\mathcal{N}=1 supersymmetric three-dimensional QED in the large-N_f limit and applications to super-graphene
PPrepared for submission to JHEP N = 1 supersymmetric three-dimensional QED in thelarge- N f limit and applications to super-graphene A. James, S. Metayer and S. Teber
Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies, LPTHE,F-75005 Paris, France
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study N = 1 supersymmetric three-dimensional Quantum Electrodynam-ics with N f two-component fermions. Due to the infra-red (IR) softening of the photon, ε -scalar and photino propagators, the theory flows to an interacting fixed point deep inthe IR, p E (cid:28) e N f /
8, where p E is the euclidean momentum and e the electric charge. Atnext-to-leading order in the 1 /N f -expansion, we find that the flow of the dimensionless ef-fective coupling constant α is such that: α → / (cid:0) N f (1 + C/N f ) (cid:1) ≈ (8 /N f )(1 − . /N f )where C = 2 (12 − π ) /π . Hence, the non-trivial IR fixed point is stable with respectto quantum corrections. Various properties of the theory are explored and related viaa mapping to the ones of a N = 1 model of super-graphene. In particular, we de-rive the interaction correction coefficient to the optical conductivity of super-graphene, C sg = (12 − π ) / (2 π ) = 0 . C g = (92 − π ) / (18 π ) = 0 . a r X i v : . [ h e p - t h ] F e b ontents N = 1 SQED action 52.2 Dimensional reduction scheme 62.3 Perturbation theory setup 72.4 Mapping to N = 1 super-graphene model 9 N = 1 super-graphene 21 N = 1 super-graphene 29 Three-dimensional Quantum Electrodynamics (QED ) is an archetypal relativisticgauge-field theory model describing strongly interacting planar fermions. In Minkowskispace, it is described by the action: S QED = (cid:90) d x (cid:18) − F µν F µν + i ¯ ψ (cid:54) Dψ (cid:19) , (1.1)– 1 –here D µ = ∂ µ +i eA µ , ψ j are N f flavours ( j = 1 , · · · , N f ) of two-component massless Diracfermions and the coupling constant e has positive mass dimension ([ e ] = 1 /
2) making thetheory super-renormalizable.This model has been studied during more than four decades now. An original moti-vation [1, 2] came from the fact QED might serve as a prototype for three-dimensionalQuantum Chromodynamics with its large- N f limit being calculable in the infra-red (IR).Since then, there has been extensive focus on dynamical (flavour) symmetry breaking andfermion mass generation in this model [3–14] (see recent progress in [15–19]). Anotheroriginal motivation was to study, in the small- N f limit, the fate of IR singularities that areubiquitous to super-renormalizable models [20–24] (see recent progress in [25–27]). In thelast three decades, a revived interest in QED also arose in relation with condensed matterphysics system exhibiting Dirac-like low-energy excitations such as high- T c superconduc-tors [28–30], planar antiferromagnets [31] and graphene [32] (for graphene, see reviews inrefs. [33–36]).In this paper, we will focus on a variant of QED , namely (minimal) N = 1 super-symmetric three-dimensional QED (SQED ) with N f two-component fermions. Originally,SQED was considered with the hope that the characteristic softer ultra-violet (UV) be-haviour of supersymmetric theories will help in solving theoretical issues related to, e.g.,dynamical flavour symmetry breaking. Note that the latter may occur without violatingsupersymmetry (SUSY) in agreement with general arguments [37] stating that SUSY isnot broken in SQED. In a seminal paper, Pisarski [3] focused on extended N = 2 SQED which is constructed by dimensional reduction from four-dimensional SQED (SQED ); thelarge- N f limit (see [38] for a review on large- N f techniques) of this model was found to be(potentially) tractable with flavour symmetry breaking taking place for all values of N f .Later, it was argued [39] that a non-perturbative non-renormalization theorem actuallyforbids dynamical mass generation in N = 1 SQED [40] which therefore extends by di-mensional reduction to N = 2 SQED . Further evidence for the absence of dynamical massgeneration in N = 2 SQED came from numerical simulations [41] and a refined analytictreatment [42].The situation in N = 1 SQED is more subtle because of the absence of non renormal-ization theorems in this case. The model was first considered by Koopmans and Steringa[39] along the lines set by Appelquist et al. for standard QED [5]. Their leading-order(LO) in the 1 /N f -expansion Schwinger-Dyson equations approach resulted in a criticalfermion flavour number, N f cr , which, in the Landau gauge ( ξ = 0), takes the value N f cr ( ξ = 0) = 32 /π = 3 . N f ≤ N = 1 SQED wasalso given in [43]. However, to the best of our knowledge, there was no improvement of thesolution proposed by [39] in the last two decades.At this point, let us note that of crucial importance for the study of QED (withrespect to, e.g., dynamical mass generation) is the existence of a non-trivial interacting IRfixed point as first noticed by [5]. Indeed, in such a super-renormalizable theory, one can– 2 –efine the following dimensionless effective charge: α ( p E ) = e p E (1 − Π ( p E )) = (cid:40) e /p E p E (cid:29) e N f / /N f p E (cid:28) e N f / , (1.2)where p E = (cid:112) − p is the Euclidean momentum, Π ( p E ) = − e N f / (16 p E ) is the one-loop polarization operator of (non-SUSY) QED and the mass scale e fixes the separationbetween the UV and IR regimes. The corresponding beta function: β QED ( α ) = p E d α ( p E )d p E = − α (cid:18) − N f α (cid:19) , (1.3)displays two stable fixed points: an asymptotically free UV fixed point ( α →
0) andan interacting IR fixed point: α → α ∗ QED = 16 /N f at the LO of the 1 /N f -expansion.The latter has some similarity with the Banks-Zaks fixed point [44] known in non-abeliangauge field theories. For the present abelian case, it originates from the softening of theQED photon propagator in the IR limit p E (cid:28) e N f /
16 [1, 2]. This in turn cures QED from its IR singularities, exchanges the dimensionful coupling e for the dimensionlessone ∼ /N f and provides QED with a four-dimensional-like power counting making iteffectively renormalizable. Extending these arguments to the next-to-leading order (NLO)of the 1 /N f -expansion yields [45]: α ∗ QED = 16 N f (cid:18) − C QED N f + O (cid:0) /N f (cid:1)(cid:19) , C QED = 4 (92 − π )9 π = 0 . . (1.4)The striking feature about this result is that the smallness of the coefficient C QED preservesthe stability of the IR fixed point with respect to radiative corrections. In the following,we shall refer to QED at the IR fixed point as the large- N f limit of QED .On the SUSY side, it is known that N = 2 and N = 4 SQED also display IR fixedpoints and that they are present for all values of N f , see the review [47] (for N = 4SQED it is even one-loop exact). In relation with the IR structure of three-dimensionalsupersymmetric gauge field theories, the cases N = 2 and N = 4 were extensively studiedby the string theory community for more than two decades now starting from the seminalpapers of Seiberg and Witten [48] and Intriligator and Seiberg [49], see also [50] for areview. The more subtle N = 1 case is still under active scrutiny, see, e.g., [51–56]. Theseworks are part of a recent revival of interest in three-dimensional quantum field theories(both supersymmetric and non-supersymmetric) in the context of the study of IR dualities,see, e.g., references in [54]. The existence of an IR fixed point in SQED at the LO of the1 /N f -expansion was readily assumed in [39] as well as in the most recent work [57] thatfits in the modern framework just described, see also [58] for a review.At the interface with condensed matter physics, there have been proposals that SUSYmay emerge in the low-energy limit of various lattice models, see, e.g., [59–65]. At thispoint, let’s recall that there is no experimental evidence so far that SUSY is realized in The result C QED was also derived in refs. [9, 10, 46]. Though it does not appear explicitly in thesepublications, the knowledge of C QED was required to perform the calculations carried out in these papers. – 3 –ature. An emergent SUSY should certainly be difficult to detect in the lab [66]. Butthis still opens the interesting possibility that some observable quantity in a particularmaterial might be affected by the IR SUSY-invariant fixed point. Alternatively, somecondensed-matter physics models describing planar relativistic Dirac fermions where su-persymmetrized. Such is the case for the super-graphene model that has been introducedin [67]. Notice that, similarly to the large- N f limit of QED , the (non-SUSY) relativis-tic model of graphene is a conformally-invariant field theory that corresponds to the IRLorentz-invariant fixed point of (non-relativistic) graphene [68]. With fermions localizedin (2 + 1)-dimensions while photons propagate in the bulk, such a (conformal) brane-worldmodel and its variants has attracted significant interest in the last years, see, e.g., [69–77].A universal quantity of interest to compute in graphene-like systems is the optical conduc-tivity in the collisionless regime. Presently, in the non-SUSY case, it is known at two-looporder in the (dimensionless) fine structure constant, α g = e g / (4 π ), and the result reads[68, 74, 78]: σ g = σ (cid:18) C g α g + O( α g ) (cid:19) , C g = 92 − π π = 0 . , (1.5)where σ = N f e /
16 is the minimal conductivity of graphene and C g the interaction cor-rection coefficient. This universal (flavour independent) coefficient can actually be relatedto C QED of (1.4) with the help of a mapping [71]. Moreover, while the most recent NLOresults show that (non-SUSY) QED has a gauge-invariant N f cr = 5 .
