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) based on

• A. van Hameren, P

.K., K. Kutak, JHEP 1212 (2012) 029 supported by LIDER/02/35/L-2/10/NCBiR/2011

Motivation

Why "one-leg off-shell" amplitudes? LHC ⇒ very high energies ⇒ collinear factorization is not sufficient for some observables

• kT-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) factorization1 The CCH was originally stated for heavy quarks production in photo-, lepto- and hadro-production.

HARD

pA pB kA kB

At high energies the single longitudinal compo- nents of momentum transfers dominate k µ

A ≃ zApµ A + k µ T A,

k µ

B ≃ zBpµ B + k µ T B.

It is then argued that dσAB→QQ ≃

• d2kT A
• dzA

zA

• d2kT B
• dzB

zB F (zA, kT A) dσg∗g∗→QQ (zA, zB, kT A, kT B) F (zB, kT B), where F are unintegrated gluon distribution functions undergo- ing BFKL evolution. The hard process dσg∗g∗→QQ is calculated by contracting an off-shell amplitude (including external off-shell propagators) with pA, pB: where

= | kT A| pµ

A

= | kT B| pµ

B

It can be shown that dσg∗g∗→QQ is gauge invariant.

1 S. Catani, M. Ciafaloni, F. Hautmann (1990), (1991), (1994)

Introduction

Remarks concerning factorization theorems

Problems with kT-factorization

• CCH factorization is not a formal theorem of kT-factorization
• detailed all-order proofs exist for1
• SIDIS (Semi Inclusive DIS)
• Drell-Yan
• back-to-back jet or hadron production in DIS or e+e− annihilation
• kT-factorization in hadron-hadron collisions does not hold (also “generalized factorization” does not hold)
• an “effective” generalized kT-factorization for proton-nuclei collision was reported2

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

. Mulders, T. C. Rogers arXiv:1102.4569 [hep-ph]

2 F. Dominguez, C. Marquet, B. Xiao, F. Yuan (2011)

Introduction

Forward processes

Kinematics For high-energy kinematics and kA + kB → k1 + k2 subprocess we have k µ

A ≃ zApµ A + k µ T A,

k µ

B ≃ zBpµ B + k µ T B

k µ

i =

• kT i
• √

S

• eηi pµ

A + e−ηi pµ B

• + k µ

T i, i = 1, 2

Therefore zA =

• kT 1
• √

S eη1 +

• kT 2
• √

S eη2, zB =

• kT 1
• √

S e−η1 +

• kT 2
• √

S e−η2 ⇒ small longitudinal fractions are probed in highly asymmetric configuration.

• large fractions → collinear approach (with on-shell partons)
• small fractions → kT-factorization (with off-shell partons)

For instance, at CMS one can go to z1 ≈ 10−4, z2 ≈ 0.2 with high-pT jets (kT > 35 GeV) in HF detector (3 < |η| < 5).

Tree-level CCH factorization for multiple jets

Additional emissions: collinear factorization

HARD

. . .

Additional emissions: TMD factorization + . . . +

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 rules1.

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, εµ

k (q).

• the gauge invariance is crucial

Change of the reference momentum q → q′ amounts for the transformation εµ

k (q) = εµ k (q′) + k µβk (q, q′).

We can adjust q freely due to the Ward identity . . . = 0 where = kµ

• proper choice of q renders rather compact expressions for helicity amplitudes, which can be squared and

summed numerically The

• ff-shell

amplitude (without bremsstrahlung contributions) is not gauge invariant:

. . . = 0 kA

Let us denote the off-shell amplitude as M (εB, ε1 . . . , εN). There exists an “amplitude” W (εB, ε1, . . . , εN) such that

• M (εB, ε1, . . . , εN) = M (εB, ε1, . . . , εN) + W (εB, ε1, . . . , εN)

satisfies

• M (εB, ε1, . . . , ki, . . . , εN) = 0.

The “gauge-restoring” amplitude W can be obtained by using the

• rdinary QCD Slavnov-Taylor identities.

Gauge-restoring amplitude

Reduction formula for CCH factorization The high-energy amplitude with the proper kinematics can be implemented via M (εB, ε1, . . . , εN) = lim

kA ·pA →0 lim k2 B →0

lim

k2 1 →0

. . . lim

k2 N→0

• kT A
• p

µA A

k 2

Bε µ1 B

k 2

1 ε µ1 1

• . . .
• k 2

Nε µN N

˜ GµA µB µ1...µN (kA, kB, k1, . . . , kN), 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

A ghost-gluon coupling in the axial gauge is

= ig fabc nµ, where n is a gauge vector.

The inverse ghost propagator is proportional to n · k.

• Usually, when squaring an amplitude one uses sum over physical gluon polarization
• λ

ε(λ) µ

k

(q) ε(λ) ν ∗

k

(q) = −gµν + qµk ν + qνk µ q · k , 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 → εk · 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 pA and summing all the gauge contributions with proper replacements of external ghosts. An exmaple for G∗G → GG Consider tree-level color-ordered ampli- tude G∗ (kA) G (kB) → G (k1) G (k2) in ax- ial gauge with gauge vector pA. =

kA kB k1 k2

Gauge contribution for replacement ε1 ↔ k1

= G1 (εB, k1, ε2) = g2

2

• kT A
• pA · εB pA · ε2

(kA − k2) · pA + g2 2

• kT A
• pA · εB pA · ε2

(kA + kB) · pA . Replacing external ghosts in favour of longitudinal gluon projection we get G1 (εB, ε1, ε2) = − g2 2

• kT A
• pA · εB pA · ε1 pA · ε2

kB · pA k2 · pA . Summing all the gauge contributions (for replacements εB ↔ kB, ε1 ↔ k1, ε2 ↔ k2) we obtain Word (εB, ε1, ε2) = G1 (εB, ε1, ε2) + GB (εB, ε1, ε2) + G2 (εB, ε1, ε2) = G1 (εB, ε1, ε2)

Gauge-restoring amplitude (cont.)

Result for any number of legs One can prove the formula for W for any number of external gluons. For a fixed color ordering (A, B, 1, . . . , N) the sum of all gauge contributions collapses into a single term Word (εB, ε1, . . . , εN) = − −g √ 2 N

• kT A
• εB · pA ε1 · pA . . . εN · pA

kB · pA (kB − k1) · pA . . . (kB − . . . − kN−1) · pA

• Those amplitudes correspond to bremsstrahlung diagrams in “embedding approach”.
• The full gauge invariant amplitude ˜

M = M + W does satisfy ordinary collinear and soft behaviour

• It corresponds to Lipatov’s R → (N + 1) G effective vertex
• If we choose the reference momentum for polarization vector of any of the external gluons to be pA the

amplitude W vanishes due to the property of polarization vectors εk (q) · q = 0.

• We have explicit analytical expressions for helicity amplitudes for G∗G → GG, G∗G → GGG, G∗G → GQQ
• Approach tested numerically up to N = 10

Summary

• Our task was to develop some methods/tools for efficient calculations of multi-partonic tree-level amplitudes

relevant for high-energy factorization

• First step – one-leg off-shell amplitudes
• We have reconstructed the gauge-restoring contribution using just the Slavnov-Taylor identities and it does

conform to Lipatov’s vertex

• The gauge-restoring amplitude allows for using the helicity method and existing tools for matrix elements
• Also a prescription for G∗G∗ → anything at tree-level for efficient numerical calculation was developed

(A. van Hameren, P .K., K. Kutak JHEP 1301 (2013) 078)