The strangeness form factors of the proton

The present empirical information on the strangeness form factors indicates that the corresponding $uuds\bar s$ component in the proton is such that the $uuds$ subsystem has the flavor spin symmetry $[4]_{FS}[22]_F[22]_S$ and mixed orbital symmetry $[31]_X$. This $uuds\bar s$ configuration leads to the empirical signs of all the form factors $G_E^s$, $G_M^s$ and $G_A^s$. An analysis with simple quark model wave functions for the preferred configuration shows that the qualitative features of the empirical strangeness form factors may be described with a $\sim$ 15% admixture of $uuds\bar s$ with a compact wave function in the proton. Transition matrix elements between the $uud$ and the $uuds\bar s$ components give significant contributions.

G s M (q 2 ) of the proton is positive [1,2,3,4], while the strangeness electric form factor G s E (q 2 ) [4,5] and the strangeness axial form factor [6] are negative. Here it is noted that there is a unique uudss configuration with at most one quark orbitally excited, which is expected to have the lowest energy, and which leads to the same signs, and for which the constituent quark model provides a good qualitative description of the empirical momentum dependence.
In this configuration thes antiquark is in the ground state, and the uuds subsystem is in the P −state, such that the flavor-spin symmetry of the uuds system is [4] [7,8].
In this configuration the strangeness magnetic moment is positive, and the strangeness contribution to the proton spin is small and negative. This configuration has the lowest energy of all uudss configurations, under the assumption that the hyperfine interaction between the quarks is spin dependent [7]. Calculation of the momentum dependence of the corresponding form factors calls for a wave function model. For a qualitative analysis the harmonic oscillator constituent quark model should do.
In this model the matrix elements of the vector and axial vector current operators lead to the following form factor contributions for the uudss configuration above: Here P ss represents the probability of the uudss component in the proton and m p and m s are the proton and strange quark masses respectively. The oscillator parameter ω will be treated entirely phenomenologically. Note that the q 2 = 0 limits of these form factors are determined by symmetry alone.
In addition to these "diagonal" matrix elements between the uudss states, there will also arise "non-diagonal" matrix elements between the uud and uudss components of the proton. These will depend both on the explicit wave function model and the model for the ss − γ vertices. If these vertices are taken to have the elementary formsv(p ′ )γ µ u(p) and v(p ′ )γ µ γ 5 u(p) and no account is taken of the interaction between the annihilating ss pair and the proton, these transition matrix elements lead to the following form factor contributions (in the Breit frame): Here P uud is the probability of the uud component of the proton. The factor C 35 is the overlap integral of the wave function of the uud and the corresponding component of the uudss configuration. In the oscillator model this factor is Here ω 3 is the oscillator constant for the uud component of the proton. In the case of compact uudss wave function, for which ω ∼ 2ω 3 , the value for C 35 is C 35 ∼ 0.4. The model parameters are the oscillator parameter ω, the probability P ss of the uudss component (here P uud = 1 − P ss ) and the phase factor δ in the non-diagonal contribution. The constituent mass of the strange quark will be taken to be 400 MeV/c 2 .
The non-diagonal contributions also depend on the relative phase δ = ±1 of the uud and uudss components of the wave functions. Below it is shown that a good description of the empirical form factors is obtained with δ = +1.
Most information on the momentum dependence of the strangeness form factors is provided by the G0 experiment [4,9] and indirectly by a combination of extant neutrino scattering data with data on parity violating electron proton scattering [10]. The former gives the momentum dependence of the combination G s E (q 2 ) + ηG s M (q 2 ), where η is a combination of kinematical variables and the ratio of nonstrange form factors [4]. The latter phenomenological combination gives values for all the three form factors G s E (q 2 ), G s M (q 2 ) and G s A (q 2 ), albeit with substantial uncertainty margins.
The empirical values for the strangeness form factors given in refs. [4,9,10] indicate that they all fall slowly with momentum transfer up to q 2 = 1 GeV 2 . This slow falloff indicates that the wave function of the strangeness component is compact relative to the proton radius.
Consider first the strangeness electric form factor shown in Fig.1. The slow falloff with q 2 may be described by taking ω as 1 GeV, which corresponds to a matter radius function The calculated values for G s M obtained with the same parameter values are shown in Fig. 2. The best description of the data is obtained by taking the probability of the uudss component to be P ss in the range 10-15% and the value of the phase factor δ in the nondiagonal contribution (4) to be positive (δ = +1). Here again the slow falloff with q 2 is noteworthy.
[10]. At q 2 = 0 G s A equals the strangeness contribution to the proton spin. The values obtained for that observable with the present parameterization are -0.03 --0.07, which fall within the empirical range of values from 0 to -0.10 [11,12,13].
In Fig.4 the calculated form factor combination G s E (q 2 ) + ηG s M (q 2 ) calculated with this parameterization is compared to the results of the G0 experiment [4]. In this case the overall features of the empirical values are best reproduced with C 35 = 0.4 and P ss = 0.10.
The quality of this comparison with the empirically obtained combination form factor combination G s E + ηG s M is not very sensitive to the precise value of the oscillator parameter ω as long as it is larger than ∼ 0.7 GeV, which corresponds to a radius of ∼ 0.3 fm for the wave function of the uudss component. for both µ s and r s and thus agrees with the current empirical values for both of these observables.