Hadronic Light-by-Light Scattering and Muon g 2: Dispersive Approach

Transcription

1 Hadronic Light-by-Light Scattering and Muon g 2: Dispersive Approach Peter Stoffer in collaboration with G. Colangelo, M. Hoferichter and M. Procura JHEP 09 (2015) 074 [arxiv: [hep-ph]] JHEP 09 (2014) 091 [arxiv: [hep-ph]] Helmholtz-Institut für Strahlen- und Kernphysik University of Bonn 15th June 2016 International School of Subnuclear Physics, Erice 1

2 Outline 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 2

3 Overview 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 3

4 1 Introduction Magnetic moment relation of spin and magnetic moment of a lepton: µ l = g l e 2m l s g l : Landé factor, gyromagnetic ratio Dirac s prediction: g e = 2 anomalous magnetic moment: a l = (g l 2)/2 helped to establish QED and QFT as the framework for elementary particle physics today: probing not only QED but entire SM 4

5 1 Introduction 286 T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 225 (g 2) µ : comparison of theory and experiment HMNT (06) JN (09) Davier et al, τ (10) Davier et al, e + e (10) JS (11) HLMNT (10) HLMNT (11) experiment BNL BNL (new from shift in λ) a µ Figure 6: World average for a µ from Hagiwara BNL compared et al to SM predictions from several groups. Figure 7: In precision fit

6 1 Introduction (g 2) µ : theory vs. experiment discrepancy between SM and experiment 3σ hint to new physics? new experiments (FNAL, J-PARC) aim at reducing the experimental error by a factor of 4 theory error completely dominated by hadronic effects 6

7 Overview 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 7

8 2 Hadronic vacuum polarisation Hadronic vacuum polarisation: O(α 2 ) had. problem: QCD is non-perturbative at low energies first principle calculations (lattice QCD) may become competitive in the future current evaluations based on dispersion relations and data, can be systematically improved 8

9 2 Hadronic vacuum polarisation Hadronic vacuum polarisation: O(α 2 ) Photon hadronic vacuum polarisation function: = i(q 2 g µν q µ q ν )Π(q 2 ) Unitarity of the S-matrix implies the optical theorem: ImΠ(s) = s e(s) 2 σ tot(e + e γ hadrons) 9

10 2 Hadronic vacuum polarisation Dispersion relation Causality implies analyticity: Im(s) Cauchy integral formula: R s0 γc γr Γ Re(s) Π(s) = 1 2πi γ Π(s ) s s ds Deform integration path: Π(s) Π(0) = s π 4M 2 π ImΠ(s ) (s s iɛ)s ds 10

11 Overview 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 11

12 3 Hadronic light-by-light scattering Hadronic light-by-light (HLbL) scattering up to now only model calculations uncertainty estimate based rather on consensus than on a systematic method will dominate theory error in a few years 12

13 3 Hadronic light-by-light scattering Dispersive approach to HLbL make use of fundamental principles: gauge invariance, crossing symmetry unitarity, analyticity relate HLbL to experimentally accessible quantities 13

14 3 Hadronic light-by-light scattering HLbL tensor: Lorentz decomposition Solution for the Lorentz decomposition: 54 Π µνλσ (q 1, q 2, q 3 ) = T µνλσ i Π i (s, t, u; qj 2 ) i=1 Lorentz structures manifestly gauge invariant scalar functions Π i free of kinematic singularities ideal quantities for a dispersive treatment: use analyticity properties 14

15 Overview 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 15

16 4 Mandelstam Representation Mandelstam representation we limit ourselves to intermediate states of at most two pions writing down a double-spectral (Mandelstam) representation allows us to split up the HLbL tensor: Π µνλσ = Π π0 -pole µνλσ + Π box µνλσ + Π µνλσ

17 4 Mandelstam Representation Mandelstam representation we limit ourselves to intermediate states of at most two pions writing down a double-spectral (Mandelstam) representation allows us to split up the HLbL tensor: Π µνλσ = Π π0 -pole µνλσ + Π box µνλσ + Π µνλσ +... one-pion intermediate state input: pion transition form factor 16

18 4 Mandelstam Representation Mandelstam representation we limit ourselves to intermediate states of at most two pions writing down a double-spectral (Mandelstam) representation allows us to split up the HLbL tensor: Π µνλσ = Π π0 -pole µνλσ + Π box µνλσ + Π µνλσ +... two-pion intermediate state in both channels input: pion vector form factor 16

19 4 Mandelstam Representation Mandelstam representation we limit ourselves to intermediate states of at most two pions writing down a double-spectral (Mandelstam) representation allows us to split up the HLbL tensor: Π µνλσ = Π π0 -pole µνλσ + Π box µνλσ + Π µνλσ +... two-pion intermediate state in first channel input: helicity partial waves for γ γ ππ 16

20 4 Mandelstam Representation Mandelstam representation we limit ourselves to intermediate states of at most two pions writing down a double-spectral (Mandelstam) representation allows us to split up the HLbL tensor: Π µνλσ = Π π0 -pole µνλσ + Π box µνλσ + Π µνλσ +... neglected so far: higher intermediate states 16

21 Overview 1 Introduction 2 Hadronic vacuum polarisation 3 Hadronic light-by-light scattering 4 Mandelstam Representation 5 Conclusion and Outlook 17

22 5 Conclusion and Outlook Summary our dispersive approach to HLbL scattering is based on fundamental principles: gauge invariance, crossing symmetry unitarity, analyticity we take into account the lowest intermediate states: π 0 -pole and ππ-cuts relation to experimentally accessible (or again with data dispersively reconstructed) quantities a step towards a model-independent calculation of a µ 18

23 19 Backup

24 6 Backup A roadmap for HLbL e + e e + e π 0 γπ ππ e + e π 0 γ ω, φ ππγ e + e ππγ Pion transition ( form factor F π 0 γ γ q 2 1, q2 2 ) ππ ππ Pion vector Partial waves for γ γ ππ e + e e + e ππ form factor F V π e + e 3π pion polarizabilities γπ γπ ω, φ 3π ω, φ π 0 γ Flowchart by M. Hoferichter 20

25 6 Backup Pion pole input: doubly-virtual and singly-virtual pion transition form factors F γ γ π 0 and F γ γπ 0 dispersive analysis of transition form factor: Hoferichter et al., EPJC 74 (2014)

26 6 Backup Pion box simultaneous two-pion cuts in two channels Mandelstam representation explicitly constructed Π i = 1 π 2 ds dt ρ st i (s, t ) (s s)(t + (t u) + (s u) t) q 2 -dependence: pion vector form factors F V π (q 2 i ) for each off-shell photon factor out 22

27 6 Backup Pion box sqed loop projected on BTT basis fulfils the same Mandelstam representation only difference are factors of Fπ V box topologies are identical to FsQED: Fπ V (q1)f 2 π V (q2)f 2 π V (q3) model-independent definition of pion loop

28 6 Backup Pion box Pion vector form factor in the space-like region: NA7 (1986) ETMC - quadratic fit ETMC - logarithmic fit Volmer et al. (Fpi coll.) (2001) Our fit VMD F ^ s (GeV 2 ) 24 Preliminary results: a π-box µ = π-box, VMD, aµ =

29 6 Backup Rescattering contribution neglect left-hand cut due to multi-particle intermediate states in crossed channel two-pion cut in only one channel expansion into partial waves 25

30 6 Backup Rescattering contribution unitarity relates it to the helicity amplitudes of the subprocess γ γ ( ) ππ dispersive integrals over the imaginary parts allow the reconstruction of Π µνλσ sum rules ensure cancellation of unphysical helicity amplitudes 26