1. On the Problem of Normal Nucleon Fermi-liquid with Fermi pseudopotential and One-Pion- Exchange Potential A. I. Sery, Brest State Pushkin University, Belarus Xth Gomel School July 2009

2. The origin of neutron stars’ magnetic fields can be explained both by the presence of proton superconducting currents, and by spontaneous nucleon spin polarization in the liquid core V. G. Baryshevsky and M. I. Podgoretzky predicted nuclear pseudomagnetism in 1964 After that V. G. Baryshevsky proposed the idea that spin-polarized state of nucleon system can turn out energetically preferrable in comparison with non-polarized one I. e. it’s an example of spontaneous symmetry violation

3. The approach to the problem of spin polarization of nucleons and its link with ordinary magnetic field before and after the idea of V. G. Baryshevsky Before 1960sThe essence of the idea of V. G. Baryshevsky external magnetic field must be present certainly (i. e. it is primary) can be absent spin polarization of nucleons owing to external magnetic field (i. e. it is secondary) can turn out energetically profitable owing to nuclear interaction between nucleons (i. e. it is primary) «polarizational» magnetic field as a correction for the external one in the absence of the external one can be the only one nuclear pseudomagnetic field (NPMF) it hadn’t been taken into consideration at all owing to spin polarization of nucleons; its effect on an individual nucleon energy is much stronger than the effect of ordinary magnetic field

4. The difference between MF and NPMF can be considered by the questions 1. What fundamental interaction is the field caused by? 2. Can it influence an electron, proton, neutron? 3. Can it be created by moving charged particles? 4. Can it exist in nucleon medium without spin polarization? 5. The relationship between field induction B, degree of particles’ polarization р 0 and their concentration N (μ i – particles’ intrins. magn. moments, μ n – intrins. magn. mom. of a moving neutron). 6. Is particle energy quantization (after Landau) in such a field possible? 7. Can it exist in vacuum (i. e. far from sources)? 8. Is it correct to speak about such a field, generated by a single particle (either a one at rest, or a moving one)?

5. The main differences between ordinary magnetic field and nuclear pseudomagnetic field Magnetic fieldNPMF 1 Electromagnetic Nuclear (namely spin dependence of nuclear forces) 2 Yes, because they have intrinsic magnetic moments No (doesn’t come in nuclear interaction); yes; yes 3 Yes (according to Biot-Savart- Laplace law) No, electric charge has absolutely nothing to do here 4 Yes, if there is an ordered motion of protons No, because the presence of spin polarization is the only factor causing NPMF formation

6. The main differences between ordinary magnetic field and NPMF (continued) Magnetic fieldNPMF 5 В ~ Nр 0, namely: В = 4πμ n Nр 0 В ~ Nр 0, namely: В = πħ 2 Nβр 0 /(mμ n ) (in case of NPMF of nucleons, which have spin ½); m – nucleons’ mass, β depends on nuclear scattering amplitudes 6 YesNo 7 Yes, because electromagnetic interaction radius is infinite No, because nuclear interaction radius is finite 8 Yes; even a single neutron at rest, owing to its intrinsic magnetic moment, creates ordinary magnetic field around itself No; an individual nucleon creates a short-range field of nuclear forces around itself, while NPMF is a collective phenomenon: it arises as a result of spatial averaging of nuclear forces’ fields, created by individual nucleons, the averaged component of potentials, which depends on directions of spins, turns out to be nonzero

7. The first papers on the problem of the possibility of spontaneous spin polarization of nucleons appeared in late 1960s, i. e. already after the discovery by V. G. Baryshevsky and M. I. Podgoretzky. The results are very different. Let’s observe the main methods and results for neutrons methodsexamples of models and potentialsspontaneous polarization is possible Fermi-gasa) hard spheres; b) NPMF in the framework of Fermi pseudopotential or OPE potential a) at n n = 0.41 fm – 3 (M. J. Rice, 1969); b) most likely, no (*2006, 2008) Fermi-liquiddifferent Skyrme models at n ~ 0.1  1 fm – 3 (A. A. Isayev et al., 2004) magnetic susceptibility calculation a) Skyrme effective forces; b) Argonne v 18 (two-body), Urbana IX (three-body); c) Stoner criterion for Fermi pseudopotential a) at n = 0.18  0.26 fm – 3 (A. Viduarre et al., 1984); b) most likely, no (S. Fantoni et al., 2001); c) most likely, no (*2006) variants of relativistic Hartree-Fock (HF) Dirac-HF with effective nucleon- meson Lagrangian at n ~ 10 38  10 39 cm – 3 (S. Marcos et al., 1991) Brueckner-HF with Nijmegen II, Nijmegen NSC97e and Reid93 most likely, no (I. Vidana et al., 2002)

