Study of the DKK and DK systems - indico.ihep.ac.cn · Outline 1 Introduction The Faddeev Equations DK Cluster: The D s0(2317) Two-Body Amplitudes 2 Formalism The Fixed Center Approximation

Study of the DKK and DKK systems

Vinıcius Rodrigues Debastiani

Instituto de Fısica Corpuscular,Universidad de Valencia - CSIC

visiting

Institute of Modern Physics,Chinese Academy of Sciences

The Seventh Asia-Pacific Conference onFew-Body Problems in Physics

(APFB 2017) Guilin, China

V. R. DEBASTIANI (IFIC, UV-CSIC / IMP, CAS) Study of the DKK and DKK systems APFB 2017 Guilin, China 1 / 30

Outline

1 IntroductionThe Faddeev EquationsDK Cluster: The D∗s0(2317)Two-Body Amplitudes

2 FormalismThe Fixed Center ApproximationDKK SystemChiral Unitary ApproachDKK SystemEnergy Distribution

3 ResultsDKK SystemKK Interaction: The a0(980) and f0(980)DKK System

4 Summary


The Faddeev EquationsIntroduction

L. D. Faddeev, Sov. Phys. JETP 12, 1014 (1961), [Zh. Eksp. Teor. Fiz. 39, 1459 (1960)].Variational Method inY. Kanada-En’yo and D. Jido, “Anti-K anti-K N molecular state in three-body calculation,”Phys. Rev. C 78, 025212 (2008).ππN system: Low energy baryon states with JP = 1/2+ inA. Martinez Torres, K. P. Khemchandani and E. Oset, “Solution to Faddeev equations withtwo-body experimental amplitudes as input and application to J**P = 1/2+, S = 0 baryonresonances,” Phys. Rev. C 79, 065207 (2009).KKN system: N∗ around 1920 MeV, made mostly of Na0(980) inA. Martinez Torres and D. Jido, “KΛ(1405) configuration of the KKN system,” Phys. Rev.C 82, 038202 (2010).φKK system: reproduce the properties of the φ(2170) inA. Martinez Torres, K. P. Khemchandani, L. S. Geng, M. Napsuciale and E. Oset, “TheX(2175) as a resonant state of the phi K anti-K system,” Phys. Rev. D 78, 074031 (2008).KKK system: associated with the K(1460) inA. Martinez Torres, D. Jido and Y. Kanada-En’yo, “Theoretical study of the KKK systemand dynamical generation of the K(1460) resonance,” Phys. Rev. C 83, 065205 (2011).πKK system: associated with the π(1300) inA. Martinez Torres, K. P. Khemchandani, D. Jido and A. Hosaka, “Theoretical support forthe π(1300) and the recently claimed f0(1790) as molecular resonances,” Phys. Rev. D 84,074027 (2011).J/ψKK system: associated with the Y (4260) inA. Martinez Torres, K. P. Khemchandani, D. Gamermann and E. Oset, “The Y(4260) as aJ/psi K anti-K system,” Phys. Rev. D 80, 094012 (2009).


The Fixed Center ApproximationIntroduction

ηKK and η′KK systems: studied inW. Liang, C. W. Xiao and E. Oset, “Study of ηKK and η′KK with the fixed centerapproximation to Faddeev equations,” Phys. Rev. D 88, no. 11, 114024 (2013).

K−pp system: calculations (including charge exchange diagrams) inT. Sekihara, E. Oset and A. Ramos, “On the structure observed in the in-flight3He(K−,Λp)n reaction at J-PARC,” PTEP 2016, no. 12, 123D03 (2016).

Experimental support for this state based onY. Sada et al. [J-PARC E15 Collaboration], “Structure near K−+p+p threshold inthe in-flight 3He(K−,Λp)n reaction,” PTEP 2016, no. 5, 051D01 (2016).

πKK∗ system: associated with the π1(1600) inX. Zhang, J. J. Xie and X. Chen, “Faddeev fixed center approximation to πKK∗

system and the π1(1600),” Phys. Rev. D 95, no. 5, 056014 (2017).


