CCNY-HEP-18/4

August 2018

Casimir Effect in (2+1)-dimensional Yang-Mills Theory as a

Probe of the Magnetic Mass

Dimitra Karabali and V.P. Nair

Department of Physics and Astronomy

Lehman College of the CUNY

Bronx, NY 10468

Physics Department

City College of the CUNY

New York, NY 10031

E-mail:

Abstract

We consider the Casimir effect in a gauge-invariant Hamiltonian formulation of nonabelian gauge theories in dimensions, for an arbitrary gauge group. We show that the result is in good agreement with recent lattice simulations. We also argue that the Casimir effect may be viewed as a good probe of magnetic screening effects in -dimensional gauge theories at high temperatures.

## 1 Introduction

Yang-Mills gauge theories in two spatial dimensions can be viewed as a guiding model for the more realistic, but also more complicated, -dimensional gauge theories. The -dimensional theories have nontrivial dynamical content and propagating degrees of freedom making them a better model than Yang-Mills theories in dimensions, yet they are still somewhat more amenable to mathematical analysis compared to their -dimensional counterparts. The Euclidean -dimensional theory, the Wick-rotated version of the -dimensional theory, can also be of direct relevance to the high temperature limit of the -dimensional theory [1]. In particular, the mass which appears as a propagator mass in dimensions can be taken as the high temperature value of the magnetic screening mass. With these motivations, for many years, we have been pursuing a Hamiltonian approach to the nonpertrubative aspects of Yang-Mills theories in dimensions [2, 3, 4]. This article will be in the nature of continued work along these lines, focusing on the Casimir effect in Yang-Mills theories in (2+1) dimensions. This was also inspired by the recent lattice simulations of the Casimir effect for the gauge theory reported in [5]. We will argue that the Casimir effect in the -dimensional Yang-Mills theory can be viewed as a probe of the magnetic screening mass in the pure QCD plasma in dimensions at high temperatures. This will also furnish a calculation for a general gauge group which can, hopefully, be tested in lattice simulations in the near future.

We begin with a brief recapitulation of the salient points of our Hamiltonian analysis. We considered the gauge, with the spatial components of the gauge potentials parametrized as

(1) |

Here we use complex coordinates , with , , and is an element of the complexified group ; i.e., it is an -matrix if the gauge transformations take values in . Gauge transformations act on via , where is an element of the group , say, for example, . Wave functions are gauge-invariant functionals of , with the inner product given as

(2) |

Here is the Wess-Zumino-Witten action given by

(3) |

In equation (2), is the Haar measure for which takes values in . Also denotes the value of the quadratic Casimir operator for the adjoint representation; it is equal to for . The Hamiltonian and other observables can be expressed as functions of the current of the WZW action, namely,

(4) |

(We have included a prefactor involving the coupling ; this is useful for later calculations.) The explicit formulae worked out in references [2, 3, 4] is given as , where

(5) | |||||

where . Regularization issues have been discussed in some detail in the cited references.

The basic strategy we used was to solve the Schrödinger equation keeping all terms in at the lowest order, treating as a perturbation. In ordinary perturbation theory (carried out using our Hamiltonian formulation), one would expand in powers of the hermitian field ; in addition, since we would also expand in powers of . In our case, we keep the term involving even at the lowest order. So even if we expand in terms of , our expansion would correspond to a partially resummed version of what would be normal perturbation expansion. Formally, we keep and as independent parameters in keeping track of different orders in solving the Schrödinger equation, only setting at the end. The lowest order computation of the wave function in this scheme was given in [3] and gave the string tension for a Wilson loop in the representation as , being the quadratic Casimir value for the representation . We have also considered corrections to this formula, taking the expansion to the next higher order (which still involves an infinity of correction terms) and found that the corrections were small, of the order of to [6]. The resulting values for the string tension agree well with the lattice estimates [7, 8].

