Searching for dark matter axions with Berry phase

We discuss novel aspects of the interaction of axions-like particles (ALP) with superfluids, superconductors in particular, and determine an induced Berry phase that is topologically singular and contributes to the generation of string-like structures. The latter are similar to vortices in superfluids. We suggest that measuring the currents generated by the Berry phase of ALP axions would enable the study of low mass regions of the ALP spectrum otherwise unobservable.


Introduction
The nature of dark matter (DM), as one of the dark components of the Universe, is still an open issue [1][2][3]. In the past years, the favourite candidates of DM have been the weakly interacting massive particles (WIMPs), because they provide an elegant explanation of the relic abundance of DM [4]. Several detection experiments, however, have not found convincing evidence of the existence of WIMPS, yielding stringent bounds on WIMPs DM particles [5][6][7], and opening up the possibility to search for other DM candidates [8,9]. Among them, axions and axion-like particles (ALPs) certainly play a preeminent role in solving, in addition, several problems in particle physics and cosmology. For instance, axion-like particles were introduced by Peccei and Quinn [64,65] in the guise of scalar-pseudoscalar fields in order to solve the strong-CP problem in QCD [10][11][12][13].
These ultra-light bosonic particles are characterized by the fact that for QCD axions the range of the parameters P = {m a , f a }, where m a is the axion mass and f a the coupling, is quite limited (the mass-coupling relation is essentially fixed), while for ALPs the parameter range P can include viable DM candidates [14]. It is worth noticing that a e-mail: lambiase@sa.infn.it (corresponding author) b e-mail: papini@uregina.ca ALPs with mass m a ∼ 10 −22 eV are predicted in string theory [15], have a relevant impact on structures formation (e.g. they could explain the small-scale structures in the cold DM Universe [16][17][18][19]), and are one of the most promising DM candidates. For these reasons, experimental searches or proposals for detecting axions or ALPs are urgent and several suggestions have been made in the literature based for example, on (1) their coupling to the electromagnetic fields, which includes the conversion of ALPs into photons [20][21][22][23], the spectral distortion of photons [24][25][26][27][28][29], a fifth force mediated by ALPs [30][31][32] and cosmological bounds [33] (these experiments have excluded part of the parameter space P of the ALPs [34]); (2) the birefringence effect (the axion field can give rise to a rotation of the photon polarization angle due to the interaction with the electromagnetic field) [35][36][37][38][39][40][41][42][43][44][45][46]; (3) resonance effects in Josephson junctions [69], quantum squeezing technology [71] (see [34,72] and references therein); (4) spin-axion coupling [61]; gravitational waves [62,63].
The birefringence effect which, on the other hand, only depends on the axion field strength, is extremely suitable for detecting in a unique way axions and ALPs located in the galactic halo center or around massive black holes. This requires high polarimetric resolution observations, such as very-long-baseline interferometry (VLBI) [47][48][49], and more recently, the Event Horizon Telescope (EHT) program, which has allowed the observation of the nearby super-massive black holes M87 and Sgr A [50][51][52]. It is worth also mentioning Haloscopes [53] (ADMX [54,55] and CARRACK II [56] Collaborations) which are direct dark matter searches exploiting microwave cavities. Other proposals use optical cavities [57]. An alternative technique relying on the QCD axion coupling to neutrons has been recently proposed in [58]. Moreover, it has been proposed in [59], to focus the radiation into a detector whose shape is a spherical cap, by making use of the electromagnetic radiation emitted by conducting surfaces excited by cold dark matter ALPs.
In this work we analyze the consequences of the coupling of axion DM to an electromagnetic field in superconductors. We show that such a coupling gives rise to an axion-induced Berry phase (for a recent review, see [68]) capable of generating a current I ∝ (v 0 · B)φ, where v 0 is the super-electron velocity, B is the magnetic field in the superconductor, and φ the axion field, along paths linking multiply connected regions of a superconductor. The current I therefore provides a possibility to consider in the search for DM in addition to those already contemplated by present experiments aimed at detecting considerably more massive axions [69,70,73,74].
We also show below that the axion induced Berry phase is itself topologically singular and apt to produce, under suitable conditions, string-like structures, similar to vortices in type II superconductors.
The paper is organized as follows. In the next section we shortly review the axion electrodynamics, recalling the main equations. In Sect. 3 we compute the Berry phase induced by ALPs and the corresponding currents they generate in a long cylindrical superconductor. An estimation of the currents is provided using the current values of axion DM and ALPs. Conclusions are drawn in Sect. 4. In the Appendix we provide some technical details.

Overview of axion electrodynamics
Axion electrodynamics [53,60] is based on the Lagrangian density where F αβ = A β,α − A α,β is the electromagnetic field tensor, F * μν = 1 2 μναβ F αβ is its dual, and φ is the axion field. The constant α is a parameter to be determined below. The classical field equations that can be derived from (1) are Because of the introduction of the potentials A μ , the relationship ∂ ν F * μν = 0 is identically satisfied and the conservation equation

Axions and Berry phase
Here we calculate Berry' phase [67]. The axion electrodynamics of Ni and Sikivie is based on the Eqs. (2)-(5). The electromagnetic field F μν and its dual F * μν satisfy Maxwell equations The quantity qφ plays the role of an effective pseudo-charge. The r.h.s. of (5) yields and By substituting Eqs. (8) and (9) into ∂ ν F μν one obtains the first two equations of (2). The superconductor is assumed to be adequately described by a gas of relativistic scalar particles (Cooper pairs )of charge q and mass m. Substituting ψ = ρ(r)e iθ(r) in the expression for the current in a lump of superconducting material we obtain Let us introduce the quantity where f a is the axion coupling constant and (α) −1 = ρq 2 /m, as the appropriate generalization of θ ,α = q A α . This equation also implies that the generalized momentum of the superlectron is where λ is the usual penetration depth of a type-IIsuperconductor superconductor. For weak fields, the Cooper pairs in the gas can be represented by the Klein-Gordon (KG) equation where Setting where φ 0 satisfies the equation and using (12) and (15), we get From (12) we obtain This result follows because the first term vanishes by choosing an electromagnetic gauge such that ∂ α A α = 0, the second term vanishes because two symmetric indices are contracted with two antisymmetric indices (assuming that φ is regular), and finally, the last term vanishes owing to the fields equations. Inserting (17) into (16) and choosing for φ 0 the plane wave solution φ 0 = e ik μ x μ we find The quantity χ ,α is subject to gauge transformations χ ,μ → χ ,μ + ,μ . On choosing ,μ = 2k μ − χ ,μ , the r.h.s. of (18) vanishes. The effect of the axion on a scalar particle with charge (here a superelectron) can therefore be contained in a phase as in (14), with which represents the Berry phase induced by the axions. For a closed path linking a multiply-connected region of the superconductor we find as required in order to make the wave function of the superconductor single-valued. Quantization of the fluxoid then occurs for the paths embracing the multiply-connected region It follows from (21) that the current φ ,α F * α ν generated by the axions, produces a magnetic field in a loop of wire, hence a current, that could, in principle, be observed. The explicit form of (21) can be calculated using for F * μν the matrix which, using (8) and (9), leads to where A α refers to the electromagnetic field in the superconductor. The fields B and E are generated by φ, and by whatever external electromagnetic fields might be present. 1 and 2 have opposite signs and tend to compensate each other.
An estimate of the current due to the term 2 can be obtained by assuming that B remains approximately constant along the integration path. Then, writing φ = f a ϑ, the current I corresponding to 2 becomes where v 0 is the velocity of the Cooper pairs along the path, L the self-inductance of the superconducting circuit, and B the magnetic field in the superconductor. The path is represented by a circle of radius R around a superconducting, cylindrical wire. The current I threads only the part of the circular sector where B is present and is given by R R−λ 2πrdr ∼ πλ 2 multiplied by N if the sector contains N loops. A direct measurement of Berry's phase, through its associated current (24), would be somewhat independent of resonance conditions and represent the analogue of a wide band detector. We have dealt so far with the quantization of fluxoids around paths surrounding a multiply connected region in superconductors. The resulting currents are estimated below. These are the currents that would be important to measure.
Berry's phase χ is itself singular and similar to the topological singularity that occurs in type II superconductors when B is greater than a critical value B c . Then B penetrates the superconducting material in the form of a vortex. We deal with this topological singularity in the wave function in the Appendix. It generates, in the case of a very long vortex with its axis along the x 3 −axis of a polar system of Fig. 1 The magnetic flux penetrates a type II superconductor in the form of vortex lines when B exceeds a critical field. In the plot is reported the vector potential A, the supercurrent j = ( j 1 , j 2 , 0), Eqs. (27) and (28), and the magnetic field B = (0, 0, B 3 ) in a vortex line, Eq. (26) coordinates, a field tensor given by where (x − x ) is the causal Green function defined in the Appendix). Thus the vortex singularity gives rise to a magnetic field along the axis of the vortex that, on using (23), becomes where r = x 2 1 + x 2 2 ,φ 0 = 1/q is the flux quantum and K 0 is a modified Bessel function . From Maxwell equations, we can then obtain by differentiation the non-vanishing components of the associated flow pattern of the supercurrent about the vortex line which vanish exponentially as r → ∞ (see Fig. 1). As indicated by (10) and (11), the quantum mechanical streamlines are parallel to the local quantum mechanical current. Under the action of ALP axions, quantum superfluids therefore acquire an additional vortical motion, however small, due to 2 (see Fig. 2). We consider DM axions and ALPs, thus the field φ is given by where DM is the DM density and while the velocity in the galaxy is v a ∼ 10 −3 . Moreover, writing the axion-photon coupling constant as the current Eq. (24) reads To give an estimate of the current I , Eq. (29), we use the bounds on the ultra-light axion mass m a and the cou-  [20], the Fermi-LAT observation of a population of SNe [22], the SN1987A [21], the X-ray observation of star clusters [23], the VLBA polarization observations of jets from active galaxies [38], pulsars [42], and Sgr A [51] Fig. 4 I vs m a for fixed g aγ [85]. The limits of the current are provided by the regions below the continue and dashed lines (grey regions) corresponding to the different values of the coupling constant g aγ . Possibilities to measure small currents have been discussed in [86,87] pling g aγ reported in Fig. 3. Results are reported in Fig.  4. For axion mass 10 −24 eV m a 10 −19 eV, and 10 −14 g aγ (GeV −1 ) 10 −10 , the current is of the order I ∈ [10 −10 , 10 −15 ]A, which could be measured, in principle, with dedicated (Lab) experiments. In the case in which the mass generation occurs if the associated global symmetry is anomalous as, for example, when a global symmetry has an anomaly with QCD, then nonperturbative dynamics at the QCD scale can generate a mass m a ∼ These results strongly suggest that although the current induced by axions with a large mass, m a 10 −19 eV is at present unobservable, the possibility to probe a region of the ALP spectrum where m a ∼ (10 −24 − 10 −19 )eV [85] is not precluded.

Conclusions
The interaction of axions with a superconductor is characterized by a Berry phase χ that appears in the superconductor's wave function. If the superconducting electrons are represented by the Klein-Gordon equation, then (14) is a solution of the KG equation and χ satisfies the equation ∂ 2 χ(x) = 0. Moreover, χ is a topologically singular function to which χ ,αβ − χ ,βα ≡ ∂ x α , ∂ x β χ = 0 applies. Wave functions with topological singularities have been considered by Berry and Nye and Berry [75][76][77] in a variety of physical problems and constitute the basis of singular optics. These singularities can be parameterized by using the world-sheet approach of Dirac, which yields χ ,α (see Eqs. (A5) and (A7) in the Appendix). The general solution for the electromagnetic field generated by the axions can be also derived when the singularity is a vortex (Eq. (A12) of the Appendix). For an infinitely long, straight vortex the explicit solution is (25), the magnetic field is represented by the Bessel function K 0 (x) and the flow pattern is proportional to K 1 (x) as expected [80]. Axions therefore contribute to the generation of string-like structures akin to vortices in type II superconductors. The vortex core is a line of normal electrons that is shielded from external electromagnetic fields.
The presence of the Berry phase induces a current I given by (29), which is related to axion coupling constant and axion mass f a − m a . This is the main result of this paper. To date, there is no direct observational confirmations of the existence of axions in the Universe, but only bounds on m a − f a parameters from laboratory, astrophysical, and other experiments. The axion-induced current I hence provides a further possibility to test the existence of axion fields. In particular, we have found that ALP axions generate a current which is measurable, in principle, in Lab experiments.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 . SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development.

Appendix A: Some technical details
Whichever the charge q, the phase (19) changes a solution φ 0 of the free Klein-Gordon (KG) equation into a solution of the KG equation in the presence of fields = e −iχ(x) φ 0 by means of a space-time dependent phase factor. That is the solution of the KG equation behaves as if a contribution ∂ α χ = −λ 2 φ ,ν F * ν α appeared. If a non-vanishing fieldB = ∇ ×Ā exists, then ∂ i χ is non-integrable and χ at a point r depends on the path from r 0 to r and is not single-valued. This is explicitly recognized by (20). By differentiation, we also obtain so that Berry phase χ finds its origin in a singular function that satisfies the equation As shown by Dirac [79], phase singularities of this type correspond to lines in space or points in the plane. They are called wavefront dislocations by Nye and Berry [75][76][77] because of their analogy with crystal dislocations. They correspond to physical objects built around a "normal" phase. Solutions of (A2) can be parameterized by means of the Dirac world-sheet function with where τ and σ are time-like and space-like parameters on the world-sheet, y μ (σ, τ ) is a point on the sheet and the corresponding Jacobian for the transformation of coordinates has been introduced. Then Assuming that ∂ σ χ is single-valued, we get from (A5) from which we obtain where ∂ 2 D(x − x ) = δ (4) (x − x ). From (A7) we can verify that ∂ 2 χ = 0 and that χ(x) can be expressed, for infinitely long and closed strings, as a path integral In order to determine the electromagnetic field tensor generated by the topological singularity, we start from (12) and use (5). We find, in the Lorentz gauge, from which we get where (x − x ) is the causal Green function for the mass M = 1/λ From (A10) we can calculate F αβ . Using (A4) we obtain which is the field tensor generated by the topological singularity in the wave function. The simplest possible singularity is that of a very long vortex with its axis along the x 3 −axis of a polar system of coordinates. Then the points on the world-sheet described by the vortex have coordinates y 0 (σ, τ ) = τ, y i (σ, τ ) = y i (σ ) and τ extends to infinity (see Fig. 2). We also find Substituting (A13) into (A12) and performing straightforward integrations, one arrives at (25). The vortex singularity gives rise to a magnetic field along the axis of the vortex that, on using (23), yields (26). A similar result can be obtained using (12) [80]. From Maxwell equations, we can then obtain by differentiation the non-vanishing components of the associated flow pattern of the supercurrent about the vortex line represented by (27)- (28).