# The NLO twist-3 contributions to form factors in factorization

###### Abstract

In this paper, we calculate the next-to-leading-order (NLO) twist-3 contribution to the form factors of transitions by employing the factorization theorem. All the infrared divergences regulated by the logarithms cancel between those from the quark diagrams and from the effective diagrams for the initial meson wave function and the final pion meson wave function. An infrared finite NLO hard kernel is therefore obtained, which confirms the application of the factorization theorem to meson semileptonic decays at twist-3 level. From our analytical and numerical evaluations, we find that the NLO twist-3 contributions to the form factors of transition are similar in size, but have an opposite sign with the NLO twist-2 contribution, which leads to a large cancelation between these two NLO parts. For the case of , for example, the NLO twist-2 enhancement to the full LO prediction is largely canceled by the negative ( about ) NLO twist-3 contribution, leaving a small and stable enhancement to the full LO prediction in the whole range of GeV. At the full NLO level, the perturbative QCD prediction is . We also studied the possible effects on the pQCD predictions when different sets of the B meson and pion distribution amplitudes are used in the numerical evaluation.

###### pacs:

12.38.Bx, 12.38.Cy, 12.39.St, 13.20.He## I Introduction

Without end-point singularity, factorization theorem npb325-62 ; npb360-3 ; prl74-4388 is a better tool to deal with the small physics when comparing with other factorization approachesplb87-359 ; prd22-2157 ; plb94-245 ; qcd1993 ; npb685-249 . Based on the factorization theorem, perturbative QCD (pQCD) factorization approachplb504-6 ; prd63-074009 ; prl65-2343 ; li2003 is a successful factorization approach to handle the heavy to light exclusive decay processes. As an effective factorization theorem, the factorization should be valid at every order expanded by strong coupling , where is the power of the expansion.

Recently, the next-to-leading-order(NLO) twist-2 (the leading twist) contributions to the form factors for the , and transitions have been evaluated prd76-034008 ; prd83-054029 ; prd85-074004 by employing the factorization theorem npb325-62 ; npb360-3 ; prl74-4388 , and an infrared finite dependent hard kernel were obtained at the NLO level for each considered process. It is worth of mentioning that a new progress about pion form factor in the scattering has been made in Ref. jhep1401-004 very recently, where the authors made a joint resummation for the pion wave function and the pion transition form factor and proved that the factorization is scheme independent. These NLO contributions could produce sizable effects to the LO hard kernels. For example, the NLO twist-2 contribution to the form factor for transition can provide enhancement to the corresponding full LO form factor prd85-074004 . In a recent paper cheng14a , we calculated the NLO twist-3 contribution to the pion electromagnetic form factor in the process by employing the factorization theorem, and found infrared finite NLO twist-3 corrections to the full LO hard kernels cheng14a .

In this paper, following the same procedure of Ref. prd85-074004 , we will calculate the NLO twist-3 contribution to the form factor of transition, which is the only missing piece at the NLO level. The light partons are also considered to be off-shell by in both QCD quark diagrams and effective diagrams for hadron wave functions. The radiation gluon from the massive quark generates the soft divergence only. Such soft divergence can be regulated either by the virtuality of internal particles or by the virtuality of other light partons, to which the emission gluons were attached. So we can replace the off-shell scale for the light parton by for the massive b quark safely to regulate the IR divergences from the massive quark line, where means the mass of the gluon radiated from the b quark. That means, the b quark remains on-shell in the framework.

We will prove that the IR divergences in the NLO QCD quark diagrams could be canceled by those in the effective diagrams, i.e., the convolution of the meson and meson wave functions with the LO hard kernel. The IR finiteness and -dependent NLO hard kernel were also derived at the twist-3 level for the transition form factor, which confirms the application of the factorization theorem to meson semileptonic decays at both the twist-2 and twist-3 level.

In our calculation for the NLO twist-3 contribution, the resummation technologyprd66-094010 ; plb555-197 is applied to deal with the large double logarithms and , where being the parton momentum fraction of the anti-quark in the meson wave functions. With appropriate choices of and , say being lower than the meson mass, the NLO corrections are under control. From numerical evaluations we find that the NLO correction at twist-3 is about of the LO part, while the NLO twist-2 contribution can provide a enhancement to the LO one. This means that the NLO twist-2 contribution to the form factor are largely canceled by the NLO twist-3 one, leaves a net small correction to the full LO form factor, around or less than enhancement.

The paper is organized as follows. In Sec. II, we give a brief introduction for the calculations of the LO diagrams relevant with the form factor of transition. In Sec. III, we calculate the NLO twist-3 contribution to the form factor. The relevant QCD quark diagrams are calculated analytically, the convolutions of wave functions and hard kernel are made in the same way as those for the evaluation of the NLO twist-2 contribution. And finally we extract out the expression of the factor , which describes the NLO twist-3 contribution to the form factor . In Sec. IV we calculate and present the numerical results for the relevant form factors and examine the -dependence of and at the LO and NLO level, respectively. A short summary was given in the final section.

## Ii LO analysis

By employing the factorization theorem, the LO twist-2 and twist-3 contributions to the form factor of transition have been calculated many years ago plb504-6 ; prd63-074009 ; prl65-2343 ; li2003 . For the sake of the readers, we here present the expressions of the leading order hard kernels directly.

The transition form factors are defined via the matrix element

(1) |

where is the meson mass, and is the transfer momentum. The momentum is chosen as with the component and . Here the parameter represents the energy fraction carried by the pion meson, and when in the large recoil region of pion. According to the factorization, the anti-quark carries momentum in the meson and in the pion meson as labeled in Fig. 1, and being the momentum fractions. The follow hierarchy is postulated in the small-x region:

(2) |

which is roughly consistent with the order of magnitude: , , GeV, and GeV prd85-074004 .

The LO hard kernels are obtained after sandwiching Fig. 1 with the meson and the pion meson wave functionsplb504-6 ; prd63-074009 ; npb592-3

(3) | |||

(4) | |||

(5) |

where is the chiral mass of pion, and denote the pion meson wave function at twist-2 and twist-3 level, the dimensionless vectors are defined by , and , and is the number of colors. Without considering the transverse component of the meson spin projector, the LO twist-3 contribution for Fig. 1(a) is of the form,

(6) |

and for Fig. 1(b) we find

(7) |

where is the color factor.

The LO twist-2 contributions for Fig. 1(a) and 1(b) are of the form,

(8) | |||

(9) |

For the LO twist-2 hard kernel , it is strongly suppressed by the small , as can be seen easily from Eqs. (8,9), and therefore the from Fig. 1(a) is the dominant part of the full LO twist-2 contribution. Consequently, it is reasonable to consider the NLO twist-2 contributions from Fig. 1(a) only in the calculation for the NLO twist-2 contributions.

For the LO twist-3 hard kernel , the first term proportional to in Eq. (7) provides the dominant contribution, while the second term proportional to is strongly suppressed by the small . The can be neglected safely when compared with , due to the strong suppression of small . We therefore consider only the component in Eq. (7) from Fig. 1(b) in our estimation for the NLO twist-3 contribution.

The LO hard kernels as given in Eqs.(6-9) are consistent with those as given in Refs. prd65-014007 ; epjc23-275 , where the meson wave function was defined as

(10) |

with the relations

(11) |

By comparing the hard kernel in Eq. (7) with in Eq. (8), one can find that the LO twist-3 contribution is enhanced by the factor and the pion chiral mass , and consequently larger than the LO twist-2 contribution which are associated with the factor . The numerical results of Eqs. (7,8) in the large recoil region also show that the LO twist-3 contribution is larger than the LO twist-2 part, by a ratio of around over . This fact means that the NLO twist-3 contribution may be important when compared with the corresponding NLO twist-2 one, this is one of the motivations for us to make the evaluation for the NLO twist-3 contribution to the form factor.

## Iii NLO corrections

Since the dominant NLO twist-3 contribution to the form factor of transition is proportional to the from the Fig. 1(b), we here consider only the NLO corrections to the Fig. 1(b) coming from the quark-level corrections and the wave function corrections at twist-3 level, to find the NLO twist-3 contribution to the form factor of transition.

Under the hierarchy in Eq. (2), only terms that don’t vanish in the limits of and are kept to simplify the expressions of the NLO twist-3 contributions greatly.

### iii.1 NLO Corrections from the QCD Quark Diagrams

The NLO corrections to Fig. 1(b) at quark-level contain the self-energy diagrams, the vertex diagrams and the box and pentagon diagrams, as illustrated by Fig. 2,3, 4, respectively. The ultraviolet(UV) divergences are extracted in the dimensional reductionplb84-193 in order to avoid the ambiguity from handling the matrix . The infrared(IR) divergences are identified as the logarithms , , and their combinations, where the dimensionless ratios are adopted,

(12) |

By analytical evaluations for the Feynman diagrams as shown in Fig. 2, we find the self-energy corrections from the nine diagrams:

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) |

where represents the UV pole, is the renormalization scale, is the Euler constant, is the number of quark color, is the number of the quarks flavors, and denotes the first term of the LO twist-3 contribution as given in Eq. (7),

(18) |

It’s easy to see that, besides for the subdiagram Fig. 2(e), the NLO self-energy corrections listed in Eqs. (14,15,17) are identical to the self-energy corrections for the NLO twist-2 case as given in Eqs. (7-8,11) in Ref. prd85-074004 . Except for a small difference in constant numbers, the for the subdiagram Fig. 2(a) in Eq. (13) is the same one as that as given in Eq. (7) of Ref. prd85-074004 for the case of the NLO twist-2 contributions. The reason for such high similarity is that the self-energy diagrams don’t involve the loop momentum flowed into the hard kernel. Only the Fig. 2(a), the self-energy correction of the b quark is emphasized here. The first term in the square brackets of required the mass renomalization, and the finite piece of the first term is then absorbed into the redefinition the b quark mass, with the relation . The second term in the square brackets of represents the correction to the b quark wave function. The involved soft divergence is regularized by the gluon mass because the valence b quark is considered on-shell, and the additional regulator will be canceled by the corresponding soft divergence in the effective diagrams Fig. 5(a). Comparing with the NLO twist-2 case, the result from the subdiagram Fig. 2(e) at twist-3 is simple, since it’s the self-energy correction to the massless internal quark line in the twist-3 case.

By analytical evaluations for the Feynman diagrams as shown in Fig. 3, we find the vertex corrections from the five vertex diagrams:

(19) | |||

(20) | |||

(21) | |||

(22) |

(23) |

The amplitude have no IR divergence due to the fact that the radiative gluon attaches to the massive b quark and the internal line in Fig. 3(a). The amplitude should have collinear divergence at the first sight because the radiative gluon in Fig. 3(b) attaches to the light valence quark, but it’s found that the collinear region was suppressed, then is IR finite. The radiative gluon in Fig. 3(c) attaches to the light valence anti-quarks, so that both the collinear and soft divergences are produced in , where the large double logarithm denoted the overlap of the IR divergences can be absorbed into the meson or the pion meson wave functions. The radiative gluon in Fig. 3(d) attaches to the light valence anti-quarks as well as the virtual LO hard gluon, so the soft divergence and the large double logarithm aren’t generated in . The radiative gluon in Fig. 3(e) attaches only to the light valence quark as well as the virtual LO hard gluon, and then just contains the collinear divergence regulated by from region.

The analytical results from the box and pentagon diagrams as shown in Fig. 4 are summarized as

(24) | |||

(25) | |||

(26) | |||

(27) | |||

(28) | |||

(29) |

Note that the amplitude of Fig. 4(a) has no IR divergence because the additional gluon is linked to the massive b quark and the virtual LO hard kernel gluon. Fig. 4(b) is two-particle reducible, whose IR contribution would be canceled by the corresponding effective diagrams for the meson function Fig. 5(c). All the other four subdiagrams Fig. 5(c,d,e,f) would generate double logarithms from the overlap region of the soft and collinear region, because the radiative gluon attached with b quark and light valence quark generate both collinear divergence and soft divergence, as well as the gluon attached two light valence partons. Fig. 4(d) is also a two-particle reducible diagram, whose contribution should be canceled completely by the corresponding effective diagrams Fig. 5(c) for the pion meson function due to the requirement of the factorization theorem. It’s found that the double logarithm in Fig. 4(c) offset with the double logarithm in Fig. 3(c), and the cancelation would also appear for the double logarithms in Fig. 4(e) and Fig. 4(f).

The NLO twist-3 corrections from all the three kinds of the QCD quark diagrams are summed into

(30) | |||||

for . The UV divergence in the above expression is the same as in the pion electromagnetic form factorprd83-054029 and in the leading twist of transition form factorprd85-074004 , which determines the renormalization-group(RG) evolution of the the coupling constant . The double logarithm arose from the reducible subdiagrams Fig. 4(b,d) would be absorbed into the NLO wave functions.

### iii.2 NLO Corrections of the Effective Diagrams

As point out in Ref. prd85-074004 , a basic argument of factorization is that the IR divergences arisen from the NLO corrections can be absorbed into the non-perturbative wave functions which are universal. From this point, the convolution of the NLO wave function and the LO hard kernel , the LO hard kernel and the NLO wave function are computed, and then to cancel the IR divergences in the NLO amplitude as given in Eq. (30). The convolutions for NLO wave functions and LO hard kernel are calculated in this subsection. In factorization theorem, the prd70-074030 collect the effective diagrams from the matrix elements of the leading Fock states , and collect the effective diagrams for the twist-3 transverse momenta dependent (TMD) light-cone wave function prd64-014019 ; epjc40-395

(31) |

(32) |

respectively, in which and are the light cone (LC) coordinates of the anti-quark field carried the momentum faction respectively, and is the effective heavy-quark field.

(33) | |||

(34) |

where is the path ordering operator. The two Wilson line and are connected by a vertical link at infinityplb543-66 . Then the additional LC singularities from the region where loop momentum appb34-3103 are regulated by the IR regulator and . The scales and are introduced to avoid the LC singularityprd85-074004 ; jhep0601-067 . It’s important to emphasize that the variation of the above scales is regarded as a factorization scheme-dependence, which would be brought into the NLO hard kernel after taking the difference between the QCD quark diagrams and the effective diagrams. And the above scheme-dependent scales can be minimized by adhering to fixed and . In Ref. jhep1401-004 , very recently, Li et al. studied the joint resummation for pion wave function and pion transition form factor, i.e., summing up the mixed logarithm to all orders. Such joint resummation can reduce the above scheme dependence effectively.

The convolution for order of meson function in Eq. (31) and over the integration variables and is

(35) |

In the evolution, the is approximated to vector with a very small plus component to avoid the LC singularity in the integration, and we choose to be positive while can be positive or negative for convenience. The NLO twist-3 corrections from the order wave function as shown in Fig. 5 are listed in the following with being the factorization scale:

(36) | |||

(37) | |||

(38) | |||

(39) | |||

(40) | |||

(41) | |||

(42) |

where the dimensionless parameter is chosen small to obtain the simple results as above. Because the two propagators in the LO hard kernel are both relevant to while only one is relevant to , there exist three 5-point integrals as shown in Fig. 5(c,e,g) need to be calculated. The reducible subdiagrams Fig. 5(c) reproduced the double logarithm as the quark subdiagram Fig. 4(b). Difference between the effective heavy-quark field employed in the meson wave function and the b quark field in the quark diagrams leads to different results in Fig. 4(b) and Fig. 5(c). It’s found that the regulator adopted to regularize the soft divergence in the reducible Fig. 5(a) will be canceled by the Fig. 2(a), while the regulators in Fig. 5(d) and Fig. 5(e) cancels each other. The large double logarithms in Fig. 5(f) and Fig. 5(g) also cancel each other. So the other IR divergences are regulated only by as the prediction because it’s just the NLO correction to the incoming meson wave function.

After summing all the contributions in Fig. 5, we obtain

(43) | |||||

The convolution of and the outgoing pion meson wave function over the integration variables and is

(44) |

The is mainly in component, and a very small minus component is kept to avoid the LC singularity. Note that the sign of is positive as while the sign of is arbitrary for convenience.

Fig. 6 collects all the NLO corrections to the outgoing pion wave function, and