Perturbative search for dead-end CFTs

To explore the possibility of self-organized criticality, we look for CFTs without any relevant scalar deformations (a.k.a dead-end CFTs) within power-counting renormalizable quantum field theories with a weakly coupled Lagrangian description. In three dimensions, the only candidates are pure (Abelian) gauge theories, which may be further deformed by Chern-Simons terms. In four dimensions, we show that there are infinitely many non-trivial candidates based on chiral gauge theories. Using the three-loop beta functions, we compute the gap of scaling dimensions above the marginal value, and it can be as small as $\mathcal{O}(10^{-5})$ and robust against the perturbative corrections. These classes of candidates are very weakly coupled and our perturbative conclusion seems difficult to refute. Thus, the hypothesis that non-trivial dead-end CFTs do not exist is likely to be false in four dimensions.


Introduction
The rule of the game is as follows: • We look for conformal field theories (CFTs) without any relevant scalar deformations. We name them dead end CFTs.
• We do not ask what will happen after introducing the relevant deformations (if any).
• We do not impose any continuous global symmetries nor discrete global/gauge symmetries. 1 • We assume dead end CFTs are unitary, causal, and have finite energy-momentum tensor. 2 • Deformations must be physical. In gauge theories, they must be BRST invariant. 3 • (Optional) In our paper, we assume the gravitational anomaly does not exist.
• (Optional) In our paper, we only discuss powercounting renormalzable weakly coupled Lagrangian field theories. 4 Let's play!

