[PPT] - Connecting Neutrinoless double beta decay to colliders. Or not. PowerPoint Presentation

SLIDE 1

Connecting Neutrinoless double beta decay to

colliders. Or not.

Amherst, Massachusetts July 2017

Michael Graesser (Los Alamos) based on: MG, arXiv:1606.04549, submitted to JHEP

V. Cirigliano, W. Dekens, MG, E. Mereghetti, (PLB 2017,1701.01443)
V. Cirigliano, W. Dekens, J. de

Vries, MG, E. Mereghetti, (1707/08.zzzz)

SLIDE 2

Neutrinoless double beta decay and TeV* scale physics Motivation

Neutrinos have mass and search is on to discover the nature of their mass. Ongoing or future experiments may detect a “neutrinoless double beta decay” signal. Such a signal arises when neutrino masses violate lepton number (i.e., Majorana) Question: is that the correct interpretation of such a signal? Are there other (new physics scenario) interpretations?

SLIDE 3

New physics scenarios for neutrinoless double beta decay

If hierarchy is “normal’’, then planned 0nubb have no chance of detecting Standard Model

Majorana neutrinos (outside of the quasi-degenerate region)

In such a circumstance, only hope is for exotic scenarios

Comparison SuperNEMO sensitivity to various admixtures of WR contribution (0%, 30%, 100%). Figure from Arnold et. al. (SuperNEMO, 2010) Should a ΔL=2 signal be detected, such exotic possibilities should be excluded before concluding that effect is due to Majorana neutrino exchange Resolving competing explanations may need a next-generation detector reconstructing both electron kinematics (e.g. NEXT, SuperNEMO)

4 2 2 4 50 100 150 200 250 300 Λ 107 mΝ meV

(a)

4 2 2 4 50 100 150 200 250 300 Λ 107 mΝ meV

(b)

4 2 2 4 50 100 150 200 250 300 Λ 107 mΝ meV

(c)

SLIDE 4

X

e− e− d u u d

BSM contributions to neutrinoless beta decay: Left-Right symmetric model

new electroweak gauge bosons couple to right-handed

currents

new right-handed or “sterile” neutrinos, electroweak

partners of Standard Model right-handed electron

possibility for type-II see-saw at TeV scale

WR WR NR

MR

Figure from Tello, Nemevsek, Nesti, Senjanovic and Vissani, 2011

lightest mN in GeV lightest neutrino mass in eV

normal inverted

MWR = 3.5 TeV largest mN = 0.5 TeV

|mee

ν+N| in eV

1 10 100 400 500 10−4 0.001 0.01 0.1 1 10−3 0.01 0.1 1.0 10 4 0.001 0.01 0.1 1 0.001 0.005 0.01 0.05 0.1 0.5 1.

mN mν

LY = 1 2⌅L MνL ↵∆L∆L⌅L + 1 2⌅R MνR ↵∆R∆R⌅R + h.c. ,

Assuming a type-II see-saw, C invariance leads

the connection between th MνR/↵∆R = M ∗

νL/↵∆L∗.

the proportionality of the

r

SLIDE 5

BSM contributions to neutrinoless beta decay: R-parity violation inspired

new charged scalar leptons (“sleptons”)
new electroweak partners of the electron
generate different contact operator at low energies

see e.g. M. Ramsey-Musolf, T. Peng and P . Winslow, 2015 for thorough LHC collider phenomenology analysis (and see M. Ramsey Musolf’s talk)

R-M P W include leading 2 pion interactions and RGE

analysis, backgrounds, detector sim.

and determine signal acceptances - very model-

dependent

X

e− e− d u u d ˜ e ˜ e

F F

Leff

LNV = C1

Λ5 O1 +h.c. , O1 = ¯ Q⌧ +d ¯ Q⌧ +d¯ LLC

see also e.g. Deppisch,

Hirsch, Pas, 2012

SLIDE 6

q/q q/q νβ να g q/q g νβ να q/q q/q g νβ να q/q

ν ν ν ν ν ν

CDF GSNP LHC lowPT LHC highPT LHC veryhighPT CDF ADD

100 101 102 103 104 105 10-4 10-3 10-2 10-1 100 MZ' @GeVD acceptance

l

w

P T C D F G S N P h i g h P T v e r y H i g h P T Broad resonance CDF ADD

100 101 102 103 104 10-3 10-2 10-1 100 101 MZ' @GeVD ∂

Sidebar: Acceptance is model-dependent

E.g.Monojet bounds on Non-standard Neutrino Interactions

(A. Friedland, MG, I. Shoemaker, L. Vecchi, ’12)

Z’ model

For fixed cuts, weaker limit for lighter mediator

can’t just use reported sigma*BR, common to

many 0nubb <-> LHC comparisons

need to determine acceptance for your

favorite model

SLIDE 7

BSM contributions to neutrinoless beta decay

u u e- e- e- A A n e- e- n e- d d p p A’ A’

existing and next-gen-multi-tonne experiments

High Energy Low Energy

SLIDE 8

e- e- e- ν

εijεmn LT

i C(DµL)j Hm(DµH)n

ν e-

εijεmn LT

i Cγµe HjHm(DµH)n e- e- e- ν d u

εij ¯ dγµu LT

i C(DµL)j

d u e- ν

εijεmn¯ dLi QT

j CLm Hn

Dimension 7 ΔL=2 LNV operators

νR νR νR

WR WR WR d u e- ν ν ν e- e-

Sample dimension -5,-7,-9 ΔL=2 LNV operators

Nice figures from E. Mereghetti, INT seminar 2017

SLIDE 9

e- e- e- ν

εijεmn LT

i C(DµL)j Hm(DµH)n

ν e-

εijεmn LT

i Cγµe HjHm(DµH)n e- e- e- ν d u

εij ¯ dγµu LT

i C(DµL)j

d u e- ν

εijεmn¯ dLi QT

j CLm Hn

Dimension 7 ΔL=2 LNV operators

νR νR νR

WR WR WR d u e- ν ν ν e- e-

Sample dimension -5,-7,-9 ΔL=2 LNV operators

Nice figures from E. Mereghetti, INT seminar 2017

Part 1

SLIDE 10

e- e- e- ν

εijεmn LT

i C(DµL)j Hm(DµH)n

ν e-

εijεmn LT

i Cγµe HjHm(DµH)n e- e- e- ν d u

εij ¯ dγµu LT

i C(DµL)j

d u e- ν

εijεmn¯ dLi QT

j CLm Hn

Dimension 7 ΔL=2 LNV operators

νR νR νR

WR WR WR d u e- ν ν ν e- e-

Sample dimension -5,-7,-9 ΔL=2 LNV operators

Nice figures from E. Mereghetti, INT seminar 2017

Part 2

SLIDE 11

Disclaimer/Philosophy for new physics scenarios for neutrinoless double beta decay

Will use effective field theory to study connection between

high-energy (below ΔL=2 mass scale) and 0nubb experiments (low-energy)

Plug-in favorite UV model to matching condition of Wilson coefficients
But it would be nice if favorite UV model had some other compelling

feature (Feynman)

Theoretical inputs: - (pQCD) anomalous dimensions of operators
lattice inputs to QCD matrix elements (becoming

increasingly under control)

nuclear matrix elements of nucleon operators
Neutrino mass generation may be sub-dominant to 0nubb experimental

signal (see Michael Ramsey-Musolf’s talk)

SLIDE 12

BSM contributions to neutrinoless beta decay

u u e- e- e- A A n e- e- n e- d d p p A’ A’ High Energy Low Energy Model -> gauge

invariant operators

RG evolution

Match at EW scale RG evolution to QCD scale

Match onto chiral EFT

(lattice input for LEC)

Neutrino potentials,

nuclear matrix element

SLIDE 13

Effective field theory analysis of BSM contributions to neutrinoless double beta decay

new particles generating ΔL=2 processes have masses in multi-TeV scale.
0nubb process generated at very short distances.
Leading effects of such TeV scale physics can be described by series of ΔL=2 violating
perators involving only quarks and leptons

Leff = LSM + Lν,M + X

i,d>4

cd

i

Λd−4 O(d)

i

dd → uue−e−

e.g.,

(collider signal: Keung, Senjanovic, PRL, 1983)

SLIDE 14

leading ΔL=2 operator with two charged leptons has a minimum of 4 quarks, in other words, dimension 9
For ΔL=2 phenomenology (e.g., 0nubb decay rates) need to know a minimal basis of operators, the set of

relevant operators that cannot be reduced by Fierz operators

Electromagnetic invariance: 24 (compared to 14=2*5+4 in prior literature):

8 scalar and 8 vector 4-quark operators

Electroweak invariance: If scale Λ of ΔL=2 violating physics is much larger than the electroweak scale, effect of

ΔL=2 physics appears as a series of higher dimension operators invariant under the full Standard Model gauge symmetry

If color + electroweak invariance is imposed, then 11 operators at LO in v/Λ: 7 scalar and 4 vector
At hadron colliders, if E << Λ, then collider only probing (color + electroweak invariant) ΔL=2 contact
perators. In this “contact limit” can classify their experimental signatures.

At “low energy” - ie QCD scale - there are a number of “short distance” operators that contribute to neutrinoless double beta decay (Prezeau, Ramsey-Musolf and Vogel (PRD, 68, 2003))

Leff = 1 Λ5

LNV

" X

i=scalar

ci,S ¯

eec + c0

i,S ¯

eγ5ec Oi + ¯ eγµγ5ec X

i=vector

ci,V Oµ

i

#

What is a minimal basis (MG, arXiv:1606.04549) ?

SLIDE 15

perator

content hadron collider signatures Low Energy PT (⇡⇡) same-sign dilepton e+MET dijet+ MET dimension 9 LM1 i(2)

ab (QaµQc)(uRµdR)(`b`C c )

√ √ √ O1LR ⊗ (LL) LO LM2 i(2)

ab (QaµAQc)(uRµAdR)(`b`C c )

√ √ √ Oλ

1LR ⊗ (LL)

LO LM3 (uRQa)(uRQb)(`a`C

b )

√ √ √ O2RL ⊗ (LL) LO LM4 (uRAQa)(uRAQb)(`a`C

b )

√ √ √ Oλ

2RL ⊗ (LL)

LO LM5 i(2)

ab i(2) cd (QadR)(QcdR)(`b`C d )

√ √ √ O2LR ⊗ (LL) LO LM6 i(2)

ab i(2) cd (QaAdR)(QcAdR)(`b`C d )

√ √ √ Oλ

2LR ⊗ (LL)

LO LM7 (uRµdR)(uRµdR)(eReC

R)

√ ¨ _ ¨ _ O3R ⊗ (RR) NNLO LM8 (uRµdR)i(2)

ab (QadR)(`bµeC R)

√ √ ¨ _ Oµ

RRLR ⊗ (LR)

LM9

(uRµAdR)i(2)

ab (QaAdR)(`bµeC R)

√ √ ¨ _ Oλµ

RRLR ⊗ (LR)

LM10

(uRµdR)(uRQa)(`aµeC

R)

√ √ ¨ _ Oµ

RRRL ⊗ (LR)

LM11

(uRµAdR)(uRAQa)(`aµeC

R)

√ √ ¨ _ Oλµ

RRRL ⊗ (LR)

Table from MG,

arXiv:1606.04549

Electroweak invariant dimension 9 operators: collider signatures

Set up systematic formalism for χPT operators in low-energy effective field theory
Applied general formalism to identify which operators contribute at LO to eeππ interactions

(i.e., which ops. in χPT dominate ΔL=2 amplitude over effects of eeπNN and eeNNNN interactions)

vector 4-quark

perators (4)

scalar 4-quark

perators (7)

LR symmetric theory RPV-inspired theory

SLIDE 16

n n n p p p e e e e e (b) (c) (a) n p p p e n n

Effective field theory analysis of BSM contributions to neutrinoless double beta decay: Weinberg power counting

Quarks couple to everything, so expect 4 quark operator to generate many multi-hadron interactions
Two pion interaction important (Faessler, S. Kovalenko, F. Simkovic, and J. Schwieger, 1996; Prezeau, Ramsey-Musolf and

Vogel (PRD, 68, 2003)) but not consistently implemented in other literature

A number of analyses comparing LHC projections and 0nubb limits only include 4-nucleon

interactions, “conservatively” suppressing limits from 0nubb experiments (unfairly promotes the competitiveness of the LHC)

Here power counting is for free field theory only - need to insert inside a nucleus and test power-

counting

O(q−1+∆O(πNN))

O(q−2+∆O(ππ))

O(q0+∆O(NNNN))

SLIDE 17

n n n p p p e e e e e (b) (c) (a) n p p p e n n

Effective field theory analysis of BSM contributions to neutrinoless double beta decay: Estimate of long-distance pion exchange

O(q−2+∆O(ππ))

chiral PT estimate: MhOii

∼ 102 (O2,3,4,5, O0

2,3)

Aππ ' 1 Λ5

LNV

Mhπ+|Oi|π−i f 2

πq2

⇠ 102 1 Λ5

LNV

MhOii 102 (100 MeV)4 f 2

πq2

ASM ' G2

F

mββ q2

SLIDE 18

Effective field theory analysis of BSM contributions to neutrinoless double beta decay (MG, arXiv:1606.04549)

O = T ab

cd (qcΓqa)(qdΓ0qb), T ab cd = (τ +) a c (τ +) b d

qL → Lq, qR → RqR, T → T ⊗ X1 ⊗ X2 ⊗ X3 ⊗ X4, Xi ∈ {L, R, L†, R†} T ab

cd ˜

Ocd

ab(π, N)

General ΔL=2 4-quark scalar operator (following Savage 1999) Transform T such that O is formally chirally invariant Construct pion and nucleon operators in chiral theory such that they are formally chirally invariant

SLIDE 19

Effective field theory analysis of BSM contributions to neutrinoless double beta decay: Weinberg power counting

With

ξ = Exp[π · τ/2Fπ], ξ → LξU † = UξR†, N → UN

Construct “proto-O” out of products of such that

ξ’s

(proto − ˜ O) → (proto − ˜ O) ⊗ Y1 ⊗ Y2 ⊗ Y3 ⊗ Y4, Yi ∈ {U, U †}

To construct invariants

only pions: takes all possible traces
pions and two nucleons: multiply by two N fields in all possible

ways, take all possible traces

Four nucleons: multiply in by 4 nucleon fields in all possible

ways

can also generate new operators involving higher chiral order

using chiral transformation properties of quark mass and covariant derivative

SLIDE 20

Example: Operators from WR exchange (Left-right-symmetric model)

O3R ≡ (qRγµτ +qR)(qRγµτ +qR) T ab

cd → T αβ ρσ Rρ cRσ dR†a α R†b β

proto − ˜ O3R = T ab

cd ξ†i a ξ†j b ξc kξd l

To construct invariants

only pions: takes all possible traces -> all vanish (in this example)
Four nucleons: multiply in by 4 nucleon fields in all possible

ways -> non-vanishing operator involving 4 nucleons

can also generate new operators involving higher chiral order

using chiral transformation properties of quark mass and covariant derivative -> Find a number of single and double trace operators, e.g.

tr(Dµξτ +Dµξ†ξτ +ξ†)

For this operator, expect first non-vanishing two-pion matrix element at NLO

- which we confirmed using chiral SU(3) -- and first non-vanishing 4 nucleon

matrix element at LO

SLIDE 21

perator

content hadron collider signatures Low Energy PT (⇡⇡) same-sign dilepton e+MET dijet+ MET dimension 9 LM1 i(2)

ab (QaµQc)(uRµdR)(`b`C c )

√ √ √ O1LR ⊗ (LL) LO LM2 i(2)

ab (QaµAQc)(uRµAdR)(`b`C c )

√ √ √ Oλ

1LR ⊗ (LL)

LO LM3 (uRQa)(uRQb)(`a`C

b )

√ √ √ O2RL ⊗ (LL) LO LM4 (uRAQa)(uRAQb)(`a`C

b )

√ √ √ Oλ

2RL ⊗ (LL)

LO LM5 i(2)

ab i(2) cd (QadR)(QcdR)(`b`C d )

√ √ √ O2LR ⊗ (LL) LO LM6 i(2)

ab i(2) cd (QaAdR)(QcAdR)(`b`C d )

√ √ √ Oλ

2LR ⊗ (LL)

LO LM7 (uRµdR)(uRµdR)(eReC

R)

√ ¨ _ ¨ _ O3R ⊗ (RR) NNLO LM8 (uRµdR)i(2)

ab (QadR)(`bµeC R)

√ √ ¨ _ Oµ

RRLR ⊗ (LR)

LM9

(uRµAdR)i(2)

ab (QaAdR)(`bµeC R)

√ √ ¨ _ Oλµ

RRLR ⊗ (LR)

LM10

(uRµdR)(uRQa)(`aµeC

R)

√ √ ¨ _ Oµ

RRRL ⊗ (LR)

LM11

(uRµAdR)(uRAQa)(`aµeC

R)

√ √ ¨ _ Oλµ

RRRL ⊗ (LR)

Table from MG,

arXiv:1606.04549

Electroweak invariant dimension 9 operators: two-pion couplings

Only one pair of scalar operators suppressed in chiPT counting (O1,O’1)
Confirm two-pion interactions from vector operators suppressed by electron mass through

NNLO (Prezeau, Ramsey-Musolf, Vogel)

4 “vector” quark

perators

7 “scalar” quark

perators

LR symmetric theory RPV-inspired theory

SLIDE 22

Effective chiral field theory analysis of BSM contributions to neutrinoless double beta decay: two pion matrix elements

V. Cirigliano, W. Dekens, MG, E. Mereghetti, 1701.01443, PLB 2017

O1 = ¯ qα

Lγµτ +qα L ¯

qβ

Lγµτ +qβ L

O2 = ¯ qα

Rτ +qα L ¯

qβ

Rτ +qβ L

O3 = ¯ qα

Rτ +qβ L ¯

qβ

Rτ +qα L

O4 = ¯ qα

Lγµτ +qα L ¯

qβ

Rγµτ +qβ R

O5 = ¯ qα

Lγµτ +qβ L ¯

qβ

Rγµτ +qα R ,

From the minimal basis, 8 scalar quark operators: + O0

1,2,3 from L ↔ R on O1,2,3

For 0nubb phenomenology, need matrix elements hπ+|Oi|π−i

SLIDE 23

Two pion matrix elements

, two pion matrix element determined by M. Savage (1999) using chiral SU(3) symmetry

to relate ππ amplitude to ΔI=3/2 K-> ππ decay

we were able to extend Savage’s analysis to all such operators, by relating two pion matrix

elements to those involving ΔS=1, 2 matrix elements which are now accurately computed on the lattice

O1

preliminary lattice computations exist for two pion matrix elements (Nicholson et. al., 2015)

O0

1

SLIDE 24

O1 ∼ 27L ⊗ 1R

O2,3 ∼ 6L ⊗ 6R

4 quark operators belong to irreducible representations of SU(3)L × SU(3)R

qL,R ∼ 3L,R

SU(3)L × SU(3)R

Two pion matrix elements

+ O‘1,2,3 by L <--> R from O1,2,3; by parity same QCD matrix element

O1 = ¯ qα

Lγµτ +qα L ¯

qβ

Lγµτ +qβ L

O2 = ¯ qα

Rτ +qα L ¯

qβ

Rτ +qβ L

O3 = ¯ qα

Rτ +qβ L ¯

qβ

Rτ +qα L

O4 = ¯ qα

Lγµτ +qα L ¯

qβ

Rγµτ +qβ R

O5 = ¯ qα

Lγµτ +qβ L ¯

qβ

Rγµτ +qα R ,

O4,5 ∼ 8L ⊗ 8R

and only 27 contains I=2

O1

O2,3 ∼ (uRdL)(uRdL)

→ I = (1L, 1R)

∼ (uLdL)(uLdL) → I = (2L, 0R) 8 ⊗ 8 = 27 + 10 + 10 + 8 + 8 + 1

and contains “symmetric component”

6L ⊗ 6R

27L ⊗ 1R

SLIDE 25

! ⇥ U = exp p 2iπ F0 ! , π = B B B @

π3 √ 2 + π8 √ 6

π+ K+ π− π3

√ 2 + π8 √ 6

K0 K− ¯ K0 2

√ 6π8

1 C C C A

Chiral perturbation theory

U → LUR† Tr T aUT bU † Only is formally invariant

First consider O2,3,4,5+ O’2,3,4,5, then return to O1, O1’

O4,5 = qLT aγµqL qRT bγµqR T a → LT aL† T b → RT bR†

SLIDE 26

! ⇥ U = exp p 2iπ F0 ! , π = B B B @

π3 √ 2 + π8 √ 6

π+ K+ π− π3

√ 2 + π8 √ 6

K0 K− ¯ K0 2

√ 6π8

1 C C C A

Chiral perturbation theory

U → LUR† Here there are two formal invariants O2,3 = qRT aqL qRT bqL T a,b → RT a,bL† Tr T aUT bU Tr T aU and Specific linear combination keeps the 6 and projects out the 3*

SLIDE 27

Oa,b

6×¯ 6 = ¯

qRT aqL ¯ qRT bqL

6×¯

6

! g6×¯

6

F 4 8 h Tr ⇣ T aUT bU ⌘ + Tr ⇣ T aU ⌘ Tr ⇣ T bU ⌘i Oa,b

8×8 = ¯

qLT aγµqL ¯ qRT bγµqR ! g8×8 F 4 4 Tr ⇣ T aUT bU †⌘ ,

O2,3 :

Matching quark operators onto chiral operators

O4,5 :

Non-perturbative dynamics encoded in each low-energy constant

g6⊗6, g8⊗8

for each chiral rep, each color contraction has its own LEC g
ΔL=2 operators T a

→ T 1 + iT 2

K-Kbar mixing ΔS=2 operators

T a → T 6 − iT 7

SLIDE 28

Oa,b

6×¯ 6 = ¯

qRT aqL ¯ qRT bqL

6×¯

6

! g6×¯

6

F 4 8 h Tr ⇣ T aUT bU ⌘ + Tr ⇣ T aU ⌘ Tr ⇣ T bU ⌘i Oa,b

8×8 = ¯

qLT aγµqL ¯ qRT bγµqR ! g8×8 F 4 4 Tr ⇣ T aUT bU †⌘ ,

Mππ

6×¯ 6 ⌘ hπ+|O1+i2,1+i2 6×¯ 6

|π−i = h ¯ K0|O6−i7,6−i7

6×¯ 6

|K0i ⌘ MK ¯

K 6×¯ 6

Mππ

8×8 ⌘ hπ+|O1+i2,1+i2 8×8

|π−i = h ¯ K0|O6−i7,6−i7

8×8

|K0i ⌘ MK ¯

K 8×8

(LO in chiPT)

O2,3 :

Matching quark operators onto chiral operators

O4,5 :

SLIDE 29

Quark masses (pion masses) break chiral symmetry. So previous relations modified at NLO. We did a loop computation to estimate the size of that splitting.

