Lattice QCD Spectroscopy for Hadronic CP Violation Andr Walker-Loud - - PowerPoint PPT Presentation

Lattice QCD Spectroscopy for Hadronic CP Violation

André Walker-Loud

Fundamental Symmetries and Low-Energy Nuclear Physics

๏ The Universe is matter dominated at roughly 1 ppb: η ≡ Xp+n Xγ = 6.19(15) × 10−10 ๏ Sources of CP-violation beyond the Standard Model (SM) are needed to generate this observed asymmetry ๏ Assuming nature is CPT symmetric, this implies T-violation which implies fermions will have permanent electric dipole moments (EDMs) ๏ This has motivated significant experimental efforts to search (or plan to search) for permanent EDMs in a variety of systems e, n, p, deuteron, triton, 3He, ..., 199Hg, 225Ra, 229Pa,...

APS/Alan Stonebraker

Fundamental Symmetries and Low-Energy Nuclear Physics

๏ The Universe is matter dominated at roughly 1 ppb: η ≡ Xp+n Xγ = 6.19(15) × 10−10

๏ There are now a number of groups working on computing EDMs from the QCD-theta term. If we can determine the couplings - they can be used in the chiral extrapolations, removing a free parameter from their analysis

APS/Alan Stonebraker

Mereghetti : Mon 10:00 Bhattacharya : Mon 14:40 Dragos : Tues 14:00 Kim : Tues 14:20 Liang : Tues 6:45 Walker-Loud : NOW Yoon : Thur 8:30 Syritsyn : Thur 8:50

Fundamental Symmetries and Low-Energy Nuclear Physics

๏ In a large nucleus, the long-range pion exchange will dominate the nuclear EDM

LCP V = − ¯ g0 2Fπ ¯ N~ ⇡ · ~ ⌧N − ¯ g1 2Fπ ¯ N⇡3N − ¯ g2 2Fπ ⇡3 ¯ N ✓ ⌧3 − ⇡3 Fπ ~ ⇡ · ~ ⌧ ◆ N

{¯ g1, ¯ g2} ∼ ¯ g0 m2

π

Λ2

χ

๏ For the QCD theta term ๏ For more generic CP Violating operators ¯ g2 ∼ {¯ g0, ¯ g1}m2

π

Λ2

χ

¯ g1 ∼ ¯ g0

๏ The nuclear EDM is proportional to the Schiff moment

Fundamental Symmetries and Low-Energy Nuclear Physics

S = X

i6=0

hΦ0|Sz|ΦiihΦi|HCP V |Φ0i E0 Ei + c.c. ๏ The Schiff parameters are computed with nuclear models under the assumption the CPV operator does not significantly distort the nuclear wave-function {a0, a1, a2} ๏ For a QCD theta term only and thus a constraint

• n can be made through the relation

¯ g1 ∼ ¯ g2 ∼ 0 ¯ θ ¯ g0 = δM md−mu

n−p

md − mu 2mdmu md + mu ¯ θ = α 2mdmu md + mu ¯ θ S = 2MNgA Fπ (a0¯ g0 + a1¯ g1 + a2¯ g2)

๏ The nuclear EDM is proportional to the Schiff moment

Fundamental Symmetries and Low-Energy Nuclear Physics

S = X

i6=0

hΦ0|Sz|ΦiihΦi|HCP V |Φ0i E0 Ei + c.c. ๏

225Ra is interesting nucleus as it is pear-

shaped (octupole deformed) ๏ “stiff ” core making nuclear model calculations more reliable ๏ nearly degenerate parity partner state ๏ 102 - 103 enhancement of a0, a1, a2 E−

1/2 − E+ 1/2 = 55 KeV

S = 2MNgA Fπ (a0¯ g0 + a1¯ g1 + a2¯ g2)

Gaffney et al. Nature 497 (2013)

๏ Sources of CP-Violation in quark sector:

Fundamental Symmetries and Low-Energy Nuclear Physics

Operator [Operator]

