98
Progress of Theoretical Physics, Vol. 75, No.1, January 1986
On the Factorization of Collinear and Infrared
Singularities in QCD Corrections to the Mikaelian Zero
]. H. REID
Physics Department, Oklahoma State University, Stillwater, OK 74078
(Received May 31, 1985)
The method of Smilga and Vysotsky is used in the Feynman gauge to extract IR/ CoIl. singularities
from one loop corrections to a general process in both scalar QED and massless QCD. These results are
used to demonstrate that a Mikaelian zero factor multiplies the IR/ CoIl. singular terms in Ora;') correc·
tions to W- .... udr and related processes containing an amplitude zero.
The subject of Mikaelianl) or amplitude zeros is of considerable experimental interest.
For example, such a zero appears in the tree level cross section for the process du~ W-y
and may eventually be detected in Pp colliding beam experiments. Mikaelian zeros were
first discovered in the process du~ W-yIl and investigated in general Yang-Mills theories
by Goebel, Halzen and Leveille 2 ) and Dongpei,3) and in a scalar model by Samue1. 4) A
general understanding of the phenomenon of zeros was given by Brodsky, Brown and
Kowalski S) who showed that it was a purely classical effect occuring in any gauge theory
with particles of spin 0, 1/2 or 1.
In general, quantum corrections will destroy the zeros, so it becomes necessary to
calculate higher order corrections. One loop corrections were calculated in an unrealistic
model with scalar quarks and a scalar W- boson by Laursen, Samuel, Sen and Tupper. 6 )
A further one loop calculation was carried out by Laursen, Samuel and Sen7 ) in a model
with scalar quarks and a vector W- boson. In both cases finite although small corrections were found which spoiled the Mikaelian zero. However it was noted that the
infrared and collinear singularities contained the Mikaelian zero as a factor and hence
these singularities were absent in the classical null zone. It was conjectured that such a
factorization would occur also in the realistic case of spin 1/2 quarks (QCD).
It is the purpose of this short note to give a general understanding of the factorization
of infrared and collinear singularities found by Laursen, Samuel and Sen and to demonstrate that the same factorization occurs in QCD.
It should be emphasized that the factorization observed in Ref. 7) is merely a very
special case of powerful and general results on factorization of IR/ ColI. singularities in
QED and QCD that have been known now for some time.S)-IS) It is known that in QED and
QCD leading and subleading IR/ ColI. singularities not only factorize "universally", i.e., in
a process independent way, but also exponentiate so that singular corrections to arbitrary
order in as are obtained by taking the exponential of the one loop result.
In our discussion we shall adopt the methods and notation of Smilga and Vysotsky.14)
Following their example we do not use dimensional regularization but instead energy (A)
and angular (7J) cutoffs so that to one loop order there are three kinds of singular terms,
namely In it In 7J, dilogarithm or leading; In it, pure IR, subleading; and In 7J, pure collinear,
On the Factorization 0/ Collinear and In/rared Singularities
99
subleading. The advantage of this approach is that the two kinds of subleading singularities are distinguished, whereas in dimensional regularization they are confused, both
being proportional to lie.
Assuming the necessary conditions for the KLN theorem, namely that a complete
summation over all degenerate initial and final states is performed, Smilga and Vysotskyl4)
have explicitly verified this theorem in both Feynman and temporal gauges for massless
QED and QCD. In such cases it is only necessary to consider singularities in the O(as)
corrected amplitude TI OOP since [TIoOP To * + TIoO P* TO]IR/COll = - [Tbrems Tbrems *]IR/COll where To
is the tree amplitude and the right-hand side contains those terms related by cutting rules l6 )
to the products of loop and tree amplitudes. A decay from an initial state not containing
quarks or gluons such as W- --> adrg would satisfy these conditions. But the crossing
related production processes such as ad--> W-rg, rd--> W-Ug do not satisfy the conditions, since gluons collinear to quarks in the initial state are not summed over. Therefore
in these cases pure collinear singularities do not cancel between virtual and real contributions, and the bremsstrahlung collinear singularities must be calculated independently.
This problem is encountered, for example, in O(as) corrections to the Drell-Yan
process l7 ) and is sidestepped by absorbing the uncancelled pure collinear singularities into
the parton distribution functions. These comments do not apply to the infrared singularities (dilog and pure infrared) which cancel exactly by virtue of the one loop BlochN ordsieck theorem. With these cautionary remarks in mind, we proceed to apply the
methods of Smilga and Vysotsky.14)
q,-g
It is well known that which loop diaor
grams contain which kinds of IR/ ColI.
singularities is a highly gauge dependent
question, and that the temporal gauge has
the very remarkable property 8).9).14) of
containing both dilog and pure collinear
or
singularities only in self-energy insertions
Fig. 1. General diagram contributing to dilog and
on external lines. This leads easily to an
pure IR singularities in the Feynman gauge.
understanding of process independent
factorization. However we wish to work in the more familiar and amenable Feynman
gauge in which factorization is a little less trivial. We calculate in both QCD and in the
scalar model used in Ref. 7).
We wish to extract dilog and pure IR singularities to which only diagrams of the
general type of Fig. 1 contribute. The respective amplitudes in the two models are
(1)
and
(2)
To extract these singularities, close the contour in the upper go plane, take the residue
of the gluon pole and ignore terms of O(g, g2) .14) One obtains an identical contribution
in both models
J H. Reid
100
(3)
where;\ is the IR cutoff, 8ap the angle between a and (3, CF=4/3 for QCD. Performing
the energy and angular integrations one finds
T,lOOP
dilog/IR -
-
aSCF
T.0 ,
7f 1n;\(1n77 -1 n (1-cos8!2))
2
(4)
where as = g/ / 47f and 77 is the angular cutoff. This exhibits process independent factorization and also model independence for dilog and pure IR singularities.
It remains to consider pure collinear singularities. To obtain factorization in the
Feynman gauge we must take a large collection of diagrams. Pure collinear (but not
dilog or pure IR) singularities are present in self-energy insertions on external lines.
They are also present in all loop diagrams in which at least one end of the gluon is
attached to an external quark.
Consider a tree graph of the kind in Fig. 2(a) with a single quark line connected to
n.e.m.! weak currents. Then all the diagrams of the type in Fig. 2(b) must be taken when
extracting the singularity due to the (q! i g) pair. In the scalar model the further set of
diagrams depicted in Fig. 2(c) must also be taken. We shall consider the external
self-energy insertion diagrams separately from those of Fig. 2.
We calculate in QCD first. Our notation is as follows:
(5)
(a)
q,
q,-g
(b)
(e)
Fig. 2. Diagrams contributing to dilog and pure collinear singularities in the Feynman gauge.
On the Factorization of Collinear and Infrared Singularities
101
Ir(f3) = Vr(f3)P r- I ((3) Vr((3)···PI((3) VI((3) ,
(6)
jr,r'= Vr(O)Pr-I(O)··· Vr'+I(O)Pr'(O) ,
(7)
where Vr=Yan P r ((3) =1/ Cf/I- z;,~~IPa-(3j!h) and g=(3ql, since ql and g are both massless
and collinear when they contribute to the collinear singularities. Then using the methods
of Smilga and Vysotsky,14) the contribution to the collinear singularities from the diagrams of Fig. 2(b) is
(8)
with
but
(10)
Using (6) and (7) it follows that
(11)
Summing Q r n over r=l···n one gets
(12)
But the tree amplitude for Fig. 2(a) is
(13)
Hence
1
= aSCF
In7Jl0 UJ-1'(3 (1-(3)
T COll
QltU
27f
(3
'T'
1.0.
(14)
N ow the total contribution from the collinear pair (q 2 i g) is exactly the same, stemming
from diagrams analogous to those in Fig. 2(b) but with the gluon attached always to the
q2 external line. Inserting an IR cutoff A at the lower end of the (3 integration one gets
T coll = - as;F In 7J(InA + 1) To,
(15)
where this includes both (ql i g) and (q2 i g) contributions but not yet the self-energy insertion terms. When these self-energy insertion contributions are added, one gets the total result
n~?f= - aSCF In7J(InA +3/4) To .
7f
(16)
The dilogarithm term agrees with that obtained previously in Eq. (4), as it must.
Combining Eqs. (4) and (16) we obtain finally the process independent factorization for
dilog, pure IR and pure ColI. singularities in QCD to one loop
,
I
102
]. H.
TNl.l'fOll = -
Reid
Q'S:F (InA ln7J -InA In( 1 -C~Se12 ) +3/4 ln7J) To .
(17)
This agrees with the result obtained by Smilga and Vysotsky who worked instead in the
temporal gauge. 14 ) This result applies only to the case in which the IR/ CoIl. gluon is not
attached to an external gluon line. The more general result where this condition is relaxed
is given in Ref. 14) but is not needed for our subsequent discussion here.
We can carry through a closely analogous argument for the scalar model but now we
must also take up the contributions from Fig. 2(c) arising from the seagull interactions.
We retain the definitions of Eqs. (5) ~ (7) but now
1
Then we get
(18)
where
(19)
The quantity inside the square bracket can be rearranged as
(20)
Substituting this in Eq. (2) we obtain
1 [ ----=--(3
/1.r ]
Q r n -- --(3
f.1.r
n,TI r -
f.1.r-1]
-(3 n,r-1 I r-1 ] .
f.1.r-1
(21)
Summing over all r values, we get
(22)
But since To = + ien]n,1 we can again express the collinear singular piece in the factorized
form
'
T~~~g=
Q'4;F ln7J 11 d(3 (2/3(3) To.
(23)
Performing the (3 integration and adding in the (q2 i g) contribution yields
Tcoll = - Q'Z;F In 7J(2lnA + 1) To .
(24)
Performing the self-energy calculation gives the results /Z -1 = - Q'sln7J/ 4Jr where Z is
the quark wavefunction renormalization constant. The total contribution including the
self-energy insertions is
On the Factorization of Collinear and Infrared Singularities
103
(25)
When this result is combined with Eq. (4) one finds
'T'lOOP
_
.1 IR/CoU-
ctsCF
(1 n/l11 n77 -1 n/l11 n (I-cos812)
+1 n77 )
J[
2
'T'
.1 0 •
(26)
Again we have process independent factorization but remark that model independence is
broken only by the pure collinear term.
As noted in the Introduction, Laursen, Samuel and Sen7) have verified factorization by
an explicit one loop calculation in massless scalar QED. In order to compare our general
result Eq. (26) with theirs one must equate InA to 1/ 2c and ln77 to 1/c. There is agreement
for the soft singularities but their result for the pure collinear singularity is in error. For
a more detailed discussion of this point see Ref. 18). It is interesting that the soft
singularities are essentially the same in scalar and QCD but that the pure collinear
singularities are not (compare Eq. (17) with Eq. (26)). in fact the coefficient of the pure
collinear singularity is related to the anomalous dimension of the appropriate field and
these numbers must in general depend in an essential way on both the field and the model
within which the calculation is performed.
Let us now apply these general results to a discussion of the process W- -+ udy to
which the three diagrams of Fig. 3 contribute at the tree level. It is well known 2l ,3l,5l that
the tree amplitude To has the Mikaelian or amplitude zero factor M = ( Qik· q2 - Qjk· (ll)
where Qi,j are quark charges and Ql,2 and k are quark and photon four momenta. Since
the IR/ ColI. singular terms in the one loop corrections to the diagrams in Fig. 3 are process
independent and multiply the tree amplitude To, then this zero factor must multiply the IR
/ ColI. singular terms in both the scalar model, as observed by Laursen, Samuel and Sen,7)
and in QCD. If one considers the partial decay rate dr/dxldx2 for W--+ udy, it is clear
ij
(b)
Fig. 3.
----- ___ w(e)
Tree diagrams for the process W - -+ duro
104
]. H. Reid
that the Mikaelian zero factor occurs quadratically multiplying both real and virtual IR
/ ColI. singularities. In fact in QCD we have from Eq. (17) and the explicit term by term
verification of the KLN theorem by Smilga and Vysotsky14) that
dr )IOOP
( dx1dxz
lR/eoll =
(dr
-
dx1dxz
)bremS
lR/eoll
(27)
where the tree level decay rate is proportional to the square of the Mikaelian zero factor.
For the crossing related processes ud~ W-y and rd~ W-u one has
aSCF (1 nil11n77 - 1nil11n 1-cos812
+3/41 n77 ) (}o,
2
loop (}IR/eOll- - - 7 [ -
(28)
where (}o is the tree level cross section.
But in view of the earlier cautionary remarks on the inapplicability of the KLN
theorem, one cannot in these latter cases make such a simple Ansatz for (}l%~t:m.
We have thus explained the factorization result obtained in Ref. 7) and have extended
this to the case of the bremsstrahlung corrections which were not considered in Ref. 7).
The general arguments given above show that the same factorization occurs in the
realistic case of QCD. In view of the well established exponentiation of leading and
subleading IR/ ColI. singularities in QCD 11 ),13),15) it follows that the Mikaelian zero factor
must multiply leading and subleading singular terms to all order in as, not just to O(as).
Acknowledgements
I thank Gary Tupper for numerous discussions on this work. r am very grateful to
Achin Sen for many discussions on the work and in particular for his collaboration in the
case of scalar QED. This work was supported by the U. S. Department of Energy under
Contract No. EY-76-S-05-5074.
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