Large logarithmic corrections in $\hat s/p_t^2$ lead to substantial variations in the perturbative predictions for inclusive $W$-plus-dijet processes at the Large Hadron Collider. This instability can be cured by summing the leading-logarithmic contributions in $\hat s/p_t^2$ to all orders in $α_s$. As expected though, leading logarithmic accuracy is insufficient to guarantee a suitable description in regions of phase space away from the high energy limit.
We present (i) the first calculation of all partonic channels contributing at next-to-leading logarithmic order in $W$-boson production in association with at least two jets, and (ii) bin-by-bin matching to next-to-leading fixed-order accuracy. This new perturbative input is implemented in \emph{High Energy Jets}, and systematically improves the description of available experimental data in regions of phase space which are formally subleading with respect to $\hat s/p_t^2$.
The perturbation series for electroweak vector boson production at small transverse momentum is dominated by large double logarithms at each order in perturbation theory. An accurate theoretical prediction therefore requires a resummation of these logarithms. This can be performed either directly in transverse momentum space or in impact parameter (Fourier transform) space. While both approaches resum the same leading double logarithms, the subleading logarithms are, in general, treated differently. We comment on two recent approaches to this problem, emphasising the particular subleading loga…
We calculate the complete-NLO predictions for $t\bar{t}W^{\pm}$ and $t\bar{t}t\bar{t}$ production in proton--proton collisions at 13 and 100 TeV. All the non-vanishing contributions of $\mathcal{O}(α_s^i α^j)$ with $i+j=3,4$ for $t\bar{t}W^{\pm}$ and $i+j=4,5$ for $t\bar{t}t\bar{t}$ are evaluated without any approximation. For $t\bar{t}W^{\pm}$ we find that, due to the presence of $tW \to tW$ scattering, at 13(100) TeV the $\mathcal{O}(α_s α^3)$ contribution is about 12(70)\% of the LO, i.e., it is larger than the so-called NLO EW corrections (the $\mathcal{O}(α_s^2 α^2)$ terms) and has opposi…
We investigate the scale dependence of transverse momentum dependent(TMD) gluon distribution in saturation regime. We found that in the Collins-2011 scheme, the scale dependence of small x gluon TMD is governed by the same renormalization group(RG) equation that holds at moderate or large $x$. Following the standard procedure, one then can resum both double leading logarithm and single leading logarithm in saturation regime by jointly solving the Collins-Soper equation and the RG equation.
We apply the strategy of regions within dimensional regularization to find functions involved in evolution equations which govern the asymptotic dynamics of the Abelian form factor and four-fermion amplitude in the SU(N) gauge theory in the Sudakov limit up to the next-to-leading logarithmic approximation. The results are used for the analysis of the dominant electroweak corrections to the fermion-antifermion pair production in the $e^+e^-$ annihilation at high energy.
Future colliders will probe the electroweak theory at energies much larger than the gauge boson masses. Large double (DL) and single (SL) logarithmic virtual electroweak Sudakov corrections lead to significant effects for observable cross sections. Recently, leading and subleading universal corrections for external fermions and transverse gauge boson lines were resummed by employing the infrared evolution equation method. The results were confirmed at the DL level by explicit two loop calculations with the physical Standard Model (SM) fields. Also for longitudinal degrees of freedom the approa…
Recent results for the asymptotic behavior of fermion scattering amplitudes in the Sudakov limit are presented including next-to-leading logarithmic corrections. These are used for the analysis of the dominant electroweak corrections to the fermion-antifermion pair production in $e^+e^-$ annihilation at high energy.
