This document presents an overview of classical Feynman graphs for the Higgs boson. It expands the classical Higgs boson solution perturbatively up to third order in the coupling constant. In the 0th order of this expansion, the first few classical Feynman graphs are constructed. Specifically, it describes how there are five classical graphs looking at terms in the equations of motion involving the W and Z fields coupled to the 0th order Higgs field. It then provides an example of writing one such solution in coordinate space and transforming it to momentum space, depicting the vertex and propagator that would represent it.

1. Basic Higgs Boson Classical Feynman Graphs
Roa, F. J. P.
Abstract
In this expository paper, first few examples of 0th
-Order classical Feynman graphs for the Higgs boson are
presented. The classical Higgs boson solution is expanded perturbatively up to third order in the coupling
constant and in the 0th
-order of this expansion, first few classical Feynman graphs are constructed.
Keywords:
1. Introduction
The Higgs boson η is a component
field in the scalar doublet called the Higgs field
φ . As a doublet the Higgs field transforms
under )1()2( XUSU non-Abelian gauge group
to which belongs the diagonal sub-group for the
scalar doublet. In its structure as a doublet, its
fundamental representation is given by[1]






+
+





=+=
εη
ϕ
β
ϕφφ
i
1
0
0
(1)
which can be considered as a field perturbed
from the ground state represented by the constant
part 





=
β
φ
0
0 [2]. Under the diagonal
)1()2( XUSU subgroup,
)1),2exp((
)exp()exp( 3
q
qq
idiag
ii
χ
χσχ
−
=−−
(2)
this scalar doublet’s hypercharge is 1=Y , and
the symmetry under this subgroup, in which the
scalar doublet transforms as
φχσχφ )exp()exp( 3 qq ii −−→ , dictates
that the covariant derivative operator be of the
form
( )
( )
( ) ασ
ασα
α
σσ
µ
µ
µµµµ
sin1
cossin
cos
3
2
3
2
)2(2)1(1
em
AQi
Z
Q
i
WWQiD
+′
+−
′
++′+∂=
(3)
In this form, the transformation
φχσχφ )exp()exp( 3 qq ii −−→ is
accompanied by a gauge transformation
q
emem
eAA χµµµ ∂+→ −1
2 (4)
given that 2/sin eQ =′ α . This gauge
transformation in
em
Aµ will cancel the extra term
picked up in the differentiation
( )φχσχµ )exp()exp( 3 qq iiD −− so as to
render the kinetic term
2
2
1
φµD invariant
under the diagonal subgroup (2). The massive
vector fields µ)1(W and µ)2(W undergo SO(2)
rotations or rather, mixing up of these vector
fields under this particular subgroup, while
em
Aµ
transforms under (4),
q
q
W
WWW
χ
χ
µ
µµµ
2sin
2cos
)2(
)1()1()1( +=′→
(5.1)
and
q
q
W
WWW
χ
χ
µ
µµµ
2cos
2sin
)2(
)1()2()2( +−=′→
(5.2)
This is so since this diagonal subgroup (2) does
not commute with the Pauli matrices 1σ and
2σ . In this particular subgroup of symmetry, the
remaining massive vector field µZ is neutral
under the transformation
φχσχφ )exp()exp( 3 qq ii −−→ .
A basic lagrangian can be constructed
for this scalar doublet that accommodates the
said symmetry under this particular diagonal
subgroup.
( )RLLR
y
VDL
22
2
][
2
1
φψψψφψ
φφµφ
+
−−=
⇓
(6)
In this lagrangian, aside from the Higgs field,
there is also the presence of Dirac fields: the

2. lefthanded spinor doublet
L
ψ and the
righthanded spinor singlet
R
2ψ . These fields
represent the first generation lightweight
fermions. The lefthanded spinor doublet
transforms in the same SU(2)XU(1) diagonal
subgroup although with a different hypercharge
1−=LY , while the righthanded spinor singlet
transforms under a unimodular symmetry
)exp()1( qRR iYU χ−= with a hypercharge
RY . Together
R
qR
R
Yi 22 )exp( ψχψ −→ (7.1)
L
qqL
L
iiY ψχσχψ )exp()exp( 3−−→
(7.2)
along with φχσχφ )exp()exp( 3 qq ii −−→ ,
these transformations will render the Yukawa
coupling
( )RLLR
Y yL 22 φψψψφψ +−= ⇓
(7.3)
invariant provided that the hypercharge
1−= LR YY . It is this Yukawa coupling that
gives masses to these basic lightweight fermions
in their interactions with the Higgs field. This
can be seen by decomposing this coupling term
into a mass term, which gives masses to the said
fermions, and an interaction term that represents
the interaction of the Higgs boson with these first
generation fermions.
( )
22
2222)(
ψψβ
ψψψψβ
y
yL RLLR
massY
−
=+−=
(8.1)
22(int) ψψηyLY −= (8.2)
In both (8.1) and (8.2) we have made use of the
combinations
( ) 2
5
222 1
2
1
ψγψψψ +=LR
( ) 2
5
222 1
2
1
ψγψψψ −=RL
(9)
given the fundamental representation of the
lefthanded spinor doublet








