Towards 3 particle correlations in the Color Glass Condensate - - PowerPoint PPT Presentation
Towards 3 particle correlations in the Color Glass Condensate framework Martin Hentschinski martin.hentschinski@gmail.com IN COLLABORATION WITH A. Ayala, J. Jalilian-Marian, M.E. Tejeda Yeomans, QCD Challenges at the LHC: from pp to AA
Martin Hentschinski
martin.hentschinski@gmail.com
IN COLLABORATION WITH
QCD Challenges at the LHC: from pp to AA
(Taxco, 18.-22. Jan. 2016)
0.2 0.4 0.6 0.8 1
10
10
10
10 1
HERAPDF2.0 NLO uncertainties: experimental model parameterisation HERAPDF2.0AG NLO
x xf
2
= 10 GeV
2 f
µ
v
xu
v
xd 0.05) × xS ( 0.05) × xg (
H1 and ZEUS
gluon g(x) and sea-quark S(x) distribution like powers ~ x-λ for x→0 → invalidates probability interpretation if continued forever (integral over x diverges) → at some x, new QCD dynamics must set in
k p X k' q
HERA collider (92-07): Deep Inelastic Scattering (DIS) of
Photon virtuality Q2 = −q2
Bjorken x = Q2 2p · q
Open Questions
The proton at high energies: saturation
theory considerations:
Geometric Scaling
Y = ln 1/x
non-perturbative region ln Q2 Q2
s(Y)
s a t u r a t i
r e g i
Λ2
QCD
αs < < 1 αs ~ 1 BK/JIMWLK DGLAP BFKL
I effective finite size 1/Q of
partons at finite Q2
I at some x ⌧ 1, partons
‘overlap’ = recominbation effects
I turning it around: system is
characterized by saturation scale Qs
I grows with energy Qs ⇠ x−∆,
∆ > 0 & can reach in principle perturbative values Qs > 1GeV
High gluon densities & heavy ions
= collisions of two Color Glass Condensate
Saturation: high densities in the fast nucleus
Expect those effects to be even more enhanced in boosted nuclei:
Boost
Q2
s ∼ # gluons/unit transverse area ∼ A1/3
pocket formula: xeff(A)= xBjorken/A
CGC and long-range rapidity correlations in high multiplicity events
scale, higher correlators from “Gaussian/dilute approximation”)
AA: glasma dominates, pp, pA also jet graph (𝞫S suppressed)
η ∆
2 4 φ ∆ 2 4
φ ∆ d η ∆ d
pair
N
2
d
trg
N 1
1.30 1.35 1.40
CMS Preliminary 110 ≥ = 7 TeV, N s pp <3 GeV/c
trig T
2<p <2 GeV/c
assoc T
1<p
φ ∆ 2 4
1.30 1.35 1.40
1 NTrig d2N d∆η d∆φ
∆φ π
1 NTrig d2N d∆φ
q p
Glasma Graph
q p
Jet Graph
p-p collisions at √s = 7 TeV. Data points are from the CMS collaboration. The curves show the results for Q2
0(x = 10−2) = 0.840 GeV2 and Q2 0(x = 10−2) = 1.008 GeV2.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.5 1 1.5 2 2.5 3 ∆φ 90 < Ntrk
1.0 < pT < 2.0 GeV jet+glasma CMS Data
0.78 0.80 0.82 0.84 0.86 0.88 0.90π −π π π ∆φ ∆φ ∆φ
pPb - data pp - data works rather good, some say too good …
What do we know really about saturated gluons? — DIS on a proton at HERA
5 30.5 1 1.5
Data Theory
r
!
2
=0.85 GeV
2
Q
0.5 1 1.5
r
!
2
=4.5 GeV
2
Q
0.5 1 1.5
r
!
2
=10.0 GeV
2
Q
5 30.5 1 1.5
r
!
2
=15.0 GeV
2
Q
−510
−410
−310
−210
0.5 1 1.5
r
!
