All-order corrections for top-mass determinations in the large - PowerPoint PPT Presentation

All-order corrections for top-mass determinations in the large number of flavours Silvia Ferrario Ravasio ∗ IPPP, Durham, Internal Seminar 14 th December 2018 *In collaboration with P. Nason and C. Oleari [arxiv:1810.10931] Silvia Ferrario Ravasio — December 14 th , 2018 1/28 Renormalons effects in top-mass measurements

Top quark phenomenology Top: last quark to be observed and heaviest elementary particle in the SM Silvia Ferrario Ravasio — December 14 th , 2018 2/28 Renormalons effects in top-mass measurements

Top quark phenomenology Top: last quark to be observed and heaviest elementary particle in the SM only quark that decays instead of hadronizing W + t b Silvia Ferrario Ravasio — December 14 th , 2018 2/28 Renormalons effects in top-mass measurements

Top quark phenomenology Top: last quark to be observed and heaviest elementary particle in the SM only quark that decays instead of hadronizing m t affects significantly many parameters of the SM, e.g. the mass of the W boson and the Higgs self-coupling λ t H t W + W + ¯ b Silvia Ferrario Ravasio — December 14 th , 2018 2/28 Renormalons effects in top-mass measurements

Top quark phenomenology Top: last quark to be observed and heaviest elementary particle in the SM only quark that decays instead of hadronizing m t affects significantly many parameters of the SM, e.g. the mass of the W boson and the Higgs self-coupling λ We want a precise determination of m t in a given renormalization scheme Silvia Ferrario Ravasio — December 14 th , 2018 2/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization Loop corrections can contain UV divergences arising from the ℓ → ∞ region � d 4 ℓ 1 1 ℓ 2 γ µ ∼ α pγ µ (2 π ) D / ℓ − / Silvia Ferrario Ravasio — December 14 th , 2018 3/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization Loop corrections can contain UV divergences arising from the ℓ → ∞ region � d 4 ℓ 1 1 ℓ 2 γ µ ∼ α pγ µ (2 π ) D / ℓ − / Regularization : d = 4 − 2 ǫ � µ 2 � I = 1 ǫ + 2 + log − p 2 Silvia Ferrario Ravasio — December 14 th , 2018 3/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization Loop corrections can contain UV divergences arising from the ℓ → ∞ region � d 4 ℓ 1 1 ℓ 2 γ µ ∼ α pγ µ (2 π ) D / ℓ − / Regularization : d = 4 − 2 ǫ � µ 2 � I = 1 ǫ + 2 + log − p 2 Renormalization : the divergent part is absorbed into the redefinition of new parameters = µ 2 ǫ Z α α r ( µ ) α b ���� � �� � bare ren dα r ( µ ) d log µ 2 = β ( α r ) = − b 0 α 2 r + . . . Silvia Ferrario Ravasio — December 14 th , 2018 3/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization In the MS scheme only the divergent part is reabsorbed. α s ( Q ) 1 b 0 = 11C A 12 π − n l T R � = α s ( k ) = � � � ; 3 π > 0 k k 1 + 2 b 0 α s ( Q ) log 2 b 0 log Q Λ QCD L = 5 . 0 fb − 1 √ s = 7 TeV CMS preliminary α s ( Q ) 0 . 22 JADE 4-jet rate LEP event shapes 0 . 20 DELPHI event shapes ZEUS inc. jets H1 DIS 0 . 18 D0 inc. jets D0 angular cor. 0 . 16 α s ( M Z ) = 0 . 1184 ± 0 . 0007 (world avg.) α s ( M Z ) = 0 . 1160 +0 . 0072 − 0 . 0031 (3-jet mass) 0 . 14 0 . 12 0 . 10 CMS R32 ratio CMS t ¯ t prod. CMS 3-jet mass 0 . 08 5 · 10 0 10 1 2 · 10 1 5 · 10 1 10 2 2 · 10 2 5 · 10 2 10 3 2 · 10 3 Q [GeV] Silvia Ferrario Ravasio — December 14 th , 2018 4/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization In the MS scheme only the divergent part is reabsorbed. α s ( Q ) 1 b 0 = 11C A 12 π − n l T R � = α s ( k ) = � � � ; 3 π > 0 k k 1 + 2 b 0 α s ( Q ) log 2 b 0 log Q Λ QCD mass of the quarks: m b = m r + δm r � � MS δm ( µ ) =Div − i p 2 = m 2 � � pole δm = − i p 2 = m 2 Silvia Ferrario Ravasio — December 14 th , 2018 4/28 Renormalons effects in top-mass measurements

UV divergencies and renormalization In the MS scheme only the divergent part is reabsorbed. α s ( Q ) 1 b 0 = 11C A 12 π − n l T R � = α s ( k ) = � � � ; 3 π > 0 k k 1 + 2 b 0 α s ( Q ) log 2 b 0 log Q Λ QCD mass of the quarks: m b = m r + δm r � � MS δm ( µ ) =Div − i p 2 = m 2 � � pole δm = − i p 2 = m 2 When the particle is unstable the renormalized mass can be chosen complex p 2 = m 2 − i Γ m Silvia Ferrario Ravasio — December 14 th , 2018 4/28 Renormalons effects in top-mass measurements

