Comments on the role of field redefinition on renormalisation of $N=\frac{1}{2}$ supersymmetric pure gauge theory

We study one loop corrections to $N=\frac{1}{2}$ supersymmetric $SU(N)\times U(1)$ pure gauge theory. We calculate divergent contributions of the 1PI graphs contain the non-anti-commutative parameter $C$ up to one loop corrections. We find the disagreement between component formalism and superspace formalism is because of the field redefinition in component case. We modify gaugino field redefinition and lagrangian. We show extra terms of lagrangian have been generated by $\lambda$ redefinition and are necessary for the renormalisation of the theory. Finally we prove $N = \frac{1}{2}$ supersymmetric gauge theory is renormalisable up one loop corrections using standard method of renormalisation


Introduction
Theories defined on non-anti-commutative superspace have been studied extensively during last ten years [1,2]. Superspace in such non-anti-commutative theories is a superspace whose fermionic supercoordinates are not anticommutative. One could construct a field theory in non-anti-commutative superspace in terms of superfields with the star-product where lagrangian is deformed from the original theory by the non-anti-commutative parameters.
Recently some renormalisability aspects of the non-anti-commutative field theories have been studies. It has been shown non-anti-commutative field theories are not power-counting renormalisable; however it has been discussed that they could be renormalisable if some additional terms have been added to the lagrangian in order to divergences to all orders [3]- [8]. The renormalisability of non-anti-commutative versions of the Wess-Zumino model has been discussed [3,4], with explicit computations up to two loops [5]. The renormalisability of non-anti-commutative gauge theory with N = 1 2 supersymmetry has been studied in [6,8]. The authors in [6] show that the theory is renormalisable to all order of perturbation theory. The conditions of the renormalisability of non-anticommutative (NAC) field theories have been studied with explicit computations up one and two loops [9]- [15].
The renormalisability of supersymmetric gauge field theories has been discussed in WZ gauge [6,7]. The explicit one loop corrections in component formalism have been done in [9]- [11]. The authors in [10,11] have claimed the precise form of the lagrangian is not preserved by renormalisation. They have shown by explicit calculation that there are problems with assumption of gauge invariance which is required to rule out some classes of divergent structure in non-anti-commutative theory.From their calculation, one can see even at one loop divergent non-gaugeinvariant terms are generated. In order to remove the non-gauge-invariant terms and restore gauge invariance at one loops they introduce a one loop divergent field redefinition in the case of pure N = 1 2 supersymmetry (i.e. no chiral matter).
On the other hand, the authors in [12,13] have started from superspace formalism and discussed renormalisability and supergauge invariance. They proved that the field redefinition is not necessary and the original effective action is not only gauge but also supergauge invariant up one loop corrections. The disagreement between two approaches put a big question mark which approaches we should relay on in N = 1 2 supersymmetric gauge theory.
In this paper we investigate the renormalisability of N = 1 2 supersymmetric pure gauge theory at one-loop perturbative corrections in component formalism. We shall show N = 1 2 supersymmetric gauge theory is renormalisable in a usual manner without any needs for field redefinition (there is not theoretical justification or interpretation for the field redefinition as mentioned by authors [10]) which leads to the lagrangian change. Therefore we shall prove two approaches lead to the same conclusion.
The paper is organized as follows: First we briefly review NAC supersymmetric gauge theories and their lagrangian. Then an explicit one-loop calculation of the three and four-point functions of the theory in the C-deformed sector is carried out to calculate the corrections. We show some anomaly terms appears in the 1PI functions which spoil the renormalisability of theory. Next we introduce extra terms to the original lagrangian in order to renormalise NAC pure gauge supersymmetric theory and calculate corrections which come from these new terms. Finally we discus the source of the extra lagrangian, and show that these new terms have hidden because of the component λ redefinition [1,20]), so in order to reproduce them one should reverse gaugino field redefinition.
2 The pure gauge supersummetric action of

