Gluonic contribution to $B\to η^{(\prime)}$ form factors

0.24 MB·English

by  Yeo-Yie Charng

Gluonic contribution to B η(′) form factors
→
Yeo-Yie Charng1, T. Kurimoto2, and Hsiang-nan Li1,3
∗ † ‡
1Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China
2Department of Physics, University of Toyama, Toyama 930-8555, Japan and
3Department of Physics, National Cheng-Kung University,
Tainan, Taiwan 701, Republic of China
We calculate the flavor-singlet contribution to the B → η(′) transition form factors from the
gluonic content of the η(′) meson in the large-recoil region using the perturbative QCD approach.
Theformulation for theη-η′ mixingin thequark-flavorand singlet-octet schemesiscompared,and
employed to determine the chiral enhancement scales associated with the two-parton twist-3 η(′)
7 meson distribution amplitudes. It is found that thegluonic contribution is negligible in the B →η
0 formfactors, andreachesfewpercentsintheB→η′ ones. Itsimpactontheaccommodation ofthe
0 measured B → η(′)K branching ratios in the perturbative QCD and QCD-improved factorization
2 approaches is elaborated.

n
a PACSnumbers: 13.25.Hw,12.38.Bx,11.10.Hi
J

3

3
I. INTRODUCTION
v
5
6 Itisstilluncertainwhethertheflavor-singletcontributionstoBmesondecaysintoη()mesonsplayanessential
′
1 role. The flavor-singlet contributions to the B η()K branching ratios, including those from the b sgg
9 → ′ →
transition [1], from the spectator scattering [2, 3], and from the weak annihilation, have been analyzed in the
0
QCD-improved factorization (QCDF) approach [4]. However, at least the piece from the weak annihilation
6
0 can not be estimated unambiguously due to the presence of the end-point singularities. The flavor-singlet
/ contribution to the B η() transition form factors from the gluonic content of the η() meson is also involved
h → ′ ′
in the annihilation amplitudes. Though this contribution seems to be most crucial among all the flavor-singlet
p
- piecesintheB η(′)K decays,ithasbeenparameterizedandvariedarbitrarily[4]. Theformfactorsassociated
p with the decays→B η()l+l were handled in a similar way recently [5]. A sizable gluonic content in the η
′ − ′
e →
meson has been indicated from a phenomenologicalanalysis of the relevant data [6]. All these previous studies
h
motivate us to make a more definite estimate of the gluonic contribution in the B η() form factors.
: ′
v InthispaperweshallcalculatethegluoniccontributiontotheB η() formfact→orsinthelarge-recoilregion
′
i →
X using the perturbative QCD (PQCD) approach [7, 8, 9]. This approach is based on kT factorization theorem
[10, 11], so the end-point singularities do not exist. It has been proposed to extract this gluonic contribution
r
a from the measured B η()lν decay spectra [12], which are, however, not available yet. To proceed with the
′
→
calculation,weneedtospecifyaschemefortheη-η mixing. Aftercomparingthequark-flavorbasis[13]andthe
′
conventionalsinglet-octetbasis,we adoptthe former,in whichfewer two-partontwist-3 η() mesondistribution
′
amplitudes are introduced. To reduce the theoretical uncertainty from the distribution amplitudes, we employ
the Gegenbauer coefficients constrained by the data of both exclusive processes [13, 14, 15], such as the η()
′
transition form factors, and inclusive processes [16], such as Υ(1S) η X. It will be shown that the gluonic
′
→
contribution is negligible in the B η form factors, and reaches few percents in the B η ones.
′
→ →
WhethertheobservedB ηK andB η K branchingratios[17]canbeaccommodatedsimultaneouslystill
′
attracts a lot of attentions→[18, 19]. We s→hall elaborate the impact of the gluonic contribution in the B η()
′
→
form factors on this issue in the QCDF and PQCD frameworks. As noticed in [4], the gluonic contribution
increases the branching ratios B(B η K), but decreases B(B ηK). Since the QCDF predictions for
′
→ →
both B(B η K) and B(B ηK) fall short compared with the data [4], the gluonic contribution does not
′
→ →
help. Onthe contrary,there is much roomfor this contribution to play in PQCD: the flavor-singletamplitudes
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
were not taken into account in the earlier PQCD analysis of the B η()K decays [20], whose predictions for
′
→
B(B η K)[B(B ηK)]arelower(higher)thanthe measuredvalues. Ifstretching the gluoniccontribution,
′
it is li→kely to accom→modate the data of the B η()K branching ratios in PQCD.
