- Email: [email protected]
CP-violation in b quark radiative inclusive decays
cm .__
1 May 1997
i!id 2%
ELSEVIER
Physics Letters B 399 (1997) 303-3 11
CP-violation in b quark radiative inclusive decays H.M. Asatrian a,1,2, G.K. Yeghiyan b~l, A.N. Ioannissian c~l a International Centre for Theoretical Physics, Strada Costiera 11, P.O. Box 586, 34100 Trieste, Italy b Deutches Electronen Synchrotron DESI: Hamburg, Germany ’ Dept. of Physics, Technion - Israel Inst. of Tech., Haifa, Israel Received
23 October
1996; revised manuscript
received 3
February 1997
Editor: P.V.Landshoff
Abstract
The direct CP-violations in SU( 2) L x su( 2)R x U( 1) model and two-Higgs doublet extension of the standard model for dy and b + sy decays are investigated. The calculated value of CP-asymmetry for these two models and for b --+ dy and b -+ sy decays for the wide range of parameters may exceed the value, predicted by the standard model and has a sign opposite to that of the latter. @ 1997 Published by Elsevier Science B.V. b +
The investigation of rare B-meson decays can give important information on new physics in the TeV region. The observation of direct CP asymmetry in B-meson decays will help to understand the CP breaking phenomenon. The first experimental evidence for the exclusive B --) K*y decay has been obtained at CLEO [ 11. More recently, the branching ratio of the inclusive B -+ X,y decay was measured [ 21. The b -+ sy decay has been investigated theoretically for the standard model and its extensions in [ 3-121. CP-violation in B - B system in sum x sum x lJ( 1) model was considered in [ 131. The problem of CP asymmetry for b + sy decay for standard model and its extensions was investigated in [ 14-161. Although the expected decay rate for b -+ dy decay is about 10 + 20 times smaller than for the b -+ sy decay, the CP asymmetry for the first decay can be about 10 times larger [ 14,171. The aim of the present paper is to consider the direct CP decay asymmetry for the b + dy and b --+ sy decays for SU( 2) L x SU( 2) R x U( 1) model and two Higgs doublet extension of the standard model. In Sum x SU(~)R x U(1) model the b -+ dy decay amplitude arises due to the interaction of quark charged weak current with the “left” and “right” W-bosons and charged Higgs field. This interaction has the following form [ 161:
1 E-mail: [email protected]. 2 Permanent address: Yerevan Physics Institute, 2 Alikhanyan 0370-2693/97/$17.00 0 1997 Published PII SO370-2693(97>00297-9
Br., 375036, Yerevan, Armenia.
by Elsevier Science B.V. All righhts reserved.
304
H.M. Asarrian et al./Physics
LCh= _!~ (i?, c, f) [@I’[-gL
cos pKLP_-
gR
Letters B 399 (1997) 303-311
sinpeisKRP+]
1
K
- tan 28KLMd + eis -MuKR cos 28
+ (tan28M,KL
p+
- eia--&KRMd
where WI is the “light” (predominantly left and right W-bosons, tan2P
>
M2 = 2sin28eA gL M&
left-handed)
charged gauge boson and /3 is the mixing
angle between
I
(2)
tan8 = -%,
KL and KR are CKM mixing matrices for left and right charged currents respectively, P& = ( 1 f- ~5)/2, M, and Md are diagonal mass matrices for quarks with Q = 2/3 and Q = -l/3 charges respectively. The matrices KL and KR can be expressed in a form, where KL has only one complex phase and KR has five complex phases [ 181. Phase S in (2) takes his origin from the vacuum expectation value of Higgs field @, connected with the SU( 2)L x U( 1) symmetry breaking:
In ( 1) the term connected with the interaction with heavy (predominantly right) W-boson is omitted, since it is not relevant for b -+ dy decay. The additional (compared with the standard model) phase factor exp(iS) in ( 1) leads to the existence of the new CP violation effects in sum x sum x U( 1) model. We define the direct CP asymmetry for b --) dy decay as [ 141:
r(6 --f dy) - T(b -+ dy) acp = u6
+&)
(3)
+r(b+dy)’
The direct CP asymmetry for b -+ dy decay arises only if the matrix element of decay has an absorptive part, which arises if the final state strong interaction effects are taken into account. In general case the amplitudes of the decays 6 --+ ~!y and b + dy can be expressed in the following form [ 141:
A(6+dy)=xtA;fiA;)r/,*,
A(bjdy)
a
a
where A& Af, are real and absorptive parts of amplitudes, 1,2,3 ,.... Then CP asymmetry is given by: acp =
Ca+b(A:&, - ALAI;)WVi*V,) Ca,b(A;A;
+ AiAi)
FWV,*V6) ’
(4)
=~(Ai+iAh)va. V, are some phases
and CKM-type
factors,
a =
(5)
To take into account QCD-corrections to radiative decays matrix elements the effective Hamiltonian approach is used. We follow [ 161 and use the results of [ 121 for the imaginary part of the amplitude, connected with the 02 operator:
H.M. Asatrian et al./Physics
30.5
Letters B 399 (1997) 303-311
= (mc/mb) 2, L = In z,. We take the ratio of c- and b-quark masses equal to 0.29 [ 121, then the imaginary parts of the amplitudes connected with cc and iiu intermediate states is approximately r = 0.145. We obtain the following expression for effective Hamiltonian of b -+ & decay in the x Sum x U( 1) model:
where z ratio of equal to SU(2)L
e 2(.+ H b+dy=-1fjT2 --rnb{(K,Ld*KiAz Jz + e-iGK~K~A$yO~
+ eiSK$K~A~y)O~
+ i[ ($(KkKkAz
+ e”‘K,“; K”,gA$) tb
KR
(6)
where oL,R I
=
iid~““( 1 f y5)ubF,,
(7)
and the functions A:$, A& ArgL, A& which include leading logarithmic strong interaction corrections, presented in [ 161. Using (5) and (6)) one obtains the following formula for CP asymmetry for b -+ dy decay in sum su(2)R x U(1) model:
2% acp(b
--+ dr) = ([q/2
- (Revrv,
+ ~C,“~“)Uplr
+ rRevTv,>
Ai$sina
{(Imv~v,+rImv~v,)
(A~~+Hco~cuAf~)~
2oc2 81 + $HsinavTu,(Ay/A&
were x
2oc2
- A$YAz)},
(8)
where
Vt = K;tdKLtb,
vu = Kiud&.ub.
