The Quantum Description of BF Model in Superspace

Hindawi Advances in High Energy Physics Volume 2018, Article ID 9291213, 11 pages https://doi.org/10.1155/2018/9291213

Research Article The Quantum Description of BF Model in Superspace

Manoj Kumar Dwivedi

Department of Physics, Banaras Hindu University, Varanasi 221005, India

Correspondence should be addressed to Manoj Kumar Dwivedi; [email protected]

Received 29 May 2017; Revised 8 August 2017; Accepted 10 August 2017; Published 10 January 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Manoj Kumar Dwivedi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We consider the BRST symmetric four-dimensional BF theory, a topological theory, containing antisymmetric tensor fields in Landau gauge and extend the BRST symmetry by introducing a shift symmetry to it. Within this formulation, the antighost fields corresponding to shift symmetry coincide with antifieldstandard ofs field/antifield formulation. Furthermore, we provide a superspace description for the BF model possessing extended BRST and extended anti-BRST transformations.

1. Introduction most powerful quantization algorithms presently available. BV formulation deals with very general gauge theories, Topological gauge field theories (TGFT) which came from including those with open or reducible gauge symmetry mathematics have some peculiar features. The examples algebras. The BV method also address the possible violations of two distinct class of TGFT are topological Yang-Mills of symmetries of the action by quantum effects. The BV theory and Chern-Simons (CS) theory, which are sometimes formulation (independently introduced by Zinn-Justin [21]) classified as Witten-type and Schwarz-type, respectively [1]. extends the BRST approach [22–36]. In fact, the BRST Besides these two types, there is another Schwarz-type symmetry [37, 38] is a very important symmetry for gauge TGFT called topological BF theory, which is an extension theories [26, 39–54]. Besides the covariant description to of CS theory [2]. The difference between CS theory and BF perform the gauge-fixing in quantum field theory, BV for- model is that action of previous theory exists only in odd- mulation was also applied to other problems like analysing dimensions while later one can be defined on manifolds of possible deformations of the action and anomalies. any dimensions. A superspace description for various gauge theories in In string theory and nonlinear sigma model, four- BV formulation has been studied extensively [55–64]. They dimensional antisymmetric (or BF) models [3–8] were intro- have shown that the extended BRST and extended anti- duced some years ago. This model is interesting due to BRST invariant actions of these theories (including some its topological nature [1] and its connection with lower shift symmetry) in BV formulation yield naturally the proper dimensional quantum gravity; for example, three space-time identification of the antifields through equations of motion. dimensional Einstein-Hilbert models with or without using The shift symmetry is important and gets relevance, for cosmological constant can be naturally formulated in terms of example, in inflation particularly in supergravity [65] as BF-models [9, 10]. Coupling of an antisymmetric tensor field well as in Standard Model [66]. In usual BV formulation, with the field strength tensor of Yang-Mills is described by theseantifieldscanbecalculatedfromtheexpressionof these models [11]. Quantization of BF model in Landau gauge gauge-fixing fermion. We extended BRST formulation and has been studied in [11]. Topological BF theory in Landau superspace description of the topological gauge (BF) model gauge has a common feature of a large class of topological is still unstudied and we try to discuss these here. models [12–16]. In the present work, we try to generalize the superspace On the other hand, the Batalin-Vilkovisky (BV) approach, formulation of BV action for BF model. Particularly, we also known as field/antifield formulation, [17–20] is one ofthe first consider BRST invariant BF model in Landau gauge 2 Advances in High Energy Physics

and extend the BRST symmetry of the theory by including The effective Lagrangian density of BF model, L = L0 + shift symmetry. By doing so, we find that the antighosts L𝑔𝑓+𝑔ℎ , possesses the following BRST symmetry: of shift symmetry get identified as antifields of standard 𝑎 𝑎 BV formulation naturally. Further, we discuss a superspace 𝑠𝐴 𝜇 =−(𝐷𝜇𝐶) , formulation of extended BRST invariant BF model. Here we 1 see that one additional Grassmann coordinate is required if 𝑠𝐶𝑎 = 𝑓𝑎𝑏𝑐𝐶𝑏𝐶c, action admits only extended BRST symmetry. However, for 2 both extended BRST and extended anti-BRST invariant BF 𝑎 𝑠𝜉𝑎 =(𝐷 𝜙) +𝑓𝑎𝑏𝑐𝐶𝑏𝜉𝑐 , model two additional Grassmann coordinates are required. 𝜇 𝜇 𝜇

This paper is framed as follows. In Section 2, we discuss 𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 the BRST invariant BF model. In Section 3, we study the 𝑠𝐵𝜇] =−(𝐷𝜇𝜉] −𝐷]𝜉𝜇) −𝑓 𝐵𝜇]𝐶 extended BRST transformation of the model. Further, we 𝑎𝑏𝑐 𝜌 𝑏𝜎 𝑐 describe extended BRST invariant action in superspace in +𝑓 𝜖𝜇]𝜌𝜎 (𝜕 𝜉 )𝜙 , Section 4. The extended anti-BRST symmetry is discussed (3) in Section 5. The superspace formulation of extended BRST 𝑠𝜙𝑎 =𝑓𝑎𝑏𝑐𝐶𝑏𝜙𝑐, and anti-BRST invariant action is given in Section 6. The last section is reserved for concluding remarks. 𝑎 𝑎 𝑠𝜉𝜇 =ℎ𝜇, 𝑎 𝑠𝐶 =𝑏𝑎, 2. BRST Invariant BF Model 𝑎 𝑠𝜙 =𝜔𝑎, In this section, we discuss the preliminaries of BF model with its BRST invariance. In this view, the BF model in flat 𝑠𝑒𝑎 =𝜆𝑎, (3+ 1) space-time dimensions is given by the following gauge 𝑎 𝑎 𝑎 𝑎 invariant Lagrangian density [11]: 𝑠(ℎ𝜇,𝑏 ,𝜔 ,𝜆 )=0.

The gauge-fixing and ghost terms of the effective Lagrangian 1 density are BRST exact and, hence, can be written in terms of L =− 𝜖𝜇]𝜌𝜎𝐹𝑎 𝐵𝑎 , 0 4 𝜇] 𝜌𝜎 (1) BRST variation of gauge-fixing fermion, Ψ (4) 𝐵𝑎 𝐹𝑎 𝑎 𝜇 𝑎 𝑎𝜇 ] 𝑎 𝑎 𝜇 𝑎 𝑎 𝑎 𝑎 𝜇 𝑎 where 𝜌𝜎 and 𝜇] aretwo-formfieldandfieldstrengthtensor =(𝐶 𝜕 𝐴𝜇 + 𝜉 𝜕 𝐵𝜇] + 𝜙 𝜕 𝜉𝜇 −𝑒 𝜔 −𝑒 𝜕 𝜉𝜇), for vector field, respectively. In order to remove discrepancy due to gauge symmetry, the gauge-fixing and ghost terms are as follows: given by L𝑔𝑓+𝑔ℎ =𝑠Ψ. (5)

In the next section, we would like to study the extended BRST L 𝑔𝑓+𝑔ℎ symmetry for the model which incorporates shift symmetry 𝑎 𝜇 𝑎 𝑎 𝜇 𝑎 𝑎] 𝜇 𝑎 𝑎 𝑎 together with original BRST symmetry. =𝑏 𝜕 𝐴𝜇 + 𝐶 𝜕 (𝐷𝜇𝐶) +ℎ (𝜕 𝐵𝜇])+𝜔 𝜕𝜉

𝑎 +ℎ𝑎 (𝜕𝜇𝑒𝑎)+𝜔𝑎𝜆𝑎 +(𝜕𝜇𝜉 )𝜆𝑎 3. Extended BRST Invariant 𝜇 𝜇 Lagrangian Density 𝜇 𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 (2) −(𝜕 𝜙 )[(𝐷𝜇𝜙) +𝑓 𝑐 𝜉𝜇] The advantage of studying the extended BRST transformations for BF model in BV formulation is that antifields get 𝜇 𝑎] 𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 −(𝜕 𝜉 )[(𝐷𝜇𝜉]) −(𝐷]𝜉𝜇) +𝑓 𝐵𝜇]𝐶 ] identification naturally. We begin with shifting all the fields from their original value as follows: 1 𝑎𝑏𝑐 𝜇]𝜌𝜎 𝑎 𝑏 𝑐 + 𝑓 𝜖 (𝜕𝜇𝜉] )(𝜕𝜌𝜉𝜎)𝜙 , 𝑎 𝑎 ̃𝑎 2 𝐵𝜇] 󳨀→ 𝐵 𝜇] − 𝐵𝜇], 𝑎 𝑎 ̃𝑎 𝐴𝜇 󳨀→ 𝐴 𝜇 − 𝐴𝜇, 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 where fields (𝐶 ,𝜉𝜇), (𝐶 , 𝜉𝜇),and(𝑏 ,ℎ𝜇) are the ghosts, 𝐶 󳨀→ 𝐶 − 𝐶̃ , antighosts, and the multipliers fields, respectively, while 𝑎 𝑎 𝑎 𝑎 𝑎 ̃𝑎 the fields 𝜙 , 𝜙 and 𝜔 are taken into account to remove 𝐶 󳨀→ 𝐶 − 𝐶 , further degeneracy due to the existence of zero modes in the 𝑎 𝑎 𝑎 transformations. 𝑏 󳨀→ 𝑏 − ̃𝑏 , Advances in High Energy Physics 3

𝑎 𝑎 ̃𝑎 𝑎 𝑎 𝑎 ̃𝑎 𝜉𝜇 󳨀→ 𝜉 𝜇 − 𝜉𝜇, 𝑠𝑒̃ =𝑁 −(𝜆 − 𝜆 ),

𝑎 𝑎 𝑎 𝑎 𝑎 ̃ 𝑠𝜉𝜇 =𝐿𝜇, 𝜉𝜇 󳨀→ 𝜉𝜇 − 𝜉𝜇, 𝑎 𝑠𝐶 = 𝜖𝑎, 𝜙𝑎 󳨀→ 𝜙 𝑎 − 𝜙̃𝑎, ̃𝑎 𝑎 ̃ ̃ 𝑎 𝑎 𝑎 ̃𝑎 𝑠𝜉 =𝐿 −[(𝐷 − 𝐷 )(𝜙−𝜙) 𝜙 󳨀→ 𝜙 − 𝜙 , 𝜇 𝜇 𝜇 𝜇 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 𝑎 𝑎 ̃𝑎 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] , ℎ𝜇 󳨀→ ℎ 𝜇 − ℎ𝜇, 𝑎 𝑎 𝑎 𝑎 𝑎 𝑒 󳨀→ 𝑒 − 𝑒̃ , 𝑠𝜉𝜇 = 𝐿𝜇,

𝜔𝑎 󳨀→ 𝜔 𝑎 − 𝜔̃𝑎, ̃𝑎 𝑎 𝑎 ̃𝑎 𝑠𝜉𝜇 = 𝐿𝜇 −(ℎ𝜇 − ℎ𝜇), 𝜆𝑎 󳨀→ 𝜆 𝑎 − 𝜆̃𝑎. (8) (6) 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 where 𝜓𝜇,𝜖 , 𝜖 ,𝜒 ,𝑀 , 𝑀 ,𝑁 ,𝐿𝜇,and𝐿𝜇 are the ghost 𝑎 𝑎 𝑎 𝑎 The effective Lagrangian density of BF model also gets shifted fields corresponding to shift symmetry for 𝐴𝜇,𝐶 , 𝐶 ,𝑏 , under such shifting of fields, respectively. This is given by 𝑎 𝑎 𝑎 𝑎 𝑎 𝜙 , 𝜙 ,𝑒 ,𝜉𝜇 and 𝜉𝜇, respectively. The nilpotency of extended BRST symmetry (8) leads to the BRST transformation for the ̃ 𝑎 ̃𝑎 𝑎 ̃𝑎 𝑎 ̃𝑎 𝑎 ̃𝑎 𝑎 following ghost fields: L = L (𝐴𝜇 − 𝐴𝜇,𝐶 − 𝐶 , 𝐶 − 𝐶 ,𝑏 − 𝑏 ,𝜉𝜇

𝑎 𝑎 𝑎 𝑠𝜓 =0, ̃𝑎 ̃ 𝑎 ̃𝑎 𝑎 ̃𝑎 𝑎 ̃𝑎 𝑎 𝑎 𝑎 𝜇 − 𝜉𝜇, 𝜉𝜇 − 𝜉𝜇,𝜙 − 𝜙 , 𝜙 − 𝜙 ,ℎ𝜇 − ℎ𝜇,𝑒 − 𝑒̃ ,𝜔 (7) 𝑠𝜖𝑎 =0, − 𝜔̃𝑎,𝜆𝑎 − 𝜆̃𝑎). 𝑠𝜖𝑎 =0,

