Conformal invariant pomeron interaction in the perurbative QCD with large N_c

An effective non-local quantum field theory is constructed, which describes interaction of pomerons in the high-coloured QCD. The theory includes both splitting and merging triple pomeron vertexes and diagrams with pomeronic loops. The Schwinger-Dyson equations for the 'physical' pomeron are written. Conformal invariance allows to reduce the theory to the old-fashioned Gribov pomeron theory with an infinite number of pomerons, one of which is supercritical.


Introduction
The high-energy behaviour in QCD with a large number of colours N c is described by the exchange of hard pomerons, which split and merge by a triple pomeron vertex.Exchange of coloured objects (single gluons) is damped by factor 1/N 2 c .In interaction with heavy nuclei the leading contribution comes from diagrams without pomeronic loops (tree diagrams).Summation of these is achieved by a closed equation for DIS (the BK equation [1,2,3]) or a closed pair of equations for nucleus-nucleus scattering [4].Some estimates of the contribution from loops were made in [5,6].To take into account the contribution from pomeronic loops in a consistent manner one has to consider the effective pomeron field theory introduced in [4] as a full-fledged quantum theory.The present study is devoted to this aim.
Note that pomeronic loops have lately been also actively studied in the framework of the colour dipole picture in the so-called JIMWLK approach (see e.g.[7] and references therin).
There evolution in rapidity of a state considered as a functional of the gluon field is governed by a certain Hamiltonian made of the field and functional derivatives in the field.Taking loops into account leads to a Hamiltonian containing functional derivatives up to the fourth order [8].In our approach a quantum Hamiltonian can also be introduced.However it contains functional derivatives only up to the second order and so is considerably simpler than in the JIMWLK approach.
In this study we restricted ourselves to a rather formal treatment of the perturbative diagrams for the pomeron interaction with or without loops.We construct the relevant Schwinger-Dyson equations for the full pomeron Green function and also discuss their conformal (Moebius) invariance, which hopefully may simplify their analysis.In fact we are not very optimistic about a realistic calculation of the amplitudes with pomeronic loops included.This problem presented enormous difficulties already for the much simpler old-fashioned Gribov local supercritical pomeron model, so that even its internal inconsistency was claimed [9] (see also [10] for a discussion of this inconsistency).Possibly the more complicated structure of the BFKL pomeron and its interaction may overcome these old troubles.However in the present study this problem is not touched but left for future investigations.

