Form factors f^{B\to\pi}_{+}(0) and f^{D\to\pi}_{+}(0) in QCD and determination of |V_{ub}| and |V_{cd}|

# Form factors fB→π+(0) and fD→π+(0) in Qcd and determination of |Vub| and |Vcd|

Zuo-Hong Li111lizh@ytu.edu.cn, Nan Zhu , Xiao-Jiao Fan  and Tao Huang 222huangtao@ihep.ac.cn Department of Physics, Yantai University, Yantai 264005, P.R.China
Institute of High Energy Physics and Theoretical Physics Enter for Science Facilities, Chinese Academy of Sciences, Beijing 100049,P.R.China
July 13, 2019
###### Abstract

We present a QCD study on semileptonic transitions at zero momentum transfer and an estimate of magnitudes of the associated CKM matrix elements. Light cone sum rules (LCSRs) with chiral correlator are applied to calculate the form factors and . We show that there is no twist-3 and-5 component involved in the light-cone expansions such that the resulting sum rules have a good convergence and offer an understanding of these form factors at twist-5 level. A detailed computation is carried out in leading twist-2 approximation and the masses are employed for the underlying heavy quarks. With the updated inputs and experimental data, we have and ; and . As a by-product, a numerical estimate for the decay constant is yielded as .

Keywords: QCD Phenomenology, NLO Computations

## I Introduction

An intensive study on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements remains a cornerstone of high energy physics programme, in testing the standard model (SM) and exploring new physics. The unitarity of the CKM matrix must be put to a test by phenomenological research on the so called unitarity triangle. Opposite to the side of the triangle whose length depends on, in addition to the CKM parameters and , the elements involving heavy-light quark mixing and , the angle is presently well-measured. So precision determination of them is central to the unitarity testing. Exclusive processes offer an indispensable avenue to understand these parameters. The decays of heavy mesons into a light pseudoscalar meson plus an electron and its antineutrino can proceed at electro-weak tree level and are much less sensitive to new physics, and accordingly they could serve as preferred exclusive channels to probe both elements that we take interest in, namely, and . Then we are confronted with calculation of the hadronic matrix elements, say, that for the transition parameterized usually as

 ⟨π(p)|¯uγμb|B(p+q)⟩=2fB→π+(q2)pμ+(fB→π+(q2)+fB→π−(q2))qμ, (1)

with the momentum assignment specified in brackets, and and being the form factors describing QCD dynamics in the decay, of which only the former is related if the small electron mass is neglected. Combining the partial rates measured in some bins with the form factor predictions of different QCD approaches, one could achieve the values for related . Another approach is to fit the experimental observations using the various form factor parameterizations. In such way, a strong constraint is imposed on distributions of the form factors such that one may obtain a precise estimate of the products , in which case theoretical task boils down to estimating the form factors at . Requiring a good knowledge of the form factors, the exclusive avenues to are theoretically more challenging than inclusive approaches. The continual data updates have aroused one’s enthusiasm for exploring heavy-to-light transitions to approach an understanding of the CKM parameters. In the wake of the recent accurate measurements of the semileptonic processes by the BaBar P.del 052011 (); P.del 032007 () and CLEO D.Besson (); B.I.Eisenstein () collaborations, new progress has been achieved in this respect. Some extent of tension, however, still holds between inclusive and exclusive extractions of . A global data-fitting from CKMFitter ckmf () and UTfit utf () is in favor of a smaller than inclusive determinations. One can be referred to CKM () for a comprehensive overview of the current status of the CKM matrix elements.

Developed from QCD sum rule technique, light cone sum rules (LCSRs)LCSR1 (); LCSR2 () have become a powerful competitor in making predictions for heavy to light transitions. Complementary to lattice QCD (LQCD) simulations, this approach is successfully applied to study decaysLCSR2 (); LCSR (); pball05 (); pballplb (); G.Duplancic08 (); A.Khodjamirian11 (); Huang1 (); Huang2 (): whereas the former are available for the high , LCSR calculation is applicable for the low and intermediate . Utilizing the LCSR predictions for , one has launched a painstaking investigation into pball05 (); pballplb (); G.Duplancic08 (); A.Khodjamirian11 (), with a consistent result with those using LQCD. The same approach has also been taken to understand decays in P.Ball06 (); A.Khodjamirian09 (), the resulting sum rules A.Khodjamirian09 () being employed to extract and .

The uncertainties in the light meson distribution amplitudes (DAs) involved in the sum rules, however, would have different degrees of impacts on the results. To gain enlightenment on how to further improve accuracy of the LCSR calculations, it is essential to look into the role played by each of the higher twist DAs. A systematic numerical analysis shows that whereas the twist-4 effects account for only a few percent of the total sum rule results, the chirally enhanced twist-3 contributions are numerically large enough to be comparable with the twist-2 ones in the meson cases, and even about twice as large as the latter for decays. As a result, there are a few problems left unsolved. To start with, one might doubt whether the potential twist-5 effects are negligible in particular while assessing decays. Secondly, the sum rule pollution by twist-3 would be serious on account of the combined uncertainties of the DAs and chiral enhancement factor. Finally, since there is an extremely different sensitivity to twist-2 between the sum rules for and , a successful LCSR application to the latter does not necessary assure, with the same inputs, a reliable LCSR prediction for the former. For the moment, these issues are difficult to essentially settle within the LCSR framework. The trick suggested in LCSR2 (); Huang1 () is available as a temporary scenario to approach them.

Focusing on and , in this work we intend to reconsider heavy to light transitions in the revised LCSR version so as to provide a calculation independent of the traditional LCSR ones and further a determination of the associated CKM parameters. We will expound that this approach does not involve the twist-3 and-5 DAs, which enables us to get an understanding of the form factors to twist-5 precision only resorting to the known twist-2 and -4 DAs and to perform a cross-check between the resulting LCSRs for and . This paper is organized as follows. In the following Section we put forward our derivation of the sum rules in question, including a detailed next-to-leading order (NLO) QCD calculation in twist-2 approximation, and elaborate on the key technical points. The modifications and improvements made are also addressed in comparison with the previous calculations Huang1 (); Huang2 (). In Section 3, after discussing assignment of the parameters for which updated and consistent findings are selected as inputs, we shift into numerical computation with a systematic error discussion included, by means of up-to-date experimental data, and present our LCSR results for and and the determination of and . Too we report on an estimate of the decay constant , as a by-product. The final Section is devoted to a summary.

## Ii QCD calculation of fB→π+(0) and fD→π+(0)

The starting point of LCSR calculation is to consider a correlation function with product of currents sandwiched between the vacuum and a light meson state . In the coordinate space and for large and negative virtuality of the current operators, the correlation function can be in form expanded, in the small light cone distance , as,

 correlation function∼∑mCm(x)⟨L(p)|Om(x,0)|0⟩, (2)

where are the Wilson coefficients, the nonlocal operators built out of quark and/or gluon fields, and the matrix elements have an expansion form in term of the light cone DAs with increasing twist . The power series ( for a large external momentum ) appearing in the expansion process are summed up effectively, which works out some of the problems with the expansion in the small distance . Switching (2) to momentum space, we have

 correlation function∼∑nT(n)H⊗Ψ(n), (3)

a factorized form with the hard kernel being convoluted with . Whereas the process-independent parameterize the long distance effects below a factorization scale , the process-dependent amplitudes describe the hard-scattering dynamics above , which are perturbatively calculable and have the following expansions in :

 T(n)H=T(n)0+αsCF4πT(n)1+⋅⋅⋅. (4)

If calculation is restricted to accuracy, we need just to estimate the leading order (LO) contributions and NLO corrections . Then the remaining procedure is standard.

Now let us take up our LCSR calculations of and . Allowing for the similarity of the two situations, for definiteness we would like to concentrate on the former. Moreover, throughout the paper the chiral limit is taken. We follow LCSR2 (); Huang1 () and adopt the following correlation function:

 Πμ(p,q) = i∫d4xeiqx⟨π(p)|T{JV+Aμ(x),JP+SB(0)}|0⟩ (5) = F((p+q)2)pμ+~F((p+q)2)qμ.

Here we substitute the chiral currents and , respectively, for the operators adopted usually and . The operator replacements do not violate renormalization group invariance of the correlation function, for both and , like the latter two, have an anomalous dimension of zero, and however make the correlation function receive an additional contribution from the set of scalar mesons. In view of that the mass of the lowest scalar meson is slightly below the one of the first excited state of the pseudoscalar mesons, we could safely isolate the pole term of the pseudoscalar ground state from the contributions of higher resonances and continuum states.

For the present purpose, it is sufficient to consider the part proportional to in (5), that is, the invariant function . It has the pole term of interest to us,

 Fpole((p+q)2)=2m2BfB→π+(0)fBm2B−(p+q)2, (6)

where and indicate, respectively, the meson mass and decay constant defined as

 ⟨B|¯biγ5d|0⟩=m2BfBmb. (7)

The spectral function is introduced to reckon in the higher state contributions in a dispersion integral starting with the threshold , which should be assigned near the squared mass of the lowest scalar meson. At this point, what remains to be done is the light cone expansion calculation on , from which the corresponding QCD spectral function is extracted in order to get the sum rule for by matching the Borel improved theoretical and phenomenological forms with the duality assumption .

The light cone expansion of (5) goes effectively in the large space-like momentum region for the channel. At tree-level and to NLO in the light-cone expansion of the quark propagator, it can be illustrated by the two Feynman diagrams as depicted in Fig.1. In comparison with Fig.1(a), which corresponds to the leading term in the quark propagator and illustrates the two-particle contribution, Fig.1(b) portrays the three-particle Fock state effect due to the soft-emission correction to the free quark propagator, which is expressed as

 −igs∫d4k(2π)4e−ikx∫10dv[12⧸k+mb(m2b−k2)2Gμν(vx)σμν+1m2b−k2vxμGμν(vx)γν]. (8)

The contribution of Fig.1(a) to is easy to estimate, using the definition of the pionic two-particle DAs:

 ⟨π(p)|¯uα(x)dβ(0)|0⟩x2→0 = (9) − (γ5)βαμπϕp3π(u)+16(σξηγ5)βαpξxημπϕσ3π(u) + 116(/pγ5)βαx2ϕ4π(u)−i12(/xγ5)βα∫u0ψ4π(v)dv],

where is the fraction of the light cone momentum of the pion carried by the constituent quark. While denotes the twist-2 DA, both and , which are accompanied by the chiral enhancement factor , have twist-3, and the other two functions are both of twist-4. From the following trace form, which emerges obviously as one works in the momentum space,

 Tr{[d(¯up)¯u(up)]wavefunctionγμ(1+γ5)(/q+u/p+mb)(1+γ5)}, (10)

we see readily that the twist-3 components make a vanishing contribution to the light cone expansion, because of the corresponding Dirac wavefunctions. In fact, the same happens to the three-particle situation, as shown from a straightforward computation with (8) and the decomposition:

 ⟨π(p)|¯uα(x)gsGμν(vx)dβ(0)|0⟩x2→0=14∫Dαieip⋅x(α1+α3v)[if3π(σρλγ5)βα +pρ(pμxν−pνxμ)p⋅x(Φ4π(αi)+Ψ4π(αi))}−ifπ2ϵμνδλ(γρ)βα ×{(pλgδρ−pδgλρ)˜Ψ4π(αi)+pρ(pδxλ−pλxδ)p⋅x(˜Φ4π(αi)+˜Ψ4π(αi))}], (11)

where is the gluonic field strength tensor and ; indicates the twist-3 component of the three-particle DAs, and the remaining functions are all of twist-4. In the usual LCSR application to decays, the chirally enhanced twist-3 terms provide a leading contribution, which engenders much negative influence as aforementioned.

At present, the two-particle contribution can be written down in a form that the DAs are convoluted with the corresponding LO hard scattering amplitudes,

 F(2p)0((p+q)2) = −fπ∫10du [T(2)0((p+q)2,u)φπ(u) (12) − T(4)0((p+q)2,u)∫u0ψ4π(v)dv−˜T(4)0((p+q)2,u)ϕ4π(u)],

with

 T(2)0((p+q)2,u)=−2m2bm2b−u(p+q)2, (13) (14) ˜T(4)0((p+q)2,u)=−u22(m2b−u(p+q)2)d2du2. (15)

The three-particle contribution is of the following convolution

 F(3p)0((p+q)2)=−fπ∫10du ¯¯¯¯T(4)0((p+q)2,u)I4π(u), (16)

with

 ¯¯¯¯T(4)0((p+q)2,u)=2u(m2b−u(p+q)2)ddu, (17)

and

 I4π(u) = ∫u0 dα1  ∫1(u−α1)/(1−α1)dvv[2Ψ4π(αi)+2˜Ψ4π(αi) (19) − Φ4π(αi)−˜Φ4π(αi)]∣∣α2=1−α1−α3α3=(u−α1)/υ.

Then we can attain the imaginary part of via estimating the ones of the hard kernels in (13–15) and (17), and further the desired QCD spectral function . The result is as follows:

 ρQCD0(s) = 2fπ∫10duδ(1−usm2b)[φπ(u)+um2b(uddu+1)∫u0ψ4π(v)dv (20) − u24m2bd2du2ϕ4π(u)−um2bdduI4π(u)].

At twist-4 level, we have provided a complete LO light cone QCD representation for . In its present form, the ensuing continuum substraction could be enforced systematically for the twist-4 as well as twist-2 parts, with the known QCD spectral function. This improves explicitly the previous treatment Huang1 (); Huang2 () in which the twist-4 contribution is written down in a form not suitable for continuum substraction.

Our main task is to evaluate the gluon emission effect on the LCSR for at one loop level. It should be sufficient for this purpose to calculate the NLO parts of the leading twist-2 and the chirally enhanced twist-3 contributions. To be specific, we are about to compute the six Feynman diagrams plotted in Fig.2, to the accuracy in question. Fig.2(a) depicts diagrammatically the hard-exchange correction between the outgoing and spectator quarks in the transition. From the nature of the correlation function, we deduce easily that there is no UV divergence in Fig.2(a) or it could not be canceled out. Of the other figures, Figs.2(b, e) and Figs.2(c, f) involve, respectively, the partial one-loop contributions to the vertex and to the one, while Fig.2(d) does the remaining loop contribution to both operators. It is conceivable that each of these five includes both UV and IR divergences, except Fig.2(d) which is merely UV divergent because obviously if any IR divergence arises it can not be reasonably absorbed into a pionic DA.

It is found that the twist-3 components still produce no effect at one-loop level, for the same reason as in the tree-level case. Hence the NLO computation is reduced to a calculation of the correction to the LO twist-2 contribution (for brevity, hereafter we indicate the LO twist-2 contribution by the symbol instead of and the corresponding NLO correction by , up to a prefactor ). We work in the Feynman gauge. In addition, we use the dimensional regularization and scheme to deal with the ultraviolet (UV) and infrared (IR) divergences appearing in the calculation, such that the LO evolution kernel of G.P.Lepage () achieved early in the same prescription is available for a proof of QCD factorization for the resulting twist-2 contribution to as we attempt to segregate the long distance contribution from the perturbative kernel. The calculation is tedious and complicated. Here we present, for the first time, some details of the diagram calculation. We summarize the divergence contribution to from each of the diagrams in Fig.2 as follows,

 Tdiv1(a)(u,r)=41¯¯¯ur2[(1−r)ln(1−r)−(1u−r)ln(1−ur)]ΔIR, (21) Tdiv1(b)(u,r)=411−ur[(r−1¯¯¯urln1−ur1−r+1)ΔIR−2ΔUV], (22) Tdiv1(c)(u,r)=411−ur[(1urln(1−ur)+1)ΔIR−12ΔUV], (23) Tdiv1(d)(u,r)=42+ur(1−ur)2ΔUV, (24) Tdiv1(e+f)(u,r)=−21−urΔIR+21−urΔUV. (25)

Here , and

 ΔIR(ΔUV)=1εIR(1εUV)−γE+ln4π (26)

with the and introduced to regularize the UV and IR divergences, respectively. Obviously, the yielded results are as expected.

Adding all the divergent and finite terms together, we have the NLO correction

 T1(u,r) = 2{11−ur(3−2 ln(1−r)1−r−urur2−2 ln(1−ur1−r)1−r−u¯¯¯ur2u¯¯¯ur2)ΔIR (27) + 6ur(1−ur)2ΔUV+1+ur(1−ur)2(3−3 lnm2bμ2+1ur) + 2[1¯¯¯ur−1r(1−ur)−(1¯¯¯ur2−11−ur)lnm2bμ2]ln(1−r) + 2(11−ur−1¯¯¯ur2)(ln2(1−r)+Li2(r)) + [41−ur+ur+u2r+¯¯¯u¯¯¯u(ur)2−2(21−ur−1−¯¯¯uru¯¯¯ur2)lnm2bμ2]ln(1−ur) − 2(21−ur−1−¯¯¯uru¯¯¯ur2)(ln2(1−ur)+Li2(ur))},

with the dilogarithm .

Keep in mind that up to now the quark mass has been treated as a bare quantity. A mass renormalization must be performed in the scheme, in order to have a UV renormalized hard-scattering amplitude via adding to . It can be done by making the parameter replacement in the related expressions, with the renormalization constant . As a result, the tree level expression (13), to the accuracy required, is modified to the form

 T0(u,r)=2ur−1−αsCF4π12ur(1−ur)2ΔUV, (28)

but the NLO term keeps its form unchanged. Here entering should be understood as the mass. The additional UV divergent contribution in (28), as it should be, precisely cancels out the one of (27). Then a complete UV renormalized result is obtained as

 T(u,r) = T0(u,r)+αsCF4πT1(u,r) (29) = 2{11−ur(3−2 ln(1−r)1−r−urur2−2 ln(1−ur1−r)1−r−u¯¯¯ur2u¯¯¯ur2)ΔIR + 1+ur(1−ur)2(3−3 lnm2bμ2+1ur) + 2[1¯¯¯ur−1r(1−ur)−(1¯¯¯ur2−11−ur)lnm2bμ2]ln(1−r) + 2(11−ur−1¯¯¯ur2)(ln2(1−r)+Li2(r)) + [41−ur+ur+u2r+¯¯¯u¯¯¯u(ur)2−2(21−ur−1−¯¯¯uru¯¯¯ur2)lnm2bμ2]ln(1−ur) − 2(21−ur−1−¯¯¯uru¯¯¯ur2)(ln2(1−ur)+Li2(ur))}.

We need to add that superior to use of the pole mass for the b quark Huang1 (); Huang2 (), employing the mass could render not only the calculation free from some element of uncertainty but the physical meaning more obvious even when the calculation is performed at QCD tree level, as shown in (28).

To proceed, we embark on handling the IR divergence term,

 TIR(u,r)=2ΔIR1−ur[3−2ln(1−r)1−r−urur2−2ln(1−ur1−r)1−r−u¯ur2u¯ur2].

If we try to subtract the divergent part from the UV renormalized hard amplitude to represent the invariant function in the form of QCD factorization, it has to abide by the form

 TIR(u,r)=−ΔIR∫10dv V0(v,u) T0(v,r), (30)

where is the kernel of the evolution equation of the pionic twist-2 DA G.P.Lepage (). As checked readily, this is indeed the case. We can therefore eliminate the divergence by defining a scale dependent DA as

 φπ(u,μ)=φπ(u)−ΔIRαsCF4π∫10dv V0(u,v) φπ(v), (31)

which is convoluted with the perturbative kernel . As a result, the twist-2 contribution to observes, at NLO, the following QCD factorization:

 FQCD((p+q)2)=−fπ∫10du TH(u,r,μ) φπ(u,μ). (32)

Up to higher order corrections in , dependence of compensates that of . It should be understood that in the above operations the factorization and renormalization scales have been set identical for simplicity.

Having in hand the hard kernel available, we can calculate the QCD spectral function to write as a dispersion integral. For , we have

 ρQCD(s)=−1πrfπ∫r0dη ImTH(u,r,μ) φπ(u,μ)|u=η/r, (33)
 12πImTH(u,r,μ)∣∣∣u=η/r = −δ(1−η)+αsCF4π{δ(1−η)[6−3lnm2bμ2 (34) − 73π2−2Li2(1−r)+2ln2(r−1)−2(lnr+1r−1)ln(r−1) − 2(4−3lnm2bμ2)(1+ddη)+2ln(r−1)(1−lnm2bμ2)] + 2θ(η−1)[4 ln(η−1)η−1∣∣∣++1η−1∣∣∣+(lnr(r−1)2+1r −2 +lnm2bμ2)−1−rηr(lnη(η−1)2+1−lnm2bμ2) − + 2θ(1−η)[(lnr(r−1)2+1r−lnm2bμ2)1η−1∣∣ ∣∣+ +

where we take the operation

 F(η)1−η∣∣∣+=F(η)−F(1)1−η, (35)

to avert the redundant divergences possibly occurring as the integral in (33) is performed over the interval .

Using (33) and counting the twist-4 contribution covered in (20), we have the final sum rule for the product

 fBfB→π+(0)e−m2BM2 = −m2bfπ2πm2B∫sB0m2bds e−sM21s∫s/m2b0dη ImT(m2bsη,sm2b,μ) φπ(m2bsη,μ) (36) + fπm2B∫1u0due−m2buM2(−u4d2ϕ4π(u)du2 + uψ4π(u)+∫u0dvψ4π(v)−dduI4π(u)) ≡ K(sB0,M2),

with indicating the Borel parameter with respect to and .

Converting (36) into the corresponding sum rule for transition by a simple replacement of the parameters, we put an end to our derivation of the LCSRs for and , to precision in twist-2 approximation and at tree-level for twist-4 contributions.

We close this Section with a few remarks. Albeit the LCSR calculations are done at twist-4 level, the results remain valid to twist-5 accuracy. The reason is simple. The twist-5 DAs as well as twist-3 ones play no role in the present context due to the Dirac structures of the related nonlocal operators, of which both and , as sandwiched between the vacuum and a pion state, bring about a chirally enhanced twist expansion. Apart from helping reduce sum rule pollution by long-distance parameters, the disappearance of twist-3 and -5 components from the light cone expansions guarantees the resulting LCSRs well convergent. We are going to return to this point in the following Section.

## Iii Choice of the inputs and numerical discussion

Presently, theoretical estimates of and with twist-5 accuracy are obtainable in the sum rules to have been given and the inputs to properly be selected. On the experimental side, from the measured shapes of the form factors for , the CKM matrix element multiplied by is numerically inferred asP.del 052011 ():

 fB→π+(0)|Vub|=(9.4±0.3±0.3)×10−4. (37)

For the semileptonic processes , a similar manipulation D.Besson () gives

 fD→π+(0)|Vcd|=0.150±0.004±0.001. (38)

Then the yielded theoretical predictions could have and extracted from these up-to-date data.

Aimed at determining and , we must do our best to enhance reliability of the LCSR assessments for the form factors in question. So special care should be taken when making our choice of the parameters entering the sum rules. The main sources of uncertainty are, of course, the related DAs, which can merely be understood at a phenomenological level. Based on the conformal symmetry of massless QCD, we can parameterize these DAs by expanding them in terms of matrix elements of conformal operators. The twist-2 DA is of the following expansion in the Gegenbauer polynomials:

 φπ(u)=6u¯u(1+a2(μ)C3/22(u−¯u)+a4(μ)C3/24(u−¯u)+⋅⋅⋅), (39)

with the even moments remaining to be determined. The Gegenbauer polynomials of higher-degree (large n) are rapidly oscillating and so one neglects usually their effects on the numerical integrals included in the sum rules by retaining only the first few terms of the expansion. Some scenarios have been put forward to examine the higher-moment effects. We are willing to mention the prescriptions suggested in BT () and in Huang2 (). In BT () Ball and Talbot (BT) presume that fall off as powers of , , in order to build a DA model. In comparison, authors of Huang2 () consider a modified transverse momentum dependent Brodsky-Huang-Lepage (BHL) wavefunction,

 Ψπ(u,K⊥) = [1+BπC3/22(2u−1)+CπC3/24(2u−1)] (40) × Aπu(1−u)exp[−β2π(K2⊥+m2qu(1−u))],

which is integrated over to give a twist-2 DA. Phenomenological studies with both models are in support of the rationality of using an expansion truncated after . We stick to such disposal. In one-loop approximation taken as default for all the renormalized parameters except QCD coupling, and