[Paper Review] Static polarizability vertex and its applications

A. Ilyichev, Светлана Анатольевна Лукашевич+1 more|ArXiv.org|Nov 27, 2006

Matrix Theory and Algorithms1 references3 citations

TL;DR

This paper derives the static polarizability vertex in effective field theory for nucleon-photon interactions, using a non-perturbative Lagrangian that incorporates electric and magnetic polarizabilities. It applies the vertex to compute real and doubly virtual Compton scattering amplitudes, deriving $Q^2$-dependent spin polarizabilities consistent with existing models and enabling extraction of forward polarizabilities from deep-inelastic scattering data via the imaginary part of the amplitude.

ABSTRACT

Using the Lagrangian that was developed on the corresponding principle between the moving medium electrodynamic and quantum field theory the explicit expression for the static polarizability vertex has been obtained. The applications of this vertex for calculations of real Compton scattering amplitude as well as the imaginary part of doubly virtual Compton scattering amplitude have been demonstrated.

Motivation & Objective

• To develop a field-theoretic framework for nucleon spin polarizabilities using effective Lagrangians incorporating static electric and magnetic polarizabilities.
• To derive the explicit form of the static polarizability vertex in momentum space using Feynman rules and Lorentz-invariant tensor structures.
• To apply the vertex to compute real Compton scattering (RCS) and doubly virtual Compton scattering (VVCS) amplitudes at leading order.
• To connect the imaginary part of the VVCS amplitude to measurable hadronic tensor components in deep-inelastic scattering (DIS), enabling extraction of $Q^2$-dependent polarizabilities.

Proposed method

• Constructs an effective Lagrangian ${\cal L}^{\text{pol}}_{\text{eff}}$ coupling nucleon currents to electromagnetic field strengths, parameterized by static polarizabilities $\alpha_0$ and $\beta_0$.
• Derives the polarizability vertex $\Gamma^{\text{pol}}_{\sigma\delta}$ in momentum space via Fourier transformation and momentum-space substitution of derivatives.
• Applies the vertex to compute the RCS amplitude $iT$ using standard Feynman rules, yielding a matrix element in terms of $P$, $K$, and $Q$ vectors.
• Computes the imaginary part of the VVCS amplitude via cuts on Feynman diagrams (Fig. 2a-d), expressing it as a sum over vertex and propagator contractions.
• Relates the imaginary part to physical observables in DIS using the standard formulae for $\sigma_T$, $\sigma_L$, $\sigma_{TT}$, and $\sigma_{LT}$, enabling $Q^2$-dependent polarizability extraction.
• Uses the relation $\alpha(Q^2) + \beta(Q^2) = \frac{1}{2\pi^2}\int \frac{K(Q^2,\nu)}{\nu} \frac{\sigma_T}{\nu^2} d\nu$ to connect amplitudes to measurable polarizabilities.

Experimental results

Research questions

• RQ1 How can the static polarizability vertex be derived from an effective Lagrangian in momentum space using field-theoretic methods?
• RQ2 What is the explicit form of the nucleon-photon vertex that incorporates both electric and magnetic polarizabilities in a Lorentz-invariant way?
• RQ3 How does the derived vertex reproduce known results for real Compton scattering (RCS) at leading order?
• RQ4 Can the imaginary part of the doubly virtual Compton scattering (VVCS) amplitude be used to extract $Q^2$-dependent polarizabilities from deep-inelastic scattering (DIS) data?
• RQ5 What is the quantitative relation between the hadronic tensor components in DIS and the forward spin polarizabilities?

Key findings

• The static polarizability vertex is derived in momentum space as $\Gamma^{\text{pol}}_{\sigma\delta}(p_1,q_1,p_2,q_2)$, incorporating both $\alpha_0$ and $\beta_0$ via a combination of tensor structures involving $q_1$, $q_2$, $p_1$, and $p_2$.
• The RCS amplitude derived from the vertex matches the known result from literature, confirming consistency with existing effective field theory descriptions.
• The imaginary part of the VVCS amplitude is computed via cuts on diagrams (Fig. 2a-d), with contributions from four distinct topologies involving the polarizability vertex and elastic vertices.
• The amplitude’s imaginary part is expressed as a sum over propagators and vertex contractions, with the $\Gamma^{\text{pol}}$ vertex coupling to nucleon and photon lines.
• The $Q^2$-dependence of forward polarizabilities is extracted via integration over the hadronic tensor components $\sigma_T$, $\sigma_L$, $\sigma_{TT}$, and $\sigma_{LT}$, using the standard formulae involving $K(Q^2,\nu)$ and energy transfer $\nu$.
• The contribution from diagram (e) is negligible ($\sim 10^{-8}$ fm$^6$) and can be safely omitted in the computation of the imaginary part.