The Forward-backward asymmetry in the angular distribution of $e^+e^-$ is
studied in the process $B\to e^+e^- and \bar{B}\to \bar{X}_d e^+e^-$ . The
possibility of observing CP violation through the asymmetries in these two
processes is examined.Comment: 5 pages, latex formatte

A. Ali

F. Krüger

G. Buchalla

S. Rai Choudhury

English

arXiv

The forward-backward asymmetry in the angular distribution of e+e- is studied in the processes B&#8594;Xde+e- and B&#x0305;&#8594;X&#x0305;de+e-. The possibility of observing CP violation through the asymmetries in these two processes is examined

Rai Choudhury, S.

Indian Academy of Sciences

ar
X
iv
:h
ep
-p
h/
97
06
31
3v
1 
 1
0 
Ju
n 
19
97
Forward-Backward Asymmetry in B → Xde+e−
S.Rai Choudhury
I.C.T.P., I-34100 TRIESTE
and
Department of Physics, University of Delhi,Delhi 110007, India.
February 1, 2008
Abstract
The Forward-backward asymmetry in the angular distribution of e+e−
is studied in the processes B → Xde
+
e
− and B̄ → X̄de
+
e
−. The
possibility of observing CP- violation through the asymmetries in these
two processes is examined.
1
The flavor changing neutral current transitions B → Xse+e− and B →
Xde
+e− offer a deeper proble for the weak interaction sector of the standard
model since they go through second order weak interactions. The basic quark
transition involved in these b → s and b → d occur through an intermediate
t, c, or u quark. These processes can be described in terms of an effective
Hamiltonian which incorporates both the results of short distance expansion
techniques as well as the effect of virtual quark-antiquark pairs [1] . For B →
Xse
+e− the transitions involving intermediate top, charm and u-quarks enter
respectively with factors VtbV
∗
ts, VcbV
∗
cs and VubV
∗
us . The last of these three is
extremely small compared to the other two ; by the unitarity relation between
the elements of the CKM matrix, the first two of these become effecively negative
of each other. Thus the CKM factors effectively act as an overall factor with the
result that there is very little chance of obtaining details of the CKM matrix,
in particular its CP-violating phase from the study of this process. Krüger
and Sehgal [2] have recently pointed out that the situation is quite different
for the process B → Xse+e−. There the three CKM factors, obtained from
above by the replacement s→d , are comparable so that the cross section for the
process will have significant interference terms, possibly opening up prospects of
meaningful estimation of the complex CKM matrix elements. Krüger and Sehgal
[2] have shown that the cross-section for the processes B → Xde+e−and B̄ →
X̄de
+e− have differences depending on the value of the CKM matrix elements
that may be experimentally significant in the near future.
For inclusive B-decays into lepton pairs, there is another asymmetry, namely
the forward-backward (FB) asymmetry, introduced by Ali,Mannel and Mo-
rozumi [3],which again is another parameter which is likely to be very useful
for comparison of theory with experimental data. For B → Xse+e− process,
this parameter is again not very sensitive to the CKM matrix elements and fur-
thermore tha magnitude of this parameter is the same for the B̄ decay as for the
B-decay. The observations made by Krüger and Sehgal[2] however show that we
may expect that these parameters will be more sensitive in the inclusive B-decay
into lepton pair with non-strange hadrons. In this brief note we obtain quanti-
tative predictions for the FB asymmetry for the processes B → Xd e+ e− and
B̄ → X̄d e+ e−. The results as expected show considerably more dependence
on the value of the CKM matrix elements and also brings out the possibility of
observing the CP-violating phase of the CKM matrix elements in a mixture of
equal numbers of B and B̄ particles.
In the lowest order of the heavy quark effective theory, the process B →
Xd e
+ e− can be equated to the QCD corrected matrix element for the process
( with the p’s and q’s representing the on-shell momentum of the particles)
b (p b) → d (pd) + e+ (q 1 ) + e− ( q 2)
The standard kinematical variables for this process are ( with all dimensional
quantities scaled to the b-quark mass):
q = q1 + q2 ; s = ( q
2 ) ; u = 2 pb. (q2 − q1)
u2( s ) = (s −
(
1 +m2d )
)
( s −
(
1 −m2d )
)
; z = u / u( s )
2
z is the cosine of the angle between ~q1 and ~pb in the rest frame of the lepton
pair. In terms of these variables and the standard Wilson coefficients Ci ’s,[1]
the matrix element for the above process can be written as:
M (pb, q1, q2 ; λ ) = 2
√
2 GF Vtb V
∗
td (α /4π) F, (1)
where
F = Ceff9 (d̄ γµ bL) (ē γµ e) + C10 (d̄ γµ bL) (ē γ
µ γ5 e)
− 2 C7 [d̄ i σµν ]( mb bR + md bL ) ( ē γµ e ))(qν /q2) (2)
The constant C’s are given by [1]
C1 = −0.249, C2 = 1.108, C3 = 1.112X10−2
C4 = −2.569X10−2 , C5 = 7.404X10−3, C6 = −3.144X10−2
C7 = −0.315, C9 = 4.227;
Ceff9 is given by:
Ceff9 = ξ1 + λ ξ2, (3)
with
ξ1 = C9 + gc ( 3C1 + C2 + 3C3 + C4 + 3C5 C6 )
− 1/2 gd (C3 + 3C4) − 1/2 gb (4C3 + 4C4 + 3C5 + C6)
+2 (3C3 + C4 + 3C5 + C6)/9 (4)
ξ2 = (gc − gu ) (3 C1 + C2 )
where
gq = −8 ln(mq)/9 + 8/27 + 4 yq/9 − (4 + 2yq) /3 +
+
√
x θ(x) [ ln(1 + x) − ln(1 − x) − iπ]
+
√
−xθ(−x)[ 2 arctan(1/
√
−x)], (5)
with yq = 4 m
2
q /s ; x = 1− yq. The parameter λ is the ratio VubV ∗ud /VtbV ∗td
and can be expressed in terms of the Wolfenstein parameters as:
λ =
ρ(1 − ρ) − η2 − i η
(1 − ρ)2 + η2 (6)
Large distance effects can also be included in this scheme by adding to the
expression for Ceff9 suitable Breit-Wigner forms corresponding to the J/ψ, ψ
′
resonances. However as we will see, the region in which we will be interested in
is below the resonance region and we will therefore neglect them. We shall also
set the mass of the d-quark and the electron zero. With the matrix elemtent
3
given as above, the differential cross-section for the process b→ d+ e+e− can
be worked out as
d2σ
dz ds
= C (1 − s )
(
(|Ceff9 |2 + C210 ) [ 2 + 2s+ − 2z2(1 − s) ]
+8 C27 [2 − ( 1 − s ) (1 − z2) ] /s
+8 Re(Ceff9 ) [2 C7 − s z C10 ]
−16 C10 C7 z
)
(7)
where C is an overall constant.
With this expression the normalized FB asymmetry A(s) defines as
A(s) =
[
∫ 1
0
−
∫ 0
−1
] dz D(z, s)
[
∫ 1
0
+
∫ 0
−1
] dz D(z, s)
=
−3[ s C10 reCeff9 + 2C10 C7 ]
(1 + 2s) [ |Ceff9 |2 + C210 ] + 4C27 (2 + 1/s) + 12 C7 reCeff9
(8)
where D(z.s) is the left hand side of equation ( 6 ).
The above expression refers to the transion b(pb) → d e+(q1)e−(q2). By
the CPT theorem, the matrix element for the process b̄(pb) → d̄ e−(q1)e+(q2)
is given by M( pb, q2, q1 ; λ
∗ ). Thus but for the imaginary part part of λ
,equation ( 6 ), the FB asymmetry of the process B̄ → X̄d e+ e− would be
exactly the negative of B - decay. The difference in the magnitude of the B-
and the B̄ FB asymmetry would thus directly measure the CP violating phase
of the CKM matrix.
Figures (1)-(3) show the calculated values of the two asymmetry parameters
for three values of the parameter ρ in the experimentally allowed range for η
=0.34. As can be seen , there is some dependence on the value of the param-
eter ρ. The study of the FB asymmetry in B-decays would thus be a useful
confirmatory data in pinning down the value of CKM matrix elements.
The difference between the B and the B̄ asymmetry is most pronounced,
below the J/ψ threshold, as expected. This raises the possibility of measuring
the asymmetry in a beam containing equal number of B and B̄ particles; this
would then be directly proportional to η. In figures (1)-(3), we have shown the
values expected for a mixed system. In Table -I , we show also the value of this
mixed asymmetry parameter averaged over a range of s from 0.05 to 0.35, well
below the region of the resonances.
The magnitude of the FB asymmetry is of the same of the order as the CP-
asymmetry in B → Xd e+ e− and will be within observational range at future
colliders. Improvement of statistics would perhaps also the make the asymmetry
observable in a beam containing equal numbers of B and B̄’s, which exper-
imentally would not require any ’tagging’ and is thus an interesting possibility .
4
Table 1: FB Asymmetry averaged between s=0.05 and s=0.35 for B, B̄ and
B+B̄ system.
ρ For B ForB̄ For B+B̄
0.30 0.086 -0.103 -0.014
-0.07 0.086 -0.094 -0.006
-0.30 0.080 -0.086 -0.005
I would like the International Centre for Theoretical Physics at Trieste where
this work was done. I would also like to thank Professor L.M.Sehgal for going
through the manuscript and for pointing out that very recently completed a
similar investigation whose results match the ones quoted here.
References
[1] For an up-to-date review see G.Buchalla,A.J.Buras and M.E.Lautenbacher,
Rev. Mod. Phys., 68,1125(1996)
[2] F.Krüger and L.M.Sehgal, Phys. Rev. D55,2799 (1997).
[3] A.Ali, T. Mannel and T. Morozumi, Phys. Lett., B 273, 505 (1991).
5
Figure 1: FB asymmetry for ρ=0.30
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B̄
For B+B̄
Figure 2: FB asymmetry for ρ=-0.07
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B̄
For B+B̄
Figure 3: FB asymmetry for ρ=-0.30
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B̄
For B+B̄
6


