Emergence of Lowenstein-Zimmermann mass terms for QED 3
Sudhaker Upadhyay, Manoj Kumar Dwivedi, Bhabani Prasad Mandal
aa r X i v : . [ h e p - t h ] S e p Emergence of Lowenstein-Zimmermann mass terms for QED ∗ , Manoj Kumar Dwivedi † , and Bhabani Prasad Mandal ‡ Department of Physics, Banaras Hindu University, Varanasi-221005, INDIA.
In this paper we consider a super-renormalizable theory of massless QED in (2+1) dimensions anddiscuss their BRST symmetry transformation. By extending the BRST transformation we derive theNielsen identities for the theory. Further, we compute the generalized BRST (so-called FFBRST)transformation by making the transformation parameter field-dependent. Remarkably, we observethat the Lowenstein-Zimmerman mass terms, containing Lowenstein-Zimmerman parameter whichplays an important role in the BPHZL renormalization program, along with the external sourcescoupled to the non-linear BRST variations appear naturally in the theory through a FFBRSTtransformation.
Keywords : QED ; Lowenstein-Zimmermann mass; BRST symmetry. PACS : 11.15.-q; 11.15.Wx.
I. INTRODUCTION
Massless QED in 2 + 1 dimensions (QED ) has very interesting and crucial features in the frontierresearch. Recently in a seminal work [1], QED has been established a super-renormalizable theoryutilizing a powerful algebraic renormalization method. Also, it is a ultraviolet finite and parity invarianttheory [1–5]. The massless QED provides an ideal platform to tackle the infrared divergence presentin the theory. The parity anomaly has been dismissed for such theories at all orders of perturbation.The Lowenstein-Zimmerman scheme plays an important role in algebraic proof of ultraviolet and infraredfiniteness and in the dismissal of parity anomaly. In spite of that the QED also gets relevance in thestudy of high- T c superconductivity [6, 7]. The dynamical fermion mass generation and chiral symmetrybreaking for QED are studied in [8]. In this context, the dynamical mass generation using Hamiltonianlattice methods has also been investigated which has been found in agreement with both the strongcoupling expansion and with the Euclidean lattice simulations [9].The Lowenstein-Zimmermann subtraction scheme plays a major role in the algebraic proof on theultraviolet and infrared finiteness, and to show the absence of a parity and infrared anomaly, in themassless QED , which is based on general theorems of perturbative quantum field theory [10–18]. Themassless QED with and without Lowenstein-Zimmermann mass terms is a gauge theory and in order todeal such theories we need to break the gauge invariance by fixing the gauge. The gauge invariance of thequantum action is also essential because the expectation values of physical quantities become independentof the choice of the gauge-fixing term. To realize the gauge independence i.e. the estimating S-matrixelements (or expectation values) of the gauge independent quantities, one utilizes the on-shell quantumeffective action (i.e., evaluated at those configurations that extremize it). Nielsen identities [19] suggestthat the variation of the quantum effective action due to changes in the functions that fix the gauge mustbe linear in the quantum corrected equations of motion for the mean fields. This follows that the on-shellquantum effective action does not depend on the choice of the gauge breaking term. Although the meanfields depend on the gauge-fixing, this dependence gets canceled by the explicit gauge-fixing dependenceof the quantum effective action [20–22].In the proof of algebraic renormalizability the BRST symmetry has an incredible importance [23]. Thegeneralizations of BRST symmetry have been studied in various contexts [24–52] since their introduction ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] in Ref. [24]. For examples, a correct