A solution to chiral fermion problem, or not

September 12, 2020

In 2013, I wrote a paper (arXiv:1305.1045) claiming solved the chiral fermion problem. It is known that some gauge-anomalous chiral fermion theory cannot be put on a lattice in the same dimension. However, for a long time, we do not even know how to put a gauge-anomaly-free chiral fermion theoy on a lattice in the same dimension without breaking the "gauge symmetry". This is the chiral fermion problem. Luscher provided a solution for U(1) gauged chiral fermion. I claimed that all gauge-and-gravity-anomaly-free chiral fermion theoies can be put on a lattice in the same dimension (i.e. the low energy sector of the lattice model reproduces the chiral fermion theory without breaking the "gauge symmetry"), by just allowing the lattice fermions to interact in certain ways. (Actually we have a stronger result: even chiral fermion gauge theories with certain type of global anomalies can be put on a lattice in the same dimension in certain sense, see arXiv:1303.1803, Section XIII C.) I also give a sufficiant condition for a chiral fermion theory to be free of all gauge and gravitational anomalies, as well as a design of the lattice fermion interaction.

The paper was rejected by two RRL referees (very strongly) and by a PRL editor. "The reason for skepticism is the long history of past failures", rather than past failures of the author. [This paper is now published in Chinese Physics Letter.]

The arguments for the main result of the paper is really simple, and can be easily presented within one page. The key assume is that a non-linear sigma-model (with no topological term) can have a gapped disordered phase even when the fluctuations have no defects (or singularities). This kind of disordered phase is discussed in more detail in a later paper. Maybe, I am too easily convinced. When I am convinced to see something, it may not be convincing enough for some other people.

The key result is a design of an symmetric interaction for certain chrial fermions in 3+1D, so that the chiral fermions are gapped out by the interaction, rather than by the mass term (the mass term always breaks the required symmetry). This phenomenon is called "mass without mass term" [see also [1] [2]). In 2015, Kitaev discussed a simialr phenomenon for a 2+1D system using a similar argument [3]. This kind of phenomena were also discussed for 1+1D systems [4][5][6] in 2012 and 0+1D systems [7] in 2009. In 1+1D and 0+1D, there are non-perturabtive methods which allow us to show "mass without mass term" more convincingly.

Since then, we posed five papers along this line: arXiv:1307.7480, arXiv:1706.04648, arXiv:1805.03663, arXiv:1807.05998, arXiv:1809.11171. Six years after the initial work, two of our papers finally get published.

The following are the referee's reports and my reply, and the editor's report.

走自己的路,欣赏自己的工作,看重自己的工作,是科学工作者对待科研的一种心态。写出来的文章,至少要得到自己的认可。最重要的也是得到自己的认可。自己看重自己的文章。这样别人不认可,也不会信心全无。写文章的目的,不是为了发表,是为了满足自己的好奇心。以此为目的,容易出好文章,容易发表。

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Lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model (arXiv:1305.1045)
by Xiao-Gang Wen

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Report of Referee A and my reply
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Referee A:
In this letter, the author is proposing a non-perturbativelattice formulation of (anomaly-free) chiral gauge theories. This problem itself is of broad theoretical interest and if a convincing solution is given, it will be of great importance.

Reply:
I agrees with the referee A that the non-perturbative lattice formulation of (anomaly-free) chiral gauge theories is of broad theoretical interest and of great importance.

Referee A:
However, what one finds in this letter are a very general description of the problem in more than 2 pages and a very brief sketch of the idea in almost half page. I myself cannot believe this naive idea does solve the problem of, for example, the breaking of the gauge symmetry with finite lattice spacings.

Reply:
In the 2 pages of the paper, I reviewed
1) a recent break through in condensed matter physics -- a classification of SPT states
2) a realization that the SPT phases classify the gauge anomalies in one lower dimension.
This two results do solve the problem of the breaking of the gauge symmetry with finite lattice spacings. To see this, we note that the anomaly-free condition imply that the SPT state in the bulk is trivial, and trivial SPT state can have a gapped boundary that do not break the symmetry. Once we understand the above two results and once we believe that the right-hand Weyl fermions coupled to SO(10) gauge theory is free of all gauge anomalies, then it is almost trivial to put right-hand Weyl fermions coupled to SO(10) gauge theory on a lattice, which I spend one page of the paper to explain. So the key progress is the above two mentioned results, which were explained in several recent (very) long papers (refs [32,41,25] in the new vertion). This short paper is a direct and natural application of those recent results.

Referee A:
In any case, one cannot even judge whether this idea works or not from the description in this letter, because no precise form of the idea (for example, what is the precise form of the underlying 4-dimensional lattice hamiltonian, how to introduce the gauge interaction, or how the anomalous cases are distinguished from the anomaly-free cases etc.) is not given. From these reasons, I think this letter is not suitable for publication in Physical Review Letters.

