PHYSICAL
PRL 114, 091801 (2015)
REVIEW
week ending 6 MARCH 2015
LETTERS
Lepton Flavor Violation in B Decays? Sheldon L. Glashow,1’ Diego Guadagnoli,2’* and Kenneth Lane1’* 1Department o f Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA 2Laboratoire d ’Annecy-le-Vieux de Physique Theorique UMR5108, CNRS et Universite de Savoie, BP 110, F-74941 Annecy-le-Vieux Cedex, France (Received 15 November 2014; published 3 March 2015) The LHCb Collaboration’s measurement of RK = B{B+ —►K +ft+/x~) / B(B+ —> K +e+e~) lies 2.6 p +p~)exp/ B(BS -* q+fi~)SM = R k = 0.75, is compatible with recent measurements of these rates. We stress the importance of searches for lepton flavor violations, especially for B -* Kqe, Kfir, and Bs -* jie, fir. DOI: 10.1103/PhysRevLett. 114.091801
PACS numbers: 12.60.Cn, 11.30.Hv, 13.20.He
The LHCb Collaboration recently measured the ratio of decay rates for B+ -*■ K+tf+tf~ ( f = fi. e), obtaining [1]
rk
=
= 0-745-o.0794°(stat) ± 0.036(syst). ( 1)
This result is a 2.6a deficit from the Standard Model (SM) expectation, RK = 1 + 0 { 10-4) [2-4], Previous measurements of RK by the Belle and BABAR Collaborations [5,6] had considerably greater uncertainties but were consistent with the SM prediction. The LHCb determination was made for 1 < q2 = M j f < 6 GeV2 in order to be well below the radiative tail of the J/yr. LHCb also measured the B -> KfB fT branching ratio in this q1 range [7]. The updated result, based on its full Run I data set of 3 fb_1, is [8] B(B+ -> K +q+ q-)[h6] = (1.19 ± 0.03 ± 0.06) x 10~7.
(2) The SM prediction [9-11] B(B+ -> K W ) f M
= (L75+q2°) x 10-7
(3)
is about 45% higher. Whether this is the cause of the R K deficit is suggestive, but not certain. Another possibility is a problem in the B -> Ke+e~ measurement, since LHCb performs better in the fifi than in the ee channel, the latter being hampered by bremsstrahlung and poorer statistics [1], On the other hand, bremsstrahlung effects largely cancel in the experimental observable, and results for B -> K e+e~ are consistent with SM expectations. The LHCb results for RK and for anomalies in the B -> K*n+n~ angular distributions (discussed below) have 0031-9007/15/114(9)/091801(4)
attracted much theoretical attention [12-29]. References [13,16-21,23-25,28] proposed the existence of particles at or above 1 TeV that induce new and nonuniversal lepton interactions. Such interactions violate lepton flavor unless the leptons involved are chosen to be mass eigenstates—an unjustifiable act of fine tuning. No known symmetry principle protects lepton flavor conservation in the presence of lepton nonuniversality. Thus, LHCb’s reported value of R K implies, e.g., that B -» and B -*■ must occur at rates much higher than would occur in the SM due to tiny neutrino masses. We urge that these and other lepton flavor violations (LFVs) be sought with renewed vigor in LHC Run II and elsewhere. To illustrate the sort of LFV processes that might be seen in B decays, the limits that currently exist on lepton mixing, and the potential for discovery of LFV, we consider a simple but well-motivated interaction that can account for the known features of the R K deficit, is consistent with existing limits on LFV, and produces effects that may be observable in LHC Run II. The LHCb data on B -* suggest that, despite the detector’s superior measurement of muons vis-a-vis electrons, the R K result is due to a ~25% deficit in the muon channel. If so, then LFV is larger for muons than for electrons. This is naturally accounted for by a third-generation interaction of the type that would be expected, e.g., in topcolor models [30], Furthermore, recent theoretical analyses [13,27,31] indicate that the Hamiltonian for B -> K ^ f i +fi~ (and B s —>p +p~) is best described by the SM and new physics (NP) terms « E M (mb)
_ 4 GY
^SM +N p (^
091801-1
SR+P )
V2 + H.c.,
tb ts
4/r (4)
© 2015 American Physical Society
PHYSICAL
PRL 114, 091801 (2015)
REVIEW
where C4 = C?SM + C((NP are Wilson coefficients with Cg = - C j 0 > 0. The NP contributions to these coefficients are opposite in sign to the SM contributions, i.e. the lepton current is approximately of V —A form. Therefore, we assume that the third-generation interaction giving rise to the NP part of this Hamiltonian is ^NP = Gb,LYxb,LT'LyxtlL,
(5)
where G is a new-physics Fermi constant (G = 1/ A^p These fields are related to the masseigenstate (unprimed) fields by unitary matrices Ud L and Up
=
b'L =
G dL2id Li,
f L3
=
t
'L
—
'2 2
1=1
1=1
(6) In particular, the NP interaction responsible for the RK deficit is ^ N
p
( ^ “ *■
$B+B
)
= G[Ud p33UdL32\U'in \2bLY>'SifiiY^RL + H.c.].
