Double-Lepton Polarization Asymmetries and Branching Ratio of the $B\rar γl^+ l^- $ transition in Universal Extra Dimension

0.18 MB·English

by  K. Azizi

Double-Lepton Polarization Asymmetries and
Branching Ratio of the B γl+l transition in
−
→
Universal Extra Dimension
2
1
0
2

n
a
J
K. Azizi1 ∗, N. K. Pak2 †, B. B. S¸irvanlı3‡
5
2
1 Department of Physics, Do˘gu¸s University, Acıbadem-Kadıko¨y, 34722 Istanbul, Turkey

]
h 2 Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
p
-
p 3 Department of Physics, Gazi University, Teknikokullar, 06100 Ankara, Turkey
e
h
[

2
v Abstract
7
2
9 We study the radiative dileptonic B γl+l− transition in the presence of a
→
2
universal extra dimension in the Applequist-Cheng-Dobrescu model. In particular,
.
2
1 using the corresponding form factors calculated via light cone QCD sum rules, we
1
analyze the branching ratio and double lepton polarization asymmetries related to
1
v: this channel and compare the results with the predictions of the standard model. We
i
X show how the results deviate from predictions of the standard model at lower values
r of the compactification factor (1/R) of extra dimension.
a
PACS number(s): 12.60-i, 13.20.-v, 13.20.He
∗e-mail: [email protected]
†e-mail:[email protected]
‡e-mail:[email protected]
1 Introduction
Although the standard model (SM) of particle physics is in perfect agreement with all
confirmedcolliderdata, therearesomeproblemsthatcannotbeaddressedbytheSM.Some
of these problems are matter-antimatter asymmetry, number of generations, unification of
the fundamental interactions, etc. Hence, we need more fundamental theories beyond the
SM (BSM) such that at low energies those theories reduce to the SM. One of the most
interesting candidates as a BSM theory is extra dimension (ED) [1–6]. A kind of ED
which permits both gauge bosons and fermions as SM fields to spread in ED’s is labeled as
universal extra dimension (UED). The simplest case of the UED is the Applequist-Cheng-
Dobrescu (ACD) model [7] which contains only one UED compactified in a circle of radius
R.
We have no experimental evidence for thenew physics effects such asED’s so far, but we
expect that the LHC will open new horizons in this respect. There are two alternative ways
to search for ED’s. In direct search, we look for Kaluza-Klein (KK) excitations directly
by increasing the center of mass energy of colling particles. In indirect search, we look for
the contributions of the KK particles to the hadronic decay channels. The flavor changing
neutral current (FCNC) transitions induced by loop level quark transitions are considered
as good tools for studying the KK effects.
The ACD model has been previously applied to many rare semileptonic decay channels
[8–17]. In the present work, we apply this model to analyze the branching ratio and double
lepton polarization asymmetries defining the radiative dileptonic B γl+l transition.
−
→
The advantage of such decay channel compared to the pure leptonic helicity suppressed
B µ+µ and B e+e channels is that due to the emission of the photon in addition to
− −
→ →
the lepton pair, we have no helicity suppression here and we expect larger branching ratio
[18,19]. The upper experimental limits, Br(B µ+µ ) < 1.5 10 8, Br(B γµ+µ ) <
− − −
→ × →
1.6 10 7, Br(B e+e ) < 8.3 10 8, Br(B γe+e ) < 1.2 10 7 [20] verify our
− − − − −
× → × → ×
expectations in this respect. The considered decay channel proceeds via FCNC transition
of b dl+l at quark level and as we previously mentioned the KK particles can contribute
−
→
to such channels. To evaluate the branching ratio and various double lepton polarization
asymmetries, we will use the form factors entering the effective Hamiltonian calculated via
light cone QCD sum rules [18,21–23].
