GGauge Invariance, Polar Coordinates and Inflation
Ali Akil a,b,c ∗ , Xi Tong a,b † a Department of Physics, The Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, P.R.China b Jockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, P.R.China c Department of Physics, Southern University of Science and Technology (SUSTech),Shenzhen 518055, P.R.China
Abstract
We point out the necessity of resolving the apparent gauge dependence in the quantumcorrections of cosmological observables for Higgs-like inflation models. We highlight the factthat this gauge dependence is due to the use of an asymmetric background current whichis specific to a choice of coordinate system in the scalar manifold. Favoring simplicity overcomplexity, we further propose a practical shortcut to gauge-independent inflationary observ-ables by using effective potential obtained from a polar-like background current choice. Wedemonstrate this shortcut for several explicit examples and present a gauge-independent pre-diction of inflationary observables in the Abelian Higgs model. Furthermore, with Nielsen’sgauge dependence identities, we show that for any theory to all orders, a gauge-invariant cur-rent term gives a gauge-independent effective potential and thus gauge-invariant inflationaryobservables.
The effective potential was introduced as an attempt to include the quantum corrections to thetree level potential, after which one can treat the background evolution in quantum field theoryusing classical variational methods [1, 2]. It is widely applied to the study of phenomena ofspontaneous symmetry breaking. For example, the effective potential plays an important role inthe determination of the quantum vacuum [1] and its decay rate [3,4], the analysis of cosmologicalphase transitions [5–9], as well as inflationary dynamics [10–13].However, as an off-shell quantity by itself, the effective potential was shown to suffer from theproblem of gauge dependence [14]. First, it was argued in [2] that the unitary gauge is the rightway to extract physical quantities, being only a field redefinition and arguably not really a choiceof gauge [15, 16]. Then, by means of Ward-Takahashi identity of BRST symmetry, [17] derived a ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ h e p - t h ] S e p rst order partial differential equation (later improved in [18]) describing the exact dependenceof the effective potential V eff ( ¯ φ, ξ ) on the gauge parameter ξ , (cid:18) ξ ∂∂ξ − C ( ¯ φ, ξ ) ∂∂ ¯ φ (cid:19) V eff ( ¯ φ, ξ ) = 0 . (1)It is argued that the physical states of the Hilbert space are represented by the different character-istic lines of the partial differential equation (1). Gauge transformations map each characteristicline to itself and therefore leaves the physical states invariant.It was later found in [19] that if one expresses the scalar field in the polar field space coordi-nates, the gauge dependence cancels out and, interestingly, the result is in exact accordance withthe effective potential in the unitary gauge with Cartesian field coordinates, thus matching theconclusions of [20]. For a U (1) -symmetric gauge theory, it is to be expected that polar-coordinatefields give a clearer picture of physics since the symmetry of the theory is made manifest. Nat-urally, after choosing field space coordinates as fundamental fields, corresponding source termsproportional to these fundamental fields must be introduced into the partition function, so as tolead the system off-shell. It then becomes clear that if one excites the system using a symmetricsource term, the off-shell effective action is automatically gauge-invariant even non-perturbatively,whereas an asymmetric source term will inevitably cause the off-shell effective action to pick upgauge-dependence.One might question the necessity of requiring gauge-invariance for off-shell quantities likethe effective potential, since processes are physical only when the background current source isremoved and the system is put on-shell. For purposes of calculating particle scattering amplitudesaround the vacuum, removing the source is equivalent to sending the field background to thequantum vacuum, where ∂V eff ∂ ¯ φ = 0 . Then the Nielsen identity (1) shows that at this minimumof the effective potential, the potential value is independent of gauge choices. Similar identitiesshow that the on-shell scattering amplitudes are also independent of ξ at the minimum. However,this is not the whole story, since in general, the field background can have a non-trivial spacetimedependence. For this purpose of an out-of-equilibrium process, removing the background currentis not simply setting the system to vacuum but letting the background to evolve freely accordingto the variational principle of the effective action. The gauge dependence issue in this scenario ismuch more subtle.Take the example of inflation, if the inflaton field is not a gauge singlet, its classical back-ground rolls according to the effective potential which receives gauge-dependent contributionsfrom quantum fluctuations. As a consequence, inflationary observables such as slow-roll param-eters and e-folding numbers appear to be polluted by a gauge-dependent quantum corrections.Since physical observables should not depend on gauge choices, this apparent gauge dependencemust be resolved. We will argue that either one has to go to the non-perturbative regime [21] orhas to analyze the slow-roll dynamics more carefully using in-in formalism. In the literature thereare preliminary attempts [22, 23] to advance in this direction, but to our knowledge, no explicitdemonstration of gauge-invariant inflationary observables has yet been made.Therefore in favor of simplicity over a heavy machinery of in-in calculations or even non-perturbative treatments, we propose the second possibility of resolution. By using a symmetricbackground current which fits naturally with polar field coordinates [19], the system is lifted up in agauge-invariant way. And the resulting effective action is manifestly independent of gauge choiceseven off-shell. Then by setting the background current to zero, one obtain a gauge-independentdescription of background evolution, along with gauge-independent inflationary observables.2his paper is organized as follows. In Sect. 2, we show the impact of apparent gauge depen-dence on inflationary observables and argue the necessity of resolving this issue. In Sect. 3, weidentify