The Effect of Spontaneous Collapses on Neutrino Oscillations

Abstract

We compute the effect of collapse models on neutrino oscillations. The effect of the collapse is to modify the evolution of the spatial part of the wave function and we will show that this indirectly amounts to a change on the flavor components. For the analysis we use the mass proportional CSL model, and perform the calculation to second order perturbation theory. As we will show, the CSL effect is very small—mainly due to the very small mass of neutrinos—and practically undetectable.

Appendices

Appendix A: Dimensional Estimate of the Decay Rate

Here we wish to discuss about the possibility of estimate the predictions of the CSL model in neutrino oscillations by using dimensional analysis. Collapse models, as discussed in Sect. 4, are described by the same type of master equations as open quantum systems (which experience decoherence due to interactions with an external environment). In such a case it is well known that the effect of decoherence is to suppress exponentially flavour oscillations. So it does not come as a surprise that for collapse models the effect is the same. Then one could try to guess the decay rate with dimensional analysis, using the relevant constants and parameters of the model. First of all is reasonable to suppose that the effect is proportional to the strength of the noise γ. Moreover, since we are using the mass proportional CSL model, for which γ is replaced by $\gamma_{m_{j}}\equiv\gamma (\frac{m_{j}}{m_{0}} )^{2}$, one expects also a factor $m_{0}^{2}$ in the denominator. Because [γ]=cm³s⁻¹, and the decay rate must have dimension s⁻¹, we need to introduce terms with dimension cm⁻³. Since the parameter r _C has the dimension of a length, then it is natural to introduce an $r_{C}^{3}$ in the denominator. Finally, we need to introduce terms with the dimension of a squared mass. The simplest choices are:

$$ \xi_{jk}^{(1)}\sim\frac{\gamma}{r_{C}^{3}m_{0}^{2}} (m_{j}-m_{k} )^{2}\quad\text{or}\quad\xi_{jk}^{(2)} \sim\frac{\gamma}{r_{C}^{3}m_{0}^{2}} \bigl(m_{j}^{2}-m_{k}^{2} \bigr). $$

(71)

Both these formulas are different compared to the correct one given by Eq. (8). If we substitute the values of the constants and the parameters and we consider, for example, the case of the cosmological neutrinos, we get:

$$\begin{aligned} \xi_{jk}^{(1)}t_{cosm} \sim&\frac{\gamma}{r_{C}^{3}m_{0}^{2}} (m_{j}-m_{k} )^{2}t_{cosm} \sim10^{-17} \end{aligned}$$

(72)

$$\begin{aligned} \xi_{jk}^{(2)}t_{cosm} \sim&\frac{\gamma}{r_{C}^{3}m_{0}^{2}} \bigl(m_{j}^{2}-m_{k}^{2} \bigr)t_{cosm}\sim10^{-11} \end{aligned}$$

(73)

The formula derived with dimensional analysis shows that the CSL effect on neutrino oscillations is very small, practically undetectable. However, it differs by many orders of magnitude from the exact (perturbative) result. We performed the lengthy calculation present here in order to arrive at a fully trustable result. As we have seen here above, dimensional analysis does not allow to reach a firm conclusion.

Appendix B: Approximation in the Calculation of $I_{j}^{ (2 )}$

In this appendix we justify the approximation we used in order to derive Eq. (62) from Eq. (61). This amounts to proving that:

$$ \frac{1}{\hbar^{3}}\int d\mathbf{p}\frac{1}{E_{p}^{ (j )}}e^{-\frac{ (\mathbf{p}-\mathbf{p}_{i} )^{2}r_{C}^{2}}{\hbar^{2}}}\simeq \frac{1}{E_{i}^{ (j )}}\frac{\pi^{3/2}}{r_{C}^{3}} $$

(74)

To see this, we can rewrite the integral in polar coordinates and integrate over the angular variables:

$$ \frac{1}{\hbar^{3}}\int d\mathbf{p}\frac{1}{E_{p}^{ (j )}}e^{-\frac{ (\mathbf{p}-\mathbf{p}_{i} )^{2}r_{C}^{2}}{\hbar^{2}}}= \frac{2\pi}{\hbar^{3}}\frac{\hbar^{2}}{2p_{i}r_{C}^{2}}\int_{-\infty}^{+\infty}dp \frac{p}{\sqrt{p^{2}c^{2}+m_{j}^{2}c^{4}}}e^{-\frac{ (p-p_{i} )^{2}r_{C}^{2}}{\hbar^{2}}} $$

(75)

