Vacuum polarization energy for general backgrounds in one space dimension

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Physics

Letters

B

www.elsevier.com/locate/physletb

Vacuum

polarization

energy

for

general

backgrounds

in

one

space

dimension

H. Weigel

PhysicsDepartment,StellenboschUniversity,Matieland7602,SouthAfrica

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received3November2016

Receivedinrevisedform9December2016 Accepted24December2016

Availableonline28December2016 Editor: A.Ringwald

Forfield theoriesinone timeand onespacedimensionsweproposean efficientmethodtocompute the vacuum polarizationenergy ofstatic field configurationsthat do not allowa decompositioninto symmetricandanti-symmetricchannels.Themethodalsoappliestoscenariosinwhichthemassesofthe quantumfluctuationsatpositiveandnegativespatialinfinityaredifferent.Asanexamplewecompute the vacuumpolarizationenergy ofthekinksolitoninthe φ6 _{model.We linkthe dependenceofthis} energyonthepositionofthesolitontothedifferentmasses.

©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Vacuumpolarizationenergies(VPE)sumtheshiftsofzeropoint energies of quantum fluctuations that interact with a (classical) background potential. Spectral methods [1] have been very suc-cessful in computingVPEs particularlyfor background configura-tionswithsufficient symmetrytofacilitateapartialwave decom-positionforthequantumfluctuations. Inthisapproach scattering dataparameterize Green functions fromwhich the VPE is deter-mined. In particular the imaginary part of the two-point Green function atcoincident points, i.e. thedensity ofstates, isrelated tothephaseshiftofpotentialscattering[2].Amongotherfeatures, thesuccessofthespectralmethods drawsfromthedirect imple-mentationofbackground independentrenormalizationconditions byidentifyingtheBornseriesforthescatteringdatawiththe ex-pansionoftheVPEinthestrengthofthepotential.Theultra-violet divergences are contained in the latter and can be re-expressed asregularized Feynman diagrams. In renormalizable theories the divergences are balanced by counterterms whosecoefficients are fullydeterminedintheperturbativesectorofthequantumtheory inwhichthepotentialiszero.

Forfield theoriesinonespacedimensionthepartialwave de-composition separates channelsthat are even or odd under spa-tialreflection.We proposea veryefficientmethod,that infactis basedon the spectral methods, to numericallycompute the VPE forconfigurationsthatevadeadecompositionintoparityevenand oddchannels.Thisisparticularlyinterestingforfieldtheoriesthat contain classical solitonsolutions connecting vacua in which the

E-mailaddress:weigel@sun.ac.za.

massesofthequantumfluctuationsdiffer.Aprimeexampleisthe

φ

6 model.Forthismodelsome analyticalresults,inparticularthe scatteringdataforthequantumfluctuations, havebeendiscussed a while ago inRefs. [3,4]. However, a full calculation ofthe VPE hasnotyetbeenreported.Adifferentapproach,basedontheheat kernel expansionwith

ζ

-function regularization [5,6] hasalready beenappliedtothismodel[7].1_{Thisapproachrequiresanintricate}

formalism ontop ofwhichapproximations(truncation ofthe ex-pansion)arerequired.Wewillseethattheybecomelessaccurate asthe background becomes sharper. We also note that a similar probleminvolvingdistinct vacua occursinscalarelectrodynamics whencomputingthequantumtensionofdomainwalls[11].

We briefly review the setting of the one-dimensional prob-lem. The dynamics of the field

φ

= φ(

t

,

x

)

is governed by the Lagrangian

L

=

1

2

∂

μ

φ ∂

μ

_φ

₋

_U

_{(φ) .}

₍₁₎

The self-interaction potential U

(φ)

typically has distinct minima andthere mayexist several staticsolitonsolutions that interlink betweentwosuchminimaasx

→ ±∞

.Wepickaspecificsoliton, say

φ

0

(

x

)

andconsidersmallfluctuationsaboutit

φ (

t

,

x

)

= φ

0

(

x

)

+

η

(

t

,

x

) .