vc = f&d&b,
The expression for CP asymmetry in b + sy (in the limit md = 0, m, = 0) decay can be obtained making replacements v,, -+ vi, v, + v:, vt 4 v:, where
u;=K;t,KLtbt v;=K&f&b, We value Let rng E
0:= K;ps&ub.
(9) from (8) by
(10)
want to stress that in [ 14-161 the approximate value for Y was used: Y = 0.12. Now we use the correct r = 0.145 [ 121, which is essential for the numerical results. us proceed to the numerical results. We take crS = 0.212, c2 N 1.1, m, = (175 f 9) GeV, mb = 4.5 GeV, mb( Mz) = (3.5f0.5) GeV [ 19,201. For CKM matrix parameters we use the Wolfenstein parametrization:
Im( vTv,> = -A2A6r],
Re(vTv,)
jvt12=A2A6((1-P)2+r,r2),
= A2A6( (1 - p)p - q2), ~z~,~~=A~A~(p~f~~),
/$I2 = A2A4( (1 + A2p)2 + h8v2), Re(vi*v:) For parameters
=-A2h6(p+A2(p2+v2)),
Im( vi*v:) = A2A6v, l~:l~=A~A*(p~+~~).
A, A, p, cq in ( 11) we use values given in [ 201.
(11)
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H.M. Asatrian et al./Physics
Letters B 399 (1997) 303-311
Table 1 Minimum and maximum values of ucp for case ( 1) for b --t dy and b --t sy decays for tan28 = 2.0, MwR = M,+ = 10 TeV and for various values of LY a!
2.2 -2.0 -1.8 1.8 2.0 2.2
WP
(b -+ dr)
0.043 0.041 0.049 0.0085 -0.0024 -0.0027
+ + + t
The CP asymmetry
0.107 0.130 0.123 0.162 + 0.172 + 0.160
WP (b -+ sy) -0.0101 f -0.0045 -0.096 t -0.0040 -0.0087 + -0.0039 -0.0037 + 0.0019 -0.0039 f 0.0039 -0.0018 f 0.0044
Table 2 Minimum and maximum values of acp for case (2a) for b --t dy and b -+ sy decays for tan20 = 3.0, MwR = Mp+ = 10 TeV and for various values of LY (Y -3.0 -2.5 -2.0 2.0 2.5 3.0
QP
(b
+
dr)
WP
-0.017 + 0.044 -0.035 + 0.062 0.008 t 0.110 -0.031 f 0.188 -0.036 f 0.108 -0.012 f 0.055
(b
+
SY)
-0.0033 t -0.0045 -0.0114 + -0.0026 -0.0145 + -0.0042 -0.0134 t 0.0102 -0.0008 + 0.0104 -0.0018 f 0.0028
depends also on parameters of SU( 2)~ x sum x U( 1) model: LY,tan 213, MwR, M,+, e will consider the following possibilities for the ratios IKi/K,“bl, IK&Sj/Kh(sj I: W
IGIGL IGfIGI. (1) IGIGd= IG&%(s)I = 1. (2) No restrictions on the ratios IKP,/Kkl,
IK~(~)/K~(,)
1, besides
those which
follow
from the unitarity
conditions for matrices KL and KR. Case ( 1) corresponds to the pseudo-manifest left-right symmetry, when the absolute values of KobayashiMaskawa mixing matrices elements in left and right sectors (KG and K$, i, j = 1,2,3 correspondingly) are equal to each other [ 181. The case (2) corresponds to non-manifest left-right symmetry when IKif;:1 # jK{ 1
[181. It is known that the experiment is in agreement with the standard model predictions for b ---f sy decay rate. Following [ 161, we will consider that in sum x SU(2)R x U( 1) model the branching of b + sy decay can differ from the standard model prediction no more than A = 10%. As for b --f dy decay rate, there is no experimental restriction for it. However, if we assume that in SU( 2)~ x 5’U( 2)~ x U( 1) model the b --+ sy decay rate is the same (with 10% accuracy) as in standard model, then for the case ( 1), the same condition will be satisfied for the b -+ dy decay rate also. For the case (2) we will consider two possibilities: (2a) b ---) dy decay rate is equal with accuracy of A = 10% to that in the standard model; (2b) b -+ dy decay rate is arbitrary. For case ( 1) and for a given Mw,, M,+ the decay asymmetry for b --+ dy and b -+ sy decays depends on CKM parameters, a and tan 28. Taking into account (8) and the equivalence of decay rates (with 10% accuracy) in standard model and SU(~)L x SU( 2)~ x U( 1) model, it is easy to understand, that for cy = 0 the absolute value of decay asymmetry for both of decays can’t exceed the standard model value by more than 10% for all the values A4wR, M+,+, tan28. The sign of the decay asymmetry will be the same as in the standard model. When we have a new source of CP violation, i.e. a # O., the terms in (8) proportional to AfY and AWLAR dr dg - AzYAz contribute to decay asymmetry and one can expect less or more significant deviations from the standard model predictions. However, the restriction for decay rate here also plays the important role. In Fig. 1 the tan26 dependence of maximum and minimum values of acp for b -+ dy and b -+ sy decays for the case ( 1) are given for various values of Mw,, M,+ . Due to the presence of the terms proportional to sin (Y in (8) there is a difference between standard model predictions (-ucp( b --f sy)/10p3 = 2.9 f 6.4, ucp( b -+ dy)/lO-* = 3.7 + 16), which practically coincide with the results obtained for Mw, = M,+ = 50 TeV and predictions of the SU( 2) L x SU( 2) R x U( 1) model for MwR, M, + < 20 TeV, tan 28 2 1. The difference is most significant for b -+ sy decay. The sign of asymmetry can be different from those in the standard model for both decays. To illustrate the cy dependence of the decay asymmetry we give in the Table 1 minimum and maximum values of decay asymmetry for two decays for tan 20 = 2, M W, = M,+ = 10 TeV and various values of (Y (for 2.30 I /aI I 3.14 and /LYE < 1.9 the condition for decay rate is not satisfied). As we have mentioned above, the difference between results obtained for ucp when taking Y = 0.12 or 0.145 in the expression (8)
H.M. Asatriun et al./Physics
Letters B 399 (1997) 303-311
307
Fig. 1. Maximum and minimum values of ncp in SU(2)r. x S’U(Z), x U( 1) model for the case (1) and a) for b + dy decay, b) for b ---f sy decay for different values of MwR and M,+: MwR = M,+ = 5 TeV (curves 1 and 2); Mw, = M,+ = 10 TeV (curves 3 and 4); MwR = M,+ = 20 TeV (curves 5 and 6); MwR = M,+ = 50 TeV (curves 7 and 8).
