Chapter 6: Radiative Corrections: Introduction

Summary of this chapter in Peskin and Schroeder by Patrick Van Esch.
Last revision August 28, 2003.

6.1 Soft Bremsstrahlung

The initial aim of the chapter is to set out to calculate 1-loop corrections to the photon-electron interaction.  In fact, we set out to calculate the correction to the electron-"heavy_charged_particle" interaction, and we can limit the corrections to the electron side.  It might at first not be clear why we only focus on such an asymmetrical situation, but it will turn out that what matters is a correction to the vertex, so we can limit ourselves to a diagram with only one vertex to be treated.  Those possible corrections are displayed in (6.1).  Two of them are external leg corrections which we shouldn't take into account to obtain an S-matrix element.  The last one will be handled in the next chapter, so the first one remains.  Why ?  The hidden program of this chapter is the vertex, and the last diagram is a correction to the photon propagator.  But at this point it is not clear yet why we make these separations.  So at first sight, for this chapter, we set out to work out the first diagram of (6.1).  This will turn out to be a scary ride !
The authors, however, anticipate one of the two problems that will occur with the working out this diagram.  So we start doing something apparently completely unrelated: we calculate the (0-loop, but same order in the coupling constant) two diagrams in (6.2), with a photon in the final state.  And we even go "back" further: we calculate classically what kind of EM radiation we expect when a classical charged particle undergoes a "kick".

Classical calculation

The computation presented is the solution to the following problem: a classical point charge has momentum p for t<0 and momentum p' for t>0.  What's the emitted EM field ?  In (6.3) is presented the general representation of the current density for moving point charges which is motivated (but not proved).  In (6.4) is presented the Fourier transform of the current density of the set-up of our classical problem.  Using this expression, we solve the equation for the 4-potential (in Lorentz gauge) in (6.5).  This expression is analysed and Coulomb contributions are separated from radiative contributions.  We are only interested in the radiative contributions as displayed in (6.7).  From the radiative part of the 4-vector potential is derived the E-field and the B-field, and the field energy contribution from these fields as an integral over k, as worked out in (6.13).  Working in a specific frame where there is only a direction change (no energy change) we work out the energy in the EM field in (6.14) and (6.15) which diverges (integral over a constant as function of k).
The argument is presented that there is of course a cut-off because it is physically impossible to have a sudden momentum change, this will be smooth on a certain level of magnification in time.  So we should cut off the k-integral somewhere, in which case this becomes a finite number, only function of the angle of k.  It turns out that there are two peak contributions as a function of angle: a contribution collinear with the initial momentum, and a contribution collinear with the final momentum ; there are negligible contributions for other angles (in the limit of highly relativistic charges).  When naively supposing that the contribution to the energy of the EM field in its Fourier representation between k and k + dk corresponds to a number of photons, we obtain expression (6.19).

Quantum computation

We calculate the contributions of the diagrams (6.2), assuming a very general interaction vertex between the electron and the virtual photon, and we're looking in the low momentum limit of the radiated photon so that we can neglect the influence on the vertex function.  In that case, the S-matrix element reduces to what is displayed in (6.22).  When switching to cross sections, we obtain in (6.24) the probability density (in d k) of having a photon emitted with momentum k, given the fact that the electron scattered from p to p'.  This expression looks a lot like the number of photons emitted as given by our naive, semiclassical calculation.
There is an interpretation problem and a mathematical one.  Both are related to the fact that for very small k-values this probability diverges: so we have probabilities that diverge.  If we introduce a small photon mass to have a cut-off on the lower side of the k-integral, we have regularized (parameterised) the divergence, and the result is displayed in (6.26).  The cross section for emitting a photon with momentum k is displayed in 6.26, and the factor accompanying the cross section for elastic scattering (no photon) is multiplied with what is called a Sudakov double logarithm.

6.2 The Electron Vertex Function: Formal Structure

In this section is derived what formal structure (tensor structure) the general vertex coupling can have.  Taking into account that it depends on p and p', can use the gamma matrices and should satisfy the Ward identity, we obtain a final structure that looks like (6.33).  There are two scalar functions of q^2 present, F1 and F2, that represent the total freedom of such a vertex function.  They are called the form factors.  These two functions describe completely the interaction of an electron with an external EM field.  We can for example extract the gross electric and magnetic properties of an electron.  
The physical charge of an electron is determined by calculating the interaction of an electron with a smooth, non zero electrostatic potential, using this vertex function containing F1 and F2.  It turns out that the electric charge (as defined by the classical Coulomb interaction) equals e  F1(0), so F1(0) = 1.  Given the fact that the tree diagram of the vertex already brings in a term 1 in F1(0), all corrections to F1 should vanish for q^2 = 0.
When calculating the interaction with an external magneto static field, we can calculate a potential (from the Born approximation of the cross section in the non relativistic limit) that is written as the typical interaction of a magnetic dipole with a B-field, and the dipole moment is expressed as a function of F1(0) and F2(0).   Finally, the magnetic dipole moment of the electron is calculated (using the Lande factor) in (6.37), and depends on F2(0) (as F1(0) = 1).

6.3 The Electron Vertex Function: Evaluation

The one loop vertex correction is worked out in (6.38) with the hope of obtaining the next order corrections to F1 and F2.  

Feynman parameters

A technique called the introduction of Feynman parameters is explained.  It consists in adding extra parameters which are integrated over to express a product of denominators as a single one to a higher power.  At first sight we complicate matters because we were supposed to solve an integral, not to add another one on top.  But it turns out that using this technique,  we can make the denominator only dependent on l^2  (l is a shifted 4-vector k + something) and integrate over l.  Working out the numerator in l, we can drop all uneven powers of l by symmetry, and the even ones are tensorial expressions only depending on l^2, so we've reduced our 4 dimensional integral over l to only a non-trivial integral over l^2 and a dummy integration over the rest.  The result is displayed in (6.47).  Analysing its form, we recognize the general form we found out in the previous section.

Wick rotation

The second technical point is the introduction of Wick rotation which simply allows us to consider the l^0 component as an imaginary variable, the reason being that the poles in the l^0 plane, using the Feynman prescription (+ i epsilon), are  not crossed if we rotate the real l^0 axis into the imaginary l^0 axis.  The advantage of doing this is that we now work with a Euclidean l "4-vector", and there it is simple to integrate over l^2 on one hand, and the "spherical" co-ordinates on the other hand.  We've previously worked out that the integrant only depends upon l^2 (now taken as a Euclidean norm).  
When applying this technique to the expression for the one loop correction to the vertex, we notice that only a few types of integrals are necessary: they are worked out in 6.49 and 6.50.  

UV divergence and Pauli-Villars regularization

However, it turns out that these integrals diverge in exactly the cases that appear in 6.47. As we're looking at the divergence caused by large k, we call it a UV divergence.   In order to regularize (have a parameter giving us the divergence) these integrals, the Pauli-Villars technique is introduced: a propagator (here the photon propagator) is replaced by the propagator minus the propagator with a large mass Lambda introduced, that's proposed in 6.51.  The advantage is that we can still use the same calculation scheme (with and without Lambda), so in the end this comes down in evaluating the difference of two "Wick" integrals, as shown in 6.53.  This gives us a result expressed in Lambda: the divergence is parametrised.
Working everything out except the integrals over the Feynman parameters, we see that our regularization has introduced Lambda in F1, and leaves F2 alone, so F1 "diverges".  The appearance of this divergence is independent of q^2, so in fact it even appears in F1(0).

Doing away the UV divergence in F1

We motivate an intervention that will be justified in chapter 7.  As we know that F1(0) = 1 at tree level, we just subtract from F1(q^2)  the correction in one loop for q^2 = 0, as expressed in (6.55).  The nice thing is that then, F1(0) = 1 at one loop level, and that nasty divergent term drops out.

Regularizing the IR divergence

When looking closer at the integrals over the Feynman parameters, we see (below 6.55) that these integrals for F1 diverge also ; the example for q^2 = 0 is taken.  In order to regularize this IR divergence, we introduce a small photon mass mu.

Final result for F1 and F2

With the UV divergence gone and the IR divergence regularized with mu, we can write out analytic expressions (still as Feynman integrals) for F1 and F2.  The nice thing is that F2 doesn't suffer at all from all these problems.  If we work out the 1-loop correction for F2 with q^2 = 0, we calculate the next leading order correction to the magnetic dipole moment of the electron, as worked out in 6.58 and 6.59.

6.4 The Electron Vertex Function: Infrared Divergence

The dominant part of the IR divergence of F1 (for small q^2) is worked out, and the result is given in 6.61 and 6.62.  The implication that this divergence has on the cross section is given in 6.63 and is catastrophic at first sight, because it implies a negative divergence !
In a first step, the factor in front of the divergent logarithm is worked out for the case where q^2 goes to - infinity and the result is displayed in 6.65.   When looking at what this does to the cross section, we see that this has the same but opposite effect as had bremsstrahlung on the cross section in 6.66 ; at least what concerns the divergent part of it.  Now we can understand why we were talking about these bremsstrahlung processes in the beginning of this chapter: if somehow we could combine both equal but opposite divergences, we would get rid of them.  The physical reason why we can only look at a combined cross section is explained: below a certain limit, no detector can detect a photon, so the cross section with and without these photons should be added together.  We proved this for large q^2, and next it is worked out for all q in 6.70..  Just notice that it is only indicated that the divergent log terms cancel against each other, but there may be a finite piece left over which is not worked out, apart from the now finite log term containing the detector limit (instead of the parameter mu).  For large q^2, however, we did work out everything, and that's shown in 6.71.

6.5 Summation and Interpretation of Infrared Divergences

In this section, we look at this relationship between divergences in vertex corrections and bremsstrahlung, to all orders.
It is first established that IR divergences only appear on the extremes of the fermion legs.  Soft photons emitted somewhere inside a diagram don't give rise to divergences.
The first structure considered is an "on shell" outgoing fermion line, to which are attached n soft photons.  After some dirac algebra (6.74) and a mathematical identity 6.75, the contribution of  such a diagram (including all permutations) is given in 6.76.  A similar calculation is carried out for an ingoing fermion line, and in 6.77 these results are combined: a pair of an ingoing and an outgoing fermion line that gives rise to n soft photons.
Next is considered the connection between two such photons into a "soft" loop.  Its contribution is given as X in 6.78, where we integrate over (the soft part of) the momentum of the connected photons.  It is made clear that X is of course also the infrared limit of the one loop correction to the vertex form factor in 6.79.  Note that this is only argumented, not proved and that many subtleties need to be worked out to do so, but that the final answer isn't affected by these.  The quantity X is regularized by a photon mass mu.
Summing over all diagrams of the kind (all numbers of loops), we obtain 6.80.
In 6.81 is considered the contribution to the cross section (the square of the matrix element) of a real, soft photon, when it is integrated over all its available phase space (and polarisations).  Because this quantity is divergent we also regularize with a photon mass mu.  It is noted that this quantity has been worked out already and equals 6.82.  Next the sum of cross sections with different numbers of soft real photons is worked out in 6.83.
Applying the same reasoning on a physical cross section being this sum of cross sections with all different numbers of soft, undetected photons (exp Y), and with all possible soft loop corrections (exp X) as for the 1-loop case in the previous section, we obtain 6.84, out of which not only the divergent factors in mu are eliminated, but which stays a positive real number (thanks to the exponential function).
At the end of the section, an interesting remark is made: if we replace our photon mass mu by a lower detection limit and consider photons up to a certain limit, we see that the conditional probability for a hard process to be accompanied by n soft (but measurable this time) photons is given by a probability distribution that fits a Poisson distribution, as in 6.86.  It turns out that the average value of the number of photons is exactly what we obtained in a semi-classical way in the beginning of this chapter.

Chapter overview

In this chapter we initially set out to calculate the 1-loop vertex correction in QED.  The general structure of a vertex function is worked out, leading to the introduction of 2 form factors F1 and F2.   It is anticipated by classical and quantum computations that the form factor F1 has IR divergences (together with a UV divergence) that are intrinsically related to bremsstrahlung.  After getting rid of the UV divergence by using a subtraction technique that will be justified only in chapter 7, the IR divergences in both processes (vertex correction and bremsstrahlung) are regularized by a small photon mass mu.  In the process of working out the 1-loop diagram in the vertex correction, basic techniques are introduced and applied,  such as Feynman parameters, Pauli-Villars regularization and Wick rotation.
Both divergences cancel in the sum of the cross sections (the loop corrected vertex diagram in elastic scattering and the bremsstrahlung diagrams) ; instead a lower limit on the detectability of photons is introduced in the cross section.
In the end, the relationship of these divergences is worked out to all orders, and shown to solve the two remaining problems: the positiveness of a cross section (no matter how sensitive the photon detector) and the statistical distribution of the number of photons emitted, which works out the same as in the semiclassical case.

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