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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

Background magnetic fields have a decisive impact on the physics of quarks and gluons and offer a wide range of applications. Strong magnetic fields appears in noncentral heavy-ion collisions

The direct influence of the external field on the phase structure of QCD has received considerable attention recently; see for instance the review

In fact, the above mentioned lattice results are based on the connected contribution to the pion correlator, which corresponds to a hypothetic meson with exclusively

Clarifying this question may also be relevant for yet another aspect of the phase structure of QCD with

Note that this is the simplest scenario for the superconductivity of the QCD vacuum when exposed to external magnetic fields. In fact, when coupled to the photon field that exhibits gauge fluctuations, the would-be massless

Resolving the above discrepancy regarding

In this paper, we expand on the ideas introduced in Ref.

Note that the use of staggered fermions for the spectroscopy of vector mesons at finite

There are further interesting quantities that are related to the spectrum, including magnetic polarizabilities and magnetic moments. These describe the leading-order response of hadrons to the external field and, thus, involve derivatives at zero field; see, e.g., Refs.

We emphasize that our results are obtained in the quenched approximation, just as all studies in the literature so far concerning the light meson spectrum at finite

This paper is organised as follows. We start with a few remarks about the meson spectrum in continuum QCD in the presence of background magnetic fields in Sec.

We begin with a few general comments about the magnetic field dependence of the pion masses in the continuum. For small magnetic fields (and at zero temperature), we may approximate the charged pion as a pointlike free scalar particle with electric charge

It turns out that the conclusion of increasing charged pion masses and an almost constant neutral pion mass remains valid if the interaction between pions is taken into account. This can be done consistently within chiral perturbation theory for nonzero magnetic fields

The independence of the quark masses on the magnetic field also becomes obvious in a completely different limit: at very high temperatures. Here, quarks become quasifree due to asymptotic freedom. Then, the quantum mechanical energy levels are the Landau levels for fermions with charge

Clearly, if QCD interactions are present, the levels will mix. Nevertheless, it turns out that the lowest Landau level can still be defined unambiguously and remains approximately

In QCD, the quark mass is also subject to a multiplicative renormalization of the form

In the following, we will also investigate the effect of the magnetic field on

Note that the

When we consider the neutral pion at

The presence of a finite external magnetic field enables the mixing between the pion and the

We consider two flavors of (quenched) unimproved Wilson fermions on the lattice with the fermion action

The term proportional to

Note that a similar equation holds for the case of the coupling to a dynamical electromagnetic field (e.g., Ref.

It is worth stressing that the presence of the additive quark mass renormalization is a lattice artifact and

At finite lattice spacing, the

We remark that, for unimproved Wilson fermions, lines of constant physics are, in general, only defined up to magnetic field–dependent or –independent

Before we move on, let us collect some theoretical expectations concerning the behavior of

The key ingredient to fixing the LCP(B) is the

See also Refs.

Results for

The correlation functions have been projected to zero momentum in the

We note in passing that ideally one would like to use the charged axial and vector WIs for the tuning of sums and differences of quark masses. The corresponding identities are worked out including electromagnetic interactions [and also

The simulations are performed in the quenched setup using the Wilson plaquette action

Details of the quenched ensembles. The results for

To determine

We have extracted

Results for

The results for

Results for

The results for

Left: Results for

The snail shell–like shape of the curves mapped out in this parameter space demands some discussion. First, all curves start with a slope which is close to 2 (indicated by the dashed black line). This is expected due to the relation given in Eq.

Using the results for

The meson masses are extracted from the large Euclidean-time behavior of the correlation functions according to

In the presence of the background magnetic field,

To determine the fit range for the function

We have measured the meson masses along the LCP(B) using the tuned values of

In Fig.

Results for the neutral (

Next, we investigate

Note that this is in contrast with the quenched results of Ref.

Results for

Left: Results for the

In Fig.

Next, we investigate the dependence of the meson masses on the lattice spacing (the final continuum extrapolation is postponed to Sec.

Results for the neutral (

Results for

Left: Results for the

We will now discuss the mixing between pions and

Due to the mixing described in Eq.

The smaller eigenvalue of the correlator matrix corresponds to the heavier state, i.e., to the

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

Results for

We proceed by performing the continuum extrapolation of the meson masses. As in the previous section, we will work at a fixed

Continuum extrapolations of the neutral (

We show the results for the continuum extrapolated meson masses in Fig.

Continuum results for pion (top) and

The mass of the

In the previous sections, we have discussed the results for meson masses obtained along the LCP(B), where the bare quark mass has been tuned in order for the renormalized quark mass to remain constant with varying

In Fig.

Results for the neutral (

Results for the masses of the neutral (

In Fig.

Results for the

In the previous sections, we have seen that the neutral pion mass decreases down to 60% of its

The results for the two different volumes are shown in Fig.

