We explore 4-dimensional SU gauge theory with a Weyl fermion in an irreducible self-conjugate representation. This theory, in general, has a discrete chiral symmetry. We use ’t Hooft anomaly matching condition of the center symmetry and the chiral symmetry, and find constraints on the spontaneous chiral symmetry breaking in the confining phase. The domain-walls connecting different vacua are discussed from the point of view of the ’t Hooft anomaly. We consider the SU gauge theory with a Weyl fermion in the rank 3 anti-symmetric representation as an example. It is argued that this theory is likely to be in the confining phase. The chiral symmetry should be spontaneously broken to under the assumption of the confinement, although there cannot be any fermion bilinear condensate in this theory.
’t Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate
Department of Physics, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan
’t Hooft anomaly matching condition  is a powerful tool to investigate the phase structure of strongly coupled quantum field theories. Since the ’t Hooft anomaly is invariant under the renormalization group flow, it often strongly constrains the vacuum structure and the low energy effective theory. ’t Hooft anomaly matching condition of discrete symmetry has also been shown to be useful .
Recently it has been shown in [3, 4] that ’t Hooft anomaly matching including higher form symmetries is also powerful. In particular the center symmetry of the gauge theory, a typical example of a 1-form symmetry, and its twist are related to confinement; the confining phase is characterized by the absence of spontaneous breaking of the center symmetry. Therefore by considering the mixed ’t Hooft anomaly of the center symmetry, one may find constraints of the phase structure of the theory in the confining phase. There has been interesting progress along this line [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].
In this paper we investigate a 4-dimensional gauge theory with a Weyl fermion in an irreducible self-conjugate representation . The phase rotation of the fermion in this theory is broken to due to the anomaly [22, 23]. We call this symmetry the “chiral symmetry.” In addition, this theory has the center symmetry, where and is the -ality of . We check the mixed ’t Hooft anomaly between the chiral symmetry and the center symmetry, and obtain constraints on the spontaneous breaking of the chiral symmetry under the assumption of confinement. In this analysis we assume that a gauge theory does not reproduce the ’t Hooft anomaly, if the theory is in the confining phase and the global symmetries are not spontaneously broken.
For example, in the gauge theory with a Weyl fermion in the rank 3 anti-symmetric representation denoted by the young diagram , the chiral symmetry is broken to if it is in the confining phase. By comparing this theory and pure super Yang-Mills theory, we find that this theory is quite likely to be in the confining phase, and thus the chiral symmetry is quite likely to be broken spontaneously.
Interestingly the above theory cannot have fermion bilinear condensate, since a Lorentz invariant and gauge invariant fermion bilinear vanishes identically in this theory. Therefore this theory is an example in which the chiral symmetry is spontaneously broken without fermion bilinear condensate. Such possibility or impossibility has been discussed in QCD [24, 25, 20] as well as the gauge theory with adjoint fermions [17, 18]. It has been argued that the chiral symmetry breaking without fermion bilinear in the gauge theory with a Weyl fermion in spin representation , in which the fermion bilinear vanishes identically.
In order to break the chiral symmetry, a four-fermi operator is likely to condense. There are two four-fermi operators which are gauge invariant, Lorentz invariant, and charged under the chiral symmetry. It will be an interesting future problem to find which four-fermi operator condenses. Other candidates which causes this spontaneous symmetry breaking are fermion bilinears including derivatives. We find that a fermion bilinear with two derivatives cannot condense. Thus only a fermion bilinear with four or more derivatives may condense.
We also discuss dynamical domain-walls connecting distinct vacua, which appears when the discrete chiral symmetry is spontaneously broken. We find that the domain-wall supports a conformal field theory or a topological field theory which reproduce the ’t Hooft anomaly for the center symmetry by a similar argument to .
The construction of this paper is as follows. In section 2, we give the detail of the setup and consider the ’t Hooft anomaly matching condition. Domain-walls are also discussed. In section 3, we discuss the SU(6) gauge theory with a Weyl fermion in . In particular we argue that this theory is in the confining phase and thus the chiral symmetry is broken spontaneously. Section 4 is devoted to summary and discussion.
2 ’t Hooft anomaly matching condition
We consider an gauge theory with a Weyl fermion in an irreducible self-conjugate representation . Let us first summarize some group theoretical conventions. It should be useful to consult [27, 28]. Let be the generators of the Lie algebra normalized as
where denotes the trace in the representation . The Dynkin index is known to be a non-negative integer. Another important quantity is the -ality, or the conjugacy class , defined as the number of boxes modulo when is expressed by a Young diagram for .
Next let us turn to the fields and the action of our theory. The gauge field is denoted by . The field strength is defined by . The fermion is denoted by and whose components are , where are two component spinor indices, and is the “color” index. The action in the Euclidean signature is written as
where is the gauge coupling constant, is the representation matrix of in , and are matrices defined in  besides .
We limit ourself to the case where is self-conjugate (real or pseudo-real) so that our theory is free from perturbative gauge anomaly since the anomaly coefficient if is self-conjugate. We also limit ourself to the case with so that our theory is free from the global gauge anomaly of .
2.2 Chiral symmetry
The action is invariant under the rotation by which and are transformed as
However this symmetry is broken quantum mechanically by the Adler-Bell-Jackiw anomaly [22, 23]. This anomaly is the change of the path-integral measure due to Fujikawa . By the transformation (4), the path-integral measure changes as
where is the Dynkin index of the representation .
In eq. (5), since the instanton number is an integer, the path-integral measure is invariant as well if is written as
Therefore the symmetry given by the following equation exists even in the quantum theory.
We call this symmetry the “chiral symmetry.”
In this paper we discuss the spontaneous breaking of this chiral symmetry.
2.3 Center symmetry
Let us discuss the center symmetry of our theory. The center symmetry group of the pure Yang-Mills theory is .
It is convenient to use the lattice gauge theory picture. In the lattice gauge theory, the gauge field is expressed by the link valuable associated to each link . The pure Yang-Mills action is written in terms of “plaquettes” or any other “small” Wilson loops.
Let us first see how the center symmetry is realized in the lattice gauge theory. First we choose an oriented co-dimension one surface in the space-time which does not cross the lattice sites. Then for an element , the transformation of the link valuables is given by
Since total intersection number of a plaquette and is , it is invariant under this transformation. Any small Wilson loop is also invariant for the same reason. Therefore the action is invariant under this transformation (8).
Next let us consider the fermion part. We assume the fermion part of the action is written as where is the representation matrix of in . If for , then is invariant under the transformation (8). For ), implies
since . This further implies
Let and . Then and are relatively prime, and eq. (10) is rewritten as
This is solved as
As a result, is invariant under the transformation (8) if .
In summary, our theory has center symmetry. Here and is the -ality of .
Actually since is a self-conjugate representation, the -ality is or . is possible only when is even. is given by when while when . Since we concentrate on the case , is always greater than and thus our theory has non-trivial center symmetry.
We can relate this center symmetry to the confinement as usual. Actually the Wilson loop in the fundamental representation exhibits area law if this center symmetry is not spontaneously broken.
2.4 ’t Hooft anomaly
Let us discuss mixed ’t Hooft anomaly between the center symmetry and the chiral symmetry, and find some constraints on the spontaneous chiral symmetry breaking. Here we look at non-invariance by the chiral symmetry transformation when the background center symmetry gauge field is introduced as in [4, 8, 7].
We formulate the gauge field for the 1-form symmetry following [4, 32, 3]. First we introduce 2-form gauge field and 1-form gauge field . They are normalized such that and take arbitrary integer values. Then we relate and by the constraint
The gauge transformation of this is given by
where a gauge transformation parameter is a gauge field, and another gauge transformation parameter is a periodic scalar field with identification . Then is almost pure gauge because of the constraint (13). However the Wilson surface can take non-trivial value. Actually taking the constraint (13) into account is calculated as
since is normalized as . So is a -th root of and it shows that this system is a formulation of the 2-form gauge field.
This 2-form gauge field couples to the gauge field as follows. First, extend the gauge field to a gauge field whose field strength is . We declare the gauge transformation (14) of as
where is the identity matrix. Then the trace part of the field strength transforms as
In order to gauge away the part of we impose the constraint
Notice that this constraint is gauge invariant.
