(1) This document discusses the vacuum-to-vacuum matrix in the presence of a cubic self-interaction term in the Hamiltonian for a scalar field.
(2) The vacuum-to-vacuum matrix, which gives the probability that a particle initially in the vacuum state remains in the vacuum state, is modified by the inclusion of the cubic self-interaction term.
(3) The modified vacuum-to-vacuum matrix is expressed in both coordinate space and momentum space, and can be depicted using Feynman diagrams involving three propagators and two vertices.
3. The resulting matrix (6), upon the setting of J = 0, consists of two groups (aside from the term
with numerical 1) of relevant identical terms. These terms are as written in coordinate space.
Let us take the group of terms with the numerical factor
1
12
and write this in momentum space
(8)
β« π4 π¦ β²π4 π¦
πΊ(π¦ β² β π¦)
(2π)2
πΊ( π¦ β π¦ β² )
(2π)2
πΊ(π¦ β² β π¦)
(2π)2
π΅
π΄
=
1
(2π)4
β« π4 π β« π4 π β²β« π4 π β²β²
1
βπ2 + π2 + ππ
1
βπ β²2 + π2 + ππ
1
βπ β²β²2 + π2 + ππ
πΏ4( π β π β²
+ π β²β² ) π¦ β² πΏ4( π β π β² + π β²β² ) π¦
The Dirac-delta functions in this expression represent vertices at the respective space-time points
π¦ β² and π¦. As indicated above there are three four-momentum integration variables with two
initial four-momentum vertices that we have just mentioned.
We can integrate over π β²β² at the space-time point π¦ with a picking π β²β² = k β β k so that we can
reduce (8) into
(9)
β« π4 π¦ β²π4 π¦
πΊ(π¦ β² β π¦)
(2π)2
πΊ( π¦ β π¦ β² )
(2π)2
πΊ(π¦ β² β π¦)
(2π)2
π΅
π΄
=
1
(2π)4
β« π4 π β« π4 π β²
1
βπ2 + π2 + ππ
1
βπ β²2 + π2 + ππ
1
β (k β β k)
2
+ π2 + ππ
πΏ4( π β π β²
+ k β β k ) π¦ β²
In this result, we have not yet written the remaining Dirac-delta function with zero four-
momentum argument because we still have to integrate over the remaining four-momentum
variabes k β and k. Integrating further over k β at y β with pickings k β = k + k β β k = k β and
kβ β k = kβ β k, we will further write (9) in the following form ( noted (k β β k )2
= (k β k β² )2
)
(10)
β« π4 π¦ β²π4 π¦
πΊ(π¦ β² β π¦)
(2π)2
πΊ( π¦ β π¦ β² )
(2π)2
πΊ(π¦ β² β π¦)
(2π)2
π΅
π΄
=
1
(2π)4
1
βπ β²2 + π2 + ππ
β« π4 π
1
βπ2 + π2 + ππ
1
β (k β k β² )
2
+ π2 + ππ
4. There is no longer a Dirac-delta function in (10) although we still have to integrate over the
remaining four-momentum variable k but we reserve this in future works as we will be dealing
with the type of integrals involved in dimensional regularization to include Feynmanβs trick.
As written in coordinate space (8) can be depicted with the following Feynman graph
(Fig.1)
where at the spacetime point π¦ β² is the Dirac-delta function πΏ4( π β π β² + π β²β² ) π¦ β², while at the
spacetime point y the Dirac-delta function πΏ4( π β π β² + π β²β² ) π¦ . (Note the symmetric property of
the Dirac-delta function, πΏ4(βπ) = πΏ4( π)). We also have the Fourier components of the
propagators
(11.1)
πΜ( π) =
1
βπ2 + π2 + ππ
(11.2)
πΜ( π β²) =
1
βπ β²2 + π2 + ππ
and
(11.3)
πΜ( π β²β²) =
1
βπ β²β²2 + π2 + ππ
In momentum space as given by (10), we have the corresponding Feynman graph
(Fig.2)
5. where
(12.1)
πΜ( π) =
1
βπ2 + π2 + ππ
(12.2)
πΜ( π β²) =
1
βπ β²2 + π2 + ππ
and
(12.3)
πΜ(k β k β²) =
1
β (k β k β² )
2
+ π2 + ππ
In (10), we are to recognize the four-momentum k as internal four-momentum and kβ as the
external four-momentum and such recognition enables us to construct the Feynman graph Fig.2.
Refβs:
[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/FTcourse.html
[3]βt Hooft, G., THE CONCEPTUAL BASIS OF QUANTUM FIELD THEORY,
http://www.phys.uu.nl/~thooft/
[4]Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
6. [5]Wells, J. D., Lectures on Higgs Boson Physics in the Standard Model and Beyond,
arXiv:0909.4541v1
[6]Cardy, J., Introduction to Quantum Field Theory
[7]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory