4.5 Superficial Degree of Divergence. Renormalization
77
Let us consider an arbitrary supergraph with L loops, V vertices, P propagators
(C of them are , ¯
¯
-propagators) and E external lines (E c of them are
(anti)chiral). We denote the SDD as ω.
Any integration over internal momentum (i.e. over d
4 k) contributes to ω with
4. Since the number of integrations over internal momenta is the number of loops,
the total contribution from all such integrations is 4L. Any propagator includes
1
k 2 +m 2 or
1
k 2 (4.91), hence contribution of all propagators is equal to −2P. Since
, ¯
¯
-propagator contains additional
1
k 2 these propagators give additional
contribution −2C. Therefore manifest dependence of momenta gives contribution
to ω equal to 4L − 2P − 2C.
Now let us consider contribution of D-factors to the SDD. Each vertex (both pure
gauge one and that one containing chiral superfields) without external chiral (antichiral) lines contains four D-factors since any superfield (contracted to propagator)
corresponds to ¯
D
2 , and ¯
—to D
2 . Therefore each vertex yields the contribution 2.
However, external chiral (antichiral) lines do not carry D-factors. As a result, any
external , ¯
line decreases ω by 1. Each , ¯
¯
-propagator contains a factor
¯
D
2 (D
2 ) with contribution 1. Then, due to the identity (4.17), contracting any loop
into a point decreases the number of D-factors which can be converted to internal
momenta by 4, consequently, ω—by 2. As a result the total contribution of D-factors
to ω is equal to 2V − E c − 2L + C (remind that each D-factor contributes to ω with
1/2).
Therefore the SDD is equal to
ω = 4L − 2P − 2C + 2V − E c − 2L + C = 2L − 2P + 2V − C − E c .
(4.108)
Using the well known topological identity L + V − P = 1 we have
ω = 2 − C − E c .
(4.109)
Really, the SDD can be lower than (4.109) if some of D-factors are transported to
external lines and do not generate internal momenta. If N D D-factors are moved to
external lines the ω is equal to
ω = 2 − C − E c −
1
2
N D .
(4.110)
This is the final expression for the SDD. As usual, at ω ≥ 0 a supergraph diverges,
and at ω < 0—converges. We note that:
1. ω ≤ 2 hence the SDD is restricted from above.
2. As the number of external lines grows (here for the sake of simplicity we consider
only external (anti)chiral legs while the presence of external gauge legs will
be treated in the Sect. 4.11), ω decreases. Therefore the number of divergent
structures is essentially limited—it is finite (really, there can be no more than
77
Let us consider an arbitrary supergraph with L loops, V vertices, P propagators
(C of them are , ¯
¯
-propagators) and E external lines (E c of them are
(anti)chiral). We denote the SDD as ω.
Any integration over internal momentum (i.e. over d
4 k) contributes to ω with
4. Since the number of integrations over internal momenta is the number of loops,
the total contribution from all such integrations is 4L. Any propagator includes
1
k 2 +m 2 or
1
k 2 (4.91), hence contribution of all propagators is equal to −2P. Since
, ¯
¯
-propagator contains additional
1
k 2 these propagators give additional
contribution −2C. Therefore manifest dependence of momenta gives contribution
to ω equal to 4L − 2P − 2C.
Now let us consider contribution of D-factors to the SDD. Each vertex (both pure
gauge one and that one containing chiral superfields) without external chiral (antichiral) lines contains four D-factors since any superfield (contracted to propagator)
corresponds to ¯
D
2 , and ¯
—to D
2 . Therefore each vertex yields the contribution 2.
However, external chiral (antichiral) lines do not carry D-factors. As a result, any
external , ¯
line decreases ω by 1. Each , ¯
¯
-propagator contains a factor
¯
D
2 (D
2 ) with contribution 1. Then, due to the identity (4.17), contracting any loop
into a point decreases the number of D-factors which can be converted to internal
momenta by 4, consequently, ω—by 2. As a result the total contribution of D-factors
to ω is equal to 2V − E c − 2L + C (remind that each D-factor contributes to ω with
1/2).
Therefore the SDD is equal to
ω = 4L − 2P − 2C + 2V − E c − 2L + C = 2L − 2P + 2V − C − E c .
(4.108)
Using the well known topological identity L + V − P = 1 we have
ω = 2 − C − E c .
(4.109)
Really, the SDD can be lower than (4.109) if some of D-factors are transported to
external lines and do not generate internal momenta. If N D D-factors are moved to
external lines the ω is equal to
ω = 2 − C − E c −
1
2
N D .
(4.110)
This is the final expression for the SDD. As usual, at ω ≥ 0 a supergraph diverges,
and at ω < 0—converges. We note that:
1. ω ≤ 2 hence the SDD is restricted from above.
2. As the number of external lines grows (here for the sake of simplicity we consider
only external (anti)chiral legs while the presence of external gauge legs will
be treated in the Sect. 4.11), ω decreases. Therefore the number of divergent
structures is essentially limited—it is finite (really, there can be no more than