Calculation of $c_\mathrm{SW}$ at one-loop order for Brillouin fermions

The Brillouin action is a Wilson-like lattice fermion action with a 81-point stencil, which was found to ameliorate the Wilson action in many respects. The Sheikholeslami-Wohlert coefficient $c_\mathrm{SW}$ of the clover improvement term has a perturbative expansion $c_\mathrm{SW}=c_\mathrm{SW}^{(0)}+g_0^2c_\mathrm{SW}^{(1)}+\mathcal{O}(g_0^4)$. At tree-level $c_\mathrm{SW}^{(0)}=r$ holds for Wilson and Brillouin fermions alike. We present the Feynman rules for the Brillouin action in lattice perturbation theory, and employ them to calculate the one-loop coefficient $c_\mathrm{SW}^{(1)}$ with plaquette or L\"uscher-Weisz gluons. Numerically its value is found to be about half that of the Wilson action.

Perturbative determinations of c SW for the Wilson fermion action in combination with various gluon actions have been carried out in Refs.[8,9,10,11,12].Our article follows Ref. [11] in methodology, except for a change in the regularization procedure.This is why we repeat the calculation for Wilson fermions (with both plaquette and Lüscher-Weisz glue), to demonstrate that we recover the known results.Our result for Brillouin fermions with plaquette glue and r = 1 was presented in Ref. [13].We now give results for a range of values of r ∈ [0.5, 1.5], also including the Lüscher-Weisz gluon action.
The remainder of this article is organized as follows.In section 2 we discuss the Feynman rules needed for the one-loop determination of c SW and how they are derived from the Brillouin action.In section 3 we use some of these Feynman rules to determine the part of the self energy Σ 0 which corresponds to the critical mass shift am crit .These calculations are carried out for plaquette gluons and Lüscher-Weisz (i.e.tree-level Symanzik-improved) gluons, as the latter are frequently used in numerical simulations.Section 4 finally outlines the calculation of the one-loop value of c SW and presents the results, while Section 5 gives a summary.

Feynman Rules
The basic idea of lattice perturbation theory is to expand the link variables U in the small coupling limit in terms of the gluon fields Inserting this series into the action yields the Feynman rules for vertices of n gluons coupling to a quark anti-quark pair at order g n 0 (contrary to continuum QCD perturbation theory where only the q qg-vertex exists).We shall see below that for the one-loop calculation of c SW we need Feynman rules up to order g 3 0 , which means the q qg, q qgg and q qggg-vertices shown in Figure 1.After extracting the relevant Feynman rules in position space, we Fourier transform the gluon and fermion fields according to where the identities (16) allow us to perform up to two integrations, thereby enforcing momentum conservation at the vertex (for more detail see Appendix A).To make the lengthy expressions for the Feynman rules more readable, we use the following shorthand notation for the trigonometric functions The q qgg and q qggg-vertices coming from the unimproved (c SW = 0) Brillouin action are given in Appendix A, in Eqs. ( 67) and (68), respectvely.They use functions which are combinations of sine and cosine functions, e.g.
The expression for the q qg-vertex from this part of the action takes the compact form Figure 1: Momentum assignments for the vertices with one, two and three gluons.
with the momentum assignments shown in Figure 1.In addition, each of the three vertices has a contribution coming from the clover term which is linear in c (0) SW .For the q qg-vertex it is and the analogous contributions to the q qgg-and q qggg-vertices are given in Appendix A.2.
The vertices given here are not symmetrised with respect to the incoming gluons.We do this at the level of the diagrams, where we sum over all possible connections of internal gluon lines.Furthermore, we need the propagator of the free Brillouin fermion with the Fourier transforms of the "free" derivative ∇ iso µ and Laplace operator ∆ bri whereupon the fermion propagator takes its final form For the gluonic part of the action we consider the standard plaquette action (c 0 = 1, c 1 = 0) as well as the tree-level Symanzik improved "Lüscher-Weisz" action (c 0 = 5  3 , where "plaq" stands for all 1 × 1 plaquettes, "rect" for all 1 × 2 rectangular loops and the coefficients fulfil c 0 + 8c 1 = 1.The gluon propagator G µν (k) and the three-gluon vertex  V abc g3 µνρ (k 1 , k 2 , k 3 ) for these actions in general covariant gauge are given in Appendix A.3 and were derived in Refs.[15,16].We follow the convention of inflowing momenta, see Figure 2, and use the Feynman gauge (α = 1) in our calculations.
We have used a computer algebra system (Mathematica [17]) to derive the Feynman rules and have additionally checked some of the calculations by hand.

