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 = |2µ B ± 2d E|. (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 ψ¯ iσµν γ ψ F − d˜ ψ¯ iσµν taψ 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 ψ¯ Γψ ψ¯ Γ0ψ , 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 ψq 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, σ|ψqiσ ψq|p, 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 |θ¯| ≤ 10−10. The 1 two dimension-five terms are respectively the quark EDM Ψ = √ (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]. Here leading-twist from the inclusive deep-inelastic scattering we use the fully self-consistent calculations in Ref. [25] and (DIS) process, which is usually the most efficient ap- include a theoretical uncertainty to account for the difference proach to measure parton distributions, and hence it among interaction models. should be measured by coupling to another chiral-odd 3

Dyson-Schwinger equation Lattice QCD Models Dyson-Schwinger equation Lattice QCD Phenomenology Future experiment Phenomenology Future experiment

gd gu g(1) = gu - gd T T T T T Pitschmann et al. (2015) Yamanaka et al. (2013) Pitschmann et al. (2015) Bhattacharya et al. (2016) Yamanaka et al. (2013) Abdel-Rehim et al. (2015) Gockeler et al. (2005) Bhattacharya et al. (2016) Cloet et al. (2008) Abdel-Rehim et al. (2015) Wakamatsu (2007) Pasquini et al. (2005) Bali et al. (2015) Gamberg, Goldstein (2001) Green et al. (2012) Schweitzer et al. (2001) Aoki et al. (2010) Ma, Schmidt (1998) Barone et al. (1997) Gockeler et al. (2005) Schmidt, Soffer (1997) Kang et al. (2016) He, Ji (1996) Kim et al. (1996) Radici et al. (2015) Kang et al. (2016) Goldstein et al. (2014) Radici et al. (2015) Anselmino et al. (2013) Goldstein et al. (2014) Anselmino et al. (2013) Ye et al. (2017) Ye et al. (2017) JLab12 SoLID JLab12 SoLID

-0.5 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2

FIG. 1. Tensor charge results. The left panel shows the values of u and d quark tensor charges, and the right panel shows the values of the isovector tensor charge. The round (green) points are from Dyson-Schwinger equation calculations [30, 43], the square (blue) points are from lattice QCD calculations [44–49], the triangle (magenta) points are from model calculations [33– 42], the filled diamond (black) points are phenomenological extractions from data [53, 54, 62–64], and the hollow diamond (red) points are the projection of JLab-12GeV SoLID experiments based on the most recent global analysis [54]. The results quoted from the references are evaluated at different scales as explained in the text. quantity. The semi-inclusive DIS (SIDIS) is one of the al. [43] and the lattice simulation results of Bhattacharya processes that can be used to measure transversity dis- et al. [44], Abdel-Rehim et al. [45], Gockeler et al. [46], tributions. At the leading twist, the transversity distri- Bali et al. [47], Green et al. [48], and Aoki et al. [49] bution can be extracted from a transverse target single are at 4 GeV2. The model calculation results of Cloet et spin asymmetry, the Collins asymmetry [51], which arises al. [33] are at 0.16 GeV2, the results of Wakamatsu [34] from the convolution of the transversity distribution and are at 0.36 GeV2, the results of Pasquini et al. [35] are at the Collins fragmentation function within the framework 0.079 GeV2, the results of Gamberg and Goldstein [36] of the transverse momentum dependent (TMD) factor- are at 1 GeV2, the results of Schweitzer et al. [37] are ization. It can also be measured in the collinear factor- at 0.36 GeV2, the results of Ma and Schmidt [38] are ization through the dihadron process [52] by coupling to at 3 ∼ 10 GeV2, the result of Barone et al. [39] are at the dihadron fragmentation function. Besides, the tensor 25 GeV2, the results of Schmidt and Soffer [40] are at charge can be estimated from generalized parton distri- 0.09 GeV2, the results of He and Ji [41] are at 1 GeV2, bution (GPD) extractions, because the transversity dis- and the results of Kim et al. [42] are at 0.36 GeV2. The tribution is the forward limit of a chiral-odd GPD. The phenomenological extraction results of Kang et al. [53] tensor charge values from some recent global analyses are are at 10 GeV2, the results of Radici et al. [62] are at shown in Figure1. The values of Anselmino et al. [64], 1 GeV2, the results of Goldstein et al. [63] are at 4 GeV2, Kang et al. [53], and Ye et al. [54] are from the global the results of Anselmino et al. [64] are at 0.8 GeV2, and fit of transversity TMDs. The TMD evolution effect is the results of Ye et al. and the SoLID projections [54] are not taken into account in Anselmino et al. [64], while it is at 2.4 GeV2. To see the size of the evolution effect, we included in the other two. The values of Radici et al. [62] quote here the results at two different scales in Ref. [54] are from the analysis of the dihadron process within the as collinear factorization. The values of Goldstein et al. [63] 0 u d are from the GPD extractions by analyzing mesons (π gT = 0.413 ± 0.133, gT = −0.229 ± 0.094, (10) and η) exclusive electroproduction data. Within the large 2 uncertainties, the results from different groups are con- at 2.4 GeV , and sistent with each other. gu = 0.395 ± 0.128, gd = −0.219 ± 0.090, (11) We should note that the tensor charge is scale depen- T T dent, and it follows the evolution equation at the leading at 10 GeV2. order as [55] As seen from Eqs. (3) and (4), the constraint on quark

4 EDMs can be obtained from the knowledge of tensor 2 33−2nf q 2 q 2 αs(Q ) charges and the nucleon EDM measurements. To have gT (Q ) = gT (Q0) 2 , (9) αs(Q0) quantitative estimations, we take the upper limit on the proton EDM derived from the most recent measurement where nf is the number of flavors. The results in Fig- of 199Hg EDM [24] with the Schiff moment method ure1 are at different scales. The Dyson-Schwinger equa- −25 tion results of Pitschmann et al. [30] and Yamanaka et |dp| ≤ 2.6 × 10 e · cm, (12) 4 and the upper limit on the neutron EDM from the direct III. THE IMPACT OF FUTURE EXPERIMENTS measurement with ultra-cold neutrons [22] The constraint on quark EDMs is affected by both the |d | ≤ 3.0 × 10−26 e · cm. (13) n sensitivity of nucleon EDM measurements and the preci- For the tensor charge, we use the results from the global sion of tensor charge extractions. To improve the quark analysis in Ref. [54], which includes the TMD evolu- EDM limit and thus its sensitivity to new physics, we tion effect. At the scale of 4 GeV2, the extracted tensor need efforts from both sides. In this section, we estimate charges for up and down quarks are the impact of the planned experiments in the next ten years. u d gT = 0.405 ± 0.130, gT = −0.225 ± 0.092, (14) One of the main goals of the 12 GeV upgraded CE- BAF Jefferson Lab is to understand the partonic struc- with the uncertainties given at 90% confidence level. In ture of the nucleon. The SIDIS experiments at Jefferson this analysis the strange quark transversity and thus its Lab, particularly those with SoLID [58] which will com- tensor charge are set to zero. Then the strange quark bine large acceptance and high luminosities, are aiming contribution to nucleon EDMs vanishes. However, since to have an unprecedented precision in measurements of the quark EDM is expected to be proportional to the quark three dimensional distributions in the momentum quark mass for a large class of models in which the La- space. The Collins asymmetry, a transverse target single grangian shares the same form for each fermion family spin asymmetry, is one of the highlighted measurements member, the contribution from the strange quark term aiming to extract quark transversity distributions and could be large even with a small tensor charge value [18]. thus the tensor charge. A quantitative study [54] shows To account for the uncertainty from the strange quark that SoLID SIDIS experiments will improve the precision contribution, we take the strange quark tensor charge of tensor charge extractions by one order of magnitude. value from the recent lattice simulation [18], The projection based on the global analysis [54] gives gs = 0.008 ± 0.009, (15) u d T gT = 0.405 ± 0.018, gT = −0.225 ± 0.008, (18) at the scale of 4 GeV2. The uncertainty is understood as with 90% C.L. uncertainties. The values are evaluated one standard deviation (1σ). For consistency, we multi- at the scale of 4 GeV2. plies it by a factor of 1.65, which corresponds to a 90% To estimate the impact of these experiments, we still C.L. based on the normal distribution assumption. Fol- use the strange quark tensor charge from the lattice sim- lowing the method in [18], we relate ds and dd with the ulation [18]. Following the same procedure, we obtain the quark mass ratio ms/md. In Ref. [18] the ratio is chosen constraint from proton and neutron EDM limits shown as 20, and in Ref. [57] the ratio is evaluated as 17 ∼ 22. in Figure2, and the combination of proton and neutron Since we are estimating the upper limit on quark EDMs, results gives the upper limits on quark EDMs as we use the value ms/md = 22 in our analysis to maximize −25 the uncertainty from the strange quark term. |du| ≤ 6.11 × 10 e · cm, (19) With the proton and neutron EDM limits and the −25 |dd| ≤ 9.70 × 10 e · cm, (20) tensor charge values above, we obtain the constraint on quark EDMs in Figure2. For flavor separation, we com- at the scale of 4 GeV2. The final results that include bine the results from proton and neutron EDM limits and the additional 10% uncertainty of the isospin symmetry obtain the constraint on up and down quark EDMs as breaking effect are listed in TableI. As observed from Figure2, although the precision of |d | ≤ 1.15 × 10−24 e · cm, (16) u tensor charge extractions is improved by one order of −24 |dd| ≤ 1.06 × 10 e · cm. (17) magnitude, the impact on the constraint on quark EDMs is not significant. Therefore as expected, more sensitive Since all scale dependent quantities in the analysis are proton and neutron EDM experiments are necessary. 2 evaluated at 4 GeV , these constraints should be under- The precision of the neutron EDM measurements has stood at the same scale. The confidence level of the con- improved by six orders of magnitude since the first ex- straints is 90%, because the nucleon EDM limits and the periment by Purcell et al. [20, 21]. The goal of the next tensor charge uncertainties applied in the evaluation are generation neutron EDM experiments [59] is to further at 90% confidence level. We should note that the isospin improve the sensitivity by two orders of magnitude. The symmetry, which will bring in additional uncertainties, statistical uncertainty of the measurement with ultra- is applied in our analysis to perform flavor separation. cold neutrons depends on the electric field E, the number With our current knowledge, it is hard to quantify this of neutrons N, and the storage time τ as [60] uncertainty. Hence we simply add a 10% uncertainty to √ account for the isospin symmetry breaking effect, and the σ ∼ (E Nτ)−1. (21) final results of up and down quark EDM limits are listed in TableI. Our current constraint on light quark EDMs The approach [60] that will be utilized in the next gen- is at 10−24 e · cm level. eration neutron EDM experiments [59] will significantly 5

