Nucleon matrix elements from Moments of Correlation Functions and - PowerPoint PPT Presentation

Nucleon matrix elements from Moments of Correlation Functions and the Proton Charge Radius Chia Cheng Chan (LBNL) Chris Bouchard (Glasgow) Kostas Orginos (JLab/WM) David Richards (JLab)* * Speaker

Proton EM form factors • Nucleon Pauli and Dirac Form Factors described in terms of matrix element of vector current F 2 ( q 2 )  � F q ( q 2 ) � µ + � µ ν q ν h N | V µ | N i ( ~ q ) = ¯ u ( ~ p f ) u ( ~ p i ) 2 m N • Alternatively, Sach’s form factors determined in experiment G E ( Q 2 ) F 1 ( Q 2 ) − Q 2 4 M 2 F 2 ( Q 2 ) = G M ( Q 2 ) F 1 ( Q 2 ) + F 2 ( Q 2 ) = Charge radius is slope at Q 2 = 0 ∂ G E ( Q 2 ) 6 h r 2 i = ∂ F 1 ( Q 2 ) � � = � 1 � F 2 (0) � � � � ∂ Q 2 ∂ Q 2 4 M 2 � � Q 2 =0 Q 2 =0

EM Form factors - II Approved expt E12-07-109 PRAD: E12-11-106 Q 2 . 8 . 2 GeV 2 Q 2 . 4 . 1 GeV 2 Nucleon Charge Radius Direct calculation of charge Boosted interpolating operators radius through coordinate- space moments Bali et al ., Phys. Rev. D 93, 094515 (2016) UKQCD, Lellouch, Richards et LHPC, Syritsyn, Gambhir, Orginos et al, Lattice 2016 al. , NPB444 (1995) 401 Distillation + Operators for hadrons in flight Bouchard, Chang, Orginos, Dudek, Edwards, Thomas, Richards, Lattice 2016 Phys. Rev. D 85, 014507 (2012) 3

Form Factor in LQCD q p p + q γ N 1 N 2 + Excited states - T sep suppressed at large T y, t ) ¯ X N ( ~ 0 , 0) | 0 i e − i ~ p · ~ x e − i ~ q · ~ y C 3pt ( t sep , t ; ~ q ) = h 0 | N ( ~ x, t sep ) V µ ( ~ p, ~ ~ x, ~ y Resolution of unity – insert states p | ¯ N | 0 i e − E ( ~ p + ~ q )( t sep − t ) e − E ( ~ p ) t � ! h 0 | N | N, ~ q ih N, ~ q | V µ | N ~ p ih N, ~ p + ~ p + ~

Electromagnetic Form Factors Wilson-clover lattices from BMW fit to experiment 1.0 Green et al (LHPC), Phys. Rev. D 90, 074507 (2014) lattice data, m π = 149 MeV 0.8 Hadron structure at nearly- G p − n 0.6 physical quark masses E 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 Q 2 (GeV 2 ) Why can’t we get rid of those excited states! Smallest non-zero Q 2 determined by spatial volume ⇒ Calculate slope of form factor directly.

Isgur-Wise Function and CKM matrix Extract V cb if know intercept at zero recoil Lattice Calculate slope at zero recoil.. UKQCD, L. Lellouch et al., Nucl. Phys. B444, 401 (1995), hep-lat/9410013

Moment Methods • Introduce three-momentum projected three-point function D E b X e � ikx 0 N a C 3pt ( t, t 0 ) = x Γ t 0 , ~ x 0 N z 0 , ~ t, ~ 0 x, ~ ~ x 0 • Now take derivative w.r.t. k 2 − x 0 D E b X z N a C 0 3pt ( t, t 0 ) = 2 k sin ( kx 0 z ) x 0 N x Γ t 0 , ~ 0 , ~ t, ~ 0 whence x, ~ ~ x 0 − x 0 2 D E b X z N a k 2 ! 0 C 0 3pt ( t, t 0 ) = lim x 0 N . x Γ t 0 , ~ 0 , ~ t, ~ 0 2 ~ x, ~ x 0 Odd moments vanish by symmetry

Moment Methods - II • Analogous expressions for two-point functions: D b E X N b e − ikx z C 2pt ( t ) = x N 0 , ~ t, ~ 0 ~ x ⇒ − x z D b E X N b C 0 2pt ( t ) = 2 k sin ( kx z ) x N 0 , ~ t, ~ 0 ⇒ ~ x − x 2 D E b X z N b k 2 ! 0 C 0 lim 2pt ( t ) = x N . 0 , ~ t, ~ 0 2 ~ x Lowest coordinate-space moment ⇔ slope at zero momentum

Lattice Details • Two degenerate light-quark flavors, and strange quark set to its physical value 0 . 12 fm ' a 400 MeV ' m π 24 3 ⇥ 64 Lattice Size : • To gain control over finite-volume effects, replicate in z direction: 24 × 24 × 48 × 64

Two-point correlator ln [ C 2pt ( t, x z )] Any polynomial moment in x z converges “Effective mass” ln C 2pt ( t, x z ) /C 2pt ( t, x z + 1)

Three-point correlator ln [ C 3pt ( t 0 , x 0 z )] “Effective mass” • Spatial moments push the peak of the correlator away from origin • Larger finite volume corrections compared to regular correlators

Fitting the data… n (0) Γ nm ( k 2 ) Z b m ( k 2 ) Z † a e � M n (0)( t � t 0 ) e � E m ( k 2 ) t 0 X C 3pt ( t, t 0 ) = 4 M n (0) E m ( k 2 ) n,m m ( k 2 ) Z b m ( k 2 ) Z b † e − E m ( k 2 ) t X C 2pt ( t ) = 2 E m ( k 2 ) m where Z † a n (0) ⌘ h Ω | N a | n, p i = (0 , 0 , 0) i b | Ω i Z b m ( k 2 ) ⌘ h m, p i = (0 , 0 , k ) | N Γ nm ( k 2 ) ⌘ h n, p i = (0 , 0 , 0) | Γ | m, p i = (0 , 0 , k ) i Allow for multi-state contributions in the fit

Fitting - II • Now look at the functional form of derivatives: ✓ 2 Z b 0 m ( k 2 ) 1 ◆ t X C 2pt C 0 2pt ( t ) = m ( t ) m ( k 2 ) − 2[ E m ( k 2 )] 2 − Z b 2 E m ( k 2 ) m Γ nm ( k 2 ) + Z b 0 nm ( k 2 ) m ( k 2 ) ⇢ Γ 0 1 t 0 � X C 3pt C 0 3pt ( t, t 0 ) = nm ( t, t 0 ) m ( k 2 ) − 2[ E m ( k 2 )] 2 − Z b 2 E m ( k 2 ) n,m spatially extended Second distance sources scale

Fitting - III In practice we use multi- exponential, Bayesian fits

F 1 Form Factor

Conclusions • Moment methods allow direct calculations of slopes of form factors at momenta allowed on lattice • Lowest (even) moment gives the slope at Q 2 = 0 . • Larger finite-volume effects than regular correlators (perhaps expected - no free lunch). • Illustrated here for u-quark contribution to EM form factor; d-quark and sea-quark contributions in progress….