###### Abstract

The consideration of quantum fields defined on a spacetime lattice provides computational techniques which are invaluable for studying gauge theories nonperturbatively from first principles.

Perturbation theory is an essential aspect of computations on the lattice, especially for investigating the behavior of lattice theories near the continuum limit. Particularly important is its role in connecting the outcome of Monte Carlo simulations to continuum physical results. For these matchings the calculation of the renormalization factors of lattice matrix elements is required.

In this review we explain the main methods and techniques of lattice perturbation theory, focusing on the cases of Wilson and Ginsparg-Wilson fermions. We will illustrate, among other topics, the peculiarities of perturbative techniques on the lattice, the use of computer codes for the analytic calculations and the computation of lattice integrals. Discussed are also methods for the computation of 1-loop integrals with very high precision.

The review presents in a pedagogical fashion also some of the recent developments in this kind of calculations. The coordinate method of Lüscher and Weisz is explained in detail. Also discussed are the novelties that Ginsparg-Wilson fermions have brought from the point of view of perturbation theory.

Particular emphasis is given throughout the paper to the role of chiral symmetry on the lattice and to the mixing of lattice operators under renormalization. The construction of chiral gauge theories regularized on the lattice, made possible by the recent advances in the understanding of chiral symmetry, is also discussed.

Finally, a few detailed examples of lattice perturbative calculations are presented.

DESY 02-185

November 2002

Lattice Perturbation Theory

Stefano Capitani ^{1}^{1}1 E-mail:

DESY Zeuthen

John von Neumann-Institut für Computing (NIC)

[0.2 cm] Platanenallee 6, 15738 Zeuthen, Germany

###### Contents

- 1 Introduction
- 2 Why lattice perturbation theory
- 3 Renormalization of operators
- 4 Discretization
- 5 Wilson’s formulation of lattice QCD
- 6 Aspects of chiral symmetry on the lattice
- 7 Staggered fermions
- 8 Ginsparg-Wilson fermions
- 9 Perturbation theory of lattice regularized chiral gauge theories
- 10 The approach to the continuum limit
- 11 Improvement
- 12 The Schrödinger functional
- 13 The hypercubic group
- 14 Operator mixing on the lattice
- 15 Analytic computations
- 16 Computer codes
- 17 Lattice integrals
- 18 Algebraic method for 1-loop integrals
- 19 Coordinate space methods
- 20 Numerical perturbation theory
- 21 Conclusions
- A Notation and conventions
- B High-precision values of and

## 1 Introduction

In a lattice field theory the quantum fields are studied and computed using a discretized version of the spacetime. The lattice spacing , the distance between neighboring sites, induces a cutoff on the momenta of the order .

A spacetime lattice can be viewed as a regularization which is nonperturbative. Since the other known regularizations, like dimensional regularization or Pauli-Villars, can be defined only order by order in perturbation theory, the lattice regularization has this unique advantage over them. It is a regularization which is not tied to any specific approximation method, and which allows calculations from first principles employing various numerical and analytical methods, without any need to introduce models for the physics or additional parameters.

In discretizing a continuum field theory one has to give up Lorentz invariance (and in general Poincaré invariance), but the internal symmetries can usually be preserved. In particular, gauge invariance can be kept as a symmetry of the lattice for any finite value of the lattice spacing, and this makes possible to define QCD as well as chiral gauge theories like the electroweak theory. The construction of the latter kind of theories on a lattice however presents additional issues due to chiral symmetry, which have been understood and solved only recently. The fact that one is able to maintain gauge invariance for any nonzero is of great help in proving the renormalizability of lattice gauge theories.

Lattice gauge theories constitute a convenient regularization of QCD where its nonperturbative features, which are essential for the description of the strong interactions, can be systematically studied. The lattice can then probe the long-distance physics, which is otherwise unaccessible to investigations which use continuum QCD. Precisely to study low-energy nonperturbative phenomena the lattice was introduced by Wilson, which went on to prove in the strong coupling regime the confinement of quarks. Confinement means that quarks, the fundamental fields of the QCD Lagrangian, are not the states observed in experiments, where one can see only hadrons, and the free theory has no resemblance to the observed physical world. The quark-gluon structure of hadrons is hence intrinsically different from the structure of other composite systems. No description in terms of two-body interactions is possible in QCD. Lattice simulations of QCD show that a large part of the mass of the proton arises from the effect of strong interactions between gluons, that is from the pure energy associated with the dynamics of confinement. Only a small fraction of the proton mass is due to the quarks. Similarly, the lattice confirms that only about half of the momentum and a small part of the spin of the proton come from the momentum and spin of the constituent quarks. Computations coming from the lattice can then give important contributions to our understanding of the strong interactions.

In this review we are interested in doing lattice calculations in the weak coupling regime. This is the realm of perturbation theory, which is used to compute the renormalization of the parameters of the Lagrangian and of the matrix elements, and to study the approach of the lattice to the continuum limit. Details of the lattice formulation that are only relevant at the nonperturbative level will not be discussed in this review. If the reader is also interested in the nonperturbative aspects of lattice field theories, they are covered at length in the books of [Creutz, 1983, Montvay and Münster, 1994, Rothe, 1997] and in the just appeared book of [Smit, 2002]. The book by Rothe is the one which contains more material about lattice perturbation theory. Useful shorter reviews, which also cover many nonperturbative aspects, sometimes with a pedagogical cut, can also be found in [Kogut, 1983, Sharpe, 1994, Sharpe, 1995, DeGrand, 1996, DeGrand, 1997, Gupta, 1999, Sharpe, 1999, Wittig, 1999, Münster and Walzl, 2000, Davies, 2002, Kronfeld, 2002] and in the recent [Lüscher, 2002]. Here we would like to explain the main methods and techniques of lattice perturbation theory, particularly when Wilson and Ginsparg-Wilson fermions are used. We will discuss, among other things, Feynman rules, aspects of the analytic calculations, the computer codes which are often necessary to carry them out, the mixing properties of lattice operators, and lattice integrals.