for b -+ sy decay it can reach 30%. For this reason values of ucp in Fig. 1 are lower than those in [ 161. Let us now proceed to the case (2). In Fig. 2a we give the dependence of acp on tan 28 for the decay b --+ dy for case (2a) and for the decay b + sy for case (2). It is obvious that for the case (2), when there are no restrictions on right current mixing matrix (besides unitarity condition), the decay asymmetry in sum x sum x U( 1) model is much more different from the standard model predictions, than for the case ( 1). Indeed, for the case (2) the minimum value of the asymmetry for b -+ dy decay can reach (for Mw, = M,+ = 5 TeV) the value -0.18, while for the previous case the minimum value of ucp for the same values Mw, and M,+ is equal to -0.02. For b -+ sy decay.the absolute value of decay asymmetry for the same values MwR and M,+ is 1.5 + 2 times higher than for the previous case. We give in Table 2 minimum and maximum values of decay asymmetry for two decays for tan28 = 3, MwR = M,+ = 10 TeV and various values of LY(for (~1 5 1.80 the conditions for decay rate is not satisfied). It is clear that for the case (2a) the deviations from the standard model predictions are more significant and can
is non-negligible:
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Letters B 399 (1997) 303-311
Fig. 2. Maximum and minimum values of ncp in sum x su(2)R x U( 1) model for the case (2a) and a) for b -+ cty decay, b) for b + sy decay for different values of MwR and M,+: M w, = M,+.t. = 5 TeV (curves 1 and 2) ; MwR= M,+ = 10 TeV (curves 3 and 4) ; Mw, = M,+ = 20 TeV (curves 5 and 6) ; A4wR= M+,+= 50 TeV (curves 7 and 8).
take place for a larger parameter space, than for the case ( 1). In Fig. 3 for the case (2b) (when we have no restriction for b --+ dy decay rate) the decay asymmetry minimum and maximum values are given. We note, that for some values of tan28, Mw,, M,+ the decay asymmetry ucp from (8) becomes abnomally large. This means, that for such values of tan 28, Mw,, M,+ (8) becomes incorrect (imaginary part of the amplitude becomes non-small in comparison with the real part and it is necessary to take into account more terms of perturbation theory on czS). Nevertheless, it is reasonably safe to suggest that in this case the difference from the standard model predictions for MwR, M,+ 5 10 TeV can be significant. Thus, for the case of non-manifest left-right symmetry for the large parameter space of the sub x SU(2)R x U(1) model (M wR, M,+ 5 (10 f 15) TeV, tan20 2 (1.5 + 2.5)) one can expect a significant deviations from the standard model predictions for acp for both decays. We discuss next the two Higgs doublet extension of the standard model. In general case Yukawa interaction of quarks with Higgs doublets ~1 and 92 is:
I::.-.-.-----..: ,
H.M. Asatrian et d/Physics
a,,/ 30
Letters B 399 (1997) 303-311
309
1 o->
. ..
,,’ __I’
,/’
20
...... 3 ./........ ‘._____._._.__..._. _..:
______5._.._.------.----. 7 __. __
tan219 Fig. 3. Maximum and minimum values of CIC,Din SU(Z)L x Sum x U( 1) model for the case (2b) and for b ---f dy decay for different values of MwR and Mp+ : MwR = M,+ = 5 TeV (curves 1 and 2); MwR = Mp+ = 10 TeV (curves 3 and 4) ; MwR = Mp+ = 20 TeV (curves 5 and 6); A4wR = M,+ = 50 TeV (curves 7 and 8).
where 9~ is the quark doublet and dR and UR are quark singlets and ys, y;, yf’, y$ are matrices in flavor space [ 151. Usually two versions of this model are considered [ 151: model I, where only one doublet (~1) interacts with quarks: $j = g = O., model II, where one of doublets interacts with up-type quarks and the second one interacts with down-type quarks: y; = yt’ = 0. In paper [ 151 a third model (III) was considered where both Higgs doublets interact with up and down quarks and all of the quantities yy, y:, yf, y$ are non-zero. Generally speaking, in this case the flavor changing neutral currents can arise [ 151. The restrictions on Higgs particles masses and other parameters in such a model were considered in [ 211. The last model (model III) is close in some respect to the SU( 2) L x SU( 2) R x U( 1) model: for this model, as for the previous one, new CP-violating phase arises. As for models I and II, there are no new sources of CP violation. Formula for CP asymmetry in b + sy decay for two Higgs doublet extension of the standard model is the following [ 151:
_ 2z (1 - r) Im(v:u,) 81 bt12
where St, h,
(CcH + Re(StSb)C$)ctaS
9
(13)
q
C&CwH, CFH, C& C,, are given in [ 151. We note, that there is a difference between formula (13) and the expression for CP asymmetry in [ 151: in [ 151 the factor 2/9 is missing. In Table 3 the numerical results for the model III for some values of charged Higgs boson masses are given. Generally speaking, values of ucp for b -+ sy decay, given by Table 3, are lower than the results [ 151 for the reason, mentioned above, but the deviation from the standard model predictions for relatively low masses of charged Higgs boson 5 200 GeV (this is within the limits given in [ 211) can be very large (more than 5
H.M. Asatrian et al./Physics Letters B 399 (1997) Xl-311
310
Table 3 Values of CP-asymmetry for b --f s + y and b ---fd + y for model III, for A < 10% and for A < 50% mH+
(GeV)
A < 10%
A < 50%
-acp(b -+ sy)/10e3 50 100 200 400 800 1600 3200 6400 12800 25600
-53 + 53 -45 + 45 -35 + 35 -25 t 25 -16 i 16 -9.7 t 9.5 0.6 t 7 2~7 2.6 + 6.8 2.8 + 6.5
acp(b --f dy)/10h2 -17 -17 -18 -18 -18 -17 0.6 3.1 3.5 3.6
+ 17 + 17 +- 18 f 18 f 18 +- 18 t 17 + 17 t 17 t 16
-acp (b ---)sy)/10W3 -71 f71 -61 e-61 -48 + 48 -34 f 34 -22 j 21 -12 f 12 -5t11 0.4 + 8.3 2.3 + 6.8 2.8 + 6.5
acp(b -+ dy)/10e2 -23 -23 -23 -24 -24 -22 -4.1 2.4 3.5 3.6
f t t + + + f
23 23 23 24 24 25 25 f 20 f 17 f 16
Table 4 Values of CP-asymmetry for the Model I for A < 10% and for A < 50% IQ+
(GeV)
A < 10% -acp(b
50 100 200 800 3200
A < 50%
-+ sy)/10v3 2.9 2.9 2.9 2.9 2.9
t t t f t
6.7 6.7 6.7 6.7 6.5
acp(b --+ dy)/10e2 3.6 3.6 3.6 3.6 3.6
+ + + + +
17 17 17 17 16
-acp (b -+ sy)/10p3 2.9 2.9 2.9 2.9 2.9
c + f c f
8.6 8.9 8.7 7.6 6.5
acp (b -+ dy)/lO-’ 3.6~21 3.6 f 22 3.6 + 22 3.6 + 19 3.6 + 17
Table 5 Values of CP-asymmetry for the Model II for A < 10% and for A < 50% mH+
(GeV)
-acp(b 400 650 1300 2600 5200
times).
A < 50%
A < 10% --+ sy)/lO_ _ 2.8 f 6.1 2.8 + 6.2 2.9 f 6.3
acp(b --f dy)/lO-’
3.4 f 15 3.5 f 16 3.6 + 16
-acp (b + s~)/lO-~ 2.4 2.5 2.7 2.8 2.9
+ f f + +
5.3 5.7 6.1 6.3 6.3
ncp(b -+ dy)/lO-’ 2.9 3.1 3.4 3.5 3.6
f f f f +
13 14 15 16 16
As for the b ---) dy decay (as follows from Table 3 for A < 10%) the restrictions on absolute value ucp are close to the predictions of the standard model. The difference is that CP-asymmetry here can have opposite sign and the minimum value of ucp can be very small. In Table 3, A < 50%, the minimum and maximum values of CP asymmetry for b + dy decay are given for the case when we use the less severe condition A < 50% for the decay rate. In this case the absolute value of CP asymmetry can be 1.5 times larger than the standard model predictions. As we have mentioned above, for the models I and II there is no new source of CP-violation and as follows from the Tables 4 and 5, the values of ucp for two decays are almost the same as for the standard model. In conclusion, the direct CP-violation in sum x SU( 2)~ x U( 1) model and two-Higgs doublet extension
H.M. Asatrian et al./ Physics Letters B 399 (1997) 303-311
of the differ much model
standard from the stronger (model
311
model for b 4 dy and b --+ sy decays was investigated. The calculated values of CP-asymmetry standard model predictions and can have a sign opposite to that of the latter. The difference is for non-manifest left-right symmetric model and two-Higgs doublet extension of the standard III).
Authors want to thank A. Ali for stimulating discussions. One of the authors (H.A.) wants to thank High Energy Group of ICTP for hospitality. The research described in this publication was made possible in part due to the contract INTAS-93-1630.
References [ 1] [2] [3] [4] [5] [6] [7] [ 81 [9]
[lo] [ 111 [ 121 [ 131 1141 [ 151 [ 161 [ 171 [ 181 [ 191 [20] [21]
R. Ammar et al. (CLEO Collaboration), Phys. Rev. L&t. 71 ( 1993) 674. M.S. Alam et al. (CLEO Collaboration), Phys. Rev. Lett. 74 (1995) 2885. B. Grinstein and M.B. Wise, Phys. Lett. B 201 (1988) 274. W.S. Hou and R.S. Willey, Phys. Lett. B 202 (1988) 591. R. Barbieri and G.F. Guidice, Phys. Lett. B 309 (1993) 86. S. Bertolini et al., Nucl. Phys. B 353 (1991) 591. L. Randal and R. Sundrum, MIT preprint MIT-CTP-2211 (1993). D. Cocolicchio et al., Phys. Rev. D 40 (1989) 1477. H.M. Asatrian and A.N. Ioannissian, Mod. Phys. Lett. A 5 (1990) 1089; Sov. 5. Nucl. Phys. 51 (1990) KS. Babu, K. Fujikawa and A. Yamada, Phys. Lett. B 333 (1994) 196. E?Cho and M. Misiak, Phys. Rev. D 49 ( 1994) 5894. C. Greub, T. Hurth and D. Wyler, Phys. Rev. D 54 (1996) 3350. G. Ecker and W. Grimus, in: Moriond 86, V. 1, p. 201. J. Soares, Nucl. Phys. B 367 (1991) 575. L. Wolfenstein and Y.L. Wu, Phys. Rev. Lett. 73 (1994) 2809. H.M. Asatrian and A.N. Ioannissian, Phys. Rev. D 54 (1996) 5242. A. Ali, DESY Report No. 96-106, HEP_PH/9606324. R.N. Mohapatra, in: CP Violation, ed., C. Jarslkog (World Scientific) p. 384. M. Bando et al., Mod. Phys. Lett. A 7 (1992) 3379. A. Ali and D. London, DESY Report No. 96-140, HEP-PH/9607392. D. Atwood, L. Reina and A. Soni, CEBAF preprint, JL-TH-96-15, hep-ph/9609279.
858.
1 May 1997
i!id 2%
ELSEVIER
Physics Letters B 399 (1997) 303-3 11
CP-violation in b quark radiative inclusive decays H.M. Asatrian a,1,2, G.K. Yeghiyan b~l, A.N. Ioannissian c~l a International Centre for Theoretical Physics, Strada Costiera 11, P.O. Box 586, 34100 Trieste, Italy b Deutches Electronen Synchrotron DESI: Hamburg, Germany ’ Dept. of Physics, Technion - Israel Inst. of Tech., Haifa, Israel Received
23 October
1996; revised manuscript
received 3
February 1997
Editor: P.V.Landshoff
Abstract
The direct CP-violations in SU( 2) L x su( 2)R x U( 1) model and two-Higgs doublet extension of the standard model for dy and b + sy decays are investigated. The calculated value of CP-asymmetry for these two models and for b --+ dy and b -+ sy decays for the wide range of parameters may exceed the value, predicted by the standard model and has a sign opposite to that of the latter. @ 1997 Published by Elsevier Science B.V. b +
The investigation of rare B-meson decays can give important information on new physics in the TeV region. The observation of direct CP asymmetry in B-meson decays will help to understand the CP breaking phenomenon. The first experimental evidence for the exclusive B --) K*y decay has been obtained at CLEO [ 11. More recently, the branching ratio of the inclusive B -+ X,y decay was measured [ 21. The b -+ sy decay has been investigated theoretically for the standard model and its extensions in [ 3-121. CP-violation in B - B system in sum x sum x lJ( 1) model was considered in [ 131. The problem of CP asymmetry for b + sy decay for standard model and its extensions was investigated in [ 14-161. Although the expected decay rate for b -+ dy decay is about 10 + 20 times smaller than for the b -+ sy decay, the CP asymmetry for the first decay can be about 10 times larger [ 14,171. The aim of the present paper is to consider the direct CP decay asymmetry for the b + dy and b --+ sy decays for SU( 2) L x SU( 2) R x U( 1) model and two Higgs doublet extension of the standard model. In Sum x SU(~)R x U(1) model the b -+ dy decay amplitude arises due to the interaction of quark charged weak current with the “left” and “right” W-bosons and charged Higgs field. This interaction has the following form [ 161:
1 E-mail: [email protected]. 2 Permanent address: Yerevan Physics Institute, 2 Alikhanyan 0370-2693/97/$17.00 0 1997 Published PII SO370-2693(97>00297-9
Br., 375036, Yerevan, Armenia.
by Elsevier Science B.V. All righhts reserved.
304
H.M. Asarrian et al./Physics
LCh= _!~ (i?, c, f) [@I’[-gL
cos pKLP_-
gR
Letters B 399 (1997) 303-311
sinpeisKRP+]
1
K
- tan 28KLMd + eis -MuKR cos 28
+ (tan28M,KL
p+
- eia--&KRMd
where WI is the “light” (predominantly left and right W-bosons, tan2P
>
M2 = 2sin28eA gL M&
left-handed)
charged gauge boson and /3 is the mixing
angle between
I
(2)
tan8 = -%,
KL and KR are CKM mixing matrices for left and right charged currents respectively, P& = ( 1 f- ~5)/2, M, and Md are diagonal mass matrices for quarks with Q = 2/3 and Q = -l/3 charges respectively. The matrices KL and KR can be expressed in a form, where KL has only one complex phase and KR has five complex phases [ 181. Phase S in (2) takes his origin from the vacuum expectation value of Higgs field @, connected with the SU( 2)L x U( 1) symmetry breaking:
In ( 1) the term connected with the interaction with heavy (predominantly right) W-boson is omitted, since it is not relevant for b -+ dy decay. The additional (compared with the standard model) phase factor exp(iS) in ( 1) leads to the existence of the new CP violation effects in sum x sum x U( 1) model. We define the direct CP asymmetry for b --) dy decay as [ 141:
r(6 --f dy) - T(b -+ dy) acp = u6
+&)
(3)
+r(b+dy)’
The direct CP asymmetry for b -+ dy decay arises only if the matrix element of decay has an absorptive part, which arises if the final state strong interaction effects are taken into account. In general case the amplitudes of the decays 6 --+ ~!y and b + dy can be expressed in the following form [ 141:
A(6+dy)=xtA;fiA;)r/,*,
A(bjdy)
a
a
where A& Af, are real and absorptive parts of amplitudes, 1,2,3 ,.... Then CP asymmetry is given by: acp =
Ca+b(A:&, - ALAI;)WVi*V,) Ca,b(A;A;
+ AiAi)
FWV,*V6) ’
(4)
=~(Ai+iAh)va. V, are some phases
and CKM-type
factors,
a =
(5)
To take into account QCD-corrections to radiative decays matrix elements the effective Hamiltonian approach is used. We follow [ 161 and use the results of [ 121 for the imaginary part of the amplitude, connected with the 02 operator:
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30.5
Letters B 399 (1997) 303-311
= (mc/mb) 2, L = In z,. We take the ratio of c- and b-quark masses equal to 0.29 [ 121, then the imaginary parts of the amplitudes connected with cc and iiu intermediate states is approximately r = 0.145. We obtain the following expression for effective Hamiltonian of b -+ & decay in the x Sum x U( 1) model:
where z ratio of equal to SU(2)L
e 2(.+ H b+dy=-1fjT2 --rnb{(K,Ld*KiAz Jz + e-iGK~K~A$yO~
+ eiSK$K~A~y)O~
+ i[ ($(KkKkAz
+ e”‘K,“; K”,gA$) tb
KR
(6)
where oL,R I
=
iid~““( 1 f y5)ubF,,
(7)
and the functions A:$, A& ArgL, A& which include leading logarithmic strong interaction corrections, presented in [ 161. Using (5) and (6)) one obtains the following formula for CP asymmetry for b -+ dy decay in sum su(2)R x U(1) model:
2% acp(b
--+ dr) = ([q/2
- (Revrv,
+ ~C,“~“)Uplr
+ rRevTv,>
Ai$sina
{(Imv~v,+rImv~v,)
(A~~+Hco~cuAf~)~
2oc2 81 + $HsinavTu,(Ay/A&
were x
2oc2
- A$YAz)},
(8)
where
Vt = K;tdKLtb,
vu = Kiud&.ub.
vc = f&d&b,
The expression for CP asymmetry in b + sy (in the limit md = 0, m, = 0) decay can be obtained making replacements v,, -+ vi, v, + v:, vt 4 v:, where
u;=K;t,KLtbt v;=K&f&b, We value Let rng E
0:= K;ps&ub.
(9) from (8) by
(10)
want to stress that in [ 14-161 the approximate value for Y was used: Y = 0.12. Now we use the correct r = 0.145 [ 121, which is essential for the numerical results. us proceed to the numerical results. We take crS = 0.212, c2 N 1.1, m, = (175 f 9) GeV, mb = 4.5 GeV, mb( Mz) = (3.5f0.5) GeV [ 19,201. For CKM matrix parameters we use the Wolfenstein parametrization:
Im( vTv,> = -A2A6r],
Re(vTv,)
jvt12=A2A6((1-P)2+r,r2),
= A2A6( (1 - p)p - q2), ~z~,~~=A~A~(p~f~~),
/$I2 = A2A4( (1 + A2p)2 + h8v2), Re(vi*v:) For parameters
=-A2h6(p+A2(p2+v2)),
Im( vi*v:) = A2A6v, l~:l~=A~A*(p~+~~).
A, A, p, cq in ( 11) we use values given in [ 201.
(11)
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Table 1 Minimum and maximum values of ucp for case ( 1) for b --t dy and b --t sy decays for tan28 = 2.0, MwR = M,+ = 10 TeV and for various values of LY a!
2.2 -2.0 -1.8 1.8 2.0 2.2
WP
(b -+ dr)
0.043 0.041 0.049 0.0085 -0.0024 -0.0027
+ + + t
The CP asymmetry
0.107 0.130 0.123 0.162 + 0.172 + 0.160
WP (b -+ sy) -0.0101 f -0.0045 -0.096 t -0.0040 -0.0087 + -0.0039 -0.0037 + 0.0019 -0.0039 f 0.0039 -0.0018 f 0.0044
Table 2 Minimum and maximum values of acp for case (2a) for b --t dy and b -+ sy decays for tan20 = 3.0, MwR = Mp+ = 10 TeV and for various values of LY (Y -3.0 -2.5 -2.0 2.0 2.5 3.0
QP
(b
+
dr)
WP
-0.017 + 0.044 -0.035 + 0.062 0.008 t 0.110 -0.031 f 0.188 -0.036 f 0.108 -0.012 f 0.055
(b
+
SY)
-0.0033 t -0.0045 -0.0114 + -0.0026 -0.0145 + -0.0042 -0.0134 t 0.0102 -0.0008 + 0.0104 -0.0018 f 0.0028
depends also on parameters of SU( 2)~ x sum x U( 1) model: LY,tan 213, MwR, M,+, e will consider the following possibilities for the ratios IKi/K,“bl, IK&Sj/Kh(sj I: W
IGIGL IGfIGI. (1) IGIGd= IG&%(s)I = 1. (2) No restrictions on the ratios IKP,/Kkl,
IK~(~)/K~(,)
1, besides
those which
follow
from the unitarity
conditions for matrices KL and KR. Case ( 1) corresponds to the pseudo-manifest left-right symmetry, when the absolute values of KobayashiMaskawa mixing matrices elements in left and right sectors (KG and K$, i, j = 1,2,3 correspondingly) are equal to each other [ 181. The case (2) corresponds to non-manifest left-right symmetry when IKif;:1 # jK{ 1
[181. It is known that the experiment is in agreement with the standard model predictions for b ---f sy decay rate. Following [ 161, we will consider that in sum x SU(2)R x U( 1) model the branching of b + sy decay can differ from the standard model prediction no more than A = 10%. As for b --f dy decay rate, there is no experimental restriction for it. However, if we assume that in SU( 2)~ x 5’U( 2)~ x U( 1) model the b --+ sy decay rate is the same (with 10% accuracy) as in standard model, then for the case ( 1), the same condition will be satisfied for the b -+ dy decay rate also. For the case (2) we will consider two possibilities: (2a) b ---) dy decay rate is equal with accuracy of A = 10% to that in the standard model; (2b) b -+ dy decay rate is arbitrary. For case ( 1) and for a given Mw,, M,+ the decay asymmetry for b --+ dy and b -+ sy decays depends on CKM parameters, a and tan 28. Taking into account (8) and the equivalence of decay rates (with 10% accuracy) in standard model and SU(~)L x SU( 2)~ x U( 1) model, it is easy to understand, that for cy = 0 the absolute value of decay asymmetry for both of decays can’t exceed the standard model value by more than 10% for all the values A4wR, M+,+, tan28. The sign of the decay asymmetry will be the same as in the standard model. When we have a new source of CP violation, i.e. a # O., the terms in (8) proportional to AfY and AWLAR dr dg - AzYAz contribute to decay asymmetry and one can expect less or more significant deviations from the standard model predictions. However, the restriction for decay rate here also plays the important role. In Fig. 1 the tan26 dependence of maximum and minimum values of acp for b -+ dy and b -+ sy decays for the case ( 1) are given for various values of Mw,, M,+ . Due to the presence of the terms proportional to sin (Y in (8) there is a difference between standard model predictions (-ucp( b --f sy)/10p3 = 2.9 f 6.4, ucp( b -+ dy)/lO-* = 3.7 + 16), which practically coincide with the results obtained for Mw, = M,+ = 50 TeV and predictions of the SU( 2) L x SU( 2) R x U( 1) model for MwR, M, + < 20 TeV, tan 28 2 1. The difference is most significant for b -+ sy decay. The sign of asymmetry can be different from those in the standard model for both decays. To illustrate the cy dependence of the decay asymmetry we give in the Table 1 minimum and maximum values of decay asymmetry for two decays for tan 20 = 2, M W, = M,+ = 10 TeV and various values of (Y (for 2.30 I /aI I 3.14 and /LYE < 1.9 the condition for decay rate is not satisfied). As we have mentioned above, the difference between results obtained for ucp when taking Y = 0.12 or 0.145 in the expression (8)
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Letters B 399 (1997) 303-311
307
Fig. 1. Maximum and minimum values of ncp in SU(2)r. x S’U(Z), x U( 1) model for the case (1) and a) for b + dy decay, b) for b ---f sy decay for different values of MwR and M,+: MwR = M,+ = 5 TeV (curves 1 and 2); Mw, = M,+ = 10 TeV (curves 3 and 4); MwR = M,+ = 20 TeV (curves 5 and 6); MwR = M,+ = 50 TeV (curves 7 and 8).
for b -+ sy decay it can reach 30%. For this reason values of ucp in Fig. 1 are lower than those in [ 161. Let us now proceed to the case (2). In Fig. 2a we give the dependence of acp on tan 28 for the decay b --+ dy for case (2a) and for the decay b + sy for case (2). It is obvious that for the case (2), when there are no restrictions on right current mixing matrix (besides unitarity condition), the decay asymmetry in sum x sum x U( 1) model is much more different from the standard model predictions, than for the case ( 1). Indeed, for the case (2) the minimum value of the asymmetry for b -+ dy decay can reach (for Mw, = M,+ = 5 TeV) the value -0.18, while for the previous case the minimum value of ucp for the same values Mw, and M,+ is equal to -0.02. For b -+ sy decay.the absolute value of decay asymmetry for the same values MwR and M,+ is 1.5 + 2 times higher than for the previous case. We give in Table 2 minimum and maximum values of decay asymmetry for two decays for tan28 = 3, MwR = M,+ = 10 TeV and various values of LY(for (~1 5 1.80 the conditions for decay rate is not satisfied). It is clear that for the case (2a) the deviations from the standard model predictions are more significant and can
is non-negligible:
308
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Letters B 399 (1997) 303-311
Fig. 2. Maximum and minimum values of ncp in sum x su(2)R x U( 1) model for the case (2a) and a) for b -+ cty decay, b) for b + sy decay for different values of MwR and M,+: M w, = M,+.t. = 5 TeV (curves 1 and 2) ; MwR= M,+ = 10 TeV (curves 3 and 4) ; Mw, = M,+ = 20 TeV (curves 5 and 6) ; A4wR= M+,+= 50 TeV (curves 7 and 8).
take place for a larger parameter space, than for the case ( 1). In Fig. 3 for the case (2b) (when we have no restriction for b --+ dy decay rate) the decay asymmetry minimum and maximum values are given. We note, that for some values of tan28, Mw,, M,+ the decay asymmetry ucp from (8) becomes abnomally large. This means, that for such values of tan 28, Mw,, M,+ (8) becomes incorrect (imaginary part of the amplitude becomes non-small in comparison with the real part and it is necessary to take into account more terms of perturbation theory on czS). Nevertheless, it is reasonably safe to suggest that in this case the difference from the standard model predictions for MwR, M,+ 5 10 TeV can be significant. Thus, for the case of non-manifest left-right symmetry for the large parameter space of the sub x SU(2)R x U(1) model (M wR, M,+ 5 (10 f 15) TeV, tan20 2 (1.5 + 2.5)) one can expect a significant deviations from the standard model predictions for acp for both decays. We discuss next the two Higgs doublet extension of the standard model. In general case Yukawa interaction of quarks with Higgs doublets ~1 and 92 is:
I::.-.-.-----..: ,
H.M. Asatrian et d/Physics
a,,/ 30
Letters B 399 (1997) 303-311
309
1 o->
. ..
