[PPT] - Intro dution A full NLLx example: Mueller-Navelet jets PowerPoint Presentation

SLIDE 1 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results First al ulation

f

Mueller Navelet jets at LHC at a

mplete

NLL BFKL

rder

Samuel W allon Universit Pierre et Ma rie Curie and Lab

ratoire

de Physique Tho rique CNRS / Universit P a ris Sud Orsa y Semina r High Energy Physi s, Depa rtment

f

Physi s & Astronomy , Universit y College London London, Ap ril 15th 2011 in

llab
ration

with D. Colferai (Firenze), F. S hw ennsen (DESY), L. Szymano wski (SINS, V a rsa w) JHEP 1012:026 (2010) 1-72 [a rXiv:1002.1365 [hep-ph℄℄ 1 / 36

SLIDE 2 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Motivations One

f

the imp

rtant

longstanding theo reti al questions raised b y QCD is its b ehaviour in the p erturbative Regge limit s ≫ −t Based

n

theo reti al grounds,

ne

should identify and test suitable

bservables

in

rder

to test this p e ulia r dynami s PSfrag repla ements

h1(M 2

1 )

h2(M 2

2 )

s → t ↓ ←

va uum quantum numb er

h′

1(M ′2 1 )

h′

2(M ′2 2 )

ha rd s ales: M 2

1 , M 2 2 ≫ Λ2 QCD

r M ′2

1 , M ′2 2 ≫ Λ2 QCD

r t ≫ Λ2

QCD

where the t− hannel ex hanged state is the so- alled ha rd P

meron

2 / 36

SLIDE 3 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Ho w to test QCD in the p erturbative Regge limit? What kind

f
bservable?

p erturbation theo ry should b e appli able: sele ting external

r

internal p rob es with transverse sizes ≪ 1/ΛQCD (ha rd γ∗ , heavy meson (J/Ψ , Υ ), energeti fo rw a rd jets)

r

b y ho

sing

la rge t in

rder

to p rovide the ha rd s ale. governed b y the "soft" p erturbative dynami s

f

QCD PSfrag repla ements

p → 0

and not b y its

llinea

r dynami s PSfrag repla ements

m = 0 m = 0

θ → 0 = ⇒

sele t semi-ha rd p ro esses with s ≫ p2

T i ≫ Λ2 QCD

where p2

T i

a re t ypi al transverse s ale, all

f

the same

rder.

3 / 36

SLIDE 4 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Ho w to test QCD in the p erturbative Regge limit? Some examples

f

p ro esses in lusive: DIS (HERA), dira tive DIS, total γ∗γ∗ ross-se tion (LEP , ILC) semi-in lusive: fo rw a rd jet and π0 p ro du tion in DIS, Mueller-Navelet double jets, dira tive double jets, high pT entral jet, in hadron-hadron

lliders

(T evatron, LHC) ex lusive: ex lusive meson p ro du tion in DIS, double dira tive meson p ro du tion at e+e−

lliders

(ILC), ultrap eripheral events at LHC (P omeron, Odderon) 4 / 36

SLIDE 5 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results The sp e i ase

f

QCD at la rge s QCD in the p erturbative Regge limit Small values

f αS

(p erturbation theo ry applies due to ha rd s ales) an b e

mp

ensated b y la rge ln s enhan ements. ⇒ resummation

f

P

n(αS ln s)n

series (Balitski, F adin, Kuraev, Lipatov)

A = + B @ + + · · · 1 C A + B @ + · · · 1 C A + · · · ∼ s ∼ s (αs ln s) ∼ s (αs ln s)2

this results in the ee tive BFKL ladder PSfrag repla ements gluon reggeon = "dressed gluon" ee tive vertex

= ⇒ σh1 h2→anything

tot

= 1 s ImA ∼ sαP(0)−1

with αP(0) − 1 = C αs

(C > 0)