2
=35 GeV
2
Q x
5 32
=2.0 GeV
2
Q
2
=8.5 GeV
2
Q
2
=12.0 GeV
2
Q
5 32
=28.0 GeV
2
Q
−410
−310
−210
2
=45 GeV
2
Q
x
σγ∗A
L,T (x, Q2) = 2
X
f
Z d2bd2r
1
Z dz
L,T (r, z; Q2)
N(x, r, b)
in Hentschinski (ICN-UNAM) The glue that binds us all November 3, 2015
[Albacete, Armesto, Milhano,Quiroga, Salgado, EPJ C71 (2011) 1705]
splitting recombination
color dipole 𝓞: all information about gluon distribution + follows non-linear evolution in ln(1/x) [JIMWLK or BK]
achieve a good description of combined (= high precision!) HERA data through rcBK fit
factorisation into photon wave function 𝜔 (ɣ*→qqbar) & color dipole 𝓞 (~dense gluon field)
pdf-fits (=DGLAP) — intrinsically dilute (virtual photon interacts with single
quark, gluon)
… and also (collinear improved) NLO BFKL evolution can fit data
[MH, Salas, Sabio Vera; PRD 87 (2013) 7, 076005]
geometric scaling
ln x
non-perturbative region
ln Q2 Q2
s(x)
saturation JIMWLK BK DGLAP BFKL
αs < < 1 αs ~ 1
Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê 0.2 0.6 1.0 1.4 10-4 10-3 10-2 F2 Hx,Q²L Q² = 1.2 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê 10-4 10-3 10-2 Q² = 1.5 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê 10-4 10-3 10-2 Q² = 2.0 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê 10-4 10-3 10-2 0.2 0.6 1.0 1.4 Q² = 2.7 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 F2 Hx,Q²L Q² = 3.5 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Q² = 4.5 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Q² = 6.5 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 Q² = 8.5 GeV² Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 F2 Hx,Q²L Q² = 10 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Q² = 12 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Q² = 15 GeV² Ê Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 Q² = 18 GeV² Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 F2 Hx,Q²L Q² = 22 GeV² Ê Ê Ê Ê Ê Ê Ê Ê Ê Q² = 27 GeV² Ê Ê Ê Ê Ê Ê Ê Q² = 35 GeV² Ê Ê Ê Ê Ê Ê Ê 0.2 0.6 1.0 1.4 Q² = 45 GeV² Ê Ê Ê Ê Ê Ê 10-4 10-3 10-2 0.2 0.6 1.0 1.4 x F2 Hx,Q²L Q² = 60 GeV² Ê Ê Ê Ê Ê 10-4 10-3 10-2 x Q² = 70 GeV² Ê Ê Ê Ê Ê 10-4 10-3 10-2 x Q² = 90 GeV² Ê Ê Ê Ê Ê 10-4 10-3 10-2 0.2 0.6 1.0 1.4 x Q² = 120 GeV² GeV2) + DGLAP fits initial conditions at small Q2
really a contradiction, but also not yet definite proof for saturation, cannot claim complete control
events through scaling of (initial) saturation scale Qs(A) = QsHERA ・A1/3, but rely on assumptions/arguments
accuracy as e.g. in pp through conventional pdfs
the world’s first eA collider: will allow to probe heavy nuclei at small x (using 16GeV electrons on 100GeV/u ions)
Brookhaven National Laboratory: supplement RHIC with Electron Recovery Linac (eRHIC)
Jefferson Lab: supplement CEBAF with hadron accelerator (MEIC)
2015: endorsed by Nuclear Science Advisory Committee (NSAC) As highest priority for new Facility construction in US Nuclear Science Long Range plan
+ plans for LHeC etc.
so far:
as IPsat, bCGC → x-dependence = assumption + fit
coefficients at LO, with a few NLO exceptions (inclusive DIS, single
inclusive jet in pA)
recent progress:
[Kovner,Lublinsky, Mulian; PRD 89 (2014) 6, 061704] known & studied + resummed &
used for first HERA fit [Iancu, Madrigal, Mueller, Soyez, Triantafyllopoulos, PLB750 (2015) 643] missing: → NLO corrections for coefficients of exclusive observables — provide strongest constraints on saturation
higher twist effects at small Q2 as signal for saturation
[Motyka, Slominski, Sadzikowski, Phys.Rev. D86 (2012) 111501]
β
1 1.5 2 2.5 3 2 3 4 5 6 7 8 9 10 χ2/N Q2
min [GeV2]
DGLAP DGLAP + MSS-Sat twist 4+6
10
10
0.02 0.04 0.06 = 0.217 β
10
10
0.02 0.04 0.06 = 0.091 β
10
10
0.02 0.04 0.06 = 0.038 β
10
10
0.02 0.04 0.06 = 0.015 β
2= 2.5 GeV
2Q
10
10
0.02 0.04 0.06 = 0.280 β
10
10
0.02 0.04 0.06 = 0.123 β
10
10
0.02 0.04 0.06 = 0.052 β
10
10
0.02 0.04 0.06 = 0.020 β
2= 3.5 GeV
2Q
10
10
0.02 0.04 0.06 = 0.333 β
10
10
0.02 0.04 0.06 = 0.153 β
10
10
0.02 0.04 0.06 = 0.066 β
10
10
0.02 0.04 0.06 = 0.026 β
2= 4.5 GeV
2Q
10
10
0.02 0.04 0.06 = 0.379 β
10
10
0.02 0.04 0.06 = 0.180 β
10
10
0.02 0.04 0.06 = 0.079 β
10
10
0.02 0.04 0.06 = 0.032 β
2= 5.5 GeV
2Q ZEUS (LRG) DGLAP DGLAP + Twist 4 DGLAP + Twist 4+6
ξ
D(3) r
σ ξ
inclusive — so far modelled using eikonal approximation [C. Marquet, Phys. Rev. D76, 094017 (2007)]
twist of GBW model
arbitrariness remains …
A popular observable in the EIC program: Di-Hadron De-correlation in DIS
, y=0.7
2=1 GeV
2Q
(rad) φ Δ
2 2.5 3 3.5 4 4.5
) φ Δ C(
0.05 0.1 0.15 0.2 0.25
ep eCa eAu 20 GeV on 100 GeV
(rad) φ Δ
2 2.5 3 3.5 4 4.5
) φ Δ C(
0.1 0.2 0.3 0.4
e+Au - no-sat eAu - sat
pT
trig > 2 GeV/c1 GeV/c < pT
assoc < pT trig0.2 < zh
trig, zh assoc < 0.41 < Q2 < 2 GeV2 0.6 < y < 0.8
20 GeV on 100 GeV
collinear factorization (dilute pQCD): gluon kT peaked at kT=0
Saturation (CGC): gluon kT peaked at saturation scale
αs < < 1 αs ∼ 1 ΛQCD
know how to do physics here
?
Qs
kT ~ 1/kT kT φ(x, kT
2)
[Mueller, Xiao, Yuan, Phys.Rev. D88 (2013) 11, 114010]
[rad] φ ∆
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
) φ ∆ C(
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
ep, No Sudakov eAu, No Sudakov ep, With Sudakov eAu, With Sudakov
10 GeV x 100 GeV
2
= 1 GeV
2
Q
[Zheng,Aschenauer, Lee, Xiao, PRD89 (2014)7, 074037]
comparison of ep and eA shows at first clear signal …. …. but Sudakov factors have a big effect …. …. signal remains, but inclusion of higher order corrections necessary for precise distinction
signal in d-Au collisions at RHIC: depletion of away side peak in central collisions described by CGC
−1 1 2 3 4 5
∆ϕ [rad]
0.16 0.17 0.18 0.19 0.20 1.1 GeV < ptrig
T
< 1.6 GeV 1.6 GeV < ptrig
T
< 2.0 GeV
theory: involves higher correlator (‘quadrupole’, not only dipole) — state-of-the art: calculate in Gaussian/ dilute approximation from dipole [Lappi, Mantysaari,
Nucl.Phys. A908 (2013) 51-72]
π0 azimuthal correlation compared to the PHENIX d-Au result (0.5GeV<pass<0.75 GeV, 3<y1,y2<3.8). solid line: QS02 = 1.51 GeV2, dashed line: QS02 = 0.72 GeV2
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
RpA y
√S = 5.02 TeV pt1>pt2>20 GeV 3.2<y1,y2<4.9 KS c=1.0 KS c=0.5 rcBK d=2.0 rcBK d=4.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 2.5 2.6 2.7 2.8 2.9 3 3.1
RpA Δφ
√S = 5.02 TeV pt1>pt2>20 GeV 3.2<y1,y2<4.9 KS c=1.0 KS c=0.5 rcBK d=2.0 rcBK d=4.0
[v. Hameren, Kotko, Kutak, Marquet, Sapeta, Phys.Rev. D89 (2014) 9, 094014]
RpA = dσp+A dO A dσp+p dO .