Counter-terms from NLO local operators have the form (V. Cirigliano, E. Golowich, 2000)

Mππ

8×8

= MK ¯

K 8×8 × F 2 π

F 2

K

× (1 + ∆8×8) = MK ¯

K 8×8 × R8×8

Mππ

6×¯ 6

= MK ¯

K 6×¯ 6 × F 2 π

F 2

K

× (1 + ∆6×¯

6) = MK ¯ K 6×¯ 6 × R6×¯ 6 ,

∆8×8 = 1 (4πF0)2 m2

π

4 (−4 + 5Lπ) − m2

K(−1 + 2LK) + 3

4m2

ηLη − a8×8

m2

K − m2 π

∆6×¯

6

= 1 (4πF0)2  −m2

π

4 (4 − 3Lπ) − m2

K(−1 + 2LK) + 5

4m2

ηLη − a6×¯ 6

m2

K − m2 π

Lπ,K,η

≡ log µ2

χ/m2 π,K,η

( )

δK ¯

K 8×8

= a8×8 m2

K + b8×8

✓ m2

K + 1

2m2

π

◆ δππ

8×8

= a8×8 m2

π + b8×8

✓ m2

K + 1

2m2

π

◆

Low-energy coefficients {a} could be extracted (in principle) from K-Kbar mixing computed

using lattice QCD at different values for the quark masses

We agree with loop corrections to K-Kbar (Becirevic,

Villadoro, 2004)

Out[13]=

600 800 1000 1200 1400

0.2
0.1

0.1 0.2

∆8

∆6

µ

SLIDE 30

Estimating central value and uncertainty

For central value for Δ’s, set renormalization scale to rho mass and counter-terms =0
Adopted two prescriptions for estimating the error due to unknown δK ¯

K 8×8, δππ 8×8

Naive-dimensional analysis : |a8×8,6×¯

6| ∼ O(1)

gives ∆8×8 = 0.02(20), ∆6×¯

6 = 0.07(20)

Mππ

8×8

= MK ¯

K 8×8 × F 2 π

F 2

K

× (1 + ∆8×8) = MK ¯

K 8×8 × R8×8

Mππ

6×¯ 6

= MK ¯

K 6×¯ 6 × F 2 π

F 2

K

× (1 + ∆6×¯

6) = MK ¯ K 6×¯ 6 × R6×¯ 6 ,

O(1) change in (log) renormalization scale (Manohar ’96): ∆(ct)

n

= ±|d∆(loops)

n

/d(log µχ)|

gives ∆8×8 = 0.02(36), ∆6×¯

6 = 0.07(16)

For final analysis, chose ∆8×8 = 0.02(30) , ∆6×¯

6 = 0.07(20)

This choice gives

R8×8 = 0.72(21) (∼ 30% uncertainty) R6×¯

6

= 0.76(14) (∼ 20% uncertainty)

SLIDE 31

hπ+|O1|π−i = (1.0 ± 0.1 ± 0.2) ⇥ 10−4 GeV4 hπ+|O2|π−i = (2.7 ± 0.3 ± 0.5) ⇥ 10−2 GeV4 hπ+|O3|π−i = (0.9 ± 0.1 ± 0.2) ⇥ 10−2 GeV4 hπ+|O4|π−i = (2.6 ± 0.8 ± 0.8) ⇥ 10−2 GeV4 hπ+|O5|π−i = (11 ± 2 ± 3) ⇥ 10−2 GeV4

Relate our operators to those defined by FLAG (Aoki et.al, 1607.00299) average central values for Nf=2+1 and Nf=2+1+1

hπ+|O2|π−i = 5 12 B2 K ⇥ R6×¯

6,

K = 2 F 2

K m4 K

(md + ms)2 hπ+|O3|π−i = 1 12 B3 K ⇥ R6×¯

6

hπ+|O4|π−i = 1 3 B5 K ⇥ R8×8 hπ+|O5|π−i = B4 K ⇥ R8×8

LQCD input: B2, B3: O(10%) error B4, B5: O(20%) error

Fractional error: O2, O3: O(20%) error O5 : O(40%) error O4 : O(35%) error

SLIDE 32

hπ+|O1|π−i

Updating M. Savage’s (1999) determination of

O1, O∆S=2, Q(27⊗1)

2

∈ 27

Observation is that

K+ → π+π0 Q(27×1)

2

→ g27×1 F 4 ✓ Lµ32Lµ

11 + 2

3Lµ31Lµ

12

◆ O∆S=2 → 5 3 g27×1 F 4

0 Lµ32Lµ 32

4 O1 → 5 3 g27×1 F 4

0 Lµ12Lµ 12

Lµ

ij = i(U †∂µU)ij

SLIDE 33

hπ+|O1|π−i = 5 3 g27×1 m2

πF 2 π

⇢ 1 + m2

π

(4πF0)2 (1 + 3Lπ) + δππ

27×1

hπ+π0|iQ2|K+i

= 5 3 g27×1 Fπ

m2

K m2 π

n 1 + ∆K+π+π0

27

for ΔS=1 part, loops are small, and counter terms found to also be small at

large Nc because of factorization of Q2 into product of currents (Cirigliano, Ecker, Neufeld, Pich, 2004)

lattice QCD computation of K-> pi pi O(10%) error (Blum et. al. 2015)
-> g27 =0.34(3)LQCD(2)chiPT
with 20% error in gives our estimate for O1 :

δππ

27×1

hπ+|O1|π−i = (1.0 ± 0.1 ± 0.2) ⇥ 10−4 GeV4

As expected from general considerations, this matrix element is

suppressed compared to other ΔL=2 two pion matrix elements

Chiral loops and counter terms again give:

SLIDE 34

Comments on validity of chiral SU(3) general comment: semi-leptonic K decay form factors agree well between lattice QCD and chiral SU(3), but need to check each example

loop factors small in our case O(30%)
Our results for g_{8x8} using K-> pi pi decay instead of K-

Kbar in reasonable agreement with method using K-Kbar

Our value for g_{27} extracted using K-Kbar and K->pi pi

agree in reasonable agreement

SLIDE 35

Summary

progress on these interactions from LQCD and chiral PT progress on these interactions from LQCD just beginning

g27×1 0.38 ± 0.08 [33] g8×8 −(3.1 ± 1.3) GeV2 [33] gmix

8×8

−(11 ± 4) GeV2 [33]

two pion matrix element results consistent with chiral PT expectations and naive dimensional analysis

SLIDE 36

New dimension-9 ΔL=2 LNV physics potentially accessibly at LHC or future hadron collider Complementarity between 0nubb and hadron colliders

(see Michael Ramsey-Musolf’s talk)

But is this generic? Not necessarily.....

SLIDE 37

Class 1 2H4 Class 5 4D OLH ✏ij✏mn(LT

i CLm)HjHn(H†H)

O(1)

LL ¯ duD

✏ij( ¯ dµu)(LT

i C(DµL)j)

Class 2 2H2D2 Class 6 4H O(1)

LHD

✏ij✏mn(LT

i C(DµL)j)Hm(DµH)n

OLL¯

eH

✏ij✏mn(¯ eLi)(LT

j CLm)Hn

O(2)

LHD

✏im✏jn(LT

i C(DµL)j)Hm(DµH)n

O(1)

LLQ ¯ dH

✏ij✏mn( ¯ dLi)(QT

j CLm)Hn

Class 3 2H3D O(2)

LLQ ¯ dH

✏im✏jn( ¯ dLi)(QT

j CLm)Hn

OLHDe ✏ij✏mn(LT

i Cµe)HjHm(DµH)n

OLL ¯

QuH

✏ij( ¯ Qmu)(LT

mCLi)Hj

Class 4 2H2X OLeu ¯

dH

✏ij(LT

i Cµe)( ¯

dµu)Hj OLHB ✏ij✏mng0(LT

i CµνLm)HjHnBµν

OLHW ✏ij(✏⌧ I)mng(LT

i CµνLm)HjHnW I µν

Dimension 7 ΔL=2 LNV operators

Lehman ’14

V. Cirigliano, W. Dekens, J. de

Vries, MG, E. Mereghetti, (1707/08.zzzz) Preliminary!

12 independent operators

special cases: i) class 1 modifies Weinberg operator scale > 1200 TeV

ii) purely leptonic operators contribute to neutrino mass (at one-loop) and neutrino magnetic moment (at tree- level)

operators involving quarks contribute to 0nubb

SLIDE 38

e- e- e- ν

εijεmn LT

i C(DµL)j Hm(DµH)n

ν e-

εijεmn LT

i Cγµe HjHm(DµH)n e- e- e- ν d u

εij ¯ dγµu LT

i C(DµL)j

d u e- ν

εijεmn¯ dLi QT

j CLm Hn

Dimension 7 ΔL=2 LNV operators

νR νR νR

WR WR WR d u e- ν ν ν e- e-

Sample dimension -5,-7,-9 ΔL=2 LNV operators

Nice figures from E. Mereghetti, INT seminar 2017

SLIDE 39

Dimension 7 ΔL=2 LNV operators:

neutrino masses, neutrino magnetic moment

purely leptonic operators contribute to neutrino mass (one-loop)

(“bound” by requiring new contribution to neutrino mass < 1 eV)

(only those more constraining than 0nubb shown)

neutrino magnetic moment (at tree-level)

constrained by solar neutrino experiments (Borexino) (Canas, Miranda, Parada, Tortola,

Valle, ‘16)

|CLHB − CLHW | . 1 4mev2 10−10 → Λ > 11 TeV . C(1)

LHD :

Λ > 280 TeV, C(2)

LHD

: Λ > 350 TeV, CLHW : Λ > 460 TeV

SLIDE 40

Dimension 7 ΔL=2 LNV quark operators:

QCD running to Electroweak scale

running either trivial (operators with no quarks, or those with

vector or axial currents) or given scalar or tensor:

d d ln µCLL ¯

QuH

= −6CF αs 4π CLL ¯

QuH,

d d ln µC(1,2),ij

S

= −6CF αs 4π C(1,2),ij

S

d d ln µC(1,2),ij

T

= 2CF αs 4π C(1,2),ij

T

,

O(1,2)

LLQ ¯ dH

combination of scalar and tensor

OLL ¯

QuH

scalar

SLIDE 41

Dimension 7 ΔL=2 LNV operators:

Integrate out W, H at electroweak scale, generate dim-6, -7, -9

perators

L∆L=2 = −1 2(mν)ijνt

L, i CνL, j + µij νt j Cσµννi eFµν + L(6) ∆L=2 + L(7) ∆L=2 + L(9) ∆L=2.

