# A formalism for the systematic treatment of rapidity logarithms in Quantum Field Theory

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## Abstract

Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the relevant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. Such observables include: transverse momentum distributions at *p* _{ T } much less then the high energy scattering scale, jet broadening, exclusive hadroproduction and decay, as well as the Sudakov form factor. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a “rapidity renormalization group”. That is, a Collin-Soper like equation is realized as a renormalization group equation, but has a more universal applicability to observables beyond the traditional transverse momentum dependent parton distribution functions (TMDPDFs) and the Sudakov form factor. This formalism has the feature that it allows one to track the (non-standard) scheme dependence which is inherent in any sce- nario where one performs a resummation of rapidity divergences. We present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form fac- tor. The formalism is then used to study observables of current interest. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. Our formalism allows one to define gauge invariant TMDPDFs which are independent of both the hard scattering amplitude and the soft function, i.e. they are universal. We present details of the factorization and re- summation of the jet broadening cross section including a renormalization in *p* _{⊥} space. We furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.

## Keywords

Resummation QCD## References

- [1]
- [2]Y.L. Dokshitzer, A. Lucenti, G. Marchesini and G. Salam,
*On the QCD analysis of jet broadening, JHEP***01**(1998) 011 [hep-ph/9801324] [INSPIRE].ADSCrossRefGoogle Scholar - [3]S. Catani, G. Turnock and B. Webber,
*Jet broadening measures in e+ e− annihilation, Phys. Lett.***B 295**(1992) 269 [INSPIRE].ADSGoogle Scholar - [4]P.E. Rakow and B. Webber,
*Transverse momentum moments of hadron distributions in QCD jets, Nucl. Phys.***B 191**(1981) 63 [INSPIRE].ADSCrossRefGoogle Scholar - [5]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].ADSGoogle Scholar - [6]A.V. Manohar and I.W. Stewart,
*The zero-bin and mode factorization in Quantum Field Theory, Phys. Rev.***D 76**(2007) 074002 [hep-ph/0605001] [INSPIRE].ADSGoogle Scholar - [7]J.C. Collins and F. Tkachov,
*Breakdown of dimensional regularization in the Sudakov problem, Phys. Lett.***B 294**(1992) 403 [hep-ph/9208209] [INSPIRE].ADSGoogle Scholar - [8]J.C. Collins and D.E. Soper,
*Back-to-back jets in QCD, Nucl. Phys.***B 193**(1981) 381 [*Erratum ibid.***B 213**(1983) 545] [INSPIRE]. - [9]J. Collins,
*Rapidity divergences and valid definitions of parton densities,*PoS(LC2008)028 [arXiv:0808.2665] [INSPIRE]. - [10]B. Grinstein and I.Z. Rothstein,
*Effective field theory and matching in nonrelativistic gauge theories, Phys. Rev.***D 57**(1998) 78 [hep-ph/9703298] [INSPIRE].ADSGoogle Scholar - [11]C.W. Bauer, S. Fleming and M.E. Luke,
*Summing Sudakov logarithms in B → X*_{s}*γ in effective field theory, Phys. Rev.***D 63**(2000) 014006 [hep-ph/0005275] [INSPIRE].ADSGoogle Scholar - [12]C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart,
*An effective field theory for collinear and soft gluons: heavy to light decays, Phys. Rev.***D 63**(2001) 114020 [hep-ph/0011336] [INSPIRE].ADSGoogle Scholar - [13]C.W. Bauer, D. Pirjol and I.W. Stewart,
*Soft collinear factorization in effective field theory, Phys. Rev.***D 65**(2002) 054022 [hep-ph/0109045] [INSPIRE].ADSGoogle Scholar - [14]C.W. Bauer, D. Pirjol and I.W. Stewart,
*On power suppressed operators and gauge invariance in SCET, Phys. Rev.***D 68**(2003) 034021 [hep-ph/0303156] [INSPIRE].ADSGoogle Scholar - [15]M.E. Luke, A.V. Manohar and I.Z. Rothstein,
*Renormalization group scaling in nonrelativistic QCD, Phys. Rev.***D 61**(2000) 074025 [hep-ph/9910209] [INSPIRE].ADSGoogle Scholar - [16]
- [17]C. Lee and G.F. Sterman,
*Momentum flow correlations from event shapes: factorized soft gluons and soft-collinear effective theory, Phys. Rev.***D 75**(2007) 014022 [hep-ph/0611061] [INSPIRE].ADSGoogle Scholar - [18]A. Idilbi and T. Mehen,
*Demonstration of the equivalence of soft and zero-bin subtractions, Phys. Rev.***D 76**(2007) 094015 [arXiv:0707.1101] [INSPIRE].ADSGoogle Scholar - [19]A. Idilbi and T. Mehen,
*On the equivalence of soft and zero-bin subtractions, Phys. Rev.***D 75**(2007) 114017 [hep-ph/0702022] [INSPIRE].ADSGoogle Scholar - [20]J.C. Collins,
*Sudakov form-factors, Adv. Ser. Direct. High Energy Phys.***5**(1989) 573 [hep-ph/0312336] [INSPIRE].Google Scholar - [21]A. Jain, M. Procura and W.J. Waalewijn,
*Parton fragmentation within an identified jet at NNLL, JHEP***05**(2011) 035 [arXiv:1101.4953] [INSPIRE].ADSCrossRefGoogle Scholar - [22]M. Beneke and T. Feldmann,
*Factorization of heavy to light form-factors in soft collinear effective theory, Nucl. Phys*.**B 685**(2004) 249 [hep-ph/0311335] [INSPIRE].ADSCrossRefGoogle Scholar - [23]A.V. Manohar, T. Mehen, D. Pirjol and I.W. Stewart
*, Reparameterization invariance for collinear operators, Phys. Lett.***B 539**(2002) 59 [hep-ph/0204229] [INSPIRE].ADSGoogle Scholar - [24]J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein
*, The rapidity renormalization group, Phys. Rev. Lett.***108**(2012) 151601 [arXiv:1104.0881] [INSPIRE].ADSCrossRefGoogle Scholar - [25]T. Becher, R.J. Hill and M. Neubert
*, Soft collinear messengers: a new mode in soft collinear effective theory, Phys. Rev.***D 69**(2004) 054017 [hep-ph/0308122] [INSPIRE].ADSGoogle Scholar - [26]G. Korchemsky,
*Sudakov form-factor in QCD, Phys. Lett.***B 220**(1989) 629 [INSPIRE].MathSciNetADSGoogle Scholar - [27]C.W. Bauer, B.O. Lange and G. Ovanesyan,
*On Glauber modes in soft-collinear effective theory, JHEP***07**(2011) 077 [arXiv:1010.1027] [INSPIRE].ADSCrossRefGoogle Scholar - [28]J.C. Collins, D.E. Soper and G.F. Sterman,
*Soft gluons and factorization, Nucl. Phys.***B 308**(1988) 833 [INSPIRE].ADSCrossRefGoogle Scholar - [29]G.T. Bodwin
*, Factorization of the Drell-Yan cross-section in perturbation theory, Phys. Rev.***D 31**(1985) 2616 [*Erratum ibid.***D 34**(1986) 3932] [INSPIRE]. - [30]A. Idilbi and A. Majumder
*, Extending soft-collinear-effective-theory to describe hard jets in dense QCD media, Phys. Rev.***D 80**(2009) 054022 [arXiv:0808.1087] [INSPIRE].ADSGoogle Scholar - [31]F. D’Eramo, H. Liu and K. Rajagopal,
*Jet quenching parameter via Soft Collinear Effective Theory (SCET), Int. J. Mod. Phys.***E 20**(2011) 1610 [arXiv:1010.0890] [INSPIRE].ADSGoogle Scholar - [32]F. D’Eramo, H. Liu and K. Rajagopal,
*Transverse momentum broadening and the jet quenching parameter, redux, Phys. Rev.***D 84**(2011) 065015 [arXiv:1006.1367] [INSPIRE].ADSGoogle Scholar - [33]V.A. Smirnov,
*Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys.***134**(1990) 109 [INSPIRE].ADSMATHCrossRefGoogle Scholar - [34]J.-Y. Chiu, A. Fuhrer, A.H. Hoang, R. Kelley and A.V. Manohar,
*Soft-collinear factorization and zero-bin subtractions, Phys. Rev.***D 79**(2009) 053007 [arXiv:0901.1332] [INSPIRE].ADSGoogle Scholar - [35]J.-Y. Chiu, A. Fuhrer, R. Kelley and A.V. Manohar,
*Factorization structure of gauge theory amplitudes and application to hard scattering processes at the LHC, Phys. Rev.***D 80**(2009) 094013 [arXiv:0909.0012] [INSPIRE].ADSGoogle Scholar - [36]J. Gatheral,
*Exponentiation of eikonal cross-sections in non-Abelian gauge theories, Phys. Lett.***B 133**(1983) 90 [INSPIRE].MathSciNetADSGoogle Scholar - [37]J. Frenkel and J. Taylor,
*Non-Abelian eikonal exponentiation, Nucl. Phys.***B 246**(1984) 231 [INSPIRE].ADSCrossRefGoogle Scholar - [38]E. Laenen, G. Stavenga and C.D. White,
*Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP***03**(2009) 054 [arXiv:0811.2067] [INSPIRE].ADSCrossRefGoogle Scholar - [39]G. Korchemsky and A. Radyushkin,
*Renormalization of the Wilson loops beyond the leading order, Nucl. Phys.***B 283**(1987) 342 [INSPIRE].ADSCrossRefGoogle Scholar - [40]A.V. Manohar,
*Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev.***D 68**(2003) 114019 [hep-ph/0309176] [INSPIRE].ADSGoogle Scholar - [41]C. Arnesen, I.Z. Rothstein and J. Zupan,
*Smoking guns for on-shell new physics at the LHC, Phys. Rev. Lett.***103**(2009) 151801 [arXiv:0809.1429] [INSPIRE].ADSCrossRefGoogle Scholar - [42]T. Becher and M. Neubert,
*Drell-Yan production at small q*_{T}*, transverse parton distributions and the collinear anomaly, Eur. Phys. J.***C 71**(2011) 1665 [arXiv:1007.4005] [INSPIRE].ADSCrossRefGoogle Scholar - [43]S. Mantry and F. Petriello,
*Factorization and resummation of Higgs boson differential distributions in soft-collinear effective theory, Phys. Rev.***D 81**(2010) 093007 [arXiv:0911.4135] [INSPIRE].ADSGoogle Scholar - [44]Y. Gao, C.S. Li and J.J. Liu,
*Transverse momentum resummation for Higgs production in soft-collinear effective theory, Phys. Rev.***D 72**(2005) 114020 [hep-ph/0501229] [INSPIRE].ADSGoogle Scholar - [45]A. Idilbi, X.-D. Ji and F. Yuan
*, Transverse momentum distribution through soft-gluon resummation in effective field theory, Phys. Lett.***B 625**(2005) 253 [hep-ph/0507196] [INSPIRE].ADSGoogle Scholar - [46]J.C. Collins, D.E. Soper and G.F. Sterman,
*Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys.***B 250**(1985) 199 [INSPIRE].ADSCrossRefGoogle Scholar - [47]I. Hinchliffe and S. Novaes,
*On the mean transverse momentum of Higgs bosons at the SSC, Phys. Rev.***D 38**(1988) 3475 [INSPIRE].ADSGoogle Scholar - [48]R.P. Kauffman,
*Higgs boson p*_{T}*in gluon fusion, Phys. Rev.***D 44**(1991) 1415 [INSPIRE].ADSGoogle Scholar - [49]D. de Florian and M. Grazzini
*, Next-to-next-to-leading logarithmic corrections at small transverse momentum in hadronic collisions, Phys. Rev. Lett.***85**(2000) 4678 [hep-ph/0008152] [INSPIRE].ADSCrossRefGoogle Scholar - [50]D. de Florian and M. Grazzini
*, The structure of large logarithmic corrections at small transverse momentum in hadronic collisions, Nucl. Phys.***B 616**(2001) 247 [hep-ph/0108273] [INSPIRE].ADSCrossRefGoogle Scholar - [51]S. Catani and M. Grazzini
*, QCD transverse-momentum resummation in gluon fusion processes, Nucl. Phys.***B 845**(2011) 297 [arXiv:1011.3918] [INSPIRE].ADSCrossRefGoogle Scholar - [52]ATLAS collaboration,
*Combination of Higgs boson searches with up to*4.9 fb−1*of pp collisions data taken at a center-of-mass energy of 7 TeV with the ATLAS experiment at the LHC,*ATLAS-CONF-2011-163, CERN, Geneva Switzerland (2011).Google Scholar - [53]CMS collaboration,
*Combination of CMS searches for a Standard Model Higgs boson,*CMS-PAS-HIG-11-032, CERN, Geneva Switzerland (2011).Google Scholar - [54]T. Inami, T. Kubota and Y. Okada,
*Effective gauge theory and the effect of heavy quarks in Higgs boson decays, Z. Phys.***C18**(1983) 69 [INSPIRE].ADSGoogle Scholar - [55]A. Djouadi, M. Spira and P. Zerwas
*, Production of Higgs bosons in proton colliders: QCD corrections, Phys. Lett.***B 264**(1991) 440 [INSPIRE].ADSGoogle Scholar - [56]C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart
*, Hard scattering factorization from effective field theory, Phys. Rev.***D 66**(2002) 014017 [hep-ph/0202088] [INSPIRE].ADSGoogle Scholar - [57]A.V. Belitsky, X. Ji and F. Yuan
*, Final state interactions and gauge invariant parton distributions, Nucl. Phys.***B 656**(2003) 165 [hep-ph/0208038] [INSPIRE].ADSCrossRefGoogle Scholar - [58]X.-D. Ji and F. Yuan
*, Parton distributions in light cone gauge: where are the final state interactions?, Phys. Lett.***B 543**(2002) 66 [hep-ph/0206057] [INSPIRE].ADSGoogle Scholar - [59]I. Cherednikov and N. Stefanis
*, Renormalization, Wilson lines and transverse-momentum dependent parton distribution functions, Phys. Rev.***D 77**(2008) 094001 [arXiv:0710.1955] [INSPIRE].ADSGoogle Scholar - [60]I. Cherednikov and N. Stefanis
*, Wilson lines and transverse-momentum dependent parton distribution functions: a renormalization-group analysis, Nucl. Phys.***B 802**(2008) 146 [arXiv:0802.2821] [INSPIRE].ADSCrossRefGoogle Scholar - [61]I. Cherednikov and N. Stefanis
*, Renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription, Phys. Rev.***D 80**(2009) 054008 [arXiv:0904.2727] [INSPIRE].ADSGoogle Scholar - [62]A. Idilbi and I. Scimemi
*, Singular and regular gauges in soft collinear effective theory: the introduction of the new Wilson line T, Phys. Lett.***B 695**(2011) 463 [arXiv:1009.2776] [INSPIRE].ADSGoogle Scholar - [63]M. Garcia-Echevarria, A. Idilbi and I. Scimemi
*, SCET, light-cone gauge and the T-Wilson lines, Phys. Rev.***D 84**(2011) 011502 [arXiv:1104.0686] [INSPIRE].ADSGoogle Scholar - [64]J.C. Collins
*, What exactly is a parton density?, Acta Phys. Polon.***B 34**(2003) 3103 [hep-ph/0304122] [INSPIRE].ADSGoogle Scholar - [65]X.-D. Ji, J.-P. Ma and F. Yuan
*, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum, Phys. Rev.***D 71**(2005) 034005 [hep-ph/0404183] [INSPIRE].ADSGoogle Scholar - [66]F. Hautmann,
*Endpoint singularities in unintegrated parton distributions, Phys. Lett.***B 655**(2007) 26 [hep-ph/0702196] [INSPIRE].ADSGoogle Scholar - [67]S. Meissner, A. Metz and M. Schlegel
*, Generalized transverse momentum dependent parton distributions of the nucleon,*arXiv:0807.1154 [INSPIRE]. - [68]B. Pasquini, S. Cazzaniga and S. Boffi
*, Transverse momentum dependent parton distributions in a light-cone quark model, Phys. Rev.***D 78**(2008) 034025 [arXiv:0806.2298] [INSPIRE].ADSGoogle Scholar - [69]A. Jain, M. Procura and W.J. Waalewijn
*, Fully-unintegrated parton distribution and fragmentation functions at perturbative k*_{T}, arXiv:1110.0839 [INSPIRE]. - [70]I.W. Stewart, F.J. Tackmann and W.J. Waalewijn
*, Factorization at the LHC: from PDFs to initial state jets, Phys. Rev.***D 81**(2010) 094035 [arXiv:0910.0467] [INSPIRE].ADSGoogle Scholar - [71]S. Frixione, P. Nason and G. Ridolfi
*, Problems in the resummation of soft gluon effects in the transverse momentum distributions of massive vector bosons in hadronic collisions, Nucl. Phys.***B 542**(1999) 311 [hep-ph/9809367] [INSPIRE].ADSCrossRefGoogle Scholar - [72]A.V. Manohar and W.J. Waalewijn, A QCD analysis of double parton scattering: color correlations, interference effects and evolution, arXiv:1202.3794 [INSPIRE].
