Exotic Glueballs and Hadrons in QCD Sum Rule
Exotic Glueballs
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Exotic hadrons?
Review of the QCD sum rule
Review of the light scalar meson nonet as the
exotic
meson
Exotic glueballs
within
the QCD sum rule
Discussion
In this talk
Hadrons of quantum numbers which are not
expected from the conventional quark model for
hadrons (Baryons of 3 quarks, mesons of quark
and antiquark).
- Examples : mesons of
J
PC
=
0
--
, 1
-+
Hadrons which are expected to have more quark
content than the conventional hadrons have.
- Example : tetraquark states for the light scalar
mesons
Exotic Hadrons?
P
=-(-1)
L
, C=(-
1
)
L+S
PDG 2016
Correlator
of the interpolating current with the
quantum number of the hadron under
consideration
Calculating it in
deeply Euclidean region
by
the
perturbative OPE
Condensates
from the
nonperturbative vacuum
QCD sum rule (SR)
Nonperturbative QCD Vacuum
is related to physical region by the
dispersion relation
Narrow resonance approx. in the phen. side
Quark-hadron duality
Borel transform
makes the contributions from the
continuum
suppressed exponentially
.
QCD sum rules
:
: Borel Mass
Must be
POSITIVE
LHS(
M
) :
Mass of Particle can be determined by
Generally, including all contributions from OPE,
the mass must be independent on
the Borel
mass,
if the current is appropriate to the hadron.
Actually, we cannot do it.
Up to a certain energy
dimension operators
,
mass plateau
appears in
some region of the Borel mass.
Borel window
Borel window must be opened in .
Light scalar meson nonet
Members :
Large decay widths
:
R
e
f
s
.
:
P
D
G
,
C
h
i
n
.
P
h
y
s
.
C
,
3
8
(
2
0
1
4
)
0
9
0
0
1
interpretation
With ideal mixing : for
P
=+1
(?1)Decays of : fraction of ?
Amsler et al,
Phys. Rep.
384(2004)61
L
=1
(?2) Mass degeneracy in
1. From
number of strange quarks
2.
gives 400MeV more mass
:
from the mass formula in a quark model
(Kochelev, H.-J. Lee, Vento,
PLB 594 (2004) 87)
,
for example :
interpretation
One
gluon
exchange &
instanton
:
strongest
attraction
in two quarks
of : scalar (S)
diquark
in two antiquarks of : S
antidiquark
-
Jaffe & Wilczek, Shuryak & Zahed
In flavor space :
Explicitly
In terms of S diquark & S antidiquark
:
L=0
Number of strange quark :
Strange quark component in :
:
Inverted mass spectrum
SRs for light scalar nonet
Interpolating currents :
energy dim.=6
After Borel transform :
Energy dimension of the correlator =
10
QCD SR : and up to
Continuum contributions :
Physical Borel region is around 1GeV :
operators up to dim=8 would be important
LHS of SRs with scalar Diquark
What we have seen…
SRs up to d=6 ops. :
- Multiquark system has large energy dim. :
SRs up to d=6 ops.
could not be enough
.
Large negative contribution from d=8 ops. :
destroys
consistency
of SRs.
Effect from Instanton?
Any possibility
for keeping consistency of SRs?
Generally, five types of
relativistic
currents :
General interpolating currents :
Chen et al., Phys.Lett.B650:369-372,2007
SR for f0(500) again
‘
t Hooft instanton induced interaction for u,d :
Fierz trans.
Including instanton effects, for
a=-b=1
, there
can be stable result!
Exotic Glueballs
Glueballs, bound states of gluons, are predicted
by QCD and have been
studied
for a long time.
There has been no clear experimental evidence.
Theoretically, the glueballs of the two or three
gluons with the quantum numbers, 0
-+
, 0
++
, 2
++
have been studied mainly.
Masses of 0
-+
, 0
++
glueballs of two gluons from
the QCD SR :
-
Mathieu, Kochelev, Vento, IJMP(2009)
These glueballs could couple to the conventional
mesons of 0
-+
, 0
++
.
If glueballs of the exotic quantum numbers exist,
those glueballs will not couple to the conventional
mesons.
One candidate for such kinds of the exotic
glueballs is the glueball with of three gluons.
This fact could be one of difficulties in
searching the glueballs in experiments!
Current of three gluons
- Under the charge conjugation and the parity
- Current will have a form, depending on
O
i
,
glueball within QCD SR
First attempt
: Qiao and Tang, PRL (2014)
- Current :
- It is checked that the imaginary part of the
correlator within OPE is
NEGATIVE. So??
- LHS of SR becomes
NEGATIVE
:
No bound
states in these kinds of three gluons.
This discussion will be appeared on PRL comm.
New current :
- Leading part is
Our analysis : PRD (2017)
Various contributions to
:
- 6 types of correlators are appeared.
- Example :
- Diagrammatic representations :
Correaltor within OPE
:
: POSITIVE
Imaginary part.
: No
Im., for J=0
: Negative
Im.
:By Hyp. Vac. Dom.
Mass and Decay constanat
:
- with
From lattice, Gregory JHEP(2012) :
Discussion
We reviewed that f0(500) could be the bound
state of the diquark-antidiquark.
It was also
discussed that interactions between the diquarks
and antidiquarks played an important role in
QCD sum rule analysis.
We showed that the exotic 0
--
glueball could be
understood within the QCD sum rule.
-
Will this exotic glueball not couple to the exotic meson?
:
0
--
tetraquark has mass ~1.6GeV (Huang et al, PRD(17))
-
It looks that we were lucky to find a good current for it.
- So, it is necessary for us to construct the current by
more systematic way as in the case of f0(500) :
u
n
d
e
r
t
h
e
p
r
o
g
r
e
s
s
.
T
h
a
n
k
y
o
u
!
EPJA(2006), PLB 642 (2006) 358, hep-ph/0702225
Delve into the world of exotic glueballs and hadrons through a review of QCD sum rule, investigation of light scalar mesons, and discussion on hadrons with unconventional quantum numbers. Learn about theoretical frameworks and calculations involving correlators, dispersion relations, Borel transforms, and mass determinations of particles. Discover the intriguing realm of exotic hadrons beyond the traditional quark model.
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Presentation Transcript
Exotic Glueballs Exotic Glueballs Hee-Jung Lee (Chungbuk Nat l Univ.) Collaborators : N. I. Kochelev (JINR) A. V. Pimikov, P. Zhang (IMP, China)
In this talk Exotic hadrons? Review of the QCD sum rule Review of the light scalar meson nonet as the exotic meson Exotic glueballs within the QCD sum rule Discussion
Exotic Hadrons? Hadrons of quantum numbers which are not expected from the conventional quark model for hadrons (Baryons of 3 quarks, mesons of quark and antiquark). - Examples : mesons of JPC=0--, 1-+ P=-(-1)L, C=(-1)L+S Hadrons which are expected to have more quark content than the conventional hadrons have. - Example : tetraquark states for the light scalar mesons
PDG 2016 PDG 2016
QCD sum rule (SR) Correlator of the interpolating current with the quantum number of the hadron under consideration S J i d x e Nonperturbative QCD Vacuum = iq x 2 4 S ( ) 0| ( ) x J (0)|0 q TJ S S Calculating it in deeply Euclidean region by the perturbative OPE Condensates from the nonperturbative vacuum 2 OPE S ( ) : q
OPE 2 ( ) q is related to physical region by the dispersion relation 1 ( ) S q 2 Im ( ) s q = 2 2 OPE S ds 2 2 s 0 Narrow resonance approx. in the phen. side Quark Quark- -hadron duality hadron duality = 4 S 0| | 2 J S f M S S = + 2 2 8 S 2 2 S 2 2 0 2 OPE S Im ( ) s 2 ( ) ( )Im ( ) s f M s M s s S S threshold = ) 0| n n J 2 2 2 n S Im ( ) ( (0)| | (0)|0 q q m J S S n
Borel transform makes the contributions from the continuum suppressed exponentially. QCD sum rules : 2 0 s 2 S 2 2 2 = / M M 2 / 2 2 8 S s M OPE S Im ( ) s 2 ds e f M e 1 S 0 LHS(M) : Must be Must be POSITIVE POSITIVE : Borel Mass M
Mass of Particle can be determined by = 3 ( /2 ) M M S M S S Generally, including all contributions from OPE, the mass must be independent on the Borel mass, if the current is appropriate to the hadron. Actually, we cannot do it. Up to a certain energy dimension operators, mass plateau appears in some region of the Borel mass. Borel window Borel window must be opened in . M s 0
Light scalar meson nonet Members : = = = 0 0 1 : , ( 980 ) I a a 0 0 0 1/2 : 0 : (500), , , f (8 980 00) ) I I ( 0 Large decay widths : = = 50 ~100MeV, = 40 ~100MeV a f 0 0 400 ~ 700MeV Refs. : PDG, Chin. Phys. C, 38(2014) 09001
interpretation qq L=1 With ideal mixing : for P=+1 1 + = = = 0 0 (980) , ( ), a u d a uu dd a du 0 0 2 + = = = = 0 0 (800) , , , us ds sd su 1 (600) = + = ( ), (980) ss uu d d f ss 0 2 0a (?1)Decays of : fraction of ? + [ (980) [ (980) a ] a = 0.85 0.02 Amsler et al, Phys. Rep. 384(2004)61 0 ] KK 0
(?2) Mass degeneracy in 1. From number of strange quarks m L = , a f 0 0 0, a m m m f 0 2. gives 400MeV more mass : from the mass formula in a quark model (Kochelev, H.-J. Lee, Vento, PLB 594 (2004) 87), for example : f 1 0(980)
interpretation [ ][ ] qq qq One gluon exchange & instanton : strongest attraction in two quarks of : scalar (S) diquark in two antiquarks of : S antidiquark -Jaffe & Wilczek, Shuryak & Zahed Jaffe & Wilczek, Shuryak & Zahed S | 3 ,3 ,1 F C S |3 ,3 ,1 F C In flavor space : = = 3 3 3 6 , 3 3 3 6 f f A = S f f A S 3 3 1 8 A A Explicitly [ ] , [ ] , [ ] ud s us d ds u A A A [ ] , [ ] , [ ] ud s us d ds u A A A
In terms of S diquark & S antidiquark : L=0 1([ 2 us ud + = = ] [ = 0 0 (980) [ ][ ], ][ ][ ]) , [ ][ ] a ds us a ds ds us us a us ds 0 0 + = = = = 0 0 (800) [ ][ ], [ ][ ], [ ][ ], [ ][ ] ds ud ud us u d ds 1 (600) = = ] [ + [ ][ ], (980) f ([ ][ ][ ] ) ud u d ds ds u s us 0 2 Number of strange quark : = m m m m : Inverted mass spectrum Inverted mass spectrum f a 0 0 , f a Strange quark component in : 0 0 , f a KK 0 0
SRs for light scalar nonet Interpolating currents : energy dim.=6 After Borel transform : Energy dimension of the correlator =10
QCD SR : and up to u m = = 0 m ( ) O m d s = + + 10 6 4 S S S ( ) M C O M E C O M E C O M E 0 0 4 4, 4, 2 6, 6, f M e 1 S i i i i 2 S 2 + + + + = / M M 2 2 2 8 S S S S / 2 C O M E C O C M 8, 8, 0 10, 10, 12, i i i i i S Continuum contributions : 1 1) 2 0 s 2 = / x M n ( ) E M dx e x n + + 2 2 n ( n M 0 Physical Borel region is around 1GeV : operators up to dim=8 would be important
LHS of SRs with scalar Diquark , f a 0 0
What we have seen SRs up to d=6 ops. : - Multiquark system has large energy dim. : SRs up to d=6 ops. could not be enough. Large negative contribution from d=8 ops. : destroys consistency of SRs. Effect from Instanton?
Any possibility for keeping consistency of SRs? Generally, five types of relativistic currents : = i S T A i T A i 3 3 : C [ ] [ ] J q q q q 1, 2, ] ( + 3, b 4, q C abc q b c ade a d e = ] ( + i S T S i T S i 6 6 : C {[ )}{[ )} J q q a b 1, 2, 3, 4, C a b a b = = = A T i A i S T i S i , , with , and , 0 0 = A i ( ), ( S ), ( ) V C C PS C 5 5 = S i ( ), ( ) T C AV C General interpolating currents : S S S J J = + + + + PS S V S AV S T S ' J vJ v J tJ Chen et al., Phys.Lett.B650:369-372,2007
SR for f0(500) again t Hooft instanton induced interaction for u,d : Fierz trans. Fierz trans. = = = = = = 1 , 1 , 0 , ' 0 , 1/4 for 3 v v t N C
Including instanton effects, for a=-b=1, there can be stable result!
Exotic Glueballs Glueballs, bound states of gluons, are predicted by QCD and have been studied for a long time. There has been no clear experimental evidence. Theoretically, the glueballs of the two or three gluons with the quantum numbers, 0-+, 0++, 2++ have been studied mainly. Masses of 0-+, 0++glueballs of two gluons from the QCD SR : - Mathieu, Kochelev, Vento, IJMP(2009)
These glueballs could couple to the conventional mesons of 0-+, 0++. This fact could be one of difficulties in searching the glueballs in experiments! If glueballs of the exotic quantum numbers exist, those glueballs will not couple to the conventional mesons. One candidate for such kinds of the exotic glueballs is the glueball with of three gluons.
glueball within QCD SR Current of three gluons - Under the charge conjugation and the parity - Current will have a form, depending on Oi,
First attempt : Qiao and Tang, PRL (2014) - Current : - It is checked that the imaginary part of the correlator within OPE is NEGATIVE. So?? - LHS of SR becomes NEGATIVE : No bound states in these kinds of three gluons. This discussion will be appeared on PRL comm.
Our analysis : PRD (2017) New current : - Leading part is
Various contributions to : - 6 types of correlators are appeared. - Example : - Diagrammatic representations :
Correaltor within OPE : : POSITIVE Imaginary part. : No Im., for J=0 : Negative Im. :By Hyp. Vac. Dom.
Mass and Decay constanat : - with From lattice, Gregory JHEP(2012) :
Discussion We reviewed that f0(500) could be the bound state of the diquark-antidiquark. It was also discussed that interactions between the diquarks and antidiquarks played an important role in QCD sum rule analysis. We showed that the exotic 0--glueball could be understood within the QCD sum rule. - Will this exotic glueball not couple to the exotic meson? : 0--tetraquark has mass ~1.6GeV (Huang et al, PRD(17)) - It looks that we were lucky to find a good current for it. - So, it is necessary for us to construct the current by more systematic way as in the case of f0(500) : under the progress. Thank you!
