Hadronic light-by-light scattering in the muon g 2

Transcription

1 Hadronic light-by-light scattering in the muon g 2 Andreas Nyffeler PRISMA Cluster of Excellence, Institut für Kernphysik, Helmholtz-Institut Mainz Johannes Gutenberg Universität Mainz, Germany nyffeler@kph.uni-mainz.de International Workshop on e + e collisions from Phi to Psi 27 (PhiPsi7) Schloss Waldthausen (near Mainz), Germany, June 27

2 Outline Introduction: Hadronic light-by-light scattering (HLbL) in the muon g 2 Current status: Model calculations Model-independent approaches:. HLbL from dispersion relations (data driven approach) (Theory Talks on Thursday by Danilkin, Kubis, Pauk, Colangelo) 2. HLbL from Lattice QCD (Talks on Thursday by Wittig, Lehner) Conclusions and Outlook

3 Hadronic light-by-light scattering HLbL in muon g 2 from strong interactions (QCD): γ π + π, η, η a had.lxl µ = = µ Coupling of photons to hadrons, e.g. π, via form factor: π γ Relevant scales ( VVVV with offshell photons): 2 GeV m µ (hadronic resonance region) View before 24: in contrast to HVP, no direct relation to experimental data size and even sign of HLbL contribution to a µ unknown! Approach: use hadronic model at low energies with exchanges and loops of resonances and some (dressed) quark-loop at high energies. Constrain models using experimental data (processes of hadrons with photons: decays, form factors, scattering) and theory (ChPT at low energies; short-distance constraints from pqcd / OPE at high momenta). Problems: Four-point function Π µνρσ(q, q 2, q 3 ) involves many Lorentz structures that depend on several invariant momenta distinction between low and high energies not as easy as for two-point function in HVP. Mixed regions: one loop momentum Q 2 large, the other Q2 2 small and vice versa. γ

4 HLbL in muon g 2 Frequently used estimates: aµ HLbL = (5 ± 26) (Prades, de Rafael, Vainshtein 9) ( Glasgow consensus ) aµ HLbL = (6 ± 39) (AN 9; Jegerlehner, AN 9) Based almost on same input: calculations by various groups using different models for individual contributions. Error estimates are mostly guesses! For comparison: a exp µ a SM µ 3 (3 4 σ) Need much better understanding of complicated hadronic dynamics to get reliable error estimate of ±2 (δa µ(future exp) = 6 ).

5 HLbL in muon g 2 Frequently used estimates: aµ HLbL = (5 ± 26) (Prades, de Rafael, Vainshtein 9) ( Glasgow consensus ) aµ HLbL = (6 ± 39) (AN 9; Jegerlehner, AN 9) Based almost on same input: calculations by various groups using different models for individual contributions. Error estimates are mostly guesses! For comparison: a exp µ a SM µ 3 (3 4 σ) Need much better understanding of complicated hadronic dynamics to get reliable error estimate of ±2 (δa µ(future exp) = 6 ). Recent new proposal: Colangelo et al. 4, 5; Pauk, Vanderhaeghen 4: use dispersion relations (DR) to connect contribution to HLbL from presumably numerically dominant light pseudoscalars to in principle measurable form factors and cross-sections (no data yet!): γ γ π, η, η ; π + π, π π Could connect HLbL uncertainty to exp. measurement errors, like HVP. Future: HLbL from Lattice QCD (model-independent, first-principle). First steps and results: Blum et al. (RBC-UKQCD) 5,..., 5, 6, 7. Work ongoing by Mainz group: Green et al. 5; Asmussen et al. 6, 7.

