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Nucleon Electric Dipole Moments from Lattice QCD Hiroshi Ohki Nara Womens University 2018, 8 9 outline Introduction (EDM) Lattice Study old formula v.s. new formula (on lattice)


  1. Nucleon Electric Dipole Moments from Lattice QCD Hiroshi Ohki Nara Women’s University 基研研究会 素粒子物理の進展 2018, 8 月 9 日

  2. outline • Introduction (EDM) • Lattice Study — old formula v.s. new formula (on lattice) numerical check using chromo-EDM •Implication to the θ -EDM • quark EDM • Summary

  3. Introduction ■ Electric Dipole Moment d 
 Energy shift of a spin particle in an electric field ■ Non-zero EDM : P&T (CP through CPT) violation T + 
 + 
 - - P → H EDM is CP-odd ! - 
 + → H EDM is P-odd

  4. ■ Origin of EDM: CP-violating (CP-odd) interactions CKM: CP violating interaction in SM But, electron and quark EDM’s are zero at 1 and 2 loop level. at least three loops to get non-zero EDM’s. EDM’s are very small in the standard model. t g W d,s,b u c nucleon EDM from CKM : ~ 10 -32 [e cm] SM contribution (3-loop diagram) t Ref: [A. Czarnecki and B. Krause ’97] CP violation (CPV) in SM is not sufficient to reproduce matter/antimatter asymmetry. Large CPV beyond SM is required. (Sakharov’s three conditions) SM prediction 10 20 : 1 photon: matter •http://www.esa.int/ESA Observation 10 10 : 1

  5. • Nucleon EDM e, µ EDM BSM physics: Higgs doublets Paramagnetic Supersymmetry Atom EDM / Molecules e-N int e-q int Left-Right Leptoquark Diamagnetic Extradimension q EDM Atom MQM N EDM EDM Composite q cEDM Schiff models (RGE) moment 4-q int N-N int Standard Model ggg (PQM) Nuclear θ -term EDM ( θ -term) ( P Q M Energy ) scale Atomic Nuclear Hadron QCD TeV : Observable available at experiment observable Important bottleneck : Sizable dependence of the EDM calculation! : Weak dependence : Matching [N. Yamanaka, et al. Eur. Phys. J. A53 (2017) 54, Ginges and Flambaum Phys. Rep. 397, 63, 2004] Role of (lattice) QCD : connect quark/gluon-level (effective) operators to hadron/nuclei matrix elements and interactions (Form factor, dn) Non-perturbative determination is important → Lattice QCD calculation! 5

  6. Neutron EDM S • Nucleon EDM Experiments 10 -18 • Previous Expts Neutron EDM Upper Limit ( e cm) Future Expts 10 -20 10 -22 Current nEDM limits: 10 -24 199Hg spin precession (UW) [Graner et al, 2016] • Ultracold Neutrons in a trap (ILL) [Baker 2006] 10 -26 Supersymmetry � | d Hg | < 7 . 4 × 10 − 30 e · cm Predictions 10 -28 | d n | < 2 . 6 × 10 − 26 e · cm 10 -30 SM nucleon EDMs expectation is Standard Model much smaller than the current bound. Predictions 10 -32 1950 1970 1990 2010 Year of Publication [B. Yoon, talk at Lattice 2017] Several experimental projects are on going. ■ nucleon, charged hadrons, lepton, PSI EDM, Munich FRMII, SNS nEDM, RCNP/TRIUMF , J-PARC

  7. • Effective CPV operators dim=4, θ QCD dim=5, chromo EDM dim=5, e, quark EDM dim=6, Weinberg three gluon C (4 q ) O (4 q ) X dim=6, Four-quark operators + i i ¯ : Strong CP problem θ ≤ O (10 − 10 ) m q / Λ 2 Dim=5 operators suppressed by -> effectively dim=6, quark EDM … the most accurate lattice data for EDM (~10% for u,d) Others are not well determined. cEDM, Weinberg ops just started.

  8. θ QCD induced Nucleon EDMs Phenomenological estimates Lattice calculations Neutron 0 method value ChPT/NDA ∼ 0 . 002 e fm QCD sum rules [1,2] 0 . 0025 ± 0 . 0013 e fm -0.05 n (e fm) 0 . 0004 +0 . 0003 QCD sum rules [3] − 0 . 0002 e fm N f =2+1 DWF, F 3 ( θ ), DSDR 32c N f =2+1 DWF, F 3 ( θ ), Iwasaki 24c d N N f =2 DWF, F 3 ( θ ) -0.1 N f =2 clover, ∆ E( θ ) [1] M. Pospelov, A. Ritz, Nuclear Phys. B 573 (2000) 177, N f =2 clover, F 3 ( θ ) [2] M. Pospelov, A. Ritz, Phys. Rev. Lett. 83 (1999) 2526, N f =2 clover, F 3 (i θ ) -0.15 N f =3 clover, F 3 (i θ ) [3] J. Hisano, J.Y . Lee, N. Nagata, Y . Shimizu, Phys. Rev. D N f =2+1+1 TM, F 3 ( θ ) 85 (2012) 114044. 0 0.2 0.4 2 ) 2 (GeV m π [E. Shintani, T . Blum, T . Izubuchi, A. Soni, PRD93, 094503(2015)] Phenomenology: |dn| ~ θ QCD 10^{-3} e fm -> | θ QCD | < 10^{-10} Lattice : |dn| ~ θ QCD 10^-2 e fm -> severer constraint on | θ QCD | Problem: a spurious mixing between EDM and magnetic moments in all previous lattice computations of nucleon form factor.

  9. Parity mixing problem on the CP-violating nucleon form factors Michael Abramczyk, HO, et al, Lattice calculation of electric dipole moments and form factors of the nucleon Phys.Rev. D96 (2017) no.1, 014501

  10. Definition of nucleon form factors Nucleon form factor in C, P-symmetric world (CP-even) F 1 ( Q 2 ) γ µ + F 2 ( Q 2 ) i σ µ ν q ν  � h p 0 , σ 0 | J µ | p, σ i = ¯ u p 0 , σ 0 u p, σ 2 m N Q 2 = − q 2 ) ( q = p 0 − p, u p : spinor wave function for the nucleon ground state |p, σ > (/ p − m N ) u p = 0 J : electromagnetic current N N

  11. Definition of nucleon form factors Nucleon form factor in CP-broken world F 1 ( Q 2 ) γ µ + F 2 ( Q 2 ) i σ µ ν q ν � F 3 ( Q 2 ) γ 5 σ µ ν q ν  � h p 0 , σ 0 | J µ | p, σ i = ¯ u p 0 , σ 0 u p, σ 2 m N 2 m N P , T even P , T odd Refs. [many textbooks, e.g. Itzykson, Zuber, “Quantum Field Theory“] CP-odd form factor F3 is introduced. the same spinor u p (F1, F2 are same as CP-even case.) Non-zero F3 is a signature of the CP violation (F3= 0 -> CP-even) permanent EDM: All previous lattice studies (prior to 2017) use a different spin structure for the form factors. (Refs. original works [S. Aoki, et al., 2005])

  12. revisit of the nucleon CP-odd (EDM) form

  13. Nucleon 2 point function in CP-even world N = u [ u T C γ 5 d ] Lattice nucleon operator for sink and source Nucleon ground state in CP-even vacuum h 0 | N | p, σ i CP − even = Zu p, σ u p is a solution spinor of the free Dirac equation: (/ p − m N ) u p = 0 p ; t ) | ¯ C 2 pt ( ~ p ; t ) CP − even = h N ( ~ N ( ~ p ; 0) i CP − even 2 3 | k, � ih k, � | 4X 5 ¯ = h N ( ~ p, t ) N ( ~ p ; 0) i CP − even + (excited states) 2 E k k, σ t →∞ | Z | 2 e − E p t X ! ( u p, σ ¯ u p, σ ) 2 E p Completeness condition for free Dirac σ spinor = | Z | 2 e − E p t m N � i / p 2 E p (From now on excited states are omitted.)

