ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 5, с. 672-680
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
EFFECTS OF FINAL-STATE INTERACTIONS IN PURE ANNIHILATION
DECAY OF B0 — n+n-
© 2014 Behnam Mohammadi*, Hossein Mehraban**
Physics Department, Semnan University, Iran Received December 14, 2012; in final form, October 1, 2013
The hadronic decay of B0 — n- is analyzed by using "QCD factorization" (QCDF) method and finalstate interaction (FSI). First, the B0 — n+n- decay is calculated via QCDF method and the annihilation graphs only exist in this method. Hence, the FSI must be seriously considered to solve the B0 — n-decay and the K+(*)K-(*) and K0(*)K0O) via the exchange of K0(*^ and K-(*) mesons are chosen for the intermediate states. To estimate the intermediate state amplitudes, the QCDF method is again used. These amplitudes are used in the absorptive part of the diagrams. The experimental branching ratio of B0 — n- decay is less than 1.2 x 10-6 and our results according to the QCDF method and FSI are 0.68 x 10-8 and 1.18 x 10-6, respectively.
DOI: 10.7868/S0044002714050183
1. INTRODUCTION
B-meson non-leptonic decays are significant for testing theoretical frameworks and searching for new physics beyond the standard model. The next-to-leading-order (NLO) low-energy effective Hamilto-nian is used for the weak interaction matrix elements and the final state interaction (FSI). The importance of the FSI in hadronic processes has been identified for a long time. Recently, its applications in D and B decays have attracted extensive interest and attention of theorists.
Since the hadronic matrix elements are fully controlled by non-perturbative QCD, the most important problem is how to evaluate them properly. Factorization method enables one to separate the non-perturbative QCD effects from the perturbative parts and to calculate the latter in terms of the field theory order by order. Several factorization approaches have been proposed to analyze B-meson decays, such as the naive factorization approach, the QCD factorization approach, the perturbative QCD approach and Soft-Collinear-Effective-Theory; none provided an estimate of the FSI at the hadronic level. These approaches successfully explain many phenomena, however there are still some problems which are not easy to describe within this framework.
These may be some hints for the need of FSI in B decays. FSI effects are non-perturbative in nature [1]. In many decay modes, the FSI may play a crucial
E-mail: b_mohammadi@sun.semnan.ac.ir
E-mail: hmehraban@semnan.ac.ir
role [2]. In this way, the CKM matrix elements and color factor are suppressed and the CKM's most favored two-body intermediate states are the only ones that have been taken into consideration [3].
The FSI can be considered as a soft re-scattering style for certain intermediate two-body hadronic channel B°s — K+K- decay. Therefore, the FSI estimated via the exchange of i°s mesones processes at the hadron loop level (HLL) is explained in Section 4.
As we know, the branching ratio of B0 — n+n-decay has already been estimated by using the perturbative QCD approach and predicted 6.52 x 10-7 [4] and the experimental result of this decay is BR(B0 —
— n+n-) < 1.2 x 10-6 [5]. We calculated the B0 —
— n+n- decay according to "QCD factorization" (QCDF) method. This process is only occurred via annihilation between b and 0. We selected the leading-order Wilson coefficients at the scale mb [6, 7] and obtained the BR(B0 — n+n-) = 0.68 x 10-8. It is therefore, expected to be very small in factorization approach. The FSI can give sizable corrections and we can include it [8—10]. Rescattering amplitude can be derived by calculating the absorptive and the dispersive part of triangle diagrams. In the FSI effects,
intermediate states are K+(*)K-(*) and Ks(*)K°(*). We calculated the B0 — n+n- decay according to HLL method. In this case, the branching ratio of B0 — n+n- is 1.18 x 10-6.
We present the calculation of QCDF for B0 —
— n+n- decay in Section 2. In Section 3, we calculate the amplitudes of the intermediate state decays.
n (n +)
u ( u)\ ./ ■■' n+(n -)
n-(n+)
u ( u )\//rc+(n )
n (n +)
n (n +)
-v a (a ) b u(u)\ //k+(n-)
Fig. 1. Diagrams describing the decay of B0 ^ n+n-.
u(u) \//k+(n )
Then, we present the calculation of HLL for B0 ^ ^ n+n- decay in Section 4. In Section 5, we give the numerical results, and in the last section, we have a short conclusion.
2. QCD FACTORIZATION OF B0 — n+n-DECAY
To compare QCDF with FSI, we explore QCDF analysis. In this case we only have current—current, penguin and electroweak penguin annihilation effects. This contributions are small, but it is interesting and necessary to discuss about them. Feynman diagrams for the B0 — n+n- decay are shown in Fig. 1. Diagrams 1a— 1c are factorizable at as order, while 1 d and 1 e are non-factorizable diagrams. When all the basic building block equations are solved, for the case that both mesons are pseudoscalar, it is found that weak annihilation Kernels exhibit endpoint divergence. Divergence terms are determined
by /0 dx/x and /0 dy/y. For the liberation of the divergence, a small e of Aqcd/Mb order was added to the denominator. So, the answer to the integral becomes ln (1 + e)/e form, which is shown with XA.
Specifically, we treat XA by using pA = 0.5 and a strong phase = —55° [11].
