Electric Dipole Moments and Sources of CP violation

σ σ

Jordy de Vries, UMass Amherst, RIKEN BNL Research Center Goals

§ Goal 1: A crash course in Electric Dipole Moments

• What are EDMs and why do people bother to find them? • Overview of EDM theory and experimental landscape (I’m a theorist…)

§ Goal 2: Put EDMs in a broader context (LHC/flavor/….)

• How do EDM measurements complement other searches ?

Excellent reviews: Pospelov/Ritz ‘05, Engel/van Kolck/Ramsey Musolf ’13 Outline

§ Part I: What are EDMs and why do (should) we care ?

§ Part II: Sources of CP violation: an EFT approach §

§ Part III: Classes of EDM searches + theoretical interpretation Magnetic dipole moments

• Let’s remind ourselves about magnetic dipole moments • Particle at rest with spin (i.e. electron) in magnetic field is described by ω ! ! eg ! ! µ = H = −µ (S ⋅ B) S = σ / 2 2m ! B ! Spin Magnetic Field σ

Magnetic dipole moment

• The B-field puts a torque on the system à spin precession ω = 2µB sinθ

• Dirac equation predicts tree-level magnetic moment Magnetic dipole moments Aµ

µ 1 eu¯(p0) u(p)A e (†)A † B µ ! 0 2m · e e 

µ ! ! eg • Dirac predicts: H = − (σ ⋅ B) µ = g =2 2 2m

• g is the gyromagnetic ratio (clasically g=1), triumph of QM ! -11 • Electron magnetic moment ~ e/me ~ 10 e cm ~ 100 e fm

• Measurements in 1940’s: ge = 2*(1.00118+-0.00003)….. • Enter Quantum Field Theory

↵ a = em + (↵2 ) e 2⇡ O em Magnetic Electric dipole moments

• Let’s remind ourselves about electric dipole moments • A particle with spin (i.e. neutron) in an electric field is described by ω ! ! ! ! H = −d (S ⋅ E) S = σ / 2 ! E! B ! Hamiltonian Spin Electric Field σ

Magnetic Electric dipole moment

• The E-field puts a torque on the system à spin precession

ω = 2µB sinθ ω = 2dE sinθ

• How large is the electric dipole moment ? Electric dipole moments

• We already exhausted the Dirac equation… No EDM there ? • Can we understand this ? ! ! ! ! H = −µ (S ⋅ B) − d (S ⋅ E)

• Perform a Time-reversal (T) transformation ! ! ! ! ! S → −S Spin like angular momentum ~ L ~ r × p ! ! ! ijk B → −B B ∝ε Fjk ~ ∂ j Ak → (+∂ j )(−Ak ) ! ! ! E → E E ∝ F0i ~ ∂0 Ai → (−∂0 )(−Ai ) ! ! ! ! H = −µ (S ⋅ B) −+ d (S ⋅ E) CP violation in the Standard Model • We can try to calculate EDMs from SM CKM phase

1 L = − Ψσ µν (µ +iγ 5d)ΨF dip 2 µν

e d e u,c,t γ γ γ W+- e u,c,t e d

e md αweak * αem µ ∝ (V V ) µ = d = 0 2 qd qd 2m e 2π mW 2π ∗ �~ �� �� = 0 SM CKM EDMs (too many acronyms!)

• At two loops: individual diagrams contribute but sum vanishes

• dq (2 loops) = 0, somewhat unexpected !

2 2 mc ↵sGF 21 • At three loops: d m 10 e fm d ' d 108⇡5 JCP ' 2 5 = c c c s s s sin 3 10 JCP 12 23 13 12 23 13 ' · • Electron EDM even at 4 loops !! � ≅ 10 � �� • Compare with magnetic moment: µ 100 e fm e ' • CKM EDMs are really very very small • Nucleon EDMs more complicated but also small Note: actual size of SM SM size of actual Note: 5 6 to orders Upperbound nEDM 10^ [e cm] With linear extrapolation: CKM neutron EDM in 2075…. EDM in CKM neutron extrapolation: linear With Standard Model suppression below nEDM upper bound bound upper has large uncertainty large has u freach! Out of “Here be dragons” “Here Baker et al ’ 06 ‘15 Strong CP violation ‘t Hooft ’76, ‘78

• Second source: QCD theta-term. (Later more detail). • Due to complicated vacuum structure of QCD

• Causes a ‘new’ CP-violating interaction with coupling constant θ

µnab ! ! q e GµnGab (in QED ~ E ⋅ B )

• Size of θ is unknown, one of the SM parameters

€ The strong CP problem If θ ~ 1

More details on calculation later Upperbound Upperbound nEDM 10^ [e cm]

Sets θ upper bound: θ < 10-10 Is there a reason for this suppression? Axions? Is θ=0 exact, or merely very small? Measurement of a nonzero EDM

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BSM sources of Standard Model: CP-violation θ-term SUSY, Left-Right, 2HDM,…

For the forseeable future: EDMs are ‘background-free’ searches for new physics Measurement of a nonzero EDM

?

