Critical behavior of gauge theories and Coulomb gases in three and four dimensions
Gauge theories with matter often have critical regions in their parameter space where gapless degrees of freedom emerge. Using controlled semiclassical calculations, we explore such critical regions in gauge theories with a topological term and fundamental fermions in four dimensions, as well as related field theories in three dimensions. In four-dimensional theories, we find that for all the critical behavior always occurs at a point in parameter space. For this is consistent with the standard QCD expectations, while for our results are consistent with recent observations concerning ’t Hooft anomalies. We also show how the -branched structure of observables transmutes into the -branched structure seen in chiral Lagrangians as the mass parameter is dialed. As a side benefit, our analysis of these 4D theories implies the unexpected result that 3D Coulomb gases can have gapless critical points. We also consider QCD-like parity-invariant theories in three dimensions, and find that their critical behavior is quite different. In particular, we show that their gapless region is an interval in parameter space, rather than a point. Our results have non-trivial implications for the infrared behavior of three-dimensional compact QED.
1 Introduction and results
This work derives new non-perturbative results in 4D and 3D QCD with fundamental Dirac fermions using using techniques such as calculable compactifications, adiabatic continuity and mixed discrete ’t Hooft anomalies. Our main results are:
It was recently argued that with some natural assumptions, 4D QCD with fundamental Dirac fermions has a critical point in the complex mass planeGaiotto:2017tne . We show that this is indeed the case in a calculable regime of the theory. This result follows from the existence of a gapless critical point in a 3d Coulomb gas. This is an interesting finding in its own right, because historically, it had been believed that 3d Coulomb gases are always gapped Polyakov:1976fu ; 0022-3719-10-19-011 ; Read:1990zza ; Fradkin:1991nr .
The transmutation of -angle dependence in the chiral Lagrangian to the dependence in pure Yang-Mills as the mass of the quarks are sent to infinity.
The existence of a critical interval (rather than a point) in 3d QCD.
We will now summarize our results in more detail, as well as place them in context relative to the various recent developments that have made them possible.
For all in the range we have studied, we find that in 4D QCD there is a single gapless point in the parameter space of the theory at zero temperature and density. For , this is of course entirely expected, because this critical point is at the massless quark point, , and is associated with spontaneous chiral symmetry breaking. For , the existence of a critical point may sound somewhat surprising, because there is no chiral symmetry breaking reason to expect a gapless mode. The lightest pseudoscalar meson, the , has historically been expected to have mass for any quark mass . However, if one turns on a term, the quark mass is naturally intepreted as a complex parameter, . Reference Gaiotto:2017tne recently pointed out there is very likely to be a point (amounting to ) where the mass must vanish. This conclusions follows from some very plausible assumptions about the CP-symmetry-breaking behavior of YM theory as a function of as well as anomaly constraintsGaiotto:2017yup . For , CP symmetry is expected to break spontaneously, while it is preserved for . One can interpret as a second-order critical point.
We would like to get some microscopic insight into this result, and related results for . Our tool will be the idea of adiabatic continuity. Over the last several years, a large number of results suggest that it is possible to continuously (that is, adiabatically) connect QCD-like theories on to a class of theories which have a weakly-coupled regime on Unsal:2007vu ; Unsal:2007jx ; Unsal:2008ch ; Shifman:2008ja ; Shifman:2009tp ; Unsal:2010qh ; Shifman:2008cx ; Cossu:2009sq ; Myers:2009df ; Simic:2010sv ; Vairinhos:2011gv ; Thomas:2011ee ; Anber:2011gn ; Poppitz:2012sw ; Poppitz:2012nz ; Unsal:2012zj ; Argyres:2012ka ; Argyres:2012vv ; Anber:2013doa ; Cossu:2013ora ; Bhoonah:2014gpa ; Anber:2014lba ; Bergner:2014dua ; Li:2014lza ; Anber:2015kea ; Anber:2015wha ; Misumi:2014raa ; Cherman:2016hcd ; Aitken:2017ayq ; Anber:2017rch . This weakly-coupled regime appears when the circumference is sufficiently small, and (approximate) center symmetry is stable. In the weakly-coupled regime we can use semiclassical methods to determine the behavior of the theory. We use this as a tool to explicitly see how the mass manages to vanish at , in a calculable setting. Our semi-classical analysis on small with appropriate stabilized symmetries (either center or color-flavor-center) is based on a systematic microscopic derivation of chiral symmetry breakingCherman:2016hcd . Our results are consistent with expectations from experimental observations for large and from continuous and discrete ’t Hooft anomaly considerations.