4 current g + current d current g + current d T p T n future g + current d future g + current d 2 T p T n future g + future d future g + future d T p T n 2 cm cm ⋅ 1 ⋅ e e

-24 -25 0 0 / 10 / 10 u u d d -1 -2

-2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -24 ⋅ -25 ⋅ dd / 10 e cm dd / 10 e cm

FIG. 2. Constraints on quark EDMs with the upper limits on nucleon EDMs and the tensor charge extractions. The left panel shows the constraints by the upper limit on the proton EDM, and the right panel shows the constraints by the upper limit u d on the neutron EDM. The current tensor charge precision refers to gT = 0.405 ± 0.130 and gT = −0.225 ± 0.092, and the u d future tensor charge precision refers to gT = 0.405 ± 0.018 and gT = −0.225 ± 0.008 [54]. The strange quark tensor charge s gT = 0.008 ± 0.015 [18] is used for both current and future tensor charge precisions. The current nucleon EDM limit refers −25 −26 to |dp| ≤ 2.6 × 10 e · cm derived from mercury atomic EDM measurement [24] and |dn| ≤ 3.0 × 10 e · cm from neutron −29 −28 EDM measurement [22], and the future nucleon EDM limit means |dp| ≤ 2.6 × 10 e · cm and |dn| ≤ 3.0 × 10 e · cm. The constraints are understood at the scale of 4 GeV2. increase E, N, and τ, and will also have better control Compared with our current knowledge, the planned on systematic uncertainties. To estimate the impact of tensor charge and proton and neutron EDM measure- these experiments, we take the future neutron EDM limit ments will improve the constraint on quark EDMs by as three orders of magnitude.

