Scale Invariance in Heavy Hadron Molecules

Manuel Pavon Valderrama

Beihang University

IPN Orsay, June 2017

With Li-Sheng Geng, Jun-Xu Lu and Xiu-Lei Ren

Contents

I Hadron Molecules: Bound States of Hadrons

I Scale Invariance and the Efimov Effect

I The Pc(4450)+ as a ΣcD∗-Λc(2590)D Molecule

I Possibility of Scale Invariance in Hadron Molecules

I Coulomb-like Hadron Molecules

I The Λc(2590)Σc Molecule

L.S. Geng, J.X. Lu, MPV arXiv:1704.06123 [hep-ph]

L.S Geng, J.X. Lu, MPV, X.L. Ren arXiv:1705.00516 [hep-ph]

Hadron Molecules

Hadron MoleculesStandard hadrons come in two varieties

But there are more types of possible hadrons...

Hadron Molecules: Early Especulations

I Heavy hadron molecules: bound states of heavy hadrons

I Theorized by Voloshin and Okun (76): the force between twoheavy mesons should be similar to the nuclear force

π, σ, ρ, ω

P

P ∗

π, σ, ρ, ω

N

N

=⇒

Like the deuteron but composed of heavy hadrons

I Early explorations by De Rujula, Georgi, Glashow (77),Tornqvist (93), Ericson and Karl (93), . . .

Hadron Molecules: the X(3872)

Hadron molecules were mere theoretical speculations until adiscovery by the Belle collaboration in B± → K±J/Ψππ (03):

... and later confirmed by D0 and CDF.

Hadron Molecules: the X(3872)

More robust molecular candidates:

Candidate Molecule IG (JPC ) / I (JP)

X (3872) DD∗ 0+(1++)

Zc(3900) DD∗ 1+(1+−)Zc(4020) D∗D∗ 1+(1+−)

Zb(10610) BB∗ 1+(1+−)Zb(10650) B∗B∗ 1+(1+−)

Pc(4450)+ ΣcD∗ 1

2

(32

−)

Later we will go back to the Pc(4450)+

Scale Invariance

Scale Invariance: What is it?

Scale invariances: invariance under dilatations

That is, it looks the same with a magnifying glass

Scale Invariance: the Two-Body System

The Two-Body System at zero energy

−u′′0 (r) + 2µV (r) u0(r) = 0

For r > r0 (the range of the potential):

u0(r)→ 1− r

a0

For a0 > r > r0 we simply have

u0(r)→ 1

and there is approximate scale invariance:

r → λr ⇒ u0 → u0

Scale Invariance: Universality

In the limit a0 →∞ we have exact scale invariance instead

r → λ r , k → 1

λk , E → 1

λ2E

In particular we have

I a0 → λa0 (because a0 =∞)

I E0 → 1λ2E0 (because E0 = 0)

The system can be explained as a series in powers of 1/a0

Universality

(Hammer, Braaten 06 Review)

Scale Invariance: Examples

I A lot of atomic systemsI Nucleon-nucleon in S-wave: 1S0 and 3S1

I mπas ' 16.7� 1I mπat ' 3.8� 1

I Unitary limit in nuclear physics(Konig, Grießhammer, Hammer, van Kolck 17)

I 8Be as an α− α resonance

I X (3872) as a D0D∗0 molecule

The Efimov Effect

Three-boson System with a0 →∞, r0 → 0

I The Schrodinger equation is scale invariant

I But the solutions are not: if there is a three body boundstate with E3 6= 0, scale invariance will be broken because

E3 →1

λ2E3

cannot be fulfilled for arbitraty λ (spectrum must be discrete)

I However scale invariance survive for a particular value of λ

E3 →1

λ20

E3

where λ0 ' 22.7, i.e. a geometric spectrum of bound states

The Efimov Effect

For a0 →∞, tower of bound states with E3(n) ' 515E3(n+1)

Experimentally confirmed: Kraemer et al., Nature 440, 315 (2006)

The Inverse Square Potential

The Two-Body System with an attractive inverse square potential

−u′′0 (r) +g

r2u0 (r) = 0

Schrodinger equation scale invariant, but not the solutions

I For g > −1/4 we have

u0(r) = c+r1/2+ν + c−r

1/2−ν

with ν =√

1/4 + g . No scale invariance

I For g < −1/4 we have

u0(r) = c r1/2 sin (ν log Λ2r)

with ν =√−1/4− g . Discrete scale invariance

r → eπ/νr , k → e−π/νk , E2 → e−2π/νE2

The Inverse Square Potential

I Rare example of anomaly in quantum mechanics

I Connection with the three-body problem:(− d2

d2ρ− s2

0 + 1/4

ρ2

)f (ρ) = 2mE3 f (ρ)

Schrodinger equation in the hyperradial coordinate ρ.

Fedorov, Jensen, PRL71, 4103 (1993)

I Are there two-body examples?

I Charged-dipole interaction in atomic physics (P-wave)Camblong et al., PRL87, 0220402 (2001)

I But until recently no known example in hadron physics

The Pc(4450) Pentaquark

The Pc(4450)+

LHCb: two hidden-charm pentaquark states where discovered

Pc(4380)+, Pc(4450)+

Re A

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.1

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

LHCb

(4450)cP

(a)

15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

(4380)cP

(b)

Pc Re APc

Im A

P c

The Pc(4450)+: Properties and Nature

Properties of Pc(4380)+ and Pc(4450)+ (Pc and P∗c ):

I Pc : M = 4380± 8± 29MeV, Γ = 205± 18± 86MeV

I P∗c : M = 4449.8± 1.7± 2.5MeV, Γ = 39± 5± 19MeV

I Opposite parities. Most likely spins are ( 32 ,

52 ) and ( 5

2 ,32 )

I Hidden charm (cc): Pc ,P∗c → J/Ψp

I Close to the thresholdsI m(Σ∗+c D0) = 4382.3± 2.4 (Pc)I m(χc1p) = 4448.93± 0.07 (P∗c )I m(Λ+

c1D0) = 4457.09± 0.35 (P∗c )

I m(Σ+c D∗0) = 4459.9± 0.5 (P∗c )

The Pc(4450)+: What it is?

Explanations for its nature

I Threshold effect(Guo et al. 15; Mikhasenko 15; Liu et al. 15; Meißner, Oller 15)

I Compact (Genuine) Pentaquark(Yuan et al. 12; Maiani et al. 15; Lebed 15)

I Baryocharmonia (Kubarovsky, Voloshin 15)

I Molecular state: Σ+c D∗0 and Σ∗+c D∗0 most popular options.

(Chen R. et al. 15; Chen H.X. et al. 15; Roca et al. 15; He 15;

Xiao, Meißner 15)

The Pc(4450)+: a proposal by Burns

P∗c ’s most usual molecular interpretation is ΣcD∗

I The S-wave molecule can have J = 12 , 3

2

But Burns proposed that there might be a Λc1D piece

I Alone Λc1D is not a good candidate:

I The S-wave molecule can only have J = 12

I But ΣcD∗-Λc1D works:

I Λc1D in P-wave provides extra attraction

The Pc(4450)+: and there is a Surprise

I Λc1D and ΣcD∗ opposite parities: vector force

e−µπr/r2, with µ2π = m2

π −∆2

I Pion almost on shell:

∆ = m(Λc1)−m(Σc) ' m(D∗)−m(D) ' mπ

Unusual long-range! The potential is 1/r2!(|µπ| = 36/16MeV for π0 / π±)

I Discrete scale invariance / Efimov Spectrum possible

The Pc(4450)+: the Potential (32

−)

If we consider the P∗c as a 32

−ΣcD

∗-Λc1D molecule:

I |P∗c (3/2−)〉 = |ΣcD∗(2D3/2 − 4S3/2 − 4D3/2)〉- |Λc1D(2P3/2)〉

I With the Schrodinger equation (for mπr > 1 > µπr):

−u′′ +

[2µP∗c VOPE +

L2

r2

]u = 0 where:

2µP∗c VOPE +L2

r2=

g( 32

−)

r2

=1

r2

6 0 0 g0 0 0 g0 0 6 −gg g −g 2

.

The Pc(4450)+: Discrete Scale Invariance (32

−)

The Schrodinger equation can be diagonalized

−u′′i +gir2

ui = 0 ,

where the eigenvalues are

gi = {6, 2, 3 +√

9 + 3g2, 3−√

9 + 3g2} ,

with

g =µP∗c g1h2ωπ

2π√

2fπ' 0.60 h2

Unlikely: g− < −1/4 requires h2 > 1.21+0.25−0.19 (h2 = 0.63± 0.07).

The Pc(4450)+: Discrete Scale Invariance (12

+)

The most promising quantum number is 12

+:

I |P∗c (1/2+)〉 = |ΣcD∗(2P1/2 − 4P1/2)〉- |Λc1D(2S1/2)〉

I With the matrix

g(1

2

+

) =

2 0 g

0 2 −√

2 g

g −√

2 g 0

.

with the attractive eigenvalue g− = 1−√

1 + 3g2

Possible: g− < −1/4 requires h2 > 0.73+0.11−0.06 (h2 = 0.63± 0.07).

But unlikely as an explanation of the Pc(4450)+

(though there could be more pentaquarks to be discovered)

The Λc1Ξb-ΣcΞ′b Molecule

We can change the DD∗π vertex by the ΞbΞ′bπ vertex:

I Ξ−b (Ξ−′

b ): M = 5794.5± 1.4 (M = 5935.02± 0.05)

I m(Ξ−′

b )−m(Ξ−b ) = 140.5± 1.4

I JP = 12

+

And consider the configurations

0+ = Σc Ξ′b(3P0)− Λc1Ξb(1S0)

0− = Σc Ξ′b(1S0)− Λc1Ξb(3P0)

1− = Σc Ξ′b(3S1 − 3D1)− Λc1Ξb(1P1 − 3P1)

which require g = µg3h2ωπ

4πf 2π

> 0.75, i.e. h2 > 0.67+0.03−0.02

(vs h2 = 0.63± 0.07).

Survival of the Geometric Spectrum

Vector force: long-range and short-range implications

I Scale invariance is approximate in 1µπ

> r > Rs

I Thus geometric spectrum requires larger couplings

ν log(1

Rsµπ) > π =⇒ |g−| >

1

4+

(π

logRsµπ

)2

which for Rsµπ ∼ 120 − 1

10 means |g−| > 1.3− 2.0� 0.25(|g−| seems to be at best about 0.25 for hadronic molecules)

I For three-boson similar idea after µπ → 1a0

substitution(but g = 1.26 leading to a higher probability for the survivalof the first excited geometric state)

Contribution to Binding

Vector force: long-range and short-range implications

Even if the chances of a geometric spectrum are low,the contribution to binding will be important

I Two-body system with short-range interactions binds if

C0 ≤ −4πRs

2µ

I Two-body system with inverse square binds if

C0 ≤ −(1− ν)4πRs

2µ

For |g−| → 0.25 we have (1− ν)→ 12 .

Half the short-range attraction is required to bind!

Contribution to Binding

Vector force: long-range and short-range implications

The specific contribution to binding at Rs = 1 fm is:

I P∗c as ΣcD∗-Λc1D: 75% of the strength for ΣcD

∗

I 0− Σc Ξ′b-Λc1Ξb: 46% of Λc1Ξb alone

I 1+ Σc Ξ′b-Λc1Ξb: 53% of Σc Ξ′b alone

The vector force can be an important contribution

Coulomb-like Baryonia

The Λc1Σc Molecule

Interesting observation: try to combine two Λc1Σcπ vertices

We obtain the coupled channel potential

VOPE(r) = ∓ h22ω

2π

4πf 2π

e−µπr

r

(0 11 0

)with -/+ for the baryon-baryon/baryon-antibaryon case (µπ � mπ)

The Λc1Σc Molecule

Baryon-antibaryon with G = (−1)L+S+1, attractive potential:

VOPE(r) = −h22ω

2π

4πf 2π

e−µπr

r

For the charged configurations:

|Y +cc〉 =

1√2

[|Λ+

c1Σ0c〉+ η |Σ++

c Λ−c1〉]

|Y−cc〉 =1√2

[|Λ+

c1Σ−−c 〉+ η |Σ0c Λ−c1〉

]we have µπ ' 18MeV, eight times smaller than mπ.

Coulomb-like if we look at standard hadronic length scales!

The Λc1Σc Molecule: Coulomb Spectrum

If µπ → 0 we will have (for both S = 0, 1)

En,l = − 1

2µY

(γB

n + l + 1

)2

with γB = 45+10−10 MeV. With µπ finite we need

γBn + l + 1

� µπ

Only the first few states can survive!

Concrete calculations: shallow S = 0, 1 states survive in S-wave.

Quantum numbers: IG (JP) = 1−(0−), 1+(1−)

The Λc1Σc Molecule: Binding

Actual binding energy depends on short-range physics:

-20

-15

-10

-5

0

-1 -0.5 0 0.5 1 1.5 2

EB [M

eV

]

c0

E0E1Contact

c0 < 0: repulsion, c0 > 0: attraction.

The Λc1Σc Molecule: Λc1 and Σc Widths

|EB | ∼ 1MeV but Γ(Λc1) = 2.6MeV and Γ(Σc) = 1.9MeV.

Does the bound state survive?

I Argument in Guo and Meissner 11: time for formation ofmolecule smaller than lifetime of components.

Γ� 1

Rwith R the range (2� 20, Okay)

I Argument in Hanhart et al. 11:

λR =ΓR

2ER� 1 (λ ∼ 0.03, 0.05, 0.24 depending on the decay)

The Λc1Σc Molecule: Decays

We have these close decay channels

5010

5020

5030

5040

5050

c+

c0 (5019.3)

c1+

c- +(5018.3)

c+

c0 0(5041.6)

Ycc+(5046.0)

Thre

shol

d[M

eV]

and in particular ΣcΣcπ might require attention

Conclusions

I Discrete Scale Invariance possible in hadron moleculesI Unlike to survive at low energies (µπ 6= 0)I Likely to play important role in bindingI Candidates: Λc1D-ΣcD

∗ (maybe the P∗c ) and Λc1Ξb-Σc Ξ′bI Coulomb-like forces in Λc1Σc

I Strange example of a hadron system where a molecularprediction seems possible: shallow baryonium at 5045MeV.

I Caution: finite widths of its components important?

The End

Thanks For Your Attention!