Loops in the Reggeon model for hA scattering
aa r X i v : . [ h e p - ph ] J un Loops in the Reggeon model for hA scattering
M. Braun, A. Tarasov ∗ Department of High Energy physics, University of S.Petersburg198904 Ulianovskaya 1, S.Petersburg, Russia
Abstract
Contribution of simplest loops is studied in the Local Reggeon Field Theory with a super-critical pomeron in the nuclear matter. It is shown that the latter transforms the supercriticalpomeron into the subcritical. Renormalization of the intercept and difficulties related to the factthat the renormalized intercept is complex are discussed. Numerical results with the conventionalparameters are reported.
With the advent of QCD much effort has been applied to study high-energy hadron-nucleus scatter-ing within its framework. However QCD can reliably describe only the hard region of the dynamics.Soft processes, contributing to the bulk of the total cross-section, are more difficult to treat. Themost sophisticated approach is to study soft processes within the perturbative QCD approach tosmall x phenomena, based on the Balitsky-Kovchegov equation, which sums all fan diagrams withself-interacting BFKL pomerons [1, 2, 3]. However this approach is based on several approxima-tions: a large number of colours N c , fixed small QCD coupling constant α s and, most seriously,neglect of loop diagrams. This latter approximation can be justified if the parameter γ = α s exp ∆ y where y is the rapidity and ∆ the pomeron intercept is small. Then for a large nuclear target, suchthat A / γ ∼
1, the tree diagrams indeed give the dominant contribution and loops may be dropped.However with the growth of y the loop contribution becomes not small and although tree diagramsare still relatively enhanced by factor A / , disregard of loops cannot be rigorously justified.Calculation of the loop contribution within the perturbative QCD appears to be a formidabletask. So it seems to be worthwhile to start with a simpler approach, that of the old local ReggeonField Theory (LRFT) with a supercritical pomeron. Such a study, apart from its possible lessons forthe modern QCD approach, also has an independent value, since the old LRFT with phenomeno-logical parameters describes not so badly the soft dynamics of high-energy strong interactions. Infact in many respects it does it better that the perturbative QCD, which encounters severe diffi-culties due to a very large value of the BFKL intercept, hardly compatible with the experimentaldata. Also in LRFT applications to hadron-nucleus [4] and nucleus-nucleus [5] collisions wererestricted to the approximation, in which the contribution from loop diagrams is neglected. Ifadditionally the slope of the pomeron α ′ is taken to be zero, then the theory is effectively livingin zero transverse dimensions and allows for the explicit analytic solution. Still remaining in zerotransverse dimension the influence of loop diagrams has also been studied, both theoretically oldago in [6, 7, 8] and numerically recently in [9]. This influence turns out to be decisive for the ∗ [email protected] ∝ − exp(1 /λ ) where λ is the triple-pomeron couplingassumed to be small.Unfortunately generalization of these beautiful results to the realistic case of two transversedimensions is prohibitively difficult. To start with, one is forced to introduce a non-zero value ofthe slope: otherwise the loop contribution is divergent in the impact parameter. However even inthe tree approximation solution of the model with α ′ = 0 is only possible numerically. Second, themodel in d T = 2 needs renormalization in the ultraviolet. And, most important, the method usedto solve the model in d T = 0, which is to study the corresponding quantum-mechanical system andthe relevant Schroedinger equation, is inapplicable in the realistic case, since instead of ordinarydifferential equations one arrives at equations with functional derivatives. In fact summation of allloop contributions is equivalent to a complete solution of the corresponding quantum field theory,the task which seems to be beyond our present possibilities. So at most one can hope to obtainsome partial results which might shed light onto the properties of the model with loops. There wereseveral attempts to study the high-energy behaviour of the LRFT with a supercritical pomeronusing different approximate techniques and giving contradicting results. In [10] it was claimed thata phase transition occurs at all values of the renormalized intercept ǫ ( r ) = α ( r ) (0) − > ǫ ( r ) greater than some critical value ǫ ( r ) c . At ǫ ( r ) < ǫ ( r ) c the theory essentially corresponds tothat with a subcritical pomeron and cross-sections vanishing at high energies. At ǫ ( r ) > ǫ ( r ) c cross-sections grow as log s . No violation of the projectile-target was found. These results were obtainedby approximating the transverse plane as a two-dimensional lattice and starting from the knownasymptotical solution at each lattice site without intersite interaction. However in this approachtransition to the continuous plane requires knowing the single-site solutions at large values of thetriple pomeron interaction constant, which are not known.In the present study we take a different approach. Instead of trying to solve the model forthe purely hadronic scattering we consider the hadron-nucleus scattering and