Light Quark Mass Dependence of the X ( 3872 ) in an Effective Field - PowerPoint PPT Presentation

Light Quark Mass Dependence of the X ( 3872 ) in an Effective Field Theory Yu Jia 1 , 2 M. Jansen 3 H.-W. Hammer 3 , 4 1 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China 2 Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China 3 Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, Darmstadt, Germany 4 ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, Darmstadt, Germany Based on: Jansen, Hammer, Jia, PRD 89 (2014)014033 International Workshop on Heavy Quarkonium 2014, CERN November 10-14, 2014 1/24

Outline Introduction and Motivation 1 D 0 D ∗ 0 Scattering Amplitude XEFT and the ¯ 2 Binding Energy and Scattering Length 3 Conclusion and Outlook 4 2/24

Introduction and Motivation First observation by the Belle Collaboration [Choi et al., 2003] Determination J PC = 1 ++ by LHCb [Aaij et al., 2013] [Chatrchyan et al., 2013] 3/24

Introduction and Motivation Interpretations: tetraquark, charmonium, hadronic molecule Mass of the X (3872) close to D 0 D ∗ 0 threshold Particle Content of the X (3872) � ¯ 1 D 0 D ∗ 0 + D 0 ¯ D ∗ 0 � X = √ 2 Recent observation of a candidate for the X on the lattice [Prelovsek and Leskovec, 2013] Performed on rather small lattices for large quark masses Previous work: Unitarized heavy meson ChpT: no sensitivity to contact interactions [Wang and Wang, 2013] Non-relativistic Faddeev-type three-body equations: contact interactions essential [Baru et al., 2013] 4/24

Basics of XEFT Universal properties due to small binding energy E X = m D ∗ + m D − M X = (0 . 17 ± 0 . 26) MeV Corrections calculable in XEFT [Fleming et al., 2007] Decay rate for X → D 0 ¯ D 0 π 0 as a function of E X 5/24

Basics of XEFT Similar to KSW theory for NN scattering [Kaplan et al., 1998] Includes pions perturbatively Unnaturally large NNLO coefficients [Fleming et al., 2000] Nearness of D 0 D ∗ 0 hyperfine splitting and pion mass induces small mass scale µ 2 = ∆ 2 − m 2 π Mass scale µ , D ( ∗ )0 and pion momenta and binding momentum of same order Q ≪ m π , m D , m D ∗ Pions and D ( ∗ )0 mesons treated non-relativistically Integrated out charged D ( ∗ ) ± mesons Effective field theory: 1 / a suppression [Braaten and Kusunoki, 2004] Charmonium- hadronic molecule hybrid: charged states small contribution [Takizawa and Takeuchi, 2013] Takes finite width of the D ∗ 0 into account 6/24

XEFT Lagrangian − → − →     ∇ 2 ∇ 2 L = D †  D + D †  D  i∂ 0 +  i∂ 0 + 2 m D ∗ 2 m D − → → − → −       ∇ 2 ∇ 2 ∇ 2 † D + ¯  ¯  π + ¯  ¯ D † D + π † D  i∂ 0 +  i∂ 0 +  i∂ 0 + + δ 2 m D ∗ 2 m D 2 m π g D D † · − → D · − → 1 � D † ¯ ∇ π † � ∇ π + ¯ + √ √ 2 m π + h.c. 2 f � ¯ � ¯ − C 0 � † · � D D + D ¯ D D + D ¯ D D 2 � ¯ � ¯ � † · + C 2 D ← → ∇ 2 D + D ← → ∇ 2 ¯ D D + D ¯ � D D + h.c. 16 � ¯ � ¯ − D 2 µ 2 � † · � D D + D ¯ D D + D ¯ D D + . . . , 2 7/24

Power Counting in XEFT ∼ Q − 1 ∼ Q − 2 - iC 0 ∼ Q 0 ∼ Q 5 - iC 2 p 2 ∼ Q 0 ∼ Q 1 - i g 1 √ 2 m π ( ε · p π ) √ - iD 2 µ 2 2 f 8/24

LO Scattering Amplitude i A - 1 = = + - iC 0 D 0 D ∗ 0 scattering amplitude to LO ¯ 2 π i 1 i A − 1 = − γ + √− 2 M DD ∗ E − i ǫ M DD ∗ 2 π γ ≡ M DD ∗ C 0 (Λ) + Λ γ 2 Pole at − E = 2 M DD ∗ 9/24

NLO Contributions to the Scattering Amplitude (I) (V) i A i A 0 0 i A 0 = + - iC 2 p 2 - iD 2 µ 2 (II) (III) (IV) i A i A i A 0 0 0 + + 2 + = + 10/24

Infrared Divergences in XEFT (VI) i A = 0 � 2 = ig 2 p ( i Λ − µ ) µ 2 1 � M DD ∗ i A (VI) A 2 0 − 1 6 f 2 2 2 π Infrared divergent Renormalization scale dependent Pion bubbles give contribution to the D ∗ self energy 11/24

Resummation for the D ∗ 0 Propagator iG = = + Σ OS Full D ∗ 0 propagator i iG = p 0 − p 2 / 2 m D ∗ + Σ OS + i ǫ = = + i Σ OS Σ OS g 2 Σ OS = 24 π f 2 i µ 3 Purely imaginary for m π < ∆, induces decay width for D ∗ 0 Real valued for m π ≥ ∆, induces mass shift for D ∗ 0 12/24

LO Scattering Amplitude i A - 1 = = + - iC 0 D 0 D ∗ 0 scattering amplitude to LO ¯ 2 π i 1 i A − 1 = − γ + √− 2 M DD ∗ E − i ǫ M DD ∗ 2 π γ ≡ M DD ∗ C 0 (Λ) + Λ γ 2 Pole at − E = 2 M DD ∗ 13/24

LO Scattering Amplitude i A - 1 = = + - iC 0 D 0 D ∗ 0 scattering amplitude to LO ¯ 2 π i 1 i A − 1 = − γ + √− 2 M DD ∗ E − 2 M DD ∗ Σ os − i ǫ M DD ∗ 2 π γ ≡ M DD ∗ C 0 (Λ) + Λ γ 2 2 M DD ∗ +Σ os Pole at − E = 13/24

NLO Contributions to the Scattering Amplitude (I) (V) i A i A 0 0 i A 0 = + - iC 2 p 2 - iD 2 µ 2 (II) (III) (IV) i A i A i A 0 0 0 + + 2 + = + 14/24

NLO Contributions to the Scattering Amplitude (I) (V) i A i A 0 0 i A 0 = + - iC 2 p 2 - iD 2 µ 2 (II) (III) (IV) i A i A i A 0 0 0 + + 2 + = + 14/24

NLO Scattering Amplitudes 2 π i 1 i A − 1 = − γ + η M DD ∗ 0 = − iC 2 � p 2 + 2 M DD ∗ Σ os − η + Λ � i A (I) A 2 − 1 C 2 − γ + Λ 0 = ig 2 1 + µ 2 1 − 4 p 2 � � �� i A (II) 4 p 2 log 0 6 f 2 µ 2 �� M DD ∗ = ig 2 ( − η + Λ) + i µ 2 � � 2 p i A (III) 2 p log 1 + A − 1 0 3 f 2 i η + µ − p 2 π � 2 = ig 2 � � � Λ � �� � M DD ∗ ( − η + Λ) 2 + µ 2 i A (IV) A 2 log + 1 + R − 1 0 6 f 2 2 η − i µ 2 π = − iD 2 µ 2 i A (V) A 2 0 − 1 C 2 0 − p 2 − 2 M DD ∗ Σ os − i ǫ � η ≡ R ≡ 1 � π + 2 � � � − γ E + log 4 3 2 15/24

One-Pion Exchange ij = ig 2 ( ε i · p π ) ( ε j · p π ) A (II) i ˆ 0 2 f 2 p 2 π − µ 2 → δ ij · ig 2 1 + µ 2 1 − 4 p 2 � � �� S-wave ≡ δ ij · i A (II) π − − − − log 0 6 f 2 4 p 2 µ 2 π Seperate amplitudes ˆ A ij = δ ij · A 16/24

