II. THÉORIE / THEORYγγ → HADRONS : ASYMPTOTIC BEHAVIOR AND DEEP INELASTIC SCATTERING

HAL Id: jpa-00215521

https://hal.archives-ouvertes.fr/jpa-00215521

Submitted on 1 Jan 1974

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

II. THÉORIE / THEORYγγ → HADRONS :

ASYMPTOTIC BEHAVIOR AND DEEP INELASTIC SCATTERING

T. Walsh

To cite this version:

T. Walsh. II. THÉORIE / THEORYγγ → HADRONS : ASYMPTOTIC BEHAVIOR AND DEEP INELASTIC SCATTERING. Journal de Physique Colloques, 1974, 35 (C2), pp.C2-77-C2-85.

�10.1051/jphyscol:1974211�. �jpa-00215521�

I THEORIEI THEORY.

y y -+ HADRONS : ASYMPTOTIC BEHAVIOR AND DEEP INELASTIC SCATTERING

T. F. WALSH

Theory Group, Deutsches Elektronen Synchrotron, Hamburg, Germany

RCsumC. - Nous divisons notre discussion sur les collisions photon-photon en trois rubriques : a) Deux photons reels ou presque reels (le processus est alors domine par le caractere tc hadronique)) de ces photons). b) Un photon reel ou presque ree1,et I'autre photon trirs virtuel (diffusion profon- dement inClastique sur un photon reel). c) Deux photons tres virtuels (dans ce cas, la rkction prksente une certaine similitude avec le processus e ' e + ;I* + hadrons). Divers moddes (en particulier celui de Regge et celui des partons) sont consideres.

Abstract. - We discuss photon-photon collisions under three rubrics : a) Two real or nearly real photons (the process is then dominated by the hadronlike ^)> character of these photons).

b) One real or nearly real photon and one far off-shell photon (deep inelastic scattering on a real photon). c) Two far off-shell photons (the reaction then has some similarity withe+ e- + y* +ha- drons). Various models (in particular Regge and partons) are considered.

1. Introduction. ^- One can ask whether the physics of photon-photon annihilation into hadrons is of really fundamental interest. An experimental program to study this process will require conside- rable effort and ingenuity. Can we expect to gain physical insight commensurate with the effort expen- ded ? I think that the answer is yes, particularly where the asymptotic behavior in energy or photon mass is concerned. This is a current reaction with two variable masses, and it is becoming clear that current-induced processes have a great simplicity and importance as contrasted to pure hadron reactions. One may hope t o learn something about the common consti- tuents of the currents and the hadrons, or at least about the algebraic and dynamical structure the constituents leave behind before their exit from theo- retical considerations.

It is convenient to discuss photon-photon reactions under three rubrics : two real or nearly real photons (q2

-

The second is at least in part like deep inelastic scattering with the target a real photon (mostly a vector meson). The last reaction has - in the proper kinematic rkgion - a similarity to e t e- + y::: -+ ha-

of physics to another in one reaction that provides the interest and the challenge where y y + hadrons is concerned.

This talk is concerned with the asymptotic behavior in one or more of the three variables of the process, q:, q:, ^.Y. See figure I . In presenting a picture of this behavior, one has to lean on a model or models.

Most of what I have to say is based on the parton model, but only in the very general sense embodied in the idea of a field theory with soft off-shell behavior or a transverse momentum cutoff. This is in most cases a physical realization of the algebra of currents on the light cone, and is of reasonable generality.

FIG. 1. - The forward photon-photon amplitude.

drons. In all these cases one can consider inclusive,

semi-inclusive or exclusive reactiolls - for example The picture to be presented may be partly false*

i n y b + y:!: + T - ( 1 ) Olle call the compton but it is simple and testable. It can be altered if this lit^&, and not just its absorptive part. I t is this is made necessary experimental developments.

possibility of continuously moving from one kind A work about the literature. There is a lot of it.

For an extensive bibliography, we refer the reader

(1) We denote by ^;$* photons ~ v i t l i q' possibly fril- from XI-0. to the review b$ Terx-awa [I], and an earlier one

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974211

C2-78 T. F. WALSH by Brodsky [2]. I will refer to work relevant to the

material of the talk and probably incompletely at that.

2. Two real photons. - This case has, according to current prejudice, the least model dependence, so we discuss it first. The reader may feel, with justice, that we also do this so as to get the subject out of the way.

