SPLITTING FUNCTIONS
These are often called Altarelli-Parisi (AP) [1] splitting functions, but to give due credit
to all who were responsible for their development they should be called DGLAP splitting
functions with credit to Dokshitzer, [2] Gribov and Lipatov [3].
They provide the mechanism for handling un-cancelled collinear divergences, which arise
from the radiation of massless partons from one of the incoming partons taking part in a
scattering process.
1
Initial State Collinear Singularities
p′
|M|2
X
p
Figure 1: A parton of type j, with momentum p scatters off a parton with momentum p′
into a final state X. Some integration over the final state-particles is implied
Denote |M|2i→X (p) as the square matrix-element for the scattering of a parton of type
i and momentum p into some final state X. The index i denotes all the properties of the
parton, i.e its type (quark, antiquark or gluon), flavour, colour, and helicity. Note that by
this quantity we could mean a partial phase-space integral over outgoing particles in order
to obtain the final state X (for example it could refer to a total cross-section - in which
case it is assumed that a complete integral has been performed over the phase-space of the
outgoing particles.)
The momentum of the particle against which it scatters is denoted by 1 p′ and in the CM
frame of the incoming partons we have
√
s
p =
(1, 0, 0, 1)
2
√
s
′
(1, 0, 0, −1)
(1)
p =
2
1
All partons are considered to be massless here
1
p′
X
|M|2
i
j
p
k
h
Figure 2: A parton, j, with momentum p radiates a parton, i, with momentum p − k, which
then scatters off a parton with momentum p′ into a final state X. In the collinear limit the
momentum of parton i is x p
where s is the total square energy, s = 2p · p′ .
The (differential) cross-section for this process is given by
σi→X (p) =
1
|M|2i→X (p)
2s
(2)
Now consider a more complicated process in which a different parton of type j and
momentum p scatters of a target parton with momentum p′ into a final state X by first
radiating off a parton of type h with momentum k and a quark of type i with (off-shell)
momentum p − k, which then scatters off the target parton to the final state X
The momentum k may be written
k µ = (1 − x)pµ + βp′ µ + kTµ
(3)
where the two-component vector kTµ is orthogonal to both p and p′ . The on-shell condition
k.k = 0 fixed β and so this may be written
k µ = (1 − x)pµ +
kT 2
p′ µ + kTµ
s(1 − x)
(4)
The parton i has off-shell momentum with
kT 2
(p − k) =
(1 − x)
2
2
(5)
Thus, the internal propagator is proportional to 1/kT 2 , so that the square matrix-element for
this processes diverges as kT → 0, i.e. in the collinear limit where the transverse momentum
of the radiated parton is small. In this limit, we can interpret the parameter x as the fraction
of the parent momentum, p, of the parent parton, j, which is carried off by the daughter
parton i, and then undergoes the scattering. Since we want to define the final state by X, we
need to integrate over the phase-space of the extra final-state parton, h and this phase-space
integral will diverge owing to the singular structure of the square matrix-element in the limit
kT → 0.
This is to be expected. We know that when a massless particle is radiated from a massless
particle, there are collinear divergences. There are also collinear divergences arising from
the radiation of a massless parton from any of the final-state particles (and also from a
massless parton radiated from the incident target parton). There are also collinear and soft
divergences (soft meaning that the energy of the emitted parton vanishes) from virtual (oneloop) corrections to the tree-level approximation to the square matrix-element, |M|2i→X (p).
The Kinoshita-Lee-Nauenberg (KLN) theorem tells us that both of these types of divergences cancel provided we sum over all possible initial and final states (to a given order in
perturbation). In the case under consideration a sum over final states (a phase-space integral
or partial phase-space integral over final-state particles) is implied in the definition of the
final-state X so that the collinear singularities associated with radiation from the final-state
particles cancels against corresponding collinear divergences in the virtual corrections, but
we are certainly not summing over all initial states - we specify that we have a two-particle
initial state with well-defined momenta. The KLN theorem does not therefore apply and we
do indeed expect to be left with some collinear divergences associated with radiation with
initial-state partons.