The top pole-mass The top is a resonance : t → Wb Complex pole scheme : p 2 = m 2 t − iΓ t m t W + 1 Inclusion of finite decay width t effects; 2 Gauge invariant; b 3 Straightforward to apply. 1 W b j @ NLO QCD [arXiv:1305.7088] 2 b ¯ ν ℓ l + ν l @ NLO QCD [arXiv:1012.4230], bℓ − ¯ NLO QCD (+PS) [arXiv:1607.04538], NLO QED [arXiv:1607.05571] 3 b ¯ bjℓ − ¯ ν ℓ l + ν l @ NLO QCD [arXiv:1710.07515] Silvia Ferrario Ravasio — December 14 th , 2018 5/28 Renormalons effects in top-mass measurements

Pole-mass renormalons The pole mass is not very well-defined for a coloured object. pole mass = location of the pole in the Feynman propagator, that corresponds to an asymptotic state . But there is confinement! Radiative corrections do not displace the location of m t : the pole mass counterterm absorbs both UV and IR contributions of the self en- ergy Σ Silvia Ferrario Ravasio — December 14 th , 2018 6/28 Renormalons effects in top-mass measurements

IR Renormalons QCD is affected by infrared slavery All orders contribution coming from low-energy region � Q � Q d k k p − 1 α s ( Q ) d k k p − 1 α s ( k ) = ⇒ 0 0 � �� � � �� � NLO=V r +R all orders � k � Q � Q ∞ � �� n � d k k p − 1 α s ( k ) = α s ( Q ) d k k p − 1 − 2 b 0 α s ( Q ) log Q 0 0 n =0 ∞ � � n n ! = Q p × α s ( Q ) � 2 b 0 p α s ( Q ) n =0 Silvia Ferrario Ravasio — December 14 th , 2018 7/28 Renormalons effects in top-mass measurements

IR Renormalons QCD is affected by infrared slavery All orders contribution coming from low-energy region � Q � Q d k k p − 1 α s ( Q ) d k k p − 1 α s ( k ) = ⇒ 0 0 � �� � � �� � NLO=V r +R all orders � k � Q � Q ∞ � �� n � d k k p − 1 α s ( k ) = α s ( Q ) d k k p − 1 − 2 b 0 α s ( Q ) log Q 0 0 n =0 ∞ � � n n ! = Q p × α s ( Q ) � 2 b 0 p α s ( Q ) n =0 Asymptotic series 1 ⇒ Minimum for n min ≈ 2 b 0 pα s ( Q ) ⇒ Size Q p × α s ( Q ) √ 2 πn min e − n min ≈ Λ p QCD We are interested in p = 1, i.e. in linear renormalons Silvia Ferrario Ravasio — December 14 th , 2018 7/28 Renormalons effects in top-mass measurements

Perturbation theory in QFT 1 Dyson, 1952 : perturbative series cannot converge in QFT ∞ � F = +( − ) α O i α i , QED: O = r 2 same (opposite) charge i = i min If we require the series to be convergent around 0, we need a convergence radius R such as the series converges ∀ α ∈ C : | α | < R , including negative values! “This instability means that electrodynamics with negative α , cannot be described by well-defined analytic functions; hence the perturbation series of electrodynamics must have zero radius of convergence.”, Adler, 1972 Silvia Ferrario Ravasio — December 14 th , 2018 8/28 Renormalons effects in top-mass measurements

Perturbation theory in QFT 1 Dyson, 1952 : perturbative series cannot converge in QFT 2 t’Hooft, 1984 : “The only difficulty is that these expansions will at best be asymptotic expansions only; there is no reason to expect a finite radius of convergence”. 3 Altarelli, 1995 : It has been known for a long time that the perturbation expansions in QED and QCD, after renormalization, are not convergent series. Silvia Ferrario Ravasio — December 14 th , 2018 8/28 Renormalons effects in top-mass measurements

Perturbation theory in QFT 1 Dyson, 1952 : perturbative series cannot converge in QFT 2 t’Hooft, 1984 : “The only difficulty is that these expansions will at best be asymptotic expansions only; there is no reason to expect a finite radius of convergence”. 3 Altarelli, 1995 : It has been known for a long time that the perturbation expansions in QED and QCD, after renormalization, are not convergent series. 4 Abel, 1828 Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever. 5 Carrier’s Rule : “Divergent series converge faster than convergent series because they don’t have to converge”, (i.e. divergent asymptotic series often yield good approximations if the first few terms are taken even when the expansion parameter is of order one, while in the case of a convergent series many terms are needed to get a good approximation). Silvia Ferrario Ravasio — December 14 th , 2018 8/28 Renormalons effects in top-mass measurements

Divergent series ∞ n max � � O n α n = ∞ O n α n O ∼ n =0 n =0 O n α n Stay away ! n 1 /α Silvia Ferrario Ravasio — December 14 th , 2018 9/28 Renormalons effects in top-mass measurements