Electric dipole moments of light nuclei Emanuele Mereghetti - - PowerPoint PPT Presentation

SLIDE 1

Electric dipole moments of light nuclei

Emanuele Mereghetti November 4th, 2016

SLIDE 2

Outline

Why EDMs of light ions?

• 1. orthogonal to the nucleon EDM

can we disentagle ¯ θ from BSM? can we disentagle BSM models?

• 2. theoretically clean

(... not too dirty...) direct connection between EDMs and high energy parameters?

• 3. experimentally feasible

just a little expensive...

SLIDE 3

Storage Ring EDM experiments

JEDI @ COSY (Julich)

• measure spin precession relative to β (η ∝ d )

ωa = e m

• aB −
• a −

1 γ2 − 1

• β × E
• ωe = ηe

m [E + β × B] , a = g − 2 2

SLIDE 4

Storage Ring EDM experiments

JEDI @ COSY (Julich) 10−16e fm by 2020?

• measure spin precession relative to β (η ∝ d )

✭✭✭✭✭✭✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤❤❤❤❤❤❤ ❤

ωa = e m

• aB −
• a −

1 γ2 − 1

• β × E
• ωe = ηe

m [E + β × B] , a = g − 2 2

a > 0: all electric ring ωa vanishes for “magic momentum” e.g. proton p = 0.7 GeV, E = 1.171 GeV a < 0: electric & magnetic ring need both B & E to cancel ωa case of deuteron, 3He

SLIDE 5

CP violation at 1 GeV

• 1. Catalog possible /

T operators in EFT

• one dimension 4 operator: QCD ¯

θ term L/

T 4 = m∗¯

θ¯ qiγ5q in principle ¯ θ = O(1) ... strong CP problem

• 9 (+ 8 w. strangeness) dimension 6 operators:
• 1. 4 (+2) quark bilinears: qEDM and qCEDM

L/

T 6

= −

• q=u,d,s

mq˜ c(q)

γ

2 ¯ qiσµνγ5q eFµν −

• q=u,d,s

mq˜ c(q)

g

2 ¯ qiσµν gsGµνγ5q v2˜ c(q)

γ

= O

• v2

Λ2

• v2˜

c(q)

g

= O

• v2

Λ2

• Λ ≫ v

SLIDE 6

CP violation at 1 GeV

• 1. Catalog possible /

T operators in EFT

• one dimension 4 operator: QCD ¯

θ term L/

T 4 = m∗¯

θ¯ qiγ5q in principle ¯ θ = O(1) ... strong CP problem

• 9 (+ 8 w. strangeness) dimension 6 operators:
• 2. 1 pure glue: Weinberg three-gluon operator (gCEDM)

L/

T 6

= C˜

G

6 f abcǫµναβGa

αβGb µρGc ρ ν

v2C˜

G = O

• v2

Λ2

• Λ ≫ v

SLIDE 7

CP violation at 1 GeV

• 1. Catalog possible /

T operators in EFT

• one dimension 4 operator: QCD ¯

θ term L/

T 4 = m∗¯

θ¯ qiγ5q in principle ¯ θ = O(1) ... strong CP problem

• 9 (+ 8 w. strangeness) dimension 6 operators:
• 3. 4 (+ 6) four-quark (LR LR & LL RR)

L/

T 6

= Σ(ud)

1

(¯ dLuR¯ uLdR − ¯ uLuR¯ dLdR) + Ξ(ud)

1

¯ dLγµuL ¯ uRγµdR + Σ(us)

1

¯ sLuR¯ uLsR + Ξ(us)

1

¯ sLγµuL ¯ uRγµsR + Ξ(ds)

1

¯ sLγµdL ¯ dRγµsR + color v2Σ = O

• v2

Λ2

• v2Ξ = O
• v2

Λ2

• Λ ≫ v
• I will neglect s quarks (but they are interesting!)

SLIDE 8

Connection to models

• new physics models induce one, a subset or all these operators

split SUSY MSSM 2 Higgs Doublet Model Leptoquarks LR symmetric models qEDM c(u,d)

γ

qCEDM c(u,d)

g

gCEDM C˜

G

LR LR Σud LL RR Ξud an incomplete list of possibilities... for more details

• M. Pospelov and A. Ritz, ‘05; W. Dekens et al, ‘14;
• J. Engel, M. Ramsey-Musolf and U. van Kolck, ‘13;
• goal: identify the distinctive manifestations of EFT operators on nuclei

SLIDE 9

Chiral Perturbation Theory

need to work here theory strongly coupled!

PDG 2016

L = −1 4Ga

µνGa µν + ¯

qLi / DqL + ¯ qRi / DqR − ¯ qLMqR − ¯ qRMqL + L/

T 4 + L/ T 6

• at the moment, we cannot compute nuclear EDMs directly from QCD

. . . maybe not too far in the future . . . e.g. NPLQCD collaboration

• chiral symmetry symmetries to the rescue!

