Model tree-level contribution. Neglecting $r_f$, $\lambda_f$ is universal, and we can define an observable phase $\phi_D$ via

$$ \lambda_f \equiv -|q/p|e^{i\phi_D}. \qquad (13.54) $$

(In the limit of CP conservation, choosing $\phi_D = 0$ is equivalent to defining the mass eigenstates by their CP eigenvalue: $|D_{\mp}\rangle = p|D^0\rangle \pm q|\bar{D}^0\rangle$, with $D_-$ ($D_+$) being the CP-odd (CP-even) state; that is, the state that does not (does) decay into $K^+K^-$.)

We define the time integrated CP asymmetry for a final CP eigenstate $f$ as follows:

$$ a_f \equiv \frac{\int_0^\infty \Gamma(D_{\text{phys}}^0(t) \to f) dt - \int_0^\infty \Gamma(\bar{D}_{\text{phys}}^0(t) \to f) dt}{\int_0^\infty \Gamma(D_{\text{phys}}^0(t) \to f) dt + \int_0^\infty \Gamma(\bar{D}_{\text{phys}}^0(t) \to f) dt}. \qquad (13.55) $$

(This expression corresponds to the $D$ meson being tagged at production, hence the integration goes from $0$ to $+\infty$; measurements are also possible with $\psi(3770) \to D^0\bar{D}^0$, in which case the integration goes from $-\infty$ to $+\infty$ giving slightly different results; see the discussion in Section 13.1.3.) We take $x, y, r_f \ll 1$ and expand to leading order in these parameters. We can then separate the contribution to $a_f$ into three parts [64],

$$ a_f = a_f^d + a_f^m + a_f^i, \qquad (13.56) $$

with the following underlying mechanisms:

1. $a_f^d$ signals CP violation in decay (similar to Eq. (13.21)):

$$ a_f^d = 2r_f \sin \phi_f \sin \delta_f. \qquad (13.57) $$

2. $a_f^m$ signals CP violation in mixing (similar to Eq. (13.30)). With our approximations, it is universal:

$$ a^m = -\frac{y}{2} \left( \left|\frac{q}{p}\right| - \left|\frac{p}{q}\right| \right) \cos \phi_D. \qquad (13.58) $$

3. $a_f^i$ signals CP violation in the interference of mixing and decay (similar to Eq. (13.31)). With our approximations, it is universal:

$$ a^i = \frac{x}{2} \left( \left| \frac{q}{p} \right| + \left| \frac{p}{q} \right| \right) \sin \phi_D. \qquad (13.59) $$

In the SM, both $a^m$ and $a^i$ are $\mathcal{O}(10^{-5})$ or less, while $a^d$ could be up to two orders of magnitude larger.

One can isolate the effects of direct CP violation by taking the difference between the CP asymmetries in the $K^+K^-$ and $\pi^+\pi^-$ modes:

$$ \Delta a_{CP} \equiv a_{K+K-} - a_{\pi+\pi-} = a_{K+K-}^d - a_{\pi+\pi-}^d, \qquad (13.60) $$

where we neglected a residual, experiment-dependent, contribution from indirect CP violation due to the fact that there may be a decay time-dependent acceptance function that can be different for the $K^+K^-$ and $\pi^+\pi^-$ channels. The current average gives [13, 62]:

$$ a_{K+K-}^d - a_{\pi+\pi-}^d = (-0.161 \pm 0.028) \times 10^{-2}, \qquad (13.61) $$

demonstrating CP violation in charm decay. While the asymmetry is somewhat larger than the theoretical predictions that preceded the measurement, it can in principle be explained by non-perturbative QCD effects.

One can also isolate the effects of indirect CP violation in the following way. Consider the time-dependent decay rates in Eq. (13.17a) and Eq. (13.17b). The mixing processes modify the time dependence from a pure exponential. However, given the small values of $x$ and $y$, the time dependences can be recast, to a good approximation, into purely exponential form, but with modified decay-rate parameters [65, 66] (given here for the $K^+K^-$ final state):

$$ \Gamma_{D^0 \to K^+ K^-} = \Gamma \times [1 + |q/p| (y \cos \phi_D - x \sin \phi_D)], \qquad (13.62a) $$

$$ \Gamma_{\bar{D}^0 \to K^+ K^-} = \Gamma \times [1 + |p/q| (y \cos \phi_D + x \sin \phi_D)]. \qquad (13.62b) $$

One can define CP-conserving and CP-violating combinations of these two observables (normalized to the true width $\Gamma$):

$$ \begin{aligned} y_{CP} &\equiv \frac{\Gamma_{\bar{D}^0 \to K^+ K^-} + \Gamma_{D^0 \to K^+ K^-}}{2\Gamma} - 1 \\ &= (y/2) (|q/p| + |p/q|) \cos \phi_D - (x/2) (|q/p| - |p/q|) \sin \phi_D, \end{aligned} \qquad (13.63a) $$

$$ \begin{aligned} A_{\Gamma} &\equiv \frac{\Gamma_{D^0 \to K+K-} - \Gamma_{\bar{D}^0 \to K+K-}}{2\Gamma} \\ &= -(a^m + a^i). \end{aligned} \qquad (13.63b) $$

In the limit of CP conservation (and, in particular, within the Standard Model), $y_{CP} = (\Gamma_+ - \Gamma_-)/2\Gamma = y$ (where $\Gamma_+(\Gamma_-)$ is the decay width of the CP-even (-odd) mass eigenstate) and $A_\Gamma = 0$. Indeed, present measurements imply that CP violation is small [62],

$$ y_{CP} - y_{CP}(K\pi) = (+0.697 \pm 0.028) \times 10^{-2}, \qquad (13.64a) $$

$$ A_{\Gamma} = (0.009 \pm 0.011) \times 10^{-2}, \qquad (13.64b) $$

where the correction $y_{CP}(K\pi)$ is necessary at high precision since experimentally the denominator of the relative widths in Eq. (13.63a) is measured with the $D^0 \to K^+\pi^+$ mode [67, 68].

The $K^\pm\pi^\mp$ states are not CP eigenstates, but they are still common final states for $D^0$ and $\bar{D}^0$ decays. Since $D^0(\bar{D}^0) \to K^+\pi^+$ is a Cabibbo-favored (doubly-Cabibbo-suppressed) process, these processes are particularly sensitive to $x$ and/or $y = \mathcal{O}(\lambda^2)$. Taking into account that $|\lambda_{K^+\pi^+}|, |\lambda_{K^+\pi^-}^{-1}| \ll 1$ and $x, y \ll 1$, assuming that there is no direct CP violation (these are Standard Model tree-level decays dominated by a single weak phase, and there is no contribution from penguin-like and chromomagnetic operators), and expanding the time-dependent rates for $xt, yt \lesssim \Gamma^{-1}$, one obtains

$$ \begin{aligned} \Gamma[D_{\text{phys}}^0(t) \to K^+\pi^-] &= e^{-\Gamma t} |\bar{A}_{K-\pi+}|^2 \\ &\quad \times \left[ r_d^2 + r_d \left|\frac{q}{p}\right| (y' \cos \phi_D - x' \sin \phi_D) \Gamma t + \left|\frac{q}{p}\right|^2 \frac{y^2 + x^2}{4} (\Gamma t)^2 \right], \end{aligned} \qquad (13.65a) $$

$$ \begin{aligned} \Gamma[\bar{D}_{\text{phys}}^0(t) \to K^-\pi^+] &= e^{-\Gamma t} |\bar{A}_{K-\pi+}|^2 \\ &\quad \times \left[ r_d^2 + r_d \left|\frac{p}{q}\right| (y' \cos \phi_D + x' \sin \phi_D) \Gamma t + \left|\frac{p}{q}\right|^2 \frac{y^2 + x^2}{4} (\Gamma t)^2 \right], \end{aligned} \qquad (13.65b) $$

where

$$ y' \equiv y \cos \delta - x \sin \delta \quad \text{and} \quad x' \equiv x \cos \delta + y \sin \delta. \qquad (13.66) $$

The weak phase $\phi_D$ is the same as that of Eq. (13.54) (a consequence of neglecting direct CP violation) and $r_d = \mathcal{O}(\tan^2 \theta_c)$ is the amplitude ratio, $r_d = |\bar{A}_{K-\pi+}/A_{K-\pi+}| = |A_{K+\pi-}/\bar{A}_{K+\pi-}|$, that is, $\lambda_{K-\pi+} = r_d|q/p|e^{-i(\delta-\phi_D)}$ and $\lambda_{K+\pi-}^{-1} = r_d|p/q|e^{-i(\delta+\phi_D)}$. The parameter $\delta$ is a strong-phase difference for these processes, that can be obtained from measurements of quantum correlated $\psi(3770) \to D^0\bar{D}^0$ decays [69, 70]. By fitting to the six coefficients of the various time-dependences, one can determine $r_d$, $|q/p|$, $(x^2 + y^2)$, $y' \cos \phi_D$, and $x' \sin \phi_D$. In particular, finding CP violation ($|q/p| \neq 1$ and/or $\sin \phi_D \neq 0$) at a level much higher than $10^{-3}$ would constitute evidence for new physics. The most stringent constraints to date on CP violation in charm mixing have been obtained with this method [71] and from the $A_\Gamma$ measurement [72].

A fit to all data [62], including also results from time-dependent analyses of $D^0 \to K_S\pi^+\pi^-$ decays, from which $x, y, |q/p|$ and $\phi_D$ can be determined directly, yields no evidence for indirect CP violation:

$$ 1 - |q/p| = +0.006^{+0.015}_{-0.016}, \qquad (13.67a) $$

$$ \phi_D = (-2.6 \pm 1.1)^\circ. \qquad (13.67b) $$
With the additional assumption of no direct CP violation in doubly-Cabibbo-suppressed $D$ decays [73–75], more stringent constraints are obtained:

$$1 - |q/p| = -0.005 \pm 0.007, \qquad (13.68a)$$

$$\phi_D = (-0.2 \pm 0.3)^\circ. \qquad (13.68b)$$

More details on various theoretical and experimental aspects of $D^0-\bar{D}^0$ mixing can be found in Ref. [32].

Searches for CP violation in charged $D_s$ decays have been performed in many modes. Searches in decays mediated by Cabibbo-suppressed amplitudes are particularly interesting, since in other channels effects are likely to be too small to be observable in current experiments. Examples of relevant two-body modes are $D^+ \to \pi^+\pi^0$, $K_S K^+$, $\phi\pi^+$ and $D_s^+ \to K^+\pi^0$, $K_S\pi^+$, $\phi K^+$. The most precise results are $A_{D^+\to K_S K^+} = +0.0011 \pm 0.0017$ and $A_{D^+\to K_S\pi^+} = +0.0038 \pm 0.0048$ [62]. The precision of experiments is now sufficient that the effect from CP violation in the neutral kaon system can be seen in $D^+ \to K_S\pi^+$ decays [76, 77].

Three- and four-body final states provide additional possibilities to search for CP violation, since effects may vary over the phase-space [78]. A number of methods have been proposed to exploit this feature and search for CP violation in ways that do not require modelling of the decay distribution [79–82]. Such methods are useful for analysis of charm decays since they are less sensitive to biases from production asymmetries, and are well suited to address the issue of whether or not CP violation effects are present. They can also be applied to tagged neutral $D$ meson as well as to charged $D_s$ decays (flavor tagging is typically achieved from the charge of the pion produced in $D^{*+} \to D^0\pi^+$ decays). The results of all searches to date are consistent with the absence of CP violation, with the most significant hint at the level of $2.7\sigma$ [83].

## 13.6 Beauty

### 13.6.1 CP violation in mixing of $B^0$ and $B_s^0$ mesons

The upper bound on the CP asymmetry in semileptonic $B$ decays [84] implies that CP violation in $B^0-\bar{B}^0$ mixing is a small effect (we use $A_{SL}/2 \approx 1 - |q/p|$, see Eq. (13.23)):

$$A_{\text{SL}}^d = (-2.1 \pm 1.7) \times 10^{-3} \implies |q/p| = 1.0010 \pm 0.0008. \qquad (13.69)$$

The Standard Model prediction is

$$A_{\text{SL}}^d = \mathcal{O} \left[ \left(m_c^2/m_t^2\right) \sin\beta \right] \lesssim 0.001. \qquad (13.70)$$

An explicit calculation gives $(-4.73 \pm 0.42) \times 10^{-4}$ [36].

The experimental constraint on CP violation in $B_s^0-\bar{B}_s^0$ mixing is somewhat weaker than that in the $B^0-\bar{B}^0$ system [84]

$$A_{\text{SL}}^s = (-0.6 \pm 2.8) \times 10^{-3} \implies |q/p| = 1.0003 \pm 0.0014. \qquad (13.71)$$

The Standard Model prediction is $A_{\text{SL}}^s = \mathcal{O} \left[ (m_c^2/m_t^2) \sin\beta_s \right] \lesssim 10^{-4}$, with an explicit calculation giving $(2.06 \pm 0.18) \times 10^{-5}$ [36].

The fit to experimental data that results in the averages quoted above has a $\chi^2$ probability of 4.5% indicating some tension between the different measurements [62]. This originates in part from a result from the D0 collaboration for the inclusive same-sign dimuon asymmetry that deviates from the Standard Model prediction by $3.6\sigma$ [85]. As yet, this has not been confirmed by independent studies.

In models where $\Gamma_{12}/M_{12}$ is approximately real, such as the Standard Model, an upper bound on $\Delta\Gamma/\Delta m \approx \mathcal{R}e(\Gamma_{12}/M_{12})$ provides yet another upper bound on the deviation of $|q/p|$ from one. This constraint does not hold if $\Gamma_{12}/M_{12}$ is approximately imaginary. (An alternative parameterization uses $q/p = (1 - \tilde{\epsilon}_B)/(1 + \tilde{\epsilon}_B)$, leading to $A_{\text{SL}} \simeq 4\mathcal{R}e(\tilde{\epsilon}_B)$.)

### 13.6.2 CP violation in interference of $B^0$ decays with and without mixing

The small deviation (less than one percent) of $|q/p|$ from 1 implies that, at the present level of experimental precision, CP violation in $B^0$ mixing is a negligible effect. Thus, for the purpose of analyzing CP asymmetries in hadronic $B^0$ decays, we can use

$$\lambda_f = e^{-i\phi_M(B^0)} (\bar{A}_f/A_f), \qquad (13.72)$$

where $\phi_{M(B^0)}$ refers to the phase of $M_{12}$ appearing in Eq. (13.28) that is appropriate for $B^0-\bar{B}^0$ oscillations. Within the Standard Model, the corresponding phase factor is given by

$$e^{-i\phi_M(B^0)} = (V_{tb}^* V_{td}) / (V_{tb} V_{td}^*) . \qquad (13.73)$$

The class of CP violation effects in interference between mixing and decay is studied with final states that are common to $B^0$ and $\bar{B}^0$ decays [86–88]. It is convenient to rewrite Eq. (13.26) for $B^0$ decays as [89–91]

$$A_f(t) = S_f \sin(\Delta mt) - C_f \cos(\Delta mt), \qquad (13.74)$$

$$S_f \equiv \frac{2 \mathcal{I} m(\lambda_f)}{1 + |\lambda_f|^2}, \quad C_f \equiv \frac{1 - |\lambda_f|^2}{1 + |\lambda_f|^2}, \qquad (13.75)$$

where we assume that $\Delta\Gamma = 0$ and $|q/p| = 1$. An alternative notation in use is $A_f \equiv -C_f$ – this $A_f$ should not be confused with the $A_f$ of Eq. (13.1), but in the limit that $|q/p| = 1$ is equivalent with the $A_f$ of Eq. (13.21).

A large class of interesting processes proceed via quark transitions of the form $\bar{b} \to \bar{q}q\bar{q}'$ with $q' = s$ or $d$. For $q = c$ or $u$, there are contributions from both tree ($t$) and penguin ($p^{q_u}$, where $q_u = u, c, t$ is the quark in the loop) diagrams (see Fig. 13.2) which carry different weak phases:

$$A_f = (V_{qb}^* V_{qq'}) t_f + \sum_{q_u=u,c,t} (V_{q_u b}^* V_{q_u q'}) p_f^{q_u}. \qquad (13.76)$$

(The distinction between tree and penguin contributions is a heuristic one; the separation by the operator that enters is more precise. A more detailed discussion of the operator product expansion approach, which also includes higher order QCD corrections, can be found in Ref. [92, 93] for example.) Using CKM unitarity, the various decay amplitudes can always be written in terms of just two CKM combinations. For example, for $f = \pi\pi$, which proceeds via a $\bar{b} \to \bar{u}ud$ transition, we can write

$$A_{\pi\pi} = (V_{ub}^* V_{ud}) T_{\pi\pi} + (V_{tb}^* V_{td}) P_{\pi\pi}^t, \qquad (13.77)$$

where $T_{\pi\pi} = t_{\pi\pi} + p_{\pi\pi}^u - p_{\pi\pi}^c$ and $P_{\pi\pi}^t = p_{\pi\pi}^t - p_{\pi\pi}^c$. CP-violating phases in Eq. (13.77) appear only in the CKM elements, so that

$$\frac{\bar{A}_{\pi\pi}}{A_{\pi\pi}} = \frac{(V_{ub}^* V_{ud}^*) T_{\pi\pi} + (V_{tb}^* V_{td}^*) P_{\pi\pi}^t}{(V_{ub}^* V_{ud}^*) T_{\pi\pi} + (V_{tb}^* V_{td}^*) P_{\pi\pi}^t}. \qquad (13.78)$$

For $f = J/\psi K$, which proceeds via a $\bar{b} \to \bar{c}\bar{c}s$ transition, we can write

$$A_{\psi K} = (V_{cb}^* V_{cs}) T_{\psi K} + (V_{ub}^* V_{us}) P_{\psi K}^u, \qquad (13.79)$$

where $T_{\psi K} = t_{\psi K} + p_{\psi K}^c - p_{\psi K}^t$ and $P_{\psi K}^u = p_{\psi K}^u - p_{\psi K}^t$. A subtlety arises in this decay that is related to the fact that $B^0$ decays into a final $J/\psi K^0$ state while $\bar{B}^0$ decays into a final $J/\psi \bar{K}^0$ state. A common final state, e.g., $J/\psi K_S$, is reached only via $K^0-\bar{K}^0$ mixing. Consequently, the phase factor (defined in Eq. (13.28)) corresponding to neutral $K$ mixing, $e^{-i\phi_M(K)} = (V_{cd}^* V_{cs}) / (V_{cd} V_{cs}^*)$, plays a role:

$$\frac{\bar{A}_{\psi K_S}}{A_{\psi K_S}} = - \frac{(V_{cb}^* V_{cs}^*) T_{\psi K} + (V_{ub}^* V_{us}^*) P_{\psi K}^u}{(V_{cb}^* V_{cs}^*) T_{\psi K} + (V_{ub}^* V_{us}^*) P_{\psi K}^u} \times \frac{V_{cd}^* V_{cs}}{V_{cd} V_{cs}^*}. \qquad (13.80)$$

For $q = s$ or $d$, there are only penguin contributions to $A_f$, that is, $t_f = 0$ in Eq. (13.76). (The tree $\bar{b} \to \bar{u}u\bar{q}'$ transition followed by $\bar{u}u \to \bar{q}q$ rescattering is included below in the $P^u$ terms.) Again, CKM unitarity allows us to write $A_f$ in terms of two CKM combinations. For example, for $f = \phi K_S$, which proceeds via a $\bar{b} \to \bar{s}s\bar{s}$ transition, we can write

$$\frac{\bar{A}_{\phi K_S}}{A_{\phi K_S}} = - \frac{(V_{cb}^* V_{cs}^*) P_{\phi K}^c + (V_{ub}^* V_{us}^*) P_{\phi K}^u}{(V_{cb}^* V_{cs}^*) P_{\phi K}^c + (V_{ub}^* V_{us}^*) P_{\phi K}^u} \times \frac{V_{cd}^* V_{cs}}{V_{cd} V_{cs}^*}, \qquad (13.81)$$
<figure><img src="image_49.png" /><figcaption>**Figure 13.2:** Feynman diagrams for (a) tree and (b) penguin amplitudes contributing to $B^0 \to f$ or $B_s^0 \to f$ via a $\bar{b} \to \bar{q}q\bar{q}'$ quark-level process.</figcaption></figure>

$$ \text{where } P_{\phi K}^{c} = p_{\phi K}^{c} - p_{\phi K}^{t} \text{ and } P_{\phi K}^{u} = p_{\phi K}^{u} - p_{\phi K}^{t}. $$

Since in general the amplitude $A_f$ involves two different weak phases, the corresponding decays can exhibit both CP violation in the interference of decays with and without mixing, $S_f \neq 0$, and CP violation in decay, $C_f \neq 0$. (At the present level of experimental precision, the contribution to $C_f$ from CP violation in mixing is negligible, see Eq. (13.69).) If the contribution from a second weak phase is suppressed, then the interpretation of $S_f$ in terms of Lagrangian CP-violating parameters is clean, while $C_f$ is small. If such a second contribution is not suppressed, $S_f$ depends on hadronic parameters and, if the relevant strong phase difference is large, $C_f$ is large.

A summary of $\bar{b} \to \bar{q}q\bar{q}'$ modes with $q' = s$ or $d$ is given in Table 13.1. The $\bar{b} \to dd\bar{q}$ transitions lead to final states that are similar to those from $\bar{b} \to \bar{u}u\bar{q}$ transitions and have similar phase dependence. Final states that consist of two vector mesons ($\psi\phi$ and $\phi\phi$) are not CP eigenstates, and angular analysis is needed to separate the CP-even from the CP-odd contributions.

The cleanliness of the theoretical interpretation of $S_f$ can be assessed from the information in the last column of Table 13.1. In case of small uncertainties, the expression for $S_f$ in terms of CKM phases can be deduced from the fourth column of Table 13.1 in combination with Eq. (13.73) (and, for $b \to \bar{q}\bar{q}s$ decays, the example in Eq. (13.80)). Here we consider several interesting examples.

For $B^0 \to J/\psi K_S$ and other $\bar{b} \to \bar{c}c\bar{s}$ processes, we can neglect the $P^u$ contribution to $A_f$, in the Standard Model, to an approximation that is better than one percent, giving

$$ \lambda_{\psi K_S} = -e^{-2i\beta} \Rightarrow S_{\psi K_S} = \sin(2\beta), \quad C_{\psi K_S} = 0. \quad (13.82) $$

It is important to verify experimentally the level of suppression of the penguin contribution. Methods based on flavor symmetries [94–97] allow limits to be obtained. All are currently consistent with the $P^u$ term being negligible. Explicit calculations [97–100] also support this conclusion.

In the presence of new physics, $A_f$ is still likely to be dominated by the T term, but the mixing amplitude might be modified. Thus, model-independentl, $C_f \approx 0$ while $S_f$ cleanly determines the mixing phase ($\phi_M - 2\arg(V_{cb}V_{cd}^*)$). The experimental measurement gave the first precision test of the Kobayashi-Maskawa mechanism. The latest world average [62] is

$$ S_{\psi K^0} = +0.709 \pm 0.011. \quad (13.83) $$

(We use $K^0$ throughout to denote results that combine $K_S$ and $K_L$ modes, but use the sign appropriate to $K_S$.) The consistency of this measurement with the predictions for $\sin 2\beta$ makes it very likely that this mechanism is indeed the dominant source of CP violation in the quark sector.

For $B^0 \to \phi K_S$ and other $\bar{b} \to \bar{s}s\bar{s}$ processes (as well as some $\bar{b} \to \bar{u}u\bar{s}$ processes), we can neglect the subdominant contributions, in the Standard Model, to an approximation that is good to the order of a few percent:

$$ \lambda_{\phi K_S} = -e^{-2i\beta} \Rightarrow S_{\phi K_S} = \sin 2\beta, \quad C_{\phi K_S} = 0. \quad (13.84) $$

A review of explicit calculations of the effects of subleading amplitudes can be found in Ref. [101]. In the presence of new physics, both $A_f$ and $M_{12}$ can have contributions that are comparable in size to those of the Standard Model and carry new weak phases. Such a situation gives several interesting consequences for penguin-dominated $\bar{b} \to \bar{q}q\bar{s}$ decays ($q = u, d, s$) to a final state $f$:

1. The value of $-\eta_f S_f$ may be different from $S_{\psi K_S}$ by more than a few percent, where $\eta_f$ is the CP eigenvalue of the final state.

2. The values of $\eta_f S_f$ for different final states $f$ may be different from each other by more than a few percent (for example, $S_{\phi K_S} \neq S_{\eta' K_S}$).

3. The value of $C_f$ may be different from zero by more than a few percent.

While a clear interpretation of such signals in terms of Lagrangian parameters will be difficult because, under these circumstances, hadronic parameters play a role, any of the above three options will clearly signal new physics. In addition, flavor symmetry relations, such as those that relate observables in $B \to K\pi$ decays [102,103] can be used to provide further tests of the Standard Model. Fig. 13.3 summarizes the present experimental results: none of the possible signatures listed above is unambiguously established, but there is definitely still room for new physics.

