超弦理論による QCD の解析 (Review) Shigeki Sugimoto （ IPMU ） 1 「量子色力学にもとづく真空構造とクォーク力学」 第３回研究会 @ 筑波大 2010/7/7
Introduction 1 Claim : Hadrons can be described by string theory without using quarks ! dual ! String theory QCD (in a certain curved background) “ Holographic QCD ” D-brane meson string D-brane with N c strings baryon 2
★ Gauge/String duality [Maldacena 1997, …] dual Gauge theory String theory ４ dim 10 dim curved space-time example dual N =4 Super Yang-Mills String theory in These two look completely different. But, they are conjectured to be equivalent! 3
★ Key idea D-brane Gauge theory String theory in 10 dim curved space-time . dual ! 4
★ D-brane open string Dp-brane (p+1) dim. plane, on which open strings can end. etc. gauge field (p+1) dim. gauge theory is realized on the Dp-brane. example N =4 Super Yang-Mills D3-brane 5
★ SUGRA description of D-brane D-brane Supergravity solution cf) particle Solution of Einstein eq. example D3-brane 6
★ holographic QCD AdS/CFT [Maldacena 1997] N =4 Super Yang-Mills dual D3-brane String theory in Note: SUSY, conformal sym. are not essential in this idea. holographic QCD QCD dual “holographic QCD” Some brane config. String theory in some curved b.g. (See for a brief review) 2 7
Plan of Talk Introduction 1 Construction of QCD 2 Applications 3 Conclusion and discussion 4 8
2 Construction of QCD ★ Brane configuration D4-brane on with fermion SUSY acquire mass [Witten 1998] D4 4 dim U(N c ) pure Yang-Mills (at low energy) To add quarks, we add D8-D8 pairs [Sakai-S.S. 2004] D8 D8 4dim QCD with N f massless quarks (at low energy) 9
★ SUGRA solution of the D4-branes [Witten 1998] Fortunately, corresponding SUGRA solution is known. where radial direction restricted to flat D4 solution D4 10 D4
★ holographic description of Yang-Mills [Witten 1998] the corresponding SUGRA solution D4-brane on (topologically) ( with ) D4 4 dim pure Yang-Mills String theory in this background dual (at low energy) 11
★ Adding quarks [Sakai-S.S. 2004] Here we assume and use “probe approximation”. [Karch-Katz 2002] D4-branes are replaced with the corresponding background. D8-D8 pairs are treated as probes. String theory in D4-brane on the D4 background + D8-D8 pairs + D8-branes D8 D8 D8 D4 Open + closed string theory QCD with massless quarks in this background dual (at low energy) 12
Applications 3 Now we are ready to discuss the applications But, don ’ t trust too much ! corrections may be large. quarks are massless in our model. The model deviates from real QCD at high energy〜M KK 〜1GeV But, don ’ t be too pessimistic. The effect of “cut off” at M KK is milder than lattice cut off. Remember “quench approximation” works in lattice QCD At least, we should not give up before trying. 13
★ Wilson (Polyakov) loop [Rey-Yee, Maldacena 1998] String world-sheet string attached on the D4 = heavy particle with a color index D4 Radial direction of is bounded from below Area For pure Yang-Mills, Area confinement Finite temperature conf./deconf. transition [Witten 1998] (see next slide) 14
★ Finite temperature [Witten 1998] Introduce temperature by temperature (If we fix M KK by ρmeson mass) Euclidean time There is a phase transition at , above which the role of and are interchanged. F-string ～ quark (for the case without D8) deconfinement confinement 15
★ Chiral symmetry breaking In QCD, it is known that the chiral symmetry is dynamically broken to the diagonal subgroup. In our model , this phenomenon is understood geometrically. D8 D8 D8 D4 and must be connected in the D4 background interpreted as the chiral symmetry breaking ! connected 16
★ Chiral symmetry restoration Low temperature high temperature D8 D8 D8 chiral sym restored chiral sym broken deconfined confined chiral sym broken deconfined [Aharony-Sonnenschein -Yankielowicz 2006] 17
