Resummation Of Logarithms Research Articles

Erratum: Next-to-next-to-leading logarithmic resummation for transverse thrust [Phys. Rev. D93, 054038 (2016)

Erratum: Next-to-next-to-leading logarithmic resummation for transverse thrust [Phys. Rev. D<b>93</b>, 054038 (2016)

Monte Carlo study of double logarithms in the smallxregion

We investigate the effect of the resummation of collinear double logarithms in the Balitsky-Fadin-Kuraev-Lipatov (BFKL) gluon Green function using the Monte Carlo event generator BFKLex. The resummed collinear terms in transverse momentum space were calculated by Sabio Vera [Nucl. Phys. B722, 65 (2005)] and correspond to the addition to the next-to-leading order BFKL kernel of a Bessel function of the first kind whose argument contains the strong coupling and a double logarithm of the ratio of the squared transverse momenta of the Reggeized gluons. We discuss how these additional terms improve the collinear convergence of the whole approach and reduce the asymptotic growth with energy of the cross sections. Taking advantage of the Monte Carlo implementation, we show how the new results reduce the diffusion of the gluon ladder into infrared and ultraviolet transverse momentum scales, while strongly affecting final state configurations by reducing the minijet multiplicity.

Mueller–Navelet jets at 13 TeV LHC: dependence on dynamic constraints in the central rapidity region

We study the production of Mueller-Navelet jets at 13 TeV LHC, within collinear factorization and including the BFKL resummation of energy logarithms in the next-to-leading approximation. We calculate several azimuthal correlations for different values of the rapidity separation $Y$ between the two jets and evaluate the effect of excluding those events where, for a given $Y$, one of the two jets is produced in the central region.

CombiningQTand small-xresummations

We analyze transverse momentum ($Q_T$) resummation of a colorless final state, e.g. Higgs production in gluon fusion or the production of a lepton pair via the Drell-Yan mechanism, in the limit where the invariant mass of the final state is much less then the center-of-mass energy, i.e. $Q^2\ll s$. We show how the traditional resummation of logarithms of $Q_T/Q$ can be supplemented with the resummation of the leading logarithmic contributions at small $x=Q^2/s$ and we compute the necessary ingredients to perform such joint resummation.

Next-to-next-to-leading logarithmic resummation for transverse thrust

We obtain a prediction for the hadron-collider event-shape variable transverse thrust in which the terms enhanced in the dijet limit are resummed to next-to-next-to-leading logarithmic accuracy. Our method exploits universality properties made manifest in the factorized expression for the cross section and only requires one-loop calculations. The necessary two-loop ingredients are extracted using known results and existing numerical codes. Our technique is general and applicable to other observables as well.

Inclusive Higgs production at large transverse momentum

We present a factorization formula for the inclusive production of the Higgs boson at large transverse momentum $P_T$ that includes all terms with the leading power of $1/P_T^2$. The cross section is factorized into convolutions of parton distributions, infrared-safe hard-scattering cross sections for producing a parton, and fragmentation functions that give the distribution of the longitudinal momentum fraction of the Higgs relative to the fragmenting parton. The infrared-safe cross sections and the fragmentation functions are perturbatively calculable. The most important fragmentation functions are those for which the fragmenting parton is the top quark, gluon, $W$, $Z$, and the Higgs itself. We calculate the fragmentation functions at leading order in the Standard Model coupling constants. The factorization formula enables the resummation of large logarithms of $P_T/M_H$ due to final-state radiation by integrating evolution equations for the fragmentation functions. By comparing the cross section for the process $q\bar{q}\to H t\bar t$ from the leading-power factorization formula at leading order in the coupling constants with the complete leading-order cross section, we infer that the error in the factorization formula decreases to less than 5\% for $P_T>600$ GeV at a future $100$ TeV collider.

Universal transverse momentum dependent soft function at NNLO

All (un)polarized transverse momentum dependent functions (TMDs), both distribution and fragmentation functions, are defined with the same universal soft function, which cancels spurious rapidity divergences within an individual TMD and renders them well-defined hadronic quantities. Moreover, it is independent of the kinematics, whether it is Drell-Yan, deep inelastic scattering, or ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}2$ hadrons. In this paper, we provide this soft function at next-to-next-to-leading order (NNLO), necessary for the calculation of all TMDs at the same order, and to perform the resummation of large logarithms at next-to-next-to-next-to-leading-logarithmic accuracy. From the results we obtain the $D$ function at NNLO, which governs the evolution of all TMDs. This work represents the first independent and direct calculation of this quantity. Given the all-order relation through a Casimir scaling between the soft function relevant for gluon TMDs and the one for quark TMDs, we also obtain the first at NNLO. The used regularization method to deal with the rapidity divergences is discussed as well.

Factorization, resummation and sum rules for heavy-to-light form factors

Precision calculations of heavy-to-light form factors are essential to sharpen our understanding towards the strong interaction dynamics of the heavy-quark system and to shed light on a coherent solution of flavor anomalies. We briefly review factorization properties of heavy-to-light form factors in the framework of QCD factorization in the heavy quark limit and discuss the recent progress on the QCD calculation of $B \to \pi$ form factors from the light-cone sum rules with the $B$-meson distribution amplitudes. Demonstration of QCD factorization for the vacuum-to-$B$-meson correlation function used in the sum-rule construction and resummation of large logarithms in the short-distance functions entering the factorization theorem are presented in detail. Phenomenological implications of the newly derived sum rules for $B \to \pi$ form factors are further addressed with a particular attention to the extraction of the CKM matrix element $|V_{ub}|$.

Hard matching for boosted tops at two loops

Cross sections for top quarks provide very interesting physics opportunities, being both sensitive to new physics and also perturbatively tractable due to the large top quark mass. Rigorous factorization theorems for top cross sections can be derived in several kinematic scenarios, including the boosted regime in the peak region that we consider here. In the context of the corresponding factorization theorem for $e^+e^-$ collisions we extract the last missing ingredient that is needed to evaluate the cross section differential in the jet-mass at two-loop order, namely the matching coefficient at the scale $\mu\simeq m_t$. Our extraction also yields the final ingredients needed to carry out logarithmic resummation at next-to-next-to-leading logarithmic order (or N$^3$LL if we ignore the missing 4-loop cusp anomalous dimension). This coefficient exhibits an amplitude level rapidity logarithm starting at $\mathcal{O}(\alpha_s^2)$ due to virtual top quark loops, which we treat using rapidity renormalization group (RG) evolution. Interestingly, this rapidity RG evolution appears in the matching coefficient between two effective theories around the heavy quark mass scale $\mu\simeq m_t$.

Resummation and matching of b-quark mass effects in b b ¯ H $$ b\overline{b}H $$ production

We use a systematic effective field theory setup to derive the $b\bar{b}H$ production cross section. Our result combines the merits of both fixed 4-flavor and 5-flavor schemes. It contains the full 4-flavor result, including the exact dependence on the $b$-quark mass, and improves it with a resummation of collinear logarithms of $m_b/m_H$. In the massless limit, it corresponds to a reorganized 5-flavor result. While we focus on $b\bar{b}H$ production, our method applies to generic heavy-quark initiated processes at hadron colliders. Our setup resembles the variable flavor number schemes known from heavy-flavor production in deep-inelastic scattering, but also differs in some key aspects. Most importantly, the effective $b$-quark PDF appears as part of the perturbative expansion of the final result where it effectively counts as an $O(\alpha_s)$ object. The transition between the fixed-order (4-flavor) and resummation (5-flavor) regimes is governed by the low matching scale at which the $b$-quark is integrated out. Varying this scale provides a systematic way to assess the perturbative uncertainties associated with the resummation and matching procedure and reduces by going to higher orders. We discuss the practical implementation and present numerical results for the $b\bar{b}H$ production cross section at NLO+NLL. We also provide a comparison to the corresponding predictions in the fixed 4-flavor and 5-flavor results and the Santander matching prescription. Compared to the latter, we find a slightly reduced uncertainty and a larger central value, with its central value lying at the lower edge of our uncertainty band.

Non-global logarithms, factorization, and the soft substructure of jets

An outstanding problem in QCD and jet physics is the factorization and resummation of logarithms that arise due to phase space constraints, so-called non-global logarithms (NGLs). In this paper, we show that NGLs can be factorized and resummed down to an unresolved infrared scale by making sufficiently many measurements on a jet or other restricted phase space region. Resummation is accomplished by renormalization group evolution of the objects in the factorization theorem and anomalous dimensions can be calculated to any perturbative accuracy and with any number of colors. To connect with the NGLs of more inclusive measurements, we present a novel perturbative expansion which is controlled by the volume of the allowed phase space for unresolved emissions. Arbitrary accuracy can be obtained by making more and more measurements so to resolve lower and lower scales. We find that even a minimal number of measurements produces agreement with Monte Carlo methods for leading-logarithmic resummation of NGLs at the sub-percent level over the full dynamical range relevant for the Large Hadron Collider. We also discuss other applications of our factorization theorem to soft jet dynamics and how to extend to higher-order accuracy.

Transverse-momentum resummation for vector-boson pair production at NNLL+NNLO

We consider the transverse-momentum (pT ) distribution of ZZ and W + W boson pairs produced in hadron collisions. At small pT , the logarithmically enhanced contributions due to multiple soft-gluon emission are resummed to all orders in QCD perturbation theory. At intermediate and large values of pT , we consistently combine resummation with the known xed-order results. We exploit the most advanced perturbative information that is available at present: next-to-next-to-leading logarithmic resummation combined with the next-to-next-to-leading xed-order calculation. After integration over pT , we recover the known next-to-next-to-leading order result for the inclusive cross section. We present numerical results at the LHC, together with an estimate of the corresponding uncertainties. We also study the rapidity dependence of the pT spectrum and we consider pT eciencies at dierent orders of resummed and xed-order perturbation theory.

QCD corrections to [formula omitted] form factors from light-cone sum rules

We compute perturbative corrections to B→π form factors from QCD light-cone sum rules with B-meson distribution amplitudes. Applying the method of regions we demonstrate factorization of the vacuum-to-B-meson correlation function defined with an interpolating current for pion, at one-loop level, explicitly in the heavy quark limit. The short-distance functions in the factorization formulae of the correlation function involves both hard and hard-collinear scales; and these functions can be further factorized into hard coefficients by integrating out the hard fluctuations and jet functions encoding the hard-collinear information. Resummation of large logarithms in the short-distance functions is then achieved via the standard renormalization-group approach. We further show that structures of the factorization formulae for fBπ+(q2) and fBπ0(q2) at large hadronic recoil from QCD light-cone sum rules match that derived in QCD factorization. In particular, we perform an exploratory phenomenological analysis of B→π form factors, paying attention to various sources of perturbative and systematic uncertainties, and extract |Vub|=(3.05−0.38+0.54|th.±0.09|exp.)×10−3 with the inverse moment of the B-meson distribution amplitude ϕB+(ω) determined by reproducing fBπ+(q2=0) obtained from the light-cone sum rules with π distribution amplitudes. Furthermore, we present the invariant-mass distributions of the lepton pair for B→πℓνℓ (ℓ=μ,τ) in the whole kinematic region. Finally, we discuss non-valence Fock state contributions to the B→π form factors fBπ+(q2) and fBπ0(q2) in brief.

A factorization approach to next-to-leading-power threshold logarithms

Threshold logarithms become dominant in partonic cross sections when the selected final state forces gluon radiation to be soft or collinear. Such radiation factorizes at the level of scattering amplitudes, and this leads to the resummation of threshold logarithms which appear at leading power in the threshold variable. In this paper, we consider the extension of this factorization to include effects suppressed by a single power of the threshold variable. Building upon the Low-Burnett-Kroll-Del Duca (LBKD) theorem, we propose a decomposition of radiative amplitudes into universal building blocks, which contain all effects ultimately responsible for next-to-leading power (NLP) threshold logarithms in hadronic cross sections for electroweak annihilation processes. In particular, we provide a NLO evaluation of the "radiative jet function", responsible for the interference of next-to-soft and collinear effects in these cross sections. As a test, using our expression for the amplitude, we reproduce all abelian-like NLP threshold logarithms in the NNLO Drell-Yan cross section, including the interplay of real and virtual emissions. Our results are a significant step towards developing a generally applicable resummation formalism for NLP threshold effects, and illustrate the breakdown of next-to-soft theorems for gauge theory amplitudes at loop level.

Mueller–Navelet jets at LHC: BFKL versus high-energy DGLAP

The production of forward jets separated by a large rapidity gap at LHC, the so-called Mueller–Navelet jets, is a fundamental testfield for perturbative QCD in the high-energy limit. Several analyses have already provided us with evidence about the compatibility of theoretical predictions, based on collinear factorization and BFKL resummation of energy logarithms in the next-to-leading approximation, with the CMS experimental data at 7 TeV of center-of-mass energy. However, the question if the same data can be described also by fixed-order perturbative approaches has not yet been fully answered. In this paper we provide numerical evidence that the mere use of partially asymmetric cuts in the transverse momenta of the detected jets allows for a clear separation between BFKL-resummed and fixed-order predictions in some observables related with the Mueller–Navelet jet production process.

