BEYOND THE EQUIVALENT-PHOTON APPROXIMATION

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BEYOND THE EQUIVALENT-PHOTON APPROXIMATION

K. Subbarao

To cite this version:

K. Subbarao. BEYOND THE EQUIVALENT-PHOTON APPROXIMATION. Journal de Physique

Colloques, 1974, 35 (C2), pp.C2-115-C2-117. �10.1051/jphyscol:1974216�. �jpa-00215526�

JOURNAL DE PHYSIQUE

Colloque C2, supplkment au

Tome

Mars 1974, page

BEYOND THE EQUIVALENT-PHOTON APPROXIMATION

K. SUBBARAO

Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14850, USA

RCsumC. -

Lorsque la section efficace d'un processus ee

eeX, oh X est un etat hadronique quelconque, est consideree de faqon plus precise que dans l'approximation du spectre de photons equivalent, des termes importants doivent Etre pris en consideration. Ces termes, qui peuvent 6tre rattaches aux amplitudes du processus

X (avec des photons virtuels), sont discutes dans le present travail.

Abstract. -

When the cross-section for the process ee

eeX for a given arbitrary hadronic state

is considered more accurately than in the equivalent-photon approximation, important terms arise. These terms, expressible in terms of the helicity amplitudes for the process

with virtual photons, are discussed.

The process ee

eeX, where X is a hadronic state, takes place a t high incident energies predominantly through the diagram shown in figure 1. Since the final electrons tend t o be in near forward directions, it is often useful to integrate over them. Let E be the enekgy

FIG. 1.

- Dominant diagram for

process

per beam of the incident beams, nz the electron mass a n d Js t h e invariant mass of the state X

F o r E

2 G e V one gets

2 x IO-'and L

1 8 ; if Js

0.3 GeV then z z 0.006 and

5. We want the cross-section da/ds which in the limit

0 with z fixed i. e., E

with

fixed, has the form

+ ( t e r m s - + O a s c - t O ) ] ( I ) The expression for

is well-known

M1 =

[(2 +

^L,

2(3 +

^(I

a,.,,,(,r) being the cross-section for the real photon process. Our main concern here is t o consider the expressions for

a n d M , in terms of off-shell helicity amplitudes for the process

X for a n arbitrarily given hadronic state X a n d thus disentangle these amplitudes. The approximation of keeping the

term only will be called the Weizsacker- Williams approximation (WWA). The Double Equi- valent-Photon Approximation (DEPA) gives [I], [2]

M , L2

+

L +

where

and M j can be read off from the formulas in reference

That

is different from

is evident from the fact that by rzot neglecting the dependence of the y y

X amplitude on even one of the virtual photons, a term of order L emerges which is absent in DEPA. Additional

terms also arise from other approximations made in DEPA as we shall see later.

Incidentally, these considerations are all valid also

d

-

for differential cross-sections -- where d T is a d s d r

differential involving variables referring to the momenta and angles of the particles in X in the

center-of-mass frame. For example, if X is a n+

d

pair, one can study where 0 is the scattering d s d cos 0

angle in the

center-of-mass frame

the integration over the final leptons can be done just as well holding

the

center-of-mass frame can be done if either the two final electrons o r all particles in

are detected and measured. (For cross-sections differential with respect to the lab momenta of the particles in X, additional constraints appear

For definiteness.

only dg1d.v will be considered.

The general expression for

is somewhat long [4]

and tli;~t fo'r

even more so [5]. However. in the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974216

C2-116 K. SUBBARAO

limit of small z i. e.,

g 4 E 2 , these expressions

simplify provided terms of order L, and lower are dropped

- 16 LL, + ^{6 L:} + ( 39 -

"1 I a,,,,

where

(k, and k2 being the virtual photon momenta) WTT

1

2 [Wllll + Wl-11-11, WLT

with

W,.

J d4x eik" < X out j T*(J,(x) Jv(0)) / ⁰ > ,

E ,

and ~j being the polarization vectors of the two

photons in the

center-of-mass frame. The func- tion WTT depends on x,

and

in general

W,,(s) stands for its value at

0.

The interesting information namely that contained in the W's has thus been disentangled in eq. (3) from the uninteresting complications arising from the leptonic vertices.

Eq. (3) is to be compared with WWA and DEPA which would give respectively

In deriving DEPA [I]

( a )

the longitudinal polarizations of the virtual photons are neglected

this is taken into account by the W,, terms

(b) the variation of the yy

X amplitude with the (transverse) photon masses is neglected

this is taken into account by the WTT(x) - W,,(O) terms

(c) the phase space is approximated by its value for zero photon masses and forward outgoing electrons, and

(d) the electron mass is neglected in the integrand

these are taken into account by the additional terms multiplying

Observe the following

(a) There are no LL: terms in eq. (3). Such terms being of order L would not necessarily have shown up in DEPA. In fact, they are not present at all.

(b) If instead of the limit

with s/E2 fixed that we have been considering, we consider the limit E

co with s fixed, then L, is to be treated .as of order L and the

4 - L: term in eq. (3) would give a

3

correction to WWA as large as

113 of the WWA result itself [6]. However for hadron production at any presently conceivable energies, it is the former limit that is relevant

the

L: term is only a few percents

3

of the WWA result, nowhere close to

113.

(c) In an explicit model WTT(x) and W,,(x) are known. For example, for pion pair production with point pions, eq. (3) agrees with the results of Baier and Fadin [7].

(d) If one is doing ee

ee + any hadrons, then WTT(x) and WLT(s) are the structure functions for deep inelastic electron scattering off a real photon target. For large

these functions are expected to scale.

(e) The terms involving helicity flip amplitudes like W,,-,-, occur only in the terms of order L,, not even in order L:.

( J ' )

The terms involving WIjmn(s,

s) correspond- ing to the scattering of highly spacelike photons also occur only in order L,,.

(g) Simple expressions analogous to eq. (3) can also be derived when the final electrons are confined to the following more restricted regions of phase space (provided

2 Em)

0 2

02

(i) 2 El s i n 2 L < c,

2

sin

2 2

2

0

(ii) 2 E l sin -I 2 > r., , 2 E, sin2 5 ^{2 2 c2}

where E l and E, are the energies and 0, and 82 the scattering angles of the outgoing electrons

c, and c2 are any numbers of order 1.

(11) While pushing the accuracy of calculation of

contributions from figure 1, the contributions from

the neglected diagrams should be kept in mind.

BEYOND THE EQUIVALENT-PHOTON APPROXIMATION

References

[l] BRODSKY, S. J., KINOSHITA, T. and TERAZAWA, H., Phys. [5] BONNEAU, G. and MARTIN, F., preprint, Laboratoire de

Rev. D 4 (1971) 1532. Physique Theorique et des Hautes Energies (1973).

[2] Note that DEPA gives (...) )) means that the cross-section SUBBARAO, K. (unpublished).

in DEPA is given by replacing the square brackets [61 BONNEAU, G.9 GOURDIN, M. and MARTIN, ^F.3 N u c ~ . P h ~ s - B 54 (1973) 573.

in eq. (1) by (...I.

[7] BAIER, V. S. and FADIN, V. N., Zh. Eksp. Teor. & Fiz. 61 [3] CHENG, H. and WU, T. T., Phys. Lett. 36B (1971) 241. (1971) 476 (~nglish Trans : Sov. Phys. JETP 34 [4] SUBBARAO, K., Phys. Rev. D 8 (1973) 4041. (1972) 253).