Let us consider the instanton number defined in eq. (5). Actually is not an integer any more in the presence of the background . Since is the traceless part of , is calculated as
Actually takes value in . Since our 4-dimensional space-time is a spin manifold and the intersection form is even, takes value in
where two relatively prime positive integers and are defined by the relation . Actually a configuration with is realized by a 4-torus with twisted boundary condition [33, 34, 35]. As a result, the fermion path-integral measure changes by the transformation (7) unless is multiple of . In other words, chiral symmetry is broken to in the presence of background. This is the mixed ’t Hooft anomaly we are looking for.
This ’t Hooft anomaly is RG invariant by the same argument as  and constrain the phase if as follows. Assume the theory is in the confining phase, i.e. gapped and the center symmetry is not broken. Then the chiral symmetry must be broken at least to since otherwise the isolated vacuum preserve the symmetry larger than but the low energy effective theory is empty and cannot reproduce the ’t Hooft anomaly.
In this argument we make the non-trivial assumption that a gauge theory with a simply connected gauge group does not have any topological order which reproduce the ’t Hooft anomaly, if the theory is in the confining phase and the global symmetries are not spontaneously broken. Actually, it has been known that a class of ’t Hooft anomalies can be produced by certain topological field theories [36, 37]. It is not easy to fully justify this assumption and we postpone it to future work.
Notice that or as discussed in the last section. When , and as well. On the other hand, when , and . In this case if is a multiple of , and if is an even number but not a multiple of . Therefore we have unless and .
When a global discrete symmetry is spontaneously broken, the theory includes dynamical domain-walls connecting distinct vacua. Here let us briefly discuss these domain-walls, when the chiral symmetry is spontaneously broken to in our theory. Such a domain-wall supports a conformal field theory or a topological field theory which reproduce the ’t Hooft anomaly for the center symmetry as discussed in .
Let us see this ’t Hooft anomaly. Suppose the low energy theory is described by a periodic scalar field which is transformed by the chiral symmetry transformation (7) as
The distinct vacua are parametrized as . Then in order to reproduce the mixed ’t Hooft anomaly (21), the effective action should include the coupling to the background field
Now let us consider the domain-wall connecting the vacua and , whose world-volume is denoted by . The term (24) includes the world-volume action
where we use the constraint (13). Therefore the partition function of the 3-dimensional worldvolume theory alone has the phase ambiguity due to the Chern-Simons term (25) at fractional level, although the total theory does not have any phase ambiguity (see eq. (24)). This is nothing but the anomaly inflow mechanism . We can conclude that the domain-wall should support a conformal field theory or a topological field theory which reproduces this ’t Hooft anomaly for 1-form symmetry. For example copies of the level Chern-Simons theory reproduce this ’t Hooft anomaly .
3 Example of chiral symmetry breaking without bilinear condensate
Here we consider the gauge theory with a Weyl fermion in for an example. This representation is pseudo-real and thus there is no perturbative gauge anomaly. The Dynkin index is , and -ality is . It would be useful to consult refs. [27, 28] to find data of various representations.
Let us apply the result of section 2 to this example. We find and and therefore . As a result if the theory is in the confining phase, the chiral symmetry is broken to . So there are 3 distinct vacua related by the broken elements in .
The ’t Hooft anomaly argument in this paper is powerful only when the theory is in the confining phase. Here we argue that our example of with is quite likely to be in the confining phase.
Let us compare our theory with pure super Yang-Mills theory, which is known to be a confining theory. This pure super Yang-Mills theory is in the class of theories we considered in section 2. Actually this pure super Yang-Mills theory is the theory with adjoint.
We observe that is a “smaller” representation than adjoint in the dimensions, the Dynkin index, and the quadratic Casimir. See table 1. This observation leads to the fact that the beta function coefficients of the theory with is larger than those of the theory with adjoint at least up to 3-loop in scheme (see for example ). In other words the theory with becomes strongly coupled in the low energy more rapidly than the theory with adjoint. This observation combined with the fact that the theory with adjoint is in the confining phase strongly suggests that the theory with is also in the confining phase.
Combined with the result of section 2, it is quite likely that in the gauge theory with , the chiral symmetry is broken to spontaneously.
