Reggeon Field Theory and Self Duality: Making Ends Meet

Motivated by the question of unitarity of Reggeon Field Theory, we use the effective field theory philosophy to find possible Reggeon Field Theory Hamiltonians $H_{RFT}$. We require that $H_{RFT}$ is self dual, reproduce all known limits (dilute-dense and dilute-dilute) and exhibits all the symmetries of the JIMWLK Hamiltonian. We find a family of Hamiltonians which satisfy all the above requirements. One of these is identical in form to the so called"diamond action"discussed in \cite{diamond,Balitsky05}. However we show by explicit calculation that the so called"diamond condition"is not satisfied beyond leading perturbative order.

Physically one expects of course that the number of gluons in the QCD wave function increases with energy, while within the JIMWLK framework the number decreases but the low gluon number states appear with negative probability. This strange behavior nevertheless produces correct energy dependence of the S-matrix but only as long as one of the colliding objects is dilute. The violation of unitarity is a precursor of the eventual breakdown of the JIMWLK evolution at high enough energy. At high energy the Pomeron loops must become important and their effect on the evolution must be significant.
This issue of the unitarity violation in the JIMWLK limit motivates us to reconsider the problem of including Pomeron loops. More precisely we take up a limited goal to try and extend H JIM W LK in a way that it becomes consistent with a very important property of RFT -the self duality. It has been established in [27] that the Hamiltonian that generates the high energy evolution must be invariant under the dense-dilute duality transformation. Physically the self duality has a very simple meaning. It expresses the fact that a scattering amplitude for a scattering of any two hadrons does not depend on which one of them is right moving and which one is left moving, i.e. which one of them we call the target and which one the projectile. As discussed many times in the literature, the JIMWLK evolution explicitly violates the self duality property which one expects to hold in RFT, since within the domain of validity of JIMWLK the target and the projectile are very different and thus are explicitly treated differently in H JIM W LK .
Although self duality alone may not be sufficient to restore unitarity of the evolution, in a zero dimensional toy model addressed in [11] it was shown that the unitary Hamiltonian is indeed seld-dual. Motivated by this, in the present paper we explore possible generalization of the JIMWLK Hamiltonian which restores self duality. Our approach here does not rely on direct derivation from QCD, but instead is akin to typical effective field theory (EFT) attitude: identify relevant degrees of freedom and impose appropriate symmetries. We also require that in the dense-dilute limit the Hamiltonian reproduces both H JIM W LK and H KLW M IJ . We find a family of such Hamiltonians which all reduce to H JIM W LK in the dense-dilute limit and are self dual. We note that one of these Hamiltonians is similar in structure to the so called "diamond action" introduced some years ago in [28] and discussed in [29]. However a more detailed analysis presented below shows that our construction does not support the condition imposed on the product of Wilson loops in [28], which was crucial in the approach of [28] to maintain self duality. Thus our current suggestion is not equivalent to the diamond action of [28]. Additionally we note that our approach relies on the development of RFT formalizm in [10], and thus provides directly an algorithm for calculation of scattering amplitudes once the Hamiltonian H RF T is specified.
We thus find a family of self-dual RFT Hamiltonians that reproduces all the known limits. Unfortunately it turns out to be technically involved to check whether the evolution generated by these Hamiltonians is unitary and we are currently unable to answer this question. We are nevertheless encouraged by many similarities with the zero dimensional toy model where the very analogous construction provided a solution to the unitarity problem. The quantitative analysis of this question is left for further research.
The plan of this paper is as follows. In Section 2 we recap the formulation of RFT, its algebra of operators and Hilbert space structure discussed in [10]. In Section 3 we present the construction of H RF T imposing the discrete symmetries of H JIM W LK in addition to self duality. In Section 4 we show that in the dense-dilute limit our H RF T reproduces the JIMWLK and KLWMIJ evolutions. In Section 5 we discuss the continuous symmetries of H RF T . This discussion is perturbative, and we conclude that the continuous symmetry group of our H RF T is somewhat surprisingly SU (N ) × SU (N ) × SU (N ) * . In Section 6 we consider the relation with the diamond action [28], and show that the so called "diamond condition" on the Wilson lines is violated at second order in g. We conclude with discussion in Section 7.
2 The Reggeon Field Theory: scattering amplitudes and field algebra.
In this section we briefly recap the general formulation of the Hamiltonian Reggeon Field Theory given in [10].