(7)
The hierarchy of the Cabibbo-Kobayashi-Maskawa matrix for quarks and the apparent preference of the new physics for muons over electrons suggest that | j12 « ; | u f 3212 «c
(Up33)2 = 1. These expectations can be tested in searches for B -» Kfie vs K ^ t . Since the coefficient of the SM term in Eq. (4) is positive, we assume that GUd L32 < 0. The reduction of the SM strength in Eq. (4) by this interaction is also supported by the LHCb measurement of the quantity P'5 in B° —> K*°b +b ~ angular distributions in the low-#2 region. Integrated over 1.0 < q2 < 6.0 GeV2, the P'5 deficit amounts to 2.5 K ^ b +R~ amplitude is
week ending
LETTERS
6 MARCH 2015
Note that the NP contribution to Cg is negligible in our model. The LHCb result for RK in Eq. (1) yields the useful ratio P np —
/?NP
—0.159 +- 00.069 . 070 '
Psm + ^NP
(9)
Then the branching ratio for B+ -+ K+B±e^ (summed over lepton charges) is given by Tjf
B(B^
K +pi± e :¥)
=
2p2
u L3\
NP tje
B(B+ -> K+n+fi"
u L32
Tjf
L3\ = (2.16+2j 04) U Tjf
x 10-
(10)
U L32
where we used B(B+ -+ K+b +b ~) = (4-29 ± 0.22) x 10-7 for the branching ratio integrated over q2 [8], The current limit B(B+ -+ K +B±e T) < 9.1 x 10"8 [33] gives the weak bound 1 ^ 3 1 /^ 3 2 ^ 3 .7 .
( 11)
Because the primary interaction 7fNP is in the third generation, the decay B + -+ K+b^ t * may be more inter esting,
j jf B{B+
K +b ± t t ) S 2
U L33
U1 L32
2b {b +
K+b +B~)-
(12) The current limit, B(B+ -+ K +b ±t4") < 4.8 x 10-5 [33], gives
K 33/U [ 32\< 8 5 .
(13)
This is a crude estimate. It is performed by keeping only the terms proportional to |C910|2 in the decay rate. It also neglects the difference in the q2 range and the phase space for the bp and PT modes. These approximations affect B -+ K t +t ~ even more. For this mode, only the weak limit B(B -* K t+t ~) < 3.3 x 10-3 has been set [34], The Bs decays to a pair of oppositely charged leptons provide an interesting correlation with B -» The only observed mode is
B(BS -> B+fOexp = (2-8/oi) x 1 0 '9 = (0-77 ± 0.20) x B(B s -> b +P -) sm > (14)
■fi J__ Tjd* Tjd ^
An \Tjg [2
2 U ^"33 u L32 1u L32 1
4 GF ----- - V W , a/ 2 tb,s
a E M (m b)
An
C9'
(8)
where the experimental value is an average of LHCb and CMS measurements with full Run I statistics [35], while the SM value is B{BS b +P~)sm = (3.65 ± 0 .2 3 ) x 10~9 [36]. The measurement is consistent with the SM prediction but also with Tipsp in Eq. (7) and the value of p NP in Eq. (9). Thus, our model implies the triple correlation
091801-2
PHYSICAL
PRL 114, 091801 (2015) „ B (B +
week ending 6 MARCH 2015
LETTERS IT* TJ?
)exp
,
,
U L3l
- B ( B + ^ K ^ - ) s u ~ t 3 ( B s - > ^ - ) SM
[
’
u L32
K + B + B
)exp ^ B ( B .s
REVIEW
This relation identifies the numerical factor on the right of Eq. (14) with Eq. (1) and stresses the importance of a more accurate measurement of B(BS -> /i+fG). The only reported LFV limit, B(BS -* ^ e *) < 1.1 x 10~8 [37], gives |f7£31/[/£ 32| < 25, weaker than the bound 3.7 from B+ -*■ K+/ie. We hope that searches for this mode and for Bs —>fit in LHC Run II can provide much improved limits. (We have not examined the interesting possibility that phases in the off-diagonal Ud-f matrix elements may induce new CP-violating effects, especially in B s decays.) Measurements exist for B(B° -* K°F+d~) ( / = e,\i) and B(BS -> c/yU/C). These are comparable to but less precise than B(B+ —►K+f +£~). Similarly, limits on B° -*■ zf+z?_ and the LFV decays B° -* and tf±tf’T, including r modes, give no more stringent limits than those for B+ decays. The amplitudes for B° -*■ f +f ~ and are suppressed by Vrd/ V ts and Ud3l/U d32 relative to the corresponding Bs decays. (Our interaction 7fNP, as well as those mentioned earlier, will induce rare LFV decays for kaons. Such a study is worthwhile because of the considerable interest worldwide in experiments with intense beams, but it is beyond the scope of this Letter.) Finally, the operator 'H-tfpiji T d — * d T c) = G \U U \2dLYW {U i*3lU{32eLY^ L + n .c .)
(16)
induces ^ -*■ e conversion in nuclei. (Our model Hamiltonian 7fNP can also induce /( ey. Simply closing up the quark line and emitting a photon from it does not produce the desired operator, oc e G m ^ e ^ fiF ^ . Virtual exchanges of electroweak or Higgs bosons and/or a mass insertion must occur between the quark and lepton lines to change the muon chirality. We expect that such an operator is too weak to give an interesting limit.) The limit on the strength of this operator from conversion in titanium is (see Table 3.6 in Ref. [38]) G\UdL3l\2\U%2U i31\ < ^ x
(8.5 x 1(T7).
(17)
Using Pnp PsM
nG Ud*33Ud L32\UeL32\V2G FaEMV*lbVtsC*9
together with \Vtb\ — 1, \UdL32\ — |V(J| = 0.043, \UdL3l \ — \Vrd\ ¥ 0.0084 [33], and aEM(mb) = 1/133, we obtain the weak limit
75 < i— r = 18,
Cg
(19)
where |Cg| = 4.07 [39]. The Mu2e experiment at Fermilab is designed to be sensitive to an interaction strength 100 times smaller than that in Eq. (17). Under our assumptions, Mu2e would then be able to probe |t/£31/t/£ 32| > 0.2. Summing up: The interesting new results on B —► K (*)li+H and R k from LHCb, if correct, tell us that there are lepton number nonuniversal interactions. Therefore, there must also be lepton-flavor-violating interactions, and there is no known reason these should be very much weaker than the nonuniversal ones. Limits from searches for B -»• and Bs -*■ are not far above interesting ranges for LFV mixing-angle parameters. LHCb’s results make searches for these and other rare processes well worth pursuing. We are grateful for stimulating conversations with Adam Martin and Edwige Toumefier. We have also benefited from discussions with Wolfgang Altmannshofer, Francesco Dettori, Federico Mescia, Jim Miller, Marco Nardecchia, Chris Quigg, David Straub, Danny van Dyk, and Roman Zwicky. S. L. G.’s research is supported in part by the U.S. Department of Energy under Grant No. DE-SC0010025. K. L. gratefully acknowledges support of this project by the Labex ENIGMASS. He thanks Laboratoire d’Annecy-leVieux de Physique Theorique (LAPTh) for its hospitality and the CERN Theory Group for support and hospitality. K. L.’s research is supported in part by the U.S. Department of Energy under Grant No. DE-SC0010106.
