S S symmetry Article Multifarious Roles of Hidden Chiral-Scale Symmetry: “Quenching” gA in Nuclei Mannque Rho Institut de Physique Théorique, Université Paris-Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France; [email protected] Abstract: I discuss how the axial current coupling constant gA renormalized in scale symmetric chiral EFT defined at a chiral matching scale impacts on the axial current matrix elements on beta decays in nuclei with and without neutrinos. The “quenched” gA observed in nuclear superallowed Gamow– Teller transitions, a long-standing puzzle in nuclear physics, is shown to encode the emergence of chiral-scale symmetry hidden in QCD in the vacuum. This enables one to explore how trace- anomaly-induced scale symmetry breaking enters in the renormalized gA in nuclei applicable to certain non-unique forbidden processes involved in neutrinoless double beta decays. A parallel is made between the roles of chiral-scale symmetry in quenching gA in highly dense medium and in hadron–quark continuity in the EoS of dense matter in massive compact stars. A systematic chiral-scale EFT, presently lacking in nuclear theory and potentially crucial for the future progress, is suggested as a challenge in the field. Keywords: nuclear quenched gA; Fermi-liquid fixed point; hidden scale and local symmetries in nuclear matter; genuine dilaton; anomalous dimension of gluon stress tensor in nuclei Citation: Rho, M. Multifarious Roles of Hidden Chiral-Scale Symmetry: 1. Introduction “Quenching” gA in Nuclei. Symmetry 2021, 13, 1388. https://doi.org/ It has been recently argued [1,2] that the long-standing mystery of “quenched” gA in 10.3390/sym13081388 nuclear super-allowed Gamow–Teller transitions can be resolved by the combination of two mechanisms: (1) strong nuclear correlations controlled by an emerging scale symmetry Academic Editors: Dubravko hidden in QCD; (2) an effect of quantum anomaly in scale symmetry inherited from QCD Klabuˇcarand Giuseppe Bagliesi at the chiral scale Lc ∼ 4p fp that defines effective field theory for nuclear dynamics. (I put the quotation marks here because what is referred to in the nuclear physics literature as Received: 24 June 2021 “quenched gA” is a misnomer. The coupling constant gA appearing in the nucleonic axial Accepted: 23 July 2021 current used to calculate nuclear Gamow–Teller matrix elements is in fact not quenched Published: 30 July 2021 in the sense used in shell-model calculations in the literature. This is explained in what follows. I continue the discussions without the quotation marks unless otherwise noted.) Publisher’s Note: MDPI stays neutral In this note, I discuss how what triggers the quenched superallowed Gamow–Teller with regard to jurisdictional claims in transitions in nuclear medium encodes the emergence of chiral-scale symmetry, hidden published maps and institutional affil- in QCD, in strong nuclear correlations and suggest in what way the anomaly-induced iations. breaking of scale symmetry—referred to by the acronym AISB—affects how the axial- current coupling constant gA can indeed be “fundamentally” renormalized by the vacuum change—as opposed to the effect of nuclear correlations. This AISB is argued to have an important impact on neutrinoless double beta decay matrix elements. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. 2. Superallowed Gamow–Teller Transitions This article is an open access article To zero-in on essentials for the quenched gA problem, first consider the superallowed distributed under the terms and Gamow–Teller decay of the doubly magic nucleus 100Sn with 50 neutrons and 50 protons. I conditions of the Creative Commons pick this case because there is what is claimed to be “accurate” data and equally importantly Attribution (CC BY) license (https:// it offers a well-defined theoretical framework. This process allows exploiting the “extreme creativecommons.org/licenses/by/ single-particle shell model (ESPSM)” description [3,4]. In the ESPSM description, the GT 4.0/). Symmetry 2021, 13, 1388. https://doi.org/10.3390/sym13081388 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1388 2 of 8 process involves, via the spin-isospin flip, the decay of a proton in a completely filled shell g9/2 to a neutron in an empty shell g7/2, which can be equated nearly precisely to the GT transition of a quasi-proton to a quasi-neutron on top of the Fermi sea formulated in the Landau Fermi-liquid fixed point theory [1]. Such a feat is not usually feasible for an ESPSM description of generic (say, non-doubly-magic) nuclei. Let us begin by defining the quenching factor q often used in the literature MGT = gAqMst. (1) Here, gA is the free-space (neutron decay) axial-vector coupling constant gA = 1.276(4) and