Lepton Flavour Violation

Lepton Flavour Violation

has to occur, [m

] says so

Sacha Davidson, IN2P3/CNRS, France 1.what is LFV ?

2.but not see LFV : bounds...

3.what to do/learn from non-detaction ?

− parametrise exptal bounds

− explore what data tells

− constrain models

4.(Non-Standardν Interactions)

What is Lepton Flavour Change=LFV ?

• three lepton flavours in the Standard Model :e, µ, τ

(flavour≡charged mass eigenstate, so CLFV tautological)

• LFV≡charged lepton flavour change, at a point=ν osc. not count.

source detector

ν e

µ e

ν

What is Lepton Flavour Change=LFV ?

• three lepton flavours in the Standard Model :e, µ, τ

(flavour≡charged mass eigenstate, so CLFV tautological)

• LFV≡charged lepton flavour change, at a point=ν osc. not count.

source detector

ν e

µ e

ν

µ e

• ex of LFV :µ→eγ

• Lepton Flavour Changeoccurswithmν,U

because all lepton flavours are unconserved⇔no symmetries ifmν renormalisable Dirac :

A ∝ m²_ν

m²_W ⇒BR^<_∼10⁻⁴⁸ GIM suppressed(like quarks)

(NB : caln diverges for non-renormalisablemν)

⇒if see LFV, lepton flavour sector different from quarks ! supposemν NOT renormalisable⇔New Physics heavy

(Calculatingµ→eγ

µ

γ W e The dipole coefficient is a challenge to calculate :

a loop, but not calculate at zero-external-momentum because, eg

¯

eσαβF^αβµ ∝ ±2qαεβu¯eσ^αβuµ

(and same diagram gives charge renormalisation)

In a model, can follow, eg, Peskin+ Schroeder (caln ofg−2 in QED) : do the loop integral with finite external momentum, express it on a simple basis, eg

{εβγ^β, εβ(Pµ+pe)^β, εβ(Pµ−pe)^β}

(first is charge renorm, last vanishes by Ward, middle becomes dipole via Gordon decomp).

)

But we don’t see LFV : only exptal bounds

some processes current constraints on BR future sensitivities µ→eγγ <7.2×10⁻¹¹

µ→eγ <4.2×10⁻¹³ 6×10⁻¹⁴ (MEG) µ→eee¯ <1.0×10⁻¹²(SINDRUM) 10⁻¹⁶(2021, Mu3e) µA→eA <7×10⁻¹³Au,(SINDRUM) 10^−(16→?) (Mu2e,COMET)

10⁻¹⁸(PRISM/PRIME) K_L⁰→µe¯ <4.7×10⁻¹²(BNL)

K⁺→π⁺µe¯ <1.3×10⁻¹¹(E865) 10⁻¹²(NA62) τ →ℓγ <3.3,4.4×10⁻⁸ few×10⁻⁹(Belle-II) τ →3ℓ <1.5−2.7×10⁻⁸ few×10⁻⁹(Belle-II, LHCb τ →e{π, ρ, φ...} ^<∼few×10⁻⁸ few×10⁻⁹(Belle-II) h→τ^±{e, µ}^∓ <4.7,2.5×10⁻³

h→µ^±e^∓ <6.1×10⁻⁵ Z →e^±µ^∓ <7.5×10⁻⁷

BR≡Branching Ratio : (rate for process)/(total decay rate)

How to parametrise all those LFV processes

... add toL : three- and four-point LFV contact interactions.

Called “operators”, should respect relevant gauge symmetries (QED*QCD at low E, SM above m_W), and can be classified by dimension.

δL= + ...+ not e A¯/ µ

ok eσ¯ ·Fµ µR eL

+...+ eL +

eL

e_L

µ q +

eL

q

µ g + h.c.

eL

g µ

• belowmW, # legs is a better operator selection criteria than operator dimension, because SM masses can appear upstairs and downstairs. Eg,

dim5: mµ

Λ²_NPeσ¯ ·Fµ , dim7: 1

mQΛ²_NPeµGG¯

• chiral leptons motivated by 1) matching to SM atm_W, and 2) mass hierarchies : lighter leptons relativistic in decays of heavier leptons, so mass-suppressed interference between chiral operators

How to parametrise all those LFV processes

... add toL : three- and four-point LFV contact interactions.

Called “operators”, should respect relevant gauge symmetries (QED*QCD at low E, SM above mW), and can be classified by dimension.

building operators eg : 1008.4884,2005.00059,...

δL = X3 n=1

1 vⁿ

X

X,ζ

C_X^ζO^ζX +h.c.

v ≈mt, 2√

2GF =1/v²

{O^ζX}=QED*QCD invar operators with 3 or 4 legs X =Lorentz structure,ζ=flavour labels.

{C_X^ζ} dimless,≡“Wilson coefficients”.

calculable in models, constrained by exptal LFV bds Now ask : what can we learn from those bounds ?

(What isµA→eA≡µ→e conversion ?

• µ⁻ captured byAl nucleus, tumbles down to 1s.(r∼Zα/mµ>

∼r_Al)

• in SM : muon captureµ+p→ν+n

(What isµA→eA≡µ→e conversion ?

