Non-perturbative pion dynamics for the X (3872) Vadim Baru Institut für Theoretische Physik II, Ruhr-Universität Bochum Germany Institute for Theoretical and Experimental Physics, Moscow, Russia Meson 2014, Krakow in collaboration with E. Epelbaum, A. Filin, C. Hanhart, Yu. Kalashnikova, A. Kudryavtsev, U.-G. Meißner and A. Nefediev related articles: PLB 726, 537 (2013), PRD 84, 074029 (2011)
X(3872). Known facts Events / ( 0.005 GeV ) 35 b) Discovered by the Belle: B ± → K ± J/ Ψ π + π − 30 25 Observed: CDF, D0, BABAR, LHCb, BESIII 20 M X = 3871.68 ± 0.17 MeV PDG (2013) 15 10 ● MeV Belle Γ X < 2 . 3 5 ● 0 X → J/ Ψ γ C = + = Belle ⇒ 3.82 3.84 3.86 3.88 3.9 3.92 M(J/ ) (GeV) � � � � ● Br ( X → J/ Ψ ω ) /Br ( X → J/ Ψ ρ ) ≈ 0 . 8 ± 0 . 3 Belle (2003), BABAR(2010) ⟹ isospin violation ● Quantum numbers: 1 ++ and 2 -+ from BABAR/Belle angular distributions ● LHCb (2013): 2 -+ is rejected with CL = 8 σ ⟹ X is 1 ++
̅ ̅ X(3872). Mechanisms η c 2 (1 1 D 2 ) ruled out by LHCb ● Conventional charmonium: c 1 (2 3 P 1 ) χ 0 ● Exotics: tetraquark, molecular, mixture.... q ̅ q ̅ compact state tetraquarks Maiani,... q q ☛ Strong hadronic interactions with q ̅ q ̅ large s-wave scatt. length D D * ☛ natural effective range molecular q q Okun, Voloshin (1976), many works ☛ specific analytic properties: D * D unitarity cut Weinberg (1963-65) M X = 3871.68 ± 0.17 MeV; M D 0 D 0 * = 3871.80 ± 0.35 MeV ⟹ X is the S-wave bound state of with E B = 0.12 ± 0.26 MeV DD *
̅ ̅ ̅ From NN system to charm sector D * D Prediction of the molecular state in analogy to the deuteron Okun, Voloshin (1976), Törnqvist (1991) D * D π + heavy mesons π + heavy mesons D D * NN force Charmonium force
̅ ̅ ̅ ̅ ̅ ̅ From NN system to charm sector D * D Prediction of the molecular state in analogy to the deuteron Okun, Voloshin (1976), Törnqvist (1991) D * D π + heavy mesons π + heavy mesons D D * NN force Charmonium force NN X(3872) 2.5 3 -7 -0.5 0 8 MeV 0 140 MeV -2 Thresholds D 0 D 0* 𝜌 ∓ D 0 D ± 𝜌 0 D - D + d NNthr NN π D ∓ D *± X 𝜌 0 D 0 D 0 20-30 MeV binding 45 MeV momentum 𝜌 0 D 0 D 0 can go on shell ⟹ Im part range of m π p µ = 2 m π ( m D ∗ � m D � m π ) ' 45 MeV interaction questionable ⟹ 3-body unitarity is spoiled good approximation static OPE Small scales << m π ⟹ NR kinematics
̅ X(3872) as a bound state. Status. D * D Pionless: ● Asymptotic behavior of the X w.f. Voloshin (2004) ● Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012)
̅ X(3872) as a bound state. Status. D * D Pionless: ● Asymptotic behavior of the X w.f. Voloshin (2004) ● Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012) Pionful: Flemming et al. (2007), ● X-EFT: resummation of LO contact operators + perturbative pions Hammer et al. (2014), Mehen, Braaten et al. —similar to KSW in NN sector (2010-2011) ● Phenomenological deuteron-like studies with nonperturb. static OPE and formfactors For example: Liu et al. (2008), Thomas and Close (2008), Törnqvist (2004)
̅ ̅ X(3872) as a bound state. Status. D * D Pionless: ● Asymptotic behavior of the X w.f. Voloshin (2004) ● Contact theory AlFiky et al. (2006), Nieves, Valderrama (2012) Pionful: Flemming et al. (2007), ● X-EFT: resummation of LO contact operators + perturbative pions Hammer et al. (2014), Mehen, Braaten et al. —similar to KSW in NN sector (2010-2011) ● Phenomenological deuteron-like studies with nonperturb. static OPE and formfactors For example: Liu et al. (2008), Thomas and Close Goals of our study (2008), Törnqvist (2004) ● Investigate the role of 3-body dynamics on a near threshold resonance 𝜌 DD ● Check the validity of the static pion approximation ● Investigate the role of coupled channel effects: D 0 ¯ D ∗ 0 , D + ¯ D ∗− Needed to explain isospin violation: Br(J/ ψρ ) ≃ Br(J/ ψω ) Gamermann, Oset (2009) ● Study the dependence of the X binding energy on the light quark masses (chiral extrapolations)
̅ ̅ ̅ ̅ Formalism. Faddeev-type integral Eqs. for 1 ++ D D * Channels: D D * | 0 i = D 0 ¯ 0 i = ¯ D ∗ 0 , | ¯ D 0 D ∗ 0 , + = | c i = D + D ∗− , c i = D − D ∗ + | ¯ D D * a 00 a 00 V V d 3 s Z a nn 0 00 ( p , p 0 , E ) = λ 0 V nn 0 0 i ( p , s ) a mn 0 X ∆ i ( s ) V nm 00 ( p , p 0 ) − ( s , p 0 , E ) λ i i 0 i =0 ,c d 3 s Z a nn 0 c 0 ( p , p 0 , E ) = λ c V nn 0 ( p , s ) a mn 0 X ∆ i ( s ) V nm c 0 ( p , p 0 ) − ( s , p 0 , E ) , λ i ci i 0 i =0 ,c isospin coefficients ● Partial waves of DD*: ⟹ projection operators 3 S 1 , 3 D 1 ● Δ 0 and Δ c — propagators of the states ∣ 0> and ∣ c> DD * ● a 0 = ( a 00 − a c 0 ) / 2 — the X-amplitude
̅ DD * potential within chiral EFT small scale: bind. momentum, range q V EF T = V (0) + χ V (1) + χ 2 V (2) + · · · χ = Λ χ P T large scale: m ρ LO: -p p’ -p-p’ 3 S 1 3 S 1 + 3 S 1 3 S 1 V (0) = 3 D 1 3 D 1 -p’ p C 0 OPE ( ~ " d )( ~ p 0 · ~ " d ) p · ~ p 0 ) = g 2 V OP E ( ~ p, ~ p 0 ) 2 M − (2 m D + m ⇡ ) − ( ~ p + ~ p 2 p 0 2 2 m D + i 0 − 2 m D − 2 m π 3-body DD ̅ π prop. ● Coupling constant g from Γ ( D ∗ 0 → D 0 π 0 ) = 42 KeV ● Same 3-body cut appears due to dressing the D * propagator by π D loops
Renormalization of OPE ● Similar to NN Lepage (1997), Nogga et al. (2005), Epelbaum et al. (2006-2009), .... ● V OPE const ⟹ divergent integrals ⟹ regularize, e.g., with cutoff Λ ⟹ p →∞ − − − − → renormalize tuning C 0 ( Λ ) to reproduce the binding energy ● Renormalization group limit cycle Nogga et al. (2005), Bedaque et al.(1999), Braaten and Phillips (2004) 2 1 -2 ] C 0 [ GeV 0 -1 -2 0 2 4 6 � [GeV] Once renormalized, observables should not depend on Λ within the range of applicability!
