Completing the supersymmetric local field theory of electric and magnetic charges: off-shell N=2 extension, the Dirac condition under renormalization, string independence, ultraviolet behavior, and charge assignments (Panagiotakopoulos-Salam-Strathdee program)

The 1982 Panagiotakopoulos-Salam-Strathdee letter supersymmetrized Zwanziger's local two-potential quantum field theory of electric and magnetic charges - the framework in which two gauge potentials bound by a fixed Dirac-string vector $n_\mu$ describe a single photon - and left its completion open: whether the attempted $S_+,S_+'$ -augmented theory is the appropriate off-shell $N=2$ extension; whether the Dirac quantization condition $gh/4\pi=j/2$ holds for renormalized charges (the authors hoped that extended supersymmetry would force the wave-function renormalization $Z$ to unity, but did not establish it); whether gauge-invariant Green functions of the supersymmetric theory are independent of $n_\mu$ (full Lorentz invariance), extending the Brandt-Neri-Zwanziger theorem; whether supersymmetry really ameliorates the ultraviolet divergences and makes mass counterterms finite and calculable; and which charge assignments and dyonic matter content a consistent extended-supersymmetric monopole theory requires. These construction-specific questions - the off-shell $N=2$ status, the Dirac condition under renormalization, all-orders string independence for the supersymmetric theory, the order-by-order ultraviolet analysis, and the dyonic charge-assignment constraint - remain unanswered; no resolution of them is known in the literature to date.

Background

A magnetic monopole is a hypothetical particle carrying magnetic charge $h$ , i.e. a source of nonzero divergence for the magnetic field. Dirac showed in 1931 that the quantum mechanics of an electric charge $g$ moving in the field of a monopole is consistent only if the product of the charges obeys the Dirac quantization condition $gh/4\pi=j/2$ with $j$ an integer - the classic explanation of why electric charge is quantized. No single, globally regular electromagnetic vector potential can describe a monopole: the potential must be singular along a semi-infinite line (the Dirac string) whose direction is conventionally fixed by a unit four-vector $n_\mu$ . A quantum field theory containing both electrically and magnetically charged matter is therefore mutually nonlocal - no one gauge potential couples locally to both kinds of current. Zwanziger (1971) circumvented this with a local Lagrangian built from two potentials $A_\mu$ and $A'_\mu$ tied together by the fixed string vector $n_\mu$ : the two potentials jointly describe a single photon, electric charges couple locally to $A$ and magnetic charges to $A'$ , and locality is recovered at the price of explicit $n_\mu$ -dependence, so the theory is not manifestly Lorentz invariant. Brandt, Neri and Zwanziger (1979) proved that in this theory gauge-invariant quantities (Green functions, i.e. vacuum correlation functions of fields, and the S-matrix) are nevertheless independent of $n_\mu$ provided the Dirac quantization condition holds, so physical Lorentz invariance survives. On the renormalization side, the theory has two coupling constants $g$ and $h$ linked by the quantization condition; whether that condition survives renormalization was later settled for the non-supersymmetric theory in the sense of opposite renormalization: the electric and magnetic charges run inversely, so the product $gh$ is invariant under changes of renormalization scale. Supersymmetry (SUSY) is a spacetime symmetry relating bosons and fermions; $N=1$ supersymmetry has one supercharge pair and $N=2$ has two, and in a supersymmetric extension of electrodynamics the photon acquires fermionic partners (photinos). In superspace notation, a $U(1)$ gauge field resides in a real vector superfield $V$ , and complex scalar-fermion matter pairs reside in chiral superfields (superfields depending on only half the Grassmann coordinates). The standard $N=2$ vector multiplet of an ordinary abelian gauge theory can be written in $N=1$ language as $V$ paired with a chiral superfield, and exhibits an off-shell global SU(2) R-symmetry - a symmetry of the full superfield formulation that holds before any equations of motion are imposed, rotating the two supercharges (Fayet; Sohnius), which is the hallmark property the construction below lacks. Panagiotakopoulos, Salam and Strathdee (1982) supersymmetrized Zwanziger's theory: two real gauge superfields $V,V'$ for the two potentials, coupled to matter, with a string-dependent kinetic structure. To build the $N=2$ vector multiplet content they augmented the theory with two chiral superfields $S_+$ and $S_+'$ whose fermionic components join the two photinos (the photon's fermionic partners, one per potential) into Dirac fermions with well-defined propagators. The construction reproduced Zwanziger's bosonic theory on eliminating the auxiliary fields (nondynamical fields introduced so that supersymmetry holds without using the equations of motion) and gave a propagator matrix for the combined photon-photino system, but the free Lagrangian does not exhibit the off-shell global SU(2) characteristic of the standard $N=2$ $U(1)$ construction, so the authors stated they do not know whether it is the appropriate $N=2$ extension. Subsequent work on this program is limited to Sazdovic's alternative supersymmetric reformulations of the two-potential theory (1985-1988) and the Blagojevic-Senjanovic review (1988); the modern $N=2$ Lagrangians carrying electric and magnetic charges were constructed by independent routes that do not build on this construction.

Problem Statement

The problem is to complete - or definitively assess - the local quantum field theory of electric and magnetic charges constructed by Panagiotakopoulos, Salam and Strathdee on the basis of Zwanziger's two-potential formalism, including its attempted extension to $N=2$ supersymmetry. The following linked questions jointly define the internal consistency of that theory: (1) Off-shell N=2 extension. Determine whether the constructed extension - the two real gauge superfields $V,V'$ augmented by chiral superfields $S_+$ and $S_+'$ whose fermionic components join the photinos into Dirac fermions - is the appropriate off-shell $N=2$ extension of the supersymmetric two-potential theory; in particular whether the absence of the off-shell global SU(2) symmetry characteristic of the standard $N=2$ $U(1)$ construction is intrinsic or removable by field redefinitions or additional auxiliary fields; if it is not appropriate, construct the correct off-shell $N=2$ extension or prove that none exists. (2) Dirac condition under renormalization. Establish whether the Dirac quantization condition $gh/4\pi=j/2$ holds for renormalized charges $g_R,h_R$ in this supersymmetric theory - or demonstrate that it fails. (3) Lorentz invariance (string independence). Determine whether gauge-invariant Green functions of the supersymmetric theory are independent of the string vector $n_\mu$ , extending the Brandt-Neri-Zwanziger result to the supersymmetric case - or demonstrate that they are not. (4) Ultraviolet behavior. Carry out the order-by-order quantum analysis establishing (or refuting) that supersymmetrization ameliorates the ultraviolet divergences of the monopole-charge theory and that mass counterterms are finite and calculable at each supersymmetric order. (5) Charge assignments and matter content. Determine the precise charge assignments and matter content required for a consistent, anomaly-free extended-supersymmetric monopole theory - in particular whether all charged matter must be dyonic, carrying equal electric- and magnetic-type charges, and whether this constraint follows from the $N=2$ algebra or from the Zwanziger gauge structure.

References

C. Panagiotakopoulos, Abdus Salam and J. Strathdee, 'Supersymmetric field theory of monopoles', Physics Letters B 115 (1982) 29-32. DOI: 10.1016/0370-2693(82)90508-1, https://doi.org/10.1016/0370-2693(82)90508-1
Daniel Zwanziger, 'Local-Lagrangian quantum field theory of electric and magnetic charges', Physical Review D 3 (1971) 880. DOI: 10.1103/PhysRevD.3.880, https://doi.org/10.1103/PhysRevD.3.880
Richard A. Brandt, Filippo Neri and Daniel Zwanziger, 'Lorentz invariance from classical particle paths in quantum field theory of electric and magnetic charge', Physical Review D 19 (1979) 1153-1167. DOI: 10.1103/PhysRevD.19.1153, https://doi.org/10.1103/PhysRevD.19.1153
P. Fayet, 'Fermi-Bose hypersymmetry', Nuclear Physics B 113 (1976) 135-155. DOI: 10.1016/0550-3213(76)90458-2, https://doi.org/10.1016/0550-3213(76)90458-2
Martin F. Sohnius, 'Supersymmetry and central charges', Nuclear Physics B 138 (1978) 109-121. DOI: 10.1016/0550-3213(78)90159-1, https://doi.org/10.1016/0550-3213(78)90159-1
B. Sazdovic, 'Supersymmetric local Lagrangian field theory of electric and magnetic charges', Physics Letters B 160 (1985) 107-110. DOI: 10.1016/0370-2693(85)91474-1, https://doi.org/10.1016/0370-2693(85)91474-1
B. Sazdovic, 'Supersymmetric one-prepotential field theory on electric and magnetic charges', Physics Letters B 161 (1985) 127-130. DOI: 10.1016/0370-2693(85)90622-7, https://doi.org/10.1016/0370-2693(85)90622-7
B. Sazdovic, 'A path-dependent supersymmetric field theory of electric and magnetic charges', Physics Letters B 200 (1988) 335-337. DOI: 10.1016/0370-2693(88)90782-4, https://doi.org/10.1016/0370-2693(88)90782-4
M. Blagojevic and P. Senjanovic, 'The quantum field theory of electric and magnetic charge', Physics Reports 157 (1988) 233-346. DOI: 10.1016/0370-1573(88)90098-1, https://doi.org/10.1016/0370-1573(88)90098-1
S. Caicedo and R. Gambini, 'Geometrical origin of supersymmetric gauge theories', Physical Review D 39 (1989) 583. DOI: 10.1103/PhysRevD.39.583, https://doi.org/10.1103/PhysRevD.39.583
Paolo Pasti, Dmitri P. Sorokin and Mario Tonin, 'Space-time symmetries in duality symmetric models', arXiv:hep-th/9509052, https://arxiv.org/abs/hep-th/9509052
Mathijs de Vroome and Bernard de Wit, 'Lagrangians with electric and magnetic charges of N=2 supersymmetric gauge theories', JHEP 0708 (2007) 064, arXiv:0707.2717, DOI: 10.1088/1126-6708/2007/08/064, https://arxiv.org/abs/0707.2717
Chandrasekhar Chatterjee, Indrajit Mitra and H. S. Sharatchandra, 'Two-potential theory of electric and magnetic charges via duality transformation', Physics Letters B 710 (2012) 223-229, arXiv:1111.6064, DOI: 10.1016/j.physletb.2012.02.069, https://arxiv.org/abs/1111.6064
John Terning and Christopher B. Verhaaren, 'Spurious poles in the scattering of electric and magnetic charges', JHEP 12 (2020) 153, arXiv:2010.02232, DOI: 10.1007/JHEP12(2020)153, https://arxiv.org/abs/2010.02232
Joshua Newey, John Terning and Christopher B. Verhaaren, 'Schwinger vs Coleman: Magnetic charge renormalization', JHEP 11 (2024) 075, arXiv:2407.13823, DOI: 10.1007/JHEP11(2024)075, https://arxiv.org/abs/2407.13823