Leading Log P omeron Balitsky , F adin, Kuraev, Lipatov 5 / 36

SLIDE 6 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Op ening the b

xes:

Impa t rep resentation γ∗ γ∗ → γ∗ γ∗ as an example Sudak

v

de omp

sition: ki = αi p1 + βi p2 + k⊥i

(p2

1 = p2 2 = 0, 2p1 · p2 = s)

write

d4ki = s

2 dαi dβi d2k⊥i

(k = Eu l. ↔ k⊥ = Mink.)

t− hannel

gluons have non-sense p

la

rizations at la rge s : ǫup/down

NS

= 2

s p2/1

PSfrag repla ements

⇒

set α1 = 0 and

R dβ1 ⇒ Φγ∗→γ∗(k1, r − k1)

impa t fa to r

⇒

set βn = 0 and

R dαn ⇒ Φγ∗→γ∗(−kn, −r + kn)

β ր α ց γ∗ γ∗ r − k1 k1 k2 kn α1 α2

M = is (2π)2 Z d2k k2 Φup(k, r − k) Z d2k′ k′2 Φdown(−k′, −r + k′) ×

δ+i∞

Z

δ−i∞

dω 2πi „ s s0 «ω Gω(k, k′, r)

αn−1

← −

multi-Regge kinemati s

β2 βn αq, ¯

q

βq, ¯

q 6 / 36

SLIDE 7 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results higher

rder
rre tions

Higher

rder
rre tions

to BFKL k ernel a re kno wn at NLL

rder

(Lipatov F adin; Cami i, Ciafaloni), no w fo r a rbitra ry impa t pa rameter

αS P

n(αS ln s)n

resummation impa t fa to rs a re kno wn in some ases at NLL

γ∗ → γ∗

at t = 0 (Ba rtels, Colferai, Giesek e, Kyrieleis, Qiao) fo rw a rd jet p ro du tion (Ba rtels, Colferai, V a a)

γ∗

L → ρL

in the fo rw a rd limit (Ivanov, K

tsky

, P apa) note: fo r ex lusive p ro esses, some transitions ma y sta rt at t wist3, fo r whi h almost nothing is kno wn. The rst

mputation
f

the γ∗

T → ρT

t wist 3 transition at LL has b een p erfo rmed

nly

re ently I. V. Anikin, D. Y. Ivanov, B. Pire, L. Szymano wski and S. W. Phys. Lett. B 688:154-167, 2010; Nu l. Phys. B 828:1-68, 2010. 7 / 36

SLIDE 8 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Mueller-Navelet jets: Basi s Mueller Navelet jets Consider t w

jets

(hadron paquet within a na rro w

ne)

sepa rated b y a la rge rapidit y, i.e. ea h

f

them almost y in the dire tion

f

the hadron lose to it, and with very simila r transverse momenta in a pure LO

llinea

r treatment, these t w

jets

should b e emitted ba k to ba k at leading

rder: ∆φ − π = 0

(∆φ = φ1 − φ2 = relative azimutal angle) and k⊥1 =k⊥2 . There is no phase spa e fo r (untagged) emission b et w een them PSfrag repla ements

p(p1) p(p2)

jet1

(k⊥1, φ1)

jet2

(k⊥2, φ2)

φ1 φ2 − π

la rge + rapidit y la rge

rapidit

y zero rapidit y

⊥

plane Beam axis 8 / 36

SLIDE 9 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Mueller-Navelet jets at LL fails Mueller Navelet jets at LL BFKL PSfrag repla ements jet1 jet2 rapidit y gap rapidit y gap

| {z }

LL BFKL Green fun tion

llinea

r pa rton (PDF)

llinea

r pa rton (PDF) Multi-Regge kinemati s (LL BFKL) in LL BFKL (∼ P(αs ln s)n ), emission b et w een these jets