′
→
In Sec. II we compare the quark-flavor and singlet-octet schemes for the η-η mixing, and obtain the chiral
′
enhancement scales associated with the two-parton twist-3 distribution amplitudes in both cases. In Sec. III
we derive the factorization formulas for the quark and gluonic contributions in the B η() form factors, and
′
→
performthe numericalevaluation,togetherwithadetaileduncertaintyanalysis. Ourresultsarethencompared
with those obtained in the literature. The impact of the gluonic contribution on the accommodation of the
measured B η()K branching ratios is discussed. Section IV is the conclusion.
′
→
II. η-η′ MIXING AND DISTRIBUTION AMPLITUDES
For the η-η mixing, the conventional singlet-octet basis and the quark-flavor basis [13] have been proposed.
′
In the latter the qq¯ (uu¯+dd¯)/√2 and ss¯ flavor states, labelled by the η and η mesons, respectively, are
q s
≡
defined. The physical states η and η are related to the flavor states through a single angle φ,
′
η η
| i =U(φ) | qi , (1)
η η
′ s
(cid:18)| i(cid:19) (cid:18) | i (cid:19)
with the matrix,
cosφ sinφ
U(φ)= − . (2)
sinφ cosφ
(cid:18) (cid:19)
It has been postulated [13] that only two decay constants f and f need to be introduced:
q s
i
0q¯γµγ q η (P) = f Pµ ,
5 q q
h | | i −√2
0s¯γµγ sη (P) = if Pµ , (3)
5 s s
h | | i −
forthelightquarkq =uord. Thispostulationisbasedontheassumptionthattheintrinsicq¯q (s¯s)component
is absentin the η (η ) meson, ie., basedon the OZI suppressionrule. The decay constants associatedwith the
s q
η and η mesons:
′
i
0q¯γµγ q η()(P) = fq Pµ ,
h | 5 | ′ i −√2 η(′)
h0|s¯γµγ5s|η(′)(P)i = −ifηs(′)Pµ , (4)
are then related to f and f via the same mixing matrix,
q s
fq fs f 0
(cid:18)fηηq′ fηsη′ (cid:19)=U(φ)(cid:18) 0q fs (cid:19) . (5)
Employing the equation of motion,
α
∂ (q¯γµγ q)=2im q¯γ q+ s G Gµν , (6)
µ 5 q 5 µν
4π
and the one corresponding to the s quark, where G is the field-strengeth tensor and G the dual field-strength
tensor, one derives the relation [13],
e
M2 =U (φ)M2U(φ). (7)
qs †
In the above expression the mass matrices are given by,
m2 0
M2 = η ,
0 m2
(cid:18) η′ (cid:19)
m2 +(√2/f ) 0α GG˜/(4π)η (1/f ) 0α GG˜/(4π)η
M2 = qq q h | s | qi s h | s | qi , (8)
qs (√2/f ) 0α GG˜/(4π)η m2 +(1/f ) 0α GG˜/(4π)η
(cid:18) q h | s | si ss s h | s | si(cid:19)
3
with the abbreviations,
√2
m2 = 0m u¯iγ u+m d¯iγ dη ,
qq f h | u 5 d 5 | qi
q
2
m2 = 0m s¯iγ sη . (9)
ss f h | s 5 | si
s
The above matrix elements define the chiral enhancement scales associated with the two-partontwist-3 η and
q
η meson distribution amplitudes. Note that the axialU(1) anomaly[21]is the only source of the non-diagonal
s
elements of M in the quark-flavorbasis, also a consequence of the postulation that leads to Eq. (3).
qs
We repeat the above formalism for the η-η mixing in the singlet-octet basis, where the (uu¯+dd¯+ss¯)/√3
′
and (uu¯+dd¯ 2ss¯)/√6 states, labelled by the η and η mesons, respectively, are considered. From Eq. (1)
1 8
−
andthe quark contentofthe η andη mesons, we havethe decompositions ofthe η andη mesonstates inthe
1 8 ′
singlet-octet basis,
η η
| i =U(θ) | 8i , (10)
η η
′ 1
(cid:18)| i(cid:19) (cid:18)| i(cid:19)
with the angle θ =φ θ , θ being the ideal mixing angle associated with the matrix,
i i
−
1 √2
U(θ )= √3 −√3 . (11)
i √2 1
!
