. After that, the integrals can be easily calculated. ( )2 This way, we obtain 1 q1' Ð k1 ------ln------2 2 2 1 2 + 2⑀ 2 2 2 (q' Ð k ) Ð (q Ð k ) (q Ð k ) dx d k x q' q 1 1 1 1 1 1 J = ------1------------1------2- (B.9) 1 ∫ ( ) ∫ 1 + ⑀ 2 1 x 1 Ð x + π Γ(1 Ð ⑀) 4k Σ 0 1 1 = dz------, 2 2 ∫ 2 2 2 2 2 2 2 z(q' Ð k ) + (1 Ð z)(q Ð k ) xq' q xq' q x(1 Ð x) q' q 0 1 1 1 1 + ------1------2------+ ------1------2------+ ------1------1 2 2 2 2  Σ  4(1 Ð x)k k 4(1 Ð x)k k 2 1 1 the integration with respect to k1 and the subsequent

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2170 FADIN, GORBACHEV integration with respect to z become trivial and give ( ⋅ )( ⋅ ) ) 2 2 2( ⋅ ) Ð q1' q2 k1 q1 + xq2' q1 Ð xk q1 q 1 2 + 2⑀ (B.13) dx d k 2 J = Ð ------1------(q ⋅ q)(1 Ð x) (1 Ð x)k Γ(⑀)Γ( ⑀) 3 ∫ ( ) ∫ 1 + ⑀ 1 2( ⋅ ) 2 2 1 1 + x 1 Ð x + π Γ(1 Ð ⑀) + q1 q1' q Ð q1' q2 ------= ------0 Σ 2Γ(2 + 2⑀)

2 ' ( ⋅ ' ) 2 ( ⋅ ) (B.10) 2 2 q1 Ð 2 k1 q1 Γ (1 + ⑀) q q ⑀ ( ) ⑀ ⑀ × ------= Ð------1------× 2( 2) q Ð q1 (( 2) ( 2 2) ( 2)1 + ) ˜ ˜ Γ ⑀ ⑀ ⑀ q1' q1 + ------q1 q1 + q2 Ð 2 q2 , t1t1' (1 + 2 ) (1 + 2 ) ( 2 2) q1 Ð q2 2 ⑀ 2 ⑀ × [( ) ( ) ] 1 2 + 2⑀ q Ð q1 . dx d k (1 + ⑀)(1 Ð x) ------1------∫ ( )∫ 1 + ⑀ 2 Analogously, we obtain x 1 Ð x π Γ(1 Ð ⑀) 2 k ˜t 0 1 1 2 + 2⑀ 2 2 dx d k1 (1 + ⑀)(1 Ð x)  2 2  ------2 2 k1 k2 q1' J4 = ∫------∫----1---+---⑀------× ( ⋅ )( ( ) ⋅ ) x(1 Ð x) π Γ( ⑀) 4  q1 Ð 2k1 q1 q1' 1 Ð x Ð 2q1' L + ------ 1 Ð  Σ  0 (B.11) 2 2 (q' Ð 2(k ⋅ q' ))(q Ð 2(k ⋅ q )) Γ(⑀)Γ( ⑀) ⑀ ⑀ × ------1------1------1------1------1------1----- 2 + ( ⑀ 2 2(( 2) ( 2) ) = ------Γ----(------⑀---)---- 2(1 + )q1 q2 q1 Ð q2 ˜t1˜t1' 4 4 + 2

⑀ 1 + ⑀ 1 + ⑀ 2 2 1 + 2 2 2 (B.14) Γ ⑀ ( ⑀)((q' ) + (q ) Ð (q ) ) ⑀ ⑀  (1 + ) 1 + 1 1 ⑀( 2 2)(( 2) + 1 ( 2) + 1)) k = Ð ------. Ð q1 + q2 q1 Ð q2 ------Γ(1 + 2⑀) 8⑀ (1 + 2⑀)(3 + 2⑀) ( 2 2)3 q1 Ð q2 2 ˜ The terms in fa(k1, k2) with the denominator k t1 and n 2 2 with x in numerators at natural n can be calculated by 2 2 q' Ð q'  ×( 2 2 2 2 ) 1 2  performing the integration with respect to k at fixed q1 + q2 + 2q1' + 2q2' Ð 2k Ð 2q Ð ------1 ( 2 2)2 Feynman parameter z and then making the change of q1 Ð q2 variable y = xz: (( 2)⑀ + 1 ( 2)⑀ + 1) q1 Ð q2 2 2 2 1 2 + 2⑀ n + ------(⑀(q' + q' Ð k ) dx d k x 2 2 1 2 ------1------(q Ð q ) ∫ ( )∫ 1 + ⑀ 2 1 2 x 1 Ð x π Γ(1 Ð ⑀)k ˜t 0 1 ( ⑀)( 2 2) ) 1 1 (B.12) --Ð 2 1 + q Ð q1 . xn + 1 = Ð ∫dx∫dz------⑀ { [ ( ) 2 ( ) 2]}1 Ð All the calculations discussed before were done xz x 1 Ð z q2 + 1 Ð x q1 0 0 exactly at arbitrary ⑀. This cannot be done for the 1 1 n remaining terms. They contribute into the function I(q1, ⑀ Ð 1 x q2; q) [see (62)]. We have used the following equality: = Ð∫dyy ∫dx------1---Ð---⑀ . [x(q 2 Ð q 2) + q 2 Ð yq 2] 0 y 2 1 1 2 1 2 + 2⑀ 2 dx d k (q2) The change of variable y = xz has been performed in the ------1------∫ ( ) ∫ 1 + ⑀ ˜ ˜ last equality. This integral can now be calculated inte- x 1 Ð x + π Γ(1 Ð ⑀)8t1t1' grating first with respect to x and then with respect to y. 0 The complete calculation for all such terms in (59) is 1 + ⑀ Γ2(⑀) (q2) 2 long, but straightforward. The integration of the terms = Ð ------Ð 2ψ(1) + 2ψ(1 Ð ⑀) Γ(2⑀) 16 ⑀ 2 2 2 2 (B.15) 1 2 2(1 Ð x)k 1 + ⑀ k k q' ------Ð q' q ------1- + ------(1 Ð x)----1------2------1-- 2 1 2 Σ 2 Σ --Ð 4ψ(1 + ⑀) + 4ψ(1 + 2⑀) k ˜t1 can be done quite analogously, since under transforma- 2 2 2 + 2⑀ 2 2 tion (49) they acquire the form of the terms discussed (q ) d k1 ln(l /q ) Ð ------∫------⑀------. above. In this way, we obtain 8 π1 + Γ( ⑀)( )2( )2 1 Ð q1 Ð l q1' Ð l 1 2 + 2⑀ dx d k 1 ------1------[2((q ⋅ q )(k ⋅ q' ) ∫ ( )∫ 1 + ⑀ 2 1 2 1 1 This equality can be obtained performing the integra- x 1 Ð x π Γ(1 Ð ⑀)k ˜t 0 1 tion with respect to x first, using the representation

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NONFORWARD COLOR-OCTET KERNEL 2171

(B.9) and the integral The integrals (B.17) and (B.18) are calculated for arbi- 2 ⑀ ⑀ trary small k and for x arbitrarily close to zero or unity. d2 + 2 l ln(l2/q2) Γ2(⑀) (q2) ------= Ð------The approximate form (64) for I(q , q ; q) can be ∫ 1 + ⑀ 2 2 Γ( ⑀) 2 1 2 π Γ(1 Ð ⑀)l (q Ð l) 2 2 q (B.16) obtained using (B.17)Ð(B.19) and the relation 1 1 × -- Ð 2(ψ(1) Ð ψ(1 Ð ⑀) + ψ(1 + ⑀) Ð ψ(1 + 2⑀)) .  2( ) 2  ⑀ xdx q1 1 Ð x + q2 x ∫------ln------ (q (1 Ð x) Ð q x)2  k2 x(1 Ð x)  The last integral can be easily obtained with the help of 0 1 2 the generalized Feynman parametrization. Using the integrals calculated above, we come to the 2 2 2 1 k + q Ð q dx representation (61) for the two-gluon contribution to = ------1------2------2 ∫( ( ) )2 the kernel. 2k q1 1 Ð x Ð q2 x ⑀ 0 (B.20) In the limit 0, the expression (64) for I(q1, q2; 2 2 2 2 q) was obtained by performing the integration with q (1 Ð x) + q x 1   q   q  × ln----1------2---- Ð ------L1 Ð ----1- Ð L1 Ð ----2- respect to k in (62) first. We have  2 ( )  2  2  2 k x 1 Ð x 2k q2 q1 2 + 2⑀ 2 d k 1 1 Ð x Γ (⑀) 2⑀ Ð 1 2 ⑀ ------1------Ӎ Ð------------x (k ) ∫ 1 + ⑀ 2 2 2 2 Γ ⑀ 2 2 π Γ(1 Ð ⑀)k ˜t k q k  (2 ) q q  1 1 1 1  1 2 Ð ------2 ln ------4---- . 2 2 4k  k  (q Ð q ) 2 + ------1------2------ln(x(q2(1 Ð x) + q 2 x) ( 2 2 ) 1 2 q1(1 Ð x) + q2 x (B.17) ⑀ APPENDIX C ( 2 ) 2 Ð q1(1 Ð x) 2k(q1(1 Ð x) + q2 x) + ------Ð ------Due to the symmetries (12) of the kernel in (61), it ⑀ ( )2 q1(1 Ð x) + q2 x is sufficient to show that the kernel vanishes at zero ⑀ transverse momentum of one of the Reggeons, say, at q 2(q 2(1 Ð x) + q 2 x) 1 Ð (q2(1 Ð x))  q = 0. But even in this special case, integration in (62) × ln----1------1------2------+ ------1------, 2 2 ⑀ cannot be performed in terms of elementary functions. xk  Fortunately, expression (62) can be simplified at q2 = 0 ⑀ and represented as d2 + 2 k k 1 1 Ӎ ( ) ∫------⑀------Ðx 1 Ð x 2 1 + 2 2 Γ (⑀) 2 2 ⑀ π Γ(1 Ð ⑀)k ˜t1k ( , ) ( ) [ψ( ) ψ(⑀) 1 I q1 q2; q = Ð------q k 1 + q2 = 0 2Γ(2⑀)  q q2(q 2(1 Ð x) + q 2 x) × ------1------ln----1------1------2------( 2)2 2( ( ) )2 2 ψ( ⑀) ] q 1 q1 q1 1 Ð x + q2 x xk Ð 2 2 Ð ------⑀------(C.1) 4 π1 + Γ(1 Ð ⑀) 2 ⑀ 1 Ð (q (1 Ð x)) k 1 2 + 2⑀ 2 2 + ------1------+ ------d l l (l Ð k)  ⑀ 2 2 2 × ------ln ------. xk (q (1 Ð x) + q x) ∫ 2 2  4  1 2 (B.18) l (l Ð q) q

2 2 2  x(q (1 Ð x) + q x) 1 Note, that at q = 0, we have q' = Ðq and q' = k Ð q. ×  ln------1------2------+ -- 2 2 1  2( ) ⑀ q1 1 Ð x Evidently, the integral term in (C.1) rules out, for the piece of the kernel (61) that does not change under the 2 2 1 q (1 Ð x) + q x  substitution q q' , the possibility of vanishing Ð ------ln----1------2---- , i i ( ( ) )2 ( ) 2 alone at q = 0. Therefore, we need to calculate I'(q , q1 1 Ð x + q2 x x 1 Ð x k  2 1 ≡ q2; q) I(q1' , q2' ; Ðq) at q2 = 0. This function can also 2 + 2⑀ 2 2 d l ln(l /q ) be simplified. Note that, as in the preceding case, sim- ∫----1---+---⑀------π Γ(1 Ð ⑀)(q Ð l)2(q' Ð l)2 plifications cannot be done for separate terms in (62) 1 1 and are possible only due to the definite combination of (B.19) them. We obtain 2 q 2q'   q2  Ӎ 1 1 2 1 1 1 2 1 2 ⑀ ----- -- + lnq  ln------ + -- ln ------ . Γ (⑀) 2 2 2 ⑀ 2 2 2 2 I(q' , q' ; Ð q) = ------q [ ( q' )(ψ(⑀) Ð ψ ( 2⑀)) q  ( )    1 2 q2 = 0 1 q q1' 2Γ(2⑀)

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2172 FADIN, GORBACHEV

( 2)⑀(ψ ψ ⑀ ) ( 2)⑀(ψ ψ ⑀ 6. V. S. Fadin and L. N. Lipatov, Nucl. Phys. B 477, 767 Ð k (1) Ð (2 ) Ð q (1) Ð (1 Ð ) (1996); V. S. Fadin, M. I. Kotsky, and L. N. Lipatov, ( 2)2 (C.2) Phys. Lett. B 415, 97 (1997); Yad. Fiz. 61, 716 (1998) ψ ⑀ ψ ⑀ ) ] q 1 [Phys. At. Nucl. 61, 641 (1998)]. + 2 ( ) Ð 2 (2 ) Ð ------⑀------4 π1 + Γ(1 Ð ⑀) 7. S. Catani, M. Ciafaloni, and F. Hautmann, Phys. Lett. B 242, 97 (1990); Nucl. Phys. B 366, 135 (1991); G. Cam- ⑀ ici and M. Ciafaloni, Phys. Lett. B 386, 341 (1996); d2 + 2 l  l2  × ∫------ln------ . Nucl. Phys. B 496, 305 (1997); V. S. Fadin, R. Fiore, l2(l Ð q)2 (l Ð k) 2 A. Flachi, and M. I. Kotsky, Phys. Lett. B 422, 287 (1998); Yad. Fiz. 62, 1066 (1999) [Phys. At. Nucl. 62, In the sum of (C.1) and (C.2), the terms with ln(l Ð k)2 999 (1999)]. cancel each other; after that, the integrals can be calcu- 8. V. S. Fadin and L. N. Lipatov, Phys. Lett. B 429, 127 (1998); M. Ciafaloni and G. Camici, Phys. Lett. B 430, lated and we obtain 349 (1998). (I(q , q ; q) + I(q' , q' ; Ð q)) 9. D. A. Ross, Phys. Lett. B 431, 161 (1998); Yu. V. Kov- 1 2 1 2 q2 = 0 chegov and A. H. Mueller, Phys. Lett. B 439, 423 (1998); 2 J. Blümlein, V. Ravindran, W. L. van Neerven, and Γ (⑀) 2 2 ⑀ 2 ⑀ A. Vogt, Preprint DESY-98-036; hep-ph/9806368; = ------q [(q' ) (ψ(⑀) Ð ψ(2⑀)) Ð (k ) (2ψ(1) 2Γ(2⑀) 1 E. M. Levin, Preprint TAUP 2501-98 (1998); hep- ph/9806228; N. Armesto, J. Bartels, and M. A. Braun, ⑀ (C.3) + ψ(⑀) Ð 3ψ(2⑀) ) Ð (q2) (2ψ(1) Phys. Lett. B 442, 459 (1998); G. P. Salam, hep- ph/9806482; M. Ciafaloni and D. Colferai, Phys. Lett. B Ð 2ψ(1 Ð ⑀) + 3ψ(⑀) Ð 3 ψ ( 2 ⑀ ) ) ] . 452 , 372 (1999); M. Ciafaloni, D. Colferai, and G. P. Salam, Preprint DFF-338-5-99 (1999); hep- With this result, it is a simple task to show that the ker- ph/9905566; R. S. Thorne, Preprint OUTP-9902P nel (61) vanishes at q2 = 0 [and, due to the symmetries (1999); hep-ph/9901331; S. J. Brodsky, V. S. Fadin, in (12), at zero transverse momentum of any of the V. T. Kim, et al., Preprint SLAC-PUB-8037; IITAP-98- Reggeons]. 010; hep-ph/9901229. 10. M. T. Grisaru, H. J. Schnitzer, and H. S. Tsao, Phys. Rev. Lett. 30, 811 (1973); Phys. Rev. D 8, 4498 (1973); REFERENCES L. N. Lipatov, Yad. Fiz. 23, 642 (1976) [Sov. J. Nucl. Phys. 23, 338 (1976)]. 1. V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, Phys. Lett. 11. Ya. Ya. Balitskii, L. N. Lipatov, and V. S. Fadin, in Pro- B 60B, 50 (1975); E. A. Kuraev, L. N. Lipatov, and V. S. Fa- ceedings of Leningrad Winter School, Physics of Ele- din, Zh. Éksp. Teor. Fiz. 71, 840 (1976) [Sov. Phys. JETP mentary Particles, Leningrad, 1979, p. 109. 44, 443 (1976)]; Zh. Éksp. Teor. Fiz. 72, 377 (1977) [Sov. Phys. JETP 45, 199 (1977)]; Ya. Ya. Balitskii and 12. J. Blümlein, V. Ravindran, and W. L. van Neerven, Phys. L. N. Lipatov, Yad. Fiz. 28, 1597 (1978) [Sov. J. Nucl. Rev. D 58, 091502 (1998); V. Del Duca and C. R. Schmidt, Phys. 28, 822 (1978)]. Phys. Rev. D 59, 074004 (1999). 13. V. S. Fadin and R. Fiore, Phys. Lett. B 440, 359 (1998). 2. L. N. Lipatov and V. S. Fadin, Pis’ma Zh. Éksp. Teor. Fiz. 49, 311 (1989) [JETP Lett. 49, 352 (1989)]; Yad. 14. M. Braun, hep-ph/9901447. Fiz. 50, 1141 (1989) [Sov. J. Nucl. Phys. 50, 712 15. V. S. Fadin, R. Fiore, M. I. Kotsky, and A. Papa, Preprint (1989)]. BUDKERINP/99-61 (Novosibirsk, 1999); UNICAL-TH 3. V. S. Fadin and L. N. Lipatov, Nucl. Phys. B 406, 259 99/3 (1999); hep-ph/9908264. (1993); V. S. Fadin, R. Fiore, and M. I. Kotsky, Phys. 16. V. S. Fadin, R. Fiore, M. I. Kotsky, and A. Papa, Preprint Lett. B 389, 737 (1996). BUDKERINP/99-62 (Novosibirsk, 1999); UNICAL-TH 99/4 (1999); hep-ph/9908265. 4. V. S. Fadin, Pis’ma Zh. Éksp. Teor. Fiz. 61, 342 (1995) [JETP Lett. 61, 346 (1995)]; V. S. Fadin, R. Fiore, and 17. M. Braun and G.P. Vacca, Phys. Lett. B 454, 319 (1999). A. Quartarolo, Phys. Rev. D 53, 2729 (1996); M. I. Ko- 18. V. S. Fadin, R. Fiore, and A. Papa, Phys. Rev. D 60, tsky and V. S. Fadin, Yad. Fiz. 59, 1080 (1996) [Phys. At. 074025 (1999). Nucl. 59, 1035 (1996)]; V. S. Fadin, R. Fiore, and 19. M. Braun and G. P. Vacca, hep-ph/9910432. M. I. Kotsky, Phys. Lett. B 359, 181 (1995); Phys. Lett. 20. J. Bartels, Nucl. Phys. B 175, 365 (1980). B 387, 593 (1996). 21. V. S. Fadin, Preprint BUDKERINP 98Ð55 (Novosibirsk, 5. V. S. Fadin and R. Fiore, Phys. Lett. B 294, 286 (1992); 1998); hep-ph/9807528; in Proceedings of the Interna- V. S. Fadin, R. Fiore, and A. Quartarolo, Phys. Rev. D 50, tional Conference “LISHEP’98,” Rio de Janeiro, 1998 5893, 2265 (1994). (in press).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

ELEMENTARY PARTICLES AND FIELDS Theory

Strong-Coupling Constant in Coordinate Space* A. M. Badalian Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117959 Russia Received April 27, 2000

Abstract—The coordinate-space behavior of (vector) strong-coupling constant in the background ﬁeld α (r) α B is compared with that in standard perturbation theory V(r). The numerically calculated two-loop coupling con- α α Շ stant B(r) is shown to exceed V(r) by 1Ð5% at very small distances, r 0.02 fm, and to be in agreement with α տ lattice measurements of the static potential. At large distances, B(r) approaches the freezing value at r 0.5 fm. α α Շ տ An analytic form of B(r) is proposed that approximates B(r) with a precision 2% in the region r 0.5 fm. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Ӷ Λ Ð1 only in the region r 0.3 fm = R , while the sizes of Precise knowledge of the static potential between quarkonium states below the open-charm or the open- heavy quarks is very important in quarkonium physics beauty threshold are much larger and change from 0.2 because this potential determines the spectrum and the to 1 fm in bottomonium and from 0.4 to 1 fm in char- wave functions of the QQ system. Usually, fundamen- monium. tal quarkonium properties are calculated in coordinate A precise determination of the strong-coupling con- space, where we need to know both perturbative and stant at relatively large distances (or at small momenta nonperturbative contributions to the static potential. As in momentum space) cannot be performed within stan- is well established now [1], the nonperturbative poten- dard perturbation theory; therefore, a large variety of tial VNP(r) has a linear behavior beginning from rather potentials were used in phenomenological calculations small distances, r տ 0.2 fm, and, at smaller distances, of quarkonium properties. Among the most popular the exact form of V (r) is not very important in potentials, we shortly discuss here only QCD-moti- NP vated potentials. quarkonium physics since VNP(r) is much smaller than the perturbative part of interaction. Still, there is the (i) For the Cornell potential [7] point of view that, at small distances, the linear poten- 4α tial σ r also exists, with σ being approximately equal V ()r =Ð ------++σrC, 0 0 Corn 3 r to the asymptotic value of the string tension σ (see [2] α and references therein). the running coupling constant s(r) is taken to be a con- ≥ The situation with the perturbative static potential, stant over the entire region r 0; yet, this potential VP(r), is much more uncertain. In perturbation theory, gives an excellent description of the QQ spectrum with Ð1 only one drawback: the wave function at the origin, V (r) is known only at small distances, r Ӷ Λ , where P R ψ(0); and therefore the leptonic widths for the Cornell Λ R is QCD constant in coordinate space which is signif- potential are found to be larger than in the case where α icantly larger than Λ in MS renormalization scheme the asymptotic-freedom behavior of s(r) is taken into MS account [8]. As a whole, one can conclude that the [3] (see also Section 3): freezing phenomenon of the strong-coupling constant () () is extremely important in quarkonium physics. nf nf a Λ = Λ exp ------1+ γ ; R MS 2β E (ii) For the Richardson potential defined in momen- 0 tum space as [9] Λ ()4 Λ()4 ≈ 44 π 1 for the number of flavors nf = 4, R = 2.536 V ()q = Ð ------; MS Rich β 2 2 2 710 ± 100 MeV if the conventional two-loop value of 3 0 q ln()1 + q / Λ ()4 2 Λ = 280 ± 40 MeV is used [4]. Therefore, the pertur- 44 π 1 Λ 2 2 MS V ()q ≈Ð------()q Ӷ Λ , Rich β 2 2 bative static potential calculated recently in the two- 3 0 q q loop approximation [5, 6] is, in strict sense, defined the nonperturbative part (~1/q4) corresponds to the lin- * This article was submitted by the authors in English. ear potential σr, and it can be subtracted, so that the

1063-7788/00/6312-2173 $20.00 © 2000 MAIK “Nauka/Interperiodica” 2174 BADALIAN α perturbative coupling constant, Rich(q), can be defi- bination (q2 + 4m 2 ), i.e., the doubled effective gluon ned as g mass, 2mg, plays a role of the background mass. In 4π 1 Λ2 recent calculations of dynamical gluon masses (which α (q) = ------Ð ----- . (1) Rich β 2 2 2 depend on chosen quantum numbers) in [22], the low- 0 ln(1 + q /Λ ) q est gluon mass mg = 0.53 GeV was calculated, so that This coupling constant is finite at q = 0: α (q = 0) = 2mg = 1.06 GeV. This number is in surprising agree- π β Rich ≈ 2 / 0—that is, it has a freezing property—and has the ment with mB 1.0 GeV obtained in [17Ð19]. correct asymptotic behavior at high momenta, q2 ӷ Λ2, In the framework of perturbation theory, an analytic effective coupling constant was constructed with the 4π q2  Ð1 α (q ∞) = ------ln----- . use of an “analytization” procedure [23]. This analytic Rich β  2 0 Λ α (q) is finite in the infrared limit and the value of αan π β an(q = 0) was found to be 4 / 0. In this approach, one- However, the nonperturbative part of the Richard- α loop and two-loop expressions of s(q) are modified, son potential is a Lorentz vector by definition, while, as but they do not contain any background mass. Note also is now established in QCD, the linear potential must be α π β that the freezing value of an(q = 0) = 4 / 0 is about a Lorentz scalar [10, 11]. three times larger than the freezing values predicted in (iii) In the calculations of meson properties by God- [16, 18] and this significant difference can give rise to frey and Isgur [12], the dependence of α (q) on q was different predictions of some observed experimental α s defined phenomenologically: s(q) was represented as values. the sum of exponentials and had also the property of The theoretical formula describing the behavior of freezing—in particular, at low momenta the strong-coupling constant in the background field, α α ( ) denoted as B(q), was deduced in [16] at large s q 0 const = 0.60. 2 տ 2 α α momenta, q mB , and to find the behavior of B(q) at (iv) In [8], the asymptotic-freedom behavior of s(r) α ≥ 2 Շ 2 at small r and the freezing of s(r) at r r0 were taken smaller momenta, q mB , an interpolation of this into account, which resulted in a good description of 2 տ 2 the quarkonium spectra and leptonic widths. expression (valid at q mB ) was used. Therefore, the precise number of the freezing coupling constant value, (v) In lattice calculations of the static potential [13, α 14], it was observed that, at distances r տ 0.2 fm, the B(q = 0), is not still theoretically fixed in BFTh. To get strong-coupling constant, taken as a constant, repro- this number, phenomenological analysis and lattice duces lattice measurements with a good accuracy. measurements of static potential at small separations can be very useful. Recently, lattice data on the static Thus, one can conclude that, in phenomenological potential in the range 0.04 ≤ r ≤ 0.4 fm were published calculations and in lattice QCD, two important proper- in [24], and from the analysis of these data the back- ties of the strong-coupling constant are needed: the ground mass m can be estimated. asymptotic-freedom behavior at small r and the freez- B ing of α (r) at relatively large distances. Our paper is mostly concentrated on the properties s α˜ On the fundamental level, the freezing phenomenon of B (r) in BFTh with a preliminary discussion of the was predicted in the framework of background field two-loop coupling constant in momentum and coordi- theory (BFTh) in [15], where the static potential in vac- nate spaces in perturbation theory, and also on the α uum background fields was considered. Later, this approximations to ˜ B (r) at large and small distances. approach was applied to the process e+e– hadrons [16], where it was shown that the presence of a back- α ground field results in the appearance of the back- 2. TWO-LOOP V(q) IN MOMENTUM SPACE ground mass mB, so that the strong-coupling constant in The perturbative potential VP(q) can be used to 2 α 2 define a vector coupling constant V(q): this case depends on the combination (q + mB ) instead 2 α ( ) of q dependence in perturbation theory. The value of ( ) π V q V P q = Ð4 CF------, (2) mB was found to be defined by the hybrid excitations 2 ≈ q [17], and the value of mB 1.0Ð1.1 GeV was also extracted from fine-structure splittings in charmonium where q2 ≡ q2. The coupling constant, expressed through [18] and bottomonium [19]. α µ MS ( ) in the renormalization scheme MS , was Similar ideas were discussed in [20, 21], where the recently calculated in the two-loop approximation [5, 6]: α 2 freezing of s(q ) was considered as a natural consequence of a picture where the gluon acquires an effec-  α (µ) α (µ) 2  α ( ) α (µ) s s  tive dynamical mass mg. In [20, 21], it was suggested V q = s 1 + C1------π----- + C2------π----- . (3) α 2  4 4  (not deduced as in [16]) that s(q ) depends on the com-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 STRONG-COUPLING CONSTANT 2175 α α µ ≡ α µ ning vector coupling constant V(q), can be expressed Here, s( ) MS ( ) and ( ) n f β through Λ [3]: C1 = a1 + 0 y, MS 2 2 (4) β (β β ) (n ) (n ) C2 = a2 + 0 y + 1 + 2 0a1 y, Λ f Λ f ( β ) V = MS exp a1/2 0 ; (12) and in (4), the variable y is or in the case n = 0 y = ln(µ2/q2). (5) f ( ) The constants a1 and a2 are given by Λ (0) Λ (0) 0.963 77 V = 1.5995 MS = ------. (13) 31 10 r0 a = ----- Ð -----n , 1 3 9 f ≈ –1 If r0 is taken to be r0 = 0.5 fm 2.54 GeV as in [25], 4 4343 2 π 22 then a = 9 ------+ 4π Ð ----- + -----ζ(3) 2 162 4 3 Λ (0) ± Λ (0) ± (6) MS = 237 19 MeV, V = 379 30 MeV. (14) 3 1798 56ζ( ) Ð --n f ------+ ----- 3  2 81 3 In our calculations below (see Section 6), we will use the following values: 1 55 ζ( ) 100 2 Ð --CFn f ----- Ð 16 3  + ------n f . 2 3 81 Λ (0) Λ (0) MS = 240.7 MeV, V = 385 MeV, (15) In (6), CF is the Casimir operator of the SU(N) group. In particular, in the quenched approximation (nf = 0) which satisfy relation (13). where the lattice measurements exist [24], … Λ β V a1 = 31/3, a2 = 456.74884 . (7) Since V and the coefficients n are known, the α In (6), ζ(3) =1.202057 denotes the Riemann ζ function. running vector coupling constant V(q) can be also On the other hand, if the definition of βV function is written in the form used, α ( ) ∞ V q n + 2 V 2∂α (µ) V α (µ) β = µ ------V------= Ð4π β ----V------, 2 V 2 ∑ n   (8) 4π  β lny˜ β 2 β β  (16) ∂µ 4π = ------1 Ð ----1------+ ------1--- ln y˜ Ð lny˜ + ----0------2- , n = 0 β ˜ 2 ˜ 4 2 2 0 y β y β y˜ β  then with the help of expression (3) the coefficients of 0 0 1 βV β V the QCD function, n (n = 0, 1, 2), can be found [6]: β V where 2 is given by number (10) and β V = β , β V = β , β V = β Ð a β + β (a Ð a 2). (9) 0 0 1 1 2 2 1 1 0 2 1 ( 2 Λ 2 ) y˜ = ln q / V . (17) Here, β ≡ β MS, and for n = 0, we have n n f Note that, in (16), the small correction coming from the β β β 3 V 4 0 = 11, 1 = 102, 2 = 2857/2. π β β last term, (4 /y˜ )( 2 / 0 ), is about three times larger Correspondingly, for the vector coupling constant than that in the MS renormalization scheme. β V β V 0 = 11, 1 = 102 are the renormalization scheme (RS) invariants, and the coefficient α˜ β V … 3. TWO-LOOP V (r) IN COORDINATE SPACE 2 = 4224.1817 (10) β MS The vector coupling constant in coordinate space, appears to be about three times larger than 2 . α ˜ V (r), is also defined through the static potential: ( ) The value of Λ 0 was recently calculated with a good MS α ( ) ˜ V r accuracy in the finite-size lattice technique [25] (nf = 0): V (r) = ÐC ------. (18) P F r ( ) Λ (0) 0.602 48 MS = ------, (11) r0 By introducing the Fourier transform VP(q), where r0 denotes the Sommer scale. With the use of 1 iq ⋅ r ( ) V (r) = ------V (q)e dq, (19) Λ n f P 3∫ P expression (3), QCD constant V , defining the run- (2π)

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2176 BADALIAN it can be shown that α˜ (r) and α (q), given by (3) or Λ (0) Λ (4) V V From here (for nf = 0 and nf = 4), R and R are (16), satisfy the relation Λ about 2.5Ð2.8 times larger than MS : 2 sin(qr) 2 sinx α˜ ( ) α ( ) α ( ) ( ) ( ) ( ) ( ) V r = π--∫ V q ------dq = π--∫ V y ------dx, Λ 0 Λ 0 Λ 4 Λ 4 q x R = 2.8488 MS , R = 2.5359 MS . (26) 2 2 (20) ( ) µ r If Λ 0 = 240.7 MeV is taken as in [25], then in y(x) = ln------. MS x2 coordinate space the QCD constant is ( ) The integral in (20) can be analytically calculated if Λ 0 = 686 MeV. (27) α R V(q) is taken in the two-loop approximation as in (3) α˜ (see Appendix), and the following expression for V (r) 4. α (q) IN BACKGROUND-FIELD THEORY µ B was obtained in [6, 26] ( = 1/r): The properties of the coupling constants in momen- α ( ) ˜ V r tum and in coordinate spaces have been discussed above in the framework of perturbation theory, which is appli- α ( ) α ( ) 2 ӷ Λ 1  1/r  1/r   (21) cable only at large momenta, q V, or at small dis- = α -- 1 + A -----M---S------+ A -----M---S------. tances, Λ r Ӷ 1. At smaller q2, the influence of vacuum MS  1 π 2 π  R r  4 4  background fields on the interaction becomes important, and, as a result, the vector coupling constant is modified Here, the coefficients A1 and A2 are the following con- [15, 16]. The static potential V (q) in the presence of stants: B background fields can be written just as VP(q) in (2): γ β A1 = a1 + 2 E 0, α (q) ( ) π -----B------3 V B q = Ð4 CF 2 . (28) 2π 2 (22) q A = a + β ----- + 4γ + 2γ (β + 2β a ). 2 2 0  3 E E 1 0 1 It is important that VB(q) does not include a purely con- Then, with the use of expansion (21), the coefficients of fining (scalar) contribution and takes into account only the two-loop βR function in coordinate space can be cal- the influence of background fields on the perturbative culated [26]: (vector) part of interaction. In this respect, VB(q) essentially differs from the Richardson potential discussed in ∞ n + 2 the Introduction and takes into account an interference of R 2∂α˜ (µ) Rα˜ (µ) β = µ ------V------= Ð4π ∑ β ----V------, (23) perturbative and nonperturbative effects at small dis- ∂µ2 n  4π  n = 0 tances. As is shown in [16], the strong-coupling constant in β R β V β β R β V β with the following result: 0 = 0 = 0, 1 = 1 = 1. the presence of a background field, denoted in (28) as They are RS invariants, while α 2 2 B(q), depends on the combination (q + mB ), where R MS 2 β = β Ð A β + β (A Ð A ). (24) mB is a background mass. This mass can in general 2 2 1 1 0 2 1 depend on a considered channel and in the case of static In the quenched approximation (n = 0), we have β R = potential mB is defined by the lowest hybrid excitation f n [16, 17] with m ≈ 1.0 GeV. Just this value of m was 8602.9962…, A = 23.032079…, and A = B B 1 2 extracted from fine-structure analysis of charmonium β R 1396.27376…. The coefficient 2 appears to be about and bottomonium in [18, 19]. β V In [16, 27], it was deduced that in BFTh the strong- two times larger than 2 in (10) and about six times α coupling constant B(q) at large momenta can be β MS obtained as a generalization of the perturbative expres- larger than 2 . It means that the three-loop correc- 2 2 2 tions to the coupling constant in coordinate space are sion with q replaced by the combination (q + mB ). significantly larger than in the MS renormalization Then the generalization of expansion (3) in momentum scheme. space can be written in the following way: ( ) 2 Λ n f MS MS QCD constant R defining the running coupling  α (µ˜ ) α (µ˜ )  α α MS µ˜  B  B   α˜ B(q) = B ( ) 1 + B1------π------+ B2 ------π------, (29) constant V (r) in coordinate space can be expressed  4  4   (n ) Λ f with through MS [3, 24] and turns out to be large: β ( ) ( ) B1 = a1 + 0t, n f n f  a  Λ = Λ exp ------1-- + γ . (25) (30) R MS  β E β 2 2 (β β ) 2 0 B2 = a2 + 0 t + 1 + 2 0a1 t.

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Here, the variable t is Λ MS Λ (4) As is shown in [18], for nf = 4, B and MS also coin- 2 µ˜ 2 2 2 2 Λ MS ------µ˜ µ ≥ cide within 1Ð2 MeV accuracy and even for nf = 3, B t = ln 2 2 , = + mB mB . (31) ( ) q + mB Λ 3 is only by about 15Ð20 MeV larger than MS ; i.e., they In (30), the coefficients a1 and a2 are the same as in (6), coincide within the theoretical and experimental errors. Therefore we assume here that, in the quenched α MS µ˜ and the running coupling constant B ( ) in the MS approximation, we have renormalization scheme now depends on t : MS (0) MS Λ ≈ Λ ( ) B MS n f = 0 . (38) µ2 2 2 It is essential that in (35) there exists a region of q2 ~ + mB µ˜ t (µ) = ln------= ln------, (32) 2 2 MS 2 2 2 ӷ Λ Λ Λ mB where the condition q V is still satisfied but MS MS the influence of background mass mB under the loga-  β  rithm becomes important. For example, if q = mB = MS 4π 1 lntMS α (µ˜ ) = ------1 Ð ------+ …. (33) Λ V Λ 2 Λ 2 B β 2 1.0 GeV and B = V = 380 MeV, then q / V = 7. This 0t  β t  MS 0 MS α region can be called a preasymptotic one. In (29), B(q) տ MS was defined at q mB. However, the coupling constant In (32), it was also assumed that in BFTh Λ coin- 2 B α 2 ≤ 2 Շ Λ B(q) was not defined at small q : 0 q V . It was cides with Λ in perturbative theory. If m = 0, then α MS B only obtained in [15, 16] that B(q) freezes at q = 0 at α α expression (29) reduces to expression (2); therefore, in some value B(0) but the exact number of B(q = 0) was BFTh the values of β V coefficients [for the coupling not calculated. Our main assumption here will be made n about α (q) behavior at small momenta: the analytic constant α (q)] turn out to be the same as in perturba- B B two-loop expression (29) or (34) for α (q) is assumed tive theory, and the running coupling constant α (q) B B to be valid over all range of momenta q ≥ 0; i.e., expres- can be written as 2 տ 2 sion (34) (obtained for q mB ) is extrapolated to the  β 2 4π 1 lntB 2 տ 2 ≥ α (q) = ------1 Ð ------region of small q : mB q 0. B β t β 2 t 0 B 0 B This assumption has a pronounced effect on the (34) behavior of the static potential in coordinate space and 2 V β 2 β β  needs a special check, in particular, by comparison of + ------1--- ln t Ð lnt Ð 1 + ----0------2- . the theoretical formula for V (r) with lattice measure- β 4 2 B B β 2 B 0 tB 1  ments. Therefore, the existing lattice data on the static potential at small and average distances appear to be of The only difference between (34) and (16) is that the great importance for checking different theoretical con- variable y˜ is replaced by cepts about the freezing of the strong-coupling constant. q2 + m 2 t = ln------B- (ΛV = Λ ). (35) B Λ 2 B V 5. α˜ (r) IN COORDINATE SPACE V B α The strong-coupling constant ˜ B (r) in coordinate Λ MS Here, some remarks about QCD constants B and space in BFTh can be defined through the static poten- V tial as in (18): Λ in BFTh are needed. B α˜ (r) In the region of very large momenta, V (r) = ÐC -----B------, (39) B F r ӷ Λ MS ӷ Λ V ӷ and with the help of the Fourier transform it can be con- q B or q B , and q mB, (36) α nected with B(q) in momentum space, as in (20), ∞ ∞ α α MS 2 sin(qr) 2 sinx and B(q) in (34) and B (q) in (33) coincide with per- α˜ α α B(r) = π--∫dq------B(q) = π--∫dq------B(x). (40) turbative expressions. Therefore, for nf = 5, QCD con- q x 0 0 Λ Λ V α stants V and B in (35) coincide: On the right-hand side of (40), B(x) depends on the variable Λ V (n = 5) = Λ (n = 5), B f V f (µ2 2 ) 2 (37) + mB r MS ˜t(x) = ln------Λ ( ) Λ ( ) 2 2 2 B n f = 5 = n f = 5 . MS x + mB r

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2178 BADALIAN γ instead of the y(x) dependence in (20). The behavior of 1(r) at large distances can be found from (46), In the two-loop approximation [with the use of rep- α˜ ÐmBr resentation (29)], B (r) can be calculated: 1 e γ (m r ӷ 1) = ------Ð ------0, 1 B 2 2 m r mBr (48) MS 1 B α˜ (r) = α µ = -- B B  r if r ∞. (41) 2 γ MS MS Thus, the function (r) is a decreasing function of r,  α (µ = 1/r) α (µ = 1/r)  ≤ 1 ∞ γ γ × ˜ B ˜ B and in the range 0 r < it changes from 1(0) = E to 1 + B1(r)------π------+ B2(r) ------π------. γ ∞  4 4  1( ) = 0: γ ≥ γ ( ) ≥ E 1 r 0. (49) In contrast to the coefficients A1 and A2 in (21), now in ˜ ˜ (41) B1 (r) and B2 (r) depend on the variable r, if the If one defines, as in perturbation theory, QCD constants µ Λ R Λ V renormalization scale = 1/r is chosen. In a general B in coordinate space in terms of B in momentum case, we have space from representation (29), ˜ ( ) γ ( )β R B1 r = a1 + 2 1 r 0, Λ = Λ exp(γ (r)), (50) (42) B V 1 ˜ ( ) γ ( )(β β ) β 2γ ( ) B2 r = a2 + 2 1 r 1 + 2 0a1 + 0 2 r . Λ R then, B depends now on the distance r and only at γ very small r coincides with the perturbative Λ [28]: Here, the functions n(r) (n = 1, 2) are defined by the R integrals (µ = 1/r) Λ R( ) Λ B r 0 = R n (51) γ ( ) 2 sinx ˜ ( ) γ n r = --∫dx------t x (43) Λ E( , γ ( ) γ ) π x = V e r 0 1 r E . ∞ γ Λ R with At large distances (r ) 1(r) 0; therefore, B Λ 2 2 coincides with QCD constant V in momentum space: 1 + mB r ˜t(x) = ln------. R 2 2 2 (44) Λ ( ∞) Λ B r = V . (52) x + mB r γ Λ R The function 1(r) (calculated analytically in the At the average distances, the values of B (r) lie Appendix) can be written in two different forms: Λ Λ Λ Λ R Λ between R and V: V < B < R. 2 2 γ 1 1 + m r The second function 2(r) in (42) is given by the γ (r) = -- ln------B------+ Ei(Ðm r) 1 2 2 2 B expression (see Appendix) mB r 2 2 2 ∞ (45) γ ( , µ ) ( ) ( )k 2 r = 1/r = ln 1 + mB r 1 ( 2 2) γ ÐmBr = -- ln 1 + mB r + E + ∑ ------, ∞ 2 k!k  (Ðm r)k k = 1 + 4ln(1 + m 2 r2)γ + ∑ ------B------ + ξ (r) B  E k!k  2 2 2 k = 1 (53) 1 + mB r γ ( ) ------2 2 2 2 2 1 r = ln 2 2     1 + mB r 1 + mB r mB r ( ) ξ˜ = ln------ + 4ln------ Ei ÐmBr + 2(r).  m 2 r2   m 2 r2  ÐmBr 1 1  1  (46) B B Ð ------+ e + ------2 + O 3 , mBr ( ) (m r) ξ mBr B Here, the function 2(r) is defined by the integral if m r ӷ 1. B ξ ( ) 2 sinx 2 ( 2 2 2) 2 r = π--∫dx------ln x + mB r , (54) γ γ x Here, E is the Euler constant. To find 1(r) at small r, it is convenient to use expression (45), from which it is while γ γ evident that 1(r 0) E and with the lowest correction taken into account γ (r) is  2  1 ˜ 2 sinx 2 x ξ2(r) = -- dx------ln 1 + ------ . (55) π∫ 2 2 γ ( ) γ x  m r  1 r 0 = E Ð mBr. (47) B

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 STRONG-COUPLING CONSTANT 2179 ξ γ The function 2(r) and therefore 2(r) can be analyti- 7. OUR CALCULATIONS cally calculated only at small and large distances: The running coupling constant α˜ (r) can be numer- γ ( ) ξ ( ) B 2 r 0 2 r 0 ically calculated. To this end, it is convenient to use for (56) α γ 2 π2 B(q) expression (34) in integral (40). Also, the last = 4 E + /3 = 4.622580. ˜Ð3 At large distances, as seen from expression (54) (see term in (34), proportional to t , will be neglected here also Appendix), for two reasons. First, its contribution to the coupling constant is less than 1Ð5% over all region q2 ≥ 0; the ξ˜ ( ∞) γ ( ∞) 2 r 0, 2 r 0; (57) second reason is that two-loop expression used here, therefore, expression (42) at large distances simplifies, (2) 4π  β lnt  α (q) = ------1 Ð ----1------B- , (66) B β 2 ˜ ( ∞) 0tB β tB B2 r a2. (58)  0  γ It is important to stress here that the dependence 1(r) with on r is quite pronounced even at rather small distances; 2 2 –1 e.g., if m = 1.0 GeV and r = 0.1 GeV = 0.02 fm, then q + mB B t = ln------, (67) the correction to the function B Λ 2 V γ ( ) γ 1 r = E Ð mBr (59) contains only RS invariant coefficients β and β . If n = Λ 0 1 f coming from the second term is about 20%, and it 0 in (67), V = 385 MeV will be taken. grows for larger distances. α(2) In the same approximation, ˜ B (r) in (40) can be written as 6. RUNNING COUPLING CONSTANT IN BFTh ∞ ( )  β ( )  ˜ ˜ 2 8 sinx 1 1 lnt x The functions B1 (r) and B2 (r) in expansion (41) α˜ ( ) ------ ------ B r = ∫dx 1 Ð 2 , BR β x t(x) β t(x) define the coefficients of β function in BFTh: 0  0  0 (68) ∞ ∂α˜ (µ˜ ) α˜ (µ˜ ) n + 2 2 2 2 βBR µ˜ 2 B π β BR B  ( ) x + mB r = ------= Ð4 ∑ n ------ , (60) t x = ln------. ∂µ˜ 2 4π Λ 2 2 n = 0 V r where β and β are RS invariants, while β BR is RS This integral was numerically calculated in the 0 1 2 quenched approximation (n = 0), and the coupling con- dependent, f (2) stant α˜ (r) is shown by the dotted curve in the figure β BR β MS β ˜ ( ) β [ ˜ ( ) ˜ 2( )] B 2 = 2 Ð 1 B1 r Ð 0 B2 r Ð B1 r , (61) over the range 0 ≤ r ≤ 1 fm, along with two different and has different limits at small and large distances. At approximate curves. ( ) Ӷ β BR β R α˜ 2 small r(mBr 1) coefficient 2 2 , given by (1) The “exact” B (r), given by integral (68) with α (24), and at large distances, r ∞, it approaches mB = 1.0 GeV, is compared to V in perturbation theory, β V which was discussed in Section 3. In the two-loop 2 —the coefficient of the vector coupling constant in α Λ Λ approximation, V(q = 1/r, = R) is given by the momentum space: expression β BR( ∞) β V 2 r 2 . (62) (2) 1  4π  β lny(r)  α q = --, Λ = ------1 Ð ----1------ (69) V  R β ( ) 2 ( ) Therefore, at large distances, the running coupling con- r 0 y r  β y r  stant can be approximated by the simple expression 0 with 4π  β lnt  α˜ (r ∞) = ------1 Ð ----1------0 , (63) B β 2  1 1 0t0 β t0  y(r) ≡ y q = -- = ln------. (70) 0  r 2Λ 2 r R where under the logarithm the variable t0 is a number, α(2) Λ ( 2 Λ 2 ) The behavior of V (q = 1/r, R) is shown by the dash- t0 = ln mB / V , (64) α (2) and QCD constant is equal to Λ . From the comparison dotted curve in the figure. The values of V (q = 1/r, V Λ of (64) and (34), one can easily get that the freezing val- R) are also given in Table 1. ues in coordinate and momentum spaces coincide: From our calculations presented in Table 1, one can α ( ∞) α ( , Λ Λ ) α (2) Λ ˜ B r = B q = 0 = V . (65) see that the perturbative V (q = 1/r, R) at small dis-

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α˜ (2) α(2) Λ Λ Table 1. Comparison of two-loop B (r) in background-field theory with V (q = 1/r, R) in perturbation theory ( R = 686 MeV, Λ = 240.7 MeV) at small distances MS

α˜ (2) α(2) Λ α˜ (2) α(2) Λ r, fm B (r)* V (q = 1/r, R)** r, fm B (r)* V (q = 1/r, R) 0.002 0.09641 0.09235 0.0355 0.19881 0.19433 0.004 0.11095 0.10505 0.041 0.20851 0.20794 0.006 0.12159 0.12089 0.008 0.13032 0.12212 0.049 0.22036 0.22636 0.012 0.14455 0.13521 0.057 0.23108 0.24549 0.016 0.15640 0.14653 0.063 0.23844 0.26053 0.020 0.16657 0.15688 0.069 0.245321 0.27635 0.024 0.17573 0.16665 0.077 0.25394 0.29894 0.030 0.18797 0.18067 0.087 0.26386 0.33026 α˜ Λ * The values of B (r) were found by calculating the integral in (68) with mB = 1.0 GeV and V = 0.385 GeV. α Λ Λ ** The approximate V (q = 1/r, R) is defined by Eq. (69), R = 0.686 GeV. tances, r Շ 0.04 fm, is systematically smaller, by about To explain this fact, it is convenient to consider the α (2) one-loop approximation when 1–4%, than the “exact” coupling constant ˜ B (r) in ∞  2 2 2 Ð1 BFTh. This fact takes place even at very small dis- (1) 8 sinx x + m r α˜ (r) = ----- dx------ln------B------ . (71) tances; e.g., at r = 0.002 fm the perturbative coupling B β ∫ 2 2 0 x  Λ r  α (2) α (2) 0 V constant is still smaller than ˜ B (r) by about 4%. V For r 0, it is usually assumed [28] that this integral However, this result is virtually independent of mB, reduces to the one-loop perturbative expression: because at such small distances the inﬂuence of the Ð1 background mass is immaterial. (1) 4π 1  α˜ (r) = ------ln------. (72) V β  2 2 0 Λ R r α(2)(r) However, it is possible to use the expansion of the func- α~(2) –1 B (r) tion t (x) [where t(x) is given in (68)] at r 0: 0.4 Ð1( ) 1 t x = Ð------(---Λ------) 2ln V r (73)  ln(x2 + m 2 r2) ln2 (x2 + m 2 r2)  0.3 × 1 + ------B------+ ------B------, 2ln(Λ r) ( (Λ ))2  V 2ln V r 

and term (m2 r2 ) can be neglected under the logarithm. 0.2 B α (1) Then, ˜ B (r) in (71) can be calculated analytically: 1 3 5 –1 r, GeV (1) 4π  1 α˜ ( )  γ B r = β------(------(--Λ------)---) 1 Ð E------(--Λ------)- 0 Ð2ln V r  ln V r Two-loop strong-coupling constant in coordinate space as a function of the distance r in the quenched approximation (74) 2 2 Ð1 (2) γ π   2  α 4 E + /3 4π 1 π (nf = 0). The three curves represent perturbative V (q = + ------ ≅ ------ln------Ð ------ , 2 β 2 2 ( (Λ )) 1/r, Λ ) with Λ = 686 MeV (dash-dotted curve); the ( (Λ )) 0 Λ 3 Ð2ln r R R 2ln V r   R r V  ( ) α˜ 2 strong-coupling constant in BFTh B (r), deﬁned by the Λ Λ γ Λ where R = V exp E. From (73), one can see that this integral in (68) with mB = 1.0 GeV and V = 385 MeV (dot- expression does not actually depend on the background α(2) Λ ted curve); and B (q = 1/r, V = 385 MeV) with mB = mass mB and in the limit r 0 it goes over to pertur- 1.0 GeV (solid curve). bative one-loop coupling constant (72). However, the

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α contribution of the second (small) term in the denomi- Table 2. Comparison of ˜ B (r) with the approximate cou- nator remains important up to very small distances: at pling constant α (q = 1/r, Λ ) at large distances × –4 B V r = 0.02 fm this correction is 3%, and at r0 = 2 10 fm α Λ this correction is still 1.4%. The importance of higher α˜ B (q = 1/r, V)** r, fm “Exact” B (r)* Λ Λ order corrections is a characteristic feature of the cou- = V = 385 MeV, mB = 1.0 GeV pling constant in coordinate space. (2) In our calculations, the background and pertur- 0.2 0.3367 0.3030 ≈ bative coupling constants coincide at the point r1 0.3 0.3710 0.3489 0.04 fm (see Table 1), and at r > r1, the perturbative 0.4 0.3914 0.3754 α(2) Λ α (2) V (q = 1/r, R) becomes larger than ˜ B (r). This 0.5 0.4041 0.3911 result of ours is in agreement with a similar observation 0.54 0.4077 0.3956 in lattice measurements of the short-distance static 0.6 0.4121 0.4009 potential [24]. In [24], it was observed that the absolute value of the lattice static potential is larger than the two- 0.7 0.4171 0.4073 ≈ loop perturbative potential up to the separation rL 0.8 0.4204 0.4117 ≈ 0.12r0 0.06 fm (r0 = 0.5 fm). This value of rL turns out 0.9 0.4225 0.4148 to be rather close to our point r = 0.04 fm discussed 1 1.0 0.4238 0.4172 above. Here, it is worth to noting that the value of rL in lattice data can depend on the chosen normalization r ∞ 0.42756 0.42756 condition and on the nonperturbative contribution to ( ) α˜ 2 the static potential and therefore can slightly differ * The values of B (r) were found by calculating the integral in (68) with m = 1.0 GeV. from the point r1 where perturbative and background B α Λ coupling constants are equal. ** The approximate coupling constant B (q = 1/r, V) is defined by Eq. (66). At larger distances, 0.1 > r > 0.04 fm, the perturba- α (2) tive coupling constant becomes larger than ˜ B (r), ≈ α (2) being about 25% larger at the point r 0.09 fm (see remaining a bit smaller than ˜ B (r) over the entire Table 1 and figure). region. The difference between them is 3.2, 2.3, and Thus, one can conclude that the perturbative two- 1.6%, respectively, at the distances 0.5, 0.7, and 1.0 fm. Λ Λ loop coupling constant (70) with = R has the accu- Still the approximate expression (74), given in the ana- racy Շ5% at the distances r Շ 0.06 fm and is signifi- lytic form, is very simple and can be useful in phenom- cantly different at larger distances. enological calculations. 3. At large distances, r տ 0.2 fm, another approxima- It is also important that both coupling constants α (2) have the same freezing value (63), which is determined tion to ˜ B (r) can be used. To get this approximation, we have taken into account that at large distances QCD only by the background mass mB since QCD constant Λ is supposed to be known. To define m , the lattice Λ Λ α˜ V B constant is close to the value V and B (r) freezes if static potential, as well as fine-structure splittings, can r ∞. Therefore, the two-loop approximation to be used [18, 19]. α (2) ˜ B (r) can be chosen as 8. CONCLUSIONS (2) 1  4π  β lnt (r)  α q = --, Λ = Λ = ------1 Ð ----1------B------, (75) B  B V β 2 In our study, we have investigated the properties of r 0tB(r) β tB(r)  0 α the strong-coupling constant ˜ B (r) in BFTh, which where was numerically calculated in the quenched approximation (nf = 0). 2 2 1 + mB r α t(r) = ln------. (76) It was found that at small distances two-loop ˜ B (r) Λ 2 2 V r is systematically 1Ð4% larger than a perturbative con- Ð4 The comparison of the approximate coupling constant stant up to very small distances, r ~ 10 fm. This property does not depend on the value of background mass α(2) Λ B (q = 1/r, V) and the “exact” coupling constant mB, playing a role of a regulator, and is a characteristic (2) α˜ (r) is illustrated by the numbers given in Table 2; feature of the coupling constant in coordinate space. B This result is in good agreement with lattice data on the see also solid and dotted curves in the figure. From static potential over small distances [24]. Table 2, one can see that the approximate coupling con- ( ) At average distances, 0.06 Շ r Շ 0.2 fm, it turns out α 2 stant is very close to ˜ B (r) at the distances r > 0.4 fm, that there is no way to define QCD constant Λ in coor-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2182 BADALIAN dinate space as a constant. As a function of the distance If n = 0, then η = J = 1. The function η (r, µ) (for n = Λ ≥ Λ ≥ Λ 0 0 1 r, it changes in the range R(r 0) (r) V(r 1) can be found by considering [29] that the integral ( ) ( ) ∞ Λ Λ β Λ 0 ≈ Λ 0 ∞ ), where V = MS exp(a1/2 0) ( V 1.6 MS for 2 sinx 2 2 2 n = 0). ξ (x) = -- dx------ln(m r + x ) f 1 π∫ x B տ α 0 At large distances, r 0.4 fm, ˜ B (r) approaches fast α ∞ = ln(m 2 r2) Ð 2 Ei(Ðm r) ≡ Ð2γ(r), (Ä.5) the freezing value ˜ B (r ), being only 3% smaller B B ∞ α˜ ∞ k than B (r ) at r = 0.5 fm. The freezing value in x since Ei(Ðx) = γ + ln(Ðx) + ------. BFTh is fully deﬁned by the background mass mB since E ∑ Λ k!k QCD constant V is supposed to be known (from lattice k = 1 data or from experiment). The value of m = 1.0 GeV, γ B Here, E is the Euler constant, and taken here from ﬁne-structure analysis in charmonium ∞ and bottomonium, gives α˜ (r ∞) = α (q = 0) = (Ðm r)k B B γ (r) = γ + ∑ ------B------. (Ä.6) 0.428, which is very close to the values used in quarko- E k!k nium phenomenology. k = 1 Then, we have ACKNOWLEDGMENTS η ( , µ) (µ2 2 2 2) γ( ) 1 r = ln r + mB r + r ; (Ä.7) This work was supported in part by RFBRÐDFG µ γ (grant no. 96-02-00088g). choosing = 1/r, one obtains expression (45) for 1(r): γ ( ) ≡ 1η  , µ 1 1 ( 2 2) γ( ) 1 r -- 1r = -- = -- ln 1 + mB r + r . (Ä.8) APPENDIX 2 r 2

To derive the coefﬁcients A1 and A2 in expansion In the case of n = 2, the integral (A.4), α (21) of perturbative coupling constant ˜ V (r), one needs ∞ 2 sinx 2 2 2 2 2 to calculate the integrals η (r, µ) = -- dx------{[ln(µ r + m r )] 2 π∫ x B ∞ (Ä.9) 0 2 dx  µ2r2 n J (µ) = ------sinx ln------(n = 0, 1, 2). (Ä.1) 2 2 2 2 2 2 2 2 2 2 2 n π∫  2  Ð 2ln(µ r + m r )ln(x + m r ) + ln (x + m r ) }, x x B B B 0 can be reduced to the expression Those integrals can be found in [29]: η ( , µ) 2 (µ2 2 2 2) 2 r = ln r + mB r J0 = 1, (Ä.10) γ( ) (µ2 2 2 2) ξ ( ) 2 2 + 4 r ln r + m r + r , J (µ, r) = [ln(µ r ) + 2γ ], (Ä.2) B 2 1 E ξ where the function 2(r), (µ, ) [ 2 (µ2 2) γ (µ2 2) γ 2 π2 ] J2 r = ln r + 4 E ln r + 4 E + /3 . ∞ 2 sinx 2 2 2 2 µ ξ (r) = -- dx------ln (x + m r ), (Ä.11) By taking the renormalization scale = 1/r in (A.2), 2 π∫ x B one gets 0 γ was not found in an analytic form. To get (A.9), expres- J1 = 2 E, ξ (Ä.3) sion (A.5) for 1(r) was used. Then choosing again in γ 2 π2 (A.9) the renormalization scale µ = 1/r, one obtains J2 = 4 E + /3; i.e., J , J , and J turn out to be constants. With the use η  , µ 1 2 ( 2 2) 0 1 2 2r = -- = ln 1 + mB r of (A.3), one can obtain the explicit expansions (22) for r (Ä.12) the coefﬁcients A and A in (21). 1 2 γ ( 2 2) ξ ( ) µ + 4 ln 1 + mB r + 2 r . In BFTh instead of integrals Jn( ), given by (A.1), one needs to calculate the integrals It was shown (see Section 5) that, in the perturbative limit, m 0, r 0, the γ(r) and ξ (r) functions ∞ B 2 2 2 2 2 n 2 sinx µ r + m r have the following limits: η (r, µ) = -- dx------ln------B------n π∫ 2 2 2 γ( ) γ x x + m r (Ä.4) r E, 0 B (Ä.13) ξ ( ) γ 2 π2 (n = 0, 1, 2). 2 r 4 E + /3;

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 STRONG-COUPLING CONSTANT 2183 γ therefore, in this case the function 2(r) in (42) is given by 15. Yu. A. Simonov, Pis’ma Zh. Éksp. Teor. Fiz. 57, 513 (1993) [JETP Lett. 57, 525 (1993)]; Yad. Fiz. 58, 113 2  1 2 π (1995) [Phys. At. Nucl. 58, 107 (1995)]. γ (r) ≡ η r, µ = -- 4γ + ----- 2 2 r E 3 (Ä.14) 16. A. M. Badalian and Yu. A. Simonov, Yad. Fiz. 60, 714 (1997) [Phys. At. Nucl. 60, 630 (1997)]. ( , ) r 0 mB = 0 . 17. Yu. A. Simonov, in QCD and Topics Physics, Lectures at the XVII International School of Physics, Lisbon, 1999; hep-ph/9911237. REFERENCES 18. A. M. Badalian and V. L. Morgunov, Phys. Rev. D 60, 1. K. G. Wilson, Phys. Rev. D 10, 2445 (1974); M. Greutz, 116008 (1999). Quarks, Gluons, and Lattices (Cambridge Univ. Press, 19. A. M. Badalian and B. G. L. Bakker, hep-ph/0004021. Cambridge, 1983). 20. J. M. Cornwall, Phys. Rev. D 26, 1453 (1982); G. Parisi 2. M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov, and and R. Petronzio, Phys. Lett. B 94B, 51 (1980). V. I. Zakharov, hep-ph/0003006; submitted to Phys. Lett. 21. A. C. Mattingly and P. M. Stevenson, Phys. Rev. D 49, 3. A. Billoire, Phys. Lett. B 104B, 472 (1981). 437 (1994). 22. A. B. Kaidalov and Yu. A. Simonov, Yad. Fiz. 63, 1507 4. Particle Data Group, Eur. Phys. J. C 3, 81 (1998). (2000) [Phys. At. Nucl. 63, 1428 (2000)]; hep- 5. M. Peter, Phys. Rev. Lett. 78, 602 (1997); M. Jezábek, ph/9912434; hep-ph/9911291; Phys. Lett. B 477, 163 M. Peter, and Y. Sumino, Phys. Lett. B 428, 352 (1998). (2000). 6. Y. Schröder, Phys. Lett. B 447, 321 (1999). 23. D. V. Shirkov and I. L. Solovtsov, Phys. Rev. Lett. 79, 7. E. Eichten et al., Phys. Rev. Lett. 34, 369 (1975). 1209 (1997). 8. A. M. Badalian and V. P. Yurov, Yad. Fiz. 51, 1368 (1990) 24. G. S. Bali, hep-ph/9905387; Phys. Lett. B 460, 170 [Sov. J. Nucl. Phys. 51, 869 (1990)]; Yad. Fiz. 56 (12), (1999). 239 (1993) [Phys. At. Nucl. 56, 1760 (1993)]. 25. S. Capitani, M. Luscher, R. Sommer, and H. Witting, 9. J. Richardson, Phys. Lett. B 82B, 272 (1979). Nucl. Phys. B 544, 669 (1999). 10. W. Buchmüller, Phys. Lett. B 112B, 479 (1982). 26. M. Melles, hep-ph/0001295. 27. Yu. A. Simonov, Lect. Notes Phys. 479, 139 (1996). 11. Yu. A. Simonov, Yad. Fiz. 60, 2252 (1999) [Phys. At. Nucl. 60, 2069 (1999)]; hep-ph/9704301. 28. W. Lucha, F. Schöberl, and D. Gromes, Phys. Rep. 200, 127 (1991). 12. S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985). 29. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, 13. S. P. Booth et al., Phys. Lett. B 294, 385 (1992). Series, and Products (Nauka, Moscow, 1971; Academic, 14. G. S. Bali and K. Schilling, Phys. Rev. D 47, 661 (1993). New York, 1980).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

ELEMENTARY PARTICLES AND FIELDS Theory

Spin Effects in Backward Deuteron Scattering and Baryonic Degrees of Freedom in the Deuteron L. S. Azhgirey* and N. P. Yudin1) Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received June 9, 1999; in ﬁnal form, December 27, 1999

Abstract—The effect of the baryon-resonance admixture to the deuteron wave function on the momentum dependences of the differential cross section, the tensor analyzing power T , and the polarization-transfer coef- κ 20 ﬁcient 0 for backward elastic deuteronÐproton scattering at high energies is investigated. The reaction in question is assumed to proceed predominantly through nucleon and nucleon-resonance exchanges. The formalism used in this study is based on light-front dynamics. The effect of various parameters of the problem on the results of the calculations is analyzed, and it is shown that, even at a 1% level of the total admixture of nucleon resonances to the deuteron wave function, a description of experimental data that is superior to that within the one-nucleon-exchange approximation can be achieved by appropriately choosing the contributions of various resonances. For qualitative agreement between the results of the calculations and experimental data to be attained, it is sufﬁcient to take into account the contributions of the lightest negative-parity baryon resonances. Upon taking into account baryon exchanges, results computed for observables with different deuteron wave functions show a smaller scatter than analogous results obtained within the one-nucleon-exchange approximation. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ing electromagnetic probes to it. In particular, such investigations resulted in establishing the existence of Among intermediate-energy-physics problems of meson-exchange currents in the deuteron [3] and in current interest, that of establishing the degrees of free- drawing the conclusion that it is necessary to take into dom that are operative in processes occurring at suffi- account nucleon isobaric states in dealing with such ciently small distances between hadrons (about 0.3Ð probes [4]. As the time passed, it became clear, how- 0.6 fm) attracts particular attention. Here, questions ever, that the problem of resonances in the deuteron is that are of prime importance for developing intermedi- very involved and that it can be solved only on the basis ate-energy physics include the following: Is it neces- of additional experimental information accumulated by sary at such distances to go over from effective had- using both electromagnetic and hadronic probes. Pro- ronic degrees of freedom, which are widely used, for tonÐdeuteron scattering into the backward hemisphere example, to describe nucleonÐnucleon interactions, to (in the c.m. frame) is one of the most appropriate reac- fundamental quarkÐgluon degrees of freedom? What tions for studying the small-distance structure of the kind of dynamics is associated with one degree of free- deuteron. dom or another at such distances? It is obvious that answers to these questions and Available experimental data on the differential cross some other questions of similar character cannot be section for this process [5Ð12] show the following spe- obtained immediately—it is sufficient to recall the long cial features: (i) The differential angular distribution way of development of nuclear physics and of the phys- has an ascending character as the c.m. scattering angle ics of nucleonÐnucleon scattering. In the present study, approaches 180°. (ii) At a fixed angle, the differential we will attempt to clarify the above questions by ana- cross section decreases fast with increasing incident- lyzing the effect of a baryon-resonance admixture in proton (incident-deuteron) energy. (iii) The cross sec- the deuteron wave function on the characteristics of tion as a function of energy has a plateau at proton backward elastic deuteronÐproton scattering. kinetic energies in the range 0.3Ð0.7 GeV (for backward elastic deuteronÐproton scattering, this approxi- The problem of isobaric states in the deuteron has mately corresponds to primary deuteron momenta in been discussed for more than two decades. A number of the range 1.6Ð2.6 GeV/c). To a great extent, these fea- reviews articles (see [1, 2]) and many original studies tures gave impetus to the original discussion of isobaric have been devoted to the subject. It could be thought states in the deuteron. that the most reliable information about the small-distance structure of the deuteron would come from apply- A radically new step in the investigation of deu- teronÐproton scattering was associated with the analy- * e-mail: [email protected] sis of the polarization characteristics of this reaction. In 1) Moscow State University, Vorob’evy gory, Moscow, 119899 Russia. particular, the tensor analyzing power T20 for backward

SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2185 elastic deuteronÐproton scattering at primary deuteron and rescattering processes do not lead to reasonable momenta between 1.1 and 3.9 GeV/c and between 3.5 agreement with experimental data [25]. and 6.5 GeV/c was measured in Saclay [13, 14] and in κ In analyzing backward deuteronÐproton scattering, Dubna [15], respectively; in addition, the coefficient 0 we attempt here to revive the u-channel pole diagram of polarization transfer from the vectorially polarized within light-front dynamics. Basically, the arguments deuteron to the forward emitted proton was measured that motivated our attempt are the following: (i) As was for the reaction 1H(d, p)X at 3.5 GeV/c [16] and for the 12 indicated above, the features of this process that are cal- reaction C(d, p)X in the deuteron-momentum range culated here significantly depend on those internal- from 6 to 9 GeV/c [17Ð19]. momentum values that are substituted into the deuteron A theoretical analysis of backward deuteronÐproton wave functions. It will be shown below that the method scattering in the energy region under investigation is of light-front dynamics opens new possibilities here. rather complicated, since it involves some unresolved (ii) Although the framework being discussed is very dynamical problems of relativistic nuclear physics. attractive, a situation where some constraints are Presently, these questions have not yet been clarified imposed on the scattering angles cannot be described in completely, and one has to rely on more or less reason- terms of relativistic quantum mechanics without invok- able approximations. If it is assumed that deuteronÐ ing very strong hypotheses. Indeed, it seems highly proton scattering is dominated by effective hadronic improbable that an exact nonperturbative description degrees of freedom, the u-channel pole mechanism can be effectively reduced to a scheme within relativis- involving the transfer of a nucleon (in general, a tic quantum mechanics such that the operator associ- baryon) from the deuteron to the proton (see the dia- ated with the spin of the system being considered gram in Fig. 1 below) appears to be the simplest and ceases to depend on interaction [26]. (iii) Previous cal- most natural reaction mechanism. A covariant treat- culations of resonance-contribution effects assumed ment of this diagram on the basis of the BetheÐSalpeter values for the resonance admixture to the deuteron equation [20, 21] leads to results that do not differ very wave function that are hardly justifiable. As is currently strongly from those obtained within old-fashioned discussed, a nucleon-resonance admixture can arise in approaches that relied on a direct use of the deuteron the deuteron wave function either via the meson mech- wave function. It is necessary in this case that a right anism [2] or via the formation of the s4p2 six-quark con- value of the internal nucleon momentum k (that allow- figuration [27, 28]. However, neither mechanism can be ing for relativistic kinematics) be substituted into the quantified precisely. It is especially important to wave function. A result that is common to such calcula- emphasize this because it was found in recent years that tions of backward deuteronÐproton scattering is that the mesonic origin of nucleonÐnucleon interaction is they lead to a noticeable quantitative (and even qualita- not universal even at low energies [29]. tive in the case of the tensor analyzing power T20) dis- The main objective that is pursued in the present crepancy with experimental data. It also appeared that study and which is suggested by the aforesaid is twofold. the required behavior of T20 could be obtained by First, we want to calculate spin effects in backward deu- assuming a P-wave admixture in the deuteron ground teronÐproton scattering within a somewhat modified for- state [22]. However, the antinucleon P wave that arises malism of light-front dynamics. Second, we will try find automatically in solving the BetheÐSalpeter equation is out—without microscopically computing the resonance overly small to explain the observed internal-momen- admixture in the deuteron wave function—whether it is tum dependence of the analyzing power T20. A further possible in principle to attain, at reasonable phenomeno- complication of the mechanism of deuteronÐproton logical values of this admixture, at least qualitative scattering within the covariant field-theoretical formal- agreement between the calculated and the measured val- ism does not seem reasonable since this would involve ues of the analyzing power T20 for backward elastic deu- introducing poorly defined form factors. teronÐproton scattering. An implicit hypothesis here is In view of this, the formalism of relativistic quan- that it is this scattering process that must be the main tum mechanics—in particular, its version in the form of source of information about the admixture of, say, the P light-front dynamics, which realizes the Poincaré group wave in the deuteron wave function. in the basis of a few particles promptly interacting with one another (that is, through an instantaneous potential) [23, 24]—has recently become a popular means for 2. DESCRIPTION OF THE FORMALISM analysis of deuteronÐproton processes. This conceptual 2.1. Baryon Exchange within Light-Front Dynamics framework seems quite appealing for the following reasons. Based on calculations in terms of wave functions, In accordance with the comments presented in the it is formally similar to nonrelativistic theory; in addi- Introduction, we will use methods of quantum field the- tion, it enables one to test the accuracy in solving the ory and consider that backward elastic deuteronÐproton scattering problem. Despite this, analyses within rela- scattering at incident-deuteron momenta about 5 GeV/c tivistic quantum mechanics that make use of a compli- is dominated by the exchange mechanism illustrated in cated mechanism taking into account, among other Fig. 1. In addition to nucleon exchange, exchange of things, resonance exchanges, delta-isobar excitations, nucleon resonances also occurs.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

2186 AZHGIREY, YUDIN

p' Expression (1) is approximate in that the spin part of the resonance propagator is taken on the mass shell, as d is usually done [33]. On one hand, this makes it possi- k, s, l ble to go over from vertices to wave functions; on the other hand, the propagator here proves to be noncovari- R, sR ant, so that relativistic invariance is violated. In order to restore covariance, it is necessary to add a contact part k', s', l' to the propagator [34]. This results in that the numera- d' tor of the propagator also involves the component R– Ð p rather than Ron . We believe that, in the approximation used here, the above violations of relativistic invariance Fig. 1. One-nucleon-exchange diagram for backward elastic deuteronÐproton scattering. are minimal. By using the variable + + The covariant form of the amplitude associated with x = p' /d , (5) this diagram requires introducing a large number of we can recast Eq. (1) into the form unknown form factors in the deuteron vertices [30] and ᏹ( )ᏹ( ) nontrivial propagators for particles of spin s = 3/2, 5/2, ᏹ 1 d p'R pR d' fi = ------∑------, (6) … [31]. Even in the particular case of only one dia- 1 Ð x M 2 Ð M 2 /x Ð M 2 /(1 Ð x) gram, it is therefore necessary to simplify radically the R d NT RT description of the process. It is of paramount impor- where tance that the formalism used be relativistically invari- 2 2 2 2 2 2 ant since, under the kinematical conditions being con- MNT = MN + RT , MRT = MR + RT , sidered, even the particles transferred in the process are MN and Md being, respectively, the nucleon and the deu- essentially relativistic. In our opinion, light-front teron mass. dynamics combines the requirements of simplicity and relativistic invariance in the most natural way [32]. In light-front dynamics, the quantity ᏹ( ) Within light-front dynamics, the invariant amplitude d R p' ϕ NR( , , µ, µ ) ᏹ ------= d x kRT R , (7) fi corresponding to the diagram in Fig. 1 can be 2 2( , ) approximately represented as the sum over poles asso- Md Ð M x RT µ µ ciated with the exchanges of a nucleon and of its where and R are the projections of, respectively, the excited states; that is, nucleon and the resonance spin onto the quantization ᏹ(d p'R)ᏹ( pR d') axis, plays the role of the wave function in the NR chan- ᏹ nel. In (7), we have used the new variable k deﬁned as fi = ∑------+------Ð------Ð------, (1) RT R 2R (R Ð R ) on k = (1 Ð x) p' Ð xR , (8) where ᏹ(d p'R) and ᏹ(pR d') are, respec- RT T T tively, the deuteron-breakup and the deuteron-forma- where the subscript T labels the momentum component tion amplitude; R is the index labeling resonances transverse with respect to the z axis. (including nucleon); R+ and R– are, respectively, the The wave function in (7) is normalized by the con- “plus” and the “minus” components of the resonance 4- dition momentum, 2 ϕ NR τ ∑∫ d d R = 1, (9) ± 1 0 3 R = ------(R ± R ), (2) R 2 τ where the integration measure d R has the form R0 and R3 being, respectively, the zeroth (energy) and dxdk R µ Ð τ T the third component of the momentum R ; and R is d R = ------. (10) on ( π)3 ⋅ ( ) the minus on-mass-shell component, 2 2x 1 Ð x In terms of the light-front-dynamics wave functions 2 2 Ð MR + RT ϕ NR R d (x, kT ), the amplitude in (6) can be represented in Ron = ------+------. (3) 2R the form Here, R is the Rµ component orthogonal to the z axis, T ᏹ 1 ϕ NR , , µ , µ ϕ NR* , , µ, µ while M is the resonance mass. In the Feynman dia- fi = ------∑ (x k ' ) (x' k' ) R 1 Ð x d RT R d RT R gram technique, which is equivalent in this case to the R (11) technique of time-ordered diagrams, we have 2 2 2 2  2 M + k M + k  × M Ð ------N------R---T- Ð ------R------R---T- , RÐ = dÐ Ð p'Ð. (4)  d x' 1 Ð x' 

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2187

where the quantity can be represented in the form

+ + NR NR 1 x' = p /d' (12) ϕ (k , µ, µ ) = ∑ϕ (k ) --νs ν sm d R R ls R 2 R R s corresponds to the kinematics of a deuteron emitted ls (17) + + from the lower vertex of the diagram in Fig. 1. × 〈 | 〉[ ( )] [ ( )]µ ν ( ) lmsms 1M R kR µν R ÐkR R RYlm nR , For the sake of convenience, we further go over from the variables x and kRT to the momentum variables where nR speciﬁes the proton-emission direction and + 3 3 where the Melosh matrices R (kR) [37] are the D func- kR = (kRT, kR ), where kR is determined by the relations tions of the spatial rotation through the Melosh angle,

3 which can be found from the explicit form of the E (k ) + k Melosh matrix in the NN channel, x = ------N------R------R------, (13) E (k ) + E (k ) N R R R ( ) 3 ⑀ σ +( ) M + EN k + k Ð i rskTr s 2 2 R kT = ------. (18) 3 EN, R = MN, R + kR . (14) [ ( ) ][ ( ) ] 2 EN k + M EN k + k ᏹ In terms of the new variables, the amplitude fi σ ⑀ Here, s stands for the Pauli matrices; r, s = 1, 2; and rs assumes the form is the unit antisymmetric tensor normalized by the con- ⑀ ⑀ 1 NR dition 12 = – 21 = 1. ᏹ = ------∑ϕ (k , µ', µ ) fi 1 Ð x d R R R (15) The problem of calculating the amplitude for the diagram in Fig. 1 in terms of the deuteron wave func- × ϕ NR*( , µ, µ )∆ 2( ) d kR' R M kR' , tions in various channels can in principle be solved with the aid of Eqs. (15)Ð(17). Hence, the problem of deter- where mining the spin characteristics of backward elastic deu- teronÐproton scattering can also be solved at the model ∆ 2( ) 2 2( ) M kR' = Md Ð M kR' , level. So far, our consideration has been quite general. Let M 2 + k 2 M 2 + k 2 M2(k' ) = ------N------R---T- + ------R------R---T- us now specify the direction of the z axis. Since the R x' 1 Ð x' effective Lagrangian of the degrees of freedom being considered must be Lorentz invariant (in particular, [ ( ) ( )]2 = EN kR' + ER kR' . invariant under rotations), the z axis can be chosen arbitrarily in general. However, the use of light-front- ϕ NR The normalization condition for the functions d dynamics wave functions of relativistic quantum then takes the form mechanics that are subjected to a constraint on angles instead of covariant vertices and the disregard of con- 3 2 tact diagrams generally lead to violation of rotational ϕ NR( , µ, µ ) d kR ∑ ∫ d kR R ------3------= 1, (16) invariance, so that the choice of the direction of the z µµ ( π) ⋅ µ ( ) R R 2 2 NR kR axis becomes significant. A general consideration of this question is beyond the scope of the present article. where Therefore, we only mention that it is common practice ( ) ( ) to align the z axis with the beam direction. This pre- µ ( ) EN kR ER kR NR kR = ------(------)------(------)-. scription is adopted, for example, in the theory of deep- EN kR + ER kR inelastic scattering [38]. We will follow this tradition. A In the ensuing analysis of polarization observables comparison with experimental data can serve as a crite- expressed in terms of the S- and D-wave components of rion of correctness of this choice of direction [39]. Our ≠ ≠ the deuteron wave function, it is necessary to use the choice leads to x x' and, hence, to kR kR' . In the case orbital angular momenta and the spin moments. In the of backward elastic deuteronÐproton scattering, we LS representation of light-front dynamics, there arise have some difficulties, since the spin operator depends, in general, on the interaction. In view of this, we further ( )2 adopt the approximation in which the wave function xx' = MN/Md . (19) used is identified, in each of the NR channels, with the two-body wave function in the relativistic quantum When the z-axis direction is chosen in this way, we do mechanics of the light front [24, 35, 36]. Within this not need to take into account Melosh matrices—in the theory, the spin operator of free particles is taken for the case of backward scattering, all transverse components of the momenta vanish, so that there are no Melosh ϕ NR µ µ spin operator S, and the wave function d (kR, , R) rotations.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2188 AZHGIREY, YUDIN 2.2. Differential Cross Section for Backward Elastic It is interesting to note that this expression is sym- DeuteronÐProton Scattering metric under the interchange of the primed and The differential cross section dσ/dΩ for backward unprimed variables x and k. This means that the differ- elastic deuteronÐproton scattering in the c.m. frame is ential cross section for backward elastic deuteronÐpro- given by ton scattering as given by (25) is invariant under the reversal of the z-axis direction. dσ 1 + ------= ------tr{ᏹ ᏹ }, (20) Ω 2 fi fi d 6(8π) s 2.3. Tensor Analyzing Power for Backward Elastic ᏹ DeuteronÐProton Scattering where s is the square of the total c.m. energy, fi is the amplitude for the process, and the trace symbol tr The tensor analyzing power T20 is given by implies summation over the spin variables of the initial + and of the ﬁnal state. After some simple, but cumber- tr{ᏹt ᏹ } T = ------20------, (26) some calculations, we arrive at 20 + tr{ᏹᏹ } + 1 RR' where the standard spinÐtensor tr{ᏹ ᏹ } = ------∑ Bκ (k , k ) fi fi 2 s R R' ( ) 1 Ð M' 1 Ð x κ RR' 〈 | | 〉 ( ) 〈 | 〉 s (21) 1M t20 1M' = Ð1 1M1 Ð M 20 (27) × RR'*( , )∆ 2 ( )∆ 2 ( ) is chosen for the spin operator T . Bκ kR' kR' ' MR kR' MR kR' ' , 20 s Substituting expression (15) for the amplitude into where (26) and taking into account (17), we obtain

RR' RR' T20 Bκ (k , k ) = ∑ bκ (ls; l's')φ (k )φ* (k ), (22) s R R' s ls R l 's' R' lsl's' 2 2 ∑ ∆M (k' )∆M (k' )Aκ (k , k )Bκ*(k' , k' ) R R' s R R' s R R' RR' sR ' + 1/2 3 κ RR' (28) bκ (ls; l's') = (Ð1) ------ˆlˆl'sˆsˆ'〈l0l'0|κ 0〉 = ---s------, s 4π s ∆ 2( )∆ 2( ) ( , ) ( , ) (23) ∑ M kR' M kR' ' Bκ kR kR' Bκ* kR' kR' '    s s l s 1 sR s 1/2 κ ×   . s RR' κ κ  s' l' s  s' sR' s  where

RR'   Aκ (k , k ) = ∑ aκ (ls; l's')φ (k )φ* (k ), (29) ˆ l s 1 s R R' s ls R l's' R' Here, ks = s + s', l = 2l + 1 ,   are Wigner lsl's'  κ  s' l' s κ RR' l' + s + s Ð sR' + 1/2 3 〈 |κ 〉 aκ (ls; l's') = (Ð1) ------sˆsˆ'ˆlˆl' 6j coefficients, and l0l'0 s0 are ClebschÐGordan s 4π coefficients. In deriving Eq. (21), we have made use of the relation   × sR s 1/2 〈 |κ 〉〈κ κ | 〉  ∑ l0l'0 l0 l0 s0 20  s' s' κ κ ϕ ( , µ, µ )ϕ ( , µ, µ ) R s lql (30) ∑ M k R M* k R' Mµ  κ  (24)  l l l'  s Ð µ × κ R' R' RR' ˆ l . = δµ µ ∑(Ð1) 〈s µ s Ð µ |κ q 〉Bκ (k , k ), 1 2 1 R R' R R R' R' s s s R R'   κ κ s  s s s'  where M is the projection of the deuteron spin onto the  κ  quantization axis (onto the z axis in our case).  l l l'  κ If we restrict our consideration to the nucleon chan- Here, l = l + l',  1 2 1  are Wigner 9j coefficients,   nel, it can be found from Eq. (20) that, in the one- κ nucleon-exchange approximation, the differential cross  s s s'  section has the form [40] and the expression for Bκ (k , k ) is given above [see s R R' 2 2 2( ) Eq. (22)]. dσ 3 Md Ð M kN' ------= ------Formula (28) determines the components of the ten- dΩ 2 2 1 Ð x (25) (64π ) s sor analyzing power T20 with the z axis directed along the beam. In a number of experiments, however, the × [φ 2( ) φ 2( )][φ 2( ) φ 2( )] 0 k + 2 k 0 k' + 2 k' . polarization of the deuteron beam was specified with

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2189 respect to the y axis that is orthogonal to the beam. It 3 l' × ϕ* (k , µ', µ ) = ------∑(Ð1) gˆ κˆ κˆ sˆsˆ'ˆlˆl' × can easily be seen that the tensor analyzing power T R M' R' R' π s s' 20 4 g with respect to this y axis is related to the above quan- × 〈 κ 〉〈κ | 〉〈κ κ | 〉 tity T20 by a simple equation. (Only the “20” compo- l0l'0 l0 lqlgmg 1q s'qs' sqs gmg (36) nent of the density matrix is nonzero for a tensorially R polarized deuteron beam.) Indeed, the spinÐtensors t2q  κ  κ   1/2 s' 1/2  l l l'  specified with respect to the y axis and the spinÐtensors ×   〈µ|tκ |µ'〉〈µ |tκ |µ 〉 s g s' 1 1 1 s'qs' R sqs R' t2q specified with respect to the z axis are related by the    κ equation  sR s sR'  s g s'  R Ð1 2 ( ) t2q = Rt2q R = ∑Dq'q R t2q', (31) × Yκ (n)φ (k )φ* (k ), q' lql ls R l's' R' where R is the rotation from the z axis to the y axis we find for backward scattering that about the x axis and Dq'q(R) are matrices of an irreduc- d + ible representation of the group of rotations. In terms of tr{t p ᏹt ᏹ } the Euler angles, we have 1q' 1q ------+------tr{ᏹᏹ } π  R = --, 0, 0 . (32) (37) 2  RR' RR' 2 2 ∑ Pκ (k , k )Bκ (k' , k' )∆M (k' )∆M (k' ) sq R R' s R R' R R' κ δ s RR' From Eqs. (31) and (32), we obtain = q'q------,-- RR' RR' 2 2 R ( , ) *( , )∆ ( )∆ ( ) ∑Bκ kR kR' Bκ kR' kR' ' M kR' M kR' ' T 20 = T 20/2. s s κ s RR' If we take into account only the nucleon channel in RR' expression (28) and set k = k', there arises the well- where the quantities Bκ (k' , k' ) are determined by known formula (see, for example, [41]) s R R' Eq. (22) and 8u(k)w(k) Ð w2(k) T = ------, (33) 20 2 2 RR'( , ) 3 3 ( )l' κ ˆˆ 〈 |κ 〉 2[u (k) + w (k)] Pκ k k = ------∑ Ð1 gˆ ˆ ssˆsˆ'll' l0l'0 0 sq R R' 4π l κ κ where u(k) and w(k) are the radial wave functions cor- ll'ss'g s l responding to the S- and the D-wave deuteron state.    1/2 1 1/2  × 〈κ | 〉〈 κ | 〉 2.4. Polarization-Transfer Coefficient in Backward l0gq 1q 1q s0 gq  s g s'  (38)   Elastic DeuteronÐProton Scattering κ  sR s sR'  The coefficient of polarization transfer from the κ vectorially polarized deuteron to the proton, 0, is given by [42]  κ   l l l'  p d + × φ ( )φ* ( ) { ᏹ ᏹ }  1 1 1  ls kR l's' kR' . κ tr 2J y J y   0 = ------+------, (34) tr{ᏹᏹ }  s g s'  p d Here, q = ±1, 0 are the spherical indices of the polariza- where Jy and J y are the y components of the proton and the neutron spin operator, respectively. In order to tion vector, determine κ , we first calculate the quantity 0 x 1 Ð1 +1 P = ------(P Ð P ), + tr{t p ᏹt d ᏹ } 2 ------1--q------1--q------, (35) + y i Ð1 +1 tr{ᏹᏹ } P = ------(P + P ), p d 2 where t1q and t1q are the standard spinÐtensors for the z 0 final proton and the initial deuteron, respectively. By P = P , using the simple, albeit cumbersome, relation and gˆ = 2g + 1 , g taking the minimal values in the ϕ ( , µ, µ )( )1 Ð M'〈 |κ 〉 | | ≤ ≤ | κ | ≤ ≤ κ ∑ M kR R Ð1 1M1 Ð M' q intervals s – s' g s + s', 1 – s g 1 + s, and |κ κ| ≤ ≤ κ κ MM' l – g l + .

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2190 AZHGIREY, YUDIN κ Returning to the definition of the coefficient 0 in effects and the D-wave admixture in the deuteron wave Eq. (34), we note that function, it was possible to improve additionally the description of experimental data [50]. p i ( p p ) J y = -- t11 + t1 Ð 1 , The tensor analyzing power T and the polariza- 2 κ 20 tion-transfer coefficient 0 calculated with the same J d = i(t d + t d ). deuteron wave functions are displayed in Figs. 2b and y 11 1 Ð 1 2c, respectively. It is important to emphasize here that With the aid of Eq. (37), we then obtain the discrepancy between the results of the calculations + and experimental data cannot be removed within the 〈2J pᏹJ d ᏹ 〉 ------z--'------z-'------one-nucleon-exchange model—in particular, it was 〈ᏹᏹ+〉 shown by Punjabi et al. [14] that, within this model, the κ (39) correlation between the T20 and 0 values associated p d + p d + with the same k value represents a circle on the (T , κ ) 1 〈t ᏹt ᏹ 〉 〈t ᏹt ᏹ 〉  20 0 = Ð --------11------11------+ ------1---Ð--1------1---Ð---1------. plane, whereas the experimental form of this correla- 2 〈ᏹᏹ+〉 〈ᏹᏹ+〉  tion is helical. A recent phenomenological analysis revealed that the description of experimental data can If we again restrict our consideration to nucleon be somewhat improved by additionally taking into exchange and set k = k', the result is account absorption effects [51]. 〈 pᏹ d ᏹ+〉 2 2 In order to calculate directly the analyzing power 2Jz' Jz' 1u (k) Ð w (k) Ð u(k)w(k)/ 2 κ ------= ------. (40) T20 and the polarization-transfer coefficient 0 within 〈ᏹᏹ+〉 3 u2(k) + w2(k) our approach, it is necessary to know the nucleon wave function of the deuteron; the wave functions φ (k ); the This expression differs from that which is used most φ ls R relation between the functions ls(kR) that are associ- often only by a trivial normalization factor. ated with the same resonance, but which correspond to different s values; and, finally, the magnitude of the 3. RESULTS OF THE CALCULATIONS admixture of baryon resonances to the deuteron wave AND DISCUSSION function. We will now discuss the choice of these parameters in some detail. We begin the discussion of our results by analyzing the potential of the pole diagram in Fig. 1 in the one- φ nucleon-exchange approximation of the present ver- 3.1. Wave Functions ls(kR) sion of light-front dynamics (that is, we take into In a consistent treatment, these functions are derived account, for the time being, only nucleonic degrees of by solving the dynamical problem of a resonance freedom). admixture in the deuteron. Since we do not aim here at For backward elastic deuteronÐproton scattering, the precisely describing experimental data, we approxi- σ Ω differential cross section (d (180°)/d )c.m. calculated mate them by oscillator functions, which were success- with the deuteron wave functions for the Paris potential fully used to calculate nuclear hadronic systems local- [43], the Reid soft-core potential [44], the A and B ver- ized in space. Specifically, we set sions of the Bonn potential [45], and the Moscow poten- φ ( ) l ( α 2 ) tial [46] (curves 1, 2, 3, 4, and 5, respectively) is displayed ls kR = CRkR exp Ð RkR , (41) in Fig. 2a as a function of the incident-deuteron momen- α tum in the laboratory frame. Also shown in this figure are where CR is a normalization factor and R is a parame- experimental data from [5Ð12] {the angular distributions ter that characterizes the spatial localization of the res- θ ° that were obtained in [5, 9] near the angle of c.m. = 180 onance. It is natural to assume that the radius of the res- were extrapolated to this angle on the basis of the depen- onance distribution in the NR channels must be smaller than the deuteron radius (that is, the radius of the NN dence (dσ(θ)/dΩ) = 3 a |cosθ|i}. c.m. ∑i = 0 i channel). This follows, in particular, from the uncer- From Fig. 2a, it can be seen that the results of the tainty relation—the resonance formed in the central calculations with the different wave functions differ region of the deuteron cannot travel a distance greater ∆ ∆ quite sizably. The figure also shows that the pole dia- than 1/ MR, where MR = MR – MN. From the same gram does not describe the plateau in the range 2Ð considerations, it follows that the heavier the reso- 2.5 GeV/c. However, this feature in the energy depen- nance, the smaller the spatial region where it must be dence of the differential cross section for elastic deu- localized. In order to avoid increasing the number of teronÐproton scattering can be explained with the aid of parameters, we approximated the oscillator radius rRj = the triangle diagram, whereby the cross section for deu- 1/ α as teronÐproton scattering is expressed in terms of the Rj cross section for the process NN dπ, the latter MR1 Ð MN being of a resonance character in the above energy rRj = r0 ------, (42) region [47Ð49]. By taking into account relativistic MRj Ð MN

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2191

σ Ω µ (d /d )c.m., b/sr T20 (a) (b) 5 0.5 2 3 1 104 4 0

–0.5

–1.0 102

κ 5 0 (c) 1.0 4

2 0.5 5 100 1 0

3 –0.5 2 3 1 4 –1.0

10–2 2.5 5.0 7.5 2.5 5.0 7.5 pd, GeV/c σ Ω κ Fig. 2. Differential cross section (d /d )c.m., tensor analyzing power T20, and polarization-transfer coefﬁcient 0 for backward elastic deuteronÐproton scattering versus the incident-deuteron momentum pd that were calculated in the one-nucleon-exchange approximation with the deuteron wave functions for (curve 1) the Paris potential [43], (curve 2) Reid soft-core potential [44], (curve 3) version A and (curve 4) version B of the Bonn potential [45], and (curve 5) the Moscow potential [46]. Experimental data were borrowed from (open stars) [5], (closed inverted triangles) [6], (closed circles) [7], (open circles) [8], (closed stars) [9], (open triangles) [10], (open diamonds) [11], and (open squares) [12] for (‡) the differential cross section; from (closed circles) [14] and (open circles) [15] for (b) the tensor analyzing power T20; and from (closed circles) [14] for (c) the polarization-transfer coefﬁcient k0.

φ where MR1 is the mass of the N(1440) resonance and r0 3.3. Relation between the Functions ls(kR) is a parameter common to all resonances. Corresponding to Different s Values

In the general case of the exchange of a spin-sR baryon, two states, that of spin s Ð 1/2 and that of spin 3.2. Magnitude of Resonance Admixtures R sR + 1/2, arise in the NR channel. The possible values of the orbital angular momentum are determined by the As was indicated above, mechanisms that can be parity of the exchanged baryon resonance. Since the responsible for the emergence of resonance admixtures φ NR in the deuteron wave function are not discussed here. functions ls (kR) that we use were not derived by solv- Both conceivable mechanisms mentioned in the Intro- ing the dynamical problem, the relations between the duction predict the magnitude of the admixture at a functions that describe NR states and which are charac- level of a few tenths of a percent to a few percent. At the terized by different s values are determined by the quality of the description of data. stage of choosing other parameter values, we ﬁxed the total magnitude of the admixture at 2%. As will be seen below, this choice is not crucial—a qualitative descrip- 3.4. Wave Function of the NN Channel tion of data can be attained with a value of about 1% or As was shown above, the results of the calculations even with a somewhat smaller value. in the one-nucleon-exchange approximation depend on

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2192 AZHGIREY, YUDIN

T20 (b) (a) 0.5 1680(5/2+) r0 = 0.7 fm 1520(3/2–) 0 0.6 1675(5/2–) –0.5 0.5

–1.0 0.3 1535(1/2–)

2.5 5.0 7.5 2.5 5.0 7.5 pd, GeV/c

Fig. 3. Tensor analyzing power T20 for backward elastic deuteronÐproton scattering versus the incident-deuteron momentum pd according to the calculations in which the deuteron wave function is assumed to include a 2% admixture of (a) one of the baryon – – – + resonances N(1520, 3/2 ), N(1535, 1/2 ), N(1675, 5/2 ), and N(1680, 5/2 ) at r0 = 0.4 fm; (b) the negative-parity baryon resonances N(1520), N(1535), N(1650), and N(1675) at various values of r0 (indicated in the figure); (c) all known resonances from N(1520) to N(2600) at various values of r0 (indicated in the figure) under the assumption that a nonvanishing contribution comes only from the φ NR wave function ls corresponding to the lower channel spin of the possible two values; and (d) the negative-parity resonances N(1520), N(1535), N(1650), and N(1675) at r = 0.4 fm and various values of the wave-function ratio φ /φ (indicated in the 0 lsmin lsmax figure). The dotted curves were computed in the one-nucleon-exchange approximation. The displayed experimental data are identical to those in Fig. 2b. the form of the NN-channel wave function. In choosing As a typical illustration, the calculated analyzing parameter values, the deuteron wave function for the B power T20 as a function of the incident-deuteron version of the Bonn potential [45] was used for the NN- momentum pd is shown in Fig. 3a for the cases where channel wave function. The eventual results were the deuteron wave function contains a 2% admixture of obtained for all wave functions with which we per- only one of the baryon resonances N(1520, 3/2–), formed the calculations in the one-nucleon-exchange N(1535, 1/2–), N(1675, 5/2–), and N(1680, 5/2+) (indi- approximation. cated in parentheses here are the baryon mass, spin, and parity). At a fixed magnitude of the resonance admix- As can be seen from Fig. 2, the data on the tensor ture, the resonance-contribution effect on the behavior analyzing power T20 show the most specific type of of T20 as a function of momentum depends on the momentum dependence; moreover, this spin observ- parameter r and on the relationship between the abso- able has been presently measured up to the highest val- 0 φ NR ues of the internal momentum k. For this reason, the lute values and phases of the wave functions ls cor- ± baryon-resonance contributions have been analyzed responding to the different spin values of s = sR 1/2. predominantly on the basis of a comparison of the The curves in Fig. 3a were computed at r0 = 0.4 fm results of the calculations precisely with these data. under the assumption that a nonvanishing contribution

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2193 comes only from the function for which the channel T20 spin takes the lower value of the possible two. As can be seen from this figure, the reversal of the ≥ sign of the analyzing power T20 in the region pd 0.4 GeV/c is the most spectacular effect of taking into 0.5 account the contributions of 1/2– resonances [N(1535) and N(1650)]. If, in addition, we include 3/2– resonances, the minimum in the analyzing power T20 calculated in the one-nucleon-exchange approximation with 0 the deuteron wave function for version B of the Bonn 0.005 potential (dotted curve) is slightly filled; concurrently, 0.015 the T values are reduced in the momentum region p ≥ 20 d –0.5 0.4 GeV/c. The contributions of the even-parity bary- 0.025 ons alone have a smaller effect on the behavior of T20. As might have been expected, the inclusion of the N(1440) and N(1710) resonances, which have the –1.0 nucleon quantum numbers, does not change the shape of T20(pd). For the analyzing power T20(pd) as a function of the incident-deuteron momentum at various values of r0, 2 4 6 pd, GeV/c the results of taking into account the total contribution of the 2% admixture of the negative-parity resonances Fig. 4. Tensor analyzing power T20 for backward elastic N(1520), N(1535), N(1650), and N(1675) to the deu- deuteronÐproton scattering versus the incident-deuteron teron wave function are shown in Fig. 3b under the momentum pd according to calculations performed at vari- above assumption that a nonvanishing contribution ous values of the total admixture (numbers on the curves) of the negative-parity resonances N(1520), N(1535), N(1650), φ NR comes only from the function ls for which the chan- and N(1675) to the deuteron wave function at r0 = 0.4 fm nel spin takes the lower value of the possible two. It can and under the assumption that a nonvanishing contribution φ NR be seen from this figure that the inclusion of the addi- comes only from the wave function ls corresponding to tional contribution from negative-parity resonances lower channel spin of the possible two values. The dotted changes substantially the result obtained within the curve was computed in the one-nucleon-exchange approxi- one-nucleon-exchange approximation by taking into mation. The displayed experimental data are identical to account only the NN-channel wave function for version those in Fig. 2b. B of the Bonn potential. Of course, it can hardly be expected that all details of the experimental depen- tion for version B of the Bonn potential. An optimal dence T20(pd) will be reproduced by using only one pole value of the parameter r0 has not yet been chosen diagram and quite a trivial form of the wave functions among those that were used in the calculations. φ NR . Nonetheless, it can be seen that, even with these ls For the analyzing power T as a function of the inci- functions at r values from the region 0.3Ð0.5 fm, we 20 0 dent-deuteron momentum, Fig. 3d illustrates the effect have achieved a better description of the data than of variations in the wave-function ratio φ /φ on within the one-nucleon-exchange approximation. lsmin lsmax Since there are no obvious reasons for taking into the results of the calculations. In the calculations, we account only the admixture of negative-parity reso- have taken into account the contributions of the nega- nances, it is of interest to explore the implications of tive-parity resonances N(1520), N(1535), N(1650), and including the exchanges of both negative- and positive- N(1675) and set the parameter r0 to 0.4 fm. As can be seen from this figure, the best agreement with the parity baryons. For the analyzing power T20, the results of taking into account the total contribution to the deu- experimental data on T20 is achieved in the case where teron wave function from the 2% admixture of all the contribution of the functions corresponding to the known resonances from N(1520) to N(2600) are shown smaller value of the possible two channel spins exceeds in Fig. 3c for various values of r0. Here, we again the contribution of the functions corresponding to the assumed that a nonvanishing contribution comes only higher spin value. In the calculations mentioned so far, NR the sign of the function φ was always taken to be φ lsmax from the function ls corresponding to the lower value identical to the sign of the function φ . It is obvious of the channel spin. As can be seen from Fig. 3c, the lsmin inclusion of the exchanges of all resonances again that, in the case where the contribution of the functions improves the description of the experimental data on corresponding to the smaller value of the channel spin T20 in relation to the description in one-nucleon- exceeds the contribution of the functions corresponding exchange approximation with the deuteron wave func- to its higher value, the reversal of the relative sign is not

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2194 AZHGIREY, YUDIN

( σ/ Ω) , µb/sr d d c.m. T20 (a) (b) 0.5 104 5 0 2

–0.5 1 3

–1.0 4 102

κ 5 0 (c) 1.0 2 1 4 3 0.5 5 2 100 1 3 0 4 –0.5

–1.0

10–2 2.5 5.0 7.5 2.5 5.0 7.5 pd, GeV/Ò σ Ω κ Fig. 5. Differential cross section (d /d )c.m., tensor analyzing power T20, and polarization-transfer coefﬁcient 0 for backward elastic deuteronÐproton scattering versus the incident-deuteron momentum pd that were calculated in the one-baryon-exchange approximation with the deuteron wave functions for (curve 1) the Paris potential [43], (curve 2) the Reid soft-core potential [44], versions (curve 3) A and (curve 4) B of the Bonn potential [45], and (curve 5) the Moscow potential [46]. In the calculations, we took into account the contribution of a 1% admixture of the negative-parity resonances N(1520), N(1535), N(1650), and N(1675) to the deu- φ NR teron wave function at r0 = 0.4 fm and assumed that a nonvanishing contribution comes only from the wave function ls corresponding to the lower channel spin of the possible two values. The displayed experimental data are identical to those in Fig. 2.

κ expected to affect signiﬁcantly the results of the calcu- fer coefﬁcient 0 calculated on the basis of the proposed lations. approach. These calculations have been performed with Figure 4 illustrates the effect that the change in the the deuteron wave functions for the Paris potential [43], magnitude of the baryon-resonance admixture to the the Reid soft-core potential [44], versions A and B of deuteron wave function can have on the quality of the Bonn potential [45], and the Moscow potential [46] description of the data on T20. In the calculations, we (curves 1, 2, 3, 4, and 5, respectively). Also shown are have taken into account the contributions of the nega- available experimental data. In the calculations, which tive-parity resonances N(1520), N(1535), N(1650), and have been performed under the assumption that a non- N(1675) and set the wave-function ratio φ /φ to vanishing contribution comes only from the functions lsmin lsmax φ NR 3/1 and the parameter r0 to 0.4 fm. It can be seen that, ls corresponding to the lower value of the channel at high pd values, qualitative agreement with experi- spin, we have taken into account the total effect of the mental data can be achieved at a relatively small mag- 1% admixture of the negative-parity resonances nitude of the admixture (about 0.5%). N(1520), N(1535), N(1650), and N(1675) to the deu- Finally, Fig. 5 presents the momentum dependences teron wave function and set r0 = 0.4 fm. A comparison σ Ω of the differential cross section [d (180°)/d )]c.m., the of these results with those calculated within the one- tensor analyzing power T20, and the polarization-trans- nucleon-exchange approximation (see Fig. 2) shows

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 SPIN EFFECTS IN BACKWARD DEUTERON SCATTERING 2195 that the inclusion of baryon-resonance exchanges, in 4. CONCLUSIONS addition to one-nucleon exchange, improves the We have investigated quantitatively the effect of the description of data in the region of high pd values. It is baryon-resonance admixture to the deuteron ground interesting to note that, upon taking into account σ Ω state on the differential cross section [d (180°)/d ]c.m., baryon exchanges in calculating the observables con- the tensor analyzing power T , and the polarization- sidered in the present study, the values predicted for κ 20 transfer coefficient 0 for backward elastic deuteronÐ these observables with the different deuteron wave proton scattering. It has been assumed that, every- functions show a smaller scatter. where, with the exception of the region of intermediate momentum values between 1.6 and 2.6 GeV/c, the pro- Let us briefly discuss the possible uncertainties that cess predominantly proceeds via the transfer of a are associated with the approximations made in deriv- baryon (exchange of a nucleon and nucleon reso- ing the formulas used in our calculations. We begin by nances). The formalism corresponding to this mecha- considering contact terms in the propagators. In instan- nism has been realized within light-front dynamics. taneous-form dynamics, a covariant field propagator The basic results of the present study can be summa- reduces to the sum of particle and antiparticle propaga- rized as follows: tors only for spinless and spin-1/2 particles. For example, the vector-particle propagator cannot be repre- (i) General formulas for the differential cross section [dσ(180°)/dΩ] , the tensor analyzing power T , sented as the sum of particle and antiparticle propaga- c.m. κ 20 tors—it is necessary to add a contact term [34]. In light- and the polarization-transfer coefficient 0 have been front dynamics, covariant propagators possess this obtained with allowance for admixtures of arbitrary property for spin-1/2 particles as well [33]. As long as resonances. the field-theory formalism is used, there is no problem (ii) Predictions for these observables have been ana- here—one can merely use a covariant propagator. But lyzed within the semiphenomenological approach that once quantities of the wave-function type are intro- consists in choosing the wave functions for various NR duced, the problem of a contact part arises, since it can- channels and phase relations between them. not be included in a wave function. At present, the (iii) The effect of various parameters of the problem question of contact terms has received virtually no on the results of the calculations have been investi- study. Following common practice, we simply disre- gated, and it has been shown that, even at a 1% level of gard the contact part since this is a purely spin effect, the nucleon-resonance admixture to the deuteron wave which is usually insignificant. function, a description of experimental data that is superior to that within the one-nucleon-exchange The second question concerns the choice of the z- approximation can be achieved by appropriately choos- axis direction. In light-front dynamics, field theory ing the contributions of various resonances. must be in general invariant under rotations, but it is not (iv) In order to fit the results of the calculations to straightforward to demonstrate this explicitly. More- experimental data, it has been sufficient to take into over, mechanisms that restore invariance under rota- account the contributions of the lightest negative-parity tions (for example, excitations associated with rota- baryon resonances, but the addition of the contributions tions of new degrees of freedom) for the manifestly from other resonances, including those of positive par- noninvariant original variables p± = (p0 ± p3)/ 2 have ity, has not led to a noticeable mismatch with data. not yet been disclosed. Usually, a formal way that con- (v) Qualitatively, the contribution of negative-parity sists in introducing additional degrees of freedom (ori- resonances slightly fills the deep minimum in the entation of the light hyperplane) is adopted in solving momentum dependence of the tensor analyzing power the problem of invariance under rotations [30, 39], but T20 calculated within the one-nucleon-exchange this significantly complicates the formalism. In our approximation and strongly diminishes this analyzing case, the results may depend on the choice of the z-axis power at deuteron-momentum values not less than direction. In all probability, however, the relevant 3.5 GeV/c, thereby reducing the discrepancy between uncertainty is insignificant (as is suggested both by the the theoretical and experimental results. results of our calculations and by the results presented (vi) It has been established that, for the observables by Karmanov [39], who studied the spectrum of for- considered in the present study, a transition from one ward emitted protons in the deuteron-breakup reac- set of the deuteron wave functions used to another leads tion). It was indicated above that, in the case of back- to a smaller scatter of the results when the baryon ward elastic deuteronÐproton scattering, the differential exchange is taken into account than when it is disre- cross section associated with nucleon exchange is garded. indeed independent of the reversal of the z-axis direction. Estimates show that uncertainties in other observables do not exceed the scatter of the results of the cal- ACKNOWLEDGMENTS culations performed with the different deuteron wave We are grateful to Yu.N. Uzikov and E.A. Stro- functions. A more comprehensive analysis of these kovsky for attention to this study and stimulating dis- questions will be performed elsewhere. cussions.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2196 AZHGIREY, YUDIN

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PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

NANP-99*

Higgs Boson Emission in Very Rare Decays of an Extra Vector Boson** G. A. Kozlov Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received September 27, 1999

Abstract—We explore the phenomenological structure of E -inspired grand uniﬁed group with the gauge × × × 6 group SU(3)c SU(2)L U(1)Y U(1), the emphasis being laid upon its implications for Higgs boson observation. In particular, we discuss the probability for the mass eigenstate Z2 to decay into a Higgs particle and a bound state composed of heavy quarks. Constraints on and relations between the Z2 and Higgs masses are presented. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION inates from the SO(10) subgroup, ⊃ ()× () Very rare decay processes provide sensitive tests of E6 SO 10 U 1 Ψ, new theories and an important testing ground for the SO()10 ⊃SU()5 ×U()1χ, Standard Model (SM). The ﬁeld of very rare decays is diverse and active since it offers a rich potential for ()⊃ ()× ()× () SU 5 SU 3 c SU 2 L U1Y, major discoveries. Theoretical interest in an extra neutral vector boson Z ' has been mainly motivated by an while the Z ' boson is a composition of ZΨ and Zχ com- experimental observation of possible deviations from ponents mixed with a free angle θ [3]: SM predictions for the decay of the SM Z boson into Z'= ZΨcosθÐ Zχsinθ. cc and bb quark pairs (Rc and Rb ratios) [1]. The deviations may be considered as one of the indications of In order to seek Z ' at LHC, it is important to know as new physics beyond the SM. The promising explana- much as possible about its decay modes both in standard sectors of the DrellÐYan (DY) type and in the tion of the observed phenomena is implied in the extra- (super)rare ones. Other channels can provide important Z ' models (see references [7Ð16] in [2]). New gauge information about the Z '-boson couplings. If we go bosons can be detected at future high-energy collid- beyond the SM, there are several possibilities for some ers—namely, Large Hadron Collider (LHC) at CERN, ≡ which can test the nature and structure of many theoret- quark bound-state resonances B {QQ } to be pro- ical models at a scale of 1 TeV, at least. Theoretical pre- duced via particle interplay accompanied by Higgs dictions of new neutral or charged gauge bosons come boson (H) emission. It is known that the Z boson is not yet an exact mass eigenstate, but that it turns out to be from various extensions of the SM [3]. New extra mixed with Z '. In the ZÐZ ' mixing scheme, the mass bosons naturally appear in the Grand Uniﬁcation The- eigenstates Z1 and Z2 are rotated with respect to the ory (GUT) models [3]. A simple and well-known ver- basis formed by Z and Z ', sion among the extensions of the SM is the minimal one, which is aimed at unifying interactions, the E   6 Z1 cosκsinκ Z GUT model [3]. Since the breakdown of E GUT into =, 6  κ κ  the SM is accompanied by at least one extra U(1) group Z2 Ðsin cos Z' [E SU(3) × SU(2) × U(1) × U(1)], there may 6 c L Y through the mixing angle κ, exist a heavy neutral boson Z ', which can mix with the 1/2 ordinary Z boson. There are two new gauge bosons M2ÐM2 Z Z1 appearing in E6 GUT models [3], where only one orig- κ=arctan------, M2ÐM2 Z2 Z * This section completes the publication of the Proceedings of the Second International Conference on Nonaccelerator New Phys- with M and M being the masses for the mass ics (NANP) held at the Joint Institute for Nuclear Research Z1 Z2 (JINR, Dubna, Russia) between June 28 and July 3, 1999. For eigenstates Z1 and Z2, respectively. the main part of these proceedings, the reader is referred to Phys. At. Nucl. 63, 621Ð1303 (2000). We study a possible extra Z2 state and its interpreta- ** This article was submitted by the author in English. tions that have direct implications for new physics at

LHC. Our interest is in Z2 production and possible pair g = αg' cosκ Ð g sinκ, V2 V V production process Z2(W, Z) with Z2 decay into pairs of (4) heavy quarks leading to QQ and QQ(W, Z) events at g = αg' cosκ Ð g sinκ, A2 A A LHC with QQ invariant mass peaked at an O(0.4 TeV) where mass. If the Z2 state is sufﬁciently heavy for producing an H boson, one can determine the effective coupling 1 2 1 for Z ÐH interaction. Since H bosons are coupled to Z g = --T Ð s e , g = --T , 2 2 V 2 3L W Q A 2 3L and quarks, this opens the possibility of ﬁnding these

Higgs bosons as products originating from the decays T3L and eQ being the third component of the weak isos- of the Z mass eigenstate. We are to estimate the ratio 2 pin and the electric charge, respectively. Both g' and of the partial decay widths Γ, V g' in (3) and (4) represent the chiral properties of the R(Z H{QQ} /QQ) A 2 s = 1 Z '-boson interplay with quarks and the relative Γ(Z H{QQ} ) (1) strengths of these interactions: ≡ ------2------s---=---1------, Γ(Z QQ) µ 2 ( γ )γ ÐLZ'Q = g'Z∑Q gV' Ð g'A 5 QZµ' . (5) where {QQ }s = 1 stands for a spin-1 quarkÐantiquark Q bound state. Among the possible decays, the process For GUT models, the free parameter g'Z in (5) is related Z2 H{QQ }s = 1 is of great interest, since it is α α ≡ ( )ω almost kinematically allowed, even for the Z2 mass as to in (3) and (4) as (g'Z /gZ) = 5/3 sW [5], low as 0.5 TeV. The main backgrounds to Z2 decays via where ω depends on the symmetry-breaking pattern Higgs boson emission will depend on the precise sec- and the fermion sector of the model, but it is usually ondary modes of the decay of the product Higgs boson. taken to be ω ~ 2/3Ð1. The choice of α Ӎ 0.62 provides In fact, background branches are most important if the the equality of both gZ and g'Z on the scale of the mass Higgs boson decays primarily to leptons or quarks Ӎ (with the exception of t quarks), as will be the case for of the unification MX MGUT into E6. Neglecting some a sufficiently light Higgs boson. Model calculations of differences in the renormalization-group evolution of α the decays of the SM Z boson via Higgs boson emission both gZ and g'Z , one can deal with at energies ~MZ' ~ to onia were performed in [4]. The rates of (super)rare M ~ O(1 TeV). Z2 decays Z2 H{QQ }s = 1 could be used as a check of the theory or to extract some of the parameters of the Suppose that the Z2 state can be produced at LHC theory that are hard to reach through other processes. via the subprocess qq Z2 and that, in the approximation of a small Z2 width, the cross section 2. DESCRIPTION OF THE FORMALISM σ( ) qq Z2 To analyze the effects of the Z2 state within the π 2 model under consideration, we focus on the Z2 cou- 2 GFMZ = K(M )------1 (g2 + g2 )δ(s Ð M2 ) plings. The interactions of the mass eigenstates Zi (i > Z2 V2 A2 Z2 1) with heavy quarks are described by the Lagrangian 3 2 density ∞ κ is dependent both on MZ and on . Here, GF is the µ 2 ÐL = g ∑∑Q(g Ð g γ )γ QZ , (2) Fermi constant, and the factor K reflects the higher ZiQ Z Vi Ai 5 iµ i = 1 Q order QCD process [6], where one of the sums is taken over all heavy quarks Q, α (M2 ) 2 ≡ ( ) s Z2 4 4π2 gZ is presented as the SM coupling g/ 1 Ð sW (sW K M = 1 + ------1 + -- . Z2 π   θ 2 3 3 sin W), Z1µ is taken to mean the SM Z-boson field, and ≥ Zi with i 2 are additional Z states in the weak-eigenstate α 2 Λ Note that the two-loop value is s(M ) ~ 0.1 at QCD = basis. We will consider the model featuring only one Z2 light Z mass eigenstate. The vector and the axial-vector 200 MeV for M < 2m (five flavors) and M > 2m 2 Z2 t Z2 t couplings g and g (i = 1, 2) in (2) are defined as Vi Ai (six flavors), mt being the top-quark mass [2]. g = g cosκ + g' αsinκ, The partial width with respect to Z decays into V1 V V 2 (3) quarks is determined by the couplings g and g V2 A2 g = g cosκ + g' αsinκ, A1 A A (4)—namely, we have (the number of colors, Nc = 3, is

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 HIGGS BOSON EMISSION 2199 taken into account) × 10 Table 1. Values of R(Z2 H{bb}s = 1/bb) 10 for var- 2 ious embedding scales M and Higgs boson masses m via Z2 H 2G M 2 Γ(Z QQ) = ------F------Z-C(M ) 2 2 Z2 the ratio xH = (mH/M ) 2π (6) Z2 × ( )1/2[ 2 ( ) 2 ( )] x MZ 1 Ð 4rq gV 1 + 2rq + gA 1 Ð 4rq . H 2 2 2 MZ , TeV 2 0 0.2 0.4 0.6 0.8 0.9 ≡ 2 2 Here, rq m / M ; MZ and m are the masses of the Z Z2 0.2 2.60 2.00 1.40 0.90 0.43 0.21 boson and a quark, respectively; and the C factor is 0.3 1.20 0.90 0.62 0.40 0.19 0.09 deﬁned by the running strong coupling constant α as s 0.5 0.42 0.32 0.23 0.15 0.07 0.03 α (µ2) α2(µ2) α3(µ2) 0.7 0.21 0.16 0.11 0.07 0.04 0.02 C(µ2) = 1 + ----s------+ 1.409----s------Ð 12.77----s------π π2 π3 × 9 µ Table 2. Values of R(Z2 H{tt }s = 1/ tt ) 10 for vari- with an arbitrary scale . The interactions of the Z2 state ous embedding scales M and Higgs boson masses mH via with quarks are expressed in terms of three parameters Z2 u d 2 x, y , and y [1], where the labels u and d mean, respec- the ratio xH = (mH/MZ ) tively, the up and the down types of quarks: 2

u u u u xH 2gV = x + y , Ð2gA = Ð x + y , MZ , TeV 2 2 2 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 2gd = x + yd, Ð2gd = Ð x + yd. V2 A2 0.6 1.00 0.81 0.61 0.49 0.8 0.33 0.30 0.26 0.22 0.18 0.10 0.06 3. RESULTS 1.0 0.17 0.15 0.14 0.12 0.11 0.09 0.07 0.05 0.02 1.2 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 The {QQ }s = 1 bound state with a 4-momentum Qµ and a mass mB may be produced in the decay of a Z2 state via the emission of an H boson with a 4-momen- The width with respect to the decay Z2 tum kµ in the heavy-quark-loop scheme. The decay H{QQ } is given by amplitude is given by [7] s = 1 Γ( { } ) Z2 H QQ s = 1  3  ( , ) 4 Γ ( ) ( , ) 2 2 3 2 ϑ 2 λ( , , ) A k Q = ∫d qtr Q q ∑ Ti q k; Q  , (7) gZgV MZ Nc cos xβ 1 xH xB   = ------2------2------i = 1 π3( ) 〈 〉 2 1 Ð xB H 0 where Γ (qµ) is the vertex function for the spin-1 quark Q  bound state—it depends on the relative momentum qµ 1 1  xβ × (1 Ð 6xβ/x )------(1 Ð x ) 1 Ð 5---- (9) B 4d 3 H  d  of Q and Q—while Ti stand for the rest of the total  0 0 matrix element. In fact, Ti in (7) carry the dependence 5  8xβ 1 of the interplay of H with heavy quarks (i = 1, 2) and + -----x 1 + ------+ --x Ð 5xβ 12 B  4 B the interplay of the Z2 state with the H boson (i = 3) via 5d0 〈 〉 g 2 〈 〉 the couplings gH = m/ H 0 and Z2 H = 2MZ / H 0, 2 1 2 xβ 1 Ð 6xβ/x  〈 〉 Ð --(x Ð x ) 1 + 4---- + ------B- , respectively, where H 0 stands for the vacuum expec- H B   Γ 3 d0 1 Ð xB  tation value of H. Generally, Q(qµ) is constructed [8] in terms of the quark and antiquark spinors u(Qµ) and 2 2 2 where x Ӎ (2m/ M ) ; x ≡ (m /M ) ; xβ ≡ β/M ; B Z2 H H Z2 Z2 v(Qµ ) in a given spin configuration accompanied by µ 2 Ӎ 1 ϑ ≡ ⑀ ⑀ ⑀µ ⑀ the covariant conﬁnement-type wave function φ (q ; β) d0 -- (1 + xH Ð xB); and cos ( á ), and Q 2 Z2 Z2 containing the model parameter β [9]: being the polarization 4-vectors of the B and Z2 states, δ j respectively. The total relative width R(Z i 2 2 Γ ( ) ( ) φ ( β ) ( ) Q qµ = u Qµ ------Uαβ Q q ; v Q µ . (8) 3 H{bb }s = 1/ bb ) (1) derived from (6) and (9) in the case where B is composed of bb , but for a DY type of bb Here, the symmetric rank-2 spinors Uαβ obey the standard α' normalization, is presented in Table 1 as a function of BargmannÐWigner equations [10] (Q/ Ð mB )α Uα'β = 0. the H boson mass mH via the ratio xH.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2200 KOZLOV

For the sake of deﬁniteness, we have considered the lower mass bound for the Z2 boson on the order of four values of M = 0.2, 0.3, 0.5, and 0.7 TeV. As can 1 TeV [12]. The Z boson with M ~ O(1 TeV) should Z2 2 Z2 be seen, the distribution is very steeply peaked toward be explored at future colliders such as LHC. low H-boson masses and drops to zero at the high-mass The possible existence of the Z ' does open the pos- end. In fact, we can render the results valid for any sibility of an additional source of Higgs boson detec- masses by merely rescaling the ratios xH, B, β. To be tion—namely, the production of a large number of Z ' understood precisely, one has to note the following: ﬁrst, the B state is treated relativistically [see Eq. (8)] bosons followed by the decays Z ' H{QQ }s = 1. In Ӎ particular, the lightest Higgs boson (mH < 120 GeV) and, in the case of zero binding energy, mB 2m; second, gluon corrections to the process have not been that is predicted by the minimal supersymmetric exten- sion of the SM should generally be observable among included. For a heavy B state such as {bb }s = 1 or Z ' decays. The active program of this investigation has {tt }s = 1 (e.g., superheavy toponium bound via a Higgs a major advancement in testing Z ' (super)rare physics boson [11]), either approximation should be accepted. within new physics outside of the SM. The estimates of R(Z H{tt } / tt ) for M = 0.6, 2 s = 1 Z2 0.8, 1.0, and 1.2 TeV are presented in Table 2. REFERENCES For a light spin-1 B state (heavier Higgs boson), the 1. G. Altarelli et al., Phys. Lett. B 375, 292 (1996). results can only be taken as a guide of an order of mag- 2. V. Barger, K. Cheung, and P. Langacker, Phys. Lett. B nitude of the rates. 381, 226 (1996). 3. J. L. Hewett and T. Rizzo, Phys. Rep. 183, 193 (1989). 4. B. Guberina et al., Nucl. Phys. B 174, 317 (1980). 4. CONCLUSION 5. R. W. Robinett and J. L. Rosner, Phys. Rev. D 25, 3036 We investigated the Z mass eigenstate of the sim- (1982); P. Langacker, R. W. Robinett, and J. L. Rosner, 2 Phys. Rev. D 30, 1470 (1984); P. Langacker and M. Luo, plest E6-inspired GUT model and explored implica- Phys. Rev. D 45, 278 (1992). tions for the production and detection of Higgs bosons 6. V. Barger and R. J. N. Philips, Collider Physics (Addi- accompanied by spin-1 heavy hadrons. It should be son-Wesley, Redwood City, 1987). emphasized that (super)rare decays Z2 H{QQ }s = 1 7. G. A. Kozlov, Nuovo Cimento A (in press). provide complementary information about potential 8. G. A. Kozlov, Int. J. Mod. Phys. A 7, 1935 (1992). new interactions. These decay processes are sensitive to 9. S. Ishida, Prog. Theor. Phys. 46, 1570, 1950 (1971); new vector and axial-vector couplings g and g . S. Ishida and M. Oda, Prog. Theor. Phys. 89, 1033 V2 A2 The physically constrained nature of the model implies (1993). that predictions need only be explored as functions of a 10. V. Bargmann and E. Wigner, Proc. Am. Acad. Sci. 34, few parameters. In this paper, we have chosen these 211 (1948). parameters to be (i) the mass of the Z eigenstate, M ; 11. H. Inazawa and T. Morii, Phys. Lett. B 247, 107 (1990); 2 Z2 K. Hagiwara, K. Kato, A. D. Martin, and C.-K. Ng, Nucl. (ii) the mass of the Higgs boson, mH; and (iii) the mass Phys. B 344, 1 (1990). of the heavy quark m; and (iv) the model parameter β 12. Gi-Chol Cho, K. Hagiwara, and Y. Umeda, Nucl. Phys. [7]. The present electroweak experiments lead to the B 531, 65 (1998).

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Physics of Atomic Nuclei, Vol. 63, No. 12, 2000, pp. 2201Ð2204. From Yadernaya Fizika, Vol. 63, No. 12, 2000, pp. 2297Ð2300. Original English Text Copyright © 2000 by Gavriljuk, Kuzminov, Osetrova, Ratkevich.

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New Limit on the Half-Life of 78Kr with Respect to the 2K(2n)-Capture Decay Mode* Ju. M. Gavriljuk, V. V. Kuzminov**, N. Ya. Osetrova, and S. S. Ratkevich1) Institute for Nuclear Research, Russian Academy of Sciences, pr. Shestidesyatiletiya Oktyabrya 7a, Moscow, 117312 Russia Received October 13, 1999

Abstract—The features of data accumulated for 1817 h in an experimental search for the 2K(2ν)-capture mode 78 ≥ × 20 of Kr decay are discussed. A new limit on the half-life for this decay is found: T1/2 2.3 10 yr (at a 90% C.L.). © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION was used to obtain information about the pulse rise time The first result on the 2K(2ν)-capture mode of 78Kr and the pulse front features. P12 signals are output decay was presented in [1, 2]. The limit derived from pulses of the additional shaping amplifier that amplified ≥ × the summary signal (PC1 + PC2) with shaping times of the data collected for 254.2 h was given by T1/2 0.9 µ 1020 yr (at a 90% C.L.). The known theoretical predic- 1.5 s. The parameter f depends on the energy space tions for the half-life with respect to the capture reac- distribution of the event in the MC volume. tion 78Kr(2e, 2ν)78Se are 3.7 × 1021 [3], 3.7 × 1022 [4], The K-shell double vacancy of the daughter 78Se** and 6.2 × 1023 yr [5]. The corresponding half-life values isotope appears as a result of the capture reaction 78Kr for the 2K(2ν)-capture decay mode are 4.7 × 1021, 4.7 × 2K(2ν) 78Se. The total energy released is 2K = 1022, and 7.9 × 1023 yr if one considers that 2K-electron ab 25.3 keV, where Kab is the binding energy of the K elec- capture accounts for 78.6% in the total number of 2e trons in the Se nucleus. One can find that the total prob- captures for 78Kr [6]. The method used in [1, 2] allows 22 ability for Se** to emit one or two characteristic x rays one to reach a sensitivity level of up to 10 yr for the is 0.837 under the assumption that this double-vacancy half-life and to test some theoretical models. The deexcitation is equivalent to the sum of deexcitations of results of the next step of measurement are presented two single vacancies. A characteristic x ray (E ≅ here. Kα ≅ 11.2 keV, EKβ 12.5 keV) has a sufficiently long path 2. EXPERIMENTAL SETUP length in krypton. Two pointlike energy releases with total energy 2Kab (total energy of the absorption peak) Our measurements were performed with aid of a will appear if the x ray is absorbed in the MC working multiwire wall-less proportional counter (MWPC) by 78 volume. One part is the x-ray energy release, while the using a krypton sample enriched in Kr. The main fea- other is the release of the Auger electron cascade tures of the counter and measurement conditions were energy accompanied by the characteristic L-shell x-ray described elsewhere [1, 2]. The MWPC comprises a energy. The x rays may leave the MC volume. A one- central main counter (MC) and a surrounding protec- or a two-point event would be detected in this case tion ring counter (RC) in the same body. A common (escape peak). All single-electron-background events, anode-wire signal (PAC) from the RC and PC1 and such as those due to Compton electrons or inner β- PC2 signals from both ends of the MC anode are read decay electrons, will have one-point energy releases. A out from the MWPC. A scheme with a signal readout multipoint-event pulse P12 would represent a sequence from two sides of the MC anode allows one to deter- of short pulses with a different time overlap. The num- mine the relative event coordinate β along the anode β × ber of pulses in the burst corresponds to the number of [ = 100 PC1/(PC1 + PC2)] and to eliminate events local regions where total ionization is distributed. The that do not correspond to the selected working length. amplitude and the duration of each pulse in a burst A shaping amplifier with an integration and differenti- µ depend on a local track length, orientation, and distance ation shaping time of 26 s was used to amplify the from the MC anode. The ADCs used to record the PAC, PC1 and PC2 pulses to have a sufficiently high energy × PC1, PC2, and P12 signals are triggered with the input- resolution. The parameter f = 1000 P12/(PC1 + PC2) pulse-amplitude maximum. The P12 signal triggering will be done for the first amplitude maximum, which * This article was submitted by the authors in English. ** e-mail: [email protected] corresponds to the energy released in the nearest anode 1) Kharkov State University, pl. Svobody 4, Kharkov, 310077 region. The peaks corresponding to one-point ampli- Ukraine. tudes P12 for the total energy of a fixed event appear in

2202 GAVRILJUK et al.

Intensity, arb. units The MWPC was placed in a low-background shield: 600 15 cm of lead, 8 cm of borated polyethylene, and 11 cm of copper. The intrinsic background of the MWPC ﬁlled up to 4.8 atm with pure xenon without radioactive contami- 400 1 nation was measured preliminary. The background energy spectrum collected for 973.9 h and conveniently 109 scaled spectrum of a Cd source (Eγ = 88 keV) are shown in Fig. 1 (curves 1 and 2, respectively). The spectra consist of (PC1 + PC2) signals from type-1 200 2 events. Curve 2 has a peak at Eγ = 88 keV and a xenon

escape peak at E = Eγ Ð EXeKα = 88 Ð 29.8 = 58.2 keV. The peak at 88 keV is not symmetric because of the radiation scattered in the counter wall. The energy res- γ 0 40 80 120 160 olution for the 88-keV line is equal to 13.7%. E, keV The background spectrum has some features. The main peaks correspond to the energy values of 16, 35, Fig. 1. Energy spectra of (1) the background and (2) the 50, 68, 82, and 92 keV. In the energy region 35Ð68 keV, 109Cd source of (PC1 + PC2) signals for type-1 events from there are initial peaks accompanied by the escape the MWPC ﬁlled with pure xenon up to 4.8 atm. peaks. The background counting rate in the energy range 20Ð100 keV is 91 hÐ1. Energy spectra of (PC1 + PC2) signals from type-1 the event-number distribution with respect to the and type-2 events for the 109Cd calibration source and parameter f (f distribution). Events with energy the krypton ﬁlling are shown in Fig. 2a (curves 1 and 2, released only in the MC and in the MC and RC simul- respectively). taneously are referred to as “type-1” and “type-2” events, respectively. One can see the 88-keV peak in the spectrum represented by curve 1. This spectrum was multiplied by a factor of 0.5 for a convenient comparison. The energy 3. RESULTS resolution is 10.8% for this peak. The highest energy Krypton enriched in 78Kr to 94% was used to seek peak on curve 2 is the krypton escape peak at E = Eγ Ð 78 ν 78 the Kr(2K, 2 ) Se capture mode. It contains an EKrKα = 88 Ð 12.6 = 75.4 keV. It appears in type-2 β 85 admixture of natural -radioactive Kr (T1/2 = 10.7 yr, events because of absorption in the RC of krypton char- Eβmax = 670 keV) with the volume activity of 0.14 Bq/l. acteristic radiation from the MC. The escape peak in Our measurements were performed in the under- the spectrum represented by curve 1 is on the left slope ground laboratory of the Baksan Neutrino Observatory of the total absorption peak. The source radiation scat- of the Institute for Nuclear Research (Russian Acad- tered in the counter body wall lies in this region too. emy of Sciences, Moscow) at a depth of 4900 mwe. The f distributions corresponding to this spectra are

Intensity, arb. units Intensity, arb. units (a) (b) 600 400

1 400 2 1 (×0.5)

200 200

0 40 80 120 160 0 500 1000 1500 2000 β f

Fig. 2. (a) Energy spectra of (PC1 + PC2) signals for (1) type-1 and (2) type-2 events for the 109Cd source and the MWPC krypton ﬁlling; (b) corresponding f distributions (1 and 2, respectively). Curves 1 are multiplied by a factor of 0.5.

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 NEW LIMIT ON THE HALF-LIFE 2203

Intensity, arb. units 15000 (a) 1 10000 2 5000

~ 60 3 40 4 20

0 40 80 E, keV (b) 150000

75000 1 2

~ 3 500

0 40 80 β

20000 (c)

1 10000

~ 150 2 3 75 4 5

0 800 1600 f

Fig. 3. (a) Background energy spectra and (b and c) corresponding β and f distributions: (1) 0 ≤ β ≤ 100, 1 ≤ f ≤ 2000; (2) 36 ≤ β ≤ 58, 1 ≤ f ≤ 2000; (3) 36 ≤ β ≤ 58, 1 ≤ f ≤ 710; (4) 36 ≤ β ≤ 58, 330 ≤ f ≤ 710; and (5) expected f distribution for the 2K-capture 78Kr events.

shown in Fig. 2b with the same scaling and notation. peak with a maximum at f2 = 920. The calculated f val- × One can see a peak on curve 1 with a maximum at f1 = ues of these maxima should be equal to f1 = 1000 × 166, which corresponds to two-point events from the 12.6/88 = 143.2 and f2 = 1000 (88 Ð 12.6)/88 = 856.8 total absorption peak when the KrKα-ray ionization is in the calibration when the P12 amplitude is equal to collected ﬁrst on the MC anode. If the photoelectron the (PC1 + PC2) one for single-point events. Actual ionization is collected ﬁrst, the relevant events form a values calculated on the basis of experimental data dif-

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000 2204 GAVRILJUK et al.

Intensity, arb. units by 36 ≤ β ≤ 58 and f ≤ 710 is shown in Fig. 3a (curve 3). The shape of this spectrum follows that of the spectrum represented by curve 1 in Fig. 1, with the exception of 20 the escape peaks. This means that almost all one-point events from the 85Krβ spectrum are eliminated by the selection used. Curve 5 in Fig. 3c shows roughly the 1 shape and the place of the f distribution expected for the 78Kr(2K, 2ν)78Se multipoint events. For a final analysis, we used events corresponding to 10 2 36 ≤ β ≤ 58 and 330 ≤ f ≤ 710 because, for such a selection, there are no peaklike distortions of the residual energy spectrum 4 in Fig. 3a in the region of interest. A low-energy part of this spectrum is shown in 3 Fig. 4 (curve 1). A sample of this spectrum and the 0 78 ν 78 20 30 E, keV location of the Kr(2K, 2 ) Se effect is shown as curve 3 (Fig. 4). The energy region 25.3 ± 3.8 keV Fig. 4. (1) Residual energy spectrum of the background; (2) includes 95% of the events. The background was fitted best fit; and (3) expected spectrum of the effect. by using points below and above this region (curve 2). The net fitted background for 25.3 ± 3.8 keV was found to be 266. The sum over experimental data is equal to fer slightly from the theoretical ones. This could be 262. The difference is Ð4 ± 23 or Ð19 ± 111 yrÐ1. Taking explained by nonideality of the experimental setup. The into account the event-detection efficiency (0.22) and parameter f depends on energy for the same reason. The the effective counter length (0.6 of working length), we energy spectrum represented by curve 2 in Fig. 2a is find that the limit on the half-life of 78Kr with respect to ν ≥ × 20 formed primarily by one-point events, and its f distribu- the 2K(2 ) capture mode is given by T1/2 2.3 10 yr tion (curve 2 in Fig. 2b) has no multipoint peaks. The (at a 90% C.L.). peaks at f = 1015 and f = 1239 are one-point peaks for, respectively, the energy region above 16 keV and the krypton 12.6-keV x-ray peak. The f distribution repre- ACKNOWLEDGMENTS sented by curve 1 has a one-point peak with a maxi- This work was supported by the Russian Foundation mum at f = 1039. for Basic Research (project nos. 94-02-05954a and 97- The background energy spectrum of the MWPC 02-16052) and in part by the International Science with krypton due to type-1 event is shown in Fig. 3a Foundation (grants RNT000 and RNT300). (curve 1). It was collected for 1817 h. The counting rate Ð1 is 1506 h for the energy range 20Ð100 keV. The cor- REFERENCES responding β and f distributions are shown in Fig. 3b (curve 1) and Fig. 3c (curve 1). One can see peaks at the 1. Ju. Gavriljuk et al., in Proceedings of the Ninth Interna- ends of the β distribution that are caused by events from tional School in Particles and Cosmology, Baksan Val- a high-energy part of the 85Krβ spectrum mainly col- ley, Kabardino-Balkaria, Russia, 1997 (Inst. Yad. lected in an ionization mode, the end-effect corrections Issled., Moscow, 1998), p. 415. associated with bugles of the anode being taken into 2. Ju. Gavriljuk et al., Yad. Fiz. 61, 1389 (1998) [Phys. At. account. In order to eliminate this background compo- Nucl. 61, 1287 (1998)]. nent, it is sufficient to perform the β selection of events 3. M. Aunola and J. Suchonen, Nucl. Phys. A 602, 133 in the range 36 ≤ β ≤ 58 (Fig. 3b, curve 2). The energy (1996). spectrum represented by curve 2 in Fig. 3a and the f dis- 4. M. Hirsch et al., Z. Phys. A 347, 151 (1994). tribution represented by 2 in Fig. 3c correspond to this 5. O. Rumyantsev and M. Urin, Phys. Lett. B 443, 51 selection. One can see from this f distribution that back- (1998). ground events corresponding to f ≤ 710 are mainly sup- 6. M. Doi and T. Kotani, Prog. Theor. Phys. 87, 1207 pressed. The energy spectrum of events characterized (1992).

PHYSICS OF ATOMIC NUCLEI Vol. 63 No. 12 2000

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Calibration of Solar-Neutrino Detectors with Artificial 51Cr and 75Se Neutrino Sources* V. N. Kornoukhov**, *** Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received October 13, 1999

Abstract—The possibility of calibrating the solar-neutrino detectors GNO, BOREXINO, and LENS with the same artiﬁcial neutrino source is discussed. For this purpose, the Russian heavy-water reactor L-2 can be used to produce a 10 MCi neutrino source based on the 51Cr isotope. For the ﬁrst time, the possibility of producing and using a neutrino source based on the 75Se isotope is demonstrated. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the same isotope [9]}. The only possibility of solving this problem is to calibrate the full-scale detector with Four pioneering experiments, Clorine [1], Kamio- an artificial neutrino source of MCi activity. kande [2], GALLEX [3], and SAGE [4], observed neutrino fluxes with a substantially lower intensity than Artificial neutrino sources based on the 51Cr isotope that predicted by the standard solar model. This dis- with an activity of a few MCi were successfully used to crepancy constitutes the so-called solar-neutrino prob- calibrate two gallium detectors, GALLEX [10] and lem (SNP), which is one of the most intriguing prob- SAGE [11]. For these detectors, an artificial neutrino lems in modern physics and astrophysics. The majority source was mainly used for an overall check of the of physicists deem that the solution to this problem lies detector; for the 176Yb detector, the calibration of the in new neutrino physics, a possibility of great impor- detector is necessary in order to measure the unknown tance for modern physics. cross section for neutrino capture with an accuracy not From the experimental point of view, the most impor- poorer than 5%. tant task for solving the solar-neutrino problem is to con- A scheme for calibrating three solar-neutrino detec- tinue GaÐGe experiments for the reason of sufficient tors (GNO, LENS, and BOREXINO) with the same accuracy (SAGE [4] and GNO [5]) and to detect low- high-activity artificial neutrino source is under devel- energy neutrino fluxes from the pp and 7Be sources with opment now. The objective of this study is to answer the real-time detectors BOREXINO [6] and LENS [7]. major questions associated with the calibration proce- The detector BOREXINO (now under construction dure and with the creation of an artificial neutrino at the Gran Sasso underground laboratory) is intended source with a very high activity for this purpose. to measure the 7Be neutrino, as well as to seek the neutrino magnetic moment [8] with an artificial neutrino source (ANS). 2. NEUTRINO EXPERIMENTS WITH ARTIFICIAL NEUTRINO SOURCES ON THE BASIS The detector LENS [7] uses a Yb target for solar- OF 51Cr AND 75Se neutrino capture. Neutrino capture by the 176Yb nucleus results in excited isomer states of 176Lu, which decay to The 51Cr isotope decays by electron capture with a 176 the ground state of Lu with a time delay of 50 ns. Q value of 751 keV and T1/2 = 27.7 d to the ground state These neutrino events have a highly specific signature: of 51V (90.14% branching ratio) and to its first excited two events produced at the same point in the detector state (9.86%), which is deexcited to the ground state with an average delay time. This signature makes it with the emission of a 320-keV γ ray. The neutrino possible to discriminate the neutrino signal against the spectrum consists of four monoenergetic lines at background by a factor of 107. The interpretation of 746 keV (81%), 751 keV (9%), 426 keV (9%), and such an experiment has one major problem: only an 431 keV (1%) [10, 11]. order of magnitude of the neutrino cross section for the 75 176 176 The Se isotope [12] decays by electron capture transition of Yb to the excited states of Lu is 75 known {this cross section is deduced from the cross with a Q value of 865 keV to the excited states of As, which are deexcited to the ground state with the emis- sections for (p, n) or (33He, 3H) scattering reactions on sion of several γ rays of different energies. Considering * This article was submitted by the author in English. the atomic levels to which transitions can occur, we find ** e-mail: [email protected] that the neutrino energies are 452 keV (83%), 465 keV *** e-mail: [email protected] (11%), 585 keV (2%), and 600 keV (3.5%). The half-

2206 KORNOUKHOV

51Cr 75Se 51Cr 75Se 746 keV 3/2– 500 keV (90%) 746 keV 600 keV (90%) BGT = 0.010 (2%) 600 keV 585 keV (2%) 585 keV (3.5%) (3.5%) 465 keV (11%) 465 keV 426 keV 175 keV 454 keV 426 keV (11%) (83%) 454 keV (10%) 5/2– (10%) 1+ 339.0 keV (83%) B = 0.12 BGT < 0.05 GT 1+ 194.5 keV – 1/2 g.s. BGT = 0.21 71 keV 1– 50.5ns BGT = 0.087 7– g.s. 233 keV 106.5 keV 3/2– 0+ 71 71 Ga Ge 176Yb 176Lu

Fig. 1. Nuclear data for GaÐGe detection of 51Cr and 75Se Fig. 2. Nuclear data for YbÐLu detection of 51Cr and 75Se neutrinos. neutrinos. life for 75Se is 119.78 d; this allows one to make the cal- To achieve the accuracy required for the GNO ibration experiment quite uninhibitedly. detector (about 5%), the activity of the 51Cr and 75Se It is suggested that the neutrino source (its volume, sources should not be less than 1.75 and 1.77 MCi, together with passive shielding, is 0.3 m3) will be respectively. For the experiment with the LENS detec- placed at the center of the GNO (or LENS) detector tor, the activity of the artificial neutrino source based on (that is, in full-angle geometry). The volume of the 51Cr should not be less than 4.30 MCi (only events pro- GNO detector is 54 m3 [3], while the volume of the duced by 746-keV neutrinos are analyzed). In view of LENS detector is 108 m3. the possible problems of delivering this comparatively short-lived isotope from the reactor to the Gran Sasso The cross section for the inverse-beta-decay reac- underground laboratory, the 51 activity should be tion was calculated according to [13]. The matrix ele- Cr 176 more than 5.1 MCi. The required activity of the artifi- ments for the transitions to the excited states of Lu 75 were evaluated by using their GamowÐTeller strengths cial neutrino source based on Se should be more than 3 3 3.84 MCi (events produced by neutrinos with energies BGT, which were measured by (p, n) and ( He, H) scattering [9]. of 452 and 465 keV are analyzed). Figures 1 and 2 show the simplified nuclear data for Thus, if GNO and BOREXINO (and LENS in the 71Ge and 176Lu detection of solar neutrinos. Tables 1 future) are placed in the same underground laboratory, and 2 list the results of the calculations of neutrino-cap- there is an opportunity to calibrate all of these detectors ture events for 51Cr (75Se) neutrinos in the GNO and with the same artificial neutrino source of very high LENS detectors. activity.

Table 1. Calculation of neutrino-capture events in the GNO detector [activity of the 51Cr (75Se) source is 1 MCi and exposure time is three months]

Neutrino Events E, keV source ground state, BGT = 0.0087 175 keV, BGT = 0.05 500 keV, BGT = 0.01 Sum 51Cr 746 205 8.2 5.4 218.6 426 10.3 0.3 Ð 10.6 Sum 215.3 8.5 5.4 229.2 75Se 452 177.7 5.4 Ð 183.1 465 24.7 0.7 Ð 25.4 585/600 17.4 0.6 Ð 18.0 Sum 219.8 6.7 Ð 226.5

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CALIBRATION OF SOLAR-NEUTRINO DETECTORS 2207

A, MCi A, MCi 9 4 7.52 MCi (a) (b) 8 3.34 MCi 7 3 6 ~ ~ 5 2 1.77 MCi 4 3.38 MCi LENS 3 LENS 1.75 MCi ~ 2 Transport 1 ~

BOREXINO Transport GNO 1 ~ ~ GNO 7 36 39 6265 155 7 113 206 T, d T, d

Fig. 3. Schedule of the GNO, LENS, and BOREXINO experiments with the same artiﬁcial neutrino source based on (a) 51Cr and (b) 75Se.

Figure 3a shows the schedule of the GNO, LENS, In Russia, there are three reactors that are suited for and BOREXINO experiments with the same artificial producing the required activity of 51Cr. These are the neutrino source based on 51Cr and the 51Cr activity at fast breeder reactor BN-600 featuring a special irradia- various stages of the procedure. The activity of 51Cr at tion assembly [14], the research reactor SM-3 [15], and the instant of the reactor shutdown should be more than the heavy-water reactor L-2 [16]. The reactor L-2 has a 8.9 MCi. The 75Se isotope can be used as an artificial very large volume with a high thermal neutron flux (up neutrino source for the GNO and LENS experiments, to 2 × 1014 cm–2 s–1), a very effective cooling system, and the activity of this source should not be less than and a continuous loadingÐunloading mode for the start- 3.48 MCi (Fig. 3b). ing material. If one takes into account the need for unloading the starting material from the reactor at an For the sake of simplicity, we assume that the time 50 it takes to deliver an artificial neutrino source from the appropriate time and the use of Cr in the form of reactor to the laboratory is 7 d and that the time it takes chips, then the choice actually has to be the heavy- to transfer it from one detector to another is 3 d. water reactor L-2. It follows from the calculation [17] (Table 3) that the expected activity of 51Cr after the irradiation of 35.6 kg of enriched chromium in the standard 3. PRODUCTION OF ARTIFICIAL NEUTRINO mode of the L-2 reactor is 8.5 MCi, and it can be SOURCES increased to 10 MCi (in a modified mode of the reactor L-2). 3.1. Artificial Neutrino Source Based on 51Cr 51Cr is produced by neutron capture on 50Cr (a 4.5% 3.2. Artificial Source on the Basis of 75Se natural abundance) with a sufficiently large cross section for thermal and epithermal neutrons (15.9 and 75Se is produced by neutron capture on 74Se with a 7.8 b, respectively). The source material must be large cross section for thermal and epithermal neutrons enriched in 50Cr and depleted in 53Cr (because of the large cross section of this isotope equal to 18.2 b). Owing to a high cost of enriched chromium, it was pro- Table 2. Calculation of neutrino-capture events in the LENS detector [activity of the 51Cr (75Se) source is 1 MCi posed to use the available material produced for the and exposure time is three months] GALLEX Collaboration [35.6 kg of chromium enriched in 50Cr to 38.6% and depleted in 53Cr to Events 0.7%)]. Neutrino E, keV 194.5 keV, 339 keV, source sum One more advantage of this material is that it is in BGT = 0.21 BGT = 0.12 the form of chips; therefore, it is easy to homogenize 51 their irradiation and to perform the sampling proce- Cr 746 72.7 30.7 103.4 dure. As a result, one can determine the activity of the 426 4.2 Ð 4.2 neutrino source by direct methods and by calorimetry Sum 76.9 30.7 107.6 75 [10]. But it should be noted that there is a need for a Se 452 70.2 26.8 97.0 large volume (about 10 l) in a reactor to place this mate- 465 9.7 4.0 13.7 rial because of its low density and a certain temperature 585/600 6.3 2.6 8.9 of a starting material to avoid sintering of Cr chips. Sum 86.2 33.4 119.6

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Table 3. Results of 51Cr production at the L-2 heavy-water The best way to produce a 51Cr source with an activ- reactor [17] ity of up to 10 MCi is to use the Russian heavy-water reactor L-2. The starting material for this purpose is Number Activity, Specific Site of target 50 of channels 35.6 kg of Cr, which was previously used by the MCi activity, Ci/g GALLEX collaboration. Standard mode. Irradiation time is 130 d The speciﬁc activity of 75Se that can be produced at Neutron traps 6 6.4 270 the L-2 reactor is 740Ð900 Ci/g, resulting in a total Core 3 2.1 180 activity sufﬁcient for the experiments in the Gran Sasso Total 9 8.5 240 laboratory, which have been discussed in this article. Standard mode. Irradiation time is 50 d (end of reactor campaign) REFERENCES Neutron traps 6 5.3 220 1. B. T. Cleveland et al., in Proceedings of IV Solar Neu- Core 3 1.7 140 trino Conference, Heidelberg, Max-Planck-Institute für Total 9 7.0 200 Kernphysik, 1997, p. 85. Modified mode with the reduced loading fuel. 2. Y. Suzuki, in Proceedings of the 17th Conference on Increasing of the reactor power 10%. Neutrino Physics and Astrophysics, Helsinki, 1997 Irradiation time is 130 d (World Sci., Singapore, 1997), p. 73. Core 9 9.7 270 3. GALLEX Collab. (P. Anselmann et al.), Phys. Lett. B 342, 440 (1995).

75 4. SAGE Collab. (J. N. Abdurashitov et al.), Phys. Lett. B Table 4. Results of Se production with the L-2 reactor (the 328, 234 (1994). isotopic composition of the starting material is 90% 74Se and 76 5. E. Bellotti et al., Proposal for a Permanent Gallium Neu- 10% Se) trino Observatory (GNO) at Laboratori Nazionali del Target type Specific activity, Ci Mass for 1 MCi, kg Gran Sasso, 1996. ∅ × 6. BOREXINO Collab. (C. Arpesella et al.), INFN BOR- 3 3 mm 875 1.26 EXINO Proposal, Ed. by G. Bellini, R. Ragavan, et al. ∅5 × 5 mm 737 1.5 (Milano Univ. Press, Milano, 1992), Vols. 1, 2. 7. R. S. Raghavan, Phys. Rev. Lett. 78, 3618 (1997); R. S. Raghavan, in Proceedings of IV Solar Neutrino (51.8 and 520 b, respectively). The source material Conference, Heidelberg, Max-Planck-Institut für Kern- should be enriched in 74Se and depleted in 76Se because physik, 1997, p. 248. of the large cross section of the latter isotope (85 b). 8. BOREXINO Collab. (G. Alimonti et al.), Proposal to the Fortunately, the enrichment can be performed by a gas National Science Foundation, Ed. by F. P. Calaprice et al. centrifugation of volatile SeF6. An optimum value of (Princeton Univ. Press, Princeton, 1996); N. Ferrari, enrichment of Se in 74Se is on the order of 90%. G. Fiorentini, and B. Ricci, Phys. Lett. B 387, 427 (1996). 9. M. Fudjiwara, Report to the LENS Collab., Grenoble, The starting material will be in the form of tablets of 1999; Ch. Goodman, Report to the LENS Collab., pressed high-purity Se powder, sealed into a hermetic Grenoble, 1999. shell of highly pure Al. The size of the tablets is less 10. GALLEX Collab. (H. Hampel et al.), Phys. Lett. B 420, than 5 to 6 mm in order to avoid the self-shielding of 114 (1998). 74 . Table 4 presents the results of 75 production Se Se 11. E. P. Veretenkin et al., in Proceedings of the IV Solar with the L-2 reactor [18]. Neutrino Conference, Heidelberg, Max-Planck-Institute für Kernphysik, 1997, p. 126. 4. CONCLUSION 12. J.-F. Cavagniac and M. Cribier, private communication. 13. G. V. Domogatsky, Candidate’s Dissertation (Lebedev The calibration of detectors involving an artificial Institute of Physics, Moscow, 1970). neutrino source with well-determined characteristics 14. J. N. Abdurashitov et al., At. Énerg. 80, 101 (1996). with to precision not poorer than 5% is the necessary 15. G. V. Kiselev, Technology of the Production of Radionu- stage of the next generation of solar-neutrino experi- clides with Nuclear Reactors (Énergoatomizdt, Moscow, ments. The experiments with the GNO, BOREXINO, 1990). and LENS detectors can be performed with the same 16. A. D. Dodonov et al., Reports to Conference “Improve- artificial neutrino source based on 51Cr if its activity at ment in Heavy Water Reactors,” Moscow, ITEP, 1998. the time of reactor shutdown is not less than 8.9 MCi. 17. B. P. Kochurov et al., Preprint No. 37-98, ITEP (Institute The activity of the artificial neutrino source based on of Theoretical and Experimental Physics, Moscow, 75Se, which can be used for the GNO and LENS exper- 1998). iments, should be more than 3.48 MCi. 18. B. P. Kochurov and V. N. Konev, private communication.

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