a) Our argument goes as the following. We reach our conclusion that there must be something light by studying the photon two point function. It turns out it’s sufficient to study this object at the zero gauge field background. If the corresponding gauge symmetry is anomalous, we can easily prove, using the anomaly equation, that there must be a massless pole in the real part of this polarization operator. Since this operator is just some two point correlation function of the theory, it’s clear that there must be a light degree of freedom propagating. As for the additional global symmetry, we can pretentiously gauge it momentarily, and repeat the same argument. Since eventually everything can be studied at the zero gauge field background, pretending to gauge it will not change the conclusions. Usually, this additional global symmetry, once gauged, will turns out anomalous, and even it it’s anomaly free, there will be mixed anomaly between this additional global symmetry (temporarily gauged) and the original gauge symmetry, and similar anomaly equation will lead to the same results.

b) When we say the state is a composite state, we just mean in order to create such a state out of the vacuum, we must use combined operators in terms of the original fields. So in this language, only states created by the original field introduced in the action are not composite states.

c) I have no idea. Non-abelian case may or may not be too much more difficult then the abelian case. The major issue is that in the non-abelian case, the topological property of the gauge field configuration space may be a lot more complicated. The main issue of the chiral gauge theories in Ginsparg-Wilson formalism is the fermion measure problem, which is deeply related to the topological properties of the U(1) bundle fibered over the gauge field configuration space that I mentioned in the end of the earlier post, which is likely a much more complicated object in the non-abelian case.

Thanks a lot, I’ve now got the idea why you need strong coupling symmetric phase. Two more questions:

a) you say that the presence of additional global symmetries in mirror sector will lead to impossibility to decouple it. Do you understand physically why it is so, why global symmetries are so important?

You also mentioned that a massless mode in the mirror sector appears in the presence of global symmetry, what’s the nature of it? Would it be local symmetry, one might have expected a spontaneous symmetry breaking and Goldstone mode, but in this case symmetry is global.

b) How do you define on the lattice that the massless mode is composite, what do you look at?

c) Do you have an impression why non-abelian case is so much more complicated than abelian one?

Cheers,
Dmitry.

1. You are absolutely right. The masses of the mirror states should be O(1) in the units of the UV cutoff of the theory, which is 1/a in the current case, so that in the continuum limit they become irrelevant.

2. The point of introducing the “mirror” fermions is so that the fermion contents of the WHOLE theory, including both “light” and “mirror”, become vector-like, meaning there are equal numbers of left and right-handed fermions of the same charges. As I explained in earlier post (or maybe not?), vector-like theories can be easily simulated on the lattice using Ginsparg-Wilson formalism. There are two reasons to use Ginsparg-Wilson formalism. First, it eliminates the fermion doubling problems. But there exist other methods to eliminate fermion doubling problem as well, such as the so called “staggered fermions”. The problem with those methods is they must couple fermions with opposite chiralities, and therefore one can not separate the action into “light” and “mirror” parts as we wish to do here. Ginsparg-Wilson formalism, to the contrary, allows us doing that, as long as we use the so called “GW chiral fermions”, as I explained in the earlier post. This is the second reason why we must use Ginsparg-Wilson fermions. Unfortunately, Ginsparg-Wilson formalism introduces new difficulties of its own when dynamic gauge fields are considered. The problem is called the “fermion measure problem”. I briefly explained this issue near the end of the earlier post. This problem only exists in chiral theories because it’s related to the phase ambiguity of the partition function, which only appears in chiral theories. In vector-like theory, there’s no phase ambiguity and therefore no “fermion measure problem” at all. The idea of “decoupling of mirror fermion” is to take advantage of this fact, start with a vector-like theory, and then try to make half of the chiral fermions, contained in S_{mirror} heavy. This way,in the IR, one is left with a chiral gauge theory, described by S_{light} only. This is a proposal to define chiral gauge theories on the lattice indirectly, if it ever works.

3. In lattice, when interactions are so strong, there exists a phase, in which all the excitations are heavy, in the unit of 1/a. Furthermore, the ground state is unique. It’s not too difficult to see, if the ground state is indeed unique, it must be a singlet of the gauge symmetry. In other words, the ground state respects the gauge symmetry. This is in contrast to the more familiar “Higgs phase”, where the ground states are not unique and in the continuum limit, there will be spontaneous breaking of the gauge symmetry. Because our goal is to find a chiral gauge theory in the IR limit, we must make sure that gauge symmetry is not spontaneously broken. That’s why we like to see the “mirror” sector of the theory in this strong coupling symmetric phase. Usually, this phase is not an interesting subject to lattice QFT people, because by the description given above, it’s obvious that in the continuum limit, there remains no interesting dynamical degrees of freedom (everything is heavy, of the order 1/a). But since our aim is to completely decouple the mirror sector, it is exactly what we need. (Only for the mirror sector though! The “light” sector is always a separate story.)

Sorry for the confusions. I hope my answer can at least improve my explanations just a little bit.

thanks again for the post, sorry for being slow to respond – I had to digest some info. Let me ask you a bunch of layman questions now

1. You probably want to make mirror fermions not simply heavy but having masses of the order , so that they don’t affect continuum limit at all, don’t you?

2. I guess this is extremely stupid question but I still want to ask it: what exactly do you want to prove introducing mirror fermions – do you try to model fermion doubling? If masses of light and mirror fermions are equal, is the overall system axial anomaly free? And you introduce axial anomaly by making the mirror sector extremely heavy?

3. What is the nature of this “strong coupling symmetric phase”? In particular, what does the word “symmetric” mean?

Cheers,
Dmitry.