Spin-dependent μ\to e conversion
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Spin-dependent µ → e conversion Vincenzo Cirigliano, Sacha Davidson, and Yoshitaka Kuno Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA IPNL, CNRS/IN2P3, Universit´e Lyon 1,Univ. Lyon, 69622 Villeurbanne, France Department 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 ordersof 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 notpreviously been considered, contribute to µ → e conversion in three ways: in nuclei with spin theymediate a spin-dependent transition; in all nuclei they contribute to the coherent ( A -enhanced)spin-independent conversion via finite recoil effects and via loop mixing with dipole, scalar, andvector operators. We estimate the spin-dependent rate in Aluminium (the target of the upcomingCOMET and Mu2e experiments), show that the loop effects give the greatest sensitivity to tensorand axial-vector operators involving first-generation quarks, and discuss the complementarity of thespin-dependent and independent contributions to µ → e conversion. Introduction – New particles and interactions be-yond the Standard Model of particle physics are requiredto explain neutrino masses and mixing angles. The searchfor traces of this New Physics (NP) is pursued on manyfronts. One possibility is to look directly for the newparticles implicated in neutrino mass generation, for in-stance at the LHC [1] or SHiP [2]. A complementaryapproach seeks new interactions among known particles,such as neutrinoless double beta decay [3] or ChargedLepton Flavour Violation (CLFV) [4].CLFV transitions of charged leptons are induced bythe observed massive neutrinos, at unobservable ratessuppressed by ( m ν /m W ) ∼ − . A detectable ratewould point to the existence of new heavy particles, asmay arise in models that generate neutrino masses, orthat address other puzzles of the Standard Model such asthe hierarchy problem. Observations of CLFV are there-fore crucial to identifying the NP of the lepton sector,providing information complementary to direct searches.From a theoretical perspective, at energy scales wellbelow the masses of the new particles, CLFV can beparametrised with effective operators (see e.g. [5]), con-structed out of the kinematically accessible StandardModel (SM) fields, and respecting the relevant gaugesymmetries. In this effective field theory (EFT) descrip-tion, information about the underlying new dynamics isencoded in the operator coefficients, calculable in anygiven model.The experimental sensitivity to a wide variety of CLFVprocesses is systematically improving. Current boundson branching ratios of τ flavour changing decays suchas τ → µγ , τ → eγ and τ → ℓ [6–8] are O (10 − ),and Belle-II is expected to improve the sensitivity byan order of magnitude [9]. The bounds on the µ ↔ e flavour changing processes are currently of order ∼ − [10, 11], with the most restrictive contraint fromthe MEG collaboration: BR ( µ → eγ ) ≤ . × − [12].Future experimental sensitivities should improve by sev- eral orders of magnitude, in particular, the COMET [13]and Mu2e [14] experiments aim to reach a sensitivityto µ → e conversion on nuclei of ∼ − , and thePRISM/PRIME proposal[15] could reach the unprece-dented level of 10 − .In searches for µ → e conversion, a µ − from the beamis captured by a nucleus in the target, and tumbles downto the 1 s state. The muon will be closer to the nucleusthan an electron ( r ∼ αZ/m ), due to its larger mass.In the presence of a CLFV interaction with the quarksthat compose the nucleus, or with its electric field, themuon can transform into an electron. This electron,emitted with an energy E e ≃ m µ , is the signature of µ → e conversion.Initial analytic estimates of