Renormalization and universality in non-abelian gauge theories PDF
Preview Renormalization and universality in non-abelian gauge theories
Renormalization and universality in non-abelian gauge theories D´aniel N´ogr´adi Motivation and outline Non-abelian gauge theories are well-known and used a lot QCD Fermion content unusual → more than just a UV fixed point (Physics Beyond Standard Model → we might need unusual fermion content) IR fixed points, more than one UV fixed point, fixed point merger, .... Non-abelian gauge theory in D = 4 dimensions R4 Let’s consider SU(N) gauge theory on coupled to N flavors of f fermions in representation R 1 (cid:90) (cid:90) 4 4 (cid:88) ¯ S = − d xTr F F + d x ψ (D + m )ψ 2 µν µν f f f 4g 0 f Gauge action + fermion action g : bare coupling 0 m : bare fermion masses f Non-abelian gauge theory in D = 4 dimensions 1 (cid:82) 4 Gauge sector, S = − d xTr F F g µν µν 2 4g 0 A (x): N × N anti-hermitian, traceless, bosonic fields µ F = ∂ A − ∂ A + [A , A ] µν µ ν ν µ µ ν Gauge transformation: g(x) ∈ SU(N) −1 −1 A → gA g − ∂ g · g µ µ µ −1 F → gF g µν µν Tr F F = invariant µν µν Non-abelian gauge theory in D = 4 dimensions (cid:82) 4 (cid:80) ¯ Fermion sector, S = d x ψ (D + m )ψ f f f f f D = γ (∂ + A ) Dirac operator µ µ µ above A acts in representation R of SU(N) µ ψ fermion field f f = 1 . . . N number of flavors (or copies) f S also gauge invariant f Non-abelian gauge theory in D = 4 dimensions Often we are interested in m = 0. f Classically S scale invariant g dimensionless 0 Classically chirally invariant ψ → ψ + iεγ ψ 5 Non-abelian gauge theory in D = 4 dimensions In QFT, formally: (cid:82) ¯ −S DADψDψ O(A , ψ) e µ (cid:104)O(A , ψ)(cid:105) = µ (cid:82) DADψDψ¯ e−S There are divergences → regularization + renormalization If m = 0 we only have 1 dimensionless coupling: g → g (µ) f 0 R Non-abelian gauge theory in D = 4 dimensions Everything defined by N, N , R. f Questions: • What about other terms in the action? • What phases and fixed points are there? 2 • What is the renormalization group trajectory g (µ)? • Relevant/irrelevant operators? • How does all this depend on N, N , R? f Non-abelian gauge theory in D = 4 dimensions Typical example: QCD: N = 3, N = 2 + 1 + 1 + 1 + 1, R = fund f 1-loop perturbation theory: 3 dg g µ = β(g) = β 1 dµ 16π2 11 2 β = − N + N 1 f 3 3 11N Asymptotic freedom: β < 0, N < 1 f 2 This is true with N = 3, N = 6 f QCD Fixed points? β(g ) = 0 → g = 0 Gaussian UV fixed point ∗ ∗ Perturbation theory trustworthy around g = 0 ∗ Operators classified as relevant/irrelevant by perturbation theory Essentially power counting, our action contains all relevant oper- ators
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