PHYSICAL REVIEW D 100, 094018 (2019)

ν solution to the strong CP problem

‡ † ∥ Marcela Carena,1,2, Da Liu,3,* Jia Liu,2, Nausheen R. Shah,4,§ Carlos E. M. Wagner,2,3, and Xiao-Ping Wang3,¶ 1Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, Illinois 60510, USA 2Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 3HEP Division, Argonne National Laboratory, 9700 Cass Avenue, Argonne, Illinois 60439, USA 4Department of Physics & Astronomy, Wayne State University, Detroit, Michigan 48201, USA

(Received 1 May 2019; published 15 November 2019)

We present a solution to the strong CP problem in which the imaginary component of the up mass, I½mu, acquires a tiny, but nonvanishing value. This is achieved via a Dirac seesaw mechanism, which is also responsible for the generation of the small neutrino masses. Consistency with the observed value of the mass is achieved via instanton contributions arising from QCD-like interactions, as is the case in the closely related massless up-quark solution to the strong CP problem. In our framework, however, the value of the is directly related to I½mu, which, due to its common origin with the neutrino masses, implies that the neutron electric dipole moment is likely to be measured in the next round of experiments. We also present a supersymmetric extension of this Dirac seesaw model to stabilize the hierarchy among the scalar mass scales involved in this mechanism.

DOI: 10.1103/PhysRevD.100.094018

I. INTRODUCTION are the quark masses. Due to the QCD , the value of θ can be modified by a phase redefinition of the The (SM) has been highly successful in chiral quark fields, but the physical value describing all experimental observations [1]. The observed flavor and CP-violating effects originate from the weak θ ¼ θ þ ½ ½ ð Þ interactions via the dependence of the charged currents on QCD arg det Mq ; 2 the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix Q V . There is, however, another potential source of CP CKM where det½Mq¼ mq, remains invariant. As will be violation in the SM, associated with the . discussed in detail later on, a non-vanishing value of After the diagonalization of the quark masses, the QCD θ QCD leads to QCD induced CP-violating effects, like Lagrangian density contains the terms the neutron electric dipole moment (nEDM), which is

2 as yet unobserved. The current bound on the nEDM, θ X −26 gs ˜ μν;a 3 0 10 L ⊃− Gμν G − ðm q¯ q þ H:c:Þ; ð1Þ dn < . × ecm [2,3], leads to the constraint 32π2 ;a q L R θ ð1 Þ ≲ 1 3 10−10 q QCD GeV . × . The dynamical origin of such θ small values of QCD is the so-called strong CP problem. where gs is the strong gauge coupling, Gμν;a is the QCD The θ term in Eq. (1) may be eliminated by a proper ˜ 1 αβ;a θ field strength tensor, Gμν;a ¼ 2 ϵμναβG is its dual, and mq phase redefinition of the quark fields. For a nonzero QCD, at least one of the quark masses, for instance the up quark * mass, would become a complex quantity, with argument Corresponding author. θ ∼ I½ j j [email protected] QCD mu = mu . Hence in such a case, all the QCD- † Corresponding author. induced CP-violating effects would be associated with [email protected] I½ ‡ mu , and would vanish in the limit of zero up quark mass. [email protected] This is the well known massless up quark solution to the §[email protected] ∥ – [email protected] strong CP problem [4 10]. ¶[email protected] We shall denote as the canonical basis, the basis in θ ¼ 0 θ which and QCD is the argument of the up Published by the American Physical Society under the terms of quark mass. Using the value of the up quark mass the Creative Commons Attribution 4.0 International license. determined in the framework of chiral perturbation theory, Further distribution of this work must maintain attribution to j ð1 Þj ≃ 5 θ the author(s) and the published article’s title, journal citation, mu GeV MeV [11], the bound on QCD becomes and DOI. Funded by SCOAP3. equivalent to

2470-0010=2019=100(9)=094018(6) 094018-1 Published by the American Physical Society CARENA, LIU, LIU, SHAH, WAGNER, and WANG PHYS. REV. D 100, 094018 (2019)

