This document provides a basic refresher on Green's functions as applied to scalar field theories. It outlines the Lagrangian for a basic Higgs boson theory and considers perturbative solutions. The Green's function is expressed as a Fourier integral, and contour integration is used to derive forms for both massless and massive scalar fields. For a massless scalar, the Green's function is shown to be proportional to the difference of two delta functions representing forward and backward traveling waves.

1. Dealing With Green’s Functions: A Basic Refresher
[Field Theory Highlights 2015 Set B]
Roa, F. J. P.
This is now the continuing part of Field Theory Highlights 2015, came the year 2016 though I
have chosen a different title for this present document that deals mainly on Green’s functions at a
basic utilitarian level as applied to scalar fields.
In [1] it is discussed that we can outline a basic Higgs boson theory from a basic SU(2)XU(1)
construction that excludes the needed fermions in the complete Electroweak theory. The
lagrangian of such a basic Higgs boson theory can be given by
(1.1)
 
3222
2
2)(
)(222
4
1
cos2
1
)(
2
1
)()(
2
1



 






 

ZZWWQxJmL T
(1.2)
222)(
)(2
cos2
1
2)( 

 


 yZZWWQxJT 





 

This is fundamentally a scalar field theory with the usual self-interaction terms that involve 2

and 3
 that come with the coupling constants  and  though in addition to these, there are
also another self-interaction terms that involve 2
 in which the massive gauge fields W and Z
partake in such terms.
Our initial approach towards the classical solutions is to consider the perturbative solution say, in
generic form
(2)
𝜂 = 𝜂0 + ∑ 𝜆2𝑙
𝜂𝑙
∞
𝑙=1
With respect to the coupling constant 𝜆 the 0th order field here is 𝜂0 and its corresponding
equation of motion is given by

2. (3)
𝜕𝜇 𝜕 𝜇
𝜂0 + 𝑚 𝜂
2
𝜂0 = 2𝑄′2
𝑊𝜇
(+)
𝑊(−)
𝜇
𝜂0 +
𝑄′2
𝑐𝑜𝑠2 𝛼
𝑍𝜇 𝑍 𝜇
𝜂0 − 𝐽 𝑇
Aside from the conventional source term that contains the source 𝐽 𝑇 we have the additional
source terms that contain the self-interacting scalar field 𝜂0 along with the massive gauge fields.
In the presence of sources, we may consider a generic Green’s function solution in the following
form
(4)
𝜑𝑐 =
1
(2𝜋)2
∫ 𝑑4
𝑥′
𝐺( 𝑥 − 𝑥′) 𝐽(𝑥′
)
(Here, note my placement of the integration variable 𝑥′
in the Green’s function and I am used to
this notation although we may resort to the symmetry of Green’s function in the integration to
interchange its place with the picking variable x but in doing so we must be specific of the
chosen causality.)
In my particular convention I write the fourier integral form of the Green’s function as
(5)
𝐺( 𝑥 − 𝑥′) = −
1
(2𝜋)2
∫ 𝑑4
𝑘
𝑒 𝑖𝑘 𝜎(𝑥−𝑥′
) 𝜎
−𝑘 𝜇 𝑘 𝜇 + 𝑀2 + 𝑖𝜖
where the fourier component is
(6)
𝐺̃( 𝑘) =
− 1
−𝑘 𝜇 𝑘 𝜇 + 𝑀2 + 𝑖𝜖
In this, I have already included the term 𝑖𝜖 to shift the poles whenever we perform the contour
integration. The metric signature used in this document is negative two so for instance, we can
write 𝑘2
= 𝑘 𝜇 𝑘 𝜇
= (𝑘0
)2
− 𝑘⃗ ∙ 𝑘⃗ .

