the heavy quark spin symmetry and su(3)-flavour partners...

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

Carlos Hidalgo, J. Nieves and M. Pavon-Valderrama

Hypernuclear and Strange Particle Physics 2012

IFIC (CSIC - Universitat de Valencia)

Outline

1 Introduction

2 Effective field theory for heavy molecules (DD,D∗D∗...)

3 Determination of the counter-terms“Isospin violation” in X(3872) decays

4 PredictionsI = 0 sector: SU(2) isoscalar and hidden strangeness.I = 1

2 sector (strangeness = ±1)I = 1 sector

5 Outlook and conclusions

Introduction

Molecular meson-antimeson systems have been predicted inspired by theexisting similarities with the nucleon-nucleon system (Voloshin, Okun; 1976).So, as the deuteron is a nucleon-nucleon bound state, it would be natural toassume the existence of meson-antimeson bound states.

Figure: Analogy between nuclear and meson-antimeson forces

Introduction

This supposition has been tested several times:

1 De Rujula, Georgi, Glashow in 1977: Ψ(4040) as a D∗D∗ resonance.

2 Jaffe in 1977: f0(980) y a0(980) as a KK bound state.

3 Belle collaboration in 2003: X (3872) as a DD∗ resonance. This is a clearcandidate to be a meson-antimeson molecule.

· · ·

Many new resonances X, Y, Z... have been detected which can also becomecandidates to be meson-antimeson molecules.

Outline

1 Introduction






HQET

The presence of a heavy quark induces a series of important simplifications in theQCD 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

mQ

). At lowest order, the QCD

Lagrangian reads:

LQ = Qv (iv · D)Qv

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(2Nh) symmetry.

Covariant representation of mesonic fields.

HQET eigenstates are “would-be” hadrons composed by a heavy quark with lightquarks, light antiquarks and gluons.

So, the fundamental state of a Qq meson [Qq antimeson] can be represented by a

field HQv [HQ

v ] which includes both the pseudoscalar isospin triplet

PQa = (P0,P+,Ps) [P Q

a = (P0,P−, Ps)] and the vector isospin triplet P∗Qa [P∗Qa ].A possible expression for this field (satisfying all the corresponding symmetries ofeach meson in the field) is:

H(Q)v (x) =

1 + /v

2

[/P∗(Q)v − P(Q)

v γ5

]H(Q)

v (x) = γ0H(Q)†v (x)γ0 =

[/P∗(Q)†v − P(Q)†

v γ5

] 1 + /v

2

H(Q)v (x) =

[/P∗(Q)v − P(Q)

v γ5

] 1− /v2

H(Q)v (x) = γ0H(Q)†

v (x)γ0

And, as HQET is a non-relativistic theory, HQv (HQ

v ) only destroys itscorresponding mesons (antimesons).

Covariant representation of mesonic fields.

HQv [HQ

v ] field transforms under spin-flavour symmetry rotations:

H(Q)a → S

(H(Q)U†

)a, H(Q)a →

(UH(Q)

)aS†

H(Q)a →(UH(Q)

)aS†, 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.

Effective field theory for meson-antimeson molecules.

Describing meson-antimeson molecular system requires an effective Lagrangian. Itshould contain all the possible strong interactions that bound themeson-antimeson system.

Figure: Possible meson-antimeson interactions at lowest order. Diagrams particularizedto the case PP∗

However, taking into account chiral symmetry of QCD, pion exchanges will besuppressed compared to contact interactions (numerically shown in Nieves andPavon PRD84, 056015). As we will work at lowest order, our effective interactionLagrangian will only consist of contact interactionsl.


At LO, the most general potential that respects HQSS takes the form:

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 based on HQSS only depends on four undetermined lowenergy constants (counter-terms)!!


So, once we have determined V, we find bound states by solving theLippmann-Schwinger equation for each spin, isospin and charge-conjugationsector:

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 removed introducing a Gaussian regulator Λ:

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

e−p′2/Λ2 ⇒ G =

∫d3~k

(2π)3e−2k2/Λ2

E−m1−m2− k2

2µ

Outline

1 Introduction






Determination of the counter-terms.

To determine the four counter-terms, we have assumed:

1 X (3917) is a D∗D∗ bound state with JPC = 0++.

2 Y (4140) is a D∗s D∗s bound state with JPC = 0++.

3 X (3872) is a DD∗ bound state with JPC = 1++.

4 The fourth condition will be obtained from the “isospin violation” in theX(3872) decays.

“Isospin violation” in X(3872) decays

It has been difficult to understand the X(3872) decay widths in different isospinchannels. In fact, if X(3872) had a well defined isospin, it would be hard toaccomodate the following experimental ratio:

B(X (3872)→ J/ψ

ω︷︸︸︷π+π−π0)

B(X (3872)→ J/ψ π+π−︸︷︷︸ρ

)' 0.8± 0.3.


To explain this ratio there are two different scenarios:

X(3872) isospin is well defined but then isospin is not conserved in thesestrong decays.

X(3872) is not well defined and strong interactions conserve isospin.

We’ll assume this latter scenario. Despite our potential conserves isospin, thekinetic term of the whole Lagrangian doesn’t because of the mass differencebetween the charged components (D+D−∗) and the neutral component (D0D0∗)which is around 8 MeV. As, the X(3872) is very close to D0D0∗ threshold, thismass difference cannot be neglected and, then, the isospin operator doesn’tcommute with the whole Hamiltonian and, thus, it is not a good quantum numberof the system.


A more detailed analysis of X(3872) decays (Hanhart et al. 2012) gives the ratiobetween the amplitudes of the decays:

RX =M(X → J/ψ ρ)

M(X → J/ψ ω)= 0.26+0.08

−0.05


And, in our model (depicted in the figure), the same ratio is given by:

RX =M(X → J/ψ ρ)

M(X → J/ψ ω)=

gρgω

(ψ1 − ψ2

ψ1 + ψ2

)being

gω =Mω(DD∗(I = 0)→ J/ψ ω) gρ =Mρ(DD∗(I = 1)→ J/ψ ρ)

and ψ1 y ψ2 an average of the neutral and charged D∗D wave functions in thevicinities of the origin.

Figure: X(3872) decay mechanism in this model.


Because of the light flavour SU(3) symmetry of QCD Hamiltonian, it is satisfied:

⇒ gω − gρ =√

2gφ

And, using OZI rule (ss pair creation is suppressed) then: gφ → 0

⇒ gω = gρ

⇒ RX =M(X → J/ψ ρ)

M(X → J/ψ ω)=

(ψ1 − ψ2

ψ1 + ψ2

)= 0.26+0.08

−0.05


If we now assume a vanishing DD∗ interaction in the isospin I = 1 sector, it isfound that the ratio RX only depends on the masses of the resonance and thethresholds of the different channels (Gammermann et al.,2010) and takes thevalue:

RX ' 0.13

We improve on this, and consider a non-vanishing I = 1 interaction that we fit tothe value for RX reported by Hanhart et al. Hence, we’ll work with two coupledchannels (neutral and charged channels) contact potential:

Vcoupled =1

2

(V0 + V1 V0 − V1

V0 − V1 V0 + V1

)being V0 and V1, 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.


Thus, we have determined the value of the different counter-terms a and we canpredict a whole family of resonances related (SU(3) and HQSS partners) to theX(3872), X(3915) and Y(4140).

aGaussian regulator dependence appears as a higher order effect. There is a relationbetween the Gaussian regulator and the counter-terms value. This relation is verysimilar to RGE of other field theories.

Outline

1 Introduction






I = 0 sector without hidden strangeness

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

I = 0 sector with hidden strangeness

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

¡¡Predictions!!

I = 12 sector

Figure: Predicted resonances for our model for the I = 12

sector.

¡¡Predictions!!

being (charge conjugation is not a good quantum number in this channel):

Vs =

(C1A −C1B

−C1B C1A

)

I = 1 sector

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

¡¡Predictions!!

Outline

1 Introduction






Outlook and conclusions

Light flavour symmetry and HQSS in heavy meson-antimeson systems,combined with the identification of three resonances, has allowed us to obtaina whole family of hidden charm molecules.

Uncertainties quoted in tables only account for the approximate nature ofHQSS (∼ 15% error) and those induced by the errors of RX .

Pion exchanges and coupled channels should be considered. However,according to previous bibliography, these effects have a smaller contributionthan expected and smaller than those expected from HQSS breaking terms.

Some of the predicted resonances have already been experimentally detectedor predicted by different models.

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