= L
L
L
2
1
ψ
ψ
ψ . In this
column matrix,
L
1ψ is the lefthanded neutrino
which is no longer included in (8.1), so it is
massless. The lefthanded electron
L
2ψ , as well
as the righthanded one
R
2ψ , gains mass through
(8.1). By gauge choice, the other components in
the Higgs field, the Goldstone Bosons ε and 1ϕ
must vanish, 01 == ϕε , leading to (8.2). This
part of the Yukawa coupling says that it is only
the Higgs boson that interacts with the fermion
fields while the Goldstones disappear. With this
gauge choice, the Higgs potential ][φV
simplifies so as to have only the Higgs boson
present and has gained mass in the same way the
fermion fields have acquired masses through
their interactions with the ground or vacuum
state 0φ .
324222
4
1
2
1
][ βηληληφ η ++= mV (10)
Note that in Heaviside units, 1==ch , so in
these units, the Higgs boson mass ηm is
proportional to λβ . The kinetic part
2
2
1
φµD ,
given this gauge choice also simplifies to
( )( )
2
0
2
2)(
)(2
2)(
)(2
2
2
1
cos2
1
cos2
1
2
2
1
2
1
φ
η
α
η
α
β
ηηφ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ
µµ
D
ZZWWQ
ZZWWQ
D
+





+′
+





+′
+∂∂=
−
+
−
+
(11)
In here, the vector fields W and Z acquire mass
through
2
0
2
1
φµD , also with the interaction
with the vacuum state,






+′= −
+ µ
µ
µ
µµ
α
βφ ZZWWQD 2)(
)(222
0
cos2
1
2
1
(12.1)
βQmW
′∝
Wz mm
αcos
1
=
( )µµµ )2()1(
)(
2
1
WiWW ±=±
(12.2)
To be noted is that the electromagnetic field
em
Aµ decouples from interaction with the vacuum
state that is, 0)1( 03 =+ em
Aµφσ . Thus, this
field does not acquire mass, and interacts only

3. with the Goldstone boson 1ϕ that it eats this
away in the term
em
Aµϕσ )1( 3+ .
2. Higgs boson Lagrangian and the
equations of motion
From the Higgs field lagrangian (6) and
with the gauge choice 01 == ϕε that earlier
led to some simplifications, we pick up the
lagrangian for the Higgs boson
( )
3222
2
2)(
)(2
22
4
1
cos2
1
)(
2
1
)()(
2
1
ηβληλ
η
α
ηηηη
µ
µ
µ
µ
η
µ
µη
−
−+′
+−−∂∂=
−
+
ZZWWQ
xJmL T
(13.1)
in the metric signature -2.
22
2)(
)(2
cos2
1
2)(
ψψ
α
β µ
µ
µ
µ
y
ZZWWQxJT
+






+′−= −
+
(13.2)
The resulting equation of motion is then given by
03
cos2
1
2)(
2232
2)(
)(2
2
=+
+





+′−
++∂∂
−
+
ηβληλ
η
α
ηη
µ
µ
µ
µ
η
µ
µ
ZZWWQxJ
m
T
(13.3)
We wish to expand the solution in terms of some
coupling constant
2
λ , so perturbatively
∑
∞
=
+=
1
2
0
l
l
l
ηληη (13.4)
and consider up to 3rd
order only in that coupling
constant. With this perturbation, our classical
equation of motion (13.3) can now be resolved
into the following equations of motion
)(
cos2
1
2 02)(
)(2
0
2
0
xJ
ZZWWQ
m
T−
=





+′
−+∂∂
−
+
η
α
ηη
µ
µ
µ
µ
η
µ
µ
(14.1)
2
0
3
0
12)(
)(2
1
2
1
3
cos2
1
2
ηβη
η
α
ηη
µ
µ
µ
µ
η
µ
µ
−−
=





+′
−+∂∂
−
+
ZZWWQ
m
(14.2)
101
2
0
22)(
)(2
2
2
2
63
cos2
1
2
ηηβηη
η
α
ηη
µ
µ
µ
µ
η
µ
µ
−−
=





