• The baryon EDMs from chiral perturbation theory

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Theta-term physics

Ulf-G. Meißner, Univ. Bonn & FZ J ¨ ulich

Supported by BMBF 05P15PCFN1 by DFG, SFB/TR-110 by CAS, PIFI by Volkswagen Stftung

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CONTENTS

• Introduction

• The baryon EDMs from chiral perturbation theory

• The neutron EDM from 2+1 flavor lattice QCD

• EDMs of light nuclei and models of CP violation

• Hadrons at finite theta

• Summary & outlook

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Introduction

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QCD with a θ-TERM

• QCD has non-trivial topological vacua: |θi = P

n

ei n θ|ni, n ∈ Z

• QCD in the presence of the θ-term

LQCD = −1

4GaµνGa,µν + X

flavors

¯

q (iD/ − M) q + θ0 g2

32π2 Gaµνa,µν

| {z }

∼ ~Ea · ~Ba

⇒ This leads to strong CP-violation

⇒ A non-vanishing vacuum angle θ0 entails dn 6= 0 (also dp 6= 0)

dN ≈ |θ0| e Mπ2

m3 ≈ 10−160| e cm → θ0 = O(10−10)

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CONNECTION to the QUARK MASSES

5

• Shift under axial rotations:

θ → ¯θ = θ − 2Nfα

• Influences the diagonalization of the quark mass matrix:

θ = θ + arg (det M)¯

⇒ QCD mass term for small values of θ:

Lmass = muuu + m¯ ddd + m¯ sss¯ + i ¯θ mumdms

mumd + mums + mdms uγ¯ 5u + ¯dγ5d + ¯sγ5s

| {z }

ps density

? no physical effect if one mq vanishes

? however, this is excluded by QCD phenomenology

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EXPERIMENTS on the NEUTRON EDM

• performed and projected experiments (UCN):

1950 1970 1990 2010

PSI MultiCell

3He−UCN

Beams UCN

d [e cm]

n

10−32 10−28 10−24 10−20

SM SUSY

year of experiment

QCD θ

10

10

10

−10

−12

−14

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The baryon EDM in chiral perturbation theory

Ottnad, Kubis, UGM, Guo, Phys. Lett. B 687 (2008) 42 Guo, UGM, JHEP 1212 (2012) 097

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BARYON EDMs - BASIC DEFINITIONS

• Baryon electromagnetic form factors in the presence of strong CP -violation [or other sources of it]

hp0| Jemν |pi = ¯u (p0) Γν q2 u (p) Γν = γνF1 q2 − i

2mB σµνqµF2 q2 − 1

2mB σµνqµγ5F3 q2

+ . . . q2 = (p0 − p)2

• Dipole moments and radii:

dB = F3,B(0)

2mB , red2 = 6dF3(q2) dq2

q2=0

• Similar formulae for light nuclei [see later]

000000 000 111111 111

p

p

q

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BARYON EDMs - CALCULATIONS

• non-perturbative methods: baryon chiral perturbation theory and/or lattice QCD

• study the nucleon/baryon electric dipole form factors in baryon CHPT

? already quite a number of investigations (mostly the neutron)

Crewther, Veneziano, Witten, Pich, de Rafael, . . ., Borasoy, Narison, Hockings, van Kolck, de Vries, . . .

? complete one-loop calculation Ottnad, Kubis, M., Guo, Phys. Lett. B 687 (2008) 42

? based on Leffa, φ0, B] supplemented by power counting

• LQCD calculations for θ and CP-violating form factors become available

⇒ must address quark mass and finite volume corrections

⇒ fruitful interplay between LQCD and CHPT/EFT practitioners

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EFFECTIVE LAGRANGIAN

• Effective Lagrangian based on U (3)L × U (3)R symmetry:

Herrera-Sikl ´ody et al., Borasoy

Leff = L[U, B] , U = expr 2 3

i F0η0

| {z }

U (1)

+ 2i Fφφ

| {z }

SU (3)



• Power counting [δ = small parameter]: Kaiser, Leutwyler, . . .

p = O (δ) , mq = O δ2

, 1/Nc = O δ2

• Determination of the low-energy constants

− meson sector: meson masses, η-η0 mixing, . . .