694 [15, 17, 19], thereis no sign of dynamical mass generation for the corresponding (non-SUSY) graphene model[71].Having set the background material together with the contemporary frame of activity,our primary concern in this paper will be related to the stability of the IR fixed point inSQED with respect to radiative corrections. Focusing on the NLO of the 1 /N f -expansion,we will therefore derive a formula analogous to (1.4) in the SUSY case. In order to achievethis, we will first provide a detailed study of the properties of SQED at the LO of the1 /N f -expansion. This will lead us to reconsider the problem of dynamical mass generationin this model following [39] but in the modern formulation initiated by [9, 10] and [15] andsuccessfully used for (non-SUSY) QED in [15, 17, 19]. Following [71], we will also find amapping between SQED and a model of super-graphene which will allow us to transcribeall the results obtained for SQED to the case of super-graphene. In particular, we willderive a formula analogous to (1.5) that will give us access to the universal interactioncorrection coefficient to the optical conductivity of N = 1 super-graphene.This paper is organized as follows. In section 2, we describe the N = 1 SQED model and its symmetries, followed by the dimensional reduction regularization scheme, theperturbative set-up and the mapping to N = 1 super-graphene. In section 3, we first obtainthe effective gauge multiplet propagators which show a softer momentum dependence inthe IR. We utilize these to calculate the field and mass anomalous dimensions of the matterfields. With the help of these results we re-examine dynamical mass generation in N = 1SQED . All of these LO results are then mapped to analogous quantities for N = 1 super-graphene. Next, in section 4, we compute the next to leading order corrections to the– 4 –auge multiplet self-energy functions. This is used to examine the stability of the infraredfixed point of this theory. These NLO results are then mapped to the optical conductivityof N = 1 super-graphene. In section 5, we conclude. Finally, in appendix A we presentour notations and conventions, in appendix B the Feynman rules of SQED , in appendixC the master integrals relevant to our calculations and in appendix D exact formula for allthe computed NLO diagrams. N = 1 SQED action The degrees of freedom of N = 1 super QED are the N f matter multiplets { φ j , ψ j , F j } and a gauge multiplet { A µ , λ } . Here, φ j are complex pseudo-scalars, ψ j are two-componentDirac fermions and F j are complex auxiliary scalar fields without any dynamics. Thegauge multiplet, after choosing the Wess-Zumino gauge, has the U (1) gauge field A µ andits superpartner the photino λ , which is a two component Majorana field, and without anyauxiliary field (unlike in 3 + 1 dimensions). Following [39] and the notation already usedin the Introduction, the microscopic action of N = 1 massless SQED is then given by S = (cid:90) d x (cid:18) − F µν F µν + i2 ¯ λ (cid:54) ∂λ + | D µ φ | + i ¯ ψ (cid:54) Dψ + | F | − i e ( ¯ ψλφ − ¯ λψφ ∗ ) (cid:19) . (2.1)Similarly to QED , this theory is super-renormalizable. Moreover, the residual gaugedegree of freedom associated to A µ can be partially fixed by adding to the above action alinear covariant gauge fixing term L gf = − (1 / ξ ) ( ∂ µ A µ ) .The supersymmetry transformations for the fields are given by δ (cid:15) φ = ¯ (cid:15)ψ , δ (cid:15) ψ = − (i (cid:54) Dφ + F ) (cid:15) , δ (cid:15) F = i¯ (cid:15) ( (cid:54) Dψ − eλφ ) , (2.2a) δ (cid:15) A µ = i¯ (cid:15)γ µ λ , δ (cid:15) λ = 12 γ µν (cid:15) F µν , (2.2b)where the parameter of the transformation (cid:15) , is an anticommuting Majorana spinor (seeappendix A for our conventions). In the absence of the gauge-fixing term, these transfor-mations leave the Lagrangian associated to (2.1) invariant up to a total derivative δ ε L = ∂ µ (cid:18) − i4 ¯ (cid:15)γ µρσ λF ρσ − i2 ¯ (cid:15)γ ν λF µν − i ¯ ψγ µ (cid:15)F + ( D µ φ ) ∗ ¯ (cid:15)ψ + D µ φ ¯ ψ(cid:15) − ( D ν φ ) ∗ ¯ (cid:15)γ ν γ µ ψ (cid:19) . (2.3)With respect to [39], we have augmented the transformation of the auxiliary field F bythe term − i e ¯ (cid:15)λφ (in agreement with the result of [79]). Next, we have to check thatthe transformations (2.2) provide a representation of the N = 1 supersymmetry algebra, { Q α , Q β } = 2( γ µ C ) αβ P µ ( α = 1 , Q α are the supercharges and C is the chargeconjugation matrix. To this end, we evaluate the following commutators[ δ (cid:15) , δ (cid:15) ] X = − (cid:15) γ µ (cid:15) D µ X , X ∈ { φ, ψ, F } , (2.4a)[ δ (cid:15) , δ (cid:15) ] λ = − (cid:15) γ µ (cid:15) ∂ µ λ , [ δ (cid:15) , δ (cid:15) ] A µ = − (cid:15) γ ρ (cid:15) ∂ ρ A µ + ∂ µ (2i¯ (cid:15) γ ρ (cid:15) A ρ ) , (2.4b)– 5 –hich result in a covariant translation (translation combined with a gauge transformation).Such closure of the algebra up to field-dependent gauge transformations (with the gaugefunction Λ = 2i¯ (cid:15) (cid:54) A(cid:15) ) is expected of supersymmetric gauge theories in the Wess-Zuminogauge.The supersymmetry invariance of (2.1) under the two supercharges Q α , together withits U ( N f ) flavour symmetry and its invariance under parity and time-reversal are actuallyenough to fix the Lagrangian once the purely bosonic part is written down. This enforcesthe (already assumed) equality of the gauge coupling e appearing in the cubic and quarticinteraction terms as well as the equality of any possible mass terms which might be addedto the model. While computing perturbative amplitudes in a supersymmetric gauge theory using theusual dimensional regularization (DREG) scheme in a d -dimensional space, one immedi-ately encounters an apparent problem that can be understood essentially in terms of themismatch of degrees of freedom in the gauge multiplet: between the gauge field (whichhas d − d = 3 − ε < d -dimensional, thefermion fields still have a 3-dimensional nature and are two-component objects, whereasthe gauge field A µ (formally) splits as A µ = A ˆ µ + A ¯ µ . Here A ¯ µ are referred to as the ε -scalars and correspond to the 2 ε amount of scalars obtained; the hatted indices refer tothe usual d dimensions. The ε -scalars thus account for the degrees of freedom lost by thegauge bosons (see appendix A for more on the conventions we use).Let’s note that some potential technical inconsistencies may arise at higher loop ordersas pointed out by Siegel [85]. These can be overcome by using certain infinite dimensionalquasi-spaces as introduced in [86] and later clarified in [87]. Such a consistent formulationof DRED was proved to be supersymmetric at low orders of perturbation theory andis presently the most convenient regularization scheme for practical calculations in thecomponent formalism. The preserved three-dimensional nature of the fields is supposed toinsure the validity of supersymmetric Ward-Takahashi or Slavnov-Taylor identities whilethe d -dimensional nature of space-time coordinates regularizes, just as in DREG, divergentloop integrals, see [83] for a detailed review on these developments in the frame of four-dimensional theories. Without entering any related mathematical construction, we willassume that they also hold in the present three-dimensional case and that, as a consequence,the DR scheme is a consistent supersymmetric regulator for SQED , at least at the loworders of perturbation theory that we shall consider in the following.– 6 –t the level of the Lagrangian, the decomposition of the gauge field yields: L d =3 = L − ε + L ε -scalars , (2.5)where L − ε takes the same form as the original Lagrangian but with all vector indicesrestricted to d = 3 − ε dimensions, and L ε -scalars = − ∂ ˆ µ A ¯ ν ∂ ˆ µ A ¯ ν − eA ¯ ν ¯ ψγ ¯ ν ψ + e A ¯ ν A ¯ ν | φ | . (2.6)Notice that the ε -scalar part of the Lagrangian, L ε -scalars , does not appear in usual dimen-sional regularization (DREG) and is specific to the DRED scheme. Thus, ε -scalars giverise to additional cubic and quartic couplings; the equality of the matter couplings to A ˆ µ and A ¯ µ is a consequence of supersymmetry [88, 89] and has been assumed. From eq. (2.5), the momentum-space bare propagators of the model are given by: S ( p ) = (cid:104) ψ ( x ) ¯ ψ (0) (cid:105) F.T. = i (cid:54) p , (2.7a) D ˆ µ ˆ ν ( p ) = (cid:104) A ˆ µ ( x ) A ˆ ν (0) (cid:105) F.T. = − i p (cid:18) g ˆ µ ˆ ν − (1 − ξ ) p ˆ µ p ˆ ν p (cid:19) , (2.7b) E ¯ µ ¯ ν ( p ) = (cid:104) A ¯ µ ( x ) A ¯ ν (0) (cid:105) F.T. = − i g ¯ µ ¯ ν p , (2.7c) σ ( p ) = (cid:104) λ ( x )¯ λ (0) (cid:105) F.T = i (cid:54) p , (2.7d)∆ ( p ) = (cid:104) φ ( x ) φ † (0) (cid:105) F.T. = i p , (2.7e)where F.T. stands for Fourier transform. These propagators are part of a set of Feyn-man rules (which also include vertices) that are displayed in appendix B and allow for asystematic study of SQED in a loop-expansion in the dimensionless parameter α/ (cid:112) − p .Upon turning on interactions, the dressed propagators and vertex functions will satisfya set of coupled Schwinger-Dyson equations. Focusing on the propagators, the generalsolutions of these equations take the form (in the massless case): S ( p ) = i (cid:54) p − Σ ψV ( p ) , (2.8a) D ˆ µ ˆ ν ( p ) = − i p (cid:18) g ˆ µ ˆ ν − (1 − ξ ) p ˆ µ p ˆ ν p (cid:19) − Π γ ( p ) , (2.8b) E ¯ µ ¯ ν ( p ) = − i g ¯ µ ¯ ν p − Π ε ( p ) , (2.8c) σ ( p ) = i (cid:54) p − Π λV ( p ) , (2.8d)∆( p ) = i p − Σ φS p ( p ) , (2.8e)– 7 –here, anticipating the study of the IR fixed point, a usual non-local gauge has beenadopted for the photon propagator [6, 90, 91]. The following parametrization has beenused for the (1-particle irreducible) self-energies entering the massless eqs. (2.8):Σ ψ ( p ) = (cid:54) p Σ ψV ( p ) , Σ ψV ( p ) = Tr[ (cid:54) p Σ ψ ( p )]2 p , (2.9a)Π ˆ µ ˆ ν ( p ) = ( p g ˆ µ ˆ ν − p ˆ µ p ˆ ν )Π γ ( p ) , Π γ ( p ) = Π ˆ µ ˆ µ ( p )( d − p , (2.9b)Π ¯ µ ¯ ν ( p ) = p g ¯ µ ¯ ν Π ε ( p ) , Π ε ( p ) = Π ¯ µ ¯ µ ( p )2 ε p , (2.9c)Π λ ( p ) = (cid:54) p Π λV ( p ) , Π λV ( p ) = Tr[ (cid:54) p Π λ ( p )]2 p , (2.9d)Σ φ ( p ) = p Σ φS p ( p ) . (2.9e)As will be shown in section 3, the photon, ε -scalar and photino propagators get IRsoftened already at the LO of the 1 /N f -expansion, i.e., D ˆ µ ˆ ν LO ( p ) = i2 a (cid:112) − p (cid:18) g ˆ µ ˆ ν − (1 − ξ ) p ˆ µ p ˆ ν p (cid:19) , (2.10a) E ¯ µ ¯ ν LO ( p ) = i g ¯ µ ¯ ν a (cid:112) − p , (2.10b) σ LO ( p ) = − i (cid:54) p a (cid:112) − p , (2.10c)where a = N f e /
