Magnetic and Electric Dipole Moments

Transcription

1 Colour meets Flavour Siegen, October 14, 2011 Magnetic and Electric Dipole Moments Arkady Vainshtein William I. Fine Theoretical Physics Institute University of Minnesota 1

2 Happy Birthday, Alex! 2

3 Outline Experimental data on g and EDM CPV scales and EDM calculations Anatomy of muon g-2, electroweak corrections Triangle amplitudes, nonrenormalization theorem Hadronic light-by-light scattering Do we see New Physics in the muon g-2? 3

4 Lepton magnetic moments The present experimental values Electron: Hanneke, Fogwell, and Gabrielse ʼ08 g/2= (28) [0.28 ppt] New value of follows H = µb = (51) [0.37 ppb] µ = g e 2mc s ich is called the Muon: BNL E821 ʼ06 g/2= (63) [630 ppt] Tau: Delphi at LEP2 ʼ04 g/2=0.982(17) 4

5 H = de Electric dipole moments Neutron: Baker et al ʼ06 d < e cm Electron: Regan et al ʼ06 d < e cm Muon: Muon g-2 Collaboration ʼ04 d < e cm 5

6 *+(,-. '() Scales hierarchy!"#$%&'#(%)* +,*-.%/'/ Pospelov, Ritz ʼ05 / (5, -24#9#":)'4#* :4"-)2#3*;*********< ='"(14#* 678*;****< / 45* -%1%&%3#'(2:* %(4&/*;******< 678/ 45* $2%&%3#'(2:** %(4&/*;*******< 6

7 Observables L nuclear eff = L edm + L πnn + L en, L edm = i 2 i=e,p,n d i ψ i (F σ)γ 5 ψ, L πnn =ḡ (0) πnn Nτ a Nπ a +ḡ (1) πnn NNπ 0 +ḡ (2) πnn( Nτ a Nπ a 3 Nτ 3 Nπ 0 ), L en = C (0) S ēiγ 5 e NN + C (0) P ēe Niγ 5 N + C (0) T ɛ µναβ ēσ µν e Nσ αβ N +C (1) S ēiγ 5 e Nτ 3 N + C (1) P ēe Niγ 5 τ 3 N + C (1) T ɛ µναβ ēσ µν e Nσ αβ τ 3 N. 7

8 QCD Low Energy Lagrangian L eff = L dim=4 + L dim=5 + L dim=6 +. L dim=4 = g2 s 32π 2 θg a µν G µν,a, L dim=5 = i 2 i=u,d,s,e,µ d i ψ i (F σ)γ 5 ψ i i 2 i=u,d,s d i ψ i g s (Gσ)γ 5 ψ i, Effective dim=6. Thus, we need to add L dim=6 = 1 3 wfabc G a µν G νβ,b G µ,c β + i,j C ij ( ψ i ψ i )( ψ j iγ 5 ψ j )+ 8

9 d PQ Examples of calculation qq d n ( θ) =(1± 0.5) (225MeV) θ e cm, 3 ± [ n (d q, d qq [ q )=(1± 0.5) 1.1e( (225MeV) 3 dd +0.5 d u )+1.4(d d 0.25d u ) ] In the Standard Model Khriplovich ʻ86 d d = e m dm 2 c α sg 2 F J CP ln 2 (m 2 108π b/m 2 c) ln(m 2 5 W /m 2 d b). KM d d KM n e cm. Gavela; Khriplovich, Zhitnitsky ʻ e cm. g d KM e e cm, u, c Khriplovich, Pospelov ʻ91 Potential for NP to show up! d b, s t W t 9

10 Anatomy of muon g-2 a QED a SM µ = a QED µ + a had µ + a EW µ µ = (1.5) Kinoshita et al a EW 11 µ = 154(1)(2) 10 Czarnecki, Marciano, AV ʻ02 a EW µ (1-loop) = 5 G µ m 2 µ 24 2π 2 ν (1 4 sin2 θ W ) 2 + O Z m 2 µ m 2 W,H = H µ µ W W µ µ µ µ µ µ γ γ γ 10

11 Two-loop corrections are more involved a EW µ (2-loop) LL = 5G µm 2 µ 24 2π α 2 π 43 3 ln m Z m µ f F N f Q 2 f I 3 f ln m Z m f µ µ Z f Fermion triangles ( Z γγ vertex) γ γ F = τ, u, d, s, c, b Kukhto, Kuraev, Schiller, Silagadze 92 Peris, Perrottet, Rafael 95 Czarnecki, Krause, Marciano 95 m u,d =0.3 GeV, m s =0.5 GeV, m c =1.5 GeV, m b =4.5 GeV Total: a EW µ = 152(4) Czarnecki, Marciano 01 11