𝑠𝜒𝑎 =0, The shifted Lagrangian density is invariant under BRST transformation together with a shift symmetry transfor- 𝑠𝑀𝑎 =0, mation, jointly known as extended BRST transformation. (9) 𝑎 The extended BRST symmetry transformations under which 𝑠𝑀 =0, Lagrangian density of BF model is invariant are written by 𝑠𝑁𝑎 =0, 𝑎 𝑎 𝑠𝐴 𝜇 =𝜓𝜇, 𝑎 𝑠𝐿 𝜇 =0, ̃𝑎 𝑎 ̃ ̃ 𝑎 𝑠𝐴𝜇 =𝜓𝜇 −(𝐷𝜇 − 𝐷𝜇)(𝐶−𝐶) , 𝑎 𝑠𝐿𝜇 =0. 𝑠𝐶𝑎 =𝜖𝑎, In order to make the theory ghost free, we need further 1 ⋆𝑎 ⋆𝑎 ⋆𝑎 𝑠𝐶̃𝑎 =𝜖𝑎 − 𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑎 − 𝜉̃𝑎), 𝐴⋆𝑎,𝐶⋆𝑎, 𝐶 ,𝑏⋆𝑎,𝜉⋆𝑎, 𝜉 ,𝜙⋆𝑎, 𝜙 𝑒⋆𝑎 2 𝜇 𝜇 antighosts 𝜇 𝜇 𝜇 ,and to be introduced corresponding to the ghost fields ̃𝑎 𝑎 ̃ 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑠𝐶 = 𝜖 −(𝑏−𝑏) , 𝜓𝜇,𝜖 , 𝜖 ,𝜒 ,𝑀 , 𝑀 ,𝑁 ,𝐿𝜇,and𝐿𝜇,respectively.The

𝑎 𝑎 BRST transformations of these antighosts are constructed as 𝑠𝑏 =𝜒 , follows: 𝑎 𝑎 𝑠̃𝑏 =𝜒 , ⋆𝑎 𝑎 𝑠𝐴 𝜇 =−𝜁𝜇, 𝑠𝜙𝑎 =𝑀𝑎, 𝑠𝐶⋆𝑎 =−𝜎𝑎,

𝑠𝜙̃𝑎 =𝑀𝑎 −𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜙𝑐 − 𝜙̃𝑐), ⋆𝑎 𝑠𝐶 =−𝜎𝑎, 𝑎 𝑎 𝑠𝜙 = 𝑀 , 𝑠𝑏⋆𝑎 =−𝜛𝑎,

̃𝑎 ⋆𝑎 𝑎 𝑠𝜙 =𝑀𝑎 −(𝜔𝑎 − 𝜔̃𝑎), 𝑠𝜙 =−𝜐 , ⋆𝑎 𝑠𝑒𝑎 =𝑁𝑎, 𝑠𝜙 =−𝜐𝑎, 4 Advances in High Energy Physics

⋆𝑎 𝑎 𝑠𝑒 =−𝜏 , Integrating the auxiliary fields of the above expression, we ⋆𝑎 𝑎 obtain 𝑠𝜉𝜇 =−𝜅 ,

⋆𝑎 𝑠𝜉 =−𝜅𝑎, ̃ 𝑎⋆ 𝑎𝜇 𝜇 𝑎 𝜇 L𝑔𝑓+𝑔ℎ =−𝐴𝜇 [𝜓 −(𝐷 𝐶) ] (10) 𝑎⋆ 1 + 𝐶 [𝜖𝑎 − 𝑓𝑎𝑏𝑐𝐶𝑏𝜉𝑐 ]+𝐶𝑎⋆ [𝜖𝑎 −𝑏𝑎] 2 𝜇 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 where 𝜁𝜇,𝜎 , 𝜎 ,𝜛 ,𝜐 ,𝜏 ,𝜅 ,and𝜅 are the Nakanishi- ⋆𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 ⋆𝑎 𝑎 𝑎 Lautrup type auxiliary fields corresponding to shifted fields −𝜙 (𝑀 −𝑓 𝐶 𝜙 )−𝜙 (𝑀 −𝜔 ) ̃𝑎 ̃𝑎 ̃𝑎 (13) ̃𝑎 ̃𝑎 ̃𝑎 ̃𝑎 𝑎 ̃𝑎 ⋆𝑎 𝑎 𝑎 ⋆𝑎 𝑎 𝐴𝜇, 𝐶 , 𝐶 , 𝑏 , 𝜙 , 𝜙 , 𝑒̃ , 𝜉𝜇,and𝜉𝜇 having the following −𝑒 [𝑁 −𝜆 ]−𝑏 𝜒 BRST transformations: ⋆𝑎 𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 +𝜉𝜇 (𝐿𝜇 −[(𝐷𝜇𝜙) +𝑓 𝐶 𝜉𝜇]) 𝑎 𝑠𝜁𝜇 =0, ⋆𝑎 𝑎 𝑎 + 𝜉𝜇 [𝐿𝜇 −ℎ𝜇]. 𝑠𝜎𝑎 =0,

𝑠𝜎𝑎 =0, The gauge-fixing and ghost terms of the Lagrangian density 𝑠𝜛𝑎 =0, are BRST exact and can be expressed in terms of a general gauge-fixing fermion Ψ as 𝑎 𝑠𝜐 =0, (11) 𝑠𝜐𝑎 =0, 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑠Ψ = 𝑠𝐴𝜇 +𝑠𝐶 +𝑠𝐶 𝑎 +𝑠𝑏 𝑎 𝛿𝐴𝑎 𝛿𝐶𝑎 𝛿𝑏𝑎 𝑠𝜏 =0, 𝜇 𝛿𝐶

𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝛿Ψ 𝑎 𝛿Ψ 𝑠𝜅 =0, +𝑠𝜉𝑎 +𝑠𝜉 +𝑠𝜙𝑎 +𝑠𝜙 𝜇 𝑎 𝜇 𝑎 𝑎 𝑎 𝛿𝜉𝜇 𝛿𝜉 𝛿𝜙 𝛿𝜙 𝑠𝜅𝑎 =0. 𝜇 𝛿Ψ +𝑠𝑒𝑎 𝑎 (14) We can recover our original BF model by fixing the shift 𝛿𝑒 symmetry in such a way such that effect of all the tilde fields 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 =− 𝜓 + 𝜖 + 𝑎 𝜖 − 𝜒 − 𝐿 will vanish. We achieve this by adding following gauge-fixed 𝛿𝐴𝑎 𝜇 𝛿𝐶𝑎 𝛿𝐶 𝛿𝑏𝑎 𝛿𝜉𝑎 𝜇 term to the shifted Lagrangian density (7): 𝜇 𝜇 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 𝛿Ψ 𝑎 − 𝑎 𝐿 − 𝑀 − 𝑎 𝑀 − 𝑁 . L̃ =−𝜁𝑎𝜇𝐴̃𝑎 −𝐴𝑎⋆ [𝜓𝑎𝜇 −(𝐷𝜇 − 𝐷̃𝜇)(𝐶 𝜇 𝛿𝜙𝑎 𝛿𝑒𝑎 𝑔𝑓+𝑔ℎ 𝜇 𝜇 𝛿𝜉𝜇 𝛿𝜙

𝑎 𝑎⋆ 1 − 𝐶)̃ ]−𝜎𝑎𝐶̃𝑎 + 𝐶 [𝜖𝑎 − 𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑎 2 𝜇 After integrating out the auxiliary fields which set the tilde ̃𝑎 − 𝜉̃𝑎)] − 𝜎𝑎𝐶 +𝐶𝑎⋆ [𝜖𝑎 −(𝑏𝑎 − ̃𝑏𝑎)] − 𝜐𝑎𝜙̃𝑎 fields to zero, we have the complete effective action for BF 𝜇 model in Landau gauge possessing extended BRST symmetry as ̃𝑎 −𝜙⋆𝑎 [𝑀𝑎 −𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜙𝑐 − 𝜙̃𝑐)] − 𝜐𝑎𝜙

⋆𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 ⋆ 𝑎 𝑎 𝑎 ̃ − 𝜙 (𝑀 −𝜔 + 𝜔̃ )−𝜏 𝑒̃ −𝑒 [𝑁 −𝜆 + 𝜆̃ ] (12) Leff = L0 + L𝑔𝑓+𝑔ℎ + L𝑔𝑓+𝑔ℎ

̃𝑎 ⋆𝑎 𝛿Ψ 𝜇𝑎 −𝜛𝑎̃𝑏𝑎 −𝑏⋆𝑎𝜒𝑎 −𝜅𝑎𝜉̃𝑎 − 𝜅𝑎𝜉 +𝜉⋆𝑎 (𝐿𝑎 = L +(−𝐴 − )𝜓 𝜇 𝜇 𝜇 𝜇 0 𝜇 𝛿𝐴𝜇𝑎 ̃ ̃ 𝑎 −[(𝐷𝜇 − 𝐷𝜇)(𝜙−𝜙) ⋆𝑎 𝛿Ψ 𝑎 ⋆𝑎 𝛿Ψ 𝑎 +(𝐶 + )𝜖 +(𝐶 + 𝑎 ) 𝜖 𝛿𝐶𝑎 𝛿𝐶 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 ⋆𝑎 𝑎 𝑎 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)]) + 𝜉𝜇 [𝐿𝜇 −(ℎ𝜇 𝛿Ψ 𝛿Ψ −(𝑏⋆𝑎 + )𝜒𝑎 +(𝜉⋆𝑎 + )𝐿𝑎 ̃𝑎 𝑎 𝜇 𝑎 𝜇 − ℎ𝜇)] . 𝛿𝑏 𝛿𝜉𝜇

⋆𝑎 𝛿Ψ 𝑎 𝛿Ψ One can easily check that this gauge-fixing Lagrangian +(𝜉 + ) 𝐿 −(𝜙⋆𝑎 + )𝑀𝑎 ̃ 𝜇 𝑎 𝜇 𝛿𝜙𝑎 density L𝑔𝑓+𝑔ℎ also admits the extended BRST invariance. 𝛿𝜉𝜇 Advances in High Energy Physics 5

⋆𝑎 𝛿Ψ 𝑎 ⋆𝑎 𝛿Ψ 𝑎 It is obvious to see that, with these anti-ghost fields, the −(𝜙 + 𝑎 ) 𝑀 +(−𝑒 − )𝑁 𝛿𝜙 𝛿𝑒𝑎 expression (15) changes to the original Lagrangian density of the BF model in Landau gauge. 𝑎⋆ 𝐶 +𝐴𝑎⋆ (𝐷𝜇𝐶𝑎)𝑎 − 𝑓𝑎𝑏𝑐𝐶𝑏𝜉𝑐 +𝐶⋆𝑎𝑏𝑎 𝜇 2 𝜇 4. Extended BRST Invariant ⋆𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑐 ⋆𝑎 𝑎𝑏𝑐 𝑏 𝑐 Superspace Description +𝜉𝜇 [(𝐷𝜇𝜙) +𝑓 𝐶 𝜉𝜇]+𝜙 𝑓 𝐶 𝜉𝜇. (15) In this section, the Lagrangian density of BF model which is invariant under the extended BRST transformations only is described in a superspace (𝑥𝜇,𝜃),where𝜃 is a Grassmann Integrating out the ghost fields associated with shift symme- 𝑥 try, we obtain coordinate and 𝜇 is the four-dimensional spect-time coordinates. In order to give superspace description for the extended 𝛿Ψ BRST invariant theory, we first define superfields of the form 𝐴𝑎⋆ =− , 𝜇 𝜇𝑎 𝑎 𝑎 𝑎 𝛿𝐴 𝐴𝜇 (𝑥,) 𝜃 =𝐴𝜇 +𝜃𝜓𝜇, ⋆𝑎 𝛿Ψ 𝑎 𝐶 =− , 𝐴̃𝑎 (𝑥,) 𝜃 = 𝐴̃𝑎 +𝜃[𝜓 −(𝐷 − 𝐷̃ )(𝐶−𝐶)]̃ , 𝛿𝐶𝑎 𝜇 𝜇 𝜇 𝜇 𝜇 𝑎 𝑎 𝑎 ⋆𝑎 𝛿Ψ 𝜒 (𝑥,) 𝜃 =𝐶 +𝜃𝜖 , 𝐶 =− 𝑎 , 𝛿𝐶 1 𝜒̃𝑎 (𝑥,) 𝜃 = 𝐶̃𝑎 +𝜃[𝜖𝑎 − 𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑐 − 𝜉̃𝑐 )] , 𝛿Ψ 2 𝜇 𝜇 𝑏⋆𝑎 =− , 𝛿𝑏𝑎 𝑎 𝜒𝑎 (𝑥,) 𝜃 = 𝐶 +𝜃𝜖𝑎, ⋆𝑎 𝛿Ψ 𝜉 =− , 𝑎 𝜇 ⋆𝑎 ̃𝑎 ̃𝑎 𝑎 ̃ 𝛿𝜉𝜇 (16) 𝜒 (𝑥,) 𝜃 = 𝐶 +𝜃[𝜖 −(𝑏−𝑏) ], ⋆𝑎 𝛿Ψ 𝑏𝑎 (𝑥,) 𝜃 =𝑏𝑎 +𝜃𝜒𝑎, 𝜉𝜇 =− ⋆𝑎 , 𝛿𝜉 𝜇 ̃𝑏𝑎 (𝑥,) 𝜃 = ̃𝑏𝑎 +𝜃𝜒𝑎, 𝛿Ψ 𝜙⋆𝑎 =− , 𝜉𝑎 (𝑥,) 𝜃 =𝜉𝑎 +𝜃𝐿𝑎 , 𝛿𝜙𝑎 𝜇 𝜇 𝜇 ̃𝑎 ̃𝑎 𝑎 ̃ ̃ 𝑎 ⋆𝑎 𝛿Ψ 𝜉𝜇 (𝑥,) 𝜃 = 𝜉𝜇 +𝜃[𝐿𝜇 −[(𝐷𝜇 − 𝐷𝜇)(𝜙−𝜙) (18) 𝜙 =− 𝑎 , 𝛿𝜙 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)]] , ⋆𝑎 𝛿Ψ 𝑒 =− . 𝑎 𝑎 𝑎 𝑎 𝛿𝑒 𝜉𝜇 (𝑥,) 𝜃 = 𝜉𝜇 +𝜃𝐿𝜇,

