Rethinking naturalness Francesco Sannino Can the Higgs be - - PowerPoint PPT Presentation

Rethinking naturalness

Francesco Sannino

Francesco Sannino

Can the Higgs be elementary ?

Plan

Degrees of (un)naturality SM ado & Triviality Interacting UV-FP for Gauge-Yukawa theories Beyond asymptotic freedom

Back to the drawing board

RG (un)-naturality

L = 1 2(∂µφr)2 − 1 2m2φ2

r − λ

4!φ4

r + δZ

2 (∂µφr)2 − δm 2 φ2

r − δλ

4! φ4

r

φB ≡ √ Zφr δZ ≡ Z − 1 m2 ≡ m2

0Z − δm

δλ ≡ λ0Z2 − λ

Z = 1 + f1(λ, gi) log Λ2 m2 + . . .

δm = f2(λ, gi)Λ2 + . . .

m2 = m2

0(1 + f1(λ, gi) log Λ2

m2 ) − f2(λ, gi)Λ2

Shades of (un)naturality

m2 = m2

0(1 + f1(λ, gi) log Λ2

m2 ) − f2(λ, gi)Λ2

Standard model: cancel m0 against cutoff Coleman-Weinberg (CW): idem with radiative EWSB Delayed naturality = Veltman Cond.

f2 = 0

Pert. CW + Delayed naturality

f2 = 0, m0 = 0

*Without a UV completion is indistinguishable from cancelling against cutoff

Classical conformality*

Λ = 0, m0 = 0

Natural theories

m2 = m2

0(1 + f1(λ, gi) log Λ2

m2 ) − f2(λ, gi)Λ2 A symmetry exists protecting

f2 = 0

Cutoff is physical as in composite models

Degrees of naturality

Natural Susy/Technicolor Perturbative quantum-CF CW + Veltman Classical CF (SSB via CW*) Higgs = pseudo-dilaton, With UV cutoff is unnatural

SM

Delayed naturality Veltman**

**Perturbative cancellation of quadratic divergences

New physics needed! Space of 4d theories

* CW = Coleman-Weinberg

95% is unknown!

Much ado for 5%

Richer than 5%? Most likely!

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

The Standard Model ado

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

Two main issues

EW scale stability UV triviality (Landau Pole)

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

The Compositeness Solution

EW scale = Composite scale UV non-interacting

• 1.0
• 0.5

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

Asymptotic freedom

Not ruled out

Arbey et al. 1502.04718

Elementary solution ?

Does an UV interacting safe 4D gauge theory exist?

Can we lose asymptotic freedom?

Exact Interacting UV Fixed Point in 4D Quantum Gauge Theories

With D. Litim, 1406.2337, JHEP

Gauge-Yukawa Template

SU(NC)

Global symmetry

SU(NF ) × SU(NF ) × UV (1)

LH = Tr ⇥ ∂µH†∂µH ⇤

LYM = −1 2 Tr [FµνF µν]

LF = i Tr [QγµDµQ]

LY = y

• Tr

⇥ QLHQR ⇤ + h.c.

• LSelf = −u Tr

⇥ (H†H)2⇤ − v Tr ⇥ (H†H) ⇤2

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 #

Computation abides Weyl consistency conditions

Antipin, Gillioz, Mølgaard, Sannino [a-theorem] 1303.1525 Antipin, Gillioz, Krog. Mølgaard, Sannino [SM vacuum stability] 1306.3234

Osborn 89 & 91, Jack & Osborn 90

NLO - Fixed Points

Gaussian fixed point Interacting fixed point

(α∗

g, α∗ y) = (0, 0)

Linearised RG Flow

Stability Matrix

ϑ = ∂β/∂α|∗

Scaling exponents: UV completion

Eigen values of M

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

NNLO - The scalars

The scalar self-couplings Only single trace effect on Yukawa

Double-trace coupling is a spectator

Single trace Double trace

NNLO - All direction UV Stable FP

Fixed point Scaling exponents

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

Ground state conditions at any Nf

Hc ∝ δij Hc ∝ δi1

α∗

h + α∗ v2 < 0 < α∗ h + α∗ v1

Stability for α∗

v1

UV critical surface

Near the fixed point (Ir)relevant directions implies UV lower dim. critical

Phase Diagram

R e l e v a n t Irrelevant

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Separatrix = Line of Physics

Globally defined line connecting two FPs

S e p a r a t r i x

Quantum Potential

The QP obeys an exact RG equation

Hc = φcδij

γ = −1 2d ln Z/d ln µ

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

Summary

Gauge + fermion + scalars theories can be fund. at any energy scale Exact results: independent on any scheme choice Higgs mass squared operator is UV irrelevant Existence of UV nontrivial Gauge-Yukawa theories Discovered UV complete Non-Abelian QED-like theories

Outlook

Composite operators critical exponents, ... Extend to other gauge theories Extensions of the Standard Model Models of DM and/or Inflation, 1412.8034 & 1503.00702 Hope for asymptotic safe quantum gravity?