3. To continue, let us tackle on (5) as applied in the scalar solution (4), deriving the form of the
Green’s function suited for the case in which the scalar is massless and resort to stationary phase
approximation to obtain for the form for the case wherein the scalar is massive.
The fourier integral form (5) assumes continuous four-momenta as the integration variables so
we may write the differential 1+3 volume as
(7.1)
𝑑4
𝑘 = 𝑑𝑘0
𝑑3
𝑘⃗
and considering the spherical momentum space we may actually write this as
(7.2)
𝑑4
𝑘 = 𝑑𝑘0
𝑑𝑘 𝑟 𝑘 𝑟
2
𝑑𝜑sin 𝜑 𝑑𝜙
0 < 𝜑 < 𝜋 , 0 < 𝜙 < 2𝜋
So in the metric signature of negative two, we can separate out the integrations indicated in (5)
and re-writing this into the following form
(8.1)
𝐺( 𝑥 − 𝑥′) =
1
(2𝜋)2
∫ 𝑑3
𝑘⃗ 𝑒−𝑖𝑘⃗ ∙ ∆𝑥
∫ 𝑑𝑘0
𝑒 𝑖𝑘0
∆𝑥0
(𝑘0)2 − 𝑏′2
(8.2)
𝑏′2
= 𝑏2
+ 𝑖𝜖 = 𝑘⃗ ∙ 𝑘⃗ + 𝑀2
+ 𝑖𝜖
(8.3)
∆𝑥0
= 𝑥0
− 𝑥′ 0
(8.4)
∆𝑥 = 𝑥 − 𝑥′

4. (8.5)
𝑘⃗ ∙ ∆𝑥 = 𝑘 𝑟 |∆𝑥| cos 𝜑
|𝑘⃗ | = 𝑘 𝑟
We perform contour integration for the integral
∫ 𝑑𝑘0
𝑒 𝑖𝑘0
∆𝑥0
(𝑘0)2 − 𝑏′2
contained in (8.1) and let us simply quote the results here as taken in the limit 𝜖 → 0. We shall
show the details of such contour integration in future attachments.
So quoting the results of the said contour integration we have
(9)
𝐺( 𝑥 − 𝑥′) = −
1
(2𝜋)2
∫ 𝑑4
𝑘
𝑒 𝑖 𝑘 𝜎( 𝑥−𝑥′)
𝜎
−𝑘 𝜇 𝑘 𝜇 + 𝑀2
= 𝑙𝑖𝑚 𝜖→0 −
1
(2𝜋)2
∫ 𝑑4
𝑘
𝑒 𝑖𝑘 𝜎( 𝑥−𝑥′)
𝜎
−𝑘 𝜇 𝑘 𝜇 + 𝑀2 + 𝑖𝜖
=
1
2
𝑖
2𝜋
∫ 𝑑3
𝑘⃗
1
𝑘0
𝑒 𝑖𝑘0 ( 𝑥0
− 𝑥′ 0)
𝑒−𝑖𝑘⃗ ∙( 𝑥− 𝑥′ )
Θ( 𝑥0
− 𝑥′ 0)
−
1
2
𝑖
2𝜋
∫ 𝑑3
𝑘⃗
1
𝑘0
𝑒 𝑖𝑘0
(𝑥′0
− 𝑥0
)
𝑒−𝑖𝑘⃗ ∙( 𝑥− 𝑥′
)
Θ(𝑥′0
− 𝑥0
)
Note here that we have set ( 𝑘0
)2
= 𝜔2
(𝑘⃗ ) = 𝑘⃗ ∙ 𝑘⃗ + 𝑀2
as evaluated in the limit as 𝜖 → 0,
where to first order in 𝜖 , 𝑏′
≈ √ 𝑘⃗ ∙ 𝑘⃗ + 𝑀2 +
𝑖𝜖
2√ 𝑘⃗ ∙ 𝑘⃗ + 𝑀2
, while inserting the Theta functions
Θ in (9) by hand to distinguish the given causality of each major term there.
Say for the moment, we have a massless scalar so that 𝑘0
= |𝑘⃗ | = 𝑘 𝑟 and specify the causality
𝑥0
> 𝑥′0
, given for | 𝑥| > |𝑥′⃗⃗⃗ | . By this, we can reduce (9) into
(10)
𝐺( 𝑥 − 𝑥′) =
𝑖
2
∫ 𝑑𝑘 𝑟 𝑒 𝑖𝑘 𝑟(𝑥0
− 𝑥′0
)
∫ 𝑑𝜑 𝑘 𝑟 sin 𝜑 𝑒−𝑖 𝑘 𝑟| 𝑥− 𝑥′| cos 𝜑

5. 0 < 𝜑 < 𝜋
Then we note the integral result
(11)
∫ 𝑑𝜑 𝑘 𝑟 sin 𝜑 𝑒−𝑖 𝑘 𝑟| 𝑥− 𝑥′| cos 𝜑
𝜋
0
=
1
𝑖| 𝑥 − 𝑥′|
( 𝑒 𝑖 𝑘 𝑟| 𝑥− 𝑥′|
− 𝑒−𝑖 𝑘 𝑟| 𝑥− 𝑥′|
)
so we may able to write (9) as
(12)
𝐺( 𝑥 − 𝑥′) =
1
2| 𝑥 − 𝑥′|
(∫ 𝑑𝑘 𝑟 𝑒
𝑖𝑘 𝑟(( 𝑥0
− 𝑥′0
) + | 𝑥− 𝑥′| )
− ∫ 𝑑𝑘 𝑟 𝑒
𝑖𝑘 𝑟(( 𝑥0
− 𝑥′0
) − | 𝑥− 𝑥′| )
)
If we are to integrate over the integration variable 𝑘 𝑟 from −∞ to ∞, then the remaining integrals
in (12) are just the integral definitions of delta functions. Thus, for a massless scalar field, we
have as its Green’s function the following form
(13)
𝐺( 𝑥 − 𝑥′) =
2𝜋
2| 𝑥 − 𝑥′|
(𝛿 ((𝑥0
− 𝑥′0
) + | 𝑥 − 𝑥′|)− 𝛿 ((𝑥0
− 𝑥′0
) − | 𝑥 − 𝑥′|) )
where the first major part refers to the backward traveling wave, while the second to the forward
traveling wave as specified with causality 𝑥0
> 𝑥′0
, given for | 𝑥| > |𝑥′⃗⃗⃗ |.
For example, we choose the wave to be a forward traveling wave so we may plug the appropriate
form of (13) in (4)
(14)
𝜑𝑐( 𝑥) =
1
(2𝜋)2
∫ 𝑑3
𝑥′
∫ 𝑑 𝑥′0
𝐺( 𝑥 − 𝑥′) 𝐽(𝑥′
)
𝐽( 𝑥′ ) = 𝐽(𝑥′
, 𝑥′0
)
Say from (13) we have

6. (15)
𝐺( 𝑥 − 𝑥′) = −
2𝜋
2| 𝑥 − 𝑥′|
𝛿 ((𝑥0
− 𝑥′0
) − | 𝑥 − 𝑥′|)
in (14) and the picking is at 𝑥′0
= 𝑥0
− | 𝑥 − 𝑥′| . Then (14) would just involve a time-retarded
source
(16)
𝜑𝑐( 𝑥) = −
1
2
2𝜋
(2𝜋)2
∫ 𝑑3
𝑥′
1
| 𝑥 − 𝑥′|
𝐽( 𝑥′
,(𝑥0
− | 𝑥 − 𝑥′|) )
In other attachments, we will continue on massive scalar.
[stopped: pp. 1H, Prep Notes: Scalar Field Theories - … ]
Ref’s
[1]Roa, F. J. P., Field Theory Highlights 2015 Set A (slideshare)
[2]W. Hollik, Quantum field theory and the Standard Model, arXiv:1012.3883v1 [hep-ph]
[3]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html
[4]’t Hooft, G., THE CONCEPTUAL BASIS OF QUANTUM FIELD THEORY,
http://www.phys.uu.nl/~thooft/
[5]Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
[6]Cardy, J., Introduction to Quantum Field Theory
[7]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory