Theory of quarkonium electromagnetic transitions Antonio Vairo - PowerPoint PPT Presentation

Theory of quarkonium electromagnetic transitions Antonio Vairo Technische Universit¨ at M¨ unchen

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

Radiative transitions: basics Two dominant single-photon-transition processes: (1) magnetic dipole transitions (M1) (2) electric dipole transitions (E1) (1) M1 transitions in the non-relativistic limit: � k γ r � ∞ k 3 2 � �� n 3 S 1 → n ′ 1 S 0 γ = 4 dr r 2 R n ′ 0 ( r ) R n 0 ( r ) j 0 γ Γ M1 3 α e 2 � � Q � � m 2 2 � 0 � j 0 ( k γ r/ 2) = 1 − ( k γ r ) 2 / 24 + . . . If k γ � r � ≪ 1 • n = n ′ allowed transitions • n � = n ′ hindered transitions

Radiative transitions: basics Two dominant single-photon-transition processes: (1) magnetic dipole transitions (M1) (2) electric dipole transitions (E1) (2) E1 transitions in the non-relativistic limit: 2   J ′ J 1 J ′ γ = 4   γ [ I 3 ( nL → n ′ L ′ )] 2 (2 J ′ +1) max { L,L ′ } Γ E1 3 αe 2 Q k 3 n 2 S +1 L J → n ′ 2 S +1 L ′ L ′ S L   where � ∞ dr r N R n ′ L ′ ( r ) R nL ( r ) I N ( nL → n ′ L ′ ) = 0 Note that, for equal energies and masses, M1 transitions are suppressed by a factor 1 / ( m � r � ) 2 ∼ v 2 with respect to E1 transitions, which are much more common.

Γ χ c (1 P ) → J/ψ γ / Γ χ b (3 P ) → Υ(3 S ) γ c k ( c ) 3 ≈ e 2 � r 2 � ( c ) Γ χ c (1 P ) → J/ψ γ γ � r 2 � ( b ) ≈ 33 +16 − 9 b k ( b ) 3 Γ χ b (3 P ) → Υ(3 S ) γ e 2 γ assuming � r 2 � ( b ) ≈ (1 . 5 ± 0 . 5) × � r 2 � ( c ) , k ( c ) ≈ 402 MeV and k ( b ) ≈ 174 MeV. γ γ ∗ from M χ c (1 P ) ≈ h c (1 P ) ≈ 3525 MeV, M J/ψ ≈ 3097 MeV, M χ b (3 P ) ≈ 10530 MeV and M Υ(3 S ) ≈ 10355 MeV.

Relativistic corrections • Relativistic corrections may be sizeable: about 30% for charmonium ( v 2 c ≈ 0 . 3 ) and 10% for bottomonium ( v 2 b ≈ 0 . 1 ). • For quarkonium radiative transitions, essentially one model/calculation has been used for over twenty years to account for relativistic corrections, based upon: relativistic equation with scalar and vector potentials; non-relativistic reduction; a somewhat imposed relativistic invariance to calculate recoil corrections. ◦ Grotch Owen Sebastian PR D30 (1984) 1924

Relativistic corrections and EFTs Nowadays, however, effective field theories (EFT) for quarkonium allow • to derive expressions for radiative transitions directly from QCD; • with a well specified range of applicability; • to determine a reliable error associated with the theoretical determinations; • to improve the theoretical determinations in a systematic way. ◦ Brambilla Pineda Soto Vairo RMP 77 (2005) 1423

Scales • p ∼ 1 E ∼ mv 2 ; in a non-relativistic system mv ≫ mv 2 r ∼ mv , • Λ QCD • k γ mv ≫ Λ QCD for weakly-coupled quarkonia ( J/ψ , η c , Υ(1 S ) , η b , ...); mv ∼ Λ QCD for strongly-coupled quarkonia (excited states); k γ ∼ mv 2 for hindered M1 transitions, most E1 transitions; ⇒ k γ r ≪ 1 k γ ∼ mv 4 for allowed M1 transitions.

Degrees of freedom • Degrees of freedom at scales lower than mv : Q states, with energy ∼ Λ QCD , mv 2 and momentum < Q - ¯ ∼ 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 .

Lagrangian − 1 µν F µν a − 1 4 F a 4 F em µν F µν em L pNRQCD = i∂ 0 − p 2 � � � LO in r � d 3 r Tr S † + m − V s S iD 0 − p 2 � � � + O † m − V o O � � O † r · g E S + S † r · g E O +Tr NLO in r + 1 � � O † r · g E O + O † O r · g E 2 Tr [if mv ≫ Λ QCD ] + · · · + L γ

L γ L M1 + L E1 L γ = + . . . γ γ � 1 � S † , σ · ee Q B em � L M1 2 m V M1 = Tr S γ 1 + 1 � O † , σ · ee Q B em � 2 m V M1 O [if mv ≫ Λ QCD ] 1 V M1 1 � r × ee Q B em ��� 2 S † , σ · � ˆ � ˆ + r × S 4 m 2 r V M1 1 � S † , σ · ee Q B em � 3 + S 4 m 2 r � 1 � S † , σ · ee Q B em � 4 m 3 V M1 ∇ 2 + r S + · · · 4 ◦ Brambilla Jia Vairo PR D73 (2006) 054005

