Low Dimensional Super-Yang-Mills theory in Aether-Superspace A. C. Lehum,1, ∗ T. Mariz,2, † J. R. Nascimento,3 and A. Yu. Petrov3 1
Faculdade de F´ısica, Universidade Federal do Par´a, 66075-110, Bel´em, Par´a, Brazil. 2
Instituto de F´ısica, Universidade Federal de Alagoas, 57072-900, Macei´o, Alagoas, Brazil
3
Departamento de F´ısica, Universidade Federal da Para´ıba,
Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil‡
arXiv:1802.01368v1 [hep-th] 5 Feb 2018
Abstract In this work we study the dynamical generation of mass in the Lorentz-violating two and threedimensional Super-Yang-Mills theory coupled to a scalar matter in the aether-superspace. PACS numbers: 11.30.Pb, 11.30.Cp, 11.15.Ex
∗
[email protected]
†
[email protected]
‡
[email protected]
1
I.
INTRODUCTION
The aether superspace is a construction of a supersymmetric field theory which allows the use the powerful techniques of the superfield formalism, by the deformation of the supersymmetry (SUSY) algebra for supergauge field theories based on the Kostelecky-Berger construction [1]. As discussed before, the extension of the usual superpace to a two [2] and threedimensional [3–5] aether- superspace is established through the deformation of the SUSY generators m Qα = i[∂α − iθβ γβα (∂m + kmn ∂ n )] m = i[∂α − iθβ γβα ∇m ],
(1)
which satisfy the anti-commutation relation m {Qα , Qα } = 2iγαβ ∇m ,
(2)
where ∇m = ∂m + kmn ∂ n , and ∂α is the derivative with respect to the Grassmannian coordinates θα . We use Latin letters to denote indices of the space-time coordinates (0, 1 for 2D and 0, 1, 2 for 3D) and Greek letters for spinorial indices. The tensor kmn is a constant tensor which in the simplest case can be chosen to assume an aether-like form kmn = um un , with um being a constant vector (cf. [6]). We can also consider a more interesting case, where kmn is a traceless tensor, i.e., kmn = um un −
1 g u2 , D mn
with D being the space-time
dimension (for details, see [7]). The superfields defined in this superspace will be called the aether-superfields. The aether-supercovariant derivative consistent with the deformed supersymmetry must anticommute with SUSY generators Qα . It is given by m Dα = ∂α + iθβ γβα ∇m ,
(3)
where the ”twisted” derivative ∇m commutes with the SUSY generators as well as with the derivative Dα . In this paper, we consider various aspects of the lower (two and three)-dimensional superYang-Mills theory, especially, the quantum corrections in this theory. In the section 2, we perform quantum calculations aimed to generate one-loop contributions to the effective action. And the section 3 is our Summary where the results are discussed. 2
II.
THE LOW-DIMENSIONAL SUPER-YANG-MILLS THEORY
As argued in Ref. [8], there is no substantial difference between conventions and notations for supersymmetric models defined in three- and two-dimensional space-times. Therefore we use the notations and conventions of Ref. [9]. Our starting point is the classical action of the two- and three-dimensional SU (N ) Super-Yang-Mills theory coupled to matter superfields defined in the aether superspace, Z n1 1 1 S = T r dD xd2 θ W α Wα − Dα Γα D2 Dβ Γβ + c¯Dα (Dα c − ie[Γα , c]) 2 4ξ 2 2 o ¯ α Φ − ΦΓ ¯ α Dα Φ , ¯ 2 + m)Φ − g ΦΓ ¯ α Γα Φ + i g Dα ΦΓ −Φ(D 2 2
(4)
ig β g2 [Γ , Dβ Γα ] − [Γβ , {Γβ , Γα }] is the gauge aether-superfield 2 6 strength which transforms covariantly, Wα0 = eiK Wα eiK , with K = K(x, θ) being a real where W α =
1 β α D D Γβ 2
−
