5757

February, 1992
(T)

Field Theory Without Feynman Diagrams:

One-Loop Effective Actions
arXiv:hep-ph/9205205v1 5 May 1992

⋆
Matthew J. Strassler

Stanford Linear Accelerator Center

Stanford University, Stanford, California 94309

In memory of Brian J. Warr.

ABSTRACT

In this paper the connection between standard perturbation theory techniques

and the new Bern-Kosower calculational rules for gauge theory is clarified. For one-
loop effective actions of scalars, Dirac spinors, and vector bosons in a background
gauge field, Bern-Kosower-type rules are derived without the use of either string
theory or Feynman diagrams. The effective action is written as a one-dimensional
path integral, which can be calculated to any order in the gauge coupling; evalu-
ation leads to Feynman parameter integrals directly, bypassing the usual algebra
required from Feynman diagrams, and leading to compact and organized expres-
sions. This formalism is valid off-shell, is explicitly gauge invariant, and can be
extended to a number of other field theories.

Submitted to Nuclear Physics B

⋆ Work supported in part by an NSF Graduate Fellowship and by the Department of Energy,
contract DE-AC03-76SF00515.
 1. Introduction

In the past year significant advances have been made in techniques for calcu-
lating one-loop scattering amplitudes in gauge theories. Following on the successes
of several authors at applying string theory and various technical innovations to
[1,2,3,4]
tree-level gauge theory calculations , Z. Bern and D. A. Kosower have derived
new rules from string theory for one-loop gauge theory scattering amplitudes. In
reference 6, they present the derivation of the rules and apply them to the com-
putation of two-to-two gluon scattering at one loop, which previously was difficult
[5]
enough to challenge the most expert calculators. In reference 7, they present their
rules in a compact form and work a simple example. Although obtained from string
theory, the Bern-Kosower rules do not refer to string theory in any way, but as they
also bear little resemblance to Feynman rules, it is of interest to derive them di-
[10]
rectly from field theory. Bern and Dunbar showed how to map the Bern-Kosower
rules onto Feynman diagrams and demonstrated that the background field method
plays an important role; in this paper I take the opposite route, deriving Bern-
Kosower rules from the field theory path integral with the use of the background
field method.

The main result of this paper is that calculational rules similar to those of Bern
and Kosower can be derived from first-quantized field theory. Unlike the “connect-
the-dots” approach of Feynman diagrams, first-quantized field theory (particle the-
ory) views a particle in a loop as a single entity, acted on by operators representing
the effects of external fields. We are all well-accustomed to this approach in atomic
physics, where electromagnetic fields are treated as operators acting on quantum
mechanical electrons, but to my knowledge it rarely been used for calculations with
relativistic particles. (Feynman presented formulas similar to those discussed in
[11]
this paper but did not use them to develop perturbation theory. ) In any case,
it will not surprise those familiar with first-quantized strings that just as string
theory amplitudes are evaluated as two-dimensional path integrals, so particle the-
ory amplitudes can be calculated using one-dimensional path integrals — the path

2
integrals of quantum mechanics.

In this paper I address the issue of the effective action at one loop. In section
2, I construct the one-loop effective action of a scalar particle in a background
gauge field, and derive rules almost identical to those of Bern and Kosower. In
sections 3 and 4 I generalize this approach to Dirac spinors and vector bosons.
Section 5 contains a study of the integration-by-parts procedure involved in the
Bern-Kosower rules, and an illustration of its relation to manifest gauge invariance.
After a short comment (section 6) on an alternative organization of color traces in
this formalism, I conclude in section 7 with some extensions of this approach to
other field theories.

2. The Effective Action of a Scalar in a Background Field

In this section, I will show that the one-loop effective action of a particle in
a background field, when written as a one-dimensional path integral, is calculable
at any order in the coupling constant g. A particle in a loop can be described
as a simple quantum mechanical system existing for a finite, periodic time, or,
alternatively, as a one-dimensional field theory on a compact space; external fields
act as operators on the particle Hilbert space, just as in usual quantum mechanics.
At any order in the external field, the effective action is a correlation function
of these operators in a free and therefore soluble theory, and can be expressed
in a compact form. By writing the effective action as a one- rather than a four-
dimensional path integral I employ quantum mechanics instead of quantum field
theory; as string theory in its present form is a first-quantized theory, it is not
especially surprising that the expressions found from string theory by Bern and
Kosower are of the same form as those found in this paper.

Working initially in Euclidean spacetime, let us first consider the one-loop

vacuum energy of a free scalar field, with Lagrangian

L = −(∂φ† ) · (∂φ) − m2 φ† φ . (2.1)

3
 [6,12]
First represent it in terms of Schwinger proper time τ :
hZ i
d4 xL
R
−
= − log det(−∂ 2 + m2 )
 
log Z = log Dφ e
Z∞ (2.2)
dT d4 p  1
Z
2 2 2 2

= − Tr log(−∂ + m ) = exp − ET (p + m ) .
T (2π)4 2
0

The parameter E (the einbein) is an arbitrary constant. Next convert this result
into a path integral over xµ (τ ):

Z∞ ZT ZT
dT 1
Z
dτ (p(τ )2 + m2 ) ]
 
log Z = Dp Dx exp dτ ip · ẋ exp[− E
T 2
0 0 0
(2.3)
Z∞ ZT
dT 1 2 E
Z
ẋ + m2 ) ,
 
= N Dx exp − dτ (
T 2E 2
0 x(T )=x(0) 0

where the normalization constant N is

Z T
− 12 dτ E p2
R
N = Dp e 0 (2.4)

and satisfies

dD p − 1 E T p 2
Z T
Z
1 2
R
− dτ 2E ẋ
N Dx e 0 = e 2 = [2πET ]−D/2 . (2.5)
(2π)D

The result of (2.3) is a one-dimensional field theory: the particle position xµ (τ )

is a set of four fields living in the one-dimensional space of proper time, called the
worldline. Eq. (2.3) contains the well-known first-order form of the action for a free
[13]
particle , which, unlike the usual Einstein action, is well defined in the massless
limit:
1 2
L= ẋ . (2.6)
2E
(Since a massless particle has no internal clock, τ is not actually proper time in this
case, though I will loosely continue to refer to it as such.) Classically, the action is

4
reparametrization invariant (that is, invariant under τ → τ ′ (τ )) when the einbein,
the square root of the one-dimensional metric, is chosen to transform in the proper
way. On the other hand, the functional integral in (2.3) is not invariant unless one
integrates over the einbein as well. In the present work I will keep E constant and
ignore the reparametrization invariance, since it is not needed for practical results.

Now let us consider the same system (massless, for simplicity) in a classical
background Abelian gauge field Aµ (x):

L = φ† D 2 φ (2.7)

where Dµ = ∂µ − igAµ . The object of interest is the one-loop effective action

generated by (2.7), as a function of Aµ . In analogy to eqs. (2.2)–(2.3),

Γ[A] = − log [det (−D 2 ) ]

Z∞
dT d4 p 1
Z
=+ 4
hp| exp[− ET (p + gA(x))2 ] |pi
T (2π) 2
0 (2.8)
∞ ZT
dT 1 2
Z Z h i
= N Dx exp − dτ ( ẋ + igA[x(τ )] · ẋ) .
T 2E
0 0

Continuing this result to Minkowski spacetime and redefining E → −E gives

Z∞ ZT
dT 1 2
Z h i
Γ[A] = N Dx exp − dτ ( ẋ − igA[x(τ )] · ẋ)
T 2E
0 0
(2.9)
Z∞
dT
Z T
I
1 2
R
dτ ( 2E ẋ )
= N Dx e− 0 exp[ig dx · A(x)] .
T
0

This expression is immediately recognizable as the expectation value of a Wilson

loop of the background field, in a certain ensemble of loops. It is therefore explicitly
gauge invariant with respect to the background gauge field, as it should be.

5
 The non-Abelian generalization of this structure is easy to guess; one merely
inserts a trace over color states:

Z∞ ZT
dT 1 2
Z h i
Γ[A] = N Dx TrR exp − dτ ( ẋ − igA[x(τ )] · ẋ) , (2.10)
T 2E
0 0

where the gauge field is a matrix Aaµ T a in the gauge group representation R of
the scalar. Notice that the usual path-ordering in the Wilson loop appears here as
proper-time–ordering, implicit in the path integral construction.

Let us now consider the expansion of this effective action to order g N , which is
equivalent to studying the one-particle-irreducible (1PI) Feynman diagrams with
N background gluons and one scalar loop. (By “gluon” I mean any non-abelian
vector boson.) In the standard Feynman graph technique there are a number
of such diagrams, involving both the one-gluon/two-scalar vertex and the two-
gluon/two-scalar vertex. Here, there is only one computation. We expand the
Wilson loop to order g N :

Z∞ N ZT
(ig)N dT
Z T Y 
1 2
R
− dτ 2E ẋ
ΓN [A] = N Dx e 0 Tr dti A[x(ti )] · ẋ(ti ) . (2.11)
N! T
0 i=1 0

Up to this point the background field is completely arbitrary. To compute ΓN [A]

as a function of momentum eigenstates, we insert for Aµ a sum of classical modes
of definite (outgoing) momentum ki , polarization ǫi , and gauge charge T ai :

N
T ai ǫµi eiki ·x
X
µ
A (x) = (2.12)
i=1

Again T ai is a matrix in the representation of the scalar. Inserting this function

into (2.11) and keeping only the terms in which each mode appears precisely once,

6
we find:

Z∞
dT
Z T 1 2
R
N dτ 2E ẋ
ΓN (k1 , . . . , kN ) = (ig) N Dx e− 0
T
0
(2.13)
N Zti+1
Y
Tr(T aN . . .T a1 ) dti ǫi · ẋ(ti )eiki ·x(ti)
i=1 0

plus terms with all other orderings of the ti and T ai . (Here tN +1 ≡ T .) Notice
that for a given integration ordering (= path-ordering around the loop = proper-
time-ordering = color-trace-ordering), the color information factors out. For pure
vector field backgrounds, only one color-ordering is actually necessary, as all other
orderings are related to it by permutation of labels; because of this, I will consider
for the remainder of this paper only one color ordering at a time, leaving the sum
over color orderings implicit.

String theorists will immediately recognize eq. (2.13); the string theory version
of this formula gives the expectation value of N “vertex operators”, which in string
theory can be interpreted as a scattering amplitude of N strings. For strings, duality
of the s and t channels implies that not only the one-particle-irreducible loop but
also the trees which are sewn onto the loop are calculated in this way. In particle
theory, however, eq. (2.13) computes only the effective action, the one-particle-
irreducible graphs with a scalar loop, at order g N . Still, it has the advantage of
being well-defined even for off-shell external gauge fields, unlike usual string theory.

To calculate this expectation value I use the standard path integral methods of
[13]
string perturbation theory. First, disregard the polarization vectors, and notice
that the momenta ki in (2.13) serve as sources of the four fields xµ (τ ):

N
i kjµ δ(τ − tj )
X
µ
J (τ ) = (2.14)
j=1

7
Using eq. (2.5), we find

ΓN (k1 , . . . , kN )
Z∞ N Zti+1
(ig)N dT Y
= Tr(T aN . . . T a1 ) dti
(4π)2 (E/2)2 T3
0 i=1 0

ZT ZT  
′ 1 µ ′ ′
exp[ dτ dτ − J (τ )GB (τ, τ )Jµ (τ ) ]
2 (2.15)
0 0
Z∞ N Zti+1 
(ig)N dT Y 
= Tr(T aN . . . T a1 ) dti
(4π)2 (E/2)2 T3
0 i=1 0
N
hX 1 i
exp ki · kj GB (ti , tj ) .
2
i,j=1

Here GB (t, t′ ) is the one-dimensional propagator on a loop, which I will discuss

later. (The B indicates that GB is the Green function of the Bosonic field xµ .)

The standard method for including the polarization vectors is to exponentiate

them, with the understanding that the only terms to be used are those which
contain one ǫi :


ẋ · Aµi (x(ti )) = T ai exp ǫi · ∂ti x(ti ) + iki · x(ti ) 

(2.16)
linear in ǫi

This leads to a new source for xµ :

N
δ(τ − ti ) (ǫµi ∂ti + ikiµ ) .
X
J µ (τ ) = (2.17)
1

8
Integration over x(τ ) gives

Z∞ N Zti+1 
(ig)N dT  Y
ΓN (k1 , . . . , kN ) = Tr(T aN . . . T a1 ) dti
(4π)2 (E/2)2 T3
0 i=1 0
N
h1 X
exp ki · kj GB (tj − ti )]
2
i,j=1
∂
− 2iki · ǫj GB (tj − ti )
∂tj
∂2
i
− ǫi · ǫj GB (tj − ti )  ;
∂ti ∂tj linear in each ǫ
(2.18)
again only terms in which each polarization vector appears exactly once are to be
[6]
used. String theorists and those familiar with the work of Bern and Kosower will
recognize this form for the amplitude.

Now let us study the Green function (one-dimensional propagator), which sat-
isfies the equation
1 2
∂ GB (t, t′ ) = δ(t − t′ ) (2.19)
E t

with appropriate boundary conditions. If we were studying this Green function on

the real line, the solution would be

E
GB (t, t′ ) = |t − t′ | + A + Bt . (2.20)
2

Notice that the Green function is finite as t approaches t′ , which is not true for
higher dimensions; thus there are no operator singularities when x fields come
together. This naturally simplifies many discussions.

To find the Green function on a circle of circumference T , one must first note
that eq. (2.19) has no solution on the loop; it is equivalent to solving Poisson’s
equation for a charge in a compact space, for which the potential is infinite unless
there is a background charge that makes the total space neutral. Since we have

9
one unit of charge at t′ , we should add a uniform background charge of density
−1/T . The new Green function equation is

1 2 1
∂ GB (t, t′ ) = δ(t − t′ ) − , (2.21)
E T

which has a solution when the condition of periodicity in t → t + T is imposed:

E 
GB (t, t′ ) = |t − t′ | − (t − t′ )2 /T + constant . (2.22)
2

It is convenient to take the arbitrary constant to be zero, as any additive constant

in GB cancels out of eq. (2.18). This function has as its derivative

E
∂t GB (t, t′ ) = sign(t − t′ ) − 2(t − t′ )/T ,


(2.23)
2

and its second derivative is given in eq. (2.21). Note that GB and ∂t2 GB are
symmetric in their arguments, while ∂t GB is antisymmetric. These functions (up
[6]
to a multiplicative constant) were found by Bern and Kosower from the one-loop
string theory bosonic Green function and its derivatives, in the limit where t − t′
is large compared to the width of the string theory torus. Roughly adhering to
their conventions, I shall use the notation Gji ji ji
B ≡ GB (tj − ti ), ĠB ≡ ∂tj GB , and
G̈ji 2 ji
B ≡ ∂tj GB .

It is useful to transform eq. (2.18) into a simpler form. First, through the use of
the crucial relations GB (t, t) = 0 and (by antisymmetry in t and t′ ) ∂t GB (t, t) ≡ 0,
the terms in (2.18) with ǫi · ki and ki2 are removed without the use of on-shell
conditions. Second, it is useful to replace ti → uiT , where ui is dimensionless; N
powers of T are thereby factored out. Next, observe that the integral over uN is
trivial; after the first N − 1 integrals no dependence on the ui remains, and so the
last integral, which contributes a factor of unity, can be dropped. It is useful to
choose the origin of proper time by fixing tN ≡ T , and as a consequence we should
sum only over color traces which are not related by cyclic permutation. A further

10
advantage is gained by choosing the (dimensionless) gauge E = 2. Lastly, antici-
pating the use of dimensional regularization, I redo the integral over momentum
in 4 − ǫ dimensions as in eq. (2.5), with µ the arbitrary mass parameter. (For the
remainder of this paper, the conventions chosen above will be used except where
explicitly noted.)

The result of all these changes is

(igµǫ/2 )N
ΓN (k1 , . . . , kN ) = Tr(T aN . . . T a1 )
(4π)2−ǫ/2
Z∞ Z1 u
ZN−1 Zu2
dT
duN −1 duN −2 · · · du1
T 3−N −ǫ/2
0 0 0 0
h N i
ki · kj Gji
X
exp B
i<j=1
N
h X
i 
exp − i(ki · ǫj − kj · ǫi ) Ġji ji
B + ǫi · ǫj G̈B  ;
linear in each ǫ
i<j=1
(2.24)
plus all other proper-time-orderings. Meanwhile the Green functions have become
 
′ ′ ′ 2
GB (t, t ) ≡ T |u − u | − (u − u ) ;
∂t GB (t, t′ ) ≡ sign(u − u′ ) − 2(u − u′ ) ;


(2.25)
2
∂t2 GB (t, t′ ) ≡ δ(u − u′ ) − 1 .


T
Comparison with reference 6 or 7 shows that the correspondence between eqs. (2.24)
and (2.25) and the Bern-Kosower rules for the one-particle-irreducible scalar loop
diagram with N gluons is exact, up to differences in conventions.
[7]
Following Bern and Kosower , let us study the result of (2.24). The overall
constant factor, the color trace and the integrals are easy to understand. The
exponential
N
h X i h XN  i
exp ki · kj Gji
B = exp T ki · kj |uj − ui | − (uj − ui ) 2
(2.26)
i<j=1 i<j=1

11
is a ubiquitous factor which, after the integration over T , becomes the usual
Feynman-parameterized denominator for a scalar loop integral (notice it contains
no polarization vectors, and is thus spin-independent):

Z∞ N
h X i Γ(α + 1)
α
dT T exp ki ·kj Gji
B = h  iα+1 .
PN 2
0 i<j=1 − i<j=1 ki · kj |uj − ui | − (uj − ui )
(2.27)
The remaining term,

N
h X i
Ġji ǫi · ǫj G̈ji


exp − i(ki · ǫj − kj · ǫi ) + , (2.28)

B B 
linear in each ǫ
i<j=1

which I shall call the “generating kinematic factor”, provides the numerator of the
Feynman parameter integral. It is the only part of (2.24) (other than the overall
normalization) which has any information about the type of particle in the loop or
the nature of the external field. It is also the only part of the result which cannot
be guessed on general grounds; we undergo the usual struggles with Feynman
diagrams and loop momentum integrals in order to obtain precisely this piece of
information.

However, the form of the generating kinematic factor causes some practical
problems. At first glance (2.24) appears to have expressed the entire result in such
a way that one has exactly one set of Feynman parameter integrals for each color
trace, but this is not quite true. The difficulties stem from the G̈B functions. The
first problem is that each term with M G̈B ’s has M fewer powers of T than terms
without G̈B ’s, so a number of different integrals over T must be performed. The
second problem is that hiding inside each G̈ji
B is a delta function in tj − ti . The
evaluation of this delta function gives the contribution of the Feynman diagram
in which gluons i and j come onto the loop via a four-point vertex. Thus the
expression in eq. (2.24) contains all of the 1PI Feynman diagrams, in fact, and
each one generates slightly different integrals and integrands. (Fortunately, these
[7]
problems can be dealt with , as I will discuss in section 5.)

12
 There is a subtle factor of two concerning the delta function in G̈ij
B . Consider
smoothing out the singularity slightly; then, in order to maintain the symmetries
of GB and its derivatives one must assign half of the delta function to ti > tj
and the other half to ti < tj . In other words, the delta function is split between
Tr(· · · T ai T aj · · ·) and Tr(· · · T aj T ai · · ·).

I now present the simplest possible example, the contribution of a massless

scalar to the gluon vacuum polarization. There are two Feynman diagrams, the
first of which involves two three-point vertices, the other of which involves a single
four-point vertex. The former is given by

dD p (i)2 ǫ1 · (2p − k1 )ǫ2 · (2p − k1 )

Z
2 a b
(ig) Tr(T T ) (2.29)
(2π)D p2 (p − k1 )2

where k1 is the momentum flowing out along gluon 1. The second diagram is given
by
dD p iǫ1 · ǫ2
Z
2 a b
2ig Tr(T T ) . (2.30)
(2π)D p2
[12]
I now use the Schwinger trick to evaluate (2.29) in a form conducive to compar-
ison with the expression in (2.24).

dD p −ǫ1 · (2p − k1 )ǫ2 · (2p − k1 )

Z

(2π)D p2 (p − k1 )2
Z∞ Z1 Z
dD p
= T dT da
(2π)D
0 0
h 2 2

− ǫ1 · (2∂v − k1 )ǫ2 · (2∂v − k1 ) e−T [p +a(k1 −2p·k1 )] ev·p 
 
v=0
Z∞ Z1 h 
ak1 ·v+v 2 /4T 
= T dT da − ǫ1 · (2∂v − k1 )ǫ2 · (2∂v − k1 )(e ) 
v=0
0 0
dD p′ −T p′2
Z
−T k12 (a−a2 )
× e e .
(2π)D
(2.31)
Carrying out the derivatives and the integral over momentum, and adding to this

13
expression the contribution of (2.30), we are left with

Z∞
−(gµǫ/2 )2 dT
Π= Tr(T a T b )
(4π)2−ǫ/2 T 1−ǫ/2
0

nh Z1  2  i (2.32)
2 −T k12 (a−a2 )
da − ǫ1 · ǫ2 − (1 − 2a) ǫ1 · k1 ǫ2 · k1 e
T
0
2 o
+ ǫ1 · ǫ2 ,
T

where ǫ = 4 − D.

