On gauge theory phase diagrams at zero and finite temperature K. - - PowerPoint PPT Presentation

On gauge theory phase diagrams at zero and finite temperature

• K. Tuominen

University of Jyväskylä & Helsinki Institute of Physics

SCGT 2012, Nagoya.

Outline:

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

xf ≡ Nf Nc SU(Nc) gauge + fund rep. matter

• 1. Vacuum phase diagrams

Confining Conformal Deconfining Quasi- conf.

xc x*

Conformal 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

• 2. Finite T phase diagrams

• 1. Vacuum Phase diagrams

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

α∗ = −β0 β1 αc = π 3C2(R) α∗ ≤ αc Fixed point from 2-loop betaf. Critical coupling for chiral breaking from SD-equ. Conformal window:

xf ≡ Nf Nc

• 1. Vacuum Phase diagrams

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

α∗ = −β0 β1 αc = π 3C2(R) α∗ ≤ αc Fixed point from 2-loop betaf. Critical coupling for chiral breaking from SD-equ. Conformal window:

Higher representations:

(Sannino, Tuominen ’04)

F 2S 2AS 2 3 4 5 6 Nc 5 10 15

Nf

xf ≡ Nf Nc

Higher representations:

(Sannino, Tuominen ’04)

• Walking with less flavors:

phenomenologically viable Technicolor models

• Study on the lattice

F 2S 2AS 2 3 4 5 6 Nc 5 10 15

Nf

Higher representations:

(Sannino, Tuominen ’04)

• Walking with less flavors:

phenomenologically viable Technicolor models

• Study on the lattice

Lots of efforts during last 4...5 years. SU(2) adjoint: (Catteral et al., Hietanen et al., Del Debbio et al.,...) SU(2) fundamental: (Del Debbio et al., Karavirta et al.,...) SU(3) fundamental: (Appelquist et al., Kuti et al.,...) SU(3) sextet: (De Grand et al.,...)

F 2S 2AS 2 3 4 5 6 Nc 5 10 15

Nf

Some history: SU(2) gauge + 2 adjoint Wilson fermions on the lattice

Strong coupling boundary at βL ∼ 2 Seems volume independent: Lattice artifact?

Lattice phase diagram and spectrum (Hietanen, Rantaharju, Rummukainen, Tuominen, JHEP 0905 (2009).

Non-QCD like continuum physics?

Some history: SU(2) gauge + 2 adjoint Wilson fermions on the lattice

Strong coupling boundary at βL ∼ 2 Seems volume independent: Lattice artifact?

At β ≥ 2 mπ ∼ mρ ∼ mq Conformal?

Lattice phase diagram and spectrum (Hietanen, Rantaharju, Rummukainen, Tuominen, JHEP 0905 (2009).

Non-QCD like continuum physics?

Measure the coupling at βL ≥ 2

SU(2) with 2 fundamentals as a control case:

5 10 15 20

L/a

3 6 9 12

g

2

L=2.2 L=2.5 L=2.8 L=3.1 L=3.5

fundamental rep.

(Hietanen, Rummukainen, Tuominen, PRD 81 (2009).)

Measure the coupling at βL ≥ 2

SU(2) with 2 fundamentals as a control case:

5 10 15 20

L/a

3 6 9 12

g

2

L=2.2 L=2.5 L=2.8 L=3.1 L=3.5

fundamental rep.

2 4 6 8

g

2

• 0.04

0.04 0.08

• fit (g*

2 = 2.2, = 7)

1-loop 2-loop 3-loop MS 4-loop MS

Similar behavior observed also in Del Debbio et al., De Grand et al.

4 8 12 16 20 24

L/a

1 2 3 4 5 6

g

2

L=2.05 L=2.2 L=2.5 L=3 L=3.5 L=4.5 L=8

g*

2 = 2.2

SU(2) with 2 adjoints:

(Hietanen, Rummukainen, Tuominen, PRD 81 (2009).)

Wilson fermions: Large (O(a)) lattice artifacts

Measure the coupling at βL ≥ 2

SU(2) with 2 fundamentals as a control case:

5 10 15 20

L/a

3 6 9 12

g

2

L=2.2 L=2.5 L=2.8 L=3.1 L=3.5

fundamental rep.

2 4 6 8

g

2

• 0.04

0.04 0.08

• fit (g*

2 = 2.2, = 7)

1-loop 2-loop 3-loop MS 4-loop MS

Similar behavior observed also in Del Debbio et al., De Grand et al.

4 8 12 16 20 24

L/a

1 2 3 4 5 6

g

2

L=2.05 L=2.2 L=2.5 L=3 L=3.5 L=4.5 L=8

g*

2 = 2.2

SU(2) with 2 adjoints:

(Hietanen, Rummukainen, Tuominen, PRD 81 (2009).)

Wilson fermions: Large (O(a)) lattice artifacts Need improved actions.

Measure the coupling at βL ≥ 2

SU(2) with 2 fundamentals as a control case:

5 10 15 20

L/a

3 6 9 12

g

2

L=2.2 L=2.5 L=2.8 L=3.1 L=3.5

fundamental rep.

2 4 6 8

g

2

• 0.04

0.04 0.08

• fit (g*

2 = 2.2, = 7)

1-loop 2-loop 3-loop MS 4-loop MS

Similar behavior observed also in Del Debbio et al., De Grand et al.

4 8 12 16 20 24

L/a

1 2 3 4 5 6

g

2

L=2.05 L=2.2 L=2.5 L=3 L=3.5 L=4.5 L=8

g*

2 = 2.2

SU(2) with 2 adjoints:

(Hietanen, Rummukainen, Tuominen, PRD 81 (2009).)

Chromoelectric background field from fixed boundaries: Coupling defined as response to changes of the background field: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

g2 g2 = ∂S ∂η ∂Scl. ∂η Measuring the coupling using the Schrödinger functional (=background field)

µ ∼ 1/L Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

Chromoelectric background field from fixed boundaries: Coupling defined as response to changes of the background field: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

g2 g2 = ∂S ∂η ∂Scl. ∂η Measuring the coupling using the Schrödinger functional (=background field)

µ ∼ 1/L Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Chromoelectric background field from fixed boundaries: Coupling defined as response to changes of the background field: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

g2 g2 = ∂S ∂η ∂Scl. ∂η Measuring the coupling using the Schrödinger functional (=background field)

µ ∼ 1/L Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep. φ1 = η − ρ φ2 = η(ν − 1/2) φ3 = −η(ν + 1/2) + ρ φ0

1 = −φ1 − 4ρ

φ0

2 = −φ3 + 2ρ

φ0

3 = −φ2 + 2ρ

η = 0, ρ = π/3, ν = 0 SU(3): Fundamental rep.

Simpr = S0 + a5csw X

x

¯ ψ(x) i 4σµνFµν(x)ψ(x) + δSG,b + δSF,b

ct

˜ ct

Two counterterms due to nontrivial boundaries

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions).

Simpr = S0 + a5csw X

x

¯ ψ(x) i 4σµνFµν(x)ψ(x) + δSG,b + δSF,b

ct

csw = 1 ct = 1 + c(1)

t g2

˜ ct = 1 + ˜ c(1)

t g2

˜ ct

Two counterterms due to nontrivial boundaries

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions).