The layout of the paper is as follows. The introduction is followed by section 2 which
encompasses the theoretical background of the decay channel under consideration and the
associated effective Hamiltonian, a brief review on the ACD model, transition matrix ele-
1
ments defining the radiative dileptonic B γl+l decay channel and explicit expressions
−
→
for the associated observables (differential decay rate and double lepton polarization asym-
metries) in the UED model. In section 3, using the fit parametrization of form factors as
the main ingredients as well as other input parameters, we numerically analyze the physical
observables both in the UED and the SM models and discuss how the results obtained from
the UED model deviate from those of the SM.
2 The radiative dileptonic B γl+l transition in the
−
→
ACD model
As we mentioned in the previous section, the B γl+l transition proceeds via the FCNC
−
→
transition of the b dl+l at the quark level. The most important contribution to the
−
→
B γl+l comes from the pure leptonic B l+l transition. The latter proceeds via
− −
→ →
the box and Z-photon mediated penguin diagrams (see for instance [18,19]). By attaching
the photon to any external and internal charged lines, we will obtain the transition matrix
elements for the B γl+l decay. In the SM, the effective Hamiltonian responsible for
−
→
b qℓ+ℓ transition can be written as
−
→
αG Ceff
= FV V Ceff(d¯γ P b)¯lγ l+C d¯γ P b¯lγ γ l 2 7 d¯iσ q (m P +m P )b¯lγ l ,
Heff √2π tb t∗d" 9 µ L µ 10 µ L µ 5 − q2 µν ν b R d L µ #
(2.1)
where PR(L) = 1+(−2)γ5, and q2 is the transferred momentum squared. The C7eff, C9eff
and C are Wilson coefficients which are the source of difference between the SM and
10
UED models. In the UED, the form of Hamiltonian remains unchanged; however, the
Wilson coefficients are modified [24–28] as a result of interactions of the KK particles with
each other as well as with the usual SM particles. In this model, each Wilson coefficient is
written in terms of the ordinarySM part andan extra part coming fromthe aforementioned
interactions. Hence,
∞
F(x ,1/R) = F (x )+ F (x ,x ), (2.2)
t 0 t n t n
n=1
X
where F (x ) is the SM part and x = m2/M2 with m and M being masses of the
0 t t t W t W
top quark and the W boson, respectively. The second part is defined in terms of the
compactification factor 1/R via
x = m2/m2 , with m = n/R, (2.3)
n n W n
2
where m is mass of the KK particles and n = 0 corresponds to the ordinary SM particles.
n
Here we should also mention that theKK sums appearing inall Wilson coefficients converge
and give finite results.
Now, we proceed to present the explicit expressions of the Wilson coefficients entering
the low energy effective Hamiltonian obtained by a renormalization group evolution from
the electroweak scale down to them scale. Inthe leading log approximation, the coefficient
b
Ceff(1/R) is written as [24–28]:
7
8
Ceff(µ ,1/R) = η1263C (µ ,1/R)+ 8 η2134 η2136 C (µ ,1/R)+C (µ ) h ηai ,
7 b 7 W 3 − 8 W 2 W i
(cid:16) (cid:17) Xi=1
(2.4)
where
α (µ )
s W
η = , (2.5)
α (µ )
s b
and
α (m )
s Z
α (x) = , (2.6)
s 1 β αs(mZ) ln(mZ)
− 0 2π x
with α (m ) = 0.118 and β = 23. The coefficients a and h , with i running from 1 to 8,
s Z 0 3 i i
are also given as [27,28]:
a = ( 14, 16, 6 , 12, 0.4086, 0.4230, 0.8994, 0.1456 ),
i 23 23 23 −23 − −
h = ( 2.2996, 1.0880, 3, 1 , 0.6494, 0.0380, 0.0186, 0.0057 ).
i − −7 −14 − − − −
(2.7)
The functions
1 1
C (µ ) = 1 , C (µ ,1/R) = D (x ,1/R) , C (µ ,1/R) = E (x ,1/R) . (2.8)
2 W 7 W ′ t 8 W ′ t
−2 −2
Also, the 1/R -dependent functions D (x ,1/R) and E (x ,1/R) are defined as
′ t ′ t
∞ ∞
D (x ,1/R) = D (x )+ D (x ,x ), E (x ,1/R) = E (x )+ E (x ,x ) , (2.9)
′ t 0′ t n′ t n ′ t 0′ t n′ t n
n=1 n=1
X X
where the SM parts are given as
(8x3 +5x2 7x ) x2(2 3x )
D (x ) = t t − t + t − t lnx , (2.10)
0′ t − 12(1 x )3 2(1 x )4 t
t t
− −
x (x2 5x 2) 3x2
E (x ) = t t − t − + t lnx , (2.11)
0′ t − 4(1 x )3 2(1 x )4 t
t t
− −
3
and the parts coming from the new interactions can be written in the forms
∞
D (x ,x )
n′ t n
n=1
X
x [37 x (44+17x )] πm R 1
= t − t t + W dy(2y1/2 +7y3/2 +3y5/2) coth(πm R√y)
72(x 1)3 12 W
t − "Z0
x (2 3x )(1+3x ) 1
t t t
− J(R, 1/2) x (1+3x )+(2 3x )[1 (10 x )x ]
− (x 1)4 − − (x 1)4 t t − t − − t t
t t
− − n o
1 (3+x )
t
J(R,1/2) [(2 3x )(3+x )+1 (10 x )x ]J(R,3/2) J(R,5/2) ,
× − (x 1)4 − t t − − t t − (x 1)4
t t #
− −
(2.12)
and
∞
E (x ,x )
n′ t n
n=1
X
x [17+(8 x )x ] πm R 1
= t − t t + W dy(y1/2 +2y3/2 3y5/2) coth(πm R√y)
24(x 1)3 4 − W
t − "Z0
x (1+3x ) 1
t t
J(R, 1/2)+ [x (1+3x ) 1+(10 x )x ]J(R,1/2)
− (x 1)4 − (x 1)4 t t − − t t
t t
− −
1 (3+x )
t
[(3+x ) 1+(10 x )x )]J(R,3/2)+ J(R,5/2) . (2.13)
− (x 1)4 t − − t t (x 1)4
t t #
− −
Here,
1
J(R,α) = dyyα coth(πm R√y) x1+αcoth(πm R√y) . (2.14)
W − t t
Z0
(cid:2) (cid:3)
The next Wilson coefficient is Ceff. In the leading log approximation and at µ scale it
9 b
is given as [27,28]:
Ceff(µ ,sˆ,1/R) = CNDR(1/R)η(sˆ)+h(z,sˆ)(3C +C +3C +C +3C +C )
9 b ′ 9 ′ ′ 1 2 3 4 5 6
1
h(1,sˆ)(4C +4C +3C +C )
′ 3 4 5 6
−2
1 2
h(0,sˆ)(C +3C )+ (3C +C +3C +C ), (2.15)
′ 3 4 3 4 5 6
−2 9
where, sˆ = q2 with the physical region 4m2 q2 m2 . The function CNDR(1/R) in the
′ m2b l ≤ ≤ B 9
naive dimensional regularization (NDR) scheme is defined as
Y(x )
CNDR(1/R) = PNDR + t 4Z(x )+P E(x ). (2.16)
9 0 sin2θ − t E t
W
4
Here we should underline that, due to smallness of P , we can neglect the contribution of
E
last term in Eq. (2.16). The constant PNDR = 2.60 0.25 [27,28], and remaining two
0 ±
functions Y(x ,1/R) and Z(x ,1/R) have the following expressions:
t t
∞
Y(x ,1/R) = Y (x )+ C (x ,x ) , (2.17)
t 0 t n t n
n=1
X
where,
x x 4 3x
t t t
Y (x ) = − + lnx , (2.18)
0 t 8 x 1 (x 1)2 t
(cid:20) t − t − (cid:21)
and,
∞ x (7 x ) πm Rx
t t W t
C (x ,x ) = − [3(1+x )J(R, 1/2)+(x 7)J(R,1/2)] .
n t n 16(x 1) − 16(x 1)2 t − t −
t t
n=1 − −
X
(2.19)
also
∞
Z(x ,1/R) = Z (x )+ C (x ,x ) , (2.20)
t 0 t n t n
n=1
X
with
18x4 163x3 +259x2 108x 32x4 38x3 15x2 +18x 1
Z (x ) = t − t t − t + t − t − t t lnx .
0 t 144(x 1)3 72(x 1)4 − 9 t
t − (cid:20) t − (cid:21)
(2.21)
To complete the presentation of the coefficient Ceff in Eq. (2.15), we define
9
α (µ )
s b
η(sˆ) = 1+ ω(sˆ), (2.22)
′ ′
π
where,
2 4 2 5+4sˆ
ω(sˆ) = π2 Li (sˆ) (lnsˆ)ln(1 sˆ) ′ ln(1 sˆ)
′ 2 ′ ′ ′ ′
−9 − 3 − 3 − − 3(1+2sˆ) − −
′
2sˆ(1+sˆ)(1 2sˆ) 5+9sˆ 6sˆ2
′ ′ ′ ′ ′
− lnsˆ + − . (2.23)
′
3(1 sˆ)2(1+2sˆ) 6(1 sˆ)(1+2sˆ)
′ ′ ′ ′
− −
The coefficients C (j = 1,...6) are given as
j
8
C = k ηai (j = 1,...6) (2.24)
j ji
i=1
X
5
and the constants k have the values
ji
k = ( 0, 0, 1, 1, 0, 0, 0, 0 ),
1i 2 −2
k = ( 0, 0, 1, 1, 0, 0, 0, 0 ),
2i 2 2
k = ( 0, 0, 1 , 1, 0.0510, 0.1403, 0.0113, 0.0054 ),
3i −14 6 − −
(2.25)
k = ( 0, 0, 1 , 1, 0.0984, 0.1214, 0.0156, 0.0026 ),
4i −14 −6
k = ( 0, 0, 0, 0, 0.0397, 0.0117, 0.0025, 0.0304 ),
5i
− −
k = ( 0, 0, 0, 0, 0.0335, 0.0239, 0.0462, 0.0112 ).
6i
− −
Finally, we should define the other functions in Eq. (2.15):
8 m 8 8 4
b
h(y,sˆ) = ln lny + + x (2.26)
′
−9 µ − 9 27 9
b
2(2+x) 1 x 1/2 ln √√11−xx+11 −iπ , for x ≡ 4sˆz′2 < 1
− −
−9 | − |  (cid:16)2arc(cid:12)tan 1(cid:12) , (cid:17) for x 4z2 > 1,
 (cid:12) √x (cid:12)1 ≡ sˆ′
(cid:12) −(cid:12)
(2.27)

where y = 1 or y = z = mc and,
mb
8 8 m 4 4
b
h(0,sˆ) = ln lnsˆ + iπ. (2.28)
′ ′
27 − 9 µ − 9 9
b
The Wilson coefficient C is scale-independent and is given as:
10
Y(x ,1/R)
t
C (1/R) = , (2.29)
10 − sin2θ
W
where, sin2θ = 0.23.
W
Once the Wilson coefficients in UED model are specified explicitly, we proceed to obtain
the amplitude for the decay channel under consideration which is obtained by sandwiching
the effective Hamiltonian between the final photon and the initial B meson state. As
previously noted, the diagrams defining the B γl+l transition are obtained attaching
−
→
the photon to any external and internal charged lines. Hence, we have three kinds of
contributions: 1)thephotonisemittedfromtheinitialquarklines, 2)thephotonisradiated
from the final charged lepton lines and 3) the photon is attached to any charged internal
line. When photon is attached to the initial quark lines (structure dependent part), the
B γl+l transition is described by three Wilson coefficients Ceff, Ceff and C and we
→ − 7 9 10
deal with the long distance effects. Therefore, the amplitude is written as
M = γ(k) B(p) (2.30)
1 eff
h |H | i
6
where k is the momentum of the photon and the p = k + q is the initial momentum. To
obtain the amplitude M , we need to define the matrix elements
1
e
γ(k) d¯γ (1 γ )b B(p) = ǫ ε νqλkσg(q2)+i ε (kq) (ε q)k f(q2) ,
µ − 5 m2 µνλσ ∗ ∗µ − ∗ µ
B
n h i o
(cid:10) (cid:12) (cid:12) (cid:11) (2.31)
(cid:12) (cid:12)
and
e
γ(k) d¯iσ qν(1+γ )b B(p) = ǫ εν qλkσg (q2)+i ε (qk) (ε q)k f (q2) ,
µν 5 m2 µνλσ ∗ 1 ∗µ − ∗ µ 1
B
n h i o
(cid:10) (cid:12) (cid:12) (cid:11) (2.32)
(cid:12) (cid:12)
where ε is the four vector polarization of the photon, and g(q2), f(q2), g (q2) and f (q2)
∗µ 1 1
are the transition form factors.
