How the small hyperfine splitting of P-wave mesons evades large loop corrections Tim Burns INFN, Roma arXiv:1105.2533
Spin-dependence in quark models Mass formula in perturbation theory, M SLJ = M + ∆ s � 1 1 2 � S + ∆ t � T � SLJ + ∆ o � L · S � SLJ , 2 for mesons with spin S , orbital L and total J angular momenta. ◮ � . . . � are model independent. ◮ M and the ∆ ’s are model dependent, but common.
Spin-dependence in quark models Mass formula in perturbation theory, M SLJ = M + ∆ s � 1 1 2 � S + ∆ t � T � SLJ + ∆ o � L · S � SLJ , 2 for mesons with spin S , orbital L and total J angular momenta. ◮ � . . . � are model independent. ◮ M and the ∆ ’s are model dependent, but common.
Spin-dependence in quark models Mass formula in perturbation theory, M SLJ = M + ∆ s � 1 1 2 � S + ∆ t � T � SLJ + ∆ o � L · S � SLJ , 2 for mesons with spin S , orbital L and total J angular momenta. ◮ � . . . � are model independent. ◮ M and the ∆ ’s are model dependent, but common.
P-wave mesons: theory Four equations, and four unknowns: M 1 P 1 = M − 3 4 ∆ s M 3 P 0 = M + 1 4 ∆ s + 2 ∆ t − 2 ∆ o M 3 P 1 = M + 1 4 ∆ s − ∆ t − ∆ o M 3 P 2 = M + 1 4 ∆ s + 1 5 ∆ t + ∆ o Hyperfine splitting: 1 � � M 3 P 0 + 3 M 3 P 1 + 5 M 3 P 2 − M 1 P 1 = ∆ s ≈ 0 9
P-wave mesons: experiment Charmonia: M χ c ( 1P ) − M h c ( 1P ) = − 0 . 05 ± 0 . 19 ± 0 . 16 MeV Bottomonia: M χ b ( 1P ) − M h b ( 1P ) = + 2 ± 4 ± 1 MeV (BaBar) M χ b ( 1P ) − M h b ( 1P ) = + 1 . 62 ± 1 . 52 MeV (Belle) M χ b ( 2P ) − M h b ( 2P ) = + 0 . 48 + 1 . 57 − 1 . 22 MeV (Belle)
Mass shifts due to channel coupling Coupling to open flavour pairs ( QQ ) ↔ ( Qq )( qQ ) ◮ unquenching causes mass shifts ◮ χ 0 , χ 1 , χ 2 and h couple to different channels and with different strengths, so their mass shifts differ ◮ expect violations to the mass formula 1 � � M 3 P 0 + 3 M 3 P 1 + 5 M 3 P 2 − M 1 P 1 = 0 9
Mass shifts due to channel coupling Charmonia Bottomonia Mass shifts of Mass shifts of ◮ χ c 0 , χ c 1 , χ c 2 and h c , ◮ χ b 0 , χ b 1 , χ b 2 and h b , due to couplings due to couplings D ∗ , D ∗ ¯ ◮ D ¯ D , D ¯ ◮ B ¯ B , B ¯ B ∗ , B ∗ ¯ D ∗ , and B ∗ , and ◮ D s ¯ D s , D s ¯ s ¯ ◮ B s ¯ B s , B s ¯ s ¯ D ∗ s , D ∗ D ∗ B ∗ s , B ∗ B ∗ s s Literature Barnes & Swanson (BT) Kalashnikova (K) Li, Meng & Chao (LMC) Yang, Li, Chen & Deng (YLCD) Ono & Törnqvist (OT) Liu & Ding (LD)
Mass shifts due to channel coupling ∆ M 3 P 0 ∆ M 3 P 1 ∆ M 3 P 2 ∆ M 1 P 1 Induced ∆ s BS (1P, cc ) 459 496 521 504 K (1P, cc ) 198 215 228 219 LMC (1P, cc ) 35 38 63 52 YLCD (1P, cc ) 131 152 175 162 OT (1P, cc ) 173 180 185 182 OT (1P, bb ) 43 44 45 44 OT (2P, bb ) 55 56 58 57 80 . 777 84 . 823 87 . 388 85 . 785 LD (1P, bb ) 73 . 578 77 . 608 80 . 146 78 . 522 LD (2P, bb ) ◮ ∆ M SLJ can be very large ◮ ∆ M S ′ L ′ J ′ − ∆ M SLJ is smaller ◮ − 1 � � ∆ M 3 P 0 + 3 ∆ M 3 P 1 + 5 ∆ M 3 P 2 + ∆ M 1 P 1 is smaller still 9
Mass shifts due to channel coupling ∆ M 3 P 0 ∆ M 3 P 1 ∆ M 3 P 2 ∆ M 1 P 1 Induced ∆ s BS (1P, cc ) 459 496 521 504 K (1P, cc ) 198 215 228 219 LMC (1P, cc ) 35 38 63 52 YLCD (1P, cc ) 131 152 175 162 OT (1P, cc ) 173 180 185 182 OT (1P, bb ) 43 44 45 44 OT (2P, bb ) 55 56 58 57 80 . 777 84 . 823 87 . 388 85 . 785 LD (1P, bb ) 73 . 578 77 . 608 80 . 146 78 . 522 LD (2P, bb ) ◮ ∆ M SLJ can be very large ◮ ∆ M S ′ L ′ J ′ − ∆ M SLJ is smaller ◮ − 1 � � ∆ M 3 P 0 + 3 ∆ M 3 P 1 + 5 ∆ M 3 P 2 + ∆ M 1 P 1 is smaller still 9
Mass shifts due to channel coupling ∆ M 3 P 0 ∆ M 3 P 1 ∆ M 3 P 2 ∆ M 1 P 1 Induced ∆ s BS (1P, cc ) 459 496 521 504 K (1P, cc ) 198 215 228 219 LMC (1P, cc ) 35 38 63 52 YLCD (1P, cc ) 131 152 175 162 OT (1P, cc ) 173 180 185 182 OT (1P, bb ) 43 44 45 44 OT (2P, bb ) 55 56 58 57 80 . 777 84 . 823 87 . 388 85 . 785 LD (1P, bb ) 73 . 578 77 . 608 80 . 146 78 . 522 LD (2P, bb ) ◮ ∆ M SLJ can be very large ◮ ∆ M S ′ L ′ J ′ − ∆ M SLJ is smaller ◮ − 1 � � ∆ M 3 P 0 + 3 ∆ M 3 P 1 + 5 ∆ M 3 P 2 + ∆ M 1 P 1 is smaller still 9
Mass shifts due to channel coupling ∆ M 3 P 0 ∆ M 3 P 1 ∆ M 3 P 2 ∆ M 1 P 1 Induced ∆ s BS (1P, cc ) 459 496 521 504 K (1P, cc ) 198 215 228 219 LMC (1P, cc ) 35 38 63 52 YLCD (1P, cc ) 131 152 175 162 OT (1P, cc ) 173 180 185 182 OT (1P, bb ) 43 44 45 44 OT (2P, bb ) 55 56 58 57 80 . 777 84 . 823 87 . 388 85 . 785 LD (1P, bb ) 73 . 578 77 . 608 80 . 146 78 . 522 LD (2P, bb ) ◮ ∆ M SLJ can be very large ◮ ∆ M S ′ L ′ J ′ − ∆ M SLJ is smaller ◮ − 1 � � ∆ M 3 P 0 + 3 ∆ M 3 P 1 + 5 ∆ M 3 P 2 + ∆ M 1 P 1 is smaller still 9
Mass shifts due to channel coupling ∆ M 3 P 0 ∆ M 3 P 1 ∆ M 3 P 2 ∆ M 1 P 1 Induced ∆ s − 1 . 8 BS (1P, cc ) 459 496 521 504 K (1P, cc ) 198 215 228 219 − 1 . 3 − 2 . 9 LMC (1P, cc ) 35 38 63 52 − 0 . 4 YLCD (1P, cc ) 131 152 175 162 − 0 . 0 OT (1P, cc ) 173 180 185 182 − 0 . 4 OT (1P, bb ) 43 44 45 44 OT (2P, bb ) 55 56 58 57 − 0 . 0 80 . 777 84 . 823 87 . 388 85 . 785 − 0 . 013 LD (1P, bb ) 73 . 578 77 . 608 80 . 146 78 . 522 − 0 . 048 LD (2P, bb ) ◮ ∆ M SLJ can be very large ◮ ∆ M S ′ L ′ J ′ − ∆ M SLJ is smaller ◮ − 1 � � ∆ M 3 P 0 + 3 ∆ M 3 P 1 + 5 ∆ M 3 P 2 + ∆ M 1 P 1 is smaller still 9
Mass shifts due to channel coupling The models differ in many ways: ◮ perturbation theory vs . coupled channel equations ◮ harmonic oscillator vs . coulomb + linear wavefunctions ◮ universal vs . flavour-dependent wavefunctions ◮ exact SU(3) vs . broken SU(3) in pair creation But have important common features: ◮ coupling ( QQ ) → ( Qq )( qQ ) has qq in spin triplet ◮ spin and spatial degrees of freedom factorise ◮ spin is conserved
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
Computing the mass shifts The mass shift ◮ of a state with S , L , J quantum numbers ◮ due to coupling with mesons spins s 1 and s 2 in partial wave l p 2 | A l ( p ) | 2 � ∆ M s 1 s 2 l SLJ = C s 1 s 2 l dp ǫ s 1 s 2 SLJ SLJ + p 2 / 2 µ s 1 s 2 ◮ ǫ s 1 s 2 SLJ and µ s 1 s 2 are binding energy and reduced mass ◮ C s 1 s 2 l SLJ depends only on the angular momenta ◮ A l ( p ) depends only on the spatial degrees of freedom ◮ A l ( p ) is common to all channels if the radial wavefunctions χ 0 = χ 1 = χ 2 = h and D = D ∗
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