Leading-logarithmic Order Research Articles

Remembrance of things past

Nonlinear sigma models on de Sitter background have proved a useful prototype for quantum gravity in summing the large logarithms which arise from loop corrections. We consider a model whose evolution is described, at leading logarithm order, by the trace of the coincident, doubly differentiated scalar propagator. An analytic approximation for this quantity on an arbitrary expansion history is applied to generalize the resummed de Sitter result to any cosmological background which has experienced primordial inflation. In addition to analytic expressions, we present explicit numerical results for the evolution in a plausible expansion history. The large scales of primordial inflation are transmitted to late times.

Pion distribution amplitude at the physical point using the leading-twist expansion of the quasi-distribution-amplitude matrix element

We present a lattice QCD determination of the distribution amplitude (DA) of the pion and the first few Mellin moments from an analysis of the quasi-DA matrix element within the leading-twist framework. We perform our study on a HISQ ensemble with $a=0.076$ fm lattice spacing with the Wilson-Clover valence quark mass tuned to the physical point. We analyze the ratios of pion quasi-DA matrix elements at short distances using the leading-twist Mellin operator product expansion (OPE) at the next-to-leading order and the conformal OPE at the leading-logarithmic order. We find a robust result for the first non-vanishing Mellin moment $\langle x^2 \rangle = 0.287(6)(6)$ at a factorization scale $\mu=2$ GeV. We also present different Ans\"atze-based reconstructions of the $x$-dependent DA, from which we determine the perturbative leading-twist expectations for the pion electromagnetic and gravitational form-factors at large momentum transfers.

Quantum kinetic theory for dynamical spin polarization from QED-type interaction

We investigate the dynamical spin polarization of a massless electron probing an electron plasma in locally thermal equilibrium via the Moller scattering from the quantum kinetic theory. We derive an axial kinetic equation delineating the dynamical spin evolution in the presence of the collision term with quantum corrections up to $\mathcal{O}(\ensuremath{\hbar})$ and the leading-logarithmic order in coupling by using the hard-thermal-loop approximation, from which we extract the spin-polarization rate induced by the spacetime gradients of the medium. When the electron probe approaches local equilibrium, we further simplify the collision term into a relaxation-time expression. Our kinetic equation may be implemented in the future numerical simulations for dynamical spin polarization.

Higher-order non-global logarithms from jet calculus

Non-global QCD observables are characterised by a sensitivity to the full angular distribution of soft radiation emitted coherently in hard scattering processes. This complexity poses a challenge to their all-order resummation, that was formulated at the leading-logarithmic order about two decades ago. In this article we present a solution to the long-standing problem of their resummation beyond this order, and carry out the first complete next-to-leading logarithmic calculation for non-global observables. This is achieved by solving numerically the recently derived set of non-linear differential equations which describe the evolution of soft radiation in the planar, large-Nc limit. As a case study we address the calculation of the transverse energy distribution in the interjet rapidity region in e+e−→ dijet production. The calculation is performed by means of an algorithm that we formulate in the language of jet-calculus generating functionals, which also makes the resummation technique applicable to more general non-global problems, such as those that arise in hadronic collisions. We find that NLL corrections are substantial and their inclusion leads to a significant reduction of the perturbative scale uncertainties for these observables. The computer code used in the calculations is made publicly available.

Two-parton scattering amplitudes in the Regge limit to high loop orders

We study two-to-two parton scattering amplitudes in the high-energy limit of perturbative QCD by iteratively solving the BFKL equation. This allows us to predict the imaginary part of the amplitude to leading-logarithmic order for arbitrary t-channel colour exchange. The corrections we compute correspond to ladder diagrams with any number of rungs formed between two Reggeized gluons. Our approach exploits a separation of the two-Reggeon wavefunction, performed directly in momentum space, between a soft region and a generic (hard) region. The former component of the wavefunction leads to infrared divergences in the amplitude and is therefore computed in dimensional regularization; the latter is computed directly in two transverse dimensions and is expressed in terms of single-valued harmonic polylogarithms of uniform weight. By combining the two we determine exactly both infrared-divergent and finite contributions to the two-to-two scattering amplitude order-by-order in perturbation theory. We study the result numerically to 13 loops and find that finite corrections to the amplitude have a finite radius of convergence which depends on the colour representation of the t-channel exchange.

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.

Soft drop

We introduce a new jet substructure technique called "soft drop declustering", which recursively removes soft wide-angle radiation from a jet. The soft drop algorithm depends on two parameters--a soft threshold $z_\text{cut}$ and an angular exponent $\beta$--with the $\beta = 0$ limit corresponding roughly to the (modified) mass drop procedure. To gain an analytic understanding of soft drop and highlight the $\beta$ dependence, we perform resummed calculations for three observables on soft-dropped jets: the energy correlation functions, the groomed jet radius, and the energy loss due to soft drop. The $\beta = 0$ limit of the energy loss is particularly interesting, since it is not only "Sudakov safe" but also largely insensitive to the value of the strong coupling constant. While our calculations are strictly accurate only to modified leading-logarithmic order, we also include a discussion of higher-order effects such as multiple emissions and (the absence of) non-global logarithms. We compare our analytic results to parton shower simulations and find good agreement, and we also estimate the impact of non-perturbative effects such as hadronization and the underlying event. Finally, we demonstrate how soft drop can be used for tagging boosted W bosons, and we speculate on the potential advantages of using soft drop for pileup mitigation.

