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Magnetic Polarizability of Virtual (ss') and (cc') Pairs in the Nucleon

We suggest 3P0 quantum state of virtual (ss') pairs in the nucleon can be polarised by the internal fields permeating the volume of the nucleon (proton or neutron). Due to the quadratic Zeeman interaction, 3P0 wavefunction of virtual (qq') pairs acquires the admixture of 1P10 quantum state in the magnetic field, which generates the antiparallel polarization of s and s' quarks (in the nucleon). Considering the internal magnetic fields of neutron and proton (originating from their measured magnetic dipole moments), we suggest the induced s-quark polarization in the neutron to be of the oposite direction compared to the proton case. We mention the influence of the internal chromo-magnetic fields on the quantum state of (qq') pairs in the nucleon and we discuss also the expected behaviour of virtual (cc') pairs.


Introduction
Internal structure of the nucleon is an important and challenging question of modern physics. Due to remaining uncertainities in the spin (angular momentum) structure of Proton (only 1/3 of the expected part of the proton spin was found [1] to be directly related to the spin orientations of quarks), the intensive experimetal efforts continue with the goal to determine the spin contribution of gluons (∆G), and quark orbital angular momentum L q . The spin of virtual (ss) pairs may also contribute to the total angular momentum (J =h/2) of proton.
The presence of virtual (ss) pairs in the nucleon is naturally expected within Parton model, and the experimental data from lepton-proton scattering as well as the observed OZI rule violations inpp annihilation [2] suggest the polarization of virtual (ss) pairs [3] in the nucleon.
In this contribution we consider a polarization of 3 P 0 (ss) quantum state, which may occur due to the interaction of quark magnetic (or chromo-magnetic) moments with the oriented internal electromagnetic or gluonic fields in the nucleon.
2. 3 P 0 quantum state in external fields Quantum state of virtual (ss) component in the proton is not fully fixed by theoretical arguments, and several possibilities (e.g. 1 S 0 , 3 S 1 , 3 P 0 or 1 P 1 type of wavefunctions) exist [3]. Experimental data indicate that 1 S 0 and 3 S 1 configurations are unlikely [3], while 3 P 0 state of (ss) pair with vacuum quantum numbers J P C = 0 ++ appears to be a reasonable assumption.
Bound state of real (cc) quark-antiquark pair in 3 P 0 configuration (0 ++ ) is observed as χ c0 meson [4], and the properties of 3 P 0 state in Positronium (e + e − ) are well understood [5]. If we presume the response of (e + e − ) and (qq) bound systems (in 3 P 0 state) to the external field to be similar, the existing knowledge about the Positronium 3 P 0 state behavior in external fields [5] can be used to make the predictions regarding the state of virtual (ss) pairs in the nucleon.
Magnetic moments (µ p , µ n ) of proton and neutron are the source of dipole magnetic field B µ , which is generated within the interior (MIT bag volume) of the nucleon. Compton length of the proton (λ c = h/m p c = 1.3 fm) is comparable to the proton radius (size), and one can imply a dipole field B µ does penetrate the interior (partonic matter) of the nucleon. Direction of internal magnetic field B µ within the proton (or neutron) volume is parallel (or antiparallel) to the spin orientation, and its magnitude is approximately |B µ | ≈ 10 13 T = 10 17 Gauss. This estimate can be obtained at semi-classical level, requiring the electric current I inside the coil of radius R p ≈ 0.82 fm to generate the magnetic dipole of size µ = πR 2 p I ≈ 2µ N . Magnetic field inside such a theoretically considered coil [6] then is:

Magnetic polarizability of 3 P 0 state
Let us consider the wavefunction of bound fermion-antifermion (ff ) pair in 3 P 0 quantum state with J P C = 0 ++ quantum numbers. Using the Positronium notation [7] we have where Y lm are orbital (spherical harmonics) functions, and χ s,sz denote the spin part of 3 P 00 wave function. Now we shall assume that quantum state of the virtual (ss) pairs in the nucleon has a similar (the same) quantum structure as given by Eq.(1). Using the uncoupled spin eigenstates χ 1−1 = | ↓↓ , and χ 1+1 = | ↑↑ , and The probability of antiparallel s,s spins (↑↓ or ↓↑) is 1/3, and parallel orienations (↑↑) and (↓↓) occur with the same probability (1/3). For the net helicities of s ands quarks in such 3 P 00 state we have ∆s = s ↑ − s ↓ = 0 and ∆s =s ↑ −s ↓ = 0, and L s + Ls = 0 also applies. It has been suggested [8] that component Y 11 | ↓↓ in 3 P 00 [ss] wave function (2) is enhanced in the proton, due to strong attraction of the valence u and virtuals quarks in the pseudoscalar channel, which results in the polarization of strange quarks in the nucleon. Magnetic fields can also modify the spin structure of 3 P 00 state. In the Positronium [7] case, 3 P 00 wavefunction acquires the admixture of 1 P 10 state. Eigenstates 3P 00 , 1P 10 , 3P 20 in small magnetic fields are [7]: where admixture parameters ε B = 2µeB ∆E 10 and ε ′ B = 2µeB ∆E 21 depend on the magnetic field B, and on eigenenergy differences ∆E 10 , ∆E 21 . Strange quarks also have their magnetic moment µ s , and therefore, a similar behavior of 3 P 00 [ss] wavefunction in external B field can be anticipated.
[f] Figure 1. Probability of the spin components in 3P 00 state in the magnetic field, as a function of the mixing angles (value φ = 45 • corresponds to Paschen-Back limit of very large fields: B → ∞).
Using 1 P 10 = Y 10 (| ↓↑ − | ↑↓ )/ √ 2, the virtual 3P 00 state of (ss) pair in small magnetic fields is: Relative imbalance of ↑↓ and ↓↑ spin configurations in the wavefunction (6) generates the induced magnetic moment of 3P 00 [ss](B) state. Such behavior corresponds to magnetic polarizability [9]. Energy of the Positronium 3P 00 state decreases in the magnetic field [5], which corresponds to ↑ µ e − ↑ µ e + orientation of e − , e + magnetic moments. In Paschen-Back limit (see Fig. 1), component |↓↑ becomes dominant in 3 P 00 state. Therefore, one can expect enhanced probability of ↑ µs ↑ µs configuration | ↓↑ for the virtual (ss) pairs in the polarized proton as well. The behavior of 3 P 00 Positronium wavefunction in the electric field is well known [7], and our observation (analysis) gives: No polarization of s,s spins occurs, if 3P 00 state is exposed to the electric fields.

Virtual ss and cc pairs in the proton and neutron
If other effects (e.g. pseudoscalar QCD interaction [8]) do not determine the polarization of (qq) pairs in the nucleon, and the polarizability of virtual (ss) pairs by the internal magnetic field is indeed significant, the following qualitative predictions can be inferred from Eq.(6): • Net spin polarization of s quarks in proton is antiparallel to the proton spin.
• Net spin polarization of s quarks in the neutron is parallel to the spin of the neutron.
• Polarization ofs quarks in the nucleon favors to be opposite to the s quark polarization.
• The polarization of c quarks in the nucleon is opposite to the polarization of s quarks.
• The polarization ofc (andū) quarks is parallel to the polarization of s quarks.
The first two statements originate from the internal magnetic dipole field B µ orientation relative to the spin of proton and neutron, and the last two statements come from the fact, that c quarks have positive charge (2/3)e, and thus their magnetic moment is parallel to their spin (contrary to the case of negatively charged s quarks). The third statement applies only if ↑↑ and ↓↓ components of the function (6) have the same probability (QCD interaction [8] is inactive).

Summary and conclusions
We have considered a simple model of 3 P 00 [qq] quantum state polarization of virtual (ss) and (cc) pairs in the nucleon, occuring due to the internal magnetic field B ≈ 10 13 T, which originates from the nucleon magnetic moment. Our estimates are based on the analogy with the behavior of 3 P 00 state of Positronium (e + e − ) in the magnetic field [7]. The angular momentum of virtual (ss) pairs remains zero, while s quarks are polarized oppositely (ands parallel) to the proton spin. The estimated polarizations are: −∆s = ∆s ≈ −0.013, and ∆c = −∆c ≈ +0.002.
However, quarks interact also via strong interactions, and QCD effects [8] can be larger than the electromagnetic interactions we have considered here. Moreover, internal volume of the polarized nucleon presumably contains an oriented chromo-magnetic fields B a [11]. The interaction H int QCD = −µ a q B a of such gluonic fields B a with chromo-magnetic moments µ a q of virtual quarks may induce a stronger polarization of 3 P 00 [qq] state. Our inferences about the (ss) polarization magnitude, the polarization orientation of (cc) pairs, and the opposite polarization ofs,d, andū quarks in neutron (relative to proton case) should thus be taken with caution.