Forward-backward asymmetry in B&#8594;Xde+e-

Crossref

Physical Review D

Forward-backward asymmetry in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mi>−</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>

Publications of the IAS Fellows

ar
X
iv
:h
ep
-p
h/
97
06
31
3v
1 
 1
0 
Ju
n 
19
97
Forward-Backward Asymmetry in B → Xde+e−
S.Rai Choudhury
I.C.T.P., I-34100 TRIESTE
and
Department of Physics, University of Delhi,Delhi 110007, India.
February 1, 2008
Abstract
The Forward-backward asymmetry in the angular distribution of e+e−
is studied in the processes B → Xde
+
e
− and B¯ → X¯de
+
e
−. The
possibility of observing CP- violation through the asymmetries in these
two processes is examined.
1
The flavor changing neutral current transitions B → Xse+e− and B →
Xde
+e− offer a deeper proble for the weak interaction sector of the standard
model since they go through second order weak interactions. The basic quark
transition involved in these b → s and b → d occur through an intermediate
t, c, or u quark. These processes can be described in terms of an effective
Hamiltonian which incorporates both the results of short distance expansion
techniques as well as the effect of virtual quark-antiquark pairs [1] . For B →
Xse
+e− the transitions involving intermediate top, charm and u-quarks enter
respectively with factors VtbV
∗
ts, VcbV
∗
cs and VubV
∗
us . The last of these three is
extremely small compared to the other two ; by the unitarity relation between
the elements of the CKMmatrix, the first two of these become effecively negative
of each other. Thus the CKM factors effectively act as an overall factor with the
result that there is very little chance of obtaining details of the CKM matrix,
in particular its CP-violating phase from the study of this process. Kru¨ger
and Sehgal [2] have recently pointed out that the situation is quite different
for the process B → Xse+e−. There the three CKM factors, obtained from
above by the replacement s→d , are comparable so that the cross section for the
process will have significant interference terms, possibly opening up prospects of
meaningful estimation of the complex CKMmatrix elements. Kru¨ger and Sehgal
[2] have shown that the cross-section for the processes B → Xde+e−and B¯ →
X¯de
+e− have differences depending on the value of the CKM matrix elements
that may be experimentally significant in the near future.
For inclusive B-decays into lepton pairs, there is another asymmetry, namely
the forward-backward (FB) asymmetry, introduced by Ali,Mannel and Mo-
rozumi [3],which again is another parameter which is likely to be very useful
for comparison of theory with experimental data. For B → Xse+e− process,
this parameter is again not very sensitive to the CKM matrix elements and fur-
thermore tha magnitude of this parameter is the same for the B¯ decay as for the
B-decay. The observations made by Kru¨ger and Sehgal[2] however show that we
may expect that these parameters will be more sensitive in the inclusive B-decay
