From Lattice QCD
gA from lattice QCD
Rajan Gupta Laboratory Fellow Theoretical Division Los Alamos National Laboratory, USA O d × u n d d p u u
Isolate the neutron e-Mn (t-|τ) Project on the proton e-Mpτ
LA-UR 17-26064 CKM 2018, September 18, 2018 PNDME collaboration (Clover-on-HISQ) • Tanmoy Bhattacharya • Vincenzo Cirigliano • Yong-Chull Jang • Huey-Wen Lin • Boram Yoon Bhattacharya et al, PRD85 (2012) 054512 Bhattacharya et al, PRD89 (2014) 094502 Bhattacharya et al, PRD92 (2015) 114026 Bhattacharya et al, PRL 115 (2015) 212002 Bhattacharya et al, PRD92 (2015) 094511 Bhattacharya et al, PRD94 (2016) 054508 Bhattacharya et al, PRD96 (2017) 114503 Gupta et al, Phys.Rev. D98 (2018) 034503 Huey-Wen Lin et al, arXiv:1806.10604 Acknowledgement: Computing Resources • Clover-on-Clover: • Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. • Institutional Computing at Los Alamos National Laboratory. • The USQCD Collaboration, funded by the Office of Science, U.S. DOE
• Clover-on-HISQ • The National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. DOE under Contract No. DE-AC02-05CH11231 • Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. • The USQCD Collaboration, funded by the Office of Science, U.S. DOE • Institutional Computing at Los Alamos National Laboratory. • The Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575 gA is known very precisely
Ultracold Neutron Decay UCNA: 1.2772(20) PERKEO II: 1.2761-+. – gA = 1.2766(20) $+,
#$% The lattice QCD calculation of !" is mature
Serves as a benchmark calculation of nucleon matrix elements using lattice QCD
Use it to demonstrate control over all systematics
4 gA is given by the matrix elements of quark bilinear operator, the axial current, within the nucleon ground state: + ! "#$%&%'( ! = *# $,%&%'$,
3 $- %&%'(| 3⟩: In the 3-point correlation function 3 5 6 7 3 8 , the operator $- %&%'( is inserted on the quark lines constituting the nonperturbative nucleon wavefunction generated by the averaging over gauge configurations in lattice QCD. Need 9 & / − 9 → ∞ to isolate the nucleon ground state
.- %&%'.
t
/ 012 Only the “Connected” diagram contributes to *# 5 matrix elements from 3-point functions
O d × u n d d p u u
Isolate the neutron e-Mn (t-|τ) Project on the proton e-Mpτ
Ω Nˆ (t, p')Oˆ(τ, p'− p)Nˆ (0, p) Ω =
− dt H − dt H Ω Nˆ (p') N e ∫ N Oˆ(τ, p'− p) N e ∫ N Nˆ (p) Ω = ∑i, j j j i i −E (t−τ ) Ω Nˆ (p') N e j N Oˆ(τ, p'− p) N e−Eiτ N Nˆ (p) Ω ∑i, j j j i i
Matrix element 6 At finite t and !, use the spectral decomposition of 2- and 3-point functions
2 2 2 2 2 −M0 t −M1 t −M2 t −M3 t Γ (t) = A0 e + A1 e + A2 e + A3 e +....
2 2 3 −M0 Δt −M1Δt Γ (t, Δt) = A0 0 O 0 e + A1 1 O 1 e +
* −M0 Δt −ΔM (Δt−t) * −ΔM t −M0Δt A0 A1 0 O 1 e e + A0 A1 1 O 0 e e +...
to extract amplitudes, energy levels, matrix elements from fits Simulations of 2+1+1 flavor lattice QCD → non-perturbative wavefunction of nucleon → ME of axial operator
➤ precise lattice data for gA(a, Mπ, Mπ L)
2 at multiple values of a, Mπ , Mπ L
• Calculations done in the isospin symmetric limit !" = !$
• !% , !' tuned to physical value using () or M-% and (1'
• Ignore bottom quarks as !2 is heavy for this physics The challenge: signal to noise The lowest state in the squared correlation function is the 3 pion state (334) which is much lighter than the 2 nucleon (237) state
!"#$%& ∼ (* +,*-./+0 1 $'"!(
Need very high statistics to reach large τ at which excited state contribution becomes negligible Systematic uncertainties in matrix elements within nucleon states
– High Statistics: O(100,000) measurements – Demonstrating control over all Systematic Errors:
• Contamination from excited states
• Non-perturbative renormalization of bilinear operators (RIsmom scheme) Ø Finite volume effects
Ø Chiral extrapolation to physical mu and md (simulate at physical point) Ø Extrapolation to the continuum limit (lattice spacing a → 0)
Perform simulations on ensembles with multiple values of
Ø Lattice size and take Mπ L→ ∞ Ø Light quark masse → physical mu and md → Mπ =135 MeV Ø Lattice spacing and take a → 0 Toolkit -1 • Multigrid Dirac invertor → propagator PF = D η • Truncated solver method with bias correction (TSM) • Coherent source sequential propagator • Deflation + hierarchical probing (for disconnected) • 3-5 values of ! with smeared sources • 2-state and 3-state fits to multiple values of !
• Non-perturbative renormalization constant "#
• Combined extrapolation in a, Mπ , MπL • Variation of results with extrapolation ansatz HISQ Ensembles Nf =2+1+1 PNDME collaboration ms tuned to the physical mass using Mss
sea val 3 val HP LP Ensemble ID a (fm) M (MeV) M (MeV) L TML ⌧/aNconf N N ⇡ ⇡ ⇥ ⇡ meas meas a15m310 0.1510(20) 306.9(5) 320(5) 163 48 3.93 5, 6, 7, 8, 9 1917 7668 122,688 ⇥ { } a12m310 0.1207(11) 305.3(4) 310.2(2.8) 243 64 4.55 8, 10, 12 1013 8104 64,832 ⇥ { } a12m220S 0.1202(12) 218.1(4) 225.0(2.3) 243 64 3.29 8, 10, 12 946 3784 60,544 ⇥ { } a12m220 0.1184(10) 216.9(2) 227.9(1.9) 323 64 4.38 8, 10, 12 744 2976 47,616 ⇥ { } a12m220L 0.1189(09) 217.0(2) 227.6(1.7) 403 64 5.49 8, 10, 12, 14 1000 4000 128,000 ⇥ { } a09m310 0.0888(08) 312.7(6) 313.0(2.8) 323 96 4.51 10, 12, 14, 16 2263 9052 144,832 ⇥ { } a09m220 0.0872(07) 220.3(2) 225.9(1.8) 483 96 4.79 10, 12, 14, 16 964 7712 123,392 ⇥ { } a09m130 0.0871(06) 128.2(1) 138.1(1.0) 643 96 3.90 10, 12, 14 883 7064 84,768 ⇥ { } a09m130W 8, 10, 12, 14, 16 1290 5160 165,120 { } a06m310 0.0582(04) 319.3(5) 319.6(2.2) 483 144 4.52 16, 20, 22, 24 1000 8000 64,000 ⇥ { } a06m310W 18, 20, 22, 24 500 2000 64,000 { } a06m220 0.0578(04) 229.2(4) 235.2(1.7) 643 144 4.41 16, 20, 22, 24 650 2600 41,600 ⇥ { } a06m220W 18, 20, 22, 24 649 2596 41,536 { } a06m135 0.0570(01) 135.5(2) 135.6(1.4) 963 192 3.7 16, 18, 20, 22 675 2700 43,200 ⇥ { } !: 0.15, 0.12, 0.09, 0.06 +, -.: 310, 220, 135 -01 3.3 < -. 3 < 5.5
Gupta et al, Phys. Rev. D98 (2018) 034503 Rajan Gupat HISQ Ensembles July 25, 2018 1 / 1 Controlling excited-state contamination
• Reduce An/A0 in an n-state fit by tuning N - Tune source smearing size σ in %& %& !" = $ '( = $ )(*(,)
• Generate data at multiple values of . • 2 → 3 → n-state fits to data at many t and .
Yoon et al, PRD D93 (2016) 114506 Yoon el al, PRD D95 (2017) 074508 Controlling excited-state contamination: n-state fit
2 2 2 2 2 −M0 t −M1 t −M2 t −M3 t Γ (t) = A0 e + A1 e + A2 e + A3 e +....
2 2 3 −M0 Δt −M1Δt Γ (t, Δt) = A0 0 O 0 e + A1 1 O 1 e +
* −M0 Δt −ΔM (Δt−t) * −ΔM t −M0Δt A0 A1 0 O 1 e e + A0 A1 1 O 0 e e +...
M0, M1, … masses of the ground & excited states A0, A1, … corresponding amplitudes O(t) n p Δt = t = t - t ti sep f i tf
Make a simultaneous fit to data at multiple t and ! ≡ Δ$
14 Toolkit: Controlling Excited State Contamination
• Better smearing • Higher statistics (10K 100K) with TSM • 4-5 values of source-sink separation τ • 4-state fits to 2-point functions • 3-state fits to 3-point functions • Full covariance error matrix
a09m220 ,tskip =3, gA 1.30 a09m220 1.3
1.25
1 2 u-d A 1.20 . g 1.15
Extrap tsep=12 1.10 1.1 tsep=10 tsep=14 τ : ∞ 10 12 14 16 -6 -4 -2 0 2 4 6 -8 -6 -4 -2 0 2 4 6 8 τ t -- tτsep/2/2 t-τ/2
Gupta et al, Phys. Rev. D98 (2018) 034503 15 Analyzing lattice data gA(a, Mπ, Mπ L): 2 Simultaneous CCFV fits versus a, Mπ , Mπ L
Take into account corrections to lattice data due to • Lattice spacing: a
2 • Dependence on light quark mass: mq ~ Mπ
• Finite volume: Mπ L
Physical result: from a simultaneous CCFV fit with leading corrections in each
#,% . . 1234 !" &, '(, '() = +, + +-a + +. '( + +/'( 0 + … Higher order terms not warranted by the Akaike Information Criteria 16 2 Simultaneous extrapolation in a, Mπ , MπL Fits using 11 clover-on-HISQ ensembles:
a15m310 a09m310 a06m135 a12m310 a09m220 extrap. a12m220L a09m130 1-extrap. a12m220 a06m310 1.3 a12m220S a06m220 1.3 1.3 d d d − − − u u A u A A g g g
1.2 1.2 1.2
00.030.060.090.120.150.02 0.04 0.06 0.08 0.1 0.12 34567 2 2 a [fm] Mπ [GeV ] MπL
#$% #0% !" = 1.218(25)(30) !" 1 = 0.575(24) 2344
PNDME collaboration: Phys. Rev. D98 (2018) 034503 17 #$% !" = 1.218(25)(30)
Statistical, Error estimate for excited-state, using the lowest order renormalization, corrections in CCFV fit error the CCFV fit
#$% !" smaller than the experimental value 1.277(2) by 5%
PNDME: Phys. Rev. D98 (2018) 034503 Status 2018: Isovector gA
gA 1.00 1.25 1.50 1.75 1 + 1 PNDME ’18 +
2 CalLat ’18 = f PNDME ’16 N
LHPC ’14 1 ’ + LHPC 10 2 RBC/UKQCD ’08 = f Lin/Orginos ’07 N
ETMC ’17 Mainz ’17 2 RQCD ’14 = f QCDSF/UKQCD ’13 N ETMC ’15 RBC ’08
AWSR ’16 COMPASS ’15 Other
Brown ’17 Mund ’13 Mendenhall ’12 Liu ’10 Abele ’02 Mostovoi ’01 Liaud ’97 Neutron Expts Yerozolimsky’97 Bopp ’86 1.250 1.275 1.300 1.325 Why do PNDME and CalLat results disagree? (Same 2+1+1 flavor HISQ ensembles used) Much is different in the 2 calculations
PNDME CalLat • Clover-on-HISQ • DWF-on-HISQ
– ZA/ZV → ~1% error – ZA/ZV =1 → no error 2 • 1-HYP smearing of lattice • Wilson flow twf/a = 1 • Sequential propagator • Sequential propagator calculated with insertion of calculated with insertion of nucleon at sink at fixed τ: operator at all t∈[0,T] : – all charges and FF in one go – Get all τ at the same time – Each τ needs new calculation – Get data for any sink smearing – Contact terms & all t outside [0,τ] • n-state fit to data at different • Data summed over all t (+ artifacts) t and τ to get gA Make 2-state fit to " $ + 1 " $ g eff = # − # A "' $ + 1 "' $
PNDME: 11 clover-on-HISQ ensembles CalLAT: 16 DWF-on-HISQ ensembles #$% !" : PNDME & CalLat agree within errors on 7 ensembles CalLat: Nature: https://doi.org/10.1038/s41586-018-0161-8 PNDME: Gupta et al, Phys. Rev. D98 (2018) 034503 PNDME CalLat a15m310 1.228(25) 1.215(12) a12m310 1.251(19) 1.214(13) a12m220S 1.224(44) 1.272(28) a12m220 1.234(25) 1.259(15) a12m220L 1.262(17) 1.252(21) a09m310 1.235(15) 1.236(11) a09m220 1.260(19) 1.253(09) CalLat uses a variant of the summation method Difference comes from the Chiral-Continuum fits: • CalLat chiral fit anchored by heavier pion masses • CalLat have not yet analyzed the a=0.06fm lattices Chiral Extrapolation
CalLat PNDME
1.3 d − u A g
1.2
0.02 0.04 0.06 0.08 0.1 0.12 2 2 Mπ [GeV ]
PNDME: small increase in gA as Mπ goes from 310 → 135MeV
3 CalLat: consistent with χPT chiral log when fit includes a large επ term The errors in their 2 physical mass points are large Continuum Extrapolation CalLat PNDME
a15m310 a09m310 a06m135 a12m310 a09m220 extrap. a12m220L a09m130 1-extrap. a12m220 a06m310 1.3 a12m220S a06m220 d − u A g
1.2
00.030.060.090.120.15 a [fm]
PNDME: decrease in gA as ! → # driven by the 3 points at a=0.06 fm
CalLat: have not simulated the a=0.06 ensembles Uncertainty in the Continuum Extrapolation
a15m310 a09m310 a06m135 a12m310 a09m220 extrap. a12m220L a09m130 1-extrap. a12m220 a06m310 1.3 a12m220S a06m220 d − u A g
1.2
00.030.060.090.120.15 a [fm] #$% Neglecting the a=0.06 ensembles, PNDME value is !" = 1.245(42) Summary ()* &' = 1.218(25)(30) • With O(105) measurements, the statistical precision reached allowed us to assess all the systematics • Obtained results at 2 physical pion mass ensembles, each with ~3% uncertainty • Largest uncertainty in PNDME result comes from the continuum extrapolation (! ≈ 0.06 fm ensembles) • PNDME error estimate more realistic • Checking with a clover-on-clover unitary calculation Extras Lattice QCD is QCD on a finite 4-d Euclidean lattice with spacing a
6 free parameters • Bare coupling g fixed by spacing a • u, d, s, c, b quark masses • Fix a, mq using 6 physical quantities
8 gluon degrees of freedom • SU(3) matrices ! = #$%&'⋅)
Dirac action gluon - • +* ,-. + / → +*!- / +(/ + 354)/3 Simulate using the path integral formalism
$%'() ' *+ • Action A = # & () , • Integrate out the fermions % $ '() ' *-. /012 $+ • A = # & () , = # • Use #$+ as Boltzmann weight to generate importance sampled gauge configurations • Feynman quark propagator 3 = 1/6 • Stitching P and U construct gauge invariant correlation functions on ensembles of gauge configurations • Calculate expectation values as ensemble averages
