arXiv:hep-ph/0505045v1 6 May 2005
Yeo-Yie Charng1∗ and Hsiang-nan Li1,2†
1Institute of Physics, Academia Sinica, Taipei,
Taiwan 115, Republic of China and
2Department of Physics, National Cheng-Kung University,
Tainan, Taiwan 701, Republic of China
We show that the leading-power B meson wave function can be extracted reliably from the photon energy spectrum of the B → γlν decay up to O(1/m2
b) and O(α2s) uncertainty, mbbeing the b quark
mass and αs the strong coupling constant. The O(1/mb) corrections from heavy-quark expansion
can be absorbed into a redefined leading-power B meson wave function. The two-parton O(1/mb)
corrections cancel exactly, and the three-parton B meson wave functions turn out to contribute at O(1/m2
b). The constructive long-distance contribution through the B → V → γ transition, V
being a vector meson, almost cancels the destructive O(αs) radiative correction. Using models of
the leading-power B meson wave function available in the literature, we obtain the photon energy spectrum in the perturbative QCD framework, which is then compared with those from other approaches.
I. INTRODUCTION
The two-parton leading-power (LP) B meson wave function (distribution amplitude) φ+plays an essential role in a
perturbative analysis of exclusive B meson decays based on kT factorization theorem [1, 2, 3] (collinear factorization
theorem [4, 5, 6, 7, 8, 9]). Its behavior certainly matters, and has been investigated in various approaches recently. Models of the distribution amplitude φ+(x) with an exponential tail in the large x region have been proposed [10],
where x is the longitudinal momentum fraction carried by the light spectator quark. Neglecting three-parton dis-tribution amplitudes in a study by means of equations of motion [11, 12], φ+(x) was found to be proportional to a
step function with a sharp drop at large x [13]. The wave function φ+(x, kT), where kT is the transverse momentum
carried by the light spectator quark, was also derived in the same framework [13]. All these models depend on at least one shape parameter, whose determination requires experimental inputs from exclusive B meson decays.
In this paper we shall show that the radiative decay B → γlν provides the cleanest information of the LP B meson wave function φ+. This mode has been widely studied in [3, 8, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]
due to different motivations: for extracting the B meson decay constant fB and the Cabibbo-Kobayashi-Maskawa
(CKM) matrix element |Vub|, for demonstrating the next-to-leading-order (NLO) calculation and the proof of QCD
factorization theorem, for deriving resummation of large logarithmic corrections, for studying long-distance effect and annihilation mechanism,.... The subject on the extraction of the B meson wave function from the B → γlν data has not yet been discussed. It will be shown that two-parton next-to-leading-power (NLP) [O(1/mb)] corrections
cancel exactly, mb being the b quark mass. The contributions from higher Fock states, the three-parton B meson
wave functions, turn out to be of O(1/m2
b). The constructive long-distance contribution through the B → V → γ
transition, V being a vector meson, almost cancels the destructive O(αs) radiative correction, αs being the strong
coupling constant. The effect from bremsstrahlung photon emissions vanishes like the lepton mass because of helicity suppression. Therefore, the extraction of φ+from the measured photon energy spectrum of the B → γlν decay suffers
only O(1/m2
b) and O(α2s) uncertainty.
We identify and discuss the higher-power corrections to the B → γlν decay in Sec. II, and calculate the long-and short-distance effects in Sec. III. Section IV is the conclusion. The hard kernel associated with the three-parton distribution amplitudes is derived in the Appendix, whose explicit expression is necessary for demonstrating the smallness of the higher-Fock-state contribution. Our conclusion differs from that drawn in [28], in which the semileptonic decay B → πlν was regarded as a more ideal process for extracting the B meson wave function. The argument is that the radiative decay B → γlν, receiving a large long-distance uncertainty, does not serve the purpose. As stated above, this long-distance effect is in fact cancelled by the O(αs) short-distance one almost exactly.
∗Electronic address: [email protected] †Electronic address: [email protected]
k P k P P − k1 P − k1 2 2 (b) (a)
FIG. 1: Lowest-order diagrams for the B → γlν decay.
