High order effects in γγ → VV

Transcription

Werner K. Sauter
UERGS/UFRGS
6 de setembro de 2007
Werner K. Sauter UERGS/UFRGS ()
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Summary
1
Introduction
2
LO & NLO & NNLO BFKL
3
Formalism
4
Results
5
Conclusions
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Summary
1
Introduction
2
3
Formalism
4
Results
5
Conclusions
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QCD & Pomeron
Understanding of high energy hadron processes from a fundamental
perspective within Quantum Chromodynamics (QCD)
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QCD & Pomeron
Regge limit (s |t|) in QCD described by Lipatov et al. QCD
Pomeron, described by BFKL equation
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QCD & Pomeron
Regge limit (s |t|) in QCD described by Lipatov et al. QCD
Pomeron, described by BFKL equation
most simple case: onium-onium scattering ⇒ heavy onium allows
perturbative expansion.
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Photon-photon collisions
off-shell photon scattering at high energy in e + e − colliders
⇒ advantage: not involve a non-perturbative target
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three variables, virtuality of the photon (Q 2 ), the momentum transfer
(t) and the quark mass (M). Real photons, Q 2 = 0 and t = 0,
perturbative calculations only if M is large
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three variables, virtuality of the photon (Q 2 ), the momentum transfer
(t) and the quark mass (M). Real photons, Q 2 = 0 and t = 0,
perturbative calculations only if M is large
Vector meson pairs production in γγ collisions as a test of BFKL
Pomeron
Calculation: convolution the amplitude of the transition γ → V with
BFKL amplitude.
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Calculation
Perturbative calculation for: heavy mesons (light meson production
only in the case of large momentum transfer), high Q 2 , t 6= 0
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Calculation
Previous work (Eur.Phys.J.C44:515-522,2005): γγ → V1 V2 in BFKL
framework with Vi = ρ, ω, φ, J/Ψ, Υ vector mesons in the large t
exchange (Q 2 = 0) ⇒ large rapidity gap in the final state.
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Calculation
Successful description of meson photo-production in HERA with the
BFKL formalism (Enberg, Motyka, Poludniowsky), even in the case of
light mesons.
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Calculation
Successful description of meson photo-production in HERA with the
BFKL formalism (Enberg, Motyka, Poludniowsky), even in the case of
light mesons.
In this work:
Use of photon virtualities, conformal spin, NLO BFKL eigenvalues (kernel,
characteristic function) with the ρ & J/ψ mesons
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Summary
1
Introduction
2
3
Formalism
4
Results
5
Conclusions
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Leading order
pQCD: hadron-hadron scattering amplitude as a sum of (αs ln s)n
(s Q 2 Λ2QCD )
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Leading order
(s Q 2 Λ2QCD )
Leading Logarithm (Order) terms: αs ' 0.2, σtot ' s 0.5
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Leading order
(s Q 2 Λ2QCD )
Leading Logarithm (Order) terms: αs ' 0.2, σtot ' s 0.5
Next Leading Order terms: αs (αs ln s)n
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DGLAP & BFKL
DGLAP:
summation of terms (αs ln Q 2 )n , αs small and ln Q 2 large,
x = Q 2 /s 1 ⇒ σtot ∝ q(x, Q 2 )
Unintegrated gluon distribution F (x, Q 2 ) as solution of a integral
equation (by Mellin transform):
p
−1/4
ln 1/x ln Q 2
Q 2 F (x, Q 2 ) ' ln(1/x) ln(Q 2 )
exp
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DGLAP & BFKL
DGLAP:
summation of terms (αs ln Q 2 )n , αs small and ln Q 2 large,
x = Q 2 /s 1 ⇒ σtot ∝ q(x, Q 2 )
Unintegrated gluon distribution F (x, Q 2 ) as solution of a integral
equation (by Mellin transform):
p
−1/4
ln 1/x ln Q 2
Q 2 F (x, Q 2 ) ' ln(1/x) ln(Q 2 )
exp
BFKL:
small Q 2 , summation of terms (αs ln 1/x)n
F (x, Q 2 ) '
α χ(1/2)
p s
Q αs χ00 (1/2) ln 1/x
χ(γ) is the BFKL kernel. In LO, χ0 (γ) = 2ψ(1) − ψ(γ) − ψ(1 − γ)
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NLO BFKL
NLO corrections: αs (αs ln s)n
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NLO BFKL
Characteristic function: χ(γ) = ᾱs χ0 (γ) + ᾱs2 χ1 (γ) + ᾱs3 χ2 + . . .
