Glueball Decay
in Holographic QCD
Seiji Terashima (YITP, Kyoto)
Based on the work (arXiv:0709.2208)
in collaboration with Koji Hashimoto (Riken)
and Chung-I Tan (Brown)
2007 Dec. 18 at NCTS
1. Introduction
2
• Excitations in QCD
– Mesons and Baryons
• found and identified in the experiments.
• Lattice QCD result and other theoretical result are consistent
with those.
– Glueballs (exist in any (confined) gauge theory)
• ex. 2-point function <Tr (F^2)(x) Tr(F^2)(y) >
• Lattice QCD calculation predicted their spectra.
3
The (glueball) spectra of
SU(3) Yang-Mills Lattice gauge theory
from Morningstar-Peardon
4
• Excitations in QCD
– Mesons and Baryons
• found and identified in the experiments.
• Lattice QCD result and other theoretical result are consistent
with those.
– Glueballs (exist in any (confined) gauge theory)
• ex. 2-point function <Tr (F^2)(x) Tr(F^2)(y) >
• Lattice QCD calculation predicted their spectra.
• However, they are not confirmed by experiment although
candidates for the glueballs are found.
5
Candidates for glueballs with J=I=0, P=C=+
(i.e. no charge. lowest mass.)
A speculation is that
• f0(σ) : artifact of final state interaction
— molecule
• f0(980): K K
• f0(1370), f0(1500), f0(1710) are glueball and 2 scalar mesons.
f0(1370) has 2-photon decay, and f0(1710) has large KK—branching ratio.
→ f0(1500) might be the glueball. Not confirmed. Other possibilities.
from Seth (2000), PDG, Armstrong et. al., Amsler
6
Pseudo scalar nonet and Scalar nonet (Nf=3)
—
Pseudo scalar nonnet (=octet+singlet) ψγψ
5
I³ = -1, - ½ , 0,
½,
1
—
—
I= ½ ,
S=-1
ds
(K0)
—
I=1,
S=0
du
(π-)
—
us
(K-)
uu-dd/√2
(π0)
—
I= ½ ,
S=1
I=0,
S=0
I=0,
S=0,
Singlet of SU(3)
—
—
ud
(π+)
s—d
—
(K0)
su
(K+)
—
Scalar nonnet (=octet+singlet) ψψ
I³ = -1, - ½ , 0,
½,
1
—
—
I= ½ ,
S=-1
ds
(K0*)
—
I=1,
S=0
du
(a0-)
—
uu+dd-2ss/√
6
(η)
—
—
—
uu+dd+ss/√
6
(η’)
—
uu-dd/√2
(a0 0)
—
I= ½ ,
S=1
—
—
—
us
(K-*)
I=0,
S=0
I=0,
S=0,
Singlet of SU(3)
—
ud
(a0+)
s—d
—
(K0*)
su
(K+*)
—
—
—
uu+dd-2ss/√
6
(f0)
—
—
—
uu+dd+ss/√
6
(f0)
7
Why it is difficult to identify the Glueballs?
•
There are several mesons which have same charges and roughly
same mass as the glueballs.
→
•
The branching ratio are needed to distinguish them.
Experimentally we do not know much about the branching ratio of the
glueball candidates.
→ We expect that LHC will give us a huge amounts of hadoronic
data and improve the experimental situation drastically.
•
However, on the theoretical side, it is very difficult to compute
reliably couplings of glueballs to ordinary mesons in QCD.
Actually, no reliable computations ever have done.
→
We need a way to compute the glueball decay
reliably!
8
Problems for existing methods to compute glueball decay
• Chiral Lagrangian approach:
The glueballs have relatively heavy (heavier than 1500 MeV). Thus no control
by the derivative expansion.
Moreover, glueballs are siglets of the flavor symmetry.
• Lattice QCD:
Spectrum is easy. To compute the decay rate is possible, but very difficult
because Lattice QCD is defined on the Euclidean space-time.
• Usual large N expansion:
Weak t’Hooft coupling is needed to compute explicitly the decay rate. But
confinement can not be seen by the weak coupling expansion.
Thus, the Holographic QCD will be useful !
(though not so reliable now)
9
→ We explicitly compute the couplings between
glueballs and mesons by using holographic QCD.
•
Holographic QCD = application of AdS/CFT to QCD
studies.
–
–
AdS/CFT → large N gauge theory at strong t’Hooft
coupling(g^2 N) = classical higher dimensional gravitational
theory.
