Radiative transitions and the quarkonium magnetic moment

Transcription

1 Radiative transitions and the quarkonium magnetic moment Antonio Vairo based on Nora Brambilla, Yu Jia and Antonio Vairo Model-independent study of magnetic dipole transitions in quarkonium PRD (2006) [arxiv:hep-ph/ ] University of Milano and INFN

2 Radiative transitions: basics Two dominant single-photon-transition processes: (1) electric dipole transitions (E1) (2) magnetic dipole transitions (M1) γ (k γ, k) H P H = (M H, 0) P H = q kγ 2 + MH 2, k H

3 Radiative transitions: basics Two dominant single-photon-transition processes: (1) electric dipole transitions (E1) (2) magnetic dipole transitions (M1) In the non-relativistic limit Γ M1 n 3 S 1 n 1 S 0 γ = 4 3 α e2 Q k 3 γ m 2 Z 0 dr r 2 R n 0(r) R n0 (r) j 0 kγ r 2 «2 If k γ r 1 j 0 (k γ r/2) = 1 (k γ r) 2 / n = n n n allowed transitions hindered transitions

4 J/ψ η c γ Only one direct experimental measure: Γ(J/ψ η c γ) = (1.14 ± 0.23) kev Crystal Ball 86 Moreover, there are several measurements of the BR J/ψ η c γ φφγ and one independent measurement of η c φφ (Belle 03). From them one obtains Γ(J/ψ η c γ) = (2.9 ± 1.5) kev Combining both Γ(J/ψ η c γ) = (1.18 ± 0.36) kev PDG 04 Γ(J/ψ ηc γ) enters into many charmonium BR. Its 30% uncertainty sets typically their experimental errors.

5 J/ψ η c γ Γ(J/ψ η c γ) Γ(J/ψ) = ± κ c m c M i:f = 0.40 ± 0.05 GeV if M i:f = 1 this implies: κc = 0, m c = 2.3 ± 0.3 GeV κc = 0.28 ± 0.09, m c = 1.8 GeV large relativistic corrections to the S-state wave functions Eichten/QWG 02

6 ψ(3770) χ c1 γ c1 Number of Events / 5 MeV c2 c0 ee E (MeV) First resolved radiative transition from a D-wave state: B(ψ(3770) χ c1 γ) = (3.2 ± 0.6 ± 0.4) 10 3 Υ(1D) transitions have been observed in the cascade: CLEO 05 Υ(3S) χ b (2P )γ, χ b (2P ) Υ(1D) γ, Υ(1D) χ b (1P )γ, χ b (1P ) Υ(1S) γ CLEO 04

7 η b search E1 Peaks E1 Peaks Number of photons / 2 % bin Search Window Number of photons / 2 % bin 8000 Search Window Eγ (MeV) Eγ (MeV) Search through Υ(3S) ηb (1S) γ (M1 hindered transition) Potentially promising due to high γ energy (k) better resolution Γ k 3 Other (observed) transitions Υ(3S) χb (2P J ) γ Υ(1S) γ CLEO 02

8 Υ(3S) η b (1S)γ 3 Zambetakis,Byers, 83 Υ(3S) γ η b (1S) Branching Ratio in units of Godfrey-Isgur, 85 B Godfrey-Isgur, 85 A 90% CL UL CLEO-III Ebert,Faustov,Galkin, 03 Lahde,Nyfalt,Riska, 99 A Lahde,Nyfalt,Riska, 99 B Eγ (MeV) CLEO/QWG 04

9 Υ(2S) η b (1S)γ Υ(2S) γ η b (1S) 1.6 Zambetakis,Byers, 83 Branching Ratio in units of Godfrey-Isgur, 85 B Grotch,Owen,Sebastian, 84 A Godfrey-Isgur, 85 A 90% CL UL CLEO-III Grotch,Owen,Sebastian, 84 B Ebert,Faustov,Galkin, 03 Lahde,Nyfalt,Riska, 99 A/B Zhang,Sebastian,Grotch, 91 A Eγ (MeV) The (non) observed transition rates are becoming problematic for most models. CLEO/QWG 04

