Two-loop supersymmetric QCD and half-maximal supergravity amplitudes
HJE
Two-loop supersymmetric QCD and half-maximal supergravity amplitudes
Henrik Johansson 0 1 2 3 5 6
Gregor Kalin 0 1 2 5 6
Gustav Mogull 0 1 2 4 5 6
Gauge Symmetry
0 The University of Edinburgh
1 Roslagstullsbacken 23 , 10691 Stockholm , Sweden
2 75108 Uppsala , Sweden
3 Nordita, Stockholm University and KTH Royal Institute of Technology
4 Higgs Centre for Theoretical Physics, School of Physics and Astronomy
5 Department of Physics and Astronomy, Uppsala University
6 Edinburgh EH9 3FD , Scotland , U.K
Using the duality between color and kinematics, we construct two-loop fourpoint scattering amplitudes in N = 2 super-Yang-Mills (SYM) theory coupled to Nf fundamental hypermultiplets. Our results are valid in D bound corresponds to six-dimensional chiral N close connection with N = 4 SYM theory | and, equivalently, six-dimensional N = (1; 1) SYM theory | we nd compact integrands with four-dimensional external vectors in both the maximally-helicity-violating (MHV) and all-chiral-vector sectors. Via the double-copy construction corresponding D-dimensional half-maximal supergravity amplitudes with external graviton multiplets are obtained in the MHV and all-chiral sectors. Appropriately tuning Nf enables us to consider both pure and matter-coupled supergravity, with arbitrary numbers of vector multiplets in D = 4. As a bonus, we obtain the integrands of the genuinely six-dimensional supergravities with N = (1; 1) and N = (2; 0) supersymmetry. Finally, we extract the potential ultraviolet divergence of half-maximal supergravity
Scattering Amplitudes; Supergravity Models; Supersymmetric Gauge Theory
-
= (1; 0) SYM theory. By exploiting a
in D = 5
2 and show that it non-trivially cancels out as expected. Keywords: Scattering Amplitudes, Supergravity Models, Supersymmetric Gauge Theory, Gauge Symmetry
1 Introduction
Review 2 3
Four dimensions
Six dimensions
3.2.1
3.2.2
Cuts
3.3.1
3.3.2
2.1
2.2
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
4
Calculation of N = 2 SQCD numerators
Four-dimensional correspondence
Six-dimensional amplitudes
One-loop example
Two-loop cuts
Master numerators
Ansatz construction
Symmetries and unitarity cuts
Additional constraints 4.4.1 4.4.2 4.4.3
Matter-reversal symmetry
Two-term identities
Matching with the N = 4 limit
All-chiral solutions
MHV solutions, bubbles and tadpoles
N = 0
5.1.1
5.1.2
N = 2
5.2.1
5.2.2
One loop
Two loops
One loop
Two loops
N = 4 construction
N = 2 construction
Color-kinematics duality with fundamental matter
Pure supergravities from the double-copy prescription
Trees and cuts in N = 2 SQCD
{ 1 {
Introduction
It is by now well established that a variety of gauge and gravity theories are perturbatively
related through the so-called double copy. In terms of the asymptotic states a squaring
relation between gauge theory and gravity follows readily from representation theory; that
such a structure is preserved by the interactions is a remarkable fact rst brought to light
by the Kawai-Lewellen-Tye (KLT) relations [1] between open and closed string amplitudes.
More recently, the double copy has been understood to arise due to the
Bern-CarrascoJohansson (BCJ) duality between color and kinematics [2, 3] that is present in many
familiar gauge theories. This realization has opened the path towards constructions of
scattering amplitudes in a multitude of di erent gravity theories [3{10] starting from the
much simpler gauge-theory amplitudes.
The duality between color and kinematics refers to the observation that many familiar
gauge theories have a hidden kinematic structure that mirrors that of the gauge-group color
structure [2, 3]. Amplitudes in, for example, pure Yang-Mills or super-Yang-Mills (SYM)
theories in generic dimensions can be brought to forms where the kinematic numerators of
individual diagrams obey Jacobi relations in complete analogue with the color factors of
the same diagrams. A natural expectation is that one (or several) unknown kinematic Lie
algebras underlie the duality [11, 12]. At tree level the duality is known to be equivalent
to the existence of BCJ relations between partial amplitudes [2], which in turn have been
proven for pure SYM theories through a variety of di erent techniques [13{17]. As of
yet there is no proof of the duality at loop level, although a number of calculations have
established its presence up to four loops in the maximally-supersymmetric N = 4 SYM
theory [3, 18{23], up to two loops for pure YM [24], and at one loop in matter-coupled YM
theories with reduced supersymmetry [4{7, 25, 26].
Color-kinematics duality has been shown to be present in weakly-coupled quantum
chromodynamics (QCD) and generalizations thereof [6, 27]; this includes Yang-Mills
theory coupled to massive quarks in any dimension, and corresponding supersymmetric
extensions. The BCJ amplitude relations for QCD were worked out in ref. [27] and proven in
ref. [28]. Using the duality for practical calculations in QCD phenomen (...truncated)