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 © 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

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[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. 8 =−φ + i [, φ]+ 1 [, [, φ]] + O(3), (9) 1 2 2 12 = Tr dV(AD2V + A D V ), (15) 8 δ =−i φ − 1( ( ))φ gauge L L cot L/2 2 2 where we introduced the auxiliary chiral superfield A, with i 1 =−φ − [, φ]+ [, [, φ]] + O(3), the following field equations 2 12 (10) δ 1 2 2 LSGF = D D V = 0, δ A 8 =[ , ] δ with L A X A X . To first order (abelian gauge limit) in 1 2 2 φ LSGF = D D V = 0. (16) this reproduces (5). δ A 8 Now, using all of the above definitions we can construct a gauge invariant N = 1 supersymmetric Stueckelberg-like Following the standard BRST procedure, the gauge fixing Yang–Mills model condition can be implemented in a BRST invariant way by defining the auxiliary field as the BRST variation of the anti- = 1 α ghost field c , namely sc A, so that we can add the fol- L =− tr dSW Wα SYM 2 lowing BRST invariant term to the action in order to fix the 128g gauge. + 2 M( H ), m tr dV V (11) 1 2 LSGF = s tr dV(c D V + c.c. with 8 1 2 2 = tr dS[(A D D2V − c D D2sV + c.c.]. 2 −V V Wα ≡ D (e Dαe ), (12) 8 (17) and Looking at the gauge fixing (17), it is important to realize that for any local quantum field theory involving dimension- M = Ha Ha + σ abc Ha Hb Hc V V 1 V V V less fields, one has the freedom of performing arbitrary re- +σ abcd Ha Hb Hc Hd + ..., 2 V V V V (13) parametrization of these fields. Examples of this are the two- dimensional non-linear sigma model [37,38] and quantum σ abc,σabcd,... where 1 2 are a set of infinite arbitrary dimen- field theories with a Stueckelberg field entering the gauge sionless parameters. As one can figure out, the fact that the fixing term [39]. In the case of the gauge fixing (17), this generalized mass term M(V H ) is an infinite power series means that we have the freedom of replacing V a by an arbi- V H follows from the dimensionless character of V H itself. trary dimensionless function of V a Though, from the pertubative point of view, only the first 2 Ha Ha a → Fa( ) = a + αabc b c + αabcd b c d quadratic terms of the series (13), i.e. m V V will enter V V V 1 V V 2 V V V + αabcde b c d e + ..., the expression of the superfield propagator. The remaining 3 V V V V (18) terms represent an infinite set of interaction vertices, a fea- αabc,αabcd,αabcde,... ture which is typical of the non-abelian Stueckelberg-like where 1 2 3 are free dimensionless coef- theories. ficients. This freedom, inherent to the dimensionless nature Notice that the first term of the action, the pure supersym- of V a, is evident at the quantum level because of the fact α metric Yang–Millsterm W Wα, is invariant under V → V H . that this field renormalizes in a non-linear way [36,40–42]. For more details about the N = 1 supersymmetry and con- Therefore, (18) is expressing precisely the freedom one has ventions, see [35,36]. in the choice of a re-parametrization for V a. 123 797 Page 4 of 14 Eur. Phys. J. C (2018) 78:797

In our case, this means that instead of Eq. (17) we could a Lagrange multiplier implementing the transversality con- have just as well started with a term straint (21), while the fields η, η are a set of ghost fields needed to compensate the Jacobian which arises from the 1 s tr dV(c D2V + c.c.) functional integral over B and in order to get a unity, see 8 [19–23,39] for the non-supersymmetric case. 1 → s tr dV(c D2F(V ) + c.c.) , (19) Thus, adopting the Landau gauge, as well as the transver- 8 sality condition, the total action becomes and this would not have affected the correlation func- 4 tions of the gauge invariant quantities. The coefficients SPYM = d x(LSYM + LSP + LSGF + LT ) (αabc,αabcd,αabcde ...) are gauge parameters, not affect- 1 2 3 1 m2 ing the correlation functions of the gauge invariant quanti- =− tr dSWa W + tr dV M(V H ) g2 a ties. The freedom that we have in the gauge fixing term will 128 2 1 2 become apparent when performing the renormalization anal- + s tr dS(c D D2V ) 8 ysis. In fact, in Sect. 5, we will use a generalized gauge-fixing term + 1 2 2 H − ηa 2 2 a (η, η) dS B D D V D D G H 8 V 1 Sgen = s tr dV(c D2F(V ) + c.c. gf 8 + c.c. ∂F( ) 1 2 2 2 2 V = tr dS A D D F(V ) − c D D sV 2 8 ∂V 1 a m H =− tr dSW Wa + tr dV M(V ) 128g2 2 + c.c., (20) 1 2 2 + dS Aa D D2V a − ca D D2Ga and by employing the corresponding Ward identities, we can 8 V handle the ambiguity that is inherent to the gauge fixing. The + 1 2 2 H − ηa 2 2 a (η, η) counterterm will then correspond to a renormalization of the dS B D D V D D G H (αabc,αabcd,αabcde ...) 8 V gauge parameters 1 2 3 , as will become clear in Sect. 6. + c.c. . (23) One of the striking features ensuring the renormalizability of the non-supersymmetric modified Stueckelberg-like mod- This action enjoys the exact BRST nilpotent symmetry els introduced in [19–23,39] was the implementation of a transversality constraint on the analogue of the gauge invari- a = a ( , ), sV GV c c ant field V H . This transversality constraint gives rise to a sV Ha = 0, deep difference between the modified models constructed in a = a ( ), [19–23,39] and the conventional non-renormalizable Stueck- s G c a = a ( ), elberg model. It is precisely the implementation of this s G c transversality constraint which ensures the UV renormaliz- i sca =− f abccbcc, ability of the modified model. We remind here the reader 2 to reference [39] for a detailed account on the differences sηa = 0 between the conventional and the modified Stueckelberg i sca =− f abccbcc action. We then pursue here the same route outlined in [19– 2 23,39] and impose the transversality constraint also in the sηa = 0 supersymmetric case. More precisely, this amounts to require sca = Aa, that the superfield V H obeys the constraint sAa = 0, 2 2 H = , D D V 0 (21) sη a = 0, which, at the level of the action, can be implemented by sBa = 0, a introducing the following terms sca = A , ηa = , L = 1 2 2 H − ηa 2 2 a (η, η) s 0 T dS B D D V D D G H 8 V with + c.c., (22) a ( , ) = ( a − a) − 1 abc b( c + c) GV c c c c f V c c , 2 Ga (η, η) = (ηa − ηa) + i f abcV H b(ηc + ηc) − with V H 2 i amr mpq p p q r 3 i f amr f mpq(η p − η p)V HqV Hr + O(V H3). The field B is − f f (c − c )V V + O(V ), 12 12 123 Eur. Phys. J. C (2018) 78:797 Page 5 of 14 797

a a 1 abc b c Table 2 Quantum numbers of the sources G(c) =−c − f c 2 a a a a a a a 1 L L − f amr f mpqc pq r + O(3), 12 Dimension2233332 a ( ) =− a + 1 abcb c − 1 amr mpq pq r c-ghost # − 10− 1 − 1 − 2 − 20 G c c f c f f c 2 12 η-ghost # 0 − 1 − 1 − 1000 3 + O( ) (24) R-weight 0 0 2 − 2 − 220 and All quantum numbers, dimensions and R-weights of all sS = 0, s2 = 0. (25) SPYM ﬁelds and sources are displayed in Tables 1 and 2.

4.1 Ward identities and algebraic characterization of the 4 Renormalizability analysis invariant counterterm

In order to analyze the renormalizability of the action (23), The complete action SPYM obeys a large set of Ward iden- we start by establishing the set of Ward identities that tities, being: will be employed for the study of the quantum corrections. Following the algebraic renormalization procedure • The Slavnov–Taylor identity: [40], we have first to add some external sources coupling to non-linear BRST transformations of the fields and of δ δ δ δ the composite operators entering the classical action. There- S() = dV + dS δa δV a δa δa fore, we need to introduce a set of external BRST invari- a a a a a δ δ δ ant sources ( , , , L , L ) coupled to the non-linear a a + + A + c.c. = 0 (29) BRST variations of (V a,a, , ca, ca) as well as sources δLa δca δca (a,a) coupled to the BRST invariant composite operators (V H , Ga ) • The gauge-fixing equations: V H , a = a = a = a = a = a = a = . δ 1 δ 1 s s s s s sL sL 0 = D¯ 2 D2V a, = D2 D¯ 2V a (30) δ a δ ¯ (26) A 8 A 8 • a We shall thus start with the BRST invariant complete The equation for the Lagrange multiplier B : action defined by δ 1 δ δ 1 δ = D¯ 2 D2 , = D2 D¯ 2 (31) = + + + + (27) δBa 8 δa δ ¯ a 8 δa SPYM SYM SP SGF T EXT B 1 m2 =− tr dSWa W + dVM(V H ) • SPYM 2 a The anti-ghost equations: 128g 2 1 + { a 2 2 a − a 2 2 a }+ . . Ga = 0, G¯a = 0, (32) dS A D D V c D D GV c c − − 8 +1 { 2 2 H − ηa 2 2 a }+ . . with dS B D D V D D GV H c c 8 δ 1 δ a a a H,a a a Ga = + ¯ 2 2 + dV{ G + V + G } − D D and V V H δc a 8 δa δ δ ¯a 1 2 2 a a i acb a b c G = + D D¯ (33) + dS{− G − f L c c }+c.c. (28) − δ ¯a δa 2 c 8

Table 1 Quantum numbers of a θa Da V a V Ha a ca c¯a ca c¯a Aa Ba ηa ηa η¯a η¯a the ﬁelds Dimension − 1/21/200 000011110101 c-ghost # − 1000 0011− 1 − 1 0 00 00 0 η-ghost # − 1000 000000001− 11− 1 R-weight − 1100 0000− 22− 2 − 20− 20 2

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• The η Ward identities [40] and perturb the classical action (23), by adding an integrated local quantity in the fields and sources, CT , that a a ( , ) F− = 0, F− = 0, (34) has R-weight 0, ghost number 0 0 , is hermitian and has dimension 3 in case of a chiral superfield, or 2 in case of a with vector superfield. We demand thus that the perturbed action, ( + εCT), where ε is an expansion parameter, fulfills, to δ δ ε a 1 ¯ 2 2 the first order in , the same Ward identities obeyed by the F− = + D D and δηa 8 δa classical action , i.e. Eqs. (29)to(39). This amounts to δ δ impose the following constraints on : a 1 2 ¯ 2 CT F− = + D D (35) δηa 8 δa Bct = 0, (40) δ δ ct = ct = • The linearly broken ghost equation [43] a 0 (41) δ Aa δ A δ δ δ δ CT = 1 2 2 , CT = 1 2 2 G+ = clas, (36) D D a D D (42) δBa 8 δa δB 8 δa Ga = Ga = with − CT − CT 0 (43) Fa = Fa = − CT − CT 0(44) δ δ abc a Ga = , G+ = dS − if c + CT 0 (45) δcc δ Ab where B is the so-called nilpotent linearized Slavnov– δ δ + dS − ifabcc a , (37) Taylor operator [40], defined as δcc δ b A δ δ δ δ B = dV + δa δV a δV a δa and δ δ δ δ δ δ + dS + + abc a b abc a b δa δa δa δa δLa δca clas = if dV V + if dSL c δ δ δ a a + + A + c.c. , (46) + ifabc dS L cb. (38) δca δLa δca

with BB = 0. From Eq. (40) one learns that CT belongs Notice that the breaking term clas is purely linear in to the cohomology [40] of the linearized Slavnov–Taylor the quantum fields. As such, it will be not affected by the operator B in the space of the integrated local quantities quantum corrections [40,43]. in the fields and sources with ghost number (0, 0), R-weight • The linear symmetries under supersymmetry, transla- 0 and dimension 3 in case of a chiral superfield, or 2 in case tions, R-transformations and rigid transformations are of a vector superfield. Therefore, we can set expressed by the Ward identities −1 CT = cohom + B , (47) δ where (−1) denotes a zero-dimensional integrated quantity W =−i δ φ = 0, X X δφ in the fields and sources with ghost number (−1, 0) and R- φ (− ) weight 0. The term B 1 in Eq. (47) corresponds to the = , , X Qα Pμ R and rigid transformations (39) trivial solution, i.e. to the exact part of the cohomology of B. On the other hand, the quantity cohom identifies the P Q Q R non-trivial solution, i.e. the cohomology of B , meaning such that δμ , δα , δα˙ , δ are defined by appendix B. Thus, we that = B Q, for any local integrated Q. can see that the covariant action satisfy the Ward identities cohom In its most general form, is given by and Lorentz invariance. cohom 2 H α m H = dSa (V ) W Wα + dV M (V ) cohom 0 g2 128 2 5 The algebraic characterization of the invariant + ab( H ) a 2 2 H,b counterterm and renormalizability dSb1 V B D D V + ab( H ) H,b 2 2 a In order to characterize the most general invariant countert- dSb2 V V D D B erm