− →

strong de o rrelation b et w een the relative azimutal angle jets, in ompatible with p¯

p

T evatron

llider

data a

llinea

r treatment at next-to-leading

rder

(NLO) an des rib e the data imp

rtant

issue: non- onservation

f

energy-momentum along the BFKL ladder. A BFKL-based Monte Ca rlo

mbined

with e-m

nservation

imp roves dramati ally the situation (Orr and Stirling) 9 / 36

SLIDE 10 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Studies at LHC: Mueller-Navelet jets Mueller Navelet jets at NLL BFKL PSfrag repla ements jet1 NLL jet vertex jet2 NLL jet vertex rapidit y gap rapidit y gap

| {z }

NLL BFKL Green fun tion

llinea

r pa rton (PDF)

llinea

r pa rton (PDF) Quasi Multi-Regge kinemati s (here fo r NLL BFKL) up to no w, the subseries αs

P(αs ln s)n

NLL w as in luded

nly

in the ex hanged

P omeron

state, and not inside the jet verti es Sabio V era, S hw ennsen Ma rquet, Ro y

n

the

mmon

b elief w as that these

rre tions

should not b e imp

rtant

10 / 36

SLIDE 11 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Jet vertex: LL versus NLL

k, k′ =

Eu lidian t w

dimensional

ve to rs LL jet vertex: PSfrag repla ements

k k

NLL jet vertex: PSfrag repla ements

k k′ k − k′ k′

11 / 36

SLIDE 12 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Jet vertex: jet algo rithms Jet algo rithms a jet algo rithm should b e IR safe, b

th

fo r soft and

llinea

r singula rities the most

mmon

jet algo rithm a re:

kt

algo rithms (IR safe but time

nsuming

fo r multiple jets

ngurations)
ne

algo rithm (not IR safe in general; an b e made IR safe at NLO: Ellis, Kunszt, Sop er) 12 / 36

SLIDE 13 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Jet vertex: jet algo rithms Cone jet algo rithm at NLO (Ellis, Kunszt, Sop er) Should pa rtons (|p1|, φ1, y1) and (p2|, φ2, y2)

mbined

in a single jet?

|pi| = transverse

energy dep

sit

in the alo rimeter ell i

f

pa rameter

Ω = (yi, φi)

in y − φ plane dene transverse energy

f

the jet: pJ = |p1| + |p2| jet axis:

Ωc 8 > > < > > :

yJ = |p1| y1 + |p2| y2 pJ φJ = |p1| φ1 + |p2| φ2 pJ

PSfrag repla ements pa rton1 (Ω1, |p1|) pa rton2 (Ω2, |p2|)

ne

axis (Ωc)

Ω = (yi, φi)

in y − φ plane If distan es |Ωi − Ωc|2 ≡ (yi − yc)2 + (φi − φc)2 < R2 (i = 1 and i = 2 )

= ⇒

pa rtons 1 and 2 a re in the same

ne Ωc
mbined
ndition: |Ω1 − Ω2| <

|p1| + |p2| max(|p1|, |p2|)R

13 / 36

SLIDE 14 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Jet vertex: LL versus NLL and jet algo rithms LL jet vertex and

ne

algo rithm

k, k′ =

Eu lidian t w

dimensional

ve to rs PSfrag repla ements

0, x k k, x S(2)

J (k⊥; x) = δ

“ 1 − xJ x ” |k| δ(2)(k − kJ)

14 / 36

SLIDE 15 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Jet vertex: LL versus NLL and jet algo rithms NLL jet vertex and

ne

algo rithm

k, k′ =

Eu lidian t w

dimensional

ve to rs

S(3,cone)

J

(k′, k − k′, xz; x) =

PSfrag repla ements

0, x k k, x

S(2)

J (k, x) Θ

„h

|k−k′|+|k′| max(|k−k′|,|k′|)Rcone

i2 − ˆ ∆y2 + ∆φ2˜«

PSfrag repla ements

0, x k k′ k − k′, x z k, x(1 − z)

+ S(2)

J (k − k′, xz) Θ

„ˆ ∆y2 + ∆φ2˜ − h

|k−k′|+|k′| max(|k−k′|,|k′|)Rcone

i2«

PSfrag repla ements

0, x k k′ k − k′, x z k, x(1 − z)

+ S(2)

J (k′, x(1 − z)) Θ

„ˆ ∆y2 + ∆φ2˜ − h

|k−k′|+|k′| max(|k−k′|,|k′|)Rcone

i2« ,

15 / 36

SLIDE 16 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Mueller-Navelet jets at NLL and niteness Using a IR safe jet algo rithm, Mueller-Navelet jets at NLL a re nite UV se to r: the NLL impa t fa to r

ntains

UV divergen ies 1/ǫ they a re abso rb ed b y the reno rmalization

f

the

upling: αS −

→ αS(µR)

IR se to r: PDF have IR

llinea

r singula rities: p

le 1/ǫ

at LO these

llinea

r singula rities an b e

mp

ensated b y

llinea

r singula rities

f

the t w

jets

verti es and the real pa rt

f

the BFKL k ernel the remaining

llinea

r singula rities

mp

ensate exa tly among themselves soft singula rities

f

the real and virtual BFKL k ernel, and

f

the jets verti es

mp

ensates among themselves This w as sho wn fo r b

th

qua rk and gluon initiated verti es (Ba rtels, Colferai, V a a) 16 / 36

SLIDE 17 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Master fo rmulas

kT

fa to

rized dierential ross-se tion

x1 x2 ↓ k1, φ1 ↓ k2, φ2 kJ,1, φJ,1, xJ,1 kJ,2, φJ,2, xJ,2

dσ d|kJ,1| d|kJ,2| dyJ,1 dyJ,2 = Z

dφJ,1 dφJ,2

Z

d2k1 d2k2

× Φ(kJ,1, xJ,1, −k1) × G(k1, k2, ˆ s) × Φ(kJ,2, xJ,2, k2)

with Φ(kJ,2, xJ,2, k2) =

R dx2 f(x2)V (k2, x2) f ≡

PDF

xJ = |kJ |

√s eyJ

17 / 36

SLIDE 18 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Master fo rmulas Angula r

e ients

Cm ≡ Z dφJ,1 dφJ,2 cos ` m(φJ,1 − φJ,2 − π) ´ × Z d2k1 d2k2 Φ(kJ,1, xJ,1, −k1) G(k1, k2, ˆ s) Φ(kJ,2, xJ,2, k2). m = 0 = ⇒

ross-se tion

dσ d|kJ,1| d|kJ,2| dyJ,1 dyJ,2 = C0 m > 0 = ⇒

azimutal de o rrelation

cos(mϕ) ≡ cos ` m(φJ,1 − φJ,2 − π) ´ = Cm C0

18 / 36

SLIDE 19 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Master fo rmulas in

nfo

rmal va riables Rely

n

LL BFKL eigenfun tions LL BFKL eigenfun tions: En,ν(k1) =

1 π √ 2

` k2

1

´iν− 1

2 einφ1

de omp

se Φ
n

this basis use the kno wn LL eigenvalue

f

the BFKL equation

n

this basis:

ω(n, ν) = ¯ αsχ0 ` |n|, 1

2 + iν

´

with χ0(n, γ) = 2Ψ(1) − Ψ

` γ + n

2

´ − Ψ ` 1 − γ + n

2

´

(Ψ(x) = Γ′(x)/Γ(x), ¯

αs = Ncαs/π

)

= ⇒

master fo rmula:

Cm = (4 − 3 δm,0) Z dν Cm,ν(|kJ,1|, xJ,1) C∗

m,ν(|kJ,2|, xJ,2)

