analysis of qcd via supergravity s. sugimoto (yitp) based on hep-th/0412141 (t. sakai @ ibaraki +...
TRANSCRIPT
![Page 1: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/1.jpg)
Analysis of QCD via Supergravity
S. Sugimoto (YITP)
based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.)
Windows to new paradigm in particle physics @ Sendai 2/14-16, 2005
![Page 2: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/2.jpg)
Sorry, my talk is not going beyond SM.
But, many features are common to models with branes, extra-dimensions, etc.
So, please don’t sleep now.
![Page 3: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/3.jpg)
etc.
Dp-brane(p+1) dimensional plane
gauge field
(p+1) dim gauge theoryis realized on the Dp-brane
open string★ D-brane① Introduction
![Page 4: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/4.jpg)
★ gauge/gravity duality
particleSolution of Einstein eq.
D-brane Supergravity solution
Conjecture The information of the gauge theory on the D-brane is somehow encoded in the supergravity solution.
cf)
![Page 5: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/5.jpg)
[Maldacena 1997]
D3-brane
Type IIB string theory on Super Yang-Mills
dual
( in R = 1 unit )
Typical example ( AdS/CFT correspondence )
※ SUGRA is good when
The (strongly coupled) gauge theory can be analyzed via (classical) SUGRA !
![Page 6: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/6.jpg)
Today, I’d like to discussan attempt to analyze
QCD using this technique.
![Page 7: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/7.jpg)
Plan of Talk
String theory
Sugra
QCD
Hadron physics
②
③
④
![Page 8: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/8.jpg)
② Construction of QCD
low energy4 dim pure Yang-Mills theory
★ Construction of 4 dim pure Yang-Mills theory[Witten 1998]
Consider D4-brane onmassless modes are
unwanted
gauge field scalar fermion
Impose
become massive( SUSY is broken )
also acquire mass via 1-loop effect
![Page 9: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/9.jpg)
★ Adding quarks
[cf) S.S.-Takahashi 2004]
4 dim QCD with massless quarks
![Page 10: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/10.jpg)
③ SUGRA description of QCD★“Probe approximation” [Karch-Katz 2002]
We assume
Only D4-brane is replaced with the SUGRA solution. pairs are treated as probes.
heavy
light
probe
cf)
![Page 11: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/11.jpg)
Fortunately, corresponding SUGRA solution is known.
restricted to
D4
D4
flat D4 solution
where radial direction
![Page 12: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/12.jpg)
④ Exploration
Now we are ready to explore !
Chiral symmetry breaking Unification of mesons Comparison to experimental data Construction of baryons
highlights
![Page 13: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/13.jpg)
★ Chiral symmetry breaking
and must be connected
in the D4 background.
As before, we replace D4 with the SUGRA sol.
interpreted as the chiral symmetry breaking !
connected
![Page 14: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/14.jpg)
★ Where are mesons? closed string
open string
glueballs
mesons∃ flavor indices
reduced to 5 dim gauge theory
The low energy effective theory on the D8-brane is a 9 dim gauge theory.
Gauge field:
There is an symmetry
(mesons in realistic QCD live in this sector)
※ Today, we only consider the singlet states, for simplicity.
![Page 15: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/15.jpg)
Insert the SUGRA solution
D8-brane action
This 5 dim Yang-Mills theory is considered as the effective theory of mesons.
[cf) Son-Stephanov 2003]
![Page 16: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/16.jpg)
mode expansion
In addition, we have one more normalizable mode
some complete sets
(first we consdier case)
★ choose as eigen functions satisfying
★ choose as
To diagonalize kinetic & mass terms of ,
![Page 17: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/17.jpg)
Using these, we obtain
eaten
massless scalar mesonmassive vector meson
eaten eaten
are unified in the 5 dim gauge field !
( is consistent with this interpretation.)
In summary,
![Page 18: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/18.jpg)
★ Comments on
is even func. are even func. are vector are odd func. are axial-vector
is pseudo-scalar
Charge conjugation : and
Parity :
★ Consistent with our interpretation.
★ Prediction: vector and axial-vector mesons appear alternately.
( … not established)
![Page 19: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/19.jpg)
★ mass spectrum
The mass of the n-th meson is given by
where is the eigen value of the equation
★ Let us solve this numerically, to see if our model is too bad or not.
We should not be too serious, since the approximation is very crude.
![Page 20: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/20.jpg)
Results :
(our model)
(experiment)
+4 %
-17 % (our model)
(experiment)
Not b ad !
![Page 21: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/21.jpg)
Residual gauge symmetry
constant
s.t.
We partially fix the gauge s.t.
This is interpreted as
★ generalization to
![Page 22: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/22.jpg)
interpreted as the pion field
transforms as
Define
![Page 23: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/23.jpg)
★ gauge
The pion field appears in the boundary condition.
Then, mode exp. is given by
where is defined by
and
with
the gauge tr.We can move to gauge by
![Page 24: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/24.jpg)
★ effective action Inserting the mode exp. into the D8-brane action
we obtain the effective action for the mesons written in terms of and .
This is the Skyrme model !
If we turn off vector mesons for simplicity,we obtain
![Page 25: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/25.jpg)
Turning on ρmeson ( ), we can calculate the couplings of π and ρ.
In particular, the 3 pt coupling
is calculated as
(our model)
(experiment, KSFR relation)
![Page 26: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/26.jpg)
Skyrme proposed
Baryon Soliton in Skyrme model( Skyrmion )
The pion field defines the winding number
This is interpreted as the baryon number charge.
[Skyrme 1961]
★ Baryon as Skyrmion
![Page 27: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/27.jpg)
★ Baryon as wrapped D-brane
Baryons in the AdS/CFT contextis constructed by wrapped D-branes
[Witten 1998]
In our case, the baryon is a D4-branewrapped on the .
D4
F1
There is a source of electric charge on D4
F-string should be attached.
Bound state of quarks
Baryon
![Page 28: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/28.jpg)
In our model, the wrapped D4 can be embedded in D8.
D4 within D8 instanton on D8
D8 D8
D4
[Douglas 1995]
Using
and the boundary condition
[Son-Stephanov 2003]
Wrapped D4 instanton on D8 Skyrmion ■∴
![Page 29: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/29.jpg)
⑤ Summary
Chiral symmetry breaking is understood geometrically.
Mesons are unified in the 5 dim gauge field.
★ The numerical results are also
encouraging.
The baryons are constructed as wrapped D4 instanton on D8 Skyrmion
★ Though the approximation is still very crude, our model catches various qualitative features of QCD and provides some new insights in the low energy hadron physics.
★ QCD is constructed in string theory.
![Page 30: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/30.jpg)
Chiral anomaly and axial anomaly are reproduced.
★ Other topics
WZW term is obtained from CS-term on the D8
The Witten-Veneziano formula for the η’meson mass is derived.
![Page 31: Analysis of QCD via Supergravity S. Sugimoto (YITP) based on hep-th/0412141 (T. Sakai @ Ibaraki + S.S.) Windows to new paradigm in particle physics @ Sendai](https://reader030.vdocuments.us/reader030/viewer/2022032701/56649c775503460f9492bf6a/html5/thumbnails/31.jpg)
Fin