NAC gauge theory
The original non-anticommutative theory defined in superfields appears to require a U(N) gauge group. Here, at first we would like to consider U(N) gauge theory for non-(anti)commutative (NAC) superspace. The action for an N = 1/2 supersymmetric U(N) pure gauge theory is given by: where and Corresponding to any index a for SU(N) we introduce the index A = (0, a), so that A runs from 0 to N 2 − 1. with R A being the group matrices for U(N) in the fundamental representation. These satisfy where f ABC is completely antisymmetric, f abc is the same as for SU(N) and f 0bc = 0, while d ABC is completely symmetric; d abc is the same as for SU(N), d 0bc = 2/N δ bc , d 00c = 0 and d 000 = 2/N . In particular, R 0 = 1 2N 1 . We have also The following identities hold in U(N) group and will be extensively used below Where Upon substituting the above relations in eq. (1), we obtain the action in the U(N) case in the form: With gauge coupling g, gauge field A µ and gaugino λ.
Beside, definition for F µν and D µ λ a are given by: C µν is related to the non-anti-commutativity parameter C αβ by: also, we have: where The useful identity is: There are some properties of C in App A. In above Eqs., C αβ is the non-anticommutativity parameter, and our conventions are consistent with ref [1]. The action for pure N = 1/2 supersymmetric gauge theory (Eq. 11) is invariant under the standard U (N ) gauge transformations and the N = 1/2 supersymmetry transformations. The standard U (N ) version of the NAC gauge theory is not renormalisable [1]. Therefore; we would like to present a SU (N ) × U (1) lagrangian which has N = 1 2 supersymmetric properties, so we introduce the following action: One beauty of the above equation is one could easily switch between the original U (N ) theory and SU (N )×U (1) theory. In our work we define γ abcde = γ 0 , γ abcd0 = γ 1 , γ 0b0de = γ a0c0e = γ 0bc0e = γ a00de = γ 2 . Indeed we give them in terms of g and g 0 . They are given by: Where h = 1. The above action is similar to the SU (N ) × U (1) action in ref [11].

One-loop corrections
In our calculation, we use standard gauge fixing term and consider the Feynman rules in the super-Fermi-Feynman gauge(α = 1).
In this section we first review the one-loop perturbative corrections to the undeformed N = 1 part of the theory. It has been shown that the quantum corrections of N = 1 part of the theory are not affected by C-deformation [9,10]. Therefore; gauge field and gaugino anomalous dimensions and gauge β-functions are the same as those in the ordinary N = 1 case. The C-independent part of the bare action can be written as: The C-independent part of the action is renormalisable if one introduce bare fields and couplings according to: that Z A , Z λ and Z g are known as a renormalisation constants. Also one can define: finally, one should add the following counter terms to the lagrangian of theory in order to renormalise theory: where, and L is given by: Here ε = 4 − D is the regulator.
(We have given here the renormalisation constants corresponding to the SU (N ) sector of the U (N ) theory; those for the U (1) sector, namely Z λ 0 , Z A 0 , Z g 0 are given by ommiting the terms in N and replacing g by g 0 . )

One-loop C deformed Corrections
In this part we will present on-loop graphs contributing to the new terms arising from C-deformed part of the action. The one-loop one-particle-irreducible (1PI) graphs of the C-deform Aλλ three point functions are depicted in Figs (2). Using Feynman rules one could compute the divergent contributions from the graphs.
As an example we calculate the one loop corrections to fig(2-a). It is given by: Using Feynman parameter in App. B: we can simplify denominator of Eq. (31) By changing variables to the denominator of integral in Eq. (33) is given by: so, the integral of Eq. (32) is given by: then we arrive: we finally have for Eq. (31): Moreover, as it be seen in Fig. (2-a), we have momentum -energy conserving in the loop: The divergent contributions up to one loop correction to diagrams in Fig. 2 are given by: Where tensor Y µν is symmetric respect to both Lorentz and spinor indices and tensor C µν is anti-symmetric. In our computation permutations has taken into account by changing the position of C as well as symmetry factors. Adding the different divergent contributions from the diagrams of fig 2 corresponding to different U (1) and SU (N ) parts, we have: Let us now continue with the relevant diagrams containing only C-deformed vertex and contributing to the four point functions ( Fig. 3 and Fig. 4 ). Using the Feynman rules, and considering all permutations between the same fields, the final result for 1PI graphs of Fig. 3 are given by: Considering all diagrams the final result for Fig. 3 is given by: Finally, The final divergence contributions ( Fig. 4) which come from the last term containing | C | 2 (λλ) 2 are given by: Note that we have considered all permutations between the same fields and changing the position of C and | C | 2 , and adds all divergences come from Fig. 4. The final result for 1PI graphs of Fig. 4 is given by: In order to renormalise the theory and remove the divergences one should rescale and redefine coupling constants. This procedure is equivalent to introduce of some counter-terms to the lagrangian. Therefore, we introduce bare couplings according to: However in the language of counter-terms, a theory is renormalisable if the counterterms are of the same form as those appearing in the original lagrangian( these counter terms are required to cancel the divergences). If we look at three and four point functions we see that the anomaly term called Y term is problematic because we can not add some kinds of counter-terms which cancel Y term, or one can say these terms spoil renormalisation. In the next section we add some extra lagrangian to the theory and prove that the NAC pure gauge theory is renormalisable.

Renormalization of N = 1 2 deformed lagrangian
In this section we shall renormalise the theory and remove the divergences. In order to renormalise the theory we should add extra term (L Extra ) to the original lagrangian. The extra lagrangian is considered as follow: The two last terms in Eq. (46) help us to make renormalisable NAC U (N ) gauge theory, the first two terms are needed because of gauge transformation rules. These terms are absent from the original lagrangian because of λ redefinition in Ref [1] as we shall explain in next part. Adding the above terms to original lagrangian, the total lagrangian is given by: Since we add some terms to original lagrangian, we have to modify the gauge transformation and SUSY transformation. It is easy to show that L total is preserved under following gauge transformation in U (N ) group: Where κ ABC is considered as arbitrary coupling which depends on A, B, C values.
The gauge transformation is not canonical because the transformation of λ depends on the non-anti-commutative parameter C. However; the SUSY transformation does not change except So yet we can refer to eqs. (21)(22)(23). In our work in order to renormalise the NAC SU (N ) × U (1) gauge theory, we choose here ξ is considered as a coefficient. Then the extra lagrangian is given by: where κ 1 = κ abc = ξγ abc , κ 3 = κ ab0 = 2ξγ ab0 , κ 0ab = κ a0b = 0. In this case the new action is given by:  Fig. 5-a, we should only change the NAC vertex iγ AJI d AJI C µν γδ (p 1 + The result for the graphs in Fig. 5-a is given by: Beside, the total divergent contribution for new graphs in Fig. 5 is given by: In order to obtain divergent contribution for 1PI graphs with both C and Y pa- γ abc )iN Ld abc αβ C µν q ν + 8γ 0bc iN Ld 0bc αβ C µν q ν − 1 2 γ a0c N Ld a0c αβ C µν q ν − 1 2 γ ab0 N Ld ab0 αβ C µν q ν There are new graphs (Fig. 6) for one loop corrections of the four point function.
The total contributions as different SU (N ) × U (1) parts corresponds to Fig. 5 is Then, the total four point 1PI divergent contribution is given by: Fortunately, new terms does not effect on four point function is containing | C | 2 (λλ) 2 . In order to compute counter terms we should decompose the lagrangian to the SU (N ) × U (1) parts because some interaction terms such as term which correspond to (U (1)) 3 receive no quantum corrections. It is given by: The C and Y -dependent part of the action are renormalisable if one introduces bare fields and couplings according to Eqs. (26) and Eqs. (44, 45) as well as: Then in order to find some counter terms we have to introduce the following identities: Adding the following counter-term terms to the part of the total action and comparing the expression with the bare action, the theory should be renormalisable.
The full C dependent part of the action can be written as: where in order to renormalise the C-dependent part of the action we have to obtain the values of δ i by solving the following equations: Where Γ C.T s come from counter terms (Appendix E). Then, using Z g , Z A , Z λ for SU (N ) sector, and Z g 0 , Z A 0 , Z λ 0 for U (1) sector, we obtain the renormalisation constants as : As it is understood because ξ is renormalised, so κ ABC should be renormalised.
Moreover we obtain It is found that our result is compatible with Ref [11,13]. Of course a natural expectation would be that Z C = Z Y , because of Eq. (36) we know that Y ∝ C.
We demonstrate that the theory is renormalisable and the N = 1/2 supersymmetric as well as NAC SU (N ) × U (1) pure gauge theory is preserved. This point also has been concluded in Ref [11] and suggested in Ref [6], where was supposed to be correct to all orders. We also arrive at the conclusion that it is not needed to renormalise the non-anticommutativity parameter C. Beside our full lagrangian is the same form as derived from non-anticommutative superspace, however Z κ ABC and Z γ ABC depends on whether A, B, C are SU (N ) indices or U (1) indices. It seems to imply that the renormalised theory is not U (N ) non-anticommutative theory any more. Because the U (N ) structure is broken by renormalisation.
In order to clarify L Extra we redefine the component λ as Then, put it in Eq. (18), and obtain Eq. (55)(in other words L total is result of λ redefinition in L original ).The λ redefinition just affects the gaugino kinetic term.
Our redefinition is opposite to Refs [1,20]. They have redefined λ in order to make gauge transformations be canonical; however it causes theory unrenormalisable because in that case some terms are been hidden in the lagrangian. In order to reverse process we should add hidden terms by hand or come back to original definition of λ Eq. (71); however we lose gauge canonical transformation. Beside, because κ ABC is obtained renormalised, the redefinition of λ A is calledλ A should be renormalised. Finally divergent field redefinition in Ref [11] is reinterpreted as

discussion on ξ −→ 0
It is worthwhile to investigate our results in the case of limit ξ −→ 0. Indeed in this case, 1PI graphs from new terms are vanished. We would like to present this claim as follow. Instance we know that Γ (1)µαβ 1−Extra (ξ) comes from extra lagrangian.
Then total divergent contribution for three point function would be finite if we add some counter terms as follow: we expect to obtain Z C = Z Y = Z |C| 2 = 1 (it means the NAC structure would be preserved under the procedure of renormalisation). Then, the above equation leads to: Note that second terms in eqs. (73) and (78) are related to divergent contribution from new 1PI graphs. In order to find Z ξ we have to benefit from eqs. (63). Hence Z ξ up to one order is given by: Now, in the case of ξ −→ 0 eqs. (73-79) result to In fact we have neglected the divergent contributions from new graphs, however; Z ξ has not been changed. This event could be checked for four point function as well. In this case we obtain the results of Ref [11,10], but because Z ξ = 0 it easy to show it is not necessary to define a nonlinear divergence field redefinition. If we Then the variation of λ A is written by Consequently, after the procedure of renormalization, the nonlinear divergent field redefinition in Ref [11] automatically is generated:

Conclusion
We have compute 1pI corrections for the pure N = 1 2 supersymmetric SU (N ) × U (1) gauge theory at one loop order. We have proved the theory is renormalisable up one loop order using a standard way of renormalisation if one adds some new terms to the original lagrangian. We have shown it is possible to interrupt these are hidden terms because of λ redefinition. it is worth to investigate if it is possible to show that the problems which arise in renormalisation of N = 1 2 supersymmetric theories comes from the redefind vector superfield.
We have shown there is not need to define divergent redefinition of λ. Moreover we suggest all works which have been done based on divergent field redefinition should be reviewed. We have used the N = 1 2 U (N ) gauge group action because as discussed in [21], just non-anticommutative theory with U (N ) gauge group is well-defined. Moreover it is worth to investigate the renormalization of theory at higher loops or including chiral matter in the standard form of renormalisation method. We guess the problem of renormalisation of non-anticommutative theory at component formalism is because of λ redefinition in [1].

Acknowledgements
It is a pleasure to thank F. Yagi for helpful discussions and N. Ghasempour for checking calculations.
A : New Algebra For Non Commutative Pa-

rameters C And Y
We have found the new properties for C and Y parameters that we have made frequent use in our calculations. C µν is related to the non-anti-commutativity parameter C αβ by: also, we have: where we have used the following notations: where in last equation the symbol of T is used for transposed. Also for Y parameter we have: B : Feynman Parameters 1 a 1 a 2 ... a n = (n − 1)! Vertices comes from each interaction in the theory so, we have some vertices as follow: three-Gauge coupling A a µ , A b ν , A c ρ with momentum k, p, q respectively: For total lagrangian we can write So, Feynman rule would be: • A A µ gauge,λ Ḃ α gaugino,λ Ċ β gaugino in NAC parameter C:
Where d A = 1 + δ 0A , c A = 1 − δ 0A . The gauge index of U(N) runs A = 0, 1, ..., N 2 − 1 where A = 0 corresponds to overall U(1) while A = a = 1, ..., N 2 − 1 corresponds to SU(N). For the simplicity of the calculation, we introduce the matrix F A and D A , whose component is given by f ABC and d ABC as Taking into account that f ABC is totally antisymmetric tensor and d ABC is totally symmetric tensor.