Let us introduce the dimensionless variable $s=\frac{ (p-p_{i} )r_{C}}{\hbar}$:

$$ \frac{1}{\hbar^{3}}\int d\mathbf{p}\frac{1}{E_{p}^{ (j )}}e^{-\frac{ (\mathbf{p}-\mathbf{p}_{i} )^{2}r_{C}^{2}}{\hbar^{2}}}= \frac{\pi}{p_{i}r_{C}r_{c}^{3}}\int_{-\infty}^{+\infty}ds \frac{ (s+p_{i}\frac{r_{C}}{\hbar} )}{\sqrt{ (s+\frac{r_{C}}{\hbar}p_{i} )^{2}c^{2}+\frac{r_{C}^{2}}{\hbar^{2}}m_{j}^{2}c^{4}}}e^{-s^{2}} $$

(76)

If p _i c≫ħc/r _C ∼10 eV (the typical range of momenta of neutrinos is between 10³ eV and 10¹⁹ eV) we can disregard s both in the numerator and denominator, obtaining:

$$ \frac{1}{\hbar^{3}}\int d\mathbf{p}\frac{1}{E_{p}^{ (j )}}e^{-\frac{ (\mathbf{p}-\mathbf{p}_{i} )^{2}r_{C}^{2}}{\hbar^{2}}} \simeq \frac{\pi}{r_{C}^{3}E_{i}^{ (j )}}\int_{-\infty}^{+\infty}dse^{-s^{2}} = \frac{\pi^{3/2}}{r_{C}^{3}E_{i}^{ (j )}}, $$

(77)

which is the desired result.

Appendix C: Approximation in the Calculation of $I_{jk}^{ (1 )}$

Here we justify the approximation we used to pass from Eq. (58) to Eq. (59). We start with Eq. (58):

$$\begin{aligned} I_{jk}^{ (1 )} (\mathbf{p}_{i},s_{i};t ) = & \sum_{s_{f}}\int d\mathbf{p}_{f} \sqrt{\gamma_{m_{j}}\gamma_{m_{k}}}\frac{m_{j}m_{k}c^{4}}{\sqrt{E_{i}^{ (j )}E_{f}^{ (j )}E_{i}^{ (k )}E_{f}^{ (k )}}} \\ &{}\times\overline{u} \bigl(p_{f}^{ (j )},s_{f}\bigr)u \bigl(p_{i}^{ (j )},s_{i}\bigr)\overline{u} \bigl(p_{i}^{ (k )},s_{i}\bigr)u\bigl(p_{f}^{ (k )},s_{f} \bigr) \\ &{} \times\frac{1-e^{\frac{i}{\hbar} (E_{f}^{ (k )}-E_{i}^{ (k )}-E_{f}^{ (j )}+E_{i}^{ (j )} )t}}{\frac{i}{\hbar}(E_{f}^{ (j )}-E_{i}^{ (j )}-E_{f}^{ (k )}+E_{i}^{ (k )})}\frac{1}{ (2\pi\hbar )^{3}}e^{-\frac{ (\mathbf{p}_{f}-\mathbf{p}_{i} )^{2}r_{C}^{2}}{\hbar^{2}}} \end{aligned}$$

(78)

where:

$$ \begin{aligned} &E_{f}^{ (j )}=\sqrt{\mathbf{p}_{f}^{2}c^{2}+m_{j}^{2}c^{4}},\quad u (p,s )=\frac{p^{\mu}\gamma_{\mu}c+mc^{2}}{\sqrt{2mc^{2} (E_{p}+mc^{2} )}}u (0,s )\quad\text{and}\\ &\sqrt{\gamma_{m_{j}}}=\sqrt{\gamma}\frac{m_{j}}{m_{0}} \end{aligned} $$

(79)

The first part of the integrand:

$$ \frac{m_{j}m_{k}c^{4}}{\sqrt{E_{i}^{ (j )}E_{f}^{ (j )}E_{i}^{ (k )}E_{f}^{ (k )}}}\sum_{s_{f}}\overline{u} \bigl(p_{f}^{ (j )},s_{f}\bigr)u \bigl(p_{i}^{ (j )},s_{i}\bigr)\overline{u} \bigl(p_{i}^{ (k )},s_{i}\bigr)u\bigl(p_{f}^{ (k )},s_{f} \bigr) $$

(80)

is a composition of polynomial functions of ${\bf p}_{f}$, and we can safely assume that it does not change too much, within the range where the Gaussian function is appreciably different from zero. Therefore we can then take ${\bf p}_{f} = {\bf p}_{i}$; by using also the relation:

$$ \overline{u}\bigl(p_{i}^{ (1 )},s_{f}\bigr)u \bigl(p_{i}^{ (1 )},s_{i}\bigr)=\delta_{s_{f},s_{i}}, $$

(81)

Eq. (78) becomes:

$$\begin{aligned} I_{jk}^{ (1 )} (\mathbf{p}_{i},s_{i};t ) \simeq{}& \sqrt{\gamma_{m_{j}}\gamma_{m_{k}}}\frac{m_{j}m_{k}c^{4}}{E_{i}^{ (j )}E_{i}^{ (k )}} \frac{1}{ (2\pi\hbar )^{3}} \\ &{}\times\underset{\equiv I}{\underbrace{\int d\mathbf{p}_{f} \frac{1-e^{\frac{i}{\hbar}(E_{f}^{ (k )}-E_{i}^{ (k )}-E_{f}^{ (j )}+E_{i}^{ (j )})t}}{\frac{i}{\hbar}(E_{f}^{ (j )}-E_{i}^{ (j )}-E_{f}^{ (k )}+E_{i}^{ (k )})}e^{-\frac{(\mathbf{p}_{f}-\mathbf{p}_{i})^{2}r_{C}^{2}}{\hbar^{2}}}}} \end{aligned}$$

(82)

Now we have to focus our attention on the integral I, which contains an oscillating term that needs special care. As before, we write the integral in polar coordinates and perform the integration over the angular variables; moreover, we introduce once again the a-dimensional variable $s=\frac{ (p_{f}-p_{i} )r_{C}}{\hbar}$. We have:

$$ I=\pi\frac{\hbar^{3}}{p_{i}r_{C}^{3}}t\int_{-\infty}^{+\infty}ds \biggl(\frac{\hbar}{r_{C}}s+p_{i} \biggr)\frac{e^{ig (s )}-1}{ig (s )}e^{-s^{2}}, $$

(83)

where we have defined:

$$\begin{aligned} g (s ) \equiv & \frac{1}{\hbar}\bigl(E_{f}^{ (k )}-E_{i}^{ (k )}-E_{f}^{ (j )}+E_{i}^{ (j )} \bigr)t \\ = & \frac{ct}{r_{C}} \bigl(\sqrt{ (s+y )^{2}+a_{k}}- \sqrt{ (s+y )^{2}+a_{j}}-\sqrt{y^{2}+a_{k}}+ \sqrt{y^{2}+a_{j}} \bigr), \end{aligned}$$

(84)

with

$$ a_{j}\equiv\frac{r_{C}^{2}}{\hbar^{2}}m_{j}^{2}c^{2}= \bigl(10^{-2}~\text{eV}^{-2} \bigr)m_{j}^{2}c^{4} \text{e} y\equiv\frac{r_{C}}{\hbar}p_{i}= \bigl(10^{-1}~\text{eV}^{-1} \bigr)p_{i}c $$

(85)

Our goal is to show that g(s) does not vary appreciably, within the range where the Gaussian term is significantly different from zero, and can be approximated with g(0)=0; in this way, the integral can be computed exactly. This kind of approximation is not obvious because the factor ct/r _C in front of Eq. (84) can be very big.

In the ultra-relativistic limit, we approximate the particle’s velocity with the speed of light. In this limit, y ²≫a _j ,a _k , and so we can expand the square roots in g(s) using the Taylor series $\sqrt{x+\epsilon}=\sqrt{x}+\frac{\epsilon}{2\sqrt{x}}$ and we get:

$$ g (s )\simeq\frac{ct}{r_{C}} (a_{j}-a_{k} ) \frac{s}{ (s+y )y}=\frac{ct}{r_{C}}\frac{ (a_{j}-a_{k} )}{ (y+\frac{y^{2}}{s} )}. $$

(86)

In order for g(s) to remain small within the interval where the Gaussian term of Eq. (83) is appreciably different from zero, we need:

$$ \biggl(y+\frac{y^{2}}{s} \biggr) \gg \frac{ct}{r_{C}} (a_{j}-a_{k} ) $$

(87)

In all physically interesting situations, the term on the right hand side of Eq. (87) is bigger than 1; moreover s is of the order of unity, because of the Gaussian in Eq. (83). Inequality (87) is verified if the following condition is true:

$$ y \gg \sqrt{\frac{ct}{r_{C}} (a_{j}-a_{k} )}. $$

(88)

Typically, cosmogenic neutrinos have energies bigger than 10¹⁸ eV and travel distances of at most 10⁹ light-years [27]. This means that, in the worst case, ct/r _C ∼10³² while (a _j −a _k )=(10⁻² eV⁻²) $(m_{j}^{2}c^{4}-m_{k}^{2}c^{4})\simeq(10^{-2}~\text{eV}^{-2})(2\times10^{-3}~\text{eV}^{2})=10^{-5}$. So we must have y≫10¹⁴, that means p _i c=y/(10⁻¹ eV⁻¹)≫10¹⁵ eV, which is satisfied.

For atmospheric neutrinos, ct/r _C is in the range 10¹¹−10¹⁴ while the range of energies is between 10⁻¹ GeV and 10⁴ GeV [25]. This means that, even in the worse case, the condition to check is y≫10⁵, which means p _i c=y/(10⁻¹ eV⁻¹)≫10⁶ eV. This is also satisfied.