Polarizability of the nucleon

For nucleons , polarizability is one of the fundamental structural constants, alongside mass , electrical charge , spin and magnetic moment .

The electrical polarizability is a measure of the displaceability of positive relative to negative charge in atoms and molecules when an external electric field is applied . Since an electric dipole moment is induced, one speaks of displacement polarization . The higher the polarizability, the easier it is to induce a dipole moment by an electric field.

A distinction is also made between paramagnetic polarizability, diamagnetic polarizability and ferromagnetism . The latter is found in solids , e.g. B. in iron and rare earths .

Overview

The proposal to measure the polarizabilities of the nucleon was first made in the 1950s. Two experimental options were considered: On the one hand, Compton scattering on the proton and, on the other hand, the scattering of slow neutrons in the Coulomb field of heavy atomic nuclei. The idea was that the pion cloud, under the influence of an electric field vector, receives an electric dipole moment that is proportional to the electric polarizability. After the discovery of the photoexcitation of the Δ resonance , it became evident that the nucleon must also have a strong paramagnetism, which is based on a virtual spin-flip transition of one of the constituent quarks. This is excited by the magnetic field vector of a real photon in a Compton scattering experiment. However, the experiments showed that the expected strong paramagnetism is absent.

Obviously there is a strong diamagnetism which compensates for the paramagnetism. Although this explanation is very obvious, it was unclear for a very long time how diamagnetism can be explained using the structure of the nucleon. A solution to the problem was only found when it was shown that diamagnetism is a property of the structure of constituent quarks. In retrospect, this explanation is actually not surprising, because the constituent quarks get their mass mainly through interaction with the QCD vacuum by exchanging a σ meson with it. This mechanism is predicted by the linear σ model at the Quark level (QLLσM), which also predicts a mass of m _σ  = 666 MeV for the σ meson . The σ-meson has the possibility to interact with two photons that are in the state of parallel linear polarization. As will be shown in the following, the σ meson, as part of the structure of the constituent quarks, contributes to Compton scattering and, as a result, generates most of the electrical polarizability and all of the diamagnetic polarizability.

Definition of the electromagnetic polarizabilities

A nucleon in an electric field receives an electric dipole moment and in a magnetic field a magnetic dipole moment : ${\displaystyle {\vec {E}}\epsilon _{0}={\vec {D}}}$ ${\displaystyle {\vec {d}}}$ ${\displaystyle {\vec {H}}\mu _{0}={\vec {B}}}$ ${\displaystyle {\vec {m}}}$

${\displaystyle {\vec {d}}\,\,=4\pi \,\alpha \,{\vec {D}}}$

${\displaystyle {\vec {m}}=4\pi \,\beta \,{\vec {H}}.}$

The constants of proportionality and (sometimes abbreviated) are called electrical and magnetic polarizability, respectively, and are measured in units of volume ; H. in units of fm ³ (1 fm = 10 ⁻¹⁵  m). ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \alpha _{m}}$

This is based on a system of units in which the elementary charge is given by (cf. fine structure constant ). The factor takes into account that the polarizabilities were defined in the Gaussian system of measurement . ${\displaystyle e}$ ${\displaystyle e^{2}/4\pi =\alpha _{e}=1/137{,}04}$ ${\displaystyle 4\pi }$

The polarizabilities can be understood as a measure of the reaction of the nucleon structure to the fields made available by a real or virtual photon. It is evident that a second photon is needed to measure the polarizabilities. This can be through the relationship

${\displaystyle \delta W=-{\frac {1}{2}}\,({\vec {d}}\,{\vec {D}}+\,{\vec {m}}\,{\vec {H}})}$

in which means the change in energy that occurs when the nucleon is present in the field. ${\displaystyle \delta W}$

Modes of two-photon reactions and experimental methods

Static electric fields of sufficient strength are made available by the coulomb field of heavy nuclei. Therefore, the electrical polarizability of the neutron can be measured by the scattering of slow neutrons in the electrical field of the Pb nucleus. The neutron has no electrical charge. Therefore, the simultaneous interaction of two electric field vectors (two virtual photons) is required in order to deflect the neutron. The electrical polarizability can then be determined from the differential effective cross section at small deflection angles. Another possibility is the Compton scattering of real photons at the nucleon, whereby two electric and two magnetic field vectors interact with the nucleon at the same time during the scattering process. ${\displaystyle {\vec {E}}}$