<figure><img src="image_50.png" /><figcaption>**Figure 13.3:** Summary of the results [62] of time-dependent analyses of $b \to \bar{q}\bar{q}s$ decays, which are potentially sensitive to new physics.</figcaption></figure>

For the $\bar{b} \to \bar{u}\bar{u}\bar{d}$ process $B \to \pi\pi$ and other related channels, the penguin-to-tree ratio can be estimated using SU(3) relations and experimental data on related $B \to K\pi$ decays. The result (for $\pi\pi$) is that the suppression is at the level of 0.2–0.3 and so cannot be neglected. The expressions for $S_{\pi\pi}$ and $C_{\pi\pi}$ to leading order in $R_{PT} \equiv (|V_{tb}V_{td}| P_{\pi\pi}^t) / (|V_{ub}V_{ud}| T_{\pi\pi})$ are:

$$ \lambda_{\pi\pi} = e^{2i\alpha} \left[ (1 - R_{PT}e^{-i\alpha}) / (1 - R_{PT}e^{+i\alpha}) \right] \Rightarrow \quad (13.85) $$

$$ S_{\pi\pi} \approx \sin 2\alpha + 2 \operatorname{Re}(R_{PT}) \cos 2\alpha \sin \alpha, \quad C_{\pi\pi} \approx 2 \operatorname{Im}(R_{PT}) \sin \alpha. \quad (13.86) $$
<table><caption>**Table 13.1:** Summary of $\bar{b} \to \bar{q}q\bar{q}'$ modes with $q' = s$ or $d$. The second and third columns give examples of hadronic final states (usually those which are experimentally most convenient to study). The fourth column gives the CKM dependence of the amplitude $A_f$, using the notation of Eqs. ((13.77), (13.79), (13.81)), with the dominant term first and the subdominant second. The suppression factor of the second term compared to the first is given in the last column. “Loop” refers to a penguin versus tree-suppression factor (it is mode-dependent and roughly $O(0.2-0.3)$) and $\lambda \simeq 0.23$ is the expansion parameter of Eq. (13.34).</caption><thead><tr><th>b&#773; &rarr; q&#773;q&#773;q&#773;'</th><th>B<sup>0</sup> &rarr; f</th><th>B<sub>s</sub><sup>0</sup> &rarr; f</th><th>CKM dependence of A<sub>f</sub></th><th>Suppression</th></tr></thead><tbody><tr><td>b&#773; &rarr; c&#773;c&#773;s&#773;</td><td>&psi;K<sub>S</sub></td><td>&psi;&phi;</td><td>(V<sub>cb</sub><sup>*</sup>V<sub>cs</sub>)T + (V<sub>ub</sub><sup>*</sup>V<sub>us</sub>)P<sup>u</sup></td><td>loop &times; &lambda;<sup>2</sup></td></tr><tr><td>b&#773; &rarr; s&#773;s&#773;s&#773;</td><td>&phi;K<sub>S</sub></td><td>&phi;&phi;</td><td>(V<sub>cb</sub><sup>*</sup>V<sub>cs</sub>)P<sup>c</sup> + (V<sub>ub</sub><sup>*</sup>V<sub>us</sub>)P<sup>u</sup></td><td>&lambda;<sup>2</sup></td></tr><tr><td>b&#773; &rarr; u&#773;u&#773;s&#773;</td><td>&pi;<sup>0</sup>K<sub>S</sub></td><td>K<sup>+</sup>K<sup>-</sup></td><td>(V<sub>cb</sub><sup>*</sup>V<sub>cs</sub>)P<sup>c</sup> + (V<sub>ub</sub><sup>*</sup>V<sub>us</sub>)T</td><td>&lambda;<sup>2</sup>/loop</td></tr><tr><td>b&#773; &rarr; c&#773;cd&#773;</td><td>D<sup>+</sup>D<sup>-</sup></td><td>&psi;K<sub>S</sub></td><td>(V<sub>cb</sub><sup>*</sup>V<sub>cd</sub>)T + (V<sub>tb</sub><sup>*</sup>V<sub>td</sub>)P<sup>t</sup></td><td>loop</td></tr><tr><td>b&#773; &rarr; s&#773;s&#773;d&#773;</td><td>K<sub>S</sub>K<sub>S</sub></td><td>&phi;K<sub>S</sub></td><td>(V<sub>tb</sub><sup>*</sup>V<sub>td</sub>)P<sup>t</sup> + (V<sub>cb</sub><sup>*</sup>V<sub>cd</sub>)P<sup>c</sup></td><td>~1</td></tr><tr><td>b&#773; &rarr; u&#773;ud&#773;</td><td>&pi;<sup>+</sup>&pi;<sup>-</sup></td><td>&rho;<sup>0</sup>K<sub>S</sub></td><td>(V<sub>ub</sub><sup>*</sup>V<sub>ud</sub>)T + (V<sub>tb</sub><sup>*</sup>V<sub>td</sub>)P<sup>t</sup></td><td>loop</td></tr><tr><td>b&#773; &rarr; c&#773;ud&#773;</td><td>D<sub>CP</sub>&pi;<sup>0</sup></td><td>D<sub>CP</sub>K<sub>S</sub></td><td>(V<sub>cb</sub><sup>*</sup>V<sub>ud</sub>)T + (V<sub>ub</sub><sup>*</sup>V<sub>cd</sub>)T'</td><td>&lambda;<sup>2</sup></td></tr><tr><td>b&#773; &rarr; c&#773;u&#773;s&#773;</td><td>D<sub>CP</sub>K<sub>S</sub></td><td>D<sub>CP</sub>&phi;</td><td>(V<sub>cb</sub><sup>*</sup>V<sub>us</sub>)T + (V<sub>ub</sub><sup>*</sup>V<sub>cs</sub>)T'</td><td>~1</td></tr></tbody></table>

Note that $R_{PT}$ is mode-dependent and, in particular, could be different for $\pi^+\pi^-$ and $\pi^0\pi^0$. If strong phases can be neglected, then $R_{PT}$ is real, resulting in $C_{\pi\pi} = 0$. The size of $C_{\pi\pi}$ is an indicator of how large the strong phase is. The present experimental average is $C_{\pi^+\pi^-} = -0.311 \pm 0.030$ [62]. As concerns $S_{\pi\pi}$, it is clear from Eq. (13.86) that the relative size or strong phase of the penguin contribution must be known to extract $\alpha$. The theoretical uncertainty stemming from $|R_{PT}| \ll 1$ is referred to in the literature as penguin pollution.

The cleanest solution involves isospin relations among the $B \to \pi\pi$ amplitudes [104]:

$$ \frac{1}{\sqrt{2}} A_{\pi+\pi-} + A_{\pi^0\pi^0} = A_{\pi+\pi^0}. \qquad (13.87) $$

The method exploits the fact that the penguin contribution to $P_{\pi\pi}^t$ is pure $\Delta I = 1/2$ (this is not true for the electroweak penguins which, however, are expected to be small), while the tree contribution to $T_{\pi\pi}$ contains amplitudes that are both $\Delta I = 1/2$ and $\Delta I = 3/2$. A simple geometric construction then allows one to find $R_{PT}$ and extract $\alpha$ cleanly from $S_{\pi+\pi^-}$. The key experimental difficulty is that one must measure accurately the separate rates for $B^0$ and $\bar{B}^0 \to \pi^0\pi^0$.

CP asymmetries in $B \to \rho\pi$ and $B \to \rho\rho$ can also be used to determine $\alpha$. In particular, the $B \to \rho\rho$ measurements are presently very significant in constraining $\alpha$. The extraction proceeds via isospin analysis similar to that of $B \to \pi\pi$. There are, however, several important differences. First, due to the finite width of the $\rho$ mesons, a final $(\rho\rho)_{I=1}$ state is possible [105]. The effect is, however, of the order of $(\Gamma_{\rho}/m_{\rho})^2 \sim 0.04$. Second, due to the presence of three helicity states for the two vector mesons, angular analysis is needed to separate the CP-even and CP-odd components. The theoretical expectation is that the CP-odd component is small. This is supported by experiments which find that the $\rho^+\rho^-$ and $\rho^\pm\rho^0$ modes are dominantly longitudinally polarized. Third, an important advantage of the $\rho\rho$ modes is that the penguin contribution is expected to be small due to different hadronic dynamics. This expectation is confirmed by the smallness of $B(B^0 \to \rho^0\rho^0) = (0.95 \pm 0.16) \times 10^{-6}$ compared to $B(B^0 \to \rho^+\rho^-) = (24.2 \pm 3.1) \times 10^{-6}$ [62]. Thus, $S_{\rho^+\rho^-}$ is not far from $\sin 2\alpha$. Finally, both $S_{\rho^0\rho^0}$ and $C_{\rho^0\rho^0}$ are experimentally accessible, which may allow a precise determination of $\alpha$. However, a full isospin analysis should allow that the fractions of longitudinal polarization in $B$ and $\bar{B}$ decays may differ, which has not yet been done by the experiments.

Detailed discussion of the determination of $\alpha$ with these methods can be found in Refs. [106, 107]. The latest world average is

$$ \alpha = (84.1 \pm 4.5) \cdot 10^{-3} \qquad (13.88) $$

The consistency between the range of $\alpha$ determined by the $B \to \pi\pi$, $\rho\pi$ and $\rho\rho$ measurements and the range allowed by CKM fits (excluding these direct determinations) provides further support to the Kobayashi-Maskawa mechanism.

All modes discussed in this Section so far have possible contributions from penguin amplitudes. As shown in Table 13.1, CP

violation can also be studied with final states, typically containing charmed mesons, where no such contribution is possible. The neutral charmed meson must be reconstructed in a final state, such as a CP eigenstate, common to $D^0$ and $\bar{D}^0$ so that the amplitudes for the $B$ and $\bar{B}$ meson decays interfere. Although there is a second tree amplitude with a different weak phase, the contributions of the different diagrams can in many cases be separated experimentally (for example by exploiting different decays of the neutral $D$ mesons) making these channels very clean theoretically. The first determination of $\sin(2\beta)$, with significance of CP violation over $5\sigma$, with this method has recently been reported [108]. Moreover, the interference between the two tree diagrams gives sensitivity to $\gamma$, as will be discussed in Section 13.6.4.

### 13.6.3 CP violation in interference of $B_s^0$ decays with and without mixing

As discussed in Section 13.6.1, the world average for $|q/p|$ in the $B_s^0$ system currently deviates from the Standard Model expectation due to an anomalous value of the dimuon asymmetry. Attributing the dimuon asymmetry result to a fluctuation, we again neglect the deviation of $|q/p|$ from 1, and use

$$ \lambda_f = e^{-i\phi_M(B_s^0)} (\bar{A}_f/A_f). \qquad (13.89) $$

Within the Standard Model,

$$ e^{-i\phi_M(B_s^0)} = (V_{tb}^* V_{ts}) / (V_{tb} V_{ts}^*). \qquad (13.90) $$

Note that in the $B_s^0$ system, and with the sign convention of Eq. (13.8b), $\Delta\Gamma/\Gamma = -0.126 \pm 0.007$ [62] and therefore $y$ should not be put to zero in Eqs. (13.17a) and (13.17b). However, $|q/p| = 1$ is expected to hold to an even better approximation than for $B^0$ mesons. One therefore obtains

$$ A_f(t) = \frac{S_f \sin(\Delta mt) - C_f \cos(\Delta mt)}{\cosh(\Delta \Gamma t/2) - A_f^{\Delta \Gamma} \sinh(\Delta \Gamma t/2)}, \qquad (13.91) $$

$$ A_f^{\Delta\Gamma} \equiv \frac{-2 \mathcal{R} e(\lambda_f)}{1 + |\lambda_f|^2}. \qquad (13.92) $$

The presence of the $A_f^{\Delta\Gamma}$ term implies that information on $\lambda_f$ can be obtained from analyses that do not use tagging of the initial flavor, through so-called effective lifetime measurements [109].

The $B_s^0 \to J/\psi\phi$ decay proceeds via the $\bar{b} \to \bar{c}c\bar{s}$ transition. The CP asymmetry in this mode thus determines (with angular analysis to disentangle the CP-even and CP-odd components of the final state) $\sin 2\beta_s$, where $\beta_s$ is defined in Eq. (13.38) [110]. The $B_s^0 \to J/\psi\pi^+\pi^-$ decay, which has a large contribution from $J/\psi f_0(980)$ and is assumed to also proceed dominantly via the $\bar{b} \to \bar{c}c\bar{s}$ transition, has also been used to determine $\beta_s$. In this case no angular analysis is necessary, since the final state has been shown to be dominated by the CP-odd component [111]. The combination of measurements yields [62]

$$ 2\beta_s = 0.049 \pm 0.019, \qquad (13.93) $$
consistent with the Standard Model prediction, assuming negligible penguin contributions, $\beta_s = 0.0184 \pm 0.0004$ [20].

A time-dependent CP asymmetry was established in $B_s^0 \to K^+K^-$ decay, which proceeds via the $\bar{b} \to \bar{u}u\bar{s}$ transition [112]:

$$C_{KK} = +0.172 \pm 0.031, \quad S_{KK} = +0.139 \pm 0.032. \quad (13.94)$$

For both $C_{KK}$ and $S_{KK}$, the hadronic ratio $(T/P^c)$ plays an important role (see Table 13.1), making a clean theoretical interpretation challenging. First results on the $\bar{b} \to \bar{q}q\bar{s}$ decays $B_s^0 \to \phi\phi$ and $K^{*0}\bar{K}^{*-0}$ have also been reported. Parameters of CP violation have also been determined from the decay-time distributions of $B_s^0 \to D_s^\mp K^\pm$ and $D_s^\mp K^{\pm}\pi^+\pi^-$ decays, involving interference between $\bar{b} \to \bar{c}u\bar{s}$ and $\bar{b} \to \bar{u}c\bar{s}$ amplitudes.

### 13.6.4 Direct CP violation in the B system

An interesting class of decay modes is that of the tree-level decays $B^\pm \to D^{(*)}K^\pm$, which allow a theoretically pristine determination of the angle $\gamma$ [113–118]. The method uses the decays $B^+ \to D^0K^+$, which proceeds via the quark transition $\bar{b} \to \bar{u}c\bar{s}$, and $B^+ \to \bar{D}^0K^+$, which proceeds via the quark transition $\bar{b} \to \bar{c}u\bar{s}$, with the $D^0$ and $\bar{D}^0$ decaying into a common final state. The decays into common final states, such $(\pi^0 K_S) D K^+$, involve interference effects between the two amplitudes, with sensitivity to the relative phase, $\delta + \gamma$ ($\delta$ is the relevant strong phase difference). The CP-conjugate processes are sensitive to $\delta - \gamma$. Measurements of branching ratios and CP asymmetries allow the determination of $\gamma$ and $\delta$ from amplitude triangle relations. The method suffers from discrete ambiguities but, since all hadronic parameters can be determined from the data, has negligible theoretical uncertainty [119].

Unfortunately, the smallness of the CKM-suppressed $b \to u$ transitions makes it difficult to use the simplest methods alone [113–115] to determine $\gamma$. These difficulties are overcome (and the discrete ambiguities are removed) by performing a Dalitz plot analysis for multi-body $D$ decays [116–118]. Detailed discussion of the determination of $\gamma$ with these methods can be found in Ref. [107].

Constraints on $\gamma$ from combinations of results on various $B \to D^{(*)}K^{(*)}$ processes have been obtained by experiments [120, 121]. The latest world average is [62, 107]

$$\gamma = (65.9_{-3.1}^{+2.9})^{\circ}. \quad (13.95)$$

The consistency between the range of $\gamma$ determined by the $B \to DK$ measurements and the range allowed by CKM fits (excluding these direct determinations) provides further support to the Kobayashi-Maskawa mechanism. As more data become available, determinations of $\gamma$ from $B_0^0 \to D_s^\mp K^\pm$ [122, 123] and $B^0 \to DK^{*0}$ [124–128] are expected to also give competitive measurements.

Decays to the final state $K^\mp\pi^\pm$ provided the first observations of direct CP violation in both $B^0$ and $B_s^0$ systems. The asymmetry arises due to interference between tree and penguin diagrams [129], similar to the effect discussed in Section 13.6.2. In principle, measurements of $A_{B^0 \to K^+\pi^-}$ and $A_{B^0 \to K^-\pi^+}$ could be used to determine the weak phase difference $\gamma$, but lack of knowledge of the relative magnitude and strong phase of the contributing amplitudes limits the achievable precision. The uncertainties on these hadronic parameters can be reduced by exploiting flavor symmetries, which predict a number of relations between asymmetries in different modes. One such relation is that the partial rate differences for $B^0$ and $B_s^0$ decays to $K^\mp\pi^\pm$ are expected to be approximately equal and opposite [130], which is consistent with current data. It is also expected that the partial rate asymmetries for $B^0 \to K^+\pi^-$ and $B^+ \to K^+\pi^0$ should be approximately equal; however, the experimental results currently show a significant discrepancy [62]:

$$A_{B^0 \to K^+\pi^-} = -0.0836 \pm 0.0032, \quad A_{B^+ \to K^+\pi^0} = +0.029 \pm 0.012. \quad (13.96)$$

It is therefore of great interest to understand whether this originates from Standard Model QCD corrections, or whether it is a signature of new dynamics. Improved tests of a more precise

relation between the partial rate differences of all four $K\pi$ final states [131–134], currently limited by knowledge of the CP asymmetry in $\bar{B}^0 \to K_S\pi^0$ decays, may help to resolve the situation.

It is also of interest to investigate whether similar patterns appear among the CP violating asymmetries in $B$ meson decays to final states containing one pseudoscalar and one vector meson. Since the vector resonance decays to two particles, such channels can be studied through Dalitz plot analysis of the three-body final state. Model-independent analyses of $B^+ \to K^+K^-K^+$, $\pi^+\pi^-K^+$, $\pi^+\pi^-\pi^+$ and $K^+K^-\pi^+$ decays have revealed large CP violation effects in certain regions of phase space [135]. For the $B^+ \to K^+K^-\pi^+$ decay, an amplitude analysis has established a large CP violation effect associated with $\pi\pi \leftrightarrow KK$ S-wave rescattering [136]. In $B^+ \to \pi^+\pi^-\pi^+$ decays, amplitude analysis has established CP violation effects in the decay amplitude involving the $f_2(1270)$ resonance, in the $\pi^+\pi^-$ S-wave at low invariant mass, and in the interference between the $\pi^+\pi^-$ S-wave and the P-wave $B^+ \to \rho(770)^0\pi^+$ amplitude [137, 138]. For the other channels it remains to be seen whether the CP violation effects are associated to particular resonances or to interference effects, which will be necessary to understand the underlying dynamics.

## 13.7 Summary and Outlook

CP violation has been experimentally established in $K, D$ and $B$ meson decays. A full list of CP asymmetries that have been measured at a level higher than $5\sigma$ is given in Section 13.8. In Section 13.1.4 we introduced three types of CP-violating effects. Examples of these three types include the following:

1. All three types of CP violation have been observed in $K \to \pi\pi$ decays:

$$ \begin{aligned} \mathcal{R}e(\epsilon') &= \frac{1}{6} \left( \left| \frac{\bar{A}_{\pi^0 \pi^0}}{A_{\pi^0 \pi^0}} \right| - \left| \frac{\bar{A}_{\pi^+ \pi^-}}{A_{\pi^+ \pi^-}} \right| \right) \\ &= (2.5 \pm 0.4) \times 10^{-6}, \end{aligned} \quad (I) \quad (13.97a) $$

$$ \begin{aligned} \mathcal{R}e(\epsilon) &= \frac{1}{2} \left( 1 - \left| \frac{q}{p} \right| \right) \\ &= (1.66 \pm 0.02) \times 10^{-3}, \end{aligned} \quad (II) \quad (13.97b) $$

$$ \begin{aligned} \mathcal{I}m(\epsilon) &= -\frac{1}{2} \mathcal{I}m(\lambda_{(\pi\pi)_{I=0}}) \\ &= (1.57 \pm 0.02) \times 10^{-3}. \end{aligned} \quad (III) \quad (13.97c) $$

2. For $D$ mesons, CP violation in decay has been established in the difference of asymmetries for $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$ decays.

$$ \begin{aligned} \Delta a_{CP} &= \frac{|\bar{A}_{K^+K^-}/A_{K^+K^-}|^2 - 1}{|\bar{A}_{K^+K^-}/A_{K^+K^-}|^2 + 1} - \frac{|\bar{A}_{\pi^+\pi^-}/A_{\pi^+\pi^-}|^2 - 1}{|\bar{A}_{\pi^+\pi^-}/A_{\pi^+\pi^-}|^2 + 1} \\ &= (-0.161 \pm 0.028) \times 10^{-2}, \end{aligned} \quad (I) \quad (13.98) $$

3. In the $B$ meson system, CP violation in decay has been observed in, for example, $B^0 \to K^+\pi^-$ transitions, while CP violation in interference of decays with and without mixing has been observed in, for example, the $B^0 \to J/\psi K_S$ channel:

$$ \begin{aligned} A_{K^+\pi^-} &= \frac{|\bar{A}_{K^+\pi^-}/A_{K^+\pi^-}|^2 - 1}{|\bar{A}_{K^+\pi^-}/A_{K^+\pi^-}|^2 + 1} \\ &= -0.0836 \pm 0.0032, \end{aligned} \quad (I) \quad (13.99a) $$

$$ \begin{aligned} S_{\psi K^0} &= \mathcal{I}m(\lambda_{\psi K^0}) \\ &= +0.709 \pm 0.011. \end{aligned} \quad (III) \quad (13.99b) $$

Based on Standard Model predictions, further observations of CP violation in $B^0, B^+$ and $B_s^0$ decays seem likely in the near future, at both LHCb and its upgrades [139–141] as well as the Belle II experiment [142]. The first observation of CP violation in $b$ baryons is also likely to be within reach of LHCb. Further improvements in the sensitivity to CP violation effects in the charm sector can also be anticipated, though uncertainty in the Standard Model predictions makes it difficult to forecast whether or
not additional discoveries will be forthcoming. A number of upcoming experiments have potential to make significant progress on rare kaon decays. Observables that are subject to clean theoretical interpretation, such as $\beta$ from $S_{\psi K_S}$, $\beta_s$ from $B_s^0 \to J/\psi\phi$, $B(K_L \to \pi^0 \nu\bar{\nu})$ and $\gamma$ from CP violation in $B \to DK$ decays, are of particular value for constraining the values of the CKM parameters and probing the flavor sector of extensions to the Standard Model. Progress in lattice QCD calculations is also needed to complement the anticipated experimental results. Other probes of CP violation now being pursued experimentally include the electric dipole moments of the neutron and electron, and the decays of tau leptons. Additional processes that are likely to play an important role in future CP studies include top-quark production and decay, Higgs boson decays and neutrino oscillations.

All measurements of CP violation to date are consistent with the predictions of the Kobayashi-Maskawa mechanism of the Standard Model. In fact, it is now established that the KM mechanism plays a dominant role in the CP violation measured in the quark sector. However, a dynamically-generated matter-antimatter asymmetry of the universe requires additional sources of CP violation, and such sources are naturally generated by extensions to the Standard Model. New sources might eventually reveal themselves as small deviations from the predictions of the KM mechanism, or else might not be observable in the quark sector at all, but observable with future probes such as neutrino oscillations or electric dipole moments. The fundamental nature of CP violation demands a vigorous search.

A number of excellent reviews of CP violation are available [143–150], where the interested reader may find a detailed discussion of the various topics that are briefly reviewed here.

We thank David Kirkby for significant contributions to earlier versions of this review.

## 13.8 Observed CP violation effects

We conclude by listing the observables where CP violation has been observed at a level above $5\sigma$ [44, 62, 84]:

* Indirect CP violation in $K \to \pi\pi$ and $K \to \pi\ell\nu$ decays, and in the $K_L \to \pi^+\pi^- e^+e^-$ decay, is given by

$$|\epsilon| = (2.228 \pm 0.011) \times 10^{-3}. \qquad (13.100)$$

* Direct CP violation in $K \to \pi\pi$ decays is given by

$$\mathrm{Re}(\epsilon'/\epsilon) = (1.65 \pm 0.26) \times 10^{-3}. \qquad (13.101)$$

* CP violation in the interference of mixing and decay in the tree-dominated $b \to c\bar{c}s$ transitions, such as $B^0 \to \psi K^0$, is given by:

$$S_{\psi K^0} = +0.709 \pm 0.011. \qquad (13.102)$$

* CP violation in the interference of mixing and decay in modes governed by the tree-dominated $b \to c\bar{d}$ transitions is given by

$$S_{D(*)h^0} = +0.71 \pm 0.09, \qquad (13.103)$$

* CP violation in the interference of mixing and decay in various modes related to $b \to c\bar{c}d$ transitions is given by

$$\begin{aligned} S_{\psi\pi^0} &= -0.86 \pm 0.14, \\ S_{D^+D^-} &= -0.84 \pm 0.12, \\ S_{D^*\pm D^\mp} &= -0.81 \pm 0.06, \\ S_{D^*+D^*} &= -0.71 \pm 0.09. \end{aligned} \qquad (13.104)$$

* CP violation in the interference of mixing and decay in various modes related to $b \to q\bar{q}s$ (penguin) transitions is given by

$$\begin{aligned} S_{\phi K^0} &= +0.74_{-0.13}^{+0.11}, \\ S_{\eta' K^0} &= +0.63 \pm 0.06, \\ S_{f_0 K^0} &= +0.69_{-0.12}^{+0.10}, \\ S_{K^+K^-K_S} &= +0.68_{-0.10}^{+0.09}. \end{aligned} \qquad (13.105)$$

* CP violation in the interference of mixing and decay in the $B^0 \to \pi^+\pi^-$ mode is given by

$$S_{\pi^+\pi^-} = -0.666 \pm 0.029. \qquad (13.106)$$

* Direct CP violation in the $B^0 \to \pi^+\pi^-$ mode is given by

$$C_{\pi^+\pi^-} = -0.311 \pm 0.030. \qquad (13.107)$$

* Direct CP violation in the $B_s^0 \to K^+K^-$ mode is given by

$$C_{K^+K^-} = 0.172 \pm 0.031. \qquad (13.108)$$

* Direct CP violation in $B^+ \to D_+^{(*)}K^+$ decays ($D_+^{(*)}$ is the CP-even neutral $D^{(*)}$ state) are given by

$$\begin{aligned} A_{B^+ \to D_+ K^+} &= +0.139 \pm 0.009 \quad \text{and} \\ A_{B^+ \to D_+^* K^+} &= -0.109 \pm 0.019, \end{aligned} \qquad (13.109)$$

while the corresponding quantity in the case that the neutral D meson is reconstructed in the suppressed $K^-\pi^+$ final state is

$$A_{B^+ \to D_{K^-\pi^+} K^+} = -0.453 \pm 0.026, \qquad (13.110)$$

* Direct CP violation has also been observed in $B^+ \to DK^+$ decays through differences between the phase-space distributions of subsequent $D \to K_S\pi^+\pi^-$ and $D \to K^-\pi^+\pi^+\pi^-$ decays.

* Direct CP violation in the $B^0 \to K^+\pi^-$ mode is given by

$$A_{B^0 \to K^+\pi^-} = -0.0836 \pm 0.0032. \qquad (13.111)$$

* Direct CP violation in the $B_s^0 \to K^-\pi^+$ mode is given by

$$A_{B_s^0 \to K^-\pi^+} = +0.225 \pm 0.012. \qquad (13.112)$$

* Direct CP violation in $B^+ \to K^+K^-\pi^+$, $B^+ \to K^+K^-K^+$ and $B^+ \to \pi^+\pi^-\pi^+$ decays is given by

$$\begin{aligned} A_{B^+ \to K^+K^-\pi^+} &= -0.115 \pm 0.008, \\ A_{B^+ \to K^+K^-K^+} &= -0.0365 \pm 0.0036, \\ A_{B^+ \to \pi^+\pi^-\pi^+} &= 0.076 \pm 0.005, \end{aligned} \qquad (13.113)$$

* Large CP violation effects have been observed model independently in certain regions of the phase space of $B^+ \to K^+K^-K^+$, $K^+K^-\pi^+$, $\pi^+\pi^-K^+$ and $\pi^+\pi^-\pi^+$ decays. An amplitude analysis has established a large CP violation effect associated with $\pi\pi \leftrightarrow KK$ S-wave rescattering in $B^+ \to K^+K^-\pi^+$ decays. In $B^+ \to \pi^+\pi^-\pi^+$ decays, amplitude analysis has established CP violation effects in the decay amplitude involving the $f_2(1270)$ resonance, in the $\pi^+\pi^-\pi^+$ S-wave at low invariant mass, and in the interference between the $\pi^+\pi^-\pi^+$ S-wave and the P-wave $B^+ \to \rho(770)^0\pi^+$ amplitude.

* Direct CP violation has been established in the difference of asymmetries for $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$ decays

$$\Delta a_{CP} = (-0.161 \pm 0.028) \times 10^{-2}. \qquad (13.114)$$

## References

[1] J. H. Christenson et al., Phys. Rev. Lett. 13, 138 (1964).

[2] B. Aubert et al. (BaBar), Phys. Rev. Lett. 87, 091801 (2001), [hep-ex/0107013].

[3] K. Abe et al. (Belle), Phys. Rev. Lett. 87, 091802 (2001), [hep-ex/0107061].

[4] H. Burkhardt et al. (NA31), Phys. Lett. B206, 169 (1988).

[5] V. Fanti et al. (NA48), Phys. Lett. B465, 335 (1999), [hep-ex/9909022].

[6] A. Alavi-Harati et al. (KTeV), Phys. Rev. Lett. 83, 22 (1999), [hep-ex/9905060].

