Ward identities for amplitudes with reggeized gluons
Starting from the effective action of high energy QCD we derive Ward identities for Green’s functions of reggeized gluons. They follow from the gauge invariance of the effective action, and allow to derive new representations of amplitudes containing physical particles as well as reggeized gluons. We explicitly demonstrate their validity for the BFKL kernel, and we present a new derivation of the kernel.
1 Generalized Ward Identities
Gauge invariance has been the guide to construct, within QCD, an effective action [1, 2] which introduces the fields of reggeized gluons and describes the high energy behavior of QCD. It automatically leads to the construction of gauge invariant amplitudes and Green’s functions of reggeized gluon and physical particles. The effective action generates a set of extended Feynman rules  with interactions which are local in rapidity and which may be used to compute amplitudes involving reggeized gluons. Since reggeized gluons are off shell and belong to unphysical polarizations111Really the reggeized gluons can be considered as gauge invariant states having the physical polarizations in the crossing channel and lying on the Regge trajectory., it is important to investigate symmetry properties derived from gauge invariance. As in normal QCD, gauge symmetry plays an important role in doing explicit calculations.
In this paper we extend the BRST invariance of QCD to Green’s functions of reggeized gluons and derive a generalized set of Ward identities (Section 1). We find it convenient to first recapitulate a few identities for amplitudes in normal QCD. We then extend these identities to the Green’s functions and amplitudes derived from the effective action case. In the second part of our paper (Section 2) we demonstrate, as a first application, the validity of these Ward identities for the 4-point function of four reggeized gluons, and we give a new derivation of the BFKL kernel.
In this section we remind which kind of simple QED-like Ward identities we may expect for a general QCD scattering amplitude.
QCD is a theory with Yang-Mills gluon fields coupled to quarks. This theory has a unitary S-matrix presented in terms of transversely polarized colored gluon and quark fields as on shell asymptotic states, even if these are not the truly physical states as hadrons and mesons of the confining phase.
Let us denote a physical transverse gluon polarization vector ( in four dimensions) . The corresponding states satisfy the Lorentz condition . The additional solutions of this equation with the longitudinal polarization are decoupled from the physical states. In other words the scattering amplitude with such longitudinal gluons is zero.
In the following we shall just consider amplitudes with external physical on shell quarks and gluons and possibly one or more longitudinally polarized gluons. They satisfy a tower of Ward-like identities. We can interpret them as saying that longitudinal polarization states () decouple from physical polarization states (). In practice one has associated to the on shell physical scattering amplitude
the tower of identities
Any single or multiple contraction with longitudinal polarizations gives zero if all the other lines are contracted with physical polarization vectors and are on shell. This is a consequence of the fact that QCD is a gauge invariant theory.
Let us now recall how to prove these identities222This proof is not original and can be found in the literature.. It is convenient to use the global BRST symmetry of the gauge fixed QCD action. Let us start from the QCD lagrangian density with the so called general Lorentz gauge fixing (Lorentz invariant) and the associated ghost terms included (restricting to the gluon sector (field ) since the quark sector is trivial in our analysis)
where is an auxiliary field which satisfies the equation of motion and is the usual covariant derivative. One can therefore write the conjugated momenta
The global BRST symmetry is defined by
where is the infinitesimal Grassmann parameter of the transformation. In terms of the momenta one can immediately write, using the Nöther theorem, the conserved BRST charge
which, after quantization, generates the quantum BRST transformation.
The BRST transformation is nilpotent and therefore . We remind that the ”large” Hilbert space can contain states with negative norm. There are, at fixed color, six different asymptotic states: are the two states with physical polarizations, then there are other two gluons states, one of them described by the auxiliary field , and finally there are the ghost and the antighost fields.
The physical states are the transverse polarized gluons which are annihilated by the BRST charge . The subspace of the large Hilbert space which belongs to the kernel of contains the physical states plus the zero norm states ( and ). The physical Hilbert space is the quotient of such a space with respect to the subspace of zero norm states, i.e. . From the BRST transformations it is easy to see that .