A simple estimate for the total y y cross section follows by using the vector-dominance and quark models. Namely, we put [3] ^,

2

o r

I

(

)

% -

hi ⁿ⁾

:)

where

via the additive quark model. This should work at high energy, and one can ask about the size of non- leading terms. In the same way one can get an estimate of about

-

y y scattering is convenient in that the non-leading (f - A,) t-channel exchanges correspond to reso- nances which can be produced in the s-channel (e. g. y y + f O --+ n+ n - ) . We can thus relate the radiative width of these mesons to the size of the non-leading Regge term. Doing this very crudely by integrating a Regge term over a broad s-range corresponding to the spacing of s-channel resonances, and equating the result to the integral over a narrow fmeson, we get a non leading term of the same order

as the leading one (namely,

where s is in GeV2) for radiative widths

This agrees roughly with the size of non-leading tra- jectory contributions to, e. g., ::a ; see figure 2.

An amusing aside is possible here. Consider the helicity structure of

there are three forward amplitudes for two real _{, ,} photons (W2::i) : ~ : f , W: 1 and W ; ; [5] (').

The first two are proportional t o cross sections for photons of total J, along the collision axis I JI I ⁼ 0 , 2 to go to hadrons. The last is an asymmetry. Let's

(2) We use parity and time reversal invaiiance everywhere.

FIG. ^2. - A schematic representation of crF(s).

consider factorization of the t-channel exchanges.

Then up t o terms with a, ⁱ³ 0,

W,; ^% constant (2b)

i. e. w;;/ W: vanishes at least as fast as 11s. These are consequences of factorization (2a) and a result of Fox and Leader (26) [6], [7]. Now suppose that we try to use local duality between .jrexchange in the t-channel and the jlmeson resonance in the s-channel.

That is, we apply all available relations from both t-channel and s-channel to the ^{J ' O} y y couplings.

We will ignore the A , in all this. For a narrow s-chan- nel resonance,

where n, is the normality of the resonance ^(tif =, + 1).

This follows from parity and s-channel factorization at a resonance. Putting all this together, we get the result that the jitrajectory decouples from y y ! The culprit here is local duality ; the flmeson can be produced in some definite helicity state and duality considerations need not apply locally to the unpo- pulated amplitude. For example, some calculations [8]

give the result that the .\-channel contribution with J= ⁼ 0 ( W i f or ^WT 5 ) is small at .v z ^inif 2 relative to w ~ I - i. e. the j'' is produced with [ J , ] = 2. One can check this by looking at the f" + n f n- decay distribution [8]. A nice test of Regge ideas call be carried out by looking to see if ^{W ; +} 6 ~f f .

The foregoing estimate for crj:! at high energy assumes that the Pomeron couples in y y scattering.

There is a well-known argument that it should not in yN + yN at t = 0 [9]. Since this is experimentally not the case, we are probably safe in assuming that no such decoupling occurs in yy + y y at t ⁼ 0.

Little is known :)bout Regge cuts, so it is perhaps best to go on the assumption that we are dealing with Fdctorizable poles. There is, however, a rule due to R. Worden which can indicate when certain Regge cuts approximately decouple [lo]. Unfortunately,

yy -> H A D R O N S : ASYMPTOTIC BEHAVIOR A N D DEEP INELASTIC SCATTERING C2-79

this indicates that the cuts at a, = 0 ( f f ; etc.) reinforce one another in y N ⁺ y N and y y ^-+ y y . This may make difficult the unambiguous test of ideas which involve singularities with r , z 0.

The semi-inclusive distributions in y y + h + any- thing should look pretty much like y + ^{p -+ h} + ^any-

thing, apart from possible parton terms. We expect a bounded transverse momentum and weak depen- dences on the azimuthal angle of I1 relative to the photon polarization direction apart from leading particle effects. T h e dependence on the Feynman scaling variable s = 11 ,,,, /I) ,,,,, should be as in the projectile fragmentation region of y + p + h + any- thing.

3. Off-shell asymptotic limits. ^{- -} The physics of y y ^-+ hadrons depends on three variables : q:, q;

and ^s = (q, + ~ 7 ~ ) ~ . We have just seen what one can expect for y , 2 = (1: ⁼ 0 and ^.s -+ CO. What about other asymptotic. limits ? One can imagine many, but for our purposes the following will be enough [I I].