Physical background of the game
This game is designed to understand a possibility of self-organized criticality [1] (see e.g. [2] for a review) in quantum field theories. In many statistical systems, it is typically the case that in order to obtain the criticality, we have to tune at least one parameter of the system (e.g. temperature). It is interesting to see if we can construct a self-tuning model so that the criticality is automatically attained by just making the size of the system larger without tuning anything else. A naive guess is unless we use some symmetry principles 1 Otherwise, we have scalars with a shift symmetry or fermions with a (discrete) chiral symmetry as trivial examples. 2 Otherwise, generalized free CFTs are trivial examples. 3 Otherwise, the ghost mass terms or gauge non-invariant mass terms give unphysical relevant defromations. 4 To the author's knowledge, there is no known example beyond perturbation theory arguably except for AdS/CFT inspired ones. See section 4.
(e.g. Nambu-Goldstone mechanism or anomaly cancellation mechanism) generic gapless systems are unstable, and the self-organized criticality is difficult to achieve.
However, we know one example in our nature. The theory of photon. The theory of photon is always at its criticality and we cannot detune the theory to make it gapped unless we introduce extra light matter fields (like Higgs mechanism). The fact that it is always at its criticality led Einstein to the discovery of the special relativity. The speed of light is absolute. It is hard to imagine if he could have come up with various gedanken experiments if the photon is massive and the propagation of light is not critical.
The criticality of the photon is not protected by any global symmetry. It is the intrinsic nature of the Maxwell theory that does not allow any relevant deformations. 5 It is an example of dead end CFTs.
Is this just a peculiar coincidence or deep feature of our particle physics in the particular space-time dimensions of four?
Putting the philosophical questions aside, one technical reason we are interested in the (non-)existence of the dead end CFT is whether we may regularize various infrared singularities in "S-matrix" of the CFTs. Strictly speaking, the S-matrix does not exist in CFTs, but once they are deformed to a massive/gapped/topological phase, the concept makes sense. Indeed, the clever use of the (regularized) S-matrix and its analyticity properties has led to many important results in quantum field theories such as the proof of a-theorem in d = 4 dimensions [3], enhancement of conformal invariance from scale invariance [4][5], the convexity properties of large twist operators in general CFTs [6], and so on.
For example, one crucial point in the argument of enhancement of conformal invariance from scale invariance is as follows. If the theory were scale invariant but not conformal invariant, the argument in [4] [5] suggests that the "c"-function (or "a"-function in d = 4 dimensions) would be decreasing forever along the RG flow. However, if the theory can be deformed to the massive/gapped/topological phase, the central charge is bounded c ≥ 0 (or a ≥ 0), and hence it is in contradiction. The argument does not apply if the theory under consideration is a dead end CFT or a dead end scale invariant field theory [7]. 5 In the BRST quantization, one may regard photon as a Nambu-Goldstone boson for the residual gauge symmetry δA µ = a µ . Since there is no way to break this symmetry in a physical manner, this fact is not important for our discussions. See also footnote 3.
In this paper, we look for candidates of dead end CFTs within powercounting renormalizable quantum field theories with weakly coupled Lagrangian description. Of course, it is desirable to give a non-perturbative argument that does not rely on perturbation theory or Lagrangian descriptions. It is, however, sufficient to give a perturbative example if we would like to disprove the claim that the dead end CFTs do not exist. We will give some further thoughts on the non-perturbative aspects of the game in section 4.
2 No non-trivial candidates in d = 3 We begin with the matter content of the renormalizable quantum field theories with We can always take the real basis for the scalar fields φ I , and they transform as real (linear) representations under the gauge groups. The existence of the kinetic term means that there exists a positive bilinear form g IJ so that the kinetic term g IJ D µ φ I D µ φ J is gauge invariant and non-degenerate. One can use the same bilinear form to construct the gauge invariant mass term for the scalars as g IJ φ I φ J . These mass terms are relevant deformations with the powercounting scaling dimension ∆ = 1.
In a similar way, we can always take the Majorana basis for the fermionic spinor fields ψ a in d = 3 dimensions, and they transform as real representations under the gauge groups. Again, the existence of the kinetic term means that there exists a positive bilinear form g ab so that the kinetic term g abψ a γ µ D µ ψ b is gauge invariant and non-degenerate. To avoid misunderstanding, we would like to comment on the non-perturbative fixed point which was claimed to be an example of self-organized criticality in certain spin 3 Non-trivial candidates in d = 4 We have seen in d = 3 dimensions, there are no non-trivial candidates of the dead end CFTs within the weakly coupled Lagrangian description. The situation is drastically different in d = 4 dimensions because mass terms of the fermions can be forbidden without using any global symmetries.
We start with the field contents. In renormalizable field theories in d = 4 dimensions with weakly coupled Lagrangian description, we have bosonic spin zero scalar fields and fermionic spin half spinor fields charged under the gauge groups with finite kinetic terms.
The argument for the scalars is the same as in d = 3 dimensions. We can always take the real basis for the scalar fields φ I , and they transform as real (linear) representations of the gauge groups. The existence of the kinetic term means that there exists a positive bilinear form g IJ so that the kinetic term g IJ D µ φ I D µ φ J is gauge invariant and nondegenerate. One can use the same bilinear form to construct the gauge invariant mass term for the scalars as g IJ φ I φ J . These are relevant deformations with the powercounting However, the situation is different in spinors. We can take the Weyl basis of the fermions ψ a so that the representations of the gauge group are complex in general. The complex conjugateψ a (with the opposite chirality) transforms under the complex conjugate representations of ψ a . The existence of the Weyl kinetic term means that there exists a Hermitian bilinear form δ a b so that the kinetic term δ a bψ a σ µ D µ ψ b is gauge invariant and non-degenerate. The crucial difference here is that unlike in d = 3 dimensions, we cannot use the bilinear form δ a b to construct the Lorentz invariant mass term becausē ψ a and ψ a have different chiralities. The gauge theories with Weyl fermions in non-real representations are called chiral gauge theories and since they do not (always) possess the mass deformations, they are good candidates for the dead end CFTs.
Not every chiral gauge theories are consistent. They may suffer gauge anomaly. The anomaly cancellation conditions are well-known. For each gauge group, we require where R F is the representation matrix and the sum is taken over all the Weyl fermions.
Note that the condition is linear in the matter representation, so we can add the anomaly free matter combinations and still it is anomaly free. We only focus on the anomaly free gauge theories.
Extreme examples are pure gauge theories. We do not have any matter at all, and we cannot add any mass terms to the gauge bosons by hand. However, we believe that the non-Abelian gauge theories in d = 4 dimensions will confine and show the mass gap.
Therefore One comment on the renormalizability is in order. One may ask if our chiral gauge theories we will discuss are really renormalizbale. At least within the power-counting renormalization, they are proved to be renormalizble, and certainly we are able to compute the physical observables in these CFTs at the three loop order we study. After all, our examples will turn out to be no more exotic than the standard model as chiral gauge theories, and if we doubt their renormalizabilities (or realizabilities in nature), we should ask the same question to the standard model. See e.g. [12] and references therein for further discussions on the non-perturbative renormalizabilities.