Mst is the proton-to-neutron single-particle GT matrix element. The quantity on the right-hand side representing nature should of course be model independent but q and M depend on how M is to be calculated, so separately model dependent. The approach adopted in [1] gives a precise meaning to what q is in the scheme and how it can be related to the experimental value. As emphasized there, given the superallowed transition with zero momentum transfer, the quantity M is just the spin-isospin factor for the Fermi-liquid as well as ESPSM descriptions with all the interaction effects, fundamental (i.e., AISB) as well as of pure nuclear correlations, lumped into the “quenching” factor q. In general, the two are of course intricately mixed, and it is difficult, if not impossible, to disentangle them in nuclear processes. However, if one assumes that the AISB effect is small, then the axial-current can be written with the anomaly factor qssb representing the scale symmetry breaking (ssb) simply multiplying the scale-invariant axial current as ± qssbgAyt¯ gmg5y (2) with b0 qssb = cA + (1 − cA)F . (3) 0 Here, cA is a (in general) density-dependent constant and b is the anomalous dimen- 2 sion of the gluon stress tensor trGmn, both of which remain, up-to-date, unknown for QCD ∗ with the flavor number Nf <∼ 3. The quantity F is the ratio fp/ fp with the ∗ indicating density dependence of the nuclear medium. F has been measured by deeply bound pionic nuclei [5], so is known up to the nuclear matter density n0. Note that, because of F, qssb is explicitly density-dependent. In the vacuum, F = 1, so there is no dependence on b0. Furthermore, if cA were equal to 1 either on symmetry grounds or by density effects in medium, there would be no dependence on b0. In both cases, the impact of scale symmetry breaking is absent in the GT transitions. In [1], resorting to an EFT that incorporates both hidden local symmetry and hid- den scale symmetry in chiral Lagrangian, the matrix element of the superallowed GT ma- trix element was calculated in the large Nc and large N¯ approximations in the Fermi- liquid approach. In QCD, the gA is proportional to Nc in the large Nc counting, and, in the Fermi-liquid theory, the Fermi-liquid fixed point is given by O(1) term in the limit N¯ = kF/(LFL − kF) ! ¥ with LFL the cut-off in the Fermi-liquid renormalization group decimation. In these double limits, it comes out that ± Landau ± hyt¯ gmg5yi f i = qsnc ht si f i (4) where the entire strong nuclear correlation (snc) effects are captured in Landau −2 qsnc = (1 − k) (5) with 1 k = FF˜p. (6) 3 1 Symmetry 2021, 13, 1388 3 of 8 ˜p Here, F1 is the pionic contribution to the Landau mass, governed by chiral symmetry −3 for any density ≤ n0 ≈ 0.16 fm . (The pionic contribution to the Landau mass is the Fock term, which is formally of O(1/N¯ ), and goes beyond the Landau Fermi-liquid fixed point approximation made in (5). However, soft theorems for both chiral symmetry (pion) and scale symmetry (dilaton) figure for the validity of the relation (5), so the pionic contribution is essential. See [2] for discussions on this matter.) Therefore, the quantity k is almost completely controlled by low-energy theorems. In addition, due to near compensation of the density effects in the two factors, k remains remarkably insensitive to density. It varies negligibly between n0 and n0/2. Therefore, it is applicable equally well to both light and heavy nuclei. From (3) and (5), we get the total quenching factor Landau q = qssbqsnc . (7) To see what we have, let us first ignore the AISB and set qssb = 1. Evaluating qsnc in the Landau Fermi-liquid theory, we get at n ≈ n0 [1] Landau qsnc ≈ 0.79. (8) Thus, what is identified as the “scale-symmetric effective gA” given in the Landau Fermi-liquid theory is ss Landau Landau gA ≡ gA ≈ gAqsnc = 1.276 × 0.79 ≈ 1.0. (9) ss Note that gA is not a quenched coupling constant. The quenching is in the nuclear Landau interactions captured in qsnc . There is more on this crucial point below. 2.1. Fermi-Liquid Fixed Point (FLFP) ≈ Extreme Single Particle Shell Model (ESPSM) In what sense can this result (9), which figures crucially in what follows, be taken as reliable? It relies on taking two limits. The first is the large Nc limit in QCD combined with soft theorems. The second is the large N¯ limit in the renormalization-group approach to Landau Fermi-liquid theory in treating strongly correlated nuclear interactions. The former is akin to the Goldberger–Treiman relation which is known in the matter-free vacuum to be accurate within a few percent discrepancy. Thus, combining the two limits in nuclear processes can be considered to be comparable to applying to the Goldberger–Treiman relation in nuclear dynamics.