(2) Up to linear order, the field equation turns into a Klein–Gordon typeequation



∂

μ

∂

μ

+

V

(

x

)



η

(

t

,

x

)

=

0

,

(3)

1 _{SeeRefs.[8–10]forreviewsofheatkernelandζ-functionmethods.}

http://dx.doi.org/10.1016/j.physletb.2016.12.055

0370-2693/©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

whereV

(

x

)

=

U

(φ

0

(

x

))

isthebackgroundpotentialgeneratedby the soliton. At spatial infinity V

(

x

)

approaches a constant to be identified as the mass (squared) ofthe quantum fluctuations. In general,ase.g. forthe

φ

6 _modelwithU

_(φ)

₌

λ

2

φ

2

(φ

2

− 

2

)

2,we allow limx→−∞V

(

x

)

=

limx→∞V

(

x

)

. This gives rise to different typesof quantum fluctuations. While

φ

0 is classical,the fluctua-tionsare subjecttocanonicalquantizationsothat theabove har-monicapproximation yieldsthe leadingquantumcorrection. Asa consequenceoftheinteractionwiththebackgroundthezeropoint energies of all modes change andthe (renormalized) sum of all thesechangesistheVPE,cf. Sec.3.

2. Phaseshifts

As will be discussed in Sec. 3 the sum of the scattering (eigen)phase shifts is essential to compute the VPE from spec-tralmethods.Weextractscatteringdatafromthestationarywave equation,

η(

t

,

x

)

→

e−i Et

_η

₍

_x

₎

_, E2

η

(

x

)

=



−∂

2 x

+

V

(

x

)



η

(

x

) .

(4)

According to the above described scenario we define m2 L

=

limx→−∞V

(

x

)

and m2_R

=

limx→∞V

(

x

)

and take the convention mL

≤

mR,otherwise we justrelabelx

→ −

x. Weintroduce a

dis-continuouspseudopotential

Vp

(

x

)

=

V

(

x

)

−

m2L

+



m2_L

−

m2_R



(

xm) (5)

with

(

x

)

beingthestepfunction.Anyfinitevaluemaybechosen forthematchingpointxm.IncontrasttoV

(

x

)

, Vp

(

x

)

→

0 as x

→

±∞

.Thenthestationarywaveequation,(4)reads



−∂

2 x

+

Vp

(

x

)



η

(

x

)

=



k2

_η

₍

_x

_{) ,}

_{for x}

_≤

_x m q2

_η

₍

_x

_{) ,}

_{for x}

_≥

_x m (6) where k

=

E2

₋

_m2 L and q

=

E2

₋

_m2 R

=

k2

₊

_m2 L

−

m2R. We

emphasize that solving Eq. (6) is equivalent to solving Eq. (4). We factorize coefficient functions A

(

x

)

and B

(

x

)

appropriate for thescatteringproblem via

η(

x

)

=

A

(

x

)

eikx for x

≤

xm and

η(

x

)

=

B

(

x

)

eiqx_forx

_≥

_x

m:

A

(

x

)

= −

2ik A

(

x

)

+

Vp

(

x

)

A

(

x

)

and

B

(

x

)

= −

2iqB

(

x

)

+

Vp

(

x

)

B

(

x

) ,

(7)

where a prime denotes a derivative with respect to x. In ap-pendix BofRef.[2]relatedfunctions, g_±

(

x

)

wereintroducedto pa-rameterizethe Jostsolutionsforimaginarymomenta.The bound-ary conditions A

(

−∞)

=

B

(

∞)

=

1 and A

(

−∞)

=

B

(

∞)

=

0 yield the scattering matrix by matching the solutions at x

=

xm.

Above threshold, k

≥

m2

R

−

m2L so that q is real, the scattering

matrixis S

(

k

)

=

e−iqxm ₀ 0 eikxm



B

−

A∗ iqB

+

B ik A∗

−

A∗



₋1

×

A

−

B∗ ik A

+

A iqB∗

−

B∗



eikxm ₀ 0 e−iqxm



,

(8)

where A

=

A

(

xm

)

,etc.are thecoefficient functionsatthe

match-ingpoint.Conventions arethat thediagonalandoff-diagonal ele-ments of S contain the transmission and reflections coefficients, respectively [12]. Below threshold we parameterize for x

≥

xm:

η

(

x

)

=

B

(

x

)

e−κx with

κ

=

m2_R

−

m2_L

−

k2

_≥

_{0 replacing}

₋

_{iq in} Eq.(7)sothatB

(

x

)

isreal.Then

S

(

k

)

= −

A



B

/

B

−

κ

−

ik

−

A A∗

(

B

/

B

−

κ

+

ik

)

−

A∗e

2ikxm ₍₉₎

isthereflectioncoefficient.Inbothcaseswecompute thesumof theeigenphaseshifts

δ(

k

)

= −(

i

/

2

)

lndetS

(

k

)

.Thenegativesignon the right handside of Eq. (9)suggests that (in mostcases)

δ(

0

)

is an odd multiple of π₂ in agreement with Levinson’s theorem. Whenthescatteringproblemdiagonalizesintosymmetric(S)and anti-symmetric ( A) channelsandtaking

δ(

k

)

→

0 ask

→ ∞

,the theorem states that

δ

S

(

0

)

=

π

(

nS

−

1₂

)

and

δ

A

(

0

)

=

π

nA, where nS and nA count the bound states in the two channels [13,14].

