- Email: [email protected]
Contributions of vector-like quarks to radiative B meson decay
17 August 2000
Physics Letters B 487 Ž2000. 321–326 www.elsevier.nlrlocaternpe
Contributions of vector-like quarks to radiative B meson decay Mayumi Aoki a , Eri Asakawa b, Makiko Nagashima b, Noriyuki Oshimo c , Akio Sugamoto b,c a
b
Theory Group, KEK, Tsukuba, Ibaraki 305-0801, Japan Graduate School of Humanities and Sciences, Ochanomizu UniÕersity, Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan c Department of Physics, Ochanomizu UniÕersity, Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan Received 17 May 2000; received in revised form 8 June 2000; accepted 21 June 2000 Editor: T. Yanagida
Abstract
™
We study the decay B X sg in a minimal extension of the standard model with extra up- and down-type quarks whose left- and right-handed components are both SUŽ2. singlets. Constraints on the extended Cabibbo–Kobayashi–Maskawa matrix are obtained from the experimental results for the branching ratio. Even if the extra quarks are too heavy to be detected in near-future colliders, the branching ratio could have a value which is non-trivially different from the prediction of the standard model. q 2000 Published by Elsevier Science B.V. PACS: 12.15.Ff; 12.60.-i; 13.25.Hw; 13.40.Hq
™™
™
The inclusive decay B X sg is well described by the free quark decays b sg and b sg g, owing to a large mass of the b quark. Since these decays are generated at the one-loop level of the electroweak interactions, the radiative B-meson decay is sensitive to new physics beyond the Standard Model ŽSM. w1x, such as the supersymmetric model w2x. Its branching ratio could deviate from the prediction of the SM. Or some constraints could be imposed on new physics. Experimentally, the branching
E-mail address: [email protected] ŽE. Asakawa..
ratio has been measured by CLEO w3x and ALEPH w4x as Br Ž B
™X g . s
s Ž 3.15 " 0.35 " 0.32 " 0.26 . = 10y4 ,
Ž 1.
s Ž 3.11 " 0.80 " 0.72 . = 10y4 .
Ž 2.
™
These results are consistent with the SM prediction BrŽ B X sg . s Ž3.29 " 0.33. = 10y4 w5x, though still show room for the contribution of new physics. The SM is minimally extended by incorporating extra colored fermions whose left-handed components, as well as right-handed ones, are singlets under the SUŽ2. gauge transformation, with electric charges being 2r3 andror y1r3. In this vector-like
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 8 1 8 - 2
322
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
quark model ŽVQM., many features of the SM are not significantly modified. However, the interactions of the quarks with the W or Z boson are qualitatively different from those in the SM. The Cabibbo–Kobayashi–Maskawa ŽCKM. matrix for the charged current is extended and not unitary. The neutral current involves interactions between the quarks with different flavors. In addition, the neutral Higgs boson also mediates flavor-changing interactions at the tree level. The VQM could thus give sizable new contributions to processes of flavorchanging neutral current ŽFCNC. w6,7x and of CP violation w8,9x. In this paper we study the radiative B-meson decay within the framework of the VQM containing one up-type and one down-type extra quarks. The decay receives contributions from the interactions mediated by the W, Z, and Higgs bosons. The effects by the Z and Higgs bosons have already been studied and found to be small w7x. Our study is concentrated on the other effects coming from the W-mediated interactions. These interactions give contributions differently from the SM at the electroweak energy scale, since an extra up-type quark is involved and the CKM matrix is not the same as that of the SM. It will be shown that the decay width can be much different from the SM prediction, even if the extra quark is rather heavy. The experimental results for the decay rate thus impose non-trivial constraints on the extended CKM matrix. We assume that there exist two extra Dirac fermions whose transformation properties are given by Ž3,1,2r3. and Ž3,1,y 1r3. for the SUŽ3. = SUŽ2. = UŽ1. gauge symmetry. The mass terms of the quarks are then expressed by 4 = 4 matrices. These mass matrices, which are denoted by M u and M d respectively for up- and down-type quarks, are diagonalized by unitary matrices AuL , AuR , A dL , and A dR as
The interaction Lagrangian for the quarks with the W and Goldstone bosons is given by
Ž m u1 ,m u2 ,m u3 ,m u4 . ,
Ž 3.
d d A d† L M A R s diag Ž m d1 ,m d 2 ,m d3 ,m d4 . .
Ž 4.
