First Order QCD Corrections to the Weak Gauge
Boson Pair Production in Hadronic Collisions
M. Y. Hussein
Department of Physics
College of science
University of Bahrain
E-mail: mhussein@sci.uob.bh
Abstract
The QCD radiative corrections to the weak gauge boson pair production in
hadron colliders pp  ZZ  X are presented. A complete next-to-leading
logarithm (NLL) calculation, which involves computing the contributions
from the 2  3 real gluon emission as well as the one loop correction to the
2  2 process. The calculation utilizes a combination of analytic and Monte
Carlo integration methods.
1. Introduction
The production of weak gauge boson pairs is an important topic to study at
future hadron colliders. The standard vector pair production can be used to
test the standard model (SM) as well as probe beyond it [1]. On the other
hand these processes are important for many reasons, such as, it can test the
trilinear coupling, which provide stringent test of the SM [2], the electroweak symmetry broken mechanism can be probed, the observation of
resonance production ZZ,W W  would be a signal for the standard model
Higgs boson [3] and finally, the standard vector boson pair production can
represent as a major background for new heave particles, such as
H  ,W , Z , q~ and g~ which can decay into weak boson pairs.
In order to test and probe the SM with hadron diboson production, it is
necessary to have a precise calculation of SM diboson production, which
means the cross section must be calculated to next-to-leading-order (NLO).
The NLO cross-section is, in general, less sensitive to the choice of the
arbitrary factorization and normalization scales.
The production of the ZZ has been calculated long times ago and the order s correction has only been estimated using soft gluon approximation [4]. In
this work we present a next-to-leading logarithm (NLL) calculation of
hadronic ZZ production. At the parton level this involves computing the
contribution for 2  3 real emission processes as well as the loop corrections
to the 2  2 process. The focus of the present calculation is on the order s correction to ZZ production. The calculated correction contain
divergences, represented in the dimensional regularization procedure by
poles O (1 /  ) and O(1 /  2 ) , where   (4  n) / 2 and n is the number of spacetime dimension. The O(1 /  2 ) terms are eliminated when real and virtual
corrections are combined and the remaining divergences will be ‘absorbed’
into the quark-momentum distribution functions.
2. Born Process
The two Feynman diagrams, which contribute to the Born amplitude for the
reaction q(k1 )  q (k 2 )  Z ( p1 )  Z ( p2 ) , are shown in Fig. 1.
k1
k2
p1
p2
k1
p2
k2
p1
Fig. 1. Born diagrams for the hadronic production of a pair of Z boson.
The matrix element for the process where the Dirac fermions annihilate to
produce a boson pair is given by:
i
u (k 2 ) 2 ( g v  g a  5 )( k 2  p 2 ) 1 ( g v  g a  5 )v(k1 )
(1)
t
i
2 
u (k 2 ) 1 ( g v  g a  5 )( k 2  p 1 ) 2 ( g v  g a  5 )v(k1 )
(2)
u
Where  1 and  2 are the polarization vector of the Z -boson and g v , g a are
1 
the weak neutral coupling. To calculate the differential cross section, we
only have to calculate
2
2
2

M Born   [ M 1  M 2  2 Re M 1 M 2 ]
(3)
Where the summation over all the polarization state of the Z and quarks and
all the kinematics invariants are defined by
s  (k1  k 2 ) , t  (k1  p1 ) , u  (k1  p2 ) , s  t  u  2mZ
2
(4)
The algebra was evaluated using the algebra manipulation program
REDUCE [5], The square amplitude becomes
g  g a  6 g v g a t u 4m Z s
1
4 1
M Born  v
[  
 mZ ( 2  2 )]
4
u t
tu
e
t
u
p p
Where       g    2 ,  5 2  1 and   e 2 / 4
mZ
4
4
2
2
2
2
(5)
The Born sub-process differential cross section, for the sub-process
qq  ZZ is given by
d  2 1

d 2 s 3
g v  g a  6 g v g a t u 4m Z s
1
4 1
[  
 m Z ( 2  2 )]
4
u t
tu
e
t
u
4
4
2
2
2
(6)
Where the differential cross section included the spin average, color average,
and identical particle factors.
As a check to the calculation, the differential cross-section for qq  ZZ is
exactly same as the differential cross-section for sub-process qq   , when
we substitute mZ  0 , g a  0 , and g v  eq .
The leading-logarithm (LL) cross-section is obtained by convoluting the
sub-process cross-section with the parton densities and summing over the
contributing partons:
Born
 ( pp  ZZ )    dˆ (qq  ZZ )[G p / q ( x1 , M 2 )Gq / p ( x2 , M 2 )  x1  x2 ]dx1dx2 (7)
q
3. O( s ) Virtual Gluon Correction to qq  ZZ
To calculate the first order virtual gluon correction to the sub-process
qq  ZZ , we have to evaluate all the diagrams of Fig. (2). We present all the
calculation in the Feynman gauge, which means that the gluon propagator
has the following form D ab  ig   ab .
Each of these diagrams contains a loop, the resulting integral over the gluon
momentum leads to ultraviolet and infrared divergences. These divergences
occur due to the behavior of the integrands at high and low virtual
momentum (high and low energy behavior of the loop integrals). The
divergences associated with high virtual momentum are called ultraviolet
divergences; the other divergences associated with the behavior of the
integrals at low momentum are called infrared divergences. In order to work
out of these diagrams, we may either introduce cutoffs or work in a space
with dimension more than four. By using the continuous dimension method,
all the poles arising from the calculation become poles of order O (1 /  ) and
O(1 /  2 ) in the expression of the cross section.
k1
p1
k1
p1
k2
p2
k2
p2
k1
p1
k1
p1
k2
p2
p1
k2
p2
p1
k1
k2
k1
p2
k2
p2
+ Crossed Z boson diagrams
Fig. (2) Virtual gluon correction diagrams of order  s for qq  ZZ .The solid
lines represent quarks, the dashed lines Z boson, and the curly ones represent
gluons.
It is clear from Fig. (2) that there are in fact three types of virtual diagrams
called self-energy, vertex and box diagrams. We shall calculate each of them
separately as follows:
a. Self-energy corrections
These consist of three diagrams; only the internal energy gives a
contribution, while the self –energy insertions on the external quark lines
vanish due to the cancellation of the UV and IR divergences [6]. The loop
integral for the self-energy has the following form:
I self  
  ( p  k ) 
i 4 s
d nk 2
g 
n 
(2 )
k ( p  k)2
(8)
Using Feynman parameterization of multiple denominators, with the use of
an n-dimensional Minkowski space integrals [7], and combining with the
lowest order amplitude, we get the following correction due to the selfenergy diagrams:
 self 
4 s
1 1
1
(4 )  [  
[ f (u, t )(ln( t )   )  (t  u)]]
3
2 2 2M 0
(9)
Where f (u, t )  8mZ 4 tu  2mZ 4 u 2  4mZ 4 t 2 u  4mZ 2 tu 2  2tu 3 and M 0 is the lowest
order matrix element for the sub-process. In the limit mZ  0 , the expression
for  self agrees with the result for  self in Ref. [8].
b) The vertex diagram
For the vertex correction, we have to write the loop integral, which has the
form:
I vertex
  ( p   k )  ( p  k ) 
i 4s
n

d k
g
(2 ) n 
k 2 ( p  k ) 2 ( p  k ) 2
(10)
It is more complicated than the self-energy case. Using Feynman
parameterization of multiple denominators, with a suitable change of
variables and taking into the account the crossed diagrams, combining with
the lowest order amplitude, we get the following correction due to the vertex
diagrams:
 vertex 
4 s
1
8
1
2
(4 )  [ (1 
) 1
[ f (u, t )(ln( mZ  t )   )  (t  u)]]
3

M0
M0
(11)
Also In the limit mZ  0 , the expression for  vertex agrees with the result for in
 vertex Ref. [8].
c) The box diagram
For the box diagram we have to work out a four-denominator integral. The
loop integral can be written in the form:
I box

i 4s
  k 2 )  2 (k  p 1  k 1 )  2 (k  k 1 ) 
n  (k

d k
g
(2 ) n 
k 2 (k  k 2 ) 2 (k  k1 ) 2 (k  ( p1  k1 )) 2
(12)
The calculation is lengthier than the previous cases. We made use of the
symbolic manipulation program to calculate the trace of the numerator
interfering with the lowest order. A double pole O(1 /  2 ) term appear as a
result of the infrared divergences as well as mass singularities in the
amplitude. Taking into account the crossed diagrams, the final form of the
correction from the box diagram has the form:
 box 
4 s
1 1
8
(4 )  [ 2  (ln( s)    2 
 f (mZ , u, t )  (t  u)]
3

M0

(13)
Where f (mZ , u, t ) are the rest finite terms. We have remind that we have only
concentrated on terms that contain a singularity. In the limit mZ  0 , the
result for box diagram is in agreement with Ref. [8].
4. O( s ) Real Gluon Correction to qq  ZZ
In order to calculate the corrections from the real gluon emission, we have to
calculate the cross-section corresponding to the Feynman diagrams of Fig.
(3)
k
k2
p2
p2
k2
p2
k2
k
k1
p1
k1
p1 k1
p1
k
+ Crossed Z Boson Diagrams
Fig. (3) Real gluon correction diagrams of order  s for qq  ZZ .
As it is clear, these diagrams contain no loop integrals, but there is an
integral over the final 3-body phase space. We again face divergences,
generated in the continuous dimension method as poles in the contribution to
the cross section. These divergences arise from two limits: First, when the
gluon is emitted in a direction parallel to either the quark or anti-quark, this
is called collinear limit, and second, when the gluon is emitted withy a very
small energy, we have what is called soft gluon emission.
a. Hard collinear gluon emission
In this case the gluon is emitted in a direction sufficiently parallel to the
quark or anti-quark. In order to regulate the divergences we have to work out
the cross section to the Feynman diagrams of Fig. (3). The matrix element
can be written in the form:
 igG
u (k 2 ) (k 2  k ) 2 (k 1  p 1 ) 1v(k1 )
4k1 . p1 k .k 2
 igG
M2 
u (k 2 ) 2 (k 2  p 2 ) ( p 1  k 1 ) 1v(k1 )
4k1 . p1 k 2 . p 2
 igG
M3 
u (k 2 ) 2 (k 2  p 2 ) 1 (k 1  k ) 1v(k 2 )
4k 2 . p 2 k .k1
 igG
M4 
u (k 2 ) (k 2  k ) 1 (k 1  p 2 ) 2 v(k1 )
4k1 . p 2 k .k1
 igG
M5 
u (k 2 ) 1 (k 2  p 1 ) ( p 2  k 1 ) 1v(k 2 )
4k 2 . p1 k1 . p 2
 igG
M6 
u (k 2 ) 1 (k 2  p 1 ) 2 (k 1  k )v(k1 )
4k 2 . p1 k1 .k
M1 
(14)
Where g is the coupling constant ( g 2  4 s ), G  ( g v 2  g a 2  2g v g a 5 ) ,  is
the polarization vector of the gluon and 1 ,  2 is the polarization vector of the
Z boson. For the case when the gluon is parallel to the antiquark k1 , only the
diagrams M 3 and M 6 leads to singularities, while when the gluon is parallel
to the quark k 2 , only diagrams M 1 and M 4 have singularities. In both cases
we have to calculate:
F    M 3  2 Re M 3 ( M 1  M 2  M 4  M 5  M 6 )  M 6  2 Re M 6 ( M 1  M 2  M 4 
2
2
*
*
M 5 )   M 1  2 Re M 1 ( M 1  M 3  M 4  M 5  M 6 )  M 4  2 Re M 4 ( M 2  M 3  M 5 
2
2
*
*
M 6 )].
(15)
Where the summation is over the polarization states of the photon, the gluon
and the quark. To calculate the traces we made use the symbolic
manipulation program (REDUCE). The gluon has treated in n-dimension,
the Z boson in 4-dimension. Working the matrix element out we get:
4
4
2
2
2 g 2 ( g v  g a  6 g v g a )s  t 2  u 2
1
t
u
2
4
M 
)  4mZ s ( )  mZ (

)  (n  4)
2[(
k .k1 k .k 2
t u 
tu 
t uu  u tt 

(t  u ) 2
t u
2
(
 2    f (mZ , u, t , u , t ))  t  u , u  t ] .
t u 
u t 
2
(16)
Where
t  ( p1  k1 ) ,
t   ( p2  k 2 ) 2 ,
2
u  ( p1  k 2 ) ,
u   ( p 2  k1 ) 2 ,
2
s  (k1  k 2 ) ,
s   ( p1  p2 ) 2 ,
2
As a check to the result we put t  t  , u  u  and s  s , then the form for the
lowest order is obtained and if we put mZ  0 , then the result for the twophoton case is obtained.
In order to get the correction due to hard collinear gluon we have to integrate
the matrix element over the gluon phase space.
After including factors for flux, Bose statistics, color sum, initial state spin
average, final state polarization sum, one calculate the cross section
contribution:
 real
( g v  g a  6 g v g a ) 4 4 s 2 4

3
2 s 3
2
4
4
2
2
d n k d 4 p1 d 4 p 2
 2  2  2  2 
n
n
4
n
 4 (k1  k 2  p1  p 2  k )2 3   (k 2 )  ( p1 2  m Z 2 )  ( p 2 2  m Z 2 ) M
2
(17)
b. Soft gluon emission
In this case the gluon energy is small so that the soft gluon matrix element is
proportional to the purely elastic cross-section. To order  s we have to
calculate the cross-section of the Fig. (4).
k
k2
p2
k2
k1
p1
k1
p2
p1
k
Fig. (4) Soft gluon emission diagrams of order  s for qq  ZZ .
The correction factor is given by the following form
 soft 
4 s 4
d n 1
1
(
2

)
s
n

2k k .k1 k .k 2
(2 ) 3
(18)
This integral has two divergences, with the gluon being soft ( k  0 ), and
with quark being mass less. We will generate both infrared and mass
singularities, working in the continuous dimension, double pole in the soft
correction appears. Integrating over the gluon energy and using some
gamma function properties, the final result for the correction can be written
in the following form:
 soft 
4 s
1 1
1
2
(4 )   [  (ln( 4 E 2 2 )   )  (ln( 4 E 2 2 )   ) 2 
]
3
 
2
4
(19)
5. Discussion
The O( s ) corrections to ZZ production have estimated previously by using
the soft gluon approximation [4]. The reason for this approach is that the  2
terms are the dominant part of the O( s ) corrections. All  2 terms are found
when the real or virtual gluon being soft k  0 which examine all graphs
that give double pole O(1 /  2 ) and deal them in the infrared limit. The soft
gluon correction is given by K  1 (8 / 9)s . A complete NLL calculation
for the total cross section for ZZ production at hadron colliders has been
reviewed in Ref. [9] , thus providing check on the calculation.
Our results have been presented for virtual and gluon emission separately. In
order to obtain results which are free from divergences, we add up the
corrections due to virtual and real gluon, the various double poles cancel,
and leftover 1 /  , which have the form
4 1  x3
3
(
  (1  x3 )
3 (1  x3 )  2
2
Pqq 
This term corresponds to a mass singularity, which is absorbed into the
structure function beyond the leading order. Since the appearance of the
mass singularities in higher order correction have universal structure,
process independed they can factorized according to a fundamental property
of factorization theorem which tells that the bare cross-section can be
factorized into universal function containing the singularities which can be
removed by subtraction procedure and a well defined short distance crosssection, which is free from mass singularities. Thus the total cross-section
can be defined as:
d  d 0  d 1
Where d 0 is the Born cross-section, and d 1 denotes the order
 s corrections which appear as large double logarithms and  2 term.
In Fig. 5 we plot the cross section for Z pair production versus energy. The
numerical results presented have obtained using the MRST parton
distribution [10]. These distribution functions have been fit at the NLL level
using the universal ( MS ) convention. It is interesting to note that, including
the soft-gluon K-factor in the LL calculation reasonably approximated the
NLL corrections.
1
Cross Section (ZZ) (pb)
10
0
10
SOLID = NLL
LONG DASH = LL
-1
10
0
5
10
15
1/2
20
s (TeV)
FIG. (5) THe total cross section for the Z pair production as function of the
center-of-mass. Caculations for LL and NLL results.
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