16. Replacing (2.7b), (2.7c) and (2.7d) by eqs. (2.10) allows for a system-atic study of SQED at the IR fixed point in a 1 /N f -expansion.At this point, let’s remark that some of the above self-energies and related anomalousdimensions might be constrained by Ward-Takahashi or Slavnov-Taylor identities. Fromthe study of four-dimensional abelian gauge field theories, see [92, 93], let us mention inparticular two important identities that will be of interest to us: (i) the polarization functions in the gauge multiplet (see (2.9)) must all be equal:Π γ ( p ) = Π ε ( p ) = Π λV ( p ) . (2.11)(ii) the mass anomalous dimensions of the selectron and the electron must coincide.We do not propose here any formal proof of these identities in the three-dimensional case.However, as will be explicitly shown in the following, both of them hold for SQED at theorders considered in this paper (LO for the anomalous mass dimensions and NLO for thepolarization functions). As noted in [93], supersymmetry is seen to realize only on the physical Hilbert space and hence un-physical quantities such as field anomalous dimensions for the matter fields (which are gauge dependent)are not constrained. In our calculations, we find that the field anomalous dimensions of the selectron andthe electron do not coincide at LO, see (3.19) and (3.25). – 8 – .4 Mapping to N = 1 super-graphene model The results that we will obtain for the large- N f limit of N = 1 SQED can be usedto study N = 1 (suspended) super-graphene. The latter may be thought of as the min-imal supersymmetric extension of the model of (suspended) graphene at its IR Lorentzinvariant fixed point [68]. It is characterized by N f electron fields localized on a threedimensional membrane and interacting via photons that are allowed to propagate in thefull four dimensional space-time. Our model for super-graphene (henceforth denoted by sgin expressions) differs from the boundary model considered in [67] in that our boundaryis a transparent interface while the model of [67] considers a purely reflecting boundary(graphene on a substrate).The action for N = 1 (suspended) super-graphene can be written as S = S bdry + S bulk where S bdry = (cid:90) d x (cid:0) | D µ φ | + i ¯ ψ (cid:54) Dψ + | F | − i e g ( ¯ ψλφ − ¯ λψφ ∗ ) (cid:1) , (2.12a) S bulk = (cid:90) d x (cid:18) − F ab F ab + i2 ¯Λ (cid:54) ∂ Λ + 12 D (cid:19) , (2.12b)where e g denotes the (dimensionless) coupling constant of graphene. In (2.12a), the nota-tion for the fields is the same as in (2.1). In the bulk action (2.12b), the indices a and b takevalues 0 , ..., (cid:54) ∂ = Γ a ∂ a where Γ a are 4 × D is a real auxiliary field. We can add a (four-dimensional) bulk gaugefixing term L gf = − / ξ ( ∂ a A a ) to write down the bulk propagators: (cid:104) A a ( p ) A b ( − p ) (cid:105) = i p (cid:18) g ab − (1 − ξ ) p a p b p (cid:19) , (2.13a) (cid:104) Λ α ( p ) ¯Λ β ( − p ) (cid:105) = i p a Γ aαβ p . (2.13b)At this point, we perform a dimensional reduction that introduces hatted and barred objectsas for SQED together with an additional propagator for the (bulk) ε -scalar. Furthermore,by integrating these propagators over the bulk degrees of freedom it is possible to deriveeffective gauge propagators on the three-dimensional membrane. This results in: D sg0;ˆ µ ˆ ν ( p ) = i2 (cid:112) − p d ˆ µ ˆ ν (cid:18) p ; η (cid:19) , D LO , SQED ˆ µ ˆ ν ( p ) = 8 i N f e (cid:112) − p d ˆ µ ˆ ν ( p ; η ) , (2.14a) σ sg0 ( p ) = − i (cid:54) p (cid:112) − p , σ LO, SQED ( p ) = − (cid:54) pN f e (cid:112) − p , (2.14b) E sg0;¯ µ ¯ ν ( p ) = i2 (cid:112) − p g ¯ µ ¯ ν , E LO, SQED ¯ µ ¯ ν ( p ) = 8 i N f e (cid:112) − p g ¯ µ ¯ ν , (2.14c)where d µν ( η ) = g µν − ηp µ p ν /p , η = 1 − ξ , η = 1 − ξ and we have also added the IR softenedpropagators of SQED (2.10a) for comparison. Similarly to the non-supersymmetric case In the case of the bulk four-component photino, Λ, this procedure is accompanied by projecting outtwo of its components to identify it with the boundary two-component photino λ . – 9 – p kk − p ˆ µ ˆ ν (a) (cid:2) p kk − p ˆ µ ˆ ν (b) Figure 1 : Leading order corrections to the photon propagator.[71], we see from (2.14) that there is a mapping between large- N f SQED and the super-graphene model which is related to the fact that the gauge multiplet propagators in bothmodels have the same form. Explicitly, the mapping reads: π N f → g = α g π , η → η (cid:18) ξ → ξ (cid:19) , (2.17)where α g = e g / (4 π ) is the (dimensionless) fine-structure constant of graphene and g thecorresponding reduced coupling constant.The mapping (2.17) will be used in the following to transcribe all the results obtainedfor large- N f SQED to the case of super-graphene. In this section we present the LO analysis of the model. We will show how severalpropagators get IR softened. We will also compute wave function renormalizations ofthe electron and scalar fields as well as their mass anomalous dimensions. Finally, thedynamical generation of a parity-even mass term will be studied.
We consider first the photon propagator, eq. (2.8b), and compute the LO photonpolarization function Π ˆ µ ˆ ν ( p ) = Π ˆ µ ˆ ν a ( p ) + Π ˆ µ ˆ ν b ( p ) , (3.1) Notice that eq. (2.17) is identical to the mapping in the non-supersymmetric case [71] except that inthe present case N f is the number of 2-component spinors while eq. (6) in ref. [71] involves the number n f of 4-component spinors. These two flavour numbers are related by: N f = 2 n f . For clarity, with thesenotations, the mapping of [71] reads:1 π n f → g = α g π (non-SUSY and n f , (2.15)or 1 π N f → g α g π (non-SUSY and N f , (2.16)while in both cases the mapping of the gauge fixing parameters is identical to the one given in (2.17). – 10 – p kk − p ¯ µ ¯ ν Figure 2 : Leading order correction to the ε -scalar propagator.which is parameterized as in (2.9b) and consists of the sum of the two diagrams displayedon figure 1. These diagrams are defined as:i Π ˆ µ ˆ ν a ( p ) = − µ ε N f (cid:90) [d d k ] Tr (cid:104) ( − i eγ ˆ µ ) S ( k − p )( − i eγ ˆ ν ) S ( k ) (cid:105) , (3.2a)i Π ˆ µ ˆ ν b ( p ) = µ ε N f (cid:90) [d d k ] (i e (2 k − p ) ˆ µ ) ∆ ( k − p ) (i e (2 k − p ) ˆ ν ) ∆ ( k ) , (3.2b)where µ is the renormalization scale and the bare propagators are given in (2.7) (see alsoappendix B). These diagrams are straightforward to compute using techniques of masslessFeynman diagram calculations, see, e.g., [94] for a review. The final results, expressed inthe DR scheme, read:Π γ a ( p ) = − N f e (4 π ) / (cid:112) − p (cid:18) µ − p (cid:19) ε d − d − e γ E ε G ( d, , , (3.3a)Π γ b ( p ) = − N f e (4 π ) / (cid:112) − p (cid:18) µ − p (cid:19) ε d − e γ E ε G ( d, , , (3.3b)where µ = 4 πe − γ E µ and G ( d, α, β ) is defined in appendix C. Performing the ε -expansion:Π γ a ( p ) = − N f e (cid:112) − p (cid:18) − (1 − L p ) ε + O( ε ) (cid:19) , (3.4a)Π γ b ( p ) = − N f e (cid:112) − p (cid:18) − L p ) ε + O( ε ) (cid:19) , (3.4b)where L p = log( − p /µ ), we see that both diagrams equally contribute in the limit ε → γ ( p ) = − N f e (4 π ) / (cid:112) − p (cid:18) µ − p (cid:19) ε e γ E ε G ( d, , , (3.5)and its expression in strictly d = 3 reads:Π γ ( p ) = − a (cid:112) − p , (3.6) Notice that, because we work in DRED, all massless tadpoles are zero. We shall therefore neglect themall, both at the LO and at the NLO of the 1 /N f -expansion. – 11 – p kk − pρ α (a) (cid:5) p k − pkρ α (b) Figure 3 : Leading order corrections to the photino propagator. The indicated momentaflow in the anti-clockwise sense.where a = N f e /
16. Interestingly, eq. (3.6) is simply twice the value for QED [1, 2] whichcoincides with the earlier result given in ref. [39], but now obtained using dimensionalreduction. Substituting (3.6) in (2.8b), the photon propagator IR softens in the large- N f limit, p E (cid:28) a = N f e / p E = (cid:112) − p is the Euclidean momentum), and takes theform already advertised in (2.10a).Next, we proceed in the same way for the ε -scalar propagator, eq. (2.8c), and computethe LO ε -scalar polarization function which is parameterized as in (2.9c) and consists of asingle non-vanishing diagram displayed on figure 2. This diagram is defined as:i Π ¯ µ ¯ ν ( p ) = − µ ε N f (cid:90) [d d k ] Tr (cid:2) ( − i eγ ¯ µ ) S ( k − p )( − i eγ ¯ ν ) S ( k ) (cid:3) , (3.7)and its computation yields:Π ε ( p ) = − N f e (4 π ) / (cid:112) − p (cid:18) µ − p (cid:19) ε e γ E ε G ( d, , , (3.8)which is exactly equal to the one-loop photon polarization function, (3.5). Exactly in d = 3we therefore have: Π ε ( p ) = − a (cid:112) − p . (3.9)Substituting (3.9) in (2.8c), the ε -scalar propagator IR softens in the large- N f limit, p E (cid:28) N f e /
8, and takes the form already advertised in (2.10b).Lastly, we proceed in the same way for the photino propagator, eq. (2.8d), and computethe LO photino self-energy which is parameterized as in (2.9d) and consists of two non-vanishing diagrams with opposite fermion number flows (or charge flows) displayed on figure3. As we briefly explain at the end of appendix B, in order to evaluate these diagrams (andother diagrams involving Majorana fermions) we need to use specific Feynman rules thatinvolve assigning a continuous fermion flow in addition to the fermion number flow. Such aprocedure reveals that the two diagrams of figure 3 are equal and the resulting contribution– 12 – p kp − k (a) (cid:7) p kp − k (b) (cid:8) p kp − k (c) Figure 4 : One loop electron self energy diagrams. The photon, ε -scalar and photinopropagators are the IR softened ones.is defined as: − i Π λ ( p ) = 2 µ ε N f (cid:90) [d d k ] (+ e ) S ( k )( − e )∆ ( k − p ) . (3.10)Evaluating the integral yields:Π λ V ( p ) = − N f e (4 π ) / (cid:112) − p (cid:18) µ − p (cid:19) ε e γ E ε G ( d, , , (3.11)which is exactly equal to both the one-loop photon (3.5) and ε -scalar (3.8) polarizationfunctions. So, exactly in d = 3 we have:Π λ V ( p ) = − a (cid:112) − p . (3.12)Substituting (3.12) in (2.8d), the photino propagator IR softens in the large- N f limit, p E (cid:28) N f e /
