When Perturbation Theory Fails...

Transcription

1 When Perturbation Theory Fails... Brian Tiburzi (University of Maryland)

2 When Perturbation Theory Fails... SU(3) chiral perturbation theory? Charm quark in HQET, NRQCD? Extrapolations of lattice QCD data? Solution? Not this talk... Literal interpretation of title Non-Perturbative Examples Toy model Hadrons in uniform electromagnetic fields Hyperons in SU(2) chiral perturbation theory

3 Toy model: 0 <x 1 F (x) = 0 e s 1+sx ds Cannot series expand about 0

4 Toy model: 0 <x 1 F (x) = 0 e s 1+sx ds Cannot series expand about 0 F (x) = ds e s ( sx) j =! 0 j=0 ( ( x) j j=0 0 ds s j e s ) Do it anyway (Physicist) Suggests approximation F N (x) = N ( ) j j! x j j=0

5 Toy model: 0 <x 1 F (x) = 0 e s 1+sx ds Cannot series expand about 0 F (x) = ds e s ( sx) j =! 0 j=0 ( ( x) j j=0 0 ds s j e s ) Do it anyway (Physicist) Suggests approximation F N (x) = N ( ) j j! x j j=0 F (x) F N (x) = x N+1 0 x N+1 (N + 1)! s N+1 e s ds 1+sx

6 Toy model: F (x) = F N (x) = 0 0 <x 1 e s 1+sx ds N ( ) j j! x j j=0 Minimize error for large N F (x) F N (x) x N+1 (N + 1)! 2πN(xN) N e N Do it anyway (Physicist) 2π x e 1 x x 1/N Include more terms: limits to smaller x Make better for larger x: dropping terms Asymptotic expansions: intuitively opposite

7 Toy model: F (x) = F N (x) = 0 0 <x 1 e s 1+sx ds N ( ) j j! x j j=0 N=1 x 1/N

8 Toy model: F (x) = F N (x) = 0 0 <x 1 e s 1+sx ds N ( ) j j! x j j=0 N=2 N=1 x 1/N

9 Toy model: 0 <x 1 F (x) = F N (x) = 0 e s 1+sx ds N ( ) j j! x j j=0 N=3 N=2 N=1 x 1/N

10 Toy model: 0 <x 1 F (x) = F N (x) = 0 e s 1+sx ds N ( ) j j! x j j=0 N=12 N=3 N=2 N=1 x 1/N Asymptotic expansions: zero radius of convergence = intuitively opposite

11 Where have I seen this behavior? N=2 N=1

12 Where have I seen this behavior? N=2 N=1 When Perturbation Theory Fails...

13 Hadrons in uniform electromagnetic fields Green s functions exist in closed form for uniform EM fields Use to study non-perturbative effects Scalar case Schwinger (1951),..., Tiburzi (2008) Magnetic field: Solution Harmonic oscillator propagator

14 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED Scalar case Power counting eb/m 2 π 1 Calculate the neutral pion energy in magnetic field

15 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED Neutral pion energy eb/m 2 π 1 Closed form Why? Background field lattice QCD computations in this regime Closed torus can leak no flux exp(ieba) eb = 2πn L 2

16 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED Neutral pion energy eb m 2 π Do it anyway (Physicist)

17 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED Neutral pion energy intuitively obvious asymptotically opposite When Perturbation Theory Fails...

18 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED Neutral pion energy When Perturbation Theory Fails...

19 Hadrons in uniform electromagnetic fields Chiral perturbation theory in strong QED IDLE AMUSEMENT Gell-Mann / Oaks / Renner Relation Electric Field: analytic continuation Schwinger mechanism Charged pions Nucleon In strong magnetic fields, proton beta decays to neutron

20 Hyperons in SU(2) Chiral Perturbation Theory Tiburzi and Walker-Loud (2008) Jiang, Tiburzi, and Walker-Loud (2009) Do it anyway (Physicist) Motivation SU(3) Heavy baryon chiral perturbation theory m η /M B 1/2 δm N (µ = Λ χ )/M N = 39% δm Λ (µ = Λ χ )/M Λ = 67% δm Σ (µ = Λ χ )/M Σ = 89% δm Ξ (µ = Λ χ )/M Ξ = 98% Kaon, eta contributions large & increase with strangeness m s Λ QCD? SU(3) expansion precarious

21 Hyperons in SU(2) Chiral Perturbation Theory Schematic SU(3) Expansion of Sigma Mass: M Σ = M SU(3) + am 2 K + bm 3 K +... Large Kaon contributions m 2 K = 1 2 m2 π m2 η s m ηs = 672 MeV Reorganize! m 2 π/m 2 η s = M Σ = M SU(3) + a m 2 η s + a m 2 π + b m 3 η s + b m ηs m 2 π + b m 3 π ( mπ m ηs ) +. M Σ = M SU(2) Σ + αm 2 π + βm 3 π +... Expansion of Sigma Mass about the SU(2) chiral limit m u,m d m s Λ QCD

22 Hyperons in SU(2) Chiral Perturbation Theory M = M SU(2) + αm 2 π +βm 3 π + β F (m π, δ) Trend opposite SU(3): greater strangeness, better convergence m π /Λ χ m π /M S g A =1.25, g ΣΣ =0.78, g ΞΞ =0.24 g N =1.48, g Σ Σ =0.76, g Ξ Ξ =0.69

23 Hyperons in SU(2) Chiral Perturbation Theory SU(2) Perturbative Expansion can FAIL! I) Perturbative expansion about SU(2) limit Duh! (Maryland Colleague) m π /Λ χ need lattice QCD II) Perturbative SU(2) expansion of SU(3)! Kaon thresholds... can study non-perturbatively

24 Hyperons in SU(2) Chiral Perturbation Theory 250 MeV K N Σ π Σ SU(2) Perturbative Expansion can FAIL! Kaon production cannot be described in SU(2) m u,m d m s Λ QCD II) Perturbative SU(2) expansion of SU(3)! Exact Solution SU(3) is theory SU(2) is asymptotically describing

25 Hyperons in SU(2) Chiral Perturbation Theory Do SU(2) expansions of hyperon masses break down because of KN thresholds? Σ, Σ K N Only virtual, but analyticities near threshold

26 Hyperons in SU(2) Chiral Perturbation Theory Do SU(2) expansions of hyperon masses break down because of KN thresholds? Σ, Σ K N Only virtual, but analyticities near threshold m 2 K = 1 2 m2 π m2 η s Do it anyway (Physicist)

27 Hyperons in SU(2) Chiral Perturbation Theory Do SU(2) expansions of hyperon masses break down because of KN thresholds? Σ, Σ K N Do it anyway (Physicist)

28 Hyperons in SU(2) Chiral Perturbation Theory N=2 Include more terms: limits to smaller pion mass Make better for larger pion mass: dropping terms Asymptotic expansions: intuitively opposite

29 Hyperons in SU(2) Chiral Perturbation Theory N=2 N=4 Include more terms: limits to smaller pion mass Make better for larger pion mass: dropping terms Asymptotic expansions: intuitively opposite

30 Hyperons in SU(2) Chiral Perturbation Theory N=2 N=6 N=4 Include more terms: limits to smaller pion mass Make better for larger pion mass: dropping terms Asymptotic expansions: intuitively opposite

31 Hyperons in SU(2) Chiral Perturbation Theory N=2 N=6 N=4 Include more terms: limits to smaller pion mass Make better for larger pion mass: dropping terms Asymptotic expansions: intuitively opposite

32 Hyperons in SU(2) Chiral Perturbation Theory N=24 N=2 N=6 N=4 Include more terms: limits to smaller pion mass Make better for larger pion mass: dropping terms Asymptotic expansions: intuitively opposite

33 When perturbation theory fails, we probably try to use it anyway Asymptotic expansions: intuitively opposite Including more terms limits to smaller range Make better for parameters by dropping terms (limited control) eb/m 2 π 1

34 Neutral pion in electric field: B -> ie