the µ → e conversion ratewere obtained by Feinberg and Weinberg [16], a widerrange of nuclei were studied numerically by Shankar [17],and relativistic effects relevant in heavier nuclei were in-cluded in Ref. [18]. State of the art conversion rates for abroad range of nuclei induced by CLFV operators whichcan contribute coherently to µ → e conversion were ob-tained in Ref. [19], while some missing operators wereincluded in Ref. [20].The calculation has some similarities with dark matterscattering on nuclei [21–23], where the cross-section canbe classified as spin-dependent (SD) or spin-independent(SI). Previous analyses of µ → e conversion [19, 20] fo-cused on CLFV interactions involving a scalar or vec-tor nucleon current, because, similarly to SI dark matterscattering, these sum coherently across the nucleus at theamplitude level, giving an amplification ∼ A in the rate,where A is the atomic number. However, other processesare possible, such as spin-dependent conversion on theground state nucleus, which we explore here, or incoher-ent µ → e conversion, where the final-state nucleus is inan excited state [17, 24].The upcoming exceptional experimental sensitivitiesmotivate our study of new contributions to µ → e conversion induced by tensor and axial vector op-erators , which were not considered in Refs. [19, 20].These operators couple to the spin of the nucleus andcan induce “spin-dependent” µ → e conversion in nu-clei with spin (such as Aluminium, the proposed tar-get of COMET and Mu2e), not enhanced by A . Inaddition, the tensor and axial operators will contributeto “spin-independent” conversion via finite-momentum-transfer corrections [25, 26], and Renormalisation Groupmixing [27, 28] . In an EFT framework, our analy-sis shows new sensitivities to previously unconstrainedcombinations of dimension-six operator coefficients, aswe illustrate below. In the absence of CLFV, this givesnew constraints on the coefficients, and when CLFV isobserved, it could assist in determining its origin. Estimating the µ → e conversion rate – Our start-ing point is the effective Lagrangian [4] δ L = − √ G F X Y (cid:16) C D,Y O D,Y + C GG,Y O GG,Y + X q = u,d,s X O C qqO,Y O qqO,Y + h.c. (cid:17) (1)where Y ∈ { L, R } and O ∈ { V, A, S, T } and the operatorsare explicitly given by ( P L,R = 1 / I ∓ γ )) O D,Y = m µ ( eσ αβ P Y µ ) F αβ O GG,Y = 932 π m t ( eP Y µ )Tr[ G αβ G αβ ] O qqV,Y = ( eγ α P Y µ )( qγ α q ) O qqA,Y = ( eγ α P Y µ )( qγ α γ q ) O qqS,Y = ( eP Y µ )( qq ) O qqT,Y = ( eσ αβ P Y µ )( qσ αβ q ) . (2)While our primary focus is on the tensor ( O qqT,Y ) and axial( O qqA,Y ) operators, we include the vector, scalar, dipoleand gluon operators because the first three are inducedby loops, and the last arises by integrating out heavyquarks.At zero momentum transfer, the quark bilinears canbe matched onto nucleon bilinears¯ q ( x )Γ O q ( x ) → G N,qO ¯ N ( x )Γ O N ( x ) (3)where the vector charges are G p,uV = G n,dV = 2 and G p,dV = G n,uV = 1, and for the axial charges we use We leave out the light-quark pseudoscalar operators and gluonoperators such as G ˜ G that can be induced by heavy-quark pseu-doscalar operators at the heavy quark thresholds. The effect ofthis class of operators in a nucleus is suppressed both by spinand momentum transfer. The analogous mixing of SD to SI dark matter interactions wasdiscussed in [29, 30]. the results inferred in Ref. [22] by using the HERMESmeasurements [31], namely G p,uA = G n,dA = 0 . G p,dA = G n,uA = − . G p,sA = G n,sA = − . µ = 2 GeV, namely G p,uT = G n,dT = 0 . G p,dT = G n,uT = − . G p,sT = G n,sT = . G p,uS = m N m u . G p,dS = m N m d . G n,uS = m N m u . G n,dS = m N m d . G p,sS = G n,sS = m N m s . µ = 2 GeV, namely m u = 2 . m d = 4 . m s = 