橋本幸士

橋本幸士    

大阪大学素粒子論研究室   H726室  

 

理化学研究所  

橋本数理物理学研究室

[Oka, Aoki, PRL 95 (2005) 137601]

Supersymmetric  QCD 1d  Mo8  insulator  

[Oka, KH 1307.7423]

Condensed  ma8er

“QCD-­‐ma8er”

What  resides  inside  neutron  stars?

Supernova  remnant  RCW103

[D.Page]

[Fukushima, Hatsuda 1005.4814]

``QCD  ma4er”  :  fron8er  in  quark  physics

Phase  diagram  of  QCD

L = 1

2E² − 1 8π²

! _∞

0

ds s³

"

eEs cot(eEs)− 1 + 1

3(eEs)²

#

Im L =

$∞ n=1

e²E² 4π³

1

n² exp

"

−nπm² eE

#

L = T^D72π²R²

!

dr r³

%

1 − (2πα^"E)² R⁴

((2πα^"m)² + r²)² z = R²/r

dz = −R²dr/r²

dzz⁻⁵ = (r⁵/R¹⁰)(−R²)dr/r² = −r³dr/R⁸ critical electric field

E_cr = 2πα^"m² R² =

√2πm²

√λ

Im L = Nc

2⁵πe²E²

&

1 + 2^5/2 m²

√λeE log m²

√λeE + higher '

Im L = 1

24πe²E²

&

1 + 6 π

m²

eE log m²

eE + higher '

L^QCD = −1

4F_µν^a F^aµν + ¯ψ_i(iγ^µD_µ − mⁱ) ψ_i

1

[藤森研講義録]

Mee8ng  point  for  …  observa8on  and  theory

EquaJon  of  state   for  nuclear  ma8er  

[Hebeler  et.al  (2010  PRL)]  

Theory  

（QCD）

Obs.

Superstringy  mathemaJcs  resolves     mysteries  in  strongly  correlated  systems  

1. Is superstring useful?

2. Duality connecting theories 3. Challenge to neutron stars

Road  map 7  pages

7  pages 5  pages

1

2

3

Superstring  gets  useful  in  these  7  years

1997

2013 My  publicaJon With  units  

Without  units   (hep-­‐th)

hep-­‐ph astro-­‐ph

nucl-­‐th hep-­‐lat

cond-­‐mat

1

^-1

Quarks  and  gluons  

Nucleons  and  hadrons  

Atomic  nuclei,  neutron  stars   Nuclear  physics  

LaZce  QCD   Superstring                Isola8on  between  par8cle  and  nuclear  physics  

1

^-2

QCD  is  strongly  coupled,  hard  to  calculate  

L = 1

2E² − 1 8π²

! _∞

0

ds s³

"

eEs cot(eEs) − 1 + 1

3(eEs)²

#

Im L =

$∞ n=1

e²E² 4π³

1

n² exp

"

−nπm² eE

#

L = T^D72π²R²

!

dr r³

%

1 − (2πα^"E)² R⁴

((2πα^"m)² + r²)² z = R²/r

dz = −R²dr/r²

dzz⁻⁵ = (r⁵/R¹⁰)(−R²)dr/r² = −r³dr/R⁸ critical electric field

E_cr = 2πα^"m²

R² =

√2πm²

√λ

Im L = N_c

2⁵πe²E²

&

1 + 2^5/2 m²

√λeE log m²

√λeE + higher '

Im L = 1

24πe²E²

&

1 + 6 π

m²

eE log m²

eE + higher '

L^QCD = −1

4F_µν^a F^aµν + ¯ψ_i (iγ^µD_µ − mⁱ) ψ_i

1

QCD  acJon  for  quarks  and  gluons:  simple  but  highly  nonlinear j = 0

j = (2πα^!E)^3/2/R

δE = T^D7π²(2πα^!E)²R⁴

!

−

" 1 0

dy y³

#1 − y² −

$

−

" _∞

0

dy y³

%

1 − y² 1 − y³

&'

y ≡ 2πα^!Ez²/R²

= c

8π² N_cE²

(c ≈ 0.856)

Γ ∼ exp

!

−

√2π²m²

√λeE '

" _∞

−∞

dx e^−mx2 =

( π m

" _∞

−∞

dx e^−mx2−gx4 = ?

" _∞

−∞

dx e^−mx2−gx4 =

" _∞

−∞

dx e^−mx2 )

1 − gx⁴ + 1

2g²x⁸ + · · ·

*

=

( π m

)

1 − 3

4m²g + 105

32m⁴ g² + · · ·

*

3

Strong  nonlinearity  (strong  coupling),  perturbaJon  impossible j = 0

j = (2πα^!E)^3/2/R

δE = T^D7π²(2πα^!E)²R⁴

!

−

" 1 0

dy y³

#1 − y² −

$

−

" _∞

0

dy y³

%1 − y² 1 − y³

&'

y ≡ 2πα^!Ez²/R²

= c

8π² N_cE²

(c ≈ 0.856)

Γ ∼ exp

!

−

√2π²m²

√λeE '

" _∞

−∞

dx e^−mx2 =

( π m

" _∞

−∞

dx e^−mx2−gx4 = ?

" _∞

−∞

dx e^−mx2−gx4 =

" _∞

−∞

dx e^−mx2 )

1−gx⁴+ g²x⁸

2 + · · ·

*

=

( π m

)

1− 3g

4m² +105g²

32m⁴ +· · ·

*

3 Linear

Nonlinear

j = 0

j = (2πα^!E)^3/2/R

δE = T^D7π²(2πα^!E)²R⁴

!

−

" 1 0

dy y³

#1 − y² −

$

−

" _∞

0

dy y³

%

1 − y² 1 − y³

&'

y ≡ 2πα^!Ez²/R²

= c

8π²N_cE²

(c ≈ 0.856)

Γ ∼ exp

!

−

√2π²m²

√λeE '

" _∞

−∞

dx e^−mx2 =

( π m

" _∞

−∞

dx e^−mx2−gx4 = ?

" _∞

−∞

dx e^−mx2−gx4 =

" _∞

−∞

dx e^−mx2 )

1−gx⁴+g²x⁸

2 + · · ·

*

=

( π m

)

1− 3g

4m² +105g²

32m⁴ +· · ·

*

3 j = 0

j = (2πα^!E)^3/2/R

δE = T^D7π²(2πα^!E)²R⁴

!

−

" 1 0

dy y³

#1 − y² −

$

−

" _∞

0

dy y³

%

1 − y² 1 − y³

&'

y ≡ 2πα^!Ez²/R²

= c

8π²N_cE²

(c ≈ 0.856)

Γ ∼ exp

!

−

√2π²m²

√λeE '

" _∞

−∞

dx e^−mx2 =

( π m

" _∞

−∞

dx e^−mx2−gx4 = ?

" _∞

−∞

dx e^−mx2−gx4 =

" _∞

−∞

dx e^−mx2 )

1−gx⁴+g²x⁸

2 + · · ·

*

=

( π m

)

1− 3g

4m² +105g²

32m⁴ +· · ·

*

3

0.5 1.0 1.5 2.0

0.5 1.0

1.5 2.0 1.0

1.5 2.0 2.5 3.0

Numerical  calculaJon?

+   +   +  …  

PerturbaJon

1

^-3

QCD Hypothetical higher dim. gravity Graviton excitation

U(N_f) gauge theory Soliton

Glueball Meson Baryon

Deconfinement Finite temp

Quark density Plasma

Black hole

Hawking temp

Electric field in U(N_f)

Event horizon formation

Useful superstring as a math tool

Solve equivalent system via duality

Problems : Strong coupling, many body, solitons, …

Note: no need for theories to be really stringy.

Exciton Insulator

Conduction Phonon

electrons Heat bath

Impurity

Thermalize

1

^-4

[Brower,Mathur,Tan (03)]

LaZce  

[Morningstar,Peardon  (99)]  

Superstring  

Superstring: better than simulations?

