Casimir force in a Lorentz violating theory

I. INTRODUCTION

The search for a theory beyond the minimal SU(3) C × SU(2) L × U(1) Y standard model (SM) is motivated by the fact that, although phenomenologically successful, the SM suffers from some theoretical inconsistencies, and from some long standing unresolved problems.

More general scenarios exist, in which SM is viewed as a low-energy limit of more fundamental theory, which should be able to provide a quantum description of gravitation.

An interesting alternative at the Planck scale is the possibility that the new physics scenario involves a violation of Lorentz symmetry. In particular, in the context of string theories, it has been shown that spontaneous breaking of Lorentz symmetry may occur with Lorentz-covariant dynamics [1]. In these theories, interactions are triggered by nonzero expectation values for Lorentz tensors, because spontaneous breaking of the higher-dimensional Lorentz invariance is expected in any realistic Lorentz-covariant fundamental theory involving more than four space-time dimensions. If the breaking extends into the four macroscopic spacetime dimensions, the Lorentz symmetry violation could appear at the level of the SM.

Because the breaking is spontaneous, Lorentz symmetry remains a property of the underlying fundamental theory. Another important property of the spontaneous breaking is that the theory remains invariant under observer Lorentz transformations, i.e., under rotations and boosts of an observer's inertial frame. In addition to that, conventional quantization, hermiticity, gauge invariance, power-counting renormalizability, and the expected microcausality and positivity of the energy are also maintained. Note also that there are other ways to break Lorentz symmetry. It can, for example, occur dynamically in quantum field theories [2], or via a CPT anomaly in compact spaces [3].

Such a framework for examining the effects of spontaneous Lorentz breaking in the context of a low-energy effective theory, known as the Standard-Model Extension (SME), has been developed explicitly by Colladay and Kostelecký first without gravity [4] and later by Kostelecky with gravity [5]. Gravitationally coupled SME is described in non-Minkowski spacetimes and leads to spacetime-dependent coefficients; and recently, the pure gravity sector of the model has been studied [5]. In this paper, we confine the framework into the minimal SME without gravity. 1 General reviews of the model and the status of the recent experimental investigations can be found in Refs. [6,7]. An extensive analysis of the implications of the model for electrostatics and magnetostatics in SME, as well as discussion of its QED sector can be found in [8,9,10].

The SME [4] has been the focus of various experimental studies, including ones with photons, from radiative corrections, photon splitting and vacuumČerenkov and synchrotron radiation [9,10,11], electrons [12,13,14], protons and neutrons, including baryosynthesis and nucleosynthesis in [15,16,17], CPT phenomena in mesons [18], muons [19], neutrinos [20], and Higgs bosons [22]. Though no evidence for Lorentz violation has yet been found, only less than a half of the possible tests involving light and ordinary matter (electrons, protons, and neutrons) have been performed, while some of the other sectors remain virtually unexplored.

Our goal here is to provide additional theoretical predictions regarding the quantum vacuum in this extended theory, in particular we concentrate on calculating the effects of the minimal SME on the Casimir force.

Zero-point fluctuations in quantum fields give rise to macroscopically observable forces between material bodies, the so-called Casimir forces [23]. In general, the Casimir effect can be defined as the stress (force per unit area) on bounding surfaces when a quantum field is confined in a finite volume of space. The boundaries can be material media, interfaces between two phases of the vacuum, or topologies of space (such as in higher dimensional theories). The simplest test is measuring the force between two parallel conducting plates.

Simply imagined, the vacuum is a sea of waves of energy of all possible length, while between the plates only waves whose wavelengths exactly fit between the plates are present. One calculates all of the zero-point energy between the plates, and the result (which is divergent) is regulated by subtracting the value of the energy when no boundaries are present [24].

Experiments testing the Casimir force are very precise tests of the predictions of the field theory. Several experimental attempts have been pursued during various decades since its theoretical prediction, for unambiguously verifying Casimir's formula [25]. Early searches have been successful only in a particular geometry, namely in a cavity constructed by a plane surface opposing a spherical one. Pioneering measurements by van Blokland and

Overbeek [26] in such a configuration resulted in the observation of the associated Casimir force, and in its detailed comparison to the Lifshitz theory [27], by taking into account finite conductivity effects. More recently, these measurements have been performed using torsion balances [28], atomic force microscopes [29], and high precision capacitance bridges [30].

