Tests of Conservation Laws

1

Tests of Conservation Laws
Revised April 2024 by A. Pich (IFIC, Valencia) and M. Ramsey-Musolf (Tsung-Dao Lee Inst.;
SJTU; U. Massachusetts).
In keeping with the current interest in tests of conservation laws, we collect together a Table ofexperimental limits on all weak and electromagnetic decays, mass differences, and moments, andon a few reactions, whose observation would violate conservation laws. The Table is given only inthe full Review of Particle Physics (RPP), not in the Particle Physics Booklet, and organizes thedata in two main sections: “Discrete Space-Time Symmetries”, i.e., C, P , T , CP and CP T ; and“Number Conservation Laws”, i.e., lepton, baryon, flavor and charge conservation. The references
for these data can be found in the Particle Listings. The following text discusses the best limits
among those included in the Table and gives a brief overview of the current status. For some topics,a more extensive discussion of the framework for theoretical interpretation is provided, particularly
where the analogous discussion does not appear elsewhere in the RPP. References to more extensive
review articles are also included where appropriate. Unless otherwise specified, all limits quoted inthis review are given at a C.L. of 90%.

DISCRETE SPACE-TIME SYMMETRIESCharge conjugation (C), parity (P ) and time reversal (T ) are empirically exact symmetries ofthe electromagnetic (QED) and strong (QCD) interactions, but they are violated by the weak forces.
Owing to the left-handed nature of the SU (2)L ⊗U (1)Y electroweak theory, C and P are maximallyviolated in the fermionic couplings of the W ± and (up to sin2 θW corrections) the Z. However,their product CP is still an exact symmetry when only one or two fermion families are considered.
With three generations of fermions, CP is violated through the single complex phase present in the
Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. An analogous CP -violating (CPV)phase appears in the lepton sector when non-vanishing neutrino masses are taken into account(plus two additional phases if neutrinos are Majorana particles). The product of the three discrete
symmetries, CP T , is an exact symmetry of any local and Lorentz-invariant quantum field theory
with a positive-definite hermitian Hamiltonian that preserves micro-causality [1, 2]. Therefore, thebreaking of CP implies a corresponding violation of T .
Violations of charge-conjugation symmetry have never been observed in electromagnetic andstrong phenomena. The most stringent limits are extracted from C-violating transitions of neutral(self-conjugate) particles such as Br(π 0 → 3γ) < 3.1 × 10−8 [3] and Br(J/ψ → 2γ) < 2.7 × 10−7[4]. P (and CP ) conservation has been also precisely tested through forbidden decays such as
Br(η → 4π 0 ) < 6.9 × 10−7 [5], but the best limits on P and T are set by the non-observationof electric dipole moments (see section 2). Obviously, the interplay of the weak interaction puts
a lower bound in sensitivity for this type of tests, beyond which violations of the corresponding
conservation laws should be detected.

1 Violations of CP and T
The first evidence of CP non-invariance in particle physics was the observation in 1964 of
KL0 → π + π − decays [6]. For many years afterwards, the non-zero ratio|η+− | ≡ |M(KL0 → π + π − )/M(KS0 → π + π − )| = (2.232 ± 0.011) × 10−3

(1)

could be explained as a K 0 –K 0 mixing effect, η+− =  (superweak CP violation), which would
imply an identical ratio η00 ≡ M(KL0 → π 0 π 0 )/M(KS0 → π 0 π 0 ) in the neutral decay mode and
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:13am

Tests of Conservation Laws2

successfully predicts the observed CP V semileptonic asymmetry (AL (e) ≈ 2 Re )
AL (e) ≡Γ (KL0 → π − e+ νe ) − Γ (KL0 → π + e− ν̄e )= (3.34 ± 0.07) × 10−3 .
Γ (KL0 → π − e+ νe ) + Γ (KL0 → π + e− ν̄e )(2)

A tiny difference between η+− and η00 was reported for the first time in 1988 by the CERN NA31
collaboration [7], and later established at the 7.2σ level with the full data samples from the NA31 [8],E731 [9], NA48 [10] and KTeV [11] experiments:
Re(0 /) =1(1 − |η00 /η+− |) = (1.66 ± 0.23) × 10−3 .
3(3)This important measurement confirmed that CP violation is associated with a ∆S = 1 transition,
as predicted by the CKM mechanism. The Standard Model (SM) prediction, Re(0 /) = (1.4±0.5)×
10−3 [12–14], is in good agreement with the measured ratio, although the theoretical uncertainty
is unfortunately large.
Much larger CP asymmetries have been later measured in B meson decays, many of them
involving the interference between B 0 –B 0 mixing and the decay amplitude. They provide manysuccessful tests of the CKM unitarity structure, validating the SM mechanism of CP violation (seethe review on CP violation in the quark sector). Prominent signals of direct CP violation in the
decay amplitudes have been also clearly established in several B ± , Bd0 and Bs0 decays, and, very
recently, in charm decays [15, 16]:
dir
0
+ −
dir
0
+ −
−4
∆adir
.
CP ≡ aCP (D → K K ) − aCP (D → π π ) = (−15.7 ± 2.9) × 10

(4a)

0
+ −
−4
adir
.
CP (D → π π ) = (23.2 ± 6.1) × 10

(4b)These direct CP asymmetries necessarily involve the presence of a strong phase-shift difference
between (at least) two interfering amplitudes, which makes very challenging to perform reliable SM
predictions for heavy-flavored mesons.
Global fits to neutrino oscillation data provide some hints of a non-zero mixing phase [17, 18].
Although the statistical significance is not yet compelling, they suggest that CP -violation effects inneutrino oscillations could be large (see the review on neutrino masses, mixings and oscillations).
The future DUNE and Hyper-Kamiokande experiments are expected to confirm the presence of CP
violation in the lepton sector or constrain the phase in the leptonic mixing matrix to be smaller
than O(10◦ ).
While CP violation implies a breaking of time-reversal symmetry, direct tests of T violation aremuch more difficult. The CPLEAR experiment observed longtime ago a non-zero difference between
the oscillation probabilities of K 0 → K 0 and K 0 → K 0 [19]. Initial neutral kaons with defined
strangeness were produced from proton-antiproton annihilations at rest, pp̄ → K − π + K 0 , K + π − K 0 ,and tagged by the accompanying charged kaon, while the strangeness of the final neutral kaon was
identified through its semileptonic decay: K 0 → e+ π − νe , K 0 → e− π + ν̄e . The average asymmetry
over the time interval from 1 to 20 KS0 lifetimes was found to be different from zero at 4σ [19]:R[K 0 (t = 0) → e+ π − νe (t)] − R[K 0 (t = 0) → e− π + ν̄e (t)]= (6.6 ± 1.3 ± 1.0) × 10−3 .
R[K 0 (t = 0) → e+ π − νe (t)] + R[K 0 (t = 0) → e− π + ν̄e (t)](5)

