Abstract
Recently Sahoo and Sen obtained a series of remarkable results concerning sub leading soft photon and graviton theorems in four dimensions. Even though the S-matrix is infrared divergent, they have shown that the subleading soft theorems are well defined and exact statements in QED and perturbative Quantum Gravity. However unlike the well studied Cachazo-Strominger soft theorems in tree-level amplitudes, the new subleading soft expansion is at the order ln w (where w is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-level counterparts. Whence it is natural to ask if these theorems are associated to asymptotic symmetries of the S-matrix.We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are indeed an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem. This shows that in the case of four dimensional QED, the leading and sub-leading soft photon theorems are equivalent to Ward identities of (asymptotic) charges.
Highlights
We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem
That we have warmed up by showing equivalence between tree-level subleading soft theorem and Ward identities associated to the subleading charges (which are conserved assuming O(e) fall-offs), we extend the above definition of subleading charges and corresponding conservation laws for general fall-offs and quantize the corresponding charges
As the hard charge is quartic in matter operators, The algebra generated by Q[V ] is not understood and we so far lack any understanding of Asymptotic symmetry associated to the conservation laws
Summary
The essential steps in which we are led to different Asymptotic conditions for gauge fields and matter currents are summarized below. Due to this slowly decaying Coulombic mode, Asymptotic behaviour of scalar field at late times is modified such that the spatial matter current components decay as ln τ τ3. This logarithmic decay of current components implies that the Maxwell fields (e.g. B(y) at large τ with y being the co-ordinates on fixed time slice. The above fall-off of radiative data at large u implies that in addition to infinity of leading charges, one has an infinity of non-trivial “sub-leading” charges which are defined at u = −∞ in terms of AA(x). We show that similar sub-leading charges are non-trivial at v = +∞ on the boundary of past Null infinity and the equations of motion of the theory at spatial infinity imply that these two sets of charges are equal, leading to new set of conservation laws
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