Transcript Scalar D meson in nuclear matter

Scalar D mesons in nuclear
matter
Taesoo Song, Woosung Park,
Kie Sang Jeong and Su Houng Lee
(Yonsei Univ.)
Procedure
1. Motivation
2. QCD sum rule
in vacuum & in nuclear matter
3. Application to Scalar D mesons
4. Conclusions
1. Motivation
• Charm-strange scalar meson Ds0+(2317) was
discovered in 2003
• Its mass is too lower than the expected values in
quark models and other theoretical predictions.
• It has been interpreted as the isosinglet state
(conventional scalar meson), a four-quark state, a
mixed state of both, a DK molecule, …
• Later charm scalar meson D0*(2400) with broad
width was discovered in 2004
• Its mass is similar to or larger than that of
D+s0(2317), even though Ds(0-) is 100 MeV higher
than D(0-).
2. QCD sum rule
1. OPE (operator product expansion)
2. Phenomenological side
3. Dispersion relation
4. Borel transformation
2. 1. OPE (operator product expansion)
lim A( x) B(0)   Cn ( x)On (0)
x 0
n
Cn ( x) : WilsonCoefficient (shortrange effect)
On (0) : Nonsingular operator w
ith dimensionn
(longrange effect)
As an example,
T hepolarization functionof charmscalar currentis

  i  d 4 xeiq x T j ( x), j  (0)

 i  d 4 xeiq x T q ( x)c( x), c (0)q (0)


 i  d 4 xeiq x : Tr S q ( x) S c ( x) :  : q ( x) S c ( x)q(0) :
: c (0) S q ( x)c( x) :  : q ( x)q (0)c (0)c( x) :
  ( 0 )   (1)   ( 2 )   (3)
, where q  u , d , s
may be ignored
Perturbative quark propagator in a weak gluonic background field
(The gluons are supposed to emerge from the ground state)


Considering  ~ : Tr Sq (x)Sc ( x) :
(0)
Perturbative part
I : dimension 0
Gluon condensates
<G2> : dimension 4
and more condensates
with higher dimension
or more strong couplings
in Fock Schwinger Gauge x  A ( x)  0
x
A ( x)  
x 1...x n ( D 1...D n F ) x 0
n n!( n  2)
1 1  n
 ( x)   x ...x ( D 1...D n  ) x 0
n n!
and
 (1)
(i ) n
 
: q D 1...D n   1... n S c (q) q :
n!
n
 is graphically
(1)
Quark condensate
<qq> : dimension 3
Quark-gluon mixed condensates
<qGq> : dimension 5
and more condensates
with higher dimension
or more strong couplings
OPE up to dimension 5 in vacuum
, where C0 (q2) is perturbative part
OPE up to dimension 5
in nuclear matter

in therest frameof thematter,v  (1,0),

and considering scalar meson at q  (q0 ,0),
 (q0 )   e (q02 )  q0  o (q02 ),
where
List of employed condensate parameters
2.2. Phenomenological side
 (q 2 )  i  d 4 xeiq x  o | T [ j ( x), j  (0)] | o 
 i  d 4 xeiq x ( x0 )  o | j ( x) j  (0) | o   ( x0 )  o | j  (0) j ( x) | o 
Insertingident it yoperat orbetween two currentoperat ors
dp3
I  
|  p   p | ,
3
(2 ) 2 p

and using translation operat ors o | j ( x) |  p  e ip x  o | j (0) |  p  ,
1
e iwx0
and thedefinit ionof step function  ( x0 ) 
dw
,

2i
w  i
  o | j (0) |    2  o | j  (0) |    2 
q
q


 (q)   

.
 E   q  i 
 E   q  i 
2
E
2
E
 
q
q
0
q
q
0


Itsimaginarypart is extractedby using
1
P
2


i

(
x
) ,
2
2
x  i x
 1 q 
2
Im      0   o | j | q   (q02  Eq2 ) 

 
 
 2 2 Eq 

1

2
  q0   o | j  |  q   (q02  E2q )
 2 2E  

q 

2
2
If  o | j | q    o | j  |  q  ,
Im     o | j | q   (q02  Eq2 )  spectralfunction!!
2

 F (q 2  mh2 )  Im   (q 2  s0 )
, which is so called ' P ole ContinuumAnsatz'
2.3. Dispersion relation
(in vacuum)
L.H.S .is a function of QCD parameters
such as g, mq, <qq>, <G2>…, obtained
from OPE.
R.H.S. is a function of physical
parameters such as mλ and s0
Dispersion relation
(in nuclear matter)
2.4. Borel transformation
n
 d 
1
2
2

Borel transformat ion Bˆ  2lim
(Q 2 ) n  
,
where
Q


q
2 
Q 
(
n

1
)!
dQ


n 
Q2 / nM 2
1) improvesOP E side by suppressing high - dimensional condensateterms
N
 1 
1  1 
Bˆ  2  
 2
( N  1)! M 
Q 
N
2) improvesthephenomenol
ogical side by suppressing continuumpart
Im  ( s )
s / M 2
Bˆ 
ds

Im

(
s
)
e
ds
2

sQ
C
C
3. Application to scalar D mesons
1. Charm scalar meson D0* in vacuum
2. Charm scalar meson D0* in nuclear matter
3. Charm-strange scalar meson Ds0 in vacuum &
in nuclear matter
3.1. Charm scalar meson D0*
in vacuum
From the dispersion relation
,
the mass of scalar D meson is