6 HLbL in muon g 2: model calculations (summary of selected results) µ (p ) µ (p) k k = p p = ρ π + π,, η, η + Exchange of other resonances + (f, a, f 2...) Q + ρ de Rafael 94: Chiral counting: p 4 p 6 p 8 p 8 N C -counting: N C N C N C

7 HLbL in muon g 2: model calculations (summary of selected results) µ (p ) k k = p p µ (p) de Rafael 94: = ρ π + π,, η, η + Exchange of other resonances + (f, a, f 2...) + ρ Chiral counting: p 4 p 6 p 8 p 8 N C -counting: N C N C N C Contribution to a µ : BPP: +83 (32) HKS: +9 (5) KN: +8 (4) MV: +36 (25) 27: + (4) PdRV:+5 (26) N,JN: +6 (39) -9 (3) -5 (8) () ud.: (9) -9 (3) +85 (3) +83 (6) +83 (2) +4 () +4 (3) +99 (6) ud.: + ud. = undressed, i.e. point vertices without form factors -4 (3) [f, a ] +.7 (.7) [a ] +22 (5) [a ] +8 (2) [f, a ] +5 (7) [f, a ] Q +2 (3) + () +2.3 [c-quark] +2 (3) ud.: +6 Recall (in units of ): δa µ(hvp) 3 4; δa µ(exp [BNL]) = 63; δa µ(future exp) = 6 Pseudoscalars π, η, η : numerically dominant contribution (according to most models!). Other contributions not negligible. Cancellation between (dressed) π-loop and (dressed) quark-loop! BPP = Bijnens, Pallante, Prades 96, 2; HKS = Hayakawa, Kinoshita, Sanda 96, 98, 2; KN = Knecht, AN 2; MV = Melnikov, Vainshtein 4; 27 = Bijnens, Prades; Miller, de Rafael, Roberts; PdRV = Prades, de Rafael, Vainshtein 9 (compilation; Glasgow consensus); N,JN = AN 9; Jegerlehner, AN 9 (compilation)

8 HLbL in muon g 2: model calculations (continued) Contribution BPP HKS, HK KN MV BP, MdRR PdRV N, JN π, η, η 85±3 82.7±6.4 83±2 4± 4±3 99 ± 6 axial vectors 2.5±..7±.7 22±5 5 ± 22 ± 5 scalars 6.8±2. 7±7 7±2 π, K loops 9±3 4.5±8. 9±9 9±3 π,k loops +subl. N C ± quark loops 2±3 9.7±. 2.3 (c-quark) 2±3 Total 83± ±5.4 8±4 36±25 ±4 5 ± 26 6 ± 39 BPP = Bijnens, Pallante, Prades 95, 96, 2; HKS = Hayakawa, Kinoshita, Sanda 95, 96; HK = Hayakawa, Kinoshita 98, 2; KN = Knecht, AN 2; MV = Melnikov, Vainshtein 4; BP = Bijnens, Prades 7; MdRR = Miller, de Rafael, Roberts 7; PdRV = Prades, de Rafael, Vainshtein 9; N = AN 9, JN = Jegerlehner, AN 9 PdRV (Glasgow consensus): Do not consider dressed light quark loops as separate contribution. Assume it is already taken into account by using short-distance constraint of MV 4 on pseudoscalar-pole contribution (no form factor at external vertex). Added all errors in quadrature. N, JN: New evaluation of pseudoscalar exchange contribution imposing new short-distance constraint on off-shell form factors. Took over most values from BPP, except axial vectors from MV. Added all errors linearly. Note that recent reevaluations of axial vector contribution lead to much smaller estimates than in MV: aµ HLbL;axial = (8 ± 3) (Pauk, Vanderhaeghen 4; Jegerlehner 4, 5). This would shift central values of compilations downwards: a HLbL µ = (98 ± 26) (PdRV) a HLbL µ = (2 ± 39) (N, JN)

9 Model calculations of HLbL: other recent developments First estimate for tensor mesons (Pauk, Vanderhaeghen 4): a HLbL;tensor µ = Dressed pion-loop Potentially important effect from pion polarizability and a resonance (Engel, Patel, Ramsey-Musolf 2; Engel 3; Engel, Ramsey-Musolf 3) Maybe large negative contribution, in contrast to BPP 96, HKS 96: a HLbL;π loop µ = ( 7) Not confirmed by recent reanalysis by Bijnens, Relefors 5, 6. Essentially get again old central value from BPP, but smaller error estimate: a HLbL;π loop µ = ( 2 ± 5) Hopefully new dispersive approaches can settle the issue without the use of models (Colangelo et al.; Danilkin, Pauk, Vanderhaeghen). Dressed quark-loop Dyson-Schwinger equation approach (Fischer, Goecke, Williams, 3) Large contribution, no damping seen, in contrast to BPP 96, HKS 96: a HLbL;quark loop µ = 7 (Incomplete calculation!) More general questions: How to get proper matching of models with perturbative QCD? How to avoid double-counting of dressed quark-loop with hadronic contributions?

