

Preprint IFUP-TH-59/95 Anomalous violation of the OZI-rule in the N _N ! \Phi \Phi ,

N _N ! \Phi fl reactions and instantons

N.I.Kochelev Dipartimento di Fisica, Universit`a di Pisa and INFN

Sezione di Pisa, I-56100 Pisa, Italy

and Bogoliubov Laboratory of Theoretical Physics,

Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia 1

Abstract It is shown, that specific properties of the instanton induced interaction between quarks leads to the anomalous violation of the OZI-rule in the N _N ! \Phi \Phi , N _N ! \Phi fl reactions. In the framework of instanton model of the QCD vacuum, the energy dependence of the cross sections of these reactions is calculated.

1Permanent address; E-mail:kochelev@thsun1.jinr.dubna.su

1

The OZI rule [1] is successfully used to explain suppression of the processes which include disconnected quark lines. However, recently, anomalous violation of this rule in some channels with creation of the \Phi -mesons was found in the N _N annihilation (see review [2] and referencies therein). The most significant diviations from OZI rule have been observed in the reactions N _N ! \Phi \Phi [3] and N _N ! \Phi fl [4], where measured values of the cross sections differ from predicted ones on the several orders.

Some explainations of this puzzle were suggested. So, in [5], the large mixture of the strange quarks inside nucleon has been supposed. The more conventional approach, based on the rescattering, was also involved to explain these results [6].

Recently, the new mechanism of the OZI-rule violation in the N _N annihilation has been proposed [7]. This mechanism is based on the specific interaction between quarks [8] which is induced by the instantons [9]. It has been shown, that taking into account of the instanton induced interaction, allow us to explain, at least qualitatively, some features of the OZI rule violation in N _N annihilation. It should be mention also, that at the begining this interaction has been used to solve the several old problems of the hadron spectroscopy, in particularly, to solve the problem of the large violation of the OZI rule in the wave functions of the pseudoscalar nonet mesons [10]. At the present time, the same interaction is considered as the evident QCD mechanism to explain so-called "spin crisis" which is connected with the large OZI rule violations in the spin-dependent structure functions [11].

The main goal of this article is the calculation of the instanton contribution to the cross sections of the processes N _N ! \Phi \Phi and N _N ! \Phi fl.

The effective Lagrangian for the interaction induced by the instantons for the number of the flavors Nf = 3 and for massless quarks has the following form [12]:

L(Nf =3)eff = Z daeae5 d(ae) 8!: Y

i=u;d;s(\Gamma

4ss2

3 ae

3 _qiRqiL)+

3 32 (

4 3 ss

2ae3)2[(ja

uj

a d \Gamma 34 j

a u_* j

a d_* )(\Gamma 43 ss2ae3 _qsRqsL) +

9 40 (

4 3 ss

2ae3)2dabcja

u_* j

b d_* j

c s + 2perm:] + 9320 ( 43 ss2ae3)3d

abcja

uj

b dj

c s +

igf abc

256 (

4 3 ss

2ae3)3ja

u_* j

b d**j

c s*_ + (R ! L) g ; (1)

where qR;L = (1\Sigma fl

5)

2 q(x); j

a i = _qiR*

aqiL; ja

i_* = _qiRoe_* *

aqiL, and d(ae) is the instanton

density and ae is their size. It should be point out that Lagrangian (1) has been obtained considering quark zero modes in the instanton field. In this case the point2

like interaction (1) appears in the limit of small size of the instantons kiae ! 0, where ki are momenta of the incoming and outcoming quarks. Taking into account the finite size of the instanton leads to some form factor in (1) which is proportional to the Fourier transformation of the quark zero modes [13]. In the regular gauge it is [14]

f (ki) = exp(\Gamma ae X

i k

i): (2)

The density of the instantons for Nf = 3 and Nc = 3 is given by [15]

d(ae) = 3:64 \Delta 10\Gamma 3( 2ssff

s(ae) )

6exp[\Gamma 2ss

ffs(ae) S(ae)]; (3)

where

S(ae) = 1 \Gamma ss

3

16ffs(ae) ! 0 j

ffs

ss G

2 _* j 0 ? ae4: (4)

The second term in the equation (4) takes into account interactions between instantons [16]. This term leads to the anomalous growth of the density of the instantons with the increasing of their size. Anomalous growth of the density is one of the main problems of practically all the calculations of the contributions of the instantons to observelable values, since the necessity to introduce some additional mechanism of density stabilization is appeared (see discussion in [17]). Recently, in [18] it was shown that unitarity condition leads to a restriction on the magnitude of the contribution of the interaction between instantons into the equation (4) by the value 1=2. Therefore, in our calculation below, we will use the following approximation for S(ae)

S(ae) = ( 1 \Gamma

ss3 16ffs(ae) ! 0 j

ffs

ss G

2 _* j 0 ? ae4; if ae ! ae0;0:5 if ae * ae

0. ; (5)

where

ae0 = ( 8ffsss3 ! 0 j ffs

ss G2 j 0 ?