−4 I½muð1 GeVÞ ≲ 6.5 × 10 eV: ð3Þ A possible ultraviolet configuration is that each generation is sensitive to a different SUð3Þ gauge interaction, with a 3 The relevant question then becomes, can one dynamically gauge group SUð3Þ ¼ SUð3Þ1 × SUð3Þ2 × SUð3Þ3 that is generate a value of I½muð1 GeVÞ consistent with such a spontaneously broken to the diagonal group SUð3Þ at a stringent bound, while the real part, R½muð1 GeVÞ,isof scale of the order of hundreds of TeV. Assuming that the the order of a few MeV? tree-level Higgs induced strange and bottom quark masses To analyze this question, one should remember that the are equal to zero, the instanton contributions in each sector up quark mass at scales of the order of 1 GeV receives would be responsible for bringing these masses to their contributions not only from its tree-level Higgs Yukawa observed values (via contributions proportional to the H interaction, which we will denote as mu , but also from charm and top quark masses, respectively). In such a case, inst instanton contributions, mu . Hence in general, the low energy CP-violating interactions will be governed by expressions similar to Eq. (6), with the only difference ð1 Þ¼ inst þ H ð Þ inst mu GeV mu mu : 4 that mu will include the ultraviolet instanton contribu- tions. Hence, any tension of the up quark mass with lattice In the case of QCD, the instanton contributions to the up determinations would be eliminated. inst quark mass depend on the masses of the other in the Irrespective of its origin, provided mu can lead to the theory. In a general basis, the light quark contributions are H observed up quark mass at scales of the order of 1 GeV, mu given by [4,6], can be arbitrarily small. One would naturally expect the Higgs induced CP-violating phase θH to be larger than expð−iθÞðmHmHÞ QCD inst ¼ d s ð Þ ∼10−2. In such a case, from Eq. (7), jmHj would be of the mu Λ ; 5 u order of or smaller than 4 × 10−2 eV. This implies values of j Hj where Λ is a scale which characterizes the size of these mu similar in magnitude to the small neutrino masses [1]. H H Our proposed solution of the strong CP problem is contributions, and md and ms are the tree-level Higgs induced down and strange quark masses. associated with the dynamical generation of precisely such inst small values of jmHj. In the canonical basis, mu is a real contribution, u H implying that I½mu ¼I½mu. The physical CP-violating phase, Eq. (2), then reads II. A DIRAC SEESAW MODEL

jmHj We present a model which realizes a seesaw mechanism θ ð1 Þ ≃ θH u ð1 Þ ð Þ H QCD GeV sin GeV ; 6 for the dynamical generation of m and of small Dirac QCD jm j u u neutrino masses [15–18] (see Refs. [19–21] for an alter- θ θH ¼ ½ H j Hj ≪ native formulation relating QCD to the neutrino masses.). where QCD arg mu and we have assumed that mu To realize this idea, we assume the presence of a Z4 jminstj. This expression is consistent with θ ¼ arg½m . u QCD u discrete symmetry that forbids the direct coupling of the up The small imaginary components of the instanton induced quark and neutrinos to the Higgs field. While the right- strange and down quarks masses, proportional to handed up quark and the right-handed neutrinos have mHðmHÞ=Λ mHðmHÞ=Λ d u and s u respectively, induce a sub- charge 1, all other SM fields carry zero charge under this dominant effect that becomes negligible in the one instan- symmetry. In addition, we introduce a heavy scalar doublet ton approximation. From Eqs. (3) and (6), we conclude Φ with Z4 charge 1 and hypercharge 1=2, and a singlet S of that a strong CP problem solution would be provided if charge -1 under the Z4 symmetry, such that νR Majorana values of masses are forbidden. The Lagrangian for the Yukawa interactions of the up jmHð1 GeVÞj sin θH ≲ 6.5 × 10−4 eV; ð7Þ u QCD quark and the neutrinos is given by: could be dynamically generated while maintaining con- ¯ ˜ ˜ L ¼ Yνl Φν þ Y q¯ Φu þ H:c:; ð8Þ sistency with the observed up quark mass. L R u L R Interestingly, it has been argued that the minst contribu- ˜ u where Φ ¼ iσ2Φ carries charge -1 under Z4. All the other tion induced by the standard QCD interactions may be as – SM have standard Yukawa interactions with the large as a few MeV [4 10], and hence be able to explain the Higgs doublet H, which are not shown here. The potential observed up quark mass value. This is allegedly in tension involving the heavy scalar fields relevant for our discussion with the lattice determination of the up quark mass at scales reads where the instanton contribution should be negligible, namely, mMSð2 GeVÞ¼2.1 MeV [12,13]. Alternatively, 2 † † † † u V ¼ mΦΦ Φ þðρSH Φ þ H:c:ÞþλΦ;1Φ ΦH H it has been postulated that similar contributions may come þ λ Φ†Φj j2 þ λ j j4 þðλ 4 þ Þþ ð Þ from instantons in some ultraviolet gauge extensions [14]. Φ;2 S S;1 S S;2S H:c : 9

