PHYSICAL REVIEW D VOLUME 42, NUMBER 6 15 SEPTEMBER 1990

Light-cone gauge and the (causal) principal-value prescription

B. M. Pimentel and A. T. Suzuki
Instituto de Fisica Teorica, Uni Uersidade Estadual Paulista, Rua Pamplona, 145, 01405-Sao Paulo, SP, Brasil

(Received 30 March 1990)

The principal-value (PV) prescription constrained with causality is used to treat the unphysical
pole (k.n) in the basic one-loop light-cone integral. It is shown that the so-modified prescription
avoids the emergence of double-pole singularities, contrary to what has been previously obtained in
the PV scheme without causality. However, there remains an overall factor —,

' which does not agree
with the results derived in the Mandelstam-Leibbrandt (ML) prescription. This overall factor is a
remnant coming from the definition of the prescription proper.

I. INTRODUCTION

The first time modern theoretical physicists were intro-
duced to the light-cone formulation for quantum field
theories goes as far back as 1949, when Dirac' discussed
the front form of relativistic dynamics. For the next two
decades or so since its debut, interest in this subject lay
dormant, with only few sporadic appearances in the
literature. (For a very nice historical survey we refer the
reader to a well-written review article on noncovariant
gauges by Leibbrandt. ) These focused not on the gauge
per se but rather on investigating the structure of relativ-
istic theories in the light-cone frame. It was only during
the 1970's that a handful of researchers started venturing
to take a fresh look at the gauge proper.

In the beginning of the last decade, the light-cone
gauge was shown to be especially appropriate for the
computation of quantum effects contributing to the
leading-logarithm approximation in deep-inelastic pro-
cesses. However, as much as this feature was attractive
and desirable, hand in hand with it there emerged an om-
inous aspect: Feynman amplitudes at the one-loop level
exhibited double-pole singularities.

Tracing back the origin of such a pathological behav-
ior, one notes that this has frequently been ascribed to
the principal-value (PV) prescription employed to treat
the unphysical poles (k n) ' of the gauge-boson propa-
gator, though direct application of dimensional regulari-
zation to one-loop light-cone integrals does also yield
double poles.

Therefore, alternative prescriptions to handle the
(k n ) singularities in the light-cone gauge were
searched for. Mandelstam and Leibbrandt authored
two seemingly different prescriptions independently;
later, Lee and Milgram' showed that they were, in fact,
equivalent. These new prescriptions were devised in such
a way so as to ensure that the location of the (k n).
poles in the k -complex plane would not hinder Wick ro-
tation nor spoil power-counting thereby; qualities which,
as is often claimed, were missing in the PV prescription.

Moreover, Basseto, Dalbosco, Lazzizzera, and Solda-
ti" demonstrated that the Leibbrandt prescription could
be recovered via canonical quantization, and more re-
cently, within the framework of functional path integrals.

II. PRELIMINARIES

At this point we introduce our notation and conven-
tion. Throughout this article, the signature for the Min-
kowski (flat) space-time metric is (+, —,—,—) with x
being the time component of a general four-vector x".
This is written in light-cone components as

x"=(x+,x,x'), i =1,2,
where

(2.1)

and

X +Xp

X v'2
Xp+X3 —X—+ (2.2)

Slavnov and Frolov' reproduced a similar result.
Successful one- and two-loop calculations employing

either of those new prescriptions have already been pub-
lished, ' enhancing their credibility as well as their suita-
bility to handle the troublesome (k n )

' factors in the
light-cone gauge.

So, with all this, the role of the PV prescription in this
particular gauge waned, while the prescription itself was
left to sink into oblivion. To add insult to injury, it went
carrying an unwelcome reputation of being the prime
culprit for all unwieldy pathologies manifested at its im-
plementation.

Here we would like to take a second look at the ill-
favored prescription and address two questions concern-
ing it: (a) %Vhy is it that the PV prescription did not work
in the early one-loop calculations in the light-cone gauge?
(b) Is it possible to spot the cause or causes either of
mathematical or physical nature that made the im-
plementation of such a prescription result in double-pole
singularities even at the one-loop level?

The answer to these questions is found in the following
sections, where it is clearly shown that the deficiency is
not only in the prescription per se, but also in its mistak-
en implementation, which used to overlook basic first
principles. In other words we will understand why those
unwieldy higher-order poles that plagued earlier calcula-
tions of quantum effects in this gauge are physically unac-
ceptable.

1990 The American Physical Society



2116 B. M. PIMENTEL AND A. T. SUZUKI 42

x'=—(x)'=(x', x ) . (2.3} effective propagator in the light-cone gauge becomes

It is convenient to define the lightlike vectors n„and n„
with Cartesian components

k„n +k n„
k +is ~ n

E)0, (2.16)

and

n„= —(1,0,0, 1)
1

v'2

n„= (1,0, 0, —1) .
1

These, therefore, satisfy

(2.4)

(2.5)

where the unphysical singularity ( k n )
' = ( k +

)
' has

yet to be defined.
In what follows, our chief concern will be the evalua-

tion of the basic one-loop light-cone integral (dk—:d k)

( ) dk
(2~)2" (k'+ie)[(k p)—+is]k+

and

n =n =0 (2.6)
where p is the mass scale produced by the renormaliza-
tion procedure and p is the external momentum. For
convenience, we leave the factor p, (2np) out and
focus our attention on the integral

n "n„=1, (2.7)

and

x =x I' n P '

The scalar product between two vectors becomes

(2.9}

while the "plus" and minus" components are con-
veniently reproduced now by

(2.8}

dk

(k +i e)[(k p) +—i e jk+
(2.17)

Until 1982, the only known way to treat the pole at
k+ =0 was to do it in the sense of the Cauchy principal
value, i.e.,

1 1 . 1 1 1P = lim——— +, g) 0 .k+ k+ 0--o 2 k++ig k+ ig—
(2.18)

x"y„(=x y)=x+y +x y+ —x'y'

=x+y +x y+ —x y .

In particular,

X =2X X X

The volume element is written

dx —dxdxdx

(2.10)

(2.11}

(2.12)

Using (2.18), Capper, Dulwich, and Litvak' arrived at
the following result for (2.17):

(p ) I (2—co)r(co —2)I (co —1)

p r(2' —3)

where the double pole for co~2 is clearly seen. This re-
sult is physically unacceptable.

On the other hand, by using the Mandelstam prescrip-
tion, that is, making the substitution

and dimensional regularization is to be implemented by
continuing the p=1, 2 components to D —2 dimensions,
i.e.,

1 . 1lim, 8)0,k+ g-o k++gk
the same authors quoted above showed that

(2.20)

d X=d X dX dx (2.13) (p2)QP 2

I=i( —m) I (2 —m)
We also employ, according to convenience, the definition

D —2=2(co —1) . (2.14)

For a vector gauge field A„, the light-cone gauge is
defined by

n"A„(=n A =A+)=0, n =0, (2.15)
(2.21)

where n" is an arbitrary lightlike constant vector.
Formally inverting the "regularized" quadratic form in

the Lagrangian density, the propagator in momentum
space reads

which, for co~2 yields

I= —S(A, ) +O(2 —co),p+ 6
(2.22)

k„n,, +k,n„G„(k)= g„„—k'+ie "~ k n

(ak +n )k„k+
(k n)

(2.23)

with

p

p

where S(k) is the Spence (or "dilogarithm") integral:

S(A)= —f dx x 'ln(1 —Ax),
0

Letting the gauge parameter a vanish and using (2.6), the
(2.24)



42 LIGHT-CONE GAUGE AND THE (CAUSAL) PRINCIPAL-. . . 2117

Note that (2.22) is finite, so that it is in harmony with the
naive power-counting assessment done on (2.17).

From the work of Lee and Milgram, ' we know that
the Leibbrandt prescription, namely,

which in the PV sense is understood as [see (2.18)]

Kic =—lim [K+(g)+K (g)], g) 0,
2 g 0

(3.7)

1 . klim, &0,k+ g-o k+k +if
(2.25)

with

K +(g) f 2ixk (k +Q )

k++ig (3.8)

when implemented in (2.17), does also yield the same re-
sult as that obtained through the Mandelstam prescrip-
tion.

The status quo for the main properties of both
prescriptions (2.20) and (2.25) can be summarized as fol-
lows (a) Poles at k+=0 do not hinder Wick rotation;
(b) the degree of divergence of momentum integrals can
be assessed by naive power counting; (c) evaluated
momentum integrals exhibit divergent parts that are local
functions of the external momentum; (d) there are no
higher-order divergences at the one-loop level; (e) Ward
and Becchi-Rouet-Stora (BRS) identities are satisfied; (f}
integrals are internally consistent.

and

f + dk
e 2ixk +(k + Q ) (3.9)

We note from (3.1) and (3.3a} that x is positive (x ) 0)
while the factor (k +Q ) can either be positive or nega-
tive. Therefore, in applying the residue theorem to evalu-
ate (3.8) and (3.9), we have the following cases to consid-
er.

(i) (k +Q ) &0 (or k ( —Q ). In this case, con-
tour C in the complex k+ plane must close in the lower
half-plane to ensure convergence at infinity. This con-
tour encloses the simple pole k+ = ig o—f (3.8), whose
residue is

III. THE PV PRESCRIPTION REVISITED
Rest-( —i g) =e "'" (3.10)

Employing the standard trick of exponentiating propa-
gators, (2.17) becomes

I= —f da f dPe'~~ M„c(a,P), (3.1)

where M„c(a,p) stands for the light-cone momentum in-
tegral

so that

Kc+(g) = —217je (" +Q )4 k ( —g (3.11)

while the pole at k+ =i g of (3.9) is outside C and its resi-
due is zero. So

(a p} ei((a+P)k 2Pp k]—dk
cx, = +e

k

For convenience, we define

x =(a+p),
pp—:—xg,
pp

—=x8.

(3.2)

(3.3a)

(3.3b)

(3.3c)

Kc (g) =0 . (3.12}

K,', (g) =0, (3.13)

while the pole at k + =i g of (3.9) contributes a residue of

(ii) (k=+Q ))0 (or k ) —
Q ). For this case, con-

tour C' must close in the upper half of the complex k+
plane. Now the pole at k+= ig of (3.8—) is outside C'
and the corresponding residue vanishes. Therefore,

With these definitions, and using (2.10)—(2.12), the
momentum integral (3.2) becomes

( p) ix(k +2k Q)G, —
+ e

Resp, (jg)=e " " +Q

so that

K; (g)=2~)e . '"'" +Q -)c, k-& —g-.

(3.14)

(3.15)

LENT

JLcixQ

t d D —2k e
—ix(k +2k Q)JLC

(3.4)

Substituting (3.11) and (3.15) into (3.7) and inserting
the result into (3.5) we obtain (the limit (~0 being un-
derstood)

2xqQ f dk
—e2(x(Q+ —ij)k

77

where the standard Gaussian integral in (D —2)-
dimensional (recall D =2') Euclidian space has been per-
formed and JLc is the integral over k+ and k

2ixQ k

—2x(Q dk
— 2ix(Q +i/)k

—
Q

(3.16)

x f + dk+(k+) —1 2ixk+(k +Q )

Let us consider the integral over k+

dk+(k+) —le2ixk (k +Q )
LC

(3.5)

(3.6)

Note that the upper (lower) limit of integration over
k in the first (second) integral of (3.16) is a direct conse-
quence of cases (i) and (ii) considered above. Straightfor-
ward integration over k (with /=0) and further integra-
tions over x and y according to the standard change of
variables



2118 S. M. PIMENTEI. AND A. T. SUZUKI

a =xy,

P=x (1—y)

(3.17a)

(3.17b)

k +is=2k+k —k +i@=0, e&0 .

This shows that there is a simple pole at

(3.18)

k+=
2k

lE e&0.
k

(3.19}

Whatever the value of k (in particular k =0), the
sign of the imaginary part of (3.19) must agree with the
sign of the poles in prescription (2.18), otherwise causali-

ty of the whole propagator would be in jeopardy, i.e., we
must haue, in Eq. (3.8),

[with (3.3a) satisfied], will lead us to the physically unac-

ceptable result (2.19).
Here is our crucial argument: We cannot and must not

overlook the pole generated by the covariant piece of the
propagator, namely, k +is=0, E &0, which, using (2.11)
reads

—2ixQ +
Q

[e(—Q )+e(Q )]
2x

( &
—2ixQ Q

2xQ+
(3.24}

which inserted into (3.4} results in

yu
—

1
(

—ixg ixg2 2

LC
X Q)~ + (3.25)

IV. COMMENTS AND CONCI. USIONS

This result is exactly half of that obtained through us-

ing the Mandelstam-Leibbrandt (ML) prescription. We
are thankful to W. Kummer for pointing this out to us. '

Finally, changing variables according to (3.17a) and
(3.17b) and integrating over x (0 & x & ao ) and y
(0&y &1), we arrive at a result which is half the one
quoted in (2.21). This overall factor —,') is a remnant from
the definition of PV prescription given in (2.18).

g&0, e&0,
2k

while in (3.9),

(3.20}

g&0, e&0.
2k

(3.21)

Since g and e are both strictly positive, (3.20) obliges us

to take only k &0; conversely, (3.21) entails us to take
only k &0. As they stand in (3.16), both integrals over
k inadvertently allow for forbidden regions of k

They include those regions of k which forces a change
in the sign of e in (3.20) and in (3.21}, thus mixing
positive-frequency (or -energy) radiation propagating for-

ward in time, with negative-frequency radiation. This
violates causality, a basic physical principle.

Therefore, physically acceptable regions of integration
over k (those which preserve causality) are confined to
the following intervals. In (3.11),

0&k & —
Q (3.22a)

while in (3.15),

—Q &k &0,

so that

(3.22b)

Jtc= —,
' lim ( —2ivr)

(~0

e( Q
—

) 2xgg f dk
— 2ix(g —ij)k

0

e( Q
—

)
—2xgg f dk

— 2ix(g +i()k
—

Q

(3.23)

where the e(+ Q ) are the usual step functions.
Carrying out the k integration and taking the limit

(~0 we obtain

By looking at the pole in the k+ complex plane gen-
erated by the covariant piece of the gauge-boson propa-
gator and noting that the sign of its imaginary part must
be concordant with the signs of the corresponding poles
at k+ =0 in the sense of Cauchy PV, we observed that (i)
the constraint defines a finite range of integration over
the k integrals, ' and (ii) the basis for this restriction
has to do with ensuing that positive-energy quanta do not
become mixed up with quanta of negative energy propa-
gating forward in time. The point is that the propagator
as a whole must be thus causal.

One should note that the poles of prescription (2.18) in
the k -complex plane, when analyzed under observation
(ii) above, are located in the second and fourth quadrants.
This ensures legitimacy to Wick rotation and thus for
power counting.

We want to emphasize that careful implementation of
the PV prescription in the light-cone gauge reproduces
(apart from an overall factor —,') the well-behaved proper-
ties ensuing from treating the (k n )

' poles via either the
Mandelstam or the Leibbrandt prescriptions. It should
be stressed that earlier attempts at applying the PV tech-
nique in the light-cone gauge computations failed because
they overlooked the fact that there is a strong constraint
over the range of the k component of momentum.
Overlooking this amounts to violating causality, and so,
there is little wonder unwieldy double poles in one-loop
calculations appeared. However, the PV prescription
per se entails an overall factor —,

' which renders it unsuit-

able for handling light-cone gauge singularities.
Let us now double back and retrace the crucial argu-

ments that entered our calculation. We started off from
the standard definition for the PV prescription; then
brought in the requirement entailed by causality of the
whole propagator (in passing we point out the fact that it
is only when this step is correctly taken into account that
the evaluation of the k integral does converge for
x ~ ao }, and then arrived at a result which equals half of
that obtained through using the M-L prescription. This



42 LIGHT-CONE GAUGE AND THE (CAUSAL) PRINCIPAL-. . . 2119

strongly suggests that causality plays a key role in the
whole calculation and that causality itself must be the
guiding factor to treat both the covariant-type pole
(q )

' and the so-called "spurious" or "unphysical" poles
of the type (q n. ) ", n =1,2, . . . .

A preliminary calculation carried out in the pure axial
gauge does also corroborate this conclusion, the results of
which will be published shortly elsewhere.

Of course, we have only evaluated the basic one-loop
integral in the light-cone gauge, (2.17). However, in corn-
puting, e.g. , the one-loop gluon self-energy, one en-
counters tadpolelike integrals as well as integrals with a
tensorial structure in the integrand. These can be evalu-

ated following the same steps as the ones used for
evaluating (2.17), or by employing techniques already
developed for the Mandelstam and/or Leibbrandt
prescriptions to compute those kinds of integral in terms
of the basic one (2.17).

ACKNOWLEDGMENTS

A.T.S. wishes to thank the kind hospitality of Instituto
the Fisica Teorica. B.M.P. was partially supported by
and A.T.S. was supported by Conselho Nacional de
Desenvolvimento Cienti'fico e Tecnologico, Brazil.

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