,,’ __I’
,/’
20
...... 3 ./........ ‘._____._._.__..._. _..:
______5._.._.------.----. 7 __. __
tan219 Fig. 3. Maximum and minimum values of CIC,Din SU(Z)L x Sum x U( 1) model for the case (2b) and for b ---f dy decay for different values of MwR and Mp+ : MwR = M,+ = 5 TeV (curves 1 and 2); MwR = Mp+ = 10 TeV (curves 3 and 4) ; MwR = Mp+ = 20 TeV (curves 5 and 6); A4wR = M,+ = 50 TeV (curves 7 and 8).
where 9~ is the quark doublet and dR and UR are quark singlets and ys, y;, yf’, y$ are matrices in flavor space [ 151. Usually two versions of this model are considered [ 151: model I, where only one doublet (~1) interacts with quarks: $j = g = O., model II, where one of doublets interacts with up-type quarks and the second one interacts with down-type quarks: y; = yt’ = 0. In paper [ 151 a third model (III) was considered where both Higgs doublets interact with up and down quarks and all of the quantities yy, y:, yf, y$ are non-zero. Generally speaking, in this case the flavor changing neutral currents can arise [ 151. The restrictions on Higgs particles masses and other parameters in such a model were considered in [ 211. The last model (model III) is close in some respect to the SU( 2) L x SU( 2) R x U( 1) model: for this model, as for the previous one, new CP-violating phase arises. As for models I and II, there are no new sources of CP violation. Formula for CP asymmetry in b + sy decay for two Higgs doublet extension of the standard model is the following [ 151:
_ 2z (1 - r) Im(v:u,) 81 bt12
where St, h,
(CcH + Re(StSb)C$)ctaS
9
(13)
q
C&CwH, CFH, C& C,, are given in [ 151. We note, that there is a difference between formula (13) and the expression for CP asymmetry in [ 151: in [ 151 the factor 2/9 is missing. In Table 3 the numerical results for the model III for some values of charged Higgs boson masses are given. Generally speaking, values of ucp for b -+ sy decay, given by Table 3, are lower than the results [ 151 for the reason, mentioned above, but the deviation from the standard model predictions for relatively low masses of charged Higgs boson 5 200 GeV (this is within the limits given in [ 211) can be very large (more than 5
H.M. Asatrian et al./Physics Letters B 399 (1997) Xl-311
310
Table 3 Values of CP-asymmetry for b --f s + y and b ---fd + y for model III, for A < 10% and for A < 50% mH+
(GeV)
A < 10%
A < 50%
-acp(b -+ sy)/10e3 50 100 200 400 800 1600 3200 6400 12800 25600
-53 + 53 -45 + 45 -35 + 35 -25 t 25 -16 i 16 -9.7 t 9.5 0.6 t 7 2~7 2.6 + 6.8 2.8 + 6.5
acp(b --f dy)/10h2 -17 -17 -18 -18 -18 -17 0.6 3.1 3.5 3.6
+ 17 + 17 +- 18 f 18 f 18 +- 18 t 17 + 17 t 17 t 16
-acp (b ---)sy)/10W3 -71 f71 -61 e-61 -48 + 48 -34 f 34 -22 j 21 -12 f 12 -5t11 0.4 + 8.3 2.3 + 6.8 2.8 + 6.5
acp(b -+ dy)/10e2 -23 -23 -23 -24 -24 -22 -4.1 2.4 3.5 3.6
f t t + + + f
23 23 23 24 24 25 25 f 20 f 17 f 16
Table 4 Values of CP-asymmetry for the Model I for A < 10% and for A < 50% IQ+
(GeV)
A < 10% -acp(b
50 100 200 800 3200
A < 50%
-+ sy)/10v3 2.9 2.9 2.9 2.9 2.9
t t t f t
6.7 6.7 6.7 6.7 6.5
acp(b --+ dy)/10e2 3.6 3.6 3.6 3.6 3.6
+ + + + +
17 17 17 17 16
-acp (b -+ sy)/10p3 2.9 2.9 2.9 2.9 2.9
c + f c f
8.6 8.9 8.7 7.6 6.5
acp (b -+ dy)/lO-’ 3.6~21 3.6 f 22 3.6 + 22 3.6 + 19 3.6 + 17
Table 5 Values of CP-asymmetry for the Model II for A < 10% and for A < 50% mH+
(GeV)
-acp(b 400 650 1300 2600 5200
times).
A < 50%
A < 10% --+ sy)/lO_ _ 2.8 f 6.1 2.8 + 6.2 2.9 f 6.3
acp(b --f dy)/lO-’
3.4 f 15 3.5 f 16 3.6 + 16
-acp (b + s~)/lO-~ 2.4 2.5 2.7 2.8 2.9
+ f f + +
5.3 5.7 6.1 6.3 6.3
ncp(b -+ dy)/lO-’ 2.9 3.1 3.4 3.5 3.6
f f f f +
13 14 15 16 16
As for the b ---) dy decay (as follows from Table 3 for A < 10%) the restrictions on absolute value ucp are close to the predictions of the standard model. The difference is that CP-asymmetry here can have opposite sign and the minimum value of ucp can be very small. In Table 3, A < 50%, the minimum and maximum values of CP asymmetry for b + dy decay are given for the case when we use the less severe condition A < 50% for the decay rate. In this case the absolute value of CP asymmetry can be 1.5 times larger than the standard model predictions. As we have mentioned above, for the models I and II there is no new source of CP-violation and as follows from the Tables 4 and 5, the values of ucp for two decays are almost the same as for the standard model. In conclusion, the direct CP-violation in sum x SU( 2)~ x U( 1) model and two-Higgs doublet extension
H.M. Asatrian et al./ Physics Letters B 399 (1997) 303-311
of the differ much model
standard from the stronger (model
311
model for b 4 dy and b --+ sy decays was investigated. The calculated values of CP-asymmetry standard model predictions and can have a sign opposite to that of the latter. The difference is for non-manifest left-right symmetric model and two-Higgs doublet extension of the standard III).
Authors want to thank A. Ali for stimulating discussions. One of the authors (H.A.) wants to thank High Energy Group of ICTP for hospitality. The research described in this publication was made possible in part due to the contract INTAS-93-1630.
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