Rapidity evolution of gluon TMD from low to
moderate x
I. Balitsky
JLAB & ODU
QCD Evolution 2017, 23 May 2017
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
1 / 34
Outline
Reminder: rapidity factorization and evolution of color dipoles
Method of calculation: shock-wave approach + light-cone expansion.
One loop: real corrections and virtual corrections.
One-loop evolution of gluon TMD
DGLAP, Sudakov and BK limits of TMD evolution equation
Gluon TMDs in particle production
Conclusions and outlook
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
2 / 34
DIS at high energy: Wilson lines and color dipoles
At high energies, particles move along straight lines ⇒
the amplitude of γ ∗ A → γ ∗ A scattering reduces to the matrix element of
a two-Wilson-line operator (color dipole):
d 2 k⊥ A
I (k⊥ )hB|Tr{U(k⊥ )U † (−k⊥ )}|Bi
4π 2
h Z ∞
i
U(x⊥ ) = Pexp ig
du nµ Aµ (un + x⊥ )
Wilson line
Z
A(s) =
−∞
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
3 / 34
DIS at high energy: Wilson lines and color dipoles
At high energies, particles move along straight lines ⇒
the amplitude of γ ∗ A → γ ∗ A scattering reduces to the matrix element of
a two-Wilson-line operator (color dipole):
d 2 k⊥ A
I (k⊥ )hB|Tr{U(k⊥ )U † (−k⊥ )}|Bi
4π 2
h Z ∞
i
U(x⊥ ) = Pexp ig
du nµ Aµ (un + x⊥ )
Wilson line
Z
A(s) =
−∞
Formally,
means the operator expansion in Wilson lines
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
3 / 34
Rapidity factorization: OPE in Wilson lines
Y> d
+
+...
Y< d
η - rapidity factorization scale
Rapidity Y > η - coefficient function (“impact factor”)
Rapidity Y < η - matrix elements of (light-like) Wilson lines with rapidity
divergence cut by η
h Z
Uxη = Pexp ig
∞
i
dx+ Aη+ (x+ , x⊥ ) ,
−∞
I. Balitsky (JLAB & ODU)
Aηµ (x) =
Z
d4 k
θ(eη − |αk |)e−ik·x Aµ (k)
(2π)4
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
4 / 34
Spectator frame: propagation in the shock-wave background.
Boosted Field
R
µ
Each path is weighted with the gauge factor Peig dxµ A . Quarks and gluons
do not have time to deviate in the transverse space ⇒ we can replace the
gauge factor along the actual path with the one along the straight-line path.
z
z’
y
x
x
y
Wilson Line
[ x → z: free propagation]×
[U ab (z⊥ ) - instantaneous interaction with the η < η2 shock wave]×
[ z → y: free propagation ]
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
5 / 34
Reminder: evolution of color dipoles at small x
To get the evolution equation for color dipoles, consider the dipole with the
rapidies up to η1 and integrate over the gluons with rapidities η1 > η > η2 .
This integral gives the kernel of the evolution equation (multiplied by the
dipole(s) with rapidities up to η2 ).
