In the $k_T$-factorization for exclusive processes, the nontrivial $k_T$-dependence of perturbative coefficients, or hard parts, is obtained by taking off-shell partons. This brings up the question of whether the $k_T$-factorization is gauge invariant. We study the $k_T$-factorization for the case $πγ^* \to γ$ at one-loop in a general covariant gauge. Our results show that the hard part contains a light-cone singularity that is absent in the Feynman gauge, which indicates that the $k_T$-factorization is {\it not} gauge invariant. These divergent contributions come from the $k_T$-dependent wave function of $π$ and are not related to a special process. Because of this fact the $k_T$-factorization for any process is not gauge invariant and is violated. Our study also indicates that the $k_T$-factorization used widely for exclusive B-decays is not gauge invariant and is violated.
This talk summarized the proof of hard-scattering factorization for hard exclusive electroproduction processes: deeply virtual Compton scattering and exclusive meson production.
We briefly review the calculational procedure for the PQCD prediction for hard exclusive quantities and reconsider the problem of the factorization scale dependence.
The effective theories for massless quarks describing exclusive and seminclusive processes are discussed, considering in particular the factorization problem.
I review the theoretical ideas and concepts along the line of factorization in the exclusive B decays. In order to understand the naive factorization, the effective field theories and the perturbative method of QCD are introduced and developed. We focus our discussions on the large energy effective theory, the QCD factorization approach and the soft-collinear effective theory.
I review recent progress on exclusive $B$ meson decays made in the perturbative QCD approach, concentrating on the evolution of the $B$ meson wave function in $k_T$ factorization, radiative decays, polarizations in $VV$ modes, and new physics effect in $B\to ϕK_S$.
In this paper, we study the application of the recently proposed soft gluon factorization (SGF) to exclusive quarkonium production or decay. We find that in the nonrelativistic QCD factorization framework there are too many nonperturbative parameters. Thanks to the factorization of kinematical physics from dynamical physics, the SGF significantly reduces the number of nonperturbative parameters. Therefore, the SGF can improve our predictive power of exclusive quarkonium production or decay. By applying to $η_c+γ$ production at B-factories, our result is the closest one to data among all theore…
I compare collinear and $k_T$ factorization theorems in perturbative QCD, and discuss their application to exclusive $B$ meson decays. Especially, I concentrate on the recently measured time-dependent CP asymmetry of the $B^0_d\toπ^+π^-$ modes, from which the unitarity angle can be extracted.
In the following, we will present a generalization of the proof of factorization for electroproduction of vector mesons to the production of several mesons such as $π^+π^-$ and a proof of factorization for exclusive $γ^* γ$ production of several hadrons for example $p \bar p$ or $π^+π^-$ to all orders in perturbation theory up to power suppressed terms.
The exclusive process e+ e- -> e+ e- rho^0_L rho^0_L allows to study various dynamics and factorization properties of perturbative QCD. At moderate energy, we demonstrate how collinear QCD factorization emerges, involving generalized distribution amplitudes (GDA) and transition distribution amplitudes (TDA). At higher energies, in the Regge limit of QCD, we show that it offers a promising probe of the BFKL resummation effects to be studied at the International Linear Collider (ILC).
The exclusive process e+ e- -> e+ e-rho0_L rho0_L allows to study various dynamics and factorization properties of perturbative QCD. At moderate energy, we demonstrate how collinear QCD factorization emerges, involving either generalized distribution amplitudes (GDA) or transition distribution amplitudes (TDA). At higher energies, in the Regge limit of QCD, we show that it offers a promising probe of the BFKL resummation effects to be studied at ILC.
The recent LHCb data for exclusive $J/ψ$ peripheral production at 13 TeV motivate an improved `NLO' analysis to estimate the gluon distribution at low $x$ in which we re-calculate the rapidity gap survival factors and use a more precise expression for the photon flux. We comment on the difference between the $k_T$ and collinear factorization approaches.
We apply the QCD factorization approach to exclusive, radiative B meson decays in the region of small invariant photon mass. We calculate factorizable and non-factorizable corrections to leading order in the heavy quark mass expansion and next-to-leading order in the strong coupling constant. Phenomenological consequences for the B -> K* gamma decay rate and the B -> K* ell+ ell- forward-backward asymmetry are discussed.
Exclusive electroproduction of hard dijets can be described within the collinear factorization. This process has clear experimental signature and provides one with an interesting alternative venue to test QCD description of hard diffractive processes and extract information on generalized nucleon parton distributions. In this work we present detailed leading-order QCD calculations of the relevant cross sections, including longitudinal momentum fraction distribution of the dijets and their azimuthal angle dependence.
In the framework of QCD factorization, we study the $B$ meson exclusive decays $B\to χ_{cJ}K$ where the spin-triplet P-wave charmonium states $χ_{cJ}(J=0,1,2)$ are described by the color-singlet non-relativistic wave functions. We find that for these decays (except $χ_{c1}$) there are infrared divergences arising from nonfactorizable vertex corrections as well as logarithmic end-point singularities arising from nonfactorizable spectator interactions at leading-twist order. The infrared divergences due to vertex corrections will explicitly break down QCD factorization within the color-singlet m…
We investigate the possibility of testing factorization hypothesis in non-leptonic exclusive decays of B-meson. In particular, we considered the non factorizable \bar{B^0} -> D^{(*)+} D_s^{(*)-} modes and \bar{B^0} -> D^{(*)+} (π^-, ρ^-) known as well-factorizable modes. By taking the ratios BR(\bar{B^0}-> D^{(*)+}D_s^{(*)-})/BR(\bar{B^0}-> D^{(*)+}(π^-,ρ^-)), we found that under the present theoretical and experimental uncertainties there's no evidence for the breakdown of factorization description to heavy-heavy decays of the B meson.
We derive a new factorization relation for the semileptonic radiative decay B -> π\ell νγin the kinematical region of a slow pion p_π~ Λand an energetic photon E_γ>> Λ, working at leading order in Λ/m_b. In the limit of a soft pion, the nonperturbative matrix element appearing in this relation can be computed using chiral perturbation theory. We present a phenomenological study of this decay, which may be important for a precise determination of the exclusive nonradiative decay.
We describe hard exclusive processes involving a transversally polarized rho meson in the twist 3 approximation, in a framework based on the Taylor expansion of the amplitude around the dominant light-cone direction.
We outline the proofs of the factorization theorems for exclusive two-body charmonium production in B-meson decay and e^+e^- annihilation to all orders in perturbation theory in quantum chromodynamics. We find that factorized expressions hold up to corrections of order m_c/m_b in B-meson decay and corrections of order m_c^2/s in e^+e^- annihilation, where m_c is the charm-quark mass, m_b is the bottom-quark mass, and root-s is the e^+e^- center-of-momentum energy.
The k_T-factorization has been widely used for exclusive decays of B-mesons. In this factorization the pertubative coefficients are extracted from scattering of off-shell partons. Because the off-shellness of partons the extracted pertubative coefficients in general are not gauge-invariant. We show that these perturbative coefficients contain gauge-dependent singularities beyond tree-level. This leads to that the k_T-factorization is gauge-dependent and violated in the general covariant gauge. This is not consistent with expectations of physics. In the factorization there exist two different w…
Exclusive, non-leptonic, two-body decays of B mesons simplify greatly in the heavy quark limit. In this talk I discuss the factorized structure that holds in this limit and some of its consequences: (a) naive factorization is recovered in a certain limit; (b) `non-factorizable' effects are hard and can be calculated; (c) strong interaction phases vanish in the heavy quark limit (and can be calculated as well). As an illustration, I compute the penguin contribution to the decay $B_d\toπ^+π^-$ and its effect on the determination of $\sin 2α$.
Within the archetypal factorization procedure, we study the gluon pole contributions manifesting in the nucleon-lepton hard processes of hadron production. We prove the dominant role of gluon pole contributions in the processes of such kind. We analyse the different sources of complexity associated with the gluon pole functions. As a practical application, we derive a new single spin asymmetry related directly to the gluon poles which can be studied experimentally.
We prove rigorously factorization in the seminclusive decay $B\to D^{(*)} + jet$ using the large energy effective theory. It is also shown that this effective theory is unable to consistently describe completely exclusive processes, such as the decay $B\to D^{(*)}+π(ρ)$, and therefore also related properties such as factorization. This is due to an oversimplification of transverse momentum dynamics. We present a variant of the large energy effective theory, i.e. a new effective theory for massless particles, which properly takes into account transverse momentum dynamics and is therefore the na…
We prove $k_T$ factorization theorem in perturbative QCD (PQCD) for exclusive processes by considering $πγ^*\to γ(π)$ and $B\toγ(π) l\barν$. The relevant form factors are expressed as the convolution of hard amplitudes with two-parton meson wave functions in the impact parameter $b$ space, $b$ being conjugate to the parton transverse momenta $k_T$. The point is that on-shell valence partons carry longitudinal momenta initially, and acquire $k_T$ through collinear gluon exchanges. The $b$-dependent two-parton wave functions with an appropriate path for the Wilson links are gauge-invariant. The …
A new method is proposed to calculate wave functions in $k_T$-factorization in \cite{LiMi} as a comment about our paper \cite{FMW}. We point out that the results obtained with the method are in conflict with the translation invariance and depend on the chosen contours for loop-integrals. Therefore, the method is in principle unacceptable and the results with the method cannot be correct.
We provide a rigorous basis for factorization for a large class of non-leptonic two-body $B$-meson decays in the heavy-quark limit. The factorization formula incorporates elements of the naive factorization approach and the hard-scattering approach, but allows us to compute systematically radiative (``non-factorizable'') corrections to naive factorization for decays such as $B\to Dπ$ and $B\to ππ$. We discuss the factorization formula for a general final state from a general point of view. We then consider factorization for decays into heavy-light final states (such as $B\to Dπ$) in more detai…
In this paper we analyze the exclusive photoproduction of heavy quarkonia pairs in the collinear factorization framework. We evaluate the amplitude of the process for $J/ψ\,-η_{c}$ quarkonia pair in the leading order of the strong coupling $α_{s},$ and express it in terms of generalized parton distributions (GPDs) of gluons in the proton. We made numerical estimates in the kinematics of the Electron Ion Collider, and found that in the photoproduction regime, when the virtuality of the photon is much smaller than the quarkonia mass, the cross-section of the process is sufficiently large for exp…
We calculate cross sections for the exclusive diffractive leptoproduction of $ρ$-mesons, $γ^*~p~\to~ρ~p$, within the framework of high-energy factorization. Cross sections for longitudinally and transversally polarized mesons are shown. We employ a wide variety of unintegrated gluon distributions available in the literature and compare to HERA data. The resulting cross sections strongly depend on the choice of unintegrated gluon distribution. We also present predictions for the proton target in the kinematics of the Brookhaven EIC.
We study the exclusive decays of $B$ meson into P-wave charmonium states $χ_{cJ}(J=0,1)$ in the QCD factorization approach with light-cone distribution functions describing the mesons in the processes. For $B \to χ_{c1} K$ decay, we find that there are logarithmic divergences arising from nonfactorizable spectator interactions even at twist-2 order and the decay rate is too small to accommodate the experimental data. For $B\to χ_{c0} K$ decay, we find that aside from the logarithmic divergences arising from spectator interactions at leading-twist order, more importantly, the factorization will…
We point out mistakes made in the one-loop calculation of some diagrams for the process $πγ^* \to γ$ in the preprint arXiv:0807.0296, and present correct results. Especially, we have difficulty to understand their argument that a highly off-shell gluon generates a light-cone (infrared) singularity. It is shown by means of the Ward identity that the gauge-dependent light-cone singularity found in arXiv:0807.0296 does not exist. It is then shown that a hard kernel derived in the $k_T$ factorization of exclusive processes is gauge invariant and free of the light-cone singularity.