Chiral symmetry is a recurrent topic in the treatment of fermions on the lattice, and we will address some issues related to it in the course of the review. We feel that a discussion of the problems connected with the realization of chiral symmetry on the lattice is needed. The reader might otherwise wonder why one should do such involved calculations like the ones required for Ginsparg-Wilson fermions. We think that it is also interesting to see how the lattice can offer fascinating solutions to the general quantum theoretical problem of defining chiral gauge theories beyond tree level.

Also discussed is an algebraic method for the reduction of any 1-loop lattice integral (in the Wilson case) to a linear combination of a few basic constants. These constants are calculable with very high precision using in a clever way the behavior of the position space propagators at large distances. The coordinate space method, which turns out to be a very powerful tool for the computation of lattice integrals, allows then the calculation of these constants with very high precision, which is also the necessary requirement in order to be able to compute two-loop lattice integrals with many significant decimal places, as we will explain in detail. A lot of nice and interesting work has been done using these techniques in the case of bosonic integrals, which thanks to them can be easily computed with extraordinary precision at one-loop, and with good precision at two loops. A nonnegligible part of this review will discuss these calculations in detail in the bosonic case.

The focus of this review is on methods rather than on results. In fact, very few numerical results will be reported. The reader, if interested, can consult many perturbative results in the references given. Instead, our objective is different. We would like to provide computational tools which can be useful for physicists who are interested in doing this kind of calculations. Technical details will be therefore explained in a pedagogical fashion. Particular attention will be paid to certain aspects that only occur in lattice computations, and that physicists expert in continuum perturbative calculations might find curious. The main objective of this review is to show how perturbation theory works on the lattice in the more common situations. It is hoped that one can learn from the material presented here.

A background in continuum quantum field theory is required, and an acquaintance with continuum perturbative calculations in gauge field theories, the derivation of Feynman rules in continuum QCD and the calculation of Feynman diagrams will be assumed. Familiarity with the path integral formalism, with the quantization of field theories through the functional integral and with the renormalization of continuum quantum field theories is also desired. The knowledge of elementary facts, such as the renormalization group equations, the running of the strong coupling, the function and asymptotic freedom of QCD, will also be taken for granted.

This review is not homogeneous. I have given more space to the topics that I believe are more interesting and more likely to be of wider use in the future. Many of the choices made and of the examples reported draw from the experience of the author in doing this kind of calculations.

To keep this review into a manageable size, not all important topics or contributions will be covered. One thing that will not be discussed in detail is perturbation theory applied to Symanzik improvement, which, although very interesting and useful, would probably require a review in itself, given also the many important result that have been produced. The Schrödinger functional is also introduced only in a very general way. I will not be able to do justice to other topics like numerical perturbation theory or tadpole improvement.

Many interesting subjects had to be entirely left out because of constraints on space. Among the topics which are not covered at all are nonrelativistic theories, heavy quarks, and anisotropic lattices. I have also omitted all what concerns finite temperature perturbation theory. Many of these things are treated in detail in the reviews and books cited above, where several topics not covered here can also be found. Moreover, we will not occupy ourselves with phenomenological results, but only with how perturbation theory is useful for the extraction from the lattice of that phenomenology. In any case, there are by now so many perturbative calculations that have been made on the lattice that it would be impossible to include all of them here.

The main reason for the introduction of the lattice was to study QCD in its nonperturbative aspects, like confinement, and we will confine this review to QCD. Although very interesting, spin models, the theory and the Higgs sector, to name a few, will be left out. Even after this restriction is made, lattice QCD is still quite a broad topic by itself, and thus to contain this review into a reasonable size we have been compelled to discuss only the main actions that have been used to study QCD on the lattice. Given the great number of different actions that have been proposed for studying lattice QCD in the past 30 years, it is necessary to limit ourselves to just the few of them that are more widely used. The Wilson and staggered formulations have been the most popular ones in all this time. Recently the particular kind of chiral fermions known as Ginsparg-Wilson (like the overlap, domain wall and fixed-point fermions) have also begun to find broader application, and they present interesting challenges for lattice physicists interested in perturbation theory. These are the actions that we will cover in this paper.

The main features of the lattice construction and of lattice perturbation theory will be discussed in detail in the context of Wilson fermions. When the other actions will be introduced the discussions will be more general, although we will try to point out the peculiarities of perturbative calculations in these particular cases. The explanations of the various lattice actions will be rather sketchy and aimed mainly at the aspects which are interesting from the point of view of perturbation theory.

### 1.1 Outline of the paper

The review is divided in three parts: Sections 2 and 3, which are a sort of motivation, Sections 4 to 12, which introduce the lattice, the various actions and their Feynman rules, and Sections 13 to 20, which in much more technical detail show how lattice computations are made.

We begin with two Sections which are meant to stress the importance of lattice perturbation theory and explain what is meant for renormalization of operators on the lattice. After having introduced these motivations, we start with the first technical points, defining in Section 4 a euclidean lattice and showing what the discretization of a continuum theory means in practice. We introduce typical lattice quantities and we pass to discuss in detail the Wilson action (which has not chiral invariance) in Section 5, explaining how to derive its Feynman rules in momentum space. All Feynman rules necessary for one-loop calculations are then explicitly given. In Section 6 we focus our attention on the relation between chirality and fermionic modes on the lattice, and the problems which arise when one tries to define chiral fermions on a lattice. After a brief interlude which discusses staggered fermions, which have some chiral symmetry and have been the major alternative to Wilson fermions (at least in the first two decades of the lattice), we then introduce in Section 8 Ginsparg-Wilson fermions, the long-awaited reconciliation of chirality with the lattice. We give some details about the overlap, the domain wall and the fixed-point actions, which are solutions of the Ginsparg-Wilson relation. In Section 9 we then explain how using Ginsparg-Wilson fermions it is possible to define on the lattice chiral gauge theories, where gauge invariance and chiral symmetry are maintained together at every order in perturbation theory and for any finite value of the lattice spacing.