  14. Nucleon 2 point function in CP-broken world h 0 | N | p, σ i � Nucleon ground state in CP-broken vacuum CP = Z ˜ u p, σ � ˜ p − m N e − 2 i αγ 5 )˜ is a solution spinor of the free Dirac equation: (/ u p = 0 u p Asymptotic state is modified: (CP-violating) γ 5 mass is allowed in general. p ; t ) | ¯ C 2 pt ( ~ p ; t ) � CP = h N ( ~ N ( ~ p ; 0) i � � � CP = | Z | 2 e − E p t u p, σ ¯ X ( ˜ u p, σ ) ˜ 2 E p Completeness condition for free Dirac spinor σ = | Z | 2 e − E p t m N e 2 i αγ 5 � i / p 2 E p u p = e i αγ 5 u p is a solution to the above Dirac equation. ˜ u p, σ ) e i αγ 5 = m N e 2 i αγ 5 − i / u p, σ ¯ X X u p, σ = e i αγ 5 ( ˜ ˜ u p, σ ¯ p σ σ [Completeness condition for free Dirac spinor with γ 5 mass]

  15. ① ② ③ Calculation of 3 point function in CP-broken world e � i ~ C 3 pt ( ~ X z, ⌧ ) ¯ p 0 · ~ y + i ~ p · ~ z h N ( ~ y, t ) J µ ( ~ p 0 , t ; ~ p, ⌧ ) � CP = N (0) i � � � CP y, ~ ~ z = | Z | 2 e � E p 0 ( t � ⌧ ) � E p ( ⌧ ) X CP h p 0 , � | J µ | p, � 0 i � h N ( p 0 ) | p 0 , � i � CP h p, � 0 | N ( p ) i � � � � CP 4 E p 0 E p � , � 0 ① & ③ : h 0 | N | p, σ i � CP = Z ˜ u p, σ � F 2 ( Q 2 ) i σ µ ν q ν F 3 ( Q 2 ) γ 5 σ µ ν q ν  � F 1 ( Q 2 ) γ µ + ˜ ② : CP = ¯ ˜ � ˜ h p 0 , σ 0 | J µ | p, σ i � ˜ ˜ u p 0 , σ 0 u p, σ � 2 m N 2 m N Refs: original works since 2005 “All” previous (prior 2017) lattice studies: F 1 , ˜ ˜ F 2 , ˜ (˜ u ) : defined in the rotated spinor basis F 3 ( F 2 ( Q 2 ) 6 = ˜ F 2 ( Q 2 ) Two form factors are different! F 3 ( Q 2 ) 6 = ˜ F 3 ( Q 2 ) ( u ) (˜ u )

  16. Relation between two spinor basis F 3 γ 5 ) i σ µ ν q ν F 3 γ 5 ) i σ µ ν q ν  �  � F 1 γ µ + ( ˜ F 1 γ µ + e 2 i αγ 5 ( ˜ ¯ ˜ F 2 + i ˜ ˜ F 2 + i ˜ ˜ u p, σ = ¯ ˜ u p 0 , σ 0 u p 0 , σ 0 u p, σ 2 m N 2 m N F 1 γ µ + ( F 2 + iF 3 γ 5 ) i σ µ ν q ν  � [conventional “lattice” parametrization ≡ ¯ u p 0 , σ 0 u p, σ 2 m N since 2005] [textbook] A simple relations between and { ˜ F 1 , ˜ F 2 , ˜ { F 1 , F 2 , F 3 } F 3 } ( ˜ = cos (2 α ) F 2 + sin (2 α ) F 3 F 2 ( F 2 + iF 3 γ 5 ) = e 2 i αγ 5 ( ˜ F 2 + i ˜ F 3 γ 5 ) , ⇔ ˜ = − sin (2 α ) F 2 + cos (2 α ) F 3 F 3 There is a spurious contribution of order ( α F2) to the previous lattice results. In other words, CP violation effects come from both tilde{F3} and α , not only tilde{F3}. This mixing angle α has to be calculated, and rotated away to get “net” CP-violation effect. Similar issues in the ChPT (perturbative) calculations? ( α may appear in the mass correction.)

  17. Numerical check using the chromo EDM operator Form factor method vs Energy shift method Computational resources : ACCC HOKUSAI greatwave, Fermilab, JLab [USQCD project]

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