Self-Energy Calculation
Let us apply the Feynman rules to the calculation of the self-energy of the fermion to first order.It is given by the "tadpole" and "sunset" diagrams shown in Figure 3, and yields the additive mass shift The contributing integrals for the two diagrams are These are simple and well known integrals in the Wilson/plaquette case (see for example Ref. [18]).Results for the Wilson/Symanzik case can be found in Ref. [12].and Brillouin fermions with plaquette glue but not any more identical with the Symanzikimproved Lüscher-Weisz gluon propagator.It also gains no additional terms from the inclusion of the clover term.The sunset contribution, however, does.It takes the form where the tree-level value of the improvement coefficient, c (0) SW = r, is to be used for a complete one-loop result.The coefficients Σ are given in Table 2. Figure 4 shows Σ 0 for values of r ranging from 0.5 to 1.5, along with quadratic fit functions for the reader's convenience.Further details are given in Appendix C. Overall we find that switching from Wilson to Brillouin fermions decreases |Σ 0 | ∝ am crit only marginally, while replacing plaquette gluons by Lüscher-Weisz gluons has a much more significant effect.

Perturbative Determination of c SW
The improvement coefficient c SW in Eq. ( 12) has a perturbative expansion in powers of the bare coupling g 2 0 = 2N c /β.It can be calculated via the quark-quark-gluonvertex function where L is the number of loops.

Tree Level
At the tree level the expression (33) is given by the lattice q qg-vertex which is the sum of (20) and 21.Expanding it to first order in a gives where SW at leading order in g 0 , and sandwiching this expression with on-shell spinors u and ū yields and hence the condition c (0) SW = r λ 1 + 6λ 2 + 12λ 3 + 8λ 4 to eliminate O(a) contributions at tree level.From Eqs. ( 9) and (11) we see that both actions fulfill λ 1 + 6λ 2 + 12λ 3 + 8λ 4 = 1.Thus the tree-level improvement coefficient is equal 2 to the Wilson parameter r for both Wilson and Brillouin fermions.

One-Loop Level: Setup
At the one-loop level the general form of the vertex function is3 where the F 2 , F 3 and H 1 terms do not contribute to on-shell quantities [11].
2 Some authors write rc SW instead of c SW in Eq. ( 12), such that c (0) SW always equals one.However, the one-loop value c (1) SW does not naturally factorise in this way, and has a complicated, non-polynomial r-dependence.
The six one-loop diagrams contributing to the vertex function.
At order g 3 0 Eq. ( 34) with c SW = c SW + . . .contributes a single term, involving c SW .Together with the one-loop vertex function (36) and going on-shell we get and this results in the condition to eliminate O(a) contributions at the one-loop level.We use the following equation (given in Ref. [11]) to extract G 1 from the off-shell vertex function The six diagrams that contribute at the one-loop level are shown in Figure 5.
We use Mathematica notebooks that take the previously generated Feynman rules as input and construct from them the relevant integrals corresponding to the six diagrams.Next, Eq. (39) gets applied to each integral, and the resulting expressions are added and numerically integrated over the four-dimensional Brillouin zone.For individual diagrams the integrals (with one exception) are divergent and need to be regularised, which is explained below.