In Section 10 we then see the approach of coupling and masses to the continuum limit and talk about the function and the parameter of the lattice theory. In Section 17 we briefly introduce the Symanzik improvement, including a short discussion about improved pure gauge actions on the lattice. We conclude the first part of the review with a brief introduction to the Schrödinger functional, which has gained a paramount place in the lattice landscape in recent years.

In Section 13 we begin the more technical part of the review, dedicated to how to actually carry out the perturbative computations on the lattice. We introduce at this point the symmetry group of the lattice, the hypercubic group. Since the lattice symmetries are not as restrictive as in the continuum theory, more mixings arise in general under renormalization, and we discuss some examples of them in Section 14. How to compute Feynman diagrams on the lattice is explained in great detail in Section 15, where we talk about the lattice power counting theorem of Reisz, which is useful for the computation of divergent integrals, and we present, step by step, the complete calculation of the 1-loop renormalization constant of the operator measuring the first moment of the momentum distribution of quarks in hadrons. This example is rather simple (compared to other operators) and contains all the main interesting features one can think of: a logarithmic divergence, a covariant derivative, symmetrized indices and of course the peculiar use of Kronecker deltas in lattice calculations. Moreover, it is an example of a calculation in which the various propagators and vertices need an expansion in the lattice spacing (in this case, at first order). Finally, it includes the computation of the quark self-energy, which is quite interesting and useful by itself. Brief discussions about overlap calculations, tadpole improvement and perturbation theory with fat link actions conclude Section 15. In Section 16 we then discuss the use of computer codes for the automated computations of lattice Feynman diagrams.

In Section 17 we explain some advanced techniques for the numerical evaluation of lattice integrals coming from Feynman diagrams (using extrapolations to infinite volume), while in Section 18 we introduce an algebraic method for the exact reduction of any Wilson 1-loop integral to a few basic constants. The bosonic case is thoroughly explained, so that the reader can learn to use it, and some applications to the exact calculations of operator tadpoles are also explicitly given. Section 18 ends with a discussion of the main points of the general fermionic case, and the expression of the 1-loop quark self-energy in terms of the basic constants.

The basic constants of the algebraic method can be computed with arbitrary precision, as explained in detail in Section 19. The values of the fundamental bosonic constants, and , are given with a precision of about 400 significant decimal places in Appendix B. In order to be able to calculate them to this precision, we need to introduce the coordinate space method of Lüscher and Weisz, which will also be used for the computation of 2-loop integrals. The 2-loop bosonic integrals are discussed at length, and the general fermionic case is also addressed. In Section 20 we then briefly introduce numerical perturbation theory, which is promising for doing calculations at higher loops. Finally, conclusions are given in Section 21, and Appendix A summarizes some notational conventions.

## 2 Why lattice perturbation theory

To some readers the words “lattice” and “perturbation theory” might sound like a contradiction in terms, but we will see that this is not the case and that lattice perturbation theory has grown into a large and established subject. Although the main reason why the lattice is introduced is because it constitutes a nonperturbative regularization scheme and as such it allows nonperturbative computations, perturbative calculations on the lattice are rather important, and for many reasons.

Perturbation theory of course cannot reveal the full content of the lattice field theory, but it can still give a lot of valuable informations. In fact, there are many applications where lattice perturbative calculations are useful and in some cases even necessary. Among them we can mention the determinations of the renormalization factors of matrix elements of operators and of the renormalization of the bare parameters of the Lagrangian, like couplings and masses. The precise knowledge of the renormalization of the strong coupling is essential for the determination of the parameter of QCD in the lattice regularization (as we will see in Section 10) and of its relation to its continuum counterpart, . In general perturbation theory is of paramount importance in order to establish the right connection of the lattice scheme which is used in practice, and of the matrix elements simulated within that scheme, to the physical continuum theory. Every lattice action defines a different regularization scheme, and thus one needs for each new action that is considered a complete set of these renormalization computations in order for the results which come out from Monte Carlo simulations to be used and understood properly.

Moreover, lattice perturbation theory is important for many other aspects, among which we can mention the study of the anomalies on the lattice, the study of the general approach to the continuum limit, including the recovery of the continuum symmetries broken by the lattice regularization (like Lorentz or chiral symmetry) in the limit , and the scaling violations, i.e., the corrections to the continuum limit which are of order . These lattice artifacts bring a systematic error in lattice results, which one can try to reduce by means of an “improvement”, as we will see in Section 11.

Perturbative calculations are thus in many cases essential, and are the only possibility for having some analytical control over the continuum limit. As we will see in Section 10, the perturbative region is the one that must be necessarily traversed in order to reach the continuum limit. Lattice perturbation theory is in fact strictly connected to the continuum limit of the discretized versions of QCD. Because of asymptotic freedom, one has when , which means . We should also point out that one cannot underestimate the role played by perturbative calculations in proving the renormalizability of lattice gauge theories.

Finally, perturbation theory will also be important for defining chiral gauge theories on the lattice at all orders of the gauge coupling, as we will see in detail in Section 9. The lattice will thus be proven to be the only regularization that can preserve chirality and gauge invariance at the same time (without destroying basic features like locality and unitarity).

We can say in a nutshell that lattice perturbation theory is important for both conceptual and practical reasons. The phenomenological numbers that are quoted from lattice computations are very often the result of the combined effort of numerical simulations and analytic calculations, usually with some input from theoretical ideas.

