4.7 Problem of Superfield Effective Potential
89
Here K
(L)
, F
(L)
, W
(L) are quantum corrections. For the Wess-Zumino model, W
(1)
=
0, however, in some quantum theories (e.g. in N = 1 SYM theory with chiral matter)
a one-loop contribution to chiral effective potential exists [26].
The structure of the effective potential presented by the Eqs. (4.147)–(4.151) is
generic describing all theories of the chiral superfields (remind that within the phenomenological context, the matter is associated with chiral superfields since namely
they involve the usual scalar fields as components) including the noncommutative
ones. However, we note that the effective potential in theories including gauge superfields must depend on these superfields in a special way. Indeed, the effective action
in such theories should be expressed in terms of some gauge invariant constructions.
For example, within the background field method, the gauge superfield is incorporated either into covariantly chiral superfields or into supercovariant derivatives and
gauge invariant superfield strengths [33, 66].
Let us give a few remarks about the method of calculating the effective potential.
The best way for it, of course, is based on the using of background dependent propagators which are expressed in terms of usual propagators and background superfields.
Background dependent propagators can be exactly found in certain cases. To calculate Kählerian effective potential and auxiliary fields’ effective potential one can
straightforwardly omit all space-time derivatives of background superfields, moreover, to study Kählerian effective potential one can omit all supercovariant derivatives
and treat background superfields as constants until the final integration. The calculation of the chiral effective potential, however, is characterized by some peculiarities.
The best example to illustrate it is the Wess-Zumino model—the simplest superfield
theory. We will consider it in the next section.
4.8 The Wess-Zumino Model and a Problem of the Chiral
Effective Potential
Now we turn our attention to considering the superfield effective potential in the
Wess-Zumino model. Here we follow the papers [27, 72, 73] and the book [33].
The superfield action of the Wess-Zumino model is given by (4.68). Following,
as usual, the loop expansion approach, we carry out background-quantum splitting
by the rule
→ +
√ φ;
¯
→ ¯
+
√ ¯
φ,
(4.152)
where again , ¯
are the background fields, and φ, ¯
φ are quantum ones. The standard expression defining the quantum contribution to the effective action, ¯
=
∞
L=1
L
L , after this splitting and introducing background fields = m + λ,
¯
= m + λ ¯
, takes the form:
89
Here K
(L)
, F
(L)
, W
(L) are quantum corrections. For the Wess-Zumino model, W
(1)
=
0, however, in some quantum theories (e.g. in N = 1 SYM theory with chiral matter)
a one-loop contribution to chiral effective potential exists [26].
The structure of the effective potential presented by the Eqs. (4.147)–(4.151) is
generic describing all theories of the chiral superfields (remind that within the phenomenological context, the matter is associated with chiral superfields since namely
they involve the usual scalar fields as components) including the noncommutative
ones. However, we note that the effective potential in theories including gauge superfields must depend on these superfields in a special way. Indeed, the effective action
in such theories should be expressed in terms of some gauge invariant constructions.
For example, within the background field method, the gauge superfield is incorporated either into covariantly chiral superfields or into supercovariant derivatives and
gauge invariant superfield strengths [33, 66].
Let us give a few remarks about the method of calculating the effective potential.
The best way for it, of course, is based on the using of background dependent propagators which are expressed in terms of usual propagators and background superfields.
Background dependent propagators can be exactly found in certain cases. To calculate Kählerian effective potential and auxiliary fields’ effective potential one can
straightforwardly omit all space-time derivatives of background superfields, moreover, to study Kählerian effective potential one can omit all supercovariant derivatives
and treat background superfields as constants until the final integration. The calculation of the chiral effective potential, however, is characterized by some peculiarities.
The best example to illustrate it is the Wess-Zumino model—the simplest superfield
theory. We will consider it in the next section.
4.8 The Wess-Zumino Model and a Problem of the Chiral
Effective Potential
Now we turn our attention to considering the superfield effective potential in the
Wess-Zumino model. Here we follow the papers [27, 72, 73] and the book [33].
The superfield action of the Wess-Zumino model is given by (4.68). Following,
as usual, the loop expansion approach, we carry out background-quantum splitting
by the rule
→ +
√ φ;
¯
→ ¯
+
√ ¯
φ,
(4.152)
where again , ¯
are the background fields, and φ, ¯
φ are quantum ones. The standard expression defining the quantum contribution to the effective action, ¯
=
∞
L=1
L
L , after this splitting and introducing background fields = m + λ,
¯
= m + λ ¯
, takes the form: