Loop corrections to soft theorems in gauge theories and gravity (original) (raw)

The soft-collinear bootstrap: Yang-Mills amplitudes at six- and seven-loops

2012

Infrared divergences in scattering amplitudes arise when a loop momentum becomes collinear with a massless external momentum p. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N = 4 super-Yang-Mills theory enjoy softer than expected behavior as p already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any loop-order, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation, we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary looporder. The six-and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively.

Loop corrected soft photon theorem as a Ward identity

Journal of High Energy Physics

Recently Sahoo and Sen obtained a series of remarkable results concerning sub­ leading soft photon and graviton theorems in four dimensions. Even though the S-matrix is infrared divergent, they have shown that the subleading soft theorems are well defined and exact statements in QED and perturbative Quantum Gravity. However unlike the well studied Cachazo-Strominger soft theorems in tree-level amplitudes, the new subleading soft expansion is at the order ln w (where w is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-level counterparts. Whence it is natural to ask if these theorems are associated to asymptotic symmetries of the S-matrix. We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are indeed an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem. This shows that in the case of four dimensi...

Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities

Physical Review Letters

We show that the soft photon, gluon and graviton theorems can be understood as the Ward-Takahashi identities of large gauge transformation, i.e., diffeomorphism that does not fall off at spatial infinity. We found infinitely many new identities which constrain the higher order soft behavior of the gauge bosons and gravitons in scattering amplitudes of gauge and gravity theories. Diagrammatic representations of these soft theorems are presented. 1 The method of [19] has been applied to flat spacetime in Ref. [21], but only the leading soft theorem was derived there. 2 Similar approach to the subleading soft graviton theorem can be found in Ref. [22] though, unlike our present work, selfinteractions of the graviton were neglected.

On Tree Amplitudes in Gauge Theory and Gravity

2008

The BCFW recursion relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum-a single copy of a "Lorentz" group for gauge theory and two copies for gravity-which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.

Gauge theory one-loop amplitudes and the BCFW recursion relations

2012

We calculate gauge theory one-loop amplitudes with the aid of the complex shift used in the Britto-Cachazo-Feng-Witten (BCFW) recursion relations of tree amplitudes. We apply the shift to the integrand and show that the contribution from the limit of infinite shift vanishes after integrating over the loop momentum, by a judicious choice of basis for polarization vectors. This enables us to write the one-loop amplitude in terms of on-shell tree and lower point one-loop amplitudes. Some of the tree amplitudes are forward amplitudes. We show that their potential singularities do not contribute and the BCFW recursion relations can be applied in such a way as to avoid these singularities altogether. We calculate in detail n-point one-loop amplitudes for n=2,3,4, and outline the generalization of our method to n>4.

More on soft theorems: Trees, loops, and strings

Physical Review D, 2015

We study soft theorems in a broader context, addressing their fate at loop level and their universality in effective field theories and string theory. We argue that for gauge theories in the planar limit, loop-level soft gluon theorems can be made manifest already at the integrand level. In particular, we show that the planar integrand for N = 4 SYM satisfies the tree-level soft theorem to all orders in perturbation theory and provide strong evidence for integrands in N < 4 SYM. We consider soft theorems for non-supersymmetric Yang-Mills theories and gravity, where no such representation of the integrand is available, and show that loop corrections to the integrated soft theorems are intimately tied to the presence of quantum symmetry anomalies. We then address the question of universality of the soft theorems for various theories. In effective field theories with F 3 and R 3 interactions, the soft theorems are not modified. However for gravity theories with R 2 φ interactions, the next-next-leading order soft graviton theorem, which is beyond what is implied by the extended BMS symmetry, requires modifications at tree level for non-supersymmetric theories, and at loop level for N ≤ 4 supergravity due to anomalies. Finally, for superstring amplitudes at finite α , via explicit calculation for lower-point examples as well as world-sheet OPE analysis for arbitrary multiplicity, we show that the superstring theory amplitudes satisfy the same soft theorem as its field-theory counterpart. This is no longer true for bosonic closed strings due to the presence of R 2 φ interactions.

From U(1) to E8: soft theorems in supergravity amplitudes

Journal of High Energy Physics

It is known that for N=8 supergravity, the double-soft-scalar limit of a n-point amplitude is given by a sum of local SU(8) rotations acting on a (n-2)-point amplitude. For N<8 supergravity theories, complication arises due to the presence of a U(1) in the U(N) isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in 4<=N<8 supergravity in four dimensions, as well as N=16 supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E8 duality group, supersymmetry ensures that the amplitude vanish in single-soft limits of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group f...

Sub-subleading soft graviton theorem in generic theories of quantum gravity

Journal of High Energy Physics

We analyze scattering amplitudes with one soft external graviton and arbitrary number of other finite energy external states carrying arbitrary mass and spin to subsubleading order in the momentum of the soft graviton. Our result can be expressed as the sum of a universal part that depends only on the amplitude without the soft graviton and not the other details of the theory and a non-universal part that depends on the amplitude without the soft graviton, and the two and three point functions of the theory. For tree amplitudes our results are valid in all space-time dimensions while for loop amplitudes, infrared divergences force us to restrict our analysis to space time dimensions five or more. With this restriction the results are valid to all orders in perturbation theory. Our results agree with known results in quantum field theories and string theory.

Amplitudes in pure Yang-Mills and MHV Diagrams

Journal of High Energy Physics, 2007

We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Bäcklund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.

Further all-loop results in softly-broken supersymmetric gauge theories

Physics Letters B, 1998

It is proven that the recently found, renormalization-group invariant sum rule for the soft scalar masses in softly-broken N = 1 supersymmetric gauge-Yukawa unified theories can be extended to all orders in perturbation theory. In the case of finite unified theories, the sum rule ensures the all-loop finiteness in the soft supersymmetry breaking sector. As a byproduct the exact β function for the soft scalar masses in the Novikov-Shifman-Vainstein-Zakharov (NSVZ) scheme for softly-broken supersymmetric QCD is obtained. It is also found that the singularity appearing in the sum rule in the NSVZ scheme exactly coincides with that which has been previously found in a certain class of superstring models in which the massive string states are organized into N = 4 supermultiplets.