Factorization model for distributions of quarks in hadrons
B. I. Ermolaev, M. Greco, S. I. Troyan
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18 pp, 7 figures. In the present version we include a discussion on rapidity divergences, correct some unclearly presented issues and eliminate misprints
We consider distributions of unpolarized (polarized) quarks in unpolarized (polarized) hadrons. Our approach is based on QCD factorization. We begin with study of Basic factorization for the parton-hadron scattering amplitudes in the forward kinematics and suggest a model for non-perturbative contributions to such amplitudes. This model is based on the simple observation: after emitting an active quark by the initial hadron, the remaining set quarks and gluons becomes unstable, so description of this colored state can approximately be done in terms of resonances, which leads to expressions of the Breit-Wigner type. for non-perturbative contributions to the distributions of unpolarized and polarized quarks in the hadrons. Then we reduce these formulae to obtain explicit expressions for the quark-hadron scattering amplitudes and quark distributions in K_T- and Collinear factorizations.
Recent works have discussed the violation of factorization and universality in hadronic hard scattering processes aimed at measurements of T-odd distributions. We use simple arguments to show that it is possible to restore an approximate factorization involving T-odd contributions if the factorization breaking interactions present a frequency spectrum dominated by by a narrow and regular peak whose maximum value corresponds to a respected factorization.
We show that hard-scattering factorization is violated in the production of high-p_T hadrons in hadron-hadron collisions, in the case that the hadrons are back-to-back, so that k_T factorization is to be used. The explicit counterexample that we construct is for the single-spin asymmetry with one beam transversely polarized. The Sivers function needed here has particular sensitivity to the Wilson lines in the parton densities. We use a greatly simplified model theory to make the breakdown of factorization easy to check explicitly. But the counterexample implies that standard arguments for fact…
Proof of transverse-momentum-dependent(TMD) factorization for hadron-hadron collision is given in this paper. We focus on processes without detected soft final hadrons or detected final hadrons that are collinear to initial hadrons. This contradicts the widely accepted viewpoint that TMD- factorization does not hold in such processes even in the generalized sense. The key point is that singular points of the type l+ = 0 can be absorbed into Wilson lines of soft gluons, where l is collinear to the plus direction. Thus one should subtract such singular points from the collinear region of l. Afte…
We study factorization of processes involving two fragmentation functions in the case of very small transverse momenta. We consider two-hadron production in inclusive $e^+ e^-$ annihilation and demonstrate a new simple and illustrative method of factorization for such processes including leading order $α_S$ corrections.
The basic ideas of kt-factorization and CCFM parton evolution is discussed. The unintegrated gluon densities, obtained from CCFM fits to the proton structure function data at HERA are used to predict hadronic final state cross sections like jet production at HERA, but also comparisons with recent measurements of heavy quark production at the Tevatron are presented. Finally, the kt-factorization approach is applied to Higgs production at high energy hadron hadron colliders and the transverse momentum spectrum of Higgs production at the LHC is calculated.
In a recent paper (arXiv:1403.5221), Kang et al.proposed the so-called "rapidity factorization" for the single inclusive forward hadron production in pA collisions. We point out that the leading small-x logarithm was mis-identified in this paper, and hence the newly added next-to-leading order correction term is unjustified and should be absent in view of the small-x factorization.
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.
We present predictions for the diffractive production of heavy quarks in deep-inelastic scattering and hadron-hadron collisions and for double diffractive dijet production in hadron-hadron collisions. With the assumption of hard scattering factorization, the predictions are made using diffractive parton densities that we have previously fitted to HERA data. Comparisons of our predictions with preliminary data from HERA and the Tevatron indicate that factorization is obeyed in diffractive DIS but fails in hadron-induced processes.
We discuss the applicability of the $k_T$ factorization theorem to low-$p_T$ hadron production in hadron-hadron collision in a simple toy model, which involves only scalar particles and gluons. It has been shown that the $k_T$ factorization for high-$p_T$ hadron hadroproduction is broken by soft gluons in the Glauber region, which are exchanged among a transverse-momentum-dependent (TMD) parton density and other subprocesses of the collision. We explain that the contour of a loop momentum can be deformed away from the Glauber region at low $p_T$, so the above residual infrared divergence is fa…
It is well-established for the vacuum case that in the limit of a large number of colors N_c the four-quark condensates factorize into products of the two-quark condensate. It is shown that in the combined large-N_c and linear-density approximation four-quark condensates do not factorize in a medium of pions (finite temperature system) but do factorize in a medium of nucleons (nuclear system).
We present hadron-level predictions from the Monte-Carlo generator CASCADE and numerical calculations of beauty quark and quarkonium production in the framework of the kt-factorization approach for LHC energies. Our predictions are compared with the CMS experimental data.
We present a factorization formula for valence quark distributions in a hadron in x-->1 limit. For the example of pion, we arrive at the form of factorization by analyzing momentum flow in the leading and high-order Feynman diagrams. The result confirms the well-known 1-x scaling rule to all orders in perturbation theory, providing the non-perturbative matrix elements for the infrared-divergence factors. We comment on re-summation of perturbative single and double logarithms in 1-x.
A previous counterexample to disprove k_T-factorization for H_1+H_2 --> H_3+H_4+X is extended calculationally to one higher order in gluon exchange. The result is that, by explicit calculation, standard k_T-factorization fails for the unpolarized cross-section for the production of hadrons of high transverse momentum in hadron-hadron collisions.
Inclusive production of a quarkonium $η_{c,b}$ in hadron collisions at low transverse momentum can be used to extract various Transverse-Momentum-Dependent(TMD) gluon distributions of hadrons, provided the TMD factorization for the process holds. The factorization involving unpolarized TMD gluon distributions of unpolarized hadrons has been examined with on-shell gluons at one-loop level. In this work we study the factorization at one-loop level with diagram approach in the most general case, where all TMD gluon distributions at leading twist are involved. We find that the factorization holds …
We present the calculation of the finite part of the heavy quark impact factor at next-to-leading logarithmic accuracy in a form suitable for phenomenological studies such as the calculation of the cross-section for single bottom quark production at the LHC within the kT-factorization scheme.
Description of hadronic reactions at high energies is conventionally done on basis of QCD factoriza- tion so that factorization convolutions involve non-perturbative inputs mimicking non-perturbative contributions and perturbative evolution of those inputs. We construct the inputs for the gluon- hadron scattering amplitudes in the forward kinematics and, using the Optical theorem, convert them into inputs for gluon distributions in the both polarized and unpolarized hadrons. Firstly, we derive general mathematical criteria which any model for the inputs should obey and then suggest a Resonance…
We examine the violation of the kT factorization approximation for quark production in high energy proton-nucleus collisions. We comment on its implications for the open charm and quarkonium production in collider experiments.
We compare the numerical predictions of the k_T-factorization approach (semihard theory) for heavy quark production in high energy nucleon-nucleon and photon-nucleon collisions with the experimental data from Tevatron-collider and HERA. Predictions for heavy quark production at Tevatron, LHC and HERA are also presented.
A detailed proof of hard scattering factorization is given with the inclusion of heavy quark masses. Although the proof is explicitly given for deep-inelastic scattering, the methods apply more generally The power-suppressed corrections to the factorization formula are uniformly suppressed by a power of Λ/Q, independently of the size of heavy quark masses, M, relative to Q.
Four-quark correlators and the factorization hypothesis are analyzed in the meson sector within Chiral Perturbation Theory. Factorization of the four-quark condensate holds to leading and next to leading order. However at next to next to leading order, a term with a nontrivial space-time dependence in the four-quark correlator yields a divergent four-quark condensate, whereas the two-quark condensate and the scalar susceptibility are finite. Such a non-factorization term vanishes only in the chiral limit. We also comment on how factorization still holds in the large $N_c$ limit, provided such …
We consider the hadronic description of the $B^0_d\to π^+π^-$ decay, with the aim to investigate the strong phases generated by the final state interactions. The derivation of the dispersion relations using the Lehmann-Symanzik-Zimmermann formalism and the Goldberger-Treiman method to include inelastic effects in the spectral function are presented. We discuss the problem of quark-hadron duality and estimate in the hadronic formalism the corrections to the factorized amplitude in the heavy quark limit.
In this work, we calculate the branching ratios and CP asymmetries of the decays of $B \to ππ$ and $πK$ in the frame of QCD factorization in the heavy quark limit. We also compare the results with the estimates by using conventional factorization and experimental measurements.
A new version of the $k_T$ factorization approach is formulated for the high energy heavy quark production. The results are in reasonable agreement with the experimental data at LHC energies.
We present a new analysis of two-jet event shape distributions in soft collinear effective theory. Extending previous results, we observe that a large class of such distributions can be expressed in terms of vacuum matrix elements of operators in the effective theory. We match these matrix elements to the full theory in the two-jet limit without assuming factorization of the complete set of hadronic final states into independent sums over partonic collinear and soft states. We also briefly discuss the relationship of this approach to diagrammatic factorization in the full theory.
