Initial State Correlations and the Ridge
Tolga Altinoluk
CENTRA, IST
Universidade de Lisboa
Various Faces of QCD, Swierk, Poland
October 8-9, 2016
T. A. , N. Armesto, G. Beuf, A. Kovner, M. Lublinsky,
Phys.Lett. B751 (2015), Phys.Lett. B752 (2016), arXiv:1610.XXXXX
CENTRA
Tolga Altinoluk
Initial state correlations and the Ridge
1/24
Motivation: The Ridge
The Ridge Structure
Tolga Altinoluk
Initial state correlations and the Ridge
2/24
Correlations within the CGC
Final state correlations carry the imprint of the partonic correlations that
exist in the initial state:
1
Local anisotropy of target fields
A.
A.
A.
A.
A.
Kovner, M. Lublinsky, Phys.Rev. D83 (2011) 034017
Kovner, M. Lublinsky, Phys.Rev. D84 (2011) 094011
Kovner, M. Lublinsky, Int. J. Mod. Phys. E Vol. 22 (2013) 1330001
Dumitru, L. McLerran, V. Skokov, Phys. Lett. B 743 (2015)
Dumitru, V. Skokov, Phys.Rev. D91 (2015) 074006
2
Spatial variation of partonic density
3
”Glasma Graph” contributions to particle production:
E. Levin, A. Rezaeian, Phys.Rev. D84 (2011) 034031
A. Dumitru, F. Gelis, J. Jalilian-Marian, T. Lappi, Phys.Lett. B697 (2011) 21
K. Dusling, R. Venugopalan, Phys.Rev.Lett. 108 (2012) 262001
K. Dusling, R. Venugopalan, Phys.Rev. D87 (2013) 051502
K. Dusling, R. Venugopalan, Phys.Rev. D87 (2013) 054014
WHAT IS THE PHYSICS BEHIND THE GLASMA GRAPH CALCULATIONS?
⇓
BOSE ENHANCEMENT OF GLUONS IN THE HADRONIC WAVE-FUNCTION!!!
Tolga Altinoluk
Initial state correlations and the Ridge
3/24
Bose enhancement in a nutshell (1)
Consider a state with fixed occupation numbers of N species of bosons at different
momenta:
!ni (p)
i
Y
ai† (p)
1
{n (p)} ≡
p
√
|0i
V
ni (p)!
i ,p
with a finite volume V and periodic boundary conditions so that momenta are
discrete. (i = 1, 2, · · · , N)
The mean particle density :
D
E X
ni (p)
n ≡ {ni (p)}|a†i (x)ai (x)|{ni (p)} =
i ,p
The 2-particle correlator:
in x-space →
D(x, y ) ≡ h{n(p)}|a†i (x)a†j (y )ai (x)aj (y )|{n(p)}i
in p-space →
D(p, k) ≡ h{n(p)}|a†i (p)a†j (q)ai (l )aj (m)|{n(p)}i
⇒ D(p, k) = δ(p − l )δ(q − m)
Tolga Altinoluk
P
i
ni (p)
P
j
nj (q) + δ(p − m)δ(q − l )
P
i
ni (p)ni (q)
Initial state correlations and the Ridge
4/24
Bose enhancement in a nutshell (2)
Using these results, the 2-particle correlator in coordinate space:
D(x, y ) = n2 +
2
X Z d 3 p
ip(x−y ) i
n (p)
(2π)3 e
|i
{z
}
the Bose enhancement term
in momentum space:
D(p, k) =
"
X
i

#
z
}|
{
X
X
nj (k) + δ(p − k)
[ni (p)]2
ni (p) 
j
i
It vanishes when the points are far away!
It gives O(1/N) enhancement when the points coincide!
The O(1/N) suppression is due to the fact the second term contains a single sum
over the species index!
The physics: Only bosons of the same species are correlated with each other.
Tolga Altinoluk
Initial state correlations and the Ridge
5/24
Bose enhancement in CGC?
The Bose enhancement is a generic phenomenon, and is not tied to the state
with fixed number of of particles.
HOWEVER,
Classical-like coherent states do not exhibit such behaviour!!!
Consider a coherent state:
Z
h
i
3
i
i
†i
|0i
|b(x)i ≡ exp i d x b (x) a (x) + a (x)
A trivial calculation in this state gives
hb(x)|a†i (x)ai (x)|b(x)i = b i (x)b i (x)
hb(x)|a†i (x)a†j (y )ai (x)aj (y )|b(x)i = b i (x)b i (x)b j (y )b j (y )
so
D(x, y ) = n(x)n(y )
Thus, in order to exhibit Bose enhancement, a state has to be nonclassical.
CGC is defined in terms of classical fields. Can they produce Bose enhancement??
YES!!
Tolga Altinoluk
Initial state correlations and the Ridge
6/24
Glasma Graphs
Aim: to show that angular collimation arising from glasma graph calculation
is due to the Bose enhancement in the projectile wave-function.
Consider inclusive two particle production and assume parton-hadron duality.
a† (k1 )
a† (k2 )
a(k2 )
a† (k1 )
a(k1 )
a† (k2 )
a(k4 )
a(k3 )
a† (k1 )
a† (k2 )
a(k4 )
q
q
q
q
q
q
p
p
p
p
p
p
N (q − k2 )
a(k3 )
N (q − k2 )
N (p − k1 )
N (p − k1 )
TYPE A
N (p − k1 )
TYPE B
N (q − k4 )
TYPE C
with N(k) ≡dipole scattering probability :
N(k) = −
Tolga Altinoluk
Z
d 2 x e i k·x
i
1 h †
tr S (x)S(0)
Nc
T
Initial state correlations and the Ridge
7/24
Averaging over the projectile
a† (k1 )
a† (k2 )
a(k2 )
a(k1 )
a† (k1 )
a† (k2 )
q
q
q
p
p
p
N (q − k2 )
N (q − k2 )
N (p − k1 )
N (p − k1 )
TYPE A1
a† (k1 )
a† (k2 )
q
q
p
p
a(k3 )
a† (k1 )
a† (k2 )
q
p
N (p − k1 )
N (q − k4 )
TYPE C1
q
q
p
a(k3 )
†
a (k1 )
†
q
q
p
p
N (p − k1 )
Tolga Altinoluk
a(k3 )
a† (k1 )
a† (k2 )
a(k1 )
q
q
a(k4 )
a(k3 )
p
p
a(k4 )
a(k3 )
N (q − k2 )
N (p − k1 )
TYPE B3
N (q − k4 )
a(k3 )
a† (k1 )
a† (k2 )
q
q
p
p
N (p − k1 )
TYPE C2
A3 is trivial contraction! (not interesting!)
B3 & C3 → A1 & A2 with T ↔ P.
B1 & C2: suppressed at high momenta.
a(k2 )
TYPE A3
a(k4 )
a (k2 )
q
p
N (q − k2 )
a(k4 )
p
q
p
N (p − k1 )
TYPE B2
a(k4 )
q
a† (k2 )
N (p − k1 )
TYPE B1
p
a† (k1 )
N (q − k2 )
N (p − k1 )
a† (k2 )
a(k1 )
TYPE A2
a(k4 )
N (q − k2 )
a† (k1 )
a(k2 )
q
p
N (q − k4 )
TYPE C3
B2 & C1:
suppressed at high momenta
∝ δ (2) (p ± q)
lead to HBT correlations
[Kovchegov, Wertepny] (2013)
Initial state correlations and the Ridge
8/24
Gluon Production
Type A ∝
R
k1 ,k2
hin|aa†i (k1 )ab†j (k2 )aak (k1 )abl (k2 )|iniP N(p − k1 ) N(q − k2 )
|
{z
}
D(k1 , k2 )
N(p − k) ≡ the probability that incoming gluon with transverse momentum k
acquires transverse momentum p after the scattering.
CGC hadronic wave-function is boost invariant.
⇓
R
i
aai (k) ≡ √1Y |η<Y /2| dη
2π aa (η, k).
with the standard commutation relations:
[aai (k), ab†j (p)] = (2π)2 δab δij δ(2) (k − p).
Tolga Altinoluk
Initial state correlations and the Ridge
9/24
Averaging over the projectile
average over soft degrees of freedom
average over the valence color charge density ρ
Averaging over the projectile state in CGC:
The wave-function for the soft fields at fixed ρ:
Z
h
i
|iniρ = exp i bai (k) aa†i (k) + aai (−k) |0i
k
ց
i
Weizsäcker-Williams field bai (k) = g ρa (k) ik
k2
Averaging over ρ ⇒ integrating over ρ with some weight functional W [ρ]!
Take MV model: h· · · iρ = N
R
D[ρ] · · · e
−
R
1
k 2µ2 (k)
ρa (k)ρa (−k)
This defines the density matrix (operator) on the soft gluon Hilbert space:
ρ̂ = N
Z
D[ρ] e
−
R
1
k 2µ2 (k) ρa (k)ρa (−k)
ei
R
q
bbi (q)φib (−q)
|0ih0| e −i
R
p
bcj (p)φjc (−p)
where φia (k) = aai (k) + aa†i (−k).
Tolga Altinoluk
Initial state correlations and the Ridge
10/24
The Enhancement
Integration over ρ gives:
ρ̂ = e
−
× h0|
R
g 2 µ2 (k) i j
k k
k
2k 4
"
n
Y
m=1
( +∞ " n Z
#
X 1 Y
g 2 µ2 (pm ) im im
pm φam (pm ) |0i
4
n!
pm
m=1 pm
n=0
#)
φib (k) φjb (−k)
jm jm
pm
φam (−pm )
e
−
R
k′
g 2 µ2 (k ′ ) ′ i ′ ′ j ′
k k
2k ′ 4
′
′
φic (k ′ ) φjc (−k ′ )
The correlator that we are interested in :
D(k1 , k2 ) = tr [ρ̂aa†i (k1 )ab†j (k2 )aak (k1 )abl (k2 )]
k i k k k j k l g 4 µ2 (k1 )µ2 (k2 )
D(k1 , k2 ) = S 2 (Nc2 − 1)2 1 12 22 2
k1 k2
k12 k22
h
i
1
(2)
(2)
δ
(k
−
k
)
+
δ
(k
+
k
)
× 1+
.
1
2
1
2
S(Nc2 − 1)
The first term is the ”classical” term (the square of the number of particles)
The second term is the typical Bose enhancement term!
Tolga Altinoluk
Initial state correlations and the Ridge
11/24
Correlated production
Consider:
Incoming projectile wave-function with saturation momentum Qs
choose k1 and k2 to be of the same order as Qs
Then:
the momentum transfer during the scattering is less than Qs
N(p − k1 )N(p − k2 ) does not have large effect!
Thus the initial correlation is transmitted to to final state.
FOR GLUONS:
INITIAL STATE BOSE ENHANCEMENT ⇒ FINAL STATE CORRELATIONS
Tolga Altinoluk
Initial state correlations and the Ridge
12/24
What about quark correlations: Pauli blocking?
Are quarks in the CGC state subject to correlations?
Within the ”glasma graph” approach:
gluons:
final state correlations are due to the Bose enhancement of the gluons in the
projectile wave function.
This effect is long range in rapidity since the CGC wave function is dominated by
the rapidity integrated mode of the soft gluon field.
quarks:
quarks should experience Pauli blocking ⇒ the probability to find two identical
quarks with the same quantum numbers in the CGC state should be supressed.
Is the effect short or long range in rapidity?
Tolga Altinoluk
Initial state correlations and the Ridge
13/24
Quark contribution to the wave function (1)
The free part of the Light Cone Hamiltonian:
Z
k 2 †a +
H0 =
a (k , k) aia (k + , k)
+ i
k + ,k 2k
i
XZ
p2 h †
dα s (p + , p) dα s (p + , p) + d¯α† s (p + , p) d¯α s (p + , p)
+
+
2p
+
p ,p
s
To zeroth order the vacuum of the LCH is simply the zero energy Fock space
¯
vacuum of the operators a, d and d:
aq |0i = 0, dp |0i = 0, d¯p |0i = 0, E0 = 0.
The normalized one-particle states to zeroth order are
|k + , k, a, i i =
hk1+ , k1 , a, i |k2+ , k2 , b, ji
=
|p + , p, α, si =
hp1+ , p1 , α, s1 |p2+ , p2 , β, s2 i
Tolga Altinoluk
=
1
aa † (k + , k) |0i,
(2π)3/2 i
δab δij δ(2) (k1 − k2 )δ(k1+ − k2+ ),
1
d † (p + , p) |0i,
(2 π)3/2 α,s
δαβ δs1 s2 δ(2) (p1 − p2 )δ(p1+ − p2+ )
Initial state correlations and the Ridge
14/24
Quark contribution to the wave function (2)
The full Hamiltonian contains several types of perturbations:
δH = δH ρ + δH g qq + · · ·
Interaction with the background field: δH ρ = δH ρ g + δH ρ qq + δH ρ gg
Quark-gluon interaction: δH g qq
using the explicit expressions of the corrections to the free Hamiltonian, we can
write the relevant matrix elements needed for the calculation of the dressed
quark wave function:
hg |δH ρ g |0i,
hq q̄|δH ρqq |0i,
hq q̄|δH g qq |g i
with all these ingredients the dressed wave function
4
|v iD
4 =g
Z
p,p̄,q,q̄
+virtual
h
i
†
†
†
†
(p) d¯ι,s
(k̄ + , q, q̄, β) dǫ,s
ζsǫι1 s2 (k + , p, p̄, α)ζrγδ
(p̄) dγ,r
(q) d¯δ,r
(q̄) |v i
1
1 r2
2
1
2
+
The amplitude ζsγδ
1 s2 (k , p, q, α) =
Tolga Altinoluk
a
τγδ
k+
R
k
ρa (k) φs1 s2 (k, p, q; α)
Initial state correlations and the Ridge
15/24
Quark pair production cross section
The formal expression for the inclusive quark pair production emission reads
dσ
dη1 d 2 k1 dη2 d 2 k2
†
†
†
= h0|Ω Ŝ † Ω† dα,s
1 (k1 ) dα,s1 (k1 ) dβ,s2 (k2 ) dβ,s2 (k2 ) Ω Ŝ Ω |0i
Ŝ: eikonal S-matrix operator
Ω: unitary operator that (perturbatively) diagonalizes the QCD Hamiltonian.
The quark pair production cross section can be written as
Z
h
dσ
g8
1 a
=
hρ (x)ρb (x̄)ρc (y )ρd (ȳ )i Φ2 (x, y ; z1 , z2 , z̄; p)Φ2 (x̄, ȳ ; z̄1 , z̄2 , w̄ ; q)
2
2
6
dη1 d p dη2 d q
(2π)
2
n
o
a
aā
× tr [τ − SA (x)SF (z1 )τ ā SF† (z̄)][τ c − SAc c̄ (y )SF (z̄)τ c̄ SF† (z2 )]
o
n
¯
¯
× tr [τ b − SAbb̄ (x̄)SF (z̄1 )τ b̄ SF† (w̄ )][τ d − SAd d (ȳ )SF (w̄ )τ d SF† (z̄2 )]
− Φ4 (x, y , x̄ , ȳ ; z1 , z2 , z̄1 , z̄2 ; z̄, w̄ ; p, q)
n
× tr [τ a − SAaā (x)SF (z1 )τ ā SF† (z̄)][τ c − SAc c̄ (y )SF (w̄ )τ c̄ SF† (z2 )]
oi
¯
¯
× [τ b − SAbb̄ (x̄)SF (z̄1 )τ b̄ SF† (w̄ )][τ d − SAd d (ȳ )SF (z̄)τ d SF† (z̄2 )]
To get the quark pair density set each S-matrix to zero!
Tolga Altinoluk
Initial state correlations and the Ridge
16/24
Quark contribution to the wave function (3)
The amplitudes:
R1
R P
Φ2 (k, p) ≡ 0 dα q s1 s2 φs1 ,s2 (k, p, q; α) φ∗s1 ,s2 (k, p, q; α)
R1
P
dβ
Φ4 (k, l , k̄, l¯; p, q) ≡ s1 s2 ,s̄1 ,s̄2 0 (β+β̄e η1 −ηdα
2 )(α+ᾱe η2 −η1 )
R
¯ q, p̄; α)
× p̄q̄ φs1 s2 (k, p, p̄; α) φs̄1 s̄2 (k̄, q, q̄; β) φ∗s1 s̄2 (l , p, q̄; β) φ∗s̄1 s2 (l,
Two different contributions to the wave function:
(p+ , p⊥ )
!1
αp
+
, k⊥
(p+ , p⊥ )
!1
"
αp
! ᾱ
αp
+
, k⊥ − p⊥
+
, k⊥
"
!1
αp
+
, k⊥
"
! ᾱ
"
αp
+
, k⊥ − p⊥
#
"
!
+
1 +
β q , k̄⊥
#
β̄ +
β q , k̄⊥
− q⊥
Φ2 Φ2
Tolga Altinoluk
ᾱ +
p , k⊥ − p⊥ + q⊥
q+ + α
"
(q , q⊥ )
$
#
$
+
(q , q⊥ )
#
p+ + ββ̄ q + , k̄⊥ − q⊥ + p⊥
$
1 +
β q , k̄⊥
$
#
1 +
β q , k̄⊥
$
#
β̄ +
β q , k̄⊥
− q⊥
$
Φ4
Initial state correlations and the Ridge
17/24
Quark pair density in the projectile wave function (1)
Φ2 Φ2 contribution after contracting the color charge densities:
p⊥
p⊥
k⊥
k⊥
k⊥
p⊥
k⊥
k⊥
k⊥ − p⊥
k⊥ − p⊥
k⊥ − p⊥
q⊥
q⊥
q⊥
k̄⊥
k̄⊥
k̄⊥ − q⊥
Uncorrelated O(Nc4 )
k̄⊥
k̄⊥
k̄⊥
k̄⊥ − q⊥
k̄⊥
k̄⊥ − q⊥
Correlated O(Nc2 )
Correlated O(Nc2 )
Φ2 Φ2 contribution to the correlated quark pair density is O(Nc2 ).
Tolga Altinoluk
Initial state correlations and the Ridge
18/24
Quark pair density in the projectile wave function (2)
Φ4 contribution after contracting the color charge densities:
p⊥
k⊥
k⊥ − p⊥
q⊥
p⊥
k̄⊥ − q⊥ + p⊥
k⊥
k⊥ − p⊥
k⊥ − p⊥ + q⊥
k̄⊥
q⊥
p⊥
k̄⊥ − q⊥ + p⊥
k⊥ − p⊥
k⊥ − p⊥ + q⊥
k̄⊥
k̄⊥ − q⊥
3
(ΦA
4 ) Correlated O(Nc )
k⊥
q⊥
k̄⊥ − q⊥ + p⊥
k⊥ − p⊥ + q⊥
k̄⊥
k̄⊥ − q⊥
k̄⊥ − q⊥
Correlated O(Nc )
3
(ΦB
4 ) Correlated O(Nc )
Φ4 contribution to the correlated quark pair density is O(Nc3 ).
In the large Nc -limit, the only contribution to the correlated quark pair density
comes from two diagrams of Φ4 .
Tolga Altinoluk
Initial state correlations and the Ridge
19/24
Quark pair density in the projectile wave function (3)
The leading Nc contribution to the correlated quark pair density in the projectile
wave function is given by
dN P (p, q; η1 , η2 )
d 2 p d 2 q dη1 dη2
=−
correlated
Z
k k̄l l¯
D
E
ρa (k)ρc (k̄)ρb (l )ρd (l¯) Φ4 (k, l , k̄, l¯; p, q) tr{τ a τ b τ c τ d }
Adopt MV model for the averaging over the color charges:
D
E
ρa (k)ρb (p) = (2π)2 µ2 (k) δab δ(2) (k + p)
For k 2 > Qs2 : µ2 (k) → µ2
For k → 0: µ2 (k) → 0.
estimate the results in the following kinematics:
(i) rapidity difference between the quarks is relatively large: η1 − η2 ≫ 1
(ii) the two transverse momenta to be of the same order and much larger than
saturation momentum: |p| ∼ |q| ≫ Qs
with this estimate answer the basic questions:
(i) what is the sign of the correlation?
(ii) how far in rapidity difference does it extend?
Tolga Altinoluk
Initial state correlations and the Ridge
20/24
Quark pair density and Pauli blocking
The final result:
(
2
4 g 8 N 3 25π 2
p 2 (2)
dN P (p, q; η1 , η2 )
c
η2 −η1
2 µ
4
−Se
(η
−
η
)
η
−
η
+
ln
δ (q − p)
q
≃
1
2
1
2
d 2 pd 2 qd η1 dη2 correlated
p4q4 4
2
Qs2
#)
"
(p 2 + q 2 ) 5p 2 q 2 − 3(p · q)2 − (p 2 + q 2 )p · q
(p − q)2
ln
+
4(η
−
η
)p
·
q
+π 3
1
2
(q − p)4
Qs2
the correlated contribution is negative! confirms our expectation based on the
physics of the Pauli blocking
the correlation is formally short range in rapidity since it decreases
exponentially as a function of the rapidity difference.
Note that the rate of the decrease in rapidity is tampered by the fourth power
of η1 − η2 , so that in practical terms the correlation may extend fairly far in
rapidity.
Tolga Altinoluk
Initial state correlations and the Ridge
21/24
Particle production and Pauli blocking
We have already the expression for the quark pair production cross section:
Z
g8
1
dσ
=
− hρa (x)ρb (x̄)ρc (y )ρd (ȳ )iΦ4 (x, y , x̄, ȳ ; z1 , z2 , z̄1 , z̄2 ; z̄, w̄ ; p, q)
dη1 d 2 p dη2 d 2 q
(2π)6
2
n
a
aā
× tr [τ − SA (x)SF (z1 )τ ā SF† (z̄)][τ c − SAc c̄ (y )SF (w̄ )τ c̄ SF† (z2 )]
o
¯
¯
× [τ b − SAbb̄ (x̄)SF (z̄1 )τ b̄ SF† (w̄ )][τ d − SAd d (ȳ )SF (z̄)τ d SF† (z̄2 )]
The same Nc counting holds for the production cross section. No Φ2 Φ2 !
S(x) = exp{igt a αa (x)} → Expand each S in α!
αa (x) =
1
(x, y )ρaT (y )
∇2
Use MV model for ρT ρT correlator:
hρaT (k)ρbT (p)i = (2π)2 λ2 (k)δab δ(2) (k + p)
contract both the target and the projectile color charge densities!
Tolga Altinoluk
Initial state correlations and the Ridge
22/24
Estimates for particle production
Kinematics:
rapidity difference between the quarks is relatively large: η1 − η2 ≫ 1
the two transverse momenta to be of the same order and much larger than
saturation momentum: |p| ∼ |q| ∼ |p − q| ≫ Qs
saturation momentum of the target is smaller than than that of projectile:
QT < Qs .
⇓
This is the regime where correlations existing in the projectile wave function are not
strongly distorted by the momentum transfer from the target.
2 3
4 4
QT π
dσ
12 5 µ λ
η2 −η1
2
−Sg
N
e
(η
−
η
)
ln
≃
1
2
c
d 2 pd 2 qd η1 dη2 correlated
Λ2 p 4
Qs2 QT2
(
2
2
2
2
2
2
4
9Qs 2(p + q ) + p q
(p − q)2
Qs
50π
(2)
ln
δ
(q
−
p)
+
ln
×
2
16
q4
(p − q)4
Qs2
QT Λ 2
)
9Q 2
q2
p2
+ 4s ln
+ ln
2q
Qs2
Qs2
Tolga Altinoluk
Initial state correlations and the Ridge
23/24
Summary
We have shown that the underlying physics in the ”glasma graph calculation”
that leads to final state correlations is Bose enhancement of the gluons in the
projectile wave function.
Another physical effect present in the glasma graph calculation is HBT
correlations between the gluons far separated in rapidity.
We have also calculated quark-quark correlated production in the CGC
approach. We find that there is a depletion of pair production at like transverse
momenta due to Pauli blocking effect.
Bose enhancement for gluons is long range in rapidity where as Pauli blocking
for quarks is short range in rapidity.
The exponential decay with rapidity difference is tempered by a factor
quadratic in rapidity difference, resulting in a dip at ∆η ∼ 2.
Quark-quark correlated production turns out to be parametrically O(α2s Nc )
realtive to gluon-gluon correlations, which for realistic values of αs ∼ 0.2 and
Nc = 3 results in a mild suppression factor.
Tolga Altinoluk
Initial state correlations and the Ridge
24/24
BACK UP SLIDES
Tolga Altinoluk
Initial state correlations and the Ridge
25/24
HBT correlations from CGC
Another physical effect present in the glasma graph calculation:
Hanbury-Brown-Twiss (HBT) correlations between gluons far separated in rapidity!
⇓
leads to a potentially observable effect in the final state mesons which may
allow a direct measurement of gluonic size of the proton.
It correlates gluons with same and opposite transverse momenta (in analogy
with the double ridge structure).
The diagrams that will lead to HBT correlations:
a† (k1 )
a† (k2 )
a(k4 )
a(k3 )
a† (k1 )
a† (k2 )
a(k4 )
q
q
q
q
p
p
p
p
a(k3 )
N (q − k2 )
N (p − k1 )
TYPE B
N (p − k1 )
N (q − k4 )
TYPE C
For translationally invariant averaging:
TYPE B ∝ δ(2) (p − q) and TYPE C ∝ δ(2) (p + q)
Relaxing the translational invariance ⇒ the δ-functions are smeared : |p ± q| ∼ R −1
Tolga Altinoluk
Initial state correlations and the Ridge
26/24
The standard HBT
The object of interest: ”Two particle correlation function”
C (p, q) =
dN
dpdq
dN dN
dp dq
consider emission of pions by a chaotic superposition of classical sources
˜ =
J(p)
N
X
i =1
e i φi e ip·xi J˜0 (p − pi )
each individual classical source is boosted by momentum pi
each individual classical source has a random phase φi
The emission function:
S(x, K ) =
Z
d 4 y −iK ·y
e
2(2π)3
1
1
J ∗a x + y J a x − y
2
2
where K is the total momenta of the emitted particles.
C (p, q) can be written in terms of the S(x, K ).
Tolga Altinoluk
Initial state correlations and the Ridge
27/24
Standard HBT vs gluon HBT
Standard HBT
˜ =
J(p)
N
X
i =1
Gluon HBT
J˜a (p) =
e i φi e ip·xi J˜0 (p − pi )
N
X
i =1
UAi
ab
e ip·xi J˜0b (p)
For gluon HBT: J˜0∗b (p) = J˜0b (−p) since J0b (x) is real!
For standard HBT: p is a three vector!
For gluon HBT: p is transverse only! [The time and longitudinal coordinate
dependence of all sources is identical ⇒ δ(x + )δ(x − )]
The emission function:
S(x, K ) =
Z
d 2 y −iK ·y
e
2(2π)3
1
1
Ja x + y Ja x − y
2
2
For a Gaussian distribution of the sources, the correlator reads:
C (q, K ) = 1 +
Tolga Altinoluk
i
1 h −R 2 q2
−R 2 K 2
e
+
e
Nc2 − 1
Initial state correlations and the Ridge
28/24
Transverse structure
Q−1
s
R
The projectile proton has a distribution of gluon fields
with spatial size R (the gluonic size of the proton).
eikonal scattering with the target → gluon fields color
rotate : bia (x) = U ab (x)bib (x)
For a nucleus with a large saturation scale Qs ≫ R −1 :
the eikonal scattering matrix U(x) varies on the spatial scale Qs−1 .
⇒ the source is color correlated only on scales of order Qs−1
color is decorrelated outside due to scattering.
The spatial structure of the source: a collection of independent color sources.
Each source have a transverse size Qs−1 .
The total transverse area of R 2 .
The number of independent sources is N ∼ Qs2 R 2 .
Tolga Altinoluk
Initial state correlations and the Ridge
29/24
Properties of the gluon HBT
The correlation is long range in rapidity - it is equally strong when the
rapidities of the two gluons are equal or when the difference between the two
rapidities is large.
The correlation is symmetric under reversal of the direction of the transverse
momentum of one of the gluons. Thus ,it is strongest when the transverse
momenta of the two gluons are either parallel or antiparallel.
The HBT radius is of the order of the inverse gluonic size of the proton R −1 .
The HBT signal dominates the correlation function (at small momentum
difference) when the number of incoherent emitters is large, NS = Qs2 R 2 ≪ 1.
Tolga Altinoluk
Initial state correlations and the Ridge
30/24
HBT or Bose Enhancement?
The ”glasma graph” calculation contains both type of correlations:
C (p, q) =
dN
dpdq
dN dN
dp dq
= 1 + CBE (p, q) + CHBT (p, q)
Both CBE (p, q) and CHBT (p, q) are rapidity independent!!
CBE (the coherent part) contribution is suppressed ∼ 1/R 2 QT2 , BUT is ”wide”
2
2
2
2
in momentum space ∼ e −(p−q) /Qs (and ∼ e −(p+q) /Qs ).
CHBT (p, q) is unsuppressed when the number of sources is large: R 2 QT2 ≫ 1
2 2
2 2
BUT gives a narrow peak ∼ e −(p−q) R (and ∼ e −(p+q) R ).
irreducible
HBT
1/R
Qs
Tolga Altinoluk
q
Initial state correlations and the Ridge
31/24
Explicit expressions - 1
φs1 s2 (k, p; α) =
1
4αᾱk 2 − [ᾱk · p + αk · (k − p)] + 2i σ 3 k × p
k 2 [ᾱp 2 + α(k − p)2 ]
∆ = g 4 tr
h
′
′
′
′
′
′
{τ a τ a [αa (x) − αa (z̄)] − τ a τ a [αa (x) − αa (z1 )]}
′
′
′
b b′
b′
b′
′
′
′
b′
b′
× {τ c τ c [αc (y ) − αc (z2 )] − τ c τ c [αc (y ) − αc (w̄ )]}
b′ b
× {τ τ [α (x̄) − α (w̄ )] − τ τ [α (x̄) − α (z̄1 )]}
i
′
′
′
′
′
′
× {τ d τ d [αd (ȳ ) − αd (z̄2 )] − τ d τ d [αd (ȳ ) − αd (z̄)]} .
∆A =
g 4 Nc5
16
∆B =
g 4 Nc5
16
Tolga Altinoluk
n
o
h[α(x) − α(z̄)][α(ȳ ) − α(z̄)] + [α(x) − α(z1 )][α(ȳ ) − α(z̄2 )]i
n
o
×
h[α(x̄) − α(w̄ )][α(y ) − α(w̄ )] + [α(x̄) − α(z̄1 )][α(y ) − α(z2 )]i .
n
o
h[α(x) − α(z̄)][α(y ) − α(w̄ )] + [α(x) − α(z1 )][α(y ) − α(z2 )]i
n
o
×
h[α(x̄ ) − α(w̄ )][α(ȳ ) − α(z̄)] + [α(x̄) − α(z̄1 )][α(ȳ ) − α(z̄2 )]i .
Initial state correlations and the Ridge
32/24
Explicit expressions - 2
δH ρ g =
Z
0
δH ρ qq =
∞
h
i
dk + d 2 k
g ki
√
ai†a (k + , k) ρa (−k) + aia (k + , k) ρa (k)
2
+
3/2
2π (2π)
2 |k |
i
X Z dk + d 2 k dp + d 2 p g 2 h
a ¯†
dα† s (p + , p) ταβ
dβ s (k + − p + , k − p) ρa (−k) + h.c. (
6
+
2
(2π)
(k )
s
X Z dp + d 2 p dk + d 2 k
θ(k + − p + ) Γis1 s2 (k + , k, p + , p)
3/2 (2π)6 (k + )1/2
2
s1 ,s2
h
i
+
a +
†
+
+
¯†
× ai (k , k) dα,
s1 (p , p) dβ,s2 (k − p , k − p) + h.c.
a
δH g qq = g ταβ
pi
ki − pi
km − pm
ki
im pm
+
−
+
2is
ǫ
Γis1 s2 (k + , k, p + , p) = δs1 s2 2 + −
1
k
p+ k + − p+
p+ k + − p+
Tolga Altinoluk
Initial state correlations and the Ridge
33/24
Explicit expressions - 3
h0|aia (k + , k) δH ρg |0i
gki
= 3/2 + 3/2 ρa (−k)
(2π)3/2
4π |k |
h0|dαs1 (q + , q) d¯βs2 (p + , p)δH ρ qq |0i
hq q̄|δH ρqq |0i =
(2π)3
2
a
g ταβ
ρa (−p − q)δs1 s2
=
(2π)3 (p + + q + )2
h0|dαs1 (p + , p)d¯βs2 (q + , q)δH q aia† (k + , k)|0i
hq q̄|δH g qq |g i =
(2π)9/2
+
+
i
a Γs1 s2 (k , k, p , p) (2)
= g ταβ
δ (p + q − k)δ(p + + q + − k + )
8π 3/2 (k + )1/2
hg |δH ρ g |0i =
Tolga Altinoluk
Initial state correlations and the Ridge
34/24
Particle production and Pauli blocking
After all possible contractions at leading Nc , there are two types of contributions to
the production cross section:
Z
Z
¯
µ2 (k)µ2 (k̄)λ2 (l )λ2 (l)
g 12 Nc5 1
dα dβ
A = −δ(2) (0)δ(2) (p − q)
η
−η
4
4
η
−η
¯
16
l l
0 (β + β̄e 1 2 )(α + ᾱe 2 1 ) k,k̄,l,l¯
i
nh
∗
∗
×tr Ψ̄(k, l , p; α)Ψ̄ (k, l , p; α) + Ψ(k, l , p; α)Ψ (k, l , p; α)
io
h
× Ψ̄(k̄, l¯, p; β)Ψ̄∗ (k̄, l¯, p; β) + Ψ(k̄, l¯, p; β)Ψ∗ (k̄, l¯, p; β)
Z
µ2 (k)µ2 (k̄)λ2 (l )λ2 (l¯)
dα dβ
η
−η
η
−η
l 4 l¯4
0 (β + β̄e 1 2 )(α + ᾱe 2 1 ) k,k̄,l,l¯
i
nh
×δ(2) (k + l − p − k̄ − l¯+ q)tr Ψ̄(k, l , p; α)Ψ̄∗ (k, l , p; β) + Ψ(k, l , p; α)Ψ∗ (k, l , p; β)
io
h
¯ q; α) + Ψ(k̄, l¯, q; β)Ψ∗ (k̄, l¯, q; α)
× Ψ̄(k̄, l¯, q; β)Ψ̄∗ (k̄, l,
B = −δ(2) (0)
g 12 Nc5
16
Z
1
with
Ψ(k, l , p; α) ≡ [φ(k + l , p; α) − φ(k, p − l ; α)]
Ψ̄(k, l , p; α) ≡ [φ(k + l , p; α) − φ(k, p; α)]
Tolga Altinoluk
Initial state correlations and the Ridge
35/24