D-dimensional Unitarity Research Articles

Gluonic evanescent operators: two-loop anomalous dimensions

Evanescent operators are a special class of operators that vanish in four-dimensional spacetime but are non-zero in d = 4−2ϵ dimensions. In this paper, we continue our systematic study of the evanescent operators in the pure Yang-Mills theory and focus on their two-loop renormalization. We develop an efficient strategy to compute the two-loop divergences of form factors of high-dimensional and high-length operators by combining the d-dimensional unitarity method and the improved tensor reduction techniques. Two-loop anomalous dimensions are obtained for the dimension-10 basis in the planar YM theory, for which both the overline{textrm{MS}} scheme and the finite-renormalization scheme are used. We verify that the two-loop anomalous dimensions are the same in these two schemes at the Wilson-Fisher conformal fixed point. Our computation shows that the evanescent operators are indispensable in order to obtain the correct two-loop anomalous dimensions. This work provides a first computation of the two-loop anomalous dimensions of the complete set of dimension-10 operators. The method we use is also expected to provide an efficient strategy for the two-loop renormalization of general high-dimensional operators.

Gluonic evanescent operators: classification and one-loop renormalization

Evanescent operators are a special class of operators that vanish classically in four-dimensional spacetime, while in general dimensions they are non-zero and are expected to have non-trivial physical effects at the quantum loop level in dimensional regularization. In this paper we initiate the study of evanescent operators in pure Yang-Mills theory. We develop a systematic method for classifying and constructing the d-dimensional Lorentz invariant evanescent operators, which start to appear at mass dimension ten. We also compute one-loop form factors for the dimension-ten operators via the d-dimensional unitarity method and obtain their one-loop anomalous dimensions. These operators are necessary ingredients in the study of high dimensional operators in effective field theories involving a Yang-Mills sector.

Color-kinematics duality for Sudakov form factor in non-supersymmetric pure Yang–Mills theory

We study the duality between color and kinematics for the Sudakov form factors of in non-supersymmetric pure Yang-Mills theory. We construct the integrands that manifest the color-kinematics duality up to two loops. The resulting numerators are given in terms of Lorentz products of momenta and polarization vectors, which have the same powers of loop momenta as that from the Feynman rules. The integrands are checked by d-dimensional unitarity cuts and are valid in any dimension. We find that massless-bubble and tadpole topologies are needed at two loops to realize the color-kinematics duality. Interestingly, the two-loop solution contains a large number of free parameters suggesting the duality may hold at higher loop orders.

The dimension-shift conjecture for one-loop amplitudes

A conjecture made by Bern, Dixon, Dunbar, and Kosower asserts a simple dimension shifting relationship between the one-loop structure of mathcal{N} = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. We prove this conjecture to all orders in dimensional regularisation using unitarity cuts, and evaluate the form of these simplest one-loop amplitudes using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.

Symmetric reduction of high-multiplicity one-loop integrals and maximal cuts

We derive useful reduction formulae which express one-loop Feynman integrals with a large number of external momenta in terms of lower-point integrals carrying easily derivable kinematic coefficients which are symmetric in the external momenta. These formulae apply for integrals with at least two more external legs than the dimension of the external momenta, and are presented in terms of two possible bases: one composed of a subset of descendant integrals with one fewer external legs, the other composed of the complete set of minimally-descendant integrals with just one more leg than the dimension of external momenta. In 3+1 dimensions, particularly compact representations of kinematic invariants can be computed, which easily lend themselves to spinor-helicity or trace representations. The reduction formulae have a close relationship with D-dimensional unitarity cuts, and thus provide a path towards computing full (all-ϵ) expressions for scattering amplitudes at arbitrary multiplicity.

All-multiplicity one-loop amplitudes in Born-Infeld electrodynamics from generalized unitarity

We initiate a study of non-supersymmetric Born-Infeld electrodynamics in 4d at the quantum level. Explicit all-multiplicity expressions are calculated for the purely rational one-loop amplitudes in the self-dual (+ + … +) and next-to-self-dual (− + … +) helicity sectors. Using a supersymmetric decomposition, d-dimensional unitarity cuts of the integrand factorize into tree-amplitudes in a 4d model of Born-Infeld photons coupled to a massive complex scalar. The two-scalar tree-amplitudes needed to construct the Born-Infeld integrand are computed using two complimentary approaches: (1) as a double-copy of Yang-Mills coupled to a massive adjoint scalar with a dimensionally reduced form of Chiral Perturbation Theory, and (2) by imposing consistency with low-energy theorems under a reduction from 4d to 3d and T-duality. The Born-Infeld integrand is integrated in d = 4 − 2ϵ dimensions at order mathcal{O}left({in}^0right) using the dimension-shifting formalism. We comment on the implications for electromagnetic duality in quantum Born-Infeld theory.

NLO QCD corrections to top quark pair production in association with one hard jet at hadron colliders

We compute the QCD corrections to the production of a t t ¯ pair in association with one hard jet at the Tevatron and the LHC, using the method of generalized D-dimensional unitarity. Top quark decays are included at leading order in perturbative QCD. We present kinematic distributions of top quark decay products in lepton plus jets and dilepton final states at the Tevatron and the LHC, using realistic selection cuts. We confirm a strong reduction of the top quark forward–backward asymmetry in p p ¯ → t t ¯ j at next-to-leading order, first observed by Dittmaier, Uwer and Weinzierl. We argue that there is a natural way to understand this reduction and that it does not imply a breakdown of the perturbative expansion for the asymmetry.

NLO QCD corrections to top quark pair production and decay at hadron colliders

We present results for the next-to-leading order QCD corrections to the production and semi-leptonic decays of a top quark pair in hadron collisions, retaining all spin correlations. To evaluate the virtual corrections, we employ generalized D-dimensional unitarity. The computation is implemented in a numerical program which allows detailed studies of t-related observables at the Tevatron and the LHC.

Masses, fermions and generalized D-dimensional unitarity

We extend the generalized D-dimensional unitarity method for numerical evaluation of one-loop amplitudes by incorporating massive particles. The issues related to extending the spinor algebra to higher dimensions, treatment of external self-energy diagrams and mass renormalization are discussed within the context of the D-dimensional unitarity method. To validate our approach, we calculate in QCD the one-loop scattering amplitudes of a massive quark pair with up to three additional gluons for arbitrary spin states of the external quarks and gluons.

One-loop amplitudes forW+3 jet production in hadron collisions

We employ the recently developed method of generalized D-dimensional unitarity to compute one-loop virtual corrections to all scattering amplitudes relevant for the production of a W boson in association with three jets in hadronic collisions, treating all quarks as massless.

Unitarity Cuts for One-Loop Amplitudes

I present analytic expressions for coefficients of the master integrals of a general one-loop amplitude. The functions are derived from D-dimensional unitarity cuts with tree-level input.

Full one-loop amplitudes from tree amplitudes

We establish an efficient polynomial-complexity algorithm for one-loop calculations, based on generalized D-dimensional unitarity. It allows automated computations of both cut-constructible and rational parts of one-loop scattering amplitudes from on-shell tree amplitudes. We illustrate the method by (re)-computing all four-, five- and six-gluon scattering amplitudes in QCD at one-loop.

D-dimensional unitarity cut method

Abstract We develop a unitarity method to compute one-loop amplitudes with massless propagators in d = 4 − 2 ϵ dimensions. We compute double cuts of the loop amplitudes via a decomposition into a four-dimensional and a − 2 ϵ -dimensional integration. The four-dimensional integration is performed using spinor integration or other efficient techniques. The remaining integral in − 2 ϵ dimensions is cast in terms of bubble, triangle, box, and pentagon master integrals using dimensional shift identities. The method yields results valid for arbitrary values of ϵ.

Perturbative Quantum Gravity and its Relation to Gauge Theory.

In this review we describe a non-trivial relationship between perturbative gauge theory and gravity scattering amplitudes. At the semi-classical or tree-level, the scattering amplitudes of gravity theories in flat space can be expressed as a sum of products of well defined pieces of gauge theory amplitudes. These relationships were first discovered by Kawai, Lewellen, and Tye in the context of string theory, but hold more generally. In particular, they hold for standard Einstein gravity. A method based on D-dimensional unitarity can then be used to systematically construct all quantum loop corrections order-by-order in perturbation theory using as input the gravity tree amplitudes expressed in terms of gauge theory ones. More generally, the unitarity method provides a means for perturbatively quantizing massless gravity theories without the usual formal apparatus associated with the quantization of constrained systems. As one application, this method was used to demonstrate that maximally supersymmetric gravity is less divergent in the ultraviolet than previously thought.

On perturbative gravity and gauge theory

We review some applications of tree-level (classical) relations between gravity and gauge theory that follow from string theory. Together with D-dimensional unitarity, these relations can be used to perturbatively quantize gravity theories, i.e. they contain the necessary information for obtaining loop contributions. We also review recent applications of these ideas showing that N = 1 D = 11 supergravity diverges, and review arguments that N = 8 D = 4 supergravity is less divergent than previously thought, though it does appear to diverge at five loops. Finally, we describe field variables for the Einstein-Hilbert Lagrangian that help clarify the perturbative relationship between gravity and gauge theory.

One-loop n-point helicity amplitudes in (self-dual) gravity

We present an ansatz for all one-loop amplitudes in pure Einstein gravity for which the n external gravitons have the same outgoing helicity. These loop amplitudes, which are rational functions of the momenta, also arise in the quantization of self-dual gravity in four-dimensional Minkowski space. Our ansatz agrees with explicit computations via D-dimensional unitarity cuts for n≤6. It also has the expected analytic behavior, for all n, as a graviton becomes soft, and as two momenta become collinear. The gravity results are closely related to analogous amplitudes in (self-dual) Yang-Mills theory.