• No. Operators

4 1 ¯ θ quark EDM 6 2 quark Chromo-EDM 6 2 Weinberg (GGG) 6 1 4-quark 6 2 4-quark induced 6 1

๏ Sources of CP-Violation in quark sector:

Fundamental Symmetries and Low-Energy Nuclear Physics

Operator [Operator]

• No. Operators

4 1 ¯ θ quark EDM 6 2 quark Chromo-EDM 6 2 Weinberg (GGG) 6 1 4-quark 6 2 4-quark induced 6 1

๏ Sources of CP-Violation in quark sector: Operator [Operator]

• No. Operators

4 1 ¯ θ quark Chromo-EDM 6 2 LCP V = − g2

s ¯

θ 32π2 ˜ GµνGµν − i 2 ¯ qσµνγ5 ⇣ ˜ d0 + ˜ d3τ3 ⌘ Gµνq

LCP V = − ¯ g0 2Fπ ¯ N~ ⇡ · ~ ⌧N − ¯ g1 2Fπ ¯ N⇡3N − ¯ g2 2Fπ ⇡3 ¯ N ✓ ⌧3 − ⇡3 Fπ ~ ⇡ · ~ ⌧ ◆ N

Fundamental Symmetries and Low-Energy Nuclear Physics

QCD Isospin Violation and CP-violating 𝜌-N

¯ g0 = δM md−mu

n−p

md − mu 2mdmu md + mu ¯ θ = α 2mdmu md + mu ¯ θ ๏ A precise determination of the strong isospin breaking contribution to Mn-Mp teaches us about CP-violation


Crewther, Vecchia, Veneziano, Witten, Phys.Lett. 91B (1980)

NNLO χPT

(gA = 1.27, fπ = 130 MeV)

δM md−mu

n−p

= δ ⇢ α  1 − m2

π

(4πfπ)2 (6g2

A + 1) ln

✓m2

π

µ2 ◆ + β(µ) 2m2

π

(4πfπ)2

• 0.0

0.1 0.2 0.3 0.4 0.5

mπ/Λχ

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

δM δ

N [MeV]

mπ = mphys

π

mπ ' 241 [MeV] mπ ' 422 [MeV] mπ ' 489 [MeV] δM δ

N : other LQCD

1.5 2.0 2.5 3.0 3.5 4.0 4.5

δM δ

n−p [MeV]

2.26(71) [hep-lat/0605014] 2.51(52) [1006.1311] 3.13(57) [1206.3156] 2.90(63) [1303.4896] 2.28(26) [1306.2287] 2.52(29) [1406.4088] 2.32(17) [1612.07733] 2.39(12) weighted average

QCD Isospin Violation and CP-violating 𝜌-N

Heffernan, Banerjee, Walker-Loud [1706.04991] Brantley, Joo, Mastropas, Mereghetti, Monge- Camacho, Tiburzi, Walker-Loud [1612.07733] Walker-Loud [0904.2404] ¯ g0 √ 2fπ = (14.7 ± 1.8 ± 1.4) · 10−3¯ θ

Using

de Vries, Mereghetti, Walker-Loud


• Phys. Rev. C92 (2015) [1506.06247]

๏ QCD Theta term LCP V = − g2

s ¯

θ 32π2 ˜ GµνGµν Lχ

CP V = − ¯

g0 2Fπ ¯ N~ ⇡ · ~ ⌧N ¯ g0 = δM md−mu

n−p

md − mu 2mdmu md + mu ¯ θ Symmetries δM md−mu

n−p

= α(md − mu) Simple spectroscopic calculation allows us to determine this long-range CP-Violating pion-nucleon coupling

de Vries, Mereghetti, Walker-Loud


• Phys. Rev. C92 (2015) [1506.06247]

