Comments on the experimental situation

The question if the process with emission of one or two photons dominates might be an impor- tant one from the experimental point of view, even if it may never be actually observed. Let

Element Schr¨odinger Dirac Ar-36 10.1 10.3 Ca-40 0.078 0.068 Cd-108 0.0011 0.00033 Gd-152 2.9· 10−7 2.2· 10−6 Dy-158 1.2· 10−4 3.0· 10−5 Er-164 4.7· 10−7 3.7· 10−9 W-180 1.2· 10−6 5.4· 10−7

Table 4.8: The ratios of the rate for ECECγγ to the rate for ECECγ. As argued in Sec. 4.3, this ratio should be independent of the NME. Furthermore, the dependences on GF, gA, R,

and |mee| cancel out, and hence this ratio is also independent of uncertainties related to those

quantities (the most dramatic of which arise from|mee|). Note that for ECECγ, we have always

used the exact calculation (without long wavelength approximation), since the approximation is expected to break down for even lower Q-values in that case.

us briefly discuss a recent measurement on the half-life of Ar-36 with respect to neutrino-less double electron capture [116, 117]. In this measurement, no photon has been seen above the background in the region of interest. The limit that has been obtained is

T1/2(0+→ 0+)¯¯_1γ ≥ 1.85 · 1018 y at 68% C.L. (4.154) This limit is directly proportional to the full energy peak detection efficiency ², which is, at the position of the peak, equal to 0.26%.

Let us do an easy estimation of the limit that could be obtained for the dominance of the 2γ-process. According to Tab. 4.8, this mode is indeed dominant by a factor of about 10. We will assume here for simplicity that this is the only possible decay mode. If ² is the probability to see a photon, the probability to miss it is (1− ²). For dominance of the 2γ-mode, the probability to see one photon is even a bit higher, because of the lower photon energy [116], but let us assume for simplicity that it is still equal to ². Then, the probability to see anything if two photons are emitted is given by 2²(1− ²) + ²2 ≈ 2² instead of ². This would increase the limit from Eq. (4.154) by a factor of 2. If the efficiency also increases to ²0, which is roughly

3.5

2.6², then the increase would even be a factor of roughly 3.

In any case, it will be useful to know the dominant decay mode well. Note, however, that these considerations will change if some other process than the 2-nucleon mechanism dominates the decay rate. Unfortunately, pinning down the exact mechanism will involve measurements for different isotopes and will be a major task in the future research on double β processes [118]. In general, an experimental detection of neutrino-less double electron capture will be very tough (if not impossible), unless some more exotic mechanism is involved that causes the decay (which might well be). Nevertheless, experiments in this direction have been done and are going on (see, for example, Refs. [119–123]), and current best limits for the half-life are around 1020years [124], which may further improve in the next years.

Lepton Flavour Violation

The last topic that we want to discuss is Lepton Flavour Violation (LFV). The term flavour essentially means the generation of a fermion. In the SM, the up-like quarks appear in three flavours, (u, c, t), exactly as the down-like quarks, (d, s, b). The same is true for the charged leptons (e, µ, τ ), as well as for the neutrinos (νe, νµ, ντ). From the experimental side, it seems

to be pretty clear that fermions indeed appear in three generations, whereas from the theory side a reason for that is still lacking [64].

In the neutrino sector, lepton flavour violating processes are well-known and are called

neutrino oscillations [24], cf. Chapter 2. These transitions like νe ↔ νµ can indeed, e.g.,

transform a state of electron-flavour into one with muon-flavour. So far, so good. The amazing point is that this flavour change does not change the charge of the particle, but rather only its flavour. In the charged current sector, flavour changing processes are well-known [125], but in the neutral current sector, the SM actually does not provide any flavour changing neutral current (FCNC) interaction at tree-level.

Even more amazing, there is actually no deeper reason for that in the SM! The absence of FCNCs is only a so-called accidental symmetry: When we take the SM gauge group and particle content and impose constraints like Lorentz or gauge invariance, we automatically end up with flavour conserving neutral currents only. This is also confirmed by experiments: A decay like, e.g., µ → eγ would be perfectly allowed by energy, momentum, and angular momentum conservation, but nevertheless we have not observed it yet (the current best limit for the branching ratio of this decay compared to ordinary muon decay µ− → e−νµνe comes

from the past MEGA-experiment [126], and amounts to Br(µ → eγ) < 1.2 · 10−11). This branching ratio will be probed by the upcoming MEG experiment that is expected to reach a sensitivity of 1.2· 10−13at 90% C.L. and a single event sensitivity of even 3.7· 10−14[127].

Since there is no reason for the absence of LFV, models beyond the SM will generically violate lepton flavour [9]. In the SM with massive neutrinos, one can actually draw a 1-loop diagram for µ→ eγ (cf. Fig. 5.1), but even an optimistic prediction (with rather large values for the neutrino masses) will only lead to a branching ratio of about 10−47 [128].1 In turn, if we can observe LFV in the near future, this will be an unambiguous signal of Physics beyond the Standard Model (BSM) and will be a major discovery.

5.1

The rare decay µ

→ eγ and other lepton flavour violating

processes

Let us now discuss the diagram in Fig. 5.1 for µ → eγ a bit closer, before we turn to other LFV-processes. An extensive treatment of this process in the SM with massive neutrinos (and

1_{The reason why this value is so tiny will become clear in a moment.}

ΜHpL eHp-kL ΓHkL W W Νi UΜi Uei*

Figure 5.1: The diagram for µ → eγ in the SM with massive neutrinos. It is a higher order process, which is additionally suppressed by the GIM-mechanism. The flavour violation happens on the neutrino line, since the neutrino mass eigenstates are no flavour eigenstates.

also beyond) can be found in Refs. [128–130]. Furthermore, there exists an excellent paper on the general process f1→ f2γ [131], which can also be used, e.g., for the calculation of an electric dipole moment of the neutrino. We will here only mention the most important points in the calculation of µ→ eγ.

First, when making a general ansatz for the amplitude, one can immediately see by applying electromagnetic gauge invariance as well as by the properties of a physical photon that the resulting transition amplitude is of magnetic type:

M(µ → eγ) = ²∗

µ(k)e(p− k) [ikνσµν(A + Bγ5)] µ(p), (5.1) with some functions A and B. In the approximation me≈ 0 (which is always fine for the above

process), one additionally obtains A = B and the amplitude can be written as

M(µ → eγ) = Ae(p − k) [2(p²) − mµ/] µ(p),² (5.2)

where we have used the Gordon decomposition and the Dirac equation. As will become impor- tant later, there will always be a chirality flip somewhere on the fermion line.2

The most important point in the calculation for the case of the SM with massive neutrinos is that terms of the following form come in:

3 X i=1 U_ei∗Uµif µ m2_i M_W2 ¶ , (5.3)

where Uαj are elements of the PMNS-matrix (cf. Eq. (4.4)), mi is the mass of the virtual

neutrino mass eigenstate νi, MW is the W -boson mass, and f is some loop function that arises

in the computation. They key point is that in the SM we have mi ¿ MW, and we can hence

expand the function f as

f µ m2_i MW ¶ = f (0) + f0(0)· m 2 i M2 W + ..., (5.4)

where the first term is independent of mi. Then, in Eq. (5.3), this leading term will be killed

by the unitarity of the PMNS-matrix U and we end up with a suppressed amplitude. This theorem is commonly known as Glashow-Iliopoulos-Maiani- (GIM-) mechanism [132].

Note that this logic can also be turned around: By a non-observation of µ→ eγ (and similar processes), one can also obtain limits on the unitarity of the PMNS-matrix [133].

The final result for the decay rate in the SM with massive neutrinos can be written as

Γ(µ→ eγ) = m 3 µ 8π ¡ |A|2_+|B|2¢_{, with A = B =} eg2mµ 256π2_M2 W · Ã ₃ X i=1 U_ei∗Uµi m2 i M_W2 ! . (5.5)

Relating this to the muon decay rate for ordinary muon decay, Γ(e−νµνe) =

m5_µG2_F

192π3, (5.6)

which is essentially equal to the total decay rate of the muon, leads to a branching ratio of

Br(µ→ eγ) = 3α 32π Ã ₃ X i=1 U_ei∗Uµi m2_i M2 W !2 . (5.7)

Of course, there can also be LFV-processes different from µ → eγ. The τ-lepton might undergo similar decays, τ → µγ or τ → eγ, where the current best limits are Br(τ → µγ) < 4.5·10−8at 90% C.L. (BELLE experiment, Ref. [134]) and Br(τ → eγ) < 1.1·10−7at 90% C.L. (Babar experiment, Ref. [135]), respectively. Further possibilities are, e.g., µ+→ e+e−e+ (with Br(µ→ 3e) < 1.0 · 10−12, SINDRUM experiment, Ref. [136]) or µ-e conversion on nuclei (with Br(µTi→ eTi) < 4.3 · 10−12, SINDRUM II experiment, Ref. [137]). Numerous other processes and limits can be found in Refs. [138] and [139].