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UV completion of extradimensional Yang-Mills theory for Gauge-Higgs unification
by Álvaro Pastor-Gutiérrez, Masatoshi Yamada
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Submission summary
| Authors (as registered SciPost users): | Masatoshi Yamada |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202207_00029v2 (pdf) |
| Date submitted: | 2022-12-19 15:50 |
| Submitted by: | Yamada, Masatoshi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The $SU(N)$ Yang-Mills theory in $\mathbb R^4\times S^1$ spacetime is studied as a simple toy model of Gauge-Higgs unification. The theory is perturbatively nonrenormalizable but could be formulated as an asymptotically safe theory, namely a nonperturbatively renormalizable theory. We study the fixed point structure of the Yang-Mills theory in $\mathbb R^4\times S^1$ by using the functional renormalization group in the background field approximation. We derive the functional flow equations for the gauge coupling and the background gauge-field potential. There exists a nontrivial fixed point for both couplings at finite compactification radii. At the fixed point, gauge coupling and vacuum energy are both relevant. The renormalization group flow of the gauge coupling describes the smooth transition between the ultraviolet asymptotically safe regime and the strong interacting infrared limit.
Author comments upon resubmission
first of all, we would like to thank you for the useful and relevant comments which have triggered corrections.
Detached from all reports, we have modified the legend of Figure 8 which now indicates that the scale setting for the dimensionless compactification radius \bar R is set at k = 1.
List of changes
Here, we list major changes:
(i) In Section 2.2 and Section 2.3, a general discussion on the residual gauge transformation with Z!=0 was added. (See around Eqs. (12) and (13) , and below Eq. (19)).
(ii) In Section 2.4, we have commented on D=4 YM theories at finite temperature and cited several related references ([50-60]).
(iii) We have commented on the understanding of the phase structure listed in Table 1 in terms of the center symmetry in the footnote 3.
(iv) We have commented on the terminologies "broken phase" and "symmetric phase".
(v) In the end of Section 3.1, we have commented on the truncation ansatz for the effective action (27).
(vi) Old figure 7 has been removed and the right panel in new Figure 8 is plotted. Besides, the expression of the IR fixed point of the effective potential has been erased.
(vii) Below Eq. (90), we have commented on the Curci-Ferrari model with some appropriate references.
Current status:
Reports on this Submission
Anonymous Report 1 on 2023-5-18 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202207_00029v2, delivered 2023-05-18, doi: 10.21468/SciPost.Report.7212
Strengths
Interesting paper regarding the asymptotic safety scenario within compact Yang-Mills theories.
Weaknesses
Misleading statements regarding "gauge symmetry breaking" and the interpretation of center-symmetry breaking, although this does not impact the main results of this work.
Report
I thank the authors for their clarifications. Most of them answer my questions. I have still some deep disagreement regarding their answer labelled “comment 3” on whether and in which sense gauge symmetry can be broken. Also, in my opinion, there is still some confusion between what the authors call the breaking of gauge symmetry and center-symmetry breaking. Since this is not the main point of the paper and since the other results (at vanishing background) are interesting on their own right I would be happy to accept the paper, but I would like to comment further on these questions as this may help the authors clarifying their text.
The classification of configurations given in Table I is certainly interesting and useful. However, I think that the terminology used in the text to refer to each of these cases as different breaking patterns of the gauge symmetry is misleading. It is acceptable as pure terminology of course but the problem is that it seems to imply that gauge symmetry can be broken spontaneously which it cannot in any case (of course it is broken explicitly via gauge fixing but this is not seen at the level of the observables).
In terms of gauge fields (which contain unphysical degrees of freedom), symmetries need always to be defined modulo gauge transformations (corresponding in the present case to transformations that are periodic at the boundaries of the compact dimension). This is crucial in particular in order to identify those configurations that comply with center symmetry because those appear as configurations that are invariant under center transformations modulo possible gauge transformations, see for instance 2009.04933. In fact this criterion applies to any physical symmetry in the problem: those configurations that comply with the physical symmetry at hand are those that are invariant under the physical transformation modulo possible gauge transformations. When applied to the particular case of gauge transformations, the criterion becomes tautological since any configuration is trivially invariant under a gauge transformation modulo a gauge transformation, see for instance 2304.00756. In this sense, any configuration complies with gauge symmetry and there is no way in which configurations of the A, B or C type could break gauge symmetry, in line with the idea (supported by Elizur theorem) that a gauge symmetry, being a mere redundancy, could not be broken in any way.
