Flavor diagonal tensor charges of the nucleon from 2+1+1 flavor lattice QCD
Abstract
We present stateoftheart results for the matrix elements of flavor diagonal tensor operators within the nucleon state. The calculation of the dominant connected contribution is done using eleven ensembles of gauge configurations generated by the MILC collaboration using the HISQ action with 2+1+1 dynamical flavors. The calculation of the disconnected contributions is done using seven (six) ensembles for the strange (light) quarks. These high statistics simulations allowed us to to address various systematic uncertainties. A simultaneous fit in the lattice spacing and the lightquark mass is used to extract the tensor charges for the proton in the continuum limit and at MeV: , and . Implications of these results for constraining the quark electric dipole moments and their contributions to the neutron electric dipole moment are discussed.
pacs:
11.15.Ha, 12.38.GcPNDME
I Introduction
High precision calculations of the matrix elements of flavor diagonal quark bilinear operators, where is one of the sixteen Dirac matrices, within the nucleon state provide a quantitative understanding of a number of properties of nucleons and their interactions with electrically neutral probes. In this paper, we present results for the tensor charges, , and , that give the contribution of the electric dipole moment (EDM) of these quark flavors to the EDM of the nucleon. They are defined as the nucleon matrix elements of the renormalized tensor operator, with , the renormalization constant and the bare quark field:
(1) 
Experimentally, they can be extracted from semiinclusive deepinelastic scattering (SIDIS) data Lin et al. (2018a); Radici and Bacchetta (2018); Ye et al. (2017). These tensor charges also provide the hadronic input to the WIMPnucleus cross section in dark matter models that generate tensor quarkWIMP operators Bishara et al. (2017).
Our first calculations of these tensor charges were reported in Refs. Bhattacharya et al. (2015a, b), and here we provide an update using significantly more data that has allowed us to control all systematic uncertainties in both the connected and disconnected contributions. We show that even though the disconnected contributions are small, , the data are precise enough to extrapolate them to the continuum limit and evaluate them at the physical pion mass MeV. In particular, we report a signal in , whose contribution to the neutron EDM can be enhanced versus by in models in which the chirality flip is provided by the Standard Model Yukawa couplings.
Ii Lattice Methodology
All the calculations were done on ensembles with 2+1+1flavors of HISQ fermions Follana et al. (2007) generated by the MILC Collaboration Bazavov et al. (2013). In order to calculate the matrix elements of flavor diagonal operators, one needs to evaluate the contribution of both the “connected” and “disconnected” diagrams. The lattice methodology and our strategy for the calculation and analysis of the twopoint and connected threepoint functions using Wilson clover fermions on the HISQ ensembles has been described in Refs. Bhattacharya et al. (2015a, 2016); Yoon et al. (2016); Gupta et al. (2017) and for the disconnected contribution in Refs. Bhattacharya et al. (2015a); Lin et al. (2018b).
The details of the calculation and analysis of the connected contributions on eleven ensembles covering the range 0.15–0.06 fm in the lattice spacing, 135–320 MeV in the pion mass, and 3.3–5.5 in the lattice size have been presented in Ref. Gupta et al. (2018) and readers are referred to it. These highstatistics calculations allowed us to analyze the three systematic uncertainties due to lattice discretization, dependence on the quark mass and finite lattice size, by making a simultaneous fit in the three variables , and . The final results for the connected contribution to the proton with this chiralcontinuumfinitevolume extrapolation, reproduced from Ref. Gupta et al. (2018), are
(2) 
In this paper, we focus on the analysis of the disconnected contributions and extracting the final results for the flavor diagonal tensor charges by combining them with the connected contributions. The lattice parameters of the seven ensembles used in the analysis of the disconnected contributions for the light and strange quarks are the same as in Ref. Lin et al. (2018b), as are the number of configurations analyzed and the number of random sources used to stochastically evaluate the disconnected quark loop on each configuration.