We consider the Higgs boson production at high energy hadron colliders in the framework of the k_T-factorization approach. The attention is focused on the dominant gluon-gluon fusion subprocess. We calculate the total cross section and transverse momentum distributions of the inclusive Higgs production using unintegrated gluon distributions in a proton obtained from the full CCFM evolution equation. We show that k_T-factorization gives a possibility to investigate the associated Higgs boson and jets production. We calculate the transverse momentum distributions and study the Higgs-jet and jet-…
We discuss the fact that $k_t$-factorization calculations for heavy-quark production include only the $gg\rightarrow Q\bar{Q}$ contribution. The cases of fixed-flavor-number scheme and variable-flavor-number scheme calculations are analyzed separately. For the latter, we show that, similarly to the collinear factorization, the main contribution is given by the $Qg\rightarrow Qg$ process. In this scheme, calculations including only the $gg$ contribution should show a large discrepancy with the data. We show that, if they do not, it is because they include (effectively) a large $K$ factor.
The NRQCD factorization approach for calculating inclusive production of heavy quarkonium gives unambiguous predictions for the polarization of quarkonium states. The factorization formula for polarized states can be obtained by using the threshold expansion method to calculate the short-distance coefficients and then using symmetries of NRQD to reduce the NRQCD matrix elements. A particularly dramatic prediction of the NRQCD factorization framework is that prompt psi's and psi-primes's produced at the Tevatron should be predominantly transversely polarized at large transverse momentum.
We review recent advances in the theory of strong-interaction effects and final-state interactions in hadronic weak decays of heavy mesons. In the heavy-quark limit, the amplitudes for most nonleptonic, two-body B decays can be calculated from first principles and expressed in terms of semileptonic form factors and light-cone distribution amplitudes. We summarize the main features of this novel QCD factorization and discuss its phenomenological applications to B -> D^(*) L decays (with L a light meson), and to the rare charmless decays B -> pi K and B -> pi pi.
We evaluate in perturbative QCD the semi-inclusive Drell-Yan cross-section for the production of a single hadron accompaining the lepton pair. We demonstrate to one loop level a collinear factorization formula within the fracture functions approach. We propose such a process as a factorization analyzer in hadronic collisions. Phenomenological implications at the hadron colliders are briefly discussed.
We consider heavy quark production in high energy pA collisions and investigate the contribution of interactions of valence quarks of proton with the nucleus. The often made assumption that valence quarks of proton can be factored out is justified only if the nucleus saturation momentum is much smaller than the heavy quark mass. This is not the case in phenomenologically relevant situations. Breakdown of factorization manifests itself in substantial decrease of the cross section at large total and small relative transverse momenta of the heavy quark -antiquark pair.
Factorization, in the sense defined for inclusive hard scattering, is discussed for diffractive hard scattering. A factorization theorem similar to its inclusive counterpart is presented for diffractive DIS. For hadron-hadron diffractive hard scattering, in contrast to its inclusive counterpart, the expected breakdown of factorization is discussed. Cross section estimates are given from a simple field theory model for non-factorizing double-pomeron-exchange (DPE) dijet production with and without account for Sudakov suppression.
Single inclusive forward hadron production in high-energy hadron collisions can provide an important test of the Color Glass Condensate picture at small $x$. Recent studies of this process at next-to-leading order have led to problematic results, with cross sections becoming negative at large transverse momenta. We study a new formulation of this quantity proposed recently by Iancu et al. We show that it leads to physical results up to large transverse momenta at fixed coupling. Taking into account running coupling effects in a way that is consistent with existing DIS calculations still poses …
We calculate and analyze Z and W production in association with quark-antiquark pair in k_T-factorization. Numerical calculations are performed using the Monte Carlo generator CASCADE for proton proton collisions at LHC energy. We compare total and differential cross sections calculated in k_T-factorization approach with total differential cross sections obtained in LO and NLO calculations in collinear factorization approach. We provide strong evidence that some of the effects of the NLO and even higher order collinear calculation are already included in the LO k_T-factorization calculation.