(i) Let ] q: 1, 1 (1: I and s ^-+ co but also demand that ^sly: approach infinity [12], [13]. If q;/q:

is fixed, this means that 4: cc s h i t h /I < +. We'll call this the scaling Regge (SR) limit. The Regge

2 2

limit proper is q,, q, fixed and s + co (R limit).

(ii) Let 1 q: 1, 1 ^4: 1 and ^{s + co} with the fixed ratios

We shall call this the scaling lilnit ( S limit) [ I l l . Limits where y: is fixed o r diverges slowly correspond to 1 4; 1 = 1, and then w is a familiar Bjorken scaling variable.

(iii) Finally, consider the limit where I I, I q: I ^-+ co with the preceding 5 fixed but

- i. e. s diverges less rapidly than y: or q:. This is the boundary w = 1 in (ii). We call this the double limit (DL limit) [14]-[I71 since it can also be repre- sented by

l i ~ n lim ~ l i m

s , , - I > / .

We shall see that in the parton model it is uniniportant precisely how one defines this limit. As a boundary here we ha\/e the interesting case when I g : l , I qf I ^{-+ a;i} ( 5 fixed) and .v = conscant. This is the light cone limit (L limit) [15].

These are all shown schematically on figure 3.

In the rest of the talk, I would like to present the picture of the asymptotic behavior which emerges from the parton model [l I]. This overlaps t o a great extent what one expects from light cone considerations (for the L and DL limits). The field theory version

FIG. 3. - The various asymptotic limits in yy scattering.

of the parton model is due to Drell, Levy and Yan [18]

and has been discussed in covariant form by Land- shoff, Polkinghorne and Short [I91 and Brodsky, Close and Gunion [20]. Why study this model rather than, say, QED ? There are two reasons ; (i) The model exhibits the physical basis of the light cone algebra results ^- the oft-mentioned finiteness of certain matrix elements of bilocals arises from the transverse momentum cutoff in the model and (ii) the disconnected pieces of the light cone algebra (giving rise to e. g.

g(e+ e - ^-+ hadrons) = Ra(e+ e - + / i f /I-)) are explicitly contained. Thus the model is a clear formulation of possibly important physics. Almost everything that follows depends only on the reality of the cutoff and tlie separation of amplitudes into connected and disconnected ones, and not on further details. T o be precise. we shall consider only fermion partons.

The elementary coupling in tlie model is of a photon to a parton pair. The basic assumption is that a connected parton-hadron or parton-parton ampli- tude vanishes rap~dly whenever the squared four- momentum carried by a parton line diverges. This is the transverse momentum cutoff of the model (usually formulated in the P = co frame), expressed in covariant form. As a consequence of this assump- tion, any diagram where hot11 partons in a current channel couple to a connected amplitude vanish [19].

This last gives rise to the asymptotic relation of the parton model that

when applied to the electromagnetic current vacuum polarization teriwr.

C2-80 T. F. WALSH Provided the I q' 1 are asymptotic, y y scattering

can be written as the sum of the diagrams of figure 4.

We are here only interested in discontinuities, which one measures in the fully inclusive reaction y + ^{y + any-}

thing. We will discuss some exclusive reactions later.

of I. = 0 indices. In addition, there is one further relation

w,': ⁼ w:: ⁽⁸⁾

which is already familiar to us from the Regge ana- lysis of real y y scattering. It is not clear from the ana- lysis of reference [I31 whether one expects W ; ; to vanish relative to w f ^:. Probably it does only in special cases (corresponding to no Regge cuts in the connected blob).

We can go a step further by assuming the dominance of factorizable Regge poles in off-shell compton scat- tering on nucleons and in y4' y'v scattering. Then from this, scaling, and the snlallness of a,/aT we can get (see also ref. [12])

= C ^{[ y f} ( N I 2 sso

R ( q i - tn;)

where y : ( ~ ) is supposed to be q2 independent pro-

c ) vided ss0/q: q: $ 1 ; in: is a mass parameter describ- ing the approach to scaling, in;

-

amplitude. the Regge residues necessary to get scaling in Off-mass shell we have, besides the three absorptive

parts of the forward y y scattering amplitude mentioned already, five more [5] : w::, w:,', w:,", w?,, w:.

The first three are cross sections and the last two are asymmetries. The y;"* cross sections and asymme- tries are (in the y:!: yy" C M )

For the pomeron contribution we can estimate

Provided, of course, that ^v wY' is really dominated by a factorizable pomeron at large w = 2 Mv/l q2 1.

In the event that s, q:, rl: are all large but ss0/(q:, q:) is not, it has been conjectured that the W' s are func- tions of this variable [I21 - something one cannot conclude simply from the Regge form. It might be interesting to investigate the functional dependences of the scaling regge limit.

This fixes the normalization of the W' s ; the total A, unexplored point of solne irnportznce is what on-shell cross section averaged over y y polarizations is happens to the diagrams 4(, as ,s/~ q 2 1 decreases from large values. One thing can be said, and tliat is that

e4 + +

a

; = - ( W + + + w,':).

4 s (6) these diagrams should vanish in the S-limit when .v,

I (1; I and I q2 2 I all diverge with constant ratios, pro- vided that we demand that all connected amplitudes 4. The scaling regge limit. - Diagrams 4b and 4c

in the model vanish as a squared parton four-monien- are expected on dimensional grounds to correspond

tum leading into a connected blob diverges. This has t o a,,,, cc 11s and should be negligible in this limit.

to include connected parton-parton amplitudes. The So we consider only 4a (see Kingsley [13]). As we

reason why the amplitudes of figure 40 vanish in this might expect in the parton model with,j = 4 partons,

limit is tliat the momentum transfer into a connected the longitudinal amplitudes are asymptotically negli-

gible [13] blob must be proportional to the dimensional quanti-

ties in the problem and these all diverge in the S-limit.

w,"," w:: w:,+ LV:, 1 ⁼ AS ;I contrast, the niomenturn transfer along a parton lim ^{---- - - - -}

WT ' WT MI, WT ' WT \ ^O ('1 line in the Bjorken limit of inelastic ep scattering is proportional to the squared nucleon mass, and only where WT is any of the transverse amplitudes w::. diverges (as t,,,i,, cc I/(w - 1 ) ) for w -+ I . It is clear W:I, W ; ; . These longitudinal amplitudes, however. that the diagrams of figure 40 must have a kind of

7 7

approach zero differently depending on the number threshold behavior as .s.s,i(r/; q ; ) decreases from the

y), -+ HADRONS : ASYMPTOTIC BEHAVIOR AND DEEP INELASTIC SCATTERING C2-81

large values where the Regge expansion might hold.

This behavior depends of the unknown off-shell behavior of the connected parton-parton amplitudes, and it is not so directly accessible as the threshold behavior of v W Y ) in inelastic e p scattering.

5. The S-limit. - We have just seen that the leading diagrams in the scaling Regge limit don't contribute to this limit. That Regge-type behavior is not to be expected is already clear when one notes that

cos 0, =.w[l - 42]-1'2

and only diverges when I 4 I + 1 or w ⁴ cz(i. e. when one goes to inelastic scattering off a real photon o r t o the S R limit). The diagram ^4c does not contribute for the same reason that the connected diagram in e C e annihilation does not (Fig. 5 ) ; one is left with a fully disconnected piece, and it can be calculated by noting that it is the box graph in QED. The helicity amplitudes are dimensionless (we constructed them that way) and it is perhaps reasonable that they become functions of 4 and w and not of &, q: and s separately [I I], [I71

Iim w;?'j2>(s, q:, q:) = w:;';:((, W) . (I 1)

S

FIG. 5. ^- The connected and disconnected amplitudes for el- e- annihilation.

From (5) we can see that the y':' y ' h r o s s section is

K I/s in this limit, as we might expect from dimen- sional analysis, whereas it goes t o a constant in the R limit. We expect that some of the amplitudes (1 1) appear in the S R and R limits in the form of Regge singularities with intercepts at o r below x, = 0.

This fully disconnected piece has its pecularities : (i) The longitudinal amplitudes d o not vanish, because that is a consequence of a transverse momen- tum cutoK which is r ~ o t present in this term. However, the longitudinal an~plitudes aren't too important ; they vanish as either to + oo o r w + I. The former is the S R limit and the latter the DL limit which we shall discuss in a moment.