We write the (differential) cross-section for parton j, with momentum p, scattering of
the target parton to a final state X as
1 Z
L.O
dσj→X (p) = dσj→X
(p) +
dP S (h)|M|2j→i→X (p),
(6)
2s
L.O
where the first term, dσj→X
(p), is the differential cross section for the process to occur
without passing though an intermediate parton, i - this is non-zero if the partons i and j are
of the same type. |M|2j→i→X (p) is the square matrix-element for the process to proceed via
an intermediate parton j 2
2
The Phase Space Integral
Since we are only concerned with a final state X, we need to perform a phase-space integral
over the extra emitted parton, h. The measure for this phase-space integral is given by
d4 k
δ(k 2 )
(2π)3
dP S (h) =
2
(7)
eq.(6 is only valid if we can neglect interferences between the two “processes”. We show later that for the
determination of the collinear singularity this is justified provided one chooses to work in a suitable gauge.
3
In terms of the variables x and kT defined in eq.(3) this may be written as
dP S
(h)
1
kT 2
2
.
=
dxd
k
dβδ
(1
−
x)β
−
T
16π 3
s
!
(8)
Using the integral over β to absorb the delta function and also performing the trivial integral
over the azimuthal angle in the transverse plane this becomes
dP S (h) =
dx
16π 2 (1
− x)
dkT 2 .
(9)
The kinematic limit on x (from the requirement that the energy of h should be positive) is
0 < x < 1
and kT 2 is integrated from zero to some (process dependent) maximum value. This process
dependence is not important as we are only concerned with the divergence as kT → 0.
3
Definition of the Splitting Functions
Inserting the expression (9) for the phase-space integral into the expression eq.(6) for the
differential cross-section we have
L.O
dσj→X (p) = dσj→X
(p) +
Z 1
1 1 Z
dx
2
dk
|M|2j→i→X (p),
T
2
2s 16π
0 (1 − x)
(10)
We define the splitting function from the collinear divergent part of this as kT → 0 by
1 1
2s 16π 2
(Z
dkT
2
Z
0
1
dx
|M|2j→i→X (p)
(1 − x)
)
div
=
Z
dkT 2
kT 2
Z
0
1
dxPij (x)dσj→X (xp)
(11)
or alternatively
n
o
1
(1 − x)
2
2
lim
k
|M|
(p)
=
Pij (x)|M|2i→X (xp)
t
j→i→X
2
16π kT →0
x
(12)
The x in the denominator of the RHS of eq.(12) arises from the fact that the flux factor for
the subprocess i → X, with the momentum of i being xp, is 2sx.
4
Factorization Scale Dependence of Parton Distribution Functions
In the classical parton picture, the parton distribution function (PDF), fN/i (x) is the probability that a nucleon, N, contains a parton of type i with a fraction x of the nucleon’s
momentum in the infinite momentum frame (nucleon moving with velocity close to c).
4
However, when quantum corrections are taken into consideration, the PDF’s acquire a
factorization scale, µF , and obey the DGLAP equation
Z 1
d
2
f
(x,
µ
)
=
Pij (µF , y)fN/j
N/i
F
dµ2F
x
!
x 2
,µ ,
y F
(13)
where we have included a dependence on µF in the splitting functions Pij , which simply
indicates that the running strong coupling used in the perturbative expansion of the splitting
functions is to be taken at renormalization scale µF .
It is convenient to think of these PDF’s as the sum of a “bare” PDF, which contains a
collinear divergence, but is factorization scale independent and a collinear divergent correction, which cancels the divergence in the “bare” PDF. This we write the factorization-scale
dependent PDF as
fN/i (x, µF ) =
bare
fN/i
(x)
+ lim
λ→ 0
(Z
µ2F
λ2
x 2
dkT 2
,µ
2 Pij (y)fN/j
y F
kT
!)