[email protected] [email protected] [email protected] [1] R. Aaij et al. (LHCb Collaboration), Test of Lepton Universality Using B+ -*■ K+tf+tf~ Decays, Phys. Rev. Lett. 113, 151601 (2014). [2] C. Bobeth, G. Hiller, and G. Piranishvili, Angular distri butions of B — K f d decays, J. High Energy Phys. 12 (2007) 040. [3] C. Bouchard, G. P. Lepage, C. Monahan, H. Na, and J. Shigemitsu (HPQCD Collaboration), Standard Model Pre dictions for B -* with Form Factors from Lattice QCD, Phys. Rev. Lett. Ill, 162002 (2013). [4] G. Hiller and F. Kruger, More model independent analysis of b -»• s processes, Phys. Rev. D 69, 074020 (2004). [5] J.-T. Wei et al. (BELLE Collaboration), Measurement o f the Differential Branching Fraction and Forward-Backward Asymmetry for B to Phys. Rev. Lett. 103, 171801 (2009). [6] J. Lees et al. (BABAR Collaboration), Measurement of branching fractions and rate asymmetries in the rare decays B -* K ^ l+ l- , Phys. Rev. D 86, 032012 (2012).
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REVIEW
[7] R. Aaij et al. (LHCb Collaboration), Differential branching fraction and angular analysis of the B+ —►K +p +p~ decay, J. High Energy Phys. 02 (2013) 105. [8] R. Aaij et al. (LHCb Collaboration), Differential branching fractions and isospin asymmetries of B -* K {^\i~pT decays, J. High Energy Phys. 06 (2014) 133. [9] C. Bobeth, G. Hiller, and D. van Dyk, More benefits of semileptonic rare B decays at low recoil: CP violation, J. High Energy Phys. 07 (2011) 067. [10] C. Bobeth, G. Hiller, D. van Dyk, and C. Wacker, The decay B -*■ Ktf+t?~ at low hadronic recoil and model-independent AB = 1 constraints, J. High Energy Phys. 01 (2012) 107. [11] C. Bobeth, G. Hiller, and D. van Dyk, General analysis of B ->■ decays at low recoil, Phys. Rev. D 87, 034016 (2013). [12] S. Descotes-Genon, J. Matias, and J. Virto, Understanding the B -* K*/i+p~ anomaly, Phys. Rev. D 88,074002 (2013). [13] W. Altmannshofer and D. M. Straub, New physics in B Eur. Phys. J. C 73, 2646 (2013). [14] F. Beaujean, C. Bobeth, and D. van Dyk, Comprehensive Bayesian analysis of rare (semi)leptonic and radiative B decays, Eur. Phys. J. C 74, 2897 (2014). [15] T. Hurth and F. Mahmoudi, On the LHCb anomaly in B -> F f + r , J. High Energy Phys. 04 (2014) 097. [16] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Quark flavor transitions in L^ - L , models, Phys. Rev. D 89, 095033 (2014). [17] R. Gauld, F. Goertz, and U. Haisch, An explicit Z'-boson explanation of the B -> K*p.+p~ anomaly, J. High Energy Phys. 01 (2014) 069. [18] R. Gauld, F. Goeitz, and U. Haisch, Minimal Z' explan ations of the B -> K*p+p~ anomaly, Phys. Rev. D 89, 015005 (2014). [19] A. Datta, M. Duraisamy, and D. Ghosh, Explaining the B -* K*p+pT data with scalar interactions, Phys. Rev. D 89, 071501 (2014). [20] A. J. Buras and J. Girrbach, Left-handed Z' and Z FCNC quark couplings facing new b -*■ sp+p~ data, J. High Energy Phys. 12 (2013) 009. [21] A. J. Buras, F. De Fazio, and J. Girrbach, 331 models facing new b —>sp+p~ data, J. High Energy Phys. 02 (2014) 112. [22] R. R. Horgan, Z. Liu, S. Meinel, and M. Wingate, Calcu lation of B° —>K*°p+p~ and B{)s —>
PRL 114, 091801 (2015)
REVIEW
week ending 6 MARCH 2015
LETTERS
Lepton Flavor Violation in B Decays? Sheldon L. Glashow,1’ Diego Guadagnoli,2’* and Kenneth Lane1’* 1Department o f Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA 2Laboratoire d ’Annecy-le-Vieux de Physique Theorique UMR5108, CNRS et Universite de Savoie, BP 110, F-74941 Annecy-le-Vieux Cedex, France (Received 15 November 2014; published 3 March 2015) The LHCb Collaboration’s measurement of RK = B{B+ —►K +ft+/x~) / B(B+ —> K +e+e~) lies 2.6 p +p~)exp/ B(BS -* q+fi~)SM = R k = 0.75, is compatible with recent measurements of these rates. We stress the importance of searches for lepton flavor violations, especially for B -* Kqe, Kfir, and Bs -* jie, fir. DOI: 10.1103/PhysRevLett. 114.091801