into lepton pair with non-strange hadrons. In this brief note we obtain quanti-
tative predictions for the FB asymmetry for the processes B → Xd e+ e− and
B¯ → X¯d e+ e−. The results as expected show considerably more dependence
on the value of the CKM matrix elements and also brings out the possibility of
observing the CP-violating phase of the CKM matrix elements in a mixture of
equal numbers of B and B¯ particles.
In the lowest order of the heavy quark effective theory, the process B →
Xd e
+ e− can be equated to the QCD corrected matrix element for the process
( with the p’s and q’s representing the on-shell momentum of the particles)
b (p b) → d (pd) + e+ (q 1 ) + e− ( q 2)
The standard kinematical variables for this process are ( with all dimensional
quantities scaled to the b-quark mass):
q = q1 + q2 ; s = ( q
2 ) ; u = 2 pb. (q2 − q1)
u2( s ) = (s − (1 +m2d )
)
( s − ( 1 −m2d )
)
; z = u / u( s )
2
z is the cosine of the angle between ~q1 and ~pb in the rest frame of the lepton
pair. In terms of these variables and the standard Wilson coefficients Ci ’s,[1]
the matrix element for the above process can be written as:
M (pb, q1, q2 ; λ ) = 2
√
2 GF Vtb V
∗
td (α /4π) F, (1)
where
F = Ceff9 (d¯ γµ bL) (e¯ γµ e) + C10 (d¯ γµ bL) (e¯ γ
µ γ5 e)
− 2 C7 [d¯ i σµν ]( mb bR + md bL ) ( e¯ γµ e ))(qν /q2) (2)
The constant C’s are given by [1]
C1 = −0.249, C2 = 1.108, C3 = 1.112X10−2
C4 = −2.569X10−2 , C5 = 7.404X10−3, C6 = −3.144X10−2
C7 = −0.315, C9 = 4.227;
Ceff9 is given by:
Ceff9 = ξ1 + λ ξ2, (3)
with
ξ1 = C9 + gc ( 3C1 + C2 + 3C3 + C4 + 3C5 C6 )
− 1/2 gd (C3 + 3C4) − 1/2 gb (4C3 + 4C4 + 3C5 + C6)
+2 (3C3 + C4 + 3C5 + C6)/9 (4)
ξ2 = (gc − gu ) (3 C1 + C2 )
where
gq = −8 ln(mq)/9 + 8/27 + 4 yq/9 − (4 + 2yq) /3 +
+
√
x θ(x) [ ln(1 + x) − ln(1− x) − iπ]
+
√−xθ(−x)[ 2 arctan(1/√−x)], (5)
with yq = 4 m
2
q /s ; x = 1− yq. The parameter λ is the ratio VubV ∗ud /VtbV ∗td
and can be expressed in terms of the Wolfenstein parameters as:
λ =
ρ(1− ρ)− η2 − i η
(1− ρ)2 + η2 (6)
Large distance effects can also be included in this scheme by adding to the
expression for Ceff9 suitable Breit-Wigner forms corresponding to the J/ψ, ψ
′
resonances. However as we will see, the region in which we will be interested in
is below the resonance region and we will therefore neglect them. We shall also
set the mass of the d-quark and the electron zero. With the matrix elemtent
3
given as above, the differential cross-section for the process b→ d+ e+e− can
be worked out as
d2σ
dz ds
= C (1 − s ) ( (|Ceff9 |2 + C210 ) [ 2 + 2s+ − 2z2(1− s) ]
+8 C27 [2 − ( 1 − s ) (1− z2) ] /s
+8 Re(Ceff9 ) [2 C7 − s z C10 ]
−16 C10 C7 z
)
(7)
where C is an overall constant.
With this expression the normalized FB asymmetry A(s) defines as
A(s) =
[
∫ 1
0
− ∫ 0
−1
] dz D(z, s)
[
∫ 1
0
+
∫ 0
−1
] dz D(z, s)
=
−3[ s C10 reCeff9 + 2C10 C7 ]
(1 + 2s) [ |Ceff9 |2 + C210 ] + 4C27 (2 + 1/s) + 12 C7 reCeff9
(8)
where D(z.s) is the left hand side of equation ( 6 ).
The above expression refers to the transion b(pb) → d e+(q1)e−(q2). By
the CPT theorem, the matrix element for the process b¯(pb) → d¯ e−(q1)e+(q2)
