Recently a manifestly gauge invariant formalism for calculating amplitudes in quantum electrodynamics was outlined in which the field strength, rather than the gauge potential, is used as the propagating field. To demonstrate the utility of this formalism we calculate the axial and gauge anomalies explicitly in theories with both electrically and magnetically charged particles. Usually the gauge anomaly is identified as an amplitude that (in certain theories) fails to be gauge invariant, so it seems particularly enlightening to understand it in a manifestly gauge invariant formalism. We find that the three photon amplitude is still anomalous in these same theories because it depends explicitly upon the choice of the Stokes surface needed to couple the field strength to sources, so the gauge anomaly arises from geometric considerations.

United States

Joshua Newey

John Terning

Christopher Verhaaren

Geometrizing the anomaly

It is usually assumed that 4D instantons can only arise in non-Abelian theories. In this paper we re-examine this conventional wisdom by explicitly constructing instantons in an Abelian gauge theory: QED 4 with N  f  flavors of Dirac fermions, in the background of a Dirac monopole. This is the low-energy effective field theory for fermions interacting with a ’t Hooft-Polyakov monopole, in the limit where the monopole is infinitely heavy (hence pointlike) and static. This theory, whose non-topological sectors were studied by Rubakov and Callan, has a far richer structure than previously explored. We show how to calculate the topological instanton number, demonstrate the existence of ’t Hooft zero modes localized around such instantons, and show how instantons in the path integral provide the underlying mechanism for the Callan-Rubakov process: monopole-catalyzed baryon decay with a cross section that saturates the unitarity bound. Our computation relies on correctly identifying the relevant 2D EFT for monopole catalysis as Axial QED 2 in an effective AdS 2 metric.

Csaba Csáki

Israel

Rotem Ovadia

Ofri Telem

Shimon Yankielowicz

Abelian instantons and monopole scattering

The kinetic mixing of two U(1) gauge theories can result in a massless photon that has perturbative couplings to both electric and magnetic charges. This framework can be used to perturbatively calculate in a quantum field theory with both kinds of charge. Here we reexamine the running of the magnetic charge, where the calculations of Schwinger and Coleman sharply disagree. We calculate the running of both electric and magnetic couplings and show that the disagreement between Schwinger and Coleman is due to an incomplete summation of topological terms in the perturbation series. We present a momentum space prescription for calculating the loop corrections in which the topological terms can be systematically separated for resummation. Somewhat in the spirit of modern amplitude methods we avoid using a vector potential and use the field strength itself, thereby trading gauge redundancy for the geometric redundancy of Stokes surfaces. The resulting running of the couplings demonstrates that Dirac charge quantization is independent of renormalization scale, as Coleman predicted. As a simple application we also bound the parameter space of magnetically charged states through the experimental measurement of the running of electromagnetic coupling.

Schwinger vs Coleman: Magnetic charge renormalization

The constructive method of determining amplitudes from on-shell pole structure has been shown to be promising for calculating amplitudes in a more efficient way. However, challenges have been encountered when a massless internal photon is involved in the gluing of three-point amplitudes with massive external particles. In this paper, we use the original on-shell method, old-fashioned perturbation theory, to shed light on the constructive method, and show that one can derive the Feynman amplitude by correctly identifying the residue even when there is an internal photon involved.

Hsing-Yi Lai

Da Liu

The constructive method for massive particles in QED

We study representations of the Poincaré group that have a privileged transformation law along a <math><mi>p</mi></math>-dimensional hyperplane, and uncover their associated spinor-helicity variables in <math><mi>D</mi></math> spacetime dimensions. Our novel representations generalize the recently introduced celestial states and transform as conformal primaries of <math><mi>S</mi><mi>O</mi><mo>(</mo><mi>p</mi><mo>,</mo><mn>1</mn><mo>)</mo></math>, the symmetry group of the <math><mi>p</mi></math>-hyperplane. We will refer to our generalized states as “partially celestial.” Following Wigner’s method, we find the induced representations, including spin degrees of freedom. Defining generalized spinor-helicity variables for every <math><mi>D</mi></math> and <math><mi>p</mi></math>, we are able to construct the little group covariant part of partially celestial amplitudes. Finally, we briefly examine the application of the pairwise little group to partially celestial states with mutually nonlocal charges.

Partially celestial states and their scattering amplitudes

We construct the Faddeev-Kulish dressed multiparticle states of electrically and magnetically charged particles, incorporating the effects of real and virtual soft photons. We calculate the properties of such dressed states under Lorentz transformations, and find that they can be identified with the pairwise multi-particle states that transform under the pairwise little group. The shifts in the dressing factors under Lorentz transformations are finite and have a simple geometric interpretation. Using the transformation properties of the dressed states we also present a novel, fully quantum field theoretic derivation of the geometric (Berry) phase obtained by an adiabatic rotation of the Dirac string, and also of the Dirac quantization condition. For half integer pairwise helicity, we show that these multiparticle states have flipped spin-statistics, reproducing the surprising fact that fermions can be made out of bosons.

China

Zi-Yu Dong

Dressed vs. pairwise states, and the geometric phase of monopoles and charges

We suggest a novel resolution for a decades old mystery—what happens when a positron scatters off a minimal grand-unification-theory monopole in an s wave, a puzzle first discussed by Callan in 1983. Using the language of on shell amplitudes and pairwise helicity we suggest that the final state contains two up quarks and a down quark in an entangled “pairwise” multiparticle state—the only particle final state that satisfies angular momentum and gauge charge conservation. The cross section for this process is as large as in the original Rubakov-Callan effect, only suppressed by the QCD scale. The final state we find cannot be seen in Callan’s truncated 2D theory, since our new pairwise state appears only in more than two dimensions.

Yuri Shirman

Pairwise Multiparticle States and the Monopole Unitarity Puzzle

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e 1 g 2 − e 2 g 1, for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

Sungwoo Hong

Michael Waterbury

Scattering amplitudes for monopoles: pairwise little group and pairwise helicity

We extend the definition of asymptotic multiparticle states of the <math><mrow><mi>S</mi></mrow></math>-matrix beyond the tensor products of one-particle states. We identify new quantum numbers called pairwise helicities, or <math><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></math>, associated with asymptotically separated pairs of particles. We first treat all single particles and particle pairs independently, allowing us to generalize the Wigner construction, and ultimately projecting onto the physical states. Our states reduce to tensor product states for vanishing <math><msub><mi>q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></math>, while for vanishing spins they reproduce Zwanziger’s scalar dyon states. This construction yields the correct asymptotic states for the scattering of electric and magnetic charges, with pairwise helicity identified as <math><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math>.

Completing Multiparticle Representations of the Poincaré Group

Theories with both electric and magnetic charges (“mutually non-local” theories) have several major obstacles to calculating scattering amplitudes. Even when the interaction arises through the kinetic mixing of two, otherwise independent, U(1)’s, so that all low-energy interactions are perturbative, difficulties remain: using a self-dual, local formalism leads to spurious poles at any finite order in perturbation theory. Correct calculations must show how the spurious poles cancel in observable scattering amplitudes. Consistency requires that one type of charge is confined as a result of one of the U(1)’s being broken. Here we show how the constraints of confinement and parity conservation on observable processes manages to cancel the spurious poles in scattering and pair production amplitudes, paving the way for systematic studies of the experimental signatures of “dark” electric-magnetic processes. Along the way we demonstrate some novel effects in electric-magnetic interactions, including that the amplitude for single photon production of magnetic particles by electric particles vanishes.