KLWMIJ Reggeon field theory beyond the large Nc limit

We extend the analysis of KLWMIJ evolution in terms of QCD Reggeon fields beyond leading order in the 1/Nc expansion. We show that there is only one type of corrections to the leading order Hamiltonian discussed in [1]. These are terms linear in original Reggeons and quadratic in conjugate Reggeon operators. All of these have the interpretation as vertices of the “‘merging”’ type 2 → 1, where two Reggeons merge into one. Importantly, the triple Pomeron merging vertex does not emerge from the KLWMIJ Hamiltonian. We show that, although in the range of applicability of the KLWMIJ Hamiltonian these merging terms are subleading in Nc, in the dense-dense regime they all become of the same (leading) order in Nc. In this regime vertices involving higher Reggeons are enhanced by inverse powers of the coupling constant.


Introduction and Conclusion
In recent papers [1,2] we have studied in some detail the relation between the CGC [3] approach to the high energy evolution of hadronic observables in the guise of JIMWLK [3] and KLWMIJ [4] equations, and the Reggeon Field Theory [5,6].In [1] we have shown that the KLWMIJ Hamiltonian at leading order in the 1/N c expansion can be written as a theory of interacting Reggeons.This theory very simply encodes the Bartels' triple Pomeon vertex, but in addition to Pomerons it contains other Reggeon states, at least some of which play an important role at high energy.In [2] we have considered a more general Hamiltonian [7] which includes the effects of large Pomeron loops, and have shown that it defines a self dual [8] Reggeon Field Theory.The analysis of [1,2] was performed in the large N c limit, although as discussed in [2] the large N c counting at high energy is peculiar, as the same Reggeon field is of different order in 1/N c in different energy regimes.
In this paper we continue our analysis of the Reggeon Field Theory representation of KLWMIJ evolution, H RF T KLW M IJ , this time including all terms subleading in N c .Our aim is to understand which type of Reggeon interactions are contained in the KLWMIJ Hamiltonian beyond leading order.This is important in order to understand whether the theory can be eventually cast into a form of reaction-diffusion process [9,10].Another reason for the study is to further clarify the evolution of higher order operators which are relevant for correlation-type observables, such as the ridge [11][12][13].Importance of such corrections subeading in 1/N c has been emphasized in [14].
Ou results can be summarized in the following way.We show that no 1/N 2 c corrections arise to Reggeon propagator terms and to any of the 1 → 2 splitting vertices involving Reggeon operators.The nonvanishing corrections to the RFT Hamiltonian are all of order 1/N 2 c , and all of them are of the type of 2 → 1 merging vertices.There is however no 2 → 1 triple Pomeron vertex which we normally associate with the JIMWLK Hamiltonian.
This large N c counting pertains to the dilute-dense regime which is the appropriate regime for the application of H KLW M IJ .It is nevertheless of some interest to rewrite the Hamiltonian in terms of the operators relevant to the dense-dense regime.This involves changing the basis from the Reggeon conjugates to dual Reggeon operators, as suggested in [2].When this transformation is performed in H RF T KLW M IJ we find that all vertices (splitting and merging) become of the same, leading order at large N c .We also find that the vertices involving higher Reggeons are enhanced by inverse powers of the 't Hooft coupling constant.
The plan of this paper is the following.In Section 2 we recall the KLWMIJ Hamiltonian, and derive H RF T KLW M IJ including all orders in 1/N 2 c , in terms of multipole operators and their conjugates.In Section 3 we discuss the definition of n-point Reggeon operators in terms of n-point multipoles, generalizing the construction of [1].In Section 4 we discuss the N c counting in the dense-dense limit.Appendices contain technical details of our derivations.

KLWMIJ Reggeon Field Theory beyond large N c
The KLWMIJ evolution equation is a functional evolution equation for the weight functional W [δ/δρ].This weight functional W represents the probability density to find a given configuration of charge density in the projectile.The expectation value of any observable O is given by Here, ρ is the color charge density of the projectile.We have allowed for the dependence of operator O on the color field of the target α, which also has to be averaged over.The averaging over the target field α is determined solely by the properties of the target, and is not directly relevant to our discussion, since we ascribe all evolution to the projectile wave function.
