Large β 0 corrections to the energy levels and wave function at N 3 LO
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1 Large β corrections to the energy levels and wave function at N LO Matthias Steinhauser, University of Karlsruhe LCWS5, March 5 [In collaboration with A. Penin and V. Smirnov] M. Steinhauser, LCWS5, March 5 p.
2 Motivation I top R q GeV σ(e + e t threshold Hoang Teubner R Beneke Signer Smirnov q GeV pole mass, NNLO: 4 different groups no stability in position of peak no stability in normalization of peak [A.H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A.A. Penin, A.A. Pivovarov, A. Signer, V.A. Smirnov, Y. Sumino, T. Teubner, O. Yakovlev, A. Yelkhovsky ] M. Steinhauser, LCWS5, March 5 p.
3 Motivation I top R q GeV σ(e + e t threshold Hoang Teubner R Beneke Signer Smirnov q GeV Threshold mass, NNLO: [A.H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A.A. Penin, A.A. Pivovarov, A. Signer, stability in position of peak no stability in normalization of peak V.A. Smirnov, Y. Sumino, T. Teubner, O. Yakovlev, A. Yelkhovsky ] M. Steinhauser, LCWS5, March 5 p.
4 Motivation I top R q GeV σ(e + e t threshold Hoang Teubner R Beneke Signer Smirnov q GeV Threshold mass, NNLO: [A.H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A.A. Penin, A.A. Pivovarov, A. Signer, stability in position of peak no stability in normalization of peak V.A. Smirnov, Y. Sumino, T. Teubner, O. Yakovlev, A. Yelkhovsky ] Resummation of logarithms (to NNLL): promising; however: not complete [Hoang,Manohar,Stewart,Teubner,Hoang 4] [Penin,Pineda,Smirnov,MS 4] M. Steinhauser, LCWS5, March 5 p.
5 Motivation I top R σ(e + e t threshold Hoang Teubner Beneke Signer Smirnov Needed: N LO prediction. for σ(e + e t t) R i.e.: N LO corrections to δe n, δψ n, δg Threshold mass, NNLO: [A.H. Hoang, M. Beneke, K. Melnikov, T. Nagano, A. Ota, A.A. Penin, A.A. Pivovarov, A. Signer, stability in position of peak no stability in normalization of peak V.A. Smirnov, Y. Sumino, T. Teubner, O. Yakovlev, A. Yelkhovsky ] Resummation of logarithms (to NNLL): promising; however: not complete [Hoang,Manohar,Stewart,Teubner,Hoang 4] [Penin,Pineda,Smirnov,MS 4] M. Steinhauser, LCWS5, March 5 p.
6 Motivation II bottom Determination of m b from bottom system: low-moment SR: NNLO: δm b = 5 MeV N LO: still missing [Kühn,Steinhauser ; Corcella,Hoang ] Υ(S)-system: M Υ(S) = m b + E p.t. + δ n.p. E needed: E p.t. to N LO [Penin,MS ] δm b = 7 MeV Υ SR: NNLO: δm b = 5 MeV N LO:??? needed: δe n and δψ n to N LO [Penin,Pivovarov 98; Beneke,Signer 99; Melnikov,Yelkhovsky ; Hoang ] M. Steinhauser, LCWS5, March 5 p.
7 Framework G(,, E) = n= ψ n () E E n ψ n () = ψn C () + δψ n () + δψ n () + δψ n () +... E n = En C + δe n () + δe n () + δe n () +... M. Steinhauser, LCWS5, March 5 p.4
8 Framework G(,, E) = n= ψ n () E E n ψ n () = ψn C () + δψ n () + δψ n () + δψ n () +... E n = En C + δe n () + δe n () + δe n () insert. expand for E E C n single pole: (n) ψ double pole: (n) E M. Steinhauser, LCWS5, March 5 p.4
9 Framework G(,, E) = n= ψ n () E E n ψ n () = ψn C () + δψ n () + δψ n () + δψ n () +... E n = En C + δe n () + δe n () + δe n () insert. expand for E E C n single pole: (n) ψ double pole: (n) E δg(,, E) G C H G C + G C H G C H G C +... M. Steinhauser, LCWS5, March 5 p.4
10 Framework G(,, E) = n= ψ n () E E n ψ n () = ψn C () + δψ n () + δψ n () + δψ n () +... E n = En C + δe n () + δe n () + δe n () insert. expand for E E C n single pole: (n) ψ double pole: (n) E δg(,, E) G C H G C + G C H G C H G C +... G C (x, y, k) P l= (l + ) P m= L l+ m (kx)ll+ m (ky)m! P (m+l+ α s C F m q /(k))(m+l+)! l((xy)/xy) single, double and triple sums analytical solution in terms of ζ function possible M. Steinhauser, LCWS5, March 5 p.4
11 Framework II potential NRQCD [Pineda,Soto 98,Brambilla,Pineda,Soto,Vairo ] expansion parameters: α s, /m q dynamical degrees of freedom: potential quarks, ultra-soft gluons effective Hamiltonian known to N LO [Kniehl,Penin,Smirnov,MS ] M. Steinhauser, LCWS5, March 5 p.5
12 Effective Hamiltonian = (π) δ( q) ( p m q p 4 + πc F α s (µ) m q [ 4m q ) + C c (α s )V C ( q ) + C /m (α s )V /m ( q ) C δ (α s ) + C p (α s ) p + p q + C s (α s ) S ] Static potential: V C ( q ) = 4πC F α s ( q ) q /m q potential: V /m ( q ) = π C F α s( q ) m q q Breit potential: /m q q = p p C C (α s ) = + α «s( q ) 4π a αs ( q ) + a + 4π q ««αs ( q ) a +8π C A ln µ 4π q +.. S: spin; L = ; no /m q (tree-level) potential M. Steinhauser, LCWS5, March 5 p.6
13 Perturbation theory for E n Energy level: E n = En C + δe n () + δe n () + δe n () +... En C = C F α s m q 4n (a) δe n () : G C δh NLO G C (b) δe n () : G C δh NNLO G C, nd iteration of δh NLO (c) δe n () : G C δh N LO G C iteration of δh NLO and δh NNLO ; rd iteration of δh NLO ultrasoft contribution: G C δ us H G C retarded ultrasoft contribution M. Steinhauser, LCWS5, March 5 p.7
14 δe () = δe () β(αs )= + δe () β(αs ) δe () = E β(α C s)= + + α s π ( a a + a π + " C A C F» `ln + L αs C» A C F " # S(S + ) ln + Lαs C F» C A C F T F n l 6» S(S + ) CF T F n l + CF LE! # " S(S + ) + CF a + + ln 6 6 ln Lαs + 7 «Lαs ln + ( ln )S(S + ) C F T F 8 δe () = E n β(α C α «s < s) π : β L µ h4β + i + a β + 8β β L µ + 4@ 6π + a + a + 8π C A C F π 6π S(S + ) A CF 8 + 4π + π4 + 64ζ() 8π ζ() + 96ζ(5) A β π + 8ζ() A a β π + 6ζ() A β a + a π4 A C A C F π4 4π + 4π4 = A S(S + ) A CF A β + a β + 4β 9 ; also known: n =, [Penin,Smirnov,MS 5; Beneke,Kiyo,Schuller 5] ) A β + a β + 4β S(S + ) C A CF + L αs 6 [Kniehl,Penin,Smirnov,MS ] + 64ζ() A β + a β 5 Lµ [Penin,MS ] Lαs = ln(c F α s), Lµ = ln(µ/(c F αsm)) # C A M. Steinhauser, LCWS5, March 5 p.8
15 Perturbation theory for ψ n Wave function: ( ) ψ n () = ψn C () + δψ n () + δψ n () + δψ n () +... δψ n () = K ln β α s + K ln α s + K }{{} known V C (r) = C F α s r ( + α s 4π (8β L r + a ) + αs 4π " +... ψ C n () = C F α s m q 8πn rem + K }{{} unknown 64β L r + (6a β + β ) L r # +a + 6π β αs + "5β L 9a r 4π + β + 64β β + 8π β + 4a β + 64a β + 8β + 6π C A L r +a + 6π a β + 4ζ()β + 6π β β # + O(α 4 s ) ) L r L r = ln(e γ E µr) M. Steinhauser, LCWS5, March 5 p.9
16 δψ () n and δψ () ln n α s ln αs δ ln α s ψ() = α s π + ( C A C F S(S + ) «4 + 7 «S(S + ) C A C F C F C F ) «β C A C F ln (C F α s ) δ ln α s ψ() = α s π +» [Kniehl,Penin ; Manohar,Stewart ] ("! "! # # + π 4π C A C F π S(S + ) C F 9 4 C AC F «# S(S + ) C F a C A ln " S(S + )» 9 4 ln S(S + ) C A C F ln» C F + + ln + ( ln ) S(S + ) » 8 6 C AC F T F n l + 9 S(S + ) 7 C F T F n l ) C F T F ln (C F α s ) β «C A C F [Kniehl,Penin,Smirnov,MS ;Hoang 4] M. Steinhauser, LCWS5, March 5 p.