which can be freely added to all order in perturbation + a H,b 2 2 ab( H ) theory, we follow the setup of the algebraic renormalization dS B V D D b3 V 123 Eur. Phys. J. C (2018) 78:797 Page 7 of 14 797 + ab ηa 2 2ηb + ab( H ) Hba dSd1 D D dVb1 V V + abc( H )ηa H,b 2 2ηc + 1 ab( H ) Hb 2 2 a + . . . dSc1 V V D D dS b1 V V D D B c c 8 + abc( H )ηaηb 2 2 H,c (50) dSc2 V D D V 2 + dScabc(V H )ηa V H,b D D2ηc Let us now discuss the trivial part of the counterterm, 3 (− ) (− ) B 1 .Theterm 1 , taking into account the quantum + ηaηb H,c 2 2 abc( H ) numbers of the fields and sources, can be parametrized in its dS V D D c4 V most general form as: + abc( H )ηaηb 2 2 H,c dSc5 V D D V −1 = ( ab ba + ab b 2 a dV F1 V F2 V D c + abc( H )ηa H,b 2 2ηc dSc6 V V D D + Fab D2V bca + D2 FabV bca + c.c.) 3 4 + ηaηb H,c 2 2 abc( H ) + . . dS V D D c7 V c c + dS(aabLacb + aabba) + c.c. , (51) 1 2 + ab( H )a H,b + abaηb dVb4 V V dVd2 where, + abc( H ) Hcaηb + abaηb dVc8 V V dVd3 Fa = Fa V,, 1,..,4 1,..,4 ab = ab + abc( H ) caηb, a1 a1 [ ] (52) dVc9 V V (48) ab = ab . a2 a2 [ ] with Imposing the constraint (41) Ha Ha abc Ha Hb Hc M = b0V V + σ V V V δ δ 1 (−1) (−1) B = B = , +σ abcd Ha Hb Hc Hd + ..., a a 0 (53) 2 V V V V (49) δ A δ A ( ( H ), ab( H ), abc( H ), ab) and a0 V bi V ci V di arbitrary coeffici- and observing from Eq. (51) that ents. Then, after implementing the constraints (42) and (43) δ(−1) δ(−1) δ(−1) we find = = 0 ⇒ B δ Aa δ a δ Aa A H (−1) a0(V ) α δ cohom = dS W Wα = B = 0, (54) 128g2 δ a A m2 + dV M (V H ) we can use the relation 2 δ δ 1 δ + dVdabaηb , B = + 2 2 , 1 D D (55) δ Aa δc a 8 δ + 1 abηa 2 2ηb + . . dS d1 D D c c to impose 8 ab a b + dVd η δ 1 2 δ 2 2 + D D2 −1 = D D2(FabV b) δca 8 δ 2 1 2 + dS dabηa D D2η b + c.c. 2 2 2 + D (Fab D2V b) + D ((D2 Fab)V b) 8 3 4 1 + abc( H )a Hbηc + 2 2( ab b) = . dVc1 V V D D F1 V 0 (56) 8 + 1 abc( H )ηa Hb 2 2ηc + . . dSc1 V V D D c c From Eq. (56) we find the relations 8 1 + dVcabc(V H )a V Hbηc ab =− ab 2 F2 F1 8 ab (57) + 1 abc( H ) ηa Hb 2 2ηc + . . F = 0 dSc2 V V D D c c 3 8 ab = δab, F4 a 123 797 Page 8 of 14 Eur. Phys. J. C (2018) 78:797 so that meaning that the correlator V (1)...... V (n) reduces to that of standard SPYM. As consequence, the dimensionless, and − 1 1 2 1 = dV FabV b(a − D2c a − D c a) thus m-independent, coefficients appearing in the countert- 1 8 8 erm CT are subject to the following additional conditions + ( ab a b + abba) + . . . dS a1 L c a2 c c (58) H a0(V ) = a0, abηb + abηb + ( abcηc + abcηc) Hb = a (η, η), d d c c V b1G H We can further reduce the number of parameters in CT 1 2 1 2 V ab( H ) = δab, by noticing that if we set [22] b1 V b1 (65) m2 = = = = = 0, (59) and for Eq. (51) a a in (28), the resulting action is F ,..., V,, = F ,..., [V ] 1 4 1 4 ab = ab 1 a a1 [ ] a1 =− tr dSW Wa SPYM 128g2 ab = ab , a2 [ ] a2 [ ] (66) +1 ( a 2 2 a − a 2 2 a ) + . . dS A D D V c D D GV c c so that 8 1 a 2 H,a a 2 a 2 + ( 2 − η 2 ) + . . a0 α m H dS B D D V D D G H c c = dS W Wα + dV M(V ) 8 V cohom 2 128g 2 a a i acb a b c Ha a + dV G + dS − f L c c + c.c. , + dV b1V V 2 (60) + 1 Ha 2 2 a − ηa 2 2 a (η, η) + . . dSb1 V D D B D D GV H c c 8 which is nothing but the super Yang–Millsgauge-fixed action + dVb a Ga (η, η), (67) in the Landau gauge (see Appendix C), with the addition of 1 V H the following terms and = 1 ( 2 2 H − ηa 2 2 a ) + . . . B dS B D D V D D G H c c 1 1 2 8 V −1 = dV Fab(V )V b a − D2ca − D ca 1 8 8 (61) + ( ab a b + ab()ba) + . . . However, upon integration over (B,η,η, ), these terms give dS a1 L c a2 c c rise to a unity. Thus, in the limit (59) the starting action takes (68) the following form: For the purpose of the analysis of the renormalization factors = + . inicial SPYM B (62) at the end of this section, we will rewrite the counterterm Let us consider now the correlation functions of the Yang- CT of the action in its parametrized form, namely as contact Mills superfield V , namely terms written in terms of the starting classical action , being given by the following expression ( )...... ( ) V 1 V n δ δ 2 2 [ ] ( )...... ( ) (− ) CT = dSa0g + dVb0m D V 1 V n exp inicial ∂g2 δm2 = , (63) [D]exp(−inicial) ∂ ∂ +b dS Ba + c.c. + b dV a 1 ∂ Ba 1 ∂a [ ] where D stands for integration over all fields. Though, 1 ∂ 1 ∂ 2 + b dVa + b dS ηa + c.c. since m = 0, one can directly perform in (63) the integration 2 1 ∂a 2 1 ∂ηa over (H, B,η,η), i.e. of the fields appearing in B. 1 ∂ + b dS ηa + c.c. This integration is easily seen to give a unity. It is in fact 2 1 ∂ηa nothing but a super Faddeev–Popov term (see [36]) which, δ δ δ + dV σ˜ abc +˜σ abcd +˜σ abcde + ... 2 = 1 δσabc 2 δσabcd 3 δσabcde due to m 0, gives a unity. 1 2 3 2 = Therefore, in the limit, m 0, it follows that δFab δ δ + dV 1 V b + Fabδbc a + FabV b δ c 1 δc 1 δ a ( )...... ( ) V V V 1 V n δ δaab(V ) δ [D]V (1)...... V (n) exp(−SPYM) + dS aabb − 2 b + aab()δbc a + c.c. = , (64) 2 δa δc 2 δc [D]exp(−SPYM) 123 Eur. Phys. J. C (2018) 78:797 Page 9 of 14 797

ab 1 δF (V ) 2 a a i abc a b c − 1 V b + Fab(V )δbc Gc (D2c a + D c a ) + dS − G + f L c c + c.c δ c 1 V 2 8 V ∂F a ( ) a a a H,a a a a a a V c 1 a 2 ab b 1 a 2 ab b + dV G + V + G + P F (V ) − R G . − A D (F V ) − A D (F V ). (69) V V H ∂ c V 8 1 8 1 V (74) 6 Analysis of the counterterm and renormalization The action SPYM obeys the following Ward identities: factors • The Slavnov–Taylor identity: Having determined the most general form of the local invari- δ δ δ ant counterterm, Eq. (69), we observe, however, that the terms S() = dV + Pa δa δV a δRa on the last line, δ δ δ δ δ 1 + + + a + . . = − ( a 2( ab b) + a 2( ab b)) dS A c c 0 S c D F1 V c D F1 V δa δa δLa δca δc a 8 ab (75) 1 δF (V ) 2 =− 1 V b + Fab(V )δbc Gc (D2ca + D ca) 8 δV c 1 V • The gauge-fixing equations: 1 1 a 2 − Aa D2(FabV b) − A D (FabV b), (70) 8 1 8 1 δ 1 δ δ 1 δ = D¯ 2 D2 , = D2 D¯ 2 cannot be rewritten in an exact parametric form in terms of δ Aa 8 δPa δ A¯ 8 δPa the starting action . This feature is due to dependence of the • a gauge fixing on the dimensionless field V . As a consequence, The equation for the Lagrange multiplier B : the renormalization of the gauge fixing itself is determined δ 1 δ δ 1 δ up to an ambiguity of the type of Eq. (18). As was mentioned = D¯ 2 D2 , = D2 D¯ 2 (76) δ a δa δ ¯ a δa before, this term can be handled by starting with the general- B 8 B 8 ized gauge fixing of Eq. (18). This means that we could have • The anti-ghost equations: equally started with a term in the action like a ¯a 1 2 G− = 0, G− = 0, (77) s tr dV(c D2V + c D V ) 8 1 2 with → s tr dV(c D2F(V ) + c D Fa(V )) , (71) 8 δ δ a 1 ¯ 2 2 G− = − D D and with Fa givenbyEq.(18). Since Fa is now a composite field, δca 8 δRa δ δ we need to introduce it into the starting action through a suit- ¯a 1 2 ¯ 2 G− = − D D (78) able external source. In order to maintain BRST invariance, δc¯a 8 δRa we make use of a BRST doublet of external sources (Ra , Pa), • η of dimension 2, R-weight 0 and ghost number (−1, 0) The Ward identities

a a a a a sR = P , sP = 0(72)F− = 0, F − = 0, (79) and introduce the term with dVs(RaFa(V )) δ δ a 1 ¯ 2 2 F− = + D D and ∂Fa δηa 8 δa = dV PaFa(V ) − Ra Gc(V ) , (73) δ δ ∂ c a 1 2 ¯ 2 V F − = + D D (80) δηa 8 δa so that the full action is now given by 2 • The linear symmetries under supersymmetry, transla- 1 a m H =− tr dS(W Wa ) + dV M(V ) SPYM 128g2 2 tions, R-transformations and rigid transformations are a 1 2 2 ∂F expressed by the Ward identities in (39). + dS Aa D D2F a (V ) − ca D D2 Gc + c.c. 8 ∂V c V + 1 { 2 2 H − ηa 2 2 a + . .} Since R and P are a pair of BRST doublet, they do not appear dS B D D V D D GV H c c 8 in the non-trivial part of the counterterm [40], so this will 123 797 Page 10 of 14 Eur. Phys. J. C (2018) 78:797

