273. Lattice Chiral Gauge Theories: What’s the Problem?

This is a guest post by Yanwen Shang. He was a graduate student of Gregory Gabadadze at NYU and is now a postdoc at the U. of Toronto working with Erich Poppitz. There will be actually two posts discussing lattice fermions – one today and another tomorrow. Dmitry.

I’d like to thank Dmitry first for the very kind invitation to write a guest post on my recent paper arXiv:0901.3402 done with E. Poppitz on lattice chiral gauge theories. Since it is not a very widely appreciated subject within the high energy theory community, I figure that, instead of going to the details about our paper, it is probably more useful to give a brief overview on why such a subject resisted decades of attacks from many extremely intelligent theorists, why we need the so called “Ginsparg-Wilson” chiral fermions, and where the current situation is at.

Intro
To high energy theorists, chiral gauge theory is not an unfamiliar concept. A Dirac spinor consists of two Weyl spinors of opposite chiralities: the left- and right-handed ones. A chiral gauge theory is a theory described by many Weyl fermions, not necessarily forming into vector multiplets, living in some complex representation of a Lie Group and minimally coupled to the corresponding gauge field.

According to our experimentally verified knowledge so far, nature is described by such a theory. It might sound a bit mysterious (but perhaps not surprising) that we don’t yet have a practical method of approximating any chiral gauge theory by latticing and then simulating it on a computer.  If one wishes to use lattice regularizations to gain some insights into its non-perturbative properties, we just don’t know how to do it.

An ambiguous phase
Before I explain why defining lattice chiral gauge theories is so hard, I should first mention that the partition function of any chiral theory is rigorously speaking not defined. This is because operators that map between two independent vector spaces don’t have a natural definition of its determinant [1].  Take for example the kinetic term of a pair of left-handed Weyl fermions:

where D stands for the usual Weyl operator and and  are two independent left-handed Weyl fermions. Attempting to define the Grassmann path integral:
,
one imagines choosing a set of orthonormal basis, say {} and {}, and expanding in them the fermion fields with Grassmann coefficients. The subscript i runs from 1 to the dimension of the fermion phase space, say it’s d, which we assume to be the same for both and for simplicity. Certainly d is infinite for theories defined in the continuum spacetime, and this discussion is rather formal at the moment. The partition function is evaluated by just defining , where is a d by d matrix (of indices i and j), whose determinant is always defined.  

Notice, however, the basis {} and {} are not unique. We can equally well choose, instead of {} for example, a different set of vectors , where is a unitary matrix. If we compute Z as above but use the new vectors {}, we find the result differs by a factor of . This factor is always a pure complex phase since is unitary. The argument presented here is totally equivalent to the trivial fact that the identification between two vector spaces of same dimension is not unique.

Such a phase ambiguity of course is usually not a severe problem since it’s sufficient to just choose and then stick to a particular set of basis vectors. The ambiguous phase of Z is always divided out in the VEV’s of any operators which are the real physical observables. 
 
The fermion doubling problem
As a most simple-minded attempt to define lattice theories of fermions, one may simply discretize the Dirac operator as we would normally do for the Laplacian operator. Not surprisingly, it won’t work, or we wouldn’t have to work so hard. The problem is when one takes the continuum limit of such a lattice theory, one finds, most intriguingly, every Weyl fermion introduced in the action is accompanied by an unexpected dude with the opposite chirality.

One can most easily see, however strange this may sound, it must happen by recalling the axial anomaly.  As we all know, in QFT with Dirac spinors, even if the action is invariant under the axial  rotation:
,
the partition function, is in general not invariant. One way to understand this is due to Fujikawa [2], who pointed out that although naively such a field rotation should certainly leave the  fermion measure unchanged (as for arbitrary ), because the fermion phase space is infinitely dimensional, d= in our notation, it needs to be regularized and there exists no method of doing so respecting both the vector and the axial symmetry simultaneously. Trying to repeat the same argument on the lattice immediately leads to problems. On a finite lattice, the theory is perfectly regularized and d is a finite integer. We apparently “do find” a regularization scheme of the measure that’s invariant under both vector and axial rotations, leaving absolutely no room for anomaly to arise. Apply this logic to gauge symmetries and gauge anomalies in chiral theories, we would similarly conclude the nonexistence of gauge anomalies on the lattice. This is only possible if every Weyl fermion that survives in the continuum limit is accompanied by another with precisely the same charge but opposite chirality. In other words, the fermion species are automatically doubled. (Remember, left- and right-handed Weyl fermions contribute to the anomaly with opposite signs and they cancel if their charges equal.)

This is the (in)famous fermion doubling problem. 

Ginsparg-Wilson formalism: an elegant solution
According to the above discussion, it should be clear that the only way to avoid the fermion doubling problem on the lattice is to explicitly break the axial (or chiral) symmetry slightly so that the action is only approximately invariant. One hopes somehow in the continuum limit, this approximate symmetry becomes exact and meanwhile the anomaly is properly generated. Methods of this kind exist and are widely used in vector-like theories, e.g. QCD.

