Meson Spectrum in Strong Magnetic Fields
We study the relativistic quark-antiquark system embedded in magnetic field (MF). The Hamiltonian containing confinement, one gluon exchange and spin-spin interaction is derived. We analytically follow the evolution of the lowest meson states as a functions of MF strength. Calculating the one gluon exchange interaction energy and spin-spin contribution we have observed, that these corrections remain finite at large MF, preventing the vanishing of the total meson mass at some , as previously thought. We display the masses as functions of MF in comparison with recent lattice data.
During the last years we have witnessed an impressive progress of the fundamental physics in ultra-intense magnetic field (MF) reaching the strength up to 1 (). Until recently magnetars 2 () were the only physical objects, where such, or somewhat weaker MF could be realized. Now MF of the above strength and even stronger is within reach in peripheral heavy ion collisions at RHIC and LHC 3 (). High intensity lasers is another perspective tool to achieve MF beyond the Schwinger limit 4 (). On the theoretical side a striking progress has been achieved along several lines. It is beyond our scope to discuss these works or even present a list of corresponding references. We mention only two lines of research which have a certain overlap with our work. The first one 5 (); 6 () is the behavior of the hydrogen atom and positronium in very strong MF. The second one 7 () is the conjecture of the vacuum reconstruction due to vector meson condensation in large MF. The relation between the above studies and our work will be clarified in what follows.
Our goal is to study from the first principles the spectrum of a meson composed of quark-antiquark embedded in MF. Use will be made of Fock-Feynman-Schwinger representation (see 8 () for review and references) of the quark Green’s function with strong (QCD) interaction and MF included. An alternative approach could have been Bethe-Salpeter type formalism. However, for the confinement originating from the area law of the Wilson loop, the use of the gluon propagator is inadequate. Numerous attempts in this direction failed because of gauge dependence and the vector character of the gluon propagator, while confinement is scalar and gauge invariant. Therefore it is sensible to use the path integral technique for QCDQED Green’s functions. This method based on the proper-time formalism allows to represent the quark-antiquark Green’s function via the Hamiltonian (see 8* () for a new derivation), and it was used in 10 () to construct explicit expressions for meson Hamiltonians without MF. In this way spectra of light-light, light-heavy and heavy-heavy mesons were computed with a good accuracy, using the string tension , strong coupling constant and quark current masses as an input 11 (),I ().
In what follows we expand this technique to incorporate the effects of MF on mesons. The latter contains: 1) direct influence of MF on quark and antiquark, and 2) the influence on gluonic fields, e.g., on the gluon propagator via loops and on the gluon field correlators determining the string tension . Since MF acts on charged objects, its influence on the gluonic degrees of freedom enters only via , however, corrections of the second type can be important, as shown in 12* (). 3) As was shown recently in the framework of our method, MF also changes quark condensate and quark decay constants etc., and in this way strongly influences chiral dynamics 12** ().
The important step in our relativistic formalism is the implementation of the pseudomomentum notion and center-of-mass (c.m.) factorization in MF, suggested in the nonrelativistic case in 12 () for neutral two particle systems. Recently the c.m. factorization was proved for the neutral 3-body system in 13 (), the situation with charged 2-body system was clarified and an approximation scheme was suggested in 13* ().
The plan of the paper is the following. Section 2 contains a brief pedagogical reminder of how the two-body problem in MF is solved in quantum mechanics. The central point here is the integral of motion (“pseudomomentum”) which allows the separation of the center of mass. Here we also show how to diagonalize the spin-dependent interaction. In section 3 we formulate the path integral for quark-antiquark system with QCDQED interaction. Then from Green’s function the relativistic Hamiltonian is obtained. Section 4 is devoted to the treatment of confining and color Coulomb terms. Here we also present the derivation of the eigenvalue equations for the relativistic Coulomb problem. In section 5 we discuss the spectrum of the system focusing on the regime of ultra-strong MF. Section 6 contains the discussion of the results, comparison with lattice calculations, drawing further perspectives and intersections of our results with those of other authors 5 (); 6 (); 7 ().
