Neutral and charged mesons in magnetic fields

Abstract

We analyze mesons in constant magnetic fields (B) within a non-relativistic constituent quark model. Our quark model contains a harmonic oscillator type confining potential, and we perturbatively treat short range correlations to account for the spin-flavor energy splittings. We study both neutral and charged mesons taking into account the internal quark dynamics. The neutral states are labelled by two-dimensional momenta for magnetic translations, while the charged states by two discrete indices related to angular momenta. For $B \ll \varLambda _\mathrm{QCD}^2$ ($\varLambda _\mathrm{QCD}\sim 200$ MeV: the QCD scale), the analyses proceed as in usual quark models, while special precautions are needed for strong fields, $B \sim \varLambda _\mathrm{QCD}^2$, especially when we treat short range correlations such as the Fermi-Breit-Pauli interactions. We compute the energy spectra of mesons up to energies of $\sim 2.5$ GeV and use them to construct the meson resonance gas. Within the assumption that the constituent quark masses are insensitive to magnetic fields, the phase space enhancement for mesons significantly increases the entropy, assisting a transition from a hadron gas to a quark gluon plasma. We confront our results with the lattice data, finding reasonable agreement for the low-lying spectra and the entropy density at low temperature less than $\sim 100$ MeV, but our results at higher energy scale suffer from artifacts of our confining potential and non-relativistic treatments.

A Some calculations

1.1 A.1 ($\hat{ \mathbf {\varPi }}^2, \hat{\mathbf {{\mathcal {K}}}}^2$) in creation and annihilation operators

When we evaluate 2D vectors such as $\hat{\mathbf {r}}_\perp$ and $\hat{\mathbf {p}}_\perp$ operators, it is more convenient to work with an algebraic method. We define two sets of the creation-annihilation operators, ($\hat{\varPi }_\pm = \hat{\varPi }_x \pm \mathrm {i}\hat{\varPi }_y$, $\hat{ {\mathcal {K}} }_\pm = \hat{ {\mathcal {K}} }_x \pm \mathrm {i}\hat{{\mathcal {K}} }_y$)

$$\begin{aligned} (\hat{a}^\dag , \hat{a} ) = \frac{1}{\, \sqrt{2|eB| } \,} \times \left\{ \begin{array}{l} ~ (\hat{\varPi }_- , \hat{\varPi }_+ ) ~~~~~(eB \ge 0) \\ ~ (\hat{\varPi }_+ , \hat{\varPi }_- ) ~~~~~(eB < 0) \end{array} \right. \end{aligned}$$

(106)

and

$$\begin{aligned} (\hat{b}^\dag , \hat{b} ) = \frac{1}{\, \sqrt{2|eB| } \,} \times \left\{ \begin{array}{l} ~ (\hat{{\mathcal {K}}}_+, \hat{{\mathcal {K}}}_-) ~~~~~(eB \ge 0) \\ ~ (\hat{{\mathcal {K}}}_-, \hat{{\mathcal {K}}}_+) ~~~~~(eB < 0) \end{array} \right. \end{aligned}$$

(107)

where $(\hat{a}, \hat{a}^\dag )$ and $(\hat{b}, \hat{b}^\dag )$ separately satisfy the usual harmonic oscillator algebra,

$$\begin{aligned} \hat{\mathbf {\varPi }}^2 = |eB| ( 2 \hat{a}^\dag \hat{a} + 1 ) ,~~~~ \hat{\mathbf {{\mathcal {K}}}}^2 = |eB| ( 2 \hat{b}^\dag \hat{b} + 1 ) , \end{aligned}$$

(108)

and $\hat{a}^\dag |n_\varPi , n_{\mathcal {K}} \rangle = \sqrt{n_\varPi +1\,} | n_\varPi +1, n_{\mathcal {K}} \rangle$, etc.

