Supersymmetric Galilean Electrodynamics

Published for SISSA by Springer Received: July 27, 2022 Accepted: September 20, 2022 Published: September 28, 2022 Supersymmetric Galilean Electrodynamics a Department of Physics, Ben-Gurion University of the Negev, David Ben Gurion Boulevard 1, Beer Sheva 84105, Israel b Università degli studi di Milano Bicocca, Piazza della Scienza 3, 20161, Milano, Italy c INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20161, Milano, Italy E-mail: , , Abstract: In 2+1 dimensions, we propose a renormalizable non-linear sigma model action which describes the N = 2 supersymmetric generalization of Galilean Electrodynamics. We first start with the simplest model obtained by null reduction of the relativistic Abelian N = 1 supersymmetric QED in 3+1 dimensions and study its renormalization properties directly in non-relativistic superspace. Despite the existence of a non-renormalization theorem induced by the causal structure of the non-relativistic dynamics, we find that the theory is nonrenormalizable. Infinite dimensionless, supersymmetric and gauge-invariant terms, which combine into an analytic function, are generated at quantum level. Renormalizability is then restored by generalizing the theory to a non-linear sigma model where the usual minimal coupling between gauge and matter is complemented by infinitely many marginal couplings driven by a dimensionless gauge scalar and its fermionic superpartner. Superconformal invariance is preserved in correspondence of a non-trivial conformal manifold of fixed points where the theory is gauge-invariant and interacting. Keywords: Extended Supersymmetry, Field Theories in Lower Dimensions, Renormalization and Regularization, Supersymmetric Gauge Theory ArXiv ePrint: 2207.06435 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP09(2022)237 JHEP09(2022)237 Stefano Baiguera,a Lorenzo Cederleb and Silvia Penatib,c Contents 1 2 Non-relativistic QFT: a short review 2.1 Null reduction and supersymmetry algebra 2.2 Review of the Galilean Wess-Zumino model 2.3 Review of Galilean Electrodynamics 3 3 5 5 3 SGED from null reduction 3.1 Action in superfield formulation 3.2 Action in components 7 7 8 4 One-loop radiative corrections 4.1 Faddeev-Popov procedure and gauge fixing 4.2 Feynman rules 4.3 Selection rules and non-renormalization theorems 4.4 One-loop corrections to the chiral self-energy 4.5 One-loop corrections to the vertices 4.6 Non-renormalizability of the theory 4.7 Non-linear sigma model 10 10 11 13 17 18 21 23 5 A renormalizable SGED 5.1 Covariant approach and background field method 5.2 Covariant Feynman rules 5.3 Original covariant one-loop radiative corrections 5.4 New covariant self-energy corrections 5.5 Renormalization of the action 24 25 26 29 31 32 6 Conclusions 35 A Galilean N = 2 superspace 36 B Dimensional analysis 41 C Mathematical tools 42 D Additional one-loop computations for the SGED action 45 E Details on the covariant approach E.1 Gauge-covariant derivatives E.2 Covariant superpropagators E.3 Convergence of vector insertions for the self-energy computation 46 46 47 49 –i– JHEP09(2022)237 1 Introduction 1 Introduction –1– JHEP09(2022)237 Symmetries play a crucial role in modern physics, in that they govern the behaviour of observables and greatly restrict the theoretical models that describe physical phenomena. In this context, it has been known from a long time that Lorentz invariance is a distinctive trait in the description of electromagnetism, whose theoretical formulation is encoded by Maxwell’s theory. While this model describes several physical phenomena with striking precision, it is still relevant to study its non-relativistic limit [1–3]. One of the reasons is that the investigation of this corner of electromagnetism could teach us lessons on the relativistic case itself. Indeed, it is sometimes not clear whether certain phenomena, not only the ones involving the electromagnetic interaction, are consequences of the Lorentz invariance of the theory, or if they would manifest even when Galilean symmetry is present. For example, time dilatation effects and a strong gravitational description of the Schwarzschild black hole can be obtained in a purely non-relativistic setting, using a torsionful connection and a vanishing Newtonian potential, without resorting to full general relativity [4, 5]. From a theoretical perspective, the Galilean version of electromagnetism provides a non-trivial example of non-relativistic QFT with massless degrees of freedom, which can be coupled in a covariant way to a curved background described by Newton-Cartan geometry [6, 7]. The renormalization properties of this theory, which was called Galilean Electrodynamics (GED), were studied in [8]. GED also arises as the linearized action on D-branes in non-relativistic open string theory [9]. Other investigations of Galilean-invariant gauge theories were considered in [10–12]. From a condensed matter perspective, there are several reasons to consider nonrelativistic limits. Emergent symmetries arising in the infrared are often different from the invariances of the microscopic description, and in particular the Lorentz group may not be present. Non-relativistic symmetries govern the realm of cold atoms [13], fermions at unitarity [14], quantum Hall effect [15], strange metallic phases [16] and quantum mechanical problems like the Efimov effect [17]. Supersymmetry (SUSY) has been studied for several years as a candidate to uncover physics beyond the Standard Model. In concrete condensed matter applications, SUSY may arise as an emergent symmetry, e.g., in the tricritical Ising model [18], in topological superconductors [19], optical lattices [20] or other settings. From a theoretical point of view, supersymmetry strongly constrains the analytic structure of the effective action, and controls the running of physical couplings along the RG flow, leading to exact results and non-renormalization theorems [21, 22]. Furthermore, even if supersymmetry plays an indirect role in holography, most of the examples where the AdS/CFT correspondence is explicitly tested are supersymmetric. Therefore, it does not appear surprising that several investigations of theories which are both supersymmetric and non-relativistic had a great revival in recent years. Starting from the first investigations involving the SUSY generalization of the Galilean algebra and limits of the relativistic models [23–25], there have been studies of superconformal anyons [26], spontaneous SUSY breaking [27], the analysis of the renormalization properties of supersymmetric Galilean or Lifshitz-invariant models [28, 29], supergravity [30, 31] and the study of non-relativistic corners of N = 4 super Yang-Mills (SYM) theory [32–36]. Quite surprisingly, we find that supersymmetry does not significantly improve the renormalization properties observed in the GED case. First of all, we derive a nonrenormaliza (...truncated)