Meson masses in electromagnetic fields with Wilson fermions

We determine the light meson spectrum in QCD in the presence of background magnetic fields using quenched Wilson fermions. Our continuum extrapolated results indicate a monotonous reduction of the connected neutral pion mass as the magnetic field grows. The vector meson mass is found to remain nonzero, a finding relevant for the conjectured $\rho$-meson condensation at strong magnetic fields. The continuum extrapolation was facilitated by adding a novel magnetic field–dependent improvement term to the additive quark mass renormalization. Without this term, sizable lattice artifacts that would deceptively indicate an unphysical rise of the connected neutral pion mass for strong magnetic fields are present. We also investigate the impact of these lattice artifacts on further observables like magnetic polarizabilities and discuss the magnetic field–induced mixing between $\rho$-mesons and pions. We also derive Ward-Takashi identities for $\mathrm{QCD}+\mathrm{QED}$ both in the continuum formulation and for (order $a$–improved) Wilson fermions.

Figure 1

Results for $a{\tilde{m}}_u(a,B;\kappa )$ (left) and $a{m}_{{\pi }^u}(a,B;\kappa )$ (right) versus $1/\kappa -1/{\kappa }_{c;u}(a,0)$ for several values of $N_b$ on the $\beta =6$ ensemble. The colored bands in the left figure are the results from linear fits, and the diamonds on the $a{\tilde{m}}_u(a,B;\kappa )=0$ line in both panels are the results for $1/{\kappa }_{c;u}(a,B)-1/{\kappa }_{c;u}(a,0)$. The error bars of the extrapolated results include the estimate for the systematic uncertainty explained in the text and are smaller than the symbols.

Figure 2

Results for $C_u$ versus $|eB|$ for different values of the lattice spacing. The dashed line indicates the continuum limit value $C_u=1$, and the grey solid line is the $B=0$ line.

Figure 3

Left: Results for $\mathrm{\Delta }{m}_{c;u}(a,B)$ as defined in Eq. (26) versus $B$ for different values of the lattice spacing. Right: Results for $\mathrm{\Delta }{m}_{c;f}(a,B)$ for different values of $B$ (starting from the black point—$B=0$—outward) and different lattice spacings. The black dashed line corresponds to the slope of 2, which we expect for $B\approx 0$ due to Eq. (21), and the colored lines simply connect the data points and are included to guide the eye. The points marked with black circles correspond to a fixed physical magnetic field of $eB=3.3{\mathrm{GeV}}^2$.

Figure 4

Results for the neutral ($\overline{u}u$) pion masses (left) and the masses of ${\pi }^{+}$ (right) versus $B$ for different values of the quark masses on the ensemble with $a=0.093\mathrm{fm}$. The filled curves correspond to the ${\pi }^{+}$ masses in the free case, Eq. (1).

Figure 5

Results for ${\rho }^{+}$-meson masses with $s_z=+1$ (left) and $s_z=-1$ (right) versus $B$ for different values of the quark masses on the ensemble with $a=0.093\mathrm{fm}$. The curves correspond to the masses in the free case, Eq. (5).

Figure 6

Left: Results for the ${\rho }^{+}$-meson masses with $s_z=+1$ versus $B$, normalized to the neutral ($\overline{u}u$) pion mass at finite values of $B$. Right: Results for ${\rho }^u$-meson masses with $s_z=\pm 1$ versus $B$ for different values of the quark masses on the ensemble with $a=0.093\mathrm{fm}$.

Figure 7

Results for the neutral ($\overline{\mathrm{u}}u$) pion masses (left) and the masses of ${\pi }^{+}$ (right) versus $B$ for different values of the lattice spacing for $m_{\pi }(0)=416\mathrm{MeV}$. The filled curves correspond to the ${\pi }^{+}$ masses in the free case, Eq. (1).

Figure 8

Results for ${\rho }^{+}$-meson masses with $s_z=+1$ (left) and $s_z=-1$ (right) versus $B$ for different values of the lattice spacing for $m_{\pi }(0)=416\mathrm{MeV}$. The curves correspond to the masses in the free case, Eq. (5).

Figure 9

Left: Results for the ${\rho }^{+}$-meson masses with $s_z=+1$ versus $B$, normalized to the neutral pion mass at finite values of $B$ for $m_{\pi }(0)=416\mathrm{MeV}$. Right: Results for ${\rho }^u$-meson masses with $s_z=\pm 1$ versus $B$ for different values of the lattice spacing for an approximately constant pion mass of 416 MeV at $B=0$.

