SU_f(3)-Symmetry Breaking Effects of the B\to K Transition Form Factor in the QCD Light-Cone Sum Rules
aa r X i v : . [ h e p - ph ] M a r SU f (3) -Symmetry Breaking Effects of the B → K Transition FormFactor in the QCD Light-Cone Sum Rules
Xing-Gang Wu ∗ , Tao Huang † and Zhen-Yun Fang ‡ Department of Physics, Chongqing University, Chongqing 400044, P.R. China Institute of High Energy Physics, Chinese Academy of Sciences,P.O.Box 918(4), Beijing 100049, P.R. China
Abstract
We present an improved calculation of the B → K transition form factor with chiral currentin the QCD light-cone sum rule (LCSR) approach. Under the present approach, the mostuncertain twist-3 contribution is eliminated. And the contributions from the twist-2 and thetwist-4 structures of the kaon wave function are discussed, including the SU f (3)-breaking effects.One-loop radiative corrections to the kaonic twist-2 contribution together with the leading-ordertwist-4 corrections are studied. The SU f (3) breaking effect is obtained, F B → K + (0) F B → π + (0) = 1 . ± . O (1 /m b ) in Ref.[8], we present a consistent analysis of the B → K transitionform factor in the large and intermediate energy regions. PACS numbers: ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] . INTRODUCTION There are several approaches to calculate the B → light meson transition form factors,such as the lattice QCD technique, the QCD light-cone sum rules (LCSRs) and the per-turbative QCD (PQCD) approach. The PQCD calculation is more reliable when the in-volved energy scale is hard, i.e. in the large recoil regions; the lattice QCD results of the B → light meson transition form factors are available only for soft regions; while, the QCDLCSRs can involve both the hard and the soft contributions below m b − m b χ ( χ is a typicalhadronic scale of roughly 500 MeV) and can be extrapolated to higher q regions. Therefore,the results from the PQCD approach, the lattice QCD approach and the QCD LCSRs arecomplementary to each other, and by combining the results from these three methods, onemay obtain a full understanding of the B → light meson transition form factors in its wholephysical region. In Refs.[1, 2], we have done a consistent analysis of the B → π transitionform factor in the whole physical region. Similarly, one can obtain a deep understandingof the B → K transition form factor in the physical energy regions by combining the QCDLCSR results with the PQCD results and by properly taking the SU f (3) breaking effectsinto account.The B → K transition form factors are defined as follows: h K ( p ) | ¯ qγ µ b | ¯ B ( p B ) i = F B → K + ( q ) ( p + p B ) µ − M B − M K q q µ ! + F B → K ( q ) M B − M K q q µ = 2 F B → K + ( q ) p µ + F B → K − ( q ) q µ , (1)where the momentum transfer q = p B − p . If we confine ourselves to discuss the semi-leptonicdecays B → Klν l , it is found that the form factors F B → K − ( q ) is irrelevant for light leptons( l = e, µ ) and only F B → K + ( q ) matters, i.e. d Γ dq ( B → Klν l ) = G F | V tb V ∗ ts | π M B λ / ( q ) | F B → K + ( q ) | , (2)where λ ( q ) = ( M B + M K − q ) − M B M K is the usual phase-space factor. So in thefollowing, we shall concentrate our attention on F B → K + ( q ).The B → K transition form factor has been analyzed by several groups under the QCDLCSR approach [3, 4, 5], where some extra treatments to the correlation function either fromthe B-meson side or from the kaonic side are adopted to improve their LCSR estimations. Itis found that the main uncertainties in estimation of the B → K transition form factor come2rom the different twist structures of the kaon wave functions. It has been found that bychoosing proper chiral currents in the LCSR approach, the contributions from the pseudo-scalars’ twist-3 structures to the form factor can be eliminated [6, 7]. In the present paper,we calculate the B → K form factor with chiral current in the LCSR approach to eliminatethe most uncertain twist-3 light-cone functions’ contributions. And more accurately, wecalculate the O ( α s ) corrections to the kaonic twist-2 terms. The SU f (3)-breaking effectsfrom the twist-2 and twist-4 kaon wave functions shall also be discussed.In Ref[8], we have calculated the B → K transition form factor up to O (1 /m b ) in thelarge recoil region within the PQCD approach [8], where the B-meson wave functions Ψ B and ¯Ψ B that include the three-Fock states’ contributions are adopted and the transversemomentum dependence for both the hard scattering part and the non-perturbative wavefunction, the Sudakov effects and the threshold effects are included to regulate the endpointsingularity and to derive a more reliable PQCD result. Further more, the contributions fromdifferent twist structures of the kaon wave function, including its SU f (3)-breaking effects,are discussed. So we shall adopt the PQCD results of Ref.[8] to do our discussion, i.e. togive a consistent analysis of the B → K transition form factor in the large and intermediateenergy regions with the help of the LCSR and the PQCD results.The paper is organized as follows. In Sec.II, we present the results for the B → K transition form factor within the QCD LCSR approach. In Sec.III, we discuss the kaonicDAs with SU f (3) breaking effect being considered. Especially, we construct a model for thekaonic twist-2 wave function based on the two Gegenbauer moments a K and a K . Numericalresults is given in Sec.IV, where the uncertainties of the LCSR results and a consistentanalysis of the B → K transition form factor in the large and intermediate energy regionsby combining the QCD LCSR result with the PQCD result is presented. The final sectionis reserved for a summary. II. F B → K + ( q ) IN THE QCD LIGHT-CONE SUM RULE
The sum rule for F B → K + ( q ) by including the perturbative O ( α s ) corrections to the kaonictwist-2 terms can be schematically written as [3, 7, 9] f B F B → K + ( q ) = 1 M B Z s m b e ( M B − s ) /M h ρ LCT ( s, q ) + ρ LCT ( q ) i ds , (3)3here ρ LCT ( s, q ) is the contribution from the twist-2 DA and ρ LCT ( q ) is for twist-4 DA, f B is the B-meson decay constant. The Borel parameter M and the continuum threshold s are determined such that the resulting form factor does not depend too much on the precisevalues of these parameters; in addition the continuum contribution, that is the part of thedispersive integral from s to ∞ that has been subtracted from both sides of the equation,should not be too large, e.g. less than 30% of the total dispersive integral. The functions ρ LCT ( s, q ) and ρ LCT ( q ) can be obtained by calculating the following correlation function withchiral currentΠ µ ( p, q ) = i Z d xe iq · x h K ( p ) | T { ¯ s ( x ) γ µ (1 + γ ) b ( x ) , ¯ b (0) i (1 + γ ) d (0) }| > = Π + [ q , ( p + q ) ] p µ + Π − [ q , ( p + q ) ] q µ . (4)The calculated procedure is the same as that of B → π form factor that has been done inRefs.[6, 7, 9, 10]. So for simplicity, we only list the main results for B → K and highlightthe parts that are different from the case of B → π , and the interesting reader may turn toRefs.[7, 9] for more detailed calculation technology.As for ρ LCT ( s, q ), it can be further written as ρ LCT ( s, q ) = − f K π Z duφ K ( u, µ )Im T T q m ∗ b , sm ∗ b , u, µ ! , (5)where T T (cid:18) q m ∗ b , sm ∗ b , u, µ (cid:19) is the renormalized hard scattering amplitude, m ∗ b stands for theb-quark pole mass [9]. Defining the dimensionless variables r = q /m ∗ b , r = ( p + q ) /m ∗ b and ρ = [ r + u ( r − r ) − u (1 − u ) M K /m ∗ b ], up to order α s , we have − Im T T ( r , r , u, µ ) π = δ (1 − ρ ) + α s ( µ ) C F π ( δ (1 − ρ ) " π − m ∗ b µ − ( r )+2Li (1 − r ) − (cid:18) ln r − − r (cid:19) + 2 (cid:18) ln r + 1 − r r (cid:19) ln ( r − − r ! + θ ( ρ − " ρ − ρ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + + 2 1 r − ρ ρ − r ! +2 ln r + 1 r − − r −
1) + ln m ∗ b µ ! ρ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + + 1 − ρρ +2(1 − r )( r − r )( r − ρ ) (cid:18) ln ρr − ρ − r − (cid:19) − ρρ − − r − r − r )( ρ − r ) ln ρ − ρ −
1) + 1 − ln m ∗ b µ ! + θ (1 − ρ ) " ln r + 1 r − r − − ln m ∗ b µ ! ρ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + − − r ) r ( r − ρ ) − − r )( r − r )( r − ρ ) r ( r − − ln m ∗ b µ ! , (6)for the case of r < r >
1. As for the coefficients of δ (1 − ρ ), the higher powersuppressed terms of order O (( M K /m ∗ b ) ) have been neglected due to its smallness. Thedilogarithm function Li ( x ) = − R x dtt ln(1 − t ) and the operation “ + ” is defined by Z dρf ( ρ ) 11 − ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + = Z dρ [ f ( ρ ) − f (1)] 11 − ρ . (7)In the calculation, both the ultraviolet and the collinear divergences are regularized bydimensional regularization and are renormalized in the M S scheme with the totally anti-commuting γ . And similar to Ref.[3], to calculate the renormalized hard scattering ampli-tude T T (cid:18) q m ∗ b , sm ∗ b , u, µ (cid:19) , the current mass effects of s -quark are not considered due to theirsmallness. By setting M K →
0, it returns to the case of B → π and it can be found thatthe coefficients of θ ( ρ −
1) and θ (1 − ρ ) agree with those of Refs.[7, 9], while the coefficientsof δ (1 − ρ ) confirm that of Ref.[9] and differ from that of Ref.[7]. The present results canbe checked with the help of the kernel of the Brodsky-Lepage evolution equation [11], sincethe µ -dependences of the hard scattering amplitude and of the wave function should becompensate to each other.As for the sub-leading twist-4 contribution ρ LCT ( q ), we calculate it only in the zerothorder in α s , i.e. R s m b e M B − sM ρ LCT ( q ) dsM B = m ∗ b f K e M BM M B (Z △ due − m ∗ b − (1 − u )( q − uM K ) uM g ( u ) uM − m b [ g ( u ) + G ( u )] u M ! + Z dv Z Dα i θ ( α + vα − ∆)( α + vα ) M e − m b − (1 − α − vα q − ( α vα M K ) M α vα × (cid:16) ϕ ⊥ ( α i ) + 2 e ϕ ⊥ ( α i ) − ϕ k ( α i ) − e ϕ k ( α i ) (cid:17)o , (8)where ϕ ⊥ ( α i ), e ϕ ⊥ ( α i ), ϕ k ( α i ) and e ϕ k ( α i ) are three-particle twist-4 DAs respectively, and g ( u ) and g ( u ) are two-particle twist-4 wave functions. Here, G ( u ) = R u g ( v ) dv , △ = √ ( s − q − M K ) +4 M K ( m b − q ) − ( s − q − M K )2 M K and s denotes the subtraction of the continuum fromthe spectral integral. By setting M K → u should be changedto be △ = m ∗ b − q s − q for the case), we return to the results of B → π [7].5 II. THE DISTRIBUTION AMPLITUDES OF KAONA. twist-2 DA moments
Generally, the leading twist-2 DA φ K can be expanded as Gegenbauer polynomials: φ K ( u, µ ) = 6 u (1 − u ) " ∞ X n =1 a Kn ( µ ) C / n (2 u − . (9)In the literature, only a K ( µ ) is determined with more confidence level and the higherGegenbauer moments are still with large uncertainty and are determined with large errors.Alterative determinations of Gegenbauer moments rely on the analysis of experimental data.The first Gegenbauer moment a K has been studied by the light-front quark model [12],the LCSR approach [13, 14, 15, 16, 17] and the lattice calculation [18, 19] and etc. InRef.[14], the QCD sum rule for the diagonal correlation function of local and nonlocalaxial-vector currents is used, in which the contributions of condensates up to dimensionsix and the O ( α s )-corrections to the quark-condensate term are taken into account. Themoments derived there are close to that of the lattice calculation [18, 19], so we shall take a K (1GeV) = 0 . ± .
02 to do our discussion. At the scale µ b = q M B − m ∗ b ≃ . a K ( µ b ) = 0 . a K (1GeV) with the help of the QCD evolution.The higher Gegenbauer moments, such as a K , are still determined with large uncertaintyand are determined with large errors [3, 13, 14, 15, 20, 21]. For example, Ref.[21] shows thatthe value of a K is very close to the asymptotic distribution amplitude, i.e. | a K (1 GeV ) | ≤ .
04; while Refs.[14, 15, 20] gives larger values for a K , i.e. a K (1 GeV ) = 0 . ± .
10 [15], a K (1 GeV ) = 0 . +0 . − . [14] and a K (2 GeV ) = 0 . ± .
065 [20]. It should be noted thatthe value of a K affects not only the twist-2 structure’s contribution but also the twist-4structures’ contributions, since the SU f (3)-breaking twist-4 DAs also depend on a K due tothe correlations among the twist-2 and twist-4 DAs as will be shown in the next subsection.Since the value of a K can not be definitely known, we take its center value to be a smallerone, i.e. a K (1 GeV ) = 0 . a K , we shall vary a K within a broader region, e.g. a K (1 GeV ) ∈ [0 . , . a K by comparing with the PQCD results.6 . Models for the twist-2 and twist-4 DAs Before doing the numerical calculation, we need to know the detail forms for the kaontwist-2 DA and the twist-4 DAs.As for the twist-2 DA, we do not adopt the Gegenbauer expansion (9), since its higherGegenbauer moments are still determined with large errors whose contributions may not betoo small, i.e. their contributions are comparable to that of the higher twist structures. Forexample, by taking a typical value a K (1 GeV ) = − .
015 [3], our numerical calculation showsthat its absolute contributions to the form factor is around 1% in the whole allowable energyregion, which is comparable to the twist-4 structures’ contributions. Recently, a reasonablephenomenological model for the kaon wave function has been suggested in Ref.[8], which isdetermined by its first Gegenbauer moment a K , by the constraint over the average value ofthe transverse momentum square, h k ⊥ i / K ≈ . B → K transition form factors up to O (1 /m b ) have been finished.In the following, we construct a kaon twist-2 wave function following the same argumentsas that of Ref.[8] but with slight change to include the second Gegenbauer moment a K ’seffect, i.e.Ψ K ( x, k ⊥ ) = [1+ B K C / (2 x − C K C / (2 x − A K x (1 − x ) exp " − β K k ⊥ + m q x + k ⊥ + m s − x ! , (10)where q = u, d , C / , (1 − x ) are the Gegenbauer polynomial. The constitute quark massesare set to be: m q = 0 . m s = 0 . A K , B K , C K and β K can be determined by the first two Gegenbauer moments a K and a K , the constraint h k ⊥ i / K ≈ . R dx R k ⊥ <µ d k ⊥ π Ψ K ( x, k ⊥ ) = 1.For example, we have A K ( µ b ) = 252 . GeV − , B K ( µ b ) = 0 . C K ( µ b ) = 0 . β K = 0 . GeV − for the case of a K (1 GeV ) = 0 .