In principle all known perturbative results of continuum QED and QCD can also be reproduced using a lattice regularization instead of the more popular ones. However, calculating in such a way the correction to the magnetic moment of the muon (to make an example) would be quite laborious. A lattice cutoff would not be the best choice in most cases, for which instead regularizations like Pauli-Villars or dimensional regularization are more suited and much easier to employ. The main virtue of the lattice regularization is instead to allow nonperturbative investigations, which usually need some perturbative calculations to be interpreted properly. As we have already mentioned, the connection from Monte Carlo results of matrix elements to their corresponding physical numbers, that is the matching with the continuum physical theory, has to be realized by performing a lattice renormalization. It is in this context that lattice perturbation theory has a wide and useful range of applications, and we will discuss this important aspect of lattice computations in more detail in the next Section. In this respect, perturbative lattice renormalization is important by itself as well as in being a hint and a guide for the few cases in which one can also determine the renormalization constants nonperturbatively according to the method proposed in [Martinelli et al., 1995] (for a recent review see [Sommer, 2002]). This is even more important in the cases in which operator mixings are present, which are generally more complex than in the continuum. In fact, mixing patterns on the lattice become in general more transparent when looked at using perturbative renormalization than nonperturbatively. We should also add that perturbative coefficients can be usually computed much more accurately than typical quantities in numerical simulations. Perturbative renormalization results can in any case be quite useful in checking and understanding results coming from nonperturbative methods (when available). When short-distance quantities can be calculated using such diverse techniques as lattice perturbation theory and Monte Carlo simulations, the comparison of the respective results, repeated in different situations, can give significant suggestions on the validity of perturbative and nonperturbative methods.

In some cases a nonperturbative determination of the renormalization constants can turn out to be rather difficult to achieve. For the method to work, it is necessary that there is a plateau for the signal over a substantial range of energies so that one can numerically extract the values of the renormalization factors. The nonperturbative renormalization methods can then fail because a window which is large enough cannot be found. Moreover, where mixings are present these methods could come out to be useless because the amount of mixing is too small to be disentangled with numerical techniques, although still not too small to be altogether ignored. It is then not clear whether all operators can be renormalized nonperturbatively in such a way. In these cases the only possibility to compute renormalization factors seems to be provided by the use of lattice perturbative methods. An important exception to this is given by the Schrödinger functional scheme, where using a particular procedure known as recursive finite-size scaling technique (which we will explain in Section 12) it is possible to carry out precise nonperturbative determinations of renormalized couplings, masses and operators for an extremely wide range of energies. Computations using the Schrödinger functional are however rather more involved than average and usually require much bigger computational resources.

We would also like to point out that in the case of Ginsparg-Wilson fermions the numerical computations required to extract nonperturbative renormalization factors (which come on top of the already substantial effort needed to determine the bare matrix elements) could turn out to be quite expensive, especially in the cases of complicated operators like the ones that measure moments of parton distributions.

We can thus say, after having looked at all these different aspects, that
lattice perturbation theory is important and fundamental. Of course sometimes
there can be issues concerning its reliability when a 1-loop perturbative
correction happens to be large, especially when the corresponding 2-loop
calculation looks rather difficult to carry out. ^{2}^{2}2In this case
mean-field improved perturbation theory, using Parisi’s boosted bare coupling,
is known to reduce the magnitude of these corrections in many situations
(see Section 15.6).

On the other hand, there are cases in which lattice perturbation theory works
rather well. As an example we show in Figs. 1 and 2
the scale evolutions of the renormalized strong coupling and masses computed
in the Schrödinger functional scheme [Capitani et al., 1999c]. ^{3}^{3}3An
explanation of the way these nonperturbative evolutions are obtained is given
in Section 12. We can see that these scale evolutions are
accurately described by perturbation theory for a wide range of energies.
The perturbative and nonperturbative results are very close to each other, and
almost identical even down to energy scales which are surprising low. The best
perturbative curves shown are obtained by including the term of
the function and the term of the function
(that is, the first nonuniversal coefficients). The other curves are lower
approximations. In Section 10 more details about these
functions can be found.

Although the coupling and masses in the figures are computed in the Schrödinger functional scheme, and their values are different from, say, a calculation done with standard Wilson fermions, this example shows how close perturbation theory can come to nonperturbative results. The case shown is particularly interesting, because these nonperturbative results are among the best that one can at present obtain. The Schrödinger functional coupled to recursive finite-size techniques allows to control the systematic errors completely. There are fewer errors in these calculations, and they are fully understood. In other situations we cannot really exclude, when we see a discrepancy between nonperturbative and perturbative results, that at least part of this discrepancy originates from the nonperturbative side.

Another nice example of the good behavior of lattice perturbation theory
is given in Fig. 3, which comes from the work
of [Necco and Sommer, 2001, Necco, 2002a]. These authors have computed the running
coupling from the static quark force or potential in three different ways,
corresponding to three different schemes: ^{4}^{4}4A few different ways
of defining a strong coupling constant on the lattice are discussed
in [Weisz, 1996].

(2.1) | |||||

(2.2) | |||||

(2.3) |

and showed that in first case, called scheme, the perturbative expansion of the coupling is rather well behaved, that is the coefficients are small and rapidly decreasing. This is what is shown in Fig. 3, which indicates that the perturbative computations can be trusted to describe with a good approximation the nonperturbative numbers for couplings up to . In the other two schemes, however, the coefficients are somewhat larger (especially in the last case), and the perturbative expansion of the coupling has a worse behavior. The perturbative couplings in these two schemes have a more pronounced difference with respect to the nonperturbative results.

The three schemes above differ only by kinematics, and the results show how the choice of one scheme or another can have a big influence on the perturbative behavior of the coupling. Discussions about the validity of lattice perturbation theory cannot then be complete until the dependence on the renormalization scheme (beside the dependence on the various actions used) is also taken into account and investigated. Not all schemes are suitable to be used in lattice perturbation theory to the same extent. In particular, the scheme is the best one among the three considered above. From this point of view the coupling in the Schrödinger functional scheme is even better behaved than , in other words the coefficient of the function (the first nonuniversal coefficient) is the smallest in this scheme, and in this sense the Schrödinger functional scheme is the closest to the scheme.

We would also like to mention the work of [Davies et al., 1997], where the QCD coupling is extracted from various Wilson loops, and which shows another instance of a good agreement between perturbation theory and simulations.

We can thus trust lattice perturbation theory, under the right conditions. It is probably not worse behaved than perturbative QCD in the continuum, which is an asymptotic expansion, and which also gives origin to large corrections in some cases. We could even say that perturbation theory can be better tested on the lattice than in the continuum, because in a lattice scheme one has also at his disposal the nonperturbative results to compare perturbation theory with. When lattice perturbation theory does not agree with nonperturbative numerical results, perhaps a look at the systematic errors coming from the numerical side can sometimes be worthwhile.

## 3 Renormalization of operators

In general, matrix elements of operators computed on the lattice using numerical simulations still require a renormalization in order to be converted into meaningful physical quantities. The Monte Carlo matrix elements can be considered as (regulated) bare numbers, and to get physical results one needs to perform a renormalization, matching the numbers to some continuum scheme, which is usually chosen to be the scheme of dimensional regularization.

In many physical problems one evaluates matrix elements of operators that appear in an operator product expansion. These matrix elements contain the long-distance physics of the system, and are computed numerically on the lattice, while the Wilson coefficients contain the short-distance physics, which can be gathered from perturbative calculations in the continuum. In this case the operators computed on the lattice must at the end be matched to the same continuum scheme in which the Wilson coefficient are known. Therefore one usually chooses to do the matching from the lattice to the scheme of dimensional regularization. A typical example is given by the moments of deep inelastic structure functions, and we will illustrate many features of perturbation theory in the course of this review using lattice operators appearing in operator product expansions which are important for the analysis of structure functions.

Thus, on the lattice one has to perform a matching to a continuum scheme, that is one looks for numbers which connect the bare lattice results to physical continuum renormalized numbers. We will now discuss how a perturbative matching can be done at one loop. Some good introductory material on the matching between lattice and continuum and on the basic concepts of lattice perturbation theory can be found in [Sachrajda, 1990, Sharpe, 1994, Sharpe, 1995]. A short review of the situation of perturbative calculations at around 1995 is given in [Morningstar, 1996].

It turns out that to obtain physical continuum matrix elements from Monte Carlo simulations one needs to compute 1-loop matrix elements on the lattice as well as in the continuum. Lattice operators are chosen so that at tree level they have the same matrix elements as the original continuum operators, at least for momenta much lower than the lattice cutoff, . Then at one loop one has

(3.1) | |||||

(3.2) |

The lattice and continuum finite constants, and
, and therefore the lattice and continuum 1-loop renormalization
constants, do not have the same value. ^{5}^{5}5We note that while
is the whole momentum-independent 1-loop correction,
does not contain the pole in and the factors
proportional to and . This happens because lattice
propagators and vertices, as will be seen in detail later on, are quite
different from their continuum counterparts, especially when the loop momentum
is of order . Therefore the 1-loop renormalization factors on the lattice
and in the continuum are in general not equal. Note however that the 1-loop
anomalous dimensions are the same.

The connection between the original lattice numbers and the final continuum physical results is given, looking at Eqs. (3.1) and (3.2), by

(3.3) |

The differences enter then in the matching factors

(3.4) |

which are the main objective when one performs computations of renormalization
constants of operators on the lattice. ^{6}^{6}6The coupling that appears in
Eq. (3.3) is usually chosen to be the lattice one, , as
advocated in [Sachrajda, 1990]. Of course choosing one coupling or the
other makes only a 2-loop difference, but these terms could still be
numerically important. The validity of this procedure should be checked by
looking at the size of higher-order corrections. Unfortunately on the lattice
these terms are known only in a very few cases and no definite conclusions can
then be reached with regard to this point.
In the work of [Ji, 1995] a generalization to higher loops was proposed,
which gives an exact matching condition to all orders. This is done using the
lattice and continuum renormalization group evolutions (see
Section 10). Using the lattice evolution, one goes to very
high energies, which means very small couplings because of asymptotic freedom,
and there he does the matching to . It thus essentially becomes a
matching at the tree level. After this, one goes back to the original scale
, using the continuum renormalization group evolution backwards. For the
matching at the scale one then obtains a factor

While and depend on the state , is independent of it and the depends only on . This is as it should be, since the renormalization factors are a property of the operators and are independent of the particular states with which a given matrix element is constructed. This is the reason why we have not specified . Furthermore, the matching factors between the lattice and the scheme are gauge invariant, and this property can be exploited to make important checks of lattice perturbative calculations.

At the end of the process that we have just explained one has computed, using both lattice and continuum perturbative techniques, the renormalization factor which converts the lattice operator into the physical renormalized operator :

(3.6) |

The reader should always keep in mind that in this way one does the matching of the bare Monte Carlo results (obtained using a lattice regulator) to the physical renormalized results in the scheme.

As for any general quantum field theory, the process at the end of which one obtains physical numbers is accomplished in two steps. First one regularizes the ultraviolet divergences, and in this case the regulator is given by the lattice itself. Afterwards one renormalizes the theory so regulated, and on the lattice this results in a matching to a continuum scheme. In the end, after renormalization the lattice cutoff must be removed, which means that one has to go to the continuum limit of the lattice theory, and that it is safe to do so at this point (if the renormalization procedure is consistent). What remains after all this is only the scale brought in by the renormalization. In our case the scale at which the matrix elements are renormalized should be in the range

(3.7) |

The lower bound ensures that perturbation theory is valid, while the upper bound ensures that the cutoff effects, proportional to positive powers of the lattice spacing, are small. One usually sets

(3.8) |

and since the 1-loop anomalous dimensions are the same on the lattice and in the continuum, only a finite renormalization connects the lattice to the scheme:

(3.9) |

Every lattice action defines a different regularization scheme, and therefore these finite renormalization factors are in principle different for different actions. The bare numbers, that is the Monte Carlo results for a given matrix element, are also different, and so everything adjusts to give the same physical result.

We conclude mentioning that Sharpe (1994) has observed that when the operators come from an operator product expansion one should multiply the 1-loop matching factors introduced above with the 2-loop Wilson coefficients, in order to be consistent. This can be seen by looking at the 2-loop renormalization group evolution for the Wilson coefficients,

(3.10) |

The term proportional to is analogous to a 1-loop matching factor, but this term contains and , which are 2-loop coefficients. So, it is only combining 1-loop renormalization matching with 2-loop Wilson coefficients that one is doing calculations in a consistent way.