Ψ ̃𝑎 ̃𝑎 𝑎 For a particular choice of gauge-fixing fermion given in (4), 𝜉 (𝑥,) 𝜃 = 𝜉 +𝜃[𝐿 −(ℎ𝑎 − ̃ℎ𝑎)] , anti-ghost fields get following identifications: 𝜇 𝜇 𝜇 𝜇 𝜇 𝜙𝑎 (𝑥,) 𝜃 =𝜙𝑎 +𝜃𝑀𝑎, 𝑎⋆ 𝑎 𝐴𝜇 =𝜕𝜇𝐶 , 𝜙̃𝑎 (𝑥,) 𝜃 = 𝜙̃𝑎 +𝜃[𝑀𝑎 −𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜙𝑐 − 𝜙̃𝑐)] , 𝑎⋆ 𝐶 =0, 𝑎 𝑎 𝑎 𝜙 (𝑥,) 𝜃 = 𝜙 +𝜃𝑀 , 𝑎⋆ 𝜇𝑎 𝐶 =−𝜕𝜇𝐴 , ̃𝑎 ̃𝑎 𝑎 𝜙 𝑥, 𝜃 = 𝜙 +𝜃[𝑀 −(𝜔𝑎 − 𝜔̃𝑎)] , 𝑏𝑎⋆ =0, ( ) 𝑎 𝑎 𝑎 𝑎 𝑒 (𝑥,) 𝜃 =𝑒 +𝜃𝑁 , 𝜉⋆𝑎 =𝜕𝜙 , 𝜇 𝜇 (17) 𝑒̃𝑎 (𝑥,) 𝜃 = 𝑒̃𝑎 +𝜃[𝑁𝑎 −𝜆𝑎 + 𝜆̃𝑎]. ⋆𝑎 ] 𝑎 𝑎 𝜉𝜇 =−𝜕𝐵𝜇] −𝜕𝜇𝑒 , The super-antifields in superspace are defined as follows: 𝜙⋆𝑎 =0, ̃⋆𝑎 ⋆𝑎 𝑎 𝐴𝜇 (𝑥,) 𝜃 =𝐴𝜇 −𝜃𝜁𝜇, ⋆𝑎 𝜙 =−𝜕𝜇𝜉𝑎, 𝜇 𝜒̃⋆𝑎 (𝑥,) 𝜃 =𝐶⋆𝑎 −𝜃𝜎𝑎, 𝑎 𝑒⋆𝑎 =𝜔𝑎 +𝜕𝜇𝜉 . ⋆𝑎 ⋆𝑎 𝜇 𝜒̃ (𝑥,) 𝜃 = 𝐶 −𝜃𝜎𝑎, 6 Advances in High Energy Physics

̃𝑏⋆𝑎 𝑥, 𝜃 =𝑏⋆𝑎 −𝜃𝜛𝑎, 𝑎⋆ 𝑎𝜇 𝜇 ̃𝜇 ̃ 𝑎 𝑎 ̃𝑎 ( ) −𝐴𝜇 [𝜓 −(𝐷 − 𝐷 )(𝐶−𝐶) ]−𝜎 𝐶

⋆𝑎 ⋆𝑎 𝑎 𝜉̃ (𝑥,) 𝜃 =𝜉 −𝜃𝜅 , 𝑎⋆ 𝑎 1 𝑎𝑏𝑐 𝑏 𝑏 𝑎 𝑎 𝑎̃𝑎 𝜇 𝜇 + 𝐶 [𝜖 − 𝑓 (𝐶 − 𝐶̃ )(𝜉 − 𝜉̃ )] − 𝜎 𝐶 2 𝜇 𝜇 ̃⋆𝑎 ⋆𝑎 𝑎 𝑎⋆ 𝑎 𝑎 𝑎 𝑎 𝑎 ⋆𝑎 𝑎 𝜉𝜇 (𝑥,) 𝜃 = 𝜉𝜇 −𝜃𝜅 , +𝐶 [𝜖 −(𝑏 − ̃𝑏 )] − 𝜐 𝜙̃ −𝜙 [𝑀

⋆𝑎 ⋆𝑎 𝑎 𝜙̃ (𝑥,) 𝜃 =𝜙 −𝜃𝜐 , ̃𝑎 ⋆𝑎 𝑎 −𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜙𝑐 − 𝜙̃𝑐)] − 𝜐𝑎𝜙 − 𝜙 (𝑀 ̃⋆𝑎 ⋆𝑎 𝑎 𝜙 (𝑥,) 𝜃 = 𝜙 −𝜃𝜐 , −𝜔𝑎 + 𝜔̃𝑎)−𝜏𝑎𝑒̃𝑎 −𝑒⋆ [𝑁𝑎 −𝜆𝑎 + 𝜆̃𝑎]−𝜛𝑎̃𝑏𝑎 𝑒̃⋆𝑎 (𝑥,) 𝜃 =𝑒⋆𝑎 −𝜃𝜏𝑎. ̃𝑎 −𝑏⋆𝑎𝜒𝑎 −𝜅𝑎𝜉̃𝑎 − 𝜅𝑎𝜉 +𝜉⋆𝑎 (𝐿𝑎 (19) 𝜇 𝜇 𝜇 𝜇 ̃ ̃ 𝑎 From the above expressions of superfields and super- −[(𝐷𝜇 − 𝐷𝜇)(𝜙−𝜙) antifields, we calculate 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 ⋆𝑎 𝑎 𝑎 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)]) + 𝜉𝜇 [𝐿𝜇 −(ℎ𝜇 𝛿(𝐴̃𝑎⋆𝐴̃𝑎𝜇) 𝜇 𝑎⋆ 𝑎𝜇 𝜇 ̃𝜇 ̃ 𝑎 =−𝐴𝜇 [𝜓 −(𝐷 − 𝐷 )(𝐶−𝐶) ] ̃𝑎 𝛿𝜃 − ℎ𝜇)] , 𝑎 ̃𝑎𝜇 (21) −𝜁𝜇𝐴 ,

𝑎⋆ 𝛿(𝜒̃ 𝜒̃𝑎) 𝑎⋆ 𝑎 1 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 which is nothing but the gauge-fixed Lagrangian density for = 𝐶 [𝜖 − 𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] ̃ 𝛿𝜃 2 shift symmetry L𝑔𝑓+𝑔ℎ given in (12). Now, one can define the general super-gauge-fixing fermion in superspace as follows: − 𝜎𝑎𝐶̃𝑎,