An example of an interesting observable is the S-matrix of a projectile dipole, given by Here R x is the unitary matrix in the fundamental representation, that represents the scattering amplitude of a quark at transverse point x, with t a is the SU (N c ) generator in the fundamental representation.A general dilute projectile is not necessarily a single dipole but is rather composed of a small number of color neutral objects.The scattering matrix of such a projectile depends on dipole, quadrupole operators and so on.These are SU L (N c ) × SU R (N c ) invariant singlets.Other observables besides the scattering matrix, such as gluon production cross section, gluon correlations, . . ., depend on the same color singlets 1 .The weight functional W is thus some functional of these color neutral objects: where the dipoles and quadrupoles are defined as 1 This statement is not strictly precise.In general the gluon production cross section depends on objects which are symmetric only under the diagonal subgroup of the SUL(Nc) × SUR(Nc) symmetry of HKLW M IJ and are not necessarily completely SUL(Nc) × SUR(Nc) symmetric, see for example [15].Nevertheless, in many cases the evolution of SUL(Nc) × SUR(Nc) symmetric operators is sufficient to also describe the energy dependence of these more general observables [2].
The evolution of any observable to higher energies in the KLWMIJ approximation is given by where the explicit form of the KLWMIJ Hamiltonian reads x,y,z Here K xyz is the Weizsäcker-Williams kernel As long as we are interested in the action of the KLWMIJ Hamiltonian on gauge invariant quantities (invariant under the SU L (N c ) × SU R (N c ) transformation), the kernel K xyz can be replaced by the dipole kernel (2.9) J a L (x) and J a R (x) are the left and right rotation generators (2.11) The action of the right and left rotation generators on the unitary matrix R(x) are: (2.12) Our goal now is to represent the action of H KLW M IJ on a functional which depends on general color singlet operators, similarly to what has been done in [16] for W which depends on dipoles only.We start off by assuming for simplicity, that W is a function of d and Q only.
The Hamiltonian is quadratic in the SU (N ) rotation generators, and one should simply act with the two rotations sequentially.For simplicity we first consider only a dependence on d and Q.The action of the first rotation operator can be represented as The integration over the coordinates 1, 2 and 1, 2, 3, 4 is implicitly assumed in Eq. (2.14).
Acting with the second generator we obtain, for example where again all coordinates appearing twice are integrated over.
The action of the left and right rotation operators on dipoles and quadrupoles is straightforwardly calculated with the help of the Eqs.(2.12) and (2.13).To simplify the resulting expressions we use the completeness relation and the identity After some algebra we find that the action of the KLWMIJ Hamiltonian on W [d, Q] can be represented as the sum of the following terms : (2.18) The leading N c terms reproduce and generalize the Mueller's dipole model [17] and have been derived in [1,16]: where ᾱs ≡ α s N c /π and (2.21) The terms subleading in N c account for color correlations and are given by where we defined the higher multiplets as ) and have introduced the notation Q T 1234 ≡ Q 4321 .Two important properties stand out in the equations above.There are no 1/N 2 c corrections to the term linear in functional derivatives (conjugate momenta) in Eq. (2.18).The only correction term is quadratic in functional derivatives, and it itself also has no further 1/N c corrections.We now show that these properties remain true even if one includes higher Reggeons in the evolution.
The generalization to higher Reggeons is straightforward, albeit lengthy.We give details of the derivation in Appendix 1, and here only present the final result.Consider a general probability density distribution, (2.28) We show in Appendix 1 that the action of H KLW M IJ on such a functional can be written as where the two terms are linear and quadratic in functional derivatives respectively : (2.31) where y 0 ≡ y 2n .Eq.(2.30) generalizes the evolution of a single dipole (or single quadrupole) to arbitrary multipoles.In Appendix 2, we present the explicit expressions for the evolution of 6-point function (X) and 8-point function (Σ).