3.3 No fermion bilinear
So far, we have seen that the chiral symmetry is spontaneously broken to . One may think that this spontaneous symmetry breaking is caused by the condensation of the fermion bilinear i.e. , but it cannot be true. Actually a Lorentz invariant and gauge invariant bilinear form in our theory identically satisfies
since both the gauge invariant bilinear form and the Lorentz invariant bilinear form are anti-symmetric. Therefore this bilinear form can never condense.
Let us show the explicit form of the gauge invariant in order to confirm that it is anti-symmetric. Since are the labels of the rank 3 anti-symmetric representation, they are expressed by the three anti-symmetrsized fundamental indices as
By this notation, the gauge invariant bilinear form is expressed as
where is the totally anti-symmetric tensor, which is gauge invariant. From the expression (28) we can confirm that is actually anti-symmetric.
We conclude that in the gauge theory with a Weyl fermion in , the chiral symmetry is spontaneously broken to , although the fermion bilinear cannot condense.
4 Summary and discussion
In this paper we consider gauge theory with a Weyl fermion in an irreducible self-conjugate representation . We obtain constraints on the chiral symmetry breaking in the confining phase in this theory. These constraints are derived from ’t Hooft anomaly matching condition of the chiral symmetry and the center symmetry. In particular in gauge theory with a Weyl fermion in , the chiral symmetry is spontaneously broken to although the fermion bilinear is identically zero.
One natural question is what causes this spontaneous symmetry breaking if it is not the fermion bilinear. One candidate is the four-fermi operator which has the correct quantum number to break to . Actually there are two possible gauge invariant and Lorentz invariant operators. As we have seen, a Lorentz invariant fermion bilinear must be symmetric in . The symmetric tensor product of is decomposed as
A gauge invariant operator is given by the gauge invariant bilinear of each irreducible representation of the right-hand side. It is an interesting future problem to find which four-fermi operator condenses.
Other candidates which causes this spontaneous symmetry breaking are fermion bilinear including derivatives. For example one may guess that causes this spontaneous symmetry breaking. However one can show as follows. Since due to the Lorentz symmetry, the derivative of this equation reads
where we use shown by the same argument as in subsection 3.3. Therefore a fermion bilinear which may condense must include four or more derivatives.
Our theory, gauge theory with a Weyl fermion in , does not have any known gauge anomaly. However it is not guaranteed to be a consistent theory, since there is possibility that it is suffered from some unknown gauge anomaly. In fact, our theory cannot have fermion mass term and this fact may suggest that our theory has some gauge anomaly. For example recently a new global anomaly is found by . One way to guarantee that a theory is anomaly free is to construct a gauge invariant and non-perturbatively regularized lattice gauge theory. This is also a challenging problem.
Interestingly there is a string theory realization of the supersymmetric gauge theory with a half hypermultiplet in . By adding mass term to all the scalars and adjoint fermions in this theory, we obtain the gauge theory with a Weyl fermion in . This fact may suggest that our theory is free from any gauge anomaly and a consistent gauge theory.
One may be curious whether there are other examples in which the chiral symmetry is spontaneously broken without fermion bilinear condensate. Actually, gauge theory with a Weyl fermion in rank anti-symmetric representation has similar ’t Hooft anomaly and a vanishing fermion bilinear. However their beta-functions are quite different. For example case, the rank 5 anti-symmetric representation is quite large, and the confinement is not quite likely to occur. Actually the Caswell-Banks-Zaks fixed point [42, 43] is found in the weakly coupled regime by looking at the 2-loop beta function in this theory. For the theory is not asymptotically free any more.
There are a lot of interesting future problems. For example, we can consider more general gauge theories including many flavors as well as different irreducible representations. It will also be an interesting problem to investigate finite temperature phase transitions.
I would like to thank Prarit Agarwal, Hidenori Fukaya, Dongmin Gang, Yuta Hamada, Naoki Kawai, Seok Kim, Yoshiyuki Matsuki, Tetsuya Onogi, Shigeki Sugimoto, Yuji Tachikawa, Seiji Terashima, Mithat Ünsal, and Shuichi Yokoyama for discussions and comments. I would also thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP workshop YITP-W-17-08 on “Strings and Fields 2017” and YITP-W-18-04 on “New Frontiers in String Theory 2018” were useful to complete this work. This work was supported in part by JSPS KAKENHI Grant Number 15K05054.
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