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4 Here the term λS;2S is allowed by the discrete symmetry similar Higgs and flavor structure to generate the proper but would not be allowed by a global Peccei-Quinn Uð1Þ CKM mixing angles, the required vanishing tree-level symmetry [22]. Hence, there is no axionlike Goldstone Yukawa coupling of strange and bottom quarks to the H [23,24]. It is easy to prove that to ensure a vacuum and Φ Higgs fields may be simply ensured by assigning sR expectation value (vev) in the real direction and stability of and bR the same Z4 charge as the one for uR. the potential, we need λS;2 < 0 and ðλS;1 þ λS;2Þ > 0.We As pointed out in Ref. [14], after the generation of the will assume that mΦ ≫ mS;mH, so that one can integrate it proper CKM mixing angles, one obtains flavor violating 1 ð3Þ3 out by the equation of motion Φ ≃− 2 ρS H, where we effects that demand the SU breaking scale to be larger mΦ than a few 100’s of TeV. Moreover, the corresponding have assumed that ρ is real. The effective Yukawa inter- off-diagonal Yukawa couplings lead to instanton correc- actions for the up quark and neutrinos, represented in tions to the imaginary component of the quark masses. Fig. 1, are given by: θ ð3Þ3 These corrections modify the value of QCD at the SU ρ ρ instanton scale, and, if they are evaluated at the scale L ≃− l¯ ˜ ν − ¯ ˜ þ ð Þ eff Yν 2 S LH R Yu 2 SqLHuR H:c: 10 Λ ∼ O 10−11 mΦ mΦ i (few 100 TeV), they are of the order of , and hence an order of magnitude smaller than the current bound θ After the singlet and the neutral componentpffiffiffi of thepffiffiffi SM on QCD. One potential problem of the formulation pre- 0 Higgs field acquire vevs, hSi¼v = 2; hH i¼v= 2, the sented is that the hierarchy between mΦ and the electro- S λ λ ρ Dirac masses of the up quark and neutrinos read: weak scale is not stable in the presence of Φ;1; Φ;2; .To address this problem, in the next section we present a ρv v ρv v ∼ S H ∼ S ð Þ supersymmetric extension of this scenario. mν Yν 2 ;mu Yu 2 : 11 2mΦ 2mΦ III. SUPERSYMMETRIC EXTENSION If vS is the order of the EW scale v ¼ 246 GeV, and ρ is of order mΦ, one gets an effective seesaw suppression of the In the case of supersymmetry (SUSY), we assume the 2 up quark and neutrino masses jmuj; jmνj ∼ v =mΦ. Hence, presence of a Z3 symmetry, and charges Φu∶−1, Φd∶1, c ∶1 νc ∶1 ∶−1 as assumed, one sees the need for large values of mΦ, uR , R , S . All other fields are neutral under the      discrete Z3 symmetry. The corresponding superpotential is 12 Yν ρ vS 0.05 eV given by mΦ ≃ 6 × 10 GeV ; 0.1 0.1mΦ v mν c c c c W ¼ −YνLΦ ν − Y QΦ u − y LH e − y QH d ð12Þ u R u u R e d R d d R κ 3 þ μHuHd þ mΦΦuΦd þ λHuΦdS þ S : ð14Þ to get an observational consistent mass for the heavier 3 neutrino, where we have assumed Yν to be real. Given the Right-handed neutrino Majorana masses generated by the bound on I½mH in Eq. (7), one obtains the bound on the up u singlet S are forbidden by the holomorphicity of the quark Yukawa at the scale of m : Z superpotential. In addition, we have imposed R-   ð c Þ3 0.1 which forbids terms like vR . The SUSY invariant jY ðm Þj < 0.05Yν ; ð13Þ u Z θH potential for the Higgs fields reads: sin QCD 2 2 2 2 V ¼jμj jH j þjμH þ λΦ Sj þjmΦΦ þ λH Sj where we have taken into account the running of the up quark SUSY u d d u u þj j2jΦ j2 þjκ 2 þ λ Φ j2 ð Þ mass due to QCD interactions jmuðmZÞj=jmuð1 GeVÞj ∼ mΦ d S Hu d ; 15 0.4. For the SUð3Þ3 instanton configuration [14], assuming a where μ is the conventional μ term. In the following, we will take mΦ ≫ μ ∼ TeV. After SUSY-breaking, we have the following soft-breaking interaction terms:

