We derive a recursion relation of the Feynman diagrams of the effective action for the third Legendre transformation in case of the bosonic field theory with cubic interaction. We apply the recursion relation to obtain the Feynman diagrams of the effective action for the third Legendre transformation up to the five-loop order. The three-particle irreducibility of the Feynman diagrams of the effective action for the third Legendre transformation is shown by induction.
We derive a new recursion relation to obtain the Feynman diagrams of the Cornwall-Jackiw-Toumboulis(CJT) effective action by using the functional derivative identities. By using this recursion relation we show the two-particle-irreducibility of the Feynman diagrams of the CJT effective action by induction. We apply the recursion relation to obtain the Feynman diagrams of the CJT effective action up to the four-loop order in case of the bosonic field theory.
We consider two seemingly unrelated problems, the calculation of the WKB expansion of the harmonic oscillator wave functions and the counting the number of Feynman diagrams in QED or in many-body physics and show that their solutions are both encoded in a single enumerative problem, the calculation of the number of certain types of ribbon graphs. In turn, the numbers of such ribbon graphs as a function of the number of their vertices and edges can be determined recursively through the application of the topological recursion of Eynard-Orantin to the algebraic curve encoded in the Schrödinger e…
Recently, it has been shown that on-shell scattering amplitudes can be constructed by the Feynman-tree theorem combined with the BCFW recursion relations. Since the BCFW relations are restricted to tree diagrams, the preceding application of the Feynman-tree theorem is essential. In this way amplitudes can be constructed by on-shell and gauge-invariant tree amplitudes. Here we want to apply this method to the electron-photon vertex correction. We present all the single, double, and triple phase-space tensor integrals explicitly and show that the sum of amplitudes coincides with the result of t…
Recently it has been shown that in gauge theories amplitudes to any perturbation order can be obtained by glueing together simple three-point on-shell amplitudes. These three-point amplitudes in turn are fixed by locality and Lorentz invariance. This factorization into three-point on-shell amplitudes follows from the BCFW recursion relations and the Feynman-tree theorem. In an explicit example, that is, the four-gluon amplitude with all plus helicities, we illustrate the method. In conventional calculation this amplitude corresponds to one-loop box diagrams.
We extend the argument presented by Benincasa, Boucher-Veronneau, and Cachazo to show that graviton tree amplitudes are well behaved under large complex deformations of the momenta of a pair of like-helicity gravitons. This shows that BCFW recursion relations for gravity amplitudes can be constructed using such shifts, providing an alternative proof to the recent one by Arkani-Hamed and Kaplan. By using auxiliary recursion relations the cancellations which are hidden when using covariant Feynman diagrams become manifest.
In this article, we use Ward identity to calculate tree and one loop level off shell amplitudes in pure Yang-Mills theory with a pair of external lines complexified. We explicitly prove Ward identity at tree and one loop level using Feynman rules, and then give recursion relations for the off shell amplitudes. We find that the cancellation details in the proof of Ward identity simplifies our derivation of the recursion relations. Then we calculate three and four point one loop off shell amplitudes as examples of our method.
The Hopf algebra of Feynman diagrams, analyzed by A.Connes and D.Kreimer, is considered from the perspective of the theory of effective actions and generalized $τ$-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory.
In this note, we study the $\mathcal{Q}$-cut representation by combining it with BCFW deformation. As a consequence, the one-loop integrand is expressed in terms of a recursion relation, i.e., $n$-point one-loop integrand is constructed using tree-level amplitudes and $m$-point one-loop integrands with $m\leq n-1$. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original $\mathcal{Q}$-cut construction, up to scale free terms.
Using functional derivatives with respect to the free correlation function we derive a closed set of Schwinger-Dyson equations in phi^4-theory. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the two- and four-point function together with their weights.
We give a proof of BCFW recursion relations for all tree-level amplitudes of gravitons in General Relativity. The proof follows the same basic steps as in the BCFW construction and it is an extension of the one given for next-to-MHV amplitudes by one of the authors and P. Svrček in hep-th/0502160. The main obstacle to overcome is to prove that deformed graviton amplitudes vanish as the complex variable parameterizing the deformation is taken to infinity. This step is done by first proving an auxiliary recursion relation where the vanishing at infinity follows directly from a Feynman diagram an…
Using functional derivatives with respect to free propagators and interactions we derive a closed set of Schwinger-Dyson equations in quantum electrodynamics. Its conversion to graphical recursion relations allows us to systematically generate all connected and one-particle irreducible Feynman diagrams for the $n$-point functions and the vacuum energy together with their correct weights.
The free energy of the Ginzburg-Landau theory satisfies a nonlinear functional differential equation which is turned into a recursion relation. The latter is solved graphically order by order in the loop expansion to find all connected vacuum diagrams, and their corresponding weights. In this way we determine the connected vacuum diagrams and their weights up to four loops.
We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams…
We show how to apply the BCFW recursion relation to Feynman loop integrals with the help of the Feynman-tree theorem. We deconstruct in this way all Feynman diagrams in terms of on-shell subamplitudes. Every cut originating from the Feynman-tree theorem corresponds to an integration over the phase space of an unobserved particle pair. We argue that we can calculate scattering amplitudes alternatively by the construction of on-shell and gauge-invariant subamplitudes.
We show how by reassembling the tree level gluon Feynman diagrams in a convenient gauge, space-cone, we can explicitly derive the BCFW recursion relations. Moreover, the proof of the gluon recursion relations hinges on an identity in momentum space which we show to be nothing but the Fourier transform of the largest time equation. Our approach lends itself to natural generalizations to include massive scalars and even fermions.
In this work, we formulate a set of rules for writing down $p$-adic Mellin amplitudes at tree-level. The rules lead to closed-form expressions for Mellin amplitudes for arbitrary scalar bulk diagrams. The prescription is recursive in nature, with two different physical interpretations: one as a recursion on the number of internal lines in the diagram, and the other as reminiscent of on-shell BCFW recursion for flat-space amplitudes, especially when viewed in auxiliary momentum space. The prescriptions are proven in full generality, and their close connection with Feynman rules for real Mellin …
Implicit Regularization (IReg) is a candidate to become an invariant framework in momentum space to perform Feynman diagram calculations to arbitrary loop order. In this work we present a systematic implementation of our method that automatically displays the terms to be subtracted by Bogoliubov's recursion formula. Therefore, we achieve a twofold objective: we show that the IReg program respects unitarity, locality and Lorentz invariance and we show that our method is consistent since we are able to display the divergent content of a multi-loop amplitude in a well defined set of basic diverge…
The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagrams with their proper multiplicities. The procedure is applied to a multicomponent scalar field theory with a phi^4-self-interaction and then to a theory of two scalar fields phi and A with an interaction phi^2 A. All Feynman diagrams with external lines are obtained from functional derivatives of t…
An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for…
We derive a new kind of recursion relation to obtain the one-particle-irreducible (1PI) Feynman diagrams for the effective action. By using this method, we have obtained the graphical representation of the four-loop effective action in case of the general bosonic field theory which have vertices higher than the four-point vertex.
Motivated by recently progresses in the study of BCFW recursion relation with nonzero boundary contributions for theories with scalars and fermions\cite{Bofeng}, in this short note we continue the study of boundary contributions of gauge theory with the bad deformation. Unlike cases with scalars or fermions, it is hard to use Feynman diagrams directly to obtain boundary contributions, thus we propose another method based on the ${\cal N}=4$ SYM theory. Using this method, we are able to write down a useful on-shell recursion relation to calculate boundary contributions from related theories. Ou…
We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator. The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluo…
We analyze the validity of BCFW recursion relations for currents of n - 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell. These currents are gauge-dependent and can be used as ingredients in the unitarity-based approach to computing one-loop amplitudes. The validity of BCFW recursion relations is well known to depend on the so-called boundary behavior of the currents as the momentum shift parameter goes to infinity. With off-shell currents, a new potential problem arises, namely unphysical poles that depend on the choice of gauge. W…
On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior …
The free energy of a multi-component scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys non-linear functional differential equations which are turned into recursion relations for the connected Greens functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into…
Motivated by new techniques in the computation of scattering amplitudes of massless particles in four dimensions, like BCFW recursion relations, the question of how much structure of the S-matrix can be determined from purely S-matrix arguments has received new attention. The BCFW recursion relations for massless particles of spin 1 and 2 imply that the whole tree-level S-matrix can be determined in terms of three-particle amplitudes (evaluated at complex momenta). However, the known proofs of the validity of the relations rely on the Lagrangian of the theory, either by using Feynman diagrams …
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.