the cause for this apparent gauge dependence and propose a simple and practical solutionby using polar-like coordinates. The examples of SU (2) in both the fundamental and the adjointrepresentations as well as SU (2) × U (1) are given to illustrate this idea. For the U (1) model, wealso present the predictions of inflationary observables using our method. Then in Sect. 4, weuse Nielsen’s gauge dependence identity to give a non-perturbative proof of the off-shell gaugeinvariance of the polar-coordinate effective action. At last, in Sect. 5, we discuss the relationbetween polar coordinates and the unitary gauge. We point out the limitations of our methodand conclude in Sect. 6. Inflation is the leading paradigm of early universe. In simplest setups, it is the exponentialexpansion period of the universe driven by a single slowly rolling scalar field called the inflaton.In principle, other quantum fields are also present during inflation. These fields usually havevanishing classical backgrounds and do not directly influence the background evolution. However,if they interact with the inflaton, quantum fluctuations of these fields will contribute to theeffective potential of the inflaton. These are the radiative corrections to the background evolution.In particular, consider an Abelian Higgs inflation model [24], S [ g µν , φ, A µ ] = (cid:90) √− g (cid:20) − M p R − α | φ | R + | Dφ | − V ( | φ | ) − F (cid:21) , (2)where D = ∂ + ieA is the covariant derivative and V ( | φ | ) = − m | φ | + λ | φ | is the tree-levelHiggs potential. The second term is a non-minimal coupling between gravity and the radial gaugesinglet component of the Higgs field. This term naturally arises as a back-reaction of fluctuationsof matter field φ to the spacetime geometry. The gauge field fluctuations contribute to both thegravitational sector and to the scalar sector. Since the gauge field does not acquire a classicalbackground, we neglect its back-reaction to the gravity sector and only consider the back-reactionto the scalar sector, which can be accounted for by replacing the classical action by the gauge-fixedeffective action computed in the spacetime background of g µν , S eff [ g µν , ¯ φ J , ξ ] = (cid:90) √− g (cid:20) − M p R − α | ¯ φ J | R (cid:21) + Γ[ ¯ φ J , ξ ] | g µν . (3)Since the quantum corrections come from fluctuations deep in the UV, we expect the spacetimecurvature to play only a subdominant role for the fluctuating matter fields and approximate Γ[ ¯ φ J , ξ ] | g µν ≈ Γ[ ¯ φ J , ξ ] | η µν . (4)Therefore, when truncated to the second order in field gradients, the action looks like S eff [ g µν , ¯ φ J , ξ ] = (cid:90) √− g (cid:20) − M p R − α | ¯ φ J | R + Z ( ¯ φ J , ξ ) | ∂ ¯ φ J | − V eff ( ¯ φ J , ξ ) (cid:21) . (5)The above expressions are all written in the Jordan frame. To obtain an inflationary solution,one usually goes to the Einstein frame by performing a conformal transformation ˆ g µν = Ω g µν , χ J = (cid:90) (cid:115) Z Ω + 12 α ¯ φ J /M p Ω d ¯ φ J , where Ω = 1 + 2 α ¯ φ J M p . (6)3inally, the action in Einstein frame becomes S eff [ˆ g µν , χ J , ξ ] = (cid:90) (cid:112) − ˆ g (cid:34) − M p ˆ R | ∂χ J | − U eff ( χ J , ξ ) (cid:35) , where U eff ( χ J , ξ ) = V eff Ω . (7)Here ˆ R is the Ricci scalar computed using the rescaled metric ˆ g µν and U eff ( χ J , ξ ) is explicitlycomputed in Appendix. A. Now, after obtaining the effective action, we set the fields on-shellby removing the background current J = 0 . Assuming a flat FRW spacetime, the Friedmannequation and the equation of motion for the inflaton (Higgs singlet after redefinition) are M p H = 12 ˙ χ + U eff ( χ , ξ ) (8) ¨ χ + 3 H ˙ χ + ∂U eff ( χ , ξ ) ∂χ = 0 . (9)This is just the normal pair of equations that governs the background dynamics during infla-tion. Usually the potential is flat such that the inflaton is drawn into the slow-roll attractorsolution. The potential gradient term is now explicitly dependent on the gauge parameter ξ . Thisfact inevitably leads to the consequence that the slow roll parameters receive a gauge-dependentquantum correction: (cid:15) ≡ − ˙ HH ≈ (cid:15) U = M p (cid:18) ∂ χ U eff ( χ , ξ ) U eff ( χ , ξ ) (cid:19) (10) η ≡ ˙ (cid:15)H(cid:15) ≈ (cid:15) U − η U = 2 M p (cid:34)(cid:18) ∂ χ U eff ( χ , ξ ) U eff ( χ , ξ ) (cid:19) + ∂ χ U eff ( χ , ξ ) U eff ( χ , ξ ) (cid:35) . (11)Furthermore, the e-folding number also becomes gauge dependent: N CMB = (cid:90) t end t CMB
Hdt ≈ (cid:90) χ end χ CMB dχ (cid:112) (cid:15) U ( χ , ξ ) , (12)where χ CMB and χ end are respectively the field values of the on-shell inflaton solution χ ( t ) at thetime when the CMB scale exits the horizon and at the time when inflation is ended. In order toexamine the impact of this apparent gauge dependence on direct inflationary observables, we canconsider the scalar spectral index and tensor-to-scalar ratio, n s − ≡ d ln P ζ d ln k ≈ − (cid:15) − η, r ≡ P γ P ζ ≈ (cid:15) , (13)where P ζ and P γ are the scalar and tensor power spectra.Using the effective action computed in the Cartesian field coordinates in the R ξ gauge, wenumerically solve the full EOM (8) and plot the ξ -dependence of the inflationary observables inFig. 1. The details of calculation is given in Appendix. A. As shown in the figure, within the validrange of one-loop result, the impact of apparent gauge dependence on inflationary observables is not insignificant. In fact, this ξ -dependence is partly degenerate with other parameters in themodel, for example, e-folding numbers.We point out that this gauge dependence problem is not only subjected to the particularAbelian Higgs inflation we mention here. It is generically present in any inflation models wherethe full inflaton field couples to gauge fields, giving a non-negligible uncertainty to the predictionof inflationary observables. Therefore in order to achieve more accurate constrains on the physicalparameters in future CMB precision measurements, the uncertainty due to unphysical ξ choicesmust be resolved. 4 - η n s r Planck 2018, 1 σ Planck 2018, 2 σ ξ = ξ = ξ = Figure 1: Left panel: The ξ dependence of the slow-roll parameters at one-loop level for dif-ferent field coordinates. The ξ -dependent solid lines are computed in conventional Cartesiancoordinates while the ξ -independent dashed lines are computed in our proposed polar field spacecoordinates. The gray region ξ (cid:38) π is where gauge dependence threatens loop expansion va-lidity and should not be taken seriously. Right panel: The apparent gauge dependence of the n s - r prediction in the Higgs inflation model at one-loop level. The smaller points correspond to N CMB = 50 while the larger points correspond to N CMB = 60 . The dashed cyan line representsthe gauge-invariant n s - r prediction in our proposed polar field coordinates. The contours are fromTT,TE,EE+lowE+lensing+BK14 constraints with Planck 2018 data. The other parameters arechosen to be λ = 0 . , e = 0 . , m = 0 , α = 17000 , µ/M p = 0 . to have a viable inflation model.See more details in Appendix A. The apparent gauge dependence of the inflationary observables can be traced to several possibleorigins.Working backwards the derivation in the previous section, we see that the ξ dependencein the final observables first appears in the potential-shape slow-roll parameters (cid:15) U , η U which areapproximations to the original ones (cid:15), η , up to errors O ( (cid:15) U , η U ) . One might question the possibilitythat higher order corrections will render (cid:15), η gauge-invariant. However, numerical solution in Fig. 1shows that with O ( (cid:15) U , η U ) corrections included, the full (cid:15), η parameters still depend on ξ .Therefore the equations of motion (8) already contain unphysical information. Apparently thisis because the inflaton potential U eff ( χ , ξ ) in the Einstein frame depends on ξ explicitly, which isnot washed out by the procedure of looking into observables. Since (8) follows from the variationalprinciple δS eff δχ J = − J = 0 , the removal of the background current should guarantee an equationof motion that gives out gauge-independent final observables. Its failure suggests that we haveused the wrong effective action (7). As field redefinitions do not change equations of motion, theeffective action in Jordan frame (5) is also problematic. Thus viewed in this way, one possiblecause for the gauge dependence problem is the procedure of expanding the originally non-localeffective action and truncating to second order in field gradients. When the inflaton backgroundis set free to roll, the whole tower of field gradients may contribute to the rolling solution.5his is the case for the scenario without gravity, where φ quickly rolls off a steep potential.When the rolling speed is too fast, i.e. , | ∂φ | (cid:38) ∂ φ V , the adiabatic condition for perturbations willbe violated and real particles will be produced. The system cannot be modeled by a uniform back-ground field φ ( t ) anymore, and the quantum corrections in the average field value φ ( t ) receivenon-local contribution from the real field quanta. To exactly evaluate the quantum correctionsand check gauge dependence, one may need a non-perturbative treatment (such as that proposedrecently in [21]).However, in the scenario with gravity, obviously the Hubble friction slows down the rollingspeed. The system evolves adiabatically and the non-local contributions are exponentially sup-pressed. In this case, the truncation procedure seems valid since the field gradients are indeedsmall. The gauge dependence issue then poses a question mark on our neglect of spacetime cur-vature in the beginning (4). Then one may need to take the expansion of spacetime into accountand use in-in formalism in a quasi-de Sitter spacetime to re-evaluate the loop diagrams that giverise to the effective potential. There are some preliminary results on this possibility. For example,it is shown in [22] that the value of the effective action for a rolling inflaton is ξ -independenton-shell, which is in agreement with the Nielsen identity. Yet to explicitly demonstrate the gaugeindependence of the whole set of inflationary observables, there is still a long way to go.In either cases, the computational work needed to resolve the gauge dependence issue is huge,if not impossible. As a result, we propose a third remedy which is much easier and more practical. J ρ J X Figure 2: Left panel: A wise coordinate choice induces a symmetric background current. Thegreen lines correspond to Cartesian coordinates while blue lines correspond to polar coordinates.In turn, J X breaks U (1) while J ρ preserves it. Right panel: In polar fields, the Goldstone directionis flat thus massless. In contrast, in Cartesian fields, the Goldstone is generally massive away fromthe minimum.In the literature, Cartesian coordinate fields are usually used to construct the effective po-tential. The physical meaning for the effective potential is then the minimal energy density ofthe system under the constraint of an average field value ¯ φ [1]. To impose the constraint, oneneeds to apply an auxiliary background current J suited to the coordinate system. Therefore,if the coordinate system does not respect the gauge symmetry, its corresponding current natu-rally breaks gauge invariance and leads to a gauge-dependent effective potential off-shell. Forthe purpose of determining the true vacuum, this is convenient since the vacuum lies at a field6onfiguration without the support of a background current. The same is true if one considers theon-shell scattering amplitudes near the vacuum. As a result, it is stated that effective potentialis only physical at its local extrema, where fields are on-shell [8, 25]. However, as we have seenabove, there is no practical solution to the sensitivity of inflationary observables to shape of theeffective potential away from the local extrema.Thus the consideration of cosmological applications leads us to the off-shell gauge-invarianteffective potential. We explore the direct use of a gauge-invariant auxiliary current, naturallyassociated with polar-like coordinates (see Fig. 2) that catch the symmetry of the theory, assuggested in [19]. We use a J i ϕ i source term for which J i δ BRST ϕ i = 0 . This guarantees a gauge-invariant effective action. This can be clearly observed in an example studied in [19], wherethe Abelian Higgs effective potential is computed using the polar expression of the scalar field.There, the gauge dependence cancels out and the resulting effective potential is clearly gauge-independent. This eventually leads to a clean and invariant effective potential which is simpleto compute. Moreover, the resulting effective potential exactly matches the one computed inthe unitary gauge, thus in agreement with the conclusions of [20]. Here, we will go furtherto check whether this prescription works for the cases of an SU (2) theory in the Fundamentalrepresentation, SU (2) in the adjoint representation, and SU (2) × U (1) (the Standard model). Forgenerality, we compute two of them in the covariant gauge and the other two in R ξ gauge. We willexplicitly compute the gauge dependent part of the effective potential, and show that it vanishesin the case of a symmetric current term. For pedagogical reasons, we will rederive first the Abelian Higgs one-loop effective potential,with polar field space coordinates as the fundamental fields. Coupling the polar fields with thecorresponding currents and choosing the covariant gauge family, the Lagrangian reads L = | Dφ | − V ( | φ | ) − F − ξ ( ∂A ) + J ρ ρ + J θ θ . (14)Setting φ = ( ¯ φ + ρ ) e iθ , then, by applying the steepest decent method (which forces J θ to 0), weget the quadratic Lagrangian L = − ρ (cid:0) ∂ + 6 λ ¯ φ − m (cid:1) ρ − ¯ φ θ∂ θ + 2 e ¯ φA µ ∂ µ θ + 12 A µ (cid:20) ( ∂ + 2 e ¯ φ ) η µν − (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) A ν , (15)where we have denoted ¯ φ ≡ (cid:104) φ (cid:105) J as the constrained radial field value. Then, after Legendretransformation and integrating over perturbations, the effective potential becomes a trace over7he Hessian matrix of the constrained Lagrangian at φ = ¯ φ , V eff = i (cid:20) − (cid:18) ∂ η µν + (1 − ξ ) ∂ µ ∂ ν + 2 e ¯ φ η µν (cid:19)(cid:21) + i (cid:20) − ∂ + 2 e ¯ φ ∂ µ ∂ ν ∂ + 2 e ¯ φ (cid:18) η µν + ∂ µ ∂ ν (1 − ξ ) − ∂ − ξe ¯ φ (cid:19)(cid:21) +gauge independent terms (16) = Γ( − d )(4 π ) d e ¯ φ ) d + Γ( − d )(4 π ) d (2 ξe ¯ φ ) d + i (cid:90) d d k (2 π ) d ln (cid:2) − k (cid:3) − Γ( − d )(4 π ) d (2 ξe ¯ φ ) d +gauge independent terms . (17)The first two terms come from the gauge fields, the last two come from the Goldstone. Here wehave only spelled out the terms with explicit ξ dependence and omitted the rest. As is clear, thegauge dependence cancels out everywhere. In the theory with Cartesian coordinates as funda-mental fields, this happens at the vacuum, anywhere else the gauge dependence is explicit.This is a clear example demonstrating how the right choice of the fundamental fields (and conse-quently, the source term), can resolve the off-shell gauge dependence of the effective potential. Onthe other hand, if we choose a current that breaks a symmetry, then we should not be surprisedthat that symmetry is broken at the level of the effective potential or effective action. In thisformalism, one can easily compute an effective action which is gauge independent on-shell andoff-shell, from which the derived physical quantities will all be gauge independent. SU (2) in fundamental representation Now we move on to explore this prescription in more complicated settings. First, we start withan SU (2) model with the scalar transforming under the fundamental representation. Upon spon-taneous symmetry breaking, all three generators of SU (2) are broken by the scalar VEV. Then,the scalar field can be parametrized as φ = e iθ a τ a (cid:18) φ + ρ (cid:19) ,τ a = 12 σ a ; j = 1 , , , (18)with ρ playing the role of the singlet radial field and θ a being the angular fields transformingnon-linearly under SU (2) . Then we turn on the gauge fields by a minimal coupling D µ φ =( ∂ µ − igA aµ τ a ) φ , where A µ = A aµ τ a ; a = 1 , , .F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ] . (19)8n the R ξ gauge, after sufficient manipulation, the relevant quadratic Lagrangian becomes L = − ρ (cid:0) ∂ + 6 λ ¯ φ − m (cid:1) ρ − ¯ φ θ a ∂ θ a − g ¯ φ A aµ ∂ µ θ a + 12 A aµ (cid:20) η µν (cid:18) ∂ + 12 g ¯ φ (cid:19) − (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) A aν . (20)This gives a one-loop effective potential which reads V eff = Tr ln (cid:20) k − g ¯ φ k − k + g ¯ φ (cid:18) − (1 − ξ ) k k − ξg ¯ φ (cid:19)(cid:21) + Tr ln (cid:20) ( − k + 12 g ¯ φ ) ( − k ξ + 12 g ¯ φ ) (cid:21) + gauge independent terms . Which in turn boils down to Γ( − d )(4 π ) d (cid:18) g ¯ φ (cid:19) d − Γ( − d )(4 π ) d (cid:18) ξg ¯ φ (cid:19) d + Γ( − d )(4 π ) d (cid:18) ξg ¯ φ (cid:19) d + gauge independent terms . (21)Where the first two terms come from the gauge fields and the last one represents the Goldstones.Finally, after cancellation, we are left solely with gauge independent terms. SU (2) in the adjoint representation We can also work out the SU (2) theory in the adjoint representation, i.e. ,the Georgi-Glashowmodel which was originally introduced in order to explain the boson mass. After the adjointscalar acquires a VEV, two of the SU (2) generators are broken and only one generator preservesthe scalar VEV, which lies on a unit sphere S ∼ = SU (2) /U (1) . Choosing the polar coordinate isequivalent to choosing a north pole on this coset manifold S . Without loss of generality, we willchoose the direction of the scalar VEV as the north pole, which is always possible by applyingsuitable similarity transformations. In its vector presentation, this theory consists of an adjointscalar field which after a suitable similarity transformation reads φ = e iα j τ j φ + ρ , j = 1 , . (22)The gauge fields are coupled minimally with φ through a covariant derivative [ D µ φ ] a = ∂ µ φ a + g(cid:15) abc A bµ φ c . (23)This time, we use R ξ gauge where the mixing between gauge bosons and Goldstones is canceled.We focus on the relevant quadratic Lagrangian, L = − ρ ( ∂ + 6 λ ¯ φ − m ) ρ − ¯ φ α j ( ∂ + 2 ξg ¯ φ ) α j + 12 A jµ (cid:20) η µν (cid:0) ∂ + 2 g ¯ φ (cid:1) − (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) A jν − ¯ c j ( ∂ + 2 ξg ¯ φ ) c j . (24)9otice that here we have only written down the terms relevant to the computation of the effectivepotential. The unbroken U (1) part can be gauge-fixed independently (for example, with a differentgauge parameter ξ (cid:48) ) and will not influence the resulting effective potential V eff ( ¯ φ, ξ ) . Then afterintegration and Legendre transformation it gives, Γ( − d )2(4 π ) d (cid:104) g ¯ φ ) d + (2 ξg ¯ φ ) d (cid:105) + Γ( − d )2(4 π ) d (2 ξg ¯ φ ) d − Γ( − d )(4 π ) d (2 ξg ¯ φ ) d . (25)Here, the first two terms are due to the gauge fields, the third to the Goldstones and the last isthe ghost contribution. Clearly, the gauge dependence cancels out, just as expected. Here we work out the effective potential for the Glashow-Salam-Weinberg (GSW) model withoutfermions, choosing polar fields as the fundamental fields. There are 4 generators for SU (2) × U (1) .When the symmetry is broken to U (1) , three generators are broken and one is left. However, as isknown in the GSW model, the unbroken generator is a linear combination of the U (1) and one ofthe SU (2) generators. After a suitable similarity transformation, the scalar field in the polar-likecoordinates can be written as φ = e i ( α + T + + α − T − ) − iα Z T Z (cid:18) φ + ρ (cid:19) , (26)where we have denoted T ± = τ ± iτ , T Z = τ − τ and T A = τ + τ .Turing on the minimal coupling with gauge fields, the scalar covariant derivative reads D µ φ = ∂ µ φ − ig √ W + µ T + + W − µ T − ) φ − igc w Z µ (cid:20)(cid:18) − s w (cid:19) T A − T Z (cid:21) φ − igs w A µ T A φ , (27)where, as usual, s w = g (cid:48) √ g + g (cid:48) and c w = g √ g + g (cid:48) . After expansion to the quadratic order, we getthe scalar kinetic term as (cid:107) Dφ (cid:107) = ( ∂ρ ) + ¯ φ ∂α + ∂α − + ¯ φ ( ∂α Z ) + 12 g ¯ φ W + W − + 14 c w g ¯ φ Z − g ¯ φ √ (cid:0) W + ∂α − + W − ∂α + (cid:1) − g ¯ φ c w Z∂α Z . (28)Now we fix the gauge into the R ξ family in order to cancel the mixed terms. Let the gauge fixingterm be G = − ξ ( ∂W + + √ ξg ¯ φ α + )( ∂W − + √ ξg ¯ φ α − ) − ζ (cid:18) ∂Z + ζg ¯ φ c w α Z (cid:19) . (29)10hen, the gauge-fixed quadratic Lagrangian reads L = − ρ (cid:0) ∂ + 6 λ ¯ φ − m (cid:1) ρ − ¯ φ α + (cid:0) ∂ + ξg ¯ φ (cid:1) α − − ¯ φ α Z (cid:18) ∂ + ζ c w g ¯ φ (cid:19) α Z + 12 W + µ (cid:20) η µν (cid:0) ∂ + g ¯ φ (cid:1) − (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) W − ν + 12 Z µ (cid:20) η µν (cid:18) ∂ + 12 c w g ¯ φ (cid:19) − (cid:18) − ζ (cid:19) ∂ µ ∂ ν (cid:21) Z ν − ¯ c + (cid:0) ∂ + ξg ¯ φ (cid:1) c − − ¯ c Z (cid:18) ∂ + ζ c w g ¯ φ (cid:19) c Z . (30)As a result, the gauge-dependent terms are Γ( − d )(4 π ) d ξg ¯ φ ) d + Γ( − d )(4 π ) d ξg ¯ φ ) d − Γ( − d )(4 π ) d × ξg ¯ φ ) d Γ( − d )(4 π ) d (cid:18) ζ c w g ¯ φ (cid:19) d + Γ( − d )(4 π ) d (cid:18) ζ c w g ¯ φ (cid:19) d − Γ( − d )(4 π ) d (cid:18) ζ c w g ¯ φ (cid:19) d . (31)In each line the first term refers to the gauge fields, the second to the Goldstones, and the thirdto the ghosts. The first line comes from the W ± ’s and their related fields while the second iscontributed by Z and its related fields. Thus we see again that they cancel each other and theeffective potential is explicitly gauge-independent.In summary, adopting polar-like scalar coordinates amounts to an application of symmetricbackground currents which preserves the gauge-invariance of the final effective potential. This,in turn, gives rise to a gauge-independent prediction of inflationary observables. For the AbelianHiggs inflation model in Sect. 2, we numerically solve the inflaton dynamics and plot our predictionof n s and r for polar coordinate choice in Fig. 1. The result for polar coordinate choice differsfrom that of the whole ξ < ∞ gauge family for traditional Cartesian coordinates and is closest tothat of the Laudau gauge ξ = 0 . In Sect. 5, we will argue that the polar coordinate result actuallymatches that of the unitary gauge, namely ξ → ∞ with the cutoff holding fixed.Before we conclude this section, we point out that in the literature, there exist other ways ofobtaining off-shell gauge-independent effective potential. For example, [26] computes the effectivepotential in the Hamiltonian formalism using a gauge-invariant order parameter. However, theirresult is different from ours. Therefore, which result is more accurate must be left for morerigorous calculations or experiments in the future. In this section we first review the derivation of the gauge dependence identities [17] using thefunctional notation introduced by [27, 28], as presented in [29, 30]. We emphasize along thederivation that the asymmetric current term is the source of all gauge dependence appearing in11he effective action. We then discuss what seems to be the most reasonable way to achieve its off-shell gauge invariance non-perturbatively, namely by using a current coupled to the gauge-singletradial field in the polar coordinates on the scalar manifold.We start by defining ϕ i to span all the fields in a given theory . The i index accounts forspacetime variables, Lorentz index, and the group index, altogether (e.g. ϕ k = A aµ ( x ) ). Onthe other hand, a Greek index (e.g. F α ≡ ∂ µ A µa ( x ) ) accounts only for spacetime and groupindices. The gauge transformation is defined by δ g ϕ i = θ α δ α ϕ i , where θ α is a transformationparameter and δ α is the gauge generators defined at a local spacetime point. The gauge algebra is [ δ α , δ β ] = f γαβ δ γ , where f γαβ consists of ordinary structure constants as well as derivatives of thespacetime Dirac delta function [31]. We will call F A the gauge fixing function, with { A } ⊂ { α } .Now, the argument goes as follows.We start with the gauge-fixed partition function Z [ J ] = (cid:90) D ϕ D η D ¯ η e i ( S − ξ F A F A + ¯ η A δ α F A η α + J i ϕ i ) . (32)The ghost propagator is defined to be the inverse of the Faddeev-Popov operator δ α F A , δ α F A G αB = − δ AB . (33)The above action is invariant under the BRST transformation, δ BRST ϕ i = ζη α δ α ϕ i , δ BRST ¯ η A = ζ F A /ξ, δ BRST η α = − ζf αβγ η β η γ . (34)Now, consider a correlation function (cid:104) O ( ϕ, ¯ η, η ) (cid:105) ≡ e − iW (cid:90) D ϕDηD ¯ η O e i ( S − ξ F A F A + ¯ η A δ α F A η α + J i ϕ i ) . (35)An easy but useful observation is that this function vanishes for an operator with odd power ofghost fields, following from ghost number conservation. An example that turns out to be useful is (cid:104) ¯ η A ( x ) G A ( x ) (cid:105) = 0 , (36)where G can be a function of any non-ghost fields. Then, one can apply a BRST transformationto this equation, to obtain another form of the same equality, so that it encodes the outcomes ofthe BRST symmetry. δ BRST (cid:104) ¯ η A G A ( x ) (cid:105) = (cid:104) δ BRST [¯ η A G A ( x )] + i ¯ η A G A ( x ) J i δ BRST ϕ i (cid:105) = 0 , (37)where the second term comes from the non-zero background current support. We can rewrite thisas (cid:10) F A G A ( x ) /ξ − ¯ η A η α δ α G A ( x ) (cid:11) = − iJ i (cid:10) η α δ α ϕ i ¯ η A G A ( x ) (cid:11) . (38)The identity (38) is of crucial importance, hence we will come back to it later. To derive theNielsen equation, we need to check how the connected-diagram-generating functional W [ J ] varieswith a change of the gauge choice. Notice that this DeWitt notation will only be used in this section to avoid clustering of irrelevant symbols.
F → F (cid:48) = F + ∆ F . Then, W becomes W (cid:48) = − i ln (cid:90) D ϕ D η D ¯ η e i ( S − ξ F (cid:48) A F (cid:48) A + ¯ η A δ α F (cid:48) A η α + J i ϕ i ) (39) = − i ln (cid:90) D ϕ D η D ¯ η e i ( S − ξ F A F A + ¯ η A δ α F A η α + J i ϕ i ) (cid:0) − i F A ∆ F A /ξ + i ¯ η A δ α ∆ F A η α (cid:1) = W − i (cid:104)F A ∆ F A /ξ − ¯ η A η α δ α ∆ F A (cid:105) , (40)with the change given by ∆ W = − i (cid:104)F A ∆ F A /ξ − ¯ η A η α δ α ∆ F A (cid:105) . (41)Now we look at (38), we find that for G A there identified with ∆ F A , it gives the RHS of (41)after integrating over x . Thus, identifying the two equations, we get ∆ W = − J i (cid:10) η α δ α ϕ i ¯ η A ∆ F A (cid:11) . (42)Now, given that the effective action Γ is the Legendre transform of W , Γ[ φ ] = W [ J ] − J i ϕ i , (43)the variation in the effective action will be ∆Γ = δ Γ δφ i (cid:10) η α δ α ϕ i ¯ η A ∆ F A (cid:11) , (44)yielding the Nielsen identity in a generalized form. Here, ∆Γ is the variation of the effective actionunder a change in of the gauge choice ∆ F . As is clear, the RHS vanishes on-shell, since the δ Γ δφ i term is equal to J i and thus vanishes. However, off-shell, it still survives. As explained in Sect. 3,there is nevertheless no easy solution to the apparent on-shell gauge dependence in the inflationaryobservables. Hence our practical shortcut turns to the usage of off-shell gauge-invariant effectiveaction, ∆Γ ≡ .Clearly (42) and (44) show that the variation of the off-shell effective action, under a changeof the gauge choice, originates from the variation of the fields J i δ BRST ϕ i ∝ J i δ α ϕ i in the currentterms. Therefore, as it is also clear by (44), that if the current term is symmetric, i.e. , J i δ α ϕ i = 0 , W and Γ are invariant under gauge transformation.For example, in the case of U (1) model, we can choose polar coordinates with the current-fieldcoupling J ρ ρ , which is clearly gauge invariant, gives ∆ W = − (cid:90) d xJ ρ ( x ) (cid:10) δ α ρ ( x ) η α ¯ η A ∆ F A (cid:11) = 0 , (45)since ρ is a gauge singlet and δ α ρ ( x ) = 0 . This leads to an off-shell gauge-invariant effectiveaction. Note that perturbativity is not assumed in this formal derivation, and the off-shell gaugeinvariance should be valid non-perturbatively, as a direct consequence of symmetry. This showsthat an easy and practical way to achieve gauge independence is to use a polar-like current termto lift up the system away from the minimum. 13 The unitary limit
In addition to the straightforward way of adding a gauge-invariant current term, there is anotherpractical way to approach off-shell gauge-invariance, namely, one can take the unitary gauge limit.For an general choice of current, the symmetry of the system might not be preserved, and theeffective potential is manifestly dependent on the gauge-fixing parameter ξ . Inspired by [19, 20],we can check what role the unitary gauge plays in the flow equations. That is, how the apparentgauge-dependence of the effective potential behaves for large values of the gauge parameter. Bychecking the behavior of Nielsen’s flow equation at large ξ , we shall show that as ξ increases, thegauge dependence introduced through an asymmetric background current is weakened, and eveninsignificant for ξ large enough. Moreover, we show that in the unitary limit it is independent ofthe arbitrary external current, which can explain why the unitary gauge result exactly matchesthe gauge-invariant current result.For simplicity, we use the U (1) model quantized in the R ξ gauge as a demonstration, L = − F + | Dφ | − V ( | φ | ) − ξ ( ∂ · A − √ eξ ¯ φY ) + L ghost , (46)By expanding φ around a background φ = ¯ φ + ( X + iY ) / √ , we extract the Lagrangian up-toquadratic order as L = − X (cid:0) ∂ + 6 λ ¯ φ − m (cid:1) X − Y (cid:0) ∂ + 2 λ ¯ φ − m + 2 ξe ¯ φ (cid:1) Y + 12 A µ (cid:20) ( ∂ + 2 e ¯ φ ) η µν − (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) A ν − ¯ c ( ∂ + 2 ξe ¯ φ ) c −√ φ (2 λ ¯ φ − m ) X . (47)The last term in the quadratic Lagrangian is linear and represents a tadpole that must be removedby the external current via the steepest descent method. We shall use a one-parameter family ofexternal current J κ ( ¯ φ + X √ + Y κ + O ( XY )) , where κ controls the background current adaptedto different local coordinate systems on the scalar manifold. The physical meaning of κ is thecurvature radius of a constant Higgs line across ¯ φ , see Fig. 3. The disappearance of an X term isbecause we require the Higgs mass to be unchanged. Another way of viewing this is because wedo not wish to further confine the fields around ¯ φ in the X direction by the external current. Theremoval of the tadpole is accompanied by a change in the Goldstone mass. For a U (1) -symmetriccurrent κ = ¯ φ , the current term becomes J ρ ρ in disguise, and the Goldstone mass is canceled asshown in the Sect. 3.1, leading to a gauge-independent effective potential. However, if one usesan asymmetric current κ (cid:54) = ¯ φ (in particular, κ → ∞ is the conventional Cartesian current), the14oldstone remains massive, leading to an effective potential that does depend on ξ and κ : V eff ( ¯ φ, ξ ) | κ = V ( ¯ φ ) − i V T (cid:40)
Tr ln (cid:2) ∂ + 6 λ ¯ φ − m (cid:3) + Tr ln (cid:20) ∂ + (2 λ ¯ φ − m ) (cid:18) − ¯ φκ (cid:19) + 2 ξe ¯ φ (cid:21) + Tr ln (cid:20) − ( ∂ + 2 e ¯ φ ) η µν + (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:21) − (cid:2) ∂ + 2 ξe ¯ φ (cid:3) (cid:41) + (counterterms) . (48) - - - - X Y κ / ϕ = ∞ , Cartesian κ / ϕ =
1, Polar κ / ϕ = κ / ϕ = - κ / ϕ = - ϕ Figure 3: The κ parameter stands for different choices of local coordinate systems on R spannedby X, Y . For a given choice of coordinate, the Goldstone mode fluctuations across the averagefield ¯ φ along a constant Higgs line, whose curvature radius is given by κ . In this figure, we haveshown different constant Higgs lines with different choices of κ .We wish to use Nielsen’s identity (1) to show that the ξ -dependence of V eff is weakened in thelarge- ξ limit, provided that the potential gradient | ∂V eff ∂ ¯ φ | is bounded as ξ → ∞ , which is indeedthe case for the one-loop potential (48).In this U (1) model, the C ( ¯ φ, ξ ) function in Nielsen’s identity (1) is given by C ( ¯ φ, ξ ) = 1 δ ln ξ (cid:28) δX √ Y δY κ + ( higher order in fields ) (cid:29) (cid:12)(cid:12)(cid:12) ¯ φ . (49)Here δX and δY are a particular field redefinition that takes the form of a gauge transformationwhich preserves (33) after sending the gauge parameter from ξ → ξ + δξ . From (33), it is notdifficult to find the required field redefinition for R ξ gauge choices is δX = − e Y ∂ + 2 ξe ¯ φ (cid:16) ∂ · Aδ ln ξ + √ e ¯ φY δξ (cid:17) (50) δY = e √ φ + X ) 1 ∂ + 2 ξe ¯ φ (cid:16) ∂ · Aδ ln ξ + √ e ¯ φY δξ (cid:17) . (51)15hese are essentially the BRST transformation δ BRST ϕ i in (44) non-localized after integrating outthe ghost field.Therefore by the Nielsen identity, the dependence of the effective potential on the gaugeparameter is given by ∂V eff ∂ ln ξ = − e √ ∂V eff ∂ ¯ φ (cid:18) − ¯ φκ (cid:19) (cid:28) Y ∂ + 2 ξe ¯ φ (cid:16) ∂ · A + √ ξe ¯ φY (cid:17)(cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ φ , (52)where the higher order terms in fields are omitted since they are also higher order in ξ − than theleading order. Notice that for a gauge-invariant current choice, κ = ¯ φ and the effective potentialis directly independent of ξ , as shown in (44). In the large ξ limit, (cid:28) Y ∂ + 2 ξe ¯ φ (cid:16) ∂ · A + √ ξe ¯ φY (cid:17)(cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ φ ξ →∞ , fixing cutoff −−−−−−−−−−→ √ e ¯ φ (cid:104) Y (cid:105) = O ( ξ − ) . (53)Therefore, if | ∂V eff ∂ ¯ φ | < O ( ξ ) , the gauge dependence vanishes in the unitary limit, | ∂V eff ∂ ln ξ | (cid:46) O ( ξ − ) → . Indeed, the one-loop result behaves as ∂V eff ∂ ¯ φ = V (cid:48) ( ¯ φ ) − i V T (cid:40)
Tr 12 λ ¯ φ∂ + 6 λ ¯ φ − m + Tr (4 λ + 4 ξe ) ¯ φ − (6 λ ¯ φ − m ) /κ∂ + (cid:0) λ ¯ φ − m (cid:1) (cid:0) − ¯ φ/κ (cid:1) + 2 ξe ¯ φ + Tr (cid:34)(cid:18) − ( ∂ + 2 e ¯ φ ) η µν + (cid:18) − ξ (cid:19) ∂ µ ∂ ν (cid:19) − (cid:0) e ¯ φη νρ (cid:1)(cid:35) − ξe ¯ φ∂ + 2 ξe ¯ φ (cid:41) = gauge independent terms + i V T
Tr 4 ξe ¯ φ∂ + 2 ξe ¯ φ + O ( ξ − ) . (54)Holding the cut off fixed, we send ξ to infinity , the second term becomes independent of ξ .Therefore, V (cid:48) eff is bounded by a ξ -independent constant O ( ξ ) .As a result, combining (52), (53) and (54), the dependence of V eff on the gauge parameter isgiven by ∂V eff ∂ ln ξ ξ →∞ , fixing κ & cutoff −−−−−−−−−−−−−→ O ( ξ − ) . (55)Thus, taking the limit ξ → ∞ , we see that the effective potential is independent of any ξ -rescalingin this unitary limit. This is for the leading order. Actually if one consider the full expansion of ρ into X, Y fields, the C functionvanish order by order in the X, Y series, since ρ itself is to all orders a gauge singlet. Note that this is necessary since the purpose of sending ξ to infinity is to kill the Goldstone degree of freedom,hence the unitary limit. If we renormalize and send the cutoff to infinity first, then however large ξ is, at asufficiently high energy scale in the loop integral, the Goldstone degree of freedom will enter and bring the gaugedependence along into the result.
16o show the independence of the effective potential on the current choice κ , we take thederivative of (48) with respect to ln κ : ∂V eff ∂ ln κ = − i V T Tr (cid:0) λ ¯ φ − m (cid:1) ¯ φ/κ∂ + (cid:0) λ ¯ φ − m (cid:1) (cid:0) − ¯ φ/κ (cid:1) + 2 ξe ¯ φ ξ →∞ , fixing κ & cutoff −−−−−−−−−−−−−→ O ( ξ − ) . (56)Thus the dependence on the current choice also vanishes in the unitary limit. In this sense, weconsider unitary gauge result as the fixed point under the gauge flow of changing ξ defined bythe Nielsen identity (1). It is invariant under both the change of ξ (gauge choice) and κ (currentchoice), and is what the gauge-dependent effective potential converges to. We have given an explicit demonstration of the apparent gauge dependence issue in Higgs-likeinflation scenario, namely its non-negligible influence on the n s - r prediction of inflation. We qual-itatively analyzed the possible underlying causes and found it might be due to either a truncationof the non-perturbative effective action to second order in gradients, or curved spacetime effectsin the loop diagram evaluation of the effective action. To resolve this problem, one can either seekmore rigorous but complicated ways in the conventional formalism, or find out -as we did- howexactly the gauge dependence infiltrates the effective potential and construct an off-shell gaugeindependent one.We found that if one chooses as fundamental fields some field space coordinates that give acurrent term which violates gauge symmetry, the resulting effective potential is spoiled. Therefore,we have used as fundamental fields those whose corresponding current terms are gauge-invariant.This is equivalent to choosing polar coordinates on the scalar manifold. In this way we haveobtained an off-shell gauge-independent effective potential. After working out various examplesand giving our n s - r prediction for the Abelian model, we used Nielsen’s identity to show why itworks even non-perturbatively. At last we discussed the relation between the polar coordinatesresult and the unitary gauge result. We found that in accordance with Nielsen’s identity, boththe gauge dependence and the current choice dependence is weakened in the unitary limit andthe resulting effective potential all converge to that of the polar coordinates.Finally we would like to mention that our method, being a practical shortcut, faces severalchallenges. • Our method is different from the mainstream approach. Therefore, even though it gives agauge independent result, a key challenge is to prove that this is the right way to get tocorrect physical predictions. • Second is the integration of the ρ field, which we treat as a Gaussian, whereas the integralbounds make it an error function. Our key assumption here is that the potential at eachpoint receives significant corrections from neighboring points only. Moreover, as we arehere interested in Higgs inflation, we expect large values of ρ rather than inflating near theorigin, where polar coordinates are ill-defined. Large field values come with large secondderivatives V (cid:48)(cid:48) ( ρ ) , hence leading to a restriction localizing ρ . This kind of field-space localityneeds further elaboration to be proven a good approximation. It is good to note that sucha difficulty does not arise in the unitary gauge. • The third shortcoming of our method is that for the angular-field integrals, the truncation tosecond order might be questionable, especially for non-Abelian theories. This is due to the17xistence of terms in the Lagrangian (e.g. schematically, terms such as ρ ( ∂θ ) in the scalarQED case, and ¯ φ sin θθ ( ∂θ ) in the SU (2) case) which are not suppressed at higher looporders, and the geometry of the coset manifold is not taken into account here by truncatingto second order in angular variables.As a result, the true accuracy of our method should in the end be determined by comparison tomore rigorous first-principle computations or future experiments. Despite the weaknesses, whichwe have left for future work, our method is highly practical for cosmological considerations. Acknowledgement
We would like to thank Henry Tye for guidance and initial collaboration. We also thank AndrewCohen for helpful discussions, comments, and constructive criticism. We are indebted to KunfengLyu, Yi Wang and Haitham Zaraket for their valuable suggestions on our manuscript.