√3 √3
The decay constants defined in terms of the SU(3) flavor-singletand flavor-octetaxial-vector currents,
0Ji η()(P) = ifi Pµ , (12)
h | µ5| ′ i − η(′)
for i=1 or 8, are related to f and f through
q s
f8 f1 f 0
fη8 fη1 =U(φ) 0q f U†(θi). (13)
(cid:18) η′ η′ (cid:19) (cid:18) s (cid:19)
Compared to the conventional parametrization,
f8 f1 f 0
η η =U 8 , (14)
f8 f1 81 0 f
(cid:18) η′ η′ (cid:19) (cid:18) 1 (cid:19)
with the matrix,
cosθ sinθ
U81 = sinθ8 −cosθ 1 , (15)
8 1
(cid:18) (cid:19)
the decay constants of the η and η mesons, f and f , respectively, then connect to f and f .
1 8 1 8 q s
Following the similar procedure, we derive the version of Eq. (7) in the singlet-octet basis,
M821 =U†(θ)M2U81 , (16)
with the mass matrix,
m2 m2 +(√3/f ) 0α GG˜/(4π)η
M2 = 88 18 1 h | s | 8i , (17)
81 m2 m2 +(√3/f ) 0α GG˜/(4π)η
(cid:18) 81 11 1 h | s | 1i(cid:19)
and the abbreviations,
2
m2 = 0m u¯iγ u+m d¯iγ d 2m s¯iγ sη ,
88 √6f h | u 5 d 5 − s 5 | 8i
8
2
m2 = 0m u¯iγ u+m d¯iγ d+m s¯iγ sη ,
18 √3f h | u 5 d 5 s 5 | 8i
1
2
m2 = 0m u¯iγ u+m d¯iγ d 2m s¯iγ sη ,
81 √6f h | u 5 d 5 − s 5 | 1i
8
2
m2 = 0m u¯iγ u+m d¯iγ d+m s¯iγ sη . (18)
11 √3f h | u 5 d 5 s 5 | 1i
1
4
The above matrix elements define the chiral enhancement scales associatedwith the two-partontwist-3 η and
1
η meson distribution amplitudes.
8
Notethatthefourhadronicmatrixelementsm2 ,m2 , 0α GG˜/(4π)η ,and 0α GG˜/(4π)η arefixedby
qq ss h | s | qi h | s | si
Eq.(7)inthequark-flavorbasis,giventhethreeinputsf ,f ,andφ. However,M2 containssixmatrixelements,
q s 81
which can not be fixed completely by Eq. (16). In fact, two OZI suppressed matrix elements 0m u¯iγ u+
u 5
m d¯iγ dη and 0m s¯iγ sη have been dropped in [13]. In the singlet-octet basis an approximha|tion can be
d 5 s s 5 q
| i h | | i
made to reduce the number of matrix elements [22, 23]: the terms proportional to the light quark masses m
u
and m are negligible compared with those proportional to m in Eq. (18), which then becomes
d s
4 f
m2 = 0m s¯iγ sη , m2 = 8 m2 ,
88 −√6f h | s 5 | 8i 18 −√2f 88
8 1
2 √2f
m2 = 0m s¯iγ sη , m2 = 1m2 . (19)
11 √3f1h | s 5 | 1i 81 − f8 11
Thethreeinputsf ,f ,andφinEq.(7)havebeenextractedfromthedataoftherelevantexclusiveprocesses
q s
[13],
f =(1.07 0.02)f , f =(1.34 0.06)f , φ=39.3 1.0 , (20)
q π s π ◦ ◦
± ± ±
which correspond, via Eqs. (13) and (14), to [24]
f 1.26f , f 1.17f , θ 21.2o , θ 9.2o , (21)
8 π 1 π 8 1
≈ ≈ ≈− ≈−
in the singlet-octet basis. The above values are well consistent with those in [23], which were also extracted
from the relevant data but based on the approximation in Eq. (19). We stress that the approximations in the
quark-flavorbasis[13] andinthe singlet-octetbasis [23]for decreasingthe number ofhadronicmatrix elements
are very different. Therefore, the above agreementimplies that these approximationsmake sense, and that the
two bases will be equivalent to each other, if one chooses the parameters obeying the constraints in Eqs. (7),
(13), (14), (16), and (19) for a calculation.