• These give O(a, mu, md, ms, mc, mb) • Take continuum limit 7 → 0 and
• mu, md, ms, mc, mb to their physical value to get Ophysical LQCD: Field theory in Euclidean time
τ
−mnτ 0 N(τ )N(0) 0 = ∑Ane n
& Any interpolating operator ! = (%$ '()+̅)%$
with the right quantum numbers creates the nucleon state + all excited and multiparticle states LQCD: Field theory in Euclidean time
τ '()* '()* '()* '()* !i…f 0 # $% & $%|$%', & … & $./,|$. & $. # 0
'57)* '5789)* '5:;9)* '()* !i…f 0 # $% & $%|$%', & … & $./,|$. & $. # 0
'5:* '5:* !i 0 # $. & $. # 0 = !i <. &
• Insert complete set of states Σ2|$.⟩ $. = 1 of the transfer matrix at each intermediate time slice • In a finite box these are discrete states • Propagate in Euclidean time: &'(* with eigenvalues &'56* • At long time τ, only the ground state survives Sequential Propagators
At sink: Fixed T, p, smear At Op: Fixed q, γµ
Prop from smeared source at t=0 Prop from smeared source at t=0
Sequential propagator with Sequential propagator with O p=0 nucleon insertion at t=T insertion at all t
Contract operator at t=τ with Contract sequential propagator prop from t=0 and sequential with original propagators to propagator from t=T form the nucleon at sink at all t Truncated solver + bias correction (TSM)
NLP NHP AMA 1 LP 1 HP HP C = ∑CLP (xi )+ ∑{CHP (xi )−CLP (xi )} NLP i=1 NHP i=1 • Use multigrid inverter with -3 - rLP = 10 -7 -10 - rHP = 10 – 10 - NLP = 64−160, NHP = 3-5 per configuration • The bias term is negligible (~1% of the error) • The TSM error is <15% larger than LP
8e-04 2e-04 2e-03 S5S5 S5S5, tsep=10 S9S9, tsep=10 S S , t =12 S S , t =12 S9S9 5 5 sep 9 9 sep 6e-04 1e-04 1e-03 > (S) > (Pion) > (Proton)
0e+00 3pt AMA 2pt AMA 4e-04 0e+00 2pt AMA / 2pt Crxn 5 5 2pt Crxn C C S S Scalar C 9 9 0e+00 -2e-04 -2e-03 0 5 10 15 20 0 2 4 6 8 10 12 14 16 -6 -4 -2 0 2 4 6 t t τ - tsep/2 Coherent Source Sequential Propagator 1 1 2 2 3 3 t i" t f" t i" t f" t i" t f" seq useq useq u" u" u1" u" u" 2" u" u" 3" d" d" d" d" d" d" P1" P2" P3" d" d" d" d" d" d" P1" P2" P3" 3 measurements being done in a single computer job 1 1 t i" t f" τ Pseq" u" P" × u" u" u" u" u" P" P" P" d" d" d" d" d" d" P" P" P" d" d" d" d" d" d" Bias = 0 after gauge integral: Tiny increase in Variance Controlling excited state contamination Ground state ME requires asymptotic values of τ: not reached 1.30 a09m220 1.25 Excited-state contamination is large u-d A 1.20 g 1.15 Extrap tsep=12 1.10 tsep=10 tsep=14 -6 -4 -2 0 2 4 6 τ - t /2 The truncated 2-state spectral decomposition of the 3-point function is: ( M M ) sep ⌘ 1 0 2 2 M0⌧ A1 1 1 ⌧ A1 0 1 ⌧/2 A 0 0 e 1+| | h |O| ie +2| | h |O| ie cosh (t ⌧/2) | 0| h |O| i A 2 0 0 A 0 0 { } | 0| h |O| i | 0| h |O| i 2 M0⌧ ⌧ ⌧/2 A 0 0 e 1+X e +2X e cosh M (t ⌧/2) (1) ⌘ | 0| h |O| i 11 01 { 1 } 2-state fit ⇥ ⇤ Summation 2 M0⌧ ⌧ ⌧/2 sinh( (⌧ 1)/2) A 0 0 e (⌧ 1)(1 + X e )+2X e (2) | 0| h |O| i 11 01 sinh( /2) Method 34 1