8. And now let’s observe the main results for neutron-proton matter methodexamples of models and potentials spontaneous polarization is possible Fermi-gasNPMF in the framework of Fermi pseudopotential or OPE potential at n n ~ n p ~ 10 35 cm – 3 (*2006); in the 2d case it can even be  - equlibrium (n p ~ 10 37 cm – 3, n p ~ 10 34 cm – 3 ) (**2008, 2009) Fermi-liquiddifferent Skyrme models at n ~ 10 38  10 39 cm – 3 (A. A. Isayev et al., 2004) magnetic susceptibility calculation a) Skyrme effective forces; b) Stoner criterion for Fermi pseudopotential a) at n ~ 10 38  10 39 cm – 3 (A. Viduarre et al., 1984); b) at n n ~ n p ~ 10 35 cm – 3 (*2006) relativistic Hartree-Fock (HF) a) Dirac-HF to strongly asymmetric matter; b) Brueckner-HF with Nijmegen II, Nijmegen NSC97e and Reid93 a) for protons if neutron- proton spin interaction exceeds some threshold value; b) most likely, no (I. Vidana et al., 2002)

9. For example, let’s see the results for beta-equlibrium polarization with oppositely directed spins of protons and neutrons in the framework of Fermi- gas approach with NPMF in case of OPE potential

10. Fermi-liquid can be either normal or superfluid. We’ll consider a normal one. Similarity of description for ideal Fermi-gas and Fermi-liquid: а) distribution functions formally have the same appearance: n(p) = ((E(p) - µ)(kT) -1 + 1) -1, but for Fermi-liquid the expression for E(p) is not known beforehand; б) Fermi momentum is determined by the same formula (according to Landau theory, 1956) p F = ħ(3  2 ) 1/3 (N/V) 1/3

11. The main differences for ideal Fermi-gas and Fermi-liquid Fermi-gas Particle Interaction function is zero Particle energy doesn’t depend on momentum distribution of other particles Fermi-liquid Quasiparticle Interaction function is nonzero Quasiparticle energy depends on momentum distribution of other quasiparticles

12. A considerable contribution to working out Fermi-liquid theory was made by Landau, Akhiezer, Peletminsky and others А. А. Isayev elaborated the algorithms of calculation of polarization conditions for normal neutron-proton (and simply neutron) Fermi-liquid ignoring beta-equilibrium in the framework of Skyrme model, where a quadratic dependence on relative momenta of 2 nucleons takes place for the Fourier transforms of nucleon-nucleon potentials However, V.G.Baryshevsky and V.V. Tikhomirov proposed to apply a superposition of Fermi pseudopotential and one-pion-exchange potential

13. Why just so? Fermi pseudopotential is very simple (delta-function, multiplied by a constant containing scattering length), and it was specially constructed to explain cross-sections of low-energy nucleon-nucleon scattering OPE potential at distances of 2-4 fm gives practically the same results, as does Hamada-Johnson potential, which regards one the most detailed and precise for nucleon-nucleon scattering description But it contains not quadratic, but more complex dependence on momenta, as a result – integrating becomes more difficult Besides, the approach assumes the absence of tensor terms in potentials, and the absence of the dependence on orbital momentum also

14. A positive result for Fermi-gas method with the same potentials allows to hope, that it will be similar for Fermi-liquid method

15. The main points of the method (see the details in posters) Neglecting the tensor part of OPE potential, because averaging over angles gives zero Expanding the Fourier transform of the potential over Pauli matrices Constructing normal Fermi-liquid amplitudes Writing expressions for 2 or 4 components of normal distribution and for normalization condition Obtaining of self-consistent system of equations Substitution of integration for summation

16. There are 4 equations and 6 unknown quantities in self- consistent system for neutron liquid (the dimensions of matrices and so-called «vectors» are 2 times larger for the neutron-proton one)

17. Temperature T can be taken as free parameter

18. There are 8 equations and 11 unknown quantities in self-consistent system for neutron-proton liquid Temperature and isospin asymmetry parameter can be taken as free parameters Then total concentration, 2 degrees of polarization, 2 chem. potentials, mathem. expectations of the squares of the transferred momenta over 4 components of Fermi-Dirac distribution function remain for neutron- proton system I’m sorry, but I had no enough time to make precise calculations; though preliminary estimates show, that if at n n ~ 10 2 n p polarization really exists, than, most likely, the spins of protons and neutrons are oppositely directed (the result is qualitatively similar to the one obtained in Fermi-gas method), besides neutron polarization is negligibly small, and protons are almost totally polarized; if it is really so, then ordinary magnetic field, generated by protons, can reach the order of 10 14 Gs in pulsars or magnetars

19. The main conclusions if we apply Fermi pseudopotential or OPE potential then the threshold density for spontaneous neutron-proton system polarization (ignoring beta-equilibrium) is about 3 orders lower (0.0001-0.001 fm -3 ) in case of Fermi-gas method with NPMF or in case of magnetic susceptibility calculation in comparison with any other method predicting nuclear ferromagnetism (0.1-1 fm -3 ) there is only one type of polarization predicted for these potentials – when proton spins are directed oppositely to neutron spins (though co-directed spins are predicted for some Skyrme models in the Fermi-liquid approach) these potentials and methods do not predict spontaneous polarization for pure neutron matter (though some Fermi- liquid methods with Skyrme models do) there is a hope that the application of these potentials to Fermi-liquid method is going to give similar results (the corresponding self-consistent systems of equations have been already obtained)

20. Thank you for your attention !