Narrow-width approximation accuracy Nikolas Kauer in collaboration with D. Berdine, D. Rainwater and C.F . Uhlemann funded by Particle Theory Seminar Paul Scherrer Institut May 29, 2008 1 / 29
Outline • Introduction • Accuracy • Limitations • Improvements • Conclusions 2 / 29
Search for the Higgs boson and BSM physics L W + L W + ◮ Unitarity-violating processes, e.g. W − L → W − L ◮ Consistent mass generation via SSB → Higgs mechanism ◮ SUSY, composite Higgs, extra dimensions, ... LHC at CERN: a powerful Terascale collider ◮ pp collisions at E CMS = 14 TeV ◮ Luminosity L = 10–100 fb − 1 /year Introduction 3 / 29
BSM physics via supersymmetry? SUSY: invariance under boson ↔ fermion transformation Particle spectrum of the Minimal Supersymmetric Standard Model (MSSM): fermions S = 1 Higgs S = 0 gauge bosons S = 1 2 ` u L ´ ` ν eL ´ ` H 0 ´ ` H + ´ , gluon, W ± , Z , γ d , d L e L u H 0 H − u d u R , d R , e R sfermions S = 0 Higgsinos S = 1 gauginos S = 1 ` e ´ ` e ´ 2 u L 2 ν eL , ` e ´ ` e ´ H 0 e H + d L e e L gluino, f W ± , e d , u Z , e γ e H 0 e H − u R , e u e d R , e d e R M X � = M e X ⇒ SUSY broken Introduction 4 / 29
Unstable particles and QFT “Ordinary” QFT: unstable particle fields → asymptotic states ( t = ±∞ ) Eliminate these states → Hilbert space of stable particle states Theory still unitary, causal and renormalizable Veltman (1963) [No discussion of gauge invariance!] Introduction 5 / 29
Unstable particles and perturbation theory LO propagators for massive gauge bosons and fermions: “ ” W − i g µν − p µ p ν i ( p / + M ) = , t t t t + t t + p 2 − M 2 p 2 − M 2 M 2 b W W Fixed-order propagator: real pole at p 2 = M 2 t t + : : : b b „ « n ∞ X i i i p 2 ≈ M 2 t : Im Π t ≈ − M t Γ t ( − i Π t ) = / − M t p / − M t p / − M t − Π t p n =0 Breit-Wigner propagator: [analytic continuation to resonant region] ! 2 i ( p / + M t ) g 2 1 − M 2 Im Π t = − 1 W (2 M 2 W + p 2 ) p p 2 − M 2 /P L t + iM t Γ t M 2 p 2 64 π W Dyson-resummed propagator: complex pole at p 2 = M 2 − iM Γ [gauge invariance?] ◮ Physical fixed-order amplitudes with unstable particles include contributions from all orders in perturbation theory Introduction 6 / 29
g g g � � b b g � e � � e t � W � � � e t � � � e W t t + � + + + t � t W W � � � � g g b b g ( b ) ( a ) g g � � b b g g g � b � b g � e � � e t � W Generic resonant enhancement g � � e t � � b � � � e W e t t � � � � + � e W e � b W + + + t t W � W + � � + � e t � � W + � W + � � b � � g g g t b b b � � b g b ( a ) ( b ) ( c ) ( d ) g g � b � � � b g b g b g g � � � b e e e � � g g W W � � � � � � � � � e W e e e b W b b + + + � � � e + � ; Z � Σ � t + H W + + W + � W W b � � � � � � t p a g b p 1 � � b g g b b g b p 2 p R ( e ) ( f ) ( c ) ( d ) p 3 � � b b g g p b p 5 p 4 � � ������ � e e 2 p R � � g g W W � � � � e e b b Z ∞ + + dp 2 ( p 2 − M 2 ) 2 + ( M Γ) 2 ∼ M � ; Z � � H M 2 + + W Generic resonant enhancement: W � � � � Γ −∞ g g b b Γ /M � 1% → nonresonant contribution typically negligible ( e ) ( f ) Theoretically consistent method to extract dominant resonant contribution? Introduction 7 / 29
Resonant production and decay factorization ◮ Off-shell MC generators with complete |M| 2 at tree level � use WHIZARD, MADEVENT, SHERPA, ... (but: often requires too much CPU time) ◮ Precise LHC/ILC predictions: need N n LO calculations, n ≥ 1 ◮ (S)particle decay chains → many-particle final states ◮ Already 4,5,6-leg one-/two-loop calculations tech. very demanding Narrow-width approximation (NWA) to the rescue Sub- and nonresonant/nonfactorizable contributions can be neglected in a theoretically consistent way (gauge inv., ...) 1 π M Γ · δ ( p 2 − M 2 ) D ( p 2 ) ≡ Γ → 0 : ∼ on-shell ( p 2 − M 2 ) 2 + ( M Γ) 2 Scales occurring in D ( p 2 ) → error estimate O (Γ /M ) Branching ratio measurement implies NWA (and spin averaging): BR X ≡ Γ X = σ NWA σ X ≈ P Γ σ p X σ X Introduction 8 / 29
