The physics of strong interactions is invariant under the exchange of
left-handed and right-handed quarks, at least in the massless limit. This
invariance is reflected in the chiral symmetry of quantum chromodynamics.
Surprisingly, it has become clear only recently how to implement this important
symmetry in lattice formulations of quantum field theories. We will discuss
realizations of exact lattice chiral symmetry and give an example of the
computation of a physical observable in quantum chromodynamics where chiral
symmetry is important. This calculation is performed by relying on finite size
scaling methods as predicted by chiral perturbation theory.Comment: Contribution to the NIC Symposium held in Juelich on December 5 and
  6, 2001. It is intended for a general audience of computational scientist

Hernandez, Pilar

Jansen, Karl

Lellouch, Laurent

English

arXiv

The physics of strong interactions is invariant under the exchange of left-handed and right-handed quarks, at least in the massless limit. This invariance is reflected in the chiral symmetry of quantum chromodynamics. Surprisingly, it has become clear only recently how to implement this important symmetry in lattice formulations of quantum field theories. We will discuss realizations of exact lattice chiral symmetry and give an example of the computation of a physical observable in quantum chromodynamics where chiral symmetry is important. This calculation is performed by relying on finite size scaling methods as predicted by chiral perturbation theory.The physics of strong interactions is invariant under the exchange of left-handed and right-handed quarks, at least in the massless limit. This invariance is reflected in the chiral symmetry of quantum chromodynamics. Surprisingly, it has become clear only recently how to implement this important symmetry in lattice formulations of quantum field theories. We will discuss realizations of exact lattice chiral symmetry and give an example of the computation of a physical observable in quantum chromodynamics where chiral symmetry is important. This calculation is performed by relying on finite size scaling methods as predicted by chiral perturbation theory

