Lepton flavor violation in decays

PHYSICAL REVIEW D 74, 035010 (2006)

Lepton flavor violation in decays Chuan-Hung Chen1,3,* and Chao-Qiang Geng2,3,† 1

Department of Physics, National Cheng-Kung University, Tainan, 701 Taiwan Department of Physics, National Tsing-Hua University, Hsin-Chu, 300 Taiwan 3 National Center for Theoretical Sciences, Taiwan (Received 7 June 2006; published 28 August 2006)

2

We study the lepton flavor violation (LFV) in tau decays in the framework of the supersymmetric seesaw mechanism with nonholomorphic terms for the lepton sector at a large tan. In particular, we analyze two new decay modes ! ‘f0 980 and ! ‘K K arising from the scalar boson exchanges contrast to ! ‘0 from the pseudoscalar ones. We find that the decay branching ratios of the two new modes could be not only as large as the current upper limits of O107 , but also larger than those of ! ‘0 . Experimental searches for the two modes are important for the LFV induced by the scalarmediated mechanism. In addition, we show that the decay branching ratios of ! ‘ are related to those of ! ‘ and ! ‘f0 980. DOI: 10.1103/PhysRevD.74.035010

PACS numbers: 12.60.Jv, 13.35.Dx

In the standard model (SM), since the neutrinos are regarded as massless particles, the processes associated with lepton flavors are always conserved. Inspired by the discoveries of nonzero neutrino masses [1,2], it has been studied enormously how to generate the neutrino masses which are less than a few eV. By supplementing with singlet right-handed Majorana neutrinos with masses MR required to be around the scale of unified theory, it is found that the seesaw mechanism is one of the natural ways [3] to obtain the small neutrino masses. Accordingly, in non-SUSY models, it is easy to understand that the effects of the lepton flavor violation (LFV) are suppressed by 1=MR . However, in models with SUSY, due to the nondiagonal neutrino mass matrix, the flavor conservation in the slepton sector at the unified scale will be violated at the MR scale via renormalization [4 –6]. The flavor violating effects could propagate to the electroweak scale so that instead of 1=MR , the suppression of the LFV could be 1=MSUSY with MSUSY O TeV being the typical mass of the SUSY particle. Consequently, the lepton flavor violating processes, such as ‘ i ! ‘j and ‘i ! ‘j ‘k ‘k become detectable at the low energy scale. The LFV has been extensively studied in the literature. For example, ! , B ! e; , and the e; conversions can be found in Refs. [7–12], while that to the detection of the LFV in colliders is given in Ref. [13]. In the large tan region, it has been pointed out that the nonholomorphic Yukawa interactions [14 –17] play very important roles for flavor changing neutral currents (FCNCs) in the quark sector. In the SUSY-seesaw model, the nonholomorphic terms [6] in the lepton sector naturally induce the LFV due to the Higgs couplings. It has been shown that the contribution to the decay of ! 3 from *Electronic address: [email protected] † Electronic address: [email protected]

1550-7998= 2006=74(3)=035010(6)

the Higgs-mediated LFV at large tan could be much larger than that from ! ! [6,18]. Recently, the experimental limits on the radiative decays of ! ‘ (‘ e; ) have been improved from O106 [19] to O107 [20,21]. Moreover, the sensitivity of probing the LFV in decays with single pseudoscalar (P) or vector (V) and double mesons in the final states, i.e., ! ‘P; V and ! ‘PP, have also reached O107 [22]. In this paper, we will simultaneously analyze ! ‘ and ! ‘X, where X are , 0 , , f0 980, 600, and K K , respectively, in the Higgs-mediated mechanism. In particular, we would like to check whether it is possible to have large rates for the processes beside the mode of ! 3. Note that the decays of ! ‘S with S f0 980 and 600 and ! ‘; K K have not been explored previously based on the Higgs-mediated mechanism in the literature, while ! ‘P have been studied in Refs. [7,9,23]. We start with the Higgs-mediated mechanism. It is known that, by the induced slepton flavor mixing, the effective Lagrangian with induced nonholomorphic terms for the Higgs bosons coupling to leptons is given by [6]

