Twisted mass lattice QCD
Introduction
Quantum Chromodynamics (QCD) is today considered the fundamental theory of strong interactions. The adimensional coupling of the theory , once it is renormalized, will depend on the energy scale of the considered physical process, and it gives a measure of the strength of the interaction at that energy scale. The outstanding property of QCD, asymptotic freedom, tells us that the coupling decreases with increasing energies. This allows the usage of perturbation theory to make phenomenological predictions for processes with large momentum transfers. However the increase of the coupling with decreasing energies does not allow one to use perturbative methods to compute physical quantities in the low energy region such as the mass spectrum or hadronic matrix elements. A possible strategy in this case is to use a non-perturbative regularization of the theory, introducing a discretized space–time (lattice) [1], [2]. This strategy has two main advantages: first it provides an ultraviolet regularization and secondly it reduces the degrees of freedom of the theory to a numerable infinity. Considering the theory in a finite volume lattice, it is possible to perform numerical simulations of QCD through Monte Carlo methods. The continuum QCD action has to be discretized in a sensible way, and a simple and attractive lattice action is the one proposed by Wilson long time ago [1], [2].
The relation between the coupling constant and the lattice spacing in physical units is given, in lattice QCD, by the renormalization group equations, and we are naturally interested in the continuum limit of the theory. The simulations of lattice QCD are always performed for values of corresponding, in the renormalized theory, to a finite and non zero value of . This introduces in the results of the simulations, using the Wilson fermion action, errors of order and numerically these O() errors could be of the order of 20%–30%. The O() discretization errors could be eliminated, in principle, following the Symanzik’s improvement program [3], [4], [5], where the O() cutoff effects in on-shell quantities are canceled by adding local O() counterterms to the lattice action and to the composite fields of interest [6], [7], [8], [9], [10]. A technical difficulty is that the improvement coefficients multiplying these counterterms are not known a priori, and they should be all computed using Monte Carlo simulations.
A new intriguing possibility is the so called automatic O() improvement [11], where none of the improvement coefficients are needed in order to have O() cutoff effects in physical observables. The basic idea is that the Wilson theory for fermions with a suitable infrared cutoff is, in the massless limit, free from O() errors. We will see that to extend automatic O() improvement to a theory in an infinite volume with a non zero mass term we have to add the so called twisted mass, keeping the standard quark mass to be zero. The twisted mass term, that in a way will act also as a sharp infrared cutoff, can be obtained in continuum QCD via a non-anomalous chiral rotation from the standard mass term. To be specific, if we consider QCD with a field describing a flavour doublet, the twisted mass term looks like where is the Pauli matrix in flavour space and is what is called the twisted mass.
The twisted mass term in a lattice QCD action appears to my knowledge for the first time in Ref. [12], where it is given an ansatz for the phase structure of Wilson fermions in the parameter space , where is the bare quark mass. Based on the analysis of the lattice Gross–Neveu model, and on the strong coupling expansion of Wilson lattice QCD, the author suggested that there are regions in the parameter space of and where the true vacuum has a non zero expectation value of signalling the spontaneous breaking of parity symmetry. It was then natural to propose, in order to pick up the real vacuum from numerical simulations, to add an external field to the original Lagrangian and to perform the limit . The is what now we would call twisted mass. The twisted mass in this case is then just an external field used to probe the structure of the vacuum of the theory and it has to be removed at the end of the computation. We will come back in detail to the chiral phase structure of the Wilson theory in Section 5. Here I just would like to mention that in the same paper a new method to improve the scaling behaviour of the chiral condensate was proposed based on the observation that the scaling violations of the condensate are odd under a change of sign of the coefficient of the Wilson term, and they can be easily averaged out. We will see in Appendix E that this is a possible starting point to understand automatic O() improvement.
The twisted mass term breaks parity and flavour symmetry. It is a natural question whether this mass term changes also the continuum action or just the discretization errors of the theory. In this report we will show that with a Wilson fermion lattice action the twisted mass term generates parity and flavour violating cutoff effects (in most cases of O()) which go away performing the continuum limit.
The fact that the twisted mass actually induces only flavour and parity breaking cutoff effects, and it is actually equivalent to QCD, can be understood considering an old remark made by Gasser and Leutwyler. In fact it was noticed many years ago [13] that the usage of what we would now call a twisted mass term is irrelevant in continuum QCD. The fermionic part of the 2 flavours QCD Lagrangian is usually given in the form where is the mass matrix. The quark masses in the standard model originate from the asymmetries of the electroweak vacuum. Since the electroweak interactions do not preserve parity there is no reason a priori for the quark mass term of QCD to be parity invariant, and can be generically written as . We assume here that is a traceless matrix to avoid an unnecessary discussion of the QCD vacuum angle. With a suitable non-anomalous chiral transformation of the quark fields the general mass term can always be brought to the standard form, where the mass matrix is diagonal with real positive eigenvalues and . The remaining part of the Lagrangian constrained by the requirement of renormalizability is left invariant by this chiral transformation. In brief, with a change of variables in the functional integral, that leaves the measure invariant, one can show that even if a general parity and flavour violating mass term is allowed, the request of having a renormalizable theory preserving gauge and Lorentz invariance, generates these “accidental symmetries”. This simple example shows that the specific form of the mass term in the 2 flavours continuum QCD Lagrangian is actually irrelevant. The reason for this is the fact that the massless theory is invariant under the chiral non-anomalous transformation that changes the form of the mass term.
This is just an example of a more general phenomenon. Renormalizable theories that describe electro-weak and strong interactions, can be considered as low energy effective theories of more general not necessarily renormalizable high energy theories. The condition of having low energy renormalizable field theories can be so stringent that the corresponding Lagrangian may turn out to obey extra accidental symmetries, that were not symmetries of the higher energy theories [14].
This observation becomes important on the lattice. If we discretize the continuum QCD action with Wilson fermions [1], the Wilson term explicitly breaks chiral symmetry and the lattice action is not invariant anymore under the field rotations mentioned before. But we still have the freedom to choose the Wilson term and the mass term to point in different relative “directions” in the Dirac and flavour space [15]. This freedom is the key to constrain the form of the cutoff effects induced by the Wilson term.
The observation that physical observables computed with the Wilson lattice action are automatically O() improved in the “infrared safe” (i.e. with no spontaneous symmetry breaking) chiral limit is relevant also for the renormalization properties of local operators that depend on the breaking of chiral symmetry induced by the Wilson term.
To summarize: Wilson twisted mass QCD is a lattice regularization that allows automatic O() improvement only tuning one parameter. The bare untwisted quark mass has to be tuned to the so called critical mass in order to maximally misalign the Wilson term and the mass term. In this approach the renormalization of local operators relevant for phenomenological applications is significantly simplified with respect to the standard Wilson regularization. The price to pay is the existence of O() cutoff effects that break parity and flavour symmetry. All these statements will be demonstrated explicitly in this report.
This is not the only review on twisted mass QCD. A set of lectures has been presented by S.Sint at the School “Perspectives in Lattice Gauge Theories” [16]. In these lectures a nice introduction on the basic setup, exceptional configurations, automatic O() improvement together with few applications of Wtm is given. Particular emphasis is also put in finite volume renormalization schemes with chirally twisted boundary conditions. In our report we enlarge the topics covered by Sint, and we elaborate on the ones already there. On the other hand we only marginally touch finite volume renormalization schemes with chirally twisted boundary conditions. For this reason we believe that the present review is in many respects complementary to the one of Sint, and together they can be used as a complete introduction to all the topics connected with twisted mass QCD.
The paper is organized as follows. In Section 2 I use classical considerations in continuum QCD to show the equivalence of twisted mass (tm) QCD and QCD. I also describe the rigorous theoretical properties of Wilson twisted mass QCD (Wtm QCD), for degenerate and non-degenerate quarks. In Section 3 I discuss the O() discretization errors of Wilson-like lattice actions. In particular I show several proofs of automatic O() improvement, with a particular emphasis on the choice of the critical mass. Numerical results confirming this property will conclude the section. In Section 4 I derive again some of the results obtained in the previous sections using a different fermion basis. Hopefully this could help the reader in a better understanding of the subject of this review. In Section 5 I analyse O() parity and isospin violating discretization errors and O() cutoff effects responsible for the non trivial chiral phase structure of the lattice theory. In Section 6 I discuss selected numerical results obtained with Wtm QCD and show different methods to ease the renormalization of local operators. In Section 7 I make a short digression on algorithms to simulate light Wilson-like quarks. This is an important prerequisite for part of the numerical results presented in this review. Conventions, notations and more technical discussions are deferred to the Appendix.
Section snippets
Basic properties
In this section we introduce twisted mass QCD in the continuum using classical arguments for a doublet of degenerate quarks. This academic exercise allows the reader to get acquainted with twisted mass QCD and to learn how to relate correlation functions from QCD to twisted mass QCD. To extend this concepts at a quantum level we discretize the theory on a lattice using Wilson fermions. The resulting theory Wilson twisted mass (Wtm) QCD is ultra-local, unitary, reflection positive and
Non-degenerate quarks
In this section we show how twisted mass QCD can be generalized to a doublet of non-degenerate quarks. We extend the topics covered in the previous section to the non-degenerate case, emphasizing the main differences with the degenerate case.
O() improvement
The continuum limit of lattice QCD is of fundamental importance to relate numerical simulations with experimental results. Practically to perform the continuum limit it is crucial to simulate at several values of the lattice spacing , and it is also possible (and sometimes mandatory) to improve the rate of the discretization errors from to , using a suitable lattice QCD action. A possibility is to apply Symanzik’s improvement program [3], [4], [5], where the O() cutoff effects in on-shell
The physical basis
In the previous two sections we have seen how renormalizability and O() cutoff effects of Wtm can be analyzed using the twisted basis. In the twisted basis the Wilson term takes the standard form while the mass term takes a “twisted” form. In this section we want to rederive some of the results already presented using the so called physical basis [11]. The physical basis is obtained from the twisted basis performing an axial rotation in the isospin direction in the bare lattice theory. If
O() cutoff effects
In this section I will be mainly concerned with the O() cutoff effects of the Wtm formulation and in particular on the cutoff effects induced by the breaking of flavour and parity symmetry. Some of these require a formal description of the quantum mechanical representation for a lattice correlator computed with the Wtm action. The first part of this section will be devoted to a brief introduction of the basic Hamiltonian formalism, while the second part will be concentrated on the typical
Renormalization and weak matrix elements
In the previous sections we have extensively discussed the issue of the continuum limit and we have analyzed cutoff effects of order and with Wtm.
Renormalization is necessary in order to perform the continuum limit and correctly evaluate hadronic matrix elements. Here we will not discuss the way how the renormalization is performed but only the mixing patterns of relevant physical quantities according to the lattice action used, i.e. according to the way the theory is regularized.
Algorithms for dynamical fermions
At present the only practical way to simulate numerically a 4-D Euclidean quantum field theory with fermions is to perform the Grassmann integral on the fermion fields analytically and then to apply Monte Carlo methods in the resulting effective bosonic theory.
After integrating out the fermion fields the partition function (2.22) of Wtm for degenerate flavors is given by with The reality of the effective action is only
Concluding remarks
I have presented an overview of the theoretical properties and numerical results of Wilson twisted mass QCD. To warm up, I started by considering a classical theory with a twisted mass term (tmQCD). In the continuum a twisted mass term can always be rotated away by a non-anomalous change of fermion variables in the functional integral.
On the lattice the twisted mass cannot be rotated away: the standard Wilson lattice action and the Wilson twisted mass (Wtm) lattice action are not related by a
Acknowledgements
I am indebted to all those people who have contributed, in one way or another, to the development and understanding of Wilson twisted mass QCD. Special thanks go to Roberto Frezzotti, Karl Jansen and Giancarlo Rossi for many enlightening discussions and suggestions in several subjects covered by this report, and for a careful reading of parts of this manuscript. I thank the mysterious referee for a careful reading and many interesting remarks. Special thanks to all the members of the European
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