Effective action and diagrams 2.1 Effective action
Our main tool will be the non-forward BFKL Green function as a function of gluon coordinates and rapidities, which satisfies the equation where r 11 ′ = r 1 − r ′ 1 etc and H is the BFKL Hamiltonian [11]: and ᾱ = α s N c /π.To economize on notations, in the following we shall denote as z the point in the space formed by rapidity y and two transverse vectors r 1 , r 2 : z = {y, r 1 , r 2 } = {y, ρ}.
In this notation we write g(y − y ′ ; r 1 , r 2 ; r ′ 1 , r ′ 2 ) = g(z, z ′ ).The Green function g(z, z ′ ) is invariant under conformal (Moebius) transformations of coordinates..The inverse function g −1 (z, z ′ ) defined by where dz = dyd 2 r 1 d 2 r 2 ≡ dydρ and δ(z) = δ(y)δ 2 (r 1 )δ 2 (r 2 ) ≡ δ(y)δρ, is however not conformal invariant, since the measure is not conformal invariant.It is not difficult to construct a conformal invariant inverse g −1 inv using the invariant measure: Then we can rewrite (3) as Now we pass to constructing the effective non-local quantum field theory, which is to describe propagation of free pomerons and their triple interaction.It has to generate all diagrams built from the pomeron propagator, which is the Green function g, and triple pomeron vertex introduced in [12,13,14].Let φ B (z) and φ A (z) be two bilocal fields for the incoming and outgoing pomerons.To reproduce the correct propagator the free action S 0 has to be We introduce interaction with an external sources as Here the sources for the projectile A and target B are different from zero at rapidities y = Y and y = 0 respectively: Finally the interaction to reproduce the 3P vertex at large N c has to be taken in the form: where z 1 = {y, r 2 , r 3 }, z 2 = {y, r 3 , r 1 }, z 3 = {y, r 1 , r 2 } and the conformal invariant operator Note that the form (10) assumes the fields to be symmetric in the two space points r 1 and r 2 .In fact the symmetry properties of the fields are determined by symmetry properties of the external sources.We assume them to be symmetric under the interchange r 1 ↔ r 2 .
This action leads to BFKL pomeron diagrams with the standard 3P interaction in the presence of an external field.Note that the signs of different building blocks of the diagrams is somewhat different from the standard ones: the propagator g and external sources enter with a minus sign as a consequence of the choice of signs in the action.This latter corresponds to the desire to make the interaction real and not pure imaginary as in the original Gribov reggeon field theory.
In absence of the external sources this action is explicitly conformal invariant provided φ B and φ A are invariant.Indeed the free part can be rewritten as and the interaction part as where γ(z ′ , z ′′ |z) is the bare interaction vertex for the incoming pomeron at z and two outgoing pomerons at z ′ and z ′′ : This is a conformal invariant function.Since both r 4 12 g −1 (z, z ′ )r ′ 12 4 and γ(z ′ , z ′′ |z) are conformal invariant, so is the action S 0 + S I .Of course in physically relevant cases conformal invariance is always broken by the external sources, which also introduce a mass scale into the theory.As a result contribution from any Feynman diagram without external sources is conformal invariant, which becomes explicit if one uses the invariant integration measure dz inv = dydτ and invariant interaction vertex γ given by Eq. ( 14).
Finally note that from (1) it follows that as an operator in the z-space Here we use the notation Note that operator ∇ 2 ρ H is symmetric: As a result S 0 can be written directly in terms of the BFKL Hamiltonian: In this form the symmetry between the projectile and target is made explicit: it has to be accompanied by changing y → −y.
Using a representation for H ( [15]) we can rewrite the free part of the action in a more explicit form

Diagrams and their order of magnitude
The quantum field theory described by the action S = S 0 + S I + S E allows to construct the perturbation theory by the standard technique, expressing the amplitude as a sum of Feynman diagrams.It is instructive to see orders of magnitude of different contributions in terms of two independent small parameters of the theory, ᾱ = α s N c /π and 1/N c .Obviously each triple interaction contributes ᾱ2 /N c .The final estimate depends on the magnitude of the external source.If we treat it perturbatively to be consistent with the whole approach then it should correspond to the quark-antiquark loop, from which we extract its order, which is ᾱ.As a result, the diagram with l E external lines (sources) and L loops has the order As one sees, for a given number of external sources, the dominant contribution comes from the tree diagrams, each loop introducing a small factor ᾱ4 /N 2 c .However this pure counting does not take into account the growth of the pomeron propagator at large y as exp ∆y where ∆ is the BFKL intercept, nor the enhancement related to the nuclear sources with large atomic numbers.For the AA collision amplitude with the overall rapidity difference Y this changes (21) to where n is the maximal number of the exchanged pomerons at a given rapidity.This number depends on the topology of the diagram and generally grows with l E and L.
So for one has to sum all tree diagrams and for rapidities Y such that ᾱ2 one has to sum also at least some of the loops.For onium-onium scattering (l E = 2) a convenient method to sum the leading contribution is to join two sums of tree diagrams starting from the projectile and target at mid-rapidity [5].
However all these perturbative estimates are to be taken with caution.The full pomeron Green function (sum of the so-called enhanced diagrams) may have an asymptotic behaviour at large y very different from the bare one.In fact there is every reason to believe that the former will grow at most as a power of y, not as an exponential.Then all the above estimates will have to be reconsidered.Unfortunately at present a reliable summation of all enhanced diagrams cannot be performed even for the old Gribov local supercritical pomeron model.

The pomeron Hamiltonian and operators
With the action fixed, one can easily construct a Hamiltonian formulation for the state evolution.Since the Lagrangian is of the first order in derivatives in rapidity the Hamiltonian is just the action without the derivative terms with a minus sign and integration over y dropped: where ρ 23 = {r 2 , r 3 } etc.To pass to the quantum theory one has to consider φ A,B (ρ) as operators.Their commutation relation can be easily established from the form of the free Green function From (26) we conclude where we have used the equation of motion for φ B .Comparison with (1) gives Thus in a representation in which φ B (ρ) is diagonal and the state vector is a functional Ψ{φ B (ρ)} the field φ A is essentially a functional derivative The state with a given field φ A (ρ) will be represented by a an exponential The state vector will satisfy the evolution equation where in H B , given by ( 25), one has to substitute the field φ A by functional derivatives.In this substitution, as always, the order of the operators is actually undetermined.If we put all the derivatives to the right ('normal ordering') then explicitly One can pass from this 'target' representation in which the field φ B is diagonal to the 'projectile' representation in which it is φ A which is diagonal and φ B is represented by a functional derivative In this representation the Hamiltonian will be obtained from (32) by interchanging A and B and changing signs of derivatives.The state vector will be obtained by a quasi-Fourier transformation using (30).
One can construct a formulation in which the symmetry between target and projectile is more explicit.To do this one can pass to slightly different field variables for which the BFKL Hamiltonian becomes symmetric.The form of these new variables is clearly seen from the commutation relation ( 28) For them the equal rapidity commutation relation takes the form One can also assume that the scalar product of state vectors is chosen to make φ and φ † Hermitian conjugate to each other.In the representation in which say φ † is diagonal with complex eigenvalues α we take (up to a normalization factor) Then indeed Thus the two quantized fields φ(y, ρ) and φ † (y, ρ) acquire the standard meaning of annihilation and creation operators for a pomeron at rapidity y and space points ρ = {r 1 , r 2 }.
In terms of these new field variables the free action takes the form where a new Hamiltonian for the pomeron is It has obviously the same eigenvalues but is Hermitian (and real).Using its hermiticity we can revert the order of operators in (37) and write S 0 in the 'normal order' form This form explicitly shows the symmetry between target and projectile, which is quite similar to the usual time reversal: one has to change φ ↔ φ † , y → −y and revert the order of all operators.
In terms of new field operators the external part of the action aquires the form where The interaction part becomes rather complicated, involving several operators T or their inverse: With the physical meaning of operators φ and φ † well established and indeed standard, the analysis of evolution becomes trivial.Let us follow it for free pomerons.Then their number is conserved and actually the only connected diagram corresponds to a single pomeron.Such a state is to be constructed as a superposition of single pomerons at different positions ρ: where Ψ 0 is the vacuum state which obeys and is normalized to unity.(we assume the Shroedinger-like picture with operators ψ and ψ † at fixed rapidity).At the initial rapidity y = 0 the pomeron wave function is determined by the external current: where we recall that J is the spatial part of J.This state evolves to the final rapidity Y at which we are interested in the amplitude A f i to pass to a specific final state determined by the current at y = Y : One has where we used (35) and (44).
The law which governs evolution of the wave function f (y, ρ) follows from the general Schroedinger equation (31) and the form of the Hamiltonian H.The free part of the latter in terms of new operators has the standard form so that from (43) one immediately finds the equation with a formal solution Here ḡ is the Green function for the operator ∂/∂y + H which can be written as an operator in the coordinate space Using this we obtain for the amplitude Returning to the initial external sources and the Green function we reproduce the standard result Indeed we have Putting this into (52) gives (53).
4 The Schwinger-Dyson equations for the pomeron Green where both operators L act on pomeron propagators (the minus sign in front of the second term is due to the propagator being in fact equal to −g).Applying to this equation operators L from the left and from the right we find where we define G = LGL, g = LgL. (57) Eq. ( 57) can be rewritten in an obvious operatorial form as which is the standard form for the Dyson equation, except for the sign.
Note that the self-mass Σ entering this equation is a conformal invariant function, due to conformal invariance of both the pomeron propagator and 3-pomeron vertex.As a result the full Green function is also conformal invariant.
To pass to pomeron 'energies' ω we have to understand how they are related in the 3-pomeron vertex.We standardly present with the inverse transform In the lowest non-trivial order, suppressing the integration over coordinates and coupling at the vertexes, we have where g 1 and g 2 are the propagators of the two intermediate pomerons.For G(ω) we find G(ω) = dydy ′ dy ′′ e −ω(y+y ′ +y ′′ ) g(y)g 1 (y ′′ )g 2 (y ′′ )g(y ′ ) = g(ω) dye −ωy g 1 (y)g 2 (y)g(ω).( 62) The integral over y can be written in the form We can always choose ω on the line Then integration over y will give 2πδ(Im (ω − ω 1 − ω 2 )).
Subsequent integration over, say, ω 2 will finally give which result should be analytically continued to arbitrary ω.This means that in terms of energies the 3-pomeron vertex formally contains that is, energies are conserved at the vertex.
Using this result, again in the lowest order, we find, explicitly showing the coordinates: (1, 2) ≡ {r 1 , r 2 } To pass to the full self-mass we have to substitute the full pomeron Green functions G for the propagators g and change one of the 3-Pomeron vertices into the full one Γ.In this way we obtain In the lowest order the vertex is given by the spatial part of γ, Eq. ( 14):