(14)
To leading order in the strong coupling αs , this is a solution to the DGLAP equation (13).
Similarly, to leading order the implicit divergence of the bare PDF cancels the log(λ) term
obtained from performing the integral over kT 2 .
The splitting functions Pij (y) are interpreted as the probability that a parton of type j
emits a (collinear) parton of type i, with a fraction y of the momentum of the parent parton.
Since the portion of the integral over kT 2 with kT < µF has been absorbed into the PDF,
it is only the perton for which kT > µF that needs to be calculated and included in the
NLO corrections to the process under consideration. This integral is free from divergences,
but depends on the factorization scale, µF - but it does so in such a way that (order by order
in perturbation theory) the µF dependence is cancelled by the µF dependence of the PDF.
This factorization scale is therefore arbitrary. However, in order to avoid large logarithms
in the NLO corrections this scale is chosen to be a “typical scale” of momenta of the process
in question. 3 .
5
Calculation of the Splitting Functions
Pqq :
This is the splitting function for a quark to radiate a quark with a fraction x of its
momentum. The square matrix element |M|2j→i→X (p) is calculated from the graph shown in
Fig.3
We first note that this is not complete. For a gauge invariant quantity, we also need
to consider interferences between graphs involving the emission of a gluon from the incom3
In many cases a process will involve more than one momentum scale. In such cases large logarithms cannot be totally removed by a judicious choice of factorization scale. Much effort ha been put into resummation
of the remaining large logarithms - but this is outside the scope of these lectures
5
|M|2 (xp)
p−k
p−k
k
p
p
Figure 3: Graph required for the calculation of |M|2j→i→X for the case of Pqq
ing quark with the emission of the gluon from any of the other colour particles (shown
schematically in Fig.4)
|M|2 (xp)
p−k
k
p
p
Figure 4: Interference between graph in which gluon is radiated from the incoming qiuark
with gluon radiated from other particles
However, if we select a physical gauge in which the polarization vector of the gluon is
transverse to p′ (as well as transverse to k) then the contribution from such interferences
will not lead to a collinear divergence.
To see this, we write out such a polarization vector explicitly
ǫ± µ =
q
2kT 2 ′ µ
kµ
lµ
p + q T ± iq T
s(1 − x)
2lT2
2kT 2
where lTµ is a vector in the transverse plane which is also orthogonal to kTµ .
6
(15)
The scalar products of this polarization vector with other momenta are
ǫ± · p′ = 0, (by construction)
ǫ± · p = √
±
ǫ · kT =
s
q
kT 2
2(1 − x)
kT 2
2
(16)
However, whereas the denominator of the graph in Fig.3 has two denominators proportional to kT 2 (from the two propagators of the internal parton with momentum (p − k)),
graphs of the type depicted in Fig.4 only have one such propagator and so the numerator,
which must involve at least one of the scalar products in eq.(16) will give rise to an integrand
whose power dependence on kT 2 is larger than -1, and therefore does not contribute to a
collinear singularity.
Before we can calculate the contribution for the diagram of Fig.3, we need to examine the
form of the insertion |M|2i→X (p). As the insertion has an incoming and out going fermion,
both with the same momentum p and the same helicity, we may write this as
|M|2i→X (p) = u(p, λ) γ · V u(p, λ).
(17)
This is the only possible structure of γ-matrix as the incoming and outgoing fermions are
massless and have the same helicity.
V µ is a general vector, which may be written
V µ = apµ + bp′ µ + vTµ
(18)
(vTµ is a vector in the transverse plane). However, we also have the relations
u(p, λ)γ · pu(p, λ) = 0 by Dirac’s equation
and
u(p, λ)γ · vT u(p, λ) = 0.
This latter relation holds because the operator γ · vT flips the component of spin in the
direction of p, and therefore flips helicity of a particle with momentum p - so its matrix
element between states of the same momentum and helicity vanishes.
Therefore we only need to consider
V µ = bp′ µ .
The coefficient b is determined by noting that
′
2
(
′
|u(p, λ)γ · p u(p, λ)| = T r γ · p γ · p γ · p γ · p
7
′ (1
+ λγ 5 )
2
)
= (2p · p′ )2
(19)
u(p, λ)γ · p′ u(p, λ) is real - so we do not need to worry about a phase and we have
V µ (p) =
p′ µ
|M|2i→X (p)
2p · p′
(20)
Note that when the fermion with momentum p is replaced by a fermion of momentum
(p − k), which in the collinear limit may be written as xp, we have
p′ µ
|M|2i→X (xp).
2xp · p′
V µ (xp) =
(21)
The insertion vertex (which carries zero momentum) is therefore
γ ·V =
γ · p′
|M|2i→X (xp)
2xp · p′
(22)
After introducing a factor of 21 for averaging overing incoming quark helicities and a factor
of 1/CA for the average over incoming quark colour we find that the graph of Fig.3 gives
γ · (k − p) γ · p′ γ · (k − p) a ∗ ±
g2 X X
u(p, λ)τ a γ·ǫ±
τ γ·ǫ u(p, λ)|M|2i→X (xp)
2CA λ ±
(k − p)2 sx (k − p)2
(23)
We may write this as
|M|2j→i→X (p) =
|M|2j→i→X (p) =
o
n
X
1
g2
±
′
∗±
T r(τ a τ a )
T
r
γ
·
p
γ
·
ǫ
γ
·
(k
−
p)
γ
·
p
γ
·
(k
−
p)
γ
·
ǫ
2CA sx
(−2p · k)2 ±
(24)
T r(τ a τ a ) = CF CA
and the sum over gluon polarization in the physical gauge we are using gives
X
±
∗±
ǫ±
µ ǫν
= −gµν
(kµ p′ν + kν p′µ )
+
k · p′
(25)
We also have the kinematic relations
kT 2
2k · p =
(1 − x)
′
2k · p = s(1 − x)
(26)
Performing the trace in eq.(24) and using the kinematic relations (26) we may express
|M|2j→i→X (p) in terms of s, x and kT 2 and get
|M|2j→i→X (p) = g 2
CF 2
(1 + x2 )|M|2i→X (xp)
sx kT 2
8
(27)
From eq.(12) and setting g 2 to 4παs , we arrive at the expression for the splitting function
for a quark to emit a quark:
αs
P̃qq (x) = CF
2π
(1 + x2 )
(1 − x)
!
(28)
The ∼ sign over Pqq indicates that this is not yet the whole story.
We notice that P̃qq (x) has a divergence as x → 1. The origin of this divergence is that
infrared divergences can be either collinear or soft or both collinear and soft. This is what is
happening here. In the limit kT → 0 and x → 1 (with kT and (1 − x) being of the same
order), the energy of the emitted gluon, k vanishes and so what we are seeing here is this
double divergence.
However, we know that soft divergences always cancel between real emissions and virtual
corrections. This means that we also have a double singularity in the virtual corrections to
the square matrix element |M|2j→X (p), which must cancel this double divergence. We can
express this as
Z
1
0
dxP̃qq (x)|M|2j→X (xp) + (1 + A)M|2j→X (p), is finite
where A is a factor of order αs , which contains this double divergent part of the virtual
corrections, converges. Let us rewrite this as
Z
1
0
dx P̃qq (x) + δ(1 − x)(1 + A) |M|2j→X (xp) is finite.
We do not need to calculate the virtual corrections. This is fortunate since it is important
in this approach to determine the splitting functions in a way that it process independent.
We can obtain the divergent factor A from the following simple proposition: Since quark
current is conserved (conservation of fermion number) the probability of a quark emitting
a gluon with some fraction (1 − x) of its momentum, summed over all allowed values of x,
plus the probability that the quark does not radiate such a gluon is unity. In other words
Z
0
1
dx P̃qq (x) + δ(1 − x)(1 + A)
= 1
(29)
so that A takes the value
αs
A = CF
lim
2π λ→0
Z
1
λ
3
αs
(1 + x2 )
− − 2 ln λ
= CF
(1 − x)
2π
2
(30)
so the complete splitting function is then
αs
(1 + x2 )
3
Pqq (x) = CF
lim
θ(1 − x − λ) + δ(1 − x)
− 2 ln λ
2π λ → 0 (1 − x)
2
(
9
)
(31)
5.1
The + prescription
This can be written in a more elegant way by realizing that δ(1 − x) and hence Pqq (x) is a
distribution, which is only defined when acting on some function, f (x). This enables us to
write
!
3
αs (1 + x2 )
,
(32)
+ δ(1 − x)
Pqq (x) = CF
2π (1 − x)+
2
where
1
(1 − x)+
is a distribution defined by its action on a function f (x) (which is regular at x = 1) by
1
(f (x) − f (1))
· f (x) =
(1 − x)+
(1 − x)
(33)
This notation eliminates the need to introduce a soft cutoff, λ and take the limit of the
difference between two diverging quantities.
Pqg :
This is the splitting function for a gluon to radiate a quark (or antiquark) with a fraction
x of its momentum. The square matrix element |M|2j→i→X (p) is calculated from the graph
shown in Fig.5
|M|2 (xp)
p−k
p−k
k
p
p
Figure 5: Graph required for the calculation of |M|2j→i→X for the case of Pqg
The calculation follows the same line as for Pqq . In this case as well as a factor of
the average over incoming gluon helicities, there is a factor of
1
2CF CA
10
1
2
for
for the gluon colour. The insertion vertex is again given by (22) and the expression for the
square-matrix element |M|2j→i→X (p) is
γ · (k − p) γ · p′ γ · (k − p) a ∗ ±
g2 X X
τ γ·ǫ u(k, λ)|M|2i→X (xp),
u(k, λ)τ a γ·ǫ±
4CF CA λ ±
(k − p)2 sx (k − p)2
(34)
which we may write as
|M|2j→i→X (p) =
|M|2j→i→X (p) =
1
1
g2
T r(τ a τ a )
4CF CA
sx (2p · k)2
X
±
o
n
T r γ · k γ · ǫ± γ · (p − k) γ · p′ γ · (p − k) γ · ǫ∗ ± |M|2i→X (xp),(35)
In this case the sum over helicities of the external gluons gives
X
±
∗±
= −gµν +
ǫ±
µ ǫν
pµ p′ν + pν p′µ
p · p′
(36)
and using the kinematic relations eq.(26) we obtain
|M|2i→X (xp) = 2g 2TR nf
(1 − x) 1 2
2
x
+
(1
−
x)
x kT 2
(37)
The factor nf arises from the sum over all possible flavours of quarks that can appear in the
loop.
Comparing this with eq.(12) we arrive at
Pqg (x) =
αs
nf TR x2 + (1 − x)2
2π
(38)
Note the symmetry under x ↔ (1 − x). This is to be expected - it tells us that the
probability for a gluon to radiate a quark with a fraction x of its momentum is the same as
the probability to radiate an antiquark with that fraction of its momentum.
Pgq :
This is the splitting function for a quark to radiate a gluon with a fraction x of its
momentum. The square matrix element |M|2j→i→X (p) is calculated from the graph shown in
Fig.6
Here we need to consider the square matrix-element |M|2i→X (p) in the case a a gluon of
momentum p (which may be off-shell) off a target parton into a final state X. This is of the
form
∗±
µν
|M|2i→X (p) = ǫ±
(p)
(39)
µ (p)ǫν (p)V
11
|M|2 (xp)
p−k
p−k
k
p
p
Figure 6: Graph required for the calculation of |M|2j→i→X for the case of Pgq
Non-Abelian Ward identities tell us that the tensor V µν must obey
pµ pν V µν = 0
even when p2 6= 0. This means that the tensor is given by
Vmuν (p) = − gµν
′ ′
(pµ p′ν + pν p′µ )
2 pµ pν
−
|M|2i→X (p)
+p
p · p′
(p · p′ )2
!
(40)
′
However since the polarization vectors ǫ±
µ (p) are orthogonal to both p and p , it is sufficient
to take only the first term and write
Vµν = −gµν |M|2i→X (p)
(41)
.
On the other hand, we need to be careful in the sum over polarizations for the internal
gluon, with momentum (p − k), which is slightly off-shell. This must be orthogonal to both
p′ and (p − k), so we write
X
±
∗±
ǫ±
µ (k − p)ǫν (k − p) = −gµ,ν +
p′µ p′ν
((k − p)µ p′ν + (k − p)ν p′µ )
2
−
(p
−
k)
(42)
(p − k) · p′
((p − k) · p′ )2
This is a projection operator, which means that
XX
±
′
∗ ± ρσ ± ∗ ±
ǫ±
ǫσ ǫν
µ ǫρ V
′
= (−gµν +
±′
((k − p)µ p′ν + (k − p)ν p′µ )
p′µ p′ν
2
|M|2i→X (xp).
− (p − k)
′
′
2
(p − k) · p
((p − k) · p )
!
12
(43)
From the graph in Fig.6 and with a factor of 1/2CA for the average over incoming quark
helicity and colour, the expression for |M|2j→i→X (p) is
XXXX
g2
1
∗ ± ρσ ±′ ∗ ±′
ǫ±
ǫσ ǫν u(p, λ)τ a γ µ u(k, λ′) u(k, λ′ )τ a γ ν u(p, λ)
µ ǫρ V
2
2
2CA ((k − p) ) λ λ′ ± ±′
(44)
Using eq.(43), we may write this as
|M|2j→→X (p) =
g2
1
T r(τ a τ a )
T r {γ · p γ µ γ · k γ ν }
2CA
2k · p2
!
((k − p)µ p′ν + (k − p)ν p′µ )
p′µ p′ν
2
|M|2i→X (xp).(45)
+
− (p − k)
(p − k) · p′
((p − k) · p′ )2
|M|2j→i→X (p) =
−gµ,ν
Performing the trace and using the kinematic relations (26) we get
i
1 (1 − x) h
2
|M|2i→X (xp).
1
+
(1
−
x)
kT 2 x2
|M|2j→i→X (p) = 2CF g 2
(46)
and from eq.(12) we get
αs
Pgq (x) = CF
2π
1 + (1 − x)2
x
!
(47)
Note that
Pgq (x) = P̃qq (1 − x)
(48)
This is to be expected. The probability of a quark to split into a quark with fraction
of its momentum x is the same as the probability for the quark to split into a gluon with
fraction of its momentum (1 − x). In this case, however, there are no virtual corrections and
no subtraction needed as x → 1
Pgg :
Finally we consider the splitting function for a gluon to radiate a gluon with a fraction
x of its momentum. The square matrix element |M|2j→i→X (p) is calculated from the graph
shown in Fig.7
With a prefactor of 21 for the average over incoming gluon helicities and 1/2CA CF for the
average over gluon colours, this graph gives
|M|2j→i→X (p) =
×
X
±
g2
1
f abc f abc
4CF CA
((k − p)2 )2
∗±
ǫ±
µ (p)ǫµ′ (p)
X
±′
×V
′
′
V3µν,σ (p, (k
′ ′ ′
p), −k)V3µ ν σ (p, (k
∗±
ǫ±
ν ((k − p))ǫρ ((k − p))
ρρ′
−
X
±′′
′′
′′
∗±
ǫ±
((k − p))
ρ′ ((k − p))ǫν
X
′′′
′′′
∗±
ǫ±
(k)
σ (k)ǫσ′
±′′′
− p), −k),
(49)
where the triple-gluon vertex, V3 is given by
V3µνσ (p1 , p2 , p3 ) = g µν (pσ1 − pσ2 ) + g νσ (pµ2 − pµ3 ) + g σµ (pν3 − pν1 ).
13
(50)
|M|2 (xp)
p−k
p−k
k
p
p
Figure 7: Graph required for the calculation of |M|2j→i→X for the case of Pgg
Using eq.(43) and eqs.(36) and (43) for the summation over polarization of the incoming and
outgoing gluon, and using also the kinematic relations (26) this gives (after some tedious
algebra)
!
2
2
(1
−
x)
g
|M|2j→i→X (p) = 4CA 2 1 + (1 − x)2 +
(51)
x2
kT
Comparison with eq.(12) gives
αs
P̃gg (x) = 2CA
2π
x
(1 − x)
+
+ x(1 − x)
(1 − x)
x
!
(52)
We note that this is symmetric under x ↔ (1 − x), as expected since for a gluon splitting
into two gluons it cannot matter which gluon carries a fraction x of the parent quark’s
momentum and which carries a fraction (1 − x).
We also note that as in the case of Pqq , Pgg contains a divergence as x → 1 corresponding to the double divergence arising which the emitted gluon, h, (momentum k) is
both collinear and soft. Such a divergence is cancelled by a corresponding divergence in the
virtual corrections to the process in question. Thus we write
Pgg (x) = P̃gg (x) + Aδ(1 − x)
(53)
where A encodes the virtual corrections and contains a divergent term. Unfortunatey, in this
case we cannot employ the argument that the sum of the probabilities for a gluon to emit
another gluon and not to emit another gluon is unity, since this assumes that gluon number
is conserved which is not the case (unlike fermion current, gluon current in not conserved).
We can, however, employ a conservation of momentum of the incoming gluon. The total
fraction of momentum carried off by an emitted gluon is xP̃gg (x) integrated over all x. The
gluon can also split into a quark-antiquark pair. The total fraction of momentum carried
off by a quark (or antiquark) is xPqg (x) integrated over all x. The sum of these, plus the
14
probabaility that the gluon does not radiate and therefore carries off all of the incoming
momentum is unity. Thus we can write
Z
0
or
1
dxx P̃gg (x) + Pqg (x) + (1 + A)δ(1 − x)
Z
1
0
= 1
(54)
dxx (Pgg (x) + Pqg (x)) = 0
(55)
This gives us
(
αs
(1 − x)
x
Pqq (x) =
lim 2CA
θ(1 − x − λ) +
+ x(1 − x)
2π λ → 0
(1 − x)
x
!)
(11CA − 4nf TR )
+ 2CA ln λ
+δ(1 − x)
6
!
(56)
or using the more convenient +-presciption
"
!
x
(1 − x)
(11CA − 4nf TR )
αs
2CA
+
+ x(1 − x) + δ(1 − x)
Pqq (x) =
2π
(1 − x)+
x
6
#
(57)
It is not an accident that the coefficient of the δ-function is equal to β0 , the first term
in the perturbative expansion for the β-function in QCD. In the gauge that we are using,
the calculation of the β-function would involve only the gluon two-point function (gluon
self-energy). The graph of Fig. 7 is this two-point function with a zero-momentum insertion
V µ,ν .
The argument concerning conservation pf momnetum can also be applied to an incoming
quark leading to the relation
Z
1
0
dxx Pqq (x) + Pqq(x)
= 0
and we can see from eqs.(32) and (47) that this is indeed the case.
References
[1] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298.
[2] Yu. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.
[3] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438.
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