Rapidity evolution of gluon TMD from low to
moderate x
I. Balitsky
JLAB & ODU
DIS 2017, 6 April 2017
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
1 / 43
Outline
Reminder: rapidity factorization and evolution of color dipoles
Definitions of small-x and “moderate-x” gluon TMDs
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 x
DIS 2017, 6 April 2017
2 / 43
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 x
DIS 2017, 6 April 2017
3 / 43
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 x
DIS 2017, 6 April 2017
3 / 43
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 x
DIS 2017, 6 April 2017
4 / 43
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 x
DIS 2017, 6 April 2017
5 / 43
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 x
DIS 2017, 6 April 2017
6 / 43
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 x
DIS 2017, 6 April 2017
7 / 43
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 x
DIS 2017, 6 April 2017
8 / 43
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 x
DIS 2017, 6 April 2017
9 / 43
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 x
DIS 2017, 6 April 2017
10 / 43
Guon TMD at low xB
At small x - Weizsacker-Williams unintegrated gluon distribution
X
trhp|U∂ i U † (z⊥ )|XihX|U∂i U † (0⊥ )}|pi
X
Rapidity factorization: each gluon has rapidity ≤ ln xB .
Rewrite (later n ≡ p1 )
Z
X
αs
αs D(xB , z⊥ ) = −
du
hp|F̃ξa (z⊥ + un)|XihX|F aξ (0)|pi
2π(p · n)xB
X
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
and define the “WW unintegrated gluon distribution”
Z
2
D(xB , k⊥ ) = d2 z⊥ e−i(k,z)⊥ D(xB , z⊥ )
xB s k⊥
Λ2QCD
NB: αs D(xB , z⊥ ) is renorm-invariant.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
11 / 43
Gluon TMD at moderate xB
D(xB , k⊥ , η) =
Z
d2 z⊥ e−i(k,z)⊥ D(xB , z⊥ , η),
αs D(xB , z⊥ , η)
Z
X
−xB−1 αs
du e−ixB u(pn)
hp|F̃ξa (z⊥ + un)|XihX|F aξ (0)|pi
=
2π(p · n)
X
There are more involved definitions with the above TMD multiplied by some
Wilson-line factors but we will discuss the “primordial” TMD.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
12 / 43
Gluon TMD at moderate xB
D(xB , k⊥ , η) =
Z
d2 z⊥ e−i(k,z)⊥ D(xB , z⊥ , η),
αs D(xB , z⊥ , η)
Z
X
−xB−1 αs
du e−ixB u(pn)
hp|F̃ξa (z⊥ + un)|XihX|F aξ (0)|pi
=
2π(p · n)
X
There are more involved definitions with the above TMD multiplied by some
Wilson-line factors but we will discuss the “primordial” TMD.
Now xB is introduced explicitly in the definition of gluon TMD.
However, because light-like Wilson lines exhibit rapidity divergencies, we
need a separate cutoff η (not necessarily equal to ln xB ) for the rapidity of the
gluons emitted by Wilson lines.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
12 / 43
Gluon TMD at moderate xB
D(xB , k⊥ , η) =
Z
d2 z⊥ e−i(k,z)⊥ D(xB , z⊥ , η),
αs D(xB , z⊥ , η)
Z
X
−xB−1 αs
du e−ixB u(pn)
hp|F̃ξa (z⊥ + un)|XihX|F aξ (0)|pi
=
2π(p · n)
X
There are more involved definitions with the above TMD multiplied by some
Wilson-line factors but we will discuss the “primordial” TMD.
Now xB is introduced explicitly in the definition of gluon TMD.
However, because light-like Wilson lines exhibit rapidity divergencies, we
need a separate cutoff η (not necessarily equal to ln xB ) for the rapidity of the
gluons emitted by Wilson lines.
