Perturbative and non-perturbative infrared behavior of ...

D 2
U D 2 φ
D 2
Figure 3.1: New vertices proportional to the external U superfield
D 2
U
D 2
D 2
3.1. Superspace approach
To compute quantum corrections at a given loop, we write down all the Feynman diagrams up
to the given order and insert an extra ¯ D 2 (D 2 ) derivative on each chiral (antichiral) line leaving
a vertex except for one of the lines at a completely (anti)chiral vertex.
The loop integrals are evaluated in the dimensional regularization (n = 4 − 2ǫ) and minimal
subtraction scheme. Divergent integrals are regularized using the so-called G-scheme
d 4
kf(k) → G(ǫ) d n kf(k) (3.1.7)
where G(ǫ) = (4π) −ǫ Γ(1 − ǫ). Factors of 4π are always neglected along the calculations and a
(4π) 2 factor is inserted for each momentum loop in the final result.
3.1.2 One-loop divergencies
At one loop divergencies appear that involve the two-, three-, and four point functions. The last
two ones are due to the deformation of the theory and also contain the spurion superfield.
The divergent two-point function is the ordinary self-energy diagram which gives the wave
function renormalization. Its contribution is
A0 → 2
ǫ g¯g
d 8 zΦ ¯ Φ (3.1.8)
No divergencies with more than one insertion of the U vertices can appear; then, the only
divergent topologies are the ones given in Fig. 3.2.
The result for the self-energy momentum integral is
d 4 k
1
(k 2 + m ¯m)[(p − k) 2 + m ¯m]
→ 1
ǫ
with which one obtain the results
A1 → − 1
ǫ g2 ¯m 2
d 8 zU(D 2 Φ) 2 = − 1
ǫ g2 ¯m 2 C 2
A2 → − 4
ǫ g2
¯g ¯m d 8 zU(D 2 Φ) 2¯
4
Φ = −
ǫ g2¯g ¯mC 2
A3 → − 4
ǫ g2¯g 2
d 8 zU(D 2 Φ) 2¯ 2 4
Φ = −
ǫ g2¯g 2 C 2
d 4 x F 2
d 4 xF 2 ¯ φ
d 4 xF 2 ¯ φ 2
(3.1.9)
(3.1.10)
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