THE EQUIVALENT PHOTON APPROXIMATION

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THE EQUIVALENT PHOTON APPROXIMATION

D. Lyth

To cite this version:

D. Lyth. THE EQUIVALENT PHOTON APPROXIMATION. Journal de Physique Colloques, 1974,

35 (C2), pp.C2-113-C2-114. �10.1051/jphyscol:1974215�. �jpa-00215525�

JOURNAL DE PHYSIQUE Colloque C2, supplPtnent au tl0 3, Tome 35, Mars 1974, page C2-113

THE EQUIVALENT PHOTON APPROXIMATION

D. H. LYTH

Department of Physics, University of Lancaster, Lancaster, U. K.

RBsumC. - Nous fixons une skrie de conditions qui doivent &tre imposees en vue de I'invariance de Lorentz du spectre equivalent de photons et de la polarisation du photon virtuel. Nous insistons en particulier sur la condition d'apr6s laquelle l'angle d'emission du photon virtue1 doit &tre petit.

Abstract. - We establish a set of conditions which must be satisfied in order to make the equi- valent photon spectrum (and also the virtual photon's polarization) Lorentz-invariant. We insist in particular on the condition that the virtual photon's emission angle must be small.

I want to summarise the approach described in the publications [I]-[2] of C. J. Brown and myself, parti- cularly ref. [2].

We use the helicity formalism for the virtual photon scattering amplitude, just as in the usual formalism for electroproduction [3]. For the process

y, + ^{p -t hadrons}

our treatment reduces to the standard one for electro- production. For the process

y v + y, ^-t hadrons

with which we are mainly concerned, our equations look almost the same, except for a doubling up of the photons indices.

For ordinary electroproduction, the one-photon exchange expression for the cross section can be written as

d a da,

-- = [ N ( E , E', k2) d ~ ' dk2] ---

dlips, dLips,,

where dn,, is the virtual photoproduction cross section and dlips, is the Lorentz invariant phase space ele- ment. The quantity in square brackets may be regarded as the number of equivalent virtual photons, usually called the cr photon flux factor )) by electroproduction people, and is a function of the initial and final electron energies E and E' and of the virtual photon mass squared k2. For electron-positron scattering, we have written down the corresponding two-photon expression [I]

diagrams. The equivalent photon approximation has the same form as the above equations. The differences are that : (i) the cross section d a , or da,, is replaced by the real photon cross section ; (ii) in evaluating this cross section, the kinematics is simplified by taking the photon momentum to be parallel with the electron beam direction ; (iii) the expression for the equivalent number of photons is usually simplified in some way.

A very important difference between the equivalent photon approximation and the (c exact

equations written down above concerns their Lorentz invariance properties. The quantity --- dn is Lorentz-invariant.

dLips,,

but the corresponding virtual cross sections are not.

and neither are the

exact

expressions for the equi- valent number of ^\ irtual photons. On the other hand, the real photon cross sections (i. e. the k 2 = 0 limits of the virtual quantities) are Lorentz-invariant. so that in the equivalent photon approximation the expressions used for the number of equivalent photons ought to be Lorentz-invariant (to maintain the Lorentz- invariance of . In practice they never are

dLips,

however. which means that the accuracy of the equivalent photon approximation is frame-dependent.

Similar considerations apply when we consider the polarisation properties of the photon. The virtual photons may be regarded as plane-polarised in the electron scattering plane (together with a longitudinal component of course). which is not a Lorentz-invariant plane. However a real polarised photon beam has the property that the plane of polarisation is invariant d o

-- under a Lorentz boost. This apparent conflict is

- [ N ( E , , E;, kl) dE', dl<:] x

dlips,, resolved by the fact that for infinitesimally small

da,,. scattering angles the electron scattering plane becomes x [N(E,, E ; , k i ) d ~ ; dl;f] x

^{-- -} . invariant. In the same way we have to consider the

dLrps,,

Lorentr-invariance properties of the quantity (usually Our treatment of the equivalent photon approxima- denoted by i : ) which describes the degree of polarisation tion starts with these equations which are of course of the v i r t ~ ~ a l photon beam.

exact evaluations of the relevant one-and two photon Thc val-iou.; conditions necessary for the validity of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974215

C2-114 D. H. LYTH

the equivalent photon approximation are not inde- This last point enables us to understand the discre- pendent, and we looked for a simple necessary and pancy between publis hed results [4]-[6] for the total sufficient set [2]. We came up with the following set. cross section

(a) The electrons are extremely relativistic.

( 6 ) The virtual photon masses are small enough so that the virtual photon helicity amplitudes reduce to their real photon limits, i. e.

- k? 42 M:,

where M, is the smallest relevant hadron mass.

(c) The virtual photon momentum is sufficiently parallel with the electron beam direction. We have shown that this last requirement is equivalent to the requirement

- k2 42 (E - E'Y

i. e. to the requirement that the virtual photon is almost real !

ee -, eC + (all hadrons of fixed mass)

in the limit of infinite beam energy. The point is that, in this limit, the ratio (E - E')/E must necessarily go to zero (for one or both electrons). Therefore the region of validity for the equivalent photon approxi- mation shrinks. On this basis, we have been able t o understand explicitly the difference between Brodsky et al. [4] and Bonneau et at. [6]. However, these consi- derations are somewhat academic, because for prac- tical beam energies the various expressions are not in serious disagreement.

Within the region of validity of the equivalent photon approximation, the following simple expression may be written down

Requirement (c) is not equivalent to the requirement

that the scattered electron goes forward. In particular, N(E, E', k2) dE' dk2 =

when the electron angle 19 is small, then _- ^- _{- ---} [ ^{+ (E'/n)2} _mf]

dk2 2 n E - E ' - k2 k4

The more complicated expressions in the literature so that the requirement on the electron angle is differ from this one only by terms vanishing like k2/(E - E')' which are therefore insignificant. The E - E'

o2 ^{-g --} . expression becomes Lorentz-invariant in the limit) E' where all our conditions are satisfied.

References

[I] BROWN, C. J. and LYTH, D. H., NucI. Phys. B 53 (1973) MANWEILER, R. and SCHMIDT, W., Phys. Rev. D 3 (197 1

323. 2752.

[2] BROWN, C. J. and LYTH, D. H., Lancaster preprint (1973), [4] BRODSKY, S. J. et al., Phys. Rev. D 4 (1971) 2927.

submitted to Nucl. Plzys. B. [5] BAIER, V. N. and FADIN, V. S., NUOVO Cimento Letters [3] JONES, H . F., NLIOVO Cimento 40A (1965) 1018. ¹ (1971) 481 ; Plzys. Lett. 35B (1971) ; JETP 34

VON GEHLEN, G., NUCI. Phy.7. B 20 (1968) 102. (1972) 253.

BERENDS, F. A., Phys. Rev. D 1 (1970) 2590. [6] BONNEAU, G . et al., Nucl. Pliys. B 54 (1973) 573.