Light and strange axial form factors of the nucleon at pion mass 317 MeV Jeremy Green Institut für Kernphysik, Johannes Gutenberg-Universität Mainz in collaboration with Nesreen Hasan, Stefan Meinel, Michael Engelhardt, Stefan Krieg, Jesse Laeuchli, John Negele, Kostas Orginos, Andrew Pochinsky, Sergey Syritsyn The 34th International Symposium on Latice Field Theory July 24–30, 2016
Outline 1. Introduction 2. Renormalization 3. Light and strange G A 4. Strange quark spin 5. Light and strange G P 6. Summary and outlook Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 2 / 25
Nucleon axial form factors Describe the strength of the coupling of a proton to an axial current: � � A ( Q 2 ) + ( p ′ − p ) µ � p ′ | A q γ µ G q G q P ( Q 2 ) u ( p ′ ) µ | p � = ¯ γ 5 u ( p ) , 2 m N where A q q γ µ γ 5 q . µ = ¯ ◮ Interaction with W boson contains (assuming isospin) isovector A u − d . µ Measured in quasielastic neutrino scatering, e.g. ¯ ν e p → e + n , and in muon capture, µ − p → ν µ n . ◮ Interaction with Z boson contains A u − d − s . µ Relevant for elastic ν p and parity-violating ep scatering. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 3 / 25
Qark spin in the proton g q A ≡ G q A ( 0 ) gives the contribution from the spin of q to the proton’s spin. This equals the moment of a polarized parton distribution function: � 1 g q dx (∆ q ( x ) + ∆ ¯ q ( x )) . A = 0 For the typical phenomenological values: ◮ g u − d is obtained from neutron beta decay. A ◮ g u + d − 2 s is obtained from semileptonic beta decay of octet baryons, A assuming SU ( 3 ) symmetry. ◮ A third linear combination is obtained from the integral of polarized PDFs measured in polarized deep inelastic scatering. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 4 / 25
Connected and disconnected diagrams We need to compute, using an interpolating operator χ , χ ( 0 ) � C 2pt ( t ) = � χ ( t ) ¯ C 3pt ( τ , T ) = � χ ( T ) A q µ ( τ ) ¯ χ ( 0 ) � . There are two kinds of quark contractions required for C 3pt : ◮ Connected, which we evaluate in the usual way with sequential propagators through the sink. ◮ Disconnected, which requires stochastic estimation to evaluate the disconnected loop , x Tr [ Γ D − 1 ( x , x ) ] . � e i � q · � T ( � q , t , Γ) = − x � We then need to compute the correlation between this loop and a two-point correlator. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 5 / 25
Stochastic estimation for disconnected loop Estimate the all-to-all propagator stochastically by introducing noise sources η that satisfy E ( ηη † ) = I . By solving ψ = D − 1 η , we get D − 1 ( x , y ) = E ( ψ ( x ) η † ( y )) . We use hierarchical probing to reduce the noise: take the component-wise product η [ b ] ≡ z b ⊙ η with a specially-constructed spatial Hadamard vector z b and then replace ηη † → 1 � η [ b ] η [ b ] † . N b b red: + 1, black: − 1 This allows for a progressively increasing amount of spatial dilution. A. Stathopoulos, J. Laeuchli, K. Orginos, SIAM J. Sci. Comput. 35(5) (2013) S299–S322 [1302.4018] Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 6 / 25
Fiting Q 2 -dependence We want to fit G A , P ( Q 2 ) with curves to characterize the overall shape of the form factor and determine the axial radius. ◮ Common approach: use simple fit forms such as a dipole. ◮ Beter: use z -expansion. Conformally map domain where G ( Q 2 ) is analytic in complex Q 2 to | z | < 1, then use a Taylor series: Q 2 t cut + Q 2 − √ t cut z � z ( Q 2 ) = , t cut + Q 2 + √ t cut � � G ( Q 2 ) = a k z ( Q 2 ) k , R. J. Hill and G. Paz, Phys. Rev. D 84 (2011) 073006 k with Gaussian priors imposed on the coefficients a k . ◮ Leave a 0 and a 1 unconstrained, so that the intercept and slope are not directly constrained. ◮ For higher coefficients, impose | a k > 1 | < 5 max {| a 0 | , | a 1 |} , and vary the bound to estimate systematic uncertainty. For G P , perform the fit to ( Q 2 + m 2 ) G P ( Q 2 ) to remove the pseudoscalar pole. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 7 / 25