## 2 The Casimir energy for parallel wires

There is an important feature which emerged from our analysis, which is very useful for the present purpose [14]. We can absorb the factor in (2) into the definition of the wave function by writing . The inner product for the ’s will involve just the Haar measure without the factor. However, the Hamiltonian acting on will now be given by . We can expand as , with the field being hermitian. As mentioned earlier, this “small ” expansion is suitable for a (resummed) perturbation theory. The Hamiltonian is then

(6) |

where . This is clearly the Hamiltonian for a field of mass with the corresponding vacuum wave function

(7) |

The Hamiltonian (6) corresponds to the action

(8) |

These results show that the propagator for the gauge-invariant component of the gluon field is the same as that of a massive scalar field with mass equal to . Further, the parametrization (1) of the gauge potentials becomes, in the small -expansion

(9) |

In the case of a perfectly conducting plate, the boundary condition is that the tangential component of the electric field should be zero. In other words, we need

(10) |

where is the unit vector normal to the plate. This is also the condition used in [5]. In terms of the parametrization in (9), focusing just on the gauge-invariant part , this means that we need

(11) |

Since the time-derivative does not affect the spatial boundary conditions, this is equivalent to imposing Neumann boundary conditions on the scalar field or, equivalently, on . The end result is that, within this approximation of keeping , but expanding the field to the lowest order in , the Casimir energy will be given by that of a massive scalar field with Neumann boundary conditions on the plates.

We now consider the standard arrangement of two parallel plates (or rather wires since we are in two spatial dimensions) which are of infinite extent in the -direction and are normal to the -direction. The wires are separated by a distance . We take the range of to be , with eventually. The fields in the region between the wires have the mode expansion

(12) |

This is consistent with the Neumann boundary conditions. We note that the Casimir energy of massive scalar fields for the parallel plate geometry with Dirichlet boundary conditions is known [15]. The result for Neumann conditions is essentially the same. Here we reproduce the result and express it in a form more suitable for comparison with lattice estimates. With the mode expansion (12), the action (8) becomes

(13) |

where . The diagonalization of the Hamiltonian is trivial, yielding the unrenormalized zero-point energy

(14) | |||||

Using the Poisson summation formula we get

(15) |

The first two terms in this expression are divergent and they have to be subtracted. The first term is independent of the distance between the wires, corresponds to a self-energy contribution, and gets subtracted when we consider the energy shift , which is the relevant renormalized quantity of interest. The second term is proportional to the spatial volume and is part of a uniform spatial density of vacuum energy which must also be subtracted out in the renormalized expression for the Casimir energy. The final renormalized expression is thus

(16) |

Doing the -integration and using the variable transformation , we find

(17) | |||||

Using the following expression for modified Bessel function ,

(18) |

we can rewrite the Casimir energy as

(19) |

where is the polylogarithm function

(20) |

We may note that, in the limit, the expression (19) agrees with the well known result for a massless scalar in dimensions,

(21) |

There are other equivalent ways to arrive at result (19). Using

(22) |

we can carry out the summation over (in (14)) to obtain

(23) | |||||

where and in the second line we used polar coordinates in the -plane and integrated over the angle and . A further substitution , and , reduces this to

(24) |

Expansion of the integrand in powers of gives the result (19) in terms of the polylogarithms.

It is useful to write the energy (19), for our case, in terms of the string tension corresponding to the fundamental representation. This has been calculated in [3]. Ignoring the small corrections discussed in [6], this is given by

(25) |

We may thus write , where . The Casimir energy is thus given by

(26) |

This is the main result of this paper. It holds for an arbitrary compact group; for the case of , we have , . There will be corrections to this formula due to the fact that we have neglected interactions involving cubic and higher powers of and due to the corrections to the string tension in the expression for in terms of . Nevertheless, the fact that string tension given in (25) to the lowest order in our expansion scheme is in good agreement with lattice calculations [7] suggests that the formula (26) can be a good estimate of the Casimir energy.

We have the Neumann boundary condition on the field for perfectly conducting wires, as mentioned before. But if we choose different boundary conditions, the result can be different. Formula (26) holds for the field obeying Neumann conditions at both wires or Dirichlet conditions at both wires. The Dirichlet condition is equivalent to the magnetic field (which is in our approximation) vanishing at the wire. If we consider the Neumann condition at one wire and the Dirichlet condition at the other, the modes involved are of the form . The Casimir energy is now given by

(27) |

The renormalized finite Casimir energy now works out to be

(28) | |||||

Notice that, as expected, this corresponds to a repulsive force because the arguments of the polylogarithms have changed sign. The two energies from (26) (same as ) and from (28) are shown in Fig. 1 for the case of .