THE HADRONIC PHYSICS OF PHOTON-PHOTON COLLISIONS

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THE HADRONIC PHYSICS OF PHOTON-PHOTON COLLISIONS

S. Brodsky

To cite this version:

S. Brodsky. THE HADRONIC PHYSICS OF PHOTON-PHOTON COLLISIONS. Journal de Physique Colloques, 1974, 35 (C2), pp.C2-69-C2-75. �10.1051/jphyscol:1974210�. �jpa-00215520�

JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 3 , Tome 35, Mars 1974, page C2-69

THE HADRONIC PHYSICS OF PHOTON-PHOTON COLLISIONS (*)

S. J. BRODSKY

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, USA

Resumk. - Les mesures des sections efficaces photon-photon ( o ( y y -. hadrons)) sont sus- ceptible~ de determiner les etats hadroniques importants, et les resonances, caracterises par C =

+

(( scaling ⁾⁾ A angle fixe, ainsi que pour le facteur de forme de transition du ^no.

Abstract. - Measurements of photon-photon cross sections (ci(.iy -. hadrons)) can determine the important C =

+

1. Introduction. - Although the first observations o f the process ee + ee

+

~ ( y y + hadrons) will be possible has stimulated a great deal of theoretical interest [3]-[5]. Photon- photon physics is enormously interesting a n d fun- damental : it allows the study of the structure and electromagnetic interactions of hadrons via a double electromagnetic probe. In principle, we can even independently ⁽⁽ tune ⁾⁾ the mass and polarization of both independent particles.

F o r the dominant part of the ee ^-+ ee

+

+

(*) Supported hy the US Atomic Encrgy Comrnis.;ion.

test of scaling laws and fixed pole behavior predicted by quark-parton models.

If a scattered lepton in ee + ee

+

In this talk I will review the theoretical predictions

for y y -t hadrons for the case of (nearly) on-shell

photons, and briefly discuss the off-shell behavior for some specific hadronic channels. A review of the applications of current algebra and soft pion theories to various exclusive y y cross sections is presented in Terazawa's contribution to this conference [ 5 ] .

2. The total photon-photon annihilation cross sec- tion. - Perhaps the most basic hadronic physics measurement accessible via the two photon process is the total y y annihilation cross section into hadrons.

We will consider here a double-tagged measurement of ee ^-+ ee

+

+

do(ec

. -+ ds ee ^~-

+

(?IN,

C2-70 S. J. BRODSKY where

is a phase-space factor obtained from integrating over the momentum w , - w 2 of the yy-center of mass, s = 4 o, w 2 is the square of the center of mass energy in the quasi-real yy-collision, so = 4 E2, and N, and N2 are the equivalent photon spectra obtained by integrating over the scattered lepton angles [7] ; e. g. for n7:/E2 6 0,$, < Om,, 2

More accurate expressions are discussed in the literature [7]-[lo]. Further, by measuring the scattering planes of the leptons, we can even separate

for parallel and perpendicular linear1 y-polarized quasi- real transverse photons. The corrections terms due to longitudinal currents and - C = - 1 hadron state contributions to a,,,,,+

,,,

Thus t o a very good approximation, we can literally convert an electron-positron or electron-electron storage ring into a facility for the collision of two real photons. In the case where the leptons are not detected, we can use

or, e. g., eq. (A.7) of reference [7], but the accuracy of this approximation (usually better than 30 %)

depends on the process, and the importance of thesC other contributions. Although somewhat harder to interpret theoretically, measurements of a,-,-

,,-,-,,,

without lepton detection as a function of so = 4 E2 will also be enormously interesting.

The tools of hadrsnic physics, specifically, our knowledge of Regge behavior, duality, and hadronic symmetries already give us a clear picture of what to expect for the total y y cross section. Here I will follow a very useful discussion given by Rosner [lo].