One-loop level: Regularisation
The sum of all diagrams is finite and can be numerically integrated without further ado ("all in one approach").As a check and to compare results with other calculations it is very helpful to calculate the contribution from each diagram separately.Five of the six diagrams are logarithmically IR-divergent.Therefore we need to regularise these integrals and extract the divergent part before we can evaluate them numerically.To this end we subtract from each divergent lattice integral the simplest logarithmically divergent lattice integral multiplied with an appropriate pre-factor.The integral B 2 is regularised by a small fictitious gluon mass µ where γ E = 0.57721566490153(1) is the Euler-Mascheroni constant and F 0 a lattice constant given in Ref. [19] as F 0 = 4.369225233874758 (1).
Because in the sum of all diagrams the divergencies cancel, we can add arbitrary constants to B 2 .We like to compare our results to those by Aoki and Kuramashi in Ref. [11], who calculated c (1) SW for the Wilson action and for various improved gauge actions.They used a slightly different method of regularisation and encoded the divergence in In essence they used the following two divergent continuum-like integrals coming from the expansion of the lattice Feynman rules to O(a).The integral L 1 appears in diagrams (a),(e) and (f), whereas L 2 appears in diagrams (b) and (c).Hence the relations will convert our results to those in the formalism of Ref. [11] (see Table 3 and Table 4

below).
With this setup the calculation is straight-forward, though tedious.As mentioned before, we use Mathematica to carry out the algebraic manipulations [17] and the files are available as ancillary material.In case of diagram (c) the calculation can be carried out by hand, the essential steps being given in Appendix B.

One-loop level: Results
The numerical results for each diagram and their sum is given in Table 3 for Wilson and Brillouin fermions, with plaquette and Lüscher-Weisz glue.Table 4 contains the same results converted into the formalism of Ref. [11].Our numbers in the "Wilson" columns agree with (and improve on) the numbers given there.Throughout the "sum" (which results from adding the six regulated single-diagram contributions) agrees with the result of the direct numerical integration of the unregularized (finite) "all in one" approach.
SW from each diagram for N c = 3 and r = 1.The second column gives the coefficients in front of the logarithmically divergent L ≡ 1 16π 2 ln π 2 /µ 2 .This corresponds to the formalism used in Ref. [11] .
SW and the full result for N c = 3 for r = 1.
A closer look at Table 3

Summary
This paper provides Feynman rules for the Brillouin fermion action, and lists the vertices q qg, q qgg, q qggg that come from the unimproved part of the fermion action.In addition, each one of these vertices receives a contribution from the clover term which is proportional to c (0) SW and known from Refs.[11,12], since it is the same for Wilson and Brillouin fermions.
As a warm-up exercise we calculate the self-energy contribution Σ 0 ∝ 1/a which encodes the critical mass am crit of Brillouin fermions.We do this for the plaquette gluon action and the Lüscher-Weisz gluon action, with the Wilson parameter in the range r ∈ {0.5, 0.6, ..., 1.5}.We find that the critical mass nearly matches the one of Wilson fermions, and the tree-level improvement in the gluon sector reduces it (for both fermion actions) by about 25%.
Last but not least the well-known tree-level result c (0) SW = r is complemented with the oneloop result c (1) SW for Brillouin fermions.The sum of the six diagrams is finite and ready for numerical integration.On a diagram-by-diagram basis, five of the diagrams are IR-divergent and need regularization.We propose a slightly different regularization technique than Ref. [11], but the "translation rules" Eqs.(45, 46) establish complete agreement of our results for the Wilson action with those in the literature, except that it is easier to reach good numerical precision with our quantity B 2 as defined in Eqs.(40, 41).
The results are given in Section 4 and Appendix D for the range r ∈ [0.5, 1.5] of the Wilson lifting parameter r and arbitrary N c .These results are ready for use in numerical simulations, and will reduce the leading cut-off effects from O(a log(a)) to O(a log 2 (a)).Of course our hope is that these results will eventually be complemented with a non-perturbative determination of c SW for a variety of couplings.We envisage that future (non-perturbative) data for c SW of the Brillouin action will be fitted with a rational ansatz of the form c SW = (1 + c 1 g 2 0 + c 2 g 4 0 + ...)/(1 + d 1 g 2 0 + ...), where the difference c 1 − d 1 is restricted to obey our 1-loop perturbative calculation.This is exactly what was done for Wilson fermions in Refs.[5,6,7], and we hope that the improvement program for Brillouin fermions will be similarly successful.

