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Progress in Particle and Nuclear Physics 61 (2008) 198204www.elsevier.com/locate/ppnp

Review

Strange quark content in the nucleon

S. Dubnickaa,, A.Z. Dubnickovaba Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovak Republic

bDepartment of Theoretical Physics, Comenius Univ., Bratislava, Slovak Republic

Abstract

Though none of the experimental evidences for the strange quark contributions to nucleon properties is explained convincinglyby an alternative, the recent experiments, even HAPPEX Collab. and A4 Collab., on a measurement of the parity-violatingasymmetries show no strangeness in the proton. Despite this conclusion we demonstrate here no accidental compatibility of ourtheoretical predictions for nucleon strange form factors with some nonzero parity violation experimental results which strengthensour belief in the strangeness in the nucleon.c 2008 Elsevier B.V. All rights reserved.

1. Introduction

In the frame of the naive quark model nucleons are composed only from up and down valence quarks. Such anidea explains well the static properties of nucleons, however, experimental results on piN sigma term, proton spin,OZI rule violation and neutrino elastic scattering on protons indicate, that the naive quark model is not complete, andmoreover, the sea strange quarkantiquark pairs can make significant contributions to nucleon structure. The lattercan be studied by consideration of the nucleon matrix elements from scalar, pseudoscalar, vector, axial vector andtensor strange quark currents. In this contribution, however, we are restricted only to the vector strange quark currentJ s = ss, which is experimentally accessible in parity-violating elastic and quasi-elastic electron scattering fromthe proton and light atomic nuclei, where the strange electric and the strange magnetic nucleon form factors (FFs) (ortheir combinations) are measured.

Recently, an analysis of the complete world set of parity-violating electron scattering data up to t = Q2 =0.3 GeV2 demonstrates [1] the strange nucleon FFs to be consistent with zero. Despite the latter conclusion, wepresent here our theoretical predictions for a behavior of the nucleon strange electric and magnetic FFs, exploitingthe idea of Jaffe [2] and our unitary and analytic approach, which are not accidentally compatible with nonzeroexperimental values [36] of the strangeness within the proton as they are obtained on the strength of the sophisticatedbackground. According to Jaffes idea the strange electric and the strange magnetic nucleon FFs can be found fromnucleon electromagnetic (EM) FFs to be obtained in the analysis of existing nucleon EM FF data by means of theUnitary and Analytic (U&A) model [7] of the nucleon EM structure. Just as a result of the isospin I = 0 value of

Corresponding author.E-mail address: dubnickova@fmph.uniba.sk (S. Dubnicka).

0146-6410/$ - see front matter c 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.ppnp.2007.12.010

S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204 199

the strange quark, the strange Dirac and Pauli nucleon FFs contribute only to the behavior of the isoscalar parts of theDirac and Pauli nucleon EM FFs and in no case to the isovector part ones. As a result there are no proton or neutron,but only common nucleon strange FFs.

From the latter it follows, that by a similar procedure to a determination of nucleon strange FFs one can determinea behavior of the strange FFs of any other hadron (even those possessing the strange valence quarks, like kaons andhyperons) the EM structure of which is described by nonzero isoscalar FFs.

Till now we have realized it practically for kaons [8]. So, there cannot be any strangeness within the charged pionsas their EM structure is completely described by the pure isovector EM FF.

Another note is concerned with the asymptotic behavior of the EM and strange FFs. In our opinion there is no reasonto construct different asymptotics of strange hadron FFs from the asymptotics of EM FFs as the great discovery of the70s of the last century [9,10] to be confirmed experimentally, refers to the FFs asymptotics and the number of valencequarks within the considered hadron.

In the next section we briefly review formalism and vectormeson coupling constant ratios prediction. TheSection 3 is devoted to the prediction of the strange nucleon FFs behaviors. Conclusions are drawn in the last section.

