Towards Exploring Parity Violation with Lattice QCD. Towards Exploring Parity Violation with Lattice QCD. Brian Tiburzi 30 July 2014 RIKEN BNL

Towards Exploring Parity Violation with Lattice QCD Brian Tiburzi 30 July 2014 RIKEN BNL Research Center Towards Exploring Parity Violation with Lattice QCD

Towards Exploring Parity Violation with Lattice QCD Hadronic Weak Interactions Lattice QCD calculations Hadronic Parity Violation isovector and isotensor Goal: provide a sense of what challenges lattice QCD computations must confront

Quark Interactions to Hadronic Couplings Textbook: gauge theories defined in perturbation theory QCD: short distance perturbative, long distance non-perturbative ψ (D/ + m q ) ψ + 1 4 G µνg µν Wilson Lattice Action Wilson Fermions Non-perturbative definition of asymptotically free gauge theories Many Technicalities a M N δ NN (k) Spectrum Interactions b (D) Ken Wilson 1936-2013 Strong interaction observables Quarks couple to other fundamental interactions: e.g. weak interaction J(x)D(x, 0)J(0) = i C i (µ)o i (x, µ) Wilson Operator Product Expansion, Wilson Coefficients, Wilson Renormalization Group Hadronic weak (& BSM) interactions require all the Wilson brand names

Weak Interactions Leptonic weak interaction Semi-leptonic weak interaction µ ν µ W ν e e µ ν µ W u s K f K Non-leptonic (hadronic) weak interaction ΔS = 1 π u d u W u s K π 10 3 fm

Example: K ππ and ΔI = 1/2 Rule Old Puzzle: I = 0 weak decay channel experimentally observed ~500x over I = 2 Amplitude level: A0 / A2 ~ 22.5 pqcd contributes a factor of ~2 Rest non-perturbative? PRL 110, 152001 (2013) A = i C i (µ)ππ O i (µ) K Lattice Almost There? A 0 /A 2 (m π = 330 MeV) = 12.0(1.7) Theoretical Challenges ΔS = 1 Processes Usual Suspects: pion mass, lattice spacing, lattice volume underway Additional Challenges: Physical Kinematics Multi-Hadron States and Normalization Operator Renormalization & Scale Invariance underway Statistically Noisy Operator Self-Contractions Can such success carry over to weak nuclear processes?

Dirtiest Corner of Standard Model Leptonic weak interaction Semi-leptonic weak interaction ν e µ W e ν µ ν e e W d u p n g A,g V Hadronic weak interaction 5 fm p 1 fm n p n n u W d A PC n d n 10 3 fm n u p A PV

Example: N (Nπ)s and ΔI = 1 Parity Violation Old Problem: hadronic neutral weak interaction is the least constrained SM current New experiments: parity violation in few-body systems, map out NN weak interaction? A = C i (µ)(πn) s O i (µ) N Lattice i PRC 85, 022501 (2012) Signal Found h 1 πnn =1.1(5) 10 7 ±?? Theoretical Challenges ΔI = 1 Processes Usual Suspects: pion mass, lattice spacing, lattice volume Additional Challenges: Physical Kinematics Multi-Hadron States and Normalization Operator Renormalization & Scale Invariance Statistically Noisy Operator Self-Contractions How many lattice advances carry over to weak nuclear processes? largely solved

Particle Physics (B=0) vs. Nuclear Physics (B>0) Statistical nature of lattice QCD two-point correlation functions (Parisi, Lepage) Pion Correlation Function qq(t)qq(0) e m πt Signal Noise^2 {A µ } {A µ } qq(t)qq(t)qq(0)qq(0) e 2m πt Signal/Noise const Nucleon Correlation Function qqq(t)qqq(0) e Mt Signal Noise^2 {A µ } {A µ } qqq(t)qqq(t)qqq(0)qqq(0) e 3m πt Baryons are statistically noisy... nuclear physics has an extra hurdle Signal/Noise e (M 3 2 m π)t Higher statistics Optimal operators

