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Twisted SUSY: twisted symmetry versus renormalizability 1 1 0 2 Marija Dimitrijević, Biljana Nikolić and n Voja Radovanović a J 6 2 University of Belgrade, Faculty of Physics ] Studentski trg 12, 11000 Beograd, Serbia h t - p e h [ 1 v 3 Abstract 2 We discuss a deformationof superspace basedon a hermitian twist. The twist 0 5 implies a ⋆-product that is noncommutative, hermitian and finite when expanded . in power series of the deformation parameter. The Leibniz rule for the twisted 1 0 SUSY transformations is deformed. A minimal deformation of the Wess-Zumino 1 action is proposed and its renormalizability properties are discussed. There is no 1 tadpole contribution, but the two-point function diverges. We speculate that the : v deformed Leibniz rule, or more generally the twisted symmetry, interferes with i renormalizability properties of the model. We discuss different possibilities to X render a renormalizable model. r a Keywords: supersymmetry,hermitiantwist,deformedWess-Zuminomodel,renormalizability eMail: dmarija,biljana,[email protected] 1 Introduction It is well known that Quantum Field Theory encounters problems at high energies and short distances. This suggests that the structure of space-time has to be modified at these scales. One possibility to modify the structure of space-time is to deform the usual commutation relations between coordinates; this gives a noncommutative (NC) space [1]. Different models of noncommutativity were discussed in the literature, see [2], [3]and [4] forreferences. Aversion of StandardModel on thecanonically deformed space-time was constructed in [5] andits renormalizability propertieswere discussed in [6]. Renormalizability of different noncommutative field theory models was discussed in [7]. A natural further step is modification of the superspace and introduction of non- (anti)commutativity. Astrongmotivationforthiscomesfromstringtheory. Namely, it was discovered that a noncommutative superspace can arise when a superstring moves in a constant gravitino or graviphoton background [8], [9]. Since that discovery there has been a lot of work on this subject and different ways of deforming superspace have been discussed. Here we mention some of them. The authors of [10] combine SUSY with the κ-deformation of space-time, while in [11] SUSY is combined with the canonical deformation of space-time. In [8] a version of non(anti)commutative superspace is defined and analyzed. The anticommutation relations between fermionic coordinates are modified in the following way θα ⋆, θβ = Cαβ, θ¯ ⋆, θ¯ = θα ⋆, θ¯ = 0 , (1.1) { } { α˙ β˙} { α˙} where Cαβ = Cβα is a complex, constant symmetric matrix. This deformation is well defined only when undotted and dotted spinors are not related by the usual complex conjugation. The notion of chirality is preserved in this model, i.e. the deformed product of two chiral superfields is again a chiral superfield. On the other hand, one half of = 1 supersymmetry is broken and this is the so-called = 1/2 super- N N symmetry. Another type of deformation is introduced in [12] and [13]. There the