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EFFECTIVE LAGRANGIAN

• U (3) × U (3) effective meson-baryon Lagragian Borasoy (2000)

LφB = i TrBγ¯ µ[Dµ, B] − ˚m Tr[ ¯BB] − D/F

2 TrBγ¯ µγ5[uµ, B]± +bD/F TrB¯χ+ − iA(U − U), B

± + b0 Tr[ ¯BB] Trχ+ − iA(U − U) +4A w100

6

F0 η0Tr[ ¯BB] + i

w13/140 θ¯0 + w13/14

6

F0 η0

TrBσ¯ µνγ5[Fµν+ , B]± +w16/17 TrBσ¯ µν[Fµν+ , B]± + w0

2 TrBγ¯ µγ5BTr[uµ]

• tree-level LECs: w13, w14, w130 , w014 [some get renormalized]

• loop LECs: w100 , w0

• further LECs are absorbed in masses, magnetic moments, etc

¯

}

more later!

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BARYON EDMS at ONE LOOP

Guo, UGM, JHEP12 (2012) 097

• Consider the ground state baryon octet (N, Λ, Σ, , Ξ)

,→ not only interesting in itself, but also provides sufficient data to fix LECs

• tree-level contributions to baryon EDMs α = 144V0(2)V3(1)/(F0FπMη0)2:

d(n) = d(Λ)/2 = −d(Σ0)/2 = d(Ξ0) = 8e ¯θ0 αw13 + w130  /3 d(p) = d(Σ+) = −4e ¯θ0 α (w13 + 3w14) + w130 + 3w140  /3

d(Σ) = d(Ξ) = −4e ¯θ0 α (w13 − 3w14) + w130 − 3w014 /3

• Charged particles feature more loop contributions than neutral ones ,→ Neutron case: {π, p} and {K+, Σ}

,→ Proton case: {π+, n}, {π08, η0), p}, {K+, Σ0(Λ)} and {K0, Σ+}

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FEYNMAN GRAPHS for F

3B

(q

2

)

• tree (a,b) and one-loop (c,d,e,f) graphs (complete one-loop)

η0

(a)

(d) (e) (f)

(b) (c)

• other loop graphs (see Ottnad et al.) mutually cancel ⊗ CP-odd

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NEUTRON LOOP CONTRIBUTIONS

• one-loop contributions to the neutron EDM (other neutral baryons similar):

F3,nloop(q2)

2mN = V0(2)e ¯θ0 π2Fπ4

(

(D + F ) (bD + bF )

×

"

1 − ln Mµ2π2 + σπ ln σσπ−1

π+1 + π(2Mπ2−q2)

2mN

−q2 arctan

−q2 2Mπ

#

−(D − F ) (bD − bF )

"

1 − ln Mµ2K2 + σK ln σσK−1

K+1

+√π

−q2

2M2

K−q2

2mN − 8 (bD − bF ) MK2 − Mπ2

× arctan

−q2 2MK

#)

h

σπ(K) = q

1 − 4Mπ(K)2 /q2i

• only known masses and couplings !

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PROTON LOOP CONTRIBUTIONS

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• one-loop contributions to the proton EDM (other charged baryons similar):

F3,ploop(q2)

2mN = −V0(2)e ¯θ02Fπ4

(

6(D + F ) (bD + bF )

×

"

1 − ln M

2 π

µ2 + σπ ln σσπ−1

π+1 + 3πM2m π

N + π(2Mπ2−q2)

2mN

−q2 arctan

−q2 2Mπ

#

+4 (DbD + 3F bF )



1 − ln M

2 K

µ2 + σK ln σσK−1

K+1 + πMm K

N



+√

−q2 arctan

−q2 2MK



(DbD+3F bF)

2mN 2MK2 − q2 +8 MK2 − Mπ2

F b2D + 3b2F  − 23DbD (bD − 3bF )



+mπ

N



6(D − F ) (bD − bF ) MK + (D − 3F ) (bD − 3bF ) Mη8 +2F

2 π

F02 (2D − 3w0) 2bD + 3b0 + 6w100 

| {z }

≡ β

Mη0

)

