Next-to-leading Logarithmic Research Articles

Analyzing high-energy factorization beyond next-to-leading logarithmic accuracy

We provide a complete and detailed study of the high-energy limit of four-parton scattering amplitudes in QCD, giving explicit results at two loops and higher orders, and going beyond next-to-leading logarithmic (NLL) accuracy. Building upon recent results, we use the techniques of infrared factorization to investigate the failure of the simplest form of Regge factorization, starting at next-to-next-to-leading logarithmic accuracy (NNLL) in ln(s/|t|). We provide detailed accounts and explicit expressions for the terms responsible for this breaking in the case of two-loop and three-loop quark and gluon amplitudes in QCD; in particular, we recover and explain a known non-logarithmic double-pole contribution at two-loops, and we compute all non-factorizing single-logarithmic singular contributions at three loops. Conversely, we use high-energy factorization to show that the hard functions of infrared factorization vanish in d = 4 to all orders in the coupling, up to NLL accuracy in ln(s/|t|). This provides clear evidence for the infrared origin of high-energy logarithms. Finally, we extend earlier studies to t-channel exchanges of color representations beyond the octet, which enables us to give predictions based on the dipole formula for single-pole NLL contributions at three and four loops.

Toward NNLL threshold resummation for hadron pair production in hadronic collisions

We investigate QCD threshold resummation effects beyond the next-to-leading logarithmic (NLL) order for the process H1 H2->h1 h2 X at high invariant mass of the produced hadron pair. We take into account the color structure of the underlying partonic hard-scattering cross sections and determine the relevant hard and soft matrices in color space that contribute to the resummed cross section at next-to-next-to-leading logarithmic (NNLL) accuracy. We present numerical results for fixed-target and collider regimes. We find a significant improvement compared to previous results at NLL accuracy. In particular, the scale dependence of the resummed cross section is greatly reduced. Use of the most recent set of fragmentation functions also helps in improving the comparison with the experimental data. Our calculation provides a step towards a systematic NNLL extension of threshold resummation also for other hadronic processes, in particular for jet production.

Resummation of Jet Shapes and Extracting Properties of the Quark-Gluon Plasma

Understanding the properties of the quark-gluon plasma (QGP) that is produced in ultra-relativistic nucleus-nucleus collisions has been one of the top priorities of the heavy ion program at the LHC. Energetic jets are produced and subsequently quenched in the collisions. Such jet quenching phenomena provide promising tools to probe the medium properties by studying the modification of jets due to the medium interactions. Significant modifications of jet shapes have been measured. In this talk we focus on the calculation of jet shapes in both proton-proton and lead-lead collisions using soft-collinear effective theory (SCET), with Glauber gluon interactions in the medium. Large logarithms in jet shapes are resummed at next-to-leading logarithmic (NLL) accuracy by the renormalization-group evolution between hierarchical jet scales. The medium interactions contribute as power corrections, and we calculate the modification of jet shapes at leading order in opacity with the static QGP model. Preliminary results are presented with good agreement with the recent CMS jet shape measurements.

Search for BFKL-evolution manifestations at high energies

A brief overview of searches for BFKL-evolution manifestations at high energies is presented. Data comparison with the predictions of BFKL-evolution in the leading logarithmic (LL) and next-to-leading logarithmic (NLL) approximations are discussed.

BFKL Amplitude Parametrisation for the Jet--Gap--Jet Events at the LHC Energies

The process of jet-gap-jet (JGJ) production is briefly described. The JGJ scattering amplitude parametrisation is discussed. On the basis of full amplitude calculations, the parametrisation formulas for the leading logarithmic (LL) and next-to-leading logarithmic (NLL) approximations are obtained. For each case a sum over all conformal spins is considered. The obtained agreement is better than 0.25% for LL and 1% for NLL.

Gaining (mutual) information about quark/gluon discrimination

Discriminating quark jets from gluon jets is an important but challenging problem in jet substructure. In this paper, we use the concept of mutual information to illuminate the physics of quark/gluon tagging. Ideal quark/gluon separation requires only one bit of truth information, so even if two discriminant variables are largely uncorrelated, they can still share the same "truth overlap". Mutual information can be used to diagnose such situations, and thus determine which discriminant variables are redundant and which can be combined to improve performance. Using both parton showers and analytic resummation, we study a two-parameter family of generalized angularities, which includes familiar infrared and collinear (IRC) safe observables like thrust and broadening, as well as IRC unsafe variants like $p_T^D$ and hadron multiplicity. At leading-logarithmic (LL) order, the bulk of these variables exhibit Casimir scaling, such that their truth overlap is a universal function of the color factor ratio $C_A/C_F$. Only at next-to-leading-logarithmic (NLL) order can one see a difference in quark/gluon performance. For the IRC safe angularities, we show that the quark/gluon performance can be improved by combining angularities with complementary angular exponents. Interestingly, LL order, NLL order, Pythia 8, and Herwig++ all exhibit similar correlations between observables, but there are significant differences in the predicted quark/gluon discrimination power. For the IRC unsafe angularities, we show that the mutual information can be calculated analytically with the help of a nonperturbative "weighted-energy function", providing evidence for the complementarity of safe and unsafe observables for quark/gluon discrimination.

Reanalysis of the BFKL Pomeron at the next-to-leading logarithmic accuracy

We apply the principle of maximum conformality (PMC) to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron intercept at the next-to-leading logarithmic (NLL) accuracy. The PMC eliminates the conventional renormalization scale ambiguity by absorbing the non-conformal $\{\beta_i\}$-terms into the running coupling, and a more accurate pQCD estimation can be obtained. After PMC scale setting, the QCD perturbative convergence can be greatly improved due to the elimination of renormalon terms in pQCD series, and the BFKL Pomeron intercept has a weak dependence on the virtuality of the reggeized gluon. For example, by taking the Fried-Yennie gauge, we obtain $\omega_{\rm MOM}^{\rm PMC}(Q^{2},0)\in [0.149,0.176]$ for $Q^2\in[1,100]\;{\rm GeV}^2$. This is a good property to apply to the high-energy phenomenology. Further more, to compare with the data, it is found that the physical ${\rm MOM}$-scheme is more reliable than the $\overline{\rm MS}$-scheme. The ${\rm MOM}$-scheme is gauge dependent, which can also be greatly suppressed after PMC scale setting. We discuss the MOM-scheme gauge dependence for the Pomeron intercept by adopting three gauges, i.e. the Landau gauge, the Feynman gauge and the Fried-Yennie gauge, and we obtain $\omega_{\rm MOM}^{\rm PMC}(Q^{2}=15\;{\rm GeV}^2,0) = 0.166^{+0.010}_{-0.017}$; i.e. about $10\%$ gauge dependence is observed. We apply the BFKL Pomeron intercept to the photon-photon collision process, and compare the theoretical predictions with the data from the OPAL and L3 experiments.

Abelian non-global logarithms from soft gluon clustering

Most recombination-style jet algorithms cluster soft gluons in a complex way. This leads to previously identified correlations in the soft gluon phase space and introduces logarithmic corrections to jet cross sections, which are known as clustering logarithms. The leading Abelian clustering logarithms occur at least at next-to leading logarithm (NLL) in the exponent of the distribution. Using the framework of Soft Collinear Effective Theory (SCET), we show that new clustering effects contributing at NLL arise at each order. While numerical resummation of clustering logs is possible, it is unlikely that they can be analytically resummed to NLL. Clustering logarithms make the anti-kT algorithm theoretically preferred, for which they are power suppressed. They can arise in Abelian and non-Abelian terms, and we calculate the Abelian clustering logarithms at O $$ \left( {\alpha_s^2} \right) $$ for the jet mass distribution using the Cambridge/Aachen and kT algorithms, including jet radius dependence, which extends previous results. We find that clustering logarithms can be naturally thought of as a class of non-global logarithms, which have traditionally been tied to non-Abelian correlations in soft gluon emission.

NLL soft and Coulomb resummation for squark and gluino production at the LHC

Abstract We present predictions of the total cross sections for pair production of squarks and gluinos at the LHC, including the stop-antistop production process. Our calculation supplements full fixed-order NLO predictions with resummation of threshold logarithms and Coulomb singularities at next-to-leading logarithmic (NLL) accuracy, including bound-state effects. The numerical effect of higher-order Coulomb terms can be as big or larger than that of soft-gluon corrections. For a selection of benchmark points accessible with data from the 2010-2012 LHC runs, resummation leads to an enhancement of the total inclusive squark and gluino production cross section in the 15-30% range. For individual production processes of gluinos, the corrections can be much larger. The theoretical uncertainty in the prediction of the hard-scattering cross sections is typically reduced to the ±10% level.

