Effective-theory methods for top-quark and squark pair production at LHC and Tevatron

Transcription

1 Effective-theory methods for top-quark and squark pair production at LHC and Tevatron Christian Schwinn Univ. Freiburg (Based on M.Beneke, P.Falgari, CS, arxiv: [hep-ph], arxiv: [hep-ph] M.Beneke, M.Czakon, P.Falgari, A.Mitov, CS arxiv: [hep-ph] M.Beneke, P.Falgari, S. Klein, CS, in progress )

2 Introduction 1 Pair production of heavy coloured particles at Tevatron/LHC NN HH + X N, N : pp, p p; HH : top-quark, squark, gluino... pairs Precise knowledge of total cross sections: top-quarks: sensitivity on mass, constraining gluon PDFs new particles: Exclusion bounds, model discrimination,... (pb) tt σ Measured σ tt Nadolsky et al., PRD 78, (2008) Cacciari et al., JHEP 09, 127 (2008) Moch and Uwer, PRD 78, (2008) DØ, L = 1 fb Top Mass (GeV) Cross Section (pb) (a) -1 DØ, L=2.1 fb m 0 =25, A =0, tan(β)=3, µ<0 0 σ NLO Observed limit Expected limit Squark Mass (GeV)

3 Total t t cross section 2 Experimental knowledge of t t cross section: Tevatron: σ t t = 6.8% ; LHC Goal: σ t t 5% Theory status: NLO + higher-order soft gluons σ t t 10%

4 Total t t cross section 2 Experimental knowledge of t t cross section: Tevatron: σ t t = 6.8% ; LHC Goal: σ t t 5% Theory status: NLO + higher-order soft gluons σ t t 10% Building blocks for NNLO: q q : Czakon 08; Bonciani et.al. 08/09) two-loop t t, (m t 0 : Czakon/Mitov/Moch 07; one-loop t t + j (Dittmaier/Uwer/Weinzierl 07) t t squared (Körner et.al , Anastasiou/Mert-Aybert 08) tree t t + jj (IR subtraction: Czakon 10)

5 Threshold resummation 3 NLO corrections enhanced for β = 1 (m H+m H ) 2 ŝ 0 ˆσ (1) pp HH = ˆσ (0) pp HH α s [a log 2 (8β 2 ) + blog(8β 2 ) + c }{{} β }{{} threshold logarithms Coulomb singularity ]

6 Threshold resummation 3 NLO corrections enhanced for β = 1 (m H+m H ) 2 ŝ 0 ˆσ (1) pp HH = ˆσ (0) pp HH α s [a log 2 (8β 2 ) + blog(8β 2 ) + c }{{} β }{{} threshold logarithms Coulomb singularity Soft corrections: (Resummation in Mellin space: Sterman 87; Catani, Trentadue 89, Kidonakis,Sterman 97, Bonciani et.al. 98,... ) ] α s log 2 (8β 2 ) α s log(8β 2 ) Coulomb gluon corrections (Fadin, Khoze 87; Peskin, Strassler 90, NRQCD,... ) α s 1 β

7 Threshold resummation 4 Basis of resummation: Factorization of soft corrections for Q 2 = (p H + p H ) 2 ŝ ˆσ pp HH = H pp HH S RGEs for factorization scale dependence of H(µ f ), S(µ f ) ( Korchemsky, Marchesini 92, Contopanagos, Laenen, Sterman 96, Manohar 03, Solve RGE in Mellin-space or momentum space Leading log resummation (LL): sums terms of the form Becher, Neubert, Pecjak 06) α n s log m (8β 2 ) = α s log 2 (8β 2 ) +... n + 1 m 2n Next-to leading log resummation (NLL): sums terms of the form (α s log(8β 2 )) n = α s log(8β 2 ) +...

8 Threshold resummation 5 Resummation of threshold logs: Estimate of higher order corrections z Accelerated convergence of perturbative series Reduced scale dependence Recent applications CDF data NLO + NNLL (Ahrens et.al. 10) total top quark cross section (Moch, Uwer; Cacciari et.al.; Kidonakis Vogt 08) t t invariant mass distribution (Ahrens, Ferroglia, Neubert, Pecjak, Yang 09/10) K X r = 0.5 r = 0.8 r = 1.2 r = 1.6 r = 2.0 Coulomb squark, gluino production (Kulesza, Moytka 08/09; Langenfeld, Moch 09; Beenakker et.al. 09/10) octet scalars (Idilbi, Kim, Mehen 09/10) Soft gluons m~ q [ TeV ] (Kulesza, Motyka 09)

9 EFT approach to heavy pair production 6 Effective theory approach for DIS, Drell-Yan, Higgs (Manohar 03; Idilbi et.al. 03/05, Becher et.al. 06/07, Ahrens et.al. 08) Construct EFT for collinear p, p, soft gluons resum threshold Logs in momentum space (Becher, Neubert 06) no numerical inverse transformation necessary

10 EFT approach to heavy pair production 6 Effective theory approach for DIS, Drell-Yan, Higgs (Manohar 03; Idilbi et.al. 03/05, Becher et.al. 06/07, Ahrens et.al. 08) Construct EFT for collinear p, p, soft gluons resum threshold Logs in momentum space (Becher, Neubert 06) no numerical inverse transformation necessary EFT approach to heavy pair production (Beneke, Falgari, CS 09/10) Consider partonic production threshold ŝ (m H + m H ) 2 EFT for collinear p, p, non-relativistic HH, soft gluons

11 EFT approach to heavy pair production 6 Effective theory approach for DIS, Drell-Yan, Higgs (Manohar 03; Idilbi et.al. 03/05, Becher et.al. 06/07, Ahrens et.al. 08) Construct EFT for collinear p, p, soft gluons resum threshold Logs in momentum space (Becher, Neubert 06) no numerical inverse transformation necessary EFT approach to heavy pair production (Beneke, Falgari, CS 09/10) Consider partonic production threshold ŝ (m H + m H ) 2 EFT for collinear p, p, non-relativistic HH, soft gluons Application to top quark, squark and gluino production Diagonal colour bases for 3 3, 3 3, 3 8, 8 8 NLL results for squark-antisquark production O(α 4 s) threshold expansion +NNLL resummation for t t

12 Pair production near threshold 7 Kinematics for p(k 1 )p (k 2 ) H(p 1 )H (p 2 ) + X near partonic threshold ŝ (m H + m H ) 2 4M 2 incoming p/p : k 1,2 = ŝ 2 (1, ± n), n2 = 1 non-relativistic H/H : p 1 = m H (1, 0) + p 1, p 1 m H (β 2, β) Real gluons soft: q X Mβ 2 (k 1 +k 2 q X ) 2 =(p 1 +p 2 ) 2 4M 2 ŝ 2 ŝ q X 4M 2 ŝ } q X } (p 1 + p 2 ) 2 4M 2 q X M 1 4M2 =Mβ 2 ŝ

13 Pair production near threshold 7 Kinematics for p(k 1 )p (k 2 ) H(p 1 )H (p 2 ) + X near partonic threshold ŝ (m H + m H ) 2 4M 2 incoming p/p : k 1,2 = ŝ 2 (1, ± n), n2 = 1 non-relativistic H/H : p 1 = m H (1, 0) + p 1, p 1 m H (β 2, β) Real gluons soft: q X Mβ 2 (k 1 +k 2 q X ) 2 =(p 1 +p 2 ) 2 4M 2 ŝ 2 ŝ q X 4M 2 ŝ } q X } (p 1 + p 2 ) 2 4M 2 q X M 1 4M2 =Mβ 2 ŝ Origin of 1/β enhancement: exchange of Coulomb gluon q m(β 2, β) p 2 p 2 p 1 p 1 q dq 0 d 3 q p p 2 2 2m t 1 q 2 1 p 0 1 p 2 2 2m t 1 β

14 Pair production near threshold 8 Combination of Coulomb- and soft effects? Heavy particles nonrelativistic near threshold: E mβ 2, p mβ soft gluon momenta of same order: q s mβ 2 E heavy particles feel soft radiation q

15 Pair production near threshold 8 Combination of Coulomb- and soft effects? Heavy particles nonrelativistic near threshold: E mβ 2, p mβ soft gluon momenta of same order: q s mβ 2 E heavy particles feel soft radiation q Factorization of cross section up to NNLL (Beneke, Falgari, CS 09/10) ˆσ pp HH ŝ 4M 2 = Hard, soft and Coulomb functions: R,i H i W R i J R H i =, W i R =, J R = Soft radiation sees only total colour charge R of heavy particles (Singlet, octet,...)

16 EFT standard lore 9 Starting point: Problem with several scales, e.g. Weak decays of b-quarks: b u cs Two scales: M W, m b. Radiative corrections α s /π log (m b /M W ) 1 Integrate out W boson four-fermion operators b u s W c O bcus s b u c

17 EFT standard lore 9 Starting point: Problem with several scales, e.g. Weak decays of b-quarks: b u cs Two scales: M W, m b. Radiative corrections α s /π log (m b /M W ) 1 Integrate out W boson four-fermion operators NLO Matching: radiative corrections to coefficients of O bcus (C(α s (µ), log(m 2 W /µ2 ))) g b u s c W C (1) W O bcus s b u O bcus c s b u c

18 EFT standard lore 9 Starting point: Problem with several scales, e.g. Weak decays of b-quarks: b u cs Two scales: M W, m b. Radiative corrections α s /π log (m b /M W ) 1 Integrate out W boson four-fermion operators NLO Matching: radiative corrections to coefficients of O bcus (C(α s (µ), log(m 2 W /µ2 ))) RGE running to sum logarithms log ( M W m b ) g b W u s W c C (1) O bcus b u b s O bcus c u s c Calculate radiative correction in EFT at scale m b g

19 EFT for heavy particles at threshold 10 Hard contributions: analogous to W in b u cs. Here: off-shell particles p 2 4M 2, MSSM particles (gluinos,...) q g q q q q C q O q q not part of EFT, effect included in C-coefficients Use RGE to sum logs log(4m 2 /µ f )

20 EFT for heavy particles at threshold 10 Hard contributions: analogous to W in b u cs. Here: off-shell particles p 2 4M 2, MSSM particles (gluinos,...) q g q q q q C q O q q not part of EFT, effect included in C-coefficients Use RGE to sum logs log(4m 2 /µ f ) EFT includes modes that can go on-shell in classical scattering gluon p H m H (1, 0) + p with ( p 0 β 2, p β) soft: (q 0, q) (β 2, β 2 ) (keep in EFT) potential : (q 0, q) (β 2, β) (integrate out) interaction of gluons with non-relativistic heavy particles Incoming partons: k 2 1/2 = 0: couple to soft gluons

21 EFT for heavy particles at threshold 11 Matching of scattering amplitude A pp HH X = i C (i) (M, µ) c(i) {α} {a} HH X φ c;a1 α 1 φ c;a2 α 2 ψ a 3 α 3 ψ a 4 α 4 pp EFT ψ, ψ : non-relativistic fields that create H and H φ c (φ c ): collinear (anti-collinear) fields that destroy p and p α i : spin, a i : colour indices, c (i) : colour basis {a}

22 EFT for heavy particles at threshold 11 Matching of scattering amplitude A pp HH X = i C (i) (M, µ) c(i) {α} {a} HH X φ c;a1 α 1 φ c;a2 α 2 ψ a 3 α 3 ψ a 4 α 4 pp EFT Example: squark-antisquark production q q q q ia (0) q i q j q il q jr ŝ=4m 2 = i2g2 sm g q m 2 q + m2 g T b a 3 a 1 T b a 2 a 4 v(m q n) ( 1 γ 5 2 ) u(m q n) q q g q q s-channel singlet/octet colour basis c (1) {a} = 1 N c δ a1 a 2 δ a3 a 4, c (2) {a} = 1 2 T β a 2 a 1 T β a 3 a 4 Short-distance coefficients: ) C (1) {α} = ( C F) 4πα sm g m 2 q + m2 g ( 1 γ 5 2, C (2) {α} = C F α 1 α 2 2N C 4πα s m g m 2 q + m2 g q q ( 1 γ 5 2 g ) q q α 1 α 2 = O(β)

23 EFT for heavy particles at threshold 12 Soft-collinear Lagrangian (Bauer et.al. 2000, Beneke et.al. 2002) for quarks with momentum p n, n 2 = 0: (n n = 2, n p = 0) ( ) L c = ξ 1 / n c in D + i /D c i /D c i n D c 2 ξ c Coupling to soft and collinear gluons: id c = i + ga c id = id c + ga s Collinear spinors satisfy /nξ c = 0, ( /n/ n 4 ) ξ c = ξ c Reproduces soft gluon approximation (k q s ) 2 2(k q s ), k Mn: k q s = ig νµ k q s qs 2 i(/k /q s ) (k q s ) 2 (ig sγ µ )u(k) ig νµ q 2 s i (k q s ) (ig sk µ )u(k) ig νµ q 2 s /n 2 i (n q s ) (ig sn µ ) / n 2 u(mn)

24 EFT for heavy particles at threshold 13 Collinear and nonrelativistic fields only connected by soft gluons Soft-gluon decoupling Diagrammatic analysis (Collins, Sterman 81; e.g. Sterman,Tasi 95) Field redefinition in EFT (Bauer, Pirjol, Stewart 01) with soft Wilson line ξ c (x) = S n (x )ξ c (0) (x) x = n x 0 ] S n (x) = P exp [ig s dt n A a s(x + nt)t a

25 EFT for heavy particles at threshold 13 Collinear and nonrelativistic fields only connected by soft gluons Soft-gluon decoupling Diagrammatic analysis (Collins, Sterman 81; e.g. Sterman,Tasi 95) Field redefinition in EFT (Bauer, Pirjol, Stewart 01) with soft Wilson line Wilson line satisfies ξ c (x) = S n (x )ξ c (0) (x) x = n x 0 ] S n (x) = P exp [ig s dt n A a s(x + nt)t a n S n (x ) = ig s n A a s (x )T a S n (x ) (c.f. Schrödinger eq. for time evolution operator i U/ t = HU) soft gluons decouple in Lagrangian: ξ n (in D s )ξ n = (0) ξ n (in )ξ n (0)

26 EFT for heavy particles at threshold 14 NRQCD Lagrangian for particles H, H in representations R, R : ) ( ) ψ + ψ L PNRQCD = ψ ( + id 0 s iγ H 2m H 2 d 3 r [ ψ T (R)a ψ ] ( r ) ( αs r id 0 s + 2 2m H ) [ψ T (R )a ψ ] (0), + iγ H 2 ψ

27 EFT for heavy particles at threshold 14 NRQCD Lagrangian for particles H, H in representations R, R : ) ( ) ψ + ψ L PNRQCD = ψ ( + id 0 s iγ H 2m H 2 d 3 r [ ψ T (R)a ψ ] ( r ) ( αs r id 0 s + 2 2m H ) [ψ T (R )a ψ ] (0), Resummation of α s β solve NR-Schrödinger equation for Green s function + iγ H 2 corrections: (Fadin, Khoze 87; Peskin, Strassler 90) ψ = 1 Cα sπ 2β + = Cπα s β ( e Cπα s β 1 ) 1

28 EFT for heavy particles at threshold 14 NRQCD Lagrangian for particles H, H in representations R, R : ) ( ) ψ + ψ L PNRQCD = ψ ( + id 0 s iγ H 2m H 2 d 3 r [ ψ T (R)a ψ ] ( r ) ( αs r id 0 s + 2 2m H ) [ψ T (R )a ψ ] (0), Resummation of α s β solve NR-Schrödinger equation for Green s function + iγ H 2 corrections: (Fadin, Khoze 87; Peskin, Strassler 90) = 1 Cα sπ 2β + = Cπα s β Soft gluon decoupling for heavy particles fields: 0 ( e Cπα s β 1 ) 1 ψ (x) = ψ (0) (x)s v (R) (x 0 ) ] S v (R) (x) = Pexp [ig s ds v A a (x + vs)t (R)a, v 2 = 1 same transformation for both heavy particles at threshold ψ

29 Derivation of factorization formula 15 Apply soft-gluon decoupling to amplitude: A pp HH X i C (i) HH ψ (0) ψ (0) 0 0 φ (0) c p 0 φ (0) p X S n S n c (i) S vs v 0 c

30 Derivation of factorization formula 15 Apply soft-gluon decoupling to amplitude: A pp HH X i C (i) HH ψ (0) ψ (0) 0 0 φ (0) c p 0 φ (0) p X S n S n c (i) S vs v 0 Inserting into formula for σ, summing over complete set of X... ŝ ω ˆσ pp (ŝ, µ) = H ii (M, µ) dω J Rα ( 2M }{{} 2 ) W R α ii (ω, µ) i,i tr C (i ) C (i) R α }{{}} {{} soft gluons c Coulomb gluons Irreducible representations R R = R α R α e.g. 3 3 = 1 8. disentangles hard, soft and Coulomb contribution (for S-wave production and up to NNLL) can perform simultaneous summation of threshold Logs and Coulomb corrections Can find colour basis that diagonalizes W ii to all orders

31 Resummation of threshold logarithms 16 Factorization scale dependence of H, W cancels against PDFs: dσ dµ = d dµ (f 1 f 2 H W J) = 0 df i dµ dh i dµ Altarelli-Parisi equation related to IR singlarities (Becher, Neubert; Ferroglia et.al. 09) get RGE for soft function d ( ( dlog µ W R α iz0 µ i (z 0, µ) = 2γ cusp (C r +C r )log 2 ) 2(γ R α H.s + as for Drell-Yan/Higgs {}}{ γ r s + γr s )) W R α i (z 0, µ) Two-loop γ H : Kidonakis; Becher/Neubert; Beneke/Falgari/CS; Czakon e.a.; Ferroglia e.a. 09

32 Resummation of threshold logarithms 16 Factorization scale dependence of H, W cancels against PDFs: dσ dµ = d dµ (f 1 f 2 H W J) = 0 df i dµ dh i dµ Altarelli-Parisi equation related to IR singlarities (Becher, Neubert; Ferroglia et.al. 09) get RGE for soft function d ( ( dlog µ W R α iz0 µ i (z 0, µ) = 2γ cusp (C r +C r )log 2 ) 2(γ R α H.s + as for Drell-Yan/Higgs {}}{ γ r s + γr s )) W R α i (z 0, µ) Two-loop γ H : Kidonakis; Becher/Neubert; Beneke/Falgari/CS; Czakon e.a.; Ferroglia e.a. 09 Resummation: evolve hard function from µ h 2M to µ f H(M, µ h ) f 1 (µ f )f 2 (µ f )H(M, µ f )J R α W R α (ω, µ f ) µ h µ f evolve soft function from µ s to µ f (Mellin space: Korchemsky/Marchesini 92, momentum space:becher/neubert 06) W R α ii (ω,µ s) µ s