is given by M( pb, q2, q1 ; λ
∗ ). Thus but for the imaginary part part of λ
,equation ( 6 ), the FB asymmetry of the process B¯ → X¯d e+ e− would be
exactly the negative of B - decay. The difference in the magnitude of the B-
and the B¯ FB asymmetry would thus directly measure the CP violating phase
of the CKM matrix.
Figures (1)-(3) show the calculated values of the two asymmetry parameters
for three values of the parameter ρ in the experimentally allowed range for η
=0.34. As can be seen , there is some dependence on the value of the param-
eter ρ. The study of the FB asymmetry in B-decays would thus be a useful
confirmatory data in pinning down the value of CKM matrix elements.
The difference between the B and the B¯ asymmetry is most pronounced,
below the J/ψ threshold, as expected. This raises the possibility of measuring
the asymmetry in a beam containing equal number of B and B¯ particles; this
would then be directly proportional to η. In figures (1)-(3), we have shown the
values expected for a mixed system. In Table -I , we show also the value of this
mixed asymmetry parameter averaged over a range of s from 0.05 to 0.35, well
below the region of the resonances.
The magnitude of the FB asymmetry is of the same of the order as the CP-
asymmetry in B → Xd e+ e− and will be within observational range at future
colliders. Improvement of statistics would perhaps also the make the asymmetry
observable in a beam containing equal numbers of B and B¯’s, which exper-
imentally would not require any ’tagging’ and is thus an interesting possibility .
4
Table 1: FB Asymmetry averaged between s=0.05 and s=0.35 for B, B¯ and
B+B¯ system.
ρ For B ForB¯ For B+B¯
0.30 0.086 -0.103 -0.014
-0.07 0.086 -0.094 -0.006
-0.30 0.080 -0.086 -0.005
I would like the International Centre for Theoretical Physics at Trieste where
this work was done. I would also like to thank Professor L.M.Sehgal for going
through the manuscript and for pointing out that very recently completed a
similar investigation whose results match the ones quoted here.
References
[1] For an up-to-date review see G.Buchalla,A.J.Buras and M.E.Lautenbacher,
Rev. Mod. Phys., 68,1125(1996)
[2] F.Kru¨ger and L.M.Sehgal, Phys. Rev. D55,2799 (1997).
[3] A.Ali, T. Mannel and T. Morozumi, Phys. Lett., B 273, 505 (1991).
5
Figure 1: FB asymmetry for ρ=0.30
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B¯
For B+B¯
Figure 2: FB asymmetry for ρ=-0.07
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B¯
For B+B¯
Figure 3: FB asymmetry for ρ=-0.30
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
For B
For B¯
For B+B¯
6


Forward-Backward Asymmetry in $B\to X_d e^+e^-$

Abstract

Similar works

Full text

Available Versions

Indian Academy of Sciences

Crossref

Publications of the IAS Fellows

Forward-Backward Asymmetry in B→Xde+e−B\to X_d e^+e^-B→Xd​e+e−

Abstract

Similar works

Full text

Available Versions

Indian Academy of Sciences

Crossref

Publications of the IAS Fellows

Forward-Backward Asymmetry in $B\to X_d e^+e^-$