̅ ̅ Partial width of the X(3872) due to the decay to 𝜌 0 D 0 D 0 VB, Filin, Hanhart, Kalashnikova, Kudryavtsev, Nefediev (2011) 𝜌 0 D 0 D 0 production rate near the X(3872) pole 4 E B =0.5 MeV D 0 ( ¯ D 0 ) D 0 ( ¯ D 0 ) -4 3 D ∗ 0 ( ¯ D ∗ 0 ) d Br/d E x 10 D ∗ 0 ( ¯ D ∗ 0 ) D ∗ 0 ( ¯ D ∗ 0 ) 2 π 0 π 0 ∆ − 1 0 ( s ) 1 a SS 00 ( s, p ) D 0 ( D 0 ) ¯ ¯ ¯ D 0 ( D 0 ) D 0 ( D 0 ) 0 -0.8 -0.6 -0.4 -0.2 E [MeV] 0.12 Conclusions: 0.1 static OPE ● Perturbative inclusion of pions is justified 0.08 � [MeV] our full with nonpert. pions ● Static approx. with nonpert. pions fails! 0.06 X-EFT band at NLO, Flemming et al.(2007) 0.04 ● Keeping 3-body dynamics in pionfull schemes is mandatory 0.02 LO, Voloshin (2004) 0 0.2 0.4 0.6 0.8 1 1.2 E B [MeV] D ̅ D for a study of the FSI see Guo et al (2014)
Chiral Extrapolations of the X(3872) with m π VB, Epelbaum Filin, Hanhart, Meißner, Nefediev (2013) m ph Strategy: vary all quantities with m π , expand around physical pion mass π ● f ( Λ) is needed to absorb extra Λ- dependence when m π 6 = m ph π 2 ! 1 + f ( Λ ) m 2 π − m ph ✓ δ m 4 ◆ C 0 ( Λ , m π ) = C 0 ( Λ ) π + O π M 2 M 4 ● could be fixed if we knew the slope from lattice � ( ∂ E B / ∂ m π ) � m π = m ph π ● Estimate f ( Λ): from two-pion exchange ⟹ ∣ f ( Λ ) ∣∼ 1, more conservatively: f ( Λ ) ∈ [-5, 5]
Chiral Extrapolations of the X(3872) with m π VB, Epelbaum Filin, Hanhart, Meißner, Nefediev (2013) m ph Strategy: vary all quantities with m π , expand around physical pion mass π ● f ( Λ) is needed to absorb extra Λ- dependence when m π 6 = m ph π 2 ! 1 + f ( Λ ) m 2 π − m ph ✓ δ m 4 ◆ C 0 ( Λ , m π ) = C 0 ( Λ ) π + O π M 2 M 4 ● could be fixed if we knew the slope from lattice � ( ∂ E B / ∂ m π ) � m π = m ph π ● Estimate f ( Λ): from two-pion exchange ⟹ ∣ f ( Λ ) ∣∼ 1, more conservatively: f ( Λ ) ∈ [-5, 5] physical pion mass 0 Conclusions: 1) If ⇒ f ( Λ )<0 ⟹ � ( ∂ E B / ∂ m π ) π < 0 -2 � m π = m ph negative slope, case 1), full bound state disappears quickly with m π -E B [MeV] negative slope, case 1), contact growth. Dominated by short-range physics -4 positive slope, case 2), full 2) If ⇒ f ( Λ )>0 ⟹ � ( ∂ E B / ∂ m π ) π > 0 positive slope, case 2), contact � m π = m ph -6 The X is bound deeper with m π growth. Strong influence of pion dynamics -8 0.5 1 1.5 2 ξ = m π / m π ph
Chiral Extrapolations of the X(3872) with m π VB, Epelbaum, Filin, Hanhart, Meißner, Nefediev (2013) m ph Strategy: vary all quantities with m π , expand around physical pion mass π ● f ( Λ) is needed to absorb extra Λ- dependence when m π 6 = m ph π 2 ! 1 + f ( Λ ) m 2 π − m ph ✓ δ m 4 ◆ C 0 ( Λ , m π ) = C 0 ( Λ ) π + O π M 2 M 4 ● could be fixed if we knew the slope from lattice � ( ∂ E B / ∂ m π ) � m π = m ph π ● Estimate f ( Λ): from two-pion exchange ⟹ ∣ f ( Λ ) ∣∼ 1, more conservatively: f ( Λ ) ∈ [-5, 5] 0 Conclusions ctd : ▶ First lattice measurement of the X(3872) -2 Prelovsek and Leskovec PRL(2013), talk on Saturday negative slope, case 1), full -E B [MeV] negative slope, case 1), contact -4 E B = -11 ± 7 MeV m π =266(4) MeV positive slope, case 2), full ⟹ scenario 2) seems to be preferred positive slope, case 2), contact -6 -8 0.5 1 1.5 2 ξ = m π / m π ph
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