αs (η1 − η2 )Kevol ⊗
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
6 / 34
Rapidity evolution of color dipoles in the leading order
d
Tr{Ûx Ûy† } = KLO Tr{Ûx Ûy† } + ... ⇒
dη
d
hTr{Ûx Ûy† }ishockwave = hKLO Tr{Ûx Ûy† }ishockwave
dη
x
x*
a
x
x*
x
x*
b
a
x*
b
x
a
b
y
x
a
b
(b)
(a)
(c)
(d)
Uzab = Tr{ta Uz tb Uz† } ⇒ (Ux Uy† )η1 → (Ux Uy† )η1 + αs (η1 − η2 )(Ux Uz† Uz Uy† )η2
⇒ Evolution equation is non-linear
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
7 / 34
Non linear evolution equation
Û(x, y) ≡ 1 −
1
Tr{Û(x⊥ )Û † (y⊥ )}
Nc
BK equation
αs Nc
d
Û(x, y) =
dη
2π 2
Z
o
d2 z (x − y)2 n
Û(x,
z)
+
Û(z,
y)
−
Û(x,
y)
−
Û(x,
z)
Û(z,
y)
(x − z)2 (y − z)2
I. B. (1996), Yu. Kovchegov (1999)
Alternative approach: JIMWLK (1997-2000)
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
8 / 34
Non-linear evolution equation
Û(x, y) ≡ 1 −
1
Tr{Û(x⊥ )Û † (y⊥ )}
Nc
BK equation
d
αs Nc
Û(x, y) =
dη
2π 2
Z
o
d2 z (x − y)2 n
Û(x,
z)
+
Û(z,
y)
−
Û(x,
y)
−
Û(x,
z)
Û(z,
y)
(x − z)2 (y − z)2
I. B. (1996), Yu. Kovchegov (1999)
Alternative approach: JIMWLK (1997-2000)
LLA for DIS in pQCD ⇒ BFKL
I. Balitsky (JLAB & ODU)
(LLA: αs 1, αs η ∼ 1)
Rapidity evolution of gluon TMD from low to moderate
QCDxEvolution 2017, 23 May 2017
9 / 34
Non-linear evolution equation
Û(x, y) ≡ 1 −
1
Tr{Û(x⊥ )Û † (y⊥ )}
Nc
BK equation
αs Nc
d
Û(x, y) =
dη
2π 2
Z
o
d2 z (x − y)2 n
Û(x,
z)
+
Û(z,
y)
−
Û(x,
y)
−
Û(x,
z)
Û(z,
y)
(x − z)2 (y − z)2
I. B. (1996), Yu. Kovchegov (1999)
Alternative approach: JIMWLK (1997-2000)
LLA for DIS in pQCD ⇒ BFKL
LLA for DIS in sQCD ⇒ BK eqn
(LLA: αs 1, αs η ∼ 1)
(LLA: αs 1, αs η ∼ 1, αs A1/3 ∼ 1)
(s for semiclassical)
NLO kernels for BK and JIMWLK are now known (even NNLO BK for N = 4SYM!)
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
10 / 34
Rapidity factorization for particle production
Sudakov variables:
k = αp1 + βp2 + k⊥ ,
p1 ' pA , p2 ' pB ,
p21 = p22 = 0
Dimensionless light-cone coordinates
r
r
s
s
x+ , x• ≡ xµ pµ1 =
x−
x∗ ≡ xµ pµ2 =
2
2
We integrate over “central” fields in the background of projectile and target
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
fields.
11 / 34
Matrix element between hadron states ⇒
P
X
=1
“Hadronic tensor”
W(pA , pB , q)
def
XZ
=
ZX
d4 x e−iqx hpA , pB |F 2 (x)F 2 (0)|pA , pB i
=
d4 x e−iqx hpA , pB |F 2 (x)|XihX|F 2 (0)|pA , pB i
Double functional integral for W
XZ
W(pA , pB , q) =
d4 x e−iqx hpA , pB |F 2 (x)|XihX|F 2 (0)|pA , pB i
X
=
tf →∞
lim
ti →−∞
Z
d4 x e−iqx
Z
Ã(tf )=A(tf )
Z
Dõ DAµ
ψ̃(tf )=ψ(tf )
Dψ̄˜Dψ̃Dψ̄Dψ Ψ∗pA (~Ã(ti ), ψ̃(ti ))
× Ψ∗pB (~Ã(ti ), ψ̃(ti ))e−iSQCD (Ã,ψ̃) eiSQCD (A,ψ) F̃ 2 (x)F 2 (y)ΨpA (~A(ti ), ψ(ti ))ΨpB (~A(ti ), ψ(ti ))
“Left” A, ψ fields correspond to the amplitude hX|F 2 (0)|pA , pB i,
“right” fields Ã, ψ̃ correspond to amplitude hpA , pB |F 2 (x)|Xi
The boundary conditions Ã(tf ) = A(tf ) and ψ̃(tf ) = ψ(tf ) reflect the sum over
intermediate states X.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
12 / 34
TMD factorization with power corrections
In the region s Q2 Q2⊥ at the tree level
Z
Z
64/s2
2
i(q,x)⊥ 2
dx• dx∗ e−iαq x• −iβq x∗
W(pA , pB , q) = 2
d x⊥ e
Nc − 1
s
n
n
n
× hpA |G∗mi (x• , x⊥ )G∗mj (0)|pA ihpB |F•i
(x∗ , x⊥ )F•j
(0)|pB i
Z
x
•
32
N 2 ∆ij,kl
2
a
b
c
+ 2 2 c
(x• , x⊥ )G∗j
(x•0 , x⊥ )G∗r
(0)|pA i
d x0 dabc hpA |G∗i
2
Q (Nc − 4)(Nc − 1) −∞ s •
Z x•
o
2
m
n 0
n
×
d x•0 dmnl hpB |F•k
(x∗ , x⊥ )F•l
(x∗ , x⊥ )F•r
(0)|pB i
−∞ s
∆ij,kl ≡ gij gkl − gik gjl − gil gjk
ab
b
G∗i
(z• , z⊥ ) ≡
[−∞• , z• ]Az ∗
a
F•i
(z∗ , z⊥ ) ≡
[−∞∗ , z∗ ]Az •
I. Balitsky (JLAB & ODU)
b
F∗i
(z• , z⊥ ),
ab
b
F•i
(z∗ , z⊥ )
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
13 / 34
Rapidity evolution: one loop
We study evolution of F̃iaη (x⊥ , xB )Fjaη (y⊥ , xB ) with respect to rapidity cutoff η
Z
2
m
(z⊥ , xB ) ≡
dz∗ eixB z∗ [∞, z∗ ]am
z F•i (z∗ , z⊥ )
s
Z
d4 k
Aηµ (x) =
θ(eη − |αk |)e−ik·x Aµ (k)
(2π)4
a(η)
Fi
At first we study gluon TMDs with Wilson lines stretching to +∞ (like in SIDIS).
0
Matrix element of F̃ia (k⊥
, xB0 )F ai (k⊥ , xB ) at one-loop accuracy:
diagrams in the “external field” of gluons with rapidity < η.
Figure : Typical diagrams for one-loop contributions to the evolution of gluon TMD.
(Fields à to the left of the cut and A to the right.)
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
14 / 34
Shock-wave formalism and transverse momenta
α α and k⊥ ∼ k⊥ ⇒ shock-wave external field
k?
k?
k?
k?
Characteristic longitudinal scale of fast fields: x∗ ∼
1
β,
Characteristic longitudinal scale of slow fields: x∗ ∼
β∼
1
β,
2
k⊥
αs
β∼
2
k⊥
αs
⇒ x∗ ∼
αs
2
k⊥
⇒ x∗ ∼
αs
2
k⊥
2
2
If α α and k⊥
≤ k⊥
⇒ x∗ x∗
⇒ Diagrams in the shock-wave background at k⊥ ∼ k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
15 / 34
Problem: different transverse momenta
α α and k⊥ k⊥ ⇒ the external field may be wide
k?
k?
k?
k?
Characteristic longitudinal scale of fast fields: x∗ ∼
1
β,
Characteristic longitudinal scale of slow fields: x∗ ∼
2
k⊥
αs
2
αs ⇒ x∗ ∼ k⊥
2
k⊥
∼ αs ⇒ x∗ ∼ kαs
2
⊥
β∼
1
βs ,
β
2
2
k⊥
⇒ x∗ ∼ x∗ ⇒ shock-wave approximation is invalid
If α α and k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
16 / 34
Problem: different transverse momenta
α α and k⊥ k⊥ ⇒ the external field may be wide
k?
k?
k?
k?
Characteristic longitudinal scale of fast fields: x∗ ∼
1
β,
Characteristic longitudinal scale of slow fields: x∗ ∼
2
k⊥
αs
2
αs ⇒ x∗ ∼ k⊥
2
k⊥
∼ αs ⇒ x∗ ∼ kαs
2
⊥
β∼
1
βs ,
β
2
2
k⊥
⇒ x∗ ∼ x∗ ⇒ shock-wave approximation is invalid
If α α and k⊥
2
2
Fortunately, at k⊥
k⊥
we can use another approximation
⇒ Light-cone expansion of propagators at k⊥ k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
16 / 34
Method of calculation
We calculate one-loop diagrams in the fast-field background
in following way:
if k⊥ ∼ k⊥ ⇒ propagators in the shock-wave background
if k⊥ k⊥ ⇒ light-cone expansion of propagators
We compute one-loop diagrams in these two cases and write down
“interpolating” formulas correct both at k⊥ ∼ k⊥ and k⊥ k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
17 / 34
Shock-wave calculation
Reminder:
F̃ia (z⊥ , xB )
2
≡
s
Z
m
dz∗ e−ixB z∗ F•i
(z∗ , z⊥ )[z∗ , ∞]ma
z
At xB ∼ 1 e−ixB z∗ may be important even if shock wave is narrow.
αs
Indeed, x∗ ∼ kαs
2 x∗ ∼ k2 ⇒ shock-wave approximation is OK,
⊥
⊥
αs
αs
but xB σ∗ ∼ xB k2 ∼ k2 ≥ 1 ⇒ we need to “look inside” the shock wave.
⊥
I. Balitsky (JLAB & ODU)
⊥
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
18 / 34
Shock-wave calculation
Reminder:
F̃ia (z⊥ , xB )
2
≡
s
Z
m
dz∗ e−ixB z∗ F•i
(z∗ , z⊥ )[z∗ , ∞]ma
z
At xB ∼ 1 e−ixB z∗ may be important even if shock wave is narrow.
αs
Indeed, x∗ ∼ kαs
2 x∗ ∼ k2 ⇒ shock-wave approximation is OK,
⊥
⊥
αs
αs
but xB σ∗ ∼ xB k2 ∼ k2 ≥ 1 ⇒ we need to “look inside” the shock wave.
⊥
⊥
Technically, we consider small but finite shock wave:
take the external field with the support in the interval [−σ∗ , σ∗ ] (where
σ∗ ∼ kαs
2 ), calculate diagrams with points in and out of the shock wave, and
⊥
check that the σ∗ -dependence cancels in the sum of “in” and “out”
contributions.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
18 / 34
One-loop corrections in the shock-wave background
(a)
(b)
Figure : Typical diagrams for production (a) and virtual (b) contributions to the
evolution kernel.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
19 / 34
Real corrections: square of “Lipatov vertex”
(a)
(b)
(c)
Figure : Lipatov vertex of gluon emission.
Definition
ab
Lµi
(k, y⊥ , xB ) = i lim k2 hT{Aaµ (k)Fib (y⊥ , xB )}i
k2 →0
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
20 / 34
Lipatov vertex in the shock-wave case
Result of calculation (in the background-Feynman gauge)
αp1µ p2µ
− 2 [Fi (xB , y⊥ ) − Ui (y⊥ )]ab
αs
k⊥
αxB s
αxB s
pi
pi
+ g(k⊥ |gµi
−U
U † + 2αp1µ
−U
U†
αxB s + p2⊥
αxB s + p2⊥
αxB s + p2⊥
αxB s + p2⊥
1
2αp1µ
2p2µ 2
∂ Upi
U† −
Ui |y⊥ )ab
+ 2ixB p2µ ∂i U − 2i∂µ⊥ Upi +
αs ⊥
αxB s + p2⊥
p2⊥
ab
Lµi
(k, y⊥ , xB ) = 2ge−i(k,y)⊥
Ui ≡ Fi (0) = i(∂i U)U † .
R
Schwinger’s notations (x⊥ |O(p̂⊥ , Xˆ⊥ )|y⊥ ) ≡ d2 pO(p⊥ , x⊥ )e−i(p,x−y)⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
21 / 34
Lipatov vertex in the light-cone case
Result of calculation (in the background-Feynman gauge)
2
2ge−i(k,y)⊥ ab
k⊥
, y⊥ )
F
(x
+
B
l
2
αs
αxB s + k⊥
h αx s k2
i
l
αxB sgµi kl
2αki kl
B
l
⊥
×
p
−
αp
δ
−
δ
k
+
+
p
2µ
1µ
i
1µ
i
µ
2
2 + αx s
2 + αx s
αs
k⊥
k⊥
k⊥
B
B
ab
Lµi
(k, y⊥ , xB )i =
NB:
ab
kµ Lµi
(k, y⊥ , xB ) = 0
for both shock-wave and light-cone Lipatov vertices.
It is convenient to write Lipatov vertex in the light-like gauge pµ2 Aµ = 0 by
µ
replacement αpµ1 → αpµ1 − kµ = −k⊥
−
2
k⊥
αs
ab
Lµi
(k, y⊥ , xB )light−like = 2ge−i(k,y)⊥
h k⊥ δ l
δµl ki + δil kµ⊥ − gµi kl
k2 gµi kl + 2kµ⊥ ki kl i ab
k2
µ i
×
−
− ⊥
Fl (xB + ⊥ , y⊥ ) + O(p2µ )
2
2
2
2
αs
k⊥
αxB s + k⊥
(αxB s + k⊥ )
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
22 / 34
Lipatov vertex at arbitrary momenta
“Interpolating formula” between the shock-wave and light-cone Lipatov
vertices
ab
Lµi
(k, y⊥ , xB )light−like
2 n αx sg − 2k⊥ k
k⊥
1
B µi
µ i
(kj U + Upj )
U†
2
αs
αxB s + k⊥
αxB s + p2⊥
2kµ⊥ o
gij
pi
− 2kµ⊥ U
U † − 2gµj U
U † + 2 gij |y⊥ )ab + O(p2µ )
2
2
αxB s + p⊥
αxB s + p⊥
k⊥
= g(k⊥ |F j xB +
This formula is actually correct (within our accuracy αfast αslow ) in the
whole range of xB and transverse momenta
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
23 / 34
Virtual corrections: similar calculation
(a)
(b)
(c)
(d)
Figure : Virtual gluon corrections.
Result of the calculation (in light-like and background-Feynman gauges)
Z σ−
←
dα
pj
hFin (y⊥ , xB )iFig. 4 = −ig2 f nkl
(y⊥ | − 2 Fk (xB )(i ∂ l +Ul )
p⊥
σ0 α
1
αxB s
× (2δjk δil − gij gkl )U
U † + Fi (xB ) 2
|y⊥ )kl
αxB s + p2⊥
p⊥ (αxB s + p2⊥ )
NB: with α < σ cutoff there is no UV divergence.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
24 / 34
Rapidity vs UV cutoff
n = p1
p
xB p2 −p
xB p2
Typical integral (n ≡ p1 , “gluon mass” m = IR cutoff)
Z 4
d p
1
xB p2 · n
I =
π 2 i (p · n − i)(p2 − m2 + i) (xB p2 − p)2 − m2 + i
Regularization # 1 (ours): n = p1 , |α| < σ
Z σ Z
Z
dβ
s
1
xB
I1 = −i 2
dα
d2 p⊥ 2
2π −σ
β − i
m + p2⊥ − αβs − i m2 + p2⊥ + α(xB − β)s − i
Z
Z
Z σ
1
1
αsxB 1
σsxB
π2
1 σ
dα
dα d2 p⊥ 2
ln 1 + 2
=
=
= ln2 2 +
2 +p2
2
m
π 0
m
2
m
6
m + p⊥ α +
⊥
0 πα
sxB
Double log of σ, no UV.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
25 / 34
Rapidity vs UV cutoff
n = p1
p
xB p2 −p
xB p2
Typical integral (n ≡ p1 , “gluon mass” m = IR cutoff)
Z 4
d p
1
xB p2 · n
I =
π 2 i (p · n − i)(p2 − m2 + i) (xB p2 − p)2 − m2 + i
Regularization # 2 (by slope of Wilson line): n = p1 + γp2 , γ 1
Z
Z
s
1
1
1
I2 = −i 2 dαdβ d2 p⊥
2π
β + γα − i m2 + p2⊥ − αβs − i m2 + p2⊥ + α(xB − β)s − i
⇒ I2 =
(p2 ·n)2 m2 n2
→
1 2 xB s2
π2
1
xB s
π2
ln 2 2 +
= ln2 2 +
2
mn
6
2
m γ
6
Double log of σ, no UV.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
25 / 34
Rapidity vs UV cutoff
n = p1
p
xB p2 −p
xB p2
Typical integral (n ≡ p1 , “gluon mass” m = IR cutoff)
Z 4
1
xB p2 · n
d p
I =
π 2 i (p · n − i)(p2 − m2 + i) (xB p2 − p)2 − m2 + i
Regularization # 3: n = p1 , β > b
Z
Z
s
1
1
xB
I3 = −i 2 dαdβ d2 p⊥
2π
β − i m2 + p2⊥ − αβs − i m2 + p2⊥ + α(xB − β)s − i
Z
Z
1 xB dβ
d2 p⊥
xB µ2
⇒ I3 =
= ln ln UV
2
2
π b β
b
m2
m + p⊥
UV × single log of the cutoff
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
25 / 34
Rapidity vs UV cutoff
n = p1
p
xB p2 −p
xB p2
Typical integral (n ≡ p1 , “gluon mass” m = IR cutoff)
Z 4
d p
1
xB p2 · n
I =
π 2 i (p · n − i)(p2 − m2 + i) (xB p2 − p)2 − m2 + i
Regularization # 1 ⇒ Regularization # 3:
change of variables β =
Z
σ
Z
dα
0
d 2 p⊥
1
1
=
m2 + p2⊥ α + m2 +p2⊥
I. Balitsky (JLAB & ODU)
sxB
Z
b
xB
xB (m2 + p2⊥ )
αsxB + m2 + p2⊥
dβ
β
Z
d2 p⊥
,
m2 + p2⊥
b =
xB (m2 + p2⊥ )
σsxB + m2 + p2⊥
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
25 / 34
Evoltuion equation for the gluon TMD operator
A. Tarasov and I.B.
ln σ
d
F̃ia (x⊥ , xB )Fja (y⊥ , xB )
d ln σ Z
= −αs d−2 k⊥ Tr{L̃i µ (k, x⊥ , xB )light−like Lµj (k, y⊥ , xB )light−like }
n
←
pm
1
− αs Tr F̃i (x⊥ , xB )(y⊥ | − 2 Fk (xB )(i ∂ l +Ul )(2δmk δjl − gjm gkl )U
U†
p⊥
σxB s + p2⊥
σxB s
|y⊥ )
+ Fj (xB ) 2
p⊥ (σxB s + p2⊥ )
1
pm
†
k l
kl
+ (x⊥ |Ũ
Ũ
(2δ
δ
−
g
g
)(i∂
−
Ũ
)
F̃
(x
)
im
k
k
l
B
i m
σxB s + p2⊥
p2⊥
o
σxB s
+ F̃i (xB ) 2
+ O(αs2 )
|x⊥ )Fj y⊥ , xB
2
p⊥ (σxB s + p⊥ )
This expression is UV and IR convergent.
It describes the rapidity evolution of gluon TMD operator in for any xB and
transverse momenta!
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
26 / 34
Evoltuion equation for the gluon TMD
ln σ
d
hp| F̃ia (x⊥ , xB )Fja (y⊥ , xB )
|pi
d ln σ
Z
k2 = −αs d−2 k⊥ hp|Tr{L̃i µ (k, x⊥ , xB )light−like θ 1 − xB − ⊥ Lµj (k, y⊥ , xB )light−like }|pi
αs
n
←
pm
1
− αs hp|Tr F̃i (x⊥ , xB )(y⊥ | − 2 Fk (xB )(i ∂ l +Ul )(2δmk δjl − gjm gkl )U
U†
p⊥
σxB s + p2⊥
σxB s
|y⊥ )
+ Fj (xB ) 2
p⊥ (σxB s + p2⊥ )
pm
1
†
k l
kl
Ũ
(2δ
δ
−
g
g
)(i∂
−
Ũ
)
F̃
(x
)
+ (x⊥ |Ũ
im
k
k
l
B
i
m
σxB s + p2⊥
p2⊥
o
σxB s
+ F̃i (xB ) 2
|x
)F
y
,
x
|pi + O(αs2 )
⊥
B
⊥
j
p⊥ (σxB s + p2⊥ )
2 k⊥
The factor θ 1 − xB − αs
reflects kinematical restriction that the fraction of
initial proton’s momentum carried by produced gluon should be smaller than
1 − xB
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
27 / 34
Light-cone limit
Z σ Z ∞ n
αs
dα
=
Nc
dβ θ(1 − xB − β)
π
σ0 α 0
h1
i
2xB
xB2
xB3
×
−
+
−
hp|F̃in (xB + β, x⊥ )
β
(xB + β)2
(xB + β)3
(xB + β)4
o
0
0
xB
hp|F̃in (xB , x⊥ )F in (xB , x⊥ )|piln σ
× F ni (xB + β, x⊥ )|piln σ −
β(xB + β)
hp|F̃in (xB , x⊥ )F in (xB , x⊥ )|piln σ
In the LLA the cutoff in σ ⇔ cutoff in transverse momenta
2
2
hp|F̃in (xB , x⊥ )F in (xB , x⊥ )|pik⊥ <µ
⇒ DGLAP equation ⇒ (z0 ≡
d
αs
αs D(xB , 0⊥ , η) =
Nc
dη
π
I. Balitsky (JLAB & ODU)
Z
1
xB
xB
xB +β )
αs
=
Nc
π
Z
0
∞
Z
dβ
µ2
βs
µ0 2
βs
dα same
α
DGLAP kernel
i
dz0 h
1 xB
1
0
0
+
−
2
+
z
(1
−
z
)
αs D 0 , 0⊥ , η
0
0
0
+
z
1−z
z
z
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
28 / 34
Low-x case: BK evolution of the WW distribution
Low-x regime: xB = 0 + characteristic transverse momenta p2⊥ ∼ (x − y)−2
⊥ s
(x−y)−2
⊥
⇒ in the whole range of evolution (1 σ s
2 k⊥
kinematical constraint θ 1 − αs
can be omitted
) we have
p2⊥
σs
1 ⇒ the
⇒ non-linear evolution equation
d a
Ũ (z1 )Uja (z2 )
dη i
Z
←
z2
g2 = − 3 Tr (−i∂iz1 + Ũiz1 ) d2 z3 (Ũz1 Ũz†3 − 1) 2 122 (Uz3 Uz†2 − 1) (i ∂jz2 +Ujz2 )
8π
z13 z23
where η ≡ ln σ and
z212
z213 z223
is the BK kernel
This eqn holds true also at small xB up to xB ∼
(x−y)−2
⊥
s
since in the whole
(x−y)−2
⊥
range of evolution 1 σ one can neglect σxB s
s
2
in the denominators (p⊥ + σxB s) ⇔ effectively xB = 0.
I. Balitsky (JLAB & ODU)
in comparison to p2⊥
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
29 / 34
Sudakov double logs
2
Sudakov limit: xB ≡ xB ∼ 1 and k⊥
∼ (x − y)−2
⊥ ∼ few GeV.
One can show that the non-linear terms are power suppressed ⇒
d
hp|F̃ia (xB , x⊥ )Fja (xB , y⊥ )|pi
d ln σ
Z −2 h
a
p2⊥
d p⊥ i(p,x−y)⊥
p2⊥
a
x
+
= 4αs Nc
hp|
F̃
x
+
e
,
x
F
, y⊥ |pi
B
B
⊥
i
j
2
σs
σs
p⊥
i
σxB s
a
−
hp|F̃i (xB , x⊥ )Fja (xB , y⊥ )|pi
2
σxB s + p⊥
(x−y)−2
⊥
s
and σxB s p2⊥ (x − y)−2
⊥
Z 2
d
αs Nc
d p⊥ D(xB , z⊥ , ln σ) = − 2 D(xB , z⊥ , ln σ)
⇒
1 − ei(p,z)⊥
d ln σ
π
p2⊥
Double-log region: 1 σ I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
30 / 34
Sudakov double logs
2
Sudakov limit: xB ≡ xB ∼ 1 and k⊥
∼ (x − y)−2
⊥ ∼ few GeV.
One can show that the non-linear terms are power suppressed ⇒
d
hp|F̃ia (xB , x⊥ )Fja (xB , y⊥ )|pi
d ln σ
Z −2 h
a
p2⊥
d p⊥ i(p,x−y)⊥
p2⊥
a
x
+
= 4αs Nc
hp|
F̃
x
+
e
,
x
F
, y⊥ |pi
B
B
⊥
i
j
2
σs
σs
p⊥
i
σxB s
a
−
hp|F̃i (xB , x⊥ )Fja (xB , y⊥ )|pi
2
σxB s + p⊥
(x−y)−2
⊥
s
and σxB s p2⊥ (x − y)−2
⊥
Z 2
d
αs Nc
d p⊥ D(xB , z⊥ , ln σ) = − 2 D(xB , z⊥ , ln σ)
⇒
1 − ei(p,z)⊥
d ln σ
π
p2⊥
Double-log region: 1 σ ⇒ Sudakov double logs
αs Nc 2 σs k2
D(xB , k⊥ , ln σ) ∼ exp −
ln 2 D(xB , k⊥ , ln ⊥ )
2π
s
k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
30 / 34
Gluon TMD with gauge links to −∞
αs
αs D(xB , z⊥ ) = −
2π(p · n)xB
Z
du e−ixB u(pn) hp|F̃ξa (z⊥ + un)[z⊥ , 0]−∞ F aξ (0)|pi
m
Fξa (z⊥ + un) ≡ [−∞n + z⊥ , un + z⊥ ]am nµ Fµξ
(un + z⊥ )
m
F̃ξa (z⊥ + un) ≡ nµ Fµξ
(un + z⊥ )[un + z⊥ , −∞n + z⊥ ]ma
Double functional integral:
p
X
y
p
x
=
y
p
p
One-loop diagrams are the same as before.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
31 / 34
Lipatov vertex for gauge links going to −∞
2 k⊥
αs
2 p⊥
2kµ⊥ gij + 2gµj pi i
k⊥
µ
+
2U
−
F
β
+
|y⊥ )ab
i
B
αs
αβB s + p2⊥
p2⊥
ab
Lµi
(k, y⊥ , βB )light−like
= g(k⊥ |UF j βB +
−∞
h αβB sgµi − 2k⊥ ki (p + k)
j
µ
2
αβB s + k⊥
αβB s + p2⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
32 / 34
Lipatov vertex for gauge links going to −∞
2 k⊥
αs
2 p⊥
2kµ⊥ gij + 2gµj pi i
k⊥
µ
+
2U
−
F
β
+
|y⊥ )ab
i
B
αs
αβB s + p2⊥
p2⊥
ab
Lµi
(k, y⊥ , βB )light−like
= g(k⊥ |UF j βB +
−∞
h αβB sgµi − 2k⊥ ki (p + k)
j
µ
2
αβB s + k⊥
αβB s + p2⊥
Compare to L. vertex for gauge links going to +∞
n αβB sgµi − 2k⊥ ki
2 (p + k)j
k⊥
µ
j
ab
F
β
+
U
U†
Lµi
(k, y⊥ , βB )light−like
= g(k⊥ |
B
+∞
2
αs
αβB s + k⊥
αβB s + p2⊥
k2 gij kµ⊥ + gµj pi † 2kµ⊥
k 2 o
− 2F j βB + ⊥ U
U + 2 Fi βB + ⊥ |y⊥ )ab
2
αs
αs
αβB s + p⊥
k⊥
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
32 / 34
Evoltuion equation for the gluon TMD
Replace
∞n → −∞n everywhere
and
xB → −xB in the virtual correction:
ln σ
d
hp| Fia (x⊥ , xB )Fja (y⊥ , xB )
|pi
d ln σ
Z
k2 = −αs d−2 k⊥ hp|Tr{Li µ (k, x⊥ , xB )light−like θ 1 − xB − ⊥ Lµj (k, y⊥ , xB )light−like }|pi
αs
n
1
pm
k l
kl
− αs hp|Tr Fi (x⊥ , xB )(y⊥ |U †
U(2δ
δ
−
g
g
)(i∂
+
U
)F
(x
)
jm
l
l
k
B
m
j
σxB s − p2⊥ + i
p2⊥
σxB s
+ Fj (xB ) 2
|y⊥ )
p⊥ (σxB s − p2⊥ + i)
←
pm
1
U
+ (x⊥ | 2 Fl (xB )(i ∂ k +Uk )(2δik δml − gim gkl )U †
p⊥
σxB s − p2⊥ − i
o
σxB s
+ Fi (xB ) 2
|x⊥ )Fj y⊥ , xB |pi + O(αs2 )
2
p⊥ (σxB s − p⊥ − i)
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
33 / 34
Conclusions and outlook
1
Conclusions
The evolution equation for gluon TMD at any xB and transverse
momenta.
Interpolates between linear DGLAP and Sudakov limits and the
non-linear low-x BK regime
2
Outlook
Conformal invariance (for N=4 SYM)?
Transition between collinear factorization and kT factorization.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
34 / 34
Conclusions and outlook
1
Conclusions
The evolution equation for gluon TMD at any xB and transverse
momenta.
Interpolates between linear DGLAP and Sudakov limits and the
non-linear low-x BK regime
2
Outlook
Conformal invariance (for N=4 SYM)?
Transition between collinear factorization and kT factorization.
Thank you for attention!
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate
QCD Evolution
x
2017, 23 May 2017
34 / 34