17 new: δψ () n β δψ () δψ () δψ () β = β = β = «β α «s»8l 5 π + 8π L 4 + 6π + π4 9 + π + ζ() 7π π6 5 + ζ() π ζ() 6ζ() 4ζ(5) «β α «s»8l π + 6π L π 6 + 7π β α s π + 4π4 9 # «L + 4ζ() 6π π ζ() 48π ζ() 8ζ() 6ζ(5) 8L + 6 8π L π + π 4 + 6ζ() # ««L L ζ(5) + 8π # [Penin,Smirnov,MS 5; Beneke,Kiyo,Schuller 5] + 5π π ζ() 8π ζ() 4ζ() L n = ln nµ C F αs(µ)mq M. Steinhauser, LCWS5, March 5 p.
18 Excited states of bottomonium M Υ(nS) = m b + E p.t. n ρ n = E n E m b +E + δ n.p. E n ( does not suffer from renormalon ambiguities) ρ p.t. =.49 ` +.79 NLO +.8 NNLO +. N LO +... = ρ exp = 5.95 ρ p.t. =.77 ` +.9 NLO +.7 NNLO +.55 N LO +... = ρ exp = 9.46 convergence not good N LO terms needed to match experimental result; impressive agreement non-perturbative contribution small!? M. Steinhauser, LCWS5, March 5 p.
19 Υ SR Compare exp. and th. moments: M th k! = M exp k Experimental input: masses and leptonic width of Υ resonances Theory input: (corrections to) energy and wave function; non-perturbative effects under control Estimate N LO corrections: k : corrections to ground state energy dominate δ m b ( m b ) N LO > small k = 4: corrections to wave function become important δ m b ( m b ) N LO MeV Wait for complete N LO result M. Steinhauser, LCWS5, March 5 p.
20 Peak for σ(e + e t t) at N LO Normalization of peak R res (e + e t t) = 6πN cq t m t Γ t c v ψ () c v : hard matching coefficient [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] ψ () : ln, α s, β terms known at N LO [Kniehl,Penin,Smirnov,MS ;Hoang 4] R res (µ)/rres LO (µ s ) R /R LO 5 µ(gev) µ (GeV) [Penin,Smirnov,MS 5] M. Steinhauser, LCWS5, March 5 p.4
21 Peak for σ(e + e t t) at N LO Normalization of peak R res (e + e t t) = 6πN cq t m t Γ t c v ψ () c v : hard matching coefficient [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] ψ () : ln, α s, β terms known at N LO [Kniehl,Penin,Smirnov,MS ;Hoang 4] R res (µ)/rres LO (µ s ) R /R µ(gev) µ (GeV) NLO LO [Penin,Smirnov,MS 5] M. Steinhauser, LCWS5, March 5 p.4
22 Peak for σ(e + e t t) at N LO Normalization of peak R res (e + e t t) = 6πN cq t m t Γ t c v ψ () c v : hard matching coefficient [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] ψ () : ln, α s, β terms known at N LO [Kniehl,Penin,Smirnov,MS ;Hoang 4] R res (µ)/rres LO (µ s ) R /R µ(gev) µ (GeV) NNLO NLO LO [Penin,Smirnov,MS 5] M. Steinhauser, LCWS5, March 5 p.4
23 Peak for σ(e + e t t) at N LO Normalization of peak R res (e + e t t) = 6πN cq t m t Γ t c v ψ () c v : hard matching coefficient [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] ψ () : ln, α s, β terms known at N LO [Kniehl,Penin,Smirnov,MS ;Hoang 4] R res (µ)/rres LO (µ s ) R /R µ(gev) µ (GeV) N LO NNLO NLO LO [Penin,Smirnov,MS 5] nice µ dependence first sign of convergence complete N LO-constant needed! M. Steinhauser, LCWS5, March 5 p.4
24 Γ(Υ(S) e + e ) Γ exp =. kev Γ(µ)/Γ LO (µ s ) Γ /Γ µ (GeV) experiment LO M. Steinhauser, LCWS5, March 5 p.5
25 Γ(Υ(S) e + e ) Γ exp =. kev Γ(µ)/Γ LO (µ s ) Γ /Γ µ (GeV) experiment NLO LO M. Steinhauser, LCWS5, March 5 p.5
26 Γ(Υ(S) e + e ) Γ exp =. kev Γ(µ)/Γ LO (µ s ) Γ /Γ µ (GeV) experiment NNLO NLO LO M. Steinhauser, LCWS5, March 5 p.5
27 Γ(Υ(S) e + e ) Γ exp =. kev α s (M Z ) = ±. Γ(µ)/Γ LO (µ s ) Γ /Γ µ (GeV) N LO experiment NNLO NLO LO M. Steinhauser, LCWS5, March 5 p.5
28 Summary N LO corrections to energy levels, δe n () : complete N LO corrections to wave function, δψ () n : ln, ln, β stabilization of pertubative series observed excited states of bottomonium Υ(S) leptonic width Υ sum rules t t threshold production Third-order corrections to δψ to be completed M. Steinhauser, LCWS5, March 5 p.6
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