Electroweak corrections in the determination of α s Stefan Dittmaier Albert-Ludwigs-Universit¨ at Freiburg Based on: A.Denner, S.Dittmaier, T.Gehrmann, C.Kurz, Phys.Lett. B679 (2009) 219 [arXiv:0906.0372] A.Denner, S.Dittmaier, T.Gehrmann, C.Kurz, Nucl.Phys. B836 (2010) 37 [arXiv:1003.0986] S.Dittmaier, A.Huss, C.Speckner, JHEP 1211 (2012) 095 [arXiv:1210.0438] S.Dittmaier, A.Huss, K.Rabbertz, to appear in the Les Houches Proceedings soon Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 1
Features of and issues in EW precision calculations Relevance and size of EW corrections generic size O ( α ) ∼ O ( α 2 s ) suggests NLO EW ∼ NNLO QCD but systematic enhancements possible, e.g. • by photon emission ֒ → kinematical effects, mass-singular log’s ∝ α ln( m µ /Q ) for bare muons, etc. • at high energies → EW Sudakov log’s ∝ ( α/s 2 W ) ln 2 ( M W /Q ) and subleading log’s ֒ EW corrections to PDFs at hadron colliders induced by factorization of collinear initial-state singularities, new: photon PDF Instability of W and Z bosons • realistic observables have to be defined via decay products (leptons, γ ’s, jets) • off-shell effects ∼ O (Γ /M ) ∼ O ( α ) are part of the NLO EW corrections Combining QCD and EW corrections in predictions • how to merge results from different calculations • reweighting procedures in MC’s Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 2
Issue of this talk • EW corrections to two process types important for α s determination: ⋄ jet event-shape observables at e + e − colliders ⋄ jet production at hadron colliders • review of the situation ⋄ EW corrections under control ? ⋄ future homework ? Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 3
Jet event-shape observables at e + e − colliders Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 4
Frequently used event-shape observables y P i | � p i · � n | • Thrust: T = max P i | � p i | � n • Normalized heavy jet mass: ρ = max { M 2 1 , M 2 2 } /s ( M i = inv. mass flowing into hemispheres H i defined by plane perpendicular to thrust axis) • Wide / total jet broadenings: P j ∈ H i | � p j × � n | B W = max { B 1 , B 2 } , B T = B 1 + B 2 , B i = 2 P k | � p k | • C parameter: 1 � p j ⊗ � p j X C = 3( λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 ) , { λ i } = eigenvalues of Θ = P i | � p i | | � p j | j • Jet transition variable: Y 3 = value of y cut at which the event turns from 3-jet to 2-jet type 2-jet configuration appears at an endpoint of d σ ( y ) Note: (e.g. at T → 1 ) d y ֒ → shapes of distributions sensitive to 3 and more jets, and thus to α s Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 5
Theory prediction for jet event shapes ( e + e − → n jets , n ≥ 3 ) 1 d σ ( y ) α s C QCD + α 2 s C QCD α 3 s C QCD = + LO NLO NNLO σ had d y | {z } | {z } R.K.Ellis, Ross, Terrano ’81; Kunszt ’81 Gehrmann-DeRidder, Gehrmann, Vermaseren, Gaemers, Oldham ’81 Glover, Heinrich ’07–’09; Weinzierl ’08,’09 Giele, Glover ’92; Catani, Seymour ’96 + NLL resummation + NLL/NNLO matching ( + NNLL resummation for T ) | {z } | {z } | {z } Catani, Turnock, Becher, Schwartz ’08 Gehrmann, Luisoni, Webber, Trentadue ’91,’93 Stenzel ’08 + non-perturbative hadronization effects | {z } Korchemsky, Sterman ’95; Dokshitzer, Webber ’95,’97 Dokshitzer, Lucenti, Marchesini, Salam ’98 + α C EW + αα s C EW NLO + α 2 α s C ISR LO LL | {z } Denner, S.D., Gehrmann, Kurz ’09,’10 • Recent NNLO QCD results already included in α s fit to event shapes Gehrmann, Luisoni, Stenzel ’08; Dissertori et al. ’08; Bethke et al. ’08; Davison, Webber ’08 • NLO EW corrections potentially of same size as NNLO QCD, since O ( α ) ∼ O ( α 2 s ) Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 6
Calculation of NLO corrections LO diagrams for e + e − → q ¯ O ( α 2 α s ) q g : q q q e e g g q = u , d , s , c , b γ, Z γ, Z q e e q q LO diagrams for e + e − → q ¯ O ( α 3 ) qγ : q q e e γ q q q e e e γ γ e q q γ, Z γ, Z q γ, Z γ, Z e e e e q q γ Comments: • q ¯ qγ final states in LO deliver contributions if γ is merged with q/ ¯ q , i.e. near 2-jet configurations • focus of our calculation: O ( α 3 α s ) = NLO EW correction to q ¯ q g production = NLO QCD correction to q ¯ qγ production Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 7
1PI loop insertions in EW one-loop corrections to e + e − → q ¯ q g O (200) diagrams q q q q e e e q q q g g g q γ, Z γ, Z γ, Z γ, Z e e e q q q q q e e q q q e g g g γ, Z γ, Z e e e q q q q q q e e e γ, Z g g q γ, Z γ, Z e e q e q g Sample QCD one-loop diagrams for e + e − → q ¯ qγ q q q q q g e q g q q e q γ, Z γ, Z γ, Z q e e g g γ e q q q e q γ, Z γ q e e γ q q q e q γ Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 8
Real emission corrections at O ( α 3 α s ) and beyond • e + e − → q ¯ q g γ = photon bremsstrahlung to q ¯ q g production = gluon bremsstrahlung to q ¯ qγ production • QCD–EW interferences for e + e − → q ¯ qq ¯ q q γ/ Z e e q Z /γ g e q e e γ/ Z q → non-singular contributions of O ( α 3 α s ) = same order as NLO EW ֒ Interferences included in our calculation ֒ → effect phenomenologically negligible ( < 0 . 1% ) • higher-order photonic ISR included via leading-log structure functions up to order LO × O ( α 3 ) Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 9
Definition of jet observables Event selection: (closely following the procedure employed by ALEPH ) 1. Discard particles too close to the beams, i.e. if | cos θ i | > cos θ cut = 0 . 965 . 2. Reject event if M visible < 0 . 9 E CM . 3. Boost to CM system of observed final-state particles. 4. Apply Durham jet algorithm with E recombination and y cut = 0 . 002 to q, ¯ q, g , γ ֒ → photons appear inside jets 5. Reject “photonic events” where photon energy fraction z > z cut = 0 . 9 in a jet. Subtleties arising at NLO EW level: • Step 3 minimizes boost effects from collinear ISR photons (otherwise two-jet configurations do not always appear at event-shape endpoints) But: at LEP two-jet events were shifted to endpoints “by hand” ֒ → renders confrontation between theory and LEP results difficult • Step 5 is not collinear safe ֒ → perturbative result is plagued by quark-mass singularities ∝ α ln m q Solution: include photon fragmentation function with non-perturbative input Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 10
Photon–jet separation via photon fragmentation function D q → γ Glover, Morgan ’94 Why does a naive γ –jet separation by a jet algorithm not work ? • collinear quarks and photons have to be recombined → ( qγ ) = jet otherwise corrections ∝ ln( m 2 q /Q 2 ) → perturbative “IR instability” • quark and gluon jets cannot be distinguished event by event ֒ → common recombination required for quarks/gluons with photons ⇒ (g hard + γ soft ) and (g soft + γ hard ) both appear as 3 jets | {z } | {z } EW corr. to 3 jets QCD corr. to 2 jets + γ Solution: E γ • exclude events with photon energy fraction z γ = E jet + E γ > z 0 for (jet + γ ) quasiparticles • subtract convolution of LO cross section with m 2 " # ˛ D MS q q → γ ( z γ , µ fact ) mass . reg . = P q → γ ( z γ ) ln + 2 ln z γ + 1 ← cancels coll. singularities ˛ µ 2 ˛ fact ( z γ , µ fact ) ← non-perturbative part fitted to + D ALEPH ALEPH data on e + e − → jet + γ q → γ 1+(1 − zγ )2 where P q → γ ( z γ ) = = quark-to-photon splitting function zγ Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 11
Numerical results Total hadronic cross section Denner, S.D., Gehrmann, Kurz ’09,’10 1000 • largest EW corrections e + e − → q ¯ q ( γ ) 100 due to ISR 10 σ had [ nb] (radiative return cut off 1 by cut M visible < 0 . 9 √ s ) 0 . 1 0 . 01 • ISR beyond one loop Born weak O ( α ) 0 . 001 relevant (some % ) full O ( α ) for √ s ∼ M Z O ( α )+h.o. LL 0 . 0001 0 . 00001 1 • weak corrs. of O (5%) , 0 . 8 increasingly negative 0 . 6 for large √ s σ i − σ Born 0 . 4 σ Born 0 . 2 • Note: σ had calculated 0 to same perturbative order − 0 . 2 − 0 . 4 as d σ/ d y to obtain 1 10 100 1000 a proper normalization √ s [ GeV] Stefan Dittmaier, Electroweak corrections ... High precision fundamental constants at the TeV scale, MITP , March 2014 – 12
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