But approximate symmetry in lattice QFT is nasty and has many disadvantages. For one, it forbids pure chiral actions or any action with a vector-like fermion content (meaning equal number of left- and right-handed Weyl fermions with the same charge so they form vector multiplets) but consisting of two separated chiral sectors. Because by the definition that chiral symmetry is explicitly broken, fermions with opposite chiralities must couple.

Ginsparg and Wilson in 1982 proposed an elegant alternative [3]. They suggest replacing the Dirac operator on the lattice by the so called GW operator, which satisfies the two conditions:
.
Here, is the lattice spacing. In the continuum limit , the rhs of the above anti-commutator vanishes and D approaches the Dirac operator.

Inserting the Ginsparg-Wilson operator in the fermion kinetic terms is sufficient to eliminate the fermion doubling problem. It works, of course, because D explicitly breaks the chiral symmetry (as it doesn’t anti commute with ). But here is what’s really beautiful. If we define a “new ” matrix by
,
the following two equations are exact:
.
The first equation says that actions like are invariant under the “GW axial rotation”:
,
which is an exact symmetry on a finite lattice (as far as the action is concerned). In the continuum limit when the lattice spacing vanishes, and the more traditional axial rotation is recovered.

The fact implies that define two projection operators, and therefore one can build, out of Dirac spinors, the Ginsparg-Wilson chiral fermions as the following:
.
This definition for spinor is identical to the usual, but for , is slightly different but the difference disappears in the continuum limit. Now, we can define “chiral” theories on a finite lattice free of fermion doubling problem using these “GW chiral fermions”. For example, a single massless free fermion can be described by the action , where is NOT the usual Weyl fermion that we are familiar with in the continuum.

The advantage of Ginsparg-Wilson formalism is it eliminates the fermion doubling problem and at the same time defines an exact chiral symmetry on the lattice. Consequently, fermions with opposite “GW chiralities” can be disentangled in the action. Let me just mentioin that exactly correct chiral anomaly is derived in this formalism because generically . In a sense, it works out just as Fujikawa pointed out in the continuum: anomaly exists because the fermion measure is not invariant.

The mystery only gets deeper: the fermion measure problem
Sadly the beauty of the GW formalism is not the end of the story and troubles arise the moment we consider dynamical gauge field. My earlier discussion about the phase ambiguity of chiral partition functions finally becomes relevant. Again, consider the partition function
,
but with being the “GW chiral fermions” now. To define Z, one chooses sets of orthonormal basis for both and , denoted as and respectively and evaluate Z as . As mentioned, the freedom of rotating either of the two basis by a unitary matrix renders the phase of Z ambiguous.

What’s changed in the Ginsparg-Wilson formalism is that with dynamic gauge field in consideration, one no longer has the option to just fix a set of {} and then stick to them. Because the definition of involves , which depends on the Ginsparg-Wilson operator D, which in turn depends on the gauge field background since D approaches the covariant Dirac operator in the continuum limit. When gauge field varies, the subspace in which lives rotates (or flips if you wish) along. The same set of ‘s inevitably fails to span it. It becomes mandatory to choose a different orthonormal basis for every gauge field configuration, and all of them are subjected to arbitrarily unitary rotations. One finds, in the Ginsparg-Wilson case, not a single ambiguous phase but a U(1) bundle fibred over the entire gauge field configuration space, or more rigorously speaking, a U(1) bundle fibred over the gauge field configuration space modular all the gauge transformations since the phase of Z should be at least gauge invariant to be considered interesting. A particular way of fixing the phase of Z for all gauge field background is fixing a global section of this U(1) bundle. It’s not obvious whether such a global section exists and when it does is unique. This is often referred to as the fermion measure problem.

Luscher proved that such a global section can be found and is unique in the case of Abelian gauge group, if and only if the chiral fermion content satisfies the gauge anomaly cancellation conditions, i.e. in 4-D [4]. Here and are the charges of left- and right-handed chiral fermions respectively. Although there is no natural choice of a connection on this U(1) bundle, as observed by Luscher (and others such as Neuberger [5]), a natural definition of its curvature, solely determined in terms of the Ginsparg-Wilson operator D, exists. Integrating this curvature over any nontrivial cycles in the gauge field configuration space, Luscher found it vanishes precisely when the gauge anomaly cancellation condition is obeyed. This result implies the statement given in the beginning of this paragraph.

Program remains open
Despite Luscher’s existence proof, a practical way of implementing the theory on a computer is still missing. At the moment of writing, a generalization of Luscher’s proof to non-Abelian gauge theories remains unknown.

References:
[1] L. Alvarez-Gaume and E. Witten, “Gravitational Anomalies,” Nucl. Phys. B 234, 269 (1984).
[2] “Path Integrals and Quantum Anomalies,” K. Fujikawa, Hiroshi Suzuki, Oxford University Press, 2004.
[3] P. H. Ginsparg and K. G. Wilson, “A Remnant Of Chiral Symmetry On The Lattice,” Phys. Rev. D 25, 2649 (1982).
[4] M. Luscher, “Abelian chiral gauge theories on the lattice with exact gauge invariance,” Nucl. Phys. B 549, 295 (1999) [arXiv:hep-lat/9811032].
[5] H. Neuberger, “Geometrical aspects of chiral anomalies in the overlap,” Phys. Rev. D 59, 085006 (1999) [arXiv:hep-lat/9802033].