Construction of Reggeons
In principle, Eqs.(2.30) and (2.31) give the complete representation of It is however useful to define Reggeon fields as irreducible representations of the discrete symmetries of the problem.This was done for the dipole and quadrupole in [1].In this section we extend this discussion to include higher Reggeons.
As explained in detail in [1] and [18], in addition to the SU L (N c )×SU R (N c ) continuous symmetry group, H KLW M IJ also posesses the discrete signature Z 2 symmetry, R → R † , and the discrete charge conjugation symmetry, R → R * .In [1] the Reggeon fields were constructed as singlets under SU L (N c ) × SU R (N c ) and either even or odd under signature and charge conjugation symmetries.In addition, in [1] we have constructed the Q ++ -Reggeon2 using a simple subtraction such that the evolution equation for the Q ++ -Reggeon does not contain a single Pomeron source term.We generalize this subtraction procedure here.
The decomposition for the two point function is straightforward [1].The Pomeron and the Odderon fields are defined as The Pomeron is even and the Odderon is odd under both signature and charge conjugation.
The evolution equations for P and O are Now let us consider the quadrupole.Decomposing it into eigenstates of discrete symmetries, we define Q ++ , Q +− , Q −− and Q −+ Reggeons : Three independent Q ++ -Reggeon and three independent Q +− -Reggeon are The ++-Reggeon has quantum numbers of the Pomeron.It is signature and charge conjugation even.+−-Reggeon is signature even and charge conjugation odd, −+ signature odd, charge conjugation even and −− has the same quantum numbers as the Odderon -odd under both signature and charge conjugation.The subtraction of the Pomeron terms in Q ++ achieves simultaneously two goals.First, when expanded in powers of δ/δρ, the field Q ++ starts with the term (δ/δρ) 4 as opposed to Q whose expansion starts with (δ/δρ) 2 .Second, the evolution equation for Q ++ does not contain a term linear in P , which is in principle allowed due to the identical quantum numbers of P and Q ++ [1].This evolution equation for completeness is presented in Appendix 2.
Clearly, the +− and −+ Reggeons do not require any subtractions, since they do not mix with either the Pomeron or the Odderon.The −− Reggeon on the other hand does mix with the Odderon, and its evolution does contain a source due to the Odderon.However, it turns out to be impossible to find a local subtraction of the type of Eq.(3.5) which eliminates this source term.In the following we will therefore only consider subtractions in the Pomeron channel.
Moving on to the six point function, we define the X ++ Reggeon in the analogous way Analogously we define and finally Note that we have included subtractions in the +− and −+ channels.These eliminate sources proportional to Q +− in the evolution of X +− and similarly for X −+ .We do not bother with subtractions in the −− channel, since the Odderon source terms cannot be eliminated already in the evolution of Q −− .The subtraction structure can be generalized to an arbitrary n-point Reggeon by inspection.Let us first define the unsubtracted signature and charge conjugation even multipole combinations: ) and, for an arbitrary 2n point function, The generalisation of Eqs.(3.5) and (3.10) for arbitrary n can then be written as (3.20) Analogously, defining we have Expressing Reggeons in terms of multipoles requires some extra algebra.For n = 1, 2, 3 this is straightforward, and we obtain It is obvious that these expressions lend themselves to a generalization for arbitrary n : We are not going to perform the substitution Eq.(4.1) explicitly.We only note that it does not change the nature of the terms in the Hamiltonian.The one important result of this substitution is that it eliminates the terms of the form which otherwise appear in Eq.(2.30) when one of the multipole operators is taken to be unity.It thus leads to partial diagonalization of the equations, in the sense that the evolution of a level 2n Reggeon does not contain a term linear in level 2(n − k) Reggeons.
The structure of the Hamiltonian thus can be described as follows.The leading N c terms are of two types.First, there are homogeneous terms of the type For ++, +− and −+ Reggeons these are diagonal, i.e. they mix only Reggeons on the same level n.