(ii) The transverse amplitudes are the most inte- resting (for the others, see [I I]). These are

and they diverge logarithmically as 15 1 + 1, for + + any w . Only W : I survives as w ^-+ oo and only W + ⁺ and W ; survive as w -+ I. In this latter limit, we see that only states of total Jz = 0 (but any total J !) are produced. Exactly the reverse happens in the Regge or S R limits for this term. I t is interesting that non-parton calculations indicate a predominant W: I

term for the.!' meson region in real y y interactions [8].

The significant feature of this disconnected term is its similarity to the parton diagram for e" e- annihi- lation. This means that in the asymptotic S limit the hadron distributions should look qualitatively like those in e' e - annihilation - only the familiar single- particle I + cos2 O angular distribution (for j =

+

tons) relative to the e + e - axis is missing. That is a consequence of one-photon exchange in the s-channel and here we have parton exchange in the t- and u- channels. The events in y'"* annihilation should involve large ^{j ) ,} secondaries just as in e + e- and other features (KIT ratios, etc.) should also be similar.

The similarity t o e + e - annihilation can be expressed in a way similar to the statement

c(e+ e - ⁺ hadrons) ⁼ Rc(e+ e- ^{+ p +} p-) with R ⁼

I

lini a;,;,(ee + ee + hadrons) =

S

= Ta,,,(ee ^-+ ee + ^{p + p-) ( 1 3)}

where T

-

For example, in the so-called colored quark model [23]

R = 2 and T = 3. In the Han-Nambu model (three integrally charged triplets [24]) one can require that the produced hadrons are singlets under the group of transformations of the triplet indices (charm). Then R = 2 and T = 2 [22]. If one had treated the Han- Nambu model as a n ordinary parton model (all hadrons, including those with charm, being produced) then T = R = x e ' = 4.

The relation between T and R can be expressed in a possibly more general way in terms of the Wilson expansions [25]. [26] except that instead of simply relating two numbers (like R and the coefficient S of the no -+ y y decay [25]) one has to calculate here a function M/;):) of 5 ^(ill1 this probably only works at w = I ) . For this one needs the entire light cone expansion [ X I . The nniplitude y'" y':' ^-t y" y':' also offers a nice example of a conformal invariant four- point function and is perhaps interesting from that point of view [27].

Actually. this pnrton term should be present in /.ccrl 11)' reactions and would lead to a class of events with unbounded transverse monientum [28]. It would only contribute - ;rpprecinbly to the total y y cross section a t low ^\ in the c\,cnt that T 2 1: however.

C2-82 T. F. WALSH 6. The fixed-s behavior (L and DL limits). -

The last asymptotic limit we need to discuss is the L limit where s is fixed and I q: I and I & 1 ^-t co with

5 fixed [29], [30]. Afterwards one can reach the thres- hold w = 1 of the S limit by taking s -+ 30 (the DL limit).

In the L limit only diagrams 46 and 4c survive ; 4b has no resonant behavior, so if we are interested in the y* y * form factors of resonances (Fig. 60) or perhaps threshold y::: y::: -+ n f TC- (Fig. 6b) we should only consider figure 4c. Since we have seen that large q2 photons (with s < 1 1, 1 q: 1) couple only to transverse photons and only then in states of total J, = 0, we expect only one surviving helicity aniplituc/e for all such states, namely T+ +. Moreover, this should be a function of 5 alone - iiidepeildent of q2 at fixed 5,

lim T* *(qI, 9:) ⁼ T* *(O

L (14)

lim T,,,, = 0 otherwise

L

FIG. 6 . - The parton diagrams for y y + resonance and y y -t z+ n-.

( T , ⁺ and T- - are related by parity). To summarize, we note two essential points : first, the fact that if the final state is a pseudoscalar resonance the coupling is of the form

introduces a factor 5 in T + + , as one can easily check.

For scalar resonances (normal parity states) we get a factor t2, ^{SO [I I ]}

lim T , +(5) =

L

Sf (0 abnormal parity resonance normal parity resonance . ⁽¹⁵⁾

This is similar to what happens when we form the combinations ~ : f T WTT in (12) for the limit w -+ 1. The above is a consequence o f , j ⁼

+

and we expect to have the latter behavior in (15) for reactions like y * 7:" rrn+ n - near s ^z 4 r r t : t3).

What happens when, at large but fixed I q: 1, I c1f I

we increase ^.s ? This is a n instructive exercise. We encounter first a set of resonances ( n o . C . f O, ...) of all J but J, = 0. At larger .v the resonance structure

( 3 ) The value of studying ^{; I *} ;,* -- z - i r - in the parton niodel has been emphasized by Brodsky P I (11. 1311.

should disappear along with two-body channels like n f T C - , whose asyn~ptotic s-behavior should be like that in e + e- annihilation. As s increases further we reach the Regge region for s -- a few times larger than ( q2 12, after which W : 1 (for example) rises linearly with s. For a fanciful picture, see figure 7, where we compare w:: and the spectral function of the photon propagator p(s).

p ( hadron) P ( P P-)

R --

L

FIG. 7. - Sketches of the two-photon and one-photon absorp- tive parts relative to those for a p+ p- final state.

The precise relation between connected and dis- connected pieces (Fig. 40, 4c) in the L limit is not specified by the model. A fascinating possibility, however, is that the leading light cone contribution at large s (Fig. 4c) overages the resonance pieces at low s ;

this would be a kind of light cone duality, and ought to hold in both e + e- annihilation and the L or DL limits of y': 7::: scattering [I I ] , [32], [33].

7 . Inclastic electron-photon scattering. - The limit where one photon has fixed ^{rl: =} 0 and the other large ( cl: ( at fixed to is related to the inelastic scattering of an electron on ^;1 photon target [34]. We can see here (where there is some similarity to inelastic eN scattering) the operation of all the preceding terms.

Suppose, to enhance the familiarity, we consider VW:" for a photon target. The connected diagrams as in 40 now look different because y f is not asymptotic : the photon connects directly to the blob and (here is

y y + HADRONS : ASYMPTOTIC BEHAVIOR A N D DEEP INELASTIC SCATTERING C2-83

no parton line running between the incident and outgoing photon with momentum q , . In fact, this set of diagrams should look mostly like y e p 0 ^-+ yep0.

We expect these connected diagrams scale and give VW?)(OO) = constant since v wiN'(co) = constant (where w = co here means o 2- 10) ; as a guess from the quark model and the assumption of pomeron dominance of v wiN',

What happens as w + 1 ? For the corrected dia- grams, the squared momentum transfer along a parton line has a minimum value t,,i,, c l / ( w - 1) and diverges as o ^-t I ; so these diagrams vanish near threshold. The rate at which they vanish is related to the y:!: pO-meson form factors by a Drell- Yan-West relation [35]. For the proton with a dipole form factor, v w?) ^cc ( o - For a meson target with a less rapid decrease of the form factor, ^v wT0'

should decrease more slowly as ^{o -t} 1 . It would be wrong to conclude from this that the y'"1-meson helicity form factors vanish as I q: I + m . All this indicates is that the y:'pO-meson vertices should vanish and then the y'9-meson vertices are eutirely given by parton or light cone considerations (namely, the term T + + ( l ) ) [29], [36]. Similarly, although the connected diagrams analogous to 4a vanish near threshold, the discoimected diagrams do not. I n fact they give [37] ⁽⁴⁾

for any fixed o ; this term is large near threshold and

v w y

connected term

\ disconnected ^term

FIG. 8. - The structure f ~ ~ n c t i o n ~ W \ ~ ' ( t o , ( 1 2 ) .

(4) See also the remark by Brodsky in [2] and, for the pure QED case [38] and [39].

vanishes as w - ' for large to, apart from the log. So there is present in inelastic electron-photon scattering a term which breaks Bjorken scaling and a simple minded threshold relation like that for the connected parton diagrams in inelastic e-N scattering. If there is any kind of duality between resonances and conti- nuum near threshold in e-y scattering, it involves the duality of resonances and this disconnected term, not a connected term as in e-N scattering.

Finally, we sketch v ~ i " ( w , q2) at a value of q2 where the disconnected term at small o is roughly of the same magnitude as the connected term at large o (Fig. 8).

8. Conclusion. - What I have sketched out here is a theoretical picture of a reaction which represents a considerable experimental program for the future.

How reliable is this picture ? Can one do experi- ments ? As to the picture : the details may be wrong in places, but in broad outline it is probably near the truth, granted that we have learned anything about deep inelastic processes from e-N and v-N experiments, and that e + e- annihilation is not too drastically diKerent from parton or light cone expectations.

Some of the details which rest on the behavior of ei e- annihilation may have to be modified if that turns out to have unexpected properties.

As to the feasibility of experiments, these are clearly most interesting with high energy e' e- and e- e- machines (E ⁼ 10-15 GeV). But one can still do a lot at lower energies. For example, the n+ n- final state in e-y scattering contains a lot of interesting information and one can study this (as well as perhaps

?::: _{+ no +} yy) in some detail in the reaction

where one detects both pions (or two photons from no + yy) and only the large angle scattered elec- tron [40], integrating over the remaining small-angle scattered electron. The helicity structure of the f 0

meson in the n+ ^71- decay channel can be studied in this way - and the possible background

e- + e- ^-t e- + e- + (C ^{= -} I hadrons) cannot contribute to ^{, f ' O} production. The reaction

+ -

y':' y -+ n n near threshold should also be interesting and can probably be separated from background.

In general, there is no reason why one cannot try to study inelastic electron-photon scattering with e- e- beams by detecting the hadrons and large-angle electron, integrating over the small angle electron.

The background of C ⁼ - 1 hadrons .can be calculated and subtracted.

So one can at least start studying deep inelastic y:': 11::: processes with the present generation of machines, and I think thaf this field olyers exciting possibilities for both the near and distant future.

T. F. WALSH

References

[ l ] TERAZAWA, H., Rev. Mod. Phys. (October, 1973).

[2] BRODSKY, S. J., Proceedings of the 1971 Symposium on Electron and Photon Interactions at High Energies, Cornell University, Ithaca, N. Y., 1972.

[3] BRODSKY, S. J., KINOSHITA, T. and TERAZAWA, H., PIIJJS.

Rev. 4 (1971) 1532.

[4] SCHREMPP, B., SCHREMPP, F. and WALSH, T. F., Phys.

Lett. 36B (1971) 463 ;

RENNER, B., NIICI. P11j.s. B 30 (1971) 634 ;

BRAMON, A. and GRECO, M., Lett. Nliovo Cittletrto 2 (1971) 522.

[5] BUDNEV, V. M. and GINZBUIIG, I. V., Sov. J. N11c.l. P I I J , ~ . 13 (1971) 198 ;

BROWN, R. W. and MUZINICH, 1. J., PIIJ's. Rev. D 4 (1971) 1496 ;

CARLSON, C. E. and TUNG, W.-K., P11y.v. Rev. D 4 (1971) 2873 ;

ARTECA-ROMERO, N., JACCARINI, A., KESSLER, P. and PARISI, J., PIIJ's. Rev. D 3 (1971) 1596.

[6] Fox, G. C. and LEADER, E., P11ys. Re)]. Lett. 18 (1967) 628.

[7] BROWN and MUZINICH, reference [5].

[8] SCHREMPP, SCHREMPP and WALSH, reference [4].

[9] ABARBANEL, H. D. I. and NUSSINOV, S., P/rj.s. Rev. I58 (1967) 1462.

[lo] WORDEN, R. P., Pltys. Lett. 40B (1972) 260.

[ I l l KOPP, G., WALSH, T. F. and ZERWAS, P., DESY report (to be published) and further references there.

[I21 EFREMOV, A. V. and GINZBURG, I. V., P/I.VS. Lett. 36B (1971) 371.

[13] KINGSLEY, R. L., Nlic.1. Phys. B 36 (1972) 575.

[I41 GROSS, D. J. and TREIMAN, S. B., Plrys. Rev. D 4 (1971) 2 105.

[15] WALSH, T. F. and ZERWAS, P., NIICI. Phjls. B 41 (1972) 551.

[I61 TERAZAWA, H., PItys. Rev. D 5 (1972) 2259.

[I71 KUNSZT, Z., Pltys. Lett. 40B (1 972) 220 ; CHERNYAK, V. L., JETP Lett. 15 (1972) 343.

[I81 DRELL, S. D., LEVY, D. J. and YAN, T.-M., Phys. Rc.11.

D 1 (1970) 1035, 1617, 2402.

[I91 LANDSHOFF, P. V., POLKINGHORNE, J. C. and SHORT, R. D., N~cc.1. Pltys. B 28 (1971) 225 ;

See also Physics Reports 5C (1972) Nr. 1.

[20] BRODSKY, S. J., CLOSE, F. and GUNION, J. F., SLAG1243 (May, 1973) and references there.