[7] B. Aubert et al. (BaBar), Phys. Rev. Lett. 93, 131801 (2004), [hep-ex/0407057].

[8] Y. Chao et al. (Belle), Phys. Rev. Lett. 93, 191802 (2004), [hep-ex/0408100].

[9] A. Poluektov et al. (Belle), Phys. Rev. D81, 112002 (2010), [arXiv:1003.3360].
[10] P. del Amo Sanchez et al. (BaBar), Phys. Rev. D82, 072004 (2010), [arXiv:1007.0504].

[11] R. Aaij et al. (LHCb), Phys. Lett. B712, 203 (2012), [Erratum-ibid. B713, 351 (2012)], [arXiv:1203.3662].

[12] R. Aaij et al. (LHCb), Phys. Rev. Lett. 110, 221601 (2013), [arXiv:1304.6173].

[13] R. Aaij et al. (LHCb), Phys. Rev. Lett. 122, 211803 (2019), [arXiv:1903.08726].

[14] See the review on "Neutrino Masses, Mixing, and Oscillations," in this Review.

[15] See the review on "Tests of Conservation Laws," in this Review.

[16] J. P. Lees et al. (BaBar), Phys. Rev. Lett. 109, 211801 (2012), [arXiv:1207.5832].

[17] J. Bernabeu, F. Martinez-Vidal and P. Villanueva-Perez, JHEP 08, 064 (2012), [arXiv:1203.0171].

[18] See, for example, R. F. Streater and A. S. Wightman, CPT, Spin and Statistics, and All That, reprinted by Addison-Wesley, New York (1989).

[19] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

[20] J. Charles et al. (CKMfitter Group), Eur. Phys. J. C41, 1 (2005), updated results and plots available at: http://ckmfitter.in2p3.fr, [hep-ph/0406184].

[21] M. Bona et al. (UTfit), JHEP 10, 081 (2006), updated results and plots available at: http://www.utfit.org/UTfit, [hep-ph/0606167].

[22] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967), [Usp. Fiz. Nauk161, no.5,61(1991)].

[23] A. Riotto, in "Proceedings, Summer School in High-energy physics and cosmology: Trieste, Italy, June 29-July 17, 1998," 326-436 (1998), [hep-ph/9807454].

[24] M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 (1986).

[25] S. Davidson, E. Nardi and Y. Nir, Phys. Rept. 466, 105 (2008), [arXiv:0802.2962].

[26] G. Aad et al. (ATLAS), Phys. Lett. B716, 1 (2012), [arXiv:1207.7214].

[27] S. Chatrchyan et al. (CMS), Phys. Lett. B716, 30 (2012), [arXiv:1207.7235].

[28] S. Okubo, Phys. Lett. 5, 165 (1963).

[29] G. Zweig (1964), An SU₃ model for strong interaction symmetry and its breaking; Version 2, CERN-TH-412.

[30] J. Iizuka, Prog. Theor. Phys. Suppl. 37, 21 (1966).

[31] V. Weisskopf and E. P. Wigner, Z. Phys. 63, 54 (1930).

[32] See the review on "$D^0\bar{D}^0$ Mixing" in this Review.

[33] O. Long et al., Phys. Rev. D68, 034010 (2003), [hep-ex/0303030].

[34] M. Gronau, Y. Grossman and J. L. Rosner, Phys. Lett. B508, 37 (2001), [hep-ph/0103110].

[35] D. Atwood and A. Soni, Phys. Rev. D68, 033003 (2003), [hep-ph/0304085].

[36] A. Lenz and G. Tetzlalmatzi-Xolocotzi, JHEP 07, 177 (2020), [arXiv:1912.07621].

[37] L. Wolfenstein, Phys. Rev. Lett. 13, 562 (1964).

[38] C. Abel et al. (nEDM), Phys. Rev. Lett. 124, 081803 (2020), [arXiv:2001.11966].

[39] R. J. Crewther et al., Phys. Lett. B88, 123 (1979), [Erratum-ibid. B91, 487 (1980)].

[40] See the review on "Cabibbo-Kobayashi-Maskawa Mixing Matrix," in this Review.

[41] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).

[42] A. J. Buras, M. E. Lautenbacher and G. Ostermaier, Phys. Rev. D50, 3433 (1994), [hep-ph/9403384].

[43] C. Jarlskog, Phys. Rev. Lett. 55, 1039 (1985).

[44] See the K-meson Listings in this Review.

[45] See the review on "CP violation in $K_S \to 3\pi$," in this Review.

[46] R. Abbott et al. (RBC, UKQCD), Phys. Rev. D102, 5, 054509 (2020), [arXiv:2004.09440].

[47] V. Cirigliano et al., JHEP 02, 032 (2020), [arXiv:1911.01359].

[48] A. J. Buras, Acta Phys. Polon. B52, 1, 7 (2021), [arXiv:2101.00020].

[49] Y. Grossman and Y. Nir, Phys. Lett. B398, 163 (1997), [hep-ph/9701313].

[50] L. S. Littenberg, Phys. Rev. D39, 3322 (1989).

[51] A. J. Buras, Phys. Lett. B333, 476 (1994), [hep-ph/9405368].

[52] G. Buchalla and A. J. Buras, Nucl. Phys. B400, 225 (1993).

[53] A. J. Buras et al., JHEP 11, 033 (2015), [arXiv:1503.02693].

[54] J. K. Ahn et al. (KOTO), Phys. Rev. Lett. 122, 021802 (2019), [arXiv:1810.09655].

[55] J. K. Ahn et al. (KOTO), Phys. Rev. Lett. 126, 121801 (2021), [arXiv:2012.07571].

[56] E. Cortina Gil et al. (NA62), JHEP 06, 093 (2021), [arXiv:2103.15389].

[57] B. Aubert et al. (BaBar), Phys. Rev. Lett. 98, 211802 (2007), [hep-ex/0703020].

[58] M. Staric et al. (Belle), Phys. Rev. Lett. 98, 211803 (2007), [hep-ex/0703036].

[59] T. Aaltonen et al. (CDF), Phys. Rev. Lett. 100, 121802 (2008), [arXiv:0712.1567].

[60] R. Aaij et al. (LHCb), Phys. Rev. Lett. 110, 101802 (2013), [arXiv:1211.1230].

[61] R. Aaij et al. (LHCb), Phys. Rev. Lett. 127, 11, 111801 (2021), [arXiv:2106.03744].

[62] Y. S. Amhis et al. (HFLAV), Eur. Phys. J. C81, 226 (2021), updated results and plots available at https://hflav.web.cern.ch/, [arXiv:1909.12524].

[63] See the D-meson Listings in this Review.

[64] Y. Grossman, A. L. Kagan and Y. Nir, Phys. Rev. D75, 036008 (2007), [hep-ph/0609178].

[65] S. Bergmann et al., Phys. Lett. B486, 418 (2000), [hep-ph/0005181].

[66] M. Gersabeck et al., J. Phys. G39, 045005 (2012), [arXiv:1111.6515].

[67] T. Pajero and M. J. Morello, JHEP 03, 162 (2022), [arXiv:2106.02014].

[68] R. Aaij et al. (LHCb), Phys. Rev. D105, 9, 092013 (2022), [arXiv:2202.09106].

[69] D. M. Asner et al. (CLEO), Phys. Rev. D78, 012001 (2008), [arXiv:0802.2268].

[70] M. Ablıkım et al. (BESIII), Phys. Lett. B734, 227 (2014), [arXiv:1404.4691].

[71] R. Aaij et al. (LHCb), Phys. Rev. D97, 3, 031101 (2018), [arXiv:1712.03220].

[72] R. Aaij et al. (LHCb), Phys. Rev. D104, 7, 072010 (2021), [arXiv:2105.09889].

[73] M. Ciuchini et al., Phys. Lett. B655, 162 (2007), [hep-ph/0703204].

[74] Y. Grossman, Y. Nir and G. Perez, Phys. Rev. Lett. 103, 071602 (2009), [arXiv:0904.0305].

[75] A. L. Kagan and M. D. Sokoloff, Phys. Rev. D80, 076008 (2009), [arXiv:0907.3917].

[76] Y. Grossman and Y. Nir, JHEP 04, 002 (2012), [arXiv:1110.3790].
[77] B. R. Ko et al. (Belle), Phys. Rev. Lett. **109**, 021601 (2012), [Erratum-ibid. **109**, 119903 (2012)], [arXiv:1203.6409].

[78] See the "Review of Multibody Charm Analyses" in this Review.

[79] B. Aubert et al. (BaBar), Phys. Rev. D78, 051102 (2008), [arXiv:0802.4035].

[80] I. Bediaga et al., Phys. Rev. D80, 096006 (2009), [arXiv:0905.4233].

[81] I. Bediaga et al., Phys. Rev. D86, 036005 (2012), [arXiv:1205.3036].

[82] M. Williams, Phys. Rev. D84, 054015 (2011), [arXiv:1105.5338].

[83] R. Aaij et al. (LHCb), Phys. Lett. B769, 345 (2017), [arXiv:1612.03207].

[84] See the B-meson Listings in this Review.

[85] V. M. Abazov et al. (D0), Phys. Rev. D82, 032001 (2010), [arXiv:1005.2757].

[86] A. B. Carter and A. I. Sanda, Phys. Rev. Lett. **45**, 952 (1980).

[87] A. B. Carter and A. I. Sanda, Phys. Rev. D23, 1567 (1981).

[88] I. I. Y. Bigi and A. I. Sanda, Nucl. Phys. B193, 85 (1981).

[89] I. Dunietz and J. L. Rosner, Phys. Rev. D34, 1404 (1986).

[90] Y. I. Azimov, N. G. Uraltsev and V. A. Khoze, Sov. J. Nucl. Phys. **45**, 878 (1987), [Yad. Fiz. **45**, 1412 (1987)].

[91] I. I. Y. Bigi and A. I. Sanda, Nucl. Phys. B281, 41 (1987).

[92] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. **68**, 1125 (1996), [hep-ph/9512380].

[93] A. J. Buras and L. Silvestrini, Nucl. Phys. B569, 3 (2000), [hep-ph/9812392].

[94] R. Fleischer, Eur. Phys. J. C10, 299 (1999), [hep-ph/9903455].

[95] M. Ciuchini, M. Pierini and L. Silvestrini, Phys. Rev. Lett. **95**, 221804 (2005), [hep-ph/0507290].

[96] S. Faller et al., Phys. Rev. D79, 014030 (2009), [arXiv:0809.0842].

[97] M. Jung, Phys. Rev. D86, 053008 (2012), [arXiv:1206.2050].

[98] H.-n. Li and S. Mishima, JHEP 03, 009 (2007), [hep-ph/0610120].

[99] K. De Bruyn and R. Fleischer, JHEP 03, 145 (2015), [arXiv:1412.6834].

[100] P. Frings, U. Nierste and M. Wiebusch, Phys. Rev. Lett. **115**, 061802 (2015), [arXiv:1503.00859].

[101] L. Silvestrini, Ann. Rev. Nucl. Part. Sci. **57**, 405 (2007), [arXiv:0705.1624].

[102] R. Fleischer et al., Phys. Rev. D78, 111501 (2008), [arXiv:0806.2900].

[103] R. Fleischer et al., Eur. Phys. J. C78, 943 (2018), [arXiv:1806.08783].

[104] M. Gronau and D. London, Phys. Rev. Lett. **65**, 3381 (1990).

[105] A. F. Falk et al., Phys. Rev. D69, 011502 (2004), [hep-ph/0310242].

[106] J. Charles et al., Eur. Phys. J. C77, 574 (2017), [arXiv:1705.02981].

[107] See the review on "Determination of CKM Angles from B hadrons," in this Review.

[108] A. Abdesselam et al. (BaBar, Belle), Phys. Rev. Lett. **115**, 121604 (2015), [arXiv:1505.04147].

[109] R. Fleischer and R. Knegjens, Eur. Phys. J. C71, 1789 (2011), [arXiv:1109.5115].

[110] A. S. Dighe, I. Dunietz and R. Fleischer, Eur. Phys. J. C6, 647 (1999), [hep-ph/9804253].

[111] R. Aaij et al. (LHCb), Phys. Rev. D89, 092006 (2014), [arXiv:1402.6248].

[112] R. Aaij et al. (LHCb), JHEP 03, 075 (2021), [arXiv:2012.05319].

[113] M. Gronau and D. London, Phys. Lett. B253, 483 (1991).

[114] M. Gronau and D. Wyler, Phys. Lett. B265, 172 (1991).

[115] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. **78**, 3257 (1997), [hep-ph/9612433].

[116] D. Atwood, I. Dunietz and A. Soni, Phys. Rev. D63, 036005 (2001), [hep-ph/0008090].

[117] A. Giri et al., Phys. Rev. D68, 054018 (2003), [hep-ph/0303187].

[118] A. Bondar, Proceedings of BINP special analysis meeting on Dalitz analysis, 24-26 Sep. 2002, unpublished.

[119] J. Brod and J. Zupan, JHEP 01, 051 (2014), [arXiv:1308.5663].

[120] J. P. Lees et al. (BaBar), Phys. Rev. D87, 052015 (2013), [arXiv:1301.1029].

[121] R. Aaij et al. (LHCb), JHEP 12, 087 (2016), [arXiv:1611.03076].

[122] R. Aleksan, I. Dunietz and B. Kayser, Z. Phys. C54, 653 (1992).

[123] R. Fleischer, Nucl. Phys. B671, 459 (2003), [hep-ph/0304027].

[124] I. Dunietz, Phys. Lett. B270, 75 (1991).

[125] M. Gronau, Phys. Lett. B557, 198 (2003), [hep-ph/0211282].

[126] T. Gershon, Phys. Rev. D79, 051301 (2009), [arXiv:0810.2706].

[127] T. Gershon and M. Williams, Phys. Rev. D80, 092002 (2009), [arXiv:0909.1495].

[128] T. Gershon and A. Poluektov, Phys. Rev. D81, 014025 (2010), [arXiv:0910.5437].

[129] M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. **43**, 242 (1979).

[130] X.-G. He, Eur. Phys. J. C9, 443 (1999), [hep-ph/9810397].

[131] D. Atwood and A. Soni, Phys. Rev. D58, 036005 (1998), [hep-ph/9712287].

[132] M. Gronau and J. L. Rosner, Phys. Rev. D59, 113002 (1999), [hep-ph/9809384].

[133] H. J. Lipkin, Phys. Lett. B445, 403 (1999), [hep-ph/9810351].

[134] M. Gronau, Phys. Lett. B627, 82 (2005), [hep-ph/0508047].

[135] R. Aaij et al. (LHCb), Phys. Rev. D90, 112004 (2014), [arXiv:1408.5373].

[136] R. Aaij et al. (LHCb), Phys. Rev. Lett. 123, 231802 (2019), [arXiv:1905.09244].

[137] R. Aaij et al. (LHCb), Phys. Rev. Lett. 124, 031801 (2020), [arXiv:1909.05211].

[138] R. Aaij et al. (LHCb), Phys. Rev. D101, 012006 (2020), [arXiv:1909.05212].

[139] A. A. Alves, Jr. et al. (LHCb), JINST 3, S08005 (2008).

[140] I. Bediaga et al. (LHCb) (2012), CERN-LHCC-2012-007.

[141] R. Aaij et al. (LHCb) (2017), CERN-LHCC-2017-003.

[142] W. Altmannshofer et al. (Belle-II), PTEP 2019, 123C01 (2019), [Erratum-ibid. **2020**, 029201 (2020)], [arXiv:1808.10567].

[143] G. C. Branco, L. Lavoura and J. P. Silva, Int. Ser. Monogr. Phys. **103**, 1 (1999).

[144] I. I. Bigi and A. I. Sanda (2000), [Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol.9.1(2009)].

[145] A. J. Bevan et al. (BaBar, Belle), Eur. Phys. J. C74, 3026 (2014), [arXiv:1406.6311].
[146] H.R. Quinn and Y. Nir, “The Mystery of the Missing Anti-matter,” Princeton University Press, Princeton (2008).

[147] T. E. Browder et al., Rev. Mod. Phys. **81**, 1887 (2009), [arXiv:0802.3201].

[148] M. Ciuchini and A. Stocchi, Ann. Rev. Nucl. Part. Sci. **61**, 491 (2011), [arXiv:1110.3920].

[149] R. Aaij et al. (LHCb), Eur. Phys. J. **C73**, 2373 (2013), [arXiv:1208.3355].

[150] T. Gershon and V. V. Gligorov, Rept. Prog. Phys. **80**, 046201 (2017), [arXiv:1607.06746].
14. Neutrino Masses, Mixing, and Oscillations

Revised September 2023 by M.C. Gonzalez-Garcia (YITP, Stony Brook; ICREA, Barcelona; ICC, U. of Barcelona) and M. Yokoyama (UTokyo; Kavli IPMU (WPI), UTokyo).

14.1 Neutrinos in the Standard Model: Massless Neutrinos 295

14.2 Extending the Standard Model to Introduce Massive Neutrinos 296

14.2.1 Dirac Neutrinos 296

14.2.2 The See-saw Mechanism 296

14.2.3 Light Sterile Neutrinos 297

14.2.4 Neutrino Masses from Generic New Physics 297

14.3 Lepton Mixing 297

14.4 Mass-Induced Flavour Oscillations in Vacuum 298

14.5 Propagation of Massive Neutrinos in Matter 299

14.5.1 The Mikheyev-Smirnov-Wolfenstein Effect for Solar Neutrinos 301

14.6 Experimental Study of Neutrino Oscillations 301

14.6.1 Solar Neutrinos 302

14.6.2 Atmospheric Neutrinos 303

14.6.3 Accelerator Neutrinos 304

14.6.4 Reactor Antineutrinos 308

14.7 Combined Analysis of Experimental Results: The 3ν Paradigm 310

14.7.1 3ν Oscillation Probabilities 311

14.7.2 3ν Oscillation Analysis 312

14.7.3 Convention-independent Measures of Lep-tonic CP Violation in 3ν Mixing 312

14.8 Beyond 3ν: Additional Neutrinos at the eV Scale 313

14.9 Laboratory Probes of ν Mass Scale and its Nature 315

14.9.1 Constraints from Kinematics of Weak Decays 315

14.9.2 Dirac vs. Majorana: Neutrinoless Double-beta Decay 316

14.9.3 Experimental Search for Neutrinoless Double-beta Decay 317

14.1 Neutrinos in the Standard Model: Massless Neutrinos

The gauge symmetry principle is one of the pillars of the great success of modern particle physics as it establishes an unambiguous connection between local (gauge) symmetries and forces mediated by spin-1 particles. In the Standard Model (SM) of particle physics the strong, weak, and electromagnetic interactions are connected to gauge symmetry under $SU(3)_C \times SU(2)_L \times U(1)_Y$ where C stands for colour, L for left-handedness, and Y for hypercharge. The SM gauge symmetry is spontaneously broken to $SU(3)_C \times U(1)_EM$ where $U(1)_{EM}$ couples to the electromagnetic charge $Q_{EM} = T_{L3} + Y(T_{L3}$ is the weak isospin which is the third generator of $SU(2)_L)$. The model explains all the interactions of the known fermions once they are assigned to a well defined representation of the gauge group. The construction and tests of the Standard Model as a gauge theory are covered in Chapter 9 “Quantum chromodynamics” and Chapter 10 “Electroweak model and constraints on new physics” of this Review. Here we emphasize that the gauge invariance principle requires that all terms in the Lagrangian, including the mass terms, respect the local symmetry. This has important implications for the neutrino and in particular for the question of the neutrino mass<fnref n="1" />.

In the SM, neutrinos are fermions that do not have strong nor electromagnetic interactions. Consequently, they are singlets of the subgroup $SU(3)_C \times U(1)_{EM}$. They are part of the lepton doublets $L_{L\ell} = \begin{pmatrix} \nu_\ell \\ \ell \end{pmatrix}_L$ where $f_L$ is the left-handed component of the fermion $f$, $f_L = P_L f \equiv \frac{1-\gamma_5}{2} f$. In what follows we will refer as active neutrinos to neutrinos that are part of these lepton doublets. In the SM there is one active neutrino for each charged

leptons, $\ell = e, \mu, \tau$. $SU(2)_L$ gauge invariance dictates the form of weak charged current (CC) interactions between the neutrinos and their corresponding charged leptons and neutral current (NC) among themselves to be:

$$-\mathcal{L}_{\text{CC}} = \frac{g}{\sqrt{2}} \sum_{\ell} \bar{\nu}_{L\ell} \gamma^{\mu} \ell_{\ell}^{-} W_{\mu}^{+} + \text{h.c.}, \quad (14.1)$$

$$-\mathcal{L}_{\text{NC}} = \frac{g}{2 \cos \theta_W} \sum_{\ell} \bar{\nu}_{L\ell} \gamma^{\mu} \nu_{L\ell} Z_{\mu}^{0}. \quad (14.2)$$

In the above equations, $g$ is the coupling constant associated with $SU(2)$ and $\theta_W$ is the Weinberg angle. Equations(14.1) and (14.2) describe all the neutrino interactions in the SM. In particular, Eq.(14.2) determines the decay width of the Z boson into light ($m_\nu \le m_Z/2$) left-handed neutrinos states. Thus from the measurement of the total decay width of the Z one can infer the number of such states. At present the measurement implies $N_\nu = 2.984 \pm 0.008$ (see Particle Listing). As a result any extension of the SM should contain three, and only three, light active neutrinos.

Sterile neutrinos are defined as having no SM gauge interactions, that is, they are singlets of the complete SM gauge group. Thus the SM, as the gauge theory able to describe all known particle interactions, contains no sterile neutrinos.

The SM with its gauge symmetry and the particle content required for the gauge interactions, that is, in the absence of SM singlets, respects an accidental global symmetry that is not imposed but appears as a consequence of the gauge symmetry and the representation of the matter fields:

$$G_{\text{SM}}^{\text{global}} = U(1)_B \times U(1)_{L_e} \times U(1)_{L_\mu} \times U(1)_{L_\tau}, \quad (14.3)$$

where $U(1)_B$ is the baryon number symmetry, and $U(1)_{L_e, L_\mu, L_\tau}$ are the three lepton flavour symmetries. The total lepton number, $L_e + L_\mu + L_\tau$, is then also an accidental symmetry since it is a subgroup of $G_{\text{SM}}^{\text{global}}$. This fact has consequences that are relevant to the question of the neutrino mass as we argue next.

In the SM, the masses of the fermions are generated via a Yukawa coupling of the scalar Higgs doublet $\phi$ with a fermion right-handed and left-handed component. The former is an $SU(2)_L$ singlet, the latter is part of a doublet. For leptons, we can build such a term coupling the left-handed lepton doublets $L_L$ with the right-handed charged lepton fields $E_R$:

$$-\mathcal{L}_{\text{Yukawa,lep}} = Y_{ij}^{\ell} \bar{L}_{Li} \phi E_{Rj} + \text{h.c.} \quad (14.4)$$

After spontaneous symmetry breaking these terms lead to charged lepton masses

$$m_{ij}^{\ell} = Y_{ij}^{\ell} \frac{v}{\sqrt{2}}, \quad (14.5)$$

where $v$ is the vacuum expectation value of the Higgs field. However, since the model does not contain right-handed neutrinos, no such Yukawa interaction can be built for the neutrinos, which are consequently massless at the Lagrangian level.

In principle, a neutrino mass term could be generated at loop level. With the particle content of the SM the only possible neutrino mass term that could be constructed is the bilinear $\bar{L}_L L_L^c$, where $L_L^c$ is the charge conjugated field, $L_L^c = C \bar{L}_L^T$ and $C$ is the charge conjugation matrix. However this term is forbidden in the SM because it violates the total lepton symmetry by two units and therefore it cannot be induced by loop corrections because it breaks the accidental symmetry of the model. Also, because $U(1)_{B-L}$ is a non-anomalous subgroup of $G_{\text{SM}}^{\text{global}}$, the bilinear $\bar{L}_L L_L^c$, cannot be induced by nonperturbative corrections either since it breaks $B-L$.

We conclude that within the SM neutrinos are precisely massless. Consequently one must go beyond the SM in order to add a mass to the neutrino.

<footnote n="1">¹The physics of massive neutrinos has been the subject of excellent books such as [1-5] and multiple review articles. The contents of the present review is built upon the structure and the contents of the review articles [6, 7].</footnote>
## 14.2 Extending the Standard Model to Introduce Massive Neutrinos

From the above discussion, we conclude that it is not possible to construct a renormalizable mass term for the neutrinos with the fermionic content and gauge symmetry of the SM. The obvious consequence is that in order to introduce a neutrino mass in the theory one must extend the particle content of the model, depart from gauge invariance and/or renormalizability, or do both.

As a matter of fact, neutrino mass terms can be constructed in different ways. In the following we shall assume to maintain the gauge symmetry and explore the different possibilities to introduce a neutrino mass term adding to the SM an arbitrary number of sterile neutrinos $\nu_{si}$ ($i = 1, \dots, m$).

In the SM extended with the addition of $m$ number of sterile neutrinos one can construct two gauge invariant renormalizable operators leading to two types of mass terms

$$-\mathcal{L}_{M_\nu} = M_{D_{ij}} \bar{\nu}_{si} \nu_{L_j} + \frac{1}{2} M_{N_{ij}} \bar{\nu}_{si} \nu_{sj}^c + \text{h.c.}, \quad (14.6)$$

where $\nu^c$ is the neutrino charge conjugated field (defined in section 14.1). $M_D$ is a complex matrix of dimension $m \times 3$ and $M_N$ is a symmetric $m \times m$ matrix.

The first term is generated after spontaneous electroweak symmetry breaking from Yukawa interactions,

$$Y_{ij}^{\nu} \bar{\nu}_{si} \tilde{\phi}^{\dagger} L_{Lj} \Rightarrow M_{D_{ij}} = Y_{ij}^{\nu} \frac{v}{\sqrt{2}}, \quad (14.7)$$

in a similar way to Eqs.(14.4) and (14.5) for the charged fermion masses. It is correspondingly called a Dirac mass term. It conserves total lepton number but it can break the lepton flavour number symmetries.

The second term in Eq.(14.6) is a Majorana mass term and it differs from the Dirac mass terms in several relevant aspects. First, it is a singlet of the SM gauge group and, as such, it can appear as a bare mass term in the Lagrangian. Second, since it involves two neutrino fields (right-handed in this case), it breaks lepton number by two units. In general, such a term is not allowed if the neutrinos carry any additive conserved charge.

