# The Evolution of Meson Masses in a Strong Magnetic Field

###### Abstract

Spectra of hadrons are investigated in the framework of the Hamiltonian obtained from the relativistic path integral in external homogeneous magnetic field. The spectra of all 12 spin-isospin s-wave states, generated by and mesons with different spin projections, are studied both analytically and numerically on the lattice as functions of (magnetic field) . Results are in agreement and demonstrate three types of behavior, with characteristic splittings predicted by the theory.

## 1 Introduction

The influence of magnetic field (MF) on the strong interacting particles is an
actively discussed topic, see, e.g. a recent review 1 (). When MF is not
ultra-intense (, where is a confinement
string tension)^{1}^{1}1We use relativistic system of units , then , the main characteristics related to the behavior of hadrons in
MF are magnetic moments and magnetic susceptibilities, while for the strong MF
limit () the hadron energy and width depend on MF
directly.

These topics are important in astrophysics of neutron stars 2 (), in cosmological theories 3 (), in atomic physics 30 (); 31 (); 31a (); 34 () in the physics of heavy ion collisions 4 (), and in the high-intensity lasers 5 ().

On the theoretical side the main directions of research in this area are the lattice studies 6 ()-13 (), 39 ()-45 (), the chiral Lagrangians with MF 14 ()-17 (), effective hadron Lagrangians 1 (); 19 (),46 ()-49 (), and recently developed path integral Hamiltonians (PIH) 20 ()-23 (), and the chiral Lagrangian with quark degrees of freedom 24 ().

The PIH method has appeared to be well suited to the inclusion of an arbitrary external MF. Here one obtains simple expressions for magnetic moments of hadrons, mesons 25 () and baryons 26 (), which are in a good agreement with available experimental and lattice data, as well as with existing model calculations. We stress at this point, that in all calculations done within the PIH framework, the final results are expressed in terms of basic QCD parameters - string tension , and current quark masses .

A sample of light neutral meson masses in MF (actually, the meson energies for zero longitudial momentum) has been calculated with PIH framework in 20 (); 21 (); 211 (), and the Nambu-Goldstone (NG) modes in MF have been studied in 24 (). In all cases the resulting values of are in reasonable agreement with lattice data from 8 ()-10 (). The three-body neutral systems in strong MF were studied with PIH in 27 (), but there is no lattice data now to compare with.

In a general case, solving the spectral problem for hadrons in MF is a cumbersome task. To proceed with analytic calculations, one should use some special techniques. One of them is the Pseudomomentum approach. It was introduced in 28 () to separate center-of-mass (c.m.) motion from the relative motion in the nonrelativistic Hamiltonian for the neutral system in MF. This approach was extended to the relativistic sector in the PIH framework for two-body systems in 20 (); 211 (); 21 () and for three-body systems in 29 (). The Pseudomomentum approach is applicable only for electrically neutral systems, and for the charged ones an exact analytical answer was obtained only in an unphysical model of charged meson with equally charged quark constituents 211 ().

Below we are suggesting a new approximate analytic method of constituent separation (CS) that allows to get a quantitative result for any meson masses with 15% accuracy for the strong MF () and with 20% accuracy for . As will be shown, the CS method allows to study charged and neutral systems in the same way. To introduce it, we first write the relativistic Hamiltonian in MF within PIH formalism and exploit the oscillator representation for the confinement interaction used before in 20 (); 211 (); 21 () with 5% accuracy. This allows to split the Hamiltonian into transversal and longitudial (with respect to the MF direction) parts analytically. All the rest interaction - one-gluon exchange, spin-dependent and self-energy interactions are studied perturbatively.

Our final results for the neutral mesons in MF are obtained in two independent ways: via Pseudomomentum and the CS methods, which allows to check the accuracy of our results.

