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

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:1506.01386 [hep-ph]] JHEP 09 (2014) 091 [arxiv:1402.7081 [hep-ph]] Helmholtz-Institut für Strahlen- und Kernphysik University of Bonn 18th May 2016 Symposium on EFT and LGT, TUM Institute for Advanced Study 1

Outline 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 2

Overview 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 3

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) 4 170 180 190 200 210 a µ 10 10 11659000 Figure 6: World average for a µ from Hagiwara BNL compared et al. 2012 to SM predictions from several groups. Figure 7: In precision fit

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 hadronic vacuum polarisation responsible for largest uncertainty, but will be systematically improved with better data input 5

1 Introduction 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 6

are shown in Table 13. We have also included some guesstimates for the total value. Note that the number a LbL;had µ =(80± 40) 10 11 written in the fourth column in Table 13 under the heading KN was actually not given in Ref. [17], but represents estimates used mainly by the Marseille group before the appearance of the paper by MV [257]. Furthermore, we have included in the sixth column the estimate a LbL;had µ = (110 ± 40) 10 Model 11 given recently in Refs. [298,41,43]. Note that PdRV [294] (seventh column) do not include the dressed light quark loops as a separate contribution. They assume them to be already covered by using calculations of HLbL the short-distance constraint from MV [257] on the pseudoscalar-pole contribution. PdRV add, however, a small contribution from the bare c-quark loop. 1 Introduction Table 13 Summary of the most recent results for the various contributions to a LbL;had µ 10 11.Thelastcolumnisourestimatebasedon our new evaluation for the pseudoscalars and some of the other results. Contribution BPP HKS KN MV BP PdRV N/JN π 0, η, η 85±13 82.7±6.4 83±12 114±10 114±13 99±16 π,k loops 19±13 4.5±8.1 19±19 19±13 π,k loops + other subleading in Nc 0±10 axial vectors 2.5±1.0 1.7±1.7 22± 5 15±10 22± 5 scalars 6.8±2.0 7± 7 7± 2 quark loops 21± 3 9.7±11.1 2.3 21± 3 total 83±32 89.6±15.4 80±40 136±25 110±40 105±26 116±39 7 Jegerlehner, Nyffeler 2009 77 pseudoscalar pole contribution most important pion-loop second most important differences between models, large uncertainties

1 Introduction How to improve HLbL calculation? lattice QCD making progress dispersive approach 8

1 Introduction Dispersive approach to HLbL make use of fundamental principles: gauge invariance, crossing symmetry unitarity, analyticity relate HLbL to experimentally accessible quantities 9

Overview 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 10

11 2 Lorentz Structure of the HLbL Tensor The HLbL tensor: definitions hadronic four-point function: Π µνλσ (q 1, q 2, q 3 ) = i dxdydze i(q1x+q2y+q3z) 0 T j em(x)j µ em(y)j ν em(z)j λ em(0) 0 σ EM current: j em µ = Q i q i γ µ q i i=u,d,s Mandelstam variables: s = (q 1 + q 2 ) 2, t = (q 1 + q 3 ) 2, u = (q 2 + q 3 ) 2 for (g 2) µ, the external photon is on-shell: q4 2 = 0, where q 4 = q 1 + q 2 + q 3

2 Lorentz Structure of the HLbL Tensor The HLbL tensor a priori 138 naive Lorentz structures: Π µνλσ = g µν g λσ Π 1 + g µλ g νσ Π 2 + g µσ g νλ Π 3 + q µ i qν j qkq λ l σ Π 4 ijkl i,k,l,m + g λσ q µ i qν j Π 5 ij +... i,j in 4 space-time dimensions: 2 linear relations among the 138 Lorentz structures Eichmann et al., 2014 six dynamical variables, e.g. two Mandelstam variables s, t and the photon virtualities q1, 2 q2, 2 q3, 2 q4 2 12

2 Lorentz Structure of the HLbL Tensor HLbL tensor: gauge invariance Ward identities {q µ 1, q ν 2, q λ 3, q σ 4 }Π µνλσ = 0 imply 95 linear relations between scalar functions Π i off-shell basis: 138 95 2 = 41 structures corresponding to 41 helicity amplitudes relations between Π i imply kinematic zeros 13

2 Lorentz Structure of the HLbL Tensor HLbL tensor: Lorentz decomposition Problem: find a decomposition Π µνλσ (q 1, q 2, q 3 ) = i T µνλσ i Π i (s, t, u; q 2 j ) with the following properties: Lorentz structures T µνλσ i manifestly gauge invariant: {q µ 1, q ν 2, q λ 3, q σ 4 }T i µνλσ = 0 scalar functions Π i free of kinematic singularities and zeros 14

2 Lorentz Structure of the HLbL Tensor HLbL tensor: Lorentz decomposition Recipe by Bardeen, Tung (1968) and Tarrach (1975): construct gauge projectors: I µν 12 = g µν qµ 2 q ν 1 q 1 q 2, I λσ 34 = g λσ qλ 4 q σ 3 q 3 q 4 gauge invariant themselves, e.g. q µ 1 I 12 µν = 0 leave HLbL tensor invariant, e.g. I µµ 12 Π µ νλσ = Π µ νλσ 15

2 Lorentz Structure of the HLbL Tensor HLbL tensor: Lorentz decomposition Following Bardeen, Tung (1968): apply gauge projectors to the 138 initial structures: 95 immediately project to 0 remove 1/q 1 q 2 and 1/q 3 q 4 poles by taking appropriate linear combinations BT basis: degenerate in the limits q 1 q 2 0, q 3 q 4 0 16

