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

Emergence of Hidden local symmetries The Cheshire 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 Lecture ILecture II

The Cheshire Cat 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 … “How hadrons transform to quarks” Damgaard, Nielsen & Sollacher 1992

Enlarge Gauge Invariance:

 Gauge fix:  Concerned with chiral symmetry: choose “Cheshire cat gauge” “quarks”“pions”   is totally arbitrary, so physics should be independent of  “CCP”  Original fermion theory  with Boson theory: Fermions arise as topological solitons

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

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 VRV

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”

Proton spin: CC in action Flavor singlet axial current (FSAC) U(1) A Anomaly: Naively: J proton = a 0 ≡ g A 0

quark sector gluon sector exp Total

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

 Going to the next energy scale, E ≈ m V, V=  (and a 1 ) Pions interact with a strong coupling and the current algebra Lagrangian breaks down at a scale 4  m V /g V ) ~ 4  f   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.  U

Most importantly local gauge invariance allows a systematic  PT expansion for m V ≈ m   ≈ 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!! EFT Current algebra a b c ……. z E ab cz  Which one is QCD?

HLS a la Harada-Yamawaki Harada and Yamawaki 2001 Although the formulas look complicated, the idea is simple and elegant and the prediction unambiguous.  Degrees of freedom:  with N F =2 or 3.  HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), F  and F  or (g, F   a≡ (F  /F  ) 2 ) (“Truncated tower”)

 The crucial next step is to Wilsonian-match HLS correlators to QCD correlators (OPE) at the matching scale   ≥ m  In 2 nd 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??

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, …

Going bottom up From effective field theory

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”  One vector meson: ; Georgi et al. 1999

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

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

o The pion field appears as a Wilson line 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. Atiyah-Manton 1989

Going top down from String theory

A short tour of string theory

Sakai-Sugimoto Theory 2005

Comes down to this procedure 5D 4D (a) Supersymmetry totally broken and dimensional reduction from 10D to 5D (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. String “QCD” AdS Holographic duality (Maldacena) Weak coupling solution in the bulk Strong coupling solution in “QCD”

Upshot Duality maps the parameters to each other. The relevant parameters are:  N c  “’t Hooft constant”  (g YM ) 2 N c  Klein-Kaluza scale M KK ~ scale in 5 th dimension   =  (f   2 /4

Holographic dual QCD 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. Supergravity solution Sakai/Sugimoto 2005

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

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

Strategy  Pretend that and N c are “huge” so terms of 1/  and 1/N c (associated with meson loops)** are ignored.  At the end of the day, put N c = 3 and determine parameters by the known properties of  and the lowest vector meson  f  ≈ 93 MeV  g 2 N c ≈ 9 M KK ≈ 0.94 GeV (** Remember Dahsen-Manohar theorem) Fixed from mesons

63 pages

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 and N c limit**. Baryon size is given by meson cloud. Back to Yukawa picture.  Baryon chiral dynamics with the  and 1/N c corrections playing the role of contact counter terms. Justification of  PT as a low-energy QCD! ** instanton size: R instanton ~ O(N c 0 ) ~ 1/(M KK  ) → 0

Mandatory vector dominance  Most relevant to this school: unequivocal prediction on vector dominance!! “All interactions, normal and anomalous, are vector-dominated.” e.g.,  →  → ,  0 →  →   →  →  No    V. Metag’s

Hong, Rho, Yee and Yi, hep-th/ Predictions  Known parameters: f   MeV, N c =3, N f = 2  Unknown parameters:  g YM ) 2 N c and M KK GeV Fit to meson spectra by Sakai and Sugimoto 05 In large  and N c approx. g A ≈ 1.32 (1.27) (1.79) (-1.91) (3.7) (A)

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

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

Vector dominance h h h = ,  v =  ’, …  “Old” (standard) vector dominance:  F 1   (Q 2 ): (a) = 1, (b) = 0, v =   F 1 N (Q 2 ): (a) ≈ (b) ≈ ½, v =  ~ pQCD ff with “intrinsic core” size ~ 0.4 fm (Brown, Rho & Weise 1986) Two-component picture : Iachello, Jackson and Lande 1973 The most prominent prediction of HDQCD In general:

 “New” (infinite-tower) vector dominance:  F 1  : (b) = 0, (a) = 1, v =  ’, …,  charge:  F 1 N : (b) = 0, (a) = 1, v =  ’, …,  Identical !! 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 !!

Interpreted in terms of HY’s HLS theory (see later): Consider nucleon as a skyrmion in HY’s HLS Lagrangian consisting of  and  Photon (A  ) 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. See also Holzwarth 1996 So what happens to the direct coupling when infinite tower intervenes??? What this means in the old picture: KSRF

5 th 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 !! Here is what happens:

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

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

HLS a la Harada-Yamawaki Harada and Yamawaki 2001  Degrees of freedom:  with N F =2 or 3.  HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), F  and F  or (g, F   a ≡ (F  /F  ) 2 ) Simple, elegant and predictive.

The crucial step: Wilsonian-match HLS correlators to the correlators of a “fundamental theory” at a matching scale   ≥ m   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  and N c limit. Major problem for the young. Nobody knows at present how to do this.

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

E Parameters Matching  M OPE of QCD EFT(HLS) (g, F   a)

EFT sector: with “counter terms” QCD sector:

Match G V,A and their derivatives at  =    → “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  M, do quantum calculations: (a) Evolve X’s by RGE to physical scale, (b) compute loop corrections in  PT+1/N c. This works out WELL in free space despite that the vector meson mass is much greater than the pion mass … Harada & Yamawaki, PR 381 (03) 1

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

Observation The RGEs expose a fixed point**, called “vector manifestaion (VM)” fixed point, constrained by QCD: Assumption : Consistency with QCD: “When chiral symmetry is restored, i.e, then the vector and axial correlators are equal to each other:” ** Among several fixed points. So

Important consequence The vector meson mass parameter (“parametric mass”) vanishes at the VM fixed point because the gauge coupling vanishes!! This is because the mass is higgsed. Basis for BR scaling.

In-Medium Parameters The bare parameters of the Lagrangian depend on medium because the QCD condensates depend on medium: Power of local gauge invariance (LGI):  One can do a systematic calculation in  PT theory with X* sliding with the background (T, n) with both  and .  LGI allows the mass parameter M  * to drop as low as m  without the difficulty encountered if LGI is absent.

Vector Manifestation In Medium RGE with the sliding X* with the same chiral symmetry condition that Assumption: In the chiral limit, as (T, n) approaches (T c, n c ), the quark condensate vanishes. As And the system “flows” to the VM fixed point! Harada, Kim, Rho & Sasaki

Prediction  “Parametric mass” near (T,n) c:  On-shell mass VERY near (T,n) c : n ≥ 0. Idem for  if U(N f ) is a good symmetry This BR scaling follows from the VM. Adami/Brown’s QCD sum rule

 Violation of the “old” vector dominance  For (T, n) = 0, a = 2 (i.e., KSRF):  coupling is vector-dominated  As (T, n)→ (T, n) c, approach the VM fixed point and a → 1 and the photon couples directly to  half of the time. “VD is violated.” Note: the factor g in the coupling to        Note: a flows from 2 in vacuum to 1 in dense/hot medium

Pion form factor is strongly affected Harada & Sasaki 2006 T=0 T ~ 0.9 T c VD violation without with What about dileptons? In this theory, one cannot assume VD!! The same story in density, perhaps even worse. Nucleon form factor!!

“VD violation” (i.e. a → 1) in the Infinite tower (a) What we found: in holographic dual QCD, all form factors are vector-dominated – albeit by  of them -- at (T, n)=0. And there is NO reason to suggest that such VD will be violated at (T, n) ≠ 0. Vector dominance must be intact but with an infinite tower of vector mesons ! (b) Nature: Pion is vector-dominated by the lowest vector  at (T, n)=0 but the VD is violated by the flow of a from 2 to its fixed point a = 1 at (T, n) c. (a)+(b) → higher members of the infinite tower must be figuring in medium. How ??

Vector dominance and anomaly process  Form factors (and also dilepton process) are governed by the “normal” component of hidden gauge action.  But processes    → 2 ,  →  etc are governed by the “anomalous” component of the action, topology These processes are also totally vector-dominated:  →  →  No direct  coupling Note: Anomalous processes are often topology-protected E.g.

All interactions, normal and anomalous, are vector-dominated. e.g.,  →  → ,  0 →  →   →  →  No    ( )

How do these “intriguing” things manifest themselves in finite nuclear systems with which experiments are done?

Going to finite nuclei is a long way!! Nuclear interactions take place near the Fermi surface Physics near the Fermi surface coming from the “matching scale”    requires “double-decimation procedure.”  1 st from  M to a nuclear physics scale  nucl ~ 2 fm -1  2 nd from  nuc to 0 MeV relative to the Fermi surface. RGE: nuclear matter saturation due to Fermi liquid fixed point explains why Walecka model works! Where does BR scaling appear?

BR scaling enters at the 1st decimation, i.e., in the intrinsic background dependent parameters; m*, g* etc.  Physical observables in nuclear matter exhibit BR scaling but compounded with (many-body) Fermi-liquid paramters. Example: Migdal formula  m  */m   contribution to Landau F 1 Given the gyromagnetic ratio in heavy nuclei (e.g., Pb), can determine  at nuclear matter density. Friman & Rho 1996

How exchange currents were confirmed Story of Isovector magnetic form factor of 3 He Before Saclay experiments in 1980’s, data were fit with the S-state wave function alone and tensor force that gives a D-state w.f. would destroy the fit. → Conclude: “No tensor force!!” Saclay experiments showed that at higher momentum transfers, the S-state w.f. could not explain the data. Exchange currents based on chiral symmetry could describe both low and high momentum transfer experiments with both S- and D-state w.f. → Restore tensor force & establish exchange currents!! Subtle is Nature

What we can say with certainty  Close to the VM fixed point, the scaling is clear-cut: Guess: T ≥ T flash ~ 125 MeV n ≥ n flash ~ (1-2) n 0  But below the “flash point” nothing much happens in temperature, while scaling in density is compounded with the Fermi liquid fixed point effect and direct connection to the quark condensate is unknown. In short, chiral symmetry effects and mundane many-body effects wage a guerrilla warfare.

Conclusion If the presently measured dilepton data were proven unequivocally to indicate that light-quark hadron masses do not undergo shift in dense/hot medium, then there would be something fundamentally wrong with the basic premise of the notion of dynamically generated mass based on chiral symmetry. This would be a serious crisis in QCD physics. Or it may be that at that density/temperature, quasi-particle notion for hadrons is wrong but then it will be at odds with the shell-model in nuclear structure.

Dense matter near the critical point n c is a lot more subtle  Current lore: “normal matter” makes a phase transition to color superconducting (CSC) matter. But is it Fermi liquid → CSC? What if kaon condensed?  If normal matter is an instanton matter as HDQCD suggests, then there can be a “deconfined quantum critical phenomenon (DQCP)”. The skyrmion-1/2 skyrmion transition in 4D can be an instanton-meron transition in 5D. Analogous to Neel magnetic-ordered state → VBS paramagnet state. Such a transition would imply that the “nomal state” is a non-Fermi liquid which would imply something like high-T superconductivity…

Half-Skyrmions f*f* Lee, Park, Rho and Vento 2004