Simple quiver type chiral gauge theories
The easiest way to solve the anomaly free condition is to study the quiver-type gauge theories of SU(N c ) K . The matter Weyl fermions are in bifundamental representations of adjacent gauge groups and represented by arrows. When the number of incoming arrows and outgoing arrows are the same at each node that represents a simple gauge group, the theory is anomaly free. In order to forbid the fermion mass term, it is sufficient to make the directions of the arrows only one way between any pair of nodes.
For simplicity, we focus on the circular quiver gauge theories of SU(N c ) K with N f generations of bifundamental Weyl fermions 8 : The beta functions of the system can be computed up to three loops by using the recent results reviewed in Appendix. The three-loop beta functions in the Modified Minimal Subtraction scheme are given by for each gauge coupling constant g i (i = 1, 2 · · · , K). The asymptotic freedom requires N f < 5.5. 9 In order to obtain the weakly coupled fixed point, it is desirable that N f is close to the upper boundary of the asymptotic freedom limit, so our main focus will be We look for the zeros of the beta functions. When N * f < N f < 5.5 with a certain critical number N * f , the zeros of the beta functions correspond to infrared stable fixed points, and we find good candidates of dead end CFTs. Once we find the zero of the beta functions, one may compute the anomalous dimensions of the field strength operators i . Up to three-loop orders, the beta functions of the gauge coupling constants actually do not depend on the number of nodes K in the quiver. This is because we need at least K numbers of fermion loops to obtain the non-trivial K dependence in the beta functions. On the other hand, the anomalous dimensions of the field strength do depend on K because we have to diagonalize the K × K Hessian matrix.
In principle, we also need to study the CP odd operators Tr i (ǫ µνρσ F µν F ρσ ) with their coupling constants θ i as theta terms beyond the perturbation theory. Actually, K − 1 out of K theta terms are redundant operators in this theory because they can get removed by the (anomalous) phase rotations of Weyl fermions. The overall theta term, however, may be non-trivial. In perturbation theory nothing depends on its value. We do not know if the theta term is non-perturbatively renormalized or it will affect the beta functions. In any way, if we have an infrared fixed point, the anomalous dimension must be positive and our discussions are still valid. In the other examples that we discuss in later subsections, all the theta terms are redundant operators in the action.
In our perturbative search, we may set g 1 = g 2 = · · · = g K . We find that the other fixed points make some of the gauge coupling constants vanish, so we end up with effectively and so on. In every cases, all the eigenvalues are positive, meaning that the fixed points are infrared stable. Each entry has an additional integer label K ≥ 3.
Although the beta functions are renormalization group scheme dependent, the anomalous dimensions at the fixed point are physical quantities and they do not depend on the choice of the renomalization scheme. Also note that the smallness of the coupling constant g i at the fixed point itself is not that important because the physical expansion parameters may be different (e.g. t' Hooft coupling g 2 i N c may be more relevant). The ratio between the two-loop predictions and the three-loop predictions may be regarded as a good barometer how the perturbation theory is reliable or not (assuming there is no accidental cancellation).
It turns out that in all the examples we have studied, the three-loop predictions actually make anomalous dimensions smaller than the two-loop predictions. We find that the loop expansion is not terribly bad for the anomalous dimensions of permutation symmetric field strength for N f = 5, which is at the percent order. For comparison, we show that the two-loop and three-loop predictions of the Banks-Zaks fixed point [14] [15] of SU(N c ) gauge theory with n f Dirac fermions in fundamental representation in table 2.
In comparison with the Banks-Zaks theories, we realize that the structure of the beta  It would be interesting to settle the conformal window, but this is not the main scope of our paper. We only attempt to offer the existence proof of dead-end CFTs so we are more interested in the weakly coupled fixed points. With this respect, we have no (known) arguments against that N f = 5 in chiral quiver gauge theories do not possess the infrared fixed point.

Anomaly free chiral matters
A more non-trivial way to obtain the anomaly free chiral gauge theories is to use the can- We may generically consider N a generations of generalized Georgi-Glashow model and N s generations of generalized Bars-Yankielowicz model. In this subsection, we focus on the single gauge group and we will discuss the quiver generalization in the next section.
We remark here that N s = 3 model is the SU(5) grand unified model of our standard model (without Higgs fields). In fact, all these chiral gauge theories are introduced in the model of our particle physics.
From the formula in Appendix, the three-loop beta functions are computed as As mentioned, the theta term in these models is redundant, so we only have to consider the non-trivial zero of the gauge coupling constant.
We can now play the game of finding very weakly coupled fixed points by changing  (see table 2). We also see that the difference between the two-loop prediction and the three-loop prediction is order of percent and the perturbation theory seems fairly reliable. We can see that some of these examples such as SU(35) with N s = 0, N f = 6 are extremely weakly coupled. Their anomalous dimensions are 10 −2 times smaller than that of QED and so are their loop corrections. It is hard to imagine that the conclusion that these models have non-trivial conformal fixed points will be refuted by any other methods. Since the loop suppression is very large, we do not have to worry about the scheme dependence of the beta function at the higher loop order, either. 10 We can find the two-loop discussions with N s N a = 0 case in [23]. When N s N a = 0, [13] also gives the estimate of the conformal window from the topological excitations. The latter claims that theirs is the first estimate of the conformal window of these models. Apparently, the existence of non-trivial conformal fixed points in chiral gauge theories have been much less studied in the literature.  We study SU(N c ) K quiver gauge theories with N f generations of Weyl fermions (arrows between nodes) in bifundamental representations. Again, for simplicity, we consider the circular quiver. In addition, at each nodes we add N a copies of generalized Georgi-Glashow model and N s copies of generalized Bars-Yankielowicz model.

Quiver with external matter
The model is chiral and dose not admit any mass term.
The two-loop beta functions at each node is given by We do not write down the three-loop terms here, which would not fit into one page length.
One may derive them from the general formula in Appendix. As in section 3.1, there is no K dependence in the beta functions at the two (or three) loop level.
We look for non-trivial zeros of the beta functions by varying N c , N f , N a and N s .
We present some examples of extremely weakly coupled fixed points together with the anomalous dimension of the permutation symmetric field strength in  Such a bound from the central charge seems interesting in higher dimensions if any.
We have found infinitely many candidates of dead end CFTs in d = 4 dimensions, but certainly, the construction based on chiral gauge theories required a large number of fields, and the infinite series we have found require more and more matter. We may conjecture that there is a lower bound on the central charge (say "a" that couples to the Euler number in trace anomaly) that is needed to construct dead end CFTs. 11 We cannot resort to the modular invariance in higher dimensions, but the recent developments in conformal bootstrap may shed some light. In particular, the study of 11 Without extra conditions, the author believes that the lowest bound for the dead end CFT comes from the free U (1) gauge theory. Unfortunately, we even do not know examples of non-free CFTs whose central charge is less than that of the free U (1) gauge theory in d = 4 dimensions. To the author's knowledge, the only non-trivial candidate is the hypothetical CFT sitting at a kink of N = 1 superconformal bootstrap discussed in [29]. It, however, possesses a relevant deformation.  12 The conformal bootstrap that we employ today cannot tell the difference between a free scalar and free Maxwell theory in d = 3 dimensions, so the Maxwell theory will be counted as a non dead end CFT.
Thus we have no candidates at all for this purpose. In relation, we should note that it is hard to exclude the possibility that the global symmetry forbids the relevant deformations from the conformal bootstrap approach.
variation of the game is as interesting as the one we discussed in this paper.

Acknowledgements
The author would like to thank the organizers of two wonderful workshops "Conformal A Three-loop beta functions of general multiple gauge theories In this appendix, we review the recent results of the three-loop beta functions for gauge coupling constants for general multiple gauge theories [33] (see also [34] for the single gauge group). We consider the direct product of simple gauge groups G i with the gauge coupling constants g i (i = 1, · · · , n). For a field transforming under the representation R of the gauge group G i with the generators R a in the matrix notation satisfying we define Casimir invariants as The following identity holds where d(R) is the dimension of the representation R and d(G) is the dimension of the group.
Explicitly for SU(N c ) group, we have In general, the matter Weyl fermion F is charged under multiple gauge groups. Following [33], we use the notation d(F i ) to specify the dimensions of the representation R with respect to the gauge group G i . Furthermore, we also define the multiplicity of For generic multiple gauge theories with arbitrary representations of Weyl fermions, the three-loop beta function of the coupling constant g i in the Modified Minimal Subtrac-tion scheme is given by In our applications, there is no matter Weyl fermions that is charged under three different gauge groups, so the last line in (15) will be dropped. In [33], one may also find the additional contributions from scalars that we do not use in this paper.