Semihard production of neutral pseudoscalar and tensor mesons in photon-photon collisions

NUCLEAR

P H VS I C S B

Nuclear Physics B388 (1992) 376—390 North-Holland

Semihard production of neutral pseudoscalar and tensor mesons in photon—photon collisions I.F. Ginzburg and D. Yu. Ivanov Department of Theoretical Physics, Institute of Mathematics, 630090, Nocosibirsl<-90, Russian Federation Received 27 February 1992 Accepted for publication 18 June 1992

We investigate the semihard production of neutral pseudoscalar and tensor mesons in high-energy yy collisions (M = P ir°,7j. 7~’ or M = T = a 2, f2, f~).We deal with the exclusive yy — MM’ or semi-exclusive yy MX reactions (X is the hadron jet with not too large mass). The considered transfer momenta are small in comparison with the photon energies and they are large in comparison with the confinement scale. The amplitudes of these processes are determined by the odderon exchange, i.e. three-gluon exchange in the lowest order of perturbative QCD. The cross sections are calculated in this approximation. The possibility of measurements at LEP and at future yy colliders is discussed.

1. Introduction This paper deals with the production of neutral pseudoscalar P and tensor T mesons in yy collisions: yy—sMM’, M=P=n~°, i~, ij’

or

(1),(2)

‘yy—*MX,

M=T=f2(1270), a2(132O),f~(1525)

in the semihard diffractive region

2, M~_<~tl (~=O.3GeV,

t=(pY_pM)2).

(3)

s>HtI>>~

The reaction (1) is the usual exclusive process. In the semi-exclusive reaction (2) the produced particles are distributed in the following way (in the c.m.s.): the meson is emitted in a small angle relative to the incident photon direction with energy close to the photon energy E. The transverse momentum q 1 VT~T<
0550-3213/92/$05.00 © 1992

—

Elsevier Science Publishers By. All rights reserved

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iF. Ginzhurg, D. Yu. hanoi

Mesons production in photon collisions

377

p,~I~T

_______

Fig. 1. Quark-exchange contribution into yy

x, ~ —~

MM’, MX.

transverse momenta of the mesons or the meson and system X are compensated with high accuracy. Their energies are practically equal to the energies of the incident photons. As we will show, the cross sections of some of the processes (2) are large enough to be investigated at LEP. The future yy colliders with c.m. energies v~>100 GeV and luminosities up to ~ iO~’cm2 s~ (cf. refs. [1,21) will open up possibilities for detailed investigation of the reactions (1), (2) in a wide range of parameters. The production of tensor mesons was investigated in ref. [31.Some results for soft production of pseudoscalar mesons were reported in ref. [141. The processes of interest occur at small distances (l/p~) ~<(1/lL), i.e. non-perturbative effects are small, and the effective QCD coupling constant a,(p~) is small too. It allows one to use QCD perturbative theory (pQCD) with the accuracy =

(q,2~+m~)

/.L~

In the lowest order of pQCD the processes (1), (2) are described by the diagrams of fig. 1 (two-quark exchange). The contribution of these diagrams to the cross section do-/dt decreases as s~2for large s. On the contrary the contribution from diagrams with gluon exchange does not decrease with s while t is fixed. Due to the difference in the quantum numbers of photon and mesons in the reaction (1), (2) the lowest nontrivial diagrams of pQCD correspond to three-gluon exchange. Therefore we deal with the diagrams of fig. 2 in which the shaded block represents the sum of diagrams of fig. 3. The calculation of these graphs is the

K 1

K2

q

,~2

2 u ( p1 p4) Fig. 2. The 3-gluon exchange (the odderon exchange) contribution into yy S~(p1+p2)S,tq2_q~2,q=P3_P1, -

—~

MM’, MX.

378

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Mesons production in photon collisions

+

K

1K2q—K1—K2

/.L

I

VS

abc +

(permutatIons

~3g—qq

M~VE

—

of5+ gluons) C~V5+D~VE+ =A~~EtB~ + (permutations)

Fig. 3. Diagrams of the virtual y3g

—.

q~process.

main subject of the paper. We treat the production of mesons consisting of light quarks. The production of mesons consisting of heavy quarks will be considered elsewhere. The transition y M extracts the amplitude with the quantum number of odderon. Therefore the investigation of reactions (1), (2) gives us an important means for the investigation of odderon in the range of validity of pQCD (cf. ref. [15]). In this paper we discuss the lowest-order pQCD contribution into odderon: the diagrams with three-gluon exchange only. The paper is organized as follows. The basic notations are introduced in sect. 2. Sect. 3 is devoted to the description of the impact representation method and to the calculation of some impact factors. The cross sections discussed are obtained in sects. 4—6. The subjects of sect. 7 are both the range of validity of our results and the possibility of corresponding measurements. -~

2. The basic notations The basic kinematical notations are given in fig. 2. We use often the c.m.s. in which the photon energies are E and their three-momenta are ~ and ~2 We choose the z-axis along ji~.Besides =

s

=

(Pt

+p

2

=

4E2,

t

=q~

—~,

u

=

2)

(p

2, 1 —p4)

q =p 3 —ps,

~i=~i~/iq~ =p3~/~ p3~I.

(5)

Next, e12 (0, ë~2,0) are the polarization vectors of the photons and e-~(0, ~T, 0) is the polarization vector of a tensor meson in the state with helicity A 1. 2/4ir 1/137, a~ g2/4~. Qqe is the quark Besides, usual we denote a ecolor group. For the numerical calculations we charge, R as NEqQ~for the SU(N) use =

=

=

=

=

=

=

a~=0.l,

R=3~Q~=2, uds

jL=O.3 GeV.

(6)

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Mesons production in photon collisions

379

In accordance with fig. 2 at the first stage each photon fragments into a q~-pair with quark four-momenta q. (i 1, 2,3,4) and with quark energies E.(-~ E); the quarks of every pair move along the momentum of their “parental” photon. Their transverse momenta are relatively small, q• .L <
Let us denote by m the quark mass and by U1 u(q1), u2 u(q2) the quark spinors. The relative motion of the quark and antiquark is described by a variable ~ which is the ratio of the difference of their energies to their sum (photon energy): =

E1—E2 _____

E

=

2(q~—q2)p2 s ,

=

—1~~1.

(7)

At the next stage the quarks of one or both pairs transfer into one or two mesons. For the description of this stage the meson “wave function” is used (cf. refs. [4,5]). The transition of a q~-pairinto the meson T with helicity A is described by the replacement: (i) for the tensor meson T with helicity A 0, 1: =

ç+1 fq~(~)Tr(...j53) d~ 4N~_1 ~ Tr(...ê1j33)

forA=0 for A 1

QT

-

Qu1...u2—+-——i

(8)

=

(The trace is performed over vector and color indices.) The production of mesons with the helicity A 2 is suppressed in the approximation in question at least by a factor (ii) for the pseudoscalar mesons P: =

5~ Qü...u2~ ~f±td~f~(~)

Tr(...y

3).

(9)

The quantity QM is the average quark charge in the meson M. According to ref. [4] we adopt for numerical estimations the following parametrization for the wave functions and the values of coupling constants: (i) for all the tensor mesons T: ~

f~=fT=85MeV; 1

Q~12=7=~, Q=

1

7~, Q~=~

(10)

(assuming here the mixing to be an ideal one, i.e. supposing f’ consists of sfl-quark only).

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Mesons production in photon collisions

(ii) for the pseudoscalar mesons P:

4,~,(fl

I5~2(~—

=

f.,~

133 MeV,

=

f.,7

=

150 MeV,

f.,~

=

110 MeV,

~ -r’

Qn”0.38, Qn014 (the standard spy’ mixing is taken into account).

(11)

3. The technique of the calculation 3.1. IMPACT REPRESENTATION

The calculation of the amplitude of the process in fig. 2 by standard methods in a difficult task due to: (i) The number of diagrams is large; (ii) The contribution of separate diagrams contains large items (‘-.~ln’~(s))which cancel each other. (iii) These contributions are not gauge invariant. To overcome these difficulties the impact representation method is useful. It dramatically simplifies calculations in comparison with the standard ones. This method was proposed in refs. [6—8]for the description of QED processes with semihard kinematics (one- or two-photon exchange). This representation was extended to similar QCD processes with two-gluon exchange in refs. [9,101. The processes with three-gluon exchange are more complicated. Nevertheless just the same method as in ref. [91results in the impact representation for them. It is written with accuracy of f/s and has the form of an integral over gluon transverse momenta: 2k~

Myy~ABJ

—

2k

)j~~Lkt,2’ —~~) d 1d(2~2 21 12~\2

.2’ 72 72 72 t.L’~2J 3±

(k3=q—k~—k2).

