decay widths of the spin-2 partners of the x(3872)decay widths of the spin-2 partners of the x(3872)...

Introduction The X2 state in the charm and bottom sector Hadronic Decays of the X2 states Radiative Decays of the X2 states Conclusions

Decay widths of the spin-2 partners of theX(3872)

M. Albaladejo, F.K. Guo, C. Hidalgo-Duque, J. Nieves, M. Pavon.

XVI International Conference on Hadron SpectroscopyMarriott at City Center, Newport News, September 14-18, 2015


Outline

1 Introduction

2 The X2 state in the charm and bottom sector

3 Hadronic Decays of the X2 states

4 Radiative Decays of the X2 states

5 Conclusions


Hadronic Molecular States

Molecular meson-antimeson systems have been predicted inspired bythe 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-antimesonbound states.


Hadronic Molecular States

The best candidate to fit this molecular interpretation is theX (3872) resonance, first observed by the Belle collaboration in 20031

with quantum numbers JPC = 1++ 2, though other interpretationssuch as charmonium or tetraquark states have also been theorized.

Therefore, it is essential to determine different experimentalobservables that allow us to distinguish among the theoreticalmodels proposed. In this sense, the decays of the X (3872) arecurrently the main source of information.

In this work, however, we suggest another approach. We propose toanalyze a JPC = 2++ D∗D∗ (and B∗B∗) state, whose existence is aconsequence of the existence of the X (3872) state according toHeavy Quark Symmetry arguments.

1Belle Collaboration, Phys.Rev.Lett. 91 (2003) 2620012LHCb Collaboration, Phys.Rev.Lett. 110 (2013) 222001


Heavy Quark Spin Symmetry

Heavy Quark Spin Symmetry is an underlying symmetry of QCDthat states that, in the infinite quark mass limit, dynamics does notdepend on neither the heavy quark mass nor the heavy quark spin.

For a heavy flavour sector, relates states/interactions with differentspin.Relates also charm and bottom sectors (assumed).Build lagrangians for the interactions of D(∗) and D(∗) mesons.

Several models make predictions for a JPC = 2++ D∗D∗ state.However, the mass splitting between the X (3872) resonance and thepredicted X2 state can vary significantly.

Charmonium models predict a mass splitting around 30-40 MeV 3.Tetraquark models predict a mass splitting around 70 MeV 4.In our model, using a molecular picture, we find thatMX2 −MX (3872) ' MD∗ −MD ' 140 MeV.

3S. Godfrey et al., Phys.Rev. D32 (1985) 189-231, B-Q. Li et al. Phys.Rev. D80(2009) 014012

4L. Maiani et al. Phys.Rev. D71 (2005) 014028


Outline

1 Introduction




5 Conclusions


Theoretical Framework

At Leading Order, the most general potential that respects HQSStakes the form5:

V4 = +CA

4Tr[H(Q)H(Q)γµ

]Tr[H(Q)H(Q)γµ

]+

+CλA4

Tr[H(Q)

a λiabH(Q)b γµ

]Tr[H(Q)

c λicd H(Q)d γµ

]+

+CB

4Tr[H(Q)H(Q)γµγ5

]Tr[H(Q)H(Q)γµγ5

]+

+CλB4

Tr[H(Q)

a λjabH(Q)b γµγ5

]Tr[H(Q)

c λjcd H(Q)d γµγ5

]being λ Gell-Mann matrices. At first order, this model only dependson four undetermined low energy constants! From now on, the fourindependent counter-terms will be rewritten, w.l.o.g., into C0X , C0Z ,C1X and C1Z .Pion exchanges and coupled channel effects are found to besubleading6.

5M. Alfiky et al. Phys.Lett. B640 (2006) 238-2456J. Nieves et al. Phys.Rev. D86 (2012) 056004 //M. Valderrama Phys.Rev. D85

(2012) 114037


Theoretical framework: fixing amplitudes

HQSS predicts the same potential for:

JPC = 1++ :

{∣∣D0D∗0⟩−∣∣D∗0D0

⟩√

2,

∣∣D+D∗−⟩−∣∣D∗+D−

⟩√

2

}

JPC = 2++ :{∣∣∣D∗0D∗0

⟩,∣∣∣D∗+D∗−

⟩}V =

1

2

(C0X + C1X C0X − C1X

C0X − C1X C0X + C1X

)

V −→ V +O(

q ' ΛQCD

mQ

)Use particle basis, since isospin is broken because of the masses:

(mD∗+ + mD+ )︸︷︷︸charged threshold

− (mD∗0 + mD0 )︸︷︷︸neutral threshold

' (mD∗0 + mD0 )−mX (3872)︸︷︷︸binding energy

X(3872) (JPC = 1++)

M = 3871.69± 0.17 MeV

R = 0.26± 0.07(X (3872)→ J/ψ(2π)(3π))

Our model: DD∗ boundstate (charged and neutralchannels)

This fixes C0X, C1X

Zb(10610) (JPC = 1+−)

BE = 2± 2 MeV

Our model: BB∗ boundstate(isospin symmetric)

This fixes C1Z

Still to be fixed: C0Z


Theoretical Framework

Once we have determined V, we find bound states by solving theLippmann-Schwinger equation for each spin, isospin andcharge-conjugation sector:

T = V + V G T

〈~p |T |~p ′〉 = 〈~p |V |~p ′〉+

∫d3~k

⟨~p |V |~k

⟩⟨~k|T |~p ′

⟩E −m1 −m2 − k2

2µ

Bound states of this model will appear as poles in the T matrix.

