arXiv:1703.02057v3 [hep-ph] 30 Aug 2018 10 h E collaboration: MEG the e h bevdmsienurns tuosral rates unobservable ( at by neutrinos, suppressed massive observed the Charged or [3] [4]. decay (CLFV) complementary beta Violation A Flavour double Lepton [2]. neutrinoless SHiP particles, as known among such or new interactions new the [1] seeks in- for approach LHC for directly the generation, at look mass stance neutrino many to in on is implicated pursued possibility is particles (NP) One Physics New fronts. this search of The traces angles. for mixing required and are masses physics neutrino explain particle to of Model Standard the yond eo h asso h e atce,CF a be can CLFV particles, (see new operators the effective with of parametrised masses the sector, below lepton the searches. of direct NP to complementary the information identifying providing there- to are crucial CLFV of as or fore Observations such masses, Model problem. Standard hierarchy neutrino the as the of generate puzzles particles, that other heavy address models that new in of arise existence may the to point would nodro antd 9.Tebud nthe on bounds The by sensitivity [9]. the magnitude improve of to order expected an is Belle-II and uueeprmna estvte hudipoeb sev- by improve should sensitivities experimental Future nbacigrto of bounds as ratios Current branching improving. on systematically is processes any in calculable coefficients, model. operator is given the dynamics new in underlying gauge descrip- encoded the (EFT) relevant about theory information the field tion, effective Standard respecting this and In accessible symmetries. fields, kinematically (SM) the Model of out structed aorcagn rcse r urnl forder of currently are processes changing flavour Introduction LVtastoso hre etn r nue by induced are leptons charged of transitions CLFV rmatertclprpcie teeg clswell scales energy at perspective, theoretical a From h xeietlsniiiyt ievreyo CLFV of variety wide a to sensitivity experimental The − τ 12 → 1,1] ihtems etitv otan from contraint restrictive most the with 11], [10, µγ 3 , eateto hsc,OaaUiest,11Machikaneyam 1-1 University, Osaka Physics, of Department fmgiuei oigyas ecnie h matof impact the consider We years. coming in magnitude of etrfu-emo prtr hc opet h pno nu to of spin contribute the considered, to been couple previously which operators four-fermion vector etroeaos eetmt h pndpnetrt nAl in rate spin-dependent vi the and estimate effects We recoil operators. finite vector via conversion spin-independent pndpnetadidpnetcnrbtosto t contributions give independent effects and qua loop spin-dependent first-generation the involving that operators show axial-vector and experiments), Mu2e and COMET eit pndpnettasto;i l ulite con they nuclei all in transition; spin-dependent a mediate τ h xeietlsniiiyto sensitivity experimental The m → 1 e atce n neatosbe- interactions and particles New – ν hoeia iiin o lmsNtoa aoaoy Los Laboratory, National Alamos Los Division, Theoretical 2 /m eγ PL NSI23 nvri´ yn1Ui.Lo,662Vil 69622 Lyon, 1,Univ. Universit´e Lyon CNRS/IN2P3, IPNL, W and τ ) BR 4 aorcagn eassuch decays changing flavour icnoCirigliano, Vincenzo ∼ τ ( µ 10 → → − 3 eγ 48 ℓ Spin-dependent eetberate detectable A . ) 68 are [6–8] ≤ 4 . 2 e.g. × µ 10 → Dtd uut3,2018) 31, August (Dated: 1 O 5) con- [5]), − e (10 ah Davidson, Sacha 13 ovrino ulii xetdt mrv yfu orders four by improve to expected is nuclei on conversion µ µ [12]. − → 8 ↔ ∼ ), e µ ovrini he as nnce ihsi they spin with nuclei in ways: three in conversion µ n ue[4 xeiet i orahasensitivity a reach to aim experiments [13] to [14] COMET the Mu2e particular, in and magnitude, of orders eral RS/RM rpsl1]cudrahteunprece- 10 the of level reach dented could proposal[15] PRISM/PRIME uncntasomit neeto.Ti electron, This electron. energy the an an field, with electric into its quarks emitted transform with the can or with nucleus, interaction muon the CLFV a compose of that presence the In down tumbles and target, 1 the the in to nucleus a by captured is hna lcrn( electron an than S) rvosaaye of analyses spin-independent Previous or can (SD) (SI). cross-section spin-dependent as the classified where be [21–23], nuclei on scattering were operators missing some [20]. Ref. while in [19], included Ref. in tained µ mltd ee,gvn namplification the an at giving nucleus level, vec- the matter amplitude across dark or coherently SI sum scalar to these similarly scattering, a because, involving current, nucleon interactions tor CLFV on cused ra ag fnce nue yCF prtr which to operators a coherently CLFV for contribute by rates induced conversion can nuclei art the of of range in- State broad were [18]. nuclei Ref. heavier in in cluded wider relevant [17], Shankar effects a by relativistic numerically [16], and studied Weinberg were nuclei and of Feinberg range by obtained were where rudsaences hc eepoehr,o incoher- or the here, explore on we ent conversion which nucleus, spin-dependent state as ground such possible, are oiaeorsuyo e otiuin to contributions new of study our motivate 24]. [17, state excited an → → → nsace for searches In h aclto a oesmlrte ihdr matter dark with similarities some has calculation The nta nltcetmtso the of estimates analytic Initial h poigecpinleprmna sensitivities experimental exceptional upcoming The e µ k,addsustecmlmnaiyo the of complementarity the discuss and rks, e µ e conversion. 2 → → µ conversion. conversion A opmxn ihdpl,saa,and scalar, dipole, with mixing loop a n ohtk Kuno Yoshitaka and rbt otechrn ( coherent the to tribute los uhoeaos hc aenot have which operators, Such cleons. → s e mnu tetre fteupcoming the of target (the uminium steaoi ubr oee,ohrprocesses other However, number. atomic the is e ,Tynk,Oaa5004,Japan 560-0043, Osaka Toyonaka, a, tt.Temo ilb lsrt h nucleus the to closer be will muon The state. ovrin hr h nlsaencesi in is nucleus final-state the where conversion, e ovrino uliof nuclei on conversion aorcagn esradaxial- and tensor flavour-changing egets estvt otensor to sensitivity greatest he lms M855 USA 87545, NM Alamos, − µ r 18 → . eran,France leurbanne, ∼ e ovrin a conversion, αZ/m E µ e 3 → ≃ µ ,det t agrmass. larger its to due ), A → e m 2 -enhanced) ovrin[9 0 fo- 20] [19, conversion µ e µ stesgaueof signature the is , ovrinwr ob- were conversion → ∼ µ − e ∼ 10 LA-UR-17-21718 ovrinrate conversion rmtebeam the from A − 2 OUHEP-17-1 16 nterate, the in n the and , 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 C˜NN = 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 . (7) p,d n,u → mN GV = GV = 1, and for the axial charges we use e e e We write the conversion rate Γ = ΓSI +ΓSD, where 2 ΓSI is the A -enhanced rate occuring in any nucleus, and

1 We leave out the light-quark pseudoscalar operators and gluon operators such as GG˜ that can be induced by heavy-quark pseu- doscalar operators at the heavy quark thresholds. The effect of 3 this class of operators in a nucleus is suppressed both by spin The gluon operators OGG,Y induce a shift in the coefficient of ˜NN and momentum transfer. the nucleon scalar density CS,Y , as discussed in Ref. [20]. We 2 The analogous mixing of SD to SI dark matter interactions was do not explicitly include this effect as it is not relevant to our discussed in [29, 30]. discussion. 3

pp nn Al ΓSD is only relevant in nuclei with spin. The usual SI where aL,± = CA,L CA,L. The SN and Sij (q) have branching ratio reads [4, 19] been calculated in the± shell model in Refs. [37, 38]. At ~q q =e 0 the conversione rate is controlled by | | ≡ Al pp pp the spin expectation values; we use Sn = 0.030 and BRSI = 2B0 [CV,R + CS,L] Z Fp(mµ) Al Sp = 0.34 [38]. At finite momentum transfer q = mµ, nn nn the structure factors provide a non-trivial correction. Us- + [C e+ C ][eA Z] Fn(mµ) V,R S,L − Al Al 2 ing dominance of the proton contribution (Sp >> Sn ) we find from Ref. [38] SAl(mµ))/SAl(0) 0.29. + 2eCD,LZeFe p(mµ) + L R , (8) { ↔ } Loop effects and the RGEs ≃ – QED and QCD loops change the magnitude of some operator coefficients, and 2 5 3 2 where B0 = GF mµ(αZ) /(π Γcap ), Γcap is the rate for QED loops can transform one operator into another. the muon to transform to a neutrino by capture on the Such Standard Model loops are neccessarily present, and 6 nucleus (0.7054 10 /sec in Aluminium [36]), and the their dominant (log-enhanced) effects are included in the × 3 −i~k·~x form factors Fp,n( ~k )= d xe ρp,n(x) can be found evolution with scale of the operator coefficients, as de- | | in Eq. (30) of Ref. [19]. scribed by the Renormalisation Group Equations (RGEs) R In the evaluation of ΓSD from (5) we treat the muon as of QED and QCD (see [5] for an introduction to the RG non-relativistic and the electron as a plane wave. Both running of operators with the scale µ). If the New Physics are good approximations for low-Z nuclei; for definite- scale is well above mW , loops involving the W, Z, and h ness we focus on Aluminium (Z = 13, A = 27,J = 5/2) could also be relevant. However, we focus here on the the proposed target for the COMET and Mu2e experi- RGE evolution from the experimental scale µN up to the ments. After approximating the muon wavefunction in weak scale mW . Since any UV model can be mapped into the nucleus to its value at the origin and taking it out- a set of operator coefficients at µ = mW , our calculation side the integral over the nucleus [16], the nuclear part of does not lose generality while remaining quite simple. the spin-dependent µ e amplitude corresponds to that We consider the one-loop RGEs of QED and QCD for → of “standard” spin-dependent WIMP nucleus scattering. µ e flavour-changing operators [27, 28]. Defining λ = At momentum transfer ~q, this is αs↔(mW ) αs(µN ) , their solution can be approximated as

− · d3xe i~q ~x Al N(x)γkγ N(x) Al . (9) e 5 aJ αeΓJI mW h | | i CI (µN ) CJ (mW )λ δJI log (11) Z ≃ − 4π µN ! The µ e amplitude is then obtained by multiplying e by the appropriate→ lepton current and coefficients 4. By where I,J represent the super- and subscripts which label analogy with WIMP scattering [22, 23, 37], we obtain: operator coefficients. The aI describe the QCD running and are only non-zero for scalars and tensors: for Nf =5 s 2 ΓII 12 4 JAl +1 Al pp Al nn SA(mµ) one has aI = = , for I = S, T,. We use this BRSD = 8B0 Sp CA,L + Sn CA,L 2β0 {− 23 23 } JAl SA(0) scaling to always give results in terms of coefficients at

+ L R . e e (10) the low scale µN = 2 GeV, where we match quarks to { ↔ } nucleons. Γe is the one-loop QED anomalous dimension Al Al The spin expectation values SN are defined as SN = matrix, rescaled [39, 40] for J, I T,S to account for z z ∈ JAl,Jz = JAl SN JAl,Jz = JAl , where SN is the z com- the QCD running: hponent of the the| total| nucleon spin,i and the expectation aI −aJ value is over the nuclear ground state. They can be im- ˜e e 1 λ λ ΓJI =ΓJI fJI , fJI = − . (12) plemented in our QFT notation (with relativistic state 1+ aJ aI 1 λ − − normalisation for Al) by setting Eqn. (9) at ~q =0 to | | In the estimates presented here, we focus on the effects of e k the off-diagonal elements of ΓJI , which mix one operator Al (JAl) 3 (3) 2SN 2mAl(2π) δ (pAl,out pAl,in) . into another, and neglect the QED running of individual JAl × − | | coefficients. e In RG evolution down to µN , photon exchange between The axial structure factor SA( ~q ) [23, 37] reads | | the external legs of a tensor operator can mix it to a scalar 2 2 operator. This contribution to the scalar coefficient is SA(q)= aL,+S00(q)+ aL+aL,−S01(q)+ aL,−S11(q) α m ∆CNN (µ ) GN,qf 24Q e log W Cqq (µ )(13) S,X N S TS q π µ T,X N ∼ q N X 4 e At finite recoil, the vector or scalar operators can also contribute where f is from Eq. (12). to the spin-dependent amplitude [26]. We neglect these contri- TS butions, because we estimate their interference with the axial The tensor operator also mixes to the dipole, when the vector is suppressed by O(mµ/mN ). quark lines are closed and an external photon is attached. 4

N,q This gives a contribution to the dipole coefficient second, the GS coefficients of eqn (3) are an order of magnitude larger than GN,q. The combination of these 2Q N m α m T eµ q c q e W qq NN > pp ∆CD,X (µN ) log CT,X (µN ) (14) gives ∆CS,X (µN ) CT,X (µN ), which respectively con- ∼ emµ π µN ∼ tribute to the SI and SD rates. Finally, the scalar coef- 2 which is suppressed by mq/mµ, due to a mass insertion ficient benefitse from ae further A enhancement in the SI on the quark line. For tensor operators involving u, d conversion rate. This shows that including the RG effects or s quark bilinears, the mixing to the scalar operator can change the branching ratio by orders of magnitude. A similar estimate for the axial operator uu gives described in Eq. (13) gives a larger contribution to SI OA,L µ e conversion than this mixing to the dipole. So → BR(µAl eAl) .12 0.84Cuu 2 +.27 .69Cuu 2 . (17) for the remainder of this letter, we do not discuss the → ∼ | A,L| | A,L| contribution of Eq. (14) to µ e conversion. We will uu uu discuss heavier quarks 5 in a later→ publication [41]. We see that the RG mixing of A,L into V,L, whose coefficient contributes to SI µ Oe conversion,O also gives Curiously, one-loop QED corrections