# Presentation: 566482

## 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*

## Example: Fermion number

- 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

##  Inside

- 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

## 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*

## exp

**quark sector**

**gluon**
**secto**r

**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 ≈ mV , V=(and a1)

- 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

- a

- b

- c

- z

- 

- 

- 

***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 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 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 

-  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

##  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

**A short tour of string theory**

## Slide 24

## Sakai-Sugimoto 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:

-  Nc
-   “’t Hooft constant”  (gYM)2 Nc
-  Klein-Kaluza scale MKK ~ scale in 5th 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 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 *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/0705.2632

- Predictions

-  Known parameters: fMeV, Nc=3, Nf = 2

-  Unknown parameters: g YM)2Nc and MKK

- 9

-    0.94 GeV

- Fit to meson spectra by
- Sakai and Sugimoto 05

-  In large and Nc 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) ≈ 1⁄2, 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 HDQC*D

- 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: 1⁄2 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

## Field redefinition

- 5th dimension

- 5D YM
- + EW

- Ext. vector
-      field

- 4D vector
-    field

**.**

**The direct coupling gets replaced by the tower of vector mesons.**
**So the *****towe*****r ****≈ *****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 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 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 GV,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:
-  *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**

## But there is a caveat ...

- unique. Without gauge invariance it’s even worse!!

- EFT

- c

- z

- E

- a

- b

- c

- 

- 

- 

***QCD picks one uniquely:**** **→**** ******HLS/VM***

## 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
- (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 mas*s” 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’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 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 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, 

-  *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.”

-  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 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 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 MeV
- n ≥ 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 effec*t 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 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 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*

- Lee, Park, Rho and Vento 2004