- [73]R.V. Harlander and W.B. Kilgore
*, Soft and virtual corrections to pp → H + x at NNLO, Phys. Rev.***D 64**(2001) 013015 [hep-ph/0102241] [INSPIRE].ADSGoogle Scholar - [74]R.P. Kauffman
*, Higgs boson p*_{T}*in gluon fusion, Phys. Rev.***D 44**(1991) 1415 [INSPIRE].ADSGoogle Scholar - [75]S.M. Aybat and T.C. Rogers
*, TMD parton distribution and fragmentation functions with QCD evolution, Phys. Rev.***D 83**(2011) 114042 [arXiv:1101.5057] [INSPIRE].ADSGoogle Scholar - [76]T. Becher and G. Bell
*, Analytic regularization in soft-collinear effective theory,*arXiv:1112.3907 [INSPIRE]. - [77]J. Collins
*, New definition of TMD parton densities, Int. J. Mod. Phys. Conf. Ser.***4**(2011) 85 [arXiv:1107.4123] [INSPIRE].CrossRefGoogle Scholar - [78]L.J. Dixon, L. Magnea and G.F. Sterman
*, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP***08**(2008) 022 [arXiv:0805.3515] [INSPIRE].ADSCrossRefGoogle Scholar - [79]G. Bozzi, S. Catani, D. de Florian and M. Grazzini,
*Higgs boson production at the LHC: transverse-momentum resummation and rapidity dependence, Nucl. Phys.***B 791**(2008) 1 [arXiv:0705.3887] [INSPIRE].ADSCrossRefGoogle Scholar - [80]J. Collins
*, Foundations of perturbative QCD,*Cambridge University Press, Cambridge U.K. (2011).CrossRefGoogle Scholar - [81]I.W. Stewart and I.Z. Rothstein
*, Glauber gluons in SCET, in SCET workshop*2010, Ringberg Germany (2010).Google Scholar - [82]M. Garcia-Echevarria, A. Idilbi and I. Scimemi
*, Factorization theorem for Drell-Yan at low q*_{T}*and transverse momentum distributions on-the-light-cone,*arXiv:1111.4996 [INSPIRE]. - [83]M. Dasgupta and G. Salam
*, Resummation of nonglobal QCD observables, Phys. Lett.***B 512**(2001) 323 [hep-ph/0104277] [INSPIRE].ADSGoogle Scholar - [84]R. Kelley, M.D. Schwartz, R.M. Schabinger and H.X. Zhu
*, The two-loop hemisphere soft function, Phys. Rev.***D 84**(2011) 045022 [arXiv:1105.3676] [INSPIRE].ADSGoogle Scholar - [85]A. Hornig, C. Lee, I.W. Stewart, J.R. Walsh and S. Zuberi
*, Non-global structure of the*\( O\left( {\alpha_s^2} \right) \)*dijet soft function, JHEP***08**(2011) 054 [arXiv:1105.4628] [INSPIRE].ADSCrossRefGoogle Scholar - [86]S. Kluth
*, Tests of quantum chromo dynamics at e+ e− colliders, Rept. Prog. Phys.***69**(2006) 1771 [hep-ex/0603011] [INSPIRE].ADSCrossRefGoogle Scholar - [87]C.F. Berger and G.F. Sterman
*, Scaling rule for nonperturbative radiation in a class of event shapes, JHEP***09**(2003) 058 [hep-ph/0307394] [INSPIRE].ADSCrossRefGoogle Scholar - [88]
- [89]S. Catani and L. Trentadue
*, Resummation of the QCD perturbative series for hard processes, Nucl. Phys.***B 327**(1989) 323 [INSPIRE].ADSCrossRefGoogle Scholar - [90]A. Hornig, C. Lee and G. Ovanesyan
*, Effective predictions of event shapes: factorized, resummed and gapped angularity distributions, JHEP***05**(2009) 122 [arXiv:0901.3780] [INSPIRE].ADSCrossRefGoogle Scholar - [91]R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart,
*Thrust at N*^{3}*LL with power corrections and a precision global fit for α*_{s}*(m*_{Z}*), Phys. Rev.***D 83**(2011) 074021 [arXiv:1006.3080] [INSPIRE].ADSGoogle Scholar - [92]S. Fleming, A.H. Hoang, S. Mantry and I.W. Stewart
*, Jets from massive unstable particles: top-mass determination, Phys. Rev.***D 77**(2008) 074010 [hep-ph/0703207] [INSPIRE].ADSGoogle Scholar - [93]C.W. Bauer, C. Lee, A.V. Manohar and M.B. Wise
*, Enhanced nonperturbative effects in Z decays to hadrons, Phys. Rev.***D 70**(2004) 034014 [hep-ph/0309278] [INSPIRE].ADSGoogle Scholar - [94]L3 collaboration, P. Achard et al.,
*Studies of hadronic event structure in e+ e− annihilation from*30*GeV to*209*GeV with the L*3*detector, Phys. Rept.***399**(2004) 71 [hep-ex/0406049] [INSPIRE].ADSCrossRefGoogle Scholar - [95]T. Becher, G. Bell and M. Neubert
*, Factorization and resummation for jet broadening, Phys. Lett.***B 704**(2011) 276 [arXiv:1104.4108] [INSPIRE].ADSGoogle Scholar - [96]J.-Y. Chiu, F. Golf, R. Kelley and A.V. Manohar
*, Electroweak corrections in high energy processes using effective field theory, Phys. Rev.***D 77**(2008) 053004 [arXiv:0712.0396] [INSPIRE].ADSGoogle Scholar - [97]B. Geshkenbein and M. Terentev
*, The enhanced power correction to the asymptotics of the pion form-factor, Phys. Lett.***B 117**(1982) 243 [INSPIRE].ADSGoogle Scholar - [98]V. Chernyak and A. Zhitnitsky
*, Asymptotic behavior of exclusive processes in QCD, Phys. Rept.***112**(1984) 173 [INSPIRE].ADSCrossRefGoogle Scholar - [99]R. Akhoury, G.F. Sterman and Y. Yao,
*Exclusive semileptonic decays of B mesons into light mesons, Phys. Rev.***D 50**(1994) 358 [INSPIRE].ADSGoogle Scholar - [100]C.W. Bauer, D. Pirjol, I.Z. Rothstein and I.W. Stewart,
*B → M*_{1}*M*_{2}*: factorization, charming penguins, strong phases and polarization, Phys. Rev.***D 70**(2004) 054015 [hep-ph/0401188] [INSPIRE].ADSGoogle Scholar - [101]C.W. Bauer, I.Z. Rothstein and I.W. Stewart
*, SCET analysis of B → K π, B →*\( {\text{K}}\overline {\text{K}} \)*B → ππ decays, Phys. Rev.***D 74**(2006) 034010 [hep-ph/0510241] [INSPIRE].ADSGoogle Scholar - [102]G.P. Lepage and S.J. Brodsky
*, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev.***D 22**(1980) 2157 [INSPIRE].ADSGoogle Scholar - [103]G. Korchemsky and G. Marchesini
*, Structure function for large X and renormalization of Wilson loop, Nucl. Phys.***B 406**(1993) 225 [hep-ph/9210281] [INSPIRE].ADSCrossRefGoogle Scholar - [104]A.H. Hoang, A.V. Manohar and I.W. Stewart
*, The running Coulomb potential and Lamb shift in QCD, Phys. Rev.***D 64**(2001) 014033 [hep-ph/0102257] [INSPIRE].ADSGoogle Scholar - [105]Z. Ligeti, I.W. Stewart and F.J. Tackmann,
*Treating the b quark distribution function with reliable uncertainties, Phys. Rev.***D 78**(2008) 114014 [arXiv:0807.1926] [INSPIRE].ADSGoogle Scholar