Alternatively we may write down the result of (2.24) for N = 2:

Z∞ Z1
(igµǫ/2 )2 dT
Γ2 (k1 , k2 ) = Tr(T a T b ) du ek1 ·k2 GB (1−u)
(4π)2−ǫ/2 T 1−ǫ/2 (2.33)
0 0
h
i
k2 · ǫ1 k1 · ǫ2 [ĠB (1 − u)]2 + ǫ1 · ǫ2 G̈B (1 − u) .

Define a = 1 − u, plug in the functions in (2.25), and the result appears:

Z∞ Z1
−(gµǫ/2 )2 a b dT T k1 ·k2 (a−a2 )
Π= Tr(T T ) da e
(4π)2−ǫ/2 T 1−ǫ/2 (2.34)
0 0
h2 i
(δ(a) − 1)ǫ1 · ǫ2 + (1 − 2a)2 ǫ1 · k2 ǫ2 · k1 .
T

Note that, as advertised, the diagram involving a four-point vertex (eq. (2.30)) is
found by evaluating the delta function in (2.34); since Tr(T a T b ) = Tr(T b T a ) this
trace receives the full contribution of the delta function. This example also makes
[6]
clear that, as explained by Bern and Kosower , the differences ui − uj are directly
related to the usual Feynman parameters.

14
 3. The Effective Action of a Spinor Particle in a Background Field

The case of a spinning particle is a simple generalization of the particle theory

used in section 2. The one-loop action of a Dirac spinor with a vector-like coupling
to a background field is
Z
S= d4 x χ(i D
6 − m)χ (3.1)

where Dµ = ∂µ −igAµ . The one-loop effective action as a function of Aµ is therefore

h
i
Γ[A] = log 6 −m
det i D
1 h

i
= log det i D6 − m det − i D 6 −m (3.2)
2
1 h ig
i
= log det D 2 1 − Fµν [γ µ , γ ν ] + m2 .
2 4

where I use det(6D ) = det(γ 5 6D γ 5 ) = det(− D

6 ). This expression for the effective
action is also associated with the second-order action for a Dirac spinor

1 † 2
Z
S= d4 x − χL (6D + m2 )χR (3.3)
m

1
where the 2 in (3.2) appears because χL,R are two-component Weyl spinors. The
relevance of these formulas to the Bern-Kosower formalism was noted by Bern and
[10]
Dunbar.

Since the gamma matrices are anticommuting operators, it is natural to in-

troduce worldline fermions to represent them. This technique has long been em-
[14,16,17] [18]
ployed to introduce spin , and even color , into quantum mechanics. There
is nothing mysterious about this; finite representations of compact groups can be
generated by a set of fermionic operators.

One may therefore implement a supersymmetric generalization of the procedure

outlined in eq. (2.8), introducing Grassmann fields ψ µ (τ ) as partners of the fields

15
xµ (τ ). I will want the usual fermionic anticommutation relations

{ψ µ , ψ ν } = g µν , (3.4)

q
1
which imply that as operators the ψ µ fields are just constants equal to 2 γµ , and
I take as the Hilbert space of the theory the four components |αi of the Dirac
fermion, which are acted on in the usual way by the ψ fields:

1 µ
ψ µ |αi = √ γαβ |βi . (3.5)
2

I will now evaluate (3.2) (in the massless case) as in section 2, taking the
worldline fermions to have the usual antiperiodic boundary conditions. (One need
[19]
consider periodic boundary conditions only for chiral fermions .) Direct construc-
tion of the particle path integral leads to

1 h ig i
Γ[A] = Tr log D 2 1 − Fµν [γ µ , γ ν ]
2 4
Z∞
1 dT X d4 p
Z
=−
2 T α (2π)4
0
 1
hα, p| exp − ET {(p + gA)2 + igFµν ψ µ ψ ν } |α, pi

2 (3.6)
Z∞
1 dT
Z
=− N Dx Dψ
2 T
0
ZT
1 2 1
ẋ + ψ · ψ̇ − igAµ ẋµ + ig(E/2)ψ µFµν ψ ν ) .
 
Tr exp − dτ (
2E 2
0

The abelian version of this action was first presented by Brink, Di Vecchia, and
[14] [15]
Howe ; the nonabelian case was discussed by several authors.

In this way, the effective action for a spinor is expressed as a supersymmetric

Wilson loop, in a free supersymmetric theory. The particle action is invariant

16
under the transformation

δη xµ = −Eηψ µ ; δη ψ µ = η ẋµ . (3.7)

This supersymmetry and the superfield formulation of this theory have been ad-
dressed by many authors, for example in reference 14; I will not discuss it further
in this work.

Now let us consider the effective action (3.6) at order g N . For the moment I
shall ignore the [Aµ , Aν ] term in Fµν ; I will return to it at the end of this section.
Expanding for the moment only the terms with a single power of the gauge field
to order N, and inserting the momentum eigenstates of eq. (2.12), one finds

Z∞ ZT
1 (ig)N dT 1 2 1
Z
0
Γ [A] = − N Dx Dψ exp[− dτ ( ẋ + ψ · ψ̇)]
2 N! T 2E 2
0 0
N ZT
Y n o
Tr dti Aµ [x(ti )] · ẋµ (ti ) − Eψ µ (ti )∂µ Aν [x(ti )] · ψ ν (ti )
i=1 0
(3.8)
Z∞ ZT
1 dT 1 2 1
Z
= − (ig)N N Dx Dψ exp[− dτ ( ẋ + ψ · ψ̇)]
2 T 2E 2
0 0
N ZT
Y
dti T ai ǫi · ∂i x(ti ) + iEǫi · ψ(ti ) ki · ψ(ti ) eiki ·x(ti )
 
Tr
i=1 0

(I write Γ0 to remind the reader that I have left out the commutator term in Fµν .)
Here string theorists will find the vertex operators for vector fields used in the
superstring.

Again we can put the polarization vectors in the exponentials; using Grassmann
variables θ and θ, we may write

V ≡ igT a ǫ · ẋ + iEǫ · ψ k · ψ eik·x

 

a
Z
 √ √  (3.9)
= igT dθdθ exp θθǫ · ẋ + θ Eǫ · ψ + iθ Ek · ψ + ik · x .

17
This leads to sources for xµ

N
δ(τ − ti )(θi θi ǫµi ∂ti + ikiµ ) .
X
µ
J (τ ) = (3.10)
1

and ψ µ
N
X √
µ
η (τ, θ, θ) = δ(τ − ti ) E(θi ǫi + iθi ki ) . (3.11)
1

The result of carrying out the x and ψ integrals (in the gauge E = 2) is

Z∞ ui+1
N Z
(ig)N dT  Y 
Γ0N (k1 , . . . , kN ) = −4 Tr(T aN . . . T a1 ) du i
2(4π)2 T 3−N
0 i=1 0
N
 X  N Z
 Y 
exp ki · kj Gji
B dθi dθi
i<j=1 i=1
N
 X
exp − i (θj θj ki · ǫj − θi θi kj · ǫi )Ġji
B (3.12)
i<j=1


+ θi θi θj θj ǫi · ǫj G̈ji
B
N
 X 
exp − θi θj ki · kj + iθi θj ki · ǫj
i<j=1

 
+ iθi θ j ǫi · kj + θi θj ǫi · ǫj Gji
F ,

plus terms involving all other proper-time/color orderings. The overall factor of
four comes from

Z 4
R T
dτ 12 ψ·ψ̇
X
−
Dψ e 0 = Trψ 1 = hα|αi . (3.13)
α=1

The generating kinematic factor (in braces) has a bosonic part identical to
(2.28), as well as terms that contain the one-loop Green functions GF (Gji
F =

18
−Gij
F ≡ GF (tj − ti )) of the fermionic ψ fields. In addition to implementing the
constraint that every polarization vector appears exactly once, the Grassmann
integrations over θ and θ ensure that in any term of the generating kinematic
factor in which ǫµi Gij ν ik
F appears, ki GF must also appear. This implies that the GF
functions always occur in closed chains of the form

d
i ,ik
Y
GFk+1 ; (id+1 ≡ i1 ) . (3.14)
k=1

(As the GF ’s are antisymmetric in their arguments, a term like G12 13 23

F GF GF is not
ruled out; on the contrary, it is equal to −G12 23 31
F GF GF which is of the form (3.14).)

The bosonic part of the action in (3.6) is the same as in section 2, so the GB
functions are again given by eq. (2.25). The GF functions satisfy

1
∂t GF (t, t′ ) = δ(t − t′ ). (3.15)
2

Since the fermions also satisfy antiperiodic boundary conditions

ψ(t → T ) = −ψ(t → 0) , (3.16)

we take the antiperiodic solution of eq. (3.15):

GF (t, t′ ) = sign(t − t′ ) = sign(u − u′ ). (3.17)

This function is double-valued, since it changes sign only at t = t′ :

GF (t, t′ ) = −GF (t + T, t′ ). (3.18)

If the theory is abelian, then the single expression (3.12) contains the entire one-
loop effective action (which is also the full photon one-loop S-matrix.) However, if

19
we are working in a non-abelian gauge theory, then in addition to the expression
given in (3.12) for the effective action we must include terms involving the quadratic
term in Fµν ,
1
− g 2 Eψ µ [Aµ , Aν ]ψ ν , (3.19)
2
which generates two-gluon vertex operators of the form

Oi,j = −g 2 E(T aj T ai ) ǫj · ψ ǫi · ψ ei(ki +kj )·x . (3.20)

In the second-order formalism for spinors in gauge fields, the usual three-point
vertex is replaced by a new three-point vertex along with a two-gluon/two-spinor
vertex, similar to the vertices of scalars in gauge fields. This can be inferred from
eq. (3.3). As in the previous section, a part of the four-point vertex is associated
with the delta function in G̈B , but because of the particle’s spin and the non-
abelian nature of the background field this vertex contains a new piece generated
by the operator Oi,j .

The contribution of this operator can be evaluated through a process known as

“pinching”, which is related to the Bern-Kosower rules for trees attached to loops.
In this process gluons i and j are brought to the same point on the loop (“pinched”),
and a subsidiary “pinched kinematic factor”, containing the contribution of Oi,j ,
is extracted from the generating kinematic factor in a systematic way. The reader
[7]
may wish to review the Bern-Kosower rules , which serve as motivation for the
following unusual manipulation of (3.20):
Z
2 aj ai
Oi,j = − g (T T ) dθi dθi dθj dθj
 √ √ 
(−θ i θ j ) exp θi Eǫi · ψ(ti ) + θj Eǫj · ψ(tj ) + i(ki + kj ) · x 
ti =tj
Z
= (ig)2 (T aj T ai ) dθi dθi dθj dθj
 √ √
exp θi Eǫi · ψ + θj Eǫj · ψ + i(ki + kj ) · x − θi θj ki · kj Gji
 
F 
ji 
ki · kj G F

ti =tj
(3.21)

20
Insertion of this operator into (3.8) to replace two operators of the type (3.9) gives
the pinched kinematic factor. Comparison with (3.12) shows that the pinched
factor consists of all the terms in the generating kinematic factor which contain
ki · kj Gji
F , with the replacement

+1, if tj > ti ;

ki · kj Gji
F → (3.22)
−1, if tj < ti ,

and with ti set equal to tj . Notice that if a term contains ǫi · ǫj Gji

F as well, it
vanishes since Gji
F (0) ≡ 0 by antisymmetry.

In order to keep track of the different pinch contributions, it is useful to write

down a simple mnemonic rule based on Bern-Kosower diagrams. While this could
be done in many ways, the particular choice presented here will eventually permit
a smoother transition from effective actions to scattering amplitudes.

Draw all (planar) φ3 graphs with one loop, N external legs and any number
NT ≤ N/2 of trees with one vertex. Consider a particular graph and a particular
color(path)-ordering; label the external legs clockwise from 1 to N following the
path-ordering. Now examine the generating kinematic factor of (3.12) term by
term. Two external gluons flow into each tree vertex; let j be the gluon lying most
clockwise, and call the other gluon i. If a given term does not contain a factor
ki · kj Gji
F for each tree vertex in the graph, then it vanishes. Even then, it must
contain exactly one Gji
F at each vertex; otherwise it vanishes. If it survives, then
replace each ki · kj Gji
F by +1, replace ti → tj in all Green functions, and eliminate
the ti integral.

As an application of the formalism of this chapter, let us consider the contri-

bution of a Dirac spinor to the gluon vacuum polarization. In the usual first-order
formalism of Dirac, the single diagram has the form

dD p −Tr[6 ǫ1 (6 p− 6 k1 ) 6 ǫ2 6 p]
Z
2 a b
g Tr(T T ) (3.23)
(2π)D p2 (p − k1 )2

21
Usually this diagram is evaluated by writing

Tr[6 ǫ1 (6 p− 6 k1 ) 6 ǫ2 6 p] = 4[ǫ1 ·(p−k1 )ǫ2 ·p+ǫ1 ·p ǫ2 ·(p−k1 )−p·(p−k1 )ǫ1 ·ǫ2 ] , (3.24)

after which the momentum integral is performed. One may also use

6 ǫi (6 p− 6 k i ) = 2ǫi · p − 6 p 6 ǫi − 6 ǫi 6 k i (3.25)

and write (after some algebra)

2Tr[6 ǫ1 (6 p− 6 k 1 ) 6 ǫ2 6 p]
= Tr[− 6 ǫ1 6 ǫ2 ](p2 + (p − k1 )2 ) + Tr[(2ǫ1 · p− 6 ǫ1 6 k 1 )(2ǫ2 · (p − k1 )− 6 ǫ2 6 k 2 )]
= −4ǫ1 · ǫ2 p2 + (p − k1 )2 + 4ǫ1 · (2p − k1 ) ǫ2 · (2p − k1 )




− 4 ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 k1 · ǫ2
(3.26)
which puts the amplitude in a second-order form. The first and second term yield
the contribution of (2.31) times a factor of −2; the last term is independent of the
loop momentum. The result is

Z∞
(gµǫ/2 )2 dT
Π=2 Tr(T a T b )
(4π)2−ǫ/2 T 1−ǫ/2
0

nh Z1  2
da − ǫ1 · ǫ2 − (1 − 2a)2 ǫ1 · k1 ǫ2 · k1 (3.27)
T
0

 −T k2 (a−a2 ) i
+ ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 k1 · ǫ2 e 1

2 o
+ ǫ1 · ǫ2 ,
T

where ǫ = 4 − D.

22
 By contrast, evaluation of (3.12) at order g 2 immediately yields

Z∞ Z1
(igµǫ/2 )2 a b dT
Γ2 (k1 , k2 ) = −2 2−ǫ/2
Tr(T T ) 1−ǫ/2
du ek1 ·k2 GB (1−u)
(4π) T
0 0
h
(3.28)
ǫ1 · k2 ǫ2 · k1 [ĠB (1 − u)]2 + ǫ1 · ǫ2 G̈B (1 − u)
i
+ ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 ǫ2 · k1 [GF (1 − u)]2 ,

which is identical to (3.27). There are no pinches to perform, since the integrand
contains no terms with a single power of G12
F .

4. The Effective Action of a Vector Particle in a Background Field

Now let us consider the case of a massless spin-one particle. There are many
ways to proceed, and among them are several directly inspired by the methods of
string theory. In a model inherited from the bosonic string, one would introduce a
single oscillator mode with a vector index, whose sole purpose would be to excite
an unphysical scalar “vacuum” (which would eventually be removed by hand) to a
vector boson state. One could then imagine projecting out all higher spin states,
either by hand or by tricks ranging from adding large masses (as in the string)
or by adding complex phases to the oscillators (along the lines of string orbifold
constructions). Another possibility is to use a supersymmetric construction; as in
the superstring, a fermionic oscillator with a vector index can be used to excite a
“vacuum” (which one projects away) to a state with vector indices. Extra states
can again be projected out in a number of ways. I will use this latter construction,
[13]
following closely both the usual superstring methodology and the work of Brink,
[14]
Di Vecchia and Howe.

The action of a Yang-Mills particle Qµ , expressed in Feynman gauge, in a

classical background Aµ is well-known to be
Z
c ρσ
S = d4 x {Qaµ [(D 2 )ab gµν − g(Fρσ J )µν f cab ]Qbν + ω(D 2 )ab ω
(4.1)
3 4
+ order(Q , Q , ωQω, etc.)} ,

23
where Dµ = ∂µ − igAµ and gFµν = i[Dµ , Dν ] are functions only of the background
field, ω is the ghost of background field Feynman gauge, and Jµν is the spin-one
(hermitean) generator of Lorentz transformations:

(Jµν )ρσ = i(δµρ δνσ − δνρ δµσ ) . (4.2)

(Feynman gauge for Qµ is appropriate in that the propagator is −1 , as we had

for scalars and spinors; background field gauge is essential since the result must
be gauge invariant with respect to the classical field Aµ . The appearance of back-
ground field gauge in this context and the following expression for the effective
[10]
action were discussed in the work of Bern and Dunbar. A useful introduction to
background field gauge is given in reference 20.) The one-loop effective action is
found from the part of (4.1) which is quadratic in the quantum fields:

1 h
2 µν
i h
2
i
Γ[A] = − log det (D − gF Jµν ) + log det (D ) . (4.3)
2

Again the structure of the effective action suggests the use of Grassmann variables,
[14]
and turning to Brink, Di Vecchia, and Howe , we find that they have discussed
the relevant theory.
µ µ
Let us consider a particle with coordinates (xµ , ψ+ , ψ− ). We will find it useful
µ µ
to consider also the real field ψ µ = (ψ− + ψ+ ). The worldline fermions satisfy

µ ν 1
{ψ+ , ψ− } = g µν = {ψ µ , ψ ν } ;
2 (4.4)
µ ν µ ν
{ψ+ , ψ+ } = {ψ− , ψ− }=0.

µ
If we define a vacuum |0i as the state such that ψ− |0i = 0 for all µ, then the full
set of sixteen states (for a given momentum) is

µ µ ν ν ρ σ µ ν ρ σ
|0i ; ψ+ |0i ; [ψ+ , ψ+ ] |0i ; ǫµνρσ ψ+ ψ+ ψ+ |0i ; ǫµνρσ ψ+ ψ+ ψ+ ψ+ |0i . (4.5)

These are antisymmetric tensors; in four-dimensions the (0,1,2,3,4)-index antisym-

metric tensors have (1,4,6,4,1) components of which only (1,2,1,0,0) are physical

24
degrees of freedom. This model therefore describes a scalar, a vector boson, and
a pseudoscalar. However, if we can implement a projection onto states with odd
fermion number, then the truncated Hilbert space

µ ν ρ σ
ψ+ |0i and ǫµνρσ ψ+ ψ+ ψ+ |0i . (4.6)

will contain only the spin-one states as physical modes.

In a complete analysis of this truncated model, one must study the super-
reparametrization ghosts in order to derive the Bern-Kosower rules; however, I
have chosen to skirt the issue of ghosts in this article. For the present paper it will
be sufficient to use a trick borrowed from string theory, in which the gluon ghosts
of field theory are accounted for by hand, and in which the three-index tensor is
given a mass which is sent to infinity at the end of the calculation.

Derivation of the superparticle Lagrangian is straightforward when one ob-

serves the following:

i i
ρ σ
= (Jµν )ρσ .

hρ| [ψµ , ψν ] |σi = ψ− [ψµ , ψν ]ψ+ (4.7)
2 2

Remembering that we will eventually do away with the spurious states, let us
extend the theory to the full set of sixteen states in (4.5). As in (3.6), we are led
to the particle Lagrangian

1 2 E
L= ẋ + ψ+ · ψ̇− − igAµ ẋµ + ig ψ µ Fµν ψ ν . (4.8)
2E 2

However, in order to carry out the trick described above we will want to make the
three-index tensor heavy. We must therefore break the degeneracy of the sixteen
states by adding a harmonic oscillator potential:

L → L − C(ψ+ · ψ− − 1) . (4.9)

For positive C the ψ’s form a fermionic harmonic oscillator whose states are spaced
by ∆m2 = C and whose vacuum is a tachyon with m2 = −C. Fortunately this

25
tachyon is unphysical; it will be removed from the theory by truncation as discussed
above, and so causes no difficulties. All other states except the vector boson will
vanish as a result of the truncation or because their masses will be taken to infinity.
[13]
(This construction is taken directly from the superstring. )

One can proceed straightforwardly with the computation of the effective ac-
tion in direct analogy to the spinor and scalar cases. The field theory ghosts in
background field gauge contribute a factor of log det D 2 ; as noted by Bern and
[10] [13]
Dunbar , and as expected from string theory , this is exactly the negative of
the effective action of a complex scalar in the adjoint representation (see eq. (2.10)):

