How to Compute One-Loop Feynman Diagrams in Lattice QCD with Wilson

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How to compute one-loop Feynman diagrams in lattice QCD with Wilson fermions Giuseppe Burgioa, Sergio Caracciolob,∗ and Andrea Pelissettoc aDipartimento di Fisica and INFN, Università degli Studi di Parma, Parma 43100, ITALIA bDipartimento di Fisica and INFN, Università degli Studi di Lecce, Lecce 73100, ITALIA cDipartimento di Fisica and INFN, Università degli Studi di Pisa, Pisa 56100, ITALIA

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilson- fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of L¨uscher and Weisz to higher-loop integrals with fermions.

In [1] we presented a general technique which four dimensions the technique can be generalized allows to express every one-loop bosonic integral to every space dimension. at zero external momentum in terms of two un- Define known basic quantities which could be computed numerically with high precision. This method al- Fδ(p, q; nx,ny,nz,nt)= 2ny 2n lows the complete evaluation of every diagram π d4k kˆ2nx kˆ kˆ2nz kˆ t x y z t (1) with gluon propagators. 4 p+δ q −π (2π) D (k, mf )D (k, mb) We have recently generalized [2] this technique Z F B to deal with integrals with both gluonic and where ni are positive integers, p and q are ar- fermionic propagators. For the fermions we use bitrary integers (not necessarily positive), kˆµ = the Wilson action [3]. Notice that our method 2sin(kµ/2) depends only on the structure of the propagator ˆ2 2 ˆ2 2 and thus it can be applied in calculations with DB(k, mb)=k+mb= ki+mb;(2) i the standard Wilson action as well as with the X improved clover action [4]. We show that every r2 D(k, m )= sin2 k + W (kˆ2)2 + m2 (3) integral at zero external momentum can be ex- F f i 4 f i pressed in terms of a small number of basic quan- X tities (nine for purely fermionic integrals, fifteen is the denominator appearing in the propagator for integrals with bosonic and fermionic propaga- for Wilson fermions2. In the following when one tors). The advantage of this procedure is twofold: 2To be precise, D (k, m ) is the denominator in the prop- first of all every Feynman diagram can be com- F f agator for Wilson fermions only for mf =0.Formf 6=0 puted in a completely symbolic way making it the correct denominator would be

easier to perform checks and verify cancellations; 2 2 rW 2 Dˆ (k, m )= sin ki + kˆ + m . (4) moreover the basic constants can be easily com- F f 2 f puted with high precision and thus the numer- i X ical error on the ﬁnal result can be reduced at However in our discussion mf will only play the role of an will. Although the presentation is restricted to infrared regulator and thus it does not need to be the true fermion mass. The deﬁnition (3) is easier to handle than ∗Speaker at the conference. (4). 2

of the arguments ni is zero it will be omitted as In all other cases we use an integration by parts. an argument of F. The parameter δ is used in It is here that we found convenient to have a non- the intermediate steps of the calculation and will vanishing δ, otherwise integration by parts in the be set to zero at the end. case in which there is only one fermionic propa- To simplify the discussion we have only consid- gator would produce a log DF . ered the case rW = 1 but the technique can be Asaresultwehavethat applied to every value of r .Moreoverwehave W F(p, q; n ,n ,n ,n )= restricted our attention to the massless case, i.e. δ x y z t p q+k we have considered the integrals Fδ in the limit ars(m, δ)Fδ(r, s)(9) mb = mf ≡ m → 0. r=p k+1 s=q k We have developed a procedure that allows to X− X− compute iteratively a generic F,whichisFδ in where k =(nx+ny+nz+nt), and ars(m, δ)isa the limit δ → 0, in terms of a finite number of polynomial in m2. them: precisely every F(p, q; nx,ny,nz,nt)with At the second step we express every Fδ (p, q)in p>0andq≤0 (purely fermionic integrals) terms of a finite number of them. We start by can be expressed in terms of F(1, 0), F(1, −1), inserting the trivial identity F(1, −2), F(2, 0), F(2, −1), F(2, −2), F(3, −2), kˆ2+m2 F(3, −3) and F(3, −4); the integral F(2, 0) ap- 0=1− i (10) D (k, m) pears only in infrared-divergent integrals and we PB write it as into the integral Fδ(p, q;1,1,1,1) to get 1 2 F(2, 0) = − log m + γ − F + Y (5) 0=I1(p, q) ≡ 16π2 E 0 0 F (p, q;1,1,1,1) − 4F (p, q +1;2,1,1,1) δ δ where Y0 is a numerical constant. If q>0the 2 result contains three additional constants which −m Fδ(p, q +1;1,1,1,1) (11) we have chosen to be Using the previous recursions we obtain from (11) 1 a non trivial relation among the Fδ(r, s). Start- Y = F(1, 1; 1, 1, 1) 1 8 ing from the analogous relation obtained from the 1 fermionic propagator we get a new set of identi- Y2 = F(1, 1; 1, 1, 1, 1) 16 ties I2(p, q) which can be used to provide addi- 1 tional relations between the remaining integrals. Y = F(1, 2; 1, 1, 1) (6) 3 16 We end up with the result we have quoted above. together with the bosonic quantities Z , Z and In order to numerically evaluate the eight ba- 0 1 sic purely-fermionic integrals we considered the F − γ . 0 E integrals As a first step in our procedure we express each integral Fδ(p, q; nx,ny,nz,nt)intermsofinte- Jq = F(1, −q)6≤q≤13 (12) grals of the form F (r, s) only. This is achieved δ which can be analytically expressed in terms of by the use of the identity the elements in the fermionic basis. We computed 4 q ˆ2 2 1 DB(k, 0) ki = DB (k, m) − m (7) Jq,L = 4 (13) L DF (k, 0) i=1 k X X in the case one of the ni’s is 1. In the case one of where k runs over the points k =(2π/L)(n1 + the ni’s is 2 we use instead 1 1 1 1 2 ,n2 + 2,n3 + 2,n4 + 2), 0 ≤ ni

The inversion of this set of equations provides a F(1, 0) 0.08539036359532067913516702879 very accurate estimate of the elements in the ba- 0.08539036359532067913516702676 F(1, −1) 0.46936331002699614475347539703 sis (see Table 1). Analogously to compute Y0, Y1, 0.46936331002699614475347539758 Y2 and Y3, we computed numerically F(1, 1, 8), (1, 2) 3.39456907367713000586008689687 F(2, 1, 9), F(3, 1, 10) and F(5, 2, 11) and then F − 3.39456907367713000586008689071 solved the corresponding equations. F(2, −1) 0.05188019503901136636490228763 A second important application of our method 0.05188019503901136636490228706 is connected with the use of coordinate-space F(2, −2) 0.23874773756341478520233613924 methods for the evaluation of higher-loop Feyn- 0.23874773756341478520233613767 man diagrams. This technique, introduced by F(3, −2) 0.03447644143803223145396188143 Lüscher and Weisz [5], is extremely powerful and 0.03447644143803223145396188128 allows a very precise determination of two- and F(3, −3) 0.13202727122781293085314731096 higher-loop integrals. One of the basic ingredi- 0.13202727122781293085314731035 ents of this method is the computation of the free F(3, −4) 0.75167199030295682253543148585 propagator in coordinate space. But this can be 0.75167199030295682253543148379 achieved by the use of our relations. Consider Table 1 dk eikx Numerical values of the constants appearing in G(p, q; x)= (15) 4 p q the fermionic integrals evaluated by the two dif- (2π) DF (k, m)DB(k, m) Z ferent methods described. The first line refers to dk cos kµxµ = µ (16) the method which goes through the evaluation (2π)4 Dp (k, m)Dq (k, m) Z F Q B of the integrals Jq, the second one goes through the propagators in coordinate space. In all cases Then express cos(k x ) as a polynomial in kˆ2 .It µ µ µ there is an agreement of at least 27 digits. follows that G is a linear combination of the F and therefore they can be expressed on our basis. Here, like in the bosonic case, the expression of G in terms of the basis become numerically un- cise estimates of G(1,q;x)atthesetofpoints stable for |x|→∞: numerical errors in the basis X(n). Evaluating the integrals by computing dis- become amplified! This problem has a standard crete sums on finite lattices, as before, with sizes way out: if the expressions are unstable going out- L = 50 – 100, we obtain the values of G(1, −3; x) ward from the origin, they will be stable in the at the points X(26) and from these numbers we opposite direction: thus, if we want to compute obtained an independent determination of the fi- the propagator for |x|