Helicity off-shell matrix elements within high energy factorization - PowerPoint PPT Presentation

Helicity off-shell matrix elements within high energy factorization Piotr Kotko Institute of Nuclear Physics (Cracow) supported by LIDER/02/35/L-2/10/NCBiR/2011 based on A. van Hameren, P .K., K. Kutak JHEP 1212 (2012) 029, JHEP 1301 (2013) 078, arXiv:1308.0452 (accepted for PRD)

Introduction Collinear factorization High-energy factorization • hard subprocess defined via • hard subprocess defined via on-shell amplitudes off-shell amplitudes • parton densities depend on x • ‘unintegrated’ parton densities and scale (UPDFs) depend also on transverse momentum • proved to all orders for large classes of processes • UPDFs are not universal (factorization does not hold to • plenty of automatic tools for all orders) tree-level amplitudes with multiple final states • tree-level amplitudes (up to five legs) obtained analytically • NLO calculations automated, some NNLO results • some HO results Motivation • automatization for gauge invariant tree-level off-shell amplitudes • unintegrated gluon density allows to include saturation • construction of ready-to-use Monte Carlo codes 1

PLAN • Introduction • high-energy factorization of Catani, Ciafaloni, Hautmann • TMD factorization vs small x • Off-shell amplitudes • general ’embedding’ approach with complex momenta • forward processes • one-leg-off-shell amplitudes • Unintegrated gluon densities • evolution with the saturation effect • Applications to LHC • forward three jet production using two new MC programs • Future plans and summary 2

High Energy Factorization • CCH factorization (Catani, Ciafaloni, Hautmann) for heavy quark production 1 Gauge choice – axial gauge with n = α p A + β p B p A The HARD part is defined by the eikonal projectors k A HARD = | � k T A | p µ k B p B A where = | � k T B | p µ B high-energy kinematics: k µ A ≃ x A p µ A + k µ T A ⇒ the amplitude g ∗ g ∗ → QQ is gauge invariant k µ B ≃ x B p µ B + k µ T B � d 2 k T A � dx A � d 2 k T B � dx B d σ AB → QQ = π x A π x B F ( x A , k T A ) d σ g ∗ g ∗ → QQ ( x A , x B , k T A , k T B ) F ( x B , k T B ) • originally F are BFKL unintegrated gluon densities 1 S. Catani, M. Ciafaloni, F. Hautmann, Nucl.Phys. B366 (1991) 135-188 3

High Energy Factorization (cont.) • for a generic multiparticle state X the amplitude g ∗ g ∗ → X is not gauge invariant k A additional terms are needed to ⇒ HARD recover the gauge invariance . . . k B · · · + . . . + HARD HARD . . . . . . · · · • one of the approaches – Lipatov’s effective action; the gauge invariant HARD sub-process corresponds to Quasi-Multi-Regge kinematics 1 • one can also find the lacking contributions without extending QCD action → by embedding HARD in a larger non-physical process → using the Slavnov-Taylor identities (when only one gluon is off-shell) → using matrix elements of straight infinite Wilson lines (today not presented) 4 1

High Energy Factorization (cont.) Transverse Momentum Dependent (TMD) factorization 1 • operator definitions for TMD gluon densities – transverse gauge links are needed; in general they are not universal • universality holds for semi-inclusive DIS, Drell-Yan, back-to-back jet production in e + e − and DIS • factorization is broken for hadro-production of hadrons TMD factorization vs small x • universality remains violated in hadron-hadron collisions • explicit NLO calculation for inclusive heavy quark production in DIS 2 • generalized factorization derived for back-to-back-like dijets in dilute-dense systems 3 • so called “hybrid” factorization (single unintegrated gluon density) might be valid 1 e.g. P 2 S. Catani, F. Hautmann, Nucl.Phys. B427 (1994) . Mulders, ArXiv:11024569 3 F. Dominguez, C. Marquet, B. Xiao, F. Yuan, Phys.Rev. D83 (2011) 105005 5

Automatic Off-shell Helicity Amplitudes An amplitude g ∗ ( k A ) g ∗ ( k B ) → X can be disentangled from q A q B → q ′ A q ′ B X . k A k A + . . . + . . . . . . . . . k B k B k B However, if we want to have explicit high-energy kinematics for k A , k B the quarks q ′ A , q ′ B cannot be on-shell ⇒ amplitude for q A q B → q ′ A q ′ B X is not gauge invariant It’s possible to have both on-shellness for all external partons and high-energy kinematics 1 : � � � � → the amplitude q A ( p A ) q B ( p B ) → q ′ p ′ q ′ p ′ X need not to be physical A A B B → introduce on-shell complex momenta for the quarks using helicity formalism (the gauge invariance is still there) 1 A. van Hameren, P . Kotko, K. Kutak, JHEP 1301 (2013) 078 6

Automatic Off-shell Amplitudes (cont.) Introduce the basis using real four vectors l 1 , l 2 and complex l 3 , l 4 defined as 3 = 1 4 = 1 2 � l 2 ; −| γ µ | l 1 ; −� , 2 � l 1 ; −| γ µ | l 2 ; −� . l µ l µ They have properties: l 2 i = 0, l 1 , 2 · l 3 , 4 = 0, l 1 · l 2 = − l 3 · l 4 . The momenta of external quarks are decomposed as follows 1 − l 4 · k T A 2 − l 3 · k T B p µ A = (Λ + x A ) l µ l µ p µ B = (Λ + x B ) l µ l µ 3 , 4 l 1 · l 2 l 1 · l 2 1 + l 3 · k T A 2 + l 4 · k T B p ′ µ A = Λ l µ l µ p ′ µ B = Λ l µ l µ 4 , 3 , l 1 · l 2 l 1 · l 2 We get both the on-shellness p 2 A , B = p ′ 2 A , B and high-energy limit for any Λ . Moreover the external spinors for quarks | p A ; −� ∝ | l 1 ; −� , | p B ; −� ∝ | l 2 ; −� etc. • in order to extract the physical amplitude take the limit Λ → ∞ , either numerically or analytically (then eikonal couplings and propagators appear) • corresponds to Lipatov’s RR → X effective vertex • implemented in fortran MC code by A. van Hameren (to be released) 7

Forward Processes and High Energy factorization Forward processes (relevant for small x) correspond to asymmetric configurations 4 10 3 jets production at √ � � = 5.02 TeV � � 3 10 � � � � p T i kinematic cuts : 35 GeV < p T 3 < p T 2 < p T 1 exp ( η i ) x A = √ 2 10 | η 1,2 | < 2.8 [ pb ] S 3.2 < | η 3 | < 4.7 i � � proton nonlinear � � 10 � � � � p T i Pb nonlinear dσ / d x a proton linear x B = exp ( − η i ) √ 1 S i -1 10 x as = | x A − x B | / ( x A + x B ) -2 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x as This accounts for a simplification: • large fractions x B → collinear approach (with on-shell parton) • small fractions x A → high energy factorization (with off-shell parton) � d 2 k T A � dx A � � dx B F ( x A , k T A ) f b / B ( x B ) d σ g ∗ a → X ( x A , x B , k T A ) 1 d σ AB → X = π x A b 1 M. Deak, F. Hautmann, H. Jung, K. Kutak, JHEP 0909 (2009) 121 8

One-leg Off-shell Helicity Amplitudes A contribution to N -jet • not gauge-invariant process: g ∗ g → gg . . . g k A . � = 0 . . k A . . . M ( ε 1 . . . , k i , . . . , ε N ) � 0 • one cannot use helicity method, ≡ M ( ε 1 , . . . , ε N ) i.e. ε µ k ( q ) = ε µ k ( q ′ ) + k µ β k ( q , q ′ ) • there exists an “amplitude” W such that � M = M + W satisfies � M ( ε 1 , . . . , k i , . . . , ε N ) = 0 • the “gauge-restoring” amplitude W can be obtained by using the ordinary QCD Slavnov-Taylor identities 1 1 A. van Hameren, P . Kotko, K. Kutak, JHEP 1212 (2012) 029 9

One-leg Off-shell Helicity Amplitudes (cont.) • introduce a reduction formula for the off-shell amplitude ( ˜ G – the Green function) �� � � � � � � ˜ � � � � � � � p µ A k 2 1 ε µ 1 k 2 N ε µ N M ( ε 1 , . . . , ε N ) = k A · p A → 0 lim lim 1 → 0 . . . lim k T A . . . G µ A µ 1 ...µ N A 1 N k 2 k 2 N → 0 • apply Slavnov-Taylor identities to ˜ G to determine gauge contributions . . . . . = . . . . + . . . . . . + + . + . . • after applying the reduction formula (and using axial gauge for internal propagators) a single term survives The r.h.s term is precisely the amount of gauge- . . . = . . . invariance violation and can be calculated. • trading the external ghosts for the longitudinal projections of the gluons and summing the gauge contributions we get the result 10

One-leg Off-shell Helicity Amplitudes (cont.) The complete color-ordered result is  � �  � �   � � � � �  A ( ε 1 , . . . , ε N ) = − k T A   k T A · J ( ε 1 , . . . , ε N ) �  � − g � N  ε 1 · p A . . . ε N · p A    +  √  k 1 · p A ( k 1 − k 2 ) · p A . . . ( k 1 − . . . − k N − 1 ) · p A 2 where (below k ij = k i + k i + 1 + . . . + k j ) � � k µ 1 N p A ,ν + k 1 N ν p µ J µ ( ε 1 , . . . , ε N ) = − i A g µ ν − k 2 k 1 N · p A 1 N   � N − 1 � �   V ναβ k 1 i , k ( i + 1 ) N J α ( ε 1 , . . . , ε i ) J β ( ε i + 1 , . . . , ε N )    3 i = 1   N − 2 � N − 1 �   V ναβγ + J α ( ε 1 , . . . , ε i ) J β ( ε i + 1 , . . . , ε j ) J γ ( ε j + 1 , . . . , ε N )   4  i = 1 j = i + 1 This result is consistent with Lipatov’s effective action. 11