• µ⁻ captured byAl nucleus, tumbles down to 1s.(r∼Zα/mµ>

∼r_Al)

• in SM : muon captureµ+p→ν+n

• if LFV, boundµ interacts with nucleus, converts toe (Ee≈mµ)

Γ p

e p

µ µ D

e

Γ ={I, γ5, γ^α, γ^αγ5, σ} Γ ={S,P,V,A, T}

Γ n

e n µ

(What isµA→eA≡µ→e conversion ?

• µ⁻ captured byAl nucleus, tumbles down to 1s.(r∼Zα/mµ>

∼r_Al)

• in SM : muon captureµ+p→ν+n

• if LFV, boundµ interacts with nucleus, converts toe (Ee≈mµ)

Γ p

e p

µ µ D

e

Γ ={I, γ5, γ^α, γ^αγ5, σ} Γ ={S,P,V,A, T}

Γ n

e n µ

≈WIMP scattering on nuclei

1) “Spin Independent” rate ∝A² (amplitude∝P

N∝A)

2) “Spin Dependent” rate ∼ΓSI/A²(Σnucleons∝spin of only unpaired N)

Why isµA→eA conversion interesting ?

1. maybe whereµ→e could be discovered ?

◮ experimental sensitivity to BR will improve :10⁻¹²→10⁻¹⁶ (expts under construction)→10⁻¹⁸ (planned expts)

(future sensitivity toBR(µ→e¯ee)∼10⁻¹⁶)

◮ ΓSI(µA→eA)∝A²,whereasΓ(µA→νA^′)is not coherently enhanced. So

BR_SI(µA→eA)≡ΓSI(µA→eA)

Γ(µA→νA′) ∝A²|X C|²

⇒sensitivityC ∼√ BR/A?

Why isµA→eA conversion interesting ?

1. maybe whereµ→e could be discovered ?

◮ experimental sensitivity to BR will improve :10⁻¹²→10⁻¹⁶ (expts under construction)→10⁻¹⁸ (planned expts)

(future sensitivity toBR(µ→e¯ee)∼10⁻¹⁶)

◮ ΓSI(µA→eA)∝A²,whereasΓ(µA→νA^′)is not coherently enhanced. So

BR_SI(µA→eA)≡ΓSI(µA→eA)

Γ(µA→νA′) ∝A²|X C|²

⇒sensitivityC ∼√ BR/A?

2. ...its coherent⇔many SI operators interfere BR_SI(µA→e_RA) =32G_F²m⁵_µ

Γcapt

eC_V^pp_,RV_A^(p)+Ce_S,L^′ppS_A^(p)+Ce_Vⁿⁿ_,RV_A⁽ⁿ⁾+Ce_S,L^′nnS_A⁽ⁿ⁾+CD,L

4 D² if observed, different nuclear targets (w/wo spin) can potentially

distinguish which coefficients,but more theory calculations required

(nuclear physics,χPT corrections to lepton-nucleon interactions...)

DKunoSaporta DKunoYamanaka

Complementarity ofµ↔e, τ↔ℓand colliders

• bds onµ→eγ, µ→eee¯ andµA→eAmuch more restrictive thanτs : 10⁻⁷∼ ^Γ(τ_Γ(τ^→_→^ℓ+...)_ℓνν)¯ ≈q

Γ(µ→e+...)

Γ(µ→eνν)¯ →10⁻⁸

(recallΓ(τ→ℓνν)¯ ≃Γτ/6 because hadronicτdecays)

becauseτµ∼10⁷ττ, so can make muon beams : PSI plans∼10¹⁰µ/sec

(allowsBR(µ→e¯ee)∼10⁻¹⁶), vs Belle-II aims to pair-produce∼100τ/sec.

⇒?Can one search forℓ→τwith intenseℓbeams on target ? Cirigliano etal 2102.06176

• more bds inτ↔ℓsector :better thanµ↔e for discriminating models ? In P^articleD^ataB^ook,BR^<_∼10⁻⁶→10⁻⁸forτ→ℓ+{π, η, ρ, φ, ω,K,l_i¯l_j, ..}. And isospin, parity/lorentz properties of mesons restrict the operators that can contribute :

BR(τ →eπ) ∝ |C_A^eµuu−C_A^eµdd|²+ #|C_P^eµuu−C_P^eµdd|²+interference BR(τ →eη) ∝ |C_A^eµuu+C_A^eµdd + #C_A^eµss|²+ #pseudoscalar

BR(τ →eρ) ∝ |C_V^eµuu−C_V^eµdd|²

to compare to

BR(µTi→eTi)∝ |C_S^eµuu+C_S^eµdd+.2(C_V^eµuu+C_V^eµdd) +.05C_S^eµss+.01C_S^eµcc+.002C_S^eµbb|²

how to calculateτ →µπ0? If add to Lagrangian :

δL = 1

v²(µγ^µP_Lτ)

C_R^uu(uγ^µP_Ru) +C_L^uu(uγ^µP_Lu) +{u↔d} obtain matrix elements like

M(τ →µπ0) = C_L^uu

v² (uµγ^αPLuτ)h0|uγ^µPLu|π(P)i+{L↔R,u↔d} A pedagogical intro to χPT is1804.05664, by A Pich, where :

h0|dγ^βγ5u|π⁺(P)i=iP^β√

2fπ ⇒ Γ(τ →πν) = G_F²f_π²m_τ³ 8π 1

2h0|(uγ^βγ5u−dγ^βγ5d)|π⁰(P)i=iP^βfπ

withfπ=92.2 MeV.^(fπdepends on√

2 conventions, can be 130 MeV). Then

δL ⊃ 1

v²(µγ^βPLτ)C_R^uu−C_L^uu

2 (uγβγ5u) +C_R^dd −C_L^dd

2 (dγβγ5d) +...