A interesting aspect of this analysis is that it yields a somewhat unexpected result concerning Coulomb gases in three dimensions. Historically it has been believed that such plasmas always have a finite correlation lengthPolyakov:1976fu ; 0022-3719-10-19-011 ; Read:1990zza ; Fradkin:1991nr . But we show that our QCD results on imply the existence of a gapless point in 3d Coulomb gases in the presence of special complex fugacities for the monopole events. In general, it is known that complex fugacities induced by Berry phases or topological angle can decrease the mass gap exponentially, from down to , , or , see, e.g. studies of VBS phase of quantum anti-ferromagnets Read:1990zza ; Fradkin:1991nr and deformed Yang-Mills at Unsal:2012zj . Our work is the first demonstration that the mass gap in magnetic Coulomb gas can actually vanish at a point in parameter space.
For , in the calculable weak coupling regime on we find that the critical point is at , consistent with QCD expectations on . In this regime, we leverage insights from our recent work with Thomas Schäfer Cherman:2016hcd , where the chiral Lagrangian was derived rigorously in the small setting by using duality and monopole operators. We find that the theory has a two-fold vacuum degeneracy for , and a unique vacuum otherwise so long as . At , the theory has a critical point with gapless excitations. This behavior is associated with spontaneous continuous chiral symmetry breaking. In our set-up, we can dial the quark mass from small to large values while keeping all approximations under control. We use this to show how the transmutation of the angle dependence of observables
takes place. This -dependence transmutation occurs as is taken from small values, where the softly broken chiral Lagrangian is a good description of the physics, to large values, where the physics becomes describable in terms of pure Yang-Mills theory.
For the theory, the analysis of Gaiotto:2017tne shows that the potential of the leading order chiral Lagrangian vanishes at , and it has not been shown whether the flavor symmetry is broken, or CP symmetry is broken. In the calculable regime on , the potential of the monopole-induced chiral Lagrangian vanishes at when . This corresponds to the expected vanishing of the leading-order chiral Lagrangian. But there are also magnetic-bion induced terms, corresponding to higher-order terms in the chiral expansion. At small , the coefficients of these higher-order terms are calculable, and hence the symmetry breaking pattern can be fixed. In particular, we find that CP is broken at .
For and , we show that there is a mixed ’t Hooft anomaly between color-flavor center (CFC) symmetry Cherman:2017tey (assuming ) and a discrete chiral symmetry. In this set-up, where is an exact symmetry on , the global symmetry is explicitly reduced to its maximal Abelian subgroup
by the quark boundary conditions on . Suppose that CFC symmetry is preserved111CFC symmetry can be stabilized for small using e.g. certain double-trace deformationsUnsal:2008ch . when is small, as well as when is largeIritani:2015ara . There are no order parameters which transform under , but are neutral under the continuous chiral symmetry . Therefore, if the CFC symmetry is unbroken for any size, then mixed discrete ’t Hooft anomaly matching actually implies that the continuous symmetry must be spontaneously broken. This is beautifully consistent with our weak-coupling derivation of chiral symmetry breaking in a regime on with unbroken CFC symmetryCherman:2016hcd . The existence of the mixed ’t Hooft anomaly imply that our results extrapolate to the strong coupling regime, without phase transitions for any size, provided that CFC symmetry is unbroken.
Finally, in order to emphasis the distinction between the above locally four-dimensional analysis, where the dynamics only becomes three-dimensional at long distances, and the dynamics of genuinely three-dimensional theories, we also analyze a three-dimensional gauge theory with Dirac fermions. (This is roughly the dimensional reduction of the 4d Dirac flavor theory.) We consider the behavior of this theory as a function of a parity-invariant mass term, and find that its behavior is very different from its four-dimensional cousins. In particular, in three dimensions, instead of finding a critical point in mass-parameter space, we find a critical interval. In our three-dimensional set-up, in contrast to an analogous 4D theory, the gapless mode seen in the critical interval is protected by a spontaneously broken continuous shift symmetry, and it is a Nambu-Goldstone boson. Our results are consistent with recent conjectures discussed in Komargodski:2017keh .
In this section we discuss the critical behavior of one-flavor QCD as a function of the complex quark mass and the number of colors by using semi-classical methods and adibatic continuity on . The Lagrangian is
where is the four-dimensional Dirac spinor in the fundamental representation of , and can be written as where are two-component (complex) Weyl spinors.
This physics of this theory depends on the combination instead of depending on and separately. In fact, one can remove the topological term by a chiral rotation, and write the mass term as
where, on the right hand side the first mass term is the parity-even Dirac mass term, while the second term is the parity-odd Dirac mass term. The latter representation makes is clear that axis is special, in that it corresponds to the presence of CP symmetry, and amounts to setting or .
This theory has a vector-like symmetry, where the quotient is associated with color gauge transformations living in the center of . For , the chiral anomaly explicitly breaks the classical symmetry to , which is actually a symmetry of the theory with a bare mass. The instanton amplitude is
Unlike theories, in which one has an anomaly-free chiral symmetry at , in the theory there is no extra symmetry at , mainly due to instanton induced soft mass term (5). Thus, as a result of non-perturbative effects, the quark mass term receives additive renormalization as opposed to multiplicative renormalization. In the semi-classical domain, we will use a formalism where we have full control over the additive non-perturbative corrections to the mass term.
Below, we study the dynamics of this theory on small on , covering first, and then discuss generic values of with some comments on large limit. All of our analysis assumes that, as mentioned in the introduction, the approximate center symmetry has been stabilized by double-trace deformations.
First, let us consider two-color QCD with quarks of a single flavor. In both pure Yang-Mills theory and one-flavor QCD at large , the distribution of the eigenvalues of the color holonomy becomes approximately center-symmetric, in the sense that the expectation value of the trace of the holonomy becomes approximately zero. In pure Yang-Mills theory the distribution becomes exactly center-symmetric for all larger than some critical value, while for one-flavor QCD the VEV of the trace of the holonomy only approaches zero as when the quarks are light. In any case, we assume that the trace of the holonomy continues to be center-symmetric at small . Justifying this assumption relies on the introduction of an appropriate double-trace deformation or heavy adjoint fermions, and we assume that this has been done.
The color holonomy has the form , where is the compact direction of circumference , and we assume that . If , then (with an obvious gauge-fixing understood) , the eigenvalues of have parametrically small fluctuations thanks to the smallness of , and take the center-symmetric vacuum expectation value (VEV) . If we integrate out Kaluza-Klein (KK) modes on , we obtain a 3D gauge theory. In this 3D effective theory, acts like a compact adjoint Higgs field. When the VEV of is center symmetric, the color gauge group is reduced down to , and theory Abelianizes at distances larger than . So in fact the long-distance theory is 3D compact QED
where parametrize the non-compact directions, and is the field strength associated with the unbroken gauge group. We will find useful to pass to the Abelian dual formulation in terms of scalar , , where is the ’t Hooft coupling. Then the action of the 3D effective field theory takes the form
2.1.1 The chiral limit
as the fermion boundary condition. The twist in the boundary conditions can be thought of as a holonomy for a background gauge field associated to the symmetry of the theory. The role of the twist is to produce a parity-invariant mass term in the 3D effective theory,
This type of mass term is also often called a “real mass term” in 3d language. If we were to set , corresponding to periodic boundary conditions, the long-distance theory would be 3D QED coupled to flavors of massless Dirac fermions. This EFT would be strongly-coupled on distances , where is the YM coupling. This issue can be avoided by keeping finite, because produces the real mass term for the 3D fermions, as written above. Consequently, so long as , our theory is weakly coupled at all distance scales, and can be treated reliably using semiclassical methods. Thus there are no infrared divergences in perturbation theory.
Since our long-distance theory is 3D compact QED, it has finite-action monopole-instanton field configurationsPolyakov:1976fu . In fact, because the theory is locally four-dimensional, the theory has two types of monopole-instantons: one is the standard BPS monopole-instanton which is present even in locally three-dimensional theories, while the other is the so-called Kaluza-Klein (KK) monopole instantonLee:1997vp ; Kraan:1998sn . We write the corresponding amplitudes as . These finite-action field configurations can be thought as the constituents of the 4d instanton on the compactified geometry, . Note that the monopole-instantons have no scale moduli, since they have a finite core size . Indeed, in this setting, a 4d BPST instanton with a large scale modulus looks like two widely separated , events. Along with our remarks concerning the gauge coupling, this implies that there are no infrared problems either perturbatively or non-perturbatively.
