PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.110.105006ARTICLESFormal aspects of field theory, field theory in curved spaceSupergeometric quantum effective actionSUPERGEOMETRIC QUANTUM EFFECTIVE ACTIONVIOLA GATTUS AND APOSTOLOS PILAFTSIShttps://orcid.org/0000-0002-7428-0514GattusViola*https://orcid.org/0000-0001-6217-2711PilaftsisApostolos†Department of Physics and Astronomy, University of Manchesterhttps://ror.org/027m9bs27, Manchester M13 9PL, United Kingdom
12November202415November2024110101050061July20242October2024Published by the American Physical Society2024authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Supergeometric quantum field theories (SG-QFTs) are theories that go beyond the standard supersymmetric framework, since they allow for general scalar-fermion field transformations on the configuration space of a supermanifold, without requiring an equality between bosonic and fermionic degrees of freedom. After revisiting previous considerations, we extend them by calculating the one-loop effective action of minimal SG-QFTs that feature nonzero fermionic curvature in two and four spacetime dimensions. By employing an intuitive approach to the Schwinger–DeWitt heat-kernel technique and a novel field-space generalized Clifford algebra, we derive the ultraviolet structure of characteristic effective-field-theory (EFT) operators up to four spacetime derivatives that emerge at the one-loop order and are of physical interest. Upon minimizing the impact of potential ambiguities due to the so-called multiplicative anomalies, we find that the EFT interactions resulting from the one-loop supergeometric effective action are manifestly diffeomorphically invariant in configuration space. The extension of our approach to evaluating higher loops of the supergeometric quantum effective action is described. The emerging landscape of theoretical and phenomenological directions for further research of SG-QFTs is discussed.
Science and Technology Facilities Council10.13039/501100000271ST/X00077X/1University of Manchester10.13039/501100000770INTRODUCTION
Covariant and differential geometric methods have been playing a prominent role in driving the development of quantum field theories (QFTs) [1]. In addition to addressing issues of gauge covariance in effective quantum actions [2–5], these methods were employed to compute transition amplitudes of chiral loops in a manifestly reparametrization invariant manner on the entire configuration space of fields and spacetime coordinates [6,7]. The same methods have also been utilized to formulate nonlinearly realized supersymmetric theories [8], as well as derive the effective field theory (EFT) limit of heterotic string theories [9]. Beyond the classical-action level, these covariant differential-geometric methods have been further developed by Vilkovisky and DeWitt (VDW) [10,11], with the aim to address the issue of gauge-fixing parameter independence in gauge and quantum gravity theories. This issue was subsequently the subject of numerous other studies [12–15].
More recently, related geometric effective actions were put forward as a potential solution to the so-called quantum frame problem [16–18] that haunts early studies of cosmic inflation. In this respect, an obstacle concerning the uniqueness of the path-integral measure of the VDW effective action was addressed beyond the tree level [19,20]. Finally, similar geometric techniques were employed to analyze new-physics phenomena within the framework of EFTs beyond the Standard Model SM [21–33], also known as SMEFT [34–38] (for a recent review, see [39]).
A major theoretical difficulty that was encountered in the formulation of geometric effective actions has been the proper inclusion of fermionic fields as independent chart variables in the path-integral configuration space [11]. Unlike supersymmetric theories in which fermions appear as tangent vectors in a field space whose geometry is determined by their bosonic counter-parts [8], the theories that were initially presented in [40–42] and are of interest to us treat scalars and fermions as independent field coordinates and so allow for general scalar-field transformations on the configuration space of a supermanifold. Since fermions are Grassmannian variables, their consistent description on this more general space would necessitate extensive use of mathematical techniques from differential supergeometry (SG) on supermanifolds[43]. Hence, the theories that we will be considering here will be called supergeometric quantum field theories (SG-QFTs) [41,42], and they should not be confused with the conventional supersymmetric theories.
In this paper we extend previous considerations and present, for the first time, analytic expressions for the one-loop effective action of minimal SG-QFT models that feature nonzero fermionic curvature. To facilitate our computations, we employ a new and more intuitive approach to the Schwinger–DeWitt heat-kernel technique based on the Zassenhaus formula. Furthermore, we use a novel field-space generalized Clifford algebra which enables us to derive the ultraviolet (UV) structure of representative EFT operators up to four spacetime derivatives that arise at the one-loop level. In particular, we find EFT operators that have new characteristic properties, since they describe magnetic-moment-type transitions between fermions induced by a nonzero field-space curvature. In our computations, we had to minimize the impact of potential ambiguities due to the so-called multiplicative anomalies by requiring that the well-known results be obtained in the flat field-space limit. An important finding of our study is that the EFT interactions resulting from the one-loop super-geometric quantum effective action (SG-QEA) are manifestly diffeomorphically invariant in both configuration and field space. The method that we present here is not limited to one loop, but we describe how it can be straightforwardly extended to evaluate higher loops of the SG-QEA.
The layout of the paper is as follows. Section II outlines basic aspects of the notion of supergeometry in QFT. Specifically, Sec. II provides a covariant formulation of a generic scalar-fermion SG-QFT, up to second order in spacetime derivatives of the fields. The respective Lagrangian of the SG-QFT is written down in terms of model functions with well-defined transformation properties under the Lorentz group and field-space reparametrizations. The same section presents our approach to deriving the supermetric that governs the SG-QFT. In Sec. III, we extend the old geometric formalism of [7] which was applied to bosons only to the more general case of a SG-QFT that includes fermions. On the basis of this formalism, we derive covariant interactions of general scalar-fermion theories, in terms of the super-Riemannian tensor in the configuration space. The covariant two-point correlation functions, along with the respective covariant propagators, are essential elements in the computation of the SG quantum effective action which will be presented in the subsequent sections. In this context, higher-order correlation functions, along with their symmetry properties in configuration space, are given in Appendix A.
In Sec. IV we briefly review the basic features of the SG-QEA by extending the VDW formalism in order to include both bosonic and fermionic degrees of freedom. In addition, we derive a master equation (4.15) where all-order SG-QEA can be recursively determined. Furthermore, we show how the Zassenhaus formula can be conveniently exploited, within the context of the Schwinger–DeWitt heat-kernel technique, in order to compute the one-loop SG-QEA in configuration space. Section V gives a practical demonstration of its use by recovering known results for a minimal Yukawa theory, as well as for a more involved scalar-fermion theory, in a flat field space at the one-loop order. Technical details pertaining to the latter case have been relegated to Appendix C. We note that all our computations are performed within the dimensional regularization (DR) scheme in the coordinate space, rather than in the usual momentum space. Therefore, useful formulas of the DR scheme in the coordinate space are collected in Appendix D.
In Sec. VI, we apply our intuitive heat-kernel method to bosonic theories that exhibit nonvanishing field-space Riemannian curvature. Such a computation of the effective action gets facilitated by taking the field-space covariant heat kernel that we derive for a general frame to a local (flat) frame, and then transforming the results back to the global field-space curved frame through general covariance. This method becomes rather advantageous, since all covariant EFT operators emerge automatically in a systematic manner, unlike other more laborious approaches that utilize ’t Hooft’s matching procedure [44]. In this respect, we clarify that higher-derivative geometric EFT interactions in the configuration space can be systematically translated into EFT operators acting on the usual field space in a manifestly covariant manner.
In Sec. VII we calculate the UV divergent contributions to the one-loop effective action for a minimal SG-QFT model that futures nonzero fermionic field-space curvature. Moreover, we develop a novel Clifford algebra, where the role of the Dirac γμ matrices is assumed by field-dependent tensors, λμ, and their adjugate counterparts, λ¯μ. In fact, λμ and λ¯μ satisfy a field-space generalized Clifford algebra, which can be used to bosonize the fermionic effective action in a strictly covariant manner in the configuration space. In order to minimize the potential impact of the so-called multiplicative anomalies due to the bosonization of the fermionic action, we require that the known UV divergences should be recovered in the flat field-space limit of the theory. In the same section, we present the UV structure of typical covariant EFT operators up to four spacetime derivatives that emerge from the one-loop SG-QEA in the configuration space and could be phenomenologically interesting. Section VII concludes with a subsection outlining potential approaches that could enable us to go beyond the minimal supergeometric framework that we will be studying here. Finally, Sec. VIII summarizes the main findings of our study and discusses new directions for future research.
SUPERGEOMETRY IN QUANTUM FIELD THEORY
In this section we give a short review on the theory of supermanifolds [43] which is relevant to the construction of SG-QFTs that are formulated on a scalar-fermion field space [40].
For an SG-QFT with N real scalars and M Dirac fermions, one may assume that all these fields live on a supermanifold of (N|2dM) dimensions in d spacetime dimensions. Hence, for d=4 dimensions, the supermanifold chart may be represented by the (N+8M)-dimensional column vector Φ≡{Φa}=(ϕAψXψ¯YT).Hereafter, we use lowercase Latin letters to denote configuration-space indices, whilst capitalized Latin letters are reserved for field-space indices, i.e., Φa≡ΦA(xA). For the ease of notation, no distinction will be made between bosonic and fermionic field-space indices. The nature of the index can always be inferred from the context considered.
A field reparametrization now corresponds to a diffeomorphism on the supermanifold, Φa→Φ˜a=Φ˜a(Φ).The above field transformation is taken to be ultralocal depending only on Φa and not on any spacetime derivatives of fields, like ∂μΦa, in line with the VDW formalism [10,11]. For simplicity, we will not consider possible relaxation of this restriction here by adopting Finslerian-type geometries in the field space [19,22,23,45].
Up to second order in ∂μΦa, the field-space diffeomorphically invariant Lagrangian L of a general SG-QFT may covariantly be written down as [40]L=12gμν∂μΦAAkB(Φ)∂νΦB+i2ζAμ(Φ)∂μΦA-U(Φ).Note that L can be expressed in terms of the three model functions: (i) a rank-2 field-space tensor, kAB(Φ), (ii) a mixed spacetime and field-space vector, ζAμ(Φ), and (iii) U(Φ), a zero-grading scalar describing the Higgs potential and Yukawa interactions. In this minimal class of SG-QFTs under study, we assume the existence of one global (defining) frame in which kAB vanishes when either or both indices are fermionic. The three model functions can be extracted unambiguously from the Lagrangian, according to the prescription outlined in [40–42].
The field-space metric of the supermanifold can be derived uniquely from the action, S, associated with the Lagrangian (2.3). To do so, we must first construct a pure field-space vector ζA to be projected out from the model function, ζAμ. This procedure was described in great detail in [41,42]. Here, we only limit ourselves to state the relevant projection operation, ζBμ(BΣ←μ)A=ζA,where Σ←μ=1d(δ←/δγμ00Γμ),with Γμ=diag(γμ,γμT). As can be seen from (2.4), the bosonic part of ζA can be extracted by differentiating ζAμ with respect to γμ (or σμ). Instead, the fermionic components of ζA are obtained by appropriately contracting ζAμ with Γμ. This projection method can be applied to a wide range of SG-QFTs, as discussed in [41,42].
After we have determined ζA following the above procedure, we can then use it to construct the rank-2 field-space tensor, λAB=12(ζA,B-(-1)A+B+ABζAB,),which is antisupersymmetric and singular in the presence of scalar fields. Here and in the following, we employ the compact convention of index notation by DeWitt [43], according to which exponents of (-1) represent the grading of the respective quantities and do not participate in index summation. The grading can be 0 or 1 for commuting or anticommuting fields, respectively. Furthermore, only adjacent indices can be contracted, and factors of (-1) must be introduced when the position of two indices is exchanged.
The scalar contribution arising from the model function kAB can be combined with λAB to form the new rank-2 field-space tensor, ΛAB=kAB+λAB.Note that ΛAB cannot play the role of the supermetric, since it is not exactly supersymmetric; it has a mixed symmetry. However, if the local form of ΛAB is known, one may still be able to compute the field-space vielbeins eB^A[46,47], where a hat (^) on a field-space index signifies local-frame coordinate. These can in turn be used to compute the supermanifold metric GBA from the local field-space metric HB^A^ as follows [40]: GBA=eM^AHN^M^eBsTN^,where the local field-space metric is given by HB^A^≡(1N00001dM0-1dM0)and the superscript sT denotes the operation of supersymmetrization. Observe that the rank-2 field-space tensor GAB is qualified to be the supermetric of the field-space supermanifold, since it is supersymmetric by construction, i.e., GAB=(GAB)sT=(-1)A+B+ABGAB.In configuration space, the supermetric and the field-space vielbeins are generalized as Gab=GAB(xA)δ(xA-xB),eb^a=eB^A(xA)δ(xA-xB^),where δ(xA-xB) is the d-dimensional δ function. The configuration-space Riemann tensor is computed from the Christoffel symbols Γabc[43], viz. Rabcd=-Γabc,d+(-1)cdΓabd,c+(-1)c(m+b)ΓamcΓmbd-(-1)d(m+b+c)ΓamdΓmbc=RABCDδ(xA-xB)δ(xA-xC)δ(xA-xD).
Finally, we introduce the useful operations of superdeterminant and supertrace. The entries of any (N+M)×(N+M)-dimensional rank-2 field-space tensor (also known as supermatrix) may be arranged in the following block structure: M=(ACDB),where A and B are N×N and M×M matrices with commuting entries, while C and D are N×M and M×N dimensional matrices of anticommuting numbers, respectively. For such a matrix, the superdeterminant may be defined in two equivalent ways: sdetM=detAdet(B-DA-1C)=det(A-CB-1D)detB,where det is the ordinary matrix determinant.
The supertrace is defined only for mixed rank-2 tensors, which have a left index as a subscript and right index as a superscript, or vice versa. If such a mixed rank-2 tensor is represented by a matrix M with the block structure as given in (2.13), its supertrace is evaluated as strM=trA-trB,where trA and trB indicate ordinary traces of the matrices A and B. The supertrace is invariant under supertransposition and cyclic permutations, in close analogy to the usual trace properties for commuting numbers. All the aforementioned definitions and identities will prove valuable for the evaluation of SG-QEAs (cf. Sec. IV).
COVARIANT INTERACTIONS FOR SCALAR-FERMION THEORIES
In this section we will extend the previous considerations in [41,42] and present the methodology for deriving two- and higher-point covariant interactions for a general SG-QFT that includes both scalars and fermions. We will also give explicit analytic results for the two lowest-point correlation functions, such as the covariant scalar-fermion propagator and the three-field interactions. Further analytic expressions for covariant higher-point vertices may be found in Appendix A.
To derive the covariant inverse propagator, three- and higher-point vertices in a relatively compact way, we adopt methodology and notation first introduced in [7]. To this end, we note that the covariant functional differentiation of ∂μΦa with respect to Φb in the configuration space is given by (∂μΦa);b=(δAB∂μ(A)+ΓABM∂μΦM)δ(xA-xB)≡(Dμ)ab.In the above, the semicolon (;) stands for covariant functional differentiation in configuration space, ∂μ(A)≡∂/∂xAμ, and ΓABC are the field-space Christoffel symbols. We should observe that in this covariant formulation (Dμ)ab as defined in (3.1) is strictly not a differential operator, but a mixed spacetime and field-space tensor in configuration space. Nevertheless, for later convenience, we term it abusively as such in this work (see also our discussion in Sec. VI).
In the covariant formalism of [7], an important role plays the identity (Dμ)ab;c=Rabcm∂μΦm,where (Dμ)ab=Gam(Dμ)mb. Most remarkably, we were able to prove that the Ecker–Honerkamp (EH) identity (3.2) will still hold true within a SG-QFT in which fermionic fields are included. A detailed proof of the EH identity (3.2) within the context of supermanifolds is given in Appendix A. Furthermore, the differential operator, (Dμ)ab, satisfies the following (anti)symmetric properties: (Dμ)ab=-(-1)ab(Dμ)ba,(Dμ)ba=-(-1)b(a+1)(Dμ)ab.Finally, in terms of the field-space vielbeins given in (2.11), another equivalent representation of (Dμ)ba in field space can be obtained: (Dμ)AB=eA^A(Dμ)A^B^esTBB^=eA^A[δB^A^∂μ(A^)+(-1)C∂μΦCωCA^B^]esTBB^,where ∂μ(A^)≡∂/∂xA^μ, and ωCA^B^ is the spin connection in the field space. More details on the role of the field-space spin connection are given in Appendix B. Here we only quote its definition in terms of field-space vielbeins and metric connection coefficients, ωCA^B^=(-1)C(A+A^)(-esTA,CA^+eMsTA^ΓMAC)eAB^.As we will see in Sec. VI, the representation of (Dμ)BA in (3.4) will prove useful to find an expression for the configuration-space heat kernel in the global curved frame.
Given the action S of the Lagrangian (2.3), we can now calculate the equation of motion of the fields Φa (with Sa≡S;a and Ua≡U;a), Sa=(∂μΦm)knm(Dμ)na+12(-1)an(∂μΦm)mkn;a(∂μΦn)+i(-1)aaλmμ(∂μΦm)-Ua.In (3.6), we have followed [41,42] and defined the mixed tensor, λabμ≡12(ζbμa,-(-1)a+b+abζaμb,).This quantity is a proper spacetime vector and rank-2 field-space tensor, which turns out to be antisupersymmetric. It can be deduced from ζaμ via ordinary functional differentiation, because the Christoffel symbols cancel.
To significantly reduce the length of our analytic expressions, we introduce the following conventions for the operations of symmetrization and cyclic permutation of configuration-space indices: [abc]=abc+(-1)abbac+(-1)c(a+b)cab(symmetrization)(abc)=abc+(-1)a(b+c)bca+(-1)c(a+b)cab(cyclicpermutation).We note that repeated indices are excluded from the operations mentioned above. With the above conventions in mind, the covariant inverse propagator is found to be Sab=(-1)am(Dμ)maknm(Dμ)nb+i(-1)a(λmμa(Dμ)mb+(-1)bmλm;bμa∂μΦm)-Uab+∂μΦm(knmRnabp∂μΦp+(-1)ankn;[am(Dμ)nb]+12(-1)n(a+b)kn;abm∂μΦn).Here and in the following, we drop the semicolon (;) and adopt the abbreviation Sabc…≡S;abc…, for covariant functional derivatives that act on the classical action S, as well as for the scalar potential, i.e., Uabc…≡U;abc…. Evidently, because the action S is a scalar, the covariant inverse propagator Sab turns out to be supersymmetric. In the homogeneous and flat field-space limit, ∂μΦa→0 and Rabcd→0, only the first two terms on the rhs of (3.9) will survive, besides Uab.
We may proceed in a similar fashion and calculate the three-point interaction vertex by taking a third field-space covariant derivative of the action S, i.e., Sabc. By making use of the supersymmetric property of the rank-2 tensor kba and the conventions of (3.8), the three vertex can be written as Sabc=∂μΦm((-1)cpknmRnabp;c+(-1)ankn;[amRnbc]p+12(-1)p(a+b+c)kp;abcm)∂μΦp+(-1)am(Dμ)m[a(knmRnbc]p+(-1)p(b+c)kp;bc]m)∂μΦp+∂μΦmknmRnabp(Dμ)pc+(-1)am+bn(Dμ)m(akn;bm(Dμ)nc)+i(-1)a(λmμaRmbcp∂μΦp+(-1)bmλm;[bμa(Dμ)mc]+(-1)m(b+c)λm;bcμa∂μΦm)-Uabc.It should be noted that in the homogeneous limit ∂μΦ→0, we recover the expressions for the vertices of the fermionic sector presented in [41,42].
The methodology that we established in this section can be used to compute higher-point correlation functions and express them in a strictly manifest covariant manner. However, the above exercise has also shown that such a task becomes increasingly more and more complex, yielding lengthier expressions. In Appendix A we display the four-point vertex which can be of direct relevance to 2→2 scattering processes.
COVARIANT SCALAR-FERMION EFFECTIVE ACTION
As mentioned in the Introduction, a rigorous differential-geometric approach to computing field-space covariant quantum effective actions, well beyond the one-loop level, is provided by the so-called VDW formalism [10,11,13,48]. For our purposes, it would be useful to first briefly review the basic aspects of this formalism by considering both bosonic and fermionic degrees of freedom, along the lines of [40,49]. Such a covariant scalar-fermion effective action will also equivalently be called the supergeometric quantum effective action.
The SG-QEA, denoted as Γ[Φ], may be evaluated iteratively in a loopwise ℏ expansion by the following functional integral equation: exp(iℏΓ[Φ])=∫|sdetG|[DΦq]exp(iℏS[Φq]+iℏ∫δ4x-gΓ[Φ],aΣa[Φ,Φq]).Here Φ (Φq) collectively denotes the mean (quantum) fields, G={Gab} is the supermetric, and the comma (,) denotes functional differentiation with respect to a mean field. Furthermore, Σa[Φ,Φq] is defined to be a superposition of geodesic tangent vectors σa[Φ,Φq] on the supermanifold, Σa[Φ,Φq]=(C-1[Φ])abσb[Φ,Φq].The rank-2 tensor Cab[Φ] may be determined by extending Vilkovisky’s condition of vanishing tadpoles [10,11] to curved field spaces, i.e., ⟨Σa[Φ,Φq]⟩Σ=0.The latter restriction ensures that the effective action can be calculated perturbatively as a sum of one-particle-irreducible (1PI) graphs [14]. By imposing (4.3) on (4.1), one is able to find the functional form of Cab[Φ] perturbatively in powers of the tangent vector σa[Φ,Φq][11]: Cab[Φ]=⟨σa;b[Φ,Φq]⟩Σ=⟨δab-13(-1)m(b+n)Rambn[Φ]σm[Φ,Φq]σn[Φ,Φq]+…⟩Σ.
After performing an ℏ expansion of the SG-QEA given in (4.1), the following one- and two-loop expressions are obtained [40]: Γ(1)[Φ]=i2lnsdet(Sba)=i2str(lnSba),Γ(2)[Φ]=-18S{abcd}ΔdcΔba+112(-1)bc+m(b+d)S{mca}ΔabΔcdΔmnS{ndb},where Sba≡∇→aS∇←b≡Δab-1 and Δab is its inverse, which is determined by the equation ΔacSbc=δba. Note that the frame-covariant propagator, Δab, and its inverse, Δab-1, are both supersymmetric. In (4.6), the notation {…} denotes the operation of supersymmetrization of all indices enclosed [40], {a1…a2}=1n!∑P(-1)PP[a1…a2],where P runs over all possible permutations of the n indices and the factor (-1)P yields -1 for an odd number of swaps between fermionic indices and +1 otherwise. Note that both (4.5) and (4.6) are superscalars, as one would expect. The presence of the prefactor (-1) in the second term of (4.6) does not spoil covariance, but it is consistent with the convention that only adjacent pairs of indices can be contracted in the standard fashion.
Master equation for all loop orders effective action
While the one- and two-loop order quantum actions, (4.5) and (4.6), can readily be obtained from a perturbative expansion of the implicit equation (4.1), it is still useful to derive a recursive master equation for determining the SG-QEA to all loops. To this end, we follow the 1PI effective action method outlined in [50] by extending it appropriately to incorporate fermionic degrees of freedom while maintaining configuration-space covariance. As we will see below, this method descends from the so-called two-particle-irreducible (2PI) effective action as introduced in [51].
We start by covariantly generalizing the instrumental identity that involves two functional derivatives of the effective action: ΓmaΓmb=-δba,where Γba≡∇→aΓ∇←b and its inverse Γab which is the full frame-covariant Feynman propagator. On the other hand, by making use of the 1PI–2PI equivalence of the respective effective actions at their extrema at the one-loop level and beyond [51], Γab that enters (4.8) can be equivalently calculated through the (n-1)-loop order as [50]Γab≡∑k=0n-1Γ(k)ab=2i∑l=1nΓ(l)δ←δΔab-1,where the covariant inverse propagator Δb-1a is defined after (4.6). Substituting (4.9) in (4.8) and using the identity Δnmδ←δΔab-1=-(-1)m(a+b)ΔnaΔbm,in order to change the variable of functional differentiation from Δb-1a to its inverse Δab by virtue of the chain rule, we eventually arrive at the important relation, 2i∑l=1nΓ(l)δ←δΔab=(-1)ab+bΔm-1bΓmnΔb-1n.Yet, the (n-1)th loop-order term, Γ(n-1)ab (with n>1), may also be written down as a power series expansion of propagators Δab and 1PI self-energies as follows (appropriate covariant index contractions implied): Γ(n-1)ab=[aΔ∑k=1n-1(-1)kΠ(l1)ΔΠ(l2)…ΔΠ(lk)Δ]b,with the constraint l1+l2+⋯+lk=n-1, for all possible combinations of li-loop 1PI self-energies Πab(li)≡Γab(li) (with i=1,2,…,k) that appear on the rhs of (4.12). Notice that we have Γ(0)ab=Δab, at zeroth loop order, or equivalently at tree level.
We may now insert (4.12) into (4.11) and utilize the well-known Jacobi’s formula which is also applicable for superdeterminants, δlnsdetM=str(M-1δM).This in turn gives rise to sdetΔ-1δ←δΔab=-(sdetΔ-1)Δab-1.Then, the following master functional differential equation for evaluating the 1PI SG-QEA to all loop orders can be derived: 2i(Γ(n)δ←δΔab)δΔba=2i(Γ(n)δ←)=str[Δ-1∑k=1n-1(-1)kΔΠ(l1)ΔΠ(l2)Δ…Π(lk)(δΔ)],which holds true for all n≥2, provided l1+l2+⋯+lk=n-1 for all terms of summation in (4.15). For n=1, the rhs of (4.15) equates to str[Δ-1δΔ]. Thus, we get the one-loop SG-QEA given in (4.5) upon functional integration of (4.15) by means of Jacobi’s formula (4.13). In this way, the master equation (4.15) allows for the nth loop-order term of the effective action to be recursively obtained from covariant 1PI self-energies of loop order (n-1) and lower.
As an illustrative example, we will now demonstrate how the two-loop SG-QEA in (4.6) can be evaluated from the master equation (4.15). Thus, we consider the case n=2 in (4.15). This yields 2i(Γ(2)δ←δΔab)δΔba=-(-1)c(cΠ(1)δΔ)c=-(-1)cΠd(1)cδΔdc,where the one-loop proper self-energy can be computed to be Πd(1)c=i2(-1)m[Δmnc;Δm;d-1n+(-1)c(m+n)ΔmnΔm;d-1c;n].It is then easy to verify that the following identity holds true: Δabc;=-(-1)c(a+m)ΔamΔn-1c;mΔnb.Substituting (4.18) in (4.17) and the resulting expression of Πd(1)c back in (4.16), we find 2i(Γ(2)δ←)=-(-1)m+ci2[-(-1)c(m+p)ΔmpΔq-1c;pΔqnΔm;d-1n+(-1)c(m+n)ΔmnΔm;d-1c;n]δΔdc.Proceeding as in [50], we may identify Δab;c-1=Sabc and Δc;d-1a;b=Scdab, and then replace these expressions back in (4.19). Finally, after integrating the rhs of (4.19) functionally over Δdc and some rearranging of superindices [43], we recover (4.6) upon supersymmetrization of indices.
The heat-kernel method and Schwinger’s proper time
The one-loop effective action (4.5) may be computed explicitly in the coordinate or x space, with the help of the well-known heat-kernel method. This method employs Schwinger’s proper time parameter t, which enables one to represent the propagator Δ of a field in the x space, or the logarithm of its propagator, i.e., lnΔ, as an integral over t. The advantage of this method is that the UV divergences in an operator expansion of the effective action show up explicitly as 1/ϵ-poles in the DR scheme, after integrating over t. These UV poles arise from the lower limit of the integration when t→0+. The standard heat-kernel method relies on an iterative solution of a Schrödinger-type equation as powers of t which give all higher-order operators of the one-loop effective action up to a given energy dimension. Here we will only briefly review the standard heat-kernel approach, but the interested reader is encouraged to consult the textbook [52] for more details.
Let us give some basic results from a simple QFT model with a real scalar field ϕ, using the heat-kernel method. The action of the model in d spacetime dimensions is S=∫ddx[12(∂μϕ)2-12m2ϕ2-U(ϕ(x))],where U is a generic scalar potential which is a polynomial of ϕ that contains a mass term, m2ϕ2, and possibly higher-order interactions, λnϕn, when n>2. We may now require that the lowest order heat kernel for this theory be the tree-level inverse propagator, K(x,y)=δ2Sδϕ(x)δϕ(y)=(-∂x2-m2)δ(x-y),where δ(x-y) is the usual Dirac-delta function in d dimensions. For such a theory, the one-loop effective action up to order t3 in the heat-kernel expansion is found to be [52]Γ(1)=i2trln(-∂2-m2-V)=-i2∫ddx∫0∞dtte-m2t(4πt)d/2[Vt+12V2t2+16(V3-(∂μV)2)t3+O(t4)],where the shorthand, V≡U′′=∂2U/∂ϕ2, was used and total derivatives were ignored in (4.22). Observe that the mass parameter m2 acts as an IR regulator for the upper limit of the t integral, t→∞. Upon integration over t, the effective action in DR is given by Γ(1)=-i2∫ddx(4π)d/2[(-m2)(-1+d/2)VΓ(1-d2)+12(-m2)(-4+d)/2V2Γ(2-d2)+16(-m2)(-6+d)/2(V3-(∂μV)2)Γ(3-d2)].In d=4-2ε dimensions, the first two terms up to O(t2) on the rhs of (4.22) exhibit UV divergences, while the last one is UV finite. Instead, when d=2-2ε, only the first term corresponding to O(t) in (4.22) is UV divergent. We note that the same results for the UV poles are obtained by standard diagrammatic methods, for a scalar QFT defined by the action S in (4.20). The explicit analytic form of such UV poles will be given in Sec. V for minimal scalar-fermion QFT models, by employing a more intuitive method based on the Zassenhaus formula which we discuss in the next subsection.
Zassenhaus formula in effective action
In order to determine the heat kernel for a scalar-fermion QFT in a curved field space, one often has to deal with so-called nonminimal operators which have non-Laplacian form [12,53]. In this case, it becomes less obvious how to calculate the polynomial coefficients of the usual heat-kernel expansion in a systematic way. In fact, these heat-kernel coefficients, known as the Hadamard–Minakshisundaram–DeWitt or Seeley–Gilkey coefficients, contain the information about the UV structure of the theory and can be evaluated straightforwardly for theories for which the inverse propagator is minimal. Minimal operators consist of a leading quadratic differential operator, the Laplacian □, plus a lower-order differential operator, F(∂)=-□δAB+P(∂).Here, □≡∂μ∂μ≡∂2 is the Laplacian in d spacetime dimensions, and P(∂) is a polynomial that depends linearly on ∂μ and possibly on other field-dependent terms.
There have been numerous studies [12,53–58], where the nonminimal or the higher-order minimal operators are expanded to have a second-order differential operator as a leading term, which allows one to compute the heat-kernel coefficients even for a curved spacetime. Alternative approaches to the heat-kernel expansion include the worldline formalism [59–61], or employing the Schwinger proper time formalism without recourse to perturbative expansion [62]. These other approaches do not address the main objective of our present work. For definiteness, in addition to recent studies [21,63], our goal is to evaluate the UV structure of the field-space covariant effective action directly in the coordinate space, rather than in the momentum space. Furthermore, we have to deal with the issue of nonminimal heat kernels for covariant scalar-fermion theories which will be discussed in more detail in Secs. VI and VII.
We will now demonstrate how the Zassenhaus formula can be used to derive the heat-kernel coefficients in a more intuitive manner. Let us therefore recall the Zassenhaus formula for two noncommutative operators or matrices, X and Y[64,65]: exp[t(X+Y)]=exp(tX)exp(tY)exp(-t22[X,Y])exp[t36(2[Y,[X,Y]]+[X,[X,Y]])]…Note that (4.25) is akin to the more known Baker—Campbell—Hausdorff formula, but the latter proves to be less useful for a heat-kernel expansion.
For illustration, let us consider the scalar QFT defined in (4.20). In this case, we may consider the operators X^=-∂2-m2 and Y^=-V. In the heat-kernel method, the following operator appears in the one-loop effective action in the x-space representation: ⟨x|e-t(-∂2-m2-V)|x′⟩=∫ddy⟨x|et(∂2+m2)|y⟩⟨y|etVe-t22[∂2,V]et36(2[V,[∂2,V]]+[∂2,[∂2,V]])eO(t4)|x′⟩,where we have used the Zassenhaus formula (4.25) up to order t3, and inserted conveniently the resolution of the unit operator, 1^≡∫ddy|y⟩⟨y|, in the same x-space representation. By doing so, we have deviated from the standard approach by separating the leading heat-kernel factor in (4.26), given by [52]⟨x|e-t(-∂2-m2)|y⟩=e-14t(x-y)2+m2t(4πt)d/2,from the remaining operators that are sandwiched within the expectation value ⟨y|…|x⟩. We should clarify here that unlike in its x-space representation, an operator, like X^ or Y^, does not include a d-dimensional delta function, e.g., ⟨x|X^|y⟩=X(x,y)=(-∂x2-m2)δ(x-y).We should therefore remark that the x-space representation of X^ in (4.28) is fully equivalent to the tree-level inverse propagator K(x,y) in (4.21) which is obtained by taking two functional derivatives of the effective action S with respect to the scalar field ϕ(x).
It is now not difficult to carry out the integral over the y coordinate in (4.26). Up to total spacetime derivatives, we find to order t3, ⟨x|e-t(-∂2-m2-V)|x′⟩=e-14t(x-x′)2+m2t(4πt)d/2[1+Vt+12V2t2+16V3t3][1-t22(∂2V+2(∂μV)∂μ)][1-23(∂μV)2t3],where all spacetime derivatives are with respect to the position x and act as usual from left to right. After subtracting its noninteracting part (V=0), the one-loop effective action Γ(1) may be computed by taking the functional trace of (4.29) and including a prefactor (i/2) according to (4.5), i.e., Γ(1)=-i2∫ddx∫0∞dtt∫ddx′δ(x′-x)⟨x|(e-t(-∂2-m2-V)-e-t(-∂2-m2))|x′⟩.Plugging (4.29) in (4.30) and integrating first over x′ and then over t yields (4.23) to order t3, as it should be expected. It is important to notice that the origin of all heat-kernel coefficients becomes evident when the Zassenhaus formula is at work, namely in effective action. In the next sections, the Zassenhaus method will be employed to compute covariant one-loop effective actions that include both scalars and fermions.
EFFECTIVE ACTIONS IN FLAT FIELD SPACE
In the previous section we discussed an intuitive heat-kernel method based on the Zassenhaus formula which has significantly aided our computation of the one-loop effective action for scalar QFTs. In this section we will present two useful applications of this Zassenhaus method to two archetypal fermionic QFT models. The first model consists of a Dirac fermion ψ that couples to a background real scalar field ϕ through a Yukawa interaction. The second model extends the previous one by including further interactions of a dynamical scalar field ϕ to the Dirac fermion ψ, beyond those contained in the Yukawa sector.
Minimal Yukawa theory
As a warm-up exercise, let us consider a simple Yukawa theory described by the action S=∫ddx[i2(ψ¯(∂ψ)-(∂ψ¯)ψ)-(mf+Y(ϕ))ψ¯ψ].To recast the fermionic kernel into the minimal operator form of (4.24), we utilize the well-known procedure of bosonization. This procedure consists of taking advantage of the symmetric property of the Dirac operator spectrum, so as to effectively “square” the inverse of the fermion propagator, (i∂-mf)(i∂+mf)=-∂μ∂μ-mf2.The above bosonization property is useful, since it can be used to obtain an explicit x-space representation for the fermionic propagator, ⟨x|(i∂-mf)-1|x′⟩=∫ddy⟨x|(i∂+mf)|y⟩⟨y|(-∂μ∂μ-mf2)-1|x′⟩=∫0∞dt(4πt)d/2e-14t(x-x′)2[mf1d-12t(x-x′)μγμ].The linear representation (5.3) of the fermionic propagator turns out to be not so convenient when computing the one-loop effective action through the heat-kernel method. Nonetheless, (5.3) will still be useful to evaluate the contributions to the effective action from mixed scalar-fermion propagators, as we will see in the next subsection.
By virtue of (5.2), the one-loop effective action Γ(1) for this theory may be evaluated as follows: Γ(1)=-ilndet[(i∂-mf-Y(ϕ))(i∂-mf)-1]=-i2trln{[-∂2-(mf+Y(ϕ))2+i∂Y](-∂2-mf2)-1}.Here, the extra minus sign arises due to the closed fermion loop, and the trace is understood to act over both the spinor and coordinate spaces. In (5.4), the effective action Γ(1) is renormalized, such that Γ(1)=0 for vanishing Yukawa interactions, i.e., for Y(ϕ)=0. Moreover, we observe that only an even number of γμ matrices contributes after taking the spinorial trace.
With the aid of Schwinger’s proper-time integral representation, we may write the one-loop effective action Γ(1) in (5.4) as Γ(1)=i2∫0∞dtttr(e-t[-∂2-(mf+Y(ϕ))2+i∂Y]-e-t[-∂2-mf2]).Following the Zassenhaus method outlined in Sec. IV C, the one-loop effective action of (5.5) that contains the UV part of the theory is found to be Γ(1)=i2∫ddxd(4π)d/2{U(-mf2)12(d-2)Γ(1-d2)+12(U2-(∂μY)2)(-mf2)12(d-4)Γ(2-d2)+16[U3-3U(∂μY)2-4(U+mf2)(∂μY)2+(∂μ∂νY)2](-mf2)12(d-6)Γ(3-d2)},where U≡2mfY+Y2. In d=4-2ε, the UV-divergent part of the one-loop effective action is given by ΓUV(1)=i∫ddx18π2[1εmf2U+12ε(U2-(∂μY)2)+finite].Setting now Y=hϕ in (5.7) yields ΓUV(1)=i∫ddx18π2(3εh2ϕ2mf2+2εhϕmf3+12εh4ϕ4+2εh3ϕ3mf-12εh2(∂μϕ)2).Including the kinetic term for the background scalar field ϕ, the Lagrangian containing the UV-divergent counterterms (CTs) associated with this minimal Yukawa model reads as LCT=12δZϕ(∂μϕ)2-12δmb2ϕ2+i(δZψψ¯∂ψ-δZψ¯∂ψ¯ψ)-δmfψ¯ψ-δhψ¯ψϕ-14!δλ4ϕ4-13!δλ3ϕ3-δcϕ,with δmb2=3h2mf24π2ε,δZϕ=h28π2ε,δc=hmf34π2ε,δλ3=3h3mf2π2ε,δλ4=3h42π2ε.Moreover, we have δZψ=δZψ¯=0. This is a consequence of the fact that the real field ϕ is not dynamical in this simple Yukawa setting.
By comparing (5.1) to (5.9), one notices that several new operators have been generated, proportional to δmb2, δZϕ, δc, δλ3, and δλ4. This is because the minimal Yukawa model of (5.1) misses these renormalizable operators at tree level which are not forbidden by some symmetry imposed on the theory. In the next subsection, we will therefore consider a minimal UV-complete QFT model with dynamical scalar and fermion fields.
Minimal scalar-fermion theory
We will now apply the Zassenhaus formula to the heat kernel method by computing the covariant one-loop effective action of a minimal QFT model with one scalar and one Dirac fermion. Specifically, the original theory of interest to us is described by the Lagrangian [40]L=12k(ϕ)∂μϕ∂μϕ-12h(ϕ)ψ¯γμψ∂μϕ+i2g(ϕ)ψ¯γμ∂μψ-i2g(ϕ)∂μψ¯γμψ-Y(ϕ)ψ¯ψ-V(ϕ).Even though this model has a nontrivial supermetric Gab, it turns out that the resulting super-Riemannian tensor vanishes [40–42], implying a flat field space for the theory. As a consequence, the Lagrangian (5.11) can be brought into a canonical Cartesian form after a suitable reparametrization of the fields, L=12(∂μϕ)2+i2(ψ¯∂ψ-∂ψ¯ψ)-Y(ϕ)ψ¯ψ-V(ϕ).More details concerning the analytic form of the required field transformations for this scenario, as well as for a similar class of multifermion models may be found in [41,42].
For our simple scalar-fermion theory defined by the Lagrangian (5.12), the field-space metric is the local 4D metric (2.9), and the covariant inverse propagator Sba has been calculated to be Sba≡SBAδ(xA-xB)=(-∂μ∂μ-Y′′ψ¯ψ-V′′-Y′ψ¯Y′ψTY′ψ¯T0(-i∂+Y)T-Y′ψi∂-Y0)δ(xA-xB),where all spacetime derivatives are with respect to the position xA and the prime (′) indicates a ϕ derivative, e.g., Y′=∂Y(ϕ)/∂ϕ. Notice that for a single scalar field Y=YT.
It proves now more convenient to use the second expression in (2.14) for the computation of the superdeterminant that appears in the one-loop SG-QEA given in (4.5). Hence, partitioning the matrix (5.13) according to the block structure (2.13), the covariant one-loop effective action under evaluation becomes Γ(1)=i2trln{-∂μ∂μ-V′′-Y′′ψ¯ψ-Y′ψ¯(i∂-Y)-1Y′ψ-Y′ψT[(-i∂+Y)T]-1Y′ψ¯T}-i2trln(-∂μ∂μ-i∂Y-Y2).When computing the above expression explicitly, we must pay attention to the fact that the first trace in (5.14) is taken over the position-space only, while the second one is taken over both position and spinorial spaces. There is a new degree of difficulty in evaluating the UV structure from (5.14). This arises from the propagatorlike operator (i∂-Y)-1 sandwiched between fermionic fields. To deal with this extra complexity, we have developed a double heat-kernel technique that makes use of two proper times and a double Schwinger integral. Technical aspects of this method are given in Appendix C.
For the sake of illustration, we consider the renormalizable potential, V=12mb2ϕ2+λ4ϕ4, which corresponds to a massive scalar field. Similarly, we choose a renormalizable Yukawa model function Y(ϕ), i.e., Y=hϕ+mf. As before, the mass terms, mb2 and mf, could be used to act as IR regulators for the integrals over Schwinger’s proper time(s). Having thus specified all interactions of our theory, the one-loop effective action given in (5.14) takes on the form Γ(1)=i2trln(-∂μ∂μ-mb2-M)-i2trln(-∂μ∂μ-mf2-N),where M=λ2ϕ2+h2ψ¯(i∂-hϕ-mf)-1ψ+h2ψT[(i∂+hϕ+mf)T]-1ψ¯T,N=ih∂ϕ+h2ϕ2+2hϕmf.
We should now observe that the second trace term in (5.15) has already been computed in the previous subsection. As for evaluating the first trace, we use the double heat-kernel method outlined in Appendix C. By employing this method along with the Zassenhaus formula, we can determine the complete UV structure of the one-loop effective action for this scalar-fermion theory. The UV-divergent part of the effective action is given by ΓUV(1)=-i2∫ddx1(4π)21ε[ih2(ψ¯∂ψ-∂ψ¯ψ)+2h2(hϕ+mf)ψ¯ψ+λ2mb2ϕ2+λ28ϕ4]+i2∫ddx4(4π)21ε[3h2ϕ2mf2+2hϕmf3-12h2(∂ϕ)2+12h4ϕ4+2h3mfϕ3].Matching this expression to the CT Lagrangian (5.9), the following CTs in the DR scheme may be deduced: δZϕ=h28π2ε,δmb2=-λmb232π2ε+3h2mf24π2ε,δZψ=-h232π2ε=δZψ¯,δmf=-h2mf16π2ε,δh=-h316π2ε,δλ4=-3λ232π2ε+3h42π2ε,δλ3=3h3mf2π2ε,δc=h2mf34π2ε.
Before concluding this section, two remarks are in order. First, the CTs computed from the covariant one-loop effective action for this scalar-fermion theory agree perfectly well with those stated in (5.10) for the minimal Yukawa model, in the two applicable limits: λ→0 and δZψ=δZψ¯→0. Second, there is no need to enforce a bosonization procedure when the one-loop SG-QEA (4.5) is used. Instead, the required bosonization is rather an outcome of the symmetric construction of the supermanifold chart given in (2.1). In the next sections, we will apply the heat-kernel techniques developed here to calculate SG-QEAs for theories with nonzero field-space curvature.
COVARIANT SCALAR EFFECTIVE ACTION WITH CURVATURE
So far, we have analyzed scenarios with vanishing field-space curvature with the aim to showcase our heat-kernel approach using the Zassenhaus formula. Here, we will apply this approach to covariant scalar QFTs, or scalar SG-QFTs, which have no fermions but a nonzero Riemannian curvature in the configuration space.
In the absence of fermions, the field-space tensor kBA that occurs in the Lagrangian (2.3) for this scalar SG-QFT plays the role of the supermetric [7], i.e., Gba≡kABδ(xA-xB). Since covariant field derivatives acting on the supermetric Gba and its inverse Gab vanish, the mixed-ranked tensor Sba is readily determined from the inverse propagator Sba as Sba=GamSbm=-(Dμ)am(Dμ)mb-Rambp(∂μϕm)(∂μϕp)-Uab,where (Dμ)ab was defined in (3.1) and Uab≡U;a;b. For such a theory with bosons only, the leading differential operator within Sba contains a Laplacian, and so it has the minimal form of (4.24). We may therefore represent the configuration-space rank-2 tensor (Dμ)ab in a Hilbert space of single off shell states as follows: (Dμ)ab=(δAB∂→∂xAμ+ΓABM[ϕ(xA)]∂μϕM(xA))δ(xA-xB)=⟨xA|(Dμ)AB|xB⟩,with (Dμ)AB=δAB∂→∂xμ+ΓABM[ϕ(x^)]∂μϕM(x^).Evidently, (Dμ)AB is a field-dependent differential operator and acts as usual from left to right on a single-state Hilbert space, which obeys the following properties: ⟨xA|xB⟩=δ(xA-xB),⟨p|x⟩=eipμxμ,x^μ|xA⟩=xAμ|xA⟩,p^μ=i∂→∂xμ.On the basis of these properties, it is easy to verify the validity of (6.2). It is important to stress here that the representation of the differential operator in (6.3) is fully consistent with our earlier description of such operators given in (4.28). Therefore, as a consequence of this representation, the product of two such operators can be written equivalently in several different ways as (Dμ)am(Dμ)mb=∫xM⟨xA|(Dμ)AM|xM⟩⟨xM|(Dμ)MB|xB⟩=⟨xA|(Dμ)AM(Dμ)MB|xB⟩.In the above, we abbreviated a spacetime integral as, ∫xM≡∫ddxM, which will be adopted from now on.
In a curved field space, the operator in (6.5) represents the covariant field-space Laplacian. By making use of the representation (3.4), it can be expressed through field-space vielbeins in terms of the local-frame Laplacian (D^2)A^B^≡(Dμ)A^M^(Dμ)M^B^ [which contains the ordinary Laplacian (∂2)A^B^≡δA^B^∂2] as (D2)AB≡(Dμ)AM(Dμ)MB=eAA^(D^2)A^B^B^eBsT.
Similarly, with the help of field-space vielbeins, one can write higher powers of D2, (D2)n (with n≥1), in terms of (D^2)n. Taking all these properties into account, the covariant heat kernel of D2 is found to be ⟨xA|etD2|xB⟩=⟨xA|eAA^(x^)A^(etD^2)B^eBsTB^(x^)|xB⟩=KABe-(xA-xB)2/4t(4πt)d/2+WAB,where KAB≡eAA^(xA)eBsTA^(xB),and WAB is a remainder that depends on the field-space spin connection given in (3.5). Further technical details are given in Appendix B. For simplicity, we do not consider contributions from the spin connection in this work, but we set WAB=0. This could be partially justified by the fact that thanks to the first equality in (6.4), WAB→0 in the UV limit for the proper time t, corresponding to t→0 in (6.7). In the coincident limit xA→xB, we have KAB→δAB, as a consequence of (2.8). Thus, (6.7) reduces to the well-known result of the heat kernel in the local frame. Also, it is not difficult to show that the covariant heat kernel in (6.7) is a solution to the covariant heat diffusion equation, ∂∂t⟨xA|etD2|xB⟩=⟨xA|D2etD2|xB⟩=⟨xA|etD2D2|xB⟩.Since we are interested in the UV structure of the covariant effective action in this work, we will only write down the leading field-independent term in (6.7) proportional to δAB, which dominates in the coincident limit: xA→xB. In other words, we omit the covariant field-space tensor KAB in (6.8) in all our covariant computations from now on, unless its explicit use is required. We only note that the presence of KAB allows us to easily switch frames, i.e., from a global to local frame and vice versa, in order to maintain global covariance in all analytic results that we will present in this and the next sections.
To facilitate our computation that follows below, we introduce the field-space matrix shorthands, R(∂ϕ)2≡RAMBP(∂μϕM)(∂μϕP),U≡UAB.We may now employ the Zassenhaus formula in the heat kernel to order t2 in the exponent and express the one-loop SG-QEA stated in (4.5) in the following way: Γ(1)[ϕ]=tr(lnSab[ϕ]-lnSab[0])=-i2∫xA,xB∫0∞dtte-(xA-xB)2/4t(4πt)d/2⟨xB|(et(R(∂ϕ)2+U)e-t22[D2,R(∂ϕ)2+U]-1)|xA⟩,where we now trace not only over position but also over field-space indices. To evaluate explicitly the covariant effective action Γ(1) in (6.11), only the leading term of the kernel (6.7) was taken into account, while all other terms are left generally covariant. This ensures that field-space covariance is maintained in all our subsequent operations.
We may now compute the commutators that occur in (6.11) in the standard fashion, e.g., [D2,R(∂ϕ)2+U]=D2(R(∂ϕ)2+U)-(R(∂ϕ)2+U)D2.Given that in the DR scheme the UV divergences emanate from O(t) and O(t2) terms in d=4-2ε dimensions, we expand the exponentials on the rhs of (6.11) through this order to obtain Γ(1)=-i2∫xA,xB∫0∞dtte-(xA-xB)2/4t(4πt)d/2[t(RMN∂νϕM∂νϕN+UAA)+t22UAMUMA+t22(RAMPN∂νϕM∂νϕNUPA+UAPRPMAN∂νϕM∂νϕN)+t22RAMPN∂μϕM∂μϕNRPSAT∂νϕS∂νϕT]δ(xB-xA)+ΔΓ(1),with ΔΓ(1)=-i2∫xA,xB∫0∞dtt⟨xA|etD2|xB⟩t22⟨xB|(RAMPN∂νϕM∂νϕN+UAP)(Dμ)PC(Dμ)CA-(Dμ)AC(Dμ)CP(RPMAN∂νϕM∂νϕN+UPA)|xA⟩.In the covariant one-loop effective action Γ(1) in (6.13), we have isolated an extra contribution, ΔΓ(1) given in (6.14), which contains derivatives of δ(xB-xA). However, these extra terms cancel against each other, after exploiting the property of D2 in the diffusion equation (6.9) under the configuration-space trace. In fact, it is easy to see that D2 is acting on the heat kernel from both its right and left side in (6.14), but with an extra negative sign arising from the commutator in (6.12). As a consequence, ΔΓ(1) vanishes identically, i.e., ΔΓ(1)=0.
We now integrate over t using the integral identity ∫0∞dtte-(xA-xB)2/4t(4πt)d/2tn=2-2nπ-d/2[(xA-xB)2]n-d/2Γ(d2-n),where n∈Z, and then employ the results from Appendix D to carry out the integration directly in the coordinate space in d=4-2ε dimensions in the DR scheme. Excluding ΔΓ(1), (6.13) now reads as ΓUV(1)=-i16π2∫xA,xB[(xA-xB)2]-1+ε(RMN∂νϕM∂νϕN+UAA)δ(xB-xA)+i64π2ε∫xA,xB[(xA-xB)2]ε{UAMUMA+RAMPN∂μϕM∂μϕNRPSAT∂νϕS∂νϕT+RAMPN∂νϕM∂νϕNUPA+UAPRPMAN∂νϕM∂νϕN}δ(xB-xA).As explained in detail in Appendix D, because of the scaling factor [(xA-xB)2]-1+ε in (6.16), the term proportional to the field-space Ricci tensor RMN and the double derivative of the potential UAA vanishes. The second integral on the rhs of (6.16) does not involve any differential operators and possesses the correct contributing factor [(xA-xB)2]ε. Thus, the only nontrivial UV structure of the covariant effective action is given by ΓUV(1)=-164π2ε∫xA(UAMUMA+RAMBN∂μϕM∂μϕNUBA+UABRBMAN∂μϕM∂μϕN+RAMBN∂μϕM∂μϕNRBSAT∂νϕS∂νϕT).The result for ΓUV(1) in (6.17) has already been presented in the literature [25,37,66] and more recently in [21,67], through more computationally extensive methods. Thus, this exercise reaffirms not only the validity of our approach, but its efficiency as well. The authors of [21,67] also presented a general expression for the divergent one-loop contribution in d=4-2ε for quadratic actions involving scalar and gauge fields. This result was first derived by ’t Hooft in [44] and was extended by [21,67] to ensure field-space covariance.
We end this section by offering a few comments on the efficiency of our heat-kernel method. This may be exemplified by noticing that by formally setting V→R(∂ϕ)2+U in the heat kernel given in (4.26), we reproduce the full covariant structure of the scalar effective action with curvature in (6.11). Interestingly enough, we find that the one-loop SG-QEA for the model under consideration has no further UV poles from higher-order EFT operators beyond those listed in (6.17). This means that the UV infinities in this nonrenormalizable theory have been organized in a finite number of EFT operators. This should be contrasted with ordinary formulations of QFT models with nonrenormalizable derivative interactions, for which one gets an infinite number of UV-divergent EFT operators already at the one-loop level. We may even conjecture that this reorganization principle of the finite number of UV infinities will persist beyond the one-loop order, even though new UV-divergent EFT operators will emerge at each higher loop order. We believe that this is a unique feature of such a geometric approach to QFT.
COVARIANT ONE-LOOP FERMIONIC EFFECTIVE ACTION
In this section we consider the minimal SG-QFT model introduced in [41,42] that features a nontrivial fermionic curvature both in two and four spacetime dimensions. After we have briefly described this minimal model, we will compute its one-loop SG-QEAs by generalizing the concept of the Clifford algebra to field space. The so-generalized field-space algebra will help us to bosonize the fermionic part of the effective action, similar to what one usually does for ordinary fermionic theories without curvature (cf. Sec. V A).
Minimal fermionic model with field-space curvature
Let us consider a minimal SG-QFT model with nonzero fermionic curvature, which includes one scalar field ϕ and one Dirac fermion, represented by a column vector ψ, in d spacetime dimensions. The Lagrangian of this minimal model is given by [41,42]L=12k(ϕ)(∂μϕ)(∂μϕ)+i2(g0+g1ψ¯ψ)[ψ¯γμ(∂μψ)-(∂μψ¯)γμψ]-U,where g0,1 are constants and γμ are the Dirac matrices, satisfying the Clifford algebra {γμ,γν}=2ημν1d,with ημν=diag(1,-1d-1). In addition, U=U(Φ) is the zero-grading scalar potential that includes the Yukawa and possibly interactions of higher powers in the chart components of Φ=(ϕ,ψT,ψ¯)T. Note that the model function ζaμ derived from (7.1) takes on the factorizable form ζaμ=ζbΓaμb, where ζa={0,(g0+g1ψ¯ψ)ψ¯,(g0+g1ψ¯ψ)ψT}and Γμ={Γaμb} is defined after (2.5). Using the method of [40] briefly outlined in Sec. II, we may derive the field-space supermetric G={Gab} in the superspace of Φ, G=(k0000fT0-f0),with f=(g0+g1ψ¯ψ)1d+g1ψψ¯. From (7.4), it is obvious that the scalar and fermionic fields form two irreducible supermanifold subspaces in this minimal SG-QFT model of (7.1). In this section, we will only focus on the fermionic part of the one-loop effective action, since its scalar part has already been analyzed in previous sections.
With this latter simplification, we may compute a key quantity by virtue of (3.7), that is the mixed Lorentz and field-space tensor λabμ, λbμa=g0(0γμbaγμab0)-g12(ψ¯a(ψ¯γμ)b+ψ¯b(ψ¯γμ)aψ¯a(γμψ)b+(ψ¯γμ)aψb-2(ψ¯ψ)γμbaψ¯b(γμψ)a+(ψ¯γμ)bψa-2(ψ¯ψ)γμab-ψ¯a(ψ¯γμ)b-ψ¯b(ψ¯γμ)a).As we will see in the next subsection, the mixed tensor λabμ will be an essential building block, in order to construct a field-space generalized Clifford algebra.
Field-space generalized Clifford algebra and bosonization
In order to calculate the fermionic part of the one-loop SG-QEA given in (4.5), we ignore in the covariant inverse propagator Sba in (5.13) all terms that depend on the model function knm, i.e., k for the minimal fermionic model of interest to us. Thus, by means of the supermetric (7.4) appropriately reduced to the fermionic subspace, or otherwise by field-space covariance, it is not difficult to write down the mixed-rank inverse propagator [41,42], Sba=λmμa(iDμ)mb+(-1)bmλm;bμai∂μΦm-Uba,which enters the one-loop effective action.
By simple comparison of (7.6) with the inverse of the flat-space fermion propagator in (5.3), it is easy to see that λabμ will be the analog of a γμ matrix. However, in order to bosonize the action as done in (5.2) but for the minimal model at hand, we introduce another set of such tensors, denoted below as λ¯bμa, which are chosen so as to obey the Clifford algebra: λmμaλ¯bνm+λamνλ¯bμm=2ημνδba.If we use the matrix representation in the fermionic field space λμ≡{λbμa}, then λ¯μ≡{λ¯bμa} turn out to be their adjugate or inverse matrices, i.e., λ¯μ={(λ0)-1,-(λi)-1},where the index i labels the d-1 spatial dimensions of the theory. Hence, the so-constructed field-dependent quantities, λμ and λ¯μ, are analogous to the usual four-vector matrix expressions, σμ=(12,σ) and σ¯μ=(12,-σ), which are defined through the Pauli matrices and obey the same Clifford algebra as in (7.7) in four dimensions.
We may now utilize the Clifford algebra to compute the one-loop effective action through the process of bosonization. To this end, the following important property was found to hold for the minimal SG-QFT model under consideration: sdet(λamμ(iDμ)mb)=sdet(λ¯mνa(iDν)mb)=(p2)d+f(ψ,ψ¯),where f(ψ,ψ¯) is some field space and Lorentz invariant function involving fermion fields and their derivatives, so that f(0,0)=0. Moreover, the differential operator (aDμ)b [cf. (6.2)] satisfies the property (aDμ)m(mDν)b=(-1)b(a+1)(bDν)m(mDμ)a.To ease our computation, we introduce the shorthands, λμDμ≡λmμa(Dμ)mb,V≡Vba≡(-1)bmλm;bμai∂μΦm-Uba.With the above abbreviations, the one-loop SG-QEA Γ(1) can be successively written as follows: Γ(1)=i2lnsdet(Sab)=i2lnsdet{(iλμDμ)[1+(iλμDμ)-1V]}=i2lnsdet(iλμDμ)+i2lnsdet[1+(iλμDμ)-1V].The last equation on the rhs of (7.12) will be used to compute the covariant one-loop effective action Γ(1).
Like in the minimal Yukawa model in Sec. V A, we employ (7.9) to bosonize the fermionic kernel in (7.12) as lnsdet(iλμDμ)=12lnsdet(-λμDμλ¯νDν).By virtue of the Clifford algebra (7.7), the differential operator λμDμλ¯νDν can then be written as λμDμλ¯νDν=D2+λμ[Dμ,λ¯ν]Dν+Σμν[Dμ,Dν],with Σμν≡14(λμλ¯ν-λνλ¯μ). As a consequence, λμDμλ¯νDν contains the covariant field-space Laplacian D2 defined in (6.6) and subleading linear differential operators proportional to ∂μ or ∂ν. As before, we expand the covariant expression in (7.13) about the leading heat kernel D2 which will enable us to determine the UV structure of the theory. With this aim in mind, we may rewrite (7.13) as 12lnsdet(-λμDμλ¯νDν)=12lnsdet(-D2-A),where we defined the subleading differential operator in (7.14) as A≡λμ[Dμ,λ¯ν]Dν+M.Note that the above expression contains a magnetic-moment-type operator M which is generated by a nonzero field-space curvature. More explicitly, in the field space, the operator M is defined as MAB≡(AΣμνRμν)B,where (ARμν)B≡[ADμ,Dν]B=(-1)MNRABMN∂μΦM∂νΦNis a differential two-form rank-2 field-space tensor. In spite of its similarity in notation, Rμν, which is sourced from a nonzero curvature on the field space, differs crucially from the mixed rank-2 spacetime Riemann tensor reported in [12], as the latter can only arise if spacetime possesses a nonzero Riemannian curvature.
We should stress here that the generation of the magnetic-moment operator M is a novel and distinct feature of SG-QFTs with nonzero fermionic curvature. We observe that the operator M describes magnetic-moment transitions in a fermionic system, potentially leading to interesting phenomenological implications which can be analyzed elsewhere.
We may now put everything together to obtain a computationally amenable form for the SG-QEA in (7.12), i.e., Γ(1)=-i4trln(-D2-A)-i2trln[1+(-iλμDμ)-1V].Since the supertrace acts on a fermionic subspace only, it can be converted to a conventional trace by multiplying it with a minus sign. Finally, the last term in (7.19) can be expanded in powers of the operator (iλμDμ)-1. In particular, with the aid of the Clifford algebra (7.7), we may express (iλμDμ)-1 in terms of D2 as (iλμDμ)-1=iλ¯μDμ(-D2-A)-1,by which the second operator (-D2-A)-1 can be more easily represented by a Schwinger proper-time integral.
We are now in a position to explicitly compute the UV structure of the SG-QEA for this theory up to fourth order in spacetime derivatives. To do so, we expand the rhs of (7.19) to second order in the Schwinger’s parameters t and s, Γ(1)=i4tr(I1-2I2-I3+…),where I1=∫xA,xB∫0∞dtt⟨xA|etD2|xB⟩⟨xB|(1+tA+t22B)|xA⟩≡I10+I1A+I1B,I2=∫xA,xB,xC∫0∞dt⟨xA|iλ¯μDμ|xB⟩⟨xB|etD2|xC⟩⟨xC|(1+tA+t22B)V|xA⟩,≡I20+I2A+I2BI3=∫xA,xB,xCxD,xE,xF∫0∞dt⟨xA|iλ¯μDμ|xB⟩⟨xB|etD2|xC⟩⟨xC|(1+tA+t22B)V|xD⟩×∫0∞ds⟨xD|iλ¯νDν|xE⟩⟨xE|etD2|xF⟩⟨xF|(1+sA+s22B)V|xA⟩,and B≡A2-[D2,A]. The integral I1 comes from the first trace term in (7.19), while I2 and I3 arise from the second trace term in (7.19) after we perform a logarithmic expansion and keep only powers of (-iλμDμ)-1V up to first and second order, respectively. We ignore further contributions to Γ(1) that arise from terms beyond the second order in t and s. However, as we will outline below, only a subset of these enter the UV structure of the theory. To better organize the exposition of our results, we have decomposed the sum of integrals within I1,2, in groups that have no dependence on the operators A and B and those that do, i.e., I1,2≡I1,20+I1,2A+I1,2B. Finally, we note that the field-space matrices λμ and λ¯μ [as defined after (7.7)] obey similar trace identities like the γμ matrices, e.g., the trace of an odd number of λμ or λ¯μ matrices vanishes.
To gain valuable insight into our analytical computation, we will first consider the leading term in the covariant heat-kernel expansion (6.7). After performing the integration over Schwinger’s proper time t using the integral identity (6.15), the integrals I1,2 given in (7.22) and (7.23) become I1=∫xA,xB⟨xB|(1π2|xA-xB|-4+2ε+14π2|xA-xB|-2+2εA-12116π2ε|xA-xB|2εB)|xA⟩,I2=∫xA,xB,xC⟨xA|iλ¯μDμ|xB⟩⟨xC|(14π2|xB-xC|-2+2ε-116π2ε|xB-xC|2εA+12164π2ε|xB-xC|2+2εB)V|xA⟩,where |x|=[(x)2]1/2.
From our analysis in Appendix D, we know that the only terms contributing to the UV poles of I1 and I2, which were obtained by a single integration over the Schwinger proper time t, are those scaling as |xA-xB|2ε. These terms can also appear, for example, when a higher order term, proportional to |xA-xB|2+2ε, happens to be differentiated twice. In view of this observation, it is easy to see that in the integral I1 in (7.25), the only potentially nonvanishing term is the one with the B operator, i.e., I1B, provided no derivative acts on |xA-xM|2ε. Instead, we have (I10,A)UV=0. With this information in mind, we once again turn our attention to the expression (7.22) with the full covariant heat kernel and write explicitly the functional form for the B operator as I1B=∫xA,xB∫0∞dttt22⟨xA|etD2|xB⟩⟨xB|12[(DμNβ)+2Σμν(DνNβ)](-Rμβ+{Dμ,Dβ})+12λμDμ(λ¯νNβ)(-Rνβ+{Dν,Dβ})+(D2)2+2MD2+λμDμ[λ¯ν(DνM)]|xA⟩,where Nμ≡λα(Dαλ¯μ) and Rμν is defined in (7.18). We make use of the covariant heat diffusion equation (6.9) in order to derive the following set of integral identities: ∫0∞dttt22⟨xA|D2etD2|xB⟩δ(xA-xB)∼1ϵIR,∫0∞dttt22⟨xA|(D2)2etD2|xB⟩δ(xA-xB)=0,where ϵIR denotes the infrared pole in the DR scheme. The covariant identities listed above are consistent with our consideration on the scaling factor |xA-xB|2ε and its derivatives, as explicitly discussed in Appendix D. Hence, upon integration over the remaining spacetime variable xB by means of the δ function, in the DR scheme the UV-divergent part of the integral I1 is (I1B)UV=164π2ε∫xA{A[Dμ(λα(Dαλ¯β))+2ΣμνDν(λα(Dαλ¯β))]Rμβ-2λμDμ(λ¯ν(DνM))+λμDμ(λ¯νλα(Dαλ¯β))Rνβ}A,whilst it is (I10,A)UV=0 as argued above. Observe that there are new fermionic EFT operators emerging as a consequence of a nonzero field-space curvature. In fact, all terms in (7.29) vanish in the flat field-space limit where λμ→γμ and any fermionic field dependence disappears.