The additional

−

π₂ in the symmetric channel arises because in that channel it is the derivative of the wave function that van-ishesatx

=

0,ratherthanthewave functionitself. Forscattering off a background that does not decompose into these channels we have

δ(

0

)

=

π

(

n

−

1₂

)

, wheren isthe totalnumberof bound states [12].There are particularcasesin which

δ(

0

)

isindeedan integer multiple of

π

.Examples are reflectionless potentialsand thecaseV

(

x

)

≡

0.Thenthereexistthresholdstatescontributing 1₂ ton.

The step potential of hight m2_R

−

m2_L centered at x

=

xm

cor-responds to Vp

≡

0.In this casethe wave equation issolved by A

(

x

)

=

B

(

x

)

≡

1 and

δ

step

(

k

)

=

⎧

⎪

⎪

⎨

⎪

⎪

⎩

(

k

−

q

)

xm

,

for k

≥

m2_R

−

m2_L kxm

−

arctan



m2 R−m2L−k2 k



,

for k

≤

m2_R

−

m2_L (10)

agreeswithtextbookresults. 3. Vacuumpolarizationenergy

FormallytheVPEisthesumoftheshiftsofthezeropoint en-ergies dueto theinteraction witha background potential that is generatedbythefieldconfiguration

φ

0,

Evac

[φ

0

] =

1 2



j



Ej

[φ

0

] −

E(_j0)



+

Ect

[φ

0

] .

(11)

Regularizationforthislogarithmicallydivergentsumisunderstood. When combined with the counterterms, Ect a unique finite re-sult arises after removing regularization. Typically there are two contributions in the sum of Eq. (11): (i) explicit bound and (ii) continuous scatteringstates.The latterpart isobtainedasan in-tegral over one particle energies weighted by the change in the density of states,



ρ

(

k

)

. We find the density

ρ

(

k

)

=

dN(k)

dk for

scattering modes incident from negative infinity by discretizing kL

+ δ(

k

)

=

N

(

k

)

π

where

δ(

k

)

is phase shift. Adopting the con-tinuum limit L

→ ∞

and subtracting the result from the non-interactingcaseyieldstheKreinformula[15],



ρ

(

k

)

=

ρ

(

k

)

−

ρ

(0)

(

k

)

=

1

π

d

dk

δ(

k

) .

(12)

The situation formodes incident from positive infinity is not as straightforward.Herewecountlevels(abovethreshold)bysetting qL

+ δ(

k

)

=

N

(

k

)

π

.Sincek is thelabelforthefree stateswe get anadditionalcontributiontothechangeinthedensityofstates

L

π

d dk[q

−

k]

=

L

π

⎡

⎢

⎣

k k2

₊

_m2 L

−

m2R

−

1

⎤

⎥

⎦

=

L

π

⎡

⎢

⎣

E2

₋

_m2 L

E2

₋

_m2 R

−

1

⎤

⎥

⎦ .

(13)

Formallyit addsaportion tothe VPEthat isnot sensitive tothe details ofthepotential. Its omissioncorresponds to the selection ofaparticular L independentpartfromthe effectivepotential as e.g. inEq. (3.42)ofRef.[11].

Then the VPE is solely extracted fromthe Krein formula. In-tegrating by parts and imposing the no-tadpole renormalization prescriptionyields Evac

=

1 2



j

(

Ej

−

mL)

−

1 2

π

∞



0 dk

k k2

₊

_m2 L



δ(

k

)

− δ

(1)

(

k

)



.

(14)

The explicit sum runs over the discrete bound states that are obtained from the solutions to Eq. (4) that exponentially ap-proachzeroatspatial infinity. The subtractionunderthe integral refers to the Born approximation with respect to the potential V

(

x

)

−

m2

L. We stress that it does not refer to Vp

(

x

)

because

the no-tadpole renormalization implements a counterterm that is local in the full potential. In general this disallows to write

δ

(1)

(

k

)

∼ −(

1

/

2k

)



dx

[

V

(

x

)

−

m2_L

]

, because the Born approxima-tionto the step potential cannot be written as thisintegral. Yet, itsphase shiftiswell defined, Eq.(10)andthelarge momentum contribution,whichisrepresentedbytheBornapproximation,can easilybecomputedfromEq.(10)

δ

step

(

k

)

−→

xm 2k



m2_R

−

m2_L



as k

−→ ∞ .