The above TMD can have double-logarithmic contributions of the type
(αs η ln xB )n while the WW distribution has only single-log terms (αs ln η)n
described by the BK evolution.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
12 / 43
Some definitions
Sudakov variables:
k = αp1 + βp2 + k⊥
Dimensionless light-cone coordinates
r
s
x∗ ≡ xµ pµ2 =
x+ ,
2
Gluon operators
x• ≡ xµ pµ1 =
(xB ≡ xB for DIS and −xB ≡
1
z
r
s
x−
2
for annihilation)
Z
d2 z⊥ e−i(k,z)⊥ Fia (z⊥ , xB ),
Z
2
m
Fia (z⊥ , xB ) ≡
dz∗ eixB z∗ [∞, z∗ ]am
z F•i (z∗ , z⊥ )
s
Fia (k⊥ , xB ) =
and similarly
Z
d2 z⊥ ei(k,z)⊥ F̃ia (z⊥ , xB ),
Z
2
m
F̃ia (z⊥ , xB ) ≡
dz∗ e−ixB z∗ F•i
(z∗ , z⊥ )[z∗ , ∞]ma
z
s
F̃ia (k⊥ , xB ) =
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
13 / 43
Some definitions
Sudakov variables:
k = αp1 + βp2 + k⊥
Dimensionless light-cone coordinates
r
s
x∗ ≡ xµ pµ2 =
x+ ,
2
Gluon operators
x• ≡ xµ pµ1 =
(xB ≡ xB for DIS and −xB ≡
1
z
r
s
x−
2
for annihilation)
Z
d2 z⊥ e−i(k,z)⊥ Fia (z⊥ , xB ),
Z
2
m
Fia (z⊥ , xB ) ≡
dz∗ eixB z∗ [∞, z∗ ]am
z F•i (z∗ , z⊥ )
s
Fia (k⊥ , xB ) =
and similarly
Z
d2 z⊥ ei(k,z)⊥ F̃ia (z⊥ , xB ),
Z
2
m
F̃ia (z⊥ , xB ) ≡
dz∗ e−ixB z∗ F•i
(z∗ , z⊥ )[z∗ , ∞]ma
z
s
F̃ia (k⊥ , xB ) =
At first we study gluon TMDs with Wilson lines stretching to +∞ (like in SIDIS).
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
13 / 43
Double fun. interval for cross sections
X
0
hp|F̃ia (k⊥
, xB0 )F ai (k⊥ , xB )|pi ≡
= −2πδ(xB −
X
xB0 )(2π)2 δ (2) (k⊥
0
hp|F̃ia (k⊥
, xB0 )|XihX|F ai (k⊥ , xB )|pi
0
− k⊥
)2πxB D(xB = xB , k⊥ , η)
Short-hand notation
hp|Õ1 ...Õm O1 ...On |pi ≡
X
X
hp|T̃{Õ1 ...Õm }|XihX|T{O1 ...On }|pi
This matrix element can be represented by a double functional integral
hÕ1 ...Õm O1 ...On i
Z
Z
=
DÃDψ̄˜Dψ̃ e−iSQCD (Ã,ψ̃) DADψ̄Dψ eiSQCD (A,ψ) Õ1 ...Õm O1 ...On
The boundary condition Ã(~x, t = ∞) = A(~x, t = ∞) (and similarly for quark
fields) reflects the sum over all intermediate states X.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
14 / 43
Gauge invariance
Due to the boundary condition Ã(~x, t = ∞) = A(~x, t = ∞) the matrix element
hF̃ia (z0⊥ , xB0 )[z0⊥ + ∞p1 , z⊥ + ∞p1 ]F ai (z⊥ , xB )i
Z
Z
−iSQCD (Ã,ψ̃)
˜
DADψ̄Dψ eiSQCD (A,ψ)
=
DÃDψ̄ Dψ̃ e
F̃ia (z0⊥ , xB0 )[z0⊥ + ∞p1 , z⊥ + ∞p1 ]F ai (z⊥ , xB )
is gauge invariant
However, the gauge link [z0⊥ + ∞p1 , z⊥ + ∞p1 ] does not contribute at least at
the one-loop level ( γcusp and self-energy diagrams vanish)
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
15 / 43
Rapidity evolution: one loop
We study evolution of F̃iaη (x⊥ , xB )Fjaη (y⊥ , xB ) with respect to rapidity cutoff η
Z
d4 k
Aηµ (x) =
θ(eη − |αk |)e−ik·x Aµ (k)
(2π)4
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 x
DIS 2017, 6 April 2017
16 / 43
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 x
DIS 2017, 6 April 2017
17 / 43
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 x
DIS 2017, 6 April 2017
18 / 43
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 x
DIS 2017, 6 April 2017
18 / 43
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 x
DIS 2017, 6 April 2017
19 / 43
One-loop corrections in the shock-wave background
(a)
(b)
Figure : Typical diagrams for one-loop evolution kernel. The shaded area
denotes shock wave of background fast fields.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
20 / 43
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 x
DIS 2017, 6 April 2017
21 / 43
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 x
DIS 2017, 6 April 2017
21 / 43
Point(s) inside the shock wave: linear terms
x?
x?
x⇤
x⇤
?
⇤
⇤
⇤
∆⊥ is small ⇒ expansion of Peig
⇒ linear evolution.
I. Balitsky (JLAB & ODU)
R
dxµ Am u
⇤
around y⊥ ⇒ same operator F(y⊥ , xB )
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
22 / 43
Point(s) outside the shock wave: non-linear terms
x?
x?
x⇤
x⇤
z?
?
⇤
⇤
⇤
R
⇤
m
∆⊥ is small ⇒ expansion of Peig dxµ A u around z⊥
⇒ Wilson line Uz = [∞∗ p1 + z⊥ , −∞∗ p1 + z⊥ ] in addition to Uy ⇒ non-linear
terms in the evolution equation
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
23 / 43
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 x
DIS 2017, 6 April 2017
24 / 43
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 x
DIS 2017, 6 April 2017
25 / 43
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 x
DIS 2017, 6 April 2017
26 / 43
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 x
DIS 2017, 6 April 2017
27 / 43
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 x
DIS 2017, 6 April 2017
28 / 43
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. 5 = −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 x
DIS 2017, 6 April 2017
29 / 43
Rapidity vs UV cutoff
Regularizing the IR divergence with a small gluon mass m2 we obtain
Z σ Z
αxB s
dα
π 2 σxB s + m2
d2 p⊥ 2
ln
'
2
2
m2
(p⊥ + m2 )(αxB s + p⊥ + m2 )
0 α
(1)
Simultaneous regularization of UV and rapidity divergence is a consequence
of our specific choice of cutoff in rapidity.
For a different rapidity cutoff we may have the UV divergence in the remaining
integrals which has to be regulated with suitable UV cutoff.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
30 / 43
Rapidity vs UV cutoff
Regularizing the IR divergence with a small gluon mass m2 we obtain
Z σ Z
αxB s
dα
π 2 σxB s + m2
d2 p⊥ 2
ln
'
2
2
m2
(p⊥ + m2 )(αxB s + p⊥ + m2 )
0 α
(1)
Simultaneous regularization of UV and rapidity divergence is a consequence
of our specific choice of cutoff in rapidity.
For a different rapidity cutoff we may have the UV divergence in the remaining
integrals which has to be regulated with suitable UV cutoff.
We calculated
Z
d−αd−βd−β 0 d−2 p⊥
(β − i)(β 0 + xB − i)(αβs − p2⊥ − m2 + i)(αβ 0 s − p2⊥ − m2 + i)
by taking residues in the integrals over Sudakov variables β and β 0 and
cutting the obtained integral over α from above by the cutofl by α < σ
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
30 / 43
Rapidity vs UV cutoff
Instead, let us take the residue over α:
Z −2
Z
0
0
d p⊥
−βd
−β 0 θ(β)θ(−β ) − θ(−β)θ(β )
ixB
d
(β 0 + xB − i)(β − i)(β 0 − β)
m2 + p2⊥
Z −2
Z −2
Z
Z ∞
d p⊥
ixB θ(β)
d p⊥
d−βd−β 0
d−β
=
=
x
B
2
2
0
0
2
2
m + p⊥ β + xB − i (β − i)(β − β + i)
m + p⊥ 0 β(β + xB )
which is integral (1) with change of variable β =
I. Balitsky (JLAB & ODU)
p2⊥
αs .
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
31 / 43
Rapidity vs UV cutoff
Instead, let us take the residue over α:
Z −2
Z
0
0
d p⊥
−βd
−β 0 θ(β)θ(−β ) − θ(−β)θ(β )
ixB
d
(β 0 + xB − i)(β − i)(β 0 − β)
m2 + p2⊥
Z −2
Z −2
Z
Z ∞
d p⊥
ixB θ(β)
d p⊥
d−βd−β 0
d−β
=
=
x
B
2
2
0
0
2
2
m + p⊥ β + xB − i (β − i)(β − β + i)
m + p⊥ 0 β(β + xB )
which is integral (1) with change of variable β =
p2⊥
αs .
A conventional way of rewriting this integral in the framework of collinear
factorization approach is
Z ∞
Z 1
Z −2
Z −2
d p⊥
dβ
dz
d p⊥
xB
=
2
2
2
2
β(β
+
x
)
1
m + p⊥ 0
m + p⊥ 0 − z
B
B
where z = xBx+β
is a fraction of momentum (xB + β)p2 of “incoming gluon”
(described by Fi in our formalism) carried by the emitted “particle” with
fraction xB p2 .
If we cut the rapidity of the emitted gluon by cutoff in fraction of momentum z,
we would still have the UV divergent expression which must be regulated by a
suitable UV cutoff.
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
31 / 43
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 x
DIS 2017, 6 April 2017
32 / 43
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 x
DIS 2017, 6 April 2017
33 / 43
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
∞
Z
dβ
0
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 x
DIS 2017, 6 April 2017
34 / 43
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)
Rapidity evolution of gluon TMD from low to moderate x
in comparison to p2⊥
DIS 2017, 6 April 2017
35 / 43
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 x
DIS 2017, 6 April 2017
36 / 43
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 x
DIS 2017, 6 April 2017
36 / 43
Gluon TMD in particle production
Suppose we produce a scalar particle (e.g.
in a gluon-gluon fusion.
R Higgs)
a
For simplicity, assume the vertex is local: dzFµν
Faµν Φ(z). Again, we integrate
over rapidities Y > η:
Y> d
Y< d
Gluons with rapidities Y < η shrink to a pancake
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
37 / 43
Gluon TMD in particle production
⇒ Rapidity factorization for particle production
Y> d
⊗
Y< d
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
38 / 43
Matrix element between hadron states ⇒
P
X
=1
+
UU =1
y
x
y
x
⇒
p
X
y
x
I. Balitsky (JLAB & ODU)
p
=
Rapidity evolution of gluon TMD from low to moderate x
y
p
DIS 2017, 6 April 2017
p
39 / 43
Gluon TMD with gauge links to −∞
+
UU =1
y
x
p
p
y
x
X
⇒
x
αs D(xB , z⊥ ) = −
αs
2π(p · n)xB
Z
=
y
y
p
p
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
I. Balitsky (JLAB & ODU)
Rapidity evolution of gluon TMD from low to moderate x
DIS 2017, 6 April 2017
40 / 43
Lipatov vertex for gauge links 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 x
DIS 2017, 6 April 2017
41 / 43
Lipatov vertex for gauge links 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 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 x
DIS 2017, 6 April 2017
41 / 43
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 x
DIS 2017, 6 April 2017
42 / 43
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 x
DIS 2017, 6 April 2017
43 / 43
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 x
DIS 2017, 6 April 2017
43 / 43