QCD at high energies and Yangian symmetry

Yangian symmetric correlators provide a tool to investigate integrability features of QCD at high energies. We discuss the kernel of the equation of perturbative Regge asymptotics, the kernels of the evolution equation of parton distributions, Born scattering amplitudes and coupling renormalization.


Gauge field theories and Quantum integrable systems
In 1993 L.N. Lipatov pointed out the relation of the perturbative QCD Regge aymptotics [1,2,3] to integrable quantum spin chains and showed that the contributions of multiple reggeized gluon exchanges can be treated by the methods of quantum integrable systems [4]. This result has great significance far beyond the particular questions of Regge asymptotics. Up to that time the application of such methods to Quantum field theory was restricted to sophisticated models in 1 + 1 dimensions.
It was clear immediately, that the Bjorken asymptotics [5,6,7,8,9,10] and the composite operator renormalization have similar features of integrability [11]. In both cases the Yangian symmetry underlying integrability is based on conformal symmetry, which is broken by loop corrections in the case of QCD. The breaking is suppressed by supersymmetry and is absent in N = 4 super Yang-Mills theory. Much work has been devoted in the last two decades to super Yang-Mills under this aspect. Quantum integrability works in the composite operator renormalization in the planar limit to all orders [12] and in the computation of scattering amplitudes and Wilson loops [13,14,15].
Yangian symmetric correlators (YSC) have been proposed as a convenient formuation of the Yangian symmetry of amplitudes and are shown to provide a tool of construction [16]. YSC depend on a set of spectral parameters, some combination of which are related to the helicities of scattering particles. The construction method of YSC are related to the method of on-shell graphs in the amplititude construction [18]. In the original form of the latter method [15] no such parameters appeared. In a number of papers the deformation of amplitude expressions by such parameters has been studied and their eventual role for regularization of loop divergencies has been discussed [17,19,20,21].
In the present contribution the notion of YSC is recalled and a few examples of their construction by R operations are given, which appear in the following applications.
We explain how the kernel of the BFKL equation in dipole form [24,25,26] emerges from a fourpoint YSC based on sℓ 2 symmetry. Based on [22,23] we discuss the relation of the parton evolution kernels and the amplitude of parton splitting [8] to three-point YSC based on sℓ 2 symmetry. The relation of the Born level scattering amplitudes of gluons and quarks to four-point YSC with sℓ 4 symmetry is considered. Finally, we consider the convolution of two three-point YSC, similar to a gluon self-energy Feynman graph, and show how the leading Gell-Mann -Low coefficient of the coupling renormalization appears. This relation is actually known in terms of the parton splitting amplitude [8] and the momentum sum rules for parton evolution kernels [9] and has been shown recently to extend to arbitrary helicity values [32].

Yangian symmetric correlators
Consider n Heisenberg canonical pairs, x a , ∂ a , a = 1, ..., n, L ab = ∂ a x b obey the general linear gℓ n Lie algebra commutation relations. We define the L operator as a n × n matrix with these entries with the unit matrix I multiplied by the spectral parameter u added, This operator acts on functions of x a , a = 1, ..., n. We consider the case of homogeneous functions with the weight denoted by 2ℓ, e.g. x 2ℓ n · ψ( x 1 xn , ... x n−1 xn ). The L operator restricted to such homogeneous functions depends additionally on the weight 2ℓ or on u + = u + 2ℓ, We need N copies of the above sets of canonical pairs, x i,a , ∂ i,a and of the L operator L i (u i , u + i ), i = 1, ..., N to define the monodromy matrix operator in terms of the matrix product It is convenient to display the parameters in the form Further, we consider functions of N points in n dimensional space, homogeneous with respect to the coordinates of each point x i of weight 2ℓ i . We define a Yangian symmetric correlator (YSC) to be such a homogeneous function of N points obeying the eigenvalue relation with the monodromy matrix The monodromy matrix can be separated into two factors, e.g. T (u) = T 1 (u 1 )T 2 (u 2 ) where T 1 denotes the product of the first L i , i = 1, ..., N 1 and T 2 the product of the subsequent L i , i = N 1 + 1, .., N .
The case of one point is trivial, but provides a convenient starting point. We have two obvious one-point YSC, L(u, u) · 1 = (u + 1)1, L(u, u − n)δ (n) (x) = uδ (n) (x), and from these the trivial product N-point YSC, The Yang-Baxter RLL relations provide a way to non-trivial YSC. We have , where the R operator can be represented by with the integration over a closed contour.
i.e. the entries at i, i + 1 in the second row are permuted. The resulting YSC are less trivial and by repeated R operations we obtain completely connected correlators characterized by the permuted parameter set of the resulting monodromy matrix, .

(2.3)
We shall use the abbreviation of writing the indices carried by the parameters only. Notice that by the substitution of parameters u σ(1) , u σ(2) , . . . , u σ(N ) → v 1 , v 2 , . . . , v N applied to both rows the first row can be put into the standard ordering.
One may draw an analogy of (2.1) to the time-independent Schrödinger equation. In this respect we have considered so far the position representation. Besides of that we need the helicity representation. It is defined in the case of even n, n = 2m by the elementary canonical transformation applied to half of the canonical pairs at each point, a = m + α = m + 1, ..., n, leaving the first half a = · α, · α= 1, ..., m unchanged, The N -point YSC can be written in the link integral form which became a standard of the modern tools of amplitude calculations [15]. In the position representation it is given by (2.5) K denotes the number of δ (n) factors.
To change to the helicity representation we apply the Fourier transformation to the dependence on the components x i,a , a = m + 1, ..., n at each point i = 1, ..., N . The Fourier variable conjugate to x m+α is denoted by λ i,α , α = 1, ..., m. The components x i,a , a = 1, ..., m are not changed and merely renamed byλ i,α , i.e.λ i,α = x i,α ,α = 1, ..., m. Thus the link integral form of a YSC in the helicity representation is Notice that the integrand function ϕ is the same in both representations. dc abbreviates dc = 3 Examples of YSC for N=2, 3, 4 We present examples of YSC encountered in the applications below and indicate their construction by R operations.
We start with the two-point YSC, N = 2, K = 1 In the second row 2 stands for u 2 and 1 + for and with N = 3, K = 2, We give also the relevant examples of four-point YSC.
As an intermediate YSC from which the following ones are constructed we write This YSC is incompletely connected with delta distributions in ϕ. It takes two more steps of R operations to arrive at the completely connected one. The sequence of operations leading to a YSC is not unique. In this case we may also choose with the indicated substitution as the last step to obtain the standard ordering in the upper row of u parameters. The permutation pattern u fixes the monodromy operator. However, the YSC is not uniquely fixed. For example, the pattern u = 1 2 3 4 4 3 2 + 1 + is the same as for Φ ∆ and is also the one of the following two YSC.
There are more YCS with the above parameter pattern. In the following we shall use Φ || (u) with It is the result of the convolution of two YSC of the form Φ X . At u 1 = u 2 it coincides with Φ ||1 and at u 3 = u 4 it coincides with Φ ||2 . The integration variables c ij may be regarded as coordinates on the corresponding Grassmannian variety G K,N and in the case of completely connected YSC the closed integration contours extend over a maximal Schubert cell. The delta distributions involving the correlator variables x orλ, λ fix some of these integration variables c ij . In the following applications we have cases where all are fixed, e.g. in the position representation for n = 2 with (K, N ) = (1, 3) and (K, N ) = (2, 4). The helicity representation has the feature that the delta distributions can be rewritten in a way with the factor δ( N 1λ i λ i ) independent of c ij . The integrations are removed in this representation for n = 4 with (K, N ) = (2, 4).

The BFKL kernel
YSC may be used as kernels of integral operators. The symmetry of the kernels implies symmetry properties like Yang-Baxter relations for these operators. At n = 2 we may change the homogeneous coordinates x i = (x i,1 , x i,2 ) to the normal coordinates x i = x i,1 x i,2 and find We define the integral operator with a 4-point YSC as kernel, In the case of the action on functions on the complex plane we consider the complex x as (x,x) and define We substitute the YSC at n = 2 in the normal coordinate form and perform the integrals over c 13 , c 14 , c 23 , c 24 .
We use the abbreviation x ij = x i − x j for the difference of normal coordinates (not to be mixed with the component notation In the case of the 4-point YSC (3.5) we obtain In the last step we have used the relation between the spectral parameters and the weight at each point i, 2ℓ i = u + σ(i) − u σ(i) , referring to the parameter permutation pattern (2.3). In our case (3.5) we have We introduce ε = u 3 − u 4 and express the exponents in terms of the independent weights 2ℓ 1 , 2ℓ 2 and the parameter ε, substituting also n = 2 for the considered case.
We substitute 1, 2, 3, 4, → 1 ′ , 2 ′ , 1, 2 . At ℓ 1 ′ = ℓ 2 ′ = ℓ the kernel for the complex plane is In the decomposition in ε the leading ε −1 term corresponds to the kernel of the permutation operator P 12 . The finite term is The subtraction is the appropriately modified + prescription. In the limiting case corresponding to the reggeized gluon exchange ℓ = 0 the action is defined on functions vanishing at coinciding arguments x 1 ′ = x 2 ′ . The kernel can be better rewritten as This is the dipole (or Moebius) form of the BFKL kernel [24,25,26,27].

The parton evolution kernels
In the helicity representation we have at n = 2, m = 1, where k i =λ i λ i have one component only in this case. In the application to parton evolution these variables k i have the physical meaning of light-cone momenta and the arguments z of the parton evolution kernels are defined as their ratios. We obtain for the YSC with N = 3 and K = 1 (3.2) or Here 2a i = 2ℓ 1 + 1 and 3 1 2a i = η, with η = ±1. The positive sign η = +1 corresponds to K = 1 (3.2) and the negative η = −1 to K = 2 (3.3 ). The amplitude of parton splitting 3 → 1 + 2 is obtained from this three-point YSC by substitutions of k i in terms of the momentum fraction z and a i by the parton helicities h i as follows The parton splitting probabilities are calculated as squares of the corresponding splitting amplitudes. The helicities h i refer to ingoing momenta, i.e. h 1 , h 2 are opposite to their physical values in the decay 3 → 1 + 2.
The expressions for the leading order parton evolution kernels are reproduced, compare e.g. [28].

Scattering amplitudes
We consider the four-point YSC at n = 4 with K = 2 in the helicity representation and do the integrals over the c variables, Here we denote (k i ) · α,α =λ i, · α λ i,α , [ij] =λ i,1λj,2 −λ i,2λj,1 . From (6.1) we obtain the explicit expressions as functions of the independent helicities and the extra parameter ε. As in sect. 4 we use the relation between the spectral parameters and the weights referring to the parameter permutation scheme u (2.3). The weights are substituted by the helicities as 2ℓ i + 2 = 2h i . In the case (3.5) we have We introduce ε = u 3 − u 4 and express the exponents in terms of the helicities h 1 , h 2 and the parameter ε.
We introduce ε = u 4 − u 3 and express the exponents in terms of the helicities h 1 , h 2 and the parameter ε. With Φ || we cover the remaining cases because here the helicities are related by In the function arguments we have abbreviated the points x i by the indices i. The result can be written in the form of a two-point correlator We substitute (3.2) and (3.3) substituting the variables correspondingly. The spectral parameters of the first factor are denoted by u 1 , u 2 ′ , u 1 ′ and the ones of the second factor by v 1 ′ , v 2 ′ , v 2 .
We calculate the weights in terms of the spectral parameters marking the weights of the second factor by a prime.
The finite term results in the contributions to the leading Gell-Mann -Low coefficient with the valuesh = 1 for the gluon loop contribution andh = 1 2 for the quark loop contribution.