According to diagrams 1b and 1c the factorzable amplitude for B0 — n+n- decay is
A(B°^7t+7t-) = - APX C3 + -U ) (ir+ir-\d(V - A)ßu\0) +
(1)
1
+ ( c5 + ^c7 ) \d(V + A)ßu\0)
x <0|b(V - ^12?°)} = -4i^fByBXp x
1
C3 + ) (ir+ir~\d(V - Ä)ßv\0) +
+ ( c5 + \c7 ) (ir+ir~\d(V + ,4)^0)
where Xp is the product of elements of the quark mixing matrix. Using the unitarity relation \u + Ac +
a
n
B
+
n
B
B
B
X
X
X
+ Xt = 0 we write
Xp = VpbVps,
p=u,c
Gf
Cf
ciVubV*s + ( c4 + ~cio ) Xp
N 2
c
X { (y)
dxdy x
+
_y(1 - xy) x2y_
+
+ rx+rx ^WiWy)-^ j + + ) Ap x
1
x J dxdy^+ (x)0n- (y)
11
+
x(1 — xy) ' xy2
+
r" =
' X
2m2,
(mb — mu)(mu + md)'
We then find the simple amplitude
M(B0 ^ n+n-) -
^fBsßi hV^VZ, +
264 + -&4;EW
Xp
where bi, b4, and b4;EW correspond to the current-current annihilation, penguin annihilation, and elec-troweak penguin annihilation. These non-singlet annihilation coefficients are given:
bi =
Nc
b4 = ^[c4A\ + c6A%
Cf
(2)
and the subscripts V ± A represent the chiral projections 7^(1 ± y5). This amplitude vanishes for ms — — 0. The as-order matrix (n+n-\d(V ± A)^u\0) also vanishes due to the cancellation between the amplitudes of diagrams 1b and 1c, so we neglect them in the future diagrams. The current—current annihilation and nonfactorizable contributions can be obtained by calculating the amplitudes of diagrams 1a, 1d, and 1e. They are
h,ew = jp lcwal + c&a2ï>
ci are the Wilson coefficients, Nc is the color number and
(7)
= 2nas
A1 w A2 =
n
9[XA-4 + -\+rfX2A
Cf =
3
N2C~ 1
2 Nr. ■
For the running coupling constant, at two-loop order (NLO) the solution of the renormalizaton group equation can always be written in the form
A(BS -»■ tt+tt ) = -j=fBaf*+U-naa(mb) x (3)
4n
a» =
ßoln
M
qcd
1
ßl V AQcd
ß2
0
ln
ß
Ä2—
^qcd
here
ßo =
11Nc — 2nf
(8)
(9)
ß1 =
34Nc2 10Nc nf
3
3
— 2Cf nf,
where x and y are the mesons momentum fractions and denote the leading-twist light-cone distribution amplitude of pseudoscalar mesons. When final-state mesons are pseudoscalar, we fix 2 (x) = 1. For n±, the ratios rX± are defined as
and running as(y) evaluated with nf = 5. There are large theoretical uncertainties related to the modeling of power corrections corresponding to weak annihilation effects, we parameterize these effects in terms of the divergent integrals Xa (weak annihilation)
XA = (l + pe^)\n^, Ah
P < 1, Ah = 0.5 GeV.
(10)
3. AMPLITUDES OF INTERMEDIATE STATES
(4)
(5)
(6)
In this section, before analyzing FSI in B0 ^ ^ n+n- decay we introduce the factorization approach in detail. The effective Hamiltonian for B decays consists of a sum of local operators Qi multiplied by QCDF coefficients ci and products of elements of the quark mixing matrix [11]. Factorization approach of the heavy meson decays can be expressed in terms of different topologies of various decay mechanism such as tree, penguin, and annihilation.
When two intermediate mesons exchange s quark and two final-state mesons exchange d quark, the Kand Kmesons are produced for intermediate state via exchange mesons of K0*. If s quark be exchanged between two intermediate mesons and u quark be exchanged between two final-state mesons, the K0(* and K0(* mesons are produced for intermediate state via Kas the exchange mesons.
1
2
x
1
o
K-
: к+ s ""*
u ""'
\k+(k -) s
u(s)
B0 w-
ч « (s) " b
••/'к-(k +)
According to quark level diagrams in the next section, the intermediate mesons cannot exchange c quark, because B0 — D+D-(D0D0) — n+n- has not been observed.
For B0 — K+K-(K+*K-*) decays, Feynman diagrams are shown in Fig. 2 and the amplitudes read
_ k+K-) = ¿^F x v2
K+(k -)
u(s) u (s)
,(un6/k-(k +)
Fig. 2. bs ^ k+k-(k+*k-*) decay diagrams.
x {(ai + a2)VubVus — K + aio +
+ X(ae + a8)]VtbVt*} - fBsfKA hV^ -
63 + 264 — -63,EW + 2^4-EW
Vtb Vt
ts
M (Bs -
(11:
where AI^sK and are form factors for Bs
K* [12], FBBsK is the form factor for Bs — K [13]
4 fKF0BsK(mBs - mK){(ai + a2)VubV*s -- [a4 + aio + rK(ae + a8)}VtbV*} +
+ fBs fk\ biVubV:s -
+ 2^4,EW
transitions and
b3 = + c5(4 + A{) + NcC6Af3],
N c
b'3,EW = + C7(4 + 4) + NcCsA(],
A3 = 0,
A3 » 6nas(rK + + rK-)(2XA - Xa), (Ц = Ci + —ci+i(i = odd),
ri<* = 2m'K* fK*
x ть /к*
(13)
63 + 264 - 2&3,ew +
Vtb Vt
ts
Mil Г -»■ K+*K~*) = -i~^iK*mK* x (12)
x {(e\ ■ e\)(msa + тк*)AfsK*(m2K*) -
2 ABsK* (m2 )
-(ег-рв)(е2-рв) 2 ^ K*} тве + тк *
N, 1
c
a-i = Ci + —Ci-i{i = even),
Nc
where i runs from i = 1,..., 10 and ai and a2 are both
+
K
в
в
(d) (к0)
'd(s) d ( ï )
S(d)\^/K0( K0) s(dy-
Fig. 3. bs ^ k°k°(k°*k) decay diagrams.
K0( K0)
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