96 MeV [35].Taking the above matching into account, the nucleon-level effective Lagrangian has the same structure of (1)with the replacements ¯ q Γ O q → ¯ N Γ O N and with effectivecouplings given by ˜ C NNO,Y = X q = u,d,s G N,qO C qqO,Y . (4)However, we remove the tensor operators, because theireffects can be reabsorbed into shifts to the axial-vectorand scalar operator coefficients. In fact, to leading orderin a non-relativistic expansion N σ ij N = ǫ ijk N γ k γ N , sothat the spin-dependent nucleon effective Lagrangian for µ → e conversion reads − √ G F X N X Y (cid:16) e C NNA,Y ( eγ α P Y µ )( N γ α γ N ) + h.c. (cid:17) (5)where N ∈ { n, p } , X, Y ∈ {
L, R } , X = Y and e C NNA,Y = X q (cid:16) G N,qA C qqA,Y + 2 G N,qT C qqT,X (cid:17) . (6)Furthermore, at finite recoil the tensor operatorinduces a contribution to the SI amplitude, since u N ( p ) σ i u N ( p − q ) contains a term proportional to q i /m N [25, 26], which contracts, in the amplitude, withthe spin of the helicity-eigenstate electron. The net effectis tantamount to replacing the coefficient of the scalaroperator with e C NNS,Y → e C NNS,Y + m µ m N e C NNT,Y . (7)We write the conversion rate Γ = Γ SI + Γ SD , whereΓ SI is the A -enhanced rate occuring in any nucleus, and The gluon operators O GG,Y induce a shift in the coefficient ofthe nucleon scalar density ˜ C NNS,Y , as discussed in Ref. [20]. Wedo not explicitly include this effect as it is not relevant to ourdiscussion. Γ SD is only relevant in nuclei with spin. The usual SIbranching ratio reads [4, 19]BR SI = 2B (cid:12)(cid:12)(cid:12)(cid:12) [ e C ppV,R + e C ppS,L ] Z F p ( m µ )+ [ e C nnV,R + e C nnS,L ] [ A − Z ] F n ( m µ )+ 2 C D,L
ZeF p ( m µ ) (cid:12)(cid:12)(cid:12)(cid:12) + { L ↔ R } , (8)where B = G F m µ ( αZ ) / ( π Γ cap ), Γ cap is the rate forthe muon to transform to a neutrino by capture on thenucleus (0 . × /sec in Aluminium [36]), and theform factors F p,n ( | ~k | ) = R d xe − i~k · ~x ρ p,n ( x ) can be foundin Eq. (30) of Ref. [19].In the evaluation of Γ SD from (5) we treat the muon asnon-relativistic and the electron as a plane wave. Bothare good approximations for low- Z nuclei; for definite-ness we focus on Aluminium ( Z = 13 , A = 27 , J = 5 / µ → e amplitude corresponds to thatof “standard” spin-dependent WIMP nucleus scattering.At momentum transfer ~q , this is Z d xe − i~q · ~x h Al | N ( x ) γ k γ N ( x ) | Al i . (9)The µ → e amplitude is then obtained by multiplyingby the appropriate lepton current and coefficients . Byanalogy with WIMP scattering [22, 23, 37], we obtain:BR SD = 8B J Al + 1 J Al (cid:12)(cid:12)(cid:12) S Alp e C ppA,L + S Aln e C nnA,L (cid:12)(cid:12)(cid:12) S A ( m µ ) S A (0)+ { L ↔ R } . (10)The spin expectation values S AlN are defined as S AlN = h J Al , J z = J Al | S zN | J Al , J z = J Al i , where S zN is the z com-ponent of the the total nucleon spin, and the expectationvalue is over the nuclear ground state. They can be im-plemented in our QFT notation (with relativistic statenormalisation for Al ) by setting Eqn. (9) at | ~q | = 0 to2 S AlN ( J Al ) k | J Al | × m Al (2 π ) δ (3) ( p Al,out − p Al,in ) . The axial structure factor S A ( | ~q | ) [23, 37] reads S A ( q ) = a L, + S ( q ) + a L + a L, − S ( q ) + a L, − S ( q ) At finite recoil, the vector or scalar operators can also contributeto the spin-dependent amplitude [26]. We neglect these contri-butions, because we estimate their interference with the axialvector is suppressed by O ( m µ /m N ). where a L, ± = e C ppA,L ± e C nnA,L . The S AlN and S ij ( q ) havebeen calculated in the shell model in Refs. [37, 38].At | ~q | ≡ q = 0 the conversion rate is controlled bythe spin expectation values; we use S Aln = 0 .
030 and S Alp = 0 .