̃𝑎 𝑎⋆ 𝛿(𝜒 𝜒̃ ) ̃𝑎 Φ (𝑥,) 𝜃 =Ψ(𝑥) +𝜃(𝑠Ψ) , (22) =−𝜎𝑎𝐶 +𝐶𝑎⋆ [𝜖𝑎 −(𝑏𝑎 − ̃𝑏𝑎)] , 𝛿𝜃 𝛿(̃𝑏𝑎⋆̃𝑏𝑎) which can further be expressed as =−𝑏𝑎⋆𝜒𝑎 −𝜛𝑎̃𝑏𝑎, 𝛿𝜃 𝛿(𝜉̃𝑎⋆𝜉̃𝑎𝜇) 𝛿Ψ 𝛿Ψ 𝛿Ψ 𝜇 𝑎⋆ 𝑎 𝜇 𝜇 𝑎 [ 𝑎 𝑎 𝑎 =𝜉 [𝐿 −[(𝐷 − 𝐷̃ )(𝜙−𝜙)̃ Φ (𝑥,) 𝜃 =Ψ(𝑥) +𝜃 − 𝜓 + 𝜖 + 𝑎 𝜖 𝛿𝜃 𝜇 𝜇 (20) 𝛿𝐴𝑎 𝜇 𝛿𝐶𝑎 𝛿𝐶 [ 𝜇 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 𝑎̃𝑎 +𝑓 (𝐶 − 𝐶 )(𝜉 − 𝜉 )]] − 𝜅 𝜉 , 𝛿Ψ 𝛿Ψ 𝛿Ψ 𝑎 𝛿Ψ 𝛿Ψ 𝑎 𝜇 𝜇 𝜇 − 𝜒𝑎 − 𝐿𝑎 − 𝐿 − 𝑀𝑎 − 𝑀 𝛿𝑏𝑎 𝛿𝜉𝑎 𝜇 𝑎 𝜇 𝛿𝜙𝑎 𝑎 (23) ̃𝑎⋆̃𝑎𝜇 𝜇 𝛿𝜉 𝛿𝜙 𝛿(𝜉 𝜉 ) 𝜇 𝜇 𝑎⋆ ̃𝑎𝜇 = 𝜉 [𝐿𝜇𝑎 −ℎ𝜇𝑎 + ̃ℎ𝜇𝑎]−𝜅𝑎𝜉 , 𝛿𝜃 𝜇 𝛿Ψ − 𝑁𝑎] . 𝛿𝑒𝑎 𝛿(𝜙̃𝑎⋆𝜙̃𝑎) ] =−𝜙̃𝑎𝜐𝑎 −𝜙𝑎⋆ [𝑀𝑎 −𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜙𝑐 𝛿𝜃 − 𝜙̃𝑐)] , From this, the original gauge-fixing Lagrangian density can be defined as the left derivation of super-gauge-fixing fermion ̃𝑎⋆ ̃𝑎 with respect to 𝜃 as [𝛿Φ(𝑥, 𝜃)/𝛿𝜃]. 𝛿(𝜙 𝜙 ) ̃𝑎 𝑎⋆ 𝑎 Hence,thecompleteeffectiveactionfortheBFmodelin =−𝜙 𝜐𝑎 − 𝜙 [𝑀 −𝜔𝑎 + 𝜔̃𝑎], 𝛿𝜃 general gauge in the superspace is now given by 𝛿(𝑒̃𝑎⋆𝑒̃𝑎) ⋆ 𝑎 𝑎 ̃𝑎 𝑎 𝑎 =−𝑒 [𝑁 −𝜆 + 𝜆 ]−𝑒̃ 𝜏 . 𝛿 𝛿𝜃 L = L + [𝐴̃𝑎⋆𝐴̃𝑎𝜇 + 𝜒̃𝑎⋆𝜒̃𝑎 + 𝜒̃𝑎𝜒̃𝑎⋆ + ̃𝑏𝑎⋆̃𝑏𝑎 eff 0 𝛿𝜃 𝜇 Addingalltheequationsof(20)sidebyside,weget (24) ̃𝑎⋆ ̃𝑎 + 𝜒̃𝑎⋆𝜒̃𝑎𝜇 + 𝜒̃𝑎⋆𝜒̃𝑎𝜇 + 𝜙̃𝑎⋆𝜙̃𝑎 + 𝜙 𝜙 + 𝑒̃𝑎⋆𝑒̃𝑎 +Φ]. 𝛿 𝜇 𝜇 (𝐴̃𝑎⋆𝐴̃𝑎𝜇 + 𝜒̃𝑎⋆𝜒̃𝑎 + 𝜒̃𝑎𝜒̃𝑎⋆ + ̃𝑏𝑎⋆̃𝑏𝑎 + 𝜉̃𝑎⋆𝜉̃𝑎𝜇 𝛿𝜃 𝜇 𝜇

̃𝑎⋆̃𝑎𝜇 ̃𝑎⋆ ̃𝑎 Next, we will study the extended anti-BRST symmetry for BF + 𝜉 𝜉 + 𝜙̃𝑎⋆𝜙̃𝑎 + 𝜙 𝜙 + 𝑒̃𝑎⋆𝑒̃𝑎)=−𝜁𝑎𝜇𝐴̃𝑎 𝜇 𝜇 model. Advances in High Energy Physics 7

𝑠𝑀𝑎 =𝜐𝑎, 5. Extended Anti-BRST Lagrangian Density 𝜇 𝑎 𝑎 In this section, we construct the extended anti-BRST trans- 𝑠𝑀𝜇 = 𝜐 , formation under which the shifted Lagrangian density for BF 𝑎 𝑎 model remains invariant as follows: 𝑠𝑁𝜇 =𝜏 .

𝑎 𝑎 𝑎⋆ ̃ ̃ (26) 𝑠𝐴𝜇 =𝐴𝜇 +(𝐷𝜇 − 𝐷𝜇)(𝐶−𝐶) , ̃𝑎 𝑎⋆ 𝑠𝐴𝜇 =𝐴𝜇 , The nilpotency of above transformations demands the auxiliary and antighost fields associated with the shift symmetry 1 𝑠𝐶𝑎 =𝐶𝑎⋆ − 𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑐 − 𝜉̃𝑐 ), transform as 2 𝜇 𝜇 ̃𝑎 𝑎⋆ 𝑎 𝑠𝐶 =𝐶 , 𝑠𝜁𝜇 =0, 𝑎 𝑎⋆ 𝑎 ̃𝑎 𝑎⋆ 𝑠𝐶 = 𝐶 −(𝑏 − 𝑏 ), 𝑠𝐴𝜇 =0, 𝑎 ̃𝑎 𝑠𝜎 =0, 𝑠𝐶 = 𝐶𝑎⋆, 𝑠𝐶𝑎⋆ =0, 𝑠𝑏𝑎 =𝑏𝑎⋆ +𝜒𝑎, 𝑠 𝜎𝑎 =0, 𝑠̃𝑏𝑎 =𝑏𝑎⋆, 𝑎⋆ 𝑠𝐶 =0, 𝑎 𝑎⋆ ̃ 𝑎 ̃𝑎 𝑠𝜉𝜇 =𝜉𝜇 −[(𝐷𝜇 − 𝐷𝜇)(𝜙 − 𝜙 ) 𝑠𝜛𝑎 =0, 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] , (25) 𝑠𝑏𝑎⋆ =0, 𝑠𝜉̃𝑎 =𝜉𝑎⋆, 𝜇 𝜇 𝑠𝜅𝑎 =0, 𝑎 𝑎⋆ 𝑠𝜉 = 𝜉 −ℎ𝑎 + ̃ℎ𝑎, 𝑎⋆ (27) 𝜇 𝜇 𝜇 𝜇 𝑠𝜉𝜇 =0, ̃𝑎 𝑎⋆ 𝑠 𝜅𝑎 =0, 𝑠𝜉𝜇 = 𝜉𝜇 , 𝑎⋆ 𝑎 𝑎⋆ 𝑎𝑏𝑐 𝑏 𝑏 𝑐 𝑐 𝑠𝜙 =𝜙 −𝑓 (𝐶 − 𝐶̃ )(𝜙 − 𝜙̃ ), 𝑠𝜉𝜇 =0, 𝑎 𝑠𝜙̃𝑎 =𝜙𝑎⋆, 𝑠𝜐 =0, 𝑎⋆ 𝑎 𝑎⋆ 𝑠𝜙 =0, 𝑠𝜙 = 𝜙 −𝜔𝑎 + 𝜔̃𝑎, 𝑠 𝜐𝑎 =0, ̃𝑎 𝑎⋆ 𝑠𝜙 = 𝜙 , 𝑎⋆ 𝑠𝜙 =0, 𝑠𝑒𝑎 =𝑒𝑎⋆ −(𝜆𝑎 − 𝜆̃𝑎), 𝑠𝜏𝑎 =0, 𝑠𝑒̃𝑎 =𝑒𝑎⋆. 𝑠𝑒𝑎⋆ =0. The ghost fields associated with the shift symmetry transform under extended anti-BRST symmetry as The gauge-fixing and ghost parts of the effective Lagrangian density are anti-BRST-exact also so it can be 𝑠𝜓𝑎 =𝜁𝑎, 𝜇 𝜇 expressed as the anti-BRST variation of this gauge-fixing Ψ 𝑠𝜖𝑎 =𝜎𝑎, fermion ( ).

𝑠 𝜖𝑎 = 𝜎𝑎, 6. Extended BRST and Anti-BRST 𝑠𝜒𝑎 =𝜛𝑎, Invariant Superspace 𝑎 𝑎 𝑠𝐿𝜇 =𝜅 , The extended BRST and anti-BRST invariant Lagrangian density for BF model can be written in superspace with the 𝑎 𝑎 𝑠𝐿𝜇 = 𝜅 , help of two additional Grassmannian coordinates 𝜃 and 𝜃. 8 Advances in High Energy Physics

̃𝑎 ̃𝑎 𝑎 𝑎𝑏𝑐 𝑏 𝑏 𝑐 Requiring the field strength to vanish along unphysical direc- 𝜙 (𝑥, 𝜃, 𝜃) = 𝜙 (𝑥) +𝜃(𝑀 −𝑓 (𝐶 − 𝐶̃ )(𝜙 tions 𝜃 and 𝜃 direction, we obtain the following superfields: − 𝜙̃𝑐)) + 𝜃𝜙⋆𝑎 +𝜃𝜃𝜐𝑎,

𝑎 𝑎 𝑎 𝑎⋆ ̃ 𝑎 𝑎 𝑎 ⋆𝑎 𝑎 𝑎 A𝜇 (𝑥, 𝜃, 𝜃) = 𝐴𝜇 (𝑥) +𝜃𝜓𝜇 + 𝜃[𝐴𝜇 +(𝐷𝜇 − 𝐷𝜇) 𝜙 (𝑥, 𝜃, 𝜃) = 𝜙 (𝑥) +𝜃𝑀 + 𝜃(𝜙 −𝜔 + 𝜔̃ )

𝑎 𝑎 ̃ 𝑎 +𝜃𝜃𝜐 , ⋅(𝐶−𝐶) ]+𝜃𝜃𝜁𝜇, ̃𝑎 ̃𝑎 𝑎 ⋆𝑎 𝜙 (𝑥, 𝜃, 𝜃) = 𝜙 (𝑥) +𝜃(𝑀 −𝜔𝑎 + 𝜔̃𝑎)+𝜃𝜙 ̃𝑎 ̃𝑎 𝑎 ̃ ̃ 𝑎 A𝜇 (𝑥, 𝜃, 𝜃) = 𝐴𝜇 (𝑥) +𝜃[𝜓𝜇 −(𝐷𝜇 − 𝐷𝜇)(𝐶−𝐶) ] +𝜃𝜃𝜐𝑎, 𝑎⋆ 𝑎 + 𝜃𝐴𝜇 +𝜃𝜃𝜁𝜇, e𝑎 (𝑥, 𝜃, 𝜃) =𝑎 𝑒 (𝑥) +𝜃𝑁𝑎 + 𝜃(𝑒⋆𝑎 −𝜆𝑎 + 𝜆̃𝑎) 1 C𝑎 (𝑥, 𝜃, 𝜃) =𝑎 𝐶 (𝑥) +𝜃𝜖𝑎 + 𝜃[𝐶𝑎⋆ − 𝑓𝑎𝑏𝑐 (𝐶𝑏 2 +𝜃𝜃𝜏𝑎,

̃𝑏 𝑐 ̃𝑐 𝑎 ̃e𝑎 (𝑥, 𝜃, 𝜃) = 𝑒̃𝑎 (𝑥) +𝜃(𝑁𝑎 −𝜆𝑎 + 𝜆̃𝑎)+𝜃𝑒⋆𝑎 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] + 𝜃𝜃𝜎 , 1 +𝜃𝜃𝜏𝑎. C̃𝑎 (𝑥, 𝜃, 𝜃) = 𝐶̃𝑎 (𝑥) +𝜃[𝜖𝑎 − 𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑐 2 𝜇 (28) ̃𝑐 ⋆𝑎 𝑎 − 𝜉𝜇)] + 𝜃𝐶 +𝜃𝜃𝜎 , With these expressions of superfields, we can calculate

