Impact of wave packet prescription on neutrino oscillations

Evan Galea, and Magdalena Zycha, b

aARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics,
The University of Queensland, St Lucia, QLD 4072, Australia
bDepartment of Physics, Stockholm University, SE 106 91 Stockholm, Sweden
 
  

Neutrino oscillations

  • Theoretically proposed by Pontecorvo in late 1950s
  • Experimental tests of solar neutrino flux in mid-1960s
  • Discrepancy with predictions, the solar neutrino problem
  • Today, discrepancy best explained by neutrino oscillations

Neutrino oscillations

  • Theoretically proposed by Pontecorvo in late 1950s
  • Experimental tests of solar neutrino flux in mid-1960s
  • Discrepancy with predictions, the solar neutrino problem
  • Today, discrepancy best explained by neutrino oscillations
Takaaki Kajita & Arthur B. McDonald, Nobel Prize in Physics 2015
for the discovery of neutrino oscillations, which shows that neutrinos have mass

How are neutrinos described in quantum mechanics?

Often see oscillations treated with plane waves

Flavour basis $\nu_\alpha$ can be related to mass basis $\nu_i$ by

$$\left| \nu_\alpha \right\rangle = \sum_i U_{\alpha i}^* \left| \nu_i \right\rangle \, ,$$

where $U_{\alpha i}$ is the mixing matrix, and its spacetime evolution is

$$\left| \nu_\alpha(t, x) \right\rangle = \sum_i U_{\alpha i}^* e^{-i (E_i t - p_i x)} \left| \nu_i \right\rangle \, .$$

How are neutrinos described in quantum mechanics?

Amplitude at detection is given by

\[ \begin{aligned} \mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) &= \left \langle \nu_\beta(x_d, t_d) | \nu_\alpha(x_p, t_p) \right\rangle \\[5px] &= \sum_i U_{\alpha i}^* e^{-i (E_i T - p_i L)} U_{\beta i} \, , \end{aligned} \]

where $T = t_d - t_p$ and $L = x_d - x_p$.

How are neutrinos described in quantum mechanics?

Amplitude at detection is given by

\[ \begin{aligned} \mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) &= \left \langle \nu_\beta(x_d, t_d) | \nu_\alpha(x_p, t_p) \right\rangle \\[5px] &= \sum_i U_{\alpha i}^* e^{-i (E_i T - p_i L)} U_{\beta i} \, , \end{aligned} \]

where $T = t_d - t_p$ and $L = x_d - x_p$.

Transition probability found by

$$P_{\nu_\alpha \to \nu_\beta}(L) = \int dT \, \left| \mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) \right|^2 \, .$$

Problems with plane wave treatment

Giunti [1] gives several reasons to consider wave packets

  • Finite lifetime of atomic transitions, so no source of waves vibrates indefinitely
  • Localisation of interactions implies that, the particle cannot be described by an unlocalized plane wave

Beuthe [2] adds that,

“this approach implies a perfectly well-known energy–momentum and an infinite uncertainty on the space–time localization of the oscillating particle. Oscillations are destroyed under these assumptions.”


[1] C. Giunti, Found. Phys. Lett. 17, 103 (2004).
[2] M. Beuthe, Phys. Rep. 375, 105 (2003).

How should we formalise the wave packet treatment?

  • Quantum mechanics (intermediate wave packet approach)
  • Quantum field theory (external wave packet approach)

How should we formalise the wave packet treatment?

  • Quantum mechanics (intermediate wave packet approach)
  • Quantum field theory (external wave packet approach)

$$\left| \nu_\alpha(t, x) \right\rangle = \sum_i U_{\alpha i}^* \psi_i(t, x) \left| \nu_i \right\rangle \, ,$$

where

$$\psi_i(t, x) = \frac{1}{\sqrt{2 \pi}} \int dp_i \, \widetilde{\psi}_i(p_i) e^{-i (E_i t - p_i x)} \, .$$

How should we formalise the wave packet treatment?

Nussinov [3] argued that wave packets cause decoherence

  • Wave packet spreads over time and decoheres when width larger than oscillation length
  • Separation of wave packets due to different group velocities of mass eigenstates

Kayser [4] first to study oscillations with wave packets, and Giunti [5] first to obtain explicit results with Gaussians


[3] S. Nussinov, Phys. Lett. B 63, 201 (1976).
[4] B. Kayser, Phys. Rev. D 24, 110 (1981).
[5] C. Giunti, C. W. Kim, and U. W. Lee, Phys. Rev. D 44, 3635 (1991).