In future collider experiments at the TeV scale, large logarithmic corrections originating from massive boson exchange can lead to significant corrections to observable cross sections. Recently double logarithms of the Sudakov-type were resummed for spontaneously broken gauge theories and found to exponentiate. In this paper we use the virtual contributions to the Altarelli-Parisi splitting functions to obtain the next to leading order kernel of the infrared evolution equation in the fixed angle scattering regime at high energies where particle masses can be neglected. In this regime the virtu…
We derive and solve renormalization group equations that allow for the resummation of subleading power rapidity logarithms. Our equations involve operator mixing into a new class of operators, which we term the "rapidity identity operators", that will generically appear at subleading power in problems involving both rapidity and virtuality scales. To illustrate our formalism, we analytically solve these equations to resum the power suppressed logarithms appearing in the back-to-back (double light cone) limit of the Energy-Energy Correlator (EEC) in $\mathcal{N}$=4 super-Yang-Mills. These logar…
Theoretical predictions show that at low values of Bjorken $x$ the spin structure function, $g_1$ is influenced by large logarithmic corrections, $ln^2(1/x)$, which may be predominant in this region. These corrections are also partially contained in the NLO part of the standard DGLAP evolution. Here we calculate the non-singlet component of the nucleon structure function, $g_1^{NS}=g_1^p-g_1^n$, and its first moment, using a unified evolution equation. This equation incorporates the terms describing the NLO DGLAP evolution and the terms contributing to the $ln^2(1/x)$ resummation. In order to …
A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum ($q_T$) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-$x$ logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consisten…
Standard dipole parton showers are known to yield incorrect subleading-colour contributions to the leading (double) logarithmic terms for a variety of observables. In this work, concentrating on final-state showers, we present two simple, computationally efficient prescriptions to correct this problem, exploiting a Lund-diagram type classification of emission regions. We study the resulting effective multiple-emission matrix elements generated by the shower, and discuss their impact on subleading colour contributions to leading and next-to-leading logarithms (NLL) for a range of observables. I…
We study in the BFKL approach the total hadronic cross section for the collision of two virtual photons for energies in the range of LEP2 and of future linear colliders. The BFKL resummation is done at the next-to-leading order in the BFKL Green's function; photon impact factors are taken instead at the leading order, but with the inclusion of the subleading terms required by invariance under changes of the renormalization scale and of the BFKL scale $s_0$. We compare our results with previous estimations based on a similar kind of approximation.
We derive high-order threshold corrections for top quark production in hadronic collisions from resummation calculations. We present analytical expressions for the cross section through next-to-next-to-next-to-next-to-leading order (N^4LO) and next-to-next-to-leading logarithmic accuracy. Special attention is paid to the role of subleading logarithms and how they relate to the convergence of the perturbation series and differences between various resummation prescriptions. It is shown that care must be taken to avoid unphysical terms in the expansions. Numerical results are presented for top q…
We consider the production, at future lepton colliders, of charged Higgs pairs in supersymmetric models. Assuming a relatively light SUSY scenario, and working in the MSSM, we show that, for c.m. energies in the one TeV range, a one-loop logarithmic Sudakov expansion that includes an "effective" next-to subleading order term is adequate to the expected level of experimental accuracy. We consider then the coefficient of the linear (subleading) SUSY Sudakov logarithm and the SUSY next to subleading term of the expansion and show that their dependence on the supersymmetric parameters of the model…
We present results from the first next-to-leading order (NLO) numerical analysis of forward hadron production in pA collisions in the small-x saturation formalism. Using parton distributions and fragmentation functions at NLO, as well as the dipole amplitude from the solution to the Balitsky-Kovchegov equation with running coupling, together with the NLO corrections to the hard coefficients, we obtain a good description of the available RHIC data in dAu collisions. We also comment on the results in the large p_T region beyond the saturation scale. Furthermore, we make predictions for forward h…
The hard-scattering contributions to heavy-to-light form factors at large recoil are studied systematically in soft-collinear effective theory (SCET). Large logarithms arising from multiple energy scales are resummed by matching QCD onto SCET in two stages via an intermediate effective theory. Anomalous dimensions in the intermediate theory are computed, and their form is shown to be constrained by conformal symmetry. Renormalization-group evolution equations are solved to give a complete leading-order analysis of the hard-scattering contributions, in which all single and double logarithms are…
The large double logarithm in loop-induced processes is one kind of logarithm at subleading power, which has a different origin from Sudakov double logarithms. We develop a method with soft-collinear effective theory to resum these large double logarithms to all orders in the strong coupling constant.
We analyse subleading electroweak effects in the top anti-top resonance production region in e+ e- collisions which arise due to the decay of the top and anti-top quarks into the W+ W- b bbar final state. These are NLO corrections adopting the non-relativistic power counting v ~ alpha_s ~ sqrt(alpha_EW). In contrast to the QCD corrections which have been calculated (almost) up to NNNLO, the parametrically larger NLO electroweak contributions have not been completely known so far, but are mandatory for the required accuracy at a future linear collider. The missing parts of these NLO contributio…
Three-jet event-shape distributions can be exploited to investigate the dependence of hadronisation effects on the colour and the geometry of the underlying hard event. We present here the first comparison of data in e+e- annihilation and state-of-the-art theoretical predictions, including resummation of large logarithms at next-to-leading logarithmic accuracy matched to exact next-to-leading order and leading non-perturbative power corrections.
Starting from the first renormalized factorization theorem for a process described at subleading power in soft-collinear effective theory, we discuss the resummation of Sudakov logarithms for such processes in renormalization-group improved perturbation theory. Endpoint divergences in convolution integrals, which arise generically beyond leading power, are regularized and removed by systematically rearranging the factorization formula. We study in detail the example of the $b$-quark induced $h\toγγ$ decay of the Higgs boson, for which we resum large logarithms of the ratio $M_h/m_b$ at next-to…
We consider the exclusive pseudoscalar heavy-quarkonium (eta_{b,c}) production in association with a photon at future lepton colliders where the collider energies of O(10^2) GeV are far greater than the quarkonium mass. At these energies, the logarithm of mass to collision energy becomes increasingly large hence its resummation becomes particularly important. By making use of the light-cone-distribution factorization formula, we resum the logarithms up to next-to-leading-logarithmic accuracy (NLL) that corresponds to order-alpha_s accuracy. We combine the resummed result with a known fixed-ord…
The perturbatively-stable scheme of Next-to-Leading order (NLO) calculations of cross-sections for multi-scale hard-processes in DIS-like kinematics is developed in the framework of High-Energy Factorization. The evolution equation for unintegrated PDF, which resums $\log 1/z$-corrections to the coefficient function in the Leading Logarithmic approximation together with a certain subset of Next-to-Leading Logarithmic and Next-to-Leading Power corrections, necessary for the perturbative stability of the formalism, is formulated and solved in the Doubly-Logarithmic approximation. An example of D…
We include a resummation of large transverse momentum logarithms in the next-to-leading order (NLO) Balitsky-Kovchegov equation. The resummed evolution equation is shown to be stable, the evolution speed being significantly reduced by NLO corrections. The contributions from NLO terms that are not enhanced by large logarithms are found to be numerically important close to phenomenologically relevant initial conditions. We numerically determine the value for the constant in the resummed logarithm that includes a maximal part of the full NLO terms in the resummation.
Factorisation theorem and pole cancellation in the resolved $Q_1^q$-$Q_{7γ}$ contribution to $\bar{B}\to X_sγ$ at NLO
Institute for Theoretical Physics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院理论物理研究所）；Institute for Astroparticle Physics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院天体粒子物理研究所）；Johannes Gutenberg University（约翰内斯·古腾堡大学）
In electroweak-boson production processes with a jet veto, higher-order corrections are enhanced by logarithms of the veto scale over the invariant mass of the boson system. In this paper, we resum these Sudakov logarithms at next-to-next-to-leading logarithmic (NNLL) accuracy and match our predictions to next-to-leading order (NLO) fixed-order results. We perform the calculation in an automated way, for arbitrary electroweak final states and in the presence of kinematic cuts on the leptons produced in the decays of the electroweak bosons. The resummation is based on a factorization theorem fo…
Recently the LHCb collaboration has announced the discovery of the $T_{cc}^+$ tetraquark. Being merely a few hundred ${\rm keV}$ below the $D^{*+} D^0$ threshold, the $T_{cc}^+$ is expected to have a molecular component, for which there is a good separation of scales that can be exploited to make reasonably accurate theoretical predictions about this tetraquark. Independently of its nature, the most important decay channels will be $D^+ D^0 π^0$, $D^0 D^0 π^+$ and $D^+ D^0 γ$. Its closeness to threshold suggests that the mass and particularly the width of the $T_{cc}^+$ tetraquark depend on th…
Double logarithms of $Q^2/m_c^2$ that appear in the cross section for $e^+e^-\to J/ψ+ η_c$ at next-to-leading order (NLO) in the strong coupling $α_s$ account for the bulk of the NLO correction at $B$-factory energies. (Here, $Q^2$ is the square of the center-of-momentum energy, and $m_c$ is the charm-quark mass.) We analyze the double logarithms that appear in the contribution of each NLO Feynman diagram, and we find that the double logarithms arise from both the Sudakov and the end-point regions of the loop integration. The Sudakov double logarithms cancel in the sum over all diagrams. We sh…
We begin a systematic investigation of the anomalous dimension of subleading power N-jet operators in view of resummation of logarithmically enhanced terms in partonic cross sections beyond leading power. We provide an explicit result at the one-loop order for fermion-number two N-jet operators at the second order in the power expansion parameter of soft-collinear effective theory.
Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process $e^+e^- \rightarrow γ^*/Z^*$ for the first three logarithmic contributions from $O(α^3 L^3), O(α^3 L^2), O(α^3 L)$ to $O(α^5 L^5), O(α^5 L^4), O(α^5 L^3)$ and compare their effects to the leading contribution $O(α^6 L^6)$ and one more subleading term $O(α^6 L^5)$. The calculation is performed in the limit of large center of mass energies squared $s \gg m_e^2$. These terms supplement the known corrections to $O(α^2)$, which were completed recently. Given the h…
We present the key features relevant to the automated computation of all the leading- and next-to-leading order contributions to short-distance cross sections in a mixed-coupling expansion, with special emphasis on the first subleading NLO term in the QCD+EW scenario, commonly referred to as NLO EW corrections. We discuss, in particular, the FKS subtraction in the context of a mixed-coupling expansion; the extension of the FKS subtraction to processes that include final-state tagged particles, defined by means of fragmentation functions; and some properties of the complex mass scheme. We combi…
We discuss recent results for the asymptotic behavior of fermion scattering amplitudes in the Sudakov limit. The results include next-to-next-to-leading logarithmic electroweak corrections and are used for the analysis of fermion-antifermion pair production in e+e- annihilation at high energy. The importance of the subleading logarithmic contributions is emphasized and the fermionic contributions to the Abelian form factor at two loops are discussed.
We present precise predictions for four-lepton plus jets production at the LHC obtained within the fully automated Sherpa+OpenLoops framework. Off-shell intermediate vector bosons and related interferences are consistently included using the complex-mass scheme. Four-lepton plus 0- and 1-jet final states are described at NLO accuracy, and the precision of the simulation is further increased by squared quark-loop NNLO contributions in the gg -> 4l, gg -> 4l+g, gq -> 4l+q, and qq -> 4l+g channels. These NLO and NNLO contributions are matched to the Sherpa parton shower, and the 0- and 1-jet fina…
Progresses towards the calculation and the understanding of NLO/NLL contributions to Deep Inelastic Scattering at low x with gluon saturation are being reviewed.
We present the evaluation of the complete set of NLO corrections to three-jet production at the LHC. To this end we consider all contributions of $\mathcal{O}(α_s^nα^m)$ with $n+m=3$ and $n+m=4$. This includes in particular also subleading Born contributions of electroweak origin, as well as electroweak virtual and QED real-radiative corrections. As an application we present results for the three- over two-jet ratio $R_{32}$. While the impact of non-QCD corrections on the total cross section is rather small, they can exceed $-10\%$ for high jet transverse momenta. The $R_{32}$ observable turns…
We review the program of the calculation of next-to-leading order corrections to the virtual photon impact factor. Following a brief introduction we present some technical aspects for the various contributions. Recently obtained results for transversely polarised virtual photons are discussed and an outline of how infrared divergences are cancelled is given. Implications of the subtraction of leading energy logarithms are discussed.
The $N$-jettiness observable $\mathcal{T}_N$ provides a way of describing the leading singular behavior of the $N$-jet cross section in the $τ=\mathcal{T}_N/Q \to 0$ limit, where $Q$ is a hard interaction scale. We consider subleading power corrections in the $τ\ll 1$ expansion, and employ soft-collinear effective theory to obtain analytic results for the dominant $α_s τ\lnτ$ and $α_s^2 τ\ln^3τ$ subleading terms for thrust in $e^+e^-$ collisions and $0$-jettiness for $q\bar q$-initiated Drell-Yan-like processes at hadron colliders. These results can be used to significantly improve the numeric…
We obtain a simple analytic expression for the high energy $γ^* γ^*$ scattering cross section at the next-to-leading order in the logarithms-of-energy power counting. To this end we employ the eigenfunctions of the NLO BFKL equation constructed in our previous paper. We also construct the eigenfunctions of the NNLO BFKL kernel and obtain a general form of the solution for the NNLO BFKL equation, which confirms the ansatz proposed in our previous paper.
A new method to construct event-generators based on next-to-leading order QCD matrix-elements and leading-logarithmic parton showers is proposed. Matrix elements of loop diagrams as well as those of a tree level can be generated using an automatic system. A soft/collinear singularity is treated using a leading-log subtraction method. Higher order re-summation of the soft/collinear correction by the parton shower method is combined with the NLO matrix-element without any double-counting in this method.
We present the status of the programme of calculating the next-to-leading order corrections to the virtual photon impact factor. In particular, we discuss new results for the transversely polarized photon. We briefly outline the definition of infrared finite terms and the subtraction of the leading logarithmic parts.
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 lambda ~ p_{\perp}/E, where p_{\perp} is the transverse momentum of a collinear particle. We construct the effective Lagrangian to first order in $λ$, and discuss its features including additional symmetries such as collinear gauge invariance and reparameterization invariance. Heavy-light currents can be m…
Motivated by the increased precision expected from LHC Run 3, equally accurate theory predictions are mandatory. As supersymmetry mass limits increase, predictions can be improved by threshold resummation. We examine the effects of including next-to-leading logarithms on associated squark-electroweakino production at the LHC and find a significant reduction in the uncertainty of factorisation and renormalisation scale dependence and a modest increase in the total cross section.
We construct an anomalous dimension for small x evolution which goes beyond standard fixed order perturbative evolution by including resummed small x logarithms deduced from the leading order BFKL equation with running coupling. Surprisingly, we find that once running coupling effects are properly taken into account, the leading approximation is very close to standard perturbative evolution in the range of x accessible at HERA, in overall agreement with the data, with no need for phenomenological parameters to summarise subleading effects. We also show that further corrections due to subleadin…
I present analytical expressions for the collinear and soft gluon corrections to Higgs production via the process b bbar -> H as well as gg -> H through next-to-next-to-next-to-leading order (NNNLO). The soft corrections are complete while the collinear corrections include leading and some subleading logarithms. Numerical results at the Tevatron and the LHC are presented, primarily for b bbar -> H. It is shown that the collinear terms greatly improve the soft and virtual approximation at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO), especially when subleading terms are …
The small-$x_B$ deep inelastic scattering in the saturation region is governed by the non-linear evolution of Wilson-line operators. In the leading logarithmic approximation it is given by the BK equation for the evolution of color dipoles. I discuss recent calculation of the next-to-leading order evolution of color dipoles in QCD and ${\cal N}=4$ SYM.
We quantify uncertainties in the Monte-Carlo simulation of inclusive and dijet final states, which arise from using the MC@NLO technique for matching next-to-leading order parton level calculations and parton showers. We analyse a large variety of data from early measurements at the LHC. In regions of phase space where Sudakov logarithms dominate over high-energy effects, we observe that the main uncertainty can be ascribed to the free parameters of the parton shower. In complementary regions, the main uncertainty stems from the considerable freedom in the simulation of underlying events.