Latice calculation Previously used for disconnected G E ( Q 2 ) , G M ( Q 2 ) . 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. ◮ N f = 2 + 1 Wilson-clover fermions ◮ a = 0 . 114 fm, 32 3 × 96 ◮ m π = 317 MeV, m π L = 5 . 9 ◮ m s ≈ m phys s ◮ 1028 gauge configurations ◮ Disconnected loops for six source timeslices (16 or 128 Hadamard vectors, plus color+spin dilution). ◮ Two-point correlators from 96 source positions. ◮ Connected three-point correlators from 12 source positions. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 8 / 25
Control over excited states C 3pt ( τ , T ) = � χ ( T ) A q Recall µ ( τ ) ¯ χ ( 0 ) � , C 2pt ( t ) = � χ ( t ) ¯ χ ( 0 ) � Connected diagrams are evaluated at fixed T / a ∈ { 6 , 8 , 10 , 12 , 14 } , which corresponds to T between 0.7 and 1.6 fm. These are obtained for all τ . ◮ For the ratio method, we compute R ( τ , T ) ∼ C 3pt ( τ , T ) / C 2pt ( T ) . For each T average over the three points near τ = T / 2. Excited-state contamination will decay asymptotically as e − ∆ E T / 2 . ◮ For the summation method, compute S ( T ) = � T − a τ = a R ( τ , T ) . Fit a line to S ( T ) at three adjacent values of T and take its slope. Excited-state contamination will decay asymptotically as Te − ∆ E T . Disconnected diagrams are evaluated at fixed τ / a ∈ { 3 , 4 , 5 , 6 , 7 } (light) or { 4 , 5 , 6 } (strange) and obtained for all T . ◮ Use the ratio method. For each T average over the two or three points near τ = T / 2. Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 9 / 25
Previous result: G E 0 . 012 0 . 010 0 . 008 0 . 006 0 . 004 G E 0 . 002 0 . 000 − 0 . 002 strange light disconnected − 0 . 004 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 Q 2 (GeV 2 ) Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 10 / 25
Previous result: G M 0 . 00 − 0 . 01 − 0 . 02 − 0 . 03 G M − 0 . 04 − 0 . 05 strange − 0 . 06 light disconnected − 0 . 07 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 Q 2 (GeV 2 ) Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 11 / 25
Renormalization of the axial current: massless case Flavour-singlet and nonsinglet axial currents renormalize differently. ◮ Nonsinglet has zero anomalous dimension and matching between “good” schemes that satisfy the axial Ward identity is trivial, e.g.: Z MS Z MS A A = 1 = , Z RI ′ -MOM Z RI-SMOM A A to all orders in perturbation theory. ◮ Singlet has an anomalous dimension starting at O ( α 2 ) . “Good” schemes should satisfy the anomalous Ward identity. We know that Z MS Z MS ∗ A = 1 + O ( α 2 ) = A . Z RI ′ -MOM Z RI-SMOM A A ∗ T. Bhatacharya, V. Cirigliano, R. Gupta, E. Meregheti and B. Yoon, Phys. Rev. D 92 , 114026 (2015) Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 12 / 25
Renormalization of the axial current: N f = 2 + 1 µ = ¯ For a single ensemble with m u = m d � m s , define A a ψγ µ γ 5 λ a ψ , where ψ = � � � � � � 1 0 0 1 0 0 u � � � � � � 1 1 1 � � � � � � λ 3 = λ 8 = λ 0 = d 0 − 1 0 0 1 0 I . , √ , √ , √ 2 6 3 � � � � � � 0 0 0 0 0 − 2 s This gives the renormalization patern A R , 3 � � = � Z 3 , 3 � � A 3 � 0 0 � � � � � � µ µ A � � � � � � Z 8 , 8 Z 8 , 0 A R , 8 A 8 0 . µ µ A A � � � Z 0 , 8 Z 0 , 0 � � � A R , 0 A 0 0 µ µ A A In the SU ( 3 ) f limit, Z 3 , 3 = Z 8 , 8 and Z 8 , 0 = Z 0 , 8 = 0. A A A A Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 13 / 25
Rome-Southampton method On Landau-gauge-fixed configurations, compute � � e − ip ′ · x + ip · y � ψ i ( x ) O ( 0 ) ¯ e − ip · x � ψ i ( x ) ¯ G O ij ( p ′ , p ) = ψ i ( 0 ) � , S i ( p ) = ψ j ( y ) � x x , y ij ( p ′ , p ) = S − 1 ij ( p ′ , p ) S − 1 Λ O i ( p ′ ) G O and j ( p ) These renormalize as Z ab A a A b A R , a = Z ab A A b S R i ( p ) = Z i R , ij ( p ′ , p ) = A ij ( p ′ , p ) . µ µ µ , q S i ( p ) = ⇒ Λ Λ µ � q Z j Z i q RI ′ -MOM or RI-SMOM schemes define a projector P ν for specific kinematics K at scale µ . The condition for Z ab A ( µ ) becomes � � λ a Λ A b � K = δ ab . Tr col , spin , flav ν R P ν ν Jeremy Green (Mainz) Light and strange axial form factors Latice 2016 14 / 25
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