The vertex equation
As in any quantum field theory with a triple interaction the full vertex is determined by an infinite sequence of skeleton diagrams, containing the vertex itself and full Green functions.
In the lowest order one has the so-called '3-gamma' equation.Its explicit form is rather complicated due to large number of variables.From the corresponding Feynman diagram we find in terms of gluon coordinates In the shorthand notation ρ = {r 1 , r 2 } and correspondingly 1 , r (2) 1 r (2) 2 ) the equation for the vertex part can be rewritten in a more compact form 5 Conformal (Moebius) invariance

Conformal basis
The basic building blocks of the pomeron perturbation theory: the propagator g, bare 3pomeron vertex γ and integration volume dτ , -are conformal invariant.As a result also the full Green function G, self-mass Σ and vertex Γ are conformal invariant functions of their arguments.It seems profitable to use this property to simplify the Schwinger-Dyson equations.To do this we have to study a general form for the conformal invariant functions for the transitions pomeron→ pomeron ('two-point functions') and pomeron→ 2 pomerons ('three-point functions').This can be achieved by expanding these functions in the conformal basis formed by functions (in complex notation) [16] where µ = {n, ν, r 0 } = {h, r 0 } with n integer, ν real and two-dimensional transverse r 0 enumerate the basis.
Functions E µ (ρ) are the proper functions of the operator where They form a complete system: where we use a notation and satisfy the orthogonality relation The coefficients b nν are given by the formula Functions E µ are not linearly independent.In fact Presenting a function f (ρ) as and using completeness (74) one gets where This gives the standard expansion in the whole overcomplete basis.
However it seems possible to limit oneself to an independent part of this basis.One possibility is to take a restricted basis with ν > 0(ν < 0).We denote this restriction by µ > 0(µ < 0).In fact we may split the integration over ν in (80) into two parts: and, say, in the second term express the eigenfunctions E µ with µ < 0 via those with µ > 0 using (78) to obtain for the second term Changing summation and integration variables n → −n and ν → −ν we get for this term Interchanging also r 0 and r ′ 0 integrations we finally find for it Summing this with the first term in (82) we get the desired expansion in states with µ > 0 Integrating this with E * µ>0 (ρ) and using (76) we find Putting this into (87) we find which means that one also has a completeness relation for half of the basis with ν > 0: Obviously the same property is valid for the second half of the basis with ν < 0.

Two-point functions
We have to deal with a conformal invariant function A(1|1 ′ ), where, as introduced in the previous sections, the arguments refer to pairs of the pomeron coordinates.Using half of the conformal basis we present A as where we use Our aim is to see which properties A µ,µ ′ should have for the function A(1, 1 ′ ) to be conformal invariant.Since E n,ν,r 0 (r 1 , r 2 ) = E n,ν,r 0 +a (r 1 + a, r 2 + a) the translational invariance requires A µµ ′ to depend only on the difference r 0 − r ′ 0 ≡ r 00 ′ .Under inversion the expression to be summed over µ > 0 and µ ′ < 0 changes as follows Here and in the following (a.f.) means 'antiholomorhic factor', that is the complex conjugate of the preceding factor.Invariance under inversion requires that However the left-hand side only depends on the product r 0 r ′ 0 , so in the right-hand side we are obliged to have either n = n ′ and ν = ν ′ or A µµ ′ (r 00 ′ ) ∝ δ(r 00 ′ ).Since ν and ν ′ have opposite signs, the first alternative cannot be realized.With A µµ ′ (r 00 ′ ) ∝ δ(r 00 ′ ).we find that n + n ′ = 0 and ν + ν ′ = 0 so that where μ = µ(n → −n, ν → −ν) and we defined A µ ≡ A nν with factor a nν separated for convenience.As a result the double sum in (91) transforms into a single one where μ = µ(n → −n, ν → −ν).A similar form with summation over µ < 0 can be obtained in the same manner.Representation (96) or a similar one with a sum over µ < 0 are valid for any conformal invariant two-point function.Note that taking an average of the sums over µ > 0 and µ < 0 one obtains a similar representation in terms of the whole overcomplete basis, which is standardly used for the BFKL Green function g(z, z ′ ) Now suppose we have conformal invariant functions B(1|1 ′ ) and C(1, 1 ′ ) and form a conformal invariant integral Each of the three functions A, B and C has the representation (96) with conformal coefficients A µ , B µ and C µ .Doing the integration with the help of orthonormalization properties of the basis functions E µ with µ > 0 we find which means that in the conformal representation (96) Applying this result to the Dyson equation ( 58) in the ω representation we immediately find Gωµ = gωµ − gωµ Σ ωµ Gωµ .
So both the energy ω and conformal quantum numbers µ of the pomeron are conserved in the interaction and its full Green function in the conformal basis is trivially expressed via its self-mass: In Eq.( 101) the free pomeron Green function g in the conformal basis is given by

3-point functions
For a 3-point function the expansion similar to (96) reads where the intermediate c.m. coordinates are R 1 , R 2 and R 3 .Conformal invariance allows to determine the dependence on them of Γ µ 1 |µ 2 µ 3 .Translational invariance requires Γ µ 1 |µ 2 µ 3 to depend only on differences R ik and from the scale invariance such a dependence should be a power one.So we seek After inversion we find an extra factor in the sum (102) times two similar factors which are obtained from (104) by cyclic permutations of 123 and conjugation.Invariance requires that each power is zero and that the sum n 1 + n 2 + n 3 be even.We get six equations to determine the α's.Their solution gives and similar expressions for the powers in the antiholomorhic factor α's with n i → −n i .

