(Asymptotic) Safety in QFT Francesco Sannino Plan Meaning of - - PowerPoint PPT Presentation

Francesco Sannino

(Asymptotic) Safety in QFT

Plan

• Meaning of fundamental
• From complete freedom to complete safety
• Controllable asymp. safe theory in 4D
• a-theorem for asymptotic safety
• Asymptotically safe thermodynamics and thermal d.o.f. count
• QCD conformal window 2.0 (adding the asymp. safe window)
• Nonperturbative results for N=1 supersymmetric safety

Gauge: SU(3) x SU(2) x U(1) at EW scale

The Standard Model works

Interactions: Gauge fields + fermions + scalars Yukawa: Fermion masses/Flavour Scalar self-interaction Fields: Culprit: Higgs

Gauge - Yukawa theories

L = −1 2F 2 + iQγµDµQ + y(QLHQR + h.c.) Tr ⇥ DH†DH ⇤ − λuTr ⇥ (H†H)2⇤ − λvTr ⇥ (H†H) ⇤2

4D: standard model, dark matter, … 3D: condensed matter, phase transitions 2D: graphene, … 4plusD: extra dimensions, string theory, …

Gauge Yukawa Scalar selfinteractions

Universal description of physical phenomena

Fundamental theory

Wilson: A fundamental theory has an UV fixed point

Irrelevant R e l e v a n t

Short distance conformality Continuum limit well defined Complete UV fixed point Smaller critical surface dim. = more IR predictiveness Mass operators relevant only for IR

The Standard Model is not a fundamental theory

Asymptotic Freedom

Trivial UV fixed point

Non-interacting in the UV UV logarithmic approach Perturbation theory in UV IR conformal or dyn. scale

• Energy

U

• *
• Energy
• Confining/chiral symmetry breaking

IR conformality/continuous spectrum

Complete Asymptotic Freedom

All marginal couplings vanish in the UV

CAF conditions obtained at 1-loop Gauge coupling drives CAF IR conformal or dyn. scale generated

Pica, Ryttov, Sannino, 1605.04712 + higher orders for IR conformality

CAF

µdαH dµ = αH [c2αH + c1αg]

c1 < 0 c2 > 0

Cheng, Eichten, Li, PRD 9, 2259 (1974) Callaway, Phys. Rept. 167, 241 Holdom, Ren, Zhang, 1412.5540 Giudice, Isidori, Salvio, Strumia, 1412.2769

Asymptotic Safety

Wilson: A fundamental theory has an UV fixed point Trivial fixed point Interacting fixed point

• 1.0
• 0.5

0.0 0.0 0.1 0.2 0.3 0.4 Log(μ/μ0) α(μ)

Asymptotic freedom

• 1.0
• 0.5

0.0 0.5 0.0 0.1 0.2 0.3 0.4 Log(μ/μ0) α(μ)

Asymptotic safety

Non-interacting in the UV Logarithmic scale depend. Integrating in the UV Power law

Does a theory like this exist?

Exact 4D Interacting UV Fixed Point

Litim and Sannino, 1406.2337, JHEP

Tr ⇥ ∂H†∂H ⇤ − uTr ⇥ (H†H)2⇤ − vTr ⇥ (H†H) ⇤2

L = −F 2 + iQγ · DQ + y(QLHQR + h.c.)+

Antipin, Gillioz, Mølgaard, Sannino 1303.1525 PRD

Veneziano Limit

Normalised couplings

At large N

NF NC 2 <+

v u = αv αhNF

Non-Asymptotically Free

t = ln µ µ0 βg = ∂tαg = −Bα2

g

0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 Log(μ/μ0) α(μ)

B < 0

• 1.0
• 0.5

0.0 0.0 0.1 0.2 0.3 0.4 Log(μ/μ0) α(μ)

B > 0

Asymptotic freedom Landau pole

Small parameters

Landau Pole ?

B < 0 ✏ > 0 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 Log(μ/μ0) αg(μ)

✏ = NF NC − 11 2

0  ✏ ⌧ 1 B = −4 3✏

Can NL help?

ϵ

αg βg

βg = −Bα2

g + Cα3 g

B = −4 3✏

↵∗

g = B

C ∝ ✏

0  α∗

g ⌧ 1

iff C < 0

Impossible in Gauge Theories with Fermions alone Caswell, PRL 1974

Add Yukawa

y = ↵y [(13 + 2✏) ↵y − 6 ↵g]

g = ↵2

g

" 4 3✏ + ✓ 25 + 26 3 ✏ ◆ ↵g − 2 ✓11 2 + ✏ ◆2 ↵y #

NLO - Fixed Points

Gaussian fixed point Interacting fixed point

(α∗

g, α∗ y) = (0, 0)

Scaling exponents: UV completion

ϑ1 < 0

Relevant direction

ϑ2 > 0

Irrelevant direction

A true UV fixed point to this order

R e l e v a n t Irrelevant

Litim and Sannino, 1406.2337, JHEP

NNLO - The scalars

The scalar self-couplings Only single trace effect on Yukawa

Double-trace coupling is a spectator

Single trace Double trace

Phase Diagram

R e l e v a n t Irrelevant

Separatrix = Line of Physics

Globally defined line connecting two FPs

S e p a r a t r i x

Complete asymptotic safety

Scalars are needed to make the theory fundamental

Gauge + fermion + scalars theories can be fund. at any energy scale

Litim and Sannino, 1406.2337, JHEP

a-theorem

gi = gi(x) γµν → e2σ(x)γµν

gi(µ) → gi(e−σ(x)µ) W = log Z DΦei

R d4xL

• L = LCF T + giOi

Quantum correct., marginal oper. Tool: Curved backgrounds Conformal transformation Variation of the generating functional

Weyl (anomaly) relations

∆σW ≡ Z d4x σ(x) ✓ 2γµν δW δγµν − βi δW δgi ◆ = σ

• aE(γ) + χij∂µgi∂νgjGµν

+ ∂µσwi ∂νgiGµν + . . .

E(γ) = RµνρσRµνρσ − 4RµνRµν + R2

Gµν = Rµν − 1 2γµνR

Euler density Einstein tensor

a, χij, ωi

βi

Functions of couplings Beta functions

Weyl relations from abelian nature of Weyl anomaly

∆σ∆τW = ∆τ∆σW

Perturbative a-theorem

a-tilde is RG monotonically decreasing if chi is positive definite

∂˜ a ∂gi = ✓ −χij + ∂wi ∂gj − ∂wj ∂gi ◆ βj ˜ a ≡ a − wiβi d dµ˜ a = −χijβiβj

Cardy 88, conjecture

True in lowest order PT

Osborn 89 & 91, Jack & Osborn 90

Analyticity: a-tilde bigger in UV

Komargodski & Schwimmer 11, Komargodski 12

Safe variation for the a-theorem function

Positive and growing with epsilon

Antipin, Gillioz, Mølgaard, Sannino 13

To leading order

∆˜ a gg = 104 171✏2 χgg = N 2

C − 1

128π2

˜ aUV

˜ aIR

Bootstrap and composite operators

Antipin, Mølgaard, Sannino 14 Sannino, in preparation

Asymptotically Safe Thermodynamics

Pressure and Entropy to NNLO

Rischke & Sannino 1505.07828, PRD ✏ = 0.08 ✏ = 0.07

✏ = 0.05 ✏ = 0.05

✏ = 0.03 ✏ = 0.07

NLO Ideal gas NNLO

Violation of the thermal d.o.f. count

Rischke & Sannino 1505.07828, PRD

F is violated Thermal d.o.f. conjecture

Appelquist, Cohen, Schmaltz, th/9901109 PRD

F does not apply to asymptotic safety? But the a-theorem works

Corrected SU(2) GB count in Sannino 0902.3494 PRD

QCD Conformal Window vs 2.0

‘If’ large Nf QCD is safe

Nf Nc

Critical Asymp. Safe Nf must exist Unsafe region in Nf-Nc Continuous (Walking) transition?

N AF

f

N IR

f

N Safe

f

On Large Nf safety of QCD Pica and Sannino 1011.5917, PRD Litim and Sannino, 1406.2337, JHEP

Supersymmetric (un)safety

Intriligator and Sannino, 1508.07413, JHEP Martin and Wells, hep-ph/0011382, PRD

Beyond perturbation theory

Unitarity constraints

Operators belong to unitary representations of the superconf. group Dimensions have different lower bounds Gauge invariant spin zero operators Chiral primary operators have dim. D and U(1)R charge R

Central charges

Positivity of coefficients related to the stress-energy trace anomaly ‘a(R)’ Conformal anomaly of SCFT = U(1)R ’t Hooft anomalies [proportional to the square of the dual of the Rieman Curvature] ‘c(R)’ [proportional to the square of the Weyl tensor] ‘b(R)’ [proportional to the square of the flavor symmetry field strength]

a-theorem

For any super CFT besides positivity we also have, following Cardy ri = dim. of matter rep. +(-) for asymptotic safety (freedom) Stronger constraint for asymp. safety, since at least one large R > 5/3

Beta functions

Gauge coupling beta function proportional to ABJ anomaly Beta function of the holomorphic Wy coupling y

SQCD with H

W = y Tr QHe Q Nf > 3Nc

AF is lost No perturbative UV fixed point

SQCD with H

Assume a nonperturbative fixed point, however

D(H) = 3 2R(H) = 3 Nc Nf < 1 for Nf > 3Nc

Violates the unitarity bound

D(O) ≥ 1

Potential loophole: H is free and decouples at the fixed point Check if SQCD without H has an UV fixed point