ϕ13
0.5 1 1.5 2 2.5 3
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
∆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
∆pT = |⃗ pT 1 + ⃗ pT 2 + ⃗ pT 3| ,
[v. Hameren, Kotko, Kutak, Phys.Rev. D88 (2013) 094001]
collinear pdfs, Pb saturated gluon
(2 jets: complete LO matrix element known in principle, 3 jets: unknown)
glue field, saturation through kT dependence
= + + +
αs < < 1 αs ∼ 1 ΛQCD
know how to do physics here
?
Qs
kT ~ 1/kT kT φ(x, kT
2)
gluon distribution obeys BK evolution
the presented studies have certain limitations uncontrolled higher order corrections (only LO in 𝞫S) dilute expansion p1t+p2t|≫QS (=probe the tail of saturation, but appropriate in certain kinematics) need to increase theory precision for establishing saturation + extracting gluon distributions (important for precision at EIC but also LHC, HERA analysis)
complete)
expect more stringent tests of CGC through more complex final state
reduce uncertainties + possibly identify overlap region between collinear factorisation and saturation physics
As a first step: limit to DIS (electron-nucleus i.e. ɣ*A collisions) but derive important general results on the way → first step for future pPb calculation in “hybrid-”formalism
Theory: quarks, gluons in the presence of high gluon densities
background gluon field
energy limit (= x →0 limit)
gluon densities multiple scatterings
γ∗
→
γ∗ x → 0: a single interaction with a strong & Lorentz contracted gluon field
φ Δ φ Δ φ Δ φ Δ
≡ ≡ A+,a(z−, z) = ↵a(z)(z−)
Theory: Propagators in background field
p q
= 2⇡(p− − q−)n
Z dd−2ze−iz·(p−q) · n ✓(p−)[V (z) − 1] − ✓(−p−)[V †(z) − 1]
q
= −2⇡(p− − q−)2p− Z dd−2ze−iz·(p−q) · n ✓(p−)[U(z) − 1] − ✓(−p−)[U †(z) − 1]
Z ∞
−∞
dx−A+,c(x−, z)tc U(z) ≡ U ab(z) ≡ P exp ig Z ∞
−∞
dx−A+,c(x−, z)T c
p q
= (2π)dδ(d)(p − q) ˜ S(0)
F (p) + ˜
S(0)
F (p)
p q
˜ S(0)
F (q)
p, µ q, ν
= (2π)dδ(d)(p − q) ˜ G(0)
µν (p) + ˜
G(0)
µα(p)
p q
˜ G(0)
αν (q)
˜ S(0)
F (p) =
ip + m p2 − m2 + i0 ˜ G(0)
µν (p) = idµν(p)
p2 + i0
dµν(p) = −gµν + n−
µ pν + pµn− ν
n− · p
interaction with the background field:
strong background field resummed into path ordered exponentials (Wilson lines)
[Balitsky, Belitsky; NPB 629 (2002) 290], [Ayala, Jalilian-Marian, McLerran, Venugopalan, PRD 52 (1995) 2935-2943], …
use light-cone gauge, with k-=n-･k, (n-)2=0, n-~ target momentum
in contrast to dilute expansion: every line interacts with dense gluon field Difference between DIS and LHC calculation: 3 parton production
each internal & each external coloured line to be split into 2 terms (-1)
LHC: q,g→3 partons
1 extra parton — can cause a lot of work! (even for DIS process)
→ 15! = 1307674368000 individual terms (not all non-zero though) necessary to achieve (potential) cancelations of diagrams BEFORE evaluation require automatization of calculation (= use of Computer algebra codes)
= + +
di-hadrons at LO: paper & pencil calculation e.g.[Gelis, Jalilian-Marian,PRD67, 074019 (2003) ]
each line & each final state splits into two terms (free + interaction) → real NLO: 16 diagrams (amp. level) → virtual NLO: 32 diagrams (amp. level)
Reduce # of Diagrams
Z ∞
−∞
dx−
i →
Z 0
−∞
dx−
i +
Z ∞ dx−
i
∆(0)
F (x) =
Z ddp (2π)d i · e−ip·x p2 − m2 + i0 = Z dp+ (2π) Z dp−dd−2p (2π)d−1 e−ip−x++ip·x 2p− · i · e−ip+x− p+ − p2+m2−i0
2p−
= Z dp−dd−2p (2π)d−1 e−ipx 2p− ⇥ θ(p−)θ(x−) − θ(−p−)θ(−x−) ⇤
p+= p2+m2
2p−
p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2p−
3x−
1x−
2k−
p−
1p−
2x−
1x−
2x−
3k−
each vertex + four momentum integral at each internal internal line
line separating positive & negative light-cone time
possible (~vertical cuts)
p−
1p−
2p−
3x−
1x−
2k−
not altered through interaction
→ interaction must be always placed at a z-=0 cut of the diagram. Note: this applies equally to configuration and momentum space
p q
∝ δ(p− − q−)
≡ ≡ A+,a(z−, z) = ↵a(z)(z−)
forbidden configurations: cannot be accommodated by vertical (s-channel type) cut
consider complete configuration space propagator (free + interacting part)
Z SF (x, y) = Z ddp (2π)d ddq (2π)d e−ipx ˜ S(0)
F (p)(2π)dδ(d)(p − q) + ˜
S(0)
F (p)τF (p, q) ˜
S(0)
F (q)
propagator proportional to complete Wilson line V (fermion)
cut at light-cone time 0 no direct translation to momentum space adding free propagation & interaction→ mixing of different mom. space diagrams but strong constraints on the structure of the full result
z− = 0 x y z− = 0 x y
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)V (x)ta
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)tbV (x)U ba(z)
p−
1p−
2p−
3x−
1x−
2k−
∝ taV †(y)V (x)
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)tbV (x)U ba(z).
Configuration Space predicts which Operators have non-zero coefficients
momentum space: necessary coefficients from only 4 (instead of 16) diagrams
(cancelation of all other contributions verified by explicit calculations)
virtual corrections: similar result, necessary coefficients from 8 (instead of 32) diagrams
Structure of Wilson correlators for 3 particle production in DIS
color dipole
quadrupole
production; finite Nc n-particle ≜n correlators
[Dominguez, Marquet, Stasto, Xiao; Phys.Rev. D87 (2013) 034007]
(in general more than one)
N (4)(x1, x2, x3, x4) = 1 Nc Tr ⇣ 1 − V (x1)V †(x2)V (x3)V †(x4) ⌘
N(r, b) = 1 Nc Tr ⇣ 1 − V (x)V †(y) ⌘
r = x − y b = 1 2(x + y)
+ express adjoint Wilson lines in terms of fundamental + make use of Fiery identities
isolate Wilson line & color generators of amplitudes + square them (Mathematica)
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)V (x)ta
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)tbV (x)U ba(z)
p−
1p−
2p−
3x−
1x−
2k−
∝ taV †(y)V (x)
p−
1p−
2p−
3x−
1x−
2k−
∝ V †(y)tbV (x)U ba(z).
tr [taAtaB] = 1 2tr [A] tr [B] − 1 2Nc tr [AB] tr [taA] tr [taB] = 1 2tr [AB] − 1 2Nc tr [A] tr [B]
U ab(zt) = tr h taV (zt)tbV †(zt) i
tr [W1W ⇤
1 ]
= (N2
c 1)SQ(xt,x0 t,y0 t,yt)
2Nc
tr [W1W ⇤
2 ]
= 1
4
⇣ SD(z0
t, x0 t)SQ(xt, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W1W ⇤
3 ]
= 1
2
⇣ SD(xt, y)SD(y0
t, x0 t) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W1W ⇤
4 ]
= 1
4
⇣ SD(z0
t, x0 t)SQ(xt, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W2W ⇤
1 ]
= 1
4
⇣ SD(xt, z)SQ(zt, x0
t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W2W ⇤
2 ]
= 1
8
⇣ SQ(xt, x0
t, z0 t, zt)SQ(z, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W2W ⇤
3 ]
= 1
4
⇣ SD(z, yt)SQ(xt, x0
t, y0 t, z) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W2W ⇤
4 ]
= 1
8
⇣ SQ(xt, x0
t, z0 t, z)SQ(zt, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W3W ⇤
1 ]
= 1
2
⇣ SD(xt, yt)SD(y0
t, x0 t) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W3W ⇤
2 ]
= 1
4
⇣ SD(y0
t, z0 t)SQ(xt, x0 t, z0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W3W ⇤
3 ]
= (N2
c 1)SQ(xt,x0 t,y0 t,yt)
2Nc
tr [W3W ⇤
4 ]
= 1
4
⇣ SD(y0
t, z0 t)SQ(xt, x0 t, z0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W4W ⇤
1 ]
= 1
4
⇣ SD(xt, zt)SQ(z, x0
t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W4W ⇤
2 ]
= 1
8
⇣ SQ(xt, x0
t, z0 t, zt)SQ(z, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W4W ⇤
3 ]
= 1
4
⇣ SD(z, yt)SQ(xt, x0
t, y0 t, z) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘ tr [W4W ⇤
4 ]
= 1
8
⇣ SQ(xt, x0
t, z0 t, zt)SQ(zt, z0 t, y0 t, yt) − SQ(xt,x0
t,y0 t,yt)
Nc
⌘
SD(x1,t, x2,t) = tr h V (x1,t)V †(x2,t) i SQ(x1,t, x2,t, x3,t, x4,t) = tr h V (x1,t)V †(x2,t)V (x3,t)V †(x4,t) i
with DIS: dipole and quadrupole sufficient even at finite NC
altogether 7 in- dependent terms
Loop integrals
something slightly strange:
technical reason:
reduction procedure
momentum integrals trivially
functions absent intuitive picture: background field = t-channel gluons interacting with the target → naturally provide a loop which is factorized & (partially) absorbed into the projectile in the high energy limit
3 particle production:
p k q l −q − k −k1 l − k1
a 1-loop and a 2-loop topology k1 and k2 are loop momenta new complication: exponentials/Fourier factors conventional: e.g. k1
+ integration by taking residues, then transverse integrals
particular for 2 loop case: complicated transverse integrals developed a new technique
★ complete exponential factors to 4 dimensions ★ evaluate integral using “standard” momentum space techniques
p k q l k2 k2 − k1 −k1 l − k1
I(p1, p2) = Z ddk1 i⇡d/2 1 [k2
1][(l − k1)2]eixt(·k1,t−p1,t)e−iyt·(k1,t+p2,t)(2⇡)2(p− 1 − k− 1 )(l− − k− 1 − p− 2 )
Z Z Z − − ✓ i k2 − m2 + i0 ◆λ = 1 Γ() Z ∞ d↵ ↵λ−1eiα(k2−m2+i0) Z − I(p1, p2) = 2⇡(l− − p−
1 − p− 2 )e−iyt·(p1,t+p2,t)
Z dr+ Z dr−(r+) Z ddk1 i⇡d/2 1 [k2
1][(l − k1)2]eir·k1
◆ Z
start with integral which contains delta functions transverse exponential factors introduce relative coordinate r=x-y represent delta function by integral introduce dummy integral over r+ →obtain 4(d) dimensional integral next step: Schwinger/𝜷-parameters complete square in exponent, Wick rotation, Gauss integral reconstruct delta functions to evaluate (some) of the 𝜷-parameter integrals to facilitate these steps for 2, 3 loops (virtual!): “developed” Mathematica package ARepCGC; implements necessary text-book methods [V. Smirnov, Springer 2006]
Ka(x) modified Bessel function of 2nd kind (Macdonald function) require f(a) for a=0,-1 and h{a,b} for a=0,-1, -2 and b = 0, -1 further reduction possible due to integration by parts identities h{a,b} can be directly evaluated for b=-1; general case into infinite sum over Bessel functions; numerics: keeping integral might be most stable massive case trivial as long one accepts one remaining integration for h{a,b}
f(a)( ¯ Q2, −r2) = Z ∞ dλλa−1e−λ ¯
Q2e
r2 4λ = 21−a
✓−r2 ¯ Q2 ◆a/2 Ka ✓q ¯ Q2(−r2) ◆ h(a,b)( ¯ Q2, r2
1, r2 3) =
Z ∞ dααa−1e−α ¯
Q2e
r2 1 4α ·
Z ¯
ρ
dρρb−1e
r2 3 4αρ
= 21−a Z ¯
ρ
dρρb−a/2−1 ✓−ρr2
1 − r2 3
¯ Q2 ◆a/2 Ka "s ¯ Q2 · ✓ −r2
1 − r2 3
ρ ◆#
From Gamma matrices to cross-sections
scalar, vector and rank 2 tensor integrals
use 2 implementations: FORM [Vermaseren, math-ph/0010025] & Mathematica packages FeynCalc and FormLink
h(a,b); result lengthy (~100kB), but manageable
by parts relation between basis function (work in progress)
their products)
[Hentschinski, Weigert, Schäfer, Phys.Rev. D73 (2006) 051501]
+ need to take care of potential soft factors work in progress related work:
[Boussarie, Grabovsky, Szymanowski, Wallon, JHEP1409, 026 (2014)] [Balitsky, Chirilli, PRD83 (2011) 031502, PRD88 (2013) 111501] [Beuf, PRD85, (2012) 034039]
energy collisions — used to fit a wealth of data (ep, pp, pA, AA)
quantitative description of data requires NLO
need precision — both experiment and theory
CGC (both eA and pA)
might have been available before, but never been exploited in a systematic way for this kind of calculation
advantage: benefit from standard techniques for higher
(important: soft- and collinear singularities!)
extends beyond →3-jets, NLO correction for saturation/ CGC observables in e.g. pA at RHIC/LHC
Electron-nucleus/-on scattering
I knowldege of scattering enery + nucleon mass
+ measure scattered electron control kinematics k p X k' q Photon virtuality Q2 = −q2 Mass of system X W = (p + q)2 = M 2
N + 2p · q − Q2
Bjorken x = Q2 2p · q Resolution λ ∼ 1
Q
Inelasticity y = 2p · q 2p · k
10
10
10
10
10 1 10
10
10
10
10
10 1 10
x x Q2 (GeV2) Q2 (GeV2)
Q2
s,quark Model-Ib=0 Au, median b Ca, median b p, median b
Q2
s,quark, all b=0Au, Model-II Ca, Model-II Ca, Model-I Au, Model-I xBJ × 300 ~ A
1/3Au Au p Ca Ca
Saturation: high densities in the fast nucleus
Expect those effects to be even more enhanced in boosted nuclei:
Boost
Q2
s ∼ # gluons/unit transverse area ∼ A1/3
conventional pQCD (make use of know techniques) inclusion of finite masses (charm mass!) intuition: interaction at t=0 with Lorentz contracted target momentum space well explored complication, but doable lose intuitive picture at first -> large # of cancelations configuration space poorly explored very difficult many diagrams automatically zero
work in momentum space, but exploit relation to configuration space to set a large fraction of all diagrams to zero
p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k− p−
1p−
2x−
1x−
2x−
3k−
geometric scaling
ln x
non-perturbative region
ln Q2 Q2
s(x)
s a t u r a t i
JIMWLK BK DGLAP BFKL
αs < < 1 αs ~ 1
Theory predictions for high & saturated gluon densities
x =Q2/2p･q→0 limit corresponds to perturbative high energy limit 2p･q→∞ for fixed resolution Q2
high energy limit
the strong coupling→resummation of finite density effects
“color dipole factor” (universal, resums ln1/x)
splits into color dipole (quark- antiquark pair) which interacts with Lorentz contracted target field
gluon densities multiple scatterings
γ∗
→
γ∗ x → 0: a single interaction with a strong & Lorentz contracted gluon field
φ Δ φ Δ φ Δ φ Δ
≡ ≡ A+,a(z−, z) = ↵a(z)(z−)
k p X k' q
σγ∗A
L,T (x, Q2) = 2
X
f
Z d2bd2r
1
Z dz
L,T (r, z; Q2)
N(x, r, b)
in Hentschinski (ICN-UNAM) The glue that binds us all November 3, 2015