10 Pion-pole contribution to a HLbL;π µ (analogously for η, η ) π, η, η a HLbL;π µ = where [Jegerlehner, AN 9] a HLbL;π () µ = a HLbL;π (2) µ = ( αe π ) ( ) 3 a HLbL;π () µ + a HLbL;π (2) µ dq dq 2 dτ w (Q, Q 2, τ) F π γ γ ( Q 2, (Q + Q 2 ) 2 ) F π γ γ ( Q 2 2, ) dq dq 2 dτ w 2 (Q, Q 2, τ) F π γ γ ( Q 2, Q2 2 ) F π γ γ ( (Q + Q 2 ) 2, ) w,2 (Q, Q 2, τ) are model-independent weight functions which are concentrated at small momenta [AN 6]. Multiply the double- and single-virtual pion transition form factors (TFF s). 3-dim. integration over lengths Q i = (Q i ) µ, i =, 2 of the two Euclidean momenta and angle θ between them: Q Q 2 = Q Q 2 cos θ with τ = cos θ. Note in arguments of form factors: (Q + Q 2 ) 2 Q 2 + 2Q Q 2 + Q 2 2 = Q2 + 2Q Q 2 τ + Q 2 2. Generalization to full HLbL tensor (Master formula from Colangelo et al. 5): aµ HLbL = 2α3 3π 2 dq dq 2 dτ 2 τ 2 Q 3 Q3 2 T i (Q, Q 2, τ) Π i (Q, Q 2, τ) i=

11 aµhlbl;p : relevant momentum regions and results from models w2 (Q, Q2, τ = ) w (Q, Q2, τ = ) Model-independent weight functions w,2 (Q, Q2, τ ) for π with θ = 9 (τ = ): Q [GeV].5.5 Q2 [GeV] Q [GeV] Q2 [GeV] 2 Model-independent weight functions w (Q, Q2, τ ) for η (left) and η (right):.25 w (Q, Q2, τ = ) w (Q, Q2, τ = ) Q [GeV].5.5 Q2 [GeV] 2.5 Q [GeV] Q2 [GeV] 2 Relevant momentum regions below GeV for π, below.5 2 GeV for η, η. w : behavior for large Q corresponds to OPE limit in first TFF /Q2. Most model calculations for light pseudoscalars (poles or exchanges) agree at level of 5%, but full range of estimates (central values) much larger: HLbL;π aµ HLbL;P aµ = = (5 8) (59 4) = = (65 ± 5) (87 ± 27) (±23%) (±3%) Hopefully soon DR approach to TFF s (Hoferichter et al. 4; Talk Kubis), data on γ γ π, η, η (AN 6; Talk Redmer) and lattice QCD calculations of TFF s (Ge rardin, Meyer, AN 6) can give precise and model-independent results.

12 Short-distance constraint from OPE on HLbL in g 2 Melnikov; Vainshtein 4, further explanations in Prades, de Rafael, Vainshtein 9. In HLbL contribution to g 2 consider OPE limit k 2 k2 2 k2 3 and k2 k2 2 m2 ρ: d 4 x d 4 x 2 e ik x ik 2 x 2 j ν(x ) j ρ(x 2 ) = 2ˆk ( ) ɛ 2 νρδγ ˆk δ d 4 z e ik3 z j γ 5 ˆk (z) + O 3 j γ 5 = q Q2 q qγγ γ 5 q is the axial current, ˆk = (k k 2 )/2 k k 2 k q q H γ γ 5 k k 2 3 k 3 Strong constraints from the AVV triangle diagram (non-renormalization theorems: Adler, Bardeen 69; t Hooft 8; Vainsthein 3; Knecht et al. 4). At large k 2, k2 2 the pseudoscalar and axial-vector exchanges dominate in HLbL. Constraints seem to imply that in pion-pole contribution there is no pion transition form factor at external vertex enhanced contribution. Saturation of this QCD short-distance constraint by pion-pole alone as suggested by Melnikov, Vainshtein 4 is, however, only a model ansatz! Only sum of all contributions to HLbL, i.e. exchanges and loops of resonances, has to match the QCD constraint from OPE / pqcd (global quark-hadron duality). See also Dorokhov, Broniowski 8; Greynat, de Rafael 2.

13 Data-driven approach to HLbL using dispersion relations Strategy: Split contributions to HLbL into two parts: I: Data-driven evaluation using DR (hopefully numerically dominant): () π, η, η poles (2) ππ intermediate state II: Model dependent evaluation (hopefully numerically subdominant): () Axial vectors (3π-intermediate state),... (2) Quark-loop, matching with pqcd Error goals: Part I: % precision (data driven), Part II: 3% precision. To achieve overall error of about 2% (δaµ HLbL = 2 ). Colangelo et al. 4, 5: Classify intermediate states in 4-point function. Then project onto g 2. Pauk, Vanderhaeghen 4: DR directly for Pauli FF F 2 (k 2 ). Colangelo et al. 7: pion-box contribution (middle diagram) using precise information on pion vector form factor and S-wave ππ-rescattering effects from pion-pole in left-hand cut (LHC) (part of right diagram): a π box µ = 5.9(2) ππ,π pole LHC aµ,j= = 8() Sum of the two = 24() HLbL sum rules to get constraints from experimental data or lattice QCD on models: Pascalutsa, Vanderhaeghen ; Pascalutsa, Pauk, Vanderhaeghen 2; Green et al. 5; Danilkin, Vanderhaeghen 7

14 Data-driven approach to HLbL using dispersion relations (continued) Intro HLbL: gauge & crossing HLbL dispersive Conclusions Hadronic light-by-light: a roadmap GC, Hoferichter, Kubis, Procura, Stoffer arxiv: (PLB 4) γπ ππ e + e π γ ω, φ ππγ e + e ππγ Pion transition ( form factor F π γ γ q 2, 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 γ Artwork by M. Hoferichter A reliable evaluation of the HLbL requires many different contributions by and a collaboration among theorists and experimentalists From talk by Colangelo at Radio Monte Carlo Meeting, Frascati, May 26

15 HLbL in muon g 2 from Lattice QCD: RBC-UKQCD approach Blum et al. 5,..., 5: First attempts: Put QCD + (quenched) QED on the lattice. Subtraction of lower-order in α HVP contribution needed, very noisy. Jin et al. 5, 6, 7: Step by step improvement of method to reduce statistical error by one or two orders of magnitude and remove some systematic errors. Calculate a HLbL µ = F 2 (q 2 = ) via moment method in position-space (no extrapolation to q 2 = needed). Use exact expression for all photon propagators. Treat r = x y stochastically by sampling points x, y. Found empirically: short-distance contribution at small r <.6 fm dominates. xsrc xop, ν z, κ y, σ x, ρ y, σ z, κ x, ρ Results (for m π = m π,phys, lattice spacing a =.73 GeV, L = 5.5 fm): a chlbl µ = (6. ± 9.6) (quark-connected diagrams) F2()/(α/π) xsnk xsrc z,κ xop,ν y,σ z,κ y,σ x,ρ a dhlbl µ = ( 62.5 ± 8.) (leading quark-disconnected diagrams) a HLbL µ = (53.5 ± 3.5) Beware! Statistical error only! Missing systematic effects: Expect large finite-volume effects from QED /L 2. Blum et al. 7: use infinite volume, continuum QED (like Mainz approach: Asmussen et al. 6). Expect large finite-lattice-spacing effects. Omitted subleading quark-disconneced diagrams (% effect?). r 48I x,ρ xsnk