V ¼ m2 jΦ j2 þ m2 jΦ j2 þ m2SS þ soft Φu u Φd d S † 3 þðλaλHuΦdS þ bλΦuHuS þ aκS þþH:c:Þ;

where we have omitted terms not relevant for our dis- cussion. Note that in the limit of κ ¼ aκ ¼ 0 there would be a Uð1Þ global symmetry which would make the singlet FIG. 1. A diagrammatic representation of the Dirac seesaw CP-odd scalar massless. More specifically, a global Uð1Þ 2 mechanism for the up quark and neutrino masses. Peccei-Quinn symmetry [22] is broken by the S ðHuΦdÞ

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3 and S terms, which are proportional to κλ or aκ. Since Φd affected by nondecoupling and CP violating contributions, CP Φif acquires a very small vev, the mass of the -odd scalar proportional to A;B [28]. Hence, if the instanton scale is predominantly originates from a negative aκ. above the supersymmetric particle mass scale, the proposed By assuming that the SUSY-invariant mass mΦ is solution to the strong CP problem will be invalidated by the much larger than all the soft masses, one can integrate appearance of new phases in the Yukawa couplings. In out the heavy scalar fields Φ : Φ ∼− λ H S, Φ ∼ u;d u mΦ u d addition, in the presence of colored Majorana gluinos, the 1 ˜ − 2 ðμλ H S þ λaλH S Þ, and obtain the low-energy instanton contribution to the up-quark Yukawa coupling jmΦj d u will be suppressed by an additional factor M3 =Λ3 com- effective Lagrangian for the Yukawa interactions in the g˜1 1 Dirac notation: pared with the non-SUSY case. Therefore, we must demand the supersymmetry particle masses to be above λ λ the instanton scale. Moreover, for the up-quark Yukawa Ly ¼ − S l¯ ˜ ν − S ¯ ˜ þ ð Þ eff Yν LHu R Yu qLHuuR 16 mΦ mΦ coupling to remain small after supersymmetry particle corrections, we should demand that the supersymmetry from which we can read off the neutrino and up quark breaking mechanism preserves the Z3 symmetry. masses: If the instanton effects come from regular SUð3Þ     interactions, the supersymmetry particle and heavy λ λ vS vu H vS vu masses are naturally much larger than the mν ∼ Yν pffiffiffi pffiffiffi;m∼ Y pffiffiffi pffiffiffi; ð17Þ 3 u u QCD instanton scale. However, for the SUð3Þ scenario our 2mΦ 2 2mΦ 2 proposed solution of the strong CP problem is only viable where we assumed v to be real, and the expression if heavy Higgs and colored SUSY particle masses are of the u ð3Þ between parenthesis on the left- and right-hand side of order of or larger than the characteristic SU i instanton Λ Eq. (17) defines the low energy Yukawa couplings yν and scales i. As discussed above, assuming a similar flavor y , respectively. The necessary values of jmΦj and jY νj and Higgs structure for the generation of the CKM mixing u u; Λ ∼O can be extracted from Eqs. (12) and (13) after replacing angles, i must be (few 