Simpr = S0 + a5csw X

x

¯ ψ(x) i 4σµνFµν(x)ψ(x) + δSG,b + δSF,b

ct

csw = 1 ct = 1 + c(1)

t g2

˜ ct = 1 + ˜ c(1)

t g2

˜ ct

Two counterterms due to nontrivial boundaries

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). Perturbative stepscaling: Σ(u, s, L/a) = g2(g0, sL/a)|g2(g0,L/a)=u = u + [Σ1,0(s, L/a) + Σ1,1(s, L/a)Nf]u2

Simpr = S0 + a5csw X

x

¯ ψ(x) i 4σµνFµν(x)ψ(x) + δSG,b + δSF,b

ct

csw = 1 ct = 1 + c(1)

t g2

˜ ct = 1 + ˜ c(1)

t g2

˜ ct

Two counterterms due to nontrivial boundaries

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). Perturbative stepscaling: Σ(u, s, L/a) = g2(g0, sL/a)|g2(g0,L/a)=u = u + [Σ1,0(s, L/a) + Σ1,1(s, L/a)Nf]u2 b0,0 = 11Nc/(48π2) b0,1 = NfTR/(12π2) δi = Σ1,i(2, L/a) 2b0,i ln 2 , i = 0, 1

Simpr = S0 + a5csw X

x

¯ ψ(x) i 4σµνFµν(x)ψ(x) + δSG,b + δSF,b

ct

csw = 1 ct = 1 + c(1)

t g2

˜ ct = 1 + ˜ c(1)

t g2

˜ ct

Two counterterms due to nontrivial boundaries

0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

a/L δ1

SU2 Improved SU2 Unimproved SU3 Improved SU3 Unimproved SU4 Improved SU4 Unimproved

SU(2) gauge + Nf = 2 fundamental

Improvement: Luscher, Narayanan, Weisz, Wolff (hep-lat/9207009) Developed and tested for QCD (fundamental rep. fermions). Perturbative stepscaling: Σ(u, s, L/a) = g2(g0, sL/a)|g2(g0,L/a)=u = u + [Σ1,0(s, L/a) + Σ1,1(s, L/a)Nf]u2 b0,0 = 11Nc/(48π2) b0,1 = NfTR/(12π2) δi = Σ1,i(2, L/a) 2b0,i ln 2 , i = 0, 1

Nontrivial calculation of the counterterms required for higher representations.

(T. Karavirta et al. JHEP 1106 (2011), 1101.0154

• T. Karavirta et al. PRD85 (1012), 1201.1883)

0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 1.1 1.2 1.3 1.4 1.5 1.6

a/L δ1

Improved Unimproved csw=1, ct=0

Nf = 2 sextet SU(3) gauge

• 1. Need to include all coefficients consistently:

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

a/L δ1

SU2 adj improved ρ = π SU2 adj unimproved ρ = π SU2 adj improved ρ = π/2 SU2 adj unimproved ρ = π/2

Nf = 2 adjoint SU(2) gauge

• 2. Also, the choice of boundary fields is nontrivial.

0.04 0.06 0.08 0.1 0.12 0.14 0.16 1 1.1 1.2 1.3 1.4 1.5 1.6

a/L δ1

Improved Unimproved csw=1, ct=0

Nf = 2 sextet SU(3) gauge

• 1. Need to include all coefficients consistently:

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

SU(2) fundamental δ1 − 1 contours:

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

SU(2) fundamental δ1 − 1 contours:

OK!

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

SU(2) fundamental δ1 − 1 contours:

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

SU(2) adjoint: η = π/8, ρ = π/2

φ1 = −η φ0

1 = η − ρ

φ0

2 = ρ − η

φ2 = η η = π/4, ρ = π SU(2): Fundamental rep.

Background field from fixed boundaries: Uk(x)|(x0=0) = exp(aCk(η)), Uk(x)|(x0=L) = exp(aC0

k(η))

Ck = i Ldiag(φ1(η), . . . , φn(η)) C0

k = i

Ldiag(φ0

1(η), . . . , φ0 n(η))

“Half of the field”

SU(2) adjoint: η = π/8, ρ = π/2

SU(3) adjoint

ρ = π/6, η = −π/9

SU(3) adjoint

ρ = π/6, η = −π/9

SU(3) sextet

ρ = 67π/150, η = −π/3

The lessons:

• Wilson fermions must be improved.
• Improvement must done consistently: all coefficients

must be fixed to their correct values.

• Optimization needed: pull between gauge and fermion

contributions; maintain large enough background field for measurements.

Alternative approaches:

• S. Sint and P

. Vilaseca, 1211.0411;

• S. Sint, 1008.4857

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Current status: SU(2) with improved fundamental Wilson fermions

(Karavirta, Rantaharju, Rummukainen, Tuominen JHEP 1205 (2012)) See also: F. Bursa et al. 1010.0901 and H.Ohki et al. 1011.0373

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Current status: SU(2) with improved fundamental Wilson fermions

1 2 3 4 5 6 g

2

• 1.2
• 0.8
• 0.4

β

*

extrapolation 2-loop step scaling 2-loop Nf = 4

confining, QCD like

(Karavirta, Rantaharju, Rummukainen, Tuominen JHEP 1205 (2012)) See also: F. Bursa et al. 1010.0901 and H.Ohki et al. 1011.0373

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Current status: SU(2) with improved fundamental Wilson fermions

1 2 3 4 5 6 g

2

• 1.2
• 0.8
• 0.4

β

*

extrapolation 2-loop step scaling 2-loop Nf = 4

confining, QCD like

2 4 6 8 10 12 g

2

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 β

*

extrapolation 2-loop step scaling 2-loop Nf = 6

Walking?

(Karavirta, Rantaharju, Rummukainen, Tuominen JHEP 1205 (2012)) See also: F. Bursa et al. 1010.0901 and H.Ohki et al. 1011.0373

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Current status: SU(2) with improved fundamental Wilson fermions

1 2 3 4 5 6 g

2

• 1.2
• 0.8
• 0.4

β

*

extrapolation 2-loop step scaling 2-loop Nf = 4

confining, QCD like

2 4 6 8 10 12 g

2

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 β

*

extrapolation 2-loop step scaling 2-loop Nf = 6

Walking?

1 2 3 4 g

2

• 0.1
• 0.05

0.05 0.1 0.15 β

*

extrapolation 2-loop step scaling 2-loop Nf = 10

IR fixed point

(Karavirta, Rantaharju, Rummukainen, Tuominen JHEP 1205 (2012)) See also: F. Bursa et al. 1010.0901 and H.Ohki et al. 1011.0373

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Current status: SU(2) with improved fundamental Wilson fermions

1 2 3 4 5 6 g

2

• 1.2
• 0.8
• 0.4

β

*

extrapolation 2-loop step scaling 2-loop Nf = 4

confining, QCD like

2 4 6 8 10 12 g

2

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 β

*

extrapolation 2-loop step scaling 2-loop Nf = 6

Walking?

1 2 3 4 g

2

• 0.1
• 0.05

0.05 0.1 0.15 β

*

extrapolation 2-loop step scaling 2-loop Nf = 10

IR fixed point

(Karavirta, Rantaharju, Rummukainen, Tuominen JHEP 1205 (2012)) See also: F. Bursa et al. 1010.0901 and H.Ohki et al. 1011.0373

SU(2) with adjoint fermions: see the poster by Jarno Rantaharju

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

What happens when you heat these theories? Suppose the theory is confining, walking/jumping or IR conformal and put it in finite T

xf ≡ Nf Nc

• 2. Finite T phase diagrams

(F. Sannino: 1205.4246)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 l

• 1.5
• 1.0
• 0.5

0.5 1.0 bHlL

β(λ) ∼ −λ2 ⇥ (λ − 1)2 + e ⇤ , e ∝ N crit

f

− Nf

e < 0

e = 0 e > 0

IR and UV fixed points annihilate. A second order zero at the lower boundary of the conformal window

0.2 0.4 0.6 0.8 1.0 1.2 1.4 l

• 1.5
• 1.0
• 0.5

0.5 1.0 bHlL

β(λ) ∼ −λ2 ⇥ (λ − 1)2 + e ⇤ , e ∝ N crit

f

− Nf

e < 0

e = 0 e > 0

IR and UV fixed points annihilate. A second order zero at the lower boundary of the conformal window

