Hadron Spectra and Quark Mass Dependence in Holographic QCD
Koji Hashimoto (RIKEN)
9th Feb. 2010@ NFQCD workshop (YITP)
arXiv/0803.4192 (JHEP) Hirayama, Lin, Yee, KH
arXiv/0906.0402 Hirayama, Hong, KHarXiv/0910.1179 Iizuka, Ishii, Kadoh, KH
Low energy effective field
theory on D-branes
Holographic QCD
QCD action
Graviry descriptionof
those D-branes
superstring theory
AdS/CFTcorrespondence
Action of hadronsHadron specturm,
interactions
?
Large Nc, strong coupling
Plan
1. A Holographic QCD
2. Mesons and Quark masses
3. Holographic Baryons
4. Baryons and Quark masses
1. A Holographic QCD
[Sakai,Sugimoto(04)]
Sakai-Sugimoto model
U(Nf) Yang-Mills-Chern-Simons theory in curved space
zImportant features:
The gauge group is flavor U(Nf).
Extra “holographic” dimension .
Chiral rotation is defined at
This is the holographic dual of massless QCD, with large and large ‘tHooft coupling
Eigen values correspond to masses of vector mesons.
Comparison with experimental data
KK modes of Vector mesonsA KK mode of Massless pion
Kaluza-Klein decomposition gives mesons
Going to gauge, integration
Chiral lagrangian
Derivation of chiral lagrangian
The action with
- Skyrm term is derived !
Good agreement with experimental data :
- Inclusion of the vector mesons, easy
Natural realization of hidden local symmetry
2. Mesons and Quark masses
arXiv/0803.4192 (JHEP) Hirayama, Lin, Yee, KH
Worldsheet instanton gives quark masses
Put a D6-brane as a probe.
D6 charge is (1,-1) underthe chiral symmetry : Explicit breaking of the chiral symmetry
Worldsheet instanton for the quark mass is given in flat spacetime
D6 spike
Derivation of the quark mass term in Chiral lagrangian
D6 spikeD4 throat
Worldsheet instantonlooks same.
: effective coupling
Chiral condensate, flavor dependence
D6
On each D8, D6 ends
Standard chiral lagrangian !
2D8
2D8
Numerics ( just for illustration )
From pion decay constant and rho meson mass,
We substitute ,
Cf) Experiments :
Pion / quark mass
Chiral condensate
Cf) Lattice :
Other terms
Vector / axial vector mesons :
Mass shift is suppressed by
Pion mass differences ? two worldsheet instantons
3. Holographic Baryons
[Sakai,Sugimoto(04)][Hata,Sakai,Sugimoto,Yamato(07)]
Cf. [Hong, Rho, Yee, Yi (07)]
Baryon = D4-brane wrapping S4
= Instanton in of SS model
Baryons = YM instantons
[Witten(98), Gross,Ooguri(98)]
[Sakai, Sugimoto(04)]
Instanton charge sources U(1)v
[Hata, Sakai, Sugimoto, Yamato (07)][Hong, Rho, Yee, Yi (07)]
Quantization of instantons Baryon spectrum
Baryon solution : dyonic instanton
The background can be approximated by flat space
Solution : BPST instanton + electrostatic potential
Inserting this back to the action leads to a potential
Size is stabilizedto be small,
Small instanton localized at
U(1) Coulomb self-repulsion Effect of curved space
Quantization of the instanton
Moduli space approximation : Moduli with small potentials
Moduli :
Lagrangian :
,
: isospin + spin
Harmonic-like potential Baryons labeled by
Baryon states are given by wave function of the QM :
Proton :
acted by
Mass spectrum of baryons
Baryon mass formula :
PDG:
(tables taken from [Hata et.al])
4. Baryons and quark masses
[Hirayama, Hong and KH, 0906.0402]
[Sakai, Sugimoto and KH, 0806.3122]
Quark mass term and baryon
Quark mass is introduced by worldsheet instantons
[Hirayama, Lin, Yee and KH, 0803.4192] [Aharony, Kutasov]
Pion mass is induced as
We substitute the BPST instanton in the singular gauge
Baryon mass shift
Then we obtain
The baryon mass shift is given by
Using the wave function for rho coordinate,
for nucleons
for Roper etc
Results and Comparisons
Pion mass dependence of nucleon mass
higher order.
Our result :[Hirayama, Hong and KH, 0906.0402]
Lattice QCD :
Pion mass dependence of Roper mass
Our result :
[Brommel et al., 0804.4706] [Bernard, 0706.0312] [QCDSF-UKQCD hep-lat/0312030][Walker-Loud et al., 0806.4549] [PACS-CS 0810.0351]
Three flavors
The computation goes exactly the same, except that we have now SU(3) chiral rotations.
We use the following three for the mass parameters,
Then the baryon mass shift is
Decouplet / Octet mass shifts
Our results :
Comparison with experiments
Our inputs :
Mass splittings of baryons with
Mass splittings of baryons with
5. Conclusion & Discussions
Conclusion
String theory technique provides quite a nice description of mesons and baryons. Spectroscopy is one of the quantities which can be computed in holographic QCD.
Derivation of quark mass term in the chiral lagrangian. Chiral condensate, GOR relation
[Hirayama, Lin, Yee, KH, 0803.4192]
Pion mass dependence of nucleon / Roper mass.[Hirayama, Hong, KH, 0906.0402]
Pion / K mass dependence of Octet / Decouplet mass.[Iizuka, Ishii, Kadoh, KH, 0910.1179]
Other quantities computed
We compute also static properties of baryons, including meson-baryon coupling.
We compute Nuclear force at short distances
Long range nuclear forceThey nicely match exp. data
= First analytic result reproducing repulsive core, from strongly coupled QCD
[Sakai, Sugimoto and KH, 0806.3122]
[Sakai, Sugimoto and KH, 0901.4449]
Three-body nuclear force.
[Iizuka, Nakatsukasa, KH, 0911.1035]
Work in progress
3-body nuclear force
Nuclear force among hyperons/nucleons
N-body in general ?
Color-flavor locking in holographic QCD
Chiral properties of hyperons
[Chen, Matsuura, KH, 0909.1296]
Static quantities of nucleons derived
input :
Glueball sector
Closed string side (gravity)
[Witten (98)]
Witten’s geometry for pure YM without SUSY
Nc D4-branes wrapping S1
Gaugino : antiperiodic
4d bosonic YM at low energy
Gravity solution
What is cleaver about this geometry :
How to break susy is specified field theory dual is clear
Double-Wick rotatedAdS7 x S4 blackhole
(written with 11 dim. supergravity notation)
Closed string side (gravity)Confinement in AdS/CFT
Spacetime is smoothly “truncated” at the core Confinement
No spacetim
e
Quark antiquark potential
Linearpotential
Computing Glueball spectrum via AdS/CFT
Supergravity fluctuation corresponding to the lightest glueball[Constable,Myers(99)] [Brower,Mathur,Tan (03)]
: Glueball field : Eigenfunction in higher dim
Witten’s geometry :
[Brower,Mathur,Tan (03)]
Glueball spectrumobtained in AdS/CFT
Lattice calculation(SU(3) pure YM)
[Morningstar,Peardon (99)]
Glueball decay
ID of QCD glueball?
: Scalar glueball
Lattice prediction of lightest glueball mass: 1600MeV
0++ state
Perturbative QCD
Lattice QCD
We need theoretical description of glueball decay!
Chiral perturbation
Mixings?Difficult Holographic QCD can compute the interaction identification of the glueball!
[Terashima, C-I Tan and KH (0709.2208)]
Computing the coupling between glueballs and mesons
Glueball → Gravity Meson → Gauge (on D8)
② In string theory, all the interactions between gravity and D8 gauge fields are encoded in D8-brane action
We substitute gravity and D8 gauge fluctuationsrepresenting the mesons and glueballs, and perform
the integration of higher dimensional space
We obtain interacting lagrangian of glueball / meson fields
① correspondence:
Glueball , Pion ,ρmeson
( This expression is for a single flavor, for simplicity )
Result
Kinetic terms
No mixing between mesons and lightest glueball
Interaction terms
Possible decay process of the lightest glueball
Interaction terms obtained via AdS/CFT :
YM
← CS
Among these, ( ii )( iii ) includes more than 5 pions after thedecay and so are negligible. Possible decay processes are
These reproduces decay products of f0(1500)
Decay width and branching ratios Decay width
of each branch
If we tune the glueball mass and eta mass by hand so thatit can fit the experimental data, then
Comparison : In experiments, f0(1500) decays as
: not produced
The results are consistent with f0(1500)
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