34 [38]. At finite momentum transfer q = m µ ,the structure factors provide a non-trivial correction. Us-ing dominance of the proton contribution ( S Alp >> S
Aln )we find from Ref. [38] S Al ( m µ )) /S Al (0) ≃ . Loop effects and the RGEs – QED and QCD loopschange the magnitude of some operator coefficients, andQED loops can transform one operator into another.Such Standard Model loops are neccessarily present, andtheir dominant (log-enhanced) effects are included in theevolution with scale of the operator coefficients, as de-scribed by the Renormalisation Group Equations (RGEs)of QED and QCD (see [5] for an introduction to the RGrunning of operators with the scale µ ). If the New Physicsscale is well above m W , loops involving the W, Z, and h could also be relevant. However, we focus here on theRGE evolution from the experimental scale µ N up to theweak scale m W . Since any UV model can be mapped intoa set of operator coefficients at µ = m W , our calculationdoes not lose generality while remaining quite simple.We consider the one-loop RGEs of QED and QCD for µ ↔ e flavour-changing operators [27, 28]. Defining λ = α s ( m W ) α s ( µ N ) , their solution can be approximated as C I ( µ N ) ≃ C J ( m W ) λ a J δ JI − α e e Γ eJI π log m W µ N ! (11)where I, J represent the super- and subscripts which labeloperator coefficients. The a I describe the QCD runningand are only non-zero for scalars and tensors: for N f = 5one has a I = Γ sII β = {− , } for I = S, T, . We use thisscaling to always give results in terms of coefficients atthe low scale µ N = 2 GeV, where we match quarks tonucleons. Γ e is the one-loop QED anomalous dimensionmatrix, rescaled [39, 40] for J, I ∈ T, S to account forthe QCD running:˜Γ eJI = Γ eJI f JI , f JI = 11 + a J − a I λ a I − a J − λ − λ . (12)In the estimates presented here, we focus on the effects ofthe off-diagonal elements of e Γ eJI , which mix one operatorinto another, and neglect the QED running of individualcoefficients.In RG evolution down to µ N , photon exchange betweenthe external legs of a tensor operator can mix it to a scalaroperator. This contribution to the scalar coefficient is∆ e C NNS,X ( µ N ) ∼ X q G N,qS f T S Q q α e π log m W µ N C qqT,X ( µ N )(13)where f T S is from Eq. (12).The tensor operator also mixes to the dipole, when thequark lines are closed and an external photon is attached.This gives a contribution to the dipole coefficient (cid:12)(cid:12)(cid:12) ∆ C eµD,X ( µ N ) (cid:12)(cid:12)(cid:12) ∼ Q q N c m q em µ α e π log m W µ N C qqT,X ( µ N ) (14)which is suppressed by m q /m µ , due to a mass insertionon the quark line. For tensor operators involving u, d or s quark bilinears, the mixing to the scalar operatordescribed in Eq. (13) gives a larger contribution to SI µ → e conversion than this mixing to the dipole. Sofor the remainder of this letter, we do not discuss thecontribution of Eq. (14) to µ → e conversion. We willdiscuss heavier quarks in a later publication [41].Curiously, one-loop QED corrections to the axial op-erator generate the vector [28] . If a New Physics modelinduces a non-zero coefficient C qqA,Y ( m W ), then photonexchange between the external legs induces a contribu-tion to the vector coefficient at the experimental scale:∆ C qqV,Y ( µ N ) ≃ − Q q α e π log m W µ N C qqA,Y ( µ N ) (15)As a result, the SI and SD processes will have comparablesensitivities to axial vector operators. Results – To interpret our results, we first estimatethe sensitivity of SD and SI µ → e conversion to the co-efficients of the tensor and axial operators of eqn (2).We allow a single operator coefficient to be non-zero at m W , and consider its various contributions to SD and SI µ → e conversion (sometimes refered to as setting bounds“one-operator-at-a-time”).Suppose first that only the tensor coefficient C uuT,L ispresent at m W . Recall that C uuT,L ( m W ) can contributeto µ → e conversion in three ways: to the SI rate viathe finite momentum transfer effects of eqn (7), to theSI rate via the RG mixing to the scalar given in eqn(13), and directly to the SD rate as given in eqn (10).It is easy to check that the RG mixing contribution to e C NNS ( µ N ) is an order of magnitude larger than the finiterecoil contribution. Furthermore, the RG mixing