Extended Grimus–Stockinger theorem and inverse-square law violation in quantum field theory

Abstract

We study corrections to the Grimus–Stockinger theorem dealing with the large-distance asymptotic behavior of the external wave-packet modified neutrino propagator within the framework of a field-theoretical description of the neutrino-oscillation phenomenon. The possibility is discussed that these corrections, responsible for breakdown of the classical inverse-square law (ISL), can lead to measurable effects at small but macroscopic distances accessible in the SBL (anti)neutrino experiments and in particular can provide an explanation of the well-known reactor antineutrino anomaly.

Appendices

Appendix A: Inverse overlap tensors

The higher order corrections are rather involved but, within the CRGP model for the external wave packets, they are constructed from the spatial components of inverse overlap tensors $\widetilde{\Re}_{s}^{\mu\nu}$ and $\widetilde{\Re}_{d}^{\mu\nu}$ defined by Eqs. (35). Here we write down the explicit expressions for these components. It is proved that for arbitrary lepton number violating processes

$$ I_s \oplus I_d \to F_s \oplus F_d $$

(A.1)

(where I _s and I _d denote the sets of incoming wave packets in the source and detector vertices, respectively; and, similarly, F _s and F _d denote the sets of outgoing wave packets which include charged leptons) the components of the inverse overlap tensors can be written as

where

the symbols and denote the sets of the in and out packets in the source and detector vertices, respectively; u _a =p _a /m _a =(Γ _a ,u _a ) and Γ _a are the (most probable) 4-velocity and Lorentz factor of the packet a, respectively. It can be verified that the determinant |ℜ _s,d | is invariant and $\widetilde{\Re}_{s,d}^{\mu \nu}$ transforms as a tensor under Lorentz transformations. Note that the above formulas were derived without taking into account the energy-momentum conservation.

As the simplest illustration, we consider here the particular case: the two-particle decay a→ℓ+ν ^∗ in the source, where ℓ is a charged lepton and ν ^∗ is a virtual (anti)neutrino with definite mass. In this case

$$\begin{aligned} \widetilde{\Re}_{s}^{\mu\nu} = &\ \bigl\{ \sigma_a^4u_{a}^{\mu}u_{a}^{\nu}+ \sigma_\ell^4u_{\ell}^{\mu }u_{\ell}^{\nu} -\sigma_a^2\sigma_\ell^2 \bigl\{ g^{\mu\nu} \bigl[(u_au_\ell)^2-1 \bigr] \\ &\ -(u_au_\ell) \bigl(u_{a}^{\mu}u_{\ell}^{\nu}+u_{\ell}^{\mu }u_{a}^{\nu} \bigr) \bigr\} \bigr\} \\ &\ \times \bigl\{ \sigma_a^2\sigma_\ell^2 \bigl(\sigma_a^2+\sigma_\ell ^2 \bigr) \bigl[(u_au_\ell)^2-1 \bigr] \bigr\} ^{-1}. \end{aligned}$$

This simple expression can also be used for estimations of the inverse overlap tensors for more involved reactions and decays in the case of strong hierarchy between the momentum spreads σ _ϰ . Let us mention, in passing, that the inequality $\mathfrak{C}_{1}<0$ (see Sect. 4) holds over the whole phase space of the two-particle decay into real (on-shell) neutrino, assuming exact energy-momentum conservation. The corresponding contribution to the inverse-squared effective momentum spread (37) is then given by the following expression:

$$ \frac{1}{4} \bigl(\widetilde{\Re}_s^{11}+ \widetilde{\Re}_s^{22} \bigr) = \frac{K}{ (2E_\nu^{\star} )^2} \biggl(\frac{m_a^2}{\sigma_a^2}+\frac{m_\ell^2}{\sigma_\ell^2} \biggr)+\frac{1}{2 (\sigma_a^2+\sigma_\ell^2 )}, $$

in which

$$ K = \biggl(1-\frac{E^{-}_{\nu}}{E_{\nu}} \biggr) \biggl(\frac{E^{+}_{\nu }}{E_{\nu}}-1 \biggr), $$