This relation holds to NNLO in the chiral expansion up to small corrections

Computational Strategy

๏ Quark Chromo-EDM Operators L6

¯ qq = − i

2 ¯ qσµνγ5( ˜ d0 + ˜ d3τ3)Gµνq − 1 2 ¯ qσµν(˜ c3τ3 + ˜ c0)Gµνq

Computational Strategy

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

๏ Quark Chromo-EDM Operators L6

¯ qq = − i

2 ¯ qσµνγ5( ˜ d0 + ˜ d3τ3)Gµνq − 1 2 ¯ qσµν(˜ c3τ3 + ˜ c0)Gµνq Symmetries ¯ g0 = δqMN ˜ d0 ˜ c3 + δMN ∆qm2

π

m2

π

˜ d3 ˜ c0 ¯ g3 = −2σπN ✓∆qMN σπN − ∆qm2

π

m2

π

◆ ˜ d3 ˜ c0

Computational Strategy

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

๏ Quark Chromo-EDM Operators L6

¯ qq = − i

2 ¯ qσµνγ5( ˜ d0 + ˜ d3τ3)Gµνq − 1 2 ¯ qσµν(˜ c3τ3 + ˜ c0)Gµνq Symmetries ¯ g0 = δqMN ˜ d0 ˜ c3 + δMN ∆qm2

π

m2

π

˜ d3 ˜ c0 ¯ g3 = −2σπN ✓∆qMN σπN − ∆qm2

π

m2

π

◆ ˜ d3 ˜ c0

δMN = nucleon mass splitting induced by O = δ ¯ q τ3 q , σπN = nucleon sigma-term induced by O = − ¯ m¯ qq , δqMN = nucleon mass splitting induced by O = −(˜ c3/2) ¯ qσµντ3Gµνq , ∆qMN = nucleon sigma-term induced by O = −(˜ c0/2) ¯ qσµνGµνq , ∆qm2

π = pion sigma-term induced by O = −(˜

c0/2) ¯ qσµνGµνq ,

Again, all that is needed are simple spectroscopic quantities

Computational Strategy

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

๏ Quark Chromo-EDM Operators L6

¯ qq = − i

2 ¯ qσµνγ5( ˜ d0 + ˜ d3τ3)Gµνq − 1 2 ¯ qσµν(˜ c3τ3 + ˜ c0)Gµνq Symmetries ¯ g0 = δqMN ˜ d0 ˜ c3 + δMN ∆qm2

π

m2

π

˜ d3 ˜ c0 ¯ g3 = −2σπN ✓∆qMN σπN − ∆qm2

π

m2

π

◆ ˜ d3 ˜ c0 Actually, there are subtle difficulties with the simple relations


Seng and Ramsey-Musolf, Phys.Rev. C96 (2017)


which require a modified relation in order to maintain the relations to NNLO (up to small NNLO corrections)

Computational Strategy

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

๏ Quark Chromo-EDM Operators L6

¯ qq = − i

2 ¯ qσµνγ5( ˜ d0 + ˜ d3τ3)Gµνq − 1 2 ¯ qσµν(˜ c3τ3 + ˜ c0)Gµνq Symmetries ¯ g0 = δqMN ˜ d0 ˜ c3 + δMN ∆qm2

π

m2

π

˜ d3 ˜ c0 ¯ g3 = −2σπN ✓∆qMN σπN − ∆qm2

π

m2

π

◆ ˜ d3 ˜ c0

Computational Strategy

de Vries, Mereghetti, Seng, Walker-Loud


• Phys. Lett. B766 (2017) [1612.01567]

¯

g0 = ˜ d0

d

d˜ c3

+ r

d d( ¯ mε)

• δmN + δmN,QCD

1 − ε2 2ε

¯ θ − ¯ θind

• ¯

g1 = −2˜ d3

d

d˜ c0

− r d

d ¯ m

• mN ,

r = 1 2

⟨0|¯

qgsσµν Gµνq|0⟩

⟨0|¯

qq|0⟩

An additional benefit of these relations - the quadratic 1/a2 mixing of the mass operator into the CMDM cancels, leaving a residual logarithmic mixing

3

• Phys. Rev. D96 (2017)

