Wolfenstein potentials for neutrinos induced by ultra-light mediators

New physics can emerge at low energy scales, involving very light and very weakly interacting new particles. These particles can mediate interactions between neutrinos and usual matter and contribute to the Wolfenstein potential relevant for neutrino oscillations. We compute the Wolfenstein potential in the presence of ultra-light scalar and vector mediators and study the dependence of the potential on the mediator mass $m_A$, taking the finite size of matter distribution (Earth, Sun, supernovae) into consideration. For ultra-light mediators with $m_{A}^{-1}$ comparable to the size of the medium ($R$), the usual $m_{A}^{-2}$ dependence of the potential is modified. In particular, when $m_{A}^{-1}\gg R$, the potential does not depend on $m_{A}$. Taking into account existing bounds on light mediators, we find that for the scalar case significant effects on neutrino propagation are not possible, while for the vector case large matter effects are allowed for $m_{A} \in [2\times10^{-17}$, $4\times10^{-14}]$ eV and the gauge coupling $g\sim 10^{-25}$.


I. INTRODUCTION
Coherent forward scattering of neutrinos on particles of medium ψ (ψ = e − , n, p) generates the Wolfenstein potential V W [1]. Being added to the neutrino evolution equation, V W can significantly affect neutrino oscillations, known as the Mikheyev-Smirnov-Wolfenstein (MSW) effect [1][2][3]. When neutrino-matter interactions are mediated by a heavy boson with the interaction radius m −1 A (where m A is the mediator mass) much smaller than the object in which neutrinos propagate (or the distance over which the density varies), the Wolfenstein potential equals where n ψ is the number density of the ψ particles, g ν and g ψ are the couplings of mediator to ν and ψ respectively. The potential depends on the local number density n ψ while the size and shape of the object are not relevant. The medium can be considered as infinite. In the standard model (SM), the mediators are the W and Z bosons, satisfying the condition for (1). New heavy particles beyond the SM can generate via non-standard interactions additional contributions to the Wolfenstein potential with the same form as (1). New neutrino interactions may be mediated by light particles as well, if the light mediators are very weakly coupled to the SM fermions. With sufficiently small values of g ν and g ψ , and correspondingly small m A , sizable g ν g ψ /m 2 A can evade various bounds from processes with large momentum transfer |q 2 | m 2 A , because the new physics contributions in such processes are typically proportional to the small quantity g ν g ψ /|q 2 | rather than g ν g ψ /m 2 A . In contrast, the Wolfenstein potential in (1), can be unchanged if m 2 A decreases proportionally with respect to g ν g ψ . This, however, is restricted by the finite size of the object R. When m A becomes smaller than 1/R, the dependence of V W on m A in (1) is modified so that the matter effect turns out to be also suppressed. In this paper we will consider this dependence and its implications in details.
The matter effects due to light mediators have been studied before [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The mediators, mostly considered in the literature, are new gauge bosons of the lepton numbers L e −L µ , L µ −L τ or L τ −L e . Long-range forces induced by these bosons can affect solar and atmospheric neutrino oscillations [4,5] as well as high energy astrophysical neutrinos interacting with electrons in the Universe [15]. Various fifth force and gravitational experiments put very strong bounds on couplings of light mediators with matter but in certain ranges it is neutrino oscillation phenomena that have the best sensitivity to couplings [17]. As for scalar interactions, it is well known that the corresponding matter effect leads to corrections to the neutrino masses. Recently this possibility has been studied in Ref. [18] and it is claimed that such scalar interactions can explain the discrepancy between the solar neutrino and KamLAND determinations of ∆m 2 21 . In this paper, we present detailed study of the Wolfenstein potentials induced by light mediators (both vector and scalar). We compute the Wolfenstein potentials for several spherically symmetric density profiles and study dependence of the potentials on the mediator mass. Taking into account existing bounds on light mediators, we assess their relevance to neutrino experiments.
The paper is organized as follows. In Sec. II we study the effects of light scalar and vector mediators on neutrino propagation, considering general matter density distributions. In Sec. III we present derivation of the effective potentials for several spherically symmetric density distributions which can be applied to the Earth, the Sun and similar celestial bodies. In Sec. IV we consider existing bounds on light mediators and apply our results to neutrinos propagating in the Sun, the Earth and supernovae. Discussion and conclusions are presented in Sec. V.