This has been applied to
(i) Glueball spectrum in large N pure Yang-Mills theory
by D4-branes compactified on a circle.
(ii) Meson spectrum/dynamics in large N QCD
by adding D6-branes or pair of D8-anti D8 branes to D4-branes.
•
We combine (i) and (ii) to compute glueball decay in
large N gauge theory.
10
Summary of the result
• Decay of any glueball to 4
is suppressed.
Prediction of the holographic QCD!
• “Vector meson dominance” for the glueball decay.
(No direct 4 pion decay.)
• Decay of glueball to a pair of photons is suppressed.
• Mixing of the lightest glueball with mesons is small.
• The decay widths and branching ratios is consistent with
the experimental data of the glueball candidate f0(1500).
11
Plan of the talk:
1.
2.
3.
4.
5.
Introduction
Review of the Holographic QCD
Glueball interaction in Holographic QCD
Decay of lightest scalar glueball
Conclusion
12
2. Review of Holographic QCD
13
• Original AdS/CFT correspondence
Low energy limit of the N D3-branes in IIB superstring
theory
→ Supersymmetric and conformal, not like QCD
• Consider type IIA superstring theory compactified on S^1
and N D4-branes wrapping the S^1 with anti-periodic
boundary conditions for fermions. (Witten)
Then, we have (bosonic) pure 4-dim. Yang-Mills
theory in the low energy limit. Close to QCD.
14
Gravity dual of the D4-brane on S^1
•
•
type IIA string = M-theory on S^1
D4-brane = M5-brane on S^1
• type IIA string on S^1= M-theory on torus
• D4-brane wrapping S^1=M5-brane on torus
Gravity dual
= Near horizon limit of the M5-branes solutions in the IIA supergravity
= doubly Wick-rotated AdS7-blackhole.
(Euclidean time → anti-periodic b.c. for fermions)
where τ= τ+
μ,ν run from 0 to 4.
L and R are the parameters.
x^4 is the M-theory circle.
15
S^4 part is not expected to be necessary in the following, so integrating out it.
Then we have 7-dimensional action:
The metric fluctuations corresponds to glueballs in Yang-Mills theory.
For example, the lightest state is the following fluctuations:
g=ḡ+h
where
and
Constable-Myers
Brower-Mathur-Tan
is the glueball filed in the 3+1 dimension (
is the mass squared of the glueball.
The fluctuations does not depends on τ and x^4.
)
16
H(r) was given by the equation of motion
which is written by new dimensionless coordinate Z:
where
and
The boundary condition should be
17
In order to compute the glueball decay, not just spectra,
we need to know the normalization of H(r), such that
We computed the normalization of the H(r) numerically
where we used the relations between IIA 10d sugra and Yang-Mills theory:
18
The solution of 11d sugra is equivalent to the solution of IIA 10d sugra.
In the Sakai-Sugimoto notation, the solution of IIA sugra is
μ,ν run from 0 to 3.
which is equivalent to the previous solution by the following identification:
τ is periodic and its Kaluza-Klein mass is
19
Other glueballs from 11d-sugra fields
from Brower-Mathur-Tan (2000)
(Note that the dilaton does not correspond to the lightest glueball.)
20
from Brower-Mathur-Tan (2000)
21
Comparison between
the holographic and lattice calculation of glueball spectra
Holographic
Lattice (SU(3) gauge group)
from Morningstar-Peardon
from Brower-Mathur-Tan (2000)
(We have dropped
state)
22
Adding quarks in AdS/CFT: holographic QCD
• Adding Nf flavors → adding another kind of D-branes as probe.
Karch-Katz
Myers et.al.
Sakai-Sugimoto
• Here, we add Nf pairs of D8-brane and anti-D8-brane.
This model has spontaneously broken chiral symmetry,
so there is massless pion.
Let us consider gravity dual, i.e. the D8-branes in the Witten’s background.
(D8-brane and anti-D8-brane are connected and
become smooth curved D8-branes as a result of the curved background.)
The D8-brane action is
where F is the field strength of the 9-dim. gauge fields on the Nf D8-branes.
23
Integrating out the S^4 part, we have
(Chern-Simons term is not relevant),
Then the massless pions and the ρ-mesons appears
as the lowest modes of KK-decomposition along z-direction:
where
Above pions and ρ-mesons are Nf xNf matrices.
We will consider Nf=2 case in the followings.
24
• So far, we have ignored (Nf x Nf) scalars corresponding
to the transverse direction of the D8-brane.