PACS numbers: 12.60.Cn, 11.30.Hv, 13.20.He
The LHCb Collaboration recently measured the ratio of decay rates for B+ -*■ K+tf+tf~ ( f = fi. e), obtaining [1]
rk
=
= 0-745-o.0794°(stat) ± 0.036(syst). ( 1)
This result is a 2.6a deficit from the Standard Model (SM) expectation, RK = 1 + 0 { 10-4) [2-4], Previous measurements of RK by the Belle and BABAR Collaborations [5,6] had considerably greater uncertainties but were consistent with the SM prediction. The LHCb determination was made for 1 < q2 = M j f < 6 GeV2 in order to be well below the radiative tail of the J/yr. LHCb also measured the B -> KfB fT branching ratio in this q1 range [7]. The updated result, based on its full Run I data set of 3 fb_1, is [8] B(B+ -> K +q+ q-)[h6] = (1.19 ± 0.03 ± 0.06) x 10~7.
(2) The SM prediction [9-11] B(B+ -> K W ) f M
= (L75+q2°) x 10-7
(3)
is about 45% higher. Whether this is the cause of the R K deficit is suggestive, but not certain. Another possibility is a problem in the B -> Ke+e~ measurement, since LHCb performs better in the fifi than in the ee channel, the latter being hampered by bremsstrahlung and poorer statistics [1], On the other hand, bremsstrahlung effects largely cancel in the experimental observable, and results for B -> K e+e~ are consistent with SM expectations. The LHCb results for RK and for anomalies in the B -> K*n+n~ angular distributions (discussed below) have 0031-9007/15/114(9)/091801(4)
attracted much theoretical attention [12-29]. References [13,16-21,23-25,28] proposed the existence of particles at or above 1 TeV that induce new and nonuniversal lepton interactions. Such interactions violate lepton flavor unless the leptons involved are chosen to be mass eigenstates—an unjustifiable act of fine tuning. No known symmetry principle protects lepton flavor conservation in the presence of lepton nonuniversality. Thus, LHCb’s reported value of R K implies, e.g., that B -» and B -*■ must occur at rates much higher than would occur in the SM due to tiny neutrino masses. We urge that these and other lepton flavor violations (LFVs) be sought with renewed vigor in LHC Run II and elsewhere. To illustrate the sort of LFV processes that might be seen in B decays, the limits that currently exist on lepton mixing, and the potential for discovery of LFV, we consider a simple but well-motivated interaction that can account for the known features of the R K deficit, is consistent with existing limits on LFV, and produces effects that may be observable in LHC Run II. The LHCb data on B -* suggest that, despite the detector’s superior measurement of muons vis-a-vis electrons, the R K result is due to a ~25% deficit in the muon channel. If so, then LFV is larger for muons than for electrons. This is naturally accounted for by a third-generation interaction of the type that would be expected, e.g., in topcolor models [30], Furthermore, recent theoretical analyses [13,27,31] indicate that the Hamiltonian for B -> K ^ f i +fi~ (and B s —>p +p~) is best described by the SM and new physics (NP) terms « E M (mb)
_ 4 GY
^SM +N p (^
091801-1
SR+P )
V2 + H.c.,
tb ts
4/r (4)
© 2015 American Physical Society
PHYSICAL
PRL 114, 091801 (2015)
REVIEW
where C4 = C?SM + C((NP are Wilson coefficients with Cg = - C j 0 > 0. The NP contributions to these coefficients are opposite in sign to the SM contributions, i.e. the lepton current is approximately of V —A form. Therefore, we assume that the third-generation interaction giving rise to the NP part of this Hamiltonian is ^NP = Gb,LYxb,LT'LyxtlL,
(5)
where G is a new-physics Fermi constant (G = 1/ A^p These fields are related to the masseigenstate (unprimed) fields by unitary matrices Ud L and Up
=
b'L =
G dL2id Li,
f L3
=
t
'L
—
'2 2
1=1
1=1
(6) In particular, the NP interaction responsible for the RK deficit is ^ N
p
( ^ “ *■
$B+B
)
= G[Ud p33UdL32\U'in \2bLY>'SifiiY^RL + H.c.].
