Two-Loop Fermionic Integrals in Perturbation Theory on a Lattice

Firstly, they are needed to determine the Λ LAT parameter of QCD in the lattice regularization and its relation to the respective value Λ QCD in the continuum theory.

Secondly, every lattice action defines a specific regularization scheme, and thus one needs a complete set of renormalization computations in order for the results obtained in Monte Carlo simulations be understood properly. Perturbation theory is required to establish the connection of the matrix elements computed on a lattice with their values in the continuum theory [2], [3].

In this connection, it should be emphasized that the use of one-loop perturbative renormalization constants gives rise to large systematic uncertainties in lattice calculations of the momenta of hadronic structure functions [3] and respective two-loop computations are needed.

Thirdly, perturbative calculations provide the only possibility for an analytical control over the continuum limit in QCD. One can also mention anomalies, proof of renormalizability, Symanzik improvement program and other fields of application of lattice perturbation theory.

Here we consider one-and two-loop diagrams with Wilson (r = 1) fermions at zero external momenta [4]. We outline the Burgio-Caracciolo-Pelissetto (BCP) method [5] of calculations of one-loop integrals and describe the respective computer algorithm [6]. This algorithm allows to compute the fermionic propagator in the coordinate representation and, therefore, to extend the Lüscher-Weisz (LW) method [7] to the fermionic case; such extension is presented in Section 4.

We use the following designations: ñ stands for the set n 1 , n 2 , n 3 , n 4 ; x = (x 1 , x 2 , x 3 , x 4 ), where x µ are integer-valued coordinates of an infinite four-dimensional lattice Λ = {x : x µ ∈ Z Z}; we also need the lattice Λ ′ = Λ{0} with removed site x=(0,0,0,0);

Then we give the expressions for the denominators of bosonic and fermionic propagators,

where µ R is the fictitious mass for infrared regularization. We also use

These propagators in the coordinate representation are defined as follows:

where BZ is the Brillouin zone, BZ = p : -

a is the lattice size. In this work we set a = 1 for the sake of simplicity.

2 The Burgio-Caracciolo-Pelissetto method

The integrals under study are defined as follows: F (q; ñ) = lim δ→0 F δ (q; ñ), where

Here δ is an infinitesimal parameter for an intermediate regularization [5]. This parameter makes it possible to derive 1 the recursion relations of the form F (q; …, n µ , …) = F (q; …, n µ -2, …) -

With these relations and similar relations for n µ ≤ 1, one can express the integrals (5) in terms of the quantities

Up to terms of the order O(µ 2 R ) and O(δ), this expression has the form

where A (-) qr (δ, ñ) have a pole singularity in δ, and

As for the function G δ (r, µ 2 R ), the domains r > 0 and r ≤ 0 should be considered separately. At r > 0, δ can be safely set to zero and the function G δ (r, µ 2 R ) should be expanded in powers of µ -2 R :

where b n are the coefficients of the asymptotic expansion at z → ∞ of the function andC is the Euler-Mascheroni constant. At r < 0, µ R can be safely set to zero and the function G δ (r, 0) should be expanded in δ as follows:

The functions J(q), in their turn, obey recursion relations of the type c 0 (q)J(q) + c 1 (q)J(q + 1) + c 2 (q)J(q + 2) + c 3 (q)J(q + 3) + c 4 (q)J(q + 4) = 0 (11) derived in [5]; the explicit expressions for the coefficients c i (q) can be found in [6]. Thus we express J(q) at q ≥ 4 and at q ≤ 0 in terms of J(0), J(1), J(2) and J(3). It should be noted that J(0) does not appear in ultimate expressions for the integrals (5). Then one can introduce the values

which are equal to [1] Z 0 ≈ 0.15493339023, Z 1 ≈ 0.10778131354, F 0 ≈ 4.369225233874758.

1 Using integration by parts 2 I 0 (z) is the Infeld function.

In the fermionic case, we consider the quantities F (p, q; ñ) = lim δ→0 F δ (p, q; ñ), where

With the recursion relations similar to (6), these integrals are expressed in terms of the functions

, which can be represented in the form

The divergent parts D(p, q; µ 2 R ) and L(p, q; µ 2 R ) in the domain of interest can be determined by a straightforward procedure [5], whereas the functions B(p, q) and J(p, q) obey recursion relations of several types. These relations and the procedure of their derivation were described in [5]; their explicit form (very cumbersome) is given in [6]. With the use of these relations, the functions F (p, q; ñ) can be represented (see [1], [5]) as linear combinations of the constants F 0 , Z 0 , Z 1 and

;

The respective codes can be found on the web page of the ITEP Lattice group http://www.lattice.itep.ru/ ∼pbaivid/lattpt/. The results stored there are as follows: (i) the program for a computation of F (p, q; ñ) at 0 ≤ p, q ≤ 9 and n 1 + n 2 + n 3 + n 4 ≤ 25; (ii) t

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