Perturbative quantum field theory in the string-inspired formalism

Abstract

We review the status and present range of applications of the "string -inspired" approach to perturbative quantum field theory. This formalism offers the possibility of computing effective actions and S-matrix elements in a way which is similar in spirit to string perturbation theory, and bypasses much of the apparatus of standard second-quantized field theory. Its development was initiated by Bern and Kosower, originally with the aim of simplifying the calculation of scattering amplitudes in quantum chromodynamics and quantum gravity. We give a short account of the original derivation of the Bern-Kosower rules from string theory. Strassler's alternative approach in terms of first-quantized particle path integrals is then used to generalize the formalism to more general field theories, and, in the abelian case, also to higher loop orders. A considerable number of sample calculations are presented in detail, with an emphasis on quantum electrodynamics.

A covariant representation of the Ball–Chiu vertex

“…As is well-known, the Dirac fermion possesses an N = 1 supersymmetric worldline path integral representation [35,36,37,38,39,40,41], and the gluon an N = 2 supersymmetric one [35,37,42,24,43]. Analogously to the original string-based derivation of those rules [21,22,23], where worldsheet SUSY was identified as the underlying symmetry, in the worldline approach the same rules can be related to this worldline SUSY [24,28].…”

“…In the present work, we will recalculate the scalar, spinor and gluon loop contributions to this "gauge-invariant" three-gluon vertex using the "string-inspired" formalism along the lines of [21,22,23,24,25,26,27] (for a review, see [28]). Our starting point is the "Bern-Kosower master formula" [21,22,23]:…”

“…The starting point of this "string-inspired worldline formalism" is the following path integral representation of the nonabelian one-loop effective action due to a scalar loop [24] (this generalizes Feynman's famous 1950 formula for scalar QED [33]), 11) where the integral Dx is over the space of all closed trajectories in spacetime with periodicity T in proper-time, x µ (T ) = x µ (0). Although this approach fell somewhat short of yielding a rederivation of the full set of Bern-Kosower rules including the non-1PI contributions, it provides a simple way to see that the master formula (1.6) as it stands is valid off-shell as a formula for the 1PI part of the N -gluon Green's function (see [24,28]). In [25], Strassler moreover started a systematic investigation of the IBP procedure, and discovered that it bears an interesting connection to gauge invariance.…”

Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes

“…It is important to note that the right-hand side represents the complete amplitude, not just a particular Feynman diagram. See [61,53] for the relation of this representation to standard Feynman parameter integrals. The exponential factor exp{−is N i<j=1 G Bij k i · k j } corresponds to the standard one-loop N -point denominator, while Q N corresponds to a numerator polynomial.…”

“…In section 3 we compute the two-loop corrections to these Lagrangians for scalar and spinor QED, using the 'string-inspired' worldline formalism [45,46] along the lines of [47,48,15,16]. This formalism is based on the representation of effective actions in terms of firstquantized particle path integrals, and has turned out to be highly convenient for computations involving constant external fields [47,49,50,51,52,15,8,16] (see [53] for a review). In section 4 we comment on the special properties of self-dual fields with respect to helicity and supersymmetry, and we use some of these properties to explain the similarity of our two-loop scalar and spinor QED results.…”