In this Section we have learned that we need perturbative lattice calculations in order to extract a physical number from Monte Carlo simulations of matrix elements of operators (unless one opts for nonperturbative renormalization, when this is possible). We will try to explain how to do this kind of calculations in the rest of the review.

## 4 Discretization

Lattice calculations are done in euclidean space. A new time coordinate is introduced by doing a Wick rotation from Minkowski space to imaginary times:

(4.1) |

In momentum space this corresponds to , so that the Fourier transforms continue in euclidean space to be defined through the same phase factor. The reason for working with imaginary times is that the imaginary factor in front of the Minkowski-space action becomes a minus sign in the euclidean functional integral,

(4.2) |

and the lattice field theory in euclidean space acquires many analogies with a
statistical system. The path integral of the particular quantum field theory
under study becomes the partition function of the corresponding statistical
system. The transition to imaginary times brings a close connection between
field theory and statistical physics which has many interesting facets.
In particular, when the euclidean action is real and bounded from below
one can see the functional integral as a probability system weighted by
a Boltzmann distribution . It is this feature that allows
Monte Carlo methods to be used. ^{7}^{7}7However, when the action is complex,
like in the case of QCD with a finite baryon number density, this is not
possible. It is this circumstance that has hampered progress in the lattice
studies of finite density QCD.
Furthermore, on a euclidean lattice of finite volume the path integral
is naturally well defined, since the measure contains only a finite number
of variables and the exponential factor, which has a negative exponent,
gives an absolutely convergent multi-dimensional integral.
One then can generate configurations with the appropriate probability
distribution and in this way sample the functional integral with Monte Carlo
techniques. This is the practical basis of Monte Carlo simulations.

From now on we will work in the euclidean space in four dimensions, with metric (1,1,1,1), and we will drop all euclidean subscripts from lattice quantities, so that is for example the time component after the Wick rotation. The Dirac matrices in euclidean space satisfy an anticommutation relation with replaced by :

(4.3) |

and they are all hermitian:

(4.4) |

The euclidean matrix is defined by

(4.5) |

is also hermitian, and satisfies . The relation between Dirac matrices in Minkowski and euclidean space is

(4.6) |

This can be inferred from the kinetic term of the Dirac action in the functional integral:

(4.7) |

The explicit euclidean Dirac matrices in the chiral representation are given in Appendix A.

We want to construct field theories on a hypercubic lattice. This is a discrete subset of the euclidean spacetime, where the sites are denoted by (with integers). We will work in this review only with hypercubic lattices, where the lattice spacing is the same in all directions. A 2-dimensional projection of such a (finite) lattice is given in Fig. 4. For convenience we will sometimes omit to indicate the lattice spacing , that is we will use . The missing factors of can always be reinstated by doing a naive dimensional counting.

In going from continuum to lattice actions one replaces integrals with sums,

(4.8) |

where on the right-hand side means now sites: . ^{8}^{8}8We use
in general the same symbols for continuum and lattice quantities, hoping that
this does not cause confusion in the reader. An exception is given by the
lattice derivatives, for which we will use special symbols.
Lattice actions are then written in terms of sums over lattice sites.
The distance between neighboring sites is , and this minimum distance
induces a cutoff on the modes in momentum space, so that the lattice spacing
acts as an ultraviolet regulator. The range of momenta is thus restricted
to an interval of range , called the first Brillouin zone, and
which can be chosen to be

(4.9) |

This is region of the allowed values of , and is the domain of integration when lattice calculations are made in momentum space. For a lattice of finite volume the allowed momenta in the first Brillouin zone become a discrete set, given by

(4.10) |

and so in principle one deals with sums also in momentum space. However, as we will see later, when doing perturbation theory one assumes to do the calculations in infinite volume and so the sums over the modes of the first Brillouin zone become integrals:

(4.11) |

The one-sided forward and backward lattice derivatives (also known as right and left derivatives) can be respectively written as

(4.12) | |||||

(4.13) |

where denotes the unit vector in the direction. It is easy to see that

(4.14) | |||||

(4.15) |

that is they are (up to a sign) conjugate to each other. Therefore or alone cannot be chosen in a lattice theory that is supposed to have a hermitian Hamiltonian. In this case one needs their sum, , which is anti-hermitian and gives a lattice derivative operator extending over the length of two lattice spacings:

(4.16) |

Note that the second-order differential operator is hermitian, and when is summed corresponds to the 4-dimensional lattice Laplacian,

(4.17) |

It is also useful to know that

(4.18) |

that is,

(4.19) |

which is valid for an infinite lattice, and also for a finite one if and are periodic (or their support is smaller than the lattice). The formula above corresponds to an integration by parts on the lattice, which in practical terms amounts to a shift of the summation variable.

There is in general some freedom in the construction of lattice actions. For the discretization of continuum actions and operators and the practical setting of the corresponding lattice theory many choices are possible. Since the lattice symmetries are less restrictive than the continuum ones, there is more than one possibility in formulating a gauge theory starting from a given continuum gauge theory. In particular, one has quite a few choices for the exact form of the QCD action on the lattice, depending on what features are of interest in the studies that one wants to carry out using the discretized version. There is not an optimal lattice action to use in all cases, and each action has some advantages and disadvantages which weigh differently in different contexts. This means that deciding for one action instead of another depends on whether chiral symmetry, flavor symmetry, locality, or unitarity are more or less relevant to the physical system under study. There is a special emphasis on the symmetry properties. One also evaluates the convenience of lattice actions by considering some balance between costs and gains from the point of view of numerical simulations and of perturbation theory. Therefore, the final choice of a lattice action depends also on the problem that one wants to study.

There are many lattice actions which fall in the same universality class, that is they have the same naive continuum limit, and each of them constitutes a different regularization, for finite , of the same physical theory. Since every lattice action defines a different regularization scheme, one needs for each action that is used a new complete set of renormalization computations of the type discussed in Section 3, in order for the results which come out from Monte Carlo simulations to be used, interpreted and understood properly. Using different actions leads to different numerical results for the matrix elements computed in Monte Carlo simulations, and also the values of the renormalization factors, and of the parameter, depend in general on the lattice action chosen. Even the number and type of counterterms required in the renormalization of operators can be different in each case. For example, a weak operator which is computed on the lattice using the Wilson action, where chiral symmetry is broken, needs in general more counterterms to be renormalized than when is computed with the overlap action, which is chiral invariant. Of course all the various differences among lattice actions arise only at the level of finite lattice spacing. The final extrapolations to the continuum limit must lead, within errors, to the same physical results.

In the case of QCD there seems to be a lot of room in choosing an action for the fermion part, although also the pure gauge action has some popular variants (but the plaquette, or Wilson, action has a clear predominance over the other ones here, except in particular situations, where for example improved gauge actions may be more convenient). The main features of perturbation theory will be first introduced in the context of Wilson fermions, which are one of the most widely used lattice formulations. Then a few other fermion actions will be discussed along the way, pointing out the differences with the standard perturbation theory made with Wilson fermions, which is generally simpler.

## 5 Wilson’s formulation of lattice QCD

One of the most popular lattice formulations of QCD is the one invented by Wilson (1974; 1977), which was also the first formulation ever for a lattice gauge theory. Its remarkable feature is that it maintains exact gauge invariance also at any nonzero values of the lattice spacing. The discretization of the (euclidean) QCD action for one quark flavor

(5.1) |

that Wilson proposed is the following:

(5.2) | |||||

where and . This action has only nearest-neighbor
interactions. ^{9}^{9}9Other actions can have more complicated interactions,
like for example overlap fermions
(see Section 8).
The derivative in the Dirac operator is
the symmetric one, Eq. (4.16) (with an integration by parts).
The fields live on the links which connect two neighboring lattice
sites, and these variables are naturally defined in the middle point of
a link. Each link carries a direction, so that

(5.3) |

These link variables, which are unitary, are not linear in the gauge potential . The fact is that they belong to the group rather than to the corresponding Lie algebra, as would be the case in the continuum. The relation of the matrices with the gauge fields , the variables which have a direct correspondence with the continuum, is then given by

(5.4) |

where the are matrices in the fundamental representation.

The Wilson action is a gauge-invariant regularization of QCD, and it has exact local gauge invariance on the lattice at any finite . The gauge-invariant construction is done directly on the lattice, extending a discretized version of the free continuum fermionic action. It is not therefore a trivial straightforward discretization of the whole gauge-invariant continuum QCD action, which would recover the gauge invariance only in the continuum limit. A naive lattice discretization of the minimal substitution rule would in fact result in an action that violates gauge invariance on the lattice, whereas with the choice made by Wilson gauge invariance is kept as a symmetry of the theory for any . It is this requirement that causes the group variables to appear in the action instead of the algebra variables . The lattice gauge transformations are given by

(5.5) |

with , and it is easy to see that they leave the quark-gluon interaction term in the Wilson action invariant. Note that also in the lattice theory the local character of the invariance group is maintained.

This form of local gauge invariance imposes strong constraints on the form of the gauge field-strength tensor . Given the above formula for the lattice gauge transformations of the ’s, it is easy to see that the simplest gauge-invariant object that one can build from the link variables involves their path-ordered product. In particular, one obtains a gauge-invariant quantity by taking the trace of the product of ’s on adjoining links forming a closed path, thanks to the unitarity of the ’s and the cyclic property of the trace.

The physical theory is a local one, and so in constructing the pure gauge action we should direct our attention toward small loops. The simplest lattice approximation of is then the product of the links of an elementary square, called “plaquette”:

(5.6) |

which is shown in Fig. 5. This form is not surprising, given
that the gauge field-strength tensor is in differential geometry the curvature
of the metric tensor. One could also take larger closed loops, but this
minimal choice gives better signal-to-noise ratios, and for the standard
Wilson action the trace of the plaquette is then used. ^{10}^{10}10Other
actions which use different approximations for , and which have
the aim of reducing the discretization errors, are discussed in
Section 11.2. This is the expression appearing in the
last line of Eq. (5.2).
The factor can be understood by looking at the expansion of the
plaquette Eq. (5.6) in powers of , which is

(5.7) |

with and hermitian fields. ^{11}^{11}11This expansion can be derived
using

(5.9) |

where we have used , because the trace of the generators is zero. The plaquette action then has the right continuum limit, and the first corrections to the continuum pure gauge action are of order . These are irrelevant terms, which are zero in the continuum limit, but they are important for determining the rate of convergence to the continuum physics. It can also be shown that in the fermionic part of the action the corrections with respect to the continuum limit are of order . In Section 11 we will see how to modify the fermion action in order to decrease the error on the fermionic part to order .

The plaquette action is also often written as

(5.10) |

where is given in Eq. (5.6), and in numerical simulations of QCD the coefficient in front of the action is

(5.11) |

The factor two comes out because here one takes the sum over the oriented plaquettes, that is a sum over ordered indices (for example, ), while in Eq. (5.2) the sum over and is free.

In the weak coupling regime, where is small, the functional integral is dominated by the configurations which are near the trivial field configuration . Perturbation theory is then a saddle-point expansion around the classical vacuum configurations, and the relevant degrees of freedom are given by the components of the gauge potential, . Thus, while the fundamental gauge variables for the Monte Carlo simulations are the ’s and the action is relatively simple when expressed in terms of these variables, in perturbation theory the true dynamical variables are the ’s. This mismatch causes a good part of the complications of lattice perturbation theory. In fact, when the Wilson action is written in terms of the ’s, using , it becomes much more complicated. Moreover, it consists of an infinite number of terms, which give rise to an infinite number of interaction vertices. Fortunately, only a finite number of vertices is needed at any given order in .

All vertices except a few of them are “irrelevant”, that is they are proportional to some positive power of the lattice spacing and so they vanish in the naive continuum limit. However, this does not mean that they can be thrown away when doing perturbation theory. Quite on the contrary, they usually contribute to correlation functions in the continuum limit through divergent () loop corrections. These irrelevant vertices are indeed important in many cases, contributing to the mass, coupling and wave-function renormalizations [Sharatchandra, 1978]. All these vertices are in fact necessary to ensure the gauge invariance of the physical amplitudes. Only when they are included can gauge-invariant Ward Identities be constructed, and the renormalizability of the lattice theory can be proven.

