t2 - updated Tuesday 8 th February 2011 11:54 2 the muon...

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Unformatted text preview: updated Tuesday 8 th February, 2011 11:54 2. the muon anomalous magnetic moment and large logs (a) muon anomalous magnetic moment (b) vacuum polarization corrections to g (c) dimensional regularization and MS (d) large logs in g µ electron g − 2 — one of the most spectacular predictions in science — if we ignore everything in the universe except electrons and photons — calculate the leading quantum correction to the Dirac value from the diagram g e = 2 + α π + 2 A (4) 1 ( α π ) 2 + ··· A (4) 1 = 197 144 + π 2 12 − π 2 2 log(2) + 3 4 ζ (3) independent determination of fine structure constant arXiv:0812.3139 α = . 00729735254 gives g e = 2 . 002319274 + ··· while experiment gives g e = 2 . 0023193043622 difference is what you would expect from ( α/π ) 3 terms — we don’t need to think about other particles because other charged particles are heavy — in fact, the theory has been done beyond ( α/π ) 4 — at this level you cannot ignore other charged stuff — in particular the muon in the effective low energy theory with only electrons and photons, what would we expect for the contribution of muons (or other heavier charged particles) to g − 2 without doing detailed diagrams??? the relevant operator looks like (where ψ is the electron field) e ¯ ψσ µν ψ F µν where σ µν = 1 2 i [ γ µ , γ ν ] dimension 5 so a very naive guess would be ξ e m µ ¯ ψσ µν ψ F µν with ξ = O (1) — this would give a δg e ≈ m e /m µ which is clearly not there chirality - L = ¯ ψ ( i ̸ ∂ − e ̸ A − m e ) ψ − 1 4 F µν F µν invariant under — “spurion analysis” ψ → γ 5 ψ m e → − m e ⇒ ξ em e m 2 µ ¯ ψσ µν ψ F µν in the effective low energy theory with only electrons and photons, what would we expect for the contribution of muons (or other heavier charged particles) to g − 2 without doing detailed diagrams??? chirality - L = ¯ ψ ( i ̸ ∂ − e ̸ A − m e ) ψ − 1 4 F µν F µν invariant under — “spurion analysis” ψ → γ 5 ψ m e → − m e ⇒ ξ em e m 2 µ ¯ ψσ µν ψ F µν muon charge - must be ∝ e 2 µ e 2 e — other charged particles can affect the electron only through diagrams involving at least two virtual photons → ξ ( α π ) 2 em e m 2 µ ¯ ψσ µν ψ F µν this is really little! — it doesn’t affect anything to this order — it is important if you want to calculate to 4th order in α — but let’s not do that!!! so now what about g µ ? — experimentally g µ = 2 . 0023318414 not quite the same as g e = 2 . 0023193043622 to calculate this, we (obviously) have to include the µ in our effective theory — let’s do this and ignore everything else — now effects of both e and µ show up g e = 2 + α π + ( 2 A (4) 1 + 2 A (4) 2 ( m e /m µ ) )( α π ) 2 + ··· g µ = 2 + α π + ( 2 A (4) 1 + 2 A (4) 2 ( m µ /m e ) )( α π ) 2 + ··· A (4) 1 = 197 144 + π 2 12 − π 2 2 log(2) + 3 4 ζ (3) A (4) 2 (1 /x ) = −...
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t2 - updated Tuesday 8 th February 2011 11:54 2 the muon...

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