The two fermion zero modes of the 4d instanton localize on one of the two types of monopole instantons, depending on the holonomy . For our choice of , we can take the fermion zero modes to be localized on . To leading non-trivial order in the semiclassical expansion, the resulting dimensionless monopole operators take the form:
where is the monopole-instanton action, which is half of the 4d-instanton action.222We have suppressed powers of (which arise from summation over bosonic zero-modes) in the prefactors of the monopole-instanton amplitudes to reduce clutter in our expressions. These prefactors are identical for , and are not important for our discussion in this paper.
To find the leading effect of the existence of monopole-instantons on the long-distance physics, we must sum over the dilute monopole-instanton “gas”Polyakov:1976fu . At leading order, the monopole-instanton contributions to the partition function simply exponentiate, so thatUnsal:2008ch
The potential renders gauge fluctuations massive, with a mass . The unique minimum of the potential is at , and this generates a “constituent quark mass” . Note that this is exponentially larger than the contribution to the constituent quark mass from instantons, which is .
One can also introduce a topological angle. But in the limit, it has no effect on the effective potential since it can be absorbed into fermion field by a field redefinition.
Turning on a small quark mass lifts the fermionic zero modes, and renders the angle physical. The contribution of the monopole-instantons to the dual photon potential is now
where and term is due to magnetic bion.333Actually, the bion potential term should have has various factors we have not shown which multiply , which involve and powers of . They could easily be restored, but have no important effect in the discussion, and we have dropped them to avoid cluttering the formulas. Here by we mean the quark-mass-dependent prefactor of the monopole-instanton amplitude, normalized so that , where the functional determinants are evaluated in the background of the monopole-instanton. It is easy to determine the behavior of in two limiting cases:
Here is an immaterial number of order one. We set it to unity to avoid clutter; it is easy to restore it in the formulas below if one wishes. We expect the cross-over value of setting the scale where changes from scaling with to becoming -independent, to be , since this is the only dimensionful scale entering the monopole-instanton field profile. However, the functional form of apart from the two limits in (15) has not yet been determined, and might not have a closed-form expression, given analogous computations for BPST instantonsDunne:2004sx . A cartoon showing the expected shape of is given in Fig. 4.
There are two crucial points to note about this potential. Despite the fact that the magnetic charge of and is the same, there is a phase difference between the induced operators, which is dictated by phase of . The other point is that the magnitudes and differ in an interesting way. In modulus, and are very different for , but become almost (but not quite) equal for . So
Only in the limit do the magnitudes of and become equal. Both of these observations will will play a crucial role in the physics at .
2.1.3 Large negative
When , the theory reduces to deformed Yang-Mills on small , which is believed to be continuously connected to pure Yang-Mills on , because all gauge invariant order parameters have the same behavior at large and small . In this limit, the prefactors of the monopole operators become equal in absolute value, and the effective potential is given by
The last term is induced by magnetic bion events, which are quasi-saddle-points which become relevant at second order in the semiclassical expansionUnsal:2007jx ; Unsal:2012zj . The first two terms dominate over the third term for , but they cancel as , where the bion-induced term becomes dominant. While the density of the monopole-instanton events of the two types are equal, their effects cancel due to destructive topological interference Unsal:2012zj . Indeed, at , all odd charge monopole induced terms of the form are absent due to topological shift-symmetry, . But thanks to the contributions of the magnetic bions, the theory always has a mass gap. For , the ground state is gapped and unique. But for , there are two gapped vacua associated with CP breaking, , which are associated with CP breaking, since CP acts on by .
Reference Gaiotto:2017yup recently showed that there exists a mixed ’t Hooft anomaly between the 1-form center symmetry and the 0-form CP symmetry at . The anomaly strongly constrains the nature of the possible ground state of the theory at . In particular, if (1) center symmetry is stable, and (2) the mass gap does not vanish, anomaly-matching implies that the CP must be broken. But of course anomaly considerations cannot determine whether the two assumptions leading to CP breaking are in fact satisfied. This requires information about the dynamics, either from lattice simulations, or from some analytic approach. But Euclidean lattice simulations suffer from a severe sign problem at .