[21] ABARBANEL, H. D. I., GOLDBERCER, M. L. and TREIMAN, S. B., Pl?)~s. Rev. Lett. 22 (1969) 500.

[22] SUURA, H., WALSH, T. F. and YOUNG, B.-L., Lett. Nliovo Cittzet7to 4 (1972) 505.

[23] FRITZSCH, H. and CELL-MANN, M., Proceedings of the XVI International Conference on High Energy Physics, Vol. 2, ed. J. D. Jackson and A . Roberts, National Accelerator Laboratory 1972.

[24] HAN, M. and N A M I ~ U , Y., Ph)~s. Re),. 139 (1965) 1006.

[25] CIIEWTHER, R., PItys. Rev. Left. 28 (1972) 1421.

[26] BARDEEN, W. A., FIIITZSCH, H. and CELL-MANN, M., CERN 1538 (J~ily, 1972).

[27] FERIIAIIA, S., GIIILLO, A. F. and PARISI, G., P11j<s. Letr.

45B (1973) 63.

[28] BEIIMAN, S. M., BJOIIKEN, J. D. and KOGUT, J. B., P11y.s.

Rev. D 4 (1971) 3388.

[29] BRANDT, R. and PREPARATA, G., PItys. Rev. Lett. 25 (1970) 1530.

[30] GEORGELIN. Y., STERN, J. and JEIISAK, J., NlicI. Plrys.

27 (1971) 493.

[31] BIIODSKY, S. J., CLOSE, F. and GUNION, J. F., Phys. Rev.

D 6 (1972) 177.

[32j JERSAK, J., LEUTWYLEII, H. and STERN, J., Nticl. Plrys.

B 57 (1973) 413.

[33] BOHM, M., JOOS, H. and KRAMMER, M., DESY 72/62 (1972) ;

See also GRECO, CERN T H 1617 (1973) and SAKURAI, J. J., UCLA preprinl ;

[34] WALSH, T. F., P I I ~ s . Lett. 36B (1971) 121 ;

BRODSKY, S. J., KINOSHITA, T. and TEIIAZAWA, H., Pl~ys.

Rev. Lett. 27 (1971) 280.

[35] DRELL, S. D., YAN, T.-M., Pltys. Rev. Lett. 24 (1970) 181 ; WEST, G., Pl~ys. Rev. Lett. 24 ( 1 970) 1206.

[36] WILSON, K., in the Proceedings of the 1971 International Symposium on Electron and Photon Interactions at High Energies Cornell University, Ithaca, N. Y. 1972.

[37] WALSH, T. F. and ZERWAS, P., P11j.s. Lett. 44B (1973) 195 ;

KINGSLEY, R. L., DAMTP preprint.

[38] FUJIKAWA, K., Nirovo Cit??etlto 12A (1972) 83.

[39] BUCHL, R. A,, Plrys. Rev. D 7 (1973) 2139.

[40] PARISI, J. and KESSLER, P., private communication.

DISCUSSION A. ZICHICHI. - If I understand what you have said,

you propose to extend the new duality of Sakurai to the process ef e- ^-, ef e- + hadrons. However, in the single-photon case, there is no problem with the reference cross section. But in the two-photon-case, for fixed beam energy, the comparison between c(e+ e- -+ e f e- hadrons) and o(ef e- ^-+ e" e- p f I ( - ) must be done for equal values of the invariant masses of the hadron and the (/if systems respectively.

This reduces the rates and makes the comparison quite difficult.

I also want to make a comment on your final conclusion. We all agree that a basic problem in hadron physics is the structure of the elementary particles. It seems to me that the most direct and

simple way to study the constituents of the hadrons is to measure (e' e- -+ hadrons), without any need to complicate our hard life with 2-photon processes. Do you agree ?

T. F. WALSH. - If ^{Y O U} detect two electrons at non-zero angle in ee -t ee + hadrons (2-photon part), then the reference variables q:, rl:, s are fixed for the comparison process ee ^-+ eep+ (?-photon part) : it is true that the cross sections are small for large ( I , , 2

: but they rise with beam energy. At E = 15 CeV, the double deep inelastic process has a cross section of the order of lo-"' cm', and should be a good cnndtdute far study by the machiiles which will come after SPEAR and DORIS. 1 hope that on will try,