Why a Gaussian?

Can obtain analytic results!

“A gaussian momentum distribution is the most convenient one for the calculation of several integrations …

Other distributions which are sharply peaked around an average momentum lead to the same results after their approximation with a gaussian …

Therefore, the gaussian momentum distributions can be taken as approximations of the real momentum distributions from the beginning.” [6]


[6] C. Giunti, JHEP 2002, 017 (2002).

Really?

A Gaussian is described by its first two moments (mean and variance).
We can have non-Gaussian wave packets!

Conditions on when a distribution $f(p) = \exp [-g(p)]$
can be approximated by a Gaussian [7].

Expanding about minimum $p = P$, we require

$$\frac{1}{4!} \left| g^{(iv)}(P) \right| \ll \frac{1}{2} \left| g''(P) \right|^2$$

[7] E. Kh. Akhmedov and J. Kopp, JHEP 2010, 8 (2010).

Mössbauer neutrinos and Lorentzian wave packets

Neutrinos in the context of the Mössbauer effect,
described by a Lorentzian wave packet [8, 9]

$$\tilde\psi(p; \bar{p}, \gamma) = \mathcal{N} \left[ \frac{\gamma}{(p - \bar{p})^2 + \gamma^2} \right]$$

Moments undefined. Cannot be approximated by a Gaussian!


[8] E. Kh. Akhmedov, J. Kopp, and M. Lindner, JHEP 2008, 005 (2008).
[9] J. Kopp, JHEP 2009, 049 (2009).

Relativistic minimum uncertainty (RMU) wave packets

Can be expressed in a Lorentz invariant form

$$\tilde\psi(p_\mu; a_\mu) = \mathcal{N} \exp \left[ -a_\mu p^\mu \right] \, ,$$

where $a_\mu = (\alpha, -\beta) \in \mathbb{C}^{2}$ and transforms as a vector.

Why RMU wave packets?

Let us reconsider Gaussian wave packets

$$\psi(p; \bar{p}, \sigma_p) = \mathcal{N} \exp \left[ -\frac{(p - \bar{p})^2}{4 \sigma_p^2} \right] \, .$$

Why RMU wave packets?

Let us reconsider Gaussian wave packets

$$\psi(p; \bar{p}, \sigma_p) = \mathcal{N} \exp \left[ -\frac{(p - \bar{p})^2}{4 \sigma_p^2} \right] \, .$$

Gaussian minimises Heisenberg-Robertson uncertainty relation

$$\sigma_x \sigma_p = \left| \frac{1}{2i} \left\langle [\hat{x}, \, \hat{p}] \right\rangle \right| = \frac{\hbar}{2} \, .$$

Assuming that position and momentum are independent

Robertson-Schrödinger uncertainty relation

More generally, can have non-vanishing covariance

$$\sigma_x^2 \sigma_p^2 - \sigma_{x p}^2 = \left| \frac{1}{2i} \left\langle [\hat{x}, \, \hat{p}] \right\rangle \right|^2$$

One obtains squeezed Gaussian wave packets

$$\psi(p; \bar{x}, \bar{p}, \sigma_p) = \mathcal{N} \exp \left[ -\frac{(p - \bar{p})^2}{4 \sigma_p^2} + i \bar{x} p \right]$$

What about RMU wave packets?

Define relativistic velocity operator

$$\hat{v} \equiv i [\hat{H}, \hat{x}] = \frac{\hat{p}}{\hat{H}}$$

Minimise uncertainty between position and velocity

$$\sigma_x^2 \sigma_v^2 - \sigma_{x v}^2 = \left| \frac{1}{2i} \left\langle [\hat{x}, \, \hat{v}] \right\rangle \right|^2$$

What about RMU wave packets?

Minimise uncertainty between position and velocity

$$\sigma_x^2 \sigma_v^2 - \sigma_{x v}^2 = \left| \frac{1}{2i} \left\langle [\hat{x}, \, \hat{v}] \right\rangle \right|^2$$

Squeezed RMU wave packets! Generalised from Ref. [10]

$$\tilde\psi(E_p, p; \alpha, \beta) = \mathcal{N} \exp \left[ -\alpha E_p + \beta p \right] \, ,$$

where $\alpha, \beta$ determined by the moments of velocity and space(time)


[10] M. H. Al-Hashimi and U.-J. Wiese, Ann. Phys. 324, 2599 (2009).