This is where semi-classical calculations enabled by adiabatic compactification are helpful. Indeed, in center-stabilized YM theory, the semiclassical analysis presented in Unsal:2012zj and reviewed above explicitly shows that CP is spontaneously broken at . If the center symmetry is stable for all , which can be guaranteed by an appropriate double-trace deformation at any , and the mass gap remains finite for any finite , which is expected at large and is shown analytically above for small , then the two-fold degeneracy of the vacua due to spontaneous CP-breaking will persist for any .
2.1.4 Small negative
We now discuss what happens as approaches with . It is convenient to do so using the shifted variable . Then the potential reads
and at we obtain
As we have just seen, when and is large, there are two degenerate vacua and a corresponding first-order phase transition. The two vacua are related by time-reversal (equivalently, CP) symmetry. When is large and positive, the potential has a unique ground state, where . As is decreased toward zero with fixed , there is a second order critical point at which mass of the fluctuation vanishes, and that the theory becomes gapless. Once , there are again two degenerate vacua, and a corresponding first order phase transition. These features are summarized in Fig. 2.
Given a smooth function with the limiting behavior given by (15), there exists a value, , at which the theory becomes gapless. Indeed, expanding to
So the mass of the fluctuations vanishes when is such that
Given that is a monotonic function with the limiting behaviors shown above, this equation will have one solution, , where the theory has a gapless second-order critical point.
In this analysis, if we wish to get an explicit estimate of , we must use a model for the precise shape of , since this shape is not currently known. For instance, if we model , then we get . But the existence of critical point is robust and model-independent, since it follows from the known limiting behaviors of the smooth function . Moreover, as we are about to see, when , it is possible to compute the value of explicitly without making ad-hoc assumptions about the particular functional form of .
For generic and to leading order in the semiclassical expansion, the low-energy EFT describing monopole instantons can be written as
Here . In writing this expression, we have assumed that the approximate center symmetry (which becomes exact as or ) has been preserved at small . Then (23) follows from essentially the same arguments as above. The gauge group is Higgsed from to , and the gauge coupling stops running at the scale . So in the Abelian dual description there are now dual scalar fields . In writing (23), however, we chose to add one extra dual scalar field, as if we had started with a gauge theory rather than an gauge theory, because this allows us to write many expressions in a simpler form. As will be clear below, this extra scalar decouples exactly from the physical fields. There are now types of monopole-instantons, with of them being standard ’t Hooft-Polyakov monopoles, while the th one is the KK monopole-instanton. The associated amplitudes are
where . Here, we have assumed that the twist angle is such that the fermion zero modes localize onto in the chiral limit. We have also exploited the freedom to perform field redefinitions to put all of the dependence onto . (Section 4.2 contains an in-depth discussions of such field redefinitions, and the related phenomenon of -dependence transmutation as goes from small to large.) These species of monopole-instantons are again the constituents of the familiar 4d BPST instanton,
Evaluating the sum over the dilute monopole-instanton gas to leading non-trivial order yields (23).
For and finite , there is no exact (axial) chiral symmetry in the limit. For generic values of , we will now see that all physical modes have mass square proportional to , but one linear combination of the modes in describes meson. However, along a ray on the complex mass plane , CP symmetry is spontaneously broken. This CP-breaking ray of first-order transitions ends at with a second order critical point.
To see this, let us set and (so that we study the theory along the real axis in the complex-mass plane), and suppose that the minimum of the potential is at . This assumption will be valid provided . Expanding around this minimum, the mass matrix is
where the upper and lower signs correspond to the effect of setting and respectively. For , the spectrum of the operator is positive definite, corresponding to the hadron spectrum of the small-circle theory. However, when , there is a point
there is an eigenvector such that , which takes the form
We identify the resulting gapless mode with the meson.444Generically, has one other vanishing eigenvalue for any , with eigenvector . The corresponding gapless excitation can be shown to be exactly free. In fact, this excitation is unphysical, and arises from our use of an -dimensional basis in writing the roots and dual photon vectors. This identification is supported by the fact that this mode is the lightest pseudoscalar excitation of the theory at , as we illustrate in Fig. 3, as well as our discussion in Section 4, where we explain how this mode enters the chiral Lagrangian. Note that the mode is not free already to leading order in the semiclassical expansion: the potential (23) generates an interaction term with coefficient .