Leff E Ri Yi ij Hd0 1 ij 2 Iij Hu0 ELj H:c: E R M0 EL H:c:; (1) ‘

where Y denotes the diagonalized Yukawa matrix of leptons, Iij m2L~ ij =m20 and 12 is related to the induced lepton flavor conserving (violating) effect, expressed by [6]

035010-1

© 2006 The American Physical Society

CHUAN-HUNG CHEN AND CHAO-QIANG GENG


1

M1 2f1 M12 ; m2‘~ ; m2‘~ f1 M12 ; 2 ; m2‘~ 2f1 M12 ; m2‘~ ; m2‘~ 2 M2 f1 2 ; m2‘~ 2f1 2 ; m2 ~‘ ; M22 ; L R L L R L 8 8

1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 M1 m0 2f2 M1 ; m‘~ ; m~L ; m~R f2 ; m‘~ ; m~L ; M1 M2 m0 f2 ; m‘~ ; m~L ; M22 2 ’ L L L 8 8 2 2 2 2 (2) 2f2 ; m ~‘ ; m ~ ; M2 ; 1 ’

where M1;2 are the masses of gauginos from the soft SUSYbreaking terms, stands for the mixing of Hu and Hd ,

and 12 g212 =4 with g12 corresponding to the gauge coupling of the U1SU2 symmetry. Because of the nonholomorphic term 2 Eij , the lepton mass matrix is not diagonal anymore. Consequently, after rediagonalizing the lepton mass matrix, the lepton flavor changing neutral interactions through the Higgs bosons appear. Since the nonholomorphic terms are expected to be much less than unity, to obtain the LFV, we take the unitary matrices used for diagonalizing lepton mass matrix to be ULR 1 LR as a leading expansion of 2 Eij , where LR are 3 3 matrices. From Eq. (15) and m2L~ ij m2L~ ji , we may set L R . Hence, the diagonal mass matrix in Eq. (1) could be obtained by UM‘0 Uy 1 M‘0 1 M‘dia ; where M‘dia is the physical mass matrix of the lepton with the diagonal elements being M‘dia ii me ; m ; m . At the leading order, we get ij

M‘0 ij M‘0 jj

M‘0 ii

ij eff

p m‘i Cij 2GF 1=2 ‘iR ‘jL sin H 0 cos2 cos h0 iA0 H:c:;

xy lnx=y yz lny=z zx lnz=x ; f1 x; y; z x yy zz x w lnw cyclic; f2 w; x; y; z w xw yw z

M‘0 ii M‘dia ii ;

H

i j:

In terms of the physical mass eigenstates of the Higgs bosons, represented by [24]

(3)

where m‘i is the mass of the ith flavor lepton and Cij 2 Iij =1 1 2 Iii tan2 . From Eq. (3), we see that the decays of ! ‘P only pick up the contributions from the pseudoscalar boson A0 , while ! ‘S and ! ‘PP are governed by both scalar bosons H 0 and h0 due to the parity properties. In our following analysis, we only concentrate on the processes associated with the productions of ss and pairs to avoid small Higgs couplings. We choose the decays of ! ‘X with X , 0 , f0 980600, and K K as the representative modes. For ! ‘ , the formalisms for the decay rates dictated by scalar and pseudoscalar bosons are given by ! ‘ ’ c‘

G2F m2 m7 jC‘ j2 3 29 3 cos6

cs sc m2h m2H

2

sin 2 ; m2A (4)

where c‘ 3=2 and 1 with ‘ and e, cs cos sin , and sc sin cos , respectively. To study the production of 0 , we adopt the quark-flavor scheme, defined by [25] cos sin q ; (5) sin cos 0 s p where q uu dd= 2 and s ss. From h0jq 0 5 q0 jq0 pi fq0 p , the mass of qs can be p 5 djq i (m2ss 5 u md d expressed by m2qq 2 h0jmu u fq

2 fs

h0jms s5 sjs i). If we neglect the q contribution due to small mu;d , the decay rates for ! ‘ can be written as