1

^-5

Superstring   Experiment  

LaZce  

[Sakai,Sugimoto,KH  (0806.3122)]  

(0.74  fm)²   0  

0.54  fm   2.2   –  1.3  

0.73   7.5   5.8   4.4   2.3   0.20   –1.9  

(0.875  fm)²   –  0.116  fm²  

0.674  fm   2.79   –1.91  

1.27   13.2   4.2  –  6.5   3.7  –  7.5  

–   –   –    

4.99   2.49   0.06   –2.45   Radii of proton/neutron

Ex) Proton radius from superstring

1

^-6

10 15 20 25 30

-0.004 -0.002 0.002 0.004 0.006 0.008 0.010

V (r)

r

(Inter-­‐nucleon  distance)  

3

S

₁

[Sakai,Sugimoto,KH  (0901.4449)]  

[Aoki,Ishii,Hatsuda  (‘07)]  

LaZce  QCD   simulaJons  :   Experiments:  

[Stoks,Klomp,Terheggen,deSwart  (‘94)]  

Superstring：   Nuclear forces

Ex) Nuclear force from superstring

1

^-7

Superstringy  mathemaJcs  resolves     mysteries  in  strongly  correlated  systems  

1. Is superstring useful?

2. Duality connecting theories 3. Challenge to neutron stars

Road  map 7  pages

7  pages 5  pages

1

2

3

Duality

theory  

A Theory  

＝ B

Strongly  coupled  (correlated) Too  nonlinear  to  solve

Too  many  DoF

Weakly  coupled,   solvable

2

^-1

Holographic  equivalence

Describe  mulJ  vorJces.

A)  Field  theory  of  order  parameter

S = dt

k

( ˙X_(k)ⁱ (t))² +

k1=k2

V (|X(k1)(t) X_(k2)(t)|)

S = d³x | ^µ (x, y, t)|² V (| |) B)  ParJcle-­‐like  vorJces

For  full  equivalence?

Topological  number  fixed?  

Near-­‐vorJces?  

VorJces  on  top  of  each  other?  

Low  energy  excitaJons  only?  

Explicit  examples:  

 ADHM  construcJon  of  instantons    Nahm  construcJon  of  monopoles  

2

^-2

           Stringy  duality:  Gauge/gravity  duality

D-­‐brane  =  vorJces  in  superstring  theory

・  Characterized  by  mass  and  charges

・  Gauge  theories  on  the  D-­‐branes Gauge/Gravity  duality:  

A)  Gauge  theory Strongly  coupled,  large  N  

B)  Gravity Weakly  coupled,  curved  higher  dimensional  space   [Maldacena  ‘98]  

S = 1

16⇡GN

Z

d⁵x p

g (R + 2⇤) + · · ·

S = 1

2g² Z

d⁴x tr F_µ⌫F^µ⌫ + · · ·

2

^-3

Quantizing strings defined in 10D spacetime

Open string Massless gauge field Closed string Massless graviton

D-branes = Object on which open strings can end

Open string theory on the Dp-brane is : SU(Nc) gauge theory in p+1 dimensions

Nc open strings

Nc parallel Dp-branes

2

D-­‐brane  giving  the  duality

2

^-4

Deform

Quantizing strings defined in 10D spacetime

Open string Massless gauge field Closed string Massless graviton

D-branes = Object on which open strings can end D-­‐brane  giving  the  duality

2

^-4

= Source of closed strings

= Source of gravity

= Extended blackhole “blackbrane” in 10D

Quantizing strings defined in 10D spacetime

Open string Massless gauge field Closed string Massless graviton

D-branes = Object on which open strings can end D-­‐brane  giving  the  duality

2

^-4

Black brane Nc D-branes

Propagation of SU(Nc) gauge theory

composite states

Propagation of graviton in near-horizon geometry

of black p-brane (Glueball)

Gluon

Large N

_c

Large λ

Deriving  the  Gauge/gravity  duality

2

^-5

Superstringy  mathemaJcs  resolves     mysteries  in  strongly  correlated  systems  

1. Is superstring useful?

2. Duality connecting theories 3. Challenge to neutron stars

Road  map 7  pages

7  pages 5  pages

1

2

3

[D.Page]

[Fukushima, Hatsuda 1005.4814]

What  resides  inside  the  neutron  stars?