The latter two experiments have reached 1% precision, more precise determinations being limited by theoretical uncertainties. Sparnaay [31] has attempted the first experiments with parallel plates configuration, while Bressi et al [32] have recently reported a measurement of the coefficient of the Casimir force between parallel conducting surfaces in the 0.5 − 3.0 µm range with 15 % precision:

In this work, we investigate the Casimir effect in the vacuum of the SME. The Casimir force is a very sensitive measurement of the quantum fluctuations of the field at the macroscopic level. Since the stress between two parallel conducting plates separated by a distance of 1 µm is K C /(1µm) 4 , where K C is of order of O(10 −27 N m 2 ), and with the hope that future experiments will improve the existing precision, this measurement should be tested against any new theory. We evaluate the stress between parallel plates, leaving evaluations of quantum fluctuations for dielectric media and different topologies for later discussions.

Our paper is organized as follows. In Section II we summarize the SME by giving only the relevant terms in the Lagrangian. In the subsequent sections, we calculate the Casimir force for scalar fields with two different frequencies of propagation, according to the SME (Section III) in the case of both CPT-even and CPT-odd Lorentz violation terms. We examine the correction for the fermion field (Section IV) and conclude and summarize our results in (Section V), where we also discuss the possibility of observing the deviations from the measured Casimir force.

II. THE LORENTZ VIOLATING MODEL

The minimal extension of the Standard Model (SME) as given in [4,5,6,7] contains all possible Lorentz-violating terms that could arise from spontaneous symmetry breaking at a fundamental level, but that preserve SU(3) × SU(2)× U(1) gauge invariance and powercounting renormalizability. All terms that are even or odd under CPT are explicitly given in Ref. [4] and will not be repeated here. We rather concentrate on the relevant part of the theory, namely the extended quantum electrodynamics drived from SME.

The general form of a Lorentz-violating term involves a part constructed from the basic fields in the standard model, whose strength is given by a coupling coefficient. This imposes various limitations on the possible structures of both the operators and the couplings. Taken together, these requirements place significant constraints on the form of allowed terms in the SME. Taken from the full SME that contains all known particles, the Lagrangian involving the Dirac field ψ of the electron and the electromagnetic field F µν can be written as

where H.c. is the hermitian conjugate of the fermion part. The Lorentz violating parameters in the photon sector are included in the components of the tensors (k F ) κλµν (CPT-even) and

(k AF ) κ (CPT-odd). For the electron, the Lorentz violating parameters are encoded in Γ ν and M as

where the parameters in Γ ν are dimensionless while the ones in M have dimension of mass.

Similarly, in the photon sector, (k F ) κλµν is dimensionless and (k AF ) κ has dimension of mass.

The parameters e ν , f ν and g κµν which are not extractable directly from SME are taken to be zero or very suppressed due to the renormalizibilty and gauge invariance requirements.

a µ can also be set to zero since it is not physical (it can be absorbed by using gauge transformations). The parameters c µν and d µν are traceless 2 and H µν is antisymmetric. In the photon sector, (k F ) κλµν has the symmetries of Riemann tensor plus a double traceless constraint, giving 19 independent components. Even though the components of (k AF ) κ are strongly constrained by cosmological observations, we allow them to be non-zero in our theoretical investigation of Casimir force.

After presenting the terms in the Lagrangian of the extended QED, we are concerned with the modified QED vacuum. So, one can re-examine the basic features of electromagnetic radiation in this framework. Working with the standard parametrization of the plane wave with wave 4-vector p α = (p 0 , p): F µν (x) = F µν (p)e −ipαx α , the equation determining the dispersion relation and the electric field E in the SME is the modified Ampère Law

3)

The dispersion relation is obtained by requiring the vanishing of determinant of M jk , and one obtains two modes for the radiation field [4]:

where

For vanishing k AF coefficients, and to leading order in the coefficients k F for Lorentz violation it becomes:

Here p 2 ρ and p 2 σ are observer Lorentz scalars, which implies ρ and σ are scalars under observer rotations. Note that in the leading order,p µ = (1, p/| p|) is assumed.

The dispersion relation (2.7) has two solutions, corresponding to electric field values E ± .

In conventional electrodynamics, the dispersion relation is p 0 = | p| and thus the propagation is independent of the polarization. However, in the present case the propagation is goverened by two specific modes E ± , with the general solution to (2.3) being any linear combination of the two, leading the phenomenon known as birefringence [4]. This represents a fundamentally different description of photon fields propagating through the vacuum from conventional QED. The vacuum Lorentz breaking extension of electrodynamics resembles more electrodynamics in moving media [4,8]. Thus we expect changes in the vacuum energy, which impact on the evaluation of the Casimir force.