Ii Pseudomomentum and Wavefunction Factorization
The total momentum of mutually interacting particles with translation invariant interaction is a constant of motion and the center of mass motions can be separated in Schroedinger equation. It was shown 12 () that a system embedded in a constant MF also possesses a constant of motion –“pseudomomentum”. As a result for the case of zero total electric charge the c.m. motion can be removed from the total Hamiltonian. The simplest example is a two-particle system with equal masses and electric charges . We define
Straightforward calculation in the London gauge yields
One can verify that the following “pseudomomentum” operator commutes with the Hamiltonian (2)
This immediately leads to the following factorization of the wave function (WF)
For the oscillator-type potential the problem reduces to a set of three oscillators, two of them are in a plane perpendicular to the magnetic field and their frequencies are degenerate, while the third one is connected solely with .
Next we briefly elucidate the spin interaction in presence of MF. The corresponding part of the Hamiltonian may be written as
where and . Diagonalization of yields the following four eigenvalues e.g. for system, comprising both and levels.
where we assume that is aligned along the positive -axis and . In a strong MF when spin-spin interaction becomes unimportant and . For the lowest level this corresponds to a configuration when the spin of negatively charged particle is aligned antiparallel to , and the spin of the positively charged one – parallel to . This means that the spin (and isospin) are no more good quantum numbers and eigenvalues (6) correspond to the mixture of spin 1 and spin 0 states. As a result the state will split into 4 states (two of them coinciding ). Till now we treated a nonrelativistic system, to incorporate relativistic effects we shall exploit the path integral form of relativistic Green’s functions 8 (); 8* ().
Iii Relativistic Green’s function and effective Hamiltonian
The derivation of the relativistic Hamiltonian of the system in MF consist of several steps. The first one is the 4d relativistic path integral for the Green’s function. The starting point is the Fock-Feynmann-Schwinger (world-line) representation of the quark Green’s function 8 (). The role of the “time” parameter along the path of the -th quark is played by the Fock-Schwinger proper time Consider a quark with a charge in a gluonic field and the electromagnetic vector potential , corresponding to a constant magnetic field B. Then the quark propagator in the Euclidean space-time is
The path-integral representation for 8 () is
Here and are correspondingly gluon and MF tensors, are ordering operators, . Eqs. (7-10) hold for the quark, , while for the antiquark one should reverse the signs of and . In explicit form one writes
Next we consider system born at the point with the current and annihilated at the point with the current . Here and denote the sets of initial and final coordinates of quark and antiquark. Using the nonabelian Stokes theorem and cluster expansion for the gluon field(see 11 () for reviews) and leaving the MF term intact, we can write
and for vector currents, for pseudoscalar currents, while
where and is an area element of the minimal surface, which can be constructed using straight lines, connecting the points and on the paths of and at the same time 8 (); 10 (). Note, that operator actually do not participate in field averaging procedure: as was shown in 13** (), the following replacement is valid: .
As a result of the first step the Green’s function is represented as a 4d path integral (including Euclidean time paths) and in addition also integrals over proper times . In the second step one introduces monotonic Euclidean time , where , so that , where is fluctuation of time trajectory around . This new variable is an ordering parameter for trajectories , and proper times transform into physical parameters – virtual and energies , so that .
Combining for simplicity all fields into one Wilson loop , one can rewrite the Green’s function in new variables as
The final step is the use of the Wilson loop dynamics to express all dynamics in terms of instantaneous interaction. Indeed, the quadratic field correlator in (14) is represented through two scalar functions and (see, e.g., 11 (); SS () for details), first of them is responsible for confinement, while the second one gives one gluon exchange (OGE) potential. So, for the case of zero quark orbital momenta with the minimal surface, discussed above, integrating over relative time in one obtains a simple instantaneous answer for spin-independent (SI) part of ,
containing and . Here is the QCD string tension, in our calculations.
