Non-derivative axionic couplings to nucleons at large and small N

Published for SISSA by Springer Received: July 16, 2019 Revised: December 11, 2019 Accepted: December 23, 2019 Published: January 17, 2020 Francesco Bigazzi,a Aldo L. Cotrone,a,b,1 Matti Järvinenc,d and Elias Kiritsise,f a INFN, Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy2 b Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy3 c Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands4 d Department of Physics and Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland5 e Crete Center for Theoretical Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece6 f APC, Université Paris 7, CNRS/IN2P3, CEA/IRFU, Obs. de Paris, Sorbonne Paris Cité, Bâtiment Condorcet, F-75205, Paris Cedex 13, France7 E-mail: , , 1 On leave at the Galileo Galilei Institute for Theoretical Physics, INFN National Center for Advanced Studies, Largo E. Fermi, 2, 50125 Firenze, Italy. 2 http://theory.fi.infn.it/index/. 3 https://www.fisica.unifi.it/mdswitch.html. 4 https://www.uu.nl/en/research/institute-for-theoretical-physics. 5 https://www.helsinki.fi/en/faculty-of-science/faculty/physics and https://www.hip.fi/. 6 http://hep.physics.uoc.gr/. 7 UMR du CNRS 7164, http://www.apc.univ-paris7.fr. Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2020)100 JHEP01(2020)100 Non-derivative axionic couplings to nucleons at large and small N Keywords: Chiral Lagrangians, Cosmology of Theories beyond the SM, Gauge-gravity correspondence, 1/N Expansion ArXiv ePrint: 1906.12132 JHEP01(2020)100 Abstract: Among the possible CP-odd couplings of the axion to ordinary matter, the most relevant ones for phenomenology are the Yukawa couplings to nucleons. We analyze such non-derivative couplings within three different approaches: standard effective field theory, the Skyrme model and holographic QCD. In all the cases, the couplings can be related to the CP-odd non-derivative couplings to nucleons of the low-lying mesons and the η ′ . Using the effective field theory approach we discuss how to derive the expressions for the CP-odd interaction terms as functions of the parameters of the effective Lagrangian at generic number of colors Nc and flavors Nf . Then, we compute the CP-odd couplings to nucleons of the axion, the η ′ and the pseudo-Goldstone mesons in both the Skyrme and the holographic QCD model with Nf = 2, 3. We present model-independent expressions for the coefficients of the non-derivative axion-nucleon couplings. This allows us to provide quantitative estimates of these couplings. Contents 2 2 The effective Lagrangian couplings of the axion to the η ′ 6 3 The effective couplings of η ′ and the axion to nucleons 3.1 Generic Nf and Nc 3.1.1 Non-derivative couplings 3.2 Results for Nf = 2 3.3 The one-loop corrections 10 10 11 14 15 4 General IR relations for the couplings 19 5 The Skyrme model picture 5.1 Nucleon mass terms and CP-odd couplings for Nf = 2 22 23 6 Large Nc estimates in the WSS holographic model 6.1 The WSS model 6.2 The derivative axion-nucleon couplings 6.3 The non-derivative axion-nucleon couplings for Nf = 2 26 26 29 29 7 Discussion and numerical estimates of the couplings 31 A The structure of axion models A.1 A typical UV model for elementary axions A.2 Other axion models 35 35 38 B Current terms and η ′ -axion mixing in chiral Lagrangians B.1 VEVs induced by CP-odd couplings B.2 Three-meson couplings 38 42 43 C Details on the nucleon couplings in chiral perturbation theory C.1 Non-derivative couplings from external CP-violation C.2 CP-odd nucleon-meson couplings C.3 Group theoretic structure of the nucleon couplings to the mesons 44 44 45 46 D On-shell nucleon vertices in the limit of large mass 47 E CP-odd couplings in the Skyrme model with Nf = 3 48 F Instanton moduli quantization 50 G CP-odd couplings in the WSS model with Nf = 3 52 H Alternative numerical estimates of the couplings 53 –1– JHEP01(2020)100 1 Introduction and results 1 Introduction and results 1 In fact it also applies to NS-NS axions that couple to world-sheet or NS5-brane instantons. There can be subtleties that arise when axionic symmetries are coupled to anomalous U(1)’s [32, 33], but the final result is analogous. 2 –2– JHEP01(2020)100 The Peccei-Quinn proposal [1] for a natural solution of the strong CP problem — the unnaturally small value of the QCD θ angle — implies the existence of a new light neutral pseudoscalar boson, the axion [2, 3]. The original theory, severely constrained by data, was not renormalizable because of the axion coupling to the QCD instanton density [4, 5]. In renormalizable axion models, successively proposed in [6–8], the axions were also made very weakly interacting with ordinary matter, as required by experimental constraints. These “invisible axions”, whose couplings to matter have been deduced using anomalies and the chiral Lagrangian [9]–[14], are nowadays considered among the most promising candidates as dark matter constituents, and also as possible realizations of the inflaton [15]–[20]. As of today, axion-like particles (ALPs) are ubiquitous and serve various purposes. The nomenclature has also evolved but still remains sometimes murky. Axions that solve the strong CP problem are typically called QCD axions. The term “axion-like particle” is often used to refer (only) to other types of axions. Generically, axions display a perturbative shift symmetry and couple to instanton densities. In any reliable quantum field theory (QFT) realization, the symmetry is broken (at best) to a discrete symmetry due to non-perturbative effects. Such effects, related to instantons in weakly-coupled models, induce a mass and, more generally, a potential term, for the QCD axion (see e.g. [13]). ALPs arise very commonly in string theory [21], the simplest example being the Ramond-Ramond (RR) axion of type IIB string theory. They can also arise, after compactification, from internal components of antisymmetric form gauge fields as well as from off-diagonal components of the metric. In these cases, ALPs are related to generalized gauge fields [22, 23]. The corresponding gauge symmetries provide the perturbative Peccei-Quinn (PQ) symmetries in string theory [24]. Continuous shift symmetries in string theory can be broken by non-perturbative effects. The argument is general:1 RR axions couple to the world volume of D-branes [25]. The same D-branes, wrapped around some appropriate Euclidean cycle, provide instanton effects in string theory [26]–[29]. The nature of these effects depends on the amount of supersymmetry. In the case of maximal supersymmetry they do not generate a potential, but affect higher derivative terms like the R4 corrections, [30, 31]. In all cases, the end result is that the original shift s (...truncated)