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

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Helicity off-shell matrix elements within high energy factorization

Piotr Kotko

Institute of Nuclear Physics (Cracow) based on

A. van Hameren, P

.K., K. Kutak JHEP 1212 (2012) 029, JHEP 1301 (2013) 078, arXiv:1308.0452 (accepted for PRD) supported by LIDER/02/35/L-2/10/NCBiR/2011

SLIDE 2

Introduction

Collinear factorization

hard subprocess defined via
n-shell amplitudes
parton densities depend on x

and scale

proved to all orders for large

classes of processes

plenty of automatic tools for

tree-level amplitudes with multiple final states

NLO calculations automated,

some NNLO results High-energy factorization

hard subprocess defined via
ff-shell amplitudes
‘unintegrated’ parton densities

(UPDFs) depend also on transverse momentum

UPDFs are not universal

(factorization does not hold to all orders)

tree-level amplitudes (up to five

legs) obtained analytically

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

SLIDE 3

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

SLIDE 4

High Energy Factorization

CCH factorization (Catani, Ciafaloni, Hautmann) for heavy quark production1

HARD pA pB kA kB

high-energy kinematics: k µ

A ≃ xApµ A + k µ T A

k µ

B ≃ xBpµ B + k µ T B

Gauge choice – axial gauge with n = αpA + βpB The HARD part is defined by the eikonal projectors where

= | kT A| pµ

A

= | kT B| pµ

B

⇒ the amplitude g∗g∗ → QQ is gauge invariant dσAB→QQ = d2kT A π dxA xA d2kT B π dxB xB F (xA, kT A) dσg∗g∗→QQ (xA, xB, kT A, kT B) F (xB, kT B)

originally F are BFKL unintegrated gluon densities

1 S. Catani, M. Ciafaloni, F. Hautmann, Nucl.Phys. B366 (1991) 135-188

3

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High Energy Factorization (cont.)

for a generic multiparticle state X the amplitude g∗g∗ → X is not gauge invariant

HARD . . . kA kB

⇒ additional terms are needed to recover the gauge invariance

+ . . . +

HARD

· · ·

HARD . . . . . .

· · ·

one of the approaches – Lipatov’s effective action; the gauge invariant HARD

sub-process corresponds to Quasi-Multi-Regge kinematics1

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)

1

4

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High Energy Factorization (cont.)

Transverse Momentum Dependent (TMD) factorization1

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 DIS2
generalized factorization derived for back-to-back-like dijets in dilute-dense

systems3

so called “hybrid” factorization (single unintegrated gluon density) might be valid

1 e.g. P

. Mulders, ArXiv:11024569

2 S. Catani, F. Hautmann, Nucl.Phys. B427 (1994) 3 F. Dominguez, C. Marquet, B. Xiao, F. Yuan, Phys.Rev. D83 (2011) 105005

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Automatic Off-shell Helicity Amplitudes

An amplitude g∗ (kA) g∗ (kB) → X can be disentangled from qAqB → q′

Aq′ BX.

. . .

kA kB

. . .

kA kB

. . .

kB

+ + . . . However, if we want to have explicit high-energy kinematics for kA, kB the quarks q′

A, q′ B cannot be on-shell ⇒ amplitude for qAqB → q′ Aq′ BX is not gauge invariant

It’s possible to have both on-shellness for all external partons and high-energy kinematics1: → the amplitude qA (pA) qB (pB) → q′

A

p′

A

q′

B

p′

B

X need not to be physical

→ 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

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Automatic Off-shell Amplitudes (cont.)

Introduce the basis using real four vectors l1, l2 and complex l3, l4 defined as l µ

3 = 1

2 l2; −| γµ |l1; −, l µ

4 = 1

2 l1; −| γµ |l2; −. They have properties: l2

i = 0, l1,2 · l3,4 = 0, l1 · l2 = −l3 · l4.

The momenta of external quarks are decomposed as follows p µ

A = (Λ + xA) l µ 1 − l4 · kT A

l1 · l2 l µ

3 ,

p µ

B = (Λ + xB) l µ 2 − l3 · kT B

l1 · l2 l µ

4

p′ µ

A = Λl µ 1 + l3 · kT A

l1 · l2 l µ

4 ,

p′ µ

B = Λl µ 2 + l4 · kT B

l1 · l2 l µ

3 ,

We get both the on-shellness p2

A,B = p′ 2 A,B and high-energy limit for any Λ.

Moreover the external spinors for quarks |pA; − ∝ |l1; −, |pB; − ∝ |l2; − 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)

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SLIDE 9

Forward Processes and High Energy factorization

Forward processes (relevant for small x) correspond to asymmetric configurations xA =

i
pT i
√

S exp (ηi) xB =

i
pT i
√

S exp (−ηi) xas = |xA − xB| / (xA + xB)

as

x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dσ/d xa [pb]

2

10

1

10 1 10

2

10

3

10

4

10

kinematic cuts : 35 GeV < pT 3 < pT 2 < pT 1 |η1,2| < 2.8 3.2 < |η3| < 4.7 3 jets production at √ = 5.02 TeV proton nonlinear Pb nonlinear proton linear

This accounts for a simplification:

large fractions xB → collinear approach (with on-shell parton)
small fractions xA → high energy factorization (with off-shell parton)

dσAB→X =

b

d2kT A π dxA xA

dxB F (xA, kT A) fb/B (xB) dσg∗a→X (xA, xB, kT A)1
1M. Deak, F. Hautmann, H. Jung, K. Kutak, JHEP 0909 (2009) 121

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One-leg Off-shell Helicity Amplitudes

A contribution to N-jet process: g∗g → gg . . . g . . . kA ≡ M (ε1, . . . , εN)

not gauge-invariant

. . . = 0 kA

M (ε1 . . . , ki, . . . , εN) 0

one cannot use helicity method,

i.e. εµ

k (q) = εµ k (q′) + k µβk (q, q′)

there exists an “amplitude” W such that

M = M + W satisfies

M (ε1, . . . , ki, . . . , εN) = 0
the “gauge-restoring” amplitude W can be obtained by using the ordinary QCD

Slavnov-Taylor identities1

1 A. van Hameren, P

. Kotko, K. Kutak, JHEP 1212 (2012) 029

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One-leg Off-shell Helicity Amplitudes (cont.)

introduce a reduction formula for the off-shell amplitude (˜

G – the Green function) M (ε1, . . . , εN) = lim

kA ·pA →0 lim k 2

1 →0 . . . lim

k 2

N→0

kT A
pµA

A

k 2

1 εµ1 1

. . .
k 2

NεµN N

˜ GµA µ1...µN

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

SLIDE 12

One-leg Off-shell Helicity Amplitudes (cont.)

The complete color-ordered result is

A (ε1, . . . , εN) = −
kT A


     kT A · J (ε1, . . . , εN) + −g √ 2 N ε1 · pA . . . εN · pA k1 · pA (k1 − k2) · pA . . . (k1 − . . . − kN−1) · pA        where (below kij = ki + ki+1 + . . . + kj) Jµ (ε1, . . . , εN) = −i k 2

1N

gµ

ν −

k µ

1NpA ,ν + k1N νpµ A

k1N · pA



     

N−1

i=1

Vναβ

3

k1i, k(i+1)N
Jα (ε1, . . . , εi) Jβ (εi+1, . . . , εN)

+

N−2

i=1

N−1

j=i+1

Vναβγ

4

Jα (ε1, . . . , εi) Jβ (εi+1, . . . , εj) Jγ (εj+1, . . . , εN)        This result is consistent with Lipatov’s effective action. 11

SLIDE 13

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 saturation1 fitted to HERA data2

F

x, k 2

T

= F0
x, k 2

T

+ αsNc

π 1

x

dz z ∞

k2 T 0

dq2

T

q2

T

               q2

TF

x

z , q2 T

θ
k2

T z − q2 T

− k 2

TF

x

z , k 2 T

q2

T − k 2 T

+

k 2

TF

x

z , k 2 T

4q4

T + k 4 T

               + αs 2πk 2

T

1

x

dz       

Pgg (z) − 2Nc

z k2

T k2 T 0

dq2

TF

x z , q2

T

+ zPgq (z) Σ

x z , k 2

T

       − 2α2

s

R2                  ∞

k2 T

dq2

T

q2

T

F

x, q2

T

       

2

+ F

x, k 2

T

∞

k2 T

dq2

T

q2

T

ln       q2

T

k 2

T

      F

x, q2

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 nuclei2

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

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Applications: forward-central three jet production

decorrelations (φ13 = |φ1 − φ3|)

ϕ13 1 2 3 4 5 6 dσ/dϕ13 [pb] 1 10

2

10

3

10

4

10

5

10

kinematic cuts : 35 GeV < pT 3 < pT 2 < pT 1 |η1,2| < 2.8 3.2 < |η3| < 4.7 3 jets production at √ = 7.0 TeV proton nonlinear Pb nonlinear proton linear

ϕ13 1 2 3 4 5 6 dσ/dϕ13 [pb]

1

10 1 10

2

10

3

10

kinematic cuts : 35 GeV < pT 3 < pT 2 < pT 1 |η1,2| < 2.8 3.2 < |η3| < 4.7 |⃗pT 1 + ⃗pT 2| < 30 GeV 3 jets production at √ = 7.0 TeV proton nonlinear Pb nonlinear proton linear

unbalanced pT of the jets

∆pT 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 dσ/d ln(∆pT) [pb]

2

10

3

10

kinematic cuts : 35 GeV < pT 3 < pT 2 < pT 1 |η1,2| < 2.8 3.2 < |η3| < 4.7 3 jets production at √ = 7.0 TeV proton nonlinear Pb nonlinear proton linear

xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx

∆pT 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 R(∆pT) 0.6 0.8 1 1.2 1.4 1.6

kinematic cuts : 35 GeV < pT 3 < pT 2 < pT 1 |η1,2| < 2.8 3.2 < |η3| < 4.7 3 jets production at √s = 7.0 TeV

13

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Applications: forward three jet production

decorrelations (φ13 = |φ1 − φ3|)

ϕ13 1 2 3 4 5 6 dσ/dϕ13 [pb] 1 10

2

10

3

10

kinematic cuts : 20 GeV < pT 3 < pT 2 < pT 1 3.2 < |η1,2,3| < 4.9 3 jets production at √ = 7.0 TeV proton nonlinear Pb nonlinear proton linear

xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx xxxxxxx

ϕ13 1 2 3 4 5 6 R(ϕ13) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

kinematic cuts : 20 GeV < pT 3 < pT 2 < pT 1 3.2 < |η1,2,3| < 4.9 3 jets production at √s = 7.0 TeV

unbalanced pT of the jets

∆pT 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 dσ/d ln(∆pT) [pb] 10

2

10

kinematic cuts : 20 GeV < pT 3 < pT 2 < pT 1 3.2 < |η1,2,3| < 4.9 3 jets production at √ = 7.0 TeV proton nonlinear Pb nonlinear proton linear

∆pT 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 R(∆pT) 0.2 0.4 0.6 0.8 1 1.2 1.4

kinematic cuts : 20 GeV < pT 3 < pT 2 < pT 1 3.2 < |η1,2,3| < 4.9 3 jets production at √s = 7.0 TeV

14

SLIDE 16

Applications: weak processes

Nuclear modification factors for bb production

gg∗ → b¯

b µ+µ−

pT q(¯

q) > 20 GeV

yq(¯

q) < 2.5

pT µ± > 20 GeV yµ± < 2.1 ∆Rq,¯

q > 0.4

∆Rq(¯

q), µ± > 0.4

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 20 40 60 80 100 120 140

pT(bb∼) ratio

g g* −− > b b∼ µ+ µ−

(proton-Pb)/(proton-proton)

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

2
1.5
1
0.5

0.5 1 1.5

rapidity(bb∼) ratio

g g* −− > b b∼ µ+ µ−

(proton-Pb)/(proton-proton)

ug∗ → b¯

b µ+νµ d

pT q(¯

q) > 20 GeV

yq(¯

q) < 2.5

20 GeV < pT µ+ < 50 GeV yµ+ < 2.1 ET > 20 GeV ∆Rq,¯

q > 0.4

∆Rq(¯

q), µ± > 0.4

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 20 40 60 80 100 120 140

pT(bb∼) ratio

u g* −− > b b∼ µ+ νµ d

(proton-Pb)/(proton-proton)

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

2
1.5
1
0.5

0.5 1 1.5

rapidity(bb∼) ratio

u g* −− > b b∼ µ+ νµ d

(proton-Pb)/(proton-proton)

15

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Future developements

Jets within hybrid factorization at NLO

virtual corrections

→ loops in axial gauge – Axiloops1

real corrections

→ dipole subtraction method for jets with massive initial states2 (prepared for ACOT (Aivazis-Collins-Olness-Tung) factorization with massive quarks) Evolution equations for unintegrated gluons

nonlinear extension of CCFM evolution3

More off-shell legs, parton showers, MPI, ...

1 O. Gituliar, gituliar.org 2 P

. Kotko, W. Slominski, Phys.Rev. D86 (2012) 094008

3 K. Kutak JHEP 1212 (2012) 033

16

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Summary

any tree-level process can be calculated within high energy factorization
two off-shell legs ⇒ an embedding approach with complex momenta,

effectively leading to eikonal Feynman rules

single-leg-off-shell amplitudes (relevant for forward processes within

“hybrid” factorization) ⇒ the Slavnov-Taylor identity-based approach

new MC programs
fortran code similar to HELAC – OSCARS (Off-Shell Currents And

Related Stuff)

C++ program using ROOT and Foam (of S. Jadach) – LxJet
phenomenological applications
three-jet forward production in p-p and p-Pb collisions with saturation
sample results for bb and lepton production