Signatures of the Conformal Window

Elisabetta Pallante
University of Groningen

Work done in collaboration with:
A. Deuzeman, M.P. Lombardo, K. Miura, T. Nunes da Silva (lattice)
A. Barranco, J. Russo (AdS/CFT)2008

Nf=8 is in the QCD phase2009

Nf=12 is in the conformal window(there is a conformal window)

Deuzeman, Lombardo EP 2008 2009

Outline

Aims and strategy
The spectrum
The strong coupling side: Bulk transitions
The fate of U(1) axial
Disappearance of conformality with AdS/CFT

Aims and Strategy I: Temperature - Flavor

TChiralPh
aseBoundary

Nf
Talk by K. MiuraAims and Strategy II: Flavor - Coupling (T=0)

FPBulkIR

Conformal PT

Miransky Yamawaki 1997PTNfgIExact chiralβ<0IIExact chiralβ>0IIIBroken chiral
UVFP?

The spectrum

QCD and non-QCD

mπ ~ √mmρ

mπ ~ constmρ

For fixed lattice spacing:(amπ)2 ~ const
am(amπ)2 ~(am)2δ →2 ~ (am)2δ-1amam

Corrections to power laws are present for an interacting theory not at FP(and finite volume)0.5<δ<1~

The Edinburgh Plot of Nf=12 and Nf=162
Nf=12: this work
Nf=12: Ref. [2]Nf=16: Ref. [3]1.8

Heavy quark limit
or free theory

mN/m!

1.6

1.4

1.2

100.2

0.4

0.6m"/m!

0.8

1Bare quark masses span a range 0.01 to 0.07 at various β for Nf=12
Bare quark masses span a range 0.025 to 0.15 at various β for Nf=16
Damgaard, Heller, Krasnitz, Olesen 1997
Fodor, Holland, Kuti, Nogradi, Schroeder 2011

QCD

↑ Ratio increases

This is compatible with a negative β functionand non-QCD
7.5

7!=4.0!=3.9!=3.8(am!)2/(am)

6.5

6

5.5

5

4.5

4
0.5

0.6

0.7(m!/m")

2

0.8

0.9

1

For a fixed mπ/mρ the inverted behavior with βL is compatible with apositive β functionMasses and Power Laws
1

0.9

m!/m"

0.8

0.7

0.6

0.5

0

0.02

0.04

0.06

0.08
am

0.1

0.12

0.14

0.16

The ratio is approximately constant in a chirally symmetric phase.
It goes to zero in the chiral (massless) limit of QCD.

The axial-vector mass splitting

Degeneracy in the chiral (massless) limit signals that chiral symmetry is restored.

Pseudo Goldstone mass and chiral condensate

2Mπcurvature opposite toFV corrections

exact chiral symmetry
anomalous dimensions ≠0broken chiral symmetry2¯<ψψ>

Kocic Kogut Lombardo 1993

Nf=12: lattice data

Exact chiral symmetry with non zero anomalous dimensions

Bulk transitions

Having already established that there is a bulk transition to a chirally broken phase...
Is it signalling the emergence of a UVFP? Or something else does happen.

The chiral condensate
Lattice fermion action: Naik improved staggered
Lattice gauge action: tree level Symanzik improved
Volumes: 16cx6, 24, 24cx8, 12, 24 (6cx6, 12cx24)
Lattice masses: 0.03, 0.025, 0.02, (0.015)

0.5

0.4
am = 0.020 V=123x24
0.35
am = 0.025
am = 0.030
0.3

0.4

PBP

PBP

0.3

V = 163 x 6
V = 243 x 8
V = 243 x 12
V = 243 x 24

0.25
0.2

0.2

0.15
0.1

0.1
02.4

2.6

2.8

βL

does not increase
with m decreasing

33.2

3.4

0.05

2.6

2.7

2.8

2.9

beta

3

Nt dependence only for Nt ≤ 12

3.1

3.2

3.3The chiral cumulant
χσ = χconn+ χdisc—χπ = <ψψ>/m

R = χπ-1/χσ-1

Mp2 = Zp χp-1

1R→1π-σ degeneracy

am = 0.020 V=123x24
am = 0.025
am = 0.030

0.8

R

0.6

0.4

R→0π Goldstone

0.2

0

2.5

2.6

2.7

2.8

2.9
βL3

3.1

3.2

3.3

Zoom-in of chiral cumulant
0.12

0.1

m=0.03m=0.025m=0.02

R

0.08

0.06

0.04

0.02

2.5

2.6

2.7!L

2.8

Inversion of mass ordering
at the chiral phase transition

2.9

The fate of U(1) axialChiral susceptibilities
mass dependence

1.6
am=0.020
am=0.025
am=0.030

4

1.4
1.2

31
χdiscχconn

am = 0.020
am = 0.025
am = 0.030

2

0.8
0.6
0.4

10.2
0

2.5

2.6

2.7

2.8

2.9
beta

3

3.1

3.2

3.3

0

2.5

2.6

2.7

2.8

2.9
beta

3

3.1

3.2

3.3

Order parameter of U(1) axial
45
χconn am=0.02
χσ

3.5

4

χconn am=0.025
χσ

3
2.5

χχ

322
1.5
11

0.5
02.6

2.8

βL

3

3.2

02.6

2.8

SU(Nf) χσ - χπ = χconn + χdisc - χπ → σ-π degeneracy
UA(1) χδ - χπ = χconn - χπ → δ-π degeneracy

βL33.2

Summary: Signatures
The study of symmetries and related phase transitions in the relevant parameter
space provides an ideal tool to explore theories inside and outside the CW.
Inside the CW, the absence of (pseudo) Goldstone bosons generates quite distinctive
features as compared with QCD.
It is appealing the possibility that a UVFP at strong coupling emerges inside the CW, inaddition to the IRFP. The first order nature of the chiral bulk transition excludes thispossibility for Nf=12. However, a UVFP (second order PT) should appear at the endpoint of a line of first order PTs --> end-point of the CW.
The appearance of an additional phase is plausibly related to the partial restorationof U(1) axial separately from the SU(Nf) chiral restoration. Additional effects such asan Aoki phase (staggered taste breaking) may add to this.

AdS/CFT
Disappearance of the CW

A new viable theory at Strong Coupling?
β(g)?
g*IRFP

g*gUVFP

It suggests the disappearance of conformality
via annihilation of a pair of FPs
see Kaplan et al 2009

AdS/CFT
CFT*

CFT
AdS5 gravity theory
warp factor ~1/z

UV brane/branch
z=0IR brane/branchz(increasing compositeness)“IR/UV correspondence”

z → 0 IR gravity
z → 0 UV field theory

Duality argument and exact β-function
Seiberg 1995

A conformal window for SQCD must be contained in the region
3/2 N
c < Nf < 3Nc
comes fromsupersymmetricgauge field theories. The renormalization gro
CD (SQCD) has been extensively studied, and the perturbative β fun
given by the well known NSVZ formula [19]NSVZ:

g 3 3Nc − Nf (1 − γ0 )βg = −,2N2g
16π
1 − 2c8π

nomalous dimension of the Nf (massless) fundamental superfields Qr anchanged, with a common anomalous dimension for all fundamental fieldpreserves a ZNf flavor symmetry. It also applies in the massive flavor ca
as it is drastically modified below the mass scale of the N flavors. ThFPs merging in “modified” SQCD?

Found a gravity solution that reproduces the NSVZ β-function of
N=1 SYM and confinement
Maldacena, Nunez 2004

Towards SQCD (large Nf, Nc: Nf/Nc fixed): SUGRA backgrounds
Massless case
Casero Nunez Paredes 2008
Massive case (smooth interpolation) Conte Gaillard Ramallo 2011
SQCD + quartic operators

Mass interpolating function00←r

Nf S(r)r→∞

NfYang Mills

Massless flavours(Maldacena-Nunez solution)(linear dilaton)

ff2�

α|Nfcr − Nf |The β-function fromψ̄ψAdS/CFT
Nf < 2Nc
πgravity solution- gauge
coupling relation
cr Ingredients:−1/Θ
Nfcr − Nf |−1/Θ exp − �(1)π- radiusrelationcr
Nfcr − Nf |−1/Θ exp energy−2� �α|N
(1)
π−N|
f 2N
Nf − N f |exp − 2�� α|Nf cr N
(1)
fN=c−|
f (r→0)Limitations: (gravity)
singularities
crf2� α|N − N |
ff

ψ̄ψ
(2)
Nf > 2Nc (2)ψ̄ψRecentψ̄ψ results: Barranco EP Russo 2011(2)
Nf < 2Nc(3)
Nf < 2Nc UV limit: β → βNSVZ (γ0 = −1/2)(3)
Nf < 2Nc IR limit: ordinary confinement(3)Nf = 2Nc(4)
β → −∞ (4)Nf = 2Nc UVFP at strong coupling
Nf = 2Nc(4)
Nf > 2Nc Seiberg dual (Nc →Nf-Nc, Nf-2N(5)c flips sign)
Nf > 2Nc(5)

Summary
Strongly coupled (quasi)conformal gauge theories might play arole in particle physics BSM

Large-Nf QCD is an instructive example
Conformal window opens around Nf ~12Spectrum and phase transitions provide distinctive
signatures of the conformal window

AdS/CFT suggests a unified view where 4D theories of particle
physics are dual to gravity theories in AdS5AdS/CFT in phenomenology: warped RS1 models
AdS/CFT as a mathematical tool: FPs merging?
A UVFP is found at strong coupling for Nf=2Nc in a class of gravity
solutions dual to SQCD plus higher dimensional operators.