8, and takes the form already advertised in (2.10c).Summarizing, we find that the photon, ε -scalar and photino self-energies are all equalat the LO of the 1 /N f -expansion:Π γ ( p ) = Π ε ( p ) = Π λ V ( p ) = − a (cid:112) − p , (3.13)in agreement with the identity discussed at the end of section 2.3 and that the corre-sponding propagators soften in the IR. Analogously to the case of QED discussed in theIntroduction, this softening gives rise to the non-trivial IR fixed point at which we have an(IR-safe) interacting N = 1 superconformal field theory. We start with the electron propagator, eq. (2.8a), and compute the LO electron self-energy Σ ψ ( p ) = Σ ψ a ( p ) + Σ ψ b ( p ) + Σ ψ c ( p ) , (3.14) For the diagrams of figure 3, we assign a fermion flow to the fermion chain going from left to right. Incase of diagram 3a, the fermion flow, fermion number flow and momentum flow are all in the same direction.Moreover, the vertices are proportional to the unit matrix. Therefore, there is no reversed propagator orvertex. In the case of diagram 3b, the fermion flow is opposite to the fermion number flow so that theDirac propagator gets reversed. However, the fermion number flow is also opposite to the momentum flowso that the momentum gets an additional − sign. All together, the Dirac propagator remains unchanged(as S ( k ) → S ( − ( − k ))) and diagram 3b is therefore equal to diagram 3a. – 13 –hich consists of the sum of the three diagrams displayed on figure 4. They are defined as: − i Σ ψ a ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ˆ µ ) S ( k )( − i eγ ˆ ν ) D LO ˆ µ ˆ ν ( p − k ) , (3.15a) − i Σ ψ b ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ¯ µ ) S ( k )( − i eγ ¯ ν ) E LO ¯ µ ¯ ν ( p − k ) , (3.15b) − i Σ ψ c ( p ) = µ ε (cid:90) [d d k ] (+ e )∆ ( k )( − e ) σ LO ( p − k ) , (3.15c)where the photon, ε -scalar and photino propagators are the IR softened ones. Computingthese diagrams with the parametrization (2.9a) yields:Σ ψ V a ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε ( d − (cid:18) d − d − − ξ (cid:19) e γ E ε G ( d, , / , (3.16a)Σ ψ V b ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε ( d − d − d − e γ E ε G ( d, , / , (3.16b)Σ ψ V c ( p ) = − π ) / N f (cid:18) µ − p (cid:19) ε d − d − e γ E ε G ( d, , / , (3.16c)where Σ ψ V b is finite while the two other contributions are singular as d →
3. The totalelectron self-energy is thus given by:Σ ψ V ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε (cid:18) d − − ( d − ξ (cid:19) e γ E ε G ( d, , / , (3.17)which, in ε -expanded form, reads:Σ ψ V ( p ) = − π N f (cid:18) µ − p (cid:19) ε (cid:18) ξε + 2 + 2(1 − log 2) ξ + O( ε ) (cid:19) . (3.18)From this result, we extract the LO electron wave-function renormalization: γ ψ = µ dΣ ψ V ( p )d µ = − ξπ N f + O(1 /N f ) . (3.19)Notice that γ ψ vanishes in the Landau gauge. This is to be contrasted with the non-supersymmetric LO electron anomalous dimension (expressed in terms of N f γ (QED ) ψ = 8 (2 − ξ ) / (3 π N f ) [46] that vanishes in the so-called Nash gauge [6], ξ = 2 / φ ( p ) = Σ φ a ( p ) + Σ φ b ( p ) , (3.20)– 14 – p kp − k (a) (cid:10) p kp − k (b) Figure 5 : One loop scalar self energy diagrams. The photon and photino propagators arethe IR softened ones. The momenta flow from left to right.which consists of the sum of the two diagrams displayed on figure 5. They are defined as: − i Σ φ a ( p ) = µ ε (cid:90) [d d k ] ( − i e ( p + k ) ˆ µ )∆ ( k )( − i e ( p + k ) ˆ ν ) D LO ˆ µ ˆ ν ( p − k ) , (3.21a) − i Σ φ b ( p ) = − µ ε (cid:90) [d d k ] Tr [( − e ) S ( k )(+ e ) σ LO ( k − p )] , (3.21b)where the photon and photino propagators are the IR softened ones and Σ φ b is a fermionloop. Computing these diagrams with the parametrization (2.9e) yields:Σ φ S p a ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε (cid:18) d − d − d − − (2 d − ξ (cid:19) e γ E ε G ( d, , / , (3.22a)Σ φ S p b ( p ) = − π ) / N f (cid:18) µ − p (cid:19) ε d − d − e γ E ε G ( d, , / . (3.22b)The total scalar self-energy is given by:Σ φ S p ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε (cid:18) d − − (2 d − ξ (cid:19) e γ E ε G ( d, , / , (3.23)which, in ε -expanded form, reads:Σ φ S p ( p ) = 2 π N f (cid:18) µ − p (cid:19) ε (cid:18) − ξε + 4(1 − log 2) + 2 ξ log 2 + O( ε ) (cid:19) . (3.24)From this result, we extract the LO scalar wave-function renormalization: γ φ = µ dΣ φ S p ( p )d µ = 4(2 − ξ ) π N f + O(1 /N f ) , (3.25)which is not equal to the one of the electron (3.19). For the diagram of figure 5b, we assign a counter-clockwise fermion flow to the fermion loop consistingof the Dirac and Majorana fermions. For the Dirac propagator, the fermion flow, fermion number flow andmomentum flow all have the same orientation. On the other hand, for the Majorana propagator, the fermionflow is opposite to the momentum flow. Hence, the Majorana propagator gets reversed: σ LO ( p − k ) → σ LO ( − ( p − k )). – 15 – .3 Mass anomalous dimensions So far we have considered a model of massless particles, (2.1). In this subsection,we add a bare mass m f to the electron in order to extract its mass anomalous dimension.Because of SUSY invariance, an equal mass should be given to the selectron, m s = m f = m .Moreover, we would like to preserve the parity invariance of our Lagrangian, (2.5). So wewill assume that these masses are parity-even.Let’s then assume that both the Dirac fermion and complex scalar fields are massiveand momentarily change their bare propagators accordingly: ˜ S ± ( p ) = i ( (cid:54) p ± m f ) p − m f , ˜∆ ( p ) = i p − m s , (3.26)where we take m f and m s as arbitrary masses for the moment (as will be shown below,we shall recover the mass degeneracy constraint from our calculations). The electron andselectron self-energies are then parameterized as:˜Σ ψ ± ( p ) = (cid:54) p Σ ψV ( p ) ± m f Σ ψS ( p ) , ˜Σ φ ( p ) = p Σ φS p ( p ) + m s Σ φS m ( p ) , (3.27)from which we define the corresponding mass anomalous dimensions: γ mψ = µ d (cid:0) ˜Σ ψS (cid:1) (cid:48) ( p )d µ , (cid:0) ˜Σ ψS (cid:1) (cid:48) ( p ) = 1 + Σ ψS ( p )1 − Σ ψV ( p ) , (3.28a) γ mφ = 12 µ d (cid:0) ˜Σ φS (cid:1) (cid:48) ( p )d µ , (cid:0) ˜Σ φS (cid:1) (cid:48) ( p ) = 1 + Σ φS m ( p )1 − Σ φS p ( p ) , (3.28b)where (cid:0) Σ ψS (cid:1) (cid:48) ( p ) and (cid:0) Σ φS (cid:1) (cid:48) ( p ) are gauge invariant combinations. Because we are interestedin anomalous dimensions, only the singular part of these self-energies will be computed.We already know Σ ψ V ( p ) and Σ φ S p ( p ) from the previous subsection. So we shall focusnow on the computations of Σ ψ S ( p ) and Σ φ S m ( p ).We first focus on the LO Σ ψS ( p ) which consists of three contributions correspondingto the scalar parts of the diagrams displayed on figure 4 and defined as in (3.15) but nowwith massive Dirac fermion and scalar propagators: − i ˜Σ ψ a ± ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ˆ µ ) ˜ S ± ( k )( − i eγ ˆ ν ) D LO ˆ µ ˆ ν ( p − k ) , (3.29a) − i ˜Σ ψ b ± ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ¯ µ ) ˜ S ± ( k )( − i eγ ¯ ν ) E LO ¯ µ ¯ ν ( p − k ) , (3.29b) − i ˜Σ ψ c ± ( p ) = µ ε (cid:90) [d d k ] (+ e ) ˜∆ ( k )( − e ) σ LO ( p − k ) , (3.29c)where the photon, ε -scalar and photino propagators are still the IR softened ones. Thescalar integrals are projected out from (3.29) with the help of:Σ ψS ( p ) = ± m f Tr (cid:2) ˜Σ ψ ± ( p ) (cid:3) . (3.30) The ± signs in the fermion propagator are related to the different fermion flavours according to (3.40)which is a parity-even mass operator. – 16 –hese integrals are then computed in the zero-mass limit ( m f = m s = 0) which is enoughto extract their UV singular structure as known from IR rearrangement [95]. The compu-tations yield: Σ ψ Sa ( p ) = 8(4 π ) / N f (cid:18) µ − p (cid:19) ε ( d − ξ ) e γ E ε G ( d, , / , (3.31a)Σ ψ Sb ( p ) = − π ) / N f (cid:18) µ − p (cid:19) ε ( d − e γ E ε G ( d, , / , (3.31b)Σ ψ Sc ( p ) = 0 , (3.31c)where the contribution of Σ ψ Sb is finite in d = 3 while that of Σ ψ Sc vanishes identically.Hence, in the limit d →
3, the LO contribution to Σ ψ S is given by:Σ ψ S ( p ) = 2(2 + ξ ) π N f (cid:18) µ − p (cid:19) ε (cid:18) ε + O(1) (cid:19) . (3.32)Together with (3.18) and (3.28a), the fermion mass anomalous dimension therefore reads: γ mψ = 8 π N f + O(1 /N f ) . (3.33)We now consider Σ φS m ( p ) which consists of two contributions corresponding to thescalar parts of the diagrams displayed on figure 5 and defined as in (3.21) but now withmassive Dirac fermion and scalar propagators: − i ˜Σ φ a ( p ) = µ ε (cid:90) [d d k ] ( − i e ( p + k ) ˆ µ ) ˜∆ ( k )( − i e ( p + k ) ˆ ν ) D LO ˆ µ ˆ ν ( p − k ) , (3.34a) − i ˜Σ φ b ( p ) = − µ ε (cid:90) [d d k ] Tr (cid:104) ( − e ) ˜ S ± ( k )(+ e ) σ LO ( k − p ) (cid:105) , (3.34b)where the photon and photino propagators are the IR softened ones. As an IR rearrange-ment, we can extract the UV singular part of the above integrals by reducing them tomassive tadpoles. This amounts to simply set the external momentum to zero, p = 0, in(3.34). Together with the parametrization (3.27), this yields:Σ φ S m a (0) = 8 ξ (4 π ) / N f (cid:18) µ m s (cid:19) ε e γ E ε B ( d, , / , (3.35a)Σ φ S m b (0) = m f m s π ) / N f (cid:32) µ m f (cid:33) ε e γ E ε B ( d, , / , (3.35b)where B ( α, β ) is the semi-massive tadpole integral defined in (C.4). In expanded form, wehave: Σ φ S m a (0) = 2 ξπ N f (cid:18) µ m s (cid:19) ε (cid:18) ε + O(1) (cid:19) , (3.36a)Σ φ S m b (0) = m f m s π N f (cid:32) µ m f (cid:33) ε (cid:18) ε + O(1) (cid:19) . (3.36b)– 17 –t this point, we recall that in minimal subtraction schemes (including DR), renormal-ization constants and anomalous dimensions cannot depend on momenta and masses [96].This enforces mass degeneracy: m f = m s = m , (3.37)which is also a requirement of SUSY invariant theories. Hence, in the limit d →
