IV. THÉORIE / THEORYTHE EQUIVALENT PHOTON APPROXIMATION IN ONE- AND TWO-PHOTON EXCHANGE PROCESSES

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IV. THÉORIE / THEORYTHE EQUIVALENT PHOTON APPROXIMATION IN ONE- AND

TWO-PHOTON EXCHANGE PROCESSES

P. Kessler

To cite this version:

P. Kessler. IV. THÉORIE / THEORYTHE EQUIVALENT PHOTON APPROXIMATION IN ONE- AND TWO-PHOTON EXCHANGE PROCESSES. Journal de Physique Colloques, 1974, 35 (C2), pp.C2-97-C2-107. �10.1051/jphyscol:1974213�. �jpa-00215523�

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IV. THEORIEI THEORY.

THE EQUIVALENT PHOTON APPROXIMATION IN ONE- AND TWO-PHOTON EXCHANGE PROCESSES

P. KESSLER

Laboratoire de Physique Corpusci~laire, College de France, Paris, France

Resume. - L'auteur rappelle le developpenient historique de I'approximation du spectre de photons equivalent (ou approxiniation de Willianis-Weizsiicker), en comnienqant par sa derivation semi-classique. L'application de cette approximation aux interactions nucleaires des muons du rayonnement cosmique (sous terre), ainsi qu'a la dinusion inelastique de diverses particules (hadroniques ou non hadroniques) par le champ coulonibien des noyaux-cibles, est evoquee. Une derivation de la formule de Williams-Weizsiicker, fondee sur la theorie des champs et utilisant I'helicite, est presentee pour les processus a Cchange d'un seul photon, puis etendue aux phenomknes a tchange de deux photons ((( collisions photon-photon n). Les conditions de validite de cette formule sont brikvement discutees, et divers tests nunieriques, dus a differents auteurs, sont prCsentCs ou du moins cites.

Abstract. - The author recalls tlie historical developnient of the equivalent photon (Williams- Weizsiicker) approximation, starting from its semi-classical derivation. Its application to the nuclear interactions of underground cosmic-ray muons and to the inelastic scattering of various (hadronic or non-hadronic) particles in the Coulomb field of nuclear targets is considered. A field-theoretical derivation of the Williams-Weizsiicker formula, based on helicity, is given for the one-photon exchange case and then extended to tlie two-photon exchange case ((( photon-photon collisions ^D).

Its conditions of validity are briefly discussed, and some numerical tests, performed by various authors, are shown or mentioned.

1. Historical background. Semi-classical derivation of the Williams-Weizsacker approximation. - T h e equivalent photon approximation, usually called

(( Williams-Weizsacker (or Weizsacker-Williams) approximation ⁾⁾in the literature, was mainly developed, in its semi-classical form, by Williams [ I ] and von Weizsacker [2] in the years 1933-35, but its origin can be traced back t o E. Fermi (1924) [3] and even t o Niels Bohr (1913-15) [4]. Now, sixty years is a long period in the history of modern science : in the mean- time, high-energy physics was born, many theories came into life and many died away ; but the Williams- Weizsacker approximation still stays alive ! N o t even has it aged. it is still fashionable, it has only become somewhat more sophisticated than it was. I cannot resist the temptation to show you briefly how really simple the criginal (semi-classical) derivation was.

As figure I shows, a n incident charged particle is passing close t o some target, and we call 11 the impact parameter between those two particles. The assump- tions we make are : E 9 nr (the incident particle is extreme-relativistic) and E $ w (the incident particle does practically not change its momentum during the interaction). c;llling E and w respectively the incident energy and the energy loss of the incident particle (in the target rest frame) and 171 its mass. Classical electro- magnetism then gives us the tl-11-ee componenls 01' the electromagnetic tield PI-oduced by tlie inciden! particle

FIG. 1 . - Classical scheme for establishing the equivalent photon approximation.

(with charge e) a t the position of the target a t a given time t :

(with y = Elm) ;

I ^Ell@)I

-

⁰

Making a Fourier transform

E,(w) = - _n1

1

^E,(t)^cos^or^{d l ;}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974213

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C2-98 P. KESSLER and making the additional assumption y % bw (which I shall discuss in a moment), we simply get :

Now we may consider that our field corresponds to a superposition of light waves of frequency w , travelling in the same direction as the incident particle. To get the equivalent photon spectrum, we equate the energy flux through a unit surface at distance b with the Poynting vector, as follows :

whence we get the spectrum

n(w) ⁼an-2 bK2 o-' .

Now we still must integrate over the surface of radius b in order to obtain the total equivalent photon spectrum

and since we must normally take [5] b = q-' (where q is the absolute value of the four-momentum transfer), we finally get

which, actually, is identical to the main term of the field-theoretical expression for the equivalent photon spectrum (in the case w << E).

Coming back to our additional assumption y b bw, it can easily be reexpressed as : q b qmi,, i. e. the lowest transfers are cut away. If one prefers to drop this restriction, then the Fourier transforms of the field components become somewhat more complicated (they then involve Hankel functions ; see [6]), and at the end of the calculation - using only the very weak condition q,,, b qnlin - one gets

where the non-logarithmic term is again almost the same as in the field-theoretical derivation (where you obtain ^-0.5).

Let us now consider the argument of the loga- rithm. It is easily shown that qnli, = n70/E. On the other hand, the adequate choice of q,n;i, is more problematic. If one takes (I,;,, z w (which means that the photon exchanged should be essentially

tr quasi-real ⁾⁾ in the target rest frame), one gets :

111 (Elni), which is frequently used nowadays. In the semi-classical derivation, however, the choice was usually made to identify bnli,, with the dimension, i. e.

the Compton wave-length, of the ⁽⁽larger ⁾⁾ of the two particles, in other words to identify q,,, with the sn~aller of the two corresponding masses. In case the

incident particle is the lighter one, one then gets : In (Elm).

Already in the thirties, many applications of this method were treated in quantum electrodynamics (see [ 6 ] , [7]). They consisted, for instance, in connect- ing bremsstrahlung in a Coulomb field with Compton scattering ; photoproduction of pairs in a Coulomb field with y y inelastic scattering (photon-photon collisions !) ; tridents with photoproduction of pairs, etc. Usually the more complicated process was derived, through the Williams-Weizsiicker approximation, from the simpler one. But there exists also, in the literature, a quite funny example of a ⁽⁽back- ward ⁾⁾use of the Williams-Weizsacker approximation : namely in the paper of Wheeler and Lamb [8] where those authors derived the cross section for photoproduction of pairs in the field of an electron (third- order process) from Racah's calculation [9] of tridents (fourth-order process). Why not ?

2. Application to nuclear interactions of underground cosmic-ray muons. - Whereas the first and most trivial applications of the Williams-Weizsacker approximation in QED may be considered by some people just as a sort of game (like a chess problem where you are requested to win in two drives when you are sure to win anyway ; it is of course more elegant to win in two drives than in ten, but who needs elegance ?), I shall now discuss an application where the approxi- _.- . -

mation was used to connect two theoretically unde- termined processes, namely muoproduction and photoproduction. Experiments on underground cosmic- ray muons were particularly popular in the fifties, before the big accelerators came into running. No theory was then available for these phenomena involving extremely high energies (up to 10"-10'2 eV), since at that time we had no quarks, no partons, no vector dominance model, and none of all those reliable theories (actually, are they ?) people use nowadays.

The first experiments of this type were done by George and Evans [lo]. The data obtained by them contained the so-called ⁽⁽stars ^)), i. e. events where, in the nuclear emulsion, at least one pion was ejected from the nucleus (minimal energy transfer :

z 150 MeV). The cross section found ( 2 10 pb) per nucleon for incident muons of mean energy = 10 GeV appeared to be well explained by the Williams-Weiz- sacker approximation, assuming an approximately constant photoproduction cross section of ²lo-'' cm' per nucleon between the threshold for pion production and several GeV (that was of course a n empirical assumption, compatible with the then existing experimental data). Later on, the Williams-Weizsacker approximation was also applied successfully to experiments involving ⁽⁽pionic showers ⁾⁾(i. e. multi-pion production, with an energy transfer of at least a few GeV) on the one hand [I I]-[I31 : and to processes involving nucleon (neutron or proton) eniission on the other hand [14]-[18], [19].

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T H E EQUIVALENT PHOTON APPROXIMATION I N ONE- A N D TWO-PHOTON EXCHANGE PROCESSES C2-99

The difference between these two formulae lies essentially in the argument of the logarithm. The WW for- mula, as written above, gives a steeper o dependence and, when integrated over o, a smaller total cross section.

In 1962, Daiyasu et al. [23] established a new field- theoretical version of the formula. slightly different from ours ; they also suggested to factorize the equivalent photon spectrum with a Hofstadter-type form factor in order to take the nucleon's structure into account. This latter suggestion does not seem to be a very good idea, as you can conclude from the fit shown by Higashi et 01. [I31 in their work of 1962.

Indeed, in figure 20, the c( point-like ⁾⁾ curve (i. e.

Daiyasu's formula without a form factor) fits the experimental histogram for da/dq"incompar-ably better than the forniula including a Hofstadter-type form factor. Now, in 1973. you would say : Sure, that is deep inelastic scattering, that's scaling, that's partons. That experinlent, however, was done long before the SLAC electroproduction experi~iients in tlie deep inelastic region.

Figure 2h shows the Iiistogsam for da/dtu. as compared to the various formulae proposed : Williams- Wei~siicker. Dniyasu ri 01. with and without a form fiic'tor. Kessler-Kessler. Here it is hard to say tvhich one gives the best fit.

G. N. Fowler [20] questioned the validity of the _{A N}

FIG, 2. - Fits shown in the work of Higashi et al. [13] (inelastic scattering of muons of energy z 100 GeV). a ) Number of events vs ⁽(= / q I>

experimental histogram,

-.-. equivalent photon approximation of Daiyasu et ul. [23], pointlike,

- - - equivalent photon approximation of Daiyas~l et al. [23], with a Hofstadter-type form factor.

Williams-Weizsasker approximation for these pro- 700 blems ; he thought that the semi-classical WW formula

might involve a gross overestimation (of about one order of magnitude). The point was important since, if that were true, those inelastic muon-nucleus scatter- 10- ing events could no longer be considered as due t o the electromagnetic field, and some new interaction bet-

ween the muon and the nuclear matter had t o be I-

considered in order to explain the data.

So my brother D a n (he was then involved in the

experiments mentioned) came and said : We must 0 1

h) N ~ ~ m b e r . of events vs co

-- expesin~ental histogram,

- . - . Kessler-Kessler formula [21], - - - Daiyasu et (11. formula (pointlike),

- - - . - - - - Daiyasu rt (d. formula (with form factor),

- . . - semi-classical Williams-Weizsacker formula.

.--%: *. ^A

\ I t

1 *'i t r ~ c e v ~

3. Application to inelastic scattering in the Coulomb field of nuclear targets. - Now we shall discuss applications of the equivalent photon approximation to the field Prof. Stodolsky treated in his talk. Such applications were studied, mainly in the sixties, by tnariy people (in particul:~r, Inany Soviet theorists).

I think one should distinguish here between t w o types of processes treated in tlie literature : (( Primakoff- type ⁾⁾processes, i. e. tliosc ones involving hudrons (like : 7 ⁺nu. ^;I ⁺1 1 . TL -> /), K + K "', A ^-t L. etc.).

wliesc of tour-sc the clcctromagnetic interaction can only he clTc~ti\c (i. c.. predominate over the strong make a field-theoretical derivation. That is what we 0.01 O.'? 1' 10 f00

did in 1956-57 [21], when we established the so-called

Kessler-Kessler formula. 1 must say that, about a t the ^{( a )} same time and of course independently, similar A N

for~nulae derived from field theory were given by loo-

Curtis and by Dalitz and Yennie [22] ; those authors were concerned by electroproduction instead of muoproduction, but in QED that is the same thing.

It then became customary for the experimentalists involved in the field t o compare their data with both the semi-classical Williams-Weizsacker formula and our formula, i. e. : 10-

WW formula :

N ( w ) = - - In - - 0.4 2 n

:

⁽

^E

⁾

K K formula :

I- ^-

N,,, = 2.1 ^(I-+ 2$) ( ~ n E ^-0.5). 7 ~b ^,^t^:^\ \\IOU

n o 171 '\'\ '

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C2-100 P. KESSLER

interaction) in the very forward direction of the particles scattered [24]-[27] ; and, on the other hand, processes involving only non-hadronic particles (photons, charged leptons, neutrinos, vector bosons W...) 1281-[35]. Mainly in connection with these studies, new formulations of the equivalent photon method were suggested by various authors [36-[38]. On several occasions, comparisons were made between the results of exact calculations for given processes and corresponding Williams-Weizsacker type approxi- mations. In some cases, a sharp disagreement was found. For instance, in the case of y + W - W + (in the Coulomb field), the approxin~ations calculated separately, and using different methods, by Bludrnan and Young [39] on the one hand, Lyagin and Tsu- kerman [40] on the other hand differed considerably (up to several orders of magnitude !) from the values obtained through an exact calculation by Williamson and Salzman [41]. Also for the process v ^-t /i W (in the Coulomb field), the approximation method used by Solov'ev and Tsukerman [42] led to results

quite different from those of Lee, Markstein and Yang [43], and also of von Gehlen [44], who performed exact calculations.

It must however be stressed that, in all those cases, different form factors were used by the various authors mentioned. Actually, comparing an exact calculation using a given type of form factor with a Williams- Weizsacker approximation involving a different type of form factor does not mean testing the Williams- Weizsacker approxinlation ; it rather means comparing the effects of two different form factors. Thus the above-mentioned examples don't really imply a breakdown of the Williams-Weizsiicker method.

Wherever the same for111 factor was used, the method actually worked quite well. Comparing their exact results for v -+ v l - I + with those of the equivalent photon approximation (used by Shabalin [45]), Czyz, Sheppey and Walecka [46] on the one hand, and Stanciu [46] on the other hand, noticed a qualitatively good agreement [47]. For v -+ p W , Uberall [48], using Badalyan's version of the Williams-Weizsacker

FIG. 3. - a ) Comparison between various approximation formulae in the calculation of the total cross section of PA' + VWJV ( M w = 2 GeV, gb~ = 1) (N = Cu63.5) [35].

- - - - - - - - Badalyan-Smorodinskii [36], - - - - - - - - Gribov et a / . [37], - , - . ^Gorge ^et ^a/.(first version) [38], Daiyasu et a/. [23].

- . . - ^{Gorge et} ^{a / .}(second version) [38].

b ) Comparison between various approximation formulae and the exact calculation, for the total cross section of

v.,\' -t jc W.\' (MIL, = 2 G e V , g l ~ = I ) (N = Cuh3.5) [35].

exact, - . - . - Badalyan-Smorodinskii,

- - - Gribov cJt 01. ; Gorge er 01. (first version), - . . ^-. . ^{Daiyasu ;}^Gorge ^et ^a/.(second version).

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THE EQUIVALENT PHOTON APPROXIMATION IN ONE- A N D TWO-PHOTON EXCHANGE PROCESSES C2-101 method, obtained an agreement within 10 % with

the exact calculation done by Bell and Veltman [49].

An interesting comparison between various versions of the Williams-Weizsacker approximation (Daiyasu et a/. [23], Badalyan and Smorodinskii 1361, Gribov et a/. [37] and both variants of GorgC et a/. [38]) was made by my former collaborator Le Guillou in his Thesis (1965) [50]. As figure 3u shows for the case ^LL + vW, all these f ~ r m u l a e give very similar results (using of course the same form factor). In figure 30, all those appl-oxirnations are also tested, for v -+ pW, versus an exact calculation, and here again the agreement seems to be very good.

To conclude this section of my talk, I shall mention some recent work, performed by Kiln and Tsai [51]

on the one hand, and by our College de France group [52] on the other hand, where - on the side of the target nucleus - incoherent inelastic (i. e.

resonant and deep inelastic) terms are included (Fig. 4, diagrams (c) and ( ( I ) ) , in addition to the coherent elastic (Fig. 40) arid incoherent elastic (Fig. 40) terms, in the equivalent photon approximation. To let you see how that works, I show you some tables extracted froni our still unpublished work. I n table I, concerning muon bremsstrahlung, we show the ratio between the approxin~ate and the exact result for all the terms (coherent elastic for three different nuclei with different form factors ^; incoherent elastic for protons and neutrons, uncorre- lated or correlated ; incoherent inelastic, i . e. resonant and deep inelastic) at various values of W (the invariant yp mass). You can see that in some cases the approximation is very bad. But actually you need not worry about those cases : indeed, wherever the equivalent photon approximation is bad, the contribution is small. That is a general rule, and it is of 'course not just a lucky accident. Bad terms are those which don't contain the pole in q 2 , or where the

FIG. 4. - Feynman diagrams for inelastic scattering in the Coulomb field of nuclear targets. a ) coherent term, b) incoherent elastic term, c ) incoherent inelastic (resonant) term, d ) inco-

herent deep inelastic term.

kinematic regions involved are far from that pole ; thus bad terms are small terms. That statement is proven by table 11, where the comparison between approximation and exact calculation is made for the sum of all (coherent and incoherent) terms, for the three nuclei considered, plus hydrogen (free protons).

You can see that the agreement is nowhere too bad, the difference between the approximation (which underestimates the process) and the exact result ranging between 2 and 27 %. In table 111, a similar comparison is shown - again summing up all coherent and incoherent terms - for photoproduction of muon pairs. You can see that here the approximation overestimates the process, and that the deviation from the exact result lies only between 0.2 and 4 "/,.

Ratio (da/d W) ,,,,,,. /(do/d W) ,,,,, ,for various tet.r?zs irr the process ph" + ~YJ\' at E, ⁼200 GeV U

coherent

- 0.929 0.969 0.986 0.995 0.998 0.999 0.999 0.999 0.999 0.999 0.998 0.997 0.996

Co cohe-

rent

-

0.892 0.951 0.976 0.99 1 0.997 0.999 0.999 0.999 0.999 0.999 0.998 0.997 0.994

C cohe-

rent

-

0.868 0.9 19 0.955 0.980 0.993 0.997 0.998 0.998 0.999 0.998 0.998 0.097 0.996

p elastic (uncor- related)

-

0.814 0.829 0.861 0.902 0.947 0.975 0.984 0.988 0.990 0.99 1 0.992 0.993 0.992

n elastic (uncor- related)

-

0.005 0.08 1 0.190 0.383 0.675 0.877 0.938 0.964 0.977 0.984 0.990 0.993 0.993

p elastic (correlated)

-

0.078 0.280 0.428 0.6 16 0.826 0.938 0.968 0.980 0.986 0.989 0.992 0.993 0.992 -

17 elastic (correlated)

-

0.00 1 0.033 0.111 0.292 0.615 0.853 0.926 0.957 0.972 0.98 1 0.989 0.992 0.993

1' or ¹¹ resonant

-

0.005 0.080 0.179 0.349 0.614 0.825 0.900 0.934 0.952 0.962 0.973 0.977 0.978

n deep inelastic

-

0.001 0.025 0.063 0.135 0.280 0.456 0.560 0.629 0.68 1 0.72 1 0.782 0.826 0.857

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C2-102 P. KESSLER

TABLE I1 The linear transverse polarization vectors of the photon are defined as unit vectors along the x and y Ratio (dald W),pp,ox/(da/d W),,,,, for the sun7 of

all terms in the process p N -+ pyN at E, = 200 GeV axes. The longitudinal polarization vector ^EO is defined as a unit vector orthogonal (in four-space) -