The paper is organized as follows: in Section 2 we write relativistic Hamiltonian and discuss the main features of CS method. (Details of this method are discussed in Appendix A). As a result we obtain in Section 2 the hadron mass and the ground state wave function as a function of and for an arbitrary meson in MF. In Section 3 a classification of meson mass trajectories with different spin and isospin projections is given with the corresponding asymptotics in high MF regime . In Section 4 the perturbative correction due to the one-gluon exchange is calculated and the absence of the color Coulomb collapse is demonstrated. The CS wave function for neutral mesons is discussed in in the Appendix B. In Section 5 the spin-spin interaction in MF and the seemingly possible “hyperfine collapse” is discussed. In Section 6 a general discussion of the we spin-isospin splitting is given. In Section 7 we study the chiral and nonchiral treatment of pion masses in MF. In Section 8 the details of our lattice calculations are given. Results of both analytic and lattice results are discussed in the concluding Section 9.

## 2 The relativistic Hamiltonian of quark systems

We start from the relativistic Hamiltonian of the N-quark system in an external homogeneous MF, which according to 20 ()-22 () is

(1) |

where are virtual quark energies to be intergrated over in the path integral, and are current quark masses. At this step we neglect any internal interactions between quarks, i.e. confinement, gluon-exchange, etc. It is convenient to choose symmetrical gauge for MF which allows to define an angular momentum projection for each quark as a quantum number. The spectrum of (1) with is

(2) |

According to 20 ()-22 () the physical spectrum is given by the stationary point value of , with respect to

(3) |

It is easy to see that this spectrum coincides with the solution of the Dirac equation for N non-interacting relativistic particles in MF.

As in 211 () we now introduce the confining interaction , which is treated nonperturbatively, while the other interactions like one-gluon exchange , spin-dependent interaction and self-energy corrections are treated perturbatively in the next sections. The Hamiltonian becomes

(4) |

with the ground state eigenvalue (nonperturbative, or dynamical mass) and the ground state wave function . The total meson mass is a sum of and the perturbative corrections

(5) |

One can note, that the contribution of the in strong MF () is negligible in the plane transverse to the MF direction and should be retained only for lowest levels, which we call “zero hadron states” (ZHS) (see below). Another feature is that in strong MF regime the translational invariance of the center-of-mass (c.m.) is broken due to magnetic forces (each quarks is placed on its own Landau level), but the confinement still defines the motion of quarks in the direction along the MF.

To simplify calculations we chose the confining term in the variable quadratic form 211 (); 21 (), restoring its original linear form at the stationary point (it was checked to be accurate within about 5%), namely

(6) |

where is variational parameter and is a confinement string tension. The dependence of the string tension on the MF is caused by the fluctuating pairs embedded to the string and provides a correction about at . This phenomenon was studied on the lattice in 21-1 () and was confirmed within PIH formalism in 21-2 (). The correction to the ground state caused by this effect is beyond the declared accuracy and is neglected in what follows. To produce an approximation for the energy, one should minimize the resulting state energy obtained from the Hamiltonian (4) with respect to and simultaneously.

The oscillator approximation (6) gives an advantage to separate motion along the axis(parallel to the MF) and in plane

(7) |

where the motion along the -axis is defined by the Hamiltonian

(8) |

where we use c.m. reference frame with .The longitudial part of the ground state energy is

(9) |

For the motion in the transversal plane one can use an approximation of decoupled quarks at large MF, making the following substitution

(10) |

where c.m. position is fixed at the origin in plane. This approximation corresponds to the configuration where the confinig string connects each quark to the c.m., i.e. the string is effectively elongated. The magnetic energy of each quark in strong MF (Landau level) is larger than the confinig interaction with the factor , which make this approximation legitimate at regime. To extend our method to the region, where the behaviour is mostly defined by confinement, one should introduce an effective sting tension and for each part of the string, connecting quarks to the c.m.

