Hidden Local Fields in Hot/Dense Matter

1. Hidden Local Fields in Hot/Dense Matter “What matters under extreme conditions” Berkeley 2007

2. TheCheshire Cat as a gauge degree of freedom Current algebra and emergence of vector mesons Dimensionally deconstructed infinite tower of vector mesons From string theory to infinite tower of vector mesons Baryons as instantons in five dimensions = Baryons as skyrmions in the infinite tower of vector mesons in four dimensions Vector dominance for ALL Harada-Yamawaki (HY) hidden local symmetry (HLS) as a truncated infinite tower Vector manifestation (VM) of chiral symmetry “Vector dominance violation” VM fixed point and the dopping of masses and coupling constants (BR scaling) Effect of VM fixed point and Landau Fermi liquid fixed point in dense medium Observations Emergence of Hidden local symmetries Lecture I Lecture II

3. The Cheshire Cat “How hadrons transform to quarks” Dual description of QCD in terms of hadronic variables In (1+1) dimensions, there is an exact bosonization of fermions: Illustration in 2D and generalization to (3+1) D. Consider fermion theory External fields Mass terms etc can be added … Damgaard, Nielsen & Sollacher 1992

4. Enlarge Gauge Invariance:

5. Gauge fix: • Concerned with chiral symmetry: choose “Cheshire cat gauge” “quarks” “pions” • D is totally arbitrary, so physics should be independent of D: “CCP” D=0 Original fermion theory D=1 with Boson theory: Fermions arise as topological solitons

6. Pick with The “Chiral Bag” (Brown-Rho 1979) Mapping “volume” physics to “surface” with boundary conditions Holography See later: AdS/QCD Example: Fermion number outside inside V R V

7. Equation of motion: • Inside • Outside • At the boundary Generates an axial vector field on the surface and gives rise to “vector anomaly” causing the fermion charge leakage. • Fermion charge: conserved V R V

8. Nature: 3+1 dimensions • QCD could in principle be “bosonized” • But • Nobody so far succeeded to accomplish it • It will have infinite number of bosons and the Lagrangian will have infinite number of terms → effective field theory • An EFT must break down at some scale and has to be “ultraviolet-completed” to a fundamental theory→ “matching” •  Cheshire Cat can be only approximate in Nature with the exception of topological quantities • Nonetheless there are intriguing predictions: e.g., “Proton spin”

9. Proton spin: CC in action Flavor singlet axial current (FSAC) U(1)A Anomaly: Naively: Jproton = a0≡ gA0

10. quark sector Total exp gluon sector

11. From soft pions to vector mesons • At E ≈ 0 , Soft pion/current algebra applies: Observe • Invariance: This local symmetry is “redundant” and arbitrary, so there is no physics by itself. But power comes with a trick.

12. Going to the next energy scale, E ≈ mV, V=r, w (and a1) Pions interact with a strong coupling and the current algebra Lagrangian breaks down at a scale 4p(mV/gV) ~ 4pfp , signaling that new degrees of freedom – the vector mesons – must figure. • How to bring in the vector degrees of freedom require an ingenuity. • Naively: But this is a mess and hopeless at high order. • Cleverly, implement local gauge invariance: e.g. S= U

13. Most importantly local gauge invariance allows a systematic cPT expansion for mV ≈ mp≈ 0. Without it, no way to handle massless vector mesons. • The strategy: Exploit the redundant degrees of freedom to render the vector mesons emergent as local gauge fields and have them propagate HLS theory • Caveat: Elevating EFT to a gauge field theory is NOT unique. Without gauge invariance it’s even worse!! Which one is QCD? b a a b c ……. z    c z E Current algebra EFT

14. HLS a la Harada-Yamawaki (“Truncated tower”) Harada and Yamawaki 2001 Although the formulas look complicated, the idea is simple and elegant and the prediction unambiguous. • Degrees of freedom: p, r (w) with NF=2 or 3. • HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), FpandFsor (g, Fp , a≡ (Fs/Fp)2)

15. The crucial next step is to Wilsonian-match HLS correlators to QCD correlators (OPE) at the matching scale LM ≥ mr In 2nd lecture I will discuss how the RGE flow picks the VM (“vector manifestation”) fixed point as rep. of QCD. “VM”=(g=0, a=1) We are sure that this theory has something to do with QCD! But is it complete?? Perhaps not??

16. Emergence of infinite tower of vector mesons • Bottom-up: Dimensional deconstruction • Top-down: Holographic dual gravity • Baryons as instantons or skyrmions-in-infinite-tower • Complete vector dominance “Strong coupled gauge theory requires fifth dimension” Polyakov, Witten, …

17. Going bottom up From effective field theory

18. Dimensional deconstruction Instead of restricting to one set of vectors as in HY, bring in towers of vector mesons as emergent gauge fields. Do this using “moose construction” Georgi et al. 1999 • One vector meson: ;

19. Two vector mesons … • Many (K=) vector mesons in “open moose”: where

20. NOTE: The “moose” construction with nearest neighbors corresponds to taking a=1: “theory space locality”↔“VM fixed point” (HY theory) Let with lattice size e And take continuum limit with K = , e→0 : →5D YM • Extention in 5th dimension, i.e., dimensional • deconstruction via infinite tower of vector mesons • which are encapsulated in YM fields in “warped” metric. • Global chiral symmetry in 4D is elevated to a local • gauge symmetry in 5D

21. The pion field appears as a Wilson line Atiyah-Manton 1989 The resulting theory, “ultraviolet completed” to QCD, is “dimensionally deconstructed QCD” Son/Stephanov 2004 • infinite tower of hidden local gauge fields • baryons are instantons in 5D YM theory.

22. Going top down from String theory

23. A short tour of string theory

25. Sakai-Sugimoto Theory 2005

26. Comes down to this procedure String “QCD” 4D (a) Supersymmetry totally broken and dimensional reduction from 10D to 5D 5D AdS (b) “Branes” are put to generate color gauge degrees of freedom and flavor degrees of freedom corresponding to “gluons” and “quarks” with suitable chiral symmetry which gets spontaneously broken. Holographic duality (Maldacena) Weak coupling solution in the bulk Strong coupling solution in “QCD”

27. Upshot Duality maps the parameters to each other. The relevant parameters are: • Nc • “’t Hooft constant” l = (gYM)2Nc • Klein-Kaluza scale MKK ~ scale in 5th dimension • k = p(fp)2/4

28. Holographic dual QCD Sakai/Sugimoto 2005 Supergravity solution Note: Same 5D action as “deconstructed QCD” with a background given by string theory in the classical limit – which is known. This amounts to an UV completion.

29. Going to 4D Wave function in z (energy scale) direction • Mode expansion: • Equation of motion:  Action with infinite tower in bulk ≡ low-energy QCD on surface: e.g.

30. In Short 5D gauge field = Infinite tower of massive vector mesons + pions Baryons as topological objects Instantons in 4D = skyrmions in infinite tower of vectors in 3D

31. Strategy • Pretend that l and Nc are “huge” so terms of 1/l and 1/Nc (associated with meson loops)** are ignored. • At the end of the day, put Nc = 3 and determine parameters by the known properties of p and the lowest vector meson r. fp ≈ 93 MeV l = g2 Nc ≈ 9 Fixed from mesons MKK ≈ 0.94 GeV (** Remember Dahsen-Manohar theorem)

32. 63 pages

33. Chiral dynamics Chiral dynamics of pions and nucleons • Point-like instanton** appears as baryon (nucleon) due to the tower of vector mesons that squeeze the soliton in the large l and Nc limit**. Baryon size is given by meson cloud. Back to Yukawa picture. • Baryon chiral dynamics with the 1/l and 1/Nccorrections playing the role of contact counter terms. Justification of cPT as a low-energy QCD! ** instanton size: Rinstanton ~ O(Nc0) ~ 1/(MKKl1/2) → 0

34. Mandatory vector dominance • Most relevant to this school: unequivocal prediction on vector dominance!! “All interactions, normal and anomalous, are vector-dominated.” e.g., pp→ r → pp, p0 → rw→ gg, w → pr → pg g g g V. Metag’s No

35. Predictions Hong, Rho, Yee and Yi, hep-th/0705.2632 • Known parameters: fp ~ 93 MeV, Nc=3, Nf = 2 • Unknown parameters: l=(g YM)2Nc and MKK 9 0.94 GeV Fit to meson spectra by Sakai and Sugimoto 05 • In large l and Nc approx. gA≈ 1.32 (1.27) (1.79) (A) (3.7) (-1.91)