0.2 0.4 0.6 0.8 1.0 1.2 1.4 l

• 1.5
• 1.0
• 0.5

0.5 1.0 bHlL

β(λ) = −b1λ2(λ + b0 b1 ), b0 < 0, b1 ∝ N crit

f

− Nf

b1 > 0 b1 < 0

A first order zero at the lower boundary of the CW. Discontinuity

Χ broken Χ symm.

xc xas

Conformal Window HΧ symm.L 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Χ broken Χ symm.

xc xas

Conformal Window HΧ symm.L 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

General features: 1) exact symmetries

Χ broken Χ symm.

xc xas

Conformal Window HΧ symm.L 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Χ broken Χ symm.

xc xas

Conformal Window HΧ symm.L 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

Miransky scaling

General features: 1) exact symmetries

General features: 2) relevant scales and dofs

Inside the conformal window:

High T: free partonic gas (like QCD) Low T: some ”unparticle” matter p = gUVT 4 p = gIRT 4

p/T 4

General features: 2) relevant scales and dofs

Inside the conformal window:

High T: free partonic gas (like QCD) Low T: some ”unparticle” matter p = gUVT 4 p = gIRT 4

p/T 4 T T

High T: free partonic gas (like QCD) p = gUVT 4 Low T: Hadronic gas (like QCD) p = 0 Intermediate T: p ' g∗T 4

Walking region:

p/T 4

2 3 4 5 6 Nc 1 2 3 4 5 6 7

xf

Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL Λ ΒHΛL

fixed Nc

xc x∗

General picture

x∗ < xf < xc xc < xf < xas

xas

“walking window” conformal window

c broken c symm.

xc xas

Conformal Window Hc symm.L

x*

Quasi- conf. 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

(K.T. 1206.5772)

S = 1 16πG5 Z d5x √−g ✓ R +  −4 3(∂µφ)2 + Vg(λ)

• − xfVf(λ, τ)

p 1 + gzzκ(λ(z)) ˙ τ 2 ◆ ,

A holographic model with fermion backreaction

Veneziano limit: Nc → ∞ and xf ≡ Nf

Nc fixed

(Järvinen, Kiritsis 1112.1261; Alho, Järvinen, Kajantie, Kiritsis, K.T. 1210.4516 )

S = 1 16πG5 Z d5x √−g ✓ R +  −4 3(∂µφ)2 + Vg(λ)

• − xfVf(λ, τ)

p 1 + gzzκ(λ(z)) ˙ τ 2 ◆ ,

A holographic model with fermion backreaction

Veneziano limit: Nc → ∞ and xf ≡ Nf

Nc fixed

(Järvinen, Kiritsis 1112.1261; Alho, Järvinen, Kajantie, Kiritsis, K.T. 1210.4516 )

ds2 = b2(z)  −f(z)dt2 + dx2 + dz2 f(z)

• b(z) ∼ L

z f(z) ∼ 1 + z4 z4

h

AdS5 black hole; i.e. conformal N=4.

S = 1 16πG5 Z d5x √−g ✓ R +  −4 3(∂µφ)2 + Vg(λ)

• − xfVf(λ, τ)

p 1 + gzzκ(λ(z)) ˙ τ 2 ◆ ,

A holographic model with fermion backreaction

Veneziano limit: Nc → ∞ and xf ≡ Nf

Nc fixed

(Järvinen, Kiritsis 1112.1261; Alho, Järvinen, Kajantie, Kiritsis, K.T. 1210.4516 )

ds2 = b2(z)  −f(z)dt2 + dx2 + dz2 f(z)

• b(z) ∼ L

z f(z) ∼ 1 + z4 z4

h

AdS5 black hole; i.e. conformal N=4.

The dilaton : , λ(z) = 1 b0 ln(1/Λz) + · · · λ(z)

Breaks conformality

β(λ) = bdλ db

S = 1 16πG5 Z d5x √−g ✓ R +  −4 3(∂µφ)2 + Vg(λ)

• − xfVf(λ, τ)

p 1 + gzzκ(λ(z)) ˙ τ 2 ◆ ,

A holographic model with fermion backreaction

Veneziano limit: Nc → ∞ and xf ≡ Nf

Nc fixed

(Järvinen, Kiritsis 1112.1261; Alho, Järvinen, Kajantie, Kiritsis, K.T. 1210.4516 )

ds2 = b2(z)  −f(z)dt2 + dx2 + dz2 f(z)

• b(z) ∼ L

z f(z) ∼ 1 + z4 z4

h

AdS5 black hole; i.e. conformal N=4.

τ(z) = mq(ln 1 Λz )az + h¯ qqi(ln 1 Λz )−az3 + · · · The tachyon

leading running of quark mass

The dilaton : , λ(z) = 1 b0 ln(1/Λz) + · · · λ(z)

Breaks conformality

β(λ) = bdλ db

S = 1 16πG5 Z d5x √−g ✓ R +  −4 3(∂µφ)2 + Vg(λ)

• − xfVf(λ, τ)

p 1 + gzzκ(λ(z)) ˙ τ 2 ◆ ,

A holographic model with fermion backreaction

Veneziano limit: Nc → ∞ and xf ≡ Nf

Nc fixed

• matching to the known UV behaviors and
• fixing the IR divergence of the tachyon

(several possibilities)

Fix Vf(λ, τ) and κ(λ) by

Vg(λ) ∼ pure YM case

(Järvinen, Kiritsis 1112.1261; Alho, Järvinen, Kajantie, Kiritsis, K.T. 1210.4516 )

ds2 = b2(z)  −f(z)dt2 + dx2 + dz2 f(z)

• b(z) ∼ L

z f(z) ∼ 1 + z4 z4

h

AdS5 black hole; i.e. conformal N=4.

τ(z) = mq(ln 1 Λz )az + h¯ qqi(ln 1 Λz )−az3 + · · · The tachyon

leading running of quark mass

The dilaton : , λ(z) = 1 b0 ln(1/Λz) + · · · λ(z)

Breaks conformality

β(λ) = bdλ db

The four functions, , solved from Einstein eqs. The action is tuned to reproduce required UV behaviors + confinement at small xf . b(z), f(z), λ(z), τ(z) As in lattice MC, the effort is to go to the limit mq → 0 Phases: black hole solutions. Stable phase: one with smallest action.

The four functions, , solved from Einstein eqs. The action is tuned to reproduce required UV behaviors + confinement at small xf . b(z), f(z), λ(z), τ(z) As in lattice MC, the effort is to go to the limit mq → 0

{λh, τh(λh)}

The solutions are parametrized by Two distinct branches of solutions: τ 6= 0 τ = 0 Phases: black hole solutions. Stable phase: one with smallest action.

Th: Transition to hadron gas Tend: 2nd order endpoint (chiral restoration)

Th: Transition to hadron gas Tend: 2nd order endpoint (chiral restoration)

p T4

xf 3.7 xf 3.9 xf 4.1

100 Ε 3 p

T4

xf 3.7 xf 3.9 xf 4.1

0.1 100 105 108 1011 1014

T

• 0.5

1.0 1.5 2.0

Inside the conformal window:

Th: Transition to hadron gas Tend: 2nd order endpoint (chiral restoration)

c broken c symm.

xc xas

Conformal Window Hc symm.L

x*

Quasi- conf. 1 2 3 4 5

xf T

SUHNL gauge theory, massless fermions

p T4

xf 3.7 xf 3.9 xf 4.1

100 Ε 3 p

T4

xf 3.7 xf 3.9 xf 4.1

0.1 100 105 108 1011 1014

T

• 0.5

1.0 1.5 2.0

Inside the conformal window:

Conclusions

Unfolding strong dynamics: Lattice and holography. Improving precision for of non-QCD like theories. Refining holographic models. Combine the two for coherent picture.