Hadronic corrections to parity violation from the latice Jeremy - - PowerPoint PPT Presentation

Hadronic corrections to parity violation from the latice

Jeremy Green

Institut für Kernphysik, Johannes Gutenberg-Universität Mainz

Physics beyond the standard model and precision nucleon structure measurements with parity-violating electron scatering ECT*, Trento, Italy August 1–5, 2016

Outline

• 1. Introduction
• 2. Latice QCD
• 3. Axial charge
• 4. Isovector electromagnetic form factors
• 5. Strange electromagnetic form factors
• 6. Light and strange axial form factors
• 7. Summary

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 2 / 27

Nucleon vector and axial form factors

Describe the strength of the coupling of a proton to a current: p′|V q

µ |p = ¯

u(p′)

• γµF q

1 (Q2) + iσµν (p′ − p)ν

2mp F q

2 (Q2)

• u(p)

p′|Aq

µ |p = ¯

u(p′)

• γµGq

A(Q2) + (p′ − p)µ

2mp Gq

P (Q2)

• γ5u(p),

where V q

µ = ¯

qγµq and Aq

µ = ¯

qγµγ5q. Electric and magnetic form factors: Gq

E(Q2) = F q 1 (Q2) − Q2 (2mp)2 F q 2 (Q2),

Gq

M(Q2) = F q 1 (Q2) + F q 2 (Q2).

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 3 / 27

Electromagnetic form factors

Elastic ep scatering has a leading contribution from single photon

• exchange. This allows the measurement of

Gγ (p)

E,M = 2 3Gu E,M − 1 3Gd E,M − 1 3Gs E,M + . . .

and in particular the charge radius r2

E = −6Gγ (p)′ E

(0), where experimental results suffer from the proton radius problem. Assuming isospin, elastic en scatering probes the same, with u ↔ d. Finally, parity-violating elastic ep scatering is sensitive to the interference between photon and Z boson exchange. This can be used to determine GZ (p)

E,M = (1 − 8 3 sin2 θW )Gu E,M − (1 − 4 3 sin2 θW )(Gd E,M + Gs E,M) + . . .

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 4 / 27

Axial form factors

Assuming isospin, the interaction between nucleons and a W boson contains the isovector axial current Au−d

µ

.

◮ Axial charge gu−d A

≡ Gu−d

A

(0) = 1.2723(23) measured from neutron beta

• decay. This has long served as a “benchmark” for latice QCD.

◮ Qasielastic neutrino scatering, e.g. ¯

νep → e+n, is sensitive to Gu−d

A

.

◮ Muon capture, µ−p → νµn, is sensitive to Gu−d P

. The interaction between a proton and a Z boson contains the axial current Au−d−s

µ

.

◮ Relevant for elastic νp and parity-violating elastic ep scatering.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 5 / 27

Latice QCD

...is a regularization of Euclidean-space QCD such that the path integral can be done fully non-perturbatively.

◮ Euclidean spacetime becomes a periodic hypercubic latice, with

spacing a and box size L3

s × Lt. ◮ Path integral over fermion degrees of freedom is done analytically, for

each gauge configuration. Solving the Dirac equation with a fixed source yields a source-to-all quark propagator.

◮ Path integral over gauge degrees of freedom is done numerically using

Monte Carlo methods to generate an ensemble of gauge configurations. The a → 0 and Ls,Lt → ∞ extrapolations need to be taken by using multiple ensembles.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 6 / 27

Precision latice QCD

Flavour Latice Averaging Group (FLAG) reviews:

http://itpwiki.unibe.ch/flag

• G. Colangelo et al., Eur. Phys. J. C 71, 1695 (2011)
• S. Aoki et al., Eur. Phys. J. C 74, 2890 (2014)
• S. Aoki et al., 1607.00299

e.g., quark masses

Collaboration Ref. publication status chiral extrapolation continuum extrapolation finite volume renormalization running mud ms RBC/UKQCD 14B⊖ [10] P ⋆ ⋆ ⋆ ⋆ d 3.31(4)(4) 90.3(0.9)(1.0) RBC/UKQCD 12⊖ [31] A ⋆

• ⋆

⋆ d 3.37(9)(7)(1)(2) 92.3(1.9)(0.9)(0.4)(0.8) PACS-CS 12⋆ [143] A ⋆

• ⋆

b 3.12(24)(8) 83.60(0.58)(2.23) Laiho 11 [44] C

• ⋆

⋆

• −

3.31(7)(20)(17) 94.2(1.4)(3.2)(4.7) BMW 10A, 10B+ [7, 8] A ⋆ ⋆ ⋆ ⋆ c 3.469(47)(48) 95.5(1.1)(1.5) PACS-CS 10 [95] A ⋆

• ⋆

b 2.78(27) 86.7(2.3) MILC 10A [13] C

• ⋆

⋆

• −

3.19(4)(5)(16) – HPQCD 10∗ [9] A

• ⋆

⋆ − − 3.39(6) 92.2(1.3) RBC/UKQCD 10A [144] A

• ⋆

⋆ a 3.59(13)(14)(8) 96.2(1.6)(0.2)(2.1) Blum 10† [103] A

• ⋆

− 3.44(12)(22) 97.6(2.9)(5.5) PACS-CS 09 [94] A ⋆

• ⋆

b 2.97(28)(3) 92.75(58)(95) HPQCD 09A⊕ [18] A

• ⋆

⋆ − − 3.40(7) 92.4(1.5) MILC 09A [6] C

• ⋆

⋆

• −

3.25 (1)(7)(16)(0) 89.0(0.2)(1.6)(4.5)(0.1) MILC 09 [89] A

• ⋆

⋆

• −

3.2(0)(1)(2)(0) 88(0)(3)(4)(0) PACS-CS 08 [93] A ⋆

• −

2.527(47) 72.72(78) RBC/UKQCD 08 [145] A

• ⋆

⋆ − 3.72(16)(33)(18) 107.3(4.4)(9.7)(4.9) CP-PACS/ JLQCD 07 [146] A

• ⋆

⋆

• −

3.55(19)(+56

−20)

90.1(4.3)(+16.7

−4.3 )

HPQCD 05 [147] A

• −

3.2(0)(2)(2)(0)‡ 87(0)(4)(4)(0)‡ MILC 04, HPQCD/ MILC/UKQCD 04 [107, 148] A

• −

2.8(0)(1)(3)(0) 76(0)(3)(7)(0)

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 7 / 27

Precision latice QCD

Flavour Latice Averaging Group (FLAG) reviews:

http://itpwiki.unibe.ch/flag

• G. Colangelo et al., Eur. Phys. J. C 71, 1695 (2011)
• S. Aoki et al., Eur. Phys. J. C 74, 2890 (2014)
• S. Aoki et al., 1607.00299

e.g., quark masses

70 80 90 100 110 120 = + + = + = pheno. MeV

Vainshtein 78 Narison 06 Jamin 06 Chetyrkin 06 Dominguez 09 PDG QCDSF/UKQCD 06 ETM 07 RBC 07 ETM 10B Dürr 11 ALPHA 12 FLAG average for = MILC 09A HPQCD 09A PACS-CS 09 Blum 10 RBC/UKQCD 10A HPQCD 10 PACS-CS 10 BMW 10A, 10B PACS-CS 12 RBC/UKQCD 12 RBC/UKQCD 14B FLAG average for = + ETM 14 HPQCD 14A FLAG average for = + +

2 3 4 5 6 = + + = + = pheno. MeV

Maltman 01 Narison 06 Dominguez 09 PDG CP-PACS 01 JLQCD 02 QCDSF/UKQCD 04 SPQcdR 05 QCDSF/UKQCD 06 ETM 07 RBC 07 JLQCD/TWQCD 08A ETM 10B Dürr 11 FLAG average for = MILC 04, HPQCD/MILC/UKQCD 04 HPQCD 05 CP-PACS/JLQCD 07 RBC/UKQCD 08 PACS-CS 08 MILC 09 MILC 09A HPQCD 09A PACS-CS 09 Blum 10 RBC/UKQCD 10A HPQCD 10 MILC 10A PACS-CS 10 BMW 10A, 10B Laiho 11 PACS-CS 12 RBC/UKQCD 12 RBC/UKQCD 14B FLAG average for = + ETM 14 FLAG average for = + +

