A conjecture made by Bern, Dixon, Dunbar, and Kosower asserts a simple dimension-shifting relationship between the one-loop structure of 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.
It has been suggested a long time ago by W. Bardeen that non-vanishing of the one-loop same helicity YM amplitudes, in particular such an amplitude at four points, should be interpreted as an anomaly. However, the available derivations of these amplitudes are rather far from supporting this interpretation in that they share no similarity whatsoever with the standard triangle diagram chiral anomaly calculation. We provide a new computation of the same helicity four-point amplitude by a method designed to mimic the chiral anomaly derivation. This is done by using the momentum conservation to rew…
We prove the conjecture made by Bern, Dixon, Dunbar, and Kosower that describes a simple dimension shifting relationship between the one-loop structure of N = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. The proof captures all orders in dimensional regularisation using unitarity cuts, by combining massive spinor-helicity with Coulomb-branch supersymmetry. The form of these amplitudes can be given in terms of pentagon and box integrals using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.
The present study illuminates the relation between null cusped Wilson loops and their corresponding amplitudes. We find that, compared to the case with no self-crossing, the one loop expectation value of a self-intersecting Wilson loop develops an additional 1/εsingularity associated to the intersection. Interestingly, the same 1/εpole exists in the finite part of the one loop amplitude, appearing in the BDS conjecture, at the corresponding kinematic limit. At two loops, we explore the behaviour of the remainder function R, encoding the deviation of the amplitude from the BDS conjecture. By an…
Two simple proofs are presented for the first order virial expansion of the self-energy of a particle moving through a medium, characterised by temperature and/or chemical potential(s). One is based on the virial expansion of the self-energy operator itself, while the other is on the analysis of its Feynman diagrams in configuration space.
In this paper we present the all-loop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in N=4 SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon tree-level amplitudes and loop integrands starting from corresponding expressions for on-shell amplitudes. The introduced gluing procedure is used to derive BCFW recursions both for tree-level reggeon amplitudes and their loop integrands. In addition we provide predictions for reggeon loop integrands written in the basis of local integrals. As a check of the cor…
A conjecture for computing all genus topological closed string amplitudes on toric local Calabi-Yau threefolds, by interpreting the associated 5-brane web as a Feynman diagram, is given. A propagator and a three point vertex is defined which allows us to write down the amplitude associated with 5-brane web. We verify the conjecture that this amplitude is equal to the closed string partition function by computing integer invariants for resolved conifold and certain curves of low degree in local del Pezzo surfaces, local Hirzebruch surfaces and their various blowups.
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 an…
We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.
In this article, we analyze the boundary behaviors of pure Yang-Mills amplitudes under adjacent and non adjacent BCFW shifts in Feynman gauge. We introduce reduced vertexes for Yang-Mills fields, prove that these reduced vertexes are equivalent to the original vertexes as for the study of boundary behaviors, which greatly simplifies our analysis of boundary behaviors. Boundary behaviors for adjacent shifts are readily obtained using reduced vertexes. Then we prove a theorem on permutation sum and use it to prove the improved boundary behaviors for non-adjacent shifts. Based on the boundary beh…
The dual formulation of planar N = 4 super-Yang-Mills scattering amplitudes makes manifest that the integrand has only logarithmic singularities and no poles at infinity. Recently, Arkani-Hamed, Bourjaily, Cachazo and Trnka conjectured the same singularity properties hold to all loop orders in the nonplanar sector as well. Here we conjecture that to all loop orders these constraints give us the key analytic information contained in dual conformal symmetry. We also conjecture that to all loop orders, while N = 8 supergravity has poles at infinity, at least at four points it has only logarithmic…
We construct two and three-line shifts for tree-level amplitude with massless and/or massive particles, and provide a method to construct general multi-line shifts for all masses. We choose the massless-massive BCFW shift from these shifts and examine its validity in renormalizable theories. Using such a shift, we find that amplitudes with at least one massless vector boson are constructible. This reveals the importance of gauge theory in the construction of amplitudes with massive particles. We also find that this kind of amplitudes have a cancellation related to group structure among differe…
In this paper, we compute the higher derivative amplitudes arising from shift symmetric-invariant actions for both the non-linear sigma model and the special galileon symmetries, and provide explicit expressions for their Lagrangians. We find that, beyond leading order, the equivalence between shift symmetries, enhanced single soft limits, and compatibility with the double copy procedure breaks down. In particular, we have shown that the most general even-point amplitudes of a colored-scalar satisfying the Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations are compatible with the no…
We derive new positivity bounds for scattering amplitudes in theories with a massless graviton in the spectrum in four spacetime dimensions, of relevance for the weak gravity conjecture and modified gravity theories. The bounds imply that extremal black holes are self-repulsive, $M/|Q|<1$ in suitable units, and that they are unstable to decay to smaller extremal black holes, providing an S-matrix proof of the weak gravity conjecture. We also present other applications of our bounds to the effective field theory of axions, $P(X)$ theories, weakly broken galileons, and curved spacetimes.
Scattering amplitudes in superconformal field theories do not enjoy this symmetry, because the definition of asymptotic states involve a notion of infinity. Concentrating on planar $\mathcal{N}=4$ Yang-Mills, we consider a generalization of scattering amplitudes which depends on twice as many Grassmann variables. We conjecture that it restores at least half of the superconformal symmetries, and all of the dual superconformal symmetries. The object arises naturally as the dual of a null polygonal Wilson loop in an $(x,θ,\barθ)$ superspace. We support the conjecture by using it to obtain the tot…
As a test of the gluon scattering amplitude/Wilson loop duality, we evaluate the hexagonal light-like Wilson loop at two loops in N=4 super Yang-Mills theory. We compare its finite part to the Bern-Dixon-Smirnov (BDS) conjecture for the finite part of the six-gluon amplitude. We find that the two expressions have the same behavior in the collinear limit, but they differ by a non-trivial function of the three (dual) conformally invariant variables. This implies that either the BDS conjecture or the gluon amplitude/Wilson loop duality fails for the six-gluon amplitude, starting from two loops. O…
I present a conjecture that all two-loop MHV amplitudes in planar $\mathcal{N} = 4$ super-Yang-Mills theory possess an antipodal symmetry when evaluated on parity-even kinematics. The symmetry acts as a change of basis on the symbol letters, followed by the antipode operation associated with the Hopf algebra structure of multiple polylogarithms. At the symbol level, I provide the symmetry map explicitly for amplitudes with up to eight external particles, and also provide evidence at all multiplicities. Intriguingly, the map acts as an isomorphism on the normal fans of the Newton polytopes of t…
The "orbifold covariance principle", or OCP for short, is presented to support a conjecture of Pradisi, Sagnotti and Stanev on the expression of the Klein- bottle amplitude.
We continue the study of the duality between super-correlators and scattering super-amplitudes in planar N=4 SYM. We provide a number of further examples supporting the conjectured duality relation between these two seemingly different objects. We consider the five- and six-point one-loop NMHV and the six-point tree-level NNMHV amplitudes, obtaining them from the appropriate correlators of strength tensor multiplets in N=4 SYM. In particular, we find exact agreement between the rather non-trivial parity-odd sector of the integrand of the six-point one-loop NMHV amplitude, as obtained from the …
We introduce a BCFW shift which can be used to recursively build the full Yang-Mills tree-level amplitude as a function of polarization vectors. Furthermore, in line with the recent results of arXiv:1612.02797, we conjecture that the Yang-Mills tree-level scattering amplitude is uniquely fixed by locality and demanding the usual asymptotic behavior under a sufficient number of shifts. Unitarity therefore emerges from locality and constructability. We prove this statement at the leading order in the soft expansion.
We conjecture a simple relationship between the one-loop maximally helicity violating gluon amplitudes of ordinary QCD (all helicities identical) and those of N=4 supersymmetric Yang-Mills (all but two helicities identical). Because the amplitudes in self-dual Yang Mills have been shown to be the same as the maximally helicity violating ones in QCD, this conjecture implies that they are also related to the maximally helicity violating ones of N=4 supersymmetric Yang-Mills. We have an explicit proof of the relation up to the six-point amplitude; for amplitudes with more external legs, it remain…
This article studies methods to obtain relations for scattering amplitudes at the loop level, with concrete examples at one loop. These methods originate in the analysis of large so-called Britto-Cachazo-Feng-Witten shifts of tree level amplitudes and loop level integrands. In particular BCFW shifts for particles which are not color adjacent and some particular generalizations of this situation are analyzed in some detail in four and higher dimensions. For generic non-adjacent shifts our results are independent of loop order for integrands and hold for generic minimally coupled gauge theories …
We prove the MHV vertex expansion for all tree amplitudes of N=4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every N^kMHV tree amplitude falls off as 1/z^k, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all N^kMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.
We study the application of BCFW recursion relations to the QED processes $0\to e^- e^+ n γ$. Based on 6-point amplitudes (both MHVA and NMHVA) computed from Feynman diagrams in the Berends-Giele gauge, we conduct a comprehensive study on all different shifts. Then we propose a new shift (LLYZ shift) which can lead to the full amplitudes for these processes and can have some realistic computation advantages. We compare the number of terms and the independent amplitudes of this novel shift with a few typical shifts.
An update is given on our long-term effort to perform a three-loop check on the Affleck-Alvarez-Manton/Lebedev-Ritus exponentiation conjecture for the imaginary part of the Euler-Heisenberg Lagrangian, using 1+1 dimensional QED as a toy model. After reviewing the history and significance of the conjecture, we present trigonometric integral representations for the single electron loop contributions to the three-loop Lagrangian, and develop a symmetry-based method for the calculation of their weak-field expansion coefficients.
We discover new hard string scattering amplitudes (HSSA) with polarizations orthogonal to the scattering plane in the NS sector of 10D open superstring theory. The corresponding HSSA in the bosonic string theory are of subleading order in energy. In addition, all HSSA are found to be proportional to each other and the ratios among these HSSA are calculated to justify Gross conjecture in 1988 on high energy symmetry of string theory. Moreover, together with these new HSSA, the total number of HSSA calculated in the first massive level of NS sector are found to agree with those of hard polarized…
We perform a comprehensive study of on-shell recursion relations for Born amplitudes in spontaneously broken gauge theories and identify the minimal shifts required to construct amplitudes with a given particle content and spin quantum numbers. We show that two-line or three-line shifts are sufficient to construct all amplitudes with five or more particles, apart from amplitudes involving longitudinal vector bosons or scalars, which may require at most five-line shifts. As an application, we revisit selection rules for multi-boson amplitudes using on-shell recursion and little-group transforma…
In a recent paper [1], three-particle interactions without invariance under Lorentz boosts were constrained by demanding that they yield tree-level four-particle scattering amplitudes with singularities as dictated by unitarity and locality. In this brief note, we show how to obtain an independent verification and consistency check of these boostless bootstrap results using BCFW momentum shifts. We point out that the constructibility criterion, related to the behaviour of the deformed amplitude at infinite BCFW parameter z, is not strictly necessary to obtain non-trivial constraints for the th…
We advance the exploration of cluster-algebraic patterns in the building blocks of scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. In particular we conjecture that, given a maximal cut of a loop amplitude, Landau singularities and poles of each Yangian invariant appearing in any representation of the corresponding Leading Singularities can be found together in a cluster. We check these adjacencies for all one-loop amplitudes up to 9 points. Along the way, we also prove that all (rational) N$^2$MHV Yangian invariants are cluster adjacent, confirming original conjectures.
We construct numerically the minimal surface in AdS spacetime surrounded by the light-like segments, which are dual to the 4, 6 and 8-point gluon scattering amplitudes in N=4 super Yang-Mills theory. We evaluate the area of the minimal surface in the radial cut-off regularization and compare these areas with the formula conjectured by Bern, Dixon and Smirnov (BDS), which is modified by the remainder function of cross-ratios of external momenta for n(\geq 6)-point amplitudes. In our momentum configuration cross-ratios are constant. We calculate the difference of areas with different conformal b…
We construct an expression for the n-point one-loop graviton scattering amplitude with a single negative helicity external leg using an augmented recursion technique. We analyse the soft-limits of these amplitudes and demonstrate that they have soft behaviour beyond the conjectured universal behaviour.
A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $ϕ^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.
We compute the four-derivative corrections to the geometry, extremality bound, and thermodynamic quantities of AdS-Reissner-Nordstr{ö}m black holes for general dimensions and horizon geometries. We confirm the universal relationship between the extremality shift at fixed charge and the shift of the microcanonical entropy, and discuss the consequences of this relation for the Weak Gravity Conjecture in AdS. The thermodynamic corrections are calculated using two different methods: first by explicitly solving the higher-derivative equations of motion and second, by evaluating the higher-derivativ…
We introduce a new set of symmetries obeyed by tree-level gauge-theory amplitudes involving at least one gluon. The symmetry acts as a momentum-dependent shift on the color factors of the amplitude. Using the radiation vertex expansion, we prove the invariance under this color-factor shift of the $n$-gluon amplitude, as well as amplitudes involving massless or massive particles in an arbitrary representation of the gauge group with spin zero, one-half, or one. The Bern-Carrasco-Johansson relations are a direct consequence of this symmetry. We also introduce the cubic vertex expansion of an amp…
Planar ${\cal N} =4$ super Yang-Mills SU(N) theory is expected to exhibit stringy behavior, anticipated by the 't Hooft genus expansion and the $AdS/CFT$ correspondence. We examine the Bern-Dixon-Smirnov (BDS) conjecture for $n$-gluon amplitudes in the context of single-Regge and multi-Regge limits and show that these amplitudes have the expected Regge form in the Euclidean region.
In this paper we consider tree-level gauge invariant off-shell amplitudes (Wilson line form factors) in $\mathcal{N}=4$ SYM. For the off-shell amplitudes with one leg off-shell we present a conjecture for their Grassmannian integral representation in spinor helicity, twistor and momentum twistor parameterizations. The presented conjecture is successfully checked against BCFW results for MHV$_n$, NMHV$_4$ and NMHV$_5$ off-shell amplitudes. We have also verified that our Grassmannian integral representation correctly reproduces soft (on-shell) limit for the off-shell gluon momentum. It is shown …
We study closed string one-loop amplitudes in string theory, in particular the average mass shift for states at given mass and Neveu-Schwarz charges. Our analysis is based only on well-defined string amplitudes and the exploitation of symmetries and unitarity properties of the torus amplitudes. We obtain the result Delta M^2 = - g_s^2 M^(2+(3-D)/2) in D spacetime dimensions for the average closed string mas-shift (Delta M^2 = - g_s^2 (M^2-Q^2)^(1+(3-D)/4) for states with non-zero Neveu-Schwarz charges Q). An interesting picture of one-loop corrections for the string in non-supersymmetric confi…
Scattering amplitudes for the simplest theory of colored scalar particles - the Tr($Φ^3$) theory - have recently been the subject of active investigations. In this letter we describe an unanticipated wider implication of this work: the Tr($Φ^3$) theory secretly contains Non-linear Sigma Model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr$(Φ^3)$ amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in potential, with extrema spontaneously breaking $U(N) \to U(N-k) \time…
The infrared singularities of gravitational amplitudes are one-loop exact, in that higher-loop divergences are characterized by the exponential of the one-loop divergence. We show that the contributions to SU(N) gauge-theory amplitudes that are most-subleading in the 1/N expansion are also one-loop exact, provided that the dipole conjecture holds. Possible corrections to the dipole conjecture, beginning at three loops, could violate one-loop-exactness, though would still maintain the absence of collinear divergences. We also demonstrate a relation between L-loop four-point N=8 supergravity and…
We study "on-shell constructibility" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles, both massless and massive. Our analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. We focus on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the MHV vertex expansion. We derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depend…
In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choic…
First-order perturbative calculation of the frequency-shifts caused by special relativity is performed for a charged particle confined in a Penning trap. The perturbed motion is approximated by the Jacobian elliptic functions which describe the periodic orbit repeating itself sinuously with a period that exceeds $2π$. We find relativistic corrections to amplitudes of oscillating modes as well as shifts of eigenfrequencies which depend on amplitudes. Besides we find the relativistic contributions to modular angles. In the low-energy limit the deformed orbit simplifies to the well-known combinat…