In a conventional Regge pole model, a,,,,,, is given by a constant (Pomeranchuk) part plus a part decreas- ing with s due to the non-leading f and A , trajectories

The factorization of the Pomeranchuk trajectory implies [7], [ I l l

and one can estimate [lo]

using exchange degeneracy and the couplings of the non-leading trajectories top, n and y in other processes.

By duality, we can also identify the non-Pomeranchuk component of a,,,,,, with the average effects of the direct channel resonances. Comparing the magni- tude of a, and a,, we can construct the following table [lo] for a,,,,,

,,,

Range of

4s

GeV Background E ⁼ 3 GeV E ⁼ 15 MeV

- - - -

0.3 to 1 2 to 1 11.0 nb 30.0 nb 1 t o 2 1 to 1 1.7 nb 6.9 nb

2 to 6 1 t o 3 0.6 nb 5.1 nb

6 to 30 1 to 5 - 1.8 nb

I D 1

[Here ace+oe is constructed by integrating over 4 m;f < s < 4 E 2 and using

From this table it is clear that the region J s < 1 GeV should be accessible to measurements with the present storage rings ; for comparison, we note that the CEA measurement at 2.5 GeV of the e + e- -+ hadron cross section is 22 t 5 nb. Note that because of the various experimental cuts and biases on angle, minimum energy, etc., the two photon cross section could be constrained to be small background of the total hadronic sections measured at CEA [12].

Figure 1 gives a simplified representation of Ros- ner's predictions for a,,,,:,,. The possible C =

+

Flnal

State Name

- -

n + n- E or a fo

[&*I

Mass

-

750 1 260 1 600 to 1 800 1 900 550 1 300 1 650 1 700 to 1 800 1 800 1 900

All of these resonances with natural JP = Of, 2 + , ...

will appear in the K + K - final state, as well as the s q

f' and their recurrences. The K f K- final state can a , = 0.27 pb. GeV

THE HADRONlC PHYSICS O F PHOTON-PHOTON COLLISIONS C2-7 1

also be used to study the fo - A , interference, to compare with SU(3) predictions for the decay ampli- tudes. The resonances seem to be fairly well separated implying that an important resonance study may be possible just from the energy dependence of a,,,,,,.

FIG. 1. - Schematic representation of the total cross sections

for y y + had based on factorizable Regge poles and two compo-

nent duality. See ROSNER, J., reference [lo]. The dashed line represents o y y + h a d = 0.24 pb i- 0.27 pb. GeV.s-'1'.

Other estimates of the hadronic channels can be given using a variety of theoretical techniques. For example, Walsh [4] simply uses the measured radiative width T( fo

-

^-

Similarly, a finite energy sum rule approach using the ony coupling leads to similar results. The analysis of Schrempp-Otto, Schrempp and Walsh [5] based on T(,f + yy) = 4 keV, T, ⁼ 150 MeV gives a(ee -+ eef)

-

r ( & + yy)

-

-

for

r, -

a,,,,, r (el2 y,J2 cpp+,,,

-

-

to get [7] o ,,,,,p,,,,,,,,,

-

Further, using Low's formula [6] we have for the narrow resonances [7] :

-

-

⁺

aee+eer,o

-

-

-

aee+ce110'(960)

-

I-) ,+ ,- ^{,+ n -} ,a

-

-

The radiative width for the is simply an estimate from broken SU(3) and may not be reliable.

From a different point of view, Gatto and Pre- parata [I I] have used the Cabibbo-Raddicati sum

rules plus a p-dominance argument to obtain the sum rule for the resonance contribution

not including correction of order (15-20 %) from isoscalar contributions.

Again, this gives a contribution to the total cross section for ee + ee

+

Because of the consistency of the various estimates for a,,, I think that a hadronic physicist would be surprised if the measured cross section differed by more than 50 '%, fi-om the duality prediction. Further, as emphasized by Bjorken and Kogut [14], the pre- dictions based on quark-parton or light-cone model for the electromagnetic current, are generally expected to agree with predictions based on vector dominance considerations of the hadronic nature of photons in processes where the photon is real. The exceptions to this rule for y y processes are : (i) when impulse approximation results are valid such as when at least one photon is highly virtual [4], [15]-[I71 ; or (ii) in processes involving two real photons which are sensitive to J = 0 fixed pole behavior arising from local two photon interactions [IS] ; and also (iii) real photon exclusive processes at large ^s and t [I91 (fixed CM angle). Thus, predictions for a,,,,,, based on hadronic physics considerations should be sound, but it would be delightfully interesting to be proved wrong ! The interesting physics of a,.,,,:,, when the photons are virtual are reviewed in Walsh's talk [4].

3. The production reactions y

+

The exclusive two photon process of perhaps the most exciting and critical interest is y

+

Among the unique features of this reaction are (1) It is perhaps the simplest production reaction in all of hadron physics : the hadronic interactions are all in the final state.

(2) It is related by crossing to the Compton ampli- tude y

+

+

(3) We can finally study n t n- interactions in the absence of other hadrons ( p , A ) in the final state.

(4) Unitarity implies to order e [2], [7]

for each angular momentum state and isotopic spin below the inelastic threshold. Therefore the phase of T,.,,,, is equal (modulo n) to the phase of nn ⁺ nn.

In fact, inelasticity is expected to be small up to the K K threshold.

(5) We can study the C =

+

C2-72 S. J. BRODSKY Linear photon polarization information from the

correlation of electron planes can also be used as a check of the spin-parity of the resonances.

( 6 ) The Compton amplitude is fundamental to the study of electromagnetic interactions and specific photon-photon physics such as J ⁼ .O fixed pole behavior [18], and scaling dependence of off-shell photons.

(7) Measurements of da/dt(y

+

+

The kinematics of the process ee ^-+ eenn and the extraction of the yy 4 nn subprocess has been dis- cussed in many places in the literature [3], [7-91, [13], [21], [22]. The general, exact, analysis of the C =

+

and Brown and Muzinich [22]. (See also ref. [9].) Specific calculations using Born approximation for

+ -

yy ^-+ n n with elementary point-like pions, which are useful as a theoretical reference point are given in references [7], [21]-[26], often with no restrictions on the final leptons assumed. Similar calculations for yy ^-t p f p- are useful for experimental norma- lization [28]-[30]. The effects of meson form factors in the case where one lepton is detected at large angles are presented in an interesting study by Kessler et al. [25].

The experiment of most direct immediate interest will be the measurement of yy -+ n + n- for nearly- real photons. Detailed discussions appropriate to experiments where both leptons are tagged in the near-forward direction so that k : , k:

-

Let us now turn to the physics of the yy + nn cross section. Of basic interest will be the extraction of n-n phase shifts using the unitarity condition. As Carlson and Tung [21] have emphasized, this can be done in a model-independent way. Using the equivalent photon approximation

da(ee ^-+ eenn) da.,,,,, f

(-q

- - 4 E

- -- --

ds d cos 0, d cos 0, 2 s N , N 2

where daYY-tnn

-- - --[I ^nu2 T + + I2

+

d cos 8, 4 s

and T+ +, T+ - are the invariant helicity amplitudes, and N, and N2 are defined in section I. Again, this result is accurate to order

o:,,.

T,l,(~,) ⁼

C

+

at?,

1 even

where dl,-,,(Or) is the usual rotation matrix, and define I

then 6: is the n-n phase shift in the region of elastic unitarity. At low s we can use the first two partial waves :

Thus the angular dependence in 0, can determine the modulus of the amplitudes and cos (So - S2), which appears linearly in the cross section. The effects of a large n-n scattering length in the s-wave, assuming all the other partial waves are given by Born approxi- mation is shown in figure 2. The solid curve with small scattering length satisfies current algebra,

FIG. 2. - n ) Two sets of 71-72 phase shifts used by Carlson and Tung [4], for the calculation of the s-wave amplitude for y y -+ nn.