It is possible to rewrite Eq.(14.6) as:

$$-\mathcal{L}_{M_\nu} = \frac{1}{2} \overline{(\bar{\nu}_L^c, \bar{\nu}_s)} \begin{pmatrix} 0 & M_D^T \\ M_D & M_N \end{pmatrix} \begin{pmatrix} \bar{\nu}_L \\ \bar{\nu}_s \end{pmatrix} + \text{h.c.} \equiv \overline{\bar{\nu}^c} M_\nu \bar{\nu} + \text{h.c.}, \quad (14.8)$$

where $\bar{\nu} = (\bar{\nu}_L, \bar{\nu}_s)^T$ is a $(3+m)$-dimensional vector. The matrix $M_\nu$ is complex and symmetric $^2$. Thus it can be diagonalized by a unitary matrix $V^\nu$ of dimension $(3+m)$, so

$$(V^\nu)^T M_\nu V^\nu = \text{diag}(m_1, m_2, \dots, m_{3+m}). \quad (14.9)$$

One can express the original weak eigenstates in terms of the resulting $3+m$ mass eigenstates

$$\bar{\nu}_{\text{mass}} = (V^\nu)^\dagger \bar{\nu}, \quad (14.10)$$

and in terms of the mass eigenstates, Eq.(14.8) takes the form:

$$-\mathcal{L}_{M_\nu} = \frac{1}{2} \sum_{k=1}^{3+m} m_k (\bar{\nu}_{\text{mass},k}^c \nu_{\text{mass},k} + \bar{\nu}_{\text{mass},k} \nu_{\text{mass},k}^c) \\ = \frac{1}{2} \sum_{k=1}^{3+m} m_k \bar{\nu}_{Mk} \nu_{Mk}, \quad (14.11)$$

where

$$\nu_{Mk} = \nu_{\text{mass},k} + \nu_{\text{mass},k}^c = (V^{\nu\dagger} \bar{\nu})_k + (V^{\nu\dagger} \bar{\nu})_k^c. \quad (14.12)$$

So these states obey the Majorana condition

$$\nu_M = \nu_M^c, \quad (14.13)$$

and are referred to as Majorana neutrinos. The Majorana condition implies that only one field describes both neutrino and antineutrino states, unlike the case of a charged fermion for which particles and antiparticles are described by two different fields. So a Majorana neutrino can be described by a two-component spinor unlike the charged fermions, which are Dirac particles, and are represented by four-component spinors.

Inverting Eq.(14.12) we can write the weak-doublet components of the neutrino fields as:

$$\nu_{Li} = P_L \sum_{j=1}^{3+m} V_{ij}^{\nu} \nu_{Mj}, \quad i = 1, 2, 3, \quad (14.14)$$

where $P_L$ is the left projector.

In the following, we will discuss some interesting particular cases of this general framework: light Dirac neutrinos in Sec.14.2.1, and light Majorana neutrinos and the see-saw mechanism in Sec.14.2.2. A special case of the second example is the possibility of light-sterile neutrinos discussed in Sec.14.2.3. In Sec.14.2.4 we shall discuss the effective generation of neutrino masses from non-renormalizable operators (of which the see-saw mechanism is a particular realization).

### 14.2.1 Dirac Neutrinos

Imposing $M_N = 0$ is equivalent to imposing lepton number symmetry on the model. In doing this only the first term in Eq.(14.6), the Dirac mass term, is allowed. If there are only three sterile ($m=3$), we can identify them with the right-handed component of a four-spinor neutrino field. In this case the Dirac mass term can be diagonalized with two $3 \times 3$ unitary matrices, $V^\nu$ and $V_R^\nu$ as:

$$V_R^{\nu\dagger} M_D V^{\nu} = \text{diag}(m_1, m_2, m_3). \quad (14.15)$$

The neutrino mass term can be written as:

$$-\mathcal{L}_{M_\nu} = \sum_{k=1}^{3} m_k \bar{\nu}_{Dk} \nu_{Dk}, \quad (14.16)$$

where

$$\nu_{Dk} = (V^{\nu\dagger} \bar{\nu}_L)_k + (V_R^{\nu\dagger} \bar{\nu}_s)_k, \quad (14.17)$$

so the weak-doublet components of the neutrino fields are

$$\nu_{Li} = P_L \sum_{j=1}^{3} V_{ij}^{\nu} \nu_{Dj}, \quad i = 1, 2, 3. \quad (14.18)$$

Let us stress that in this case both the low-energy matter content and the assumed symmetries are different from those of the SM. Consequently, the SM is not even a good low-energy effective theory. Furthermore, this scenario does not explain the fact that neutrinos are much lighter than the corresponding charged fermions, because all of them acquire their mass via the same mechanism.

### 14.2.2 The See-saw Mechanism

If the mass eigenvalues of $M_N$ are much higher than the scale of electroweak symmetry breaking $v$, the diagonalization of $M_\nu$ leads to three light neutrinos, $\nu_l$, and $m$ heavy neutrinos, $N$:

$$-\mathcal{L}_{M_\nu} = \frac{1}{2} \bar{\nu}_l M^l \nu_l + \frac{1}{2} \bar{N} M^h N, \quad (14.19)$$

with

$$M^l \approx -V_l^T M_D^T M_N^{-1} M_D V_l, \quad M^h \approx V_h^T M_N V_h, \quad (14.20)$$

and

<footnote n="2">²Notice that Eq.(14.8) corresponds to the tree-level neutrino mass matrix. Corrections are induced at the loop level, which in particular lead to non-vanishing $\bar{\nu}_L^c \nu_L$ entry [8].</footnote>
$$V^\nu \approx \left[ \begin{pmatrix} 1 - \frac{1}{2} M_D^\dagger M_N^* - 1 M_N^{-1} M_D & V_l \\ -M_N^{-1} M_D V_l & 1 - \frac{1}{2} M_N^{-1} M_D M_D^\dagger M_N^* - 1 & V_h \end{pmatrix} \right], \quad (14.21)$$

where $V_l$ and $V_h$ are $3 \times 3$ and $m \times m$ unitary matrices respectively. From Eq.(14.20) we see that the masses of the heavier states are proportional to $M_N$ while those of the lighter ones to $M_N^{-1}$, hence the name see-saw mechanism [9–13]. Also, as seen from Eq.(14.21), the heavy states are mostly right-handed while the light ones are mostly left-handed. Both the light and the heavy neutrinos are Majorana particles. Two well-known examples of extensions of the SM leading to a see-saw mechanism for neutrino masses are SO(10) Grand Unified Theories [10, 11] and left-right symmetry models [13].

In this case, the SM is a good effective low energy theory. Indeed the see-saw mechanism is a particular example of a full theory whose low energy effective realization is the SM with three light Majorana neutrinos which we describe in Sec.14.2.4.

### 14.2.3 Light Sterile Neutrinos

If the scale of some $n_s \le m$ eigenvalues of $M_N$ are not higher than the electroweak scale, the low energy spectrum contains $n_s$ additional light states with a large admixture of sterile component. As in the case with Dirac Neutrinos, the SM is not a good low energy effective theory: there are more than three (3+$n_s$) light neutrinos, and they are admixtures of doublet and singlet fields. As in the general case, both light and heavy neutrinos are Majorana particles.

### 14.2.4 Neutrino Masses from Generic New Physics

Under the generic hypothesis that new physics (NP) beyond the SM only manifests itself directly above some scale $\Lambda_{NP}$, we can consider that the SM is an effective low energy theory which is valid to describe the physical world at energies well below $\Lambda_{NP}$ with the same gauge group, fermionic spectrum, and the pattern of spontaneous symmetry breaking of the SM. However, this is an effective theory, holding only till energy below $\Lambda_{NP}$, and consequently does not need to be renormalizable. In this case, the low energy Lagrangian can contain non-renormalizable higher dimensional terms whose effect will be suppressed by powers $1/\Lambda_{NP}^{\text{dim}-4}$.

In this approach, the least suppressed NP effects at low energy are expected to come from dim=5 operators. With the SM fields and gauge symmetry one can only construct the following set of dimension-five terms

$$O_5 = \frac{Z_{ij}^\nu}{\Lambda_{\text{NP}}} (\bar{L}_{Li} \tilde{\phi}) (\tilde{\phi}^T L_{Lj}^c) + \text{h.c.} \quad (14.22)$$

This set violates (14.3), which poses no problem since, in general, there is no reason for the NP to respect the accidental symmetries of the SM. In particular, it violates the total lepton number by two units, and after spontaneous symmetry breaking it generates a bilinear neutrino field term:

$$- \mathcal{L}_{M_\nu} = \frac{Z_{ij}^\nu}{2} \frac{v^2}{\Lambda_{\text{NP}}} \bar{\nu}_{Li} \nu_{Lj}^c + \text{h.c.} \quad (14.23)$$

This is a Majorana mass term (see Eq.(14.8)). It is built with the left-handed neutrino fields and with mass matrix:

$$(M_\nu)_{ij} = Z_{ij}^\nu \frac{v^2}{\Lambda_{\text{NP}}}. \quad (14.24)$$

We conclude that Eq.(14.24) would arise in a generic extension of the SM and that neutrino masses are very likely to appear if there is NP. Comparing Eq.(14.24) and Eq.(14.5), we also find that the scale of neutrino masses is suppressed by $v/\Lambda_{NP}$ when compared to the scale of charged fermion masses, which provides an explanation for their smallness. Furthermore, both total lepton number and the lepton flavour symmetry $U(1)_e \times U(1)_\mu \times U(1)_\tau$ are broken by Eq.(14.24), which means that, generically, in the absence of additional symmetries on the coefficients $Z_{ij}$, we can expect lepton flavour mixing and CP violation as we discuss in the next section.

Finally, we notice that, as mentioned in Sec.14.2.2, a theory where the NP is composed of $m$ heavy sterile neutrinos, provides an specific example of a theory which at low energy contains three light mass eigenstates with an effective dim-5 interaction of the form (14.22) with $\Lambda_{NP} = M_N$. In this case, the NP scale is the characteristic mass scale of the heavy sterile neutrinos.

## 14.3 Lepton Mixing

Let us start by considering $n = 3 + m$ massive neutrino states and denote the neutrino mass eigenstates by $(\nu_1, \nu_2, \nu_3, \dots, \nu_n)$. The neutrino interaction eigenstates are denoted by $\bar{\nu} = (\nu_{Le}, \nu_{L\mu}, \nu_{L\tau}, \nu_{s1}, \dots, \nu_{sm})$. We label the corresponding mass and interaction eigenstates for the charged leptons as $(e, \mu, \tau)$ and $(e^I, \mu^I, \tau^I)$, respectively. The Lagrangian for the leptonic charged current interactions in the mass basis takes the form:

$$- \mathcal{L}_{CC} = \frac{g}{\sqrt{2}} (\bar{e}_L, \bar{\mu}_L, \bar{\tau}_L) \gamma^\mu U \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \\ \vdots \\ \nu_n \end{pmatrix} W_\mu^+ + \text{h.c.}, \quad (14.25)$$

where $U$ is a $3 \times n$ matrix [14–16]. It satisfies the unitary condition

$$UU^\dagger = I_{3 \times 3}. \quad (14.26)$$

However, in general $U^\dagger U \ne I_{n \times n}$.

In the interaction basis, the mass terms for the leptons are:

$$- \mathcal{L}_M = [(\bar{e}_L^I, \bar{\mu}_L^I, \bar{\tau}_L^I) M_\ell \begin{pmatrix} e_R^I \\ \mu_R^I \\ \tau_R^I \end{pmatrix} + \text{h.c.}] - \mathcal{L}_{M_\nu}, \quad (14.27)$$

with $\mathcal{L}_{M_\nu}$ given in Eq.(14.8). $M_\ell$ can be diagonalized with two $3 \times 3$ unitary matrices $V^\ell$ and $V_R^\ell$ which satisfy

$$V^\ell \dagger M_\ell V_R^\ell = \text{diag}(m_e, m_\mu, m_\tau). \quad (14.28)$$

Then for the charged leptons we have

$$- \mathcal{L}_{M_\ell} = \sum_{k=1}^{3} m_{\ell_k} \bar{\ell}_k \ell_k, \quad (14.29)$$

with

$$\ell_k = (V^{\ell \dagger} \ell_L^I)_k + (V_R^{\ell \dagger} \ell_R^I)_k. \quad (14.30)$$

Inverting the equation above we find that the weak-doublet components of the charged lepton fields are

$$\ell_{Li}^{I} = P_{L} \sum_{j=1}^{3} V_{ij}^{\ell} \ell_{j}. \quad i = 1, 2, 3 \quad (14.31)$$

From Eqs.(14.14), (14.18) and (14.31) we find that the mixing matrix $U$ can be expressed as:

$$U_{ij} = \mathcal{P}_{\ell,ii} V_{ik}^{\ell \dagger} V_{kj}^{\nu} (\mathcal{P}_{\nu,jj}). \quad (14.32)$$

The matrix $V^{\ell \dagger} V^\nu$ contains a number of phases that are not physical. Three of them are eliminated by the diagonal $3 \times 3$ phase matrix $\mathcal{P}_\ell$ that absorbs them in the charged lepton mass eigenstates. If neutrinos are Dirac states, further $n-1$ are similarly eliminated by absorbing them in the neutrino mass eigenstates with the diagonal $n \times n$ phase matrix $\mathcal{P}_\nu$. For Majorana neutrinos, $\mathcal{P}_\nu = I_{n \times n}$ because one cannot rotate by an arbitrary phase a Majorana field without physical effects. If one rotates a Majorana neutrino by a phase, this phase will appear in its mass term, which will no longer be real. Consequently, the number of phases that can be absorbed by redefining the mass eigenstates
depends on whether the neutrinos are Dirac or Majorana particles. Altogether for $n \ge 3$ Majorana [Dirac] neutrinos, the $U$ matrix contains a total of $6(n-2)$ [5$n-11$] real parameters, of which $3(n-2)$ are angles, and $3(n-2)$ [$2n-5$] can be interpreted as physical phases.

The possibility of arbitrary mixing between massive neutrino states was first discussed in the context of two neutrinos in Ref. [17] (the possibility of two mixed massless flavour neutrino states had been previously considered in the literature [18], and the possibility of mixing between neutrino and antineutrino states even

earlier, in the seminal paper of Pontecorvo [19]). For the case where only mixing between two generations is considered with $n = 2$ distinct neutrino masses, the $U$ matrix is $2 \times 2$ and contains one mixing angle if the neutrinos are Dirac and an additional physical phase if they are Majorana.

If there are only $n = 3$ Majorana neutrinos, $U$ is a $3 \times 3$ matrix analogous to the CKM matrix for the quarks [20, 21], but due to the Majorana nature of the neutrinos it depends on six independent parameters: three mixing angles and three phases. In this case the mixing matrix can be conveniently parameterized as:

$$ U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \cdot \begin{pmatrix} c_{13} & 0 & s_{13}e^{-i\delta_{CP}} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta_{CP}} & 0 & c_{13} \end{pmatrix} \cdot \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} e^{i\eta_1} & 0 & 0 \\ 0 & e^{i\eta_2} & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad (14.33) $$

where $c_{ij} \equiv \cos \theta_{ij}$ and $s_{ij} \equiv \sin \theta_{ij}$. The angles $\theta_{ij}$ can be taken without loss of generality to lie in the first quadrant, $\theta_{ij} \in [0, \pi/2]$ and the phases $\delta_{CP}$, $\eta_i \in [0, 2\pi]$. This is to be compared to the case of three Dirac neutrinos. In this case, the Majorana phases,

$\eta_1$ and $\eta_2$, can be absorbed in the neutrino states so the number of physical phases is one (similar to the CKM matrix). Thus we can write $U$ as:

$$ U = \begin{pmatrix} c_{12} & c_{13} & s_{12} & c_{13} \\ -s_{12} & c_{23} & -c_{12} & s_{13} \\ s_{12} & s_{23} & -c_{12} & s_{23} \end{pmatrix} \begin{pmatrix} e^{i\delta_{CP}} & & & \\ & c_{12} & c_{23} & -c_{13} \\ & & -c_{12} & c_{23} \end{pmatrix} \begin{pmatrix} s_{13} & e^{-i\delta_{CP}} & & \\ & & c_{13} & s_{23} \\ & & & c_{13} \\ & & & c_{23} \end{pmatrix} \cdot \begin{pmatrix} e^{i\eta_1} & 0 & 0 & 0 \\ 0 & e^{i\eta_2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \quad (14.34) $$

This matrix is often called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix.

Notice that when the charged leptons have no other interactions than the SM ones, one can identify their interaction eigenstates with the corresponding mass eigenstates up to phase redefinition. This implies that, in this case, $U$ is just a $3 \times n$ sub-matrix of the unitary neutrino mass diagonalizing matrix $V^\nu$.

Finally, let us point out that for the case of 3 light Dirac neutrinos, the procedure above leads to a unitary $U$ matrix for the light states. But for three light Majorana neutrinos, this is not the case since the full spectrum contains states which are heavy and are not in the low energy spectrum as seen, for example, in Eq.(14.21). This implies that, strictly speaking, the parametrization in Eq.(14.33) is not valid to describe the flavour mixing of the three light Majorana neutrinos in the see-saw mechanism. The violation of unitarity, however, is rather small, of the order $O(M_D/M_N)$ as seen in Eq.(14.21). It is also severely constrained experimentally [22,23]. For all these reasons, for all practical purposes, we will consider the $U$ matrix for the $3\nu$ mixing case to be unitary independently of whether neutrinos are Dirac or Majorana particles.

## 14.4 Mass-Induced Flavour Oscillations in Vacuum

If neutrinos have masses and lepton flavours are mixed in the weak CC interactions, lepton flavour is not conserved in neutrino propagation [19, 24]. This phenomenon is usually referred to as neutrino oscillations. In brief, a weak eigenstates, $\nu_\alpha$, which by default is the state produced in the weak CC interaction of a charged lepton $\ell_\alpha$, is the linear combination determined by the mixing matrix $U$

$$ |\nu_\alpha\rangle = \sum_{i=1}^{n} U_{\alpha i}^* |\nu_i\rangle, \quad (14.35) $$

where $\nu_i$ are the mass eigenstates, and here $n$ is the number of light neutrino species (implicit in our definition of the state $|\nu\rangle$ is its energy-momentum and space-time dependence). After traveling a distance $L$ ($L \simeq ct$ for relativistic neutrinos), that state evolves as:

$$ |\nu_\alpha(t)\rangle = \sum_{i=1}^{n} U_{\alpha i}^* |\nu_i(t)\rangle. \quad (14.36) $$

This neutrino can then undergo a charged-current (CC) interaction producing a charge lepton $\ell_\beta$, $\nu_\alpha(t)N' \rightarrow \ell_\beta N$, with a

probability

$$ P_{\alpha\beta} = |\langle\nu_{\beta}|\nu_{\alpha}(t)\rangle|^2 = \left| \sum_{i=1}^{n} \sum_{j=1}^{n} U_{\alpha i}^* U_{\beta j} \langle\nu_j|\nu_i(t)\rangle \right|^2. \quad (14.37) $$

Assuming that $|\nu\rangle$ is a plane wave, $|\nu_i(t)\rangle = e^{-iE_i t}|\nu_i(0)\rangle$, $^3$ with $E_i = \sqrt{p_i^2 + m_i^2}$ and $m_i$ being, respectively, the energy and the mass of the neutrino mass eigenstate $\nu_i$. In all practical cases neutrinos are very relativistic, so $p_i \simeq p_j \equiv p \simeq E$. We can then write

$$ E_i = \sqrt{p_i^2 + m_i^2} \approx p + \frac{m_i^2}{2E}, \quad (14.38) $$

and use the orthogonality of the mass eigenstates, $\langle\nu_j|\nu_i\rangle = \delta_{ij}$, to arrive to the following form for $P_{\alpha\beta}$:

$$ P_{\alpha\beta} = \delta_{\alpha\beta} - 4 \sum_{i<j}^{n} \mathrm{Re}[U_{\alpha i} U_{\beta i}^* U_{\alpha j}^* U_{\beta j}] \sin^2 X_{ij} \\ + 2 \sum_{i<j}^{n} \mathrm{Im}[U_{\alpha i} U_{\beta i}^* U_{\alpha j} U_{\beta j}] \sin 2X_{ij}, \quad (14.39) $$

where

$$ X_{ij} = \frac{(m_i^2 - m_j^2)L}{4E} = 1.267 \frac{\Delta m_{ij}^2}{\mathrm{eV}^2} \frac{L/E}{\mathrm{m}/\mathrm{MeV}}. \quad (14.40) $$

If we had made the same derivation for antineutrino states, we would have ended with a similar expression but with the exchange $U \rightarrow U^*$. Consequently, we conclude that the first term in the right-hand-side of Eq.(14.39) is CP conserving since it is the same for neutrinos and antineutrinos, while the last one is CP violating because it has opposite signs for neutrinos and antineutrinos.

Equation (14.39) is oscillatory in distance with oscillation lengths

$$ L_{0,ij}^{\text{osc}} = \frac{4\pi E}{|\Delta m_{ij}^2|}, \quad (14.41) $$

and with amplitudes proportional to products of elements in the mixing matrix. Thus, neutrinos must have different masses

<footnote n="3">³ For a pedagogical discussion of the quantum mechanical description of flavour oscillations in the wave package approach see for example Ref. [3]. A recent review of the quantum mechanical aspects and subtleties on neutrino oscillations can be found in Ref. [25].</footnote>
<table><caption>**Table 14.1:** Characteristic values of *L* and *E* for experiments performed using various neutrino sources and the corresponding ranges of |Δ*m*²| to which they can be most sensitive to flavour oscillations in vacuum. SBL stands for Short Baseline, VSBL stands for Very Short Baseline, MBL stands for Medium Baseline, and LBL for Long Baseline.</caption><thead><tr><th>Experiment</th><th></th><th><i>L</i> (m)</th><th><i>E</i> (MeV)</th><th>|Δ<i>m</i><sup>2</sup>| (eV<sup>2</sup>)</th></tr></thead><tbody><tr><td>Solar</td><td></td><td>10<sup>10</sup></td><td>1</td><td>10<sup>-10</sup></td></tr><tr><td>Atmospheric</td><td></td><td>10<sup>4</sup> - 10<sup>7</sup></td><td>10<sup>2</sup>-10<sup>5</sup></td><td>10<sup>-1</sup> - 10<sup>-4</sup></td></tr><tr><td rowspan="2">Reactor</td><td>VSBL-SBL-MBL</td><td>10 - 10<sup>3</sup></td><td>1</td><td>1 - 10<sup>-3</sup></td></tr><tr><td>LBL</td><td>10<sup>4</sup> - 10<sup>5</sup></td><td></td><td>10<sup>-4</sup> - 10<sup>-5</sup></td></tr><tr><td rowspan="2">Accelerator</td><td>SBL</td><td>10<sup>2</sup></td><td>10<sup>3</sup>-10<sup>4</sup></td><td>> 0.1</td></tr><tr><td>LBL</td><td>10<sup>5</sup> - 10<sup>6</sup></td><td>10<sup>3</sup> - 10<sup>4</sup></td><td>10<sup>-2</sup> - 10<sup>-3</sup></td></tr></tbody></table>

($\Delta m_{ij}^2 \neq 0$) and they must have not vanishing mixing ($U_{\alpha_i}U_{\beta_i} \neq 0$) in order to undergo flavour oscillations. Also, from Eq.(14.39) we see that the Majorana phases cancel out in the oscillation probability. This is expected because flavour oscillation is a total lepton number conserving process.

Ideally, a neutrino oscillation experiment would like to measure an oscillation probability over a distance *L* between the source and the detector, for neutrinos of a definite energy *E*. In practice, neutrino beams, both from natural or artificial sources, are never monoenergetic but have an energy spectrum $\Phi(E)$. In addition, each detector has a finite energy resolution. Under these circumstances what is measured is an average probability

$$ \begin{aligned} \langle P_{\alpha\beta} \rangle &= \frac{\int dE \frac{d\Phi}{dE} \sigma(E) P_{\alpha\beta}(E) \epsilon(E)}{\int dE \frac{d\Phi}{dE} \sigma_{CC}(E) \epsilon(E)} \\ &= \delta_{\alpha\beta} - 4 \sum_{i<j}^{n} \mathrm{Re}[U_{\alpha i} U_{\beta i}^* U_{\alpha j}^* U_{\beta j}] \langle \sin^2 X_{ij} \rangle \quad (14.42) \\ &\quad + 2 \sum_{i<j}^{n} \mathrm{Im}[U_{\alpha i} U_{\beta i}^* U_{\alpha j}^* U_{\beta j}] \langle \sin 2X_{ij} \rangle. \end{aligned} $$

$\sigma$ is the cross-section for the process in which the neutrino flavour is detected, and $\epsilon(E)$ is the detection efficiency. The minimal range of the energy integral is determined by the energy resolution of the experiment.

It is clear from the above expression that if $(E/L) \gg |\Delta m_{ij}^2|$ ($L \ll L_{0,ij}^{\text{osc}}$) so $\sin^2 X_{ij} \ll 1$, the oscillation phase does not give any appreciable effect. Conversely, if $L \gg L_{0,ij}^{\text{osc}}$, many oscillation cycles occur between production and detection, so the oscillating term is averaged to $\langle \sin^2 X_{ij} \rangle = 1/2$.

We summarize in Table 14.1 the typical values of $L/E$ for different types of neutrino sources and experiments and the corresponding ranges of $\Delta m^2$ to which they can be most sensitive.

Historically, the results of neutrino oscillation experiments were interpreted assuming two-neutrino states so there is only one oscillating phase; the mixing matrix depends on a single mixing angle $\theta$; and no CP violation effect in oscillations is possible. At present, as we will discuss in Sec.14.7, we need at least the mixing among three-neutrino states to fully describe the bulk of experimental results. However, in many cases, the observed results can be understood in terms of oscillations dominantly driven by one $\Delta m^2$. In this limit $P_{\alpha\beta}$ of Eq.(14.39) takes the form [24]

$$ P_{\alpha\beta} = \delta_{\alpha\beta} - (2\delta_{\alpha\beta} - 1)\sin^2 2\theta \sin^2 X. \quad (14.43) $$

In this effective $2 - \nu$ limit, changing the sign of the mass difference, $\Delta m^2 \to -\Delta m^2$, and changing the octant of the mixing angle, $\theta \to \frac{\pi}{2} - \theta$, is just redefining the mass eigenstates, $\nu_1 \leftrightarrow \nu_2$: $P_{\alpha\beta}$ must be invariant under such transformation. So the physical parameter space can be covered with either $\Delta m^2 \ge 0$ with $0 \le \theta \le \frac{\pi}{2}$, or, alternatively, $0 \le \theta \le \frac{\pi}{4}$ with either sign for $\Delta m^2$.

However, from Eq.(14.43) we see that $P_{\alpha\beta}$ is actually invariant under the change of sign of the mass splitting and the change of octant of the mixing angle separately. This implies that there is a two-fold discrete ambiguity since the two different sets of physical parameters, $(\Delta m^2, \theta)$ and $(\Delta m^2, \frac{\pi}{2} - \theta)$, give the same

transition probability in vacuum. In other words, one could not tell from a measurement of, say, $P_{e\mu}$ in vacuum whether the larger component of $\nu_e$ resides in the heavier or in the lighter neutrino mass eigenstate. This symmetry is broken when one considers mixing of three or more neutrinos in the flavour evolution and or when the neutrinos traverse regions of dense matter as we describe in Sec.14.7.1 and Sec.14.5, respectively.

## 14.5 Propagation of Massive Neutrinos in Matter

Neutrinos propagating in a dense medium can interact with the particles in the medium. The probability of an incoherent inelastic scattering is very small. For example the characteristic cross section for $\nu$-proton scattering is of the order

$$ \sigma \sim \frac{G_F^2 s}{\pi} \sim 10^{-43} \text{ cm}^2 \left( \frac{E}{\text{MeV}} \right)^2, \quad (14.44) $$

where $G_F$ is the Fermi constant and $s$ is the square of the center of mass energy of the collision.

But when neutrinos propagate in dense matter, they can also interact coherently with the particles in the medium. By definition, in coherent interactions, the medium remains unchanged so it is possible to have interference of the forward scattered and the unscattered neutrino waves which enhances the effect of matter in the neutrino propagation. In this case, the effect of the medium is not on the intensity of the propagating neutrino beam, which remains unchanged, but on the phase velocity of the neutrino wave, and for this reason the effect is proportional to $G_F$, instead of the $G_F^2$ dependence of the incoherent scattering. Coherence also allows decoupling the evolution equation of the neutrinos from those of the medium. In this limit, the effect of the medium is introduced in the evolution equation for the neutrinos in the form of an effective potential which depends on the density and composition of the matter [26].

As an example, let us consider the evolution of $\nu_e$ in a medium with electrons, protons, and neutrons with corresponding $n_e, n_p$, and $n_n$ number densities. The effective low-energy Hamiltonian describing the relevant neutrino interactions at point $x$ is given by

$$ H_W = \frac{G_F}{\sqrt{2}} \left[ J^{(+)}(x) J_{\alpha}^{(-)}(x) + \frac{1}{4} J^{(N)}(x) J_{\alpha}^{(N)}(x) \right], \quad (14.45) $$

where the $J_\alpha$'s are the standard fermionic currents

$$ J_{\alpha}^{(+)}(x) = \bar{\nu}_e(x)\gamma_{\alpha}(1 - \gamma_5)e(x), \quad (14.46) $$

$$ J_{\alpha}^{(-)}(x) = \bar{e}(x)\gamma_{\alpha}(1 - \gamma_5)\nu_{e}(x), \quad (14.47) $$

$$ \begin{aligned} J_{\alpha}^{(N)}(x) = \bar{\nu}_e(x)\gamma_{\alpha}(1 - \gamma_5)\nu_e(x) & \\ & - \bar{e}(x)[\gamma_{\alpha}(1 - \gamma_5) - 4 \sin^2 \theta_W \gamma_{\alpha}]e(x) & \\ & + \bar{p}(x)[\gamma_{\alpha}(1 - g_A^{(p)}\gamma_5) - 4 \sin^2 \theta_W \gamma_{\alpha}]p(x) & \\ & - \bar{n}(x)\gamma_{\alpha}(1 - g_A^{(n)}\gamma_5)n(x), & \end{aligned} \quad (14.48) $$

and $g_A^{(n,p)}$ are the axial couplings for neutrons and protons, respectively.
Let us focus first on the effect of the charged current inter-

actions. The effective CC Hamiltonian due to electrons in the medium is

$$
\begin{align}
H_C^{(e)} &= \frac{G_F}{\sqrt{2}} \int d^3 p_e f(E_e, T) \times \left\langle \langle e(s, p_e) | \bar{e}(x) \gamma^\alpha (1-\gamma_5) \nu_e(x) \bar{\nu}_e(x) \gamma_\alpha (1-\gamma_5) e(x) | e(s, p_e) \rangle \right\rangle \tag{14.49} \\
&= \frac{G_F}{\sqrt{2}} \bar{\nu}_e(x) \gamma_\alpha (1-\gamma_5) \nu_e(x) \int d^3 p_e f(E_e, T) \left\langle \langle e(s, p_e) | \bar{e}(x) \gamma_\alpha (1-\gamma_5) e(x) | e(s, p_e) \rangle \right\rangle . \nonumber
\end{align}
$$

In the above equation, we denote by $s$ the electron spin, and by $p_e$
its momentum, and $f(E_e, T)$ is the energy distribution function of
the electrons in the medium which is assumed to be homogeneous
and isotropic and is normalized as

$$
\int d^3 p_e f(E_e, T) = 1. \tag{14.50}
$$

We denote by $\langle \dots \rangle$ the averaging over electron spinors and sum-
ming over all electrons in the medium. Coherence dictates that
$s$, $p_e$ are the same for initial and final electrons. The axial cur-
rent reduces to the spin in the non-relativistic limit and therefore
averages to zero for a background of non-relativistic electrons.
The spatial components of the vector current cancel because of
isotropy. Therefore, the only non-trivial average is

$$
\int d^3 p_e f(E_e, T) \left\langle \langle e(s, p_e) | \bar{e}(x) \gamma_0 e(x) | e(s, p_e) \rangle \right\rangle = n_e(x), \quad (14.51)
$$

which gives a contribution to the effective Hamiltonian

$$
H_C^{(e)} = \sqrt{2} G_F n_e \bar{\nu}_{eL}(x) \gamma_0 \nu_{eL}(x) . \quad (14.52)
$$

This can be interpreted as a contribution to the $\nu_{eL}$ potential
energy

$$
V_C = \sqrt{2} G_F n_e . \qquad (14.53)
$$

Should we have considered antineutrino states, we would have
ended up with $V_C = -\sqrt{2}G_F n_e$. For a more detailed derivation of
the matter potentials see, for example, Ref. [3].

With an equivalent derivation, we find that for $\nu_\mu$ and $\nu_\tau$, the potential due to its CC interactions is zero for most media since neither $\mu$'s nor $\tau'$s are present, while the effective potential for any active neutrino due to the neutral current interactions is found to be

$$
V_{NC} = \frac{\sqrt{2}}{2} G_F \left[ -n_e(1 - 4 \sin^2 \theta_w) + n_p(1 - 4 \sin^2 \theta_w) - n_n \right]. \quad (14.54)
$$

In neutral matter, $n_e = n_p$ and the contribution from electrons and protons cancel each other. So we are left only with the neutrino contribution

$$
V_{NC} = -1/\sqrt{2} G_F n_n . \qquad (14.55)
$$

After including these effects, the evolution equation for $n$ ultra-
relativistic neutrinos propagating in matter written in the mass
basis is (see for instance Ref. [27–29] for the derivation):

$$
i \frac{d\vec{\nu}}{dx} = H \vec{\nu}, \quad H = H_m + U^{\nu\dagger} V U^{\nu}. \quad (14.56)
$$

Here $\vec{\nu} \equiv (\nu_1, \nu_2, \dots, \nu_n)^T$, $H_m$ is the the kinetic Hamiltonian,

$$
H_m = \frac{1}{2E} \operatorname{diag}(m_1^2, m_2^2, \dots, m_n^2), \quad (14.57)
$$

and $V$ is the effective neutrino potential in the interaction basis. $U^\nu$ is the $n \times n$ submatrix of the unitary $V^\nu$ matrix corresponding to the $n$ ultrarelativistic neutrino states. For the three SM active neutrinos with purely SM interactions crossing a neutral medium with electrons, protons and neutrons, the evolution equation takes the form (14.56) with $U^\nu \equiv U$, and the effective potential:

$$
V = \operatorname{diag}(\pm\sqrt{2}G_F n_e(x), 0, 0) \equiv \operatorname{diag}(V_e, 0, 0). \quad (14.58)
$$

The sign + (−) in Eq.(14.58) applies to neutrinos (antineutrinos), and $n_e(x)$ is the electron number density in the medium, which in general is not constant along the neutrino trajectory so the potential is not constant. The characteristic value of the potential at the Earth core is $V_e \sim 10^{-13}$ eV, while at the solar core $V_e \sim 10^{-12}$ eV. Since the neutral current potential Eq.(14.55) is flavour diagonal, it can be eliminated from the evolution equation as it only contributes to an overall unobservable phase.

The instantaneous mass eigenstates in matter, $\nu_i^m$, are the eigenstates of the Hamiltonian $H$ in (14.56) for a fixed value of $x$, and they are related to the interaction basis by

$$
\vec{\nu} = \bar{U}(x) \vec{\nu}^{\mathrm{m}}. \qquad (14.59)
$$

The corresponding instantaneous eigenvalues of $H$ are
$\mu_i(x)^2/(2E)$ with $\mu_i(x)$ being the instantaneous effective
neutrino masses.

Let us take for simplicity a neutrino state which is an admixture
of only two neutrino species $|\nu_\alpha\rangle$ and $|\nu_\beta\rangle$, so the two instanta-
neous mass eigenstates in matter $\nu_1^m$ and $\nu_2^m$ have instantaneous
effective neutrino masses

$$
\mu_{1,2}^2(x) = \frac{m_1^2 + m_2^2}{2} + E[V_\alpha + V_\beta] \quad (14.60)
$$

$$
\mp \frac{1}{2} \sqrt{[\Delta m^2 \cos 2\theta - A]^2 + [\Delta m^2 \sin 2\theta]^2},
$$

and $\tilde{U}(x)$ is a 2x2 rotation matrix with the instantaneous mixing
angle in matter given by

$$
\tan 2\theta_m = \frac{\Delta m^2 \sin 2\theta}{\Delta m^2 \cos 2\theta - A}. \quad (14.61)
$$

In the Eqs.(14.60) and (14.61) A is

$$
A \equiv 2E(V_{\alpha} - V_{\beta}), \quad (14.62)
$$

and its sign depends on the composition of the medium and on
the flavour composition of the neutrino state considered. From
the expressions above, we see that for a given sign of A the mix-
ing angle in matter is larger(smaller) than in vacuum if this last
one is in the first (second) octant. We see that the symmetry
about $\theta = 45^\circ$ which exists in vacuum oscillations between two
neutrino states is broken by the matter potential in propagation
in a medium. The expressions above show that significant effects
are present when A, is close to $\Delta m^2 \cos 2\theta$. In particular, as seen
in Eq.(14.61), the tangent of the mixing angle changes sign if,
along its path, the neutrino passes by some matter density region
satisfying, for its energy, the resonance condition

$$
A_R = \Delta m^2 \cos 2\theta. \qquad (14.63)
$$

This implies that if the neutrino is created in a region where
the relevant potential satisfies $A_0 > A_R$ ($A_0$ here is the value
of the relevant potential at the production point), then the ef-
fective mixing angle in matter at the production point is such
that sgn(cos 2θ<sub>m,0</sub>) = −sgn(cos 2θ). So the flavour component
of the mass eigenstates is inverted as compared to their composi-
tion in vacuum. In particular, if at the production point we have
$A_0 = 2A_R$, then θ<sub>m,0</sub> = π/2 − θ. Asymptotically, for $A_0 \gg A_R$,
θ<sub>m,0</sub> → π/2. In other words, if in vacuum the lightest (heaviest)
mass eigenstate has a larger projection on the flavour α (β), in-
side a matter with density and composition such that $A > A_R$,
the opposite holds. So if the neutrino system is traveling across a monotonically varying matter potential, the dominant flavour component of a given mass eigenstate changes when crossing the region with $A = A_R$. This phenomenon is known as level crossing.

Taking the derivative of Eq.(14.59) with respect to $x$ and using Eq.(14.56), we find that in the instantaneous mass basis the evolution equation reads:

$$i \frac{d\tilde{\nu}^m}{dx} = \left[ \frac{1}{2E} \operatorname{diag}(\mu_1^2(x), \mu_2^2(x), \dots, \mu_n^2(x)) - i \tilde{U}^\dagger(x) \frac{d\tilde{U}(x)}{dx} \right] \tilde{\nu}^m. \quad (14.64)$$

The presence of the last term, Eq.(14.64) implies that this is a system of coupled equations. So, in general, the instantaneous mass eigenstates, $\nu_i^m$ are not energy eigenstates. For constant or slowly enough varying matter potential this last term can be neglected and the instantaneous mass eigenstates, $\nu_i^m$, behave approximately as energy eigenstates, and they do not mix in the evolution. This is the adiabatic transition approximation. On the contrary, when the last term in Eq.(14.64) cannot be neglected, the instantaneous mass eigenstates mix along the neutrino path. This implies there can be level-jumping [30–33], and the evolution is non-adiabatic.

For adiabatic evolution in matter, the oscillation probability take a form very similar to the vacuum oscillation expression, Eq.(14.39). For example, neglecting CP violation:

$$P_{\alpha\beta} = \left| \sum_i \tilde{U}_{\alpha i}(0) \tilde{U}_{\beta i}(L) \exp \left( -\frac{i}{2E} \int_0^L \mu_i^2(x') dx' \right) \right|^2. \quad (14.65)$$

To compute $P_{\alpha\beta}$ in a varying potential, one can always solve the evolution equation numerically. Also, several analytic approximations for specific profiles of the matter potential can be found in the literature [34].

### 14.5.1 The Mikheyev-Smirnov-Wolfenstein Effect for Solar Neutrinos

The matter effects discussed in the previous section are of special relevance for solar neutrinos. As the Sun produces $\nu_e$'s in its core, we consider the propagation of a $\nu_e - \nu_X$ neutrino system ($X$ is some superposition of $\mu$ and $\tau$, which is arbitrary because $\nu_\mu$ and $\nu_\tau$ have only and equal neutral current interactions) in the matter density of the Sun.

The density of solar matter is a monotonically decreasing function of the distance $R$ from the center of the Sun, and it can be approximated by an exponential for $R < 0.9 R_\odot$

$$n_e(R) = n_e(0) \exp(-R/r_0), \quad (14.66)$$

with $r_0 = R_\odot / 10.54 = 6.6 \times 10^7$ m = $3.3 \times 10^{14}$ eV$^{-1}$.

As mentioned above, the nuclear reactions in the Sun produce electron neutrinos. After crossing the Sun, the composition of the neutrino state exiting the Sun will depend on the relative size of $\Delta m^2 \cos 2\theta$ versus $A_0 = 2 E G_F n_{e,0}$ (where 0 refers to the neutrino production point, which is near but not exactly at the center of the Sun, $R=0$).

If the relevant matter potential at production is well below the resonant value, $A_R = \Delta m^2 \cos 2\theta \gg A_0$, matter effects are negligible. With the characteristic matter density and energy of the solar neutrinos, this condition is fulfilled for values of $\Delta m^2$ such that $\Delta m^2 / E \gg L_{\text{Sun-Earth}}$. So the propagation occurs as in vacuum with the oscillating phase averaged to 1/2 and the survival probability at the exposed surface of the Earth is

$$P_{ee}(\Delta m^2 \cos 2\theta \gg A_0) = 1 - \frac{1}{2} \sin^2 2\theta > \frac{1}{2}. \quad (14.67)$$

If the relevant matter potential at production is only slightly below the resonant value, $A_R = \Delta m^2 \cos 2\theta \gtrsim A_0$, the neutrino does not cross a region with resonant density, but matter effects are sizable enough to modify the mixing. The oscillating phase is

averaged in the propagation between the Sun and the Earth. This regime is well described by an adiabatic propagation, Eq.(14.65). Using that $\tilde{U}(0)$ is a 2x2 rotation of angle $\theta_{m,0}$ – the mixing angle in matter at the neutrino production point–, and $\tilde{U}(L)$ is the corresponding rotation with vacuum mixing angle $\theta$, we get

$$\begin{aligned} P_{ee}(\Delta m^2 \cos 2\theta \ge A_0) &= \cos^2 \theta_{m,0} \cos^2 \theta + \sin^2 \theta_{m,0} \sin^2 \theta \\ &= \frac{1}{2} [1 + \cos 2\theta_{m,0} \cos 2\theta]. \end{aligned} \quad (14.68)$$

This expression reflects that an electron neutrino produced at $A_0$ is an admixture of $\nu_1$ with fraction $P_{e1,0} = \cos^2 \theta_{m,0}$ and $\nu_2$ with fraction $P_{e2,0} = \sin^2 \theta_{m,0}$. On exiting the Sun, $\nu_1$ consists of $\nu_e$ with fraction $P_{1e} = \cos^2 \theta$, and $\nu_2$ consists of $\nu_e$ with fraction $P_{2e} = \sin^2 \theta$ so $P_{ee} = P_{e1,0}P_{1e} + P_{e2,0}P_{2e} = \cos^2 \theta_{m,0} \cos^2 \theta + \sin^2 \theta_{m,0} \sin^2 \theta$ [35–37], exactly as given in Eq.(14.68). Since $A_0 < A_R$ the resonance is not crossed so $\cos 2\theta_{m,0}$ has the same sign as $\cos 2\theta$ and still $P_{ee} \ge 1/2$.

Finally, in the case that $A_R = \Delta m^2 \cos 2\theta < A_0$, the neutrino can cross the resonance on its way out. In the convention of $\Delta m^2 > 0$ this occurs if $\cos 2\theta > 0$ ($\theta < \pi/4$), which means that in vacuum $\nu_e$ is a combination of $\nu_1$ and $\nu_2$ with a larger $\nu_1$ component, while at the production point $\nu_e$ is a combination of $\nu_1^m$ and $\nu_2^m$ with larger $\nu_2^m$ component. In particular, if the density at the production point is much higher than the resonant density, $\Delta m^2 \cos 2\theta \ll A_0$,

$$\theta_{m,0} = \frac{\pi}{2} \Rightarrow \cos 2\theta_{m,0} = -1, \quad (14.69)$$

and the produced $\nu_e$ is purely $\nu_2^m$.

In this regime, the evolution of the neutrino ensemble can be adiabatic or non-adiabatic depending on the particular values of $\Delta m^2$ and the mixing angle. The oscillation parameters (see Secs.14.6.1 and 14.7) happen to be such that the transition is adiabatic in all ranges of solar neutrino energies. Thus the survival probability at the exposed surface of the Earth is given by Eq.(14.68) but now with mixing angle (14.69) so

$$P_{ee}(\Delta m^2 \cos 2\theta < A_0) = \frac{1}{2} [1 + \cos 2\theta_{m,0} \cos 2\theta] = \sin^2 \theta. \quad (14.70)$$

So, in this case, $P_{ee}$ can be much smaller than $1/2$ because $\cos 2\theta_{m,0}$ and $\cos 2\theta$ have opposite signs. This is referred to as the Mihheev-Smirnov-Wolfenstein (MSW) effect [26, 38], which plays a fundamental role in the interpretation of the solar neutrino data.

The resulting energy dependence of the survival probability of solar neutrinos is shown in Fig.14.3 (together with a compilation of data from solar experiments). The plotted curve corresponds to $\Delta m^2 \sim 7.5 \times 10^{-5}$ eV$^2$ and $\sin^2 \theta \sim 0.3$ (the so-called large mixing angle, LMA, solution). The figure illustrates the regimes described above. For these values of the oscillation parameters, neutrinos with $E \ll 1$ MeV are in the regime with $\Delta m^2 \cos 2\theta \gg A_0$ so the curve represents the value of vacuum averaged survival probability, Eq.(14.67), and therefore $P_{ee} > 0.5$. For $E > 10$ MeV, on the contrary, $\Delta m^2 \cos 2\theta \ll A_0$ and the survival probability is given by Eq.(14.70), so $P_{ee} = \sin^2 \theta \sim 0.3$. In between, the survival probability is given by Eq.(14.68) with $\theta_0$ changing rapidly from its vacuum value to the asymptotic matter value (14.69), 90°.

## 14.6 Experimental Study of Neutrino Oscillations

Neutrino flavour transitions, or neutrino oscillations, have been experimentally studied using various neutrino sources and detection techniques. Intense sources and large detectors are mandatory because of a large distance necessary for observable oscillation effects in addition to the tiny cross-sections. Also, the relevant neutrino flux before oscillations should be known with sufficient precision for a definitive measurement. Here, the experimental status of neutrino oscillations with the different neutrino sources: the Sun, Earth’s atmosphere, accelerators, and nuclear reactors are reviewed.
14.6.1 Solar Neutrinos

14.6.1.1 Solar neutrino flux

In the Sun, electron neutrinos are produced in the thermonu-
clear reactions which generate solar energy. These reactions occur
via two main chains, the pp chain and the CNO cycle. The pp
chain includes reactions p+p → d+e⁺+ν(pp), p+e⁻+p → d+ν
(pep), ³He + p → ⁴He + e⁺ + ν(hep), ⁷Be + e⁻ → ⁷Li + ν(+γ)
(⁷Be), and ⁸B → ⁸Be* + e⁺ + ν(⁸B). The CNO cycle involves
¹³N → ¹³C + e⁺ + ν(¹³N), ¹⁵O → ¹⁵N + e⁺ + ν(¹⁵O), and
¹⁷F → ¹⁷O + e⁺ + ν(¹⁷F). Those reactions result in the overall
fusion of protons into ⁴He, 4p → ⁴He + 2e⁺ + 2νₑ, where the
energy released in the reaction, Q = 4mₚ - m₄He - 2mₑ ~ 26
MeV, is mostly radiated through the photons and only a small
fraction is carried by the neutrinos, ⟨E₂νₑ⟩ = 0.59 MeV. In addi-
tion, electron capture on ¹³N, ¹⁵O, and ¹⁷F produces line spectra
of neutrinos called ecCNO neutrinos. Dividing the solar lumi-
nosity by the energy released per neutrino production, the total
neutrino flux can be estimated. At Earth, the pp solar neutrino
flux is about 6 × 10¹⁰ cm⁻²s⁻¹.

The solar neutrino flux has been calculated based on the Stan-
dard Solar Model (SSM). The SSM describes the structure and
evolution of the Sun based on a variety of inputs such as the mass,
luminosity, radius, surface temperature, age, and surface elemen-
tal abundances. In addition, the knowledge of the absolute nuclear
reaction cross sections for relevant fusion reactions and radiative
opacities are necessary. John Bahcall and his collaborators contin-
uously updated the SSM calculations over several decades [39,40].
Figure 14.1 shows the solar neutrino fluxes predicted by the SSM
calculation in [41] and ecCNO neutrinos in [42].

<figure><img src="image_51.png" /><figcaption>**Figure 14.1:** Spectrum of solar neutrino fluxes predicted by SSM calculation in [41]. In addition to standard fluxes, ecCNO neutrinos have been added based on [42]. Electron capture fluxes are given in cm⁻²s⁻¹. Taken from [43].</figcaption></figure>

14.6.1.2 *Detection of solar neutrinos and the solar neutrino problem*

Experiments that observed solar neutrinos are summarized in Table 14.2. A pioneering solar neutrino experiment was carried out by R. Davis, Jr. and collaborators at Homestake starting in the late 1960s [44]. The Davis’ experiment utilizes the reaction $\nu_e + {}^{37}\text{Cl} \rightarrow e^- + {}^{37}\text{Ar}$. Because this process has an energy threshold of 814 keV, the most relevant fluxes are the $^7\text{Be}$ and $^8\text{B}$ neutrinos. The detector contained ~ 615 t of C$_2$Cl$_4$. The produced $^{37}\text{Ar}$, which has a half-life of 34.8 days, was chemically extracted and introduced into a low-background proportional chamber every few months. The Auger electrons from electron capture of $^{37}\text{Ar}$ were counted to determine the reaction rate.

From the beginning, the observed number of neutrinos in the
Homestake mine experiment was significantly smaller than the
prediction by SSM — it was almost one-third. After thorough
checks of both experimental and theoretical work, the discrep-
ancy remained. This became to be known as the solar neutrino
problem. The final result from the Homestake experiment is
2.56 ± 0.16 ± 0.16 SNU [45], where SNU (solar neutrino unit)

is a unit of event rate, 1 SNU = $10^{-36}$ captures/(s atom). On the
other hand, prediction based on SSM is $8.46^{+0.87}_{-0.88}$ SNU [46].

The detection of neutrinos from other production processes was
recognized as an important input to investigate the origin of the
solar neutrino problem. In particular, the pp neutrino is most
abundant, and its flux prediction has the smallest uncertainty.
Using the radiochemical technique with gallium, the reaction $\nu_e +
{}^{71}\text{Ga} \rightarrow e^- + {}^{71}\text{Ge}$ has an energy threshold of 233 keV and can be
used for the pp neutrino detection. According to the SSM, more
than half of the events on ${}^{71}\text{Ga}$ are due to the pp neutrinos, with
the second dominant contribution coming from the ${}^{7}\text{Be}$ neutrinos.
${}^{71}\text{Ge}$ decays via electron capture with a half-life of 11.4 days. The
SAGE experiment in Baksan [47] used about 50 t of liquid metallic
gallium as a target. The GALLEX experiment in LNGS [48] used
101 t of GaCl$_3$, containing 30.3 t of gallium. Both experiments
used natural gallium, containing 39.9% of ${}^{71}\text{Ga}$ isotope. GALLEX
was followed by its successor GNO experiment. The measured
capture rate is $69.3 \pm 4.1 \pm 3.6$ SNU for GALLEX+GNO [49] and
$65.4^{+3.1}_{-3.0} \pm 2.6$ SNU for SAGE [50]. A SSM prediction is $127.9^{+8.1}_{-8.2}$
SNU [46].

The radiochemical detectors measure the reaction rate inte-
grated between extractions. The real-time measurement of so-
lar neutrinos was realized by the Kamiokande experiment [51].
The Kamiokande detector was a 3,000-t water-Cherenkov detec-
tor in the Kamioka mine. Super-Kamiokande, the successor of
Kamiokande, started operation in 1996. It is a large upright
cylindrical water Cherenkov detector containing 50 kt of pure
water <fnref n="4" />. An inner detector volume corresponding to 32 kt wa-
ter mass is viewed by more than 11,000 inward-facing 50 cm di-
ameter photomultiplier tubes (PMTs). Kamiokande and Super-
Kamiokande can observe solar neutrinos using $\nu_e$ elastic scatter-
ing (ES), $\nu_x + e^- \rightarrow \nu_x + e^-$. The ES reaction occurs via both
charged and neutral current interactions. Consequently, it is sen-
sitive to all active neutrino flavours, although the cross-section
for $\nu_e$, which is the only flavour to interact via charged current, is
about six times larger than that for $\nu_\mu$ or $\nu_\tau$. Because the energy
threshold is 6.5 MeV for Kamiokande and 3.5 MeV for the present
Super-Kamiokande (for the kinetic energy of recoil electron), these
experiments are sensitive to primarily to $^8$B neutrinos.

The results from Kamiokande [52] and Super-Kamiokande [53] showed significantly smaller numbers of observed solar neutrinos compared to the prediction. The latest solar neutrino flux observed by Super-Kamiokande is (2.345±0.014±0.036)×10⁶ cm⁻²s⁻¹ [54], while a prediction based on the SSM is (5.46 ± 0.66) × 10⁶ cm⁻²s⁻¹ [55]. In addition, no significant zenith angle variation nor spectrum distortion were observed in the initial phase of Super-Kamiokande, which placed strong constraints on the solution of the solar neutrino problem [56].

14.6.1.3 *Solution of the solar neutrino problem*

The SNO experiment in Canada used 1,000 t of heavy wa-
ter (D₂O) contained in a spherical acrylic vessel which was sur-
rounded by an H₂O shield. An array of PMTs installed on a
stainless steel structure detected Cherenkov radiation produced in
both the D₂O and H₂O. The SNO detector observed ⁸B neutrinos
via three different reactions. In addition to the ES scattering with
an electron, with D₂O target the CC $\nu_e + d \rightarrow e^- + p + p$ and
the NC $\nu_x + d \rightarrow \nu_x + p + n$ interactions are possible. The CC
reaction is sensitive to only $\nu_e$, while NC reaction is sensitive to
all active flavours of neutrinos with equal cross-sections. There-
fore, by comparing the measurements of different reactions, SNO
could provide a model-independent test of the neutrino flavour
change [57].

