1 Introduction

Abstract

Using plaquette and Symanzik improved gauge action and stout link clover fermions we determine the improvement coefficient in one-loop lattice perturbation theory from the off-shell quark-quark-gluon three-point function. In addition, we compute the coefficients needed for the most general form of quark field improvement and present the one-loop result for the critical hopping parameter . We discuss mean field improvement for and and the choice of the mean field coupling for the actions we have considered.

DESY 08-034

Edinburgh 2008/12

Leipzig LU-ITP 2008/001

Liverpool LTH 792

Perturbative determination of for plaquette and

[0.25em] Symanzik gauge action and stout link clover fermions

R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz and A. Schiller


School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK

Institut für Theoretische Physik, Universität Leipzig,

D-04109 Leipzig, Germany

Theoretical Physics Division, Department of Mathematical Sciences,

University of Liverpool, Liverpool L69 3BX, UK

Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany


QCDSF Collaboration

1 Introduction

Simulations of Wilson-type fermions at realistic quark masses require an improved action with good chiral properties and scaling behavior. A systematic improvement scheme that removes discretization errors order by order in the lattice spacing has been proposed by Symanzik [1] and developed for on-shell quantities in [2, 3]. improvement of the Wilson fermion action is achieved by complementing it with the so-called clover term [3], provided the associated clover coefficient is tuned properly.

Wilson-type fermions break all chiral symmetries. This introduces an additive negative mass renormalization term in the action, which gives rise to singularities in the quark propagator at small quark masses and makes the approach to the chiral regime difficult. A chiral improvement of the action is expected to reduce the additive mass renormalization and the spread of negative eigenvalues. Surprisingly, this is not accomplished by the clover action.

While the magnitude of the additive mass term decreases with increasing clover term, the problem of negative eigenvalues is more severe for the clover than for the standard Wilson action. It is well known that via a combination of link fattening and tuning of the clover coefficient, it is possible to reduce both the negative mass term and the spread of negative eigenvalues [4, 5, 6].

The focus of this investigation is to determine the clover coefficient and the additive mass renormalization for plaquette and Symanzik improved gauge action and stout link clover fermions in one-loop lattice perturbation theory.

The Symanzik improved gauge action reads [1]

(1)

with and

(2)

This reduces to the standard plaquette action for .

Clover fermions have the action for each quark flavor [3]

where

(4)

being the critical hopping parameter, is the additive mass renormalization term, and is the field strength tensor in clover form with . We consider a version of clover fermions in which we do not smear links in the clover term, but the link variables in the next neighbor terms have been replaced by (uniterated) stout links [7]

(5)

with

(6)

denotes the sum over all staples associated with the link and is a tunable weight factor. Stout smearing is preferred because (5) is expandable as a power series in , so we can use perturbation theory. Many other forms of smearing do not have this nice property. Because both the unit matrix and the terms are smeared, each link is still a projection operator in the Dirac spin index.

The reason for not smearing the clover term is that we want to keep the physical extent in lattice units of the fermion matrix small which is relevant for non-perturbative calculations. In that respect we refer to these fermions as SLiNC fermions, from the phrase Stout LinkNon-perturbative Clover.

The improvement coefficient as well as the additive mass renormalization are associated with the chiral limit. So we will carry out the calculations for massless quarks, which simplifies things, though it means that we cannot present values for the mass dependent corrections.

For complete improvement of the action there are five terms which would have to be added to the effective action, they are listed, for example, in [8]. Fortunately, in the massless case only two remain,

(7)
(8)

The first is the clover term, the second is the Wilson mass term. We have both in our action, there is no need to add any other terms to the action.

In perturbation theory

(9)

The one-loop coefficient has been computed for the plaquette action using twisted antiperiodic boundary conditions [9] and Schrödinger functional methods [10]. Moreover, using conventional perturbation theory, Aoki and Kuramashi [11] have computed for certain improved gauge actions. All calculations were performed for non-smeared links and limited to on-shell quantities.

We extend previous calculations of to include stout links. This is done by computing the one-loop correction to the off-shell quark-quark-gluon three-point function. The improvement of the action is not sufficient to remove discretization errors from Green functions. To achieve this, one must also improve the quark fields. The most general form consistent with BRST symmetry is [12]1

(10)

From now we denote improved quark fields and improved Green functions by an index . These are made free of effects by fixing the relevant improvement coefficients.

There is no a priori reason that the gauge variant contribution vanishes. The perturbative expansion of has to start with the one-loop contribution [12]. As a byproduct of our calculation we determine that coefficient

(11)

and find that it is indeed nonvanishing.

2 Off-shell improvement

It is known [11] that the one-loop contribution of the Sheikoleslami-Wohlert coefficient in conventional perturbation theory can be determined using the quark-quark-gluon vertex sandwiched between on-shell quark states. () denotes the incoming (outgoing) quark momentum. In general that vertex is an amputated three-point Green function.