The DKK and DKK systemsIntroduction

Our work:

V. R. Debastiani, J. M. Dias and E. Oset,“Study of the DKK and DKK systems,”Phys. Rev. D 96, no. 1, 016014 (2017).

Similar study of the DKK system using two different methods:QCD Sum Rules and full Faddeev equations:

A. Martinez Torres, K. P. Khemchandani, M. Nielsen and F. S. Navarra,“Predicting the Existence of a 2.9 GeV Df0(980) Molecular State,”Phys. Rev. D 87, no. 3, 034025 (2013).


DK Cluster: The D∗s0(2317)Introduction

The D∗s0(2317) molecule will act as the cluster in both systems we have studied: theDKK and the DKK .

The D∗s0(2317) is strongly bound (about 45 MeV of binding energy), and verynarrow, with almost zero width.

It is essentially a DK molecule.

Its existence has already been established both from chiral unitary approach andLattice simulations:

F. K. Guo, P. N. Shen, H. C. Chiang, R. G. Ping and B. S. Zou, “Dynamicallygenerated 0+ heavy mesons in a heavy chiral unitary approach,” Phys. Lett. B 641,278 (2006).

D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, “Dynamicallygenerated open and hidden charm meson systems,” Phys. Rev. D 76, 074016 (2007).

A. Martinez Torres, E. Oset, S. Prelovsek and A. Ramos, “Reanalysis of lattice QCDspectra leading to the D∗s0(2317) and D∗s1(2460),” JHEP 1505, 153 (2015).

L. Liu, K. Orginos, F. K. Guo, C. Hanhart and U. G. Meißner, “Interactions ofcharmed mesons with light pseudoscalar mesons from lattice QCD and implicationson the nature of the D∗s0(2317),” Phys. Rev. D 87, no. 1, 014508 (2013).


DK Cluster: The D∗s0(2317)Introduction

We will follow the description of the D∗s0(2317) of the following work, where anextension of chiral unitary approach to SU(4) was made:

D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, “Dynamicallygenerated open and hidden charm meson systems,” Phys. Rev. D 76, 074016 (2007).

Dynamically generated from the DK interaction in isospin 0.Other channels can be included but they are negligible if compared to the role of theDK channel in our study.

It has a “Delta” shape in DK channel (zero width), which we adjust to get the peakat 2317 MeV.


Two-Body AmplitudesIntroduction

The three-body amplitude will be written in terms of the two-body interactions:

DKK system, described through the interaction of the external K with thecomponents of the cluster [DK ]:

KK Using the chiral unitary approach [1].(Generates the a0(980) in I = 1 and the f0(980) in I = 0).

DK obtained from the same formalism of the DK interaction [2].(Repulsive in isospin 1; Attractive in isospin 0; No bound state inthe energy region corresponding to the DKK interaction).

[1] J. A. Oller and E. Oset, “Chiral symmetry amplitudes in the S wave isoscalar andisovector channels and the σ, f0(980), a0(980) scalar mesons,” Nucl. Phys. A 620, 438(1997), Erratum: [Nucl. Phys. A 652, 407 (1999)].

[2] D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, “Dynamicallygenerated open and hidden charm meson systems,” Phys. Rev. D 76, 074016 (2007).


Two-Body AmplitudesIntroduction

The three-body amplitude will be written in terms of the two-body interactions:

DKK system, described through the interaction of the external K with thecomponents of the cluster [DK ]:

KK which we derive from the same approach used in the KK [1].(Repulsive in isospin 1).

DK which is the same used to generate the D∗s0(2317) [2].(Attractive in isospin 0).

[1] J. A. Oller and E. Oset, “Chiral symmetry amplitudes in the S wave isoscalar andisovector channels and the σ, f0(980), a0(980) scalar mesons,” Nucl. Phys. A 620, 438(1997), Erratum: [Nucl. Phys. A 652, 407 (1999)].

