Theta dependence in holographic QCD

Published for SISSA by Springer Received: December 20, 2016 Accepted: January 30, 2017 Published: February 7, 2017 Theta dependence in holographic QCD a Dipartimento di Fisica “E. Fermi”, Università di Pisa and INFN, Sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy b INFN, Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy c Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy d Institute of Physics, EPFL, Rte de la Sorge, BSP 728, CH-1015 Lausanne, Switzerland E-mail: , , , , Abstract: We study the effects of the CP-breaking topological θ-term in the large Nc QCD model by Witten, Sakai and Sugimoto with Nf degenerate light flavors. We first compute the ground state energy density, the topological susceptibility and the masses of the lowest lying mesons, finding agreement with expectations from the QCD chiral effective action. Then, focusing on the Nf = 2 case, we consider the baryonic sector and determine, to leading order in the small θ regime, the related holographic instantonic soliton solutions. We find that while the baryon spectrum does not receive O(θ) corrections, this is not the case for observables like the electromagnetic form factor of the nucleons. In particular, it exhibits a dipole term, which turns out to be vector-meson dominated. The resulting neutron electric dipole moment, which is exactly the opposite as that of the proton, is of the same order of magnitude of previous estimates in the literature. Finally, we compute the CP-violating pion-nucleon coupling constant ḡπN N , finding that it is zero to leading order in the large Nc limit. Keywords: Gauge-gravity correspondence, AdS-CFT Correspondence, D-branes ArXiv ePrint: 1611.00048 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP02(2017)029 JHEP02(2017)029 Lorenzo Bartolini,a Francesco Bigazzi,b Stefano Bolognesi,a Aldo L. Cotroneb,c and Andrea Manentid Contents 2 2 Witten-Sakai-Sugimoto model 2.1 The background 2.2 Adding probe flavor branes 2.3 Holographic mesons 4 5 7 8 3 The U(1)A anomaly and flavor effects on θ 3.1 Horava-Witten solution of the anomalous Bianchi identity 9 11 4 WSS model with massive fermions 13 5 θ dependence of the vacuum energy 14 6 Holographic baryons 6.1 Quantization 6.2 Baryon Hamiltonian with quark mass and θ 16 18 20 7 Mass and θ perturbations to holographic baryons 7.1 Abelian field: space components 7.2 Non Abelian field: time component 7.2.1 The solution in the “flat region” 7.3 Non Abelian field: space components 7.4 Abelian field: time component 22 25 26 27 27 29 8 The neutron electric dipole moment 8.1 NEDM state of the art 8.2 The currents 8.3 Quantization reloaded 8.4 The holographic computation of the NEDM 8.5 The electric dipole form factor 30 30 32 34 36 40 9 The CP-breaking pion-nucleon coupling 9.1 The axial form factors 9.2 A more direct argument 43 43 46 10 Conclusions 46 A Meson sector 47 B The C7 and F̃2 action 50 C Alternative choice of parameters 51 –1– JHEP02(2017)029 1 Introduction 1 Introduction 1 An extension of these results to any order in θ/Nc and an analysis of the θ-dependent behavior of various relevant Yang-Mills observables can be found in [16]. –2– JHEP02(2017)029 In the electroweak sector of the Standard Model, parity (P), time reversal (T) and charge conjugation (C) can be separately broken, while their combination (CPT) is preserved. Whether some of these discrete symmetries are separately broken also in QCD remains to be experimentally verified. Instantons in the model naturally induce a P- and T-violating topological term proportional to θ TrF ∧ F , where F is the SU(3) field strength and θ is a parameter. In principle, nothing forbids θ from taking a generic value. However, experiments tell us that it should be extremely small. The strongest bound on its value comes from measurements of the neutron electric dipole moment (NEDM) dn . Recent experiments [1, 2] give |dn | ≤ 2.9 × 10−26 e · cm (90% CL). The topological θ angle in QCD could provide the main contribution to the NEDM, since CP-violating effects from the electroweak sector give rise to a dipole moment which is orders of magnitude smaller than the above mentioned experimental bound. A tentative order-of-magnitude theoretical −3 estimate [3, 4] gives |dn | ≈ |θ|e m2π MN ≈ 10−16 |θ|e · cm where mπ (resp. MN ) is the pion (resp. nucleon) mass. Put together with the above mentioned experimental bound, this gives an unnaturally small value |θ| ≤ 10−10 for the topological parameter. This is the so called strong CP problem, a possible theoretical resolution of which (a θ angle relaxing to zero dynamically) is provided by the Peccei-Quinn mechanism [5] which would imply the existence of axions [6, 7]. From a theoretical perspective, studying how the θ parameter affects the physics of QCD requires going beyond perturbation theory. Lattice techniques find some limitations in this case, since the topological term is imaginary in the Euclidean Lagrangian and a sign problem arises. While relevant results have been obtained expanding, up to few terms, around θ = 0 in the pure Yang-Mills case (see e.g. [8] for a detailed review on the subject), lattice estimates of CP-breaking observables in full QCD, notably estimates of the NEDM (see e.g. [9–12]), are still plagued by quite large systematic and statistical errors. In this perspective it is important to compare lattice results with model calculations. Famous results arise within chiral perturbation theory, where both the θ-dependent ground state energy density [13] and the NEDM — which turns out to be proportional to the nonderivative CP-violating pion-nucleon coupling ḡπ N N [14] — have been computed. Within this approach only the pion cloud contributes to the NEDM, since massive (axial) vector mesons have been integrated out. Another model approach, complementary to the one above, consists in taking ’t Hooft’s large Nc limit where Nc is the number of colors. This limit is known not to commute in general with the small quark mass one in which chiral perturbation theory is organized. In the unflavored Yang-Mills case, relevant features of the θ-dependent ground state energy density have been first discussed in [13] and then explicitly realized, to leading order in θ/Nc , in a holographic Yang-Mills model in [15].1 When Nc = ∞ mesons (and glueballs) are non-interacting and stable. At large, finite √ Nc , meson-meson couplings are found to be of order 1/ Nc , while baryon masses scale as –3– JHEP02(2017)029 Nc . This suggests that baryons can be seen as solitons in the effective large Nc mesonic Lagrangian [17]. This picture is actually realized within the chiral effective theory (the Skyrme model [18]), whose solitons are identified with the baryons. Static properties of nucleons with Nf = 2 massless (resp. massive) flavors have been studied in the semina (...truncated)