NLO QCD for the Practitioner Barbara J¨ ager Institute for Theoretical Physics University of Karlsruhe
Plan ✘ Vector Boson Fusion: Basics, Details & Examples · basics: LHC physics and VBF · details: methods, implementation, . . . and all the dirty tricks · phenomenological applications ✘ The Dipole Subtraction Method – An Introduction · sketching the method · details & formulae · basic examples: e + e − → 2 jets qq → qqH via VBF Barbara J¨ ager @ KEK, October 2006, p. 0/1
Schedule plan: ✘ Basic Concepts of an NLO QCD Calculation ✘ The Dipole Subtraction Method – An Introduction ✘ Vector Boson Fusion: Basics, Details & Examples adapted to your needs Barbara J¨ ager @ KEK, October 2006, p. 0/2
Outline of a perturbative calculation � |M| 2 σ ab → ... ∼ ab → cd... F J ( P f ) dP S f d ˆ ✘ calculation of |M| 2 at LO and NLO (in α s or α ) - regularization - renormalization ✘ handling of infrared singularities ✘ phase space integration and convolution with PDFs I. NLO QCD for the Practitioner / p. 1 Barbara J¨ ager @ KEK, October 2006
Settle the stage: the leading order let’s focus on pp → jje + ν e µ − ¯ ν µ (short: “ pp → jjW + W − ”) need to compute numerical value for 2 |M B | 2 = + . . . + at each generated phase space point in 4 dim (finite) . . . altogether 92 diagrams for CC, 181 diagrams for NC processes in principle two approaches for computing matrix elements squared: – trace techniques – amplitude techniques I. NLO QCD for the Practitioner / p. 2 Barbara J¨ ager @ KEK, October 2006
Evaluation of Feynman Diagrams amplitude techniques: evaluate M first (numerically) for specific helicities of external particles, then square it: |M| 2 = ( M 1 + M 2 + M 3 + . . . ) · ( M 1 + M 2 + M 3 + . . . ) ⋆ – reduced number of terms → complexity ∼ # graphs – fast numerical programs and many implementations approach proposed by Hagiwara, Zeppenfeld (1986,1989) : · implemented in HELAS (Murayama et al., 1992) · employed by MadGraph (Stelzer et al., 1994; and updates) I. NLO QCD for the Practitioner / p. 3 Barbara J¨ ager @ KEK, October 2006
Amplitude Techniques basic approach of HELAS/MadGraph : – each phase space point → numerical values of external 4-momenta p µ i , k µ i – polarization vectors ε µ ( k, λ ) and spinors u ( p, σ ) ≃ complex 4-arrays – products like 1 / ε k − mu ( p, λ ) / p − / of momenta, polarization vectors, spinors, and γ µ -matrices are computed via numerical 4 × 4 matrix multiplication ☞ perfect for LO amplitudes (results completely finite) I. NLO QCD for the Practitioner / p. 4 Barbara J¨ ager @ KEK, October 2006
Some Complications at NLO obvious: meaningful observables theoretical prediction: finite result but: how is finite result obtained in practice? generally: perturbative calculation beyond LO → singularities encountered in intermediate steps even though they will eventually cancel, divergencies need to be treated properly throughout! I. NLO QCD for the Practitioner / p. 5 Barbara J¨ ager @ KEK, October 2006
Regularization ☞ regularization needed to manifest singularities in intermediate steps of a calculation different prescriptions on the market for a nice review see, e.g., T. Muta, “Foundations of Quantum Chromodynamics” (1986) ✘ momentum cut-off: UV and / or IR divergent loop integrals � ∞ � Λ ∞ d 4 q d 4 q 1 1 ( q 2 ) n → (2 π ) 4 (2 π ) 4 ( q 2 ) n 0 Λ 0 simple to implement but: violates translation and gauge invariance I. NLO QCD for the Practitioner / p. 6 Barbara J¨ ager @ KEK, October 2006
Regularization ✘ mass regularization: introduce auxiliary mass m for massless gauge bosons 1 1 e . g ., photon : propagator q 2 + iδ → q 2 − m 2 + iδ · calculation more complicated due to additional mass scale · problems with gauge invariance in Non-Abelian case (QCD) · frequently used for EW calculations ✘ many other schemes (Pauli Villars, analytical regularization, lattice regularization, . . . ) · often problematic if Lorentz / gauge symmetries are to be preserved · may be useful for specific applications I. NLO QCD for the Practitioner / p. 7 Barbara J¨ ager @ KEK, October 2006
Regularization ✘ dimensional regularization: dimension of space-time d = 4 → d = 4 − 2 ε � ∞ � ∞ d 4 q d d q 1 1 ( q 2 ) n → (2 π ) 4 (2 π ) d ( q 2 ) n 0 0 ε > 0 . . . UV regulator, ε < 0 . . . IR regulator divergencies → poles in ε · preserves Lorentz and gauge invariance · problem: have to perform Dirac algebra in d dimensions; ε µνρσ and γ 5 a priori undefined in d � = 4 still: THE method of choice in QCD I. NLO QCD for the Practitioner / p. 8 Barbara J¨ ager @ KEK, October 2006
Dimensional Regularization different (but finally equivalent) implementations: · “genuine” dimensional regularization: polarization vectors/spinors of external particles and internal loop momenta d -dimensional · dimensional reduction: polarization vectors/spinors of external particles 4-dimensional, internal loop momenta d -dimensional well-defined transformation rules between different schemes our method of choice: dimensional reduction I. NLO QCD for the Practitioner / p. 9 Barbara J¨ ager @ KEK, October 2006
Dimensional Regularization: An Example let’s calculate the quark selfenergy in d dim ( MS scheme): k, a Σ b il ( p ) = (un-renormalized) p, i p, l ( p − k ) , j compute color factor � a,j T a ij T a jl = C F δ il and 4 π µ 2 � ε g 2 � e γ replace coupling by dimensional one g 2 s → s � d d k k ) γ µ γ µ ( / p − / s µ 2 ε C F δ il Σ b il ( p ) = − g 2 pC F δ il Σ b ( p 2 ) = − i/ (2 π ) d k 2 ( k − p ) 2 I. NLO QCD for the Practitioner / p. 10 Barbara J¨ ager @ KEK, October 2006
Quark Selfenergy for evaluation of Σ b we need scalar integral � − ε � − p 2 � � � d d k B 0 = 1 1 1 2 + 1 ˜ k 2 ( k − p ) 2 = Γ(1+ ε ) (2 π ) d i 16 π 2 4 π ε and find after some algebra (details on computation of loop integrals: see below) � µ 2 � ε � � Σ b ( p 2 ) = − α s 1 + 1 − p 2 4 π ε UV pole! remove by renormalization I. NLO QCD for the Practitioner / p. 11 Barbara J¨ ager @ KEK, October 2006
Quark Selfenergy ☞ renormalized selfenergy for off-shell quarks: �� µ 2 � � ε � � Σ( p 2 � = 0) = − α s 1 + 1 − 1 − p 2 4 π ε ε � µ 2 � � � = − α s 1 + ln + O ( ε ) − p 2 4 π note: · result finite as ε → 0 · introduced arbitrary mass scale µ for on-shell quarks: subtraction different ☞ I. NLO QCD for the Practitioner / p. 12 Barbara J¨ ager @ KEK, October 2006
Quark Selfenergy ☞ renormalized selfenergy for on-shell quarks: �� µ 2 � � ε � � Σ( p 2 ) = − α s 1 + 1 − 1 . − p 2 4 π ε 2 ε need to replace ε → − ε and find �� − p 2 � � ε � � Σ( p 2 ) = − α s 1 − 1 + 1 µ 2 4 π ε 2 ε now the quark can safely be put onto the mass-shell ( p 2 = 0 ): Σ( p 2 = 0) = − α s 1 2 ε . 4 π note: UV pole transformed into IR pole (sign of ε changed) I. NLO QCD for the Practitioner / p. 13 Barbara J¨ ager @ KEK, October 2006
Cancelation of divergencies at NLO collinear singularities UV divergencies � � factorization renormalization of at scale µ f α s at scale µ r soft singularities sum of all real and � virtual contributions to cancel in sum of well-defined virtual and real observable: emission contributions finite I. NLO QCD for the Practitioner / p. 14 Barbara J¨ ager @ KEK, October 2006
Cancelation of divergencies at NLO intermediate collinear singularities steps: regularize � all divergencies by factorization d → 4 − 2 ε at scale µ f soft singularities sum of all real and � virtual contributions to cancel in sum of well-defined virtual and real observable: emission contributions finite for ε → 0 I. NLO QCD for the Practitioner / p. 15 Barbara J¨ ager @ KEK, October 2006
Cancelation of divergencies at NLO cancelation of ε poles can be performed explicitly in collinear singularities analytical calculation, but � how can divergencies be factorization handled in numerical at scale µ f calculation? soft singularities sum of all real and � virtual contributions to cancel in sum of well-defined virtual and real observable: emission contributions finite for ε → 0 I. NLO QCD for the Practitioner / p. 16 Barbara J¨ ager @ KEK, October 2006
Cancelation of divergencies at NLO typical NLO QCD calculation up to 1990ies: · compute |M NLO | 2 (i.e. |M R | 2 and 2 M V M ⋆ B ) analytically in d dimensions (by hand or with the help of algebraic computer programs like tracer , FeynCalc , Form , etc.) · perform phase-space integration for m + 1 and m final state particles (for |M R | 2 and 2 M V M ⋆ B ) analytically in d dim (considering polarization, cuts, etc.) σ V and d ˆ σ R explicitly · cancel matching poles in d ˆ · perform factorization of remaining collinear singularities ana- lytically, set ε → 0 , and convolute d ˆ σ with PDFs (and/or FFs) numerically in 4 dimensions I. NLO QCD for the Practitioner / p. 17 Barbara J¨ ager @ KEK, October 2006
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