∂meff

λ

(t, τ) ∂λ

• λ=0

= 1 τ −∂λCλ(t + τ) Cλ(t + τ) − −∂λCλ(t) Cλ(t)

• λ=0

“Feynman-Hellmann” correlation function

∂λEn|λ=0 = hn|Hλ|ni

Feynman-Hellmann theorem = gλ + O(e−∆t)

2015 INT: http://www.int.washington.edu/talks/WorkShops/int_15_3/People/Walker- Loud_A/Walker-Loud.pdf
 2015 Fall DNP Meeting Santa Fe: http://www.lanl.gov/conferences/dnp-2015/ 2016 Lattice 2016: https://conference.ippp.dur.ac.uk/event/470/contributions/2449/

• Phys. Rev. D96 (2017)

∂meff

λ

(t, τ) ∂λ

• λ=0

= 1 τ −∂λCλ(t + τ) Cλ(t + τ) − −∂λCλ(t) Cλ(t)

• λ=0

“Feynman-Hellmann” correlation function

O(tO)

= Z dtO

“Feynman-Hellmann”
 propagator

∂λEn|λ=0 = hn|Hλ|ni

Feynman-Hellmann theorem standard 2-point function

t t

derivative correlation function = gλ + O(e−∆t)

2015 INT: http://www.int.washington.edu/talks/WorkShops/int_15_3/People/Walker- Loud_A/Walker-Loud.pdf
 2015 Fall DNP Meeting Santa Fe: http://www.lanl.gov/conferences/dnp-2015/ 2016 Lattice 2016: https://conference.ippp.dur.ac.uk/event/470/contributions/2449/

Key features of this method The correlation function is given by excited state contamination is demonstrably controlled we can access very early Euclidean time, allowing the use of exponentially more precise numerical points

excited-state-subtracted result

5 10 15 t/a 1.15 1.20 1.25 1.30 1.35 1.40 ˚ geff

A (t/a)

a09m220 SS PS SS: excited-state subtracted PS: excited-state subtracted

∂λmeff

λ

(t)

• λ=0 = g00 + z(e−(t+1)∆10 − e−t∆10) + · · ·

“raw” correlation functions

∂meff

λ

(t, τ) ∂λ

• λ=0

= 1 τ ∂λCλ(t + τ) C(t + τ) ∂λCλ(t) C(t)

• ur unconventional method

t+1 t+1 t t

arXiv:1612.06963

2015 INT: http://www.int.washington.edu/talks/WorkShops/int_15_3/People/Walker- Loud_A/Walker-Loud.pdf
 2015 Fall DNP Meeting Santa Fe: http://www.lanl.gov/conferences/dnp-2015/ 2016 Lattice 2016: https://conference.ippp.dur.ac.uk/event/470/contributions/2449/

• Phys. Rev. D96 (2017)

O(tO)

= Z dtO

“Feynman-Hellmann” propagator

= SF H(y, x) = X

z

S(y, z)Γ(z)S(z, x)

Our unconventional method is similar too

traced back to Maiani, Martinelli, Paciello and Taglienti Nucl. Phys. B293 (1987): Güsken, Low, Mutter, Sommer, Patel, Schilling PLB227 (1989) first computed Bulava, Donnellan, Sommer, JHEP 1201 (2012): combined above with GEVP de Divitiis, Petronzio, Tantalo, PLB718 (2012): computed derivatives of form factors Chambers et al. PRD90 (2014), PRD92 (2015) Savage et al. PRL199 (2017); used unconventional method with background field (λ) varying strength of field to extract derivative Our method: uses analytic representation of derivative correlator instead of background field (cheaper) uses complete spectral decomposition of correlator, including contact operators analysis was pushed to greater detail, showing stability of analysis (PRD96 [1612.06963], [1704.01114], Nature 558 [1805.12130])

−∂λCλ(t)

∂meff

λ

(t, τ) ∂λ

• λ=0

= 1 τ ∂λCλ(t + τ) C(t + τ) ∂λCλ(t) C(t)

• ur unconventional method

t+1 t+1 t t

t

2015 INT: http://www.int.washington.edu/talks/WorkShops/int_15_3/People/Walker- Loud_A/Walker-Loud.pdf
 2015 Fall DNP Meeting Santa Fe: http://www.lanl.gov/conferences/dnp-2015/ 2016 Lattice 2016: https://conference.ippp.dur.ac.uk/event/470/contributions/2449/

arXiv:1612.06963

• Phys. Rev. D96 (2017)

O(tO)

= Z dtO

“Feynman-Hellmann” propagator

= SF H(y, x) = X

z

S(y, z)Γ(z)S(z, x)

Our unconventional method is similar too

traced back to Maiani, Martinelli, Paciello and Taglienti Nucl. Phys. B293 (1987): Güsken, Low, Mutter, Sommer, Patel, Schilling PLB227 (1989) first computed Bulava, Donnellan, Sommer, JHEP 1201 (2012): combined above with GEVP de Divitiis, Petronzio, Tantalo, PLB718 (2012): computed derivatives of form factors Chambers et al. PRD90 (2014), PRD92 (2015) Savage et al. PRL199 (2017); used unconventional method with background field (λ) varying strength of field to extract derivative Our method: uses analytic representation of derivative correlator instead of background field (cheaper) uses complete spectral decomposition of correlator, including contact operators analysis was pushed to greater detail, showing stability of analysis (PRD96 [1612.06963], [1704.01114], Nature 558 [1805.12130])

−∂λCλ(t)

∂meff

λ

(t, τ) ∂λ

• λ=0

= 1 τ ∂λCλ(t + τ) C(t + τ) ∂λCλ(t) C(t)

• ur unconventional method

t+1 t+1 t t

t

2015 INT: http://www.int.washington.edu/talks/WorkShops/int_15_3/People/Walker- Loud_A/Walker-Loud.pdf
 2015 Fall DNP Meeting Santa Fe: http://www.lanl.gov/conferences/dnp-2015/ 2016 Lattice 2016: https://conference.ippp.dur.ac.uk/event/470/contributions/2449/

arXiv:1612.06963

• Phys. Rev. D96 (2017)

We have improved our method to allow for arbitrary current insertion see talk by Arjun Gambhir

• Thurs. 11:00, Structure

18

Our Lattice QCD Action

Möbius Domain Wall Fermions on gradient flowed 2+1+1 HISQ ensembles Berkowitz et al. PRD96 (2017) [1701.07559]

Approximate chiral symmetry, many finite lattice spacing operators not allowed Leading discretization errors begin at O(a2)

To control the three standard systematics for LQCD calculations, need

multiple lattice spacings multiple volumes pion masses at/near the physical pion mass

The only set of publicly available ensembles which satisfy these criteria are the

Nf=2+1+1 Highly Improved Staggered Quark (HISQ: Follana et al. PRD75 (2007) [hep-lat/0610092]) ensembles generated by the MILC Collaboration Bazavov et al. PRD82 (2010) [1004.0342], PRD87 (2013) [1212.4768]

The DWF on asqtad action (Renner et al. [LHPC] NPPS 140 (2005) [hep-lat/0409130])

was used very successfully: LHPC; NPLQCD; Aubin, Laiho, Van de Water; … Fully developed Mixed-Action EFT: Bar, Bernard, Rupak, Shoresh; Tiburzi; 
 Chen, O’Connell, Van de Water, Walker-Loud; … This motivated us to use an improved version of this action

19

Our Lattice QCD Action

Möbius Domain Wall Fermions on gradient flowed 2+1+1 HISQ ensembles Berkowitz et al. PRD96 (2017) [1701.07559] Gradient Flow smearing of HISQ cfgs more effective at reducing residual chiral symmetry breaking than the HYP smearing used in DWF on asqtad
 mres < 0.1 ml on all ensembles for small-to-moderate L5 and M5≤1.3

FLAG 1.0 0.8 0.6 0.4 0.2

tgf/a2

1.16 1.18 1.20 1.22 1.24

FK±/Fπ±

FLAG 2+1+1 tgf = 1.0 tgf = 0.8 tgf = 0.6 tgf = 0.4 tgf = 0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a/w0)2

9.5 10.0 10.5 11.0 11.5 12.0 12.5

mN/Fπ

mπ ∼ 310 MeV

tgf = 1.0 tgf = 0.8 tgf = 0.6 tgf = 0.4 tgf = 0.2

tgf

ZA

mπ ∼ 310 MeV

4 6 8 10 12 14 16 18

t

0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32

meff

π (t)

a15m310 tgf = 0.2 tgf = 0.4 tgf = 0.6 tgf = 0.8 tgf = 1.0

tgf

mres

l

mπ ∼ 310 MeV

Our Lattice QCD Action

additional HISQ ensembles generated @ LLNL available to interested parties

Our Lattice QCD Action

Results: Preliminary

4 6 8 10 12 14 16 t/a −1 1 2 3 4 5 6 7 8 ∂˜

c0meff |˜ c0=0

a09m400 4 6 8 10 12 14 16 t/a 0.15 0.20 0.25 0.30 0.35 ∂˜

c3meff |˜ c3=0

a09m400

1 2hp|¯ qσµνGµντ3q|pi

1 2hp|¯ qσµνGµνq|pi

4 6 8 10 12 14 t/a 5 10 15 20 ∂ ¯

mmeff | ¯ m

a09m400

hp|¯ qq|pi

4 6 8 10 12 14 16 t/a 0.5 0.6 0.7 0.8 0.9 1.0 ∂ ¯

m✏meff | ¯ m✏

a09m400

hp|¯ qτ3q|pi

Disconnected Diagrams

Iso-vector CPV pion-nucleon coupling dependent on iso-scalar quantities.

¯ g1 = −2 ˜ d3 ✓ d d˜ c0 − r d d ¯ m ◆ ∆mN

∂ ∂˜ c0 C˜

c0(t)

• ˜

c0=0

= Z dt0hΩ|T{O(t) ✓1 2 ¯ qσµνGµνq(t0) ◆ O†(0)}|Ωi

t0 X

t0

= − =

Requires information of all-to-all

−tr[σµνGµνS(x|x)]

see talk by Arjun Gambhir

• Thurs. 11:00, Structure

Results: Preliminary

0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ 1 2 3 4 5 6 7 @˜

c0meff |˜ c0=0

a09 a12 a15

0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 r = h¯

uσ·Gui 2h¯ uui

Bare matrix element

hp|¯ qσµνGµνq|pilatt

r-subtracted matrix element

2 fπ (hp|¯ qσµνGµνq|pilatt rhp|¯ qq|pi)

r = hΩ|¯ qσµνGµνq|Ωi 2hΩ|¯ qq|Ωi

Putting it all Together

0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ 1 2 3 4 5 6 7 @˜

c0meff |˜ c0=0

a09 a12 a15 0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ −0.4 −0.2 0.0 0.2 @˜

c3meff |˜ c3=0

a09 a12 a15 0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ 4 6 8 10 12 14 16 @ ¯

mmeff | ¯ m

a09 a12 a15 0.10 0.15 0.20 0.25 0.30 0.35 ✏⇡ = m⇡/4⇡f⇡ 0.0 0.2 0.4 0.6 0.8 @ ¯

m✏meff | ¯ m✏

a09 a12 a15 0.10 0.15 0.20 0.25 0.30 0.35 ✏π = mπ/4⇡fπ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 r = h¯

uσ·Gui 2h¯ uui

hp|¯ qq|pi

1 2hp|¯ qσµνGµνq|pi

hp|¯ qτ 3q|pi

1 2hp|¯ qσµνGµντ 3q|pi r = 1 2 hΩ|¯ qσµνGµνq|Ωi hΩ|¯ qq|Ωi

Renormalization

O1 ≡ C = ¯ ψσµνgGµνtaψ, O2 ≡ ∂2S = ∂2( ¯ ψtaψ), O3 ≡ E = e 2 ¯ ψσµνFµν{Q, ta}ψ, O4 ≡ mFF = Tr[MQ2ta]FµνF µν, O5 ≡ mGG = Tr[Mta]Gb

µνGb µν,

O8 ≡ (m2S)1 = 1 2 ¯ ψ

• M2, ta

ψ, O9 ≡ (m2S)2 = Tr[M2] ¯ ψtaψ, O10 ≡ (m2S)3 = Tr[Mta] ¯ ψMψ, O11 ≡ SEE = ¯ ψEtaψE, O12 ≡ (∂ · V )E = i∂µ ¯ ψγµtaψE − ¯ ψEtaγµψ

• ,

O13 ≡ V∂ = ¯ ψta(i / − → ∂ )ψE + ¯ ψE(−i / ← − ∂ )taψ, O14 ≡ VAγ = e 2 ¯ ψ {Q, ta} / AγψE + e 2 ¯ ψE {Q, ta} / Aγψ, O15 ≡ (mSE)1 = 1 2 ¯ ψ {M, ta} ψE + ¯ ψE {M, ta} ψE

• ,

O16 ≡ (mSE)2 = Tr[Mta] ¯ ψψE + ¯ ψEψ

• ,

O17 ≡ (mDG) = Tr[Mta]

• Dbc

µ Gb µν

• Aν c.

ψE ≡ (iDµγµ − M)ψ , D ¯ ψE ≡ − ¯ ψ (i← − D µγµ + M) ,

, Dµ = ∂µ − igAa

µT a − ieQA(γ) µ

← − D µ = ← − ∂ µ + igAa

µT a + ieQA(γ) µ

RI/SMOM Renormalization: MANY extra operators needed for complete renormalization

Renormalization

Coordinate space renormalization? (Thanks Sergey Syritsyn)

Gimenez, Giusti, Guerriero, Lubicz, Martinelli, Petrarca, Reyes, Taglienti, Trevigne Phys.Lett. B598 (2004) [hep-lat/0406019]

Only two operators needed qq-qq graphs already determined in perturbation theory Imposing tree-level condition -

• ff-diagonal graphs vanish

Tree-level renormalization condition for CMDM-CMDM is already a two-loop calculation

✓¯ qq(x) ¯ qq(0) ¯ qσµνGµνq(x) ¯ qσµνGµνq(0) ◆

R

= Zx ✓ ¯ qq(x) ¯ qq(0) ¯ qq(x) ¯ qσµνGµνq(0) ¯ qσµνGµνq(x) ¯ qq(0) ¯ qσµνGµνq(x) ¯ qσµνGµνq(0) ◆

bare

Zx

๏ By exploiting symmetries, we can study the modification of the nucleon spectrum in the presence of CP conserving operators which will then determine the values of the CP-violating 𝜌-N couplings which arise from the quark chromo-EDM operators ๏ The long-range CPV pion-nucleon couplings could allow for a direct connection with fundamental coefficients in dimension-6 quark

• perators and real nuclear physics, 225Ra

๏ The calculations are simplified through a non-standard matrix-element calculation technique ๏ We need to decide upon and complete a renormalization scheme ๏ We plan to compute at smaller pion masses to control the physical pion mass extrapolation

Outlook

plus a few friends

DOE Topical Collaboration Double Beta Decay

Lattice QCD Team

David Brantley: almost single handedly wrote the code and performed all the calculations for the project (he would be here giving this talk except for life- interference)

Jason Chang Jordy de Vries Arjun Gambhir Nicolas Garron Emanuele Mereghetti Sergey Syritsyn Evan Berkowitz Kate Clark Bálint Joó Thorsten Kurth Henry Monge-Camacho Amy Nicholson Kostas Orginos Enrico Rinaldi Pavlos Vranas AWL