Iii Controlling ExcitedState Contamination (ESC)
The first step in the analysis is to understand and remove the excited state contamination (ESC) in the disconnected contribution. A number of features stand out in the data shown in Fig. 1. First, for a given value of the sourcesink separation , the data are much more noisy compared to the corresponding connected contribution analyzed in Ref. Gupta et al. (2018). Second, within statistical uncertainties, there is no clear distinction between results at different values. In fact, the data at the various sourcesink separations overlap for both and , and no ESC is apparent. Lastly, the magnitude, in most cases, is smaller than , which is smaller than the statistical uncertainty in the connected contribution. Thus, the ESC is expected to be even smaller, so for the estimate on each ensemble, we take a simple average over the multiple data. The results for the bare charges, and , obtained from the average shown in Fig. 1 are given in Table. 1.
Note that, since there is no evidence for ESC we average the data over the various values, and the disconnected contribution is very small, the uncertainty due to analyzing the connected and disconnected contributions separately using the QCD spectral decomposition, as discussed in Refs. Gupta et al. (2018); Lin et al. (2018b), is expected to be even smaller.




ID  

a15m310  0.0057(16)  0.0015(8)  0.0054(15)  0.0014(8)  0.0054(15)  0.0014(8) 
a12m310  0.0086(14)  0.0023(9)  0.0081(14)  0.0021(8)  0.0084(14)  0.0022(9) 
a12m220  0.0063(24)  0.0018(14)  0.0059(23)  0.0017(13)  0.0061(23)  0.0017(14) 
a09m310  0.0057(10)  0.0016(9)  0.0056(10)  0.0016(9)  0.0058(10)  0.0017(9) 
a09m220  0.0070(21)  0.0016(9)  0.0069(21)  0.0016(9)  0.0071(21)  0.0016(9) 
a09m130  0.0016(21)  0.0016(21)  0.0016(21)  
a06m310  0.0055(11)  0.0033(11)  0.0057(12)  0.0034(11)  0.0059(12)  0.0035(12) 
Iv Renormalization of the operators
Flavor diagonal light quark operators, , can be written as a sum over isovector () and isoscalar () combinations which renormalize differently–isovector with and isoscalar with . The isovector renormalization constants are given in Ref. Gupta et al. (2018). The difference between and starts at twoloops in perturbation theory and is expected to be small. For the tensor operators, this expectation has been checked for the twisted mass action; explicit nonperturbative calculations have shown that to within a percent Alexandrou et al. (2017a, b). Since the errors in our data are larger, we take and renormalize all three charges, , and , using . Using these factors, both the connected and disconnected contributions are renormalized in two ways:
(3) 
The second definition uses the conserved vector charge condition . These two results for the renormalized disconnected contributions on each ensemble are also given in Table 1. They are extrapolated separately to the continuum limit and MeV, and the extrapolated results are given in Table 2.
V The ContinuumChiral Extrapolation
The last step in the analysis is to evaluate the results at MeV and in the continuum and infinite volume limits, and . Since the range of spanned by our disconnected data is small, , and the connected contributions showed no significant finite volume corrections, we neglect these in the analysis of the disconnected contributions. Thus, we fit the renormalized data given in Table 1 keeping just the leading correction terms in and :
(4) 
The data with the renomalization method and the results of the fits are shown in Fig. 2. The dependence of both and on and is small and the extrapolated value is consistent with what we obtain by just averaging the six (seven) points. We consider the final errors from the fit reasonable as they are larger than those in most individual points and cover the total range of variation between the points. The extrapolated results for the two ways of doing the renormalization are given in Table 2 along with the DOF of the fit using Eq. (4). The final value is taken to be the average of the two and summarized in both Tables 2 and 3.
light  strange  

/DOF  /DOF  /DOF  /DOF  
0.0062(33)  0.85  0.0066(33)  0.91  0.0064(33)  0.0026(15)  0.31  0.0027(16)  0.29  0.0027(16) 
Connected  0.790(27)  0.198(10)  

Disconnected  0.0064(33)  0.0064(33)  0.0027(16) 
PDNME’18 (Sum)  0.784(28)  0.204(11)  0.0027(16) 
ETMC’17 Alexandrou et al. (2017a)  0.782(21)  0.219(17)  
PNDME’15 Bhattacharya et al. (2015a)  0.774(66)  0.233(28)  0.008(9) 
Vi Comparison with Previous Work
In Table 3, we also give results obtained by the ETMC collaboration Alexandrou et al. (2017a) using a single physical mass ensemble generated with 2flavors of maximally twisted mass fermions with a clover term at fm, MeV and . The two sets of results agree. Such consistency is expected if the differences due to the number of dynamical flavors and possible discretization corrections in the ETMC’17 results are small or cancel.
Vii Implications for neutron electric dipole moment
The tensor charges for the neutron, in the isospin symmetric limit, are obtained by interchanging the light quark labels, , in the results for the proton given in Table 3. Using these and the experimental bound on the nEDM ( Baker et al. (2006)), the relation
(5) 
provides constraints on the CP violating quark EDMs, , arising in BSM theories, assuming that the quark EDM is the only CPviolating BSM operator. The bounds on are shown in the left panel of Fig. 3. Of particular importance is the reduction of the error in compared to our previous result in Ref. Bhattacharya et al. (2015b). The new results lets us bound . Conversely, the overall error in is reduced even if is enhanced versus by as occurs in models in which the chirality flip is provided by the Standard Model Yukawa couplings.
In general, BSM theories generate a variety of CPviolating operators that all contribute to with relations analogous to Eq. (5). As discussed in Ref. Bhattacharya et al. (2015b), in the “split SUSY” model ArkaniHamed and Dimopoulos (2005); Giudice and Romanino (2004); ArkaniHamed et al. (2005), the fermion EDM operators provide the dominant BSM source of CP violation. In Fig. 3 (right), we update the contour plots for in the gaugino () and Higgsino () mass parameter plane with the range GeV to TeV. For this analysis, we have followed Ref. Giudice and Romanino (2006) and set .
Thanks to the greatly reduced uncertainty in the tensor charges (factor of for and for ), the ratio is much more precisely known in terms of SUSY mass parameters. This allows for stringent tests of the split SUSY scenario with gaugino mass unification ArkaniHamed and Dimopoulos (2005); Giudice and Romanino (2004); ArkaniHamed et al. (2005). In particular, our results and the experimental bound cm Baron et al. (2014), imply the splitSUSY upper bounds cm. This limit is falsifiable by the next generation experiments.
Viii Conclusions
The data in Table 3 show that the PNDME’18 results (this work) with much higher statistics are a significant improvement over the PNDME’15 Bhattacharya et al. (2015a) values, in which we had neglected the disconnected contribution to and because the value on each ensemble was consistent with zero and the quality of the data was insufficient to perform a chiralcontinuum fit. The reduced uncertainty has tightened the constraints on the quark EDM couplings and on the ratio in the split SUSY scenario with gaugino mass unification ArkaniHamed and Dimopoulos (2005); Giudice and Romanino (2004); ArkaniHamed et al. (2005) as shown in Fig. 3.
Acknowledgements.
We thank the MILC Collaboration for providing the 2+1+1flavor HISQ lattices used in our calculations. The calculations used the Chroma software suite Edwards and Joo (2005). Simulations were carried out on computer facilities of (i) the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC0205CH11231; and, (ii) the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC0500OR22725; (iii) the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy, and (iv) Institutional Computing at Los Alamos National Laboratory. T. Bhattacharya and R. Gupta were partly supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Contract No. DEAC5206NA25396. T. Bhattacharya, V. Cirigliano, R. Gupta, YC. Jang and B. Yoon were partly supported by the LANL LDRD program. The work of H.W. Lin is supported by the US National Science Foundation under grant PHY 1653405 “CAREER: Constraining Parton Distribution Functions for NewPhysics Searches”.References
 Lin et al. (2018a) H.W. Lin, W. Melnitchouk, A. Prokudin, N. Sato, and H. Shows, Phys. Rev. Lett. 120, 152502 (2018a), arXiv:1710.09858 [hepph] .
 Radici and Bacchetta (2018) M. Radici and A. Bacchetta, Phys. Rev. Lett. 120, 192001 (2018), arXiv:1802.05212 [hepph] .
 Ye et al. (2017) Z. Ye, N. Sato, K. Allada, T. Liu, J.P. Chen, H. Gao, Z.B. Kang, A. Prokudin, P. Sun, and F. Yuan, Phys. Lett. B767, 91 (2017), arXiv:1609.02449 [hepph] .
 Bishara et al. (2017) F. Bishara, J. Brod, B. Grinstein, and J. Zupan, JHEP 11, 059 (2017), arXiv:1707.06998 [hepph] .
 Bhattacharya et al. (2015a) T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, A. Joseph, H.W. Lin, and B. Yoon (PNDME), Phys. Rev. D92, 094511 (2015a), arXiv:1506.06411 [heplat] .
 Bhattacharya et al. (2015b) T. Bhattacharya, V. Cirigliano, R. Gupta, H.W. Lin, and B. Yoon, Phys. Rev. Lett. 115, 212002 (2015b), arXiv:1506.04196 [heplat] .
 Follana et al. (2007) E. Follana et al. (HPQCD Collaboration, UKQCD Collaboration), Phys.Rev. D75, 054502 (2007), arXiv:heplat/0610092 [heplat] .
 Bazavov et al. (2013) A. Bazavov et al. (MILC Collaboration), Phys.Rev. D87, 054505 (2013), arXiv:1212.4768 [heplat] .
 Bhattacharya et al. (2016) T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, H.W. Lin, and B. Yoon, Phys. Rev. D94, 054508 (2016), arXiv:1606.07049 [heplat] .
 Yoon et al. (2016) B. Yoon et al., Phys. Rev. D93, 114506 (2016), arXiv:1602.07737 [heplat] .
 Gupta et al. (2017) R. Gupta, Y.C. Jang, H.W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D96, 114503 (2017), arXiv:1705.06834 [heplat] .
 Lin et al. (2018b) H.W. Lin, R. Gupta, B. Yoon, Y.C. Jang, and T. Bhattacharya, (2018b), arXiv:1806.10604 [heplat] .
 Gupta et al. (2018) R. Gupta, Y.C. Jang, B. Yoon, H.W. Lin, V. Cirigliano, and T. Bhattacharya, (2018), arXiv:1806.09006 [heplat] .
 Alexandrou et al. (2017a) C. Alexandrou et al., Phys. Rev. D95, 114514 (2017a), [Erratum: Phys. Rev.D96,no.9,099906(2017)], arXiv:1703.08788 [heplat] .
 Alexandrou et al. (2017b) C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero AvilÃ©sCasco, and C. Wiese, Phys. Rev. Lett. 119, 142002 (2017b), arXiv:1706.02973 [heplat] .
 Baker et al. (2006) C. Baker, D. Doyle, P. Geltenbort, K. Green, M. van der Grinten, et al., Phys.Rev.Lett. 97, 131801 (2006), arXiv:hepex/0602020 [hepex] .
 ArkaniHamed and Dimopoulos (2005) N. ArkaniHamed and S. Dimopoulos, JHEP 0506, 073 (2005), arXiv:hepth/0405159 [hepth] .
 Giudice and Romanino (2004) G. Giudice and A. Romanino, Nucl.Phys. B699, 65 (2004), arXiv:hepph/0406088 [hepph] .
 ArkaniHamed et al. (2005) N. ArkaniHamed, S. Dimopoulos, G. Giudice, and A. Romanino, Nucl.Phys. B709, 3 (2005), arXiv:hepph/0409232 [hepph] .
 Giudice and Romanino (2006) G. Giudice and A. Romanino, Phys.Lett. B634, 307 (2006), arXiv:hepph/0510197 [hepph] .
 Baron et al. (2014) J. Baron et al. (ACME), Science 343, 269 (2014), arXiv:1310.7534 [physics.atomph] .
 Edwards and Joo (2005) R. G. Edwards and B. Joo (SciDAC Collaboration, LHPC Collaboration, UKQCD Collaboration), Nucl.Phys.Proc.Suppl. 140, 832 (2005), arXiv:heplat/0409003 [heplat] .