05 and a K (1 GeV ) = 0 . B K , C K and β K decreases with the increment of a K ; β K decreaseswith the increment of a K , while B K and C K increase with the increment of a K . Under suchmodel, the uncertainty of the twist-2 DA mainly comes from a K and a K . It can be found thatthe SU f (3) symmetry is broken by a non-zero B K and by the mass difference between the s quark and u (or d ) quark in the exponential factor. The SU f (3) symmetry breaking effectof the leading twist kaon distribution amplitude has been studied in Refs.[14, 23] and refer-7nces therein. The SU f (3) symmetry breaking in the lepton decays of heavy pseudoscalarmesons and in the semileptonic decays of mesons have been studied in Ref.[24]. After doingthe integration over the transverse momentum dependence, we obtain the twist-2 kaon DA, φ K ( x, µ ) = Z k ⊥ <µ d k ⊥ π Ψ K ( x, k ⊥ )= A K π β h B K C / (2 x −
1) + C K C / (2 x − i × exp " − β K m q x + m s − x ! − exp − β K µ x (1 − x ) ! , (11)where µ = µ b for the present case. Then, the Gegenbauer moments a Kn ( µ ) can be definedas a Kn ( µ ) = R dxφ K (1 − x, µ ) C / n (2 x − R dx x (1 − x )[ C / n (2 x − , (12)where φ K (1 − x, µ ) other than φ K ( x, µ ) is adopted to compare the moments with thosedefined in the literature, e.g. [13, 14, 15], since in these references x stands for the momentumfraction of s -quark in the kaon ( ¯ K ), while in the present paper we take x as the momentumfraction of the light q -(anti)quark in the kaon ( K ).The twist-3 contribution is eliminated by taking proper chiral currents under the LCSRapproach, so we only need to calculate the subleading twist-4 contributions. The neededfour three-particle twist-4 DAs that are defined in Ref.[25] can be expressed as [26] ϕ ⊥ ( α i ) = 30 α ( α − α ) (cid:20) h + h α + 12 h (5 α − (cid:21) , e ϕ ⊥ ( α i ) = − α (cid:20) h (1 − α ) + h h α (1 − α ) − α α i + h h α (1 − α ) −
32 ( α + α ) i(cid:21) ,ϕ k ( α i ) = 120 α α α [ a ( α − α )] , ˜ ϕ k ( α i ) = 120 α α α [ v + v (3 α − , (13)where h = v = − M K η = − δ ,a = 21 M K η ω − a K M K = δ ǫ − a K M K ,v = 21 M K η ω = δ ǫ,h = 7 M K η ω − a K M K = 23 δ ǫ − a K M K Similar to Ref.[3], we adopt the results that only include the dominant meson-mass corrections. The lessimportant meson-mass correction terms are not taken into consideration. h = 7 M K η ω + 320 a K M K = 43 δ ǫ + 320 a K M K , with η = δ /M K , ω = 8 ǫ/
21 and δ (1 GeV ) = 0 . GeV and ε (1 GeV ) = 0 .
53 [26]. Withthe help of QCD evolution, we obtain δ ( µ b ) = 0 . GeV and ε ( µ b ) = 0 .
34. It can be foundthat the dominant meson-mass effect are proportional to a K and M K , so if setting M K → a K is quite small, then we return to the results of Ref.[25]. For the remainingtwo-particle twist-4 wave functions, their contributions are quite small in comparison to theleading twist contribution and even to compare with those of the three-particle twist-4 wavefunctions. And by taking the leading meson-mass effect into consideration only, they can berelated to the three-particle twist-4 wave functions through the following way: g ( u ) = Z u dα Z ¯ u dα α [2 ϕ ⊥ ( α i ) − ϕ k ( α i )] (14)and g ( u ) + Z u dvg ( v ) = 12 Z u dα Z ¯ u dα α (¯ uα − uα )[2 ϕ ⊥ ( α i ) − ϕ k ( α i )] , (15)which lead to g ( u ) = ¯ uu − uu (9 h + 3 h − h + 4¯ uh u + 10¯ uh u ) + a (6 + ¯ uu (9 + 80¯ uu ))] + a ¯ u (10 − u + 6¯ u ) ln ¯ u + a u (10 − u + 6 u ) ln u, (16) g ( u ) = 5¯ uu ( u − ¯ u )2 [4 h + 8 a ¯ uu − h (1 + 5¯ uu ) + 2 h (1 − ¯ uu )] . (17)Similarly, it can be found that when setting a K →
0, the above expressions of g ( u )and g ( u ) return to those of Ref.[25]. Here by adopting the relations ddu g ( u ) = − lim M K → M K [ g K ( u ) − φ K ( u )] and g ( u ) − R u dvg ( v ) = lim M K → M K A ( u ), one canconveniently obtain the higher mass-correction terms for g ( u ) and g ( u ) on the basis of g K ( u ) and A ( u ) derived in Refs.[13, 26], and numerically, it can be found that these terms’contributions are indeed small. IV. NUMERICAL RESULTSA. basic input
In the numerical calculations, we use M B = 5 . GeV, M K = 494 M eV, f K = 160 M eV, f π = 131 M eV. (18)9
Total Twist-2 Twist-4 F + B - > K ( q ) M (GeV ) FIG. 1: F B → K + ( q ) as a function of Borel parameter M at q = 6 GeV , where s = 33 . GeV , a K (1 GeV ) = 0 . a K (1 GeV ) = 0 . m ∗ b = 4 . GeV . The solid line stands for the total contribu-tions, the dashed line is for NLO result of the twist-2 kaonic wave function and the dotted line isfor the LO result of twist-4 kaonic wave functions.
Next, let us choose the input parameters entering into the QCD sum rule. In general,the value of the continuum threshold s might be different from the phenomenological valueof the first radial excitation mass. Here we set the threshold value of s to be smallerthan s max ≃ GeV , whose root is slightly bigger than the mass of the B-meson firstradial excitation predicted by the potential model [27]. The pole quark mass m ∗ b is takenas 4 . − . GeV . Another important input is the decay constant of B meson f B . To keepconsistently with the next-to-leading order calculation of twist-2 contribution, we need tocalculate the two-point sum rule for f B up to the corrections of order α s . And in doing thenumerical calculation, we shall adopt the NLO f B to calculate the NLO twist-2 contributionand LO f B for the LO twist-4 contributions for consistence.The reasonable range for the Borel parameter M is determined by the requirement thatthe contributions of twist-4 wave functions do not exceed 10% and those of the continuumstates are not too large, i.e. less than 30% of the total dispersive integration. At a typical q = 6 GeV , we draw F B → K + ( q ) versus M in Fig.(1). It can be found that the contributionfrom the kaonic twist-2 wave function slightly increases with the increment of M while thecontributions from the kaonic twist-4 wave functions decreases with the increment of M ,as a result, there is a platform for F B → K + ( q ) as a function of the Borel parameter M for10he range of 8 GeV < M < GeV . For convenience, we shall always take M = 12 GeV to do our following discussions. B. uncertainties for the LCSR results
In the following we discuss the main uncertainties caused by the present LCSR approachwith the chiral current.The present adopted chiral current approach has a striking advantage that the twist-3light-cone functions which are not known as well as the twist-2 light-cone functions areeliminated, and then it is supposed to provide results with less uncertainties. In fact, ithas been pointed out that the twist-3 contributions can contribute ∼ −
40% to the totalcontribution [28] by using the standard weak current in the correlator, e.g.Π µ ( p, q ) = i Z d xe iq · x h K ( p ) | T { ¯ s ( x ) γ µ b ( x ) , ¯ b (0) iγ d (0) }| > . (19)If the twist-3 wave functions are not known well, then the uncertainties shall be large .So in the literature, two ways are adopted to improve the QCD sum rule estimation onthe twist-3 contribution: one is to calculate the above correlator by including one-loopradiative corrections to the twist-3 contribution together with the updated twist-3 wavefunctions [3]; the other is to introduce proper chiral current into the correlator, cf. Eq.(4),so as to eliminate the twist-3 contribution exactly, which is what we have adopted. Weshall make a comparison of these two approaches in the following. For such purpose, weadopt the following form for the QCD sum rule of Ref.[3], which splits the form factor intocontributions from different Gegenbauer moments: F B → K + ( q ) = f as ( q ) + a K ( µ ) f a K ( q ) + a K ( µ ) f a K ( q ) + a K ( µ ) f a K ( q ) , (20)where f as contains the contributions to the form factor from the asymptotic DA and allhigher-twist effects from three-particle quark-quark-gluon matrix elements, f a K ,a K ,a K con-tains the contribution from the higher Gegenbauer term of DA that is proportional to a K , a K and a K respectively. The explicit expressions of f as,a K ,a K ,a K can be found in Table V A better behaved twist-3 wave function is helpful to improve the estimations, e.g. Ref.[29] provides suchan example for the pionic case. F B - > K + ( q ) q (GeV ) FIG. 2: F B → K + ( q ) for a K (1 GeV ) ∈ [0 . , . a K (1 GeV ) ∈ [0 . , .
15] and m ∗ b ∈ [4 . , . GeV .The solid line is obtained with a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
15 and m ∗ b = 4 . GeV ; the dashedline is obtained with a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
05 and m ∗ b = 4 . GeV , which set the upperand the lower ranges of F B → K + ( q ) respectively. As a comparison, the shaded band shows the resultof Ref.[3] together with its 12% theoretical uncertainty. and Table IX of Ref.[3]. And in doing the comparison, we shall take the same DA momentsfor both methods, especially the value of a K ( µ ) is determined from Eq.(12).We show a comparison of our result of F B → K + ( q ) with that of Eq.(20) in Fig.(2) by varying a K (1 GeV ) ∈ [0 . , . a K (1 GeV ) ∈ [0 . , .
15] and m ∗ b ∈ [4 . , . GeV . In Fig.(2) thesolid line is obtained with a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
15 and m ∗ b = 4 . GeV ; thedashed line is obtained with a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
05 and m ∗ b = 4 . GeV , whichset the upper and the lower ranges of F B → K + ( q ) respectively. The shaded band in the figureshows the result of Eq.(20) within the same a K and a K region and with its 12% theoreticaluncertainty [3]. It can be found that our present LCSR results are consistent with those ofRef.[3] within large energy region q ∈ [0 , GeV ]. In another words these two treatmentson the most uncertain twist-3 contributions are equivalent to each other, while the chiralcurrent approach is simpler due to the elimination of the twist-3 contributions. One mayalso observe that in the lower q region, different from Ref.[3] where F B → K + ( q ) increases withthe increment of both a K and a K , the predicted F B → K + ( q ) will increase with the incrementof a K but with the decrement of a K . This difference is caused by the fact that we adoptthe model wave function (10) to do our discussion, whose parameters are determined by the12 O result NLO result- s M f B s M f B m b = 4 . m b = 4 . m b = 4 . f B , where m b and f B are given in GeV , s and M in GeV . combined effects of a K and a K ; while in Ref.[3], a K and a K are varied independently andthen their contributions are changed separately.Next we discuss the main uncertainties caused by the present LCSR approach with thechiral current. Firstly, we discuss the uncertainties of F B → K + ( q ) caused by the effectivequark mass m ∗ b by fixing a K (1 GeV ) = 0 . GeV and a K (1 GeV ) = 0 . GeV . Under suchcase, the value of s , the LO and NLO vales of f B should be varied accordingly and bedetermined by using the two-point sum rule with the chiral currents, e.g. to calculate thefollowing two-point correlator:Π( q ) = i Z d xe iqx h | q ( x )(1 + γ ) b ( x ) , b (0)(1 − γ ) q (0) | i . (21)The sum rule for f B up to NLO can be obtained from Ref.[30] through a proper combinationof the scalar and pseudo-scalar results shown there , which can be schematically written as f B M B e − M B /M = Z s m b ρ tot ( s ) e − s/M ds, (22)where the spectral density ρ tot ( s ) can be read from Ref.[30]. The Borel parameter M andthe continuum threshold s are determined such that the resulting form factor does notdepend too much on the precise values of these parameters; in addition, 1) the continuumcontribution, that is the part of the dispersive integral from s to ∞ , should not be too large,e.g. less than 30% of the total dispersive integral; 2) the contributions from the dimension-six condensate terms shall not exceed 15% for f B . Further more, we adopt an extra criteriaas suggested in Ref.[3] to derive f B : i.e. the derivative of the logarithm of Eq.(22) with One needs to change the c -quark mass to the present case of b -quark mass and we take h α s π G aµν G aµν i =2 × (0 . GeV ) [31] and α s h q ¯ q i = 0 . × − GeV [30] to do the numerical calculation. F B - > K ( q ) q (GeV ) FIG. 3: F B → K + ( q ) as a function of q with varying m ∗ b . The solid, dashed line and the dash-dotline are for m ∗ b = 4 . GeV , 4 . GeV and 4 . GeV respectively, where a K (1 GeV ) = 0 . GeV and a K (1 GeV ) = 0 . GeV . respect to 1 /M gives the B-meson mass M B , M B = Z s m b ρ tot ( s ) e − s/M sds , Z s m b ρ tot ( s ) e − s/M ds, and we require its value to be full-filled with high accuracy ∼ . m ∗ b . Some typical values of f B are shown in TAB.I,where f B is taken as the extremum within the reasonable region of ( M , s ) and the valueof m ∗ b is taken as [32]: m ∗ b ≃ . ± . GeV . f B decreases with the increment of m ∗ b . TheNLO result agrees with the first direct measurement of this quantity by Belle experiment f B = 229 +36 − (stat) +34 − (syst) MeV from the measurement of the decay B − → τ ¯ ν τ [33].The value of F B → K + ( q ) for three typical values of m ∗ b , i.e. m ∗ b = 4 . GeV , 4 . GeV and4 . GeV respectively, are shown in Fig.(3). F B → K + ( q ) increases with the increment of m ∗ b . Itcan be found that the uncertainty of the form factor caused by m ∗ b ∈ [0 . GeV, . GeV ] is ∼
5% at q = 0 and increases to ∼
9% at q = 14 GeV . By taking a more accurate m ∗ b , e.g. m ∗ b = (4 . ± . GeV as suggested by Ref.[3], the uncertainties can be reduced to ∼ q = 0 and ∼
5% at q = 14 GeV .Secondly, we discuss the uncertainties of F B → K + ( q ) caused by the twist-2 wave functionΨ K , i.e. the two Gegenbauer moments a K (1 GeV ) and a K (1 GeV ). For such purpose, wefix s = 33 . GeV and m ∗ b = 4 . GeV . To discuss the uncertainties caused by a K (1 GeV ),we take a K (1 GeV ) = 0 . F B → K + ( q ) for three typical a K (1 GeV ), i.e. a K (1 GeV ) = 0 . F B - > K + ( q ) q (GeV ) FIG. 4: F B → K + ( q ) as a function of q with varying a K (1 GeV ), where a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
03, 0 .
05 and 0 .
07 respectively. F B - > K + ( q ) q (GeV ) FIG. 5: F B → K + ( q ) as a function of q with varying a K (1 GeV ), where a K (1 GeV ) = 0 .
05. The solidline, the dashed line and the dash-dot line are for a K (1 GeV ) = 0 .
05, 0 .
10 and 0 .
15 respectively. .
05 and 0 .
07 respectively, are shown in Fig.(4). F B → K + ( q ) decreases with the increment of a K . It can be found that the uncertainty of form factor caused by a K (1GeV) ∈ [0 . , . q = 0 and becomes even smaller for larger q . Similarly,to discuss the uncertainties caused by a K (1 GeV ), we fix a K (1GeV) = 0 .
05. Since thevalue of a K is less certain than a K , so we take three typical values of a K (1 GeV ) withbroader separation to calculate F B → K + ( q ), i.e. a K (1 GeV ) = 0 .
05, 0 .
10 and 0 .
15 respectively.The results are shown in Fig.(5). It can be found that the uncertainty of the form factor15aused by a K (1GeV) ∈ [0 . , .
15] is also small, i.e. it is about 5% at q = 0 and becomessmaller for larger q . F B → K + ( q ) increases with the increment of a K in the lower energyregion q < GeV and decreases with the increment of a K in the higher energy region q > GeV .As a summary, a more accurate values for m ∗ b , a K and a K shall be helpful to derive a moreaccurate result for the form factor. Our results favor a smaller a K to compare with the formfactor in the literature, e.g. a K (1 GeV ) ≤ .
15. And under such region, the uncertaintiesfrom a K is small, i.e. its uncertainty is less than 5% for a K (1 GeV ) ∈ [0 . , . a K (1 GeV ) ∈ [0 . , .
07] and a K (1 GeV ) ∈ [0 . , . q = 0. Theuncertainties of a K shows that the SU f (3)-breaking effect is small but it is comparableto that of the higher twist structures’ contribution. So the SU f (3) breaking effect andthe higher twist’s contributions should be treated on the equal footing. Using the chiralcurrent in the correlator, as shown in Eq.(4), the theoretical uncertainty can be remarkablyreduced. And our present LCSR results are consistent with those of Ref.[3] within largeenergy region q ∈ [0 , GeV ], which is calculated with the correlator (19) and includesone-loop radiative corrections to twist-2 and twist-3 contributions together with the updatedtwist-3 wave functions. In another words these two approaches are equivalent to each otherin some sense, while the chiral current approach is simpler due to the elimination of themore or less uncertain twist-3 contributions. For higher energy region q > GeV , theLCSR approach is no longer reliable. Therefore the lattice calculations, would be extremelyuseful to derive a more reliable estimation on the high energy behaviors of the form factors. C. SU f (3) breaking effect of the form factor within the LCSR To have an overall estimation of the SU f (3) breaking effect, we make a comparisonof the B → π and B → K form factors: F B → π + ( q ) and F B → K + ( q ). The formulaefor F B → π + ( q ) can be conveniently obtained from that of F B → K + ( q ) by taking the limit M K →
0. In doing the calculation for F B → π + ( q ), we directly use the Gegenbauer ex-pansion for pion twist-2 DA, because different to the kaonic case, now the higher Gegen-bauer terms’ contributions are quite small even in comparison to the twist-4 contributions,e.g. by taking a π (1 GeV ) = − .
015 [3], our numerical calculation shows that its abso-16 F B - > K + ( q ) q (GeV ) FIG. 6: Comparison of F B → K + ( q ) and F B → π + ( q ), where m ∗ b = 4 . GeV , s = 33 . GeV , f LOB =0 . GeV , f NLOB = 0 . GeV , a K (1 GeV ) = 0 .
05 and a π/K (1 GeV ) = 0 . F B → K + ( q ) and F B → π + ( q ) respectively. lute contributions to the form factor is less than 0 .
5% in the whole allowable energy re-gion. We show a comparison of F B → K + ( q ) and F B → π + ( q ) in Fig.(6) with the parame-ters taken to be m ∗ b = 4 . GeV , s = 33 . GeV , f LOB = 0 . GeV , f NLOB = 0 . GeV , a K (1 GeV ) = 0 .
05 and a π/K (1 GeV ) = 0 . m ∗ b ∈ [4 . , . GeV , a K (1 GeV ) ∈ [0 . , .
07] and a π/K (1 GeV ) ∈ [0 . , . F B → π + (0) ∈ [0 . , . F B → K + (0) ∈ [0 . , . F B → K + (0) F B → π + (0) = 1 . ± .
03, which favors a small SU f (3) breaking effect and is consistent with the PQCD estimation 1 . ± .
02 [8], theQCD sum rule estimations, e.g. [ F B → K + (0) /F B → π + (0)] ≈ .
16 [3] , 1 . +0 . − . [34] and 1 . +0 . − . [14] respectively, and a recently relativistic treatment that is based on the study of theDyson-Schwinger equations in QCD, i.e. [ F B → K + (0) /F B → π + (0)] = 1 .
23 [35].
D. consistent analysis of the form factor within the large and the intermediateenergy regions
Recently, Ref[8] gives a calculation of the B → K transition form factor up to O (1 /m b )in the large recoil region within the PQCD approach [8], where the B-meson wave functions To estimate the ratio [ F B → K + (0) /F B → π + (0)] from Ref.[3], we take a K (1 GeV ) = 0 . ± . q (GeV ) F B - > K + ( q ) FIG. 7: LCSR and PQCD results for F B → K + ( q ). The solid line is for our LCSR result, the dash-dotline is for the LCSR result of Ref.[3] with a K (1 GeV ) = 0 .
07 and a K (1 GeV ) = 0 .
05. The shadedband is the PQCD result with ¯Λ ∈ [0 . , .
55] and δ ∈ [0 . , . .
525 and δ = 0 . .
50 and δ = 0 . .
55 and δ = 0 . Ψ B and ¯Ψ B that include the three-Fock states’ contributions are adopted and the transversemomentum dependence for both the hard scattering part and the non-perturbative wavefunction, the Sudakov effects and the threshold effects are included to regulate the endpointsingularity and to derive a more reliable PQCD result. Further more, the uncertainties forthe PQCD calculation of the B → K transition form factor has been carefully studied inRef.[8]. So we shall adopt the PQCD results of Ref.[8] to do our discussion. Only we needto change the twist-2 kaon wave function Ψ K used there to the present one as shown inEq.(10).We show the LCSR results together with the PQCD results in Fig.(7). In drawingthe figure, we take a K (1 GeV ) = 0 . a K (1 GeV ) = 0 .
05 and m ∗ b = 4 . GeV . And theuncertainties of these parameters cause about ∼
10% errors for the LCSR calculation. Whilefor the PQCD results, we should also consider the uncertainties from the B-meson wavefunctions, i.e. the values of the two typical parameters ¯Λ and δ , and we take ¯Λ ∈ [0 . , . δ ∈ [0 . , .
30] [8]. It can be found that the PQCD results can match with the LCSRresults for small q region, e.g. q < GeV . Then by combining the PQCD results withthe LCSR results, we can obtain a consistent analysis of the form factor within the large and18he intermediate energy regions. Inversely, if the PQCD approach must be consistent withthe LCSR approach, then we can obtain some constraints to the undetermined parameterswithin both approaches. For example, according to the QCD LCSR calculation, the formfactor F B → K + ( q ) increases with the increment of b-quark mass, then the value of m b can notbe too large or too small , i.e. if allowing the discrepancy between the LCSR result and thePQCD results to be less than 15%, then m ∗ b should be around the value of 4 . ± . GeV . V. SUMMARY
In the paper, we have calculated the B → K transition form factor by using the chiralcurrent approach under the LCSR framework, where the SU f (3) breaking effects have beenconsidered and the twist-2 contribution is calculated up to next-to-leading order. It is foundthat our present LCSR results are consistent with those of Ref.[3] within large energy region q ∈ [0 , GeV ], which is calculated with the conventional correlator (19) and includes one-loop radiative corrections to twist-2 and twist-3 contributions together with the updatedtwist-3 wave functions. And our present adopted LCSR approach with the chiral current issimpler due to the elimination of the more or less uncertain twist-3 contributions.The uncertainties of the LCSR approach have been discussed, especially we have foundthat the second Gegenbauer moment a K prefers asymptotic-like smaller values. By varyingthe parameters within the reasonable regions: m ∗ b ∈ [4 . , . GeV , a K (1 GeV ) ∈ [0 . , . a π/K (1 GeV ) ∈ [0 . , . F B → π + (0) = 0 . ± .
026 and F B → K + (0) = 0 . ± . F B → K + (0) F B → π + (0) = 1 . ± .
03, which favors a small SU f (3)breaking effect. Also, it has been shown that one can do a consistent analysis of the B → K transition form factor in the large and intermediate energy regions by combining the QCDLCSR result with the PQCD result. The PQCD approach can be applied to calculate the B → K transition form factor in the large recoil regions; while the QCD LCSR can beapplied to intermediate energy regions. Combining the PQCD results with the QCD LCSR,we can give a reasonable explanation for the form factor in the low and intermediate energyregions. Further more, the lattice estimation shall help to understand the form factors’ Another restriction on m b is from the experimental value [33] on f B . q > GeV . So, we suggestsuch a lattice calculation can be helpful. Then by comparing the results of these threeapproaches, the B → K transition form factor can be determined in the whole kinematicregions. Acknowledgements
This work was supported in part by the Natural Science Foundation of China (NSFC)and by the Grant from Chongqing University. This work was also partly supported bythe National Basic Research Programme of China under Grant NO. 2003CB716300. Theauthors would like to thank Z.H. Li, Z.G. Wang and F.Zuo for helpful discussions on thedetermination of f B . [1] T. Huang and X.G. Wu, Phys.Rev. D , 034018(2005).[2] T. Huang, C.F. Qiao and X.G. Wu, Phys.Rev. D , 074004(2006).[3] P. Ball and R. Zwicky, Phys.Rev. D , 014015(2005); hep-ph/0406232.[4] P. Ball, J.High Energy Phys. , 005(1998).[5] A. Khodjamirian, T. Mannel and N. Offen, Phys.Rev. D , 054013(2007).[6] T. Huang, Z.H. Li and X.Y. Wu, Phys.Rev. D , 094001(2001).[7] Z.G. Wang, M.Z. Zhou and T. Huang, Phys.Rev. D , 094006(2003).[8] X.G. Wu, T. Huang and Z.Y. Fang, Eur.Phys.J. C , 561(2007).[9] A. Khodjamirian, R. Ruckl, S. Weinzierl and Oleg I. Yakovlev, Phys.Lett. B , 154(1998).[11] G.P. Lepage and S.J. Brodsky, Phys.Lett. B87 , 359(1979); Phys.Rev. D
22, 2157(1980).[12] C.R. Ji, P.L. Chung and S.R. Cotanch, Phys.Rev. D , 4214(1992); H.M. Choi and C.R. Ji,Phys.Rev. D , 034019(2007).[13] P. Ball, V.M. Braun and A. Lenz, J.High Energy Phys. , 004(2006).[14] A. Khodjamirian, Th. Mannel and M. Melcher, Phys.Rev. D , 094002(2004).[15] P. Ball and M. Boglione, Phys.Rev. D , 094006(2003).[16] V.M. Braun and A. Lenz, Phys.Rev. D , 074020(2004).
17] P. Ball and R. Zwicky, JHEP , 034(2006); V.M. Braun and A. Lenz, Phys.Rev. D ,074020.[18] V.M. Braun et al. , Phys.Rev. D , 074501(2006).[19] P.A. Boyle et al. , Phys.Lett. B , 67(2006); hep-lat/0610025.[20] V.M. Braun, etal. , QCDSF/UKQCD collaboration, hep-lat/0610055; Phys.Rev. D , 076005(2006).[22] X.H. Guo and T. Huang, Phys.Rev. D , 2931(1991).[23] P. Ball and R. Zwicky, Phys.Lett. B , 289(2006).[24] S.S. Gershtein and M.Yu. Khlopov, JETP Lett. , 338 (1976); M.Yu. Khlopov, Yad. Fiz. 18,1134 (1978).[25] V.M. Braun and I.E. Filyanov, Z.Phys. C
48, 239(1990).[26] P. Ball, J.High Energy Phys. , 010(1999).[27] M.Di Pierro and E. Eichten, Phys.Rev. D , 114004(2001).[28] V.M. Belyaev, A. Khodjamirian and R. Ruckl, Z.Phys. C , 349(1993).[29] T. Huang and X.G. Wu, Phys. Rev. D
70, 093013(2004).[30] A. Khodjamirian and R. Ruckl, hep-ph/9801443.[31] S. Narison, “QCD as a Theory of Hadrons, From Partons to Confinement”, Cambridge Uni-versity Press, Cambridge (2004); and references therein.[32] P. Colangelo and A. Khodjamirian, hep-ph/0010175, in
At the Frontier of Particle Physics ,edited by M. Shiftman (World Scientific, Singapore, 2001), Vol.3, p. 1495.[33] K. Ikado et al. , Belle Collaboration, Phys.Rev.Lett. , 251802(2006).[34] A. Khodjamirian, T. Mannel and M. Melcher, Phys.Rev. D , 114007(2003).[35] M.A. Ivanov, J.G. Korner, S.G. Kovalenko and C.D. Roberts, Phys.Rev. D , 034018(2007)., 034018(2007).