★ Hadrons in the model The topology of the background is D8-branes are extended along glueballs Closed strings [Csaki-Ooguri-Oz-Terning 1998, Koch-Jevicki-Mihailescu-Nunes 1998, A.Hashimoto-Oz 1998, Brower-Mathur-Tan 2000, etc] Open strings on D8 mesons [Sakai-S.S. 2004,2005, Imoto-Sakai-S.S. 2010] D4 wrapped on baryons [Witten, Gross-Ooguri 1998, Sakai-S.S. 2004, etc] 18
★ Glueball spectrum QCD (Lattice) SUGRA Morningstar-Peardon Brower-Mathur-Tan hep-lat/9901004 hep-th/0003115 19
★ Quantitative tests for pure Yang-Mills (SUGRA vs Lattice) Studied around 1998 ～ [For review, see e.g. Aharony-Gubser-Maldacena-Ooguri-Oz 1999] Taken from A.Hashimoto-Oz hep-th/9809106 20
★ Meson effective theory We have D8-branes extended along The effective theory on the D8 is a 9 dim gauge theory Here we only consider the states invariant under ( SO(5) non-inv. states are unwanted artifact of the model ) The effective theory of mesons is reduced to 5 dim gauge theory 21
D8-brane action Inserting the SUGRA solution, D4 charge CS 5-form This 5 dim YM-CS theory is considered as the effective theory of mesons. [cf) Son-Stephanov 2003] 22
★ mode expansion complete sets Chosen to diagonalize kinetic & mass terms of massless Using these, we obtain scalar meson eaten massive vector meson We interpret are unified in the 5 dim gauge field ! 23
★ Quantitative tests for the meson sector (Our model vs Experiment) mass input coupling 24
★ More lessons Structure of interaction ・ hidden local symmetry [Bando-Kugo-Uehara-Yamawaki-Yanagida 1985] consistent with ・ vector meson dominance [Gell ‐ Mann-Zachariasen 1961, Sakurai 1969] ・ GSW model [Gell ‐ Mann -Sharp-Wagner 1962] next slide Anomalies in QCD is reproduced an easy derivation of WZW term [Wess-Zumino 1971, Witten 1983] Witten-Veneziano formula [Witten, Veneziano 1979] Baryon Wrapped D4 instanton on D8 Skyrmion [Skyrme 1961] [Witten, Gross-Ooguri 1998] [Atiyah-Manton 1989] after the next slide 25
★ ω meson decay ( and ) Our model predicts that the relevant diagrams for and are Exactly the same as the GSW model ! [Gell-Mann -Sharp-Wagner 1962] Furthermore, we find reproduces the proposal given by Fujiwara et al ! [Fujiwara-Kugo-Terao-Uehara-Yamawaki 1985] 26
★ Baryon as wrapped D4-brane Baryons in the AdS/CFT context are constructed by wrapped D-branes [Witten 1998, Gross-Ooguri 1998] In our case , Baryon D4-brane wrapped on the RR flux forces F-strings to be attached on it. D4 on Bound state of quarks F1 Baryon D8 Baryon mass (∝ vol. of S 4 ) is generated by the geometry! 27
★ Baryon as instanton [Sakai-S.S. 2004, Hata-Yamato-Sakai-S.S. 2007] In our model, the wrapped D4 can be embedded in D8. D4 within D8 instanton on D8 [Douglas 1995] D8 D8 D4 Using the techniques to quantize soliton, we can analyze the baryon spectrum. Mass spectrum 28
Baryon spectrum Experiment Theory ( states from PDG) *) Evidence for existence is poor mass *) *) *) Note: We only consider the mass difference, since term in is not known. (fixed by ρ ‐ meson mass) is a bit too large. 29 It looks better if were around .
Results for properties of nucleons [Hashimoto-Sakai-S.S. 2008] [See also Hong-Rho-Yee-Yi 2007, Hata-Murata-Yamato 2008] We can also evaluate these for excited baryons such as 30
4 Conclusion and discussion Though the approximation is still very crude, our model catches various features of QCD and provides new insights in hadron physics. “ much better than expected ! ” A lot of qualitative properties in QCD can be understood from the geometry of the background. • Confinement • Chiral symmetry breaking • Phase transition • Origin of baryon mass • etc … 31
It is in principle possible to improve the approximation. QCD String theory correction loop correction correction correction : string length To make M KK large, we have to go beyond SUGRA approximation : fixed to be determined 32
questions Connection to perturbative QCD Extension to standard model + gravity Proof of gauge/string duality 33
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