In 2001, SNO reported the initial result of CC measure-
ment [60]. Combined with the high statistics measurement of
$\nu-e$ elastic scattering from Super-Kamiokande [61], it provided a
direct evidence for the existence of non-$\nu_e$ component in solar
neutrino flux. The result of the NC measurement in 2002 [62]
established it with 5.3$\sigma$ of statistical significance. Figure 14.2
shows the fluxes of electron neutrinos ($\phi(\nu_e)$) and muon and tau

<footnote n="4">⁴Recently, gadolinium (0.01% by weight in 2020, 0.03% in 2022) is loaded in the water.</footnote>
<table><caption>**Table 14.2:** List of solar neutrino experiments</caption><thead><tr><th>Name</th><th>Target material</th><th>Energy threshold (MeV)</th><th>Mass (ton)</th><th>Years</th></tr></thead><tbody><tr><td>Homestake</td><td>C<sub>2</sub>Cl<sub>4</sub></td><td>0.814</td><td>615</td><td>1970–1994</td></tr><tr><td>SAGE</td><td>Ga</td><td>0.233</td><td>50</td><td>1989–</td></tr><tr><td>GALLEX</td><td>GaCl<sub>3</sub></td><td>0.233</td><td>100 [30.3 for Ga]</td><td>1991–1997</td></tr><tr><td>GNO</td><td>GaCl<sub>3</sub></td><td>0.233</td><td>100 [30.3 for Ga]</td><td>1998–2003</td></tr><tr><td>Kamiokande</td><td>H<sub>2</sub>O</td><td>6.5</td><td>3,000</td><td>1987–1995</td></tr><tr><td>Super-Kamiokande</td><td>H<sub>2</sub>O</td><td>3.5</td><td>50,000</td><td>1996–</td></tr><tr><td>SNO</td><td>D<sub>2</sub>O</td><td>3.5</td><td>1,000</td><td>1999–2006</td></tr><tr><td>KamLAND</td><td>Liquid scintillator</td><td>0.5/5.5</td><td>1,000</td><td>2001–</td></tr><tr><td>Borexino</td><td>Liquid scintillator</td><td>0.19</td><td>300</td><td>2007–2021</td></tr></tbody></table>

<figure><img src="image_52.png" /><figcaption>**Figure 14.2:** Fluxes of $^8$B solar neutrinos, $\phi(\nu_e)$, and $\phi(\nu_{\mu,\tau})$, deduced from the SNO's CC, ES, and NC results [58]. The Super-Kamiokande ES flux is from [59]. The BS05(OP) standard solar model prediction [40] is also shown. The bands represent the $1\sigma$ error. The contours show the 68%, 95%, and 99% joint probability for $\phi(\nu_e)$ and $\phi(\nu_{\mu,\tau})$. The figure is from [58].</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>**Figure 14.3:** Electron neutrino survival probability as a function of neutrino energy. The points represent, from left to right, the Borexino pp, $^7$Be, pep, and $^8$B data (red points) and the SNO+SK $^8$B data (black point). The three Borexino $^8$B data points correspond, from left to right, to the low-energy (LE) range, LE+HE range, and the high-energy (HE) range. The electron neutrino survival probabilities from experimental points are determined using a high metallicity SSM from [55]. The error bars represent the $\pm 1\sigma$ experimental + theoretical uncertainties. The curve corresponds to the $\pm 1\sigma$ prediction of the MSW-LMA solution using the parameter values given in [64]. This figure is provided by A. Ianni.</figcaption></figure>

neutrinos ($\phi(\nu_{\mu,\tau})$) with the 68%, 95%, and 99% joint probability contours obtained with the SNO data. Finally, together with the reactor neutrino experiment KamLAND (see Sec.14.6.4), the solution of solar neutrino problem was found to be the MSW adiabatic flavour transitions in the solar matter, the so-called large mixing angle (LMA) solution. From a combined result of three phases of SNO [63], the total flux of $^8$B solar neutrino is found to be $(5.25 \pm 0.16^{+0.11}_{-0.13}) \text{ cm}^{-2}\text{s}^{-1}$, consistent with the SSM prediction. This consistency is one of the major accomplishments of SSM.

In order to understand the SSM as well as to study the MSW effect for the solar neutrino, measurements of solar neutrinos other than $^8$B are important. The Borexino experiment at Gran Sasso detected solar neutrino via $\nu$-e scattering in real-time with a low energy threshold. The Borexino detector consisted of 300 t of ultra-pure liquid scintillator, which achieved 0.19 MeV of energy threshold and 5% energy resolution at 1 MeV. Borexino reported the first real-time detection of $^7$Be solar neutrinos [65]. They also measured the fluxes of pep [66], pp [67], and CNØ [68] neutrino for the first time. Together with $^8$B [69] neutrino measurement, Borexino provides important data to study the MSW effect. The KamLAND experiment also measured solar neutrinos [70, 71].

Figure 14.3 shows the survival probability of solar $\nu_e$ as a function of neutrino energy. The data points are from the Borexino results [72, 73] and the SNO+SK $^8$B data. The theoretical curve shows the prediction of the MSW-LMA solution. All the data shown in this plot are consistent with the theoretically calculated curve. This indicates that these solar neutrino measurements are consistent with the MSW-LMA solution of the solar neutrino problem.

The matter effects can also be relevant to the propagation of solar neutrinos through the Earth. Because solar neutrinos go

through the Earth before interaction in the detector during the nighttime, a comparison of measured event rate between daytime and nighttime provides a clean and direct test of matter effects on neutrino oscillations. Super-Kamiokande reported the indication of the day/night asymmetry in $^8$B solar neutrinos [54]. The measured asymmetry, defined as the difference of the average day rate and average night rate divided by the average of those two rates, is $(-3.3 \pm 1.0 \pm 0.5)\%$, corresponding to a statistical significance of $2.9\sigma$.

### 14.6.2 Atmospheric Neutrinos

#### 14.6.2.1 Atmospheric neutrino flux

Atmospheric neutrinos are produced by the decays of pions and kaons generated in the interaction of cosmic rays and nucleons in the Earth's atmosphere. They have a broad range of energy ($\sim 0.1$ GeV to $> \text{TeV}$) and long travel distances before detection ($\sim 10$ to $\sim 10^4$ km), and contain all the flavours of neutrinos and antineutrinos.

Considering their dominant production modes, some generic relations for flux ratios of different flavours of neutrinos can be derived without detailed calculations. From the decay chain of a charged pion $\pi^+ \rightarrow \mu^+\nu_\mu$ followed by $\mu^+ \rightarrow e^+\nu_e\bar{\nu}_\mu$ (and the charge conjugate for $\pi^-$), the ratio $(\nu_\mu + \bar{\nu}_\mu)/(\nu_e + \bar{\nu}_e)$ is expected to be around 2 at low energies ($\sim 1$ GeV), where most muons decay in the atmosphere. For higher energies, some muons reach the Earth before they decay and the ratio increases. One can also
expect that the zenith angle distributions of atmospheric neu-
trinos are symmetric between upward-going and downward-going
neutrinos. It is true for the energy above 1 GeV, but at lower
energies, the Earth's geomagnetic field induces up-down asymme-
tries in the primary cosmic ray. The zenith angle corresponds
to the flight length of atmospheric neutrinos. Vertically upward-
going neutrinos come from the other side of the Earth with flight
lengths of ~10⁴ km, while downward-going neutrinos produced
just above the experimental site travel ~10 km before detection.

The atmospheric neutrino fluxes are calculated in detail based
on the energy spectrum and composition of primary cosmic rays
and their hadronic interactions in the atmosphere. The effects of
solar activity and geomagnetic field are also taken into account.
Results of calculations by several groups are available [74–77]. A
typical uncertainty of the absolute flux is 10–20%, while the ratio
of fluxes between different flavour has much smaller uncertainty
(< 5%).

14.6.2.2 Observation of atmospheric neutrino oscillations

The first detection of atmospheric neutrinos was reported in the
1960's by the underground experiments in the Kolar Gold Field
experiment in India [78] and in South Africa [79]. In the 1980's,
experiments searching for nucleon decays started operation. They
used large underground detectors which could also observe atmo-
spheric neutrinos that were studied as backgrounds to nucleon
decays. Among the early experiments were Kamiokande [80] and
IMB [81] using water Cherenkov detectors, and Frejus [82] and
NUSEX [83] using iron tracking calorimeters.

The flavour of an atmospheric neutrino can be identified in
charged current interaction with nuclei, which produces the corre-
sponding charged lepton. Those detectors originally designed for
nucleon decay search had the capability to distinguish muons and
electrons. For example, a water Cherenkov detector can utilize
the information from Cherenkov ring patterns for particle iden-
tification; e-like particles (e<sup>±</sup>, γ) produce more diffuse ring than
μ-like particles (μ<sup>±</sup>, π<sup>±</sup>) because of electromagnetic cascades and
multiple Coulomb scattering effects.

To reduce the uncertainty, in early results the flux ratio $\nu_{\mu}/\nu_{e} \equiv (\nu_{\mu} + \bar{\nu}_{\mu})/(\nu_{e} + \bar{\nu}_{e})$ was measured, and the double ratio between observation and expectation $(\nu_{\mu}/\nu_{e})_{\text{obs}}/(\nu_{\mu}/\nu_{e})_{\text{exp}}$ was reported. The Kamiokande experiment reported an indication of a deficit of $(\nu_{\mu} + \bar{\nu}_{\mu})$ flux [80]. IMB also observed a similar deficit [81], but measurements by Frejus [82] and NUSEX [83] were consistent with the expectations. This was called the atmospheric neutrino anomaly. Kamiokande reported studies with an increased data set of the sub-GeV ($< 1.33 \text{ GeV}$) [84] as well as the multi-GeV ($> 1.33 \text{ GeV}$) [85] samples. In the latter, they reported an analysis of zenith angle distributions, which showed an indication that the muon disappearance probability is dependent on the zenith angle, hence the travel length of neutrinos. However, the statistical significance was not sufficient to provide a conclusive interpretation.

The solution to the atmospheric neutrino anomaly was brought
by Super-Kamiokande, which reported compelling evidence for
neutrino oscillations in atmospheric neutrinos in 1998 [86]. The
zenith angle ($\theta_z$, with $\theta_z = 0$ for vertically downward-going) dis-
tributions of μ-like events showed a clear deficit of upward-going
events, while no significant asymmetry was observed for e-like
events. The asymmetry is defined as $A = (U - D)/(U + D)$, where
$U$ is the number of upward-going ($-1 < \cos \theta_z < -0.2$) events and
$D$ is the number of downward-going ($0.2 < \cos \theta_z < 1.0$) events.
With multi-GeV (visible energy $> 1.33 \text{ GeV}$) μ-like events alone,
the measured asymmetry was $A = -0.296 \pm 0.048 \pm 0.001$, deviat-
ing from zero by more than $6\sigma$. The sub-GeV ($< 1.33 \text{ GeV}$) μ-like,
upward through going, and upward stopping μ samples which cor-
respond to different energy ranges of neutrinos, showed the con-
sistent behaviour which strengthens the credibility of the obser-
vation. Super-Kamiokande's results were confirmed by other at-
mospheric neutrino observations MACRO [87] and Soudan2 [88].

Although the energy and zenith-angle-dependent muon neu-
trino disappearance observed with atmospheric neutrinos could be
consistently explained by the neutrino oscillations predominantly
between vμ and ντ, other exotic explanations such as neutrino
decay [89] or decoherence [90] were not initially ruled out. By
using a selected sample from Super-Kamiokande's atmospheric

data with good L/E resolution, the L/E dependence of the sur-
vival probability was measured [91]. The observed dip in the L/E
distribution was consistent with the expectation from neutrino
oscillation, while alternative models were strongly disfavored.

As an experimental proof of $\nu_\mu-\nu_\tau$ oscillation, an appearance signal of $\nu_\tau$ was searched for in the atmospheric neutrino data. Because of the high energy threshold ($> 3.5 \text{ GeV}$) of $\nu_\tau$ CC interaction and the short lifetime of $\tau$ lepton (0.3 ps), identifying the appearance of $\nu_\tau$ experimentally is challenging. Super-Kamiokande reported evidence of tau neutrino appearance using atmospheric neutrino data with $4.6\sigma$ significance [92]. The definitive observation of $\nu_\tau$ appearance was made by the long-baseline experiment, OPERA (See Sec.14.6.3.3), and recently IceCube also reported the $\nu_\tau$ appearance analysis [93] using atmospheric neutrinos.

14.6.2.3 Neutrino oscillation measurements using atmospheric neutrinos

Figure 14.4 shows the zenith angle distributions of atmospheric neutrino data from Super-Kamiokande. For a wide range of neutrino energy and path length, the observed distributions are consistent with the expectation from neutrino oscillation. Atmospheric neutrinos in the energy region of a few to ~10 GeV provide information for the determination of the neutrino mass ordering [94].

The neutrino telescopes primarily built for high-energy neutrino
astronomy such as ANTARES and IceCube can also measure neu-
trino oscillations with atmospheric neutrinos. ANTARES consists
of a sparse array of PMTs deployed under the Mediterranean Sea
at a depth of about 2.5 km to instrument a 10⁵ m³ volume. Ice-
Cube is a detector deployed in ice in Antarctica at the South Pole,
at a depth between 1.45 and 2.45 km. In the bottom center of
IceCube there is a region of ~ 10⁷ m³ volume with denser PMT
spacing called DeepCore to extend the observable energies to the
lower energy region. By observing the charged current interaction
of up-going $\nu_\mu$, they measure the $\nu_\mu$ disappearance. ANTARES
reported a measurement of $\nu_\mu$ disappearance with 20 GeV thresh-
old [95]. The $\nu_\mu$ disappearance measurements from IceCube Deep-
Core [96] provided a precision comparable to the measurements
by Super-Kamiokande and long-baseline accelerator neutrino ex-
periments.

There are several projects for atmospheric neutrino observa-
tions either proposed or under preparation. The atmospheric
neutrino observation program is included in the plans for future
neutrino telescopes such as ORCA in the KM3NeT project [97] in
the Mediterranean Sea and the IceCube Upgrade [98]. In India,
a 50 kt magnetized iron tracking calorimeter ICAL is planned
at the INO [99]. Future large underground detectors, Hyper-
Kamiokande in Japan [100] and DUNE in US [101] can also study
the atmospheric neutrinos.

14.6.3 Accelerator Neutrinos

14.6.3.1 Accelerator neutrino beams

A comprehensive description of the accelerator neutrino beams is found in [102]. Conventional neutrino beams from accelerators are produced by colliding high-energy protons onto a target, producing $\pi$ and $K$, which then decay into neutrinos. Undecayed mesons and muons are stopped in a beam dump and soil. Because pions are the most abundant product of the high energy collisions, a conventional neutrino beam contains a dominant amount of muon-type neutrinos (or antineutrinos).

Focusing devices called magnetic horns are used to concentrate the neutrino beam flux towards the desired direction [103]. A magnetic horn is a pulsed electromagnet with toroidal magnetic fields to focus charged particles that are parents of neutrinos. One can choose the dominant component of the beam to be either neutrinos or antineutrinos by selecting the direction of current in the magnetic horns. Even with the focusing with horns, wrong sign neutrinos contaminate in the beam. Also, there is a small amount of contamination of $\nu_e$ and $\bar{\nu}_e$ coming primarily from kaon and muon decays.

In order to maximize the sensitivity of the experiment, the ratio
of baseline and neutrino energy (L/E) should be chosen to match
the oscillation effects to be studied. In addition to maximizing
the flux of neutrinos with relevant energy, neutrinos with irrele-
<figure><img src="image_54.png" /><figcaption>Figure 14.4: The zenith angle distributions of Super-Kamiokande atmospheric neutrino events. A data set corresponding to 328 kton-years of exposure is used. Fully contained 1-ring e-like and μ-like events with visible energy < 1.33 GeV (sub-GeV) and > 1.33 GeV (multi-GeV), as well as upward stopping and upward through going μ samples are shown. Partially contained (PC) events are combined with multi-GeV μ-like events. The blue histograms show the non-oscillated Monte Carlo events, and the red histograms show the best-fit expectations for neutrino oscillations. (This figure is provided by the Super-Kamiokande Collaboration)</figcaption></figure>

vant energy that result in unwanted background process should be suppressed. The energy of a neutrino from a pion decay is

$$E_{\nu} = \frac{[1 - (m_{\mu}/m_{\pi})^2]E_{\pi}}{1 + \gamma^2\theta^2}, \qquad (14.71)$$

where $E_ν$ and $E_π$ are the energy of neutrino and pion, respectively, $θ$ is the angle between the pion and neutrino direction, and $γ = E_π/m_π$. For $θ = 0$, the energy of neutrino is linearly proportional to the energy of pion. In this case, a narrow band beam can be made by selecting the momentum of pions. On the other hand, for $θ \neq 0$, the energy of neutrino is not strongly dependent on the parent energy for a wide range of pion energy, but dependent on the off-axis angle $θ$. Using this relation, a neutrino beam with narrow energy spectrum, around the energy determined by $θ$, can be produced. This off-axis beam method was first introduced for BNL E889 proposal [104] and adopted in T2K and NOvA experiments. For a list of neutrino beamlines, see also Chapter 33 of this Review, “Neutrino Beam Lines at High-Energy Proton Synchrotrons.”

As indicated in Table 14.1, there are two different scales of baselines for accelerator-based experiments to study different ranges of $Δm^2$. The atmospheric mass splitting $Δm^2 \sim 2.5 \times 10^{-3}$ eV$^2$ gives rise to the first oscillation maximum at $L/E \sim 500$ GeV/km. In order to study this parameter region with a $\sim 1$ GeV accelerator neutrino beam, a long baseline of a few hundred to a thousand km is necessary. On the other hand, there have been reports of possible neutrino oscillations at the $\sim 1$ eV$^2$ scale, which can be studied at $\sim 1$ km baseline with neutrinos from accelerators. These experiments are called short-baseline oscillation experiments.

The flux of a neutrino beam is calculated using Monte Carlo simulation based on the configuration of the beamline. An important ingredient of the neutrino flux prediction is the hadron production cross-section. Data from dedicated hadron production experiments [105–108] are used to tune the beam simulation and constrain the uncertainty. The uncertainty of predicted neutrino flux for the most relevant energy region is 5–10% with the latest hadron production data [109–111].

### 14.6.3.2 Near detectors and neutrino interaction cross sections

Many long-baseline experiments use two detectors to reduce the systematic uncertainties arising from neutrino flux and neutrino-nucleus interactions. The near detectors either use the same technology as the far detector or consist of sub-detectors with complementary functions to obtain detailed information of the neutrino beam and interactions. The near detectors provide information for the neutrino flux, energy spectrum, and the interaction cross-sections, which is used as input to make predictions of observables at the far detector. However, even with the two-detector configuration, one should note that the neutrino flux is inevitably different between the near and the far detectors. In addition to the fact that the neutrino source looks like a line source for the near detector while it looks like a point source for the far detector, the neutrino oscillations alter the flavour composition of the neutrino beam quite significantly, as the design of a neutrino oscillation experiment requires.

For the precision measurements of neutrino oscillations with long-baseline experiments, the understanding of the neutrino-nucleus interaction becomes crucial. Because heavy nuclei are used as the interaction target, the nuclear effects complicate the understanding of the neutrino-nucleus interaction. For more information on the neutrino cross-sections, see also Chapter 52 of this Review, “Neutrino Cross Section Measurements.”

### 14.6.3.3 Long-baseline experiments

Long-baseline neutrino oscillation experiments are summarized in Table 14.3. The first long-baseline experiment was the K2K experiment which used a neutrino beam from the KEK 12 GeV proton synchrotron directed towards Super-Kamiokande with a baseline of 250 km. The beam had an average energy of 1.3 GeV. The K2K near detectors, located 300 m downstream of the production target, consisted of a combination of a 1 kt water Cherenkov detector and a set of fine-grained detectors. K2K confirmed the muon neutrino disappearance originally reported by Super-Kamiokande atmospheric neutrino observation [112].

The MINOS experiment used a beam from Fermilab and a de-
<table><caption>Table 14.3: List of long-baseline neutrino oscillation experiments</caption><thead><tr><th>Name</th><th>Beamline</th><th>Far Detector</th><th>L (km)</th><th>E<sub>ν</sub> (GeV)</th><th>Year</th></tr></thead><tbody><tr><td>K2K</td><td>KEK-PS</td><td>Water Cherenkov</td><td>250</td><td>1.3</td><td>1999–2004</td></tr><tr><td>MINOS</td><td>NuMI</td><td>Iron-scintillator</td><td>735</td><td>3</td><td>2005–2013</td></tr><tr><td>MINOS+</td><td>NuMI</td><td>Iron-scintillator</td><td>735</td><td>7</td><td>2013–2016</td></tr><tr><td>OPERA</td><td>CNGS</td><td>Emulsion hybrid</td><td>730</td><td>17</td><td>2008–2012</td></tr><tr><td>ICARUS</td><td>CNGS</td><td>Liquid argon TPC</td><td>730</td><td>17</td><td>2010–2012</td></tr><tr><td>T2K</td><td>J-PARC</td><td>Water Cherenkov</td><td>295</td><td>0.6</td><td>2010–</td></tr><tr><td>NOvA</td><td>NuMI</td><td>Liquid scint. tracking calorimeter</td><td>810</td><td>2</td><td>2014–</td></tr><tr><td>DUNE</td><td>LBNF</td><td>Liquid argon TPC</td><td>1300</td><td>2–3</td><td></td></tr><tr><td>Hyper-Kamiokande</td><td>J-PARC</td><td>Water Cherenkov</td><td>295</td><td>0.6</td><td></td></tr></tbody></table>

tector in the Soudan mine 735 km away. The neutrino beam is produced in the NuMI beamline with a 120 GeV proton beam from the Main Injector. The MINOS detectors were both iron-scintillator tracking calorimeters with toroidal magnetic fields. The far detector was 5.4 kt, while the near detector had a total mass of 0.98 kt and was located 1 km downstream of the production target. The energy spectrum of the NuMI beamline can be varied by changing the relative position of the target and horns. Most of MINOS data were taken with the “low energy” configuration with a peak energy of around 3 GeV. Since 2013, NuMI was operated with a “medium energy” configuration with the peak neutrino energy of around 7 GeV. The experiment for this period was called MINOS+. MINOS and MINOS+ combined accelerator and atmospheric neutrino data in both disappearance and appearance modes to measure oscillation parameters [113, 114].

In Europe, the CNGS neutrino beamline provided a beam with mean energy of 17 GeV from CERN to LNGS for long-baseline experiments with about 730 km of baseline. The beam energy was chosen so that CC interaction of $\nu_{\tau}$ can occur for direct confirmation of $\nu_{\tau}$ appearance. There was no near detector in CNGS because it was not necessary for the $\nu_{\tau}$ appearance search. The OPERA experiment used a detector consisting of an emulsion/lead target with about 1.25 kt total mass, complemented by electronic detectors. The excellent spatial resolution of the emulsion enabled the event-by-event identification of $\tau$ leptons. OPERA observed ten $\nu_{\tau}$ CC candidate events with $2.0 \pm 0.4$ expected background [115] and confirmed $\nu_{\mu} \to \nu_{\tau}$ oscillation in appearance mode with a statistical significance of $6.1\sigma$. Another neutrino experiment, ICARUS [116], which used 600 t liquid argon time projection chambers (TPCs), was operated in Gran Sasso from 2010 to 2012.

The first generation of long-baseline experiments confirmed the existence of neutrino oscillation. The major initial goal of second-generation experiments was the observation of $\nu_{\mu} \to \nu_{e}$ oscillation. Using this appearance mode, and by comparison of neutrino and antineutrino oscillation probabilities, search for CP violation in the neutrino mixing and measurement of the mass ordering and the octant of $\theta_{23}$ become possible.

The T2K experiment started in 2010 using a newly constructed high-intensity proton synchrotron J-PARC and the Super-Kamiokande detector. It is the first long-baseline experiment to employ the off-axis neutrino beam. The off-axis angle of $2.5^\circ$ was chosen to set the peak of the neutrino energy spectrum at 0.6 GeV, matching the first maximum of oscillation probability at the 295 km baseline for $\Delta m^2 \sim 2.5 \times 10^{-3}$ eV$^2$. T2K employs a set of near detectors at about 280 m from the production target. In 2011, T2K reported the first indication of $\nu_{\mu} \to \nu_{e}$ oscillation with a statistical significance of $2.5\sigma$ [117]. In the framework of 3ν mixing, it corresponds to detecting non-zero amplitude generated by the mixing angle $\theta_{13}$ (see Eq.14.33). Later $\nu_{\mu} \to \nu_{e}$ oscillation was established by T2K with more than $7\sigma$ in 2014 [118]. Figure 14.5 shows the observed kinematic distributions from T2K, for neutrino and antineutrino beam mode and also for muon and electron candidates. By a combined analysis of the neutrino and antineutrino data, T2K reported a hint of CP violation at the $2\sigma$ level [119].

The NOvA experiment uses the upgraded NuMI beamline with an off-axis configuration. The 14 kt NOvA far detector is lo-

cated near Ash River, Minnesota, 810 km away from the source. At 14.6 mrad off-axis from the central axis of the NuMI beam, the neutrino energy spectrum at the far detector has a peak of around 2 GeV. The near detector, located around 1 km from the source, has a functionally identical design to the far detector with a total mass of 290 t. Both detectors are tracking calorimeters consisting of planes of polyvinyl chloride cells alternating in vertical and horizontal orientation filled with liquid scintillator. The physics run of NOvA was started in 2014. After confirmation of $\nu_e$ appearance from $\nu_\mu$ beam [120], NOvA started data taking with antineutrino beam in 2016. Using the antineutrino beam data, NOvA has reported the observation of $\bar{\nu}_e$ appearance from $\bar{\nu}_\mu$ beam with $4.4\sigma$ significance [121]. Figure 14.6 shows the reconstructed neutrino energy distributions from NOvA [122]. The NOvA data do not show a strong asymmetry in the rate of $\nu_e$ versus $\bar{\nu}_e$ appearance, excluding $\delta_{CP} = 90^\circ$ for the inverted mass ordering ($m_3 < m_2 < m_1$, see Sec.14.7 for definitions) with $> 3\sigma$ and disfavouring $\delta_{CP} = 270^\circ$ for the case of normal mass ordering ($m_1 < m_2 < m_3$, see Sec.14.7 for definitions) with $> 2\sigma$.

Two large-scale long-baseline experiments are under construction. DUNE [101] will be a 1,300 km long-baseline experiment based in the US. The DUNE far detector will consist of four modules of at least 10 kt fiducial mass liquid argon time projection chambers, located 1.5 km underground at the Sanford Underground Research Facility in South Dakota. The beamline for DUNE, 1.2 MW at start and upgradable to 2.4 MW, as well as the facility for near detectors will be newly constructed at Fermilab. The Hyper-Kamiokande detector [100] in Japan will be the successor of the Super-Kamiokande detector. It will be a water Cherenkov detector with 260 kt total water mass. With upgrades to the existing accelerator and beamline, J-PARC will provide a 1.3 MW neutrino beam to Hyper-Kamiokande with a baseline of 295 km. Both DUNE and Hyper-Kamiokande will have a rich physics program besides the long-baseline experiment, such as searches for nucleon decays and study of supernova neutrinos.

### 14.6.3.4 Short-baseline experiments

The LSND experiment searched for neutrino oscillation using neutrinos from stopped pions at Los Alamos. A 800 MeV linac was used to produce pions that stopped in the target. Most of $\pi^-$s are absorbed by the nuclei inside the target, while $\pi^+$s and their daughter $\mu^+$s decay and produce neutrinos. Therefore, the produced neutrinos are mostly $\nu_\mu$, $\bar{\nu}_\mu$, and $\nu_e$ with minimal contamination of $\bar{\nu}_e$. The detector was a tank filled with 167 t of diluted liquid scintillator, located about 30 m from the neutrino source. LSND searched for $\bar{\nu}_\mu \to \bar{\nu}_e$ appearance using the inverse beta decay process, $\bar{\nu}_e + p \to e^+ + n$, and found an excess of $87.9 \pm 22.4 \pm 6.0$ events over the expected background [123].