Consider an S matrix element S f i for scattering from the initial QCD state |Ψ i = |x 1 , a 1 ; ...; x N , a N T |y 1 , c 1 ; ...; y M , c M P to the final state |Ψ f = |x 1 , b 1 ; ...; x N , b N T |y 1 , d 1 ; ...; y M , d M P . Here the target state (subscript T ) contains N gluons, and the projectile state (subscript P ) contains M gluons. The states are labeled by the transverse coordinates and color indexes of the gluons. At high energy in the eikonal approximation this is given by where the left and right RFT Fock vacuum states satisfy L|Ū ab = δ ab L|; U ab |R = δ ab |R (2. 2) The projectile and target adjoint Wilson line operators are defined in terms of the projectile color charge denstity ρ a (x) asŪ (x) = e T a δ δρ a (x) ; U (x) = e igT a y φ(x−y)ρ a (y) The scale L is arbitrary and does not enter calculations of any physical quantities. The SU (N ) generators in the adjoint representation are defined in terms of the SU (N ) structure constants as These equations imply non-trivial commutation relations, between U andŪ , which constitute the algebra of the RFT in analogy with Heisenberg algebra of fields in the ordinary QFT. In order to calculate the scattering amplitude eq.(2.1) one uses the algebra of U andŪ to commute the factors of U to the right ofŪ , at which point they disappear by virtue of Eq. (2.2). * We have abused the notation here somewhat. The symmetry group is not in fact a direct product of three factors of SU (N ). The more appropriate way to characterize it is to say that the generators contain three linearly independent sets of generators of SU (N ). The commutation relations between some of these generators are quite complicated to calculate and thus the full group structure is not known. We will expand on this in the body of the paper. This algebra encodes the diagrammatic calculation of scattering amplitudes in the operator language. Consider for example the scattering of one gluon on one gluon. The scattering amplitude up to second order in α s is given by This corresponds to the sum of one and two gluon exchange diagrams in Fig. 1-a. In fact as was shown in [10], higher order terms organize themselves into all possible diagrams where the relative order of the vertices on the target gluon line is permuted in all possible ways. These are the relevant diagrams for eikonal scattering in the Lorentz gauge. The O(α 3 s ) contributions correspond to the three gluon exchange diagrams ( Fig.1-b).
With the algebra encoded in Eq. (2.3) and the rule for calculating scattering amplitudes Eq. (2.1), the framework of the QCD RFT is defined. To complete the RFT framework one needs to specify the Hamiltonian H RF T that generates the evolution of the scattering amplitude in energy. We will spend some time discussing this Hamiltonian below. But before setting along this route let us recap unitarity constraints on any RFT state as derived in [10]. These constraints must be preserved by energy evolution of the scattering amplitudes. This implies a non-trivial constraint on H RFT [10].
Eq.(2.1) is easily extended for scattering of a state which is a superposition of states with fixed number of gluons. For example, starting with the initial QCD projectile state ..an |x 1 , a 1 ; ...; x n , a n (2.7) the eikonal scattering can only produce a state of the form The same holds for the target The S-matrix element is given by As shown in [10] some these conditions are violated in JIMWLK evolution, which leads to negative probabilitiesF when evolving the state of a dense target.

The RFT Hamiltonian
The subject of RFT is the evolution of scattering amplitudes with energy. In general the energy evolution is generated by the action of the RFT Hamiltonian H RF T [U,Ū ]. The S-matrix element of eq.(2.1) evolved to rapidity Y is given by

JIMWLK/KLWMIJ Hamiltonians.
Exploring the functional form of H RF T is the subject of this paper. Ideally we would like to derive it directly from a QCD calculation. This has been achieved in the dense-dilute limit, where one of the scattering objects is dense and the other one is dilute. The two versions of the Hamiltonian related by the duality transformation have been derived in [20][21][22].
When the target is dense and the projectile dilute, the relevant limit is the JIMWLK Hamiltonian: Here the right and left rotation operators are defined as [30] J a L (x) = 3) The function on the right hand side as usual should be understood as a power series expansion. For a single variable t we have  One seemingly peculiar feature of these definitions is that when considered as operators on the standard Hilbert space of functions of ρ, the operators J L(R) are not Hermitian However one has to keep in mind that the operation of Hermitian conjugation of the operators in QCD Hilbert space does not correspond to naive Hermitian conjugation in the RFT space. Without going into detailed discussion here, we refer the reader to [31] where it was shown that the RFT transformation that corresponds to Hermitian conjugation in the QCD Hilbert space is Under this transformation indeed we have as is required for Hermitian operators in the QCD Hilbert space.
