The spin-dependent quark beam function at NNLO Ulrich Schubert - PowerPoint PPT Presentation

The spin-dependent quark beam function at NNLO Ulrich Schubert Argonne National Laboratory In collaboration with R. Boughezal, F. Petriello and H. Xing arXiv:1704.05457

Proton Spin Puzzle • Proton spin sum rule 2 = 1 1 2 ∆Σ + ∆ G + L q + L g Z 1 Z 1 X ∆Σ = dx ∆ f q i ( x ) ∆ G = dx ∆ f g ( x ) 0 0 i • Contribution from quarks much smaller then expected ∆Σ ≈ 0 . 25 • Helicity parton distributions are probed by DIS$ SIDIS$ pp$(RHIC)$

Current Status • Current data is not well described 0.4 LO E155 Θ =2.75 ° Θ NLO E155-DATA; 0.3 0.2 A LL 0.1 θ 0 σ -0.1 -0.2 30 21 22 23 24 25 26 27 28 29 30 P h [GeV] [Ringer, Vogelsang] [Hinderer, Schlegel, Vogelsang] • We need more data and more accurate theoretical predictions => Extent techniques from unpolarized collision

N-Jettiness [Boughezal, Focke, Liu, Petriello; Gaunt Stahlhofen Tackmann, Walsh] virtual real virtual real-real

N-Jettiness [Boughezal, Focke, Liu, Petriello; Gaunt Stahlhofen Tackmann, Walsh] Θ ( τ cut − τ ) τ cut Θ ( τ − τ cut )

N-Jettiness [Boughezal, Focke, Liu, Petriello; Gaunt Stahlhofen Tackmann, Walsh] Θ ( τ cut − τ ) τ cut Θ ( τ − τ cut ) => Use factorisation theorem => NLO N+1 jet calculation derived from SCET � N � d σ � = H ⊗ B ⊗ S ⊗ + J n Power corrections d T N n [Stewart, Tackmann, Waalewijn] Hard function (H): virtual corrections, process dependent Soft function (S): describes soft radiation Jet function (J): describes radiation collinear to final state jets Beam function (B): describes collinear initial state radiation

Polarized Collisions • Above cut piece can simply be polarised • Similar factorization theorem for the below cut piece � N � d σ LL � = ∆ H ⊗ ∆ B ⊗ S ⊗ + · · · J n d T N n Soft function: unchanged from unpolarized version [Boughezal, Liu, Petriello] Jet function: unchanged from unpolarized version [Becher, Neubert; Becher, Bell] Hard function: known for DIS and DY ∆ H = H + − H − Beam function: previously unknown, discussed here ∆ B = B + − B −

Beam function [Stewart, Tackmann, Waalewijn] Z 1 ✓ ◆ d ξ t, x X ∆ B i ( t, x, µ ) = ∆ f j ( ξ , µ ) ξ ∆ I ij ξ x j • Parton j with momentum distribution determined by PDF emits collinear radiation, which builds up jet described by I ij • These emissions might change the parton i entering the hard scattering (type, momentum fraction) • can be calculated perturbatively I ij

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij • Reverse Unitarity [Anastasiou, Melnikov; Anastasiou, Dixon, Melnikov, Petriello] Integration-by-parts(IBP) • n [Chetyrkin,Tkachov] X ∆ B bare ( t, z ) = c i ( t, z ) I i ( t, z ) ij i =1

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij • Reverse Unitarity [Anastasiou, Melnikov; Anastasiou, Dixon, Melnikov, Petriello] Integration-by-parts(IBP) • n [Chetyrkin,Tkachov] X ∆ B bare ( t, z ) = c i ( t, z ) I i ( t, z ) ij Differential Equations(DEQ) i =1 • [Kotikov;Gehrmann,Remiddi]

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij • Reverse Unitarity [Anastasiou, Melnikov; Anastasiou, Dixon, Melnikov, Petriello] Integration-by-parts(IBP) • n [Chetyrkin,Tkachov] X ∆ B bare ( t, z ) = c i ( t, z ) I i ( t, z ) ij Differential Equations(DEQ) i =1 • [Kotikov;Gehrmann,Remiddi] � • UV renormalization ∆ B bare dt ′ Z i ( t − t ′ , µ ) ∆ B ij ( t ′ , z, µ ) , ( t, z ) = ij

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij • Reverse Unitarity [Anastasiou, Melnikov; Anastasiou, Dixon, Melnikov, Petriello] Integration-by-parts(IBP) • n [Chetyrkin,Tkachov] X ∆ B bare ( t, z ) = c i ( t, z ) I i ( t, z ) ij Differential Equations(DEQ) i =1 • [Kotikov;Gehrmann,Remiddi] � • UV renormalization ∆ B bare dt ′ Z i ( t − t ′ , µ ) ∆ B ij ( t ′ , z, µ ) , ( t, z ) = ij • Matching on PDF � ∆ B ij ( t, z, µ ) = ∆ I ik ( t, z, µ ) ⊗ ∆ f kj ( z ) k

Outline of Calculation Generate squared amplitude • ∆ B bare ( t, z ) = + . . . ij • Reverse Unitarity [Anastasiou, Melnikov; Anastasiou, Dixon, Melnikov, Petriello] Integration-by-parts(IBP) • n [Chetyrkin,Tkachov] X ∆ B bare ( t, z ) = c i ( t, z ) I i ( t, z ) ij Differential Equations(DEQ) i =1 • [Kotikov;Gehrmann,Remiddi] � • UV renormalization ∆ B bare dt ′ Z i ( t − t ′ , µ ) ∆ B ij ( t ′ , z, µ ) , ( t, z ) = ij • Matching on PDF � ∆ B ij ( t, z, µ ) = ∆ I ik ( t, z, µ ) ⊗ ∆ f kj ( z ) k ⇣ Z 5 ⌘ ⇣ ⌘ Z 5 ⊗ ∆ ˜ ∆ ˜ I ⊗ ¯ • Additional renormalization for γ 5 ∆ B = f ⊗

Master Integrals • Initially integrals O (100) − O (1000) • 9 MIs in real-real channel • 3 MIs in real-virtual channel • Generate DEQ @ x ~ f = A x ~ f , x = t, z

Calculation of Master Integrals • Bring DEQ in canonical form with Magnus algorithm [Henn; Argeri, Di Vita, Mastrolia, Mirabella, Schlenk, Tancredi, U.S.] ˆ ˆ ˆ A 1 A 2 A 3 g = ✏ ˆ ˆ @ x ~ A x ~ g A z = z + 1 + z + 1 − z • Matrices have only numeric entries A i • Simple alphabet { 1 − z, z, 1 + z } • Solution can be written in terms of Harmonic Polylogarithms Z z dtH a 2 ,...a n ( t ) H a 1 ,...,a n ( z ) = a i ∈ 0 , − 1 , 1 , t − a 1 0 H 0 ,..., 0 ( z ) = 1 n ! log n ( z )

Calculation of Master Integrals (1 − z ) − 2 ✏ F ( z ) • MI for RR channel behave like when z → 1 [Gaunt, Stahlhofen, Tackmann] => fixes 7 out of 9 boundary constants • One MI is easily obtained by direct integration • Last boundary constant obtained by • Introduce extra scale • Solve DEQ with extra scale • Here all boundaries can be fixed easily • take scale carefully to zero (1 − z ) − 2 ✏ , − ✏ F ( z ) • MI for RV behave like when z → 1 => fixes one boundary constant • Taking carefully fixes second boundary constant z → 0 • Last boundary can be easily obtained by direct integration

UV renormalisation and Matching • Use standard renormalization e MS � ∆ B bare (2) ( t, z ) = ∆ B (2) ij ( t, z, µ ) + Z (2) dt ′ Z (1) ( t − t ′ , µ ) ∆ B (1) ij ( t ′ , z, µ ) . ( t, µ ) δ ij δ (1 − z ) + ij i i ∆ B (1) O ( ✏ 2 ) • Requires calculation of up to ij ( t, z, µ ) • Match beam function on PDFs I (2) ij ( t, z, µ ) = ∆ B (2) f (2) I (1) f (1) ∆ ˜ ij ( t, z, µ ) − 4 δ ( t ) ∆ ˜ � ∆ ˜ ik ( t, z, µ ) ⊗ ∆ ˜ ij ( z ) − 2 kj ( z ) . k ij ( z ) = − 1 f (1) P (0) ∆ ˜ � ∆ ˜ ij ( z ) , ij ( z ) = 1 ij ( z ) − 1 kj ( z ) + β 0 f (2) P (0) P (0) P (0) P (1) ∆ ˜ � ∆ ˜ ik ( z ) ⊗ ∆ ˜ 4 � 2 ∆ ˜ 2 � ∆ ˜ ij ( z ) , 2 � 2 k • Cancellation of poles provides consistency check

Treatment of Gamma5 • We use HVBM scheme � 5 ≡ i { γ 5 , ˜ γ µ } = 0 , [ γ 5 , ˆ γ µ ] = 0 . 4! ✏ µ νρσ � µ � ν � σ � ρ • Result of Dirac traces depends on d- and 4-d-dimensional momenta • Map 4-d momenta to auxiliary vectors k 2 ] = − 2 � I d [ˆ k 1 · ˆ I d [( k 1 · v ⊥ )( k 2 · v ⊥ ))] , v 2 ⊥ • But: HVBM breaks helicity conservation Z 5 => Must be restored with additional renormalization ⇣ Z 5 ⌘ ⇣ ⌘ Z 5 ⊗ ˜ ∆ ˜ I ⊗ ¯ ∆ B = f ⊗ • can be obtained by demanding helicity conservation Z 5 ∆ I (2 ,V ) = I (2 ,V ) ∆ I (2 ,V ) = − I (2 ,V ) qq qq q ¯ q ¯ q q

Consistency checks • HVBM scheme implemented in public code Tracer and in-house Form routine [Jamin,Lautenbacher] • MIs calculated by DEQ and direct integration • Cancellation of poles during renormalization and matching • Confirmed polarised LO and NLO splitting functions [Vogelsang] • Confirmed UV renormalisation constant [Stewart, Tackmann, Waalewijn; Ritzmann, Waalewijn] • Confirmed unpolarised quark beam function calculation at NLO and NNLO [Stewart, Tackmann, Waalewijn; Gaunt, Stahlhofen, Tackmann] Z 5 • consistent with Literature [Ravindran, Smith, van Neerven]

Conclusions & Outlook • Calculated spin-dependent quark beam function • Last missing ingredient to apply N-jettiness subtraction to many polarized processes • Provided independent check on: - unpolarized quark beam function up to NNLO - polarised splitting function up to NLO • Ready for phenomenological studies

Conclusions & Outlook • Calculated spin-dependent quark beam function • Last missing ingredient to apply N-jettiness subtraction to many polarized processes • Provided independent check on: - unpolarized quark beam function up to NNLO - polarised splitting function up to NLO • Ready for phenomenological studies Thank you for your attention