Lectures on. Holographic QCD. Shigeki Sugimoto (Kavli IPMU) 1/53. holography 2013 APCTP, Pohang 2013/6/19 and 20

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1 Lectures on Holographic QCD Shigeki Sugimoto (Kavli IPMU) holography 2013 APCTP, Pohang 2013/6/19 and 20 1/53

2 1 Introduction Claim : Hadrons can be described by string theory without using quarks and gluons! QCD dual! (at low energy) String theory (in a certain curved background) Holographic QCD D-brane meson baryon string D-brane with Nc strings 2

3 Gauge/String duality [Maldacena 1997, ] Gauge theory 4 dim dual String theory 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

4 Key idea open string gauge field D-brane corresponding curved background Gauge theory on the D-brane 4 dim dual! String theory in a curved background 10 dim Note: SUSY, conformal sym. are not essential in this idea. 4

5 Holographic QCD D4-D8-D8 system QCD [Sakai-S.S. 2004] with massless quarks (at low energy) dual String theory in a non-susy D4 background with probe D8-branes (assuming ) The meson sector is described by 5 dim Yang-Mills-Chern-Simons theory in a curved background. A lot of interesting results 5

6 What can we learn? Non-trivial tests of gauge/string duality Beyond SUSY and conformal symmetry. We can use experimental data to test the duality! New techniques in hadron physics The calculation is very simple, easy and fun! Geometric interpretation of QCD phenomena Confinement, Chiral symmetry breaking etc. New perspective in particle physics QCD is not the only way to describe hadrons. Revival of the old ideas in string theory The old ideas of string theory around 1970 were correct! New ingredients: D-brane, curved space-time and holography. 6 6

7 Plan of Talk Introduction Construction of QCD Applications Conclusion and discussion Blackboard 7

8 2 Construction of QCD pure Yang-Mills theory [Witten 1998] D4-brane on with fermion SUSY D4 acquire mass (low energy) 4 dim U(Nc) pure Yang-Mills 8

9 Brane configuration for QCD [Sakai-S.S. 2004] To add quarks, we add D8-D8 pairs chiral symmetry D8 D8 D4 (low energy) 4dim QCD with Nf massless quarks (at low energy) 9

10 SUGRA solution of the D4-branes [Witten 1998] The curved background corresponding to the D4-brane is explicitly known. D4 in flat space-time corresponding curved background D4 (topologically) S 4 radial coordinate Holographic description 10

11 Holographic Yang-Mills [Witten 1998] D4-brane on ( with ) the corresponding SUGRA solution (topologically) D4 4 dim pure Yang-Mills (at low energy) : t Hooft coupling String theory dual in this background Holographic Yang-Mills : string coupling Good description when : string length (in MKK=1 unit) 11

12 Adding quarks [Sakai-S.S. 2004] Here we assume and use probe approximation. D4-branes are replaced with the corresponding background. D8-D8 pairs are treated as probes. [Karch-Katz 2002] D4-brane on + D8-D8 pairs D8 D8 D4 String theory in the D4 background + D8-branes D8 QCD with massless quarks (at low energy) dual Open + closed string theory in this background Holographic QCD 12

13 3 Applications How realistic our QCD is? Our analysis is only reliable when: large large ( difficult to make large) scale of 5 th dimension (probe approximation) SUGRA approximation Hong s talk for mq 0 (We want to take in the following.) We should not be too serious in the quantitative comparison with the experiments. 13

14 Hadrons in the model The topology of the D4 background is D8-brane D8-branes are extended along particles in : Closed strings glueballs Open strings on D8 D4 wrapped on D8-brane mesons baryons D4-brane 14

15 Plan Glueballs Mesons Baryons Toward QCD Phase Structure 15

16 Glueballs 16

17 Glueball spectrum QCD (Lattice) SUGRA Morningstar-Peardon hep-lat/ Brower-Mathur-Tan hep-th/ [See a review by Aharony-Gubser-Maldacena-Ooguri-Oz 1999 for more results for pure Yang-Mills] 17

18 Mesons D8-brane 18

19 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 19

20 D8-brane action Inserting the SUGRA solution, constant D4 charge CS 5-form 20

21 Therefore, we obtain 5 dim U(Nf) YM-CS theory in a curved background This is considered as the effective theory of mesons. cf) traditional meson effective action [cf) Son-Stephanov 2003] Very complicated, Many parameters 21

22 5 dim YM-CS theory = 4 dim meson theory complete sets Chosen to diagonalize kinetic & mass terms of Reproduces old phenomenological models Vector meson dominance Gell-Mann Sharp Wagner model [Gell-Mann -Sharp-Wagner 1962] Hidden local symmetry [Bando-Kugo-Uehara-Yamawaki-Yanagida 1985] Skyrme model [Skyrme 1961] [Sakurai 1960, Gell-Mann -Zachariasen 1961, ] Masses and couplings roughly agree with experiments. 22

23 Mesons masses To diagonalize Kinetic & mass terms, we choose: Then, we obtain eaten massless pion Mass massive vector meson other candidates 1570? 1720? see later discussion input 23

24 Coupling constants Pion decay constant Couplings including ρmeson 24

25 Chiral symmetry The gauge transformation at corresponds to the chiral symmetry D8 D4 D8 Define D8 transforms as interpreted as the pion field in chiral Lagrangian 25

26 Pion effective action Inserting the mode expansion into the 5 dim action, we obtain This is the Skyrme model action. Note: If we fix to fit, we obtain 26

27 WZW term If we want to `gauge the chiral symmetry, The corresponding (external) gauge fields appear as In the gauge, the CS-term becomes terms with vector mesons reproduces the well-known WZW term. 27

28 Pion form factor pion form factor is computed as charge radius vector meson dominance [Sakurai 1960, Gell-Mann Zachariasen 1961, ] If we fix by fitting the meson mass, we obtain (our model) (experiment) 28

29 Couplings from CS-term Inserting the mode exp. into the CS-term, we obtain Moreover, one can show Complete vector meson dominance Terms with more than one pion field vanish. 29

30 ω meson decay ( and ) Our model predicts that the relevant diagrams for and are Exactly the same as the GSW model! Furthermore, we find [Gell-Mann -Sharp-Wagner 1962] reproduces the proposal given by Fujiwara et al! [Fujiwara-Kugo-Terao-Uehara-Yamawaki 1985] 30

31 Excited open strings Other mesons, including higher spin mesons, are obtained as excited string states. Strategy rotating open string higher spin meson [Imoto-Sakai-S.S. 2010] 1. Consider excited states in the flat background 2. Include the effect of non-trivial background perturbatively. Results J PC for the n=0 states (n>0 states are heavier) 1 st excited (N=1): 2 nd excited (N=2): 31

32 Comparison to observed mesons mass (MeV) (Nf = 2, Isovector)... not established : ground state : 1 st excited (~1300 MeV) : 2 nd excited (~1700 MeV), we also find : no good candidates considered to be 4 quark states 32

33 Regge trajectory highest spin subleading term not a straight line, but seems to work curve with Note: If we use to fix, we get This is unfortunately too small. 33

34 Baryons D4-brane (See also Hong s talk for this topic) 34

35 Baryon as wrapped D4-brane Consider a D4-brane wrapped on the S 4 Eq. of motion for U(1) gauge field on the D4-brane implies RR flux D4 on forces F-strings to be attached on it. [Witten, Gross-Ooguri 1998] D8 F1 Bound state of quarks Baryon Baryon D4-brane wrapped on the Baryon mass ( vol. of S 4 ) is generated by the geometry! 35

36 Baryon as instanton [Sakai-S.S. 2004] In our model, the wrapped D4 can be embedded in D8. D4 within D8 instanton on D8 D8 D8 D4 [Douglas 1995] Various descriptions of baryons are now connected. Wrapped D4 instanton on D8 Skyrmion [Witten, Gross-Ooguri 1998] [Atiyah-Manton 1989] [Skyrme 1961] Bound state of Nc quarks [Gell-Mann, Zweig 1964] Skyrmion instanton 36

37 Three approaches 1 This instanton behaves as a particle in the 5 dim gauge theory. write down an effective field theory for this particle. [Hong-Rho-Yee-Yi 2007, ] Hong s talk 2 Quantize the fluctuations around the instanton and follow the idea developed for Skyrmions. [Hata-Yamato-Sakai-S.S. 2007, Hashimoto-Sakai-S.S. 2008, ] This talk 3 The D4-brane world-volume field theory is reduced to a quantum mechanics by reducing the S 4 direction. quantize this system. (becomes a matrix model for multiple D4-branes) [Hashimoto-Iizuka-Yi 2010, ] 37

38 Quantization of solitons Consider a slowly moving (rotating) baryon configuration. Use the moduli space approximation method : Instanton moduli time Quantum Mechanics for For SU(2) one instanton, [Hata-Yamato-Sakai-S.S. 2007] position size SU(2) orientation Note : genuine moduli : massive modes. We will keep them, since they are lighter than the other massive modes. 38

39 Solving the Schrodinger equation for this quantum mechanics, we obtain the wave functions for the baryons. Example Nucleon wave function: for etc. Evaluating the currents, we can calculate various quantities, e.g., charge radius, magnetic moments, axial coupling, ele-mag form factors, etc. etc. baryon state 39

40 Baryon spectrum [Hata-Yamato-Sakai-S.S. 2007] Theory mass Experiment ( states from PDG) *) Evidence for existence is poor *) *) *) Note: We only consider the mass difference, since term in is not known. (fixed by ρ-meson mass) is a bit too large. It looks better if were around. 40

41 Properties of nucleons [Hashimoto-Sakai-S.S. 2008] [See also, Hong-Rho-Yee-Yi 2007, Hata-Murata-Yamato 2008, Kim-Zahed 2008, Pomarol-Wulzer 2008, Panico-Wulzer 2008,...] nucleon ele-mag form factor See also Hong s talk dipole ( experiment) our result [GeV 2 ] 41

42 Toward QCD Phase Structure 42

43 Confinement in Yang-Mills theory Consider a potential between quark and anti-quark [Rey-Yee, Maldacena 1998] heavy quark heavy anti-quark D4 String Strings cannot end must be connected D4 linear potential confinement Finite temperature conf./deconf. transition (see next slide) [Witten 1998] 43

44 Finite temperature [Witten 1998] Introduce temperature by temperature Euclidean time (If we fix MKK by ρmeson mass) There is a phase transition at, above which the role of and are interchanged. String world-sheet (not allowed without D-brane) confinement deconfinement 44

45 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 45

46 G=O(Nc), USp(Nc) cases O6 ± QCD with G=O(Nc), USp(Nc) D8 : O6 ± : Unbroken global gauge symmetry on D8-brane O6 + O(Nf) O6 - USp(Nf) for O(Nc) QCD for USp(Nc) QCD D8 D8 with O6 ± as expected in QCD! 46

47 Chiral symmetry restoration Low temperature D8 high temperature D8 D8 chiral sym broken confined chiral sym restored deconfined chiral sym broken deconfined [Aharony-Sonnenschein -Yankielowicz 2006] 47

48 Discussion D4 In the deconfined phase, the thermal circle is smaller than the 5 th direction along the D4-brane. It is not clear whether it is smoothly connected to physics in QCD. In principle, we want to make the cutoff large: (*) What is the interpretation of the L dependence? QCD doesn t have such parameter. It is not clear what happens to the phase structure in the previous slide, when we take the above limit (*). Mandal-Morita (2011) argued that there is a phase transition in the limit (*) for case and proposed another scenario of the conf./deconf. transition in holographic QCD using Gregory-Laflamme instability. 48

49 Finite density (Just a few comments... ) What happens if we have many baryons? Baryon = soliton in 5dim U(Nf) theory Repulsive at short distance mainly because of U(1) charge. U(1) part Baryon # : Size of the soliton is small Behaves like a point charge Solitons want to sit near z=0 49

50 Finite density higher density spread in z direction [Rozali-Shieh-Van Raamsdonk -Wu 2007, Kaplunovsky-Melnikov -Sonnenschein 2012, de Boer -Chowdhury-Heller-Jankowski 2012] Quarkyonic phase? Interpretation is not clear to me yet. 50

51 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 51

52 It is in principle possible to improve the approximation. QCD correction correction String theory loop correction correction : string length To make MKK large, we have to go beyond SUGRA approximation : fixed to be determined 52

53 Questions Connection to perturbative QCD Complete understanding of QCD phase diagram Extension to standard model + gravity Proof of gauge/string duality 53