100 TeV) [14]. This suppresses 3 all the CP-violating and flavour-changing effects induced jρ=mΦj by jλj, and v by vu. For the SUð3Þ case, as in the non-SUSY scenario, the required vanishing tree-level by the heavy Higgs and SUSY particles in Eq. (18). On the Yukawa coupling of strange and bottom quarks to the other hand, it introduces a little hierarchy problem, which will not be addressed further in this work. Hd and Φd Higgs fields may be simply ensured by c c Z c assigning sR and bR the same 3 charge as the one for uR. Generically supersymmetric extensions lead to addi- IV. NEUTRON ELECTRIC DIPOLE MOMENT tional contributions to the electric dipole moments. In A notable outcome of our framework is that a nonzero the absence of flavor violation in the scalar mass param- θ Φif ¼ ½ nEDM is induced by the nonvanishing value of QCD.We eters, they are proportional to the phases A arg MiAf ; can calculate the contribution to the nEDM from current Φ ¼arg½M μ ðBμÞ, where y A are the scalar trilinear B g˜ f f algebra [29,30]; the result reads: couplings, Mg˜ is the mass of the gluino, Mi the masses, and Bμ the H H bilinear mass parameter. The ¯ u d dn ∼ gπNNgπNN MN ð Þ one-loop SUSY corrections to the nEDM, controlled by 2 ln ; 19 e 4π M mπ Φif Φ N A and B, may be parametrized as [25]   where MN ∼ 940 MeV is the nucleon mass, mπ ∼ 100 GeV 2 140 j j ∼ 13 4 SUSY ≃ 2 Φif 10−23 ð Þ MeV is the pion mass and gπNN . is the usual dn A;B ecm; 18 mSUSY CP conserving pion-nucleon coupling. The CP violating coupling g¯πNN is given by: where mSUSY denotes a common soft supersymmetry breaking mass scale. There are also relevant contributions ¯ ∼ θ meff ð Þ gπNN QCD ; 20 at the two-loop level, that lead to a somewhat more Fπ complicated dependence on the SUSYand Higgs spectrum, ≡ j j ðj jþj jþj jÞ as well as to possible cancellations between one and two with meff mumdms = mumd mums mdms , ∼ 93 loop contributions [26,27]. These contributions will be Fπ MeV is the pion decay constant, and the masses suppressed well below the current bounds without fine- of the quarks and the strong CP phase are evaluated at the ∼ 1 tuning the CP-violating phases if the masses of the scale Q GeV. Using the currently determined values j j gluino, squark and heavy Higgs boson masses are larger for mu;d;s [1], this result becomes consistent with the than 10 TeV. calculation of Refs. [31,32] by using the QCD sum rules, An important consideration is that after integrating out ∼ θ ð2 4 0 7Þ 10−16 ð Þ the SUSY particles, the low energy Yukawa couplings are dn QCD × . . × ecm; 21

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–25 10 new particle masses, the latter are about an order of Current magnitude smaller than the current bound on the nEDM. Although these corrections may potentially break the