It is also interesting to note that when is large, we can use the known behavior of for small to conclude that, in terms of the complexified quark mass , when we get
So at large , the location of the fixed point can be self-consistently calculated using only the known properties of . Moreover, in the limit, the meson becomes gapless at , just as one would expect from the fact that an unbroken symmetry must emerge at in the large limit.
3 How can a 3d Coulomb gas ever fail to generate screening?
The gauge theory on maps to a (magnetic) Coulomb gas. A Coulomb gas in is known to have a finite Debye length. For example, in the Polyakov model, this finite Debye length is the inverse mass gap of the system. There is a rigorous renormalization group argument due to Kosterlitz 0022-3719-10-19-011 which also shows that the Coulomb gas is always in a screening phase for . In this case, the term that generates the mass gap is relevant. As a result, at long distance one obtains Debye screening.
The Kosterlitz argument is also valid for systems which are effectively generalized Coulomb gases, as discussed in Fradkin:1991nr . In some systems, in the monopole-gas representation, the action possesses an imaginary part, coming either from (lattice/cut-off scale) Berry phases or a topological theta angle. Examples include the valence-bond-solid (VBS) phase of quantum anti-ferromagnets Read:1990zza ; Fradkin:1991nr and center-stabilized Yang-Mills at Unsal:2012zj . In these cases, in order to derive the long-distance effective field theory, one can do a succession of “coarse-graining” steps, as is common in statistical mechanics treatments of renormalization. This process gets rid of type operators, due to destructive interference induced by Berry phases or the topological interference due to -angles among monopoles with the same magnetic charge, but differing phases. Namely, one has a relation of the form
In the statistical-mechanics examples, such relations are possible because the magnitudes of the distinct monopoles amplitudes with identical magnetic charge are forced to be identical by exact lattice symmetries. In Yang-Mills theory, the equality of magnitudes of monopole-instanton amplitudes is guaranteed by center symmetry. However, there are always some monopoles with higher charge which do not carry Berry phases, or any dependence on the topological angle. Such monopoles survive the coarse-graining procedure. In gauge theory on , these are the magnetic bions, which have magnetic charge and vanishing topological charge, while in the VBS phase of quantum anti-ferromagnets, the surviving contributions are from certain multi-charge monopoles. Generically, the surviving operators induce a potential for the dual photon of the form
for some . Then the Kosterlitz argument goes through, because this potential always includes relevant terms. The fluctuations of the dual photon field become suppressed, since the relevant cosine operator pins the fluctuations down to the bottom of one of its minima. And hence, one “always” obtains a finite screening length for generalized Coulomb gas as well. The arguments described above have lead to intuition in the literature that Coulomb gases in three dimensions are always gapped.
The only historically known exception appeared in the work of Affleck, Harvey, and WittenAffleck:1982as , which added massless adjoint fermions to the non-Abelian gauge theory UV completion of a compact QED system, and observed that this rendered the system gapless. But in the case considered in Affleck:1982as , the system is not simply a magnetic Coulomb gas with bare interactions of the form , because the exchange of fermion zero modes between monopoles immediately produces a logarithmic contribution to the potential . As a result, this case does not fit into the paradigm discussed above.
In light of the above discussion, one may wonder how can one ever obtain a Coulomb gas which can be gapless. If a Coulomb gas with infinite correlation length can exist, where is the flaw in the above classic arguments?
The answer of course is already present in the analysis of the previous section, both for and . Let us focus on . We have a Coulomb gas of magnetic monopoles at , and there is a relative phase between the two distinct monopoles of the same magnetic charge. If there were to be a precise symmetry between monopole and , their contributions would cancel each other out and leave room for contributions from higher charge operators, such as . However, in our system, the contributions of and to the effective potential do not cancel each other out exactly, because the symmetry relating them — center symmetry — is explicitly broken. The amount by which these leading-order contributions fail to cancel can be dialed by changing the fermion mass parameter . In particular, the sign of the leading-order contribution to the mass of the dual photon can be arranged to be negative. Then one can tune the negative leading-order contribution against the higher-order contributions, in such a way that the coefficient of vanishes exactly. Consequently, one finds that there is a second-order critical point where the theory is gapless. Note that here there is no shift symmetry for , and at the critical point.
