Corrections to propagators of quantum electrodynamics

: Introduction/purpose: e problem of quantum corrections to propagators in Quantum Electrodynamics (QED) is discussed. Methods: e Dyson–Schwinger equation is employed for correcting propagators in QED. Results: e observable quantities in QED are ﬁnite. Conclusions: QED divergencies can be avoided by redeﬁning physical quantities in a suitable manner.

obtained, roughly speaking, by inverting the term F 2 in the Lagrangian (5) of (Fabiano, 2021).In Minkowskian metric the vacuum polarisation is given by (2) e physical or renormalised photon propagator is obtained by considering all possible corrections to the photon line, as illustrated in eq. ( 3). (3) As we can see, the physical photon propagator is obtained by repeated insertions of vacuum polarisation diagrams at oneloop level, in the following manner: (4) Recalling the geometric series for which this expression holds true (5) one could immediately recognise the same pattern in eq. ( 4) and rewrite it as (Dyson, 1949), (Schwinger, 1951) (6)

Corrected electron propagator
Proceeding in a manner completely analogous to previous section we could calculate the physical electron propagator.e bare electron propagator is given by (7) while the physical propagator S P (p) is obtained by repeated insertions of Σ(p) calculated in (Fabiano, 2021), formula ( 27): (8) e expression for S P is pictorially represented in eq. ( 8), this translates to: (9) and using eq.( 5) we end with the expression (10)

Counterterms
Up to now, we have computed all possible fundamental divergencies in QED.ose are necessary to build the necessary counterterms in order to renormalise QED.ose counterterms are suitably constructed terms in the Lagrangian in order to cancel out divergencies and make results finite.To recap, we started with this classical Lagrangian in D dimensions (11) and we add a counterterm Lagrangian with the same form of the present Lagrangian of (11) where we have introduced Dyson's Z notation (Dyson, 1952), and bare quantities, which do not depend on the scale µ, are denoted by a 0 subscript.Oen, eq. ( 14) is called wave function renormalisation.e renormalised Lagrangian is (18) or in Dyson's notation (19) e covariant derivative in transforms as (20) and, in order not to spoil gauge invariance of the Lagrangian it needs to be Z 1 = Z 2 .It is possible to show that this is actually the case to all orders of perturbation theory.
e counterterms can be read off the one-loop calculations encoun-tered in (Fabiano, 2021).Starting with fermion line correction, from eq. ( 37) of (Fabiano, 2021) we extract the term (21) and comparing to the inverse of the bare electron propagator, eq. ( 7) (22) one could infer that the term in is related to Z 2 , while the term proportional to m is related to Z m .erefore ( 23) and (24) where functions F 2 and F m are arbitrary finite parts depending upon ε and m/µ, and are analytical as ε → 0. It means that the counterterms contain just the part proportional to 1/ε necessary to cancel the overall divergencies.
e second correction we tackle is the one for the photon line encountered in (Fabiano, 2021).From eq. ( 22) of (Fabiano, 2021) we have (25) and using the relation of eq. ( 4) we have for the one-loop propagator (26) so that (27) where F 3 is an arbitrary dimensionless finite function.Last comes the vertex correction, from (Fabiano, 2021) eq. ( 50) we have (28) that gives (29) where, once more, F 1 is a finite function.In terms of the Z notation, we summarise our results as We remark once more that Z 1 = Z 2 is satisfied to this order in perturbation theory.So using the relation of eq. ( 16) and remembering that ε = (4 − D)/2, for D → 4 we have (34) If we ignore the finite part of the counterterms by adopting a mass independent prescription, also known as the minimal subtraction scheme, or MS scheme ('t Hoo, 1973), (Weinberg, 1973), for which the finite part is zero, we can compute the so-called beta function due to Gell-Mann and Low (Gell-Mann and Low, 1954) defined in the following way: (35) which is an analytic function in ε.Compute the beta function from eq. ( 34) by differentiating with respect to µ, remembering that µ0 is constant taking the prescribed limit ε → 0, and obtain (36) which is actually a differential equation for electric charge e as a function of a mass scale µ: (37) where µ 0 is an arbitrary scale.e explicit solution to this equation is (38) which can be written in an explicit form for e 2 (µ): (39) A few comments on eq. ( 39).It has a singularity at the point (40) better known as the Landau pole (Landau et al, 1954), (Landau and Pomeranchuk, 1955).A careful evaluation in QED shows that the Landau pole is of order of 10 284 eV, a huge scale much larger than anything envisaged so far -for instance the Large Hadron Collider (LHC) works at about 10 13 eV, while the Planck scale, that is a scale at which quantum gravity effects should become relevant, is at "only" 10 28 eV.As the energy scale increases, or conversely, the distance decreases, the electron charge increases.

Running coupling constant
e formalism of the beta function and the existence of a so-called running coupling constant (An oxymoron!) is not a peculiarity of QED but it is standard behaviour in any quantum field theory.We have seen that in the minimal subtraction scheme the counterterms in the Lagrangian have no finite parts, therefore can be expanded in a Laurent series in ε containing only divergent parts.Call the generic renormalised coupling constant g and its bare version g 0 , then the above statement could be written as (hereaer, ε = 4 − D) (41) where g k are regular functions in g.Analogous expansions exist for bare mass m 0 and bare fields ψ 0 , .Now, a crucial observation is that all bare quantities are independent of the scale by definition.As the bare coupling constant is not dependent upon µ, dg 0 /dµ = 0. Applying the derivative to eq. ( 41), one obtains (42) We have already discussed that µ∂g/∂µ is an analytical function in ε, so we can write it as follows: (43) and insert this form into eq.( 42).We obtain the equation for coefficients d of the beta function: (44) and observe that only the first two d terms survive, d 0 and d 1 , so that eq. ( 43) is only linear in ε.We group different powers of ε, and each one of them has to vanish separately, so we have (50) Solving eqs.( 45) and plugging it back in eq. ( 43) we end up with (46) and taking the limit ε → 0: (47) We also found the recurrence relation for the coefficients of the counterterms: (48) is recursion relation is very important because it shows that the coefficients of higher order poles can, at least in principle, be computed from just the knowledge of the simple pole term.So, in the minimal subtraction scheme we have seen that the beta function depends only on the coupling constant g, and the latter depends only on the scale µ; therefore, we can write (49) is equation is known as the Callan-Symanzik equation (Callan, 1970), (Symanzik, 1970).
PDF generated from XML JATS4R by Redalyc Project academic non-profit, developed under the open access initiative 933(13) could be expressed in terms of the bare quantities defined in the following way: