Perturbative corrections to B → D form factors in QCD

Journal of High Energy Physics, Jun 2017

We compute perturbative QCD corrections to B → D form factors at leading power in Λ/m b , at large hadronic recoil, from the light-cone sum rules (LCSR) with B-meson distribution amplitudes in HQET. QCD factorization for the vacuum-to-B-meson correlation function with an interpolating current for the D-meson is demonstrated explicitly at one loop with the power counting scheme \( {m}_c\sim \mathcal{O}\left(\sqrt{\Lambda {m}_b}\right) \). The jet functions encoding information of the hard-collinear dynamics in the above-mentioned correlation function are complicated by the appearance of an additional hard-collinear scale m c , compared to the counterparts entering the factorization formula of the vacuum-to-B-meson correction function for the construction of B → π from factors. Inspecting the next-to-leading-logarithmic sum rules for the form factors of B → Dℓν indicates that perturbative corrections to the hard-collinear functions are more profound than that for the hard functions, with the default theory inputs, in the physical kinematic region. We further compute the subleading power correction induced by the three-particle quark-gluon distribution amplitudes of the B-meson at tree level employing the background gluon field approach. The LCSR predictions for the semileptonic B → Dℓν form factors are then extrapolated to the entire kinematic region with the z-series parametrization. Phenomenological implications of our determinations for the form factors f BD +,0 (q 2) are explored by investigating the (differential) branching fractions and the R(D) ratio of B → Dℓν and by determining the CKM matrix element |V cb | from the total decay rate of B → Dμν μ .

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29062.pdf

Perturbative corrections to B → D form factors in QCD

Received: January Perturbative corrections to Yu-Ming Wang 0 1 3 6 7 Yan-Bing Wei 0 1 2 3 5 7 Yue-Long Shen 0 1 3 4 7 Cai-Dian Lu 0 1 2 3 5 7 power in 0 1 3 7 Open Access 0 1 3 7 c The Authors. 0 1 3 7 0 Yuquan Road 19 , 100049 Beijing , P.R. China 1 Boltzmanngasse 5 , 1090 Vienna , Austria 2 School of Physics, University of Chinese Academy of Sciences 3 Weijin Road 94 , 300071 Tianjin , P.R. China 4 College of Information Science and Engineering, Ocean University of China 5 Institute of High Energy Physics , CAS 6 School of Physics, Nankai University 7 Songling Road 238 , Qingdao, 266100 Shandong , P.R. China We compute perturbative QCD corrections to B ! D form factors at leading =mb, at large hadronic recoil, from the light-cone sum rules (LCSR) with Bmeson distribution amplitudes in HQET. QCD factorization for the vacuum-to-B-meson correlation function with an interpolating current for the D-meson is demonstrated explicitly at one loop with the power counting scheme mc encoding information of the hard-collinear dynamics in the above-mentioned correlation function are complicated by the appearance of an additional hard-collinear scale mc, compared to the counterparts entering the factorization formula of the vacuum-to-B-meson the subleading power correction induced by the three-particle quark-gluon distribution amplitudes of the B-meson at tree level employing the background gluon eld approach. The LCSR predictions for the semileptonic B ! D` form factors are then extrapolated to the entire kinematic region with the z-series parametrization. Phenomenological implications of our determinations for the form factors fB+D;0(q2) are explored by investigating the (di erential) branching fractions and the R(D) ratio of B ! D` and by determining the CKM matrix element jVcbj