⊃ 1

v²(µγβPLτ)C_R^uu−C_L^uu−C_R^dd+C_L^dd

4 [(uγβγ5u)−(dγβγ5d)] +...

so can calculate BR, or _Γ(τ^Γ(τ_→^→_π^π−^0µ)ν)

Complementarity, ctd : and colliders ?

• sufficiently energetic colliders can produce new heavy particles

• colliders probe LFV processes with heavy legs :h,t,Z τ↔ℓ: LHC competitive with low energy ;

µ↔e: better sensitivity at low energy,but could be cancellations.eg δL= _c^g_WC_Z^eµe Z¯/ P_Lµ→ ^Γ(ZΓ(Z^→→^e±µ^µµ)¯^∓) ≃ ^|C_|g^Zeµ_L^e_|^2|2 ≤^7.53.4^××¹⁰10⁻⁷⁻² ⇒ |C_Z^eµ|∼^<10⁻³

+ eL

µL

eL

eL

Z

eL

µL

eL

eL

C^eµee

BR(µ→3e)∼ |g_L^eC^eµ+C^eµee|²≤10⁻¹²

⇒ |g_L^eC_Z^eµ+C^eµee| ≤10⁻⁶ but hesitate to use “naturalness” notions for EFT concellations(coefficients are functions of model parameters, and cancellations depend on operator basis choice, eg could use gradient operator at Z vertex)

(How small can we seevsHow big could it be ?)

sensitivity≡how small a coefficient could one see ?

⇔“setting bounds one operator at a time”

|g_L^eC_Z^eµ+C^eµee| ≤10⁻⁶⇔ |g_L^eC_Z^eµ|,|C^eµee| ≤10⁻⁶

µ ee

Ce

Z C

constraints

≡values of a coefficients excluded by the data

|g_L^eC_Z^eµ+C^eµee| ≤10⁻⁶

|C_Z^eµ| ≤10⁻³

⇔ |C_Z^eµ|,|C^eµee|^<∼10⁻³

Are those bounds restrictive ? What doesBR(µ→eee)¯ <10⁻¹² mean ?

BRs ofµ, τ normalised toweakdecay, mediated by δL=2√

2GF(¯eγ^αP_Lν)(¯νγ^αP_Lµ), 2√

2GF =1/v²

mµ=.105 GeV v=174 GeV

BR(µ→eee¯ )≡ Γ(µ→eee¯ )

Γ(µ→e¯νν) , Γ(µ→e¯νν) = G_F²m⁵_µ

192π³ = m⁵_µ 1536π³v⁴ so if δLLFV = (¯eγ^αP_Lµ)(¯eγ^αP_Le)/Λ²_LFV ⇔C=v²/Λ²_LFV

Are those bounds restrictive ? What doesBR(µ→eee)¯ <10⁻¹² mean ?

BRs ofµ, τ normalised toweakdecay, mediated by δL=2√

2GF(¯eγ^αPLν)(¯νγ^αPLµ), 2√

2GF =1/v²

mµ=.105 GeV v=174 GeV

BR(µ→eee¯ )≡ Γ(µ→eee¯ )

Γ(µ→e¯νν) , Γ(µ→e¯νν) = G_F²m⁵_µ

192π³ = m⁵_µ 1536π³v⁴ so if δLLFV = (¯eγ^αP_Lµ)(¯eγ^αP_Le)/Λ²_LFV ⇔C=v²/Λ²_LFV

Γ(µ→eee¯ )≃ m_µ⁵

1536π³Λ⁴_LFV ⇒ BR^<_∼

10⁻¹²⇒ΛLFV ∼10³v ≃200TeV 10⁻¹⁶⇒ΛLFV ∼10³v ≃2000TeV

vsΛLNV ∼v²/mν∼3×10¹¹TeV

Are those bounds restrictive ? What doesBR(µ→eee)¯ <10⁻¹² mean ? BRs ofµ, τ normalised toweakdecay, mediated by

δL=2√

2GF(¯eγ^αPLν)(¯νγ^αPLµ), 2√

2GF =1/v²

mµ=.105 GeV v=174 GeV

BR(µ→eee¯ )≡ Γ(µ→eee¯ )

Γ(µ→e¯νν) , Γ(µ→e¯νν) = G_F²m⁵_µ

192π³ = m⁵_µ 1536π³v⁴ so if δLLFV = (¯eγ^αPLµ)(¯eγ^αPLe)/Λ²_LFV ⇔C=v²/Λ²_LFV

Γ(µ→eee¯ )≃ m_µ⁵

1536π³Λ⁴_LFV ⇒ BR^<_∼

10⁻¹²⇒ΛLFV ∼10³v ≃200TeV 10⁻¹⁶⇒ΛLFV ∼10³v ≃2000TeV

Compare to ^(g−₂^2)µ ≡a≃αem/π (electromagneticamplitude) :

torque~τ=~µ×B~;~µ=g_2m^e ~S

∆a ≡ a^SM−a^exp≃3×10⁻⁹

∼ m²_µ 16π²Λ²_NP

⇒ΛNP∼mt.

Are those bounds restrictive ? What doesBR(µ→eee)¯ <10⁻¹² mean ? BRs ofµ, τ normalised toweakdecay, mediated by

δL=2√

2GF(¯eγ^αPLν)(¯νγ^αPLµ), 2√

2GF =1/v²

mµ=.105 GeV v=174 GeV

BR(µ→eee¯ )≡ Γ(µ→eee¯ )

Γ(µ→e¯νν) , Γ(µ→e¯νν) = G_F²m⁵_µ

192π³ = m⁵_µ 1536π³v⁴ so if δLLFV = (¯eγ^αPLµ)(¯eγ^αPLe)/Λ²_LFV ⇔C=v²/Λ²_LFV

Γ(µ→eee¯ )≃ m_µ⁵

1536π³Λ⁴_LFV ⇒ BR^<_∼

10⁻¹²⇒ΛLFV ∼10³v ≃200TeV 10⁻¹⁶⇒ΛLFV ∼10³v ≃2000TeV Compare to decays of Υ(2s)electromagnetically decayingbb¯ state

BR(Υ→τµ) =¯ Γ(Υ→τµ)¯ Γ(Υ→µ¯µ)

∼

1/Λ²_LFV e²Qb/m²_Υ

2

≤ 6×10⁻⁶ 2.5×10⁻² which excludes ΛLFV <

∼ 100 GeV.

Sensitivity to New Physics in loops

Two dipole operators contribute to µ→eγ:

δLmeg = −4GF

√2mµ C_R^DµRσ^αβeLFαβ+C_L^DµLσ^αβeRFαβ

BR(µ→eγ) = 384π²(|C_R^D|²+|C_L^D|²)<4.2×10⁻¹³

⇒ |C_X^D|∼^<10⁻⁸ MEG expt, PSI

Sensitivity to New Physics in loops

Two dipole operators contribute to µ→eγ:

δLmeg = −4GF

√2mµ C_R^DµRσ^αβeLFαβ+C_L^DµLσ^αβeRFαβ

BR(µ→eγ) = 384π²(|C_R^D|²+|C_L^D|²)<4.2×10⁻¹³

⇒ |C_X^D|∼^<10⁻⁸ MEG expt, PSI

How big does one expectC to be ?

n=1 n=2 Cmµ

v² ∼ emµ

(16π²)ⁿΛ² ⇒probes Λ_∼^< 100TeV 10TeV Cmµ

v² ∼ ev

(16π²)ⁿΛ² ⇒probes Λ_∼^< 3000TeV 300TeV 2-loop sensitivity to New Particles that are beyond the reach of the LHC...

But QED loops areO(α/4π)...negligeable in discovery mode for LFV ?

But QED loops areO(α/4π)...negligeable in discovery mode for LFV ?

Work top-down = suppose model that gives only tensor op. atmW : δL(mW) =2√

2GF CT(uσu)(eσPYµ) 1 : forget loopsMatch to nucleonsN∈ {n,p}as Ce_T^NN=hN|uσu¯ |NiC_T^uu_∼^{< 3}₄C_T^uu

⇒BR ≈BR_SD ≈ ¹₂|C_T|²

nuclear matrix elements : EngelRTO, KlosMGS

But QED loops areO(α/4π)...negligeable in discovery mode for LFV ?

Work top-down = suppose model that gives only tensor op. atmW : δL(mW) =2√

2GF CT(uσu)(eσPYµ) 1 : forget loopsMatch to nucleonsN∈ {n,p}as Ce_T^NN=hN|uσu¯ |NiC_T^uu_∼^{< 3}₄C_T^uu

⇒BR ≈BR_SD ≈ ¹₂|C_T|²

nuclear matrix elements : EngelRTO, KlosMGS

2 : include loops T

e

µ u

u

+... ⇒

C_T^uu(uσu)(eσPYµ) S

e

µ q

q

64^α_4π^elog^m_m^W

τ C_T^uu (uu)(ePYµ)

∆C_S^uu(mτ)∼ ¹₇C_T^uu(mW)

But QED loops areO(α/4π)...negligeable in discovery mode for LFV ?

Work top-down = suppose model that gives only tensor op. atmW : δL(mW) =2√

2GF CT(uσu)(eσPYµ) 1 : forget loopsMatch to nucleonsN∈ {n,p}as Ce_T^NN=hN|uσu¯ |NiC_T^uu_∼^{< 3}₄C_T^uu

⇒BR ≈BR_SD ≈ ¹₂|C_T|²

nuclear matrix elements : EngelRTO, KlosMGS

2 : include loops T

e

µ u

u

+... ⇒

C_T^uu(uσu)(eσPYµ) S

e

µ q

q

64^α_4π^elog^m_m^W

τ C_T^uu (uu)(ePYµ)

∆C_S^uu(mτ)∼ ¹₇C_T^uu(mW) Then match to nucleons : Ce_S^NN=hN|uu¯ |Ni∆C_S^uu ∼C_T^uu soCe_S^pp _∼^>Ce_T^pp,

BR ≈BR_SI ∼Z²|2C_T^uu|²∼8Z²BR_SD

loops can change Lorentz structure/external legs⇒different operator whose coefficient better constrained

Recall Outline :

1.SM leptons,mν and LFV 2.but not see LFV : bounds...

3.what to do/learn from non-detaction ?

− parametrise exptal bounds

− explore what data tells

− constrain models 4.Non-Standardν Interactions

Assume New LFV physics heavy : M_NP≫m_SM ∼Λexpt

Wilsonian interpretation of renormalisation says that translations in scale mean

add/remove loops of dynamical ( = SM) particles

data↔models : how to include loops ?

1. you have intuition to find the right model

⇒calculateM= matrix element for observable ! but :SM calns conceptually+ technically difficult

= expand in loops and many couplings,

many different mass scales in integrals+ need to resum QCD...

also, have to repeat as many times as your taste in models changes

data↔models : how to include loops ?

1. you have intuition to find the right model

⇒calculateM= matrix element for observable ! but :SM calns conceptually+ technically difficult

= expand in loops and many couplings,

many different mass scales in integrals+ need to resum QCD...

also, have to repeat as many times as your taste in models changes 2. only dynamics belowMNP is SM : use EFT

(already met EFT as parametrisation-via-contact-interactions of heavy New Physics)

EFT simplifies loop calculations by expanding in scale ratios eg replace NP by contact interactions

(and allows to get right=Wilsonian logs in dim reg ! See Georgi, “EFT”, ARNPS, 1993)

◮ match model to EFT atM_NP by equating matrix elements M^model=M(C_I)

◮ SM loop corrections toMobtained via Renormalisation Group running of operator coefficients(coefficients “run” like all coupling constants)

Once operator RGEs are calculated, its done !

EFT to translate in scale ;top-down or bottom-up match model to{CI} atMNP (equateMs)

run{CI} toΛexpt with RGEs

M(CI) =M(model) MNP >TeV

{Z,W, γ,g,h,t,f}

M(Cabove) =M(Cbelow) m_W ∼m_h∼m_t {γ,g,eµ, τ,u,d,c,s,b}

RGEs : µ_∂µ^∂ C~ =CΓ~

⇒C(m~ τ)∼C(m~ W)exp{−Γlog}

GeV ∼m_c,m_b,mτ

{γ,e, µ,p,n,(π)} data

Many models, data fixed : lets takeµ→e data toM_NP

Counting constraints :µ→eγ+µ→ee e¯

µ e

Two dipole operators contribute to µ→eγ: δLmeg = −4GF

√2mµ CD,RµRσ^αβe_LFαβ+CD,LµLσ^αβe_RFαβ BR(µ→eγ) = 384π²(|CD,R|²+|CD,L|²)<5.7×10⁻¹³⇒ |C_X^D|^<∼10⁻⁸

Counting constraints :µ→eγ+µ→ee e¯

µ e

Two dipole operators contribute to µ→eγ: δLmeg = −4GF

√2mµ CD,RµRσ^αβe_LFαβ+CD,LµLσ^αβe_RFαβ BR(µ→eγ) = 384π²(|CD,R|²+|CD,L|²)<5.7×10⁻¹³⇒ |C_X^D|^<∼10⁻⁸ 2dipoles + 6 4-f-ops contribute toµ→eee,¯ (most interference between ops

∝m²_e/m²_µ)

(ePLµ)(ePLe)

s e +

e e µ

(ePRµ)(ePRe)

s e +

e e µ

(eγPLµ)(eγPLe)

v e +

e e µ

(eγPRµ)(eγPRe)

v e +

e e µ

(eγPLµ)(eγPRe)

v e +...

e e µ

BR(µ→eee¯ ) = |CS,LL|²+|CS,RR|²

8 +2|CV,RR|²+2|CV,LL|²+|CV,LR|²+|CV,RL|²

≤ 10⁻¹² ⇒ |C_X|^<_∼10⁻⁶ q

BR/10⁻¹²

see nothing in µ→eγ,µ→eee¯ ,⇒all 8 Cs small see something⇒distinguish operator via angular distributions ?

adding up constraints fromµA→eA,µ→eγ+µ→eee¯ 1.constrain 2 dipoles +6 4ℓcoeffs withµ→eγ +µ→eee¯ 2.µA→eA, Spin Indep, now : two targets{Au, Ti }

two constraints/target^(outgoingeL,eR)

SI future : 3 independent targets ?

3.Spin-Dependent, now : ( ?) 2 counstraints ? (Ti ?) SD future : 4 constraints ?

⇒

now 12→14

future ^<_∼20

constraints

≪ #operators

∼< #parameters in many models

adding up constraints fromµA→eA,µ→eγ+µ→eee¯ 1.constrain 2 dipoles +6 4ℓcoeffs withµ→eγ +µ→eee¯ 2.µA→eA, Spin Indep, now : two targets{Au, Ti }

two constraints/target^(outgoingeL,eR)

SI future : 3 independent targets ?

