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High energy QCD at NLO: from light-cone wave function to JIMWLK evolution
Received: October
High energy QCD at NLO: from light-cone wave function to JIMWLK evolution
Michael Lublinsky 0 1 2
Yair Mulian 0 2
Open Access 0
c The Authors. 0
0 Beer-Sheva 84105 , Israel
1 Physics Department, University of Connecticut
2 Department of Physics, Ben-Gurion University of the Negev
Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to the third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background eld for the soft modes and is represented by a valence charge distribution that accounts for non-linear density e ects in the projectile. When scattered o a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evolution, the JIMWLK Hamiltonian. This way we provide a new direct derivation of the JIMWLK Hamiltonian at the Next-to-Leading Order.
NLO Computations; QCD Phenomenology
Contents
1 Introduction and summary
Basics of JIMWLK
Light cone QCD Hamiltonian
Field quantisation
Light cone wave-function of a fast hadron
Eigenstates of free Hamiltonian
Eikonal approximation for QCD Hamiltonian
Eikonal scattering
LO JIMWLK Hamiltonian
NLO JIMWLK Hamiltonian
The light cone wave function at NLO
Third order perturbation theory
Matrix elements
Technical aspects of the calculation
Computation of the NLO wave function
Quark anti-quark state
Two gluon state
One gluon state
The nal result
Computation of qq
Computation of
Computation of
Computation of
Computation of
Computation of
JJSSJJ and
JJJSJ
Virtual contributions
The NLO JIMWLK Hamiltonian assembled
Reduction of the (LO)2 contribution
C Integrals and Fourier transformations
D Properties of the NLO JIMWLK kernels
The phase of the wave function
G Supplementary for section 3
G.1 Supplement for computation of
G.2 Supplement for computation of
G.3 Supplement for computation of
G.4 Supplement for computation of
G.5 Supplement for computation of
G.6 Supplement for computation of
H Supplement for section 4
H.1 Supplement for computation of qq
H.2 Supplement for computation of
H.3 Supplement for computation of
H.4 Supplement for computation of JJSSJJ
Colour Glass Condensate (CGC).
functional equation of the form
Introduction and summary
dO =
HJIMWLK O :
Here the rapidity Y
JIMWLK was argued to be incomplete).
to the lowest order (LO) in
recent progress in
s corrections to the triple
Pomeron vertex.
framework.
rst part, the light
coordinate).
Soft gluons with momenta smaller than the cuto
do not participate in scattering.
at NLO schematically has the form
j i = (1
) j no soft gluonsi +
) j one soft gluoni
) j two soft gluonsi + gs2 6 j quark
antiquarki :
rather than being sourced by any topology.
in which
s corrections enhanced by the density
were resummed.
the cuto
space. In fact,
is a rapidity evolution operator
= e
Y HJIMWLK
Y HJIMWLK +
Y 2 HJ2IMWLK : : :
de ned as
HJIMWLK
d Y j Y =0
our direct calculation.
check on our calculation.
devoted to the calculation of
Basics of JIMWLK
tonian formalism.
Light cone QCD Hamiltonian
nates, four-vectors are x
= (x+; x ; x), where x+
0 + x3 and x
x3 stand
metric [86]. The light-cone gauge:
Aa+ = Aa0 + Aa3 = 0:
sketched in appendix A:
HLC QCD
dx d2x
1 a(x ; x) a(x ; x) +
gf abcAibAjc:
i 2 (1; 2) is a transverse component index;
denotes Dirac's 4-component quark spinor,1
with free Hamiltonian H0 given by:
(@iAja)2 + i +y @@i+@i
The interaction Hamiltonian Hint reads
gf abcAibAjc@iAja +
f abcf adeAibAjcAidAje
@+ (Aib@+Aic) @1+ (Ajd@+Ae)
j
@+ (Aib@+Aic) @1+
ig2 +ytatb iAia @1+
( +yta +) + 2g2 1
gauge group.
Field quantisation
gauge elds:
Aia(x) =
Z 1 dk+ Z
aia(k+; k)e ik x + aiay(k+; k)eik x :
p+) (2)(k
Transforming to coordinate space,
aia(k+; k) =
e ik z aia(k+; z) ;
aiay(k+; k) =
eik z aiay(k+; z);
1The avour index is suppressed in this section.
p+) (2)(x
b (k+; k)e ik x + d y(k+; k)eik x
1 =
2 = 2 1 1 2 :
b 1 (k+; k); b y(p+; p)o = n
2
d 1 (k+; k); d y(p+; p)o
2
= (2 )
b 1 (k+; x); b y(p+; y)o = n
2
d 1 (k+; x); d y(p+; y)o
2
= 2
H0 =
Z 1 dk+ Z
(2 )2 2k+
aiay(k+; k) aia(k+; k)
+ X hb y(k+; k) b (k+; k)
d (k+; k) d y(k+; k)
the commutation relation becomes:
For the quark elds:
+(x) =
= 12
The polarisation vectors are:
Z 1 dk+ Z
+ 12 =
2 =
The anti-commutation relations:
Light cone wave-function of a fast hadron
, which is implicitly related
are not
are also referred
as valence modes.
valence modes, and A for the soft gluon eld.
of partons in the LCWF get shifted up ( gure 1):
j iY0 = j i :
j iY = j i
get lifted above
Ignoring the modes below
momenta.
be to
Eigenstates of free Hamiltonian
aia(k+; k) j0i = 0;
bi (k+; k) j0i = 0;
di (k+; k) j0i = 0;
for any
One gluon state.
jgia(k)i
(2 )3=2 j0i
gia(k+; z)
(2 )1=2 j0i ;
These is a normalised state with the normalisation
Two gluon state.
Dgjb(p) jgia(k) E
= ab ij (2)(k
p) (...truncated)