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

SLIDE 1

Selection of recent theory and phenomenology developments in forward physics 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), arXiv:1308.0452 078 supported by LIDER/02/35/L-2/10/NCBiR/2011

SLIDE 2

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

SLIDE 3

High Energy Factorization

Production of a state X in a collision of hadrons A, B at high energies dσAB→X = d2kT A π dxA xA d2kT B π dxB xB F (xA, kT A) dσg∗g∗→X (xA, xB, kT A, kT B) F (xB, kT B)

motivated by CCH factorization 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

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

originally F are BFKL unintegrated gluon densities

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

2

SLIDE 4

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

in terms of the Lipatov’s effective action the correct HARD part corresponds to

Quasi-Multi-Regge kinematics; one can use the resulting Feynman rules1

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 kT-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

SLIDE 5

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) → more details on A. van Hameren’s talk

1 A. van Hameren, P

. Kotko, K. Kutak, JHEP 1301 (2013) 078

4

SLIDE 6

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

5

SLIDE 7

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

6

SLIDE 8

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 7

SLIDE 9

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. 8

SLIDE 10

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

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

9

SLIDE 11

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

10

SLIDE 12

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

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 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) 0.2 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

11

SLIDE 13

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

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)

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)

12

SLIDE 14

Summary and outline

any tree-level process can be calculated within high energy factorization
at small x non-linearities may be important – new evolution equations

relevant for large pT

new MC programs: LxJet (C++, ROOT, FOAM) and OSCARS (fortran code

similar to HELAC) Future developments:

nonlinear extension of CCFM evolution equation1,2

F

x, k 2

T, p

= F0
x, k 2

T , p

+ αsNc

π 1

x/x0

dz z

d2qT

πq2

T

θ (p − qTz) ∆NS (z, kT, qT)         F x z ,

kT +

qT

, qT
−

πα2

s

4NcR2 q2

Tδ

q2

T − k 2 T

∇2

qT

        ∞

q2 T

dl2

T

l2

T

ln       l2

T

k 2

T

      F

x, l2

T , lT

       

2

       

final state parton shower
better understanding of ’factorization’ in a nonlinear regime
NLO corrections to the hard process

1 K. Kutak JHEP 1212 (2012) 033, 2 currently investigated by D. Toton