The factorization theorem for the dijet cross section is presented in hadron-hadron collisions with a cone-type jet algorithm. We also apply the beam veto to the beam jets consisting of the initial radiation. The soft-collinear effective theory is employed to see the factorization structure transparently when there are four distinct lightcone directions involved. There are various types of divergences such as the ultraviolet and infrared divergences. And when the phase space is divided to probe the collinear and the soft parts, there appears an additional divergence called rapidity divergence.…
We consider the production of $J/ψ$ and $ψ(2S)$ mesons originating from the decays of $b$-flavored hadrons at the LHC using the $k_T$-factorization approach. Our analysis covers both inclusive charmonia production and production of $J/ψ$ mesons in association with $Z$ bosons. We apply the transverse momentum dependent (or unintegrated) gluon density in a proton derived from Catani-Ciafaloni-Fiorani-Marchesini (CCFM) evolution equation and adopt fragmentation functions based on the non-relativistic QCD factorization to describe the inclusive $b$-hadron decays into the different charmonium state…
A typical factorization formula for production of a particle with a small transverse momentum in hadron-hadron collisions is given by a convolution of two leading-twist TMD parton densities with cross section of production of the final particle by the two partons. For practical applications at a given transverse momentum, though, one should estimate at what momenta the power corrections to TMD factorization formula due to higher-twist operators become essential. In this paper we calculate the first power corrections to TMD factorization formula for Z-boson production and Drell-Yan process in h…
Recent theoretical investigations of two-body hadronic B decays have provided justification for the use of the factorization ansatz to evaluate the B decay matrix elements. Motivated by this, we present simplified formulae for a wide range of B decays into the lowest mass pseudoscalar and vector mesons. These formulae, valid in the heavy quark limit, involve a reduced set of soft QCD parameters and, although resulting in some loss of accuracy, should still provide an adequate and transparent tool with which to confront data for some time to come. Finally, we confront these formulae with data o…
We review recent advances in the theory of strong-interaction effects and final-state interactions in hadronic weak decays of heavy mesons. In the heavy-quark limit, the amplitudes for most nonleptonic, two-body B decays can be calculated from first principles and expressed in terms of semileptonic form factors and light-cone distribution amplitudes. We summarize the main features of this novel QCD factorization and discuss its phenomenological applications to B->D^(*)+L decays (with L a light meson), and to the rare charmless decays B->pi+K and B->pi+pi.
In standard Regge theory with a pomeron intercept a(0)=1+ε, the contribution of the tripe-pomeron amplitude to the t=0 differential cross section for single diffraction dissociation has the form dσ/dM^2(t=0) \sim s^{2ε}/(M^2)^{1+ε}. For ε>0, this form, which is based on factorization, does not scale with energy. From an analysis of p-p and p-pbar data from fixed target to collider energies, we find that such scaling actually holds, signaling a breakdown of factorization. Phenomenologically, this result can be obtained from a scaling law in diffraction, which is embedded in the hypothesis of po…
We investigate the factorization hypothesis of the four-quark condensate $\langle q \bar{q} q \bar{q} \rangle = \, A \, \langle q \bar{q} \rangle^2$ with the help of the Nambu Jona-Lasinio Model supplemented with eighth order interactions. For that purpose we use the bosonization method with multiple auxiliary variables. We find that in a simplified U(1) version of the model factorization holds, whereas in the full SU(3)-flavor version of the model factorization is broken by terms which are related to the 't Hooft interactions.
We construct a theoretical framework to match the formulas for forward inclusive hadron productions in pA collisions in the small-x saturation formalism and collinear factorization. The small-x calculation can be viewed as a power series in $Q_s^2/k_\perp^2$, in which the collinear factorization result corresponds to the leading term. At high transverse momentum, the subleading correction terms are insignficant, whereas at low $p_\perp$, the power corrections become important and the small-x resummation is essential to describe the differential cross section. We show that the familiar collinea…
We calculate transverse momentum dependent quark splitting kernels $P_{gq}$ and $P_{qq}$ within $k_T$-factorization, completing earlier results which concentrated on gluon splitting functions $P_{gg}$ and $P_{qg}$. The complete set of splitting kernels is an essential requirement for the formulation of a complete set of evolution equations for transverse momentum dependent parton distribution functions and the development of corresponding parton shower algorithms.
The application of k_t - factorization, supplemented with the CCFM small-x evolution equation, to heavy quark production is discussed. Differential cross sections of b-b_bar production and also inelastic J/psi production as measured at HERA are compared to the hadron level CCFM Monte Carlo generator CASCADE, using the unintegrated gluon density obtained within the CCFM evolution approach from a fit to HERA F_2 data.
Quit recently, two sets of new experimental data from the $LHCb$ and the $CMS$ collaborations have been published, concerning the production of the $Z^0$ vector boson in hadron-hadron collisions with the center-of-mass energy $E_{CM}= \sqrt{s}=13\;TeV$. On the other hand, in our recent work, we have conducted a set of $NLO$ calculations for the production of the electroweak gauge vector bosons, utilizing the unintegrated parton distribution functions ($UPDF$) in the frameworks of $Kimber$-$Martin$-$Ryskin$ ($KMR$) or $Martin$-$Ryskin$-$Watt$ ($MRW$) and the $k_t$-factorization formalism, concl…
We present calculations of heavy quark and quarkonium production at CERN LEP2 in the k_T-factorization QCD approach. Both direct and resolved photon contribution are taken into account. The conservative error analisys is performed. The unintegrated gluon distribution in the photon is taken from the full CCFM evolution equation. The traditional color-singlet mechanism to describe non-perturbative transition of heavy quark pair into a final quarkonium is used. Our analisys covers polarization properties of heavy quarkonia at moderate and large transverse momenta. We find that the total and diffe…
Using an eikonal structure for the scattering amplitude, Block and Kaidalov have derived factorization theorems for nucleon-nucleon, $γp$ and $γγ$ scattering at high energies, using only some very general assumptions. We present here an analysis giving experimental confirmation for factorization of cross sections, nuclear slope parameters B and $ρ$-values (ratio of real to imaginary portion of forward scattering amplitudes), showing that: i) the three factorization theorems hold, ii) the additive quark model holds to ~1%, and iii) vector dominance holds to better than ~4%.
In the framework of the k_T-factorization approach, the production of prompt photons in association with a heavy (charm or beauty) quarks at high energies is studied. The consideration is based on the O(αα_s^2) off-shell amplitudes of gluon-gluon fusion and quark-(anti)quark interaction subprocesses. The unintegrated parton densities in a proton are determined using the Kimber-Martin-Ryskin prescription. Our numerical predictions are compared with the D0 and CDF experimental data. Also we extend our results to LHC energies.
We discuss a higher-twist factorization procedure of the hadron tensor in deep inelastic scattering, proposed by Qiu, and extend it to include quark mass terms. Its property that the hard scattering parts are electromagnetic gauge invariant separately, is manifestly preserved. Using an auxiliary parton, a so-called spurion, to generate the quark mass terms, a simple parton model interpretation is also retained.
Exclusive nonleptonic bottom meson decays are studied in the covariant osillator quark model using the factorization assumption. The main feature of this model is that it can simultaniously be applied to both heavy to heavy and heavy to light transitions, satisfying the constraints of the heavy quark effective theory (HQET) in the appropriate limit. The results obtained are in overall agreement with the present experimental data for various B decays.
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 Drell-Yan hadronic tensor for electromagnetic (EM) current is calculated in the Sudakov region $s\gg Q^2\gg q_\perp^2$ with ${1\over Q^2}$ accuracy, first at the tree level and then with the double-log accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting tensor for unpolarized hadrons is EM gauge-invariant and depends on two leading-twist TMDs: $f_1$ responsible for total DY cross section, and Boer-Mulders function $h_1^\perp$. The order-of-magnitude estimates of ang…
The application of k_t-factorization supplemented with the CCFM small-x evolution equation to heavy quark production at the TEVATRON and at HERA is discussed. The bb_bar production cross sections at the TEVATRON can be consistently described using the k_t-factorization formalism together with the unintegrated gluon density obtained within the CCFM evolution approach from a fit to HERA F_2 data. Special attention is drawn to the comparison with measured visible cross sections, which are compared to the hadron level Monte Carlo generator CASCADE
We consider the processes of compare the heavy quark production using the unintegrated gluon distributions. The numerical predictions for high energy nucleon-nucleon and photon-nucleon collisions of the $k_T$-factorization approach (semihard theory) are compared with the experimental data from Tevatron-collider and HERA. The total production cross sections and $p_T$ distributions are considered and they are in reasonable agreement with the data for rather value of QCD scale.
We further analyze the definition and the calculation of the heavy quark impact factor at next-to-leading (NL) $\log s$ level, and we provide its analytical expression in a previously proposed k-factorization scheme. Our results indicate that k-factorization holds at NL level with a properly chosen energy scale, and with the same gluonic Green's function previously found in the massless probe case.
We investigate the heavy quark photoproduction based on the k_T factorization approach, focusing on the results from the saturation model. The deviations in the results using the unintegrated gluon distribution considering the saturation model and the derivative of the collinear gluon distribution are analysed. Total cross sections and p_T distributions are analysed in detail, setting the deviations between the color dipole approximation and the complete semihard approach.
It has by now been established that standard QCD factorization using transverse momentum dependent parton distribution functions fails in hadro-production of nearly back-to-back hadrons with high transverse momentum. The essential problem is that gauge invariant transverse momentum dependent parton distribution functions cannot be defined with process-independent Wilson line operators, thus implying a breakdown of universality. This has led naturally to proposals that a correct approach is to instead use a type of "generalized" transverse momentum dependent factorization in which the basic fac…
