Experimental Constraint on Quark Electric Dipole Moments

Experimental constraint on quark electric dipole moments Tianbo Liu,1, 2, ∗ Zhiwen Zhao,1 and Haiyan Gao1, 2 1Department of Physics, Duke University and Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA 2Duke Kunshan University, Kunshan, Jiangsu 215316, China The electric dipole moments (EDMs) of nucleons are sensitive probes of additional CP violation sources beyond the standard model to account for the baryon number asymmetry of the universe. As a fundamental quantity of the nucleon structure, tensor charge is also a bridge that relates nucleon EDMs to quark EDMs. With a combination of nucleon EDM measurements and tensor charge extractions, we investigate the experimental constraint on quark EDMs, and its sensitivity to CP violation sources from new physics beyond the electroweak scale. We obtain the current limits on quark EDMs as 1:27×10−24 e·cm for the up quark and 1:17×10−24 e·cm for the down quark at the scale of 4 GeV2. We also study the impact of future nucleon EDM and tensor charge measurements, and show that upcoming new experiments will improve the constraint on quark EDMs by about three orders of magnitude leading to a much more sensitive probe of new physics models. I. INTRODUCTION one can derive it from nucleon EDM measurements. A bridge that relates the quark EDM and the nucleon EDM Symmetries play a central role in physics. Discrete is the tensor charge, which is a fundamental QCD quan- symmetries, charge conjugate (C), parity (P), and time- tity defined by the matrix element of the tensor cur- reversal (T ), were believed to be conserved until the dis- rent. It also represents the transverse spin carried by covery of parity violation in weak interactions suggested the quarks in a transversely polarized nucleon in the par- by Lee and Yang [1] and then confirmed in nuclear beta ton model. The double role of the tensor charge in the decays [2] and successive meson decays [3]. It is a cor- understanding of strong interaction and the search for nerstone of the standard model (SM) of particle physics. new physics received great interest in recent years [17]. Subsequently, the violation of the combination of charge Nowadays, the progress on lattice QCD gives a better conjugate and parity, CP, was discovered in neutral kaon control on the uncertainty of the tensor charge calcula- decays [4] and also observed in B-meson and strange B- tion [18]. On the experimental side, the proposed exper- meson decays in recent years [5{7]. The CP violation is iments at the 12 GeV upgraded Jefferson Lab will have one of Sakharov conditions for generating a baryon num- an unprecedented precision in measurements of the ten- ber asymmetry from a symmetric initial state [8]. Al- sor charge [19]. though the Kobayash-Maskawa (KM) mechanism [9] pro- The history of nucleon EDM experiments can be traced vides a consistent and economical SM description of all back to the 1950s, and the first experiment was pro- observed CP violating phenomena in collider physics, the posed by Purcell and Ramsey using the neutron-beam CP violating phase in the Cabibbo-Kobayashi-Maskawa magnetic resonance method [20, 21]. The current upper (CKM) matrix is not enough to account for the observed limit on the neutron EDM from direct measurements is −26 asymmetry of matter and antimatter in the visible uni- 3:0×10 e·cm (90% C.L.) [22], which was obtained with verse. Therefore, additional CP violation sources are re- ultra-cold neutrons permeated in uniform electric and quired, unless one accepts the fine tuning solution of the magnetic fields by measuring the difference of Larmor initial condition prior to the Big Bang. precession frequencies when flipping the electric field, A permanent electric dipole moment (EDM) of any hν = j2µ B ± 2d Ej: (1) particle with a non-degenerate ground state violates both n n parity and time-reversal symmetries. Assuming CPT in- The current upper limit on the proton EDM is de- arXiv:1704.00113v2 [hep-ph] 17 Jan 2018 variance, a consequence of local quantum field theories rived from the mercury atomic EDM limit with the with Lorentz invariance [10{13], it is a signal of CP vio- Schiff moment method [23], because the existence of lation. As the CKM complex phase requires the partic- an electric monopole usually overwhelms the dipole sig- ipation of three fermion generations, the EDM of light nal. The most recent measurement of the EDM of 199Hg quarks is highly suppressed by the flavor changing inter- atoms sets the upper limit on the mercury atomic EDM actions at the three-loop level, and hence the KM mech- to 7:4 × 10−30 e · cm (95% C.L.) [24]. Following the anism only results in an extremely small EDM [14{16]. Schiff moment method [23] and the theoretical calcula- Therefore, the quark EDM is one of the most sensitive tions [25], we obtain the upper limit on the proton EDM probes to new physics beyond the SM. as 2:6 × 10−25 e · cm 1. As mentioned in Ref. [23], the Due to the confinement property of strong interaction, uncertainty in this derivation is not statistical and we the quark EDM is not directly measurable, but instead 1 The derived upper limit on the proton EDM in Ref. [24] is ∗ [email protected] 2:0 × 10−25 e · cm, which was obtained with the relation from the 2 cannot give the probability distribution. As a conserva- EDM limit is no worse than a 90% confidence level. tive estimation, we expect the derived value of the proton The effective Lagrangian that contributes to the nucleon EDM can be expressed up to dimension-six as [26] g2 1 X 1 X L = −θ¯ s µνρσGa Ga − d ¯ iσµν γ F − d~ ¯ iσµν ta Ga eff 64π2 µν ρσ 2 q q 5 q µν 2 q q q µν q q (2) 1 X + d f abcµνρσGa Gb Gc λ + C ¯ Γ ¯ Γ0 ; 6 W µν ρλ σ ijkl i j k l i;j;k;l a u;d;s where gs is the strong coupling constant, Gµν is the coefficient gT is the tensor charge, which is defined by gluon field, Fµν is the electromagnetic field, and q is the matrix element of a local operator as the quark field. The first term, a dimension-four op- ¯ µν q µν erator, is allowed in the standard model as the QCD hp; σj qiσ qjp; σi = gT u¯(p; σ)iσ u(p; σ): (5) θ-term, where the overall phase of the quark mass matrix is absorbed into θ¯. It could in principle generate In the naive nonrelativistic quark model, it can be ob- large hadronic EDMs, but the upper limit on the neu- tained from the SU(6) spin-flavor wave function [31] tron EDM constrains the coefficient to jθ¯j ≤ 10−10. The 1 two dimension-five terms are respectively the quark EDM Ψ = p (2u u d − u u d − u u d + permutations) ~ p " " # " # " # " " dq and the quark chromoelectric dipole moment dq. In 18 order to restore the SU(2) × U(1) symmetry above the (6) electroweak scale, a Higgs field insertion should be in- as cluded in these two terms [27]. Therefore, they are essen- u 4 d 1 tially dimension-six operators, and are often in practice gT = ; gT = − : (7) 2 3 3 supplied by an insertion of the quark mass as mq=Λ , where Λ represents a large mass scale. For consistency, Due to relativistic effects, the tensor charge values reduce other dimension-six operators, the three-gluon Weinberg from the prediction of the naive quark model [32], and dif- operator and the four-fermion interactions, should also fer from the axial-vector charge, which is defined by the be introduced. matrix element of the axial-vector current. As shown in In this paper, focusing on the quark EDM term, we Figure1, the tensor charge has been calculated in many present the experimental limit on quark EDMs with the phenomenological models [33{42], and with some non- combination of nucleon EDM measurements and tensor perturbative methods, such as Dyson-Schwinger equation charge extractions, and the impact of the next genera- calculations [30, 43] and lattice QCD simulations [44{49]. tion EDM experiments and the planned precision mea- In the quark-parton model, the tensor charge is equal surements of the tensor charge. The constraint on new to the first moment of quark transversity distribution, physics is also discussed. Z 1 q q q¯ gT = dx[h1(x) − h1(x)]; (8) 0 II. TENSOR CHARGE AND QUARK EDM where x represents the longitudinal momentum fraction The nucleon EDM is related to the quark EDM as [28{ carried by the quark. The transversity distribution h1(x) 30] as a leading-twist parton distribution function is inter- preted as the net density of transversely polarized quarks u d s dp = gT du + gT dd + gT ds; (3) in a transversely polarized proton. Unlike its longitudinal counter part, the helicity distribution, which mea- d = gd d + gu d + gs d ; (4) n T u T d T s sures the density of longitudinally polarized quarks in a where the isospin symmetry is applied in Eq. (4). In longitudinally polarized proton, the transversity distri- this study we neglect heavy flavor contributions. The bution is a chiral-odd quantity, which results in a sim- pler QCD evolution effect without mixing with gluons and as such is dominated by valence quarks [50]. How- ever, the chiral-odd property makes it decouple at the random-phase approximation with core polarization [23].

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