(15)

By definition, the Born approximation is linear in the potential. WeuseEq.(5)towrite V

(

x

)

−

m2_L

=

Vp

(

x

)

+



m2_R

−

m2_L

(

xm

)

and

obtaintheBornapproximation

δ

(1)

(

k

)

= −

1 2k ∞



−∞ dx Vp(x

)





xm

+

xm 2k



m2_R

−

m2_L



= −

1 2k ∞



−∞ dx Vp(x

)





0

.

(16)

Thesubscript recalls that Vp

(

x

)

isdefinedwithrespect toa

spe-cific matching point xm. However, the final Born approximation

doesnot depend on xm.This is a step towards establishing that

theVPE doesnot depend on the matching point. We stress that thisindependencedoesnot reflecttranslational invariance ofthe systemasdescribedbyshiftingthecoordinatex

→

x

−

x0 inV

(

x

)

. Onthecontrary,Eq.(16)showsthatatleasttheBorn approxima-tionvariesunderthistransformation.2

Whenthepotentialisreflectionsymmetricthescattering prob-lemseparatesintoevenandoddchannels.Thissymmetryalso im-pliesq

=

k andallowstoanalyticallycontinue toimaginaryk

=

it witht

≥

0 straightforwardly. Integratingovert collectsthebound statecontribution[1]andtheVPEis

E(vacS)

=

∞



mL dt 2

π

t

t2

₋

_m2 L



ln



g

(

t

,

0

)

g

(

t

,

0

)

−

1 t g 

₍

_t

_,

₀

₎

 !

1

.

(17)

2 _{ItseemssuggestivethattheBornapproximationshouldhaveastepfunction} factor(k−

m2

R−m2L).InthelimitmR→mL itsmodificationoftheVPEis

pro-portionalto xm mL(m 2 R−m 2 L)

3/2_{.Itisthusofhigherorderandalso violatesthe x}

m

independence.HencethisfactorisnotpartoftheBornapproximation.

Table 1

NumericalVPEsforthesymmetricbackgroundbasedonthesolitonofthe(φ2₊ a2_)(φ2₋₁₎2_model.

a Heat kernel, Ref.[5] Jost, Eq.(17) Present, Eq.(14) 0.1 −1.349 −1.461 −1.462 0.2 −1.239 −1.298 −1.297 1.0 −1.101 −1.100 −1.102 1.5 −1.293 −1.295 −1.297 AgaintheBornapproximationhasbeensubtractedasindicatedby thesubscript.Here g

(

t

,

x

)

isthenon-trivialfactoroftheJost solu-tionontheimaginaryaxisthatsolvestheDEQ

g

(

t

,

x

)

=

2t g

(

t

,

x

)

+

V

(

x

)

g

(

t

,

x

)

(18)

withtheboundarycondition g

(

t

,

∞)

=

1 and g

(

t

,

∞)

=

0. Above wehave usedheuristic arguments tocompute the VPE fromscatteringdata.Westressthatitcanbederived from funda-mentalconceptsofquantumfieldtheory[2].

4. Numericalresults

Forsimplicitywescaletodimensionlesscoordinatesandfields such that asmany as possiblemodel parameters, for example

λ

and



fromtheintroduction,areunity.

In all considered cases we have ensured that the phase shift does not vary withthe choice of xm; that Levinson’s theorem is

reproduced;andthatattachingfluxfactors S11

→

q ke i(q−k)xm_S 11 andS22

→

k qe i(k−q)xm_S

22tothetransmissioncoefficientsalways producesaunitaryscatteringmatrix.WhenmL

=

mR wehavealso

numericallyverified that thesumofthe eigenphaseshiftsequals thephaseofthetransmissioncoefficient S11

=

S22[16].

4.1. Symmetricbackground

Wefirstcomparetheresultfromthenovelmethodforcasesin whichV

(

x

)

isreflectionsymmetricandtheapproachviaEq.(17)is applicable.Analyticresultsareavailable forthe

φ

4 _kinkand sine-Gordonmodelsthathavebackgroundpotentials[asinEq.(3)]

VK

(

x

)

=

6tanh2

(

x

)

−

2 and VSG

(

x

)

=

8tanh2

(

2x

)

−

4

,

(19)

withmL

=

mR

=

2. The numerical simulation forEq. (14)agrees

with the respective VPEs, Evac,K

=

√

2

/

4

−

3

/

π

and Evac,SG

=

−

2

/

π

[17],tobetterthanoneinathousand.

We next compute the vacuum polarization energies of the U

(φ)

=

1

2

(φ

2

₊

_a2

_)(φ

2

₋

₁

₎

2 _{model,where a is a realparameter.} Fora

=

0 thereisonlyasingle solitonsolutionthatinterlinksthe vacua3

_φ

vac

= ±

1[3]:

φ

0

(

x

)

=

a X

−

1

"

4X

+

a2

₍

₁

₊

_X

₎

2 where X

=

e 2"1+a2_x

.

(20)

Forthismodel VPE resultsfroma heatkernel calculation[5]are available.Bycomparingtoourresults,weestimate thevalidityof the approximations applied in the that approach. This compari-son is essential because(toour knowledge)the only estimate of the VPEin thepure (a

=

0)

φ

6 model,whichis amain target of thepresentinvestigation,utilizesthistechnique[7].Theresultsare presentedinTable 1andweobservethatthevariouscomputations

3 _{ThepotentialU}

(φ)hastwoglobalminimaatφ= ±1 fora2

>12.Whena 2

<12 athird(local)minimumexists.Thethreeminimaaredegeneratefora=0.

Table 2

ComparisonofdifferentmethodstocomputetheVPEforanon-symmetricbackground.TheR dependentdataare halftheVPEofthebackground,Eq.(21)computedviaEq.(17).

R 1.0 1.5 2.0 2.5 3.0 3.5 Present, Eq.(14)

A=2.5,σ=1.0 −0.0369 −0.0324 −0.0298 −0.0294 −0.0293 −0.0292 −0.0293

R 4.0 5.0 6.0 7.0 8.0 9.0 Present, Eq.(14)

A=0.2,σ=4.0 −0.0208 −0.0188 −0.0170 −0.0161 −0.0158 −0.0157 −0.0157

Fig. 1. Potentials (left panel) and phase shift (right panel) for scattering off a soliton in theφ6_{model. The pseudo potential V}

p(x)is shown for xm=0.

agreewell formoderateandlargea.Themethodsbased on scat-teringdataagreewithin numericalprecision. Butwhena issmall deviations ofabout10–15% are observed forthe (approximative) heatkernelmethod.

4.2. Asymmetricbackground,identicalvacua

Forthe lack ofa (simple) solitonmodelwe consider the two parameter( A,

σ

)pseudopotentialVp

(

x

)

=

Axe−x

2_/σ2

.Thepresent method can be applied directly but also the standard spectral methods,Eq.(17)canemployedaftersymmetrizing

VR

(

x

)

=

A



(

x

+

R

)

e−(x+R) 2 σ2

− (

_x

−

_R

₎

_e−(x−R) 2 σ2

!

(21)

sothatthelimit R

→ ∞

shouldgivetwice theVPEof Vp

(

x

)

[18]. Table 2 verifies that agreement is obtained, but large values for R are needed to avoid interference effects for wide background potentials.

4.3. Asymmetricbackground,unequalvacua,

φ

6_model Wenowturntothepure

φ

6_modelwithU

_(φ)

₌

1

2

φ

2



φ

2

−

1

2. For a

=

0 the soliton of Eq. (20) ceases to be a solution. How-ever, there are solitons that interlink the degenerate vacua at

φ

vac

=

0 and

φ

vac

= ±

1. The curvatures of U

(φ)

at these vacua differ so that the masses of the corresponding fluctuations are unequal. The soliton that corresponds to mL

=

1 and mR

=

2 is

φ

0

(

x

)

=



1

+

e−2x

−1/2 _{[3].Theresultingpotentialsforthe} fluctu-ations are shown in the left panel of Fig. 1. Also shown is the resultingsum,

δ(

k

)

,oftheeigenphaseshiftsasobtainedfromthe scatteringmatrix,Eqs.(8)and(9).Thedirectnumericalcalculation provides a discontinuous function between

−π

/

2 and

π

/

2. The discontinuitiesareremoveduniquelybyaddingappropriate multi-plesof

π

anddemandingthat

δ(

k

)

→

0 ask

→ ∞

.Inthatlimitit agrees withthe Bornapproximation, Eq.(16).However, thecusp, which is typical forthreshold scattering, remains. Note also that

Table 3

VPEsforVα(x)=32[1+tanh(αx)].Theentry’step’referstousingδstepfromEq.(10) withxm=0 inEq.(14).

α 1.0 2.0 5.0 10.0 30.0 Step

Evac 0.1660 0.1478 0.1385 0.1363 0.1355 0.1355

δ(

0

)

=

π

2 complieswithLevinson’s theoremin onespace dimen-sion asthere is only a single bound state: the translational zero modeofthesoliton.

Our resultsforthe momentumdependence ofthephase shift (and reflection coefficient) agree with the formulas given in Refs. [3,4] up to overall signs. We are confident about our signs from Levinson’s theorem and the Born approximation. Putting things together we find the vacuum polarization energy of the kinkinthe

φ

6 _model

Evac

= −

0

.

5

+

0

.

4531

= −

0

.

0469

,

(22) wherethe summands denotethebound stateand(renormalized) continuumpartsasseparatedinEq.(14).

In Ref. [7] the VPE of the

φ

6 model kink was estimated rel-ative to V α

(

x

)

=

3

2[1

+

tanh

(

α

x

)

] for

α

=

1. In Table 3 we give our results for various values of

α

. For

α

=

1 our relative VPE is



Evac

= −

0

.

0469

−

0

.

1660

= −

0

.

2129 to be compared with

−

0

.

1264

√

2

= −

0

.

1788 fromRef.[7].Inviewoftheresultsshown inTable 1,especiallyforsmalla, thesedatamatchwithin the va-lidityoftheapproximationsappliedintheheatkernelcalculation.

4.4. Translationalvarianceandsymmetrization

We complete the discussion of the numerical results with a contemplation on translational invariance. In Sec. 3 we have already seen that the Born approximation changes when the center of the configuration is shifted by a finite amount. To investigate this further, we compute the VPE for the

φ

6 kink



1

+

e−2(x+x0)



−1/2 _{in U}

_(φ)

_{and V}

₍

_x

₎

=

3

2tanh

[

α

(

x

+

x0

)

]

as a generalizationoftheabovestudy.Thedependenceonx0originates

Table 4

VPEsasafunctionofthecenteroftheconfigurationsmentionedinthetext.The twoentriesα=2 andα=5 refertothechoicesintanh[α(x+x0)].

x0 Evac

−2 −1 0 1 2

φ6 0.154 0.053 −0.047 −0.148 −0.249 α=2 0.351 0.250 0.148 0.046 −0.057 α=5 0.341 0.240 0.139 0.037 −0.064 solelyfromthephaseshiftpartbecauseboundstatesmovewithx0 withoutchangingtheir energyeigenvalues.In Ref.[4]this x0 de-pendence was removed aspart of therenormalization condition. This is not fully acceptable since the renormalization conditions shouldnotdependonthefieldconfiguration.

For both potentials the numerical results from Table 4 show thattheVPEdecreasesbyabout0.101perunitofshiftingthe cen-tertowards negative infinity. We can build up a similar scenario in form of a symmetric barrier V(x0)

SB

(

x

)

=

v0





x0

2

− |

x

|

whose VPE can be straightforwardly computed fromEq. (17). Substitut-ingV(x0)

SB intotheDEQ,Eq.(18)yields

g

(

t

,

0

)

=

κ

1e −κ2x0/2

−

_κ

_2e−κ1x0/2

κ

1

−

κ

2 and g

(

t

,

0

)

=

κ

1

κ

2

κ

1

−

κ

2



e−κ2x0/2

−

_e−κ1x0/2



,

(23) with

κ

1,2

=

t

±

"

t2

₊

_v

0.Sincewe onlyconsiderthebarrierwith v0

>

0,the

κ

1,2 arealways real.The relevantBornapproximation isparticularlysimple ln



g

(

t

,

0

)

g

(

t

,

0

)

−

1 t g 

₍

_t

_,

₀

₎



₌

v0x0 2t

+

O



v2₀



.

(24)

We theseingredients we have evaluated the integral in Eq.(17)

using v0

=

m2R

−

m2L

=

3 as suggestedby the

φ

6 modelkink and

find lim x0→∞ Evac

[

V_SB(x0)

]

x0

≈ −

0

.

1015

.

(25)

Wecanrelate thisresultto theenergydensityofa stepfunction potentialatspatialinfinityusingthephaseshiftfromEq.(10)

Evac

[

Vstep(xm)

]

|

xm

|

→ −

sign

(

xm)

#

√



v0 0 dk 4

π

2k2

₋

_v 0

k2

₊

_m2 L

+

∞



√ v0 dk 4

π

2k2

−

2k

"

k2

₋

_v 0

−

v0

k2

₊

_m2 L

$

as

|

xm

| → ∞ .

(26)

FormL

=

1 and v0

=

3 the expressionis square bracketshasthe numerical value

−

0

.