33 Squark-antisquark production at NLL 17 Squark -antisquarks at LHC Two production channels: q i q j q k q l, gg q k q l Simplified setup: equal squark masses, no stop Matching to NLO result (Beenakker et.al. 96, PROSPINO ) Resummed Results: NLL: full Coulomb res. soft nobs: NLL without bound states NLL s+h : resummation of H and W C: Coulomb resummation Σ Σ NLO NLL NLL nobs NLL s h C NLL s h m q GeV ( s = 14 TeV, m g /m q = 1.25MSTW08NLO) (NLL s +C consistent with results from Mellin-approach: Kulesza/Motyka; Beenakker et.al. 09)

34 Squark-antisquark production at NLL 18 Scale uncertainty reduced by combined resummation NLO m q 2 < µ f < m q NLL: vary all scales µ i 2 < µ i < 2 µ i, add in quadrature significant reduction for combined resummation! NLL Σ X Μ Σ X Μ NLL s h NLO Σ pb LO NLO 0.5 EFT NLL BS m q GeV Μ f m q ( s = 14 TeV, MSTW08NLO, m g /m q = 1.25) (m q = 1 TeV, µ 0 s /2 < µ s < 2µ 0 s)

35 Total top-pair production cross-section 19 NLO corrections (Nason, Dawson Ellis 88, Beenakker et.al. 89/91) Resummations, expansions for various observables: NNLO approx (Moch, Uwer 08, Beneke, Czakon, Falgari, Mitov CS, 09 ) log β-resummation (Bonciani et.al. 98) σ (2) q q = 3.61 β β ( ln β ln β ) ln β ln β ln β ln β + C (2) qq Pair invariant mass cross sections (Kidonakis,Sterman 97) ] dσ(t t) dm t t [ log n (1 z) 1 z +, z = M2 t t ŝ NNLL resummation of log n (1 z): (Ahrens et.al. 10) One particle inclusive cross sections: (Laenen, Oderda, Sterman 98) [ ] dσ(t + X) log n (s 4 /m 2 t ), s 4 = p 2 X s m2 t 4 ds 4

36 t t cross section at Tevatron and LHC 20 σ t t(pb) Tevatron LHC7 LHC10 LHC14 NLO NNLO approx (β) NNLO approx (β) + NNLL (Beneke, Falgari, Klein, CS in progress ) NLO + NNLL (M t t) (Ahrens et.al. 10) NNLO approx (s 4 ) (mt=173; Kidonakis 10) (m t = GeV, µ f = mt, MSTW08NNLO) Application: (Moch, Langenfeld, Uwer 09) Comparison to Tevatron data m t (MS) = GeV σ [pb] Tevatron MSTW 2008 NNLO NLO NNLO approx m t (pole) = GeV m(m)

37 Summary and Outlook 21 Threshold resummation Threshold logarithms, Coulomb correction EFT approach use SCET+NRQCD to factorize soft and Coulomb gluons log β resummation from momentum space solution to RGEs Application to squark-antisquark production combined Soft and Coulomb resummation total corrections 4 10% for m q = 300 GeV-2 TeV Threshold expansion to O(α 2 s) of t t cross section NNLL resummation for t t dominant higher-order corrections included in NNLO approx discrepancy to NNLL from integrated dσ dm 2 tt? (Ahrens et.al. 10)

38 Bonus slides 22

39 Colour structure 23 Construction of colour basis tenors c (i) for all squark/gluino production processes q q, q q, g g, q g soft radiation off total colour charge of (HH ) Rα (Bonciani et.al. 98) one-loop basis (Kidonakis/Sterman 97; Kulesza/Moytka 08, Beenakker et.al. 09) all-order diagonalization of soft function (Beneke, Falgari, CS 09) pairs of equivalent initial- and final state representations: e.g : P i {(1, 1), (8 S, 8), (8 A, 8)} Clebsch-Gordan coefficients e.g A, : construct basis tensors: C (8 A) αa 1 a 2 = i 3 f a 2αa 1, C (8) αa 1 a 2 = 2T α a 2 a 1 c (i) {a} = 1 dim(rα ) Cr α αa1 a 2 C R β αa 3 a 4 e.g. c (3) {a} = i 12 f a 2αa 1 T α a 3 a 4 Combine final-state Wilson lines: C R (R) β S v S (R ) v = S R β v C R β soft function for single final-state particle in R α

40 Subleading corrections 24 Subleading PNRQCD and SCET interactions: ψ x E ( ) us (x 0, 0)ψ, ξ x µ nν W c gfµν us W c n+ 2 ξ... Soft gluons not decoupled by field redefinitions. Possibly relevant at NNLL in soft potential corrections :, α s β α sβ log β α 2 s log β Related to three-parton colour correlations in IR singularities of amplitudes (Ferroglia et.al. 09) σ tot : effects vanish at NNLL! (Beneke, Czakon,Falgari, Mitov, CS 09) no collinear/potential correction β for k = 0 no potential/soft corrections due to rotational invariance (no heavy particle three-momentum available)

41 Squark-antisquark production at NLL 25 Choice of scales for resummation in momentum space Soft scale µ s that minimizes hadronic σ NLO soft µ s /m q for m q = 0.5,... 2 TeV (Becher, Neubert, Xu 07) Hard scale: µ h = 2m q Dependence on scale choices: Σ NLL Μ s Σ NLL Μ s 1.2 m q 500 GeV m q 1 TeV m q 2 TeV Σ NLL Μ h Σ NLL Μ h 1.8 m q 500 GeV m q 1 TeV m q 2 TeV Μ s Μ s Μ h Μ h ( s = 14 TeV, m g /m q = 1.25) Coulomb scale: µ C = max{2m q β, C F m q α s (µ C )}

42 Squark-antisquark production at NLL 26 Comparison to Mellin-approach: (Kulesza, Motyka 08/09, Beenakker et.al. 09) Good agreement for appropriate choice of scales (µ h = µ f : NLL s ): m q [GeV] NLO[pb] NLL Mellin [pb] NLL s [pb] NLL [pb] (1%) (1%) (2%) (1.2%) (1.3%) (3%) (1.7%) (1.7%) (4.4%) (3.4%) (3.1%) (9%) (6.4)% (LHC 14 TeV, m g = m q )

43 Top-pair production at NNLL 27 All threshold enhanced O(α 2 s) terms (Beneke, Czakon, Falgari, Mitov, CS 09) Pure soft corrections: σ (2) s α 2 s(c (2) LL ln4 β + c (2) NLL ln3 β + c (2) NNLL,2 ln2 β + c (2) ln β NNLL,1 ) Potential corrections: 2nd Coulomb, NLO potentials }{{} 2-loop γ H,s σ p (2) α 2 s ( c C β 2 β (c(2) + C,0 c(2) C,1 log β) + c(2) ln β n-c ) }{{} spin-dependent (extracted from Beneke, Signer, Smirnov 99, Czarnecki/Melnikov 97/01) mixed Coulomb/soft, hard corrections: σ (2) p sh α s β α s(c (1) ln LL β2 + c (1) NLLln β + c + H(1) ) }{{} process dependent

44 Threshold expansion at O(α 2 s) 28 Potential corrections: 2nd Coulomb correction NLO Coulomb potentials: ( ) Ṽ (1) C (p, q) = D R α α 2 s a q 2 1 β 0 ln q2 µ 2 Non-Coulomb potential: [ Ṽ (1) nc (p, q) = 4πD R α α s q 2 πα s q 4m ( DRα 2 + C A ) ] + p2 m + q2 2 m v 2 spin, (v spin = 0 (singlet); 2/3 (triplet)) Corrections to cross section: ˆσ nc = ˆσ (0) α 2 s ln β [ 2D 2 R α (1 + v spin ) + D Rα C A ] (extracted from Beneke, Signer, Smirnov 99, Pineda, Signer 06)