HADRONIC LIGHT-BY-LIGHT FOR THE MUON ANOMALOUS MAGNETIC MOMENT

Transcription

1 1/56 HADRONIC LIGHT-BY-LIGHT FOR THE MUON ANOMALOUS MAGNETIC MOMENT Lund University bijnens bijnens/chpt.html Uppsala 31 October 2012

2 Overview 1 2 QED Electroweak Lowest order hadronic HO hadronic /56

3 Literature 3/56 Final experimental paper: G. W. Bennett et al. [Muon G-2 Collaboration], Final Report of the Muon E821 Anomalous Magnetic Moment at BNL, Phys. Rev. D 73 (2006) [hep-ex/ ]. Review 1: F. J. M. Farley and Y. K. Semertzidis, The 47 years of muon g-2, Prog. Part. Nucl. Phys. 52 (2004). Review 2: J. P. Miller, E. de Rafael and B. L. Roberts, Muon (g-2): Experiment and theory, Rept. Prog. Phys. 70 (2007) 795 [hep-ph/ ]. Review 3: F. Jegerlehner and A. Nyffeler, The Muon g-2, Phys. Rept. 477 (2009) 1 [arxiv: [hep-ph]].

4 Literature 4/56 Lectures: M. Knecht, Lect. Notes Phys. 629 (2004) 37 [hep-ph/ ]. Final HLBL number: J. Bijnens and J. Prades, Mod. Phys. Lett. A 22 (2007) 767 [hep-ph/ ]. J. Prades, E. de Rafael and A. Vainshtein, Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment, (Advanced series on directions in high energy physics. 20) [arxiv: [hep-ph]].

5 Muon g 2: measurement 5/56

6 Muon g 2: measurement 6/56

7 Muon g 2: measurement 7/56 Phys.Rev. D73 (2006) [hep-ex/ ]

8 Muon g 2: measurement 8/56 Phys.Rev. D73 (2006) [hep-ex/ ] 2001: µ, others µ +

9 Muon g 2: overview 9/56 in terms of the anomaly a µ = (g 2)/2 Data dominated by BNL E821 (statistics)(systematic) a exp µ = (6)(5) a exp = (8)(3) µ aµ exp = (5.4)(3.3) Theory is off somewhat (electroweak)(lo had)(ho had) aµ SM = (0.2)(4.2)(2.6) a µ = aµ exp aµ SM = 28.7(6.3)(4.9) (PDG) E821 goes to Fermilab, expect factor of four in precision Note: g agrees to 10 8 with theory Many BSM models CAN predict a value in this range (often a lot more or a lot less) Numbers taken from PDG2012, see references there

10 Muon g 2: QED 10/56 a QED µ = α 2π (27)( α π) (43) ( α π (80) ( α π) (20) ( α π) 5 + First three loops known analytically four-loops fully done numerically Five loops estimate Kinoshita, Laporta, Remiddi, Schwinger,... α fixed from the electron g 2: α = 1/ (51) a QED µ = (0.015) surprisingly large: ) 3 QED Electroweak Lowest order hadronic HO hadronic e = 20.95,µ = 0.37,τ = 0.002

11 Muon g 2: Electroweak 11/56 W ν W e,µ,τ,u,d,s,c,t,b a EW µ [1 loop] = a EW µ [2 loop] = 4.07(0.10)(0.18) a EW µ = 15.4(0.1)(0.2) (triangle)(higgs mass) 2-loop large because of the large logarithms in the triangle anomaly present here SM anomaly cancellation very needed anomaly as part of the anomaly Z QED Electroweak Lowest order hadronic HO hadronic

12 Muon g 2: LO hadronic 12/56 = aµ LOhad = 1 ( α ) 2 K(s) 3 π m ds π 2 s R (0) (s) R (0) (s) bare cross-section ratio σ(e+ e hadrons) σ(e + e µ + µ ) Bare, many different evaluations,... e + e versus τ-decays a LOHad µ = 692.3(4.2)(0.3) (exp)(pert. QCD) QED Electroweak Lowest order hadronic HO hadronic

13 LO hadronic 13/56 QED Electroweak Lowest order hadronic HO hadronic Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ]

14 LO hadronic 14/56 QED Electroweak Lowest order hadronic HO hadronic Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ]