Heavy dark matter annihilation from effective field theory.

We formulate an effective field theory description for SU(2)_{L} triplet fermionic dark matter by combining nonrelativistic dark matter with gauge bosons in the soft-collinear effective theory. For a given dark matter mass, the annihilation cross section to line photons is obtained with 5% precision by simultaneously including Sommerfeld enhancement and the resummation of electroweak Sudakov logarithms at next-to-leading logarithmic order. Using these results, we present more accurate and precise predictions for the gamma-ray line signal from annihilation, updating both existing constraints and the reach of future experiments.

Precision calculations for supersymmetric Higgs bosons

Recent progress is presented on higher-order calculations for the mass spectrum of Higgs particles in the CP-conserving and CP-violating MSSM, covering diagrammatic two-loop calculations for neutral and charged Higgs bosons as well as all-order resummation of large logarithms arising from the strong and Yukawa coupling sectors.

Exclusive radiative decays of W and Z bosons in QCD factorization

We present a detailed theoretical analysis of very rare, exclusive hadronic decays of the electroweak gauge bosons V=W, Z from first principles of QCD. Our main focus is on the radiative decays V->M+gamma, in which M is a pseudoscalar or vector meson. At leading order in an expansion in powers of Lambda_{QCD}/m_V the decay amplitudes can be factorized into convolutions of calculable hard-scattering coefficients with the leading-twist light-cone distribution amplitude of the meson M. Power corrections to the decay rates arise first at order (Lambda_{QCD}/m_V)^2. They can be estimated in terms of higher-twist distribution amplitudes and are predicted to be tiny. We include one-loop O(alpha_s) radiative corrections to the hard-scattering coefficients and perform the resummation of large logarithms [alpha_s log(m_V^2/mu_0^2)]^n (with mu_0=1 GeV a typical hadronic scale) to all orders in perturbation theory. Evolution effects have an important impact both numerically and conceptually, since they reduce the sensitivity to poorly determined hadronic parameters. We present detailed numerical predictions and error estimates, which can serve as benchmarks for future precision measurements. We also present an exploratory study of the weak radiative decays Z->M+W. Some of the decay modes studied here have branching ratios large enough to be accessible in the high-luminosity run of the LHC. Many of them can be measured with high accuracy at a future lepton collider. This will provide stringent tests of the QCD factorization formalism and enable novel searches for new physics.

Solution of the NLO BFKL Equation and γ*γ* Cross-Section at NLO

The analytic NLO cross section of the γ*γ* scattering process at high-energy is calculated. The result is obtained factorizing the scattering amplitude in rapidity space using the high-energy Operator Product Expansion in terms of composite Wilson line operators. The next-to-leading logarithmic resummation is included employing the solution of the NLO BFKL equation which obtained from the linearization of the evolution in rapidity space of the composite Wilson line operators.

Soft collinear effective theory for heavy WIMP annihilation

In a large class of models for Weakly Interacting Massive Particles (WIMPs), the WIMP mass $M$ lies far above the weak scale $m_W$. This work identifies universal Sudakov-type logarithms $\sim \alpha \log^2 (2\,M/m_W)$ that spoil the naive convergence of perturbation theory for annihilation processes. An effective field theory (EFT) framework is presented, allowing the systematic resummation of these logarithms. Another impact of the large separation of scales is that a long-distance wave-function distortion from electroweak boson exchange leads to observable modifications of the cross section. Careful accounting of momentum regions in the EFT allows the rigorously disentanglement of this so-called Sommerfeld enhancement from the short distance hard annihilation process. The WIMP is modeled as a heavy-particle field, while the light, energetic, final-state electroweak gauge bosons are treated as soft and collinear fields. Hard matching coefficients are computed at renormalization scale $\mu \sim 2\,M$, then evolved down to $\mu \sim m_W$, where electroweak symmetry breaking is incorporated and the matching onto the relevant quantum mechanical Hamiltonian is performed. The example of an $SU(2)_W$ triplet scalar dark matter candidate annihilating to line photons is used for concreteness, allowing the numerical exploration of the impact of next-to-leading order corrections and log resummation. For $M \simeq 3$ TeV, the resummed Sommerfeld enhanced cross section is reduced by a factor of $\sim 3$ with respect to the tree-level fixed order result.