(−1) δ δ δ remain as in Eq. (67). On the other hand, the -term + dV σ˜ abc +˜σ abcd +˜σ abcde + ... 1 δσabc 2 δσabcd 3 δσabcde becomes 1 2 3 δFab δ −1 = ( ab ba + ab b 2 a − dV 1 V b + Fabδbc a dV F1 V F2 V D c δV c 1 δc δ δ δ + ab 2 b a + 2 ab b a + ab b a + . ) + ab b + ( ) a + ( ) a F3 D V c D F4 V c F8 V R c c F V F2 0 P F2 0 A a 1 δV a δPa δ A + ( ab a b + abab) + . . . δ δ dS a1 L c a2 c c (81) +F (0)Aa + F (0)Ra 2 δ a 2 δ a A R δ δ In a similar fashion as in the analysis of Eqs. (53)–(56), +F (0)c a + F (0)c a 2 δca 2 δca [ ∂ − 1 2 2 δ , ]= δ − we can use the relation ∂ Aa 8 D D δPa S δca δ δ 2 δ + dS aabLa − aabca 1 2 1 δ b 1 δ b 8 D D δR , to find L c δaab() δ δ δ 1 2 δ − 2 − 2 b + aab()δbc a + aab()b + c.c. − 2 1 = 2( ab b) δc 2 δc 2 δa D D D D F2 V δc a 8 δR δ 2 ab 2 b 2 2 ab b + dV(αabc − F (0)aabc) + D (F D V ) + D ((D F )V ) 1 2 1 δαabc 3 4 1 1 2 δ − D D2(FabV b) = 0, (82) +(αabcd − F (0)αabcd) + (αabcde 8 2 2 2 δαabcd 3 8 2 δ from which we obtain the relations −F (0)αabcde) + ..., (86) 2 3 δαabcde 1 3 Fab = Fab 2 8 8 where the dots ... in the last line denote the infinite set of ab = F3 0 terms of the kind ab = δab, F4 a (83) .. .. δ (αabcde − F (0)αabcde ) , j = 4,...,∞. j 2 j ∂αabcde so that (81) becomes j j (87) 1 1 2 −1 = dV FabV ba + FabV b(Ra + D2ca + D ca) 1 2 8 8 The usefulness of rewriting the counterterm (86) in the para- + ( a a + abba) + . . . metric form becomes clear by casting it into the form dS a1 L c a2 c c (84) R = 0, (88) We now set with Fab(V ) = Fab(0) + Fab(V ), 2 2 2 ab( ) = αabc c + αabcd c d δ δ F2 V 1 V 2 V V R = dSa g2 + b dV m2 0 δg2 0 δm2 +αabcdeV cV d V e + ..., (85) 2 δ δ +b dS Ba + c.c. + b dVa ab( ) = ( )δab 1 δBa 1 δa where F2 0 F2 0 is the first, V -independent term of the Taylor expansion of Fab(V ) in powers of V , and 1 δ 1 δ 2 + b dVa + b dS ηa + c.c. Fab(V ) denotes the remaining V -dependent terms. We find 2 1 δa 2 1 δηa 2 that Fab(0) is connected to the renormalization of the fields δ 2 + 1 ηa + . . ab( ) b1 dS a c c within the gauge-fixing, while F2 V renormalizes the 2 δη gauge parameters in Eq. (18). Employing the generalized δ δ δ + dV σ˜ abc +˜σ abcd +˜σ abcde + ... gauge-fixing (71), we can now write the full counterterm in 1 δσabc 2 δσabcd 3 δσabcde 1 2 3 a complete parametric form, namely δFab δ δ δ δ − dV 1 V b + Fabδbc a + FabV b = dSa g2 + b dV m2 δV c 1 δc 1 δV a CT 0 δg2 0 δm2 δ δ δ δ + ( ) a + ( ) a +b dS(Ba ) + c.c. + b dVa F2 0 P F2 0 A a 1 δBa 1 δa δPa δ A δ δ 1 δ 1 δ a a + b dVa + b dS ηa + c.c. +F2(0)A + F2(0)R 2 1 δa 2 1 δηa δ Aa δRa δ δ δ 1 a + ( ) a + ( ) a + b1 dS η + c.c. F2 0 c F2 0 c a 2 δηa δc a δc 123 Eur. Phys. J. C (2018) 78:797 Page 11 of 14 797 δ δ + dS aabLa − aabca and 1 δLb 1 δcb (αabc) = ( + ( ))αabc + εαabc δaab() δ 1 0 1 F2 0 1 1 − 2 b + aab()δbc a δc 2 δc (αabcd) = ( + ( ))αabcd + εαabcd 2 0 1 F2 0 2 2 abcde abcde abcde δ (α )0 = (1 + F2(0))α + εα +aab()b + c.c. 3 3 3 2 δa ... (94) δ + dV(αabc − F (0)αabc) + (αabcd 1 2 1 δαabc 2 F(V ) 1 This shows that the inclusion of the generalized field δ in the gauge fixing leads to the standard renormalization of −F (0)αabcd) + (αabcde 2 2 δαabcd 3 the fields, parameters and sources. The renormalization of 2 δ F(V ) itself is encoded in the renormalization of the infinite abcde −F2(0)α ) + ... (89) (αabc,αabcd,αabcde,...) 3 δαabcde set of gauge parameters 1 2 3 ,asinEq. 3 (94). Now, in order to determine the renormalization factors we Note that both V and , as well as their sources and can use that , are renormalized in a non-linear way through a power series in V and , respectively. This is expected, due to the () + εCT() = () + εR() fact that both superfields are dimensionless. However, one 2 = (0) + O(ε ), (90) has to note that the dimensionless superfield V contains a massive supermultiplet (Aμ,λ). Despite the fact that V itself with renormalizes in a non-linear way due to its dimensionless nature, the component fields (Aμ,λ)do renormalize in fact = = ( + εR) + O(ε2), 0 Z 1 (91) in a standard multiplicative way through a constant ( i.e. field independent) renormalization factors, a feature which can be where 0 is a short-hand notation for all renormalized quan- checked out by employing the the Wess–Zumino gauge. tities: fields, parameters and external sources. We thus find the the following renormalization factors

Zg = 1 + a0 7 Conclusion = + Zm2 1 b0 2 2 2 2 2 In this work we took a first step towards the understanding Z B = Z = Z = Z = Zη = Zη = Zη = Z = 1 + b1 B η of Stueckelberg-like models in supersymmetric non-abelian δFac ab = δab − 1 c + ab gauge theories. The gauge invariant transverse field configu- Z V F1 δV b ration V H has been investigated in supersymmetric Yang- ab = δab + ab ZV F1 Mills theory with the Landau gauge. An auxiliary chiral = = = = = = + ( ) superfield was introduced that compensates the gauge vari- Z P Z A Z A Z R Zc Zc 1 F2 0 ation of the vector superfield V , thus preserving gauge invari- Z ab = Z ab = Z ab = Z ab = δab + aab L L c c 1 ance of the composite field V H . This gauge invariant com- δaac Z ab = δab − 2 c + aab() posite field allows the construction of a local BRST-invariant δb 2 massive model, summarized by the action (23). Both V and ac δa c are dimensionless, which leads to ambiguities in defin- Z ab = δab − 2 + aab() δb 2 ing both the mass term and the gauge fixing term. However, ab = δab + ab() working with a generalized gauge fixing term, we find that Z a2 the model turns out to be renormalizable to all orders of per- Z ab = δab + aab(), (92) 2 turbation theory, as was discussed in Sects. 5 and 6. As a possible future application of the present result, as well as a multiplicative renormalization of the infi- let us mention that the possibility of having constructed a nite set of gauge parameters (σ abc,σabcd,σabcde) and 1 2 3 manifestly BRST invariant supersymmetric renormalizable (αabc,αabcd,αabcde) of Eqs. (13) and (18), being 1 2 3 version of the modified Stueckelberg models introduced in (σ abc) = ( + εσ abc)σ abc [19–23,39] can open the possibility to investigate the impor- 1 0 1 1 1 tant issue of the non-perturbative phenomenon of the Gribov (σ abcd) = (1 + εσ abce)σ abcd 2 0 2 2 copies directly in superspace, by generalizing to N = 1 (σ abcde) = ( + εσ abcde)σ abcde 3 0 1 3 3 the Gribov–Zwanziger setup. This would enable us to study ... (93) aspects of the non-perturbative region of N = 1 confin- 123 797 Page 12 of 14 Eur. Phys. J. C (2018) 78:797 ing supersymmetric theories, see also [44] for a preliminary Using the supersymmetry transformations as a Lie algebra attempt in this direction. we can define objects

μ i(−x Pμ+θ Q+θ Q) Acknowledgements The Conselho Nacional de Desenvolvimento G(x,θ,θ) = e (B5) Científico e Tecnológico (CNPq-Brazil), the Faperj, Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, the SR2-UERJ called superfields which transform covariantly under super- and the Coordenação de Aperfeiçoamento de Pessoal de Nível Supe- symmetry transformations. Superfields can be of the gen- rior (CAPES) are gratefully acknowledged for financial support. S. eral type, or chiral type. A chiral superfield A(x,θ,θ) and P. Sorella is a level PQ-1 researcher under the program Produtivi- ( ,θ,θ) dade em Pesquisa-CNPq, 300698/2009-7; M. A. L. Capri is a level anti-chiral superfield A x , is a superfield obeying the PQ-2 researcher under the program Produtividade em Pesquisa-CNPq, constraint 302040/2017-0. M. S. Guimaraes is supported by the Jovem Cientista do Nosso Estado program - FAPERJ E-26/202.844/2015, is a level Dα˙ A = 0, Dα A = 0, (B6) PQ-2 researcher under the program Produtividade em Pesquisa-CNPq, 307801/2017-9 and is a Procientista under SR2-UERJ. where ∂ ∂ = − σ μ θ α˙ ∂ , =− + θ ασ μ ∂ Open Access This article is distributed under the terms of the Creative Dα α i αα˙ μ Dα˙ α˙ i αα˙ μ (B7) Commons Attribution 4.0 International License (http://creativecomm ∂θ ∂θ ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit are the covariant superspace derivatives. The superfields are to the original author(s) and the source, provide a link to the Creative functions of superspace which should be understood in com- Commons license, and indicate if changes were made. ponents by series power in θ and θ. The transformation laws 3 Funded by SCOAP . (translation, supersymmetry and R-symmetry) of a superfield φ are respectively defined P δ φ = ∂μφ, (B8) Appendix A: Notation μ ∂ Q μ α˙ H Ha a δ φ = + iσ θ ∂μ φ, V = V T α ∂θα αα˙ (B9) ∂ T a, T b =−ifacbT c Q α μ δα˙ φ = − − iθ σαα˙ ∂μ φ, (B10) ∂θ α˙ dV = d4xd2θd2θ¯ (A1) and ∂ ∂ dS = d4xd2θ δ Rφ = + θ α − θ α˙ φ. i n α α˙ (B11) ∂θ ∂θ ¯ 4 2 ¯ dS = d xd θ The n number in (B11) is the “R-weight” of the superfield φ. The R-weights are opposite to each other in complex con- jugates superfields. These operators obey the super-Poincaré Appendix B: N = 1 superfields algebra μ {δQ,δQ}=−2iσ δ P , (B12) In the N = 1 case, we have a Poincaré algebra with spinor α α˙ αα˙ μ Q R Q Q R Q charges that anticommutes as [δα ,δ ]=iδα , [δα˙ ,δ ]=−iδα˙ . (B13) μ {Qα, Qα˙ }=2σαα˙ Pμ, {Qα, Qβ }=0, {Qα˙ , Qβ˙}=0, (B1) Appendix C: N = 1 supersymmetric Yang–Mills and commutes 1 β α˙ 1 α˙ β˙ For the benefit of the reader, we provide in this appendix a [Qα, Mμν]= (σμν)α Qβ , [Q , Mμν]=− (σ μν) ˙ Q , 2 2 β short overview of the well known renormalizability of pure (B2) N = 1 standard massless Super Yang–Mills in the Landau gauge. Let us start by giving the complete BRST invariant [Qα, Pμ]=0, [Qα˙ , Pμ]=0, (B3) action, namely and to = + + SYM SY M SGF EXT [Qα, R]=−Qα, [Qα˙ , R]=Qα˙ (B4) 1 a =− tr dS (W Wa) 128g2 The R is a symmetry transforming different charges in a theory into each other and that is isomorphic to a global 1 2 +s dV(ca D2V a + ca D V a) U(1) group. 8 123 Eur. Phys. J. C (2018) 78:797 Page 13 of 14 797 i + a a ( , ) + abc a b c 1 1 2 dV GV c c dSf L c c −1 = dV FabV ba + FabV b(Ra + D2ca + D ca) 2 1 2 8 8 i abc ¯ a b c + dSf L c¯ c¯ . (C1) + ( ab a b) + . . . 2 dS a1 L c c c (C9) The full action SPYM obeys the Ward identities (30), (32), Defining (36), (39) as well as the Slavnov–Taylor identity: Fab(V ) = F (0)δab + Fab(V ) δ δ 2 2 2 S() = dV Fab(V ) = αabcV c + αabcdV d + αabcdeV e + ..., (C10) δa δV a 2 1 2 3 δ δ δ we find the counterterm to be + dS + Aa + c.c. = 0. (C2) δLa δca δca δ δ δ = 2 + ( ) a + ( ) a CT dSa0g dV F2 0 P a F2 0 A a As usual, the counterterm CT can be written as ∂g2 δP δ A δ δ δ + F (0)Aa + F (0)Ra + F (0)c a (−1) 2 δ a 2 δ a 2 δ a CT = + B , (C3) A R c δ + F (0)c a 2 δca with δ δFab δ + ab b − 1 b + ac a α dV F1 V V F1 = a0 Tr dS W Wα, (C4) δV a δV c δc ∂ ∂ + dS α La − α ca + c.c. and 1 ∂ La 1 ∂ca δ − 1 2 abc 1 = ab( ) ba − ab( ) b( 2 a + a) + dV (α − F2(0)) dV F1 V V F1 V V D c D c 1 ∂αabc 8 1 δ + ( ab a b) + . . . +(αabcd − F (0)) dS a1 L c c c (C5) 2 2 ∂αabcd 2 δ +(αabcde − F (0)) + ... . (C11) When constructing the counterterm, we will run into to same 3 2 ∂αabcde parametrizing problem as in Sect. 5. Repeating the procedure 3 of Sect. 6, we introduce a doublet (Ra, Pa) and the full action Following the analysis at the end of Sect. 6, for the renor- is now given by malization factors we obtain 1 Z = 1 + a =− tr dS W a W g 0 SPYM 2 a δ ac 128g ab ab F1 c ab Z = δ − V + F i δV b 1 + dS f abcLacbcc + c.c. (C6) 2 ab = δab + ab ZV F1 ∂Fa 1 a 2 a a 2 c = = = = = = + ( ) + dV A D F − c D G + c.c. Z P Z A Z A Z R Zc Zc 1 F2 0 8 ∂V c V Z ab = Z ab = Z ab = Z ab = δab + aab . (C12) ∂Fa L L c c 1 + dV a Ga (c, c) + PaFa(V ) − Ra Gc (c, c) . V ∂V c V Thereby concluding the proof of the renormalizability of (C7) N = 1 pure massless Super Yang–Mills action.

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