Confining and chiral properties of QCD in extremely strong magnetic fields

We investigate, by numerical lattice simulations, the static quark-antiquark potential, the flux tube properties and the chiral condensate for $N_f=2+1$ QCD with physical quark masses in the presence of strong magnetic fields, going up to $eB=9{\mathrm{GeV}}^2$, with continuum extrapolated results. The string tension for quark-antiquark separations longitudinal to the magnetic field is suppressed by 1 order of magnitude at the largest explored magnetic field with respect to its value at zero magnetic background, but is still nonvanishing; in the transverse direction, instead, the string tension is enhanced but seems to reach a saturation at around 50% of its value at $B=0$. The flux tube shows a consistent suppression/enhancement of the overall amplitude, with mild modifications of its profile. Finally, we observe magnetic catalysis in the whole range of explored fields with a behavior compatible with a lowest Landau level approximation, in particular with a linear dependence of the chiral condensate on $B$ which is in agreement, within errors, with that already observed for $eB\sim 1{\mathrm{GeV}}^2$.

Figure 1

The representation of the $SU(3)$ path ${\rho }_{\mathrm{conn}}^{\mu t}$ defined in Eq. (13). Since we are interested in the longitudinal chromoelectric field only, $P$ and $W$ always lie on parallel planes.

Figure 2

Change of the renormalized average light quark condensate due to the magnetic field for $eB=4$ and $9{\mathrm{GeV}}^2$ as a function of the lattice spacing, together with continuum extrapolations obtained assuming $O(a^2)$ corrections.

Figure 3

Continuum extrapolated results for the change of the chiral condensate due to the magnetic background field: results obtained in this study are compared to those reported in Ref. [22]. The colored band is the result of a linear fit to the data of Ref. [22] (${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.\simeq 1/3$), which after extrapolation turns out to be in nice agreement, within errors, with our present determinations.

Figure 4

Logarithm of Wilson loop ratio according to Eq. (12) as a function of $n_t$. Data have been extracted from the ${48}^3\times 96$ lattice ($a=0.0572\mathrm{fm}$) at $eB=9{\mathrm{GeV}}^2$. They are displayed for three choices of the APE smearing level in both the T- and L-cases. Continuum lines correspond to the determination of the plateau for $N_{\mathrm{APE}}=30$.

Figure 5

Static potential $V(r)$ between the quark-antiquark pair as a function of the distance $r$, for the explored values and orientations of $\overrightarrow{B}$. Results refer to the finest lattice spacing $a=0.0572\mathrm{fm}$, i.e., the ${48}^3\times 96$ lattice.

Figure 6

Continuum limit of the string tension at $eB=0$, together with the results of Ref. [44].

Figure 7

Continuum limit of the $\sigma$-ratios for both the values of $B$. Dashed/continuum lines correspond to the best extrapolations performed in $\mathrm{T}/\mathrm{L}$-cases.

Figure 8

Continuum limit of the $\sigma$-ratios in the T- and L-cases, for both the values of the background field $eB=4$, $9{\mathrm{GeV}}^2$. The dashed gray regions correspond to the continuum extrapolations of Ref. [44].

Figure 9

Ratio $E_l(B)/E_l(B=0)$ at $a=0.0572\mathrm{fm}$ for different numbers of smearing steps. Data are evaluated at $x_t=0$, so that the transverse configurations TT and TL are equivalent (see Table 1 for details). The physical distance between the color charges is $d=0.68\mathrm{fm}$.

Figure 10

Chromoelectric field at $eB=4{\mathrm{GeV}}^2$ for two choices of the lattice spacings $a=0.0858$, 0.0572 fm, in the physical range $x_t\in [-1,+1]\mathrm{fm}$. The relative distance of the quark-antiquark pair is fixed to $d=0.68\mathrm{fm}$.

Figure 11

Chromoelectric field at $eB=9{\mathrm{GeV}}^2$ for two choices of the lattice spacings $a=0.0858$, 0.0572 fm, in the physical range $x_t\in [-1,+1]\mathrm{fm}$. The relative distance of the quark-antiquark pair is fixed to $d=0.68\mathrm{fm}$.

Figure 12

Ratio $E_l(B,x_t)/E_l(0,x_t)$ as a function of the transverse distance $x_t$. Data have been computed for $eB=4$ and $9{\mathrm{GeV}}^2$ and for each orientation class at $a=0.0858\mathrm{fm}$. The physical distance between the pair is fixed to $d=0.68\mathrm{fm}$.

Figure 13

Ratio $E_l(B,x_t)/E_l(0,x_t)$ computed in L configurations. Data for $eB=4$ and $9{\mathrm{GeV}}^2$ have been computed at $a=0.0858\mathrm{fm}$ and $d=0.68\mathrm{fm}$. Values at weaker fields were computed in Ref. [45] at $a=0.0989\mathrm{fm}$ and $d\simeq 0.7\mathrm{fm}$.

Figure 14

Dependence of the color flux tube profile on the intensity of the magnetic field in L configurations ($a=0.0572\mathrm{fm}$ and $d=0.68\mathrm{fm}$). The dashed lines represent the best fits according to the Clem ansatz of Eq. (19).

Figure 15

Continuum limits of the ratio $\epsilon (B)/\epsilon (0)$ at $eB=4$, $9{\mathrm{GeV}}^2$.

Figure 16

Comparison of $\sigma (B)/\sigma (0)$ and $\epsilon (B)/\epsilon (0)$ computed at $eB=4$ and $9{\mathrm{GeV}}^2$. Points are slightly shifted for readability.