We develop and discuss an infrared-finite factorization and optimized renormalization scheme for calculating exclusive processes which enables the inclusion of transverse degrees of freedom without entailing suppression of calculated observables, like form factors. This is achieved by employing an analytic, i.e., infrared stable, effective coupling $α_{s}(Q^{2})$ which removes the Landau singularity at $Q^{2}=Λ_{\rm QCD}^{2}$ by a power-behaved correction. The ensuing contributions to the cusp anomalous dimension, related to the Sudakov form factor, and to the quark anomalous dimension, which …
Exclusive processes in hard electroproduction is one of the best place for understanding the factorization properties of QCD. The HERA experiment recently provided precise data for rho electroproduction, including all spin density matrix elements. From QCD, it is expected that such a process should factorize between a hard (calculable) coefficient function, and hadronic (p and rho) matrix elements. Such a factorization is up to now only proven for a longitudinaly polarized rho. Within the kt-factorization approach (valid at large s_{gamma* p}), we evaluate the impact factor of the transition g…
The structure of the nonperturbative vector meson vertex function complicates the proof of the factorisation theorem for the reaction $γ^*p\to Vp$. It leads to additional contributions but, in a simple model for the vertex function, gauge invariance ensures that they cancel and factorisation is preserved.
We study the exclusive production of $J^{PC}=0^{++}, 0^{--}$ charmonium states in proton-proton collisions at the LHC energies The $pp \to ppη_c$ reaction is discussed for the first time. We observe a substantial contribution from the nonperturbative domain of gluon virtualities, especially for $η_c$ production. To model the nonperturbative region better, we utilize models of the unintegrated gluon distribution based on parametrizations of the color dipole cross-section.
Belle has observed surprisingly copious production of $ψ(3770)$ in $B$ meson decay $B\to ψ(3770)K$, of which the rate is comparable to that of $B\to ψ(3686)K$. We study this puzzling process in the QCD factorization approach with the effect of S-D mixing considered. We find that the soft scattering effects in the spectator interactions play an essential role. With a proper parametrization for the higher twist soft end-point singularities associated with kaon, and with the S-D mixing angle $θ=-12^{\circ}$, the calculated decay rates can be close to the data. Implications of these soft spectator…
The exclusive processes $e^+e^-\to VP$, in the region of which the final state meson momentum is much larger than the hadronic scale $Λ_{QCD}$, are studied in the framework of PQCD approach based on the $k_T$ factorization. Including the transverse momentum distribution in the light cone wave functions, our results are consistent with the experimental measurements. According to our results, many processes have large enough cross sections to be detected at $\sqrt s=10.58$ GeV. The $s$ dependence of the cross section has been directly studied and our result indicates that the $1/s^3$ scaling is …
We outline a proof of factorization in exclusive processes, taking into account the presence of soft and collinear modes of arbitrarily low energy, which arise when the external lines of the process are taken on shell. Specifically, we examine the process of e^+e^- annihilation through a virtual photon into two light mesons. In an intermediate step, we establish a factorized form that contains a soft function that is free of collinear divergences. In contrast, in soft-collinear effective theory, the low-energy collinear modes factor most straightforwardly into the soft function. We point out t…
We formulate and prove a QCD factorization theorem for hard exclusive electroproduction of mesons in QCD. The proof is valid for the leading power in Q and all logarithms. This generalizes previous work on vector meson production in the diffractive region of small x. The amplitude is expressed in terms of off-diagonal generalizations of the usual parton densities. The full theorem applies to all kinds of meson and not just to vector mesons. The parton densities used include not only the ordinary parton density, but also the helicity density (g_1 or $Δq$) and the transversity density ($h_1$ or …
We present factorization theorems for two exclusive heavy-quarkonium production processes: production of two quarkonia in e^+e^- annihilation and production of a quarkonium and a light meson in B-meson decays. We describe the general proofs of factorization and supplement them with explicit one-loop analyses, which illustrate some of the features of the soft-gluon cancellations. We find that violations of factorization are generally suppressed relative to the factorized contributions by a factor v^2m_c/Q for each S-wave charmonium and a factor m_c/Q for each L-wave charmonium with L>0. Here, v…
We study analytical properties of the hard exclusive process amplitudes. We found that QCD factorization for deeply virtual Compton scattering and hard exclusive vector meson production results in the subtracted dispersion relation with the subtraction constant determined by the Polyakov-Weiss $D$-term.
Factorization violating effects in hadron scattering are due mainly to spectator-spectator interactions. While it is known that these interactions cancel in inclusive cross sections, like for the Drell-Yan process, not much is known about for what classes of observables factorization is violated. We show that for pure Glauber ladder graphs, all amplitude-level factorization violating effects completely cancel at cross section level for any single-scale observable (such as hadronic transverse energy or beam thrust). This result disproves previous claims that these pure Glauber graphs are factor…
We calculate next-to-leading-order (NLO) corrections to exclusive processes in $k_T$ factorization theorem, taking $πγ^*\toγ$ as an example. Partons off-shell by $k_T^2$ are considered in both the quark diagrams from full QCD and the effective diagrams for the pion wave function. The gauge dependences in the above two sets of diagrams cancel, when deriving the $k_T$-dependent hard kernel as their difference. The gauge invariance of the hard kernel is then proven to all orders by induction. The light-cone singularities in the $k_T$-dependent pion wave function are regularized by rotating the Wi…
Usually, B->pi+pi- decays are expressed in terms of weak amplitudes explicitly dependent on the CKM weak phase alpha or gamma. In this letter, we show that the weak amplitudes can be rewritten such that a manifest dependence on beta emerges instead. Based on this, we constrain new-physics contributions to the CP-violating phase phi_d in B_0 anti-B_0 mixing. Further, we apply reparametrization invariance and use QCD factorization predictions to investigate the bounds on an additional new-physics amplitude in B->pi+pi-.
We address the question as to whether data for J/ψmesons produced exclusively in the forward direction at the LHC can be used in global parton analyses (based on collinear factorization) to pin down the low x gluon PDF. We show that it may be possible to overcome the problems that (i) the process is described by a skewed or Generalized Parton Distribution (GPD), (ii) it is very sensitive to the choice of factorization scale and (iii) there is bad LO, NLO,... perturbative stability to the predictions. However, we start by briefly explaining how the alternative k_T factorization approach has bee…
We prove factorization and gauge invariance of the twist-3 single-spin-dependent cross section in the leading order perturbative QCD, which was missing in the previous literature. We emphasize that the consistency relation from the Ward identities for color gauge invariance is crucial to guarantee the cancelation among the various gauge-noninvariant contributions in the cross section. This relation also proves the absence of the "derivative" terms in the cross section corresponding to the hard-pole and soft-fermion-pole contributions. Applying the formalism to SIDIS, $ep^\uparrow\to eπX$, we h…
We describe a consistent approach to factorization of scattering amplitudes for exclusive processes beyond the leading twist approximation. The method involves the Taylor expansion of the scattering amplitude in the momentum space around the dominant light-cone direction and thus naturally introduces an appropriate set of non-perturbative correlators which encode effects not only of the lowest but also of the higher Fock states of the produced particle. The reduction of original set of correlators to a set of independent ones is achieved with the help of equations of motion and invariance of t…
We study exclusive production of scalar $χ_{c0}\equiv χ_c(0^{++})$ and pseudoscalar $η_c$ charmonia states in proton-proton collisions at the LHC energies. The amplitudes for $gg \to χ_{c0}$ as well as for $gg \to η_c$ mechanisms are derived in the $k_{T}$-factorization approach. The $p p \to p p η_c$ reaction is discussed for the first time. We have calculated rapidity, transverse momentum distributions as well as such correlation observables as the distribution in relative azimuthal angle and $(t_1,t_2)$ distributions. The latter two observables are very different for $χ_{c0}$ and $η_c$ case…
We discuss the very rare, exclusive hadronic decays of a Z boson into a meson and a photon. The QCD factorization approach allows to organize the decay amplitude as an expansion in powers of $Λ_{\rm QCD}/m_Z\,$, where the leading terms contain convolutions of perturbatively calculable hard functions with the leading-twist light-cone distribution amplitudes of the meson. We find that power corrections to these leading terms are negligible since they are suppressed by the small ratio $(Λ_{\rm QCD}/m_Z)^2\,$. Renormalization-group effects play a crucial role as they render our theoretical predict…
