Anapole Dark Matter Quantum Mechanics

The dynamics of an anapole seen as dark matter at low energies is studied by solving the Schr\"odinger-Pauli equation in a potential involving Dirac-delta and its derivatives in three-dimensions. This is an interesting mathematical problem that, as far as we know, has not been previously discussed. We show how bound states emerge in this approach and the scattering problem is formulated (and solved) directly. The total cross section is in full agreement with independent calculations in the standard model.


I. INTRODUCTION
In the context of dark matter it is widely believed that some discrete symmetries are not conserved, in the same way that in the visible sector, for example, the parity is broken in the weak interactions. In both cases -dark and visible matter -if parity is violated an anapole term emerges as the result of the interaction between the spin of the fermions and an electromagnetic source.
This last fact is obtained taking the non-relativistics limit of where M is a mass scale,m is the mass of fermions χ (electrically neutral), γ µ are the Dirac matrices (with µ = 0, · · · , 3), F µν is the electromagnetic strength tensor and g is the dimensionless coupling constant. The interaction term in the Lagrangian (1) was proposed by Zeldovich [1] sixty years ago as a way to produce photons by neutral particles such as neutrinos, and extending the ideas of parity violation previously proposed by Lee and Yang [2]. This term was named by Zeldovich himself as an anapole interaction because it is the natural extension of the multipolar expansion for truly neutral interactions (for a review see [3]).
Even more, in the context of Lee and Yang's parity violation [2], Zeldovich pointed out [1] that the process νν could also produce a virtual photon and therefore violate parity assuming an effective vertex as in the Figure 1. Since the incoming particles are neutral, the interaction is described through the anapole which is, technically, the analog of the next to quadrupolar term in the multipolar expansion of 1 |x−y| in electrodynamics. However, although this effect is very weak, it is measurable and its detection was announced in 1997 [4] for the transition 6S to 7S in cesium atoms (see also [5]). * jorge.gamboa@usach.cl † fernando.mendez@usach.cl ‡ natalia.tapia@usach.cl The idea of anapoles has been used recently by several authors [6] and mainly by Ho and Schrerrer [7] who have proposed that the anapole could be considered a form of dark matter.
In this paper we study some properties of this anapole interaction from the point of view of quantum mechanics and we show the emergence of non-trivial properties such as the formation of bound states and a notorious simplicity in scattering processes.
The paper is organized as follows: in section II we discuss the non-relativistic anapole and we formulate the problem in general, in section III we discuss the bound state and scattering problem in three dimensions and we provide of exact solution of the Schödinger-Pauli equation, in section IV we discuss our results. Two appendices of technical issues are included.

II. NON-RELATIVISTIC ANAPOLE
The non-relativistic limit [9] of the Hamiltonian (1) is where the first term is the kinetic energy of the leptons with three-momentum p and H anap = − g M 2 S · J is the Hamiltonian describing the anapole interaction, with S = 1 2 σ its spin and J is the electromagnetic source (coming from the Ampere's law).
In the case of dark matter the above issue is a little bit more involved because the interaction between visible and dark matter is indirect and the concept of kinetic mixing is necessary [8].
The interaction between the dark fermions and the gauge fields is produced by the diagonalization of the kinetic mixing implying that instead of (2), for the case of dark matter one has to consider where, now, g ′ = ξg is an effective coupling constant, ξ ≪ 1 is the kinetic mixing parameter and the anapole describes the interaction between dark matter and the electromagnetic source. The change g → g ′ is a nontrivial consequence of the diagonalization of the kinetic mixing and the gauge group enlargement U (1) to U (1) × U (1).
Then as J is a current density of particles of the standard model, we can assume that the particles are millicharged and the Ohm's law implies J = ρ v where v is the velocity of the charge carriers moving in a volume of V with the charge density ρ.
As the charge carriers are point particles ρ ∼ δ(x),and classical anapole Hamiltonian can be written as which is an "electromagnetic" interaction J · A where A is formally identified with σδ(x) which is a Aharonov-Bohm-like effect. In other words, we have a toroidal cylinder with a magnetic field confined inside in a similar way to the Tonomura experiments for the Aharonov-Bohm effect [12]. Then, in our non-relativistic quantum mechanics model, particles with momentum p interact with dark matter through an anapole term, giving rise to the total Hamiltonian operator iŝ and the solution to this problem will provide a complete quantum mechanical picture, including bound and scattering states.
In order to solve the eigenvalue problem, let us start considering the following Schrödinger-Pauli equation where k 2 = 2 m E. Previous equation (6), can be recast as an integral equation 1 with ψ 0 (x) = A e ık·x , the homogeneous solution of the operator ∇ 2 + k 2 , and the Green function G[x, x ′ ] given 1 By convenience we will do the calculation in D dimensions. by An instructive example is to consider the onedimensional case, where previous solution reduces to It is interesting to determine the bound states of this problem. Then ψ 0 = 0 and and we evaluate previous expression at x = 0, that is We impose Robin's boundary condition where γ ∈ ℜ is the parameter that defines the self-adjoint extension of the Hamiltonian, to find The functions G[0, 0] and G ′ [0, 0], from (9), are and therefore and therefore there are not bounds states in the onedimensional case.

III. THREE-DIMENSIONAL NON-RELATIVISTICS ANAPOLE; BOUND AND SCATTERING STATES
The three-dimensional case is more complicated and has important differences with the previous onedimensional example, such as the existence of bound state and renormalization of the coupling constant as a consequence of the three-dimensional scale invariance [10,11] Then for bound states we put ψ 0 = 0 and (7) becomes We are interested in the case x = 0 = x ′ and due to the spherical symmetry and the explicit form of the Green function one gets ∂ r G → 0.
Therefore we write (16) as follows Integral (17) can be done straightforwardly. To do that, we write explicitly the spinors and look for solutions with the form where Φ ± are functions of r, while spinors carry the angular dependence.
It is interesting to note that in usual cases the radial functions are equals (Φ + = Φ − ), but in our case the interaction changes the orbital angular momentum states according to (A9), what support our choice and then the state is a superposition of the two different orbital angular momentum ℓ = j ± 1 2 for a fixed total angular momentum j.
Now we use the explicit form of the Green function and integrate (17) for the solution in (18) and we evaluate in x = 0. One gets where g ′ has been rescaled through g ′ → g ′ (1 + θ(0)) as a consequence of scale invariance [10,11].
The Green function G Λ , instead, has been regularized through an ultraviolet cutoff Λ for x, x ′ → 0 (see appendix B).
Previous equations can be recast as where subscript 0 stand for x = 0. In order to find the bound states we posit a generalization of the Robin's boundary conditions, as follows with G a 2 × 2 matrix with dimensions of mass. With this choice, condition (21) read and the condition for bound sates is We look for imaginary solutions for k, since then k 2 < 0, corresponding to negative energy states. For the Green function (see appendix B) with {α, β} ∈ ℜ, α 2 + β 2 = 1 and κ, a constant with dimensions of mass. Note that G † G = κ 2 I. By doing this, we get and, therefore, the energy of the bound states turn out to be implying that, for this particular choice of the Robin's boundary conditions, there exists bound states for κ ∈ ℜ. Note also that, for the particular choice of κ ∓ = ∓ M 2 Λ g ′ , the bound states have zero energy.
For scattering processes, a similar behavior is verified. Indeed, the problem can be formulated as follow: first we rewrite (8) iteratively However, the last term in RHS of (26) is energetically less relevant than the first one and we can write the last equation as The appropriate Green function for the boundary conditions of the scattering problem is and therefore scattering state (27) is where, formally, is our definition of the scattering amplitude.
We prepare the initial state as ψ 0 = ξ + ψ + + ξ − ψ − (in principle we can take ψ + = ψ − = e ıkz ) and we get and, therefore, the total scattering amplitude σ TOT = A † A turn out to be We impose now the Robin's boundary condition (21) with G † G = κ 2 I, and therefore total cross section is We note that, up to the normalization factor |ψ + (0)| 2 + |ψ − (0)| 2 , this result is in full agreement with independent calculations reported in [13] if we interpret κ as the mass of the DM-nucleon.
From the above results we could conclude that having sufficiently reliable bounds for Λ and M could be argued about observability (or not) of anapole dark matter.
A careful analysis of the Xenon-100 data [14] and the bounds for hidden photons from [16] could help clarify these issues.