It would be very interesting to reexamine the physics of VBS phases of quantum anti-ferromagnets discussed in Read:1990zza in view of the above remarks. In Read:1990zza , there are lattice symmetries which ensure the equality of the magnitude of the unit-charge monopole amplitudes. It might be possible to explicitly break these lattice symmetries in such a way that, as a function of the perturbation, the system can be tuned to a gapless critical point. This would be analogous to the critical point in one-flavor 4d QCD as a function of , where the relevant broken symmetry is center symmetry.
We now let , and consider what happens when , with the assumption that the quark flavors have a common mass , so that the global symmetry of the theory on is
Here the and factors in the quotient act by , , and come from the action of center elements of and respectively. In the chiral limit the anomaly-free internal global symmetry is enhanced to
which breaks spontaneously breaks to .
When these boundary conditions preserve a subgroup of , where is the global center symmetry group of gauge theory, while is the group of cyclic flavor permutations, which act by . Following Cherman:2017tey we refer to as the color-flavor-center (CFC) symmetry. 555In the setting where , the same boundary conditions were discussed in Kouno:2012zz ; Sakai:2012ika ; Kouno:2013zr ; Kouno:2013mma ; Iritani:2015ara ; Kouno:2015sja ; Hirakida:2016rqd ; Larsen:2016fvs ; Hirakida:2017bye . In these references the construction is interpreted as defining a “QCD-like” theory termed “ QCD”. In contrast, in Cherman:2017tey and here, we interpret the CFC symmetry as a bona fide symmetry of QCD itself. As pointed out in Cherman:2017tey , this symmetry has interesting order parameters whose behavior depends non-trivially on the parameters of the theory. Its order parameters include Polyakov loops wrapping the compactified direction, as well as some local operators built out of the quark bilinears Cherman:2017tey .
Note that the symmetry relevant to the CFC symmetry is different from the action of the factor in (33). The transformations relevant to the CFC symmetry are related to the global center symmetry of pure YM theory arising from gauge transformations which are aperiodic by an element of the center, while the in (33) comes from the action of standard periodic gauge transformations.
The boundary conditions in (35) break to
where is the Weyl group of , which is the group of permutations of the quark flavors, while is the non-anomalous remnant of the classical symmetry acting as . The cyclic flavor permutation group is a subgroup of . In any case, the flavor-center-symmetric boundary conditions give the charged Nambu-Goldstone bosons masses , but the neutral Nambu-Goldstone bosons remainmassless when . Lattice simulations and continuity arguments in imply that CFC symmetry is not spontaneously broken at large Iritani:2015ara for any , provided that is not large enough to put the theory into the conformal windowCherman:2017tey . The CFC symmetry can be preserved at small either by adding heavy adjoint fermions or certain double-trace terms, just as in pure YM theoryCherman:2016hcd , and we assume that this has been done from here onward.
Let us now discuss the behavior of the theory at small . When , a 4D BPST instanton has zero modes. These zero modes are distributed over the monopole-instantons in a manner which is dictated by the quark boundary conditionsNye:2000eg ; Poppitz:2008hr . In particular, with the flavor-twisted boundary conditions in (35), which are equivalent to turning on a flavor-center-symmetric holonomy for a background gauge field, the zero modes are distributed among the monopoles as evenly as possible Cherman:2016hcd .
It is sensible to split the leading monopole-instantons with action into two sets. One set, , consists of monopole-instantons which carry two fermionic zero modes each. The other set, , consists of monopole-instantons which do not carry any fermionic zero modes. The associated amplitudes take the form
The monopole operators in play a major role in the low energy dynamics, and especially in the emergence of the chiral Lagrangian.
This set-up actually provides a microscopic semi-classical derivation of chiral Lagrangian on Cherman:2016hcd , which we now briefly review. In the absence of the monopole operators, the dual photon has topological shift symmetry. In general there is no such symmetry in the original 4D gauge theory, so we would expect it to be broken by non-perturbatively induced interactions. And indeed, the operators in break explicitly to a subgroup as they induce a potential for certain modes. However, the effect of the operators in is more subtle. To see this, note that the flavor-twisted boundary preserve a anomaly-free chiral symmetry, as seen in (36). As a result, the monopole operators in must be invariant under this symmetry. But this is impossible unless the dual scalars transform by shifts under , since transforms under the chiral symmetry. In particular, for the set of monopole-instantons in , we have:
The parameters obey one constraint, , which can be realized as where are the simple roots of the Lie algebra . This implies that the symmetry gets intertwined with the part of the would-be topological shift symmetry of the dual photons. This happens in such a way that their diagonal combination is an exact symmetry of the theory in the limit.