‘.

I

~

I

(12)

The impact factor J~ (J~)corresponds to the upper (lower) block in fig. 2. It is s-independent. 3.2. THREE-GLUON IMPACT FACTOR J~

The three-gluon impact factor J~ is calculated in the same way as the two-gluon one J.~ in ref. [9]. These impact factors have a similar form. The

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I.E. Ginzhurg, D. Yu. iranor

Mesons production in photon collisions

381

expression obtained is d”~ J~

=

—

~

eg3Qq( ~

[mRêi

—

—

Q=

Q=(0,Q,0),

R=

+

qt

(1

Qetl

~)Qe1+

(k1±_~i

2

—U2,

2+(k

-

i=t

m

- 1±

2

—(~l~~2).

_~

+

i=

±

+

q± m 2 +q~1 +

+

—

m

+

P

q~)2

(13)

~)

This equation is valid only if q,~>> p~.Therefore, we shall neglect the quark masses in our case, but we reverse them in eq. (13) in order to have an opportunity in the future to calculate the cross section for the production of mesons consisting of heavy quarks. Let us emphasize some properties of the above impact factors: (i) The impact factor vanishes when the transverse momentum of the gluon in fig. 2 tends to zero: Jyq~(~l,2I

,

~k11

~

at ~

(14)

(ii) With ~ (or q21) being small, the impact factor does not depend on ~ and k12 ~. The main contribution in these cases comes from diagrams of type A (or D) in fig. 3 in which the quark line between three gluons is effectively contracted: 2<~ 1/q11

at~

1<
~E Jyq~

3.3. THE IMPACT FACTORS

U1

U2

~

2

1—

~-~_

12 ~

~2

=

.~V

2

~yT’ ‘yP

To produce meson M, both quarks should move almost collinear (with relative transverse momentum < p). In other words, their transverse momenta are related

382

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i.E. Ginzhurg, D. Yu. lianor

Mesons production in photon collisions

q21/~2(with accuracy ~~2/~j21) i.e.

as ~

q~1—~(1+~)q~, q21~(1~)q1,

s12=~(1±~)E. (16)

Taking into account these relations we insert eq. (13) into eqs. (8 and 9) to obtain impact factors ~yT’ J~~: t f(~)~Qe~ forA=0 — ~ Qr ç~ ~ ~ 21,q11—eg (1 ) 4N

2

f.4(~)mRëtë~

~—t

forA=1

d°’~

~

(18)

The subsequent calculation is performed for mesons consisting of light quarks, for which the quark masses are neglected everywhere. Eq. (17) shows that tensor mesons can be produced only in the state with helicity 2/t.) 0. (The production of mesons with helicity 1 is suppressed by a factor ~t Let

us introduce dimensionless vectors k 11=

?,.

,

~+~q1

instead of k, ~:

~=~~/Iq~

I.

(19)

Using these variables the impact factor (17 and 18) have the form

f

d”~

(e~T) [ë1x~p~

JYM(k.2I,~I)=eg(~)QM11

—

forM=T

forM=P

(20)

2

~

FM=L1d~ i—~-~+,~ (F-~) forM=T forM=P

~1~T(~)

~ In accordance with eq. (14) FM

(i,,

~i)

—~0

at

~,

—s

~

3.4. THE QUARK IMPACT FACTOR

The impact representation (12) is valid for the photoproduction on quark with the replacement of ~yB by the quark three-gluon impact factor. It is calculated in

IF. Ginzburg, D.Yu. Ivanov / Mesons production in photon collisions

entire analogy with the calculation of the electron impact factor in ref. [8] impact factor is

383

This

~.

d~i~w 3

qq Here Aq and ~

4N

121) A

5A5,~

are the helicities of the initial and final quarks.

