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

Rethinking naturalness Francesco Sannino

Can the Higgs be elementary ? Francesco Sannino

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( ∂ µ φ r ) 2 − 1 r − λ r + δ Z 2 ( ∂ µ φ r ) 2 − δ m r − δ λ 2 m 2 φ 2 4! φ 4 2 φ 2 4! φ 4 r √ m 2 ≡ m 2 δ λ ≡ λ 0 Z 2 − λ φ B ≡ Z φ r δ Z ≡ Z − 1 0 Z − δ m Z = 1 + f 1 ( λ , g i ) log Λ 2 δ m = f 2 ( λ , g i ) Λ 2 + . . . + . . . m 2 0 0 (1 + f 1 ( λ , g i ) log Λ 2 m 2 = m 2 ) − f 2 ( λ , g i ) Λ 2 m 2 0

Shades of (un)naturality 0 (1 + f 1 ( λ , g i ) log Λ 2 m 2 = m 2 ) − f 2 ( λ , g i ) Λ 2 m 2 0 Standard model: cancel m 0 against cutoff Coleman-Weinberg (CW): idem with radiative EWSB Classical conformality* Λ = 0 , m 0 = 0 Delayed naturality = Veltman Cond. Pert. f 2 = 0 CW + Delayed naturality f 2 = 0 , m 0 = 0 *Without a UV completion is indistinguishable from cancelling against cutoff

Natural theories 0 (1 + f 1 ( λ , g i ) log Λ 2 m 2 = m 2 ) − f 2 ( λ , g i ) Λ 2 m 2 0 A symmetry exists protecting f 2 = 0 Cutoff is physical as in composite models

Degrees of naturality Classical CF (SSB via CW*) SM Higgs = pseudo-dilaton, With UV cutoff is unnatural Space of 4d theories Delayed naturality Perturbative quantum-CF Veltman** CW + Veltman Natural Susy/Technicolor New physics needed! * CW = Coleman-Weinberg **Perturbative cancellation of quadratic divergences

Much ado for 5% 95% is unknown! Richer than 5%? Most likely!

The Standard Model ado Fields: Gauge fields + fermions + scalars Interactions: Gauge: SU(3) x SU(2) x U(1) at EW scale Yukawa: Fermion masses/Flavour Culprit: Higgs Scalar self-interaction

Two main issues EW scale stability UV triviality (Landau Pole) 0.4 Landau pole 0.3 α ( μ ) 0.2 0.1 0.0 0.0 0.5 1.0 Log ( μ / μ 0 )

The Compositeness Solution EW scale = Composite scale UV non-interacting 0.4 Asymptotic freedom 0.3 α ( μ ) 0.2 0.1 Not ruled out 0.0 - 1.0 - 0.5 0.0 Arbey et al. 1502.04718 Log ( μ / μ 0 )

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 ( N C ) L YM = − 1 L F = i Tr [ Q γ µ D µ Q ] 2 Tr [ F µ ν F µ ν ] ∂ µ H † ∂ µ H ⇥ ⇤ L H = Tr ⇤ 2 ( H † H ) 2 ⇤ ( H † H ) ⇥ ⇥ L Self = − u Tr − v Tr � ⇥ ⇤ � L Y = y Tr Q L HQ R + h . c . Global symmetry SU ( N F ) × SU ( N F ) × U V (1)

Veneziano Limit Normalised couplings v α v u = α h N F N F At large N 2 < + N C

Non-Asymptotically Free t = ln µ β g = ∂ t α g = − B α 2 g µ 0 B < 0 B > 0 0.4 0.4 Landau pole Asymptotic freedom 0.3 0.3 α ( μ ) α ( μ ) 0.2 0.2 0.1 0.1 0.0 0.0 - 1.0 - 0.5 0.0 0.0 0.5 1.0 Log ( μ / μ 0 ) Log ( μ / μ 0 )

Small parameters B = − 4 ✏ = N F − 11 3 ✏ N C 2 B < 0 ✏ > 0 0  ✏ ⌧ 1 0.4 0.3 α g ( μ ) 0.2 Landau Pole ? 0.1 0.0 0.0 0.5 1.0 Log ( μ / μ 0 )

Can NL help? B = − 4 β g = − B α 2 g + C α 3 3 ✏ g 0  α ∗ i ff C < 0 g ⌧ 1 β g g = B ϵ ↵ ∗ C ∝ ✏ α g Impossible in Gauge Theories with Fermions alone Caswell, PRL 1974

Add Yukawa " # ◆ 2 ✓ ◆ ✓ 11 4 25 + 26 � g = ↵ 2 3 ✏ + 3 ✏ ↵ g − 2 2 + ✏ ↵ y g � y = ↵ y [(13 + 2 ✏ ) ↵ y − 6 ↵ g ] Computation abides Weyl consistency conditions Osborn 89 & 91, Jack & Osborn 90 Antipin, Gillioz, Mølgaard, Sannino [a-theorem] 1303.1525 Antipin, Gillioz, Krog. Mølgaard, Sannino [SM vacuum stability] 1306.3234

NLO - Fixed Points Gaussian fixed point ( α ∗ g , α ∗ y ) = (0 , 0) Interacting fixed point

Linearised RG Flow ϑ = ∂β / ∂α | ∗ Stability Matrix

Scaling exponents: UV completion Eigen values of M Irrelevant Relevant direction ϑ 1 < 0 t n a v e l e Irrelevant direction R ϑ 2 > 0 A true UV fixed point to this order

NNLO - The scalars The scalar self-couplings Single trace Double trace Only single trace effect on Yukawa Double-trace coupling is a spectator

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(N f )xU(N f ) symmetry If V vanishes on H c it will vanish for any multiple of it Litim, Mojaza, Sannino 1501.03061

Ground state conditions at any Nf H c ∝ δ ij H c ∝ δ i 1 α ∗ h + α ∗ v 2 < 0 < α ∗ h + α ∗ v 1 Stability for α ∗ v 1

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

Phase Diagram Irrelevant t n a v e l e R Apple Thunderbolt Cable (2.0 m) - White

Separatrix = Line of Physics Globally defined line connecting two FPs x i r t a r a p e S

Quantum Potential The QP obeys an exact RG equation γ = − 1 H c = φ c δ ij 2 d ln Z/d ln µ

Resumming logs Dimensional analysis

The Potential Lambert Function Effective gauge coupling

Visualisation 1.0 � ��� ( ϕ � ) � ��� ( ϕ � ) 0.8 � �� ( μ � ) 1.15 � �� ( ϕ � ) 0.6 1.10 0.4 NLO 1.05 NNLO 0.2 0.0 1.00 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?