The photon wave equation proposed in terms of the Riemann-Silberstein vector is derived from a first-order Dirac/Weyl-type action principle. It is symmetric w.r.t. duality transformations, but the associated Noether quantity vanishes. Replacing the fields by potentials and using instead a quadratic Klein-Gordon-type Lagrangian allows us to recover the double-Chern-Simons expression of conserved helicity and is shown to be equivalent to recently proposed alternative frameworks. Applied to the potential-modified theory the Dirac/Weyl-type approach yields again zero conserved charge, whereas the Klein-Gordon-type approach applied to the original setting yields Lipkin's "zilch".
The theorem which says that helicity is the conserved quantity associated with the duality symmetry of the vacuum Maxwell equations is proved by viewing electromagnetism as an infinite dimensional symplectic system. In fact, it is shown that helicity is the moment map of duality acting as an $SO(2)$ group of canonical transformations on the symplectic space of all solutions of the vacuum Maxwell equations.
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 …
I review some recent work that describes the close analogy between self-dual Yang--Mills amplitudes and QCD amplitudes with external gluons of positive helicity. This analogy is carried at tree level for amplitudes with two external quarks and up to one-loop for amplitudes involving only external gluons.
I show that helicity plays an important role in the development of rules for computing higher loop effective Lagrangians. Specifically, the two-loop Heisenberg-Euler effective Lagrangian in quantum electrodynamics is remarkably simple when the background field has definite helicity (i.e., is self-dual). Furthermore, the two-loop answer can be derived essentially algebraically, and is naturally expressed in terms of one-loop quantities. This represents a generalization of the familiar ``integration-by-parts'' rules for manipulating diagrams involving free propagators to the more complicated cas…
We give a proof of the equivalence of the electric-magnetic duality on one side and helicity conservation of the tree level amplitudes on the other side within general models of nonlinear electrodynamics. Using modified Feynman rules derived from generalized normal ordered Lagrangian we discuss the interrelation of the above two properties of the theory also at higher loops. As an illustration we present two explicit examples, namely we find the generalized normal ordered Lagrangian for the Born-Infeld theory and derive a semi-closed expression for the Lagrangian of the Bossard-Nicolai model (…
The Euler-Heisenberg effective action in a self-dual background is remarkably simple at two-loop. This simplicity is due to the inter-relationship between self-duality, helicity and supersymmetry. Applications include two-loop helicity amplitudes, beta-functions and nonperturbative effects. The two-loop Euler-Heisenberg effective Lagrangian for QED in a self-dual background field is naturally expressed in terms of one-loop quantities. This mirrors similar behavior recently found in two-loop amplitudes in N=4 SUSY Yang-Mills theory.
The spinor-helicity formalism has proven to be very efficient in the calculation of scattering amplitudes in quantum field theory, while the loop tree duality (LTD) representation of multi-loop integrals exhibits appealing and interesting advantages with respect to other approaches. In view of the most recent developments in LTD, we exploit the synergies with the spinor-helicity formalism to analyse illustrative one- and two-loop scattering processes. We focus our discussion on the local UV renormalisation of IR and UV finite amplitudes and present a fully automated numerical implementation th…
We show that in D=4 AdS, $s\ge 3/2$ partially massless (PM) fermions retain the duality invariances of their flat space massless counterparts. They have tuned ratios $ {m^2}/{M^2}\ne 0$ that turn them into sums of effectively massless unconstrained helicity $\pm(s,\cdots,\frac{3}{2})$ excitations, shorn of the lowest (non-dual)\helicity $\pm \frac{1}{2}$-rung and -more generally- of succeeding higher rung as well. Each helicity mode is separately duality invariant, like its flat space counterpart.
It is well known that not every symmetry of a classical field theory is also a symmetry of its quantum version. When this occurs, we speak of quantum anomalies. The existence of anomalies imply that some classical Noether charges are no longer conserved in the quantum theory. In this paper, we discuss a new example for quantum electromagnetic fields propagating in the presence of gravity. We argue that the symmetry under electric-magnetic duality rotations of the source-free Maxwell action is anomalous in curved spacetimes. The classical Noether charge associated with these transformations acc…
In $D=2+1$ dimensions there are two dual descriptions of parity singlets of helicity $\pm 1$, namely the self-dual model of first-order (in derivatives) and the Maxwell-Chern-Simons theory of second-order. Correspondingly, for helicity $\pm 2$ there are four models $S_{SD\pm}^{(r)}$ describing parity singlets of helicities $\pm 2$. They are of first-, second-,third- and fourth-order ($r=1,2,3,4$) respectively. Here we show that the generalized soldering of the opposite helicity models $S_{SD+}^{(4)}$ and $S_{SD-}^{(4)}$ leads to the linearized form of the new massive gravity suggested by Bergs…
The amplitudes of purely photonic and photon{2-fermion processes of non- commutative QED (NCQED) are derived for different helicity configurations of photons. The basic ingredient is the NCQED counterpart of Yang-Mills recursion relations by means of Berends and Giele. The explicit solutions of recursion relations for NCQED photonic processes with special helicity configurations are presented.
We study large gauge transformations for soft photons in quantum electrodynamics which, together with the helicity operator, form an ISO(2) algebra. We show that the two non-compact generators of the ISO(2) algebra correspond respectively to the residual gauge symmetry and its electromagnetic dual gauge symmetry that emerge at null infinity. The former is helicity universal (electric in nature) while the latter is helicity distinguishing (magnetic in nature). Thus, the conventional large gauge transformation is electric in nature, and is naturally associated with a scalar potential. We suggest…
We calculate the complete set of two-loop leading-colour QCD helicity amplitudes for $γγj$-production at hadron colliders. Our results are presented in a compact, fully-analytical form.
