Hadronic electroweak processes in a finite volume

Transcription

1 Hadronic electroweak processes in a finite volume Andria Agadjanov, University of Bonn Dr. Klaus Erkelenz Kolloquium November 29, 2016 Collaborators: V. Bernard, Ulf-G. Meißner and A. Rusetsky 1/34

2 Outline Introduction Radiative decays of resonances: the Nγ transition Nucl. Phys. B 886 (2014) 1199 the B K (892)l + l decay Nucl. Phys. B 910 (2016) 387 Compton scattering at low energies Summary and Outlook arxiv: /34

3 Standard Model Standard Model is a triumph of theory Higgs particle - a great discovery of the 21 st century Nobel Prize 2013 along with gravitational waves by LIGO and Virgo Electroweak sector Strong sector 3/34

4 Experimental tensions Proton radius puzzle Anomalous magnetic moment of muon ( muon g 2 ) Anomalies in B meson decays Strong sector of SM has to be better understood! 4/34

5 Strong interaction Quantum chromodynamics (QCD): non-abelian gauge theory α s (Q 2 ) τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) e.w. precision fits (NNLO) ( ) pp > jets (NLO) pp > tt (NNLO) October QCD α s (M z ) = ± Q [GeV] 1000 The measurements of α s as a function of the energy scale Q High energies Asymptotic Freedom Nobel Prize 2004 Low energies ( 1GeV) Confinement Perturbation theory breaks down lattice QCD, disp. relations,... 5/34

6 Lattice QCD Path integral formulation in Euclidean space-time L 3 L t L = Na a L L t = N t a t Space-time is discretized and finite natural UV cut-off 1/a K. G. Wilson, Phys. Rev. D 10 (1974) 2445 Integration Monte Carlo methods Correlation functions energy levels, current matrix elements FLAG: S. Aoki et al., arxiv: (2016) 6/34

7 Stable hadrons The nucleon two-point function Two-point correlation function: C(τ) = [ 0 O N (x, τ)o N (x, 0) 0 = Z O 2 e M N τ 1+ x n Z n 2 e Enτ ] O N (x, τ) -nucleonfield,z O, Z n -overlapfactors, E n > 0 Effective mass: M eff (τ) = 1 a log C(τ +a) C(τ ), M eff(τ )=M N 7/34

8 Resonances Riemann sheets on the complex s-plane Energy levels do not correspond to any resonance Resonances complex poles of the scattering amplitude: E R = m R i Γ R 2, s R = E 2 R m R -mass,γ R -width PDG: Chin. Phys. C 40, (2016) 8/34

9 Lüscher method A standard tool to study resonances on the lattice scattering phase shift finite volume energy spectrum M. Lüscher, Nucl. Phys. B 354 (1991) 531. cot δ(p n )+cotφ(q n )=0 (Lüscher equation) m 1 p 2 p 1 m2 no partial wave mixing p 2 = λ(s, m 2 1,m2 2 )/4s, cot φ(q) = 1 2π 2 q q = pl 2π 1 n Z 3 n 2 q 2 (l =0,m=0) energy levels Lüscher equation scattering phase effective-range expansion (ERE) resonance pole position E R 9/34

10 An example: ρ resonance Source: Hadron Spectrum Collaboration, Phys. Rev. D 92 (2015) m π [MeV] m R [MeV] Γ R [MeV] (2) 85(2) (2) 12.4(6) I=1, P-wave ππ scattering phase shift rapid change of phase shift resonance 10 / 34

11 Work of K. Erkelenz N N H N The nucleon-nucleon interaction N M = π, σ, ρ, ω,... one-boson exchange models K. Erkelenz, Phys. Rept. 13 (1974) 191 Bonn Model: R. Machleidt, et al., Phys.Rept.149 (1987) 1 Resonances play a crucial role! 11 / 34

12 Radiative decays of resonances 12 / 34

13 N Nγ transition M 1+ E 1+,S 1+ Amplitudes E 1+, S 1+ are small: E 1+, S 1+ M 1+ R. Beck et al., Phys. Rev. Lett. 78 (1997) 606 Form factors the transition probability (transverse densities) The recent lattice measurement (with a stable ) : C. Alexandrou et al., Phys. Rev. D 83 (2011) EMR ImE 1+ ImM 1+, CMR ImS 1+ ImM / 34

