Some Recent Results on Physics Beyond the Standard
Model

Robert Shrock
C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy,
Stony Brook University, email: robert.shrock@stonybrook.edu, robertshrock@gmail.com

20’th Lomonosov Conference, Moscow State University, Aug. 19-25, 2021

Outline• Some constraints on sterile neutrinos and related charged lepton flavor violation• n − n̄ oscillations: general formalism and current experimental limits• Models in which n − n̄ oscillations are dominant manifestation of baryon number
violation• Conclusions

Some Constraints on Sterile Neutrinos and Related Charged Lepton
Flavor Violation
Neutrino masses and lepton mixing have been established by ν oscillation experiments
and are of great importance as physics beyond the original Standard Model (SM).
The active
neutrino weak eigenstates νℓ are expressed in terms of mass eigenstates asP3+n|νℓi = j=1 s Uℓj |νj i, where ℓ = e, µ, τ , and ns refers to possible additional
mass eigenstates occurring as small admixtures in |νℓi, and U is the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton mixing matrix. Denote the SM
extended to include ν masses as the νSM, allow for ns sterile ν eigenstates.
Neutrino masses lead to a number of effects besides neutrino oscillations, e.g., decays
with charged lepton flavor violation (CLFV) such as µ → eγ, µ → eeē, etc.
However, in the minimal νSM, the rates are much too small to be observable:
BR(µ → eγ) =3+n3αem Xs

32π

j=1

∗
Uej
Uµj

 m2ν  2j

m2W

(Marciano-Sanda 1977, Petcov 1977, B. W. Lee, Pakvasa, Sugawara, RS 1977). In theminimal νSM, BR(µ → eγ) ∼ 10−52, far below observable levels, and similarly with
BR(µ → eeē) and CLFV τ decays such as τ → ℓγ.

The suppression arises due to a leptonic GIM mechanism operating at the one-loop
level; necessary and sufficient conditions are that leptons of the same chirality andcharge have the same weak isospin I and I3 (B. W. Lee and RS, PRD16, 1444 (1977)).
With a mostly sterile neutrino, denoted ν4, having, e.g., mν4 ≃ 350 MeV, using upper
−32
−12
−9
2 <
from NA62, BR(µ → eγ) <bounds |Ue4|2 <∼ 10 ,∼ 10 and |Uµ4| ∼ 10
still far below detectable BR; similar comments for CLFV τ decays.
A massive Dirac neutrino has a magnetic moment (Fujikawa and RS, 1980)m 
3eGF mνi
νiµ νi =µB√ = (3.2 × 10−19)
2
1 eV
8π 2
where µB = e/(2me). Current limits from astrophysics and laboratory exps.
−11− 10−12 (recent reviews by Giunti and Studenikin, 2015; Balantekin andµ νi <∼ 10
Kayser, 2018)
Recent progress on obtaining a more sensitive search for µν using coherent elastic
neutrino-atom scattering using ν̄e from tritium decay and a liquid He target (Cadeddu,
Dordei, Giunti, Kouzakov, Picciau, Studenikin, PRD 100, 073014 (2019); talks here.

Tests for massive neutrino emission, via lepton mixing, in nuclear and particle decays(RS, PLB 96, 159 (1980); re neutrino mass limits, RS, in PDG, Rev. Mod. Phys. 52,
S63 (1980)). Among these tests is the search for heavy neutrinos emitted in two-body
leptonic decays of charged pseudoscalar mesons M + → ℓ+νℓ, where ℓ = e, µ andM + = π +, K +, D +, Ds+, B +; one searches for a peak in dN/dEℓ at
Eℓ = (m2M + m2ℓ − m2ν4 )/(2mM ) due to the ℓ recoiling opposite a massive ν4.
The sensitivity of this test stems mainly from the (a) monochromatic signal, (b) theremoval of helicity suppression for decay to a heavy neutrino in the M + → e+νe+
decays, which involves a relative enhancement factor up to ∼ 104 in πe2
decay and+∼ 105 in Ke2
decay with little phase space suppression for moderate ν4 massesmν4 < mM − mℓ. (We consider mν4 >> eV scale discussed w.r.t.
LSND/miniBooNE and reactor data.)
This “peak search” test was applied retroactively to data in 1980 to obtain first upperbounds in the PLB paper. It has been applied in a series of dedicated experiments on+• πℓ2
decay at IUCF, TRIUMF and SIN/PSI+• Kℓ2
decay at KEK, Serpukhov, BNL, and CERN±
decay at Belle• Bℓ2
to set very stringent upper bounds on |Uej |2 and |Uµj |2 as function of mνj for aheavy neutrino νj .

Recent results from:+PIENU experiment at TRIUMF (Bryman et al.) in PRD 97, 072012 (2018) (πe2); PLB+
798, 134980 (2019) (πµ2)

NA62 experiment at CERN (including Lazzeroni, Goudzovski, Duk, Ceccucci, Bryman,
Kudenko...) in PLB 772, 712 (2017); PLB 778, 137 (2018); PLB 807, 135599 (2020);++
PLB 816, 136259 (2021) (Kµ2
and Ke2).
Massive neutrino emission would also change(π)
Re/µ =
(K)

(D )

(D )

BR(π + → e+νe)

BR(π + → µ+νµ)

Re/µ , Re/τs , and Rµ/τs from their SM values, so the agreement of the measured
(π)

(K)

(D )

(D )

ratios Re/µ, Re/µ , and Rµ/τs and the consistency of the upper limit on Re/τs withSM predictions provide further constraints.

In the SM
2
2 2
2
m
m
|V
|
f
G
M
ab
ℓ
M
Γ(M + → ℓ+νℓ )SM = F
8π



m2ℓ
1− 2
mM

where Vab = Vud for M + = π +, Vab = Vus for M + = K +, etc.

2

For the decay to a ν4 of non-negligible mass,
3
2
2 2
2
m
|V
|
|U
|
f
G
ab
ℓ4
(M )
M
M
)
ρ(δℓ , δν(M),Γ(M + → ℓ+ν4) = F
4
8π

where
and

ρ(x, y) = [x + y − (x − y)2] [1 + (x − y)2 − 2(x + y)]1/2 ,
(M )
δℓ
=

m2ℓ
m2M

,

)
δν(M
=
4

m2ν4
m2M

,
(π)

For a massless neutrino, ρ(x, 0) = x(1 − x)2 with x = δℓ . Thus,
Γ(M + → ℓ+ν4)

Γ(M + → ℓ+νℓ)SM(M )= |Uℓ4|2ρ̄(δℓ), δν(M)4

where ρ̄(x, y) = ρ(x, y)/ρ(x, 0) = ρ(x, y)/[x(1 − x)2]. Recent analysis:
Bryman and RS, Phys. Rev. D 100, 053006, 073011 (2019):

These are direct kinematic limits. Other bounds can be obtained from searches for
production and decay of heavy neutrinos in fixed-target exps. and at colliders(dependent on ν decay channels).
Given the Pontecorvo-Maki-Nakagawa-Sakata lepton mixing, it follows that massive
neutrinos can also be emitted, via lepton mixing, in nuclear beta decay; with one ν4,


h
i1/2
dN
= C (1−|Ue4|2)pE(E0−E)2 +|Ue4|2pE(E0−E) (E0−E)2 −m2ν4
θ(E0−E−mν4 ) ,dE

where E = e energy; C = G2F |Vud|2F |M|2/(2π 3) (F = Fermi function, M =nuclear matrix element).
Signature: kink in the Kurie plot at the endpoint of the massive ν4 distribution;suggestion to search for such kinks, and initial bounds obtained from retroactive data
analysis (RS,PLB,1980).

This kink search has been performed by many exps.with 3H (Mainz, Troitsk), 20F, 35S,
63
Ni, 64Cu, 187Re, etc., getting upper limits |Ue4|2 ranging from ∼ 10−2 from∼ 10−3 for mν4 from ∼ 0.3 keV eV to ∼ 2 MeV.
The kink search test can be performed in future in KATRIN (looking away fromendpoint) (e.g., Drewes, Lasserre, Merle, Mertens, et al., arXiv:1602.04816).

Another method is to measure both electron and recoil ion in nuclear beta decay (e.g.,
Cook et al., 1992; Finocchiaro, RS, 1992); used with 38mK β + decay by Trinczek et al.
(2003), obtaining |Ue4|2 . 10−2 for 0.5 < mν4 < 3 MeV.
Analysis of recoil ion spectrum in e-capture in 7Be (Likhovid and Pantuev, 2021)
yielding |Ue4|2 . few × 10−4 for 300-750 keV.
R+D work for an exp. to measure ion recoil after e-capture in 131Cs (Martoff, Meyers,
Smith et al.)Set of superallowed Fermi nuclear beta decays have very good mutual agreement on F t
rate values, with Q values ranging over several MeV. Emission of a massive ν4 wouldchange rates and remove this mutual agreement. This agreement can thus be used to
set an upper limit |Ue4|2 . few × 10−4 for this mass range (Bryman and RS, 2019).

Further bounds can be obtained from cosmology, e.g., with assumptions about early
universe history, agreement of primordial nucleosynthesis predictions for H, He
2
abundances suggest lifetime τν4 <∼ O(1) sec, yielding lower bounds on |Uℓ4| ,depending on mν4 . Thus, cosmology disfavors some regions of mν4 and |Uℓ4|2.
Considerable recent interest in heavy neutrinos by theorists as well as experimentalists,
e.g. Asaka and Shaposhnikov 2005; Gorbunov and Shaposhnikov 2007; Kusenko,
Pascoli, and Semikoz 2008; Kusenko 2009; Boyarsky et al. 2009; Helo, Kovalenko,
Schmidt 2011; Abada et al. 2014; SHIP Proposal at CERN (Alekhin et al., 2016) deGouvêa and Kobach 2016; Fernandez-Martinez et al. 2016; Drewes and Garbrecht
2017; Batell et al. 2018; Bondarenko, Boyarsky, Gorbunov, Ruchayskiy 2018; Coloma etal. 2020; Bondarenko et al. 2021; and many other theoretical papers.

Current upper bounds on some CLFV µ and τ decays:• BR(µ → eγ) < 4.2 × 10−13 (MEG at PSI)

• BR(µ → eeē) < 1.0 × 10−12 (SINDRUM II at SIN/PSI)• BR(τ → eγ) < 3.3 × 10−8 (limits on CLFV τ decays from Belle and BABAR)• BR(τ → µγ) < 4.4 × 10−8

• BR(τ → eeē) < 2.7 × 10−8

• BR(τ → eµµ̄) < 2.7 × 10−8

• BR(τ → µeē) < 1.8 × 10−8

• BR(τ → µµµ̄) < 2.1 × 10−8
BSM physics that could yield CLFV at observable levels includes supersymmetry, Z ′vector bosons with flavor-nondiagonal couplings, etc.
So searches for CLFV processes such as decays and µ − e conversion in the field of a
nucleus are interesting as probes of BSM physics.

Some current/future exps. on CLFV processes• MEG II (PSI), planning to achieve sensitivity to BR(µ+ → e+γ) ∼ 10−14(EPJC 78, 380 (2018); arXiv:1912.08656)• Mu3e (PSI), planning to achieve sensitivity to BR(µ+ → e+ e+e−) ∼ 10−14(arXiv:2009.11690),• Belle II (KEK) improving upper limits on CLFV τ decays (arXiv:1011.0352)• Mu2e (Fermilab) searching for µ− + (Z, A) → e− + (Z, A) (talks by Miller and
Di Falco)
An analysis of BABAR data from search for τ → ℓγ yields the first upper bounds on
two other CLFV decays: BR(τ → eγγ) < 2.4 × 10−4 andBR(τ → µγγ) < 5.8 × 10−4 (Bryman, Ito, RS, arXiv:2106.02451).

There is also interest in processes violating total lepton number, L. Upper limits onneutrinoless double beta decay (Z, A) → (Z + 2, A) + 2e− are very sensitive
probes of |∆Le| = 2 interactions, and K decays yield complementary limits for modes
involving µ, e.g., K + → π −µ+µ+ (see fig.) and K + → π −µ+e+ .
From retroactive data analysis, first upper limit on BR(K + → π −µ+µ+) set in1992 (Littenberg, RS); dedicated search by BNL E865 (2000, 2005) obtained• BR(K + → π −µ+µ+) < 3.0 × 10−9

• BR(K + → π −µ+e+) < 5.0 × 10−10

By using the upper limit on µ− + (Z, A) → e+ + (Z − 2, A) with estimate for the
nuclear matrix element, one can get an indrect upper limit−11
O(10) (Littenberg, RS, 2000).
BR(K + → π −µ+e+) <∼Recent progress by CERN NA62 (2019, 2021):• BR(K + → π −µ+µ+) < 4.2 × 10−11• BR(K + → π −e+µ+) < 4.2 × 10−11

n − n̄ Oscillations: General Formalism
Motivations:
Producing the observed baryon asymmetry in the universe requires
baryon-number-violating (BNV) interactions (as well as CP violation and deviation fromthermal equilibrium) (Sakharov, 1967).
Suggestion of n − n̄ transitions as a mechanism involved in generating baryon
asymmetry in the universe (Kuzmin, 1970).
Standard Model (SM) conserves B perturbatively. SU(2)L instantons produce
nonperturbative violation of B and L, while conserving B − L (’t Hooft, 1976), but
this is negligible (exponentially small) at temperatures T low compared with theelectroweak scale (finite-T baryogenesis: Kuzmin, Rubakov, Shaposhnikov, 1985).
Grand unified theories (GUT’s) also predict violation of B and L. Besides proton decay(∆B = −1 BNV), n − n̄ oscillations (|∆B| = 2 BNV) can occur.
There are good motivations for new experimental searches for n − n̄ oscillations andassociated ∆B = −2 dinucleon decays as well as proton decay.

General Formalism for n − n̄ Oscillationsn − n̄ Oscillations in Field-Free Vacuum:
CPT: hn|Hef f |ni = hn̄|Hef f |n̄i = mn − iλn/2, where Hef f denotes3
relevant Hamiltonian and λ−1
=
τ
=
0.88
×
10
sec. Hef f may also mediatennn ↔ n̄ transitions: hn̄|Hef f |ni ≡ δm. Consider the matrix in (n, n̄) basis:M=mn − iλn/2δmδmmn − iλn/2Diagonalizing M yields mass eigenstates1|n±i = √ (|ni ± |n̄i)2with mass eigenvalues m± = (mn ± δm) − iλn/2.
So if start with pure |ni state at t = 0, then there is a finite probability P for it to be
an |n̄i at t 6= 0:
P (n(t) = n̄) = |hn̄|n(t)i|2 = [sin2(t/τnn̄)]e−λnt

where τnn̄ = 1/|δm|.

Most sensitive reactor n − n̄ exp. done with ILL High Flux Reactor (HFR) at Grenoble(Baldo-Ceolin, Fidecaro,.., 1985-1994, obtaining limit τnn̄ ≥ 0.86 × 108 sec.
A future n − n̄ search experiment could significantly improve this limit or observe a
signal.
Plan for n − n̄ search exp. at European Spallation Source, ESS, under construction in
Lund, Sweden; white paper: J. Phys. G 48, 070501 (2021) [arXiv:2006.04907].
R+D for improved sensitivity with ultra-cold neutrons (UCNs), e.g., Fomin, Serebrov etal., Kurchatov Inst. PNPI, Gatchina.
n − n̄ oscillations lead to matter instability ( m.i.) via annihilation of n̄ withneighboring n or p in a nucleus (producing mainly multipion final state); resultant
lifetime τm.i..
This matter instability can be searched for with large proton decay detectors; current
bound from Super-K: τm.i. > 3.6 × 1032 yrs, giving τnn̄ > 4.7 × 108 sec (90 %CL) (Abe et al., PRD 103, 012008 (2021)).

Question: are there theoretical models that provide motivation for a new n − n̄ searchexperiment?
Answer: Yes; we proceed to discuss such models.
references: S. Girmohanta and RS, PLB 803, 135296 (2020) [arXiv:1910.08356]; PRD
101, 015017 (2020) [arXiv:1911.05102]; PRD 101, 095012 (2020) [arXiv:1911.05102];
S. Nussinov and RS, PRD 102, 035003 (2020) [arXiv:2005.12493]; S. Girmohanta, R.
Mohapatra, RS, PRD 103, 015021 (2021) [arXiv:2011.01237].

n − n̄ Oscillations in an Extra-Dimensional Model
We discuss models in which proton decay can easily be suppressed well below
experimental limits while n − n̄ oscillations can occur at a level comparable to existing
limits (Girmohanta and RS, 2020, 2021; Nussinov and RS, PRL 88, 171601 (2002)).
Consider models in a d = 4 + n dimensional spacetime, with n extra spatial
dimensions. Denote usual spacetime coords. as xν , ν = 0, 1, 2, 3 and consider nextra compact coordinates, yλ with 0 ≤ yλ ≤ L, i.e., size of extra dimension(s) is L.
Each SM fermion f has the form Ψf (x, y) = ψf (x)χf (y) with strong localization
at a point yf in the extra dimensions, with a Gaussian profile of half-widthσ ≡ µ−1 << L:22

χf (y) = Ae−µ ky−yf k = Ae−kη−ηf k2Pn2)1/2, A is a normalization constant, and we define awhere kyf k = ( λ=1 yf,λconvenient dimensionless variable ηf = µyf = yf /σ.
Such models are of interest partly because they can provide a mechanism for obtaining
a generational hierarchy in fermion masses and quark mixing by placement of fermion
wave function centers in extra dimensions (Arkani-Hamed + Schmaltz;
Mirabelli+Schmaltz, 2000).

We use a low-energy effective field theory (EFT) approach with an ultraviolet cutoff
M∗, where M∗ > µ for self-consistency. It suffices here to consider the lowest
Kaluza-Klein (KK) modes; effects of higher KK modes are discussed in Girmohanta,
Mohapatra, RS, PRD 103, 015021 (2021). Gauge and Higgs fields have flat profiles inthe extra dimensions.
Starting from the Lagrangian in the d-dimensional spacetime, one obtains the resultant
low-energy EFT in 4D by integrating over the extra n dimensions.
For canonical normalization of the 4D fermion kinetic term, A = (2/π)n/4µn/2.
Define ΛL = 1/L; take ΛL ≃ 102 TeV, i.e., L ≃ 2 × 10−19 cm, andσ/L = 1/30; this gives adequate separation of fermions while fitting in thecompactification interval [0, L], consistent with precision electroweak data, collider
bounds, flavor-changing neutral current constraints.

A Yukawa interaction in the d-dimensional space with coefficients of order unity andmoderate separation of localized fermion wavefunction centers yields a strong hierarchyin the low-energy 4D Yukawa interaction,Z

dny χ̄(yfL )χ(yfR ) ∼

Z

2

2

dnη e−kη−ηfL k e−kη−ηfR k ∼ e−(1/2)kηfL −ηfR k

2

Resultant fermion masses mf :h 1iv
mf ≃ h(f ) √ exp − kηfL − ηfR k2 ,22√where v/ 2 is SM Higgs VEV. With h(f ) ≃ 1, produce fermion generational
hierarchy via different separation distances kηfL − ηfR k for different generations.
Leading nucleon decay operators are of the form qqqℓ. Hence, one can suppress
nucleon decay well below experimental limits by arranging that the wavefunction centersof the u and d quarks are separated far from those of the leptons.
Key point: this does not suppress n − n̄ oscillations because the n − n̄ transition
operators do not involve leptons.

For example, one nucleon decay operator is (with ℓ = e, µ)
(N d)

β
γ T
T
O1
= ǫαβγ [uαCd][uR
R CℓR ]
R
where α, β, γ are SU(3)c color indices.

The product of y-dependent fermion wavefunctions in this operator is
h

n



1n

A4 exp − 2kη − ηuR k2 + kη − ηdR k2 + kη − ηℓR k2
The integral over y yields
(N d)

I1

= b4 exp −4√
where b4 = (µ/ π)n.

oi

2kηuR − ηdR k2 + 2kηuR − ηℓR k2 + kηdR − ηℓR k2

One can guarantee that this is sufficiently small by taking the distances between
wavefunction centers kηuR − ηℓR k and/or kηdR − ηℓR k2 sufficiently large.

o

R 3
(nn̄)
(nn̄)
Analyze n − n̄ oscillations: with Hef f = d x H(nn̄), δm = hn̄|Hef f |ni.
In d = 4 dims., effective Lagrangian(nn̄)
Lef f (x) =Xrcr(nn̄) Or(nn̄)(x) + h.c. .

Correspondingly, in d = 4 + n dimensions,(nn̄)
Lef f,4+n(x, y) =X

κr(nn̄)Or(nn̄)(x, y) + h.c. .

rwhere the Or(nn̄)(x) and Or(nn̄)(x, y) are 6-quark operators in d = 4 and
5+2nd = 4 + n dims. Coeffs. κ(nn̄)= κ̄r(nn̄)/Mnn̄
, where Mnn̄ is an effective massr
scale of physics producing the n − n̄ oscillations.
Integration of fermion wavefunctions in the Or(nn̄)(x, y) over y yield the coeffs. cr(nn̄)
in terms of κr(nn̄)

Relevant six-quark operators in SM EFT:
(nn̄)

ρT
γT
β
σ
δ
Cd
Cd
][d
][d
= (Ts)αβγδρσ [uαT
Cu
R
R
R
R]
R
R

O2

(nn̄)

β
γT
ρT
δ
σ
= (Ts)αβγδρσ [uαT
Cd
][u
Cd
][d
Cd
R
R
R
R
R
R]

(nn̄)

jβ
γT
ρT
δ
σ
= (Ta)αβγδρσ ǫij [QiαT
CQ
][u
Cd
][d
L
R
R CdR ]
L
R

O1

O3
(nn̄)

jβ
kγT
ρT
mδ
σ
= (Ta)αβγδρσ ǫij ǫkm[QiαT
CQ
][Q
CQ
][d
L
L
R CdR ]
LLu
where QL = d L, i, j... are SU(2)L indices, and color SU(3)c tensors are

O4(Ts)αβγδρσ = ǫραγ ǫσβδ + ǫσαγ ǫρβδ + ǫρβγ ǫσαδ + ǫσβγ ǫραδ(Ta)αβγδρσ = ǫραβ ǫσγδ + ǫσαβ ǫργδsymmetries: (Ts)αβγδρσ = T(αβ)(γδ)(ρσ) ;(Ta)αβγδρσ = T[αβ][γδ](ρσ).

(nn̄)(nn̄)

The integrals of these operators over y: operators O1
and O24(nn̄)
I1,2 = b6 exp − kηuR − ηdR k2 ,
3
(nn̄)

O3

(nn̄)

I3

(nn̄)

O4

yield the integral

yields the integral= b6 exp −1n
62kηQL − ηuR k2 + 6kηQL − ηdR k2 + 3kηuR − ηdR k2yields the integral


(nn̄)

4



= b6 exp − kηQL − ηdR k2 .
3where b6 = (2 · 3−1/2 π −1µ2)n.
I4

(nn̄)

κ̄r
(nn̄)
Then coefficients. cr(nn̄) = (M
5 Ir
)
nn̄

o

.

Consider, e.g., case n = 2: one can fit data on quark masses, mixing withkηQL − ηuR k = 4.75; kηQL − ηdR k ≃ 4.60; kηuR − ηdR k ≃ 7(nn̄)

We find |cr(nn̄)| for r = 1, 2, 3 are << |c4
(nn̄)
(nn̄)
δm ≃ c4 hn̄|O4 |ni ≃



4µ

(nn̄)

|, and hence focus on c4493π 2Mnn̄



2

1/2

md

v

!8/3

. Then(nn̄)

hn̄|O4|ni

Requiring that τnn̄ = 1/|δm| agree with the lower limit from Super-K,τnn̄ > 4.7 × 108 sec. yields the lower bound on the mass scale of n − n̄ oscillations:
Mnn̄ > (47 TeV)τnn̄
4.7 × 108 sec

1/9 

µ
3 × 103 TeV

4/9  |hn̄|O (nn̄)|ni| 1/9
4
Λ6QCD

Hence, for relevant values of Mnn̄ in this model, n − n̄ oscillations could occur at alevel that is close to the current limit.

.

This model was constructed using the SM gauge group
GSM = SU(3)c ⊗ SU(2)L ⊗ U(1)Y . It is also of interest to analyze an
extra-dimensional model using the extended gauge group
GLRS = SU(3)c ⊗ SU(2)L ⊗ SU(2)R ⊗ U(1)B−L
in which B − L is gauged.
We have carried out this analysis in Girmohanta and RS, PRD 101, 095012 (2020) andGirmohanta, Mohapatra, and RS, PRD 103, 015021 (2021) with similar results.
There are also other models that can predict n − n̄ oscillations to occur near tocurrent limits (reviewed in the white paper J. Phys. G J. Phys. G 48, 070501 (2021)[arXiv:2006.04907]).

Conclusions• There is much progress in studies of neutrino masses and lepton mixing, and theireffects, including charged lepton flavor-violating processes. We have obtained new
constraints on massive neutrino emission via lepton mixing in particle decays, yielding
upper bounds on |Ue4|2 and |Uµ4|2, where ν4 is a heavy neutrino.
• We have discussed models that show how new physics beyond the SM can producen − n̄ oscillations at rates comparable with current limits. These models also showthat n − n̄ oscillations can be the main manifestation of baryon number violation
with proton decay being strongly suppressed.
Thank you to Alexander Studenikin and the organizers, and to the participants in this20’th Lomonosov Conference.

Backup Slides: General Formalism for n − n̄ Oscillations
In the (n, n̄) basis, writeM=M11 δmδm M22



Diagonalization yields mass eigenstates

where



|n1i
|n2i



=



cos θ sin θ− sin θ cos θtan(2θ) =



|ni
|n̄i



2δm
∆M

and ∆M = M11 − M22. The energy eigenvalues areE1,2 =

12



M11 + M22 ±p(∆M )2 + 4(δm)2Let ∆E = E1 − E2 =p(∆M )2 + 4(δm)2; transition probability:

P (n(t) → n̄) = |hn̄|n(t)i|2 = sin2(2θ) sin2[(∆E)t/2] e−λnt
"

2

(δm)
=
(∆M/2)2 + (δm)2#

hpi(∆M/2)2 + (δm)2 t e−λnt
sin2

pN.B.: if (∆M/2)2 + (δm)2 t << 1, then by expanding the sin, the quantity(∆M/2)2 + (δm)2 cancels, so
P (n(t) → n̄) ≃ [(δm)t]2 e−λnt = (t/τnn̄)2 e−λnt~ where B~ is a small residual magnetic field in a reactor
Although ∆M = 2~
µn · B,exp., this inequality enables exp. to be sensitive to δm.

n − n̄ Oscillations in Matter:
For n − n̄ oscillations involving a neutron bound in a nucleus, considermn,ef f. δmM=δm mn̄,ef f.
withmn,ef f = mn + Vn ,mn̄,ef f. = mn + Vn̄

where the nuclear potential Vn is real, Vn = VnR, but Vn̄ has an imaginary part
representing the n̄N annihilation: Vn̄ = Vn̄R − iVn̄I with
VnR, Vn̄R, Vn̄I ∼ O(100) MeV (Dover, Gal, Richard; Friedman; recently work by
Barrow, Golubeva, Ladd, Paryev, Richard for 12C (ESS) and 40Ar (DUNE)).
Mixing is thus strongly suppressed; tan(2θ) is determined by
2δm
2δm
p
<< 1
=
2
2
|mn,ef f. − mn̄,ef f.|(VnR − Vn̄R) + Vn̄I

−31
Using the reactor exp. bound on |δm|, this gives |θ| <∼ 10 . This suppression in
mixing is compensated for by the large number of nucleons in a nucleon decay detector,∼ 1033 n’s in Super-K.

Eigenvalues:q1
m1,2 =
mn,ef f. + mn̄,ef f. ± (mn,ef f. − mn̄,ef f.)2 + 4(δm)22Expanding m1 for the mostly n mass eigenstate |n1i ≃ |ni,m1 ≃ mn + Vn − i(δm)2 Vn̄I
2(VnR − Vn̄R)2 + Vn̄I

Imaginary part leads to matter instability, mainly via n̄n, n̄p → π’s, with rate2(δm)2|Vn̄I |=Γm.i. =
2
τm.i.
(VnR − Vn̄R)2 + Vn̄I
1

2
So τm.i. ∝ (δm)−2 = τnn̄.
2
, one has R ∼ O(100) MeV ∼ 1023 sec, dependent on
Writing τm.i. = R τnn̄
nucleus.

On integration calculations for the n − n̄ model: Given the localization of fermion
wavefunctions on scale Lµ << L, in the integration over the extra dimensions, canR∞
RL
extend 0 → −∞ to good approximation.
Integrals over extra dimensions have the general form (with

Z

n

h

d η exp −

m
X
i=1

2

aikη − ηfi k

i



π

= Pm

i=1 ai

n/2

exp

"

−

Pm

R

n

d η=

R∞

n
d
η)
−∞

2

j,k=1; j<k aj ak kηfj − ηfk k
Pm
s=1 as#.

For example, for m = 3,
Z


i
2
2
2
d η exp − a1kη − ηf1 k + a2kη − ηf2 k + a3kη − ηf3 k
=

" 
n/2

2
2
2 #
− a1a2kηf1 − ηf2 k + a2a3kηf2 − ηf3 k + a3a1kηf3 − ηf1 k
π
.
exp
=
a1 + a2 + a3
a1 + a2 + a3nhRecall result of calculation in n − n̄ model:Mnn̄ > (47 TeV)

1/9 
4/9µ
τnn̄
4.7 × 108 sec
3 × 103 TeV



(nn̄)

|hn̄|O4 |ni|
Λ6QCD

1/9

where ΛQCD = 0.25 GeV. This bound is not very sensitive to the precise size of(nn̄)
hn̄|O4 |ni because of the 1/9 power in the exponent. Lattice calculation:(nn̄)|hn̄|O4 |ni| ≃ 2Λ6QCD (Buchoff et al., 2019); substituting this yields factor
21/9 = 1.08, so lower bound is (1.08)47 TeV = 51 TeV.

.