NWA accuracy: what to compare to? Tree-level finite-width schemes [safe → agreement up to O ((Γ /M ) 2 ) ] Running-width scheme (Dyson-resummed propagator) 1 / ( p 2 − M 2 ) → 1 / ( p 2 − M 2 + ip 2 Γ /M ) not gauge invariant, typically unsafe Fixed-width scheme (FWS) 1 / ( p 2 − M 2 ) → 1 / ( p 2 − M 2 + iM Γ) not gauge invariant, typically safe Overall-factor scheme M Γ=0 · ( p 2 − M 2 ) / ( p 2 − M 2 + iM Γ) gauge invariant, unsafe for complex resonance structure Baur, Vermaseren, Zeppenfeld (1992) Complex-mass scheme √ M 2 − iM Γ M → ⇒ complex cos θ W , y t gauge invariant, no known practical problems Lopez Castro, Lucio, Pestieau (1991); Denner, Dittmaier, Roth, Wackeroth (1999) Accuracy 9 / 29
Complex-mass scheme at one-loop level Denner, Dittmaier, Roth, Wieders (2005) For each unstable particle: → complex renormalized mass real bare mass + complex counterterm | {z } | {z } → free propagator (resummed) → CT vertex (not resummed) ◮ Gauge invariance relations fulfilled order by order ◮ No double counting ◮ 1-loop integrals with complex internal masses required ◮ Potential unitarity violations are of higher order Applications: radiative corrections to ◮ e + e − ( → W + W − ) → 4 fermions Denner, Dittmaier, Roth, Wieders (2005) ◮ H → W + W − /ZZ → 4 fermions Bredenstein, Denner, Dittmaier, M. Weber (2006-7) q ) 2 → ( q/ ¯ ◮ ( q/ ¯ q ) 2 H Ciccolini, Denner, Dittmaier (2008) Accuracy 10 / 29
Unstable-particle effective field theory Chapovsky, Vy. Khoze, Signer, Stirling (2002) Beneke, Chapovsky, Signer, Zanderighi (2004) Γ ≪ M → hierarchy of scales → Expand cross section in powers of α and δ ≡ ( p 2 − M 2 ) /M 2 ∼ Γ /M 1. Integrate out hard momenta k ∼ M in underlying theory: underlying theory → effective theory with short-distance matching coefficients 2. Identify remaining dynamical modes (corresponding to momentum configurations near mass shell) → field operators → effective Lagrangian 3. Match coefficients (up to a certain order in α and δ ) to underlying theory by calculating and comparing on-shell n -point functions ◮ Gauge invariance, resummation of self-energy insertions � ◮ Systematic extension to higher orders � , no double counting ◮ Well suited for inclusive calculations close to threshold ◮ Not well suited for fully differential calculations ◮ Underlying theory results with “ Γ → 0 ” required for matching Corrections to e + e − → u ¯ dµ − ¯ ν µ at WW threshold Beneke, Falgari, Schwinn, Signer, Zanderighi (2008) Accuracy 11 / 29
Limitations of the O (Γ /M ) uncertainty estimate Relative deviation R ≡ σ FWS /σ NWA − 1 = R (1) + O (Γ 2 ) for scalar process: m d m p m d m p M, Γ M M ( " R (1) = M ( s − m 2 d ) 2 ln − 1 + [ πM ( m 2 d − M 2 )( s − M 2 )( s + m 2 p )( s − M 2 + m 2 p )] m 2 d s ( s − M 2 + m 2 p ) s π ( m 2 d − M 2 )( s − M 2 ) m 2 d ) „ « “ ” 3 ` ´ s − m 2 d + m 2 M 2 − m 2 p · Γ + ( s + m 2 p ) m 2 s + m 2 − 2 M 2 s + M 4 − M 4 m 2 d + M 4 m 2 p ( s − m 2 d + m 2 p ) ln 5 s ln d p p s − M 2 m 2 p Note: additional scales ( √ s , particle masses in production or decay, . . . ) Not { . . . } ≈ 1 /M when √ s → M (production threshold � ) or m d → M Limitations 12 / 29
Scalar process 10 2 √ s m d m p M 1.1 M, Γ 1.01 0.01 0.1 1 1 − m d /M Contour lines (dashed, solid, dot-dashed) for R/ (Γ /M ) ∈ {− 10 , − 3 , − 1 , 0 , 1 , 3 , 10 } with Γ /M = 1% and m p = 0 . 02 M Dash length increases with magnitude Limitations 13 / 29
√ ˆ s Scalar process with distributed Constant PDFs � ˆ � s max s max ) ∝ 1 s ) · ln s � σ � σ [ NWA ] ( � ˆ d ˆ s σ [ NWA ] (ˆ R ≡ − 1 and s s ˆ � σ NWA 0 20 17.5 15 12.5 � R 10 Γ 7.5 M 5 2.5 0 0 0.5 1 1.5 2 2.5 3 � s max /M ˆ Cross sections � σ NWA (dotted) and � σ (dashed) and relative deviation � R/ (Γ /M ) for m d = 0 . 1 M (solid) [and m d = 0 . 001 M (dot-dashed)], √ s = 3 M , Γ = 0 . 02 M , m p = 0 . 01 M Limitations 14 / 29
20 15 10 � R 5 Γ M 0 � 5 � 10 1 1.5 2 2.5 3 � s max /M ˆ As before, but with m d = 0 . 9 M . Limitations 15 / 29
Nonscalar process 10 2 √ s m d m p M 1.1 M, Γ 1.01 0.01 0.1 1 1 − m d /M Contour lines (dashed, solid, dot-dashed) for R/ (Γ /M ) ∈ {− 10 , − 3 , − 1 , 0 , 1 , 3 , 10 , 30 , 100 , 300 } with Γ /M = 1% and m p = 0 . 02 M Dash length increases with magnitude Limitations 16 / 29
√ s ˆ Nonscalar process with distributed Constant PDFs 100 80 60 � R 40 Γ M 20 1 1.5 2 3 5 7 10 � � 20 s max /M ˆ � R/ (Γ /M ) for m d = 0 . 95 M (solid) and m d = 0 . 9 M (dashed), √ s = 10 M , Γ = 0 . 02 M , m p = 0 . 01 M Limitations 17 / 29
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