276. Lattice Chirality and (non)Decoupling of the Mirror Fermions
Save This Post as PDFFollowing my previous post on a brief overview of the current situation of lattice chiral gauge theories and the Ginsparg-Wilson formalism, I would like to say just a few words about what J. Giedt, E. Poppitz and I did in the recent series of papers.
The idea of decoupling of the mirror fermions
Our idea (not really, to our best knowledge, this idea was first proposed in [1] by E. Eichten and J. Preskill) to tackle the long-standing problem of defining lattice chiral gauge theories is to try to work around the toughest part. Realizing that defining such a theory explicitly is beyond our reach, we propose to start with a vector-like theory instead, which is always unambiguously defined. It should be a theory described by an action S which splits into two separate “chiral parts”:
,
which we call the “light” and “mirror” sectors. The “light” sector describes a chiral gauge theory that is the target theory we wish to obtain in the end, and the “mirror” sector contains all the fermions with the “wrong” chiralities. As explained in my earlier post, such a clear separation is only possible on the lattice using the Ginsparg-Wilson formalism and the so called “chiral fermions” refer to the Ginsparg-Wilson chiral fermions. The hope is to design the “mirror” sector in such a way that all the “mirror fermions” of “wrong chiralities” contained in there become heavy and therefore disappear in the IR limit. This way, we obtain automatically a chiral gauge theory described by the action 
, circumventing the difficulty of defining the fermion measure explicitly (never do we have to worry about choosing the ambiguous phase explained in my earlier post). We call this method “decoupling of the mirror fermions”.
Some encouraging numerical results
A 2-D toy model, at zero gauge field background, was studied by J. Giedt and E. Poppitz in [2]. The model is called the 1-0 model because it contains a pair of Dirac fermions of charge 1 and 0 respectively. The “light sector” of the action is given by nothing but a free theory of two Ginsparg-Wilson chiral fermions. The “mirror” sector, containing all the chiral fermions with “wrong chiralities”, is more complicated. Besides the similar kinetic terms, all possible Yukawa as well as Majorana type Yukawa coupling terms were introduced. A unitary Higgs field was also needed in order to form these interactions while still respecting the gauge symmetry. Numerical simulations presented in [2] indicate that all the local composite states of the charged “mirror” fermions are indeed heavy provided that the Yukawa coupling constants are large, suggesting the idea of “decoupling of mirror fermions” might be working.
Two important ingredients that made this study possible is the Ginsbarg-Wilson formalism which allows for a clear splitting of the action into different chiral sectors without introducing the fermion doubling problem, and the existence of the so called “strong coupling symmetrical phase” on the lattice. I do not have the opportunity to explain the second ingredient here, and would just note that this phase is usually considered uninteresting lattice artifact because it leads to triviality in the continuum limit, but for our purpose of the entire decoupling of the “mirror fermions”, it would be ideal should this phase arise in the mirror sector.
Smooth splitting of the partition function
The findings in [2] were certainly encouraging, but several issues remained mysterious. First of all, while the action can be clearly split into different chiral sectors thanks to the Ginsbarg-Wilson formalism, it wasn’t obvious that the partition function does so too since the path integrals defined for vector-like and chiral theories are different. This is very particularly true in the GW formalism because in this formalism the fermion measure of a vector-like theory doesn’t factorize to a product of those of two chiral theories. We studied this issue carefully in [3]. Among many useful analytic results derived there, we proved a “splitting-theorem”. This theorem implies that a “smooth separation” (in a particular sense) of the partition function of a vector-like theory into its “light” and “mirror” sectors is only possible when both “light” and “mirror” part of the action are free from gauge anomalies if each is considered as a chiral gauge theory on its own. (Notice that the full theory, being vector-like, is always gauge anomaly free.)
This condition is violated by the toy model studied in [2], which already suggests that things would go wrong when dynamical gauge field is turned on. More puzzling, however, is the following consideration. Using various formulae derived in [3], one “easily” reaches the conclusion that any anomalous chiral gauge theory in the Ginsbarg-Wilson formalism must contain at least one light degree of freedom that’s coupled to the gauge field. This result can be found by studying the photon polarization operator at the zero gauge field background and realizing that a real massless pole must appear in the continuum limit. This appeared to contradict the results presented in [2].
A “non-local” massless degree of freedom
Our most recent paper [4] is mostly devoted to a detailed explanation to the above statement and an extensive numerical study verifying this “prediction”. Making use of the various formulae derived in [3] and after some tedious calculations, we managed to simulate the photon polarization operator in the 2-D toy model and found clear evidence that our theoretical “prediction” pans out. There indeed is a massless pole in the real part of the polarization operator, indicating that a massless degree of freedom exists in the mirror section and is coupled to the gauge field. While no such states were found in [2] where all possible charged local composite states of the mirror fields were analyzed, this “theoretically predicted” and then “experimentally observed” massless fermion must be a non-local composite state in terms of the original fields. We would be very happy to make more concrete this statement by actually finding the corresponding operator, but are unable to do so by now.
The results in [4] can be generalized and lead to even stronger conclusions. We claim that any models designed for the purpose of “decoupling of the mirror fermions” will necessarily fail, even when the anomaly free condition is satisfied, if the proposed “mirror sector” contains any additional global symmetry besides the global part of the targeted gauge symmetry, in which case there exists at least one massless degree of freedom in the mirror sector.
As for more sophisticated models that is both gauge anomaly and extra global symmetries free, all we can say right now is that we don’t have evidence to argue that it won’t work. Further numerical “experiments” for these kinds of models are on the way. We hope to report more on this issue in the not so far future if no unexpected technical issues defeat our attempt.
I hope this little introduction would raise your curiosity about this seemingly trivial but surprisingly subtle problem. Lattice by itself is not the toughest part here. Rather, it’s the subtlety of regularizing QFT but maintaining the correct anomalies that is the major difficulty of the whole program. Defining chiral gauge theories through the trick of “decoupling of mirror fermions” remains a promising venture which may soon lead to the first example of a chiral gauge theory simulated on a computer. But those who are really smart should try to challenge the proof of the (in)existence of lattice chiral gauge theories for non-abeliean gauge groups.
References:
[1] E. Eichten and J. Preskill, “Chiral gauge theories on the lattice,” Nucl. Phys. B 268, 179 (1986).
[2] J. Giedt and E. Poppitz, “Chiral lattice gauge theories and the strong coupling dynamics of a Yukawa-Higgs model with Ginsparg-Wilson fermions,” JHEP 0710, 076 (2007) [arXiv:hep-lat/0701004].
[3] E. Poppitz and Y. Shang, “Lattice chirality and the decoupling of mirror fermions,” JHEP 0708, 081 (2007) [arXiv:0706.1043 [hep-th]].
[4] E. Poppitz and Y. Shang, “Lattice chirality, anomaly matching, and more on the (non)decoupling of mirror fermions,” arXiv:0901.3402 [hep-lat].
If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:
Loading ...
Dear Yanwen,
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.
Hi, Dmitry, sorry for responding slowly. Let me answer your questions as ordered.
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.
Hi Yanwen,
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.
Hi, Dmitry.
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.