[2] D. Gamermann, E. Oset, D. Strottman and M. J. Vicente Vacas, “Dynamicallygenerated open and hidden charm meson systems,” Phys. Rev. D 76, 074016 (2007).


The Fixed Center ApproximationFormalism

T1 = t1 + t1 G0 t2 + t1 G0 t2 G0 t1 + ...

T2 = t2 + t2 G0 t1 + t2 G0 t1 G0 t2 + ...(1)

= + + + . . .t1T1t1

t2G0

t1

G0

t2

G0t1

Figure: Diagrams of Fixed Center Approximation to Faddeev equations.

T1 = t1 + t1 G0 T2

T2 = t2 + t2 G0 T1

T = T1 + T2

(2)


The Fixed Center ApproximationFormalism

We use the following channels to account for all the possible interactions in the DKKsystem, where the [DK ] acts as the cluster and the K will interact with one of itscomponents:

(1) K−[D+K 0], (2) K−[D0K+], (3) K 0[D0K 0], when the K interacts with the D ofthe cluster;

(4) [D+K 0]K−, (5) [D0K+]K−, (6) [D0K 0]K 0, when the K interacts with the K ofthe cluster.

The channels (3) and (6) are used as intermediate charge-exchange steps


DKK systemFormalism

For the K−[D+K 0]→ K−[D+K 0] amplitude, denoted by TFCA11 , we have

=

K−

K−

D+ K0

D+ K0

D+

D+

K0

K0K−

K−

+

K− D+ K0

K− D+ K0

K0D0+

K− D+ K0

K− D+ K0

K−D+

Figure: Diagrams for the K− multiple scattering of the process K−[D+K0]→ K−[D+K0]. Thewhite circles indicate two-body amplitudes of DK → DK , while the gray bubbles are associatedwith three-body amplitudes of DKK .

TFCA11 (s) = t1 + t1 G0 T

FCA41 + t2 G0 T

FCA61 , (3)

where t1 and t2 are the D+K− → D+K− and D+K− → D0K 0 two-body scatteringamplitudes, respectively.


DKK systemFormalism

This way we can systematically write the three-body amplitudes in terms of the two-bodyamplitudes and the propagator of K inside the cluster:

TFCAij (s) = V FCA

ij (s) +6∑

l=1

V FCAil (s)G0(s)TFCA

lj (s) , (4)

where Vij and Vil are the elements of the matrices below:

V FCA =

t1 0 t2 0 0 00 t3 0 0 0 0t2 0 t4 0 0 00 0 0 t5 0 00 0 0 0 t6 t7

0 0 0 0 t7 t8

, V FCA =

0 0 0 t1 0 t2

0 0 0 0 t3 00 0 0 t2 0 t4

t5 0 0 0 0 00 t6 t7 0 0 00 t7 t8 0 0 0

, (5)

which are the two-body scattering and transition amplitudes:

t1 = tD+K−→D+K− t5 = tK0K−→K0K−

t2 = tD+K−→D0K0 t6 = tK+K−→K+K−

t3 = tD0K−→D0K− t7 = tK+K−→K0K0

t4 = tD0K0→D0K0 t8 = tK0K0→K0K0 ,

(6)


DKK systemFormalism

Isolating the T FCA matrix we get:

TFCAij (s) =

6∑l=1

[1− V FCA(s)G0(s)

]−1

ilV FCA

lj (s) . (7)

Finally, to get the DKK → DKK amplitude we need to consider that the DK clustergenerates the D∗s0(2317) in isospin 0:

|DK(I = 0) 〉 = (1/√

2) |D+K 0 + D0K+ 〉, (8)

and then sum up all the channels corresponding to initial and final state of K interactingwith the |DK(I = 0) 〉 cluster:

TDKK =1

2

(TFCA

11 + TFCA12 + TFCA

14 + TFCA15 + TFCA

21 + TFCA22 + TFCA

24 + TFCA25

+ TFCA41 + TFCA

42 + TFCA44 + TFCA

45 + TFCA51 + TFCA

52 + TFCA54 + TFCA

55

). (9)


DK Cluster: The G0 propagatorFormalism

The propagator of K between the particles of the cluster, denoted by G0, is calculated by

G0(s) =1

2MD∗s0

∫d3q

(2π)3

FR(q)

(q0)2 − ω2K (q) + iε

, q0 =s −m2

K −M2D∗s0

2MD∗s0

, (10)

where MD∗s0

is the mass of the cluster D∗s0(2317), and ω2K (q) = q2 + m2

K , while q0 is the

energy carried by K in the cluster rest frame. The form factor F (q), describing anS-wave bound state [1], is given by

FR(q) =1

N

∫|p|,|p−q|<Λ

d3p1

MD∗s0− ωD(p)− ωK (p)

1

MD∗s0− ωD(p− q)− ωK (p− q)

, (11)

where ωD(p) =√

p2 + m2D and the normalization factor N is

N =

∫|p|<Λ

d3p( 1

MD∗s0− ωD(p)− ωK (p)

)2

. (12)

The upper integration limit Λ has the same value of the cutoff used to regularize the DKloop, adjusted in order to get the D∗s0(2317) molecule from the DK interaction.[1] F. Aceti and E. Oset, “Wave functions of composite hadron states and relationship to

couplings of scattering amplitudes for general partial waves,” Phys. Rev. D 86, 014012 (2012).


Chiral Unitary ApproachFormalism

We use an effective chiral Lagrangian where the pseudoscalar mesons are the degrees offreedom:

L2 =1

12 f 2π

Trace[ (∂µΦ Φ− Φ ∂µΦ)2 + MΦ4 ] , (13)

Φ =

π0√

2+ η8√

6π+ K+

π− − π0√

2+ η8√

6K 0

K− K 0 − 2 η8√6

; M =

m2π 0 0

0 m2π 0

0 0 2m2K −m2

π

, (14)

where in M we take the isospin limit (mu = md) and η8 = η.

+ ++ . . .

Figure: Diagrams representing meson-meson loops.

[1] J. A. Oller and E. Oset, “Chiral symmetry amplitudes in the S wave isoscalar and isovectorchannels and the σ, f0(980), a0(980) scalar mesons,” Nucl. Phys. A 620, 438 (1997), Erratum:[Nucl. Phys. A 652, 407 (1999)].


Chiral Unitary ApproachFormalism

From this Lagrangian we extract the kernel of each channel which are then inserted intothe Bethe-Salpeter equation, summing the contribution of every meson-meson loop.

Tij = Vij +∑l

VilGlTlj =⇒ T = (1− VG)−1 V , (15)

where G is the loop-function, which we regularize with a cutoff. In these works we useqmax ∼ 600 MeV. After the integration in q0 and cos θ we have:

G =

∫ qmax

0

q2dq

(2π)2

ω1 + ω2

ω1ω2[(P0)2 − (ω1 + ω2) + iε], (16)

ωi =√

q2 + m2i , (P0)2 = s (17)

Each contribution is projected in S-wave and a normalization factor is included whenidentical particles are present.

The matrix T gives us the scattering amplitude and transitions between each channel,which in charge basis are: 1) π+π−, 2) π0π0, 3) K+K−, 4) K 0K 0, 5) ηη and 6) π0η.


DKK SystemFormalism

In analogy to the DKK system, we have also studied the DKK System, where instead ofan external K we added another K meson. Therefore, we will have the following channels:

(1) K+[D+K 0], (2) K+[D0K+], (3) K 0[D+K+], when the external K interacts withthe D of the cluster;

(4) [D+K 0]K+, (5) [D0K+]K+, (6) [D0K 0]K 0, when the external K interacts withthe K of the cluster.

As before, the channels (3) and (6) are used as intermediate charge-exchange steps

We use the same cluster, the D∗s0(2317) generated from the DK interaction in isospin 0.Then the procedure to obtain the DKK amplitude is completely analogous.


