Massless Composite Fermions Research Articles

Phase structure of linear quiver gauge theories from anomaly matching

We consider the phase structure of the linear quiver gauge theory, using the 't Hooft anomaly matching condition. This theory is characterized by the length $K$ of the quiver diagram. When $K$ is even, the symmetry and its anomaly are the same as those of massless QCD. Therefore, one can expect that the spontaneous symmetry breaking similar to the chiral symmetry breaking occurs. On the other hand, when $K$ is odd, the anomaly matching condition is satisfied by the massless composite fermions. We also consider the thermal partition function under the twisted boundary conditions. When $K$ is even, from the anomaly at finite temperature, we estimate the relation between the critical temperatures associated with the confinement/deconfinement and the breaking of the global symmetry. Finally we discuss the anomaly matching at finite temperature when $K$ is odd.

Strong anomaly and phases of chiral gauge theories

We present a simple argument which seems to favor, when applied to a large class of strongly-coupled chiral gauge theories, a dynamical-Higgs-phase scenario, characterized by certain bifermion condensates. Flavor symmetric confining vacua described in the infrared by a set of baryonlike massless composite fermions saturating the conventional ’t Hooft anomaly matching equations, appear instead disfavored. Our basic criterion is that it should be possible to write a strong-anomaly effective action, analogous to the one used in QCD to describe the solution of the U(1)A problem in the low-energy effective action, by using the low-energy degrees of freedom in the hypothesized infrared theory. We also comment on some well-known ideas such as the complementarity and the large N planar dominance in the context of these chiral gauge theories. Some striking analogies and contrasts between the massless QCD and chiral gauge theories seem to emerge from this discussion.

Dynamics from symmetries in chiral SU(N) gauge theories

The symmetries and dynamics of simple chiral SU(N) gauge theories, with matter Weyl fermions in a two-index symmetric tensor and N + 4 anti-fundamental representations, are examined, by taking advantage of the recent developments involving the ideas of generalized symmetries, gauging of discrete center 1-form symmetries and mixed ’t Hooft anomalies. This class of models are particularly interesting because the conventional ’t Hooft anomaly matching constraints allow a chirally symmetric confining vacuum, with no condensates breaking the U(1) × SU(N + 4) flavor symmetry, and with certain set of massless baryonlike composite fermions saturating all the associated anomaly triangles. Our calculations show that in such a vacuum the UV-IR matching of some 0-form−1-form mixed ’t Hooft anomalies fails. This implies, for the theories with even N at least, that a chirally symmetric confining vacuum contemplated earlier in the literature actually cannot be realized dynamically. In contrast, a Higgs phase characterized by some gauge-noninvariant bifermion condensates passes our improved scrutiny.

Possible new phase for adjoint QCD

We discuss an exotic phase that adjoint QCD possibly exhibits in the deep infrared (IR). It is a confining phase, with a light spectrum consisting of massless composite fermions. The discrete chiral symmetry is broken, with unbroken continuous chiral symmetry. We argue that it may give a description of the IR of adjoint QCD with three massless Weyl flavors and that it passes all consistency checks known to us.

On the baryon-color-flavor (BCF) anomaly in vector-like theories

We consider the most general fractional background fluxes in the color, flavor, and baryon number directions, compatible with the faithful action of the global symmetry of a given theory. We call the obstruction to gauging symmetries revealed by such backgrounds the baryon-color-flavor (BCF) ’t Hooft anomaly. We apply the BCF anomaly to vector- like theories, with fermions in higher-dimensional representations of arbitrary N-ality, and derive non-trivial constraints on their IR dynamics. In particular, this class of theories enjoys an independent discrete chiral symmetry and one may ask about the fate of this symmetry in the background of BCF fluxes. We show that, under certain conditions, an anomaly between the chiral symmetry and the BCF background rules out massless composite fermions as the sole player in the IR: either the composites do not form or additional contributions to the matching of the BCF anomaly are required. We can also give a flavor-symmetric mass to the fermions, smaller than or of order the strong scale of the theory, and examine the θ-angle periodicity of the theory in the BCF background. Interestingly, we find that the conditions that rule out the composites are the exact same conditions that lead to an anomaly of the θ periodicity: the massive theory will experience a phase transition as we vary θ from 0 to 2π.

Patterns of symmetry breaking in chiral QCD

We consider $SU(N)$ Yang-Mills theory with massless chiral fermions in a complex representation of the gauge group. The main emphasis is on the so-called hybrid psi-chi-eta model. The possible patterns of realization of the continuous chiral flavor symmetry are discussed. We argue that the chiral symmetry is broken in conjunction with a dynamical Higgsing of the gauge group (complete or partial) by bi-fermion condensates. As a result a color-flavor locked symmetry is preserved. The 't Hooft anomaly matching proceeds via saturation of triangles by massless composite fermions or, in a mixed mode, i.e., also by the "weakly" coupled fermions associated with dynamical Abelianization, supplemented by a number of Nambu-Goldstone mesons. Gauge-singlet condensates are of the multifermion type and, though it cannot be excluded, the chiral symmetry realization via such gauge invariant condensates is more contrived (requires a number of four-fermion condensates simultaneously and, even so, problems remain), and less plausible. We conclude that in the model at hand, chiral flavor symmetry implies dynamical Higgsing by bifermion condensates.

Renormalization-group evolution and nonperturbative behavior of chiral gauge theories with fermions in higher-dimensional representations

We study the ultraviolet to infrared evolution and nonperturbative behavior of a simple set of asymptotically free chiral gauge theories with an SU($N$) gauge group and an anomaly-free set of $n_{S_k}$ copies of chiral fermions transforming as the symmetric rank-$k$ tensor representation, $S_k$, and $n_{\bar A_\ell}$ copies of fermions transforming according to the conjugate antisymmetric rank-$\ell$ tensor representation, $\bar A_\ell$, of this group with $k, \ \ell \ge 2$. As part of our study, we prove a general theorem guaranteeing that a low-energy effective theory resulting from the dynamical breaking of an anomaly-free chiral gauge theory is also anomaly-free. We analyze the theories with $k=\ell=2$ in detail and show that there are only a finite number of these. Depending on the specific theory, the ultraviolet to infrared evolution may lead to a non-Abelian Coulomb phase, or may involve confinement with massless composite fermions, or fermion condensation with dynamical gauge and global symmetry breaking. We show that $S_k \bar A_k$ chiral gauge theories with $k \ge 3$ are not asymptotically free. We also analyze theories with fermions in $S_k$ and $\bar A_\ell$ representations of SU($N$) with $k \ne \ell$ and $k, \ \ell \ge 2$.

Phases of chiral gauge theories

We discuss the behavior of two non-supersymmetric chiral SU(N) gauge theories, involving fermions in the symmetric and antisymmetric two-index tensor representations respectively. In addition to global anomaly matching, we employ a recently proposed inequality constraint on the number of effective low energy (massless) degrees of freedom of a theory, based on the thermodynamic free energy. Several possible zero temperature phases are consistent with the constraints. A simple picture for the phase structure emerges if these theories choose the phase, consistent with global anomaly matching, that minimizes the massless degree of freedom count defined through the free energy. This idea suggests that confinement with the preservation of the global symmetries through the formation of massless composite fermions is in general not preferred. While our discussion is restricted mainly to bilinear condensate formation, higher dimensional condensates are considered for one case. We conclude by commenting briefly on two related supersymmetric chiral theories.

Massless composite fermions in two dimensions and the overlap

There exist chiral gauge models in two dimensions that have massless composite fermions. Two examples are presented and it is suggested that they be accepted as benchmark test-cases for generic proposals of non-perturbatively regulating chiral gauge theories in any dimension. We apply the overlap to the two benchmarks and present the results of a numerical simulation of modest size.

Massless composite fermions in two dimensions and the overlap

There exist chiral gauge models in two dimensions that have massless composite fermions. Two examples are presented and it is suggested that they be accepted as benchmark test-cases for generic proposals of non-perturbatively regulating chiral gauge theories in any dimension. We apply the overlap to the simpler of the two benchmarks and present the results of a numerical simulation of modest size.

Exotic nonsupersymmetric gauge dynamics from supersymmetric QCD.

We extend Seiberg's qualitative picture of the behavior of supersymmetric QCD to nonsupersymmetric models by adding soft supersymmetry breaking terms. In this way, we recover the standard vacuum of QCD with $N_f$ flavors and $N_c$ colors when $N_f < N_c$. However, for $N_f \geq N_c$, we find new exotic states---new vacua with spontaneously broken baryon number for $N_f = N_c$, and a vacuum state with unbroken chiral symmetry for $N_f > N_c$. These exotic vacua contain massless composite fermions and, in some cases, dynamically generated gauge bosons. In particular Seiberg's electric-magnetic duality seems to persist also in the presence of (small) soft supersymmetry breaking. We argue that certain, specially tailored, lattice simulations may be able to detect the novel phenomena. Most of the exotic behavior does not survive the decoupling limit of large SUSY breaking parameters.

CHIRAL GAUGE E6 AS A BINDING GROUP FOR COMPOSITE LEPTONS, QUARKS AND HIGGS BOSONS

The uniqueness of the chiral gauge E6 symmetry in providing the mechanism of binding for composite models is stressed. A maximally symmetric pattern of chiral symmetry breaking, consistent with dynamical mass generation along with preservation of the strongly coupled E6 gauge symmetry, is considered. Chiral anomaly matching conditions for the residual chiral symmetry are studied and likely massless composite fermions are found. The possibility for these fermions as well as Goldstone bosons to be treated eventually as leptons, quarks and Higgs bosons is discussed. The scheme possesses the generic realistic-like features and could serve as a prototype for a realistic composite model.

A new anomaly matching condition?

