108
4 Four-Dimensional Superfield Supersymmetry
analogous to that one contributing to the chiral effective potential in the Wess-Zumino
model.
After D-algebra transformations and loop integrations completely analogous to
those ones carried out in the previous section we find that two-loop contribution to
the chiral effective potential in this model looks like
W
(2)
=
1
2(16π 2 )
2
ζ(3)λ
2
W
(z)
K
2
¯
(z)
3
.
(4.212)
One reminds that λ = ¯
W
( ¯
)| ¯
=0 and K ¯
(z) =
∂
2 K ( ¯
,,)
∂∂ ¯
| ¯
=0 here. We see that the
correction (4.212) is finite and does not require renormalization in any case despite
the theory is non-renormalizable in general case. In the case of the Wess-Zumino
model it reduces to (4.195).
We note that the calculation of two-loop Kählerian effective potential can be
carried out with help of matrix superpropagator (4.208). The results are given in
[62, 82].
A coupling of the general chiral superfield model to the gauge superfield can be
introduced as well, and corresponding calculations are performed along the same
lines as here. The details of calculations are given in [70]. Qualitatively the one- and
two-loop Kählerian effective potentials in this theory are rather similar to the results
obtained for the Wess-Zumino model and supersymmetric QED which have been
discussed in this chapter.
Now let us consider some phenomenological applications of the theory characterized by the action (4.205). Let us suppose that the column vector
describes two
superfields: the light (massless) one φ and the heavy one , so,
=
φ
. For this
case, we find the one-loop effective action and eliminate heavy superfields with use
of their effective equations of motion. As a result we arrive at the effective action
of light superfields. There is a decoupling theorem [71, 84] according to which this
effective action after redefining of parameters, such as fields, masses, couplings, can
be expressed in the form of a sum of effective action of the theory obtained from
initial one by putting heavy fields to zero and terms proportional to different powers
of
1
M
where M is mass of heavy superfield (which in the case under consideration
is put, by phenomenological reasons, to be equal to the characteristic string mass
M String [80, 82]).
We study such a theory in the one-loop approximation. The low-energy leading one-loop contribution to effective action is given by (4.211). In principle, one
can proceed with diagonalization of the matrices K ¯
=
∂
2 K
∂ i ∂ ¯
j and W
=
∂
2 W
∂ i ∂ j .
However, within this review let us restrict ourselves by a qualitative description of
the situation only.
We consider, as a didactic example, the minimal theory whose action is slightly
different from that one used [85], so,
4 Four-Dimensional Superfield Supersymmetry
analogous to that one contributing to the chiral effective potential in the Wess-Zumino
model.
After D-algebra transformations and loop integrations completely analogous to
those ones carried out in the previous section we find that two-loop contribution to
the chiral effective potential in this model looks like
W
(2)
=
1
2(16π 2 )
2
ζ(3)λ
2
W
(z)
K
2
¯
(z)
3
.
(4.212)
One reminds that λ = ¯
W
( ¯
)| ¯
=0 and K ¯
(z) =
∂
2 K ( ¯
,,)
∂∂ ¯
| ¯
=0 here. We see that the
correction (4.212) is finite and does not require renormalization in any case despite
the theory is non-renormalizable in general case. In the case of the Wess-Zumino
model it reduces to (4.195).
We note that the calculation of two-loop Kählerian effective potential can be
carried out with help of matrix superpropagator (4.208). The results are given in
[62, 82].
A coupling of the general chiral superfield model to the gauge superfield can be
introduced as well, and corresponding calculations are performed along the same
lines as here. The details of calculations are given in [70]. Qualitatively the one- and
two-loop Kählerian effective potentials in this theory are rather similar to the results
obtained for the Wess-Zumino model and supersymmetric QED which have been
discussed in this chapter.
Now let us consider some phenomenological applications of the theory characterized by the action (4.205). Let us suppose that the column vector
describes two
superfields: the light (massless) one φ and the heavy one , so,
=
φ
. For this
case, we find the one-loop effective action and eliminate heavy superfields with use
of their effective equations of motion. As a result we arrive at the effective action
of light superfields. There is a decoupling theorem [71, 84] according to which this
effective action after redefining of parameters, such as fields, masses, couplings, can
be expressed in the form of a sum of effective action of the theory obtained from
initial one by putting heavy fields to zero and terms proportional to different powers
of
1
M
where M is mass of heavy superfield (which in the case under consideration
is put, by phenomenological reasons, to be equal to the characteristic string mass
M String [80, 82]).
We study such a theory in the one-loop approximation. The low-energy leading one-loop contribution to effective action is given by (4.211). In principle, one
can proceed with diagonalization of the matrices K ¯
=
∂
2 K
∂ i ∂ ¯
j and W
=
∂
2 W
∂ i ∂ j .
However, within this review let us restrict ourselves by a qualitative description of
the situation only.
We consider, as a didactic example, the minimal theory whose action is slightly
different from that one used [85], so,