Table Of ContentNew Physics effects on decay B γγ in Technicolor Model
s
→
Qin XiuMei, Wujun Huo, and Xiaofang Yang
Department of Physics Department,
Southeast University, Nanjing, Jiangsu 211189, China
Abstract
1 In this paper we calculate the contributions to the branching ratio of B γγ from the charged
s
1 →
0 Pseudo-Goldstone bosons appeared in one generation Technicolor model. We find that the theo-
2
±
n retical values of the branching ratio, BR(Bs γγ), including the contributions of PGBs, P and
→
a
J P± , are much different from theSM prediction. The new physics effects can beenhance 2-3 levels
8
2
1 to SM result. It is shown that the decay B γγ can give the test the new physics signals from
s
→
] the technicolor model.
h
p
-
p
e
h
[
1
v
7
3
4
2
.
1
0
1
1
:
v
i
X
r
a
1
I. INTRODUCTION
As is well known, the rare radiative decays of B mesons is in particular sensitive to
contributions from those new physics beyond the standard model(SM). Both inclusive and
exclusive processes, such as the decays B Xγ, B γγ and B X γ have been
s s s
→ → →
received some attention in the literature[1−14]. In this paper, we will present our results in
Technicolor theories.
The one generation Technicolor model (OGTM)[15−16]is the simplest and most frequently
studied model which contained the parameters are less than SM. Same as other models, the
OGTM has its defects such as the S parameter large and positive[17]. But we can relax the
constraints on the OGTM form the S parameter by introducing three additional parameters
(V,W,X)[18]. The basic idea of the OGTM is: we introduce a new set of asymptotically
free gauge interactions and the Technicolor force act on Technifermions. The Technicolor
interaction at 1Tev become strong and cause a spontaneous breaking of the global flavor
symmetry SU(8) SU(8) U(1) . The result is 82 1 = 63 massless Goldstone bosons.
L R Y
× × −
Three of the these objects replace the Higgs field and induce a mass of W± and Z0 gauge
bosons. And at the new strong interaction other Goldstone bosons acquire masses. As for
the B γγ, only the charged color single and color octets have contributions. The gauge
s
→
couplings of the PGBs are determined by their quantum numbers. In Table 1 we listed the
relevant couplings[19] needed in our calculation, where the V is the corresponding element
ud
of Kobayashi Maskawa matrix . The Goldstone boson decay constant F [20] should be
π
−
F = v/2 = 123GeV, which corresponds to the vacuum expectation of an elementary Higgs
π
field .
P+P−γµ ie(p+ p−)µ
− −
P8+aP8−bγµ −ie(p+−p−)µδab
P+ u d iVud 2[M (1 γ ) M (1+γ )]
2Fπ 3 u − 5 − d 5
q
P+ u d iVudλ [M (1 γ ) M (1+γ )]
8a 2Fπ a u − 5 − d 5
P+P−g gf (p p )
8a 8b cµ − abc a− b µ
TABLE I: The relevant gauge couplings and Effective Yukawa couplings for the OGTM.
2
At the LO in QCD the effective Hamiltonian is
4G 8
F ∗ − −
= − V V C (M )O (M ). (1)
Heff √2 tb ts i W i W
i=1
X
∗
Where, as usual, G denotes the Fermi coupling constant and V V indicates the Cabibbo-
F tb ts
Kobayashi-Maskawa matrix element.And the current-current, QCD penguin, electromag-
netic and chromomagnetic dipole operators are of the form
O = (c γµb )(s γ c ) (2)
1 Lβ Lα Lα µ Lβ
O = (c γµb )(s γ c ) (3)
2 Lα Lα Lβ µ Lβ
O = (s γµb ) (q γ q ) (4)
3 Lα Lα Lβ µ Lβ
q=u,d,s,c,b
X
O = (s γµb ) (q γ q ) (5)
4 Lα Lβ Lβ µ Lα
q=u,d,s,c,b
X
O = (s γµb ) (q γ q ) (6)
5 Lα Lα Rβ µ Rβ
q=u,d,s,c,b
X
O = (s γµb ) (q γ q ) (7)
6 Lα Lβ Rβ µ Rα
q=u,d,s,c,b
X
O = (e/16π2)m s σµνb F (8)
7 b L R µν
O = (g/16π2)m s σµνTab Ga (9)
8 b L R µν
where α and β are color indices, α = 1,...,8 labels SU(3)c generators, e and g refer to the
electromagnetic and strong coupling constants, while F and Ga denote the QED and
µν µν
QCD field strength tensors, respectively.
The Feynman diagrams that contribute to the matrix element as the following In Fig.2
b b b
= +
s
s s
b b
+
+
s
s
FIG. 1: Examples of Feynman diagrams that contribute to the matrix element.
± ±
the shot-dash lines represent the charged PGBs P and P of OGTM. We at first integrate
8
out the top quark and the weak W bosons at µ = M scale, generating an effective five-
W
quark theory and run the effective field theory down to b-quark scale to give the leading
3
b s b s b s
p p t
8
t t p p
8
FIG. 2: The Feynman diagrams that contribute to the Wilson coefficients C7,C8.
log QCD corrections by using the renormalization group equation. The Wilson coefficients
are process independent and the coefficients C (µ) of 8 operators are calculated from the
i
Fig.2.The Wilson coefficients are read[21]
C (M ) = 0, i = 1,3,4,5,6, C (M ) = 1, (10)
i W 2 W
B(x) 8B(y)
C (M ) = A(δ)+ + (11)
7 W − 3√2G F2 3√2G F2
F π F π
D(x) 8D(y)+E(y)
C (M ) = C(δ)+ + (12)
8 W
− 3√2G F2 3√2G F2
F π F π
with δ = M2 /m2, x = (m(P±)/m )2 and y = (m(P±)/m )2.From the Eq(11),(12) , we
W t t 8 t
can see the situation of the color-octet charged PGBs is more complicate than that of the
color-singlet charged PGBs ,because of the involvement of the color interactions. where
1 + 5 δ 7 δ2 3δ 1δ2
A(δ) = 3 24 − 24 + 4 − 2 log[δ] (13)
(1 δ)3 (1 δ)4
− −
11 + 53y 25y2
B(y) = −36 72 − 72
(1 y)3
−
1y + 2y2 1y3
+ −4 3 − 3 log[y] (14)
(1 y)4
−
1 5δ 1δ2 3δ2
C(δ) = 8 − 8 − 4 4 log[δ] (15)
(1 δ)3 − (1 δ)4
− −
5 + 19y 5y2
D(y) = −24 24 − 6
(1 y)3
−
1y2 1y3
+ 4 − 2 log[y] (16)
(1 y)4
−
3 15y 15y2 9y 9y2
E(y) = 2 − 8 − 8 + 4 − 2 log[y] (17)
(1 y)3 (1 y)4
− −
By caculate the graphs of the exchanged W boson in the SM we gained the function A
and C;And by caculate the graphs of the exchanged color-singlet and color-octet charged
PGBs in OGTM we gained the function B, D and E. when δ < 1, x,y >> 1, the OGTM
contribution B, D and E have always a relative minus sign with the SM contribution A
4
and C. As a result, the OGTM contribution always destructively interferes with the SM
contribution.
The leading-order results for the Wilson coefficients of all operators entering the effective
Hamiltonian in Eq.(1) can be written in an analytic form. They are
8
Ceff(m ) = η16/23C (M )+ (η14/23 η16/23)
7 b 7 W 3 − ×
8
C (M )+C (M ) h ηai. (18)
8 W 2 W i
i=1
X
With η = α (M )/α (m ),
s W s b
626126 56281 3 1
h = ( , , , , 0.6494,
i
272277 −51730 −7 −14 −
0.0380, 0.0186, 0.0057). (19)
− − −
14 16 6 12
a = ( , , , ,
i
23 23 23 −23
0.4086, 0.4230, 0.8994,0.1456). (20)
− −
To calculate B γγ , one may follow a perturbative QCD approach which includes
s
→
a proof of factorization, showing that soft gluon effects can be factorized into B meson
s
wave function; and a systematic way of resumming large logarithms due to hard gluons with
energies between 1Gev and m . In order to calculate the matrix element of Eq(1) for the
b
B γγ , we can work in the weak binding approximation and assume that both the b and
s
→
the s quarks are at rest in the B meson, and the b quarks carries most of the meson energy,
s
and its four velocity can be treated as equal to that of B . Hence one may write b quark
s
momentum as p = m v where is the common four velocity of b and B . We have
b b s
1
p k = m v k = m m = p k ,
b · 1 b · 1 2 b Bs b · 2
p k = (p k k ) k =
s 1 1 2 1
· − − ·
1
m (m m ) = p k , (21)
−2 Bs Bs − b s · 2
We compute the amplitude of B γγ using the following relations
s
→
0 s¯γ γ b B (P) = if P ,
h | µ 5 | s i − Bs µ
0 s¯γ b B (P) = if M , (22)
h | 5 | s i Bs B
where f is the B meson decay constant which is about 200 MeV .