w,, (MeV) H C Co U t o E,, E~ and q (the photon's four-momentum). As

- ^- - - - usual, one introduces the circular combinations

120 0.768 0.830 0.882 0.925

Ratio (da/d W),,p,o,/(da/d W),,,,, for the sum of all terms in the process yN -+ ji,ZAp at E., = 200 GeV

wit, _-^(MeV) ^H_- _-^C ^Co_- _-^U _z

220 1.004 1.003 1.002 1.001 300 1.008 1.007 1.003 1.001 500 1.013 1.012 1.005 1.001 1 000 1.024 1.018 1.007 1.002

2 000 1.029 1.021 1.009 1.004 FIG. 5. - 0) Feynman diagram for t h e reaction AB -t AC 3 000 1.040 1.022 1.012 1.008 occurring via one-photon exchange. b) Corresponding kine- 4 000 1.038 1.031 1.022 1.020 matic scheme in the yB center-of-mass frame.

5 000 1.038 1.036 1.033 1.032 6 000

8 000 and one has the relation

1.032 1.039 1.040 1.040

The conclusions of Kim and Tsai (considering other differential cross sections, other nuclei with other form factors, and partly other processes) are qualitatively similar.

4. Derivation of the equivalent photon approximation for one-photon exchange. - Many field-theoretical derivations have been presented by many authors : but in any case, lielicitj~ is the key to the equivalent photon approximation. So 1 shall show you the kind of helicity treatment we use [53].

Let us consider the Feynman diagram of figure 5n, where we assuriie A to be the incident particle and B the target (there is no loss of generality involved in this assumption, since we can always go to the inverse Lorentz frame). I n the yB center-of-mass frame (see Fig. 5b), let us call : the virtual photon's nionientum axis. Then we single out some particle C , f r o n ~ the final system C of the right-hand vertex, in order to define the :s plane ((( scattering plane ))).

To make our derivation, let us start from the matrix element

J L

-

^1,^ghV^I-"

where I, is the current of the left-hand vertex and r, that of the right-hand vertex. Using the well-known closure relation

where the last term at the rhs drops out because of vector current conservation, we then get

(with summation over ¹⁷¹ implicit), defining :

We then get

-

d a

-

^CA(,&,* = L _ll1l'I^-R'""'

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T H E EQUIVALENT PHOTON APPROXIMATION IN ONE- A N D TWO-PHOTON EXCHANGE PROCESSES C2-103

with

where C means summing over the spin states of all external particles at the vertex involved.

Using various symmetry properties (hermiticity, parity conservation combined with rotational invariance, and T-invariance), one easily gets :

and the analogous relations for the matrix R,;.

The off-diagonal elements of L,; contain factors ei("-")q, where cp is the azimuthal angle between A and C, in the yB cm frame. Integrating over cp (we here assume that the full phase-space of the system C is integrated over), the contribution of all off-diagonal matrix elements thus vanishes, and we stay with

where N t r and N , are the virtual photon spectra, transverse and longitudinal respectively, and a,, and a, are the corresponding virtual photoproduction cross sections ; s is the total energy squared in the overall c. m. frame ; t = I q2 I ; and W is the invariant mass of the system C. Such a formula was actually given already a long time ago by Hand [54].

The equivalent photon approximation now consists in making

where a, means the real photoproduction cross section, and where the last identity is obviously due to gauge invariance. We can fix empirically the most reasonable condition of validity for this approximation, namely : t 6 w * ~ , where to* is the photon's energy in the yB cm frame ; in other words, the photon should be ⁽⁽ quasi-real ⁾⁾in that frame. Under this condition, one gets

where M is the target mass and w the photon's energy in the target rest frame. Since w > w * , the photon is then quasi-real a fortiori in the target rest frame.

' The transverse virtual photon spectrum is (in the case A

-

^{e or}^p),integrating over t :

x [ ( t + 4 1 7 7 ' ) sinh' x + 2 t ] .

This formula is rigourous, except for the fact that we neglect the incident particle's mass m with respect to its energy E (in the target rest frame). x is a funda- mental parameter in the helicity treatment : it is the rotation angle, in four-space, between the left- hand and the right-hand ⁽⁽vertex plane ^)), turning around the photon line (Fig. 6 ; see also [53]).

FIG. 6 . - Space-time diagram for AB + AC occurring via one-photon exchange.

We get [53]

k i n -

where t,,, IS the highest valu.: t is allowed to reach

k i n

by the kinematics ; since tmax is usually of the order of s, we may assume essentially : t Q ::t ; using also our above-defined condition for t , we get

k i n 2 ~ ( t - tmin) tmax sinh x ^2:-

w2 ~ * ~ ( t + ^{4 m2) '}

and using the kinematic relation

stmin tz:x = m2(w2 - M212 = 4 m2 W' w * ~ , one finally gets

2 4 m2(t - tmin) sinh x

-

_tmin(t₊_{4 m2)}'

Thus

Using t,,,, 9 tmin, and

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C2- 104 P. KESSLER and the above-given relation between W2 and o, we are led to the standard formula for the equivalent photon spectrum :

The main term in this formula is the logarithmic one, and all authors agree on the coefficient of this term. There are, however, some differences appearing in the literature, concerning (i) the non-logarithmic part, and (ii) the argument of the logarithm. I shall discuss these two points briefly.

(i) Since the error involved in the approxinlation is of the order of the non-logarithmic term, some authors (for instance, F. Low [ 5 5 ] ) leave this term out. Other authors (for instance, Dalitzand Yennie [ 2 2 ] ) include small additional terms in tlie formula.

(ii) The choice of t,,,;,, can be made in different ways. Only in the case where the calculation is done for an experiment involving a cut-off Or,;,, on the scattering angle of the incident particle A, this choice is unambiguous, namely : t ,,,:,, -4 E(E-w) sin2 (0,,;,,/2).

Otherwise, one may take :

k i n

- either t ,;,, = t ,,,:,, ; one thus includes high transfer values, where tlie approximation is comple- tely wrong ; but since both the exact calculation and the approxinlation give very small contributions in this high transfer region, their difference should also be very small, and the integration over t should still give the right order of magnitude ;

- or tlnnx a w:i:2 , according to the condition we have defined ;

- or t,,;,, z w Z , which has the advantage of leading to the very simple form : In (Elm) ;

- or t,,;,, a A , where A would be some cut-off parameter (for instance, A = mi) accounting for some structure at the right-hand vertex.

There is thus some freedom in the equivalent photon method : the c h o i c ~ of the formula used depends largely on the taste of the physicist and of course also on the physical probleln considered. I shall make a remark which is quite trivial, but perhaps nevertheless worth while making, since there has been some controversy on the question which equivalent photon formula is better than which : Wherever the equivalent photon approximation tends to over- estimate a process, the formula giving the smallest spectrum will be the best one ; wherever it tends to underestimate, the formula involving the largest spectrum will give the best result. In conclusion, I would say that we must be very flexible and tole- rant in this field, and that all fornlulae given by the

(( good ⁾⁾ authors (i. e. those who made a correct field-theoretical derivation) stand more or less on equal footing.

A last remark to conclude this section of my talk.

Similar equivalent photon spectra can be calculated

in the case of incident hadrons (bosons, baryons, nuclei) instead of leptons, but in that case one should in principle include form factors [ 5 6 ] .

5. The equivalent photon approximation for two- photon exchange processes (a photon-photon colli- sions ^D).^-I shall here stick to two-photon processes occurring in e- e + (or e- e-) storage rings, although many authors also studied their occurrence in lepton- hadron and hadron-hadron collisions [57]. We thus consider the Feynman diagram of figure 7a and the corresponding kinematic diagram of figure 76 where the : axis is now chosen to be the momentum axis in the ^y;lc. m. frame. Again we single out some final particle X , in order to define the z . ~ plane. The transverse polarization vectors of both photons are defined as in figure 7b, and again we makecircular combina- tions of them in order to connect them with the

