Emerging Translational Variance: Vacuum Polarization Energy of the $\mathbf{\phi^6}$ kink

We propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow a decomposition into symmetric and anti-symmetric channels in one space dimension. In particular we compute the vacuum polarization energy of the kink soliton in the $\phi^6$ model. We link the dependence of this energy on the position of the center of the soliton to the different masses of the quantum fluctuations at negative and positive spatial infinity.

Although the φ 6 model is not fully renormalizable, at one loop order the ultra-violet divergences can be removed unambiguously. However, another very interesting phenomenon emerges. The distinct topological structures induce non-equivalent vacua that manifest themselves via different dispersion relations for the quantum fluctuations at positive and negative spatial infinity. At some intermediate position the soliton mediates between these vacua. Since this position cannot be uniquely determined the resulting VPE exhibits a translational variance. This is surprising since, after all, the model is defined through a local and translational invariant Lagrangian. In this paper we will describe the emergence of this variance and link it to the different level densities that arise from the dispersion relations. To open these results for discussion 2 it is necessary to review in detail the methods developed in Ref. [20] to compute the VPE for backgrounds in one space dimension that are not (manifestly) invariant under spatial reflection.
Following this introductory motivation we will describe the φ 6 model and its kink solutions. In chapter III we will review the spectral method that ultimately leads to a variant of the Krein-Friedel-Lloyd formula [21] for the VPE. The novel approach to obtain the relevant scattering data will be discussed in chapter IV and combined with the one-loop renormalization in chapter V. A comparison with known (exact) results will be given in chapter VI while chapter VII contains the predicted VPE for the solitons of the φ 6 model. Translational variance of the VPE that emerges from the existence of non-equivalent vacua will be analyzed in chapter VIII. We conclude with a short summary in chapter IX.

II. KINKS IN φ 6 MODELS
In D = 1 + 1 dimensions the dynamics for the quantum field φ are governed solely by a field potential U (φ) that is added to the kinetic term For the φ 6 model we scale all coordinates, fields and coupling constants such that the potential contains only a single dimensionless parameter a From figure 1 we observe that there are three general cases. For a 2 > 1 2 two degenerate minima at φ = ±1 exist. For 0 < a 2 ≤ 1 2 an additional local minimum emerges at φ = 0. Finally, for a = 0 the three minima at φ = 0 and φ = ±1  are degenerate. Soliton solutions connect different vacua between negative and positive spatial infinity. For a = 0 the vacua are at φ = ±1 and the corresponding soliton solution is [12] φ The case a = 0 is actually more interesting because two distinct soliton solutions do exist. The first one connects φ = 0 at x → −∞ to φ = 1 at x → ∞, while the second one interpolates between φ = −1 and φ = 0, These soliton configurations are shown in figure 2. In either case the classical mass is E cl = 1 4 = 1 2 lim a→0 E cl (a). This relation for the classical energies reflects the fact that as a → 0 the solution φ K (x) disintegrates into two widely separated structures one corresponding to φ K1 (x) the other to φ K2 (x).
The computation of the VPE requires the construction of scattering solutions for fluctuations about the soliton. In the harmonic approximation the fluctuations experience the potential generated by the soliton (φ sol = φ K , φ K1 or φ K2 ). These three potentials are shown in figure 3. For a = 0 the potential is invariant under x ↔ −x. But the particular case a ≡ 0 is not reflection symmetric, though x ↔ −x swaps the potentials generated by φ K1 and φ K2 . The loss of this invariance disables the separation of the fluctuation modes into symmetric and anti-symmetric channels, which is the one dimensional version of partial wave decomposition.
Even more strikingly, the different topological structures in the a = 0 case cause lim x→−∞ V (x) = lim x→∞ V (x), which implies different masses (dispersion relations) for the fluctuations at positive and negative spatial infinity.

