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QUANTUM FIELD THEORY
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Putting everything together, the soft photon emission cross section reads
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(da) 21X 1 - { (2cp coth 2cp -; 1) In 211E + -In 1 + [3 dO. Mott n J1 2[3 1 [3
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-"2 cosh 2cp [3 sin (()12)
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Ilcos (1J/2) d~ (1- [32e)[~2 -1 cos2 (()12)] 1/2 In 11 -+ [3~ } [3~
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The physically measurable quantity, which we denote by (daldo.)(I1E) is the sum of elastic and inelastic contributions
da (I1E) = (da) + [da (I1E)J (7-89) dO. dO. elastic dO. inelastic Both terms are evaluated to the same order 1X3 and are given respectively by Eqs. (7-86) and (7-88). Schwinger, who first computed these corrections, has introduced the notations (j Rand (j B for the relative corrections arising from virtual or real photon emission and second-order Born approximation. In this way
(7-90) When adding the results of Eqs. (7-86) and (7-88) the infrared cutoff drops out altogether, as expected. The fictitious mass J1 has only been useful to give a meaning to the intermediate steps of the calculation and the J1 -+ 0 limit is now perfectly legitimate when care has been taken to allow for the experimental resolution I1E. The final result is, of course, sensitive to 11E. When adding the two contributions we may use the following identity which can be 0 btained by comparing the changes of variables performed when integrating (daldo.)elastic and [(daldo.)(I1E)]melastlc. Recalling that sinh cp = [[31(1 - [32)1/2] x sin (()12), we have
dljJ IjJ tanh IjJ = -
In (1 - [32)
+ 4(!i:(!;~) sinh 2cp
~ln(1-[32~2)
x cos(IJ/2) d~ (1 - [3 2~ 2) [ ~ 2 - cos 2 (()I2)] 1/2 Hence we find
(jR = 21X { ---;:
(1 - 2cp coth 2cp) (2I1E) 1 + In ---;;:;-
+ cp tanh cp + (1
- cp tanh cp)
1 1 - [3 2 coth2 cp) 1 - - + -In - - - cp coth 2cp In (1 - [3 ) 3 9 2[3 1 + [3
RADlATIVE CORRECTIONS
f32 sin 2 (012) (() 2 (012) sinh 2({) 1 - f32 sin
1 - f32 + 2" cosh 2({) f3 sin (012)
cos (8/2)
cos 2 (OI2)]1/2
[In(I+f3~)_ln(I-f3~)J} 1- f3~ 1 + f3~
(7-91)
(7-92)
f3 sin (OI2)[I-sin(OI2)] 1 _ f32 sin 2 (012)
Numerical investigation reveals that these corrections are far from negligible as they increase with energy. When liE/m ..... 0 the previous formulas become invalid. A criterion for validity is
-ln~<
m liE
To go beyond this result, it is necessary to include higher-order terms involving the emission of several or even an infinite number of photons. The contributions of soft photons is then known to factorize in exponential form, showing that the process has zero probability in the limit I:J.Elm -+ O.
7-3 NEW EFFECTS
Higher-order terms in the perturbation series induce new effects, some of which will now be briefly presented.
7-3-1 Photon-Photon Scattering
Four photon interactions have no classical counterpart and arise through quantum fluctuations of virtual charged particle pairs. From a theoretical point of view it is of interest to demonstrate how gauge invariance and current conservation cure the remaining potentially divergent amplitude, as was anticipated in the discussion of Sec. 7-1. Let us first sketch some dimensional arguments. The basic amplitude (Fig. 7-15) is of order a 2 so that the cross section takes the form 3 3 4 k 3 k4 a 1 MI2 J4(k 1 + k2 - k3 - k 4) a~ (7-93)
kl 'k 2
with obvious notations for the momenta and energies of the photons. The dimen-
Figure 7-15 Basic diagram for photon-photon scattering.
QUANTUM FIELD THEORY
sional parameters are, for instance, the common energy w in the center of mass frame and the electron mass m. Here M stands for a dimensionless matrix element so that (J behaves correctly as a surface. We expect gauge invariance to allow for the extraction of four powers of the momenta from M. This will be subsequently confirmed. Consequently, 1 12 must M behave at least as (w/m)8 for small w. In this limit kl . k2 is proportional to w 2 . We therefore predict that at low energies
w - 1 m (7-94)
where a is a numerical constant. Since rx 4 /m 2 ~ 10 ,ubarns, this cross section is exceedingly small up to the hundred-keY region. On the other hand, for w/m 1, due to the convergence of the process we do not encounter mass singularities so that simple dimensional counting indicates that
w - 1 m (7-95)
with b a second numerical constant. In other words, it is likely that (J peaks for typical values of w/m of order one. We shall not present a fully fledged calculation, but instead will show that the results of the last section in Chap. 4 allow us to compute the numerical constant a of the low-energy behavior. For that matter we observe that the EulerHeisenberg lagrangian [Eq. (4-123)] involves a summation of all one-loop diagrams, each one integrated over the same constant electromagnetic field as many times as there are external lines. Note that E and B are linear in the external momenta. Furthermore, at a given order in e the sum of all diagrams (corresponding to permutations of the external line indices) is gauge invariant. When contracted with a polarization vector 8P(k) for an on-shell process it should be invariant in the substitution 8p(k) -+ 8p(k) + Ak p. This means that this contracted quantity can only depend on 8(k) through the combination 8p(k)k a - 8a(k)kp- Up to a factor i this is the Fourier transform fpa(k) of the corresponding electromagnetic field. The term proportional to e4 will then involve at least four powers of the momenta. In the low-ener,p l.imit it will then be sufficient to study the coefficient of the combination f~:~, at zero frequency. If the external electromagnetic fields coincide, the corresponding quantity is the fourth-order term in the effective lagrangian computed in a constant field, which was given by Eq. (4-125) as 2 i(jit'4 = i [(E2 - B2 + 7(E . B ] = +7 (7-96)
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