10–26 correlation between the nEDM and the neutrino masses, barring an unlikely strong cancellation, they reinforce the

ecm] expectation of a measurement of the nEDM in the near [

n future. Moreover, as pointed out in Ref. [41],ifθ is the

d QCD 10–27 dominant source of CP violation in the strong sector, there

Future will be strong correlations between the nEDM and the measurable EDMs of light nuclei and atoms, which can be used to distinguish different contributions to the nEDM. 10–28 10–6 10–5 10–4 10–3 [mH] [eV] u V. CONCLUSION FIG. 2. The neutron EDM as a function of the imaginary part of In this article, we have explored the Dirac seesaw the up quark mass. We have also shown the current 90% C.L. mechanism to provide a common origin of two small bound [2] and prospective sensitivity from the future neutron scales I½m and mν, which are related to two different EDM measurements [34–40]. u physical phenomena: strong CP-violation and neutrino oscillations. We propose a dynamical generation of the and also with a recent lattice calculation [33]. In the small nonzero imaginary part of the up quark mass, θ ∼ I½ H j j canonical basis, where QCD mu = mu , and normaliz- naturally solving the strong CP problem. Similar to the ing the value of the nEDM to the present bound [2],we case of the related massless up-quark solution to the strong obtain CP problem, the real part of the up quark mass obtains additive renormalization from instanton effects above the I½mH chiral symmetry breaking scale ∼1 GeV. However, irre- d ¼ u × 3.0 × 10−26 ecm: ð22Þ n ð6.5 2.0Þ × 10−4 eV spective of the detailed origin of this additive instanton contribution, the novel part of our construction is that the Figure 2 shows the nEDM as a function of the imaginary neutrino mass scale is strongly correlated with the static part of the up quark mass. While the current measurement nonzero value of the neutron EDM, with predicted values H −4 leads to a bound on I½mu < ð6.5 2.0Þ × 10 eV, that are expected to be probed by the next generation of future nEDM experiments [34–40] will be able to improve experiments. the present sensitivity by two orders of magnitude ∼3 × 10−28 ecm [37], and hence will be able to probe H −6 ACKNOWLEDGMENTS I½mu up to about 6 × 10 eV. Note that even for a phase θH ≃ 10−2 j Hj We would like to thank P. Agrawal, W. Jay, A. Kronfeld QCD , the values of mu that will be probed are much smaller than the ones that naturally arise from the and L. T. Wang for useful discussions and comments. This H manuscript has been authored by Fermi Research Alliance, relation of mu and the neutrino masses. Hence, it is natural to expect a measurement of the nEDM by the next LLC under Contract No. DE-AC02-07CH11359 with the generation of experiments within this framework. U.S. Department of Energy, Office of Science, Office of Finally, we should comment on additional contributions High Energy Physics. Work at University of Chicago is to the nEDM. As discussed above, they can either come supported in part by U.S. Department of Energy Grant from sources of CP violation associated with the new No. DE-FG02-13ER41958. Work at ANL is supported in physics introduced to stabilize the scale hierarchies, part by the U.S. Department of Energy under Contract Eq. (18), or, in the SUð3Þ3 scenario [14], from instanton No. DE-AC02-06CH11357. N. R. S. is supported by contributions to the imaginary part of the quark masses, Wayne State University and by the U.S. Department of arising after the generation of off-diagonal Yukawa cou- Energy under Contract No. DESC0007983. J. L. acknowl- plings. While the former are suppressed by the square of the edges support by an Oehme Fellowship.