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 7 / 27

Nucleon matrix elements using latice QCD

To find matrix elements, compute using an interpolating operator χ: C2pt(t, p) =

• x

e−i

p· xχ(

x,t) ¯ χ ( 0,0)

t→∞

−→ e−E(

p)t|p| ¯

χ|Ω|2 C3pt(T,τ; p, p′) =

• x,

y

e−i

p′· xei( p′− p)· yχ (

x,T)O( y,τ ) ¯ χ ( 0,0)

τ →∞ T−τ →∞

−→ e−E(

p′)(T−τ )e−E( p)τ Ω|χ|p′p′|O|pp| ¯

χ|Ω Then form ratios to isolate p′|O|p.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 8 / 27

Systematic error: excited states

With interpolating operator χ, compute, e.g., C2pt(t) = χ (t)χ†(0) =

• n

e−Ent

• n|χ†|0
• 2
• For a nucleon, the signal-to-noise asymptotically decays as e−(mN− 3

2 mπ )t. Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 9 / 27

Systematic error: excited states

With interpolating operator χ, compute, e.g., C2pt(t) = χ (t)χ†(0) =

• n

e−Ent

• n|χ†|0
• 2
• For a nucleon, the signal-to-noise asymptotically decays as e−(mN− 3

2 mπ )t. Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 9 / 27

Connected and disconnected diagrams

Two kinds of quark contractions required for C3pt(τ,T) = χ(T)¯ qΓq(τ ) ¯ χ (0):

◮ Connected, which we evaluate in the usual way

with sequential propagators through the sink.

◮ Disconnected, which requires stochastic estimation to evaluate the

disconnected loop, T ( q,t,Γ) = −

• x

ei

q· x Tr[ΓD−1(x,x)].

Introduce noise sources η that satisfy E(ηη†) = I. By solving ψ = D−1η, we get D−1(x,y) = E(ψ (x)η†(y)), so that T ( q,t,Γ) = E

• −
• x

ei

q· xη†(x)Γψ (x)

• .

We then need to compute the correlation between this loop and a two-point correlator.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 10 / 27

Fiting Q2-dependence

We want to fit GE,M,A,P (Q2) with curves to characterize the overall shape of the form factor, and determine the intercepts and/or radii.

◮ Common approach: use simple fit forms such as a dipole. ◮ Beter: use z-expansion. Conformally map domain where G(Q2) is

analytic in complex Q2 to |z| < 1, then use a Taylor series:

Q 2 z

e.g. R. J. Hill and G. Paz, Phys. Rev. D 84 (2011) 073006

z(Q2) =

• tcut + Q2 − √tcut
• tcut + Q2 + √tcut

, G(Q2) =

• k

akz(Q2)k, with Gaussian priors imposed on the coefficients ak.

◮ Leave a0 and a1 unconstrained, so that the intercept and slope are not

directly constrained.

◮ For higher coefficients, impose |ak>1| < 5 max{|a0|,|a1|}, and vary the

bound to estimate systematic uncertainty. For GP, perform the fit to (Q2 + m2)GP (Q2) to remove the pseudoscalar pole.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 11 / 27

Latice 2016 conference

Occurred last week in Southampton. 2 plenary + 31 parallel talks on nucleon or octet baryon structure.

◮ Isovector charges and form factors. ◮ Disconnected diagrams. ◮ Decomposition of proton spin. ◮ Neutron electric dipole moment. ◮ Parton distribution functions. ◮ Form factors at high Q2. ◮ New methods for direct calculation of charge radii.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 12 / 27

Axial charge gA

0.9 1.0 1.1 1.2 1.3 1.4 1.5 0.10 0.15 0.20 0.25 0.30 gA mπ (GeV) Nf=2+1+1 TM (ETMC) Nf=2 TM+clover (ETMC) Nf=2 TM (ETMC) Nf=2+1+1 mixed (PNDME) Nf=2+1 DW (RBC/UKQCD) Nf=2+1 clover (CSSM) Nf=2+1 clover (LHPC) Nf=2 clover (Mainz) Nf=2 clover (RQCD) Nf=2 clover (QCDSF) PDG

Works that extend below 300 MeV.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 13 / 27

Axial charge gA

0.9 1.0 1.1 1.2 1.3 1.4 1.5 0.10 0.15 0.20 0.25 0.30 gA mπ (GeV) Nf=2+1+1 mixed (PNDME) Nf=2+1 clover (CSSM) Nf=2+1 clover (LHPC) Nf=2 clover (Mainz) PDG

Works that extend below 300 MeV, have mπ L ≥ 4, and control exc. states.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 13 / 27

Isovector electromagnetic form factors

talk by Nesreen Hasan (LHP collaboration), Latice 2016

Single ensemble:

◮ Physical pion mass. ◮ mπ L = 4 ◮ a = 0.093 fm

New method for directly computing the momentum derivative of the contribution from quark-connected diagrams to nucleon correlators:

• 1. Set up a calculation with twisted boundary conditions to access

arbitrary non-Fourier momenta.

• 2. Take the derivative with respect to the twist angle, which gives ∂/∂qj.

The evaluation of this on the latice was worked out in G. M. de Divitiis,

• R. Petronzio, N. Tantalo, Phys. Let. B 718, 589 (2012) .

First and second momentum derivatives yield direct calculations of the isovector nucleon magnetic moment and charge radius.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 14 / 27

Isovector Pauli form factor

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Isovector F2 Q2 (GeV2)

PRELIMINARY

z-expansion fit standard method derivative method Kelly

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 15 / 27

Isovector Dirac form factor

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Isovector F1 Q2 (GeV2)

PRELIMINARY

derivative method z-expansion fit standard method Kelly

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 16 / 27

Strange electromagnetic form factors

JG, S. Meinel, M. Engelhardt, S. Krieg, J. Laeuchli, J. Negele, K. Orginos, A. Pochinsky, S. Syritsyn,

• Phys. Rev. D 92, (2015) 031501(R) [1505.01803]

◮ Ensemble generated by JLab / William & Mary. ◮ Nf = 2 + 1 Wilson-clover fermions ◮ a = 0.114 fm, 323 × 96 ◮ mπ = 317 MeV, mπ L = 5.9 ◮ ms ≈ mphys s

We use hierarchical probing to reduce the noise: take the component-wise product η[b] ≡ zb ⊙ η with a specially-constructed spatial Hadamard vector zb and then replace ηη† → 1 Nb

• b

η[b]η[b]†.

red: +1, black: −1

This allows for progressively removing the noise from more distant points.

• A. Stathopoulos, J. Laeuchli, K. Orginos, SIAM J. Sci. Comput. 35(5) (2013) S299–S322 [1302.4018]

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 17 / 27

Disconnected GE(Q2)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.004 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 GE

strange light disconnected

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 18 / 27

Disconnected GM(Q2)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0.00 GM

strange light disconnected

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 19 / 27

Strange magnetic moment

−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 µs (µN) lattice QCD (this work, mπ = 317 MeV) lattice QCD (this work, physical point) lattice QCD [17] connected LQCD + octet µ from expt. [16] ...same, with quenched lattice QCD [29] finite-range-regularized chiral model [30] light-front model + deep inelastic scattering data [31] perturbative chiral quark model [32] dispersion analysis [33] parity-violating elastic scattering [34]

See also recent result at mphys

π

: µs = −0.073(17)(8) µN

• R. S. Sufian, Y.-B. Yang et al. (χQCD Collaboration), 1606.07075

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 20 / 27

Strange magnetic moment

−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 µs (µN) lattice QCD (this work, mπ = 317 MeV) lattice QCD (this work, physical point) lattice QCD [17] connected LQCD + octet µ from expt. [16] ...same, with quenched lattice QCD [29] finite-range-regularized chiral model [30] light-front model + deep inelastic scattering data [31] perturbative chiral quark model [32] dispersion analysis [33] parity-violating elastic scattering [34]

See also recent result at mphys

π

: µs = −0.073(17)(8) µN

• R. S. Sufian, Y.-B. Yang et al. (χQCD Collaboration), 1606.07075

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 20 / 27

Forward-angle scatering experiments

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.05 0.00 0.05 0.10 0.15 Gs

E +ηGs M

G0 HAPPEX A4 lattice η ≈ AQ2, A = 0.94 GeV−2

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 21 / 27

Light and strange axial form factors

talk by JG (LHP Collaboration), Latice 2016

Same mπ = 317 MeV dataset as for electromagnetic form factors. Additional challenge: renormalization, which differs between flavour singlet and nonsinglet axial currents.

◮ Nonsinglet has zero anomalous dimension and is fairly straightforward. ◮ Singlet has nonzero anomalous dimension starting at O(α2).