)

1 4 ; (6)

and ! 0 j ffsss G2 j 0 ?= (330M ev)4 is the gluon condensate. It should be mentioned, that a similar approximation was used to estimate contribution of the instanton to DIS structure functions in the paper [19].

Contribution of the instanton to the N _N ! \Phi fl and N _N ! \Phi \Phi reactions is presented on the diagrames in the Fig.1. The part of the Lagrangian (1) which has

3

the right quantum numbers and can contribute to the cross sections of these processes is

\Delta L(Nf=3)eff = Z daeae5 d(ae) ! 0 j _qq j 0 ?4 f (ki)( 4ss

2

3 ae

3)3((_uifl5u + _difl5d)(_sifl5s)): (7)

In the formula (7) the appearence of the quark condensate ! 0 j _qq j 0 ? follows from the contraction of the two quark legs from six-fermionic vertex. Form factor f (ki) is determined by the energy of the colliding particles and equals

f (ki) = e\Gamma 2aeps: (8) The matrix elements from Lagrangian (7) for the reactions N _N ! \Phi fl, N _N ! \Phi \Phi are

M (N _N ! \Phi fl) = *inst(s) ! N _N j (_uifl5u + _difl5d)_sifl5s j \Phi fl ?; (9)

M (N _N ! \Phi \Phi ) = *inst(s) ! N _N j (_uifl5u + _difl5d)_sifl5s j \Phi \Phi ? : (10) where

*inst(s) = Z daeae5 d(ae) ! 0 j _qq j 0 ?4 ( 4ss

2

3 ae

3)3e\Gamma 2aeps: (11)

We will use the axial anomaly and vector dominance model (VDM) to calculate the matrix elements (9), (10). So one can reduce the equations (9), (10) to the product of the matrix elements

! N _N j (_uifl5u + _difl5d)_sifl5s j \Phi fl ?= (12) ! N _N j (_uifl5u + _difl5d) j 0 ?! 0 j _sifl5s j \Phi fl ?; (13)

! N _N j (_uifl5u + _difl5d)_sifl5s j \Phi \Phi ?= (14) ! N _N j (_uifl5u + _difl5d) j 0 ?! 0 j _sifl5s j \Phi \Phi ?; (15)

Some of these matrix elements can be determined by means of the VDM connection [20]:

! 0 j _sifl5s j \Phi fl ?= g\Phi e ! 0 j _sifl5s j flfl ?; (16)

! 0 j _sifl5s j \Phi \Phi ?= ( g\Phi e )2 ! 0 j _sifl5s j flfl ?; (17) and the relation given by axial anomaly (see for example [21])

! 0 j _sifl5s j flfl ?= 1m\Lambda

s

ffs 2ss Nce

2 sF (1)_* eF (2)_* : (18)

4

In the equations (16), (17), (18) g2\Phi =4ss = 13:3 is \Phi -meson coupling constant with photon [20],

F (1)_* = k(1)_ ffl(1)* \Gamma k(1)* ffl(1)_ ; (19)e

F (2)_* = 12 ffl_*aeoe(k(2)_ ffl(2)* \Gamma k(2)* ffl(2)_ ); (20) k(i)_ (ffl(i)_ ) is the momentum (polarization) of the photons (\Phi -mesons),

m\Lambda s = mcurs + m\Lambda q; m\Lambda q = \Gamma 23 ss2ae2c ! 0 j _qq j 0 ? (21) is the mass of the strange quark (m\Lambda s; mcurs = 150M eV ) and the masses of the nonstrange quarks (m\Lambda u = m\Lambda d = m\Lambda q), aec = 1:6GeV \Gamma 1 is average size of the instanton in the QCD vacuum [14].

One can estimate the remaining matrix element ! N _N j (_uifl5u + _difl5d) j 0 ? by using the relation given by gluon axial anomaly:

! N _N j (_uifl5u + _difl5d) j 0 ? = \Gamma 12m\Lambda

q ! N

_N j ffs4ss G_* eG_* j 0 ? (22)

= \Gamma MNm\Lambda

q g

0 A _N (p1)ifl5 fN (p2): (23)

To get equation (23), the connection between matrix element of topological charge density and singlet axial vector coupling constant which determines of determines of the part of the proton spin carried by quarks, was used [22]. In the paper [23] from analysis of the experimental data on spin-dependent structure functions [24] the value of the g0A was extracted

g0A = 0:31 \Sigma 0:07: (24)

By taking into account all factors, we get the final formulas for cross sections:

oeN _N!\Phi \Phi (S) = 0:398ss ( *inst(S)MN g

0 A

12m\Lambda sm\Lambda qss (

g2\Phi 4ss ))

2 \Delta

((S \Gamma 2m2\Phi )2 \Gamma 4m4\Phi )(S \Gamma 4m2\Phi )

1 2

(S \Gamma 4M 2N )

1 2 (mb) (25)

5

and

oeN _N!\Phi fl(S) = 0:394ss ( *inst(S)MN g

0 A

12m\Lambda sm\Lambda qss )

2ff( g2\Phi

4ss ) \Delta

(S \Gamma m2\Phi )3 (S \Gamma 4M 2N )

1 2 S

1 2 (mb): (26)

For numerical calculation we used NLO approximation for strong coupling constant:

ffs(ae) = \Gamma 2ssfi

1t(1 +

2fi2logt

fi1t ); (27)

where

fi1 = \Gamma 33 \Gamma 2Nf6 ; fi2 = \Gamma 153 \Gamma 19Nf12 (28)

and

t = log( 1ae2\Lambda 2 + ffi): (29)

In equation (29) the parameter ffi ss 1=ae2c \Lambda 2 provides a smooth interpolation the value of the ffs(ae) from perturbative (ae ! 0) to the nonperturbative region (ae ! 1) [25].

