Light-cone sum rules for form factors revisited
We reconsider and update the QCD light-cone sum rules for form factors. The gluon radiative corrections to the twist-2 and twist-3 terms in the correlation functions are calculated. The -quark mass is employed, instead of the one-loop pole mass used in the previous analyses. The light-cone sum rule for is fitted to the measured -distribution in , fixing the input parameters with the largest uncertainty: the Gegenbauer moments of the pion distribution amplitude. For the vector form factor at zero momentum transfer we predict . Combining it with the value of the product extracted from experiment, we obtain . In addition, the scalar and penguin form factors and are calculated.
The form factors of heavy-to-light transitions at large energies of the final hadrons are among the most important applications of QCD light-cone sum rules (LCSR) . In this paper we concentrate on the transition form factors , and of the electroweak vector and penguin currents, respectively. Previously, these form factors have been calculated from LCSR in [2, 3, 4, 5, 6, 7, 8, 9, 10], gradually improving the accuracy.
The main advantage of LCSR is the possibility to perform calculations in full QCD, with a finite -quark mass. In the sum rule approach, the matrix element is obtained from the correlation function of quark currents, rather than estimated directly from a certain factorization ansatz. This correlation function is conveniently “designed”, so that, at large spacelike external momenta, the operator-product expansion (OPE) near the light-cone is applicable. Within OPE, the correlation function is factorized in a series of hard-scattering amplitudes convoluted with the pion light-cone distribution amplitudes (DA’s) of growing twist. To obtain the form factors from the correlation function, one makes use of the hadronic dispersion relation and quark-hadron duality in the -meson channel, following the general strategy of QCD sum rules . More details can be found in the reviews on LCSR, e.g., in [12, 13, 14]. A modification of the method, involving -meson distribution amplitudes and dispersion relation in the pion channel was recently suggested in ; the analogous sum rules for form factors in soft-collinear effective theory (SCET) were derived in .
LCSR provide analytic expressions for the form factors, including both hard-scattering and soft (end-point) contributions. Because the method is based on a calculation in full QCD, combined with a rigorous hadronic dispersion relation, the uncertainties in the resulting LCSR are identifiable and assessable. These uncertainties are caused by the truncation of the light-cone OPE, and by the limited accuracy of the universal input, such as the quark masses and parameters of the pion DA’s. In addition, a sort of systematic uncertainty is brought by the quark-hadron duality approximation adopted for the contribution of excited hadronic states in the dispersion relation. Importantly, form factors are calculable from LCSR in the region of small momentum transfer (large energy of the pion), not yet directly accessible to lattice QCD.
The decays, with continuously improving experimental data, provide nowadays the most reliable exclusive determination. Along with the lattice QCD results, the form factor obtained  from LCSR is used for the extraction. Furthermore, the LCSR form factors can provide inputs for various factorization approaches to exclusive decays, such as QCD factorization , whereas the penguin form factor is necessary for the analysis of the rare decay. Having in mind the importance of form factors for the determination and for the phenomenological analysis of various exclusive decays, we decided to reanalyze and update the LCSR for these form factors. One of our motivations was to recalculate the gluon radiative correction to the twist-3 part of the correlation function. Only a single calculation of this term exists [9, 10], whereas the corrections to the twist-2 part have been independently obtained in  and . In what follows, we derive and present the explicit expressions for all hard-scattering amplitudes and their imaginary parts for the twist-2 and twist-3 parts of the correlation function and some of these expressions are new.
In the OPE of the correlation function the mass is used, a natural choice for a virtual -quark propagating in the hard-scattering amplitudes, calculated at large spacelike momentum scales . Importantly, in the resulting sum rules we keep using the mass. Note that the value of is rather accurately determined from the bottomonium sum rules. In previous analyses, the one-loop pole mass of the -quark was employed in LCSR. The main motivation was that the pole mass was used also in the two-point sum rule for the -meson decay constant , needed to extract the form factor from LCSR. In the meantime, the sum rule is available also in -scheme , and we apply this new version here.
Furthermore, we fix the most uncertain input parameters, the effective threshold and simultaneously, the Gegenbauer moments of the pion twist-2 DA, by calculating the -meson mass and the shape of from LCSR and fitting these quantities to their measured values. In addition, the nonperturbative parameters of the twist-3,4 pion DA’s entering LCSR are updated, using the results of the recent analysis .
The paper is organized as follows. In sect. 2 the correlation function is introduced and the leading-order (LO) terms of OPE are presented, including the contributions of the pion twist-2,3,4 two-particle DA’s and twist-3,4 three-particle DA’s. In sect. 3 the calculation of the twist-2 and twist-3 parts of the correlation function is discussed. In sect. 4 we present LCSR for all three form factors. Sect. 5 contains the discussion of the numerical input and results, as well as the estimation of theoretical uncertainties, and finally, the determination of . Sect. 6 is devoted to the concluding discussion. App. A contains the necessary formulae and input for the pion DA’s. The bulky expressions for the hard-scattering amplitudes and their imaginary parts are collected in App. B, and the sum rule for is given in App. C.
2 Correlation function
The vacuum-to-pion correlation function used to obtain the LCSR for the form factors of transitions is defined as:
for the two different transition currents, For definiteness, we consider the flavour configuration and, for simplicity we use instead of in the penguin current, which does not make difference in the adopted isospin symmetry limit. Working in the chiral limit, we neglect the pion mass () and the -, -quark masses, whereas the ratio remains finite.
At and , that is, far from the -flavour thresholds, the quark propagating in the correlation function is highly virtual and the distances near the light-cone dominate. It is possible to prove the light-cone dominance, following the same line of arguments as in . Contracting the -quark fields, one expands the vacuum-to pion matrix element in terms of the pion light-cone DA’s of growing twist. The light-cone expansion  of the -quark propagator is used (see also ):
where only the free propagator and the one-gluon term are retained. The latter term gives rise to the three-particle DA’s in the OPE.
Diagrammatically, the contributions of two- and three-particle DA’s to the correlation function are depicted in Fig. 1. In terms of perturbative QCD, these are LO (zeroth order in ) contributions. The Fock components of the pion with multiplicities larger than three, are neglected, as well as the twists higher than 4. This truncation is justified by the fact that the twist-4 and three-particle corrections to LCSR obtained below turn out to be very small.
In addition we include the gluon radiative corrections to the dominant twist-2 and twist-3 parts of the correlation function. The OPE result for the invariant amplitude is then represented as a sum of LO and NLO parts:
and the same for and . The leading-order (LO) invariant amplitudes , , and including twist 2,3,4 contributions have been obtained earlier in [3, 6, 7, 21]. We present them here switching to the new notations  of the twist-3,4 DA’s:
where , and the definitions of the twist-2 (), twist-3 (, , ) and twist-4 (, , , , , ) pion DA’s and their parameters are presented in App. A. Note that all twist-4 terms are suppressed with respect to leading twist-2 terms, with an additional power of the denominator compensated by the normalization parameter of the twist-4 DA’s.
The calculation of the NLO amplitudes will be discussed in the next section.
3 Gluon radiative corrections
In the light-cone OPE of the correlation function (1) each twist component receives gluon radiative corrections. To obtain the desired NLO terms, one has to calculate the one-loop diagrams shown in Fig. 2, convoluting them with the twist-2 and two-particle twist-3 DA’s, respectively. The diagrams are computed using the standard dimensional regularization and scheme. In addition, in our calculation the reduction method from  is employed.
The invariant amplitude in (3) is obtained in a factorized form of the convolutions:
where the hard-scattering amplitudes , result from the calculation of the diagrams in Fig.2. The two other NLO amplitudes and have the same expressions with , , and , , respectively. The resulting expressions for all hard-scattering amplitudes are presented in App. B. Note that the LO expressions for the correlation functions in (4)-(6) also have a factorized, albeit a much simpler form, with the zeroth-order in hard-scattering amplitudes stemming from the free propagator of the virtual -quark. In particular, the twist-2 component in is a convolution of with .
Let us mention some important features of the terms of OPE. The currents and in the correlation function are physical and not renormalizable. Hence, the ultraviolet singularities appearing in and are canceled by the renormalization of the heavy quark mass. For an additional renormalization of the composite operator has to be taken into account. Furthermore, in the twist-2 term in (7) the convolution integral is convergent due to collinear factorization. As explicitly shown in [4, 5], the infrared-collinear divergences of the diagrams are absorbed by the well known one-loop evolution  of the twist-2 pion DA. As a result of factorization, a residual dependence on the factorization scale enters the amplitude and the twist-2 DA . This scale effectively separates the long- and short (near the light-cone) distances in the correlation function. In the twist-3 part of , the complete evolution kernel has to include the mixing of two- and three-particle DA’s. To avoid these complications, and following , the twist-3 pion DA’s in (7) are taken in their asymptotic form: and , whereas the nonasymptotic effects in these DA’s are only included in the LO part . We checked that the infrared divergences appearing in the amplitudes and cancel in the sum of the and contributions with the one-loop renormalization of the parameter (i.e., of the quark condensate density). Finally, in accordance with [9, 10], all renormalized hard-scattering amplitudes are well behaved at the end-points , regardless of the form of the DA’s.
After completing the calculation of OPE terms with the LO (NLO) accuracy up to twist-4 (twist-3), we turn now to the derivation of the sum rules.
4 LCSR for form factors
In the LCSR approach the matrix elements are related to the correlation function (1) via hadronic dispersion relation in the channel of the current with the four-momentum squared . Inserting hadronic states between the currents in (1) one isolates the ground-state -meson contributions in the dispersion relations for all three invariant amplitudes:
where the ellipses indicate the contributions of heavier states (starting from ). The three form factors entering the residues of the pole in (8) are defined as:
and is the -meson decay constant.
Substituting the OPE results for , and in l.h.s. of (8), one approximates the contributions of the heavier states in r.h.s. with the help of quark-hadron duality, introducing the effective threshold parameter . After the Borel transformation in the variable , the sum rules for all three form factors are obtained. The LCSR for the vector form factor reads:
where originates from the OPE result for the LO (NLO) invariant amplitude .
The LO part of the LCSR has the following expression:
where , and the short-hand notations introduced for the integrals over three-particle DA’s are:
The NLO term in (11) is cast in the form of the dispersion relation:
where the bulky expressions for the imaginary parts of the amplitudes ,, are presented in App. B.
The LCSR following from the dispersion relation for the invariant amplitude in (8) reads:
Finally, the LCSR for the penguin form factor obtained from the third dispersion relation in (8) has the following expression:
For entering LCSR we use the well known two-point sum rule  obtained from the correlator of two currents. The latest analyses of this sum rule can be found in [18, 25]; here we employ the version . For consistency with LCSR, the sum rule for is taken with accuracy. For convenience, this expression is written down in App. C.
Note that the expressions for LCSR in LO are slightly modified as compared to the ones presented in the previous papers. We prefer not to use the so-called “surface terms”, which originate from the powers of with in the correlation functions. Instead, we use a completely equivalent but more compact form, with derivatives of DA’s.
The twist-2 NLO part of LCSR for , hence, the expressions for and in App. B, after transition to the pole scheme (the additional expressions necessary for this transition are also presented in App. B) coincide with the ones obtained in . We have also checked an exact numerical coincidence with the twist-2 NLO part of the sum rule in , written in a different analytical form. The explicit expressions for the amplitudes , , , and , and their imaginary parts presented in App. B are new. The spectral density entering the LCSR for is given in  in a different form, that is, with the -integration performed, making an analytical comparison of our result with this expression very complicated. The numerical comparison is discussed below, in sect. 6. Furthermore, in  the LCSR for the form factor was obtained, and the imaginary part of was presented. A comparison with our expression for reveals, however, some differences.
Since the imaginary parts of the hard-scattering amplitudes have a very cumbersome analytical structure, we carried out a special check of these expressions. Each hard-scattering amplitude taken as a function of was numerically compared with its dispersion relation in the variable , where the expression for was substituted. Note that one has to perform one subtraction in order to render the dispersion integral convergent.
In addition, we applied a new method which completely avoids the use of explicit imaginary parts of hard-scattering amplitudes, allowing one to numerically calculate the NLO parts of LCSR, e.g., in (14), analytically continuing integrals to the complex plane. We make use of the fact that the hard-scattering amplitudes are analytical functions of the variable in the upper half of the complex plane, because of ’s in Feynman propagators. Consider, as an example the twist-2 part of given by the integral over in the second line of (14). Since the integration is performed along the real axis, the operation of taking the imaginary part can be moved outside the integral. To proceed, one has to shift the lower limit of the -integration to any point at . This is legitimate because all ’s are real at . Then one deforms the path of the -integration, replacing it by a contour in the upper half of the complex plane, as shown schematically in Fig. 3, so that all poles and cuts are away from the integration region. Only when is approaching the upper limit , one nears the pole at while performing the integration over . Because this pole does not touch the limits , it is possible to avoid it by moving the contour of the -integration into the upper half of the complex -plane (see Fig. 3). After that, both numerical integrations become completely stable. Note, that in both - and -integrations, we integrate over the semi-circle, but the contour of the integration can be deformed in an arbitrary way in the upper half of the complex plane. The numerical integrations of over these contours yield an imaginary part which represents the desired answer for . We have checked that the numerical results obtained by this alternative method coincide with the ones obtained by the direct integration over the imaginary parts, thereby providing an independent check.
5 Numerical results
The value of the -quark mass is taken from one of the most recent determinations :
based on the bottomonium sum rules in the four-loop approximation. Note that (21) has a smaller uncertainty than the average over the non-lattice determinations given in : However, as we shall see below, the uncertainty of does not significantly influence the “error budget” of the final prediction. Furthermore, in our calculation, the scale-dependence is taken into account in the one-loop approximation which is sufficient for the -accuracy of the correlation function. Note that using the mass inevitably introduces some scale-dependence of the lower threshold in the dispersion integrals in both LCSR and sum rule. However, this does not create a problem, because the imaginary part of the OPE correlation function obtained from a fixed-order perturbative QCD calculation is not an observable, but only serves as an approximation for the hadronic spectral density.
The QCD coupling is obtained from , with the NLO evolution to the renormalization scale . In addition to and , one encounters the factorization scale in the correlation function, at which the pion DA’s are taken. In what follows, we adopt a single scale in both LCSR and two-point SR for . The numerical value of will be specified below.
|twist||Parameter||Value at GeV||Source|
|2||average from |
|form factor |
|GeV||GMOR relation; from |
|3||GeV||2-point QCD SR |
|2-point QCD SR|
|4||GeV||2-point QCD SR |
|2-point QCD SR |
The input parameters of the twist-2 pion DA include MeV  and the two first Gegenbauer moments and normalized at a low scale 1 GeV. For the latter we adopt the intervals presented in Table 1. The range for is an average  over various recent determinations, including, e.g., calculated from the two-point sum rule in . For we use, following , the constraint , obtained  from the analysis of form factor. Having in mind, that at large scales the renormalization suppresses all higher Gegenbauer moments, we set in our ansatz for specified in App. A. The uncertainties of remain large, hence we neglect very small effects of their NLO evolution taken into account in .
The normalization parameter of the twist-3 two-particle DA’s presented in Table 1 is obtained adopting the (non-lattice) intervals  for the light quark masses: , . Correspondingly, the quark-condensate density given by GMOR relation is:
where very small corrections are neglected. We prefer to use the above range, rather than a narrower “standard” interval employed in the previous analyses. In fact, (22) is consistent with quoted in the review , as well as with the recent determination of the light-quark masses from QCD sum rules with accuracy : , .
The remaining parameters of the twist-3 DA’s (, ) and twist-4 DA’s (, ) presented in Table 1 are taken from , where they are calculated from auxiliary two-point sum rules. The latter are obtained from the vacuum correlation functions containing the local quark-gluon operators that enter the matrix elements (34), (35) and (40), (41). The one-loop running for all parameters of DA’s is taken into account using the scale-dependence relations presented in App. A. Note that the small value of effectively suppresses all nonasymptotic and three-particle contributions of the twist-3 DA’s. Furthermore, the overall size of the twist-4 contributions to LCSR is very small. Hence, although the parameters of the twist-3,4 DA’s have large uncertainties, only the accuracy of plays a role in LCSR111We also expect that the use of the recently developed renormalon model  for the twist-4 DA’s, instead of the “conventional” twist-4 DA’s  used here, will not noticeably change the numerical results.. Finally, in the sum rule (86) for the gluon condensate density GeV and the ratio of the quark-gluon and quark-condensate densities  are used, the accuracy of these parameters playing a minor role.
The universal parameters listed above determine the “external” input for sum rules. The next step is to specify appropriate intervals for the “internal” parameters: the scale , the Borel parameters and and the effective thresholds and . In doing that, we take all external input parameters at their central values, allowing only and to vary within the intervals given in Table 1.
From previous studies [4, 5, 8, 10] it is known that an optimal renormalization scale is (where does not scale with the heavy quark mass), and simultaneously, has the order of magnitude of the Borel scales defining the average virtuality in the correlation functions. In practice, and are varied within the “working windows” of the respective sum rules, hence one expects that also has to be taken in a certain interval.
Calculating the total Borel-transformed correlation function (that is, the limit of LCSR) we demand that the contribution of subleading twist-4 terms remains very small, of the LO twist-2 term, thereby diminishing the contributions of the higher twists, that are not taken into account in the OPE. This condition puts a lower bound GeV. In addition, in order to keep the -expansion in the Borel-transformed correlation function under control, both NLO twist-2 and twist-3 terms are kept of their LO counterparts, yielding a lower limit GeV. Hereafter a “default” value GeV is used.
Furthermore, we determine the effective threshold parameter in LCSR for each . We refrain from using equal threshold parameters in LCSR and two-point sum rule for , as it was done earlier, e.g. in [4, 8]. Instead, we control the duality approximation by calculating certain observables directly from LCSR and fitting them to their measured values. Importantly, we include in the fitting procedure not only , but also the two least restricted external parameters and , under the condition that both Gegenbauer moments remain within the intervals of their direct determination given in Table 1.
The first observable used in this analysis is the -meson mass. In a similar way, as e.g., in [10, 18], is calculated taking the derivative of LCSR over and dividing it by the original sum rule. The -meson mass extracted from LCSR has to deviate from its experimental value GeV by less than 1 %. Secondly, we make use of the recent rather accurate measurement of the -distribution in by BABAR collaboration . We remind that LCSR for form factors are valid up to momentum transfers , typically at GeV. To be on the safe side, we take the maximal allowed slightly lower than in the previous analyses and calculate the slope from LCSR at GeV. The obtained ratio is then fitted to the slope of the form factor inferred from the data. We employ the result of , where various parameterizations of the form factor are fitted to the measured -distribution. Since all fits turn out to be almost equally good, we adopt the simplest BK-parameterization :