#### 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) (k+
gia(k+; z) gjb(p+; z0)E
gia(k) gjb(p)
aiay(k) ajby(p)
aiay(k+; z) ajby(p+; z0)
b y(k) d y(p)
b y(k+; z) d y(p+; z0)
with the energy Egg(k; p)
Quark anti-quark state.
q 2 (p+; z0) q 1 (k+; z)
elds with the bar contain modes with k+ > e Y
this decomposition into Hint is done in appendix B.
+(x) =
+(x) where the underlined elds contain
(soft modes) while
(valence modes). Substitution of
+). The action of @1+ is
employ the eikonal approximation. The result is
The rst eight contributions de ned by (B.5)
(B.12), account for soft-soft and soft-valence
a background for the soft elds, and is de ned by:
( p) + qaq( p);
if abc Z 1 dk+ Z
a Z 1 dk+ Z d2k
p) b (k+; k) + d (k+; k) d y(k+; k
where a(x) is a Fourier transform of a(p):
a(x) =
( k) =
d2x e ik x a(x) :
(B.9) in terms of
further details and write the nal results,
Hg =
Hg qq =
1; 2= 12
dk+ Z
dk+ dp+
2jk+j3=2
haiay(k+; k) a( k) + aia(k+; k) a(k)i ;
(2 )2 (2 )2
2(p+
dp+ dq+ Z
hb y(p+; p) d y(q+; q) + h:c:i
p+ + q+
p+ + q+
dk+ dp+
(2 )2 (2 )2
igf abc
2p2k+p+q+
i aia(k)ajby(p)ajcy(q) (3)( k + p + q)
i aiay(k)ajby(p)ajc(q) (3)(k + p
q); + h:c:
dp+ dq+ Z
d2q ig2f abc
(2 )2 (2 )2 pp+q+
2(p+ + q+)q2) aiby(p+; p) aicy(q+; q)
( p
(2 )2 (2 )2
(p+ + q+)2
Hg gg =
Hgg inst =
Hqq inst =
functional Lie derivative:
on the state jSvi as a
S^ ga(x) j i = JL; adj (x) S^ E
a
Eikonal scattering
S^ bi y(x+; x) j0i = S
(x) di y(x+; x) j0i : (2.36)
SAab(x) and S
SAab(x) =
(x) =
dx+ T cA c(x+; x)
dx+ tcA c(x+; x)
tr SA(x)T a
SAy(x)
[SA(x) JR(x)]a =
tr T aSA(x)
SAy(x)
S (x)
S (x)
taSy(x) ;
Sy(x)ta :
In a similar manner we can also express qq:
JLa; F (x)
ST (x)
ST (x)
S(x)ta
taS(x)
hJLa(x); J Lb(y)i =
if abcJLc (x) (x
y); (2.45)
hJLc (x); SAab(z)i =
if cadSAdb(z) (x
z): (2.46)
LO JIMWLK Hamiltonian
where N
LO is determined by the normalisation condition,
= 1:
= N
The emission vertex is de ned by
vanishing matrix-element is that of Hg, (2.29):
Therefore, (2.47) becomes:
= N
hgia(k) j Hg j 0i =
2 3=2pk+k2
using the relations (2.44), we arrive at:
X Y h JLa(x) JLa(y) + JRa (x) JRa (y)
2JLa(x) SAab(z) J Rb(y) i :
LO . The coordinate
densities a
= N
dk+ Z
x; z 2 3=2X2
a(x) gia(k+; z) ;
where Xi
zi. From (2.48):
where Y i
zi. The scattered wave function reads:
LO = 1
2 Z e Y
dk+ Z
X Y
a(y) a(x) + O(g4);
2 Z e Y
dk+ Z
x; z 2 3=2X2
S^ a(x) gia(k+; z) :
function, while the
gluon from the valence current.
HJLIOMWLK =
X Y h JLa(x) JLa(y) + JRa (x) JRa (y)
YLO , which gives (see
2JLa(x) SAab(z) J Rb(y) :
transformation, S ! Sy together with JL !
and the charge conjugation symmetry S ! S .
JR, which in [84] was identi ed as signature,
NLO JIMWLK Hamiltonian
x; y;z;z0
x; y; z; z0
w ;x; y; z; z0
w ;x; y; z
HJNILMOWLK =
KJSJ(x; y; z) hJLa(x)JLa(y) + JRa (x)JRa (y)
2JLa(x)SAab(z)J Rb(y)
KJSSJ(x; y; z; z0) hf abcf def JLa(x)SAbe(z)SAcf (z0)JRd (y)
NcJLa(x)SAab(z)J Rb(y)
Kqq(x; y; z; z0) h2 JLa(x) tr[Sy(z) ta S(z0)tb] J Rb(y)
JLa(x) SAab(z) J Rb(y)
JLa(w) SAcd(z) SAbe(z0) JRd (x) J Re(y) +
(JLc (x) J Lb(y) JLa(w)
J Rc(x) J Rb(y) JRa (w))
KJJSJ(w ; x; y; z) f bde hJLd(x) JLe (y) SAba(z) JRa (w)
JLa(w) SAab(z) JRd (x) J Re(y) +
(JLd(x) JLe (y) J Lb(w)
JRd (x) J Re(y) J Rb(w)) :
z; X0
KJJSJ(w; x; y; z) =
KJSSJ(x; y; z; z0) =
KJJSSJ(w; x; y; z; z0) =
(Y 0)j Xi
(Y 0)2X2
Y i(X0)j
(X0)2Y 2
(W 0)j Zi
W iZj
W 2Z2
z0; Y
X2W 2
W i(W 0)j
W 2(W 0)2
Y 2W 2
z0; W
(W 0)2
X2(Y 0)2 + (X0)2Y 2
Y )2Z2
Z4(X2(Y 0)2
(X0)2Y 2)
X2(Y 0)2
(X0)2Y 2
Y 2(X0)2
X2(Y 0)2
Y 2(X0)2
X2(Y 0)2
(X0)2Y 2
2I(x; z; z0)
2I(y; z; z0)
+ Ke (x; y; z; z0);
z ; W 0
I(x; z; z0)
(X0)2
(X0)2
X2 + (X0)2
2(X0)2
X2 + (X0)2
(X0)2
(X0)2
(X0)2
(X0)2
Ke (x; y; z; z0)
KJJSSJ(x; x; y; z; z0)
KJJSSJ(y; x; y; z; z0)
KJJSSJ(x; y; x; z; z0) + KJJSSJ(y; y; x; z; z0) :
z0 ; Z
; (2.58)
Explicitly:
Ke (x; y; z; z0) =
(X0)2Z2(Y 0)2 +
(X0)2Z2Y 2
(X0)2X2(Y 0)2
X2(X0)2Y 2
Z2(Y 0)2
(X0)2
X2Z2Y 2
+ (x $ y) : (2.62)
The following useful equality holds:
Ke (x; y; z; z0) =
Ke (x; y; z; z0):
The kernel KJSJ reads:2
KJSJ(x; y; z) =
ln 2) +
-function:
KJSJ(x; y; z)
X Y
Kqq(x; y; z; z0) =
The quark sector:
An alternative representation for KJSJ:
ln 2) +
Ke (x; y; z; z0):
If (x; z; z0)
Z2(X2
(X0)2Y 2 + (Y 0)2X2
Y )2Z2
Z4 (X2(Y 0)2
(X0)2Y 2)
X2(Y 0)2
(X0)2Y 2
If (x; z; z0)
If (y; z; z0)
(X0)2
relations among them can be found in appendix D.
The light cone wave function at NLO
2The term 2b(
there as is clear from [92].
{ 16 {
! jii) and also
normalising the state when dividing by its norm:
= N
Ei Ej
Ei Ej Ek
2Ei2 Ej2
sation of the state:
NLO is determined from
normali= N
LO is the LO contribution (see section 2.7):
g
d2k gki a( k)
k+k2 jgia(k)i =
ig a(x)Xi
x;z 2 3=2X2
Order g2 states:
NLO
+ h0 jHintj ii hi jHintj ji hj jHintj 0i
2Ei Ej
h0 jHintj ii hi jHintj ji hj jHintj ki hk jHintj 0i
2Ei Ej Ek
3 jhj jHintj 0ij2 jhi jHintj 0ij2 :
8Ei2 Ej2
can be nevertheless uniquely
cording to its soft component content:
Our objective is to compute
NLO . It is convenient to split the wave function
ac
Order g3 states:
The following contributions vanish:
Matrix elements
expressions are computed based on (2.29)
(2.34). At NLO there are two matrix elements
Gluon splits into quark and anti-quark pair.
8 3=2pk+
Triple gluon interaction.
igf abc (3)(k
8 3=2pk+p+q+
kn + q
k+ + q+
k+ + p+
Dglc(q) gjb(p) jHgj gia(k)
E =
The denominator in the rst line of (E.3):
(Ei(0))2 j6=0
j iN = j0i
n(i)E n(i) jHintj 0
+ n(i)E n(i) jHintj n(j)
n(i)E n(i) jHintj n(j)
n(j) jHintj n(k)
E(0)Ej(0)E(0)
i k
(Ei(0))2Ej(0)
E(0)E(0)
The phase of the wave function
NLO
d(w)
= 0:
NLO (3.84) is inserted in (F.1).
of the LCWF and the phase:
d(w)
d(w)
NLO
= h gg j
d(w) j gg i + h gg j
d(w) j g i + h0j
d(w)
i NLO j0i = 0
KJJSSJ(y; w; x; z; z0)] + h0j
d(w)
i NLO j0i = 0:
in section 4.7.
i NLO =
v ;x; y; z; z0
KJJSSJ(v; x; y; z; z0)f abc b(y) a(x) c(v):
x; y; z; z0
Supplementary for section 3
Supplement for computation of
d2k d2p
g3 a( k) tr[ta td] ki
= I ;
(k+)2
By using
y =
= 2
2peipej
(k+)2
pi(kj
pj ) + pj (ki
pi)(kj
pj ) + "il"jk(ki
pi)(kj
k+p+
"il"jk(kk
p+(k+
4 + 1)peipej + "il"jkpekpel :
)2(k+)2
2"il"jkpekpel
(k+)2
32 9=2 (1
)k4 ( (1
4 + 1)peipej + "il"jkpekpel k
kipj + kj pi
ki(kj
pj ) + kj (ki
pipj + "il"jkpkpl
k+(k+
pk)pl
"il"jk(kl
pl)pk
(p+)2
i j = i"ij 3 + ij I ;
tr htatdi =
"il"jk = ij kl
{ 58 {
2 Z e Y
g1 =
g1 =
Integration over pe is done with the aid of (C.31):
After expanding with (C.33) and taking
! 0 limit, the result becomes:
(2 )d (2 )d
k+k4 ( (1
2 Z e Y
(4 )d=2 2
jgia(k)i :
4 + d) ki
k+k2d
Finally, integrating over
according to (C.1):
64 7=2pk+k2
which we can write as in (3.50).
Supplement for computation of
d2k d2p
g2f abcf dbc
(k p)2
4 + 2)
64 7=2pk+k2
1 + 2 jgia(k)i :
4 3=2jk+j3=2
k+ + p+
k+ + p+
jgia(k)i :
After some algebra:
2 Z e Y
(2 )d (2 )d
g2 =
g3f abcf dbc a( k)ki
8(2 )3 3=2 (1
)k4 (k2 (1
By replacing the measure according to (3.28):
16 9=2k4 (k2 (1
g3Nc a( k)kipe2
g3Nc a( k)kipe2
MS
g3Nc a( k)ki
16 7=2k2pk+
jgia(k)i :
2 Z e Y
2(4 )d=2
g3Nc a( k)ki
)k2 d=2 1
After expanding with the aid of (C.33) taking the
! 0 limit and, the last result becomes:
+ 4 ln
2 g3Nc a( k)ki
32 7=2k2pk+
MS
+ 2 ln2
jgia(k)i :
d2k d2p
ig2f abc(p+ + k+) c( k + p) ji !
2(2 )3pk+p+(k+
4 3=2jp+j3=2
d2k d2p
ig3f abc(1 + ) c( k + p) a( p)pi
16 9=2(1
operators according
to (3.27), we notice that the contribution with one
operator vanishes after integraion
over p. The two
part reads:
d2k d2p
ig3f abc(1 + )pi f c( k + p); a
32 9=2(1
The relevant integral for the integration over
is (C.7), the result after the integration
appears in (3.56).
Supplement for computation of
1 Z e Y
d2k d2p d2q
ig2f abc(k+
2(2 )3p(k+
2q+) ij a( k) !
q+)q+(k+)2
gqj b(q)
qi) c(k
4 3=2jk+
q+j3=2
After integration over q:
g4 =
d2k d2p
ig3f abc(2p+
k+) c(k
p) a( k)(ki
64 9=2
(k p)2
p+))2pp+
After changing variables according to (3.34):
d2k d2p
ig3f abc c(k
16 9=2 ( (k
p) a( k)(ki
pi)(2
p)2 + (1
)p2) p2(1
1) 3=2
)pk+ gib( k+; p)E :
Now let us work out the case of
matrix elements, (2.49), (3.23) and (3.22),
g5 as de ned in (3.61). By inserting the relevant
d2k d2p d2q
igf abc
16 3=2pk+q+(k+
4 3=2jk+j3=2
k+ + q+
gqj b(q)
qk) c(k
4 3=2jk+
q+j3=2
By adding together
g5 , we arrive at:
d2k d2p
ig3f abc c(k
p) a( k) 3=2
16 9=2k2p2 ( (k
p)2 + (1
)p2) (1
k2 + 2k p p
arrive at:
ig3f abc c(k
p) a( k) 3=2
16 9=2k2p2 ( (k
p)2 + (1
)p2) (1
2k p + 2 k2 (kj
pj ) +
2kj +
Equivalently, we can write the last result as:
g5 =
k p +
ig3f abc c(k
p) a( k) 3=2
16 9=2k2p2 ( (k
p)2 + (1
)p2) (1
We rewrite the denominator of
following de nitions:
4+5 by using Feynman parameter. We introduce the
g
x + x ;
x + x
and perform the shift k
! k + x p. Then we arrive at:
d2k d2p
32 9=2pk+(1
g3Nc a( p) 3=2
)p2 (k2 +
+ 2 k +
d2k d2p
k2 + 2k p p
g3Nc a( p) 3=2
32 9=2k2p2 ( (k
p)2 + (1
)p2) (1
operators.
Below, we will focus on the part which involve one
operator only, which can be isolated
via the prescription (3.27).
2 Z e Y
(2 )d (2 )d
g3Nc a( p) 3=2pj
) 2p2 (k2 +
(1 + )(1
x) x(1
2 1 + 1
x 1 +
g3Nc a( p) 3=2pj
{ 63 {
After taking the
x + x
x 1 +
x + x
2 1 +
x + x )2
x 1 +
(4 )d=2
(4 )d=2 2
x + x
MS
2 !#!
x + x
k+ + p+
g(kn
pn) c( k + p)
4 3=2jk+
p+j3=2
x + x
x + x
2) ln ( ) + ( + 1) ln (1
Supplement for computation of
we arrive at:
dp+ dq+
d2k d2p d2q
igf bcd
(k p)2
4 3=2jq+j3=2
k + p)
After simpli cations the last result becomes:
d2k d2p
ig3f bad b( p) a( k + p)
16 9=2 (k
p)2 ((1
)p2 + (k
{ 64 {
k2 + p
and two
via the prescription in (3.27).
operator:
d2k d2p
)k p +
32 9=2
g3f badf bac c( k)
k)2 +
p)2 +
We introduce again the variable
according to p
k we arrive at:
d2k d2p
)k (p +
p)2 +
32 9=2 (p2 +
jgla(k)i :
last result as:
k2 pl +
2) + 2 2(1
)2(1 + )
ddk ddp Z 1 k+
(2 )d (2 )d
2)(d + 2) + 2d(1
)(1 + )
)2(1 + )
(2 + d) 2(
k+ (p2 +
)k2)2 k2
2) + 4 (1
)(1 + )
(1 + )d
jgla(k)i :
(4 )d=2
{ 65 {
2)d + 2 (
After taking the
(1 + )d
jgla(k)i :
2) + 2 2(1
32 7=2 k2pk+
)2(1 + )
)(1 + )
jgla(k)i :
(1 + )
2(4 )d=2
)(1 + )
(1 + )
d2k d2p
)k p +
After integration over x:
g3Nc a( k)kl
32 7=2 (1
2) ln ( )
(1 + ) ln (1
(1 + )
2 g3Nc a( k)ki
32 7=2k2pk+
+ 2 ln2
jgia(k)i :
This part can be deduced directly from (3.68):
p and dividing
by 2, we can write
equivalently as:
ig3f bad
( k + p)
32 9=2 (k
p)2 ((1
)p2 + (k
p)2 +
d2k d2p
ig3f bad
64 9=2 (1
( k + p)
)p2(k
MS
2 !#!
jgla(k)i :
After integration over
( k + p)
32 9=2p2(k
k pkl +
Notice that under the change p ! k
p the second summand inside the rectangled
obtained after this change in (3.71).
Supplement for computation of
1 Z e Y
dk+ dp+ dq+ dr+ Z
d2k d2p d2q d2r
4 3=2jp+j3=2
grj b(r)
gqk c(q)
4 3=2jq+j3=2
4 3=2jr+j3=2
g7 =
d2k d2p
g3p2ki b(p)
64 9=2
jgia(k)i :
By using the algebra of
g7 =
d2k d2p
2g3ki b(p)
64 9=2
jgia(k)i :
{ 67 {
g3Nc a( k)ki
(4 )d=2
g3Nc a( k)ki "
16 7=2pk+ k2
Expanding with the aid of (C.33) taking the
! 0 limit and, the last result becomes:
After integration over
MS
+ ln2
jgia(k)i :
jgia(k)i :
2 !#
MS
jgia(k)i :
2 Y
k+ according to (C.9):
2 g3Nc a( k)ki
32 7=2pk+k2
we obtain:
Supplement for section 4
Supplement for computation of
g7 =
d2k d2p
2 Z e Y
ddk ddp
Integration over p by using (C.31) yields:
We can now isolate the contribution to one
via the prescription in (3.27).
operator reads:
k2 + p2
g3ki b(p)
8 9=2pk+ k2
jgia(k)i ;
g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y)
512 10 ( (1
d2k d2pe d2u d2ve
)k2 + pe2) ( (1
pej vei kiuj
k2u2
simplify the denominators using:
)Z2 + (X0
Z)2 = (1
)(X0)2 + X2:
x;y;z;z0 0
g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y)
32 6Z4 ((1
)(X0)2 + X2) ((1
)(Y 0)2 + Y 2)
(X0)2 + (Y 0)2
x;y;z;z0 0
g4Nf JLa(x) T r Sy(z)taS(z0)tb J Rb(y)
64 6Z4 ((1
)(X0)2 + X2) ((1
)(Y 0)2 + Y 2)
)(X0)2 + X2 + (1
)(Y 0)2 + Y 2
+2 (1
It is possible to integrate over
Supplement for computation of
JSSJ
Then, we arrive at:
Rewriting the scalar products:
X0 Z =
X0 Y 0 =
we arrive at:
is shown in (4.15).
to (4.19), we get:
JNSLSOJ =
(Y 0)2Y 2((Y 0)2
k+ f abcf def JLa(x) SAbe(z) SAcf (z0) JRd (y)
(x; y; z; z0)
(X0)2X2((X0)2
X2)2((X0)2Y 2
(y; x; z; z0)
Y 2)2((X0)2Y 2
(Y 0)2
(X0)2
x; y; z; z0
(x; z; z0) T (y; z; z0)
(X0)2(Y 0)2
k+ f abcf def JLa(x) SAbe(z) SAcf (z0) JRd (y)
(x; z0; z) T (y; z0; z) !
X2Y 2
The following de nitions were used:
jl(x; z; z0) =
(x; y; z; z0)
Xl(X0)j
jl(y; z; z0) X2 + lj(y; z0; z) (X0)2 (X2 (X0)2)
jl(y; z; z0) X2(X0)2
jl(x; z; z0) X2 + lj(x; z0; z) (X0)2 (X2 (X0)2)
jl(x; z; z0) X2(X0)2 :
from which we deduce:
X2(X0)2(X2
(x; y; z; z0)
X2(X0)2
(X0)2Z2
(X0)2Y 2 + X2(Y 0)2
X2(X0)2
2Z2(X2
(X0)2)
X2 Z2 X0 Y 0
X2 X0 Z (X0)2 X Z
+ X X0
(X0)2 Y 0 Z X Z
+ X0 Y 0 X Z
X2Z2
X2 X0 Y 0 Y Z (X0)2 X Y Y 0 Z
X Z =
X X0 = 2
1 (X0)2 X2 Z2 ;
12 X2 + (X0)2 Z2 :
Y Z =
1 (Y 0)2 Y 2 Z2 ;
(x; y; z; z0)
X2(X0)2(X2
(X0)2)2
= 18
X2(Y 0)2 (X0)2Y 2 2
X2(Y 0)2 + (X0)2Y 2 4(X
Y )2Z2
Z4(X2(Y 0)2
(X0)2Y 2)
(X0)2Y 2
(X2 (X0)2)Z2
X2(Y 0)2 + Y 2(X0)2 +
X2 + (X0)2 + 2(X0)2 +
Z2
Z2
X2 Z2 (X0)2
X2(Y 0)2
Y 2(X0)2
+( X(Y0)02)Z4X2Y2 2 + Z2X2(Y 0)2
Y 4(X0)2
(X0)2(X
X2Z2
Z2(Y 0)2
(X0)2Y 2
(X0)2
X2(X
X2(Y 0)2
(x; y; z; z0)
(X0)2X2((X0)2
X2)2((X0)2Y 2
(y; x; z; z0)
Y 2)2((X0)2Y 2
(Y 0)2
(X0)2
Supplement for computation of
JJSJ
introduced in (4.28). Let us start with
JJSJ
g4f bde
JLa(w) SAab(z) JRd (x) J Re(y)
d2k d2p e ik X+ip (X Y )
p2(k
k2(k
k2p2
Ke (x; y; z; z0) =
16 4 ((X0)2Y 2
X2(Y 0)2)
Z2(Y 0)2
X2(X
Z2(Y 0)2
(Y 0)2(X
(X0)2Y 2
(X0)2Z2
(X0)2
(X0)2
X2(Y 0)2
X2(Y 0)2
+ (x $ y) ;
we arrive at the conclusion that:
(Y 0)4X2
(X0)2Z2Y 2
Y 4(X0)2
Z2X2(Y 0)2
(X0)2(X
(Y 0)2(X
X2Z2
(X0)2
following way:
KJSSJ(x; y; z; z0):
JNJLSOJ and
After the change p ! k
p, we nd:
JNJLSOJ =
g4f bde
We can now use integral (C.27), and write:
JNJLSOJ =
ig4f bde
hJLd(x) JLe (y) SAba(z) JRa (w)
JLa(w) SAab(z) JRd (x) J Re(y) :
JLa(w) SAab(z) JRd (x) J Re(y)
p2(k
k2(k
d2k d2p e ik Y ip (X Y )
JJSJ
g4f bde
d2k d2p e ik X+ip (X Y )
p (p
k2p2(k
k) ki
p2(k
JLa(w) SAab(z) JRd (x) J Re(y)
JJSJ =
ig4f bde
2X0 Z Y
(X0)2Z2Y 2W 2
w; x; y; z; z0
JLa(w) SAab(z) JRd (x) J Re(y) :
Supplement for computation of
JJSSJJ
From (4.39) we deduce that:
2X0 Y 0 Y
(X0)2(Y 0)2Y 2W 2 +
X0 Y 0 Z
(X0)2(Y 0)2Z2W 2
(X $ Y ) hJLd(x) JLe (y) SAba(z) JRa (w)
JJSSJJ =
w; v; x; y; z; z0
128 6 (1
X Y W 0 V 0
X Y W 0 V 0
X Y W 0 V 0
X Y W 0 V 0
Integrating over
and reordering the J operators:
JJSSJJ =
4X Y W 0 V 0
x; y; z; z0 X2Y 2(X0)2(Y 0)2
f bar f cdeJLr (x) SAbc(z0) SAad(z)J Re(y) :
Open Access.
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