A.1.1 The qqg-vertex
As a short example we show in some detail how to derive the part of the q qg-vertex V a 1 µ (p, q) proportional to λ 2 and ρ 2 , i.e. we start with the following term of the Brillouin action First we rewrite the sum, such that it is over positive indices only x,y where γ −µ = −γ µ has been used.We insert expanded to order g 0 , hence we replace where we have used Next we insert the Fourier transforms of ψ, ψ, A and δ.One example term reads x,y The sum over y introduces (2π) 4 δ(r − p), which lets us perform the r-integral.Then the sum over x introduces (2π) 4 δ(q − p − k), which enforces momentum conservation at the vertex and allows us to perform the k-integral.As a result we are left with and together with the term − − ρ 2 γ µ − r λ 2 4 A a µ (x − μ)δ(x − μ − ν, y) this yields the contribution to the q qg vertex.Adding all the other terms and some more trigonometric manipulations will finally result in what is shown in Eq. ( 20).
The procedure for the q qgg-and q qggg-vertices is basically the same.After Fourier transforming the fields, we integrate out the gluon momentum k 1 and find ourselves left with expressions containing p, q, k 2 and p, q, k 2 , k 3 respectively.At this point some diligence is needed to rename the indices in terms containing for example

A.1.3 The qqgg-vertex
The q qgg vertex for the Brillouin action takes the form and needs to be complemented with the piece linear in c (0) SW = r that comes from the clover term.

A.1.4 The qqggg-vertex
The q qggg vertex for the Brillouin action takes the form and needs to be complemented with the piece linear in c SW = r that comes from the clover term.

A.2 Feynman Rules of the Clover Term
Here we give the contributions to the qqgg-and qqggg-vertices coming from the clover term.Similar expressions can be found in Refs.[11,12].
A.3 Feynman Rules of the Gauge Action The following gluon propagator and three-gluon vertex have been derived in Refs.[15,16].The gluon propagator in a general covariant gauge is and we shall use the Feynman gauge with α = 1.The symmetric matrix A is given by with the denominator taking the form Similarly, the three-gluon vertex depicted in Figure 2 is found to be with V

B Example calculation of diagram (c)
The integral corresponding to diagram (c) is The pre-factor 6 = 2 • 3 is due to the possible permutations of the internal gluon lines.The simplest form of this integral is in the case of the Wilson fermion action and the plaquette gluon action.In order to compute G (c) 1 by Eq. (39), we need the derivatives Here we may use that under the integral only even functions will survive, therefore terms of the form s(k µ )s(k ν ) and s(k µ )s(k ν ) with µ ̸ = ν do not contribute.This yields = − 1 8 and this integral is divergent and cannot be directly calculated numerically.In order to regularize it we subtract the simple lattice integral where all we need to know is the appropriate coefficient c.We can determine it easily by restoring factors of the lattice spacing k → ak, expanding the first term to lowest order in a π −π and reading off c = 3/4.Thus the contribution to c

C Self-Energy as a function of r
To complement the results presented in Section 3, Table 6 shows our results for the self-energy Σ 0 /a for eleven values of the Wilson parameter r between 0.5 and 1.
SWBrillouin for N c = 3 from each diagram as a function of r.

Figure 3 :
Figure 3: Tadpole (left) and sunset (right) diagrams of the quark self energy.

Figure 4 :
Figure 4: Self energy Σ 0 of Wilson and Brillouin fermions as a function of r.

2 Figure 6 :
Figure 6: The one-loop values of c (1) SW for Wilson and Brillouin fermions with N c = 3 as a function of r.
= r = 1 and N c = 3 in the last step.

Table 1 :
Complete results for Wilson or Brillouin fermions with a clover term and plaquette or Symanzik glue are given in Table1.The tadpole contribution is exactly the same for Wilson Contributions to the self energy Σ 0 for r = 1 and N c = 3 coming from the tadpole and sunset diagrams, along with the sum.

Table 2 :
Contributions to the self energy Σ

Table 3 :
Divergent and constant contributions to c

Table 4 :
Divergent and constant contributions to c

Table 6 :
5. The four options of the overall action (Wilson/Brillouin fermions on plaquette/Lüscher-Weisz glue) are given with a split-up between the tadpole and sunset contributions and the combination of the two diagrams.Self energy Σ 0 of Wilson and Brillouin fermions for N c = 3 with plaquette and treelevel Symanzik improved ("Lüscher-Weisz") gauge action as a function of r.

Table 10 :
Contributions to c