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DESY 01-204CPT-2001/P.4272From enemies to friends: chiral symmetry on the latticePilar Herna´ndez1, Karl Jansen2, and Laurent Lellouch31 CERN, Theory Division,CH-1211 Geneva 23, SwitzerlandE-mail: Pilar.Hernandez@cern.ch2 NIC/DESY Zeuthen,Platanenallee 6, D-15738 Zeuthen, Germany,E-mail: Karl.Jansen@desy.de3 Centre de Physique The´orique, Case 907, CNRS Luminy,F-13288 Marseille Cedex 9, FranceE-mail: lellouch@cpt.univ-mrs.frThe physics of strong interactions is invariant under the exchange of left-handed and right-handed quarks, at least in the massless limit. This invariance is reflected in the chiral symmetryof quantum chromodynamics. Surprisingly, it has become clear only recently how to implementthis important symmetry in lattice formulations of quantum field theories. We will discuss real-izations of exact lattice chiral symmetry and give an example of the computation of a physicalobservable in quantum chromodynamics where chiral symmetry is important. This calculationis performed by relying on finite size scaling methods as predicted by chiral perturbation theory.1 IntroductionNature as well as physicists like symmetries. The –somewhat amusing– reason for this isthat symmetries can be broken. A very important concept is the spontaneous breakdown ofa symmetry: here some symmetry is broken down to a situation with less symmetry whentuning a parameter (e.g. the temperature or the coupling strength) of the theory to a criticalvalue. In the process a non-vanishing vacuum expectation value is developed that breaksthe symmetry of the interaction. The process is accompanied by the appearance of so-called Goldstone particles (spinwaves) that are massless. This phenomenon comes underthe name of the Goldstone theorem. An example is the spontaneous magnetization: a metalat high temperature is in a symmetric state – the elementary magnets or spins can point inany direction such that the net magnetization vanishes. Decreasing the temperature belowsome critical value, a spontaneous magnetization occurs, the spins point into a prefereddirection and the metal becomes magnetic.Within the standard model of elementary particle interactions we know two placeswhere such a spontaneous symmetry breaking is supposed to have occurred: in the elec-troweak sector spontaneous symmetry breaking manifests itself in the Higgs phenomenonwith the development of a Higgs field expectation value hΦi, giving mass to the elementaryparticles, and the appearance of Goldstone particles leading to the W- and Z-bosons.1In our theory of strong interactions, quantum chromodynamics (QCD), it is a chiralsymmetry that is assumed to be spontaneously broken. This symmetry allows for an in-terchange of left handed and right handed quarks while leaving physics invariant – at leastwhen these quarks are massless. In this case a scalar quark-antiquark qq¯ condensate isdeveloped and the Goldstone particles are identified with the light pions that are observedin nature.As stated above the occurrence of spontaneous symmetry breaking is an assumption.The phenomenon is inherently non-perturbative and cannot be addressed with approxima-tive methods like perturbation theory. However, even with numerical simulations it is dif-ficult to test, whether a certain model exhibits spontaneous symmetry breaking (SSB). Thereason for this becomes clear when the way to detect SSB is considered. Let us choose asystem that has a finite physical volume V as would be required for numerical simulations.Further, we couple the system to an external magnetic field. Spontaneous symmetry break-ing is tested in a double limit, where first the volume of the system is sent to infinity andthen the external magnetic field is sent to zero. If a non-vanishing magnetization remains,spontaneous symmetry breaking is identified. Obviously, such a procedure is unfeasiblewithin the approach of numerical simulations.The way out is the use of chiral perturbation theory 1. In this approach chiral symme-try breaking is taken as an assumption with the consequences of the appearance of non-vanishing field expectation values and Goldstone particles. A special situation arises whenthe size of the box becomes comparable to or even smaller than the Compton-wavelengthof the Goldstone particle. Then the corresponding field can be considered as being uni-form and it is possible to set up a systematic expansion that starts in the lowest order withan effective lagrangian of this constant mode and then taking systematically higher orderfluctuations into account 2.2 Example of Φ4-theoryLet us give an example of the Ginsburg-Landau or Φ4 theory withO(N)-symmetry in fourdimensions. The action of this theory is defined byS =Zd4x12(@µΦ(x))2 +12m20Φ(x)2 + 0Φ(x)4 + jαΦα(x) (1)with Φ a N -component vector, m0 the bare mass, 0 the bare quartic coupling and jα aconstant external source in direction  of the groupO(N).Expectation values of observables are computed through the partition function or pathintegral in finite volume V = L4,hΦij,V =ZDΦΦe−S : (2)Symmetry breaking is detected via a spontaneous magnetization hΦi 6= 0 in the doublelimithΦi = limj!0limV!1hΦij,V : (3)2Chiral perturbation theory can provide now a prediction for the behaviour of hΦij,V , i.e.the expectation value in finite volume and at non vanishing external field. In a situationwhen the external field j and 1=V are very small, hΦij,V is given in terms of the expectationvalue of hΦi at j = 0 and V = 1:hΦij,V = f(u) (4)where u is a scaling variableu = hΦijV (5)that contains the infinite volume and j = 0 spontaneous magnetization hΦi.Let us give for completeness the function f(u):f(u) =u2(u)jVwith (u) =1uI2(u)I1(u)(6)with I1, I2 modified Bessel functions. The important point here is that these functions onlydepend on a single scaling variable u and hence on the quantity of interest, the magnetiza-tion hΦi.In an already quite old work 3 the prediction of chiral perturbation theory, eq. (4), wasconfronted with numerical data. In this test the data obtained through numerical simula-tions were described very well by the theoretical formulae of chiral perturbation theoryand the method of chiral perturbation theory turned out to be very fruitful, at least in thisexample of a relatively simple model. The lesson that is to be learnt from this exampleis that finite size effects can actually be used to determine properties of an infinitely largesystem. In this sense, the finite volume system can be regarded as a probe of the target the-ory in infinite volume. This gives an entirely new perspective on finite size effects: insteadof being afraid of them, they can be used to determine important physical information.3 The example of quantum chromodynamicsChiral perturbation theory as well as the lattice method was developed for quantum chro-modynamics in order to deepen our understanding of the strong interactions. Still, untilrelatively recently, both approaches could not really come together, at least in the regionof very small quark masses where chiral symmetry starts to get restored. The reason wasthat the lattice seemed to be lacking the concept of chiral symmetry and for many years theinfamous Nielson-Ninomiya theorem 4 was telling us that it would even be impossible toimplement chiral symmetry in a consistent way on a lattice.The situation only changed a few years ago, when an old work by Ginsparg and Wil-son 5 was rediscovered 6. The Ginsparg-Wilson paper contained actually a clue for an-swering the problem of chiral fermions on the lattice. The interaction of the fermions isdescribed by some particular operator, the Dirac operator, the details of which should notbe discussed here. Now, in the continuum theory this Dirac operator anticommutes witha certain 4-dimensional matrix γ5 = diag(1; 1;−1;−1). On a lattice with non-vanishinglattice spacing a, such an anti-commutation property cannot be demanded. If the anti-commutation property is insisted on, the fermion spectrum of the lattice theory does notcorrespond to the one of the target continuum theory.3The suggestion of Ginsparg and Wilson was to replace the anticommutation condi-tion by a relation (now known as the Ginsparg-Wilson (GW) relation) for a lattice DiracoperatorD:γ5D +Dγ5 = aDγ5D : (7)Clearly, in the limit that the lattice spacing vanishes the usual anti-commutation relation ofthe continuum theory is recovered.The fact that renders the relation eq. (7) conceptually extremely fruitful is that it impliesan exact chiral symmetry on the lattice even if the value of the lattice spacing does notvanish 7. The notion of a chiral symmetry on the lattice is a conceptual breakthrough andrenders the lattice theory in many respects to behave like its continuum counterpart withfar reaching consequences.However, as nice as the theoretical progress that followed the rediscovery of theGinsparg-Wilson relation was as much of a challenge are realizations of operators D thatsatisfy the Ginsparg-Wilson relation (see 8,9,10 for reviews). Let us give a particular exam-ple for such a solution as found by H. Neuberger 11 from the overlap formalism 12, basedon the pioneering work of D. Kaplan 13. To this end, we first consider the standard WilsonDirac operator on the lattice:Dw =12γµ(rµ +rµ)− arµrµ} (8)with rµ, rµ the lattice forward, backward derivatives, i.e. nearest neighbour differences,acting on a field Φ(x)rµΦ(x) = Φ(x+ )− Φ(x)rµΦ(x) = Φ(x)− Φ(x+ ) :We then defineA = 1 + s−Dw (9)with 0 < s < 1 a tunable parameter. Then Neuberger’s operatorDN with mass m is givenbyDN =1− m2(1 + s)D(0)N +m(10)whereD(0)N = (1 + s)h1−A(AyA)−1/2i: (11)What is important here in the definition of Neuberger’s operator is the appearance ofthe square root of the operator AyA. This means that DN connects all points of the latticewith each other. Note, however, that despite this the operator is a local operator in thefield theoretical sense 15. In practice, the operator AyA is represented by a matrix thatis, unfortunately, very large. Having a physical volume of size L4, the number of sites isN4 = (L=a)4. As the internal number of degrees of freedom per lattice point is 12, weend up with A being a (12N4) ⊗ (12N4) complex matrix with N typically in the range10 < N < 30 for present days simulations. Hence we have to construct the square root of a4Figure 1. The low-lying end of the eigenvalue spectrum of the matrix A†A.very large matrix. What is worse, to compute relevant physical observables, we don’t needthe operator DN itself but its inverse. Such an inverse is constructed by iterative methodslike the conjugate gradient algorithm and its relatives 14.The square root can be constructed by a polynomial expansion, normally based on aChebyshev approximation, or by rational approximations. Both methods give comparableperformances in practice. The convergence of the approximation to the square root isdetermined by the condition number of the (positive definite) matrixAyA. When the matrixAyA is normalized such that the largest eigenvalue is one, the condition number is givenby the inverse of the lowest eigenvalue. We show in figure 1 the low-lying eigenvaluesof AyA for different values of  where the parameter  is inversely proportional to thecoupling strength of the theory.It can clearly be observed that very small eigenvalues can occur resulting in large con-dition numbers. In such situations the convergence of the approximations chosen can berather slow and special tricks have to be implemented to accelerate the convergence. Themost fruitful improvement is to treat a part of the low-lying end of the spectrum exactlyby projecting this part out of the matrix AyA 16,17. Further improvements can be imple-5mented, examples of which are discussed in ref. 17.Despite all these technical improvements it is found that a typical value for the degreeof a polynomial is O(100) and a typical value for the number of iterations to compute theinverse of DN is again O(100). Since in each iteration to compute D−1N the Chebyshevpolynomial has to be evaluated, this means that for a value of a physical observable ona single configuration ten thousand applications of a huge matrix on a vector has to beperformed. To compute the expectation value of some observable, this observables has tobe averaged over many gluonic configurations. Clearly, this results in a very demandingcomputational effort and gives rise to a numerical challenge well suited for NIC.The only thing that helps a lot in this problem is that the matrixA is sparse. Only thediagonal and a few sub-diagonals are actually filled, a fact that finally makes the problemmanageable – although, still a very large amount of computer resources are needed totackle it.4 The scalar condensateThe existence of an exact lattice chiral symmetry allows the use of finite size effects to testfor SSB in QCD as in the case of the Φ4-theory. Using a chiral invariant formulation oflattice QCD, it is possible to reach the region of very small quark masses where it is to beexpected that chiral symmetry starts to get restored.The “magnetization” in the case of quantum chromodynamics is the condensate of aquark-antiquark state Σ = h  ¯i. The role of the external magnetic field is played by thequark mass mq . We have developed a fully parallelized code with all technical improve-ments implemented. This allowed us to compute the scalar condensate as a function ofthe quark mass at several volumes. A (standard) caveat here is the fact that all compu-tations are done in the so-called quenched approximation where all internal quark loopsare neglected. In QCD there is a peculiarity: the field configurations can have topolog-ical properties, characterized by the so-called topological charge which can be measured–unambiguously– through the number of zero modes of the operator DN. In fact, theformulae from quenched chiral perturbation theory are parametrized by the topologicalcharge and it is hence very important to be able to identify the topological charge of thegauge field configurations. Without the special properties of lattice Dirac operators thatsatisfy the Ginsparg-Wilson relation such an identification would be very difficult.Let us give the complete theoretical formula from quenched chiral perturbation theoryin lowest order:Σν=1(mq; V ) = Σ z [Iν(z)Kν(z) + Iν+1(z)Kν−1(z)] + C mq=a2 : (12)The only important thing to notice here is that this relatively involved combination ofBessel functions do, as in the case of the Φ4-theory, only depend on one scaling variablez = ΣmqV (13)that contains the quantity of interest, namely the infinite volume, chiral limit scalar con-densate Σ. The additional term C  mq=a2 is a power divergence that comes from therenormalization properties of the theory. We will not discuss this field theoretical aspecthere but just notice that this term has to be included in the fit.6Figure 2. The scalar condensate computed on lattices of various size as a function of the quark mass. The solidlines are 2-parameter (Σ and the constant C of eq. (12)) fits according to the prediction of chiral perturbationtheory.In figure 2, we show the result of our numerical computation of the scalar condensate 18in a fixed topological charge sector jj = 1 as a function of the quark mass at severalvolumes. The solid line is a fit to the prediction of chiral perturbation theory, eq.(12). Wefind that the simulation data are described by this prediction very well. This means thatwe find evidence for the basic assumption on which the theoretical prediction relies: theappearance of spontaneous chiral symmetry breaking in (quenched) QCD.We want to remark that this work that has been performed at NIC was the first of thiskind. The project consumed 1400 CPU hours on a typical distribution of the lattice on 128nodes. After this work, a number of other groups repeated such an analysis 21,22,23 andit was reassuring to observe that very consistent results were found. In a subsequel work19,20 we developed also a quite general method for renormalizing the value of the barescalar condensate as extracted from the finite size scaling analysis performed here.75 ConclusionIn this contribution we have demonstrated that by a combination of theoretical ideas, im-proved numerical methods and the use of powerful supercomputer platforms it is possibleto test basic properties of field theories. Of particular interest was the question of whetherthe phenomenon of spontaneous symmetry breaking does occur in certain field theoriesimportant in elementary particle physics. The phenomenon of SSB leads to far reachingconsequences in theories like QCD or the scalar sector of the electroweak interactions. Inthe work performed here, we found strong evidence for the appearance of spontaneouschiral symmetry breaking in quenched lattice QCD. This conclusion is the result of thefact that two theoretical concepts, the lattice approach to quantum field theories and chiralsymmetry, met finally – and that enemies became friends.We are very much indebted to Hartmut Wittig for numerous discussions, suggestionsand finally participating in the project at the stage when the non-perturbative renormal-ization of the scalar condensate was computed. L.L. thanks the INT at the Universityof Washington for its hospitality and the DOE for partial support during the completionof this work. This work was supported in part by the EU TMR program under contractFMRX-CT98-0169 and the EU HP program under contract HPRN-CT-2000-00145.References1. S. Weinberg, Physica A96 (1979) 327; J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.)158 (1984) 142.2. J. Gasser and H. Leutwyler, Phys. Lett. B188 (1987) 477;H. Neuberger, Phys. Rev. Lett. 60 (1988) 889; Nucl. Phys. B 300 (1988) 180;F. C. Hansen, Nucl. Phys. B345 (1990) 685; F. C. Hansen and H. Leutwyler, Nucl.Phys. B 350 (1991) 201;P. Hasenfratz and H. Leutwyler, Nucl. Phys. B 343 (1990) 241.3. A. Hasenfratz, K. Jansen, J. Jersa´k, H.A. Kastrup, C.B. Lang, H. Leutwyler andT. Neuhaus, Nucl. Phys. B356 (1991) 332.4. H. B. Nielsen and M. Ninomiya, Phys.Lett.B105 (1981) 219.5. P.H. Ginsparg and K.G. Wilson, Phys. Rev. D25 (1982) 2649.6. P. Hasenfratz, Nucl. Phys. B (Proc.Suppl.) 63A-C (1998) 53; P. Hasenfratz, V.Laliena and F. Niedermayer, (1998), hep-lat/9801021.7. M. Lu¨scher, Phys. Lett. B428 (1998) 342, hep-lat/9802011.8. F. Niedermayer, Nucl. Phys. Proc. Suppl. 73 (1999) 105, hep-lat/9810026.9. T. Blum Nucl. Phys. Proc. Suppl. 73 (1999) 167, hep-lat/9810017.10. P. Herna´ndez hep-lat/0110218.11. H. Neuberger, Phys. Lett. B417 (1998) 141, hep-lat/9707022.12. R. Narayanan, H. Neuberger, Nucl.Phys. B412 (1994) 574, hep-lat/9307006.13. D.B. Kaplan, Phys.RLett. B288 (1992) 342.14. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. 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From enemies to friends: chiral symmetry on the lattice

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