L(6)

∆L=2

= 2GF √ 2 ( C(6)

VL,ij ¯

uLγµdL ¯ eR,i γµ C¯ νt

L,j + C(6) VR,ij ¯

uRγµdR ¯ eR,i γµ C¯ νt

L,j

+C(6)

SR,ij ¯

uLdR ¯ eL,i C¯ νt

L,j + C(6) SL,ij ¯

uRdL ¯ eL,i C¯ νt

L,j + C(6) T,ij ¯

uLσµνdR ¯ eL,iσµν C¯ νt

L,j

) + h.c. L(7)

∆L=2

= 2GF √ 2v ( C(7)

VL,ij ¯

uLγµdL ¯ eL,i C i← → ∂ µ¯ νt

L,j + C(7) VR,ij ¯

uRγµdR ¯ eL,i Ci← → ∂ µ¯ νt

L,j

) + h.c. L(9)

∆L=2

= ¯ eL,iC¯ et

L,j

v5 ⇢ C(9)

1,ij ¯

uLγµdL ¯ uLγµdL + C(9)

4,ij ¯

uLγµdL ¯ uRγµdR + C(9)

5,ij ¯

uα

Lγµdβ L ¯

uβ

Rγµdα R

ld

ld sd

SLIDE 42

Dimension 7 ΔL=2 LNV operators:

QCD running below electroweak scale

d d ln µC(6)

SL (SR)

= −6CF αs 4π C(6)

SL (SR) ,

d d ln µC(6)

T

= 2CF αs 4π C(6)

T .

d d ln µC(9)

1

= 6 ✓ 1 − 1 Nc ◆ αs 4π C(9)

1

, d d ln µ C(9)

4

C(9)

5

! = αs 4π ✓6/Nc −6 −12CF ◆ C(9)

4

C(9)

5

!

SLIDE 43

Dimension 7 ΔL=2 LNV operators

Long-distance contributions : single nucleon couplings use SU(2) chiral EFT and external source method

(a) (b) (d) (c)

long-distance 0nubb = top row ^2 (SM) + top row * bottom row

e ν

SM dim-6, dim-7

SLIDE 44

Dimension 7 ΔL=2 LNV operators:

Long-distance and short-distance contributions total 0nubb contribution = SM + long-distance neutrino (middle) + short distance (right)

LD: 1 unknown LEC: SD: 2 unknown LECs:

gπN

27×1, gNN 27×1

∼ O(1) g0

T

∼ O(1)

g27×1 0.38 ± 0.08 [33] gA 1.272 ± 0.002 [34] g8×8 −(3.1 ± 1.3) GeV2 [33] gS 0.97 ± 0.13 [35] gmix

8×8

−(11 ± 4) GeV2 [33] gT 0.99 ± 0.06 [35]

“known” LEC inputs (right) + some NLO ops fixed by reparameterization invariance

SLIDE 45

Dimension 7 ΔL=2 LNV operators:

Matrix elements and general formula Fortunately, 15 matrix elements needed for computing rate have been computed by several groups, using different methods

Muto, Bender and Klapdor, 1989 Barea and Iachello, 2009 Hyvarinen and Suhonen, 2015 Horoi and Neacsu, 2016, 2017

⇣ T 0ν

1/2

⌘−1 = g4

A

n G01 |MSM|2 + 4G02 |ME|2 + 2G04 ⇥ |Mme|2 + Re

M∗

meMSM

⇤ + G09 |MM|2 2G03 Re (MSMM∗

E + 2MmeM∗ E) + G06 Re (MSMM∗ M)

,

(43)

G0k = 1 ln 2 G4

F m2 e

64π5R2

A

Z dE1dE2|k1||k2|d cos θ b0k F(Z, E1)F(Z, E2)δ(E1 + E2 + Mf Mi).

SLIDE 46

Dimension 7 ΔL=2 LNV operators:

Bounds on operators from 0nubb experiments

Λ

76Ge 82Se 130Te 136Xe

mββ (eV) 0.19 1.4 0.49 0.1 Λ (TeV) C6

SL

210. 110. 150. 260. C6

SR

210. 110. 150. 260. C6

T

180. 92. 140. 240. C6

VL

150. 74. 110. 190. C6

VR

26. 15. 20. 34. C7

VL

6.4 3.3 4.6 7.8 C7

VR

6.4 3.3 4.6 7.8 C9

1

14. 7.4 11. 19. C9

4

41. 21. 31. 53. C9

5

63. 32. 47. 81.

76Ge 82Se 130Te 136Xe

mββ (eV) 0.17 1.6 0.32 0.084 Λ (TeV) C6

SL

270. 130. 220. 350. C6

SR

270. 130. 220. 350. C6

T

220. 100. 180. 280. C6

VL

180. 83. 150. 220. C6

VR

33. 17. 28. 44. C7

VL

8.1 3.8 6.8 11. C7

VR

8.1 3.8 6.8 11. C9

1

13. 5.9 12. 18. C9

4

54. 26. 48. 69. C9

5

84. 40. 73. 110.

Bounds on: effective dim-6, dim-7, dim-9 couplings (right)

* bounds weaker or stronger depending on whether contribution is chirally suppressed or enhanced by large magnetic moment

SM C(6)

SL, SR

C(6)

T

C(6)

VL

C(6)

VR

C(7)

VL, VR

C(9)

1

C(9)

4,5

meMSM mββ Λχ Λχ✏2

χ

− −

Λ2

χ

v ✏2 χ Λ2

χ

v ✏2 χ Λ2

χ

v

meMM − − − Λχ✏2

χ

− − − − meME − − − Λχ✏3

χ

Λχ✏3

χ

− − − meMme − − − Λχ✏3

χ

Λχ✏3

χ

− − − Table 4: Power-counting estimates of the contribution of low-energy dimension-six, -seven and

nine operators to the matrix elements in Eq. (42). Here ✏χ ≡ mπ/Λχ, where Λχ ∼ mN ∼ 1

GeV is the symmetry-breaking scale. For the power counting, we consider the electron energies and mass to be small, E1 ∼ E2 ∼ me ∼ Λχ ✏3

χ.