II. EFFECTS OF LIGHT MEDIATORS ON NEUTRINO PROPAGATION
Let us consider interactions between neutrinos (ν) and particles in matter (ψ) mediated by a new light vector boson A µ or scalar boson φ. The relevant part of the Lagrangian reads in the vector case. In the case of a scalar mediator, the last three terms in (2) should be replaced by We assume that neutrinos are Dirac particles. For Majorana neutrinos, though the interaction forms are slightly different, the results are the same. Also we consider neutrinos of a single flavor. It can be straightforwardly generalized to the case of three-neutrino mixing. The Lagrangian (2) determines the equations of motion (EOM) of ν and A µ : According to Eq. (4), the effect of A µ on neutrino propagation can be described as the displacement i / ∂ → i / ∂ − g ν / A, which in the momentum space corresponds to where p µ is the 4-momentum of the neutrino. In particular, the neutrino energy E receives the correction: In the scalar case, the EOM from Eq.
Since the neutrino number density is much smaller than the number density of electrons or nucleons, we take νν ψψ and νγ µ ν ψγ µ ψ in Eqs. (9) and (5). This means that φ and A µ are dominantly induced by ψ.
For the vector case, Eq. (11) implies that the spatial components of A µ vanish (up to gauge uncertainties): Furthermore, since the ψ particles are at rest, A 0 has no temporal dependence (∂ t A 0 = 0). Therefore, Eq. (5) becomes Given a distribution of n ψ , Eq. (13) determines A 0 . All the above analyses can be straightforwardly applied to a scalar mediator. Starting from Eq. (9), we obtain the equation similar to Eq. (13): and hence a similar solution. From Eqs. (13) and (14), we can see that φ and A 0 are determined by n ψ in the same way. However, the effects of φ and A 0 on neutrino propagation are very different.
In the neutrino evolution equation φ gives correction to the mass and therefore appears as (m ν + δm ν ) 2 /2E. Approximately(for δm ν m ν ), this corresponds to adding to (m ν ) 2 /2E. In comparison, A 0 appears in the equation as an addition of V = g ν A 0 to m 2 ν /2E. Thus, the scalar matter effect enters the flavor evolution equation with the additional suppression factor m ν /E. This factor is due to chiral suppression. Since the helicity is conserved in neutrino oscillations, the chirality-flipping terms in the Lagrangian, such as mass terms or scalar interactions, have to change it twice, which is the reason that m 2 ν appears instead of m ν in neutrino oscillations. Because the scalar interaction flips the chirality, another flip is needed which is given by m ν /2E. In other words, only m ν /2E fraction of the chirality-flipped state contains the original helicity. This means that, to obtain effects of the same size, y ν φ should be E mν times larger than g ν A 0 . That is, y ν y ψ /m 2 φ should be E mν times larger than g ν g ψ /m 2 A . This strongly affects the relevance of the scalar case to the oscillation phenomenology.
In the case of pseudo-scalar and axial-vector mediators, the corresponding fields are produced by interactions with ψγ 5 ψ and ψγ µ γ 5 ψ. In unpolarized medium, these two quantities vanish due to γ 5 . For polarized medium, the pseudo-scalar and axial-vector potentials decrease as r −3 , being dipole effects (see, e.g., Ref. [23]). Therefore these potentials are small compared to scalar and vector potentials.

III. EFFECTIVE POTENTIALS FOR SPHERICALLY SYMMETRIC DENSITY DISTRIBUTIONS
In what follows, we consider the vector case, while for the scalar case, the results can be obtained immediately with the substitutions: In many applications (e.g., the Earth, the Sun), the matter density distribution is, to a good approximation, spherically symmetric. The spherical symmetry allows us to reduce Eq. (13) to a radial differential equation: For any given profile n ψ (r), Eq. (16) can be solved by a standard method known as variation of parameters, which gives The above computations of potentials are essentially classical. Therefore for an arbitrary density distribution n ψ (r) (not necessarily spherically symmetric), A 0 can be found by performing summation (integration) of the Yukawa potentials produced by individual particles [see also Eq. (23)]: For the spherically symmetric case, one can integrate over angular variables in Eq. (18), which also leads to Eq. (17). Using Eq. (17) we compute the Wolfenstein potentials for several matter density profiles, which can be used to approximately describe the density distributions of the Earth, the Sun, and supernovae. Some interesting limits will be discussed.

Constant density within a sphere
For a constant density distribution within a sphere of radius R: we find from (17) where describes deviation from the infinite medium potential. Then according to Eq. (7), the effective neutrino potential produced by A 0 equals Several important limits are in order.
• m A → 0. In the limit of massless mediator, Eqs. (21) and (22) lead to which coincides with the r dependence of the Coulomb potential of a charged sphere. In this limit, the potential for r > R is determined by the total particle number N ψ inside the sphere, independently of the radial distribution. If m A is nonzero but small, the first order correction in m A to the potential equals (for both r ≤ R and r > R) • m A → ∞. In this limit, Eqs. (21) and (22) give Inside medium, the potential is a constant and outside it vanishes. This reproduces the standard Wolfenstein potential for infinite medium.
• |R − r| m −1 A R. The limit means that the radius of interaction is much smaller than R but much larger than the depth of the trajectory R − r. The absolute sign "| |" indicates that this limit applies not only to underground neutrino trajectories but also to above-the-surface neutrino beams. The limit can be realized for reactor and accelerator experiments. In this limit, we obtain It differs from the standard Wolfenstein potential by the additional factor of 1/2, which reflects that only half of the space produces the potential.
In the left panel of Fig. 1, we show the dependence of V on r for different values of m A according to (22) and (21). For m A R −1 (green curve) the r-dependence is close to that of the standard Wolfenstein potential, which is essentially a step function. Near the surface (r ≈ R) the shortrange potential (green curve) is roughly half the standard Wolfenstein potential (the plateau of this curve), as expected from Eq. (27). When m A decreases, it becomes smoother (yellow line). For very small m A , the long-range force leads to a Column-like potential (blue curve).
In the right panel of Fig. 1, we show the dependence of V at r = R on m A according to the exact formula (blue solid curve) as well as the small-and large-m A limits in Eqs. (24)- (26). The inverse-square dependence of the standard Wolfenstein potential on m A deviates significantly from the exact dependence for small values of m A .

Multi-layer density profile
A multi-layer density profile with constant densities within layers is a good approximation for the Earth density distribution. Using result (21) for constant density spheres one can obtain results for the Earth. For instance, for two layers (a simplified mantle-core profile) with constant average densities n M ψ and n C ψ and radii R M and R C respectively, the Wolfenstein potential in the mantle is and in the core is where F r<R and F r>R are given by the lower and the upper lines of Eq. (21). For more realistic profiles with many layers, the generalization is straightforward.

Exponential density distribution
Exact analytic results can be obtained for an exponential density distribution, This distribution can be used to partially describe the matter density of the Sun and supernovae. Computing the integral in (17) gives Several important limits are in order.
• m A κ. In this limit, the radius of force is much larger than the scale of density change. From Eq. (31), we obtain which does not depend on m A . For r κ −1 , Eq. (32) reduces to that is, the potential at large r decreases as 1/r, i.e., slower than the matter density (∝ e −κr ) decreases. This can produce interesting phenomena such as new level crossing for solar neutrinos [24].
• m A ≈ κ. In this limit, Eq. (31) gives Exponential density distribution with a cut-off The density profile of the Sun can be described more accurately than in (30) by an exponential distribution with a cut-off at the solar radius: After straightforward calculations, we obtain the potential: with K in and K out given by All the above results can be applied to the scalar case with the simple substitution: A 0 → φ, m A → m φ , g ν → y ν , and g ψ → y ψ .