IV. FINAL COMMENTS
As final comments we would like to point out the following; a) the anapole observables in scattering processes are only dependent on the mass scale and coupling constant that appear in the Zeldovich term and, therefore, the number of adjustable parameters needed to extract bounds are minimal and this is an advantage with respect to other approaches; b) the approach proposed here not only provides a systematic way of carrying out parity violation effects as a consequence of the standard model, but can also be considered as a starting point to study possible corrections associated with dark matter and its interactions.
We would like to thank Manuel Asorey, Paola Arias, and Jose G. Esteve by discussions. This work was supported by Dicyt 041831GR (J.G) and 041931MF (F. M.). One of us (J.G.) thanks to DESY theory group and the Alexander von Humboldt Foundation for the hospitality and support.

Appendix A: The eigenspinors and the equation of motion
In order to built the solution of the equation of motion, let us review some facts related to the spinor factorization [17].
Consider the Hamiltonian with the vector operators L = r × p, the orbital angular momentum, and J = L+S, the total angular momentum. The spin operator is S and we take S = 1 2 σ, with σ, the Pauli matrices.
It is direct to prove that which will be useful in what follows. On the other hand, the eigenvalue problem has solution κ = · · · , −2, −1, 1, 2, · · · (A4) with κ = 0. For a given κ, the eigenvalues of L 2 and J 2 ( ℓ(ℓ + 1) and j(j + 1), respectively) satisfy In the usual notation we have, therefore, with j = ℓ ± 1 2 and the projection m j (eigenvalue of J 3 ) are given by the 2j + 1 values m j = −j, −j + 1, −j + 2, . . . , j − 2, j − 1, j Eigenfunctions of the commuting set of operators { L 2 , J 2 , J 3 } are the spinor spherical harmonics (Pauli spinors) For a fixed total angular momentum j, previous spinors correspond to two angular momentum ℓ, namely, ℓ = j ± 1 2 . Let us introduce the notation ξ + for the spinor in (A7) and ξ − for the one in (A8) omitting labels ℓ, j, m j . The following relation holds This relation is useful because the interaction term can be written in terms of previous operator and H only. Indeed, from L = −ı r × ∇, the following identity holds ∇ =r(r · ∇) + ı r 2 r × L, and therefore The last term in previous expression can be written as σ · (r × L) = ı(r · σ)(σ · L), once the relation σ i σ j = δ ij + ı ǫ ijk σ k is used. Then we get for the interaction term σ · ∇ = (r · σ) (r · ∇) − 1 r (σ · L) .
Finally, we can use (A2) in order to replace the last term by H + 1. Since (r · ∇) = ∂ r , it is possible to write the interaction term as follows σ · ∇ = (r · σ) ∂ r + 1 r (H + 1) .
The action of this operator on the spinors ξ ± is obtained from (A9) Then, the interaction terms changes the orbital angular momentum ℓ when the total angular momentum j is fixed.
Finally, let us show a simpler way to obtain previous results. In this approach, we take the limit ∆ → 0 before integration, that is In the limit Λ → ∞, only the first two terms are relevant. Finally (restoring also ǫ = −ık) we find which is the same result as before and therefore The derivative, in this approach, is direct to calculate since ∂ ∆ p sin(p∆) ∆(ǫ 2 + p 2 ) = p ∆(ǫ 2 + p 2 ) p cos(p∆) − sin(p∆) ∆ and it is direct to prove that in the limit ∆ → 0 previous expression vanishes.