(7)
The hierarchy of the Cabibbo-Kobayashi-Maskawa matrix for quarks and the apparent preference of the new physics for muons over electrons suggest that | j12 « ; | u f 3212 «c
(Up33)2 = 1. These expectations can be tested in searches for B -» Kfie vs K ^ t . Since the coefficient of the SM term in Eq. (4) is positive, we assume that GUd L32 < 0. The reduction of the SM strength in Eq. (4) by this interaction is also supported by the LHCb measurement of the quantity P'5 in B° —> K*°b +b ~ angular distributions in the low-#2 region. Integrated over 1.0 < q2 < 6.0 GeV2, the P'5 deficit amounts to 2.5 K ^ b +R~ amplitude is
week ending
LETTERS
6 MARCH 2015
Note that the NP contribution to Cg is negligible in our model. The LHCb result for RK in Eq. (1) yields the useful ratio P np —
/?NP
—0.159 +- 00.069 . 070 '
Psm + ^NP
(9)
Then the branching ratio for B+ -+ K+B±e^ (summed over lepton charges) is given by Tjf
B(B^
K +pi± e :¥)
=
2p2
u L3\
NP tje
B(B+ -> K+n+fi"
u L32
Tjf
L3\ = (2.16+2j 04) U Tjf
x 10-
(10)
U L32
where we used B(B+ -+ K+b +b ~) = (4-29 ± 0.22) x 10-7 for the branching ratio integrated over q2 [8], The current limit B(B+ -+ K +B±e T) < 9.1 x 10"8 [33] gives the weak bound 1 ^ 3 1 /^ 3 2 ^ 3 .7 .
( 11)
Because the primary interaction 7fNP is in the third generation, the decay B + -+ K+b^ t * may be more inter esting,
j jf B{B+
K +b ± t t ) S 2
U L33
U1 L32
2b {b +
K+b +B~)-
(12) The current limit, B(B+ -+ K +b ±t4") < 4.8 x 10-5 [33], gives
K 33/U [ 32\< 8 5 .
(13)
This is a crude estimate. It is performed by keeping only the terms proportional to |C910|2 in the decay rate. It also neglects the difference in the q2 range and the phase space for the bp and PT modes. These approximations affect B -+ K t +t ~ even more. For this mode, only the weak limit B(B -* K t+t ~) < 3.3 x 10-3 has been set [34], The Bs decays to a pair of oppositely charged leptons provide an interesting correlation with B -» The only observed mode is
B(BS -> B+fOexp = (2-8/oi) x 1 0 '9 = (0-77 ± 0.20) x B(B s -> b +P -) sm > (14)
■fi J__ Tjd* Tjd ^
An \Tjg [2
2 U ^"33 u L32 1u L32 1
4 GF ----- - V W , a/ 2 tb,s
a E M (m b)
An
C9'
(8)
where the experimental value is an average of LHCb and CMS measurements with full Run I statistics [35], while the SM value is B{BS b +P~)sm = (3.65 ± 0 .2 3 ) x 10~9 [36]. The measurement is consistent with the SM prediction but also with Tipsp in Eq. (7) and the value of p NP in Eq. (9). Thus, our model implies the triple correlation
091801-2
PHYSICAL
PRL 114, 091801 (2015) „ B (B +
week ending 6 MARCH 2015
LETTERS IT* TJ?