III. SPECTRAL METHODS AND VACUUM POLARIZATION ENERGY
The formula for the VPE, Eq. (12) below, can be derived from first principles in quantum field theory by integrating the vacuum matrix element of the energy density operator [22]. It is, however, also illuminative to count the energy levels when summing the changes of the zero point energies. This sum is O( ) and thus one loop order ( = 1 for the units used here). We call the single particle energies of fluctuations in the soliton type background ω n while the ω (0) n are those for the trivial background. Then the VPE formally reads where the subscript indicates that renormalization is required to obtain a finite and meaningful result. On the right hand side we have separated the explicit bound state (sum of energies ǫ j ) and continuum (integral over momentum k) contributions. The latter involves ∆ ρ ren. (k) which is the (renormalized) change of the level density induced by the soliton background. Let L be a large distance away from the localized soliton background. For x ∼ L the stationary wave-function of the quantum fluctuation is a phase shifted plane wave ψ(x) ∼ sin [kx + δ(k)], where δ(k) is the phase shift (of a particular partial wave) that is obtained from scattering off the potential, Eq. (6). The continuum levels are counted from the boundary condition ψ(L) = 0 and subsequently taking the limit L → ∞. The number n(k) of levels with momentum less or equal to k is then extracted from kL + δ(k) = n(k)π. The corresponding number in the absence of the soliton is n (0) (k) = kL/π, trivially. From these the change of the level density is computed via which is often referred to as the Krein-Friedel-Lloyd formula [21]. Note that ∆ ρ(k) is a finite quantity; but ultraviolet divergences appear in the momentum integral in Eq. (7) and originate from the large k behavior of the phase shift. This behavior is governed by the Born series where the superscript reflects the power at which the potential, Eq. (6) contributes. Though this series does not converge 3 for all k, it describes the large k behavior well since δ (N +1) (k)/δ (N ) (k) ∝ 1/k 2 when k → ∞. Hence produces a finite integral in Eq. (7) when N is taken sufficiently large. We have to add back the subtractions that come with this replacement. Here the spectral methods take advantage of the fact that each term in the subtraction is uniquely related to a power of the background potential and that Feynman diagrams represent an alternative expansion scheme for the vacuum polarization energy The full lines are the free propagators of the quantum fluctuations and the dashed lines denote insertions of the background potential, Eq. (6), eventually after Fourier transformation. These Feynman diagrams are regularized with standard techniques, most commonly in dimensional regularization. They can thus be straightforwardly combined with the counterterm contribution, E CT [V ] with coefficients fully determined in the perturbative sector of the theory.
This combination remains finite when the regulator is removed.
The generalization to multiple channels is straightforward by finding an eventually momentum dependent diagonalization of the scattering matrix S(k) and summing the so-obtained eigenphase shifts. This replaces 4 δ(k) −→ (1/2i)lndet S(k) and analogously for the Born expansions, Eqs. (9) and (10). Since after Born subtraction the integral converges, we integrate by parts to avoid numerical differentiation and to stress that the VPE is measured with respect to the translationally invariant vacuum. We then find the renormalized VPE to be, with the sum over partial waves re-inserted, Here D ℓ is the degree of degeneracy, e.g. D ℓ = 2ℓ + 1 in three space dimensions. The subscript N refers to the subtraction of N terms of the Born expansion, as e.g. in Eq. (10). We stress that, with N taken sufficiently large, both the expression in curly brackets and the sum E N are individually ultra-violet finite and no cut-off parameter is needed [23].