(11) |

to compensate an effective string elongation. As shown in Appendix A, the appropriate values of are

(12) |

Using this ”-renormalization” procedure, one can show that the dynamical mass of the ground state , calculated in 211 () with the Pseudomomentum technique for neutral mesons, exactly coincides with the dynamical mass obtained in the above CS formalism for the arbitrary value of MF.As a result this approximation make quarks effectively decoupled in plane and one can write

(13) |

The transversal part of the hamiltonian has the ground state energy

(14) |

where are given by (12). The total dynamical mass is given by the sum

(15) |

The actual trajectories for the dynamical mass in MF, are obtained using the stationary point conditions in a similar way as (3)

(16) |

The corresponding wave function for the ground state is

(17) |

## 3 Meson trajectories in MF

We turn now to the general structure of the meson spectrum and the limits of weak () and strong () MF.

For small MF both and are independent of MF in the leading order and the lowest order the correction to the dynamical mass is

(19) |

where is the magnetic moment of the hadron, and the term is c.m. energy contribution (the lowest Landau level) in MF for the charged mesons (note, that in this paper we discuss only s-wave hadrons and all orbital momenta are zero).

Magnetic moments in PIH formalism have been calculated in 25 () for mesons and are in good agreement with experiment and available lattice data. It is easy to see there that for massless quarks the expansion in (19) is actually done in powers of .

For the strong MF limit the situation is more complicated. Confining ourselves to the lowest Landau levels (LLL) for all quarks and antiquarks, i.e. in (9) and (14), we can separate out the hadrons, which consist of only LLL states of both quarks with . These states are MF-independent at and we shall call them “zero hadron states” (ZHS). Note, that ZHS do not possess definite total spin and isospin quantum numbers.

All other hadron states, except for ZHS, will have energies growing with MF as and therefore thermodynamically suppressed at large MF. In the limit of strong MF the dynamical masses for ZHS can be written as

(20) |

The stationary point analysis according to (16) for , yields

(21) |

The same result was obtained in 20 (); 211 (); 21 (); 27 () with the Pseudomomentum technique.

We turn now to the meson states still with zero orbital momentum and not belonging to the ZHS states, i.e. violating the equality . The resulting meson energy according to (15)-(16) for is

(22) |

which yields

(23) |

The same result occurs when with replacement Now we turn to the case when both products and are negative. In this case one obtains

(24) |

Thus we have three different asymptotic modes for s-wave meson dynamical masses in MF, classified with respect to spin projections

(25) |

We shall return to this classification later in Section 6 in our study of spin-isospin splittings in weak MF regime.

## 4 One-gluon exchange in MF

The first order perturbation correction for one-gluon exchange potential(OGE, or color Coulomb interaction) in MF entering in (5) according to 21 () is

(26) |

where , QCD parameter , and the preventing Landau singularity was calculated in 456 (). Form (26) includes screening of the OGE potential by the quark-antiquark pairs created in MF. This effect prevents the “fall-to-the-center” phenomenon for ZHS hadrons in MF, as shown in the Fig.1 and 7. One can see that the matrix element for meson saturates at and the system becomes “asymptotically free” in limit when . The driving force of the Coulomb collapse is an uncontrollable growth of the Coulomb interaction when the system is squeezed by MF forces. The role of screening of the Coulomb interaction in MF has a long story, see e.g. 30 (); 31 (); 31a () for atomic systems.

The next step is to average the potential (26) over the wave function (17) obtained by the CS method.

(27) |

Separating the integration in plane and in -direction, one has

(28) |

In the momentum space one obtains

(29) |

where are Fourier images

(30) |

(31) |

where and are given by (18). Comparing this result in case of the neutral meson with the exact one, obtained with Pseudomomentum procedure, one has to make a correction for the wave function, see Appendix B for details.

## 5 Spin-dependent corrections

A detailed review of the spin-dependent forces in PIH framework is given in 32 (). Here we only emphasize that the spin-dependent perturbative corrections arise from the correlators, where are Clifford for i-th quark constituent and are non-abelian field strength tensors.

Averaging over the stochastic gluonic background field, one has two types of corrections – the self-energy term for and color-magnetic spin-spin interaction terms for , where are quark numbers

(32) |

The self-energy correction in (32) was used in a large number of calculations 33 (), confirmed by the experimental data and lattice simulations. In case of an external MF we retain in the value , instead of , which does not change appreciably .

A different story is for the spin-spin interaction in (32). As it was shown in 211 (); 32 (); 34 (), the wave function of hadronic and atomic systems becomes “focused” at the origin by MF, i.e. for large MF value. This “magnetic focusing” phenomenon could induce the fall-to-the-center phenomenon for the lowest lying ZHS states. However, as shown in 32 (), the colormagnetic fields cannot violate the positivity of the spectra, implying that some sort of the cut-off parameter must occur in the whole perturbative series with nonperturbative background. Moreover, PIH method has a natural dimensional cutoff parameter for color field – correlation length of the vacuum gluonic background, which should be used to smear -function in (32) 32a ()

(33) |

and after the averaging with the CS meson wave function (17) one obtains the spin-spin matrix element

(34) |

Smearing procedure prevents the collapse of the meson in strong MF and it stops the unbounded fall of the total mass value in increasing MF.

It is important to notice here that the approximation of the confinement potential by the harmonic oscillator potential (6) gives too small value for and the hyperfine splitting between the non-chiral and mesons at is too small (see Fig. 3 and 7) as compared with realisitc case of linear interaction. Moreover, the pion mass at is additionally shifted down by chiral dynamics, which we shall take into account in Section 7.

## 6 Spin-isospin splittings in MF

As pointed out in Section 1, MF violates spin and isospin symmetries, therefore split into 8 states and each and states split into 4 states in MF correspondingly. Using the asymptotics (25), obtained in Section 3 for strong MF regime, one has

(35) |

Here on the l.h.s we have the standard spin-isospin configurations for mesons at zero MF, and on the r.h.s we have asymptotic classification according to (25) in strong MF for the corresponding states. The states 1)-4), 5)-8) and 9)-12) are composed of quarks and antiquarks in the combinations which yield the required spin and isospin values of and mesons at .

With increasing MF the eigenvalues of the total Hamiltonian (5) at nonzero MF demonstrate two types of phenomena: a) the mixing effect, due to spin-spin forces, equivalent to the Stern–Gerlach phenomenon, when the MF eigenstate can be expanded in two eigenstates; b) the splitting effect, when the zero MF state composed of and components, splits into two trajectories due to isospin flavor. Finally, the trajectories for charged mesons like and starting at the same mass at , split into two for .

To take into account the spin-spin interaction, we choose the basis states , , , in spin space. The states 1) and 2), that corresponds to and mesons at correspondingly, are diagonal and their dynamical masses are

(36) |

After the stationary point analysis (16) one has two sets of parameters and . The total mass of these states according to PIH formalism are given by

(37) |

gives rise to two trajectories in MF starting at meson mass at .

The behavior of states 3) and 4) corresponding to and at zero MF is more complicated. These states are composed of and combinations at . When the MF increases, the states start to mix in the mutually orthogonal combinations

(38) |

The basis vectors are equal to the and states at . The mixing phenomenon is defined by the non–diagonal spin–spin matrix elements

(39) |

The dynamical masses and the parameters , are defined by the stationary point analysis for the and the diagonal elements

(40) |

The final step is to diagonalize the total mass matrix

(41) |

and to calculate mixing coefficients

(42) |

The eigenvalues for (41) are

(43) |

The trajectory with ”+” sign in (43) starts from the mass at and grows with MF, and the trajectory with with ”-” sign corresponds to the at zero MF. The states and are mixtures of and at with mixing coefficients defined by (42).

One can define the states 1)-4) with the same isospin structure as quartet . States 9)-12) also form the quartet with the dynamics in MF the same as for if one changes spins and charge signs to the opposite. The states 5)-8) are composed of and configurations in isospin. Since the relativistic Hamiltonian is diagonal in isospin, one can split these states into two independent quartets and . The diagonal state 5) splits into two trajectories from quartet and from quartet