Effective Range Expansion Relation between scattering amplitude and S-matrix S − 1 = e 2 i δ s − 1 = i pM DD ∗ A π Apply effective range expansion M DD ∗ A = − 1 2 π + 1 2 r s p 2 + . . . p cot δ s = ip + a s OPE in coordinate space oscillatory [Suzuki, 2005] ig 2 ( ε i · p π ) ( ε j · p π ) 2 f 2 p 2 π − µ 2 → ig 2 r )) cos ( µ r ) + µ r sin ( µ r ) F.T. − − 8 π f 2 ( ε i · ε j − 3 ( ε i · ˆ r ) ( ε j · ˆ + . . . r 3 Effective range expansion only valid up to order p 0 17/24

Suppression of the Two-Pion Exchange ∼ g 2 M DD ∗ µ × 4 πf 2 Expansion factor in KSW for NN scattering � ph � g 2 A M N m π ∼ 0 . 5 [Kaplan et al., 1998] 8 π f 2 Expansion factor in XEFT for DD ∗ scattering � ph � g 2 M DD ∗ µ ∼ 0 . 05 [Fleming et al., 2007] 4 π f 2 Quark mass dependent → estimate range of validity 18/24

Expansion factor 0.15 � 4 πf 2 � 0.1 g 2 M DD ∗ | µ | / 0.05 m 2 m π = ∆ π 0 π ) 2 ( m ph 1.0 1.5 2.0 19/24

Results for the Binding Energy 0.4 0.3 E X [MeV] 0.2 0.1 0 m 2 m π = ∆ π π ) 2 ph ( m 1.0 1.5 2.0 Red: LO contact interaction and OPE only Bounds: Natural ranges for NLO coefficients Green: Unnaturally large NLO coefficient 20/24

Results for the Scattering Length 13 11 a s [fm] 9 10.5 7 10.4 5 10.3 1.102 1.108 1.114 m 2 m π = ∆ π 3 π ) 2 ( m ph 1.0 1.5 2.0 Red: LO contact interaction and OPE only Bounds: Natural ranges for NLO coefficients 21/24

Conclusion and Outlook Conclusion XEFT applicable to calculate chiral extrapolations analytically Quark mass dependent contact interaction essential for renormalization X (3872) should be observable on the lattice High sensitivity of scattering length (cusp effect) Qualitative agreement with results from non-relativistic Faddeev-type three-body equations [Baru et al., 2013] Discrepancy with results from unitarized heavy meson ChpT [Wang and Wang, 2013] Outlook Extension to NNLO; Inclusion of charged D -mesons Relativistic pion fields for extrapolation to chiral limit Calculation of finite volume effects 22/24

Outlook Outlook: finite volume effects Binding energy of the X , E X � 0 . 5 MeV ⇒ S -wave Scattering length a s � 5 fm Recent simulation on lattice with a spatial size L ≈ 2 fm [Prelovsek and Leskovec, 2013] Finite volume corrections essential Periodic boundary conditions ⇒ allowed loop momenta q = 2 π n L , n ∈ Z d 3 q (2 π ) 3 → 1 � � Replace integrals by sums q = 2 π n L 3 L For m π ≫ ∆ use pw expansion to include effects of pions ⇒ analogous procedure as in pionless EFT [Beane et al., 2004] Close to and below threshold evaluate diagrams with pions explicitly 23/24

Outlook E X � MeV � 4 3 2 1 0 25 L � fm � 5 10 15 20 m π = 135MeV Dots: Binding energy in the finite volume Lines: Binding energy in the infinite volume 24/24

Renormalization of C 2 and D 2 C 2 = M DD ∗ r 0 2 ( C 0 ) 2 ≡ c 2 ( C 0 ) 2 2 π � Λ � 2 � D 2 = 6 f 2 � 2 π � � ( C 0 ) 2 d 2 + log − R g 2 µ ph M DD ∗ 25/24