Selection of recent theory and phenomenology developments in forward - PowerPoint PPT Presentation

Selection of recent theory and phenomenology developments in forward physics 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), arXiv:1308.0452 078

PLAN • High-energy factorization • off-shell amplitudes and gauge invariance • automated calculation of tree-level off-shell amplitudes • forward processes ⇒ one-leg-off-shell amplitudes • Unintegrated gluon densities • evolution with the saturation effect • nonlinear extension of CCFM • Applications to LHC • Monte Carlo implementations • results for three jet production • bb production in weak processes 1

High Energy Factorization Production of a state X in a collision of hadrons A , B at high energies � d 2 k T A � dx A � d 2 k T B � dx B d σ AB → X = π x A π x B F ( x A , k T A ) d σ g ∗ g ∗ → X ( x A , x B , k T A , k T B ) F ( x B , k T B ) • motivated by CCH factorization for heavy quark production 1 The HARD part is defined by the eikonal projectors p A k A HARD = | � k T A | p µ A k B p B where = | � k T B | p µ B high-energy kinematics: k µ A ≃ x A p µ A + k µ ⇒ the amplitude g ∗ g ∗ → QQ is gauge invariant T A k µ B ≃ x B p µ B + k µ T B • originally F are BFKL unintegrated gluon densities 1 S. Catani, M. Ciafaloni, F. Hautmann, Nucl.Phys. B366 (1991) 135-188 2

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 • in terms of the Lipatov’s effective action the correct HARD part corresponds to Quasi-Multi-Regge kinematics; one can use the resulting Feynman rules 1 • one can also find the lacking contributions by demanding the gauge invariance → suitable for automated calculation of HARD for multiple final states → two new methods (using helicity method and implemented in MC codes) Comments • this is not a theorem of PQCD (actually k T -factorization is broken) • extremaly useful in phenomenology studies, even with HARD at tree level 1 E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov, Nucl.Phys. B721 (2005) 111-135 3

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) → more details on A. van Hameren’s talk 1 A. van Hameren, P . Kotko, K. Kutak, JHEP 1301 (2013) 078 4

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 5

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 6

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 7

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. 8

Unintegrated Gluon Densities • in the high-energy factorization orginally BFKL gluon evolution was used ⇒ why not to try to include more subtle effects relevant to small x? • nonlinear evolution with saturation 1 � �   � x � � x � k 2  � x �     q 2 z , q 2 θ T z − q 2 − k 2 z , k 2  � 1 � ∞  T F T F   k 2 z , k 2  � � � � dq 2  T T T T F  + α s N c dz   T x , k 2 x , k 2 T F = F 0 � � +  �   � �  T T   π z q 2 � q 2 T − k 2 k 2  �    x 4 q 4 T + k 4   T  T  T 0 T  � � 1 � � � k 2  � x � � x    + α s P gg ( z ) − 2 N c   T dq 2 z , q 2 z , k 2 dz T F + zP gq ( z ) Σ     T T 2 π k 2   z k 2 x T T 0    �    � ∞ � � ∞  2     dq 2 � � dq 2 q 2 � � − 2 α 2                 s T x , q 2 x , k 2 T T x , q 2   + F     F  ln   F       T T T  R 2 q 2 q 2 k 2  k 2 k 2  T T T T T → includes kinematic constraints → includes nonsingular pieces of the splitting functions → the parameter R has an interpretation of a target radius ⇒ one may attempt to use it for nuclei 2 • warning: at large densities factorization issue is much more complicated (CGC) 3 1 K. Kutak, J. Kwiecinski, Eur.Phys.J. C29, 521 (2003); K. Kutak, A. Stasto, Eur.Phys.J. C41, 343 (2005) 2 K. Kutak, S. Sapeta, Phys.Rev. D86, 094043 (2012) 3 F. Dominguez, C. Marquet, B. Xiao, Feng Yuan, Phys.Rev. D83 (2011) 105005 9