The second type of terms are 1 → 2 splitting vertices, contained in Eq.(2.30).These are akin to the 1 → 2 Pomeron splitting vertices and generalize it to higher Reggeons.The most interesting such vertex corresponds to splitting of the Q ++ Reggeon into two Pomerons, and contributes to the effective four Pomeron vertex as discussed in [2].
Finally the third type of terms is of order 1/N 2 c , and is of a different nature.These are "merging" vertices of the 2 → 1 type.Note that there is no vertex corresponding to transition from two Pomeron to one Pomeron.This is straightforward to see examining Eqs.(2.22), (2.23), (2.24).In Eq.( 2.22) all the single Pomeron terms cancel in the factor that multiply the functional derivatives, while Eqs.(2.23) and (2.24) do not contain a term with two functional derivatives with respect to the Pomeron.This is natural, since we expect to find this vertex in the JIMWLK Hamiltonian which describes the dense projectile regime, and not in the KLWMIJ Hamiltonian.On the other hand Eqs.(2.22), (2.23), (2.24) do give rise, for example to a vertex of the type P P → Q ++ .
The order in 1/N c assigned to the vertices is based on large N c counting for the dilute projectile, dense target regime.In this regime all the Reggeons are close to saturation, and thus are of order one.The Reggeon conjugate operators δ/δO 2n are also assigned O(1) at large N c .However, as discussed in [2] the large N c limit is tricky, and the order in N c of a given operator depends on whether the target or the projectile is saturated, or both.For example, for a dense projectile a dipole scattering on it scatters with probability of order one.This means that the following dual Pomeron operator is of order one in the large N c limit : where S F (x) = e ig 2 t a α a P (x) (4.5) and α P is the color field created by the color charges of the projectile.The dual Pomeron operator is a nonlinear function of the projectile color sources.However, for weak fields it is related to the Pomeron conjugate operator by a simple linear relation In the dense projectile limit P remains finite even at large N c .Thus, if this relation can be taken as a guide of the correct N c behaviour, in the limit of dense projectile, where P ∼ O(1), one has The dual Reggeons with the nonvacuum quantum numbers are not important in the projectile saturation regime, since the limiting saturated value of Q Substituting this into the expression for the Hamiltonian, we see that in the regime where both the projectile and the target are dense, all terms in the Hamiltonian are of the order N 2 c .Another interesting feature of this regime is that the vertices involving higher Reggeons are in fact enhanced by inverse powers of the ᾱs , when 't Hooft coupling is small.The vertices scale as3 It is possible that Eq.(4.8) is not correct parametrically in the dense limit, since strictly speaking it was deduced in the dilute regime.However if it holds, this suggests an interesting albeit complicated picture.At weak 't Hooft coupling, the higher Reggeons which are less important in the dilute-dilute and dense-dilute regime (since their value is small at small ᾱs ) take over the quantum evolution in the dense-dense regime.To make sense of this regime one would then have to somehow resum the contributions of higher Reggeons.In other words one would have to deal with an essentially nonlinear regime of the Reggeon field theory, a level higher than nonlinear QCD which leads to the Reggeon field theory in the first place.

Appendix 1: The derivation of the KLWMIJ Reggeon theory
In this Appendix, we derive the action of H KLW M IJ on a generic functional of color singlet operators of the type The simplest such operator is the dipole.The dipole evolution operator H d given in Eq.(2.19), picks up a dipole from the weight functional W and evolves that dipole via the KLWMIJ Hamiltonian i.e.H d can equivalently be written as (5.2) In the KLWMIJ evolution of a single dipole d 12 the subleading piece in 1/N c cancels between the real and virtual terms in the Hamiltonian and the exact KLWMIJ evolution of a dipole is all leading N c .The same is true for the evolution of a quadrupole.The color correlators in Eq.(2.18) that are subleading in N c arise from the evolution of a product of two color neutral objects (two dipoles, one dipole and one quadruple and two quadrupoles).In this case, one of the two color rotation operators acts on one object, and the other acts on the second object generating the order 1/N 2 c term.The terms of order 1/N 3 c , which a priori could be present, cancel in the final result between the real and virtual contributions of the KLWMIJ Hamiltonian.Hence, the expression given in Eq. (2.18) is the result of exact color algebra.In order to generalise this expression one should show that a similar cancellation happens for the generic color neutral object O 2n .