DKK SystemFormalism

However, in this case the V FCA and V FCA matrices will be written in terms of the DKand KK two-body amplitudes.

V FCA =

t1 0 0 0 0 00 t2 t3 0 0 00 t3 t4 0 0 00 0 0 t5 0 t5

0 0 0 0 t6 00 0 0 t5 0 t5

, V FCA =

0 0 0 t1 0 00 0 0 0 t2 t3

0 0 0 0 t3 t4

t5 0 t5 0 0 00 t6 0 0 0 0t5 0 t5 0 0 0

. (18)

t1 = tD+K+→D+K+ t4 = tD+K0→D+K0

t2 = tD0K+→D0K+ t5 = tK+K0→K+K0

t3 = tD0K+→D+K0 t6 = tK+K+→K+K+

(19)

This time the DK two-body amplitudes are explicitly taken into account due to theinteraction of the external K with the D inside the cluster. This interaction is veryattractive, and the DKK system could bind.

On the other hand, now we have the KK interaction, which is repulsive.


KK InteractionFormalism

Following the same formalism of the chiral unitary approach developed to study the KKinteraction, we have derived the KK kernels from the tree-level amplitudes for K+K 0 andK+K+, which after projection in s-wave read as

vK+K0 ,K+K0 =1

2f 2π

(sKK − 2m2

K

);

vK+K+ ,K+K+ =1

f 2π

(sKK − 2m2

K

),

(20)

From these equations one finds that v I=0KK = 0 (therefore t I=0

KK = 0).

For isospin 1 we need to take the unitary normalization appropriate for identicalparticles |K+K+, I = 1〉 = |K+K+〉/

√2. So

v I=1KK =

1

2vK+K+ ,K+K+ ⇒ t I=1

KK = (1− v I=1KK GKK )−1 v I=1

KK (21)

Then t I=1KK has to be multiplied by two to restore the good normalization.

We have used a cutoff of 600 MeV to regularize the KK loops, the same that wasused in the KK and coupled channels system.


Energy DistributionFormalism

The partitions TFCAij (s) are written into the three-body center-of-mass energy

√s.

While the scattering amplitudes ti (si ) in the V FCA and V FCA matrices are writteninto the two-body center-of-mass energy

√si .

We use two sets of transformations to obtain the√si ’s in terms of

√s:

We call this set by “Prescription I”, standard transformation from 3-body kinematicsused in most works.

sDK(DK) = m2K + m2

D +1

2M2D∗s0

(s −m2K −M2

D∗s0

) (M2D∗s0

+ m2D −m2

K ), (22)

sKK(KK) = 2m2K +

1

2M2D∗s0

(s −m2K −M2

D∗s0

) (M2D∗s0

+ m2K −m2

D) . (23)


Energy DistributionFormalism

In order to estimate the uncertainties we also use “Prescription II”, assuming that thekinetic energy in the DK cluster is of the order of the binding energy.

sDK(DK) =( √

s

MD∗s0

+ mK

)2 (mK +

mD MD∗s0

mD + mK

)2

− P22 , (24)

sKK(KK) =( √

s

MD∗s0

+ mK

)2 (mK +

mK MD∗s0

mD + mK

)2

− P21 , (25)

where P1 and P2 stand for the momenta of the D and K mesons in the cluster.We take P2

1 = P22 = 2µBD∗

s0= 2µ (mD + mK −MD∗

s0), with µ the reduced mass of DK .

This prescription is based on another one [1], which shares the binding energy among thethree-particles proportionally to their respective masses.

[1] M. Bayar, P. Fernandez-Soler, Z. F. Sun and E. Oset, “States of ρB∗B∗ with J = 3within the fixed center approximation to Faddeev equations,” Eur. Phys. J. A 52, no. 4,106 (2016).


Energy distributionFormalism

2760 2780 2800 2820 2840 2860 2880 2900

1000

1500

2000

2500

Figure: Energy distribution in the center-of-mass of each two-body system as a function of thetotal energy of the three-body system, using prescriptions I and II. Here s1 = sDK(DK) and

s2 = sKK(KK). The lower curves are for KK or KK and the upper curves are for DK or DK .


DKK System: D∗s0(2317) K or D [a0(980) / f0(980)] ?Results

2800 2810 2820 2830 2840 28500

2

4

6

8

(a)

2830 2840 2850 2860 2870 2880 2890

0.0

0.5

1.0

1.5

2.0

(b)

Figure: Results for the total DKK amplitude squared using prescriptions I (left) and II (right).

Narrow bound state around D f0(980) threshold (∼ 2855 MeV) in both prescriptions.


DKK System: mostly D f0(980).Results

2800 2820 2840 2860 2880 29000

2

4

6

8

10

12

Figure: Results for the DKK amplitude squared after removing the f0(980) contribution, usingprescriptions I and II.

Table: Comparison between position and intensity of the peaks found in the DKK amplitude.

Prescription I Prescription II√s |T |2

√s |T |2

Total 2833 6.8 × 106 2858 1.8 × 107

I = 1 only 2842 7.7 × 104 2886 7.8 × 104


KK Interaction: The a0(980) and f0(980)Results

(a) KK amplitude in I = 1; couples to a0(980). (b) KK amplitude in I = 0; couples to f0(980).

Figure: Comparison between KK amplitude squared in isospin 1 and 0.

a0(980) couples to KK in I = 1, but the coupling to πη is the dominant one.

f0(980) couples strongly to KK in I = 0, and with less intensity to ππ. (The σmeson, f0(500), is dominated by ππ in I = 0).

In KK the f0(980) amplitude squared is more than 2 orders of magnitude strongerthan a0(980) around their peaks.


DKK System: mostly D f0(980).Results

Our Result [1]:

MDKK = 2833− 2858 MeV.

QCD Sum Rules [2]:MDf0 = (2926± 237) MeV.

Full Faddeev equations [2]:MDf0 = 2890 MeV.

[1] V. R. Debastiani, J. M. Dias and E. Oset,“Study of the DKK and DKK systems,”Phys. Rev. D 96, no. 1, 016014 (2017).

[2] A. Martinez Torres, K. P. Khemchandani, M. Nielsen and F. S. Navarra,“Predicting the Existence of a 2.9 GeV Df0(980) Molecular State,”Phys. Rev. D 87, no. 3, 034025 (2013).


DKK System: D∗s0(2317)K ?Results

2760 2780 2800 2820 2840 2860 2880 2900

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure: Results for the total DKK amplitude squared using prescriptions I and II.

Broad, irregular structures. Both amplitudes decrease around 2812 MeV whichcorresponds to the D∗s0(2317)K threshold.

KK repulsion vs. DK attraction.

Enhancement below threshold with “Prescripiton I”.

Small strength if compared to the DKK system. No clear bound state or resonance.


Summary

Fixed Center Approximation to Faddeev Equations. Charge exchange diagramsincluded.

DKK and DKK three-body interactions.

Cluster DK in the I = 0: D∗s0(2317).

Two-body scattering: DK and KK in DKK system; DK and KK in DKK .

Uncertainties were estimated taking into account two different prescriptions toobtain

√sDK and

√sKK from the total energy of the system

√s.

DKK system binds: I (JP) = 1/2(0−) state, with mass about 2833− 2858 MeV,dominated by a Df0(980) component (from KK in I = 0).

Results compatible with other methods using QCD Sum Rules (MDf0 = (2926± 237)MeV) and full Faddeev equations (MDf0 = 2890 MeV.)

This state could be seen in the π πD invariant mass distribution (fromf0(980)→ ππ decay).

DKK does not seem to bind. KK repulsion seems to be of the same magnitude asthe attraction on the DK interaction.


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Study of the DKK and DK systems - indico.ihep.ac.cn · Outline 1 Introduction The Faddeev Equations DK Cluster: The D s0(2317) Two-Body Amplitudes 2 Formalism The Fixed Center Approximation