Glueball Decay in Holographic QCD

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Glueball Decay in Holographic QCD

• Seiji Terashima (YITP, Kyoto)
• Based on the work (arXiv0709.2208)
• in collaboration with Koji Hashimoto (Riken)

2007 Nov. 20 at KIAS
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1. Introduction
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• 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 ltTr (F2)(x) Tr(F2)(y) gt
• Lattice QCD calculation predicted their spectra.

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The (glueball) spectra of SU(3) Yang-Mills
Lattice gauge theory
from Morningstar-Peardon
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• 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 ltTr (F2)(x) Tr(F2)(y) gt
• Lattice QCD calculation predicted their spectra.
• However, they are not confirmed by experiment
  although candidates for the glueballs are found.

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Candidates for glueballs with JI0, PC (i.e.
no charge. lowest mass.)
A speculation is that

• f0(s) artifact of final state interaction
• f0(980) K K molecule
• 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
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Pseudo scalar nonet and Scalar nonet (Nf3)


Pseudo scalar nonnet (octetsinglet) ???
Scalar nonnet (octetsinglet) ??
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I³ -1, - ½ , 0, ½, 1
I³ -1, - ½ , 0, ½, 1




d s (K0)
d s (K0)
u s (K-)
I ½ , S-1
u s (K-)
I ½ , S-1



d u (p-)

d u (a0-)


u d (p)

I1, S0
u d (a0)

I1, S0
uu-dd/v (p0)
uu-dd/v (a0 )
2
2
0




s u (K)
s d (K0)
s u (K)
s d (K0)
I ½ , S1
I ½ , S1








uudd-2ss/v (?)
uudd-2ss/v (f0)
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I0, S0
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I0, S0






uuddss/v (?)
I0, S0, Singlet of SU(3)
uuddss/v (f0)
I0, S0, Singlet of SU(3)
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6
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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!

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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 tHooft 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)

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? 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 tHooft
  coupling(g2 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.

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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).

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Plan of the talk

• Introduction
• Review of the Holographic QCD
• Glueball interaction in Holographic QCD
• Decay of lightest scalar glueball
• Conclusion

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2. Review of Holographic QCD
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• 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 S1 and N D4-branes wrapping the S1 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.

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Gravity dual of the D4-brane on S1

• type IIA string M-theory on S1
• D4-brane M5-brane on S1
• type IIA string on S1 M-theory on torus
• D4-brane wrapping S1M5-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)

µ,? run from 0 to 4. L and R are the
parameters. x4 is the M-theory circle.
where t t
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S4 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
Constable-Myers Brower-Mathur-Tan
where is the glueball filed
in the 31 dimension ( )
and is the
mass squared of the glueball. The fluctuations
does not depends on t and x4.
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H(r) was given by the equation of motion
which is written by new dimensionless coordinate
Z
and
where
The boundary condition should be
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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
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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,? run from 0 to 3.
which is equivalent to the previous solution by
the following identification
t is periodic and its Kaluza-Klein mass is
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Other glueballs from 11d-sugra fields
from Brower-Mathur-Tan (2000)
(Note that the dilaton does not correspond to the
lightest glueball.)
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from Brower-Mathur-Tan (2000)
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Comparison between the holographic and lattice
calculation of glueball spectra
Lattice (SU(3) gauge group)
Holographic
from Morningstar-Peardon
from Brower-Mathur-Tan (2000)
(We have dropped state)
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Adding quarks in AdS/CFT holographic QCD

• Adding Nf flavors ? adding another kind of
  D-branes as probe.
• 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 Wittens 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

Karch-Katz Myers et.al.
Sakai-Sugimoto
where F is the field strength of the 9-dim. gauge
fields on the Nf D8-branes.
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Integrating out the S4 part, we have
(Chern-Simons term is not relevant),
Then the massless pions and the ?-mesons appears
as the lowest modes of KK-decomposition along
z-direction
where
Above pions and ?-mesons are Nf xNf matrices. We
will consider Nf2 case in the followings.
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• So far, we have ignored (Nf x Nf) scalars
  corresponding to the transverse direction of the
  D8-brane.
• ? They are the scalar nonet (for Nf3)
  including
• chageless scalars.
• They will mix the lowest glueball.
• If this mixing is large, we have trouble to
  identify the glueball.

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3. Glueball interaction in Holographic QCD
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Computation of the interaction

• 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.
• These two sectors are coupled in the combined
  system of supergravity plus D8-branes.
• 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.
• 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.

Basically just evaluate the D8-brane action. Very
simple.
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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 is
  suppressed. Prediction of the holographic QCD!
• (2) Direct couplings of a glueball with more than
  five meson are suppressed. (implies vector meson
  dominance) No

These are from Holographic gauge choice
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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
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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 tHooft
  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.
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As we have seen, the direct glueball decay
vs
Direct meson decay
x
times the mixing
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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 t.
• 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.
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4. Decay of lightest scalar glueball
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Lightest glueball mass is
We have . Thus no
2 ?-meson decay in holographic QCD
(In the experiment, M1507MeV, m?775MeV)
We will use
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Possible decay process (from kinematics)
Branching ratio for f0(1500) (a) 35
(b)(c) 49 (d) 7
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From the effective action we have, we can compute
the decay width.
For
Experimentally,
Good agreement.
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and
This is too small, but if we set to
the experimental value by hand, we have
Thus
Experimentally,
Consistent (In particular, taking into accont
the masslessness of the pions)
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5. Conclusion
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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!
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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 (J1,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, FB. 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.
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Fin.
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