IV. PHENOMENOLOGY
Let us consider experimental bounds on the couplings and masses of light mediators and check whether light scalar and vector mediators, that satisfy these bounds, can produce observable matter effects in neutrino oscillations. For brevity we introduce the couplings g ≡ √ g ν g ψ and y ≡ √ y ν y ψ .
For m A,φ O(10 2 ) keV, g 2 /m 2 A G F and y 2 /m 2 φ G F have been excluded by numerous laboratory experiments including the elastic neutrino-electron scattering [25][26][27], neutrino-nucleus scattering [28][29][30], fixed target experiments [31][32][33], collider searches [34,35], etc. For 1 eV m A,φ 10 2 keV, the astrophysical observations provide much stronger constraints. For instance, the expected amount of energy loss via neutrinos in the Sun and globular clusters excludes g and y down to 10 −14 , corresponding to an upper bound of g 2 /m 2 A or y 2 /m 2 φ above 10 −5 G F [36]. Below 1 eV, the constraints mainly come from searches of fifth forces and precision tests of gravity [37,38]. These constraints are only applicable to g ψ or y ψ . To obtain the bounds on √ g ν g ψ or √ y ν y ψ , we can use the cosmological bounds on neutrino self-interactions, [39]. In addition, for certain ranges of m A,φ , the black hole super-radiance provides robust constraints [15,40] which are independent of the couplings. These constraints are combined and presented in Fig. 2. Now let us determine the required values of g or y to generate significant matter effects. The latter can be quantified by where V SM ≡ √ 2G F n ψ , and F presented in Eqs. (21) describes the deviation from the standard Wolfenstein potential. According to Eq. (39), for given values of V V SM and δm ν , g 2 and y 2 are determined by: In Fig. 2, we plot the dependence g = g(m A ) and y = y(m φ ) from Eq. (40) for V /V SM ∈ {10 −2 , 10 −1 , 1} and δm ν ∈ {10 −3 , 10 −2 , 10 −1 } eV correspondingly. The curves determine the bands of strong (observable) matter effects in the mediator parameter space. The values of F have been computed numerically for the center of the Sun (r = 0) using the solar density distributions from [41], and for the surface of the Earth (r = R ⊕ ) using the density distribution from [42]. The value of F has been computed numerically using the density distributions for the Sun from [41] and for the Earth from [42]. The value of r is set to zero for the Sun, and to the Earth radius for the Earth.
The dependence g = g(m A ) and y = y(m φ ) can be understood from our analytic results. For large m A (short range forces), F ≈ 1. According to Eq. (40), we have g ∝ m 2 A for m A 1/R, where R is the radius of the object. For very small m A , F should be proportional to m 2 A so that g is independent of m A . In the intermediate range, m A ∼ 1/R, the dependence is more complicated.
According to Fig. 2, the solid curves turn at the values of mediator masses 2.8 × 10 −16 eV for the Sun, and 3.1 × 10 −14 eV for the Earth that are determined by inverse of the solar radius R = 7.0 × 10 5 km and the Earth radius R ⊕ ≈ 6.4 × 10 3 km correspondingly. Below these masses at m A , m φ < 10 −16 eV (for the Sun) and m A , m φ < 10 −14 eV (for the Earth) the curve becomes horizontal, i.e. dependence on the mediator masses disappears. This corresponds essentially to the Coulomb-like potentials generated by massless mediators.
According to Fig. 2 (left) there are several m A -g regions of observable matter effects which are generated by the vector mediator and allowed by the present bounds. In the long-range forces domain they include for neutrinos in the Sun, and both for neutrinos in the Sun and in the Earth. The unexcluded regions in the short range forces domain are the same for the neutrinos in the Sun and the Earth: This could motivate further phenomenological studies. As follows from Fig. 2 (right panel) the scalar mediators cannot generate significant matter effects, in contrast to the claim in Ref. [18]. For comparison, a black dotted line corresponds to δm ν = 0.001 eV generated in the infinite size medium with the same density as in the center of the Sun. (For large m φ it coincides with the corresponding line for the Sun.) In this case, significant matter effects would be possible for m φ 10 −20 eV). However, this possibility is excluded when F (r) (the correction) due to the finite size of medium is taken into account.
One may be interested in the allowed values of scalar matter effect. So, we plot two dashed curves that correspond to values of corrections δm ν = 10 −11 eV for the Earth and δm ν = 10 −8 eV which have small sections below the excluded region. These values are far beyond the precision of any realistic experiments.
Finally, strong constraints on the scalar mediator parameters can be obtained from supernova neutrinos, which have not been considered in the literature. Let us estimate the effect in the core of a supernova with a typical radius of R SN = 20 ∼ 30 km and a density of n SN ψ = 10 13 ∼ 10 14 g/cm 3 . For relatively heavy scalar mediators with m φ > 1/R SN , the mass correction is about 13 orders of magnitude larger than the correction in the Earth δm earth ν . For δm earth ν = 0.01 eV, we obtain δm SN ν ∼ 100 GeV, which is certainly excluded since in this case neutrinos can not even be produced in supernovae. For very light mediators with m φ < 1/R SN , Eq. (45) is modified to Taking δm earth ν = 0.01 eV, we obtain δm SN ν = 5 MeV and correspondingly, y = 10 −20 (see Fig. 2). Since neutrino masses m ν > 5 MeV would strongly affect production of neutrinos and their energy spectrum, values of couplings y > 10 −20 can be excluded. More detailed analyses will be given elsewhere [24].

V. CONCLUSION
We studied the effects of new neutrino-matter interactions mediated by ultra-light scalar and vector bosons on neutrino propagation, taking into account the finite size and density distribution of the medium. We compute the Wolfenstein potentials explicitly for several spherically symmetric density profiles which can be approximately applied to the Earth and the Sun. The Wolfenstein potentials induced by ultra-light mediators have very different dependence on the mediator masses (m A ) from the standard case. In infinite medium, due to the 1/m 2 A dependence in Eq. (1), the Wolfenstein potentials can be enhanced by reducing the mediator masses. In finite medium of size R, when m A decreases down to the region where m −1 A becomes comparable or larger than R, the 1/m 2 A dependence is modified. In this case the matter effect is determined by the geometry (including the size) and density distribution of the object, rather than the local density. In particular, the potential can extend outside of the medium. For m A R, the potential does not depend on m A .
With correct expressions for the Wolfenstein potentials and existing bounds on light mediators, we find that scalar mediators cannot produce observable effects in realistic experiments, which implies that the scenario considered in Ref. [18] is not viable. Vector mediators, however, can produce significant matter effect, in particular, within the parameter space m A ∈[2 × 10 −17 , 4 × 10 −14 ] eV and g ∼ 10 −25 .