2. Formalism and vectormeson coupling constant ratios prediction

The momentum dependence of the nucleon matrix element of the strange quark vector current J s = ss iscontained in the strange Dirac F s1 (t) and Pauli F

s2 (t) nucleon FFs

N |ss|N = u(p)[F

s1 (t)+ i

q

2mNF s2 (t)

]u(p) (1)

by means of which the strange electric and strange magnetic nucleon FFs are defined

GsE (t) = F s1 (t)+t

4m2NF s2 (t), G

sM (t) = F s1 (t)+ F s2 (t) (2)

and to be measured in parity-violating elastic and quasi-elastic scattering of electrons on protons and light atomicnuclei.

The main idea of a prediction of strange nucleon FFs behaviors from the known isoscalar parts of the Dirac andPauli nucleon EM FFs is based on two assumptions: the mixing is also valid for coupling constants between EM current (the strong strange quark current as well)and vectormeson

1f

= 1f0

cos 1f0

sin ; 1f

= 1f0

sin + 1f0

cos , (3)

where = 3.70 is a deviation from the ideal mixing angle 0 = 35.30. the quark current of some flavor couples with universal strength exclusively to the vectormeson wave functioncomponent of the same flavor

0|qr qr |(qtqt )V = m2V r t, (4)where mV and are the mass and the polarization vector of the considered vectormeson.Starting from a definition of the virtual-photon vectormeson transition coupling constants 1/ f eV by the relation

0|J e|V =m2Vf eV

(5)

and the second assumption for the isoscalar EM current J I=0 to be expressed by quark fields, one comes to theequations

0|J I=0 |0 = 0|16(uu + dd) 13 ss|

12(|uu + |dd)

= 16

(12+ 1

2

)m20

m20f e0

(6)

200 S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204

0|J I=0 |0 = 0|16(uu + dd) 13 ss|ss

= 13m20

m20f e0

(7)

from where expressions for EM coupling constants follow

1f e0

= 16

(12+ 1

2

) = 1

6

13; 1

f e0= 1

3 = 1

6

23. (8)

Substituting the latter into the first assumption, together with identities 13= sin 0 and

23 = cos 0, one obtains

coupling constants of real and

1f e

= 6sin( + 0); 1f e

= 6cos( + 0). (9)

These relations, together with 1f e =12 following from

0|J I=1 | = 0|12(uu dd)| 1

2(|uu |dd)

= 12

(12+ 1

2

)m2

m2f e

, (10)

give for the ratios of the universal vectormeson coupling constants the values 1f e :1f e: 1f e = 0.71 : 0.25 : (0.32) in

a very good agreement with experimental values 1f e :1f e: 1f e = 0.79 : 0.23 : (0.31) obtained from leptonic widths

(V e+e) of considered vectormesons. Just this agreement demonstrates the previous two assumptions to becompatible with physical reality and one can extend their validity also for strong strange quark current vectormesontransition coupling constants 1/ f sV .

Then analogically one can write for the strong strange quark current the equations

0|J s|0 = 0|(ss

12(|uu + |dd) = 0 m

20

f s0 (11)

0|J s|0 = 0|(ss

|ss = 1.m20 m20f s0 (12)from where one gets 1f s0

= 0 and 1f s0 = 1. . Substituting them into mixing relations one comes to the strangecoupling constants of the real and

1f s

= sin ; 1f s

= + cos . (13)

Bringing these expressions for and vector mesons into ratios with EM coupling constants (9), respectively, onecan get rid of the unknown parameter and comes to the relations

( f (i)NN/ fs) =

6

sin sin( + 0) ( f

(i)NN/ f

e)

( f (i)NN/ fs ) =

6

cos cos( + 0) ( f

(i)NN/ f

e ) (i = 1, 2) (14)

giving a possibility to calculate the unknown strange coupling constant ratios (parameters of strange nucleon FFs)from the known EM coupling constant ratios (parameters of EM nucleon FFs) to be determined in a description of allexisting nucleon EM FF data by a suitable model of the EM structure of nucleons.

S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204 201

The derived relations (14) are also valid for any pairs of excited states ,; ,; etc. of the ground state of and isoscalar vectormesons.

3. Prediction of strange nucleon FFs behaviors

For this aim we will use U&A model of the nucleon EM structure [7], which comprises all known nucleon FFproperties to be contained in the following models of isoscalar parts of Dirac and Pauli nucleon EM FFs

F I=01 [V (t)] =(1 V 21 V 2N

)4 {12L(V)L(V)+

[L(V)L(V)

(C C)(C C)

L(V)L(V) (C C)(C C) L(V

)L(V)

]( f (1)NN/ f

e)

+[L(V)L(V)

(C C)(C C)

L(V)L(V) (C C)(C C) L(V

)L(V)

]( f (1)NN/ f

e )

}(15)

F I=02 [V (t)] =(1 V 21 V 2N

)6 {L(V)L(V)L(V)

[1 C

(C C)((C C)

C (C C)

C

)] ( f (2)NN/ f e)+ L(V)L(V)L(V)[1 C

(C C)((C C)

C (C C)

C

)]( f (2)NN/ f

e )

}(16)

and of the Dirac and Pauli strange nucleon FFs

F s1 [V (t)] =(1 V 21 V 2N

)4 {[L(V)L(V)

(C C)(C C) L(V

)L(V)(C C)(C C)

L(V)L(V)]( f (1)NN/ f

s)+

[L(V)L(V)

(C C)(C C)

L(V)L(V) (C C)(C C) L(V

)L(V)

]( f (1)NN/ f

s )

}(17)

F s2 [V (t)] =(1 V 21 V 2N

)6 {L(V)L(V)L(V)

[1 C

(C C)((C C)

C (C C)

C

)] ( f (2)NN/ f s)+ L(V)L(V)L(V){[

1 C(C C)

((C C)

C (C C)

C

)]( f (2)NN/ f

s )

}}(18)

where

L(Vr ) = (VN Vr )(VN Vr )(VN 1/Vr )(VN 1/V r )

(V Vr )(V V r )(V 1/Vr )(V 1/V r ),

Cr = (VN Vr )(VN Vr )(VN 1/Vr )(VN 1/V r )

(Vr 1/Vr )(V r 1/V r ),

VN = V (t)|t=0; Vr = V (t)|t=(mrir /2)2; (r = , , , ),

V (t) = i

[ tN Nt I=00

t I=00]1/2 + [ tt I=00

t I=00]1/2

[ tN Nt I=00

t I=00]1/2 [ tt I=00

t I=00]1/2

[ tN Nt I=00t I=00

]1/2 + [ tt I=00t I=00

]1/2 +[ tN Nt I=00

t I=00]1/2 [ tt I=00

t I=00]1/2

(19)

tN N = 4m2N is a square-root branch point corresponding to N N threshold.

202 S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204

Fig. 1. Predicted strange electric and magnetic nucleon FFs behaviors.

Fig. 2. Predicted behaviors of the strange nucleon FFs in the time-like region.

The expressions (15) and (16) for F I=01 , FI=02 , respectively, together with similar expressions for F

I=11 , F

I=12 [7]

were used for a description of all solid nucleon EM FF data and a reasonable description of them has been achieved.The found numerical values of ( f (i)NN/ f

e), ( f

(i)NN/ f

e ) i = 1, 2 in the relations (15) and (16) are used to calculate the

unknown strange coupling constant ratios ( f (i)NN/ fs), ( f

(i)NN/ f

s ) i = 1, 2 in (17) and (18) by means of the relations

(14). As a result the behaviors of the strange nucleon FFs are predicted (see Fig. 1) without the use of any experimentalpoint obtained in parity-violating elastic and quasi-elastic scattering of electrons on protons and light atomic nuclei.

Due to the fact, that expressions (17) and (18) for the strange nucleon FFs represent also the U&A model and so,both FFs are analytic functions for < t < +, naturally we are predicting the time-like behavior of strangenucleon FFs (see Fig. 2), though there is no known method of their experimental determination until now.