Viewing Eqs.(9) and(18), it is obviousthatfewer two-partontwist-3distribution amplitudes areintroduced
in the quark-flavor scheme [50]. Hence, we adopt this scheme for the η-η mixing, and specify the η and
′ q
η meson distribution amplitudes. Their two-parton quark components are defined via the nonlocal matrix
s
elements,
i 1
hηq(P)|q¯γa(z)qβb(0)|0i = −2√N δab dxeixP·z [γ5 6P]βγφAq(x)+[γ5]βγmq0φPq (x)
c Z0
+mq[γ (n n 1)] (cid:8)φT(x) ,
0 5 6 + 6 −− βγ q
η (P)s¯a(z)sb(0)0 = i δab 1dxeixP z [γ P(cid:9)] φA(x)+[γ ] msφP(x)
h s | γ β | i −√2N · 5 6 βγ s 5 βγ 0 s
c Z0
+ms[γ (n n 1)] (cid:8)φT(x) , (22)
0 5 6 + 6 −− βγ s
where P = (P+,0,0 ) is the η meson momentum, the light-like(cid:9)vector z = (0,z ,0 ) the coordinate of
T q,s − T
the q and s quarks, the dimensionless vector n = (1,0,0 ) parallel to P, n = (0,1,0 ) parallel to z, the
+ T T
superscripts a and b the color indices, the subscripts γ and β the Dirac indice−s, N = 3 the number of colors,
c
and x the momentum fraction carried by the q and s quarks. The chiral enhancement scales mq and ms have
0 0
been fixed by Eq. (7), whose explicit expressions are
m2 1 √2f
mq0 ≡ 2mqqq = 2mq "m2ηcos2φ+m2η′sin2φ− fq s(m2η′ −m2η)cosφsinφ# ,
m2 1 f
ms0 ≡ 2msss = 2ms (cid:20)m2η′cos2φ+m2ηsin2φ− √2qfs(m2η′ −m2η)cosφsinφ(cid:21) , (23)
respectively, assuming the exact isospin symmetry m m = m (For the inclusion of the isospin symmetry
q u d
≡
breaking effect, refer to [25]).
5
The distribution amplitudes φA aretwist-2,and φP and φT twist-3. As explained in[26], both the twist-2
q,s q,s q,s
andtwist-3two-partondistributionamplitudescontributeatleadingpowerintheanalysisofexclusiveB meson
decays. We follow the parametrization for the pion distribution amplitudes proposed in [27],
f
φA (x) = q(s) 6x(1 x) 1+aq(s)C3/2(2x 1) ,
q(s) 2√2N − 2 2 −
c
h i
f 5
φP (x) = q(s) 1+ 30η ρ2 C1/2(2x 1)
q(s) 2√2N 3− 2 q(s) 2 −
c(cid:20) (cid:18) (cid:19)
9
3 η ω + ρ2 (1+6aq(s)) C1/2(2x 1) ,
− 3 3 20 q(s) 2 4 −
(cid:26) (cid:27) (cid:21)
f 1 7 3
φT (x) = q(s) (1 2x) 1+6 5η η ω ρ2 ρ2 aq(s) (1 10x+10x2) , (24)
q(s) 2√2N − 3− 2 3 3− 20 q(s)− 5 q(s) 2 −
c (cid:20) (cid:18) (cid:19) (cid:21)
with the mass ratios ρ =2m /m and ρ =2m /m , and the Gegenbauer polynomials,
q q qq s s ss
1 3 1
C1/2(t) = 3t2 1 , C3/2(t) = 5t2 1 , C1/2(t) = 3 30t2+35t4 . (25)
2 2 − 2 2 − 4 8 −
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
The values of the constant parameters aq, η and ω in Eq. (24) will be given in the next section.
2 3 3
The leading-twist gluonic distribution amplitudes of the η and η mesons are defined by [16]
q s
√2f C δab nρPσ 1 φG(x)
hηq(P)|Aa[µ(z)Abν](0)|0i = √3q 4√F3N2 1ǫµνρσ n− P dxeixP·zx(1q x) ,
c − −· Z0 −
f C δab nρPσ 1 φG(x)
η (P)Aa (z)Ab (0)0 = s F ǫ dxeixP z s , (26)
h s | [µ ν] | i √34√3N2 1 µνρσ n P · x(1 x)
c − · Z0 −
with the notation Aa (z)Ab (w) [Aa(z)Ab(w) Aa(z)Ab(w)]/2 and the function [14, 28, 29, 30, 31],
[µ ν] ≡ µ ν − ν µ
φG (x)=x2(1 x)2Bq(s)C5/2(2x 1), C5/2(t)=5t. (27)
q(s) − 2 1 − 1
According to the above definition, the gluon labelled by the subscript µ carries the fractional momentum xP.