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MAKING PREDICTIONS

• Close inspection of the tree and one-loop expressions reveals

⇒ only two combinations of LECs appear at NLO in all baryon EDMs:

wa(µ) ≡ αw13 + w130 r(µ)

wb(µ) ≡ 3[αw14 + w140 r(µ)] + V0(2)β

4πF02Fπ2maveMη0

• use this formalism to analyze lattice data → next section

• there are relations free of LECs such as dΣ0 + dΛ ∼ MK2 − Mπ2

F b2D + 2DbDbF + 3F b2F  + O(δ4) dn − dΞ0 ∼ (DbD + F bF) 

2 ln M

2 K

Mπ2 + πMπm−MK

ave

 +M

K MK2 − Mπ2

Db2D + 2F bDbF + Db2F  + O(δ4)

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The neutron EDM from 2+1 flavor lattice QCD

Guo, Horsley, UGM, Nakamura, Rakow, Schierholz, Zanotti, Phys. Rev. Lett. 115 (2015) 062001 [arXiv:1502.02295]

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LATTICE FORMULATION: GENERALITIES

• work around the SU(3) symmetry point, keeping the singlet mass ¯m fixed:

¯

m = (mu + md + ms)/3 Bietenholz et al., Phys. Rev. D84 (2010) 054059

⇒ keeps the kaon mass low for varying strange quark masses constrained polynomials fits

⇒ works fine for Nf = 2 + 1 [mu = md = m`]

0.00 0.25 0.50 0.75 1.00 1.25

0.8 0.9 1.0 1.1 1.2

MNO/XN [Octet]

experiment N(lll) Λ(lls) Σ(lls) Ξ(lss) sym. pt.

π K η ρ K* φ N Λ Σ Ξ Σ* Ξ* Ω 0

500 1000 1500 2000

M [MeV]

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LATTICE FORMULATION at FIXED TOPOLOGY

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• The θ-term on the lattice [at fixed topological charge Q]:

S = S0 + Sθ, Sθ = i θ Q, Q = − 1

64π2 µνρσ a4 X

x

GbµνGbρσ ∈ Z

• rotate the θ-term into the fermionic part of the action Baluni (1979)

Sθ = − i

3θ ˆma4 X

x

¯

5u + ¯dγ5d + ¯sγ5s ˆ

m−1 = 1 3



m−1u + m−1d + m−1s 

= 1 3



2m−1` + m−1s 

• take the vacuum angle imaginary: θ = i ¯θ [th’y analytic at θ = 0]

⇒ Sθ = ¯θ m` ms

2ms + m` a4 X

x

¯

5u + ¯dγ5d + ¯sγ5s

⇒ real action, vanishes at m and m

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LATTICE SET-UP

• Wilson fermions with a clover term & Symanzik improved gluon action (SLiNC) Sq = S0q + Sθq = a4 X

x

¯ q



D − 1

4cSW σµν Gµν + mq + λ 2a γ5

 q

• lattice set-ups (β = 5.50)

# κ` κs V Mπ [MeV] MK [MeV] λ

1 0.1209 0.1209 243 × 48 465 465 0.003 2 0.1209 0.1209 243 × 48 465 465 0.005 3 0.1210 0.1206 243 × 48 360 505 0.003 4 0.1210 0.1206 243 × 48 360 505 0.005 5 0.1210 0.1206 323 × 64 360 505 0.003 6 0.1211 0.1205 323 × 64 310 520 0.003

• a = 0.074(2)m from the average baryon octet mass

• only ensembles 1 to 4 in the PRL publication

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LATTICE SET-UP: TOPOLOGICAL CHARGE

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• keep the singlet mass ¯m fixed at its physical value and vary δmq = mq − ¯m

λ = ¯θ 2a m` ms

2ms + m`, amq = 1

q 1

0,c, κ0,c = 0.1211

• Ensembles carry topological charge, hQi ∼ ¯m

topological charge distribution #4 average topological charge

0.05 0.1 0.15 0.2

-8 -6 -4 -2 0 2 4

p(Q)

Q 0.05

0.1 0.15 0.2

-8 -6 -4 -2 0 2 4

p(Q)

Q 0.05

0.1 0.15 0.2

-8 -6 -4 -2 0 2 4

p(Q)

