Rapidity resummation for B -meson wave functions
aa r X i v : . [ h e p - ph ] S e p The Journal’s name will be set by the publisherDOI: will be set by the publisherc (cid:13)
Owned by the authors, published by EDP Sciences, 2018
Rapidity resummation for B -meson wave functions Yue-Long Shen , a and Yu-Ming Wang , , b College of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266100,P.R. China Physik Department T31, Technische Universität München, James-Franck-Straße 1, D-85748 Garching,Germany InstitutfürTheoretische Teilchenphysik und Kosmologie, RWTHAachen University, D-52056 Aachen, Ger-many
Abstract.
Transverse-momentum dependent (TMD) hadronic wave functions developlight-cone divergences under QCD corrections, which are commonly regularized by therapidity ζ of gauge vector defining the non-light-like Wilson lines. The yielding rapid-ity logarithms from infrared enhancement need to be resummed for both hadronic wavefunctions and short-distance functions, to achieve scheme-independent calculations ofphysical quantities. We briefly review the recent progress on the rapidity resummationfor B -meson wave functions which are key ingredients of TMD factorization formulae forradiative-leptonic, semi-leptonic and non-leptonic B -meson decays. The crucial observa-tion is that rapidity resummation induces a strong suppression of B -meson wave functionsat small light-quark momentum, strengthening the applicability of TMD factorization inexclusive B -meson decays. The phenomenological consequence of rapidity-resummationimproved B -meson wave functions is further discussed in the context of B → π transitionform factors at large hadronic recoil. QCD factorization theorems are indispensable to the theoretical descriptions of high-energy processesprobed at worldwide collider experiments, and the predictive power lies on the process-independenceof non-perturbative hadronic functions, entering the factorization formulae. B -meson light-cone dis-tribution amplitudes (LCDAs) and transverse-momentum dependent (TMD) wave functions are fun-damental inputs for QCD calculations of exclusive B -meson decays applying collinear factorizationand TMD factorization theorems. Great e ff orts have been devoted to the studies of renormalizationproperties and perturbative constraints of B -meson LCDAs in both momentum space [1–4] and dualspace [5, 6], and to the explorations of QCD evolution equations of B -meson TMD wave functions[7, 8] with the standard renormalization-group (RG) method and the QCD resummation technique(for a recent review, see [9]).Providing three-dimensional profile of a given hadron, TMD wave functions are more complicateddue to the emergence of light-cone (or rapidity) singularities in the end-point region, which cancel in a e-mail: [email protected] b e-mail: [email protected] he Journal’s name the QCD corrections to LCDAs. Such rapidity divergences can be regularized by the rapidity ( ζ )of non-light-cone Wilson lines, at the price of generating infrared enhanced double logarithm ln ζ [10, 11] for B -meson wave functions and single logarithm ln ζ [12, 13] for pion wave functions.Resummation of these rapidity logarithms in both hadronic wave functions and hard functions areessential to make scheme-independent predictions for physical observables. In the following, wewill discuss the construction of rapidity evolution equation for B -meson TMD wave functions in theMellin and impact parameter spaces in section 2, and present the resummation improved B -mesonwave functions in momentum space by performing the inverse Mellin transformation in section 3where the resummation e ff ects on B → π transition form factors are also reported. The TMD wave functions of B -meson are defined by the non-local vacuum-to-hadron matrix elementin coordinate space [8] h | ¯ q ( y ) W y ( n ) † I n ; y, W ( n ) Γ h (0) | ¯ B ( v ) i = − i f B m B + /v Φ + B ( t , y ) + Φ − B ( t , y ) − Φ + B ( t , y ) t /y γ Γ , (1)which depends on longitudinal ( t = v · y ) and transverse ( y ) variables as well as the non-light-likegauge vector n defining the Wilson lines.It is demonstrated in [10] that both the double rapidity logarithm ln ζ with ζ = v · n ) / n , dueto the overlap of the collinear enhancement from a loop momentum l collimated to the gauge vector n and the soft enhancement, and the mixed logarithm ln µ f ln ζ ( µ f being the renormalization scale)which has the same origin of cusp divergence in B -meson LCDAs [1] appear in the next-to-leading-order (NLO) QCD corrections to B -meson wave functions. Simultaneous resummation of two distincttypes of rapidity logarithms for B -meson wavefunctions makes it subtler than that of the traditionalSudakov resummation for fast-moving light hadrons (see, however, [14] for recent progress).Recalling that the Sudakov resummation of ln ζ for pion wave function can be achieved byconstructing the evolution equation from varying the gauge vector n , because the collinear divergencearises from the region with a loop momentum collimated to the pion momentum and varying thevector n does not result in an additional collinear divergence. This trick however does not apply to theresummation of B -meson wave function due to the nature of rapidity logarithm explained in the above.To resum the rapidity logarithm for B -meson wave functions, we utilize the fact that the resultingcollinear divergence is insensitive to the heavy-quark velocity v and the e ff ect from varying v can thenbe factorized from the TMD wave functions. Trading the rapidity derivative for the di ff erential ofvelocity yields ζ dd ζ Φ ( b ) B ( x , k T , ζ , µ f ) = v · n ǫ αβ v α n β v + dd v + Φ ( b ) B ( x , k T , ζ , µ f ) , (2)where ǫ αβ is an anti-symmetric tensor with ǫ + − = − ǫ − + = v + is the plus component of the b -quarkvelocity and x is the longitudinal momentum fraction of the light quark. Here, the crucial point is thatthe velocity derivative will be only applied to the Feynman rules involving an e ff ective heavy b -quark v · n ǫ αβ v α n β v + dd v + v µ v · l = ˆ v µ v · l , (3)inducing the special vertex ˆ v µ ≡ v · n ǫ αβ v α n β ǫ ρλ v ρ g µλ − v µ l λ v · l ! . (4) CD@Work 2014
BB vk
Figure 1.
Leading-order soft kernel for the rapidity evolution equation, where the square denotes the specialvertex of Eq. (4). Taken from [8].
The rapidity evolution equation of B -meson wave function can then be written as ζ dd ζ Φ B ( x , k T , ζ , µ f ) = K ( b , ⊗ Φ B ( x , k T , ζ , µ f ) − Z Φ ζ dd ζ Z Φ ! Φ B ( x , k T , ζ , µ f ) , (5)where Z Φ is the counterterm for the ultraviolet renormalization of TMD wave function. The one-loopkernel K ( b , collects the soft gluon dynamics as displayed in figure 1.Computing the two e ff ective diagrams yields K ( b , = − α s C F π Γ ( ǫ ) πµ f λ ǫ v · n ǫ αβ v α n β ! , (6)˜ K (1)2 ( N , b , ζ ) = α s C F π v · n ǫ αβ v α n β ! " K ( λ b ) − K q ζ m B bN ! , (7)where the gluon mass λ is introduced to regularize the infrared divergence in each diagram, Mellin andFourier transformations for the B -meson wave function are performed in the evaluation of irreduciblecontribution illustrated in the second diagram. It is then straightforward to derive ζ dd ζ ˜ Φ B ( N , b , ζ , µ f ) = ˜ K (1) ( N , b , ζ , µ f ) ˜ Φ B ( N , b , ζ , µ f ) , (8)with the renormalized evolution kernel˜ K (1) ( N , b , ζ , µ f ) = − α s C F π " ln µ f b + γ E + K q ζ m B bN ! , (9)where the infrared divergence cancels exactly between the two e ff ective diagrams. The single loga-rithm of soft kernel ˜ K (1) can be organized by the RG resummation K (1) ( N , b , ζ , µ f ) = ˜ K (1) ( N , b , ζ , µ c ) − Z µ f µ c d µµ λ K ( α s ( µ )) θ ( µ f − µ c ) , (10)where the scale µ c = a p ζ m B / N , a being an order-unity constant, is adjusted to diminish the loga-rithmic enhancement in the initial condition.In addition, the factorization scale evolution of TMD wave function itself is computed as˜ Φ B ( N , b , ζ , µ f ) = exp " − Z µ f µ d µµ α s ( µ )2 π C F (cid:16) ln ζ − (cid:17) ˜ Φ B ( N , b , ζ , µ ) . (11) he Journal’s name Combining the rapidity and scale evolutions and choosing µ f = a ζ m B , we obtain˜ Φ B ( N , b ) = exp Z N ζ ζ d ˜ ζ ˜ ζ K (1) ( N , b , ˜ ζ , µ f ) − Z µ f µ d µµ α s ( µ )2 π C F (cid:16) ln ζ − (cid:17) × ˜ Φ B ( N , b , ζ , µ ) , (12)with the simplified rapidity kernel K (1) ( N , b , ζ , µ f ) = − α s ( µ c )2 π C F ln a − Z µ f µ c d µµ α s ( µ )2 π C F θ ( µ f − µ c ) (13)which is valid in the large N limit. B -meson TMD Wave Functions We are now in a position of discussing the resummation e ff ect on B -meson wave functions in x -spaceby performing the inverse Mellin transformation Φ ± B ( x , k T ) = Z c + i ∞ c − i ∞ dN π i (1 − x ) − N ˜ Φ ± B ( N , k T ) , (14)for the solution given by Eq. (12). To achieve this purpose, we need to know the initial conditions ofTMD wave functions which are postulated to have the factorized form Φ ± B ( x , k T , ζ ) = φ ± B ( x , ζ ) φ ( k T ) , (15)to reduce the numerical analysis. The longitudinal parts are further taken from the so-called “free-parton" model [15] φ + B ( x , ζ ) = x x θ (2 x − x ) φ − B ( x , ζ ) = x − x x θ (2 x − x ) , (16)which in Mellin space correspond to˜ φ + B ( N , k T , ζ ) = − (1 − x ) N (1 + x N )2 x N ( N + , ˜ φ − B ( N , k T , ζ ) = (1 − x ) N + + x N + x − x N ( N + . The inverse Mellin transformation will be firstly performed for the resummation improved wavefunctions ˜ Φ ± B ( N , k T ) with frozen strong coupling constant α s , to make the analytical behavior moretransparent, and then with running α s . The resummation e ff ects on the x dependence of B -mesonwave functions φ ± B ( x ) = Φ ± B ( x , k T ) /φ ( k T ) are illustrated in figure 2. The primary observations aresummarized as follows: • Faster than linear attenuation of φ ± ( x ) in the small x region is realized after the resummation im-provement, albeit with the non-vanishing φ − ( x ) at x = • Resummation improved B -meson wave functions become smooth and develop radiative tails. • The normalization conditions R dx φ ± B ( x )( x ) = ff ected by the rapidity resummation. CD@Work 2014 Φ B - H x L Φ B + H x L Figure 2.
The dashed, dotted and solid curves correspond to the x dependence of initial condition φ ± B ( x , ζ ), andthe resummation improved φ ± B ( x ) for fixed α s = . α s with ζ = e /
10 and a =
1. Taken from [8].
It will be interesting to inspect the rapidity resummation e ff ect for phenomenological observableswhich are relevant for the precision test of CKM mechanism (for recent reviews, see [16, 17]) and forthe hunting of beyond standard model physics. In this respect, we consider the B → π transition formfactors f ± B π ( q ) at large hadronic recoil as illuminative examples, which are essential to the golden-channel determination of matrix element | V ub | exclusively. Numerically the resummation e ff ects arefound to decrease the above two form factors by approximately 25% at q =
0, due to the strongsuppression of B -meson wave functions at the end-point. Notice that such improvement is sizeableand must be taken into account in the future calculations of B → π form factors applying TMDfactorization theorem to catch up with the precision achieved in the calculations from light-cone QCDsum rules [18, 19]. Applying the QCD resummation technique with non-light-like Wilson lines, we construct rapidityevolution equation for TMD wave functions of B -meson collecting the double logarithm ln ζ andmixed logarithm ln µ f ln ζ . Technically, the rapidity resummation of B -meson wave functions isnovel due to the di ff erent nature of collinear divergence in QCD correction from that of energeticlight mesons. The resummation improved B -meson wave functions induce a strong suppression in thesmall x region and hence enhance the applicability of TMD factorization in exclusive B -meson decays.Sizeable corrections to the B → π transition form factors are also observed due to the resummationimprovement. Many open questions concerning the TMD wave functions of B -meson remain to beanswered, and these include (a) What are the relations between TMD wave functions and LCDAsat large transverse momentum? (b) What are the operator-product-expansion constraints of TMDwave functions? (c) Can we gain some insights of TMD wave functions at low energy scale fromnon-perturbative approaches? As a final remark, the rapidity resummation technique presented herecan be generalized immediately to the TMD wave functions of Λ b - baryon entering the factorizationformulae of many exclusive decays [20–22] which are of increasing interest at the LHC and Tevatron. he Journal’s name Acknowledgements
We are grateful to Hsiang-nan Li for a very fruitful collaboration. YMW would like to thank theorganizers of QCD@work 2014 for inviting him to this stimulating workshop and for the generousfinical support. This research is also partially supported by the National Science Foundation of Chinaunder Grant No. 11005100 (YLS), and by the DFG Sonderforschungsbereich / Transregio 9 “Com-putergestützte Theoretische Teilchenphysik" (YMW).
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