propagation of thepomeron inside the heavy nucleus target. Moreover to avoid using numerical solution of the treediagrams contribution with diffusion in the impact parameter, we concentrate on the case of aconstant nuclear density which allows to start with the known analytical solutions. The point whichis decisive for the following derivation is that the nuclear surrounding transforms the pomeron fromthe supercritical one with intercept ǫ > − ǫ . Then Reggecuts, corresponding to loop diagrams, start at branch points located to the left of the pomeronpole and their contribution is subdominant at high energies. As a result the theory acquires theproperties similar to the standard LRFT with a subcritical pomeron and allows for application ofthe perturbation theory. For a finite nucleus it leads to cross-sections with tend to a constant valueat large energies. For the physical nucleus with a long distance tail it seems to lead to cross-sectionsgrowing as log s in agreement with [11].In fact we cannot prove all these statements rigorously, since we are not able to sum all theloops but only a certain simplest subset of them. However the results we find seem to be rathergeneral.The paper is organized as follows. In the next section we introduce the model and remind somerelevant known results for the zero-slope case α ′ = 0. In Section 3 we reformulate the model todescribe the loops in the nuclear surrounding. Then we calculate the contribution from simplestloops and discuss their renormalization in Section 4. Next in Sections 5 and 6 we sum insertions2f any number of simplest loops into the scattering amplitude and Green function by solving therelevant Dyson equations. Section 7 gives some numerical illustration of the loop influence for theamplitude and Green function with physically chosen parameters of the theory. Finally Section 8draws some conclusions. We study the LRFT model based on two pomeron fields φ ( y, b ) and φ † ( y, b ) depending on therapidity y and impact parameter b , with a Lagrangian density L = φ † Sφ + λφ † φ ( φ + φ † ) + gρφ. (1)Here in the free part S = 12 ( −→ ∂ y − ←− ∂ y ) + α ′ ∇ b + ǫ, (2)where ǫ is the intercept minus unity and α ′ is the slope. The source term describing interactionwith the nuclear target is ρ = AT ( b ) δ ( y ) . (3)It is assumed that for a supercritical pomeron ǫ > λ <
0. The action is A = Z dy Z d bL ( y, b ) (4)and the generating functional is Z = Z DφDφ † e A . (5)The free Green function in the momentum space is G (0) ( y, k ) = θ ( y ) e y ( ǫ − α ′ k ) . (6)The classical equation of motion are δLδφ = − ∂ y φ † + α ′ ∇ b φ † + ǫφ † + λφ † + 2 λφφ † + gρ = 0 (7)and δLδφ † = ∂ y φ + α ′ ∇ b φ + ǫφ + λφ + 2 λφ † φ = 0 . (8)From the latter equation we find φ = 0 and the equation for φ † takes the form ∂ y φ † = α ′ ∇ b φ † + ǫφ † + λφ † , (9)with an initial condition φ † ( y = 0) = gAT ( b ) . (10)Equation (9) describes evolution of the pomeron field in rapidity and its diffusion in the impactparameter inside the nucleus. In the approximation α ′ = 0 equation for φ † simplifies to ∂ y φ † = ǫφ † + λφ † , (11)3hich is trivially solved for each given b [4]: φ † ( y, b ) = gAT ( b ) e ǫy − λgAT ( b ) ǫ (cid:16) e ǫy − (cid:17) ≡ ξ ( y, b ) . (12)The scattering matrix is T ( y, b ) = gξ ( y, b ) . (13) In this section we shall analyze the structure of the model beyond the tree approximation in thenuclear background. To locate loops we make a shift in field φ † : φ † ( y, b ) = φ † ( y, b ) + ξ ( y, b ) (14)and reinterpret our theory in terms of fields φ and φ † .The Lagrangian density becomes L = ( φ † + ξ ) Sφ + λ ( φ † + ξ ) φ + λφ ( φ † + ξ ) + gρφ. (15)Terms linear in φ vanish due to equation of motion for ξ . We are left with L = φ † ( S + 2 λξ ) φ + λξφ + λφ † φ ( φ † + φ ) . (16)This Lagrangian corresponds to a theory in the vacuum with the pomeron propagator in the externalfield f ( b, y ) = 2 λξ ( y, b ) G (0) f = − ( S + 2 λξ ) − , (17)the standard triple interaction and an extra interaction described by the term λξφ . This newinteraction corresponds to the transition of a pair of pomerons into the vacuum at point ( y, b ) witha vertex λξ ( y, b ), see Fig. 1.Loops in the Green function may be formed both by the standard interaction and the new one.In the latter case they are to be accompanied by at least a pair of of standard interactions. Diagramswith a few simple loops in the Green function are illustrated in Fig. 2. One immediately observesthat a loop formed by the standard interaction has the order λ /α ′ and requires renormalization.A loop formed by a new interaction has the order λ /α ′ and is finite.The amplitude is obtained as a tadpole g < φ † ( y, b ) > . The simplest diagrams for it containsone loop and are shown in Fig. 3 a, b . Diagrams with more loops are shown in Figs. 3 c, d . λφ † ( y ) In the construction of loops there appears a Green function G (0) f ( y, b | y ′ b ′ ) in which the pomeronfields are coupled to the external field f ( y, b ) = 2 λξ ( y, b ) . (18)4his Green function satisfies the equation dG (0) f ( y, b | y ′ b ′ ) dy = ( ǫ + α ′ ∇ b ) G (0) f ( y, b | y ′ , b ′ ) + f ( y, b ) G (0) f ( y, b | y ′ , b ′ ) (19)with the boundary conditions G (0) f ( y, b | y ′ , b ′ ) = 0 , y − y ′ < , G (0) ( y ′ , b | y ′ , b ′ ) = δ ( b − b ′ ) . (20)In the general case the Green function G (0) f can only be