 
~  m~ 2 / M 2
2

F
e
/

1
/
M
~2  
m
~ m~ 2 / M 2
Fe

Borel window
The range of M2 satisfying below two conditions
1) continuum/total < 0.3
/
< 0.3
2) Power correction/total < 0.3
/
<0.3
Borel curve for D0*(2400) in vacuum
(adjust s0 to obtain flattest curve
within Borel window)
Mass (GeV)
2.45
2.44
2.43
S0=7.7 GeV2
2.42
2.41
2.4
2.39
2.38
1
1.5
2
M2 (GeV2)
2.5
3
3.2. Charm scalar meson D0*
in nuclear matter
• Two dispersion relations for πe and πo exist
•
•
≡f(s0+,s0-)
≡g(s0+,s0-)
Afterproperrecombination of two functions,
d
d




f
(
s
,
s
)

m
g
(
s
,
s
)
0
0

0
0
2
2
d (1 / M )
d (1 / M )
m2  
f ( s0 , s0 )  m g ( s0 , s0 )
d
d




f
(
s
,
s
)

m
g
(
s
,
s
)
0
0

0
0
2
2
d (1 / M )
d (1 / M )
m2  
.




f ( s0 , s0 )  m g ( s0 , s0 )
T hatis,
m  m (m , s0 , s0 )
m  m (m , s0 , s0 ) ,
and theyare solved by iterativemethod.
For example,in thecase of 0.1 timesnuclear saturationdensity,
iteration No.
0
1
2
3
4
D+ s0
D+ mass
7.55
2.373
7.5
D- s0
7.7
D- mass
2.392
7.8
2.391
7.8
2.391
2.371
Mass shift of D0*±(2400) in nuclear matter
(The conditions for Borel window are
continuum/total<0.5 & power correction/total <0.5)
2.4
2.38
D+
Mass (GeV)
2.36
D-
2.34
2.32
2.3
2.28
2.26
2.24
0
0.2
0.4
0.6
Nuclear density (0.17 nucleon/fm3)
Continuum threshold s0 (GeV2)
8
7.5
s0+
s0-
7
6.5
6
5.5
5
0
0.2
0.4
Nuclear denstiy (0.17 nucleon/fm3)
0.6
Physical interpretation of the result
T hequark componentof D0* is d c, and thatof D0* is c d .
Nuclear matteris rich in u and d quarks.
As a result,d c D0*  is moreattractivethanc d D0* .
T hatis thereason whythemass of D0* decreasesmore than
thatof D0* in nuclear matter.
3.3. Charm-strange scalar meson Ds0
in vacuum & in nuclear matter
T hemass of strangequark is ignoredfor simplicity.
Condensateparametersare changedas belows.
d d  s s  0.8 d d
vacuum
 y dd
matter
d gFd  s gFs  0.8 GeV 2 s s
d d  ss  0
d  iD0 d  s  iD0 s  0.018GeV n
1
1
1






d  D02  gF  d  s  D02  gF  s  y d  D02  gF  d
8
8
8






d  D02 d  s  D02 s  y d  D02 d
d  gFd  s  gFs  y d  gFd
and y  0.36 fromlatticecalculations.
Borel curves for Ds0 (2317) in vacuum
(The conditions for Borel window are
continuum/total<0.5 & power correction/total <0.5)
Mass (GeV)
2.33
2.32
S0=6.9 GeV2
2.31
2.3
2.29
2.28
1.1
1.6
2.1
M2 (GeV2)
2.6
Mass shift of Ds0±(2317) in nuclear matter
(The conditions for Borel window are
continuum/total<0.5 & power correction/total <0.5)
2.31
Mass (GeV)
2.3
2.29
D+
D-
2.28
2.27
2.26
2.25
Continuum threshold s0 (GeV2)
7
6.9
6.8
6.7
s0+
6.6
s0-
6.5
6.4
6.3
6.2
0
0.5
1
1.5
Nuclear density (0.17 nucleon/fm3)
0
0.5
1
1.5
Nuclear density (0.17 nucleon/fm3)
4. Conclusion
1. We successfully reproduced the mass of charm scalar meson D0*
in vacuum and that of charm-strange scalar meson Ds0 by using
QCD sum rule
2. Based on this success, their mass shifts in the nuclear matter are
estimated by considering the change of condensate parameters
and adding new condensate parameters which do not exist in
vacuum state.
3. The mass of D0*+ decreases in the nuclear matter more than that
of D0*-, as naively expected.
4. But the mass of Ds0+ increases a little in the nuclear matter, while
that of Ds0- decreases.
5. The quark component of these scalar particles are still in
question, whether they are conventional mesons or quark-quark
states or their combinations or others.
6. We expect that the behavior of their masses in nuclear matter
serves to determine the exact quark component of those scalar
particles.
4.1. Future work
• QCD sum rule for pseudoscalar D meson,
which is the chiral partner of charm scalar
meson – the mass difference between scalar
and pseudoscalar meson is expected to
decrease in nuclear matter due to the partial
restoration of chiral symmetry.
• QCD sum rule for charm scalar meson as four
quark state – The comparison of its mass shift
in nuclear matter with that of conventional
meson will reveal the exact component of
charm scalar meson.