KLWMIJ Evolution of a generic single trace operator
First, we derive the evolution equation of a single trace operator with arbitrary number of entries and show that the large N c approximation for the KLWMIJ evolution coincides with the exact evolution i.e. all the subleading N c terms cancel in the action of H KLW M IJ on a single trace operator.We write the generic single trace operator as In order to find the action of H KLW M IJ , we need the calculate the action of right and left rotation generators on O 2n .For J L we have Here, we introduced a short hand notation for the δ-functions as (5.5) Moreover, when writing Eq.( 5.4) we have adopted the following conventions : Similarly, one can write the action of J R on O 2n : Now using Eqs.(5.4) and (5.8) we obtain (5.10) (5.11) These expressions can be simplified by using the color algebra relations ) Then Eqs.(5.12) and (5.13) can be rewritten as (5.14) Note that the N c suppressed terms can be further simplified by using the identity n j=1 Now, combining all the pieces Eqs.(5.16), (5.17) and (5.19), one obtains Eq.(5.20) gives the complete KLWMIJ evolution of a generic single trace operator and it is equivalent to Eq.(2.30).All the terms subleading in N c cancel in this equation.

KLWMIJ evolution of a generic double trace operator
We now derive the evolution equation of a double trace operator.Consider two generic operators O 2n and O 2 m : In the evolution of the product of these two objects, the two SU (N c ) rotation generators appearing in H KLW M IJ can either act on the same object keeping the other one untouched, or one rotation generator can act on one object and the second one on the other object.Therefore, generically By using Eq.(2.12), the first term in Eq.( 5.24) can be written as Further, using we have (5.27)By using Eq.(2.13) the action of the virtual term can be written as Performing the color algebra we get (5.29) The real term in H KLW M IJ contributes to the "mixed" term: The relevant piece of color algebra here is (5.31) Using Eq.( 5.31), the real "mixed" term, Eq.(5.30), can be written as can be written as (5.34) Combining all the terms we see that all the order 1/N 3 c terms cancel.Hence, The delta functions can be now realized performing integrations over x and y.Combining Eqs.(5.20) and (5.35), we arrive at Eq.(2.29).
The evolution equation of a quadrupole is [19,20] The evolution equation of a 6-point function can be calculated by taking the action of the KLWMIJ Hamiltonian.The resulting expression is In the evolution of the 6-point function the terms with six points are either the hexapole or the breaking of the hexapole into two lower color singlet states which in this case it is a dipole and a quadrupole.The second structure which introduces two more points, breaks into two possible lower color singlet states; either a dipole and a hexapole or two quadrupoles.This is the same trend that was seen in the evolution of the 4-point function.
By using Eq.(3.10), one can write the evolution equation of the where is introduced as a shorthand notation.
We have also calculated the evolution of the octupole The evolution equation for 8-point ++-Reggeon is a long expression.
where χ (1) is the term that arises from the direct Σ ++ subtraction and comes with the same kernel as the first term in Eq.(6.7).It is given as M 12z +M 23z +M 34z +M 45z +M 56z +M 67z +M 78z +M 18z Σ ++ 12345678 .(6.9) χ (2) is the term that arises from dipole times hexapole terms in the evolution for Σ whose explicit form reads χ (2) = L 1823z (P 2z The χ (3) part of the evolution of 8-point ++-Reggeon originates from the quadrupole square terms in the evolution of Σ.Its explicit form is Finally, χ (5) are the terms that come from the dipole times hexapole terms in the evolution of the 8-point function.Its explicit form reads