OpenLoops 2 M. F. Zoller in collaboration with F. Buccioni, J.-N. - PowerPoint PPT Presentation

OpenLoops 2 M. F. Zoller in collaboration with F. Buccioni, J.-N. Lang, J. Lindert, P. Maierhöfer, S. Pozzorini and H. Zhang [arxiv:1907.13071] LoopFest XVIII – Fermilab – 08/14/2019

OpenLoops OpenLoops is a fully automated numerical tool for the tree and one-loop computation of hard scattering amplitudes required in Monte-Carlo simulations of scattering events • Full NLO QCD and NLO EW corrections available • Strong CPU performance and excellent numerical stability [Höche] Scattering probability densities in perturbation theory   |M 0 | 2 ,  M ∗ |M 1 | 2 W 00 = � � W 01 = � � 0 M 1 W 11 = � � 2 Re  , hel col hel col hel col computed from sums of l -loop Feynman diagrams: M 0 = M 1 = + + . . . + + . . . 1

Applications of OpenLoops Interfaces to many Monte Carlo programs (unchanged from OpenLoops 1) Sherpa [Höche, Krauss, Schönherr, Siegert et al.] → NLO matching and merging , Munich/Matrix [Grazzini, Kallweit, Rathlev, Wiesemann] → (N)NLO parton level MC, Powheg [Nason, Oleari et al.], Herwig [Gieseke, Plätzer et al.] Geneva [Alioli, Bauer, Tackmann et al.], Whizard [Kilian, Ohl, Reuter et al.] Many OpenLoops applications: • NLO QCD and NLO+PS for any 2 → 2 , 3 , 4 SM process at LHC and future colliders • NLO EW for any 2 → 2 , 3 , 4 SM process [Kallweit, Lindert, Maierhöfer, Pozzorini, Schönherr] • NNLO QCD for pp → V, V V with Matrix [Grazzini, Kallweit, Rathlev, Wiesemann] First OpenLoops 2 applications (2019): • NNLO QCD Spin correlations in t ¯ t production [Behring, Czakon, Mitov, Papanastasiou, Poncelet] tb ¯ • NLO QCD t ¯ b +jet production [Buccioni, Kallweit, Pozzorini, M.Z.] 2

Outline I. Program structure of OpenLoops 2 II. Full NLO QCD and EW corrections in the SM – Power counting in α S and α – Input schemes and parameters – Evaluation of amplitudes → Automated scale variations – Colour- and Spin-correlators III. The on-the-fly algorithm – Recursive construction of tree and loop diagrams – On-the-fly reduction, merging and helicity summation – On-the-fly stability system → Numerical stability and performance IV. Summary and Outlook 3

I. The structure of OpenLoops 2 • OpenLoops program (public): User interfaces and process-independent OpenLoops routines. Available as tar from https://openloops.hepforge.org or from git repository: git clone https://gitlab.com/openloops/OpenLoops.git • Process generator (not public): Perform analytical steps (e.g. colour factors) and generate process-dependent code for numerical calculation → stored in process libraries • Process libraries (public) automatically downloaded by the user. A process a library contains all partonic channels for a process class. Example: ppjj contains d ¯ d → d ¯ u → d ¯ d , d ¯ d → gg , gg → gg , etc d , u ¯ and real corrections d ¯ d → d ¯ u → d ¯ dg , u ¯ dg , etc Also: particle permutations + channel maps, e.g. b ¯ b → gg mapped to d ¯ d → gg for M B = 0 . More than 200 process libraries available for all relevant SM processes (+ HEFT) → see https://openloops.hepforge.org Additional libraries provided upon user request • Third party tools for integral evaluation (included): Collier 1.2.2 [Denner, Dittmaier, Hofer ’16] , OneLoop 3.6.1 [van Hameren ’10] 4

II. Full NLO QCD and EW corrections in the SM EW corrections enhanced by soft/collinear logarithms from virtual EW bosons: α w ln 2 ( Q 2 /M 2 • Order W ) ∼ 25% > α S in observables at the TeV scale πs 2 ⇒ EW corrections crucial for SM tests and BSM searches at the LHC But also more challenging than NLO QCD! γ, Z , W q i ℓ − ℓ + • Virtual corrections involve more particles and masses ( γ, Z, W, H, b, t ) γ, Z , W q i g g • V → lepton decays: Effective particle multiplicity increased due to final-state interactions and non-factorisation, e.g. pp → ZZ is 2 → 2 in QCD and 2 → 4 with EW ( × 400 more diagrams) ℓ − ℓ − ℓ − Z/γ ∗ Z/γ ∗ Z/γ ∗ p p p ℓ + ℓ + ℓ + p p p 5

Power counting: Nontrivial QCD-EW interplay q at Born level: M 0 ∼ O ( e 2 ) + O ( g 2 q → q ¯ Simple example: q ¯ S ) q ∼ W 00 ∼ O ( α 2 O ( α 1 S α 1 ) + O ( α 2 ) ⇒ σ q ¯ S ) + q → q ¯ � �� � � �� � � �� � QCD EW − QCD interf . EW NLO EW corrections of O ( α 2 S α 1 ) for q ¯ q → q ¯ q : • EW corrections to QCD Born → only full O ( α 2 S α 1 ) IR finite γ γ γ, Z → O ( α ) corrections can involve • QCD corrections to EW–QCD interference emissions of γ and g, q, ¯ q γ, Z γ, Z γ, Z n q ¯ ˜ � q e m +2 k M ( k ) S e m M (0) k =0 g n − 2 k � g n In general (e.g. pp → X + jets): M 0 = = M 0 where � � � 0 0 S � � LO QCD n q ¯ ˜ q = n q ¯ q − 1 , if n q ¯ q ≥ 1 (number of external q ¯ q pairs), else ˜ n q ¯ q = 0 6

NLO EW corrections in OpenLoops 2 n q ¯ ˜ q e m +2 k M ( k ) g n − 2 k M 0 = � S 0 k =0 �M 0 |M 0 � ∼ O ( α n S α m ) + O ( α n − 1 α m +1 ) + . . . + O ( α n − k α m + k ) ⇒ W 00 = S S alternating series of dominant contributions involving | M ( k ) 0 | 2 and suppressed pure interference terms 0 | M ( k ′ ) involving � M ( k ) � with k � = k ′ . 0 α n S α m α n − 1 α m +1 α n − 2 α m +2 S S LO α S α NLO α n +1 α m α n S α m +1 S ⇒ Mixed α α S power counting with non-trivial interference contributions ⇒ OpenLoops provides any desired order O ( α n S α m ) in a fully automated way 7

Input schemes and parameters • Three EW schemes implemented: scheme input parameters value of 1 /α derived parameters: ≈ 137 α (0) α (0) , M W , M Z , M H + fermion masses cos 2 ( θ w ) = µ 2 W Z , . . . G µ , ≈ 132 G µ (default) M W , M Z , M H + fermion masses µ 2 α ( M Z ) α ( M Z ) , M W , M Z , M H + fermion masses ≈ 128 ⊲ α (0) -scheme: pure QED interactions at scales Q 2 ≪ M 2 W , production of on-shell photons ⊲ G µ -scheme: optimal description of W -interactions at EW scale ⊲ α ( M Z ) -scheme: hard EW interactions at EW scale (optimal for QED, decent for SU (2) ) γ ∗ • External photons in process A → B + n γ + n ∗ (+ γ ) ���� ���� ���� on-shell off-shell real emission  α | G µ  if α = α (0) ,     ⇒ rescale with ratios of input α and α on = α (0) , α off = if α = α | G µ or α = α ( M Z )  α      n   n ∗ W     α off  α on ⇒ W → (No rescaling for real emission) α α Optimal scale choice for external on-shell, off-shell and real-emission photons 8

Complex masses, Scale variations and Renormalisation • Consistent treatment of resonances with complex mass scheme at 1-loop [Denner, Dittmaier] → complex mass µ 2 p = M 2 p − i M p Γ p from real physical mass M p and width Γ p as input ⇒ implemented in a flexible way , i.e. mix between on-shell and off-shell massive particles tl + l − with off-shell Z at NLO EW allowed ⇒ Consistent calculation of e.g. pp → t ¯ tZ → t ¯ • Different Renormalisation schemes implemented, e.g. on-shell or MS for quark masses; different flavour schemes for α S • Efficient QCD scale variations: If scattering amplitudes are re-evaluated multiple times with different values of µ r and α S (all other input and kinematic parameters fixed) → For each new phase-space point, matrix elements are computed and stored in a cache. → For ( µ r , α S ) variations, only µ r -dependent QCD counterterms are explicitly re-computed and the bare amplitude from the cache is re-scaled according to its α S -dependence. ⇒ Highly efficient algorithm for scale variations fully automated 9

More OpenLoops 2 features • Colour and charge correlators , → IR subtraction methods � � � � � � e.g. �M L | T a j T a � � k |M L �M L | Q j Q k |M L � and � S α q for L = 0 , 1 � � � � � α p � α p � � S α q (exchange of soft gluon/photon between external legs j, k ) • Spin and Spin-colour correlators , → IR subtraction methods � � B ( p,q | jk | µν ) e.g. B µν = �M L | T a j T a �� � � = �M| µ, j � � ν, j |M� and k | µ, j ν, j |M L � � j LL, LO � � α p � S α q (soft-collinear radiation of external gluons/photons) for L=0,1 (all L=0 correlators already available in OpenLoops 1) • Catani-Seymour I-operator → subtraction of IR poles • Selection of helicity states → polarised initial or final states ⇒ Ingredients for a wide range of applications available 10

III. The OpenLoops algorithm: Tree level Tree-level amplitudes constructed recursively from sub-trees M 0 = → split into sub-trees For example + . . . Numerical recursion step: sub-tree w c = X α βγ ( k b , k c ) w β w α b w γ × a = = c k 2 a − m 2 a sub-tree w b � �� � universal building block from Feynman rules w b k b k a w a Generic depiction: = ( k i external momenta) α α w c k c Highly efficient: Sub-trees constructed only once for multiple tree and loop diagrams 11