We formulate ``Witten'' matching conditions for confining gauge theories. The conditions are analogous to 't Hooft's, but involve Witten's global SU(2) anomaly. Using a group theoretic result of Geng, Marshak, Zhao and Okubo, we show that if the fourth homotopy group of the flavor group $H$ is trivial ($\Pi_4(H) = 0$) then realizations of massless composite fermions that satisfy the 't Hooft conditions also satisfy the Witten conditions. If $\Pi_4 (H)$ is nontrivial, the new matching conditions can yield additional information about the low energy spectrum of the theory. We give a simple physical proof of Geng, et. al.'s result.

Preon model with three SU3 hypercolours and three SU5 hyperflavours

From ‘vertical’ preons V with SU5V-5, ‘horizontal’ preons H with SU5H-5 and « techni »-preons T withSU5T- 5, where eachSU5 gauge group is broken asSU5 → SU3 xSU2 X U1 at λGUT ~ 1015 GeV, we obtain a set of massless composite fermions which satisfies ’t Hooft’s anomaly matching conditions at an energy scale λHC ~ 106 GeV, where SU3XSU3XSU3″]-hypercolours become strong. Then, at λSTC ~ 104 GeV, where [SU2XSU2′]- supertechnicolours become strong, we obtain quarks and leptons fL = (5 + 10, 3, 1)L and technifermions FR = (5 + 10, 1, 3)R of SU5,VXSU3,HXSU3,TC as three-preon states (VHH) + (VVH) and (VTT) + (VVT), respectively. TechnicolourSU3,TC is broken intoSO3,TC at λSTC and it becomes strong at λTC ~ 102 GeV. The condensate of the technifermionsFR, , provides reasonable quark and lepton masses.

Chiral gauge theories in the 1/ N expansion

We study the large- N limit of various (SU( N) gauge theories with chiral fermion content. Assuming that the leading N → ∞ behavior is given by a sum of planar diagrams, we find that the gauge interactions must fail to confine color in some models. Other models, assuming both a planar diagram limit and confinement, must contain massless composite fermions.

A tool kit for builders of composite models

I motivate and discuss a dynamical picture of strong gauge interactions in a simple class of models, some of which yield massless composite fermions. I show how to put simple components together into models that can describe quarks with color and flavor.

Composite fermions without U(1)'s

We discuss a class of models that may produce masslesscomposite fermions which carry nonabelian unbroken chiral symmetries, but no chiral U(1) symmetries. We analyze these models both in the confining picture and in a variant of the most-attractive-channel picture.

Light composite fermions in a dynamical subquark model of pregauge interactions

The masses of composite leptons and quarks are discussed in a “dynamical subquark model of pregauge interactions”. In this model, the leptons and quarks are made of a spinor and scalar subquark with equal mass, M, and the gauge bosons and Higgs scalar of the SU(3) c×SU(2) L×U(1) Y model are made of a subquark-antisubquark pair. The SU(2) L×U(1) Y symmetry is spontaneously broken by the composite Higgs scalar and the (scalar) subquark mass parameter is in turn bounded as M > 5.4 TeV (=2π( 2 G F −1) 1 2 where G F is the Fermi coupling constant). The spontaneously generated mass of a lepton or quark, m i ( n) ( i = 1, 2; n = 1 ∼ N g), is calculated to be: m i (n) = r i (n) = r i (n) × (4+3N g )α e.m. ( 2 G F −1) 1 2 /3 6 (=0.35r i (n) (4+3N g ) GeV), where r i (n) are the parameters satisfying that 0 ⩽ r i ( n) ⩽ 1 and Σ ( r i ( n) ) 2 = 1; N g is the total number of generations of the leptons and quarks; α e.m. is the fine structure constant. The appearance of light composite fermions is related to a specific mechanism of generating global chiral symmetries of the leptons and quarks. Global symmetries of scalar subquarks yield chiral symmetries of the leptons and quarks. Our model turns out to satisfy 't Hooft's anomaly conditions on massless composite fermions.

Color Singlet Composite Superfields Containing Massless Composite Fermions Obtained by the 't Hooft Equation -- Supermetacolor. II

Further study on supermetacolor, a supersymmetric extension of 't Hooft's metacolor, is made. In this scheme, both chiral symmetry and supersymmetry at the preon level are assumed to be taken over the composite level. As a consistency of this requirement, massless composite fermions should emerge and they should be members of chiral supermultiplets. In this paper, we develop a systematic prescription for finding superpartners of massless composite fermion. By making use of this method, superpartners of massless composite fermions are given explicitly in terms of original preon fields. In view of this result, it is argued that supermetacolor scheme is consistent one. At the same time this result leads to the suggestion that the above-mentioned symmetry realization (unbroken chiral symmetry and supersymmetry) is in fact possible.

Composite models of quarks and leptons and strong-coupling lattice gauge theories

We extend a previous variational block-spin analysis of the realization of chiral symmetry in a strong-coupling lattice gauge theory to models which have been suggested as possibly describing massless composite fermions. In all cases we find massive fermion composites and spontaneous breaking of chiral symmetry. We also discuss the relevance of this result to the continuum limit.