Results for the masses of the neutral (

Further interesting features of hadrons are their magnetic moments and the electric and magnetic polarizabilities. The latter describe the indirect response of a bound state to external electromagnetic fields. The background field method gives access to both the magnetic

The energy of a relativistic particle (hadron) with mass

The additional terms including derivatives of the quark masses with respect to the external field are lattice artifacts and thus vanish in the continuum limit. They are not present for actions where the quark mass is protected by a remnant of chiral symmetry. For Wilson fermions, however, those terms are present and contaminate the results for magnetic moments and polarizabilities. These contaminations can be removed by tuning the quark mass along LCP(B)s as discussed above, resulting in a cancellation of the derivatives of the quark masses with respect to the external field. Even when the tuning is only approximate, it leads to a strong reduction of the additional terms in Eq.

We proceed by investigating the strength of the effect mentioned above for polarizabilities and

Let us start with the polarizability of the neutral and the charged pion. Concerning the neutral pion, we can only determine the polarizability of pions with

Results for the polarizabilities of the neutral pions with

Moving on to the

Since the relative size of the change in quark mass depends on the bare quark mass we have at

Results for the polarizabilities of the neutral pions with

In this paper, we investigated the meson spectrum in QCD at zero temperature in the presence of background magnetic fields

We emphasize that the improvement only differs from the naive approach by lattice artifacts, which, however, may be large. In particular, we demonstrated that without the improvement, meson masses suffer from enormous discretization effects for strong magnetic fields. Besides the impact for the strong field limit, we also considered the effect of the improvement for derivatives with respect to the magnetic field at

Our most important results about the spectrum involve the mass of the neutral (connected) pion, which was found to decrease monotonously as the magnetic field grows, and the mass of the lightest charged

Finally, we elaborate on the connection between the

The details of the staggered simulation setup are described in Refs.

Comparison of Wilson and staggered continuum extrapolated results for the lightest pion mass (normalized by its

Regarding the QCD transition temperature, we considered the parametrization of the crossover transition temperature

This research was funded by the DFG (Emmy Noether Programme EN 1064/2-1 and SFB/TRR 55). The majority of the simulations was performed on the iDataCool cluster of the Institute for Theoretical Physics at the University of Regensburg. The authors are grateful for the useful correspondence with Pavel Buividovich, Maxim Chernodub, Davide Giusti, Rainer Sommer, and Arata Yamamoto. We thank Max Theilig for a careful reading of the Appendixes.

To demonstrate the presence of a

The massless Wilson Dirac operator may be written schematically as

The lowest eigenvalue (in lattice units) of the Wilson Dirac operator with

This increase of the lowest eigenvalue is equivalent to an additive shift in the quark mass, implying a

The effect of this

Our numerical results for the energies at fixed

Results for the energies associated with free correlation functions in the pseudoscalar channel without (left) and with (right) tuning of

The shift in the bare quark mass basically follows from the magnetic field dependence of the lowest energy state of a noninteracting charged scalar particle. For a homogeneous background magnetic field, this dependence can be found analytically and is given by Eq.

In this Appendix, we will derive axial and vector WIs for QCD in the presence of electromagnetic interactions with two light quark flavors, i.e., for

We thank Davide Giusti for drawing our attention to this.

In the continuum, the Euclidean fermion action including

Ward-Takahashi identities can be derived by varying the expectation value of some test operator

Using Eqs.

We use the notation

Considering vector transformations, the additional terms are given by

With axial vector transformations, we obtain

We will now derive similar identities for (unimproved) Wilson fermions using the fermion action from Eq.

Note that this choice corresponds to a particular convention for the definition of point-split bilinear operators. An alternative convention includes the Pauli matrices to the left of the link variables. Both definitions lead to equivalent results as long as one strictly follows one and the same convention analytically and numerically.

When the linksLet us start again with the vector transformations. Using the results for Wilson fermions from Ref.

Similarly, we obtain for the axial WI

Let us briefly discuss the properties and the nature of the terms denoted by (1) and (2). These two types of terms are due to the presence of the electromagnetic interactions in the

This is true up to the anomalous terms for the isosinglet axial WI.

The terms denoted by (1) arise from the Wilson term (consequently, they are multiplied by a factor ofTo compute current quark masses, terms of type (2) should be included explicitly in their definition, since these terms are also present in the continuum WI, while terms of type (1) should be left out (similar to

Note that the electromagnetic link variables that need to be included in the point-split currents depend on the particular flavor matrix for which the WI is evaluated. As an example, let us once more consider the WIs for the matrix

For

For vector transformations, we only get a contribution from the electromagnetic clover term. In particular, we get terms which are proportional to the commutator

For axial vector transformations, it is the anticommutator of

In this Appendix, we demonstrate that the QCD mass renormalization constant is independent of the background magnetic field. On general grounds, it is expected that the ultraviolet divergent renormalization constants of the theory do not depend on physical parameters like the magnetic field. Here, we show this using one-loop perturbation theory in continuum QCD.

For a similar calculation at nonzero temperature, see Ref.

In the Schwinger proper time representation

In fact, Eq.

The propagator can be expanded in powers of the magnetic field. The consecutive orders read

The

We proceed with the

Finally, we consider the