Q

-3 -2.5 -2 -1.5 -1 -0.5 0

0 0.5 1 1.5 2 2.5 3

hQi

θ¯ -3

-2.5 -2 -1.5 -1 -0.5 0

0 0.5 1 1.5 2 2.5 3

hQi

θ¯ -3

-2.5 -2 -1.5 -1 -0.5 0

0 0.5 1 1.5 2 2.5 3

hQi

θ¯

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i

EVALUATION

• at non-vanishing θ, Dirac spinors pick up a phase ∼ α(θ)

→ can be obtained from a ratio of two-point functions

Tr[GθN N(t; 0)Γ4γ5]

Tr[GθN N(t; 0)Γ4] = i sin 2α(θ)

1 + cos 2α(θ) -0.2

-0.15 -0.1 -0.05 0

0 0.5 1 1.5 2 2.5 3

¯α(¯θ)

θ¯ -0.2

-0.15 -0.1 -0.05 0

0 0.5 1 1.5 2 2.5 3

¯α(¯θ)

θ¯

mπ= 465MeV mπ= 360MeV

• flavor-singlet ps density interacts with the nucleon through quark-line disconnected diagrams only

*

γ5 γµ

+

• Form factor F3(q2) extracted from a ratio of 3-point and 2-point functions

generalizing the method of Capitani et al., Nucl. Phys. Proc. Suppl. 73 (1999) 294

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FORM FACTOR RATIOS

• Extracting F3θ,n¯ (q2)/F1θ,p¯ (q2) at finite θ:

easier extrapolation to q2 = 0

-0.3 -0.2 -0.1 0

0 0.1 0.2 0.3

¯ θ,n 3¯ F/F

¯ θ,p 1

(aq)2 -0.3

-0.2 -0.1 0

0 0.1 0.2 0.3

¯ θ,n 3¯ F/F

¯ θ,p 1

(aq)2 -0.3

-0.2 -0.1 0

0 0.1 0.2 0.3

¯ θ,n 3¯ F/F

¯ θ,p 1

(aq)2

mπ= 465MeV , 243× 48

-0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

-0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

-0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

mπ= 360MeV , 323× 64

-0.8 -0.6 -0.4 -0.2 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

-0.8 -0.6 -0.4 -0.2 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

-0.8 -0.6 -0.4 -0.2 0

0 0.05 0.1 0.15 0.2

¯ θ,n 3¯ F

/F

¯ θ,p 1

mπ= 310MeV , 323× 64

Mπ = 465 MeV

Mπ = 360 MeV Mπ = 310 MeV

new

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RESULTS

• Form factor as a fct of ¯θ

• continue θ and F3θ(0) to real values

• expand around θ = 0 : F3θ(0) = F3(1)(0)θ + . . .

→ dn = eF3(1)(0)θ/2mN

• chiral extrapolation with 2MK2 + Mπ2 =const.

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.5 1 1.5 2 2.5 3

¯ θ,n¯ F

R 3(0)

θ¯ -0.7

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.5 1 1.5 2 2.5 3

¯ θ,n¯ F

R 3(0)

θ¯ -0.7

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0 0.5 1 1.5 2 2.5 3

¯ θ,n¯ F

R 3(0)

θ¯

mπ= 465MeV mπ= 360MeV

→ wa(µ = 1 GeV) = 0.04(1) GeV−1

→ dn = −0.0039(2)(9) [e fm θ]

→ |θ| . 7.4 × 10−11

• proton under analysis

-0.05 -0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m2π[GeV2] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m2π[GeV2] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m2π[GeV2] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m2π[GeV2] new

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OTHER RESULTS

• Neutron EDM using Nf = 2 + 1 + 1 twisted mass fermions

|dn| = 0.045(6)(1)|θ| e fm, one ensemble @ Mπ = 373 MeV

Alexandrou et al., Phys. Rev. D 93 (2016) 074503

• Neutron EDM and tensor charges from LQCD

dn < 4 × 10−28 e cm in split SUSY w/ gaugino mass unification

Bhattacharya et al., Phys. Rev. Lett. 115 (2015) 212002

• other groups are close to announce numbers

domain wall fermions Nf = 2 + 1, low pion masses, various volumes

Shintani et al., Phys. Rev. D 93 (2016) 094503

nucleon EDM from gradient flow, tested in quenched QCD

Shindler et al., Phys. Rev. D 92 (2015) 094518

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EDMs of light nuclei and models of CP violation

