Greiner W., Schrammm S., Stein E. Quantum Chromodynamics
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6.1 The Drell–Yan Process 389
Finally we must know the elementary cross section q+
¯
q → µ
+
+µ
−
. Calculat-
ing this is very easy and will be done in Exercise 6.1. We obtain
σ(q +
¯
q → µ
+
+µ
−
) =
1
3
4πα
2
3M
2
. (6.5)
The additional factor 1/3 stems from averaging over quark colors, since the
structure functions of the nucleon are already known.
The Drell–Yan cross section for quark–antiquark annihilation is thus to
lowest order
d
2
σ
DY
dτ dy
=
4πα
2
9M
2
q
e
2
q
q
a
(x
a
)
¯
q
b
(x
b
) +q
b
(x
b
)
¯
q
a
(x
a
)
(6.6)
or, with
dy
dx
a
=
1
2x
a
, M
2
= τs ,
σ
DY
=
4πα
2
9
1
0
dτ
1
τ
dx
a
2x
a
1
τs
q
e
2
q
q
a
(x
a
)
¯
q
b
τ
x
a
+q
b
τ
x
a
¯
q
a
(x
a
)
.
(6.7)
To calculate Drell–Yan cross sections one has in addition to calculate the
Compton process q +G →γ
∗
+q → µ
+
µ
−
+q, which is usually even more
important. We shall not do this, since all the problems encountered are exactly
the same for both processes. We shall simply state the corresponding result for
the Compton process at the very end.
Equation (6.7) and the corresponding Compton cross section yield unique
prediction for lepton-pair production in high-energy proton–proton collisions.
When this was measured, it turned out that the experimental cross section ex-
ceeded the prediction by a factor of about two. This missing factor, i.e., the
quotient of the measured cross section over the calculated one, is termed the
“K factor”. Surprisingly it is independent of x
a
and x
b
, i.e., a true constant.
This can, in the framework of perturbative QCD, only be due to higher-order
corrections. Since such higher-order perturbative corrections are of the same
order as the lowest-order contribution, convergence of the perturbation series is
exceedingly questionable.
To gain a deeper understanding of the problems in Drell–Yan reactions, in the
following we shall discuss the calculation of some corrections to the annihilation
graph q +
¯
q → γ
∗
. The relevant processes are displayed in Fig. 6.2. However, we
can avoid much work by using some results from Chap. 5. The graphs in Fig. 6.3
differ from those in Fig. 6.2 only by the exchange of the Mandelstam variables s
and u. But the graphs in Fig. 6.3 are just those we calculated in Chap. 5, namely
the term with µ = µ
. Accordingly one gets for the processes in Fig. 6.3 the result
(see (5.12))
F
2
= S
µ
µ
=−8
s
t
+
t
s
+
2M
2
u
st
. (6.8)
390 6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics
Fig. 6.2. First-order gluonic
corrections to the Drell–Yan
process
Fig. 6.3. First-order gluonic
corrections to the muon–
quark scattering
Here we have replaced −Q
2
by M
2
. We must then, according to the “crossing
symmetry”, exchange s and u:
F
2
= 8
u
t
+
t
u
+
2M
2
s
ut
. (6.9)
Here an additional minus sign had to be taken into account. This phase is related
to the definition of particle states (we replaced a quark by an antiquark). Now
the combinatorial factors remain to be determined. The different charges give
e
2
e
2
q
g
2
, spin averaging yields 1/4, and color averaging
a
1
9
c,c
λ
a
2
cc
λ
a
2
c
c
=
8
9
1
2
=
4
9
.
We therefore have
dσ
=
1
2s
d
3
q
1
(2π)
2
s
e
2
e
2
q
g
2
8
9
δ
q
0
1
+q
0
2
− p
0
1
− p
0
2
u
t
+
t
u
+
2M
2
s
ut
.
(6.10)
With dt = dΩ|q|
2
/π (Exercise 2.6, (26)), we obtain
dσ
=
1
2s
dt
8πs
8
9
αα
s
e
2
q
16π
2
u
t
+
t
u
+
2M
2
s
ut
=
αα
s
e
2
q
π
s
2
8
9
u
t
+
t
u
+
2M
2
s
ut
dt . (6.11)
It should be noted that the relation between the Mandelstam variables connects
u, s,andt:
u = M
2
−s −t .
6.1 The Drell–Yan Process 391
The additional contribution to the cross section is thus, at given s, a function of t
only. We find that
M
2
−s ≤ t ≤ 0
holds and that the integral diverges at the upper bound of the integration, i.e.,
for t → 0. The appearance of infrared divergences is typical for all calcula-
tions of perturbative QCD. Its reason is that in the framework of the parton
model all small dimensional quantities like quark masses and gluon virtuali-
ties are neglected. These quantities normally inhibit the appearance of infrared
divergences. The simplest solution to the problem would therefore be to intro-
duce finite masses and virtualities for all particles. But then the results would
exhibit a strong dependence on these parameters and thus be more or less mean-
ingless. The only chance of getting reliable predictions lies in the hope that
the infrared divergences of different contributions cancel. This is indeed what
happens.
At this point it is useful to recall QED bremsstrahlung, where similar prob-
lems surface owing to the vanishing mass of the photon. It turned out there that
the infrared divergence canceled with the diverging radiative corrections. This
effect, known as the Bloch–Nordsieck theorem
1
, is of basic relevance for the
consistency of QED, and even more so for that of QCD. Its physical background
is related to the fact that massless particles with arbitrarily small energy, e.g.,
photons of infinitely long wavelengths, are, strictly speaking, unphysical since
they cannot be detected by any means. The transition from some state to the same
state with an additional undetectable photon is not well defined. In perturbation
theory, however, all states can be classified by their occupation numbers and such
states can be strictly distinguished. Since this distinction is unphysical, it can
very well lead to spurious divergences in different contributions that cancel each
other.
Next we face the question of how a radiative correction whose amplitude is
proportional to g
2
can cancel an amplitude that is of first order in the coupling.
This is explained by noting that all radiative correction graphs interfere with
the lowest-order graph since they have the same initial and final states. Fig. 6.4
depicts this.
The validity of the Bloch–Nordsieck theorem is of great relevance for the
whole of perturbative QCD. Nonetheless a general complete proof seems still to
be missing and is clearly confronted with problems for the Drell–Yan process in
particular. We are unable to explore this theoretical question in more detail here,
but the Bloch–Nordsieck theorem holds for the lowest-order term, investigated
in the following, as will be shown.
The procedure to show that two divergences cancel is as follows. First
a regulator is introduced to render the contributions finite. Then, after summing
the contibutions, we send the regulator to zero or infinity depending on the nature
of the divergences (infrared or ultraviolet).
In our case we choose a finite gluon mass m
G
as a regulator. Then the cal-
culation of the graphs in Fig. 6.2 must be repeated with a finite gluon mass. We
1
F. Block and A. Nordsieck: Phys. Rev. 52, 54 (1937)
392 6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics
Fig. 6.4. All graphs lead-
ing to the same final state
interfere with one another.
Therefore both photon +
gluon graphs and radia-
tive corrections contribute
to the order α
s
. The Bloch–
Nordsieck theorem states
that infrared divergencies of
these contributions cancel
shall skip this and give the final result only. The result is
dσ
(q +
¯
q → γ
∗
+G)
=
αα
s
e
2
q
π
s
2
8
9
'
u
t
+
t
u
+
2(M
2
+m
2
G
)s
ut
−M
2
m
2
G
1
t
2
+
1
u
2
(
dt (6.12)
with
u = M
2
+m
2
G
−s −t
and
d
2
σ
DY
(q +
¯
q → µ
+
+µ
−
+G) =
α
3πM
2
dσ
(q +
¯
q → γ
∗
+G) dM
2
.
Obviously the changes in the cross section are minor. In particular, the trouble-
some terms with 1/t are unchanged, and an even more troublesome term in 1/t
2
has appeared. The crucial question is thus: what are the bounds of the integral for
finite m
G
? We choose the center-of-momentum frame to calculate these bounds.
Here k
γ
=−k
G
= k, and thus
s = (E
γ
+E
G
)
2
=k
2
+M
2
+2
"
k
2
+M
2
k
2
+m
2
G
+k
2
+m
2
G
, (6.13)
(s −M
2
−m
2
G
−2k
2
)
2
= 4(k
2
+M
2
)(k
2
+m
2
G
). (6.14)
6.1 The Drell–Yan Process 393
From this, k is calculated:
k =
6
(s −M
2
−m
2
G
)
2
−4M
2
m
2
G
4s
. (6.15)
Bounds for t are calculated from
t = ( p
q
−k
γ
)
2
= M
2
−2(E
γ
E
q
−E
q
k cos θ) , (6.16)
t
min/max
= M
2
−2E
γ
E
q
∓2E
q
k . (6.17)
We put 2E
q
=
√
s and we use (6.15):
t
min/max
= M
2
−
!
s
4s
(s −M
2
−m
2
G
)
2
−4M
2
m
2
G
+4sM
2
±
(s −M
2
−m
2
G
)
2
−4M
2
m
2
G
= M
2
−
1
2
(s +M
2
−m
2
G
)
2
±
(s −M
2
−m
2
G
)
2
−4M
2
m
2
G
=−
1
2
(s −M
2
−m
2
G
) ±
(s −M
2
−m
2
G
)
2
−4M
2
m
2
G
. (6.18)
Obviously t
max
is now nonzero, so the integral is finite. As claimed above, t
max
goes to zero as m
G
→0. This result can be rewritten by introducing the quantities
a = M
2
/s and b = m
2
G
/M
2
:
t
min/max
=−
M
2
2a
1 −a −ab ±
"
(1 −a)
2
+ba(ba −2 −2a)
. (6.19)
The integral over t is explicitly calculated in Exercise 6.2, leading to
σ
=
8παα
s
e
2
q
9s
⎡
⎣
2
1 + B
2
/s
2
1 − B/s
ln
⎛
⎝
s −B +
(s −B)
2
−4M
2
m
2
G
s −B −
(s −B)
2
−4M
2
m
2
G
⎞
⎠
−4
6
1 −
B
s
2
−4
M
2
m
2
G
s
2
⎤
⎦
(6.20)
with
B = M
2
+m
2
G
.
To get the total Drell–Yan contribution we have to substitute τs = x
a
x
b
s for s,
where s is now the total four-momentum squared of the colliding hadrons. We
also have to insert the relation between σ(q +
¯
q → γ
∗
) and dσ/dM
2
(q +
¯
q →
γ
∗
→µ
+
µ
−
), which is derived in Exercise 6.3. For negligible lepton masses
this is
dσ
dM
2
(q +
¯
q → µ
+
µ
−
) = σ(q +
¯
q → γ
∗
)
α
3πM
2
dM
2
. (6.21)
394 6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics
In our problem, i.e., the first-order gluon correction, M
2
corresponds to the
invariant mass of the γ
∗
and gluon. We therefore replace it by τs. Thus the
correction to the annihilation part of the Drell–Yan cross section is
d∆σ
DY
dM
2
=
α
3π
dx
b
dx
a
σ
(s → τs)
τs
q
[q
a
(x
a
)
¯
q
b
(x
b
) +
¯
q
a
(x
a
)q
b
(x
b
)]
=
α
3π
dτ
dx
a
x
a
σ
(τs)
τs
q
[q
a
(x
a
)
¯
q
b
(x
b
) +
¯
q
a
(x
a
)q
b
(x
b
)] .
(6.22)
We still have to specify the integration boundaries. Obviously it holds that
τs ≥ (M +m
G
)
2
≈ M
2
1 +2
m
G
M
. (6.23)
On the other hand τs can be very large. We write
τs ≤ M
2
N (6.24)
and later on set 1/N ≈ 0. Because q
a
(x
a
),
¯
q
b
(x
b
), and so on are smooth func-
tions of x
a
and x
b
, we can neglect the small shifts in x
a
and x
b
induced by the
difference between τs and M
2
, i.e., we can neglect the τ dependence of the x
a
x
b
:
d∆σ
DY
dM
2
=
α
3π
1
0
dx
a
x
a
q
q
a
(x
a
)
¯
q
b
(x
b
) +
¯
q
a
(x
a
)q
b
(x
b
)
×
M
2
N/s
M
2
(1+2m
G
/M)/s
dτ
8παα
s
e
2
q
9(τs)
2
⎡
⎢
⎢
⎢
⎢
⎣
2
1 +
M
2
+m
2
G
τs
2
1 −
M
2
+m
2
G
τs
×ln
⎛
⎜
⎜
⎜
⎜
⎝
1 −
M
2
+m
2
G
sτ
+
6
1 −
M
2
+m
2
G
sτ
2
−4
M
2
m
2
G
s
2
τ
2
1 −
M
2
+m
2
G
sτ
−
6
1 −
M
2
+m
2
G
sτ
2
−4
M
2
m
2
G
s
2
τ
2
⎞
⎟
⎟
⎟
⎟
⎠
−4
/
0
0
1
1 −
M
2
+m
2
G
sτ
2
−4
M
2
m
2
G
s
2
τ
2
⎤
⎥
⎥
⎥
⎥
⎦
. (6.25)
6.1 The Drell–Yan Process 395
The τ integral is expanded in β = m
2
0
/M
2
. To this end we substitute τ = M
2
/sr:
I =
8παα
s
e
2
q
9M
2
1−2
√
β
1
N
∼0
dr
2
1 +r
2
(1 +β)
2
1 −r(1 +β)
×ln
⎛
⎝
1 −r(1 +β) +
[
1 −r(1 +β)
]
2
−4βr
2
1 −r(1 +β) −
[
1 −r(1 +β)
]
2
−4βr
2
⎞
⎠
−4
[
1 −r(1 +β)
]
2
−4βr
2
. (6.26)
We study the limit β →∞. We shall now take this limit, and in doing so we shall
make sure that we keep all the constant terms as well as those proportional to
logarithms of β.Powersofβ are, however, neglected.
We make the substitution r →r/(1 +β). The factors 1/(1 +β) lead only to
corrections proportional to β, which we neglect. The leading term, which we
have to take care to treat correctly, is that proportional to 2
√
β.
I =
8παα
s
e
2
q
9M
2
1−2
√
β
0
dr
2
1 +r
2
1 −r
ln
2(1−r)
2βr
2
/(1 −r)
−2
1 +r
2
1 −r
ln
2(1−r)
1 −r +
"
(1 −r)
2
−4βr
2
+2
1 +r
2
1 −r
ln
⎛
⎝
2βr
2
(1 −r)
2
1 −
"
1 −4βr
2
/(1 −r)
2
⎞
⎠
−4
(1 −r)
2
−4βr
2
. (6.27)
In the terms on the right-hand side it is tempting to neglect 4βr
2
. This would be
incorrect, however, since it is of the same order as (1 −r)
2
at the upper integra-
tion boundary. To treat this carefully we have split up the logarithmic term into
three parts. The first contains the approximation valid for (1 −r)
2
> 4βr
2
.The
other two terms contain the ratio between the approximate expressions and the
exact ones. We investigate these first:
I
1
=2
1−2
√
β
0
dr
1 +r
2
1 −r
⎡
⎣
ln
⎛
⎝
2βr
2
(1 −r)
2
1 −
"
1 −4βr
2
/(1 −r)
2
⎞
⎠
−ln
2(1−r)
(1 −r) +
"
(1 −r)
2
−4βr
2
⎤
⎦
. (6.28)
396 6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics
Obviously these terms vanish unless r ≈ 1. Thus we set r
2
=1 except in (1 −r).
Next we substitute 1−r = t:
I
1
= 4
1
2
√
β
dt
1
t
⎡
⎣
ln
⎛
⎝
2β
t
2
1 −
"
1 −4β/t
2
⎞
⎠
−ln
2
1 +
"
1 −4β/t
2
⎤
⎦
.
(6.29)
Now we substitute t = 2
√
β/u,dt =−2
√
β du /u
2
=−t du /u:
I
1
=−4
2
√
β
1
du
u
⎡
⎣
ln
⎛
⎝
u
2
2
1 −
√
1 −u
2
⎞
⎠
−ln
2
1 +
√
1 −u
2
⎤
⎦
. (6.30)
In this expression it is now safe to let
√
β go to zero. We are left with finite
integrals only. To do these we further substitute u =
√
1 −z
2
,du =−z dz /u:
I
1
= 4
1
0
dz
ln
1 −z
2
2(1−z)
−ln
2
1 +z
z
1 +z
2
= 2
1
0
dz 2ln
1
2
(1 +z)
1
1 −z
−
1
1 +z
=−2ln
2
1 +z
2
1
0
+4
1
0
dz
1
1 −z
ln
1 +z
2
. (6.31)
We finally substitute z = 2u −1 and use the fact that
−
1
1−x
dt
ln(t)
1 −t
= Li
2
(x), Li
2
1
2
=
π
2
12
−
1
2
ln
2
(2). (6.32)
Here Li
2
(x) =
∞
n=1
x
2
/n
2
, |x| < 1 is the dilogarithm, also called the Struve
function,
I
1
= 2ln
2
(2) +8
1
1/2
du
1
2(1−u)
ln(u)
= 2ln
2
(2) −4
π
2
12
−
1
2
ln
2
(2)
=−
π
2
3
+4ln
2
(2). (6.33)
The last integral in (6.27) is trivial, since one can directly set β = 0:
I
2
=−4
1−2
√
β
0
dr
(1 −r)
2
−4βr
2
→−4
1
0
dr (1 −r) =−2 . (6.34)
6.1 The Drell–Yan Process 397
The remaining integral leads to an expression found in good integration tables:
I
3
=−2
1−2
√
β
0
dr
1 +r
2
1 −r
ln(β) +2ln(r) −2ln(1 −r)
=−2ln(β)
1−2
√
β
0
dr
(r −1)
2
+2(r −1) +2
1 −r
−4
1−2
√
β
0
dr
1 +r
2
1 −r
ln(r) +4
1−2
√
β
0
dr
1 +r
2
1 −r
ln(1−r)
=−2ln(β)
1
2
−2 −2ln
2
"
β
+O
"
β
−4
−
π
2
3
+
5
4
+4
−ln
2
2
"
β
+
7
4
=2 −4ln
2
2
"
β
+
4
3
π
2
+3ln(β) +4ln(β) ln
2
"
β
=
4
3
π
2
+2 +3ln(β) +4ln
2
"
β
ln(β) −ln
2
"
β
=
4
3
π
2
+2 +3ln(β) +4
ln
"
β
+ln(2)
ln
"
β
−ln(2)
=
4
3
π
2
+2 +3ln(β) +4
ln
2
"
β
−ln
2
(2)
=ln
2
(β) +3ln(β) +
4
3
π
2
+2 −4ln
2
(2). (6.35)
Finally we put the results (6.33), (6.34), and (6.35) together to get
I =
ln
2
(β) +3ln(β) +π
2
8παα
s
e
2
q
9M
, (6.36)
or
d∆σ
DY
dM
2
=
α
3π
1
0
dx
a
x
a
q
q
a
(x
a
)
¯
q
b
(x
b
) +
¯
q
a
(x
a
)q
b
(x
b
)
×
8παα
s
e
2
q
9M
2
ln
2
(β) +3ln(β) +π
2
. (6.37)
We hope that this explicit example shows how the logarithms associated with
the infrared divergences can be isolated and where one has to be careful to avoid
errors. We shall next discuss how the Block–Nordsieck theorem works for this
specific example, i.e., how the logarithmic terms in (6.37) are cancelled by those
of the radiative corrections. However, we shall give only an overview of the
explicit calculation of the radiative corrections, since it contains nothing new.
398 6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics
Fig. 6.5a,b. First-order ra-
diative vertex corrections
According to Fig. 6.4 the graphs we have to calculate are those contributing
to the vertex correction to first order (see Fig. 6.5). If we introduce a momentum
cutoff Λ, after the introduction of Feynman parameters the two relevant integrals
are, for the quark self-energy,
Σ( p) =−g
2
s
Λ
2
m
2
g
ds
1
0
dx
d
4
k
(2π)
4
2(1−x)
−2(1 −x)
k
2
+ p
2
x(1 −x) −s(1 −x)
3
=...=−
α
s
4π
ln
Λ
2
m
2
g
(6.38)
and, for the vertex correction,
α →α
⎛
⎜
⎜
⎝
1 +
2α
s
3π
Λ
2
m
2
g
ds
1
0
dy
1
0
dx
)
x(1 −x)q
2
2 +2x
2
y(1−y) −2x
−y(1−y)x
2
q
2
+s(1 −x)
2
+
2x(1−x)
−y(1−y)x
2
q
2
+s(1 −x)
*
⎞
⎟
⎟
⎠
=α
)
1 +
2α
s
3π
'
−ln
2
m
2
g
−q
2
−3ln
m
2
g
−q
2
−
7
2
−
2π
2
3
(*
. (6.39)
In contrast to deep inelastic scattering the four-momentum squared
q
2
= M
2
is now positive:
ln
m
2
g
−q
2
= Re
'
ln
−
m
2
g
M
2
(
= Re
ln
(
β
)
−iπ
= ln(β) (6.40)
ln
2
m
2
g
−q
2
= Re
'
ln
2
−
m
2
g
M
2
(
= Re
ln
2
(
β
)
−2iπ ln(β) −π
2
= ln
2
(β) −π
2
, (6.41)