15 LO hadronic 15/ GeV ρ, ω ρ, ω 0.0 GeV, Υ ψ 9.5 GeV 3.1 GeV φ, GeV φ, GeV Contribution Error Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ] Direct measurements 0.0 GeV, 3.1 GeV 2.0 GeV QED Electroweak Lowest order hadronic HO hadronic Radiative return: radiate hard photon to get lower E cms Radiative corrections τ-decay: isospin breaking can be improved experimentally

16 A comment on models 16/56 a LOHad µ = 692.3(4.2) ds Rewrite: aµ LOHad = 8α 2 s K(s)1 π ImΠ(s) 1 Rho: π ImΠ(s) = 2 3 f V 2 M2 V δ(s M2 V )+ 2 N c 312π 2θ(s s 0) K(s) m2 µ 3t (s mµ 2 ) aµ LOHad = (f 16 9 α2 V 2 mµ 2 MV 2 + N c 12π 2 m 2 µ s 0 Put in f V 0.17, M V 0.77 GeV and s GeV 2 a LOHad µ ( ) Many models have this within 30% ) QED Electroweak Lowest order hadronic HO hadronic

17 Muon g 2: HO hadronic 17/56 Two main types of HO HVP HO HVP is like LO Had but a more complicated function K(s) aho HVP µ = 9.84(0.06) is the real problem: best estimate now: aµ = 10.5(2.6) Note that the sum is very small: but not an indication of the error QED Electroweak Lowest order hadronic HO hadronic

18 of Muon g 2 18/ a µ exp theory QED EW LO Had HO HVP difference Error on LO had all e + e based OK τ based 2 σ Error on Errors added quadratically 3.5 σ Generic SUSY: ( 100 GeV M SUSY ) 2tanβ M SUSY 66 GeV tanβ Difference: 4% of LO Had 270% of 1% of leptonic LbL QED Electroweak Lowest order hadronic HO hadronic

19 Our object 19/56 p 3 β p 2 α p 1 ν q ρ p p p 4 p 5 Muon line and photons: well known The blob: fill in with hadrons/qcd Trouble: low and high energy very mixed Double counting needs to be avoided: hadron exchanges versus quarks

20 A separation proposal: a start E. de Rafael, to the muon g-2 and low-energy QCD, Phys. Lett. B322 (1994) [hep-ph/ ]. Use ChPT p counting and large N c p 4, order 1: pion-loop p 8, order N c : quark-loop and heavier meson exchanges p 6, order N c : pion exchange Does not fully solve the problem only short-distance part of quark-loop is really p 8 but it s a start 20/56

21 A separation proposal: a start E. de Rafael, to the muon g-2 and low-energy QCD, Phys. Lett. B322 (1994) [hep-ph/ ]. Use ChPT p counting and large N c p 4, order 1: pion-loop p 8, order N c : quark-loop and heavier meson exchanges p 6, order N c : pion exchange Implemented by two groups in the 1990s: Hayakawa, Kinoshita, Sanda: meson models, pion loop using hidden local symmetry, quark-loop with VMD, calculation in Minkowski space JB, Pallante, Prades: Try using as much as possible a consistent model-approach, calculation in Euclidean space 20/56

22 Papers: BPP and HKS JB, E. Pallante and J. Prades Comment on the pion pole part of the light-by-light contribution to the muon g-2, Nucl. Phys. B 626 (2002) 410 [arxiv:hep-ph/ ]. Analysis of the Light-by-Light Contributions to the Muon g 2, Nucl. Phys. B 474 (1996) 379 [arxiv:hep-ph/ ]. light by light to the muon g-2 in the large N c limit, Phys. Rev. Lett. 75 (1995) 1447 [Erratum-ibid. 75 (1995) 3781] [arxiv:hep-ph/ ]. Hayakawa, Kinoshita, (Sanda) Pseudo pole terms in the hadronic light by light scattering contribution to muon g - 2, Phys. Rev. D57 (1998) [hep-ph/ ], Erratum-ibid.D66 (2002) [hep-ph/ ]. light by light scattering contribution to muon g-2, Phys. Rev. D54 (1996) [hep-ph/ ]. light by light scattering effect on muon g-2, Phys. Rev. Lett. 75 (1995) [hep-ph/ ]. 21/56