The mass eigenstates are expressed by u a and d a Ž a s 1 y 4., a being the generation index, which are also called as Ž u,c,t,U . and Ž d, s,b, D ..
u aVa bg m
Ý
'2
4
u aVa b
Ý
'2
a, bs1
mdb
y
1 y g5 2
a, bs1
g q
MW
ž
1 q g5 2
/5
d b Wm†
mua
1 y g5
MW
2
½ ž
/
d b G † q h.c.
Ž 5.
Here the 4 = 4 matrix V stands for an extended Cabibbo–Kobayashi–Maskawa matrix, which is defined by 3
Va b s
Ý Ž Au†L . ai Ž AdL . i b .
Ž 6.
is1
It should be noted that V is not unitary:
Ž V † V . a b s d a b y A dL4)a A dL4 b .
Ž 7.
The interaction Lagrangian for the down-type quarks with the Z, Higgs, and Goldstone bosons is given by 4
g Lsy
=
Ý
cos u W 1 y g5
q
a, bs1
MW
ž
1 q g5 2
/5
a, bs1
mdb MW
ž
1 q g5 2
/5
1 y g5
MW
2
/
d bH 0
d aŽ V †V . ab
Ý
mda
½ ž
4
g 2
d aŽ V †V . ab
Ý
mdb
qi
5
q 13 sin2u W d a b d b Zm
4
g y 2
½
d ag m y 12 Ž V † V . a b
a, bs1
2
y u u Au† L M A R s diag
4
g Ls
mda
1 y g5
MW
2
½ ž
d bG0 .
/ Ž 8.
Since V is not a unitary matrix, there appear interactions of FCNC at the tree level. The Lagrangians in Eqs. Ž5. and Ž8. contain new sources of CP violation w8x. The decay B X sg is approximated by the radiative b-quark decays, which are mediated by the W,
™
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
323
Fig. 1. The diagrams which give contributions to C 7 and C8 . The photon or gluon line should be attached appropriately.
Z, and Higgs bosons. The relevant effective Hamiltonian with five quarks is then written as 6
4GF
He f f s y
Ý
'2
js1
½C Ž m . O Ž m . j
qC˜j Ž m . O˜j Ž m .
j
J1 Ž r .
5
1
8
Ý C j Ž m . Oj Ž m .
q
The functions I1Ž r . and J1Ž r . are defined as w11x 1 I1 Ž r . s 2 q 3r y 6 r 2 q r 3 q 6 r ln r . , 4 Ž 12 Ž 1 y r . Ž 13 .
,
Ž 9.
s 12 Ž 1 y r .
Ž 1 y 6 r q 3r 2 q 2 r 3 y 6 r 2 ln r . .
4
js7
where Oj , O˜j represent operators for the D B s 1 transition, with C j , C˜j being their Wilson coefficients. The evaluated energy scale is denoted by m. The four-quark operators induced by the gauge boson interactions are denoted by Oj Ž j s 1 y 6. w10x. The Higgs boson interactions induce new four-quark operators, which are denoted by O˜j Ž j s 1 y 6.. The dipole operators for b sg and b sg are denoted by O 7 and O 8 , respectively, which are generated by the one-loop diagrams shown in Fig. 1. Hereafter, we only take the W boson interactions into consideration, since the contributions coming from the Z and Higgs boson interactions are known to be much smaller than the SM contribution. At the leading order ŽLO., the Wilson coefficients C2 , C 7 , and C8 have non-vanishing values at m s MW , which are given by
™
™
C2 Ž MW . s V32) V33 q V42) V43 y Ž V † V . 23 , C 7 Ž MW . s
23 36
Ž 10 .
Ž 14 . The non-unitarity of the CKM matrix V yields the terms proportional to Ž V † V . 23 for C2 , C 7 , and C8 . The Wilson coefficients at m s m b are obtained by solving the renormalization group equations. Using the LO anomalous dimension matrix, the coefficients are given by C2 Ž m b . s
1 2
6
12 y
žh
23
qh
23
/
C2 Ž MW . ,
Ž 15 .
16
C 7 Ž m b . s h 23 C 7 Ž MW . 14
16
q 83 h 23 y h 23 C8 Ž MW .
ž
/
8
Ý h ih a C2 Ž MW . ,
q
Ž 16 .
i
is1 14
8
C8 Ž m b . s h 23 C8 Ž MW . q
Ý h ih a C2 Ž MW . , i
is1
†
Ž V V . 23
Ž 17 .