Having recalled these basic facts now we can see how to obtain the Ward Identities given in eq. (1.2). Consider the reduction formula for a matrix element where gluon asymptotic states have been removed and replaced by momentum contraction, leaving incoming and outgoing physical states. Taking the Fourier transform () one has
The ’s are introduced to keep track of the standard relation between Green’s functions and S matrix elements but they do not play any role in deriving the identity. Let us stress that the momenta corresponding to the dependence in are not on shell. The expression is zero thanks to the nilpotent property of and to the fact that .
1.2 Effective action with reggeized gluons
Let us now consider the case of our interest, the Effective Action  which includes reggeized gluons. This action is non local, and it has been constructed in such a way that it is gauge invariant, including the reggeized gluons (which are not on mass shell) as external states. In order to achieve this one has to introduce, order by order in perturbation theory, a well-defined set of induced interactions of reggeized gluons with normal gluons. All these induced interactions, as well as the conventional QCD interactions, are contained in the gauge invariant effective action for gluons and reggeized gluons.
Let us write the effective action which describes the coupling of the reggeized gluons to a cluster centered at rapidity where all the produced particles belong to a rapidity interval , which means that . We denote the gluon field as and the reggeized gluons fields as , which satisfy the kinematical constraints
according to the quasi-multi-reggeon kinematics. The reggeized gluons are characterized by and polarization vectors.
The effective action for a given rapidity interval reads 
Here the first term coincides with the gauge fixed QCD lagrangian in (1.3), and the second one describes the interaction of the reggeized gluons fields with the gluons through the induced terms:
In contrast to these interactions terms which are local in rapidity, the last term, the kinetic term for the reggeized gluons, describes the interaction of particles with different rapidities. We note that, in this action, the reggeized gluons play a role similar to classical fields, i.e. within each rapidity cluster they do not appear in loops. For the last two terms in Eq. (1.9) we can also introduce a more compact notation. We define
with the normalization and . Note that we have , due to Eq. (1.8), which is similar to the Lorentz condotion for the real gluons. This allows to write, in Eq. (1.9), the kinetic term of the reggeized gluon as and the induced interaction part as .
Under gauge transformations one has which implies on the induced terms a variation . Therefore, recalling the kinematical constraints in Eq. (1.8), one notes that, after integration by parts, the variation of the terms in the traces is zero provided the function which describes the gauge transformation vanishes as and . Apart from the gauge fixing and ghost terms we have therefore a fully gauge invariant effective action by considering gauge invariant reggeized gluons. After introducing gauge fixing and ghost terms this action enjoys, similarly to the normal QCD case, a global BRST symmetry with associated conserved canonical charge such that .
The objects in which we are interested are the gauge invariant scattering amplitudes which, in general, involve quarks, on shell physical gluon states and reggeized gluons (the extensions to the SYM case is straightforward). Such scattering amplitudes are constructed from the Green’s functions using the LSZ reduction with respect to the lines with the physical quark and gluon states. For example, an amplitude with a physical gluon and two reggeized gluons (the BFKL production vertex) is related to the following Green function
Choosing the unphysical polarizations for each of the reggeized gluons and the physical polarization for the normal gluon, taking the Fourier transform in the LSZ reduction formula and removing for the latter the physical pole, we have the effective BFKL production vertex
where the polarization vector belongs to the reggeized gluon with momentum (with a large ”” component), and the vector to the gluon with momentum (with a large ”” component).
Note that this amplitude contains induced terms which couple the reggeized gluons to the usual gluons (rhs of (1.10)). These contributions are necessary for restoring the gauge invariance of the amplitude. We shall use two different equivalent notations to write the Ward identities. In the notation of , starting from (1.10), not all Feynman diagrams are proportional to the product of two external polarization vectors, . Induced terms which do not contain the vector have to be left out in the Ward identity in , terms without do not participate in the Ward identity in (such terms come from the second, third,… terms in Eq. (1.10), i.e. from those interactions where the external reggeized gluon couples to two or more elementary gluons). In Eq. (1.13) we have introduced a simplified notation which allows to write the full amplitude using a uniform contraction with the vectors. This follows from having introduced the general objects of Eq. (1.11) containing both polarizations, which, after contractions with , give the specific polarized terms.