calculated numerically, just as theexternal field f ( y, b ). Its analytic form can be found in two cases.If the slope α ′ = 0 then obviously we have G (0) f ( y, b | y ′ , b ′ ) = δ ( b − b ′ ) G (0) f ( y, y ′ , b ) , (21)where G (0) f ( y, y ′ , b ) is the Green function in the zero-dimensional world which satisfies dG (0) f ( y, y ′ , b ) dy = ǫG (0) f ( y, y ′ , b ) + f ( y, b ) G (0) f ( y, y ′ , b ) (22)with the boundary conditions G (0) f ( y, y ′ , b ) = 0 , y − y ′ < , G (0) f ( y ′ , y ′ , b ) = 1 (23)and f ( y, b ) given by (18) and (12). Solution of (22) for G (0) f is trivial. It is easy to find that G (0) f ( y, y ′ , b ) = e R yy ′ ds ( ǫ + f ( s,b )) = e − ǫ ( y − y ′ ) a − ( a − e − ǫy ′ a − ( a − e − ǫy ! = e ǫ ( y − y ′ ) p ( y ′ ) p ( y ) , (24)where a = − λgAT ( b ) ǫ > p ( y ) = 1 + a (cid:16) e ǫy − (cid:17) . (26)It is remarkable that at large y the Green function G (0) f ( y, y ′ , b ) ≃ exp( − ǫy ), that is behaves as thefree Green function with the opposite sign of ǫ and so vanishes at y → ∞ . In contrast to the freeGreen function, it corresponds to a subcritical pomeron.The second case which admits an analytic solution for the Green function is that of the nuclearmatter, that is the case when the profile function is constant in all transverse space T ( b ) = T . (27)Physically this case corresponds to the behaviour of the Green function at values of of the impactparameter well inside the nucleus, where the variation of T ( b ) is small. Then the classical field ξ and the external field f = 2 λξ become b -independent and the equation for the Green function inthis field takes the form dG (0) f ( y, b | y ′ , b ′ ) dy = ( ǫ + α ′ ∇ b ) G (0) f ( y, b | y ′ , b ′ ) + f ( y ) G (0) f ( y, b | y ′ , b ′ ) (28)5ith the same initial condition (20). This equation can also be easily solved. We present G (0) f ( y, b | y ′ , b ′ ) = X ( y − y ′ , b, b ′ ) e R yy ′ dy ( ǫ + f ( y )) = X ( y − y ′ , b, b ′ ) G (0) f ( y, y ′ , b ) , (29)where G (0) f ( y, y ′ , b ) is the Green function in zero-dimensional world (24) satisfying G (0) f ( y ′ , y ′ , b ) = 1Then we obtain an equation for X : ∂ y X ( y, b, b ′ ) = α ′ ∇ b X f ( y, b, b ′ ) (30)with an initial condition X (0 , b, b ′ ) = δ ( b − b ′ ) (31)and thus with the solution X ( y − y ′ , b, b ′ ) = 14 πα ′ ( y − y ′ ) e − ( b − b ′ ) / (4 α ′ ( y − y ′ )) . (32)So in the nuclear matter the Green function in the external field takes a simple form G (0) f ( y, b | y ′ , b ′ ) = 14 πα ′ ( y − y ′ ) e − ǫ ( y − y ′ ) − ( b − b ′ ) / (4 α ′ ( y − y ′ )) a − ( a − e − ǫy ′ a − ( a − e − ǫy ! , (33)where we used (24). In the momentum space we find G (0) f ( y, y ′ , k ) = e − ( y − y ′ )( ǫ + α ′ k ) a − ( a − e − ǫy ′ a − ( a − e − ǫy ! = e ( y − y ′ )( ǫ − α ′ k ) p ( y ′ ) p ( y ) . (34)Note an especially simple case when a = 1 and the Green function in the external field coincideswith the free Green function with the opposite sign of ǫ : G (0) f ( y, y ′ , k ) (cid:12)(cid:12)(cid:12) a =1 = e − ( y − y ′ )( ǫ + α ′ k ) . (35)Then the theory formally corresponds to a subcritical pomeron model with an additional interactionshown in Fig. 1 Let us study the simplest contribution from loops, the single loop in the Green function. In thesecond order in λ we get for the Green function G (2) ( y, k ) = − λ Z dy dy d k (2 π ) G (0) ( y − y , k ) G (0) ( y − y , k ) G (0) ( y − y , k − k ) G (0) ( y , k ) . (36)The minus sign comes from the fact that the simple Green function is in fact − G (0) , so that inthe second order we get − G (0) ∗ ( − G (0)2 )( − G (0) ) = − G (2) . Factor 2 comes from the contraction of φ † ( y , b ) φ ( y , b ). Integration over k is done triviallyΣ( y, k ) ≡ − λ Z d k (2 π ) G (0) ( y, k ) G (0) ( y, k − k ) = − C y e ǫy − (1 / α ′ yk , (37)6here C = λ πα ′ . (38)Σ( y, k ) is finite as it stands but leads to divergence at y = 0 when substituted into (36). So itrequires renormalization, which is achieved by introducing an extra mass term ∆ ǫφ † φ into theLagrangian. The renormalized self-mass is thereforeΣ ( r ) ( y, k ) = Σ( y, k ) + ∆ ǫδ ( y ) (39)and we have to require that Z y dy ′ Σ ( r ) ( y ′ , k ) = finite terms − C ln 1 y min + ∆ ǫ < ∞ , (40)So we conclude that ∆ ǫ = − C ln( c R y min ) , (41)where y min is the cutoff at small values of y and c R is an arbitrary finite constant. Thus we findthat the renormalized self mass isΣ ( r ) ( y, k ) = − C (cid:16) y e ǫy − (1 / α ′ yk + δ ( y ) ln( c R y min ) (cid:17) . (42)or, in the impact parameter space,Σ ( r ) ( y, b ) = − C (cid:16) πα ′ y e ǫy − b / (2 α ′ y ) + δ ( y ) δ ( b ) ln( c R y min ) (cid:17) . (43)The Dyson equation for the Green function becomes G ( y, , k ) = G (0) ( y, , k ) + Z y dy G (0) ( y − y , k ) Z y dy Σ ( r ) ( y − y , k ) G ( y , k ) . (44)It is trivially solved by the Laplace transformation. We introduce Laplace transforms G ( E, k ) = Z ∞ dye − Ey G ( y, , k ) (45)and similarly G (0) ( E, k ) and Σ ( r ) ( E, k ). In terms of the Laplace transforms G ( E, k ) = 1 (cid:16) G (0) ( E, k ) (cid:17) − − Σ ( r ) ( E, k ) (46)From (6) and (42) we have G (0) ( E, k ) = 1 E − ǫ + α ′ k . (47)and Σ ( r ) ( E, k ) = − C (cid:16) Z ∞ y min dyy e − y ( E − ǫ + α ′ k / + ln( c R y min ) (cid:17) = C (ln( E − ǫ + α ′ k /
2) + C E − ln c R ) . (48)7Note that terms with ln( y min ) cancel). So the Laplace transform for the Green function is G − ( E, k ) = E − ǫ + α ′ k − C [ln( E − ǫ + α ′ k /
2) + C E − ln c R ] . (49)Its singularities in the complex E -plane consist of a branchpoint at E = 2 ǫ − α ′ k / E ( k ) determined by the equation: G − ( E , k ) = 0 (50)At small λ Eq. (50) gives two complex conjugate poles E ( ± )0 = ǫ − α ′ k + C (ln( | ǫ + α ′ k / | ) + C E − ln c R ± iπ ) . (51)These poles appear on the physical sheet due to the abnormal sign of the self-mass contribution,which is a consequence of the imaginary coupling in the original Lagrangian of the LRFT. Theasymptotic of the Green function is ∼ exp(2 µy ) in accordance of the contribution of the Regge cutfor a supercritical pomeron. According to (41) the unrenormalized pomeron intercept (dependingon the cutoff) is ǫ ( y ) = ǫ − C ln( c R y min ) . (52)It goes to infinity as y min →
0, since
C >
0. As to the renormalized intercept its definition may bechosen in different ways. Traditionally it may be taken as the position of the pole in E in the Greenfunction at k = 0. Then it is given by (50), complex and c R -dependent. One observes the difficultyin doing renormalization in the standard way by requiring the zero order propagator to have thepole at a chosen energy E ( k = 0): unless E > ǫ there are two complex conjugate values of E and if one chooses E > ǫ contributions from higher order self mass will again make it complex.So the structure of perturbative singularities prohibits choosing the zero-order propagator to carrythe pole singularity, since there are two. However from the physical point of view such a choice isnot necessary Inclusion of more pomeron exchanges shifts the dominating singularity further to theright, so that the simple pole contribution to any physical process becomes completely obliterated.It is more reasonable to relate the value of the renormalized intercept (and thus fix the arbitraryconstant c R ) to some physical observables. In the following we shall see that in the scattering onthe nucleus there appears a possibility to define the renormalized intercept from experimental dataat high energies, with a certain choice of c R . The contribution from the lowest order loop (Fig. 2 a ) in the nuclear matter to the Green functionin is G (2) f ( y, , k ) = Z dy dy G (0) f ( y, y , k )Σ ( y , y , k ) G (0) f ( y , , k ) . (53)where the Green functions G (0) f are given by (24) andΣ ( y , y , k ) = − C (cid:16) y − y e − ǫ ( y − y ) − (1 / y − y ) α ′ k e R y y dsf ( s ) + δ ( y − y ) ln( c R y min ) (cid:17) . (54)is the renormalized 2nd order self mass in the nuclear background. The subtraction term is easilycalculated: G (2) f,subtr = − C ln( c R y min ) Z dy G (0) f ( y, y , k ) G (0) f ( y , , k ) = − C ln( c R y min ) yG (0) f ( y, , k ) . (55)8he main term is given by − CG (0) f ( y, , k ) Z y dy Z y dy y − y e ( y − y )( ǫ + α ′ k ) p ( y ) p ( y ) . (56)Changing to integration variable z = y − y we find G (2) f,main ( y, , k ) = − CG (0) f ( y, , k ) Z y dzz W ( z ) , (57)where W ( z ) = e z ( ǫ + α ′ k ) Z yz dy p ( y − z ) p ( y ) ≡ e zα ′ k ˜ W ( z ) . (58)We note that W (0) = y so that we can write G (2) f,main ( y, , k ) = − CG (0) f ( y, , k ) n Z y dzz (cid:16) W ( z ) − y (cid:17) + y ln yy min o . (59)The second term just substitutes in the subtraction contribution y min by y . The first term can beintegrated numerically since it converges at z = 0.Function ˜ W ( z ) can be calculated analytically. Passing to integration variable u = e ǫy anddoing simple integrals we find˜ W ( z ) = e − ǫz (cid:16) y − z + q − q ǫq ln u ( u + q ) u ( u + q ) − ( q − q ) ǫq u − u ( u + q )( u + q ) (cid:17) . (60)where u = e ǫz , u = e ǫy , q = 1 /a − q = qu .Note that after renormalization the 2nd order contribution to the Green function is also finitein the nuclear background. b The new loop shown in Fig. 2 b , which appears in the nucleus, has formally order λ . However itis proportional to AT ( b ), so that the extra λ combines into the characteristic parameter a , whichis not small. So in fact it has the same order of magnitude as the simple loop of Fig. 2 a .The contribution to the pomeron self-mass corresponding to Fig. 2 b isΣ ( y , y , k ) = 4 λ Z y dy Z d k (2 π ) G (0) f ( y , y , k − k ) G (0) f ( y , y , k ) G (0) f ( y , y , k ) ξ ( y ) . (61)Here it is assumed that y > y . The final factor 4 combines factor 4 from contractions of φ ( y )with φ † ( y ) and φ † ( y ), factor 2 coming from interchange 2 ↔ / ξ and definition (25) we rewrite (61) asΣ ( y , y , k ) = − λ aǫ Z y dy p ( y ) p ( y ) e ǫ (2 y − y ) − ( y − y ) α ′ k Z d k (2 π ) e − α ′ k ( y − y )+2( y − y ) α ′ kk . (62)9he integral over k is 18 πα ′ ( y − y ) e α ′ k y − y y − y , so that in the end we obtainΣ ( y , y , k ) = − λ aǫ πα ′ p ( y ) e ǫy − ( y − y ) α ′ k Z y dy y − y p ( y ) e − ǫy + α ′ k y − y y − y (63)It is trivial to see that Σ is finite and does not need any more renormalization. We start from the lowest order approximation, when at fixed b the scattering amplitude is T (0) ( y, b ) = g AT ( b ) e ǫy p ( y ) . (64)The total forward scattering amplitude is obtained after integration over all b : T (0) ( y ) = Z d bT (0) ( y, b ) . (65)This expression can be rewritten in the momentum space, once we introduce the Fourier transformof T ( y, b ) by T (0) ( y, b ) = Z d k (2 π ) e ikb ˜ T (0) ( y, k ) , (66)as T (0) ( y ) = Z