II. HIGHER-POWER CORRECTIONS
In this section we identify and discuss higher-power corrections to the B → γlν decay. The B meson momentum P1and the photon momentum P2are parameterized, in the light-cone coordinates, as
P1= mB √ 2(1, 1, 0T) , P2= mB √ 2(0, η, 0T) , (1)
respectively, where η ≡ 2Eγ/mB, mB being the B meson mass, denotes the photon energy fraction. The decay
amplitude is decomposed into 1
eγ (P2, ǫT) |¯uγµ(1 − γ5)b| ¯B (P1) = ǫµναβǫ
∗ν
T vαP2βFV(q2) + iǫT µ∗ (v · P2) − (ǫ∗T · v) P2µ FA(q2) , (2)
where e is the electron charge, ǫT the polarization vector of the photon, v = P1/mB the B meson velocity, and
q2≡ (P
1− P2)2= (1 − η)m2B the lepton-pair invariant mass. The decay spectrum is then given, in terms of the form
factors FV,A, by dΓ dη = αG2 F|Vub|2 96π2 m 5 B(1 − η)η 3F2 V(q 2) + F2 A(q 2) , (3)
with α ≡ e2/(4π) and the Fermi constant G F.
The collinear factorization theorem for the form factors FV,Ain the large η region has been proved in [23, 26], which
are expressed as the convolution of hard kernels with the B meson distribution amplitudes in the momentum fractions x of the light spectator quark. A hard kernel, being infrared-finite, is calculable in perturbation theory. The B meson distribution amplitudes, collecting the soft dynamics in exclusive B meson decays, are not calculable but universal. In the framework of factorization theorem, there are five sources of higher-power corrections to the B → γlν decay:
1. The heavy-quark expansion of the heavy-light current in Eq. (2), ¯
uγµ(1 − γ5)b → ¯uγµ(1 − γ5)h + 1
2mb
¯
uγµ(1 − γ5)i 6Dh + O(1/m2b) , (4)
where the operator D represents the covariant derivative, and the rescaled b quark field h is related to the full field b by
h(z) =1+ 6v 2 e
imbv·z
b(z) . (5)
The factorization of the transition matrix element associated with the first (second) term in the above expansion leads to the LP (NLP) B meson distribution amplitudes.
2. The higher-power interactions in the Lagrangian of the heavy-quark effective theory (HQET). The insertion of the HQET interactions,
O1= 1 mb ¯ h(iD)2h , O2= g 2mb ¯ hσµνG µνh , (6)
into the transition matrix element associated with the first term in Eq. (4) yields O(1/mb) corrections. We
mention that there exists an alternative heavy-quark effective theory, in which the higher-power corrections are formulated in a different way [29].
3. The higher Fock states of the B meson. The nonlocal matrix element,
h0|¯u(z)gGαβ(uz)h(0)| ¯B(P1)i , (7)
defines the three-parton distribution amplitudes, where Gαβ(uz) is the gluon field strength evaluated at the
coordinate uz, 0 ≤ u ≤ 1. The additional valence gluon, attaching internal off-shell quark lines, introduces one more hard propagator, i.e., one more power of 1/mb.
4. The subleading parton-level diagrams (hard kernels). The two-parton lowest-order hard kernels are displayed in Fig. 1, where the upper quark line represents a b quark. It is easy to observe that Fig. 1(a) [(b)] represents the LP (NLP) hard kernel, since the internal quark line is off-shell by mbΛ (m¯ 2b) with ¯Λ being a hadronic scale,
such as the mass difference mB− mb.
