TMD splitting functions unabridged: real contributions Mirko Serino - PowerPoint PPT Presentation

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions TMD splitting functions unabridged: real contributions Mirko Serino Ben Gurion University of the Negev, Be’er Sheva, Israel & Institute of Nuclear Physics, Cracow, Poland Resummation, Evolution, Factorization 2017, Madrid, Spain, November 13–16 2017 Work in collaboration with Martin Hentschinski, Aleksander Kusina and Krzysztof Kutak arXiv: 1711.04587 ! Supported by NCN grant DEC-2013/10/E/ST2/00656 & the Kreitman School of the Ben Gurion University TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions 1 Gauge invariant amplitudes in High Energy Factorisation 2 The C.F.P. approach to splitting kernels 3 Transverse Momentum Dependent splitting functions TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions What is your purpose: the TMD splitting kernels Short term: connecting DGLAP and low-x evolutions. Long-term: Monte Carlo evolution bridging between DGLAP and BFKL Squared matrix elements for the determination of the real contributions to the splitting functions à-la Curci-Furmanski-Petronzio (CFP) q p 0 k ⊥ + q 2 + q 2 k µ = yp µ + k µ q µ = xp µ + q µ p ′ = q − k n µ ⊥ 2 x p n p µ = ( 1 , 0 , 0 , 1 ) n µ = ( 1 , 0 , 0 , − 1 ) First three computed and consistent with DGLAP Catani and Hautmann Nucl.Phys. B427 (1994) Gituliar, Hentschinski, Kutak JHEP 1601 (2016) 181 So far, a computation of the ˜ P gg kernel which could successfully reproduce the DGLAP AND BFKL limit not done: let me show one ! TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions Gauge invariant amplitudes in High Energy Factorisation TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions High Energy Factorization: more degrees of freedom High Energy Factorization (Catani,Ciafaloni,Hautmann, 1991 / Collins,Ellis, 1991) � dx 1 dx 2 d 2 k 1 ⊥ d 2 k 2 ⊥ F g ( x 1 , k 1 ⊥ , µ 2 ) F g ( x 2 , k 2 ⊥ , µ 2 ) ˆ σ h 1 , h 2 → q ¯ q = σ gg ( m , x 1 , x 2 , s , k 1 ⊥ , k 2 ⊥ ) x 1 x 2 � d 2 k T F g ( x , k t , µ 2 ) = f g ( x , µ 2 ) . F g ’s: unintegrated gluon densities, Non negligible transverse momentum ⇔ small x physics. Exact initial state kinematics ⇒ collinear higher order effects ab initio . Progress to connect DGLAP and low-x evolution (this approach) Momentum parameterization: k µ 1 = x 1 l µ 1 + k µ k µ 2 = x 2 l µ 2 + k µ , 1 ⊥ 2 ⊥ l 2 k 2 i = − k 2 i = 0 , l i · k i = 0 , i ⊥ , i = 1 , 2 TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions High Energy Factorization in Lipatov’s approach High Energy Factorization (Catani,Ciafaloni,Hautmann, 1991 / Collins,Ellis, 1991) � dx 1 dx 2 d 2 k 1 ⊥ d 2 k 2 ⊥ F g ( x 1 , k 1 ⊥ , µ 2 ) F g ( x 2 , k 2 ⊥ , µ 2 ) ˆ σ h 1 , h 2 → q ¯ q = σ gg ( m , x 1 , x 2 , s , k 1 ⊥ , k 2 ⊥ ) x 1 x 2 Usual tool for hard matrix elements: Lipatov’ effective action Lipatov, Nucl.Phys. B721 (1995) 111-135 Antonov, Cherednikov, Kuraev, Lipatov, Nucl.Phys. B452 (2005) 369-400 � d 4 x ( W − [ v ] − A − ) ∂ 2 ⊥ A + + ( W + [ v ] − A + ) ∂ 2 S eff = S QCD + � tr � �� ⊥ A − W ± [ v ] = − 1 1 1 1 v ± + g 2 v ± g ∂ ± U [ v ± ] = v ± − g v ± v ± v ± + . . . ∂ ± ∂ ± ∂ ± v µ ≡ − i t a A a A ± ≡ − i t a A a µ , gluon field ± , reggeized gluon fields � x ± � � − g dz ± v ± ( z ± , x ⊥ ) U [ v ± ] = P exp , x ⊥ = ( x ± , x ) 2 −∞ Sudakov parameterisation of initial state for for HEF: k µ 1 = x 1 l µ 1 + k µ k µ 2 = x 2 l µ 2 + k µ l 2 k 2 i = − k 2 , 2 ⊥ , i = 0 , l i · k i = 0 , i ⊥ , i = 1 , 2 1 ⊥ TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions Gauge invariant amplitudes with off-shell gluons Kutak, Kotko, van Hameren, JHEP 1301 (2013) 078 Problem: general partonic processes must be described by gauge invariant amplitudes ⇒ ordinary Feynman rules are not enough ! THE IDEA: on-shell amplitudes are gauge invariant, so off-shell gauge-invariant amplitudes could be got by embedding them into on-shell processes... 1) For off-shell gluons: represent g ∗ as coming from a ¯ ...first result...: qqg vertex, with the quarks taken to be on-shell p A p A ′ p A p A ′ p A p A ′ p A p A ′ k 1 = + + + · · · p B ′ k 2 k 2 p B p B ′ p B p B ′ p B p B ′ p B embed the scattering of the off-shell gluons in the scattering of two quark pairs carrying momenta p µ A = k µ 1 , p µ B = k µ 2 , p µ A ′ = 0 , p µ B ′ = 0 k 2 → i p ik / / 1 Assign the spinors | p 1 � , | p 1 ] to the A -quark and p 1 · k to the A-propagators; same for the B -quark line. � − k 2 ordinary Feynman elsewhere and factor x 1 ⊥ / 2 to match to the collinear limit Big advantage: Spinor helicity formalism TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions Prescription for off-shell quarks Kutak, Salwa, van Hameren, Phys.Lett. B727 (2013) 226-233 ... and second result: 2) for off-shell quarks: represent q ∗ as coming from a γ A ¯ q A q vertex, where γ A and ¯ q A are on shell; γ A is an artificial flavour-changing neutral boson coupling only to q ! q A γ A q A q A γ A q A γ A γ A u u u u ( k 1 ) u = + + + · · · X g g g g embed the scattering of the quark with whatever set of particles in the scattering of an auxiliary quark-photon pair, q A and γ A carrying momenta p µ q A = k µ 1 , p µ γ A = 0 i p / 1 Let q A -propagators of momentum k be p 1 · k and assign the spinors | p 1 � , | p 1 ] to the A -quark. Assign the polarization vectors ǫ µ + = � q | γ µ | p 1 ] 2 � p 1 q � , ǫ µ − = � p 1 | γ µ | q ] √ √ to the auxiliary 2 [ p 1 q ] photon, with q a light-like auxiliary momentum. � − k 2 Multiply the amplitude by x 1 1 ⊥ / 2 and use ordinary Feynman rules everywhere else. TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions Novel results on Amplitudes in High Energy Factorization QCD With growing number of legs, it is necessary to figure out practical ways to compute amplitudes efficiently. A promising possibility is the BCFW (Britto-Cachazo-Feng-Witten) recursion relation, originally discovered for on-shell QCD amplitudes and extended to off-shell gluon amplitudes in A. van Hameren, JHEP 1407 (2014) 138 A general analysis extending the modified BCFW to amplitudes with fermion pairs has been developed in A. van Hameren, MS JHEP 1507 (2015) 010 and A. van Hameren, K. Kutak, MS, JHEP 1702 (2017) 009 Numerical implementation and cross-checks are done and always successful. A program exists implementing Berends-Giele recursion relation, A. van Hameren, M. Bury, Comput.Phys.Commun. 196 (2015) 592-598 Loops in HEF, a proof of concept: Andreas van Hameren, arXiv:1710.07609 ; In order to apply CFP, we need to switch amplitudes ⇒ open-index vertices Let us extract them ! TMD splitting functions unabridged: real contributions Mirko Serino

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions Open-index vertices in HEF: Γ µ g ∗ q ∗ q ( q , k , p ′ ) Simple Idea: write down Feynman diagrams and then "deconstruct" by removing the "polarisation vectors", i.e. ǫ µ ( p ) for an on-shell particle vs. p µ for an off-shell particle + radiated quark p ′ = k − q p ′ 2 = 0 HEF gluon q = y p + q ⊥ , HEF quark k = x p + k ⊥ , � p | γ µ | p ] γ ν − γ ν d µν ( q ) � p | ǫ / p + / / / p + ǫ k p A ( q , k , p ′ ) = √ √ √ √ √ | n ] q 2 k 2 2 p · p ′ 2 2 2 2 2 � k p ν � = n µ d µν ( q ) k 2 | p ] = n µ d µν ( q ) / γ ν − [ n | Γ µ g ∗ q ∗ q ( q , k , p ′ ) | p ] [ n | p · p ′ k / q 2 q 2 d µν ( q ) = − g µν + n µ q ν + n ν q µ N.B. not invertible for n 2 = 0 , i.e in light-cone gauge , q 2 TMD splitting functions unabridged: real contributions Mirko Serino