SUL(2) × SUR(2) & spontaneous breaking to SUV(2)

• pions are Goldstone boson
• strong constraints on low-energy interactions of pions with nucleons, photons

etc.

SLIDE 10

Chiral Perturbation Theory

L = −1 4Ga

µνGa µν + ¯

qLi / DqL + ¯ qRi / DqR − ¯ qLMqR − ¯ qRMqL + L/

T 4 + L/ T 6

• at the moment, we cannot compute nuclear EDMs directly from QCD
• chiral symmetry to the rescue!

SUL(2) × SUR(2) → SUV(2) pions are Goldstone boson mπ ≪ Λχ ∼ 1 GeV strong constraints

• n pion-N interactions

SLIDE 11

Chiral Perturbation Theory

Chiral Perturbation Theory

• low-energy theory of nucleon and pions
• consistent with the symmetries of QCD
• organized in an expansion in powers of Q, mπ/Λχ

L[π, N] =

• f, ∆

L(∆)

f

[π, N] +

• f, ∆/

T

L

(∆/

T)

/ T, f [π, N]

• ∆: # of inverse powers of Λχ in coefficients

∆ = d + 2m + f/2 − 2 ≥ 0

• f = 0, 2, 4: # of nucleon legs
• d: # of derivatives or photon fields
• m: # of quark mass insertions

SLIDE 12

Chiral Perturbation Theory

Chiral Perturbation Theory

• low-energy theory of nucleon and pions
• consistent with the symmetries of QCD
• organized in an expansion in powers of Q, mπ/Λχ

L(0)

f=2+4 = ¯

Niv · DN − gA Fπ ¯ Nτ SµDµπN + C11 ¯ NN ¯ NN + Cττ ¯ NτN · ¯ NτN

• ∆: # of inverse powers of Λχ in coefficients

∆ = d + 2m + f/2 − 2 ≥ 0

• f = 0, 2, 4: # of nucleon legs
• d: # of derivatives or photon fields
• m: # of quark mass insertions
• e.g. ∆ = 0
• 1. interactions fixed by symmetry/power counting
• 2. low-energy constants (LECs) contain non-perturbative info

O(1) numbers, fixed by experiments

SLIDE 13

Chiral Perturbation Theory. A = 1

A ≤ 1:

• only one relevant scale Q ∼ mπ
• perturbative expansion of the amplitudes

T ∼ Q Λχ ν ν = 2L +

• i

∆i, Λχ = 2πFπ ν = 4

• more loops
• and/or insertions of subleading vertices

= ⇒ Q/Λχ suppression

• at a given ν, only finite number of diagrams

SLIDE 14

Chiral Perturbation Theory. A ≥ 2

∼

g2

A

F2

π

• another relevant scale: binding energy Q2/mN

SLIDE 15

Chiral Perturbation Theory. A ≥ 2

∼

g2

A

F2

π

∼

Q5 4πmN

×

g4

A

F4

π

Q4 (Q2+m2

π)2

×

m2

N

Q4

integration measure vertices & pion prop. nucleon prop.

• another relevant scale: binding energy Q2/mN
• in loops

nucleon energy EN ∼ Q2/mN nucleon momentum pN ∼ Q pion momentum pπ ∼ Q

SLIDE 16

Chiral Perturbation Theory. A ≥ 2

∼

g2

A

F2

π

∼

Q5 4πmN

×

g4

A

F4

π

Q4 (Q2+m2

π)2

×

m2

N

Q4

integration measure vertices & pion prop. nucleon prop.

• another relevant scale: binding energy Q2/mN
• in loops

nucleon energy EN ∼ Q2/mN nucleon momentum pN ∼ Q pion momentum pπ ∼ Q loop ∼ g2

A

F2

π

g2

AmNQ

4πF2

π

∼ g2

A

F2

π

Q MNN MNN ∼ 300 MeV

• loop not (at best barely) suppressed

SLIDE 17

Chiral Perturbation Theory. A ≥ 2

+ . . . Weinberg’s recipe:

• identify “irreducible diagram”:

nucleon prop. not “pinched” EN ∼ Q follow χPT power counting define the potential V

SLIDE 18

Chiral Perturbation Theory. A ≥ 2

Weinberg’s recipe:

• identify “irreducible diagram”:

nucleon prop. not “pinched” EN ∼ Q follow χPT power counting define the potential V

• iterate the nucleon-nucleon potential

i.e. solve Lippmann-Schwinger equation

SLIDE 19

Chiral Perturbation Theory. A ≥ 2

Weinberg’s recipe:

• identify “irreducible diagram”:

nucleon prop. not “pinched” EN ∼ Q follow χPT power counting define the potential V

• iterate the nucleon-nucleon potential

i.e. solve Lippmann-Schwinger equation

• “non-perturbative pions”
• 1. pion exchange leading effect

Q/MNN ∼ 1

SLIDE 20

Chiral Perturbation Theory. A ≥ 2

Weinberg’s recipe:

• identify “irreducible diagram”:

nucleon prop. not “pinched” EN ∼ Q follow χPT power counting define the potential V