5. Power corrections induced by infrared renormalons, which have been known to start with 1/m2 b [30].
A. Heavy-quark Expansion
The factorization of soft dynamics from the transition matrix element associated with the first term on the right-hand side of Eq. (4),
γ (P2, ǫT) |¯uγµ(1 − γ5)h| ¯B (P1) , (8)
leads to the nonlocal matrix element [10], Z dz−d2z T (2π)3 e i(k+ z−−kT·zT) h0|¯uρ(z)hδ(0)| ¯B(P1)i = i fB √ 2{(6P1+ mB)γ5[6n+Φ+(k)+ 6n−Φ−(k)]}δρ , (9) which defines the two-parton LP B meson wave functions Φ±, with the null vectors n+= (1, 0, 0T) and n−= (0, 1, 0T),
and the light quark momentum k. Because the photon momentum P2has been chosen in the minus direction, the hard
kernels for the form factors FV,A are independent of the component k−, which becomes irrelevant. We construct the
B meson distribution amplitudes φ±(x), x ≡ k+/P1+, from the B meson wave functions φ±(x, kT) ≡ P1+Φ±(xP1+, kT)
by integrating the latter over kT,
φ±(x) =
Z
d2kTφ±(x, kT) . (10)
The dependence of φ±(x) and of φ±(x, kT) on the renormalization scale µ has been suppressed.
Define the moments of the B meson distribution amplitude φ+(x),
Λ0≡ Z dxφ+(x) x , Λ1≡ Z dxφ+(x) . (11)
The asymptotic behavior of φ+(x) has been extracted from a renormalization-group equation, which exhibits a decrease
slower than 1/x [31, 32]. That is, the normalization Λ1of the B meson distribution amplitude is divergent after taking
into account the evolution effect. It has been argued that a non-normalizable B meson distribution amplitude does not cause a trouble in practice [33], since only the inverse moment Λ0 matters at LP [24, 34], which is convergent.
Note that a hard kernel would not be as simple as 1/x at higher orders in αs, and information of more moments is
also necessary. In the following discussion we shall neglect the evolution effect, and assume that φ+(x) is normalized
to unity, i.e., Λ1= 1. Since the B meson distribution amplitudes absorb soft dynamics, the light quark momentum k
is of O(¯Λ). We then have the relative importance Λ1/Λ0∼ ¯Λ/mb for x ∼ O(¯Λ/mb).
The factorization of soft dynamics from the transition matrix element associated with the second term on the right-hand side of Eq. (4) gives the nonlocal matrix element,
h0|¯uρ(z)i 6Dhδ(0)| ¯B(P1)i . (12)
The factorization of the transition matrix elements with the insertion of the O(1/mb) interactions in Eq. (6) into
Eq. (8) leads to,
h0|i Z
k
2k
1P
2P − k − k
1 1 2FIG. 2: Three-parton contribution to the B → γlν decay.
The contributions from Eqs. (12) and (13) can be absorbed into the nonlocal matrix element,
h0|¯uρ(z)bδ(0)| ¯B(P1)i , (14)
where the rescaled b quark field h has been replaced by the full field b. It is easy to check that the heavy-quark expansion of Eq. (14) generates Eqs. (12) and (13). This absorption makes sense, because Eqs. (12) and (13), concerning only the initial b quark, are universal for all exclusive B meson decays. The decomposition in Eq. (9) still holds, but the B meson distribution amplitudes Φ±, redefined by Eq. (14) in terms of the full field b, exhibit a
renormalization-group evolution different from that in Eq. (9) [35].
B. Three-parton Distribution Amplitudes
We explain that the nonlocal matrix element in Eq. (7) is negligible in the current accuracy: the three-parton distribution amplitudes, whose contributions to the form factors are supposed to be of O(1/mb), turn out to appear
at 1/m2
b. The three-parton distribution amplitudes ˜ΦV, ˜ΦA, ˜XA, and ˜YA in coordinate space are defined via the
decomposition, h0|¯uρ(z)gGαβ(uz)nβ−hδ(0)| ¯B(P1)i = fB (6P1+ mB)γ5 (vα6n−− v · n−γα) ˜ΦV(t, u) − ˜ΦA(t, u) −iσαβnβ−Φ˜V(t, u) −n−αX˜A(t, u) + n−α v · n− 6n− ˜ YA(t, u) δρ , (15)
with the variable t = v · z. The corresponding hard kernels arise from the contraction of all the structures Γ = vα6n−,
v · n−γα,..., in Eq. (15) with Fig. 2, written as
M(3)a ∝
tr{6ǫ∗
T[u 6n+γα(6P2− 6k1− u 6k2) − ¯u(6P2− 6k1− u 6k2)γα6n+]γµ(1 − γ5)(6P1+ mB)γ5Γ}
[(P2− k1− uk2)2]2
, (16)
where k1(k2) is the momentum carried by the light quark (gluon). The derivation of the above expression is referred
to the Appendix.
For Γ = v · n−γα, Eq. (16) vanishes because of ǫ∗T· n+= ǫ∗T· (P2− k1− uk2) = 0. Express σαβnβ−= i(n−α− 6n−γα),
in which the first term has the same structure as of ˜XA. The second term 6n−γαrenders Eq. (16) vanish for the same
reason. For the other structures vα 6 n−, n−α, and n−α 6 n−, we always have γα = γ+. Once γα = γ+, Eq. (16) is
proportional to
M(3)a ∼
P1· (k1+ uk2)
[(P2· (k1+ uk2)]2
. (17)
Note that k+1 and k2+are of O(¯Λ), and that the moments of the three-parton B meson distribution amplitudes are at
most of O(¯Λ2) [10]. Therefore, when convoluting Eq. (17) with the three-parton distribution amplitudes, the resultant
contribution to the form factors FV,A is of O(¯Λ2/m2b) compared to the LP contribution in Eq. (20). With a similar
reasoning, the three-parton B meson wave functions also contribute at O(1/m2
b) in kT factorization theorems. We
emphasize that the three-parton B meson wave functions are relevant in the NLP analysis of the B → π transition form factors. This is the reason the B → γlν decay is a cleaner mode than the B → πlν decay for determining the LP B meson wave function.
C. NLP Hard Kernels
Contracting Fig. 1 with the two structures in Eq. (9), we get the quark-level amplitudes, M+ a = i 4√2 tr[6ǫ∗(6P 2− 6k)γµ(1 − γ5)(6P1+ mB)γ56n+] (P2− k)2 , M+b = i 4√2 tr[γµ(1 − γ5)(6q− 6k + mb) 6ǫ∗(6P1+ mB)γ56n+] (q − k)2− m2 b , (18)
and M−a,b with the null vector n+ in M +
a,b being replaced by n−. As stated above, Fig. 1(a) is LP, because of
(P2− k)2 = −2P2· k ∼ O(mbΛ), and Fig. 1(b) is NLP, because of (q − k)¯ 2− m2b = −2P1· P2 ∼ O(m2b). We shall
neglect the mass difference between the B meson and the b quark in M+,−b in our analysis accurate up to NLP.
The collinear factorization formulas for FV,A are written as
FV(A)(q2) = fB
Z
dxhφ+(x)HV+(A)(x, η) + φ−(x)HV−(A)(x, η)
i
, (19)
where the hard kernels H are extracted according to Eq. (2) by keeping only the longitudinal component k+ in
Eq. (18). In terms of the LP and NLP moments in Eq. (11), Eq. (19) becomes FV,A(q2) = fB ηmB Λ0± 1 +1 η Λ1 , (20)
in which the coefficient 1 of Λ1comes from Fig. 1(a) and 1/η from Fig. 1(b). It has been mentioned that the equality
of FV and FAat LP is attributed to the spin symmetry in the large-recoil region [20]. The coefficient 1/η implies the
increase of the subleading-power correction with the decrease of the photon energy. This is why a perturbation theory is reliable only in the large η region. The distribution amplitude φ−(x), contributing only through the normalization
of the combination,
Z
dx[φ+(x) − φ−(x)] = 0 , (21)
disappears from Eq. (20). As shown in Eq. (20), the first moment Λ1does appear at NLP, which is divergent under
the evolution. This is another example that the QCD-improved factorization (QCDF) approach based on collinear factorization theorem breaks down at NLP [34, 36].
The decay spectrum in Eq. (3) becomes dΓ dη = αG2 F|Vub|2 48π2 f 2 Bm3B(1 − η)η " Λ2 0+ 1 + 1 η 2 Λ2 1 # . (22)
The above expression indicates that the NLP terms for the spectrum have cancelled, and only the O(1/m2
b) term Λ21
is left. In this case we can estimate the O(1/m2
b) effect using the models for the B meson distribution amplitudes
available in the literature [13, 37],
φ±(x) = λ ± (x − λ)
2λ2 θ(x)θ(2λ − x) , (23)
with the shape parameter λ ≡ ¯Λ/mb. The value of ¯Λ has been found to range between 0.5 and 0.7 GeV [24, 38, 39],
which corresponds to λ = 0.1 ∼ 0.15 approximately. Certainly, there are other models of the B meson distribution amplitudes (see [40]).
Employing the inputs α = 1/137, GF = 1.16639 × 10−5GeV−2, |Vub| = 3.9 × 10−3, fB = 190 MeV, and mB = 5.28
GeV, we derive the photon energy spectra of the B → γlν decay for λ = 0.1 and for λ = 0.15 in Fig. 3. The specific models in Eq. (23) lead to the relation Λ1/Λ0 = λ. Therefore, the subleading-power term is indeed negligible at
large η, whose contribution is around 5%. However, this term diverges quickly at small η, breaking the perturbative expansion in 1/mb. The form factors FV,Ain Eq. (20) contain a dominant monopole component proportional to Λ0/η,
and a small dipole component proportional to Λ1/η2, which is important only at small η. This is the reason one
always obtains a symmetric spectrum in η at LP from a perturbation theory [20] as shown in Fig. 3. To generate an asymmetric spectrum, the dipole component must be enhanced as postulated in [16, 25]. Therefore, an asymmetric spectrum signals an important NLP contribution, ie., a breakdown of factorization theorem.
0.2
0.4
0.6
0.8
1
Η
1
·
10
-182
·
10
-183
·
10
-184
·
10
-185
·
10
-18d
G
d
Η
Λ=
0.15
Λ=
0.1
FIG. 3: Spectra Spectra in units of GeV−1 from collinear factorization with the solid (dashed-dotted)line corresponding to
the LP contribution for λ = 0.1 (λ = 0.15), and the dashed (dotted)line to the inclusion of the NLP contribution for λ = 0.1 (λ = 0.15).
It has been explained that the undesirable feature of the B meson distribution amplitude under evolution is a consequence of collinear factorization, which can be removed in kT factorization [35]. The evolution effect on the
kT-dependent B meson wave function was also studied in [41]. Moreover, applying kT factorization theorem to the
B → γlν decay, which has been proved in [3], we can extend the spectrum to lower η as demonstrated below. Keeping both the longitudinal momentum k+and the transverse momentum k
T in Eq. (18), the hard kernels in kT factorization
theorem are derived. Defining the LP and NLP functions, Λ0(η) ≡ m2B Z dx Z d2k T φ+(x, kT) ηxm2 B+ kT2 , Λ1(η) ≡ m2B Z dx Z d2kT φ+(x, kT) ηm2 B+ kT2 + xφ−(x, kT) η(ηxm2 B+ kT2) , (24)
respectively, we obtain the form factors,
FV,A(q2) =
fB
mB
[Λ0(η) ± Λ1(η)] . (25)
Because of kT ∼ O(¯Λ) in the B meson, Λ1(η) is of O(¯Λ/mb) relative to Λ0(η) in the large η region. Again, only a
single B meson wave function is relevant in the LP analysis of the B → γlν decay, consistent with the observation in [48]. Compared to Eq. (20), both φ± appear in the kT factorization theorem at NLP.
The decay spectrum is then given, according to Eq. (3), by dΓ dη = αG2 F|Vub|2 48π2 f 2 Bm 3 B(1 − η)η 3Λ2 0(η) + Λ21(η) . (26)
Similarly, the NLP terms have cancelled, and only the O(¯Λ2/m2b) term Λ21(η) is left. We adopt the models for the B
meson wave functions in [13], whose kT dependence is coupled to the x dependence through a δ-function,
φ±(x, kT) =λ ± (x − λ) 2λ2 θ(x)θ(2λ − x) 1 πδ k 2 T − x(2λ − x)m2B . (27)
Using the same input parameters, we obtain the photon energy spectra from kT factorization theorem in Fig. 4
for λ = 0.1 and for λ = 0.15. These spectra are symmetric in η, and modified only slightly by the higher-power correction. Hence, the higher-power correction is under control in kT factorization theorem compared to that in
collinear factorization theorem: the power behavior 1/η of the spectrum in the small η region has been smeared into η ln2η. It implies that the perturbative QCD (PQCD) approach based on kT factorization theorem [42, 43, 44, 45]
0.2
0.4
0.6
0.8
1
Η
5
·
10
-191
·
10
-181.5
·
10
-182
·
10
-182.5
·
10
-183
·
10
-183.5
·
10
-18d
G
d
Η
Λ=
0.15
Λ=
0.1
FIG. 4: Spectra Spectra in units of GeV−1from k
T factorization with the solid (dashed-dotted)line corresponding to the LP
contribution for λ = 0.1 (λ = 0.15), and the dashed (dotted)line to the inclusion of the NLP contribution for λ = 0.1 (λ = 0.15).
III. LONG- AND SHORT-DISTANCE CORRECTIONS
In this section we discuss the long-distance and short-distance corrections to the B → γlν decay spectrum. For this purpose, the form factors are written, in kT factorization theorem, as
FV,A(q2) = fB mB h Λ0(η) + Λ(1)0 (η) i + FV,ALD(q2) , (28) where Λ(1)0 and FLD
V,A denote the O(αs) and long-distance correction to the leading result, respectively. We shall
estimate the latter by considering the B → V → γ transition. This correction is certainly significant in the small η (large q2) region, where the internal quark becomes soft, and easily form a resonance with the spectator quark. Hence,
it could break the QCD factorization of the form factors FV,A at small η. At large η, the long-distance contribution
may be suppressed by the values of the B → V transition form factors [15]. The long-distance amplitude is written as [46]
1 eγ (P2, ǫT) |¯uγµ(1 − γ5)b| ¯B (P1) = X V hγ (P2, ǫT) |Jemα |V (P2, ǫT)i −iǫ ∗ T α P2 2 − m2V + imVΓV ×V (P2, ǫT) |¯uγµ(1 − γ5)b| ¯B (P1) , (29)
with the vector mesons V = ρ, ω, · · ·, and their masses mV and widths ΓV. Take the B meson transition into a
transversely polarized ρ meson as an example, for which the first matrix element on the right-hand side of Eq. (29) gives hγ(P2, ǫT)|Jemα |ρ(P2, ǫT)i = − i 2mρfρǫ α T , (30)
fρ being the ρ meson decay constant. The second matrix element is decomposed into
hρ(P2, ǫT)|¯uγµ(1 − γ5)b| ¯B(P1)i = − 2V (q2) mB+ mρ ǫµνρσǫ∗νT P ρ 1P2σ− i(mB+ mρ)A1(q2)ǫ∗T µ, (31)
with the B → ρ form factors V (q2) and A
1(q2). Combining Eqs. (30) and (31), we extract from Eq. (29),
FVLD(q2) = fρ mρ− iΓρ mB mB+ mρ V (q2) , FLD A (q2) = fρ mρ− iΓρ (mB+ mρ) ηmB A1(q2) . (32)
For the long-distance contribution through the B → ω transition, we have the similar expressions to Eq. (32), but with the charge factor 1/2 in Eq. (30) being replaced by 1/6, and the appropriate replacement of the vector meson mass and of the decay constant. The B → ψ transitions do not contribute in this case.
For the ρ and ω mesons, we employ the inputs [46]
mρ= 0.771 GeV , Γρ/mρ= 0.21 , fρ= 0.217 GeV ,
mω= 0.783 GeV , Γω/mω≈ 0 , fω= 0.195 GeV . (33)
For the B → ρ, ω form factors, we adopt the models derived from the light-front QCD [47], which have been parameterized as
F (q2) = F (0) 1 − a(q2/m2
B) + b(q2/m2B)2
, (34)
with the constants,
V (q2) : F (0) = 0.27 , a = 1.84 , b = 1.28 ,
A1(q2) : F (0) = 0.22 , a = 0.95 , b = 0.21 . (35)
We restrict the above formalism in the region,
η > 1 −q 2 max m2 B = 0.275 , (36) q2
maxbeing the maximal value of q2in the B → ω transition, in which Eq. (34) holds. The long-distance contribution
increases FV,A by about 30 ∼ 50% for λ = 0.1 ∼ 0.15 at large η, consistent with the observation in [15, 24]. Its effect
to the decay spectrum is quite important, especially for η < 0.8, as shown in Fig. 5.
The B → ρ, ω transition form factors at large recoil could be regarded as an O(αs) object [48]. This observation
hints that we should attempt to take into account the NLO short-distance correction to FV,A. The O(αs) term in
Eq. (28) is quoted from [20],
Λ(1)0 (η) = −αs(2E4πγ)CF Z dx φ+(x) ηx + x(2λ − x) × ln2η x− 5 2ln η x+ 4π2 3 − ln 2 1 + 2λ − x x + 2πi ln 1 + 2λ − x x . (37)
The weaker evolution of fB will be neglected for simplicity. Due to the large negative double logarithm, the NLO
correction to the form factors FV,A is destructive, and about 30% of the leading result for both λ = 0.1 and λ = 0.15
at large η. This double logarithm has also been discussed in [27].
We emphasize that the NLO hard kernel depends on a factorization scheme, in which the B meson wave function is defined. Therefore, it is not very legitimate to adopt an expression straightforwardly from some other works in the literature. The calculation of the NLO hard kernel for the B → γlν decay in the factorization scheme specified in [35] is in progress, which will be published elsewhere. Similarly, the model-dependent estimate of the long-distance contribution also suffers large uncertainty. Hence, we just intend to point out the potential strong cancellation between the long-distance and NLO corrections in this mode. As shown in Fig. 5, after combining both subleading contributions, the net effect has been greatly reduced. Especially, for the shape parameter λ = 0.1, the cancellation is almost exact for η > 0.8. We conclude that the leading result in the large η region is stable under these corrections.
Using the lifetime of a charged B meson τB± = 1.674 × 10−12s and considering only the leading contribution, we
obtain the branching ratios for λ = 0.15 ∼ 0.1,
B(B → γlν) = (1.8 ∼ 4.8) × 10−6, (38)
from Eq. (26) in kT factorization theorem (PQCD), with only the O(¯Λ2/m2b) and O(α2s) uncertainty. Note that the
spectrum in collinear factorization theorem (QCDF) leads to a logarithmically divergent total decay rate. The values in Eq. (38) are more or less consistent with other estimates in the literature: a model-dependent evaluation of the structure-dependent photon emission contribution gave the branching ratio 10−7 ∼ 10−6 [14]. Using the B meson
bound-state wave function from a Salpeter equation, 0.9 × 10−6 has been obtained [17, 21]. Both a simple
non-relativistic model and light-front QCD led to 3.5 × 10−6[18, 19]. Light-cone sum rules and the pole-model calculation
gave 2 × 10−6[16] and 2.26 × 10−6 [25], respectively. At last, the experimental upper bound at 90% confidence level
was [49]
0.2 0.4 0.6 0.8 1 Η 1·10-18 2·10-18 3·10-18 4·10-18 5·10-18 6·10-18 dG dΗ LP+LD LP+SD Total LP Λ=0.1 0.2 0.4 0.6 0.8 1 Η 5·10-19 1·10-18 1.5·10-18 2·10-18 2.5·10-18 3·10-18 3.5·10-18 dG dΗ LP+LD LP+SD Total LP Λ=0.15
FIG. 5: Spectra in units of GeV−1for λ = 0.1 and λ = 0.15 with the solid lines corresponding to the LP contribution only, the
dashed-dotted lines to the inclusion of long-distance contribution, the dashed lines to the inclusion of the NLO correction, and the dotted lines to the inclusion of both the long-distance and NLO contributions.
IV. CONCLUSION
In this paper we have studied the B → γlν decay in the PQCD approach based on kT factorization theorem. This
formalism is well-defined at subleading level, since the two-parton LP B meson wave functions remain normalizable even after including the evolution effect. Note that the QCDF approach based on collinear factorization theorem fails at NLP. We have shown that the O(1/mb) corrections from the heavy-quark expansion can be absorbed into the
LP B meson wave functions redefine by the nonlocal matrix element in Eq. (14). The NLP contributions from the hard kernels to the decay spectrum cancel. The three-parton B meson wave functions turn out to be suppressed by 1/m2
b in this special mode. The constructive long-distance contribution almost cancels the destructive NLO radiative
correction for both the form factors FV and FA. The B meson wave function φ+ can then be extracted from the
observed B → γlν decay spectrum using the leading formalism, which suffers only the O(1/m2
b) and O(α2s) uncertainly.
We conclude that the B → γlν decay is the cleanest mode for determining this important nonperturbative input for the perturbation theories of exclusive B meson decays.
Measuring the B → γlν spectrum in the lepton and photon energies [20], d2Γ dηdy = αG2F|Vub|2m3B 64π2 (1 − η)[F 2 V(q2) + FA2(q2)][2(1 − y)(1 − y − η) + η2] (40) −2FV(q2)FA(q2)η(2 − 2y − η) , (41)
with the lepton energy fraction y = 2El/mB, 1 − η ≤ y ≤ 1, we can extract the information of the form factors FV
and FA separately. It is then possible to fix the two two-parton B meson wave functions φ± simultaneously from
Eq. (25). At this NLP level, the three-parton wave functions are still absent following the reasoning in Sec. II.B. The long-distance contribution and the NLO corrections also cancel each other as indicated in Eq. (28). With the B → γlν branching ratio around 10−6, the above experimental determination is possible.
We thank I. Bigi, Y. Kwon, Z. Ligeti, M. Neubert, T. Onogi, and A.I. Sanda for helpful discussions. HNL acknowl-edges the hospitality of Nagoya University during his visit, where this work was initiated. This work was supported by the National Science Council of R.O.C. under Grant No. NSC-93-2112-M-001-014.
APPENDIX A: THREE-PARTON CONTRIBUTION
We start with Eq. (1.3) in Ref. [50]: G(1)(z) = Z d4w i(6z− 6w) 2π2(z − w)4ig 6A(w) i 6w 2π2w4 , (A1)
which describes the interaction of a quark with a gluon. In momentum space the above expression becomes G(1)(z) = Z d4l (2π)4 Z d4k 2 (2π)4e i(k2+l)·zi(6k2+ 6l) (k2+ l)2 γαi 6l l2ig ˜Aα(k2) , (A2)
where l (k2) is the momentum carried by the incoming quark (gluon). The Feynman parametrization gives G(1)(z) = − Z du Z d4l (2π)4e il·z Z d4k 2 (2π)4e iuk2·z(6l + u 6k2)γ α(6l − ¯u 6k 2) (l2)2 ig ˜Aα(k2) , (A3)
where the variable change l + ¯uk2→ l, ¯u ≡ 1 − u, has been applied.
In the case we are considering, the gluon momentum k2is of O(¯Λ), since the B meson is dominated by soft dynamics.
We expand the above expression up to O(k2):
G(1)(x) = − Z du Z d4l (2π)4e il·zZ d4k2 (2π)4e iuk2·z ( 6lγα6l (l2)2 + u 6k2γα6l (l2)2 − ¯ u 6lγα6k 2 (l2)2 ) ig ˜Aα(k2) , = − Z du Z d4l (2π)4e il·z ( 6lγα6l (l2)2igAα(uz) + u 6n+γα6l − ¯u 6lγα6n+ (l2)2 ig∂βAα(uz)n β − ) . (A4)
The first term on the right-hand side of Eq. (A4), contributing to a phase factor [50], will be dropped. For convenience, we work in the light-cone gauge A+= 0, in which the second and third terms are rewritten as
G(1)(z) = i Z dugGαβ(uz)nβ− Z d4l (2π)4e il·zu 6n+γα6l − ¯u 6lγα6n+ (l2)2 . (A5)
It is clear that the field strength gGαβ(uz)nβ−can be factored together with the rescaled b quark field h and the light
quark field ¯u into the nonlocal matrix element in Eq. (15). The integrand depending on l is then identified as the hard kernel in momentum space for the three-parton contribution. Employing Eq. (A5) for Fig. 2, and substituting P2− k1− uk2for l, we obtain Eq. (16).
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