First we need to find the Hamiltonian of the system at . To this end we define the Euclidean Lagrangian . We write . Then all terms in the exponents in (12), (14) and (17) can be represented as and thus we arrive at the following representation:
with the action
Here is the –th component of the QED vector potential. The next step is the transition to the Minkowski metric. This is easy, since confinement is already expressed in terms of string tension. We have , and
Explicit expression for Hamiltonian without spin-dependent terms is
The c.m. projection of the Green’s function yields
where and are eigenfunctions and eigenvalues of Hamiltonian . At large the integral over can be taken by the stationary point method, and hence the effective energies are to be found from the minimum of the total mass , as it was suggested in 10 (). To introduce the minimization procedure and to check its accuracy we shall begin by the calculation of the eigenvalues of one and two quarks in MF, and the energy of the ground state of a relativistic charge in the atom in the next section, reproducing the known exact results.
We have the following equations defining from the total mass
For a single quark in MF the first of the above equations gives
Then the minimization over yields the correct answer
Now we turn to the case of system and introduce the coordinates which are the generalization of (1):
It is convenient to introduce the following two additional parameters
Let us consider the case of neutral meson, so that . Then the total Hamiltonian may be written as
and contains self–energy and spin–spin contributions, which come from unaccounted spin-dependent terms of . One can verify, that the “pseudomomentum” operator in (3), introduced in section II, commutes with and hence we can again separate the c.m. motion according to the ansatz (4):
Then the problem reduces to the eigenvalue problem for with the Hamiltonian having the following form:
For the system has a rotational symmetry and the c.m. is freely moving along the -axis. Here we shall consider a state with zero orbital momentum . As a result is replaced by a purely internal space operator
To test our method we put and arrive at the equation
Consequent minimization of in , as in (26), yields the expected answer for the two independent quarks,
We turn now to the particular case of charged two-body system in MF, and also , when exact factorization of and can be done. In this case, for and , the Hamiltonian has the following form 8* ()
Iv Treating confinement and gluon exchange terms. The absence of the magnetic QCD collapse
From(33), (36) it is clear, that inclusion of and in leads to a differential equation in variables which can be solved numerically. However, in order to obtain a clear physical picture, we shall represent in a quadratic form. This will allow to get an exact analytic solution in terms of oscillator functions with eigenvalue accuracy of the order of . The OGE contribution will be estimated as an average , thus yielding an upper limit for the total mass.
For we choose the form
Here is a positive variational parameter; minimizing w.r.t. , one returns to . We shall determine corresponding to , and to define an additional condition
will be added to (24). As a result will be the final answer for the mass of the system, neglecting the contribution. The difference of the exact numerical solution from that obtained with the genuine potential does not exceed . The solution of the equation for the ground state is
where . As we shall see below, for the lowest mass eigenvalue with , one has and the system acquires the form of an elongated ellipsoid. Similar quasi–one–dimensional picture was observed before for the hydrogen–like atoms in strong MF 5 (); 6 (). In such geometrical configuration manifests itself in a peculiar way, again similar to what happens in hydrogen, or positronium atoms, and as was shown in 12* () in QCD the outcome is also similar to the case of QED, with the screening of the diverging effects.
We turn now to the OGE term to treat it in our formalism. As a starting point we present another check of our approach, namely we shall obtain the ground state energy of two relativistic particles with opposite charges without MF interacting via the Coulomb potential. The corresponding Hamiltonian reads then , and for we have
Minimizing in in the limit (the hydrogen atom), one obtains
which coincides with the known eigenvalue of the Dirac equation.
In our case one can calculate the expectation value of with the asymptotic freedom and IR saturation behaviour in –space (see 16 () for a derivation and a short review)
where is the Fourier transform of squared wave function and . Estimating the integral in (46), for , i.e. for one obtains for massless quarks
With increasing the upper bound for the mass is boundlessly decreasing. The exact eigenvalue should lie even lower.
This situation is similar to the hydrogen atom case, where diverges as , and in this case loop contribution to the photon line stabilizes the result (the “screening effect” 6 (), 7 ()). In our case the loop contribution to the OGE term can by written in a similar way, adding to the gluon loop also the Lowest Landau Level (LLL) of the in the MF,
where . Calculating now the average value of (48),
one obtains saturation of at large , as shown in Fig. 1, eliminating in this way the possible “Color Coulomb catastrophe”, discussed in the first version of this paper Simonov:2012if ().
V Meson masses in magnetic field
Our next task is to calculate analytically the mass of a meson. We have to solve the equation
The resulting mass for neutral meson without spin-dependent contribution from is