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D) nswt infiac f24sadr deviations standard 2.4 of significance a with ents j [1] % 1 , oa to eII. le 1 ig ml hthge opcnrbtoscan contributions loop higher that imply lings d h ffc fteLFV the of effect the udy . ev uroi a esseaial oein done systematically be can quarkonia heavy Zl 3 × n h l uroimtcnqe described techniques quarkonium old the ing eeainof generation hn aiso h osdrdpoessand processes considered the of ratios ching e a leptons: tau roi ea tde nRf 1] Another [10]. Ref. in studied decay arkonia × i 10 igcoupling hing tluprlmt for limits upper ntal f Hτµ nldcy novn nycagdleptons charged only involving decays nnel l fteepce rnhn ai r still are ratio branching expected the if n 10 j 0 − nl u hsi eodtesoeo the of scope the beyond is this but nnels al o h orsodn branching corresponding the for I Table h coupling the F opig toelo and loop one at couplings LFV and − 10 H 9 α opigadw ytmtclystudy systematically we and coupling → s µ and hr h experimental the where µτ exico c oes[] hsvleo the of value this [4], models ics B v 2,wt measured with [2], Hµτ atre sBleII. Belle as factories expansion. Hµτ fteorder the of ob fteodrof order the of be to τ H → → µγ τµ and opigin coupling h scale the t . 6 τ l W W × → 10 by n 3 (1) − i µ l 3 j . 2 present work. 2S+1 The invariant amplitude for the annihilation of color-singlet quarkonium in a LJ angular momentum configu- ration QQ[2S+1L ] X is given by [13, 14] J → d4q [QQ[2S+1L ] X]= T r[ (Q, q)χ(Q, q)], (2) M J → (2π)4 O Z where (Q, q) is the operator entering amplitude for the corresponding free quarks transition O Q Q Q Q [Q( q,s2),Q( + q,s1) X]= v( q,s2) (Q, q)u( + q,s1), (3) M 2 − 2 → 2 − O 2

2S+1 and χ(Q, q) denotes the wave function for the QQ[ LJ ] bound state

2 0 q χ(Q, q)= 2πδ(q )ψLM (q)PS,Sz (Q, q) LM; SSz JJz . (4) − 2mQ h | i M,SXz

Here, PS,Sz stands for the spin projectors

Nc Q Q 1 1 PS,Sz (Q, q) = u( + q,s1)v( q,s2) s1; s2 SSz (5) mQ 2 2 − h2 2 | i s s1,s2 X 5 Nc 1 /Q γ /Q S =0 = ( + /q + mQ) ( + /q mQ) for , (6) smQ 2√2mQ ! 2 ( /ε(Q,Sz) ) 2 − ( S =1 ) where ε(Q,Sz) denotes the polarization vector of the spin one system. For s-wave quarkonium the wave function is rapidly damped in the relative momentum q and the leading terms are given by PS,Sz (Q, 0) and (Q, 0). In the zero-binding approximation the quarkonium mass M is given by M 2mQ and the amplitude reads O ≈

5 2S+1 R(0) 3 γ S =0 [QQ[ SJ ] X]= T r (Q, 0) ( /Q M) for , (7) M → √4π r4M "O ( /ε(Q,Sz) ) − # ( S =1 ) with M denoting the quarkonium physical mass and

d3q R(0) ψ00(q)= . (8) (2π)3 √4π Z 2S+1 A similar calculation of the invariant amplitude for the production of color singlet quarkonium, X QQ[ SJ ]+Y yields →

5 2S+1 R(0) 3 γ S =0 (X QQ[ SJ ]+ Y )= T r (Q, 0) ( /Q + M) for . (9) M → − √4π r4M "O ( /ε(Q,Sz) ) # ( S =1 ) For p-wave quarkonium, the wave function at the origin vanishes and the leading term for the annihilation amplitude is given by the first term in the expansion in q of Eq. (2). A straightforward calculation yields

3 ′ [QQ[2S+1P ] X]= i 1M; SS JJ ε (M) R (0)T r α(Q, 0)P (Q, 0)+ (Q, 0)P α (Q, 0) , M J → − h z| zi α 4π O S,Sz O S,Sz M,Sz r X  (10) where ∂A(Q, q) Aα(Q, q) , (11) ≡ ∂qα and in this case

3 d q α 3 ′ q ψ1 (q)= i R (0)ε (M). (12) (2π)3 M − 4π α Z r 3

The polarization vector εα(M) satisfies the following relations Q Q 1M;1S 00 ε (M)ε (S ) = g + α β , (13) h z| i α β z − αβ M 2 M,SXz i 1 µ ν 1M;1Sz 1Jz εα(M)εβ(Sz) = − εαβµν Q ε (Jz), (14) h | i M √2 M,SXz 1M;1S 2J ε (M)ε (S ) = ε (J ). (15) h z| zi α β z αβ z M,SXz 2S+1 The Higgs to QQ[ LJ ] quarkonium coupling is obtained from the diagram in Fig. (1) which yields the following operator m (Q, q)= i Q , (16) O v where v stands for the Higgs vacuum expectation value. Using this operator in the previous formulae it is easy to

Q 2 + q,s1

2S+1 Q LJ H

Q 2 − q,s2

FIG. 1: Feynman diagrams for the Higgs-quarkonium coupling. show that the only non-vanishing coupling of the Higgs to quarkonium is to S = 1, J = 0 p-wave quarkonium, in which case we obtain

′ 3 3R (0) 3M [QQ[ P0] H]= . (17) M → v r π Notice that this coupling is proportional to the derivative of the wave function at the origin, which according to the NRQCD rules is suppressed by a v2 factor with respect to the wave function at the origin. This makes the radiative transitions involving s-wave quarkonium configurations of the same order as the non-radiative ones involving p-wave quarkonium configurations. The radiation changes the quarkonium quantum numbers allowing the corresponding quarkonium to couple to the Higgs. The calculation of Higgs-mediated lepton flavor violating radiative transitions involving s-wave quarkonium requires to work out Higgs-quarkonium-photon coupling. This transition is induced by the diagrams in Fig. (2).

Q k, µ Q 2 + q,s1 2 + q,s1 H

2S+1 2S+1 LJ LJ H k, µ Q Q 2 − q,s2 2 − q,s2

FIG. 2: Feynman diagrams for the Higgs-quarkonium-photon coupling.

From these diagrams, we identify the transition operator as

/Q /Q m + /q + k/ + mQ /q + k/ mQ (Q, q)= iee Q ε(k) 2 2 − − ε(k) . (18) Q / Q 2 2 Q 2 2 / O v " ( 2 + q + k) m − ( 2 q + k) m # − Q − − Q where eQ stands for the heavy quark charge in units of e. For s-wave J = 0 from Eq. (7) we obtain

/Q /Q 1 eeQMR(0) 3 /ε(k)( 2 + k/) ( 2 + k/)/ε(k) 5 [H QQ[ S0]γ]= i T r − γ ( /Q M) =0. (19) Q 2 2 M → − 4v 4πM " ( 2 + k) m ! − # r − Q 4

Similarly for s-wave J =1 we get

/Q /Q 3 eeQmQR(0) 3 /ε(k)( 2 + k/) ( 2 + k/)/ε(k) [H QQ[ S1]γ]= i T r Q − 2 η/(Q)( /Q M) , (20) M → 2v 4πM " ( + k)2 m ! − # r 2 − Q where η(Q) stands for the polarization vector of the quarkonium. A straightforward calculation yields

3 eeQR(0) 3M µ ν [H QQ[ S1]γ]= Tµν ε η (21) M → v r π with Qµkν T = g . (22) µν µν − Q k · 3 Now we focus on the Higgs mediated LFV processes. We start with the QQ[ P0] µτ decay through the diagram in Fig (3). A direct calculation yields the following decay width →