Let us look at the expansion of tree-level which is derived from action (1)

(12)

For simplicity we omit in all three-point Green functions the common overall color matrix . That tree-level expression between on-shell quark states is free of order if the expansion of starts with one, as indicated in (9)

(13)

Therefore, at least a one-loop calculation of the is needed as necessary condition to determine .

The off-shell improvement condition states that the non-amputated improved quark-quark-gluon Green function has to be free of terms in one-loop accuracy. In position space that non-amputated improved quark-quark-gluon Green functions is defined via expectation values of improved quark fields and gauge fields as

(14)

Since the gluon propagator is -improved already, we do not need to improve gauge fields. Using relation (10) we can express the function by the unimproved quark fields

(15)

where is the unimproved Green function.

Taking into account

(16)

and setting (unless there is an unexpected symmetry breaking), we obtain the following relation between the improved and unimproved Green function

(17)

From (17) it is obvious that tuning only to its optimal value in , there would be an contribution left in the improved Green function. The requirement that should be free of terms leads to an additional condition which determines the constant . It has not been calculated before.

Taking into account the expansion (11) of we get in momentum space ( denotes the Fourier transform)

(18)

or its amputated version

(19)

The relation between non-amputated and amputated unimproved and improved three-point Green functions are defined by

(20)
(21)

denotes the full gluon propagator which is -improved already, and the corresponding quark propagators.

With the definition of the quark self energy

(22)

the unimproved and improved inverse quark propagators are given by

(23)
(24)

Using the Fourier transformed (17) with (19) and amputating the Green function (20), taking into account the inverse quark propagators (23), we get the off-shell improvement condition in momentum space

(25)

This expression should hold to order by determining both and correctly. It is clear from (25) that the improvement term does not contribute if both quarks are on-shell.

3 The one-loop lattice quark-quark-gluon vertex

The diagrams contributing to the amputated one-loop three-point function are shown in Fig. 1.

Figure 1: One-loop diagrams contributing to the amputated quark-quark-gluon vertex.

The calculation is performed with a mixture of symbolic and numerical techniques. For the symbolic computation we use a Mathematica package that we developed for one-loop calculations in lattice perturbation theory (for a more detailed description see  [14]). It is based on an algorithm of Kawai et al. [15]. The symbolic treatment has several advantages: one can extract the infrared singularities exactly and the results are given as functions of lattice integrals which can be determined with high precision. The disadvantage consists in very large expressions especially for the problem under consideration. In the symbolic method the divergences are isolated by differentiation with respect to external momenta.

Looking at the general analytic form of the gluon propagator for improved gauge actions [16] one easily recognizes that a huge analytic expression would arise. As discussed in  [16] we split the full gluon propagator

(26)

where is the covariant gauge parameter ( corresponds to the Feynman gauge). The diagrams with only contain the logarithmic parts and are treated with our Mathematica package. The diagrams with at least one are infrared finite and can be determined safely with numerical methods. The decomposition (26) means that we always need to calculate the plaquette action result, as part of the calculation for the improved gauge action. Therefore, we will give the results for both plaquette gauge action and Symanzik improved gauge action using the corresponding gluon propagators and , respectively.

Because the numerical part determines the accuracy of the total result we discuss it in more detail. There are several possibilities to combine the various contributions of the one-loop diagrams as given in Fig. 1. In view of a later analysis we have decided to group all coefficients in front of the independent color factors and and the powers of the stout parameter

(27)

where the have to be computed numerically. In order to obtain we first add all contributions of the diagrams shown in (1) and integrate afterwards. We have used a Gauss-Legendre integration algorithm in four dimensions (for a description of the method see [14]) and have chosen a sequence of small external momenta to perform an extrapolation to vanishing momenta.

Let us illustrate this by an example: the calculation of the coefficient . We know the general structure of the one-loop amputated three-point function as (we set )

(28)

From this we can extract the coefficients by the following projections

(29)

Relations (3) show that one has to compute the three-point function for all four values of . Further they suggest choosing the external momenta orthogonal to each other: . A simple choice is and .

We discuss the determination of and in more detail. For small momenta they can be described by the ansatz

(30)

The choice of the momenta is arbitrary except for two points. First, they should be sufficiently small in order to justify ansatz (30). Second, they should not be integer multiples of each other in order to avoid accidental symmetric results. The symmetry of the problem demands the relation which must result from the numerical integration also. Performing the integration at fixed and we obtain complex matrices for and from which the quantities and are extracted via (3).

