Magnetic field induced anomalous pion couplings

1. Problem Statement

The paper addresses the impact of external magnetic fields on hadron structure and dynamics, specifically focusing on pion-constituent quark interactions. While strong magnetic fields ($10^{11}–10^{18}$ T) are expected in environments like magnetars and relativistic heavy-ion collisions (r.h.i.c.), their effects on virtual quark lines usually require complex numerical simulations.

The authors investigate a specific regime: relatively weak magnetic fields ($eB \ll M^2$, where $M$ is the constituent quark mass). In the vacuum, standard pion couplings to quarks are pseudoscalar (and axial). However, the authors hypothesize that an external magnetic field can induce anomalous couplings that vanish in the vacuum limit ($B=0$). Specifically, they seek to calculate scalar and vector single-pion couplings to constituent quark currents, which are forbidden by conservation rules in the vacuum but may arise due to magnetic field symmetry breaking.

2. Methodology

The authors employ a semi-analytical approach based on Weinberg's Large $N_c$ Effective Field Theory (EFT).

• Theoretical Framework:

  • They utilize the Large $N_c$ EFT, which describes constituent quarks and gluons.
  • To facilitate the calculation of anomalous couplings, they redefine the pion field to transition from a non-linear realization (standard in Chiral Perturbation Theory) to a linear realization, allowing the emergence of a pseudoscalar pion-quark coupling ($g_{ps}$).
  • The Lagrangian includes the constituent quark field ($\psi$), the pion field ($\pi$), and an effective gluon propagator to model confinement and non-Abelian effects.

• Diagrammatic Calculation:

  • The calculation involves three-point Green's functions (Feynman diagrams with one pion and two quark legs).
  • A background magnetic field is inserted into the internal quark lines (represented by a wavy line with a dot).
  • The quark propagator in a magnetic field is expanded to the first order in $eB$ (weak field limit), utilizing the Schwinger phase which is gauged away for these specific diagrams.
  • An effective gluon propagator with a mass parameter ($M_g$) is used to account for confinement.

• Projection and Form Factors:

  • The vertex functions are projected onto scalar and vector Dirac-flavor channels using specific projection operators ($P_s$ and $P_v$).
  • In the vacuum, these projections vanish. The non-zero results obtained with the magnetic field insertion represent the induced form factors.
  • The resulting effective Lagrangian terms are proposed as:
    $$\mathcal{L}_1 \sim \frac{g_1}{\bar{M}^3} \tilde{F}^{\mu\nu} \partial_\mu \pi^i \bar{q} \tau^i \gamma_\nu q$$
    $$\mathcal{L}_2 \sim \frac{g_2}{\bar{M}^4} \tilde{F}^{\mu\nu} \partial_\mu \pi^i \bar{q} \tau^i \partial_\nu q$$
    where $\tilde{F}^{\mu\nu}$ is the dual electromagnetic tensor.

• Kinematical Regions:
  Numerical evaluations are performed for two distinct kinematical scenarios:

  1. On-shell quarks, off-shell pion: Representative of pion exchange potentials in nucleons.
  2. Off-shell quarks, on-shell massless pion: Representative of pion emission/absorption processes.

3. Key Contributions

• Discovery of Anomalous Couplings: The paper demonstrates that weak magnetic fields induce scalar and vector pion-constituent quark couplings that are strictly zero in the vacuum.
• Linear Dependence: The induced form factors are shown to be linearly proportional to the magnetic field strength ($eB$).
• Analytical Framework: The authors provide a semi-analytical derivation of these form factors using Large $N_c$ EFT, avoiding the need for full Lattice QCD simulations for this specific weak-field limit.
• Extension to Other Mesons: The formalism is noted to be applicable to $\eta$ and $\eta'$ mesons, provided appropriate flavor mixing is considered.

4. Results

The numerical results are presented for constituent quark mass $M = 350$ MeV, effective gluon mass $M_g = 400$ MeV, and coupling $g_{ps} = 13$.

• Form Factor Behavior:

  • Scalar Form Factors ($F_S$):
    • For on-shell quarks (off-shell pion): The charged pion form factor is stronger than the neutral one. The values are negative and decrease with increasing momentum transfer ($q^2$).
    • For off-shell quarks (on-shell pion): The neutral pion form factor is stronger than the charged one. The values are negative.
  • Vector Form Factors ($F_V$):
    • For on-shell quarks: The neutral pion form factor is stronger than the charged one. The values are positive and decrease rapidly with $q^2$.
    • For off-shell quarks: The neutral pion form factor is initially larger but decays faster than the charged pion form factor.
  • Magnitude: Generally, the scalar form factors are one order of magnitude smaller than the vector form factors ($F_S \sim 0.05$ vs $F_V \sim 0.2$), though they become comparable when multiplied by the kinematic factor $\tilde{Q}^\mu$.

• Physical Implications:

  • Repulsive Potential: The opposite signs of the scalar and vector form factors in the on-shell quark region suggest that an anomalous pion exchange between scalar and vector currents could generate a weak repulsive component in the Yukawa-type potential.
  • Range: The vector form factors decay faster than scalar ones, indicating shorter-range interactions for the vector channel.
  • Mean Square Radii: The authors calculated the averaged quadratic radii ($\langle r^2 \rangle$). For $eB = 0.1 M^2$, the radii are approximately $0.20–0.30$ fm, with vector radii being slightly larger than scalar radii.
  • $\pi^0 \to \gamma$ Transition: The authors investigated a potential neutral pion decay into a photon via a vector quark current. They found the amplitude proportional to $\tilde{Q} \cdot Q$, which vanishes, meaning this specific transition is forbidden without external momentum modification.

5. Significance

• Phenomenological Probes: These anomalous couplings offer a potential theoretical signature for detecting magnetic fields in the confined (hadronized) phase of relativistic heavy-ion collisions. If hadrons formed in strong magnetic fields leave the field region in excited states or decay via these anomalous channels, it could be observed experimentally.
• Hadron Structure: The results imply that external magnetic fields modify the internal structure of pions, introducing scalar and vector components that are otherwise absent.
• Experimental Feasibility: While the effects are small (requiring high-precision measurements), the identification of isospin-breaking differences between charged and neutral pion interactions in magnetic fields could serve as a diagnostic tool for the magnetic field strength in r.h.i.c. experiments.

In conclusion, the paper establishes a theoretical basis for magnetic-field-induced scalar and vector pion couplings, providing explicit form factors and suggesting that these effects, though small, could be observable signatures of strong magnetic fields in high-energy nuclear physics.