The KARMEN experiment was performed at the neutron spallation facility ISIS of the Rutherford Appleton Laboratory. The KARMEN 2 detector was a segmented liquid scintillation calorimeter with a total volume of 65 m$^3$ located at a mean distance of 17.7 m from the ISIS target. KARMEN found a number of events consistent with the total background expectation, showing no signal for $\bar{\nu}_\mu \to \bar{\nu}_e$ oscillations [124]. The resulting limits exclude large regions of the parameter area favored by LSND.

The MiniBooNE experiment at Fermilab used a conventional neutrino beam to search for $\nu_e$ and $\bar{\nu}_e$ appearance in the same
<figure><img src="image_55.png" /><figcaption>**Figure 14.5:** Observed kinematic distributions from T2K compared to the expectations with the best-fit parameters [119]. Top and bottom panels correspond to data from neutrino and antineutrino beam mode, respectively. Left, middle, and right panels show distributions for single ring $\mu$-like events, single ring $e$-like events with no associated decay electron, and single ring $e$-like events with one associated decay electron (only for neutrino beam mode data), respectively.</figcaption></figure>

<figure><img src="image_56.png" /><figcaption>**Figure 14.6:** Reconstructed neutrino energy distributions from the NOvA far detector [122]. Top plots are for neutrino beam mode, and bottom plots are for antineutrino beam mode. Left: muon-type candidates. Right: electron-type candidates, split into a low and high purity sample as well as the event counts in the peripheral sample which occurred near the edge of the detector.</figcaption></figure>
parameter region as LSND. The booster neutrino beamline (BNB) with a single magnetic horn uses an 8 GeV proton beam from the Fermilab booster to produce a neutrino (antineutrino) beam with an energy spectrum peak of 600 (400) MeV. The MiniBooNE detector consists of a 12.2 m diameter sphere filled with 818 t of mineral and oil located 541 m from the target. MiniBooNE reported $\nu_e$ and $\bar{\nu}_e$ event excess in both neutrino and antineutrino running modes. In total, $638.0 \pm 52.1 \pm 122.2$ excess events are observed over the expected backgrounds, corresponding to $4.8\sigma$ significance [125].

Both LSND and MiniBooNE are single detector experiments. The short-baseline neutrino (SBN) program at Fermilab BNB [126] is further investigating the reported excess with a multi-detector configuration. The SBN program comprises three liquid argon time projection chambers at different baselines in the same neutrino beamline. The 85 t MicroBooNE detector was operated at 470 m from the target. The 112 t Short-Baseline Near Detector is located at 110 m from the target. The ICARUS detector was transported from Europe after refurbishment at CERN and is located at a baseline of 600 m as the SBN Far Detector. As of the writing of this review, MicroBooNE is the only experiment that has presented results from SBN, observing no excess of $\nu_e$ events [127] and no evidence of light sterile neutrino oscillations [128].

JSNS² experiment at J-PARC has started the search for neutrino oscillations with $\Delta m^2 \sim 1 \text{ eV}^2$ [129]. 1MW proton beam from the 3 GeV Rapid Cycling Synchrotron of J-PARC produces neutrinos from muon decay at rest. With a detector filled with a gadolinium-loaded liquid scintillator of 17 t fiducial mass at 24 m from the target, JSNS² is aiming to provide a direct test of the LSND anomaly. The second phase with an additional second detector at 48 m is under preparation.

### 14.6.4 Reactor Antineutrinos

#### 14.6.4.1 Reactor antineutrino flux

Nuclear reactors are very intense sources of $\bar{\nu}_e$'s in the MeV energy region, which are generated in nuclear fission of heavy isotopes (mainly $^{235}$U, $^{238}$U, $^{239}$Pu, and $^{241}$Pu). The $\bar{\nu}_e$ flux from a reactor can be estimated based on the thermal power output and fuel composition as a function of time. On average, about six $\bar{\nu}_e$'s are emitted, and about 200 MeV of energy is released per fission. Therefore, a 1 GWth (thermal power) reactor produces about $2 \times 10^{20}$ $\bar{\nu}_e$'s per second.

The detailed estimate of $\bar{\nu}_e$ flux and energy spectrum can be obtained by either summing up the spectra of beta decays involved using available nuclear data information of each fission fragment and its decays, or by using measurements of cumulative electron spectra associated with the beta decays of fission fragments. Because the fission of four main fuel isotopes involves thousands of beta-decay branches, a complete *ab initio* calculation is challenging. The cumulative electron spectra for $^{235}$U, $^{239}$Pu, and $^{241}$Pu were measured at the Institut Laue-Langevin (ILL) reactor in Grenoble, France in the 1980s [130–132]. For the prediction of $\bar{\nu}_e$ flux from $^{238}$U, a summation calculation in [133] was often used together with the ILL results.

A recent calculation of the reactor $\bar{\nu}_e$ flux [134] uses an improved *ab initio* approach for $^{238}$U and combined information from nuclear databases and electron spectra measured at ILL for $^{235}$U, $^{239}$Pu, and $^{241}$Pu. Another calculation [135] is provided for $^{235}$U, $^{239}$Pu, and $^{241}$Pu based on the ILL measurement of electron spectra, taking into account higher-order corrections and minimizing the use of nuclear databases. Both calculations predict a few percent higher normalization for the energy-averaged antineutrino fluxes of $^{235}$U, $^{239}$Pu, and $^{241}$Pu compared to the original analyses of ILL data. However, the reactor antineutrino flux measurement at Daya Bay [136] is consistent with the old flux predictions and the flux measurement results. Also, an excess of $\bar{\nu}_e$ flux around 5 MeV, compared to the prediction, has been observed by recent reactor experiments [137–139].

#### 14.6.4.2 Reactor antineutrino oscillation experiments

Charged current interaction cannot happen if a reactor $\bar{\nu}_e$ changes its flavour to $\bar{\nu}_\mu$ or $\bar{\nu}_\tau$, because its energy is not sufficient to produce heavier charged leptons. Thus, $\bar{\nu}_e$ disappear-

ance is the only channel to study neutrino flavour change with reactor experiments. The inverse beta decay $\bar{\nu}_e + p \rightarrow e^+ + n$ provides a way to detect $\bar{\nu}_e$ in the relevant energy region. The energy of prompt signal from $e^+$, $E_p$, is related to the energy of $\bar{\nu}_e$, $E_{\bar{\nu}} \sim E_p + 0.8$ MeV. The delayed coincidence with the signal from $\gamma$ ray emitted by neutron capture on nucleus after thermalization very efficiently suppresses the backgrounds. A liquid scintillator is often used to realize large detectors containing hydrogen as the target of inverse beta decay. In order to increase the neutron detection efficiency, a liquid scintillator is sometimes loaded with gadolinium because of a large neutron capture cross-section and higher energy of emitted $\gamma$ rays, the total energy of about 8 MeV, by gadolinium, in contrast to 2.2 MeV for the capture by hydrogen.

Table 14.4 shows a list of reactor antineutrino experiments measuring neutrino oscillations. As was also shown in Table 14.1, experiments are designed with different baselines because of the different scale of mass splittings found by solar and atmospheric neutrino experiments. Experiments with O(100) km baseline are sensitive to $\Delta m^2$ of $10^{-4}-10^{-5}$ eV$^2$, while $\sim 1$ km of baseline results in a sensitivity in the range of $10^{-2}-10^{-3}$ eV$^2$.

The KamLAND detector consists of 1,000 t of ultra-pure liquid scintillator contained in a 13-m diameter spherical balloon [140]. The detector is located in the original Kamiokande cavern, where the $\bar{\nu}_e$ flux was dominated by a few reactors at an average distance of $\sim 180$ km until 2011. KamLAND reported the first results in 2002 showing that the ratio of the observed number of $\bar{\nu}_e$ events and expectation without disappearance is $0.611 \pm 0.085 \pm 0.041$, evidence for reactor $\bar{\nu}_e$ disappearance at the 99.95% confidence level [140]. It confirmed a large value of the mixing angle corresponding to the LMA solution, which was reported by solar neutrino experiments. KamLAND also showed the evidence of $\bar{\nu}_e$ spectrum distortion consistent with the expectation from neutrino oscillations [141]. Figure 14.7 shows the ratio of observed $\bar{\nu}_e$ spectrum to the expectation for no-oscillation as a function of $L_0/E$ ($L_0$ = 180 km) for the KamLAND data. A clear oscillatory signature can be seen.

<figure><img src="image_57.png" /><figcaption>**Figure 14.7:** Ratio of the observed $\bar{\nu}_e$ spectrum to the expectation for no-oscillation versus $L_0/E$ for the KamLAND data. $L_0$ = 180 km is the flux-weighted average reactor baseline. The 3-$\nu$ histogram is the best-fit survival probability curve from the three-flavour unbinned maximum-likelihood analysis using only the KamLAND data. This figure is taken from [142].</figcaption></figure>

Following the establishment of neutrino oscillations with atmospheric, solar, accelerator, and reactor experiments, the measurement of the remaining mixing angle $\theta_{13}$ was recognized as the next major milestone. A reactor neutrino experiment with a baseline of $\sim 1$ km can make an almost pure measurement of $\sin^2 2\theta_{13}$ from the disappearance of $\bar{\nu}_e$. To be sensitive to a small value of $\theta_{13}$, experiments with two detectors were proposed. Among several proposals, three experiments have been realized: Double Chooz
<table><caption>Table 14.4: List of reactor antineutrino oscillation experiments</caption><thead><tr><th>Name</th><th>Reactor power (GW<sub>th</sub>)</th><th>Baseline (km)</th><th>Detector mass (t)</th><th>Year</th></tr></thead><tbody><tr><td>KamLAND</td><td>various</td><td>180 (ave.)</td><td>1,000</td><td>2001–</td></tr><tr><td>Double Chooz</td><td>4.25×2</td><td>1.05</td><td>8.3</td><td>2011–2018</td></tr><tr><td>Daya Bay</td><td>2.9×6</td><td>1.65</td><td>20×4</td><td>2011–2020</td></tr><tr><td>RENO</td><td>2.8×6</td><td>1.38</td><td>16</td><td>2011–</td></tr><tr><td>JUNO</td><td>26.6 (total)</td><td>53</td><td>20,000</td><td></td></tr></tbody></table>

in France, Daya Bay in China, and RENO in Korea.

These three experiments employ a similar detector design optimized for the precise measurement of a reactor antineutrino. An antineutrino detector consists of a cylindrical stainless steel vessel that houses two nested acrylic cylindrical vessels. The innermost vessel is filled with gadolinium-doped liquid scintillator as the primary antineutrino target. It is surrounded by a liquid scintillator layer to contain γ rays from the target volume. A buffer layer of mineral oil is placed outside to shield inner volumes from radioactivity of PMTs and surrounding rock. The light from liquid scintillator is detected by an array of PMTs mounted on the stainless steel vessel. Optically separated by the stainless steel vessel, the outside region is instrumented as a veto detector with either liquid scintillator (Double Chooz) or water Cherenkov (Daya Bay and RENO) detector.

The Double Chooz detector has a gadolinium-doped liquid scintillator with a mass of 8.3 t. The far detector at a baseline of ~1050 m from the two 4.25 GW<sub>th</sub> reactors started physics data taking in 2011. The near detector, located at ~400 m from the reactors, was completed in the end of 2014. Double Chooz finished data taking in early 2018. Daya Bay has two near (flux-weighted baseline 470 m and 576 m), and one far (1648 m) underground experimental halls near six reactors with 2.9 GW<sub>th</sub> each. Daya Bay has eight antineutrino detectors in total; two detectors in each of the near detector halls, and four detectors in the far detector hall. Each detector contains 20 t of gadolinium-loaded liquid scintillator. RENO has two identical detectors located at 294 m and 1383 m from the center of an array of six 2.8 GW<sub>th</sub> reactors. The mass of the gadolinium-loaded liquid scintillator is 16 t per detector. RENO started data taking with both near and far detectors in August 2011.

All three reactor neutrino experiments published their first results in 2012. First, Double Chooz reported an indication of reactor electron antineutrino disappearance with the ratio of observed to expected events of $R = 0.944 \pm 0.016 \pm 0.04$, ruling out the no-oscillation hypothesis at the 94.6% CL [143]. Daya Bay observed $R = 0.940 \pm 0.011 \pm 0.004$, corresponding to $5.2\sigma$ significance of non-zero value of $\theta_{13}$ [144]. RENO also reported $R = 0.920 \pm 0.009 \pm 0.014$, indicating a non-zero value of $\theta_{13}$ with a significance of $4.9\sigma$ [145]. These results established a non-zero value of $\theta_{13}$.

Both Daya Bay [146] and RENO [137] report results constraining mass-squared difference as well as the mixing angle by using both relative $\bar{\nu}_e$ rate and energy spectra information. Double Chooz has reported the analysis based on both far and near detectors [138] for the mixing angle, using neutron capture on any elements (primarily gadolinium and hydrogen) to increase the effective target mass. Figure 14.8 shows the energy spectra of the prompt signals observed in the far detector of three experiments.

In all three experiments, an excess of $\bar{\nu}_e$ events over the expected energy spectrum has been observed around 5 MeV, as mentioned earlier. This excess is observed in both near and far detectors and scales with the reactor power. Thanks to the cancellation between the near and far detectors, the neutrino oscillation measurements are not affected in a multi-detector setup.

With a baseline of ~50 km and an excellent energy measurement, reactor antineutrino experiments have significant sensitivity to the mass ordering. The JUNO detector, which is under construction, will consist of a 20 kt liquid scintillator and be located at 53 km from two nuclear power plants in China [147]. The JUNO experiment aims to determine the mass ordering with this technique as its primary goal. It can also provide precision

<figure><img src="image_58.png" /><figcaption>Figure 14.8: Energy spectra for prompt events at the far detectors for Daya Bay [146], RENO [137], and Double Chooz [138].</figcaption></figure>

measurements of neutrino mixing parameters as well as a broad non-oscillation science program.

### 14.6.4.3 Reactor experiments sensitive to $O(1) eV^2$ oscillations

Possible hints of neutrino oscillation at a scale of $\Delta m^2 \sim 1 \text{ eV}^2$ (see Sec.14.8) have motivated reactor experiments at a distance of ~10 m from the core. Recent experiments searching for ~1 eV$^2$ oscillation at reactors are summarized in Table 14.5.

As the antineutrino source, some use commercial reactors, which can provide a large flux leading to high statistical preci-
<table><caption>Table 14.5: List of reactor antineutrino experiments for O(eV²) oscillations</caption><thead><tr><th>Name</th><th>Reactor power (MW<sub>th</sub>)</th><th>Baseline (m)</th><th>Detector mass (t)</th><th>Detector technology</th><th>σ<sub>E</sub>/E @1 MeV(%)</th><th>S/B</th></tr></thead><tbody><tr><td>NEOS</td><td>2,800</td><td>24</td><td>1</td><td>Gd-LS</td><td>5</td><td>22</td></tr><tr><td>DANSS</td><td>3,100</td><td>10–13</td><td>0.9</td><td>Gd-PS</td><td>34</td><td>~30</td></tr><tr><td>STEREO</td><td>57</td><td>9–11</td><td>1.7</td><td>Gd-LS</td><td>10</td><td>0.9</td></tr><tr><td>PROSPECT</td><td>85</td><td>7–9</td><td>4</td><td><sup>6</sup>Li-LS</td><td>4.5</td><td>1.3</td></tr><tr><td>NEUTRINO-4</td><td>100</td><td>6–12</td><td>1.5</td><td>Gd-LS</td><td>16</td><td>0.5</td></tr><tr><td>SoLid</td><td>80</td><td>6–9</td><td>1.6</td><td><sup>6</sup>Li-PS</td><td>14</td><td></td></tr></tbody></table>

sion. On the other hand, though the flux is orders of magnitude smaller, a research reactor could have favorable conditions such as relatively easier access to a short baseline, simpler fuel composition, and compact core size.

The detectors are based on organic scintillators, either liquid scintillator (LS) or solid plastic scintillator (PS), which contain hydrogen as the target for inverse beta decay ($\bar{\nu}_e + p \rightarrow e^+ + n$). To identify the signal, neutron capture on either gadolinium (Gd) or $^6$Li is detected with delayed coincidence. When a neutron is captured by Gd, $\gamma$ rays with a total energy of 8 MeV are emitted. After neutron capture, $^6$Li decays into triton and $\alpha$. The effect of neutrino oscillation appears as a distortion of energy spectrum. To be independent from the reactor neutrino spectrum uncertainties, some experiments compare the spectra at different baselines by using a segmented detector or moving the detector.

NEOS [139,148] uses about 1 t of gadolinium-loaded liquid scintillator in an unsegmented detector. It is located at 23.7 m from the center of a commercial reactor and covered by an overburden of about 20 meters of water equivalent.

DANSS [149] is another experiment using a commercial reactor. The detector is highly segmented, consisting of 2,500 plastic scintillator strips, each with the size of $1 \times 4 \times 100$ cm³ and coated with a thin gadolinium-loaded reflective layer. The detector is placed on a movable platform below the reactor core, which varies the distance to the core from 10.7 to 12.7 m.

The STEREO detector [150,151] has six identical target cells of 37 cm length, ~2m³ of volume in total, filled with gadolinium-loaded liquid scintillator. They are placed from 9.4 to 11.1 m from the compact (80 cm high, 40 cm diameter) core of the ILL research reactor.

The PROSPECT detector [152] consists of a segmented 4 t $^6$Li-doped liquid scintillator detector. It covers a baseline range of 7–9 m from the highly $^{235}$U enriched reactor core of the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory. Thin reflecting panels divide the LS volume into an $11 \times 14$ two-dimensional array of 154 optically isolated rectangular segments ($14.5 \times 14.5 \times 117.6$ cm³).

NEUTRINO-4 [153,154] uses a gadolinium-loaded liquid scintillator detector segmented in $10 \times 5$ sections with a total volume of 1.8 m³. The detector is installed on a movable platform and moved to various positions with baselines of 6–12 m from the SM-3 reactor core in Russia.

The SoLid detector [155] is a finely segmented detector made of $5 \times 5 \times 5$ cm³ plastic scintillator cubes and $^6$LiF:ZnS sheets. Using the difference of time constant of scintillation between ZnS and the plastic scintillator a $n-\gamma$ separation capability is achieved. A detector with 1.6 t of active volume is installed at a distance of 6–9 m from the research reactor core in the BR2 reactor, Belgium.

## 14.7 Combined Analysis of Experimental Results: The 3ν Paradigm

From the experimental situation described in Sec.14.6 we conclude that

* Atmospheric $\nu_\mu$ and $\bar{\nu}_\mu$ disappear most likely converting to $\nu_\tau$ and $\bar{\nu}_\tau$. The results show an energy and distance dependence perfectly described by mass-induced oscillations.

* Accelerator $\nu_\mu$ and $\bar{\nu}_\mu$ disappear over distances of ~200 to 800 km. The energy spectrum of the results show a clear oscillatory behavior also in accordance with mass-induced oscillations with a wavelength in agreement with the effect observed in atmospheric neutrinos.

* Accelerator $\nu_\mu$ and $\bar{\nu}_\mu$ appear as $\nu_e$ and $\bar{\nu}_e$ at distances ~200 to 800 km.

* Solar $\nu_e$ convert to $\nu_\mu$ and/or $\nu_\tau$. The observed energy dependence of the effect is well described by massive neutrino conversion in the Sun matter according to the MSW effect

* Reactor $\bar{\nu}_e$ disappear over distances of ~200 km and ~1.5 km with different probabilities. The observed energy spectra show two different mass-induced oscillation wavelengths: at short distances in agreement with the one observed in accelerator $\nu_\mu$ disappearance, and a long distance compatible with the required parameters for MSW conversion in the Sun.

The minimum scenario to describe these results requires the mixing between the three flavour neutrinos of the standard model in three distinct mass eigenstates. In this case $U$ in Eq. (14.32) is a $3 \times 3$ matrix analogous to the CKM matrix for the quarks [21], but due to the possible Majorana nature of the neutrinos it can depend on six independent parameters: three mixing angles and three phases. There are several possible conventions for the ranges of the angles and ordering of the states. The community finally agreed to a parametrization of the leptonic mixing matrix as in Eq. (14.33). The angles $\theta_{ij}$ can be taken without loss of generality to lie in the first quadrant, $\theta_{ij} \in [0, \pi/2]$, and the phase $\delta_{CP} \in [0, 2\pi]$. Values of $\delta_{CP}$ different from 0 and $\pi$ imply CP violation in neutrino oscillations in vacuum [156–158]. The Majorana phases $\eta_1$ and $\eta_2$ play no role in neutrino oscillations [157, 159]. Hence for the study of neutrino oscillations in the 3ν mixing scenario one can use the parametrization in Eq. (14.34) irrespective of whether neutrinos are Dirac or Majorana particles. Indeed, Majorana phases are very hard to measure since they are only physical if neutrino mass is non-zero, and therefore, the amplitude of any process involving them is suppressed by a factor $m_\nu/E$ to some power where $E$ is the energy involved in the process, which is typically much larger than the neutrino mass. The most sensitive experimental probe of Majorana phases is the rate of neutrinoless $\beta\beta$ decay discussed in Secs. 14.9.3 and 14.9.2.

In this convention there are two non-equivalent orderings for the spectrum of neutrino masses:

- Spectrum with Normal Ordering (NO) with $m_1 < m_2 < m_3$

- Spectrum Inverted ordering (IO) with $m_3 < m_1 < m_2$.

Furthermore, the data show a hierarchy between the mass splittings, $\Delta m_{21}^2 \ll |\Delta m_{31}^2| \simeq |\Delta m_{32}^2|$ with $\Delta m_{ij}^2 \equiv m_i^2 - m_j^2$.

In this section, we follow the convention used in the listing section of the PDG and discuss the results for both, NO and IO, using $\Delta m_{21}^2$, which is always the smallest mass splitting, and $\Delta m_{32}^2$, which up to a sign, is the largest mass splitting for IO, while for NO the largest mass splitting is $\Delta m_{31}^2 = \Delta m_{32}^2 + \Delta m_{21}^2$.

With what we know of the mass differences (see table 14.7) and the neutrino mass scale (see Sec. 14.9), depending on the value of the lightest neutrino mass, the neutrino mass spectrum can be further classified in:

- Normal Hierarchical Spectrum (NH): $m_1 \ll m_2 < m_3$, $\Rightarrow m_2 \simeq \sqrt{\Delta m_{21}^2} \sim 8.6 \times 10^{-3} \text{eV}$, $m_3 \simeq \sqrt{\Delta m_{32}^2 + \Delta m_{21}^2} \sim 0.05 \text{eV}$,
- Inverted Hierarchical Spectrum (IH): $m_3 \ll m_1 < m_2$, 
  $\Rightarrow m_1 \approx \sqrt{|\Delta m_{32}^2 + \Delta m_{21}^2|} \sim 0.0492\,\text{eV}, m_2 \approx \sqrt{|\Delta m_{32}^2|} \sim 0.05\,\text{eV},$

- Quasidegenerate Spectrum (QD): $m_1 \approx m_2 \approx m_3 \gg \sqrt{|\Delta m_{32}^2|}$.

Sometimes in the literature the determination of the neutrino mass spectrum is referred to as determination of the neutrino hierarchy. However, as described above, with what we know so far of the neutrino mass scale, the neutrino spectrum may or may not be hierarchical. Therefore determination of neutrino mass ordering is a more precise expression, and it is the one used in this review.

In total, the 3-ν oscillation analysis of the existing data involves six parameters: 2 mass differences (one of which can be positive or negative), 3 mixing angles, and the CP phase. The different experiments described in Sec.14.6 provide information on different subsets of these parameters. The precise statistical analysis of the data requires the numerical evaluation of the corresponding oscillation probabilities by solving the evolution equation of the neutrino ensemble from their source to the experiment. Nevertheless, the dominant effects in the different experiments can be qualitatively understood in terms of approximate expressions for the oscillation probabilities which, for convenience, we briefly summarize here.

### 14.7.1 3ν Oscillation Probabilities

The relevant survival probabilities for solar and KamLAND experiments in the framework of three neutrino oscillations can be written as:

$$P_{ee}^{3\nu} = \sin^4 \theta_{13} + \cos^4 \theta_{13} P_{ee}^{2\nu} (\Delta m_{21}^2, \theta_{12}), \quad (14.72)$$

where we have used the fact that $L_{0,32}^{\text{osc}} = 4\pi E_\nu / \Delta m_{32}^2$ is much shorter than the distance traveled by both solar and KamLAND neutrinos, so that the oscillations related to $L_{0,32}^{\text{osc}}$ are averaged. In the presence of matter effects $P_{ee}^{2\nu}(\Delta m_{21}^2, \theta_{12})$ should be calculated, taking into account the evolution in an effective matter density $n_e^{\text{eff}} = n_e \cos^2 \theta_{13}$. For $10^{-5} \lesssim \Delta m_{32}^2 / \text{eV}^2 \lesssim 10^{-4}$, $P_{ee}^{2\nu}(\Delta m_{21}^2, \theta_{12})$ presents the following asymptotic behaviors [160]:

$$P_{ee}^{2\nu, \text{sun}} \approx 1 - \frac{1}{2} \sin^2(2\theta_{12}) \quad \text{for } E_\nu \lesssim \text{ few } \times 100 \text{ keV}, \quad (14.73)$$

$$P_{ee}^{2\nu, \text{sun}} \approx \sin^2(\theta_{12}) \quad \text{for } E_\nu \gtrsim \text{ few } \times 1 \text{ MeV}, \quad (14.74)$$

$$P_{ee}^{2\nu, \text{kam}} = 1 - \sin^2(2\theta_{12}) \sin^2 \frac{\Delta m_{21}^2 L}{4E_\nu}. \quad (14.75)$$

At present most of the precision of the solar analysis is provided by SNO and SK, for which the relevant MSW survival probability provides a direct measurement of $\sin^2 \theta_{12}$, as seen in Eq. (14.74). In the MSW regime, the determination of $\Delta m_{21}^2$ in solar experiments comes dominantly from the ratio between the solar potential and the $\Delta m_{21}^2$ term required to simultaneously describe the CC/NC data at SNO and the undistorted spectra of $^8$B neutrinos as measured in both SK and SNO. Conversely, KamLAND $\bar{\nu}_e$ survival probability proceeds dominantly as vacuum oscillations and provides a most precise determination of $\Delta m_{21}^2$ via the strong effect of the oscillating phase in the distortion of the reactor energy spectrum. On the contrary, it yields a weaker constraint on $\theta_{12}$ as the vacuum oscillation probability depends on the double-valued and “flatter” function $\sin^2(2\theta_{12})$.

