Naser Ahmadiniaz

A covariant representation of the Ball–Chiu vertex ☆

In nonabelian gauge theory the three-gluon vertex function contains important structural information, in particular on infrared divergences, and is also an essential ingredient in the Schwinger-Dyson equations. Much eort has gone into analyzing its general structure, and at the one-loop level also a number of explicit computations have been done, using various approaches. Here we use the string-inspired formalism to unify the calculations of the scalar, spinor and gluon loop contributions to the one-loop vertex, leading to an extremely compact representation in all cases. The vertex is computed fully o-shell and in dimensionally continued form, so that it can be used as a building block for higher-loop calculations. We nd that the Bern-Kosower loop replacement rules, originally derived for the on-shell case, hold o-shell as well. We explain the relation of the structure of this representation to the low-energy eective action, and establish the precise connection with the standard Ball-Chiu decomposition of the vertex. This allows us also to predict that the vanishing of the completely antisymmetric coecient function S of this decomposition is not a one-loop accident, but persists at higher loop orders. The sum rule found by Binger and Brodsky, which leads to the vanishing of the one-loop vertex in N = 4 SYM theory, in the present approach relates to worldline supersymmetry.

String { inspired representations of photon/gluon amplitudes

A bstractThe string-based Bern-Kosower rules provide an efficient way for obtaining parameter integral representations of the one-loop N -photon/gluon amplitudes involving a scalar, spinor or gluon loop, starting from a master formula and using a certain integration-by-parts (“IBP”) procedure. Strassler observed that this algorithm also relates to gauge invariance, since it leads to the absorption of polarization vectors into field strength tensors. Here we present a systematic IBP algorithm that works for arbitrary N and leads to an integrand that is not only suitable for the application of the Bern-Kosower rules but also optimized with respect to gauge invariance. In the photon case this means manifest transversality at the integrand level, in the gluon case that a form factor decomposition of the amplitude into transversal and longitudinal parts is generated naturally by the IBP, without the necessity to consider the nonabelian Ward identities. Our algorithm is valid off-shell, and provides an extremely efficient way of calculating the one-loop one-particle-irreducible off-shell Green’s functions (“vertices”) in QCD. It can also be applied essentially unchanged to the one-loop gauge boson amplitudes in open string theory. In the abelian case, we study the systematics of the IBP also for the practically important case of the one-loop N -photon amplitudes in a constant field.

Multiphoton amplitudes and generalized Landau-Khalatnikov-Fradkin transformation in scalar QED

We apply the worldline formalism to amplitudes in scalar quantum electrodynamics involving open scalar lines, with an emphasis on their nonperturbative gauge dependence. At the tree level, we study the scalar propagator interacting with any number of photons in configuration space as well as in momentum space. At one loop we rederive, in an efficient way, the off-shell vertex in an arbitrary dimension and any covariant gauge. Generalizing the Landau-Khalatnikov-Fradkin transformation for the nonperturbative propagator, we find simple nonperturbative transformation rules for arbitrary x-space amplitudes under a change of the covariant gauge parameter in terms of conformal cross ratios.

Dressed scalar propagator in a non-Abelian background from the worldline formalism

We study the propagator of a colored scalar particle in the background of a non-Abelian gauge field using the worldline formalism. It is obtained by considering the open worldline of a scalar particle with extra degrees of freedom needed to take into account the color charge of the particle, which we choose to be in the fundamental representation of the gauge group. Specializing the external gauge field to be given by a sum of plane waves, i.e. a sum of external gluons, we produce a master formula for the scalar propagator with an arbitrary number of gluons directly attached to the scalar line, akin to similar formulas derived in the literature for the case of the scalar particle performing a loop. Our worldline description produces at the same time the situation in which the particle has a color charge given by an arbitrarily chosen symmetric or antisymmetric tensor product of the fundamental.

One-particle reducible contribution to the one-loop spinor propagator in a constant field

Abstract Recently, Gies and Karbstein showed that the two-loop Euler–Heisenberg Lagrangian receives a finite one-particle reducible contribution in addition to the well-known one-particle irreducible one. Here, we demonstrate that a similar contribution exists for the propagator in a constant field already at the one-loop level, and we calculate this contribution for the scalar QED case. We also present an independent derivation of the Gies–Karbstein result using the worldline formalism, treating the scalar and spinor QED cases in a unified manner.

Master formulas for the dressed scalar propagator in a constant field

Abstract The worldline formalism has previously been used for deriving compact master formulas for the one-loop N-photon amplitudes in both scalar and spinor QED, and in the vacuum as well as in a constant external field. For scalar QED, there is also an analogous master formula for the propagator dressed with N photons in the vacuum. Here, we extend this master formula to include a constant field. The two-photon case is worked out explicitly, yielding an integral representation for the Compton scattering cross section in the field suitable for numerical integration in the full range of electric and magnetic field strengths.

Noncommutative U(1) gauge theory from a worldline perspective

A bstractWe study pure noncommutative U(1) gauge theory representing its one-loop effective action in terms of a phase space worldline path integral. We write the quadratic action using the background field method to keep explicit gauge invariance, and then employ the worldline formalism to write the one-loop effective action, singling out UV-divergent parts and finite (planar and non-planar) parts, and study renormalization properties of the theory. This amounts to employ worldline Feynman rules for the phase space path integral, that nicely incorporate the Fadeev-Popov ghost contribution and efficiently separate planar and non-planar contributions. We also show that the effective action calculation is independent of the choice of the worldline Green’s function, that corresponds to a particular way of factoring out a particle zero-mode. This allows to employ homogeneous string-inspired Feynman rules that greatly simplify the computation.

QCD gluon vertices from the string-inspired formalism

The Bern–Kosower formalism, developed around 1990 as a novel way of obtaining QCD amplitudes as the limit of infinite string tension of the corresponding string amplitudes, was originally designed as an on-shell formalism. Building on early work by Strassler, the authors have recently shown that this “string-inspired formalism” is extremely efficient also as a tool for the study of off-shell amplitudes in QCD, and in particular for achieving compact form factor decompositions of the N-gluon vertices. Among other things, this formalism allows one to achieve a manifestly gauge invariant decomposition of these vertices by way of integration-by-parts, rather than the usual tedious analysis of the non-abelian off-shell Ward identities, and to combine the spin zero, half and one cases. Here, we will provide a summary of the method, as well as its application to the three- and four-gluon vertices.

Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators

A bstractIn this work we have studied the Kleiss-Kuijf relations for the recently introduced Parke-Taylor factors at one-loop in the CHY approach, that reproduce quadratic Feynman propagators. By doing this, we were able to identify the non-planar one-loop Parke-Taylor factors. In order to check that, in fact, these new factors can describe non-planar amplitudes, we applied them to the bi-adjoint Φ3 theory. As a byproduct, we found a new type of graphs that we called the non-planar CHY-graphs. These graphs encode all the information for the subleading order at one-loop, and there is not an equivalent of these in the Feynman formalism.

Form factor decomposition of the off-shell four-gluon amplitudes

We show how to use the Bern-Kosower master formula, originally a generating functional for on-shell gluon matrix elements, to derive well-organized form factor decompositions of the off-shell one-particle-irreducible N - gluon vertices. Two such algorithms are presented which can be used for any N, the first one optimized with respect to the nonabelian gauge invariance, the second one with respect to transversality. We give explicit results for the three- and four-gluon cases. The second algorithm in the three-point case reproduces precisely the well-known Ball-Chiu decomposition, and in the four-point case a natural generalization thereof. A particularly simple structure emerges in the N=4 SYM case.