)exp
,
,
U L3l
- B ( B + ^ K ^ - ) s u ~ t 3 ( B s - > ^ - ) SM
[
’
u L32
K + B + B
)exp ^ B ( B .s
REVIEW
This relation identifies the numerical factor on the right of Eq. (14) with Eq. (1) and stresses the importance of a more accurate measurement of B(BS -> /i+fG). The only reported LFV limit, B(BS -* ^ e *) < 1.1 x 10~8 [37], gives |f7£31/[/£ 32| < 25, weaker than the bound 3.7 from B+ -*■ K+/ie. We hope that searches for this mode and for Bs —>fit in LHC Run II can provide much improved limits. (We have not examined the interesting possibility that phases in the off-diagonal Ud-f matrix elements may induce new CP-violating effects, especially in B s decays.) Measurements exist for B(B° -* K°F+d~) ( / = e,\i) and B(BS -> c/yU/C). These are comparable to but less precise than B(B+ —►K+f +£~). Similarly, limits on B° -*■ zf+z?_ and the LFV decays B° -* and tf±tf’T, including r modes, give no more stringent limits than those for B+ decays. The amplitudes for B° -*■ f +f ~ and are suppressed by Vrd/ V ts and Ud3l/U d32 relative to the corresponding Bs decays. (Our interaction 7fNP, as well as those mentioned earlier, will induce rare LFV decays for kaons. Such a study is worthwhile because of the considerable interest worldwide in experiments with intense beams, but it is beyond the scope of this Letter.) Finally, the operator 'H-tfpiji T d — * d T c) = G \U U \2dLYW {U i*3lU{32eLY^ L + n .c .)
(16)
induces ^ -*■ e conversion in nuclei. (Our model Hamiltonian 7fNP can also induce /( ey. Simply closing up the quark line and emitting a photon from it does not produce the desired operator, oc e G m ^ e ^ fiF ^ . Virtual exchanges of electroweak or Higgs bosons and/or a mass insertion must occur between the quark and lepton lines to change the muon chirality. We expect that such an operator is too weak to give an interesting limit.) The limit on the strength of this operator from conversion in titanium is (see Table 3.6 in Ref. [38]) G\UdL3l\2\U%2U i31\ < ^ x
(8.5 x 1(T7).
(17)
Using Pnp PsM
nG Ud*33Ud L32\UeL32\V2G FaEMV*lbVtsC*9
together with \Vtb\ — 1, \UdL32\ — |V(J| = 0.043, \UdL3l \ — \Vrd\ ¥ 0.0084 [33], and aEM(mb) = 1/133, we obtain the weak limit
75 < i— r = 18,
Cg
(19)
where |Cg| = 4.07 [39]. The Mu2e experiment at Fermilab is designed to be sensitive to an interaction strength 100 times smaller than that in Eq. (17). Under our assumptions, Mu2e would then be able to probe |t/£31/t/£ 32| > 0.2. Summing up: The interesting new results on B —► K (*)li+H and R k from LHCb, if correct, tell us that there are lepton number nonuniversal interactions. Therefore, there must also be lepton-flavor-violating interactions, and there is no known reason these should be very much weaker than the nonuniversal ones. Limits from searches for B -»• and Bs -*■ are not far above interesting ranges for LFV mixing-angle parameters. LHCb’s results make searches for these and other rare processes well worth pursuing. We are grateful for stimulating conversations with Adam Martin and Edwige Toumefier. We have also benefited from discussions with Wolfgang Altmannshofer, Francesco Dettori, Federico Mescia, Jim Miller, Marco Nardecchia, Chris Quigg, David Straub, Danny van Dyk, and Roman Zwicky. S. L. G.’s research is supported in part by the U.S. Department of Energy under Grant No. DE-SC0010025. K. L. gratefully acknowledges support of this project by the Labex ENIGMASS. He thanks Laboratoire d’Annecy-leVieux de Physique Theorique (LAPTh) for its hospitality and the CERN Theory Group for support and hospitality. K. L.’s research is supported in part by the U.S. Department of Energy under Grant No. DE-SC0010106.
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