Determination of electric dipole transitions in heavy quarkonia - PowerPoint PPT Presentation

Determination of electric dipole transitions in heavy quarkonia using potential non-relativistic QCD arxiv:1701.02513, arxiv:1708.08465 Jorge Segovia, Sebastian Steinbeißer and Antonio Vairo Physik-Department-T30f Technische Universität München 07.11.2017 T30f Theoretische Teilchen- und Kernphysik E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 1/19

The heavy quarkonia The Charmonium and bottomonium systems were discovered in the 1970s Experimentally clear spectrum of narrow states below the open-flavor threshold Eichten et al. Rev. Mod. Phys. 80 (2008) ☞ Heavy quarkonia are bound states of a heavy quark and its antiquark c charmonium and b ¯ ( c ¯ b bottomonium). ☞ They can be classified in terms of the quantum numbers of a non-relativistic bound state → Similar to the positronium [ ( e + e − ) -bound state] in QED. ☞ Heavy quarkonia are a very well established multi-scale system which can serve as an ideal laboratory for testing all regimes of QCD. E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 2/19

The nonrelativistic expansion ☞ Heavy quarkonium is a non-relativistic system: v c ∼ 0 . 55 , v b ∼ 0 . 32 ( v light = 1 . 0 ) ☞ Heavy quarkonium is a multi-scale system: M ≫ p ∼ 1 / r ∼ Mv ≫ E ∼ Mv 2 ☞ Scales are entangled in full QCD ☞ Systematic expansions in the small heavy-quark velocity v may be implemented at the Lagrangian level by constructing suitable effective field theories (EFTs): Expanding QCD in p / M , E / M leads to NRQCD . → Bodwin, Braaten, and Lepage. Phys. Rev. D51 (1995) Expanding NRQCD in E / p leads to pNRQCD . → Brambilla et al. Nucl. Phys. B566 (2000) E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 3/19

There is another scale in QCD: Λ QCD ☞ The matching of QCD to NRQCD M ≫ Λ QCD → Perturbative matching. ☞ The matching of NRQCD to pNRQCD p ∼ 1 / r ≫ Λ QCD → Weak coupling regime. Perturbative matching. p ∼ 1 / r � Λ QCD → Strong coupling regime. Nonperturbative matching. E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 4/19

Potential nonrelativistic QCD at weak coupling In summary... ☞ Provides a QM description from FT: the matching coefficients are the interaction potentials and the leading order dynamical equation is of the Schrödinger type. ☞ The degrees of freedom in pNRQCD (at weak coupling) are color singlet and octet fields and ultra-soft gluon fields. ☞ Account for non-potential terms as well. Singlet to Octet transitions via ultra-soft gluons provide loop corrections to the leading potential picture. ☞ The Quantum Mechanical divergences are canceled by the NRQCD matching coefficients. ☞ Poincaré invariance is realized via exact relations between different matching coefficients. Potential non-relativistic QCD is the state-of-the-art tool for addressing Quarkonium bound state properties. ☞ Conventional meson spectrum: higher order perturbative corrections in v and α s . ☞ Inclusive and semi-inclusive decays, E1 and M1 transitions, EM line-shapes. ☞ Doubly- and triply-heavy baryons. ☞ Precise extraction of Standard Model parameters: m c , m b , α s , ... ☞ Exotic states such as gluelumps and hybrids. ☞ Properties of Quarkonium systems at finite temperature. E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 5/19

Electromagnetic transitions γ Aim: Compute the electric dipole (E1) transitions: ( k γ , � k ) χ bJ ( 1 P ) → γ Υ( 1 S ) with J = 0 , 1 , 2 h b ( 1 P ) → γη b ( 1 S ) P H = ( M H ,� 0 ) using the EFT called potential non-relativistic QCD. H �� � H ′ , − � k 2 γ + M 2 P H ′ = k H ☞ Electromagnetic transitions are often significant decay modes of heavy quarkonium states that are below the open-flavor threshold. ☞ They can be classified in a series of electric and magnetic multipoles. The electric dipole (E1) and the magnetic dipole (M1) are the most important ones. ☞ Large set of accurate experimental data taken by B-factories, τ -charm facilities and proton-proton colliders ask for a systematic and model-independent analysis. E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 6/19

Decay rate of electric dipole transitions ☞ The LO decay width, which scales as ∼ k 3 γ / ( mv ) 2 , is E1 = 4 � 2 Γ ( 0 ) � I ( 0 ) 9 α e/m e 2 Q k 3 ( n 1 → n ′ 0 ) γ 3 ☞ A probe of the internal structure of hadrons: � d k � ∞ � I ( k ) N ( n ℓ → n ′ ℓ ′ ) = d r r N R ∗ n ′ ℓ ′ ( r ) dr k R n ℓ ( r ) 0 ☞ Up to order k 3 γ / m 2 , the expressions we use for the decay rates under study are: I ( 0 ) � k 2 ( n 1 → n ′ 0 ) 1 + R S = 1 ( J ) − k γ Γ( n 3 P J → n ′ 3 S 1 + γ ) = Γ ( 0 ) γ 5 6 m − E 1 I ( 0 ) 60 ( n 1 → n ′ 0 ) 3 � J ( J + 1 ) � k γ ) I ( 1 ) ( n 1 → n ′ 0 ) + 2 I ( 0 ) � � �� ( n 1 → n ′ 0 ) 1 � 1 + κ e/m m 2 ( 1 + 2 κ e/m 2 1 + − 2 − 2 m + Q Q I ( 0 ) 2 ( n 1 → n ′ 0 ) 3 I ( 0 ) � k 2 � ( n 1 → n ′ 0 ) 1 + R S = 0 − k γ Γ( n 1 P 1 → n ′ 1 S 0 + γ ) = Γ ( 0 ) γ 5 6 m − E 1 I ( 0 ) 60 ( n 1 → n ′ 0 ) 3 R S = 1 ( J ) and R S = 0 include the initial and final state corrections due to higher order potentials and higher order Fock states. The remaining corrections within the brackets are the result of taking into account O ( v 2 ) -suppressed electromagnetic interaction terms in the Lagrangian. The terms proportional to the anomalous magnetic moment, κ em Q , go beyond our accuracy and are therefore not considered in the numerical analysis. E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 7/19

Coulomb-like potential in the LO Hamiltonian ☞ The states are solutions of the Schrödinger equation: H ( 0 ) ψ ( 0 ) r ) = E ( 0 ) ψ ( 0 ) n ℓ m ( � n ℓ m ( � r ) n ☞ Only the Coulomb-like term of the static potential is exactly included in the LO Hamiltonian ( C F = 4 / 3): � � ∇ 2 ∇ 2 − C F α s H ( 0 ) = − + V ( 0 ) ( r ) = − s 2 m r 2 m r r ☞ Therefore, ψ ( 0 ) r ) and E ( 0 ) n ℓ m ( � can be written in the hydrogen-like form: n n e − ρ n ψ ( 0 ) r ) = R n ℓ ( r ) Y ℓ m (Ω r ) = N n ℓ ρ ℓ 2 L 2 ℓ + 1 n ℓ m ( � n − ℓ − 1 ( ρ n ) Y ℓ m (Ω r ) = − m r C 2 F α 2 E ( 0 ) s n 2 n 2 �� 2 � 3 ( n − ℓ − 1 )! ρ n = 2 r / na , a = 1 / m r C F α s , N n ℓ = 2 n [( n + ℓ )!] . na These states are not eigenstates of the complete Hamiltonian due to higher order potentials and the presence of ultra-soft gluons that lead to singlet-to-octet transitions. ⇓ One has to consider corrections to the wave function which can contribute to the decay rate at the required order of precision! E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 8/19

Corrections due to higher order potentials (I) ☞ To account for O ( v 2 ) -corrections to the decay width, we need to consider the complete Hamiltonian: � ∇ 2 H = − + V s ( r ) + δ H 2 m r ☞ The static potential is: � α s � ∞ � � n V s ( r ) = V ( 0 ) � ( r ) 1 + a n ( ν, r ) s 4 π n = 1 ☞ The known O ( α n s ) radiative corrections to the LO static potential are: a 1 ( ν, r ) = a 1 + 2 β 0 ln ( ν e γ E r ) a 2 ( ν, r ) = a 2 + π 2 0 ln 2 ( ν e γ E r ) 3 β 2 0 + ( 4 a 1 β 0 + 2 β 1 ) ln ( ν e γ E r ) + 4 β 2 0 π 2 + 5 π 2 a 3 ( ν, r ) = a 3 + a 1 β 2 6 β 0 β 1 + 16 ζ 3 β 3 0 � 0 + 6 a 2 β 0 + 4 a 1 β 1 + 2 β 2 + 16 � 2 π 2 β 3 3 C 3 A π 2 ln ( ν e γ E r ) + � � ln 2 ( ν e γ E r ) + 8 β 3 0 ln 3 ( ν e γ E r ) 12 a 1 β 2 + 0 + 10 β 0 β 1 + δ a us 3 ( ν, ν us ) . E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 9/19

Corrections due to higher order potentials (II) ☞ The term δ H encodes the relativistic corrections to the static potential and to the non-relativistic kinetic operator: + V ( 2 ) m 2 + V ( 2 ) � 4 m 3 + V ( 1 ) ∇ 4 SI SD δ H = − m 2 m ☞ At order 1 / m 2 , we can split the contributions into spin-independent (SI) and spin-dependent (SD) terms: ( r ) + 1 V ( 2 ) SI ( r ) = V ( 2 ) 2 { V ( 2 ) ∇ 2 } + V ( 2 ) L 2 ( r ) � p 2 ( r ) , − � L 2 r V ( 2 ) SD ( r ) = V ( 2 ) S + V ( 2 ) S 2 + V ( 2 ) LS ( r ) � L · � S 2 ( r ) � S 12 ( r ) S 12 ☞ In the weak-coupling case, the above potentials read at leading (non-vanishing) order in perturbation theory: V ( 1 ) ( r ) = − C F C A α 2 V ( 2 ) s ( r ) = π C F α s δ ( 3 ) ( � , r ) , r 2 r 2 p 2 ( r ) = − C F α s L 2 ( r ) = C F α s V ( 2 ) V ( 2 ) , , 2 r 3 r LS ( r ) = 3 C F α s S 2 ( r ) = 4 π C F α s V ( 2 ) V ( 2 ) δ ( 3 ) ( � , r ) , 2 r 3 3 S 12 ( r ) = C F α s V ( 2 ) . 4 r 3 E1 transitions in heavy quarkonia using pNRQCD Sebastian Steinbeißer ( sebastian.steinbeisser@tum.de ) 10/19