2 Lorentz Structure of the HLbL Tensor HLbL tensor: Lorentz decomposition According to Tarrach (1975): degeneracies in the limits q 1 q 2 0, q 3 q 4 0: k c i kt µνλσ k = q 1 q 2 X µνλσ i + q 3 q 4 Y µνλσ i extend basis by additional structures X µνλσ i, Y µνλσ i taking care of remaining kinematic singularities equivalent: implementing crossing symmetry 17

2 Lorentz Structure of the HLbL Tensor HLbL tensor: Lorentz decomposition Solution for the Lorentz decomposition: Π µνλσ (q 1, q 2, q 3 ) = 54 i=1 T µνλσ i Π i (s, t, u; q 2 j ) Lorentz structures manifestly gauge invariant crossing symmetry manifest: only 7 distinct structures, 47 follow from crossing scalar functions Π i free of kinematic singularities ideal quantities for a dispersive treatment 18

Overview 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 19

3 Master Formula for (g 2) µ Master formula: contribution to (g 2) µ from gauge invariance: Π µνλρ = q4 σ q ρ Π µνλσ 4 for (g 2) µ : afterwards take q 4 0 no kinematic singularities in scalar functions: perform these steps with the derived Lorentz decomposition only 12 linear combinations of the scalar functions Π i contribute to (g 2) µ 20

3 Master Formula for (g 2) µ Master formula: contribution to (g 2) µ d a HLbL µ = e 6 4 q 1 d 4 q 2 (2π) 4 (2π) 4 12 i=1 ˆT i (q 1, q 2 ; p)ˆπ i (q 1, q 2, q 1 q 2 ) q 2 1q 2 2(q 1 + q 2 ) 2 [(p + q 1 ) 2 m 2 µ][(p q 2 ) 2 m 2 µ] ˆTi : known integration kernel functions five loop integrals can be performed with Gegenbauer polynomial techniques Wick rotation possible even in the presence of anomalous thresholds 21

3 Master Formula for (g 2) µ Master formula: contribution to (g 2) µ a HLbL µ = 2α3 3π 2 0 1 dq 1 dq 2 dτ 1 τ 2 Q 3 1Q 3 2 0 12 i=1 1 T i (Q 1, Q 2, τ) Π i (Q 1, Q 2, τ), T i : known integration kernels Πi : linear combinations of the scalar functions Π i Euclidean momenta: Q 2 i = qi 2 Q 2 3 = Q 2 1 + Q 2 2 + 2Q 1 Q 2 τ 22

Overview 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 23

4 Mandelstam Representation Analytic properties of scalar functions right- and left-hand cuts in each Mandelstam variable double-spectral regions (box topologies) anomalous thresholds for large photon virtualities 24

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 +... 25

4 Mandelstam Representation 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) 3180 26

4 Mandelstam Representation 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 27

4 Mandelstam Representation 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) 2 + + 28 model-independent definition of pion loop

4 Mandelstam Representation Pion box Very simple expressions for box contributions in terms of Feynman parameter integrals Π π-box with e.g. i (q 2 1, q 2 2, q 2 3) = F V π (q 2 1)F V π (q 2 2)F V π (q 2 3) I 7 (x, y) = 4 3 1 16π 2 1 0 dx 1 x (1 2x) 2 (1 2y) 2 y(1 y), 3 123 0 dy I i (x, y), ijk = M 2 π xyq 2 i x(1 x y)q 2 j y(1 x y)q 2 k. 29

4 Mandelstam Representation Pion box Pion vector form factor in the space-like region: 1 0.9 0.8 0.7 0.6 NA7 (1986) ETMC - quadratic fit ETMC - logarithmic fit Volmer et al. (Fpi coll.) (2001) Our fit VMD F ^2 0.5 0.4 0.3 0.2 0.1 0-1.6-1.5-1.4-1.3-1.2-1.1-1 -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 s (GeV 2 ) 30 Preliminary results: a π-box µ = 15.9 10 11 π-box, VMD, aµ = 16.4 10 11

4 Mandelstam Representation Pion-box saturation with photon virtualities Pion-box saturation in % 100 90 80 70 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 cutoff on the virtualities in GeV 31

4 Mandelstam Representation 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 32

4 Mandelstam Representation 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 33

4 Mandelstam Representation The subprocess Helicity amplitudes for γ γ ππ: dispersive solution as Roy-Steiner equations γγ ππ: Moussallam 2010, Hoferichter, Phillips, Schat 2011 γ γ ππ: Moussallam 2013 γ γ ππ: work in progress Hoferichter, Colangelo, Procura, PS 2013 34

Overview 1 Introduction 2 Lorentz Structure of the HLbL Tensor 3 Master Formula for (g 2) µ 4 Mandelstam Representation 5 Conclusion and Outlook 35

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 36

5 Conclusion and Outlook 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 37

38 Backup

6 Backup Wick Rotation and Anomalous Threshold Wick rotation Trajectory of triangle anomalous threshold: q 2 2 Im(s) Re(s) q 2 2 = 0 q 2 2 = 4m 2 q 2 2 0 < q 2 1 < 4m 2 39

6 Backup Wick Rotation and Anomalous Threshold Wick rotation Trajectory of triangle anomalous threshold: Im(s) q 2 2 q 2 2 = 4m 2 q 2 2 = 0 q 2 2 Re(s) 4m 2 < q 2 1 40