„ ˆ s s0 «ω(m,ν)

with

Cm,ν(|kJ|, xJ) = Z dφJ d2k dx f(x)V (k, x)Em,ν(k) cos(mφJ)

at NLL, same master fo rmula: just hange ω(m, ν) and V 19 / 36

SLIDE 20 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results BFKL Green's fun tion at NLL NLL Green's fun tion: rely

n

LL BFKL eigenfun tions NLL BFKLk ernel is not

nfo

rmal inva riant LL En,ν a re not anymo re eigenfun tion this an b e

ver ome

b y

nsidering

the eigenvalue as an

p

erato r with a pa rt

ntaining

∂ ∂ν

it a ts

n

the impa t fa to r

ω(n, ν) = ¯ αsχ0 „ |n|, 1 2 + iν « + ¯ α2

s

" χ1 „ |n|, 1 2 + iν « − πb0 2Nc χ0 „ |n|, 1 2 + iν «  −2 ln µ2

R − i ∂

∂ν ln Cn,ν(|kJ,1|, xJ,1) Cn,ν(|kJ,2|, xJ,2) ff # , | {z } 2 ln |kJ,1| · |kJ,2| µ2

R

20 / 36

SLIDE 21 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results LL substra tion and s0

ne

sums up P(αs ln ˆ

s/s0)n + αs P(αs ln ˆ s/s0)n

(ˆ

s = x1 x2 s )

at LL s0 is a rbitra ry natural hoi e: s0 = √s0,1 s0,2 s0,i fo r ea h

f

the s attering

bje ts

p

ssible

hoi e: s0,i = (|kJ| + |kJ − k|)2 (Ba rtels, Colferai, V a a) but dep end

n k ,

whi h is integrated

ver

ˆ s

is not an external s ale (x1,2 a re integrated

ver)

w e p refer

s0,1 = (|kJ,1| + |kJ,1 − k1|)2 → s′

0,1 =

x2

1

x2

J,1

k2

J,1

s0,2 = (|kJ,2| + |kJ,2 − k2|)2 → s′

0,2 =

x2

2

x2

J,2

k2

J,2

9 > > > > > = > > > > > ; ˆ s s0 → ˆ s s′ = xJ,1 xJ2 s |kJ,1| |kJ,2| = eyJ,1−yJ,2 ≡ eY

s0 → s′

ae ts the BFKL NLL Green fun tion the impa t fa to rs:

ΦNLL(ki; s′

0,i) = ΦNLL(ki; s0,i) +

Z d2k′ ΦLL(k′

i) KLL(k′ i, ki)1

2 ln s′

0,i

s0,i

(1) numeri al stabilities (non azimuthal averaging

f

LL substra tion) imp roved with the hoi e s0,i = (ki − 2kJ,i)2 (then repla ed b y s′

0,i

after numeri al integration) (1) an b e used to test s0 → λ s0 dep enden e 21 / 36

SLIDE 22 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Collinea r imp rovement at NLL Collinea r imp roved Green's fun tion at NLL

ne

ma y imp rove the NLL BFKLk ernel fo r n = 0 b y imp

sing

its

mpatibilit

y with DGLAP in the

llinea

r limit Salam; Ciafaloni, Colferai usual (anti) ollinea r p

les

in γ = 1/2 + iν (resp. 1 − γ ) a re shifted b y ω/2

ne

p ra ti al implementation: the new k ernel ¯

αsχ(1)(γ, ω)

with shifted p

les

repla es

¯ αsχ0(γ, 0) + ¯ α2

sχ1(γ, 0)

ω(0, ν)

is

btained

b y solving the impli it equation

ω(0, ν) = ¯ αsχ(1)(γ, ω(0, ν))

fo r ω(n, ν) numeri ally . there is no need fo r any jet vertex imp rovement b e ause

f

the absen e

f

γ

and 1 − γ p

les

(numeri al p ro

f

using Cau hy theo rem ba kw a rd) 22 / 36

SLIDE 23 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Numeri al implementation In p ra ti e MSTW 2008 PDF s (available as Mathemati a pa k ages)