3, the LOcontribution to Σ φ S m is given by:Σ φ S m (0) = 2(2 + ξ ) π N f (cid:18) µ m (cid:19) ε (cid:18) ε + O(1) (cid:19) . (3.38)Together with (3.24) and (3.28b), the scalar mass anomalous dimension therefore reads: γ mφ = 8 π N f + O(1 /N f ) , (3.39)and is equal to the fermion mass anomalous dimension (3.33). Hence, not only the barefermion and scalar masses are the same, the corresponding renormalized masses are alsoequal. This also agrees with general constraints arising from SUSY invariance as discussedat the end of section 2.3. Here, we have explicitly checked that such constraints do hold atthe LO of the 1 /N f -expansion for SQED . In the previous section, bare (parity-even) mass terms for the electron and selectronfields were added to the Lagrangian. Starting from a model with zero bare masses ( m f = m s = 0), it is possible that, at strong coupling, masses be dynamically generated. Againbecause of SUSY invariance, if the electron acquires a dynamical mass then the selectronhas to acquire an equal dynamical mass. Let’s recall that parity-odd masses cannot bedynamically generated [97]. On the other hand, a parity-even but flavour breaking masscan be generated. This corresponds to the condensation of the following operator (expressedin terms of 2-component spinors): N f / (cid:88) i =1 ( ¯ ψ i ψ i − ¯ ψ i +[ N f / ψ i +[ N f / ) , (3.40)which indeed breaks U ( N f ) to U ( N f / × U ( N f / / ( π N f ), i.e., for small values of N f . Of importance is to determine the critical flavourfermion number, N f cr , which is such that dynamical symmetry breaking takes place at N f < N f cr . We therefore focus here on an estimation of N f cr at the LO of the 1 /N f -expansion.In order to compute N f cr , it is enough to focus on the critical properties of SQED ,i.e., to work at the non-trivial IR fixed point. We then have two ways to proceed further.The first one is via Schwinger-Dyson (SD) equations for either the electron or the selectron A parity-odd mass term would correspond to the condensation of: (cid:80) N f i =1 ¯ ψ i ψ i . – 18 – kp − k (a) (cid:12) kp − k (b) (cid:13) kp − k (c) Figure 6 : The three diagrams contributing to the electron self energy in terms of dressedpropagators and vertices. The photon, ε -scalar and photino propagators are also intendedto be dressed quantities.(because of SUSY invariance, both sets of SD equations yield the same critical coupling formass generation). This has already been considered in [39] but we will follow a simpler andhopefully clearer approach following [9, 10], see also [19] and references therein for morerecent works. Another approach is based on using anomalous mass dimensions of eitherthe electron or selectron (they are equal as we found in the last subsection) following [15].We consider first the SD equation approach. Following the notations of the previoussubsection, we set the bare masses to zero: m f = m s = m = 0, and parameterize theelectron and scalar self-energies as:Σ ψ ± ( p ) = (cid:54) p Σ ψV ( p ) ± Σ ψS ( p ) , Σ φ ( p ) = p Σ φS p ( p ) + Σ φS m ( p ) , (3.41)where now Σ ψS ( p ) and Σ φS m ( p ) are dynamically generated, i.e., they are obtained as non-trivial solutions of the SD equations. The dressed electron and selectron propagators thenread: S ± ( p ) = 11 − Σ ψV ( p ) i (cid:54) p ∓ (cid:0) Σ ψS (cid:1) (cid:48) ( p ) , (cid:0) Σ ψS (cid:1) (cid:48) ( p ) = Σ ψS ( p )1 − Σ ψV ( p ) , (3.42a)∆( p ) = 11 − Σ φS p ( p ) i p − (cid:0) Σ φS (cid:1) (cid:48) ( p ) , (cid:0) Σ φS (cid:1) (cid:48) ( p ) = Σ φS m ( p )1 − Σ φS p ( p ) , (3.42b)where (cid:0) Σ ψS (cid:1) (cid:48) and (cid:0) Σ φS (cid:1) (cid:48) are analogous to the ones defined in (3.28) but in the absenceof a bare mass. We then consider the SD equations for the electron propagator (similarderivations hold for the selectron SD equations). They read: − i Σ ψa ± ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ˆ µ ) S ± ( k )( − i e Γ ˆ ν ) D ˆ µ ˆ ν ( p − k ) , (3.43a) − i Σ ψb ± ( p ) = µ ε (cid:90) [d d k ] ( − i eγ ¯ µ ) S ± ( k )( − i e Γ ¯ ν ) E ¯ µ ¯ ν ( p − k ) , (3.43b) − i Σ ψc ( p ) = µ ε (cid:90) [d d k ] ( − e )∆( k )(+ ef ( p, k )) σ ( p − k ) , (3.43c)corresponding to the three diagrams displayed on figure 4 in terms of dressed propagatorsand vertices. The unknown function (cid:0) Σ ψS (cid:1) (cid:48) may further be parameterized as: (cid:0) Σ ψS (cid:1) (cid:48) ( p ) = B ( − p ) − α , (3.44)– 19 –with an arbitrary constant B ) and where the index α has to be self-consistently deter-mined.In the LO approximation to the 1 /N f -expansions, we take the photon, ε -scalar andphotino propagators as the IR softened ones and Σ ψV = 0, Γ ˆ ν = γ ˆ ν , Γ ¯ ν = γ ¯ ν and f ( p, k ) =1. Together with (3.44), the scalar part of eqs. (3.43) in the linearized approximationsignificantly simplify and read:Σ ψ aS ( p ) = B ( − p ) − α ξπ N f α (1 / − α ) , (3.45a)Σ ψ bS ( p ) = B ( − p ) − α − dπ N f α (1 / − α ) , (3.45b)Σ ψ cS ( p ) = 0 , (3.45c)where Σ bS vanishes in the limit d → cS vanishes identically. The total LO scalarself-energy is therefore given by Σ ψ aS which is equal to (3.44). From this identity, wededuce the LO gap equation: α (1 / − α ) = 2 + ξπ N f . (3.46)Solving the gap equation yields two values for the index α : α ± = 14 (cid:18) ± (cid:115) −
16 (2 + ξ ) π N f (cid:19) . (3.47)Dynamical symmetry breaking takes place for complex values of α , i.e., for N f < N f cr with N f cr = 16 (2 + ξ ) π . (3.48)In the Landau gauge ( ξ = 0), we recover from (3.48) the result first obtained in [39].The gauge-dependence of (3.48) and (3.58) is not satisfactory especially that criticalcouplings are physical observables. Following [15], we therefore consider an alternativederivation based on using the anomalous mass dimensions of either the electron or theselectron. This approach is based on noting that theses self-energies have two asymptoticbehaviours [15, 98] as a function of the external momentum p Σ( p ) ∼ m ( − p ) − γ m / + m dyn ( − p ) − ( d − − γ m ) / , (3.49)where m is the bare mass (that we have momentarily restored for the sake of generality)and m dyn the dynamically generated one. In particular, eq. (3.49) shows that in d = 3 andin the limit of large euclidean momenta, p E = − p → ∞ , the bare mass dominates theasymptotic behaviour for γ m < / γ m > /
2. Actually, γ m is related to the scaling dimension of the quarticfermion operator: ∆[( ¯ ψψ ) ] = 2 d − − γ m = 4 − γ m where the last equality is valid in d = 3. For γ m < /
2, we have ∆[( ¯ ψψ ) ] >
3, e.g., the four-fermion operator is irrelevant.On the other hand, for γ m < / ψψ ) ] <
3, e.g., the four-fermion operator isrelevant. These arguments [99] therefore relate the relevancy of the four-fermion operator– 20 –o the regime where the dynamical mass dominates the asymptotic behaviour of the fermionpropagator. Assuming that the bare mass is zero, the critical regime separating the masslessand (dynamically) massive phases occurs when the four-fermion operator is marginal. Inorder to find the marginality crossing relation, let’s further note that for d = 3 and m = 0,we have Σ( p ) ∼ m dyn ( − p ) − (1 − γ m ) / , (3.50)which is similar to (3.44) and implies a relation between α and γ m : α + ( N f ) = 12 (1 − γ m ( N f )) = 12 − α − ( N f ) , (3.51)where the N f dependence has been explicited. From (3.47), we saw that dynamical sym-metry breaking takes place for complex values of α and in particular for: ( α − / < γ m , this relation reads: ( γ m − / <
0. Therefore, the marginality crossingrelation is given by: (cid:18) γ m ( N f cr ) − (cid:19) = 0 . (3.52)Within a 1 /N f -expansion, this equation has to be truncated. In particular, the LO criticalcoupling constant is given by: γ m ( N f cr ) = 14 ⇒ N f cr = 32 π = 3 . , (3.53)where either (3.33) or (3.39) can be used and the result is gauge-invariant because γ m isa gauge-invariant quantity. Interestingly, (3.53) coincides with the result of (3.48) in theLandau gauge which was advocated by [39]. This is not so surprising because for SQED the Landau gauge is the “good gauge” where to work as it is the gauge where the LO wavefunction renormalization of the fermion vanishes, see (3.19). N = 1 super-graphene All the results so far derived for N = 1 SQED can be mapped to N = 1 super-graphene as discussed in section 2.4.With the help of (2.17), the mapping of the results (3.19) and (3.25) to the ones validfor super-graphene yields: γ (sg) ψ = − g (1 + ξ ) + O( g ) . (3.54a) γ (sg) φ = +2 g (3 − ξ ) + O( g ) . (3.54b)Similarly, from (3.33) and (3.39) we obtain: γ (sg) mψ = γ (sg) mφ = 8 g + O( g ) . (3.55)As for dynamical mass generation, in the case of super-graphene, the dimensionlesscoupling is the (reduced) fine structure constant so dynamical mass generation may takeplace for g > g cr and of interest is to compute the critical reduced coupling constant, g cr .– 21 –his can be done either via SD equations or via the mass anomalous dimension. In theformer case, from (2.17), the gap equation (3.46) is mapped to: α (1 / − α ) = g ξ ) . (3.56)Solving this gap equation yields two values for the index α : α ± = 14 (cid:18) ± (cid:113) − g (5 + ξ ) (cid:19) . (3.57)Dynamical symmetry breaking takes place for complex values of α , i.e., for g > g cr with g cr = 18 (5 + ξ ) . (3.58)In the gauge ξ = − ξ ), the critical reduced coupling is given by g cr = 1 /
32. A fully gauge-invariant answer can be obtained by mapping (3.53) which yields: γ (sg)1 m ( g cr ) = 14 ⇒ g cr = 132 = 0 . , (3.59)where (3.55) was used. Comparing (3.59) to (3.58), we see that the gauge-invariant resultcoincides with the result in the result for ξ = − α g = 1 /
137 whichyields a reduced fine structure constant g = α g / (4 π ) = 0 . g cr = 0 .