The evolution in the reverse situation (dilute target and dense projectile) is governed by the so called KLWMIJ Hamiltonian, where I L(R) are defined as with α a (x) defined in eq.(2.4). These satisfy (3.11) The two sets of operators satisfy two copies of SU (N ) × SU (N ) commutation relations: and (3.13) The commutation retaions between J and I are rather complicated and we will not attempt to derive them here.
The Hamiltonian of RFT must possess a property of self duality, i.e. it has to be invariant under the transformation that interchanges the projectile and the target. This is obvious from the point of view of QCD, since it is immaterial which one of the colliding objects we call the target, and which one the projectile. Thus scattering of an N gluon projectile on an M gluon target is the same as scattering of an M gluon projectile on an N gluon target. The JIMWLK (and likewise KLWMIJ) Hamiltonian is not self dual, since it is only meant to be valid in the very asymmetric regime where one of the colliding objects is dense and one is dilute. This lack of self duality means among other things, that JIMWLK cannot be used at asymptotically high energies, where the projectile becomes dense as well. It is thus clearly desirable to find a self dual extension of H JIM W LK . Some years ago a considerable effort has been dedicated to a search for a self dual extension of the Hamiltonian. One such extension in the context of large N c Pomeron theory was suggested by Braun [16]. The solutions to the Braun theory however exhibit a nonphysical bifurcating behavior [32] which was an original motivation for the study of [11]. It was shown in [11] that Braun's theory suffers from unitarity violation. Other attempts based on the QCD path integral approach were reported in [28,29]. Those works have proposed the so called "diamond action" as a self dual effective action of RFT. Although the question has not been settled, in recent years this effort has only been simmering on a back burner.
Here we return to this problem motivated by considerations of unitarity. As we showed in [10], the JIMWLK Hamiltonian violates QCD unitarity constraints when acting on the dense target wave function. In view of the discussion in [11] of the zero dimensional toy model, it seems likely that the self duality of H RF T is necessary in order to restore unitarity. In this section we present a self dual H RF T and show that it reduces to H JIM W LK and H KLW M IJ in the appropriate dense-dilute limit.

The symmetries
Our strategy in this paper is similar to that of EFT: we are not going to attempt to derive H RF T from first principles, but will rather construct a family of Hamiltonians which on the one hand reduce to H JIM W LK and H KLW M IJ in the appropriate limits, and on the other hand are symmetric under the known symmetries of H JIM W LK in addition to being self dual.
The symmetries of H JIM W LK have been analyzed for example in [23] and [33]. H JIM W LK possesses the continuous symmetry group SU L (N ) × SU R (N ) generated by J L(R) . In addition it has the discrete Z S 2 × Z C 2 symmetry group with the two discrete transformations acting in the following way: 2. The charge conjugation Z C 2 . For simplicity we choose to work in the basis where the generators in the fundamental representation t a are either real and symmetric or imaginary and antisymmetric. In this basis the charge conjugation symmetry corresponds to changing the sign of the real generators since this has the effect t a → −t a * which interchanges the generators in fundamental and anti fundamental representations. Defining the matrix the "second quantized" form of the transformation is The eikonal factors in fundamental (U F ) and adjoint (U ) representations transform as We expect both the discrete symmetries of H JIM W LK to remain the symmetries of the general H RF T since they directly reflect the symmetries of QCD. The situation with SU L (N ) × SU R (N ) is less clear. It is certainly true that we expect the diagonal vector subgroup SU V (N ) to be a symmetry of H RF T , since it descends directly from the global color group of QCD as it rotates simultaneously the initial and final scattering states. The left rotation acts only on the initial states and may be an accidental symmetry of the dense-dilute limit. Thus we will not insist on SU L (N ) and SU R (N ) to be separate symmetries but will return to this question later.
In addition to these symmetries which are symmetries of JIM W LK limit, we will also require H RF T to be invariant under the dense dilute duality Z D 2 . To understand how the duality transformation acts on the field variables in the current RFT setup, we recall that physically it simply interchanges the projectile and the target. In other words for basic scattering amplitude we should have Self duality, or invariance, under Z D 2 is a realization of the fact that the two amplitudes must be equal at any collision enery When considered as a transformation acting on a function of the basic fields ρ and δ δρ , the Z D 2 transformation can be written as In terms of individual operators this is [27] However, in addition to this action one has to take an overall Hermitian conjugation of the whole expression which is being transformed. Note that due to this additional action of Hermitian conjugation the duality transformation Z D 2 cannot be represented by an action of a unitary operator on the RFT Hilbert space. This is similar to time reversal in quantum mechanics, which is not a unitary but an anti unitary transformation. Recall that anti unitary transformation involves complex conjugation of an operator function in addition to the transformation of basic variables. The duality is not an anti unitary transformation either, since it involves hermitian conjugation rather than a simple complex conjugation of a function F . Nevertheless, just like the time reversal in quantum mechanics, it is a bona fide linear transformation in the Hilbert space and thus should be considered on par with other symmetries of the theory.