14 Experiment vs. Lattice Source: V. Pascalutsa, M. Vanderhaeghen, Phys. Rev. D 73 (2006) the chiral extrapolation is not reliable for m π 300 MeV 14 / 34

15 Theoretical issues E πn (1232) πk ηk K (892) and K are resonances Lüscher method Lattice data might be affected by another threshold (ηk) Resonance matrix elements should be properly defined Form factors: real axis vs. complex resonance pole 15 / 34

16 Electroweak processes Seminal work on K ππ by L. Lellouch and M. Lüscher, Commun.Math.Phys. 219, 31(2001) = Lellouch-Lüscher formula ( A(K ππ) ππ H weak K The range of applicability: dδ(p) dp p 2 + dφ(q) dp large volumes (m π L 4), below 3(4)-particle threshold ) 1/2 16 / 34

17 The framework: non-relativistic EFT Ideally suited for the problems we study = Bubble-chain diagrams T 1+cG + c 2 G 2 + = 1 1 cg J. Gasser, B. Kubis and A. Rusetsky, Nucl. Phys. B 850 (2011) 96 In a finite volume: G G L (the loop function) A bridge: finite volume spectrum scattering sector 17 / 34

18 Form factors on the real energy axis model- and process-dependent 8π Im A(E BW, q ) = p BW Γ F A(E BW, q ) F A (E BW, q ) current matrix elements, Γ resonance width at the resonance pole process-independent favourable! P, resonance J(0) Q, stable = lim Z 1/2 P 2 s R, Q 2 M 2 R Z 1/2 (s R P 2 )(M 2 Q 2 )F (P, Q) F (P, Q) 3-point Green s function Z, Z R wave-function renormalization constants generalization of: S. Mandelstam, Proc. Roy. Soc. Lond. A 233 (1955) / 34

19 B K (892)l + l P q 2 (GeV 2 /c 4 ) LHCb and Belle data on the observable P 5 A. Abdesselam et al., arxiv: [hep-ex] Example: observable P 5 3σ deviation from Standard Model Observed tensions in same Wilson coefficients (C 7, C 9, C 10 ) 19 / 34

20 Source: C. Hambrock et al., Phys. Rev. D 89 (2014), Pattern of deviations observed by LHCb and Belle Collaboration BSM or QCD? Hadronic uncertainties: form factors, long-distanceeffects In the low recoil region, lattice QCD simulations are reliable 20 / 34

21 Low recoil For q 2 Λ QCD,thedecayamplitudeA is approximately 7 A c M (C 7,...)f M (q 2 ) M=1 see, however, J. Lyon and R. Zwicky, arxiv: [hep-ph] Seven semileptonic form factors f M (q 2 ), M =1,...7, K (k, λ) sγ µ b B(p) = 2iV (q2 ) m B + m V ϵ µνρσ ϵ ν k ρp σ, etc. The first unquenched lattice calculation (with a stable K ): R. R. Horgan et al., Phys. Rev. D 89, (2014) 21 / 34

22 Lellouch-Lüscher formula for B K (892)l + l γ K K O B O F 1 M + X 1 X 2 π η F 1 M + + O X 1 F M 2 O + F X 2 M 2 F M (E n, q ) = V 1 M v1 F 8πE 1 + v 2 F M 2 E=En V (asymmetric)volume v 1 = v 1 (E, L), v 2 = v 2 (E, L) knownfunctions(gener.ofllfactor) F M (E n, q ) current matrix elements, measured on the lattice Two-particle irreducible vertices F 1 M(E n, q ), F 2 M(E n, q ) decay amplitudes A M (B πkl + l ) Watson s theorem, agrees with S. Sharpe and M. Hansen, Phys. Rev. D 86 (2012) / 34

23 Form factors at the K pole (q 0, q) ( m 2 B + q 2, q) (E R, 0) Suppose, the K pole is on the Riemann sheet II Analytic continuation: vary E with q fixed (analog of the ERE) FR M (E R, q ) = i ( ) M M w1 F 8πE 1 w 2 F 2 E=ER w 1 = w 1 (E), w 2 = w 2 (E) volume-independentquantities Form factors at the pole complex 23 / 34

