A ug 2 01 6 Ultraviolet cutoffs and the photon mass

The momentum UV cutoff in Quantum Field Theory is usually treated as an auxiliary device allowing to obtain finite amplitudes satisfying all physical requirements. It is even absent (not explicit) in the most popular approach the dimensional regularization. We point out that the momentum cutoff treated as a bona fide physical scale, presumably equal or related to the Planck scale, would lead to unacceptable predictions. One of the dangers is a non-zero mass of the photon. In the naive approach, even with the cutoff equal to the Planck scale, this mass would grossly exceed the existing experimental bounds. We present the actual calculation using a concrete realization of the physical cutoff and speculate about the way to restore gauge symmetry order by order in the inverse powers of the cutoff scale. PACS numbers: 12.60.Fr,1480.Ec,14.80Va In usual applications of Quantum Field Theory (QFT) the momentum cutoff (explicit or, as in the Dimensional Regularization, implicit) is treated as an auxiliary parameter and sent to infinity at the end of the renormalization procedure. However in the context of a quest for a fundamental theory unifying elementary particle interactions with gravity, QFT models should be viewed as only effective theories with a real momentum cutoff which, as in QFT applications to statistical physics problems, should have a concrete physical interpretation, most probably of the intrinsic scale Λ of the underlying fundamental theory. In this short note, based on the previous work [1] and on the accompanying paper [2] (where all the relevant references can be found) we would like to point out some important aspects of treating the cutoff scale as a bona fide physical scale Λ (the problem was partly analyzed in connection with quadratic divergences in QFT [3–5]). The most spectacular danger of keeping Λ finite is, unless the effective field theory is of very special form, generation of the photon mass proportional to inverse powers of Λ. This is because the gauge symmetry ensuring the vanishing of the photon mass for Λ → ∞, for finite Λ remains generically broken. Since the bounds are extremely stringent, even the natural assumption Λ ≈ MPl (MPl being the Planck scale) could lead to unacceptably large photon mass, bigger than the experimental limit. In this note we illustrate this on a simple example and speculate how the problem could possibly be avoided in the context of an underlying more fundamental finite theory. To define the framework we consider first renormalization of a general YM theory choosing (out of many other possibilities) the momentum cutoff regularization which consists of modifying every derivative in the Lagrangian (including the recursively generated counterterms see below) according to the rule ∂μ → exp(∂ /2Λ)∂μ . (1) In the momentum space this prescription corresponds to the replacement kμ → Rμ(k) ≡ exp(−k /2Λ) kμ . (2) For instance, the regularized ghost contribution to the vacuum polarization tensor (diagram C in Fig. 1) reads Γ̃ αβ(l) = −tr(eαeβ) ∫ dk (2π) 4 i R(k)R(k + l) R2(k)R2(k + l) . (3) where eα are the antihermitian generators of the adjoint representation with included coupling constants (i.e. eα = g T ADJ α for a simple gauge group). With the replacement (2) the Wick rotation is, strictly speaking, not justified and neglecting the integral over the contour at infinity must be regarded a part of the regularization prescription (alternatively, the prescription can be formulated directly in the Euclidean version of the theory). As the standard analysis carried out in [2] shows, in the regularization (1) all diagrams of a renormalizable theory are convergent with the exception of one-loop vacuum graphs (which anyway cannot appear in physically interesting amplitudes as divergent subdiagrams). Computation of diagrams regularized in this way is more complicated than in the Dimensional Regularization but still manageable. For example, each one-loop diagram can be expressed in terms of the confluent hypergeometric function