1013. These data suggest that translational varianceoriginatesfromthepresenceoftheregionsinwhichthe quantumfluctuationshavedifferentmasses.Thenumericalresults inTable 4andEqs.(25)and(26)show thattherateatwhichthe VPE changes isnot sensitive to theparticular shapeof the back-ground;butitdependsonv0.Formallywecouldaddtheomission ofEq.(13)



_dk 2

π

k2

₊

_m2 L d dk

"

k2

₋

_v 0

−

k



∼



_dk 2

π

k

k2

₊

_m2 L



k

−

"

k2

₋

_v 0



totheenergydensitytoeliminatethe(leading)translational vari-ance.Theaboveintegrationbypartsmissesasurfacetermwhose divergence is regularized by the Born subtraction in the actual calculationofEq.(26).Weseethat translationalvariance is qual-itatively linked to the difference between the densities of states at positive and negative infinity, yet quantitative conclusions are notpossiblebecausethatdifferencecannotbeexplicitlyrelatedto the centerofthe backgroundpotential. The pictureemerges that shiftingthe regionwiththelargermasstowards negative infinity removes modes from the spectrum and thus decreases the VPE. On the other hand it is not surprisingthat the bound state en-ergies are translationally invariant becausethe bound state wave functionsdonotreachtospatialinfinity.

ByshiftingtheargumentsinEq.(19)wehaveverifiedthatthe proposed numerical approach indeedproduces translationally in-variant VPEs (actually phase shifts) for the

φ

4 and sine-Gordon solitons. In the present formalism that verification is simple. In contrast,decouplingevenandoddparitychannels, asrequiredto obtain Eq.(17),distinguishes x

=

0 and doesnot leave spacefor varyingthecoordinateargument.

Substitutingthesymmetrizedkink–antikinkbarrier

φ

0

(

x

)

=



1

+

e2(x−¯x)



₋1/2

+



1

+

e−2(x+¯x)



₋1/2

−

1 (27)

intoU

(φ)

producesasymmetricbackgroundthatisavariationof abarrierwithapproximatewidth2x.

¯

Thevacuumischaracterized bymL

=

1.Numericallywefind lim ¯ x→∞

%

Evac

[

U

(φ

0

)

] −

2Evac

[

V_SB(2¯x)

]

&

= −

0

.

340

=

2

× (−

0

.

170

)

(28)

whichisintherightballparkincomparisonwiththedatainthe

φ

6 rowofTable 4.Unfortunately,itisnot clearwhichvalue ofxs

in V(2xs)

SB to useforthesubtractioninEq.(28).Forexample,it is sensibletodefinethecenterofthesoliton



1

+

e2(x−¯x)



−1/2 _viaits classicalenergydensity

(

x

)

=

1

2

φ

02

+

U

(φ

0

)

: xs

=



dxx



(

x

)



dx



(

x

)

= ¯

x

+

1 2 and subtract V(2xs)

SB in Eq. (28). This changes that result to

−

0

.

239

=

2

× (−

0

.

120

)

.WhenattemptingtoextractthekinkVPE fromtheantikink–kinkconfiguration

φ

0

(

x

)

=



1

+

e−2(x−¯x)



−1/2

+



1

+

e2(x+¯x)



−1/2_{, a well of depth v}

0 and width 2x is

¯

generated yieldingacompletelydifferentVPEduetothemanyboundstates thatemergeforlargeantikink–kinkseparation.

5. Conclusion

We have developed a method to compute the VPE for local-ized configurations in one spacedimension. It is based on spec-tralmethodsbutgeneralizespreviousapproachestoconfigurations that are not amenable to a partial wave decomposition.Being a generalizationofthespectralmethod,thenovelapproachalso nat-urally inherits the renormalization from the perturbative sector. The proposed method is very efficient: For a given background potentialthenumericalsimulationsonlytakeonlyafewCPU min-utes on a standard desktop computer. We solve two uncoupled second orderordinarydifferentialequations,Eq.(7),forthe com-plexvaluedfunctionsA

(

x

)

andB

(

x

)

thatdeterminethescattering matrix.Anequallysimpleequation(4)yieldstheboundstate ener-gies.Herewehaveonlyconsideredasinglebosonfield,buttaking A

(

x

)

and B

(

x

)

to be matrix valuedstraightforwardly generalizes

themethodto multiplefieldsand/orfermions. Theefficiencycan alsobeestablishedwhenconfrontingitwiththeheavymachinery neededfortheheatkernelapproach[5,7]thatwasearlierusedto findtheVPEofconfigurationslackingthesymmetriesforapartial wave decomposition.Weconsiderthe presentmethodatleastas efficientasthat used inRef. [11], whichis basedon a particular techniquetocomputefunctionaldeterminants[19].Bothmethods solveadifferentialequationforsingleparticleenergies.Integrating overtheseenergiesyieldstheVPE.