(Un)Physical Kinematics in N (Nπ)s Lattice states are created on-shell G(τ) = x e ip x N(x, τ)n (0, 0) = Ze p 2 +M 2 N τ + ground-state saturation Hadronic transition matrix elements have energy insertion E N = M N E (πn)s = M N + m π (πn) s O i (µ) N Lattice = h 1 πnn( E) Partial solution implemented (due to Beane, Bedaque, Parreno, Savage, NUPHA:747, 55 (2005)) p nπ + nπ + p T-invariance h 1 πnn(m π ) h 1 πnn( m π ) h 1 πnn = 1 2 h 1 πnn (m π )+h 1 πnn( m π ) + +O(m 2 π) Consequence: remove via chiral extrapolation but then only can determine chiral limit coupling Likely small: only~10% at 400 MeV pion mass. Precision demands in nuclear physics typically not as great as particle physics Full solution: determine energy dependence, extrapolate to zero, e.g. TwBCs

Multi-Hadron States and Normalization Multi-Hadron operator not used... Matrix element evaluated by a trick G (τ) =0 N (τ)n (0) 0 = Ze E (Nπ)s τ + M N >M N + m π ground-state saturation (Nπ) s three-quark operator for odd-parity N four-quarks + antiquark = Ze E (Nπ)s τ + Z e E τ + Method requires this condition to hold for lattice parameters Unfortunately likely Z << Z... requires extremely long time to resolve ground state Finite volume and infinite volume states have different normalizations Lellouch, Lüscher, Commun. Math. Phys. 291, 31 (2001) 1 = N 1 1 V n n = Nn 2 V n n V = Nne 2 Enτ + Not needed for spectrum 2 = N 2 2 V 2 O 1 = N 2 N 1 V 2 O 1 V = N 2 N 1 (h 1 πnn) V Lellouch-Lüscher factor requires two-particle energy Not Computed Computed

Lellouch-Lüscher Factor Single Particle Energy Quantization: E = p 2 + M 2 p = 2π L n Two Particle Energy Quantization: E total = k 2 + M 2 + k 2 + m 2 P =0 nπ δ 0 (k) =φ(k) ρ V (E) = dn de = φ (k)+δ (k) E 4πk (known function for a torus) One-to-Two Particle Amplitude: 2 =4π VEρV (E) k 2 V M 2 = 8πV 2 ME 2 total k 2 [δ (k)+φ (k)] M V 2 Not Computed Computed (h 1 πnn) V 2 Generalization for energy insertion: Lin, Martinelli, Pallante, Sachrajda, Villadoro NuPhB:650, 301 (2003) Kim, Sachrajda, Sharpe NuPhB:727, 218 (2005)

Operator Renormalization and Scale Invariance A = i C i (µ)(πn) s O i (µ) N µ = 90 GeV µ =1 2 GeV computable in pqcd at high scale computable on lattice at low scale Tree Level M Z,M W O i m Q Λ QCD

Operator Renormalization and Scale Invariance A = i C i (µ)(πn) s O i (µ) N µ = 90 GeV µ =1 2 GeV computable in pqcd at high scale computable on lattice at low scale Tree Level One Loop M Z,M W log M 2 Z p 2 O i m Q Λ QCD log µ2 p 2 log M Z 2 µ2 µ2 = log log p2 p2 M 2 Z δc(µ) α s (µ) log µ2 M 2 Z 70 s Donoghue, McKellar,..., 90 s Dia Savage Liu Springer

Operator Renormalization and Scale Invariance Tree Level L I=1 PV = i C i (µ)o i (µ) sin 2 θ W Non-Strange 1 vs. Strange One Loop Results C i (µ =1GeV) /C Tree 1 Discrepancies DSLS provide only ratios α s (m c )/α s (m b )=1.44 O 5 =(uu dd) A (ss) V O 6 =(uu dd] A [ss) V O 7 =(uu dd) V (ss) A O 8 =(uu dd] V [ss) A (Fierz) **Using their ratios, I get their values** No heavy quark masses quoted in 1990 PDG Dia, Savage, Liu, Springer PLB 271, 403 (1991) Tiburzi, PRD 85 054020 (2012)

Operator Renormalization and Scale Invariance Tree Level L I=1 PV = i C i (µ)o i (µ) sin 2 θ W Non-Strange 1 vs. Strange One Loop Results O 5 =(uu dd) A (ss) V O 6 =(uu dd] A [ss) V O 7 =(uu dd) V (ss) A O 8 =(uu dd] V [ss) A C i (µ =1GeV) /C Tree 1 (Fierz) LO: 1992 PDG 0.54(4) 0.55(6) 0.35(3) 0 5.35(7) 1.57(10) 4.45(8) 2.12(15) Dia, Savage, Liu, Springer PLB 271, 403 (1991) Tiburzi, PRD 85 054020 (2012)

Operator Renormalization and Scale Invariance A = i C i (µ)(πn) s O i (µ) N µ = 90 GeV µ =1 2 GeV computable in pqcd at high scale computable on lattice at low scale Tree Level One Loop log M 2 Z p 2 O i log µ2 p 2 Two Loop α s (1 GeV) 0.4

QCD Renormalization of Isovector Parity Violation Results ( t Hooft-Veltman scheme) L I=1 PV = i C i (µ)o i (µ) ΔI = 1 Alleged: 95% probe of hadronic neutral current C i (µ =1GeV) /C Tree 1 sin 2 θ W Non-Strange vs. (Fierz) (Fierz) (Fierz) 1 Strange 80-100% Dynamical Question! Tiburzi, PRD 85 054020 (2012)

QCD Renormalization of Isovector Parity Violation Results ( t Hooft-Veltman scheme) L I=1 PV = i C i (µ)o i (µ) ΔI = 1 Alleged: 95% probe of hadronic neutral current Additional finding: chiral basis reveals only 5 independent operators (consequence of non-singlet chiral symmetry) C i (µ =1GeV) /C Tree 1 L L R R L R R L sin 2 θ W Non-Strange vs. (Fierz) (Fierz) (Fierz) 1 Strange 80-100% Dynamical Question! Tiburzi, PRD 85 054020 (2012)

Operator Renormalization and Scale Invariance A = i C i (µ)(πn) s O i (µ) N µ = 90 GeV µ =1 2 GeV computable in pqcd at high scale computable on lattice at low scale Scale Invariance: requires same renormalization scheme pqcd t Hooft-Veltman scheme Anisotropic Lattice Regularization + Wilson Fermions 5 independent PV operators in chiral basis 14 independent PV operators Unphysical + unphysical chiral mixing Matching calculation required...

Statistically Noisy Operator Self-Contractions G(τ, τ) =0 N(τ )O i (τ)n (0) 0 Another notorious difficulty quark disconnected diagrams sin 2 θ W Vector and Axial-Vector self-contractions (a)+(b) Flavor dependence? ~m q Extend to SU(3) + chiral corrections? Utilize Fierz redundancy? 1 O 5 =(uu dd) A (ss) V O 6 =(uu dd] A [ss) V O 7 =(uu dd) V (ss) A (b) ss sγ µ s small nucleon strangeness sγ µ sqγ µ q? Wilson coeffs. O 8 =(uu dd] V [ss) A 0.16 from Adelaide

Isotensor Parity Violation O =(qτ 3 q) A (qτ 3 q) V 1 3 (qτ q) A (qτ q) V Only one operator & without self-contractions L I=2 PV = G F 2 C(µ)O(µ) Operator Renormalization Tiburzi, PRD86: 097501 (2012) 1992 PDG 0.78(1) Better proving ground for Lattice QCD? L NN =[ p σ σ 2 p ] [n T σ 2 n]+... s- to p-wave NN interaction Operator matrix element between 2 hadrons (one step beyond multi-hadron calculations done) πn interactions L ππn + L πγn External fields could ``substitute for pions πpv Isotensor 3 pion interaction exists [15] Kaplan Savage, NuPhA 556 (1993) Wilson fermions still to do... Easier for lattice compute parameters in DDH model?

Summary Lattice QCD: Wilsonian machinery turns high-scale interactions (both SM & Beyond) into QCD-scale hadronic couplings After decades of dedicated work, trustworthy results emerging e.g. K ππ Some of this success will carry over to weak nuclear processes! Challenges = Opportunity Hadronic Parity Violation: πn-coupling more or less challenging than K ππ? Use external axial fields for coupling to pions? Develop technology for isotensor NN-interaction? Isovector parity-violating lattices from auxiliary fields?