A pnrqcd approach to t t near threshold

Transcription

1 LoopFest V, SLAC, 20. June 2006 A pnrqcd approach to t t near threshold Adrian Signer IPPP, Durham University BASED ON WORK DONE IN COLLABORATION WITH A. PINEDA AND M. BENEKE, V. SMIRNOV LoopFest V p. 1/2

2 outline A NNLL computation Electroweak effects Outlook introduction outline of calculation using pnrqcd results and theoretical error QED effects from e + e t t to e + e W + bw b unstable particle effective theory further improvements towards NNNLO LoopFest V p. 2/2

3 introduction e t R σ(e+ e QQ) σ(e + e µ + µ ) γ/z... problem with three scales: hard: m e + t soft: p mv mαs ultrasoft: E = s 2m mv 2 mα 2 s fixed order: σ(= R) = v X n αs v hierarchy of scales: m mv mv 2 Λ QCD n n o 1 (LO); α s, v (NLO); α 2 s, v 2, α s v (NNLO) resummed: σ = v X n αs v n X (α s log v) l l n o 1 (LL); α s, v (NLL); α 2 s, v 2, α s v (NNLL) LoopFest V p. 3/2

4 introduction exploit αs 1 and v 1 double expansion identify modes [Beneke, Smirnov] asymptotic expansion (method of regions) hard soft potential k µ m k µ mv k 0 mv 2 ; k mv ultrasoft k µ mv 2 integrate out unwanted modes (final state described by potential quarks and ultrasoft gluons): QCD (h,s,p,u) NRQCD (s,p,u) pnrqcd (p q,u) matching of currents done to NNLO [Beneke et.al; Hoang et.al; Melnikov et.al; Yakovlev;...] use threshold mass, not pole mass [Bigi et.al; Beneke; Hoang et.al; Pineda] LoopFest V p. 4/2

5 introduction k underlying theory µ h m h QCD L QCD (ψ h, ψ s, ψ p, g h, g s, g p, g us ) effective theory I µ s mv p s NRQCD L NRQCD (ψ s, ψ p, g s, g p, g us ) effective theory II µ us mv 2 us pnrqcd L pnrqcd (ψ p, g us ) LoopFest V p. 5/2

6 calculation In full QCD: q 2 = s = (E + 2m) 2 R(s) = 4π e2 q s Im» Z i d 4 x e iqx 0 T {j µ (x)j µ (0)} 0 current: (Z exchange not included) j µ Qγ µ Q c 1 χ σ i ψ c 2 6m 2 χ σ i (id) 2 ψ +... in pnrqcd : R(E) = 24πe2 q N c s c 2 1 c 1 c 2 E 3 m «Im G(0, 0, E) LoopFest V p. 6/2

7 calculation NRQCD Lagrangian [Caswell, Bodwin, Braaten, Lepage]! D L NRQCD = ψ id c k 2m ψ + c 4 8m 3 ψ D 4 ψ g c F 2m ψ σ i B i ψ + g c D 8m 2 ψ ˆD i, E i ψ + ig c S 8m 2 ψ σ ij ˆD i, E j ψ + (ψ χ) + d ss m 2 ψ ψ χ χ + d sv m 2 ψ σ i ψ χ σ i χ + d vs m 2 ψ T a ψ χ T a χ + d vv m 2 ψ σ i T a ψ χ σ i T a χ + L light all calculations in momentum space, using dimensional regularization in D = 4 2ɛ dimensions thus e.g: σ i B i = (i/4)[σ i, σ j ] G ij resum log(µh /µ s ) in c i and d ij using renormalization group RGI: single heavy quark sector as in HQET [Bauer, Manohar,...] RGI: four heavy quark operators [Pineda] LoopFest V p. 7/2

8 calculation pnrqcd Lagrangian [Pineda, Soto] «L pnrqcd = ψ id ψ + χ id 0 2 2m 2m Z + d 3 r ψ T a 4πCF α s ψ q 2 χ T a χ + Z d 3 r ψ T a ψ δv χ T a χ «χ + ψ 4 ««8m 3 g s x E ψ + χ 4 8m 3 g s x E χ leading order Coulomb potential is LO effect remaining terms in potential, δv (Breit-Fermi potential, static potential [Schröder, Peter], non-analytic potential...) included perturbatively (some) matching coefficients in δv have to be known in D dimensions ultrasoft effects enter at NNNLO LoopFest V p. 8/2

9 calculation The renormalization group improved pnrqcd potential: [Pineda, AS] V NNLL = 4πC F αṽs q 2 C F C A D (1) s 2πC F D (2) 1,s p 2 + p 2 m 2 q 2 π 2 m q 1+2ɛ (1 ɛ) (4π)ɛ Γ 2 ( 1 2 ɛ)γ( ɛ) π 3/2 Γ 2 (1 2ɛ) p 2 p 2 + πc (2) F D2,s m 2 q 2 «2 1! + 3πC F D (2) d,s m 2 4πC F D (2) S 2,s d m 2 [S i 1, S j 1 ][Si 2, S j 2 ] +... use renormalization-group equations to evolve potentials DX from µ s to µ us, resumming log µ s /µ us. [Pineda] LL running of DX known potential known at NNLL LoopFest V p. 9/2

10 calculation We use dimensional regularization throughout, perform all calculations in momentum space and always use MS-subtraction [Beneke, AS, Smirnov] G c ( r, r, E) r= r =0 Z d d p d d p G (2π) d (2π) d c ( p, p, E) G c (0, 0, E) = α s C F m 2 4π 1 2λ mE ln µ 2 1 «2 + γ E + ψ(1 λ) where λ C F α s /(2 p E/m) ; This sums all potential gluon (ladder) diagrams for higher-order corrections evaluate single and double insertions δg c (0, 0, E) = Z Y d d p i (2π) d G c ( p 1, p 2, E) δv ( p 2, p 3 ) G c ( p 3, p 4, E) LoopFest V p. 10/2

11 calculation current c1 needed at two loop [Czarnecki, Melnikov; Beneke, AS, Smirnov] higher dimensional operators of single heavy quark sector mix into lower dimensional operators in heavy quark-antiquark sector through potential loops need NLL matching coefficients of NRQCD to obtain NLL current done [Pineda; Hoang, Manohar, Stewart] d µ s c 1 = C2 F dµ s 4 α s α s 2 «3 D(2) S 2,s 3D(2) d,s + 4D(2) 1,s C AC F 2 D (1) s however NNLL current not complete, only partial results available [Kniehl et.al; Hoang; Penin et.al.] NNLL NNLL these are the only missing NNLL terms LoopFest V p. 11/2

12 calculation vnrqcd vs. pnrqcd resummation of log v done before at NNLL using vnrqcd [Hoang, Manohar, Stewart, Teubner] in vnrqcd there is only one step in the matching procedure and the correlation between the scales is fixed from the start µ us = µ 2 s /m in the pnrqcd approach the correlation between the scales is taken into account in the RG solutions done (so far) only for the spin dependent term [Penin, Pineda, Steinhauser, Smirnov], thus NNLL NNLL independent variation of µs and µ h more conservative error estimate, µ h dependence is now larger than µ s dependence. ideally, we also would like to independently vary the ultrasoft scale µus, i.e. µ us = µ 2 s/µ h µ us µ 2 s/µ h ; has not been done so far LoopFest V p. 12/2

13 results µ s dependence of fixed-order results R Σtt ΣΜΜ LL NLL NNLL µ 2 s 4m t qe 2 + Γ 2 t m P S = 175 GeV Γ t = 1.4 GeV µ F = 20 GeV Μ s GeV E PS s 2 m PS normalization of cross section has a large theoretical error, scale dependence increases from NLO to NNLO and NNLO corrections as large as NLO corrections! no top width / Yukawa coupling measurement LoopFest V p. 13/2

14 results µ s dependence of renormalization-group improved results [Pineda, AS] 1 LL NLL NNLL µ 2 s 4m t qe 2 + Γ 2 t m P S = 175 GeV R Σtt ΣΜΜ Γ t = 1.4 GeV µ F = 20 GeV Μ s GeV 0.4 Μ s 20 GeV E PS s 2 m PS normalization of cross section much more stable, confirms previous results by [Hoang, Manohar, Stewart, Teubner] µs scale-dependence bands do not overlap estimate of theoretical error?? LoopFest V p. 14/2

15 results µ s dependence of renormalization-group improved results 1.4 [Pineda, AS] RMAX NNLL LL LO NNLO µ 2 s 4m t qe 2 + Γ 2 t m P S = 175 GeV Γ t = 1.4 GeV µ F = 20 GeV NLL NLO Μ S problem with small scales solved by including multiple insertions of Coulomb potentials [Beneke, Kiyo, Schuller] reliable region for soft scale: 30 GeV µs 80 GeV LoopFest V p. 15/2

16 results µ h dependence of renormalization-group improved results [Pineda, AS] Μ h GeV 1 µ 2 s 4m t qe 2 + Γ 2 t R Σtt ΣΜΜ LL m P S = 175 GeV Γ t = 1.4 GeV µ F = 20 GeV NLL 0.4 NNLL E PS s 2 m PS µh scale dependence larger than µ s scale dependence µh scale-dependence bands do overlap more reliable estimate of theoretical error for normalization: 10% LoopFest V p. 16/2

17 electroweak leading order top quark propagator (E p 2 ) 2m 1 1 scales as t mv 2 1 mα 2 s the width Γt mα ew is a LO effect, E E + iγ [Fadin, Khoze] 1 mα ew E 1 p2 2m t E 1 p2 2m t + iγ t Coulomb singularity v 0 propagator pole Γ 0 resum (α s /v) n (potential gluon exchange) resum (Γ/m) n (self-energy insertions) systematic expansion in α and v systematic expansion in α and Γ LoopFest V p. 17/2

18 electroweak higher-order electroweak corrections for stable top NLO QED corrections: single potential photon exchange suppressed by α/v α 2 s/v v V V 4πα e2 q q 2 NNLO QED corrections: double potential photon exchange (α/v) 2 v 2 and hard corrections c 1 c 1 2e2 qα π beyond NNLO: many corrections of order α αs e.g. Higgs mass dependence δm t up to MeV [Eiras, Steinhauser] LoopFest V p. 18/2

19 electroweak NLO and NNLO QED corrections for stable top [Pineda, AS] 1 LL NLL µ 2 s 4m t q E 2 + Γ 2 t NNLL m P S = 175 GeV R Σtt ΣΜΜ with QED Γ t = 1.4 GeV µ F = 20 GeV no QED 0.4 Μ s 40 GeV E PS s 2 m PS shift in position of peak, i.e. δmt 100 MeV, about the same size as NNLL corrections LoopFest V p. 19/2

20 electroweak electroweak corrections beyond stable top Strictly speaking, it does not make sense to talk about σ(e + e t t) (or any cross section with an unstable particle in the final state). for threshold scan, δmt Γt, thus σ(e + e t t) σ(e + e W + W b b) QCD and electroweak effects W W b b t t e+ e electroweak effects are important! partially computed δm t = MeV [Hoang, Reisser] LoopFest V p. 20/2

21 electroweak effective theory approach to unstable particles use effective theory methods (again!) to systematically expand in small parameter δ (p 2 m 2 )/m 2 Γ/m [Chapovsky, Khoze, AS, Stirling] identify relevant modes (soft/resonant modes from HQET and NRQCD, collinear modes from SCET) asymptotic expansion [Beneke, Chapovsky, AS, Zanderighi] integrate out unwanted modes tower of effective theories (Unstable Particle Effective Theory) hard effects correspond to factorizable corrections non-factorizable corrections due to still dynamical modes this is neither a quick-fix nor a free lunch, it is a method to identify the minimal amount of calculation to be done for a systematic expansion in the small parameters (as for NRQCD) gauge invariance is automatic since the split into the various contributions respects gauge invariance LoopFest V p. 21/2

22 outlook needed: c1 fully NNLL and all electroweak effects ultrasoft effects (retardation effects) due to chromoelectric dipole operator x E NNNLO effects α 3 s (NNLL part α 3 s ln α s already included) potentially particularly important: α 3 s α 2 s (µ s) α s (µ us ) full NNNLO... compute all insertions (up to triple insertions), some results available: [Beneke, Kiyo, Schuller] compute all potentials, some results available: [Kniehl, Penin, Steinhauser, Smirnov] bottleneck: three-loop static potential and current matching coefficient more exclusive quantities LoopFest V p. 22/2

23 conclusions the theory for t t production near threshold is in good shape and further progress is on its way achieving δmt 100 MeV and δr max 3% relies on further theoretical progress (and the patience to actually do a threshold scan!!) full NNLL!! at least ultrasoft (if not full) NNNLO fully take into account instability of top quark more exclusive final states? tools are set up, but a lot of (tedious) additional work required this is one of the rare problems that is very fascinating from a theoretical point of view and extremely relevant from an experimental point of view LoopFest V p. 23/2