$E^{\pm}_{\nu} = E_{\nu}^{\star}\varGamma_{a}(1\pm|\mathbf{v}_{a}|)$ are the kinematic boundaries of E _ν , $E_{\nu}^{\star}= (m_{a}^{2}-m_{\ell}^{2} )/(2m_{a})$ is the neutrino energy in the rest frame of the decaying particle a, and v _a is the velocity of a. The neutrino mass has been neglected in derivation. The kinematic factor K varies within wide limits at relativistic energies but vanishes when the particle a is at rest. For the median neutrino energy, $E_{\nu}=\overline{E}_{\nu}= (E^{+}_{\nu}+E^{-}_{\nu} )/2=E_{\nu}^{\star}\varGamma_{a}$,

$$K = |\mathbf{v}_a|^2 = 1- \bigl({E_\nu^{\star}}/{ \overline{E}_\nu} \bigr)^2. $$

Appendix B: Effective neutrino wave packet

Here we consider some kind of dualism between the propagator and effective wave-packet approaches. Namely we show that in the asymptotic regime L→∞, the generalized neutrino Green function (1) can be represented as a product of outgoing and incoming effective wave functions of the on-mass-shell neutrino. Below, we use the results of Ref. [30] (obtained within the CRGP model) but with a different and—as we believe—more adequate interpretation which, in particular, allows us to get another look at the corrections to the GS asymptotic.

Let us consider the factor

$$ \frac{1}{L}e^{-\varOmega(T,L)}P_- ( \hat{p}_{\nu}+m )P_+ $$

(B.1)

originated in the full amplitude after applying the GS theorem 1 to Eq. (1) and using the saddle-point integration in variable q ₀ [30]. Here

$$ \varOmega(T,L) = i(p_{\nu}X)+ \frac{2\mathfrak{D}^2}{E_\nu^2} \bigl[(p_{\nu}X)^2-m^2X^2 \bigr], $$

(B.2)

p _ν =(E _ν ,p _ν ) is the mean neutrino 4-momentum (p _ν ≈E _ν l in the ultrarelativistic case) and $P_{\pm}=\tfrac{1}{2}(1\pm\gamma_{5})$ are the chiral projectors involved into the Standard Model amplitude. The 4-vector X=(X ₀,X)=X _d −X _s is the difference between the so-called impact points X _s and X _d which determine the coordinates of the space-time overlap regions of the external wave packets in the source and detector vertices; note that X=L=L l and X ₀=T (see Ref. [30] for more details). The function $\mathfrak{D}$, which can be treated as the uncertainty of the virtual neutrino energy, is defined by

(B.3)

where $q_{0}^{\text{st}} \approx E_{\nu}$ is the stationary point. So, in the first approximation the function $\mathfrak{D}$ does not depend on the neutrino mass and in the coordinate frame where l=(0,0,1) is determined by the 00, 03 and 33 components of the inverse overlap tensors. By applying the identity

$$P_- (\hat{p}_{\nu}+m )P_+=P_-u(\mathbf {p}_{\nu})\overline{u}(\mathbf {p}_{\nu})P_+, $$

in which u(p _ν ) is the usual Dirac bispinor for a free ultrarelativistic left-handed neutrino, we can rewrite Eq. (B.1) as follows:

$$\begin{aligned} &{u(\mathbf {p}_{\nu})\frac{e^{-\varOmega(T,L)}}{L} \overline{u}( \mathbf {p}_{\nu})} \\ &{\quad = \frac{\psi_{X_d}(\mathbf {p}_{\nu},X_s-X_d)\overline{\psi}_{X_s}(\mathbf {p}_{\nu},X_d-X_s)}{|\mathbf {X}_d-\mathbf {X}_s|},} \end{aligned}$$

(B.4)

where

$$\begin{aligned} \psi_{y}(\mathbf {p}_{\nu},x) = &\ \exp \biggl\{ -i(p_{\nu}y)-\frac{\mathfrak{D}^2}{E_\nu^2} \bigl[(p_{\nu}x)^2-m^2x^2 \bigr] \biggr\} u(\mathbf {p}_{\nu}) \\ \equiv&\ \varXi_{y}(\mathbf {p}_{\nu},x)u(\mathbf {p}_{\nu}) \end{aligned}$$

and

$$ \overline{\psi}_{y}(\mathbf {p}_{\nu},x) = \psi^{\dagger}_{y}(\mathbf {p}_{\nu},x) \gamma_0 = \overline{u}(\mathbf {p}_{\nu})\varXi_{y}^*( \mathbf {p}_{\nu},x). $$

Let us now compare the function ψ _y (p _ν ,x) with the wave function

(B.5)

describing a generic fermion wave packet |p,s,y〉 in the coordinate representation (here Ψ(x) is the free fermion field and s is the spin projection). Within the CRGP approximation, valid under the conditions

$$\begin{aligned} &{\sigma^2\ll m^2, \qquad\sigma^4(px)^2 \ll m^4,}\\ &{ \sigma^4 \bigl[(px)^2-m^2x^2 \bigr] \ll m^4,} \end{aligned}$$

the function (B.5) is [30]

We see therefore that the spinor multiplier $\psi_{X_{d}}(\mathbf {p}_{\nu },X_{s}-X_{d})$ in Eq. (B.4) can be interpreted as the wave function describing the wave packet of a real (on-shell) neutrino incoming to the detector, while the multiplier $\psi_{X_{s}}^{\dagger}(\mathbf {p}_{\nu },X_{d}-X_{s})/|\mathbf {X}_{d}-\mathbf {X}_{s}|$ can naturally be treated as the spherical wave function of the same neutrino packet escaping the source. From this interpretation it in particular follows that the effective neutrino momentum spread, $\sigma_{\nu} = m\mathfrak{D}/E_{\nu}$, is not a constant but an invariant function of the neutrino energy as well as of the mean momenta, masses, and momentum spreads of the external wave packets; due to the factor m/E _ν it is extremely small in all realistic situations. The main distinctive feature of the effective neutrino wave packet from the conventional quantum-mechanical wave packets is that it depends on both the source and detector variables. Therefore the duality under discussion should not be understood in a literal sense.

The case becomes even more complicated when one takes into account the higher-order corrections to the GS asymptotics. These corrections yield an additional factor

$$ \varSigma= 1+\sum_{k=1}^\infty \frac{a_k}{L^k} $$

(B.6)

with the complex-valued coefficient functions a _k dependent on the contributions from both the source and detector. Except for very special or trivial cases (when, i.g., one of the contributions is negligible) the functions a _k cannot be factorized to the product of the source and detector dependent multipliers. Therefore, if the spacing between the impact points is short the incoming and outgoing neutrino wave packets are not yet separated from each other as it is the case at the asymptotically large distances. In other words, the duality between the propagator and wave-packet treatments is destroyed at short distances and such a case cannot be adequately described in terms of the asymptotically free in and out neutrino wave packets.

Considering now that the modulus-squared full amplitude incorporates the factors $|\varXi_{X_{s}}|^{2}$ and $|\varXi_{X_{d}}|^{2}$ which represent (up to a normalization) the probability densities to obtain the in and out neutrino wave packets near the corresponding impact points in the source and detector, we conclude that the factor |Σ|² affects both these densities or, to put this another way, at relatively short distances between the impact points the factor |Σ|² changes the probabilities of neutrino production and detection.

It is important to note that, in contrast with the function $\mathfrak {D}$, the coefficient functions a _k include the components $\widetilde{\Re}_{s,d}^{11}$ and $\widetilde{\Re }_{s,d}^{22}$ transversal with respect to the neutrino momentum direction l=(0,0,1) (see Eq. (37) as the simplest example). These components are not suppressed by the Lorentz factor E _ν /m and thus the ISL violation effect can be large even for massless neutrinos.