Motivation Basic formalism E1 transitions Electric dipole transitions of heavy quarkonium Piotr Pietrulewicz TU München, T30f in collaboration with N. Brambilla and A. Vairo Hadron 2011 14.06.2011 Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Basic formalism E1 transitions Outline Motivation 1 Basic formalism 2 Effective Field Theory approach to heavy quarkonium Quarkonium states and transitions E1 transitions 3 Definition & non-relativistic limit Matching of the Lagrangian Wave-function corrections Results Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Basic formalism E1 transitions Why should one study EM transitions? information about the quarkonium spectrum and the wave-functions significant contributions to the decay rate (at least for E1) new experimental data provided in the last and next few years (CLEO, BES, B factories) Figure: K. Nakamura et al. (PDG), J. Phys. G 37 (2010) Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Basic formalism E1 transitions What has been done? phenomenological approach: QCD motivated potential models Grotch et al., Phys. Rev. D 30 (1984) Eichten et al., Rev.Mod.Phys. 80 (2008) → Cornell potential, Buchmüller-Tye potential, ... BUT: strict model-independent derivation missing, systematic procedure for relativistic corrections desirable Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Basic formalism E1 transitions What has been done? phenomenological approach: QCD motivated potential models Grotch et al., Phys. Rev. D 30 (1984) Eichten et al., Rev.Mod.Phys. 80 (2008) → Cornell potential, Buchmüller-Tye potential, ... BUT: strict model-independent derivation missing, systematic procedure for relativistic corrections desirable lattice QCD (quenched): Dudek et al., Phys. Rev. D 73, 074507 (2006) Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Basic formalism E1 transitions What has been done? phenomenological approach: QCD motivated potential models Grotch et al., Phys. Rev. D 30 (1984) Eichten et al., Rev.Mod.Phys. 80 (2008) → Cornell potential, Buchmüller-Tye potential, ... BUT: strict model-independent derivation missing, systematic procedure for relativistic corrections desirable lattice QCD (quenched): Dudek et al., Phys. Rev. D 73, 074507 (2006) EFT treatment of radiative decays: pNRQCD → M1 transitions Brambilla et al., Phys. Rev. D 73 (2006) → still missing: treatment of E1 transitions Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions Basic formalism EFT for heavy quarkonium Description of decay processes Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions Scales in quarkonium separation of scales in heavy quarkonium m ≫ p ∼ mv ≫ E ∼ mv 2 where v 2 ≪ 1 ( v 2 ≈ 0 . 1 for b ¯ b , v 2 ≈ 0 . 3 for c ¯ c ) → systematic treatment of relativistic corrections in powers of v → language of effective field theories appropriate Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions Scales in quarkonium separation of scales in heavy quarkonium m ≫ p ∼ mv ≫ E ∼ mv 2 where v 2 ≪ 1 ( v 2 ≈ 0 . 1 for b ¯ b , v 2 ≈ 0 . 3 for c ¯ c ) → systematic treatment of relativistic corrections in powers of v → language of effective field theories appropriate weakly coupled quarkonia ( E � Λ QCD ) → perturbative treatment with Coulomb potential at leading order (valid for the ground states J /ψ , Υ( 1 S ) , η c , η b ) α s ( m ) v 2 ∼ α s ( mv ) v ∼ α s ( mv 2 ) ∼ 1 Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions Effective field theories for quarkonium LONG−RANGE SHORT−RANGE QUARKONIUM QUARKONIUM / QED QCD/QED m perturbative matching perturbative matching µ mv NRQCD/NRQED µ mv 2 non−perturbative perturbative matching