Three-parton contribution to the $B\to\pi$ form factors in $k_T$ factorization

We calculate the three-parton twist-3 contribution to the $B\to\pi$ transition form factors in the $k_T$ factorization theorem. Since different mesons are involved in the initial and final states, two(three)-parton-to-three(two)-parton amplitudes do not vanish. It is found that the dominant contribution arises from the diagrams with the additional valence gluon attaching to the leading-order hard gluon. Employing the three-parton meson distribution amplitudes from QCD sum rules, we show that this subleading piece amounts only up to few percents of the form factors at large recoil of the pion. The framework for analyzing three-parton contributions to $B$ meson decays in the $k_T$ factorization is established.

the gauge-dependent amplitudes cancel in each of the two sources. The proof for the amplitudes with three partons from the B meson side is the same. The B meson momentum P 1 and the pion momentum P 2 are parameterized as where the energy fraction η = 1 − q 2 /m 2 B carried by the pion ranges between 0 and 1. The momenta of the antiquarks in the B meson and in the pion, represented by the lower fermion line, are parameterized as respectively, x 1 and x 2 being the momentum fractions. It is understood that the components k − 1 and k + 2 have been dropped in hard kernels, and integrated out of the B meson and pion wave functions, respectively. The gluon propagator of momentum l is written as in the covariant gauge, where the parameter λ is used to identify sources of gauge dependence. We sandwich Fig. 1(a) with the spin projectors from the initial and final states, respectively, where N c = 3 is the number of colors, γ 5 γ β is a higher-twist projector [15] selected for the proof below, and the subscript β takes the transverse components. The resultant hard kernel contains the gauge-dependent piece with the b quark mass m b . In the small x region where the k T factorization applies, we keep the transverse momentum dependence in the denominator [22]. The transverse momentum dependence in the numerator belongs to the twist-3 contribution. Inserting the identity for the gluon vertex on the b quark line, we find that the second term vanishes at leading-twist accuracy on the B meson side, as it is multiplied by P 1 + m B . The derivative of the numerator with respect to k β 2T gives The LO hard kernel from Fig. 1(b) contains the gauge-dependent amplitude The similar differentiation with respect to k β 2T leads to For the first term in the above expression, k 1 ( k 2 ) implies one more derivative of the spectator field on the B meson (pion) side, so it is neglected. The second term, after employing where the term P 2 − k 2 , implying the derivative of the energetic quark field on the pion side, has been dropped. Apparently, Eqs. (6) and (8) cancel each. That is, the gauge dependence associated with the derivative of quark fields disappears at 1/m B . We then compute the gauge-dependent amplitudes from Fig. 2, in which any contributions from the derivatives of quark fields should be ignored. For the attachment A of the valence gluon to the virtual quark line, the spin projector for the pion in Eq. (4) is replaced by γ 5 γ β /2 [15]. The color factor associated with this attachment is given by tr[T a T b T b T c ] = C F δ ac /2, where the index a labels the color of the valence gluon. Summing over the index c, the corresponding amplitude is written as which vanishes because k 1 ( k 2 ) in the factor k 1 − k 2 implies one more derivative of the spectator field on the B meson (pion) side. Using a similar argument, the attachment B does not generate the gauge dependence either. The gauge-dependent amplitude from the attachment C is given by According to the above explanation, if the vertex on the spectator line contains k 1 − k 2 , the associated term comes from the derivative of the spectator field, and should be dropped. Hence, the gauge dependence can appear only in the first term linear in λ, which leads to The evaluation of the gauge-dependent pieces for the rest of attachments is similar. With the color factor for the attachment D, tr[T b T a T b T c ] = −δ ac /(4N c ), the corresponding amplitude is written as The gauge-dependent amplitudes from the attachments E and F diminish. The attachments G and H give respectively. The cancellation between Eqs. (11) and (13), and between Eqs. (12) and (14) is observed. That is, the gauge dependence from the three-parton Fock state also disappears. This completes the proof of the gauge invariance of the k T factorization for the B → π transition form factors at leading order in α s and at three-parton twist-3 level.

III. THREE-PARTON CONTRIBUTIONS
In this section we calculate the B → π transition form factors F + and F 0 involved in the semileptonic decay B(P 1 ) → π(P 2 )ℓν, where q = P 1 − P 2 is the lepton-pair momentum. Another equivalent definition is given by in which the form factors f 1 and f 2 are related to F + and F 0 via We start with the hard kernels from the two-parton-to-three-parton diagrams in the Feynman gauge (λ = 0). The following matrix element [23] defines the three-parton twist-3 pion wave function T (z, z ′ ), with the chiral scale m 0 = m 2 π /(m u + m d ), m π , m u , and m d being the pion, u quark and d quark masses, respectively. The three momenta P 2 − k 2 − l 2 , k 2 , and l 2 are assigned to the final-state quark, antiquark, and gluon, respectively. For the calculation, we replace the projector for the pion in Eq. (4) by γ 5 P 2 γ T β m 0 /(4y 2 ) [15], where the valence gluon momentum fraction is defined by y 2 = l − 2 /P − 2 , the gamma matrix γ T contains only transverse components, and the pion decay constant has been absorbed into the wave function T (z, z ′ ).
The amplitudes from the attachments A, B, · · · , H in Fig. 2 are collected as follows: The denominators of Eqs. (19) and (21) indicate that the contribution from the former is down by a power of k + 1 /m B ∼ Λ QCD /m B . That is, the attachments to the b quark line and to the energetic parton line of the pion give power-suppressed contributions in the dominant region with soft spectator momenta. This observation is similar to that obtained in the study of the three-parton twist-3 contribution to the pion form factor [15]. Equation (20) vanishes, since the γ matrix associated with the valence gluon attachment takes only the transverse components. One can then flip the b quark propagator and this γ matrix, and apply ( P 1 − k 1 − m b )( P 1 + m B ) ≈ 0 at leading-twist accuracy on the B meson side. The attachments E, F , and H do not contribute as shown in Eq. (23), simply because of γ ν γ 5 P 2 γ T β γ ν = 0. The k 1µ term in Eq. (24) is of higher-power and negligible. It is found that all the above amplitudes are proportional to m B , namely, diminish as m B → 0. This must be the case, since the two(three)-parton-to-three(two)-parton diagrams do not contribute to the pion form factor [15]. In the numerical analysis below we shall not differentiate m B and m b , whose difference gives an additional power of 1/m B . Ignoring Eq. (19) and the second term in Eq. (24), the two-parton-to-three-parton amplitudes are summed into To derive the above expression, we have followed the hierarchy among the relevant scales xm 2 B ≫ k 2 T [13,15], under which the k T -dependent terms in the denominators of the b quark and energetic quark propagators are dropped.
The three momenta P 1 −k 1 −l 1 , k 1 and l 1 are assigned to the initial-state b quark, antiquark, and gluon, respectively. Equation (29) corresponds to the spin projector γ ± γ T α γ 5 √ 2λ 2 ± /(4y 1 ) from the B meson side, where the decay constant f B has been absorbed into the three-parton B meson distribution amplitude, and the valence gluon momentum fraction is defined by y 1 = l + 1 /P + 1 . For the attachments of the valence gluon in the B meson to the lines in Fig. 1(a), only the one to the hard gluon contributes, because of γ ν γ ± γ T α γ 5 γ ν = 0. All attachments to the lines in Fig. 1(b) do not contribute, since the corresponding Feynman rules have no m B dependence. As explained before, the three-partonto-two-parton amplitudes must be proportional to m B . Assuming the same three-parton distribution amplitudes associated with the normalization constants λ 2 ± , we derive It has been known that the form factor f 1 is suppressed by m 0 /m B compared to f 2 [8]. Therefore, it is natural that the three-parton contribution corrects only f 2 at the accuracy considered here, which is summarized as with the factorization formulas We have neglected the intrinsic b dependence of the pion distribution amplitudes, because the suppression of the Sudakov factor exp[−s(P − 2 , b)] is strong enough in the large b region [5,[26][27][28]. On the contrary, the Sudakov effect associated with the B meson is weak, since it is dominated by soft dynamics. For the B meson distribution amplitudes, the intrinsic b dependence is more effective. The hard kernels are written as The functional form of the three-parton B meson distribution amplitude is still unknown in the literature, though there are already studies of its relation to the two-parton ones [9,29]. Below we shall postulate a simple form for an order-of-magnitude estimate. The involved two-parton and three-parton meson distribution amplitudes are chosen as with the parameters ω B = 0.4 GeV [27] and η 3 = 0.015 [30], and the Gegenbauer polynomials The normalization constants N B and N ′ B are determined through the relations The two-parton B meson and pion distribution amplitudes have been chosen as in [8] in order to have an appropriate comparison of numerical outcomes.
Equation (32) represents the three-parton contribution to the form factor f 2 , which then corrects the form factors F + and F 0 via Eq. (17). The numerical results derived from Eq. (32) for f B = 0.2 GeV, f π = 0.13 GeV, m B = 5.28 GeV, m 0 = 1.4 GeV and α s = 0.5 are listed in Table I, which confirm the ratio of the three-parton-to-two-parton contribution over the two-parton-to-three-parton one, 2λ 2 + /(m B m 0 η 3 ) ≈ 2.6 (0.8) from Eq. (27) [Eq. (28)]. The dominant contribution arises from the diagrams with the additional valence gluon attaching to the leading-order hard gluon, i.e., from Eqs (21) and (31). Figure 3 shows that the three-parton contribution amounts only up to few percents of the B → π transition form factors F + (0) = F 0 (0) ≈ 0.3 at large recoil of the pion. The relative importance is obvious from the order-of-magnitude estimate η 3 m 0 /t ∼ λ 2 + /(m B t) ∼ 1%, in which the scale η 3 m 0 (λ 2 + /m B ) is associated with the spin projector of the three-parton pion (B meson) distribution amplitude, and t ∼ 1.7 GeV denotes   the characteristic scale involved in B meson decays at large recoil [8]. Figure 3 also indicates that the three-parton contribution is of the same order as the third piece in the following projector associated with the two-parton B meson distribution amplitudes [31,32] ( with the dimensionless vectors n + = (1, 0, 0 T ) and n − = (0, 1, 0 T ). Collecting the observations obtained in the literature, we summarize the various contributions to the B → π transition form factors: the first term in the above projector, which has been considered in [8], gives the leading contribution. The second termφ B , proportional to the difference of the two leading-power B meson wave functions, contributes 30% [32]. The third term, proportional to the integration ofφ B in the momentum fraction, and the three-parton Fock state contribute only few percents.

IV. CONCLUSION
In this letter we have extended the investigation of the B → π transition form factors in the k T factorization theorem to the three-parton twist-3 level. It was demonstrated that the gauge-dependent pieces cancel each other in the two(three)-parton-to-three(two)-parton diagrams, so the gauge invariance of this formalism is verified. The contributions from the above diagrams were then calculated, and found to be few percents at most, considering the normalization inputs for the three-parton B meson distribution amplitudes from QCD sum rules. The theoretical framework for analyzing three-parton contributions to B meson decays was established in this work, which can be compared to other approaches, such as light-cone sum rules [33], the QCD (collinear) factorization [34], and the soft-collinear effective theory [35].