• iterate the nucleon-nucleon potential

i.e. solve Lippmann-Schwinger equation

• “non-perturbative pions”
• “perturbative pions”
• 1. LO potential: contact S-wave operator
• 2. pion exchange as perturbation: Q/MNN ≪ 1

SLIDE 21

The T-violating Chiral Lagrangian

• include dim-four and dim-six /

T in χPT Lagrangian L/

T

= −2¯ N ¯ d0 + ¯ d1τ3

• SµvνNFµν − ¯

g0 Fπ ¯ Nπ · τN − ¯ g1 Fπ π3 ¯ NN + ¯ C1 F2

π

¯ NN ∂µ (¯ NSµN) + ¯ C2 F2

π

¯ NτN ∂µ (¯ NSµτN)

• at LO, nucleon/nuclear EDMs expressed in terms of a few couplings

¯ d0, ¯ d1 neutron & proton EDM,

• ne-body contribs. to A ≥ 2 nuclei

¯ g0, ¯ g1 pion loop to nucleon & proton EDMs leading / T OPE potential ¯ C1, ¯ C2 short-range / T potential

relative size of the coupling (∆/

T)

depends on chiral/isospin properties of / T source

SLIDE 22

Low energy couplings. ¯ θ term

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Q ¯ C1,2 × Q ¯ θ ×

m2

π

FπΛχ

1 ε

m2

π

Λ2

χ

NDA ∆/

T

1 3 ¯ θ term: LQCD = ¯ qi / Dq − ¯ m¯ qq + ¯ mε ¯ qτ3q + m∗¯ θ ¯ qiγ5q ¯ m = mu + md 2 , ¯ mε = md − mu 2 , m∗ = mumd mu + md

• ¯

θ breaks chiral symmetry = ⇒ non-derivative π-N couplings

• but not isospin

= ⇒ need extra mu − md to generate ¯ g1

SLIDE 23

Low energy couplings. ¯ θ term

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Q ¯ C1,2 × Q ¯ θ ×

m2

π

FπΛχ

1 ε

m2

π

Λ2

χ

Q2 Λ2

χ

Q2 Λ2

χ

NDA ∆/

T

1 3 3 3 ¯ θ term: LQCD = ¯ qi / Dq − ¯ m¯ qq + ¯ mε ¯ qτ3q + m∗¯ θ ¯ qiγ5q ¯ m = mu + md 2 , ¯ mε = md − mu 2 , m∗ = mumd mu + md

• ¯

θ breaks chiral symmetry = ⇒ non-derivative π-N couplings

• but not isospin

= ⇒ need extra mu − md to generate ¯ g1

• higher dimensional vertices

= ⇒ Λχ ∼ (2πFπ) suppression ¯ g0 dominates the / T potential short-distance effects are suppressed

SLIDE 24

Low energy couplings. qCEDM & qEDM

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Q ¯ C1,2 × Q

• ˜

c(0)

g v2

×

m2

πΛχ

Fπv2

1 ε

m2

π

Λ2

χ

Q2 Λ2

χ

Q2 Λ2

χ

NDA

• ˜

c(3)

g v2

×

m2

πΛχ

Fπv2

ε

m2

π

Λ2

χ

1

Q2 Λ2

χ

Q2 Λ2

χ

NDA, PQ

• ˜

cγv2 ×

m2

πΛχ

Fπv2

— —

Q2 Λ2

χ

— NDA L/

T 6

= −

• q=u,d

mq˜ c(q)

γ

2 ¯ qiσµνγ5q eFµν −

• q=u,d

mq˜ c(q)

g

2 ¯ qiσµν gsGµνγ5q qCEDM

• break chiral symmetry
• & isospin

qEDM

• break chiral symmetry
• & isospin
• ... but requires photons

large ¯ g1 & ¯ g0! short-range suppressed

• nly EDM operators

are relevant

SLIDE 25

Low energy couplings. gCEDM & four-quark

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Q ¯ C1,2Q

• {C˜

G, Σ(ud) 1,8 }v2

×

m2

πΛχ

Fπv2

1 ε 1 1 NDA

• Ξud

1,8v2

× FπΛχ

v2

ε Q2

Λ2

χ

1

Q2 Λ2

χ

Q2 Λ2

χ

NDA Weinberg & LR LR operator L/

T 6

= C˜

G

6 f abcǫµναβGa

αβGb µρGc ρ ν + 1

4Σ(ud)

1

(¯ qq¯ qiγ5q − ¯ qτq · ¯ qτ iγ5q)

• respects chiral symmetry & isospin

= ⇒ need extra ¯ m, ¯ mε to generate ¯ g0,1

• higher dimensional vertices

= ⇒ no Λχ ∼ (2πFπ) suppression comparable short & long distance contrib. to / T potential LL RR operator L/

T 6

= Ξ(ud)

1

¯ dLγµuL ¯ uRγµdR

• dominantly isospin breaking

SLIDE 26

Low energy couplings. Summary

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Fπ ¯ C1,2 × Fπ ¯ θ 1 0.01 0.05 0.05

• ˜

c(0)

g v2,˜

c(3)

g v2

1 1 0.05 0.05 ˜ cγv2 — — 0.05 — (C˜

Gv2, Σ(ud) 1,8 v2)

1 1 1 1 Ξud

1,8v2

0.01 1 0.05 0.05

• relative size depends on /

T source different signals for one, two, three-nucleon EDMs e.g.

• enhanced deuteron EDM from qCEDM
• single body dominance for qEDM

SLIDE 27

Low energy couplings. Summary

¯ g0/Fπ ¯ g1/Fπ ¯ d0,1 × Fπ ¯ C1,2 × Fπ ¯ θ 1 0.01 0.05 0.05

• ˜

c(0)

g v2,˜

c(3)

g v2

1 1 0.05 0.05 ˜ cγv2 — — 0.05 — (C˜

Gv2, Σ(ud) 1,8 v2)

1 1 1 1 Ξud

1,8v2

0.01 1 0.05 0.05

• relative size depends on /

T source different signals for one, two, three-nucleon EDMs WARNING need more than NDA for quantitative statements e.g.

• enhanced deuteron EDM from qCEDM
• single body dominance for qEDM

SLIDE 28

Low energy couplings beyond NDA

¯ g0/F2

π

¯ g1/F2

π

dn ¯ θ ×10−3 {13, 18} ⋆ {−7, −1} ⋆ {−7.7, −0.9} ˜ c(u)

g v2

×10−8 {−5, 14} ◦ {9, 55} ◦ {−0.7, −0.2} ◦ ˜ c(d)

g

v2 ×10−8 {−10, 30} ◦ {20, 119} ◦ {−3.0, −1.0} ◦ ˜ c(u)

γ v2

×10−8 — — {−0.14, −0.11} ˜ c(d)

γ v2

×10−8 — — {−0.50, −0.41} C˜

Gv2

×10−8 ??? ??? {−7, 7} ◦ Ξud

1,8v2

×10−6 — {−162, −54} ⋆ {0.1, 0.3} ⋆ * these are theory, not statistical, uncertainties ** PQ always assumed

• well under control, ∼ 10% errors
• somewhat under control ... but not

quite

• no idea...

⋆ chiral trickery lattice QCD

• QCD sum rules

Message 1: way too much red and orange more theory effort required! Message 2: ¯ θ differs a bit from NDA ¯ g0 too small, ¯ g1&dn too big

SLIDE 29

Low energy couplings beyond NDA

¯ g0/F2

π

¯ g1/F2

π

dn ¯ θ ×10−3 {13, 18} ⋆ {−7, −1} ⋆ {−7.7, −0.9} ˜ c(u)

g v2

×10−8 {−5, 14} ◦ {9, 55} ◦ {−0.7, −0.2} ◦ ˜ c(d)

g

v2 ×10−8 {−10, 30} ◦ {20, 119} ◦ {−3.0, −1.0} ◦ ˜ c(u)

γ v2

×10−8 — — {−0.14, −0.11} ˜ c(d)

γ v2

×10−8 — — {−0.50, −0.41} C˜

Gv2

×10−8 ??? ??? {−7, 7} ◦ Ξud

1,8v2

×10−6 — {−162, −54} ⋆ {0.1, 0.3} ⋆ * these are theory, not statistical, uncertainties ** PQ always assumed

• well under control, ∼ 10% errors
• somewhat under control ... but not

quite

• no idea...

⋆ chiral trickery lattice QCD

• QCD sum rules

Message 3: NDA expectations well respected by qCEDM & LL RR Message 4: ¯ g0 always on the small side agrees with large Nc arguments

SLIDE 30

The / T nucleon-nucleon potential. ¯ θ term

V(0)

¯ θ

= −¯ g0gA F2

π

τ (1) · τ (2) σ(1) − σ(2) · ∇ e−mπr 4πr

• LO: isoscalar OPE

NLO: TPE & short range, same structure as LO additional short distance ¯ C1 O

• m2

π

Λ2

χ

SLIDE 31

The / T nucleon-nucleon potential. ¯ θ term

V(2)

¯ θ

= τ (1) · τ (2) σ(1) − σ(2) · ∇

• −¯

g0gA F2

π

fTPE(r) + ¯ C2δ(r)

• + 1

2 ¯ C1

• σ(1) − σ(2)

· ∇δ(r)

LO: isoscalar OPE NLO: TPE & short range, same structure as LO additional short distance ¯ C1 O

• m2

π

Λ2

χ

SLIDE 32

The / T nucleon-nucleon potential. ¯ θ term

V(2)

¯ θ

= −¯ g0gA 2F2

π

¯ g1 ¯ g0 − β1 2gA

• (τ (1)

3

+ τ (2)

3

)(σ(1) − σ(2)) + ¯ g1 ¯ g0 + β1 2gA

• (τ (1)

3

− τ (2)

3

)(σ(1) + σ(2))