The integral I2 in (7.26) differs crucially from the integral I1. It has an extra differential operator Dμ and the potential-like term V which contains the usual potential U and a term that depends on a covariant field-space derivative acting on the λμ matrix. Since the operators A and B both have an even number of λμ matrices, the term involving U vanishes since it involves the trace of an odd number of λμ matrices. The term depending on the operator A, denoted as I2A, is accompanied with the proper scaling factor, |xB-xC|2ε, leading to the following covariant UV-divergent contribution: (I2A)UV=i32π2ε∫xA[A(DμV)λ¯νRμν+λμDμ(λ¯νV)λ¯αRνα+2Σμν(DνV)λ¯αRμα]A.The term with the B operator gives a UV-divergent contribution when acting on the covariant derivative of λμ: (I2B)UV=i64π2ε∫xA{AλμDμ[λ¯νλαDα(λ¯βDβV)]λ¯ν+λμDμ[λ¯νDν(λαDα(λ¯βV))]λ¯β+Dμ[λνDν(λ¯αDαV)]λ¯μ+2ΣμνDν[λαDα(λ¯βDβV)]λ¯μ+λμDμ[λ¯νDν(λαλ¯βDβV)]λ¯α}A.As in (7.29), we obtain new EFT operators that vanish in the flat field-space limit.
Let us finally turn our attention to the integral I3 in (7.32). After carrying out two integrations over the Schwinger’s proper times t and s, I3 can be written as I3=∫xA,xB,xCxD,xE,xF⟨xA|iλ¯μDμ|xB⟩⟨xC|(14π2|xB-xC|-2+2ε-116π2ε|xB-xC|2εA+12164π2ε|xB-xC|2+2εB)V|xD⟩⟨xD|iλ¯νDν|xE⟩⟨xF|(14π2|xE-xF|-2+2ε-116π2ε|xE-xF|2εA+12164π2ε|xE-xF|2+2εB)V|xA⟩.As a consequence of the double insertion of the heat kernel, we get different forms of spacetime integrands that do not involve a residual δ function, like δ(xA-xB), in the last step of integration over xB. The complete set of all terms that occur in I3 is quite lengthy to list them all here. But, in terms of the spacetime integrations involved and the number of Dμ derivatives, they can be organized as follows: (i) one integral with at least two derivatives, (ii) two integrals with four and eight derivatives, (iii) three integrals with six derivatives, and (iv) one integral with at least ten derivatives. The terms that feature at least two derivatives give rise to a UV-divergent contribution to I3, which was computed to be (I3)UV=-132π2ε∫xA[Aλ¯μVλ¯μ(D2V)+(D2λ¯μ)Vλ¯μV+2(Dνλ¯μ)Vλ¯μ(DνV)]A.Notice that only the first term in the integrand survives in the flat field-space limit which stems from the wave function renormalization of the scalar fields in the theory. Instead, the other two terms in the integrand of (7.33) represent new EFT operators that can only emerge from a curved fermionic field space.
In addition to the above EFT interactions, other UV-divergent EFT operators with at least six spacetime covariant derivatives can also be generated. These sixth-order EFT operators can be worked out in a methodical and systematic way within our approach. But listing all these EFT operators is an exercise that goes beyond the scope of the present work. Nonetheless, it is important to stress here that the one-loop SG-QEA for the model under consideration has no further UV poles beyond the sixth order of spacetime derivatives. Hence, exactly as was the case for the scalar QFT model discussed in Sec. VI, we observe that the UV infinities in this nonrenormalizable fermionic theory have reorganized themselves in a finite number of EFT operators, as opposed to ordinary formulations of QFT models with nonrenormalizable derivative interactions, for which an infinite number of UV-divergent EFT operators get generated at the one-loop level. Consequently, this unique feature seems to persist, at least within the framework of minimal SG-QFTs with nonzero fermionic curvature.
Beyond the minimal supergeometric framework
In this and previous sections, we have been analyzing minimal SG-QFTs that feature either a scalar or a fermionic curvature, without including gauge interactions. Going beyond this minimal framework is not a trivial task, as the complexity of the problem increases drastically. For this reason, we will only briefly outline potential approaches that could be followed to compute SG-QEAs for specific nonminimal scenarios.
If a SG-QFT contains both scalar and fermionic fields but without scalar-fermion mixing in the kinetic terms of the covariant inverse propagator Sba given in (3.9), one may then still be able to use the techniques that we have developed in this paper. In this case, the grading-one (fermionic) part of Sba can be bosonized independently from the grading-zero (scalar) part. Specifically, the Laplacian describing the scalar kinetic term may be written in the local frame as (Dμ)amknm(Dμ)nb=ea^a(a^D^μ)m^kn^m^(n^D^μ)b^ebsTb^,where kn^m^ is a diagonal matrix in its scalar block entry, that is {kN^M^}=diag(1N,02dM). The latter is a consequence of the approach introduced in [19] to determine the field-space vielbeins eB^A and the local supermetric HB^A^ (see also our discussion in Sec. II). As discussed in Sec. VI (with more details given in Appendix B), this would lead to an obvious modification of the leading term in the heat kernel in (6.7), where the prefactor KAB in (6.8) will now read as KBA=eA^A(xA)kB^A^eBsTA^(xB).Observe that in the coincident limit xA→xB, we have KBA→kAB, which is in general field dependent, and it does no longer equate to δAB.
If the fermionic kinetic term happens to have a nonfactorizable form for ζaμ, i.e., ζaμ≠ζbΓaμb, we can always find the closest minimal SG-QFT model, for which ζaμ=ζbΓaμb, so that its kinetic term can be bosonized using the Clifford algebra of the λμ’s in (7.7). Then, the SG-QEA for such a nonminimal model will consist in computing deviations from the minimal one, through the linear decomposition: λ˜bμa=λabμ+nabμ.Evidently, this approach inevitably leads to the introduction of an extra covariant mixed tensor nabμ which acts as a remainder of λ˜bμa. Unlike λabμ, the field-dependent tensors, λabμ and nabμ, do not satisfy in general the Clifford algebra. It should be clarified here that the role of the remainder tensor nabμ will be to induce new covariant EFT operators beyond those obtained in the minimal SG-QFT model discussed in Sec. VII B.
We note that SG-QFTs may also include gauge fields as extra bosonic chart variables following the standard VDW formalism [11,68]. In a gauge-invariant extension of an SG-QFT, the new aspect is that the generators of the gauge group become the Killing vectors of the field-space manifold (e.g., see [30] and references therein). To what extent this property can carry over to supermanifolds is an open question that would require further investigation in another dedicated study. In this respect, we find the earlier works which considered the application of the VDW formalism to Yang–Mills theories [11,48,68] and the SM [69] encouraging, but on flat field-space manifold. They have shown that one can define a projected field-space metric, such that the resulting VDW effective action is independent of the gauge-fixing condition through two-loop order. We therefore envisage being able to employ the techniques developed in these previous works, in order to construct the projected field-space metric in a unique manner.
We should reiterate here that for the nonrenormalizable extensions of SG-QFTs discussed above and in the previous section, the metric of the field space can unambiguously be determined from the classical action S as outlined in Sec. II. This can be done in a systematic and self-consistent manner by keeping track of the powers of ℏn, e.g., as ℏnΓ(n)[Φ], where n=0,1,2,… indicates the loop order of the 1PI SG-QEA, according to our discussion in Sec. IV A. For n=0, we have S[Φ]=Γ(0)[Φ], and so taking the limit ℏ→0 for determining the field-space metric should be considered to be smooth in the 1PI formulation of the quantum effective action. Moreover, any CTs of renormalization needed to get rid of the UV infinities in the one-loop SG-QEA, which we have computed in this study, should be added to Γ(0)[Φ], but with an extra factor ℏ, in line with the earlier work in [13]. One may even require that the classical form of the field-space metric be preserved at some reference point μ of the RG scale. This is exactly the renormalization scheme that we have tacitly adopted in all our computations.
We conclude this section by commenting on the issue of multiplicative anomalies which usually haunt algebraic manipulations with matrices of infinite dimensionality and infinite trace. We have encountered such manipulations in the process of bosonization of the effective action, before we applied the Schwinger–DeWitt heat-kernel technique. Specifically, for two infinite dimensional matrices or operators, e.g., A and B, the following expression, a(A,B)=lndetAB-lndetA-lndetB,does not need to vanish in general. Only for trace class operators (which possess a finite trace), one can show that their trace is independent of the chosen orthonormal basis, obeying the desirable algebraic constraint: a(A,B)=0[70]. However, as demonstrated in [71] within simple QFT examples, multiplicative anomalies do not encode any novel physics but they merely illustrate potential ambiguities in the Schwinger proper time regularization scheme, especially for mass-dependent regularization methods, such as the ζ-function regularization scheme. For this reason, all our computations were done in the massless DR scheme. Moreover, when an IR mass regulator was introduced, we have always verified that all our results agree with well-known results in the flat field-space limit which were derived independently through diagrammatic methods. In this way, potential ambiguities that might originate from multiplicative anomalies were kept under good control.
CONCLUSIONS
Supergeometry in quantum field theory offers a new concept in going beyond the standard supersymmetric framework. Unlike conventional supersymmetric theories, SG-QFTs treat scalars and fermions on equal footing, as independent field variables, and as such, they permit for general scalar-fermion field transformations on the configuration space of a supermanifold. In particular, SG-QFTs do not require equality of the degrees of freedom between bosons and fermions, at least in their minimal setting. Clearly, an on shell S-matrix amplitude is usually field-reparametrization independent. Nonetheless, the diffeomorphic structure of the supergeometric quantum effective action reduces the appearance of independent UV divergences that occur in the emergent covariant EFT operators in the configuration space. Consequently, the computation of the SG-QEA, along with its covariant UV structure at the one-loop order, was one of the central objectives of the present work.
Before embarking on the calculation of the SG-QEA, we briefly reviewed the old covariant field-space formalism [6] for bosonic theories, including other more recent developments within the context of the geometric SMEFT [21–38]. To go beyond the tree level, we devised a new approach to the Schwinger–DeWitt heat-kernel technique based on the Zassenhaus formula where the emergence of all possible covariant EFT operators can be systematically tracked. For a given SG-QFT, our intuitive approach provides a comprehensive account of all possible covariant EFT operators that can occur in the configuration space, unlike other more laborious methods that utilize ’t Hooft’s matching procedure. Hence, our covariantly improved heat-kernel method enables one to identify all covariant EFT operators in a consistent and systematic manner.
The aforementioned covariant heat-kernel approach was further extended to include SG-QFTs that feature nonzero fermionic curvature in two and four spacetime dimensions. In order to compute the resulting SG-QEA by the process of bosonization, we had to introduce a novel Clifford algebra which is obeyed by a set of field-dependent γ matrices, denoted as λμ, and their adjugate counterparts, λ¯μ. In this respect, we have attempted to minimize the impact of potential ambiguities due to the so-called multiplicative anomalies by matching the UV divergences of the one-loop SG-QEA to those obtained in the flat field-space limit of the theory. In this way, we find that the EFT interactions resulting from the one-loop SG-QEA are manifestly diffeomorphically invariant in configuration and field space alike. Although our explicit computations have been restricted to the one-loop level, it is important to note that the extension of our approach to evaluating higher loops can in principle proceed recursively through the master equation derived in (4.15).
The present study opens up new horizons for further theoretical and phenomenological exploration within the framework of SG-QFTs. One obvious direction will be to include gauge and gravitational interactions to chiral fermions which could lead to a complete geometrization of realistic theories of microcosmos, such as the SM and its gravitational sector. Furthermore, one might consider including additional global or local symmetries in SG-QFTs which will act as isometries [10,11] on the supermanifold [43]. In this way, one may be tempted to investigate whether the famous S-matrix theorem by Haag et al.[72] could be evaded by defining off shell less restrictive forms of scalar-fermion symmetries in a SG-QFT which does not rely on ordinary supersymmetry.
An important advantage of a Riemannian geometry that we have adopted in this work for the field space over other Finslerian-type geometries emanates from their powerful embedding theorems. These embedding theorems due to Nash and Kuiper (NK) [73–76] ensure that a curved manifold can always be isometrically embedded into a Euclidean (flat) field space of sufficiently higher dimensionality. The NK theorems hold not only for compact spaces, but also for noncompact hyperbolic spaces, like the field space of the minimal Hilbert–Einstein action [19], in which the field-space Ricci scalar turns out to be negative. It is crucial to emphasize here that such embeddings lead to renormalizable QFTs, at least as far as their kinetic sector is concerned. In fact, the famous O(N) nonlinear chiral model [6,77] and the embedding of its compact curved field-space manifold into a Euclidean field space by including the σ, or the Higgs field, as an extra dimension along the radial direction, constitutes one of the classic examples of UV completion (e.g., see discussion of Sec. V. C of [78]). On the other hand, the NK theorems do not provide a precise recipe on how to explicitly construct the mapping function for such topological embeddings, even though their existence is guaranteed. Nonetheless, it is interesting to note that although in their infancy, studies of such embeddings of supermanifolds into higher-dimensional Euclidean-type, orthosymplectic geometries [43] do exist and this topic remains an active field of research by the mathematics community [79–81].
