Resummation in High Energy
Scattering
-- BFKL vs Sudakov
Feng Yuan
Lawrence Berkeley National Laboratory
Refs: Mueller, Xiao, Yuan, PRL110, 082301 (2013); Phys.Rev. D88
(2013) 114010.
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1
x0>>…>>xn
High energy scattering
kT
kT
kT
Un-integrated gluon distribution
BFKL:
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Non-linear term at high
density

Balitsky-Fadin-Lipatov-Kuraev, 1977-78

Balitsky-Kovchegov: Non-linear term, 98
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Hard processes at small-x
x0>>…>>xn
Q, P⊥

kT
kT
kT
Manifest dependence on un-integrated
gluon distributions
 Dominguiz-Marquet-Xiao-Yuan,
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2010
4
Additional dynamics comes in
Soft gluon
x0>>…>>xn
Q, P⊥

kT
kT
kT
BFKL vs Sudakov resummations (LL)
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Sudakov resummation at small-x

Take massive scalar particle production
p+A->H+X as an example to demonstrate
the double logarithms, and resummation
p
H,MH
WW-gluon distribution
A
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Explicit one-loop calculations
Collinear divergence  DGLAP evolution
 Small-x divergence  BK-type evolution

Dominguiz-Mueller-Munier-Xiao, 2011
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Soft vs Collinear gluons

Radiated gluon momentum
Soft gluon, α~β<<1
 Collinear gluon, α~1, β<<1
 Small-x collinear gluon, 1-β<<1, α0

 Rapidity
divergence
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Examples

Contributes to
1. Examples
of one-loop
from t he real gluon radiat ion (L) an
FIG.
Collinear
gluon
fromdiagrams
the proton
(R) for t he scalar part icle product ion in pA collisions in t he sat urat ion form
int
Collinear
fromt arget
nucleus
eract ions witgluon
h t he nucleus
t aken int o account . Here t he vert ical gl
number of t he WW small-x gluons which are summed int o Wilson lines.
 Soft
gluon to Sudakov double logs
where k⊥ and k2⊥ represent the transverse momenta for the final st
2
the radiated gluon,
respectively,
k
=
k
+
k
=
k
+
k
,
g
⊥
g1
⊥
g2
⊥
⊥
2
⊥
f =
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⊗

⊗
(a)
(b)
⊗
(d)
⊗
(e)
(
Only contributes to small-x collinear gluon
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Final result

Double logs at one-loop order

Include both BFKL (BK) and Sudakov
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Photon-Jet correlation

Leading order
(a)
(b)
Dipole gluon distribution
FIG. 7. T wo LO amplit udes.
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One gluon radiation (real)
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BK-evolution
(a)
(a)
(b)
(b)
(b)
These two do not
Contribute to soft
Gluon radiation
iat ion cont ribut ion t o t he BK -evolut ion, but not t o t he leading double
(c) of t he phase space
(a) (d)
(b)
. All t he int egral
result ing int o t he leading
double
he previous sect ion holds in t he above funct ional forms as well.
(d)
focus on two different regions of t he radiat ed gluon: (1) soft gluon
ne loop
graphs
whichion
contcont
ribut
e
t o nucleus,
tion
he leading
power
amplit
ude
FI
G.real
9. emission
Real
radiat
t o t he
BK -evolut
but
(2)
collinear
t o tgluon
he
moment
um
of tribut
he
where
α g ion,
1 for
but
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ct
or,logarit
whilehms.
two
graphs which
have tlogarit
he radiat
ed gluon
at t aching
o t he red
cont
ribut
e
t oott her
he Sudakov
double
hms,
whereas
t hetregion
(2)
Soft gluon radiation
(a)
(b)
(c)
(d)
A2 from (a,b) contribute to CF/2 (jet)
 A2 from (c,d) contribute to CF
d u d v d u d v
contribute
to 1/2Nc
e x q
(x Interference
)2 1 + (1 − z) (1 − z)
e
(2π)

oop real emission graphs which cont ribut e t o t he leading power amplit ude for
, while t wo ot her graphs which have t he radiat ed gluon at t aching t o t he red
⊥
2
2
f
p f
p
2
⊥
2
⊥
2
6
⊥
2
⊥
− i q⊥ ·( v⊥ − v⊥ ) − i P⊥ ·( u ⊥ − u ⊥ )
S(2) (b⊥ , b⊥ ) + S(2) (v⊥ , v⊥ ) − 4/8/2015
S(2) (v⊥ , b⊥ ) − S(2) (b⊥ , v⊥ ) ,
(43)
15
ggqq
|A1|2CA, |A2|2CF/2, |A3|2CF/2
 2A1*(A2+A3)-Nc/2
 2A2*A3, 1/Nc suppressed

Soft gluon radiat ion in gg → q¯
q process in t he sat urat ion formalism. T he blobs in t he
diagrams represent t he mult iple gluon int eract ion wit h nucleus formulat ed in Ref. [1] in
t ion limit .
in we have t aken t he correlat ion limit . T he amplit ude squared will be,
|A 0 |2 =
d2x ⊥ y⊥ ei q⊥ ·(x ⊥ − y⊥ ) Γ β (k1⊥ ))Γ β (k1⊥ ))
1
(1 − z) 2 + z2 Tr U(x ⊥ )U †(y⊥ ) Tr ∂⊥ U(x ⊥ )∂⊥ U †(y⊥ )
2
− 2z(1 − z)Tr U(x ⊥ )∂⊥ U †(y⊥ ) Tr U(x ⊥ )∂⊥ U †(y⊥ )
.
×
nsist ent wit h what we have found in Ref. [1].
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(67)
16
qgqg
|A1|2CF, |A2|2CF/2, |A3|2CA/2
 2A3*(A1+A2)-Nc/2
 2A1*A2, large Nc suppressed

FIG. 11. Same as Fig. 10 for qg → qg process.
n we have t aken t he correlat ion limit . T he amplit ude squared will be,
d2x ⊥ d2y⊥ ei q⊥ ·(x ⊥ − y⊥ ) Γ β (k1⊥ ))Γ β (k1⊥ ))
|A 0|2 =
1
Tr U(x ⊥ )U †(y⊥ ) Tr ∂⊥ U(x ⊥ )U †(x ⊥ )∂⊥ U(y⊥ )U †(y⊥ )
2
+ z2 CF ∂⊥ U(x ⊥ )∂⊥ U †(y⊥ )
.
×
nsist ent wit h what we have found in Ref. [1].
t ial st at e radiat ion cont ribut ion can be writ
t en as
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µ
(76)
17
,3)
gggg
2C
2process.
2C /2
as Fig. 10,for|A
gg → gg
 FIG.
|A12.1|Same
|
C
/2
|A
|
A
2
A ,
3
A
2A1*(A2+A3)+2A2*A3-Nc
represent t he following Wilson lines,

= N cTr[∂⊥ U(x ⊥ )∂⊥ U †(y⊥ )]Tr[U(x ⊥ )U †(y⊥ )] ,
= N cTr[∂⊥ U(x ⊥ )U †(y⊥ )]Tr[U(x ⊥ )∂⊥ U †(y⊥ )] ,
= Tr[∂ ⊥ U(x ⊥ )U †(x ⊥ )∂⊥ U(y⊥ )U †(y⊥ )]Tr[U(x ⊥ )U †(y⊥ )]Tr[U(x ⊥ )U †(y⊥ )] .
(87)
expressions, we will obt ain t he same different ial cross sect ions as t hat in Ref. [1].
al and final st at e radiat ion cont ribut ions can be writ t en as
kg⊥ − kg2⊥ ) µ β
1
Γ
(k
)
Tr[T d U(x 2)T eU †(x 2)]
1
⊥
2
−
k
)
N
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g⊥
g2⊥
F
1
18
Sudakov leading double logs

Each incoming parton contributes to a half
of the associated color factor
 Initial
gluon radiation, aka, TMDs
Sudakov
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Phenomenological applications
Di-hadron azimuthal
Correlations at the
Electron-ion Collider
Zheng, Aschenauer,Lee,Xiao, Phys.Rev. D89 (2014) 074037
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Beyond leading logs

Additional nuclear effects, such as energy
loss will come in
 Liou-Mueller,
1402.1647
Interference between
Them, leads to energy
Loss calculated in
Liou-Mueller
0. Soft gluon radiat ion in gg → q¯
q process in t he sat urat ion formalism. T he blobs in t he
wo diagrams represent t he mult iple gluon int eract ion wit h nucleus formulat ed in Ref. [1] in
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Next-to-leading-logarithms (NLL)

Matrix form in the Sudakov resummation
 Sun,Yuan,Yuan,1405.1105
D0@Tevatron
 Kidonakis-Sterman,
NPB 1997
 Banfi-Dasgupta-Delenda,
PLB2008
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Dijet with large rapidity gap
P1⊥
P2⊥
Sudakov resummation will dominate
Small angle distribution
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CMS@LHC
Ducloue,Szymanowski,Wallon
1309.3229, only take into account
BFKL
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Conclusion



Sudakov double logs can be re-summed in
the small-x saturation formalism
Soft gluon and collinear gluon radiation is
well separated in phase space
Shall provide arguments to apply the
effective kt-factorization to describe dijet
correlation in pA collisions
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