We demonstrate that in the region of Bjorken x ~m_π^2/(4πF_π)^2 and/or x ~|t|/(4πF_π)^2 the standard ChPT for the pion GPDs fails and one must perform all order resummation of ChPT. We perform such resummation in the large-N limit of the O(N+1) extension of the chiral theory. Explicit resummation allows us to reveal novel phenomena -- the form of the leading chiral correction to pion PDFs and GPDs depends on the small-x asymptotic of the pion PDFs. In particular, if the pion PDF in the chiral limit has the Regge-like small x behaviour q(x)~1/x^ω, the leading large impact parameter ($b_\perp\to\infty$) asymptotics of the quark distribution in the transverse plane has the form ($m_π=0$) $q(x,b_\perp)\sim 1/x^ω \ln^ω(b_\perp^2)/b_\perp^{2{(1+ω)}}$. This result is model independent and it is controlled completely by the all order resummed ChPT developed in this paper. This asymptotic interweaves with small-$x$ behaviour of usual PDFs, hence it depends on the scale, at which the corresponding PDF is defined. This is a new and interesting result in which the chiral expansion meets the QCD evolution.
We consider a general (beyond $T\bar T$) deformation of the 2D $O(N+1)$ $σ$-model by the irrelevant dimension-four operators. The theory deformed in this most general way is not integrable, and the $S$-matrix loses its factorization properties. We perform the all-order summation of the leading infrared logs for the $2 \to 2$ scattering amplitude and provide the exact result for the $2 \to 2$ $S$-matrix in the leading logarithmic approximation. These results can provide us with new insights into the properties of the theories deformed by irrelevant operators more general than the $T\bar T$ defo…
When parton distributions are falling steeply as the momentum fractions of the partons increases, there are effects that occur at each order in $α_s$ that combine to affect hard scattering cross sections and need to be summed. We show how to accomplish this in a leading approximation in the context of a parton shower Monte Carlo event generator.
In a companion publication, we have explored how to examine the summation of large logarithms in a parton shower. Here, we apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. The method is to work with an appropriate integral transform of the distribution for the observable of interest. Then, we reformulate the parton shower calculation so as to obtain the transformed distribution as an exponential for which we can compute the terms in the perturbative expansion of the exponent.
We propose a method to examine how a parton shower sums large logarithms. In this method, one works with an appropriate integral transform of the distribution for the observable of interest. Then, one reformulates the parton shower so as to obtain the transformed distribution as an exponential for which one can compute the terms in the perturbative expansion of the exponent. We apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. Of the approaches that we use, the most generally applicable is to compute some of the perturbati…
A model for the parton distributions of hadrons in impact parameter space has been constructed using soft gluon summation. This model incorporates the salient features of distributions obtained from the intrinsic transverse momentum behaviour of hadrons. Under the assumption that the intrinsic behaviour is dominated by soft gluon emission stimulated by the scattering process, the b-spectrum becomes softer and softer as the scattering energy increases. In minijet models for the inclusive cross-sections, this will counter the increase from $σ_{jet}$ .
A microscopic treatment of cumulative phenomena based on perturbative QCD calculations of the corresponding quark diagrams near the threshold is presented. To sum all diagrams like these the special technique based on the recurrence relations was developed. The x-behaviour of the nuclear structure function $F_{2}(x)$ in the cumulative region x>1 was found to be roughly exponential, governed by an effective coupling constant, which depends on the QCD coupling constant and quark mass. Two mechanisms for cumulative particle production, direct and spectator ones, were analysed. It was shown that d…
We present an alternative method to calculate cross sections for multi-parton scattering processes in the Standard Model at leading order. The helicity amplitudes are computed using recursion relations in the number of particles, based on Dyson-Schwinger equations whereas the summation over colour and helicity configurations is performed by Monte Carlo methods. The computational cost of our algorithm grows asymptotically as 3^n, where n is the number of particles involved in the process, as opposed to the n!-growth of the Feynman diagram approach. Typical results for the total cross section, d…
We explore jet physics in hadron collisions using the parton shower event generator Deductor. Of particular interest is the one jet inclusive cross section dsigma/dpT for jets of very high pT. Compared to the Born level, the cross section decreases substantially because of pT loss from the jet during showering. We compare to the same effect in Pythia and Dire. The cross section then increases substantially because of the summation of threshold logarithms included in Deductor. We also study the cross section to have a gap with no jets between two hard jets that are widely separated in rapidity.…
We investigate enhanced EW corrections to inclusive hard processes in the TeV energy region with emphasis on the small-x situation, in which the hard scale Q is significantly smaller than the available energy \sqrt{s}= Q/x. We first propose and justify a general factorization formula in which the (double-log) EW form factor at scale Q^2 is factorized from EW parton distribution functions, which satisfy evolution equations of DGLAP type. We then investigate the small-x behavior of the EW parton distributions including the novel ones for non-vanishing t-channel weak isospin T and we compare it w…
The Bloch-Nordsieck model for the parton distribution of hadrons in impact parameter space, constructed using soft gluon summation, is investigated in detail. Its dependence upon the infrared structure of the strong coupling constant $α_s$ is discussed, both for finite as well as singular, but integrable, $α_s$. The formalism is applied to the prediction of total proton-proton and proton-antiproton cross-sections, where screening, due to soft gluon emission from the initial valence quarks, becomes evident.
In this paper we discuss the systematics of quarkonium production at the LHC. In particular, we focus on the necessity to sum logs of the form log(Q/p_perp) and log(p_perp/m_Q). We show that the former contributions are power suppressed, while the latter, whose contribution in fragmentation is well known, also arise in the short distance (i.e., non-fragmentation) production mechanisms. Though these contributions are suppressed by powers of m_Q/p_perp, they can be enhanced by inverse powers of v, the relative velocity between heavy quarks in the quarkonium. In the limit p_perp >> m_Q short dist…
The interpretation of experimental measurements at the LHC requires accurate theoretical predictions for exclusive observables, and in particular the summation of soft and collinear radiation to all orders in perturbation theory. We report on recent progress towards the automated calculation of multi-parton LHC cross sections at next-to-leading order in QCD, including the summation of next-to-leading logarithmic corrections through the combination with parton showers.
We demonstrate that all the known single- and double-logarithm summations for a parton distribution function can be unified in the Collins-Soper resummation technique by applying soft approximations appropriate in different kinematic regions to real gluon emissions. Neglecting the gluon longitudinal momentum, we obtain the $k_T$ (double-logarithm) resummation for two-scale QCD processes, and the Balitsky-Fadin-Kuraev-Lipatov (single-logarithm) equation for one-scale processes. Neglecting the transverse momentum, we obtain the threshold (double-logarithm) resummation for two-scale processes, an…
We show how perturbation theory may be reorganized to give splitting functions which include order by order convergent sums of all leading logarithms of $x$. This gives a leading twist evolution equation for parton distributions which sums all leading logarithms of $x$ and $Q^2$, allowing stable perturbative evolution down to arbitrarily small values of $x$. Perturbative evolution then generates the double scaling rise of $F_2$ observed at HERA, while in the formal limit $x\to 0$ at fixed $Q^2$ the Lipatov $x^{-λ}$ behaviour is eventually reproduced. We are thus able to explain why leading ord…
Predictions for total inelastic cross-sections for photon induced processes are discussed in the context of the QCD-inspired minijet model. Large theoretical uncertainties exist, some of them related to the parton distributions of hadrons in impact parameter space. A model for such distribution is presented, based on soft gluon summation. This model incorporates (the salient features of distributions obtained from) the intrinsic transverse momentum behaviour of hadrons. Under the assumption that the intrinsic behaviour is dominated by soft gluon emission stimulated by the scattering process, t…
When the energy of the heavy quark is comparable with its mass, it is natural to attribute this heavy quark to the hard part of the reaction. At large energies, this approach is impractical due to large logarithms from intensive QCD radiation affecting both inclusive and differential observables. We present a formalism for all-order summation of such logarithms and reliable description of heavy-quark distributions at all energies. As an illustration, we calculate angular distributions of B-mesons produced in neutral-current events at large momentum transfers at the ep collider HERA.