to the axial op- uu → erator generate the vector [28] 6. If a New Physics model the best sensitivity to CA,L. However, the ratio of SI to qq SD contributions is smaller than in the tensor case, due induces a non-zero coefficient CA,Y (mW ), then photon exchange between the external legs induces a contribu- to the smaller anomalous dimension in eqn (15). tion to the vector coefficient at the experimental scale: SI µ e conversion will also give the best sensitivity to tensor→ and axial operators involving d quarks. How- qq αe mW qq ever, in the case of strange quarks, the vector current ∆CV,Y (µN ) 3Qq log CA,Y (µN ) (15) ss ≃− π µN vanishes in the nucleon, so A,Y only contributes to SD µ e conversion. The largestO contribution of the strange As a result, the SI and SD processes will have comparable tensor→ operator is via its mixing to the scalar, with a sen- sensitivities to axial vector operators. sitivity to Css reduced by a factor GNs/2GNu with Results – To interpret our results, we first estimate T,X T T respect to Cuu . The strange tensor∼ also mixes signifi- the sensitivity of SD and SI µ e conversion to the co- T,X cantly to the dipole (see eqn (14)) which contributes to efficients of the tensor and axial→ operators of eqn (2). µ eγ; we estimate that the sensitivity to Css of the We allow a single operator coefficient to be non-zero at → T,Y MEG experiment with BR 2 10−14 (as expected after m , and consider its various contributions to SD and SI W their upgrade), would be comparable∼ × to that of COMET µ e conversion (sometimes refered to as setting bounds or Mu2e with BR few 10−16. “one-operator-at-a-time”).→ Let us now focus∼ on× the complementarity of SD and Suppose first that only the tensor coefficient Cuu is T,L SI contributions to the µ e conversion rate, which de- present at m . Recall that Cuu (m ) can contribute W T,L W pend on different combinations→ of operator coefficients. to µ e conversion in three ways: to the SI rate via So once a signal is observed, measuring µ e conversion the finite→ momentum transfer effects of eqn (7), to the in targets with and without spin could assist→ in differ- SI rate via the RG mixing to the scalar given in eqn entiating among operators or models. To illustrate this (13), and directly to the SD rate as given in eqn (10). complementarity, we restrict to scalar and tensor opera- It is easy to check that the RG mixing contribution to tors involving u quarks, whose coefficients we would like CNN (µ ) is an order of magnitude larger than the finite S N to determine. Figure 1 represents the allowed parame- recoil contribution. Furthermore, the RG mixing effect is uu uu uu ter space for CT,L and CS,L evaluated at µN (dotted dominante contribution of C (mW ) to µ e conversion, T,L blue) and m (solid red). We see that, irrespective of as can been seen numerically by calculating→ the SD and W the operator scale, SD µ e conversion always gives an SI contributions to the branching ratio: independent constraint. In→ its absence, there would be an BR(µAl eAl) .12 1.54Cuu 2 + .27 47Cuu 2 (16) unconstrained direction in parameter space, correspond- → ∼ | T,L| | T,L| uu ing to CT,Y at the experimental scale, or the diagonal red where the coefficients are at the experimental scale, and band at mW . The figure also shows that the enhanced the second term is the A2-enhanced SI contribution. sensitivity of SI conversion illustrated in eqn (16) requires The RG mixing is the largest contribution of the (model-dependent) assumption that the model does uu CT,L(mW ) to µ e conversion due to three enhance- not induce a scalar contribution which cancels the mixing → e ments: first, the anomalous dimension ΓTS is large, and of the tensor into the scalar, which would correspond to venturing along the red ellipse in the plot. Prospects – In this letter, we followed the pragmatic low-energy perspective of parametrising charged Lepton 5 The heavy quark scalar contribution to µ → e conversion[20] is Flavour Violating interactions with effective operators, ∝ suppressed 1/mQ, so the tensor mixing to the dipole could and considered the contribution of axial vectors and ten- dominate. 6 µ µ sors to µ e conversion. To our knowledge, this has If the lepton current contained γ , rather than γ PY , this would → not occur. not been studied previously. We found that the Spin- 5

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