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 I Lecture 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

V R V

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 riseto “vector anomaly” causing the fermion charge leakage.

Fermion charge: conserved

V R V

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: Jproton = a0 ≡ gA0

quark sector

gluonsector

exp

Total

From soft pions to vector mesons

At E ≈ 0 , Soft pion/current algebra applies:

Invariance:

This local symmetry is “redundant” and arbitrary, sothere is no physics by itself. But power comes with a trick.

Observe

Going to the next energy scale, E ≈ mV , V=(and a1)

Pions interact with a strong coupling and the currentalgebra Lagrangian breaks down at a scale4mV/gV) ~ 4fsignaling 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 systematicPT expansion for mV ≈ m≈ 0. Without it, no way to handlemassless vector mesons.

The strategy: Exploit the redundant degrees of freedomto render the vector mesons emergent as local gauge fieldsand have them propagate HLS theory

Caveat: Elevating EFT to a gauge field theory is NOTunique. Without gauge invariance it’s even worse!!

EFT Current algebra

a b c ……. z

E

a bc z

Which one is QCD?

HLS a la Harada-YamawakiHarada and Yamawaki 2001

Although the formulas look complicated, the idea is simpleand elegant and the prediction unambiguous.

Degrees of freedom: with NF=2 or 3.

HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), F and For (g, Fa≡ (F/F)2)

(“Truncated tower”)

The crucial next step is to Wilsonian-match HLS correlatorsto QCD correlators (OPE) at the matching scale ≥ m

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??

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 theoryrequires fifth dimension”

Polyakov, Witten, …

Going bottom upFrom effective field theory

Dimensional deconstruction Instead of restricting to one set of vectors as in HY, bringin 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 5th dimension, i.e., dimensional deconstruction via infinite tower of vector mesonswhich 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 Theory2005

Comes down to this procedure

5D

4D

(a) Supersymmetrytotally broken anddimensional reductionfrom 10D to 5D

(b) “Branes” are put togenerate color gauge degreesof freedom and flavor degrees of freedomcorresponding 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:

Nc

“’t Hooft constant” (gYM)2

Nc

Klein-Kaluza scale MKK ~ scale in 5th dimension

= (f2/4

Holographic dual QCD

Note: Same 5D action as “deconstructed QCD” with a backgroundgiven 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 Nc are “huge” so terms of 1/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 and the lowest vector meson

f≈ 93 MeV

g2 Nc ≈ 9

MKK ≈ 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 Nc limit**. Baryon size is given by meson cloud. Back to Yukawa picture. Baryon chiral dynamics with the and 1/Nc corrections playing the role of contact counter terms. Justification of PT as a low-energy QCD!

** instanton size:

Rinstanton ~ O(Nc0)

~ 1/(MKK ) → 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/0705.2632Predictions

Known parameters: fMeV, Nc=3, Nf = 2

Unknown parameters: g YM)2Nc and MKK

9 0.94 GeV

Fit to meson spectra bySakai and Sugimoto 05• In large and Nc approx.

gA ≈ 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

gNN = 4.8 ± 0.4 < 2 OBE fit: 4.2 – 6.5

gNN =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 skyrmionin 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

hh

h = ,v = ’ , …

“Old” (standard) vector dominance:

F1 (Q2): (a) = 1, (b) = 0, v =

F1N (Q2): (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:

F1 : (b) = 0, (a) = 1, v = ’ , …,

charge:

F1N

: (b) = 0, (a) = 1, v = ’ , …, Identical !!

charge

There is no direct photon coupling to theSkyrmion 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 HLSLagrangian consisting of and

Photon (A) coupling to pion and nucleon:

Quark charge matrixPion 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 wheninfinite tower intervenes???

What this means in the old picture:

KSRF

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 !!

Here is what happens:

Universality restored

The sum rule is saturated by the lowest 4 vector mesonsto 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 andadopt Harada-Yamawaki approach

HLS a la Harada-YamawakiHarada and Yamawaki 2001

Degrees of freedom: with NF=2 or 3.

HLS Lagrangian in the chiral limit: 3 parameters g (gauge coupling), F and For (g, Fa ≡ (F/F)2)

Simple, elegant and predictive.

The crucial step: Wilsonian-match HLS correlatorsto 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 Nc 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 bothapplicable.

Match physical quantities: current-current correlators

E

Parameters

Matching M

OPE of QCD

EFT(HLS)

(g, Fa)

EFT sector:

with

“counter terms”

QCD sector:

Match GV,A and their derivatives at =→ “Bare” parametersof 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/Nc .

This works out WELL in free space despite that the vectormeson mass is much greater than the pion mass …

Harada & Yamawaki, PR 381 (03) 1

Elevating EFT to a gauge field theory is NOTunique. Without gauge invariance it’s even worse!!

EFT

czE

a bc

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 toeach other:”

** Among several fixed points.

So

Important consequence

The vector meson mass parameter (“parametric mass”)vanishes at the VM fixed point because the gaugecoupling vanishes!! This is because the mass is higgsed.

Basis for BR scaling.

In-Medium ParametersThe bare parameters of the Lagrangian depend on mediumbecause 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 symmetrycondition that

Assumption: In the chiral limit, as (T, n) approaches(Tc, nc), the quark condensate vanishes.

As

And the system “flows” to the VM fixed point!Harada, Kim, Rho & Sasaki 01-03

0

Prediction

“Parametric mass” near (T,n)c:

On-shell mass VERY near (T,n)c:

n ≥ 0.

Idem for if U(Nf) is a good symmetry

This BR scaling follows from the VM.

Adami/Brown’sQCD 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 Tc

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 factorsare 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 withan 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 governedby the “normal” component of hidden gauge action.

But processes → 2, → etc are governed bythe “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” thingsmanifest themselves in finite nuclearsystems with which experiments aredone?

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.”

1st from M to a nuclear physics scale nucl ~ 2 fm -1

2nd from nuc to 0 MeV relative to the Fermi surface.

RGE: nuclear matter saturation due to Fermi liquidfixed 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 F1

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 3He

• 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 ≥ Tflash ~ 125 MeVn ≥ nflash ~ (1-2) n0

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 provenunequivocally to indicate that light-quark hadron massesdo not undergo shift in dense/hot medium, then there wouldbe something fundamentally wrong with the basic premiseof the notion of dynamically generated mass based on chiralsymmetry. This would be a serious crisis in QCD physics.

Or it may be that at that density/temperature, quasi-particlenotion for hadrons is wrong but then it will be at odds with the shell-model in nuclear structure.

Dense matter near the critical point nc 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 bean instanton-meron transition in 5D. Analogous toNeel magnetic-ordered state → VBS paramagnet state.Such a transition would imply that the “nomal state” isa non-Fermi liquid which would imply something likehigh-T superconductivity…

Half-Skyrmions

f*

Lee, Park, Rho and Vento 2004