The "left" and "right" Wilson lines
To construct H RF T let us introduce the following Wilson line like operators in the fundamental representation These expressions resemble our reggeized gluon operators U andŪ . However they are defined in terms of SU (N ) generators J L(R) and I L(R) rather than commuting variables ρ.
The reason to introduce these operators is that they look like appropriate building blocks for H RF T . Recall that we need H RF T to reduce to H JIM W LK in the dense dilute limit, i.e. in the leading order of expansion in powers of ρ. Now H JIM W LK is a simple function when written in terms of J L(R) rather than the regular Wilson line operators U . It therefore seems likely that in order to extend it beyond the densedilute limit the basic building blocks also should be simple function of J 's. On the other hand H JIM W LK is also a simple function ofŪ . Given that we want to impose self duality on H RF T it is reasonable to choose our building blocks to be in some way similar to Wilson lines. Hence the motivation to introduce the operators in Eq. (3.23). We chose to discuss these operators in fundamental representation for simplicity. As we will show later, the construction we propose works with an arbitrary representation of SU (N ), thus providing an infinite set of Hamiltonians that satisfy our requirements.
When calculating the RFT "correlators" of these operators with U andŪ , the ordering of the vertices is important, unlike in the calculation of correlators of U 's andŪ 's among themselves. For example consider the simplest correlator where the ellipsis denotes contributions of order g 6 and higher, i.e. three and higher gluon exchange diagrams. For comparison, a similar correlator for the fundamental Wilson line defined as  At the two gluon exchange level the difference between the two is which corresponds to the diagram in Fig. 2. Note that this difference is a two gluon exchange in the octet channel, and may be viewed simply as the reggeization correction to a single gluon exchange.
In general if one thinks about V L as representing a fundamentally charged parton in the target wave function, the parton in question would be something of a black sheep. It would always scatter on the projectile only after all the other partons have had their day. As an example, a sample diagram corresponding to the calculation of the correlator L|U (x 1 )U (x 2 )V L (z)Ū (y)|R is depicted on Fig. 3. Note that all the gluons exchanged betweenŪ and V L attach to theŪ line to the left of any gluon exchanged betweenŪ and any of the U 's. This follows since V L contains only left rotation generators ofŪ . Similarly, V R only contains right rotation operators, and therefore in a scattering diagram always exchanges gluons with the projectile before any other exchanges with target gluons.
Also note that operatorially V L and V R do not commute with U , although they commute with each other. Similar comments apply toV L(R) .

Constructing H RF T
Let us now consider the following expression where in the last line we have integrated by parts assuming that the boundary terms vanish. Note that the order of factors is important, since the operators V andV do not commute with each other. In (3.27) all factors V L , V R are understood as positioned to the right of any factorV L ,V R . The diagram that schematically represents the color flow between the four Wilson lines is shown in Fig. 4.
We start with this expression since as we will see shortly it reproduces both, the JIMWLK and the KLWMIJ Hamiltonians in the appropriate dense-dilute limit. Following our EFT like strategy we would like to impose on H RF T the discrete symmetries discussed above. It turns out that it is quite easy to do.
We start with the duality transformation Z D 2 . We perform the transformation in two steps. First we perform the canonical transformation or equivalently, Second, in accordance with Eq. (3.21) we take the Hermitian conjugation of the transformed Hamiltonian to obtain Thus we find that H RF T is self dual already. The next in line is the signature transformation Eq. (3.14) It is easily seen that H Applying the charge conjugation on H It is easy to see that H (1)c RF T is by itself invariant under the signature and duality transformations. Therefore, the following Hamiltonian is invariant under all relevant discrete symmetries: So far we have not discussed the continuous symmetries of H RF T . We will postpone this discussion to Section 5 after we consider the dense-dilute limit.
We have found a candidate RFT Hamiltonian which is self dual. In fact the construction above defines a family of self dual Hamiltonians. In particular rather than using the fundamental representation for defining V L(R) andV L(R) we could have used any representation of the color group. Any one of these variations is self dual and, as we will see later reduces to the JIMWLK Hamiltonian in the dense-dilute limit. We do not have any a priori reason to prefer one of these versions to another, although it may seem unnatural to involve very high representations of the color group. One should also note that for representations that have vanishing N -ality, like the adjoint representation one has H (1)

RF T = H
(1)c RF T which is a simplifying feature.
In this paper we will be working with the fundamental representation defined in Eq. (3.23) when deriving the JIMWLK and KLWMIJ limits so that not to loose generality. We will show that H

The dense-dilute limit.
The most important test for H RF T is that it must reproduce H JIM W LK in the dense-dilute limit. In this section we demonstrate explicitly that this is indeed the case.
The dense-dilute limit arises when the number of gluons in the projectile is of order one, while the number of gluons in the target is large, parametrically n ∼ O(1/α 2 s ). Thus we are considering the amplitude in Eq. (2.1) and Eq. (3.1) where the number of factorsŪ is of order one, and the number of factors U is of order 1/α 2 s . In this limit several simplifications occur. We will first give a simplified argument, and then complete the mathematical details of the demonstration.
First of all, note that at weak coupling any given projectile gluon can exchange at most two gluons with any given target gluon. However, since the number of gluons in the target is large, a projectile gluon can multiply scatter on many gluons of the target. A representative diagram for scattering of a single projectile gluon is depicted on Fig. 5. The diagram in Fig. 5 contains single and double gluon exchanges between individual pairs of gluons. If a single gluon exchange is present such a diagram contributes to an inelastic amplitude as the final state of the scattering process is necessarily different from the initial state. The elastic amplitude has contribution only from two gluon exchanges where the two gluons are in the color singlet. Since every two gluon exchange carries a factor α 2 s , and there are in total O(1/α 2 s ) target partons that can participate in the scattering, the total elastic scattering amplitude in the dense-dilute limit is of order unity † . † Single gluon exchanges behave a little differently. One does not add single gluon exchange amplitudes between a given projectile gluon and different target gluons since those lead to different final states of the target and do not contribute to the same S matrix element. Instead the single gluon exchanges with distinct target gluons lead to appearance of many On the other hand since the projectile is dilute, every target gluon can only scatter either on one or two projectile gluons. The appropriate diagrams are represented on Fig. 6. Technically this means that in the dense-dilute limit all factors of U have to be expanded to second order in ρ. This insures that once two gluons are exchanged between a target gluon and the projectile, the target gluon does not participate in any further scattering. Now consider the diagrams as in Fig. 6 but which, instead of one of the factorsŪ contain a factorV L that appears in the RFT Hamiltonian.
As we have discussed above, the only difference between these two sets of diagrams is that all the gluons exchanged betweenV L and any given factor U connect to the left of any other gluons that might be exchanged by this U and a different factor ofŪ present in the amplitude. However any given U can exchange at most two gluons. If these two gluons are exchanged between U andV L , no further gluons are exchanged and the action ofV L is identical to the action ofV . If U exchanges only one gluon withV L and another gluon with some other factor ofŪ , it is still true that as far as elastic amplitude is concerned the action ofV L andV is identical. The difference only appears in the inelastic amplitude, but here again it appears as α s suppressed correction through a diagram analogous to that of Fig. 2, see Fig. 7. This correction is not enhanced by the number of target gluons, and thus is indeed negligible in the dense-dilute limit. We therefore conclude that in the dense dilute limit we can safely replaceV L byV . The same is obviously true forV R . Thus in the dense-dilute limit in H RF T we can replacē  Another simplification follows since any factor of U , V L or V R can be expanded to second order as only two gluons can be exchanged by any of the target gluons. Thus in the dense-dilute limit we have With these simplification we now consider the RFT Hamiltonian. Let us concentrate on H with the understanding that V L , V R are expanded to second order. The zeroth order in expansion, the product V L V R is a constant and does not contribute to the Hamiltonian due to derivative acting on it. The first order also vanishes because it involves a factor Tr(t a ) = 0. At second order there are three terms Substituting the above expression into H RF T , one obtains Note that the spatial derivatives generate other terms However, with ∂ 2 x φ(x − y) = gδ(x − y) and ∂ 2 x φ(x − z) = gδ(x − z), performing the integration over x and using the relationsŪ ed (y)J e L (y) = J d R (y) andŪ ed (z)J d R (z) = J e L (z), these addtional terms cancel each other. Thus only the term where the two derivatives separately act on φ(x − y) and φ(x − z) survives. Performing the same calculation for H (1)c RF T we find to this order an identical result. Thus in the dense-dilute approximation we get There is one subtlety in this derivation which we need to address, i.e. at what order does the correction to Eq. (4.1) affect the calculation. To answer this we need to develop a controlled expansion of H RF T in the dense-dilute limit. To do this we note that although we have justified Eqs.(4.1) and (4.2) by analyzing the contributions to the S-matrix generated by exchanges of at most two gluon, the same result can be obtained formally by taking the limit of small ρ. It is obvious that at small ρ, the operators V L and V R should be simply expanded in power series in J L(R) to the leading order to which the Hamiltonian does not vanish, leading to Eq. (4.2). On the other hand at small ρ we should also expand I L(R) to leading order in ρ, which gives In fact expansion in powers of ρ is the proper formal way to derive the form of the Hamiltonian in the dense-dilute limit.
