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TRANSCRIPT
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!