Pentaquark decay width in QCD sum rules
F.S. Navarra , M. Nielsen and R.R da Silva
University of São Paulo, USP
Brazil
Introduction
Pentaquark mass
 decay width
Conclusions
( mass) Phys. Lett. B578 (2004) 323
( mass) Phys. Lett. B602 (2004) 185
( decay width) hep-ph/0503193
LC 2005
CAIRNS
Introduction
Something new in Hadron Physics:
july/2003
+ (1540 MeV) (u d u d s )
september/2003
-- (1860 MeV) (d s d s u )
april/2004
c (3099 MeV) (u d u d c )
Exotic baryons: can not be three-quark states
contain an antiquark !
may/2005
vanishing...
 decay width
Resonance in the s channel  peak in the cross section
K+ d scattering: Sibirtsev et al., PLB 599 (2004) 230
No peak !
  1 MeV
Extremely narrow !
Pentaquark structure
Meson-baryon molecules?
n
K+
Five-quark bags?
Strottman,
PRD 20 (1979) 748
Topological solitons?
Diakonov, Petrov, Poliakov
ZP A359 (1997) 305
“Diamonds”?
(non-planar flux tubes)
Song and Zhu,
MPL A 19(2004)2791
Triquark-Diquark?
Karliner and Lipkin
PLB 575 (2003) 249
Diquark-Diquark-Antiquark?
Jaffe and Wilczek,
PRL 91 (2003) 232003
QCD Sum Rules
Method for calculations in the non - perturbative regime of QCD
Identities between correlation functions written with hadron
and quark – gluon degrees of freedom
Two - point function: hadron masses
Three - point function: form factors and decay width
Results are functions of the quark masses and vacuum expectation
values of QCD operators : condensates
 mass
q   i  d 4 x eiqx  0 | T  ( x) (0) | 0 


= current (interpolating field)

hadronic fields
composite quark fields
+ : d u d u s
-- : d s d s u
How to combine quark fields in a DDA arrange ?
Matheus, Navarra, Nielsen, Rodrigues da Silva, PLB 578 (2004) 323



1
1 ( x) 
 abc daT ( x) C  5 sb ( x) dcT ( x) C  5 se ( x) C ueT
2
2 scalar diquarks
 2 ( x) 
1
2
 abc d aT ( x) C sb ( x) d cT ( x) C se ( x) C u eT
2 pseudoscalar diquarks
Sugiyama, Doi, Oka PLB 581 (2004) 167
 ( x)   abc  def  cfg saT ( x) C db ( x)sdT ( x) C  5 de ( x)C ugT ( x)
pseudoscalar diquark
scalar diquark
Current contains contribution from the pole (particle)
and from the continuum (resonances)
s0
1 1
 (q )   d s
Im  ( s)
2
sq 
0
2
S0 = continuum
threshold
parameter
Im  =  = spectral density
Combination of 1 and 2
 ( x)  t 1 ( x) 2 ( x)
Insert  in the correlation function
Operator product expansion (OPE)
mq  ab
i  ab
S ab ( x)   0 | T [ qa ( x) qb (0) ]| 0  
x
2 4 
2 x
4 2 x2


i
32  2 x 2
 ab
12
A
t abA g s G
( x      x )
q q
mq
32 
A
A

2
t
g
G

ln
(

x
)