10 Υ(3S) η b (2S)γ 3 Υ(3S) γ η b (2S) From the top dashed line: Zambetakis,Byers, 83 Godfrey-Isgur, 85A Branching Ratio in units of Godfrey-Isgur, 85B Ebert,Faustov,Galkin, 03 Lahde,Nyfalt,Riska B, 99 Lahde,Nyfalt,Riska A, 99 90% CL UL CLEOIII Eγ (MeV) CLEO/QWG 04

11 Several model determinations exist. Grotch Owen Sebastian 84

12 Several model determinations exist. Grotch Owen Sebastian 84 We would like to understand to which extent these determinations are consistent with QCD; what is their range of applicability. In practice, we would like to have theoretical determinations with a realistic error bar attached to them; improvable in a systematic way.

13 Several model determinations exist. Grotch Owen Sebastian 84 We would like to understand to which extent these determinations are consistent with QCD; what is their range of applicability. In practice, we would like to have theoretical determinations with a realistic error bar attached to them; improvable in a systematic way. These are provided by Effective Field Theories.

14 Scales 1 p r mv, ΛQCD E mv2 kγ In a non-relativistic system mv mv 2. k γ mv 2 for hindered transitions; k γ mv 4 for allowed transitions. As a consequence k γ r 1.

15 Degrees of freedom Degrees of freedom at scales lower than mv: Q- Q states, with energy Λ QCD, mv 2 and momentum < mv (i) singlet S (ii) octet O (if mv Λ QCD ) Gluons with energy and momentum Λ QCD, mv 2 (if mv Λ QCD ) Photons of energy and momentum lower than mv. Power counting: p 1 r mv; all gauge fields are multipole expanded: A(R, r, t) = A(R, t) + r A(R, t) +... and scale like (Λ QCD or mv 2 ) dimension.

16 Lagrangian L pnrqcd = 1 4 F a µνf µν a 1 4 F em +Tr µν em µν F js i 0 p2 m V s + O id 0 p2 m V o ««S ff O n o +Tr O r ge S + S r ge O + 1 o no 2 Tr r ge O + O Or ge + +L γ (if mv Λ QCD ) LO in r NLO in r Pineda Soto 97, Brambilla Pineda Soto Vairo 99

17 L γ L γ = Tr ( V em A + 1 2m V m V V 2 4m 2 r + 1 V 3 4m 2 r + 1 4m 3 V 4 S r ee Q E em S n S, σ ee Q B emo S n O, σ ee Q B emo O (if mv Λ QCD ) ns em o, σ ˆˆr `ˆr ee Q B S n S, σ ee Q B emo S n S, σ ee Q B emo 2 rs + )

18 Matching The matching consists in the calculation of the coefficients V. They get contributions from hard modes ( m): From HQET: ψ(id/ m)ψ ψ id 0 + D2 2m + cem F 2m σ ee QB em + c em F is the quark magnetic moment. soft modes ( mv). 1 + κ = α s 3π + O(α2 s ) «ψ

19 M1 operator at O(1) j V 1 S, σ ff ebem S 2m V 1 =!! hard soft hard! = c em F = 1 + 2α s(m c ) 3π + Since σ eb em (R) behaves like the identity operator to all orders V 1 does not get soft contributions.

20 t f Z tf t i dt t 1 t t c em F σ ee Q B em 2m t i = Z tf t i dt Diagrammatic factorization of the magnetic dipole coupling in the SU(3) f limit.

21 M1 operator at O(1) j V 1 S, σ ff ebem S 2m V 1 = 1 + 2α s(m c ) 3π + No large quarkonium anomalous magnetic moment! (see also the lattice calculation of Dudek Edwards Richards 06)

22 M1 operators at O(v 2 ) V 4 =!! hard soft j V 4 S, σ ff ebem 4m 3 2 rs hard! = 1 due to reparametrization invariance Manohar 97 V 4 = 1 + O(α s soft contributions)

23 M1 operators at O(v 2 ) 1 V 2 4m 2 r ns em o, σ ˆˆr `ˆr ee Q B S c F σ B/m!! = hard soft A A em /m c s σ (A em E)/m 2!! to all orders hard = 2c F c s = 1 ; soft = r 2 V s /2 (due to reparametrization/poincaré invariance) Brambilla Gromes Vairo 03 Therefore V 2 = r 2 V s /2 and V 3 = 0 No scalar interaction!

24 J/ψ η c γ Γ J/ψ ηc γ = Z d 3 k (2π) 3 (2π)δ(EJ/ψ p k E η c k ) γ(k)η c L γ J/ψ 2

25 O(v 2 ) corrections to the quarkonium states j Coupling of photons with octets: V 1 O, σ ff ebem O (if mv Λ QCD ) 2m δz H = 0 r ge If mv 2 Λ QCD the above graphs are potentially of order Λ 2 QCD /(mv)2 v 2. The contribution vanishes because σ eb em (R) behaves like the identity operator. There are no non-perturbative contributions at O(v 2 )!

26 J/ψ η c γ Up to order v 2 the transition J/ψ η c γ is completely accessible by perturbation theory. Γ(J/ψ η c γ) = 16 3 αe2 c k 3 γ M 2 J/ψ» 1 + C F α s (M J/ψ /2) π 2 3 (C F α s (p J/ψ )) 2 The normalization scale for the α s inherited from κ c is the charm mass (α s (M J/ψ /2) 0.35 v 2 ), and for the α s, which comes from the Coulomb potential, is the typical momentum transfer p J/ψ mc F α s (p J/ψ )/2 0.8 GeV mv. Γ(J/ψ η c γ) = (1.5 ± 1.0) kev.

27 Γ Υ(1S) ηb γ and Γ Υ(2S) ηb (2S) γ Γ Υ(1S) ηb γ (ev) Γ Υ(2S) ηb (2S) γ (ev) k γ (MeV) k γ (MeV) B Υ(1S) ηb γ = (6.8 ± 5.5) 10 5

28 M1 hindered transitions Two new operators contribute: 1 16m 2 cem S hs em i, σ ˆ i, ee Q E S and 1 16m 2 cem S hs, σ ˆ i r, r i ( i ee Q E ) i em S Two new wave function corrections contribute: (1) induced by the spin-spin potential; (2) recoil correction induced by the spin-orbit potential; Due to the recoil, the final state develops a nonzero P -wave component suppressed by a factor v k γ /m, which, in a n 3 S 1 n 1 S 0 γ transition, can be reached from the initial 3 S 1 state through a 1/v enhanced E1 transition.

29 Γ Υ(2S) ηb γ and Γ ηb (2S) Υ(1S) γ Γ Υ(2S) ηb γ (kev) 5 4 Γ ηb (2S) Υ(1S) γ (kev) k γ (MeV) k γ (MeV)

30 Γ hb (1P ) χ b0 (1P ) γ, Γ hb (1P ) χ b1 (1P ) γ and Γ χb2 (1P ) h b (1P ) γ Γ hb (1P ) χ b0 (1P ) γ (kev) Γ hb (1P ) χ b1 (1P ) γ (kev) Γ χb2 (1P ) h b (1P ) γ k γ (MeV) (kev) k γ (MeV) Γ hb (1P ) χ b0 (1P ) γ = 1 ± 0.2 kev Γ hb (1P ) χ b1 (1P ) γ = 17 ± 4 ev 0.2 Γ χb2 (1P ) h b (1P ) γ = 90 ± 20 ev k γ (MeV)

31 Conclusions We confirm the results of Grotch Owen Sebastian 84 under the following conditions: There is no scalar interaction. The quarkonium anomalous magnetic moment is small and positive: 2α s /(3π) +... The expressions are valid only in the weak coupling regime (i.e. for the lowest quarkonium resonances). They are valid up to relative order α 2 s. * In the strong coupling regime (which applies to most of the charmonium resonances) at relative order v 2 much more terms than those predicted by naive potential models appear. The EFT allows to express them as Wilson loop amplitudes to be calculated eventually on the lattice.