A few more expressions are used for charged mesons discussed in the main text. We note that $\hat{\mathbf {p}}_\perp = ( \hat{\mathbf {{\mathcal {K}}}} + \hat{\mathbf {\varPi }} )/2$ and $e \mathbf {B}\times \hat{\mathbf {r}} = \hat{\mathbf {{\mathcal {K}}}} - \hat{\mathbf {\varPi }}$. Finally, using Eq. (14),

$$\begin{aligned} 2 e\mathbf {B}\cdot \mathbf {l}= \hat{ \mathbf {\varPi }}^2 - \hat{ \mathbf {{\mathcal {K}}}}^2 = 2 |eB| \big ( \hat{b}^\dag \hat{b} - \hat{a}^\dag \hat{a} \big ) . \end{aligned}$$

(109)

1.2 A.2 Rearrangement of $\mathbf {\varPi }_r$

To compute Eqs. (75) and (78), we note

$$\begin{aligned} \hat{ \tilde{{\varvec{\varPi }}} }_r = \hat{ \mathbf {p}}_r - \frac{1}{\, 2 \,} \mathbf {{\mathcal {B}}} \times \hat{ \mathbf {r}} = \hat{ \mathbf {\varPi }}_r - \frac{1}{\, 2 \,} ( \mathbf {{\mathcal {B}}} - \mathbf {B}) \times \hat{ \mathbf {r}} , \end{aligned}$$

(110)

and $\hat{ \tilde{\varvec{{\mathcal {K}}}} }_r = \hat{ \tilde{{\varvec{\varPi }}} }_r + \mathbf {{\mathcal {B}}} \times \hat{\mathbf {r}}$. Eliminating $\mathbf {{\mathcal {B}}} \times \hat{\mathbf {r}}$,

$$\begin{aligned} \hat{ \mathbf {\varPi }}_r= & {} \frac{1}{\, 2 \,} \bigg (1 + \frac{\, B \,}{{\mathcal {B}}} \bigg ) \hat{ \tilde{{\varvec{\varPi }}} }_r + \frac{1}{\, 2 \,} \bigg (1 - \frac{\, B \,}{{\mathcal {B}}} \bigg ) \hat{ \tilde{\varvec{{\mathcal {K}}}}}_r \nonumber \\\equiv & {} f_+ \hat{ \tilde{{\varvec{\varPi }}} }_r + f_- \hat{\tilde{\varvec{{\mathcal {K}}}}} _r. \end{aligned}$$

(111)

The last expression will be used when we evaluate the coupling $\hat{ \mathbf {\varPi }}_R \cdot \hat{ \mathbf {\varPi }}_r$ which will appear in computations of charged mesons.

1.3 A.3 $N_{\mathcal {K}}$ and the density of states

We have not discussed any constraints on $N_{\mathcal {K}}$ (except $N_{\mathcal {K}} \ge 0$), which would give an impression that $N_{\mathcal {K}}$ has no upper bound. At this stage we have to be careful about the counting of the density of states (for the detailed discussions, e.g. Ref. [48]). For this purpose we consider the system size of $V_2 = \pi R^2$. The momenta ${\mathcal {K}}_R$ characterizes the guiding center of the cyclotron orbit measured from the origin, and its radius is $|\mathbf {{\mathcal {K}}}_R/B_R | = \sqrt{2 N_{\mathcal {K}}/|B_R|}$ which must be smaller than R. Thus the maximum of $N_{\mathcal {K}}$ for a given volume $V_2$ is $N_{\mathcal {K}}^\mathrm{max} = R^2 |B_R|/2 = V_2 \times | B_R |/2\pi$. Taking this into account, the sum of states per volume is

$$\begin{aligned} \frac{1}{\,V_2 \,} \sum _{ N_{\mathcal {K}} = 0}^{N_{\mathcal {K}}^\mathrm{max} } = \frac{\, |B_R| \,}{\, 2\pi \,} . \end{aligned}$$

(112)