1 𝜕 𝜕 𝑎 𝜇𝑎 𝑎 ̃𝑎 𝑎 𝑎 ̃𝑎 ̃𝜇𝑎 ̃𝑎 ̃𝜇𝑎 𝑎 𝑎 𝑎 ⋆𝑎 ̃ 𝑎 − (Ã Ã + 𝜒̃ 𝜒 + b̃ b̃ + 𝜉 𝜉 + 𝜉 𝜉 C (𝑥, 𝜃, 𝜃) = 𝐶 (𝑥) +𝜃𝜖 + 𝜃[𝐶 −(𝑏−𝑏) ] 2 𝜕𝜃 𝜕𝜃 𝜇 𝜇 𝜇

𝑎 ̃𝑎 ̃𝑎 ̃𝑎 ̃𝑎 𝑎 𝑎 𝑎𝜇 ̃𝑎 𝑎⋆ 𝑎𝜇 +𝜃𝜃𝜎 , + 𝜙 𝜙 + 𝜙 𝜙 + ̃e ̃e )=−𝜁 𝐴𝜇 −𝐴𝜇 [𝜓

̃𝑎 ̃𝑎 𝑎 ⋆𝑎 𝑎 𝑎⋆ 1 C (𝑥, 𝜃, 𝜃) = 𝐶 (𝑥) +𝜃[𝜖𝑎 −(𝑏−̃𝑏) ]+𝜃𝐶 −(𝐷𝜇 − 𝐷̃𝜇)(𝐶−𝐶)̃ ]−𝜎𝑎𝐶̃𝑎 + 𝐶 [𝜖𝑎 − 2 𝑎 +𝜃𝜃𝜎 , 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑎 ̃𝑎 𝑎̃𝑎 𝑎⋆ 𝑎 ⋅𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] − 𝜎 𝐶 +𝐶 [𝜖 b𝑎 (𝑥, 𝜃, 𝜃) = 𝑏𝑎 (𝑥) +𝜃𝜒𝑎 + 𝜃(𝑏⋆𝑎 +𝜒𝑎)+𝜃𝜃𝜛𝑎, 𝑎 ̃𝑎 𝑎 ̃𝑎 ⋆𝑎 𝑎 𝑎𝑏𝑐 𝑏 ̃𝑏 −(𝑏 − 𝑏 )] − 𝜐 𝜙 −𝜙 [𝑀 −𝑓 (𝐶 − 𝐶 ) (29) b̃𝑎 (𝑥, 𝜃, 𝜃) = ̃𝑏𝑎 (𝑥) +𝜃𝜒𝑎 + 𝜃𝑏⋆𝑎 +𝜃𝜃𝜛𝑎, ̃𝑎 ⋆𝑎 𝑎 ⋅(𝜙𝑐 − 𝜙̃𝑐)] − 𝜐𝑎𝜙 − 𝜙 (𝑀 −𝜔𝑎 + 𝜔̃𝑎)−𝜏𝑎𝑒̃𝑎 𝑎 𝑎 𝑎 𝑎⋆ 𝜉𝜇 (𝑥, 𝜃, 𝜃) =𝜇 𝜉 (𝑥) +𝜃𝐿𝜇 + 𝜃(𝜉𝜇 −𝑒⋆ [𝑁𝑎 −𝜆𝑎 + 𝜆̃𝑎]−𝜛𝑎̃𝑏𝑎 −𝑏⋆𝑎𝜒𝑎 −𝜅𝑎𝜉̃𝑎 ̃ 𝑎 ̃𝑎 𝜇 −[(𝐷𝜇 − 𝐷𝜇)(𝜙 − 𝜙 ) ̃𝑎 𝑎 − 𝜅𝑎𝜉 +𝜉⋆𝑎 (𝐿𝑎 −[(𝐷 − 𝐷̃ )(𝜙−𝜙)̃ 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 𝑎 𝜇 𝜇 𝜇 𝜇 𝜇 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)]) + 𝜃𝜃𝜅 , ⋆𝑎 𝑎 +𝑓𝑎𝑏𝑐 (𝐶𝑏 − 𝐶̃𝑏)(𝜉𝑐 − 𝜉̃𝑐 )]) + 𝜉 [𝐿 −(ℎ𝑎 ̃𝑎 ̃𝑎 𝑎 ̃ ̃ 𝑎 𝜇 𝜇 𝜇 𝜇 𝜇 𝜉𝜇 (𝑥, 𝜃, 𝜃) = 𝜉𝜇 (𝑥) +𝜃[𝐿𝜇 −(𝐷𝜇 − 𝐷𝜇)(𝜙−𝜙) − ̃ℎ𝑎)] , 𝑎𝑏𝑐 𝑏 ̃𝑏 𝑐 ̃𝑐 𝑎⋆ 𝑎 𝜇 +𝑓 (𝐶 − 𝐶 )(𝜉𝜇 − 𝜉𝜇)] + 𝜃𝜉𝜇 +𝜃𝜃𝜅 ,

𝑎 𝑎 𝑎 ⋆𝑎 which is nothing but the gauge-fixed Lagrangian density for 𝜉 (𝑥, 𝜃, 𝜃) = 𝜉 (𝑥) +𝜃𝐿 + 𝜃(𝜉 −ℎ𝑎 + ̃ℎ𝑎) 𝜇 𝜇 𝜇 𝜇 𝜇 𝜇 shift symmetry. Being the 𝜃𝜃 component of a superfield, this 𝑎 Lagrangian density is manifestly invariant under both the +𝜃𝜃𝜅 , extended BRST and the anti-BRST transformations. Now, we define the general super-gauge-fixing fermion in ̃𝑎 ̃𝑎 𝑎 𝑎 ̃𝑎 ⋆𝑎 𝜉𝜇 (𝑥, 𝜃, 𝜃) = 𝜉𝜇 (𝑥) +𝜃(𝐿𝜇 −ℎ𝜇 + ℎ𝜇)+𝜃𝜉𝜇 superspace as

+𝜃𝜃𝜅𝑎, Φ(𝑥,𝜃,𝜃) = Ψ (𝑥) +𝜃(𝑠Ψ) + 𝜃 (𝑠Ψ) +𝜃𝜃 (𝑠𝑠Ψ) , (30)

𝜙𝑎 (𝑥, 𝜃, 𝜃) =𝑎 𝜙 (𝑥) +𝜃𝑀𝑎 + 𝜃(𝜙𝑎⋆ −𝑓𝑎𝑏𝑐 (𝐶𝑏 which yields the original gauge-fixing and ghost part of the effective Lagrangian density upon differentiation as follows: ̃𝑏 𝑐 ̃𝑐 𝑎 − 𝐶 )(𝜙 − 𝜙 )) + 𝜃𝜃𝜐 , Tr[(𝜕/𝜕𝜃)[𝛿(𝜃)Φ(𝑥, 𝜃, 𝜃)]]. Advances in High Energy Physics 9

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