Transcript Document

Four-quark spectroscopy within the hyperspherical formalism
J. Vijande 1,2, N. Barnea 3, A. Valcarce 2
1Theoretical
Physics Department, Universidad de Valencia
2Nuclear Physics Group, Universidad de Salamanca
3The Racah Institute of Physics, The Hebrew University.
PRD73 (2006) 054004
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
1
• Motivation
• Hyperspherical formalism
• Constituent quark model
• Four-body results (cccc)
• Summary
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
2
Motivation
Why multiquarks?
| Meson  B  0    | qq    | gg    | qqg    | qqqq   ...
Most of the meson spectra can be described assuming that the qq component is
not only the dominant one, but also the only one. However, there are some
exceptions like mesons with exotic quantum numbers or those with properties
which are difficult to explain (like the light scalars, the X(3872) or the opencharmed mesons).
• The hyperspherical harmonic method, widely applied and tested in nuclear
physics for N ≤ 7, has only been applied to quark physics for N = 3.
•Therefore, its generalization would be ideally suited for the study of the
properties of multiquark systems for all quantum numbers.
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
3
Hyperspherical harmonic formalism
• The idea is to generalize the simplicity of the spherical harmonic expansion
for the angular functions of a single particle motion to a system of particles.
r
LK
YLM  ,    K , L , M , Sym 3 N 4 
• The main disadvantage of this method is that the number of HH needed to
obtaing a good convergence is very large. So, one needs to construct HH
functions of proper symmetry for any value of K. This has been overcome by
means of a HH formalism based on the symmetrization of the N−body wave
function with respect to the symmetric group using the Barnea and
Novoselsky algorithm (Ann. Phys. 256, 192).
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
4
Color-Spin basis with well defined C-parity
•
1)
2)
3)
The difficult part is to construct a symmetrized color-spin basis for the N-body
system with well defined C-parity.
–
Spin part → One make use of the SU(2) Clebsh-gordan coefficients.
–
Color part → One could be temped to use the SU(3) Clebsh-gordan
coefficients, however this is not feasible.
To construct the color part we have used a method based on an algorithm by
Novoselsky, Katriel and Gilmore (J. Math. Phys. 29, 1368), obtaining states with
well defined permutational symmetry, spin and color proyection.

Evaluating the quadratic casimir SU(3) operator one can determine the specific
representation of each state, picking only those belonging to the SU(3) singlet.

Evaluating the charge conjugation operator one can choose only those states with
well defined C-parity, c = ±1.
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
5
Technical details
6500
4He
ccc c
6400
E (MeV)
6300
6200
6100
• Usually unbound systems
6000
0
5
10
15
20
K
• Confinement color structure
PRC61 (2000) 054001
  
Conf  cc   Confine/ Deconfine





7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
6
Improving convergence
We extrapolate the energies using
E K   E K     
K
where E(K=∞), α, and β are fitted
7 July, 2015
(K0,Kf )
E(K=∞) 0−+
(3,21)
7018
(5,21)
7007
(7,21)
7004
(9,21)
7003
(11,21)
7000
(13,21)
7000
(15,21)
6993
(17,21)
6990
Four-quark spectroscopy within the
hyperspherical formalism
[MeV]
7
Comparison with other approaches
Bhaduri Model (ccnn) S=1 I=0 L=0
4120
• HH expansion (This work)
•Variational [FBS 35 (2004) 175]
• HO basis [Z. Phys. C 57 (1993) 273]
4080
EPJA19 (2004) 383. (ccnn)
(S,I)
Variational
(li=0)
HH
(li=0)
(0,1)
4155
4154
(1,0)
3927
3926
(1,1)
4176
4175
(2,1)
4195
4193
E (MeV)
4040
4000
3960
3920
3880
0
5
10
15
20
25
K
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
8
Constituent quark model
• This model has been applied to the description of the barion-barion
interaction and the meson and baryon spectra.
• Confinement:


Unquenched
lattice QCD


1


 r
Coulomb



 1 e
cc
c c


r

One Pion Exchange
   

 
- magnet ic   c c  
• One gluon exchange:  Color

T ensor
 
Sigma
Exchange
One
and spin
- orbit

 



One Eta Exchange








• Goldstone Boson exchange: 
One Kaon Exchange 
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
9
Multiquark spectrosopy
4600
4400
Since we are going to study cccc systems we need to
describe properly the spectra of the charmonium
4200
E (MeV)
4000
3800
3600
3400
3200
3000
2800
7 July, 2015
c0-+
J/(1--)
c(0++)
c(1++)
c(2++)
Four-quark spectroscopy within the
hyperspherical formalism
hc(1+-)
(2--)
10
Two-meson thresholds
 Bound  M (ccc c )  M 1 (cc )  M 2 (cc )

Unbound M (ccc c )  M 1 (cc )  M 2 (cc )
cc
cc c c
cc
• Quantum number conservation
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
11
L=0 Four-body results
7400
Exotic
Exotic
Non-exotic
7100
E (MeV)
6800
6500
6200
5900
0++
7 July, 2015
0-+
1++
1+-+-
1----
++
2++
22-+-+
22----
Four-quark spectroscopy within the
hyperspherical formalism
00+-+-
00----
11-+-+
22+-+-
12
Four-quark state Vs. Quark mass.
200
200
P-wave thresholds
S-wave thresholds
100
2+−
MeV
(MeV)
100
1+−
0
0+−
0
2++
-100
-100
0
1000
2000
3000
0
1000
2000
3000
mq (MeV)
mq (MeV)
E  E(ccc c )  T (M1M 2 )
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
13
• Estimation in the light sector (Work in progress).
2++ (I=2)
1−+ (I=2)
M (MeV) ≈
1500
2900
PDG (Exp)
X(1600) 1600 ± 100
(I=1) 1(1400) 1376 ± 17
(I=1) 1(1600) 1653 ± 17
• Only four set of quantum numbers seems promising to be detected
mq < 1 GeV
mq ≈1.7 GeV
mq > 3 GeV
2++
Bound
Bound
Bound
0+−
Small Width
Almost bound
Bound
1+−
Unbound
Almost bound
Bound
2+−
Unbound
Small Width
Small Width
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
14
Summary and outlook
• We have generalized a numerical technique widely used in nuclear physics to study the
two-quark two-antiquark systems.
• We have applied it to the L=0 cccc multiquarks, obtaining that only three(four) states
are good candidates to be observed, the 2++, the 0+−, the 1+−, and maybe the 2+−.
• Improve the method to:
• Include isospin degree of freedom.
• Consider non identical quarks.
• We want to apply the method to
• L different from 0.
• Heavy-light tetraquarks. QQq q
• Hidden-charm/bottom tetraquarks. QqQ q
• General heavy tetraquarks. Q1qQ2q
• Light sector (scalar, exotics….) qqq q
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
15
7 July, 2015
Four-quark spectroscopy within the
hyperspherical formalism
16
Two-body spectroscopy
11200
2000
More than 110 states reproduced
11000
1800
10800
1600
10600
1400
E (MeV)
E (MeV)
10400
1200
10200
1000
10000
800
9800
600
Light I=1
9600
400
Bottomonium
9400
200
9200
0
7 July, 2015
--(1
)
-b+(0-+)
+
0 
2 
++
b0--(0++)
b1--(1++) b2(2
(2--)
+- )
b1(1 )
a2(2++)
(1 )
3 
Four-quark spectroscopy within the
hyperspherical formalism
a1(1++)
17