An example of this fact is given by the diagrams contributing to the gluon self-energy at one loop (Fig. 6). If one would only consider the diagrams on the upper row, that is the ones that would also exist in the continuum, the lattice results would contain an unphysical divergence. This divergence is canceled away only when the results of the diagrams on the lower row are added, that is only when gauge invariance is fully restored. Notice that for this to happen also the measure counterterm is needed (see Section 5.2.1). In a similar way, terms of the type , which are not Lorentz covariant and are often present in the individual diagrams, disappear only after all diagrams have been considered and summed.

From what we have seen so far, we can understand that a lattice regularization does not just amount to introducing in the theory a momentum cutoff. In fact, it is a more complicated regularization than just setting a nonzero lattice spacing, because one has also to provide a lattice action. Different actions define different lattice regularizations. Because of the particular form of lattice actions, the Feynman rules are much more complicated that in the continuum, and in the case of gauge theories new interaction vertices appear which have no analog in the continuum. The structure of lattice integrals is also completely different, due to the overall periodicity which causes the appearance of trigonometric functions. The lattice integrands are then given by rational functions of trigonometric expressions.

At the end of the day, lattice perturbation theory is much more complicated than continuum perturbation theory: there are more fundamental vertices and more diagrams, and these propagators and vertices, with which one builds the Feynman diagrams, are more complicated on the lattice than they are in the continuum, which can lead to expressions containing a huge number of terms. Finally, one has also to evaluate more complicated integrals. Lattice perturbative calculations are thus rather involved. As a consequence, for the calculation of all but the simplest matrix elements computer codes have to be used (see Section 16).

Matrix elements computed in euclidean space do not always correspond to the analytic continuation of matrix elements of a physical theory in Minkowski space. For this to happen, the lattice action has to satisfy a property known as reflection positivity, which involves time reflections and complex conjugations (roughly speaking is the analog of hermitian conjugation in Minkowski space). In this case the reconstruction theorem of Osterwalder and Schrader (1973; 1975) says that it is possible to reconstruct a Hilbert space in Minkowski space in the usual way starting from the lattice theory.

The Wilson action with is reflection positive, and therefore
corresponds to a well-defined physical theory in Minkowski
space [Lüscher, 1977, Creutz, 1987].
For instead the lattice theory contains additional time doublers,
which disappear in the continuum limit. ^{12}^{12}12This is at variance with the
doublers which appear for (naive fermion action), which do not disappear
in the continuum limit, as we will see in Section 6.
In the following we will only work with .
This is what is usually meant for Wilson action.

### 5.1 Fourier transforms

To perform calculations of Feynman diagrams in momentum space (the main topic of this review) we need to define the Fourier transforms on the lattice. They are given in infinite volume (which is the standard setting in perturbation theory) by

(5.12) |

where with abuse of notation we indicate and its Fourier transformed function with the same symbol. The inverse Fourier transforms are given by

(5.13) |

and this means that

(5.14) |

This lattice delta function is zero except at . The Kronecker delta in position space is

(5.15) |

Of course on a lattice of finite volume the allowed momenta are a discrete set. However, in perturbation theory we will always consider the limit of infinite volume.

Notice that the Fourier transform of is taken at the point , that is in the middle of the link. This comes out naturally from its definition. This choice turns out to be quite important also for the general economy of the calculations, as we can see from the following example.

Let us consider the gauge interaction of the quarks in the Wilson action at first order in the gauge coupling, which gives rise to the quark-quark-gluon vertex. Going to momentum space we have

(5.17) | |||||

We can notice at this point that all exponential phases cancel with each other, because of the function expressing the momentum conservation at the vertex (where ). We are then left with

(5.18) | |||||

which gives us the lattice Feynman rule for this vertex. It is easy to see that in the continuum limit this Wilson vertex reduces to the familiar vertex of QCD,

(5.19) |

Notice that had we chosen for the Fourier transform of the gauge potential the expression

(5.20) |

the exponential phases would not have canceled, and the terms would still be present in the final expression of the vertex. This is a general feature of lattice perturbation theory: if one uses the Fourier transforms in Eq. (5.12), all terms of the type coming from the various gluons exactly combine to cancel all other phases floating around, and then only sine and cosine functions remain in the Feynman rules in momentum space. This cancellation becomes rather convenient in the case of complicated vertices containing a large number of gluons.

We are now going to give the explicit expressions for the propagators and for the vertices of order and of the Wilson action, which is all what is needed for 1-loop calculations. In the following we will not explicitly write the function of the momenta present in each vertex and propagator. In our conventions for the vertices, all gluon lines are entering, and when there are two quarks or ghosts one of them is entering and the other one is exiting.

### 5.2 Pure gauge action

As we have seen, in the Wilson formulation gauge invariance is imposed directly on the free fermion lattice action, so that the group elements appear instead of the algebra elements , which are the fundamental perturbative variables. To derive the gluon vertices from the pure gauge action, one has then to expand the ’s in the plaquette in terms of the ’s. As a consequence, an infinite number of interaction vertices are generated, which express the self-interaction of gluons, with any . Since the power of the coupling which appears in these vertices grows with the number of gluons, only a finite number of them is needed at any given order in .

In lattice QCD the ’s are also nontrivial color matrices, and therefore they do not commute with each other. The expansion of the plaquette in terms of the ’s can be carried out using the Baker-Campbell-Hausdorff formula

(5.21) |

Since the color matrices are traceless and are closed under commutation,
the exponent in the expansion of the plaquette obtained using the
Baker-Campbell-Hausdorff formula is also traceless, so that the knowledge of
the terms of this expansion which are cubic in is sufficient to
calculate all vertices with a maximum of four gluons, that is to order
, which is all that is needed for 1-loop calculations. ^{13}^{13}13For
computing vertices of higher order it is useful to know that the
Baker-Campbell-Hausdorff formula can be written as