product of two chiral superfields is not a chiral superfield but the model is invariant under the full supersymmetry. Renormalizability of different models (both scalar and gauge theories) has been discussed in [14], [15], [16] and [13]. The twist approach to nonanticommutativity was discussed in [17]. In our previous paper [18] we introduced a hermitian deformation of the usual superspace. The non(anti)commutative deformation was introduced via the twist = e12Cαβ∂α⊗∂β+21C¯α˙β˙∂¯α˙⊗∂¯β˙. (1.2) F Here Cαβ = Cβα is a complex constant matrix, C¯α˙β˙ its complex conjugate and ∂ = ∂ are fermionic partial derivatives. The twist (1.2) is hermitian under the α ∂θα usual complex conjugation. Due to this choice of the twist, the coproduct of the SUSY transformations becomes deformed, leading to the deformed Leibniz rule. The inverse of (1.2) defines the ⋆-product. It is obvious that the ⋆-product of two chiral fields will 1 not be a chiral field. Therefore we have to use the method of projectors to decompose the ⋆-products of fields into their irreducible components. Collecting the terms invariant under the twisted SUSY transformations we construct the deformed Wess-Zumino action. Beinginterested inimplications ofthetwistedsymmetryonrenormalizability properties, in this paper we calculate the divergent part of the one-loop effective action. More precisely, we calculate divergent parts of the one-point and the two-point functions. The plan of the paper is as follows: In the next section we summarize the most important properties of our model, more details of the construction are given in [18]. In Section 3 we describe the method we use to calculate divergent parts of the n-point Green functions: the background field method and the supergraph technique. In Sections 4 the tadpole diagram and the divergent part of the two-point function are calculated. Finally, wediscussrenormalizability ofthemodel. Wegive somecomments andcompareourresultswiththeresultsalready presentintheliterature. Somedetails of our calculations are presented in the Appendix. 2 Construction of the model There are different ways to realize a noncommutative and/or a nonanticommutative space and to formulate a physical model on it, see [2] and [4]. We shall follow the approach of [3] and [18]. Let us first fix the notation and the conventions which we use. The superspace is generated by supercoordinates xm, θα and θ¯ which fulfill α˙ [xm,xn] = [xm,θα]= [xm,θ¯ ]= 0, θα,θβ = θ¯ ,θ¯ = θα,θ¯ = 0, (2.3) α˙ { } { α˙ β˙} { α˙} with m = 0,...,3 and α,β = 1,2. To xm we refer as bosonic and to θα and θ¯ α˙ we refer as fermionic coordinates. We work in Minkowski space-time with the metric ( ,+,+,+) and xmx = (x0)2+(x1)2+(x2)2+(x3)2. m − A general superfield F(−x,θ,θ¯) can be expanded in powers of θ and θ¯, F(x,θ,θ¯) = f(x)+θφ(x)+θ¯χ¯(x)+θθm(x)+θ¯θ¯n(x)+θσmθ¯v (x) m +θθθ¯λ¯(x)+θ¯θ¯θϕ(x)+θθθ¯θ¯d(x). (2.4) Under the infinitesimal = 1 SUSY transformations it transforms as N δ F =(ξQ+ξ¯Q¯)F, (2.5) ξ whereξα and ξ¯ are constant anticommuting parameters and the SUSYgenerators Qα α˙ and Q¯ are given by, α˙ Q = ∂ iσm θ¯α˙∂ , Q¯ = ∂¯ +iθασm ∂ . (2.6) α α− αα˙ m α˙ − α˙ αα˙ m Transformations (2.5) close in the algebra [δ ,δ ]= 2i(ησmξ¯ ξσmη¯)∂ . (2.7) ξ η m − − 2 The product of two superfields is a superfield again; its transformation law is given by δ (F G) = (ξQ+ξ¯Q¯)(F G), ξ · · = (δ F) G+F (δ G). (2.8) ξ ξ · · The last line is the undeformed Leibniz rule for the infinitesimal SUSY transformation δ . ξ Nonanticommutativity is introduced following the twist approach [3]. For the twist we choose F = e12Cαβ∂α⊗∂β+12C¯α˙β˙∂¯α˙⊗∂¯β˙, (2.9) F with the complex constant matrix Cαβ = Cβα. Note that Cαβ and C¯α˙β˙ are related by the usual complex conjugation. It can be shown that the twist (2.9) satisfies all necessary requirements [20]. The inverse of the twist (2.9) −1 = e−21Cαβ∂α⊗∂β−12C¯α˙β˙∂¯α˙⊗∂¯β˙, (2.10) F defines a new product in the algebra of superfields called the ⋆-product. For two arbitrary superfields F and G the ⋆-product is defined as follows F⋆G = µ F G ⋆ { ⊗ } = µ −1F G {F ⊗ } = µ e−21Cαβ∂α⊗∂β−21C¯α˙β˙∂¯α˙⊗∂¯β˙F G { ⊗ } 1 1 = F G ( 1)|F|Cαβ(∂ F) (∂ G) ( 1)|F|C¯ (∂¯α˙F)(∂¯β˙G) · − 2 − α · β − 2 − α˙β˙ 1CαβCγδ(∂ ∂ F) (∂ ∂ G) 1C¯ C¯ (∂¯α˙∂¯γ˙F)(∂¯β˙∂¯δ˙G) −8 α γ · β δ − 8 α˙β˙ γ˙δ˙ 1CαβC¯ (∂ ∂¯α˙F)(∂ ∂¯β˙G) −4 α˙β˙ α β 1 + ( 1)|F|CαβCγδC¯ (∂ ∂ ∂¯α˙F)(∂ ∂ ∂¯β˙G) 16 − α˙β˙ α γ β δ + 1 ( 1)|F|CαβC¯ C¯ (∂ ∂¯α˙∂¯γ˙F)(∂ ∂¯β˙∂¯δ˙G) 16 − α˙β˙ γ˙δ˙ α β + 1 CαβCγδC¯ C¯ (∂ ∂ ∂¯α˙∂¯γ˙F)(∂ ∂ ∂¯β˙∂¯δ˙G), (2.11) 64 α˙β˙ γ˙δ˙ α γ β δ where F = 1 if F is odd (fermionic) and F = 0 if F is even (bosonic) and the | | | | pointwise multiplication µ is the bilinear map from the tensor product to the space of superfields (functions). The definition of the multiplication µ is given in the second ⋆ line. No higher powers of Cαβ and C¯ appear since the derivatives ∂ and ∂¯α˙ are α˙β˙ α Grassmanian. Expansion of the ⋆-product (2.11) ends after the fourth order in the deformation parameter. This ⋆-product is different from the Moyal-Weyl ⋆-product [21] where the expansion in powers of the deformation parameter leads to an infinite power series. One should also note that the ⋆-product (2.11) is hermitian, (F⋆G)∗ = G∗⋆F∗, (2.12) 3 where denotes the usual complex conjugation. ∗ The ⋆-product (2.11) implies θα ⋆, θβ = Cαβ, θ¯ ⋆, θ¯ = C¯ , θα ⋆, θ¯ = 0, { } { α˙ β˙} α˙β˙ { α˙} [xm ⋆, xn] = 0, [xm ⋆, θα]= 0, [xm ⋆, θ¯ ]= 0. (2.13) α˙ Relations (2.13) enable us to define the deformed superspace or ”nonanticommutative superspace”. Itis generated bytheusualbosonicandfermioniccoordinates (2.3)while the deformation is contained in the new product (2.11). The next step is to apply the twist (2.9) to the Hopf algebra of SUSY transformations. We will not give details here, they can be found in [18]. We just state the most important results. The deformed infinitesimal SUSY transformation is defined in the following way δ⋆F = (ξQ+ξ¯Q¯)F. (2.14) ξ Thetwist(2.9)leadstoadeformedLeibnizruleforthedeformedSUSYtransformations (2.14). This ensures that the ⋆-product of two superfields is again a superfield. Its transformation law is given by δ⋆(F⋆G) = (ξQ+ξ¯Q¯)(F⋆G), (2.15) ξ = (δ⋆F)⋆G+F⋆(δ⋆G) ξ ξ i + Cαβ ξ¯γ˙σm (∂ F)⋆(∂ G)+(∂ F)⋆ξ¯γ˙σm (∂ G) (2.16) 2 αγ˙ m β α βγ˙ m iC¯ (cid:16)ξασm εγ˙α˙(∂ F)⋆(∂¯β˙G)+(∂¯α˙F)⋆ξασm εγ˙β˙(cid:17)(∂ G) . −2 α˙β˙ αγ˙ m αγ˙ m (cid:16) (cid:17) Note that we have to enlarge the algebra (2.7) by introducingthe fermionic derivatives ∂ and ∂¯ . Since these derivatives commute with the generators of Poincaré algebra α α˙ ∂ and M , the super Poincaré algebra does not change. Especially, the Leibniz rule m mn for ∂ and M does not change. m mn Being interested in a deformation of the Wess-Zumino model, we need to analyze properties of the ⋆-products of chiral fields. A chiral field Φ fulfills D¯ Φ = 0, with the α˙ supercovariant derivative D¯ = ∂¯ iθασm ∂ . In terms of component fields the α˙ − α˙ − αα˙ m chiral superfield Φ is given by Φ(x,θ,θ¯) = A(x)+√2θαψ (x)+θθF(x)+iθσlθ¯∂ A(x) α l i 1 θθ∂ ψα(x)σm θ¯α˙ + θθθ¯θ¯(cid:3)A(x). (2.17) −√2 m αα˙ 4 It is easy to calculate the ⋆-product of two chiral fields from (2.11). It is given by Φ⋆Φ = A2 C2F2+ 1CαβC¯α˙β˙σm σl (∂ A)(∂ A)+ 1 C2C¯2((cid:3)A)2 − 2 4 αα˙ ββ˙ m l 64 +θα 2√2ψ A 1 CγβC¯α˙β˙ε (∂ ψρ)σm σl (∂ A) α − √2 γα m ρβ˙ βα˙ l (cid:16) (cid:17) 4 i C2θ¯ σ¯mα˙α(∂ ψ )F +θθ 2AF ψψ α˙ m α −√2 − (cid:16) (cid:17) C2 1 +θ¯θ¯ (F(cid:3)A (∂ ψ)σmσ¯l(∂ ψ)) m l − 4 − 2 (cid:16) 1 (cid:17) +iθσmθ¯ (∂ A2)+ CαβC¯α˙β˙σ σl ((cid:3)A)(∂ A) m 4 mαα˙ ββ˙ l (cid:16) 1 (cid:17) +i√2θθθ¯ σ¯mα˙α(∂ (ψ A))+ θθθ¯θ¯((cid:3)A2), (2.18) α˙ m α 4 where C2 = CαβCγδε ε and C¯2 = C¯ C¯ εα˙γ˙εβ˙δ˙. One sees that due to the θ¯-term αγ βδ α˙β˙ γ˙δ˙ and the θ¯θ¯-term (2.18) is not a chiral field. But, in order to write an action invariant under the deformed SUSY transformations (2.14) we need to preserve the notion of chirality. This can be done in different ways. One possibility is to use a different ⋆- product, the one which preserves chirality [8]. However, chirality-preserving ⋆-product implies working in a space where θ¯ = (θ)∗. Since we want to work in Minkowski 6 space-time and keep the usual complex conjugation, we use the ⋆-product (2.11) and decompose the ⋆-products of superfields into their irreducible components using the projectors defined in [22]. In that way (2.18) becomes Φ⋆Φ = P (Φ⋆Φ)+P (Φ⋆Φ)+P (Φ⋆Φ), (2.19) 1 2 T with antichiral, chiral and transversal projectors given by 1 D2D¯2 P = , (2.20) 1 16 (cid:3) 1 D¯2D2 P = , (2.21) 2 16 (cid:3) 1DD¯2D P = . (2.22) T −8 (cid:3) Finally, the deformed Wess-Zumino action is constructed requiring that the action is invariant under the deformed SUSY transformations (2.14) and that in the commutative limit it reduces to the undeformed Wess-Zumino action. In addition, we require that deformation is minimal: We deform only those terms that are present in the commutative Wess-Zumino model. We do not, for the time being, add the terms whose commutative limit is zero. Taking these requirements into account we propose the following action m S = d4x Φ+⋆Φ + P (Φ⋆Φ) 2 θθθ¯θ¯ 2 θθ Z n (cid:12) h (cid:12) λ (cid:12) (cid:12) + P2 Φ⋆P2(Φ(cid:12) ⋆Φ) + c.c ,(cid:12) (2.23) 3 θθ (cid:16) (cid:17)(cid:12) io where m and λ are real constants. To rewrite (2.2(cid:12)3) in terms of component fields and (cid:12) as compact as possible, we introduce the following notation C = K (σabε) , (2.24) αβ ab αβ C¯ = K∗ (εσ¯ab) , (2.25) α˙β˙ ab α˙β˙ 5 where K = K is an antisymmetric self-dual complex constant matrix. Then we ab ba − have C2 = 2K Kab, C¯2 = 2K∗ K∗ab, KabK∗ = 0. (2.26) ab ab ab K∗ K (σnσ¯cdσ¯mσab) β = 4δβKmaK∗n +8KmaK∗nb(σ ) β, (2.27) cd ab α − α a ba α CαβC¯α˙β˙σm σl = 8KamK∗ l. (2.28) αα˙ ββ˙ a Using these formulas and expanding (2.23) in component fields we obtain S = d4x Φ+⋆Φ θθθ¯θ¯ Z n (cid:12) m (cid:12) λ + P2(Φ⋆Φ)(cid:12) + P2 Φ⋆P2(Φ⋆Φ) + c.c 2 θθ 3 θθ h (cid:12) (cid:16) (cid:17)(cid:12) io = d4x A∗(cid:3)A(cid:12)+i(∂ ψ¯)σ¯mψ+F∗F (cid:12) (cid:12) m (cid:12) Z m n + (2AF ψψ)+λ(FA2 Aψψ) 2 − − hλ KmK∗naψ(∂ ψ) 2KmK∗n(∂ ψ)σbaψ (∂ A) −3 a n − a b n m λ(cid:16) λ (cid:17) KmnK F3+ KmK∗nlF(∂ A)(∂ A) (2.29) −12 mn 6 l m n λ 1 λ + KmK∗nlF ∂ (∂ A)(cid:3)A + KabK K∗cdK∗ F((cid:3)A)2 +c.c . 3 l (cid:3) m n 192 ab cd (cid:16) (cid:17) io Partial integration was used to rewrite some of the terms in (2.29) in a more compact way. Note that this it the complete action; there are no higher order terms in the deformation parameter Kab. However, for simplicity in the following sections we shall keep only terms up to second order in the deformation parameter. 3 One-loop effective action In this section we look at the quantum properties of our model. We calculate the one- loop divergent part of the one-point and the two-point functions up to second order in the deformation parameter. We use the background field method, dimensional regularization and the supergraph technique. The supergraph technique significantly simplifies calculations. However, we cannot directly apply this technique since our action (2.29) is not written as an integral over the whole superspace and in terms of thechiralfieldΦanditsderivatives. Thisisaconsequenceoftheparticulardeformation (2.4) and differs from [13]. In order to be able to use the supergraph technique we notice the following: From (2.17), see also [22], it follows that the fields A, ψ and F can be written as 1 1 A = Φ , ψ = D Φ , F = D2Φ . (3.30) |θ,θ¯=0 α √2 α |θ,θ¯=0 −4 |θ,θ¯=0 6 Inserting this in (2.29) we obtain m D2 D2 S = d8z Φ+Φ+ Φ Φ λΦ2 Φ − 8 (cid:3) − 12(cid:3) Z n 1 h +λθθθ¯θ¯ KmnK (D2Φ)3 mn 768 1 (cid:16) KmK∗na(DαΦ)(∂ D Φ) 2KmK∗n(∂ DαΦ)(σba) βD Φ (∂ Φ) −6 a n α − a b n α β m 1(cid:16) (cid:17) KmK∗na(D2Φ)(∂ Φ)(∂ Φ) −24 a m n 1 1 KmK∗na(D2Φ) ∂ ((∂ Φ)((cid:3)Φ)) +c.c. , (3.31) −12 a (cid:3) m n (cid:17) io with f(x)1g(x) = f(x) d4y G(x y)g(y). Notice that two spurion fields (cid:3) − UmnR = KmK∗nθθθ¯θ¯, U = KmnK θθθ¯θ¯ (3.32) (1) ab a b (2) mn appear in (3.31). This is a consequence of rewriting the action (2.29) as an integral over the whole superspace. Now we can start the machinery of the background field method. First we split the chiral and antichiral superfields into their classical and quantum parts Φ Φ+Φ , Φ+ Φ++Φ+ (3.33) → q → q and integrate over the quantum superfields in the path integral. Since Φ and Φ+ are q q chiral and antichiral fields, they are constrained by D¯ Φ = D Φ+ = 0. α˙ q α q To simplify the supergraph technique we introduce the unconstrained superfields Σ and Σ+, 1 1 Φ = D¯2Σ, Φ+ = D2Σ+ . (3.34) q −4 q −4 Note that we do not express the background superfields Φ and Φ+ in terms of Σ and Σ+, only the quantum parts Φ and Φ+. After the integration of quantum superfields, q q the result is expressed in terms of the (anti)chiral superfields. This is a big advantage of the background field method and of the supergraph technique. The unconstrained superfields are determined up to a gauge transformation Σ Σ+D¯ Λ¯α˙, Σ+ Σ++DαΛ , (3.35) α˙ α → → with the gauge parameter Λ. This additional symmetry has to be fixed, so we add a gauge-fixing term to the action. For the gauge functions we choose χ = D Σ, χ¯ = D¯ Σ+ . (3.36) α α α˙ α˙ 7 The product (χ)(χ¯) in the path integral is averaged by the weight e−iξRd8zf¯Mf: . . dfdf¯(χ f )(χ¯α˙ f¯α˙)e−iξR d8zf¯α˙Mα˙αfα (3.37) . α− α . − Z where 1 3 f¯α˙M fα = f¯α˙(D D¯ + D¯ D )fα (3.38) α˙α α α˙ α˙ α 4 4 and the gauge-fixing parameter is denoted by ξ. The gauge-fixing term becomes 3 1 S = ξ d8z (D¯ Σ¯)( D¯α˙Dα+ DαD¯α˙)(D Σ). (3.39) gf α˙ α − 16 4 Z One can easily show that the ghost fields are decoupled. After the gauge-fixing, the part of the classical action quadratic in quantum superfields is given by S(2) = S(2)+S(2), (3.40) 0 int with 1 Σ S(2) = d8z Σ Σ+ (3.41) 0 2 M Σ+ Z (cid:18) (cid:19) (cid:0) (cid:1) and 1 Σ S(2) = d8zd8z′ Σ Σ+ (z) (z,z′) (z′). (3.42) int 2 V Σ+ Z (cid:18) (cid:19) (cid:0) (cid:1) Kinetic and interaction terms are collected in the matrices and respectively. The M V matrix is given by M m(cid:3)1/2P (cid:3)(P +ξ(P +P )) = − − 2 1 T , (3.43) M (cid:3)(P +ξ(P +P )) m(cid:3)1/2P 1 2 T + (cid:18) − (cid:19) with D2 D¯2 P = , P = . (3.44) + 4(cid:3)1/2 − 4(cid:3)1/2 The interaction matrix is V F 0 = . (3.45) V 0 F¯ (cid:18) (cid:19) There are two types of elements in , local and nonlocal. We split them into F and 1 V F 2 F(z,z′) = F (z)δ(z z′)+F (z,z′). (3.46) 1 2 − Elements of F are given by 1 10 F (z) = F(i) 1 i=0 X λ λ = ΦD¯2 KmK∗na←D¯−2−D−−α(∂ Φ)θθθ¯θ¯∂ D D¯2 −2 − 48 a m n α 8 λ KmK∗na←D¯−2−D−−α(∂ D Φ)θθθ¯θ¯∂ D¯2 −48 a m α n λ KmK∗na←∂−D−¯−2(DαΦ)θθθ¯θ¯∂ D D¯2 −48 a m n α λ + KmK∗n←D¯−2−D−−α(σab) β(∂ Φ)θθθ¯θ¯∂ D D¯2 24 a b α m n β λ + KmK∗n←∂−D−¯−2(DαΦ)(σab) βθθθ¯θ¯∂ D D¯2 24 a b m α n β λ + KmK∗n←∂−D−¯−2(∂ DαΦ)(σba) βθθθ¯θ¯D D¯2 24 a b m n α β λ KmnK ←D¯−2−D−2Φθ¯θ¯D2D¯2 (3.47) mn −512 λ KmK∗na←∂−D−¯−2(∂ Φ)θθθ¯θ¯D2D¯2 −96 a m n λ KmK∗na←∂−D−¯−2(D2Φ)θθθ¯θ¯∂ D¯2 −192 a m n λ 1 + KmK∗na←(cid:3)−D¯−2 d8z′ (∂ D2Φ)(z′) δ(z′ z) θθθ¯θ¯∂ D¯2, 96 a m (cid:3)z′ − n (cid:16)Z (cid:17) while the elements of F read 2 12 F (z,z′) = F(i) 2 i=11 X λ 1 = KmK∗na←∂−−D¯−2−D−−2 δ(z′ z)θθθ¯θ¯((∂ Φ)(cid:3)D¯2)(z′) 96 a m (cid:3)z′ − n λ 1 + KmK∗na←∂−−D¯−2−D−−2 δ(z′ z)θθθ¯θ¯((cid:3)Φ∂ D¯2)(z′). (3.48) 96 a m (cid:3)z′ − n The one-loop effective action is then i Γ =S +S + Trlog(1+ −1 ). (3.49) 0 int 2 M V The last term in (3.49) is the one-loop correction to the effective action and −1 is M the inverse of (3.43) given by −1 = A B = 4(cid:3)(m(cid:3)D−2m2) 16(cid:3)D((cid:3)2D−¯2m2) + D¯2D126−ξ2(cid:3)D¯2D2D¯ . M (cid:18) B¯ A¯ (cid:19) 16(cid:3)D¯((cid:3)2D−2m2) + D2D¯126−ξ2(cid:3)D2D¯2D 4(cid:3)(m(cid:3)D−¯2m2) ! (3.50) Expansion of the logarithm in (3.49) leads to the one-loop corrections i ∞ ( 1)n+1 ∞ Γ = Tr − ( −1 )n = Γ(n). (3.51) 1 2 n M V 1 n=1 n=1 X X 9