Two-loop resummation in (F)APT A. P. Bakulev Bogoliubov Lab. Theor. - PowerPoint PPT Presentation

Two-loop resummation in (F)APT A. P. Bakulev Bogoliubov Lab. Theor. Phys., JINR (Dubna, Russia) Two-loop resummation in (F)APT – p. 1 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

OUTLINE Intro: Analytic Perturbation Theory (APT) in QCD Problems of APT and their resolution in FAPT: Technical development of FAPT: thresholds Resummation in APT and FAPT Applications: Higgs decay H 0 → b ¯ b Conclusions Two-loop resummation in (F)APT – p. 2 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Collaborators & Publications Collaborators: S. Mikhailov (Dubna) and N. Stefanis (Bochum) Publications: A. B., Mikhailov, Stefanis — PRD 72 (2005) 074014 A. B., Karanikas, Stefanis — PRD 72 (2005) 074014vspace*2mm A. B., Mikhailov, Stefanis — PRD 75 (2007) 056005 Mikhailov — JHEP 0706 (2007) 009 [arXiv:hep-ph/0411397] A. B.&Mikhailov — arXiv:0803.3013 [hep-ph] A. B. — Phys. Part. Nucl. 40 (2009) 715 A. B., Mikhailov, Stefanis — JHEP 1006 (2010) 085 Two-loop resummation in (F)APT – p. 3 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Analytic Perturbation Theory in QCD Two-loop resummation in (F)APT – p. 4 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � pQCD + RG: resum π 2 -terms RG + Analyticity ghost-free α QED ( Q 2 ) Arctg ( s ) , UV Non-Power Series Bogoliubov et al. 1959 Radyush.,Krasn. &Pivov. 1982 pQCD + renormalons DispRel + renormalons IR finite α eff s ( Q 2 ) Arctg ( s ) at LE region Ball, Beneke & Braun 1994-95 Dokshitzer et al. 1995 RG + Analyticity Integral Transformation: ghost-free α E ( Q 2 ) R [ α s ] → Arctg ( s ) Shirkov & Solovtsov 1996 Jones & Solovtsov 1995 Two-loop resummation in (F)APT – p. 5 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � RG + Analyticity Integral Transformation: ghost-free α E ( Q 2 ) R [ α s ] → Arctg ( s ) Shirkov & Solovtsov 1996 Jones & Solovtsov 1995 pQCD + RG + Analyticity − 1 Transforms: ˆ D = ˆ R Couplings: α E ( Q 2 ) ⇔ α M ( s ) Milton & Solovtsov 1996–97 Analytic (global) pQCD + Analyticity Global couplings: A n ( Q 2 ) ⇔ A n ( s ) Non-Power perturbative expansions Shirkov 1999–2001 Two-loop resummation in (F)APT – p. 6 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Global Fractional APT (FAPT) Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) A. B. & Mikhailov & Stefanis 2005–2006 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) s × Log m : L ν,m ( Q 2 ) ⇔ L ν,m ( s ) Analytization of α ν A. B. & Mikhailov & Stefanis 2005–2006 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) s × Log m : L ν,m ( Q 2 ) ⇔ L ν,m ( s ) Analytization of α ν A. B. & Mikhailov & Stefanis 2005–2006 Resummation in 1-loop APT S. Mikhailov 2004 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) s × Log m : L ν,m ( Q 2 ) ⇔ L ν,m ( s ) Analytization of α ν A. B. & Mikhailov & Stefanis 2005–2006 Resummation in 1-loop global FAPT A. B. & Mikhailov 2008 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) s × Log m : L ν,m ( Q 2 ) ⇔ L ν,m ( s ) Analytization of α ν A. B. & Mikhailov & Stefanis 2005–2006 Resummation in 1-loop global FAPT A. B. & Mikhailov 2008 s (1 + c 1 α s ) ν ′ : B ν,ν ′ ( Q 2 ) ⇔ B ν,ν ′ ( s ) Analytization of α ν A. B. 2008–2009 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

History of F(ractional)APT Euclidean Minkowskian Q 2 = � q 2 − q 2 q 2 ≥ 0 s = q 2 0 ≥ 0 0 − � Analytization of α ν s : A ν ( Q 2 ) ⇔ A ν ( s ) s × Log m : L ν,m ( Q 2 ) ⇔ L ν,m ( s ) Analytization of α ν A. B. & Mikhailov & Stefanis 2005–2006 Resummation in 1-loop global FAPT A. B. & Mikhailov 2008 s (1 + c 1 α s ) ν ′ : B ν,ν ′ ( Q 2 ) ⇔ B ν,ν ′ ( s ) Analytization of α ν A. B. 2008–2009 Resummation in 2-loop global FAPT with 2-loop evolution factors B ν,ν ′ ( Q 2 ) ⇔ B ν,ν ′ ( s ) A. B. & Mikhailov & Stefanis 2010 Two-loop resummation in (F)APT – p. 7 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Intro: PT in QCD coupling α s ( μ 2 ) = (4 π/b 0 ) a s [ L ] with L = ln ( μ 2 / Λ 2 ) RG equation d a s [ L ] = − a 2 s − c 1 a 3 s − . . . d L 1-loop solution generates Landau pole singularity: a s [ L ] = 1 /L Two-loop resummation in (F)APT – p. 8 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Intro: PT in QCD coupling α s ( μ 2 ) = (4 π/b 0 ) a s [ L ] with L = ln ( μ 2 / Λ 2 ) RG equation d a s [ L ] = − a 2 s − c 1 a 3 s − . . . d L 1-loop solution generates Landau pole singularity: a s [ L ] = 1 /L 2-loop solution generates square-root singularity: a s [ L ] ∼ 1 / √ L + c 1 ln c 1 Two-loop resummation in (F)APT – p. 8 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Intro: PT in QCD coupling α s ( μ 2 ) = (4 π/b 0 ) a s [ L ] with L = ln ( μ 2 / Λ 2 ) RG equation d a s [ L ] = − a 2 s − c 1 a 3 s − . . . d L 1-loop solution generates Landau pole singularity: a s [ L ] = 1 /L 2-loop solution generates square-root singularity: a s [ L ] ∼ 1 / √ L + c 1 ln c 1 PT series: D [ L ] = 1 + d 1 a s [ L ] + d 2 a 2 s [ L ] + . . . Two-loop resummation in (F)APT – p. 8 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Intro: PT in QCD coupling α s ( μ 2 ) = (4 π/b 0 ) a s [ L ] with L = ln ( μ 2 / Λ 2 ) RG equation d a s [ L ] = − a 2 s − c 1 a 3 s − . . . d L 1-loop solution generates Landau pole singularity: a s [ L ] = 1 /L 2-loop solution generates square-root singularity: a s [ L ] ∼ 1 / √ L + c 1 ln c 1 PT series: D [ L ] = 1 + d 1 a s [ L ] + d 2 a 2 s [ L ] + . . . � � RG evolution: B ( Q 2 ) = Z ( Q 2 ) /Z ( μ 2 ) B ( μ 2 ) reduces in 1-loop approximation to � � Z ∼ a ν [ L ] � ν = ν 0 ≡ γ 0 / (2 b 0 ) Two-loop resummation in (F)APT – p. 8 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Problem in QCD PT: Minkowski region? � � � Quantities in Minkowski region = f ( z ) D ( z ) dz . Im z − s + iε • • • Re z = Q 2 − s − iε Two-loop resummation in (F)APT – p. 9 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Problem in QCD PT: Minkowski region? � � � � � � d m α m In f ( z ) D ( z ) dz one uses D ( z ) = s ( z ) . m Im z •− s + iε • •− s − iε Re z = Q 2 Two-loop resummation in (F)APT – p. 9 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Problem in QCD PT: Minkowski region? This change of integration contour is legitimate if D ( z ) f ( z ) is analytic inside Im z − s + iε • • • Re z = Q 2 − s − iε Two-loop resummation in (F)APT – p. 9 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Problem in QCD PT: Minkowski region? But α s ( z ) and hence D ( z ) f ( z ) have Landau pole singularity just inside! Im z − s + iε • • � • Re z = Q 2 − s − iε Two-loop resummation in (F)APT – p. 9 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Problem in QCD PT: Minkowski region? In APT effective couplings A n ( z ) are analytic functions ⇒ Problem does not appear! Equivalence to CIPT for R ( s ) . Im z − s + iε • • • Re z = Q 2 − s − iε Two-loop resummation in (F)APT – p. 9 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Equivalence CIPT and APT for R ( s ) �� � � �� � � � � D ( z ) dz D ( z ) dz CIPT = APT z z Γ 2 Γ 3 Im z Γ 2 Γ 3 Γ 1 − s + iε × • • R e z = Q 2 • − s − iε Two-loop resummation in (F)APT – p. 10 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Basics of APT Different effective couplings in Euclidean (S&S) and Minkowskian (R&K&P) regions Two-loop resummation in (F)APT – p. 11 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010

Basics of APT Different effective couplings in Euclidean (S&S) and Minkowskian (R&K&P) regions Based on RG + Causality ⇓ ⇓ UV asymptotics Spectrality Two-loop resummation in (F)APT – p. 11 Bogoliubov Readings@JINR, Dubna (Russia), Sept. 22–25, 2010