NLL+NNLO predictions for jet-veto efficiencies in Higgs-boson and Drell-Yan production

Abstract Using the technology of the caesar approach to resummation, we examine the jet-veto efficiency in Higgs-boson and Drell-Yan production at hadron colliders and show that at next-to-leading logarithmic (NLL) accuracy the resummation reduces to just a Sudakov form factor. Matching with NNLO calculations results in stable predictions for the case of Drell-Yan production, but reveals substantial uncertainties in gluon-fusion Higgs production, connected in part with the poor behaviour of the perturbative series for the total cross section. We compare our results to those from powheg with and without reweighting by hqt, as used experimentally, and observe acceptable agreement. In an appendix we derive the part of the NNLL resummation corrections associated with the radius dependence of the jet algorithm.

Non-global logs and clustering impact on jet mass with a jet veto distribution

There has recently been much interest in analytical computations of jet mass distributions with and without vetos on additional jet activity [1-6]. An important issue affecting such calculations, particularly at next-to-leading logarithmic (NLL) accuracy, is that of non-global logarithms as well as logarithms induced by jet definition, as we pointed out in an earlier work [3]. In this paper, we extend our previous calculations by independently deriving the full jet-radius analytical form of non-global logarithms, in the anti-$\kt$ jet algorithm. Employing the small-jet radius approximation, we also compute, at fixed-order, the effect of jet clustering on both $\CF^{2}$ and $\CF\CA$ colour channels. Our findings for the $\CF\CA$ channel confirm earlier analytical calculations of non-global logarithms in soft-collinear effective theory [5]. Moreover, all of our results, as well as those of [3], are compared to the output of the numerical program \texttt{EVENT2}. We find good agreement between analytical and numerical results both with and without final state clustering.

Timelike dipole-antenna showers with massive fermions

We present a complete formalism for final-state (timelike) dipole-antenna showers including fermion masses, but neglecting polarization and finite-width effects. We make several comparisons of tree-level expansions of this shower algorithm to fixed-order matrix elements for hadronic $Z$ decays, up to and including $Z\ensuremath{\rightarrow}6$ partons, to which the algorithm can be consistently matched over all of phase space. We also compare to analytical resummations at the next-to-leading logarithmic (NLL) level. The shower algorithm has been implemented in the publicly available VINCIA plug-in to the PYTHIA 8 event generator, which enables us to compare to experimental data at the fully hadronized level. We therefore also include comparisons to selected observables in $b$-tagged $Z$ decays.

NLL Order Contributions for Exclusive Processes in Jet-Calculus Scheme

We investigate the next-to-leading logarithmic (NLL) order contributions of the quantum chromodynamics (QCD) for exclusive processes evaluated by Monte Carlo methods. Ambiguities of the Monte Carlo calculation based on the leading-logarithmic (LL) order approximations are pointed out. To remove these ambiguities, we take into account the NLL order terms. In a model presented in this paper, interference contributions due to the NLL order terms are included for the generation of the transverse momenta in initial-state parton radiations. Furthermore, a kinematical constraint due to parton radiation, which is also a part of the NLL order contributions, is taken into account. This method guarantees a proper phase space boundary for hard scattering cross sections as well as parton radiations. As an example, cross sections for lepton pair productions mediated by a virtual photon in hadron-hadron collisions are calculated, using the jet-calculus scheme for flavor nonsinglet quarks.

Next-to-leading ultrasoft running of the heavy quarkonium potentials and spectrum: Spin-independent case

We compute the next-to-leading logarithmic (NLL) ultrasoft running of the spin-independent singlet potentials up to ${\cal O}(1/m^2)$, and the corresponding contribution to the spectrum. This includes the static energy at next-to-next-to-next-to-leading logarithmic (NNNLL) order. As a byproduct of these results one could obtain the heavy quarkonium spectrum with N$^3$LL accuracy for $l\not=0$ (angular momentum) and $s=0$ (spin) states, setting the stage for the full analytic and numerical N$^3$LL analysis of such quantity. We also compute the next-to-next-to-next-to-next-to-leading order (N$^4$LO) ultrasoft spin-independent contribution to the heavy quarkonium mass and static energy.

Ultrasoft NLL running of the nonrelativistic O(v) QCD quark potential

Using the nonrelativistic effective field theory vNRQCD, we determine the contribution to the next-to-leading logarithmic (NLL) running of the effective quark-antiquark potential at order v (1/mk) from diagrams with one potential and two ultrasoft loops, v being the velocity of the quarks in the c.m. frame. The results are numerically important and complete the description of ultrasoft next-to-next-to-leading logarithmic (NNLL) order effects in heavy quark pair production and annihilation close to threshold.

Transverse momentum distributions from effective field theory with numerical results

We derive a factorization theorem for the differential distributions of electroweak gauge bosons in Drell-Yan processes, valid at low transverse momentum, using the Soft-Collinear Effective Theory (SCET). We present the next-to-leading logarithmic (NLL) transverse momentum distribution for the Z-boson and find good agreement with Tevatron data collected by the CDF and D0 collaborations. We also give predictions for the Higgs boson differential distributions at NLL based on a factorization theorem derived in earlier work. We derive formulae for all quantities needed to study low transverse momentum production of color-neutral particles within the effective theory. This effective field theory approach is free of Landau poles and can be formulated entirely in momentum space. Consequently, our results are free of the Landau-pole prescriptions necessary in the standard approach.

Jet shapes and jet algorithms in SCET

Jet shapes are weighted sums over the four-momenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its partonic origin. In this work we make predictions for quark and gluon jet shape distributions in N-jet final states in e+e- collisions, defined with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of Soft-Collinear Effective Theory, we prove a factorization theorem for jet shape distributions and demonstrate the consistent renormalization-group running of the functions in the factorization theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size R of the jets, as long as R is much smaller than the angular separation between jets. We calculate the jet and soft functions for angularity jet shapes \tau_a to one-loop order (O(alpha_s)) and resum a subset of the large logarithms of \tau_a needed for next-to-leading logarithmic (NLL) accuracy for both cone and kT-type jets. We compare our predictions for the resummed \tau_a distribution of a quark or a gluon jet produced in a 3-jet final state in e+e- annihilation to the output of a Monte Carlo event generator and find that the dependence on a and R is very similar.

Phenomenology of event shapes at hadron colliders

We present results for matched distributions of a range of dijet event shapes at hadron colliders, combining next-to-leading logarithmic (NLL) accuracy in the resummation exponent, next-to-next-to leading logarithmic (NNLL) accuracy in its expansion and next-to-leading order (NLO) accuracy in a pure alpha_s expansion. This is the first time that such a matching has been carried out for hadronic final-state observables at hadron colliders. We compare our results to Monte Carlo predictions, with and without matching to multi-parton tree-level fixed-order calculations. These studies suggest that hadron-collider event shapes have significant scope for constraining both perturbative and non-perturbative aspects of hadron-collider QCD. The differences between various calculational methods also highlight the limits of relying on simultaneous variations of renormalisation and factorisation scale in making reliable estimates of uncertainties in QCD predictions. We also discuss the sensitivity of event shapes to the topology of multi-jet events, which are expected to appear in many New Physics scenarios.

Consistent factorization of jet observables in exclusive multijet cross sections

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 kT-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 μ. The anomalous dimensions of the various factorized pieces, however, depend on the color representation of jets, choice of jet observable, the number of jets whose shapes are measured, and the jet algorithm, making it highly nontrivial to satisfy the consistency condition. We demonstrate the intricate cancellations between anomalous dimensions that occur at the NLL level, so that, up to power corrections that we identify, our factorization of the jet shape distributions is consistent for any number of quark and gluon jets, for any number of jets whose shapes are measured or unmeasured, for any angular size R of the jets, and for any of the algorithms we consider. Corrections to these results are suppressed by the SCET expansion parameter λ (set by the ratio of the typical momentum in a jet transverse to the jet axis to the total jet energy) and in the jet separation measure 1/t2=tan2(R/2)/tan2(ψ/2), where ψ is the angular separation between jets. Our results can be used to calculate a wide variety of jet observables in multijet final states to NLL accuracy.

Soft-gluon resummation for squark and gluino hadroproduction

We consider the resummation of soft gluon emission for squark and gluino hadroproduction at next-to-leading-logarithmic (NLL) accuracy in the framework of the minimal supersymmetric standard model. We present analytical results for squark-squark and squark-gluino production and provide numerical predictions for all squark and gluino pair-production processes at the Tevatron and at the LHC. The size of the soft-gluon corrections and the reduction in the scale uncertainty are most significant for processes involving gluino production. At the LHC, where the sensitivity to squark and gluino masses ranges up to 3 TeV, the corrections due to NLL resummation over and above the NLO predictions can be as high as 35% in the case of gluino-pair production, whereas at the Tevatron, the NLL corrections are close to 40% for squark-gluino final states with sparticle masses around 500 GeV.