Formally expanding H RF T in powers of ρ we see that H JIM W LK arises at order ρ 2 by multiplying the O(1) term inV LVR and O(ρ 2 ) term in V L V R . However we also have to consider a possible contribution arising from O(ρ) term inV LVR (the first order correction to Eq. (4.8)) multiplied by O(ρ) term in V L V R . We write this additional term as HereV L andV R are understood as expanded to O(g), however we will not need the explicit from of this expansion, since we will show that this expression vanishes.
We use the two identitiesV † λα t e αβVβγ =Ū ed t d λγ (4.10) Here, as beforeV † = exp{−t a δ δρ a } andV = exp{t a δ δρ a } are defined in the fundamental representation whilē U = exp{T a δ δρ a } is defined in the adjoint representation. We then calculate Thus the four terms in Eq. (4.9) pairwise cancel.
We have thus proved that when expanded to second order in ρ, the Hamiltonian H

Continuous symmetries
Let us now discuss the continuous symmetries of H RF T . As we have mentioned above, both the JIMWLK and the KLWMIJ Hamiltonians have a continuous SU (N ) × SU (N ) symmetry, albeit those are distinct symmetry transformations. The SU (N ) × SU (N ) symmetry of H JIM W LK is generated by the charges It is an interesting question which of these symmetries are also the symmetries of the self dual H RF T Eq. (3.35). The question is not completely straightforward to answer even though we do have an explicit representation of the charge operators on the RFT Hilbert space. The reason is that the commutation relations between J L(R) andV L(R) as well as between I L(R) and V L(R) are quite complicated. We will nevertheless try to answer this question, using a perturbative expansion. Our answer is somewhat surprising: the symmetry of H RF T appears to be SU (N ) × SU (N ) × SU (N ) ‡ .
We start with discussing the vector part of the group, which is the easiest and can be analyzed without recourse to perturbation theory.
To better organize the calculation, we rescale the charge densityρ a (x) = gρ a (x) and also introducẽ φ(x − y) = 1 g φ(x − y). Then J a L , J a R , I a L , I a R can be Taylor expanded by counting the powers of the coupling constant g. We will use this expansion in this and the next sections. We will refer to this counting in powers of the coupling constant as the "BFKL counting", since it is equivalent to simultaneous expansion in powers of ρ and δ/δρ.

5.1
The vector SU V (N ) symmetry The analysis of the vector symmetry is facilitated by the following simple observation To prove this we note that Integrating by parts we find It is now straightforward to check that the vector SU V (N ) transformation generated by Q L − Q R is the symmetry of H RF T . By virtue of Eq. (5.3) the charge Q a V ≡ Q a L − Q a R acts as a rotation generator on all the currents, i.e.
(5.7) ‡ To be precise, while SU (N ) × SU (N ) is there, the third SU (N ) does not necessary form a direct product with the first two. We have not attempted to write down the full algebra of the currents, which appears to be quite complicated.

It then follows that for a finite group transformation
with the fundamental representation matrix W κβ F = e iλ e t e κβ (5.12) The same transformation as in Eq. (5.11) applies to V R (x), as well as toV L(R) . It is now obvious that H RF T is invariant under SU V (N ).

Is SU L (N ) there?
Let us now consider other transformations generated by the left and right charges. The analysis for all of them is similar, and we will concentrate on Q L . The question we are asking, does Q L commute with H RF T ?
What is the action of Q a L on the building blocks of H RF T ? The answer for V L and V R is obvious.
What is the transformation ofV L andV R ? Examining the expression for H RF T we see that if the transformation waŝ (true or false ?) (5.18) the Hamiltonian would be invariant under SU L (N ). Indeed if instead ofV L andV R we hadV andV † , this would be the case. This is precisely what happens in the JIMWLK limit.
The transformation Eq. (5.18) is equivalent to the commutation relation (true or false ?) (5.19) and similarly forV R .