Q + q,s1 2 µ 3 H P0 τ Q 2 − q,s2

3 FIG. 3: Diagram for the QQ[ P0] → µτ decay.

2 ′ 2 2 2 3 − + 27y R (0) M mτ Γ[QQ[ P0] µ τ ]= | | 1 (23) → 8π2 v2m4 − M 2 H   where we neglected the muon mass and y stands for the Hµτ coupling. 3 Next we go through the corresponding radiative process QQ[ S1] µτγ. This decay proceeds through the diagrams in Fig. (4). →

Q Q µ 2 + q,s1 k, µ 2 + q,s1 H

3 3 τ S1 µ S1 H k, µ Q Q 2 − q,s2 τ 2 − q,s2

3 FIG. 4: Feynman diagrams for the S1 → µτ decay.

Neglecting the muon mass we obtain the following decay width

2 2 2 4 2 3 − + αeQy R(0) M mτ Γ[QQ[ S1] µ τ γ]= | | f , (24) → 12π3v2 m4 M 2 H   where

f(x)=1 6x +3x2 +2x3 6 ln(x). (25) − − The Hµτ coupling can also mediate LFV decays of the tau meson involving light quarkonium. Although this is beyond the scope of the systematic NRQCD expansion due to the light quark mass, we still can use the quarkonium techniques taking care of extracting the corresponding non-perturbative pieces from the appropriate experimental 3 data. The first possible decay is τ µ QQ[ P0] which goes through the diagram shown in Fig. (5). The decay width is given by →

2 ′ 2 2 − − 3 27y R (0) mτ M M Γ(τ µ QQ[ P0]) = | | 1 , (26) → 16π2v2 m4 − m2 H  τ  5

Q 2 + q,s1 3 H P0 τ Q − q,s2 µ 2

3 FIG. 5: Feynman diagram for the τ → µQQ[ P0] decay.

3 S1 H H τ 3 τ S1 µ µ

3 FIG. 6: Feynman diagram for the τ → µQQ[ S1]γ decay. where we neglected the muon mass. 3 The corresponding radiative decay is τ µ QQ[ S1]γ. The Feynman diagrams for this process are given in Fig. (6). Neglecting the muon mass we obtain the→ following decay width

2 2 2 3 2 − − 3 αeQy R(0) mτ M M Γ(τ µ QQ[ S1]γ)= | | h , (27) → 32π3v2 m4 m2 H  τ  where 3 h(x)=(1 x)3 + x[(1 x)(3 x) + 2 ln(x)]. (28) − 2 − − Finally, although the calculation does not require to use the quarkonium techniques, it is interesting to estimate the effects of the Hµτ coupling in LFV decay of gauge bosons. As a sample we calculate W µτπ. The non-perturbative piece of this decay is related to the pion decay constant. This decay is induced by the→ diagram in Fig.(7).

W π W H µ

τ

FIG. 7: Feynman diagram for the W → τµπ decay.

The amplitude for W (Q,ε) µ(p1)τ(p2)π(p3) is → 2 yg Vudfπ p3 ε(Q) = · 2 2 u(p2)v(p1), M 2√2m (p1 + p2) m W − H where g is the weak coupling constant. The resulting decay width is

2 2 2 2 − − + − fπg λ Vud Γ(W µ τ π )= 3 f(a,b), (29) → 73728π mW where 1 f(a,b) = a2 b2 1 24a6 6a4 4b2 +7 + a2 b2 + 17 2b2 +1 6b2 a4 − − − 2  2


a 1

6 a2 1 4a6 a4 6b2 +1 +2a2b4 + b4 ln − 12b4 ln(b) , (30) − − − a2 b2 −  −  


6

2 3 ′ 2 5 Process Γexp(GeV ) |R(0)| (GeV ) |R (0)| (GeV ) Υ → e+e− 1.28 × 10−6 4.856 - − − J/ψ → e+e 5.54 × 10 6 0.560 - − − − φ → e+e 1.26 × 10 6 5.53 × 10 2 - 0 −6 −2 χc → γγ 2.34 × 10 - 3.10 × 10 −6 −4 f0 → γγ 0.29 × 10 - 1.08 × 10

TABLE II: Numerical values of the non-perturbative matrix elements extracted from the leptonic and two photon decays of quarkonia.

mH mτ with a = mW , b = mW and we neglected the pion and muon masses. 3 ′ The results for the studied decays depend on the color-singlet matrix elements R(0) for S1 quarkonium and R (0) 3 for P0 quarkonium. The same matrix elements appear in the leptonic decay of the first and two photon decays of the latter. We use the available experimental results on these decays to extract the phenomenological value of the ′ 3 matrix elements. The only matrix element that cannot be calculated this way is R (0) for the P0 ¯bb quarkonium χb0. ′ There is no available experimental data on the χb0 γγ transition, but R (0) has been calculated in several potential models summarized in Ref. [15] yielding R′(0) 1→GeV 5 and we will use this value in our calculations. The leptonic decay of vector quarkonia is induced≈ by the diagram in Fig. (8). The corresponding decay width is

Q + q,s1 2 l+ 3 γ S1 − Q l 2 − q,s2

3 + − FIG. 8: Diagram for the QQ[ S1] → l l decay.

2 2 2 3 + − 4α eQ R(0) Γ(QQ[ S1] l l )= | | (31) → M 2 3 where we neglected the lepton mass. The two photon decay of P0 quarkonium proceeds through the diagrams in Fig. (9). The decay width is

Q k1, µ Q 2 + q,s1 2 + q,s1 k2,ν

3 3 P0 P0 k2,ν k1, µ Q Q 2 − q,s2 2 − q,s2

3 FIG. 9: Feynman diagrams for the two photon decay of P0 quarkonium.

2 2 ′ 2 3 432α eQ R (0) Γ( P0 γγ)= | | (32) → M 4 In Table II. we give the results for the matrix elements of the different quarkonia. As a first approximation we use an ss¯ configuration for the f0(980). There is no available information on the total width of the χb0, thus we report the ′ 5 decay width when we use Rχb (0) = 1 GeV from quark model calculations [15]:

−17 Γ(χ 0 µτ)=5.5 10 GeV. (33) b → × The branching ratios of the remaining decays are calculated using the estimates for the non-perturbative matrix elements in Table II. We list in Table III the so obtained branching ratios. Here, the branching ratios include a factor 2 − + + − to account for the two charge states where appropriate, e.g. BR(χ 0 µτ)= BR(χ 0 µ τ )+ BR(χ 0 µ τ ). c → c → c → In general these branching ratios are small. The most promising decay is the τ µf0. We recall that we assumed anss ¯ configuration for this meson. The nature of the low lying scalar mesons is an old→ problem (see [16] and references therein) and it would be desirable to have a closer approximation to the non-perturbative effects in this decay. 7

Process Branching Ratio Exp. bound −17 χc0 → µτ 1.5 × 10 − Υ → µτγ 5.7 × 10 14 − J/ψ → µτγ 5.1 × 10 17 −12 −8 τ → µf0(980) 8.4 × 10 < 3.4 × 10 − τ → µφγ 1.7 × 10 14 − W → µτπ 3.2 × 10 17

TABLE III: Branching ratios for lepton flavor violation decays involving the Hµτ coupling.

Acknowledgments

We acknowledge financial support from CONACYT and SNI (M´exico).D. D. is grateful to Conacyt (M´exico) S.N.I. and Conacyt project (CB-156618), DAIP project (Guanajuato University) and PIFI (Secretaria de Educacion Publica, M´exico) for financial support.

[1] V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 749, 337 (2015) [arXiv:1502.07400 [hep-ex]]. [2] G. Aad et al. [ATLAS Collaboration], arXiv:1508.03372 [hep-ex]. [3] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214 [hep-ex]]. S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012) [arXiv:1207.7235 [hep-ex]]. S. Chatrchyan et al. [CMS Collaboration], JHEP 1306, 081 (2013) [arXiv:1303.4571 [hep-ex]]. [4] J. D. Bjorken and S. Weinberg, Phys. Rev. Lett. 38, 622 (1977). J. L. Diaz-Cruz and J. J. Toscano, Phys. Rev. D 62, 116005 (2000) [hep-ph/9910233]. T. Han and D. Marfatia, Phys. Rev. Lett. 86, 1442 (2001) [hep-ph/0008141]. G. F. Giudice and O. Lebedev, Phys. Lett. B 665, 79 (2008) [arXiv:0804.1753 [hep-ph]]. S. Davidson and G. J. Grenier, Phys. Rev. D 81, 095016 (2010) [arXiv:1001.0434 [hep-ph]]. A. Celis, V. Cirigliano and E. Passemar, Phys. Rev. D 89, 013008 (2014) [arXiv:1309.3564 [hep-ph]]. M. D. Campos, A. E. C. Hern´andez, H. P¨as and E. Schumacher, Phys. Rev. D 91, no. 11, 116011 (2015) [arXiv:1408.1652 [hep-ph]]. J. Heeck, M. Holthausen, W. Rodejohann and Y. Shimizu, Nucl. Phys. B 896, 281 (2015) [arXiv:1412.3671 [hep-ph]]. A. Crivellin, G. D’Ambrosio and J. Heeck, Phys. Rev. Lett. 114, 151801 (2015) [arXiv:1501.00993 [hep-ph]]. I. Dorˇsner, S. Fajfer, A. Greljo, J. F. Kamenik, N. Koˇsnik and I. Niˇsandˇzic, JHEP 1506, 108 (2015) [arXiv:1502.07784 [hep-ph]]. I. de Medeiros Varzielas and G. Hiller, JHEP 1506, 072 (2015) [arXiv:1503.01084 [hep-ph]]. A. Crivellin, G. D’Ambrosio and J. Heeck, Phys. Rev. D 91, no. 7, 075006 (2015) [arXiv:1503.03477 [hep- ph]]. D. Aristizabal Sierra, F. Staub and A. Vicente, Phys. Rev. D 92, no. 1, 015001 (2015) [arXiv:1503.06077 [hep-ph]]. A. Vicente, arXiv:1503.08622 [hep-ph]. C. X. Yue, C. Pang and Y. C. Guo, J. Phys. G 42, 075003 (2015) [arXiv:1505.02209 [hep-ph]]. [5] B. McWilliams and L. F. Li, Nucl. Phys. B 179, 62 (1981). [6] R. Harnik, J. Kopp and J. Zupan, JHEP 1303, 026 (2013) [arXiv:1209.1397 [hep-ph]]. [7] S. Chatrchyan et al. [CMS Collaboration], Nature Phys. 10, 557 (2014) [arXiv:1401.6527 [hep-ex]]. [8] S. Chatrchyan et al. [CMS Collaboration], JHEP 1405, 104 (2014) [arXiv:1401.5041 [hep-ex]]. [9] A. Goudelis, O. Lebedev and J. h. Park, Phys. Lett. B 707, 369 (2012) [arXiv:1111.1715 [hep-ph]]. [10] A. Abada, D. Beˇcirevi´c, M. Lucente and O. Sumensari, Phys. Rev. D 91, no. 11, 113013 (2015) [arXiv:1503.04159 [hep-ph]]. [11] G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Phys. Rev. D 55, 5853 (1997)] [hep-ph/9407339]. [12] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38, 090001 (2014). [13] J. H. Kuhn, J. Kaplan and E. G. O. Safiani, Nucl. Phys. B 157, 125 (1979). [14] B. Guberina, J. H. Kuhn, R. D. Peccei and R. Ruckl, Nucl. Phys. B 174, 317 (1980). [15] A. K. Likhoded, A. V. Luchinsky and S. V. Poslavsky, Phys. Rev. D 86, 074027 (2012) [arXiv:1203.431]. [16] E. van Beveren, G. Rupp and M. D. Scadron, Phys. Lett. B 495, 300 (2000) [Phys. Lett. B 509, 365 (2001)] [hep-ph/0009265]; M. Napsuciale and S. Rodriguez, Phys. Rev. D 70, 094043 (2004) [hep-ph/0407037]; A. A. Osipov, B. Hiller and J. da Providencia, Phys. Lett. B 634, 48 (2006) [hep-ph/0508058]; S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 90, 114005 (2014) [arXiv:1408.4921 [hep-ph]].