36. (B) Coupling constants figuring crucially in modern OBE NN potentials grNN = 4.8 ± 0.4 < 2p OBE fit: 4.2 – 6.5 gwNN =17.0 ± 1.5 OBE fit: 1.1 – 1.5 First theoretical prediction!! These quantities have Never been predicted before

37. Hint for a “Core” Deviation from Cheshire Cat ? Baryon size: Predicted: Empirical: The nucleon given by instanton in 5D or skyrmion in an infinite tower of vector mesons lacks size of The “core” seen in elastic electron scattering at JLab ? Core size ~ 0.2 fm Petronzio et al 2003

38. Vector dominance The most prominent prediction of HDQCD v = r , r’ , … In general: h = p , r , ... ,N , .. h h • “Old” (standard) vector dominance: • F1p(Q2): (a) = 1, (b) = 0, v = r ~ pQCD ff • F1N (Q2): (a) ≈ (b) ≈ ½, v = r with “intrinsic core” size ~ 0.4 fm (Brown, Rho & Weise 1986) Two-component picture: Iachello, Jackson and Lande 1973

39. “New” (infinite-tower) vector dominance: • F1p : (b) = 0, (a) = 1, v = r, r’ , …,  charge: Identical !! • F1N : (b) = 0, (a) = 1, v = r, r’ , …,  charge There is no direct photon coupling to the Skyrmion or “bag” or other extended object. Direct photon coupling is eaten up by the infinite tower !!

40. What this means in the old picture: Interpreted in terms of HY’s HLS theory (see later): Consider nucleon as a skyrmion in HY’s HLS Lagrangian consisting of p and r. Photon (Am) coupling to pion and nucleon: Quark charge matrix Pion current • Pion: a=2: Direct coupling = 0, • Nucleon: a ≈ 1: ½ direct coupling to the skyrmion. KSRF See also Holzwarth 1996 So what happens to the direct coupling when infinite tower intervenes???

41. Here is what happens: . 5th dimension 5D YM + EW Ext. vector field 4D vector field Field redefinition The direct coupling gets replaced by the tower of vector mesons. So the tower ≈ instanton≈ chiral bag !!

42. Universality restored The sum rule is saturated by the lowest 4 vector mesons to less than 1% accuracy. Sakai & Sugimoto 2005 charge Hong, Rho, Yee & Yi 2007 “New Universality” Cf. “Old universality”:

43. What happens to the infinite tower in hot/dense matter ? Nobody knows …. So we will truncate the tower and adopt Harada-Yamawaki approach

44. HLS a la Harada-Yamawaki Harada and Yamawaki 2001 • Degrees of freedom: p, r (w) with NF=2 or 3. • HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), FpandFsor (g, Fp , a ≡ (Fs/Fp)2) Simple, elegant and predictive.

45. The crucial step: Wilsonian-match HLS correlators to the correlators of a “fundamental theory” at a matching scale LM ≥ mr. What is the “fundamental theory”? • HDQCD: we do not know how the quantities of the theory change as a function of temperature/density. E.g., the quark condensate does not depend on temperature (and density) in the large l and Nc limit. Major problem for the young. Nobody knows at present how to do this.

46. Matching to OPE of QCD A la Harada and Yamawaki Basic assumption: In the vicinity of LM, there is an overlap region where EFT and OPE of QCD are both applicable. Match physical quantities: current-current correlators

47. OPE of QCD Matching LM E EFT(HLS) Parameters (g, Fp , a)

48. EFT sector: “counter terms” with QCD sector:

49. Match GV,A and their derivatives at L=LM. → “Bare” parameters of the EFT Lagrangian expressed in terms of the QCD variables that are known at that scale by pQCD, lattice etc: • Given the bare Lagrangian at L=LM, do quantum calculations: • Evolve X’s by RGE to physical scale, (b) compute loop • corrections in cPT+1/Nc . This works out WELL in free space despite that the vector meson mass is much greater than the pion mass … Harada & Yamawaki, PR381 (03) 1

50. But there is a caveat … • Elevating EFT to a gauge field theory is NOT unique. Without gauge invariance it’s even worse!! a b c c    z E EFT QCD picks one uniquely:→HLS/VM