What about RMU wave packets?

Squeezed RMU wave packets! Generalised from Ref. [10]

$$\tilde\psi(E_p, p; \alpha, \beta) = \mathcal{N} \exp \left[ -\alpha E_p + \beta p \right] \, ,$$

where $\alpha, \beta$ determined by the moments of velocity and space(time)

Reduce to Gaussians in non-relativistic limit… and to Lorentzians in the ultra-relativistic limit! (In configuration space and neglecting mass)


[10] M. H. Al-Hashimi and U.-J. Wiese, Ann. Phys. 324, 2599 (2009).

Propagators for different wave packets

Plane wave: $$\mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) = \sum_i U_{\alpha i}^* e^{-i (E_i T - p_i L)} U_{\beta i}$$

Gaussian: $$\mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) \sim \sum_i U_{\alpha i}^* \exp \left[ -\frac{(L - \bar{v}_i T)^2}{4 \sigma_x^2} - i (\bar{E}_i T - \bar{p}_i L) \right] U_{\beta i}$$

RMU: $$\mathcal{A}_{\nu_\alpha \to \nu_\beta}(T, L) \sim \sum_i U_{\alpha i}^* \frac{(T - i \alpha) m_i K_1\left( -m_i \sqrt{(T - i \alpha)^2 + (L - i \beta)^2} \right)}{\sqrt{(T - i \alpha)^2 + (L - i \beta)^2}} U_{\beta i}$$

Implications for decoherence

  1. Gaussian wave packets spread over evolution,
    while RMU wave packets maintain their localisation [11]

    2. In general, $\sigma_x^{(RMU)}(t) \leq \sigma_x^{(G)}(t)$ for all time

  2. RMU wave packets follow semi-classical trajectories,
    so different mass eigenstates do not separate!

[11] C. E. Wood and M. Zych, Phys. Rev. Research 3, 013049 (2021).

Can neutrinos propagate with equal velocities?

Controversy in early 2000s whether group velocities can be equal [12-14]

Consensus is debate settled as “no” from kinematical arguments

  1. If $v_1 = v_2$, then $p_1 / E_1 = p_2 / E_2$ and $p_1 / p_2 = E_1 / E_2$
  2. Since $E_i = \gamma m_i$, then $p_1 / p_2 = E_1 / E_2 = m_1 / m_2$
  3. $E_1 / E_2 \simeq 1$ in ultra-relativistic regime, not generally true for $m_1 / m_2$

[12] Y. Takeuchi, Y. Tazaki, S. Y. Tsai, and T. Yamazaki, Mod. Phys. Lett. A 14, 2329 (1999).
[13] S. De Leo, G. Ducat, and P. Rotelli, Mod. Phys. Lett. A 15, 2057 (2000).
[14] L. B. Okun and I. S. Tsukerman, Mod. Phys. Lett. A 15, 1481 (2000).

Can neutrinos propagate with equal velocities?

Let's be more careful!

Should one take $\bar{v} = \langle p \rangle / \langle E \rangle$?

In general, one has $\bar{v} = \langle \partial_p E \rangle \neq \langle p \rangle / \langle E \rangle$!

RMU wave packets have $\bar{v} = \langle \partial_p E \rangle = \mathrm{Re}(\beta) / \mathrm{Re}(\alpha)$, while $$\frac{\langle p \rangle}{\langle E \rangle} = \frac{\mathrm{Re}(\beta)}{\mathrm{Re}(\alpha)} \left( \frac{1}{1 - \chi_{m, \mathrm{Re}(\alpha), \mathrm{Re}(\beta)}} \right) \, .$$

Only agrees in semi-classical regime, when the wave packets are sufficiently spread with respect to the Compton scale

Summary

  • Neutrino wave packets could have a non-Gaussian profile
    •  
  • If described by RMU wave packets, then neutrinos are highly localised, and decoherence is heavily suppressed
    •  
  • Propagation at equal velocities should be taken seriously,
    and could be experimentally tested

Thank you!


Evan Gale | e.gale@uq.edu.au