Σ and Γ in the conformal basis
To illustrate the simplifications introduced by transition to the conformal basis in this section we study the pomeron self-mass and triple interaction vertex in this basis In a shorthand notation for the gluon coordinates 1 = (r 1 , r1 ) the pomeron self mass is given by The actual number of integrations is in fact smaller due to δ-functions in the conformal vertex Γ (0) defined by Eq. (68).We expand both the Green function and the vertexes in the conformal basis with µ > 0 (see 96) and (103).In the following we also suppress the ω-dependence to economize on subindexes.We have Integrations over the gluon coordinates are done with the help of (76) and give the product of δ-functions: where and the suppressed dependence on ω is obvious from its conservation at the vertexes.Thus we have found for Σ(1, 2|1 ′ , 2 ′ ) an expansion in the conformal basis (91).From its conformal invariance it follows that where Σ µ is the desired pomeron self mass in the conformal representation.It can be found from (115) after summation over µ ′ 1 This gives The sum over µ i , i = 1, 2, 3 includes integrations over three c.m coordinates R i on which only the vertexes depend.We get an integral depending on the four conformal weights Here R 0 is arbitrary since the integral is independent of it.The powers are given by and α 02 and α 32 are obtained from α 01 and α 31 by interchange 1↔2.The integral (118) is convergent both in the infrared and ultraviolet.However its calculation does not look simple.
Once this integral is known, the self-mass in the conformal basis is given by a sum over three conformal weights: Here we made explicit the ω dependence introducing it in the arguments; Ω is the Korchemski vertex (107), Γh 1 , h2 |h 3 is defined by Eq. ( 104) and h given by (74) without integration over r 0 .
In the same manner one can obtain expressions for the vertex part Γ.We shall limit ourselves with the '3-gamma' approximation, Eq. ( 79).As before we expand the vertex parts and Green functions in the conformal basis.Suppressing again the ω-dependence we have Integrations over the double primed coordinates will give 6 δ-functions in µ.So finally we are left with 6 summations over µ i , i=1,...6.The result can be presented in the form (with where is the desired vertex part in the conformal basis.Its dependence on the c.m. coordinates R i , i = 1, 2, 3, is determined according to Eq. ( 104).The bare vertex Γ (0) µ 2 ,µ 3 |µ 1 is here given by the formula analogous to (107).
In (121) summations over µ i , i = 4, 5, 6, include integrations over c.m. coordinates R i , i = 4, 5, 6.In the integrand the R dependence comes only from the vertex parts and in its turn is defined by Eq. ( 104).Thus the expression

Conclusions
We have presented a formalism which allows to study interaction of pomerons in the QCD with N c → ∞ using the standard methods of quantum field theory.In particular we constructed the Schwinger-Dyson equations which sum the so-called enhanced graphs and carry information of the 'physical' pomeron as compared to the 'bare' one.
Conformal symmetry of the theory leads to certain simplifications.As a result we obtain a picture very similar to old Gribov local supercritical pomeron.The difference is reduced to an (infinite) number of pomerons with varying n = 0, ±2, ±4, ... and more complicated form of the 'energy' ω nν as a function of ν, which plays the role of the pomeron momentum in the old theory, and of the bare triple pomeron vertex, which now depends both on n and ν.If however one selects the supercritical pomeron with n = 0 and small values of ν the formal similarity becomes almost complete, since the energy becomes a quadratic function of ν and the bare vertex then reduces to a well-know constant [17].Unfortunately with this similarity also the problems of the old theory, mentioned in the Introduction, return together with the question of the internal consistency of the model.At present we do not know the answer to this question and leave it for future studies.

Acknowledgments
The author has benefited from fruitful discussions with J.Bartels.He is indebted for hospitality to Hamburg University where this paper was finished.This work has been supported by the NATO grant PST.CLG.980287.

function 4 . 1
The pomeron self-massThe pomeron self-mass operator starts and finishes with the 3-pomeron vertex, which contains operator L acting on the incoming and outgoing pomeron propagator.As a result, the Dyson equation for the full Pomeron Green function G(z, z ′ ) takes the form

9 2 c − 1 ) 3 /2 f a 1 a 2 c f ca 2 d f da 3 e f ea 3 a 1 = N 2 c (N 2 c − 1 ) 3 /2 δ a 1 d δ a 1
Appendix 1. Color factors Let the color wave function of a pair of gluons be |ab , where a, b = 1, ...N 2 c − 1.Then the vacuum color state is obviously onto the vacuum color state.The color structure of the vertex for the transition 2→4 reggeized gluons, with initial and final color variables a 1 , b 1 → a 2 , b 2 , a 3 , b 3 is given by the expressionV c = f a 1 a 2 c f cb 2 d f da 3 e f eb 3 b 1 (126)We want the projection of this color vertex onto the 3 vacuum color states formed by the gluons with colors a 1 b 1 , a 2 b 2 and a 3 b 3 .Applying three corresponding projectors P 1 , P 2 and P 3 we obtain P 2 P 3 V c P 1 = 1 (N however that the quarks quark loop which represents the external source has its color factor δ ab = N 2 c − 1P ≃ N c P , so that each external source contributes a factor N c .