HADRONIC CONTRIBUTIONS TO THE MUON ANOMALOUS MAGNETIC MOMENT: overview and some new results for the hadronic Light-by-light part

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1 1/72 HADRONIC CONTRIBUTIONS TO THE MUON ANOMALOUS MAGNETIC MOMENT: overview and some new results for the hadronic Light-by-light part Lund University bijnens bijnens/chpt.html

2 1 2 QED Electroweak Lowest order hadronic HO hadronic 3 : new stuff is here /72

3 Literature 3/72 Final paper: G. W. Bennett et al. [Muon G-2 Collaboration], Final Report of the Muon E821 Anomalous Magnetic Moment Measurement 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 4/72 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]]. New stuff here: JB, Mehran Zahiri Abyaneh, Johan Relefors pion loop contribution arxiv: , arxiv: , arxiv: and to be published

5 Muon g 2: measurement 5/72

6 Muon g 2: measurement 6/72

7 Muon g 2: measurement 7/72 Phys.Rev. D73 (2006) [hep-ex/ ]

8 Muon g 2: measurement 8/72 Phys.Rev. D73 (2006) [hep-ex/ ] 2001: µ, others µ +

9 Muon g 2: overview 9/72 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 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 of Muon g 2 10/ 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

11 Muon g 2: QED 11/72 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) Light-by-light surprisingly large: ) 3 e = 20.95,µ = 0.37,τ = QED Electroweak Lowest order hadronic HO hadronic

12 Muon g 2: Electroweak 12/72 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

13 Muon g 2: LO hadronic 13/72 = 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

14 LO hadronic 14/72 QED Electroweak Lowest order hadronic HO hadronic Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ]

15 LO hadronic 15/72 QED Electroweak Lowest order hadronic HO hadronic Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ]

16 LO hadronic 16/ GeV ρ, ω ρ, ω 0.0 GeV, 0.0 GeV, Υ ψ 9.5 GeV 3.1 GeV 3.1 GeV 2.0 GeV φ, GeV φ, GeV Contribution Error Jegerlehner, Nyffeler Phys.Rept. 477 (2009) [ ] Direct measurements Radiative return: radiate hard photon to get lower E cms Radiative corrections τ-decay: isospin breaking can be improved ly QED Electroweak Lowest order hadronic HO hadronic

17 A comment on testing models on a LOHad µ 17/72 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% ρ right then mostly a LOHad µ right ) QED Electroweak Lowest order hadronic HO hadronic

18 Muon g 2: HO hadronic 18/72 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

19 : the main object to calculate 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, 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

21 A separation proposal: a start E. de Rafael, 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, ENJL, calculation in Euclidean space

22 Π ρναβ (p 1,p 2,p 3 ) = p 1 Actually we really need δπρναβ (p 1,p 2,p 3 ) δp 3λ q p3 =0 p 2 p 3

23 Π ρναβ (p 1,p 2,p 3 ): In general 138 Lorentz structures (but only 28 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

24 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 Relefors P 2 1, P2 2 and Q2 remain study aµ X = dl P1 dl P2 aµ XLL = dl P1 dl P2 dl Q aµ XLLQ l P = ln(p/gev), to see where the are Study the dependence on the cut-off for the photons

25 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 Relefors P 2 1, P2 2 and Q2 remain study aµ X = dl P1 dl P2 aµ XLL = dl P1 dl P2 dl Q aµ XLLQ l P = ln(p/gev), to see where the are Study the dependence on the cut-off for the photons

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

27 π 0 exchange BPP: Nonlocal quark model: a π0 µ = 5.9(0.9) a π0 µ = A. E. Dorokhov, W. Broniowski, Phys.Rev.D78 (2008) [ ] DSE model: a π0 µ = Goecke, Fischer and Williams, Phys.Rev.D83(2011)094006[ ] LMD+V: a π0 µ = ( ) M. Knecht, A. Nyffeler, Phys. Rev. D65(2002)073034, [hep-ph/ ] Formfactor inspired by AdS/QCD: a π0 µ = Cappiello, Cata and D Ambrosio, Phys.Rev.D83(2011) [ ] Chiral Quark Model: a π0 µ = D. Greynat and E. de Rafael, JHEP 1207 (2012) 020 [ ]. Constraint via magnetic susceptibility: a π0 µ = A. Nyffeler, Phys. Rev. D 79 (2009) [ ]. All in reasonable agreement

28 MV short-distance: π 0 exchange K. Melnikov, A. Vainshtein, light-by-light scattering contribution 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 =

29 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

30 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 Which momentum regions do what: volume under the plot a µ 1 P P 1

31 Pseudoscalar exchange 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-scalar exchange is about a PS µ = AdS/QCD estimate (includes excited pseudo-scalars) a PS µ = D. K. Hong and D. Kim, Phys. Lett. B 680 (2009) 480 [ ]

32 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 The HLS model used has problems with π + -π 0 mass difference (due to not having an a 1 ) Final numbers quite different: 0.45 and 1.9 ( ) For BPP stopped at 1 GeV but within 10% of higher Λ

33 π and K-loop Cut-off a µ GeV π bare π VMD π ENJL π HLS K ENJL (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 K-loop neglected in the remainder

34 π loop: Bare vs VMD 2e-10 π loop 1.5e-10 VMD bare LLQ -a µ 1e-10 5e Q P 1 = P 2 plotted a LLQ µ for P 1 = P 2 a µ = dl P1 dl P2 dl Q a LLQ µ l Q = log(q/1 GeV)

35 π loop: VMD vs HLS π loop 1e-10 VMD HLS a=2 8e-11 6e-11 LLQ -a µ 4e-11 2e e-11-4e-11 Q P 1 = P 2 Usual HLS, a = 2

36 π loop: VMD vs HLS 1.2e-10 π loop 1e-10 8e-11 6e-11 LLQ -a 4e-11 µ 2e-11 0 VMD HLS a=1 Q HLS with a = 1, satisfies more short-distance constraints 0.1 P 1 = P 2

37 π loop 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

38 π loop 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

39 π loop 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

40 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

41 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

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

43 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 P 1 = P 2 L 9 +L 10 0 gives an enhancement of 10-15% To do it fully need to get a model: include a 1

44 Include a 1 L 9 +L 10 effect is from a 1 But to get gauge invariance correctly need a 1 a 1

45 Include a 1 Consistency problem: full a 1 -loop? Treat a 1 and ρ classical and π quantum: there must be a π that closes the loop Argument: integrate out ρ and a 1 classically, then do pion loops with the resulting Lagrangian To avoid problems: representation without a 1 -π mixing Check for curiosity what happens if we add a 1 -loop

46 Include a 1 Use antisymmetric vector representation for a 1 and ρ Fields A µν, V µν (nonets) Kinetic terms: 1 2 λ V λµ ν V νµ 1 2 V µνv µν λ A λµ ν A νµ 1 2 A µνa µν 1 2 Terms that give to the L r i : F V 2 f 2 +µνv µν + ig V 2 V µν u µ u ν + F A 2 f 2 µνa µν L 9 = F VG V, L 2MV 2 10 = F2 V + F2 4MV 2 A 4MA 2 Weinberg sum rules: (Chiral limit) F 2 V = F2 A +F2 π VMD for ππγ: F 2 V M2 V = F2 A M2 A F V G V = F 2 π

47 V µν only Π ρναβ (p 1,p 2,p 3 ) is not finite (but was also not finite for HLS) But δπρναβ (p 1,p 2,p 3 ) also not finite δp 3λ (but was finite for HLS) p3 =0 Derivative one finite for G V = F V /2 Surprise: g 2 identical to HLS with a = F2 V F 2 π Yes I know, different representations are identical BUT they do differ in higher order terms and even in what is higher order Same comments as for HLS numerics

48 V µν only Π ρναβ (p 1,p 2,p 3 ) is not finite (but was also not finite for HLS) But δπρναβ (p 1,p 2,p 3 ) also not finite δp 3λ (but was finite for HLS) p3 =0 Derivative one finite for G V = F V /2 Surprise: g 2 identical to HLS with a = F2 V F 2 π Yes I know, different representations are identical BUT they do differ in higher order terms and even in what is higher order Same comments as for HLS numerics

49 V µν and A µν Add a 1 Calculate a lot δπ ρναβ (p 1,p 2,p 3 ) δp 3λ p3 =0 finite for: G V = F V = 0 and F 2 A = 2F2 π If adding full a 1 -loop G V = F V = 0 and F 2 A = F2 π Clearly unphysical (but will show some numerics anyway)

50 V µν and A µν Add a 1 Calculate a lot δπ ρναβ (p 1,p 2,p 3 ) δp 3λ p3 =0 finite for: G V = F V = 0 and F 2 A = 2F2 π If adding full a 1 -loop G V = F V = 0 and F 2 A = F2 π Clearly unphysical (but will show some numerics anyway)

51 V µν and A µν Start by adding ρa 1 π vertices λ 1 [V µν,a µν ]χ +λ 2 [V µν,a να ]h µ ν +λ 3 i [ µ V µν,a να ]u α +λ 4 i [ α V µν,a αν ]u µ +λ 5 i [ α V µν,a µν ]u α +λ 6 i [V µν,a µν ]f α ν +λ 7 iv µν A µρ A ν ρ All lowest dimensional vertices of their respective type Not all independent, there are three relations Follow from the constraints on V µν and A µν (thanks to Stefan Leupold)

52 V µν and A µν : big disappointment Work a whole lot δπ ρναβ (p 1,p 2,p 3 ) δp 3λ not obviously finite p3 =0 Work a lot more Prove that δπρναβ (p 1,p 2,p 3 ) finite, only same solutions as before δp 3λ p3 =0 Try the combination that show up in g 2 only Work a lot Again, only same solutions as before Small loophole left: after the integration for g 2 could be finite but many funny functions of m π,m µ,m V and M A show up.

53 π loop: add a 1 and F 2 A = 2F2 π 2e e-10 bare F A 2 = -2 F 2 π loop 1e-10 LLQ -a µ 5e Q Lowers at low energies, L 9 +L 10 < 0 here P 1 = P 2 funny peak at a 1 mass

54 π loop: add a 1 and F 2 A = F2 π plus a 1 -loop 2e e-10 bare F A 2 = -F 2 π loop 1e-10 LLQ -a µ 5e Q P 1 = P 2 Lowers at low energies, L 9 +L 10 < 0 here funny peak at a 1 mass canceled Still unphysical case

55 a 1 -loop: cases with good L 9 and L 10 2e e-10 1e-10 5e-11 0 LLQ -5e-11 -a µ -1e e-10-2e Q 1 π loop bare Weinberg no a 1 -loop Add F V, G V and F A Fix values by Weinberg sum rules and VMD in γ ππ no a 1 -loop P 1 = P 2

56 a 1 -loop: cases with good L 9 and L 10 2e e-10 1e-10 5e-11 0 LLQ -5e-11 -a µ -1e e-10-2e Q 1 π loop bare Weinberg with a 1 -loop Add F V, G V and F A Fix values by Weinberg sum rules and VMD in γ ππ With a 1 -loop (is different plot!!) P 1 = P 2

57 a 1 -loop: cases with good L 9 and L 10 2e-10 π loop bare a 1 no a 1 -loop,vmd 1.5e-10 1e-10 LLQ -a µ 5e Q P 1 = P 2 Add a 1 with F 2 A = +F2 π Add the full VMD as done earlier for the bare pion loop

58 a 1 -loop: cases with good L 9 and L 10 2e-10 π loop bare a 1 with a 1 -loop,vmd 1.5e-10 1e-10 LLQ -a µ 5e P 1 = P Q Add a 1 with FA 2 = +F2 π and a 1-loop Add the full VMD as done earlier for the bare pion loop

59 Integration results -a µ Λ 5e e-10 4e e-10 3e e-10 2e e-10 1e-10 5e-11 sqed sqed π 0 VMD HLS HLS a=1 ENJL P 1,P 2,Q Λ Λ

60 Integration results -a µ Λ 4e e-10 3e e-10 2e e-10 1e-10 5e-11 a 1 F A 2 = -2F 2 a 1 F A 2 = -1 a1 -loop HLS HLS a=1 VMD a 1 VMD a 1 Weinberg P 1,P 2,Q Λ Λ

61 Integration results with a 1 Problem: get high energy behaviour good enough But all models with reasonable L 9 and L 10 fall way inside the error quoted earlier ( 1.9±1.3) Tentative conclusion: Use hadrons only below about 1 GeV: a π loop µ = ( 2.0±0.5) Note that Engel and Ramsey-Musolf, arxiv: is a bit more pessimistic quoting numbers from ( 1.1 to 7.1) 10 10

62 Pure quark loop 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

63 Pure quark loop: momentum area 10 1 Q This plots a ql µ = dl P1 dl P2 dl Q a LLQ µ Succeeded in 3D plot but was useless P P 1 7e-10 6e-10 5e-10 4e-10 3e-10 2e-10 1e-10 0

64 Pure quark loop: momentum area 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

65 ENJL quark-loop Cut-off a µ a µ a µ a µ Λ sum GeV VMD ENJL masscut ENJL+masscut Very stable ENJL 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

66 ENJL: scalar 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 : ENJL VMD legs 1 M 2 S (r2 ) r 2 In ENJL only scalar+quark-loop properly chiral invariant

67 ENJL: scalar/ql Cut-off a µ a µ a µ Λ GeV VMD ENJL Exchange ENJL only scalar+quark-loop properly chiral invariant Note: ENJL+scalar (BPP) VMD (HKS) M S 620 MeV certainly an overestimate for real scalars If scalar is σ: related to pion loop part? quark-loop: a ql µ bare a ql µ =

68 Quark loop DSE DSE model: a ql µ = 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 a lot larger than bare quark loop with constituent mass I am puzzled: this DSE model (Maris-Roberts) does reproduces a lot of low-energy phenomenology. I would have guessed that it would give numbers very similar to ENJL. Can one find something in between full DSE and ENJL that is easier to handle? Error found in calculation, still not finalized: preliminary a ql µ = 10.7(0.2) T. Goecke, C. S. Fischer and R. Williams, arxiv:

69 Other quark loop de Rafael-Greynat Boughezal-Melnikov Masjuan-Vanderhaeghen ( ) ( ) ( ) Various interpretations: the full calculation or not All (even DSE) have in common that a low quark mass is used for a large part of the integration range

70 Axial-vector exchange exchange 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-scalar 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 =

71 : ENJL vc PdRV BPP PdRV arxiv: quark-loop (2.1±0.3) pseudo-scalar (8.5±1.3) (11.4±1.3) axial-vector (0.25±0.1) (1.5±1.0) scalar ( 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

72 What can we do more? 65/72 The ENJL model can certainly be improved: Chiral nonlocal quark-model (like nonlocal ENJL): 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 : difficulties with infinities

73 What can we do more? 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 Some tentative steps from dispersion theory Pauk-Vanderhaeghen 66/72

74 What can we do more? 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 Some tentative steps from dispersion theory Pauk-Vanderhaeghen 66/72

75 BNL magnet has moved to Fermilab 67/72 Goal ± Credit: Brookhaven National Laboratory

76 BNL magnet has moved to Fermilab 68/72 Goal ± Credit: Brookhaven National Laboratory

77 BNL magnet has moved to Fermilab 69/72 Credit: Fermilab

78 BNL magnet has moved to Fermilab 70/72 Goal ± Credit: Fermilab

79 JPARC with a very different method Ultracold muons at low energy g 2 (Credit: JPARC) 3 GeV proton beam ( 333 ua) Silicon Tracker Graphite target (20 mm) Surface muon beam 66 cm diameter (28 MeV/c, 4x108/s) Muonium Production (300 K ~ 25 mev) Super Precision Magnetic Field (3T, ~1ppm local precision) Resonant Laser Ionization of Muonium (~106 µ+/s) Muon LINAC (300 MeV/c) 19 71/72

80 of Muon g 2 72/ 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