Ultraviolet divergences are treated introducing a Gaussian regulatorΛ. We use two different values for the Gaussian regulatorΛ = 0.5, 1.0 GeV to estimate the errors obtained by thisregularization method.

〈~p|V |~p ′〉 = V (~p, ~p ′) = v e−~p2/Λ2

e−~p′2/Λ2

⇒ G =∫

d3~k(2π)3

e−2~k2/Λ2

E−m1−m2− k2

2µ


The X2 state in the charm sector.

Charm Sector: The X2 state.

Dis

trib

utio

n

MX(4013) (MeV)

BX(4013) (MeV)

Λ = 0.5 GeVw HQSS error

w/o HQSS error

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

4006 4007 4008 4009 4010 4011 4012 4013 4014 4015

-101234567

Dis

trib

utio

n

MX(4013) (MeV)

BX(4013) (MeV)


w/o HQSS error

0

0.2

0.4

0.6

0.8

1

1.2

1.4

4002 4004 4006 4008 4010 4012 4014

0246810

Considering only the experimental error of the X (3872) mass, a clearloosely bound state can be generated (blue). When taking into accountthe uncertainty caused by the HQSS expansion, the existence of thisbound state is unsure (red).


The X2 state in the bottom sector.

Bottom Sector: The X2b state.

Dis

trib

utio

n

MXb2 (MeV)

BXb2 (MeV)


w/o HQSS error

0

0.1

0.2

0.3

0.4

0.5

10610 10615 10620 10625 10630 10635 10640 10645 10650

510152025303540

Dis

trib

utio

n

MXb2 (MeV)

BXb2 (MeV)


w/o HQSS error

0

0.05

0.1

0.15

0.2

0.25

10520 10540 10560 10580 10600 10620 10640 10660

020406080100120

In the bottom sector, a clear bound state with a binding energy of tens ofMeV is generated in both cases.

EX2b= 10631+8

−7 MeV EX2b= 10594+22

−26 MeV

for Λ = 0.5 and 1.0 GeV, respectively.


Outline

1 Introduction




5 Conclusions


Hadronic Decays of the X2 states.

The hadronic decays of this bound state should almost saturate thewidth of the resonance. Thus, this is the most likely channel wherethe predicted state can be found.

In the charm sector, where we are distinguishing between differentisospin channels, there are four different decay channels: D0D0,D+D−, D∗0D0 and D∗+D−.

In the bottom sector, where isospin symmetry makes sense, only twodecay channels are discussed: BB and BB∗.


Hadronic decays

D∗Dπ and D∗D∗π interactions

L =g

2〈H(Q)uTµH

(Q)γµγ5〉+g

2〈H(Q)uTµ H

(Q)γµγ5〉 g = 0.570± 0.006

g from D∗+ decay width (PDG), Γ = 83.4± 1.8 keV

X2 decays through the coupling with D∗0D∗0 (gN) and D∗+D∗−

(gC )

d-wave decay: no DD(∗) FSI (only for s-wave)

We can study four decays in the charm sector: D0D0, D+D−,D0D∗0, D+D∗−; and two decays in the bottom sector (whereisospin symmetry holds): BB, BB∗.

X(4013)D+

D−

D∗+

D∗−

π0

X(4013)D+

D−

D∗0

D∗0

π−


Hadronic decays


L =g

2〈H(Q)uTµH

(Q)γµγ5〉+g

2〈H(Q)uTµ H

(Q)γµγ5〉 g = 0.570± 0.006



(gC )


We can study four decays in the charm sector: D0D0, D+D−,D0D∗0, D+D∗−; and two decays in the bottom sector (whereisospin symmetry holds): BB, BB∗.