23 Differences 22/56 HK(S) Purely hadronic exchanges quark-loop with hadronic VMD Studied dependence of everything on m V BPP Use the as an overall model to have a similar uncertainty on all low-energy parts repair some of the worst short-comings Add the short-distance quark-loop Study of cut-off dependence Sign mistake HKS: Euclidean versus Minkowski ε µναβ BPP: notes all correct sign, program had wrong sign, probably minus sign from fermion loop not removed

24 Differences 22/56 HK(S) Purely hadronic exchanges quark-loop with hadronic VMD Studied dependence of everything on m V BPP Use the as an overall model to have a similar uncertainty on all low-energy parts repair some of the worst short-comings Add the short-distance quark-loop Study of cut-off dependence Sign mistake HKS: Euclidean versus Minkowski ε µναβ BPP: notes all correct sign, program had wrong sign, probably minus sign from fermion loop not removed

25 The overall 23/56 aµ = 1 tr[( p +m µ )M λβ (0)( p +m µ )[γ λ,γ β ]]. 48m µ d M λβ (p 3 ) = e 6 4 p 1 d 4 p 2 1 (2π) 4 (2π) 4 q 2 p1 2p2 2 (p2 4 m2 µ )(p2 5 m2 µ [ ) δπ ρναβ ] (p 1,p 2,p 3 ) γ α ( p 4 +m µ )γ ν ( p 5 +m µ )γ ρ. δp 3λ We used: Π ρναλ δπ ρναβ (p 1,p 2,p 3 ) (p 1,p 2,p 3 ) = p 3β. δp 3λ Can calculate at p 3 = 0 but must take derivative derivative in quark-loop: each permutation finite Four point function of V µ i (x) i Q i [ q i (x)γ µ q i (x)] Π ρναβ (p 1,p 2,p 3 ) i 3 d 4 x d 4 y d 4 z e i(p1 x+p2 y+p3 z) ) 0 T (V a ρ (0)Vν b (x)vα c (y)vβd (z) 0

26 24/56 Π ρναβ (p 1,p 2,p 3 ): In general 138 Lorentz structures (but only 32 contribute to g 2) Using q ρ Π ρναβ = p 1ν Π ρναβ = p 2α Π ρναβ = p 3β Π ρναβ = 0 43 gauge invariant structures Bose symmetry relates some of them All depend on p 2 1, p2 2 and q2, but before derivative and p 3 0 also p 2 3,p 1 p 2,p 1 p 3,p 2 p 3 Compare HVP: one function, one variable calculation from experiment difficult to see how In four photon measurement: lepton contribution

27 25/56 d 4 p 1 d 4 p 2 plus loops inside the hadronic part (2π) 4 (2π) 4 8 dimensional integral, three trivial, 5 remain: p 2 1,p2 2,p 1 p 2,p 1 p µ,p 2 p µ Rotate to Euclidean space: Easier separation of long and short-distance Artefacts (confinement) in models smeared out. More recent: can do two more using Gegenbauer techniques Knecht-Nyffeler, Jegerlehner-Nyffeler,JB Zahiri-Abyaneh P 2 1, P2 2 and Q2 remain study a X µ = dl P1 dl P2 a XLL µ = dl P1 dl P2 dl Q a XLLQ µ l P = ln(p/gev), to see where the are

28 25/56 d 4 p 1 d 4 p 2 plus loops inside the hadronic part (2π) 4 (2π) 4 8 dimensional integral, three trivial, 5 remain: p 2 1,p2 2,p 1 p 2,p 1 p µ,p 2 p µ Rotate to Euclidean space: Easier separation of long and short-distance Artefacts (confinement) in models smeared out. More recent: can do two more using Gegenbauer techniques Knecht-Nyffeler, Jegerlehner-Nyffeler,JB Zahiri-Abyaneh P 2 1, P2 2 and Q2 remain study a X µ = dl P1 dl P2 a XLL µ = dl P1 dl P2 dl Q a XLLQ µ l P = ln(p/gev), to see where the are