Asanapplicationwehaveconsideredconfigurationsforwhich the quantum fluctuations have different masses at positive and negative spatial infinity. Then thebackground can be interpreted as a modification of a step function potential that interpolates betweendifferent vacua. Thoughthe parameterization ofthe so-lutions to the stationary wave equation differs on the left and righthalf lines (joined at the matchingpoint xm) we stress that

wealways solvethewaveequationforthefullproblem.We have ensured that all results for the VPE (actually for the eigenphase shifts) do not depend on xm. We did not explicitly compute the

VPE versus anotherconfiguration;butthe stepfunction potential featuredessentialwhen(i) identifyingtheBornapproximationfor renormalizationand(ii) establishingindependencefromtechnical parameterslikexm.

Though we may freely choose xm for computing the

scatter-ing matrix, translational invariance withrespect to the center of the solitonis lost when the massesof the quantum fluctuations differatpositive andnegativespatial infinity.Thislossof transla-tionalinvariancesignalsthattheglobalvacuumstructureislocally relevant. We have also collected numerical and formal evidence thatthispositiondependenceis(mainly)duetothedifferencesof thedensitiesofstatesforscatteringmodesincidentfrompositive ornegative infinity. Evenifthiswasthe onlycause, themultiple bywhichthecorrespondingspatialenergydensityshouldbe sub-tractedisnotuniqueleavingaresidualpositiondependence.

Inthe

φ

6 _{modelthe exactno-tadpole renormalizationscheme} is required. Any additional, though finite, renormalizationof the countertermcoefficientisnotwelldefinedasthemultiplying spa-tialintegral is infinite.However, this isnot too surprisingasthe modelisnotfullyrenormalizable.

We wish to extend the present approach in the framework of the interface formalism [20] anduse it to investigate domain wall dynamics. This will allow a comparison with the results of Ref.[11].Alsoothersolitonmodelsinonespacedimensioncanbe investigated. Forexample,the

φ

8 model[21] has solitonswithin differenttopologicalsectors. ComparingtheirVPE willshedsome lightontherelevanceofquantumcorrectionstothebinding ener-giesofsolitonsthatrepresentnuclei[22].

Acknowledgements

Helpfuldiscussions withN.Graham andM.Quandtare grate-fully acknowledged. This work is supported in parts by the NRF undergrant 77454.

References

[1]N.Graham,M.Quandt,H.Weigel,Lect.NotesPhys.777(2009)1.

[2]N.Graham,R.L.Jaffe,V.Khemani,M.Quandt,M.Scandurra,H.Weigel,Nucl. Phys.B645(2002)49.

[3]M.A.Lohe,Phys.Rev.D20(1979)3120.

[4]M.A.Lohe,D.M.O’Brien,Phys.Rev.D23(1981)1771.

[5]A.Alonso-Izquierdo,J.MateosGuilarte,Nucl.Phys.B852(2011)696. [6]A.Alonso-Izquierdo,J.MateosGuilarte,Ann.Phys.327(2012)2251. [7]A.Alonso-Izquierdo,W.GarciaFuertes,M.A.GonzalezLeon,J.MateosGuilarte,

Nucl.Phys.B635(2002)525.

[8]E.Elizalde,Lect.NotesPhys.Monogr.35(1995)1.

[9]E.Elizalde,S.D.Odintsov,A.Romeo,A.A.Bytsenko,S.Zerbini,Zeta Regulariza-tionTechniqueswithApplications,WorldScientific,Singapore,1994. [10]K.Kirsten,AIPConf.Proc.484(1999)106.

[11]A.Parnachev,L.G.Yaffe,Phys.Rev.D62(2000)105034. [12]K.Kiers,W.vanDijk,J.Math.Phys.37(1996)6033. [13]G.Barton,J.Phys.A18(1985)479.

[14]N.Graham,R.L.Jaffe,M.Quandt,H.Weigel,Ann.Phys.293(2001)240. [15]J.S.Faulkner,J.Phys.C10(1977)4661.

[16]A.R.Aguirre,G.Flores-Hidalgo,arXiv:1609.07341[hep-th]. [17]R.Rajaraman,SolitonsandInstantons,NorthHolland,1982. [18]N.Graham,R.L.Jaffe,Phys.Lett.B435(1998)145.

[19]S.Coleman,AspectsofSymmetry,CambridgeUniversityPress,1985. [20]N.Graham,R.L.Jaffe,M.Quandt,H.Weigel,Phys.Rev.Lett.87(2001)131601. [21]V.A.Gani,V.Lensky,M.A.Lizunova,J.HighEnergyPhys.1508(2015)147. [22]D.T.J.Feist,P.H.C.Lau,N.S.Manton,Phys.Rev.D87(2013)085034.