When photon is radiated from the final charged leptons (Bremsstrahlung part) the
corresponding amplitude is called M . From the helicity arguments it follows that the
2
amplitude M should be proportional to the lepton mass m ; for the cases of the l = e, µ
2 l
we can safely ignore fromsuch contributions. For τ lepton case, this amplitude is calculated
in [19]. Finally, when the photon is attached to any charged internal line, the amplitude
of such contributions (M ) is proportional to m2b ; so these contributions for all leptons are
3 m2W
strongly suppressed and we can safely ignore those contributions (see for instance [18,19]).
Now we proceed to present the 1/R-dependent physical observables defining the radia-
tive dileptonic B γl+l transition. Considering the aforementioned contributions, the
−
→
differential decay rate for the l = e or µ case as a function of the compactification factor is
obtained as [18]:
dΓ α3G2 mˆ 2
(sˆ,1/R) = F V V 2m5 sˆ(1 sˆ)3 1 4 l
dsˆ 768π5 | tb t∗d| B − s − sˆ ×
1 1
A 2 + B 2 + C (1/R) 2 f2(q2)+g2(q2) , (2.33)
× m2 | | | | m2 | 10 |
( B B )
(cid:2) (cid:3) (cid:2) (cid:3)
where sˆ= q2 , mˆ = ml ,
m2B l mB
m
A = A(sˆ,1/R) = Ceff(sˆ,1/R)g(q2) 2Ceff(1/R) b g (q2) ,
9 − 7 sˆm2 1
B
and
m
B = B(sˆ,1/R) = Ceff(sˆ,1/R)f(q2) 2Ceff(1/R) b f (q2) .
9 − 7 sˆm2 1
B
7
In the case of τ, the differential decay width is obtained as [19]:
dΓ
(sˆ,1/R) =
dsˆ
αG 2 α 1 1 4r 4r
F V V m5 π − x3dx 1 m2 A 2 + B 2 (1 x+2r)
(cid:12)2√2π tb t∗d(cid:12) (2π)3 B (12 Zδ r − 1−x B"(cid:16)| ′| | ′| (cid:17) −
(cid:12) (cid:12)
(cid:12) (cid:12) 4r
(cid:12) (cid:12) 1+ 1
1 4r
+ C 2 + D 2 (1 x 4r) 2C10(1/R)fBr − x2dxRe(A′) ln r − 1−x
| | | | − − #− Zδ 1 1 4r
(cid:0) (cid:1) − − 1 x
r −
4r
1+ 1
4 f C (1/R) 2r 1 1−4rdx 2+ 4r 2 x ln r − 1−x
− | B 10 | m2B Zδ "(cid:18) x − x − (cid:19) 1 1 4r
− − 1 x
r −
2 4r
+ (1 x) 1 , (2.34)
x − − 1 x
r − #)
2E
γ
where the f is the leptonic decay constant of the B meson, x = is a dimensionless
B
m
B
m2
parameter with E being the photon energy and r = τ . The lower limit of integration
γ m2
B
over x comes from imposing a cut on the photon energy (for details see [19]). Considering
the experimental cut imposed on the minimum energy for detectable photon, we demand
the energy of the photon to be larger than 50 MeV, i.e., E am with a 0.01. As
γ B
≥ ≥
a result, the lower limit is set as δ = 2a and we will take δ = 0.02 for the lower limit of
integration over x. In Eq. (2.34), we have introduced the following coefficients:
A(sˆ,1/R)
A = A(sˆ,1/R) = ,
′ ′
m2
B
B(sˆ,1/R)
B = B (sˆ,1/R) = ,
′ ′
m2
B
C (1/R)
C = C(1/R) = 10 g(q2) ,
m2
B
C (1/R)
D = D(1/R) = 10 f(q2) . (2.35)
m2
B
At the end of this section we would like to present various 1/R-dependent double–lepton
polarization asymmetries for the transition under study. Note that using the most general
model-independent form of the effective Hamiltonian including all possible forms of the
interactions, these double–lepton polarization asymmetries are calculated in the [29]. To
calculate the double–polarization asymmetries in our case, we consider the polarizations
8
of both lepton and anti-lepton, simultaneously and suggest the following spin projection
operators for the lepton ℓ and the anti-lepton ℓ+:
−
1
Λ = (1+γ s ) ,
1 2 56 −i
1
Λ = (1+γ s+) , (2.36)
2 2 56 i
where i = L,N and T correspond to the longitudinal, normal and transversal polarizations,
respectively. Then, we introduce the following orthogonal vectors sµ in the rest frame of
the lepton and anti-lepton:
~p
s−Lµ = 0,~eL− = 0, ~p1 ,
(cid:18) | 1|(cid:19)
(cid:0) (cid:1)
~
k p~
s−Nµ = 0,~eN− = 0, ~ × 1  ,
k p~
1
(cid:0) (cid:1) ×
s−Tµ = 0,~eT− = 0,~eN(cid:12)(cid:12)(cid:12)− ×~eL−(cid:12)(cid:12)(cid:12),
~p
s+µ = (cid:0)0,~e+(cid:1) = (cid:0) 0, 2 ,(cid:1)
L L ~p
(cid:18) | 2|(cid:19)
(cid:0) (cid:1)
~
k p~
s+µ = 0,~e+ = 0, × 2 ,
N N  ~ 
k p~
2
(cid:0) (cid:1) ×
s+µ = 0,~e+ = 0,~e(cid:12)+ ~e+(cid:12), (2.37)
T T N(cid:12)(cid:12) × L(cid:12)(cid:12)
where ~p are the three–momenta (cid:0)of the(cid:1)lept(cid:0)ons ℓ (+) a(cid:1)nd ~k is three-momentum of the
1(2) −
final photon in the center of mass (CM) frame of ℓ ℓ+. The longitudinal unit vectors are
−
boosted to the CM frame of ℓ ℓ+ via Lorenz transformations
−
p~ E~p
s−µ = | 1| , 1 ,
L CM m m p~
(cid:18) ℓ ℓ| 1|(cid:19)
(cid:0) (cid:1) p~ E~p
s+µ = | 1| , 1 , (2.38)
L CM m −m ~p
(cid:18) ℓ ℓ| 1|(cid:19)
(cid:0) (cid:1)
while the other two vectors remain unchanged. Finally, we define the double–lepton polar-
ization asymmetries as:
dΓ dΓ dΓ dΓ
(~s ,~s+) ( ~s ,~s+) (~s , ~s+) ( ~s , ~s+)
dsˆ −i j − dsˆ − −i j − dsˆ −i − j − dsˆ − −i − j
! !
P (sˆ) = , (2.39)
ij
dΓ dΓ dΓ dΓ
(~s ,~s+)+ ( ~s ,~s+) + (~s , ~s+)+ ( ~s , ~s+)
dsˆ −i j dsˆ − −i j dsˆ −i − j dsˆ − −i − j
! !
where the subindex j also stands for the L, N or T polarization. The subindexses, i and
j correspond to the lepton and anti-lepton, respectively. Using the above definitions, the
various 1/R-dependent double lepton polarization asymmetries are obtained as:
9