→ They are the scalar nonet (for Nf=3) including
chageless scalars.
They will mix the lowest glueball.
If this mixing is large, we have trouble to identify the
glueball.
25
3. Glueball interaction
in Holographic QCD
26
Computation of the interaction
1. We would like to compute the couplings between the glueballs and
mesons. In the gravity dual, they correspond to the supergravity
fluctuations and the Yang-Mills fluctuations on the D8-branes,
respectively.
2. These two sectors are coupled in the combined system of
supergravity plus D8-branes.
3. The couplings between them are only in the D8-brane action
through the background metric and dilaton, which includes the
fluctuations corresponding to the glueball.
4. We substitute the fluctuations of the supergravity fields
(corresponding to the glueball) and the D8-brane massless fields
(mesons) into the D8-brane action and integrate over the extra
dimensions, to obtain the desired couplings in 4d.
Basically just evaluate the D8-brane action. Very simple.
27
Generic feature of holographic glueball decay
•
Glueballs are obviously flavor-blind. Thus couplings to mesons
are universal against flavors.
•
From the D8-brane action,
we see that
(1) No glueball interaction involving more than two pions. because
Decay of any glueball to 4
holographic QCD!
is suppressed. Prediction of the
(2) Direct couplings of a glueball with more than five meson are
suppressed. (implies “vector meson dominance”): No
These are from “Holographic gauge” choice
28
Interaction of the lightest scalar glueball
First, we rewrite the metric fluctuations in the 10d IIA sugra fields:
Substituting them into the D8-brane action, we have
where we have kept
only the relevant terms
for the decay of glueballs.
The constants c
are calculated
numerically as
29
Mixing of glueball with mesons
• In large N expansion, we know the mixing between G
(glueball) and X (meson) is
. But we want to
know the dependence on the t’Hooft coupling also.
• Actually, our leading order caluculation in the
holographic QCD show that it is just
:
It is suppressed in large N limit.
However, for a generic glueball, direct decay process
is comparable to the decay through mixing.
30
As we have seen, the direct glueball decay:
vs
Direct meson decay:
x
times the mixing
31
But, we can show that
no mixing between lightest glueball and meson
• Scalar mesons = transverse scalar of D8-branes,
denoted by y, which is essentially τ.
• Terms linear in y in the D8-action is:
All of these vanish for the lightest glueball.
No mixing with mesons at order
This is very important to distinguish the glueball and meson.
32
4. Decay of lightest scalar glueball
33
Lightest glueball mass is
We have
in holographic QCD
. Thus no 2 ρ-meson decay
(In the experiment, M=1507MeV, mρ=775MeV)
We will use
34
Possible decay process (from kinematics)
Branching ratio for f0(1500):
(a) 35%
(b)+(c) 49%
(d) 7 %
35
36
From the effective action we have, we can
compute the decay width.
For
Experimentally,
Good agreement.
37
and
This is too small, but if we set
we have
to the experimental value by hand,
Thus
Experimentally,
Consistent
(In particular, taking into accont the masslessness of the pions)
38
5. Conclusion
39
First attempt in computing decays of glueballs to
mesons using a holographic QCD (Sakai-Sugimoto model).
The holographic QCD is, in principle, equivalent to QCD.
We therefore expect that the holographic approach should provide
interesting information on strong coupling physics of QCD.
Explicit couplings between the lightest glueball and the mesons are given,
and the associated decay products/widths are calculated.
Our results are consistent with the experimental data of the decay for the f0(1500)
which is thought to be the best candidate of a glueball in the hadronic spectrum.
We have shown that there is no mixing with the mesons at the leading order.
Decay of any glueball to 4
is suppressed.
This is a prediction of the holographic QCD!
40
Other interesting directions
• Multi-glueball couplings.
Self-couplings of the glueballs can be computed in the
supergravity sector. Emission of mesons from a propagating glueball can be
described by the D8-brane action similarly.
• Universally narrow width of glueballs.
If one can show in the holographic
QCD that the total decay width of any glueball state is narrow, that would
provide support for this widely-held belief. We have shown the
narrowness only for the lightest glueball.
• Other glueballs.
For example, the (J=1,P=+,C=−) glueballs reside in the NS-NS 2-form field,
and it should have a large mixing with the meson fields
because F always appeared in the action as a combination, F+B.
• Thermal/dense QCD.
• Computation of glueball couplings in other models of holographic QCD.
For example, the flavor D6-branes enable one to introduce easily the quark mass.
41
Fin.
42
43