from the total decay rate of B ! D Heavy Quark Physics; Perturbative QCD; Resummation - correction function for the construction of B ! from factors. Inspecting the next-to Contents 1 Introduction The framework 2 3 Two-particle contributions to the LCSR at O( s) Perturbative matching functions at NLO Weak vertex diagram D-meson vertex diagram Wave function renormalization Box diagram The NLL hard and jet functions The NLL LCSR for B ! D form factors Numerical analysis Theory input parameters Predictions for B ! D form factors Phenomenological implications Concluding discussion A Loop integrals B Spectral representations C The coe cient functions of Introduction QCD and heavy-particle e ective eld theories. The long-standing tension between the BR(B ! function performed in [10]. correlation function at leading power in eld appeared in the infunction, following closely [15, 19, 20]. the sum rules for B ! in the background eld approach. power counting mb , which was further updated in [25] recently. Albeit with lish the power counting scheme for the external momenta in =mb with the power The framework (n p; n p) = i b(0)g jB(pB)i ; where pB frame of the B-meson and introduce two light-cone vectors n and n satisfying n v = n p = m2B + m2D O(mB) ; p is the transfer momentum and is a hadronic scale of order the diagram in gure 1 with the light-cone OPE and we obtain ; 2P(n p; n p) = i f~B( ) mB n i f~B( ) mB n !c + i 0 !0 + i 0 pion interpolating current in the mc ! 0 limit. B ! D transition form factors h0jqn= 5 ujD(p)i = i n p fD ; (n p; n p) = i fD mB 2 (m2D=n p meson correlation function (n p; n p) de ned in (2.1) at tree level. h0j (q Ys) (t n) Ysybv (0)jB(v)i if~B( ) mB 1 + v= h2 ~B+(t; ) + ~B+(t; ) n= the soft gauge link is given by Ys(t n) = P Exp i gs dx n As(x n) f~B( ) = n p fB+D(q2) + fB0 D(q2) fB0 D(q2) at leading power in the LCSR for the two B ! D form factors at tree level fB+D; 2P(q2) = d!0 Exp fB+D; 2P(q2) + O( s) ; B(!0) + O( s) ; leading power in the sum rules for B ! 3=2=mb1=2 ; 3=2=mb1=2 : Based upon the canonical behaviour for the B-meson DA level sum rules (2.9), di erent from the scaling law fB+D(q2) O(1) [14] obtained with the B(!0), one can readily deduce counting scheme mc Two-particle contributions to the LCSR at O( s) ; 2P(n p; n p) = i f~B( ) mB !c + i0 Ci;n(n p) Ji;n(n p; !) n + Ci;n(n p) Ji;n(n p; !) n ; k = s manifestly and the convenience. Perturbative matching functions at NLO gure 2 in the following. Weak vertex diagram in gure 2 (a) ; weak= (p k)2 mc2 +i0 (2 )D [(p k +l)2 mc2 +i0][(mbv +l)2 mb2 +i0][l2 +i0] q(k)n= 5(p= k= + mc) k= + =l + mc) (mbv= + =l + mb) b(v) ; (3.2) meson correlation function (n p; n p) de ned in (2.1) at O( s). to the above-mentioned QCD diagrams at leading power in =mb will be cancelled out O(mb) I1 = (2 )D [(p k + l)2 mc2 + i0][(mbv + l)2 mb2 + i0][l2 + i0] the hard contribution from the weak vertex diagram yields where the two hard functions are given by ; weak = Ch(w;neak)(n p) = Ch(w;neak)(n p) = 2 Li2 1 ln r + + 2 ln d(k)n= 5 b(v) + 2 ln + 2 r1 ln 2 n (p+l) n J (w;neak)(n p)= n p (! n p (! 1 + r1 2 ln(1 + r1) ln 2 Li2 1 + r1 ln (1 + r1) + 1 + ln2 n p (! n p (! + 1 + ln2(1 + r1) with r = n p=mb. =mb ; weak = n p n k !c (2 )D [n (p+l) n (p k +l)+l2 mc2 +i0][n l+i0][l2 +i0] with r1 = mc2= [n p n (k p)] and ! n k. It is apparent that our result of J (w;neak) of [15], for constructing the sum rules of B ! form factors in the mc ! 0 (i.e., r1 ! 0) diagram by expanding (3.2) in the soft region ; weak = q(k)n= 5 b(v) ; (2 )D [n (p k + l) !c + i0][v l + i0][l2 + i0] D-meson vertex diagram in gure 2(b) ; D = of the partonic DA of the B-meson at NLO in s, calculable from the Wilson-line Feynman terized by the B-meson DA at leading power in n p (n p (2 )D [(p =ln= 5 (p= =l + mc) k= + mc) mc2 + i0][(l k)2 + i0][l2 + i0] =mb, in [35, 36] yields (1;)D = (1); hc = i f~B( ) mB B(!) J ( D;n)(n p) + n B+(!) J +(D;n)(n p) (3.11) nmcp B+(!) n J +(D;n)(n p) B(!) + n where we have introduced the following jet functions J ( D;n)(n p)= s CF 1 + r2 + r3 1 + r2 2(1 + r2) 1 + r2 1 + r2 + r3 1 + r2 +2 ln (1+r2) + r2 [r2(1+r3) 2] 1 + r3 1 + r2 + r3 1 + r2 + r3 1 + r2 + ln (1 + r2) 2(2 + r3)(1 + r2 + r3) (1 + r2)(1 + r3) (4+r2) ; 1 + r3 r2 ; (3.13) J +(D;n)(n p)= s CF r3(1 + r3) (1 + r2 + r3)2 r2 [2(1 + r3) + r2(2 + r3)] (1 + r3)2 r3(1+r3) 1 + r3 1+r2 +r3 1 + r2 1 + r2 + r3 1 + r2 1 + r2 1 + r3 r2 = r3 = vertex diagram presented in (3.11) are in order. scheme mc mb and n p O( ). This observation can be readily unO(pmb= ) of mc= (1); mc = mc n q(k) n= 5 b(v) 2) (n l)2 (2 )D [(p mc2 + i0][(l k)2 + i0][l2 + i0] energetic charm quark and the light spectator quark yields hD(p)jc + : : : ; (3.18) contribution to B ! D form factors. at tree level, the NLO jet function J +(D;n)(n p) must be infrared nite to validate the diagram as displayed in (38) of [15]. bution to the D-meson vertex diagram, at leading power in =mb, is cancelled out precisely by the corresponding soft subtraction term. Wave function renormalization we obtain ; wfc = { 10 { J (w;nfc)(n p)= J (w;nfc; 1)(n p)= J (w;nfc; 2)(n p)= 1 + r1 2 r1 J (w;nfc; 2)(n p) ; +r12 ln r1(r1 +4) ln ln (1+r1)+1 r1 ; 3 ln (1+r1)+r1 +5 ; (3.22) T (0) = T (0) = 0 ; renormalization of the external quarks Box diagram computed as ; box= igs2CF given by C(b;wnf)(n p) = + 3 ln (2 )D [(pb +l)2 =l)n= 5 (p= mb2 +i0][(p k +l)2 k= + =l + mc) mc2 +i0][(k l)2 +i0][l2 +i0] (p=b + =l + mb) hard-collinear region to the leading power in =mb yields (1;)b;ohxc = 2) n l n= 2 mb n=] 5 b(v) n (p + l) n [n (p + l) n (p k + l) + l2 mc2 + i0][n l n(l k) + l2 + i0][l2 + i0] projector of the B-meson leads to (1;)b;ohxc = J (b;onx) = i f~B( ) mB s CF (1 + r1)(1 + r3) 1 + r1 1 + r1 J +(b;onx) = s CF (1 + r1)(1 + r3) r r12) ln (1 + r1) 1 + r1 1 + r1 as presented in (52) of [15], in the mc ! 0 limit. =mb can be further extracted from (3.26) as follows B(!) J (b;onx)(n p) + B+(!) J +(b;onx)(n p) ; ln (1 + r1) + 1 + 1 ln (1 r4 + r1) n p (! 1 + r1 (1;)b;osx = (2 )D [v l + i0][n (p k + l) !c + i0][(k l)2 + i0][l2 + i0] =l) n= 5 n= which is precisely the same as the soft subtraction term T (0) computed with the of the B-meson in HQET. The NLL hard and jet functions renormalized hard coe cients including the O( s) corrections C+;n = C+;n = 1 ; C ;n = 1 + Ch(w;neak) + = 1 C ;n = Ch(w;neak) = + 5 ln 2 Li2 1 1+r2 +r3 r2r3 ln ; (3.33) r2(1+r3) J+;n = s CF (1+r2 +r3)2 r3(1+r3) mc (1+r2 +r3)2 1+r2 +r3 r2 [2(1+r3)+r2(2+r3)] (1+r3)2 r (1+r2 +r3) (1+r3)2 1+r2 +r3 +r22(r3 +2) ln J ;n = 1 ; n p (! n p) (1+r2 +r3)2 (1+r2)(1+r3) 1+r2 +r3 +2 ln2 (1+r2 +r3) 4 ln (1+r2 +r3) ln (1+r3) = 1+ s CF n p (! n p) + ln2 (1+r3)+ +2 ln (1+r2) ln2 (1+r2)+r2 4 Li2 1+r3 1+r2 +r3 2(1+r2) + r2 (r2 +2(1+r3)) 1 ln (1+r2 +r3) 1+r2 +r3 +2 Li2 (1+r3)2 ln (1+r3)+ 1+r2 +r3 (1+r3)2 1+r3 (1+r2)(r3(1+r2) 2) r2 1+r2 +r3 + 1+r3 ln (1+r2) matching coe cients of the weak current q 5) b from QCD onto SCET, as disjet functions for the massless hard-collinear quark. it is then straightforward to write down ; 2P(n p; n p) = i f~B( ) mB 1 + r2 + r3 + 4 ln n p (! i0 d ln (1 + r2 + r3)2 (1 + r2)(1 + r3) 1 + r2 + r3 evolution equation of the B-meson DA B(!; ) in the absence of the light-quark mass e ect hf~B( ) B(!; )i = (1)(!; !0; ) can be found in [39, 40], we can readily where the renormalization kernel H compute the last term of (3.38) as = i f~B( ) mB i0 d ln Substituting (3.40) into (3.38) immediately leads to { 14 { ; 2P(n p; n p) = O( s2) ; the RG evolution of the B-meson DA as a hard-collinear scale hc 0 being a hadonic scale of O( ), from B(!; ), since we will take the factorization scale mb which is quite close to the hadronic scale 0 static B-meson decay constant f~B( ) dC ;n(n p; ) df~B( ) cusp( s) ln + ( s) C ;n(n p; ) ; = ~( s) f~B( ) ; (1 + r2 + r3)2 (1 + r2)r3 accuracy, the cusp anomalous dimension cusp( s) needs to be expanded at the three-loop computed as torization formula for the correlation function ; 2P(n p; n p) = i hU2(n p; h2; ) f~B( h2)i mB !c +i0 C+;n(n p; ) J+;n(n p; !; ) n +C+;n(n p; ) J+;n(n p; !; ) n taken at a hard-collinear scale O(pmb ). where h1 and h2 are the hard scales of O(mb) and the factorization scale needs to be The NLL LCSR for B ! D form factors collected in appendix B yields spectral representation of the factorization formula for obtained in the above, which ; 2P(n p; n p) = i hU2(n p; h2; ) f~B( h2)i mB C+;n(n p; ) e+;n(!0; ) n + C+;n(n p; ) e+;n(!0; ) n where we have introduced the \e ective" DA hard-collinear QCD corrections to the correlation function ie;k(!0; ) (i = e+;n(!0; ) = ! + !c !c) (!0 (! + !c)2 (! + !c + r (!0 !c) (!0 !c) (!0 !c) + e+;n(!0; ) = e ;n(!0; ) = e ;n(!0; ) = B+(!0 !c) (!0 !c) !c(2! + !c) 1 !(! + !c)2 !0 !c) (!0 !c) (2;)n(!0) + B(!0) (3;)n(!0) + B(0) (4;)n(!0) d! B(!) (5;)n(!; !0) + fD Exp n p fB+D; 2P(q2) ; fB0 D; 2P(q2) U2(n p; h2; ) f~B( h2) d!0 Exp { 16 { meson correlation function (n p; n p) de ned in (2.1) at tree level. C+;n(n p; ) e+;n(!0; ) + C ;n(n p; ) e ;n(!0; ) gluon eld method [46{48] (k= + mc) (u x) ; with G ; 3P(n p; n p) = ~2;n(u; !; ) ~2;n(u; !; ) ~3;n(u; !; ) where the coe cient functions ~i;k(u; !; ) are given by ~2;n(u; !; ) = 2 (u 1) [ V (!; ) + A(!; )] ; ~2;n(u; !; ) = V (!; ) + (2 u ~3;n(u; !; ) = 2 (2 u 1) XA(!; ) 2 YA(!; ) : element on the light cone [49, 50] h0jq (x) G (u x) bv (0)jB(v)i x2=0 f~B( ) mBZ 1 operator on the left-hand side. The following conventions XA(!; ) = YA(!; ) = twist-3 DA V (!; ) which was shown to be completely integrable tion function the Borel transformation lead to the following sum rules fD Exp F3P;n(!s; !M ) fB+D; 3P(q2) ; fBD; 3P(q2) 0 F3P;n(!s; !M ) ; F3P;n(!s; !M ) = F3P;n(!s; !M ) = ~2;n(u; !; ) ~2;n(u; !; ) ~3;n(u; !; ) ~3;n(u; !; ) u= !0 !c ! ~2;n(u; !; ) u= !s !c ! u= !s !c ! ~2;n(u; !; ) u= !0 !c ! three-particle B-meson DA, already at tree level. three-particle B-meson DA in the end-point region [7] XA(!; ) YA(!; ) by a factor of fB0 D(q2) = fBD; 2P(q2) + fB0 D; 3P(q2) ; 0 and (4.6), respectively. Numerical analysis { 19 { 1j according to the conformal transformation [64] z(q2; t0) = mD)2 will be tion leads to the suggested parametrization [32, 61] fB+D(q2) = fB+D(0) q2=m2Bc z(0; t0)k z(q2; t0)N z(0; t0)N 0:009) GeV [65], and the z-series expanwe will employ the following parametrization fB0 D(q2) = fB0 D(0) q2=m2Bc(0) 1 + X ~bk z(q2; t0)k z(0; t0)k with the aid of the LCSR predictions at qm2in 2 qmax and the z-series parametrizakinematic range 0 mD)2 with theory uncertainties in gure 7 where recent 0:009) GeV [5], and we will only keep indicated by the shaded regions. fB+D(z) = fB+D(0) 1 fB0 D(z) fB+D(z) 1:0036 1 + 0:0068 ! + 0:0017 !2 + 0:0013 !3 ; (5.12) { 26 { Central value fB+D(0) from the variations of theory inputs. where we have introduced the following conventions z = p 1 + ! ! = m2B + m2D 2 mB mD the form factors of B ! D` at qm2in leads to 2 qmax onto the CLN parametrization (5.12) fB+D(z = 0) = 1:22 = 1:07+00::0181 ; large theory uncertainties for the slop parameter . Phenomenological implications ! = 1 r = mD=mB : (5.13) m2B jp~Dj where jp~Dj = the magnitude of the three-momentum of the D-meson and 2bc is EW = 1 + ' 1:0066 for the B ! D` decays [69, 70]. (pink band) and B ! D (blue band) as a [t1; t2] GeV2 [0:00; 0:98] [0:98; 2:16] [2:16; 3:34] [3:34; 4:53] [4:53; 5:71] (t1; t2) (10 12 GeV) (t1; t2) (10 12 GeV) [t1; t2] GeV2 this work Belle [71] this work Belle [71] 1:00+00::2281 1:09+00::2271 0:97+00::2116 0:85+00::1163 0:72+00::1019 [5:71; 6:90] [6:90; 8:08] [8:08; 9:26] [9:26; 10:45] [10:45; 11:63] 0:57+00::0086 0:43+00::0054 0:28+00::0022 0:14+00::0011 0:03+00::0000 compared with the Belle measurements from [71]. { 28 { measurements [71, 72]. can be obtained straightforwardly In particular, our predictions for the total decay width of B ! D in units of 1=jVcbj2 (0; 11:63 GeV2) = 6:06+10::1982 10 12 GeV ; from which the exclusive determinations of jVcbj are achieved jVcbj = < 39:2+33::43 th [BaBar 2010] [Belle 2016] q2 shape of B ! D displayed in table 1 are plotted in gure 8, including also the recent experimental measurements for the combination of B+ `(t1; t2) = R(t1; t2) = Rtt12 dq2 d (B ! D Rtt12 dq2 d (B ! D )=dq2 )=dq2 accuracy of form factors at di erent q2 for future work. { 29 { GeV2 [4:00; 4:53] [4:53; 5:07] [5:07; 5:60] [5:60; 6:13] [6:13; 6:67] [6:67; 7:20] [7:20; 7:73] [7:73; 8:27] [8:27; 8:80] [8:80; 9:33] [9:33; 9:86] [9:86; 10:40] [10:40; 11:63] this work { 30 { this work and for the binned distributions of calculations presented in [74]. branching fractions of two semileptonic decay channels BR(B ! D BR(B ! D = 0:305+00::002225 ; (SM) at the 1 space region 0 in the phase Concluding discussion form factors with the power counting scheme mc mb , at leading power in tion between the vector and scalar B ! D form factors. 0 numerically decay rates and of the ratio R(t1; t2) by applying our determinations of the form factors { 31 { with the HFAG average value in [2]. ties of the B-meson DA B(!; ) at two loops (e.g., the eigenfunctions and the analytical with X = ( ; ) the general properties of e ective eld theories. Acknowledgments Loop integrals vacuum-to-B-meson correlation function (2.1) at O( s). [n (p + l) n (p k + l) + l2 mc2 + i0][n l + i0][l2 + i0] n (p + l) { 32 { I2;a = 2 r3(1 + r3) I2;b = 21r33 (2 + 2r2 r3)r3 2(1 + r2)2 ln 1 + r2 + r3 1 + r2 p2 ln(1 + r2 + r3) + 3 2r32 (1 + r3)2 2(1r+2 r3) ; (A.5) (1 + r2)2 ln2 1 + r2 + r3 1 + r2 + 12+r32r2 ln2 1 + r2 + r3 1 + r2 r2 [22r+3(r13+(2r+3)r22)] ln 1 + r2 r2 2r32(1 + r3)2 r32 + r2 + 2r2r3) ln 1 + r2 + r3 1 + r2 1 + r2 + r3 1 + r2 1 + r3 1 + r2 + r3 [d l] [l2 + i0][(p l)2 mc2 + i0][(l k)2 + i0] p2 [k I4;a + p I4;b] ; 1 + r2 r3 (1 + r3) = I5;a (p= k=) + I5;b mc ; ln [(1 + r2)(1 + r2 + r3)] + 2 ln (1 + r2) 1 + r2 + r3 ; (A.10) I4; = I4;a = 1 + r3 1 + r2 I4;b = r3(1 + r3) 1 + r2 + r3 ln (1 + r2 + r3) 1 + r2 ln (1 + r2) + 1 + r3 I5 = k + l)2 k= + =l) + D mc mc2 + i0][l2 + i0] (n p)2 (1+r2 +r3)2 1 + r2 (1 + r3)2 I5;a = I5;b = 4 I6;a = I6;b = +Li2 1 ln (1+r1)+1 r1 ; ln (1+r1)+ 1+r1 r4 n (p+l) n p (! n p) n l n (p+l) Here, D = 4 2 , the integration measure is de ned as follows 1 ln (1 r4 +r1)+ (1 r12) ln (1+r1) ln (1+r1) : (A.15) d! (!0 !c) !+!c !02 !0 ! !c +i0 r3(1+r3) !0 ! !c +i0 1+r3 (1+r2 +r3)2 1+r2 +r3 (!+!c)2 P !0 ! !c !0 and we also introduce the conventions r1 = r2 = r3 = r4 = Spectral representations + !!c0 B+(!0)+ 0(!0) !0 ! !c +i0 r2 (r3 r2) ln r2 B+(!) d! (!0) !+!c P !0 ! !c !(!+!c) !0 ! !c +i0 r2 B+(!) !c (!0 !c) B+(!0 !c) { 35 { 1 (1+r2 +r3)2 d! (!0 !c) P !0 ! (!0 !c) : d! (!0 !c) !(!+!c) !02 + 0(!0) ln = (!0) (!c !0) (!0) (!c !0) B+(!0) (!0) + (!0) r3(1+r3) r2 (1+r2 +r3) ln(1+r2 +r3) B+(!) d! (!c +! !0) (!0) !c r2 (1+r2) ln(1+r2) B+(!) B+(!0 !c) (!0 !c) !+!c !0 ! !c !0 ! !c +i0 !0 (1+r3)2 ! (!+!c)2 !0 !0 ! !c +i0 1+r3 (!0) (!c !0) !+!c P !0 ! !c B+(!0 !c) (!0 !c) 1+r2 +r3 r22 (1+r3)2 { 36 { = !c2 n p (! !0) d! d! (!0) (!c !0) (! !0) d! (!0 !) P !0 ! !c !0 ! !c +i0 (1+r2 +r3)2 (1+r2)(1+r3) n p (! !0) !0 ! !c +i0 1+r2 +r3 n p (! !0) !c (!c !0) ln !0 ! !c +i0 r22 (r3 +2) 0(!0) ln r2(1+r2 +r3) B+(!) !0 ! !c +i0 !!c0 B+(!0) !c (!0) B(!0 !c) (!0 !c) : 2 (!0 !c) (!+!c !0) n p (! !0) B(!0 !c) (!0 !c) !0 ! !c +i0 ln2(1+r3) B(!) B(!0 !c) (!0 !c) : (! !0) (!0) B(!) !0 ! !c +i0 d! ln2 !+!c !0 d! (!+!c !0) (!0) ln ln(1+r2 +r3) ln(1+r3) B(!) (! !0) (!0) !+!c !0 1 Z 1 !0 ! !c +i0 !+!c !0 !0 ! !c +i0 !+!c 1+r2 +r3 (!+!c !0) (!0 !c) (!c !0) (1+r2)2 ln (1+r2) B(!) B(!0 !c) (!0 !c) B(!0 !c) (!0 !c) !c ln !0 ! !c +i0 ln2(1+r2 +r3) B(!) (!c !0) (!0) B(0) d! ln2 !+!c !0 (!+!c !0) (!0) (!0) (!c !0) (!0) (!c !0) (!c !0) (!0) ln +P ! !0 +P ! !0 d! = (!c !0) (!0) ln !0 ! !c +i0 !0 ! !c +i0 (1+r3)2 (!0) ln B+(!0) (!0)+ B+(!0 !c) (!0 !c) d! (!+!c !0) (!0) ln !+!c !0 !0 ! !c +i0 d! (!c !0) (!0) d! (! !0) (!0) d! (!+!c !0) (!0) ln !+!c !0 !0 ! !c +i0 1+r2 +r3 ln (1+r3) B(!) B(!0 !c) (!0 !c) B(!0) (!0) (!0 !c) !c ln d! (! !0) (!0) !0 ! !c +i0 !0 ! !c d! B(!) : ln2 (1+r2) B(!) B(!0 !c) (!0 !c) d! (!c !0) (!0) ln r2 ln r2 !0 ! !c +i0 1+r2 +r3 (!0 !c) ln !0 d!0 B(!0 !c) ln (1+r2 +r3) B(!) ln (1+r2) ln (1+r3) B(!) B(!0 !c) (!0 !c) !0 d!0 B(!0 !c) +r2 +2 ln r2 B(!) (!c !0) (!0) (1+r3)2 1+r3 (!0 !c) + 0(!0) ln (!c !0) (!0) (!c !0) (!0) !02 +2 !0!c !2 c d! (!0) P !0 ! !c B(!0 !c) (!0 !c) !2 +4 ! !c +2 !c2 (!+!c)2 !+!c !0 (!+!c !0) (!0 !c) (!+!c !0) !0 ! !c +i0 !+!c !0 1+r2 +r3 + (!0 ! !c) ln (!+!c !0) (!0 !c) Li2 !0 ! !c +i0 !+!c !0 (!0 ! !c) ln = Li2 d! (!0 !c) ln !0 !c P !0 ! !c B(!) : (!0 !c) B(0)+ d!0 B(!0 !c) (!0 !c) : { 40 { B(!0 !c) (!0 !c) !0 +!c B(!0) (!0)+ B(!0 !c) (!0 !c) e ;n(!0; ) de ned in (3.49). (1;)n(!0) = ln 4 (!+!c !0) (!0) ln + (!0) (!0 ! !c) (2;)n(!0) = 2 !c 3 ln (3;)n(!0) = 2 ln2 ln2 !c + 2 (!c !0) ; n p (!0 !c) 2 (!0 !) ln + (!c !0) (!0) d!0 f (!0) + g(!0) = d!0 f (!0) ++ g(!0) = The coe cient functions of 2 ln2 !c +ln2 !0 !c (!+!c)2 +4 (!+!c !0) (!0 !c) ln 2 (!0) +2 (!0) + (!0 !c) (! !0) (!c !0)+ (! !0) !c +2 ! 1 !+!c !0 (!0) !c ln !+!c !0 !+!c !+!c { 41 { (!c !0) (!0) ; !+!c !0 n p (! !0) +4 (!0 !c) ln !+!c n p (! +2 (!0) (! +!c !0) (!0 !) ln2 ! +!c !0 4 (! +!c !0) (!0) ln 2 (!c +! !0) (!0) ln 4 (! +!c !0) (!0 !c) Li2 + 4 (!0 ! !c) ln ! +!c !0 ! +!c !0 !c) (!0) ! +!c !0 (!0) (! +!c !0) ; (!0 !) (!0 !c) (! +!c !0)+ (! !0) (!0) ! +!c !0 Open Access. [INSPIRE]. [INSPIRE]. [INSPIRE]. Phys. C 38 (2014) 090001 [INSPIRE]. arXiv:1612.07233 [INSPIRE]. [INSPIRE]. form factors, JHEP 02 (2008) 031 [arXiv:0711.3999] [INSPIRE]. Rev. D 73 (2006) 036003 [hep-ph/0504005] [INSPIRE]. JHEP 05 (2006) 056 [hep-ph/0512157] [INSPIRE]. Phys. B 591 (2000) 313 [hep-ph/0006124] [INSPIRE]. form factors from light-cone sum (2004) 054024 [hep-ph/0301240] [INSPIRE]. 564 (2003) 231 [hep-ph/0303099] [INSPIRE]. Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE]. form factors from QCD (2016) 159 [arXiv:1606.03080] [INSPIRE]. [INSPIRE]. [hep-ph/9412340] [INSPIRE]. [INSPIRE]. functions, JHEP 02 (2013) 008 [arXiv:1210.2978] [INSPIRE]. ` decays 099902] [arXiv:0807.2722] [INSPIRE]. (1997) 272 [hep-ph/9607366] [INSPIRE]. [hep-ph/9703278] [INSPIRE]. [INSPIRE]. [INSPIRE]. [INSPIRE]. bound states, JHEP 04 (2008) 061 [arXiv:0802.2221] [INSPIRE]. 71 (2011) 1818 [arXiv:1110.3228] [INSPIRE]. and B ! K twist, Phys. Rev. D 72 (2005) 034023 [hep-ph/0502239] [INSPIRE]. [INSPIRE]. [63] P. Ball and E. Kou, B ! e transitions from QCD sum rules on the light cone, JHEP 04 189 (1981) 157 [INSPIRE]. jVcbj, Phys. Rev. D 93 (2016) 032006 [arXiv:1510.03657] [INSPIRE]. Phys. Lett. B 771 (2017) 168 [arXiv:1612.07757] [INSPIRE]. , Phys. Lett. B 716 [INSPIRE]. [1] Particle Data Group collaboration , K.A. Olive et al., Review of particle physics, Chin. [2] Y. Amhis et al., Averages of b-hadron, c-hadron and -lepton properties as of summer 2016 , [3] M. Neubert , Heavy quark symmetry , Phys. Rept . 245 ( 1994 ) 259 [hep-ph/9306320] [4] MILC collaboration , J.A. Bailey et al., B ! D` form factors at nonzero recoil and jVcbj from 2 + 1- avor lattice QCD, Phys. Rev. D 92 ( 2015 ) 034506 [arXiv:1503.07237] [5] HPQCD collaboration, H. Na , C.M. Bouchard , G.P. Lepage , C. Monahan and [6] A. Khodjamirian , T. Mannel and N. O en, B-meson distribution amplitude from the B ! [7] A. Khodjamirian , T. Mannel and N. O en, Form-factors from light-cone sum rules with [8] F. De Fazio , T. Feldmann and T. Hurth , Light-cone sum rules in soft-collinear e ective theory, Nucl . Phys . B 733 ( 2006 ) 1 [Erratum ibid . B 800 ( 2008 ) 405] [hep-ph/0504088] [9] F. De Fazio , T. Feldmann and T. Hurth , SCET sum rules for B ! P and B ! V transition [10] S. Faller , A. Khodjamirian , C. Klein and T. Mannel , B ! D( ) form factors from QCD [11] H. Boos , T. Feldmann , T. Mannel and B.D. Pecjak , Shape functions from B ! Xc` `, Phys . [12] H. Boos , T. Feldmann , T. Mannel and B.D. Pecjak , Can B ! Xc`nu` help us extract jVubj? , [13] P. Gambino , T. Mannel and N. Uraltsev , B ! D zero-recoil formfactor and the heavy quark [14] M. Beneke , G. Buchalla , M. Neubert and C.T. Sachrajda , QCD factorization for exclusive , [15] Y.-M. Wang and Y.-L. Shen , QCD corrections to B ! [16] I.Z. Rothstein , Factorization, power corrections and the pion form-factor , Phys. Rev. D 70 [17] A.K. Leibovich , Z. Ligeti and M.B. Wise , Comment on quark masses in SCET, Phys. Lett . B [18] M. Beneke and V.A. Smirnov , Asymptotic expansion of Feynman integrals near threshold , [19] Y.-M. Wang and Y.-L. Shen , Perturbative corrections to b ! [20] Y.-M. Wang , Factorization and dispersion relations for radiative leptonic B decay , JHEP 09 [21] H.-B. Fu , X.-G. Wu , H.-Y. Han , Y. Ma and T. Zhong , jVcbj from the semileptonic decay B ! D` ` and the properties of the D meson distribution amplitude, Nucl . Phys . B 884 [22] R.- H. Li , C.-D. Lu  and Y.-M. Wang , Exclusive Bs decays to the charmed mesons [23] H.-N. Li , Applicability of perturbative QCD to B ! D decays, Phys. Rev. D 52 (1995) 3958 [24] T. Kurimoto , H.-N. Li and A.I. Sanda , B ! D( ) form-factors in perturbative QCD , Phys. [26] H.-N. Li , Y.-L. Shen , Y.-M. Wang and H. Zou , Next-to-leading-order correction to pion form factor in kT factorization , Phys. Rev. D 83 ( 2011 ) 054029 [arXiv:1012.4098] [INSPIRE]. [27] H.-N. Li , Y.-L. Shen and Y.-M. Wang , Next-to-leading-order corrections to B ! [28] H.-N. Li , Y.-L. Shen and Y.-M. Wang , Resummation of rapidity logarithms in B meson wave [29] H.-N. Li , Y.-L. Shen and Y.-M. Wang , Joint resummation for pion wave function and pion [30] Y.-M. Wang , Non-dipolar gauge links for transverse-momentum-dependent pion wave [31] H.-N. Li and Y.-M. Wang , Non-dipolar Wilson links for transverse-momentum-dependent [32] C. Bourrely , I. Caprini and L. Lellouch , Model-independent description of B ! and a determination of jVubj , Phys. Rev . D 79 ( 2009 ) 013008 [Erratum ibid . D 82 ( 2010 ) [33] I. Caprini , L. Lellouch and M. Neubert , Dispersive bounds on the shape of B ! D( )` [34] A.G. Grozin and M. Neubert , Asymptotics of heavy meson form-factors , Phys. Rev. D 55 [35] M. Beneke and T. Feldmann , Symmetry breaking corrections to heavy to light B meson [36] G. Bell , T. Feldmann , Y.-M. Wang and M.W.Y. Yip , Light-cone distribution amplitudes for [37] K.G. Chetyrkin , Quark mass anomalous dimension to O( S4), Phys . Lett . B 404 ( 1997 ) 161 [38] J.A.M. Vermaseren , S.A. Larin and T. van Ritbergen , The four loop quark mass anomalous [39] G. Bell and T. Feldmann , Modelling light-cone distribution amplitudes from non-relativistic [40] S. Descotes-Genon and N. O en, Three-particle contributions to the renormalisation of [41] M. Beneke and J. Rohrwild , B meson distribution amplitude from B ! ` , Eur . Phys . J. C [42] M. Beneke and T. Feldmann , Factorization of heavy to light form-factors in soft collinear [43] M. Beneke and D. Yang , Heavy-to-light B meson form-factors at large recoil energy : [44] C.W. Bauer , S. Fleming , D. Pirjol and I.W. Stewart , An e ective eld theory for collinear [45] C.W. Bauer , D. Pirjol and I.W. Stewart , Factorization and endpoint singularities in heavy to light decays , Phys. Rev. D 67 ( 2003 ) 071502 [hep-ph/0211069] [INSPIRE]. [46] I.I. Balitsky and V.M. Braun , Evolution equations for QCD string operators, Nucl . Phys . B [47] A. Khodjamirian , T. Mannel , A.A. Pivovarov and Y.-M. Wang , Charm-loop e ect in [48] A. Khodjamirian , T. Mannel and Y.M. Wang , B ! K` +` decay at large hadronic recoil , [49] H. Kawamura , J. Kodaira , C.-F. Qiao and K. Tanaka , B-meson light cone distribution amplitudes in the heavy quark limit , Phys. Lett . B 523 ( 2001 ) 111 [Erratum ibid . B 536 [51] V.M. Braun , A.N. Manashov and N. O en, Evolution equation for the higher-twist B-meson [52] S.J. Lee and M. Neubert , Model-independent properties of the B-meson distribution [53] T. Feldmann , B.O. Lange and Y.-M. Wang , B-meson light-cone distribution amplitude : [55] V.M. Braun , D. Yu . Ivanov and G.P. Korchemsky , The B meson distribution amplitude in [58] S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. [59] M. Beneke , A. Maier , J. Piclum and T. Rauh , The bottom-quark mass from non-relativistic [60] B. Dehnadi , A.H. Hoang and V. Mateu , Bottom and charm mass determinations with a [61] A. Khodjamirian , C. Klein , T. Mannel and N. O en , Semileptonic charm decays D ! [64] C. Bourrely , B. Machet and E. de Rafael, Semileptonic decays of pseudoscalar particles (M ! M 0 + ` + `) and short distance behavior of quantum chromodynamics, Nucl . Phys . B [65] E.B. Gregory et al., A prediction of the Bc mass in full lattice QCD , Phys. Rev. Lett. 104 [66] C.G. Boyd , B. Grinstein and R.F. Lebed , Model independent extraction of jVcbj using [67] D. Bigi and P. Gambino , Revisiting B ! D` , Phys . Rev . D 94 ( 2016 ) 094008 [69] A. Sirlin , Large mW , mZ behavior of the O( ) corrections to semileptonic processes mediated [70] N. Carrasco et al., QED corrections to hadronic processes in lattice QCD , Phys. Rev. D 91 [68] C. DeTar , LQCD: avor physics and spectroscopy, PoS(LeptonPhoton2015) 023 [71] Belle collaboration , R. Glattauer et al., Measurement of the decay B ! D` ` in fully [72] BaBar collaboration , J.P. Lees et al ., Measurement of an excess of B ! D( ) and implications for charged Higgs bosons, Phys. Rev. D 88 (2013) 072012 [74] A. Celis , M. Jung , X.-Q. Li and A. Pich , Scalar contributions to b ! c(u) [75] J.F. Kamenik and F. Mescia , B ! D hadron colliders, Phys. Rev. D 78 ( 2008 ) 014003 [arXiv:0802.3790] [INSPIRE]. [76] D. Becirevic , N. Kosnik and A. Tayduganov , B ! D [78] V.M. Braun , Y. Ji and A.N. Manashov , Higher-twist B-meson distribution amplitudes in [79] Z. Ligeti , M. Papucci and D.J. Robinson , New physics in the visible nal states of


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP06%282017%29062.pdf

Yu-Ming Wang, Yan-Bing Wei, Yue-Long Shen, Cai-Dian Lü. Perturbative corrections to B → D form factors in QCD, Journal of High Energy Physics, 2017, 1-47, DOI: 10.1007/JHEP06(2017)062