Factorization and NNLL Resummation for Higgs production with a jet veto

Using methods of effective field theory, we derive the first all-order factorization theorem for the Higgs-boson production cross section with a jet veto, imposed by means of a standard sequential recombination jet algorithm. Like in the case of small-q_T resummation in Drell-Yan and Higgs production, the factorization is affected by a collinear anomaly. Our analysis provides the basis for a systematic resummation of large logarithms log(m_H/p_T^veto) beyond leading-logarithmic order. Specifically, we present predictions for the resummed jet-veto cross section and efficiency at next-to-next-to-leading logarithmic order. Our results have important implications for Higgs-boson searches at the LHC, where a jet veto is required to suppress background events.

A simplified quantum gravitational model of inflation

Inflationary quantum gravity simplifies drastically in the leading logarithm approximation. We show that the only counterterm which contributes in this limit is the 1-loop renormalization of the cosmological constant. We go further to make a simplifying assumption about the operator dynamics at leading logarithm order. This assumption is explicitly implemented at 1- and 2-loop orders, and we describe how it can be implemented nonperturbatively. We also compute the expectation value of an invariant observable designed to quantify the quantum gravitational back-reaction on inflation. Although our dynamical assumption may not prove to be completely correct, it does have the right time dependence, it can naturally produce primordial perturbations of the right strength, and it illustrates how a rigorous application of the leading logarithm approximation might work in quantum gravity. It also serves as a partial test of the ‘null hypothesis’ that there are no significant effects from infrared gravitons.

SHEAR VISCOSITY OF HOT QGP AT FINITE CHEMICAL POTENTIAL

The shear viscosity of quark–gluon plasma (QGP) at leading logarithm order at finite temperature and chemical potential in weakly coupled limit is studied by solving transport equations. The result shows that the chemical potential effect adds a positive μ2/T2 correction to the pure temperature case and thus increases the shear viscosity.

Universality of QCD traveling waves with running coupling

``Geometric scaling'', i.e. the dependence of DIS cross-sections on the ratio Q/Q_S, where Q_S(Y) is the rapidity-dependent \saturation scale, can be theoretically obtained from universal ``traveling wave'' solutions of the nonlinear Balitsky-Kovchegov (BK) QCD evolution equation at fixed coupling. We examine the similar mean-field predictions beyond leading-logarithmic order, including running QCD coupling.

Stability of subsequent-to-leading-logarithm corrections to the effective potential for radiative electroweak symmetry breaking

We demonstrate the stability under subsequent-to-leading logarithm corrections of the quartic scalar-field coupling constant λ and the running Higgs boson mass obtained from the (initially massless) effective potential for radiatively broken electroweak symmetry in the single-Higgs-doublet Standard Model. Such subsequent-to-leading logarithm contributions are systematically extracted from the renormalization group equation considered beyond one-loop order. We show λ to be the dominant coupling constant of the effective potential for the radiatively broken case of electroweak symmetry. We demonstrate the stability of λ and the running Higgs boson mass through five orders of successively subleading logarithmic corrections to the scalar-field-theory projection of the effective potential for which all coupling constants except the dominant coupling constant λ are disregarded. We present a full next-to-leading logarithm potential in the three dominant Standard Model coupling constants ( t-quark-Yukawa, α s , and λ) from these coupling constants' contribution to two loop β- and γ-functions. Finally, we demonstrate the manifest order-by-order stability of the physical Higgs boson mass in the 220 – 231 GeV range. In particular, we obtain a 231 GeV physical Higgs boson mass inclusive of the t-quark-Yukawa and α s coupling constants to next-to-leading logarithm order, and inclusive of the smaller SU ( 2 ) × U ( 1 ) gauge coupling constants to leading logarithm order.

Radiative electroweak symmetry breaking beyond leading logarithms

The top-quark Yukawa coupling is too large to permit radiative electroweak symmetry breaking to occur, to leading-logarithm order, for small values of y — the Higgs self-coupling. However, a large y solution leading to a viable Higgs mass of approximately 220 GeV does exist, and differs from conventional symmetry breaking by an approximately five-fold enhancement of the Higgs self-coupling. This scenario for radiative symmetry breaking is reviewed, and the order-by-order perturbative stability of this scenario is studied within the scalar-field theory projection of the Standard Model in which the Higgs self-coupling y represents the dominant Standard-Model coupling.PACS Nos.: 11.30.Qc, 11.10.Hi, 11.15.Tk, 12.15.Lk

Factorization Algorithm for Parton Showers beyond the Leading Logarithmic Order of QCD

A factorization algorithm for a patron shower model based on the evolution of momentum distributions proposed in a previous work is studied. The scaling violation of initial state parton distributions is generated using parton showers to an accuracy of the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD) using the information from only splitting functions and initial parton distributions at some fixed low energy. In the algorithm proposed in this paper, the total momentum of the initial state partons is conserved in any factorization scheme for mass singularities. As an example, the scaling violation of the parton distributions and the transverse momentum distributions due to initial state parton radiation are calculated with the jet calculs scheme.

Initial State Parton Evolution beyond the Leading Logarithmic Order of QCD

A Monte-Carlo model based on the evolution of momentum distributions proposed in a previous work is extended to the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD). Scaling violation of initial state parton distributions are generated using parton showers to accuracy of the NLL order by using the information from only splitting functions and initial parton distributions at some fixed low energy. Interference contributions due to the NLL order terms are taken into account for the generation of the transverse momenta in initial state parton radiations.

Collinear effective theory at subleading order and its application to heavy-light currents

We consider a collinear effective theory of highly energetic quarks with energy E, interacting with collinear and soft gluons by integrating out collinear degrees of freedom to subleading order. The collinear effective theory offers a systematic expansion in power series of a small parameter $\ensuremath{\lambda}\ensuremath{\sim}{p}_{\ensuremath{\perp}}/E,$ where ${p}_{\ensuremath{\perp}}$ is the transverse momentum of a collinear particle. We construct the effective Lagrangian to first order in $\ensuremath{\lambda}$ and discuss its features, including additional symmetries such as collinear gauge invariance and reparametrization invariance. Heavy-light currents can be matched from the full theory onto the operators in the collinear effective theory at one loop and to order $\ensuremath{\lambda}.$ We obtain heavy-light current operators in the effective theory, calculate their Wilson coefficients at this order, and the renormalization group equations for the Wilson coefficients are solved. As an application, we calculate the form factors for decays of B mesons to light energetic mesons to order $\ensuremath{\lambda}$ and at leading-logarithmic order in ${\ensuremath{\alpha}}_{s}.$

Sources ofbquarks at the Fermilab Tevatron and their correlations

The leading-logarithm order QCD hard scattering Monte Carlo models of HERWIG, ISAJET, and PYTHIA are used to study the sources of b quarks at the Fermilab Tevatron. The reactions responsible for producing b and $\overline{b}$ quarks are separated into three categories: flavor creation, flavor excitation, and parton-shower or fragmentation. Flavor creation corresponds to the production of a $b\overline{b}$ pair by gluon fusion or by annihilation of light quarks, while flavor excitation corresponds to a b or $\overline{b}$ quark being knocked out of the initial state by a gluon or a light quark or antiquark. The third source occurs when a $b\overline{b}$ pair is produced within a parton shower or during the fragmentation process of a gluon or a light quark or antiquark (includes gluon splitting). The QCD Monte Carlo models indicate that all three sources of b quarks are important at the Tevatron and when combined they qualitatively describe the inclusive cross-section data. Correlations between the b and $\overline{b}$ quark are very different for the three sources and can be used to isolate the individual contributions.

Jet rates at small x to single-logarithmic accuracy

We present predictions of jet rates in deep inelastic scattering at small x to leading-logarithmic order in x, including all sub-leading logarithms of Q^2/m_R^2 where m_R is the transverse momentum scale at which jets are resolved. We give explicit results for up to three jets, and a perturbative expansion for multi-jet rates and jet multiplicities.

Threshold production of heavy top quarks: QCD and the Higgs boson.

We calculate the threshold cross section for ${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}tt\ifmmode\bar\else\textasciimacron\fi{} to leading-logarithmic order in QCD, using a nonrelativistic approximation suggested by Fadin and Khoze. We study the mass range 100\ensuremath{\le}${\mathit{m}}_{\mathit{t}}$\ensuremath{\le}250 GeV, and show that the cross section is an excellent measure of ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$ for the lower portion of the mass range, while for a heavier top quark it is sensitive to the mass and couplings of the Higgs boson. We argue that a precise determination of ${\mathit{m}}_{\mathit{t}}$ and a measurement of ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{t}}$ are possible. We also show that nonperturbative effects are small, confirming that the tt\ifmmode\bar\else\textasciimacron\fi{} threshold is a detailed perturbative test of the standard model.

QCD radiative corrections to the transverse spin structure function g2(x,Q2): Nonsinglet operators.

We study QCD radiative corrections to the nucleon's transverse spin-dependent structure function ${g}_{2}(x,{Q}^{2})$ in the leading-logarithmic order for large ${Q}^{2}$. The anomalous-dimension matrix for the nonsinglet twist-3 operators appearing in the operator-product expansion for ${g}_{2}(x,{Q}^{2})$ is calculated and the experimental implications are discussed.