6 ) Integrated cross section for ee ⁺ eerrrr at E = 3 GeV as a function of p, the magnitude of the pion momentum in the dipion C M frame. The dashed line corresponds to large scattering

length.

unitarity and crossing symmetry and has the form for I ⁼ 0, S0(.r) ⁼ 67.6 n(s - 4)/(s

+

Both have resonances at

,

T H E HADRONIC PHYSICS O F PHOTON-PHOTON COLLISIONS C2-73

length, but Carlson and Tung find much less sensi- tivity to the asymptotic behavior of do, and, in their model, minimal sensitivity to - the behavior of the phase shift in the ^E region (,/s

-

A great challenge to theoretical physics is whether we can predict the amplitudes for y y + 7cz and y y + K K before the data comes in. There have been several valiant attempts [21], [26]-[34]. The general strategy is to apply all of the relevant knowledge from hadronic physics on direct and cross channel resonance or Regge contributions, and the constraints of analyticity, unitarity, the low energy theorem, as well as assumptions derived from current algebra and possibly, superconvergence relations. In Yudurain's phenomenological approach [33], the y y + nn ampli- tude is constructed from E and f l Breit-Wigner reso- nances in the direct channel, with crossed channel contributions from the Born terms (7c exchange), and elementary p and w exchange (see also ref. [26]). In the papers of Lyth [31], Carlson and Tung [21], Isaev and Kleskov [35], Schierholz and Sunder- meyer [32], and Gensini [34], a partial-wave dispersion relation is used for the s-wave incorporating n-n data, and the higher waves are assumed to be given by Born approximation, plus in some cases, vector meson Regge exchange. Figure 3, which is taken from Schierholz and Sundermeyer's paper, shows the effects of using the ⁽⁽ down ⁾⁾ n-n partial wave solution favored by Protopopesev et al. [36] (curve I) which has a very broad inelastic resonance at m, = 420 MeV,

r

contribution satisfying :he constraints of current algebra is discussed by Goble and Rosner [26].

At larger ^s the theoretical ambiguities become even worse, since the inelastic effects of other channels,

FIG. 3. - The y y + nn cross section calculated by Schierholz and Sudermeyer [32]. Curves 1 and 11 correspond to n-n phase shifts with inelastic and elastic c resonances. respectively.

FIG. 4. - The y y -> nn cross section calculated by Gensini [34].

The quantity s do/dQ is plotted versus C M energy. The effects of the K K threshold and S* resonance are included. Curve ( a )

is Born approximation.

especially KK becomes severe, and a multichannel analysis is required. Model calculations have been given by Isaev and Kleskov [35], Gensini [34], tand Sundermeyer [37]. In Gensini's work a two channel (nn, K K ) problem is solved, and the results are calcu- lated assuming partial wave dispersion relations for the s-waves with poles at 177, = 660 - i 320 MeV and 117~. = 997 - i 27 MeV, and Born approximation for the other partial waves. A rather dramatic beha- vior at K f K - threshold is predicted (see Fig. 2), but little sign of the E is seen - in fact, the cross section is depressed below Born approximation in the region

,

-

An important question is how reliable the Born approximation is as an estimate of the crossed channel contribution to the .s-wave, as well as to the higher partial waves. As one indicator, Lyth [38] points out that the cross-channel contribution to the s-wave from p and w exchange alone exceed that of the Born term for .c 2 0.8 GeV2, so there must be an inherent smooth background uncertainty in all of the model estimates. But beyond this there is a more basic question. The Born contribution gives a scale- invariant cross section

for s 9 4 ni:, at fixed cos (I,,. This clearly must be unreliable at large s (unless the pion were a point- like elementary particle with no form factor). In fact, there is strong reason to believe that the asymp- totic behavior of d a j,,-,, /d cos U,, is . s - ~ at fixed angle. This is the prediction of the parton interchange model of reference [18], assuming the meson form

C2-74 S. J. BRODSKY

factors behave asymptotically at t-'. More recently, Glennys Farrar and I [20] have derived a general scaling law for any electromagnetic or strong exclusive process :

where the cross section is integrated over a fixed cm angular region with finite pi .pj/s. Here NM and ^' N , are the total number of external mesoas and baryons.

Thus we predict (modulo factors of log s) s - l o PP +,PP

s p a np + np, K p ⁺ k p , Kp ⁺ nZ.

d a YP ^-' EP

s e' e- ^-+ n+ n - , K + K -

at fixed t/s or cos O,,, and F,,

-

-

and F,, F, -- t-' for the asymptotic dependence of spacelike or timelike form factors. All of these pre- dictions are consistent with experiment. For y)] physics we predict at fixed CM angle

s-'yy ^-t nn, KK

.'-I

d t

s - ~ ey + e n 0 .

The last result is for no production in the two photon process where the electron is detected at large angles and the other lepton is at small angles. If we define the y(q) y(k) ^-+ no transition form factor (q2 large, k2 0) via

this implies Fno(q2)

-

The scaling laws of reference [20] are derived from renormalizable field theories based on quark-field degrees of freedom, assuming two conditions : (i) asymptotically scale-invariant interactions among

the constituents within the hadrons, and (ii) Bethe- Salpeter wavefunctions for the composite hadrons which are finite at the origin. The asymptotic behavior of yy + n + n - is clearly a crucial test of this scheme.

We also note the prediction for the exclusive process

d a 1

(ee ^-t eenn) + -- at fixed cos OCM d cos QcM S,,³

where the electron angles need not be constrained.

In this case the scaling comes from the dimension of the yy + nn amplitude. The equivalent photon approach is analogous to Feynman scaling ; the elec- tron fragmentation into photons is scale invariant.

On the other hand, the inclusive process

d o I

(e- e- ^-+ nX) + --; at fixed cos 0,,

d cos U,, P I

is predicted to be asyn~ptotically scale invariant (see also ref. [ l l ] ) .

The scaling law for y y + nn also implies that the .+wave cross section decreases asymptotically as do/d cos OCM

-

I n addition, the quark-parton and light-cone models also can give predictions on the spin and angular dependence of the yy -t nn asymptotic ampli- tude. The angular dependence will reflect the angular and fixed pole behavior of the underlying y+y+q+q amplitudes [18], [40]. Also, there is no off-shell photon mass dependence for s -t oo, f/s fixed, q2 < s.

On the other hand, for a virtual photon

do/dt(y

+

References

[ I ] Hadron events have been reported by BARBIELLINI, G., ORITO, S. a / . , Rome-Frascati preprint INFN-471, May 1973.

[2] Leptonic events produced by the two photon process have been observed at Novosibirsk, Frascati and CEA.

[3] For reviews, see BRODSKY, S., Proceedings of the 1971 Symposium on Electron and Photon Interactions, Cornell University, New York, and Proceedings of the XVI International Conference on High Energy Physics, Chicago University, NAL, 1972.

TERAZAWA, H., RCV. Mod. Phj's. October (1973) ; ARTEGA-ROMEIIO, N., JACCARINI, A,, KESSLER, P. and

PARIS], J., Plry.~. Rev. D 3 (1971 ) 1569.

BUDNEV, V., G I N Z B U R G , I . , MELEDIN, C., S E R ~ O , V., YnrlErrrayrr Fiz. 16 (1972) 362 ;

BAIER, V. N. and FADIN, V. S., JETP 34 (1972) 253, and references thercin.

[4] WALSH, T., J. Plrysiylie 35 (1974) (22-77.

[5] TERAZAWA, H., J. Plrysiytre 35 ( 1 974) C2-61.

[6] Low, F., Plrys. Rev. 120 (1960) 582.

[7] BRODSKY, S., KINOSHITA, T. and TERAZAWA, H., P11y.v.

Rev. D 4 (1971) 1532.

[8] KESSLER, P., J. Pllysiyue 35 (1974) C2-97.

[9] BROWN, C. J. and LYTH, D. H., NUCI. Phys. B 53 (1973) 323, and University of Lancaster preprint (1973).

[lo] ROSNER, J., BNL preprint CRISP71 -26(1971) (unpublished).

The identification of the non-Pon~eranchuk trajectory with the average effect of direct channel resonances is due to P. G . 0 . Fre~lnd. Phj1.s. Rev. Let,. 20 (1968) 235 :

H A R A R I , H., PIIJJS. Rev. Lett. 20 ( I 968) 1395.

[ I l l GATTO, R. and PREPARATA, G., INFN reports 441, 479 11 973).

[I21 STRAUCH, C., ~ ~ n p u b l i s h e d .

THE HADRONlC PHYSICS OF PHOTON-PHOTON COLLISIONS C2-75 [I31 SCHREMPP-OTTO, B., SCHREMPP, F. and W A L S I ~ , T. F.,

Plrj..\. Lett. 36B ( 1 971) 463.

[I41 BJORKEN, J . and KOGUT, K., Plrys. Rev. D 8 (1973) 1341.

[I51 BRODSKY, S., KINOSHITA, T. and TERAZAWA, H., P I ~ J ~ s . RPY. Lett. 27 (1971) 280.

[I61 WALSH, T. F., PIIJJS. Lett. 36B (1971) 121.

[I71 CARLSON, C. and TUNG, W., Plrys. Rev. D 4 (1971) 2873.

[I81 BRODSKY, S., CLOSE, F. and GUNION, J., P11y.s. Rev. D 6 (1972) 177, and SLAC-PUB-1243 (1973).

[I91 BLANKENBECLER, R., BRODSKY, S. and GUNION, J., P11yx.

Rev. D 8 (1973) 289, and Plys. Lett. 39R (1972) 549 ; LANDSHOFF, P. V. and POLKINGHORNE, J . C., PIIJ~s. Rev.

D 8 (1973) 927.

[201 BRODSKY, S. and FARRAR, G., SLAC-PUB-I290 (1973) (to be published in Pl~ys. RPV. Lett.) ;

MATREEV, V., MURADYAN, R. and TAVKHELIDZE, A,, Dubna preprint D2-7110 (1973).

[21] CARLSON, C. and TUNG, W., P11y.s. Rev. D 6 (1972) 147.

[22] BROWN, R. W. and MUZINICH, I., P11y.s. Rev. D 4 (1971) 1496.

[23] BAIER, V. N. and FADIN, V. S., JETP Lett. 13 (1971) 208.

[24] CHENG, H. and Wu, T. T., Nlrcl. Plrys. B 32 (1971) 461.

[25] ARTEGA-ROMERO, N. et nl., references [3] and [a], and P11y.s. Rev. D 4 (1971) 2927.

[26] G O ~ L E , R. and ROSNER, J., PIIYS. Rev. D 5 (1972) 2345.

[27] BONNEAU, G., GOURDIN, M. and MARTIN, F., NIICI. Phys.

B 54 (1973) 573 ;

BONNEAU, G. and MARTIN, F., Preprint LPTHE.73.4.

1281 CRAMMER, G. and KINOSHITA, T., Cornell University preprint (1973).

[29] FUJIKAWA, K., Nriovo Cil~retrto 12A (1972) 83, 117.

[30] CHEN, M. S., MUZINICH, I., TERAZAWA, H. and CHENG, T. P., BNL preprints 17161 and 17557 (1973).

[31] LYTI-I, D. H., Phys. Lett. B 30 (1971) 195.

[32] SCHIERHOLZ, H . and SUNDERMEYER, K., NucI. Phys. B 40 (1972) 125.