ThetwoGegenbauercoefficientsBq andBscouldbedifferentinprinciple. However,duetothelargeuncertainty
2 2
in their values, it is acceptable to assume Bq = Bs B . As shown later, the contribution from the above
2 2 ≡ 2
gluonic distribution amplitudes is smaller than that from the quark distribution amplitudes. Therefore, the
subleading-twist gluonic distribution amplitudes of the η meson will be dropped below.
q,s
The gluonic distribution amplitude of the η meson defined in [32] is related to those in Eq (26) through
′
Eq. (1),
η Aa (z)Ab (0)0 η Aa (z)Ab (0)0
h | [µ ν] | i =U(φ) h q| [µ ν] | i , (28)
η Aa (z)Ab (0)0 η Aa (z)Ab (0)0
h ′| [µ ν] | i! h s| [µ ν] | i!
which also defines the gluonic distribution amplitude of the η meson. Besides, our parametrization of the
gluonicdistributionamplitudesareidenticaltothosein[4]exceptadifferentdefinitionofB : theirGegenbauer
2
coefficient is 2/9 times of ours.
III. B→η(′) FORM FACTORS
Afterdefiningthequarkandgluonicdistributionamplitudesoftheη andη mesons,wearereadytocalculate
q s
the B η() transition form factors at leading order of the strong coupling constant α . In the B meson rest
′ s
frame,→we choose the B meson momentum P and the η() meson momentum P in the light-cone coordinates:
1 ′ 2
m m
P = B(1,1,0 ), P = B(ρ,0,0 ), (29)
1 T 2 T
√2 √2
6
where the energy fraction ρ carried by the η() meson is related to the lepton-pair momentum q = P P
′ 1 2
−
via q2 = (1 ρ)m2, m being the B meson mass. The η() meson mass, appearing only in power-suppressed
− B B ′
terms, has been neglected. The spectator momenta k on the B meson side and k on the η() meson side are
1 2 ′
parameterized as
m m
k = 0,x B,k , k = x ρ B,0,k , (30)
1 1 1T 2 2 2T
√2 √2
(cid:18) (cid:19) (cid:18) (cid:19)
x and x being the parton momentum fractions, and k and k the parton transverse momenta.
1 2 1T 2T
We first compute the B η form factors defined by the local matrix elements,
q(s)
→
m2 m2
η (P )¯b(0)γ u(0)B(P ) = FBηq(s)(q2) (P +P ) B− ηq(s)q
h q(s) 2 | µ | 1 i + " 1 2 µ− q2 µ#
m2 m2
+FBηq(s)(q2) B − ηq(s)q ,
0 q2 µ
FBηq(s)(q2)
η (P )¯b(0)iσµνq d(0)B(P ) = T (m2 m2 )qµ q2(Pµ+Pµ) , (31)
h q(s) 2 | ν | 1 i mB+mηq(s) h B − ηq(s) − 1 2 i
where the η meson mass m will be set to zero eventually. The form factors F are associatedwith the
q(s) ηq(s) +,0
semileptonic decay B η()lν, and F with B η()l+l . For the involved ¯b u¯,d¯transitions, the above
′ T ′ −
→ → →
form factors are decomposed into
FBηq =FBηq +FBηq , FBηs =FBηs . (32)
+(0,T) q+(0,T) g+(0,T) +(0,T) g+(0,T)
That is, the η meson state contributes only through the flavor-singlet pieces FBηs . The B η() form
s g+,g0,gT → ′
factors are then obtained from the mixing,
FBη FBηq
F++B((η00′,,TT)) !=U(φ) F++B((η00s,,TT)) ! . (33)
It is then expected that the gluonic contribution is more significant in the B η form factors than in the
′
→
B η ones, since those from the η and η mesons add up in the former, but partially cancel in the latter.
q s
→
7
B η( )
FIG.1: Gluonic contribution totheB→η(′) form factors. Anotherdiagram with thetwo gluons crossed issuppressed.
