ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
THE PROPERTY OF MAXIMAL TRANSCENDENTALITY OF ANOMALOUS DIMENSIONS OF THE WILSON OPERATORS
IN THE N = 4 SYM
©2011 A. V. Kotikov*
Joint Institute for Nuclear Research, Dubna, Russia Received November 22,2010
We present results for the universal anomalous dimension 7uni(j) of Wilson twist-2 operators in the N = 4 Supersymmetric Yang—Mills theory in the first four orders of perturbation theory. These expressions are obtained by extracting the most complicated contributions from the corresponding anomalous dimensions in QCD.
1. INTRODUCTION
The anomalous dimensions of the twist-2 Wilson operators govern the Bjorken scaling violation for parton distributions in a framework of Quantum Chromodynamics (QCD). These quantities are given by the Mellin transformation (the simbol is used for spin-dependent case and as = as/(4n))
Yab(j) = / dxxj Wb^a(x) =
(1)
Yab(j )as + tb' (j К + tb(j К + O(a%
J1)/
(2),
Yab(j) = I dxxj 1Wb^a(x) =
= (j )as + tl> (j )a2 + tf (j )a3 + O(a4)
of the splitting kernels Wb^a(x) and Wb^a(x) for the Dokshitzer— Gribov—Lipatov—Altarelli—Parisi (DGLAP) equation [1] which evolves the parton densities fa(x,Q2) and fa(x,Q2) (hereafter a = = for the spinor, vector, and scalar particles,
respectively1)) as follows:
(1)
(2)
(2)
dy
YJWb^a(x/y)fb(y,Q2 ),
E-mail: kotikov@theor.jinr.ru
!)In the spin-dependent case a = X,g.
d ln Q
:fa(x,Q2) =
y V
i(x/y)fb(y,Q2).
The anomalous dimensions and splitting kernels in QCD are known up to the next-to-next-to-leading order (NNLO) of the perturbation theory (see [2] and references therein).
The QCD expressions for anomalous dimensions can be transformed to the case of the N-extended Supersymmetric Yang—Mills theories (SYM) if one will use for the Casimir operators CA, CF, Tf the following values Ca = Cf = Nc, Tf nf = NNc/2. For N = 2 and N = 4-extended SYM the anomalous dimensions of the Wilson operators get also additional contributions coming from scalar particles [3]. These anomalous dimensions were calculated in the next-to-leading order (NLO) [3, 4] for the N = 4 SYM.
However, it turns out that the expressions for eigenvalues of the anomalous dimension matrix in the N = 4 SYM can be derived directly from the QCD anomalous dimensions without tedious calculations by using a number of plausible arguments. The method elaborated in [3] for this purpose is based on special properties of the integral kernel for the Balitsky—Fadin—Kuraev—Lipatov (BFKL) equation [5—7] in this model and a new relation between the BFKL and DGLAP equations (see [3]). In the NLO approximation this method gives the correct results for anomalous dimensions' eigenvalues, which was checked by direct calculations in [4]. Its properties will be reviewed below only briefly and a more extended discussion can be found in [3]. Using the results for the NNLO corrections to anomalous dimensions in QCD [2] and the method of [3] we derive the eigenvalues of the anomalous dimension matrix for the N = 4 SYM in the NNLO approximation [8].
9 ЯДЕРНАЯ ФИЗИКА том 74 № 6 2011
1
d
x
1
О
1
О
1
y
x
Starting from four loops, i.e. above existing QCD calculations, the corresponding results for the anomalous dimensions can be obtained (see [9—11]) from the long-range asymptotic Bethe equations together with some additional terms, so-called wrapping corrections, coming in agreement with Luscher approach2).
The obtained result is very important for the verification of the various assumptions [13—17] coming from the investigations of the properties of conformal operators in the context of AdS/CFT correspondence [18].
2. EVOLUTION EQUATION IN N = 4 SYM
The reason to investigate the BFKL and DGLAP equations in the case of supersymmetric theories is related to a common belief that the high symmetry may significantly simplify their structure. Indeed, it was found in the leading logarithmic approximation (LLA) [19], that the so-called quasi-partonic operators in N =1 SYM are unified in supermultiplets with anomalous dimensions obtained from the universal anomalous dimension Yuni(j) by shifting its argument by an integer number. Further, the anomalous dimension matrices for twist-2 operators are fixed by the superconformal invariance [19]. Calculations in the maximally extended N = 4 SYM, where the coupling constant is not renormalized, give even more remarkable results. Namely, it turns out that here all twist-2 operators enter in the same multiplet, their anomalous dimension matrix is fixed completely by the super-conformal invariance and its universal anomalous dimension in LLA is proportional to ^(j — — 1) — ^(1) (see the following section), which means that the evolution equations for the matrix elements of quasi-partonic operators in the multicolor limit Nc ^ to are equivalent to the Schrodinger equation for an integrable Heisenberg spin model [20, 21]. In QCD the integrability remains only in a small sector of these operators [22] (see also [23]). In the case of N = 4 SYM the equations for other sets of operators are also integrable [14, 24, 25].
Similar results related to the integrability of the multicolor QCD were obtained earlier in the Regge limit [26]. Moreover, it was shown [7] that in the N = 4 SYM there is a deep relation between the BFKL and DGLAP evolution equations. Namely, the j-plane singularities of anomalous dimensions of the Wilson twist-2 operators in this case can be obtained
2)The three- and four-loop results for the universal anomalous dimension have been reproduced (see [12]) also by solution of so-called Baxter equation, which can be obtained from the long-range asymptotic Bethe equations.
from the eigenvalues of the BFKL kernel by their analytic continuation. The NLO calculations in N = 4 SYM demonstrated [3], that some of these relations are valid also in higher orders of perturbation theory. In particular, the BFKL equation has the property of the Hermitian separability, the linear combinations of the multiplicatively renormalized operators do not depend on the coupling constant, the eigenvalues of the anomalous dimension matrix are expressed in terms of the universal function Yuni(j) which can be obtained also from the BFKL equation [3].
3. LLA ANOMALOUS DIMENSION MATRIX IN N = 4 SUSY
In the N = 4 SYM theory [27] one can introduce the following color and SU(4) singlet local Wilson twist-2 operators [3, 4]:
O9 — QGa T) T) — p^i . . . T) rm Djj-1 Gpjj, (3)
Q 9 — Dj2 ... T) G a Djj-1 Gpjj, (4)
ox ...,jj — Dj2 . ..Dj Xai, (5)
q\ j — SX aY5 Ym Dj2 ...Dj Xai, (6)
Q<P ...jj = S<Xa-D^1 Dj2 ...Dj tf, (7)
where V^ are covariant derivatives. The spinors Xi and field tensor GPI1 describe gluinos and gluons, respectively, and are the complex scalar fields. For all operators in Eqs. (3)—(7) the symmetrization of the tensors in the Lorentz indices and
a subtraction of their traces is assumed. Due to the fact that all twist-2 operators belong to the same supermultiplet, the eigenvalues of anomalous dimensions matrix can be expressed through one universal anomalous dimension Yuni(j) with shifted argument.
The elements of the LLA anomalous dimension matrix in the N = 4 SUSY have the following form (see [21]):
for tensor twist-2 operators
Yg (j)= 4(^(1) - - 1) -
2 1 1 ---1----
j j + 1 j + 2
(8)
Yg) (j) = 12
J 1
YgxÜ) = 2 (j
j+1 j+2
2 2 1
. —T - - + -—-
j - 1 j j +1
7$Ü) = 7»
8 j'
j + i
j + i
7$ (j) = 4(m - + i))
7
$ j)=4
i
i
j-i j
for the pseudo-tensor operators:
2
yXü) = 4(^(1) + + -
+ -
(9)
j + 1
DrD
-1
N=4
unpol
-4Si(j - 2) 0 0 -4S1(j)
0
0
0 -4Si(j + 2)
N=4
DrD
-1
pol
-4Si(j - 1) 0
0 -4Si(j + 1)
Thus, the leading order (LO) anomalous dimension of all multiplicatively renornalized operators can be extracted through one universal function
7Ä (j) = -4S(j - 2) =
= -4[*(j - 1) - *(1)) = -4^
j-2
4. METHOD OF OBTAINING THE EIGENVALUES OF THE ANOMALOUS DIMENSION MATRIX IN N = 4 SYM
As it was already pointed out in the Introduction, the universal anomalous dimension can be extracted directly from the QCD results without finding the scalar particle contribution. This possibility is based on the deep relation between the DGLAP and BFKL dynamics in the N = 4 SYM [3, 7].
To begin with, the eigenvalues of the BFKL kernel turn out to be analytic functions of the conformal spin \n\ at least in two first orders of perturbation theory [3]. Further, in the framework of the DR scheme [28] one can obtain from the BFKL equation (see [7]), that there is no mixing among the special functions of different transcendentality levels i3), i.e. all special functions at the NLO correction contain only sums of the terms -1/j1 (i = 3). More precisely, if we introduce the transcendentality level i for the eigenvalues u(j) of integral kernels of the BFKL equations in accordance with the complexity of the terms in the corresponding sums
V - 1/j, V - 0 - Z(2) - 1/y2 ,
The matrices, based on the anomalous dimensions (8) and (9), can be diagonalized [3, 21]. They have the following remarkable form:
•ß" - z(3) -1/73,
r=l
then for the BFKL kernel in the LO and in NLO the corresponding levels are i = 1 and i = 3, respectively.
Because in N = 4 SYM there is a relation between the BFKL and DGLAP equations (see [3, 7]), the similar properties should be valid for the anomalous dimensions themselves, i.e. the basic functions
Y^(j), Jm 01 and yS(j) are assumed to be of
the types -1/j1 with the levels i = 1, i = 3, and i = = 5, respectively. An exception could be for the terms appearing at a given order from previous orders of the perturbation theory. Such contributions could be generated and/or removed by an approximate finite renormalization of the coupling constant. But these terms do not appear in the DR scheme.
It is known, that at the LO and NLO approximations (with the SUSY relation for the QCD color factors CF = CA = Nc) the most complicated contributions (with i = 1 and i = 3, respectively) are the
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