One-leg off-shell helicity amplitudes in high-energy factorization - PowerPoint PPT Presentation

One-leg off-shell helicity amplitudes in 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

Motivation Why "one-leg off-shell" amplitudes? LHC ⇒ very high energies ⇒ collinear factorization is not sufficient for some observables • k T -factorization (or TMD factorization – transverse-momentum dependent factorization) is relevant • a “hard-process” involves amplitudes with initiating partons being off-shell • for low- x probing a typical kinematic configuration is asymmetric, i.e. one probes one of the PDFs at very small x whereas the second is probed at relatively large x ⇒ DGLAP and BFKL formalisms (one parton is approximated as being on-shell) • for on-shell tree-level amplitudes there are plenty of efficient tools (based on helicity method); this is not the case for off-shell ones • for one-leg off-shell amplitudes in high energy factorization the solution for any number of gluons is simple and uses only standard QCD (in particular the Slavnov-Taylor identities)

Introduction High-energy factorization CCH (Catani, Ciafaloni, Hautmann) factorization 1 The CCH was originally stated for heavy quarks production in photo-, lepto- and hadro-production. It is then argued that � � � � dz A dz B d 2 k T A d 2 k T B d σ AB → QQ ≃ z A z B p A F ( z A , k T A ) d σ g ∗ g ∗→ QQ ( z A , z B , k T A , k T B ) F ( z B , k T B ) , k A where F are unintegrated gluon distribution functions undergo- HARD ing BFKL evolution. The hard process d σ g ∗ g ∗→ QQ is calculated by contracting an off-shell amplitude (including external off-shell k B p B propagators) with p A , p B : = | � k T A | p µ At high energies the single longitudinal compo- A where nents of momentum transfers dominate = | � k T B | p µ B k µ A ≃ z A p µ A + k µ k µ B ≃ z B p µ B + k µ T A , T B . It can be shown that d σ g ∗ g ∗→ QQ is gauge invariant. 1 S. Catani, M. Ciafaloni, F. Hautmann (1990), (1991), (1994)

Introduction Remarks concerning factorization theorems Problems with k T -factorization • CCH factorization is not a formal theorem of k T -factorization • detailed all-order proofs exist for 1 • SIDIS (Semi Inclusive DIS) • Drell-Yan • back-to-back jet or hadron production in DIS or e + e − annihilation • k T -factorization in hadron-hadron collisions does not hold (also “generalized factorization” does not hold) • an “effective” generalized k T -factorization for proton-nuclei collision was reported 2 Despite the above difficulties, tree-level high energy amplitudes within CCH approach are well defined and do not generate conceptual problems (except gauge invariance and technical issues). The issue regarding different unintegrated PDFs and particular evolution equations is disconnected from the present study. 1 P 2 F. Dominguez, C. Marquet, B. Xiao, F. Yuan (2011) . Mulders, T. C. Rogers arXiv:1102.4569 [hep-ph]

Introduction Forward processes Kinematics For high-energy kinematics and k A + k B → k 1 + k 2 subprocess we have k µ A ≃ z A p µ A + k µ k µ B ≃ z B p µ B + k µ T A , T B � � � � � � � � k T i � � � k µ e η i p µ A + e − η i p µ + k µ i = √ T i , i = 1 , 2 B S Therefore � � � � � � � � � � � � � � � � k T 1 k T 2 � � e η 1 + e η 2 , z A = √ √ S S � � � � � � � � � � � � � � � � k T 1 k T 2 � � e − η 1 + e − η 2 z B = √ √ S S ⇒ small longitudinal fractions are probed in highly asymmetric configuration. • large fractions → collinear approach (with on-shell partons) • small fractions → k T -factorization (with off-shell partons) For instance, at CMS one can go to z 1 ≈ 10 − 4 , z 2 ≈ 0 . 2 with high- p T jets ( k T > 35 GeV ) in HF detector (3 < | η | < 5).

Tree-level CCH factorization for multiple jets Additional emissions: collinear factorization Additional emissions: TMD factorization · · · + . . . + HARD HARD HARD . . . . . . . . . Two standard approaches to HARD • Consider an on-shell process with hadron replaced by a quark and eventually perform the high-energy limit. The following structure emerges . . . + . . . + + HARD HARD HARD . . . . . . ⇒ the bremsstrahlung diagrams are necessary in order to maintain gauge invariance. • Use Lipatov’s effective action and resulting Feynman rules 1 . An off-shell gluon contracted with eikonal vector (+ gauge contributions) ≡ reggeon (R), ⇒ R → QQGG . . . G effective vertex 1 E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov (2005)

Off-shell amplitudes and helicity method Helicity method for on-shell amplitudes • uses the spinor representation for polarization vectors of gluons For a gluon with momentum k the polarization vector is defined with the help of a reference vector q , ε µ k ( q ) . • the gauge invariance is crucial Change of the reference momentum q → q ′ amounts for We can adjust q freely due to the Ward identity the transformation ε µ k ( q ) = ε µ where k ( q ′ ) + k µ β k ( q , q ′ ) . . . . = 0 = k µ • proper choice of q renders rather compact expressions for helicity amplitudes, which can be squared and summed numerically Let us denote the off-shell amplitude as M ( ε B , ε 1 . . . , ε N ) . The off-shell amplitude (without bremsstrahlung contributions) is not There exists an “amplitude” W ( ε B , ε 1 , . . . , ε N ) such that gauge invariant: � M ( ε B , ε 1 , . . . , ε N ) = M ( ε B , ε 1 , . . . , ε N ) + W ( ε B , ε 1 , . . . , ε N ) k A . . � = 0 satisfies . � M ( ε B , ε 1 , . . . , k i , . . . , ε N ) = 0 . The “gauge-restoring” amplitude W can be obtained by using the ordinary QCD Slavnov-Taylor identities.

Gauge-restoring amplitude Reduction formula for CCH factorization The high-energy amplitude with the proper kinematics can be implemented via �� � � � � ˜ � � � � � � µ A µ 1 µ 1 µ N � � � � k 2 k 2 k 2 M ( ε B , ε 1 , . . . , ε N ) = kA · pA → 0 lim lim lim . . . lim k T A � p B ε 1 ε . . . N ε G µ A µ B µ 1 ...µ N ( k A , k B , k 1 , . . . , k N ) , A B 1 N k 2 k 2 k 2 B → 0 1 → 0 N → 0 where ˜ G is the momentum-space Green function. Slavnov-Taylor (S-T) identity We apply the S-T identity to ˜ G : . . . . . . . . + . . . . = . + . + . + . . . After applying the reduction formula most of the terms vanish, except one The r.h.s term is precisely the amount of gauge-invariance violation and can be calculated (note however, this is not . . . = . . the “gauge-restoring” amplitude yet, as it contains the ex- . ternal ghost line).

Gauge-restoring amplitude (cont.) Remarks concerning gauges and ghosts • It is allowed to use two different gauges for on-shell lines and internal off-shell lines. • Ghosts do exist in the axial gauge (but usually decouple) 1 = ig f abc n µ , where n is a gauge vector. A ghost-gluon coupling in the axial gauge is The inverse ghost propagator is proportional to n · k . • Usually, when squaring an amplitude one uses sum over physical gluon polarization � ( q ) = − g µν + q µ k ν + q ν k µ ε ( λ ) µ ( q ) ε ( λ ) ν ∗ , k k q · k λ with some light-like momentum q . Alternatively, one can use external gluons in the Feynman gauge −→ and cut ghost loops . • The last remark allows us to trade an external ghost with momentum k to a gluon projected onto some light-like momentum q external ghost → ε k · q k · q 1 e.g. G. Leibrandt, Rev. Mod. Phys. (1987)

Gauge-restoring amplitude (cont.) It turns out that the gauge-restoring amplitude can be easily obtained by using axial gauge with gauge vector p A and summing all the gauge contributions with proper replacements of external ghosts. An exmaple for G ∗ G → GG Consider tree-level color-ordered ampli- k A k 2 tude G ∗ ( k A ) G ( k B ) → G ( k 1 ) G ( k 2 ) in ax- = ial gauge with gauge vector p A . k B k 1 Gauge contribution for replacement ε 1 ↔ k 1 = � � � � G 1 ( ε B , k 1 , ε 2 ) = g 2 + g 2 � � p A · ε B p A · ε 2 � � p A · ε B p A · ε 2 � � � � � � � � . k T A � k T A � 2 ( k A − k 2 ) · p A 2 ( k A + k B ) · p A Replacing external ghosts in favour of longitudinal gluon projection we get � � G 1 ( ε B , ε 1 , ε 2 ) = − g 2 � � p A · ε B p A · ε 1 p A · ε 2 � � � � k T A . � 2 k B · p A k 2 · p A Summing all the gauge contributions (for replacements ε B ↔ k B , ε 1 ↔ k 1 , ε 2 ↔ k 2 ) we obtain W ord ( ε B , ε 1 , ε 2 ) = G 1 ( ε B , ε 1 , ε 2 ) + G B ( ε B , ε 1 , ε 2 ) + G 2 ( ε B , ε 1 , ε 2 ) = G 1 ( ε B , ε 1 , ε 2 )