The factorization formulas for FBηq are similar to those for the B π form factors [26, 33]:
q+,q0,qT →
8
FBηq(q2) = πm2C dx dx b db b db φ (x ,b )
q+ √2 B F 1 2 1 1 2 2 B 1 1
Z Z
2
(1+x ρ)φA(x )+r 1 2x φT(x )+r (1 2x )φP(x ) E(t(1))h(x ,x ,b ,b )
× ((cid:20) 2 q 2 q(cid:18)ρ − − 2(cid:19) q 2 q − 2 q 2 (cid:21) 1 2 1 2
+2r φPE(t(2)h(x ,x ,b ,b ) ,
q q 2 1 2 1
)
8
FBηq(q2) = πm2C ρ dx dx b db b db φ (x ,b )
q0 √2 B F 1 2 1 1 2 2 B 1 1
Z Z
2
(1+x ρ)φA(x )+r (1 2x )φT(x )+r 1 2x φP(x ) E(t(1))h(x ,x ,b ,b )
× ((cid:20) 2 q 2 q − 2 q 2 q(cid:18)ρ − − 2(cid:19) q 2 (cid:21) 1 2 1 2
+2r φPE(t(2)h(x ,x ,b ,b ) ,
q q 2 1 2 1
)
8
FBηq(q2) = πm2C dx dx b db b db φ (x ,b )
qT √2 B F 1 2 1 1 2 2 B 1 1
Z Z
2
φA(x )+r +x φT(x ) r x φP(x ) E(t(1))h(x ,x ,b ,b )
× ((cid:20) q 2 q(cid:18)ρ 2(cid:19) q 2 − q 2 q 2 (cid:21) 1 2 1 2
+2r φPE(t(2)h(x ,x ,b ,b ) , (34)
q q 2 1 2 1
)
with the color factor C =4/3, the B meson wave function φ , the impact parameter b (b ) conjugate to the
F B 1 2
parton transverse momentum k (k ), the mass ratio r = mq/m , the hard function h, and the evolution
1T 2T q 0 B
factor,
E(t)=αs(t)e−SB(t)−Sηq(t) . (35)
The choice of the hard scale t, and the explicit expressions for h, for the Sudakov exponent S associated
B
with the B meson, and for the Sudakov exponent S associated with a light meson are referred to [26]. The
ηq
threshold resummation factor associated with the hard function is the same as in the B π form factors [34].
→
The coefficient 1/√2 appears, because only the u or d quark component of the η meson is involved. We point
q
out that the term proportional to 2/ρ in FBηq is missed in [35].
qT
For the gluonic contribution, it has been argued [4] that the diagram in Fig. 1 with the two gluon emitted
from the light spectator quark of the B mesonis leading. Another diagramwith the two gluons crossed,giving
the identical contribution, is not displayed. The third diagram vanishes, in which the virtual gluon from the
u and u¯ quark annihilation couples the two valence gluons in the η() meson. The flavor-singlet pieces in the
′
8
B η form factors are written as
q
→
8 C2√N φG(x )
FBηq(q2) = πm2f F c dx dx b db b db φ (x ,b ) 2
g+ −3 B q N2 1 1 2 1 1 2 2 B 1 1 x (1 x )
c − Z Z 2 − 2
x [1+(1 ρ)x )]E(t(2)h(x ,x ,b ,b ),
1 2 2 1 2 1
× −
8 C2√N φG(x )
FBηq(q2) = πm2f F cρ dx dx b db b db φ (x ,b ) 2
g0 −3 B q N2 1 1 2 1 1 2 2 B 1 1 x (1 x )
c − Z Z 2 − 2
x [1 (1 ρ)x )]E(t(2)h(x ,x ,b ,b ),
1 2 2 1 2 1
× − −
8 C2√N φG(x )
FBηq(q2) = πm2f F c dx dx b db b db φ (x ,b ) 2
gT −3 B q N2 1 1 2 1 1 2 2 B 1 1 x (1 x )
c − Z Z 2 − 2
x (1+x )E(t(2)h(x ,x ,b ,b ). (36)
1 2 2 1 2 1
×
A factor 2 has been included, which is attributed to the identical contribution from the second diagram men-
tioned above. The calculation is similar to that of the η g g( ) vertex in [36, 37]. The expressions for FBηs
′ ∗ ∗ g+,g0,gT
arethesameasinEq.(36),butwithf beingreplacedbyf /√2. ItisobservedthatEq.(36)isproportionalto
q s
the small momentum fraction x Λ/m [38], Λ being a hadronic scale, compared to the quark contribution.
1 B
∼
Becausethe abovefactorizationformulasdonotdeveloptheend-pointsingularities,ifremovingk ,thegluonic
T
contribution can in fact be computed in collinear factorization theorem.
The evolution factor in Eq. (36) is given by
E(t)=αs(t)e−SB(t)−SG(t) , (37)
where the Sudakov exponent S is associated with the gluonic distribution amplitudes of the η mesons.
G q,s
Following the studies in [39], its expression is, up to the leading-logarithm accuracy, similar to S , but with
ηq
the anomalous dimension α C /π being replace by α C /π, C =3 being a color factor:
s F s A A
S (t) = s (x P+,b )+s ((1 x )P+,b ),
G G 2 2 2 G − 2 2 2
Q dµ Q α
s
s (Q,b) = ln A(α (µ)) , A= C . (38)
G s A
µ µ π
Z1/b (cid:20) (cid:18) (cid:19) (cid:21)
Thatis,theSudakovsuppressionisstrongerinthegluonicdistributionamplitudesthaninthequarkones. There
is no point to include the next-to-leading-logarithm resummation, since the gluonic distribution amplitudes
are not very certain yet. For consistency, we neglect the single-logarithm renormalization-group summation
governed by the anomalous dimension of the gluon wave function. We adopt the one-loop expression of the
running coupling constant α , when evaluating the above Sudakov factors.
s
We then performa detailednumericalanalysis,including that oftheoreticaluncertainties [40]. TheB meson
wavefunction is the sameas in[41]with the shape parametervariedbetweenω =(0.40 0.04)GeV.There is
B
±
anotherB mesonwavefunctionintheheavy-quarklimit[42],whosecontributiontotransitionformfactorsmay
be finite. However, it has been shown that its effect can be well mimicked by a single B meson wave function,
if a suitable ω is chosen [40]. Therefore, we adopt this approximation, and vary ω in the above range. To
B B
obtain the chiral enhancement scale mq0, we need the masses mη = 0.548 GeV and mη′ = 0.958 GeV, and
the inputs in Eq. (20). Because meson distribution amplitudes are defined at 1 GeV, we take the light quark
mass m (1 GeV) = (5.6 1.6) MeV [43]. We have confirmed that the twist-2 and twist-3 quark distribution
q
±
amplitudes of the η′ meson in [32] are consistent with Eqs. (16) and (22), if their η′ meson decay constant fη′
is regardedasf1 inthe singlet-octetbasis. Hence,it is legitimateto adoptaq = 0.008 0.054extractedfrom
η′ 2 − ±
the relevant data [16] for the twist-2 distribution amplitude. The twist-3 distribution amplitudes have not yet
been constrained experimentally, so we choose η = 0.015 and ω = 3 the same as for the pion distribution
3 3
−
amplitudes [27]. It is not necessary to specify the parameters involved in the quark distribution amplitudes of
the η meson here. The overallcoefficients in Eq. (26) have been arrangedin the way that the decay constants
s
in [32] and in Eq. (26) satisfy Eq. (13). Following Eq. (28), the range of B = 4.6 2.5 extracted in [16] can
2
±
alsobe adopteddirectly. The theoreticaluncertainty arisingfromthe variationof the aboveparameterswillbe
investigated.
9
FBη FBη FBη′ FBη′
+,0 T +,0 T
F(0) 0.147 0.139 0.121 0.114
ratio 0.0031 0.0028 0.023 0.021
F 0.252 0.237 0.252 0.237
1
F 0.0034 0.0029 0.0034 0.0029
2
TABLE I: Form factor values at maximal recoil, the ratios of their gluonic contributions, and values of F1,2 defined in
Eq (39).
The following parametrization for the B η() form factors was proposed in [4],
′
→
fq √2fq +fs
FBη(′)(0)=F η(′) +F η(′) η(′) , (39)
0 1 √2f 2 √6f
π π
where F (F ) corresponds to the quark (gluonic) contribution, and a factor 1/√2 is included to match our
1 2
convention. F has been set to the B π form factor, and the unknown F varied arbitrarily between [0,0.1]
1 2
→
[4]. The above parametrization also applies to the other form factors F and F . It is easy to identify, from
+ T
Eq. (33), the relations of F to our form factors,
1,2
√2f √3f
F = πFBηq(0), F = πFBηq(0). (40)
1 f q 2 f g
q q
To have a picture of the magnitude of the gluonic contribution, we present in Table I the central values of the
form factors at maximal recoil, the ratios of their gluonic contributions, and the values of F . It is indicated
1,2
that the gluonic contribution is about 0.3% in the B η form factors, and about 2% in the B η ones. The
′
centralvalueofF fortheformfactorsFBη(′) isindeed→closetotheB πformfactorsFBπ(0)[2→6]. Thecentral
1 +,0 → +,0
valueofF =0.0034forFBη(′) islocatedwithin[0,0.1][4],butnearthe lowerendofthe interval(to beprecise,
2 +,0
F runs between [0.0016,0.0052],consideringthe range ofthe parameterB ). We notice the difference between
2 2
F for FBη(′)(0) and F for FBη(′)(0), which is obvious from the factorization formulas in Eq. (36): FBηq(0)
2 +,0 2 T gT
contains an additional term proportional to x compared with FBηq (0). Our observation is thus contrary to
2 g+,g0
the assumption FBηq (0) = FBηq(0) postulated in the analysis of the B η()l+l decays [5]. Nevertheless,
g+,g0 gT → ′ −
this difference is not crucial for the estimate in [5], because the gluonic contribution is not dominant.