031 found in (3.59). Hence, alreadyat the LO of the 1 /N f -expansion, we find that no dynamical mass is generated in super-graphene. In this section, we study how the IR softening of the photon, ε -scalar and photinopropagators obtained at the LO of the 1 /N f -expansion in section 3 is affected by NLOcorrections. The results will be applied to the study of the stability of the IR interactingfixed point and to the computation of the interaction correction coefficient to the opticalconductivity of super-graphene. At the NLO of the 1 /N f -expansion, 20 Feynman diagrams contribute to the photonpolarization function. Taking into account of the fact that mirror conjugate graphs takethe same value, we are left with 11 distinct graphs to evaluate. This can be done exactlyfor all of the diagrams and the final result for the total NLO photon polarization functionin the DR scheme reads:Π γ ( p ) = e (4 π ) (cid:112) − p (cid:18) µ − p (cid:19) ε d − e γ E ε (cid:26) ( d − G (cid:0) d, , , , , / (cid:1) + 6 (3 d − G (cid:0) d, , / (cid:1) G (cid:0) d, , (3 − d ) / (cid:1)(cid:27) , (4.1)– 22 –here the 2-loop master integral G (cid:0) d, , , , , / (cid:1) is defined in appendix C and its ex-pansion is also given there. The result is finite and, in d = 3, simplifies to:Π γ ( p ) = − e (cid:112) − p − π ) π . (4.2)Together with the LO result of (3.6), the total contribution to the photon polarizationfunction up to NLO in d = 3 reads:Π γ ( p ) = Π γ ( p ) (cid:18) CN f + O (cid:0) /N f (cid:1)(cid:19) , C = 2 (12 − π ) π = 0 . . (4.3)This result will be commented on in section 4.4.As for the calculational details, the exact results for the 11 distinct diagrams con-tributing to (4.1) are all given in appendix D. For clarity, we here provide an explicit viewon the graphs together with their ε -expansion with accuracy O( ε ) (we use the notations of(D.1)): (cid:1) : ˆΠ γ a = − ξ ) π + O( ε ) , (4.4a)4 × (cid:2) : ˆΠ γ bcde = 163 π (cid:18) ε + 193 + 3 ξ ε ) (cid:19) , (4.4b)2 × (cid:3) : ˆΠ γ fg = − π (cid:18) − ξε + 1283 − ξ + O( ε ) (cid:19) , (4.4c) (cid:4) : ˆΠ γ h = − π (cid:18) ξε + 709 − π ξ + O( ε ) (cid:19) , (4.4d)2 × (cid:5) : ˆΠ γ ij = − π (cid:18) − ξε + 143 − ξ + O( ε ) (cid:19) , (4.4e) (cid:6) : ˆΠ γ k = 23 π (cid:18) − ξε −
323 + 3 π − ξ + O( ε ) (cid:19) , (4.4f)2 × (cid:7) : ˆΠ γ lm = 43 π + O( ε ) , (4.4g)– 23 – : ˆΠ γ n = − π + O( ε ) , (4.4h)2 × (cid:9) : ˆΠ γ op = 43 π (cid:18) ε + 133 + O( ε ) (cid:19) , (4.4i)2 × (cid:10) : ˆΠ γ qr = 43 π (cid:18) ε + 193 + O( ε ) (cid:19) , (4.4j)2 × (cid:11) : ˆΠ γ st = − π (cid:18) ε + 113 + O( ε ) (cid:19) . (4.4k)From these results, we first notice that individual graphs may diverge and/or depend onthe gauge fixing parameter. Though the sum of all these contributions is finite and gauge-invariant, in agreement with (4.2) and general properties of the polarization function in d = 3, a proper use of the DRED scheme is crucial to regularize the divergences which takethe form of 1 /ε poles.In order to further clarify the diagrammatic analysis, we can classify the 11 distinctdiagrams into 3 distinct groups depending on the nature of the internal propagators enteringthe graph. Accordingly, we refer to the first group as the “scalar QED”-like diagrams and itconsists of the first 4 diagrams in (4.4). Summing them yields a finite and gauge-invariantresult: ˆΠ γ = ˆΠ γ a + ˆΠ γ bcde + ˆΠ γ fg + ˆΠ γ h = 9 π − π + O( ε ) . (4.5)Upon further using (D.1), we have (in d = 3):Π γ ( p ) = Π γ ( p ) C scalar N f , C scalar = 164 − π π = 0 . . (4.6)The next 2 diagrams are referred to as the “spinor QED”-like diagrams. Their sum isalso finite and gauge-invariantˆΠ γ = ˆΠ γ ij + ˆΠ γ k = 9 π − π + O( ε ) . (4.7)Upon further using (D.1), we have (in d = 3):Π γ ( p ) = Π γ ( p ) C spinor N f , C spinor = 92 − π π = 0 . . (4.8)At this point, we may perform a useful check of our results by relating them to the onesof (non-SUSY) spinor QED with N f two-component fermions. In order to do that, we– 24 –ave two elements to take into account. Firstly, as we saw in the previous section, theLO polarization operator of SQED is twice the value for QED . Secondly, such differencein LO softening for the two models in turn affects the internal photon lines of the NLOdiagrams and implies that the NLO contribution to the polarization operator of SQED is twice smaller than the corresponding contribution for QED . Hence, altogether, the(non-SUSY) spinor QED coefficient is actually given by: C QED = 4 C spinor = 0 . and (non-SUSY) graphene at its IR Lorentz invariant fixed point [71],we have that C g = ( π/ C QED = ( π/ C spinor = 0 . results, we recover theresults of [45] and [68] for (non-supersymmetric) spinor QED in the large- N f limit and(non-supersymmetric) graphene at its IR Lorentz invariant fixed point, respectively.The next 2 diagrams are referred to as the “ ε -scalar QED”-like diagrams. Their sumvanishes (in d = 3) ˆΠ γ ε -scalar = ˆΠ γ lm + ˆΠ γ n = O( ε ) , (4.9)so: C ε -scalar = 0 . (4.10)The last 3 diagrams are referred to as the “photino QED”-like diagrams. Summingthese gauge-invariant contributions again yields a finite result:ˆΠ γ = ˆΠ γ op + ˆΠ γ qr + ˆΠ γ st = 409 π + O( ε ) . (4.11)Upon further using (D.1), we have (in d = 3):Π γ ( p ) = Π γ a ( p ) C photino N f , C photino = − π = − . , (4.12)which is opposite in sign with respect to both C scalar and C spinor .Hence, the following decomposition holds: C = C scalar + C spinor + C ε -scalar + C photino , (4.13)where C scalar has an overwhelming contribution to the screening which is counter-balancedby the anti-screening brought by C photino . On the other hand C ε -scalar does not have anycontribution while the one of C spinor is negligible. At the NLO of the 1 /N f -expansion, 9 Feynman diagrams contribute to the ε -scalarpolarization function. Taking into account of the fact that mirror conjugate graphs takethe same value, we are left with 6 distinct graphs to evaluate. Proceeding along the same We use (2.16) to derive this relation. – 25 –ay as for the photon polarization function, the NLO ε -scalar polarization function in theDR scheme reads:Π ε ( p ) = e (4 π ) (cid:112) − p (cid:18) µ − p (cid:19) ε d − e γ E ε (cid:26) ( d − G (cid:0) d, , , , , / (cid:1) + 6 (3 d − G (cid:0) d, , / (cid:1) G (cid:0) d, , (3 − d ) / (cid:1)(cid:27) , (4.14)which is exactly equal to (4.1). Hence, in d = 3, we have:Π ε ( p ) = Π ε ( p ) (cid:18) CN f + O (cid:0) /N f (cid:1)(cid:19) , (4.15)where (3.9) has been used and the coefficient C is identical to the one in (4.3).The exact results for the 6 distinct diagrams contributing to (4.14) are all given inappendix D. In the following, we will display the corresponding graphs together with their ε -expansion with accuracy O( ε ). Let’s note that, contrary to the case of the photon polar-ization function, there is no advantage here in grouping diagrams according to the natureof the internal propagators entering each graph. This is because there is no completecancellation of singularities and/or gauge dependence within each group.The diagrams are given by: (cid:12) : ˆΠ ε a = − π + O( ε ) , (4.16a)2 × (cid:13) : ˆΠ ε bc = 83 π + O( ε ) , (4.16b) (cid:14) : ˆΠ ε d = 8 π (cid:18) ε + 2 + O( ε ) (cid:19) , (4.16c)2 × (cid:15) : ˆΠ ε ef = − π (cid:18) − ξ ε + 209 − ξ + O( ε ) (cid:19) , (4.16d) (cid:16) : ˆΠ ε g = − π (cid:18) ξε + 10 − π ξ + O( ε ) (cid:19) , (4.16e)2 × (cid:17) : ˆΠ ε hi = 83 π (cid:18) ε + 163 + O( ε ) (cid:19) , (4.16f)– 26 –umming all of these contributions and using (D.1) yields back the NLO term in (4.15). At the NLO of the 1 /N f -expansion, 14 Feynman diagrams contribute to the photinoself-energy. Taking into account of the fact that mirror conjugate graphs take the samevalue, we are left with 7 distinct graphs to evaluate. Proceeding along the same way asfor the photon and ε -scalar polarization functions, the NLO photino self-energy in the DRscheme reads:Π λ V ( p ) = e (4 π ) (cid:112) − p (cid:18) µ − p (cid:19) ε d − e γ E ε (cid:26) ( d − G (cid:0) d, , , , , / (cid:1) + 6 (3 d − G (cid:0) d, , / (cid:1) G (cid:0) d, , (3 − d ) / (cid:1)(cid:27) , (4.17)which is exactly equal to both (4.1) and (4.14). Hence, in d = 3, we have:Π λV ( p ) = Π λ V (cid:18) CN f + O (cid:0) /N f (cid:1)(cid:19) , (4.18)where (3.12) has been used and the coefficient C is identical to the one in (4.3).The exact results for the 7 distinct diagrams contributing to (4.17) are all given inappendix D. We display the corresponding graphs together with their ε -expansion withaccuracy O( ε ):2 × (cid:18) : ˆΠ λ V ab = − π (cid:18) − ξε + 803 − ξ + O( ε ) (cid:19) , (4.19a)2 × (cid:19) : ˆΠ λ V cd = − π (cid:18) − ξε + 83 − ξ + O( ε ) (cid:19) , (4.19b)2 × (cid:20) : ˆΠ λ V ef = − π (cid:18) ξε + 6 − π ξ + O( ε ) (cid:19) , (4.19c)2 × (cid:21) : ˆΠ λ V gh = 43 π + O( ε ) , (4.19d)2 × (cid:22) : ˆΠ λ V ij = 43 π (cid:18) ε + 133 + O( ε ) (cid:19) , (4.19e)– 27 – × (cid:23) : ˆΠ λ V kl = 43 π (cid:18) ε + 103 + O( ε ) (cid:19) , (4.19f)2 × (cid:24) : ˆΠ λ V mn = 4 π (cid:18) ε + 2 + O( ε ) (cid:19) . (4.19g)Summing all of these contributions and using (D.1) yields back the NLO term in (4.18). From eqs. (4.3), (4.15) and (4.18), we see thatΠ γ ( p ) = Π (cid:15) ( p ) = Π λV ( p ) ≡ Π( p ) , (4.20)whereΠ( p ) = − N f e (cid:112) − p (cid:18) CN f + O (cid:0) /N f (cid:1)(cid:19) , C = 2 (12 − π ) π = 0 . . (4.21)The equality (4.20) is in agreement (up to and including NLO corrections) with the identitydiscussed at the end of section 2.3. Moreover, eq. (4.21) shows that, for N f = 1, the NLOcontribution shifts the LO result by ∼ N f are allowed which further reduces this shift to ∼
20% for N f = 2. Therefore,the IR softening of the photon, ε -scalar and photino propagators is only weakly affectedby NLO corrections.By analogy with QED , (1.2), a single effective dimensionless coupling constant canthen be defined for SQED as follows α ( p E ) = e p E (1 − Π( p E )) = (cid:40) e /p E p E (cid:29) e N f / / (cid:0) N f (1 + C/N f ) (cid:1) p E (cid:28) e N f / , (4.22)where we have used (4.21). As a result and again similarly to the QED result, (1.3), thecorresponding beta function: β ( α ) = p E d α ( p E )d p E = − α (cid:18) − N f C/N f ) α (cid:19) (4.23)displays two stable fixed points: an asymptotically free UV fixed point ( α →
0) and aninteracting IR fixed point ( α → α ∗ ) such that: α ∗ = 8 N f (cid:18) − . N f + O (cid:0) /N f (cid:1)(cid:19) . (4.24)Hence, the NLO corrections only weakly shift the non-trivial IR fixed point SQED albeitwith a coefficient C which is approximately 3 times larger than C QED , see (1.4).– 28 – .5 Optical conductivity of N = 1 super-graphene The coefficient C = 0 . can also be related to the interaction correctioncoefficient affecting the optical conductivity of N = 1 super-graphene. In order to dothis, we again use (2.17) which allows to map (4.3) to the corresponding two-loop photonpolarization function of N = 1 super-graphene:Π γ sg ( p ) = Π γ ( p ) (cid:18) C sg α g + O (cid:0) α g (cid:1)(cid:19) , C sg = π C . (4.25)This can be related to the optical conductivity via: σ ( q ) = lim (cid:126)q → i q | (cid:126)q | Π ( q , (cid:126)q ) , (4.26)where q µ = ( q , (cid:126)q ). So: σ sg = σ (cid:18) C sg α g + O( α g ) (cid:19) , C sg = 12 − π π = 0 . , (4.27)where σ = N f e / C sg is also seento be significantly larger than in the non-supersymmetric case, C sg ≈ C g , see (1.5). In this paper, we have presented a detailed study of massless N = 1 supersymmetricQED with N f matter flavours. Due to its super-renormalizable nature, we have studiedit in the 1 /N f -expansion in the IR limit, where an effective scale invariance emerges. Oneof our goals has been to analytically investigate the stability of this IR fixed point in thepresence of quantum corrections.Our analysis has shown that this is indeed the case, up to NLO in the 1 /N f -expansion.To this end, we have employed the dimensional reduction with modified minimal subtrac-tion (DR) scheme, which also introduces 2 ε amount of real scalars called ε -scalars into thegauge multiplet. The gauge multiplet propagators soften their divergent behaviour from1 /p to 1 / (cid:112) − p already at the leading order, due to the bubble summation. We haveutilized these softened propagators for studying the corrections to the matter multipletpropagators at LO and the gauge multiplet propagators at the NLO.Our work extended the analytic results for gauge multiplet to NLO and calculatedanomalous dimensions for matter fields to LO. Although the choice of the Wess-Zuminogauge combined with the R ξ covariant gauge fixing breaks the supersymmetry of the model,there are still constraints that follow from supersymmetric Slavnov-Taylor identities anal-ogous to the ones known for SQED (see discussion at the end of subsection 2.3). Foran abelian gauge field theory, they predict that the gauge multiplet propagators must allreceive identical quantum corrections and that the renormalized masses in each multipletmust be the same. – 29 –ndeed, we have demonstrated that the polarization functions in the gauge multipletΠ γ ( p ) , Π ε ( p ) and Π λV ( p ) for the photon, ε -scalars and the photino, respectively, all co-incide in SQED , up to NLO as given in (4.20) and (4.21). Our focus has been on thepropagators of the theory, since they are sufficient to extract the quantities of interest tous. At leading order, we have verified that the mass anomalous dimensions for the selectron γ mφ (3.39) and the electron γ mψ coincide (3.33), whereas the field anomalous dimensions γ φ (3.25) and γ ψ (3.19) do not. The effective coupling at the IR interacting fixed point wasfound to shift as in (4.24) suggesting that it remains stable.Furthermore, we have explored the possibility of dynamical mass generation in ourtheory by closely examining the Schwinger-Dyson equations at LO. This led to a gaugeparameter dependent value for the critical flavour number generalizing the result of [39].Interestingly, a refined analysis based on mass anomalous dimensions (following the modernapproach of [15], see also [57]) has yielded a gauge-invariant value: N f,cr ≈ . ξ = 0) value from the former method. Aspreviously stated, this is not so surprising because for SQED the Landau gauge is the“good gauge” where to work as it is the gauge where the LO wave function renormalizationof the fermion vanishes, see (3.19). Though this suggests a mass generation, higher ordercorrections may strongly affect this result, as we shall remark below.Along the way, we were able to study the critical behaviour of an N = 1 super-graphenemodel by relating it to the large- N f limit of SQED , see (2.17). This allowed us to obtainthe anomalous dimensions for the matter multiplet and a critical coupling for dynamicalmass generation. We have found that super-graphene does not exhibit mass generation atthe leading order. An interesting outcome of this mapping has been to analytically obtainthe 2-loop interaction correction coefficient of our super-graphene model, which turns outto be C sg ≈ . are actually more important than in QED .In closing, we make some brief remarks about possible extensions of our study. First,as we just summarized above, our explicit calculations have revealed that supersymmetricSlavnov-Taylor identities analogous to the ones known for SQED do hold at the lowestorders of perturbation theory for SQED . Because they represent stringent constraints onpresent and future calculations, it would be worth demonstrating that such identities dohold (to all orders in perturbation theory) in the present three-dimensional case (possiblyalong the lines of the four-dimensional proof [92, 93]). Second, it would be quite interest-ing to reformulate (and extend) the present study using the three-dimensional superfieldformalism [100, 101] which fully utilizes supersymmetry. This could throw light on theadditional structures that may arise. Third, it would be interesting to calculate the criticalflavour number at the NLO of the 1 /N f -expansion, either in the component or the super-field formalism. This could help one clarify whether flavour symmetry breaking is indeedforbidden as per the duality between N f = 2 SQED and the so called 7-field Wess-Zuminomodel [54, 55, 57, 58]. We leave these tasks for future investigations.– 30 – cknowledgments We would like to thank Pietro Slavich for valuable discussions and Dominik St¨ockingerfor detailed correspondences and careful explanations. S.T. thanks Francesco Benini, SergioBenvenuti and Hrachya Khachatryan for many interesting exchanges related to [58] (whereSQED was studied) during the summer 2019. The work of A.J. is supported by the ILPLABEX (under reference ANR-10-LABX-63) through French state funds managed by theANR within the Investissements d’Avenir programme under reference ANR-11-IDEX0004-02. A Notations, conventions and useful identities
We work in a three-dimensional Minkowski space with metric g µν = diag(+ , − , − ).The three 2 × γ -matrices satisfy the usual Clifford algebra: { γ µ , γ ν } = 2 g µν . Asan explicit representation, we take the Majorana basis where all γ -matrices are purelyimaginary: γ = σ , γ = i σ , γ = i σ . (A.1)The following relations are useful in practice: γ µ γ ν = g µν + i ε µνρ γ ρ , (A.2) γ µν = +i ε µνρ γ ρ , (A.3) γ µ γ ν γ ρ = γ µνρ + g µν γ ρ + g νρ γ µ − g µρ γ ν , (A.4)where γ µν = [ γ µ , γ ν ] / ε = +1.The identity matrix, , together with the three γ µ ( µ = 0 , ,
2) span the vector spaceof 2 × ψ ¯ ψ = −
12 ¯ ψ ψ − (cid:0) ¯ ψ γ µ ψ (cid:1) γ µ , (A.5)which holds for two (Dirac or Majorana) spinors ψ and ψ .Majorana spinors are defined as: ψ = C ¯ ψ (cid:62) = ψ ∗ , ¯ ψ = ψ † γ = − ψ (cid:62) C , (A.6)where the charge conjugation matrix is anti-symmetric: C (cid:62) = − C , and such that: CC † = , γ (cid:62) µ = − C − γ µ C and C = − γ . For two Majorana spinors (cid:15) and (cid:15) , we have thefollowing Majorana flip identities:¯ (cid:15) (cid:15) = ¯ (cid:15) (cid:15) , ¯ (cid:15) γ µ (cid:15) = − ¯ (cid:15) γ µ (cid:15) , ¯ (cid:15) γ µν (cid:15) = − ¯ (cid:15) γ µν (cid:15) , ¯ (cid:15) γ µνρ (cid:15) = ¯ (cid:15) γ µνρ (cid:15) . (A.7)Let us also note that the field strength satisfies the Bianchi identity ∂ [ µ F νρ ] = 0 . (A.8)– 31 –n the DRED scheme, the metric tensor and γ -matrices are decomposed as (we followthe notations of [83]): g µν = g ˆ µ ˆ ν + g ¯ µ ¯ ν ( g ˆ µ ¯ ν = 0) , γ µ = γ ˆ µ + γ ¯ µ , (A.9)and the following properties hold: g µµ = 3 , g ˆ µ ˆ µ = d, g ¯ µ ¯ µ = 2 ε, (A.10a) { γ ˆ µ , γ ˆ ν } = 2 g ˆ µ ˆ ν , { γ ¯ µ , γ ¯ ν } = 2 g ¯ µ ¯ ν , { γ ˆ µ , γ ¯ ν } = 0 . (A.10b)where d = 3 − ε . The following trace formula is useful in practice:Tr[ γ ˆ ν · · · γ ˆ ν n γ ¯ µ · · · γ ¯ µ m ] = 12 Tr[ γ ˆ ν · · · γ ˆ ν n ]Tr[ γ ¯ µ · · · γ ¯ µ m ] , (A.11)with Tr[ ] = 2. Some simple examples of contraction and trace identities include: γ ˆ µ γ ˆ µ = d, γ ˆ µ γ ˆ α γ ˆ µ = − ( d − γ ˆ α , γ ¯ µ γ ˆ α γ ¯ µ = − ε γ ˆ α , (A.12a)Tr[ γ ˆ µ γ ˆ ν ] = 2 g ˆ µ ˆ ν , Tr[ γ ˆ µ γ ˆ α γ ˆ β γ ˆ ν ] = 2 (cid:0) g ˆ µ ˆ α g ˆ β ˆ ν − g ˆ µ ˆ β g ˆ α ˆ ν + g ˆ µ ˆ ν g ˆ α ˆ β (cid:1) . (A.12b) B Feynman rules
In this appendix, we present the Feynman rules (in Minkowski space) that follow fromeq. (2.5) before and after taking the large N f limit. As known from the main text, thepropagators of eqs. (B.2), (B.4) and (B.6) (see below) are subject to an IR softening. Theyare given by (2.7b), (2.7c) and (2.7d), respectively, before taking the large N f limit and by(2.10) after taking the large- N f limit where the softening takes place. For simplicity, wewill not make any graphical distinction between the plain propagators and the IR softenedones. It will be clear within the main text which ones are used. Moreover, we will drop theflavour indices, i and j , in the following since they always lead to a factor of δ ij wheneverthey appear.The rules are given by:(i) S βα ( p ) = i (cid:54) p βα p = (cid:25) pβ α (B.1)(ii) D ˆ µ ˆ ν ( p ) = − i p (cid:18) g ˆ µ ˆ ν − (1 − ξ ) p ˆ µ p ˆ ν p (cid:19) = (cid:26) p ˆ µ ˆ ν (B.2) D ˆ µ ˆ ν LO ( p ) = i2 a (cid:112) − p (cid:18) g ˆ µ ˆ ν − (1 − ξ ) p ˆ µ p ˆ ν p (cid:19) = (cid:27) p ˆ µ ˆ ν (B.3)– 32 –iii) E ¯ µ ¯ ν ( p ) = − i g ¯ µ ¯ ν p = (cid:28) p ¯ µ ¯ ν (B.4) E ¯ µ ¯ ν LO ( p ) = i g ¯ µ ¯ ν a (cid:112) − p = (cid:29) p ¯ µ ¯ ν (B.5)(iv) σ βα ( p ) = i (cid:54) p βα p = (cid:30) pβ α (B.6) σ βα LO ( p ) = − i (cid:54) p βα a (cid:112) − p = (cid:31) pβ α (B.7)(v) ∆ ( p ) = i p = p (B.8)(vi) ! ˆ µ αβ = − i eγ ˆ µβα (B.9)(vii) " pk ˆ µ = − i e ( p + k ) ˆ µ (B.10)(viii) ˆ µ ˆ ν = +2i e g ˆ µ ˆ ν (B.11)(ix) $ ¯ µ αβ = − i eγ ¯ µβα (B.12)– 33 –x) % ¯ µ ¯ ν = +2i e g ¯ µ ¯ ν (B.13)(xi) & α β = eδ αβ (B.14)(xii) ’ α β = − eδ αβ (B.15)Additionally, notice that we have adopted the compact Feynman rules of [102, 103] thatare based on assigning a fermion flow to each graph along fermion lines and involve onlyone kind of propagator together with vertices without explicit charge-conjugation matricesfor Majorana fermions. We find that these rules allow for a rather simple evaluation ofFeynman diagrams with Majorana fermions and are helpful to fix some sign ambiguities.They may be contrasted with those of [104, 105] that lead to multiple propagators andvertices as compared to the purely Dirac fermion case. C Master integrals
We consider an Euclidean space of dimension d . Following the notations of the review[94], the one-loop massless propagator-type master integral is given by: J ( d, p, α, β ) = (cid:90) [d d k ] k α ( k − p ) β = ( p ) d/ − α − β (4 π ) d/ G ( d, α, β ) , (C.1)where [d d k ] = d d k/ (2 π ) d , α and β are the so-called indices of the propagators and thedimensionless function G ( d, α, β ) has a simple expression in terms of Euler Γ-functions: G ( d, α, β ) = a ( α ) a ( β ) a ( α + β − d/ , a ( α ) = Γ( d/ − α )Γ( α ) . (C.2)Our calculations will also require the knowledge of the one-loop semi-massive tadpole in-tegral which is defined as: M ( d, m, α, β ) = (cid:90) [d d k ]( k + m ) α k β = ( m ) d/ − α − β (4 π ) d/ B ( d, α, β ) , (C.3)– 34 –ith B ( d, α, β ) = Γ( α + β − d/
2) Γ( d/ − β )Γ( d/
2) Γ( α ) . (C.4)The two-loop massless propagator-type master integral is given by: J ( d, p, α , α , α , α , α ) = (cid:90) [d d k ][d d k ]( k − p ) α ( k − p ) α k α k α ( k − k ) α = ( p ) d − (cid:80) i =1 α i (4 π ) d G ( d, α , α , α , α , α ) , (C.5)where G ( d, α , α , α , α , α ) is dimensionless and unknown for arbitrary indices { α i } i =1 − .In our calculations, we encountered two master integrals at NLO that we shall denote as: T , = e γ E ε G ( d, , / G (cid:0) d, , (3 − d ) / (cid:1) , T , = e γ E ε G ( d, , , , , / , (C.6)where T , is a product of two simple one-loop G -functions (C.2). The more interesting T , is part of a class of complicated two-loop master integrals of the type (C.5) (with upto 3 arbitrary indices) that have been computed in [106]. With the help of the result of[106], T , can be expanded to high orders in ε ( ε = (3 − d ) /
2) and we only provide herethe first terms of this expansion for completeness: T , π = 1 + (cid:18) π ζ − (cid:19) ε + (cid:18) π Li (1 /
2) + 14310 ζ + 6 π log − (cid:19) ε + O( ε ) , (C.7)where ζ n = Li n (1) and Li n are polylogarithms. D Feynman diagrams at NLO
In this appendix, we provide the interested reader with the exact expressions (validfor any d ) of all the NLO diagrams we have computed. They were explicitly displayed inthe main text in section 4.We introduce the notation:Π X a ··· a n ( p ) = e (cid:112) − p (cid:18) µ − p (cid:19) ε ˆΠ X n ··· n , (D.1)where X = { γ, ε, λ } and a , · · · , a n is a collection of n indices referring to graphs takingthe same value; hence: Π X a ··· a n ( p ) = n Π X a i ( p ) ( i = 1 , · · · , n ) (it is also understood thatΠ λ a ··· a n ( p ) ≡ Π λ V a ··· a n ( p )). For example, Π γ fg ( p ) = 2Π γ f ( p ) because, for the photonpolarization function, NLO diagrams (f) and (g) are mirror conjugate graphs and take thesame value. The same notation holds for the momentum-independent function ˆΠ X n ··· n .The results below will further be expressed in terms of the two master integrals that wereintroduced in (C.6). – 35 –or the 11 distinct photon polarization diagrams, our results read:ˆΠ γ a = 4 π (cid:20) ξd − (cid:21) T , , (D.2a)ˆΠ γ bcde = 8 π (cid:20) d − d − d − − ξd − (cid:21) T , , (D.2b)ˆΠ γ fg = − π (cid:20) d − d − d − d − − (2 d − ξd − (cid:21) T , , (D.2c)ˆΠ γ h = 2( d − π (cid:20) d − d + 106 d + 21 d − d − (2 d − − d − d + 11 d − ξ (cid:21) T , + (5 d − d − d − d − π T , , (D.2d)ˆΠ γ ij = − π ( d − d − (cid:20) d − − ξd − (cid:21) T , , (D.2e)ˆΠ γ k = − π d − d − d − (cid:20) d − d + 1077 d − d + 3405 d − d − (2 d − d − ξ (cid:21) T , + ( d − (cid:0) d − d + 41 d − (cid:1) ( d − d − d − π T , , (D.2f)ˆΠ γ lm = − d − ( d − d − π T , , (D.2g)ˆΠ γ n = 2( d − (cid:0) d − d + 204 d − (cid:1) ( d − d − π T , − ( d − ( d − d − d − π T , , (D.2h)ˆΠ γ op = 4( d − ( d − d − π T , , (D.2i)ˆΠ γ qr = 4( d − d − d − π T , , (D.2j)ˆΠ γ st = − d − (cid:0) d − d + 69 (cid:1) ( d − d − d − (2 d − π T , + 8( d − d − d − d − d − π T , . (D.2k)For the 6 distinct ε -scalar polarization diagrams, our results read:ˆΠ ε a = 4 π T , , (D.3a)ˆΠ ε bc = − d − (2 d − π T , , (D.3b)ˆΠ ε d = 2( d − (cid:0) d − d + 41 (cid:1) ( d − d − π T , − ( d − d − d − π T , , (D.3c)ˆΠ ε ef = 4( d − ( d − π (cid:18) ξ − d − d − (cid:19) T , , (D.3d)ˆΠ ε g = − d − d − π (cid:18) d − d + 72 d − d − ξ (cid:19) T , + ( d − d − d − π T , , (D.3e)ˆΠ ε hi = 4( d − d − d − d − π T , . (D.3f)– 36 –or the 7 distinct photino self-energy diagrams, our results read:ˆΠ λ V ab = − d − d − π (cid:20) d − d − d − − (2 d − ξ (cid:21) T , , (D.4a)ˆΠ λ V cd = − d − d − d − π (cid:20) d − d − − ξ (cid:21) T , , (D.4b)ˆΠ λ V ef = 4( d − π (cid:20) d − d − − (2 d − d − ξ (cid:21) T , + 2( d − d − π T , , (D.4c)ˆΠ λ V gh = − d − d − d − π T , , (D.4d)ˆΠ λ V ij = 4( d − ( d − d − π T , , (D.4e)ˆΠ λ V kl = 2( d − d − d − d − π T , , (D.4f)ˆΠ λ V mn = 2(3 d − d − d − d − π T , − d − d − π T , . (D.4g) References [1] T. Appelquist and R.D. Pisarski,
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