+′
−+∂∂
−
+
ZZWWQ
m
(14.3)
)2(3)(3
cos2
1
2
20
2
120
2
10
32)(
)(2
3
2
3
ηηηβηηηη
η
α
ηη
µ
µ
µ
µ
η
µ
µ
+−+−
=





+′
−+∂∂
−
+
ZZWWQ
m
(14.4)
These in the perturbative expansion
3
6
2
4
1
2
0 ηληληληη +++= (14.5)
3. Classical Feynman graphs in the
0th
order
We proceed to constructing the
Feynman graphs but only in the 0th
order,
corresponding to the solutions of (14.1). Looking
up to (13.2), this source contains three terms plus
those terms in (14.1) involving the fields W and
Z coupled to 0η so, there will be five classical
graphs. We count those terms in (14.1)
containing the W and Z fields that are coupled to
0η as another source in addition to source, TJ .
In coordinate space, we write one
solution to (14.1) as
)()()(
2)(
)2(
1
)(
0)(
)(
2
0
4
210
xxWxW
QxxGxdx
′′′
′′−′=
−
+
∫
η
π
η
µ
µ
(15.1)
To be noticed in here is that the field 0η itself is
part of the source but only existing at a different
location x′ , and 10 )(xη is associated with the
propagator )(0 xxG ′−= .
Through fourier transformation, we can
write (15.1) in momemtum space

4. )exp(
)(~)(
~
)(
~
2
1
)2(
1
)(
3
1
)(
)3(
0
)2(
)(
)1()(2
2
3
1
4)(4
610
σ
σ
µ
µ
η
µ
µ
δ
η
π
η
xikkk
kkW
kWQ
imkk
kdkdx
l
l
i
i






−
′
∈++−
=
∑
∫ ∫∏ ∫
=
−
+
=
L
(15.2)
We have already shifted the poles in (15.2) by
including ∈i , which puts causal significance to
the propagator in (15.1). We will note the
symmetry in the delta function that is,
)()( aa −=δδ in writing for the vertex that this
delta function represents in momentum space.
Say for example, 10 )(xη propagates from x′ to
x , where it is given that xx ′>
rr
for
00
xx ′> , and at x′ the sources
)(+
µW ,
)(−
µW
and 0η have emerged that caused the
propagation of the field 10 )(xη from x′ to x .
At x′ and in momentum space the graph looks
like this
In here, all incoming four-momenta are
)1(
k ,
)2(
k and
)3(
k , while )(~
0 kη is outgoing,
carrying the four-momentum k . The vertex is
represented by the delta function






−∑=
3
1
)(
l
l
kkδ , wherein the outgoing four-
momentum is positive while the incoming ones
are negative but since delta functions are
symmetric, we can switch signs between the
incoming and outgoing four-momenta. The delta
function in (15.2) corresponds to a scalar field
propagator in (15.1), written in momentum space
as
∫ ∈++−
′−
=′−
imkk
xxik
kd
xxG
2
4
2
0
))(exp(
)2(
1
)(
η
µ
µ
σσ
σ
π
(15.3)
This is already shifted in poles. One complex
pole is at
2
20
1
2 η
η
mkk
i
mkkkq
+⋅
∈
++⋅≈ rr
rr
to 1st
order in ∈. The upper half contour in the
complex plane encloses this pole and in the limit
0∈→ , this leads to the propagator
∫
′−
′−Θ−
=′−+
0
300 ))(exp(
)(
4
)(
k
xxik
kdxx
i
xxG
σσ
σ
π
r
(15.4)
defined causally for xx ′>
rr
and
00
xx ′> at
20
ηmkkk +⋅=
rr
for a field 10 )(xη that is
traveling forward in time from x′ to x .
Because (15.3) has two poles, one of which leads
to (15.4), our solution (15.1) contains a part that
corresponds to the other pole enclosed by the
lower half contour but we no longer dwell into
this other part in which 10 )(xη travels
backward, from x to x′ although carrying
negative spatial momenta.
The part of the lagrangian (13.1) that
the above graph depicts is given by
2
)(
)(2
ηµ
µ −
+
′ WWQ (15.5)
The next one solution to (14.1) as
represented in coordinate space is
)()()(
cos
)(
)2(
1
)(
0
2
2
0
4
220
xxZxZ
Q
xxGxdx
′′′
′
′−′= ∫
η
απ
η
µ
µ
(16.1)
and in momentum space,

5. )exp(
)(~)(
~
)(
~
cos
1
)2(
1
)(
3
1
)(
)3(
0
)2()1(
2
2
2
3
1
4)(4
620
σ
σ
µ
µ
η
µ
µ
δ
η
α
π
η
xikkk
kkZkZ
Q
imkk
kdkdx
l
l
i
i






−
′
∈++−
=
∑
∫ ∫∏ ∫
=
=
L
(16.2)
The interaction involved in this solution is
depicted by the following graph,
The same vertex as in (15.2), where all incoming
four-momenta are
)1(
k ,
)2(
k ,
)3(
k , while k is
the outgoing one carried out by )(~
0 kη as
20 )(xη propagates forward in time from x′ to
x . This causality has the same propagator (15.4)
and all sources µZ , µZ and 0η have emerged
at x′ prior to the arrival of 20 )(xη at some
point x . The part of the Higgs boson lagrangian
depicted in this graph represents the interaction
2
2
2
cos2
η
α
µ
µ ZZ
Q′
(16.3)
The other remaining graphs are those
for the fields contained in the source TJ in its
interaction with the Higgs boson η . This
interaction is included in the Higgs boson
lagrangian and is represented by the term
ηTJ− . We have from this the 3rd
solution
)()(
2)(
)2(
1
)(
)(
)(
2
0
4
230
xWxW
QxxGxdx
′′
′′−′=
−
+
∫
µ
µ
β
π
η
(17.1)
or in momentum space
)exp(
)(
~
)(
~
2
1
)2(
1
)(
2
1
)(
)2(
)(
)1()(
2
2
2
1
4)(4
430
σ
σ
µ
µ
η
µ
µ
δ
β
π
η
xikkk
kWkW
Q
imkk
kdkdx
l
l
i
i






−
′
∈++−
=
∑
∫ ∫∏ ∫
=
−
+
=
L
(17.2)
The vertex 





−∑=
2
1
)(
l
l
kkδ in (17.2) has two
incoming four-momenta,
)1(
k and
)2(
k , and an
outgoing one k . The causality in this interaction
is given by the propagator (15.4), which also
applies here as it is given the causal order that
30 )(xη propagates from x′ to x . At x′ the
sources
)(+
µW and
)(−
µW have emerged prior to
the arrival of 30 )(xη at some point x . In
momentum space, we can depict this interaction
at x′ by the following graph
The term in ηTJ− that this interaction is
pictured out is
ηβ µ
µ )(
)(2
2 −
+
′ WWQ (17.3)
For the fourth one as contained in
ηTJ− , we have
)()(
cos
)(
)2(
1
)( 2
2
0
4
240
xZxZ
Q
xxGxdx
′′
′
′−′= ∫
µ
µ
α
β
π
η
(18.1)
as also in momentum space

6. )exp(
)(
~
)(
~
cos
1
)2(
1
)(
2
1
)(
)2()1(
2
2
2
2
1
4)(4
440
σ
σ
µ
µ
η
µ
µ
δ
α
β
π
η
xikkk
kZkZ
Q
imkk
kdkdx
l
l
i
i






−
′
∈++−
=
∑
∫ ∫∏ ∫
=
=
L
(18.2)
This depicts the interaction drawn on the
Feynman graph
Corresponding to the term in ηL that is given by
η
α
β µ
µ ZZ
Q
2
2
cos
′
(18.3)
The fifth remaining one in this 0th
order
solution involves the interaction between two
first generation lightweight fermions from which
interaction the Higgs boson arises
In coordinate space this is given by
)()(
)()(
)2(
1
)(
22
0
4
250
xx
yxxGxdx
′′
−′−′= ∫
ψψ
π
η
(19.1)
while in momentum space
)exp(
)(~)(
~
)(
1
)2(
1
)(
2
1
)(
)2(
2
)1(
2
2
2
1
4)(4
450
σ
σ
η
µ
µ
δ
ψψ
π
η
xikkk
kk
y
imkk
kdkdx
l
l
i
i






−
−
∈++−
=
∑
∫ ∫∏ ∫
=
=
L
(19.2)
In ηL , this is the term given by
ηψψ 22y− (19.3)
(stopped: pp. 19, draft)
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[1]W. Hollik, Quantum field theory and the
Standard Model, arXiv:1012.3883v1 [hep-ph]
[2]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTco
urse.html
’t Hooft, G., THE CONCEPTUAL BASIS OF
QUANTUM FIELD THEORY,
http://www.phys.uu.nl/~thooft/
Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
Wells, J. D., Lectures on Higgs Boson Physics
in the Standard Model and Beyond,
arXiv:0909.4541v1
Dittmar, M., Searching for the Higgs and other
Exotic Objects (A “How to” Guide from LEP to
the LHC), arXiv:hep-ex/9901009v1
Cardy, J., Introduction to Quantum Field Theory
Gaberdiel, M., Gehrmann-De Ridder, A.,
Quantum Field Theory II