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Soft Binary Processes, NJL Model and Absolute Values of the Amplitudes of Reactions pi- p -->pi0 n and pi- p -->eta n

In the model the entire high-energy part of the amplitude (Regge power dependence from the total energy) comes as consequence of Gribov's diffusion of the constituent quarks in the space of rapidities and impact parameters. Charge exchange of the quarks, which goes at small quark anti-quark invariant masses, is described by low-energy Nambu - Jona-Lasinio model. It is shown that spin structure of the amplitudes and dominating spin-orbit (spin-flip) amplitudes are well described by this model without free parameters.


Introduction
I will consider the charge-exchange reactions 0 − 1 2 → 0 − 1 2 π − p → π 0 n(1), π − p → ηn (2), π − p → K 0 Λ(Σ) (3) in the Regge region of s and t. Figs.1 a,b and 2 show some data on differential cross sections of these reactions [1,2,3]. The amplitudes M and differential cross sections are expressed via elements M 0 (non-flip) and M 1 (spin-flip): where σ are Pauli matrices, n is an orth of normal of the reaction plane, q ⊥ is transverse momentum transfer, p c is momentum in the c.m.s, R is the radius of the residue of Regge poles which dominates in the reactions. Our choice is due to simple spin structure of these reactions and to presence of exact data on differential cross sections of reactions 1,2 within the interval s = 8 − 400GeV 2 ; q 2 ⊥ = 0.004 − 0.3( GeV c ) 2 [1,2]. The simplest scheme of reactions 1,2,3 is presented by the dual diagram of Fig.3b, which is interpreted by A.B.Kaidalov in [4] as a three-stage process [5] of Fig.3c. In stage I choose configurations of initial hadrons with small rapidities of annihilating constituents. These configurations fuse into intermediate flux-tube (f.t) where the transitionūu →dd orūu → ss (stage II) appear, after that f.t. disintegrates into two parts which evolve into finite hadrons (stage III) or continue disintegrating giving rise to a multiple process.
This picture is based on experimental facts: 1. Power decrease of cross-sections with s increasing where α R -is trajectory of the secondary Regge pole. The power decrease of cross sections agrees with the parton concept providing the rapidities of annihilating constituents to be small and independent of the rapidity of initial hadrons [4,6].
2. Mass relations [4] which follow from factorization of the binary amplitude in s channel are well fulfilled. Factorization in s channel is a good argument in the favour of existence of an intermediate object -f.t. 3. The appearance of f.t. with increase of colour dipole moment of a pairqq static test quarks is confirmed in QCD on lattices [7].
The model [4] results in a correct Regge asymptotics of binary processes [8].
In the framework of the dual diagram of Fig.3b the spin structure and the value of residues of physical amplitude poles is determined by the spin structure of soft 4-quark interaction in a subprocess accompanied by a change of flavourūu →dd,ūu → ss which holds in f.t. in the II stage and may be represented by five 4-fermionic bilinear amplitudes [6] (2p e √ s) −1 ImM b = f.t| where symbol f.t.|T |f.t. implies the soft pair charge-exchange amplitude averaged over relative moment ofq and q. The pair results (by analogy with [4]) from diffusion of partons of interacting hadrons in the space of rapidities and impact parameters b [9]. The weights and signs of bilinears T n are given by the effective chiral Nambu and Jona-Lasinio Lagrangian (the NJL model [10]), broadened by 't Hooft [13] (see reviews [11], [12].
In this paper I will show that such a construction well reproduces the experimental data of reactions 1 and 2.
Since in the NJL model, in the lowest in 1/N c order (N c is the colour number) there are only bilinears of scalars (s) and pseudoscalars (p.s.) [12]: where D are propagators of intermediate states. The sign − in eq.(4) is determined by the sign in the 't Hooft determinant not conserving SU(3) singlet current (ūu +dd +ss). The account of bilinear (ūu)(dd) in eqs. (3), (4) makes it possible to explain qualitatively a large spin-orbital effect in reactions 1,2 , their planarity and smallness of spin-orbital interaction at the Pomeranchuk pole [6,14]. Let us remind this explanation. Fig.3d shows the 3 P 0 = O + scheme of reaction π − p → π 0 n on a "proton" consisting of a totally polarized along +z axis soft u quark and a leading spectator ud with the impact parameter +b ud . The impact parameter of mesonic spectator -b d = 0. The sign denotes the transversal spin of quark with s z = + 1 2 . Z axis is directed along the normal of reaction plane. In 3 P 0 state the transverse orbital moment l z = −1 denoted as ⇄ and the summary transversal spinūu S z = +1 are antiparallel. That is why the quark charge-exchange, under a nontriviall condition of S z and l z conserving in 3 P 0 = 0 + state in theūu →dd process follows predominantly at +b, since at −b annihilation in 3 P 0 state is realized with a smaller probability (l z and S z are parallel). Thus, l σ interaction appears. In the third part of the paper I show that it is T (S) = T ( 3 P 0 ) but not T ( 3 P 2 ) which determines M 1 in reaction 1. At P pole effects due to soft quark annihilation are of no importance and spin-orbital interaction is small.
Large spin-orbital amplitude M 1 in reaction 1 is a result of l z and S z conservation in 3 P 0 = 0 + state in theūu →ūd process. We deal with a "hidden orbital moment". Conservation of l z corresponds to the planary processūu →dd (Fig.4). Thus the presence of a dip in reaction is an experimental evidence for planarity and, consequently, for that dual diagram is appropriate to describe π − p → π 0 n. Some arguments according to which all the mentioned on reaction 1 is also valid for reaction 2, are presented in the third part of the paper.
The next discussion is planned as follows: In the 2-d part we consider some consequences of extended NJL model for theūu →dd amplitude. One may expect them to reveal in the M 0,1 amplitudes in reactions 1 and 2. We note 1. Dominance of S and P S bilinears in (3). 2. Their strong destructive interference in M 0 provided that the intermediate particle mass in P.S. channel is small.
3. Independence ofūu →dd soft amplitudes on the coupling constant of the NJL effective Lagrangian stemming from spontaneous violation of chiral symmetry. This fact is an argument in the favour of existence of the "blackness condition" (12), the physical meaning of which is that the softūu →dd amplitude is determined by transversal dimension of R region where the pair charge-exchange occurs, but not by the coupling constant. In the 4-th section we see that R enters the expression for M as a radius of residue of the Regge pole.
In the 3-d section we perform the analysis of experimental data based on the Fiertz expansion of the t channel 4-quark amplitude in the dual diagrams of reactions 1 and 2 taking into account their exchange degeneration. This analysis will confirm the 1-st and the 2-d consequence of the NJL model for reactions 1 and 2 (strong destructive interference of M 0 ( 3 P 0 ) and M 0 ( 1 S 0 ) and, consequently, the smallness of the intermediate particle mass in P.S.channel and shows that in the framework of dual diagram M 1 = M 1 ( 3 P 0 ). The experimental |M 1 |/|M 0 | = 5; 10 for reactions 1 and 2 ,respectively,are compared with the estimate (|M 1 |/|M 0 |) s < 1) in reactions π − p → K 0 Λ(Σ) obtained on the basis of the differential cross sections data.It is stated that the dynamics ofūu →dd andūu → ss is different. There is a strong SU(3) symmetry violation in M 1 . Possible reasons for this phenomenon are discussed within the NJL model.
In the 4-th section the relation (3) is written in explicit form (29) basing on the unitarity condition of the f.t. model in s channel. With the account of the "blackness" condition (12) this enables one to calculate absolute values of the M 1 ( 3 P 0 ) amplitudes which are dominant in reactions 1 and 2. The model expression M 1 ( 3 P 0 ) (39) contains: the colour number N c = 3, the radius of the residue of the Regge pole R 2 and α(−t). Since R 2 and α(−t) are determined from the energy dependence of the slope of reactions cone, the expressions for the absolute value M 1 ( 3 P 0 ) has no free parameters M 1 ( 3 P 0 ) is compared with the experim ental values M 1 (exper.) (24). As a result: We have refrained from estimating the M 0 amplitudes because of their theoretical ambiguity: M 0 = M 0 ( 3 P 0 − M 0 ( 1 S 0 ) is a small difference of large values. The 5-th concluding part deals with some problems following from the results of our analysis.
The minimal U L (1) ⊗ U R (1) symmetric (chiral) NJL Lagrangian of theqq light pair system with one flavour in Dirac vacuum has the form where G is the positive interaction constant (quarks undergo attraction), m = 5MeV -is the current quark mass, γ is the Dirac matrix, q is the quark field operator. The form of this bare Lagrangian is determined by requirement of chiral invariance. The current quark mass is small as compared with the cut-off energy of nonrenormalizable NJL model Λ ≃ 1GeV . The value Λ = 0.631GeV [12] is agreed with the low-energy data array. The equality of interaction constants with which bilinears of scalar and pseudoscalar currents enter L N JL is the fact following from requirement of chiral invariance. This equality takes place also for extended NJL model. When G increases up to a critical value G c ≥ π 2 /N c Λ 2 the current quark system becomes energy-unstable. The chiral symmetry spontaneously violates as a result of the instanton transition and phase transition into minimal energy state occurs. Here current quarks of each flavour acquire mass (in the extended NJL model due to scalar virtualqq pairs of three flavours) and go over to dynamical (constituent) quarks with the mass M c ≃ 0.3GeV which are paired into scalar pairs with J π = 0 + producing a new "QCD" vacuum of chiral condensates <qq > with the energy density ǫ < 0.
An energetic gap |∆E| ≥ |ǫ| + M c appears between external constituent quarks and the new vacuum. Since the NJL model considers only fermionic degrees of freedom chiral condensate is a condensate of constituent 3 P 0 pairs (by analogy with super-conductor).
Production of massive 3 P 0 pairs in which transversal spin of aqq is compensated by transversal orbital moment, from light pairs of longitudinal current quarks with opposite moments and equal chiralities (bilinear scalar in (5)) is connected with quark spin flip described by the Bogolyubov-Valatin transformation and should be accompanied by production of pseudoscalar Goldstone bosons (bilinear pseudoscalar in (5)) which are associated with an octet of pseudoscalar mesons.
On the background of new vacuum (chiral condensate) external constituent quarks undergo residual interaction which may be represented, under mean field approximation, by the sum of two terms of the extended NJL model: The first term has the structure of (5), it does not mix flavours, if small effects due to current mass difference will be excluded. The determinant term constructed by 't Hooft violates U A (1) symmetry, does not conserve singlet SU 3 current and consequently gives transitions uu →dd,ūu → ss. In the case of three flavours L res det has six-fermion structure. L res det can be reduced to the effective 4-quark representation [15] which imitates instantonous transition (Fig.5).
where λ 0 = − 2/3 · I, λ n (n = 1 − 8) are Gell-Mann matrices [16], (7) gives the sign − in (4). For the case of two flavours (u and d) and up to 1/N 2 c L res tH (2) has the form [11,12] where τ are Pauli matrices. The transitionūu →dd corresponds to 180 o rotation ofū and u around the second axis in the isotopical spin space (u → −d,ū → −d) [17]. Since at such rotation bilinears of scalars and pseudoscalars go over into each another we restrict ourselves when analysing reactions 1 and 2, to the term in (8) averaged over isospin ūu(dd) are superpositions of states with I = 0, 1 that is why in intermediate two-quark states we fix the third component of isospin I 3 = 0, but not I. The extended NJL model does not consider the problem on conservation of l z , S z in (ūu)(dd), (ūu)(ss).
In order that to turn from contact amplitudes to amplitudes (4) one should take in to account propagators of scalar and pseudoscalar states D S,P S . To this end the authors of [11,12] used random phase approximation [18] where Π S,P S are loop integrals depending on the total energy of charge-exchanged quarks k 2 = 4m 2 q + 4 k 2 , on the intermediate state mass m * S,P S and Λ. Note three, important for our following analysis, consequences of the NJL model: 1. It is shown in [11] (eqs.4.14 -4.32) and furthermore 6.49) that where I ′ -a is fluent, weakly depending on k 2 function. Substituting (11) into (10) we see that after spontaneous violation of chiral symmetry (i.e. at G ≥ G c , ' m mq = m Mc ≃ 0, D S,P S stops depending on G provided that k 2 = m * 2 S,P S . G-independence of D S,P S at G ≥ G c suggests that as constituent quarks at the edges of two f.t. appear to be in 3 P 0 or 1 S 0 states they annihilate with a unity probability (two f.t. fuse into one with a unity probability): We call the equality γ 2 = 1 as "blackness" condition. The "blackness" condition (12) will be used to estimate the absolute value of M 1 in the 4-th section. The "blackness" effect is a consequence of spontaneous violation of chiral symmetry. Note that G-independence of amplitudes at G ≥ G c by no means implies that in T (S) and T (P S) (4) the values G S and G P S are arbitrary. In fact, G S = G P S = G since (11) follows from (5).
