The Heavy Quark Spin Symmetry and SU(3)-Flavour Partners of the X(3872)

Transcription

1 The Heavy Quark Spin Symmetry and SU(3)-Flavour Partners of the X(3872) Carlos Hidalgo, J. Nieves and M. Pavón-Valderrama Hypernuclear and Strange Particle Physics 2012 IFIC (CSIC - Universitat de València)

2 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

3 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

4 Introduction Molecular meson-antimeson systems have been predicted inspired by the existing similarities with the nucleon-nucleon system (Voloshin, Okun; 1976). So, as the deuteron is a nucleon-nucleon bound state, it would be natural to assume the existence of meson-antimeson bound states. Figure: Analogy between nuclear and meson-antimeson forces

5 Introduction This supposition has been tested several times: 1 De Rújula, Georgi, Glashow in 1977: Ψ(4040) as a D D resonance. 2 Jaffe in 1977: f 0 (980) y a 0 (980) as a K K bound state. 3 Belle collaboration in 2003: X (3872) as a D D resonance. This is a clear candidate to be a meson-antimeson molecule. Many new resonances X, Y, Z... have been detected which can also become candidates to be meson-antimeson molecules.

6 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

7 HQET The presence of a heavy quark induces a series of important simplifications in the QCD Lagrangian, which are consequence of a new approximate symmetry: HQSS. This symmetry can be used to derive an ( effective field theory from the QCD Lagrangian by expanding it in powers of ΛQCD m Q ). At lowest order, the QCD Lagrangian reads: L Q = Q v (iv D)Q v HQET and its symmetries So, at lowest order: Lagrangian doesn t depend on the heavy quark mass. Lagrangian doesn t depend on the heavy quark spin. HQET has a spin-flavour SU(2N h ) symmetry.

8 Covariant representation of mesonic fields. HQET eigenstates are would-be hadrons composed by a heavy quark with light quarks, light antiquarks and gluons. So, the fundamental state of a Q q meson [ Qq antimeson] can be represented by a field Hv Q [H Q v ] which includes both the pseudoscalar isospin triplet Pa Q = (P 0, P +, P s ) [P Q a = ( P 0, P, P s )] and the vector isospin triplet Pa Q [P Q a ]. A possible expression for this field (satisfying all the corresponding symmetries of each meson in the field) is: H (Q) v H ( Q) v (x) = H v (Q) (x) = 1 + /v [ /P (Q) v 2 (x) = γ 0 H v (Q) (x)γ 0 = [ ] /P ( Q) v P ( Q) 1 /v v γ 5 2 P (Q) v γ 5 ] [ ] /P (Q) 1 + /v v P v (Q) γ 5 2 H ( Q) v (x) = γ 0 H ( Q) v (x)γ 0 And, as HQET is a non-relativistic theory, Hv Q corresponding mesons (antimesons). (H Q v ) only destroys its

9 Covariant representation of mesonic fields. H Q v [H Q v ] field transforms under spin-flavour symmetry rotations: H (Q) a H (Q)a ( S H (Q) U ) ( ), H ( Q)a UH ( a Q) S a ( U H (Q)) a ( S, H( Q)a S H( Q) U ) a being S a heavy quark spin transformation and U a light quark flavour rotation, with a the corresponding SU(3) index.

10 Effective field theory for meson-antimeson molecules. Describing meson-antimeson molecular system requires an effective Lagrangian. It should contain all the possible strong interactions that bound the meson-antimeson system. Figure: Possible meson-antimeson interactions at lowest order. Diagrams particularized to the case P P However, taking into account chiral symmetry of QCD, pion exchanges will be suppressed compared to contact interactions (numerically shown in Nieves and Pavon PRD84, ). As we will work at lowest order, our effective interaction Lagrangian will only consist of contact interactionsl.

11 Effective field theory for meson-antimeson molecules. At LO, the most general potential that respects HQSS takes the form: V 4 = + C [ ] [ A 4 Tr H(Q) H (Q) γ µ Tr H ( Q) H( Q) γ µ] + + C A λ [ ] [ 4 Tr H(Q) a λ i abh (Q) b γ µ Tr H ( Q) c λ i ( Q) cd H d γ µ] + + C [ ] ] B 4 Tr H(Q) H (Q) γ µ γ 5 Tr [H ( Q) H( Q) γ µ γ C B λ [ 4 Tr H(Q) a being λ Gell-Mann matrices. λ j ab H(Q) ] [ b γ µ γ 5 Tr H ( Q) c λ j H ] ( Q) cd d γ µ γ 5 At first order, this model based on HQSS only depends on four undetermined low energy constants (counter-terms)!!

12 Effective field theory for meson-antimeson molecules. So, once we have determined V, we find bound states by solving the Lippmann-Schwinger equation for each spin, isospin and charge-conjugation sector: T = V + V G T p T p = p V p + d 3 k p V k k T p E m 1 m 2 k2 2µ Bound states of this model will appear as poles in the T matrix. Ultraviolet divergences are removed introducing a Gaussian regulator Λ: p V p = V ( p, p ) = v e p2 /Λ 2 e p 2 /Λ 2 G = d 3 k (2π) 3 e 2k 2 /Λ 2 E m 1 m 2 k2 2µ

13 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

14 Determination of the counter-terms. To determine the four counter-terms, we have assumed: 1 X (3917) is a D D bound state with J PC = Y (4140) is a D s D s bound state with J PC = X (3872) is a D D bound state with J PC = The fourth condition will be obtained from the isospin violation in the X(3872) decays.