−28 |dn| ≤ 3.0 × 10 e · cm, (22) at 90% confidence level. The result of its constraint on IV. THE PROBE OF NEW PHYSICS quark EDMs is shown in Figure2. For the proton EDM measurement, a storage ring ex- Since the SM CKM complex phase produces an experiment [61] is proposed apart from the indirect mea- tremely small quark EDM [14], which can be viewed as a surements. The new method, which is based on the ap- background within the experimental precisions at present proach of minimizing the g−2 precession in the horizontal and even in the next ten years, the quark EDM is one plane by using a radial electric field, can reach a sensi- of the most sensitive probes of new physics models that tivity of 10−29 e · cm for the EDM measurement of the provide additional CP violation sources. proton [61]. To estimate the impact of the experiment, For new physics beyond the electroweak scale, the we take the future proton EDM limit as quark EDM is suppressed by the quark mass [26, 27]. −29 A simple dimensional analysis gives the quark EDM as |dp| ≤ 2.6 × 10 e · cm, (23) 2 dq ∼ emq/(4πΛ )[56], where Λ represents the scale of at 90% confidence level. The result of its constraint on new physics. As presented in the previous section, the quark EDMs is shown in Figure2. constraint on quark EDMs in the next ten years will be Combining the results estimated with the precision of improved by three orders of magnitude. Therefore we all these future experiments, we obtain the constraint on expect it will be able to probe new physics at a scale quark EDMs as 30 ∼ 40 times higher than that of its current reach. As a very rough estimation, we take light quark mass in [57] −27 2 |du| ≤ 1.09 × 10 e · cm, (24) evaluated at 4 GeV , and then with the current quark −28 EDM limit, which is about 10−24 e · cm, the Λ is about |dd| ≤ 6.53 × 10 e · cm, (25) 1 TeV. This energy scale is directly reached by the LHC. at the scale of 4 GeV2. The limits that include the 10% Future precise measurements of the tensor charge and isospin symmetry breaking uncertainty are listed in Ta- the nucleon EDM will allow us to probe new physics up bleI. The results of other combinations of current and to 30 ∼ 40 TeV, which is above the LHC energy. future precisions of tensor charge and nucleon EDM mea- As a specific example, we study the constraint on the surements are also listed in TableI. parameter space of the split-supersymmetric extension 6

TABLE I. Limits on quark EDMs. All EDM values are given as 90% C.L. upper limits at the scale of 4 GeV2 in unit of e · cm. 10% uncertainties are added to account for the isospin symmetry breaking. The current tensor charge precision refers u d u to gT = 0.405 ± 0.130 and gT = −0.225 ± 0.092, and the future tensor charge precision refers to gT = 0.405 ± 0.018 and d s gT = −0.225 ± 0.008 [54]. The strange quark tensor charge gT = 0.008 ± 0.015 [18] is used for both current and future tensor −25 charge precisions. The current nucleon EDM limits refer to |dp| ≤ 2.6 × 10 e · cm derived from the mercury atomic EDM −26 measurement [24] and |dn| ≤ 3.0 × 10 e · cm from the neutron EDM measurement [22], and the future nucleon EDM limits −29 −28 refer to |dp| ≤ 2.6 × 10 e · cm and |dn| ≤ 3.0 × 10 e · cm.

tensor charge precision proton EDM limit neutron EDM limit du limit dd limit current current current 1.27 × 10−24 1.17 × 10−24 future current current 6.72 × 10−25 1.07 × 10−24 current future current 1.27 × 10−25 6.39 × 10−26 future future current 1.14 × 10−25 6.18 × 10−26 current current future 1.16 × 10−24 1.12 × 10−24 future current future 5.60 × 10−25 1.01 × 10−24 current future future 1.36 × 10−27 7.41 × 10−28 future future future 1.20 × 10−27 7.18 × 10−28

104 future d + current g future d + current g n T p T future d + future g future d + future g n T p T

-29 4 10 10 10-30

-29 10 10-28

(GeV) (GeV) 3 2.6×10-29

µ µ 10 3.0×10-28 3 -28 10 10

10-27

10-27 103 104 103 104 M (GeV) M (GeV) 2 d 2

FIG. 3. The constraint on the split-supersymmetric model. The left panel is the constraint from the proton EDM, and the right panel is the constraint from the neutron EDM. The dotted (black) curves are constraints with precise tensor charge values, i.e. zero tensor charge uncertainties, the solid (blue) curves are those with current tensor charge uncertainties, and the dashed (red) curves are those with future tensor charge uncertainties. The nucleon EDM limit values marked on curves are in unit of e · cm. The area below the curves represents the excluded parameter space. of the SM, which is also investigated by the PNDME in an approximate linear way as sin φ sin 2β [66], where lattice QCD group [18]. In the split-supersymmetric tan β = vu/vd is the ratio between Higgs vacuum expec- model [65], one-loop contributions are highly suppressed tation values. In our estimation, we choose sin φ = 1 by heavy sfermions [66, 67] and thus it avoids the usual and tan β = 1. The results are shown in Figure3. Our supersymmetric CP problem with current experimental current limits have no sensitivity to this model. The com- limit on nucleon EDMs. In this model, the dominant parison of the dotted (black) curves shows the impact of contribution is from the quark EDM term arising from the improvement on nucleon EDM measurements, and the two-loop level. With the theoretical calculation and the comparison between the solid (blue) and the dashed the setup in Ref. [66], namely the unified framework of (red) curves shows the impact of the improvement on gaugino masses at the grand unified theory scale and the tensor charge extractions. So the combination of future sfermion mass of 109 GeV, we estimate the constraints on nucleon EDM and tensor charge measurements will be a M2 and µ, which are the mass parameters in the gaugino- powerful tool for the search of beyond SM new physics higgsino sector. At the split limit with gaugino mass uni- and provide more stringent constraint on the parameter fication, quark EDMs are controlled by a single phase φ space of new physics models. 7

V. DISCUSSIONS AND CONCLUSIONS the leading order. In general, it is possible to have cancel- lations among different sources. We leave more complete In this paper, we investigate the experimental con- investigations of all sources to future studies. straint on quark EDMs by combining nucleon EDM The isospin symmetry is another assumption we have measurements and tensor charge extractions. With the used in our analysis. The 10% uncertainty that we have present sensitivity of the proton and neutron EDM ex- added based on our empirical estimation is more or less periments and the current precision of tensor charge ex- arbitrary. With our current knowledge, we cannot quan- tractions, we obtain the upper limit on quark EDMs as tify this uncertainty. Better understandings of the nu- 1.27×10−24 e·cm for the up quark and 1.17×10−24 e·cm cleon structure both theoretically and experimentally in for the down quark at the scale of 4 GeV2. It corresponds the future may help us to have more accurate estimation to a probe of new physics roughly up to the energy scale of this uncertainty. of 1 TeV, which is directly reached by the LHC. The validation of the neglect of the charm as well as In the next ten years, both the sensitivity of nucleon heavier flavor contributions depends on the size of its EDM experiments and the precision of tensor charge ex- tensor charge. Since the charm quark is about 600 times tractions are expected to be dramatically improved. The heavier than the up quark, if we require the charm quark planned SIDIS experiments at Jefferson Lab will improve contribution to be one order of magnitude smaller than the uncertainty of the determination of the tensor charge the up quark contribution, the charm quark tensor charge by one order of magnitude [54]. The next generation is required to be smaller than 7 × 10−5 at the scale of neutron EDM experiments aim to improve the precision 4 GeV2. To our best knowledge, there are no theoreti- to 10−28 e · cm [59]. The proposed storage ring proton cal calculations or experimental extractions of the charm EDM experiment is capable of reaching a sensitivity of quark tensor charge up to now. Future studies may tell 10−29 e·cm [61]. Our analysis shows that the combination us if the drop of heavy flavor terms is allowed. of all these experiments is expected to improve the limit In conclusion, the quark EDM is a sensitive probe to on quark EDMs by about three orders of magnitude. It additional CP violation sources beyond the SM. The up- means the energy scale of the quark EDM probe of new coming experiments will significantly improve the con- physics models will increase by 30 ∼ 40 times, and thus is straint on the quark EDM and hence make it a much above the LHC energy. Taking the split-supersymmetric more powerful tool to test the SM and to search for new model as an example, we show the impact of the im- physics models. provements on nucleon EDM measurements and tensor extractions with the future experiments. Therefore it will become an important approach for us to explore the new source of CP violating effects and hence the baryogenesis ACKNOWLEDGMENTS mechanism of our universe. Our analysis in this study is based on the sole contribu- We are grateful to Kalyan Allada, Jian-Ping Chen, tion assumption. The strong CP violation θ-term is set to Bradley W. Filippone, Robert Golub, Zhong-Bo Kang, zero. Other contributions such as the chromo-EDM and Alexei Prokudin, Nobuo Sato, Peng Sun, Zhihong Ye, the Weinberg term are also neglected. In some model and Feng Yuan for useful discussions. This work is sup- like the split-supersymmetric model we investigated, the ported in part by U.S. Department of Energy under con- quark EDM term dominates and one can neglect the tract number DE-FG02-03ER41231, and by the Duke chromo-EDM and the Weinberg term contributions at Kunshan University.

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