The Neutrino Casimir Force

In the low energy effective theory of the weak interaction, a macroscopic force arises when pairs of neutrinos are exchanged. We calculate the neutrino Casimir force between plates, allowing for two different mass eigenstates within the loop. We also provide the general potential between point sources. We discuss the possibility of distinguishing whether neutrinos are Majorana or Dirac fermions using these quantum forces.

1 Introduction on the order of a micron, much larger than the atomic scale. In the case of planar geometry, the potential is dominated by long wavelength contributions and therefore it is not obvious how the confusion theorem applies. It is with these motivations that we study the neutrino Casimir force in the plate-plate and point-plate configurations. These can then serve as approximations of the force in more evolved geometries [25].
For completeness, we study the general point-point neutrino-induced quantum force in Sec. 2. The evaluation uses the standard momentum-space formalism. We then introduce a mixed position-momentum space formalism and present the plate-plate and plate-point calculations in 3. We discuss the results in Sec. 4. for Majorana neutrinos. The link to the 2-component fermion notation is given in App. A. The g V ij and g A ij coupling matrices depend on the SM field and on the neutrino generation. They are given in App. A for completeness.
In this section, we present the force between two nonrelativistic fermions ψ arising from the exchange of two neutrinos, ν i and ν j . Similar results have already been presented in the literature, see for instance [15,16]. Here we present the most complete result including the spin-dependent part of the potential.
The calculation starts from the scattering amplitude in 4-momentum space. This formalism has been used in the literature (see e.g. [19,26] for details), so we only present results here. See App. B for additional details.
We introduce a discrete variable to distinguish between the Majorana and Dirac cases: The full potential can be written using discontinuities (noted D [f ]) across branch cuts of a basis of f mn functions which come from evaluating the loop integral. In this basis, the full potential is given by where σ A denotes the Pauli matrices acting on the spinors of source fermion A. Latin indices, such as a, b, ..., are summed over 1, 2, 3. We refer to the V ij as partial potentials. Summing over all combinations of neutrinos from the 3 generations yields the full quantum potential from neutrinos.
We can perform the integral exactly for the diagonal terms (m i = m j ≡ m). These partial potentials take on the form where we have introduced the Meijer G-function The spin-independent piece of (2.6) is given by which corresponds to a repulsive force and is consistent with the literature (e.g. [15,16]). At short distances mr 1, Dirac and Majorana predictions converge to as expected from the confusion theorem.

The Neutrino Casimir Force
Here we consider the quantum force between extended sources. Focusing on nonrelativistic, unpolarized sources formed by the SM fermions, we haveψγ µ γ 5 ψ ≈ 0,ψγ µ ψ ≈ δ µ0 ψ † ψ = δ µ0 n(x) were n(x) is the number density operator. We denote by J(x) the density expectation value in the presence of matter, J(x) = Ω|n(x)|Ω . We can write effective neutrino Lagrangians in the presence of such nonrelativistic static matter, We assume the matter density is compound of two pieces with density J 1 , J 2 , separated by a distance L. The full matter density is J = J 1 + J 2 . The potential between these two sources can be obtained by varying the quantum vacuum energy of the system with respect to L. In case of weak coupling, the potential is given by the leading term of the one-loop functional determinant. 3 See App. C and Ref. [27] for details. We find the potential induced by the neutrinos between extended sources J 1 and J 2 to be is the Feynman propagator of 4-component fermions. The trace is in spinor space. Notice that one of the integrals is in 3d space while the other is in spacetime. This reflects the fact that the quantum force is intrinsically relativistic.
In the limit of pointlike sources

Potential Between Plates
We consider the sources are infinite plates with separation L. The plates are taken to have number densities n 1 and n 2 and are orthogonal to the z direction, The two transverse spatial coordinates are denoted by x , hence A naive method to obtain the plate-plate potential would be to directly integrate the general point-point result (2.5). This is however rather challenging in the case of different masses. We show here a simpler path to the general result.
Since the sources are Lorentz-invariant along x , we introduce Fourier transforms along these coordinates and time. This introduces the 3-momentum conjugate of the x α coordinates. In this mixed position-momentum space, the fermion propagators are found to be Introducing the mixed space propagator in Eq. (3.3) gives A momentum redefinition makes appear the loop integral, the external momentum q α and the overall Fourier transform in q α , In the case of planar geometry considered here, it turns out that the external 3momentum is set to zero because of (3.10) The fact that q 0 = 0 is a mere consequence of the nonrelativistic limit. The fact that q = 0 is specific of the planar geometry and indicates that the force is dominated by fluctuations with infinite transverse wavelengths. The remaining transverse integral is factored as a surface d 2 x = S, and the potential is given by Performing both remaining position integrals and evaluating the trace, we have (3.12) The only remaining integral is the loop integral. We Wick rotate the momentum integral from 2 + 1 Lorentzian to 3-dimensional Euclidian space, where we has defined ω Ei = k 2 E + m 2 i . We go to spherical coordinates and perform the angular integrals. For the remaining radial integral we introduce a dimensionless variable (3.14) The potential between plates is found to be The rest of the integral cannot be performed analytically in general. Notice the loop integral is finite by construction because the two sources have finite separation. In this calculation there is no need for any loop integral regularization, expressions are finite at every step.
The pressure between the plates is given by (3.16) The neutrino Casimir pressure is thus repulsive.
Finally, at short distance i.e. in the limit of m i , m j 1/L, the integrals can be done exactly, (3.17) In this regime the Majorana and Dirac predictions have become equal, as expected from the confusion theorem.

Potential Between a Plate and a Point Source
To obtain the plate-point potential, we consider sources of the form with (3.9) and (3.10). Performing the remaining position integrals and evaluating the trace, we have We follow the steps of the plate-plate calculation-Wick rotating, performing the angular integral, and using the definitions (3.14). We obtain At short distances, m i , m j 1/L for all (i, j), the integral can be done exactly, yielding (3.21) We again find that any trace of the mass generation mechanism has vanished from the short-distance result.

Discussion
The neutrino Casimir force has not previously been determined in the literature, to the best of our knowledge. In this section, we elucidate its properties. The expressions for the neutrino Casimir force (3.15), (3.20) contain only one numerical integral, just as for the point-point result (2.5). This property generalizes to plates with an arbitrary number of layers, which is easily obtainable in our formalism. In our calculation, we take into account loops with two different mass eigenstates, that we denote below as m > = max (m i , m j ) , m < = min (m i , m j ). The Dirac and Majorana partial potentials V ij converge to each other in the limit of short distance, L 1/m > . The convergence holds for all configurations of sources considered and is shown for the case of equal masses (m i = m j ) in Fig. 3. This is the fingerprint of the confusion theorem-only the ν L neutrino contributes to the pressure and thus any trace of the mass generation mechanism vanishes (see Fig. 1). The plateplate, plate-point, and point-point potentials scale as 1/L, 1/L 2 , and 1/L 5 in this limit respectively (see (3.17) and (3.21)).
For distances L 1/m > , the partial potentials are exponentially suppressed for all configurations. We find that for L 1/m < , the Dirac and Majorana partial potentials have distinct L-dependencies with 1 − V M /V D ∼ O (1). The latter effect occurs when the partial potential is already exponential suppressed, since 1/m < ≥ 1/m > . Hence we find that the contributions from the cross-term partial potentials (m i = m j ) are not helpful in making a Dirac/Majorana distinction.
Experiments probing point-point potentials at atomic-scale distances currently have the best sensitivity for the neutrino forces. Bounds from muonium spectroscopy currently place experimental limits just two orders of magnitude shy of being able to detect quantum forces from neutrinos [24]. Unfortunately, even with a positive detection, the confusion theorem renders a Dirac/Majorana distinction nearly impossible by this probe, as 1 − V M /V D ∼ O 10 −11 for r Bohr ∼ L 1/m. Conversely, for plates at a separation of L ∼ 1/m where 1 − V M /V D ∼ O(1), the current sensitivity to neutrino forces remains very low. With data from a recent Casimir force experiment [28] (whose result is recast in [17] to bound the relevant quantum force), it turns out that 20 orders of magnitude still remain between current experimental limits and the quantum neutrino forces.
Hence for both atomic and micron scale experiments, we conclude that there are still many orders of magnitude in sensitivity needed to make a Dirac/Majorana distinction with quantum neutrino forces.

A Lagrangians
Here we give more details on Lagrangians in the 2 and 4-component formalisms. The 2component neutrino charged under SU (2) L is denoted ν L , the singlet neutrino is denoted ν R . The L and R labels only refer to the gauge charge. ν L and ν R are left-handed i.e. transform as the (1/2, 0) representation of the Lorentz group.
The free Lagrangian for ν L in case of Dirac and Majorana masses are given by Integrating out the Z boson in the electroweak Lagrangian gives the effective interaction where J ψµ is the weak neutral current for fields other than neutrinos. Integrating out the W bosons gives We used a Fierz rearrangement in the last step. The ν L field can be described as a 4-component Majorana fermion The ν L , ν R can be combined into a Dirac fermion This provides the Dirac and Majorana fields used in our calculations. The neutrino bilinear in the various representations is expressed as Using this and the definitions (A.5), (A.6) in L D/M,kin + L int gives the 4-component Lagrangians (2.1), (2.2). In these 4-component Lagrangians, the relevant couplings to SM fermions in case of unpolarized matter are the vector ones. We find (A.10)

B Point-Point Derivation
For this calculation, we follow the steps outlined in [17,19]. The scattering amplitude corresponding to the loop diagram in Fig. 2 is given by When both point sources are nonrelativistic and polarized, the spin structure simplifies tō We introduce Feynman parameters to simplify the loop integral. Upon dimensional regularization, the resulting integrals are given by The remaining function can be decomposed into the basis of These functions have a branch cut when ∆ ij < 0. The discontinuity across this branch cut is The amplitude is related to the spatial potential by Inside the Fourier transform, we identify the transfer momentum with a gradient, q = −i∇. This gives an expression for the potential that is a Fourier transform of a function that only depends on the magnitude |q| and the gradient. 4 The magnitude is analytically continued as |q| = iλ, and after some manipulations we find Summing the partial potentials from three generations of neutrinos then yields (2.5).

C Casimir Force from the Path Integral
We show how to derive the potential between generic extended sources, shown in (3.3). Start from an effective Lagrangian with a bilinear coupling between a Dirac fermion Ψ and a nonrelativistic density of matter J, where Γ can be any Lorentz structure. We are interested in calculating the energy of a configuration involving two objects J 1 , J 2 acting as sources, both described by the distribution J = J 1 + J 2 . The relevant information is contained in the generating functional of connected correlators W [J], given by where Det/Tr is the determinant/trace in the functional sense. E[J] contains infinities-the observable quantity is rather the variation ∂ L E[J], which gives the Casimir force. In the limit where the ΓJ contribution can be treated perturbatively, the leading contribution to ∂ L E[J] is from the n = 2 term, where tr is the trace on spinor indexes. The piece of potential associated to this term is found to be Restoring the coupling constant yields (3.3) in the Dirac case.