4
y
Ý
Va2) Va3 32 r a 23 I1
Ž r a . q J1 Ž r a . 4 ,
as3
Ž 11 . 4
C8 Ž MW . s 13 Ž V † V . 23 y
Ý
Va2) Va3 32 r a I1 Ž r a . ,
as3
Ž 12 . ra s
m2u a MW2
.
with h s a s Ž MW .ra s Ž m b . which is set for h s 0.56 in the following numerical study. The constants h i ,h i , and a i are listed in Table 1 w12x. The branching ratio for B X sg is obtained by normalizing the decay width to that of the semileptonic decay B X c en , leading at the LO to 6 a EM 2 Br Ž B X sg . s C7 Ž m b . 2 p f Ž z . < V23 <
™
™
™
=Br Ž B
™ X en . , c
Ž 18 .
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
324
Table 1 The values of h i , h i , and a i in Eqs. Ž16. and Ž17. i
1
2
3
14 23 626126 27 2277 313063 i 36 3036
16 23 56281 51730
6 23 3 7
ai hi h
y
0
4
5
6
7
8
y 12 0.4086 y0.4230 y0.8994 0.1456 23 y y 141 y0.6494 y0.0380 y0.0186 y0.0057 0
0 y0.9135
0.0873 y0.0571
0.0209
with z s m2crm2b and f Ž z . s 1 y 8 z q 8 z 3 y z 4 y 12 z 2 ln z. The obtained branching ratio at the LO has nonnegligible perturbative uncertainties, which are reduced by taking into account corrections at the next leading order ŽNLO.. For the numerical evaluation, therefore, we incorporate NLO corrections for the matrix elements at m s m b w13x and the anomalous dimensions w14x. Our calculations follow formulae given in Ref. w5x, which also include QED corrections. The decay width of B X sg depends on the U-quark mass mU and the CKM matrix elements V32) V33 , V42) V43 , Ž V † V . 23 . The value of Ž V † V . 23 determines the FCNC interactions at the tree level in Eq. Ž8., which is constrained from non-observation of B K mq my w15x as
™
™
†
< Ž V V . 23 < - 8.1 = 10y4 .
Ž 19 .
The CKM matrix elements connecting light ordinary quarks, which are directly measured in experiments, have the same values as those in the SM. From the values of V12 , V13 , V22 , and V23 w15x, we obtain a constraint 0.03 - < V32) V33 q V42) V43 y Ž V † V . 23 < - 0.05.
Fig. 2. The values of A1 , A 2 , and A 3 in Eq. Ž21..
Expressing explicitly the dependence on the CKM matrix elements, the coefficient C 7 Ž m b . in Eq. Ž16. is written as C 7 Ž m b . s A1 Ž V † V . 23 q A 2 V32) V33 q A 3V42) V43 ,
Ž 21 . where A 3 is a function of mU while A1 and A 2 are constants. We show the mU dependency of A 3 in Fig. 2, where A1 and A 2 are also depicted. For mU R 200 GeV, the value of A 3 does not vary much with mU and is comparable with A 2 . Unless V42) V43 is much smaller than V32) V33 , the coefficient C 7 Ž m b . can be predicted differently from the SM value.
Ž 20 .
The mass mU should be heavier than the t-quark mass. In principle, the U-quark mass and the CKM matrix elements are not independent of each other, their relations being determined by the mass matrices M u and M d. However, these relations depend on many unknown factors for the mass matrices. Furthermore, the values of mU and V42) V43 are thoroughly unknown phenomenologically except for the above constraints. We therefore take them for independent parameters. The decay width is mainly determined by the Wilson coefficient C 7 Ž m b . as seen from Eq. Ž18..
Fig. 3. The allowed regions for V32) V33 and V42) V43 . Ž V † V . 23 s 8.1=10y4 .
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
325
values of V32) V33 and V42) V43 are constrained. These constraints do not much depend on the mass of the extra quark U. The VQM could make the branching ratio of B X sg different from the SM prediction. If precise measurements in the near future show a difference between the experimental value and the SM prediction, the VQM may become one candidate for physics beyond the SM.
™
Acknowledgements
™
Fig. 4. The branching ratio of B X sg . V42) V43 sy0.006,y 0.002,0.004,0.006, V32) V33 s 0.04, Ž V † V . 23 s8.1=10y4 .