12 Perturbative and nonperturbative ~ j "! " 5 " µ " triangle amplitudes j µ = qvγ µ q, T µγν = T µν = T µγν e γ (k) =i j 5 ν = qaγ νγ 5 q d 4 xd 4 y e iqx iky 0 T {j µ (x) j γ (y) j 5 ν (0)} 0 d 4 x e iqx 0 T {j µ (x) j 5 ν (0)} γ(k) 5 j! T µν = i 4π 2 j µ [ wt (q 2 ) ( q 2 fµν + q µ q σ fσν q ν q σ fσµ ) + wl (q 2 ) q ν q σ fσµ ] f µν = 1 2 ɛ µνγδf γδ, f µν = k µ e ν k ν e µ. w 1 loop L [m = 0] = 2 w 1 loop T [m = 0] = 2N c Tr (AV Ṽ ) Q 2 12

13 j 5 ν Nonrenormalization theorem ~ j γ γ ν γ 5 γ µ j µ AV ʻ02 Czarnecki, Marciano, AV ʻ02 Knecht, Peris, Perrottet, de Rafael ʼ03 No pertubative corrections both in longitudinal and transversal parts in the chiral limit. Pole in the longitudinal part corresponds to massless pion. But it should be no massless pole in the transversal part. A shift from zero is provided by nonperturbative effects. Four-fermion operators in OPE. q! Z q w T [u, d] = 1 m 2 a 1 m 2 ρ m2 a 1 m 2 π Q 2 + m 2 ρ m2 ρ m2 π Q 2 + m 2 a 1 13

14 Hadronic contributions a had µ = a had,lo µ + a had,ho µ + a LBL µ q!! µ µ q µ µ γ γ γ q q Lowest order hadronic contribution represented by a quark loop An example of higher order hadronic contribution γ Light-by-light scattering contribution In theory a had,lo µ = α mµ 2 3π 4m 2 π ds s 2 K(s)R(s) K(s) is the known function, K(s) 1, s m 2 µ R(s) is the cross section of e + e annihilation into hadrons in units o σ(e + e µ + µ ). e K sof 14

15 In difference with a had,lo µ there is no experimental input for the light-by-lig contribution.what are possible theoretical parameters to exploit? 1 Smallness of chiral symmetry breaking, m 2 ρ/m 2 π 1 ght α n a (n) m 2 µ µ c 1 π " +!! µ µ " # m 2 π, LO : n =2, LbL : n =3 µ µ!!! " " The Goldstone nature of pion implies m 2 π m q much less than typical M 2 had m2 ρ. Thus, the threshold range in pion loops produces the 1/m2 π enhancement.! 15

16 Large number of colors, N c Quark loops clearly give a µ N c. Dual not to pion loops but to meso exchanges.! "! µ µ! µ!! "! on No continuum in the large N c limit. M = ρ 0, ω, φ, ρ,...for the polarization operator M = π 0, η, η,a 0,a 1,...(and any C-even meson) for the light-by-light α n a µ (n) m 2 µ c 2 Nc π m 2 ρ 16

17 We can check for a had,lo µ Two regions. The threshold region s 4m 2 π where R(s) 1 1 4m2 π 4 s and the resonance region s m 2 ρ 3/2 where by quark-hadron duality on average R(s) N c Q 2 q The chirally enhanced threshold region gives numerically a had,lo µ (4m 2 π s m 2 ρ/2) Compare with the N c enhanced ρ peak, a had,lo µ (ρ) = m2 µ Γ(ρ e + e ) π m 3 ρ This contribution is enhanced by N c, 2 Nc m 2 µ a µ (ρ) c 2 α π m 2 ρ What is a lesson from this exercise? We see that the large N c enhancement prevails over chiral one. 17

18 In the chiral perturbation theory a 2π µ = 1 α 2 m 2 µ 40 3π m 2 π = 1 α 2 m 2 µ 40 3π m 2 π m 2 πfππγ (0) ln m ρ 2m π m2 π m 2 ρ ln m ρ 2m π Chiral perturbation theory does not work. The leading term is suppressed by p-wave nature. 18

19 WORKSHOP ON (g 2)µ, GLASGOW, OCTOBER 25, 2007 A. Vainshtein Determination of the light-by-light contributions 7 In light-by-light Light-by-light In light-by-light In light-by-light π + π + π + π 0, πa 0, 1 a 1 π + + π 0, a 1 π π 0, 0 a, a 1 1 a a b b a b e pion box a b b The chirally enhanced pion box contribution does does not not result result in in large large number, it is it is The chirally rather small, enhanced pion box contribution does not result in large number, it is actually rather small, The actually chirally rather enhanced small, pion box contribution doesnot not result in in large large number, it isit is a LbL a LbL a LbL actually rather small, µ (pion box) 11 µ (pion box) µ (pion box) Hayakawa, Kinoshita, Sanda; Melnikov Hayakawa, Kinoshita, Sanda; Melnikov similarly a to to the the hadronic polarization case case above. above. larger Asimilarly larger a LbL value value to the (-19) (-19) hadronic for for the the pion polarization pion box box was was case obtained above. by by Bijnens, Pallante, Prades A larger value (-19) for the pion box was obtained by Bijnens, Pallante, Prades similarly to the hadronic polarization case above. similarly A larger to value the (-19) hadronic for the polarization pion box was case obtained above. by Bijnens, Pallante, Prades A larger value (-19) for the pion box was obtained by Bijnens, Pallante, Prades µ µ (pion box) Hayakawa, Kinoshita, Sanda; Melnikov

20 Instability of the number is due to relatively large pion momenta in the loop, of order of 4m π as we estimated. Then details of the model becomes important and theoretical control is lost. In HSL model few first terms of m 2 π/m 2 ρ expansion are a µ (charged pion loop) = = 4.9 If momenta were small compared with m ρ the result would be close to the leading term free pion loop. In case of polarization operator the suppression of the leading term in the chiral expansion (larger momenta) can be related to the p-wave p 3 suppression. There is a suppression for s-wave in two-pion intermediate state near threshold in the case of LbL. 20

21 0 Hayakawa, Kinoshita, Sanda Bijnens, Pallante, Prades Barbieri, Remiddi Pivovarov Bartos, Dubničkova,Dubnička, Kuraev, Zemlyanaya Knecht, Nyffeler Knecht, Nyffeler, Perrotttet, de Rafael Ramsey-Musolf, Wise Blokland, Czarnecki, Melnikov Melnikov, A.V. Different models: constituent quark loop, extended Nambu Jano-Lasinio model (ENJL), hidden local symmetry (HLS) model... The π 0 pole part of LbL contains besides N c the chiral enhancement in the logarithmic form, leading to the model-independent analytical expression a LbL µ (π 0 )= α π 3 Nc m 2 µ N c 48π 2 F 2 π ln 2 m ρ m π +... However next, model dependent, terms are comparable with the the leading on Numerically a LbL µ (π 0 ) = 58(10) Knecht, Nyffeler 21

22 Massive quark loop (Laporta, Remiddi 91) a HLbL (quark loop) = ( α π ) 3 N c Q 4 q { [3 ] 19 ζ(3) 2 16 }{{} 0.62 m 2 µ m 2 q + O [ m 4 µ m 4 q log 2 m2 µ m 2 q ]} For c-quark with m c 1.5 GeV, a HLbL (c) = Light quark estimate for the constituent mass 300 MeV ough estimate for the light quark a HLbL (u, d, s) = Together with the neutral pion exchange it gives Adding up this contribut a HLbL

23 Models HLS model is a modification the Vector Meson Dominance model. ENJL model is represented by the following graphs $ p 3! p " 2! q# & p 1 ' (a) " p 2! $ p 3 %! r! q# & p 1 ' (b) 23

24 X H q k k k2 1 3 μ p 1 p 2 OPE constraints and hadronic model µ i (q i), i =1, 2, 3, 4, qi =0 4 represents the external magnetic field f γδ = q γ 4 δ 4 q4 δ γ 4, q 4 0. The LbL amplitude M = α 2 N c Tr [ ˆQ 4 ] A = α 2 N c Tr [ ˆQ 4 ] A µ1 µ 2 µ 3 γδ µ 1 1 µ 2 2 µ 3 3 f γδ = e 3 d 4 x d 4 y e iq 1x iq 2 y µ 1 1 µ 2 2 µ T {j µ 1 (x) j µ2 (y) j µ3 (0)} γ The electromagnetic current j µ = q ˆQγ µ q, q = {u, d, s} Three Lorentz invariants: q 2 1,q 2 2,q 2 3 Consider the Euclidian range q 2 1 q 2 2 q 2 3 Λ 2 QCD 24

25 We can use OPE for the currents that carry large momenta q 1,q 2 i d 4 x d 4 y e iq 1x iq 2 y T {j µ1 (x),j µ2 (y)} = d 4 z e i(q 1+q 2 )z 2i ˆq 2 µ 1 µ 2 δρ ˆq δ j ρ 5 (z)+. ˆq =(q 1 q 2 )/2, the axial current j ρ 5 = q ˆQ 2 γˆ ρ γ 5 q is the linear combination of j (3) 5ρ = q λ 3γ ρ γ 5 q isovector j (3) 5ρ = q λ 8γ ρ γ 5 q hypercharge j (3) 5ρ = q γρ γ 5 q singlet k 1 j 5ρ = a=3,8,0 q 0 Tr [λ a ˆQ2 ] Tr [λ 2 a] j (a) 5ρ q 0 H γ γ 5 k k 2 3 k 3 25

26 The triangle amplitude T µ (a) 3 ρ = i 0 d 4 z e iq 3z T {j (a) 5ρ (z) j µ 3 (0)} γ kinematically is expressed via two scalar amplitudes T µ (a) 3 ρ = ie N ctr [λ ˆQ2 a ] 4π 2 w (a) L (q2 3) q 3ρ q3 σ f σµ3 + +w (a) T (q2 3) q3 2 f µ3 ρ+q 3µ3 q3 σ f σρ q 3ρ q3 σ f σµ3 Longitudinal w L : Transversal w T : pseudoscalar mesons exchange pseudovector mesons exchange In perturbation theory for massless quarks Nonvanishing w w (a) L (q2 )=2w (a) T (q2 )= 2 q 2 is the signature of the axial Adler Bell Ja 26

27 Nonvanishing w L is the signature of the axial Adler Bell Jackiw anomaly. Moreover, for nonsinglet w (3,8) it is the exact QCD result, no perturbative as well L as nonperturbative corrections. So the pole behavior is preserved all way down to small q 2 where the pole is associated with Goldstone mesons π 0, η. Comparing the pole residue we get the famous ABJ result g πγγ = N ctr [λ 3 ˆQ2 ] 16π 2 F π There exists the nonrenormalization theorem for w T as well but only in respect to perturbative corrections. A.V. 02; Knecht, Peris, Perrottet, de Rafael 03 Higher terms in the OPE does not vanish in this case, they are responsible for shift of the pole 1/q 2 1/(q 2 m 2 V,P V ) Combining we get at q 2 1 q 2 2 q 2 3 A µ1 µ 2 µ 3 γδf γδ = 8ˆq 2 µ 1 µ 2 δρˆq δ a=3,8,0 W (a) w (a) L (q2 3) q ρ 3 qσ f 3 σµ3 + w (a) T (q2 3) q3 2 f ρ µ3 +q 3µ3 q3 σ f σ ρ q ρ 3 qσ f 3 σµ3 + where the weights W (3) =1/4, W (8) =1/12, W (0) =2/3. 27

28 The model Melnikov, AV ʻ03 A = A PS + A PV + permutations, A PS = W (a) φ (a) L (q2 1,q2) 2 w (a) L (q2 3) {f 2 f1 }{ ff 3 }, A PV = a=3,8,0 a=3,8,0 W (a) φ (a) +{q 1 f 1 f2 ff3 q 3 }+ q2 1 + q2 2 4 For π 0 w (3) 2 L (q2 )= q 2 + m 2, π φ 3 L(q 2 1,q 2 2)= N c 4π 2 F 2 π T (q2 1,q2) 2 w (a) T (q2 3) {q 2 f 2 f1 ff3 q 3 } F πγ γ (q 2 1,q 2 2) {f 2 f1 }{ ff 3 }. Knecht, Nyffeler V, =693 G = q2 1q 2 2(q q 2 2) h 2 q 2 1q h 5 (q q 2 2)+(N c M 4 1 M 4 2 /4π 2 F 2 π) (q M 2 1 )(q2 1 + M 2 2 )(q2 2 + M 2 1 )(q2 2 + M 2 2 ) 28

29 The model results in a π0 µ = , a PS µ = 114(10) A similar analysis for pseudovector exchange gives a PV µ = 22(5) and finally a LbL µ = 136(25) of 29

30 Summary for LbL In our 08 mini-review with Prades, de Rafael we combined different calculations with some educated guesses about possible errors to come to: a HLbL = (105 ± 26) However the error estimates are quite subjective and further study of different exchanges is certainly needed. Experimental data on radiative decays can be a help. M & permutations M pseudoscalar mesons π 0, η, η ; scalars f 0,a 0 ;vectorsπ 0 1 ; pseudovectors a0 1,f 1,f 1 ; nd pseudotensor mesons f, a, η, π. spin 2 f 2,a 2, η 2, π 2 30

31 Do we see NP in the muon g-2? QED (1.5) Electroweak 154(2)(1) Hadronic LO 6 901(42)(19)(07) Hadronic HO Hadronic LbL Total SM Experimental a (0.9)(0.3) (26) (52) (63) a 300 (82) σ Both experimental and theoretical uncertainty should be reduced to be sure of NP. 31

32 e γ f Γ(f 1 (1285) γγ ) = (2.8 ± 0.8) kev γ e This is compatible with our model of pseudovector exchange. However, Γ(f 1 (1285) γρ 0 ) Γ total = (5.5 ± 1.3) 10 2 ± leads to a strong enhancement (of order of 5) for PV exchange. Work in progress. 32

33 Conclusions Having in mind the new g-2 experiment more theoretical efforts are needed to improve accuracy for the hadronic light-by-light contribution. In my view it should also involve new measurements of hadronic two-photon production which provides a good test of theoretical models for HLbL. 33

34 Many happy returns of the day! 34