◮ Perform matching and running to MS at µ = 2 GeV. ◮ Can reuse the previously-computed disconnected diagrams when

evaluating the relevant observables for nonperturbative renormalization.

Singlet-nonsinglet difference causes mixing between As

µ and Au+d µ

under renormalization.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 22 / 27

Effect of mixing: Gs

A

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.05 −0.04 −0.03 −0.02 −0.01 0.00 Gs

A

full renormalization no mixing Largest effect: mixing of (large) Gu+d

A

into (small) Gs

A.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 23 / 27

GA, disconnected

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −0.05 −0.04 −0.03 −0.02 −0.01 0.00 GA (disconnected)

PRELIMINARY

strange light disconnected

Fit (using z-expansion) produces more precise result at Q2 = 0. Systematic uncertainty for Gu,disc

A

= Gd,disc

A

dominated by excited states.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 24 / 27

Strange quark spin

0.1 0.2 0.3 0.4 0.5 mπ (GeV) −0.05 −0.04 −0.03 −0.02 −0.01 0.00 gs

A

QCDSF Engelhardt ETMC CSSM and QCDSF/UKQCD this work (preliminary)

Comparison with published results.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 25 / 27

GP, isoscalar

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q2 (GeV2) −6 −4 −2 2 4 6 8 10 GP

PRELIMINARY

u+d connected u+d u+d disconnected 2s

Connected u + d seems to have a pion pole, cancelled by disconnected part. Disconnected contributions are too large to neglect.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 26 / 27

Summary and outlook

Calculations of quark-connected nucleon matrix elements and form factors are approaching maturity:

◮ Multiple collaborations are working with near-physical pion masses. ◮ Control over excited states and exponential growth of noise remain a

challenge.

◮ Remaining systematics (a → 0, L → ∞) remain to be controlled.

There is a serious effort to calculate disconnected diagrams:

◮ Modern techniques seem effective at controlling the noise from

stochastic estimation.

◮ Non-perturbative renormalization can also be done.

Exploratory calculations are ongoing for many more challenging

• bservables:

◮ Decompositions of (angular) momentum into quarks and gluons. ◮ Form factors at high Q2, and direct computation of charge radii. ◮ Parton distribution functions and transverse momentum-dependent

PDFs.

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 27 / 27

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 28 / 27

Plateau plot: gA

1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 −8 −6 −4 −2 2 4 6 8 gA (τ − T/2)/a T/a = 6 8 10 12 14 summation

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 29 / 27

Plateau plot: GP(Q2

min) 6 8 10 12 14 16 18 20 22 24 −8 −6 −4 −2 2 4 6 8 GP (Q2

min)

(τ − T/2)/a T/a = 6 8 10 12 14 summation

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 30 / 27

Axial charge gA systematics: excited states

0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.4 0.6 0.8 1.0 1.2 1.4 1.6 gA(T)/gA(T 1 fm) T (fm) ETMC 373 MeV PNDME 310 MeV RQCD 294 MeV Mainz 340 MeV LHPC 317 MeV

gA vs. Tsrc-snk, normalized so gA(T 1 fm) = 1 (ratio method)

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 31 / 27

Axial charge gA systematics: excited states

0.90 0.95 1.00 1.05 1.10 1.15 1.20 0.4 0.6 0.8 1.0 1.2 1.4 1.6 gA(T)/gA(T 1 fm) T (fm) ETMC 373 MeV PNDME 310 MeV RQCD 294 MeV Mainz 340 MeV LHPC 317 MeV ...summation

gA vs. Tsrc-snk, normalized so gA(T 1 fm) = 1 (ratio, summation methods)

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 31 / 27

Axial charge gA systematics: infinite volume extrapolation

0.80 0.85 0.90 0.95 1.00 1.05 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 gA(mπL)/gA(mπL = ∞) mπL RQCD 151 MeV RQCD 290 MeV PNDME 217 MeV LHPC 356 MeV QCDSF 261 MeV QCDSF 290 MeV Lt = 48a LHPC 254 MeV Lt = 24a LHPC 254 MeV

Few fully-controlled studies; fit them with floating norm gA(mπ L,. . . ) = A(. . . )(1 + Be−mπ L) → implies −0.8(5)% shif at mπ L = 4 (χ2/dof = 19/9).

Jeremy Green (Mainz) Hadronic corrections to parity violation from the latice ECT*, August 1–5, 2016 32 / 27