effect isdominant contribution of C uuT,L ( m W ) to µ → e conversion,as can been seen numerically by calculating the SD andSI contributions to the branching ratio: BR ( µ Al → e Al) ∼ . | . C uuT,L | + . | C uuT,L | (16)where the coefficients are at the experimental scale, andthe second term is the A -enhanced SI contribution.The RG mixing is the largest contribution of C uuT,L ( m W ) to µ → e conversion due to three enhance-ments: first, the anomalous dimension Γ eT S is large, and The heavy quark scalar contribution to µ → e conversion[20] issuppressed ∝ /m Q , so the tensor mixing to the dipole coulddominate. If the lepton current contained γ µ , rather than γ µ P Y , this wouldnot occur. second, the G N,qS coefficients of eqn (3) are an order ofmagnitude larger than G N,qT . The combination of thesegives ∆ e C NNS,X ( µ N ) > ∼ e C ppT,X ( µ N ), which respectively con-tribute to the SI and SD rates. Finally, the scalar coef-ficient benefits from a further A enhancement in the SIconversion rate. This shows that including the RG effectscan change the branching ratio by orders of magnitude.A similar estimate for the axial operator O uuA,L gives BR ( µ Al → e Al) ∼ . | . C uuA,L | + . | . C uuA,L | . (17)We see that the RG mixing of O uuA,L into O uuV,L , whosecoefficient contributes to SI µ → e conversion, also givesthe best sensitivity to C uuA,L . However, the ratio of SI toSD contributions is smaller than in the tensor case, dueto the smaller anomalous dimension in eqn (15).SI µ → e conversion will also give the best sensitivityto tensor and axial operators involving d quarks. How-ever, in the case of strange quarks, the vector currentvanishes in the nucleon, so O ssA,Y only contributes to SD µ → e conversion. The largest contribution of the strangetensor operator is via its mixing to the scalar, with a sen-sitivity to C ssT,X reduced by a factor ∼ G NsT / G NuT withrespect to C uuT,X . The strange tensor also mixes signifi-cantly to the dipole (see eqn (14)) which contributes to µ → eγ ; we estimate that the sensitivity to C ssT,Y of theMEG experiment with BR ∼ × − (as expected aftertheir upgrade), would be comparable to that of COMETor Mu2e with BR ∼ few × − .Let us now focus on the complementarity of SD andSI contributions to the µ → e conversion rate, which de-pend on different combinations of operator coefficients.So once a signal is observed, measuring µ → e conversionin targets with and without spin could assist in differ-entiating among operators or models. To illustrate thiscomplementarity, we restrict to scalar and tensor opera-tors involving u quarks, whose coefficients we would liketo determine. Figure 1 represents the allowed parame-ter space for C uuT,L and C uuS,L evaluated at µ N (dottedblue) and m W (solid red). We see that, irrespective ofthe operator scale, SD µ → e conversion always gives anindependent constraint. In its absence, there would be anunconstrained direction in parameter space, correspond-ing to C uuT,Y at the experimental scale, or the diagonal redband at m W . The figure also shows that the enhancedsensitivity of SI conversion illustrated in eqn (16) requiresthe (model-dependent) assumption that the model doesnot induce a scalar contribution which cancels the mixingof the tensor into the scalar, which would correspond toventuring along the red ellipse in the plot. Prospects – In this letter, we followed the pragmaticlow-energy perspective of parametrising charged LeptonFlavour Violating interactions with effective operators,and considered the contribution of axial vectors and ten-sors to µ → e conversion. To our knowledge, this hasnot been studied previously. We found that the Spin- uuT,L C -1.5 -1 -0.5 0 0.5 1 1.5 uu S , L C -0.1-0.0500.050.1 FIG. 1: The horizontal dotted blue (diagonal red) areas arethe allowed parameter space at the experimental scale (at m W ), if BR ( µ → e conversion) ≤ − . This plot assumesthat CLFV only occurs in up-quark operators. Dependent process depends on different operator coef-ficients from the Spin-Independent case, so comparing µ → e conversion rates in targets with and without spinwould give additional constraints, and could allow toidentify axial or tensor operators coefficients. 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