• · ∇

e−mπr 4πr

• LO: isoscalar OPE

NLO: TPE & short range, same structure as LO additional short distance ¯ C1 O

• m2

π

Λ2

χ

• isospin breaking terms

+ relativistic corrections O

• ε

m2

π

Λ2

χ ,

m2

π

m2

N

SLIDE 33

The / T potential. Dimension-six operators

V6 = −¯ g0gA F2

π

τ (1) · τ (2) σ(1) − σ(2) · ∇ e−mπr 4πr

• −¯

g1gA 2F2

π

• (τ (1)

3

+ τ (2)

3

)(σ(1) − σ(2)) + (τ (1)

3

− τ (2)

3

)(σ(1) + σ(2))

• · ∇

e−mπr 4πr

• + 1

2

• ¯

C1 + ¯ C2τ (1) · τ (2) σ(1) − σ(2) · ∇δ(r)

• qCEDM: isoscalar & isovector OPE
• LL RR: isovector OPE
• gCEDM & LR LR : OPE & short range
• qEDM: photon-exchange (negligible)

SLIDE 34

/ T currents

• only one-body currents are relevant
• two-body currents are in general suppressed

J0 = J0

T + J0 / T

= e 2(1 + τ (i)

3 ) − i(d0 + d1τ (i) 3 ) σ · q

SLIDE 35

EDMs of Light Nuclei. Power Counting

d0,1 ¯ g0 m2

N

Q MNN ¯ g0,1 Q2 , ¯ C1,2F2

π ×

Q MNN ¯ g0,1 m2

N

Q2 M2

NN

MNN ∼

4πF2

π

mN , typical nuclear scale

SLIDE 36

EDMs of Light Nuclei. Power Counting

d0,1 ¯ g0 m2

N

Q MNN ¯ g0,1 Q2 , ¯ C1,2F2

π ×

Q MNN ¯ g0,1 m2

N

Q2 M2

NN

• Theta, qCEDM & LL RR: pion-exchange

dominates

• qEDM: one-body dn, dp dominate
• gCEDM & LR LR: one-body, pion-exchange &

short range equally important. selection rules! beware of ¯ g0 suppression

SLIDE 37

EDMs of Light Nuclei. Power Counting

d0,1 ¯ g0 m2

N

Q MNN ¯ g0,1 Q2 , ¯ C1,2F2

π ×

Q MNN ¯ g0,1 m2

N

Q2 M2

NN

• Theta, qCEDM & LL RR: pion-exchange

dominates

• qEDM: one-body dn, dp dominate
• gCEDM & LR LR: one-body, pion-exchange &

short range equally important. selection rules! beware of ¯ g0 suppression

SLIDE 38

Deuteron EDM and MQM

Spin 1, Isospin 0 particle H/

T = −2ddD†S · ED − MdD†{Si, Sj}D∇(iB j)

dd: deuteron EDM Md: deuteron magnetic quadrupole moment (MQM). dEDM

• isoscalar (¯

g0, ¯ C1,2) TV corrections to wavefunction vanish at LO. dMQM

• both isoscalar & isovector corrections contribute

SLIDE 39

Deuteron EDM and MQM

Spin 1, Isospin 0 particle H/

T = −2ddD†S · ED − MdD†{Si, Sj}D∇(iB j)

dd: deuteron EDM Md: deuteron magnetic quadrupole moment (MQM). dEDM

• isoscalar (¯

g0, ¯ C1,2) TV corrections to wavefunction vanish at LO. dMQM

• both isoscalar & isovector corrections contribute

SLIDE 40

Deuteron EDM. Perturbative Pions

One-body / T corrections to wavefunction

• only sensitive to isoscalar nucleon EDM

FD(q2) = 2d0 4γ |q| arctan |q| 4γ

• = 2d0
• 1 − 1

3 |q| 4γ 2 + . . .

• ,

γ = √ BmN

• sensitive to isobreaking ¯

g1 FD(q2) = −2 3egA¯ g1 m2

π

mNmπ 4πF2

π

1 + ξ (1 + 2ξ)2

• 1 − 0.45

|q| 4γ 2 + . . .

• ,

ξ = γ mπ

SLIDE 41

Deuteron EDM. Perturbative Pions

One-body / T corrections to wavefunction

• only sensitive to isoscalar nucleon EDM

dd = dn + dp

• sensitive to isobreaking ¯

g1 dd = −0.22 ¯ g1 F2

π

• good agreement with ptb. pion power counting Q/MNN ∼ 1/3

SLIDE 42

Deuteron EDM

from EM and U.van Kolck, ‘15

• several calculations with non-perturbative iteration of OPE potential
• using pheno & chiral T-conserving potentials
• C. P. Liu and R. Timmermans, ‘05; J. de Vries et al, ‘11;
• J. Bsaisou et al, ‘13, J. Bsaisou et al, ‘15;
• N. Yamanaka and E. Hiyama, ‘15
• one-body & /

T OPE contribution not affected by different potentials

• results agree well with ptb. pion estimates

SLIDE 43

Implications of the deuteron EDM

v2˜ c(u)

g

v2˜ c(d)

g

v2Ξud ¯ θ v2C˜

G

(dd − dn − dp)/dn {2, 50} {1, 22} {30, 300} {−1.4, −0.02} 1

• qCEDM & LL RR: strong enhancement of dd
• LL RR is particularly enhanced
• ¯

θ term : ratio at most O(1)

• gCEDM & LR LR : ratio 1

WARNING need better LECs!