Bounds more or less consistent with chiral expectations:

SLIDE 47

Dimension 7 ΔL=2 LNV operators:

Bounds on operators from 0nubb experiments

Λ

76Ge 82Se 130Te 136Xe

mββ (eV) 0.19 1.4 0.49 0.1 Λ (TeV) C6

SL

210. 110. 150. 260. C6

SR

210. 110. 150. 260. C6

T

180. 92. 140. 240. C6

VL

150. 74. 110. 190. C6

VR

26. 15. 20. 34. C7

VL

6.4 3.3 4.6 7.8 C7

VR

6.4 3.3 4.6 7.8 C9

1

14. 7.4 11. 19. C9

4

41. 21. 31. 53. C9

5

63. 32. 47. 81.

76Ge 82Se 130Te 136Xe

mββ (eV) 0.17 1.6 0.32 0.084 Λ (TeV) C6

SL

270. 130. 220. 350. C6

SR

270. 130. 220. 350. C6

T

220. 100. 180. 280. C6

VL

180. 83. 150. 220. C6

VR

33. 17. 28. 44. C7

VL

8.1 3.8 6.8 11. C7

VR

8.1 3.8 6.8 11. C9

1

13. 5.9 12. 18. C9

4

54. 26. 48. 69. C9

5

84. 40. 73. 110. Λ

76Ge 82Se 130Te 136Xe

C(1)

LHD

17. 8.7 13. 22. CLHDe 130. 65. 98. 160. CLHW 27. 14. 21. 35. C(1)

LLduD

70. 36. 53. 91. C(1)

LLQdH

200. 100. 140. 250. C(2)

LLQdH

93. 48. 72. 120. CLLQuH 250. 130. 180. 300. CLeudH 23. 14. 18. 30. Λ

76Ge 82Se 130Te 136Xe

C(1)S

LHD

15. 7.2 15. 22. CLHDe 160. 73. 130. 200. CLHW 24. 11. 23. 35. C(1)S

LLduD

94. 44. 82. 120. C(1)S

LLQdH

240. 110. 210. 320. C(2)S

LLQdH

110. 53. 94. 150. CLLQuH 310. 150. 260. 410. CLeudH 29. 15. 25. 39.

C(1)

LHD

CLHDe CLHW C(1)

LL ¯ duD

C(1)

LLQ ¯ dH

C(2)

LLQ ¯ dH

CLL ¯

QuH

CLeu ¯

dH

101 102 103 22 160 35 91 250 120 300 30 22 200 35 120 320 150 410 39

Λ (TeV) Horoi et al. [44] Hyv¨ arinen et al. [41]

Bounds on: effective dim-6, dim-7, dim-9 couplings (right) electro-weak inv. dim-7 couplings (below): Single-coupling bounds* on dim-7 couplings from 0nubb (Kamland-Zen)

* constraints on C_LHD, C_LHW from neutrino mass much stronger * bounds weaker or stronger depending on whether contribution is chirally suppressed or enhanced by large magnetic moment

SLIDE 48

Dimension 7 ΔL=2 LNV operators:

Simultaneous contributions of neutrino masses and dim-7

perators: m_eff
10-4

10-3 10-2 10-1 1 10 10-4 10-3 10-2 10-1 1 |mββ(eff)| (eV)

Inverted Hierarchy

LLQuH > 0 LLQuH → 0 LLQuH < 0

mνlightest (eV)

10-4

10-3 10-2 10-1 1 10 10-4 10-3 10-2 10-1 1 |mββ(eff)| (eV)

Normal Hierarchy

LLQuH > 0 LLQuH → 0 LLQuH < 0

mνlightest (eV)

Regions correspond to varying phases in dim-7 operator and PMNS matrix
Size of region set by size of dim=7 operator (plots for 600 TeV)

SLIDE 49

Dimension 7 ΔL=2 LNV operators:

Simultaneous contributions of neutrino masses and dim-7

perators: energy and angular dependence

1 2 3 4 5 0.10 0.15 0.20 0.25 0.30 0.35

2 3 4 5 0.1 0.2 0.3 0.4 0.5

1.0
0.5

0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cos(θ) 1/Γ dΓ/dcos(θ)

LHDe LHW LLQuH LeudH

E/me E/me

Standard Majorana neutrino mass Mechanism: blue

G02 G03 G01 G01 G04 G06 G09

SLIDE 50

New sources of ΔL=2 LNV could dominate “standard non-standard” contribu;on (i.e., long-

distance Majorana neutrino mass contribu;on)

If neutrino hierarchy is “normal”*, such non-conven;onal sources for ΔL=2 LNV and 0nubb only

physics case for discovery

Discussed possibili;es, from both model-dependent and effec;ve field theory descrip;ons. In

contact limit reduced set of electroweak invariant operators: dim-7 and dim-9 operators.

first chiral es;mates of all two pion matrix elements arising from scalar 4-quark operators,

necessary ingredient for leading 0nubb matrix elements arising from such non-conven;onal short-distance sources

expect error to be improved only through direct LQCD computa;ons. QCD input increasingly

becoming under control for end-to-end computa;on. LaVce input for πNN and NNNN s;ll needs to be developed (hard).

dimension-7 ΔL=2 LNV operators constrained by 0nubb to be O(100 TeV) scale. Probably not

accessible at LHC, but future 100 TeV collider possible opportunity

big inverse problem if ΔL=2 LNV discovered, but that is a good situa;on to be in