IV. SCATTERING DATA IN ONE SPACE DIMENSION
In this section we obtain the scattering matrix for general one dimensional problems and develop an efficient method for its numerical evaluation. This will be at the center of the novel approach to compute the VPE.
We first review the standard approach that is applicable when V (−x) = V (x), e.g. left panel of figure 3. Then the partial wave decomposition separates symmetric ψ S (−x) = ψ S (x) and anti-symmetric, ψ A (−x) = −ψ A (x) channels.
The respective phase shifts can be straightforwardly obtained in a variant of the variable phase approach [24] by parameterizing ψ(x) = e i[kx+β(k,x)] and imposing the obvious boundary conditions ψ ′ S (0) = 0 and ψ A (0) = 0. (The prime denotes the derivative with respect to x.) The wave-equation turns into a non-linear differential equation for the phase function β(k, x). When solved subject to lim x→∞ β(k, x) = 0 and lim x→∞ β ′ (k, x) = 0 the scattering matrix given by [11] Linearizing and iterating the differential equation for β(k, x) yields the Born series, Eq. (9). At this point it is advantageous to use the fact that scattering data can be continued to the upper half complex momentum plane [25].
That is, when writing k = it, the Jost function, whose phase is the scattering phase shift when k is real, is analytic for Re[t] ≥ 0. Furthermore the Jost function has simple zeros at imaginary k = iκ j representing the bound states.
Formulating the momentum integral from Eq. (12) as a contour integral automatically collects the bound state contribution and we obtain a formula as simple as [11,22] for the VPE. Here g(t, x) is the non-trivial factor of the Jost solution whose x → 0 properties determine the Jost function. The factor function solves the differential equation with the boundary conditions g(t, ∞) = 1 and g ′ (t, ∞) = 0; iterating g(t, x) = 1 + g (1) (t, x) + g (2) (t, x) + . . . produces the Born series.
In general, however, the potential V (x) is not reflection invariant and no partial wave decomposition is applicable.
Even more, there may exist different masses for the quantum fluctuations as it is the case for the φ 6 model with a = 0, cf. right panel of figure 3. We adopt the convention that m L ≤ m R , otherwise we simply swap x → −x. Three different cases must be considered. First, above threshold both momenta To formulate the variable phase approach we introduce the matching point x m and parameterize Observe that the pseudo potential vanishes at positive and negative spatial infinity. The differential equations (17) By equating the solutions and their derivatives at x m the scattering matrix is obtained from the factor functions as where A = A(x m ), etc.. The second case refers to k ≤ m 2 R − m 2 L still being real but q = iκ becoming imaginary with κ = m 2 R − m 2 L − k 2 . The parameterization of the wave function for x > x m changes to ψ(x) = B(x)e −κx yielding the differential equation B ′′ (x) = κB ′ (x) + V p (x)B(x). The scattering matrix then is a single unitary number It is worth noting that V p ≡ 0 corresponds to the step function potential. In that case the above formalism obviously yields A ≡ B ≡ 1 and reproduces the textbook result In the third regime also k becomes imaginary and we need to identify the bound states energies ǫ ≤ m L that enter Eq. (12). We define real variables λ = m 2 L − ǫ 2 and κ(λ) = m 2 R − m 2 L + λ 2 and solve the wave equation subject to the initial conditions where x min and x max represent negative and positive spatial infinity, respectively. Continuity of the wave function requires the Wronskian determinant to vanish. This occurs only for discrete values λ j that in turn determine the bound state energies 5 ǫ j = m 2 L − λ 2 j .

V. ONE LOOP RENORMALIZATION IN ONE SPACE DIMENSION
To complete the computation of the VPE we need to substantiate the renormalization procedure. We commence by identifying the ultra-violet singularities. This is simple in D = 1 + 1 dimensions at one loop order as only the first diagram on the right hand side of Eq. (11) is divergent. Furthermore, this diagram is local in the sense that where ǫ is the regulator (e.g. from dimensional regularization). Hence a counterterm can be constructed that not only removes the singularity but the diagram in total. This is the so-called no tadpole condition and implies In the next step we must identify the corresponding Born term in Eq. (9). To this end it is important to note that the counterterm is a functional of the full field φ(x) that induces the background potential, Eq. (6). Hence we must find the Born approximation for V (x) − m 2 L rather than the one for the pseudo-potential V P (x), Eq. (18). The standard formulation of the Born approximation as an integral over the potential is, unfortunately, not applicable to V (x) − m 2 L since it does not vanish at positive spatial infinity. However, we note that V (x)−m 2 L = V P (x)+(m 2 L −m 2 R )Θ(x−x m ) = V p (x) + V step (x) and that, by definition, the first order correction is linear in the background, and thus additive. We may therefore write The Born approximation for the step function potential has been obtained from the large k expansion of δ step (k) in FD and E CT require further regularization when m L = m R . In that case no further finite renormalization beyond the no tadpole condition is realizable.