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[1] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [20] P. Q. Hung, in Proceedings, Meeting of the APS Division of 030001 (2018). Particles and Fields (DPF 2017): Fermilab, Batavia, [2] J. M. Pendlebury et al., Phys. Rev. D 92, 092003 (2015). Illinois, USA (2017), http://arxiv.org/abs/arXiv:1710.00498. [3] C. A. Baker et al., Phys. Rev. Lett. 97, 131801 (2006). [21] P. Q. Hung, arXiv:1704.06390. [4] H. Georgi and I. N. McArthur, Report No. HUTP-81/A011, [22] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 1981. (1977). [5] D. B. Kaplan and A. V. Manohar, Phys. Rev. Lett. 56, 2004 [23] F. Wilczek, Phys. Rev. Lett. 40, 279 (1978). (1986). [24] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978). [6] K. Choi, C. W. Kim, and W. K. Sze, Phys. Rev. Lett. 61, 794 [25] C. F. Kolda, Nucl. Phys. B, Proc. Suppl. 62, 266 (1998). (1988). [26] T. Ibrahim and P. Nath, Rev. Mod. Phys. 80, 577 (2008). [7] T. Banks, Y. Nir, and N. Seiberg, in Yukawa Couplings and [27] J. R. Ellis, J. S. Lee, and A. Pilaftsis, J. High Energy Phys. the Origins of Mass. Proceedings, 2nd IFT Workshop, 10 (2008) 049. Gainesville, USA (1994), pp. 26–41, http://arxiv.org/abs/ [28] M. Carena, J. R. Ellis, A. Pilaftsis, and C. E. M. Wagner, hep-ph/9403203. Nucl. Phys. B586, 92 (2000). [8] H. Davoudiasl and A. Soni, Phys. Rev. D 76, 095015 (2007). [29] R. J. Crewther, P. Di Vecchia, G. Veneziano, and E. Witten, [9] M. Dine, P. Draper, and G. Festuccia, Phys. Rev. D 92, Phys. Lett. 88B, 123 (1979); 91B, 487(E) (1980). 054004 (2015). [30] V. Baluni, Phys. Rev. D 19, 2227 (1979). [10] W. A. Bardeen, arXiv:1812.06041 [Phys. Rev. Lett. (to be [31] M. Pospelov and A. Ritz, Nucl. Phys. B573, 177 (2000). published)]. [32] M. Pospelov and A. Ritz, Phys. Rev. Lett. 83, 2526 (1999). [11] J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982). [33] J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif, [12] A. Bazavov et al. (Fermilab Lattice, MILC, and TUMQCD arXiv:1902.03254. Collaborations), Phys. Rev. D 98, 054517 (2018). [34] C. Abel et al.,inInternational Workshop on Particle [13] S. Aoki et al. (Flavour Lattice Averaging Group), arXiv: Physics at Neutron Sources 2018 (PPNS 2018) Grenoble, 1902.08191. France (2018), https://arxiv.org/abs/1811.02340. [14] P. Agrawal and K. Howe, J. High Energy Phys. 12 (2018) [35] R. Picker, J. Phys. Soc. Jpn. Conf. Proc. 13, 010005 (2017). 035. [36] W. Schreyer (TUCAN Collaboration), in CIPANP [15] P.-H. Gu and H.-J. He, J. Cosmol. Astropart. Phys. 12 Conference, Palm Springs, California, USA (2018), (2006) 010. https://arxiv.org/abs/1809.10337. [16] C. Bonilla, J. M. Lamprea, E. Peinado, and J. W. F. Valle, [37] S. Slutsky et al., Nucl. Instrum. Methods Phys. Res., Sect. A Phys. Lett. B 779, 257 (2018). 862, 36 (2017). [17] S. C. Chuliá, E. Ma, R. Srivastava, and J. W. F. Valle, Phys. [38] A. Serebrov, Proc. Sci., INPC2016 (2017) 179. Lett. B 767, 209 (2017). [39] T. M. Ito et al., Phys. Rev. C 97, 012501 (2018). [18] S. C. Chuliá, R. Srivastava, and J. W. F. Valle, Phys. Lett. B [40] F. Kuchler et al., Hyperfine Interact. 237, 95 (2016). 781, 122 (2018). [41] J. de Vries, P. Draper, K. Fuyuto, J. Kozaczuk, and D. [19] P. Q. Hung, arXiv:1712.09701. Sutherland, Phys. Rev. D 99, 015042 (2019).

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