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

(Asymptotic) Safety in QFT Francesco Sannino

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

The Standard Model works 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

Gauge - Yukawa theories L = − 1 2 F 2 + iQ γ µ D µ Q + y ( Q L HQ R + h . c . ) Yukawa ⇤ 2 DH † DH (H † H) 2 ⇤ (H † H) ⇥ ⇤ ⇥ ⇥ Tr − λ u Tr − λ v Tr Gauge Scalar selfinteractions 4D: standard model, dark matter, … 3D: condensed matter, phase transitions 2D: graphene, … 4plusD: extra dimensions, string theory, … Universal description of physical phenomena

Fundamental theory Wilson: A fundamental theory has an UV fixed point Irrelevant Short distance conformality Continuum limit well defined Complete UV fixed point t n Smaller critical surface dim. = more a v e l IR predictiveness e R Mass operators relevant only for IR The Standard Model is not a fundamental theory

Asymptotic Freedom Confining/chiral symmetry breaking � Trivial UV fixed point Non-interacting in the UV UV logarithmic approach Energy � Perturbation theory in UV � IR conformal or dyn. scale IR conformality/continuous spectrum * � Energy � U

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 µd α H dµ = α H [ c 2 α H + c 1 α g ] CAF c 1 < 0 c 2 > 0 Cheng, Eichten, Li, PRD 9, 2259 (1974) Pica, Ryttov, Sannino, 1605.04712 + Callaway, Phys. Rept. 167, 241 higher orders for IR conformality 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 Non-interacting in the UV Integrating in the UV Logarithmic scale depend. Power law 0.4 0.4 Asymptotic safety 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 - 1.0 - 0.5 0.0 0.5 Log ( μ / μ 0 ) Log ( μ / μ 0 )

Does a theory like this exist?

Exact 4D Interacting UV Fixed Point Litim and Sannino, 1406.2337, JHEP Antipin, Gillioz, Mølgaard, Sannino 1303.1525 PRD L = − F 2 + iQ γ · DQ + y ( Q L HQ R + h . c . )+ ⇤ 2 ∂ H † ∂ H (H † H) 2 ⇤ (H † H) ⇥ ⇤ ⇥ ⇥ Tr − uTr − vTr

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 ]

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

Scaling exponents: UV completion Irrelevant t n a v e l e R Relevant direction ϑ 1 < 0 Irrelevant direction ϑ 2 > 0 A true UV fixed point to this order Litim and Sannino, 1406.2337, JHEP

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

Phase Diagram Irrelevant t n a v e l e R

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

Complete asymptotic safety Litim and Sannino, 1406.2337, JHEP Gauge + fermion + scalars theories can be fund. at any energy scale Scalars are needed to make the theory fundamental

a-theorem L = L CF T + g i O i Quantum correct., marginal oper. g i = g i ( x ) Tool: Curved backgrounds γ µ ν → e 2 σ ( x ) γ µ ν Conformal transformation g i ( µ ) → g i ( e − σ ( x ) µ ) Z � d 4 x L R D Φ e i Variation of the generating functional W = log

Weyl (anomaly) relations ✓ ◆ Z δ W δ W + ∂ µ σ w i ∂ ν g i G µ ν + . . . d 4 x σ ( x ) � aE ( γ ) + χ ij ∂ µ g i ∂ ν g j G µ ν � ∆ σ W ≡ 2 γ µ ν − β i = σ δγ µ ν δ g i E ( γ ) = R µ νρσ R µ νρσ − 4 R µ ν R µ ν + R 2 Euler density G µ ν = R µ ν − 1 Einstein tensor 2 γ µ ν R Beta functions β i χ ij , ω i Functions of couplings a, Weyl relations from abelian nature of Weyl anomaly ∆ σ ∆ τ W = ∆ τ ∆ σ W

Perturbative a-theorem − χ ij + ∂ w i − ∂ w j ✓ ◆ ∂ ˜ a a ≡ a − w i β i β j = ˜ ∂ g i ∂ g j ∂ g i d a = − χ ij β i β j dµ ˜ a-tilde is RG monotonically decreasing if chi is positive definite 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 To leading order ∆ ˜ = 104 a 171 ✏ 2 � gg χ gg = N 2 C − 1 128 π 2 ˜ ˜ a IR a UV Positive and growing with epsilon Sannino, in preparation Antipin, Gillioz, Mølgaard, Sannino 13 Bootstrap and composite operators Antipin, Mølgaard, Sannino 14

Asymptotically Safe Thermodynamics

Pressure and Entropy to NNLO Rischke & Sannino 1505.07828, PRD Ideal gas NLO NNLO ✏ = 0 . 03 ✏ = 0 . 05 ✏ = 0 . 05 ✏ = 0 . 08 ✏ = 0 . 07 ✏ = 0 . 07

Violation of the thermal d.o.f. count Thermal d.o.f. conjecture Appelquist, Cohen, Schmaltz, th/9901109 PRD Corrected SU(2) GB count in Sannino 0902.3494 PRD F is violated Rischke & Sannino 1505.07828, PRD F does not apply to asymptotic safety? But the a-theorem works

QCD Conformal Window vs 2.0

‘If’ large Nf QCD is safe N f N Safe f Critical Asymp. Safe Nf must exist Unsafe region in Nf-Nc N AF f Continuous (Walking) transition? N IR f On Large Nf safety of QCD Pica and Sannino 1011.5917, PRD Litim and Sannino, 1406.2337, JHEP N c

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 r i = 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 W y coupling y

SQCD with H AF is lost N f > 3 N c W = y Tr QH e Q No perturbative UV fixed point

SQCD with H Assume a nonperturbative fixed point, however D ( H ) = 3 2 R ( H ) = 3 N c < 1 for N f > 3 N c N f 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 Can be ruled out via a-theorem a UV − safe − a IR − safe < 0 Non-abelian SQED with(out) H cannot be asymptotically safe 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

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Phenomenological Applications