Thus we find that dual photons transform by shifts under a continuous chiral symmetryCherman:2016hcd . Therefore they cannot develop a mass either perturbatively or non-perturbatively. Moreover, at any given point on the vacuum manifold, symmetry is spontaneously broken. The vacuum alignment relevant to the chiral limit can be worked in the same way it is always done in chiral perturbation theory: after finding the vacuum at finite positive , which is , one can expand around this point and take the limit to find the spectrum. The result is that the gapless dual photons are the interpolating fields for the gapless neutral Nambu-Goldstone bosons expected from the symmetries of the compactified theory. The remaining dual photons acquire a non-perturbative mass of order , as in pure center-stabilized YM theory at small .666It is not currently known how to ensure adiabatic continuity when .
Now, when we introduce a bare mass term in the QCD action, the angle becomes physical. One can absorb the theta angle into the mass term by a chiral rotation to yield . Let us suppose that the quark mass is small, , so that we can write the dependence of the monopole operators on explicitly. Then, soaking up the fermion zero modes with mass operator, the monopole operators are modified into
The effective potential induced by monopole-events is now
This action is invariant under the global CFC symmetry . We have already analyzed this action for in a previous section, so now we focus on .
First, for , modes are gapless in the limit. So is the critical point of the theory, as one would expect. These gapless modes are the Nambu-Goldstone bosons associated to the spontaneously broken abelianized chiral symmetry. When , these modes have mass squares proportional to , i.e, they are light compared to . There is also another set of heavier modes whose mass squared is , as well as other heavy modes with masses we have not explicitly included in our 3D effective field theory. (They are discussed in detail in Aitken:2017ayq .) Note that at the critical point , the meson is not massless, except in the limit with fixed .
For , the effective theory has two CP breaking vacua for . In Gaiotto:2017tne , it is shown that there is a mixed anomaly between and CP symmetry when . In our case, there is an symmetry at short distances , but it is broken to its maximal torus by the boundary conditions. But there still exists a mixed anomaly between and CP symmetry when , similarly to the easy axis model Komargodski:2017smk . Our results are consistent with the constraints of this anomaly. As we mentioned earlier, the anomaly does not determine the vacuum structure, but in this case the dynamical information encoded in (42) implies that, of the several choices for the vacuum structure allowed by the anomaly, the theory chooses to have broken CP symmetry.
When , modes are gapless in the limit. In this case, in our effective Lagrangian based on dual photons, there are no degrees of freedom corresponding to an mode. Of course, when , there is no reason to expect the mode to be parametrically light either at finite or large . Indeed, the full theory contains many more degrees of freedom beyond the dual photons, such as the fields corresponding to W-bosons and heavy quark modes. These heavier modes have their own very rich dynamics discussed in Aitken:2017ayq . We expect that the can be identified with the lightest isoscalar parity-odd heavy-mode bound state in the spectrum discussed in Aitken:2017ayq , but have not explored this quantitatively.
It is important to note that the truncation of the monopole operators to the set is equivalent to the reduction of our effective theory down to chiral Lagrangian. However, keeping heavier modes has its advantages. In particular, in Gaiotto:2017tne , the vacuum structure remains undetermined at leading order in chiral Lagrangian for , as the associated effective potential term vanishes identically. It is posited that sub-leading terms in the chiral Lagrangian renders vacuum two-fold degenerate. Moreover, in exploring the vacuum structure at generic , Gaiotto:2017tne assumes that symmetry is not broken spontaneously at . There is no known systematic argument ensuring this, since the Vafa-Witten theoremVafa:1983tf does not apply at . In our case, since our effective theory has both light and heavy modes, the leading order in semi-classics actually suffice to prove that is broken spontaneously for all , while the vector-like part of flavor symmetry is not spontaneously broken.
To see the connection of (42) to chiral Lagrangian, construct the matrix :