A Effective potential in Abelian Higgs inflation
We first work in the conventional Cartesian field coordinates. In the Jordan frame, we neglectthe spacetime curvature and approximate g µν ≈ η µν . Then the Abelian Higgs model (46) yields a ξ -dependent effective potential regularized in the MS-scheme, V eff ( ¯ φ, ξ ) = − m ¯ φ + λ ¯ φ + 14(4 π ) (cid:40) (cid:88) b = X,Y,A i m b ( ¯ φ, ξ ) (cid:20) ln m b ( ¯ φ, ξ ) M − (cid:21) − m c ( ¯ φ, ξ ) (cid:20) ln m c ( ¯ φ, ξ ) M − (cid:21)(cid:41) , (57)where m X = 6 λ ¯ φ − m m Y = 2 λ ¯ φ − m + 2 ξe ¯ φ m A = 2 e ¯ φ m c = 2 ξe ¯ φ , (58)and M is an arbitrary scale to be determined by renormalization. Note that in this appendixwe drop subscript J in the classical field ¯ φ J for convenience and ¯ φ is understood to be held by abackground current. For simplicity, let us consider the massless case m = 0 . And we impose therenormalization condition for V eff as ∂ V eff ∂ ¯ φ (cid:12)(cid:12)(cid:12) ¯ φ = µ = 4! λ . (59)Solving M in terms of µ , we obtain V eff ( ¯ φ, ξ ) = λ ¯ φ + 3 e + 10 λ + 2 ξλe (4 π ) ¯ φ (cid:20) ln ¯ φ µ − (cid:21) , (60)in agreement with [1] in the Landau gauge ξ → .18igure 4: The loop diagrams contributing to the finite part of Z ( ¯ φ, ξ ) in Cartesian coordinates,which is dominated by the gauge-boson-Goldstone loop in the parameter regime we are interestedin. The field strength factor Z ( ¯ φ, ξ ) receives contributions from five diagrams shown in Fig. 4,of which the dominating diagram is the logarithmically divergent gauge-boson-Goldstone loop.Taking their momentum derivatives at zero external momenta, we obtain the MS-scheme fieldstrength factor Z − π ) (cid:40) λ ¯ φ m X + 2 λ ¯ φ m Y − ξe ¯ φ m A + e ¯ φ (11 ξ + 4 ξ − ξ ln ξ − ξ − m A + e ξ m A ( ξ m A − ξm A m Y − m Y ) ln ξ ( m Y − ξm A ) + e ( m A − m Y ) ( m Y − ξm A ) × (cid:104) m A ln m A M (cid:16) ( ξ − ξ m A − ξ (cid:0) ξ + ξ − (cid:1) m A m Y + ξ (( ξ − ξ − m A m Y + (cid:0) ξ + 3 (cid:1) m Y (cid:17) + m Y ln m Y M (cid:16) ξ m A + ξ (2 ξ − m A m Y − ξ − ξm A m Y + ( ξ − m Y (cid:17)(cid:105)(cid:41) . (61)In the massless case, we impose the renormalization condition Z (cid:12)(cid:12)(cid:12) ¯ φ = µ − (62)and solve M in terms of µ . The result simplifies considerably, Z ( ¯ φ, ξ ) = 1 + e (4 π ) (3 − ξ ) ln ¯ φ µ . (63)In the polar-coordinate case, φ = ( ¯ φ + ρ ) e iθ , we also work in the R ξ gauge family and add agauge-fixing term − ξ ( ∂ · A − eξθ ) . the effective potential looks the same as (57) but withoutthe Goldstone and ghost terms, V eff, polar ( ¯ φ, ξ ) = − m ¯ φ + λ ¯ φ + 14(4 π ) (cid:88) b = X,A i m b ( ¯ φ, ξ ) (cid:20) ln m b ( ¯ φ, ξ ) M − (cid:21) , (64)19sing the same renormalization condition (59), the massless limit gives V eff, polar ( ¯ φ, ξ ) = λ ¯ φ + 3 e + 9 λ (4 π ) ¯ φ (cid:20) ln ¯ φ µ − (cid:21) . (65)The field-strength factor receives contribution from four diagrams shown in Fig. 5. After applyingFigure 5: The loop diagrams contributing to the finite part of Z polar ( ¯ φ, ξ ) in polar coordinates.There is no ghost loop because the polar-coordinate ghost is free and does not couple to the ρ field.the renormalization condition (62), we obtain Z polar ( ¯ φ, ξ ) = 1 + 3 e (4 π ) ln ¯ φ µ (66)in the massless case. The agreement of Z polar ( ¯ φ, ξ ) with that computed in the covariant gaugefamily in [19] shows again the gauge-family-independence of polar-coordinate quantities.Now, the combined field redefinition (6) can be performed separately in two steps. Namely wefirst compute the canonical field in Jordan frame, taking into account of the radiative corrections,and then we go to the Einstein frame.The canonical field in Jordan frame is given by ¯ φ c = √ (cid:82) Z ( ¯ φ, ξ ) / d ¯ φ . For all reasonableparameter choices, | δZ | = | Z − | (cid:46) e (4 π ) × O (1) (cid:28) , thus we can approximate ¯ φ = 1 √ (cid:34) ¯ φ c − (cid:90) ¯ φ c √ µ δZ ( y/ √ , ξ ) dy (cid:35) . (67)The effective potential expressed in the canonical field ¯ φ c becomes then V eff ( ¯ φ ( ¯ φ c , ξ ) , ξ ) through(67). After canonicalizing the Higgs field in the Jordan frame, the second step is going to theEinstein frame by a field redefinition χ = (cid:90) (cid:115) Ω + 6 α ¯ φ c /M p Ω d ¯ φ c ≈ √ M p ln √ α ¯ φ c M p , where Ω = 1 + α ¯ φ c M p . (68)Here we have approximated the integral by considering our assumed parameter regime Ω (cid:29) and (cid:28) √ α (cid:28) (see [24] for more discussions). The shape of the final potential U eff ( χ, ξ ) = V eff ( ¯ φ ( ¯ φ c ( χ ) ,ξ ) ,ξ )Ω is shown in Fig. 6.Notice that in principle, a Renormalization Group (RG) improvement of V eff is still neededto resum large logarithms. In this work, for simplicity, we did not perform such an analysis.Instead, we try to avoid large logarithms by renormalizing the system around its typical scaleduring inflation, namely χM p ∼ . ⇔ ¯ φM p ∼ . , thus keeping ¯ φ ∼ µ for µM p = 0 . . 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