L γ � L E1 V E1 S † r · ee Q E em S = Tr γ 1 + V E1 O † r · ee Q E em O [if mv ≫ Λ QCD ] 1 + 1 24 V E1 S † r · [( r · ∇ ) 2 ee Q E em ]S 2 + i 4 m V E1 S † { ∇ · , r × ee Q B em } S 3 i 12 m V E1 S † { ∇ r · , r × [( r · ∇ ) ee Q B em ] } S + 4 + 1 4 m V E1 [S † , σ ] · [( r · ∇ ) ee Q B em ]S 5 � i [S † , σ ] · ( ee Q E em × ∇ r )S + · · · 4 m 2 V E1 − 6 ◦ Brambilla Jia Vairo PR D73 (2006) 054005

Matching The matching consists in the calculation of the coefficients V . They get contributions from • hard modes ( ∼ m ): iD 0 + D 2 2 m + c em � � 2 m σ · ee Q B em + · · · ¯ / − m ) ψ → ψ † F ψ ( iD ψ From HQET: ≡ 1 + κ em = 1 + 2 α s c em 3 π + . . . F is the quark magnetic moment. ◦ Grozin Marquard Piclum Steinhauser NP B789 (2008) 277 (3 loops) • soft modes ( ∼ mv ).

M1 operator at O (1) S † , σ · e B em � � V M1 S 1 2 m � � � � V M1 = hard × soft 1 � � = 1 + 2 α s ( m ) = c em • hard + · · · F 3 π • Since σ · e B em ( R ) behaves like the identity operator to all orders V M1 does not get soft contributions. 1

t f t 1 � t f + + dt t t i t 2 t i σ · ee Q B em c em F 2 m � t f = dt t i Diagrammatic factorization of the magnetic dipole coupling in the SU(3) f limit. • The argument is similar to the factorization of the QCD corrections in b → u e − ¯ ν e , which leads to √ u L γ µ b L to all orders in α s . L eff = − 4 G F / 2 V ub ¯ e L γ µ ν L ¯

M1 operator at O (1) S † , σ · e B em � � V M1 S 1 2 m = 1 + 2 α s ( m ) V M1 • + · · · 1 3 π • No large quarkonium anomalous magnetic moment! ◦ Dudek Edwards Richards PR D73 (2006) 074507 (lattice)

M1 operators at O ( v 2 ) V M1 V M1 1 1 � r × ee Q B em ��� � S † , σ · ee Q B em � 2 S † , σ · 3 � ˆ � ˆ r × S and S 4 m 2 4 m 2 r r c F σ · B /m � � � � � � � � + + ... = hard × soft � � � � � � c s σ · ( A em × E ) /m 2 A · A em /m � � � � = r 2 V ′ • to all orders hard = 2 c F − c s = 1 ; soft s / 2 ◦ Brambilla Gromes Vairo PL B576 (2003) 314 (Poincar´ e invariance) Luke Manohar PL B286 (1992) 348 (reparameterization invariance) V M1 = r 2 V ′ s / 2 and V M1 • = 0 2 3 • No scalar interaction!

M1 operators at O ( v 2 ) S † , σ · e B em � � ∇ 2 V M1 r S 4 4 m 3 � � � � V M1 = hard × soft 4 � � • hard = 1 ◦ Manohar PR D56 (1997) 230 (reparameterization invariance) � � • soft = 1 to all orders ◦ Brambilla Pietrulewicz Vairo PRD 85 (2012) 094005 V M1 • = 1 4

O ( v 2 ) corrections to weakly-coupled quarkonia O † , σ · e B em � � Coupling of photons with octets: V M1 O [if mv ≫ Λ QCD ] 1 2 m × δZ H + + + = 0 r · g E • If mv 2 ∼ Λ QCD the above graphs are potentially of order Λ 2 QCD / ( mv ) 2 ∼ v 2 . • The contribution vanishes, for σ · e B em ( R ) behaves like the identity operator. • There are no non-perturbative contributions at O ( v 2 ) ! • This is not the case for strongly-coupled quarkonia: V M1 1 � S † , σ · ee Q B em � 5 non-perturbative corrections affect the operator S . m 3 r 2

J/ψ → η c γ d 3 k � (2 π ) 3 (2 π ) δ ( E J/ψ − k − E η c k ) |� γ ( k ) η c |L γ | J/ψ �| 2 Γ J/ψ → η c γ = p

J/ψ → η c γ Up to order v 2 the transition J/ψ → η c γ is completely accessible by perturbation theory. k 3 α s ( M J/ψ / 2) � � Γ J/ψ → η c γ = 16 − 32 γ 3 αe 2 27 α s ( p J/ψ ) 2 1 + 4 c M 2 3 π J/ψ The normalization scale for the α s inherited from κ em 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 ≈ 2 mα s ( p J/ψ ) / 3 ≈ 0 . 8 GeV ∼ mv . Γ J/ψ → η c γ = (1 . 5 ± 1 . 0) keV to be compared with the non-relativistic result ≈ 2 . 83 keV. ◦ Brambilla Jia Vairo PR D73 (2006) 054005

J/ψ → η c γ (experimental status) • Only one direct experimental measurement existed for long time: Γ J/ψ → η c γ = (1 . 14 ± 0 . 23) keV ◦ Crystal Ball coll. PR D34 (1986) 711 • The situation changed in the last few years: Γ J/ψ → η c γ = (1 . 85 ± 0 . 08 ± 0 . 28) keV ◦ CLEO coll. PRL 102 (2009) 011801 Γ J/ψ → η c γ = (2 . 17 ± 0 . 14 ± 0 . 37) keV (preliminary?) ◦ KEDR coll. Chin. Phys. C34 (2010) 831