matching pNRQCD/pNRQED Figure: A. Vairo, arXiv 0902.3346 (2009) Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions NRQCD integrate out energy & momentum modes of order m from QCD Lagrangian � � iD 0 + D 2 2 m + D 4 ϕ † L = 8 m 3 + . . . ϕ + g ϕ † � c F 2 m σ · B + i c s � 8 m 2 σ · [ D × , E ] + . . . ϕ � c em 2 m σ · B em + i c em � F 8 m 2 σ · [ D × , E em ] + . . . s + ee Q ϕ † ϕ + c . c . + L light + L YM coefficients by matching with QCD Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions pNRQCD (for weak coupling) integrate out → quarks with energy & momentum ∼ mv → gluons & photons of energy or momentum ∼ mv new degrees of freedom: Q ¯ Q color singlet and octet fields Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions pNRQCD (for weak coupling) integrate out → quarks with energy & momentum ∼ mv → gluons & photons of energy or momentum ∼ mv new degrees of freedom: Q ¯ Q color singlet and octet fields Lagrangian � � � i ∂ 0 + ∇ 2 4 m + ∇ 2 � r d 3 r Tr S † m − V S S L pNRQCD = � � 4 m + ∇ 2 iD 0 + D 2 r + O † m − V O O + gV A ( O † r · E S + S † r · E O ) { O † , r · E } � + gV B O + . . . 2 + L γ pNRQCD + L light + L YM Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions pNRQCD (for weak coupling) now: Only relevant degrees of freedom present high energy dynamics encoded in Wilson coefficients (obtained by matching with NRQCD at energy mv ) definite power counting of operators r 1 / mv ∼ ( mv 2 ) 2 E , B ∼ E em , B em k 2 ∼ γ mv 2 , k γ ∇ = ∂/∂ R ∼ Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Motivation Effective Field Theory approach to heavy quarkonium Basic formalism Quarkonium states and transitions E1 transitions Quarkonium states and transitions quarkonium state (leading Fock space component): � � d 3 r e i P · R Tr � � | H ( P , λ ) � = d 3 R φ H ( λ ) ( r ) S † ( r , R ) | 0 � , at leading order: � � 2 − ∇ r H ( 0 ) S φ ( 0 ) + V ( 0 ) φ ( 0 ) H ( λ ) = E ( 0 ) H ( λ ) φ ( 0 ) H ( λ ) = m S H ( λ ) at higher orders: wave-function corrections due to higher order potentials and singlet-octet transitions → calculation of decay rates for H → H ′ γ in CM frame Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Definition & non-relativistic limit Motivation Matching of the Lagrangian Basic formalism Wave-function corrections E1 transitions Results E1 Transitions Work in progress Formalism as for M1 transitions in N. Brambilla et al. (2006) Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Definition & non-relativistic limit Motivation Matching of the Lagrangian Basic formalism Wave-function corrections E1 transitions Results General properties definition: ∆ S = 0 , | ∆ L | = 1 change in parity, no change in C parity Examples 1 3 P J → 1 3 S 1 ( χ c → J /ψγ , χ b → Υ( 1 S ) γ ) 1 1 P 1 → 1 1 S 0 ( h c → η c γ , h b → η b γ ) for the considered transitions: k γ ∼ mv 2 Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
Definition & non-relativistic limit Motivation Matching of the Lagrangian Basic formalism Wave-function corrections E1 transitions Results Nonrelativistic limit leading order operator for E1 transitions � S † r · E em S L E 1 = ee Q d 3 r Tr � � Nonrelativistic decay rate k 3 Γ n 3 P J = 0 , 1 , 2 → n ′ 3 S 1 γ = 4 9 α em e 2 Q k 3 γ I 2 3 ( n 1 → n ′ 0 ) ∼ γ m 2 v 2 � ∞ I 3 ( n 1 → n ′ 0 ) = dr r 3 R n ′ 0 ( r ) R n 1 ( r ) 0 differences to M1 transitions: → leading order amplitude depends on the wave-function → enhancement of E1 transitions by factor 1 / v 2 now: relativistic corrections of O ( v 2 ) Piotr Pietrulewicz Electric dipole transitions of heavy quarkonium
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