d bd k (2 π ) e ikb ˜ T (0) ( y, k ) = ˜ T (0) ( y, . (67)In the nuclear matter T ( y, b ) does not depend on b : T (0) ( y, b ) = g AT e ǫy p ( y ) , (68)so that its Fourier transform is ˜ T (0) ( y, k ) = (2 π ) δ ( k ) g AT e ǫy p ( y ) . (69)From (65) the forward scattering amplitude is T (0) ( y ) = Z d bt ( y ) = πR A g AT e ǫy p ( y ) = g A e ǫy p ( y ) , (70)where we used that T = 1 / ( πR A ). Due to (67) this of course implies that (2 π ) δ ( k = 0) → πR A . In the limit y → ∞ the lowest order amplitude tends to a finite value T (0) ( y ) y →∞ = g Aa = πR A gǫ | λ | . (71)10 .2 2nd order In the next order we have˜ T (2) ( y, k ) = Z dy dy G (0) f ( y, y , k )Σ ( y , y , k ) ˜ T (0) ( y , k )= (2 π ) δ ( k ) g AT Z dy dy G (0) f ( y, y , ( y , y , e ǫy p ( y ) (72)so that the forward scattering amplitude is T (2) ( y ) = πR A Z dy dy G f ( y, y , ( y , y , T (0) ( y , b ) . (73)The subtraction term in Σ ( r ) gives a contribution T (2) sub ( y ) = − CπR A ln( c R y min ) Z y dy G (0) f ( y, y , T (0) ( y , b ) . (74)Using (68) and (24) we get T (2) sub ( y ) = − Cg AG (0) f ( y, ,
0) ln( c R y min ) Z y dy p ( y ) . (75)We define I ( y, z ) ≡ Z yz dy p ( y ) = aǫ (cid:16) e ǫy − e ǫz (cid:17) + ( y − z )(1 − a ) . (76)Then we find T (2) sub ( y ) = − Cg A e ǫy p ( y ) ln( c R y min ) I ( y, . (77)The main term is T (2) main ( y ) = − Cg A e ǫy p ( y ) Z y dy Z y dy e ǫ ( y − y ) y − y p ( y ) p ( y ) . (78)Passing to integration variables y and z = y − y we find T (2) main ( y ) = − Cg A e ǫy p ( y ) Z y dzz Ψ( z ) , (79)where Ψ( z ) = e ǫz Z yz dy p ( y − z ) p ( y ) . (80)Note that Ψ(0) = I ( y, . (81)Thus we find T (2) main ( y ) = − Cg A e ǫy p ( y ) n Z y dzz (cid:16) Ψ( z ) − Ψ(0) (cid:17) + ln yy min I ( y, o . (82)The second term just changes y min to y in the subtraction term.11unction Ψ( z ) can be calculated analytically in the same way as ˜ W ( z ) (see Eq. (58)). Onefinds Ψ( z ) = I ( y, z ) u + 3 aq u − u ( y − z ) + aq (2 + u ) ( u − ǫu ln u ( u + q ) u ( u + q )+ aq ( u − ǫu u − u ( u + q )( u + q ) . (83)where as before u = e ǫz , u = e ǫy , q = 1 /a − q = qu . The remaining integration over z has to be done numerically.It is not difficult to find the asymptotic behaviour of T (2) ( y ) at large y . At y → ∞ , asymptoti-cally p ( y ) = ae ǫy , I ( y,
0) = aǫ e ǫy (84)so that in this limit T (2) ( y ) sub = − C g Aaǫ ln( c R y min ) (85)The asymptotic of the main term comes from large values of z inside the integral. At large z Ψ( z ) − Ψ(0) = − aǫ e ǫy (cid:16) − e − ǫz (cid:17) (86)and so Z y dzz (cid:16) Ψ( z ) − Ψ(0) (cid:17) ≃ aǫ e ǫy (cid:16) Ei( − ǫy ) − ln(2 ǫy ) − C E (cid:17) ≃ − aǫ e ǫy (cid:16) ln(2 ǫy ) + C E (cid:17) . (87)Thus asymptotically T (2) main ( y ) = C g Aaǫ (cid:16) ln(2 ǫy min ) + C E (cid:17) . (88)In the sum the terms with ln y min cancel and we find that in the limit y → ∞ the loopcontribution also tends to a constant T (2) ( y ) y →∞ = − C g Aaǫ (cid:16) ln c R ǫ − C E (cid:17) . (89)The ratio first to second order is at y → ∞ r (2) ( y ) = T (2) ( y ) T (0) ( y ) (cid:12)(cid:12)(cid:12) y>> = − λ πα ′ ǫ (cid:16) ln c R ǫ − C E (cid:17) . (90)Remarkably it does not depend on A . In the third order the forward scattering amplitude is T (3) ( y ) = πR A Z dy dy G f ( y, y , ( y , y , T (0) ( y , b ) . (91)Using (63) at k = 0 and interchanging the order of integration in y and internal integration in y inside Σ we find T (3) ( y ) = − g A λ aǫ πα ′ e ǫy p ( y ) Z y dy e ǫy p ( y ) Z y dy y − y p ( y ) e − ǫy Z y y dy e ǫy p ( y ) . (92)12t is not difficult to find the asymptotic of T (3) ( y ) at large y . We present (92) in the form T (3) ( y ) = − g A λ aǫ πα ′ e ǫy p ( y ) Z y dy e ǫy p ( y ) F ( y ) , (93)where taking y = y β and y = y β F ( y ) = y Z dβ − β p ( y β ) e − ǫy β Z β dβ e ǫy β p ( y β ) . (94)We are interested in the behaviour of F ( y ) as y >>
1, since this governs the behaviour of theintegral over y in (93). As y → ∞ we have p ( y β ) → a exp( ǫy β ) so that F ( y ) y >> = a y Z dβ − β e ǫy β Z β dβ = a ǫ (cid:16) e ǫy − (cid:17) . (95)Putting this asymptotic in (93) we obtain T (3) ( y ) y>> = − g A λ πaα ′ e − ǫy Z y dy e ǫy ≃ − g A λ πaα ′ ǫ . (96)Recalling that at y >> T (0) = g A/a we find the ratio r (3) ( y ) = T (3) ( y ) T (0) ( y ) (cid:12)(cid:12)(cid:12) y>> = − λ πα ′ ǫ (97)Note that in the particular case a = 1, T (3) can easily be found explicitly at all y . In fact trivialintegrations give T (3) a =1 ( y ) = − g A λ πα ′ ǫ (cid:16) − e − ǫy (cid:17) . (98)The full ratio T /T (0) from both loops, Fig. 2 a and b , turns out to be r ( y ) = r (2) ( y ) + r (3) ( y ) = − λ πα ′ ǫ (cid:16) ln c R ǫ + 1 − C E (cid:17) . (99)The bracket is universal. So one may define the renormalized self-mass by requiring that thisbracket is zero, which implies that the loop correction vanishes at large y :ln c R ǫ − C E + 1 = 0 , or c R = 2 ǫe C E − . (100)This allows to experimentally determine a and hence ǫ from the behaviour of the amplitude at large y . In this way we may define the renormalized intercept in a physically reasonable manner.Note that with this choice of c R the renormalized intercept ǫ ( r ) formally defined as the positionof the pole of the propagator in the vacuum at k = 0 is given by the solution of the equation (50)which reads ǫ ( r ) − ǫ − C [ln ǫ ( r ) − ǫ ǫ + 1] = 0 (101)At small λ it is complex and close to ǫ . 13 .4 Random phase approximation One may try to sum all primitive loop insertions into the amplitude (’random phase approxima-tion’). In this approximation the amplitude T ( y ) in the nuclear matter satisfies the Dyson equation T ( y ) = T (0) ( y ) + Z y dy Z y dy G (0) f ( y, y , (cid:16) Σ ( y , y ,