BSM sources of Standard Model: CP-violation θ-term SUSY, Left-Right, 2HDM,…

Collider/flavor input Matter/Antimatter asymmetry (electroweak baryogenesis) Very active experimental field

System Group Limit C.L. Value Year 205Tl Berkeley 1.6 × 10−27 90% 6.9(7.4) × 10−28 2002 e YbF Imperial 10.5 × 10−28 90 −2.4(5.7)(1.5) × 10−28 2011 ThO ACME 1.1 × 10−29 90 4.3(3.1)(2.6) × 10−30 2018 HfF+ Boulder 1.3 × 10−28 90 0.9(7.7)(1.7) × 10−29 2017 n Sussex-RAL-ILL 3.0 × 10−26 90 0.2(1.5)(0.7) × 10−26 2006 129Xe UMich 4.8 × 10−27 95 0.26(2.3)(0.7) × 10−27 2019 199Hg UWash 7.4 × 10−30 95 -2.2(2.8)(1.5) × 10−30 2016 225Ra Argonne 1.4 × 10−23 95 4(6.0)(0.2) × 10−24 2016

muon E821 BNL g−2 1.8 × 10−19 95 0.0(0.2)(0.9) × 10−19 2009

+ new electron, muon, neutron, proton, Xe, Ra, Rn ….. experiments

• Why do experiments on so many different systems? • How do the different limits compare? • How to interpret the limits or a future signal? Measuring EDMs • General idea now. Later more.

σ B E B E σ

T/CP

= + d ~ E d ~ B Hd e · m ·

−28 • Note d =10 e cm E =100 kV / cm δω ~ 10−7 rad / s ~ 1 rad / year CP violation beyond the SM

• We saw that CKM EDMs are too small to be measured • We have the theta term, but that seems small for some reason… • Why bother with measuring further ?

• EDMs can be much larger in models of new physics, even at high scales • CKM EDMs are small because of the peculiar SM flavor structure • Need at least 3 loops to get an EDM (big suppression)

• NOT GENERAL at all. In many BSM models, EDMs at tree- or one- loop level. Much larger EDMs ! Example: the MSSM

Slide from Altmannshofer, 2014 The SUSY/LR CP problem

• CPV phase already at one-loop ! • Typical size of EDM

↵ n m d em e sin e ⇠ ⇡ ⇤2 ⇣ ⌘ If phase = O(1): Λ > 30 TeV (n=1) There are clever ways to avoid this !

uL d + L • In left-right symmetric models WL

+ L = i⌅(¯uRµdR)(¯uLµdL)+h.c. W d R u R R Ξ~ sin � /Λ € € € • Tree-level CP violation, EDMs probe ~50 TeV scale € € € The EDM metromap Outline

§ Part I: What are EDMs and why do (should) we care ?

§ Part II: Sources of CP violation: an EFT approach §

§ Part III: Classes of EDM searches + theoretical interpretation Heavy BSM physics and the SM EFT

• Assume BSM fields exists but are heavy à Integrate them out

Fermi’s theory:

• We don’t need ‘high-energy details’, the W boson, at low energies ! Energy SM fields Λ BSM fields

Effective operators E << Λ 1 exp ~ Λn Buchmuller & Wyler ‘86 Standard Model EFT Gradzkowski et al ’10 Many others • To stay model independent: add all EFT operators (infinite…)

1) Degrees of freedom: Full SM field content 2) Symmetries: Lorentz, SU(3)xSU(2)xU(1) 1 1 L = L + L + L +! new SM Λ 5 Λ2 6 • Low-energy effects suppressed by (E/Λ)^n

Advantages: General framework, no need to specify a BSM model Keep constraints from gauge invariance and EWSB Great tool to analyze experiments at different scales

Disadvantages: Many operators at relevant order Might not be valid for LHC physics if scale too low Not applicable for light BSM physics Typical Energy Scale Beyond-the-SM ? Λ physics ? TeV EFT operators

LEFF RG flow

M ~ v ~ M EW Z,W,H,t Integrate out heavy SM fields 100 GeV L' EFF RG flow

~ 2 F ~ M Λ χ π π N Nonperturbative QCD regimes 1 GeV

L'''EFF Lattice QCD + chiral effective field theory

100 MeV - eV EDMs Typical Energy Scale Beyond-the-SM ? Λ physics ? TeV EFT operators

LHC physics LEFF RG flow

M ~ v ~ M EW Z,W,H,t Integrate out heavy SM fields 100 GeV L' EFF RG flow B + D flavor physics

~ 2 F ~ M Λ χ π π N Nonperturbative QCD regimes 1 GeV

L'''EFF Lattice QCD + chiral effective field theory

100 MeV - eV EDMs + Kaons + beta + …. Examples of EFT operators: dipoles

EDMs and MDMs appear in the SMEFT Lagrangian at dimension-six

MCP � � �� � + ℎ. �. → � �� � +ℎ. �. ? TeV

€ B, W One-loop QCD mixing q, l q’, l’ H H

Electron (muon, tau) Quark chromo-EDM Quark EDM EDM

1 GeV e e Weinberg PRL ’89 Gluon chromo-EDM Braaten et al PRL ’90

d f abcε µναβ Ga Gb Gc λ w αβ µλ ν Weinberg operator

MCP ? TeV

€ QCD RGE mixing

1 GeV Weinberg PRL ’89 Gluon chromo-EDM Braaten et al PRL ’90

d f abcε µναβ Ga Gb Gc λ w αβ µλ ν Weinberg operator

µν MCP CG tLσ tR Gµνϕ + h. c. ? TeV Threshold contributions from heavy CEDMs

€ mQ Q QCD RGE mixing

1 GeV Modified Yukawa couplings

† C y t t ! ( ) h.c. L = C m t t (vh + h2 + )+ h.c. Λ Y t L Rϕ ϕ ϕ + Y t L R !

g, γ g t Barr, Zee ’90 t Barr-Zee diagrams g, γ g mt g QCD RGE mixing

γ γ 1 GeV e e Many more… But when the dust settles….. γ

Few GeV QCD + + + + (θ-term) Four-quark Quark EDM Quark C-EDM Gluon C-EDM operators (semi-)leptonic interactions JdV et al ‘12 ‘13 Ramsey-Musolf/Chupp ‘13 γ e e

` e e Electron EDM q q

• Any BSM model at high scales with CP-violation can be described at low energies by O(10) CPV operators (integrated out high-energy details !) • Different BSM model, different relevant operators Intermediate summary • Parametrized BSM CP violation in terms of dim6 operators • Evolved them to lower energies to ~ 1 GeV • O(10) operators left: theta, (C)EDMs, Weinberg, Four-fermion • Important: different BSM models à different EFT operators

1. Standard Model: only theta has a chance to be measured 2. 2-Higgs doublet model: quark+electron EDM, CEDMs, Weinberg (exact hierarchy depends on detail of models) 3. Split SUSY: only electron + quark EDMs (ratio fixed) 4. Left-right symmetric models: FQ operators, way smaller (C)EDMs 5. Leptoquark: FQ + semi-leptonic operators

• Now we must calculate EDMs. Not easy, since EDMs involve horrible objects (for a particle physicist) such as nucleons, atoms, and molecules Outline

§ Part I: What are EDMs and why do (should) we care ?

§ Part II: Sources of CP violation: an EFT approach §

§ Part III: Classes of EDM searches + theory interpretation The ‘easy case’: (semi-)leptonic CP violation

(semi-)leptonic interactions Few GeV γ e e

` e e Electron EDM q q

Hadronization (known matrix elements

e e N = nucleon ` < GeV e e (neutron, proton) Electron EDM N N Probing semi-leptonic operators System Group Limit C.L. Value Year 205Tl Berkeley 1.6 × 10−27 90% 6.9(7.4) × 10−28 2002 e YbF Imperial 10.5 × 10−28 90 −2.4(5.7)(1.5) × 10−28 2011 ThO ACME 1.1 × 10−29 90 4.3(3.1)(2.6) × 10−30 2018 HfF+ Boulder 1.3 × 10−28 90 0.9(7.7)(1.7) × 10−29 2017

• Why these complicated systems ? Cannot use free electrons…. • Look at neutral systems with unpaired electron (paramagnetic) • Why not simply use Hydrogen ? Probing semi-leptonic operators System Group Limit C.L. Value Year 205Tl Berkeley 1.6 × 10−27 90% 6.9(7.4) × 10−28 2002 e YbF Imperial 10.5 × 10−28 90 −2.4(5.7)(1.5) × 10−28 2011 ThO ACME 1.1 × 10−29 90 4.3(3.1)(2.6) × 10−30 2018 HfF+ Boulder 1.3 × 10−28 90 0.9(7.7)(1.7) × 10−29 2017

Schiffs theorem: the electron cloud will rearrange so that the nucleus does not ‘feel’ an external E-field. The EDMs of the nucleus and electron are screened (clasical argument but works in QM as well)! E Probing the electron EDM

System Group Limit C.L. Value Year 205Tl Berkeley 1.6 × 10−27 90% 6.9(7.4) × 10−28 2002 e YbF Imperial 10.5 × 10−28 90 −2.4(5.7)(1.5) × 10−28 2011 ThO ACME 1.1 × 10−29 90 4.3(3.1)(2.6) × 10−30 2018 HfF+ Boulder 1.3 × 10−28 90 0.9(7.7)(1.7) × 10−29 2017

Schiff Theorem: EDMs of charged constituents are screened in a neutral atom Schiff, ‘63 • Assumption : non-relativistic constituents • Invalid in heavy atoms/molecules 3 2 dA (de ) = KAde KA∝ Z αem Sandars ’65

−25 Bound on atomic Tl EDM : d 205Tl < 9 ⋅10 e cm Regan et al ’02 Strong enhancement! dA (de ) = KAde KTl= −(570±20) −27 de <1.6 ⋅10 e cm State-of-the-art: polar molecules Liu,Kelly ‘92, Dzuba, ‘09, Porsev et al ’12, Mayer ’09 Polar molecules: Convert small external to huge internal field

# & de ΔEYbF = (15± 2)⋅GeV % (+O(CS ) $ e cm ' Ε eff # & Εext de ΔEThO = (80 ±10)⋅GeV % (+O(CS ) $ e cm '

Current world record d < 1.1 10 e cm ACME ’18 Table-top experiment ! O(10) people. � � d ~( ) If EDMs at one loop ~ à roughly 30 TeV � Λ Fuyuto et al ‘18 4 Cs can be induced at tree-level and no me à roughly 10 TeV Constraining Higgs physics Brod et al ’13 ‘18 Cirigliano et al ‘16

† CY yt tLtRϕ! (ϕ ϕ )+ h.c.

g, γ t g, γ

• Electron EDM limit tells us imaginary part of Top Yukawa < 0.3 % • Or new physics only enters there Λ > 7 TeV • Note LHC constraints on real part ~< 10-20% (1 TeV roughly) Onwards to hadronic CPV γ Few GeV QCD + + + (θ-term)

Quark EDM Quark chromo- Weinberg Four-quark EDM operator operators

Hadronic/Nuclear CP-violation Theoretically more difficult

Goal: Electric dipole moments of nucleons, nuclei, and diamagnetic atoms Onwards to hadronic CPV γ Few GeV QCD + + + (θ-term)

Quark EDM Quark chromo- Weinberg Four-quark EDM operator operators

Intermediate step Lattice/Chiral perturbation theory

π 0,± γ N N

N N N N N N

Goal: Electric dipole moments of nucleons, nuclei, and diamagnetic atoms Classes of hadronic EDM experiments

§ Class 1: neutron EDM experiments (traditional, easiest to interpret, but running out of steam? )

§ Class 2: Diamagnetic atoms (powerful but hard to understand, nuclear physics uugh)

§ Class 3: Storage ring experiments (the future? ) (very cool idea but expensive and perhaps science fiction) Experiments

Running out of steam? Experiments suffer from lack of neutrons Ongoing experiments at LANL/Oak Ridge/PSI (goal 10-27,-28 e cm) Ideal to illustrate the ‘strong CP problem’ Strong CP violation ‘t Hooft ’76, ‘78

QCD theta term We finally understand CP violation…..

• Second SM source: QCD theta-term • Unlike CKM-phase: many open questions • Technical subject: here main idea. Introductory remarks • Start by considering the QED Lagrangian

µ 1 µν L = q(iγ D − m )q − F F Dµ = ∂µ −iQq Aµ µ q 4 µν • SM = all renormalizable terms that obey SU(3)xSU(2)xU(1) gauge invariance involving known degrees of freedom

• Fµν is a gauge-invariant quantity.... So we could have added a dim-4 term: e2 e2 θ εαβµν F F µν ≡θ F! F µν 32π 2 αβ 32π 2 µν

2 e αβµν µν P or T transformation θ ε Fαβ F ~ � � − � � 32π 2

So it describes a CP-odd interaction! Total derivatives ‘t Hooft ’76, ‘78 • But.... This term has no physical consequences ! Why ?

• αβµν µν αβµν Cause its a total derivative: ε Fαβ F = ∂µ (ε Aν Fαβ )

• QCD is more complicated, non-Abelian group, but still total derivative

αβµν µν αβµν ε GαβG = ∂µε (Aν Fαβ + Aα Aβ Aν ) Total derivatives ‘t Hooft ’76, ‘78 • But.... This term has no physical consequences ! Why ?

• αβµν µν αβµν Cause its a total derivative: ε Fαβ F = ∂µ (ε Aν Fαβ )

• QCD is more compliate but d, non-Abelian group, still total derivative

αβµν µν αβµν ε GαβG = ∂µε (Aν Fαβ + Aα Aβ Aν )

• `t Hooft: for non-Abelian gauge groups, there are instanton solutions that still contribute to the action

• Lecture on its own, but theta has physical effects ! Complex masses ‘t Hooft ’76, ‘78 • Let us look at 2-flavor QCD with masses + theta term g2 µ s αβµνG Gµν L = ∑q(iγ Dµ − mq )q +θ 2 ε αβ u,d 32π

• In principle, the quark mass can be complex 5 iα5γ • To make the masses real, we do a UA(1) transformation q → e q • The axial U(1) transformation is a classical symmetry, but broken by QM Complex masses ‘t Hooft ’76, ’78 Fujikawa ‘79 • Let us look at 2-flavor QCD with masses + theta term g2 µ s αβµνG Gµν L = ∑q(iγ Dµ − mq )q +θ 2 ε αβ u,d 32π

• In principle, the quark mass can be complex 5 iα5γ • To make the masses real, we do a UA(1) transformation q → e q • The axial U(1) transformation is a classical symmetry, but broken by QM • Action (or Lagrangian) invariant but path-integral measure is not (anomaly) Complex masses

g2 µ s αβµνG Gµν L = ∑q(iγ Dµ − mq )q +θ 2 ε αβ u,d 32π

• In principle, the quark mass can be complex

• To make the masses real, we do a UA(1) transformation 5 q → eiα5γ q

• But this induces a theta term from the anomaly ! • So we can ‘trade’ the theta term for a complex mass or vice versa 2 ⎛ ⎞ gs αβµν µν mumd 5 +θ 2 ε GαβG −⎜ ⎟θ qiγ q 32π ⎝ mu + md ⎠ • One ‘physical’ combination � = � + arg det � How do we measure the theta term ? • Electric dipole moments of hadrons and nuclei

• Problem: low-energy QCD is nonperturbative. • How to calculate the nucleon EDM from CPV at quark-gluon level ?

dn µν 5 ⎛ m m ⎞ L = − Ψ σ iγ Ψ F u d 5 * 5 dip n µν from −⎜ ⎟θ qiγ q = −(m )θ qiγ q 2 ⎝ mu + md ⎠

* • Let’s guess something: dn should be proportional to ~ (m θ) • There should be a coupling to a photon somewhere ~ e • To get dimensions right we need 1/mass^2, let’s say nucleon mass....

* m 10 MeV −3 dn ~ e 2 θ ~ e 2 θ ~ 10 θ e fm mN (1GeV) Limiting theta

* m 10 MeV −3 dn ~ e 2 θ ~ e 2 θ ~ 10 θ e fm mN (1GeV)

If θ ~ 1, just like the CKM phase Upperbound Upperbound nEDM 10^ [e cm]

Sets θ upper bound: θ < 10-10 Perhaps the estimate is stupid….

• We can do better: chiral perturbation theory L →L = L + L + L +! QCD chiPT ππ πN NN

• Quark masses = 0 à QCD has SU(2)LxSU(2)R symmetry § Spontaneously broken to SU(2)-isospin (pions are Goldstone) € § Explicit breaking (quark mass) à pion mass

• ChPT gives systematic expansion in Q ~ m Λ χ π Λ χ Λ χ ≅1GeV § Form of interactions fixed by symmetries § Each interactions comes with an unknown constant (LEC)

• Extended to include P and CP violation Review: JdV & Ulf-G. Meissner ‘15

Weinberg, Gasser, Leutwyler, and many many others m + m mu md Example m¯ = u d " = 2 mu + md

2 flavor QCD: = m¯ qq¯ "m¯ q¯⌧ 3q LQCD Lkin ChPT gives you the form of the effective low-energy terms � = � �

gA µ 5 Zero quark mass: = kin mN NN¯ + Dµ~⇡ N¯~⌧ N L L f⇡

Size mN=930 MeV, gA=1.27 we must measure or use lattice QCD !

2 m⇡ 2 3 Turn on quark mass: 0 = ⇡ m N¯⌧ N L L 2 N Small quark masses à small pion mass and nucleon mass splitting Theta and chiral perturbation theory

What does the theta-induced complex quark mass do ? Crewther et al’ 79 Baluni ‘79 3 5 = m¯ qq¯ "m¯ q¯⌧ q +m? ✓¯qi¯ q LQCD Lkin mumd m? = mu + md

2 m⇡ 2 3 0,± 0 = ⇡ mN N¯⌧ N +¯g N¯⌧ ⇡N π L L 2 0 ·

g0 CP-odd pion-nucleon interaction Theta and chiral perturbation theory

After axial U(1) and SU(2) rotations, complex CP-odd quark mass: Crewther et al’ 79 3 5 = m¯ qq¯ "m¯ q¯⌧ q +m? ✓¯qi¯ q Baluni ‘79 LQCD Lkin mumd m? = mu + md Linked via SUA(2) rotation 2 m⇡ 2 3 0,± 0 = ⇡ mN N¯⌧ N +¯g N¯⌧ ⇡N π L L 2 0 ·

g0 Nucleon mass splitting CP-odd pion-nucleon (strong part, no EM!) interaction

2 1−ε 3 Use lattice for mass splitting g = δm θ = (15.5± 2.5)⋅10− θ 0 N 2ε Walker-Loud ‘14, Borsanyi ’14, Aoki (FLAG) ’13 JdV et al‘15 The Neutron EDM γ π ±

Neutron EDM € g0 gA eg g m2 A 0 π 16 ¯ dn = − 2 Log 2 € dn 2.5 10 ✓ e cm 8π fπ mN ' · Crewther et al’ 79 • Very close to the naïve estimate Mereghetti et al’ 10 ‘15 • Recent developments. Use lattice€- QCD (difficult !) Shindler et al ‘19 Lattice � = − 1.5 ± 0.7 10 �̅ � ��

10 • So everything agrees it seems ✓¯ < 10

• Why is theta term so small ? ‘Strong-CP Problem’

• Note other small numbers in the SM: ye , Vcb • No anthropic argument for small theta (as far as I know) • Most popular solution: Axions (not in this lecture) And dim-6 sources ?

• Quark EDM accurately determined Bhattacharya et al ‘15 ‘16 d = (0.22 0.03)d +(0.74 0.07)d +(0.008 0.01)d n ± u ± d ± s • Other dim-6 operators no accurate numbers available. Model calculations or estimates only. See e.g. Pospelov/Ritz ‘05 • For example: Weinberg operator Weinberg ‘89 d ~ d ~ ±(50 ± 40)MeV ed Demir et al ’03 n p W JdV et al ‘10

• If Weinberg induced at 1 (2) loop: nEDM probes 20 (2) TeV

• Uncertainties dilute EDM constraining/discriminating power • A lot of recent lattice-QCD interest Classes of hadronic EDM experiments

§ Class 1: neutron EDM experiments (traditional, easiest to interpret, but running out of steam? )

§ Class 2: Diamagnetic atoms (powerful but hard to understand, nuclear physics uugh)

§ Class 3: Storage ring experiments (the future? ) (very cool idea but expensive and perhaps science fiction) Neutrons are hard …. • Would be easier to use something stable • Protons are charged and would fly out of the experimental setup • Diamagnetic atoms, closed electron shell and nonzero nuclear spin.

Schiff Theorem: EDMs of charged constituents are screened in a neutral atom Neutrons are hard …. • Would be easier to use something stable • Protons are charged and would fly out of the experimental setup • Diamagnetic atoms, closed electron shell and nonzero nuclear spin.

Schiff Theorem: EDMs of charged constituents are screened in a neutral atom

• Assumption : non-relativistic constituents (nucleus is non-rel…) • Assumption : point particles ! (nucleus has finite size !)

2 d " R % Typical suppression: Atom ∝10Z 2 $ N ' ≈ 10−3 dnucleus # RA & • Atomic part well under control Dzuba et al, ’02, ‘09 Sing et al, ‘15 −4 2 d199 Hg = (2.8 ± 0.6)⋅ 10 SHg e fm • So the atomic limit gets screened by a factor 1000 roughly

€ Huge collaboration…..

Slide from B. Heckel, KITP ‘16

Graner et al, ‘16

−30 Best EDM limit in the world ! d199 Hg < 8.7⋅10 e cm Theory for nuclear CP violation γ g 0 π ± π ±,0 € Nucleon EDM Nuclear EDM € • New contribution from CP-odd pion exchange: no loop suppression • Nuclear CP-violation typically dominated by CP-odd nuclear force

π 0,± π 0 Tree LO LO g0,1 g 0 g1 • Theory input 1: g in terms of theta, qCEDM, Weinberg, etc NLO 0,1 NLO • Theory input 2: Nuclear Schiff moment in terms of g0,1 • Lots of progress on both theoretical aspects in recent years.

N2LO N2LO Back to pion-nucleon couplings • 2 CP-odd structures π 0,± π 0

0 L = g0 Nπ ⋅τ N + g1 Nπ N g0 g1 Key idea • The theta-term and dim-6 operators have different chiral properties • Different models à Different g0/g1 ratios

Theta 2HDM mLRSM JdV et al ’12 ‘14 Theta term Quark FQLR Quark EDM CEDMs and Weinberg g Both 1 − 0.2 ≈ 1 couplings are +50 suppressed ! g0

• But how to experimentally probe these ratios ? Nuclear theory…. Uuugh…

• Very complicated many-body calculation • Use nuclear model and mean-field theory

3 S = g(a0g0 + a1g1) e fm g =13.5

a0 range (best) a1 range (best)

199Hg 0.03±0.025 (0.01) 0.030±0.060 (±0.02) 225Ra -3.5±2.5 (-1.5) 14±10 (6) 129Xe -0.03±0.025 (-0.008) -0.03±0.025 (-0.009)

Flambaum, de Jesus,Engel, Dobaczewski,,…. • Based on calculations from various groups • Hg & Xe: spread >100% (unclear why, difficult ‘soft’ nuclei) • Difficult to interpret the limits on BSM parameters. table from review: Engel et al, ‘13 Link to flavor Cirigliano et al ‘16

d

• There has been some tension between lattice- QCD predictions for CP- violation in Kaon decays and experiments. (probably just a fluke) .

See e.g. Buras ‘17 ‘18

Lectures by Ulrich Nierste tomorrow • This tension easily explained by a right-handed charged currents at ~ 100 TeV • But the extra CP violation also generates EDMs • Using central values of matrix element, dHg rules out this explanation • But matrix elements uncertainty makes this difficult JdV et al ‘18 Link to flavor

• Certain models that for the RK(*) and RD(*) also affected by EDM constraints

Probably covered extensively at this school

• Model based on a leptoquark solution to flavor puzzles Fafjer et al ‘18 • This particular model in principle ruled out by Hg EDM • Uncertainties are too big to be sure …. • Future nEDM would test the model. Classes of hadronic EDM experiments

§ Class 1: neutron EDM experiments (traditional, easiest to interpret, but running out of steam? )

§ Class 2: Diamagnetic atoms (powerful but hard to understand, nuclear physics uugh)

§ Class 3: Storage ring experiments (the future? ) (very cool idea but expensive and perhaps science fiction) EDMs of light nuclei

Anomalous magnetic moment Electric dipole moment ! ! ! dS ! q * ! # 1 & ! !- ! ! ! = S ×Ω Ω = aB + − a v × E + 2d E + v × B , % 2 ( / ( ) dt m+ $ v ' .

All-purpose ring (1H, 2H, 3He, … ) ~ 10−28,29 e cm 100-1000 x current neutron EDM sensitivity! (takes a while tough….)

Already used for muon EDM −19 dµ ≤1.8⋅ 10 e cm (95% C. L.) Bennett et al (BNL g-2) PRL ‘09

Major progress in: € JEDI collaboration, PRL ’15, ’16 ‘18 Why EDMs of light nuclei

• Several advantages of using light instead of heavy systems

• No Schiff screening ! No suppression associated to Hg/Ra/Xe • This means a measurement at 10-26 level would be world-leading • Theory is well under control. Few-body equations are ‘easy’

JdV et al ‘11, Bsaisou et al 12’14, Viviani et al ‘19

dD = 0.9(dn + dp )+ [(0.18± 0.02) g1 + (0.0028± 0.0003) g0 ] e fm

d3He = 0.9 dn − 0.05 dp + [(0.14 ± 0.04) g1 + (0.10 ± 0.03) g0 ] e fm +....

• Disadvantage: Expensive, requires a big storage ring

• Under development in Germany/Korea/CERN? but funding not guaranteed The chiral filter

• The theoretical accuracy can be used to isolate the source of CP violation

dD = 0.9(dn + dp )+ [(0.18± 0.02) g1 + (0.0028± 0.0003) g0 ] e fm

Theta term Quark Four-quark Quark EDM CEDMs operator and Weinberg

d − d − d D n p 0.5± 0.2 7± 4 14 ± 7 ≅ 0 dn

• EDM ratio hint towards underlying CP-odd operator! • Separate theta term from BSM physics, unravel different BSM models !

Dekens, JdV et al, ’14 ‘Unraveling models of CP violations with EDMs’ Complementary measurements § Class 1: Paramagnetic atoms and molecules • Very sensitive to electron EDM, but not to hadronic sources • No SM background, not even theta. • Atomic theory is under control

§ Class 2: Neutron EDM experiments • Theta + BSM CP violation involving quarks and gluons • Difficult to improve a lot

§ Class 3: Diagmagnetic atoms • Sensitive to nuclear CPV and thus complements nEDM • Very good measurements but Schiff screening and theory…

§ Class 4: Light nuclear EDMs • New idea, no Schiff screening and good theory • Expensive and requires new technology The EDM metromap Conclusion/Summary/Outlook

EDMs ü CP-violation but no CKM contribution ü Probe of strong CP violation: QCD theta term ü Very powerful search for BSM physics (probe the highest scales) ü Heroic experimental effort and great outlook

Classes of EDM experiments ü Basically 4 ‘classes: neutron, paramagnetic, diamagnetic, storage rings ü Probe complementary sources of CP-violation ü Measurements in several classes à unravel CPV source

EFT framework ü Framework exists for CP-violation (EDMs) from 1st principles ü Keep track of symmetries (gauge/CP/chiral) from multi-Tev to atomic scales ü Can be combined to LHC and flavor experiments ! Very complementary ! ü Still need much better theory control of hadronic and nuclear theory Backup Other electric dipole moments

• Take a classical dipole configuration ! r • Electric dipole ~ d ~ q r +q • Does not violate anything −q

• So we mean with an EDM: the coupling of spin and the E-field.

• For electron, neutron, atom, the only quantity available is the spin. So there is no ‘r’ around

• So where does the non-CPV EDM of molecules come from ? Double-well potential

• Analogy take a double-well potential V

V0 • If V0 is very small, get usual solutions

0 b a Double-well potential

• Analogy take a double-well potential b

V0 • If V0 is very small, get usual solutions

0 b a

• With nonzero V0 , two solutions appear with different parity and a small enery difference (tunneling effect !). E+ - E- ~ b

• A molecule like water has indeed a nearly-degenerate ground state with opposite parity Fake EDMs

• So we have 2 states which we call ± • Turn on Electric Field E (mixing of states)

+ 0 0 Eb H = E + 0 Eb 0 ✓ E ◆ ✓ ◆ • Diagonalize matrix to get energy eigenvalues

1 2 2 2 1,2 = ( + + ) ( + ) /4+E b E 2 E E ± E E p Fake EDMs

• So we have 2 states which we call ± • Turn on Electric field E (mixing of states)

+ 0 0 Eb H = E + 0 Eb 0 ✓ E ◆ ✓ ◆ • Diagonalize matrix to get energy eigenvalues

1 2 2 2 1,2 = ( + + ) ( + ) /4+E b E 2 E E ± E E • If the E field is smaller than the energyp gap 1 1 2E2b2 1,2 = ( + + ) ( + ) 1+ 2 E 2 E E ± 2 E E ( + ) ✓ E E ◆ • The energy shift is quadratic in the E field !! So no P or T violation • If the E field is larger than the gap: degenerate ground state 1 1,2 = ( + + ) Eb E 2 E E ± But there is more than de e e

e e `

N CS N de 2HDM Leptoquark

• Find a signal: what is responsible? eEDM or Cs ? • Need at least two measurements Dzuba et al ‘11 E d C M. Jung ’13 Δ = α e + β S Chupp, Ramsey-Musolf ‘15

YbF ThO HfF+ α/β 1.09 0.67 1.1

• Unfortunately: Probing roughly same combination Cancellations ?

Single source −29 de < 8.7⋅10 e cm −9 CS < 5.9 ⋅10

Allow for cancellations

−27 de < 5.4⋅10 e cm −7 CS < 4.5⋅10

• Also more difficult to pinpoint CS and de from 2 measurements! • Difficult to disentangle different models + • 2017 HfF measurement helps Fleig, Jung ‘18 Plots and numbers from Chupp, Ramsey-Musolf PRC ‘15 Including diamagnetic atoms Fig from Fleig, Jung ‘18

Single source −29 de < 8.7⋅10 e cm −9 CS < 5.9 ⋅10

Allow for cancellations

−28 de < 3.8⋅10 e cm −8 CS < 4.9 ⋅10

• This would allow to separate de from Cs • But… not realistic that Hg EDM solely from leptonic sources 4 • Without cancellations: The Cs limit roughly suggests 10 TeV scales. • But there can be small Yukawa couplings.