Spin-Dependent $\Mu\To E $ Conversion

$Spin-Dependent $\Mu\To E $ Conversion$ LA-UR-17-21718 OUHEP-17-1 Spin-dependent µ → e conversion Vincenzo Cirigliano,1 Sacha Davidson,2 and Yoshitaka Kuno3 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 2 IPNL, CNRS/IN2P3, UniversitéLyon 1,Univ. Lyon, 69622 Villeurbanne, France 3Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan (Dated: August 31, 2018) The experimental sensitivity to µ e conversion on nuclei is expected to improve by four orders of magnitude in coming years. We consider→ the impact of µ e flavour-changing tensor and axial- vector four-fermion operators which couple to the spin of nucleons.→ Such operators, which have not previously been considered, contribute to µ e conversion in three ways: in nuclei with spin they mediate a spin-dependent transition; in all→ nuclei they contribute to the coherent (A2-enhanced) spin-independent conversion via finite recoil effects and via loop mixing with dipole, scalar, and vector operators. We estimate the spin-dependent rate in Aluminium (the target of the upcoming COMET and Mu2e experiments), show that the loop effects give the greatest sensitivity to tensor and axial-vector operators involving first-generation quarks, and discuss the complementarity of the spin-dependent and independent contributions to µ e conversion. → Introduction – New particles and interactions be- eral orders of magnitude, in particular, the COMET [13] yond the Standard Model of particle physics are required and Mu2e [14] experiments aim to reach a sensitivity to explain neutrino masses and mixing angles. The search to µ e conversion on nuclei of 10−16, and the for traces of this New Physics (NP) is pursued on many PRISM/PRIME→ proposal[15] could reach∼ the unprece- fronts. One possibility is to look directly for the new dented level of 10−18. particles implicated in neutrino mass generation, for in- In searches for µ e conversion, a µ− from the beam stance at the LHC [1] or SHiP [2]. A complementary is captured by a nucleus→ in the target, and tumbles down approach seeks new interactions among known particles, to the 1s state. The muon will be closer to the nucleus such as neutrinoless double beta decay [3] or Charged than an electron (r αZ/m), due to its larger mass. Lepton Flavour Violation (CLFV) [4]. In the presence of a∼ CLFV interaction with the quarks CLFV transitions of charged leptons are induced by that compose the nucleus, or with its electric field, the the observed massive neutrinos, at unobservable rates muon can transform into an electron. This electron, 4 −48 suppressed by (m /m ) 10 . A detectable rate emitted with an energy Ee mµ, is the signature of ν W ≃ would point to the existence∼ of new heavy particles, as µ e conversion. → may arise in models that generate neutrino masses, or Initial analytic estimates of the µ e conversion rate that address other puzzles of the Standard Model such as were obtained by Feinberg and Weinberg→ [16], a wider the hierarchy problem. Observations of CLFV are there- range of nuclei were studied numerically by Shankar [17], fore crucial to identifying the NP of the lepton sector, and relativistic effects relevant in heavier nuclei were in- providing information complementary to direct searches. cluded in Ref. [18]. State of the art conversion rates for a From a theoretical perspective, at energy scales well broad range of nuclei induced by CLFV operators which below the masses of the new particles, CLFV can be can contribute coherently to µ e conversion were ob- → parametrised with effective operators (see e.g. [5]), con- tained in Ref. [19], while some missing operators were structed out of the kinematically accessible Standard included in Ref. [20]. arXiv:1703.02057v3 [hep-ph] 30 Aug 2018 Model (SM) fields, and respecting the relevant gauge The calculation has some similarities with dark matter symmetries. In this effective field theory (EFT) descrip- scattering on nuclei [21–23], where the cross-section can tion, information about the underlying new dynamics is be classified as spin-dependent (SD) or spin-independent encoded in the operator coefficients, calculable in any (SI). Previous analyses of µ e conversion [19, 20] fo- given model. cused on CLFV interactions→ involving a scalar or vec- The experimental sensitivity to a wide variety of CLFV tor nucleon current, because, similarly to SI dark matter processes is systematically improving. Current bounds scattering, these sum coherently across the nucleus at the on branching ratios of τ flavour changing decays such amplitude level, giving an amplification A2 in the rate, ∼ as τ µγ, τ eγ and τ 3ℓ [6–8] are (10−8), where A is the atomic number. However, other processes and Belle-II→ is expected→ to improve→ the sensitivityO by are possible, such as spin-dependent conversion on the an order of magnitude [9]. The bounds on the µ ground state nucleus, which we explore here, or incoher- e flavour changing processes are currently of order ↔ ent µ e conversion, where the final-state nucleus is in → 10−12 [10, 11], with the most restrictive contraint from∼ an excited state [17, 24]. the MEG collaboration: BR(µ eγ) 4.2 10−13 [12]. The upcoming exceptional experimental sensitivities Future experimental sensitivities→ should≤ improve× by sev- motivate our study of new contributions to µ → 2 e conversion induced by tensor and axial vector op- the results inferred in Ref. [22] by using the HERMES 1 p,u n,d erators , which were not considered in Refs. [19, 20]. measurements [31], namely GA = GA = 0.84(1), These operators couple to the spin of the nucleus and p,d n,u p,s n,s GA = GA = 0.43(1), and GA = GA = .085(18). can induce “spin-dependent” µ e conversion in nu- − − → For the tensor charges we use the lattice QCD re- clei with spin (such as Aluminium, the proposed tar- sults [32] in the MS scheme at µ = 2 GeV, namely get of COMET and Mu2e), not enhanced by A2. In p,u n,d p,d n,u GT = GT = 0.77(7), GT = GT = 0.23(3), and addition, the tensor and axial operators will contribute p,s n,s − GT = GT = .008(9), Finally, for the scalar charges to “spin-independent” conversion via finite-momentum- induced by light quarks we use a precise dispersive deter- transfer corrections [25, 26], and Renormalisation Group mination [33], Gp,u = mN 0.021(2), Gp,d = mN 0.041(3), mixing [27, 28] 2. In an EFT framework, our analy- S mu S md Gn,u = mN 0.019(2), and Gn,d = mN 0.045(3), and an sis shows new sensitivities to previously unconstrained S mu S md combinations of dimension-six operator coefficients, as average of lattice results [34] for the strange charge: Gp,s = Gn,s = mN 0.043(11). In all cases, we take cen- we illustrate below. In the absence of CLFV, this gives S S ms new constraints on the coefficients, and when CLFV is tral values of the MS quark masses at µ = 2 GeV, namely observed, it could assist in determining its origin. mu =2.2 MeV, md =4.7 MeV, and ms = 96 MeV [35]. Estimating the µ e conversion rate – Our start- Taking the above matching into account, the nucleon- ing point is the effective→ Lagrangian [4] level effective Lagrangian has the same structure of (1) with the replacementsq ¯ΓOq N¯ΓON and with effective 3 → δ = 2√2GF CD,Y D,Y + CGG,Y GG,Y couplings given by L − O O Y X CÑN = GN,q Cqq . (4) + Cqq qq + h.c. (1) O,Y O O,Y O,Y OO,Y q=u,d,s q u,d,s O X =X X However, we remove the tensor operators, because their where Y L, R and O V,A,S,T and the operators ∈{ } ∈{ } effects can be reabsorbed into shifts to the axial-vector are explicitly given by (PL,R =1/2(I γ )) ∓ 5 and scalar operator coefficients. In fact, to leading order ij ijk k αβ in a non-relativistic expansion Nσ N = ǫ Nγ γ5N, so D,Y = mµ(eσ PY µ)Fαβ O that the spin-dependent nucleon effective Lagrangian for 9 αβ µ e conversion reads GG,Y = 2 (ePY µ)Tr[Gαβ G ] O 32π mt → qq α V,Y = (eγ PY µ)(qγαq) √ NN α O 2 2GF CA,Y (eγ PY µ)(Nγαγ5N)+ h.c. (5) qq α − = (eγ PY µ)(qγαγ q) N Y OA,Y 5 X X qq e = (ePY µ)(qq) OS,Y where N n,p , X, Y L, R , X = Y and qq αβ ∈{ } ∈{ } 6 = (eσ PY µ)(qσαβq) . (2) OT,Y CNN = GN,qCqq +2GN,qCqq . (6) qq A,Y A A,Y T T,X While our primary focus is on the tensor ( T,Y ) and axial q qq O X ( A,Y ) operators, we include the vector, scalar, dipole e andO gluon operators because the first three are induced Furthermore, at finite recoil the tensor operator by loops, and the last arises by integrating out heavy induces a contribution to the SI amplitude, since 0i quarks. uN (p)σ uN (p q) contains a term proportional to i − At zero momentum transfer, the quark bilinears can q /mN [25, 26], which contracts, in the amplitude, with be matched onto nucleon bilinears the spin of the helicity-eigenstate electron. The net effect is tantamount to replacing the coefficient of the scalar N,q q¯(x)ΓO q(x) G N¯(x)ΓO N(x) (3) operator with → O p,u n,d NN NN mµ NN where the vector charges are GV = GV = 2 and CS,Y CS,Y + CT,Y .

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