+ ¹helicities. The longitudinal polarization vectors are again given by the orthogonality condition. Now we start our helicity treatment as above. We write

M - ^I,,^gPY^t1iyp^gp"^ro

FIG. 7. - a ) Feynman diagram for the reaction ee' + ee' X occurring via two-photon exchange (photon-photon collision).

b) Corresponding kinematic scheme in the yy' center-of-mass frame.

where I, and r, are again the left-hand and right- hand currents, and m,, is the second-rank electromagnetic tensor in the middle. The closure relation for both gp and gP" (using the left-hand and the right-hand photon's polarization vectors respectively) leads us to

with

(10)

THE EQUIVALENT PHOTON APPROXIMATION IN ONE- AND TWO-PHOTON EXCHANGE PROCESSES C2-105 whence we get

with

(1 ⁼sum over external spin states).

-

Here L,,, contains a factor ei(n'-n')'P and R,,; a factor - ~ P P ' , cp being the azimuthal angle between the left-hand electron and X,, and cp' being the azimuthal angle between X I and the right-hand electron, both in the y y c. m. frame. Setting cp + ^cp' ⁼cp,, (azimuthal angle between both electrons in the y y c. m. frame), integrating over cp or cp' (we here suppose again that the full phase space of the system X is integrated over), the product of the factors containing the azimuthal angles finally becomes

Thus

where the factors with azimuthal dependence were now taken out from the left-hand and right-hand matrices.

Using all symmetry properties as in the previous section, one gets :

d o

-

2 L + + ( M + + , + + + M + - , + - ) R + + +

+ 2 L + + M + o , + o Roo + 2 Loo Mo+,o+ R + +

+ Loo Moo Roo

+ 4 cos c p 1 2 L + o ( M + + , o o - M+o,o-1 R + o

+ 2 cos 2 q I 2 L + - M + + . - - R + - .

Such a helicity formula has been obtained independently by many authors [58]-[63].

Now the small-transfer approxiniation is made in the following way. F o r its validity, we fix the empirical condition

(with t = 1 q2 1, t' = 1 q r 2 1, W = invariant mass of the system X). Actually, this condition is practically equivalent t o that set for the one-photon case in section 4, since (once tlie above condition is satisfied) one has for both photon energies in the y y c. m. frame :

ot:/i

-

^{W / 2 .}^Then

(i) All terms containing longitudinal amplitudes drop out.

(ii) Using the relation (which has been proven C641)

where f$,, is the azimuthal angle between the outgoing electrons in the lab frame (ee c. m. frame), m the elec- tron mass and E, the various electron lab energies, the last term in the above-given formula drops out when integrating over $,,.

(iii) Jn the first term,

Using also

where o and w' are the photon energies in the lab frame, we get

d o

-

N(o) d o N(o') dm' a,, (4 ww') with (calling E the beam energy) :

and the analogous formula for N(wl). The expression of sinh2 x is simplified as in section 4 (only replacing M 2 by - t' ; and the condition t<w*, by t, t' < W2).

and one gets again the standard formula obtained in that section for the equivalent photon spectrum.

Also the whole discussion given there on the formula is still valid here.

Let us notice that the double equivalent photon method may also be used to compute angular distributions (more precisely : distributions with respect to orbital angles) of particles produced. However, for that purpose, we must require that both photon momenta are essentially parallel to the beam axis, and this requirement can be shown (see [65]) to be equivalent to the conditions : t < w2 ; t ' 6 w',.

This set of conditions, combined with the restriction t.

t' < w2, is tilore stringent than the latter restriction alone.

Numerical tests on the validity of the double equivalent photon approximation have been performed by various authors [62], [66]-[69], using various versions of tlie approximation formula. They are all excellent, in the sense that the error involved in the approximation practically never exceeds f 50 %.

T o conclude my talk, I would like to say how deeply I regret that our colleagues from Novosibirsk are not here t o participate in tlie discussion ; all the mot-c since both theorists' groups there (Baier and Fndin [70]. and Budncv and co-workers [71]) have done co .much of valuable work in that field.