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NLO BFKL
Contributions to χ1 : running coupling, splitting functions, energy
scale. Collinear corrections:
χcoll
1 (γ) =
A1
A1 − b
1
1
+
−
−
γ2
(1 − γ)2 2γ 3 2(1 − γ)2
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NLO BFKL
χcoll
1 (γ) =
A1
A1 − b
1
1
+
−
−
γ2
(1 − γ)2 2γ 3 2(1 − γ)2
Full solution: lengthy expression (see in the following)
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NLO BFKL
χcoll
1 (γ) =
A1
A1 − b
1
1
+
−
−
γ2
(1 − γ)2 2γ 3 2(1 − γ)2
Full solution: lengthy expression (see in the following)
Consequences:
Large corrections
Chance of the structure of the characteristic function: 2 saddle points
(γ̄, γ̄ ∗ )
ᾱs χ(γ̄) 2 γ̄
Q
s
Cross sections:σ(s, Q 2 , Q02 ) ∼ Q12 QQ
+ (γ̄ ↔ γ̄)∗ ,
Q2
0
0
oscillates as functions of ln Q 2 /Q02 .
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NNLO BFKL
Beyond NLO: long calculations without guarantee of convergence.
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NNLO BFKL
Approximation to the NNLO kernel (see Marzani et al.,
arXiv:0704.2404)
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NNLO BFKL
arXiv:0704.2404)
Slow convergence of the perturbative expansion. Expectation: NNLO
aprox. with same qualitative shape as the LO term
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NNLO BFKL
arXiv:0704.2404)
Inclusion of the collinear & anti-collinear singularities based in the
duality between BFKL & DGLAP kernel anomalous dimension
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NNLO BFKL
arXiv:0704.2404)
Contributions:
Running coupling effects;
dependence of the
factorization scheme;
kinematic variables.
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NNLO BFKL
arXiv:0704.2404)
Contributions:
Running coupling effects;
dependence of the
factorization scheme;
kinematic variables.
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Summary
1
Introduction
2
3
Formalism
4
Results
5
Conclusions
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Factorization
Scattering amplitude to cross sections
16π
3αs
s
|t|
Im A(s, t) = 2 F(z, τ ), z =
ln 2 , τ = 2
9t
2π
Λ
MV + Qγ2
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Factorization
16π
3αs
s
|t|
Im A(s, t) = 2 F(z, τ ), z =
ln 2 , τ = 2
9t
2π
Λ
MV + Qγ2
F(z, τ ) =
Z
t2 X
ν 2 + m2
2
dν
e Y ωm (Q ,ν)
1
1
3
2
2
2
2
(2π) m
[ν + (m + 2 ) ][ν + (m − 2 ) ]
γVa
γVb
Im,ν
(Q⊥ )Im,ν
(Q⊥ )∗
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Factorization
16π
3αs
s
|t|
Im A(s, t) = 2 F(z, τ ), z =
ln 2 , τ = 2
9t
2π
Λ
MV + Qγ2
F(z, τ ) =
Z
t2 X
ν 2 + m2
2
dν
e Y ωm (Q ,ν)
1
1
3
2
2
2
2
(2π) m
[ν + (m + 2 ) ][ν + (m − 2 ) ]
γVa
γVb
Im,ν
(Q⊥ )Im,ν
(Q⊥ )∗
dσ
1
=
|A(s, t)|2 ,
dt
16π
Z
∞
σ(γγ → V1 V2 ) =
d|t|
|t|min
dσ(γγ → V1 V2 )
d|t|
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Impact factors
γV
Im,νi (Q⊥ )
=
8Ci
1
Q3
⊥
„
2
Q⊥
4
«iν „
∗
Q⊥
Q⊥
«m
Γ(1/2−iν+|m|)
Γ(1/2+iν+|m|)
1 − i (ξ−ν))Γ( 1 − i (ξ+ν))Γ2 (1+iξ+|m|)
Γ( 4
2
4 2
,
i
3 i
Γ( 3
4 + 2 (ξ−ν)+|m|)Γ( 4 + 2 (ξ+ν)+|m|)
R +∞
−∞
Ci2 =
dξ
1+iξ+|m|
( τ4i )
V
3
3Γeei MV
αem
i
×
.
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Impact factors
γV
Im,νi (Q⊥ )
=
8Ci
1
Q3
⊥
„
2
Q⊥
4
«iν „
∗
Q⊥
Q⊥
«m
Γ(1/2−iν+|m|)
Γ(1/2+iν+|m|)
1 − i (ξ−ν))Γ( 1 − i (ξ+ν))Γ2 (1+iξ+|m|)
Γ( 4
2
4 2
,
i
3 i
Γ( 3
4 + 2 (ξ−ν)+|m|)Γ( 4 + 2 (ξ+ν)+|m|)
R +∞
−∞
Ci2 =
dξ
1+iξ+|m|
( τ4i )
V
3
3Γeei MV
αem
i
×
.
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Impact factors
γV
Im,νi (Q⊥ )
=
8Ci
1
Q3
⊥
„
2
Q⊥
4
«iν „
∗
Q⊥
Q⊥
«m
Γ(1/2−iν+|m|)
Γ(1/2+iν+|m|)
1 − i (ξ−ν))Γ( 1 − i (ξ+ν))Γ2 (1+iξ+|m|)
Γ( 4
2
4 2
,
i
3 i
Γ( 3
4 + 2 (ξ−ν)+|m|)Γ( 4 + 2 (ξ+ν)+|m|)
R +∞
−∞
Ci2 =
dξ
1+iξ+|m|
( τ4i )
V
3
3Γeei MV
αem
i
×
.
In the case of the NLO
Calculation, the form factor is
remains LO. NLO result have a
huge expression (See Papa,
Kotsky paper)
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Impact factors
γV
Im,νi (Q⊥ )
=
8Ci
1
Q3
⊥
„
2
Q⊥
4
«iν „
∗
Q⊥
Q⊥
«m
Γ(1/2−iν+|m|)
Γ(1/2+iν+|m|)
1 − i (ξ−ν))Γ( 1 − i (ξ+ν))Γ2 (1+iξ+|m|)
Γ( 4
2
4 2
,
i
3 i
Γ( 3
4 + 2 (ξ−ν)+|m|)Γ( 4 + 2 (ξ+ν)+|m|)
R +∞
−∞
Ci2 =
dξ
1+iξ+|m|
( τ4i )
V
3
3Γeei MV
αem
i
×
.
In the case of the NLO
Calculation, the form factor is
remains LO. NLO result have a
huge expression (See Papa,
Kotsky paper)
The results estimate the effects
of NLO contributions
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The BFKL eigenvalue ω
In leading order
ωLO = 2ᾱs <e [ψ(1) − ψ(1/2 + |m| + iν)]
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In leading order
ωLO = 2ᾱs <e [ψ(1) − ψ(1/2 + |m| + iν)]
The NLO BFKL eigenvalue are given using BLM energy scale setting
within momentum space subtraction (MOM) renormalization scheme
[Brodsky et al.]:
MOM (Q 2 ,ν)=N χ
ωBLM
c LO (ν)
αMOM (Q̂ 2 )
π
»
–
α
(Q̂ 2 )
1+r̂ (ν) MOM
π
Q̂ 2 (ν)=Q 2 exp[ 21 χLO (ν)− 53 +2(1+ 23 I )],
r̂ (ν)
=
−
β0
4
h
χLO (ν)
− 53
2
i
− 4χ Nc (ν) {
LO
I ≈2.3439
π 2 sinh(πν)
[3+
2ν cosh2 (πν)
„
1+
Nf
Nc3
«
×
π 2 −4
11+12ν 2
π3
]−χ00
LO (ν)+ 3 χLO (ν)− cosh(πν) −6ζ(3)+4φ(ν)}
16(1+ν 2 )
+7.471−1.281β0
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11Nc − 2Nf
,
3
Z 1
cos(ν ln(x)) π 2
√
φ(ν) = 2
dx
− Li2 (x) ,
6
(1 + x) x
0
Z x
ln(1 − t)
dt
.
Li2 (x) = −
t
0
β0 =
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ω comparison
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Choice of parameters
Running coupling (based in the previous works):
NLO (µ2 )=
Nc =3, Nf =4, αLO
s =0.2, αs
4π
β0 ln µ2 /Λ2
QCD
(
)
, µ2 =Qa QB , ΛQCD =0.1 GeV
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NLO (µ2 )=
Nc =3, Nf =4, αLO
s =0.2, αs
4π
β0 ln µ2 /Λ2
QCD
(
)
Rapidity:
In the present work, the choice is
s
Ypw = 2 , Λ2ab = γ|t|+λa (ma2 +Qa2 )+λb (mb2 +Qb2 ), γ = 0, λa,b = 1/2
Λab
In Enberg et al. work (EPSW) the choice is
s
Yepsw = ln CY
, CY = 0.3
Qa Qb
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NLO (µ2 )=
Nc =3, Nf =4, αLO
s =0.2, αs
4π
β0 ln µ2 /Λ2
QCD
(
)
Rapidity:
s
Λab
s
Yepsw = ln CY
, CY = 0.3
Qa Qb
Cut in the t integration: in the present work, we use tmin = 0 (in
previous results also tmin = 1) or as in EPSW, tmin = (Qa2 Qb2 )/s
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NLO (µ2 )=
Nc =3, Nf =4, αLO
s =0.2, αs
4π
β0 ln µ2 /Λ2
QCD
(
)
Rapidity:
s
Λab
s
Yepsw = ln CY
, CY = 0.3
Qa Qb
Cut in the t integration: in the present work, we use tmin = 0 (in
previous results also tmin = 1) or as in EPSW, tmin = (Qa2 Qb2 )/s
Difference in the longitudinal and transverse parts of cross section:
2 2
dσT
ma
mb
dσL
1 dσL
=
∼
dt
Qa
Qb
dt
hQ 2 i dt
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Summary
1
Introduction
2
3
Formalism
4
Results
5
Conclusions
19 / 27
Comparison with LO order
dσ/dt × t
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Comparison with LO order
σtot ×
√
s
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Choice of scale combination & parameters
Choice of scales in rapidity and coupling
For fixed coupling, zero virtuality: s0 = EMP
For fixed coupling, non-zero virtuality: s0 = EMP or EPSW
For running coupling, zero virtuality: s0 = EMP; µ0 = BLM
For running coupling, non-zero virtuality: s0 = EMP; µ0 = EMP or s0
= EPSW; µ0 = EPSW
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= EPSW; µ0 = EPSW
Cut-off in integration: ρρ, tmin = 1, others tmin = 0
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= EPSW; µ0 = EPSW
Cut-off in integration: ρρ, tmin = 1, others tmin = 0
Results are complete = total + longitudinal parts
c = t + l,
t=
l = fl
1
c,
1+f
f =
l=
Q12 Q22
⇒
M12 M22
f
c
1+f
In general,
t c,
l 'c
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Differential cross sections
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Total cross sections
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Characteristics
conformal spin: very small difference (not displayed);
Overall result: decrease of the cross sections ⇒ ωLO ≥ ωNLO ;
increase of the mass and the virtuality: flatter cross section, decrease
of the cross sections;
The LO and NLO results are distinguishable.
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Summary
1
Introduction
2
3
Formalism
4
Results
5
Conclusions
26 / 27
Conclusions
Probe QCD dynamics in a new & unexplored kinematic regime
27 / 27
Conclusions
γγ processes have a important rôle in background contributions in
other processes.
27 / 27
Conclusions
other processes.
γγ processes can constrain the QCD dynamics in more clear process
→ heavy ion collisions (Eur.Phys.J.C46:219-224,2006); e + e −
γγ
(Phys.Rev.D73:077502,2006); σtot
(hep-ph/0611171. To app. in J.
Phys. G).
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Conclusions
other processes.
γγ
(Phys.Rev.D73:077502,2006); σtot
Phys. G).
Future photon colliders can be experimentally check/confront this
predictions.
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Conclusions
other processes.
γγ
(Phys.Rev.D73:077502,2006); σtot
Phys. G).
Future photon colliders can be experimentally check/confront this
predictions.
Forthcoming studies: application on peripheral heavy ion collisions or
other photon processes; use of other summations schemes for NLO
BFKL; NLO meson form factors.
27 / 27

High order effects in γγ → VV

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