SQCD

Unitarity bound is not sufficient

Non-abelian SQED with(out) H cannot be asymptotically safe

Can be ruled out via a-theorem

aUV−safe − aIR−safe < 0

Generalisation to several susy theories using a-maximisation*

Key points

Gauge + fermion + scalars theories can be fund. at any energy scale Precise results: independent on scheme choice Discovered UV complete Non-Abelian QED-like theories N = 1 Susy cousin-theories are unsafe Asymptotically safe thermodynamics and violation of F-theorem Conjectured conformal window for QCD vs 2.0 (asymp. safe side)

Scalars needed for perturbative asymptotic safety

Higgs as shoelace

Outlook

Extend to other (chiral) gauge theories/space-time dim

[Ebensen, Ryttov, Sannino,1512.04402 PRD, Codello, Langaeble, Litim, Sannino, JHEP 1603.03462, Bond and Litim 1608.00519, Mølgaard and Sannino to appear]

N=1 Susy GUTs safety [Bajc and Sannino, to appear] Wilson loops, critical exponents, MHV Similarities and differences w.r.t. N=4 Go beyond P .T. [Lattice, dualities, holography, truncations] New ways to unify flavour? Models of DM and/or Inflation Hope for asymptotic safe quantum gravity*? * Weinberg

Backup slides

Phenomenological Applications

Safe QCD

QCD

QCD is not IR conformal because Asymptotic freedom verified < TeV

Hadronic spectrum/dyn. mass Pions <-> Spont. ChSB

If above TeV asymptotic freedom is lost, then what?

Asymptotic safety

• 0.0

0.1 0.2 0.3 0.4 μ μ α μ

• 0.0

0.1 0.2 0.3 0.4 μ μ α μ

ChSB/Confinement 1 GeV ~ TeV Before Planck

αs µ

New coloured states Higgs mechanism Light quarks Top Top partners Colorons Gluino-like Unexpected

Safe QCD scenario

Sannino, 1511.09022

Cosmology Cosmic rays LHC

Asymptotic safety

• 0.0

0.1 0.2 0.3 0.4 μ μ α μ

• 0.0

0.1 0.2 0.3 0.4 μ μ α μ

ChSB/Confinement 1 GeV ~ TeV Before Planck

αs µ

Is the safe QCD scenario testable?

Sannino, 1511.09022

Asymptotic freedom is not a must for UV complete theories

Large Nf, QCD, Holdom 1006.2119 PLB & Pica & Sannino,1011.5917 PRD

Safe Dark Matter

Safe DM

σ ∝ αqαX m4

V

µ2

X X SM SM V

hσannvi / αqαX m4

V

m2

X

X X SM SM

V

Offset direct detection

Sannino & Shoemaker, 1412.8034, PRD

Anomalous dimensions

HB = Z

1 2

HH

∆H = 1 + γH

γH = −1 2 d ln ZH d ln µ H = 4✏ 19 + 14567 − 2376 √ 23 6859 ✏2 + O(✏3)

Mass dimensions

∆F = 3 − γF γF = d ln M d ln µ

F = 4 19✏ + 4048 √ 23 − 59711 6859 ✏2 + O(✏3)

MQQ Fermion

Mass dimensions

Small perturb., hence m2= 0 at the UV-FP Scalar m2Tr ⇥ H†H ⇤ γ(1)

m = 2αy + 4αh + 2αv

γm = 1 2 d ln m2 d ln µ

UV critical surface

(Ir)relevant directions implies UV lower dim. critical

`

Near the fixed point

Double - trace and stability

Is the potential stable at FP? Which FP survives?

Moduli

Classical moduli space Use U(Nf)xU(Nf) symmetry If V vanishes on Hc it will vanish for any multiple of it

Litim, Mojaza, Sannino 1501.03061 JHEP

Ground state conditions at any Nf

Hc ∝ δij Hc ∝ δi1

α∗

h + α∗ v2 < 0 < α∗ h + α∗ v1

Stability for α∗

v1

Quantum Potential

The QP obeys an exact RG equation

Hc = φcδij

γ = −1 2d ln Z/d ln µ

Litim, Mojaza, Sannino 1501.03061, JHEP

Resumming logs

Dimensional analysis

The Potential

Lambert Function

Effective gauge coupling

Visualisation

0.0 0.2 0.4 0.6 0.8 1.0 1.00 1.05 1.10 1.15 ϕ/μ (ϕ) (ϕ)

NLO NNLO

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ϕ/μ (ϕ) (μ)

QFT is controllably defined to arbitrary short scales

Gauge - Yukawa theories/Gradient Flow

Relations among the modified β of different couplings Precise prescription for expanding beta functions in perturb. theory

∂˜ a ∂gi = ✓ −χij + ∂wi ∂gj − ∂wj ∂gi ◆ βj ⇒ ∂˜ a ∂gi = −βi , βi ≡ χijβj

∂βj ∂gi = ∂βi ∂gj ,

Gradient flow fundamental relation

Antipin, Gillioz, Mølgaard, Sannino 13

• mega is an exact form

Osborn 89 & 91, Jack & Osborn 90 Jack and Poole 15