EFT for Jets with Massive Quarks André H. Hoang University of Vienna EFT ERC Workshop Mainz, November 10-13, 2014
Why complete mass dependence for jets? Aims : • Full quark mass dependence of jet observables. • Theory description for all kinematic regions ( “decoupling limit” ⇔ “massless limit” ) • Understanding factorization with quark masses • Account for initial state and final state jets Possible applications : • Quark mass effects in precision QCD analyses e.g. • Event shapes in e + e - (bottom effects for low Q data) • Top quark mass measurements in reconstruction • Understanding of Monte-Carlo top mass parameter • Role of massive quark vacuum polarization effects • Intrinsic charm ? This talk: I will mostly talk about final state jets. Show that SCET (+ extensions) is a good framework to address the problem of quark masses. EFT ERC Workshop Mainz, November 10-13, 2014
Outline • Motivation and Aims • Factorization for massless quarks • Effective theories including massive quark effects • Flavor number dependent renormalization • Rapidity logs • Running short-distance mass scheme • Conclusions & Outlook * In collaboration with: P. Pietrulewicz, V. Mateu, I. Jemos, S. Gritschacher B. Dehnadi, M. Butenschön arXiv:1302.4743 (PRD 88, 034021 (2013)) arXiv:1309.6251 (PRD 89, 014035 (2013)) arXiv:1405.4860 (PRD ..) More to come … EFT ERC Workshop Mainz, November 10-13, 2014
Thrust → consider: dijet in e + e - annihilation e.g. Thrust: Q ALEPH, DELPHI, L3, OPAL, SLD peak τ = 0 2 jets + soft radiation tail τ = 0 . 5 2 jets, 3 jets → Mass mode treatment of this talk applicable to any SCET-1-type observable → We use thrust to be definite and as a first important application. EFT ERC Workshop Mainz, November 10-13, 2014
Massless Quark Thrust in FO ⇣ ln( 1 d σ h 1 − 2 τ ) ⌘ i τ ( π 2 2 ) δ ( τ ) + − 3+9 τ +3 τ 2 − 9 τ 3 − 2 − 3 τ +3 τ 2 δ ( τ ) + C F α s 6 − 1 d τ = σ Born 2 τ (1 − τ ) (1 − τ ) π τ + tot h i ( π 2 τ ) + − 2( ln( τ ) 6 − 1 2 ) δ ( τ ) − 3 2 ( 1 δ ( τ ) + C F α s = ) + + { non-sing. terms } π τ singular terms Strongly dominate in kinematic regions where jets are produced EFT ERC Workshop Mainz, November 10-13, 2014
Singular vs. Non-singular EFT ERC Workshop Mainz, November 10-13, 2014
Massless Quark SCET Bauer, Fleming, Luke Bauer, Fleming, Pirjol, → consider: dijet in e + e - annihilation, all quarks are light (m q < Λ ) Stewart n µ = (1 , 0 , 0 , 1) n µ = (1 , 0 , 0 , − 1) ¯ p µ = p − n µ 2 + p + ¯ n µ 2 + p ⊥ p 2 = p − p + + p 2 ⊥ Korchemsky, Sterman EFT ERC Workshop Mainz, November 10-13, 2014
Factorization for Massless Quarks Schwartz Fleming, AH, Mantry, Stewart Bauer, Fleming, Lee, Sterman → evolution with n l light quark flavors observable-dependent → consistency conditions w.r. to profile functions different evolution choices → top-down evolution considered in the following � d � ⇥ sing ⇤ d ⇤ d ⇤ � U J ( Q ⇥ − ⇤ − ⇤ � , µ Q , µ s ) J T ( Q ⇤ � , µ j ) S T ( ⇤ − ∆ , µ s ) ∼ � 0 H ( Q, µ Q ) U H ( Q, µ Q , µ s ) d ⇥ part EFT ERC Workshop Mainz, November 10-13, 2014
Final State Jets with a Massive Quark Gritschacher, AH, → consider: dijet in e + e - annihilation, n l light quarks ⊕ one massive quark Jemos, Pietrulewicz → obvious: (n l +1)-evolution for µ ≳ m and (n l )-evolution for µ ≲ m → obvious: different EFT scenarios w.r. to mass vs. Q – J – S scales Aims: “profile functions” • Full mass dependence (little room for any strong hierarchies): decoupling, massless limit • Smooth connections between different EFTs • Determination of flavor matching for current-, n l + 1 jet- and soft-evolution m • Reconcile problem of SCET 2 -type rapidity divergences n l “Variable Flavor Number Scheme” (VFNS) → Deal with collinear and soft “mass modes” → Additional power counting parameter EFT ERC Workshop Mainz, November 10-13, 2014
VFNS for Hadron Collisions d σ ( e − p → e − + X ) Q 2 = − q 2 e.g. Deep Inelastic Scattering: dQ dx → consider all quarks as as light (m q < Λ ) → quark number operators with an anomalous dimension between proton states → DGLAP equations → Hadronic tensor: X Q W µ ν ( Q, x ) ∼ f a ( µ ) ⊗ w µ ν ( Q, x, µ ) partons a → µ-dependence with DGLAP equations for (light) parton distribution functions Λ α 2 d α s ( Q ) s ( Q ) m light β 0 = 11 − 2 d ln Q 2 = − β 0 + . . . 3 n light (4 π ) EFT ERC Workshop Mainz, November 10-13, 2014
VFNS for Hadron Collisions d σ ( e − p → e − + X ) e.g. Deep Inelastic Scattering: dQ dx → realistic case: massive quarks with Q > m > Λ (charm, bottom [top]) → Hadronic tensor: X f ( n l +1) W µ ν ( m, Q, x ) ∼ ( µ ) ⊗ w µ ν ( m, Q, x, µ ) a a = q,g,Q Q ACOT scheme: • DGLAP evolution for n l flavors for µ ≲ m (only light quarks) • DGLAP evolution for n l +1 flavors for µ ≳ m (light quarks + massive quark) • Flavor matching for α s and the pdfs at µ m ~ m m f ( n l +1) X F q,g,Q | a ( m, µ m ) ⊗ f ( n l ) q,g,Q ( µ m ) = ( µ m ) a a = q,g → hard coefficient w µ ν (m,Q,x) approaches massless w µ ν (Q,x) for m → 0 Λ → calculations of w µ ν (m,Q,x) involves subtraction of pdf IR mass singularities → full dependence on m/Q without any large logarithms m light EFT ERC Workshop Mainz, November 10-13, 2014
Fully Massive Thrust p p → fully massless • Full N 3 LL ’ (u.t. 4-loop cusp)+ 3-loop non-singular • Gap scheme for soft function Only SCET authors: Becher, Schwartz, Fleming, AH, Mantry, Stewart Bauer, Fleming, Lee, Sterman p ′ p ′ p p → secondary massive • Full N 2 LL ’ /N 3 LL m → New: In detail in this talk. m p ′ p ′ p p → primary massive • bHQET: full N 2 LL ’ /N 3 LL → valid for: Δ m jet << m • NLL ’ /N 2 LL for other cases m m Δ m p ′ p ′ p p → primary massive secondary massive m New: complete and systematic description m → Only briefly in this talk. m m p ′ p ′ EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Simplest non-trivial case to study: → massless primary quark dijet production in e + e - annihilation: n l light quarks ⊕ one massive quark arise only through secondary production → does not lead to bHQET-type theory when the jet scale approaches the quark mass → only SCET-type theories EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Simplest non-trivial case to study: → massless primary quark dijet production in e + e - annihilation: n l light quarks ⊕ one massive quark arise only through secondary production → field theory: close relation to the problem of massive gauge boson radiation → dispersion relation: massive quark results can be obtained directly from massive gluon calculations when quark pair treated inclusively (e.g. hard coefficient, jet function) → separation of conceptual issues to be resolved and calculations issues related to gluon splitting. → explicit two-loop calculation needed when quarks are treated exclusively (e.g. soft function → hemisphere prescription) Gritschacher, AH, Jemos, Pietrulewicz 2013 EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Scenario 1: λ m > 1 > λ > λ 2 ( m > Q > J > S ) • EFT only contains light quarks • Massive quark only in current matching coeff. • Decoupling for m/Q → ∞ EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks U (0) stands for: (a) massive gluon integrated out i (b) (n l )-evolution EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Scenario 2: 1> λ m > λ > λ 2 ( Q > m > J > S ) • Massive modes only virtual • Jet and soft function as in massless case • Hard coefficient must have massless limit • Known Sudakov problem for massive gauge boson Chiu, Golf, Kelley, Manohar Chiu, Führer, Hoang, Kelley EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Contains all mass-singularities U (0) stands for: (a) massive gluon integrated out i (b) (n l )-evolution U (1) stands for: (a) massive gluon dynamical i (b) (n l +1)-evolution EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Scenario 2: mass mode SCET calculation + soft-bin subtractions (collinear for k 2 =M 2 ) rapidity logarithms EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks Scenario 3: 1 > λ > λ m > λ 2 ( Q > J > m > S ) • Current evolution unchanged w.r. to Scen. 2 • Hard coefficient must have massless limit • Jet function has massless limit • Massive and massless collinear in same sector • Collinear mass modes integrated out at m EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks • Soft-bin subtraction • Rapidity singularities cancel • UV divergences agree with massless case • finite • sum of virtual and real: rapidity logs cancel • sum of virtual and real: approaches massless jet function for m → 0 EFT ERC Workshop Mainz, November 10-13, 2014
VFN Scheme: Secondary Massive Quarks EFT ERC Workshop Mainz, November 10-13, 2014
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