ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 5, с. 871-878
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
ANOMALOUS DIMENSIONS OF THE WILSON OPERATORS IN N = 4 SUPERSYMMETRIC YANG-MILLS THEORY
© 2009 A. V. Kotikov1), L. N. Lipatov2)
Received October 14, 2008
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.
PACS: 12.38.Bx
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 "tilde" is used for spin-dependent case and as = as/(4n))
Yab(j) = / dxxj 1Wb^a(x) =
(1)
Ydb)(j)as + la* (ja + tb(jИ + O(a4),
о
V
(2),
1
Yab(j) = I dxxj 1Wb^a(x) =
Yab (j)as + ti> (j)a2 + fab) (jК + O(a4)
(1)
s
(2)
dy У
Y,Wb^a(x/y)fb(y,Q2 ),
1)1 Joint Institute for Nuclear Research, Dubna, Russia.
2)Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina; and Institut für Theoretische Physik, Universityt Hamburg, Germany.
3)In the spin-dependent case a = X,g.
d ln Q
:fa(x,Q2) =
dJLE Wb-
У V
-(x/y)fb(y, Q2).
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 = X,g, and fi for the spinor, vector, and scalar particles, respectively3)) as follows:
The anomalous dimensions and splitting kernels in QCD are known up to the next-to-next-to-leading order (NNLO = N2LO) 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, Tfnf = 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) [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 eigenvalues of the anomalous dimensions which was checked by direct calculations in [4]. Its properties will be reviewed below only shortly 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 N2LO approximation [8]. Moreover, the method of [3] together with long-range asymptotic Bethe ansatz [9, 10] allows
1
d
x
1
о
1
x
to predict the next-to-next-to-next-to-leading order (NNNLO = N3LO) anomalous dimension in [11].
The obtained results are very important for the verification of the various assumptions [12—15] coming from the investigations of the properties of conformal operators in the context of AdS/CFT correspondence [16].
2. EVOLUTION EQUATION IN n = 4 SYM
The possibility 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) [17], that the so-called quasi-partonic operators in n = 1 SYM are unified in supermultiplets with anomalous dimensions obtained from the universal anomalous dimension 7un¡ (j) with shifting its argument by an integer number. Further, the anomalous-dimension matrices for twist-2 operators are fixed by the superconformal invariance [17]. 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 superconformal invariance and its universal anomalous dimension in LLA is proportional to Si(j - 2) = ^(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 ^^ are equivalent to the Schrodinger equation for an integrable Heisenberg spin model [18, 19]. In QCD the integrability remains only in a small sector of these operators [20]. In the case of n = 4 SYM the equations for other sets of operators are also integrable [21-24].
Similar results related to the integrability of the multicolor QCD were obtained earlier in the Regge limit [25]. 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 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
7uni(j) which can be obtained also from the BFKL equation [3]. The results for 7uni(j) were checked by direct calculations in [4].
3. LLA ANOMALOUS-DIMENSION MATRIX IN n = 4 SUSY
In the n = 4 SYM theory [26] one can introduce the following color and SU(4) singlet local Wilson twist-2 operators [3, 4]:
G<pvj, (3)
1,...,jj = SGPH1 Dj2 Dj3
1 ,...,jj = SGaji Dj2Dj3 • • • Djj-i Gajj ,
Ox
Dj Xa *,
sxiYji vj2
= s~Ki5YvV,2 xai,
oti= StfV^v,2 ...vj
where v^ are covariant derivatives. The spinors Xi and field tensor GPj describe gluinos and gluons, respectively, and are the complex scalar fields. For all operators in Eq. (3) the symmetrization of the tensors in the Lorentz indices ¡i\,... 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-dimension matrix can be expressed through one universal anomalous dimension Yuni (j) with a shifted argument.
Indeed, the elements of the LLA anomalous-dimension matrix in the n = 4 SUSY have the following form (see [19]):
for tensor twist-2 operators,
(4)
+
1
1
j + 1 j + 2
1
Y$¡¡ (j) = 12
7£)Ü) = 4(-51Ü-D + 7-771)'
(j) = — 4Si (j),
yS (j) =
j+1
y,
$ (j) = 4
j - 1 j
for the pseudo-tensor operators,
tJ0')= 4 + f)
6
1
1
1
2
^> = ^7 + 771
(j)
= 2
j + 1 1
i+T
After transform of the above Wilson operators (3) to ones with a multiplicative renormalization (i.e., after diagonalization), we have the following diagonal anomalous dimensions:
7i0) (j) = -4Si(j T 2), fo'ij ) = -4Si(j )
7i0)(j) = -S(j T 1). This procedure is considered in detail in [3].
(o),
4. METHOD OF OBTAINING THE EIGENVALUES OF THE ANOMALOUS-DIMENSION MATRIX IN n = 4 SYM
As it was already pointed out in 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 the two first orders of perturbation theory [3]. Further, in the framework of the DR scheme [27] one can obtain from the BFKL equation (see [7]) that there is no mixing among the special functions of different transcendentality levels i4), i.e., all special functions at the NLO correction contain only sums of the terms -1/j% (i = 3). More precisely, if we introduce the transcendentality level i for the eigenvalues u(y) of integral kernels of the BFKL equations in accordance with the complexity of the terms in the corresponding sums
V - 1/Y, V - 0 - Z(2) - 1/y2 ,
V - 0" - Z(3) - 1/y3,
then for the BFKL kernel in the leading order (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^i (j), Y^i(j), and Y^i(j) are assumed to be of the types -1/jz 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 same for all LO and NLO anomalous dimensions in QCD (see [3] and discussions therein) and for the LO and NLO scalar—scalar anomalous dimensions [4]. This property allows one to find the universal anomalous dimensions Y^i (j) and Y^i (j) without knowing all elements of the anomalous-dimension matrix [3], which was verified by the exact calculations in [4].
Using above arguments, we conclude that at the NNLO level there is only one possible candidate
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