We prove collinear factorization theorem for the process $πγ^*\toπ$ at the twist-3 level in the covariant gauge by means of the Ward identity, concentrating on the two-parton case. It is shown that soft divergences cancel and collinear divergences are grouped into the pseudo-scalar and pseudo-tensor two-parton twist-3 pion distribution amplitudes. The delicate summation of a complete set of diagrams for achieving factorization in momentum, spin, and color spaces is emphasized. The proof is then extended to the exclusive semileptonic decay $B\toπl\barν$, assuming the hard scale to be of $O(\sqr…
In this study, we present continuum limit results for the unpolarized parton distribution function of the nucleon computed in lattice QCD. This study is the first continuum limit using the pseudo-PDF approach with Short Distance Factorization for factorizing lattice QCD calculable matrix elements. Our findings are also compared with the pertinent phenomenological determinations. Inter alia, we are employing the summation Generalized Eigenvalue Problem (sGEVP) technique in order to optimize our control over the excited state contamination which can be one of the most serious systematic errors i…
We present analytical calculations of the distribution of non-global jet shapes in Higgs/vector boson + jet production at hadron colliders. Within the eikonal-limit framework and implementing various jet algorithms, we compute the full distribution of the particular jet mass shape observable at 2-loops, including the large single-logarithms known as non-global logs and clustering logs. We compare our next-to-leading-log analytical resummation to parton showers and, after matching and including non-perturbative effects, to experimental data. A good agreement is shown for all comparisons.
The thrust distribution in e+e- annihilation is calculated exploiting its exponentiation property in the two-jet region t = 1-T << 1. We present a general method (DGE) to calculate a large class of logarithmically enhanced terms, using the dispersive approach in renormalon calculus. Dressed Gluon Exponentiation is based on the fact that the exponentiation kernel is associated primarily with a single gluon emission, and therefore the exponent is naturally represented as an integral over the running coupling. Fixing the definition of Lambda is enough to guarantee consistency with the exact expon…
I review recent theoretical progress in the resummation of small x contributions to the evolution of parton distributions, in view of its potential significance for accurate phenomenology at future colliders. I show that a consistent perturbative resummation of collinear and energy logs is now possible, and necessary if one wishes to use recent NNLO results in the HERA kinematic region.
The present work is devoted to study the high-energy $QCD$ events, such as the di-jet productions from proton-proton inelastic collisions at the $LHC$ in the forward-center and the forward-forward configurations, using the $unintegrated$ parton distribution functions ($UPDF$) in the $k_t$-factorization framework. The $UPDF$ of $Kimber$ et. al. ($KMR$) and $Martin$ et.al. ($MRW$) are generated in the leading order ($LO$) and next-to-leading order ($NLO$), using the $Harland-Lang$ et al. ($MMHT2014$) $PDF$ libraries. While working in the forward-center and the forward-forward rapidity sectors, o…
In a series of papers, we have investigated the compatibility of the $Kimber$-$Martin$-$Ryskin$ ($KMR$) and $Martin$-$Ryskin$-$Watt$ ($MRW$) $unintegrated$ parton distribution functions ($UPDF$) as well as the description of the experimental data on the proton structure functions. The present work is a sequel to that survey, via calculation of the transverse momentum distribution of the electro-weak gauge vector bosons in the $k_t$-factorization scheme, by the means of the $KMR$, the $LO\;MRW$ and the $NLO\;MRW$ $UPDF$, in the next-to leading order ($NLO$). To this end, we have calculated and …
The Standard Approach (SA) for description of the structure function g_1 combines the DGLAP evolution equations and Standard Fits for the initial parton densities. The DGLAP equations describe the region of large Q^2 and large x, so there are not theoretical grounds to exploit them at small x. In practice, extrapolation of DGLAP into the region of large Q^2 and small x is done with complementing DGLAP with special, singular (~x^{-a}) phenomenological fits for the initial parton densities. The factors x^{-a} are wrongly believed to be of the non-perturbative origin. Actually, they mimic the res…
Differential distributions for heavy quark production depend on the heavy quark mass and other momentum scales, which can yield additional large logarithms and inhibit accurate predictions. Logarithms involving the heavy quark mass can be summed in heavy quark parton distribution functions in the ACOT factorization scheme. A second class of logarithms involving the heavy-quark transverse momentum can be summed using an extension of Collins-Soper-Sterman (CSS) formalism. We perform a systematic summation of logarithms of both types, thereby obtaining an accurate description of heavy-quark diffe…
We present a calculation of the pion valence quark distribution extracted using the formalism of reduced Ioffe time pseudo-distributions or more commonly known as pseudo-PDFs. Our calculation is carried out on two different 2+1 flavor QCD ensembles using the isotropic-clover fermion action, with lattice dimensions $24^3\times 64$ and $32^3\times 96$ at the lattice spacing of $a=0.127$ fm, and with the quark mass equivalent to a pion mass of $m_π\simeq 415$ MeV. We incorporate several combinations of smeared-point and smeared-smeared pion source-sink interpolation fields in obtaining the lattic…