prescription for poles in the gauge field propagators in noncovariantgauges has been derived by connecting covariant gauges and noncovariant gauges of the theory by usingFFBRST transformation [29]. The long outstanding problem of divergent energy integrals in Coulombgauge has been regularized using FFBRST transformation [26]. The Gribov-Zwanziger theory [53, 54],a limiting case of Yang-Mills theory , which plays a crucial role in the non-perturbative low-energyregion while it can be neglected in the perturbative high-energy region, has also been related to the YMtheory in Euclidean space through FFBRST transformation [55, 56]. The FFBRST formulation has alsobeen established at quantum level utilizing BV formulation [25]. Recently, the field-dependent BRSTtransformation has also been considered with same philosophy and goal as in original work [24] eventhough in slightly different manner to calculate the explicit Jacobian for such transformation [57–60]. Asystematic Hamiltonian formulation of such theories have also been done [61, 62].In this paper we analyse the massless QED , which is a super-renormalizable and free from parityand infrared anomaly, within generalized BRST (FFBRST) framework. To implement the FFBRSTformulation, we first make the infinitesimal parameter of transformation field dependent through contin-uous interpolation of a parameter κ : (0 ≤ κ ≤ κ in its extreme limit to obtain the FFBRST transformation.Such FFBRST transformation leads a non-trivial Jacobian for the path integral measure of functionalintegral. Remarkably, we realize that the Lowenstein-Zimmerman mass terms along with the externalsources for QED emerges naturally through Jacobian calculation under FFBRST transformation. Inproof of renormalizability of the theory such source term play an important role.The plan of the paper is as follows. We start with a brief discussion of massless QED theory inSec. II. The section III is devoted to derive the Nielsen identities. Further, in section IV, we sketch thegeneralized BRST transformation. In Sec. V, we show the emergence of Lowenstein-Zimmermann massterms along with external source term for QED . The last section is reserved for conclusions. II. THE MASSLESS QED In this section, we recapitulate the massless QED with and without Lowenstein-Zimmermann massterms. Let us begin with the gauge invariant massless action for QED defined asΣ = Z d x (cid:20) − F µν F µν + i ¯ ψγ µ D µ ψ (cid:21) , (1)where the covariant derivative is defined by D µ = ∂ µ + ieA µ and e is a dimensionful coupling constant.Since, the gauge invariant theory can be quantized correctly only after choosing a particular gauge forthe theory. We define the gauge-fixing and induce ghost terms for QED as follows:Σ gf + gh = Z d x (cid:20) b∂ µ A µ + ξ b + ¯ c∂ µ ∂ µ c (cid:21) , (2)where b, c and ¯ c are Nakanishi-Lautrup auxiliary field, ghost field and anti-ghost field respectively. There-fore, the effective action can be written easily byΣ eff = Σ + Σ gf + gh . (3)This quantum action remains invariant under following nilpotent BRST transformations: δ b A µ = s b A µ ǫ = 1 e ∂ µ c ǫ,δ b ψ = s b ψ ǫ = icψ ǫ,δ b ¯ ψ = s b ¯ ψ ǫ = − ic ¯ ψ ǫ,δ b c = s b c ǫ = 0 ,δ b ¯ c = s b ¯ c ǫ = 1 e b ǫ,δ b b = s b b ǫ = 0 , (4)where ǫ is an infinitesimal anticommuting parameter of transformation.Now, we introduce a gauge invariant Lowenstein-Zimmermann mass term for the massless QED whichmakes the theory super-renormalizable [1]:Σ m = Z d x h µ s − ǫ µνρ A µ ∂ ν A ρ + m ( s −
1) ¯ ψψ i , (5)where 0 ≤ s ≤ with the Lowenstein-Zimmermann mass termsis given by Σ meff = Σ eff + Σ m . (6)It is desirable to define an external source term for the theory [1],Σ ext = Z d x [ ¯Ω s b ψ − s b ¯ ψ Ω] . (7)This source terms play an important role in demonstrating the Slavnov-Taylor identity which guaranteesthe renormlizability of the theory. With this source term the final action reads,Σ T = Σ eff + Σ m + Σ ext . (8)This final action (Σ T ) is invariant under the same set of BRST transformations (4). Now we would liketo derive the Nielsen identities for the QED with the Lowenstein-Zimmermann mass terms in order todiscuss the gauge independence of the physical quantity. III. NIELSEN IDENTITIES
To study the Nielsen identities we extend the action by introducing a global Grassmannian variable, χ as follows Σ m ′ eff = Σ meff + Z d x χ cb. (9)Here we remark that the addition of this term does not change the dynamics of the theory because ofthe Grassmannian nature of χ . This extended action (9) remains unchanged under the following set ofextended BRST transformations: δ + b A µ = 1 e ∂ µ c ǫ, δ + b ψ = icψ ǫ,δ + b ¯ ψ = − ic ¯ ψ ǫ, δ + b c = 0 ,δ + b ¯ c = 1 e b ǫ, δ + b b = 0 δ + b ξ = − e χ ǫ, δ + b χ = 0 , (10)where ǫ is a transformation parameter. Now, to derive the Nielsen identities we define the following pathintegral with sources Z = Z D φ exp i (cid:20) Σ meff + Z d x (cid:0) J µ A µ + ¯ J ψ ψ + ¯ ψJ ¯ ψ + bJ b + ¯Ω( − icψ ) + ic ¯ ψ Ω (cid:1)(cid:21) , (11)where various J are the sources respective to associated fields and composite fields. Now we define thevertex functional as follows∆( A µ , ψ, ¯ ψ, b, c, ¯ c, χ, ξ, ¯Ω , Ω) = W ( J µ , J ¯ ψ , ¯ J ψ , J b , Ω , ¯Ω , χ, ξ ) − Z d x ( J µ A µ + ¯ J ψ ψ + ¯ ψJ ¯ ψ + bJ b ) , (12)where the generating functional W generates only connected Green’s function. To study the gaugedependence of the propagators, we now introduce the functional integral of proper Green functions δ + b A µ δ ∆ δA µ + δ + b ψ δ ∆ δψ + δ + b ¯ ψ δ ∆ δ ¯ ψ + δ + b ¯ c δ ∆ δ ¯ c + δ + b ξ δ ∆ δξ = 0 . (13)Now we demand the invariance of the above functional under the extended BRST transformations whichyields 1 e ∂ µ c δ ∆ δA µ + δ ∆ δ ¯Ω δ ∆ δψ + δ ∆ δ Ω δ ∆ δ ¯ ψ + 1 e b δ ∆ δ ¯ c − χ δ ∆ δξ = 0 . (14)Differentiation of above equation with respect to χ and then set χ equal to zero results δ ∆ δξ + 1 e ∂ µ c δ ∆ δχδA µ − δ ∆ δχδ ¯Ω δ ∆ δψ + δ ∆ δ ¯Ω δ ∆ δχδψ − δ ∆ δχδ Ω δ ∆ δ ¯ ψ + δ ∆ δ Ω δ ∆ δχδ ¯ ψ − e b δ ∆ δχδ ¯ c = 0 . (15)This is the most general expression for the Nielsen identities for the QED with the Lowenstein-Zimmermann mass terms. From these expression we can generate the Nielsen identities for the two-pointfunctions. With this result one can show that the pole mass of the electron is gauge independent andthat the photon self-energy can be simply shown to be gauge parameter independent. IV. FFBRST TRANSFORMATION
Let us review the FFBRST formulation [24] in brief. To do so, we first write the usual BRST trans-formation for a generic field φ written collectively for massless QED theory, δ b φ = s b φ ǫ = R [ φ ] ǫ, (16)where R [ φ ] = s b φ is Slavnov variation of φ and ǫ is infinitesimal parameter of transformation. Theimportance of the BRST transformation does not alter by considering (i) the finite or infinitesimal and(ii) the field-dependent or field-independent versions of the parameter δ Λ provided the parameter must beanticommuting and space-time independent. This observation gives us a freedom to generalize the BRSTtransformation by making the parameter, ǫ , finite and field-dependent. We first define the infinitesimalfield-dependent transformation as follows [24] dφ ( x, κ ) dκ = R [ φ ( x, κ )]Θ ′ [ φ ( x, κ )] , (17)where the Θ ′ [ φ ( x, κ )] is an infinitesimal field-dependent parameter. The FFBRST transformation ( δ f )with the finite field-dependent parameter then can be obtained by integrating the above transformationfrom κ = 0 to κ = 1, as follows: δ f φ ( x ) ≡ φ ( x, κ = 1) − φ ( x, κ = 0) = R [ φ ( x )]Θ[ φ ( x )] , (18)where Θ[ φ ( x )] is the finite field-dependent parameter constructed from its infinitesimal version. Undersuch FFBRST transformation with finite field-dependent parameter the measure of generating functionwill not be invariant and will contribute some non-trivial terms to the generating function in general [24].The Jacobian of the path integral measure ( D φ ) in the functional integral for such transformations isthen evaluated for some particular choices of the finite field-dependent parameter, Θ[ φ ( x )], as follows D φ ′ = J ( κ ) D φ ( κ ) . (19)Now, we replace the Jacobian J ( κ ) of the path integral measure as J ( κ ) e i Σ [ φ ( x,κ )] , (20)iff the following condition [24] Z D φ ( x ) (cid:20) ddκ ln J ( κ ) − i d Σ [ φ ( x, κ )] dκ (cid:21) exp [ i (Σ eff + Σ )] = 0 (21)is satisfied where Σ [ φ ] is some local functional of fields satisfying initial boundary condition Σ [ φ ] | κ =0 =0. Moreover, the infinitesimal change in Jacobian, J ( κ ), is calculated by [24] ddκ ln J ( κ ) = − Z d y " ± X i R [ φ i ( y )] ∂ Θ ′ [ φ ( y, κ )] ∂φ i ( y, κ ) , (22)where, for bosonic fields, + sign is used and for fermionic fields, − sign is used.Therefore, by constructing an appropriate Θ, we can calculate the non-trivial (local) Jacobian whichextends the effective action by a term Σ . V. EMERGENCE OF LOWENSTEIN-ZIMMERMANN MASS TERMS
In this section, we explicitly show the emergence of Lowenstein-Zimmermann mass terms for the mass-less QED theory under FFBRST formulation. To implement these notions, we construct the FFBRSTtransformation corresponding to Eq. (4) following the techniques outlined in Sec. III: δ f A µ = 1 e ∂ µ c Θ[ φ ] ,δ f ψ = icψ Θ[ φ ] ,δ f ¯ ψ = − ic ¯ ψ Θ[ φ ] ,δ f c = 0 ,δ f ¯ c = 1 e b Θ[ φ ] ,δ f b = 0 , (23)where Θ[ φ ] is an arbitrary field-dependent parameter of transformation. It is easy to check that theeffective action for QED given in (8) is invariant under these set of transformations. Now, we constructa particular field-dependent parameter (following the procedure given in Ref. [24]) asΘ[ φ ] = Z d x e ¯ cbb h exp (cid:16) i µ s − ǫ µνρ A µ ∂ ν A ρ + im ( s −
1) ¯ ψψ + iψ ¯Ω − i ¯ ψ Ω (cid:17) − i . (24)Now, following the method discussed in section III, we calculate the Jacobian for path integral measureunder finite field-dependent BRST transformation with above parameter as follows J [ φ ( x )] = e i Σ = e i R d x [ µ ( s − ǫ µνρ A µ ∂ ν A ρ + m ( s −
1) ¯ ψψ +¯Ω s b ψ − s b ¯ ψ Ω ] . (25)Here to obtain this expression we have utilized the relation (21).As a consequence of performing FFBRST transformation on path integral measure of functional integralwe see that the effective action of the theory gets extendend (within functional integral) as follows:Σ eff + Σ = Z d x (cid:20) − F µν F µν + i ¯ ψDψ + b∂ µ A µ + ξ b + ¯ c∂ µ ∂ µ c + µ s − ǫ µνρ A µ ∂ ν A ρ + m ( s −
1) ¯ ψψ + ¯Ω s b ψ − s b ¯ ψ Ω i . (26)which exactly coincides with effective action given in (8). This justifies our claim of emergence ofLowenstein-Zimmermann mass terms naturally under the celebrated FFBRST technique. We also empha-sized that the external source terms for the non-linear BRST variations which are required to prove therenormalizability of the theory are automatically generated through the same FFBRST transformation.It means that under FFBRST transformation with appropriately constructed parameter (24) the ef-fective action (within functional integral) changes asΣ eff + Σ = Σ T . (27)Hence, the whole mechanism is, precisely, given by Z D φ e i Σ eff F F BRST − − −− −→ Z D φ e i (Σ eff +Σ ) = Z D φ e i (Σ eff +Σ m +Σ ext ) , (28)where the generic path integral measure ( D φ ) is explicitly given by D φ = DA µ D b D c D ¯ c D ψ D ¯ ψ . Therefore,the Lowenstein-Zimmermann mass terms and the external source term are generated through the Jacobianof the path integral measure under generalized BRST transformations with appropriate transformationparameter. VI. CONCLUSIONS