Reply:
In the new version, I added a section in supplemental material to give a more detailed description of the lattice model in 4D space. Such a construction is well known, so I did not include it in the main text. I also explained how to introduce the gauge interaction (by simply gauging the SO(10) global symmetry).

"how the anomalous cases are distinguished from the anomaly-free cases" is a very good question. To stress this issue, I added a generalization of the SO(10) result as conjecture in the new version (in a box on page 2). I also added a few examples on page 4, which demonstrate that our approach does not apply for the known anomalous chiral theories.

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Report of Referee B and my reply
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Referee B:
I regret having to say this, but the manuscript fails to satisfy the most minimal requirements for a scientific publication. All the author can offer is, literally, wishful thinking. The manuscript contains no study of any kind -- neither analytic nor numerical --to support the author's proposal. That proposal is not even well defined (see below). I hate to say this but this is a rare case of "not even wrong" -- the manuscript does not even contain any attempt to do scientific work of any kind that would support the claims, or hopes, of the author.

Reply:
According to Referee A, the non-perturbative lattice formulation of (anomaly-free) chiral gauge theories is of broad theoretical and of great importance. In this paper, I claim that right-hand Weyl fermions in 16-dimensional representation of SO(10) coupled to SO(10) gauge field can be put on a lattice of the same dimension (without breaking the SO(10) gauge symmetry on lattice), if we just allow lattice fermion to directly interact. Such a claim can be wrong (and the referee B try to argue that the claim is indeed wrong). So this paper is not "not even wrong". In the new version, the key claims of the paper are put into three boxes.

It is also not fair to say "the manuscript does not even containany attempt to do scientific work of any kind that would support the claims, or hopes, of the author". In this paper, I showed that
1) There is a (SO(10) symmetry breaking) Higgs field that give all 16 Weyl fermions a mass
2) The target space of the Higgs field generated by the SO(10) rotation is a 9D sphere, S_9, which have a trivial homotopy group pi_d(S_9) for d<9.
The above two results allow me to argue that the Higgs field can be in a disordered phase (that do not break the SO(10) symmetry) while still give the fermions (the doublers) a mass. (See the discussion on the first half of page 4 in the new version.) These two simple arguments lead to the result (the claim) of this paper.

Referee B:
The problem of constructing lattice chiral gauge theories is a long-standing one. The proposal put forward by the author is basically a variant of the Eichten-Preskill model [14].

Reply:
Indeed, the proposal put forward is basically a variant of the Eichten-Preskill model. The new features of this paper are the two simple arguments mentioned above. This leads to a new way to design the fermion interaction, which, in turn, leads to the result of this paper.

(added later: Eichten-Preskill's approach and my approach both start with lattice fermions with non-weak interactions. But the proposed designs of the interactions are different. Eichten-Preskill's design some times works and some times does not work, since valid designs exist even for theories with perturbative anomaly. This paper provides a sufficient condition and a design for interaction to gap out the mirror sector without breaking the symmetry. For details, see Appendix C of the arXiv version arXiv:1305.1045)

Referee B:
This model was studied in detail by Golterman, Petcher and Rivas[20], where the main questions were identified, and extensive evidence pointed to the failure of the model. Other models were proposed that were found to fail for similar reasons later on, whereas completely different approaches lead to at least partial success.

Reply:
In the paper by Golterman, Petcher and Rivas, it was stated that "because of the presence of a symmetry breaking phase transition, the scenario envisioned by Eichten and Preskill will most likely not be realized." The claim of Golterman, Petcher and Rivas, "most likely not be realized", does not logically contradict with the claim of this paper, "be realized". Golterman, Petcher and Rivas studied a particular fermion interaction. They showed that, for such an interaction, to make all the doublers massive, one has to break the symmetry. Based on the new ideas of SPT phases and their relation to gauge anomalies, this paper propose a new way to design a fermion interaction. Such a fermion interaction should give all the doublers a mass without break the symmetry. The new way to design the fermion interaction is the new result of this paper.

In other words, Golterman, Petcher and Rivas considered one particular interaction, and show that it does not works. In this paper, we argue that there is another interaction that should work. There is no contradition.

Referee B:
As far as I can tell from this manuscript, the author is not even aware of what are the main issues. Just like Eichten and Preskill, the author wishes to find a symmetric phase with a chiral spectrum. This cannot possibly be the weak-coupling symmetric (PMW) phase, so the hope is for a strong-coupling symmetric (PMS) phase in which the scalar field will have zero vacuum expectation value, while the spectrum will remain chiral in the continuum limit. The mechanism that can, and does, leads to a failure is the formation of bound states of the lattice fields that become additional elementary fermions in the continuum limit. In particular, ref [20] provided conclusive evidence that this is what happens in the PMS phase of the Eichten-Preskill model: because of the bound-states formation, the continuum-limit spectrum consists of Dirac fermions; it is not chiral.

As I said, the author does not even seem to be aware that this is the main question. That this is the question he should be addressing would have been clear to anyone who really studied the literature -- the original papers and/or review articles. References to all this literature can be found for example in the recent series of papers by Poppitz et al [17,22].

Reply:
It is indeed clear to every one that weak-coupling symmetric (PMW) phase does not work. Golterman-Petcher-Rivas and other papers provided conclusive evidence that a strong-coupling symmetric (PMS) phase also does notwork, for a particular form of fermion interaction. This paper suggests a new way to design fermion interaction which I believe should work (ie to give all doublers an energy gap without breaking the symmetry), based on the insights from the SPT states and their relation to gauge anomalies. I showed that
1) There is a (SO(10) symmetry breaking) Higgs field that give all16 Weyl fermions a mass
2) The target space of the Higgs field generated by the SO(10) rotation is a 9D sphere, S^9, which have a trivial homotopy group pi_d(S^9) for d<9.
The above two results allow me to argue that the Higgs field can be in a disordered phase (that do not break the SO(10) symmetry) while still give the fermions (the doublers) a mass. (See the discussion on the first half of page 4 in the new version.) These two simple arguments lead to the result (the claim) of this paper.

(Added later: The weak interaction limit (PMW) does not work. The infinite interaction limit (PMS) does not work. But the proper interaction proposed in this paper is between the two limits. The proper interaction strength is of the lattice cut-off energy scale. Strong interation can mean (1) non-weak interaction, or (2) infinite interaction. I used definition (1), while some lattice gauge paper used definition (2).)

Referee B:
The author talks about a hamiltonian approach instead of the common euclidean path integral. However, the physics issues are the same, and as I noted, the author did not do any work what so ever, let alone any work that would suggest that a hamiltonian approach would help in any way.

Reply:
I agree with the referee B that hamiltonian lattice approach and space-time lattice approach are similar. I stressed the hamiltonian lattice approach in order to state the result (the claim) of this paper clearly.

As I mention above, I did do some new work in this paper: the two simple arguments listed above. I like to add that I am surprised that such simple arguments can lead to a solution of the long standing problem of putting some anomaly-free chiral gauge theories on lattice. I guess the insights from SPT states and the new understanding of anomalies help a lot. I try to explain those insights in the first two pages of the paper.

Referee B:
The author adds a fourth space dimension so that there are chiral fields on the three dimensional spatial boundaries in the free theory case. This putting together of domain-wall fermions and the Eichten-Preskill proposal had been tried in the past, and once again it was found to fail for similar reasons. Because of the extra space dimension, the author has to make up his mind whether or not the gauge field depends on this extra dimension. There is no word on this, so that the proposal is not even well defined. In the literature, both approaches had been tried, and the existing evidence points to a failure in both cases.

Reply:
I have stressed in the paper that the extra dimension is finite. The 4+1D lattice theory is really a 3+1D lattice theory. In this case, I do not have
to make up my mind whether or not the gauge field depends on this extra dimension, since both assumptions lead to local 3+1D lattice gauge theory. I like to repeat that this paper suggests a new way to design fermion interaction which I believe should work (ie to give all doublers anenergy gap without breaking the symmetry), based on the insights from the SPT states and their relation to gauge anomalies.

Referee B:
It is the author's burden to prove by scientifically sound calculations that he can do better.

Reply:
The two arguments listed above are scientifically sound calculations, although they are very simple. There is indeed a logical gap between the two arguments and the claim of this paper. I myself is convinced that the two arguments lead to the claim of this paper. The approach in this paper lead to a design of fermion interaction. One can confirm or disapprove my judgement by future numerical calculations. I myself plan do some numerical calculations in the future, motivated by the approach of this paper.

I am a condensed matter physicist, and I am sorry that I cannot write a paper from an angle of lattice gauge physicists. But I feel that a new angle to look at the long standing chiral-fermion problem should be useful, that may lead to a breakthrough as I try to argue in this paper.

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Report of the Divisional Associate Editor
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This paper claims to have solved the important, long-standing problem of formulating a chiral gauge theory non-perturbatively on the lattice. It has been reviewed by two referees, who both recommend rejection. I have then been contacted, because the author has appealed the rejection.

I have hesitated for a long time about what to recommend, and I apologize for this delay. In the end, I agree with the previous referees: it seems to me that, in spite of its novelty, the paper should not be published in Phys. Rev. Letters. A publication in PRL would mislead the readers by conveying the notion that this paper presents a solution to the formulation of a chiral lattice gauge theory. In my opinion, the paper presents a proposal to be tested, rather than a solution.

The reason for skepticism is the long history of past failures. The proposal put forward by the author belongs to the family of "mirror fermions", with the new ingredient of symmetry-protected topological order. It is not at all clear to me that this new ingredient guarantees success. In particular, the most recent attempt,Ref.22 (arXiv:1211.6947v3), shows failure of the mirror fermions to decouple, for no obvious reason. This finding should motivate all of us, including the author, to consider new claims with caution.