We were unable to calculatie the commutation relation in Eq. (5.19) in a closed form. However we were able to calculate first several orders in perturbative expansion in g. We performed the calculation in the BFKL counting of orders of g. The details of the calculation are presented in the Appendix A. Our results are the following.
We have calculated the commutator between Q a L andV L up to order g 3 and found that relation Eq. (5.19) holds up to order g 2 , but is violated at order g 3 .
We have also calculated the commutator of Q a L with the Hamiltonian [Q a L , H RF T ] up to order g 6 . We have found that this commutator vanishes up to this order. This leads us to believe that even though Eq. (5.19) is not satisfied, the SU L (N ) is indeed a symmetry of H RF T . We stress that we do not have a closed form proof of this, but only perturbative calculation to order g 6 .
The analysis ofQ a L is identical, since Q andQ are related by duality transformation. Thus we believe thatQ a L also commutes with the Hamiltonian. If this is indeed the case, the continuous symmetry of H RF T is at least SU (N ) × SU (N ) × SU (N ). In fact the symmetry could be even larger since we have not calculated the commutators [Q L ,Q L ]. If this commutator does not close on any of the four charges (or their products) Q L(R) ,Q L(R) the symmetry group is larger. We have not investigated this question any further.
6 Is this the "Diamond action"?
The family of Hamiltonians that we have identified carries uncanny resemblance to the so called "Diamond action" suggested in [28] and also discussed in [29]. There is of course a host of differences between our approach and that of [28] and [29]. On the technical level we are dealing with the Hamiltonian formulation of RFT together with the accompanying field algebra and the structure of the RFT Hilbert space, while these references strive to derive the effective action in terms of certain Wilson line functions. On the other hand [28] and [29] derive the action directly from QCD (although in both cases certain not entirely straightforward approximations are utilized) whereas our expression is an ansatz constrained by the expected symmetries and the appropriate limiting forms.
Nevertheless, abstracting ourselves from these differences we can compare H RF T with the effective action of [28]. We concentrate on the Hamiltonian Eq. (3.35) defined with Wilson line in the adjoint representation.
In this case the two terms in Eq. (3.35) are equal and we have It is easily checked that with the correspondence our Eq. (6.2) looks identical to the effective action suggested in [28]. However beyond the looks there are significant differences between the two. In particular in [28] the four Wilson lines are not independent, but satisfy the so called diamond condition This relation was essential in the derivation of [28] and only using this relation the effective action obtained in [28] could be written in the form Eq. (6.2). On the other hand in our framework, although all four Wilson line operators are expressible in terms of ρ and δ/δρ, there is no such condition that constrains the four.
We can check Eq. (6.4) explicitly, expanding all the operators U L,R andŪ L,R to first order in the respective left and right charge densities. In our notations Eq. (6.4)) corresponds tō We will calculate the LHS of Eq. (6.5) to second order in g. To this order we need From the definition of I a L and I a R , one obtains (6.10) We have used integration by parts. As a consequencē (6.12) On the other hand, from (6.13) one obtains (6.14) (6.15) At order O(g) it is obvious that Eq. (6.5) is satisfied, and the first nontrivial check of the relation is at O(g 2 ). At this order we obtain Thus we have established that at order O(g 2 ), the diamond condition is not satisfied by our Wilson line like operators.
We thus conclude that in spite of certain similarities, the self dual RFT Hamiltonian Eq. (6.2) is not the same as the effective action of [28,29].

Discussion
In this paper we have revisited the problem of constructing a self dual Reggeon Field Theory Hamiltonian H RF T . We have followed the EFT strategy by imposing the relevant symmetries and also required that H RF T reduces to H JIM W LK (or H KLW M IJ ) in the dense-dilute limit.
As a result we have found a family of Hamiltonians that satisfy these requirements. These Hamiltonians are constructed from Wilson line -like operators in different representations of the SU (N ) group. We have analyzed the continuous symmetries of H RF T . This is an interesting question since both H JIM W LK and H KLW M IJ possess an SU L (N ) × SU R (N ) symmetry group, but the generators of these transformations are not the same in the two dense-dilute cases. For H RF T we are able to show nonperturbatively the existence of one SU V (N ) symmetry, which is the diagonal subgroup of the symmetry group in both JIMWLK and KLWMIJ limits. We established the fact that the two diagonal subgroups are identical explicitly using the algebra of the generators in the RFT Hilbert space. We have also shown that H RF T is invariant under the left and right rotations at least to O(g 6 ) in perturbative expansion. This is a strong indication that the continuous symmetry group is at least SU (N ) × SU (N ) × SU (N ).
