Abstract
We examine interacting Abelian theories at low energies and show that holomorphically normalized photon helicity amplitudes transform into dual amplitudes under SL(2,Z) as modular forms with weights that depend on the number of positive and negative helicity photons and on the number of internal photon lines. Moreover, canonically normalized helicity amplitudes transform by a phase, so that even though the amplitudes are not duality invariant, their squares are duality invariant. We explicitly verify the duality transformation at one loop by comparing the amplitudes in the case of an electron and the dyon that is its SL(2,Z) image, and extend the invariance of squared amplitudes order by order in perturbation theory. We demonstrate that S-duality is property of all low-energy effective Abelian theories with electric and/or magnetic charges and see how the duality generically breaks down at high energies.
Highlights
In general SL(2, Z) duality maps a particle with electric charge q to a dual particle with electric charge q and magnetic charge g
We will show how S-duality is implemented in the Zwanziger formalism [11] as a local field redefinition with a change of coupling constant, which means that S-duality is a property of any lowenergy effective theory with electric and/or magnetic charges
The N = 4 theory has a perturbative coupling and a hierarchy of mass scales and a weakly coupled low-energy effective theory for the light charged gauge bosons and photon, the duality can produce another weakly coupled theory similar to what we describe in this paper
Summary
In order to check the SL(2, Z) duality transformation of helicity amplitudes we will need to perturbatively calculate the photon helicity amplitude generated by a dyon loop. This calculation can performed using the Zwanziger two potential formulation [11, 28,29,30] of QED. While there are still only two propagating photon degrees of freedom this formulation allows for local couplings to both electric and magnetic charges. This type of formulation was later rediscovered and generalized by Schwarz and Sen [31]. The fundamental domain tiles H with congruent hyperbolic triangles, as shown in figure 1, and we can see there is a complex pattern of weakly coupled theories
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