As shown above, two photons are required for the experimental measurement of the polarizabilities of the nucleon, which simultaneously interact with the electrical components of the nucleon. These photons can be linearly polarized in parallel or perpendicular planes. These two modes in accordance with the measurement of the polarizabilities , or the spin polarizabilities . The spin polarizabilities are only different from zero if the particle has a spin. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$

In total, these experimental options provide six combinations of two electric and two magnetic field vectors. These are described in the following two equations:

• Photons in parallel planes of linear polarization:

${\displaystyle ({\text{Fall}}\,1)\,\,\alpha :\,\,\,\,{\vec {E}}\uparrow \uparrow {\vec {E}}'\quad \quad ({\text{Fall}}\,2)\,\,\beta :\,{\vec {H}}\rightarrow \rightarrow {\vec {H}}'\quad \,({\text{Fall}}\,\,3)\,\,-\beta :\,{\vec {H}}\rightarrow \leftarrow {\vec {H}}'}$

• Photons in perpendicular planes of linear polarization:

${\displaystyle ({\text{Fall}}\,\,4)\,\,\gamma _{E}:{\vec {E}}\uparrow \rightarrow {\vec {E}}'\quad \,\,({\text{Fall}}\,\,5)\,\,\gamma _{H}:{\vec {H}}\rightarrow \downarrow {\vec {H}}'\quad \,\,({\text{Fall}}\,\,6)\,\,-\gamma _{H}:{\vec {H}}\rightarrow \uparrow {\vec {H}}'}$

Case (1) corresponds to the measurement of the electrical polarizability by means of two electrical field vectors . These parallel electric field vectors can either be made available as longitudinal photons from the Coulomb field of a heavy nucleus, or by Compton scattering in the forward direction, or by reflection of the photon by 180 °. Real photons simultaneously provide transverse electrical and magnetic field vectors. This means that in a Compton scattering experiment, linear combinations of electrical and magnetic polarizabilities and linear combinations of electrical and magnetic spin polarizabilities are measured. The combination of case (1) and case (2) leads to the measurement of and is observed in the Compton scattering in the forward direction. The combination of case (1) and case (3) leads to the measurement of and is observed with the Compton scattering in the backward direction. The combination of case (4) and case (5) leads to the measurement of and is observed in the Compton scattering in the forward direction. The combination of case (4) and case (6) leads to the measurement of and is observed with the Compton scattering in the backward direction. ${\displaystyle \alpha }$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {H}}}$ ${\displaystyle \alpha +\beta }$ ${\displaystyle \alpha -\beta }$ ${\displaystyle \gamma _{0}\equiv \gamma _{E}+\gamma _{H}}$ ${\displaystyle \gamma _{\pi }\equiv \gamma _{E}-\gamma _{H}}$

Compton scattering experiments exactly in the forward direction and exactly in the backward direction are not possible for technical reasons. Therefore, the corresponding measured variables must be taken from Compton scattering experiments at medium angles.

Experimental results

The experimental polarizabilities of the proton and the neutron can be summarized as follows: ${\displaystyle (p)}$ ${\displaystyle (n)}$

${\displaystyle \alpha _{p}=12{,}0\pm 0{,}6,\quad \beta _{p}=1{,}9\mp 0{,}6,\quad \alpha _{n}=12{,}5\pm 1{,}7,\quad \beta _{n}=2{,}7\mp 1{,}8\,{\text{ in Einheiten von}}\,\,10^{-4}\,{\rm {fm}}^{3}}$.

The spin polarizabilities of the proton and the neutron are

${\displaystyle \gamma _{\pi }^{(p)}=-36{,}4\pm 1{,}5,\quad \gamma _{\pi }^{(n)}=58{,}6\pm 4{,}0\,{\text{ in Einheiten von}}\,\,10^{-4}\,{\rm {fm}}^{4}}$.