From a more phenomenological perspective, one may envisage that SG-QFTs offer a new portal to dark-sector dynamics. For instance, dark-sector fermions through their nonlinear dynamics can change the dispersion properties of weakly interacting particles, like SM neutrinos and axions. In particular, the emergence of the new field-space-induced magnetic-moment operator M, as defined in (7.17), can induce transitions between fermions with interesting phenomenological consequences. Likewise, it would be interesting to go beyond the minimal fermionic setting that we have studied here and investigate the dynamics of nonminimal SG-QFTs as well. We plan to explore some of the above ideas in future works.
ACKNOWLEDGMENTS
We thank Fedor Bezrukov for enlightening discussions. The work of A. P. is supported in part by the STFC Research Grant No. ST/X00077X/1. V. G. acknowledges support by the University of Manchester through the President’s Doctoral Scholar Award.
HIGHER-POINT COVARIANT INTERACTIONS
In deriving higher-point covariant interactions, like Sabc…, the identity given in (3.2) by Ecker and Honerkamp (EH) [7] plays an instrumental role. Since we have not seen, to the best of our knowledge, a sufficiently detailed proof of the EH identity in the literature, we give in Sec. A 1 of this appendix all intermediate steps. In particular, we show how the EH identity can be generalized to a supermanifold that includes both bosonic and fermionic field coordinates. Then, in Sec. A 2 we apply this identity to obtain the four-point correlation function Sabcd.
Proof of the Ecker–Honerkamp identity
Here we will provide the intermediate steps needed to prove the important identity due to Ecker and Honerkamp [7]: (Dμ)ab;c=Rabcm∂μΦm.The EH identity has proven to be fundamental in writing down compact expressions for the supervertices, as discussed in Sec. III and will be further applied for the covariant scalar-fermion four-vertex in the next subsection.
To start with, we treat the covariant derivative, (Dμ)ab=[GAB(xA)∂μ(A)+ΓABM(xA)∂μΦM(xA)]δ(xA-xB),which was defined by (3.1) in Sec. III, as a rank-2 tensor in the configuration space. As such, we use the standard rules of covariant derivatives for rank-2 tensors with lower indices to obtain (Dμ)ab;c=(Dμ)ab,c-(Dμ)amΓmbc-(-1)b(m+a)(Dμ)mbΓmac.Here we reiterate that repeated configuration-space indices imply two things: (i) summation over their corresponding field-space indices, and (ii) integration over their respective spacetime coordinates. Our goal is to prove the equivalence between (A1) and (3.2).
To this end, we first write (Dμ)ab,c explicitly as (Dμ)ab,c=GAB,C(xA)δ(xA-xC)∂μ(A)δ(xA-xB)+ΓABM(xA)δ(xA-xB)δMC∂μ(A)δ(xA-xC)+(-1)CMΓABM,C(xA)∂μΦM(xA)δ(xA-xB)δ(xA-xC),and then the two metric-connection dependent terms that occur on the rhs of (A1), (Dμ)amΓmbc=∫xM[GAM(xA)∂μ(A)δ(xA-xM)+ΓAMN(xA)∂μΦN(xA)δ(xA-xM)]ΓMBC(xM)δ(xM-xB)δ(xM-xC),(Dμ)mbΓmac=∫xM[GMB(xM)∂μ(M)δ(xM-xB)+ΓMBN(xM)∂μΦN(xM)δ(xM-xB)]ΓMAC(xM)δ(xM-xA)δ(xM-xC).
Our next step is to change the spacetime dependence of the metric GAM in (A3) from xA→xM, which will enable us to contract GAM with the metric connection ΓMBC(xM). This step can be accomplished by making a Taylor series expansion about xM, GAM(xA)=GAM(xM)+(xA-xM)μ∂μGAM(xM)+12(xA-xM)2∂2GAM(xM)+…,together with the metric identity, GAB,C=ΓABC+(-1)ABΓBAC,and the following δ-function identities: zδ(z)=0,zδ′(z)=-δ(z),zδ′′(z)=-2δ′(z),z2δ′′(z)=2δ(z),where z=xA-xM and the symbol prime (′) stands for differentiation with respect to the first argument xA of the δ function.
With the help of (A5)–(A7), we may then rewrite (A3) as ∫xM[GAM(xA)∂μ(A)δ(xA-xM)+ΓAMN(xA)∂μΦN(xA)δ(xA-xM)]ΓMBC(xM)δ(xM-xB)δ(xM-xC)=-∫xM∂μ(M)[ΓABC(xM)δ(xM-xB)δ(xM-xC)]δ(xM-xA)+(2ΓAMN(xA)+(-1)AMΓMAN(xA))∂μΦN(xA)ΓMBC(xA)δ(xA-xB)δ(xA-xC).By combining (A2), (A4), and (A8), Eq. (A3) can be brought into the form (Dμ)ab;c=((-1)MCΓABM,C(xA)-ΓABC,M(xA))∂μΦM(xA)δ(xA-xB)δ(xA-xC)+(-1)AM(ΓMAN(xA)∂μΦN(xA)ΓMBC(xA)-(-1)C(M+B)ΓMAC(xA)ΓMBN(xA)∂μΦN(xA))δ(xA-xB)δ(xA-xC).This last expression can in turn be written exactly in the form stated in (3.2) by swapping the dummy indices M and N for the last two terms in (A9) and also noting that the super-Riemann tensor with lower indices in the configuration space is given by Rabcd=RABCD(xA)δ(xA-xB)δ(xA-xC)δ(xA-xD),where RABCD=GAMRMBCD=(-1)CDΓABD,C-ΓABC,D+(-1)M(A+D)+D(B+C)ΓMADΓMBC-(-1)(M+A)(B+D)+CDΓMBDΓMAC.This completes our proof of the EH identity for supermanifolds.
Covariant scalar-fermion four vertex
We may now use successively (3.1), along with the EH identity (3.2), to carry out covariant functional differentiations on the action S≡∫ddxL in the configuration space. This is the route we followed to obtain from Sa in (3.6) the covariant inverse propagator Sab in (3.9) and the three-point scalar-fermion interaction Sabc in (3.10). By taking one more field derivative and after some tedious superalgebra, we arrive at the scalar-fermion four-point interaction, Sabcd=∂μΦm((-1)p(c+d)knmRnabp;cd+knmRnabqRqcdp+(-1)an+dpkn;[amRnbcp;d]+(-1)n(a+d)+d(b+c)kn;[ad_mRnbc]p+(-1)p(a+b+c+d)kp;abcdm)∂μΦp+(-1)am(Dμ)m[a((-1)dpknmRnbcp;d]+(-1)bnkn;bmRncd]p+(-1)p(b+c+d)kp;bcd]m)∂μΦp+∂μΦm((-1)cpknmRnabp;[c+(-1)ankn;[amRnbcp)(Dμ)pd]+(-1)am+p(b+c)(Dμ)m[akp;bc]m(Dμ)pd+(-1)b(n+a)+d(m+a+b+c)(Dμ)m[dkn;bmRnac]p∂μΦp+(-1)b(n+a)+c(m+a+b)(Dμ)m[ckn;bmRnad]p∂μΦp+(-1)p(b+c)+am+d(m+b+c)Rm[ad_q∂μΦqkp;bc]m∂μΦp+(-1)am+n(b+d)+cd(Dμ)m(akn;bd_m(Dμ)nc)+(-1)am(Dμ)m[aknmRnbcp(Dμ)pd]+(-1)m(a+d)+d(b+c)Rm[ad_q∂μΦqknmRnbc]p∂μΦp+i(-1)a(λmμaRmbcp(Dμ)pd+(-1)bm+d(m+c)λm;[bd_μa(Dμ)mc]+(-1)m(b+c)λm;bcμa(Dμ)md)+i(-1)a((-1)dpλmμaRmbcp;d+(-1)bmλm;[bμaRmcd]p+(-1)m(b+c+d)λp;bcdμa)∂μΦp-Uabcd.We should note that in (A12) the underlined index d, i.e., d_, albeit being free, still does not enter the symmetrization process. It only keeps its placing with respect to the other free indices, a, b, and c, which get symmetrized or cyclically permuted according to (3.8). As we discuss in Sec. IV, it is the supersymmetrized part of Sabcd, i.e., S{abcd}, that enters the SG-QEA beyond the one-loop level. As we did for Sabc in (3.10), we have checked that in the homogeneous limit ∂μΦ→0, the four supervertex Sabcd in (A12) reproduces the analytic result presented in [41,42]. Clearly, the complexity of the expressions increases drastically, as we are calculating more and more derivatives in the configuration space.
SPIN CONNECTION IN SG-QFTS
In Sec. II, we have introduced the field-space vielbeins eB^A, where hatted (^) indices refer to local-frame quantities. In what follows, we assume that unhatted indices are raised and lowered by the covariant supermanifold metric GBA and its inverse, while hatted indices are raised and lowered by the local field-space metric HB^A^. Given that vielbeins have a mixed coordinate and noncoordinate nature, it is possible to define two types of metric connections: (i) ΓABC for the usual global tensors and (ii) ωBA^C^ when applied to tensors that carry local-frame indices. The latter connection is termed the spin connection. The vielbeins in question satisfy the so-called tetrad postulate which is just the requirement that the total covariant derivative of the vielbeins should vanish, i.e., esTA;BA^=esTA,BA^-esTMA^ΓMAB+(-1)B(A^+A)ωBA^B^esTAB^=0.One can then solve (B1) to obtain the definition of the field-space spin connection reported in (3.5). Hence, ωBA^C^ is a vector in the unhatted indices, a nontensor in the hatted index, and is antisymmetric with respect to the local-frame indices, i.e., ωCA^B^=-(-1)A^B^ωCB^A^.
Another equivalent definition of the field-space spin connection can be obtained in terms of the standard connection with all local-frame indices ωBA^B^=(-1)B(A^+B^)ΓA^B^C^esTBC^.The spin connection generally encodes the inertial effects associated to a given frame. Hence, in inertial frames where the standard connection is set to zero, inertial effects are absent and the spin-connection vanishes. From (B2), one can also conclude that in the absence of torsion the following identity holds ωBA^B^esTCB^=(-1)CA^+B(A^+C)ωCA^B^esTBB^.We will use the equality above to obtain an expression for the operator (Dμ)AB in the local frame. Since (Dμ)AB is a proper tensor in field space, its local frame counterpart is obtained by making use of the vielbeins as usual, i.e., (Dμ)AB=eA^A(Dμ)A^B^esTBB^=eA^A(∂μΦA^);B^esTBB^,where we have used definition (3.1) to write the expression on the rhs. Inserting the definition of (∂μΦA^);B^ in the local frame in (B4) and using (B3) and (B2), we obtain an expression for (Dμ)AB in terms of the vielbeins and the spin connection, (Dμ)AB=eA^A[δB^A^∂μ+ωμA^B^]esTBB^=eM^AesTBM^∂μ+eM^A(esTB,CM^)∂μΦC+eA^AωμA^B^esTBB^,where in configuration space all the quantities above depend on the spacetime variable xA. In addition, we have introduced the shorthand notation, ωμA^B^≡(-1)C∂μΦCωCA^B^.Alternatively, one can preserve the different spacetime dependence of the vielbeins in configuration space and write the following more compact expression: (Dμ)ab=eA^A(xA)[δB^A^∂μ(A)δ(xA-xB)+ωμA^B^(xA)δ(xA-xB)]esTBB^(xB).To recover the original definition (6.2), it is sufficient to expand the rightmost vielbein field as prescribed in (A6) in order to change the spacetime dependence from xB→xA.
One can proceed in a similar fashion to obtain an expression for D2 in the local field-space frame, i.e., (D2)AB=eA^A[δB^A^∂2+2ωμA^B^∂μ+(∂μωμA^B^)+ωμA^M^ωμM^B^]esTBB^.As before, the corresponding configuration space operator reads as (D2)ab=(D2)AB(xA)δ(xA-xB),which can be equivalently written as (D2)ab=eA^A(xA)[δB^A^∂2δ(xA-xB)+2ωμA^B^∂μδ(xA-xB)+(∂μωμA^B^+ωμA^M^ωμM^B^)δ(xA-xB)]esTBB^(xB).To compute the covariant effective action, we require for the operator D2 to be exponentiated. This involves an extra degree of difficulty since now the splitting of the Laplacian operator and the remaining terms in (B10) has to be done via the Zassenhaus formula. Upon performing a series expansion in t, one recovers (6.7)⟨xA|etD2|xB⟩=KABe-(xA-xB)2/4t(4πt)d/2+WAB,where KAB is defined in (6.8), and up to order t3, WAB=∫xMe-(xA-xM)2/4t(4πt)d/2eAA^(xA)⟨xM|tΩB^A^+t22(ΩM^A^ΩB^A^-[∂2,ΩB^A^])|xB⟩esTBB^(xB).In the above expression, we have used the shorthand notation, Ω≡2ωμ∂μ+(∂μωμ)+ωμωμ,where the field-space indices have been suppressed. Finally, it should be commented that the heat kernel, given in (6.7) for a curved field space, differs fundamentally in its analytic structure from the one derived for gravitational theories of curved spacetime (cf. [12,53]).