To be more specific, consider the discussion in chapters 4 and 5 of 2009.04933 that can be mapped to the present one via the identification θ1=(r3+r8/√3)/2 and θ2=(−r3+r8/√3)/2. The potential V(θ1,θ2) is invariant under G, the group of periodic transformations modulo a center element, which includes in particular the invariance under G0, the group of strictly periodic transformations. The group G0 is peculiar, however, for it is the group of actual gauge transformations that do not alter the physical state of the system (that is, they do not alter any of the observables, unlike G that alters at least one observable, the Polyakov loop). The interpretation of the invariance of V(θ1,θ2) with respect to G0 is that the space of the variables theta is subdivided into physically equivalent regions (connected by actual gauge transformations in G0) known as Weyl chambers. For any configuration theta in one of the Weyl chambers, there are equivalent configurations within the other Weyl chambers and the whole collection of such configurations forms an orbit under G0. Now, when studying whether a given configuration is invariant or not under a particular symmetry (thus determining whether or not that configuration realizes or breaks that symmetry), what matters is how this whole orbit transforms under the symmetry. The states invariant under some symmetry are those corresponding to invariant G0-orbits. Yet, the configurations that compose this invariant G0-orbit are in general not themselves invariant under that symmetry but invariant only modulo G0. In particular the center-symmetric configurations referred to above are obtained by imposing the invariance under G modulo G0. Given these general considerations, one may ask whether a given configuration breaks G0 or not. Since invariance needs to be though modulo transformations in G0, the answer is clearly no, no matter which configuration one is considering. This is because, any configuration is trivially invariant under G0 modulo G0. Phrased differently, the G0-orbit of a given configuration is always invariant under G0 and thus none of the cases A, B or C should be seen as an actual breaking of G0, in line with the other observations above.
This being said, when analyzed in the usual sense of strict invariance (that is without the “modulo a gauge transformation” decoration), the configurations of type A, B and C differ in that some of them are more invariant than the others under certain gauge transformations. If a transition occurs between any of those configurations, this can have observable consequences, although, once again, these should not be interpreted as the breaking of gauge invariance but rather as the system exploring orbits whose representatives are less invariant under certain gauge transformations (without this affecting the gauge-invariance of the orbit itself). Interestingly enough, within the Curci-Ferrari model (which amounts to adding a gluon mass to the presently discussed approach), and at least in the case of 4d YM with one compact dimension (related to temperature), a transition between C and A has been observed at one-loop order at some temperature (distinct from the center breaking temperature). However, it is believed to be an artifact as it disappears at two-loop order, see 1412.5672, at least within the range of temperatures analyzed in that reference.
In the presence of a gluon mass, one typically finds (again, within 4d compact YM) that the system is in case C and the actual relevant question is whether it takes any arbitrary configuration within the case C or the particular center-symmetric configurations (that also belong to the case C) corresponding to the center of the red dots in Fig.3. So, if there is a distinction to be drawn, it is actually between the generic configurations in case C, and the particular center-symmetric configurations which one could call D. Center-symmetry breaking occurs (when it occurs) between D and C and has nothing to do with what the authors call gauge symmetry breaking, see for instance their paragraph starting section 2.4, where it is written “such a breaking of gauge symmetry is understood as different realizations of center symmetry”. This stated connection between center-symmetry and "breaking of gauge symmetry" is incorrect I believe.
Author: Masatoshi Yamada on 2023-05-27 [id 3690]
(in reply to Report 1 on 2023-05-18)Thank you very much for very valuable comments. We would reply to some comments. We agree that no spontaneous gauge symmetry breaking takes place thanks to the Elitzur’s theorem, while the well-known Higgs mechanism states that non-zero expectation value of the scalar field breaks the gauge symmetry. Indeed, the statement “SU(2)× U(1) gauge symmetry is broken into U(1)” is widely accepted as the standard jargon although this statement is actually incorrect in sense of the Elitzur’s theorem. The terminology in Table I is also standard in the Gauge-Higgs theories. Therefore, we would like to keep it. We agree that a transition between C and A does not take place. Indeed, phases B and C are observed only when the adjoint fermion with finite mass is introduced. Therefore, pure YM theories may show only phases X and A where phase X originates from the distribution of the Haar measure which shows the same configuration as phase B. Thus the phase transition between X and A would be observed. See Ref. [1309.4198] “Breaking of gauge symmetry” in this paper is classified by evaluating the commutation between the Wilson line with configuration of θi and the generators of SU(3). For instance, in phase C, only T3 and T8 within 8 generators commute with the Wilson line and then the gauge fields A3μ and A8μ are massless and the others have mass proportional to θi. We observe this fact as long as the functional method (including perturbation theory) is used although this conflicts with the Elitzur’s theorem. In order to understand this phenomenon in terms of gauge invariant quantities the Froehlich-Morchio-Strocchi mechanism should be employed, but this is out of the purpose in this paper and then we do not intend to discuss it. In order to avoid the confusion, we erase “such a breaking of gauge symmetry is understood as different realizations of center symmetry”, and write simply “As a similar system, D = 4 YM theories at finite temperature (corresponding to R3 × S1) have been discussed in Refs. [50-60]”. We think actually that the Curci-Ferrari model in R4 ×S1 may be interesting as a Gauge- Higgs unification model and may bring a novel direction for understanding the dynamics of the Gauge-Higgs unification theory. We will consider this possibility as a future work. We would thank reviewer 4 very much again for detailed discussions.