[33] Y U D U R A I N , F., N~iovo Cit?~etlto 7A (1972) 687.

1341 GENSINI, P., Lecce report UL/IF-7-72/73 (1973), submitted to the International Symposium on Electron-Photon Interactions, Bonn (1973).

[35] ISAEV, P. and KLESHKOV, V., J l N R reports E 2-6257 and E 2-6666 (Dubna, 1972).

[36] PROTOPOPESEV, S. D. et (I/., Plrys. Rev. D 7 (1973) 1279, 1429, 259 1.

[37] SUNDERMEYER, K. (private communication).

[38] LYTH, D., submitted to the XVI International Conference on High Energy Physics, Chicago-NAL (1972).

[39] SARKER, A,, P11ys. Rev. Lett. 25 (1970) 1527.

[40] BRODSKY, S. and CHEN, M. S. (in preparation).

[41] BLANKENBECLER, R., BRODSKY, S., GUNION, J., SAVIT, R.

(in preparation).

D I S C U S S I O N

G . S A L V I N I . - I) Y o u r formula a.,? = a,,

+

is valid f o r quasi-real a n d real photons. I suppose (referring t o t h e talk given by J. Parisi) t h a t t h e contri- bution o f t h e o t h e r diagrams t o hadfonic production is small. This m e a n s t h a t t h e C E A correction is valid. Is t h a t correct ? 11) T h e behaviour a t high s-values should b e even better. S o t h e a,, formula may b e a p p r o p r i a t e t o plan future experiments. Could we have a n y surprise a t very high energies ?

S. BRODSKY. - 1) T h e C = - I h a d r o n production c o m p o n e n t is o f course comparatively small unless o n e o f t h e incident lepton scatters a t large angles ; its contribution c a n be calculated directly from t h e experimentally measured annihilation cross section.

T h e C E A correction is consistent with t h e theoretical expectations of hndronic physics. 11) A s Terazawa has e m p h a s i ~ e d , there could be surprises from a specific t w o p h o t o n parton modcl o n light cone contributions t o o,,, considering t h a t theory has not d o n e very well in predicting t h e annihilation cross section. T h e parton model expectations a r e discussed in Walsh's talk. Surprisingly, t h e novel parton model contributions t o o,.,, based o n virtual quark-pair productton may only be apparent a t low energies since these contributions fall as . r - ' relative t o the hadron-like components. Duality arguments a n d t h e Bjorken-Kogut correspondence principle, t h o u g h , lead o n e t o anticipate that t h e parton a n d hadron physics estimates will not differ appreciably.

R. MADARAS. - Y O U m a d e a (( best h a d r o n phy- sics estimate )) f o r cr,,+,;,,(s), namely

H o w accurate d o you think t h a t this estimate is ? S. BRODSKY. - I believe theorists experienced in hadron physics, including Rosner, would b e very surprised if this estimate could b e off a s much a s 50 %. F r o m t h e parton model point of view there could be surprises specific t o t w o p h o t o n interactions, a s discussed in Walsh's talk. b u t t h e general experience is t h a t t h e hadron-like aspect of p h o t o n interactions dominates processes involving real photons.

S. ORITO. - Y o u m a d e an experimental suggestion o n t h e tagging system. namely t o stay away from the beam line in order t o get rid o f t h e bremsstrahlung background. T h e n you apply a n extrapolation t o c12 = 0. I think that, if you d o t h a t with a machine like S P E A R o r D O R I S , t h e final accuracy y o u would get will be similar t o o r worse t h a n w h a t y o u c a n expect from A D O N E . T h e reason is t h e steeply decreasing counting rate ( p h o t o n propagator I /y2) a n d t h e uncertainty involved in t h e extrapolation.

All that is true especially in t h e threshold region of yy -+ hadrons. Practical experimental conditions a r e much more favourable with allower enefgy machine

+ -

if you want t o investigate y y -+ 71 71 near threshold.