David C. Dunbar, Warren B. Perkins, Joseph M. W. Strong
We present an explicit expression for a particular $n$-gluon two loop scattering partial amplitude. Specifically we present an analytic form for the single trace $N_c$ independent colour partial amplitude for the case where all external gluons have positive helicity.
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We calculate the expectation value of one circular Wilson loop and the correlator of two concentric circular Wilson loops in AdS/QCD using the modified AdS_5-metric given in Ref.[1]. The confinement properties of this metric in AdS/QCD are analyzed and compared with QCD and Nambu-Goto theory in four dimensions.
We use AdS/QCD to analyze the quark and gluon scalar and pseudo-scalar condensates around static color sources described by a circular Wilson loop. We also derive the static dipole-dipole interactions between rectangular Wilson loops in AdS/QCD and discuss their relevance for static string interactions in QCD at strong coupling.
We review some of the modern approaches to scattering amplitude computations in QCD and their application to precision LHC phenomenology. We emphasise the usefulness of momentum twistor variables in parameterising general amplitudes.
We present the decomposition of QCD partial amplitudes into primitive amplitudes at one-loop level and tree level for arbitrary numbers of quarks and gluons. Our method is based on shuffle relations. This method is purely combinatorial and does not require the inversion of a system of linear equations.
I review the derivation of large-N QCD meson scattering amplitudes in the Regge regime, where the effective theory of long strings applies in d=4. A special attention is payed to the reparametrization path integral which plays a crucial role in the consistency of off-shell amplitudes. I show how the linear Reggeon trajectory is obtained for QCD string in the mean-field approximation, which turns out to be exact for the Nambu-Goto string, and discuss the interrelation with perturbative QCD.
Using the recent solution of the triple Pomeron coupling in the QCD dipole picture as a closed string amplitude with six legs, its analytical form in terms of hypergeometric functions and numerical value are derived.
We show that all Born amplitudes in QCD can be calculated from scalar propagators and a set of three- and four-valent vertices. In particular, our approach includes amplitudes with any number of quark pairs. The quarks may be massless or massive. The proof of the formalism is given entirely within quantum field theory.
We calculate the complete set of two-loop leading-colour QCD helicity amplitudes for $γγj$-production at hadron colliders. Our results are presented in a compact, fully-analytical form.
In this talk we discuss the color decomposition of tree-level and one-loop QCD amplitudes with arbitrary numbers of quarks and gluons. We present a method for the decomposition of partial amplitudes into primitive amplitudes, which is based on shuffle relations and is purely combinatorial. Closed formulae are derived, which do not require the inversion of a system of linear equations.
We analyse the high-energy limit of the gluon-gluon scattering amplitude in QCD, and display an intriguing relation between the finite parts of the one-loop gluon impact factor and the finite parts of the two-loop Regge trajectory.
We use the infrared consistency of one-loop amplitudes in N=4 Yang-Mills theory to derive a compact analytic formula for a tree-level NNMHV gluon scattering amplitude in QCD, the first such formula known. We argue that the IR conditions, coupled with recent advances in calculating one-loop box coefficients, can give a new tool for computing tree-level amplitudes in general. Our calculation suggests that many amplitudes have a structure which is even simpler than that revealed so far by current twistor-space constructions.
This paper has been withdrawn by the author due to an error made in the analysis of MHV amplitudes with external fundamental matter fields in the superconformal N=2 SQCD.
I present results for soft anomalous dimensions through three loops for many QCD processes. In particular, I give detailed expressions for soft anomalous dimensions in various processes with electroweak and Higgs bosons as well as single top quarks and top-antitop pairs.
We link the FDR treatment of ultraviolet (UV) divergences to dimensional regularization up to two loops in QCD. This allows us to derive the one-loop and two-loop coupling constant and quark mass shifts necessary to translate infrared finite quantities computed in FDR to the MSbar renormalization scheme. As a by-product of our analysis, we solve a problem analogous to the breakdown of unitarity in the Four Dimensional Helicity (FDH) method beyond one loop. A fix to FDH is then presented that preserves the renormalizability properties of QCD without introducing evanescent quantities.
We derive the analytic expression of the two one-loop dipole contributions to the elastic 4-gluon amplitude in QCD. The first one corresponds to the double QCD pomeron exchange, the other to an order alpha^2 correction to one-pomeron exchange. Both are expressed in terms of the square of the recently derived triple QCD pomeron vertex and involve a summation over all conformal Eigenvectors of the BFKL kernel.
We derive color decompositions of arbitrary tree and one-loop QCD amplitudes into color ordered objects called primitive amplitudes. Furthermore, we derive general fermion flip and reversion identities spanning the null space among the primitive amplitudes and use them to prove that all color ordered tree amplitudes of massless QCD can be written as linear combinations of color ordered tree amplitudes of $\mathcal{N}=4$ super Yang-Mills theory.
We review recent developments for the calculation of Born amplitudes in QCD. This includes the computation of gluon helicity amplitudes from MHV vertices and an approach based on scalar propagators and a set of three- and four-valent vertices. The latter easily generalizes to amplitudes with any number of quark pairs. The quarks may be massless or massive.
We elaborate on the radiative behaviour of tree-level scattering amplitudes in the soft regime. We show that the sub-leading soft term in single-gluon emission of quark-gluon amplitudes in QCD is controlled by differential operators, whose universal form can be derived from both on-shell recursion relation and gauge invariance, as it was shown to hold for graviton- and gluon-scattering.
In this note we discuss a generalization of the Lipatov's effective action approach, \cite{LipatovEff,LipatovEff1}, for the case of description of gluon and quark production amplitudes in the quasi-multi-Regge kinematics. Following to \cite{Faddeev}, we define the S-matrix elements of high energy QCD processes in this kinematics and discuss applications of the obtained results.
A general formalism for computing only the rational parts of oneloop QCD amplitudes is developed. Starting from the Feynman integral representation of the one-loop amplitude, we use tensor reduction and recursive relations to compute the rational parts directly. Explicit formulas for the rational parts are given for all bubble and triangle integrals. Formulas are also given for box integrals up to two-masshard boxes which are the needed ingredients to compute up to 6-gluon QCD amplitudes. We use this method to compute explicitly the rational parts of the 5- and 6-gluon QCD amplitudes in two ac…
Splitting amplitudes at N$^3$LO in QCD
本文利用基于区域方法的按子图展开新技术计算三圈阶 QCD 分裂振幅，给出任意类时分裂及三部分子类空分裂的普适结果。
We compute the two-loop massless QCD corrections to the four-point amplitude $g+g \rightarrow H+H$ resulting from effective operator insertions that describe the interaction of a Higgs boson with gluons in the infinite top quark mass limit. This amplitude is an essential ingredient to the third-order QCD corrections to Higgs boson pair production. We have implemented our results in a numerical code that can be used for further phenomenological studies.
We review the extension of modern techniques for the calculation of helicity amplitudes in QCD to massive particles. The focus is on the use of supersymmetric Ward identites that relate amplitudes with massive quarks to those with massive scalars, the application of on-shell recursion relations to amplitudes with massive quarks and an extension of the CSW rules to massive scalars.
We analytically compute all four-loop QCD corrections to the photon-quark and Higgs-gluon form factors involving a closed massless fermion loop. Our calculation of non-planar vertex integrals confirms a previous conjecture for the analytical form of the non-fermionic contributions to the collinear anomalous dimensions of quarks and gluons.
Two-loop QCD Amplitudes from the Chiral Algebra Bootstrap
本文利用手征代数自举法，结合超对称Ward恒等式，计算了无质量QCD中两圈全正螺旋度胶子振幅，并证明全色两圈振幅可由扭量理论形状因子及低圈振幅得到，从而简化计算。
We calculate the small angle expansion of the four-loop QCD cusp anomalous dimension. As a byproduct of our calculation, we also obtain the four-loop anomalous dimension of the heavy-quark field in HQET. The validity of the calculational setup is cross-checked by the independent calculation of the four-loop QCD beta-function from heavy-quark-gluon vertex renormalization in HQET. We check the obtained results for the cusp anomalous dimension and heavy-quark field anomalous dimension against available analytical and numerical results. Finally, we find that the maximal transcendentality contribut…
Using the QCD dipole picture of BFKL dynamics and the conformal invariance properties of the BFKL kernel in transverse coordinate space, we show that the 1--to--p dipole densities can be expressed in terms of dual Shapiro- Virasoro amplitudes B{2p+2} and their generalization including non-zero conformal spins. We discuss the possibility of an effective closed string theory of interacting QCD dipoles.
We find a one to one mapping between low energy string dilaton states in AdS bulk and high energy glueball states on the corresponding boundary. This holographic mapping leads to a relation between bulk and boundary scattering amplitudes. From this relation and the dilaton action we find the appropriate momentum scaling for high energy QCD amplitudes at fixed angles.
The effective action for hard thermal loops in QCD is related to a gauged WZNW theory. Some of the technical issues of this approach are clarified and the Hamiltonian formulation is presented. The two-point correlation function for the induced current in QCD is obtained; some simplifications of the dynamics of the longitudinal modes are also pointed out.
In this talk we review relations and representations of primitive QCD tree amplitudes. Topics covered include the BCJ relations, the CHY representation, and the KLT relations. We will put a special emphasis on how these relations and representations generalise from pure Yang-Mills theory to QCD. The generalisation of the KLT relations from pure Yang-Mills to QCD includes the case of massive quarks. On the gravity side we then obtain hypothetical particles interacting with gravitational strength, which can be massive and non-relativistic.
Many Texo's have been corrected and a reference added.
We review progress in calculating one-loop scattering amplitudes required for next-to-leading-order corrections to QCD processes. The underlying technical developments include the spinor helicity formalism, color decompositions, supersymmetry, string theory, factorization and unitarity. We provide explicit examples illustrating these techniques.
We calculate the four-loop massless QCD corrections with three closed quark lines to quark and gluon form factors. We apply a novel integration by parts algorithm based on modular arithmetic and compute all relevant master integrals for arbitrary values of the space-time dimension. This is the first calculation of a gluon form factor at this perturbative order in QCD.
We consider on-shell recursion relations for all Born QCD amplitudes. This includes amplitudes with several pairs of quarks and massive quarks. We give a detailed description on how to shift the external particles in spinor space and clarify the allowed helicities of the shifted legs. We proof that the corresponding meromorphic functions vanish at z --> infinity. As an application we obtain compact expressions for helicity amplitudes including a pair of massive quarks, one negative helicity gluon and an arbitrary number of positive helicity gluons.
We calculate the collinear anomalous dimensions in massless four-loop QCD and $\mathcal{N} = 4$ supersymmetric Yang-Mills theory from the infrared poles of vertex form factors. We give very precise numerical approximations and a conjecture for the complete analytic results in both models we consider.
Asymptotic Padé Predictions up to Six Loops in QCD and Eight Loops in $λϕ^4$
本文基于后续精确结果评估了QCD五阶β函数和夸克质量反常维数的渐近Padé预测精度，发现低阶系数在1%以内准确，并展示了渐近Padé方法在更高阶循环中预测精度提升，从而给出六阶QCDβ函数、夸克质量反常维数及八阶λϕ⁴理论β函数的预测。
We review techniques simplifying the analytic calculation of one-loop QCD amplitudes with many external legs, for use in next-to-leading-order corrections to multi-jet processes. Particularly useful are the constraints imposed by perturbative unitarity, collinear singularities and a supersymmetry-inspired organization of helicity amplitudes. Certain sequences of one-loop helicity amplitudes with an arbitrary number of external gluons have been obtained using these constraints.
We generalize modern ideas about the duality between Wilson loops and scattering amplitudes in ${\cal N}$=4 SYM to large-N (or quenched) QCD. We show that the area-law behavior of asymptotically large Wilson loops is dual to the Regge-Veneziano behavior of scattering amplitudes at high energies and fixed momentum transfer, when quark mass is small and/or the number of particles is large. We elaborate on this duality for string theory in a flat space, identifying the asymptotes of the disk amplitude and the Wilson loop of large-N QCD.
We present examples of on-shell recurrence relations for determining rational functions appearing in one-loop QCD amplitudes. In particular, we give relations for one-loop QCD amplitudes with all legs of positive helicity, or with one leg of negative helicity and the rest of positive helicity. Our recursion relations are similar to the tree-level ones described by Britto, Cachazo, Feng and Witten. A number of new features arise for loop amplitudes in non-supersymmetric theories like QCD, including boundary terms and double poles. We show how to eliminate the boundary terms, which would interfe…
We explicitly construct the dominant saddle-point trajectory in the sum-over-path representation of meson scattering amplitudes in large $N$ QCD for area-behaved Wilson loops and show that it dominates in the Regge regime. The graphic representation of the leading trajectory is very similar to the diagrams widely used to illustrate meson scattering.
QCD string is formed at the distances larger than the confinement scale and can be described by the Polchinski--Strominger effective string theory with a nonpolynomial action, which has nevertheless a well-defined semiclassical expansion around a long-string ground state. We utilize modern ideas about the Wilson-loop/scattering-amplitude duality to calculate scattering amplitudes and show that the expansion parameter in the effective string theory is small in the Regge kinematical regime. For the amplitudes we obtain the Regge behavior with a linear trajectory of the intercept (d-2)/24 in d di…
Color-factor symmetry is used to derive a KLT-type relation for tree-level QCD amplitudes containing gluons and an arbitrary number of massive or massless quark-antiquark pairs, generalizing the expression for Yang-Mills amplitudes originally postulated by Bern, De Freitas, and Wong. An explicit expression is given for all amplitudes with two or fewer quark-antiquark pairs in terms of the (modified) momentum kernel. We also introduce the bicolor scalar theory, the "zeroth copy" of QCD, containing massless biadjoint scalars and massive bifundamental scalars, generalizing the biadjoint scalar th…
Renormalization of the pseudoscalar operator at four loops in QCD
本文计算了QCD中采用非反交换$γ_5$定义的赝标算符直至四圈阶的重整化常数，并利用重整化群不变性预测了其五圈阶的$\overline{\mathrm{MS}}$因子。
The integrand form of infrared singularities of two-loop QCD scattering amplitudes
本文提出用局部有限主积分基表达二阶QCD散射振幅的红外奇异性，通过系统重构积分形式，为双胶子产生振幅的有限部分提供显式表达。
The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants $2 p_i p_j$. The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in ${\mathcal N}=4$ super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive …
We compute the complete one-loop corrections to the simplest class of QCD gluon amplitudes, those with two color-adjacent opposite-helicity external particles. We present results for an arbitrary number of external legs. The computation uses the recently-developed on-shell recursion relations for the rational parts, along with older unitarity-based results for the cut-containing terms.
We generalize modern ideas about the duality between Wilson loops and scattering amplitudes in ${\cal N}=4$ SYM to large $N$ QCD by deriving a general relation between QCD meson scattering amplitudes and Wilson loops. We then investigate properties of the open-string disk amplitude integrated over reparametrizations. When the Wilson loop is approximated by the area behavior, we find that the QCD scattering amplitude is a convolution of the standard Koba-Nielsen integrand and a kernel. As usual poles originate from the first factor, whereas no (momentum dependent) poles can arise from the kerne…
We present the two-loop pure gauge contribution to the gluon-gluon scattering amplitude with maximal helicity violation. Our construction of the amplitude does not rely directly on Feynman diagrams, but instead uses its analytic properties in 4 - 2 epsilon dimensions. We evaluate the loop integrals appearing in the amplitude through O(epsilon^0) in terms of polylogarithms.
Two-Loop Spacelike Splitting Amplitudes in Full-Color QCD
本文首次完整计算了全色QCD中双重环面色分裂幅度，验证了其在N=4超Yang-Mills理论中的一类贡献的普遍性，并识别了新的色因子破坏效应源。
For the first time, we compute three-loop contributions to all triple vertices in QCD at the symmetric point. The analytic results are obtained in massless QCD with an arbitrary color group in the Landau gauge. All new loop integrals are expressed in terms of harmonic polylogarithms at the sixth root of unity. These corrections allow us to derive expressions for the four-loop QCD beta function in a set of momentum-subtraction schemes.
We discuss divergences of loop functions in thermal QCD and compute perturbatively the Polyakov loop, the Polyakov loop correlator and the cyclic Wilson loop. We show how these functions get mixed under renormalization.
We report on a technique for evaluating finite unitarity cut for one-loop amplitudes in gauge theories, and discuss its application to the cut-constructible part of six-gluon amplitude in QCD.
We evaluate the static QCD potential to two--loop order. Compared to a previous calculation a sizable reduction of the two--loop coefficient $a_2$ is found.
In this talk we describe the recursive Approach to One-loop QCD Matrix Elements.
The B-meson distribution amplitude is calculated using QCD sum rules. In particular we obtain an estimate for the integral relevant to exclusive B-decays λ_B = 460 \pm 110 MeV at the scale 1 GeV. A simple QCD-motivated parametrization of the distribution amplitude is suggested.
Reply to Comments on ``Asymptotic Estimate of the {\it n}-Loop QCD Contribution to the Total $e^{+}e^{-}$ Annihilation Cross Section''
We discuss the structure of infrared and ultraviolet singularities in on-shell QCD and supersymmetric QCD amplitudes at one-loop order. Previous results, valid for massless partons, are extended to the case of massive partons. Using dimensional regularization, we present a general factorization formula that controls both the singular epsilon-poles and the logarithmic contributions that become singular for vanishing masses. We introduce generalized Altarelli--Parisi splitting functions and discuss their relations with the singular terms in the amplitudes. The dependence on the regularization sc…
It is known that the correlator of one axial and two vector currents, that receives leading contributions through one-loop fermion triangle diagrams, is not modified by QCD radiative corrections at two loops. It was suggested that this non-renormalization of the VVA correlator persists in higher orders in perturbative QCD as well. To check this assertion, we compute the three-loop QCD corrections to the VAA-correlator u sing the technique of asymptotic expansions. We find that these corrections do not vanish and that they are proportional to the QCD beta-function.
I discuss recent progress in the calculation of two-loop QCD corrections in the eikonal approximation. I present specific results for the UV structure of the corrections.
We give a short overview of the renormalization properties of rectangular Wilson loops, the Polyakov loop correlator and the cyclic Wilson loop. We then discuss how to renormalize loops with more than one intersection, using the simplest non-trivial case as an illustrative example. Our findings expand on previous treatments. The generalized exponentiation theorem is applied to the Polyakov loop correlator and used to renormalize linear divergences in the cyclic Wilson loop.
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