arXiv:hep-ph/9603318v1 16 Mar 1996
Y ER PH I-1449(19)-95
T EC H N IO N -PH -96-1
hep-ph/9603318
CP-Violation in theD ecay b! s in theLeft-RightSym m etric
M odel
H .M .A satriana,A .N .Ioannissiana;b
a Yerevan Physics Institute,A likhanyan Br. 2,Yerevan,A rm enia
b D ept. ofPhysics,Technion - IsraelInst. ofTech.,H aifa,Israel
e-m ail: asatryan@ vx1.yerphi.am ,ioannissyan@ vxc.yerphi.am
A bstract
T he directC P-violation in the left-rightsym m etric SU (2) SU (2) U (1) m odelis
investigated for the decay b ! s . T he calculated C P-asym m etry for the w ide range
ofparam eters can be larger than in standard m odeland can have an opposite sign.
Yerevan Physics Institute
Yerevan 1995
T he experim ental and theoretical investigation of the decay b ! s can give a
sign for a new physics in the TeV region [1]. T his decay has been extensively studied
during the last years. T he rst experim entalevidence was obtained at C LEO for the
exclusive decay B ! K
[2]. T he decay b ! s has been investigated theoretically
for the standard m odeland its extensions [3]-[11]. C P-violation in B B system was
considered in [12].
In this paper we consider the C P asym m etry in the decay b! s forthe left-right
sym m etric SU (2) SU (2) U (1) m odel,w hich is one of the sim plest extensions of
the standard m odel. C alculated value ofC P-asym m etry forsom e range ofparam eters
ofthe m odel(the m ass ofthe right W -boson,the ratio oftwo H iggs doublet vacuum
expectation values,phasesand m ixing angles)isalm ost2 tim eslargerthan in standard
m odeland can have an opposite sign,w hile the decay rate is alm ost the sam e.
T he Lagrangian of interaction of quarks w ith scalar and SU (2) SU (2) U (1)
gauge elds has the follow ing form :
L = (A ik
igR
~
R i + B ik L i
^a
R iW R a R i + igL
Li
Ri+
c:c)+
^a
L iW L a
(1)
Li
w here i,k= 1,2,3 and
e
=
+
o
!
~=
o
+
o
!
!
i;L ;R
=
U
D
i;L ;R
T he sym m etry SU (2) SU (2) U (1) can be broken to SU (2)L
U (1) by m eans
of vacuum expectation values (vev) of doublet or triplet elds [9, 10, 11]. A s for
SU (2) U (1) sym m etry breaking we assum e that it takes place w hen the scalar eld
acquires the vev:
!
k
0
;
(2)
=
0
ei k0
T he interaction ofquark charged currentw ith W gauge boson and charged H iggs elds
has the form :
i
+ h
1
p (u;c;t) W^1
gL cos K L P
gR sin ei K R P + +
2
1
gL
M u K R P+ +
tan 2 K L M d + ei
+ ’+ p
cos2
2M W L
1
0
d
1
C
B
+
tan 2 M u K L ei
K RM d P
@ sA
cos2
b
L ch =
w here W 1 is the "light" (predom inantly left-handed) charge gauge boson and
m ixing angle between left and right W -bosons,
tan 2 = 2sin 2
gR M
gL M
1
2
W L
2
W R
tan =
k0
k
(3)
is the
K L and K R areK obayashi-M askawa m ixing m atricesforleftand rightcharged currents
respectively,P = (1
5)=2,
M
u
1
0
1
0
md 0 0
B
M d = @ 0 m s 0 CA
0 0 mb
mu 0 0
B
m c 0 CA ;
= @ 0
0
0 mt
T he m atrices K L and K R can be expressed in such a form w here K L has only one
com plex phaseand K R has vecom plex phases[13].In (3)weom ittheterm ,connected
w ith the interaction w ith heavy (predom inantly right)W -boson,w hich isnotrelevant
for b! s decay.
T he e ective lagrangian for b ! s decay has the follow ing form [9,10,11]:
H b!
s
e 2G F
m t W LR L
p m b K tLs K tLbA Ws L O 7L + ei K tLs K tRb
A
O7 +
2
16
mb s
2
sin 2 m t ’ + L m s L L W L R
A O +
K K A O7
+ ei K tLs K tRb 2
cos 2 m b s 7 m b ts tb s
!
i
R
L m t W RL R
i
R
L sin 2 m t ’ +
R
+ e K ts K tb
A
O 7 + e K ts K tb 2
A O
mb s
cos 2 m b s 7
=
(4)
w here
O 7L ;R = us
(1
5)ubF
; O 8L ;R = us
(1
2
8
232
23
A Ws L + A WsgL
1
A Ws L =
3
513
2
16
8
W
W LR
W
23
As
= 23 A s L R + A sgL R
1
3
2
16
8 +
+
+
23
1
A ’s = 23 A ’s + A ’sg
3
16
23
W
5)ubG
19
23
W
+
1
(5)
+
w here = ss(m(m Wb)) and the functionsA Ws L ,A s L ;R ,A s A WsgL ,A sgL ;R ,A sg waspresented
in [9,10,11,14,15,16]:
A Ws L = Q tF 1(x)+ G 1(x);
A WsgL = F 1(x)
A Ws L ;R = Q tF 2(x)+ G 2(x); A WsgL ;R = F 2(x)
+
A s = Q tF 3(y)+ G 3(y);
(6)
+
A sg = F 3(y)
and
"
F 1(x)=
G 1(x)=
F 2(x)=
G 2(x)=
#
x4 3 3 3 2 x 3 2
x + x + + x log(x)
x)4 4
2
4
2 2
"
#
4
3 3 3 2 x 3 3
x
1
+ x
x +
x log(x)
x)4 2
4
2
4 2
"
#
x3 3
1
x + 2 + 3x log(x)
x)3
2
2
"
#
1
x3
15
2
2
+ 6x
x + 2 3x log(x)
x)3
2
2
1
(1
(1
(1
(1
2
(7)
"
#
y3
3
+ 2y2
y ylog(y)
3
(1 y)
2
2
"
#
1
y3 1
2
+ y + y log(y)
(1 y)3
2
2
1
F 3(y)=
G 3(y)=
w here Q t = 2=3 is the electric charge ofthe top quark,x = m 2t=m 2W ,y = m 2t=m 2’ + .
T he direct C P- asym m etry for b ! s decay arises only w hen one take into account
the nalstate interaction e ects,w hen the absorptive parts arise.
A bsorptive parts of the decay am plitude arise from rescattering b ! suu ! s ,
b! scc ! s ,b! sg ! s :
(
X
e 2G F
absorbt
L
L W L
H b!
= i 2 p m b K tLs K tLbA WsgL O 8L tsg! s +
K qs
K qb
A sqq tsg! s +
s
16
2
q= u;c
sin 2 m t ’ + L
m t W RL L
A sg O 8 tsg! s + ei K tLs K tRb 2
A O tsg! s +
(8)
+ ei K tLs K tRb
mb
cos 2 m b sg 8
)
i
R
L m t W RL R
i
R
L sin 2 m t ’ +
R
+ e K ts K tb
A
O 8 tsg! s + e K ts K tb 2
A O tsg! s
m b sg
cos 2 m b sg 8
w here
+
A ’sg =
14
23
+
A ’sg ; A WsgL =
14
23
h
i
A WsgL
0:1687 ; A WsgL R =
14
23
A WsgL R
(9)
In the standard m odelonly for the rescattering b ! suu ! s and b ! scc ! s
one obtains nonegligible contribution [14]. For the two H iggs doublet extension ofthe
standard m odelorleft-right sym m etric m odelthe rescattering b! sg ! s also m ust
be taken into consideration. Taking into account the standard m odelresult [14]we
obtain the follow ing result for the absorptive part of the decay b ! s in left-right
sym m etric m odel:
KR
2
e 2G F s
m bfO 7L ( (K tLs K tLbA WsgL + ei K tLs K tLb tLb A Rsg)+
i 2 p
16
9
K tb
2
R
1 L L
K
2
L
L
+ (K us
(10)
K ub + 0:12K cs
K cb
)c1)+ O 7R e i K tLs K tLb tLs A Rsgg
4
9
K ts
absorbt
H b!
’
s
w here
A Rsg
m t W LR
m t sin 2 ’ +
A sg +
A
mb
m b cos2 2 sg
(11)
W e note that our result is di erent from the one obtained in [15]: the contribution
of the rescattering b ! sg ! s di ers from that in [15] by the factor 2/9. T he
C P-asym m etry for the decay b! s and b! s is de ned as:
(b! s ) (b! s )
(b! s )+ (b! s )
T he resulting C P-asym m etry is equalto:
2 s
f(Im vt vu + 0:12Im vt vc)
acp =
L 2
(jC 7 j + jC 7R j2)vt vt
c1
(R evt vu + 0:12R evt vc)
(A Ws L + H cos A Rs )
4
c1 2
A Rs H sin
+ H sin vtvt(A Ws L A Rsg A Rs A WsgL )
4 9
acp =
3
(12)
(13)
w here
m t sin 2 ’ +
m t W LR
K R tb
As +
; A Rs
A
K L tb
mb
m b cos2 2 s
vt K L tsK L tb;
vc K L csK L cb;
vu K L usK L ub
!
