• The baryon EDMs from chiral perturbation theory

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

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

Introduction

QCD with a θ-TERM

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

n

e^{i n θ}|ni, n ∈ Z

• QCD in the presence of the θ-term

L_QCD = −1

4G^a_µνG^a,µν + X

flavors

¯

q (iD/ − M) q + θ₀ g²

32π² G^a_µνG˜^a,µν

| {z }

∼ ~E^a · ~B^a

⇒ This leads to strong CP-violation

⇒ A non-vanishing vacuum angle θ₀ entails d_n 6= 0 (also d_p 6= 0)

d_N ≈ |θ₀| e M_π²

m³ ≈ 10⁻¹⁶|θ₀| e cm → θ₀ = O(10⁻¹⁰)

CONNECTION to the QUARK MASSES

• Shift under axial rotations:

θ → ¯θ = θ − 2N_fα

• Influences the diagonalization of the quark mass matrix:

θ = θ + arg (det M)¯

⇒ QCD mass term for small values of θ:

L_mass = m_uuu + m¯ _ddd + m¯ _sss¯ + i ¯θ m_um_dm_s

m_um_d + m_um_s + m_dm_s uγ¯ ₅u + ¯dγ₅d + ¯sγ₅s

| {z }

ps density

? no physical effect if one mq vanishes

? however, this is excluded by QCD phenomenology

EXPERIMENTS on the NEUTRON EDM

• performed and projected experiments (UCN):

1950 1970 1990 2010

PSI MultiCell

3He−UCN

Beams UCN

d [e cm]

10⁻³² 10⁻²⁸ 10⁻²⁴ 10⁻²⁰

SM SUSY

year of experiment

QCD θ

10

10

10

−10

−12

−14

The baryon EDM in chiral perturbation theory

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

BARYON EDMs - BASIC DEFINITIONS

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

hp⁰| J_em^ν |pi = ¯u (p⁰) Γ^ν q²
u (p) Γ^ν = γ^νF₁ q²
− i

2m_B σ^µνq_µF₂ q²
− 1

2m_B σ^µνq_µγ₅F₃ q²

+ . . . q² = (p⁰ − p)²

• Dipole moments and radii:

d_B = F_3,B(0)

2m_B , r_ed²  = 6dF₃(q²) dq²

q²=0

• Similar formulae for light nuclei [see later]

000000 000 111111 111

p

p

q

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 L_eff[φ_a, 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

EFFECTIVE LAGRANGIAN

• Effective Lagrangian based on U (3)_L × U (3)_R symmetry:

Herrera-Sikl ´ody et al., Borasoy

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

i F₀η₀

| {z }

U (1)

+ 2i F_φφ

| {z }

SU (3)



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

p = O (δ) , m_q = O δ²

, 1/N_c = O δ²

• Determination of the low-energy constants

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

EFFECTIVE LAGRANGIAN

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

L_φB = i TrBγ¯ ^µ[D_µ, B] − ˚m Tr[ ¯BB] − D/F

2 TrBγ¯ ^µγ₅[u_µ, B]_± +b_D/F TrB¯χ₊ − iA(U − U^†), B

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

√6

F₀ η₀Tr[ ¯BB] + i

w_13/14⁰ θ¯₀ + w_13/14

√6

F₀ η₀

TrBσ¯ ^µνγ₅[F_µν⁺ , B]_± +w_16/17 TrBσ¯ ^µν[F_µν⁺ , B]± + w₀

2 TrBγ¯ ^µγ₅BTr[u_µ]

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

• loop LECs: w₁₀⁰ , w₀

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

¯

}

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⁾/(F₀F_πM_η0)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}, {π⁰(η₈, η₀), p}, {K⁺, Σ⁰(Λ)} and {K⁰, Σ⁺}

FEYNMAN GRAPHS for F

(q

)

• 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

NEUTRON LOOP CONTRIBUTIONS

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

F_3,n^loop(q²)