The experimental polarizabilities of the proton were determined as the mean value of the results of a large number of Compton scattering experiments. The experimental electrical polarizability of the neutron is the mean value of a result of electromagnetic scattering of neutrons in the Coulomb field of a lead nucleus and a Compton scattering experiment on a quasi-free neutron, i.e. H. a neutron that is replaced by a deuteron during the scattering process. The two results are:

${\displaystyle \alpha _{n}=12{,}6\pm 2{,}5}$ from the electromagnetic scattering of slow neutrons in the electric field of a Pb nucleus, and

${\displaystyle \alpha _{n}=12{,}5\pm 2{,}3}$ from the quasi-free Compton scattering on a neutron that was initially bound in a deuteron.

The mean value given above was calculated from these two results.

There are also ongoing experiments at Lund University (Sweden) in which the electrical polarizability of the neutron is determined by Compton scattering on the deuteron .

Calculation of the polarizabilities

Recently, great advances have been made in breaking down the total photoabsorption cross-section into parts, which are differentiated according to the spin, the isospin and the parity of the intermediate state. This decomposition is based on the meson photoproduction amplitudes from Drechsel et al. The spin of the intermediate state can be or is, depending on the spin directions of the photon and nucleon in the initial state. The parity change during the transition from the basic state to the intermediate state is for the multipoles and for the multipoles . If one now calculates the partial cross sections for the photo absorption from the photomeson data, the following sum rules can be evaluated: ${\displaystyle s=1/2}$ ${\displaystyle s=3/2}$ ${\displaystyle \Delta P={\text{yes}}}$ ${\displaystyle E1,\,M2,\ldots }$ ${\displaystyle \Delta P={\text{no}}}$ ${\displaystyle M1,\,E2,\ldots }$

${\displaystyle \alpha +\beta ={\frac {1}{2\pi ^{2}}}\int _{\omega _{0}}^{\infty }{\frac {\sigma _{\rm {tot}}(\omega )}{\omega ^{2}}}d\omega }$,

${\displaystyle \alpha -\beta ={\frac {1}{2\pi ^{2}}}\int _{\omega _{0}}^{\infty }{\sqrt {1+{\frac {2\omega }{m}}}}\left[\sigma (\omega ,E1,M2,\ldots )-\sigma (\omega ,M1,E2,\ldots )\right]{\frac {d\omega }{\omega ^{2}}}+(\alpha -\beta )^{t}}$,

${\displaystyle \gamma _{0}=-{\frac {1}{4\pi ^{2}}}\int _{\omega _{0}}^{\infty }{\frac {\sigma _{3/2}(\omega )-\sigma _{1/2}(\omega )}{\omega ^{3}}}d\omega }$,

${\displaystyle \gamma _{\pi }={\frac {1}{4\pi ^{2}}}\int _{\omega _{0}}^{\infty }{\sqrt {1+{\frac {2\omega }{m}}}}\left(1+{\frac {\omega }{m}}\right)\sum _{n}P_{n}[\sigma _{3/2}^{n}(\omega )-\sigma _{1/2}^{n}(\omega )]{\frac {d\omega }{\omega ^{3}}}+\gamma _{\pi }^{t}}$,

${\displaystyle P_{n}=-1\,{\text{bei}}\,E1,M2,\ldots \,{\text{Multipolen und}}\,P_{n}=+1\,{\text{bei}}\,M1,E2,\ldots \,{\text{Multipolen}}}$.

${\displaystyle (\alpha -\beta )^{t}={\frac {1}{2\pi }}\left[{\frac {g_{\sigma NN}{M}(\sigma \to \gamma \gamma )}{m_{\sigma }^{2}}}+{\frac {g_{f_{0}NN}{M}(f_{0}\to \gamma \gamma )}{m_{f_{0}}^{2}}}+{\frac {g_{a_{0}NN}{M}(a_{0}\to \gamma \gamma )}{m_{a_{0}}^{2}}}\tau _{3}\right]}$,

${\displaystyle \gamma _{\pi }^{t}={\frac {1}{2\pi m}}\left[{\frac {g_{\pi NN}{M}(\pi ^{0}\to \gamma \gamma )}{m_{\pi ^{0}}^{2}}}\tau _{3}+{\frac {g_{\eta NN}{M}(f_{0}\to \gamma \gamma )}{m_{\eta }^{2}}}+{\frac {g_{\eta 'NN}{M}(a_{0}\to \gamma \gamma )}{m_{\eta '}^{2}}}\right]}$,

where is the photon energy in the laboratory system. ${\displaystyle \omega }$

The sum rules for and depend only on the degrees of freedom (excited states) of the nucleon, while the sum rules for and about the terms or have to be supplemented. These terms are the -channel contributions. They can be understood as contributions of the scalar and pseudoscalar mesons, which are present as components of the constituent quarks in the nucleon. The sum rule for depends on the total photo-absorption cross-section and therefore does not require a quantum number breakdown. The sum rule for requires a decomposition with regard to the parity change of the transition into the intermediate state. The sum rule for requires a decomposition with regard to the spin of the intermediate state. The sum rule for requires a decomposition with regard to the spin and the parity change. The -channel contributions depend on the scalar and pseudoscalar mesons, which (i) are part of the structure of the constituent quarks and (ii) can couple to two photons. These are the mesons , and in the case of , and the mesons , and in the case of . By far the largest shares are contributed by the σ- and the - meson, while the other mesons only provide small corrections. ${\displaystyle \alpha +\beta }$ ${\displaystyle \gamma _{0}}$ ${\displaystyle \alpha -\beta }$ ${\displaystyle \gamma _{\pi }}$ ${\displaystyle (\alpha -\beta )^{t}}$ ${\displaystyle \gamma _{\pi }^{t}}$ ${\displaystyle t}$ ${\displaystyle \alpha +\beta }$ ${\displaystyle \alpha -\beta }$ ${\displaystyle \gamma _{0}}$ ${\displaystyle \gamma _{\pi }}$ ${\displaystyle t}$ ${\displaystyle \sigma (600)}$ ${\displaystyle f_{0}(980)}$ ${\displaystyle a_{0}(980)}$ ${\displaystyle (\alpha -\beta )^{t}}$ ${\displaystyle \pi ^{0}}$ ${\displaystyle \eta }$ ${\displaystyle \eta '}$ ${\displaystyle \gamma _{\pi }^{t}}$ ${\displaystyle \pi ^{0}}$

Results of the calculations

The results of the calculations are summarized in the following eight equations:

${\displaystyle \alpha _{p}=\,+4{,}5\,{\text{(nucleon)}}+7{,}5\,{\text{(const. quark)}}=+12{,}0}$

${\displaystyle \beta _{p}=\,+9{,}4\,{\text{(nucleon)}}-7{,}5\,{\text{(const. quark)}}=\,+1{,}9}$

${\displaystyle \alpha _{n}=\,+5{,}1\,{\text{(nucleon)}}+8{,}3\,{\text{(const. quark)}}=+13{,}4}$

${\displaystyle \beta _{n}=+10{,}1\,{\text{(nucleon)}}-8{,}3\,{\text{(const. quark)}}=\,+1{,}8\,{\text{in Einheiten von}}\,10^{-4}\ {\rm {fm}}^{3}}$

${\displaystyle \gamma _{0}^{(p)}=\,-0{,}58\pm 0{,}20\,{\text{(nucleon)}}}$

${\displaystyle \gamma _{0}^{(n)}=\,+0{,}38\pm 0{,}22\,{\text{(nucleon)}}}$

${\displaystyle \gamma _{\pi }^{(p)}=\,+8.5\,{\text{(nucleon)}}-45{,}1\,{\text{(const. quark)}}=-36{,}6}$

${\displaystyle \gamma _{\pi }^{(n)}=+10{,}0\,{\text{(nucleon)}}+48{,}3\,{\text{(const. quark)}}=+58{,}3\,{\text{in Einheiten von}}\,10^{-4}\ {\rm {fm}}^{4}}$