R
KR
K
C 7L = A Ws L + ei tLb A Rs ; C 7R = e i tLs A Rs
K tb
K ts
H ei
ei
(14)
In the num ericalresults we take s = 0:24,c1 ’ 1:1,m t = 175G eV ,m b = 4:5G eV ,
mb
m b(m t) = 3G eV [10]. T he C P-asym m etry acp depends on the param eters of
K obayashi-M askawa m ixing m atrix in W olfensteins param etrization: = 0.221, ,
[14,17,18,19],and also it depends on the param eters ofleft-right sym m etric m odel:
,tan 2 ,M W R ,M ’ + ,H .W eassum ethatjH j= jK tRb=K tLbj’ 1.Forthe xed m asseswe
vary therem aining param etersand obtain theallowed region ofacp values.W etakeinto
accountthatin the left-rightsym m etric m odelthe decay rate cannotdi ersu ciently
from those obtained in the standard m odel. T he point is that the standard m odel
predictions for decay rate are reasonable agreem ent w ith experim ent. W e present in
Fig 1,2,3 the m inim aland m axim alvaluesofthe asym m etry forthe variousM W R ,M ’ +
and 0 < tan 2 < 3:5 (the m axim aland m inim alvalues ofaC P are sym m etric under
changing ofsign oftan 2 ). It is easy to see that the aC P can be about 2 tim es larger
than the m axim alvalue ofasym m etry in the standard m odel. For the m asses ofright
sector 5TeV the asym m etry is rather sensitive on changes ofthe H iggs boson m ass
than therightW -boson m ass.In contrary to thestandard m odel,w heretheasym m etry
have a negative sign,in our case the sign ofthe asym m etry can also be positive. T he
results for aC P in Fig 1,2,3 are obtained under the assum ption that the decay rate
in left-right sym m etric m odelcan di er from the standard m odelpredictions no m ore
than = 10% .O urcalculationsshow thatthe resultsform axim aland m inim alvalues
of aC P practically does not change w hen we vary from 10% to 50% . T his fact is
illustrated in Fig 4,w here we com pare the results for = 10% and = 50% for m asses
M W R = M ’ + = 10TeV .
In table we present the aC P m inim aland m axim alvalues in left-right sym m etric
m odelforsom e valuesofthe param etersM W R ,M ’ + ,tan 2 and for , "best t" [19]:
( ; )= ( 0:05;0:37). For = 0:37 the standard m odelprediction for aC P is -0.64% .
Forthe sam e the value ofaC P in left-rightsym m etric m odelcan be alm ost2.5 tim es
larger.
In conclusion, we have calculated the C P asym m etry in the decay b ! s for
the left-right sym m etric m odelSU (2) SU (2) U (1). W e have show n that the C Pasym m etry forthereasonablerangeofparam eterscan belargerthan in standard m odel
and can have an opposite sign.
T he research described in this publication was m ade possible in part by G rant N
M V U 000 from theInternationalScience Foundation.A .I.hasbeen supported by Lady
D avis Trustship.
4
TA BLE
M W R = 1.5TeV ,M ’ + = 10TeV
M W R = 10TeV ,M ’ + = 1.5TeV
M W R = 10TeV ,M ’ + = 10TeV
M W R = 20TeV ,M ’ + = 20TeV
M W R = 50TeV ,M ’ + = 10TeV
M W R = 10TeV ,M ’ + = 50TeV
M W R = 50TeV ,M ’ + = 50TeV
tan 2 = 1
( 1:51 0:33)%
( 0:90
0:37)%
( 0:73
0:55)%
( 0:93
0:35)%
( 0:65
0:63)%
( 0:66
0:62)%
tan 2 = 2
( 0:76 0:16)%
( 1:27 0:25)%
( 0:93
0:35)%
( 1:31 0:28)%
( 0:67
0:61)%
( 0:70
0:58)%
tan 2 = 3
( 0:39 0:41)%
( 1:21
0:06)%
( 0:48 0:44)%
( 0:74
0:54)%
( 0:77
0:51)%
T he m inim aland m axim alvalues ofaC P for = -0.05, = 0.37 and som e values of
M W R ,M ’ + and tan 2 .
5
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6
Figure C aptions
Fig1. T he m axim aland m inim alvalues ofaC P in % for M W R = 5TeV ,M ’ + = 5TeV
(curves 1 and 2);for M W R = 10TeV ,M ’ + = 10TeV (curves 3 and 4);for M W R = 20TeV ,
M ’ + = 20TeV (curves 5 and 6);for M W R = 50TeV ,M ’ + = 50TeV (curves 7 and 8).
Fig2.T hem axim aland m inim alvaluesofaC P in % forM W R = 1.5TeV and M ’ + = 10TeV
(curves 1 and 2);M W R = 5TeV and M ’ + = 10TeV (curves 3 and 4);M W R = 10TeV and
M ’ + = 10TeV (curves 5 and 6);M W R = 20TeV and M ’ + = 10TeV (curves 7 and 8).
Fig3.T hem axim aland m inim alvaluesofaC P in % forM W R = 10TeV and M ’ + = 1.5TeV
(curves 1 and 2);M W R = 10TeV and M ’ + = 5TeV (curves 3 and 4);M W R = 10TeV and
M ’ + = 10TeV (curves 5 and 6);M W R = 10TeV and M ’ + = 20TeV (curves 7 and 8).
Fig4.T he m axim aland m inim alvaluesofaC P in % forM W R = 10TeV ,M ’ + = 10TeV
and allowed di erencefrom standard m odel10% (curves1 and 2);M W R = 10TeV ,M ’ + = 10TeV
and allowed di erence from standard m odel50% (curves 3 and 4).
7
8
9
10