3.Spin-Dependent, now : ( ?) 2 counstraints ? (Ti ?) SD future : 4 constraints ?

⇒

now 12→14

future ^<_∼20

constraints

≪ #operators

∼< #parameters in many models

• Inµ→esector, expt probes∼12-dim subspace of∼100- dim space of operators. ( = Many “flat directions” ; this is feature of data, not problem of EFT).

Have exptal subspace @MNP, just need to project models onto this subspace ...lots to do !

• “basis vectors” of exptal subspace have projection on most operators⇔ most operators mix via loops into observables. So expts havesensitivity to most operators

Can still calculate sensitivities...

sensitivity :“one at a time bound” = below, parameter is to small to see in expt. (But larger possible, if cancelled by another contribution.)

coefficient µ→eγ µ→e¯ee µ−econv.

|CD,X| 1.12×10⁻⁸ 4.30×10⁻⁷ 2.35×10⁻⁷

|C_V^ee_,XX| 1.10×10⁻⁴ 7.80×10⁻⁷ 1.86×10⁻⁵

|C_V^ee_,XY| 2.55×10⁻⁴ 9.34×10⁻⁷ 3.77×10⁻⁵

|C_S,XX^ee | 1.73×10⁻⁴ 2.8×10⁻⁶ (3.64×10⁻³)

|C_V^µµ_,XX| 1.10×10⁻⁴ 5.60×10⁻⁵ 1.85×10⁻⁵

|C_V^µµ_,XY| 2.56×10⁻⁴ 1.12×10⁻⁴ 3.77×10⁻⁵

|C_S,XX^µµ | 8.24×10⁻⁷ (1.58×10⁻⁵) (1.73×10⁻⁵)

|C_V^{τ τ}_,XX| 3.80×10⁻⁴ 1.95×10⁻⁴ 1.24×10⁻⁵

|C_V^{τ τ}_,XY| 4.40×10⁻⁴ 1.91×10⁻⁴ 1.25×10⁻⁵

|C_S,XX^{τ τ} | 5.33×10⁻⁶ 1.02×10⁻⁴ 1.12×10⁻⁴

|C_S,XY^{τ τ} | — — —

|C_T,XX^{τ τ} | 1.10×10⁻⁸ (4.20×10⁻⁷) (2.30×10⁻⁷)

Table –Current sensitivities ofµ→eγ,µ→ee e¯ , andµ−econv.to the coefficients, atm_W, of QCD×QED-invariant 2- and 4-lepton operators.

X,Y ∈ {L,R},X 6=Y.

Summary

Lepton Flavour Violation is BSM that exists. Not yet detected, but experimental sensitivities for µ↔e should improve by orders of magnitude in coming years (→BR∼10⁻¹⁶→10⁻¹⁸ ).

Current experimental constraints on µ↔e are few, but restrictive⇒ sensitivity to loop-induced LFV at scales beyond the LHC. Butτ ↔ℓ maybe better for discriminating models ?

If assume LFV induced by heavy New Physics, can parametrise LFV interactions with non-renormalisable operators, and systematically include SM loop effects via RGEs for operator coefficients. Find that, in µ↔e sector

• (many) more operators than constraints

• but constraints probe independent directions in operator space

• It would be interesting to explore the ability ofµ↔e observables to discriminate models...

But we already found some BSM in neutrino oscillations, maybe we could find some more = NSI !

NSI

BSM to find inν oscillations

(not neccessarily BSM where to learn about mass mechanism)

Non-Standard Interactions

Wolfenstein, Valle, GuzzoMasieroPetcov

NSI: δL=−2√

2GFε^ρσ_f (νργαPLνσ)(fγ^αf) , f ∈ {e,d,u} εmatrix QED×QCD invariant.

Non-Standard Interactions

Wolfenstein, Valle, GuzzoMasieroPetcov

NSI: δL=−2√

2GFε^ρσ_f (νργαPLνσ)(fγ^αf) , f ∈ {e,d,u} εmatrix QED×QCD invariant. At finite density

hmedium|fˆγαfˆ(x)|mediumi →δα0nf , contributes to forward scattering amplitude ⇔“effective

∆m²/E ∼√

2GFne” to oscillation Hamiltonian

Non-Standard Interactions

Wolfenstein, Valle, GuzzoMasieroPetcov

NSI: δL=−2√

2GFε^ρσ_f (νργαPLνσ)(fγ^αf) , f ∈ {e,d,u} εmatrix QED×QCD invariant. At finite density

hmedium|fˆγαfˆ(x)|mediumi →δα0nf , contributes to forward scattering amplitude ⇔“effective

∆m²/E ∼√

2GFne” to oscillation Hamiltonian

source, CC,messy clean, quantum, NC

probe (little known)ν propagator detector, CC, messy ν e

µ e

ν

interest : ν oscillations= quantum mechanics on macroscopic scales = very sensitive window to probe poorly knownν propagator (⇔mν + NC ν interactions)

Current constraints on NSI from oscillation data + COHERENT

EstebanGonzalezGarciaMaltoniEtal

Add NSI to low-EL^ν (add no new CC), suppose ε^αβ_f =ε^αβεf.

−.008< ε^ee_u < .62 −.06< ε^eµ_u < .05 −.25< ε^eτ_u < .11

−.01< ε^ee_d < .56 −.06< ε^eµ_d < .05 −.21< ε^eτ_d < .11

−.01< ε^ee_e <2.0 −.18< ε^eµ_e < .15 −.86< ε^eτ_e < .35

−.11< ε^µµ_u < .40 −.012< ε^µτ_u < .009

−.10< ε^µµ_d < .36 −.011< ε^µτ_d < .009

−.36< ε^µµ_e <1.3 −.035< ε^µτ_e < .35

−.11< ε^{τ τ}_u < .40

−.10< ε^{τ τ}_d < .36

−.35< ε^{τ τ}_e <1.40

≈constraints= bigger is incompatible with data.

Comments...

◮ ?oscillations (maybe) have separate sensitivity to NSI onuanddbecause the sun is made of protons ?

◮ Oscillations only sensitive toε^αα−ε^ββ, but COHERENT lifts degeneracy (NC scattering, sensitive toε^σρ)

◮ rangesneglectother solutions where SM parameters disconnected from bestfit values (LMA-Dark solution) !

◮ ε^αα_e ∼1 allowed because flips sign of SM(¯νγPLν)( ¯fγPLf)(oscillations sensitive to signs, but only of flavour differences...)

◮ not matched onto SMEFT, so not accounting for potential contribution to flav-diagonal

“SM” inputs by CC or charged-lepton components of the SMEFT operator.

◮ Energy scales :q²→0 in matter effect, 30-70 MeV at COHERENT.

But Standard Model neutrinos are in a doubletℓρ= νρ

eρ

...LFV ? New Physics must respect SM gauge symmetries : given bounds on (charged) Lepton Flavour Violation, can NSI be detectably large ?

But Standard Model neutrinos are in a doubletℓρ= νρ

eρ

...LFV ?

New Physics must respect SM gauge symmetries : given bounds on (charged) Lepton Flavour Violation, can NSI be detectably large ?

• ex : SU(2) invariant dimension 6 operators that induceντ→νµ NSI one ε^{τ µ}_(3)ℓℓ(ℓτγατ^aℓµ)(ℓeγ^ατ^aℓe), ε^{τ µ}_ℓℓ(ℓτγαℓµ)(ℓeγ^αℓe), ε^{τ µ}_ee(ℓτγ^αℓµ)(eeγµee)

NSI∝ε^{τ µ}_(3)ℓℓ+ε^{τ µ}_ℓℓ, ε^{τ µ}_ee

BR(τf →3l)≃ |ε^{τ µ}_(3)ℓℓ−ε^{τ µ}_ℓℓ|²+|ε^{τ µ}_ee|^2<∼10⁻⁷ ...

⇒LFV constraints, applied at tree level, exclude several (combinations of) dim 6 operators from inducing observable NSI.

But Standard Model neutrinos are in a doubletℓρ= νρ

eρ

...LFV ?

New Physics must respect SM gauge symmetries : given bounds on (charged) Lepton Flavour Violation, can NSI be detectably large ?

• ex : SU(2) invariant dimension 6 operators that induceντ→νµ NSI one ε^{τ µ}_(3)ℓℓ(ℓτγατ^aℓµ)(ℓeγ^ατ^aℓe), ε^{τ µ}_ℓℓ(ℓτγαℓµ)(ℓeγ^αℓe), ε^{τ µ}_ee(ℓτγ^αℓµ)(eeγµee)

NSI∝ε^{τ µ}_(3)ℓℓ+ε^{τ µ}_ℓℓ, ε^{τ µ}_ee

BR(τf →3l)≃ |ε^{τ µ}_(3)ℓℓ−ε^{τ µ}_ℓℓ|²+|ε^{τ µ}_ee|^2<∼10⁻⁷ ...

⇒LFV constraints, applied at tree level, exclude several (combinations of) dim 6 operators from inducing observable NSI.

• To avoid LFV constraints, build NSI at dim 8f ∈ {e,u,d,q1, ℓe} : C_f^ρσ

Λ⁴ (ℓρH)γα(H^†ℓσ)(fγ^αf)^H−→^→v C_f^ρσv²

Λ⁴ (νργανσ)(fγ^αf), ε^ρσ_f = C_f^ρσv⁴ Λ⁴ ε^ρσ_f ^>_∼10⁻²⇔Λ_∼^<.3→1 TeV⇒is there a model ?

Is there a model ?

1. 10⁻²_∼^<ε^<_∼1 suggests feebly-coupled mediator,m≪mW?

• ∼10 MeVZ^′, flav.diag. couplingg^′∼10⁻⁴toℓµ, ℓτ,qL,1,u_R,d_R.

Farzan

• light Z’ feebly coupled to quarks andνsterile, smallmνsνSM.

PospelovPradler

avoid someν scattering bounds ifm_mediator² ≪ hq²i avoid inducing LFV by chosing couplings...

2.