15 Isospin violation in X(3872) decays It has been difficult to understand the X(3872) decay widths in different isospin channels. In fact, if X(3872) had a well defined isospin, it would be hard to accomodate the following experimental ratio: ω {}}{ B(X (3872) J/ψ π + π π 0 ) B(X (3872) J/ψ π } + {{ π 0.8 ± 0.3. } ) ρ

16 Isospin violation in X(3872) decays To explain this ratio there are two different scenarios: X(3872) isospin is well defined but then isospin is not conserved in these strong decays. X(3872) is not well defined and strong interactions conserve isospin. We ll assume this latter scenario. Despite our potential conserves isospin, the kinetic term of the whole Lagrangian doesn t because of the mass difference between the charged components (D + D ) and the neutral component (D 0 D 0 ) which is around 8 MeV. As, the X(3872) is very close to D 0 D0 threshold, this mass difference cannot be neglected and, then, the isospin operator doesn t commute with the whole Hamiltonian and, thus, it is not a good quantum number of the system.

17 Isospin violation in X(3872) decays A more detailed analysis of X(3872) decays (Hanhart et al. 2012) gives the ratio between the amplitudes of the decays: R X = M(X J/ψ ρ) M(X J/ψ ω) =

18 Isospin violation in X(3872) decays And, in our model (depicted in the figure), the same ratio is given by: being R X = M(X J/ψ ρ) M(X J/ψ ω) = g ρ g ω ( ˆψ 1 ˆψ 2 ˆψ 1 + ˆψ 2 g ω = M ω (D D (I = 0) J/ψ ω) g ρ = M ρ (D D (I = 1) J/ψ ρ) and ˆψ 1 y ˆψ 2 an average of the neutral and charged D D wave functions in the vicinities of the origin. ) Figure: X(3872) decay mechanism in this model.

19 Isospin violation in X(3872) decays Because of the light flavour SU(3) symmetry of QCD Hamiltonian, it is satisfied: g ω g ρ = 2g φ And, using OZI rule (s s pair creation is suppressed) then: g φ 0 R X = g ω = g ρ M(X J/ψ ρ) M(X J/ψ ω) = ( ˆψ1 ˆψ 2 ˆψ 1 + ˆψ 2 ) =

20 Isospin violation in X(3872) decays If we now assume a vanishing D D interaction in the isospin I = 1 sector, it is found that the ratio R X only depends on the masses of the resonance and the thresholds of the different channels (Gammermann et al.,2010) and takes the value: R X 0.13 We improve on this, and consider a non-vanishing I = 1 interaction that we fit to the value for R X reported by Hanhart et al. Hence, we ll work with two coupled channels (neutral and charged channels) contact potential: ( V0 + V 1 V 0 V 1 ) V coupled = 1 2 V 0 V 1 V 0 + V 1 being V 0 and V 1, the well defined isospin potentials I = 0 and I = 1, respectively. Therefore, the experimental ratio of the ρ and ω decay widths of the X(3872) provides further constraints to the counter-terms.

21 Isospin violation in X(3872) decays Thus, we have determined the value of the different counter-terms a and we can predict a whole family of resonances related (SU(3) and HQSS partners) to the X(3872), X(3915) and Y(4140). a Gaussian regulator dependence appears as a higher order effect. There is a relation between the Gaussian regulator and the counter-terms value. This relation is very similar to RGE of other field theories.

22 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

23 I = 0 sector without hidden strangeness Figure: Predicted resonances for our model for the I = 0 sector without hidden strangeness. In every case, expect for 1 ++ y 2 ++, isospin is I = 0, and it is well defined.

24 I = 0 sector with hidden strangeness Figure: Predicted resonances for our model I = 0 sector with hidden strangeness. Predictions!!

25 I = 1 2 sector Figure: Predicted resonances for our model for the I = 1 2 sector. Predictions!! being (charge conjugation is not a good quantum number in this channel): ( C1A C V s = 1B C 1B C 1A )

26 I = 1 sector Figure: Predicted resonances for our model for the I = 1 sector. Predictions!!

27 Outline 1 Introduction 2 Effective field theory for heavy molecules (D D, D D...) 3 Determination of the counter-terms Isospin violation in X(3872) decays 4 Predictions I = 0 sector: SU(2) isoscalar and hidden strangeness. I = 1 2 sector (strangeness = ±1) I = 1 sector 5 Outlook and conclusions

28 Outlook and conclusions Light flavour symmetry and HQSS in heavy meson-antimeson systems, combined with the identification of three resonances, has allowed us to obtain a whole family of hidden charm molecules. Uncertainties quoted in tables only account for the approximate nature of HQSS ( 15% error) and those induced by the errors of R X. Pion exchanges and coupled channels should be considered. However, according to previous bibliography, these effects have a smaller contribution than expected and smaller than those expected from HQSS breaking terms. Some of the predicted resonances have already been experimentally detected or predicted by different models.