Result of the calculation of the cross section of the process N _N ! \Phi \Phi is presented in Fig.2 together with the preliminary experimental data from JETSET Collaboration [4], for energies from threshold up to pS = 2:4GeV , and the data of the R-704 Collaboration at pS = 3GeV [26]. There is a good agreement between model and experimental data. The small discrepance, probably, can be connected with uncertainties in the total normalization of the value of the cross sections in the JETSET Collaboration (see [27]). Instanton model correctly discribes observable strong energy dependence of the violation of the OZI rule. This dependence is due, in fact, to the strong sensibility of the value of the integral (11) the value of the initial energy pS, through the instanton density. Therefore, precise measurement of this dependence can give the very important information on the distribution of the instantons in the QCD vacuum.

It should be stressed, that OZI rule predicts the value of the cross section oeOZI ss 10nb which does not depend on the value of energy and is about two order smaller then the experimental data.

In Fig.3 the prediction of the cross section oe(N _N ! \Phi fl)(S) is given. By using the experimental interpolation of the data on the total cross section of the N _N annihilation

6

[28]

oetotalN _N = \Gamma 30:1 + 46:6p + 60:3p (nb); (30)

where p is the antiproton momentum, one can estimate the branching ratio of the p_p ! \Phi fl reaction at rest

Br(p_p ! \Phi fl) = 1:8 \Delta 10\Gamma 5: (31)

The value (31) is in very good agreement with the Crystal Barrel Collaboration result [3]

Br(p_p ! \Phi fl) = (1:7 \Sigma 0:4) \Delta 10\Gamma 5: (32)

The OZI rule contradics these experimental data. So, if one uses the value of the branching ratio for the reaction p_p ! !fl

Br(p_p ! !fl) = (6:8 \Sigma 1:8) \Delta 10\Gamma 5; (33) given by the same collaboration [29], then this rule predicts

Br(p_p ! \Phi fl) = tan2\Theta Br(p_p ! !fl) = 2:8 \Delta 10\Gamma 7(10\Gamma 8) (34) for the quadratic (linear) Gell-Mann-Okubo mass formula 2.

Thus, taking into account the instanton induced interaction between quarks allows us to explain the anomalous violation of the OZI rule in the reaction p_p ! \Phi \Phi , p_p ! \Phi fl.

We can conclude that significant violation of the OZI rule, which was revealed in the N _N annihilation, is the nontrivial manifestation of the complex structure of the QCD vacuum connected with existence of the instantons.

Therefore, the investigation of the violation of the OZI rule in the N _N annihilation can allow to obtain the very useful information on the structure of the ground state of the theory of the strong interaction, the QCD vacuum.

The author is sincerely thankful P.N.Bogolubov, A.E.Dorokhov, A.F"assler, M.F"assler, H.Petry, E.Klempt, D.Kharzeev, S.B.Gerasimov, B.Mench, M. Mintchev, M.G.Sapozhnikov, Yu.A.Simonov, B-S.Zou, for useful discussions, Prof. A. Di Giacomo for warm hospitality at Universit`a di Pisa and INFN for support.

2In the equation (34) \Theta is the deviation from the ideal mixing angle (35

:30).

7

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10 \Gamma \Gamma @@

++

Ifflffi fifloe_s

s.^

\Phi fl\Lambda ! p j ! _p j

a)

\Gamma \Gamma @@

++

Ifflffi fifl oe_s

s- \Phi

\Phi ! p j ! _p j

b)

Fig.1 Contribution of the instantons to the reactions a) N _N ! \Phi fl and b) N _N ! \Phi \Phi . The instanton is denoted as I and crosses mark the quark legs which are connected through vacuum.

11 2Error: /typecheck in --put--
Operand stack:
--nostringval-- --nostringval-- --nostringval-- --nostringval-- --nostringval-- 65.6231 197.849 --nostringval-- 5.9998 5.9998 (2) 0 5.9998
Execution stack:
%interp_exit .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- false 1 %stopped_push 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- --nostringval-- 1 1 0 --nostringval-- %for_pos_int_continue --nostringval-- --nostringval-- (.2) --nostringval-- %string_continue --nostringval-- --nostringval-- --nostringval-- --nostringval--
Dictionary stack:
--dict:1100/1123(ro)(G)-- --dict:0/20(G)-- --dict:74/200(L)-- --dict:120/250(L)-- --dict:16/200(L)-- --dict:88/100(L)--
Current allocation mode is local
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