4. Reactions ~yq—s Tq, yq

—~

Pq

Let2>> us calculate at first amplitudes for the reactions yq Mq at Syq (p~ t I. In order to the do this, we substitute J~+Pq) 1-, J~.,and J~ into eq. (12). Besides we use the identity 2—1)(N2—4) 40 (N —~

=

______________________

=

=

Then M yq

-.

Mq

s~ ~ eg

—

2

-

[etxn] 1

i’~

[LIT

-

fM q

M

(22)

forM=P, F21’M(r~,2,n)

~.

23

1)(N2 —4) QM 16N3 ~2 forM=T

(et~i)

x

1= ~‘

—

6(N

....2

(r—

2_

2 2

—

2

drdr

1 +n) (r2-i-n) (rt +r2) The integration over ~ ‘~2 is performed as follows: the x-axis is chosen along ~i, the integrals over r51, r~2are performed by means of residues. The subsequent integrals over r~1,r52 are elementary ones, though very awkward. The result is 1

~

d~

2J_~ 1_~2

2

in

~

IS~~4)T(Se) forM=T

~

—

~

forM=P

Using the wave functions (10 and ii) we obtain 2 IT

l+~~ =9.11.

‘M’T’P5 *

The similar method was used in ref.

lii]

(23)

for the description of the three-gluon-exchange (with

massive gluons) contribution in qq scattering at

t

=

0.

I.E. Ginzhurg, D. Yu. honor

384

The cross section is dt7yq_.,M

dtd4

/

Mesons production in photon collisions

(4 is the azimuthal angle)

=2ITaa~ 6

2

2—1)(N2—4) (N

~

QM’MfM

N

forM=T

(__)2

[e

2

1xñ}

forM=P (24)

5. Reactions ~y~y —s TX, ‘y’y —p PX These processes are favorable for observation due to the fact that their cross sections are larger than that for y —s TT’, PP’, TP. The yy —s TX, PX amplitudes can be found by substitution of the impact factors J~-,J~ and Jyq0 into eq. (12). Due to properties (15) the main contribution arises from the region 72 ~,—2 I.2J.

q1.

To obtain the corresponding cross sections one should integrate over the quark momenta q3 and q4 (with the constraint q3 +q4 =p2 —q). These cross sections decrease strongly at >> i.e. at M~> t I. Thus, the limitation M~ t in eq. (3) is obvious. (When M~>~t the essential contribution to the cross section arises from other diagrams, e.g. with additive production of the gluon jets by gluons.) To extract system X from background the additive condition (M~markedly smaller t ) may be useful. The above integration includes the regions ~ <~2 where eq. (13) for the impact factor is invalid (but the impact representation (12) is valid). Fortunately, the contribution of these regions does not increase with the growth of t On the other hand, the regions q~~ >> l~give a logarithmically increasing contribution. Hence the calculations of the yy —~TX, PX reactions performed 2)only. This can facebeallows us to in usepQCD here with a logarithmic accuracy 1/ln(t/~i instead of eq. (12) a simple approximation method (with the same accuracy) the equivalent quark approximation (cf. ref. [9]). ~

~,

.

—

5.1. EQUIVALENT QUARK APPROXIMATION

One can see from eq. (15) that the amplitudes considered are relatively large only in the regions ~ (25)

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Mesons production in photon collisions

385

where one of the virtual quarks in the lower block becomes quasi-real. Here the diagrams of type A and D in fig. 3 dominate. In these regions ~yM does not depend on k121 (see eq. (15)) just as Jyq~. So the integration over k~21yields here the yq —s Mq (or y~—p M~)amplitude, multiplied by a factor 1/q41 (or 1/q31). Therefore, the processes discussed can be described as two subsequent subprocesses: (i) The photon p2 fragments into the q~j pair with relatively small (in comparison with q1) transverse momenta; (ii) The —s Mq or y~—s M~process takes place. Such an approximation is called the equivalent quark approximation. In this approximation the cross section for the -yy Mq~reaction is the product of the yq —s cross section by the number of 7iq,Mq just as in the equivalent photon the equivalent quarks dnq and antiquarks d approximation in QED (the quark energy Eq =XqE s -~

3)

do~~.Mx =

L(dflq

+

dn4)

dO~yq_.Mq,

dflq=NQ~~~’ [(1~xq)2+x~1

(26)

dXq. 3qI

—~

The

~.)

(The transverse momentum of (25) virtual quark in region is ~ integration over 13q1 in region gives (together with (25) the contribution of dn 0 in region (25)) _R1(l

E(q+dfl0)

ln(~).

dXq

_Xq)2+X~j

(27)

In our approximation the yq —s Mq cross sections (24) are s-independent. Therefore, one can integrate over Xq (from 0 to 1): 2a ~(dn11+dn0)=—Rln37T

—t —~-

(28)

.

q

5.2. THE REACTIONS

yy

-÷

TX, PX

Substituting eqs. (24) and (28) into eq. (26), we obtain —t /i

do~~,Mx=Q~ In

5

AM

=

~ITRa

—i-

2

6

AM dt d~ ((e~i)2 ~ t~

—~

2IT

~,[ë~xii]

(N21)(N24)

N3

forM=T forM=P,

,

2

/MfM

(29)

386

1. F. Ginzburg, D. Yu. Iranoi

/

Mesons production in photon collisions

The yy —s MX cross section for unpolarized photons integrated over the region It I t0 is obtained from eq. (29) by adding the U-exchange contribution:

=Q~~~ln(_~).

~YY,MX(tt0)

(30)

2: In particular, at to

3 GeV

=

=

Uyy

4.3 X i035 cm2,

a

:

2X ~yy~~(IX=

~

fX

1.1 x

:

:

~

io~cm : ~=n’x

6. Reactions yy The yy ~yM

—~

1: 1/9:2/9,

=

2,

—~

=

(

31

1:0.37:0.02.

TT’, PP’, PT

MM’ amplitude is calculated by using eq. (12) with impact factors

(17), (18). Replacing k 11 by F1, eq. (19), one gets M~~MM’ =

~DMMIMM’QMQM’,

2— 1)(N2—4) N3

For

‘MM’

there are three cases: 1

d2r~d2r 2 + 2 ~ (F~+ n)2(F 2(F,

12ir

I

F2)2

2 + n)

(et.~M(Ft,

F 2, ~))(~2~M’(—Ft~ —F2, —i))

X

(32)

(N

DMM=l6IT2aa~fMfM

(e~x

~M(Fl,

F2, ~))(ë2~M’(—Ft,

(e~x

~M(Ft,

F2,

n))(e2 X FM(

for M, M’

—F2, —ii))

—F1, —F2,

—a))

for M

=

=

T, T’

P, M’

for M, M’

=

=

T

F, P’ (33)

I.E. Ginzburg, D.Yu. Ivanoc

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Mesons production in photon collisions

387

It is convenient to represent the quality ‘MM’ in the form —

(~~~2)B1 [2(~1~)(e2~i) for M, M’

‘MM’

=

—

(e1e2)B2

(F1ë2)]B2

—

—

T, T’;

=

[2(e1n)(e2n) (e1e2)] B1

—

(34)

—

for M, M’ P, P’; [(ii xe1)(e2n) + =

—(e1 xe2)B1

(n xe2)(e1n)]B2

—

for M

T, M’

=

=

P.

The cumbersome integration over F1 is performed in the same manner as in sect. 4. The result can be written in the form

~

P~

~

B~=

for M,M~T’ forMM’ PP’

~2){~2

1 ~

~

/3~= In

~1~2

In

+ln

1+~

1+~

/32=ln

~2(’~1)

~1~2 ~

(~2_~)

in

5~t~2

ln 1+~

~~(1—~)

1+~1 —

In

(~_~)

~

1+~2 .

(35)

1~

The integration over ~, with the wave functions (10) and (11) is performed by using formulas from ref. [121. The results are (~(3) 1.202) 2)~26.8,

B

Bt=-~(3+IT

2=~(87_IT2_~-~(3))s~10.9.

(36)

The constants B1, B2 were found identical in all cases MM’ TT’, PP’ and TP. These relations demonstrate a strong dependence of the obtained cross section on the photons polarization. Really, the helicity amplitudes, M~÷ M_) and Mt... (=M.~) differ by a factor of B1/B2 2.46. For unpolarized photon beams one has =

(=

=

2

dr~~,MM~ —

dt

—

G(fMfM’QMQM’)

t4

G=8IT3a2a~(B?+B~) The cross sections integrated over the region

(N2— 1)(N2—4) N3

It

I

.

(37)

t 0 are 7.3pb.

~

2

0yy.,.,,oa=

(38)

388

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Mesons production in photon collisions

Other cross sections are smaller. (For the production of non-identical mesons the u-contribution is taken into account too.) 7. Discussion 7.1. COMPARISON WITH THE VECTOR MESON PRODUCTION

The two-gluon impact factors J~ and J~jfor production of vector mesons have been obtained in ref. [9]. It has the form of eq. (13). The corresponding vector Q and scalar R are constructed from the items which are similar to that in eq. (13). The expressions for Q (or R) in two-gluon and three-gluon exchanges have the opposite parities relative to the q1 ~-sq2 substitution. It is caused by the opposite parities of systems with 2 and 3 gluons. The cross sections obtained are surprisingly large in comparison with the ones for vector meson production (from refs. [4,9]). For example one can naturally expect that 2

°~yy 2TX 0.7_s p~X .,

2 a~

f~

4

1

o~ ,

_.,i.,..ij

2 a~

~

Really ~~yy —, ~T0X 0.77

p°X

1 3

~ ~yy

~.(I~I

p°p°

This fact needs to an explanation. 7.2. THE RANGE OF VALIDITY OF THE RESULTS AND THE EXPERIMENTAL POSSIBILITIES

In the reactions discussed the transverse momenta of the mesons or the meson and system X compensate each other with high accuracy. This high accuracy is important to exclude the processes with one-gluon exchange. In such a case soft discolour is accompanied by production of a soft gluon which would break the exact compensation of transverse momenta of the mesons in reaction (1) or the meson and system X in reaction (2). For the processes considered the diagrams of the lowest order in a~,are given in fig. 1 (diagrams with two-quark exchange in the t channel). Their contribution to the cross section decrease for large s as 52 The three-gluon contribution considered dominates in the region a2s

2

>1.

(39)

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Mesons production in photon collisions

389

On the other hand the results obtained are valid until the contribution of the next diagrams is not too large, i.e. c—ln(s/t) <1

(40)

iT

(the quantity c is unknown up to now). The limitations (39) and (40) determine the narrow region of s/t values where the expressions obtained can be compared with experiment straightforwardly. Our results can be treated as estimations of order of cross section values out of this narrow region. Lipatov gives some arguments for the growth of the cross sections of discussed type with s. It is the case our results can be treated as the estimations of cross sections from below. We predict that the mesons in reactions (1) and (2) will be produced in the state with helicity A 0 only. Taking into account higher orders of pQCD should preserve this conclusion. This is in contrast with the energy dependence of cross sections which should deviate due to high orders of pQCD series. The detection of pions and kaons from the decay of mesons M is simplified by the fact that their production angles are not very small; they are close to the production angle of M, i.e. 9.,~= q1/E (cf. ref. [9]). =

(i) LEP, TRISTAN. Using for LEP and TRISTAN the well-known values for their luminosities and energies we obtain the luminosities of yy collisions at 10 ±2 GeV (cf. ref. [13]): 2 7 x 10~cm 5~ for LEP, =

=

L77

=

7 X 10~°° cm s~ for TRISTAN.

Using eq. (31), one gets the following values for the number of events per year for the processes (2): LEP:N= 800

(‘yy—sir°X); 300 TRISTAN:N=80

(yy—sa2X); (yy—siT°X).

300

(yy—*s~X) (41)

These quantities show how feasible it is to investigate our processes on these colliders. (ii) The yy colliders with the real photons The yyprocesses luminosity these 2 ~ [1,2]. Therefore (1) ofand (2) colliders should be L77 10~_10M cm should give thousands of events per hour. It will be possible to study these processes up to transfers It I 100 GeV with the admissible statistics. The investigation of processes (1) and (2) at these colliders will be a task of great importance. ‘~

‘~

390

IF. Ginzburg, D.Yu. Ivanov / Mesons production in photon collisions

We are grateful to V.L. Chernyak, A.G. Grozin, S.L. Panfil, V.G. Serbo, B.V. Struminsky and K.A. Ter-Martirosjan for useful discussions.

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