29 : our main model L = q α {iγ µ ( µ iv µ ia µ γ 5 ) (M+s ipγ 5 )}q α )( ) +2g S (q α Rq β L q β L qα R )( ) ( )( )] g V [(q α Lγ µ q β L q β L γ µql α + q α Rγ µ q β R q β R γ µqr α q ( u,d,s ) v µ, a µ, s, p: external vector, axial-vector, and pseudo matrix sources M is the quark-mass matrix. g V 8π2 G V (Λ) N cλ 2, g S 4π2 G S (Λ) N cλ 2. G V, G S are dimensionless and valid up to Λ No confinement but has good pion, vector meson and OK axial vector-meson phenomenology 26/56

30 : our main model 27/56 (this) JB, Bruno, de Rafael, Nucl. Phys. B390 (1993) 501 [hep-ph/ ]; JB, Phys. Rep. 265 (1996) 369 [hep-ph/ ] (review) Gap equation: chiral symmetry spontaneously broken = + Generates poles, i.e. mesons via bubble resummation 2

31 : our main model 28/56 Can be thought of as a very simple rainbow and ladder approximation in the DSE equation with constant kernels for the one-gluon exchange Parameters fit via F π, L r i, vector meson,... G S = 1.216, G V = 1.263, Λ = 1.16 GeV has M Q = 263 MeV Has a number of decent matchings to short-distance, e.g. Π V Π A but fails in others. Generates always VMD in external legs (but with a twist) Hook together general processes by one-loop vertices and bubble-chain propagators

32 Separation of 29/56 p α 2 p 3 β qρ Quark loop with external bubble-chains Quark-loop with VMD α p 2 (a) β p 3 r p 1 ν qρ Also internal bubble chain meson exchange Note that vertices have structure p 1 ν Off-shell effect in model included (b)

33 30/56 π 0 exchange π 0 π 0 = 1/(p 2 m 2 π ) The blobs need to be modelled, and in e.g. contain corrections also to the 1/(p 2 m 2 π) Pointlike has a logarithmic divergence Numbers π 0, but also η,η

34 π 0 exchange 31/56 a µ Cutoff Pointlike Transverse CELLO- (GeV) Point-like VMD VMD VMD VMD (2) 3.29(2) 3.46(2) 3.60(3) 3.53(2) (4) 4.24(4) 4.49(3) 4.73(4) 4.57(4) (7) 4.90(5) 5.18(3) 5.61(6) 5.29(5) (2) 5.63(8) 5.62(5) 6.39(9) 5.89(8) (5) 6.22(17) 5.58(5) 6.59(16) 6.02(10) BPP: All in reasonable agreement a π0 µ =

35 π 0 exchange BPP: a π0 µ = 5.9(0.9) Nonlocal quark model: a π0 µ = A. E. Dorokhov, W. Broniowski, Phys. Rev. D78 (2008) [ ]] DSE model: a π0 µ = T. Goecke, C. S. Fischer and R. Williams, Phys. Rev. D 83 (2011) [arxiv: [hep-ph]] LMD+V: a π0 µ = ( ) M. Knecht, A. Nyffeler, Phys. Rev. D65(2002)073034, [hep-ph/ ] Formfactor inspired by AdS/QCD: a π0 µ = L. Cappiello, O. Cata and G. D Ambrosio, Phys. Rev. D 83 (2011) [arxiv: [hep-ph]] Constraint via magnetic susceptibility: a π0 µ = A. Nyffeler, Phys. Rev. D 79 (2009) [arxiv: [hep-ph]]. All except possibly last in good agreement 32/56

36 33/56 MV short-distance: π 0 exchange K. Melnikov, A. Vainshtein, light-by-light scattering contribution to the muon anomalous magnetic moment revisited, Phys. Rev. D70 (2004) [hep-ph/ ] take P 2 1 P2 2 Q2 : Leading term in OPE of two vector currents is proportional to axial current Π ρναβ Pρ 0 T (J P1 2 Aν J Vα J Vβ ) 0 J A comes from + AVV triangle anomaly: extra info Implemented via setting one blob = 1 π 0 = π 0 a π0 µ =

37 34/56 π 0 exchange The pointlike vertex implements shortdistance part, not only + Are these part of the quark-loop? See also in Dorokhov,Broniowski, Phys.Rev. D78(2008)07301 BPP quarkloop + MV

38 35/56 π 0 exchange Which momentum regimes important studied: JB and J. Prades, Mod. Phys. Lett. A 22 (2007) 767 [hep-ph/ ] a µ = dl 1 dl 2 a LL µ with l i = log(p i /GeV) LL a µ (VMD) 7e-10 6e-10 5e-10 4e-10 3e-10 2e-10 1e P P 1 LL a µ (KN) 7e-10 6e-10 5e-10 4e-10 3e-10 2e-10 1e P P 1 Checking which momentum regions do what (but would need three dimensional)

39 Pseudo exchange 36/56 Point-like VMD: π 0 η and η give 5.58, 1.38, Models that include U(1) A breaking give similar ratios Pure large N c models use this ratio The MV argument should give some enhancement over the full VMD like models Total pseudo- exchange is about a PS µ = AdS/QCD estimate (includes excited pseudo-s) a PS µ = D. K. Hong and D. Kim, Phys. Lett. B 680 (2009) 480 [arxiv: [hep-ph]]

40 Axial-vector exchange exchange 37/56 Cut-off a µ from Λ Axial-Vector (GeV) Exchange O(N c ) (0.01) (0.01) (0.01) (0.02) (0.07) There is some pseudo- exchange piece here as well, off-shell not quite clear what is what. a axial µ = MV: short distance enhancement + mixing (both enhance about the same) a axial µ =

41 Pure quark loop 38/56 Cut-off a µ 10 7 a µ 10 9 a µ 10 9 Λ Electron Muon Constituent Quark (GeV) Loop Loop Loop (8) 2.41(3) 0.395(4) (10) 3.09(7) 0.705(9) (7) 3.76(9) 1.10(2) (6) 4.54(9) 1.81(5) (9) 4.60(11) 2.27(7) (6) 4.84(13) 2.58(7) Known Results (4) (16) M Q : 300 MeV now known fully analytically Us: 5+(3-1) integrals extra are Feynman parameters Slow convergence: electron: all at 500 MeV Muon: only half at 500 MeV, at 1 GeV still 20% missing 300 MeV quark: at 2 GeV still 25% missing

42 Pure quark loop: momentum area 39/ Q This plots aµ ql = dl P1 dl P2 dl Q aµ LLQ Succeeded in 3D plot but was useless JB-Zahiri-Abyaneh, work in progress P P 1 7e-10 6e-10 5e-10 4e-10 3e-10 2e-10 1e-10 0

43 Pure quark loop: momentum area 40/56 quark loop m Q = 0.3 GeV P 2 = P 1 P 2 = P 1 /2 P 2 = P 1 /4 P 2 = P 1 /8 7e-10 6e-10 5e-10 4e-10 3e-10 2e-10 1e Q Most from P 1 P 2 Q, sizable large momentum part P 1

44 quark-loop 41/56 Cut-off a µ a µ a µ a µ Λ sum GeV VMD masscut +masscut Very stable cuts off slower than pure VMD masscut: M Q = Λ to have short-distance and no problem with momentum regions Quite stable in region 1-4 GeV

45 : 42/56 α p 2 p 3 β r qρ Π ρναβ p 1 ν = Π VVS ab (p 1,r)g S ( 1+gS Π S (r) ) Π SVV cd (p 2,p 3 )V abcdρναβ +permutations g S (1+g S Π S ) = g A(r 2 )(2M Q ) 2 2f 2 (r 2 ) V abcdρναβ : VMD legs 1 M 2 S (r2 ) r 2 In only +quark-loop properly chiral invariant

46 : /QL 43/56 Cut-off a µ a µ a µ Λ Quark-loop Quark-loop Scalar GeV VMD Exchange Note: + (BPP) Quark-loop VMD (HKS) M S 620 MeV certainly an overestimate for real s If is σ: related to pion loop part? quark-loop: a ql µ bare a ql µ =

47 Quark loop DSE DSE model: aµ ql = 13.6(5.9) T. Goecke, C. S. Fischer and R. Williams, Phys. Rev. D 83 (2011) [arxiv: [hep-ph]] Not a full calculation (yet) but includes an estimate of some of the missing parts Note: a lot larger than bare quark loop with constituent mass I am puzzled: this DSE model (Maris-Roberts) does reproduce a lot of low-energy phenomenology. I would have guessed that it would be very similar to in its results. Can one find something in between full DSE and that is easier to handle? Error found in calculation, still not finalized: preliminary aµ ql = 10.7(0.2) T. Goecke, C. S. Fischer and R. Williams, arxiv: [hep-ph] 44/56

48 π and K-loop 45/56 The ππγ vertex is always done using VMD ππγ γ vertex two choices: Hidden local symmetry model: only one γ has VMD Full VMD Both are chirally symmetric Check if they live up to MV short distance (Full VMD does, HLS does not) The HLS model used has problems with π + -π 0 mass difference (due to not having an a 1 ) Final numbers quite different: and 0.19 For BPP stopped at 1 GeV but within 10% of higher Λ

49 π and K-loop 46/56 Cut-off a µ GeV π bare π VMD π π HLS K (7) 1.16(3) 1.20(0.03) 1.05(0.01) 0.020(0.001) (8) 1.41(4) 1.42(0.03) 1.15(0.01) 0.026(0.001) (9) 1.46(4) 1.56(0.03) 1.17(0.01) 0.034(0.001) (9) 1.57(6) 1.67(0.04) 1.16(0.01) 0.042(0.001) (12) 1.59(15) 1.81(0.05) 1.07(0.01) 0.048(0.002) (18) 1.70(7) 2.16(0.06) 0.68(0.01) 0.087(0.005) (18) 1.66(6) 2.18(0.07) 0.50(0.01) 0.099(0.005) HLS JB-Zahiri-Abyaneh note the suppression by the propagators

50 π loop: Bare vs VMD 47/56 π loop 1.6e-10 LLQ a µ 1.4e e-10 1e-10 8e-11 6e-11 4e-11 2e Note: plotted a LLQ µ Q bare VMD P 1 = P 2

51 π loop: VMD vs HLS 48/56 1e-10 8e-11 6e-11 4e-11 LLQ a µ 2e e-11-4e-11 Q Note: plotted a LLQ µ π loop VMD HLS a= P 1 = P 2

52 π loop 49/56 ππγ γ for q1 2 = q2 2 has a short-distance constraint from the OPE as well. HLS does not satisfy it full VMD does: so probably better estimate Ramsey-Musolf suggested to do pure ChPT for the π loop K. T. Engel, H. H. Patel and M. J. Ramsey-Musolf, light-by-light scattering and the pion polarizability, Phys. Rev. D 86 (2012) [arxiv: [hep-ph]]. So far ChPT at p 4 done for four-point function in limit p 1,p 2,q m π (Euler-Heisenberg plus next order) Polarizability (L 9 +L 10 ) up to 10%, charge radius 30% Both HLS and VMD have charge radius effect but not polarizability

53 π loop 49/56 ππγ γ for q1 2 = q2 2 has a short-distance constraint from the OPE as well. HLS does not satisfy it full VMD does: so probably better estimate Ramsey-Musolf suggested to do pure ChPT for the π loop K. T. Engel, H. H. Patel and M. J. Ramsey-Musolf, light-by-light scattering and the pion polarizability, Phys. Rev. D 86 (2012) [arxiv: [hep-ph]]. So far ChPT at p 4 done for four-point function in limit p 1,p 2,q m π (Euler-Heisenberg plus next order) Polarizability (L 9 +L 10 ) up to 10%, charge radius 30% Both HLS and VMD have charge radius effect but not polarizability

54 π loop 49/56 ππγ γ for q1 2 = q2 2 has a short-distance constraint from the OPE as well. HLS does not satisfy it full VMD does: so probably better estimate Ramsey-Musolf suggested to do pure ChPT for the π loop K. T. Engel, H. H. Patel and M. J. Ramsey-Musolf, light-by-light scattering and the pion polarizability, Phys. Rev. D 86 (2012) [arxiv: [hep-ph]]. So far ChPT at p 4 done for four-point function in limit p 1,p 2,q m π (Euler-Heisenberg plus next order) Polarizability (L 9 +L 10 ) up to 10%, charge radius 30% Both HLS and VMD have charge radius effect but not polarizability

55 π loop: L 9,L 10 ChPT for muon g 2 at order p 6 is not powercounting finite so no prediction for a µ exists. But can be used to study the low momentum end of the integral over P 1,P 2,Q The four-photon amplitude is finite still at two-loop order (counterterms start at order p 8 ) Add L 9 and L 10 vertices to the bare pion loop JB-Zahiri-Abyaneh Program the Euler-Heisenberg plus NLO result of Ramsey-Musolf et al. into our programs for a µ Bare pion-loop and L 9, L 10 part in limit p 1,p 2,q m π agree with Euler-Heisenberg plus next order analytically Numerics very preliminary 50/56

56 π loop: L 9,L 10 ChPT for muon g 2 at order p 6 is not powercounting finite so no prediction for a µ exists. But can be used to study the low momentum end of the integral over P 1,P 2,Q The four-photon amplitude is finite still at two-loop order (counterterms start at order p 8 ) Add L 9 and L 10 vertices to the bare pion loop JB-Zahiri-Abyaneh Program the Euler-Heisenberg plus NLO result of Ramsey-Musolf et al. into our programs for a µ Bare pion-loop and L 9, L 10 part in limit p 1,p 2,q m π agree with Euler-Heisenberg plus next order analytically Numerics very preliminary 50/56

57 π loop: VMD vs charge radius 51/56 LLQ a µ 1e-10 8e-11 6e-11 4e-11 2e e-11-4e VMD L 10 = -L 9 Q 0.2 π loop Note: plotted a LLQ µ, low scale, charge radius effect well reproduced P 1 = P 2

58 52/56 π loop: VMD vs L 9 and L 10 LLQ a µ 1.2e-10 1e-10 8e-11 6e-11 4e-11 2e VMD L 10,L 9 Q 0.2 π loop Note: plotted a LLQ µ, gives an enhancement of 10-15% P 1 = P 2

59 : vc PdRV 53/56 BPP PdRV arxiv: quark-loop (2.1±0.3) pseudo- (8.5±1.3) (11.4±1.3) axial-vector (0.25±0.1) (1.5±1.0) ( 0.68±0.2) ( 0.7±0.7) πk-loop ( 1.9±1.3) ( 1.9±1.9) errors linearly quadratically sum (8.3±3.2) (10.5±2.6) 10 10

60 What can we do more? 54/56 The model can certainly be improved: Chiral nonlocal quark-model (like nonlocal ): so far only done DSE: similar to everyone else, quark-loop very different, looking forward to final results More resonances models should be tried, AdS/QCD is one approach, RχT (Valencia et al.) possible,... Note short-distance matching must be done in many channels, there are theorems JB,Gamiz,Lipartia,Prades that with only a few resonances this requires compromises : HLS smaller than double VMD (understood) models with ρ and a 1 (in progress)

61 What can we do more? 55/56 Constraints from experiment: J. Bijnens and F. Persson, hep-ph/hep-ph/ Studying three formfactors Pγ γ in P l + l l + l, e + e e + e P exact tree level and for g 2 (but beware sign): Conclusion: possible but VERY difficult Two γ off-shell not so important for our choice of form-factor All information on hadrons and off-shell photons is welcome: constrain the models More short-distance constraints: MV, Nyffeler integrate with all, not just Need a new overall evaluation with consistent approach. Lattice has done first steps

62 What can we do more? 55/56 Constraints from experiment: J. Bijnens and F. Persson, hep-ph/hep-ph/ Studying three formfactors Pγ γ in P l + l l + l, e + e e + e P exact tree level and for g 2 (but beware sign): Conclusion: possible but VERY difficult Two γ off-shell not so important for our choice of form-factor All information on hadrons and off-shell photons is welcome: constrain the models More short-distance constraints: MV, Nyffeler integrate with all, not just Need a new overall evaluation with consistent approach. Lattice has done first steps

63 of Muon g 2 56/ a µ exp theory QED EW LO Had HO HVP difference Error on LO had all e + e based OK τ based 2 σ Error on Errors added quadratically 3.5 σ Generic SUSY: ( 100 GeV M SUSY ) 2tanβ M SUSY 66 GeV tanβ Difference: 4% of LO Had 270% of 1% of leptonic LbL