2m_N = V₀⁽²⁾e ¯θ₀ π²F_π⁴

(

(D + F ) (b_D + b_F )

×

"

1 − ln ^M_µ2^π2 + σ_π ln ^σ_σ^π−1

π+1 + ^π(^2Mπ²−q²)

2m_N

√

−q² arctan

√−q² 2M_π

#

−(D − F ) (b_D − b_F )

"

1 − ln ^M_µ2^K2 + σ_K ln ^σ_σ^K−1

K+1

+√^π

−q²

_2M2

K−q²

2m_N − 8 (b_D − b_F ) M_K² − M_π²


× arctan

√−q² 2M_K

#)

h

σ_π(K) = q

1 − 4M_π(K)² /q²i

• only known masses and couplings !

PROTON LOOP CONTRIBUTIONS

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

F_3,p^loop(q²)

2m_N = −V₀⁽²⁾e ¯θ₀ 6π²F_π⁴

(

6(D + F ) (b_D + b_F )

×

"

1 − ln ^M

2 π

µ² + σ_π ln ^σ_σ^π−1

π+1 + ^3πM_2m ^π

N + ^π(^2Mπ²−q²)

2m_N√

−q² arctan

√

−q² 2M_π

#

+4 (Db_D + 3F b_F )



1 − ln ^M

2 K

µ² + σ_K ln ^σ_σ^K−1

K+1 + ^πM_m ^K

N



+√^4π

−q² arctan

√−q² 2M_K



(Db_D+3F b_F)

2m_N 2M_K² − q²
+8 M_K² − M_π²

F b²_D + 3b²_F
− ²₃Db_D (b_D − 3b_F )



+_m^π

N



6(D − F ) (b_D − b_F ) M_K + (D − 3F ) (b_D − 3b_F ) M_η8 +^2F

2 π

F₀² (2D − 3w₀) 2b_D + 3b₀ + 6w₁₀⁰

| {z }

≡ β

M_η0

)

MAKING PREDICTIONS

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

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

w_a(µ) ≡ αw₁₃ + w₁₃^{0 r}(µ)

w_b(µ) ≡ 3[αw₁₄ + w₁₄^{0 r}(µ)] + V₀⁽²⁾β

4πF₀²F_π²m_aveM_η0

• use this formalism to analyze lattice data → next section

• there are relations free of LECs such as d_Σ⁰ + d_Λ ∼ M_K² − M_π²

F b²_D + 2Db_Db_F + 3F b²_F
+ O(δ⁴) d_n − d_Ξ⁰ ∼ (Db_D + F b_F) 

2 ln ^M

2 K

M_π² + π^Mπ_m^−MK

ave

 +_M^8π

K M_K² − M_π²

Db²_D + 2F b_Db_F + Db²_F
+ O(δ⁴)

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]

LATTICE FORMULATION: GENERALITIES

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

¯

m = (m_u + m_d + m_s)/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 N_f = 2 + 1 [m_u = m_d = 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]

LATTICE FORMULATION at FIXED TOPOLOGY

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

S = S₀ + S_θ, S_θ = i θ Q, Q = − 1

64π² _µνρσ a⁴ X

x

G^b_µνG^b_ρσ ∈ Z

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

S_θ = − i

3θ ˆma⁴ X

x

¯

uγ₅u + ¯dγ₅d + ¯sγ₅s
ˆ

m⁻¹ = 1 3



m⁻¹_u + m⁻¹_d + m⁻¹_s 

= 1 3



2m⁻¹_` + m⁻¹_s 

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

⇒ S^θ = ¯θ m_` m_s

2m_s + m_` a⁴ X

x

¯

uγ₅u + ¯dγ₅d + ¯sγ₅s

⇒ real action, vanishes at m and m

LATTICE SET-UP

• Wilson fermions with a clover term & Symanzik improved gluon action (SLiNC) S^q = S₀^q + S_θ^q = a⁴ X

x

¯ q



D − 1

4c_SW σ_µν G_µν + m_q + λ 2a γ₅

 q

• lattice set-ups (β = 5.50)

