VBFNLO NLO QCD corrections for processes with electroweak bosons in the final state giuseppe bozzi universit` a di milano e infn sezione di milano in collaboration with: D. Zeppenfeld and the VBFNLO group • VBF processes • • NLO for triboson production • • phenomenology and results for LHC • • conclusions •
QCD corrections to VBF processes Precise predictions require QCD corrections qq → qqH Han, Valencia, Willenbrock; Figy, Oleari, Zeppenfeld; Campbell, Ellis, Berger • Higgs coupling measurements • qq → qqZ and qq → qqW Oleari, Zeppenfeld • Z → ττ as background for H → ττ • • measure central jet veto acceptance at LHC • qq → qqWW , qq → qqZZ , qq → qqWZ J¨ ager, Oleari, Bozzi, Zeppenfeld • qqWW is background to H → WW in VBF • • underlying process is weak boson scattering: WW → WW , WW → ZZ , WZ → WZ etc. • giuseppe bozzi VBFNLO 1
Generic features of QCD corrections to VBF t -channel color singlet exchange = ⇒ QCD corrections to different quark lines are independent real emission contributions: upper line Born and vertex corrections to upper line No t -channel gluon exchange at NLO Features are generic for all VBF processes giuseppe bozzi VBFNLO 2
Real emission Calculation is done using Catani-Seymour subtraction method Consider q ( p a ) Q → g ( p 1 ) q ( p 2 ) QH . Subtracted real emission term x 2 + z 2 p 1 · p 2 p 1 · p a C F |M emit | 2 − 8 πα s ( 1 − x )( 1 − z ) |M Born | 2 with 1 − x = , 1 − z = Q 2 ( p 1 + p 2 ) · p a ( p 1 + p 2 ) · p a is integrable = ⇒ do by Monte Carlo Integral of subtracted term over d 3 p 1 can be done analytically and gives � � ǫ � 2 � 4 πµ 2 ǫ 2 + 3 ǫ + 9 − 4 α s R Γ ( 1 + ǫ ) |M Born | 2 3 π 2 δ ( 1 − x ) 2 π C F Q 2 after factorization of splitting function terms (yielding additional “finite collinear terms”) The divergence must be canceled by virtual corrections for all VBF processes only variation: meaning of Born amplitude M Born giuseppe bozzi VBFNLO 3
Higgs production Most trivial case: Higgs production virtual amplitude proportional to Born Virtual correction is vertex correction only � � ǫ 4 πµ 2 α s ( µ R ) R M V = M Born Γ ( 1 + ǫ ) C F Q 2 4 π � � ǫ + π 2 − 2 ǫ 2 − 3 3 − 7 + O ( ǫ ) • • Divergent piece canceled via Catani Seymour algorithm Remaining virtual corrections are accounted for by trivial factor multiplying Born cross section � � C F |M Born | 2 1 + 2 α s 2 π c virt • • Factor 2 for corrections to upper and lower quark line • Same factor to Born cross section absorbs most of the virtual corrections for other VBF • processes
W and Z production ν ν W W l + l + u u d d γ ,Z γ ,Z c c c c (a) (b) • 10 · · · 24 Feynman graphs • u u d d • ⇒ use amplitude techniques, i.e. nu- • W W l + W merical evaluation of helicity ampli- c c l + ν W γ ,Z tudes c c ν (c) (d) • However: • numerical evaluation works in d=4 dimensions only u u d d W W l + ν l + ν Z γ ,Z c c c c (e) (f) giuseppe bozzi VBFNLO 5
Virtual contributions For each individual pure vertex graph Vertex corrections: same as for Higgs case M ( i ) the vertex correction is proportional to the corresponding Born graph � � ǫ 4 πµ 2 α s ( µ R ) V + + + . . . M ( i ) M ( i ) R = Γ ( 1 + ǫ ) C F V B Q 2 4 π � � V ǫ + π 2 − 2 ǫ 2 − 3 3 − 7 New: Box type graphs (plus gauge related diagrams) Vector boson propagators plus attached quark currents are effective polarization vectors V V V + + + . . . build a program to calculate the finite part of the sum of the graphs giuseppe bozzi VBFNLO 6
Boxline corrections Divergent terms in 4 Feynman graphs Virtual corrections for quark line with 2 EW combine to multiple of corresponding gauge bosons Born graph k 1 k 2 k 1 k 2 M ( i ) M ( i ) = B F ( Q ) boxline � � ǫ + π 2 − 2 ǫ 2 − 3 q 1 q 2 q 1 q 2 3 − 7 (a) (b) α s ( µ R ) M τ ( q 1 , q 2 )( − e 2 ) g V 1 f 1 g V 2 f 2 C F � + τ τ 4 π k 1 k 2 k 1 k 2 + O ( ǫ ) q 1 q 2 q 1 q 2 C F ( 4 πµ 2 with F ( Q ) = α s ( µ R ) Q 2 ) ǫ Γ ( 1 + ǫ ) (c) (d) R 4 π M τ ( q 1 , q 2 ) = � � M µν ǫ µ 1 ǫ ν 2 is universal vir- The external vector bosons correspond to tual qqVV amplitude: use like HELAS V → l 1 ¯ l 2 decay currents or quark currents calls in MadGraph giuseppe bozzi VBFNLO 7
Virtual corrections Born sub-amplitude is multiplied by same factor as found for pure vertex corrections ⇒ when summing all Feynman graphs the divergent terms multiply the complete M B Complete virtual corrections � � ǫ + π 2 − 2 ǫ 2 − 3 + � M V = M B F ( Q ) 3 − 7 M V where � M V is finite, and is calculated with amplitude techniques. The interference contribution in the cross-section calculation is then given by � � � � ǫ + π 2 − 2 ǫ 2 − 3 � 2 Re [ M V M ∗ B ] = |M B | 2 F ( Q ) M V M ∗ 3 − 7 + 2 Re B The divergent term, proportional to |M B | 2 , cancels against the subtraction terms just like in the Higgs case. giuseppe bozzi VBFNLO 8
3 weak bosons on a quark line: qq → qqWW , qqZZ , qqWZ at NLO e + • example: WW production via VBF with • µ - ν e α Γ V ν e leptonic decays: pp → e + ν e µ − ¯ γ ,Z ν µ + 2 j e + ν µ ν µ W - W + µ - u u u u • Spin correlations of the final state leptons • γ ,Z γ ,Z c c c c • All resonant and non-resonant Feynman • (a) (b) diagrams included µ - u u W - • • NC = ⇒ 181 Feynman diagrams at LO γ ,Z ν µ e + u u ν e αβ W T VV • CC = • ⇒ 92 Feynman diagrams at LO ν µ c c ν e µ - γ ,Z W + e + c c Use modular structure, e.g. leptonic tensor (c) (d) W + W + W + e + e + ν e µ - ν e W - W + e + ν e ν e e + ν µ u u u u (a) (b) (c) Z γ ,Z γ ,Z W - W + µ - ν e αβ αβ T W + V T W - V e + Calculate once, reuse in different processes ν µ γ ,Z γ ,Z c c c c Speedup factor ≈ 70 compared to MadGraph (e) (f) for real emission corrections giuseppe bozzi VBFNLO 9
New for virtual: pentline corrections Virtual corrections involve up to pen- The sum of all QCD corrections to a single quark tagons line is simple � � ǫ 4 πµ 2 α s ( µ R ) M ( i ) M ( i ) R k 1 k 2 k 1 k 2 k 1 k 2 = Γ ( 1 + ǫ ) C F V B Q 2 4 π q 1 q 2 q 3 q 1 q 2 q 3 q 1 q 2 q 3 � � V 1 V 2 V 3 V 1 V 2 V 3 V 1 V 2 V 3 − 2 ǫ 2 − 3 (a) (b) (c) ǫ + c virt k 1 k 2 k 1 k 2 k 1 k 2 M ( i ) � + V 1 V 2 V 3 , τ ( q 1 , q 2 , q 3 ) + O ( ǫ ) q 1 q 2 q 3 q 1 q 2 q 3 q 1 q 2 q 3 V 1 V 2 V 3 V 1 V 2 V 3 V 1 V 2 V 3 (d) (e) (f) • • Divergent pieces sum to Born amplitude: canceled via Catani Seymour algorithm k 1 k 2 k 1 k 2 q 1 q 2 q 3 q 1 q 2 q 3 • Use amplitude techniques to calculate finite • V 1 V 2 V 3 V 1 V 2 V 3 remainder of virtual amplitudes (h) (g) The external vector bosons correspond to Pentagon tensor reduction with Denner- V → l 1 ¯ l 2 decay currents or quark currents Dittmaier is stable at 0.1% level giuseppe bozzi VBFNLO 10
Gauge invariance tests Numerical problems flagged by gauge invariance test: use Ward identities for pentline and boxline contributions q µ 2 2 � E µ 1 µ 2 µ 3 ( k 1 , q 1 , q 2 , q 3 ) = � D µ 1 µ 3 ( k 1 , q 1 , q 2 + q 3 ) − � D µ 1 µ 3 ( k 1 , q 1 + q 2 , q 3 ) With Denner-Dittmaier recursion relations for E ij functions the ratios of the two expressions agree with unity (to 10% or better) at more than 99.8% of all phase space points. Ward identities reduce importance of computationally slow pentagon contributions when contracting with W ± polarization vectors J µ ± = x ± q µ ± + r µ ± choose x ± such as to minimize pentagon contribution from remainders r ± in all terms like J µ 1 + J µ 2 − � E µ 1 µ 2 µ 3 ( k 1 , q + , q − , q 0 ) = r µ 1 + r µ 2 − � E µ 1 µ 2 µ 3 ( k 1 , q + , q − , q 0 ) + box contributions Resulting true pentagon piece contributes to the cross section at permille level = ⇒ totally negligible for phenomenology giuseppe bozzi VBFNLO 11
Phenomenology Study LHC cross sections within typical VBF cuts • Identify two or more jets with k T -algorithm ( D = 0.8) • p T j ≥ 20 GeV , | y j | ≤ 4.5 • Identify two highest p T jets as tagging jets with wide rapidity separation and large dijet • invariant mass ∆ y jj = | y j 1 − y j 2 | > 4, M jj > 600 GeV • Charged decay leptons ( ℓ = e , µ ) of W and/or Z must satisfy • p T ℓ ≥ 20 GeV , | η ℓ | ≤ 2.5 , △ R j ℓ ≥ 0.4 , m ℓℓ ≥ 15 GeV , △ R ℓℓ ≥ 0.2 and leptons must lie between the tagging jets y j , min < η ℓ < y j , max For scale dependence studies we have considered weak boson virtuality : Q 2 µ = ξ m V fixed scale µ = ξ Q i i = 2 k q 1 · k q 2
WW production: pp → jje + ν e µ − ¯ ν µ X @ LHC Stabilization of scale dependence at NLO
WZ production in VBF, WZ → e + ν e µ + µ − Transverse momentum distribution of the softer tagging jet • Shape comparison LO vs. NLO • depends on scale • • = Scale choice µ Q pro- duces approximately constant K -factor • Ratio of NLO curves for differ- • ent scales is unity to better than 2%: scale choice matters very little at NLO Use µ F = Q at LO to best approxi- mate the NLO results giuseppe bozzi VBFNLO 14
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