Systematics in nucleon matrix element calculations Jeremy Green - PowerPoint PPT Presentation

Systematics in nucleon matrix element calculations Jeremy Green NIC, DESY, Zeuthen The 36th International Symposium on Latice Field Theory East Lansing, MI, USA July 22–28, 2018

Motivation Three reasons for studying structure of protons and neutrons: ◮ Understanding the quark and gluon substructure of a hadron. ◮ Understanding nucleons as tools in experiments. ◮ Validation of latice QCD using “benchmark” observables. Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 2

Neutron beta decay Coupling to W boson via axial current depends on nucleon e ν e p ¯ axial charge, � p ( P , s ′ ) | ¯ u γ µ γ 5 d | n ( P , s ) � = д A ¯ u p ( P , s ′ ) γ µ γ 5 u n ( P , s ) . W − Interpreted as isovector quark spin contribution. → Yi-Bo Yang plenary PDG 2018: д A = 1 . 2724 ( 23 ) . n WEIGHTED AVERAGE -1.2724 ± 0.0023 (Error scaled by 2.2) -1.16 -1.18 -1.20 χ 2 -1.22 DARIUS 17 SPEC MENDENHALL 13 UCNA 1.1 MUND 13 SPEC 3.6 -1.24 SCHUMANN 08 CNTR MOSTOVOI 01 CNTR 0.7 LIAUD 97 TPC 2.5 YEROZLIM... 97 CNTR 11.7 -1.26 BOPP 86 SPEC 4.3 23.8 (Confidence Level = 0.0002) -1.28 1970 1980 1990 2000 2010 -1.29 -1.28 -1.27 -1.26 -1.25 -1.24 λ ≡ g A / g V Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 3

Other observables Precision β -decay experiments may be sensitive to BSM physics; leading contributions are controlled by the scalar and tensor charges: T. Bhatacharya et al. , Phys. Rev. D 85 , 054512 (2012) [1110.6448] � p ( P , s ′ ) | ¯ u p ( P , s ′ ) u n ( P , s ) , u d | n ( P , s ) � = д S ¯ � p ( P , s ′ ) | ¯ u σ µν d | n ( P , s ) � = д T ¯ u p ( P , s ′ ) σ µν u n ( P , s ) . Sigma terms control the sensitivity of direct-detection dark mater searches to WIMPs that interact via Higgs exchange. uu + ¯ σ π N = m ud � N | ¯ dd | N � σ q = m q � N | ¯ qq | N � Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 4

Toward precision nucleon structure Trustworthy latice calculations need control over all systematics: ◮ Physical quark masses ◮ Isolation of ground state ◮ Infinite volume ◮ Continuum limit Want to transition from this ... N f = 2 + 1 + 1 TM (ETMC) N f = 2 + 1 clover (LHPC) 1.6 N f = 2 TM+clover (ETMC) N f = 2 + 1 clover (NME) N f = 2 TM (ETMC) N f = 2 clover (Mainz) 1.5 N f = 2 + 1 + 1 mixed (PNDME) N f = 2 clover (RQCD) N f = 2 + 1 + 1 mixed (CalLat) N f = 2 clover (QCDSF) N f = 2 + 1 DW (RBC/UKQCD) PDG 1.4 N f = 2 + 1 clover (CSSM) 1.3 g A 1.2 1.1 1.0 0.9 0.10 0.15 0.20 0.25 0.30 m π (GeV) Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 5

Toward precision nucleon structure Trustworthy latice calculations need control over all systematics: ◮ Physical quark masses ◮ Isolation of ground state ◮ Infinite volume ◮ Continuum limit Want to transition from this ...to something like this. / + FLAG average for = + + FNAL/MILC 14A + ETM 14 = FLAG average for = + RBC/UKQCD 14B RBC/UKQCD 12 PACS-CS 12 Laiho 11 + BMW 10A, 10B RBC/UKQCD 10A = Blum 10 PACS-CS 09 MILC 09A MILC 09 PACS-CS 08 RBC/UKQCD 08 MILC 04, HPQCD/MILC/UKQCD 04 FLAG average for = ETM 14D = ETM 10B RBC 07 ETM 07 QCDSF/UKQCD 06 PDG pheno. Oller 07 Narison 06 Kaiser 98 Leutwyler 96 Weinberg 77 22 24 26 28 30 32 34 Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 5

Outline 1. Methodologies for nucleon matrix elements 2. Excited-state effects: theory and practice 3. Other systematics: L , ( a ), m q 4. Outlook Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 6

Hadron correlation functions Compute two-point and three-point functions, using interpolator χ and operator insertion O . In simplest case: C 2pt ( t ) ≡ � χ ( t ) χ † ( 0 ) � � | Z n | 2 e − E n t = 0 t n 1 + O ( e − ∆ Et ) → | Z 0 | 2 e − E 0 t � � , where Z n = � Ω | χ | n � , C 3pt ( τ , T ) ≡ � χ ( T ) O ( τ ) χ † ( 0 ) � 0 τ T � Z n ′ Z ∗ n � n ′ |O| n � e − E n τ e − E n ′ ( T − τ ) = n , n ′ 1 + O ( e − ∆ Eτ ) + O ( e − ∆ E ( T − τ ) ) → | Z 0 | 2 � 0 |O| 0 � e − E 0 T � � Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 7

Hadron matrix elements Ratio method R ( τ , T ) ≡ C 3pt ( τ , T ) = � 0 |O| 0 � + O ( e − ∆ Eτ ) + O ( e − ∆ E ( T − τ ) ) C 2pt ( T ) Midpoint yields R ( T 2 , T ) = � 0 |O| 0 � + O ( e − ∆ ET / 2 ) . Summation method L. Maiani et al. , Nucl. Phys. B 293 , 420 (1987); д A in S. Güsken et al. , Phys. Let. B 227 , 266 (1989) d dT S ( T ) = � 0 |O| 0 � + O ( Te − ∆ ET ) � S ( T ) ≡ R ( τ , T ) , τ Sum can be over all timeslices or from τ 0 to T − τ 0 . Improved asymptotic behaviour noted in talks at Latice 2010. S. Capitani et al. , PoS LATTICE2010 147 [1011.1358]; J. Bulava et al. , ibid. 303 [1011.4393] In practice noisier than ratio method at same T . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 8

Ratio and summation method 1.30 1.25 g A (bare) 1.20 1.15 R ( T/ 2 , T ) S ( T + a ) − S ( T ) 2 4 6 8 10 12 T/a physical m π , a = 0 . 116 fm N. Hasan, JG, et al. , in preparation Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 9

Feynman-Hellmann approach If we add a term to the Lagrangian L ( λ ) ≡ L + λ O , then � ∂ � ∂ λE n ( λ ) = � n |O| n � . � � � λ = 0 Discrete derivatives sometimes used, particularly for sigma terms: ∂ m N σ q ≡ m q � N | ¯ qq | N � = m q . ∂ m q Exact derivatives lead directly to summation method (neglecting � Ω |O| Ω � ): � − ∂ � ∂ λ log C 2pt ( t ) = S ( t ) . � � � λ = 0 L. Maiani et al. , Nucl. Phys. B 293 , 420 (1987) A. J. Chambers et al. (CSSM and QCDSF/UKQCD), Phys. Rev. D 90 , 014510 (2014) [1405.3019] C. Bouchard et al. , Phys. Rev. D 96 , 014504 (2017) [1612.06963] Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 10

Controlling excited states Three approaches: ◮ Go to large T . Need ∆ ET ≫ 1 . 2 m π ) T . But signal-to-noise decays as e − ( E 0 − 3 ◮ Improve the interpolating operator. ◮ Fit correlators to remove excited states. Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 11

Excited-state energies First approximation: noninteracting stable hadrons in a box. Nπ , Nππ , ... 1 . 0 0 . 8 ∆ E (GeV) 0 . 6 0 . 4 0 . 2 N π 0 . 0 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 m π L Physical m π . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

Excited-state energies First approximation: noninteracting stable hadrons in a box. Nπ , Nππ , ... 1 . 0 0 . 8 ∆ E (GeV) 0 . 6 0 . 4 0 . 2 N π N ππ 0 . 0 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 m π L Physical m π . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

Excited-state energies First approximation: noninteracting stable hadrons in a box. Nπ , Nππ , ... 1 . 0 0 . 8 ∆ E (GeV) 0 . 6 0 . 4 0 . 2 N π N ππ N πππ 0 . 0 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 m π L Physical m π . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

Excited-state energies First approximation: noninteracting stable hadrons in a box. Nπ , Nππ , ... 1 . 0 0 . 8 ∆ E (GeV) 0 . 6 0 . 4 0 . 2 N π N ππ N πππ N ππππ 0 . 0 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 m π L Physical m π . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

Excited-state energies First approximation: noninteracting stable hadrons in a box. Nπ , Nππ , ... 1 . 0 0 . 8 ∆ E (GeV) 0 . 6 0 . 4 0 . 2 N π N ππ N πππ N ππππ 0 . 0 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 0 . 35 0 . 40 0 . 45 0 . 50 m π (GeV) m π L = 4 . Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 12

Excited-state energies M. T. Hansen and H. B. Meyer, Nucl. Phys. B 923 , 558 (2017) [1610.03843] Focus on Nπ sector and apply finite-volume quantization using scatering phase shif from experiment. I ( J P ) =1 / 2(1 / 2 + ) 1800 1600 E (MeV) 1400 1200 1000 3.0 3.5 4.0 4.5 5.0 5.5 6.0 M π L Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 13

Going to large T Statistical errors: Excited state systematics: δ exc ∼ e − ∆ ET / 2 (ratio method) δ stat ∼ N − 1 / 2 e ( E 0 − 3 2 m π ) T Suppose we want these to be equal, i.e. δ stat = δ exc ≡ δ . Required statistics are given by 2 + 4 E 0 − 6 mπ � � N ∝ δ − . ∆ E At the physical point with ∆ E = 2 m π , the exponent is ≈ − 13 . Multi-level methods could potentially improve this. Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 14

Predicting excited-state contributions Need to know: 1. energies E n 2. matrix elements � n ′ |O| n � 3. overlaps Z n Have been studied in ChPT. B. C. Tiburzi; O. Bär Key insight: at leading order a single LEC controls coupling of local interpolator to N and Nπ states. Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 15

ChPT prediction of excited-state effects 1.04 physical point 1.03 m π L = 4 m eff ( T ) / m 1.02 truncated to five Nπ states 1.01 1.00 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T (fm) O. Bär, Int. J. Mod. Phys. A 32 no. 15, 1730011 (2017) [1705.02806] generalization to axial FFs → Oliver Bär parallel, Wed. 14:00 Jeremy Green | DESY, Zeuthen | Latice 2018 | Page 16