Although A1 is larger than A 2 and A 3 in magnitude, the smallness of Ž V † V . 23 makes the term A1Ž V † V . 23 less important. In Fig. 3 we show allowed regions for V32) V33 and ) V42 V43 , assuming for simplicity that these values are real. The shaded regions are compatible with the experimental results of both Eq. Ž2. for B X sg and Eq. Ž20. for the CKM matrix elements. The regions between the solid lines satisfy the latter. We have taken the U-quark mass for 200 GeV - mU - 1 TeV and Ž V † V . 23 for its maximal value 8.1 = 10y4 . The branching ratio of B X sg sizably constrains the CKM matrix elements of the VQM. The allowed regions are slightly altered for Ž V † V . 23 s y8.1 = 10y4 . In Fig. 4 the branching ratio of B X sg is depicted as a function of mU for V42) V43 s y0.006, y0.002,0.004,0.006. For definiteness, we put V32) V33 s 0.04 and Ž V † V . 23 s 8.1 = 10y4 . The experimental bounds Eqs. Ž1. and Ž2. are also shown. For < V42) V43rV32) V33 < R 0.1, the predicted value is non-trivially different from that of the SM. The branching ratio could have any value within the experimental bounds. In summary, we have studied the effects of the VQM on the branching ratio for the radiative B-meson decay. Among the possible new contributions, the W-mediated diagrams yield sizable effects. From the experimental results for the branching ratio, the
™
™
™
The authors thank G.C. Cho for discussions. This work is supported in part by the Grant-in-Aid for Scientific Research on Priority Areas ŽPhysics of CP Violation, No. 12014205. from the Ministry of Education, Science and Culture, Japan.
References w1x For reviews, see Y. Nir, H.R. Quinn, Annu. Rev. Nucl. Part. Sci. 42 Ž1992. 211; M. Misiak, S. Pokorski, J. Rosiek, Heavy Flavours II, p. 795, World Scientific, Singapore, 1998; J.L. Hewett, hep-phr9803370. w2x N. Oshimo, Nucl. Phys. B 404 Ž1993. 20. w3x CLEO Collaboration, S. Ahmed et al., CLEO CONF 99–10, 1999. w4x ALEPH Collaboration, R. Barate et al., Phys. Lett. B 429 Ž1998. 169. w5x A.L. Kagan, M. Neubert, Eur. Phys. J. C 7 Ž1999. 5, and references therein. w6x G.C. Branco, L. Lavoura, Nucl. Phys. B 278 Ž1986. 738; Y. Nir, D. Silverman, Phys. Rev. D 42 Ž1990. 1477; D. Silverman, Phys. Rev. D 45 Ž1992. 1800; L. Lavoura, J.P. Silva, Phys. Rev. D 47 Ž1993. 1117; G.C. Branco, T. Morozumi, P.A. Parada, M.N. Rebelo, Phys. Rev. D 48 Ž1993. 1167; K. Fujikawa, Prog. Theor. Phys. 92 Ž1994. 1149; G. Bhattacharyya, G.C. Branco, D. Choudhury, Phys. Lett. B 336 Ž1994. 487; V. Barger, M.S. Berger, R.J.N. Phillips, Phys. Rev. D 52 Ž1995. 1663. w7x L.T. Handoko, T. Morozumi, Mod. Phys. Lett. A 10 Ž1995. 309; Mod. Phys. Lett. A 10 Ž1995. 1733 ŽE.; C.-H.V. Chang, D. Chang, W.-Y. Keung, Phys. Rev. D 61 Ž2000. 053007. w8x E. Asakawa, M. Marui, N. Oshimo, T. Saito, A. Sugamoto, Eur. Phys. J. C 10 Ž1999. 327. w9x J. McDonald, Phys. Rev. D 53 Ž1996. 645; T. Uesugi, A. Sugamoto, A. Yamaguchi, Phys. Lett. B 392 Ž1997. 389; G.C. Branco, D. Delepine, D. Emmanuel-Costa, R. Gonzalez ´ ´ Felipe, Phys. Lett. B 442 Ž1998. 229. w10x See e.g. G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 Ž1996. 1125.
326
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
w11x M. Aoki, G.C. Cho, N. Oshimo, Nucl. Phys. B 554 Ž1999. 50. w12x A.J. Buras, M. Misiak, M. Munz, ¨ S. Pokorski, Nucl. Phys. B 424 Ž1994. 374. w13x A. Ali, C. Greub, Phys. Lett. B 361 Ž1995. 146; N. Pott,
Phys. Rev. D 54 Ž1996. 938; C. Greub, T. Hurth, D. Wyler, Phys. Lett. B 380 Ž1996. 385; Phys. Rev. D 54 Ž1996. 3350. w14x K. Chetyrkin, M. Misiak, M. Munz, ¨ Phys. Lett. B 400 Ž1997. 206; Phys. Lett. B 425 Ž1998. 414 ŽE.. w15x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 1.
Physics Letters B 487 Ž2000. 321–326 www.elsevier.nlrlocaternpe
Contributions of vector-like quarks to radiative B meson decay Mayumi Aoki a , Eri Asakawa b, Makiko Nagashima b, Noriyuki Oshimo c , Akio Sugamoto b,c a
b
Theory Group, KEK, Tsukuba, Ibaraki 305-0801, Japan Graduate School of Humanities and Sciences, Ochanomizu UniÕersity, Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan c Department of Physics, Ochanomizu UniÕersity, Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan Received 17 May 2000; received in revised form 8 June 2000; accepted 21 June 2000 Editor: T. Yanagida
Abstract
™
We study the decay B X sg in a minimal extension of the standard model with extra up- and down-type quarks whose left- and right-handed components are both SUŽ2. singlets. Constraints on the extended Cabibbo–Kobayashi–Maskawa matrix are obtained from the experimental results for the branching ratio. Even if the extra quarks are too heavy to be detected in near-future colliders, the branching ratio could have a value which is non-trivially different from the prediction of the standard model. q 2000 Published by Elsevier Science B.V. PACS: 12.15.Ff; 12.60.-i; 13.25.Hw; 13.40.Hq
™™
™
The inclusive decay B X sg is well described by the free quark decays b sg and b sg g, owing to a large mass of the b quark. Since these decays are generated at the one-loop level of the electroweak interactions, the radiative B-meson decay is sensitive to new physics beyond the Standard Model ŽSM. w1x, such as the supersymmetric model w2x. Its branching ratio could deviate from the prediction of the SM. Or some constraints could be imposed on new physics. Experimentally, the branching
E-mail address: [email protected] ŽE. Asakawa..
ratio has been measured by CLEO w3x and ALEPH w4x as Br Ž B
™X g . s
s Ž 3.15 " 0.35 " 0.32 " 0.26 . = 10y4 ,
Ž 1.
s Ž 3.11 " 0.80 " 0.72 . = 10y4 .
Ž 2.
™
These results are consistent with the SM prediction BrŽ B X sg . s Ž3.29 " 0.33. = 10y4 w5x, though still show room for the contribution of new physics. The SM is minimally extended by incorporating extra colored fermions whose left-handed components, as well as right-handed ones, are singlets under the SUŽ2. gauge transformation, with electric charges being 2r3 andror y1r3. In this vector-like
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 8 1 8 - 2
322
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
quark model ŽVQM., many features of the SM are not significantly modified. However, the interactions of the quarks with the W or Z boson are qualitatively different from those in the SM. The Cabibbo–Kobayashi–Maskawa ŽCKM. matrix for the charged current is extended and not unitary. The neutral current involves interactions between the quarks with different flavors. In addition, the neutral Higgs boson also mediates flavor-changing interactions at the tree level. The VQM could thus give sizable new contributions to processes of flavorchanging neutral current ŽFCNC. w6,7x and of CP violation w8,9x. In this paper we study the radiative B-meson decay within the framework of the VQM containing one up-type and one down-type extra quarks. The decay receives contributions from the interactions mediated by the W, Z, and Higgs bosons. The effects by the Z and Higgs bosons have already been studied and found to be small w7x. Our study is concentrated on the other effects coming from the W-mediated interactions. These interactions give contributions differently from the SM at the electroweak energy scale, since an extra up-type quark is involved and the CKM matrix is not the same as that of the SM. It will be shown that the decay width can be much different from the SM prediction, even if the extra quark is rather heavy. The experimental results for the decay rate thus impose non-trivial constraints on the extended CKM matrix. We assume that there exist two extra Dirac fermions whose transformation properties are given by Ž3,1,2r3. and Ž3,1,y 1r3. for the SUŽ3. = SUŽ2. = UŽ1. gauge symmetry. The mass terms of the quarks are then expressed by 4 = 4 matrices. These mass matrices, which are denoted by M u and M d respectively for up- and down-type quarks, are diagonalized by unitary matrices AuL , AuR , A dL , and A dR as
The interaction Lagrangian for the quarks with the W and Goldstone bosons is given by
Ž m u1 ,m u2 ,m u3 ,m u4 . ,
Ž 3.
d d A d† L M A R s diag Ž m d1 ,m d 2 ,m d3 ,m d4 . .
Ž 4.
The mass eigenstates are expressed by u a and d a Ž a s 1 y 4., a being the generation index, which are also called as Ž u,c,t,U . and Ž d, s,b, D ..
u aVa bg m
Ý
'2
4
u aVa b
Ý
'2
a, bs1
mdb
y
1 y g5 2
a, bs1
g q
MW
ž
1 q g5 2
/5
d b Wm†
mua
1 y g5
MW
2
½ ž
/
d b G † q h.c.