24 Infinitely narrow width Results are simplified in the limit Γ 0(K is above ηk) Assume the Breit-Wigner form in the vicinity of E = E BW Real axis: F M (E n, q ) = V 1 2En F M A (E n, q ) +O(Γ 1/2 ), E n = E BW +O(Γ) Complex plane (E R E BW ): F M R (E R, q ) Γ 0 = F M A (E BW, q ) + O(Γ 1/2 ) similar to the one-channel case of the Nγ transition: AA, V. Bernard, U.-G. Meißner and A. Rusetsky, Nucl. Phys. B 886 (2014) / 34

25 Summary Extraction of the B K form factors on the lattice is studied Possible admixture of the ηk to πk channels is taken into account Equation for the B K l + l amplitude at low recoil is derived Form factors at the K pole are determined Infinitely-narrow width approximation of the results is considered Similar work on the Nγ transition: Nucl. Phys. B 886 (2014) 1199 An open issue: long-distance effects non-local matrix elements T µ i d 4 xe iqx K Tj µ em(x)h 4q (0) B 25 / 34

26 External fields on the lattice: Compton scattering 26 / 34

27 γ (q) γ (q) N(p, s) N(p, s) Compton scattering Two invariant amplitudes: T 1 (ν, q 2 )andt 2 (ν, q 2 ), ν p q/m Why important? the proton-neutron mass difference Lamb shift in the muonic hydrogen (low q 2 ) the existence of a fixed pole in Regge theory 27 / 34

28 Cottingham formula q p p Electromagnetic contribution to the proton-neutron mass difference: (m p m n ) em = ie 2 Λ 2m(2π) 4 d 4 qd(q 2 ){3q 2 T 1 +(2ν 2 +q 2 )T 2 }+counter terms D(q 2 )-photonpropagator Electroproduction cross sections T 1, T 2 (q 2 < 0) Aproblem:S 1 (q 2 ) T 1 (0, q 2 )isnotfixedbyexperiment 28 / 34

29 Theoretical approaches 5 4 NS1 inel p n WCM ESTY This work Prediction at Q 2 =0 = 0.5(1.6) exp β p n M Q 2 Source: J. Gasser et al., Eur. Phys. J. C 75, 375(2015) Chiral EFTs Hill, Meißner, Pascalutsa,... Phenomenological Ansatzes Pachucki, Walker-Loud,... Reggeon dominance hypothesis Gasser, Leutwyler,... Lattice QCD devoid of any model dependence 29 / 34

30 BMW 2014 HCH Neutron-proton mass splitting 10 8 Σ Ξ experiment QCD+QED prediction M [MeV] 6 4 D Ξ cc 2 N CG 0 Source: S. Borsanyi et al., Science 347 (2015) 1452 m [MeV] QCD [MeV] QED [MeV] N = n p 1.51(16)(23) 2.52(17)(24) -1.00(07)(14) The fully unquenched lattice QCD + QED computation 4non-degenerateflavors,m π 195 MeV BMW Collaboration 30 / 34

31 External field method Source: W. Detmold et al., Phys. Rev. D 73 (2006) Two-point function in an external electromagnetic field A µ (x) O(A 2 ) term Compton scattering amplitude Uniform magnetic field polarizabilities NPLQCD Static magnetic field energy levels, measured on the lattice 31 / 34

32 Field configuration N V 0 Static periodic magnetic field B =(0, 0, B 3 ): A 1 = B ω sin(ωx 2), A 0 = A 2 = A 3 =0 Frequency ω 0 photon virtuality q 2 = ω 2 Magnetic flux is quantized ω = 2πN L, N Z\{0} 32 / 34

33 Energy shift For ν =0andq 2 < 0 S 1 (q 2 )isreal If V 0 = e 2 B 2 /2mω 2 is small perturbation theory The free energy spectrum: w(k n )= m 2 + k 2 n, k n = 2πn L, n Z3 Spin-averaged energy shift of the ground state (k n =0): δe = 1 2 s δe s = B 2 4m S 1( ω 2 ) A remarkably simple expression! 33 / 34

34 Outlook It is planned to test the formula on the lattice One can start with Compton scattering off pions The limit ω 0 should be thoroughly studied Disconnected contributions have to be estimated Finite-volume effects should be considered Work Ahead! 34 / 34