The theoretical framework of hard exclusive reactions is reviewed with special emphasis on the Elfe project program. Perturbative QCD studies have shown that factorization properties allow to separate well-defined non perturbative objects which are crucial in the understanding of confinement dynamics from perturbatively calculable hard processes. The applicability of this factorization in a definite energy domain is controlled by some definite statements, as the dimensional counting rules, the helicity conservation law and the appearance of color transparency. The few data available indicate t…
We demonstrate the consistency at the next-to-leading-logarithmic (NLL) level of a factorization theorem based on Soft-Collinear Effective Theory (SCET) for jet shapes in e+e- collisions. We consider measuring jet observables in exclusive multijet final states defined with cone and k_T-type jet algorithms. Consistency of the factorization theorem requires that the renormalization group evolution of hard, jet, and soft functions is such that the physical cross-section is independent of the factorization scale mu. The anomalous dimensions of the various factorized pieces, however, depend on the …
We present the main results of our recent analyses of exclusive production of vector charmonia ($J/ψ$ and $ψ'$) in $k_t$-factorization approach and for $γγ$ production of charged dilepton pairs in exclusive and semiinclusive processes in a new approach, similar in spirit to $k_t$-factorization. The results for charmonia are compared with recent results of the LHCb collaboration. We include some helicity flip contributions and quantify the effect of absorption correction. The effect of $c \bar c$ wave function is illustrated. We present uncertainties related to $F_2$ structure function which ar…
Exclusive electroproduction of vector mesons is a crucial way to probe QCD factorization properties. Such a factorization is up to now only proven, at the twist 2 level, for a longitudinaly polarized meson. It is crucial to extend our understanding to the case of transversely polarized vector mesons. As a first step in this direction, we evaluate the impact factor of the transition gamma* -> rhoT, which is the relevant part of the amplitude within the kt-factorization approach valid at large energies, taking into account the twist 3 contributions, coming both from quark antiquark and from quar…
Using the renormalon calculus, we study the asymptotic behavior of the perturbative expansion of the hard-scattering kernels entering the QCD factorization formula for the nonleptonic weak decays B->D M, where M is a light meson. In the ``large-beta_0 limit'', the kernels are infrared finite and free of endpoint singularities to all orders of perturbation theory. The leading infrared renormalon singularity corresponding to a power correction of order Lambda/m_b vanishes if the light meson has a symmetric light-cone distribution amplitude. We calculate the Borel transforms and the corresponding…
We present a detailed computation of the fully exclusive cross section of p + antip --> W + photon + X with X = 0 and 1 jet in the framework of the factorization theorem and dimensional regularization. Order alpha-strong and photon bremsstrahlung contributions are discussed in the MS-bar mass factorization scheme. The resulting expressions are ready to be implemented numerically using Monte Carlo techniques to compute single and double differential cross sections and correlations between outgoing pairs of particles.
We discuss exclusive radiative decays in QCD factorization within the Standard Model. In particular, we consider the decays B -> V gamma, with a vector meson K* or rho in the final state, and the double radiative modes B_s -> gamma gamma and B_d -> gamma gamma. At quark level, all these decays are governed by the flavour-changing neutral-current b -> s gamma or b -> d gamma transitions, which appear at the one-loop level in the Standard Model. Such processes allow us to study CP violation and the interplay of strong and electroweak interactions, to determine parameters of the CKM matrix, and t…
We study the factorization of the dijet cross section in $e^+ e^-$ annihilation using the generalized exclusive jet algorithm which includes the cone-type, the JADE, the $k_T$, the anti-$k_T$ and the Cambridge/Aachen jet algorithms as special cases. In order to probe the characteristics of the jet algorithms in a unified way, we consider the generalized $k_T$ jet algorithm with an arbitrary weight of the energies, in which various types of the $k_T$-type algorithms are included for specific values of the parameter. We show that the jet algorithm respects the factorization property for the para…
We present the detailed analysis of genuine twist-3 contributions to the exclusive electroproduction amplitude of transversely polarized vector mesons. Using the formalism based on the QCD factorization in the momentum representation we calculated all the genuine twist-3 terms and found the total expression of this amplitude at 1/Q level. Generally speaking, these terms violate standard factorization owing to the existence of the infrared divergencies in the amplitude of hard sub-processes, although the strongest divergencies cancel due to the QCD equations of motion. We discuss the possible t…