Bsaisou, Hanhart, Liebig, UGM, Nogga, Wirzba, Eur. Phys. J. A 49: 31 (2013)

Bsaisou, de Vries, Hanhart, Liebig, UGM, Minosi, Nogga, Wirzba, JHEP 03 (2015) 104 Bsaisou, UGM, Nogga, Wirzba, Annals Phys. 359 (2015) 317

Wirzba, Bsaisou, Nogga, Int. J. Mod. Phys. E 26 (2017) 1740031

see also: de Vries, Hockings, Mereghetti, van Kolck, Timmermans (2005 - 2015) Yamanaka, Hiyama (2015-2017)

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MOTIVATION

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• Why nuclei? ⇒ non-trivial test of the θ-scenario and other models of CPV

⇒ allow access to other CP-violating couplings

⇒ allow to disentangle various model of CPV

• two-flavor effective Lagrangian in standard heavy baryon formulation

Bernard, UGM, Kaiser, Int.J.Mod.Phys. E4 (1995) 193

LπN

CP = − dnN(1 − τ3)SµvνN Fµν − dpN(1 + τ3)SµvνN Fµν + (mN∆) π3π2 + g0N~π · ~τ N + g1Nπ3N

+ C1NN Dµ(NSµN ) + C2N~τ N · Dµ(N~τ SµN )

• various contributions at next-to-leading order:

∗ single nucleon EDMs – dn, dp

∗ CP-violating 3-pion and pion-nucleon couplings – ∆, g0, g1

∗ CP-violating nucleon-nucleon contact interactions – C , C

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CP-VIOLATING NUCLEAR OPERATORS

• EDM matrix element (A =2H, 3He, 3H) [Breit frame]:

iqF3A(q2)

2mA = D

A; MJ = J

0(q)

A; MJ = JE

, dA = lim

q2→0

F3A(q2) 2mA

• CP-violating transition current (only linear terms):

µ = Jµ

CP+ V

CPG Jµ + Jµ G V

CP+ · · · [G = 2N, 3N Greens function]

= + + +

J˜µ

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

• single nucleon current (a):

Jiµ = 2e 

1 + τ(i)3 

vµ , Jµ

CP,i = 12 h

dn 

1 − τ(i)3 

+ dp 

1 + τ(i)3 i

i~q · ~σ(i) vµ

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CP-VIOLATING NUCLEAR OPERATORS continued

irreducible 2N potential (b) + (c):

V N N

CP,ij (~ki) = i gA 2Fπ

~ki

~ki2 + Mπ2 g0(ij)(i) · ~τ(j) + i gA

4Fπ

~ki

~ki2 + Mπ2 h

g1 + ∆ fg1(|~ki|)i 

~

σ(ij)+ τ(ij) + ~σ(ij) τ(ij)+  + i

2

β2Mπ2~ki

~ki2 + β2Mπ2 h

C1(ij) + C2(ij)(i) · ~τ(j) i

fg1(k) = −15 32

gA2 MπmN πFπ2

"

1 + 1 + 2~k2/(4Mπ2)

3|~k |/(2Mπ) arctan |~k | 2Mπ

!

− 1 3

!#

• ∆ fg1 induced from 3-pion vertex

• β → ∞ in calcs, used as diagnostic tool









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CP-VIOLATING NUCLEAR OPERATORS continued

• irreducible 3N potential (d):

V 3N

CP (~k1, ~k2, ~k3) = −i∆mNgA3

4Fπ3 δabδc3 + δacδb3 + δbcδa3 τ(1)a τ(2)b τ(3)c

× (~σ(1) · ~k1)(~σ(2) · ~k2)(~σ(3) · ~k3) h~k12 + Mπ2i h~k22 + Mπ2i h~k32 + Mπ2i

• evaluate 2N and 3N bound states using Faddeev equations hψA| ˜JµAi = hψA|Jµ

CP + V

CPG Jµ + Jµ G V

CP + · · · |ψAi

• employ wave functions from chiral EFT at N2LO (precise enough)

• uncertainty from varying the cut-off in the chiral EFT, better for 2H

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RESULTS for LIGHT NUCLEI

• evaluating the nuclear matrix elements gives:

d2H = (0.936±0.008)(dn+dp)+(0.183±0.002)g1−(0.646±0.023) ∆ fg1 e fm d3He = (0.90 ± 0.01) dn − (0.03 ± 0.01)dp − (0.017 ± 0.006) ∆

− (0.61 ± 0.14) ∆ fg1 − (0.11 ± 0.01)g0 − (0.14 ± 0.02)g1

− [(0.04 ± 0.02)C1 − (0.09 ± 0.02)C2] × fm−3 e fm

d3H = −(0.03 ± 0.01) dn + (0.92 ± 0.01)dp − (0.017 ± 0.006) ∆

− (0.61 ± 0.14) ∆ fg1 − (0.11 ± 0.01)g0 − (0.14 ± 0.02)g1 + [(0.04 ± 0.02)C1 − (0.09 ± 0.02)C2] × fm−3 e fm

⇒ various models of CP violation give different predictions for the various coupling constants

[Note: terms ∼ C3,4 only relevant for L-R symm. models not given]

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BSM OPERATORS at LOW ENERGIES

• translate (B)SM sources of P- and T-odd interactions into tailored EFTs

qEDM qCEDM gCEDM

...

N, π, γ, ...

q, G, γ, H

H H

q, γ, G

100GeV

10GeV

≪ 1GeV

4q

q

q

N

g

Fig. courtesy of A. Wirzba, J. de Vries

EFT

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GLOSSARY OF CPV HADRONIC VERTICES

33

• Leading dimension-6 terms plus dimension-4 θ-term [beyond CKM]:

LCP = − ¯θ gs2

64π2µνρσGaµνGaρσ

| {z }

θ-term

− i 2

X

q=u,d

dqqγ¯ 5σµνF µνq

| {z }

qEDM

− i 2

X

q=u,d

qqγ¯ 5 1

aσµνGaµνq

| {z }

qCEDM +dW

6 fabcµναβGaαβGbµρGc ρν

| {z }

gCEDM

+ X

i,j,k,l=u,d

Cijkl4qiΓqjkΓ0ql

| {z }

4qEDM

• Note that the 4q-terms from L-R symmetric models are treated separately

2 2 4q 2

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SCALING OF CPV HADRONIC VERTICES

• from the θ term to BSM sources

coupling g0 g1 d0, d1 (mN∆) C1,2(C3,4)

CP, isospin

CP, IC

CP, IV

CP, IC+IV

CP, IV

CP, IC (IV) θ-term O(1) O(Mπ/mN) O(Mπ2/m2N) O(Mπ2/m2N) O(Mπ2/m2N)

qEDM O(αEM/(4π)) O(αEM/(4π)) O(1) O(αEM/(4π)) O(αEM/(4π))

qCEDM O(1) O(1) O(Mπ2/m2N) O(Mπ2/m2N) O(Mπ2/m2N) gCEDM O(Mπ2/m2N)? O(Mπ2/m2N)? O(1) O(Mπ2/m2N) O(1)

4qLR O(Mπ2/m2N) O(1) O(Mπ3/m3N) O(Mπ/mN) O(Mπ2/m2N) 4q O(Mπ2/m2N)? O(Mπ2/m2N)? O(1) O(Mπ2/m2N) O(1)

?) Goldstone theorem → relative O(Mπ2/m2N) suppression of πN interactions

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SPECIFIC CALCULATIONS

• Nuclear contribution from the QCD θ-term:

dθ2H − 0.94 dθp + dθn

= θ · (0.89 ± 0.30) · 10¯ −16 e cm dθ3He − 0.90 dθn + 0.03 dθp = − ¯θ · (1.01 ± 0.42) · 10−16 e cm dθ3H − 0.92 dθp + 0.03 dθn = θ · (2.37 ± 0.42) · 10¯ −16 e cm.

• Nuclear contribution from the FQLR-term:

dLR2H − 0.94 dLRp + dLRn 

= −∆ · (2.1 ± 0.5) e fm dLR3He − 0.90 dLRn + 0.03 dLRp = −∆ · (1.7 ± 0.5 e fm dLR3H − 0.92 dLRp + 0.03 dLRn = −∆ · (1.7 ± 0.5 e fm

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TESTING STRATEGIES

• Deuteron EDM might distinguish between ¯θ and other scenarios allows extraction of the g1 coupling through dD − 0.94(dp + dn)

3He (or 3H) EDM necessary for a proper test of ¯θ and LR scenarios

• a2HDM scenario: both helion & triton EDMs would be needed

• Deuteron & helion work as complementary isospin filters of EDMs

• gCEDM, 4q chiral singlet: disentanglement difficult, may be lattice calcs?

• ultimate progress may come from combining LQCD and experiments

• of course, these various models also predict EDMs for leptons etc.

⇒ precision calcs in hadronic physics are an absolute must!

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Hadrons at finite theta

Acharya, Guo, Mai, UGM, Guo, Phys. Rev. D 92 (2015) 054023 Guo, UGM, Phys. Lett. B 749 (2015) 728

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WHY FINITE THETA?

• A tiny θ poses problems to anthropic reasoning

Banks et al. (2008), ..., Kaloper, Terning (2017)

• A small but not tiny θ significantly alters element generation

Ubaldi (2010)

• Lattice QCD requires an understanding of QCD at fixed topology

Brower, Chandasekhan, Negele, Wiese (2003), and many follow-ups

• Interesting QCD phase structure at θ ∼ π

di Vecchia, Veneziano, Witten (1980), Creutz (1995), Smilga (1999)

⇒ Study the pion, the sigma and the rho-meson in (unitarized) CHPT at NLO Step 1: Pion properties and scattering amplitude at finite θ

Step 2: Unitarize Tππ(θ) to get θ-dep. mass & width of σ(500), ρ(770)

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PION PROPERTIES at FINITE THETA

• In the presence of θ, vacuum alignment: U (x) = U0 U (x)˜

| {z }

GBs

• Ground state form minimizing the potential energy (2 flavors):

V2 = −Σ

2 Tr n

U0 eiθ/2 + U0 e−iθ/2

Mo

• Parametrization of U (x):

U0 = diag{e, e−iϕ}, U = e˜ i

2Φ/F, Φ = 1

2

π0

+

√2π −π0

!

⇒ tan ϕ = − tan θ2,  = md − mu

mu + md , ¯m = 12(mu + md)

• LO pion mass: Brower et al. (2003)

˚2 θ q

2 2 θ

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PION PROPERTIES at FINITE THETA

• NLO pion mass:

Mπ2+(θ) = ˚M2(θ)+

4(θ) F 2

1

32π2 ln

2(θ)

µ2 + 2lr3 + 2l7  (1 − 2) tan(θ/2) 1 + 2 tan2(θ/2)

2!

Mπ20(θ) = Mπ2+(θ) − 2l7

4(θ) F2

2

cos4(θ/2) 1 + 2 tan2(θ/2)2

• LO and NLO pion mass in the isospin limit mu = md: LO: Mπ2(θ) = 2B ¯m cos θ2

NLO: Mπ2(θ) = M2(θ) + M4(θ) F2

1

32π2 ln M2(θ)

µ2 + 2lr3 + 2l7 tan2 θ 2

| {z }

extra term

!

? extra term ∼ l7 vanishes at θ = 0 Gasser, Leutwyler (1985)

2

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PION-PION SCATTERING at FINITE THETA







(b) 

(a) (c) (d) (e)

• Scattering to one loop: T (s, t, u) = A(s, t, u)

| {z }

tree,O(p2)

+ B(s, t, u)

| {z }

loop,O(p4)

+ C(s, t, u)

| {z }

tree,O(p4)

A(s, t, u) = s − ˚Mθ2

F2 , B(s, t, u) = . . ., C(s, t, u) = . . .

• LECs are not θ-dependent, see e.g. F

• but use here θ-dependent l1,2 (defined at the pion mass) lri = γi

32π2

¯l + ln

θ2 µ2

!

, γ1 = 13, γ2 = 23

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GENERATING the SIGMA and the RHO

• Two lightest SU(2) non-GB mesons: σ(500) and ρ(770)

• Can be generated by resummation (unitarization) of CHPT ππ amplitudes

Truong (1988) and many others

• Use IAM (Inverse Amplitude Method), enforce Adler zeros (suppress indices) T (s) = (T2(s))2

T2(s) − T4(s) + A(s)

A(s) = T4(s2) − (s2 − sA)(s − s2)(T20(s) − T40(s)) s − sA

? T (sA) = 0 → T2(s2) = 0, T2(s2 + s4) + T4(s2 + s4) = 0

? li from Hanhart et al. (2008)

? σ and ρ poles: √

sσ = (443.1 − i 217.4) MeV

√sρ = (751.9 − i 75.4) MeV

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SIGMA and RHO PROPERTIES at FINITE THETA

• Use the θ-dependent Tππ in the IAM

⇒ θ-dependent σ and ρ properties:

Mass [GeV] Width [GeV]

Mρ Γσ

Mσ Γρ

• only visible changes for sizeable values of θ

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SUMMARY & OUTLOOK

• Baryon EDMs evaluated in U(3) × U(3) CHPT at NLO ,→ Chiral extrapolation formulae worked out

,→ Only two LECs at complete one-loop order ,→ Finite volume corrections available

• LQCD calculation of the neutron EDM for 2+1 flavors ,→ simulation at various pion masses & lattice volumes

,→ working with an imaginary θ [th’y assumed to be analytic at θ = 0]

,→ using CHPT, a precise value of dn at the physical point emerges:

dn = −0.0039(2)(9) [e fm θ]

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SUMMARY & OUTLOOK

• Theory of nuclear EDMs

,→ general formulas as functions of all NLO CP-violating operators ,→ explicit calculations for 2H, 3He, 3H

,→ testing strategies for various specific models in and beyond the SM ,→ UCN experiments on-going

,→ protons and charged light nuclei: storage ring measurements

,→ JEDI collaboration performs proof-of-principle exp. for the proton at COSY for details, see http://collaborations.fz-juelich.de/ikp/jedi/

• Hadrons at finite theta

,→ Pion mass and pion-pion scattering amplitude at NLO for fixed θ ,→ θ-dependent σ(500) and ρ(770) mass and width

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OUTLOOK: AXION PHYSICS

46

• Peccei-Quinn mechanism: a(x) → a(x) − θfa Peccei, Quinn (1977)

SCP/ = 1

2 (∂µa(x))2 + i 3

a(x)

fa mˆ X

x

¯

5u + ¯dγ5d + ¯sγ5s

• Axion mass (mean field): m2afa2 ' 2

t , χt = hQ2i

V ' (190 MeV)4

• Evaluate as before with: λ = a fa

m` ms

2ms + m` ≡ a finv

• Surprising first result:

0 0.02 0.04 0.06 0.08 0.1

-0.01 -0.1 -1 -10 -100 ma

0.01 0.1 1 10 100 0 0.02 0.04 0.06 0.08 0.1

Lattice χt/fa

0.01 0.1 1 10 100 0 0.02 0.04 0.06 0.08 0.1

Lattice χt/fa

→ stay tuned

PRELIMINARY

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SPARES

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Finite volume effects for baryon EDMs

Akan, Guo, UGM, Phys.Lett. B 736 (2014) 163

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CALCULATIONAL SCHEME

• Lattice QCD operates in a finite volume

⇒ must consider finite volume corrections

⇒ LO nEDM corrections known O’Connell, Savage (2006)

• work in the limit of infinite time and large volumes L3

• momenta quantized pn = 2π~n/L

→ mode sums: i

Z d4p

(2π)4 → i L3

X

~ n

Z dp0

• finite volume corrections: δL[Q] = Q(L) − Q(∞)

,→ these are entirely generated by loops

• here: perform NLO calculations for all baryon EDMs Guo, UGM (2012)

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RESULTS for the NEUTRON EDM

• At LO, recover the results of O’Connell and Savage

• Find sizeable NLO corrections → must be included

δL[dn]/dloopn δLNLO[dn]/δLLO[dn]

MΠ=138 MeV MΠ=200 MeV MΠ=300 MeV MΠ=400 MeV

3 4 5 6 7

10-4 0.001 0.01 0.1 1

L @fmD

L@dndnloop

3 4 5 6 7

0.0 0.5 1.0 1.5 2.0 2.5 3.0

L @fmD

L@dnL@dnLO D

• available also for the proton and the hyperons

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Figure

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References

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