Phase  diagram  of  QCD

L = 1

2E² − 1 8π²

! _∞

0

ds s³

"

eEs cot(eEs)− 1 + 1

3(eEs)²

#

Im L =

$∞ n=1

e²E² 4π³

1

n² exp

"

−nπm² eE

#

L = T^D72π²R²

!

dr r³

%

1 − (2πα^"E)² R⁴

((2πα^"m)² + r²)² z = R²/r

dz = −R²dr/r²

dzz⁻⁵ = (r⁵/R¹⁰)(−R²)dr/r² = −r³dr/R⁸ critical electric field

E_cr = 2πα^"m² R² =

√2πm²

√λ

Im L = Nc

2⁵πe²E²

&

1 + 2^5/2 m²

√λeE log m²

√λeE + higher '

Im L = 1

24πe²E²

&

1 + 6 π

m²

eE log m²

eE + higher '

L^QCD = −1

4F_µν^a F^aµν + ¯ψ_i(iγ^µD_µ − mⁱ) ψ_i

1

3

^-1

From  hypothe8cal  QCD  to  real  QCD

Gauge  theory  with  4  supersymmetries  (N=4  Super  Yang-­‐Mills)

Supersymmetric  gauge  theory  +  quarks    (N=2  Super  QCD)

Non-­‐supersymmetric  SU(N)  gauge  theory  +  quarks  (Large  N    QCD)

Non-­‐supersymmetric  SU(3)  gauge  theory  +  quarks  (QCD) (1)

(2)

(3)

(4)

3

^-2

Gauge  theory  with  4  supersymmetries  (N=4  Super  Yang-­‐Mills)

The  road  to  real  QCD:    (1)

(1)

Gluon  sector:    gluons  +  4  gluinos  +  6  scalars Quark  sector:  Not  allowed

Temp

Baryon  chemical  potenJal Gluon  plasma  

Black hole in

higher dimensions

Closed string

3

^-3

(2)

Gluon  sector:    gluons  +  2  gluinos  +  2  scalars Quark  sector:    quarks  +  scalar  quarks

Quark-­‐gluon   plasma  

Gluon  plasma   +  mesons  

Unstable   phase

[Ghoroku, Ishihara, Nakamura ‘07]

Closed string

Open string Temp

Baryon  chemical  potenJal

The  road  to  real  QCD:    (2)

Supersymmetric  gauge  theory  +  quarks    (N=2  Super  QCD)

3

^-4

(3)

Gluon  sector:    gluons  +  heavy  gluinos     Quark  sector:    quarks

Quark  gluon  plasma  

Hadron Instability?

Gluon  plasma   +  meson  

[Aharony, Sonnenschein, Yankielowicz ‘06]

Temp

Baryon  chemical  potenJal

The  road  to  real  QCD:    (3)

Non-­‐supersymmetric  SU(N)  gauge  theory  +  quarks  (Large  N    QCD)

3

^-5

(4)

Gluon  sector:    gluons Quark  sector:    quarks

？ Challenge ？

Temp

Baryon  chemical  potenJal

The  road  to  real  QCD:    (4)

Non-­‐supersymmetric  SU(3)  gauge  theory  +  quarks  (QCD)

3

^-6

Various  applica8ons  to  cond-­‐mat

Holographic  superconducJvity

Scalar  condensaJon

[Hartnoll, Herzog, Horowitz ‘08]

Holographic  viscosity

Gravity  perturbaJon

[Kovtun, Son, Starinets ‘04]

3

^-7

Superstringy  mathemaJcs  resolves     mysteries  in  strongly  correlated  systems  

1. Is superstring useful?

2. Duality connecting theories 3. Challenge to neutron stars

Road  map 7  pages

7  pages 5  pages

1

2

3