2 ab s
i  ab
x 2  ab

mq  q q  x  6  g s q   G q 
48
2 .3
i x 2  ab
 7 2 mq  g s q   G q  x
2 .3
x 4  ab
 10 3  q q   g s2 G 2 
2 .3
Parameters:
Numerical inputs: (standard)
 q q    (0.23)3 GeV 3
t
s0
 s s   0.8  q q 
 s g s   G  s   m02  s s 
m02  0.8 GeV 2
 gs2 G G    0.5 GeV 4
What is good sum rule?
Borel stability
Good OPE convergence
Dominance of the pole contribution
Reasonable value of S0
ms
M
ms=0.1 GeV
t=1
s0=2.3 GeV
m=1.87 ± 0.22 GeV
OPE
perturbative
dimension 4
dimension 6
continuum
pole
 decay width
Extremely narrow width: < 10 MeV or even < 1 MeV
 (1540)  K  (493)  n(938)
Mass excess of 100 MeV (no problem with phase space)
Possible reasons for a narrow width:
Spatial configuration
Color configuration
Non-trivial string rearrangement
Destructive interference between almost degenerate states
Chiral symmetry
...
 decay in QCDSR:
n
(p´)
K
(q)
Θ
(p)
Three-point function:
 ( p, q)   d 4 x d 4 y ei p x e i q y ( x, y )
( x, y )   0 | T { N ( x) jK ( y )  (0) }| 0 
Phenomenological side
 ( p´, q)  
s , s´
L = ig nK Kn
(negative parity)
L = ig nK   5 Kn
(positive parity)
 i  0 |  N | n( p´, s´)  V ( p, p´)  K (q) | jK | 0  ( p, s) |  | 0 
( p´2 mN2 )( q 2  mK2 )( p 2  m2 )
 0 |  N | n( p´, s´)  N u( p´, s´)
 K (q) | jK | 0  K
 ( p, s) |  | 0  u ( p, s)
(positive parity)
 ( p, s) |  | 0  u ( p, s) 5
(negative parity)
V ( p, p´)   gnK u ( p´, s´) 5u( p, s)
(positive parity)
V ( p, p´)   gnK u ( p´, s´)u( p, s)
(negative parity)
f K mK2
K 
mu  ms
 ( p, q) 
N

 g nK  N K
E
2
2
2
2
2
2
( p´ mN )( q  mK )( p  m )
from QCD sum rules
+
E     5 q p´ im N q    i (mN  m ) p´    5
 i 5 ( p´2  m mN  q. p´)
continuum
Theoretical side (OPE side):
jK ( y)  s ( y)i 5u( y)
currents
 N ( x)   abc (daT ( x)C  db ( x)) 5  uc ( x)
 (0)    abc  def  cfg sgT (0) C [de (0)  5 C ugT (0)] [db (0) C ugT (0)]
correlator
( x, y)  2i abc def  cfg a´b´c´ [ N 2 ( x)  N1 ( x)] K ( y)
N1 ( x)   5  Sc´d ( x)CSaT´e ( x)C  Sb´b ( x) 5
N2 ( x)   5  Sc´d ( x) 5CSaT´e ( x)C  Sb´b ( x)
T
T
K ( y)  CSha
( y)C 5CSgh
( y, ms )C
mq  ab
i  ab
S ab ( x)   0 | T [ qa ( x) qb (0) ]| 0  
x
2 4 
2 x
4 2 x2
i
A
A



t
g
G
(
x



x )
 
2 2 ab s
32  x

 ab
12
q q
mq
32 
A
A

2
t
g
G

ln
(

x
)

2 ab s
i  ab
x 2  ab

mq  q q  x  6  g s q   G q 
48
2 .3
i x 2  ab
 7 2 mq  g s q   G q  x
2 .3
x  ab
 10 3  q q   g s2 G 2 
2 .3
4

x 2 ln(  x 2 ) mq  ab
29 . 3  2
 q q   g s2 G 2 
OPE
color
disconnected
color
connected
Continuum and pole-continuum transitions
continuum
pole
pole
pole
continuum
continuum
Continuum and pole-continuum transitions


(q 2 , p 2 , p´2 )   ds  du
0
0
 ( s, u , p 2 )
( s  p´2 )(u  q 2 )
 ( s, u, p 2 )  a( p 2 ) ( s  mN2 ) (u  mK2 )
 cc ( s, u, p 2 ) ( s  s0 ) (u  u0 )
 b1 (u, p 2 ) ( s  mN2 ) (u  mK2 )
 b2 ( s, p 2 ) (u  mK2 ) ( s  mN2 * )
s0  ( m N   N ) 2
u 0  ( mK   K ) 2
Continuum and pole-continuum transitions
(q 2 , p 2 , p´2 )  pole pole (q 2 , p 2 , p´2 )  c c (q 2 , p 2 , p´2 )
 pc1 (q 2 , p 2 , p´2 )  pc2 (q 2 , p 2 , p´2 )
 g nK  N K
pole pole (q , p , p´ ) 
( p´2 mN2 )( q 2  mK2 )( p 2  m2 )
2
2
cc (q 2 , p 2 , p´2 )
2
cc ( s, u, p 2 )  OPE ( s, u, p 2 )
(quark-hadron duality)

b1 (u, p 2 )
pc1 (q , p , p´ )   du 2
2
2
(
m

p
´
)(
u

q
)
2
N
m
2
2
2
K*

b2 ( s, p 2 )
pc 2 (q , p , p´ )   ds 2
2
2
(
m

q
)(
s

p
´
)
2
K
m
2
2
2
N

A
B
~
b ( )
b1 (u, p 2 )  b1 (u )  d 1 2
(  p )
m2


~
b2 ( )
2
b2 ( s, p )  b2 ( s )  d
2
(


p
)
2
m
~
b (u )
b1 (u, p 2 )  21 2
(m  p )
~
b
2
2 ( s)
b2 ( s, p )  2
(m  p 2 )
Borel transform schemes
n
1 2 n 1 
d 
2
2


 ( M )  lim
(
Q
)


(
Q
)
2 
2

n ,Q  n !
 dQ 
I)
q2  0
p 2  p´2
p2  P2  M 2
p2  P2  M 2
II)
q2  0
p 2  p´2
2
2
2
2
III) q  p  p´   P
q 2  Q 2  M ´2
(unstable sum rule)
Sum rules
0
e  m / M  e  mN / M
 mN2 * / M 2
s / M 2
G
 Ae
  ds OPE ( s) e
2
2
m  mN
0
2
IA
IB
G
e
2
2
 m2 / M 2
2
 m N2 / M 2
e
m2  mN2
G e
II A
s0
/M´
G e
/M
2
s0
  ds OPE ( s) e
s / M 2
e  m / M  e  mN / M
 mK2 / M ´2  mN2 * / M 2
 Ae
e
2
2
m  mN
2
u0
  ds  du OPE ( s, u ) e
0
II B
2
 Ae
 m2
0
2
 mK2
s
2
s / M 2
2
e
u / M ´2
0
 mK2
s0
u0
0
0
2
/M´
e
 m2 / M 2
 mN2 / M 2
e
m2  mN2
 Ae
  ds  du OPE ( s, u ) e  s / M e u / M ´
2
2
 mK2 / M ´2
e
 m2 / M 2
G   gnK N K
Numerical evaluation of the sum rules
From each sum rule and its derivative determine G and A
 ´s are known from the mass sum rules : G 
K  0.37 GeV 2
N  (2.4  0.2)  102 GeV 3
  (2.4  0.3)  105 GeV 6
f K  160 MeV
ms  100 MeV
mu  5 MeV
g nK
Sum rules with color connected diagrams
N = 0.4 GeV
N = 0.5 GeV
N = 0.6 GeV
IA
g nK
N = 0.4 GeV
II A
N = 0.5 GeV
N = 0.6 GeV
g nK
N = 0.4 GeV
N = 0.5 GeV
g nK
N = 0.6 GeV
M´ = 1.0 GeV
M = 1.5 GeV
IB
N = 0.4 GeV
N = 0.5 GeV
g nK
N = 0.6 GeV
M´ = 1.0 GeV
M = 1.5 GeV
II B
g nK
Results
(negative parity)
color
connected
all
diagrams
IA
0.7
2.6
IIA
0.8
3.6
IB
0.8
3.2
IIB
1.0
4.5
g nK  0.83  0.42
g nK  3.48  1.8
 = 8.6 MeV implies that gnK = 0.4
Decay width
 
1
2
2
2
2
2
2
g
[(
m

m
)

m
]

(
m
,
m
,
m
nK
N

K

N
K)
3
8  m
 (m2 , mN2 , mK2 )  (m2  mN2  mK2 ) 2  4m2 mN2
Negative parity:
All diagrams:
 = 650 MeV
Color connected:
 = 37 MeV
Conclusions
We have used QCDSR to study pentaquark properties
QCDSR for pentaquarks are not as satisfying as for other hadrons
It is possible to obtain reasonable values for the  and  masses
However: the continuum contribution is large !
the OPE has irregular behavior !
The  narrow width is more difficult to understand :
With all diagrams we can not obtain a narrow width!
With only the color connected diagrams we obtain a smaller width
Negative parity  strongly disfavored
Pentaquarks properties
 mass
Constituent quark mass:
mu  md  340 MeV
ms  510 MeV
p (uud )   (udu d s )
Adition of two quarks
One unit of
angular momentum


mq = 340 + 510 = 860 MeV
E  mN *  m p  600 MeV
M   mp  mq  E  2380 MeV
Non-trivial atraction mechanism