Figure 10

Results for the effective masses for the two different eigenvalues (blue circles for the ground state and red boxes for the first excited state) obtained from the GEVP in the channel relevant for ${\pi }^{+}$ and ${\rho }_0^{+}$ mixing. The results have been obtained on the ensemble with $a=0.093\mathrm{fm}$ with a $B=0$ pion mass of 416 MeV. The colored areas are the results from a fit to the correlation function. The magenta areas are the ones for the analysis of the ${\pi }^{+}$ correlation function alone, without the use of the GEVP.

Figure 11

Results for ${\rho }_{s_z=0}^{\pm }$ versus $B$ for different values of the quark masses on the ensemble with $a=0.093\mathrm{fm}$ (left) and for different values of the lattice spacing at a constant $B=0$ pion mass of 416 MeV (right). The curves correspond to the masses in the free case, Eq. (5).

Figure 12

Continuum extrapolations of the neutral ($uu$) pion masses (left) and the ${\rho }_{-}^{+}$ masses for different values of $B$, normalized by the mass at $B=0$. The filled curves represent linear fits including all data points. The error bar at $a=0$ includes in addition the systematic error obtained by varying the fit range.

Figure 13

Continuum results for pion (top) and $\rho$-meson (bottom) masses versus $B$ for a $B=0$ pion mass of 415 MeV, normalized to the respective $B=0$ values. Note, that the scaling of the $y$ axes changes at the dashed horizontal line.

Figure 14

Results for the neutral ($\overline{u}u$) pion masses versus $B$ for a pion mass of 415 MeV at $B=0$ and different values of the lattice spacing, normalized to their $B=0$ value. The open symbols correspond to results obtained with constant $\kappa$, while the filled symbols correspond to the LCP(B). We have slightly shifted the results horizontally for better visibility of the different data sets.

Figure 15

Results for the masses of the neutral ($\overline{u}u$) pions versus the lattice spacing in units of $r_0$, normalized by the mass at $B=0$. The filled points correspond to the results obtained along the LCP(B), and the filled gray curves represent their continuum extrapolation. The open symbols denote the results obtained without $B$-dependent improvement. In this case, a continuum extrapolation is only possible for $eB=0.66{\mathrm{GeV}}^2$ (blue curve). The black points on the $a=0$ line, indicated by the vertical dashed line, are the results from the continuum extrapolations, including the estimate for systematic uncertainties. In the figures, we have slightly shifted some of the results horizontally (including the associated curves) for better visibility of the different data sets.

Figure 16

Results for the ${\rho }_{+}^{+}$-meson masses versus $B$ for a pion mass of 415 MeV at $B=0$ and different values of the lattice spacing, normalized to their $B=0$ value. The open symbols correspond to results obtained with constant $\kappa$, while the filled symbols show results along the LCP(B). We have slightly shifted the results horizontally for better visibility of the different data sets.

Figure 17

Results for the masses of the neutral ($\overline{u}u$) pions (left) and the ${\rho }_{+}^{+}$-mesons (right) in units of the vacuum Sommer parameter $r_0=0.5\mathrm{fm}$. Shown are two different spatial volumes of $16^3$ and $24^3$ at $\beta =6.0$, corresponding to $m_{{\pi }^u}(B)L\approx 2$ and 3, respectively, for the largest value of $B$.

Figure 18

Results for the polarizabilities of the neutral pions with $\overline{u}u$ and $\overline{d}d$ flavor content and for the magnetic moment of the (charged) $\rho$-meson versus the lattice spacing. The filled symbols correspond to the results obtained from LCP(B)s, and the open symbols are the results from measurements with a constant $\kappa$ value.

Figure 19

Results for the polarizabilities of the neutral pions with $uu$ and $dd$ flavor content versus $m_{\pi }$ at $B=0$. The filled symbols correspond to the results obtained from LCP(B)s, and the open symbols are the results from measurements with a constant $\kappa$ value. All results were obtained on the same lattice ensemble with $\beta =6.0$.

Figure 20

Comparison of Wilson and staggered continuum extrapolated results for the lightest pion mass (normalized by its $B=0$ value). Also shown is the QCD transition temperature defined via the inflection point of the average light quark condensate—also in units of its $B=0$ value. Note the difference in the $B=0$ pion mass, which is 415 MeV for the Wilson results and 135 MeV for the staggered data.

Figure 21

The lowest eigenvalue (in lattice units) of the Wilson Dirac operator with $r=1$ against the magnetic field (also in lattice units), calculated on a two-dimensional $32^2$ lattice.

Figure 22

Results for the energies associated with free correlation functions in the pseudoscalar channel without (left) and with (right) tuning of $\kappa$ with $B$ along the trajectory (a1). The colored dashed lines in the left plot show the analytic expectations $E_f(B)=E(0)+a|q_{f}B|/2$ for energies associated with neutral correlation functions, and the solid line is the expectation for charged pions.