One member of the family of the Hamiltonians we found is very similar to the "diamond action" [28,29]. Our Hamiltonian RFT framework is different from the effective action approach of [28,29] which somewhat hampers direct comparison. Nevertheless if we juxtapose our H RF T defined in terms of adjoint Wilson lines directly with the effective action of [28,29], the two look identical. There is however one significant difference between our result and that of [28]. Namely the action in [28] is written in terms of four Wilson loops that satisfy the diamond condition, Eq. (6.4). This condition played a very important role in [28]. In fact the effective action derived in [28] directly from QCD is equivalent to the "KLWMIJ+" Hamiltonian suggested in [34,35], whereby KLWMIJ Hamiltonian is generalized by including nonlinear corrections in the solution for classical field. This Hamiltonian is not explicitly self dual, and only with the help of the diamond condition it was recast in [28] in the form which looks self dual, at least superficially. However whether the "diamond action" is in fact self dual or not remained an open question. To check the self duality one has to verify that the duality transformation is canonical, or in the quantum sense a linear transformation on the RFT Hilbert space. This was not possible to do with the tools of [28], as no operator realization of the algebra of Wilson lines was explicitly presented. In the present paper we operate within the RFT Hilbert space with well defined operator algebra; and therefore we have explicit realization of the duality transformation in the Hilbert space. We find within this consistent framework that the diamond action (RFT Hamiltonian) is self dual, but the diamond condition between the Wilson lines is not satisfied. The condition is violated starting with order O(g 2 ) in perturbative expansion. In this sense our paper is closer to [29], where the diamond action is derived as a self dual form of the action in the dense-dilute limit without assuming the diamond constraint between the Wilson lines. In [29] the constraint was shown to hold in the first order in perturbation theory, which is consistent with our conclusion here, but was not checked at higher orders.
Our "bottom up" approach does not allow us to decide which one of the candidate hamiltonians we have found is the right one, and in fact whether any one of them is the correct QCD RFT Hamiltonian. Even though we have used the EFT methodology to determine possible terms in H RF T , we are at a disadvantage here compared to standard applications of EFT in quantum field theory. The generic situation is that one is searching for local operators that can be incorporated into the EFT Lagrangian (or Hamiltonian) in the situation where there is only a finite number of possible operators of a given dimension. The higher the dimension of the operator the stronger the suppression of its contribution to low energy observables. Thus EFT organizes the possible operators according to their importance in the interesting kinematics. In our case the situation appears to be different. Although RFT is the effective theory of QCD at high energy, all the operators we have found may contribute at leading order in E −1 . We do not see any obvious parameter which would order the possible contributions. The similarity with the diamond action may suggest that one should work with the Wilson lines in the adjoint representation. However as is clear from the derivation in [29] the diamond action is not the full story, but is only a leading term in an expansion away from the abelian limit. Thus it is possible that the other candidate terms we have found also play a role in the full RF T Hamiltonian.
It would be interesting to find a criterion which could discriminate between the possible terms. One possibility is to compare H RF T with NLO JIMWLK. Although we have no reason to expect that H RF T contains all, or even most NLO terms, it does contain some such terms. Comparing those to NLO JIMWLK could be instructive and possibly discriminatory.
Another interesting question is the unitarity of H RF T . As we have mentioned in the introduction, our main motivation to search for the self dual H RF T was the unitarity violation in H JIM W LK . Given H RF T one can in principle follow the procedure explained in [10] to determine whether its action corresponds to unitarity evolution of QCD states in energy. Unfortunately analyzing the unitarity conditions beyond the JIMWLK limit is technically a complicated problem which at this point we are not able to solve. We believe it is a very important question and are planning to address it in future work.
We start by trying to verify the conjectured commutation relation: We calculate the commutator perturbatively using the BFKL counting. We express To expandV L we need ThenV L is expanded as In terms of coupling constant g, we check the commutator Eq. (A.1) order by order. (A.10) • O(g), the relation to be checked is First note that each individual term is (1) . (A.13) We have one additional term but it vanishes.
This vanishes due to ∂ 2 y z δ(y − z) = 0. where the subscript "(n)" indicates the n-th order in g.
The first term vanishes due to Eq. (A.49). The second and third terms add up to zero because of Eq. (A.55). Now we focus on the fourth, fifth and sixth terms.
For the sixth term, note that and The sum of these two terms vanishes due to equality d L(3) = d R (3) .
For the fourth term These two terms also cancel each other due to d L(3) = d R (3) . We therefore proved that [Q a L , H RF T ] (6) = 0.
Thus we see that up to order O(g 6 ) the left rotation generator Q a L commutes with the Hamiltonian.