3. heavy New Physics,m_mediator∼^>m_W recipe :GavelaHernandezOtaWinter

tune NP masses/cplgs so tree LFV coefficients vanish(dim 6 and 8) : eg oneat dimension 6, need

ε^{τ µ}_(3)ℓℓ−ε^{τ µ}_ℓℓ =ε^{τ µ}_ee =0

ex : scalar + vector leptoquark with tuned masses/couplings.

or scalar bileptonS, withL=2,Qem=1,Sℓ^α_iǫ^ijℓ^β_j, induces only 2e2ν

⋆can do EFT = results that apply to many models

Summary

Lepton Flavour Violation is BSM that exists. Not yet detected, but experimental sensitivities for µ↔e should improve by orders of magnitude in coming years (→BR∼10⁻¹⁶→10⁻¹⁸ ).

Current experimental constraints on µ↔e are few, but restrictive⇒ sensitivity to loop-induced LFV at scales beyond the LHC. Butτ ↔ℓ maybe better for discriminating models ?

If assume LFV induced by heavy New Physics, can parametrise LFV interactions with non-renormalisable operators, and systematically include SM loop effects via RGEs for operator coefficients. Find that, in µ↔e sector

• (many) more operators than constraints

• but constraints probe independent directions in operator space

• It would be interesting to explore the ability ofµ↔e observables to discriminate models...

But we already found some BSM in neutrino oscillations, maybe we could find some more = NSI !

Some Phenomenological LFV/EFT Reviews :

µ↔e: Kuno and Okada, Rev.Mod.Phys. 73(2001)151, hep-ph/9909265 BelleII : The BelleII Physics Book, 1808.10567

EFT : Georgi, Ann.Rev.Nucl.Part.Phys. (1993)

Les Houches school, July 2017, Neubert, Manohar, Silvestrini(quarks)

Definitions, some references...

I use chiral Dirac spinors, 4 degrees of freedom(dof) ψ=

ψL

ψR

, {γ^α}=

0 I I 0

,

0 σi

−σi 0

{σi}= 0 1

1 0

,

0 −i i 0

,

1 0 0 −1

ψL= PLψ avec PL= (1−γ5)

2 , ψR = PRψ

chirality notobservable (→helicity =±sˆ·kˆ=±1/2 in relativistic limit), butPL,R simple to calculate with :)

notation :(ψR) = (PRψ)^†γ0=ψ^†P_Rγ0=ψ^†γ0P_L= (ψ)L

(ψ^c)L=P_L(−iγ0γ2γ0ψ^∗) =−iγ0γ2γ0ψ_R^∗

operator basis forµA→eA,µ→eee,¯ µ→eγ atΛexpt

Kuno Okada

µ interaction with nucleonN∈ {n,p}parametrised by 20 4-f operators : S, V ePXµNN eγ^αPXµNγαN X ∈ {L,R} A, T eγ^αP_XµNγαγ5N eσ^αβP_XµNσαβN

P, Der eP_XµNγ5N eγ^αP_Xµ(Ni ∂^↔αγ5N)

Matching inχPT gives Derivative. But absorb in matching intoG_O^N,q= quark matrix elements in nucleons. and 2 dipoles

D eσ^αβPXµFαβ

which also contribute inµ→eγ,µ→eee¯ . Forµ→eee¯ V (eγ^αPYµ)(eγαPYe) (eγ^αPYµ)(eγαPXe)

S (ePYµ)(ePYe)

chiral basis for the lepton current (relativistice), but not for the non-rel. nucleons.

28 operators

µ↔e operators (at scalemW ↔mτ) ; otherwise flav.diag.

Kuno-Okada

mµ(eσ^αβP_Yµ)Fαβ dim5

(eγ^αP_Yµ)(eγαP_Ye) (eγ^αP_Yµ)(eγαP_Xe) (ePYµ)(ePYe) (ePYµ)(µPYµ)

(eγ^αP_Yµ)(µγαP_Yµ) (eγ^αP_Yµ)(µγαP_Xµ) dim6 (eγ^αP_Yµ)(fγαP_Yf) (eγ^αP_Yµ)(fγαP_Xf)

(ePYµ)(f PYf) (ePYµ)(f PXf) (eσPYµ)(fσPYf)

1 mt

(ePYµ)GαβG^αβ 1 mt

(ePYµ)GαβGe^αβ dim7 1

m_t(ePYµ)FαβF^αβ 1

m_t(ePYµ)FαβFe^αβ

...zzz...

f ∈ {τ,u,d,c,s,b},P_X 6=P_Y = (1±γ5)/2

82 operators.+ 80 more if allow quark flavour-changing.×3 to account forµ↔e,τ ↔e, andτ↔µ. + 48∆Li=2.

To calculate SIµA→eA(at LO inχPT...)

intro for QFTists +SDep in DKunoSaporta

differs from WIMP scattering in that µand nucleus charged

1. suppose start with µ↔e operators involvinggluons,γ,u,d,s,c,b 2. match quark/gluon operators onto nucleon(N∈ {n,p})operators :

Gs in Appendix

¯

q(x)ΓOq(x)→G_O^N,qN(x¯ )ΓON(x)

eg, hN|q(x)q(x)¯ |Ni=G_O^N,qhN|N(x¯ )N(x)|Ni=G_O^N,qu_N(Pf)uN(Pi)e^{−i(Pf−Pi)x}