VI. COMPARISON WITH KNOWN RESULTS
Before presenting detailed numerical results for VPEs, we note that all simulations were verified to produce S † S = 1 after attaching pertinent flux factors to the scattering matrix, Eq. (19). These flux factors are not relevant for the VPE as they multiply to unity under the determinant in Eq. (12). In addition the numerically obtained phase shifts, i.e. (1/2i)lndet S, have been monitored to not vary with x m . Since this is also the case for the bound energies, the VPE is verified to be independent of the unrestricted choice for the matching point.
The VPE calculation based on Eq. (12) has been applied to the φ 4 kink and sine-Gordon soliton models that are defined via the potentials respectively. The soliton solutions φ K = tanh(x − x 0 ) and φ SG (x) = 4arctan e −2(x−x0) induce the scattering In  14) can straightforwardly applied [11]. However, this method singles out x 0 (typically set to x 0 = 0) to determine the boundary condition in the differential equation and therefore cannot be used to establish translational invariance of the VPE. On the contrary, the boundary conditions for Eq. (17) are not at all sensitive to x 0 and we have applied the present method to compute the VPE for various choices of x 0 , all yielding the same numerical result.
The next step is to compute the VPE for asymmetric background potentials that have m = m L = m R . For the lack of a soliton model that produces such a potential we merely consider a two parameter set of functions for the pseudo potential in Eq. (17). Although Eq. (14) is not directly applicable, it is possible to relate V R,σ (x) to the symmetric potential and apply Eq. (14). In the limit R → ∞ interference effects between the two structures around x = ±R disappear resulting in twice the VPE of Eq. (29). The numerical comparison is listed in table I. Indeed the two approaches produce identical results as R → ∞. The symmetrized version converges only slowly for wide potentials (large σ) causing obstacles for the numerical simulation that do not at all occur in the present approach.

VII. VACUUM POLARIZATION ENERGIES IN THE φ 6 MODEL
We first discuss the VPE for the a = 0 case. A typical background potential is shown in the left panel of figure 1.
Obviously it is reflection invariant and thus the method based on Eq. (14) is applicable. In table II we also compare  our results to those from the heat kernel expansion of Ref. [15] since, to our knowledge, it is the only approach that has also been applied to the asymmetric a = 0 case in Ref. [14]. Not surprisingly, the two methods based on scattering data agree within numerical precision for all values of a. The heat kernel results also agree for moderate and large a; but for small values deviations of the order of 10% are observed. The heat kernel method relies on truncating the expansion of the exact heat kernel about the heat kernel in the absence of a soliton. Although in Ref. [15] the expansion has been carried out to eleventh(!) order, leaving behind a very cumbersome calculation, this does not seem to provide sufficient accuracy for small a.
We are now in the position to discuss the VPE for a = 0 associated with the soliton φ K1 (x) from Eq. (4). The potentials for the fluctuations and the resulting scattering data are shown in figure 4. By construction, the pseudo potential jumps at x m = 0. However, neither the phase shift nor the bound state energy (the zero mode is the sole bound state) depends on x m . As expected, the phase shift has a threshold cusp at m 2 R − m 2 L = √ 3 and approaches π 2 at zero momentum. This is consistent with Levinson's theorem in one space dimension [27] and the fact that there is only a single bound state. In total we find a significant cancellation between bound state and continuum contributions The result 6 −0.1264 √ 2 = −0.1788 of Ref. [14] was estimated relative to V α (x) = 3 2 [1 + tanh(αx)] for α = 1. Our results for various values of α are listed in table III. These results are consistent with V α (x) turning into a step function for large α. For the particular value α = 1 our relative VPE thus is ∆E vac = −0.0469 − 0.1660 = −0.2129. In view of the results shown in table II, especially for small a, these data match within the validity of the approximations applied in the heat kernel calculation.