µR = µF

(this is imp

sed

b y the MSTW 2008 PDF s) t w

-lo
p

running

upling αs(µ2

R)

W e use a ν grid (with a dense sampling a round 0) all numeri al al ulations a re done in Mathemati a w e use Cuba integration routines (in p ra ti e V egas): p re ision 10−2 fo r 500.000 max p

ints

p er integration mapping |k| = |kJ| tan(ξπ/2) fo r k integrations ⇒ [0, ∞[ → [0, 1] although fo rmally the results should b e nite, it requires a sp e ial grouping

f

the integrand in

rder

to get stable results

= ⇒

14 minimal stable basi blo ks to b e evaluated numeri ally 23 / 36

SLIDE 24 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Results: symmetri

nguration

(|kJ,1| = |kJ,2| = 35 GeV ) Cross-se tion pure LL LL verti es + imp roved

llinea

r NLL Green's fun tion NLL verti es + NLL Green's fun tion NLL verti es + imp roved

llinea

r NLL Green's fun tion

6 7 8 9 10 0.01 0.1 1

PSfrag repla ements

C0 ˆ

nb GeV2

˜ = σ Y

Dierential ross se tion in dep enden e

n Y

fo r |kJ,1| = |kJ,2| = 35 GeV . erro r bands=erro rs due to the Monte Ca rlo integration (2% to 5%) The ee t

f

NLL vertex

rre tion

is very sizeable,

mpa

rable with NLL Green's fun tion ee ts 24 / 36

SLIDE 25 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Results: symmetri

nguration

(|kJ,1| = |kJ,2| = 35 GeV ) Cross-se tion: stabilit y with resp e t to µR = µF and s0 hanges pure LL LL verti es + imp roved

llinea

r NLL Green's fun tion NLL verti es + NLL Green's fun tion NLL verti es + imp roved

llinea

r NLL Green's fun tion

7 8 9 10 0.5 0.5 1.0 1.5 7 8 9 10 0.2 0.1 0.1 0.2

PSfrag repla ements

δC0 ˆ

nb GeV2

˜ δC0 ˆ

nb GeV2

˜ Y Y

Relative ee t

f

hanging µR = µF b y fa to rs 2 (thi k) and 1/2 (thin) Relative ee t

f

hanging √s0 b y fa to rs 2 (thi k) and 1/2 (thin) 25 / 36

SLIDE 26 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

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Results Results: symmetri

nguration

(|kJ,1| = |kJ,2| = 35 GeV ) Cross-se tion: PDF and Monte Ca rlo erro rs pure LL LL verti es + imp roved

llinea

r NLL Green's fun tion NLL verti es + NLL Green's fun tion NLL verti es + imp roved

llinea

r NLL Green's fun tion

7 8 9 10 0.2 0.1 0.1 0.2 0.3 7 8 9 10 0.06 0.04 0.02 0.02 0.04 0.06

PSfrag repla ements

δC0 ˆ

nb GeV2

˜ δC0 ˆ

nb GeV2

˜ Y Y

Relative ee t

f

the PDF erro rs Relative ee t

f

the Monte Ca rlo erro rs 26 / 36

SLIDE 27 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: symmetri

nguration

(|kJ,1| = |kJ,2| = 35 GeV ) Azimuthal

rrelation

pure LL LL verti es + imp roved

llinea

r NLL Green's fun tion NLL verti es + NLL Green's fun tion NLL verti es + imp roved

llinea

r NLL Green's fun tion

6

7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ

Y

erro r bands = erro rs due to the Monte Ca rlo integration dots = results

btained

with Pythia (DGLAP LL MC) squa res = results

btained

with Herwig (DGLAP LL MC) LL → NLL verti es hange results dramati ally A t NLL, the de o rrelation is very lose to LL DGLAP t yp e

f

Monte Ca rlo 27 / 36

SLIDE 28 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: symmetri

nguration

(|kJ,1| = |kJ,2| = 35 GeV ) Azimuthal

rrelation:

dep enden y with resp e t to µR = µF and s0 hanges pure LL LL verti es + imp.

llinea

r NLL Green's fn. NLL verti es + NLL Green's fn. NLL verti es + imp.

llinea

r NLL Green's fn.

6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2 6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ C1 C0 = cos ϕ

Y Y

Ee t

f

hanging µR = µF b y fa to rs 2 (thi k) and 1/2 (thin)

6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2 6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ C1 C0 = cos ϕ

Y Y

Ee t

f

hanging √s0 b y fa to rs 2 (thi k) and 1/2 (thin)

cos ϕ

is still rather µR = µF and s0 dep endent

llinea

r resummation an lead to cos ϕ > 1(!) fo r small µR = µF based

n

NLL double-ρ p ro du tion (Ivanov, P apa)

ne

an exp e t that small s ales a re disfavo red (Cap

rale,

P apa, Sabio V era) 28 / 36

SLIDE 29 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Motivation fo r asymmetri

ngurations

Initial state radiation (unseen) p ro du es divergen ies if

ne

tou hes the

llinea

r singula rit y q2 → 0 PSfrag repla ements

pJ,1 pJ,2 p3 q

they a re

mp

ensated b y virtual

rre tions

this

mp

ensation is in p ra ti e di ult to implement when fo r some reason this additional emission is in a

rner

f

the phase spa e (dip in the dierential ross-se tion) this is the ase when p1 + p2 → 0 this alls fo r a resummation

f

la rge remaing logs ⇒ Sudak

v

resummation PSfrag repla ements

pJ,1 pJ,2

29 / 36

SLIDE 30 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Motivation fo r asymmetri

ngurations

sin e these resummation have never b een investigated in this

ntext,
ne

should b etter avoid that region note that fo r BFKL, due to additional emission b et w een the t w

jets,
ne

ma y exp e t a less severe p roblem (at least a smea ring in the dip region

|p1| ∼ |p2|

) PSfrag repla ements

pJ,1 pJ,2

this ma y ho w ever not mean that the region |p1| ∼ |p2| is p erfe tly trustable even in a BFKL t yp e

f

treatment w e no w investigate a region where NLL DGLAP is under

ntrol

30 / 36

SLIDE 31 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: asymmetri

nguration

(|kJ,1| = 35 GeV , |kJ,2| = 50 GeV ) Cross-se tion pure LL LL verti es + imp roved

llinea

r NLL Green's fun tion NLL verti es + NLL Green's fun tion NLL verti es + imp roved

llinea

r NLL Green's fun tion

6

7 8 9 10 1.000 0.500 0.100 0.050 0.010 0.005

PSfrag repla ements

C0 ˆ

nb GeV2

˜ = σ Y

bands = erro rs due to the Monte Ca rlo integration dots = based

n

the NLO DGLAP pa rton generato r Dijet (thanks to M. F

ntannaz)

31 / 36

SLIDE 32 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: symmetri

nguration

(|kJ,1| = 35 GeV , |kJ,2| = 50 GeV ) Azimuthal

rrelation: cos ϕ

pure LL LL verti es + imp.

llinea

r NLL Green's fn. NLL verti es + NLL Green's fn. NLL verti es + imp.

llinea

r NLL Green's fn.

6

7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ

Y

bands = erro rs due to the Monte Ca rlo integration dots = based

n

the NLO DGLAP pa rton generato r Dijet (thanks to M. F

ntannaz)

Both NLL and imp roved NLL results a re almost at in Y no signi ant dieren e b et w een NLL BFKL and NLO DGLAP 32 / 36

SLIDE 33 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: asymmetri

nguration

(|kJ,1| = 35 GeV , |kJ,2| = 50 GeV ) Azimuthal

rrelation: cos 2ϕ

pure LL LL verti es + imp.

llinea

r NLL Green's fn. NLL verti es + NLL Green's fn. NLL verti es + imp.

llinea

r NLL Green's fn.

6

7 8 9 10 0.2 0.4 0.6 0.8

PSfrag repla ements

C2 C0 = cos 2ϕ

Y

bands = erro rs due to the Monte Ca rlo integration dots = based

n

the NLO DGLAP pa rton generato r Dijet (thanks to M. F

ntannaz)

Same

n lusions:

Both NLL and imp roved NLL results a re almost at in Y no signi ant dieren e b et w een NLL BFKL and NLO DGLAP 33 / 36

SLIDE 34 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: asymmetri

nguration

(|kJ,1| = 35 GeV , |kJ,2| = 50 GeV ) Azimuthal

rrelation:

dep enden y with resp e t to µR = µF and s0 hanges pure LL LL verti es + imp.

llinea

r NLL Green's fn. NLL verti es + NLL Green's fn. NLL verti es + imp.

llinea

r NLL Green's fn.

6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2 6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ C1 C0 = cos ϕ

Y Y

Ee t

f

hanging µR = µF b y fa to rs 2 (thi k) and 1/2 (thin)

6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2 6 7 8 9 10 0.2 0.4 0.6 0.8 1.0 1.2

PSfrag repla ements

C1 C0 = cos ϕ C1 C0 = cos ϕ

Y Y

Ee t

f

hanging √s0 b y fa to rs 2 (thi k) and 1/2 (thin) Again:

cos ϕ

is still rather µR = µF and s0 dep endent

llinea

r resummation an lead to cos ϕ > 1(!) fo r small µR = µF 34 / 36

SLIDE 35 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Results: asymmetri

nguration

(|kJ,1| = 35 GeV , |kJ,2| = 50 GeV ) Ratio

f

azimuthal

rrelations cos 2ϕ/cos ϕ

pure LL LL verti es + imp.

llinea

r NLL Green's fn. NLL verti es + NLL Green's fn. NLL verti es + imp.

llinea

r NLL Green's fn.

6

7 8 9 10 0.2 0.4 0.6 0.8

PSfrag repla ements

C2 C1 = cos 2ϕ/cos ϕ

Y

bands = erro rs due to the Monte Ca rlo integration dots = based

n

the NLO DGLAP pa rton generato r Dijet (thanks to F

ntannaz)

NB: NLL

llinea

r imp roved hanged nothing wrt pure NLL This is the

nly
bservable

whi h might still dier noti eably b et w een NLL BFKL and NLO DGLAP s ena rii 35 / 36

SLIDE 36 Intro du tion A full NLLx example: Mueller-Navelet jets Pra ti al implementation

f

the

mputation

Results Con lusion W e have p erfo rmed fo r the rst time a

mplete

NLL analysis

f

Mueller-Navelet jets the

rre tion

due to NLL jets

rre tions

have a dramati ee t, simila r to the NLL Green fun tion

rre tions

fo r the ross-se tion: it mak es the p redi tion mu h mo re stable with resp e t to va riation

f

pa rameters (fa to rization s ale, s ale s0 entering the rapidit y denition, P a rton Distribution F un tions) it is lose to NLO DGLAP (although surp risingly a bit b elo w!) the de o rrelation ee t is very small: it is very lose to NLO DGLAP it is very at in rapidit y Y it is still rather dep endent

n

these pa rameters pure NLL BFKL and

llinea

r imp roved NLL BFKL leads to simila r results

llinea

r imp roved NLL BFKL fa es some puzzling b ehaviour fo r the azimuthal

rrelation

ex ept fo r cos 2ϕ/cos ϕ, there is almost no dieren e b et w een NLL BFKL and NLO DGLAP based

bservables

Mueller Navelet jets a re thus p robably not su h a

n lusive
bservable

to see the p erturbative Regge ee t

f

QCD to

mpa

re with data, a serious study

f

Sudak

v

t yp e

f

ee ts is still missing, b

th

in DGLAP and BFKL app roa hes 36 / 36