Ž 5.
Here the 4 = 4 matrix V stands for an extended Cabibbo–Kobayashi–Maskawa matrix, which is defined by 3
Va b s
Ý Ž Au†L . ai Ž AdL . i b .
Ž 6.
is1
It should be noted that V is not unitary:
Ž V † V . a b s d a b y A dL4)a A dL4 b .
Ž 7.
The interaction Lagrangian for the down-type quarks with the Z, Higgs, and Goldstone bosons is given by 4
g Lsy
=
Ý
cos u W 1 y g5
q
a, bs1
MW
ž
1 q g5 2
/5
a, bs1
mdb MW
ž
1 q g5 2
/5
1 y g5
MW
2
/
d bH 0
d aŽ V †V . ab
Ý
mda
½ ž
4
g 2
d aŽ V †V . ab
Ý
mdb
qi
5
q 13 sin2u W d a b d b Zm
4
g y 2
½
d ag m y 12 Ž V † V . a b
a, bs1
2
y u u Au† L M A R s diag
4
g Ls
mda
1 y g5
MW
2
½ ž
d bG0 .
/ Ž 8.
Since V is not a unitary matrix, there appear interactions of FCNC at the tree level. The Lagrangians in Eqs. Ž5. and Ž8. contain new sources of CP violation w8x. The decay B X sg is approximated by the radiative b-quark decays, which are mediated by the W,
™
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
323
Fig. 1. The diagrams which give contributions to C 7 and C8 . The photon or gluon line should be attached appropriately.
Z, and Higgs bosons. The relevant effective Hamiltonian with five quarks is then written as 6
4GF
He f f s y
Ý
'2
js1
½C Ž m . O Ž m . j
qC˜j Ž m . O˜j Ž m .
j
J1 Ž r .
5
1
8
Ý C j Ž m . Oj Ž m .
q
The functions I1Ž r . and J1Ž r . are defined as w11x 1 I1 Ž r . s 2 q 3r y 6 r 2 q r 3 q 6 r ln r . , 4 Ž 12 Ž 1 y r . Ž 13 .
,
Ž 9.
s 12 Ž 1 y r .
Ž 1 y 6 r q 3r 2 q 2 r 3 y 6 r 2 ln r . .
4
js7
where Oj , O˜j represent operators for the D B s 1 transition, with C j , C˜j being their Wilson coefficients. The evaluated energy scale is denoted by m. The four-quark operators induced by the gauge boson interactions are denoted by Oj Ž j s 1 y 6. w10x. The Higgs boson interactions induce new four-quark operators, which are denoted by O˜j Ž j s 1 y 6.. The dipole operators for b sg and b sg are denoted by O 7 and O 8 , respectively, which are generated by the one-loop diagrams shown in Fig. 1. Hereafter, we only take the W boson interactions into consideration, since the contributions coming from the Z and Higgs boson interactions are known to be much smaller than the SM contribution. At the leading order ŽLO., the Wilson coefficients C2 , C 7 , and C8 have non-vanishing values at m s MW , which are given by
™
™
C2 Ž MW . s V32) V33 q V42) V43 y Ž V † V . 23 , C 7 Ž MW . s
23 36
Ž 10 .
Ž 14 . The non-unitarity of the CKM matrix V yields the terms proportional to Ž V † V . 23 for C2 , C 7 , and C8 . The Wilson coefficients at m s m b are obtained by solving the renormalization group equations. Using the LO anomalous dimension matrix, the coefficients are given by C2 Ž m b . s
1 2
6
12 y
žh
23
qh
23
/
C2 Ž MW . ,
Ž 15 .
16
C 7 Ž m b . s h 23 C 7 Ž MW . 14
16
q 83 h 23 y h 23 C8 Ž MW .
ž
/
8
Ý h ih a C2 Ž MW . ,
q
Ž 16 .
i
is1 14
8
C8 Ž m b . s h 23 C8 Ž MW . q
Ý h ih a C2 Ž MW . , i
is1
†
Ž V V . 23
Ž 17 .