Bs s
5
The total amplitude is now separated into a CP-even and a CP-odd part
T(B γγ) = M+F Fµν +iM−F F˜µν. (23)
s µν µν
→
We find that
4√2αG
M+ = − Ff m V∗V
9π Bs bs ts tb ×
m 3C
b BK(m2)+ 7 . (24)
mBs b 8Λ¯ !
with B = (3C +C )/4, Λ¯ = m m , and
− 6 5 Bs − b
4√2αG
− F ∗
M = f m V V
9π Bs bs ts tb ×
m 3C
A J(m2)+ b BL(m2)+ 7 . (25)
q q q mBs b 8Λ¯ !
X
where
A = (C C )N +(C C )
u 3 5 c 4 6
− −
1
A = [(C C )N +(C C )]
d 3 5 c 4 6
4 − −
A = (C +C C )N +(C +C C )
c 1 3 5 c 2 4 6
− −
1
A = [(C +C C )N +(C +C C )] (26)
s 3 4 5 c 3 4 6
4 − −
1
A = [(C +C C )N +(C +C C )]. (27)
s 3 4 5 c 3 4 6
4 − −
The functions J(m2), K(m2) and L(m2) are defined by
J(m2) = I (m2),
11
K(m2) = 4(I (m2) I (m2)),
11 00
−
L(m2) = I (m2), (28)
00
with
1 1−x xpyq
I (m2) = dx dy (29)
pq m2 2xyk k iε
Z0 Z0 − 1· 2 −
The decay width for B γγ is simply
s
→
m3
Γ(B γγ) = Bs( M+ 2 + M− 2). (30)
s
→ 16π | | | |
In SM, with C = C (M ) = 1 , and the other Wilson coefficients are zero, we find
2 2 W
Γ(B γγ) = 1.3 10−10 eVwhichamountstoabranchingratioBr(B γγ) = 3.5 10−7,
s s
→ × → ×
6
for the given Γtotal = 4 10−4 eV. In numerical calculations we use the corresponding
Bs ×
input parameters M = 80.22 GeV, α (m ) = 0.117, m = 1.5 GeV, m = 4.8 GeV
W s Z c b
and V V∗ 2/ V 2 = 0.95 , respectively. The present experimental limit[22] on the decay
| tb ts| | cb|
B γγ is
s
→
Br(B γγ) 8.6 10−6, (31)
s
→ ≤ ×
which is far from the theoretical results. So, we can not put the constraint to the masses
of PGBs. The constraints of the masses of P± and P± can be from the decay[24] B sγ :
8 →
m ± > 400GeV.
P8
8
200Gev
600Gev
1000Gev
6 1400Gev
SM
)
6
- 0
1
r( 4
B
2
0
600 700 800 900 1000
m (Gev)
p8
±
FIG. 3: the Br(Bs → γγ) about the mass of P8 under different values of mP±.
± ±
Fig.3(4) denotes the Br(B γγ) about the mass of P (P ) under different values of
s → 8
±
mP± (P8 ). From Fig.3 and 4, we find the the curves are much different from the the SM
one. It can be enhanced about 1-2 levels to the SM prediction in the reasonable region of
the masses of PGBs. This gives the strong new physics signals from the Technicolor Model.
± ±
The branching ratio of B γγ decrease along with the mass of P and P reduce. This is
s → 8
±
from the decoupling theorem that for heavy enough nonstandard boson. When m(P ) and
±
m(P ) have large values, the contributions from OGTM is small.From the Eq(16),(17),(18)
8
,we can see the functions B, D and E go to zero, as x, y .The branching ratio in the
→ ∞
7
8 600Gev
1000Gev
1400Gev
1600Gev
6 SM
)
6
- 0
1
(
r
B 4
2
0
200 400 600 800 1000
m (Gev)
p
± ±
FIG. 4: the Br(B γγ) about the mass of P under different values of P .
s → 8
Fig.(3) is changed much faster than that in the Fig.(4).This is because the contribution to
±
B γγ from the color octet P is large when compared with the contribution from color
s → 8
±
singlet P .
As a conclusion, the size of contribution to the rare decay of B γγ from the PGBs
s
→
strongly depends on the values of the masses of the charged PGBs. This is quite different
from the SM case. By the comparison of the theoretical prediction with the current data
± ±
one can derived out the the contributions of the PGBs: P and P to B γγ and give
8 s →
the new physics signals of new physics.
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