Pion in a uniform background magnetic field with clover fermions

Background-field methods provide an important nonperturbative formalism for the determination of hadronic properties which are complementary to matrix-element calculations. However, new challenges are encountered when utilizing a fermion action exposed to additive mass renormalizations. In this case, the background field can induce an undesired field-dependent additive mass renormalization that acts to change the quark mass as the background field is changed. For example, in a calculation utilizing Wilson fermions in a uniform background magnetic field, the Wilson term introduced a field-dependent renormalization to the quark mass which manifests itself in an unphysical increase of the neutral-pion mass for large magnetic fields. Herein, the clover-fermion action is studied to determine the extent to which the removal of $\mathcal{O}(a)$ discretization errors suppresses the field-dependent changes to the quark mass. We illustrate how a careful treatment of nonperturbative improvement is necessary to resolve this artifact of the Wilson term. Using the ${32}^3\times 64$ dynamical-fermion lattices provided by the PACS-CS Collaboration we demonstrate how our technique suppresses the unphysical mass renormalization over a broad range of magnetic-field strengths.

Figure 1

Pion energies from Wilson-fermion correlation functions as a function of background magnetic-field strength. The colored curves are the expected energies for Wilson fermions based on Eqs. (10) and (11).

Figure 2

Pion energies from clover-fermion correlation functions as a function of background magnetic-field strength. The colored lines are the expected energies in the absence of the Wilson background-field additive mass renormalization, constant for the neutral pions and Eq. (12) for the charged pion.

Figure 3

Neutral-pion energy shifts from Eqs. (31) and (32) for $E_{[{\pi }^0]}(B)+m_{{\pi }^0}$ (left) and $E_{[{\pi }^0]}(B)-m_{{\pi }^0}$ (right), respectively, using a nonperturbatively improved clover-fermion action on the $m_{\pi }=296\mathrm{MeV}$ ensemble. The three smallest field strengths are illustrated. Shaded regions illustrate the fit windows selected through the consideration of the full covariance matrix ${\chi }_{\mathrm{dof}}^2$, the extent of the fit window and the desire to select the same fit window for all effective-energy shifts.

Figure 4

Fits of the magnetic-field-induced energy shift to the magnetic-field quanta for the nonperturbatively improved clover-fermion action. The full covariance-matrix-based ${\chi }_{\mathrm{dof}}^2$ provides evidence of a nontrivial value for fit coefficient $c_1$, indicating the presence of unwanted Wilson-like additive mass renormalizations in the nonperturbatively improved clover-fermion action.

Figure 5

Neutral-pion energy shifts from Eqs. (31) and (32) for $E_{{\pi }^0}(B)+m_{{\pi }^0}$ (left) and $E_{{\pi }^0}(B)-m_{{\pi }^0}$ (right), respectively, using the BFC clover-fermion action on the $m_{\pi }=296\mathrm{MeV}$ ensemble. The three smallest field strengths are illustrated. Shaded regions illustrate the fit windows selected through the consideration of the full covariance matrix ${\chi }_{\mathrm{dof}}^2$, the extent of the fit window and the desire to select the same fit window for all effective-energy shifts.

Figure 6

Fits of the magnetic-field-induced energy shift to the magnetic-field quanta for the BFC clover action. The full covariance-matrix-based ${\chi }_{\mathrm{dof}}^2$ for the simple $c_2{k}_B^2$ quadratic fit provides evidence of the elimination of Wilson-like additive mass renormalizations in the BFC clover action. Allowing for a nontrivial value of $c_1$ produces a value consistent with zero.

Figure 7

Fits of the magnetic-field-induced energy shift to the first five magnetic-field quanta for the BFC clover action. Here a term cubic in the magnetic-field strength is required to describe the largest field strength.