Since this asymmetry violates also CP , its interpretation as direct evidence of T violation requires
a detailed analysis of the underlying K 0 –K 0 mixing process [20–22].
More recently, the exchange of initial and final states has been made possible in B decays, taking
advantage of the entanglement of the two daughter mesons produced in the decay Υ (4S) → B B̄
31st May, 2024

respectively, equivalent to 17σ and 0.3σ, consistent withCP violation and CP T invariance.
Tests of Conservation Laws

3

∗

∆CT±

1

0.5

0

-0.5

-1

-1

0

1

∆S±T

Figure 1: Measured values of ∆ST+ , ∆CT+ (blue point, dashed lines) and ∆ST− , ∆CT− (red square,solid lines) [26]. The two-dimensional contours correspond to 1 − CL = 0.317, 4.55 × 10−2 , 2.70 ×
FIG. 3: (color online). The central values (blue point
10−3 , 6.33 × 10−5 , 5.73 × 10−7 , and 1.97 × 10−9 . The + sign indicates the T -invariant point.

and red square) and two-dimensional CL contours for 1 −−3
CL allows= 0.317,4.55(B×0 →
10`−2
6.33
× 100 −5 , 5.73 ×+ X,, B2.70
0 → `−×0
which
for both flavorX) 10
and CP, (B+ → J/ψKL , B− → J/ψKS )−7−9tagging.
one B candidate
is reconstructed
CP eigenstateand inthe flavor
10 ,Selecting
and events
1.97where× 10, calculated
fromin athechangetheof the other B is identified, one can compare the rates of the B 0 → B± and B 0 → B± transitions
value of −2∆ ln L compared with its value at maximum
with their T -reversed B± → B 0 and B± → B 0 processes, as a function of the time difference∆tbetweenlnthe
decays6.2,[23–25].
Neglecting
the small
width difference
betweenthe two(−2∆
L two=B 2.3,
11.8,
19.3, 28.7,40.1),
for thepairs
of+ has a time-dependent+
Bd0 mass eigenstates, each of these eight transitions
decay rate of the form
T -asymmetry parameters (∆ST ,where∆CΓT is) the(blue
dashed curves)e−Γ ∆t {1 + Sα± β sin (∆md ∆t) + Cα± β cos (∆md ∆t)},average decay width, ∆md thed−−0(∆ST , ∆C) (redαsolid
Band
the T
subindices= `+ , `−curves).
and β = KS0Systematic, KL0 stand for theuncertainties
reconstructed final
d mass difference,statesofthetwoB
mesonsandthesuperindex+
or−indicates
whetherthedecay
the flavor
are included. The T -invariance point is shown as a +tosign.
d,,final state α occurs before or after the decay to the CP final state β. Figure 1 shows confidence-level
contours for the T -asymmetry parameters ∆ST± ≡ S`∓− ,K 0 − S`±+ ,K 0 and ∆CT± ≡ C`∓− ,K 0 − C`±+ ,K 0 ,
L

S

L

S

reported by the BABAR experiment [26], which clearly demonstrate a violation of T in ∆ST± , witha significance of 14σ.

2 Electric dipole moments
Among the most powerful tests of CP invariance is the search for a permanent electric dipole
moment (EDM) of an elementary fermion or non-degenerate quantum system. The EDM of anelementary spin-1/2 fermion f is defined by the effective, non-renormalizable interaction

In summary, we have measured T -violating parametersiµν
in the time evolutionLEDM
of =neutral− df f¯σµν γB(6)5 f Fmesons, by comparing2
the µνprobabilities of B 0 → B− , B+ → B 0 , B 0 → B+ ,where F is the QED field strength tensor. The values for df are conventionally expressed in units
B−interaction→ B(6)0 istransitions,
their
T non-relativistic
conjugate.
We
ofande cm. The
separately odd underto
T and
P . In thelimit, Eq.
(6)determine for the main31stTMay,-violating
parameters ∆ST+ =2024−1.37 ± 0.14 (stat.) ± 0.06 (syst.) and ∆ST− = 1.17 ±0.18 (stat.) ± 0.11 (syst.), and observe directly for the

N
A
†
A
P
‡
N
3
§
D
¶
N
3
∗∗
A
[1] J
[2] B
8
P
[3] B
9
P
[4] N
M
6
[5] G
V
t
t
[6] R
H
g
[7] S
[8] K
0
[9] A
L
[10] P[11] L[12] L[13] HrTests of Conservation Laws

4
reduces to~
LEDM → df χ†f ~σ χf · E(7)~ is the electric field. Note the interaction (7) is
where χ is a two-component Pauli spinor and E
manifestly T -odd and carries no direct information on CP . The observation of a non-zero EDM
of a non-relativistic (and non-degenerate) quantum system, such as the mercury atom (see below)
would imply CP violation under the assumption of CP T invariance.
To date, no experimental observation of an EDM of an elementary particle or non-degeneratebound quantum system has been observed. The most stringent limits have been obtained for the
EDMs of the electron, mercury atom, and neutron. A selection of the representative, most stringent
limits is given in Table 1. The limits on the electron EDM are inferred from experiments involving
polar molecules, paramagnetic systems with an unpaired electron spin. In contrast, the neutronand 199 Hg atom are diamagnetic. A variety of experimental efforts aimed at improved sensitivities
are underway. For reviews of the experimental and theoretical situation, see, e.g. [27–30].
Table 1: Most stringent limits on electric dipole moments.
EDM
Electron
Muon
Neutron
199 Hg Atom
129 Xe Atom

Limit (e cm)
1.1 × 10−29 (90% C.L.)
4.1 × 10−30 (90% C.L.)
1.8 × 10−19 (95% C.L.)
1.8 × 10−26 (90% C.L.)
7.4 × 10−30 (95% C.L.)
1.5 × 10−27 (95% C.L.)

SourceThO [31]
HfF+ [32]
[33]
[34]
[35]
[36]

EDMs in the Standard Model
The SM provides two sources of df : the CP V phase in the CKM matrix and the P - and T -odd
‘θ term’ in the QCD Lagrangian. The former is characterized by the Jarlskog invariant [37]∗J = Im(Vus Vcs∗ Vcb Vub) ∼ A2 λ6 η < 10−4 ,(8)while the latter is given byLθ̄ = −



g32
µν
θ̄
Tr
G
G̃
,
µν
16π 2

(9)

where Gµν (G̃µν = µναβ Gαβ /2) is the QCD field strength tensor (dual).
The CKM-induced EDMs of quarks and charged leptons arise at three- and four-loop orders,
respectively [38–41]. The resulting numerical impact for the experimental observables (see below)
falls well below present and prospective experimental sensitivities. The most important impact ofJ for the EDMs of the neutron and diamagnetic atoms arise via induced hadronic interactions.
The resulting theoretical expectations for the electron, neutron and 199 Hg EDMs are

|dA (

199

|de |CKM ≈ 10−44 e cm [41],(10a)|dn |CKM ≈ (1 − 6) × 10−32 e cm [42],

(10b)

−34

(10c)

Hg)|CKM . 4 × 10

e cm [27].

For dn and dA (199 Hg), the dominant CKM contributions arise from four-quark operators (generated
after integrating out the electroweak gauge bosons) rather than from the EDMs of the individual
31st May, 2024

Tests of Conservation Laws

5quarks. The corresponding sensitivities to the QCD θ̄ parameter are given by|dn |θ̄ ≈ (0.9 − 1.2) × 10−16 θ̄ e cm [42],
199

|dA (

−20

Hg)|θ̄ ≈ (0.07 − 8) × 10

θ̄ e cm [27, 28],

(11a)
(11b)

where the ranges quoted include the impacts of hadronic, nuclear, and atomic theory uncertainties.
The neutron EDM puts then a stringent limit on ‘strong’ CP violation: θ̄ . 2 × 10−10 . The
corresponding limit from dA (199 Hg) is weaker due to the large theoretical uncertainty.
EDMs Beyond the Standard Model
It is possible that the next generation of EDM searches will yield a non-zero result, arisingfrom the θ̄-term interaction and/or physics beyond the SM (BSM). Most of the considered BSM
scenarios involve new particles with masses well above the electroweak scale. At energies much
lower than the BSM mass scale Λ, the dynamics can be described through an effective field theory(d)(SMEFT) involving an infinite set of non-renormalizable operators Ok , with dimensions d > 4,
that are invariant under the SM gauge group:
LSMEFT = LSM +

X

(d)



αk

k,d

1
Λ

d−4

(d)

Ok .

(12)

The operators contain only SM fields, while all short-distance information on the BSM physics is(d)encoded in their Wilson coefficients αk . The d = 4 term corresponds to the SM Lagrangian.
For the systems of Table 1 and for many BSM scenarios of recent interest, it suffices to consider
the leading contributions from d = 6 operators. Considering only the first-generation SM fermions,
there exist 12 independent CP V pertinent operators. For a complete listing, see e.g., Refs. [28,43].
For a given elementary fermion f , two of these operators reduce to the EDM interaction in Eq. (6).
Of the remaining, the most relevant include the chromo-electric dipole moments (cEDMs) of thequarks; a CP -odd three gluon operator; three semileptonic, four-fermion operators; two four-quark
operators; and a CP V interaction involving two Higgs fields and a right-handed quark current. For(6)the dipole operators, it is useful to define a rescaled Wilson coefficient αf Vj ≡ gj Cf Vj , where Vj(j = 1, 2, 3) denote the gauge bosons for the three SM gauge groups with corresponding couplings(6)
gj ; for all other d = 6 operators we correspondingly identify αk ≡ Ck . In this case, one has for
the EDM (df ) and cEDM (d˜q )df = −(1.13 × 10−16 e cm)





v
Λ

d˜q = −(1.13 × 10−16 cm)

v
Λ

2

Im Cf γ ,

(13a)

2

Im CqG ,(13b)with Im Cf γ = Im Cf B + 3I3f Im Cf W . As the expressions (13a,13b) illustrate, the magnitude of
the BSM contributions scales with two inverse powers of the scale Λ. A similar conclusion holds
for the contributions from the other d = 6 operators to the EDMs of Table 1.
It is important to emphasize that if the BSM mediators are light, with masses below the weak
scale, the effective field theory description of Eq. (12) does not apply. For recent studies along
these lines, see, e.g. [44, 45].
EDM Interpretation: From Short Distances to the Atomic Scale
The EDM limits in Table 1 are obtained using composite quantum systems, wherein the relevant
dynamics involve physics at the hadronic, nuclear, atomic and molecular scales. The manifestation
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Tests of Conservation Laws

6

Table 2: Pertinent dimension-six EDM and cEDM sources (first generation fermions only).
SystemParamagnetic
Diamagnetic∗

d = 6 Source
Electron EDM
Electron-quark
Quark EDM
Quark cEDM
Three gluon
Four quark
Quark-Higgs
Electron-quark tensor∗

Wilson Coefficient
Im Ceγ(±)
Ceq
Im CqγIm CqGCG̃
(1,8)
Im Cquqd
Im Cϕud(3)
Im C`equApplicable only to atoms.

of a given CP V source (CKM, θ̄ term, BSM) involves an interplay of these dynamics. In all cases,
one must first evolve the Wilson coefficients from the weak scale to the hadronic scale, then match
onto the relevant low-energy degrees of freedom (electrons, nucleons, pions, etc.). At this level,
the most straightforward interpretation involves the paramagnetic systems, for which two sources
dominate: the electron EDM and the electron spin-dependent semileptonic interaction ēγ5 e q̄q. The
latter gives rise to an spin-independent Hamiltonian, for an atom with Z electrons/protons and N
neutrons,
h
i
iGF
(0)
(1)
ĤS = √ δ(~r) (Z + N ) CS + (Z − N ) CS γ0 γ5 ,
(14)
2
(0)

(1)

(+)

(−)

(0,1)

where CS (CS ) is proportional to Ceq (Ceq ). The computation of CS
is relatively free fromtheoretical uncertainty since the operator q̄q essentially counts the number of quarks of flavor q in
the nucleus. Experimental results for paramagnetic systems, thus, often quote bounds on(0)CS ≡ CS +



Z −N
Z +N



(1)

CS

(15)

as well as on de , assuming only one of these two sources is non-vanishing. Combining results fromThO and HfF+ (see Figure 2) allows one to obtain the global, 90% C.L. bounds|de | < 2.1 × 10−29 e cm ,|CS | < 1.9 × 10−9 .

(16)Note that the limits on de given in Table 1 have been obtained assuming CS = 0.
For the diamagnetic systems, the situation is considerably more involved. For the neutron, avariety of approaches – including lattice QCD, chiral perturbation theory, QCD sum rules, andthe quark model – have been employed to compute the relevant hadronic matrix elements of the
CP V sources (see, e.g., [28,29,47,48]). For diamagnetic atoms, the non-leptonic sources of Table 2
give rise to the EDM of the nucleus as well as other P - and T -odd nuclear moments, as allowedby the nuclear spin. However, according to a theorem by Schiff [49], the nuclear EDM generates
no contribution to the neutral-atom EDM due to screening by atomic electrons. The leading~ an r3 -weighted
contribution from these sources, instead, arises via the nuclear Schiff moment, S,
moment of the T - and P -odd component of the nuclear charge density. The resulting effective
atomic Hamiltonian is~ e (0) · S~,
ĤSchiff = −4π ∇ρ(17)31st May, 2024

7

Tests of Conservation Laws

Figure 2: Constraints on de and CS from EDM searches using polar molecules [32]. The green
and grey bands are the 90% confidence regions from HfF+ and ThO, while the ellipse shows the
90% CL limits from the global fit. Parameters used in fits are from Ref. [46].
~ e (0) is the gradient of the electron density at the nucleus. To date, computations of thewhere ∇ρ
nuclear Schiff moment have assumed that the leading contribution arises from a pion-exchange
induced nuclear force, with the P - and T -odd πN interaction given by
h

i

(0)
τ · ~π + ḡπ(1) π 0 + ḡπ(2) (3τ3 π 0 − ~τ · ~π ) N .
LT,P
πN = N̄ ḡπ ~

(18)
(2)

Chiral effective field theory power counting implies that in general the magnitude of ḡπ is suppressed with respect to the isoscalar and isovector couplings. The CP V sources then generate adiamagnetic atom EDM dA via the sequence~ −→ dA .
CPV source −→ ḡπ(i) −→ S

(19)

The steps in this sequence involve dynamics at the hadronic, nuclear, and atomic scales, respectively.
In addition, dA may receive contributions from the nuclear spin-dependent interaction generated by
the semileptonic tensor interaction listed in Table 2, with the corresponding atomic HamiltonianiX h (0)2iGF(1)ĤT = √ δ(~r)
CT + CT τ3 ~σN · ~γ ,
2
N
(0,1)

(3)

(20)

where σN is the nucleon spin Pauli matrix and CT∝ Im C`equ .
Given the large number of CP V sources and existing diagmagnetic EDM limits, it is not possible
to obtain a set of global constraints on the former. One may, however, do so for the low-energy
(0,1)
(0,1)
effective parameters ḡπ , CTand d¯srn , where the latter denotes a ‘short-range’ contribution to
the neutron EDM [27, 50]. In this context, the dominant source of theoretical uncertainty involves
31st May, 2024

Tests of Conservation Laws

8

computations of the nuclear Schiff moment. From the bounds on the low-energy parameters, one
may then derive constraints on the CPV sources by utilizing computations of the hadronic matrix
elements. Reducing the degree of theoretical hadronic and nuclear physics uncertainty is an areaof active effort.

3 Tests of CP T
CP T symmetry implies the equality of the masses and widths of a particle and its antiparticle.
The most constraining limits are extracted from the neutral kaons [51, 52]:2

|mK 0 − mK 0 |< 6 × 10−19 ,(mK 0 + mK 0 )2|ΓK 0 − ΓK 0 |= (8 ± 8) × 10−18 .
(ΓK 0 + ΓK 0 )(21)

The limit on the K 0 –K 0 mass difference assumes that there is no other source of CP T violation.
An upper bound on CP T breaking in KL0 → 2π has been also set through the measured phasedifference of the CP V ratios η00 and η+− , φ00 − φ+− = (0.34 ± 0.32)◦ , thanks to the small value of(1 − |η00 /η+− |) (see the review on CP violation in KL0 decays).
The measured masses and electric charges of the electron, the proton and their antiparticles
provide also strong limits on CP T violation [53–56]:2|me+ − me− |< 8 × 10−9 ,me+ + me−|qe+ + qe− |< 4 × 10−8 ,eqp̄ /mp̄− 1 = (0.3 ± 1.6) × 10−11 .
qp /mp(22)

Worth mentioning are also the tight constraints derived from the lepton and antilepton magnetic
moments [57, 58],2ge+ − ge−= (−0.5 ± 2.1) × 10−12 ,ge+ + ge−2gµ+ − gµ−= (−0.11 ± 0.12) × 10−8 ,gµ+ + gµ−(23)

those of the proton and antiproton [59],(µp + µp̄ ) /µp = (2 ± 4) × 10−9 ,(24)

and the measurement of the 1S-2S atomic transition in antihydrogen which agrees with the corresponding frequency spectral line in hydrogen at a relative precision of 2 × 10−12 [60].
A violation of CP T in an interacting local quantum field theory would imply that Lorentz
symmetry is also violated [61]. Signatures of Lorentz-invariance violation have been searched forwith atomic clocks, penning traps, matter and antimatter spectroscopy, colliders and astroparticle
experiments, with so far negative results [62]. A compilation of experimental bounds is given in
Ref. [63], parametrized through the coefficients of the so-called Standard Model Extension (SME)
Lagrangian which contains all possible Lorentz- and CP T -violating operators preserving gauge
invariance, renormalizability, locality and observer causality [64].

QUANTUM-NUMBER CONSERVATION LAWS
Conservation laws of several quantum numbers have been empirically established with a very
high degree of confidence. They are usually associated with some global phase symmetry. However,while some of them are deeply rooted in basic principles such as gauge invariance (charge conservation; local symmetry implies global symmetry) or Lorentz symmetry (fermion number conservation), others appear to be accidental symmetries of the SM Lagrangian and could be broken by
new physics interactions.
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Tests of Conservation Laws

9

In fact, if one only assumes the SM gauge symmetries and particle content, the most general
dynamics at energies below the BSM mass scale is described by the SMEFT Lagrangian in Eq. (12).
All d = 4 operators (i.e., the SM) happen to preserve the B and L quantum numbers, but this is nolonger true for the gauge-invariant structures of higher dimensionality. There is only one operator
with d = 5 (up to hermitian conjugation and flavor assignments), and it violates lepton number by
two units [65], giving rise to Majorana neutrino masses after the electroweak spontaneous symmetry
breaking. With d = 6, there are five operators that violate B and L [66, 67]. Thus, violations ofthese quantum numbers can be generically expected, unless there is an explicit symmetry protectingthem.

4 Electric charge
The conservation of electric charges is associated with the QED gauge symmetry. The mostprecise tests are the non-observation of the decays e → νe γ (lifetime larger than 6.6 × 1028 yr [68])and n → pνe ν̄e (Br < 8 × 10−27 , 68% C.L. [69]). The neutrality of matter can be also interpreted asa test of electric charge conservation. Worth mentioning are the experimental limits on the electriccharge of the neutron, qn /e = (−0.2 ± 0.8) × 10−21 , and on the sum of the proton and electron
charges, |qp + qe |/e < 1 × 10−21 [70].
The isotropy of the cosmic microwave background has been used to set stringent limits on
a possible charge asymmetry of the Universe [71]. Assuming that charge asymmetries produced
by different particles are not anticorrelated, this implies upper bounds on the photon (|qγ |/e <1 × 10−35 ) and neutrino (|qν |/e < 4 × 10−35 ) electric charges. A much stronger upper bound on thephoton charge (|qγ |/e < 1 × 10−46 ) has been derived from the non-observation of Aharonov-Bohm
phase differences in interferometric experiments with photons that have traversed cosmologicaldistances, under the assumption that both positive and negative charged photons exist [72].

5 Lepton family numbers
In the SM with massless left-handed neutrinos there is a separate conservation number for each
lepton family. However, neutrino oscillations show that neutrinos have tiny masses and there aresizable mixings among the different lepton flavors. Compelling evidence from solar, atmospheric,
accelerator and reactor neutrino experiments has established a quite solid pattern of neutrino massdifferences and mixing angles [17,18]. (see the review on neutrino masses, mixings and oscillations).
Nevertheless, flavor mixing among the different charged leptons has never been observed.
If neutrino masses and mixings among the three active neutrinos were the only sources of
lepton-flavor violation (LFV), neutrinoless transitions from one charged lepton flavor to anotherwould be heavily suppressed by powers of mνi (GIM mechanism), leading to un-observably small
rates; for instance [75–80],
Br(µ → eγ) =2m2 − m2
3α X ∗Uµi Uei νi 2 ν1 < 10−54 ,
32π i
MW

(25)

where Uia are the relevant elements of the PMNS mixing matrix. This contribution is clearly too
small to be observed in any realistic experiment, so any experimentally accessible effect would arise
from BSM physics with sources of LFV not related to mνi . The search for charged LFV (CLFV)
remains an area of active interest, which has the potential to probe physics at scales much higher
than the TeV.
Among the most sensitive probes are searches for the CLFV decays of the muon, µ → eγ andµ → 3e, as well as the conversion process µ− + A(N, Z) → e− + A(N, Z), where A(N, Z) denotes
a nucleus with N neutrons and Z protons. Searches for rare τ decays such as τ → `γ (` = e, µ)also provide interesting probes of CLFV. A variety of BSM scenarios predict that rates for these
31st May, 2024

Tests of Conservation Laws
Λ (TeV)Λ (TeV)

10

7000
6000
5000
4000

B(µ → e conv in
10

27

-18

Al)=10B(µ → eee)=10-16

3000

4

B(µ → eγ)=10-14

2000

B(µ → e conv in 27Al)=10-16

B(µ → eee)=10-14
1000
900
800
700
600

B(µ → eγ)=10-14

EXCLUDED (90% CL)

500

10

3

400
300

EXCLUDED (90% CL)
10

-2

10

-1

1

10

10
κ

2

10

-2

10

-1

1

10

10
κ

2

Figure 3: Model-independent CLFV sensitivities based on Eq (31). Left panel shows the comparison of present constraints with prospective future sensitivities for µ → eγ and µ → e conversion.
Right panel gives analogous comparison for µ → eγ and µ → 3e. Updated by [73] from Ref. [74].
CLFV processes could be sufficiently large to be observed in the present or planned searches. To
date, no observation has been reported, and the resulting null results place strong constraints on
BSM scenarios. For extensive reviews of the experimental and theoretical status and prospects, see
Refs. [74, 81, 82].
A detailed set of upper bounds on CLFV branching rations is given in the listings for the muonand tau leptons. Here we emphasize those with the strongest limits:
Br(µ → eγ) < 4.2 × 10−13
andBµ→e ≡

Br(µ → 3e) < 1.0 × 10−12

[83],

[84]

Γ (µ− + A(N, Z) → e− + A(N, Z)
Γ (µ− + A(N, Z) → ν + A(N + 1, Z − 1)

(26)

(27)

with the best limit so far, Bµ→e < 7 × 10−13 [85], obtained with gold. Several proposed experiments
aim to improve these limits by several orders of magnitude with different atoms.
One may interpret both µ → eγ and µ → e conversion in terms of the amplitudes to emit a realor virtual photon:hMµ→eγ (∗) = eGµ εα∗ ē(p − q) q 2 γα − 6 qqαL+ imµ σαβ q β ÃR
2 PR + Ã2 PL

i



L
ÃR
1 PR + Ã1 PL

µ(p) ,(28)

where it is conventional to normalize the amplitude to the Fermi constant. One then has




2
L 2
Br(µ → eγ) = 48π 3 α |ÃR.
2 | + |Ã2 |31st May, 2024(29)

5.2.1 Lepton Flavour Violation in τ decays

Iγ

IP0

IS0

IV0

III

Λh

Ihh

−5

10

10− 6

CLEO
BaBar
Belle
LHCb
ATLAS
Belle II
LHC-HL

10− 7
10− 8
−9

10

-

KΛ

-

πΛ
KΛ

-

K e +K
K µ+K
π-Λ

-

π-e+K
π-µ+K

e-K 0SK 0S
0 0µ-K SK S
π-e+ππ-µ+π-

-

+

e-K K
µ-K +K- +e K πµ-K +π-

-

e-π+K
µ-π+K

-

e π +πµ-π+π-

e-e+eµ-e+ee µ+µµ-µ+µe µ+eµ-e+µ-

-

e K*
µ K*
eφ
µ-φ
e-ω
µ-ω

e-K
µ-K*

0

e-K S
µ-K 0S
e-f0
µ-f0
eρ
0
µ-ρ

e-γ
µγ

0
*

− 10

10

e-π0
µ-π0
e-η
µ-η
e η'
µ-η'

22

90% C.L. upper limits for LFV τ decaysTausearch for
The τ lepton
is heavy enough to decay into hadrons.
11 decays offer a rich landscape to Tests
of CLFV.
Conservation
Laws
Until now already 48 LFV modes have been bounded at the level of 10−8 [196], as can be seen in Fig. 35.

FigureCurrent
neutrinolessτ decays[86]. Also
are theFig.
35: 4:Bounds
onexperimental
Tau Lepton limits
FlavouronData
from theLFVexisting
experiments
areshown
compiledbyfutureprojectionsatBelle-II
[87]
and
at
the
HL-LHC[88].
HFLAV [196]; projections of the Belle-II bounds were performed by the Belle-II collaboration assuming
50 ab−1 of integrated luminosity [195].
-1

CMS

|Yτe|

|Yτµ|

τ→3µ

−1

-1

CMS

137 fb (13 TeV)

−1

137 fb (13 TeV)

τ→3e

Yτ
τ
|Y µ

10The B factories, BaBar and Belle, have improved by
10more than an order of magnitude [411–423]
the previous
CLEObounds [424–426] for a significant number of modes. Some of the modes, for
τ→µ
γ
τ→eγ
instance,−2 τ → `ω, have been bounded for the first time [412].
−2
10
10Table 23 shows a list of limits obtained for the τ → 3µ channel by different experiments. The
strongest limits
comethe B-factories, with a competitive limit
obtained
expected
H→eτ by LHCb [427]. Table 23
expected
H→µfromτ−3−3
also10contains the recent measurement by ATLAS [428], 10
as well as the expected limit from the BelleII experiment at the SuperKEKB collider, which will improve current limits by almost two orders ofmagnitude [195]. Finally, Table 23 also summarizes the expected limits from the HL-LHC that we
10−4in more detail below.
10−4discuss
B<50%

B<10%

B<1%

B<0.1%

observed H→eτ

2

B<50%

B<10%

B<1%

/v B<0.01%
mτ
me
|=
Y τe
τ
|Y e

2

/v
mτ
mµ
|=

µ

B<0.1%

observed H→µτ

B<0.01%The physics reach and model-discriminating power of LFV tau decays is most efficiently analyzed
−the
above
scale using SMEFT, and in a corresponding
EFT when below the weak
10 5 −5electroweak
10−5 −5 low-energy
−3
−3
−4
−2
−1
−4
scale [429].
Several
to10LFV tau
at−2the low-scale,10
10 classes
10 of dimension-six
1010 operators contribute
10
10decays
10
10−1
2
|
with effective couplings denoted by Ci /Λ|Y.µτ|Loop-induced dipole operators mediate radiative|Ydecayseττ → `γ as well as purely leptonic τ → 3` and semi-leptonic decays. Four-fermion – both four-lepton±upperFigure8:semi-leptonic
Expected
line)
andHiggs
observed
solid
95%
CL
limitsthe LFV
Figure5: Current
limitsonwiththeLFV
τ (black
YukawasfromdirectH0 →τ ∓ decays(` = one, µ),
and–(red
operators
differentDiracstructures
canline)be induced
at`tree-levelor loop-level,
Yukawacouplings,| fromvs.
and |YeτAs| vs.
|Yτe |example,(right). we
The|Yµτ|Yeτ limits| couplingsand contributeindirectconstraints
andto τ|Y→and|Yττ τdecays→` +[89].
hadrons.
a typical
notethat| orcurrentµτ3`µ | (left)p
correspondto probeleft chiralmuon
or electron
and rightτ lepton,whileYτ µ | or|Yττe | couon τ → µγ
scales on
the orderof Λ/ CDipole∼ 500chiral
TeV. Besides
probinghigh |scales,
LFV
decaystwo mainhandlesamong
underlying
models
ofin
NP,
to In
identify
whichplingstoprocess,leftchiralτdiscriminate
leptonandchiralmuon
orelectron.
theleftplot, theForcorrespondtheofferconversion
thetovirtualphotonis right
absorbedby thequarksthei.e.,nucleus,yielding
anoperatorsarepresent
at
lowenergy
and
what
is
their
relativestrength:(i)
correlations
among
different
expectedlimit
is coveredby the In
observedlimit
as theyofhaveThe
effective
four-fermion
operator.
general, theexchange
other similar
particlesvalues.
could leadto flavor
similar diagonal
or
LFV
τ
decay
rates
[429];(ii)
differential
distributions
inhighermultiplicity
decays,
such
as
the
ππ
alternate
Lorentz
structures,
and
it
is
not
possibletodistinguishbetweentheexchangeofavirtual
Yukawa couplings are approximated by their SM values. The green and yellow bands indicate
invariantτ → µππIt[429]and the Dalitz
plot intheτ→
3µ general[430, 431].
photon thatormassotherparticle.
is conventional
to writemostfour-fermion
valid
the rangeisinexpectedtocontain
68%
and
95%
of
all
observed
limitamplitude,
variations
from theforenergies
below
theelectroweakscaleas(adapted
from
Ref.
[90])
expected limit. The shaded regions are constraints obtained from null searches for τ → 3µ or
X 89
(n)
n
τ → 3e (dark blue) [92] and τM→µ→eµγorµ τ →aa,qeγēΓ(purple)
The blue diagonal (30)line is the=G
Pa µ q̄Γn q[93].
,2theoretical naturalness limit |Yij Yji | = mi mn,a,q
j /v [11].

where Pa (a = L, R) denote the left and right-handed projectors and Γ n denotes 1, γ5 , γ µ , γ µ γ5 ,(n)and σµν . If any of the coefficients aa,q are generated by physics at a scale Λ > v, then their
31st May, 2024

Tests of Conservation Laws

12

effects would be encoded in the SMEFT Lagrangian (12). For scenarios in which the leading CLFV(n)operators occur at d = 6, the aa,q will scale as (v/Λ)2 . The corresponding decay and conversion
rates will then scale as (v/Λ)4 . Note that the scalar and time component of the vector interactions
are coherent over the nucleus, essentially counting the number of quarks. Consequently, theseinteractions typically yield the greatest sensitivities to high BSM mass scales.
It is sometimes convenient to compare the relative sensitivities of the decay and conversion
processes using the following simplified effective Lagrangian [74]:
LCLFV
=
eff

X
mµ
κ
µν
µ̄
σ
e
F
+
µ̄γ
e
q̄γ µ q + h.c. .
µνµRL(κ + 1)Λ2(κ + 1)Λ2q(31)

Note that one may replace the second term in Eq. (31) by any one of the other four-fermion
interactions given in Eq. (30). An analogous expression applies to the process µ → 3e whenreplacing the sum over quarks by the corresponding electron bilinear. A comparison of the present
and prospective sensitivities for various muon CLFV searches in this framework is shown in Figure 3.
Stringent limits have been also set on the LFV decay modes of the τ lepton [91]. As shown in
Figure 4, the large τ data samples collected at the B factories have made possible to reach a 10−8
sensitivity for many of its leptonic (τ → `γ, τ → `0 `+ `− ) and semileptonic (τ → `P 0 , τ → `V 0 ,τ → `P 0 P 0 , τ → `P + P 0− ) neutrinoless LFV decays, and BELLE-II is expected to push theselimits beyond the 10−9 level [87]. Being a third generation lepton, the τ could be more sensitive
to heavier new-physics scales, which makes his LFV decays particularly interesting. Compared tothe muon, the τ decay amplitudes could be enhanced by a chirality ratio (mτ /mµ )2 ∼ 280 and/or
by lepton-mixing factors such as |Uτ 3 /Ue3 |2 ∼ 20, but the exact relation is model dependent. In
any case, the τ LFV decays provide a rich data set that is very complementary to the µ bounds. If
LFV is finally observed, the correlations between µ and τ data, and among different LFV τ decays
will allow to probe the underlying mechanism of lepton flavor breaking.
Interesting limits on LFV are also obtained in meson decays. The best bounds come from kaon
experiments, e.g., Br(KL0 → e± µ∓ ) < 4.7 × 10−12 [92], Br(K + → π + µ+ e− ) < 1.3 × 10−11 [93].
Quite strong limits have also been set in decays of B and D mesons, the best upper bounds being
Br(B 0 → e± µ∓ ) < 1.0 × 10−9 [94] and Br(D0 → e± µ∓ ) < 1.3 × 10−8 [95].
The LFV decays of the Z boson were probed at LEP at the 10−5 to 10−6 level. Stronger (95%
C.L.) limits have been set recently by the LHC ATLAS collaboration [96, 97]:
Br(Z → e± µ∓ ) < 2.62 × 10−7 ,Br(Z → e± τ ∓ ) < 5.0 × 10−6 ,Br(Z → µ± τ ∓ ) < 6.5 × 10−6 .
(32)

LHC is now testing LFV in Higgs decays, within the available statistics. From the current (95%
C.L.) experimental upper bounds [89, 98–102],Br(H 0 → e± µ∓ ) < 4.4 × 10−5 ,Br(H 0 → e± τ ∓ ) < 0.20% ,Br(H 0 → µ± τ ∓ ) < 0.15% ,(33)

one can derive direct limits on the LFV Yukawa couplings of the Higgs boson,
LY = −H 0XY`i `j `¯iL `jR + h.c. .

(34)

i6=j

q

2 + Y 2 < 1.9 × 10−4 , which is not yet competitive with the
From H 0 → e± µ∓ , one obtains Yµeeµindirect limit set by µ → eγ through a (one-loop) virtual Higgs exchange:q

2 + Y 2 < 3.6 × 10−6 .
Yµeeµ31st May, 2024

(35)

Tests of Conservation Laws

13

However, the LHC data provide at present the strongest bounds on the LFV τ Yukawas [89, 101]:q

Yeτ2 + Yτ2e < 1.3 × 10−3 ,q

2 + Y 2 < 1.11 × 10−3 .
Yµτ
τµ

(36)

Figure 5 compares the Higgs exclusion limits on the τ Yukawas with the current indirect constraints
from LFV τ decays.

6 Baryon and Lepton Number
The transitions discussed in the previous section preserve the total lepton number L = Le +Lµ + Lτ . In the SM, conservation of B − L is an accidental symmetry of the Lagrangian. At
the classical level, B + L is also conserved, though it is violated at the loop level by the anomaly.
The latter is a topological effect that is highly suppressed at zero temperature and, moreover, does
not contribute to the processes discussed in the review. Going beyond renormalizable interactions,
there exists a tower of operators in the SMEFT Lagrangian (12), containing only SM fields, thatbreak one or both of these symmetries. We briefly review these possibilities in turn.
Lepton Number
The lowest-dimension operator containing only SM fields that breaks baryon or lepton number
is the d = 5, lepton-number-violating (LNV) ‘Weinberg’ neutrino-mass operator [65]:
LLNV =y C
L̄ HH T L .
Λ(37)

When the neutral component of the Higgs field obtains its vacuum expectation value, this ∆L =2 interaction yields a Majorana mass for the light, active neutrinos. The most comprehensive
approach for probing this effect is the search for neutrinoless double-beta decay (0νββ) of atomic
nuclei, (Z, A) → (Z +2, A)+e− +e− [103,104] (see the review on neutrinoless double-β decay). The
detection of a non-zero 0νββ signal could represent a spectacular evidence of Majorana neutrinos.
The current best limit, τ1/2 > 1.07 × 1026 yr, was obtained by the KamLAND-Zen experiment with
136 Xe [105].
Theoretically, the interaction (37) can arise from BSM interactions in the well-known see-saw
mechanism for neutrino mass (for a review, see [106]). In this context, the conventional choice forthe scale Λ is of order the GUT scale, yielding light neutrino masses of order eV and below when
the couplings y are of order the charged elementary fermion Yukawa couplings. BSM theories may
also give rise to LNV observables in other contexts. In these scenarios, if the LNV scale is of order
1 TeV, one may observe signatures of LNV not only in 0νββ but also in collider searches for final
states containing same sign dileptons. Searches for same sign dileptons plus a di-jet pair at the
LHC have placed constraints on TeV-scale LNV [107, 108] that in some cases complement thoseobtained from 0νββ.
Stringent constraints on violations of L have been also set in µ− → e+ conversion in muonicatoms, the best limit being σ(µ− Ti → e+ Ca)/σ(µ− Ti → all) < 3.6 × 10−11 [109], and at the flavor
factories through L-violating decays of the τ lepton and K, D and B mesons. Some representative
examples are Br(τ − → e+ π − π − ) < 2.0 × 10−8 [110], Br(K + → π − µ+ µ+ ) < 4.2 × 10−11 [111],
Br(D+ → π − µ+ µ+ ) < 1.4 × 10−8 [112] and Br(B − → π + µ− µ− ) < 4.0 × 10−9 (95% CL) [113]. All
these |∆L| = 2 processes could be mediated by a massive Majorana neutrino. They provide usefulPbounds on the effective Majorana neutrino mass matrix m``0 ∼ i U`i U`0 i mνi [114], although not
as strong as the 0νββ constraint on mee .
Baryon Number
Grand Unified Theories (GUTs) combine leptons and quarks in the same symmetry multiplets
and, therefore, predict the violation of the baryon and lepton quantum numbers. Many experiments
31st May, 2024

Tests of Conservation Laws

14

have searched for B-violating transitions, but no positive signal has been identified so far. Proton
decay would be the most relevant violation of B, as it would imply the unstability of matter. The
current lower bound on the proton lifetime is 9 × 1029 yr [115]. Stronger limits have been set forparticular decay modes, such as τ (p → e+ π 0 ) > 2.4 × 1034 yr [116]. For a discussion of proton
decay in the context of GUTs, see the review on Grand Unified Theories.
Another spectacular signal would be neutron-antineutron oscillations. Searches have been performed for quasi-free n–n̄ oscillations and for nn̄ annihilation products in a nucleus. The latter
would arise when the n̄ produced through oscillations annihilates with another neutron in the nuclear medium. The corresponding best limits, expressed in terms of the free and bound oscillation
times, τnn̄ and τm , respectively, are:τnn̄ > 0.86 × 108 s [117],(38a)

τm > 3.6 × 1032 yr [118](38b)

From the latter, one may infer a bound τnn̄ > 4.7 × 108 s, as discussed below. See Ref. [119] for a
recent review.
The theoretical interpretation of these bounds starts with an assumed, effective Hamiltonian
for the free (anti-)neutron, Heff that contains a B-violating part, yielding matrix elements
λhn|Heff |ni = hn̄|Heff |n̄i = m − i ,2hn|Heff |n̄i = hn̄|Heff |ni ≡ δm ,

(39a)
(39b)

where CP T is assumed to be conserved, the neutron lifetime τn = 1/λ and τnn̄ = 1/|δm|. The rate
for a neutron to oscillate into an antineutron after a time t is given byPnn̄ (t) = sin

2



t
τnn̄



e−λt .

(40)

For t << τn << τnn̄ , one has
Pnn̄ (t) → (t/τnn̄ )2 .

(41)

In realistic experiments, there exist effects, such as background magnetic fields, that split theenergies of the neutron and antineutron. One must ensure that the observation time is sufficiently
short so that these effects do not overwhelm the small B-violating term δm and that Eq. (40)applies.
In nuclei, the interactions of neutrons and antineutrons with the surrounding medium are sufficiently distinct that one must take the corresponding matter potentials into account. In particular,
the matrix elements in Eq. (39a) become
hn|Heff |ni = m + Vn ,hn̄|Heff |n̄i = m + Vn̄ ,(42)with Vn being essentially real (Vn ≡ VnR ) and Vn̄ = Vn̄R −iVn̄I . The imaginary part Vn̄I characterizes
the annihilation of the antineutron with bound nucleons into secondary hadrons. The rate for abound neutron to disappear is given by
Γm =


−1
2(δm)2 |Vn̄I |
2
≡
R
τ
.
nn̄
2
(VnR − Vn̄R )2 + Vn̄I

(43)

For the nuclei of experimental interest, nuclear theory computations yield R ∼ 1023 s−1 . Null resultsof bound n-n̄ oscillation searches thus allow one to infer a bound on τnn̄ via Eq. (43).
31st May, 2024

Tests of Conservation Laws

15

From an elementary particle standpoint, n-n̄ oscillations involve the conversion of three quarks
into three antiquarks (and vice-versa). The lowest-dimension operators mediating such processarise at dimension nine in the SMEFT:
Ln-n̄ =

1 X (9) BNV
α Oj.
Λ5 j j(44)6(9) ΛHAD
,
Λ5

(45)

Consequently, one expects
δm ∼ αjwhere ΛHAD is a hadronic scale set by the n-n̄ matrix elements in Eq. (39b). Taking ΛHAD to be oforder the QCD scale and using the present bounds on τnn̄ yields a lower bound on the B-violating
mass scale of ∼ 100 TeV.
The search for B-violating decays of short-lived particles such as Z bosons, τ leptons and B
mesons provides also relevant constraints. The best limits are Br(Z → pe, pµ) < 1.8 × 10−6 (95%
C.L.) [120], Br(τ − → p̄µ− µ+ ) < 1.8 × 10−8 [121] and Br(B 0 → pµ− ) < 2.6 × 10−9 [122]

7 Quark flavors
While strong and electromagnetic forces preserve the quark flavor, the charged-current weak interactions generate transitions among the different quark species (see the review on the CKM quarkmixing matrix). Since the SM flavor-changing mechanism is associated with the W ± fermionicvertices, the tree-level transitions satisfy a ∆F = ∆Q rule where ∆Q denotes the change incharge of the relevant hadrons. Remember that the flavor quantum number F is defined to
be +1 for positively charged quarks (F = U, C, T ) and −1 for quarks with negative charges(F = D, S, B). The strongest tests on this conservation law have been obtained in kaon decays suchas Br(K + → π + π + e− ν̄e ) < 1.3 × 10−8 [123], and (Re x, Im x) = (−0.002 ± 0.006, 0.0012 ± 0.0021)[124, 125] where x ≡ M(K 0 → π − `+ ν)/M(K 0 → π − `+ ν).
The ∆F = ∆Q rule can be violated through quantum loop contributions giving rise to flavorchanging neutral-current transitions (FCNCs). Owing to the GIM mechanism, processes of thistype are very suppressed in the SM, which makes them a superb tool in the search for new physics
associated with the flavor dynamics. Within the SM itself, these transitions are also sensitive tothe heavy-quark mass scales and have played a crucial role identifying the size of the charm (K 0 –K 0 mixing) and top (B 0 –B 0 mixing) masses before the discovery of those quarks. In addition
to the well-established ∆F = 2 mixings in neutral K and B mesons, ∆MK 0 ≡ MK 0 − MK 0 =S
L(0.5293 ± 0.0009) × 1010 s−1 , ∆MB 0 ≡ MB 0 − MB 0 = (0.5069 ± 0.0019) × 1012 s−1 and ∆MBs0 ≡H
L
MB 0 − MB 0 = (17.765 ± 0.006) × 1012 s−1 , the mixing of the D0 meson and its antiparticle hassH
sL
been recently observed with a significance of more than seven standard deviations [126], showingthat there is a nonzero mass difference between the two neutral charm-meson eigenstates, of the
expected size:
MD0 − MD0 = (0.997 ± 0.116) × 1010 s−1 .
(46)
H

L

0 –D 0 mass difference is dominated by long-distance physics, because
The SM prediction of the DHL

it involves virtual loops with down-type light quarks, and has unfortunately quite large uncertainties [127].
The FCNC kaon decays into lepton-antilepton pairs put stringent constraints on new flavorchanging interactions. The measured KL0 → µ+ µ− rate, Br(KL0 → µ+ µ− ) = (6.84 ± 0.11) × 10−9 , iscompletely dominated by the known 2γ absorptive contribution, leaving very little room for new−12 [128] (the tiniest branching ratio ever measured)
physics, and Br(KL0 → e+ e− ) = (9+6
−4 ) × 10
also agrees with the SM expectation [129]. The experimental KS0 upper bounds on the electron,
31st May, 2024

16

Tests of Conservation Laws

Br(KS0 → e+ e− ) < 9 × 10−9 [130], and muon, Br(KS0 → µ+ µ− ) < 2.1 × 10−10 [131], modes arestill five and two orders of magnitude, respectively, larger than their SM predictions [129]. Anothervery clean test of FCNCs is provided by the decay K + → π + ν ν̄. The CERN NA62 experiment has
already observed 20 signal candidates, providing the first evidence of this decay [132]. This leads to
0.40−10 , in agreement with the predicted SM branching fraction ofBr(K + → π + ν ν̄) = (1.14 +− 0.33 ) × 10
(7.73±0.61)×10−11 [133,134]. Even more interesting is the CP -violating neutral mode KL0 → π 0 ν ν̄,
expected at a rate of (2.59 ± 0.29) × 10−11 [133, 134] that is still far away from the current upperbound of 3.0 × 10−9 [135]. The KOTO experiment at KEK is expected to substantially increase
the sensitivity to this mode.
The strongest bound on FCNC transitions in charm decays is Br(D0 → µ+ µ− ) < 3.1 × 10−9[136], while in B decays the LHC experiments have recently reached the SM sensitivity: Br(Bd0 →µ+ µ− ) < 1.5 × 10−10 [137] and Br(Bs0 → µ+ µ− ) = (3.34 ± 0.27) × 10−9 .
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[2] W. Pauli, in L. Rosenfelf and V. Weisskopf, editors, “Niels Bohr and the Development of
Physics,” 30–51, McGraw-Hill, New York (1955).
[3] J. McDonough et al., Phys. Rev. D38, 2121 (1988).
[4] M. Ablikim et al. (BESIII), Phys. Rev. D90, 9, 092002 (2014), [arXiv:1409.4040].
[5] S. Prakhov et al. (Crystal Ball), Phys. Rev. Lett. 84, 4802 (2000).
[6] J. H. Christenson et al., Phys. Rev. Lett. 13, 138 (1964).
[7] H. Burkhardt et al. (NA31), Phys. Lett. B206, 169 (1988).
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