A nonlinear regression fit with ansatz (30) gives

(31)

It shows that the symmetry is fulfilled up to an error of which sets one scale for the overall error of our numerical calculations. In Fig. 2 we

Figure 2: (circles) and (squares) as function of together with their corresponding linear fits in .

show the almost linear dependence of and on . (In the integration we haven chosen so that we can restrict the plot to one variable.)

Another source of errors is the numerical Gauss-Legendre integration routine itself. We have chosen a sequence of , , , and nodes in the four-dimensional hypercube and have performed an extrapolation to infinite nodes with an fit ansatz. Both procedures, Gauss-Legendre integration and the fit , give a combined final error of .

The third error source are the errors of the lattice integrals of our Mathematica calculation for the terms containing the plaquette propagator only. These integrals have been calculated up to a precision of . Therefore, their errors can be neglected in comparison with the others.

Summarizing we find that the error of our numerical procedure is of . Additionally, we have checked our results by an independent code which completely numerically computes the one-loop contributions for each diagram including the infrared logarithms. Both methods agree within errors.

The Feynman rules for non-smeared Symanzik gauge action have been summarized in [11]. For the stout smeared gauge links in the clover action the rules restricted to equal initial and final quark momenta are given in [6]. As mentioned in the introduction we perform a one-level smearing of the Wilson part in the clover action. The corresponding Feynman rules needed for the one-loop quark-quark-gluon vertex are much more complicated than those in [6]. The qqgg-vertex needed in diagrams (c) and (d) of Fig. 1 receives an additional antisymmetric piece. The qqggg-vertex in diagram (e) does not even exist in the forward case. The Feynman rules are given in Appendix A. The diagrams which are needed for the calculation of the quark propagator are shown in Fig. 3. We have performed our calculation in general covariant gauge.

Figure 3: One-loop diagrams contributing to the quark self energy.

4 Results for the improvement coefficients and critical hopping parameter

The anticipated general structure for the amputated three-point function in one-loop is

(32)

is the universal part of the three-point function, independent of the chosen gauge action, computed in the -scheme

(33)

We have calculated the complete expressions for and .

The contribution, , simplifies if we set as in (9). After some algebra we find

where is the mass scale (not to be confused with the index ). Therefore, we only need to present the one-loop result (33). is given in Appendix B.

If we insert (32) and (33) with (4) into the off-shell improvement relation (25) we get the following conditions that all terms of order have to vanish

(35)
(36)

with defined from (23) as

(37)

and

(38)

The constant depends on the chosen lattice action.

It should be noted that equations (35) and (36) are obtained by using the general structure (4) only – we do not need to insert the complete calculated result for . In order to get momentum independent and gauge invariant improvement coefficients we see from (35) that itself has to be constant and gauge invariant. From (36) and (38) we further conclude that the -terms from and have to cancel those from . The same is true for the corresponding gauge terms. The terms () coming from (37) are canceled by the corresponding terms in (4).

Therefore, the relation between and as given in (4) is a nontrivial result. Once more, it should be emphasized that this relation only holds if we use at leading order in . If (4) were not true, we would not be able to improve the Green functions by adding the simple terms we have considered.

For completeness we also give the corresponding one-loop values for the quark field improvement coefficient as defined in (10). They can be derived from the improvement of the quark propagator. The one-loop improvement coefficient is related to the quark self energy by

(39)

has been calculated for ordinary clover fermions and plaquette gauge action in [13].

Now we present our numerical results for general covariant gauge and as function of the stout parameter . For the plaquette action with stout smearing the quantities , and are obtained as

For the stout smeared Symanzik action we get

As shown in (25) (or equivalently (36)) we need the self energy parts and as defined in (23) to solve the off-shell improvement condition. They have the general form

(42)

For the plaquette and Symanzik actions we obtain

(43)

This results in the following expressions for as defined in (38)

Inserting the corresponding numbers into (35), (36) and (39), we obtain the one-loop contributions of the clover improvement coefficient

(45)
(46)

the off-shell quark field improvement coefficient

(47)
(48)

and the on-shell quark field improvement coefficient

(49)
(50)

for the plaquette and Symanzik action, respectively. For both the plaquette result (45) and the Symanzik result (46) agree, within the accuracy of our calculations, with the numbers quoted in [9, 10] and [11].

From Ward identity considerations it is known that the coefficient has to be proportional to only. Additionally, and should be gauge invariant. Both conditions are fulfilled within the errors which have been discussed in the previous section. It should be noted that (47) and (48) are the first one-loop results for the quark field improvement coefficient . The gauge dependent improvement coefficient depends only on the color factor because it is determined by improvement of the quark propagator.

The additive mass renormalization is given by

(51)

This leads to the critical hopping parameter , at which chiral symmetry is approximately restored,

(52)

Using the plaquette or Symanzik gauge actions, we obtain

(53)
(54)

This leads to the perturbative expression for