Z∞ ZT
dT 1 2
Z h i
Γ[A]ghosts = − N Dx Tr exp − dτ ( ẋ − igA · ẋ) . (4.10)
T 2E
0 0

The gauge boson contribution may be calculated by projecting out the even fermion
states in the theory and by letting C → ∞. The projection, which is the GSO
[21]
projection well-known from string theory , is implemented by the operator

1
1 − (−1)F ,

PGSO = (4.11)
2

where F = (ψ+ )µ · (ψ− )µ is the fermion number of a state. Clearly only the states
[13]
of (4.6) survive. It is well-known that the operator (−1)F is implemented in the
path-integral by choosing periodic boundary conditions for fermions:

ψ(t → T ) = ψ(t → 0) . (4.12)

26
We may therefore write

1 h i
Γ[A] = − Tr log D 2 1 − gF µν Jµν
2
Z∞ 1
1 dT d4 p
X Z
= lim
2 C→∞ T (2π)4
0 s0 ,s1 ,s2 ,s3 =0
 1
hsρ , p| PGSO exp − ET {(p + gA)2
2
− C(ψ+ · ψ− − 1) + igFµν ψ µ ψ ν } |sρ , pi


Z∞ (4.13)
1 dT 1h
Z Z Z i
= lim N Dx Dψ − Dψ
2 C→∞ T 2
0 ( 21 ) (0)

ZT
 1 2 E
Tr exp − dτ ( ẋ + ψ+ · ψ̇− − C(ψ+ · ψ− − 1)
2E 2
0
E
− igAµ ẋµ + ig ψ µ Fµν ψ ν ) ,

2

where the subscripts ( 12 ) and (0) indicate antiperiodic and periodic boundary con-
ditions on the worldline fermions.

Proceeding as in the previous section (eqs. (3.6)–(3.12)), we find (in the gauge
E = 2)

27
 Z∞ ui+1
N Z
(ig)N dT  Y 
Γ0N (k1 , . . . , kN ) = Tr(T aN . . . T a1 ) du i
2(4π)2 T 3−N
0 i=1 0
N
 X  N Z
 Y 
exp ki · kj Gji
B dθi dθi
i<j=1 i=1
N
 X
exp − i (θj θj ki · ǫj − θi θi kj · ǫi )Ġji
B
i<j=1


+ θi θi θj θj ǫi · ǫj G̈ji
B
1 N
X
p+1
Z p2  X 
(−) exp 2 − θi θj ki · kj + iθi θj ki · ǫj
2
p=0 i<j=1

 ( p2 )ji 
+ iθi θ j ǫi · kj + θi θj ǫi · ǫj GF ;
(4.14)
again the symbols ( 12 ) and (0) indicate antiperiodic and periodic fermions. No-
tice the factor of two relative to (3.12) in the exponential of the fermionic Green
functions. The Z factors are given (in Minkowski spacetime) by

( )
Z( 1 )
Z R T
dτ [ψ+ ·ψ̇− −C(ψ+ ·ψ− −1)]
2
= Dψ e− 0 = Trψ (±1)F e−H[ψ]T
Z(0)
ψ(T )=(∓)ψ(0)
1
cosh4
( )
X
4
=e−CT hs| (±eCT )s |si = 16eCT (−CT /2)
s=0 sinh4
=e−CT ± 4 + 6eCT + . . .
(4.15)
When continued to Euclidean spacetime, the arguments of the exponentials change
sign; cancellations remove all growing exponentials, as I will explain below.

The bosonic green functions are identical to those used for the scalar and spinor
particle (eq. (2.25)), since the free bosonic action

1 2
LB = ẋ (4.16)
2E

28
is independent of the particle’s spin. The free fermionic action is

LF = ψ+ · ψ̇− − Cψ+ · ψ− ; (4.17)

however, this leads in Minkowski spacetime to Green functions which blow up as

C → ∞. It is therefore necessary to analytically continue to Euclidean spacetime
to study this limit.

Moving to Euclidean spacetime, and being careful to define the number oper-
ator properly, we have
µ
LEucl
F = ψ+ ν
gµν (∂t + C)ψ− , (4.18)

Let us first compute the Green functions on the line. Define

G+− ′ ′


F (t, t ) = ψ+ (t)ψ− (t ) ; (4.19)

this function satisfies

(∂t + θ(t − t′ )C)G+− ′ ′

F (t, t ) = δ(t − t ) . (4.20)

where θ(t) is a step function which is zero for negative t. This equation implies

G+− ′ ′ ′
F (t, t ) = θ(t − t ) exp(−C|t − t |) . (4.21)

Similarly

G−+ ′ ′ ′
F (t, t ) = −θ(t − t) exp(−C|t − t |) . (4.22)

Since G++ −−
F and GF both vanish,

GF (t, t′ ) = ψ(t)ψ(t′ ) = sign(t − t′ ) exp(−C|t − t′ |) .


(4.23)

(0)
On the circle of circumference T , we will need to find functions, one periodic (GF ),
( 21 )
another antiperiodic (GF ) in t → t + T , which reduce to eq. (4.23) in the limit

29
that T → ∞. An analysis analogous to the above yields

(1) 1  1
GF2 (t − t′ ) = 2 sign(t − t′ )e− 2 CT cosh C( T − |t − t′ |) ;

2 (4.24)
(0) 1 1
GF (t − t′ ) = 2 sign(t − t′ )e− 2 CT sinh C( T − |t − t′ |) .
 
2

Again these are precisely the functions found by Bern and Kosower in the derivation
[6]
of their field theory rules .

The next task is to discard the three-index tensor by sending C to infinity. We

must carefully analyze the effective action (4.14) to see what terms remain in this
[6]
limit. The following discussion is almost identical to that of Bern and Kosower ;
I repeat it here for the sake of completeness.

It is necessary to study separately terms with and without GF chains. For

terms in (4.14) that contain no GF ’s, the only dependence on C is given in the
prefactors Z p2 , which in Euclidean spacetime take the form

cosh4
( ) ( )
Z( 1 )
2
= 16eCT 4
(CT /2) = eCT ± 4 + 6e−CT + . . . (4.25)
Z(0) sinh

The first term, associated with the propagation of the tachyon, blows up as C → ∞;
fortunately it cancels in the expression

1
Z( 1 ) − Z(0) = 4 + O(e−2CT ) ,


(4.26)
2 2

leaving us with an overall factor of 4. This factor stems from the sum over the four
µ
states ψ+ |0i which can propagate around the loop. These purely bosonic terms
are partially cancelled by the contribution of the ghosts (eq. (4.10)); the removal
of the timelike and longitudinal modes of the vector boson reduces the number of
states, and the overall factor, from 4 to 2. (In the usual dimensional regularization
schemes, this number becomes 2 − 21 ǫ; however it is natural in this formalism to
[6,7]
use dimensional reduction or the variant of it developed by Bern and Kosower ,
in which the number of states is left at 2.)

30
 Consider next the expansion in powers of eC of a chain product of antiperiodic
(1) (0)
GF2 ’s, minus the same chain of periodic GF ’s. This is precisely the sort of ex-
pression we obtain from (4.14) as a result of the GSO projection. From (4.24) we
find that

d d
1 h Y ( 12 ) Y (0) i
GF (tik+1 , tik ) − GF (tik+1 , tik )
2
k=1 k=1
d
Y d
X
−CT


=[ sign(tik+1 − tik )]e exp − C |tik+1 − tik | × (4.27)
1 k=1
d
X
exp 2C|tin+1 − tin | + O(e−CT ) .



n=1

(Here id+1 ≡ i1 .) The leading term in (4.27) is of the form

d
Y d
X
[ sign(tik+1 − tik )]e−CT


exp − Cf (ti ; tn ) (4.28)
1 n=1

where
d
X
f (ti ; tn ) = |tik+1 − tik | − 2|tin+1 − tin | ≥ 0 . (4.29)
k=1

Unless f (ti ; tn ) = 0 for some n, (4.28) will contribute too strong a power of e−C ,
and a term containing it will vanish in the limit C → ∞.

Since the expressions above are cyclic in k, one can rotate the k’s to make
tid = tmax ≡ max[tik ]; let tmin ≡ min[tik ]. Then

d
X
2|tin+1 − tin | ≤ 2(tmax − tmin ) ≤ |tik+1 − tik | . (4.30)
k=1

For f (ti ; tn ) = 0, both equalities must obtain. Notice that the second equality can

31
be satisfied only when

tmax = tid > tid−1 > · · · > ti2 > ti1 = tmin (4.31)

or

tmax = tid > ti1 > ti2 > · · · > tid−1 = tmin . (4.32)

(I will call a chain satisfying (4.31) or (4.32) a path-ordered chain since the ordering
is with respect to proper time. I remind the reader that the color trace is ordered
in the same way.) The first equality in (4.30) can only hold when tn = tmax and
tn+1 = tmin , or vice versa. Thus the condition f (ti ; tn ) = 0 can only occur either
when (4.31) holds and n = d, or when (4.32) holds and n = d − 1. (In the case
d = 2, both (4.31) and (4.32) hold.) It follows that a path-ordered chain of GF ’s
contributes

−1, if (4.31) holds;


d
Y


[ sign(tik+1 − tik )]e−CT = e−CT −(1)d−1 , if (4.32) holds; (4.33)

1 
−2, if d = 2.

Of course, as this derivation is essentially the same as that of reference 6, the result
(4.33) agrees with that of Bern and Kosower.

The exponential in (4.33) cancels the overall factor of eCT which was found in
eq. (4.25), leaving only the numerical factor −2 or ±1. All other terms from such
a chain, as well as those from chains which are not path-ordered, have additional
decaying exponentials which vanish in the limit C → ∞. Using the above argument
twice, it is easy to see that a term with more than one GF chain will always vanish
in the limit C → ∞. We therefore find that out of the expression (4.14), only
terms with single path-ordered chains of GF ’s of length 0 to N contribute, and
then are simply replaced by the factor ±1 or ±2. At this point all dependence on
C has vanished and we may return to Minkowski spacetime.

32
 How should one interpret these rules? It is easiest to do so from an operator
standpoint. Since we are throwing away all states of (4.5) except the spin-one
tensor, we require that the application of a ψ+ operator, which moves us out of
the space of spin-one states, be accompanied by the simultaneous application of
a ψ− operator in order to bring us back to it. This translates into a requirement
that the Wick contractions which generate the Green functions do not overlap one
another; hence the GF ’s must be path-ordered.

We now have enough information to write down a set of rules for the unpinched
diagram, starting with the same formula we had in the spinor case (eq. (3.12)).
To obtain the generating kinematic factor of the vector boson, manipulate the
kinematic factor of (3.12): throw away all terms except those with no GF ’s and
those with a single GF chain, and multiply terms without GF ’s by 2. Next, replace
the GF chains by


 −2d , if (4.31) holds;
d

−(−2)d , if (4.32) holds;


i ,i
Y
[ GFk+1 k ] → (4.34)
1


 −8, if d = 2;


0 otherwise.

where the powers of two account for the slight differences between equations (4.14)
and (3.12). Finally, substitute the bosonic Green functions of (2.25), plug the
result back into (3.12), multiply by − 14 and evaluate the integral.

The non-Abelian part of Fµν contributes to amplitudes for vectors just as it

does for spinors. The resulting pinch rules are almost as described in the previous
section, but one must decide whether to perform pinches before or after requiring
that all chains be path-ordered. The relevant consideration is that the pinch tech-
nique is just a trick to generate the correct set of GF ’s; one could drop the trick
and calculate directly the pinched kinematic factor by inserting Oi,j (eq. (3.20))
into the path integral, just as is done in (3.8) with the usual V’s (eq. (3.9)). Only
after the whole set of GF chains in the pinched kinematic factor is known should
one apply the analysis of eqs. (4.27)–(4.33) to determine which chains survive in

33
the limit C → ∞. Therefore, one should perform all pinches before requiring that
GF chains be path ordered; for example, the chain

2,i+1
G12
F GF ki · ki+1 Gi+1,i
F Gi1
F (4.35)

for tN > tN −1 > · · · > t1 will contribute to the diagram in which gluons i + 1
and i are pinched, even though in the evaluation of the unpinched Bern-Kosower
diagram it is discarded. (Notice that pinching cannot change the number of GF
chains in a given term, and so one may safely discard from the original generating
kinematic factor any term with more than one such chain.)

Thus, the rule for pinched diagrams is the following: Return to the generating
kinematic factor for the vector boson, and carry out the pinches as explained in
section 3. Next, apply the path-ordering requirement to GF chains, replacing them
with the factors in eq. (4.34). Finally, substitute the usual functions for the GB ’s,
insert the kinematic factor into (3.12), multiply by − 41 and compute the integrals.

As an example, consider the pure SU(N) Yang-Mills vacuum polarization in

background field gauge. The reader may check that if the algebra of Feynman
[10]
diagrams is organized as explained by Bern and Dunbar , it is straightforward to
obtain
Z∞
(gµǫ/2 )2 f acd f bdc dT
Π=
(4π)2−ǫ/2 T 1−ǫ/2
0

nh Z1  2
da − ǫ1 · ǫ2 − (1 − 2a)2 ǫ1 · k1 ǫ2 · k1 (4.36)
T
0

 2 2
i
+ 4 ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 k1 · ǫ2 e−T k1 (a−a )
2 o
+ ǫ1 · ǫ2 ,
T

where ǫ = 4 − D. I have included the ghosts in this expression, using dimensional

reduction in which the number of physical helicity states is exactly 2.

34
 According to the above rules for vector bosons, this result can be extracted
from the result of (3.28) by replacing (G21 2 21 12
F ) = −GF GF with +8, multiplying
1
the terms with (Ġ21 2 21
B ) and G̈B by 2, and multiplying the entire expression by − 4 .
Indeed this gives

Z∞ Z1
(gµǫ/2 )2 a b dT
Π=− 2−ǫ/2
Tr(T T ) 1−ǫ/2
du ek1 ·k2 GB (1−u)
(4π) T
0 0
h (4.37)
ǫ1 · k2 ǫ2 · k1 [ĠB (1 − u)]2 + ǫ1 · ǫ2 G̈B (1 − u)

i
+ 4 ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 ǫ2 · k1 ,

a )cd = −if acd .) There are no pinches

which is identical to (4.36) (recall that (Tadj
to perform; this is the complete result.

It is amusing to combine the results of (2.33), (3.28) and (4.37). Consider the
gluon vacuum polarization in a theory with nf Dirac fermions and ns complex
scalars in the adjoint representation:

Z∞ Z1
(gµǫ/2 )2 dT
Π=− Tr(T a T b ) du ek1 ·k2 GB (1−u)
2(4π)2−ǫ/2 T 1−ǫ/2
0 0
 h i (4.38)
(2 − 4nf + 2ns ) ǫ1 · k2 ǫ2 · k1 [ĠB (1 − u)]2 + ǫ1 · ǫ2 G̈B (1 − u)



+ 4(2 − nf ) ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 ǫ2 · k1 ,

(Since γ 5 does not play a role in vacuum polarizations, the contribution of a chiral
fermion to the above expression is exactly half that of a Dirac fermion.) Notice
that the factor multiplying the bosonic Green functions counts degrees of freedom,
and therefore cancels for all supermultiplets. With appropriate choices of matter
supermultiplets in various representations, it is possible to make the remainder
of (4.38) vanish, leaving the theory one-loop finite. When all particles are in the
adjoint representation, complete cancellation occurs for the case nf = 2 and ns = 3;

35
this is the famous N = 4 spacetime supersymmetric Yang-Mills theory, which is
[22]
known to be finite. Notice that this result requires no integrations; it follows
directly from the rules for obtaining the generating kinematic factors from (3.12)
and from the overall normalizations.

5. Integration by Parts and Manifest Gauge Invariance

[6,8]
Bern and Kosower showed that there are benefits associated with perform-
ing an integration-by-parts (IBP) on all terms involving a G̈B ; when the G̈B ’s
are completely eliminated, it is possible to derive a much simpler set of rules for
[10]
scattering amplitudes. As discussed by Bern and Dunbar , this IBP causes an
interesting and intricate reshuffling of terms. Essentially, the delta-functions which
produce the four-point vertices of field theory are removed by the IBP, allowing a
scattering amplitude to be expressed in terms of Bern-Kosower graphs, which have
only φ3 vertices. Each Bern-Kosower graph is related to the “unpinched diagram”
– the one with all gluons attached directly to the loop – through the systematic
pinch prescription.

In the effective action, the reorganization from the IBP is not much of a simpli-
fication, as it leads to as many or more diagrams than Feynman graphs. Nonethe-
less it is worthwhile in many cases: the additional diagrams are easier to calculate
than usual Feynman graphs due to the systematic “pinch” rules, and the number
of types of Feynman parameter integrals is reduced. Furthermore, and perhaps
most importantly, it makes possible a direct analysis of individual gauge invariant
contributions to the effective action. Still, the IBP is not essential for effective
actions, and the casual reader may safely skip this section at a first reading.

The reader intending to study this section should be warned that the IBP, while
necessary for a complete picture of the possibilities opened by the work of Bern and
Kosower, represents the weakest link in the present paper. A full understanding
of the IBP requires a clarification of the role of string duality, which permits the
reorganization which I will outline below. In the absence of this clarification it is

36
only possible to present the IBP and the associated pinch rules as a trick, motivated
[6,7]
by the Bern-Kosower rules for scattering amplitudes and the work of Bern and
[10]
Dunbar. Specifically, these rules match on to the Bern-Kosower rules when the
external gluons are on-shell. I will demonstrate the validity of this trick in a simple
case; however, while I have checked that it works in more complicated cases, I do
not know a complete proof. For this reason these effective-action pinch rules appear
completely ad hoc at the present time, and the reader is urged to familiarize herself
with the Bern-Kosower rules outlined in reference 7 to help put the present section
in context.

To illustrate the trick, I present the simplest case. Consider a term from the
generating kinematic factor of (3.12) of the form

ǫi · ǫj G̈ij
B × F (ǫm , kn ) , (5.1)

where F contains neither ki nor kj and therefore has no dependence on either ti

or tj . The IBP of (5.1) can be done with respect to ti , tj , or ti − tj ; different
results will be found in the different cases, the variations among them being total
derivatives. For simplicity let us IBP with respect to ti ; for a particular color
ordering, the initial expression from (3.12) is

ZT Zti+1 Zti Zt2

X
kr · ks Gsr

dtN −1 · · · dti dti−1 · · · dt1 ǫi · ǫj G̈B (ti − tj ) F exp B (5.2)
0 0 0 0 r<s

which becomes

ZT Zti+1 Zti Zt2

X
kr · ks Gsr

dtN −1 · · · dti dti−1 · · · dt1 ǫi · ǫj ĠB (ti − tj ) F exp B
0 0 0 0 r<s
 X 
× δ(ti+1 − ti ) − δ(ti − ti−1 ) − ki · km ĠB (ti − tm ) .
m6=i
(5.3)
The last term now fits in neatly with the terms in the generating kinematic factor
which lack G̈B ’s, but the delta functions — the surface terms from the IBP —

37
are an annoyance. (These delta functions contribute only to one color trace, so
there are no subtle factors of two associated with them.) Essentially they are color
commutators; they would cancel against surface terms from other proper-time
orderings were the theory abelian, but cannot do so here since different proper-
time orderings have independent color traces. Fortunately these surface terms bear
a simple relationship to the last term in (5.3). Specifically, take the terms in the
sum over m with m = i ± 1:

ZT Zti+1 Zti Zt2

− dtN −1 · · · dti dti−1 · · · dt1
0 0 0 0
X X
kr · ks Gsr

ǫi · ǫj ĠB (ti − tj ) ki · km ĠB (ti − tm ) F exp B .
m=i±1 r<s
(5.4)
[10]
Now, motivated by the pinch rules of section 3 and the work of Bern and Dunbar ,
i,i±1
replace ki · ki±1 ĠB with ∓1 and set ti = ti±1 ; in this way the surface terms are
reproduced.

The case j = i ± 1 is special: one of the surface terms contains Ġjj

B ≡ 0, and so
the pinch ti = tj does not get a contribution from the IBP. This leads to a modifi-
i,i±1 2
cation of the rule for “pinching”: the pinch of a term containing (ĠB ) vanishes.
(Again this matches with Bern and Kosower and with section 3.) Recall that G̈ij
[7]
B
contains a delta function, which accounts for the Feynman graph in which a four-
point vertex connects gluons i and j; the missing surface term is cancelled by the
half of this delta function that contributes to the color trace under consideration.

In addition to terms like (5.1), the kinematic factor of eq. (3.12) has terms
in which F (ǫm , kn ) contains ĠB functions dependent on ti and tj , or in which
there are several G̈B ’s; these cases must be dealt with in turn. It appears that the
resulting pinches are governed by simple rules, which I will now present. However,
as mentioned above, no proof exists for these rules; their main feature is their
similarity to the Bern-Kosower rules.

The first stage of the IBP reorganization involves the elimination of all G̈B ’s

38
in analogy to eqs. (5.2)–(5.3). Specifically, carry out the IBP of the generating
kinematic factor, dropping all surface terms, until no G̈B ’s remain. (Bern and
[8]
Kosower have proven that this is always possible. ) The result is the “improved
generating kinematic factor”, associated with the unpinched diagram. Every term
in this improved kinematic factor contains a certain number of factors of ki · kj ,
where i and j are arbitrary. The number of these factors cannot exceed N/2, since
the maximum number of G̈ij ij
B ’s and ki ·kj GF ’s in any term in the original generating
kinematic factor is also N/2. Each pinch absorbs one of these factors, as well as
one of the integrals over ti , and so the maximum number of pinches which must
be performed simultaneously is N/2.

If the theory is abelian, then no further calculation is necessary, as all surface

terms do in fact vanish. However, in a non-abelian theory, it is necessary to use the
following pinch rules in order to account for the IBP surface terms. The procedure
is closely related to the Bern-Kosower rules for scattering amplitudes; the reader
is again urged to review reference 7.

Draw all (planar) φ3 graphs with one loop, N external legs and any number NT
of trees, such that although each tree may have several vertices, the total number of
tree vertices NV is at most N/2. (Diagrams with trees may seem out of place in the
construction of a 1PI object like an effective action, but the trees used here, unlike
those for scattering amplitudes, do not contribute the usual propagator poles; they
serve only as a mnemonic for ensuring all surface terms are accounted for.) The
gluons which flow into a tree before entering the loop are said to be pinched;
the number of these is NV + NT . Consider a particular graph and a particular
color(path)-ordering; label the external legs clockwise from 1 to N following the
path-ordering. Each tree vertex, since it is a three-point vertex, is characterized
by one line pointing toward the loop and two outward pointing lines I and J, with
two sets of external legs i1 , ..., im and j1 , ..., jn that flow into them. Let J be the
line lying most clockwise. Now examine the improved generating kinematic factor
term by term. If a given term does not contain a factor ki · kj Ġji ji
B or ki · kj GF
for each tree vertex, where i belongs to the set of gluons flowing into line I and j

39
flows into J, then it vanishes. Even then, it must contain exactly one Ġji ji
B or GF
at each vertex; otherwise it vanishes. If it survives, then replace Ġji ji
B or GF by +1,
replace ti → tj in all Green functions, and eliminate the ti integral. Finally, for
every internal tree line (into which flows momentum from gluons r, r + 1, . . . , s),
divide by
s s
1h X 2 X i
( kq ) − (kq )2 , (5.5)
2 q=r q=r

which becomes the expected intermediate-state pole only when all external gluons
are on-shell. The effect of this procedure is to produce contact terms; no actual
poles are ever generated.

It is useful to review the arguments of Bern and Kosower for carrying out the
[6,7,9]
IBP. After the IBP, the improved generating kinematic factor is made up of
only ĠB ’s and GF ’s; it has no singularities and contains no dependence on T .
This simpler form leads to fewer separate integrations, and also allowed Bern and
Kosower to construct a formalism in which one needs only φ3 graphs to compute
scattering amplitudes. In addition, since the kinematic factor is independent of
T , the overall power of T is given by the number of ti integrations; a diagram
with N gluons and k pinches has an integral dT /T 3−N +k . As a consequence,
R

the ultraviolet infinities of gauge theory appear only in terms with N − 2 pinches,
R
since dT /T is the only possible source of ultraviolet divergences. Indeed one
may interpret this reorganized amplitude using gauge invariant structures. I will
illustrate this in a simple example below, and will discuss this further in later work.

To see the IBP in action, let us apply it to the vacuum polarization in (2.33):

(gµǫ/2 )2 a b
h i
Π = Γ2 (k1 , k2 ) = Tr(T T ) ǫ1 · ǫ2 k1 · k2 − ǫ1 · k2 ǫ2 · k1
(4π)2−ǫ/2
Z∞ Z1 (5.6)
dT
du ek1 ·k2 GB (1−u) [ĠB (1 − u)]2 .
T 1−ǫ/2
0 0

This expression has the remarkable property of being explicitly transverse. In usual
techniques this property is not visible until the full set of integrations is complete.

40
(This is the full result; since the integrand contains two powers of Ġ12
B , there is no
pinch contribution. Of course this will always be true for a two-point function.)
In fact, (5.6) represents precisely the (Aµ )2 piece of F µν Fµν , which appears as the
only infinite term in the unrenormalized effective action. In light of the previous
paragraph, it will not surprise the reader that other infinities, namely the one-pinch
piece of the (Aµ )3 term and the two-pinch piece of the (Aµ )4 term of the effective
action, reproduce explicitly the remaining pieces of F µν Fµν . Additionally, since
one may perform at most N/2 pinches, there are no infinities beyond N = 4 in the
effective action. Thus, even though the complicated process of pinching replaces
the many diagrams of Feynman rules, the IBP and the Bern-Kosower-type pinch
rules allow for a clearer separation of the different types of contributions to the
effective action. This may prove useful in the analysis of the divergence structure
of more complex theories.

Another interesting feature of this reorganization is illustrated through the IBP

of (3.28):

(gµǫ/2 )2 a b
h i
Π = −2 Tr(T T ) ǫ1 · ǫ2 k 1 · k 2 − ǫ1 · k 2 ǫ2 · k 1
(4π)2−ǫ/2
Z∞ Z1 (5.7)
dT k1 ·k2 GB (1−u)

2 2

du e [Ġ B (1 − u)] − [G F (1 − u)] .
T 1−ǫ/2
0 0

[6,13]
As pointed out by Bern and Kosower , the IBP allows use of worldline super-
symmetry in a clever way. Were the system truly worldline supersymmetric, the
effective action would vanish. Supersymmetry would require that both xµ and ψ µ
satisfy periodic boundary conditions, so that Ġij ij
B and GF would be equal. It follows
that every supersymmetric amplitude expressed as a function of only ĠB and GF
would vanish under the formal replacement Ġij ij
B → GF . However, in (3.12) the only
dependence on boundary conditions is hidden in the Green functions themselves;
the functional dependence on the Green functions is the same in all cases. As a
result, even when xµ and ψ µ have different boundary conditions the replacement

41
Ġij ij
B → GF everywhere in the improved kinematic factor (and use of momentum
conservation) leads to a complete cancellation. In particular, the result of (5.7)
has this property. This trick can be used as a check on the algebra of the IBP.

To find the vacuum polarization for a vector boson loop, follow the rules in
section 4. Specifically, take eq. (5.7), replace (G21 2 21 12
F ) = −GF GF by +8, multiply
1
the term with (Ġ21 2
B ) by 2, and multiply the entire expression by − 4 :

(gµǫ/2 )2 a b
h i
Π= Tr(T T ) ǫ 1 · ǫ2 k 1 · k 2 − ǫ1 · k 2 ǫ2 · k 1
(4π)2−ǫ/2
Z∞ Z1
dT T k1 ·k2 (a−a2 )

2

da e (1 − 2a) − 4
T 1−ǫ/2
0 0
(5.8)
(gµǫ/2 )2 h
ab
i
= N
2−ǫ/2 c
δ ǫ1 · ǫ k
2 1 · k 2 − ǫ1 · k ǫ
2 2 · k 1
(4π)
 Z∞ Z1 
dT 
2

da (1 − 2a) − 4 + finite.
T 1−ǫ/2
0 0

The reader may easily check that the same result is obtained by integrating (4.37)
by parts, and that the divergent term yields the usual 11/3 associated with the
Yang-Mills beta function.

6. Colorful Comments

It is often desirable to use a formulation, referred to as “color-ordered”, in

which only group matrices in the fundamental representation appear; the useful-
[1,2,8]
ness of this approach for scattering amplitudes is detailed in the literature. In
particular, the utility of computing color-ordered tree-level partial amplitudes using
[2]
color-ordered Feynman diagrams was emphasized by Mangano, Parke and Xu .
[8]
A study of color-ordering in loop graphs was performed by Bern and Kosower ,
using the techniques of open-string theory, in which these color traces, known as
[23]
Chan-Paton factors , appear automatically.

42
 To arrive at such a formulation in the language of this paper, one should write
the effective action as a product of parallel or antiparallel Wilson loops. Since in
U(Nc ) the U(1) photon decouples from the SU(Nc ) gluons, one-loop amplitudes
[1]
for SU(Nc ) can be calculated using U(Nc ) ; working with the full unitary group
[1,8]
allows the use of a number of useful tricks. If the particle in the loop lies in the
adjoint representation of U(Nc ), one may consider it as a sort of “bound state” of
a fundamental Nc and an antifundamental Nc representation; some of the external
vector bosons couple to the Nc while others couple, independently, to the Nc . For
a scalar particle, the effective action is

Z∞ ZT
dT 1 2 i
Z h
Γ[A] = N Dx exp − dτ ( ẋ )
T 2E
0 0
(6.1)
h ZT i  h ZT i†
TrNc exp dτ (igA · ẋ) TrNc exp dτ (igA · ẋ) ,
0 0

where the gauge field is a matrix in the fundamental representation. The first
trace is path-ordered, while the second is anti-path-ordered. In such an expression
it becomes immediately obvious that one expects contributions with one or two
group traces at the one loop-level, as is well-known to those familiar with the
[24] [8]
double line formalism of ’t Hooft or with open string theory. Rewriting (3.12)
in this form changes only the trace structure: letting X a (T a ) be the group matrices
in the adjoint (fundamental) representation, we replace

N
X
aN a1
Tr(X ···X ) → (−1)m Tr(T bN−m · · · T b1 )Tr(T c1 · · · T cm ) (6.2)
m=1

where tbi+1 > tbi and tcj+1 > tcj . Thus we divide the gluons into two sets, writing
down a path-ordered trace for one and an anti-path-ordered trace for the other,
and sum over all sets and all orderings. If m = 0 or N the trace of the unit matrix
yields a factor of Nc . Notice that for N = 2 the traces with m = 0 and m = 2 are
equal, while the case N = 4, m = 2 appears twice in this sum since it is invariant

43
under proper-time-reversal; this accounts for the factors of two which appear for
[6]
these traces in the Bern-Kosower rules.

Each color trace in (6.2) is internally path-ordered, but operators in different

traces may be integrated past each other without altering the color structure. As a
result, surface terms from the IBP and the operator Oi,j (eq. (3.19)) only appear for
gluons lying adjacent to each other in the same color trace; we must therefore only
pinch gluons in the same trace. Again this is in agreement with the Bern-Kosower
[6]
rules. (For vector particles, the rules for GF chains are unaffected by changes in
the organization of color; for a chain to contribute it must still be path-ordered as
in (4.31) or (4.32).)

It may have occurred to the reader educated in string theory that although I
treated color using a Wilson-loop formalism related to the open string, I might have
introduced color via the use of internal currents as in the closed string. This has
[18]
been discussed in the literature. Such a treatment can easily be implemented, and
[6]
rules can be derived using an approach very similar to that of Bern and Kosower ;
however this is somewhat more complicated than the technique used in this paper.

7. Some Extensions

There are a number of additional theories that are simple to construct. For
example, to study massive scalars or spinors in a background gauge field, add a
mass term to the particle Lagrangian, as in eq. (2.3):

1
L → L − Em2 (7.1)
2

where E is the einbein, and I work in Minkowski spacetime. From the point of
view of one-dimensional general relativity, this is just a cosmological constant. In

44
the gauge E = 2, the scalar effective action becomes

Z∞ ZT
dT 1 2
Z h i
2
Γ[A] = − N Dx exp − dτ ( ẋ − m − igA · ẋ)
T 4
0 0
(7.2)
Z∞ ZT
dT 1
Z h i
2
=− N e+m T Dx exp − dτ ( ẋ2 − igA · ẋ) .
T 4
0 0

2
Thus the effect is merely to add a factor of e+m T to the integrand of the integral
over T. Exactly the same factor occurs for massive spinors. In Euclidean spacetime
2
the factor is e−m T , which illustrates the decoupling of particles as m → ∞.

Another straightforward modification is the inclusion of background scalars.

Consider the theory
1
L = (∂φ)2 − V (φ) (7.3)
2

The one-loop particle Lagrangian of a scalar particle in a background scalar field

can be found by letting φ = Φ + δφ, where δφ is a quantum fluctuation around the
classical field Φ, and keeping only the terms quadratic in δφ.

1 2 1
L= ẋ − EV ′′ (Φ). (7.4)
2E 2

A prime denotes a derivative with respect to Φ. Notice that mass terms for the
scalar arise correctly from this formula.

Spinors interact with this field in a slightly more complex way; the Yukawa
interaction hΦΨΨ is easily incorporated in analogy to eq. (3.2):

h
i
Γ[A] = log 6 − hΦ
det i D
1 h

i
= log det i D 6 − hΦ − i D 6 − hΦ (7.5)
2
1 h
i
= log det D 6 2 1 − ih D
6 Φ + h2 Φ2
2

45
The associated spinor particle has Lagrangian
1 2 1
L= ẋ + ψ ψ̇ − h2 Φ2 + ihψ µ Dµ Φ . (7.6)
2E 2
Notice that the one-scalar vertex operator for Φ = eik·x is VΦ = −ih(ik · ψ)eik·x ,
as in string theory. If we let the scalar field have a vacuum expectation value v,
and let Φ′ = Φ − v, then (7.6) becomes
1 2 1
L= ẋ + ψ ψ̇ − (hv)2 − 2h2 vΦ′ − h2 Φ′2 + ihψ µ ∂µ Φ′ . (7.7)
2E 2
Of course the particle picks up a mass mΨ = hv, and the scalar vertex operator
becomes VΦ = −ih(ik · ψ − 2imΨ )eik·x .

More interesting is the interaction of a vector boson with a scalar. At this

point we should remember that a single background scalar can change the particle
in the loop from a vector into a scalar! We must therefore build a theory which
consistently describes a particle that can be either scalar or vector. Again string
theory is a guide; simply use dimensional reduction. Extend the vector theory of
section 4 to a fifth dimension (add fields x4 , ψ±
4 ) but insist that the fifth component

of all momenta of all particles or fields must vanish. Since the momentum of the
particle must lie in the usual spacetime, a polarization vector pointing solely in the
x4 direction will always satisfy the physical condition ǫ · k = 0; thus the particle’s
new physical mode is a Lorentz scalar, while its others are unchanged. In short,
we have a theory of gauge bosons and a Higgs boson in the adjoint representation.

The reduction of (4.1) from five to four dimensions, with Φ ≡ A4 and φ ≡ Q4 ,

is
Z
S= d4 x {Qaµ [(D 2 + g 2 ΦΦ)ab gµν − g(Fρσ
c ρσ
J )µν f cab ]Qbν

+ gQaµ(Dµ Φ)c φb f abc − gφa (Dµ Φ)c Qbµ f abc

(7.8)
a 2 2 ab b a 2 2 ab b
− φ [(D + g ΦΦ) φ + ω (D + g ΦΦ) ω
+order(Q3 , Q4 , DQφ2 , ωQω, etc.)} .
This formula stems from the gauge D µ Qµ +ig[Φ, φ] = 0, called background ’t Hooft-
Feynman gauge. Notice that this gauge contains a new, gauge dependent Φ2 φ2 in-

46
teraction, different from the Φ2 φ2 interaction present in usual ’t Hooft-Feynman


[25]
gauge , in which ∂ µ Qµ + ig[hΦi , φ] = 0. It is clear from (7.8) that if Φ acquires
a vacuum expectation value the gluons, ghosts, and Goldstone bosons associated
with spontaneously broken generators have the same mass matrix:
D E
(M 2 )ab = g 2 hΦΦiab = g 2 f ace f bde Φc Φd . (7.9)

It is straightforward to add in the symmetry breaking potential for the Higgs boson,
and to extend this approach to Higgs bosons in other representations.

The particle Hamiltonian for this theory is

H = (pµ − gAµ )2 − (p4 − gΦ)2 − igψ µ Fµν ψ ν + 2igψ µ Dµ Φψ 4 ; (7.10)

when p4 is set to zero, the resulting Lagrangian is

1 2 4 4
L= ẋ + ψ+ · ψ̇− − ψ+ ψ̇− − g 2 Φ2 − igAµ ẋµ
2E (7.11)
+ igψ µ Fµν ψ ν + 2igψ µ Dµ Φψ 4 .

The last term is the one that turns a scalar in the loop into a vector, and vice
versa. When hΦi is non-zero the mass matrix of (7.9) is clearly generated. To add
in a Higgs potential V (Φ), use

4 4
L → L + V ′′ (Φ)(ψ+ ψ− ) ; (7.12)

the oscillator potential for ψ 4 assures that of the physical states only ψ+
4 |0i, the

scalar, will feel the potential. This sort of theory can be used — perhaps profitably
— for calculations in the standard model; a set of rules is in preparation.

Particles in background gravity may also be treated in this way. Consider a

theory of a scalar boson in a background metric Gµν :

1 E
L= Gµν ẋµ ẋν − m2 . (7.13)
2E 2
This is generally covariant with respect to both worldline and spacetime coordinate
redefinitions. One may extend this theory to particles with spin. The relevant

47
 [14]
Lagrangians were again written down by Brink, Di Vecchia and Howe , and I
[26]
shall not repeat them here. However, quantization of such a Lagrangian is subtle.
The technique for constructing internal gravitons also appears in reference 14:
instead of one complex set of worldline fermions, use two. Define a particle with
µ
an N=4 worldline supersymmetry, described by coordinates (xµ , ψ± , χµ± ). The
allowed states can be written down as in (4.6); projections onto odd ψ and χ
number and onto states which are even under ψ → χ leaves a rank-two symmetric
tensor as the propagating modes of the theory. While not particularly elegant, this
example illustrates that it is straightforward to construct a tensor of any arbitrary
rank and symmetry. It may be hoped that useful rules can be obtained from this
theory as well.

Finally, I should point out that every theory described in this paper is part
[27]
of the mode expansion of a string in a background string field. The possible
connection of this construction to the Bern-Kosower rules was noted by Bern and
[10]
Dunbar.

8. Conclusion

In this paper, I have shown that it is possible to construct one-loop effective

actions perturbatively without the use of Feynman diagrams, and with a method
that has certain conceptual and practical advantages over the standard technique.
By viewing a one-loop computation as a system of a particle (or superparticle)
in a background field, one can construct formulas and rules valid to all orders
in the background field which closely match the string-derived rules of Bern and
Kosower for gauge theory. It is now evident that one reason for the simplicity of
the Bern-Kosower rules compared to Feynman diagrams is that string theory is a
first-quantized system; the ease of one-dimensional as opposed to four-dimensional
calculations is clearly demonstrated both in this paper and in the work of Bern and
Kosower. The formalism developed in this paper represents a technical and con-
ceptual shift away from the standard techniques of path integral perturbative field

48
theory and back to basic quantum mechanics and the background field method.

Appendix: Conventions

In this paper I have used conventions which are appropriate for particles and
Wilson loops and which generate expressions that are simple to compare with those
of Feynman diagrams. Unfortunately they are not the most convenient from all
points of view, and indeed Bern and Kosower have chosen a very different set of
conventions. It is straightforward to convert from one to the other, and in this
appendix I explain how to do so.

First, let me review my conventions. I use

1
gµν = diag{+ − −−} ; Tr[T a T b ] = δ ab ;
2 (8.1)
Dµ = ∂µ − igAµ ; gFµν = i[Dµ , Dν ]

and for Grassmann integrations

Z
dθdθ θθ = 1 . (8.2)

[7]
To convert my expressions to those of Bern and Kosower:

1. Reverse the order of the color trace.

R
2. Write the Grassmann integral of (3.12) as dθi dθi (but keep eq. (8.2).

3. Replace ĠB with −ĠB .

4. Divide all ĠB and GF functions by 2.

√
5. Multiply all group matrices by 2.

6. Account for these factors of two by multiplying the entire amplitude by 2N/2 .

As a result,

7. The improved kinematic factor vanishes under ĠB → −GF .

49
 8. Pinches at a vertex with gluons j and i, j the most clockwise, result in the
replacement ki · kj Ġji ji 1
B (GF ) → +(−) 2 .

Acknowledgements

I have benefited enormously from discussions with a number of physicists;

their ideas and insights appear throughout this paper. Z. Bern and D. A. Kosower
answered many questions and helped me to understand the relation of their rules
to usual field theory concepts. I especially thank Z. Bern for explaining to me the
role of Schwinger proper time and for discussions on gauges, tree diagrams, and the
integration-by-parts procedure. I thank L. J. Dixon for explaining many aspects of
string theory, and particularly for important discussions about the mode expansion
of the string. R. Kallosh clarified certain issues concerning supersymmetry and
the background field method, and also pointed me toward the work of Brink, Di
Vecchia and Howe. D. C. Lewellen advised looking at first-quantized field theory
and suggested several useful papers. In addition to helping me with the many
subtleties of string theory, M. E. Peskin repeatedly pointed out the value of Wilson
loops in gauge theory, and made useful observations regarding the integration-
by-parts procedure and manifest gauge invariance. I also had useful discussions
with S. Ben-Menachem, A. W. Peet, Y. Shadmi, L. Susskind, L. Thorlacius and
B. J. Warr.

50
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52