1 Re Hd0 vd p cos H0 sin h0 ; 2 1 ReHu0 vu p sin H0 cos h0 ; 2 1 ImHd0 p cosG0 sinA0 ; 2 1 ImHu0 p sinG0 cosA0 ; 2

! ‘ ’

where is the mixing angle of the two CP-even neutral scalars, the interactions for the LFV via the Higgsmediated mechanism are expressed by

m2 2 G2F m3 jC‘ j2 6 m2 2 tan sinfs ss : (6) 1 64 m2A m2

Similarly, the rate for ! ‘0 is given by 1 m2 0 =m2 2 ! ‘0 cot2 : 2 2 ! ‘ 1 m =m

(7)

For ! ‘f0 980600 decays, although the quark contents of f0 980 and 600 are still uncertain, we

035010-2

LEPTON FLAVOR VIOLATION IN DECAYS


adopt two quark contents to describe the states. In terms of the notations in Refs. [26,27], the isoscalar states f0 980 and 600 are described by jf0 980i cos jssi and j600i with sin jnni sin jssi cos jnni p nn uu dd= 2 and being the mixing angle. The decay constants are defined as hf0s jssj0i

mf0 f~sf0 ;

hs jssj0i

m f~s ;

(8)

d ! ‘K K G2F m3 jC‘ j2 sc 2 K K 2 cs m ’ f s s dQ2 m2h m2H 28 3 cos6 Q2 2 4m2 1=2 1 2 1 2K : (13) m Q From Eqs. (4), (6), and (9), it is interesting to see that the various decay rates mediated by the Higgs bosons have the relationship

where f0s and s represent the ss component in f0 980 and 600, respectively. As a result, the decay rates of ! ‘f0 980 are given by ! ‘f0 980 ’

G2F m3 jC‘ j2 ms mf0 f~sf0 cos 2 16cos6 m2f 2 cs sc 2 2 2 1 20 : mh mH m

(9)

On the other hand, the rates for ! ‘600 can be obtained by ~s m f tan 2 1 m2 =m2 2 ! ‘600 : ’ ! ‘f0 980 1 m2f0 =m2 mf0 f~sf0 (10) For the three-body decays of ! ‘K K , the associated hadronic effects are much more complicated and unclear. Nevertheless, the uncertainties could be fixed by the B decays, such as B ! KKK. The related form factor including resonant and nonresonant effects is defined by [28] hK p1 K p2 jssj0i fsK

K

X S

! ‘

where C sin2 sinfs m2ss =22 1 m2 =m2 2 and Cf0 ms mf0 f~sf0 cos 2 1 m2f0 =m2 2 . We now consider the radiative modes of ! ‘. At the large tan scenario, the dominant contributions to the decays are illustrated in Fig. 1. To simplify the estimations, we use the mass insertion method to formulate the decay amplitudes. The induced LFVs in the slepton mass matrix can be approximately written as [4,5,31] 1 2 Y y Y 2Ay A ln MU ; (15) m2L~ ij ’ 6m ij 0 MR 42 where m0 , Y , and A denote the typical initial soft SUSYbreaking mass of the slepton, the neutrino Yukawa couplings, and the trilinear soft SUSY-breaking effects, respectively, at the unified scale of MU . From Fig. 1 and Eq. (15), the effective interactions for ! ‘ are given by

Q2

G ki k AR 1 5 p; T pF em k‘p 2 (16)

mS f~sS gS!KK fsNR ; m2S Q2 imS S (11)

where

where S stands for the possible scalar meson state, mS f~sS hSjssj0i, gS!KK denotes the strong coupling for S ! KK, and v Q2 1 NR 1 2 fs 3FNR 2FNR v 2 ln 2 ; 3 Q (12) 12 x1 x12 Q2 1 12 2 FNR 4 ln 2 ; Q2 Q

AR

X M2 m2W 2 tanm GS ; ‘ ~ L 42 M22 ~ S‘;~

(17)

‘

1 tan2 W fn x‘~L fn x~L ; m2‘~ m2‘~ m2~L m2~L L L fc x ~‘ fc x ~ 4 ; 2 m ~‘ m2 ~ m2 ~‘ m2 ~

G‘~ G ~‘

v m2K m2 =ms md , x11 3:26 GeV2 , 5:02 GeV2 , x21 0:47 GeV2 , and x22 0. It is found

with x12 that only f0 980 and f0 1530 have the largest couplings to the KK pair [29]. Note that in calculating B ! KKK [28], the factorization approach in Ref. [30] has been used. In our numerical estimations, we will only consider these two scalar contributions. The differential decay rates as a function of the invariant mass in the KK system are given by

c‘ m2 m4 ! ‘ ! ‘f0 980 ; C Cf0 3 25 2 (14)

(18)

1 fn x 1 x2 2x lnx; 1 x3 1 fc x 3 4x x2 2 lnx; 21 x3 with xS M22 =m2S [5]. Here, we have set the masses of Higgsinos and gauginos to be the same, denoted as M2 . Subsequently, the decay rates of ! ‘ are given by

035010-3



˜L

˜L R

L

H˜ d0

H˜ u0

˜

˜ R

L

H˜ d−

˜ 0 W L

˜ − W L

H˜ u−

(a)

(b)

FIG. 1. The Feynman diagrams for ! ‘ with a large tan. The crosses represent the various mixing effects.

(19)

The diagrams in Fig. 1 can also induce ! ‘ , ! ‘, and ! ‘K K when the photon is off-shell. From Eq. (16), it is easy to estimate the ratios of branching ratios (BRs) to be [5] BR ! ‘X

O em 103 ; R‘ BR ! ‘ (20) X ; ; K K : Note that it is impossible to produce modes with X being a single pseudoscalar or scalar by the dipole operators in Eq. (16). In our estimations for the modes with X and K K , we have used the hadronic matrix elements qji im f k defined by h0jq and K K Q2 , with qjK p1 K p2 i p h0jq p F q 1 2 the form factors given in Refs. [28,32], respectively. It is clear that from the current limits on BR ! ‘, BR ! ‘X are too small to be observed. We remark that other loop contributions to the decays, such as those from box diagrams, are expected to be small due to the light fermion final states. For the numerical estimations on ! ‘ and ! ‘X, we assume that M1 M2 m0 m‘~ m~ to simplify our discussions. Consequently, Eqs. (2) and (17) become 3 em

em 1 1 ; 1

; 2 16 3cos2 W sin2 W 4sin2 2 W AR

1 m2W 642 m2~

m2L~ ‘ m20

tan1 tan2 W ;

f~sf0 0:33 GeV [27] for ! ‘f0 980; 600; v 2:87 GeV, 10:4 GeV4 , f~sf0 1530 f~f0 980 gf0 1530!KK 0:33 GeV, gf0 980!KK 1:50 GeV, 3:18 GeV [28], f0 980 80 MeV, and f0 1530 1:16 GeV [29] for ! ‘K K . For simplicity, we do not distinguish the difference between Y y Y e and Y y Y , i.e., m2L~ e m2L~ . In Fig. 2, we present the BRs for ! ‘ as a function of the slepton mass. In comparison with the BELLE and BABAR results of BR ! < 3:1 107 [20] and 0:68 107 [21], we see clearly that m~ > 1 TeV is favorable. The BRs of ! ‘ as a function of the pseudoscalar mass are displayed in Fig. 3(a). From Eq. (7), we have BR ! ‘0 0:93BR ! ‘. The BRs of ! ‘f0 980 and ! ‘K K as a function of MH cs=m2h sc=m2H 1=2 are shown in Figs. 3(b) and 3(c), respectively. In terms of Eq. (10), we get BR ! ‘600 0:2BR ! ‘f0 980. In addition, from Eq. (14), we obtain that BR ! ‘ ’ 0:33BR ! ‘ 1:6BR ! ‘f0 980 . Clearly, all ! ‘X modes except ! ‘600 are suitable to search for the LFV. Finally, it is worth mentioning that if we take the decoupling limit, i.e., mH mA and ! =2 [24], leading to MH mH , we get ! ‘f0 980: ! ‘ : ! ‘ 1:3:0:36c‘ :1.

−7

em 2 5 G m jA j2 : 2 F R

BR(τ→lγ)10

! ‘

(21)

respectively. If we regard Ay A in Eq. (15) as Ay A‘ m20 Y y Y ‘ m20 O1, we get m2L~ ‘ =m20 2 8=4 lnMu =MR . Thus, we find that C‘ are insensitive to the SUSY-breaking scale and the decays of ! ‘ and ! ‘X are only sensitive to the masses of the slepton and Higgs bosons, respectively. In calculating the numerical values, we set GUR 1019 1014 GeV and tan 60. Other parameters in various modes are taken to be as follows: 39 , fs 0:17 GeV, and mss 0:69 GeV for ! ‘0 [25]; 30 , ms 0:15 GeV, and f~s

60 45 30 15 0

0.6

0.9

1.2

1.5 mτ∼ (TeV)

FIG. 2. Branching ratios (in units of 107 ) for ! ‘ as a function of the stau mass.

035010-4

LEPTON FLAVOR VIOLATION IN DECAYS 2

5

1

−7

2.5

+ −

1.5

BR(τ→lK K )10

BR(τ→l f0(980))10

−7

−7

BR(τ→lη)10

PHYSICAL REVIEW D 74, 035010 (2006) 3

2 1.5

200 (a)

250 300 mA (GeV)

2 1

0.5

150

3

1

0.5

100

4

100

150

200 (b)

250 300 MH (GeV)

100

150

200 (c)

250 300 MH (GeV)

FIG. 3. Branching ratios (in units of 107 ) for (a) ! ‘ and (b)[(c)] ! ‘f0 980K K as functions of the pseudoscalar and scalar Higgs masses, respectively.

In summary, we have studied the lepton flavor violating decays through the Higgs-mediated mechanism with the nonholomorphic terms from the couplings between the Higgs bosons and leptons at the large tan. By assuming that all masses associated with SUSY breaking are the same, we have demonstrated that BRs of ! ‘ only depend on the stau mass. In the Higgs-mediated mechanism, we have shown that the BRs of the new proposed decays of ! ‘f0 980 and ! ‘K K arising from the scalar exchanges can be as large as the upper limits O107 of the current data and, moreover, they can be

larger than those of ! ‘ from pseudoscalar exchanges. We have also pointed out that ! ‘ are related with ! ‘ and ! ‘f0 980. It is clear that future experimental searches for the LFV in the leptonic and semileptonic tau flavor violating decays are important for us to identify the Higgs-mediated mechanism.

[1] S. Fukuda et al. (SuperKamiokande Collaboration), Phys. Rev. Lett. 85, 3999 (2000). [2] S. Fukuda et al. (SuperKamiokande Collaboration), Phys. Rev. Lett. 86, 5656 (2001); Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 87, 071301 (2001); 89, 011301 (2002). [3] M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity, edited by P. van Nieuwenhuizen and D. Z. Freedman (North-Holland, New York, 1979); T. Yanagida, in Proceedings of the Workshop on Unified Theory and Baryon Number in the Universe, edited by O. Sawada and A. Sugamoto (KEK, Tsukuba, 1979); R. N. Mohapatra and G. Senjanovic´, Phys. Rev. Lett. 44, 912 (1980). [4] F. Borzumati and A. Masiero, Phys. Rev. Lett. 57, 961 (1986). [5] J. Hisano et al., Phys. Rev. D 53, 2442 (1996); J. Hisano and D. Nomura, Phys. Rev. D 59, 116005 (1999). [6] K. S. Babu and C. Kolda, Phys. Rev. Lett. 89, 241802 (2002). [7] M. Sher, Phys. Rev. D 66, 057301 (2002); D. Black et al., Phys. Rev. D 66, 053002 (2002). [8] A. Dedes, J. R. Ellis, and M. Raidal, Phys. Lett. B 549, 159 (2002); A. Brignole and A. Rossi, Nucl. Phys. B701, 3 (2004). [9] A. Ilakovac, B. A. Kniehl, and A. Pilaftsis, Phys. Rev. D 52, 3993 (1995); A. Ilakovac, Phys. Rev. D 54, 5653

(1996); A. Atre, V. Barger, and T. Han, Phys. Rev. D 71, 113014 (2005); T. Fukuyama, A. Ilakovac, and T. Kikuchi, hep-ph/0506295; V. Cirigliano and B. Grinstein, hep-ph/ 0601111. R. Kitano et al., Phys. Lett. B 575, 300 (2003). A. Masiero, P. Paradisi, and R. Petronzio, Phys. Rev. D 74, 011701 (2006). M. Sher and I. Turan, Phys. Rev. D 69, 017302 (2004). A. Brignole and A. Rossi, Phys. Lett. B 566, 217 (2003); S. Kanemura et al., Phys. Lett. B 599, 83 (2004); E. Arganda et al., Phys. Rev. D 71, 035011 (2005). C. Hamzaoui, M. Pospelov, and M. Toharia, Phys. Rev. D 59, 095005 (1999). T. Banks, Nucl. Phys. B303, 172 (1988); E. Ma, Phys. Rev. D 39, 1922 (1989); R. Hempfling, Phys. Rev. D 49, 6168 (1994); K. S. Babu, B. Dutta, and R. N. Mohapatra, Phys. Rev. D 60, 095004 (1999); L. J. Hall, R. Rattazzi, and U. Sarid, Phys. Rev. D 50, 7048 (1994); F. Borzumati, G. R. Farrar, N. Polonsky, and S. D. Thomas, Nucl. Phys. B555, 53 (1999). A. Dedes and A. Pilaftsis, Phys. Rev. D 67, 015012 (2003); A. J. Buras, P. H. Chankowski, J. Rosiek, and L. Slawianowska, Phys. Lett. B 546, 96 (2002). G. Isidori and A. Retico, J. High Energy Phys. 11 (2001) 001; 09 (2002) 063. J. K. Parry, hep-ph/0510305.

This work is supported in part by the National Science Council of R.O.C. under Grants No. NSC-94-2112-M-006009 and No. NSC-94-2112-M-007-004.

[10] [11] [12] [13]

[14] [15]

[16]

[17] [18]

035010-5



[19] S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592, 1 (2004). [20] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 92, 171802 (2004); K. Hayasaka et al., Phys. Lett. B 613, 20 (2005). [21] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 95, 041802 (2005); 96, 041801 (2006). [22] Y. Enari et al. (Belle Collaboration), Phys. Lett. B 622, 218 (2005); Y. Yusa et al., hep-ex/0603036. [23] W. J. Li, Y. D. Yang, and X. D. Zhang, Phys. Rev. D 73, 073005 (2006). [24] J. F. Gunion et al., The Higgs Hunter’s Guide (AddisonWesley, Reading, MA, 1990); hep-ph/9302272 [25] T. Feldmann, P. Kroll, and B. Stech, Phys. Rev. D 58, 114006 (1998).

[26] C. H. Chen, Phys. Rev. D 67, 014012 (2003); 67, 094011 (2003). [27] H. Y. Cheng, C. K. Chua, and K. C. Yang, Phys. Rev. D 73, 014017 (2006). [28] H. Y. Cheng, C. K. Chua, and A. Soni, Phys. Rev. D 72, 094003 (2005). [29] V. V. Anisovich, V. A. Nikonov, and A. V. Sarantsev, Phys. At. Nucl. 65, 1545 (2002). [30] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999); Nucl. Phys. B591, 313 (2000). [31] J. A. Casas and A. Ibarra, Nucl. Phys. B618, 171 (2001); K. Cheung et al., Phys. Rev. D 72, 036003 (2005). [32] P. Ball, V. M. Braun, Y. Koike, and K. Tanaka, Nucl. Phys. B529, 323 (1998).

035010-6