Volume 65B, number 3 PHYSICS LETTERS 22 November 1976

P H O T O P R O D U C T I O N O F C H A R M E D P A R T I C L E S A N D

A S Y M P T O T I C A L L Y F R E E F I E L D T H E O R I E S

M.A. SHIFMAN, A.I. VAINSHTEIN and V,I. ZAKHAROV Institute of Theoretical and Experimental Physics, Moscow, USSR

Received 3 September 1976

It is argued that photoproduction of charmed particles can be considered within the same theoretical framework as deep inelastic scattering off the usual hadrons. The role of large Q2 now plays the charmed quark mass m2c . In par- ticular, the standard technique of the moments from the structure functions is applicable. In this way we get a sum rule which relates an integral from the charmed particles photoproduction cross section to the heavy quark mass and effective coupling constant of strong interactions.

Since the discovery of the J/$-particles the mecha- nisms of charmed hadrons production have been wide- ly discussed in the literature (for a review see, e.g., refs. [ 1 ]). Experimentally, only production of hidden charm has been observed so far. It is important for our further consideration that if-mesons are more copiously produced in the photon- rather than in the hadron-induced reactions. This experimental observa- tion is in accord with theoretical expectations since the photon is directly coupled to charm through the term -d~luc in the electromagnetic current. In this note we will confine ourselves just to the consideration of photoproduction of charmed particles in this, so to say, quasidiffractive way.

Thus far, the S-meson photoproduction has been considered within the conventional models of strong interactions, such as vector meson dominance, Regge poles, meson exchanges and so on (for the most recent discussion of such approaches see ref. [2]). One of the basic motivations for all the theoretical specula- tions is the calculation of the cross section of charmed particles photoproduction.

In this paper we would like to use quite a different approach to the same problem which links photopro- duction of charmed particles to deep inelastic scatter- ing. The close analogy between the two processes fol- lows in fact from an elementary consideration based on the uncertainty principle [3].

Indeed, the pair production of charmed particles can be viewed as a fluctuation of photon into acE pair with a subsequent scattering of one of the quarks on the target. The time of fluctuation is of the order

r ~ v / 4 m ~ (1)

where m c is the charmed quark mass and v is the pho- ton energy. The similar time of fluctuation of a virtual photon into usual quarks, which is relevant to deep inelastic scattering, is of the order

r "" u/Q 2 (2)

where _Q2 is the photon four-momentum squared. 2 plays the role of Q2 Thus, we see that m c

In the other words, the charmed quark can "exist" within the usual hadrons only for a short time and at short distances. On the other hand, the strong interac- tions at short distances are consistently described by quantum chromodynamics which we use throughout this paper. Thus, photoproduction of charmed par- ticles belongs to asymptotic freedom.

In a formal way, the argument implies that only the graphs with the c-quark loops are to be kept in the case considered (see, e.g., the graph in fig. 1). In other words, there is no contribution to the amplitude of the forward Compton scattering from the graphs with the c-quark lines coming in or going out. Let us mention in this connection that similar graphs were considered in refs. [4,5] for weak interactions and electroproduction of charmed particles, respectively.

Our main result is the following sum rule for the c of charmed particles photoproduction cross section O~

f d_vv ~(v) = 22n aCZs(mc 2)

threshold V2 ~ O m4 , (3) C

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Volume 65B, number 3 PHYSICS LETTERS 22 November 1976

/1/

\ i

Fig. 1. The simplest graph for the ampli tude of forward Compton scattering with charmed quarks in the intermediate state.

where v = pq and p, q stand for the nucleon and pho- ton momenta, respectively; parameter p denotes the fraction of the nucleon's momentum carried by gluons as is measured in deep inelastic scattering at Q2 =mc .2 As is well known partons carry about half of the total momentum so that

p~0.5 .

Cross section o,~ in eq. (3) does not include charm production via excitation of intermediate state of light quarks alone (such as 7 ~ w ~ ~ transition). As was already mentioned above there exist good reasons to expect that the direct coupling of photons to c- quarks dominate in the total cross section (from the theoretical point of view this is due to thevalidity of the Zweig rule).

From the existing experimental data [6,7] we can estimate the contribution of the q/-meson photopro- duction into the l.h.s, ofeq. (3). If we put m c = 2 GeV and as(m 2) = 0.2 then this contribution is about of 1/20 of the total.

It is worth emphasizing that we do not put any in- formation on the cross section of the qJN interaction which is crucial for all the conventional models of strong interactions. As is well known the o(ffN) turns out to be anomalously small. In our approach the smallness of the cross section of charmed particles photoproduction is entirely due to the mass of the heavy quark. The photoproduction of heavy particles is viewed as a short distance process here while in the model of vector meson dominance, e.g., all the off- mass-shell effects are neglected. Nevertheless we come to a cross section which seems to be very reasonable and is rather close to that predicted by the VDM model, although it seems to be smaller somewhat (for a critique of the VDM model see ref. [3]). Thus, the

calculation within the asymptotically free field theo- ry may give some clue to the phenomenological ob- servation of the smallness of the interaction cross sec- tion.

After discussing in brief the implications of the result obtained let us proceed with some details of its derivation.

The cross section of charmed particles photopro- duction is related to the imaginary part of the Compton forward scattering amplitude T#v

Tuv = ~ 47rai f exp( iqy )d4y (4)

× (N[ T ?(.y) 7uc (y), ?(0) 7v c (0) IN)spin averaged

The general form of Tuv is as follows

e(1)A2) T = (F(1)- "~(F(2)po)A(v ) /~ cv ,uv ~- ~tv b 'v lk /~p (s)

F ~ v = e t a q v - e v q ~ •

where e (1) (e (2)) is the initial (final) photon polariza- tion vector, q and p are the photon and nucleon four- momenta, respectively, A(v) is the invariant amplitude.

In the limit of the small photon momentum, q ~ 0, the integral in eq. (4) is determined by the distances of the ordery ~< 1/m c. Since we consider m c to be large the Wison expansion for the T-product in the r.h.s, of eq. (4) can be used. In particular, the value of A(0) is determined by the operators of twist two.

In the lowest order in 'the strong interactions the loop graph gives rise to the operator of the energy- momentum tensor of the gluon field

O~ v_ a ,a - GuaGow - trace, (6)

e (1) e (2) T.v =(Nlc (1) oG^ F(1)F (2) + ... IN) ¢x~ a~/ ,,/p

where dots denote the terms irrelevant to the present consideration and G~a (a = I ..... 8) is the gluon field strength tensor. The corresponding coefficient can be found from an explicit calculation of the graph which enters as the upper block into the diagram in fig. 1 and turns out to be equal to

c(1)_ 44 CZas (7) 405 4

m c

where a s = g2/4rr is the quark-gluon coupling constant. When the higher orders in a s is replaced by the ef-

fective coupling constant of strong interactions, oq(m2). Moreover, operator oGv mixes with the opera-

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Volume 65B, number 3 PHYSICS LETTERS 22 November 1976

tors constructed from the quark fields:

e~,-c0,=m~l)e~~tC’~=m,1)e~, (8)

where c, c’ are some coefficients. All the consideration of the effects of mixing is

quite similar to the consideration of the moments from the structure functions of deep inelastic scatter-

ing at Q2 = mf. Therefore, we can use at this point the empirical observation that at Q2 = 4 GeV2 the

gluons carry about a half of the nucleon momentum. This condition fixes the matrix element from (8) to

be 2pUpv*p and we come to the conclusion that

88 ..,k$) A(O) = - -$jy ___

rnz ” which by means of the dispersion relation in the pho- ton energy can be transformed into eq. (3).

Let us notice in conclusion that the higher mo- ments from the cross section can be’treated in the same way. There arises, however, a new problem of finding the matrix elements from the relevant opera- tors containing fluon fields. These matrix elements cannot be extracted from the existing experimental data in a direct way and for their estimate we use an

additional assumption that the gluons come from the bremsstrahlung of the valence quarks. The correspond- ing procedure and results obtained are described else- where.

The authors are grateful to V.A. Novikov for valu- able discussions.

References

[I] M.K. Gaillard, B.W. Lee and J. Rosner, Rev. Mod. Phys.

47 (1975) 277; V.I. Zakharov, B.L. Ioffe and L.B. Okun, Uspekhi Fiz.

Nauk, 117 (1975) 227.

[ 21 KG. Boreskov and B.L. Ioffe, preprint ITEP-102 (1976).

[3] V.I. Zakharov, B.L. Ioffe and L.B. Okun, Zh. Teor. i Exp.

Fiz. 68 (1975) 1635. [4] M.A. Shifman, AI. Vainshtein and V.I. Zakharov, preprint

ITEP-59 (1975);preprint ITEP-63 (1976).

[5] E. Witten, Nucl. Phys. 8104 (1976) 445.

[6] W.Y. Lee, Proc. Intern. Symposium on Lepton and pho-

ton interactions at high energy, Stanford, Stanford Uni-

versity (1975) p. 213.

[7] R.L. Anderson, Report to the Wisconsin Intern. Conf.

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