2. In the NJL model m * S ≡ m σ = 2M c (see experimental confirmation in [25] and references therein. Substituting m 2 σ into (11) we get (accounting for (13) and (10)) (4) is determined by the 3-moment of charge-exchanged quarks , i.e. by the dimension of the region where charge exchange occurs. In the 4-th section we shall see that the transversal part of this region R 2 = k 2 ⊥ −1 enters the expression for the amplitude M as the radius of the residue of the Regge pole M which dominates in the reaction.
In papers of the High Energy Physics Institute -CERN [1,2] the world data on differential cross sections of reactions are well parametrized in the region s = 8 − 400GeV 2 and q 2 The parameters of (16) are listed in Table 1. We assume that Whence For reactions 1 and 2, correspondingly. The assumption (17) is confirmed by the following data: 1. A direct amplitude analysis of πN → πN including data on spin flip at s ∼ = 12GeV 2 in the region q 2 ⊥ ≤ 0.5( GeV c ) 2 gives for the charge exchange amplitude M 1 /M 0 = −8 in accordance with the M 0,1 calculations on the basis of dispersion sum rules at finite energy [19,20].
If we restrict ourselves in s channel of the dual diagram of Fig.3b to S and P combinations of charged exchanged quarks (minimum sufficient for self-consistent description of reactions 1 and 2), then the subprocessūu →dd may involve the following states: Only 3 P 0 and 3 P 2 may contribute to M 1 since spin-orbital amplitude in the subprocess appears if there is a difference in cross sections in pair annihilation with l z = ∓1 at S z = +1. Sign inversion in l z transfers 3 P 0 ( 3 P 2 ) into another state -3 P 2 ( 3 P 0 ). In the rest cases there are no changes, as either l z = 0 or S z = 0.
M 0 can be , in principle, contributed by all 6 states.
"s"-channel spin structures of the amplitudesūu →dd with the opposite "t" channel signature (ρ and A 2 ) differ by their "u" channel singularities. But in "u" channel (ūd →ūd) repulsion forces predominate and there are no physical poles; for this reason "u" channel gives no large contribution into "s" channel that is confirmed by exchange degeneration of ρ and A 2 trajectories. This argumentation allows one to narrow the set of "s" channel bilinears of Table 2 in reactions 1 and 2 up to two general ( 3 P 0 ) 2 = (qq) 2 and ( 1 S 0 ) 2 = (qiγ 5 q) 2 . Thus, reactions 1 and 2 involve the same bilinear forms as in the minimal NJL model. Now eq. (19) is open: From (20) we determine the relative contributions M 0 ( 3 P 0 ) and M 0 ( 1 S 0 ): see Table 3. The data of Table 3 agree with predictions of the extended NJL model -destructive interference of S and P S bilinears in M 0 and their close absolute values (relic of equality G S = G P S = G in the bare Lagrangian of NJL (5).
To calculate |M 1 ( 3 P 0 )| we need the unitarity conditions of f.t. model of binary processes. Multiple processes do not enter the unitarity conditions of f.t. model since they occur after the first disruption of f.t. and causally is almost independent of the binary process. The intermediate stage contain only single f.t. which may differ only by transformation properties of the first disruption mode. The unitarity conditions in s channel in "b" representation have the form [6] where c is the probability of production of a coloured f.t. at annihilation of a colouredqq pair. Since we estimate only 3 P 0 mode of disruption, index i in (27) and hereafter is omitted. Up to exchange degeneration of reactions π − p → f.t. → π − p and π 0 n → f.t. → π 0 n where only contributions from the ρ and f pole, respectively, are taken into account From (28) and (27) where W (b, y d − y ud ) is a dimensionless probability of production of a coloured f.t. with coloured quark and diquark at the edges with the rapidity difference y d − y ud = ξ and with projection b = b d − b ud onto the plane of the impact parameters y, z (Fig.3d); is a normalized [6] density of a probability to find in the impact parameter plane a parton at the point b k with rapidity ∼ 0 if it comes out from the point b i with rapidity y i [9].
The dimensionless probability of production of a unite coloured f.t. at annihilation of a coloured pair from 3 P 0 state: is the square of the Fourier-image of scalar current with transformation properties 3 P 0 = J P = O + . In the given model the current J c S corresponds to annihilation (production) of a constituent pair of converging (diverging) waves with moments k and −k and with unit summary spin and orbital moment with conserved S Z = +1 and l Z = −1, correspondingly: where χ 1,m are spin functions of a pair. The parameter R 2 determines the scale of annihilating quark momenta. It enters the final expression of the amplitude as the radius of the Regge pole residue. Integration over k = k x is performed within −∞ < k < 0 (converging wave) Integration over dk and d 2 k ⊥ and squaring results in: where ϕ b ′ is counted from the axis Y . F (b ′ ) -the b ′ -dependence of the pair production probability in 3 P 0 state is determined such that 0 ≤ F ≤ 1. γ 2 = 1 in accordance with the blackness condition (12). 4 . The scalar current j nc S with transformation properties J P = 3 P 0 = 0 + but not conserving l z , S Z gives no contribution into amplitude M(q 2 ⊥ ) (1) since in Y 1−1 Y * 1+1 χ 1+1 χ 1−1 = 0 by virtue of orthogonality, while (χ 10 Y 10 ) 2 is out when going over into "q ⊥ " representation [14].

Conclusion
Charge-exchange reactions π − p → π 0 n, ηn; π − p → K 0 Λ(Σ) occur through two phases. 1. Diffusion of constituent quarks of each interacting hadrons in the space of rapidities and impact parameters involving flux-tube production, one edge of this f.t. having fast spectator and another -slow reagent. Confinement forces (α ′ in the diffusion equation (31)) are explicitly taken into account in this phase); it determines the Regge power decrease of the amplitude with energy increase.
2. Charge-exchange of slow reagents:ūu →dd,ūu → ss. This phase determines the spin structure and the absolute value of the Regge pole residue.
The spin-orbital amplitude M 1 , caused by spin-orbital interaction (SOI) in the process uu →dd, dominates in reactions π − p → π 0 n, ηn (more than %90 of cross section of the reaction). SOI stems from quark charge-exchange in the state J π = 0 + = 3 P 0 provided l z conservation in this process. Consequently, the processūu →dd is planary (Fig.4a). This fact is an argument in the favour of that to consider the simplest planary dual diagram of Fig.3b as an experimentally substantiated model of reactions.
The NJL model gives certain predictions on relations between different 4-quark invariants inūu →dd and, consequently, in charge-exchange physical amplitudes. In the minimal NJL model only bilinears of scalars ( 3 P 0 ) 2 and of pseudoscalars ( 1 S 0 ) 2 are effective. They enter the physical non-flip amplitude M 0 with the opposite signs and with approximately equal weights (strong destructive interference). The weights are independent of the coupling constant of NJL Lagrangian (the "blackness" condition (12)) and are determined by the dimensions of the region of charge exchangeūu →dd. The transversal radius of this region R 2 enters the finite expression of the physical amplitude as a radius of the Regge pole residue and determines its absolute value. These predictions are direct consequences of dynamical violation of the chiral invariance and of non conservation of the SU(3) singlet current , underlying the extended NJL model.
Luckily, the most completely experimentally studied reactions π − p → π 0 n(ηn) contain the same bilinears ( 3 P 0 ) 2 and ( 1 S 0 ) 2 as the minimal NJL model. This conclusion resulted from comparison of the Fiertz expansion of the t-channel soft 4-fermion amplitude in reactions π − p → π 0 n(ηn) with the account of their exchange degeneration. The analysis of experimental data had confirmed the consequences of the NJL model -the strong destructive interference 3 P 0 and 1 S 0 in M 0 . We have refrained from estimating the absolute value of M 0 in view of its theoretical instability (a small difference of large values).
A question arises why in the scalar channel of the soft instanton transitionūu →dd l z is conserved, while inūu → ss it is not?
Reactions π − p ↑ → KΛ ր (Σ ր ) are very interesting objects for studying transitionsūu → ss since the non conserving parity decays Λ ր (Σ ր ) → π − p provide an experimental possibility for the total and exact amplitude analysis of these reactions in the whole region of q ⊥ .
In this work we based on: 1. The parton concept of binary processes. 2. Non conservation of the SU(3) singlet current (gluon anomaly consequence). 3.The "blackness" condition which follows from dynamical violation of chiral symmetry. In view of a general character of these elements, the above mentioned give, as it seems to us, grounds to believe soft binary processes as a good laboratory of binary quark reactions in the nonperturbative region of QCD.
The work has been performed under a partial sponsorship of the Russian Found of Fundamental Investigations.  (16). b) Differential crossections of the reaction π − p → π 0 n at small q 2 ⊥ . The data [1] V.D. Apel et al. The solid curve (16). Fig. 2. Differential crossections π − p → K 0 Λ 0 -black circles and π − p → K 0 σ 0 -white circles. Fig. 3. Space-time scheme of the reaction π − p → π 0 n in 3 P 0 model with flux-tube. The arrow corresponds to the spin direction along axis +Z.