Let us now formulate, for this example, the Ward identity in the external reggeon line with momentum . The argument is analogous to the one used in the previous section for pure QCD and is based on the gauge invariance of the Effective Action for reggeized gluons. Since we have seen that under a BRST transformation the reggeized gluons do not change, the corresponding quantum states are also annihilated by the BRST operator and we can extend the Hilbert space corresponding to gauge invariant states adding also the reggeized gluon states, which can be considered on the mass shell as physical states in the -channel. From it one may construct the multi-particle Fock space. As before, therefore, in order to obtain a Ward identity starting from the Green’s function in Eq. (1.12) we consider a new Green’s function, obtained replacing at point the reggeized gluon field operator by the operator :
Proceeding now in the same way as in Eq. (1.7), using the relation , applying the LSZ reduction to the physical gluon line and fixing the polarization of the reggeized gluon at point , we end up with the following identity
It is important to note that the replacement eliminates all those induced contributions in Eq. (1.10), where the reggeized gluon ends on two gluons (for example at tree level here one cannot have induced terms of higher order). In the same time the reggeized gluon with momentum interacts with all vertices in Eq. (1.10).
Let us generalize these arguments. Using the same notation as in Eq. (1.13), the amplitudes are of the form
where the tensor amplitude is constructed from fields of reggeized gluons, , , and physical gluon fields . At this stage, the amplitude contains all the induced interaction associated to the ”” polarized reggeized gluons and to the ”” polarized reggeized gluons. By suitable replacements one can now write a tower of Ward-like identities for the reggeized gluons which are analogous to the ones in eq. (1.2). They are obtained from Eq. (1.16) by replacing the corresponding polarization vectors of the reggeized gluons by the contraction with the momentum, taking into account that the tensor amplitude must be substituted by a new one, , since the terms corresponding to the induced interactions of the line with contractions must be removed. Indeed, as in our example, the replacement eliminates those induced graphs where the reggeized gluon couples to two (or more) gluon fields. In the notation of , these are exactly those graphs which do not have a polarization vector for the reggeized gluon and do not participate in the Ward identity. In a similar way one obtains also Ward identities for the gluon fields.
Our Ward identities therefore take the form:
where the subsets , and of physical polarization states, ”-” polarized and ”+” polarized reggeized gluons, respectively, have been replaced by the corresponding momentum contractions. We stress that is related to the modified product of field operators where some of the reggeized gluon fields have been replaced by gluon fields .
In the following we will show that these Ward identities can be used to obtain a new (and useful) representation for amplitudes and Green’s functions, in which the unphysical polarizations of reggeized gluons are substituted by transverse momentum vectors.
2 An application: the BFKL kernel
As a first application, we investigate the use of Ward identities for the real part of the BFKL kernel. As a result, we will present a new derivation of the kernel. We proceed as follows. Making use of the effective action we begin with the LO 4-point function of four reggeized gluons where, in contrast to the usual derivation based on the -channel unitarity, the produced -channel gluon is off shell, and its longitudinal momenta (Sudakov or light cone variables) are not yet integrated. At this stage, the reggeized gluons carry unphysical polarizations. Beginning with one of the reggeized gluons, we derive a Ward identity and use it to substitute the unphysical polarization vector by the corresponding transverse momentum vector. Repeating this procedure, step by step, for the remaining gluons we arrive at a representation of the BFKL kernel in which for all four reggeized gluons the unphysical polarization vectors or are replaced by transverse momentum vectors. We then show that after the integration over the longitudinal momentum of the -channel gluon this new form of the BFKL kernel coincides with the standard expression. Finally, we re-formulate our results and show that they are in agreement with the Ward identities stated in section 1.
2.1 Derivation from the effective action
We begin with the notation for the effective diagrams shown in Fig.1:
Fig.1: the BFKL kernel.