• qEDM: one solid prediction!

dd = dn + dp In summary:

• dd ≫ dn + dp indicates isobreaking /

T & non-¯ θ origin

• dd = dn + dp clear sign of qEDM

SLIDE 44

Deuteron MQM

Corrections to wavefunction

nucleon magnetic moment κ0 = −0.12, κ1 = 3.71

• deuteron MQM does not have single nucleon backgroud
• clean probe of ¯

g0 and ¯ g1 mdMd = −2egA¯ g0 m2

π

mNmπ 2πF2

π

• (1 + κ0) + ¯

g1 3¯ g0 (1 + κ1)

• 1 + ξ

(1 + 2ξ)2 = −1.43(1 + κ0) ¯ g0 Fπ e fm − 0.48(1 + κ1) ¯ g1 Fπ e fm,

• for qCEDM, LL RR
• mdMd

2dd

• = (1 + κ1) + 3¯

g0 ¯ g1 (1 + κ0) can discriminate LL RR (¯ g0 = 0) from qCEDM (¯ g0 ¯ g1)

SLIDE 45

EDM of 3He and 3H

• one-body not affected by different potentials

dh = 0.9dn dt = 0.9dp

• OPE agrees well with ptb. pion counting

< 10% error on ¯ g1 ∼ 20% error on ¯ g0

• short-range ¯

C1,2 much harder to estimate

• ... keep going ... recent interesting results on 6Li, 9Be, 13C
• N. Yamanaka and E. Hiyama, ‘15, N. Yamanaka, ‘16

SLIDE 46

Implications of 3He EDM

v2˜ c(u)

g

v2˜ c(d)

g

v2Ξud ¯ θ (dh − dn)/dn {1, 44} {1, 19} {25, 225} {0.01, 1.4} (dh − dn)/(dd − dp − dn) {0.5, 1.4} {0.5, 1.4} 0.7 {−6.8, −0.04}

• three-nucleon system provides extra texture
• qCEDM: dh ≫ dn, but similar to dd
• LL RR: dh ≫ dn and one solid prediction!
• ¯

θ term : dh ∼ dn, but larger than dd need to reduce uncertainties on ¯ g1(¯ θ)

• gCEDM & LR LR : ∼ 1, but large uncertainties from ¯

C1,2

• qEDM: one solid prediction!

dh = 0.9dn

SLIDE 47

EDMs of light nuclei. Summary

✤ ✣ ✜ ✢

• 1. Identify qEDM
• EDMs of nuclei does not deviate

from nucleon EDM

• clean connection to quark level

... boring...

✤ ✣ ✜ ✢

• 2. Identify qCEDM & LL RR
• clean signal: large dd & dh
• prediction

(dh − dn)/(dd − dn − dp)

• “easy” to improve ¯

g0 & ¯ g1 Measure dn, dp, dd & dh

✤ ✣ ✜ ✢

• 3. Pinpoint ¯

θ?

• hard to do with dd, dh

¯ g1 poorly known, hard to improve

• best shot:

precise LQCD calc. of dn, dp

☛ ✡ ✟ ✠

• 4. in any case

need more work on LECs!

SLIDE 48

SLIDE 49

Backup Slides

SLIDE 50

/ T pion-nucleon couplings. QCD ¯ θ term

L4 = −¯ m¯ qq + ¯ mε ¯ qτ3q + m∗¯ θ ¯ qiγ5q

• ¯

θ term and mass splitting are chiral partners ¯ qiγ5q ¯ qτq

• SUA(2)

− − − − → −¯ qα · τq α ¯ qiγ5q

• i.e. one spurion enough to construct iso- and T-breaking couplings

T violation isospin breaking = 1 − ε2 2ε ¯ θ ≡ ρ¯

θ

• powerful at LO
• breaks down at O(Q2/Λ2

χ)

× ignorance of CP-even LECs × too many operators when adding EM

SLIDE 51

QCD Theta Term. ¯ g0

courtesy of

• A. Walker-Loud

L(1) = ∆mN ¯ NN+1 2δmN

• ¯

Nτ3N − 2ρ¯

θ ¯

N π · τ Fπ N

• =

⇒ ¯ g0 = δmN 1 − ε2 2ε ¯ θ

• δmN = (mn − mp)st, strong mass splitting
• well respected beyond LO

¯ g0 Fπ = (15.5 ± 2 ± 1.6) · 10−3¯ θ lattice error conservative estimate of theory error

SLIDE 52

/ T pion-nucleon couplings. Quark CEDM

• same trick works for other chiral-breaking op.

L6 = −1 2 ¯ qσµνgsGµν

• ˜

c0 + iγ5τ3˜ d3

• q − 1

2 ¯ qσµνgsGµν

• ˜

c3τ3 + iγ5˜ d0

• q

+r ¯ qiγ5˜ d3 (τ3 − ε) q

• qCEDM has CP-even chiral partner

1 2

• ¯

qσµνgsGµνq −¯ qσµνiγ5τgsGµνq

• isovector qCEDM

& isoscalar qCMDM 1 2

• ¯

qσµνiγ5gsGµνq ¯ qσµντgsGµνq

• isoscalar qCEDM

& isovector qCMDM

SLIDE 53

/ T pion-nucleon couplings. Quark CEDM

• same trick works for other chiral-breaking op.

L6 = −1 2 ¯ qσµνgsGµν

• ˜

c0 + iγ5τ3˜ d3

• q − 1

2 ¯ qσµνgsGµν

• ˜

c3τ3 + iγ5˜ d0

• q

+r ¯ qiγ5˜ d3 (τ3 − ε) q

• qCEDM has CP-even chiral partner

1 2

• ¯

qσµνgsGµνq −¯ qσµνiγ5τgsGµνq

• isovector qCEDM

& isoscalar qCMDM 1 2

• ¯

qσµνiγ5gsGµνq ¯ qσµντgsGµνq

• isoscalar qCEDM

& isovector qCMDM

• ˜

d3 causes vacuum misalignment

• re-alignment causes the appearance of a mass term

r = 1 2 ¯ qσµνgsGµνq ¯ qq = dm2

π

d˜ c0 d¯ m dm2

π

= 0.4 GeV2

SLIDE 54

Quark CEDM. ¯ g0, and ¯ g1

• chiral partners of qCEDM modify the meson and baryon spectrum

L6 = −1 2 ¯ qσµνgsGµν˜ c0q − 1 2 ¯ qσµνgsGµν˜ c3τ3q − → −1 2 ˜ ∆mππ2 + ˜ ∆mN ¯ NN + ˜ δmN ¯ Nτ3N

• chiral symmetry relates /

T couplings to masses ¯ g0 = dδmN d ˜ c3 + r dδmN d¯ mε

• ˜

d0 + δmN 1 − ε2 2ε ¯ θ − ¯ θinduced

• ¯

g1 = −2 dmN d˜ c0 − r dmN d¯ m

• ˜

d3,

• ¯

θinduced = rTr(M−1˜ d), term vanishes if Peccei-Quinn

• can estimate the “tadpole” piece
• not yet the “direct” pieces

these calculations can be done on the lattice!

SLIDE 55

SLIDE 56

Nucleon EDM. Theta Term & qCEDM.

• EDM depends on ¯

g0, and short-distance LECs ¯ d0,1

• neutron EDM

|dn| = |d0 − d1| > ∼ egA¯ g0 (2πFπ)2

• ln m2

N

m2

π

+ π 2 mπ mN

• ≃ (0.130 + 0.008) ¯

g0 Fπ e fm

• NLO bound on isoscalar EDM

|d0| > ∼ egA¯ g0 (2πFπ)2 π 3mπ 4mN ≃ 0.012 ¯ g0 Fπ e fm.

• S′

0,1 only depends on ¯

g0 S′ = − egA¯ g0 12(2πFπ)2 πδmN m2

π

= −0.3 · 10−3 ¯ g0 Fπ e fm3, S′

1

= egA¯ g0 6(2πFπ)2 1 m2

π

• 1 − 5π

4 mπ mN

• = (11.2 − 6.5) · 10−3 ¯

g0 Fπ e fm3,

• contribs. to Schiff moment

relevant for atomic EDMs

SLIDE 57

Electromagnetic and T-violating operators

• chiral properties of (P3 + P4) ⊗ (I + T34)
• lowest chiral order ∆ = 3
• P3 + P4

L(3)

/ χ,f=2,em = c(3) 1,em

1 D 2π3 Fπ + ρ

• 1 − π2

F2

π

• ¯

N (Sµvν − Sνvµ) N eFµν

• (P3 + P4) ⊗ T34

L(3)

/ χ,f=2,em = c(3) 3,em ¯

N

• −

2 FπDπ · t − ρ

• t3 − 2π3

F2

πDπ · t

• (Sµvν − Sνvµ) N eFµν

+ tensor

• isoscalar and isovector EDM related to pion photo-production.

SLIDE 58

Electromagnetic and T-violating operators

At the same order S4 ⊗ (1 + T34)

• S4

L(3)

/ χ,f=2,em = c(3) 6,em

• −

2 FπD

• ¯

Nπ · t (Sµvν − Sνvµ) N eFµν

• S4 ⊗ T34

L(3)

/ χ,f=2,em = c(3) 8,em

2π3 FπD ¯ N (Sµvν − Sνvµ) N eFµν + tensor

• same chiral properties as partners of /

T operator

• pion-photoproduction constrains only c(3)

1, em + c(3) 6, em and c(3) 3, em + c(3) 8, em

• but /

T only depends on c(3)

1, em and c(3) 3, em

no T-conserving observable constrains short distance contrib. to nucleon EDM

• true only in SU(2) × SU(2)
• larger symmetry of SU(3) × SU(3) leaves question open

SLIDE 59

Deuteron EDM and MQM. KSW Power Counting

T-even sector Lf=4 = −C

3S1

0 (NtPiN)†NtPiN+C

3S1

2

8

• (NtPiN)†NtD2

−PiN + h.c.

• +. . . ,

Pi = 1 √ 8 σ2σiτ

• enhance C0 to account for unnaturaly large scattering lengths. In PDS scheme

C

3S1

= O 4π mNµ

• ,

µ ∼ Q

• iterate C0 at all orders

C0 C0

mNQ 4π C0

C0 mNQ

4π C0

2

SLIDE 60

Deuteron EDM and MQM. KSW Power Counting

T-even sector Lf=4 = −C

3S1

0 (NtPiN)†NtPiN+C

3S1

2

8

• (NtPiN)†NtD2

−PiN + h.c.

• +. . . ,

Pi = 1 √ 8 σ2σiτ

• enhance C0 to account for unnaturaly large scattering lengths. In PDS scheme

C

3S1

= O 4π mNµ

• ,

µ ∼ Q

• iterate C0 at all orders
• operators which connect S-waves get enhanced C

3S1

2

= O

• 4π

mNΛNN 1 µ2

• C0

Q ΛNN

C0

Q ΛNN mNQ 4π C0

C0

Q ΛNN

mNQ

4π C0

2

SLIDE 61

Deuteron EDM and MQM. KSW Power Counting

• treat pion exchange as a perturbation

C0

g2

AmNQ

4πF2

π

C0

g2

AmNQ

4πF2

π

mNQ 4π C0

C0

g2

AmNQ

4πF2

π

mNQ

4π C0

2

• identify ΛNN = 4πF2

π/mN ∼ 300 MeV.

Perturbative pion approach:

• expansion in Q/ΛNN, with Q ∈ {|q|, mπ, γ = √mNB}
• competing with the mπ/MQCD of ChPT Lagrangian
• successful for deuteron properties at low energies

Kaplan, Savage and Wise, Phys. Rev. C 59, 617 (1999);

• problems in 3S1 scattering lenghts,
• ptb. series does not converge for Q ∼ mπ

Fleming, Mehen, and Stewart, Nucl. Phys. A 677, 313 (2000);

SLIDE 62

Deuteron EDM and MQM. KSW Power Counting

• treat pion exchange as a perturbation

g2

A

F2

π

g2

A

F2

π

g2

AmNQ

4πF2

π

• identify ΛNN = 4πF2

π/mN ∼ 300 MeV.

Perturbative pion approach:

• expansion in Q/ΛNN, with Q ∈ {|q|, mπ, γ = √mNB}
• competing with the mπ/MQCD of ChPT Lagrangian
• successful for deuteron properties at low energies

Kaplan, Savage and Wise, Phys. Rev. C 59, 617 (1999);

• problems in 3S1 scattering lenghts,
• ptb. series does not converge for Q ∼ mπ

Fleming, Mehen, and Stewart, Nucl. Phys. A 677, 313 (2000);

SLIDE 63

Deuteron EDM and MQM. KSW Power Counting

T-odd sector

• a. four-nucleon T-odd operators

L/

T,f=4 = C1,/ T ¯

NS · (D + D†)N ¯ NN + C2,/

T ¯

Nτ S · (D + D†)N · ¯ N τN.

• in the PDS scheme
• 1. Theta
• 2. qCEDM
• 3. qEDM
• 4. gCEDM

Ci,/

T 4π µmN ¯

θ

m2

π

MQCDΛ2

NN

4π µmN ˜

δ

m2

π

M2

/ TMQCD

4π µmN w M2

/ T ΛNN

• b. four-nucleon T-odd currents

L/

T, em,f=4 = C1,/ T, em ¯

N(Sµvν − Sνvµ)N ¯ NNFµν.

• in the PDS scheme
• 1. Theta
• 2. qCEDM
• 3. qEDM
• 4. gCEDM

Ci,/

T,em 4π µ2mN ¯

θ

m2

π

MQCDΛ2

NN

4π µ2mN ˜

δ

m2

π

M2

/ TMQCD

4π µ2mN δ m2

π

M2

/ TMQCD

4π µ2mN w M2

/ T ΛNN

SLIDE 64

Deuteron EDM. Formalism

• crossed blob: insertion of interpolating field Di(x) = N(x)P

3S1

i

N(x)

• two-point and three-point Green’s functions expressed in terms of irreducible

function irreducible: do not contain C

3S1

• by LSZ formula

p′j|Jµ

em,/ T|p i = i

• Γµ

ij (¯

E, ¯ E′, q) dΣ(¯ E)/dE

• ¯

E,¯ E′=−B

• two-point function

dΣ(1) d¯ E

• ¯

E=−B

= −i m2

N

8πγ