X(4013)D+

D∗−

D∗+

D∗−

π0

X(4013)D+

D∗−

D∗0

D∗0

π−


Hadronic decays


L =g

2〈H(Q)uTµH

(Q)γµγ5〉+g

2〈H(Q)uTµ H

(Q)γµγ5〉 g = 0.570± 0.006



(gC )


without pion FF with pion FF

Λ = 0.5 GeV Λ = 1 GeV Λ = 0.5 GeV Λ = 1 GeV

Γ(X2 → D+D−) [MeV] 3.5+4.0−1.3 7.4+8.5

−1.9 0.5+0.5−0.2 0.8+0.7

−0.2

Γ(X2 → D0D0) [MeV] 2.9+3.8−1.2 5.8+8.5

−1.7 0.4+0.5−0.2 0.6+0.7

−0.2

Γ(X2 → D+D∗−) [MeV] 2.5+2.5−0.9 4.5+3.3

−1.1 0.7+0.6−0.3 1.0+0.5

−0.2

Γ(X2 → D0D∗0) [MeV] 2.1+2.5−0.9 3.5+3.8

−1.0 0.5+0.6−0.2 0.7+0.5

−0.2

Γ(Xb2 → BB) [MeV] 26+1−3 8+15

−3 4.4+0.1−0.4 0.7+1.4

−0.6

Γ(Xb2 → B∗B) [MeV] 7+3−4 – 2.0+0.9

−1.0 –


Hadronic decays


L =g

2〈H(Q)uTµH

(Q)γµγ5〉+g

2〈H(Q)uTµ H

(Q)γµγ5〉 g = 0.570± 0.006



(gC )


Partial widths including all channels:

Γ(X2 → DD

)= 1.2± 0.3︸︷︷︸

sys (Λ)

+1.3−0.4 MeV

Γ(X2 → DD∗ + D∗D

)= 2.9± 0.5︸︷︷︸

sys (Λ)

+2.0−1.0 MeV


Outline

1 Introduction




5 Conclusions


Radiative decays

(Electro)magnetic interaction D∗Dγ

L =eβ

4Qa

b

(〈H(Q)bH

(Q)a σµν〉+ 〈H(Q)bH

(Q)a σµν〉

)Fµν

+eQ′

4mQ

(〈H(Q)aσµνH

(Q)a 〉+ 〈H(Q)aσµν H

(Q)a 〉

)Fµν

Q = (2/3,−1/3) (u, d)Q′ = (2/3,−1/3) (c, b)

We fix mc = 1.5 GeV, fit β−1 ' mu,d

Exp. Fit

Γ(D∗0 → D0γ) (keV) 22.7± 2.6 22.9± 1.8

Γ(D∗+ → D+γ) (keV) 1.3± 0.3 1.3± 0.2

Γ(B∗0 → B0γ) (keV) – 0.23± 0.02

Γ(B∗− → B−γ) (keV) – 0.49± 0.05

=⇒ β−1 = 293± 11 MeV


Radiative decays: Tree level

Width Λ = 0.5 GeV Λ = 1 GeV

Γ(X2 → D0D∗0γ) (keV) 18+2−6 16+2

−9

Γ(X2 → D+D∗−γ) (keV) 0.10+0.10−0.05 0.09+0.06

−0.03

Γ(Xb2 → B−B∗+γ) (eV) 6+10−5 –

Γ(Xb2 → B0B∗0γ) (eV) 13+23−10 –

...but... DD∗ is produced in s-wave, so FSI are important


Radiative decays: Final State Interactions

For the case of the X2 → D0D∗0γ process, we must consider thefollowing diagrams.

The DD∗ FSI include the undetermined C0Z constant. Decay widths arenow a function of this parameter.


Radiative decays: Final State InteractionsΓ(k

eV)

C0Z (fm2)

FSIno FSI

0

10

20

30

40

50

60

-6 -5 -4 -3 -2 -1 0 1 2 3 4

X2→ D0D∗0γ

Λ = 0.5 GeV

C0Z (fm2)

FSIno FSI

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-6 -5 -4 -3 -2 -1 0 1 2 3 4

X2→ D+D∗−γ

Λ = 0.5 GeV

We compute widths for a sensiblerange of C0Z

Neutral mode: For many values of C0Z

FSI are not important

Charged mode: FSI are alwaysimportant, since it brings in themagnetic D∗0 → D0γ coupling, muchlarger than the D∗+ → D+γ coupling

In the range C0Z = [−2,−1] (fm)2 wesee large interferences: bound (orvirtual) DD∗ state appears


Outline

1 Introduction




5 Conclusions


Conclusions

We assume X (3872) to be a D∗D bound state

By means of HQSFS we predict the existence of a X2 (charm) and aXb2 (bottom) state (JPC = 2++)

Their properties of these states are related to those of X (3872) bymeans of the LECs (C0X ,C1X )

We compute the decay widths X2 → DD(∗) and the analogous ofthe bottom sector. These widths have values of a few MeV

We compute the widths X2 → DD∗γ which are of the order of keV,and find that the width of the neutral mode is larger than that ofthe charged mode

In the bottom sector, these widths are of the order of eV

FSI corrections have been computed for these decays. Thismeasurement would help to determine the LECs of the theory andthus in the understanding of the spectrum

decay widths of the spin-2 partners of the x(3872)decay widths of the spin-2 partners of the x(3872)...

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