THE DOUBLE HEAT-KERNEL METHOD
In evaluating the first term of the effective action Γ(1) in (5.15), we have employed the Zassenhaus formula in the following way: Γ(1)⊃i2trln(-∂2-mb2-M)⊃-i2∫x,y∫0∞dtt⟨x|et(∂2+mb2+M)|y⟩⟨y|etMe-t22[∂2,M]|x⟩=-i2∫x,y∫0∞dtt⟨x|et(∂2+mb2+M)|y⟩⟨y|[1+tM+t22(M2-[∂2,M])]|x⟩+…,where the field-space operator M is given in (5.16) and the ellipses represent powers of t3 and higher in the Schwinger proper time t.
In the last equation given in (C1), the first term within the expectation value, ⟨y|⋯|x⟩, is the unity operator 1, which leads to an integral that can be computed using the standard heat-kernel method. However, the computation of its second term, ⟨y|M|x⟩, becomes more elaborate, as it also contains a fermion-field dependent operator OF(x,y) in the configuration space: ⟨y|M|x⟩⊃h2⟨y|ψ¯(i∂-hϕ-mf)-1ψ|x⟩≡h2OF(x,y).In this appendix, we give further technical details related to its computation. As we will see below, such a calculation necessitates the introduction of an additional Schwinger time s (besides t), and as such, it was termed the double heat-kernel method.
Upon inserting an identity of states to isolate the kernel, we may compute the fermionic operator OF(x,y) in (C2) by making use of the bosonization property (5.2): OF(x,y)=∫v,z,w⟨y|ψ¯|v⟩⟨v|(i∂+hϕ+mf)|z⟩⟨z|[-∂2+ih∂ϕ-(hϕ+mf)2]-1|w⟩⟨w|ψ|x⟩.The expectation value of the new bosonic inverse operator in the integrand of (C3) can be represented as an integral over a new Schwinger proper time parameter, which we call s, i.e., ⟨z|[-∂2+ih∂ϕ-(hϕ+mf)2]-1|w⟩=∫0∞ds⟨z|e-s[-∂2+ih∂ϕ-(hϕ+mf)2]|w⟩.This in turn can be evaluated by means of the Zassenhaus formula, ∫0∞ds⟨z|e-s[-∂2-mf2+A]|w⟩=∫0∞dsesmf2-(z-w)2/4s(4πs)d/2[1-sA+12s2(A2-∂2A)-16s3A3+s3(13(∂A)2+12A∂2A)+12s(z-w)μ(s2∂μA-s3A∂μA)],where A=ih∂ϕ-h2ϕ2-2hϕmf. We may substitute (C5) in (C3) and notice that the expression in the square brackets is a function of w, as well as that A does not contain any differential operators. Then, upon integration over v and w, the fermionic operator OF(x,y) stated in (C3) becomes OF(x,y)=∫z∫0∞dsψ¯(y)[(i∂y+hϕ(y)+mf)δ(y-z)]esmf2-(z-x)2/4s(4πs)d/2×[1-sA+12s2(A2-∂2A)-16s3A3+s3(13(∂A)2+12A∂2A)+12s(z-x)μ(s2∂μA-s3A∂μA)]ψ(x).
If we insert (C6) in (C2) and the resulting expression of ⟨y|M|x⟩ back in the last equation for the effective action in (C1), we arrive at a multivariable integral which has the form I=∫x,y∫0∞dtt∫0∞dste-(x-y)2/4t+tmb2(4πt)d/2[f(y)-i2t(x-y)μγμl(y)]e-(y-x)2/4s+smf2(4πs)d/2[sngn(x)+12s(y-x)μsmqmμ(x)].Here we should clarify that I=∫x,y∫0∞dt⟨x|et(∂2+mb2+M)|y⟩OF(x,y),and the four functions, f(x),l(x),gn(x),qmμ(x), that appear in (C7) acquire their spacetime dependence implicitly through the scalar field ϕ(x) and the fermion fields, ψ(x) and ψ¯(x). Their explicit analytic form is given by f(x)=-i∂ψ¯(x)+(mf+hϕ(x))ψ¯(x),l(x)=ψ¯(x),gn(x)={1forn=0,-A(x)ψ(x)forn=1,(A2(x)-∂2A(x)ψ(x)forn=2,-16A3(x)ψ(x)+(13(∂A)2+12A∂2A)ψ(x)forn=3,qmμ(x)={0form=0,1,∂μA(x)ψ(x)form=2,-A(x)∂μA(x)ψ(x)form=3,as they can be read off from (C6).
After changing the integration variables (x,y)→(x-y,y) and using the properties of Fourier transforms, (C7) may be recast into the form: I=∫x∫0∞dt∫0∞dsetmb2+smf2(4πt)d/2(4πs)d/2{f(x)(πa)d/2e14a∂2/∂x2[sngn(x)-a-d8ssm∂μqmμ(x)]+(-i4t)l(x)e14a∂2/∂x2[12(πa3)d/2sn∂gn(x)-116s(πa5)d/2sm(2a+∂2)qm(x)]},with a=14t+14s. While this last expression may look complicated in d=4-2ε dimensions, only terms O(s0) dictate the UV behavior of the theory. All terms with higher powers of s lead to UV-finite contributions. After this observation, the UV-divergent part of (C10) was found to be IUV=116π2ε∫ddx[-i(∂ψ¯)ψ+(hϕ+mf)ψ¯ψ].A similar UV-divergent term is found when computing the trace of the operator: ψT[(i∂+hϕ+mf)T]-1ψ¯T,where one has to replace h and mf with their negatives. Note that terms containing higher powers of t will be UV finite, with the exception of the ϕ3 and ϕ4 contributions.
DIMENSIONAL REGULARIZATION IN COORDINATE SPACE
In this appendix, we will give useful formulas that permit us to evaluate loop integrals in DR, but in the coordinate or x space. Before doing so, we briefly review how DR is applied to loop integrals in momentum or k space. The translation of loop integrals from k to x space goes through the so-called Gegenbauer method, which we will not present here explicitly. The interested reader may consult Appendix C of the lecture notes by Pascual and Tarrach in [82]. In the following, we will consider two types of coordinate-space integrals for the heat kernels: (i) the massless heat kernel and (ii) the massive heat kernel.
Massless heat kernels
To extract the UV behavior of a theory when complicated differential operators are involved, such as in (6.14), it might be useful to evaluate firstly the integral over the Schwinger proper time t and then carry out any remaining integrals over position space variables. For massless heat kernels, we encounter the following type of integral: ∫0∞dte-(xA-xB)2/4t(4πt)d/2tn=2-2+2nπ-d/2|xA-xB|-d+2+2nΓ(-1+d2-n),with |x|≡[(x)2]1/2 and n∈Z. Then, we must evaluate a coordinate-space integral over vanishing spatial separations, i.e., when |xA-xB|→0. To accomplish this task, one needs to perform this integration directly to x space in the DR, rather than in the often-considered k space.
Let us therefore start by reiterating the well-known one-loop result for d-dimensional momentum integrals in Minkowski space: ∫ddk(2π)d1(k2-Δ)α=i(-1)α(4π)d/2Γ(α-d/2)Γ(α)(1Δ)α-d/2,with α∈R. The condition of vanishing tadpoles in d=4-2ε dimensions is imposed by requiring the momentum integral (D2) with α=1 to go to zero, i.e., ∫ddk(2π)d1k2=-i(4π)d/2Γ(1-d/2)Γ(1)limΔ→0(1Δ)1-d/2=0.
Our aim is now to obtain an expression equivalent to (D2), but in x space. Using Schwinger parameters, one may represent the power law of an x2 function as 1[x2]d/2-α=1(d/2-α-1)!∫0∞duud/2-α-1e-ux2.Taking the Fourier transform of the expression above and integrating over the d-dimensional x space yields ∫ddxeik·x[x2]d/2-α=1Γ(d/2-α)∫0∞duπd/2e-k2/4uu-α-1,where we have performed the integration over position space by completing the square. After a suitable change of variables, like t=k2/4u, one may write the rhs of (D5) in terms of the integral representation of the usual Γ(x) function as ∫ddxeik·x[x2]d/2-α=πd/2Γ(d/2-α)4α[k2]α∫0∞dte-ttα-1.Finally, integrating both sides of (D6) with respect to the d-dimensional momentum k and making use of (D2), we obtain ∫ddxδ(d)(x)[x2]d/2-α=i(-1)α4α-d/2Γ(α-d/2)Γ(d/2-α)limΔ→0(1Δ)α-d/2,which can be used to investigate the presence of 1/ε poles directly from position space integrals. Hence, depending on the coefficient α, (D7) can either be zero, finite, or exhibit IR divergences when limΔ→0(1/Δ) appears. We observe that the integral (D7) vanishes for 0≤α<2, it is equal to -i for α=2, and becomes IR divergent when α>2.
Massive heat kernels
Let us first remark that even if a mass term is present in the potential U, like in (6.10) and in (7.11), the theory is effectively massless when a kernel of the form (D1) is used. This in turn implies that in this massless scheme, the only nonzero contributions will arise from the wave function renormalization, the correction to the Yukawa coupling, and the scalar quartic interactions, as all other counterterms are mass dependent; cf. (5.18). However, we may access the UV structure of the latter counterterms if we use a massive heat kernel.
For a massive heat kernel, its x-space Schwinger proper-time representation yields a more complicated expression than (D1). In this case, the proper-time integral can be written in terms of the modified Bessel function of the second kind Kν of order ν=(d/2-n), ∫0∞dtte-tμ2-(x-y)2/4t(4πt)d/2tn=(μ2)14(d-2n)21-d/2-nπ-d/2|x-y|n-d/2Kν(μ2|x-y|).To obtain the analytic result in (D8), one must use the specific integral representation of Kν[83], Kν(z)=12(z2)ν∫0∞exp(-t-z24t)dttν+1,which also satisfies the standard property: Kν(z)=K-ν(z). In this massive case, the simple power law function, 1/|x|d-2α in (D7), gets replaced with Kν(μ2|x-y|). To carry out the integral in x space, one must first expand the Bessel function Kν(z) for small arguments z[83], Kν(z)=12[Γ(ν)(z2)-ν(1+z24(1-ν)+⋯)+Γ(-ν)(z2)ν(1+z24(1+ν)+⋯)].For convenience, one can then define a reduced Bessel function K^ν(z) having the property z-νKν(μz)=12μν2-νK^ν(μz),so that if ν<0, limz→0K^ν(μz)=Γ(-ν).
For illustration, we will now consider an example where modified Bessel functions appear. Let us compute the following integral over Schwinger proper time: I=tr∫x,y∫0∞dt(i∂+mf)δ(x-y)e-(y-x)2/4t+mf2t(4πt)d/2Y(x),where Y(x)=hϕ(x). Such an expression arises, for example, when computing the effective action for the simple Yukawa model, as given in V A. In d=4-2ε dimensions, the UV-divergent part of the integral I in (D13) may explicitly be computed as IUV=-14π2∫x,yδ(x-y)mf(-mf2)(1-ε)K^1-ε(-mf2|y-x|)Y(x)=-14π2∫xmf(-mf2)(1-ε)Γ(-1+ε)Y(x)=-14π2ε∫xmf3hϕ(x).This result exactly reproduces the counterterm δc in (5.10).
Let us now elucidate how Bessel function techniques can be used to evaluate the Schwinger integrals in the double heat-kernel method in (7.32), e.g., to recover the flat field-space limit. In this limit, we have λabμ=(-γμ00(γμ)T),(Dμ)ab=(-∂μ00∂μ).In a flat field space, the potential like term V in (7.11) reduces to the scalar potential U and both the A and B operators appearing in the definitions of I1,2,3 vanish, since the λμ matrices are now field independent. The only term that survives arises from (7.24), where we choose now to include the mass term in the kernel so that U=Y. By doing so, the integral I3 takes on the form I3flat=∫xA,xB,xC,xD(i∂+mf)δ(xA-xB)∫0∞dte-(xB-xC)2/4t+mf2t(4πt)d/2Y(xC)(i∂+mf)δ(xC-xD)∫0∞dse-(xD-xA)2/4s+mf2s(4πs)d/2Y(xA).Notice that each of the Schwinger proper-time integrals in (D16) is evaluated by means of (D8). In d dimensions, the expression resulting from (D16) becomes then an integral over a product of two Bessel functions Kν(z), i.e., I3flat=2dd(4π)d(-mf2)ν∫xA,xB,xC,xD[(-∂(A)μ∂μ(C)+mf2)δ(xA-xB)δ(xC-xD)]|xB-xC|-ν|xD-xA|-ν×Kν(-mf2|xB-xC|)Y(xC)Kν(-mf2|xD-xA|)Y(xA),with ν=d/2-1. The product of two Bessel functions of the same ν order, Kν(z)Kν(ζ), can be written as a t integral over a single ν-order Bessel function, Kν(zζ/t), with an appropriate weighted function. To be specific, we utilize the integral identity [83], Kν(z)Kν(ζ)=12∫0∞dttexp(-t22-z2+ζ22t)Kν(zζt).Likewise, by making use of the differential recursive relation, ∂Kν(z)∂z=-Kν-1(z)-νzKν(z),and the recurrence identities for consecutive neighbors, Kν(z)=Kν+2(z)-2(ν+1)zKν+1(z)=Kν-2(z)+2(ν-1)zKν-1(z),one can obtain the wave function and mass counterterms listed in (5.10).
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