scalar aether-superfield. D stands for the dimension of space-time, equal to 2 or 3. To quantize the model, we added the usual gauge fixing and the corresponding Faddeev-Popov terms, with c¯, c are the corresponding ghosts. In order to obtain the gauge aether superfield propagator, it is convenient to write the quadratic part of the gauge aether superfield action as Z n 1 o 1 D 2 α γ β α 2 β S2 = T r d xd θ − Γγ D D D Dα Γβ − Γα D D D Γβ 8 4ξ Z o n D 1 p˜βγ D2 1 d p 2 2 α 2 β , (5) d θ − Γ (˜ p , θ)˜ p C + Γ (−˜ p , θ) − Γ D D D Γ = Tr γ βγ β α β (2π)2 4 p˜2 4ξ where p˜βγ = (γ m )βγ p˜m = (γ m )βγ (pm + kmn pn ) is the twisted moment, p˜2 = p2 + 2kmn pm pn + k mn kml pn pl and D2 = ∂ 2 − θβ (γ m )βα p˜m ∂ α + θ2 p˜2 . The propagators obtained from Eq. (4), can be cast as iδab D2 (Dβ Dα − ξDα Dβ ) δ12 2 (˜ p2 )2 iδab (1 + ξ)Cβα p˜2 + (1 − ξ)˜ pβα D2 δ12 , = 2 (˜ p2 )2 D2 hca (˜ p, θ1 )¯ cb (−˜ p, θ2 )i = iδab 2 δ12 , p˜ D2 − m ¯ hΦa (˜ p, θ1 )Φb (−˜ p, θ2 )i = −iδab 2 δ12 , p˜ + m2
hΓαa (−˜ p, θ1 )Γβb (˜ p, θ2 )i =
(6)
(7)
where δ12 = δ 2 (θ1 − θ2 ). Without loss of generality, we choose to work in the Feynman gauge, i.e. ξ = 1. 3
The effective action receives one-loop pure gauge sector contributions from the diagrams drawn in Figs. 1 (a), (b) and (c). Performing the D-algebra manipulations with the help of the computer package SusyMath [10], we get the following results. The supergraph depicted at Fig. 1 (a) vanishes, while other contributions can be cast as Z Z dD p 2 dD q α (˜ pαβ D2 + 2Cβα q˜2 ) β g2 d θ Γ (˜ p , θ) Γa (−˜ p, θ); S1(b) =−N 4 (2π)2 (2π)2 a q˜2 (˜ q + p˜)2 S1(c)
g2 =−N 4
Z
dD p 2 dθ (2π)2
Z
dD q α (˜ p2 − 2˜ q 2 ) Cβα β Γ (˜ p , θ) Γa (−˜ p, θ). a (2π)2 q˜2 (˜ q + p˜)2
(8)
(9)
Adding the two diagrams above, after some algebraic manipulations, we have Z Z β dD q dD p 2 γ 1 g2 2 2 Γ (−˜ p , θ) S1loop = −N d θ Γ (˜ p , θ) p ˜ D + C p ˜ αβ βα a a 4 (2π)2 (2π)2 q˜2 (˜ q + p˜)2 Z Z dD q 1 dD p 2 α N g2 p˜βγ D2 2 β =− p ˜ Γ (−˜ p , θ) d θ Γ (˜ p , θ) C + . (10) βγ a a 4 (2π)2 p˜2 (2π)2 q˜2 (˜ q + p˜)2 Summing up the one-loop correction (10) to the classical part of the effective action (5), the pure gauge sector contributions to the effective action can be cast as Z dD p 2 γ p˜βγ D2 1 d θ Γa (˜ p, θ) Cβγ + p˜2 × Sgauge = − 2 2 4 (2π) p˜ Z D 2 d q Ng × 1+ Γβ (−˜ p, θ). (11) 2 2 (2π) q˜ (˜ q + p˜)2 a p˜βγ D2 p˜2 = Dβ Dγ D2 , this action is proportional to Γγa Dβ Dγ D2 Γβa , We note that since Cβγ + p˜2 thus being perfectly gauge invariant. In D = 3, the one-loop contribution to the above effective action is just a non-local correction to the − 18 Γγ Dα Dγ Dβ Dα Γβ term, that is, the Maxwell term. But, in two dimensions the situation is different. In D = 2, the integral over q gives Z 1 d2 p 2 γ p˜βγ D2 N g2 2 Sgauge = − p, θ) Cβγ + p˜ 1 + Γβ (−˜ p, θ) d θ Γa (˜ 4 (2π)2 p˜2 4π p˜2 a Z 1 d2 p 2 γ p˜βγ D2 2 =− d θ Γa (˜ p, θ) Cβγ + p˜ + M 2 Γβa (−˜ p, θ) 2 2 4 (2π) p˜
(12)
where M 2 = N ∆g 2 /4π. Here, in the integral over q we have changed the variable q to q˜, R R so that we can write d3 q = ∆ d3 q˜, where ∆ = det−1 (δnm + knm ) is the Jacobian of the transformation. In the usual notations, we can rewrite this expression as Z d2 p 2 M2 1 α d θ Wa (˜ p, θ) 1 + 2 Waα (−˜ p, θ), Sgauge = 2 (2π)2 p˜ 4
(13)
that is, the nonlocal extension of the Maxwell action. However, it is easy to check that this nonlocality essentially affects only the longitudinal sector, while in the transversal one, or as is the same, under the condition Dα Γaα = 0, it completely goes away. For kmn = um un with small um , we have ∆ ≈ (1 − u2 ). Alternatively, we can consider kmn to be traceless, i.e., kmn = um un − D1 gmn u2 as above, where now ∆ = det−1 (δnm + 43 un um ) det−3 (δnm − 14 un um ). In this case, for the small um , one evidently has ∆ = 1. Moreover, in principle nobody forbids to consider an antisymmetric kmn which evidently yields ∆ = 1 as well, i.e., there is no modification of the measure. Even though the pure gauge sector of the model is massless at classical level, the charge g is a dimensionful parameter with mass dimension one. Therefore, the pure aether-SYM theory in two dimensions exhibit a dynamical generation of mass, where M is a parameter dependent on the Lorentz breaking properties of the aether-superspace. Let us compute the one-loop corrections to the effective action of the gauge aethersuperfield due to matter interactions, Figs. 1 (d) and 1 (e). The contributions Fig. 1 (d) and Fig. 1 (e) are given by Sm1d = −g
2N
2
Z
dD p 2 d θ Γaβ (−˜ p, θ) (2π)2
Z
dD q C αβ Γaα (˜ p, θ), (2π)2 q˜2 + m2
(14)
and dD q 1 Sm1e = g 2 2 2 (2π) [(˜ q + p˜) + m2 ](˜ q 2 + m2 ) 1 γβ γβ α 2 2 αβ αβ αβ 2 q + mC )Dγ D Γaα (˜ × (˜ q + m )C + (˜ q + mC )D + (˜ p, θ), 2 2N
Z
dD p 2 d θ Γaβ (−˜ p, θ) (2π)2
Z
respectively. Adding the two contributions above, we have Z Z dD p 2 dD q q˜βγ − mCβγ β 2N ˜ γ (˜ Sm = g d θ Γ (−˜ p , θ) W p, θ). a 2 (2π)2 (2π)2 [(˜ q + p˜)2 + m2 ](˜ q 2 + m2 ) a
(15)
(16)
˜ aα = 1 Dβ Dα Γaβ is the linear part of the Yang-Mills aether-superfield strength W α . where W 2 Through the identity q˜αβ dD q (2π)2 [(˜ q + p˜)2 + m2 ](˜ q 2 + m2 ) Z 1 p˜αβ dD q =− = 2 2 2 (2π) [(˜ q + p˜) + m2 ](˜ q 2 + m2 ) p˜αβ =− f (˜ p), 2 Z
5
(17)
the Eq.(16) can be rewritten as Z h i dD p 2 2N α ˜ aα (−˜ ˜aα (˜ ˜ Sm = −g d θ f (˜ p ) W p , θ) W p , θ) + 2mΓ (−˜ p , θ) W (˜ p , θ) . aα a 2 (2π)2
(18)
It is interesting to consider the low-energy limit of this expression, that is, to keep only the terms leading at p → 0. In this case one has f (˜ p)|p→0 = 2N
∆ Sm ' −g 2 8π|m|
Z
∆ . 8π|m|
So, our result takes the form
i dD p 2 h ˜ α α ˜ ˜ (−˜ p , θ) W (˜ p , θ) . (19) d θ (−˜ p , θ) W (˜ p , θ) + 2mΓ W aα aα a a (2π)2
We conclude that we succeeded to generate the aether-like supersymmetric Maxwell-ChernSimons term. It is well-known that the presence of Chern-Simons (CS) term Γα W0α generates a topological massive pole of the gauge aether superfield propagator. In two dimensions, the above expression represents the effective action for a massive gauge invariant aether superfield, just as discussed in Refs. [11, 12] in a two-dimensional ordinary superspace. We note that generalizing of this calculation to three- and four-point functions of the gauge superfield Γαa , in the case when we consider gauge-matter coupling only, naturally will imply in arising the aether generalization of the full-fledged non-Abelian supersymmetric Chern-Simons term. Actually, to do it, one can simply repeat all sequence of the calculations performed in the paper [13] by some of us, with the only difference will consist in using of deformed supercovariant derivatives instead of the usual ones, with the result will be the same as that one obtained in [13], with the only modifications of the result will be the presence of the Jacobian ∆ multiplier and the replacement of usual supercovariant derivatives acting on the background superfields by the aether-deformed ones (and similarly, all momenta will be replaced by twisted ones). The details of this calculation will be presented elsewhere.
III.
FINAL REMARKS
In this work we studied the perturbative generation of the linearized Yang-Mills and Chern-Simons terms in the aether superspace. We explicitly demonstrated that this generation requires no more difficult calculations than in the usual superspace. Unlike the earlier studies [2, 4], we, for the first time, considered the contributions from essentially non-Abelian vertices. We explicitly showed that the nonlocality arising in the term arising 6
from the purely gauge sector is nonphysical as it does not affect the transversal part of the gauge superfield. Finally, we argued that the same approach can be applied to generation of the full-fledged non-Abelian super-Yang-Mills theory as well, the details will be presented in our next work.
ACKNOWLEDGMENTS
This work was partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). The work by A. Yu. P. has been supported by the CNPq project 303783/2015-0. A. C. L. has been partially supported by the CNPq projects 307723/2016-0 and 402096/2016-9.
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(a)
(b)
(d)
(e)
(c)
Figure 1. One-loop contributions to the gauge superfield effective action. Wiggly, dashed and continuous lines represent the gauge, ghost and matter superfield propagators, respectively.
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