Download Exotics and hybrids from lattice

Exotics and hybrids from lattice QCD
Nilmani Mathur
Department of Theoretical Physics,
TIFR, INDIA
Non-quark model states
 States with excited gluon
•
•
•
Hybrid mesons ( qq meson + excited glue)
Hybrid baryons ( qqqbaryon + excited glue)
Glue balls (consitutent glue)
•
Tetraquark, Pentaquark and higher number of quark states
 Multi-quark states
 So far there is no conclusive experimental evidence
of such a state
 These states are not well understood
•


Quark model fails to explain these states
Lack of understanding makes experimental identification difficult.
Lattice QCD calculations can provide crucial information of such a state
Strong Interactions, TIFR
S = 0, 1
L = 0, 1, 2, 3…
  
J  L  S , P  ( 1) L1 , C  ( 1) L S
Allowed :
J PC  0  ,0  ,1  ,1  ,1  ,2  ,2  ,2  ,....
Forbidden (Exotics) :
Strong Interactions, TIFR
J PC  0  ,0  ,1  ,2  ,3  ,4  ,....
Example of an Exotic
 States with quantum number : 1-+
 It is not possible to write an
interpolating field for this state with a


form :
1
 qq, for any 
 Possible operators :

b
q a  4 E ab
q
,
j

i jkl q a  k Blabq b    B

i jkl q a  4 k Blabq b

 jkl q a  5 4 k E labq b
 
q  4 Dq


qa  5 q a q b  5 i qb    a1


 ijk q  5 4 j Dk q
 

 
 ijk q  j Bk q , Bi   ijk D j Dk


 ijk q  4 j Bk q
Strong Interactions, TIFR
Strong Interactions, TIFR
Lattice results for 1-+
Strong Interactions, TIFR
Decay states
Strong Interactions, TIFR
Decay states
Strong Interactions, TIFR
Octahedral group and lattice operators
Construct operator which transform irreducibly under the symmetries of the lattice
J
Λ
G1
G2
H
1/2⊕7/2⊕9/2⊕11/2 …
5/2⊕7/2⊕11/2⊕13/2 …
3/2⊕5/2⊕7/2⊕9/2 …
Baryon
Strong Interactions, TIFR
Λ
A1
A2
E
T1
T2
J
0⊕4⊕6⊕8
3⊕6⊕7⊕9
2⊕4⊕5⊕6
1⊕3⊕4⊕5
2⊕3⊕4⊕5
…
…
…
…
…
Meson
…R.C. Johnson, Phys. Lett.B 113, 147(1982)
Strong Interactions, TIFR
Dudek, Edwards, Mathur, Richards :
Phys. Rev. D77, 034501 (2008)
Strong Interactions, TIFR
Dudek, Edwards, Mathur, Richards :
Phys. Rev. D77, 034501 (2008)
Strong Interactions, TIFR
Overlap Factor (Z)
Strong Interactions, TIFR
Renaissance in Charmonium physics
S. Olsen arXiv:0801.1153v1 (hep-ex)
Strong Interactions, TIFR
Z+(4430)
Belle,
S-K. Choi et al, PRL 100, 142001 (2008)
Since charged, it cannot be either charmonium or hybrid
Recent Babar data do not support it
Small Charm state
However,
So, tetraquark?
more recent Belle inside
analysis
a large still
light quark hadron?
claims it
Molecule?
Study in baryon is needed.
Strong Interactions, TIFR
…….M. Voloshyn, arXiv:0711.4556v2
Dudek et. al
Strong Interactions, TIFR
Hadron spectrum Collaboration
Dudek et. al.
Strong Interactions, TIFR
A more recent calculation
Mπ = 700 MeV Nf = 3, as = 0.12 fm, at-1 = 5.6 GeV, L = 2 fm
Dudek et al, Phys.Rev.Lett.103, 262001 (2009)
Strong Interactions, TIFR
Status of exotic and hybrids
• There is suggestive, though not conclusive, evidence of
the existence of exotics and hybrid both in light and
charm quark sectors.
• Lattice calculations find evidence of existence of exotics
and hybrid at non-realistic quark masses. More detail
calculations with controlled systematic and proper
understanding of decay channels are necessary.
Strong Interactions, TIFR
Glueball
A glueball is a purely gluonic bound state.
In the theory of QCD glueball self coupling admits
the existence of such a state.
Problems in glueball calculations :
‫٭‬
Glueballs are heavy – correlation functions die rapidly at
large time separations.
‫ ٭‬Glueball operators have large vacuum fluctuations
‫ ٭‬Signal to noise ratio is very bad
Strong Interactions, TIFR
Glueball
On Lattice, continuum rotational symmetry becomes discrete cubic symmetry
with representation
A1, A2 , E, T1 ,T2 etc.
of different quantum numbers.
Typical gluon operators :
Moringstar and Peardon ….
hep-lat/9901004
Fuzzed operator :
Try to make the
overlap of the ground
state of the operator to
the glueball as large as
possible by killing
excited state
contributions.
Strong Interactions, TIFR
Generalized Wilson loops
• Gluonic terms required for glueballs and hybrids can be
extracted from generalized Wilson loops
Strong Interactions, TIFR
SU(3) Spectrum
Exotic Glueball
Oddballs!
Y. Chen…N.Mathur.. et al. Phys. Rev. D73, 014516 (2006)
In current PDG
Strong Interactions, TIFR
E.B. Gregory et al. PoS LATTICE2008:286,2008
Strong Interactions, TIFR
f0(1710)
a0(1450)
M (MeV)
a1(1230)
f0(1500)
f0(1370)
a2(1320)
a0(980)
K0*(1430)
f0(980)
κ(800)
ρ(770)
σ(600)
π(137)
0¯ ¯(1)
JPG(I)
1+ ¯(1)
1¯+(1)
2+ ¯(1)
0+ ¯(1)
0++(0)
0+ (1/2)

G
I (J
PC

)  1 (0



), 1 (1
)
Our results shows scalar mass around 1400-1500 MeV, suggesting
a0(1450) is a two quark state … hep-ph/0607110
Strong Interactions, TIFR
C. Mcneile : Nucl.Phys.Proc.Suppl.186:264-267,2009
Strong Interactions, TIFR
q 2q 2 nonet
q q nonet
R. L. Jaffe
Strong Interactions, TIFR
ππ four quark operator (I=0)


1
 
 
o o

      ,
3

  u  5d

  d  5u
1
o
  ( u  5 u  d  5d )
2
Strong Interactions, TIFR
1
1



 (t )  (0)  2  D(t )  C (t )  3  A(t )  G (t )   ,
2
2




 D (t )  C (t ) ,
Strong Interactions, TIFR
I2
I0 ,
  5   5 [ , I (J
G
PC

)  0 (0

)]
N. Mathur et. al.. Phys.Rev.D76, 114505 (2007)
u
u
d
u
u
d
d
d
t
t0
Scattering states
E ( p  1)  E ( p  1)
Possible BOUND state
σ(600)?
π
π
u
d
u
d
π E ( p  0)  E ( p  0)
d u
d u
π
t0
t
Evidence for a tetraquark state?
Strong Interactions, TIFR
Scattering states
(Negative scattering
length)
d
u
d
u
Volume dependence of spectral weights
N. Mathur et.al…Phys.Rev.D76,114505 (2007)
W0
W1
Volume independence suggests the observed state is an one particle state
Strong Interactions, TIFR
Interpolating fields for Tetraquarks
Strong Interactions, TIFR
Prelovsek….NM …et. al (2010)
Tetraquark States
Prelovsek….NM …
et. al (2010)
Strong Interactions, TIFR
Tetraquark States
Prelovsek….NM …
et. al (2010)
Strong Interactions, TIFR
f0(1710)
N. Mathur et. al.. Phys.Rev.D76, 114505 (2007)
f0(1500)
a0(1450)
K0*(1430)
f0(1370)
a2(1320)
a1(1230)
M (MeV)
qq ?
a0(980)
f0(980)
κ(800)
ρ(770)
σ(600)
Kπ Mesonium?
KK Mesonia ?
ππ Mesonium
π(137)
0¯ ¯(1)
JPG(I))
1+ ¯(1)
1¯+(1)
2+ ¯(1)
0+ ¯(1)
0++(0)
d
d
u
u
0+ (1/2)
Θ+ on the Lattice
Quark content :
uudd s
Two possible states
N
u
d u
d
K
u
u
d
S
S d
Θ+ bound state
Two-particle NK scattering state
m(Θ+) ~ 1540 MeV
S-wave : mK+ mN ~ 1432 MeV
P-wave :
Strong Interactions, TIFR
mK2  p2  mN2  p2
Θ+ on the Lattice
Most of the experiments do not see any
evidence of a pentaquark state
N
u
d u
S
d
K
u
u
d
S d
Almost all detail quenched lattice calculations do not see
any evidence of Θ+ pentaquark state
Strong Interactions, TIFR
Future study on lattice
• Future calculations with dynamical fermions and realistic
quark masses.
• Calculations for spin-3/2 pentaquarks as claimed by a
lattice study  Phys. ReV D72, 074507 (2005).
• Study of charm pentaquarks as predicted by models.
Strong Interactions, TIFR
How do you distinguished various states?
•
•
•
•
•
Local :
Tetraquark :
Molecular :
Hybrid :
Glueball :
q ( x )q( x )
q ( x )1q( x )q ( x )2q( x )
q ( x )1q( x )q ( y)2q( y)
q ( x )Dq( x )
• Solve generalized eigenvalue
problem including all operators and
find contribution from each state
Strong Interactions, TIFR
G. Bali : arXiv:0911.1238
Strong Interactions, TIFR
Strong Interactions, TIFR
PANDA
Antiproton Annihilation at Darmstadt at the High
Energy Storage Ring at GSI
An Experiment at FAIR “Facility of Antiproton and Ion Research"
Special detector, high luminosity (L~ 2x1032 cm-2s-1) and phase space
cooled antiproton beam.
http://www-panda.gsi.de
Energy resolution ~50 keV
Physics :
 Charonium spectroscopy
 Excited Glue
(Glueballs and Hybrids)
 Charm in Nuclei
 Charmonium Hypernuclei
 D and DS-Physics
 Other Topics
The Panda Experiment is
collaboration of more then 45
institutes in 15 countries with
Strong
Interactions,
TIFR
more than
300 collaborators.
Physics
 Strangeness nuclear physics,
hypernuclei, kaonic nuclei
Nuclei with strangeness
 Exotic hadron search, Chiral
dynamics and meson
properties in nuclear medium
 Structure function, hard
exclusive processes, spin
structure of the nucleon with
target polarization
 Hadron physics in neutrino
Strong Interactions, TIFR
scattering
Strong Interactions, TIFR
Conclusion
Lattice QCD is entering an era where it can make
significant contributions in nuclear and particle physics.
Exotic and Multiquark (>3) states :
 Exotic and Multiquark states may exist in nature and
lattice QCD can contribute significantly by predicting
masses and quantum numbers.
 One need to be careful to distinguish a bound state from
a scattering state.
 Various laboratories is (will be) searching for these
states.
Strong Interactions, TIFR
Variational Analysis
ψi : gauge invariant fields on a timeslice t that corresponds to
Hilbert space operator ψj whose quantum numbers are also carried
by the states |n>.
Construct a matrix
 Need to find out variational coefficients
such that the overlap to a state is maximum
 Variational solution  Generalized eigenvalue problem :
 Eigenvalues give spectrum :
 Eigenvectors give the optimal operator :
Strong Interactions, TIFR


n
2  1
1

t
 ( t )  e Ht  (0)e  Ht
  



 ip .( x  x 0 )
G ( t , p)   e
0

(
x
)
n
,
q
n
,
q
 ( x0 ) 0


x

n ,q

x

n ,q
  
  
  


  e  ip .( x  x0 )  0 e H ( t  t 0) ip'.( x  x0 ) ( x0 )e  H ( t  t 0) ip'.( x  x0 ) n, q n, q  ( x0 ) 0
 e
  
 ip .( x  x 0 )

x
e


0  ( x 0 ) n , q n, q  ( x 0 ) 0
  
iq .( x  x 0 )  E qn ( t  t 0 )

n ,q
  i ( p  q ). x 0  Eqn ( t  t 0)


   ( p  q )e
0  ( x 0 ) n, q n , q  ( x 0 ) 0

n ,q
 e
 E np ( t  t 0 )

0  ( x 0 ) n, p
2
n
  Wn e
n
Strong Interactions, TIFR
 E np ( t  t 0 )
t
 W1e  E1 ( t  t 0)

n
Source : N. Christ
Strong Interactions, TIFR
Source : N. Christ
Strong Interactions, TIFR
Mass in Euclidean space
Fourier transform in Euclidean time
 d e
ip4 τ
e -M n | |
1
1


2M n
2 M n ( M n  ip4 ) 2 M n ( M n  ip4 )

1
M n2  p42



p4   iE
1
M n2  E 2
Mn : location of poles in the propagator of |n>.
pole masses of physical state
Strong Interactions, TIFR
Strong Interactions, TIFR
What is a resonance particle?
 Resonances are simply energies at which differential cross-section of a particle
reaches a maximum.
 In scattering expt. resonance  dramatic increase in cross-section with a
corresponding sudden variation in phase shift.
 Unstable particles but they exist long enough to be recognized as having a
particular set of quantum numbers.
 They are not eigenstates of the Hamiltonian, but has a large overlap onto a single
eigenstates.
 They may be stable at high quark mass.
 Volume dependence of spectrum in finite volume is related to the two-body
scattering phase-shift in infinite volume.
 Near a resonance energy : phase shift rapidly passes through pi/2, an abrupt
rearrangement of the energy levels known as avoided “level crossing” takes place.
Strong Interactions, TIFR
C. Morningstar, Lat08
Strong Interactions, TIFR
Solution in a finite box

1
1
L  x  L,
2
2
VL ( x ) 
Strong Interactions, TIFR

 V ( x  nL)
n  
C. Morningstar, Lat08
Identifying a Resonance State
• Method 1 :
 Study spectrum in a few volumes
 Compare those with known multi-hadron decay channels
 Resonance states will have no explicit volume dependence
whereas scattering states will have inverse volume dependence.
• Method 2 :
 Relate finite box energy to infinite volume phase shifts by
Luscher formula
 Calculate energy spectrum for several volumes to evaluate
phase shifts for various volumes
 Extract resonance parameters from phase shifts
• Method 3 :
 Collect energies for several volumes into momentum bin in
energy histograms that leads to a probability distribution which
shows peaks at resonance position.
….V. Bernard et al, JHEP 0808,024 (2008)
Strong Interactions, TIFR
Observables
1
ˆ > = Lim
ˆ (U, ψ, ψ)]
<O
Tr[e -βH O
β
Z
-S g [U]-S F [U,ψ,ψ]
DU
Dψ
Dψ
O[U,
ψ,
ψ]
e

= Lim β
-S g [U]-S F [U,ψ,ψ]
DU
Dψ
Dψ
e

Integrating out the Grassmann variables is possible since
Quark
Jungle
Gym
S = ψDψ
F
-1 -Sg [U]
DU detD O[U, D ] e
1
nf -Sg [U]

-1
ˆ
< O >=
=
dU
det
D(U)
e
O[U,D
]



n
nf -Sg [U]

Z
DU detD e
n
nf

1
ˆ
< O >=
N

n
1 -Sg[U]+ln detD f
U e
Z
Parameters : gauge coupling
and quark masses
Strong Interactions, TIFR
O[U, D-1]
ChPT
D
-1
La t t i c e
(U)D-1(U)...D-1(U)
QCD
pQCD
Quark
(on Lattice
sites)
Gluon
Quark
Jungle
Gym
(on
Links)
quark propagators :
Inverse of very large
matrix of space-time,
spin and color
Strong Interactions, TIFR


n
2  1
1

t
 ( t )  e Ht  (0)e  Ht
  



 ip .( x  x 0 )
G ( t , p)   e
0

(
x
)
n
,
q
n
,
q
 ( x0 ) 0


x

n ,q

x

n ,q
  
  
  


  e  ip .( x  x0 )  0 e H ( t  t 0) ip'.( x  x0 ) ( x0 )e  H ( t  t 0) ip'.( x  x0 ) n, q n, q  ( x0 ) 0
 e
  
 ip .( x  x 0 )

x
e


0  ( x 0 ) n , q n, q  ( x 0 ) 0
  
iq .( x  x 0 )  E qn ( t  t 0 )

n ,q
  i ( p  q ). x 0  Eqn ( t  t 0)


   ( p  q )e
0  ( x 0 ) n, q n , q  ( x 0 ) 0

n ,q
 e
 E np ( t  t 0 )

0  ( x 0 ) n, p
2
n
  Wn e
n
Strong Interactions, TIFR
 E np ( t  t 0 )
t
 W1e  E1 ( t  t 0)

n
Analysis (Extraction of Mass)
N
G ( )   Wi e
i 1
 m i
 W1 e
 m1
 
Correlator
decays
exponentially
Effective mass :
m1
G ( )
 e  m1  m1 ( 1)
G (  1)
m1, m2
 G ( ) 
m ( )  ln 

 G (  1) 


|w 1 |2 e  E1  |w 2 |2 e  E2  ...
 ln  2  E1 (  a )

 |w 2 |2 e  E2 (  a )  ... 
 |w 1 | e
 a E1[1   (|w 2 |2 / |w 1 |2 e ( E 2  E1 ) / a )
Strong Interactions, TIFR
Chiral extrapolation
Continuum Limit
Physical pion mass
M
M
m
( m2 )
q
a (a2)
Input parameter
Finite volume Effect
M
Strong Interactions, TIFR
L