The electrical polarizabilities and consist mainly of a smaller component, which can be traced back to the “pion cloud” ( nucleon ), and a larger component, which is based on the meson as part of the structure of the constituent quarks ( const. Quark ). The magnetic polarizabilities and have a paramagnetic part, which is related to the spin structure of the nucleon ( nucleon ) and an only slightly smaller diamagnetic part, which is contributed by the meson as part of the structure of the constituent quarks ( const. Quar k). The contributions of the meson are supplemented by small corrections made by the mesons and . ${\displaystyle \alpha _{p}}$ ${\displaystyle \alpha _{n}}$ ${\displaystyle \sigma }$ ${\displaystyle \beta _{p}}$ ${\displaystyle \beta _{n}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle f_{0}(980)}$ ${\displaystyle a_{0}(980)}$

The spin polarizabilities and are determined by a destructive interference of a component based on the pion cloud with a component based on the nucleon spin . The different signs for the proton and the neutron are based on this destructive interference. The spin polarizabilities and have a smaller component on the structure of the nucleon ( nucleon is based) and a larger component, which on the pseudoscalar mesons , and as part of the structure of the constituent quark ( const quark. Based). ${\displaystyle \gamma _{0}^{(p)}}$ ${\displaystyle \gamma _{0}^{(n)}}$ ${\displaystyle \gamma _{\pi }^{(p)}}$ ${\displaystyle \gamma _{\pi }^{(n)}}$ ${\displaystyle \pi ^{0}}$ ${\displaystyle \eta }$ ${\displaystyle \eta '}$

The agreement with the experimental data is excellent in all eight cases.

literature

• Wolfgang Demtröder: Experimental Physics 4: Nuclear, Particle and Astrophysics . (Springer textbook). 3. Edition. Springer Berlin Heidelberg, Berlin and Heidelberg 2009, ISBN 978-3-642-01597-7 , pp. 554 .

Individual evidence

1. ↑ ^{a b c} Schumacher Martin: Polarizability of the nucleon and Compton scattering . In: Progress in Particle and Nuclear Physics . tape 55 , no. 2 , October 2005, p. 567-646 , doi : 10.1016 / j.ppnp.2005.01.033 , arxiv : hep-ph / 0501167 .
2. ↑ ^{a b c} Schumacher Martin: Polarizabilities of the nucleon and spin dependent photo-absorption . In: Nuclear Physics A . tape 826 , no. 1–2 , July 15, 2009, pp. 131–150 , doi : 10.1016 / j.nuclphysa.2009.05.093 , arxiv : 0905.4363 .
3. ↑ ^{a b c} Martin Schumacher, MI Levchuk: Structure of the nucleon and spin-polarizabilities . In: Nuclear Physics A . tape 858 , no. 1 , May 15, 2011, p. 48-66 , doi : 10.1016 / j.nuclphysa.2011.04.002 , arxiv : 1104.3721 .
4. ↑ D. Drechsel, SS Kamalov, L. Tiator: Unitary isobar model --MAID2007 . In: The European Physical Journal A . tape 34 , no. 1 , October 22, 2007, p. 69-97 , doi : 10.1140 / epja / i2007-10490-6 , arxiv : 0710.0306 .
5. ↑ Martin Schumacher: Observation of the Higgs Boson of strong interaction via Compton scattering by the nucleon . In: The European Physical Journal C . tape 67 , no. 1–2 , March 30, 2010, pp. 283-293 , doi : 10.1140 / epjc / s10052-010-1290-x , arxiv : 1001.0500 .
6. ↑ Martin Schumacher,: Structure of scalar mesons and the Higgs sector of strong interaction . In: Journal of Physics G: Nuclear and Particle Physics . tape 38 , no. 8 , August 2011, p. 083001 , doi : 10.1088 / 0954-3899 / 38/8/083001 , arxiv : 1106.1015 .
7. ↑ Martin Schumacher, Michael D. Scadron: Dispersion theory of nucleon Compton scattering and polarizabilities . In: Fortschr. Phys. tape 61 , no. 7-8 , April 2, 2013, pp. 703 , doi : 10.1002 / prop.201300007 , arxiv : 1301.1567 .