16 HLbL in muon g 2 from Lattice QCD: Mainz approach Developed independently (Asmussen, Green, Meyer, AN 5, 6, 7) Master formula in position-space: aµ HLbL = me6 3 i Π ρ;µνλσ (x, y) = d y[ 4 d 4 x L [ρ,σ];µνλ (x, y) }{{} QED i Π ρ;µνλσ (x, y) }{{} QCD d 4 z z ρ j µ(x) j ν(y) j σ(z) j λ () ] z x y Semi-analytical calculation of QED kernel. QED kernel computed in continuum and in infinite volume (Lorentz covariance manifest, no power law /L 2 finite-volume effects). Kernel parametrized by 6 weight functions (and derivatives thereof), calculated on 3D grid in x, y, x y and stored on disk. Test of QED kernel function: pion-pole contribution, lepton loop. Result for pion-pole contribution with VMD model agrees with known results for m π > 3 MeV. Need already rather large lattice size > 4 fm for smaller pion masses. Analytical result for lepton loop (QED) with m loop = m µ, 2m µ is reproduced at the percent level. 8 f( y ) Integrand of lepton loop contribution to a µ HLbL m l = m µ m l =2 m µ m µ y y = 7.5 fm

17 Conclusions and Outlook a µ: Test of Standard Model, potential window to New Physics. Current situation: a exp µ a SM µ = (39 ± 82) [3.8 σ] Hadronic effects? Sign of New Physics? Frequently-used model-estimates for HLbL (updated axial-vectors): a HLbL µ = (98 ± 26) (PdRV ( Glasgow consensus )) a HLbL µ = (2 ± 39) (N, JN) Two new g 2 experiments at Fermilab (E989) and J-PARC (E34) with goal of δa exp µ = 6 (factor 4 improvement) Theory for HLbL needs to match this precision! Concerted effort needed of experiments (measuring processes with hadrons and photons), phenomenology / theory (data-driven using dispersion relations, matching with QCD short-distance constraints and modelling) and lattice QCD to improve HLbL estimate with reliable uncertainty. Muon g 2 Theory Initiative (Working Group on HLbL) (Talk El-Khadra) First Workshop recently near Fermilab, more to come in future.

18 Backup slides

19 Muon g 2: current status Contribution a µ Reference QED (leptons) ±.36 Aoyama et al. 2 Electroweak 53.6 ±. Gnendiger et al. 3 HVP: LO ± 35.2 Jegerlehner 5 NLO ±. Jegerlehner 5 NNLO 2.4 ±. Kurz et al. 4 HLbL 2 ± 39 Jegerlehner 5 (JN 9) NLO 3 ± 2 Colangelo et al. 4 Theory (SM) ± 53 Experiment ± 63 Bennett et al. 6 Experiment - Theory 39 ± σ Discrepancy a sign of New Physics? Hadronic uncertainties need to be better controlled in order to fully profit from future g 2 experiments at Fermilab (E989) and J-PARC (E34) with δa µ = 6.

20 Numerical test of QED kernel: Pion-pole contribution to a HLbL µ VMD model for pion transition form factor for illustration. Result for arbitrary pion mass can easily be obtained from 3-dimensional momentum-space representation (Jegerlehner + AN 9). 3-dim. integration in position space: y 2π2 d y y 3 x 4π d x x 3 π dβ sin2 β (cutoff for x integration: x max = 4.5 fm) Integrand after integration over x, β: Result for a HLbL µ ( y max ): 4 m π = 3 MeV m π = 6 MeV m π = 9 MeV 3 d d y ahlbl µ y max /fm a Hlbl µ ( y max ) y max /fm m π = 3 MeV m π = 6 MeV m π = 9 MeV All 6 weight functions contribute to final result, some only at the percent level. x max, y max > 4 fm needed for m π < 3 MeV. For the physical pion mass, one needs to go to very large values of x and y, i.e. very large lattice volumes, to reproduce known result of 5.7.