B. Nikolic - Renormalizability of the D-Deformed Wess-Zumino Model

Non(anti)commutative deformation via the twist
Construction of the D-deformed Wess-Zumino model
Renormalizability of the model
Summary

Renormalizability of the D-deformed
Wess-Zumino model

Marija Dimitrijevi´ , Biljana Nikoli´ and Voja Radovanovi´
c c c

Faculty of Physics, University of Belgrade, Serbia

Balkan Workshop, 29th-31st August 2011
Donji Milanovac, Serbia

B. Nikoli´
c Renormalizability of the D-deformed Wess-Zumino model

Summary

Outline

1 Non(anti)commutative deformation via the twist

2 Construction of the D-deformed Wess-Zumino model

3 Renormalizability of the model

B. Nikoli´

Summary

Superspace and superfields

¯˙
General superfield is a function on superspace (x m , θα , θα )
¯ ¯¯ ¯¯
F (x, θ, θ) = f (x) + θφ(x) + θχ(x) + θθm(x) + θθn(x)
¯ ¯¯ ¯¯ ¯¯
+ θσ m θvm (x) + θθθλ(x) + θθθϕ(x) + θθθθd(x)

Infinitesimal supersymmetry transformations
¯¯
δξ F = ξQ + ξ Q F

Generators of supersymmetry
m ¯˙
Qα = ∂α − iσαα θα ∂m , ¯˙ ¯˙
Qα = −∂α + iθα σ m αα∂m
˙
˙

B. Nikoli´

Summary

Hopf algebra of supersymmetry transformations

algebra
¯˙ ¯˙ ¯˙ m
{Qα , Qβ } = {Qα , Qβ } = 0, {Qα , Qα } = 2iσαα ∂m
˙
¯ α] = 0
[∂m , ∂n ] = [∂m , Qα ] = [∂m , Q ˙
coproduct
∆Qα = Qα ⊗ 1 + 1 ⊗ Qα
¯˙ ¯˙ ¯˙
∆Qα = Qα ⊗ 1 + 1 ⊗ Qα
∆∂m = ∂m ⊗ 1 + 1 ⊗ ∂m
counit and antipode
¯˙
ε(Qα ) = ε(Qα ) = ε(∂m ) = 0
S(Qα ) = −Qα , ¯˙ ¯˙
S(Qα ) = −Qα , S(∂m ) = −∂m
B. Nikoli´

Summary

Deformed Hopf algebra of supersymmetry
transformations

Twist
1 αβ D ⊗D
F = e2C α β
,
m ¯˙
where C αβ = C βα ∈ C and Dα = ∂α + iσαα θα ∂m
˙
This twist is associative, but not hermitian.
Algebra, coproduct, counit and antipode remain the same.
The full supersymmetry is preserved.

B. Nikoli´

Summary

The -product

Inverse of the twist deﬁnes -product.
For arbitrary superﬁelds F and G the -product reads

F G = µ {F ⊗ G}
= µ{F −1 F ⊗ G}
1
= F · G − (−1)|F | C αβ (Dα F ) · (Dβ G)
2
1 αβ γδ
− C C (Dα Dγ F ) · (Dβ Dδ G),
8
where |F | = 1 if F is odd (fermionic) and |F | = 0 if F is
even (bosonic).

B. Nikoli´

Summary

Deformed superspace

¯˙
-commutation relations for coordinates x m , θα , θα

{θα , θβ } = C αβ , ¯˙ ¯ ˙ ¯˙
{θα , θβ } = {θα , θα } = 0
¯¯
[x m , x n ] = −C αβ (σ mn ε)αβ θθ
m ¯ ˙ ¯˙
[x m , θα ] = −iC αβ σβ β θβ ,
˙ [x m , θα ] = 0

Both the bosonic and fermionic part of the superspace are
deformed.

B. Nikoli´

Summary

Deformed supersymmetry transformations

Under deformed infinitesimal supersymmetry
transformations superfield F transforms
¯¯
δξ F = ξQ + ξ Q F

-product of two superfields is a superfield

δξ (F G) = ¯¯
ξQ + ξ Q (F G)
= (δξ F ) G + F (δξ G)

Undeformed coproduct gives usual Leibniz rule.

B. Nikoli´

Summary

Undeformed Wess-Zumino model

¯˙
Chiral field Φ is a superfield that fulfills Dα Φ = 0, with
¯ α = −∂α − iθα σ m ∂m .
D˙ ¯˙
αα
˙
Product of two chiral fields is a chiral field.
θθ-component of chiral field is SUSY invariant.
Action is given by

m λ
S= d4 x Φ+ Φ + ΦΦ + ΦΦΦ + c.c. ,
¯¯
θθθθ 2 θθ 3 θθ

where Φ is chiral superfield and m and λ real constants.

B. Nikoli´

Summary

Deformed Wess-Zumino model

Deformation is realized by changing pointwise
multiplication by -multiplication.
Problem: Action is not SUSY invariant!
The -product of two chiral ﬁelds is not a chiral ﬁeld.
1 2 2
Φ Φ = ΦΦ − C (D Φ)(D 2 Φ),
32
where C 2 = C αβ C γδ εαγ εβδ .
We use projectors to separate chiral and antichiral parts of
-products
¯
1 D2D2 ¯
1 D2D2 ¯
1 DD2D
P1 = , P2 = , PT = −
16 16 8
B. Nikoli´

Summary

Invariants under supersymmetry transformations

Construct -products of two or three (anti)chiral superﬁelds
that are SUSY invariant and have good commutative limit
Collecting all the appropriate terms we come to
D-deformed SUSY invariant Wess-Zumino action

m
S= d4 x Φ+ Φ + P2 (Φ Φ) + 2a1 P1 (Φ Φ)
¯¯
θθθθ 2 θθ ¯¯
θθ
λ
+ P2 (P2 (Φ Φ) Φ) + 3a2 P1 (P2 (Φ Φ) Φ)
3 θθ ¯¯
θθ
+
+2a3 (P1 (Φ Φ) Φ) + 3a4 P1 (Φ Φ) Φ
¯¯
θθθθ ¯¯
θθ
¯2
+3a5 C P2 (Φ Φ) Φ +
+ c.c.
¯¯
θθθθ

B. Nikoli´

Summary

Divergent part of the one-loop effective action

Use the background ﬁeld method, supergraph technique
and dimensional regularization
For the supergraph technique the classical action must be
written as an integral over the whole superspace

m D2 ma1 C 2 2
S0 = d8 z Φ+ Φ + −Φ Φ+ (D Φ)Φ + c.c.
8 8
D2 a2 C 2
Sint = λ d8 z − Φ2 Φ+ ΦΦ(D 2 Φ)
12 8
a3 C 2 2 a4 C 2 ¯
− (D Φ)(D 2 Φ)Φ + Φ(D 2 Φ)Φ+ + a5 C 2 ΦΦΦ+ c.c.
48 8

B. Nikoli´

Summary

The background field method

Split the chiral and antichiral fields into the classical and
quantum parts
Φ → Φ + Φq , Φ+ → Φ+ + Φ+
q

Introduce unconstrained superfields Σ and Σ+
1¯ 1
Φq = − D 2 Σ, Φ+ = − D 2 Σ+
q
4 4
Action quadratic in quantum superfields
1 Σ
S (2) = d8 z Σ Σ+ M+V ,
2 Σ+
where matrices M and V corespond to the kinetic and
interaction terms
B. Nikoli´

Summary

The n-point Green functions

The one-loop effective action

Γ = S0 + Sint + Γ1
i
Γ1 = Tr log(1 + M−1 V)
2
∞ ∞
i (−1)n+1 (n)
= Tr (M−1 V)n = Γ1
2 n
n=1 n=1

B. Nikoli´

Summary

The results

(1)
Γ1 = 0,
λ2 (1 − (2a1 + a4 ¯
+ 2a5 )m2 (C 2 + C 2 ))
(2)
Γ1 = 2 m
d8 zΦ+ (z)Φ(z)
dp 4π 2 ε
λ2 (m2 a3 − ma4 − a5 )
+ d8 z C 2 Φ(z)D 2 Φ(z) + c.c.
16π 2 ε
(3) λ3 (ma1 + ma4 + 2a5 ) ¯
Γ1 =− 2ε
d8 z C 2 Φ(z)Φ(z)Φ+ (z) + c.c.
dp 2π
λ3 (2ma3 − a4 )
+ d8 z C 2 Φ(z)Φ+ (z)D 2 Φ(z) + c.c.
8π 2 ε

B. Nikoli´

Summary

The results

(4) λ4 ¯
Γ1 = d8 z C 2 Φ(z)Φ(z)Φ+ (z)
dp 8π 2 ε
¯
a3 D 2 Φ+ (z) − 2a4 Φ(z) + c.c.
(n)
Γ1 = 0, n≥5
dp

B. Nikoli´

Summary

(4)
Divergences in Γ1 cannot be absorbed, for arbitrary
constants a3 and a4 the model IS NOT renormalizable
Analyze special case a3 = a4 = 0
(4)
Γ1 =0
dp
¯
λ2 (1 − (2a1 + 2a5 )m2 (C 2 + C 2 ))
(2)
Γ1 = m
d8 z Φ+ (z)Φ(z)
dp 4π 2 ε
λ2 a5
− d8 z C 2 Φ(z)D 2 Φ(z) + c.c.
16π 2 ε
(3) λ3 (ma1 + 2a5 ) ¯
Γ1 =− d8 z C 2 Φ(z)Φ(z)Φ+ (z) + c.c.
dp 2π 2 ε
Nonminimal deformation needed for canceling divergences
(3)
in Γ1
B. Nikoli´

Summary

Renormalization of the model with a2 = a3 = a4 = 0

The bare action
(2) (3)
SB = S0 + Sint − Γ1 − Γ1
dp dp

Superﬁeld, mass and the couplings are renormalized as
√
Φ0 = Z Φ, m = Zm0 , λ = Z 3/2 λ0
λ2 ¯
Z = 1 − 2 1 − 2m(C 2 + C 2 )(ma1 + a5 )
4π
λ2 a5 ¯2 ¯ λ2 (ma1 + 2a5 )
a10 C0 = a1 C 2 1 + 2
2
, a50 C0 = a5 C 2 1 +
2π a1 m 2π 2 a5
Model with a2 = a3 = a4 = 0 IS renormalizable!
B. Nikoli´

Summary

Summary

A deformation of the Hopf algebra of supersymmetric
transformations is introduced via the twist.
The nonhermitian -product is deﬁned.
The SUSY Hopf algebra is undeformed, the Leibniz rule
remains undeformed.
Requiring the SUSY invariance and good commutative
limit of the -products of (anti)chiral ﬁelds the D-deformed
Wess-Zumino model is formulated.
The general model is not renormalizable.
Special choice a2 = a3 = a4 = 0 renders the
renormalizability.

B. Nikoli´

B. Nikolic - Renormalizability of the D-Deformed Wess-Zumino Model

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B. Nikolic - Renormalizability of the D-Deformed Wess-Zumino Model