We identify a means to explicitly construct primary operators of free conformal field theories (CFTs) in spacetime dimensions $d=2,~3$, and $4$. Working in momentum space with spinors, we find that the $N$-distinguishable-particle Hilbert space $\mathcal{H}_N$ exhibits a $U(N)$ action in $d=4$ ($O(N)$ in $d=2,3$) which dually describes the decomposition of $\mathcal{H}_N$ into irreducible representations of the conformal group. This $U(N)$ is a natural $N$-particle generalization of the single-particle $U(1)$ little group. The spectrum of primary operators is identified with the harmonics of $…
We calculate all planar contributions to the two-loop massless helicity amplitudes for the process $q\bar q\to γγγ$. The results are presented in fully analytic form in terms of the functional basis proposed recently by Chicherin and Sotnikov. With this publication we provide the two-loop contributions already used by us in the NNLO QCD calculation of the LHC process $pp\to γγγ$ [Chawdhry et al. (2019)]. Our results agree with a recent calculation of the same amplitude [Abreu et al. (2020)] which was performed using different techniques. We combine several modern computational techniques, nota…
We identify momentum/helicity probability amplitudes for the photon and find their relativistic transformation properties. We also find their behaviour under space inversion and time reversal. The discussion begins with a review of the unitary, irreducible representations of the Poincare group for massless particles. The little group, the set of Lorentz transformations that leave the momentum unchanged, is distinctly different for massless particles compared to the little group of rest frame rotations for a massive particle. We give a physical interpretation of the little group for a massless …
Employing Maxwell's equations as the field theory of the photon, quantum mechanical operators for spin, chirality, helicity, velocity, momentum, energy and position are derived. The photon ``Zitterbewegung'' along helical paths is explored. The resulting non-commutative geometry of photon position and the quantum version of the Pythagorean theorem is discussed. The distance between two photons in a polarized beam of given helicity is shown to have a discrete spectrum. Such a spectrum should become manifest in measurements of two photon coincidence counts. The proposed experiment is briefly des…
Starting from a Hamiltonian description of the photon within the set of Bargmann-Wigner equations we derive new semiclassical equations of motion for the photon propagating in static gravitational field. These equations which are obtained in the representation diagonalizing the Hamiltonian at the order $\hbar $, present the first order corrections to the geometrical optics. The photon Hamiltonian shows a new kind of helicity-magnetotorsion coupling. However, even for a torsionless space-time, photons do not follow the usual null geodesic as a consequence of an anomalous velocity term. This ter…
A kinetic mixing term, which generalizes the duality symmetry of Dirac, is studied in a theory with two photons (visible and hidden). This theory can be either CP conserving or CP violating depending on the transformation of fields in the hidden sector. However if CP is violated, it necessarily occurs in the hidden sector. This opens up an interesting possibility of new sources of CP violation.
Motivated by the recent DELPHI report on the anomalous photon production in e^+e^- annihilation, we exactly calculate the inclusive cross section of soft photons in the strong coupling limit of N=4 super Yang-Mills. We find that the energy distribution is that of the Bremsstrahlung, while the angular distribution is spherical. Our result elucidates a new non-perturbative source of soft photons not associated with the final state hadronic Bremsstrahlung.
The axion photon system in an external magnetic field, when for example considered with the geometry of the experiments exploring axion photon mixing, displays a continuous axion-photon duality symmetry in the limit the axion mass is neglected. The conservation law that follows from this symmetry is obtained. The magnetic field interaction is seen to be equivalent to first order to the interaction of a complex charged field with an external electric potential, where this ficticious "electric potential" is proportional to the external magnetic field. This allows to solve for the scattering ampl…
We compute the leading probability for a gluon to flip helicity state upon traversing a background plane wave gauge field in pure Yang-Mills theory and QCD, with an arbitrary number of colours and flavours. This is a one-loop calculation in perturbative gauge theory around the gluonic plane wave background, which is treated without approximation (i.e., to all orders in the coupling). We introduce a background-dressed version of the spinor helicity formalism and use it to obtain simple formulae for the flip amplitude with pure external gluon polarizations. We also give in-depth examples for gau…
We study the production of prompt photons in strongly coupled out-of-equilibrium Super Yang-Mills plasma using the AdS/CFT correspondence. Our goal is to determine the photon emission spectrum at different stages of a thermalization process, which is modeled via the gravitational collapse of a thin spherical shell in AdS$_5$ space. Particular emphasis is placed on the limit of large frequencies, which we are able to treat analytically.
Vacuum birefringence is governed by the amplitude for a photon to flip helicity or polarisation state in an external field. Here we calculate the flip and non-flip amplitudes in arbitrary plane wave backgrounds, along with the induced spacetime-dependent refractive indices of the vacuum. We compare the behaviour of the amplitudes in the low energy and high energy regimes, and analyse the impact of pulse shape and energy. We also provide the first lightfront-QED derivation of the coefficients in the Heisenberg-Euler effective action.
We review the status of electric/magnetic duality for free gauge field theories in four space-time dimensions with emphasis on Maxwell theory and linearized Einstein gravity. Using the theory of vector and tensor spherical harmonics, we provide explicit construction of dual photons and gravitons by decomposing the fields into axial and polar configurations with opposite parity and interchanging the two sectors. When the theories are defined on AdS(4) space-time there are boundary manifestations of the duality, which for the case of gravity account for the energy-momentum/Cotton tensor duality …