Moreover, as the U&A model represents compatible unification of the pole and continuum (given by cuts onthe positive real axis of the analytic strange nucleon FFs) contributions, one can predict even the imaginary partsbehaviors of the strange nucleon FFs (see Fig. 3) to be given by unitarity conditions of FFs under consideration.

Since the first results reported by the SAMPLE Collab. in 1997 [11], cca ten independent measurements of theparity-violating contribution to the elastic EM FFs of the nucleons have now been completed. But only four of them[36] declare clearly nonzero experimental values of the strangeness within the proton and our theoretical predictionsare compatible with them.

S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204 203

Fig. 3. Imaginary parts of the complex electric and magnetic strange nucleon FFs.

Fig. 4. Compatibility of our theoretically predicted curve with G0 Collab. data.

The A4 Collab. [3] result at Q2 = 0.230 GeV2 is GsE + 0.225GsM = 0.039 0.034 to be in accordance with ourprediction 0.055; the SAMPLE Collab. [4] result at Q2 = 0.1 GeV2 is GsM = 0.37 0.34 also to be in accordancewith our result 0.18; another A4 Collab. [6] result at Q2 = 0.108 GeV2 is GsE + 0.225GsM = 0.039 0.034again to be in accordance with our theoretical prediction 0.030. Maybe the most impressive is a compatibility ofour theoretically predicted curve (see Fig. 4) with the recent data [5] obtained by G0 Collab. on the combinationGsE (Q

2) + (Q2)GsM (Q2) for the interval 0.12 GeV2 < Q2 < 1.0 GeV2 of momentum transfer squared values. Ofcourse, we did not reproduce the remarkable structure in the data around Q2 = 0.2 GeV2, which, however, cannotbe comprehended theoretically as all data are located in the space-like region, far away from any singularities of thestrange nucleon FFs to be able to cause similar structure in the behavior of FFs.

4. Conclusions

Some evidences are roughly reviewed, which indicate the naive quark model of hadrons not to be complete anda real existence of nonzero admixture of sea strange quark ss pairs to the nucleon structure is justified. Especially,

204 S. Dubnicka, A.Z. Dubnickova / Progress in Particle and Nuclear Physics 61 (2008) 198204

the nonaccidental compatibility of our theoretical predictions for nucleon strange FFs with some parity violationexperimental results strengthen our belief in the strangeness in the nucleon.

Acknowledgement

The work was supported in part by the Slovak Grant Agency for Sciences, Gr. No 2/7116/27.

References

[1] R.D. Young, et al., Phys. Rev. Lett. 97 (2006) 102002.[2] R.L. Jaffe, Phys. Lett. B 229 (1989) 275.[3] F.E. Maas, et al., [A4 Collab.], Phys. Rev. Lett. 93 (2004) 022002.[4] D.T. Spayde, et al., [SAMPLE Collab.], Phys. Lett. B 583 (2004) 79.[5] D.S. Armstrong, et al., [G0 Collab.], Phys. Rev. Lett. 95 (2005) 092001.[6] F.E. Maas, et al., [A4 Collab.], Phys. Rev. Lett. 94 (2005) 152001.[7] S. Dubnicka, A.Z. Dubnickova, P. Weisenpacher, J. Phys. G 29 (2003) 405.[8] S. Dubnicka, A.Z. Dubnickova, J. Phys. G 28 (2002) 2137.[9] V.A. Matveev, R.M. Muradyan, A.N. Tavkhelidze, Lett. Nuovo Cimento 7 (1973) 719.[10] S.J. Brodsky, G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153.[11] B. Mueller, et al., [SAMPLE Collab.], Phys. Rev. Lett. 78 (1997) 3824.

Strange quark content in the nucleonIntroductionFormalism and vector--meson coupling constant ratios' predictionPrediction of strange nucleon FFs behaviorsConclusionsAcknowledgementReferences