VIII. TRANSLATIONAL VARIANCE
So far we have computed the VPE for the φ 6 model soliton centered at x 0 = 0. We have already mentioned that there is translational invariance for the VPE of the kink and sine-Gordon solitons. It is also numerically verified for the asymmetric background, Eq. (29). In those cases the two vacua at x → ±∞ are equivalent and q = k in Eq. (19). When shifting x → x + x 0 , the transmission coefficients (s 11 and s 22 ) remain unchanged relative to the 6 The factor √ 2 is added to adjust the datum from Ref. [14] to the present scale. The effect is immediately linked to varying the width of a symmetric barrier potential with height m 2 R − m 2 L = 3: For this potential the Jost solution, Eq. (15) can be obtained analytically [20] and the VPE has the limit which again reveals the background independent slope observed above.
Having quantitatively determined the translation variance of the VPE, it is tempting to subtract E vac V (x0) SB .
Unfortunately this is not unique because x 0 is not the unambiguous center of the soliton. For example, employing the classical energy density ǫ(x) to define the position of the soliton 1/ √ 1 + e −2(x−x) , that is formally centered at x, as an expectation value leads to  This changes the VPE by approximately 0.050. This ambiguity also hampers the evaluation of the VPE as half that of a widely separated kink-antikink pair similarly to the approach for Eq. (30). The corresponding background potential V B is shown in figure 5. For computing the VPE, the large contribution from the constant but non-zero potential in the regime |x| x should be eliminated.
The above considerations lead to When the VPE from V Now we also understand why the VPE for a = 0 diverges as a → 0, cf. table II. In that limit kink and antikink structures separate and the "vacuum" in between produces an ever increasing contribution (in magnitude).
Finally, we discuss the link between the translational variance and the Krein-Friedel-Lloyd formula, Eq. (8). We have already reported the VPE for the step function potential when x m = 0. We can also consider x m → ∞: reproducing the linear dependence on the position from above. Formally, i.e. without Born subtraction, the integral, Essentially this is that part of the level density that originates from the different dispersion relations at positive and negative spatial infinity.

IX. CONCLUSION
We have advanced the spectral methods for computing vacuum polarization energies (VPE) to also apply for static localized background configurations in one space dimension that do not permit a parity decomposition for the quantum fluctuations. The essential progress is the generalization of the variable phase approach to such configurations. Being developed from spectral methods, it adopts their amenities, as for e.g. an effective procedure to implement standard renormalization conditions. A glimpse at the bulky formulas for the heat kernel expansion (alternative method to the problem) in Refs. [14][15][16] immediately reveals the simplicity and effectiveness of the present approach. The latter merely requires to numerical integrate ordinary differential equations and extract the scattering matrix thereof, cf.
Eqs. (17) and (20). Heat kernel methods are typically combined with ζ-function regularization. Then the connection to standard renormalization conditions is not as transparent as for the spectral methods, though that is problematic only when non-local Feynman diagrams require renormalization, i.e. in larger than D = 1 + 1 dimensions or when fermion loops are involved.
We have verified the novel method by means of well established results, as, e.g. the φ 4 kink and sine-Gordon solitons.
For these models the approach directly ascertains translational invariance of the VPE. Yet, the main focus was on the VPE for solitons in φ 6 models because its soliton(s) may connect in-equivalent vacua leading to background potentials that are not invariant under spatial reflection. This model is not strictly renormalizable. Nevertheless at one loop order a well defined result can be obtained from the no-tadpole renormalization condition albeit no further finite renormalization is realizable because the different vacua yield additional infinities when integrating the counterterm. The different vacua also lead to different dispersion relations for the quantum fluctuations and thereby induce translational variance for a theory that is formulated by an invariant action. We argue that this variance is universal, as it is not linked to the particular structure of the background and can be related to the change in the level density that is basic to the Krein-Friedel-Lloyd formula, Eq. (8).
Besides attempting a deeper understanding of the variance by tracing it from the energy momentum tensor, future studies will apply the novel method to solitons of the φ 8 model. Its elaborated structure not only induces potentials that are reflection asymmetric, but also leads to a set of topological indexes [28] that are related to different particle numbers. Then the novel method will progress the understanding of quantum corrections to binding energies of compound objects in the soliton picture. Furthermore the present results can be joined with the interface formalism [29], that augments additional coordinates along which the background is homogeneous, to explore the energy (densities) of domain wall configurations [30].