III. THE SCALAR FIELD

Two main methods exist in the literature to calculate the vacuum energy of a field confined between parallel plates. In the simplest and most transparent method, one simply sums over all the modes of the fields in between the plates and then regularizes the result by subtracting the energy in the free space. A more rigorous technique uses consistent Green's functions and is presented in the following subsection. Summing the modes correctly produces the finite, observable force starting from a divergent formal expression, without any explicit subtractions, and is therefore of great utility in practice. In what follows, we assume that the photon field can be approximated by two scalar fields of frequency

. Following normal procedure, we evaluate the correction to the usual Casimir energy by taking first (k AF ) κ = 0, (k F ) κλµν = 0 (Case I), then (k AF ) κ = 0, (k F ) κλµν = 0 (Case II).

A. Case I: k AF = 0

Summing of the Modes

For simplicity, the electromagnetic field between parallel, uncharged, perfectly conducting plates in the SME due to the CPT-even Lorentz-violating parameter in the Lagrangian is seen as two massless scalar fields φ confined between two parallel plates separated by a distance D. In conventional QED, one accounts for the two polarization modes by simply multiplying the scalar result by 2. In SME, the two scalar fields will propagate with different frequencies, due to birefringence. The scalar field satisfies Dirichlet boundary conditions on the plates, that is:

The Casimir force between the plates results from the zero-point energy per unit transverse

where ω n,1 and ω n,2 represent the two zero-point energies associated with the surface modes, labeled by the positive integer n, [33] of the electromagnetic fields in the SME. The formalism bears some resemblance to the treatment of the force in a dispersive medium of effective "dielectric" constant ǫ where

The so-called surface modes are associated with the zero of the wave number:

where we set = c = 1, where k is the transverse momentum. for the simple case where ω ± ≡ p 0 ± = (1 + ρ ± σ)ω, ω being the frequency in the vacuum in the absence of dispersion. In this case (which corresponds to the conventional QED vacuum), the expression for the frequency of the field would be ω = | k| for both surface modes; while in SME the frequencies of the two surface modes are ω 1,2 = ω √ ǫ ± , and the energy of the zero modes becomes:

where we denoted Λ = 1 2

and k z is obtained from the boundary conditions.

To evaluate Eq. (3.4) we employ dimensional regularization. We let number the transverse dimensions be d, which we will treat as a continuous, complex variable. We also employ the Schwinger proper-time representation for the square root, so that we have

where we have used the Euler representation for the Gamma function. We follow the standard procedure and carry out the Gaussian integration over k, use the Euler representation, and use he definition of the Riemann zeta function ζ to obtain:

When d is an odd integer, this expression is indeterminate, but we use Γ z

and obtain the final result for the energy per unit area in the transverse direction in SME

where we used ζ(4) = π 4 /90. The force per unit area between the plates is obtained by taking the negative derivative of u with respect to D:

The above result represents the Casimir force due to two scalar fields of frequency ω + and ω − . Comparing this to the classic result of Casimir [23]:

the deviation in SME is given by the multiplicative correction factor Λ. For ρ, σ ≪ 1, Λ ∼ = 1 − ρ, so the expression is simplified to

Thus the lowest-order prediction of the SME is a reduction in the Casimir force by ρf em .

Scalar Green's Function

We derive the result of the previous subsection by the Green's function approach for a scalar field. A more general indexed Green function for the photon in the SME framework, in the Coulomb gauge, has been given in [8]. We start from the equation of motion of a massless scalar field φ produced by a source K

from which we deduce the equation satisfied by the corresponding Green's function

We introduce a reduced Green's function g(z, z ′ ) using the Fourier transformation

where we have suppressed the dependence of g on k and ω, and have allowed z on the right hand side to represent the coordinate perpendicular to the plates. The reduced Green's function satisfies 15) where

represent the two scalar modes. Equation (3.34) is to be solved subject to the boundary conditions (3.1), or

The form of the solution, obtained by the discontinuity method [24] for each of the two scalar modes, is (3.17) where g is continuous at z = z ′ , but its derivative has a discontinuity, which is:

The solution to this system of equations is 21) which gives for the reduced Green's function 22) where z > (z < ) is the greater (lesser) of z and z ′ .

From the Green's function we can calculate the force on the bounding surfaces by evaluating the stress tensor. For a scalar field, the stress tensor obtained from 23) where the Lagrange density is

The normal-normal component of the Fourier transform of the stress tensor on the boundaries is (3.25) However this expression will not give rise to a finite result [24], because we did not consider the discontinuity in T zz . The corresponding normal-normal component of the stress tensor 26) Thus the correct expression for the force per unit area on the conducting surface is obtained integrating over all possible frequencies and momenta, and using complex frequencies ω → iξ, λ ± → iκ ± :

. (3.27) We evaluate this integral using polar coordinates:

Here A d+1 is the surface area of a unit sphere in d + 1 dimensions. The force per unit transverse area in the SME is

which, for d = 2, using Γ(2) = 1 and ζ(4) = π 4 90 gives 30) which confirms the formula obtained before, (3.9).

B. Case II:

If one starts from the photon part of Lagrangian (2.1) by setting k F = 0, the equation of motion is modified to:

This expression is similar to the one obtained in the analysis of the Casimir Effect in two dimensions (the so-called Maxwell-Chern-Simons Casimir Effect [24]).The gauge field is effectively massive, and it satisfies the equation:

with the factor k AF playing the role of the mass. The reduced Green function 33) satisfies the equation: 34) where λ 2 I,II = ω 2 I,II − k 2 , ω I,II = (1 ± τ )p 0 . This gives, for the reduced Green function:

where z > (z < ) is the greater (lesser) of z and z ′ and κ I,II = λ 2 I,II − (k AF ) 2 . The Casimir force becomes (with the usual Schwinger change ω → iξ, κ → iρ):

(3.36)

Neglecting terms of higher order in σ, the equation can be transformed, working in polar coordinates, into:

Setting x = 2D ρ 2 + (k AF ) 2 , d = 2, and approximating, for small values of (k AF D)

which gives, expressed in terms of the usual Riemann zeta functions:

(3.40)

One recognizes in the first term the usual scalar Casimir force, while the second represents the correction from the CPT-odd terms in the photon Lagrangian, which has and different dependence on the separation between the plates (1/D 2 ). However, it is very unlike to have any observable affect since k AF is constrainted to have very tiny values and appears squared in f s ; also the weaker separation dependence makes the sensivity to k AF even weaker.

IV. THE FERMION FIELD

By taking the fermion part of the Lagrangian (2.1) and setting e ν , f ν and g κµν to zero, the modified Dirac equation can be obtained as:

with D µ = i∂ µ + ieA µ , and we assume a conventional photon sector.

To reproduce the boundary set for the Casimir effect between two plates for massless fermions, take V = V 0 θ(x) and A µ = 0, and set ψ(r, t) = ψ(r)e iωt .

where n µ represents an outward normal at a boundary surface. Note that neither a µ , nor b µ appears in the boundary condition. Taking V 0 → ∞, we obtain the boundary condition corresponding to the Dirac Equation for the SME:

Because the parameter a µ is not physical and can be eliminated (which is equivalent to shifting the zero-point energy) and that Dirac equation is still complicated and difficult to disentangle when all parameters are swiched on at the same time, we consider the simplest non-trivial case by analyzing the effect of nonzero b µ . Under these simplifying conditions, the Dirac equation becomes: A. Summing the modes

The easiest, but not the most rigorous method, is to sum the modes. We introduce a Fourier transform in time and the transverse spatial directions,

In what follows we choose a representation of the Dirac matrices in which iγ 5 is diagonal.

In 2 × 2 block form, the representation for the γ 5 , γ 0 is: 8) and the explicit form of all the other Dirac matrices follow from γ = iγ 0 γ 5 σ. We obtain two equations (taking for simplicity )

where the subscripts indicate the eigenvalues of iγ 5 and u and v are eigenvectors of σ 3 with eigenvalue +1 or −1, respectively. In the above, we used the notation

x ±ib y . The equations simplify considerable if we take k along the y axis and b = (0, b x , 0, 0), (or similarly, k along the x-axis, and b along the y axis) and we obtain: 11) where

x . From the boundary conditions

When we compute the zero-point energy, we must sum over odd integers, remember that there are two modes, and taking into account the minus sign (for fermion energy):

and ζ(−3) = −B 4 /4 = 1/120, we obtain the Casimir energy for a fermion in the SME: 4.14) and correspondingly, the Casimir force

The first term in this expression represents the conventional fermion Casimir force. The second term represents the shift due to the Lorentz-violating terms in SME.

B. Green's Function Method

A more controlled calculation starts from the equation satisfied by the Dirac Green's function, 16) subject to the boundary condition

We introduce a reduced, Fourier-transformed, Green's function, 18) which satisfies

Introducing the components of g corresponding to the +1 or −1 eigenvalues of iγ 5 , 20) we insert these Green's functions into the vacuum expectation value of the energy-momentum tensor. The latter is

where L is the fermionic part of L QED in (2.1) with only b µ nonzero. We take the vacuum expectation value by the usual replacement ψψγ 0 → −iG, where G is the fermionic Green's function, and obtain 22) where

x . The force per unit area is

Introducing polar coordinates k = κ sin θ and (ξ 24) we obtain:

As in [24], we omit the first term in the last square bracket, as the same term is present in the vacuum energy outside the plates. Using 26) we find

x (4π) (2π) 3 1 (2D) 2 ζ(2)Γ 2( 4.27) with A 3 the area of the unit sphere in 3 dimensions; which is, indeed, 28) the same as (4.15).