In what respects the interpretation of $\nu_\mu$ disappearance data, at LBL experiments, the $\nu_\mu$ survival probability can be expanded in the small parameters $\sin\theta_{13}$ and $\alpha \equiv \Delta m_{21}^2 / \Delta m_{31}^2$ to good

accuracy as [161, 162]

$$\begin{align}
P_{\nu_{\mu} \to \nu_{\mu}} &\approx 1 - \sin^2 2\theta_{\mu\mu} \sin^2 \frac{\Delta m_{\mu\mu}^2 L}{4E_{\nu}} \nonumber \\
&\approx 1 - \cos^2 \theta_{13} \sin^2(2\theta_{23}) \sin^2 \frac{\Delta m_{32}^2 L}{4E_{\nu}} + \mathcal{O}(\alpha, s_{13}^2), \tag{14.76}
\end{align}$$

with

$$\begin{align*}
\sin^2 \theta_{\mu\mu} &= \cos^2 \theta_{13} \sin^2 \theta_{23}, \\
\Delta m_{\mu\mu}^2 &= \sin^2 \theta_{12} \Delta m_{31}^2 + \cos^2 \theta_{12} \Delta m_{32}^2 \\
&\phantom{=} + \cos \delta_{\text{CP}} \sin \theta_{13} \sin 2\theta_{12} \tan \theta_{23} \Delta m_{21}^2.
\end{align*}$$

At present $\nu_\mu$ disappearance results at LBL provide the best determination of $|\Delta m_{32}^2|$ and $\theta_{23}$, but as seen above, the probability is symmetric with respect to the octant of $\theta_{\mu\mu}$, which implies symmetry around $s_{23}^2 = 0.5/c_{13}^2$.

The relevant oscillation probability for $\nu_e$ appearance at LBL experiments can be expanded at the second order in the small parameters $\sin\theta_{13}$ and $\alpha$, and assuming a constant matter density it takes the form [163–165]:

$$\begin{align}
P_{\nu_{\mu} \to \nu_e, (\bar{\nu}_{\mu} \to \bar{\nu}_e)} &\approx 4 \sin^2 \theta_{13} \sin^2 \theta_{23} \frac{\sin^2 \Delta(1-A)}{(1-A)^2} \nonumber \\
&\quad + \alpha^2 \sin^2 2\theta_{12} \cos^2 \theta_{23} \frac{\sin^2 A\Delta}{A^2} \nonumber \\
&\quad + 8\alpha J_{\text{CP}}^{\text{max}} \cos(\Delta \pm \delta_{\text{CP}}) \frac{\sin \Delta A}{A} \frac{\sin \Delta(1-A)}{1-A}, \tag{14.77}
\end{align}$$

with

$$J_{\text{CP}}^{\text{max}} = \cos \theta_{12} \sin \theta_{12} \cos \theta_{23} \sin \theta_{23} \cos^2 \theta_{13} \sin \theta_{13}, \quad (14.78)$$

and

$$\Delta \equiv \frac{\Delta m_{31}^2 L}{4E_\nu}, \quad A \equiv \frac{2E_\nu V}{\Delta m_{31}^2}, \quad (14.79)$$

where V is the effective matter potential in the Earth’s crust. Results on $\nu_e$ appearance at LBL provide us with the dominant information on leptonic CP violation. Furthermore, $\alpha$, $\Delta$, and A are sensitive to the sign of $\Delta m_{32}^2$ (i.e., the type of the neutrino mass ordering). The plus (minus) sign in Eq. (14.77) applies for neutrinos (antineutrinos), and for antineutrinos $V \to -V$, which implies $A \to -A$. Numerically one finds for a typical Earth crust matter density of 3 g/cm$^3$ that at T2K with $E \sim 0.7$ GeV, matter effects cause $\sim \pm 10\%$ differences in the rates, whereas in NOvA with $E \sim 2$ GeV, we can have $|A| \sim 0.2$. Also, $\alpha^2 \approx 10^{-3}$, which implies that the second term in the first line of Eq. (14.77) gives a very small contribution compared to the other terms. Also, the first term in Eq. (14.77) (which dominates for large $\theta_{13}$) depends on $\sin^2 \theta_{23}$ and therefore is sensitive to the octant.

The $\nu_e$ survival probability relevant for reactor experiments with medium baseline (MBL), $L \sim 1$ km, can be approximated as [162, 166]:

$$P_{\nu_e \to \nu_e} = 1 - \sin^2 2\theta_{13} \sin^2 \frac{\Delta m_{ee}^2 L}{4E_\nu} + \mathcal{O}(\alpha^2), \quad (14.80)$$

where

$$\Delta m_{ee}^2 = \cos^2 \theta_{12} \Delta m_{31}^2 + \sin^2 \theta_{12} \Delta m_{32}^2. \quad (14.81)$$

These MBL reactor experiments provide the most precise determination of $\theta_{13}$. Furthermore there is an additional effect sensitive to the mass ordering when comparing the disappearance of $\nu_\mu$ at LBL experiments – which is symmetric with respect to the sign of $\Delta m_{\mu\mu}^2$ given in Eq. (14.7.1) – with that of $\nu_e$ disappearance at MBL reactors which is symmetric with respect to the slightly different effective mass-squared difference $\Delta m_{ee}^2$ given in Eq. (14.81).

Finally, for atmospheric neutrinos, the fluxes contain $\nu_e$, $\nu_\mu$, $\bar{\nu}_e$, and $\bar{\nu}_\mu$, and for a good fraction of the events, neutrinos travel through the Earth’s matter. In the context of 3ν mixing, the
<table><caption>Table 14.6: Experiments contributing to the present determination of the oscillation parameters.</caption><thead><tr><th>Experiment</th><th>Dominant</th><th>Important</th></tr></thead><tbody><tr><td>Solar Experiments</td><td>θ<sub>12</sub></td><td>Δm<sup>2</sup><sub>21</sub>, θ<sub>13</sub></td></tr><tr><td>Reactor LBL (KamLAND)</td><td>Δm<sup>2</sup><sub>21</sub></td><td>θ<sub>12</sub>, θ<sub>13</sub></td></tr><tr><td>Reactor MBL (Daya-Bay, Reno, D-Chooz)</td><td>θ<sub>13</sub>, |Δm<sup>2</sup><sub>31,32</sub>|</td><td>θ<sub>23</sub>, |Δm<sup>2</sup><sub>31,32</sub>|, θ<sub>13</sub>, δ<sub>CP</sub></td></tr><tr><td>Atmospheric Experiments (SK, IC-DC)</td><td>|Δm<sup>2</sup><sub>31,32</sub>|, θ<sub>23</sub></td><td>θ<sub>13</sub>, θ<sub>23</sub></td></tr><tr><td>Accel LBL ν<sub>μ</sub>, ν̄<sub>μ</sub>, Disapp (K2K, MINOS, T2K, NOνA)</td><td>δ<sub>CP</sub></td><td>θ<sub>13</sub>, θ<sub>23</sub></td></tr><tr><td>Accel LBL ν<sub>e</sub>, ν̄<sub>e</sub> App (MINOS, T2K, NOνA)</td><td></td><td></td></tr></tbody></table>

dominant oscillation channel of atmospheric neutrinos is $\nu_{\mu} \rightarrow \nu_{\tau}$ driven by $|\Delta m_{32}^2|$ with an amplitude controlled by $\theta_{23}$ with sub-leading oscillation modes, triggered by $\Delta m_{21}^2$ and/or $\theta_{13}$, which depend on the octant of $\theta_{23}$, on the mass ordering, and on $\delta_{CP}$. In that respect, an interesting observable is the deviation of e-like events relative to the no-oscillation prediction $N_e^0$, since in the two-flavour limit one expects $N_e = N_e^0$. Such deviation can be written in the following way (see, e.g., [167]):

$$ \frac{N_e}{N_e^0} - 1 \approx (r \sin^2 \theta_{23} - 1) P_{2\nu}(\Delta m_{32}^2, \theta_{13}) \\ + (r \cos^2 \theta_{23} - 1) P_{2\nu}(\Delta m_{21}^2, \theta_{12}) \\ - \sin \theta_{13} \sin 2\theta_{23} r \Re(A_{ee}^* A_{\mu e}). \quad (14.82) $$

Here $r \equiv \Phi_{\mu}/\Phi_e$ is the flux ratio with $r \approx 2$ in the sub-GeV range and $r \approx 2.6 \rightarrow 4.5$ in the multi-GeV range. $P_{2\nu}(\Delta m^2, \theta)$ is an effective two-flavour oscillation probability, and $A_{ee}$, $A_{\mu e}$ are elements of a transition amplitude matrix. The three terms appearing in Eq. (14.82) have a well-defined physical interpretation. The first term is important in the multi-GeV range and is controlled by the mixing angle $\theta_{13}$ in $P_{2\nu}(\Delta m_{32}^2, \theta_{13})$. This probability can be strongly affected by resonant matter effects [168–173]. Depending on the mass ordering, the resonance will occur either for neutrinos or antineutrinos. The second term is important for sub-GeV events, and it takes into account the effect of oscillations due to $\Delta m_{21}^2$ and $\theta_{12}$ [174–177]. Via the pre-factor containing the flux ratio $r$ both, the first and second terms in Eq. (14.82) depend on the octant of $\theta_{23}$, though in opposite directions: the multi-GeV (sub-GeV) excess is suppressed (enhanced) for $\theta_{23} < 45^{\circ}$. Finally, the last term in Eq. (14.82) is an interference term between $\theta_{13}$ and $\Delta m_{21}^2$ amplitudes and this term shows also dependence on the CP phase $\delta_{CP}$ [167,177].

Subdominant three neutrino effects can also affect μ-like events. For example, for multi-GeV muon events one can write the excess in μ-like events as [178,179]

$$ \frac{N_{\mu}}{N_{\mu}^{0}} - 1 \approx \sin^2 \theta_{23} \left( \frac{1}{r} - \sin^2 \theta_{23} \right) P_{2\nu}(\Delta m_{32}^2, \theta_{13}) \\ - \frac{1}{2} \sin^2 2\theta_{23} [1 - \Re(A_{33})]. \quad (14.83) $$

The first term is controlled by $\theta_{13}$ and is subject to resonant matter effects, similar to the first term in Eq. (14.82), though with a different dependence on $\theta_{23}$ and the flux ratio. In the second term, $A_{33}$ is a probability amplitude satisfying $P_{2\nu}(\Delta m_{32}^2, \theta_{13}) = 1 - |A_{33}|^2$. In the limit $\theta_{13} = 0$ we have $\Re(A_{33}) = \cos(\Delta m_{32}^2 L/2E)$, such that the second term in Eq. (14.83) just describes two-flavour $\nu_{\mu} \rightarrow \nu_{\mu}$ vacuum oscillations.

### 14.7.2 3ν Oscillation Analysis

We summarize in Table 14.6 the different experiments which dominantly contribute to the present determination of the different parameters in the chosen convention.

The table illustrates that the determination of the leptonic parameters requires global analyses of the data from the different experiments. Over the years, these analyses have been in the hands of a few phenomenological groups. We show in Table 14.7 the results from the latest analyses in Refs. [180–183]. For the sake of comparison, all results are presented in the convention of the listing section as described above.

The table illustrates the dependence of the present determination of the parameters on variations of the statistical analysis

performed by the different groups and on the data samples included. In that last respect, the main difference resides in the results from Super-Kamiokande atmospheric data [94] which, at present, can only be included in this analysis by directly adding the $\chi^2$ tabulated $\chi^2$ map provided by the experiment.

Altogether the different analyses find consistent results, in particular on the better-known parameters, $\theta_{12}$, $\theta_{13}$ and $\Delta m_{21}^2$ and $|\Delta m_{32}^2|$. The issues which still require clarification are the mass ordering discrimination, the determination of $\theta_{23}$ and the leptonic CP phase $\delta_{CP}$:

* In all analyses the best fit is for the normal mass ordering. Inverted ordering is disfavoured with a $\Delta\chi^2$, which ranges from slightly above $1.5\sigma$ – driven by the interplay of long-baseline accelerator and short-baseline reactor data – to $2.5\sigma$ when adding the atmospheric $\chi^2$ table from Ref. [184].

* There is no statistically significant preference for the octant of $\theta_{23}$ in any of the analysis.

* The best fit for the complex phase in NO is at $\delta_{CP} \sim 200^{\circ}$. CP conservation (for $\delta_{CP} \sim 180^{\circ}$) is allowed at a confidence level (CL) of $1-2\sigma$.

### 14.7.3 Convention-independent Measures of Leptonic CP Violation in 3ν Mixing

In the framework of 3ν mixing leptonic CP violation can also be quantified in terms of the leptonic Jarlskog invariant [185], defined by:

$$ \Im[U_{\alpha i} U_{\alpha j}^* U_{\beta i}^* U_{\beta j}] \equiv \sum_{\gamma=e,\mu,\tau} \sum_{k=1,2,3} J_{CP} \epsilon_{\alpha\beta\gamma} \epsilon_{ijk} \quad (14.84) \\ \equiv J_{CP}^{\max} \sin \delta_{CP}. $$

With the convention in Eq. (14.33) $J_{CP}^{\max}$ is the combination of mixing angles in Eq. (14.78). For example, from the analysis in Ref. [180,181]

$$ J_{CP}^{\max} = 0.0330 \pm 0.0006 (\pm 0.0019), \quad (14.85) $$

at $1\sigma$ ($3\sigma$) for both orderings, and the preference of the present data for non-zero $\delta_{CP}$ implies a small non-zero best fit value $J_{CP}^{\text{best}} = -0.009$.

The status of the determination of leptonic CP violation can also be graphically displayed by projecting the results of the global analysis in terms of leptonic unitarity triangles [186–188]. Since in the analysis $U$ is unitary by construction, any given pair of rows or columns can be used to define a triangle in the complex plane. There are a total of six possible triangles corresponding to the unitary conditions

$$ \sum_{i=1,2,3} U_{\alpha i} U_{\beta i}^* = 0 \text{ with } \alpha \neq \beta, \quad \sum_{\alpha=e,\mu,\tau} U_{\alpha i} U_{\alpha j}^* = 0 \text{ with } i \neq j. \quad (14.86) $$

As an illustration we show in Fig. 14.9 the recasting of the allowed regions of the analysis in Ref. [180,181] in terms of one leptonic unitarity triangle. We show the triangle corresponding to the unitarity conditions on the first and third columns (after the shown rescaling), which is the equivalent to the one usually shown for the quark sector. In this figure, the absence of CP violation would imply a flat triangle, i.e., $\Im(z) = 0$. So the CL at which leptonic CP violation is being observed would be given by the CL at which the region crosses the horizontal axis. Notice however, that this representation is made under the assumption
<table><caption>**Table 14.7:** 3ν oscillation parameters obtained from different global analyses of neutrino data. In all cases, the numbers labeled as NO (IO) are obtained assuming NO (IO), i.e., relative to the respective local minimum. SK-ATM makes reference to the tabulated χ² map from the Super-Kamiokande analysis of their data in Ref. [184].</caption><thead><tr><th rowspan="3">NO</th><th colspan="2">Ref. [181] w/o SK-ATM</th><th colspan="2">Ref. [181] w SK-ATM</th><th colspan="2">Ref. [182] w SK-ATM</th><th colspan="2">Ref. [183] w SK-ATM</th></tr><tr><th colspan="2">Best Fit Ordering</th><th colspan="2">Best Fit Ordering</th><th colspan="2">Best Fit Ordering</th><th colspan="2">Best Fit Ordering</th></tr><tr><th>bfp ±1σ</th><th>3σ range</th><th>bfp ±1σ</th><th>3σ range</th><th>bfp ±1σ</th><th>3σ range</th><th>bfp ±1σ</th><th>3σ range</th></tr></thead><tbody><tr><td>sin<sup>2</sup> θ<sub>12</sub></td><td>3.03<sup>+0.12</sup><sub>-0.11</sub></td><td>2.70 → 3.41</td><td>3.03<sup>+0.12</sup><sub>-0.12</sub></td><td>2.70 → 3.41</td><td>3.03<sup>+0.13</sup><sub>-0.13</sub></td><td>2.63 → 3.45</td><td>3.18<sup>+0.16</sup><sub>-0.16</sub></td><td>2.71 → 3.69</td></tr><tr><td>θ<sub>12</sub>/°</td><td>33.41<sup>+0.75</sup><sub>-0.72</sub></td><td>31.31 → 35.74</td><td>33.41<sup>+0.75</sup><sub>-0.72</sub></td><td>31.31 → 35.74</td><td>33.40<sup>+0.80</sup><sub>-0.82</sub></td><td>30.85 → 35.97</td><td>34.3<sup>+1.0</sup><sub>-1.0</sub></td><td>31.4 → 37.4</td></tr><tr><td>sin<sup>2</sup> θ<sub>23</sub></td><td>5.72<sup>+0.18</sup><sub>-0.23</sub></td><td>4.06 → 6.20</td><td>4.51<sup>+0.19</sup><sub>-0.16</sub></td><td>4.08 → 6.03</td><td>4.55<sup>+0.18</sup><sub>-0.15</sub></td><td>4.16 → 5.99</td><td>5.74<sup>+0.14</sup><sub>-0.14</sub></td><td>4.34 → 6.10</td></tr><tr><td>θ<sub>23</sub>/°</td><td>49.1<sup>+1.0</sup><sub>-1.3</sub></td><td>39.6 → 51.9</td><td>42.2<sup>+1.1</sup><sub>-0.9</sub></td><td>39.7 → 51.0</td><td>42.4<sup>+1.0</sup><sub>-0.9</sub></td><td>40.2 → 50.7</td><td>49.3<sup>+0.8</sup><sub>-0.8</sub></td><td>41.2 → 51.3</td></tr><tr><td>sin<sup>2</sup> θ<sub>13</sub></td><td>2.203<sup>+0.056</sup><sub>-0.059</sub></td><td>2.029 → 2.391</td><td>2.225<sup>+0.056</sup><sub>-0.059</sub></td><td>2.052 → 2.398</td><td>2.23<sup>+0.07</sup><sub>-0.06</sub></td><td>2.04 → 2.44</td><td>2.200<sup>+0.069</sup><sub>-0.062</sub></td><td>2.00 → 2.405</td></tr><tr><td>θ<sub>13</sub>/°</td><td>8.54<sup>+0.11</sup><sub>-0.12</sub></td><td>8.19 → 8.89</td><td>8.58<sup>+0.11</sup><sub>-0.11</sub></td><td>8.23 → 8.91</td><td>8.59<sup>+0.13</sup><sub>-0.12</sub></td><td>8.21 → 8.99</td><td>8.53<sup>+0.13</sup><sub>-0.12</sub></td><td>8.13 → 8.92</td></tr><tr><td>δ<sub>CP</sub>/°</td><td>197<sup>+1.2</sup><sub>-25</sub></td><td>108 → 404</td><td>232<sup>+0.36</sup><sub>-26</sub></td><td>144 → 350</td><td>223<sup>+0.32</sup><sub>-23</sub></td><td>139 → 355</td><td>194<sup>+2.24</sup><sub>-22</sub></td><td>128 → 359</td></tr><tr><td>Δm<sup>2</sup><sub>21</sub></td><td>7.41<sup>+0.21</sup><sub>-0.20</sub></td><td>6.82 → 8.03</td><td>7.41<sup>+0.21</sup><sub>-0.20</sub></td><td>6.82 → 8.03</td><td>7.36<sup>+0.16</sup><sub>-0.15</sub></td><td>6.93 → 7.93</td><td>7.50<sup>+0.22</sup><sub>-0.20</sub></td><td>6.94 → 8.14</td></tr><tr><td>10<sup>-5</sup> eV<sup>2</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δm<sup>2</sup><sub>32</sub></td><td>2.437<sup>+0.028</sup><sub>-0.027</sub></td><td>2.354 → 2.523</td><td>2.433<sup>+0.026</sup><sub>-0.027</sub></td><td>2.353 → 2.516</td><td>2.448<sup>+0.023</sup><sub>-0.031</sub></td><td>2.367 → 2.521</td><td>2.47<sup>+0.02</sup><sub>-0.03</sub></td><td>2.40 → 2.46</td></tr><tr><td>10<sup>-3</sup> eV<sup>2</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>IO</td><td colspan="2">Δχ<sup>2</sup> = 2.3</td><td colspan="2">Δχ<sup>2</sup> = 6.4</td><td colspan="2">Δχ<sup>2</sup> = 6.5</td><td colspan="2">Δχ<sup>2</sup> = 6.4</td></tr><tr><td>sin<sup>2</sup> θ<sub>12</sub></td><td>3.03<sup>+0.12</sup><sub>-0.11</sub></td><td>2.70 → 3.41</td><td>3.03<sup>+0.12</sup><sub>-0.11</sub></td><td>2.70 → 3.41</td><td>3.03<sup>+0.13</sup><sub>-0.13</sub></td><td>2.63 → 3.45</td><td>3.18<sup>+0.16</sup><sub>-0.16</sub></td><td>2.71 → 3.69</td></tr><tr><td>θ<sub>12</sub>/°</td><td>33.41<sup>+0.75</sup><sub>-0.72</sub></td><td>31.31 → 35.74</td><td>33.41<sup>+0.75</sup><sub>-0.72</sub></td><td>31.31 → 35.74</td><td>33.40<sup>+0.80</sup><sub>-0.82</sub></td><td>30.85 → 35.97</td><td>34.3<sup>+1.0</sup><sub>-1.0</sub></td><td>31.4 → 37.4</td></tr><tr><td>sin<sup>2</sup> θ<sub>23</sub></td><td>5.78<sup>+0.16</sup><sub>-0.21</sub></td><td>4.12 → 6.23</td><td>5.69<sup>+0.16</sup><sub>-0.21</sub></td><td>4.12 → 6.13</td><td>5.69<sup>+0.13</sup><sub>-0.21</sub></td><td>4.17 → 6.06</td><td>5.78<sup>+0.10</sup><sub>+0.17</sub></td><td>4.33 → 6.08</td></tr><tr><td>θ<sub>23</sub>/°</td><td>49.5<sup>+0.9</sup><sub>-1.2</sub></td><td>39.9 → 52.1</td><td>49.0<sup>+1.0</sup><sub>-1.2</sub></td><td>39.9 → 51.5</td><td>49.0<sup>+0.7</sup><sub>-1.4</sub></td><td>40.2 → 51.1</td><td>49.5<sup>+0.6</sup><sub>-1.0</sub></td><td>41.2 → 51.2</td></tr><tr><td>sin<sup>2</sup> θ<sub>13</sub></td><td>2.219<sup>+0.060</sup><sub>-0.057</sub></td><td>2.047 → 2.396</td><td>2.223<sup>+0.058</sup><sub>-0.058</sub></td><td>2.048 → 2.416</td><td>2.23<sup>+0.06</sup><sub>-0.06</sub></td><td>2.03 → 2.45</td><td>2.225<sup>+0.064</sup><sub>-0.070</sub></td><td>2.02 → 2.42</td></tr><tr><td>θ<sub>13</sub>/°</td><td>8.57<sup>+0.12</sup><sub>-0.11</sub></td><td>8.23 → 8.90</td><td>8.57<sup>+0.11</sup><sub>-0.11</sub></td><td>8.23 → 8.94</td><td>8.59<sup>+0.13</sup><sub>-0.12</sub></td><td>8.19 → 9.00</td><td>8.58<sup>+0.12</sup><sub>-0.14</sub></td><td>8.17 → 8.96</td></tr><tr><td>δ<sub>CP</sub>/°</td><td>286<sup>+2.7</sup><sub>-32</sub></td><td>192 → 360</td><td>276<sup>+2.2</sup><sub>-29</sub></td><td>194 → 344</td><td>274<sup>+2.2</sup><sub>-27</sub></td><td>193 → 342</td><td>284<sup>+2.6</sup><sub>-28</sub></td><td>200 → 353</td></tr><tr><td>Δm<sup>2</sup><sub>21</sub></td><td>7.41<sup>+0.21</sup><sub>-0.20</sub></td><td>6.82 → 8.03</td><td>7.41<sup>+0.21</sup><sub>-0.20</sub></td><td>6.82 → 8.03</td><td>7.36<sup>+0.16</sup><sub>-0.15</sub></td><td>6.93 → 7.93</td><td>7.50<sup>+0.22</sup><sub>-0.20</sub></td><td>6.94 → 8.14</td></tr><tr><td>10<sup>-5</sup> eV<sup>2</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δm<sup>2</sup><sub>32</sub></td><td>-2.498<sup>+0.032</sup><sub>-0.025</sub></td><td>-2.581 → -2.408</td><td>-2.486<sup>+0.028</sup><sub>-0.025</sub></td><td>-2.570 → -2.406</td><td>-2.492<sup>+0.025</sup><sub>-0.030</sub></td><td>-2.578 → -2.413</td><td>-2.52<sup>+0.03</sup><sub>-0.02</sub></td><td>-2.60 → -2.44</td></tr><tr><td>10<sup>-3</sup> eV<sup>2</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

of a unitary U matrix and therefore does not provide any test of unitarity in the leptonic sector.

<figure><img src="image_59.png" /><figcaption>**Figure 14.9:** Leptonic unitarity triangle for the first and third columns of the mixing matrix. After scaling and rotating the triangle so that two of its vertices always coincide with (0, 0) and (1, 0) the figure shows the 1σ, 90%, 2σ, 99%, 3σ CL (2 dof) allowed regions of the third vertex for the NO from the analysis in Ref. [180, 181].</figcaption></figure>

## 14.8 Beyond 3ν: Additional Neutrinos at the eV Scale

Besides the huge success of three-flavour oscillations described in Sec.14.7, as mentioned in Secs.14.6.3 and 14.6.4, there are some anomalies which cannot be explained within the 3ν framework and which have been interpreted as hints for the existence of additional neutrino states with masses at the eV scale. In brief:

* the LSND experiment [123] reported evidence for $\bar{\nu}_{\mu} \to \bar{\nu}_{e}$ transitions with $E/L \sim 1$ eV$^2$, where E and L are the neu-

trino energy and the distance between source and detector, respectively (see Short Baseline Experiments subsection of Sec.14.6.3).

* this effect was further explored by the MiniBooNE experiment [189], which reported a yet unexplained event excess in the low-energy region of the electron neutrino and antineutrino event spectra. No significant excess is found at higher neutrino energies. Interpreting the data in terms of oscillations, parameter values consistent with the ones from LSND were obtained, though the test was not definitive;

* radioactive source experiments at the Gallium solar neutrino experiments both in SAGE and GALLEX/GNO obtained an event rate that is somewhat lower than expected. This result was recently confirmed by the BEST experiment [190]. If not due to uncertainties in the interaction cross-section, this effect by itself can be explained by the hypothesis of $\nu_e$ disappearance due to oscillations with $\Delta m^2 \gtrsim 1$ eV$^2$ ("Gallium anomaly") [191–193];

* the calculations of the neutrino flux emitted by nuclear reactors in Refs. [134, 135] predicted a neutrino rate that is a few percent higher than observed in short-baseline ($L \lesssim 100$ m) reactor experiments. This "reactor anomaly" could be explained by assuming $\bar{\nu}_e$ disappearance due to oscillations with $\Delta m^2 \sim 1$ eV$^2$ [194]. The flux predictions from [134, 135] were not supported by the reactor antineutrino flux measurement at Daya Bay [195, 196] which was consistent with the older (lower) flux predictions and the flux measurement results in the short-baseline reactor neutrino oscillation experiments. In the last years the reactor flux calculations have been revisited [197–199] and in most cases lower fluxes are predicted, in agreement with the flux measurement at Daya Bay. These new fluxes, correspondingly reduce the significance of the anomaly [200].