Four-fermion production near the W pair production threshold with - PowerPoint PPT Presentation

Four-fermion production near the W pair production threshold with unstable particle effective field theory M. Beneke (RWTH Aachen) Outline • Introduction • Unstable particle EFT and the Born cross section • Radiative corrections • Result, δ M W • Dominant NNLO corrections and cuts MB, Chapovsky, Signer, Zanderighi, PRL93:011602,2004; NPB686:205-247,2004 (EFT formalism) MB, Kauer, Signer, Zanderighi, Nucl.Phys.Proc.Suppl.152:162-167,2006 (WW) MB, Falgari, Schwinn, Signer, Zanderighi, NPB792:89, 2008 (0707.0773 [hep-ph]) (NLO WW threshold) Actis, MB, Falgari, Schwinn, 0807.0102 [hep-ph] (WW threshold beyond NLO) M. Beneke (RWTH Aachen) PSI, 18 September 2008 1 / 31

Motivation “Fundamental question in QFT” – Perturbation expansions do not work for the production of resonances (“unstable particles”) even for weak coupling, because M 2 / ( s − M 2 ) ∼ M 2 / ( M Γ) ∼ 1 / g 2 Systematic expansion? The electroweak gauge bosons W , Z , the top quark and, 20 20 20 perhaps, the Higgs boson (if m H ≥ 2 m W ) decay rapidly ( τ < 10 − 25 s ) such that Gentle YFSWW3 σ WW [ pb ] RacoonWW 15 15 15 mass ≡ Γ width M ∼ O ( α EW ) ≪ 1 but non-negligible. 10 10 10 18 17 5 5 5 “Electroweak precision tests” – Measurements of M W 16 and m t determine M H or M new through virtual effects. 15 Accurate W mass from 0 0 0 160 160 160 170 170 170 180 180 180 190 190 190 200 200 200 E cm [ GeV ] pp → WX → ℓν X e − e + → W + W − X → µ − ¯ ν µ u ¯ d X M. Beneke (RWTH Aachen) PSI, 18 September 2008 2 / 31

What’s the problem? “Kinematical” breakdown of perturbation theory. Propagators become singular near resonance g 2 p 2 − M 2 ∼ M Γ ∼ ( gM ) 2 p 2 − M 2 + i ǫ ∼ 1 when Process involves two very different scales: short-distance production (1 / √ s , 1 / M ) and the lifetime 1 / Γ ≫ 1 / M (unless the contour can be deformed away from the singularity). “Dyson resummation” of self-energy insertions 1 1 s − M 2 → s − M 2 − Π( s ) regularizes the singularity, since Π( M 2 ) ≈ δ M 2 − iM Γ , but: gauge-dependence of Π( s ) and the propagator of a gauge boson resonance. Need a systematic approximation in g 2 and Γ / M to the scattering amplitude/cross section. Note: unstable particles have no asymptotic states and their lines are never cut in Cutkosky’s rules ( Veltman, 1963 ). Theory is unitary in the Hilbert space of asymptotic states. “On-shell” production of unstable particles corresponds to the leading-order approximation M Γ Γ → 0 → πδ ( p 2 − M 2 ) ( p 2 − M 2 ) + M 2 Γ 2 M. Beneke (RWTH Aachen) PSI, 18 September 2008 3 / 31

Methods/approaches Mainly to deal with gauge invariance. Often more pragamtic than systematic. “Fermion-loop scheme” ( Argyres et al., 1995 ) “Pinch technique” ( Papavassilou et al., 1994 ) “(Double) Pole approximation” ( Stuart, 1991; Aeppli, van Oldenborgh, Wyler, 1994 ) Expansion of scattering amplitude around the complex pole of the resonance(s). Exploits Γ ≪ M . Diagrammatic, never beyond NLO. Breaks down for pair production near threshold (?) “Complex mass scheme” ( Denner, Dittmaier, Roth, Wackeroth, 1999 ) Standard perturbative calculation with complex mass counterterms, so p 2 − M 2 is never zero. With M Z , M W and G F as inputs for the renormalized electroweak parameters → sin θ W and coupling constants become complex (essential for Ward identities to hold). Complete NLO calculation of e − e + → 4 f has been performed ( Denner, Ditt- maier, Roth, Wieders, 2005 ) in the complex mass scheme. Rather challenging – first 1-loop calculation of a 2 → 4 process. M. Beneke (RWTH Aachen) PSI, 18 September 2008 4 / 31

Matching kinematic regions Consider line-shape A + B → resonance → X 0.01 δ ≡ s − M 2 combined effective theory m 2 full theory Off resonance, δ ∼ 1, conventional perturbation 0.0001 theory applies σ = g 4 f 1 ( δ ) + g 6 f 2 ( δ ) + . . . 1e-06 50 100 150 Near resonance, δ ≪ 1, expand in δ and reorganize „ g 2 n « X × { 1 ( LO ); g 2 , δ ( NLO ) , . . . } = h 1 ( g 2 /δ ) + g 2 h 2 ( g 2 /δ ) + . . . σ ∼ δ n The two approximations can be matched in an intermediate region, where δ and g 2 /δ are small. In the following we concentrate on the resonance region (threshold for pair production). M. Beneke (RWTH Aachen) PSI, 18 September 2008 5 / 31