The 4-point function (which, after integration over the longitudinal component of the momentum , will become the BFKL kernel) is obtained from squares of two effective production vertices plus the quartic interaction. In contrast to the usual derivation of the BFKL kernel, we take the produced gluon to be off-shell and do the longitudinal integration later. The upper effective production vertex in Fig.1a has the form (disregarding, for the moment, the color structure):
Within the effective action the production vertex is derived from the triple gluon Yang-Mills vertex and from the two induced vertices:
It will be convenient to introduce a short-hand notation for the induced terms333The subscripts ’L’ and ’R’ follow the notation of : ’left central’ and ’right central’:
such that (2.19) takes the form:
Contraction with the momentum of the produced gluon leads to:
Including the induced terms, and , gauge invariance of the production vertex is restored:
Including the color structure, Figs.1a and b have the color coefficients
resp. The quartic interaction in Fig.1c provides the same two color structures with the following coefficients:
The third color structure of the quartic coupling, , will be disregarded since it correspond to the color octet -channel. Putting everything together we find, from Fig.1a,
(note that, on the rhs of this equation, denotes the square of a four vector; all other squared momenta are effectively purely transverse). Without the piece of the quartic coupling there would be the additional term inside the bracket. We write (2.29) also in another form:
Here the numerator of the second term has the property that it vanishes whenever any of the transverse momenta , , , or goes to zero. In order to make this ’zero property’ explicit, we list a few alternative forms of the vertex:
In the first two lines we have introduced a cross product-like notation , while in the last line we have used complex notation where (note that, following the Sudakov notation, we use ). Because of , one easily recognizes the cancellation of the first term, once we add the crossed graph. We shall show in sections and that, after the use of the Ward identities, the first piece, , which destroys the zero property will be absent even before adding the crossed graph.
Before we study the integration over the longitudinal variable, , we mention another important feature of the representations (2.31). In this form, the BFKL kernel is closely related to the Weizsäcker-Williams approximation in which the interaction of the reggeized gluons, at vanishing transverse momenta, can be expressed in terms of on-shell gluon scattering amplitudes.
Next we turn to the integration over the longitudinal variable, . Let us fix, in (2.30), at some positive value. As we said before, the first term, , cancels after adding the contribution from the crossed graph. The second term with the additional denominator converges for large , but the singularity at needs to be regularized. From the analysis of Feynman graphs we find an infrared cutoff which implies the following definition of the integral:
where denotes a momentum scale of the order of the transverse momenta. Adding and subtracting the contribution of a small semicircle in the upper half complex plane, in the first case the upper half plane has no singularity, and the integral vanishes. From the subtraction of the semicircle we are left with
The same result is obtained if we add and subtract a semicircle in the lower half plane. Obviously, this result is independent of any description of the pole at . Proceeding in the same way for the region , we find
Including the integration over we find, for the sum over both regions,
The same result is obtained if we use the principal value prescription which follows from the derivation of the effective BFKL production vertex from Feynman graphs: the two graphs which describe the radiation of the produced -channel gluon from the vertices to the left of the BFKL vertex imply:
For the crossed graph we interchange and and replace by . Using we obtain:
In this way, the BFKL kernel is crossing symmetric under the exchange .
We conclude by mentioning that we would have obtained the same result for the longitudinal integrations by using the identity
The principal value term vanishes since we integrate over positive and negative values of and .
2.2 Ward identities on the rhs, and
In the following we search for an alternative expression which improves the convergence in the longitudinal component, , To this end we replace the and vectors of the -channel gluons by transverse momenta. We begin with the two gluons on the rhs, and in second step, apply the same procedure to the gluons on the lhs. We begin with the Ward identity in the upper -channel gluon on the rhs with momentum . We find:
(if the produced gluon were on mass shell, and we would multiply with a physical polarization vector the rhs would vanish). When contracting the rhs, in Fig.1a, with the lower production vertex, the piece proportional to vanishes because of the gauge invariance property of the lower vertex. We thus are left with:
The first term cancels if we add the contribution from the quartic coupling, obtained from (2.27) by replacing by and observing . Similarly, the crossed graph in Fig.1b yields:
As a result, we have verified the following Ward identity:
we can substitute ()
Taking into account that a purely transverse vector, contracted with or with the quartic coupling gives zero contribution, the 4-point function can written in the following form: