Renormalizability of Super Yang–Mills Theory in Landau Gauge with A

Eur. Phys. J. C (2018) 78:797 https://doi.org/10.1140/epjc/s10052-018-6239-5 Regular Article - Theoretical Physics Renormalizability of N = 1 super Yang–Mills theory in Landau gauge with a Stueckelberg-like field M. A. L. Capria, D. M. van Egmondb, M. S. Guimaraesc, O. Holandad, S. P. Sorellae,R.C.Terinf, H. C. Toledog Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro 20550-013, Brazil Received: 9 March 2018 / Accepted: 11 September 2018 / Published online: 1 October 2018 © The Author(s) 2018 Abstract We construct a vector gauge invariant transverse due to the high energy behavior of the longitudinal vector field configuration V H , consisting of the well-known super- degree of freedom. In the abelian case it is perfectly compen- field V and of a Stueckelberg-like chiral superfield .The sated by the dynamics of the Stueckelberg field but in non- renormalizability of the Super Yang Mills action in the Lan- abelian theories this seems to be not so, resulting in incur- dau gauge is analyzed in the presence of a gauge invariant able divergent interacting amplitudes or unbounded cross mass term m2 dVM(V H ), with M(V H ) a power series sections. in V H . Unlike the original Stueckelberg action, the resulting Nevertheless there have been recent interests in the study action turns out to be renormalizable to all orders. of massive vector models without the Higgs. The main moti- vation comes here from the continuous efforts to understand the low energy behavior of strongly interacting gauge the- 1 Introduction ories, such as QCD. Confinement is a very important phenomenon in this context, but the physical mechanism behind In this work we study the renormalizability properties of a it is still an open problem. A way to obtain information about N = 1 non-abelian gauge theory defined by a multiplet con- this phenomenon is through lattice investigations which have taining a massive vectorial excitation. The model we study revealed that the gluon propagator shows a massive behav- is the supersymmetric version of a Stueckelberg-like action, ior in the deep infrared non-perturbative region, while also in the sense that the massive gauge field is constructed by displaying positivity violations which precludes a proper par- means of a compensating scalar field, thus preserving gauge ticle propagation interpretation [7–13]. Therefore, in a con- invariance. fining theory, the issue of the physical unitarity is a quite The history of Stueckelberg-like models is very well complex and difficult topic. Of course, physical unitarity reviewed in [1]. Traditionally, most of the investigations must hold in terms of the physical excitations of the spec- have studied such models as potential alternative to the trum which are bound states of quarks and gluons like, for Higgs mechanism of mass generation, but as discussed in instance, mesons, barions, glueballs, etc. Though, the posi- [2] there seems to be an unavoidable clash between renor- tivity violation of the two-point gluon correlation functions malizability and unitarity in non-abelian Stueckelberg-like is taken as a strong evidence of confinement, signalling that models. The original Stueckelberg model is abelian and has gluons are not excitations of the physical spectrum of the been rigorously proved [3] to be renormalizable and uni- theory. Nevertheless, renormalizability should be expected tary, but its non-abelian version is known to be perturba- to hold since one wants to recover the good UV behavior of tively non-renormalizable [4–6]. Physically, the problem is QCD. This trend of investigations led to many works that proposed modifications of the Yang-Mills theory to accommo- a e-mail: [email protected] date the lattice results [14–18]. Recent developments along b e-mail: [email protected] these lines involve the introduction of modified Stueckelberg- c e-mail: [email protected] like models [19–23] constructed as a generalization of a class d e-mail: [email protected] of confining effective theories known as Gribov–Zwanziger e e-mail: [email protected] scenarios [24,25], see [26] for a review. Unlike the stan- f e-mail: [email protected] dard Stueckelberg action, these modified models enjoy the g e-mail: [email protected] pleasant property of being renormalizable to all orders, see 123 797 Page 2 of 14 Eur. Phys. J. C (2018) 78 :797 [19–23] for a detailed account on the construction of these 2PureN = 1 SUSY Stueckelberg-like Yang–Mills modified models and on their differences with the standard theory Stueckelberg theory. Let us also also mention here that, recently, a BRST invariant reformulation of the Gribov– In order to define the N = 1 Supersymmetric Stueckelberg- Zwanziger theory has been achieved [21,23], allowing its like Yang–Mills theory, we start with a real abelian gauge extension from the Landau gauge to an arbitrary covariant superfield, gauge. The present model is intended only to construct a renor- μ i V (x,θ,θ) = C + iθχ − iθχ + θσ θ Aμ + θθ(M + iN) malizable theory which generalizes the non-supersymmetric 2 construction given in [22]. Issues like the perturbative uni- i i μ − θθ(M − iN) + iθθ θ¯ λ¯ + σ¯ ∂μχ tarity of the models so obtained are not explicitly addressed. 2 2 As far as we know, the non-supersymmetric model is not per- i μ − iθθθ λ − σ ∂μχ turbative unitary. Though, it can be successfully employed 2 as an effective renormalizable model in order to investigate 1 1 the non-perturbative infrared region of confining Yang–Mills + θθθθ D − ∂2C , (1) theories. So far, the prediction of the non-supersymmetric 2 2 model are in good agreement with the actual lattice data on and with a massless chiral superfield that acts as a Stueckel- the correlation functions of the theory, like the two-point berg field gluon propagator. Our aim here is to construct a sypersimmetric generaliza- √ ( ,θ,θ) = ξ( ) + θψ( ) + θσμθ∂¯ ξ( ) − θθ ( ) tion of this model for a future investigation of the confine- x x 2 x i μ x f x ment aspects of pure STM, which is known to be a confining i μ ¯ 1 2 − √ θθ∂μψ(x)σ θ − θθθθ∂ ξ(x) (2) theory. This is the main purpose of the present model. 4 2 √ μ In a confining YM theory, the issue of the unitarity has (x,θ,θ) = ξ(x) + 2θψ(x) − iθσ θ∂μξ(x) to be faced through the study of suitable colorless bound- i μ state, a topic which is still too far from the goal of the present − θθ f (x) + √ θθθσ ∂μψ(x) work, whose aim is that of obtaining a renormalizable mas- 2 1 sive SPYM theory which generalizes the model of [22]. − θθθθ∂2ξ(x). (3) In this work we will carry out a supersymmetric general- 4 ization of the Stueckelberg-like model proposed in [19–23]. It is then possible to construct a gauge-invariant superfield We prove that the present supersymmetric generalization is renormalizable, a task that will be done by means of a set of H ( ,θ,θ) = ( ,θ,θ)+ ( ,θ,θ)− ( ,θ,θ), suitable Ward identities. Supersymmetric generalizations of V x V x i x i x (4) Stueckelberg-like models was studied since very early [27] but mostly concentrated on the better behaved abelian mod- which is invariant under the abelian gauge transformations els (see [28–30], for instance, for a proposal of an abelian Stueckelberg sector in MSSM), with some constructions of V → V + iφ − iφ, → − φ, → − φ. (5) non-abelian theories with tensor multiplets [31–33] and also with composite gauge fields [34]. We now need a generalization of the definition of V H to the The work is organized as follows. In Sect. 2 we con- non-abelian case. We start with the gauge-invariant superfield struct the N = 1 Supersymmetric massive classical action. (4) with every component now in the adjoint representation In Sect. 3 we discuss the gauge fixing and the ensuing BRST of the gauge group G, V H ≡ V HaT a, (a = 1,...,dim G) symmetry. Sections 4 and 5 are devoted to the derivation where the T a are the generators in the adjoint. Now, the H of a set of suitable Ward identities and to the characteri- fundamental object is eV instead of V H . The non-abelian zation of the most general invariant local counterterm fol- generalization of (4)is lowing the setup of the algebraic renormalization. In Sect. 6 we provide a detailed analysis of the counterterm by H eV = HeV H, (6) showing that it can be reabsorbed into the starting classical action through a redefinition of the fields and parame- = i = iφ ters, thus establishing the all orders renormalizability of the where H e , U e and V is a usual gauge superfield. model. Section 7 contains our conclusion. The final Appen- The gauge transformations then have to be dices collect the conventions and a few additional technical − −1 details. eV → UeV U, H → U 1 H, H → H U , (7) 123 Eur. Phys. J. C (2018) 78 :797 Page 3 of 14 797 such that V H is gauge invariant. For infinitesimal transfor- 3 Supersymmetric gauge invariant Stueckelberg-like mations, this explicitly yields Yang–Mills action in the Landau gauge i i The supersymmetric extension of the Landau gauge is [1] δgaugeV = L V (φ + φ) + (L V coth(L V/ ))(φ − φ) 2 2 2 2 i i D D2V = 0. (14) = i(φ − φ) + [V,φ+ φ]+ [V, [V,φ− φ]] 2 12 +O(V 3), (8) We thus need to add the following terms to the action δ = i φ − 1( ( ))φ gauge L L cot L/2 1 2 2 2 2 LSGF = Tr dSAD D V + c.c.

Load more