# κ_` κ_s V M_π [MeV] M_K [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

LATTICE SET-UP: TOPOLOGICAL CHARGE

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

λ = ¯θ 2a m_` m_s

2m_s + m_`, am_q = 1

2κ_q − 1

2κ_0,c, κ_0,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¯

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[G^θ_{N N}(t; 0)Γ₄γ₅]

Tr[G^θ_{N 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 γ_µ

+

• 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

i 23

FORM FACTOR RATIOS

• Extracting F₃^θ,n¯ (q²)/F₁^θ,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

M_π = 465 MeV

M_π = 360 MeV M_π = 310 MeV

new

RESULTS

• Form factor as a fct of ¯θ

• continue θ and F₃^θ(0) to real values

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

→ d_n = eF₃⁽¹⁾(0)θ/2m_N

• chiral extrapolation with 2M_K² + M_π² =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

→ w_a(µ = 1 GeV) = 0.04(1) GeV⁻¹

→ d_n = −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]

m²_π[GeV²] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m²_π[GeV²] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m²_π[GeV²] -0.05

-0.04 -0.03 -0.02 -0.01 0

0 0.1 0.2 0.3

dn[efmθ]

m²_π[GeV²] new

OTHER RESULTS

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

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

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

• Neutron EDM and tensor charges from LQCD

d_n < 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

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)

MOTIVATION

• 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

L^πN

^CP = − d_nN^†(1 − τ³)S^µv^νN F_µν − d_pN^†(1 + τ³)S^µv^νN F_µν + (m_N∆) π₃π² + g₀N^†~π · ~τ N + g₁N^†π₃N

+ C₁N^†N D_µ(N^†S^µN ) + C₂N^†~τ N · D_µ(N^†~τ S^µN )

• various contributions at next-to-leading order:

∗ single nucleon EDMs – d_n, d_p

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

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

CP-VIOLATING NUCLEAR OPERATORS

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

iqF₃^A(q²)

2m_A = D

A; M_J = J   

J˜⁰(q)  

 A; M_J = JE

, d_A = lim

q²→0

F₃^A(q²) 2m_A

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

J˜^µ = J^µ

^CP+ V

^CPG J^µ + J^µ G V

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

= + + +

J˜^µ

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

• single nucleon current (a):

J_i^µ = ₂^e 

1 + τ_(i)³ 

v^µ , J^µ

CP,i = ¹₂ h

d_n 

1 − τ_(i)³ 

+ d_p 

1 + τ_(i)³ i

i~q · ~σ_(i) v^µ

CP-VIOLATING NUCLEAR OPERATORS continued

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

V ^{N N}

^CP,ij (~k_i) = i g_A 2F_π

~k_i

~k_i² + M_π² g₀ ~σ_(ij)⁻ ~τ_(i) · ~τ_(j) + i g_A

4F_π

~k_i

~k_i² + M_π² h

g₁ + ∆ f_g1(|~k_i|)i 

~

σ_(ij)⁺ τ_(ij)⁻ + ~σ_(ij)⁻ τ_(ij)⁺  + i

2

β²M_π²~k_i

~k_i² + β²M_π² h

C₁~σ_(ij)⁻ + C₂ ~σ_(ij)⁻ ~τ_(i) · ~τ_(j) i

f_g1(k) = −15 32

g_A² M_πm_N πF_π²

"

1 + 1 + 2~k²/(4M_π²)

3|~k |/(2M_π) arctan |~k | 2M_π

!

− 1 3

!#

• ∆ f_g1 induced from 3-pion vertex

• β → ∞ in calcs, used as diagnostic tool









CP-VIOLATING NUCLEAR OPERATORS continued

• irreducible 3N potential (d):

V ^3N

^CP (~k₁, ~k₂, ~k₃) = −i∆m_Ng_A³

4F_π³ δ^abδ^c3 + δ^acδ^b3 + δ^bcδ^a3
τ₍₁₎^a τ₍₂₎^b τ₍₃₎^c

× (~σ₍₁₎ · ~k₁)(~σ₍₂₎ · ~k₂)(~σ₍₃₎ · ~k₃) h~k₁² + M_π²i h~k₂² + M_π²i h~k₃² + M_π²i

• evaluate 2N and 3N bound states using Faddeev equations hψ_A| ˜J^µ|ψ_Ai = hψ_A|J^µ

^CP + V

^CPG J^µ + J^µ G V

^CP + · · · |ψ_Ai

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

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

RESULTS for LIGHT NUCLEI

• evaluating the nuclear matrix elements gives:

d²_H = (0.936±0.008)(d_n+d_p)+(0.183±0.002)g₁−(0.646±0.023) ∆ f_g1 e fm d³_He = (0.90 ± 0.01) d_n − (0.03 ± 0.01)d_p − (0.017 ± 0.006) ∆

− (0.61 ± 0.14) ∆ f_g1 − (0.11 ± 0.01)g₀ − (0.14 ± 0.02)g₁

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

d³_H = −(0.03 ± 0.01) d_n + (0.92 ± 0.01)d_p − (0.017 ± 0.006) ∆

− (0.61 ± 0.14) ∆ f_g1 − (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 ∼ C_3,4 only relevant for L-R symm. models not given]

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

GLOSSARY OF CPV HADRONIC VERTICES

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

L^CP = − ¯θ g_s²

64π²^µνρσG^a_µνG^a_ρσ

| {z }

θ-term

− i 2

X

q=u,d

d_qqγ¯ ₅σ_µνF ^µνq

| {z }

qEDM

− i 2

X

q=u,d

d˜_qqγ¯ ₅ 1

2λ^aσ^µνG^a_µνq

| {z }

qCEDM +d_W

6 f_abc^µναβG^a_αβG^b_µρG^{c ρ}_ν

| {z }

gCEDM

+ X

i,j,k,l=u,d

C_ijkl^4q q¯_iΓq_jq¯_kΓ⁰q_l

| {z }

4qEDM

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

2 2 4q 2

SCALING OF CPV HADRONIC VERTICES

• from the θ term to BSM sources

coupling g₀ g₁ d₀, d₁ (m_N∆) C_1,2(C_3,4)

CP, isospin

CP, IC

CP, IV

CP, IC+IV

CP, IV

CP, IC (IV) θ-term O(1) O(M_π/m_N) O(M_π²/m²_N) O(M_π²/m²_N) O(M_π²/m²_N)

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

qCEDM O(1) O(1) O(M_π²/m²_N) O(M_π²/m²_N) O(M_π²/m²_N) gCEDM O(M_π²/m²_N)^? O(M_π²/m²_N)^? O(1) O(M_π²/m²_N) O(1)

4qLR O(M_π²/m²_N) O(1) O(M_π³/m³_N) O(M_π/m_N) O(M_π²/m²_N) 4q O(M_π²/m²_N)^? O(M_π²/m²_N)^? O(1) O(M_π²/m²_N) O(1)

?) Goldstone theorem → relative O(M_π²/m²_N) suppression of πN interactions

SPECIFIC CALCULATIONS

• Nuclear contribution from the QCD θ-term:

d^θ2H − 0.94 d^θ_p + d^θ_n

= θ · (0.89 ± 0.30) · 10¯ ⁻¹⁶ e cm d^θ3He − 0.90 d^θ_n + 0.03 d^θ_p = − ¯θ · (1.01 ± 0.42) · 10⁻¹⁶ e cm d^θ3H − 0.92 d^θ_p + 0.03 d^θ_n = θ · (2.37 ± 0.42) · 10¯ ⁻¹⁶ e cm.

• Nuclear contribution from the FQLR-term:

d^LR2H − 0.94 d^LR_p + d^LR_n

= −∆ · (2.1 ± 0.5) e fm d^LR3He − 0.90 d^LR_n + 0.03 d^LR_p = −∆ · (1.7 ± 0.5
e fm d^LR3H − 0.92 d^LR_p + 0.03 d^LR_n = −∆ · (1.7 ± 0.5
e fm

TESTING STRATEGIES

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

• 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!

Hadrons at finite theta

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

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 T_ππ(θ) to get θ-dep. mass & width of σ(500), ρ(770)

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^† e^iθ/2 + U₀ e^−iθ/2

Mo

• Parametrization of U (x):

U₀ = diag{e^iϕ, e^−iϕ}, U = e˜ ⁱ

√2Φ/F, Φ = ^√1

2

π⁰ √

2π⁺

√2π⁻ −π⁰

!

⇒ tan ϕ = − tan ^θ₂,  = m_d − m_u

m_u + m_d , ¯m = ¹₂(m_u + m_d)

• LO pion mass: Brower et al. (2003)

˚^{2 θ} q

2 2 θ

PION PROPERTIES at FINITE THETA

• NLO pion mass:

M_π²+(θ) = ˚M²(θ)+

M˚⁴(θ) F ²

1

32π² ln

M˚²(θ)

µ² + 2l^r₃ + 2l₇  (1 − ²) tan(θ/2) 1 + ² tan²(θ/2)

²!

M_π²0(θ) = M_π²₊(θ) − 2l₇

M˚⁴(θ) F²

²

cos⁴(θ/2) 1 + ² tan²(θ/2)
²

• LO and NLO pion mass in the isospin limit m_u = m_d: LO: M_π²(θ) = 2B ¯m cos ^θ₂

NLO: M_π²(θ) = M²(θ) + M⁴(θ) F²

1

32π² ln M²(θ)

µ² + 2l^r₃ + 2l₇ tan² θ 2

| {z }

extra term

!

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

2

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 − ˚M_θ²

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

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

• but use here θ-dependent l_1,2 (defined at the pion mass) l^r_i = γ_i

32π²

¯l_iθ + ln

M˚_θ² µ²

!

, γ₁ = ¹₃, γ₂ = ²₃

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 − s_A)(s − s₂)(T₂⁰(s) − T₄⁰(s)) s − s_A

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

? l_i from Hanhart et al. (2008)

? σ and ρ poles: √

s_σ = (443.1 − i 217.4) MeV

√s_ρ = (751.9 − i 75.4) MeV

SIGMA and RHO PROPERTIES at FINITE THETA

• Use the θ-dependent T_ππ in the IAM

⇒ θ-dependent σ and ρ properties:

Mass [GeV] Width [GeV]

M_ρ Γ_σ

M_σ Γ_ρ

• only visible changes for sizeable values of θ

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 d_n at the physical point emerges:

d_n = −0.0039(2)(9) [e fm θ]

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

OUTLOOK: AXION PHYSICS

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

S_CP/ = 1

2 (∂_µa(x))² + i 3

a(x)

f_a mˆ X

x

¯

uγ₅u + ¯dγ₅d + ¯sγ₅s

• Axion mass (mean field): m²_af_a² ' 2

9χ_t , χt = hQ²i

V ' (190 MeV)⁴

• Evaluate as before with: λ = a f_a

m_` m_s

2m_s + m_` ≡ a f_inv

• 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

SPARES

Finite volume effects for baryon EDMs

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

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 p_n = 2π~n/L

→ mode sums: i

Z d⁴p

(2π)⁴ → i L³

X

~ n

Z dp⁰ 2π

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

,→ these are entirely generated by loops

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

RESULTS for the NEUTRON EDM

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

• Find sizeable NLO corrections → must be included

δ_L[d_n]/d^loop_n δ_L^NLO[d_n]/δ_L^LO[d_n]

M^Π=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@dnDdnloop

3 4 5 6 7

0.0 0.5 1.0 1.5 2.0 2.5 3.0

L @fmD

∆L@dnD∆L@dnLO D

• available also for the proton and the hyperons