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
2 4 6 8 10 2 4 6 8 10
q2(GeV 2 ) q2(GeV 2 )
(a) (b)
FIG. 2: q2 dependence of (a) FBη and (b) FBη′ , corresponding to the central values of the inputs with the solid,
+,0,T +,0,T
dotted and dash-dotted lines representing theF+, F0, and FT, respectively.
The q2 dependence of the B η and B η form factors corresponding to the central values of the inputs
′
→ →
aredisplayedinFigs.2(a)and(b). Theratiosofthegluoniccontributionstothetotalvaluesoftheformfactors
areshowninFig. 3. It is found that the gluonic contributionremainsnegligible inthe B η formfactors,and
→
10
0.03
η’
0.025
0.02
0.015
0.01
η
0.005
2 4 6 8 10
q2(GeV 2 )
FIG.3: q2 dependenceoftheratioofthegluoniccontributiontothetotalvaluesofFBη(η′) (solidlines),FBη(η′) (dotted
+ 0
lines), and FBη(η′) (dash-dotted lines) corresponding to the central valuesof theinputs.
T
about 2% in the B η ones in the whole large-recoil region. The form factors FBη(′)(q2) exhibit a smaller
→ ′ 0
slope with q2, because the overallcoefficients of FBηq contain the energy fraction ρ as shown in Eqs. (34) and
q0,g0
(36), consistent with the large-energy form-factor relation in [44]. The B η form factors are proportional to
→
cosφFBηq, and the B η form factors to sinφFBηq plus some amount of gluonic contributions. Because of
q ′ q
the small gluonic contr→ibution indicated in Fig. 3, we have FBη(q2)>FBη′(q2) shown in Fig. 2 simply due to
cosφ > sinφ for φ 39.3o. This observation is in agreement with the tendency exhibited in the measurement
≈
of the semileptonic branching ratios [45],
B(B+ ηl+ν) = (0.84 0.27 0.21) 10 4 <1.4 10 4 (90%C.L.),
l − −
→ ± ± × ×
B(B+ η l+ν) = (0.33 0.60 0.30) 10 4 <1.3 10 4 (90%C.L.). (41)
′ l − −
→ ± ± × ×
More precise data will provide the information on the importance of the gluonic contribution. Our analysis
impliesthattheq2 dependenceofthegluoniccontributionisweakerthanthatofthequarkcontribution. Hence,
the assumption [5, 12] that both pieces show the same q2 dependence is not appropriate.
q
m
0
q
m
2 0
1.5 4
1 2
0.5
1.15 1.25 1.3 1.35
38 39 40 41
φ -2 fs fq
(a) (b)
FIG. 4: Dependenceof mq0 (a) on φ in units of degrees, and (b) on fs/fq.
We then investigate the theoretical uncertainty for the B η() form factors from each of the input pa-
′
rameters, and the results are presented in Table II. The form→factors FBηq in Eq. (34) receive substantial
q+,q0,qT
contributions from the twist-3 distribution amplitudes φP and φT, which appear together with the mass ratio
q q
r = mq/m . The chiral enhancement scale mq changes rapidly with the mixing angle φ and with the decay
q 0 B 0
constantsf asillustratedinFig.4. Thisexplainsthesensitivityoftheformfactorstotheseinputs. Wenotice
q,s
that mq runs into the unrealistic negative region easily, as increasing f , leading to the negative form factor
0 s
values in Table II. Therefore, we prefer not to vary f . The dependence of the form factors on the Gegenbauer
s
coefficients aq and B is weak, because aq is close to zero and the gluonic contribution is subdominant. The
2 2 2
variation of the ratios of the gluonic contributions in the B η() form factors with each of the parameters
′
→
is listed in Table III. Ignoring the effect from changing f for the reason stated above, we conclude that the
s