Abstract
In principle, there is no obstacle to gapping fermions preserving any global symmetry that does not suffer a 't Hooft anomaly. In practice, preserving a symmetry that is realised on fermions in a chiral manner necessitates some dynamics beyond simple quadratic mass terms. We show how this can be achieved using familiar results about supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking. We present simple models that gap fermions while preserving a symmetry group under which they transform in chiral representations. For example, we show how to gap a collection of 4d fermions that carry the quantum numbers of one generation of the Standard Model, but without breaking electroweak symmetry. We further show how to gap fermions in groups of 16 while preserving certain discrete symmetries that exhibit a mod 16 anomaly.
Highlights
What symmetries are lost when fermions gain a mass? Naively, one might think that chiral symmetries are broken, while vectorlike symmetries survive
Preserving a symmetry that is realized on fermions in a chiral manner necessitates some dynamics beyond simple quadratic mass terms
We show how this can be achieved using familiar results about the strong coupling dynamics of supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking
Summary
What symmetries are lost when fermions gain a mass? Naively, one might think that chiral symmetries are broken, while vectorlike symmetries survive. For someone steeped in the Wilsonian perspective on continuum quantum field theory, relying on dynamics at the UV cutoff to drive the low-energy physics of interest might induce a level of anxiety Any such nervousness is likely to be compounded by the observation that, on closer inspection, the Eichten-Preskill mechanism seems not to work, with no hint of the gapped chiral phase appearing as one explores some (admittedly finite-dimensional) parameter space [13,14]. [Tachikawa and Yonekura [17] previously advocated the use of supersymmetric gauge dynamics to explore symmetric mass generation Their interest was in gapping 16 fermions in d 1⁄4 2 þ 1 dimensions while preserving time reversal, albeit viewed from the perspective of the bulk d 1⁄4 ð3 þ 1Þ-dimensional symmetry protected topological phase.] Usually in s-confining theories one has massless fermions in both the UV and IR, but with the ’t Hooft anomalies for unbroken symmetries realized in startlingly different fashions. A small tweak of this idea allows us to gap fermions preserving chiral symmetries, including the example highlighted above of fermions in the standard model [18]
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