QED loops

Corrected photon propagator

In (Fabiano, 2021) we have computed the correction to the photon line at one-loop level in QED. Remembering that the bare photon propagator is given by the expression

obtained, roughly speaking, by inverting the term F² in the Lagrangian (5) of (Fabiano, 2021). In Minkowskian metric the vacuum polarisation is given by

The physical or renormalised photon propagator is obtained by considering all possible corrections to the photon line, as illustrated in eq. (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:

Recalling the geometric series for which this expression holds true

one could immediately recognise the same pattern in eq. (4) and rewrite it as (Dyson, 1949), (Schwinger, 1951)

Corrected electron propagator

Proceeding in a manner completely analogous to previous section we could calculate the physical electron propagator. The bare electron propagator is given by

while the physical propagator S^P(p) is obtained by repeated insertions of Σ(p) calculated in (Fabiano, 2021), formula (27):

The expression for S^P is pictorially represented in eq. (8), this translates to:

and using eq. (5) we end with the expression

Counterterms

Up to now, we have computed all possible fundamental divergencies in QED. Those are necessary to build the necessary counterterms in order to renormalise QED. Those 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

and we add a counterterm Lagrangian with the same form of the present Lagrangian of (11)

The obtained renormalised Lagrangian

could be expressed in terms of the bare quantities defined in the following way:

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. Often, eq. (14) is called wave function renormalisation. The renormalised Lagrangian is

or in Dyson’s notation

The covariant derivative in transforms as

and, in order not to spoil gauge invariance of the Lagrangian it needs to be Z₁ = Z₂. It is possible to show that this is actually the case to all orders of perturbation theory.

The 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

and comparing to the inverse of the bare electron propagator, eq. (7)

one could infer that the term in is related to Z₂, while the term proportional to m is related to Z_m. Therefore

and

where functions F₂ 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.

The second correction we tackle is the one for the photon line encountered in (Fabiano, 2021). From eq. (22) of (Fabiano, 2021) we have

and using the relation of eq. (4) we have for the one–loop propagator

so that

where F₃ is an arbitrary dimensionless finite function.

Last comes the vertex correction, from (Fabiano, 2021) eq. (50) we have

that gives

where, once more, F₁ is a finite function. In terms of the Z notation, we summarise our results as

We remark once more that Z₁ = Z₂ 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

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 Hooft, 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:

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

which is actually a differential equation for electric charge e as a function of a mass scale µ:

where µ₀ is an arbitrary scale. The explicit solution to this equation is

which can be written in an explicit form for e²(µ):

A few comments on eq. (39). It has a singularity at the point

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 10²⁸⁴ eV, a huge scale much larger than anything envisaged so far – for instance the Large Hadron Collider (LHC) works at about 10¹³ eV, while the Planck scale, that is a scale at which quantum gravity effects should become relevant, is at “only” 10²⁸ eV.

As the energy scale increases, or conversely, the distance decreases, the electron charge increases.

Running coupling constant

The 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 g₀, then the above statement could be written as (hereafter, ε = 4 − D)

where g_k are regular functions in g. Analogous expansions exist for bare mass m₀ and bare fields ψ₀, .

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, dg₀/dµ = 0. Applying the derivative to eq. (41), one obtains

We have already discussed that µ∂g/∂µ is an analytical function in ε, so we can write it as follows:

and insert this form into eq. (42).

We obtain the equation for coefficients d of the beta function:

and observe that only the first two d terms survive, d₀ and d₁, 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

Solving eqs. (45) and plugging it back in eq. (43) we end up with

and taking the limit ε → 0:

We also found the recurrence relation for the coefficients of the counterterms:

This 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

This equation is known as the Callan–Symanzik equation (Callan, 1970), (Symanzik, 1970).

References

Callan, C.G. 1970. Broken Scale Invariance in Scalar Field Theory. Physical Review D, 2(8), pp.1541­-1547. Available at: https://doi.org/10.1103/PhysRevD.2.1541.

Dyson, F.J. 1949. The S Matrix in Quantum Electrodynamics. Physical Review, 75(11), p.1736-1755. Available at: https://doi.org/10.1103/PhysRev.75.1736.

Dyson, F.J. 1952. Divergence of Perturbation Theory in Quantum Electrody- namics. Physical Review, 85(4), pp.631-632. Available at: https://doi.org/10.1103/PhysRev.85.631.

Fabiano, N. 2021. Quantum electrodynamics divergencies. Vojnotehnički glasnik/Military Technical Courier, 69(3), pp.656-675. Available at: https://doi.org/10.5937/vojtehg69-30366.

Gell–Mann, M. & Low, F.E. 1954. Quantum Electrodynamics at Small Distances. Physical Review, 95(5), pp.1300-1312. Available at: https://doi.org/10.1103/PhysRev.95.1300.

’t Hooft, G. 1973. Dimensional regularization and the renormalization group. Nuclear Physics B, 61, p.455-468. Available at: https://doi.org/10.1016/0550-3213(73)90376-3.

Landau, L.D., Abrikosov, A.A. & Khalatnikov, I.M. 1954. An asymptotic expression for the photon Green function in quantum electrodynamics.Dokl. Akad. Nauk SSSR, 95, 497, 773, 1177 (in Russian).

Landau, L.D. & Pomeranchuk, I.Ya. 1955. On point interactions in quantum electrodynamics. Dokl. Akad. Nauk SSSR, 102, 489 (in Russian).

Schwinger, J. 1951. On the Green’s functions of quantized fields. I. PNAS, 37(7), pp.452-455. Available at: https://doi.org/10.1073/pnas.37.7.452.

Symanzik, K. 1970. Small distance behaviour in field theory and power counting. Communications in Mathematical Physics, 18, pp.227-246. Available at: https://doi.org/10.1007/BF01649434.

Weinberg, S. 1973. New Approach to the Renormalization Group. Physical Review D, 8(10), pp.3497-3509. Available at: https://doi.org/10.1103/PhysRevD.8.3497.

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