0) + Σ ( y , y , (cid:17) T ( y ) , (102)where Σ and Σ are the 2nd and 3d order self-mass contributions studied above.We present T ( y ) = T (0) ( y ) r ( y ) . (103)The equation for r ( y ) reads r ( y ) = 1 + X ( y ) + X ( y ) , (104)where X and X are parts coming from Σ and Σ respectively.We find X ( y ) = − Cp ( y ) ln( c R y min ) Z y dy p ( y ) r ( y ) − Cp ( y ) Z y dy Z y dy y − y e ǫ ( y − y ) p ( y ) p ( y ) r ( y ) . (105)Again we pass to variable z = y − y to rewrite the last term as − Cp ( y ) Z y dzz ω ( y, z ) = − Cp ( y ) Z y dzz (cid:16) ω ( y, z ) − ω ( y, (cid:17) − Cp ( y ) ω ( y, Z yy min dzz , (106)where ω ( y, z ) = e ǫz Z yz dy p ( y − z ) p ( y ) r ( y − z ) . (107)Obviously ω ( y,
0) = Z y dy p ( y ) r ( y ) , (108)so that the last term in (106) changes y min in (105) to y . As a result we find that the part X ( y )in the equation for r takes the form X = − Cp ( y ) ln( c R y ) Z y dy p ( y ) r ( y ) − Cp ( y ) Z y dzz (cid:16) ω ( y, z ) − ω ( y, (cid:17) . (109)The part X ( y ) is X ( y ) = − aǫCp ( y ) Z y dy e ǫy p ( y ) Z y dy y − y p ( y ) e − ǫy Z y y dy e ǫy p ( y ) r ( y ) . (110)In the case a = 1 and neglecting X we can solve the Dyson equation passing to the Laplacetransforms in rapidity. We find T ( E ) = T (0) ( E ) (cid:16) G (0) f (cid:17) − ( E, (cid:16) G (0) f (cid:17) − ( E, − Σ ( r ) f ( E, , (111)where G (0) f ( E,
0) and Σ ( r ) f ( E,
0) have the same form as (47) and (48) with the opposite sign of ǫ and T (0) ( E ) = g A E . (112)14o the Laplace transform for the amplitude is T ( E ) = g AE E + ǫE + ǫ − C [ln( E + 2 ǫ ) + C E − ln c R ] . (113)From the expression for T ( E ) we immediately conclude that in the limit y → ∞ the amplitudetends to T ( y ) | y →∞ = g AZ, (114)where Z = ǫǫ − C [ln(2 ǫ ) + C E − ln c R ] . (115)Factor Z − Z = 1, which implies for this case ( X = 0)ln(2 ǫ ) + C E − ln c R = 0 , (116)With this choice Σ ( r ) ( E, k ) = C ln E + 2 ǫ + α ′ k / ǫ (117)and T ( E ) = g AE E + ǫE + ǫ − C ln E +2 ǫ ǫ . (118) In this approximation the full Green function in the nuclear matter G f ( y, , k ) satisfies the Dysonequation G f ( y, , k ) = G (0) f ( y, , k ) + Z y dy Z y dy G (0) f ( y, y , k ) (cid:16) Σ ( y , y , k ) + Σ ( y , y , k ) (cid:17) G f ( y , , k )(119)We present G f ( y, , k ) = G (0) f ( y, , k ) R ( y, k ) (120)to obtain an equation for R R ( y, k ) = 1 + Y + Y . (121)Here Y comes from Σ : Y ( y, k ) = − C ln( c R y min ) Z y dy R ( y , k ) − C Z y dy Z y dy y − y e β ( y − y ) p ( y ) p ( y ) R ( y , k ) (122)and β = ǫ + α ′ k /
2. Passing to integration variable z = y − y we rewrite the last term in Eq.(122) as − C Z y dzz Ω( y, z ) = − C Z y dzz (cid:16) Ω( y, z ) − Ω( y, (cid:17) − C Ω( y, Z yy min dzz , (123)where Ω( y, z ) = e βz Z yz dy p ( y − z ) p ( y ) R ( y − z, k ) . (124)15bviously Ω( y,
0) = Z y dy R ( y , k ) , (125)so that the last term in (123) changes y min in (122) to y . As a result we find Y ( y, k ) = − C ln( c R y ) Z y dy R ( y , k ) − C Z y dzz (cid:16) Ω( y, z ) − Ω( y, (cid:17) . (126)The part Y comes from Σ : Y ( y, k ) = − aǫC Z y dy e ǫy p ( y ) Z y dy y − y p ( y ) e − ǫy Z y y dy e ǫy p ( y ) R ( y , k ) exp (cid:16) α ′ k ( y − y ) y − y (cid:17) . (127)In the general case Eq. (126) can only be solved numerically. In a particular case a = 1 boththe Green function G (0) and Σ depend only on the rapidity difference: p ( y ) = e ǫy (128)and G (0) f ( y , y , k ) = e − ( y − y )( ǫ + α ′ k ) = G (0) f ( y − y , k ) , (129)Σ ( y , y , k ) = − C (cid:16) y − y e − ( y − y )(2 ǫ + α ′ k / + δ ( y − y ) ln( c R y min (cid:17) = Σ ( y − y , k ) . (130)They have the same form as in the vacuum with the opposite sign of ǫ . So if one additionallyneglects the formally 3d-order contribution Σ the Dyson equation (119) can be again analyticallysolved by the Laplace transform in the same way as in the vacuum case. Moreover, the solution isexactly the same as in the vacuum case with the change ǫ → − ǫ : G − f ( E, k ) = E + ǫ + α ′ k − C [ln( E + 2 ǫ + α ′ k /
2) + C E − ln c R ] . (131)Singularities of G f ( E, k ) in the complex E − plane consist of a left cut starting at E c = − ǫ − α ′ k / − ǫ anda possible pole whose position depends on the renormalization constant c R and which determinesthe asymptotic at large rapidities. With the choice (116) corresponding to Σ = 0 we have G − f ( E, k ) = E + ǫ + α ′ k − C ln E + 2 ǫ + α ′ k / ǫ (132)The Green function vanishes at y → ∞ as e − yǫ where for α ′ k < ǫǫ + α ′ k < ǫ ≤ ǫ + α ′ k / . (133)If α ′ k ≥ ǫ , the two-pomeron cut moves to the right of the pole, which splits into two complexconjugate poles on the physical sheet, and the asymptotic is determined by the cut so that y =2 ǫ + α ′ k / spoils invariance with respect to translations in rapidity, andsolution of the Dyson equation by means of the Laplace transform becomes impossible.16 Numerical illustration
In this section we report on numerical results for the amplitude T ( y ) and Green function G f ( y, k )which follow from the Dyson equations (102) and (119) with the pomeron self-mass Σ = Σ + Σ given by loops of Fig. 2 a and b and realistic values of ǫ , α ′ , λ and g For the latter we take thestandard values, which correspond to the experimental data at comparatively low energies ǫ = 0 . , α ′ = 0 . GeV − , λ = − . GeV − , g = 5 . GeV − (134)For the transverse nuclear density we choose that in the center of the nucleus with the constantnuclear density within a sphere of radius R A = A / · .
15 fm.With ǫ , α ′ and λ given by (134) we obtain from Eq. (101) the renormalized values for theintercept and slope in the vacuum: ǫ ( r ) = 0 . ± i . , α ′ ( r ) = 0 . ∓ i .
043 (135)Both are complex. The real part of the pomeron trajectory is found to be linear in k with a goodprecision. The imaginary part is not, the value of Im α ′ ( r ) diminishing from 0.043 at k = 0 to 0.020at k = 2 GeV/c. So we are dealing with a supercritical pomeron with a complex intercept andslope.Next we pass to the amplitude and pomeron Green function in the nuclear background.Fig. 4 shows the ratio r ( y ) of the calculated amplitude to the lowest order one as a functionof y for two values of the atomic number A = 64 and 207. As one observes as rapidity grows theratio initially goes up reaching at its maximum values of the order 2.2 ÷ A andthen gradually goes to unity, as determined by the chosen value of c R . The A -dependence is infact weak, so that the two curves are close to one another. The rather large values of the ratioat intermediate rapidities show that for the chosen parameters the loop contribution is not at allsmall. In other words the strength of the triple pomeron interaction is relatively large although iteffectively goes down at large rapidities.In Figs. 5-7 we show the R ( y, k ) of the Green function to its lowest order value as a functionof y for values of k = 0, 1 and 2 GeV/c and A = 64 and 207. For convenience for k = 2 GeV/cwe plot R ( y, k ) = − R ( y, k ) exp (cid:16) y ( ǫ − α ′ k / (cid:17) , since for such large k the asymptotic is negativeand governed by the cut contribution. Again we observe that the A -dependence is weak. The y -dependence strongly depends on the chosen value of k , which is to be expected, since the cutcontribution vanishes at k → ∞ much slower, with a twice smaller slope. For all momenta theloop contribution is relatively large at all rapidities, which again indicates that with our choice λ is not small. Oscillations observed in Figs. 5 and 6 illustrate that the ’physical’ intercept ǫ ( r ) formoderate k in fact splits into two complex conjugate values. At larger k the asymptotic is takenover by the cut contribution and oscillations disappear. We studied the contributions of the two simplest loops in the local reggeon field theory with asupercritical pomeron in the nucleus. Our results show that the nuclear surrounding effectivelytransforms the supercritical pomeron with the intercept α (0) − ǫ > − ǫ . As a result, at high energies the pomeron Green function vanishes and17ontributions from multipomeron exchanges vanish still faster according to the standard predic-tions for the subcritical pomeron. With that, for a finite nucleus with a constant profile function,the scattering amplitude tends to a constant value at high energies. Contribution from self-massinsertions do not change this behaviour, since in the rapidity space they are dominated by the con-figuration in which these insertions enter at small rapidities and the bulk of the rapidity is coveredby the lowest order amplitude. With an appropriate choice of the renormalization constant onecan make the contribution from loops vanish at high energies. The renormalized pomeron inter-cept in this theory (in vacuum) is complex with a positive real part. This makes the conventionalrenormalization technique inapplicable.These results have been obtained for a simple case of a constant nuclear profile function T ( b ) andwith only lowest order loops taken into account. The limitation to a constant T ( b ) is technical andwe believe that the same conclusions can be obtained for a realistic T ( b ). However this requires usingnon-trivial pomeron Green functions in the external b -dependent field, which can only be calculatednumerically. Construction of loops in this case presents a formidable calculational problem, whichseems to be outside our technical possibilities. The only difference we expect with a realistic T ( b )is due to its long distance tail. Then at large b > b ( y ) the amplitude and the Green functionwill no more be damped by the nucleus and become purely perturbative, that is growing as e ǫy at large y . Very crudely we expect b to be determined by the relation T ( b ) e ǫy ∼ b ( y ) ∼ ay where a is an A -independent parameter of the dimension of length ( a = 0 .
545 fm withthe Woods-Saxon nucleus density). Then, again very crudely, the cross-section obtained by theintegration over all b will become proportional to y in accordance with the conclusions in [11].Inclusion of more complicated loops does not present any difficulty of principle either. For smallenough λ their contribution will be small at all rapidities and we do not expect any qualitativechange.Note that with the conventional values (134) for ǫ , α ′ and λ the loop contribution proves to benot at all small, in spite of the rather small value of the effective loop parameter C ∼ .
09. Therenormalized intercept (135) obtained as the solution of Eq.(50) is quite different from its zero-order value ǫ = 0 .
08, which testifies that the coupling constant λ is in fact quite large. Becauseof that we cannot claim that with the conventional parameters (134) our results are complete .More complicated loops have to be included to make the results reliable and fit to be used for thedescription of physical observables.This is unfortunate, but we consider our results to be interesting mostly from the purely theo-retical point of view. They may serve as a basis for treating loop contributions in the perturbativeQCD. It seems to be advantageous to study them in the nuclear surrounding, which makes thehigh-energy behaviour much more tractable. Appendix A: Possible poles of propagators
Let the denominator in (49) be D ( E, k ) = E − ǫ + α ′ k − | C | [ln( E − ǫ + α ′ k /
2) + C E − ln c R ] , (A1)where we have taken into account that C <
0. We denote δ = | C | ( C E − ln c R ) and rewrite (A1) as D ( E, k ) = E − ǫ + α ′ k − δ − | C | ln( E − ǫ + α ′ k / . (A2)Function D ( E, k ) has a left cut starting at E = 2 ǫ − α ′ k /
2, so that it can vanish only to theright of the branchpoint. At E → ǫ − α ′ k / D ( E, k ) → + ∞ . At E → + ∞ also18 ( E, k ) → + ∞ . The single minimum of D ( E, k ) as a function of E occurs at ∂D ( E, k ) ∂E = 1 − | C | E − ǫ + α ′ k / , (A3)that is at E min = e | C | + 2 ǫ − α ′ k . (A4)The minimal value of D ( E, k ) is D min ( E, k ) = e | C | + ǫ + 12 α ′ k − δ − | C | . (A5)Obviously if D min > D ( E, k ) does not vanish on the physical sheet of E . If D min < E determines the asymptotic of thepropagator at large y . The value of D min depends on the choice of δ . So if δ > e | C | + ǫ + 12 α ′ k − | C | (A6)the propagator develops two poles to the right of the branchpoint at E = 2 ǫ − α ′ k /
2, the larger ofwhich is located to the right of E min and thus to the right of the unperturbed pole at E = µ − α ′ k .This larger pole takes the function of the renormalized intercept. If, on the other hand, relation(A6) is not satisfied, the propagator has no poles on the physical sheet and its asymptotic at y → ∞ is determined by the branchpoint at E = 2 ǫ − α ′ k / a = 1, at k = 0 we find a denominator D ( E ) = E + ǫ − | C | ln E + 2 ǫ ǫ (A7)We put E + 2 ǫ = 2 ǫξ , | C | / (2 ǫ ) = η <<
1, so that zeros are determined by the equation for ξf ( ξ ) = ξ − η ln ξ −
12 = 0 . (A8)The minimum of f ( ξ ) occurs at ξ min = η and f min = η − η ln η −
12 (A9)At η < f min < η < . < ξ < η < ξ <
12 (A10)(the limiting value 1 / η → η and are located between the unperturbed pole at E = − ǫ andbranchpoint at E = 2 ǫ . One can easily see that this result is also true for k >
0. So the poles donot change the asymptotic of the amplitude, but diminish the intercept of the propagator, makingit still more subcritical. 19 eferences [1] I.I.Balitsky, Nucl. Phys. B , 99 (1996)[2] Yu.V.Kovchegov, Phys. Rev. D , 034008 (1999); , 074018 (2000)[3] M.A.Braun, Eur. Phys. J. C , 337 (2000)[4] A.Schwimmer, Nucl.Phys. B , 445 (1975)[5] D.Amati, L.Caneschi, R.Jengo, Nucl. Phys. , 397 (1975)[6] V.Alessandrini, D.Amati, R.Jengo, Nucl.Phys. B , 425 (1976)[7] R.Jengo, Nucl. Phys. B , 425 (1976)[8] M.Ciafaloni, M.Le Bellac, G.C.Rossi, Nucl. Phys. B , 388 (1977)[9] M.A.Braun, G.P.Vacca, Eur. phys. J. C , 857 (2007)[10] H.D.Abarbanel, J.B.Bronzan, A.Schwimmer, R.Sugar, Phys. Rev. D , 632 (1976)[11] D.Amati, M.Le Bellac, G.Marchesini, M.Ciafaloni, Nucl. Phys. B , 107 (1976) Figure captions
Figure 1: The new vertex for two-pomeron annihilation, which appears after the shift in field φ † .Figure 2: Some simple loop diagrams for the pomeron Green function.Figure 3: Diagrams with one loop ( a, b and two loops ( c and d ) for the scattering amplitude.Figure 4: The ratio r ( y ) of forward scattering amplitude to its lowest order for Cu and Pb targets.Figure 5: The ratio R ( y ) of the pomeron Green function to its lowest order at k = 0 for Cu andPb targets.Figure 6: Same as Fig. 5 at k = 1 GeV/c.Figure 7: The scaled ratio R ( y, k ) = − R ( y, k ) exp (cid:16) y ( ǫ − α ′ k / (cid:17) at k = 2 GeV/c for Cu and Pbtargets. 20igure 1: The new vertex for two-pomeron annihilation, which appears after the shift in field φ † Figure 2: Some simple loop diagrams for the pomeron Green functionFigure 3: Diagrams with one loop ( a, b and two loops ( c and d ) for the scattering amplitude21 r( y ) y PbCu Figure 4: The ratio r ( y ) of forward scattering amplitude to its lowest order for Cu and Pb targets -4-3-2-101234 0 5 10 15 20 25 30 35 40 R ( y ) y PbCuk=0.0 GeV/c Figure 5: The ratio R ( y ) of the pomeron Green function to its lowest order at k = 0 for Cu andPb targets 22 R ( y ) yPbCu k=1.0 GeV/c Figure 6: Same as Fig. 5 at k = 1 GeV/c R ( y ) yPb Cuk=2.0 GeV/c Figure 7: The scaled ratio R ( y, k ) = − R ( y, k ) exp (cid:16) y ( ǫ − α ′ k / (cid:17) at kk