4
y
Ý
Va2) Va3 32 r a 23 I1
Ž r a . q J1 Ž r a . 4 ,
as3
Ž 11 . 4
C8 Ž MW . s 13 Ž V † V . 23 y
Ý
Va2) Va3 32 r a I1 Ž r a . ,
as3
Ž 12 . ra s
m2u a MW2
.
with h s a s Ž MW .ra s Ž m b . which is set for h s 0.56 in the following numerical study. The constants h i ,h i , and a i are listed in Table 1 w12x. The branching ratio for B X sg is obtained by normalizing the decay width to that of the semileptonic decay B X c en , leading at the LO to 6 a EM 2 Br Ž B X sg . s C7 Ž m b . 2 p f Ž z . < V23 <
™
™
™
=Br Ž B
™ X en . , c
Ž 18 .
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
324
Table 1 The values of h i , h i , and a i in Eqs. Ž16. and Ž17. i
1
2
3
14 23 626126 27 2277 313063 i 36 3036
16 23 56281 51730
6 23 3 7
ai hi h
y
0
4
5
6
7
8
y 12 0.4086 y0.4230 y0.8994 0.1456 23 y y 141 y0.6494 y0.0380 y0.0186 y0.0057 0
0 y0.9135
0.0873 y0.0571
0.0209
with z s m2crm2b and f Ž z . s 1 y 8 z q 8 z 3 y z 4 y 12 z 2 ln z. The obtained branching ratio at the LO has nonnegligible perturbative uncertainties, which are reduced by taking into account corrections at the next leading order ŽNLO.. For the numerical evaluation, therefore, we incorporate NLO corrections for the matrix elements at m s m b w13x and the anomalous dimensions w14x. Our calculations follow formulae given in Ref. w5x, which also include QED corrections. The decay width of B X sg depends on the U-quark mass mU and the CKM matrix elements V32) V33 , V42) V43 , Ž V † V . 23 . The value of Ž V † V . 23 determines the FCNC interactions at the tree level in Eq. Ž8., which is constrained from non-observation of B K mq my w15x as
™
™
†
< Ž V V . 23 < - 8.1 = 10y4 .
Ž 19 .
The CKM matrix elements connecting light ordinary quarks, which are directly measured in experiments, have the same values as those in the SM. From the values of V12 , V13 , V22 , and V23 w15x, we obtain a constraint 0.03 - < V32) V33 q V42) V43 y Ž V † V . 23 < - 0.05.
Fig. 2. The values of A1 , A 2 , and A 3 in Eq. Ž21..
Expressing explicitly the dependence on the CKM matrix elements, the coefficient C 7 Ž m b . in Eq. Ž16. is written as C 7 Ž m b . s A1 Ž V † V . 23 q A 2 V32) V33 q A 3V42) V43 ,
Ž 21 . where A 3 is a function of mU while A1 and A 2 are constants. We show the mU dependency of A 3 in Fig. 2, where A1 and A 2 are also depicted. For mU R 200 GeV, the value of A 3 does not vary much with mU and is comparable with A 2 . Unless V42) V43 is much smaller than V32) V33 , the coefficient C 7 Ž m b . can be predicted differently from the SM value.
Ž 20 .
The mass mU should be heavier than the t-quark mass. In principle, the U-quark mass and the CKM matrix elements are not independent of each other, their relations being determined by the mass matrices M u and M d. However, these relations depend on many unknown factors for the mass matrices. Furthermore, the values of mU and V42) V43 are thoroughly unknown phenomenologically except for the above constraints. We therefore take them for independent parameters. The decay width is mainly determined by the Wilson coefficient C 7 Ž m b . as seen from Eq. Ž18..
Fig. 3. The allowed regions for V32) V33 and V42) V43 . Ž V † V . 23 s 8.1=10y4 .
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
325
values of V32) V33 and V42) V43 are constrained. These constraints do not much depend on the mass of the extra quark U. The VQM could make the branching ratio of B X sg different from the SM prediction. If precise measurements in the near future show a difference between the experimental value and the SM prediction, the VQM may become one candidate for physics beyond the SM.
™
Acknowledgements
™
Fig. 4. The branching ratio of B X sg . V42) V43 sy0.006,y 0.002,0.004,0.006, V32) V33 s 0.04, Ž V † V . 23 s8.1=10y4 .
Although A1 is larger than A 2 and A 3 in magnitude, the smallness of Ž V † V . 23 makes the term A1Ž V † V . 23 less important. In Fig. 3 we show allowed regions for V32) V33 and ) V42 V43 , assuming for simplicity that these values are real. The shaded regions are compatible with the experimental results of both Eq. Ž2. for B X sg and Eq. Ž20. for the CKM matrix elements. The regions between the solid lines satisfy the latter. We have taken the U-quark mass for 200 GeV - mU - 1 TeV and Ž V † V . 23 for its maximal value 8.1 = 10y4 . The branching ratio of B X sg sizably constrains the CKM matrix elements of the VQM. The allowed regions are slightly altered for Ž V † V . 23 s y8.1 = 10y4 . In Fig. 4 the branching ratio of B X sg is depicted as a function of mU for V42) V43 s y0.006, y0.002,0.004,0.006. For definiteness, we put V32) V33 s 0.04 and Ž V † V . 23 s 8.1 = 10y4 . The experimental bounds Eqs. Ž1. and Ž2. are also shown. For < V42) V43rV32) V33 < R 0.1, the predicted value is non-trivially different from that of the SM. The branching ratio could have any value within the experimental bounds. In summary, we have studied the effects of the VQM on the branching ratio for the radiative B-meson decay. Among the possible new contributions, the W-mediated diagrams yield sizable effects. From the experimental results for the branching ratio, the
™
™
™
The authors thank G.C. Cho for discussions. This work is supported in part by the Grant-in-Aid for Scientific Research on Priority Areas ŽPhysics of CP Violation, No. 12014205. from the Ministry of Education, Science and Culture, Japan.
References w1x For reviews, see Y. Nir, H.R. Quinn, Annu. Rev. Nucl. Part. Sci. 42 Ž1992. 211; M. Misiak, S. Pokorski, J. Rosiek, Heavy Flavours II, p. 795, World Scientific, Singapore, 1998; J.L. Hewett, hep-phr9803370. w2x N. Oshimo, Nucl. Phys. B 404 Ž1993. 20. w3x CLEO Collaboration, S. Ahmed et al., CLEO CONF 99–10, 1999. w4x ALEPH Collaboration, R. Barate et al., Phys. Lett. B 429 Ž1998. 169. w5x A.L. Kagan, M. Neubert, Eur. Phys. J. C 7 Ž1999. 5, and references therein. w6x G.C. Branco, L. Lavoura, Nucl. Phys. B 278 Ž1986. 738; Y. Nir, D. Silverman, Phys. Rev. D 42 Ž1990. 1477; D. Silverman, Phys. Rev. D 45 Ž1992. 1800; L. Lavoura, J.P. Silva, Phys. Rev. D 47 Ž1993. 1117; G.C. Branco, T. Morozumi, P.A. Parada, M.N. Rebelo, Phys. Rev. D 48 Ž1993. 1167; K. Fujikawa, Prog. Theor. Phys. 92 Ž1994. 1149; G. Bhattacharyya, G.C. Branco, D. Choudhury, Phys. Lett. B 336 Ž1994. 487; V. Barger, M.S. Berger, R.J.N. Phillips, Phys. Rev. D 52 Ž1995. 1663. w7x L.T. Handoko, T. Morozumi, Mod. Phys. Lett. A 10 Ž1995. 309; Mod. Phys. Lett. A 10 Ž1995. 1733 ŽE.; C.-H.V. Chang, D. Chang, W.-Y. Keung, Phys. Rev. D 61 Ž2000. 053007. w8x E. Asakawa, M. Marui, N. Oshimo, T. Saito, A. Sugamoto, Eur. Phys. J. C 10 Ž1999. 327. w9x J. McDonald, Phys. Rev. D 53 Ž1996. 645; T. Uesugi, A. Sugamoto, A. Yamaguchi, Phys. Lett. B 392 Ž1997. 389; G.C. Branco, D. Delepine, D. Emmanuel-Costa, R. Gonzalez ´ ´ Felipe, Phys. Lett. B 442 Ž1998. 229. w10x See e.g. G. Buchalla, A.J. Buras, M.E. Lautenbacher, Rev. Mod. Phys. 68 Ž1996. 1125.
326
M. Aoki et al.r Physics Letters B 487 (2000) 321–326
w11x M. Aoki, G.C. Cho, N. Oshimo, Nucl. Phys. B 554 Ž1999. 50. w12x A.J. Buras, M. Misiak, M. Munz, ¨ S. Pokorski, Nucl. Phys. B 424 Ž1994. 374. w13x A. Ali, C. Greub, Phys. Lett. B 361 Ž1995. 146; N. Pott,
Phys. Rev. D 54 Ž1996. 938; C. Greub, T. Hurth, D. Wyler, Phys. Lett. B 380 Ž1996. 385; Phys. Rev. D 54 Ž1996. 3350. w14x K. Chetyrkin, M. Misiak, M. Munz, ¨ Phys. Lett. B 400 Ž1997. 206; Phys. Lett. B 425 Ž1998. 414 ŽE.. w15x Particle Data Group, Eur. Phys. J. C 3 Ž1998. 1.