Inclusive e − e + → 4 f Consider e − e + → µ − ¯ ν µ u ¯ d X near threshold. Dominated by nearly on-shell W − W + . Large sensitivity to M W . ILC with GIGAZ option: δ M W ≈ 6 MeV experimentally (Wilson, 2001). Rule of thumb: δσ ≈ 1 % ⇔ δ M W ≈ 15 MeV. Calculate totally inclusive final state, except for flavour quantum numbers. Extract cross section from the forward-scattering amplitude σ = 1 s Im A ( e − e + → e − e + ) | µ − ¯ ˆ ν µ u ¯ d Perform a “QCD-style” calculation of the short-distance cross section with massless electrons in the MS scheme, then Z 1 σ ( s ) = dx 1 dx 2 f e / e ( x 1 ) f e / e ( x 2 ) ˆ σ ( x 1 x 2 s ) . 0 MS electron distribution function depends on m e , but not on √ s , M , Γ . M. Beneke (RWTH Aachen) PSI, 18 September 2008 6 / 31

Scales, parameters, power counting – WW and t ¯ t WW pair production near threshold is dominated by electroweak interactions (in leading orders), top pair production by the strong interaction. t ¯ WW t δ 2 α ew δ (def.) δ 2 α em δ √ α s δ δ (def.) Γ / M δ δ 2 v 2 ≡ ( √ s − [ 2 M + i Γ]) / M δ 2 δ √ g 2 / v (Coulomb) δ 1 Both require non-relativistic + unstable particle EFT, but for top the former is more essential, while for W unstable particle effects are more important, and the Coulomb interaction does not have to be summed. √ δ : LO, N 1 / 2 LO, NLO, ... Expansion runs in M. Beneke (RWTH Aachen) PSI, 18 September 2008 7 / 31

Unstable particle EFT (I) For simplicity, consider SM with α s = 0. Integrate out short-distance fluctuations, such that only virtualities k 2 ≪ M 2 W are left. What are the fields and interactions in the EFT? Fields No top, Z , Higgs. Two non-relativistic spin-1 fields Ω i ∓ . Photon and light fermion fields (soft and collinear). Interactions The nearly on-shell W bosons can interact with soft photons ( k ∼ Γ W ) or potential k ∼ √ M W Γ W ) (Coulomb interaction) photons ( k 0 ∼ Γ , � Soft and collinear ( k 0 ∼ M W , k 2 ≪ M 2 W ) interactions with the high-energy, initial state electron (positron). The production of the W bosons is short-distance and must be incorporated into the EFT by local operators (more precisely, local modulo collinear Wilson lines). M. Beneke (RWTH Aachen) PSI, 18 September 2008 8 / 31

Unstable particle EFT (II) Matching of the leading production operator W W W e e e ν i W W γ Z W e e e = πα ew O ( 0 ) “ ” “ ” e c 2 , L γ [ i n j ] e c 1 , L Ω † i − Ω † j ¯ p + M 2 W At LO in the expansion around threshold, only the t -channel diagram contributes. General formula for the forward-scattering amplitude including non-resonant produc- tion Z d 4 x � e + e − | T ( i O ( k ) p ( 0 ) i O ( l ) � e + e − | i O ( k ) X p ( x )) | e + e − � + X 4 e ( 0 ) | e + e − � . i A = k , l k The local four-electron operator includes off-shell WW or single W intermediate states. M. Beneke (RWTH Aachen) PSI, 18 September 2008 9 / 31

Unstable particle EFT (III) ∂ 2 − M W ∆) 2 " � ! ( � # ∂ 2 − ∆ Ω † i ∓ + Ω † i X iD 0 Ω i Ω i L eff = s + ∓ ∓ ∓ 8 M 3 2 M W 2 W ∓ − α QED Z h i “ ” h i Ω † i Ω † j + Ω j d 3 r − Ω i + − ( x + � r ) ( x ) + . . . + r ∆ is a short-distance coefficient determined by matching the W two-point function. Let s ≡ M 2 s − ˆ M 2 W − Π W ¯ W − iM W Γ W be the complex pole position, ¯ T (¯ s ) = 0. Then s − ˆ M 2 ∆ ≡ ¯ pole scheme W = − i Γ W ˆ M W For the massless fields obtain terms familiar from the soft-collinear effective theory (SCET). Not much of this is needed explicitly at NLO. Propagator i δ ij “ 2 M W − ∆ ( 1 ) � ” k 0 − k 2 2 This accomplishes the reorganisation of PT, since ∆ ( 1 ) = − Π W ( 1 ) ( ˆ W ) / ˆ M 2 M W ∼ g 2 M W . T M. Beneke (RWTH Aachen) PSI, 18 September 2008 10 / 31

Unstable particle EFT (IV) Gauge invariance is automatic, since The full electroweak SM is SU(2) × U(1) Y gauge-invariant The effective Lagrangian is U(1) em gauge-invariant. The matching equations are formulated as “on-shell” equations at the complex pole of the W propagator (including a complex LSZ residue factor) which are (SU(2) × U(1) Y ) gauge-independent, e.g. √ √ R W ] 2 A ( e − e + → W − W + ) | p 2 s = C i [ ̟ − 1 / 2 ] 2 A i ( e − e + → W − W + ) eff , tree , R e ] 2 [ [ W =¯ where 1 / 2 1 + M W ∆ + � ! k 2 ̟ − 1 ≡ i M 2 W is a field normalization factor for non-relativistic fields (usually E / M ). M. Beneke (RWTH Aachen) PSI, 18 September 2008 11 / 31