Spectral functions in the φ4-theory from the spectral DSE

Jan Horak,1 Jan M. Pawlowski,1, 2 and Nicolas Wink1

1Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany2ExtreMe Matter Institute EMMI, GSI, Planckstr. 1, 64291 Darmstadt, Germany

We develop a non-perturbative functional framework for computing real-time correlation functionsin strongly correlated systems. The framework is based on the spectral representation of correlationfunctions and dimensional regularisation. Therefore, the non-perturbative spectral renormalisationsetup here respects all symmetries of the theories at hand. In particular this includes space-timesymmetries as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectralrenormalisation can be applied within general functional approaches such as the functional renor-malisation group, Dyson-Schwinger equations, and two- or n-particle irreducible approaches. As anapplication we compute the full, non-perturbative, spectral function of the scalar field in the φ4-theory in 2+1 dimensions from spectral Dyson-Schwinger equations. We also compute the s-channelspectral function of the full φ4-vertex in this theory.

I. INTRODUCTION

The study of the dynamics and resonance structureof strongly correlated systems requires the knowledge ofreal-time (time-like) correlation functions. In particularthe evolution of slow modes is dominated by the low-energy regime and is crucial for transport approachesand hydrodynamics. Spectral functions encode the full,non-perturbative, information of the respective degrees offreedom and open the door to additional real-time quanti-ties such as transport coefficients. They are also a partic-ularly useful tool when discussing resonances and boundstates, since they give direct access to the spectrum ofexcitations in a given theory. Emergent composite statesare of central interest not just in particle and nuclearphysics, but in most physics areas.

The treatment of strongly correlated systems asks fornon-perturbative techniques. In recent years functionaland lattice approaches have been applied very success-fully to the low-energy regime of theories ranging fromQCD to condensed matter systems. The general frame-work for the quantum-field theoretical description ofbound states with functional methods was introduced byBethe and Salpeter [1, 2]. For a recent review see [3], fora review of respective lattice results see [4].

Despite this rapid and very impressive progress inparticular in the last decade, the reliable direct non-perturbative calculation of real-time correlation func-tions is still in its infancy. Functional approaches havematured in recent years and are by now quantitativelycompetitive in QCD in Euclidean space-time. The ex-tension of such calculations to Minkowski space-time isyet hindered by technical and conceptual complications,for recent advances see [5–32].

In this work we develop a novel approach for the di-rect computation of real-time (Minkowski) observablesthat is based on spectral representations of correlationfunctions. The approach comes with the advantage thatwe can use dimensional regularisation for the analyticcomputation of momentum integrals in fully numericalnon-perturbative calculations. Accordingly, the respec-

tive renormalisation scheme, spectral renormalisation, isbased on standard dimensional regularisation and re-spects the space-time symmetries, internal symmetriessuch as chiral symmetry, and gauge symmetries of thetheory at hand. Clearly, this method is applicable to abroad range of theories, including non-Abelian gauge the-ories, within a regularisation and renormalisation schemewhich is manifestly gauge-invariant.

In the present work we apply our novel approach tothe scalar φ4-theory in d = 2 + 1 dimensions. This the-ory is a simple strongly correlated system and serves asa good benchmark for new techniques before applyingthem more involved theories such as non-Abelian ones.It is also interesting in its own right and has many ap-plications as a model theory for perturbative and non-perturbative phenomena.

The numerical application in the present work is donewithin the spectral Dyson-Schwinger approach. We com-pute the spectral function of the scalar field from the gapequation for the propagator, using its Kallen-Lehmannspectral representation. In a first step all vertices are ap-proximated with the classical ones, and the two-loop di-agrams in the Dyson-Schwinger equations (DSE) are in-cluded. In a second step a use a skeleton expansion of theDSE with a bubble-resummation of the s-channel four-point function. The non-perturbative s-channel spectralfunction of the vertex is computed and used in the DSE.The setup gives us direct numerical access to Minkowskispace-time correlation functions due to the spectral rep-resentation. This also allow for a (physical) on-shellrenormalisation via the spectral renormalisation scheme.

The paper is organized as follows: In Section II weintroduce spectral renormalisation at the example ofthe Dyson-Schwinger approach along with the necessarytechnical tools. In particular we discuss dimensionalspectral renormalisation and a Bogoliubov-Parasiuk-Hepp-Zimmermann–type (BPHZ) spectral renormalisa-tion. Section III works out the calculational details ofsolving the DSE in the developed scheme, followed by ourresults in Section IV A for classical vertices and in Sec-tion IV B-Section IV E for the skeleton expansion. Fi-nally, our conclusion is presented in Section V.

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FIG. 1. Diagrammatic notation used throughout this work:Lines stand for full propagators, small black dots stand forclassical vertices, and larger grey dots stand for full vertices.

II. SPECTRAL RENORMALISATION

The general real-time renormalisation scheme devel-oped here aims at combining a practical numerical imple-mentation in non-perturbative applications while main-taining all underlying symmetries including gauge sym-metries. This is achieved by utilising dimensional reg-ularisation, which respects all space-time, internal andgauge symmetries of the theory at hand. We also de-velop a BPHZ-type subtraction scheme which facilitatesthe analytical computations significantly. If such a sub-traction schemes does not violate any symmetries in thetheory at hand, it is the scheme of choice.

A practical implementation of dimensional regulari-sation requires an analytic momentum structure of thepropagators and vertices in the given loop integrals.While this allows for its use in perturbation theory, non-perturbative applications, with their necessarily numeri-cal computation of propagators and vertices, usually relyon ultraviolet momentum cutoffs. The latter are neitherconsistent with space-time symmetries nor with gaugesymmetries. It is well-known that in gauge theories andsupersymmetric theories such a regularisation requiressymmetry-breaking counterterms. This is not a concep-tual problem, but it typically triggers additional power-counting relevant terms that may leads to additional fine-tuning tasks, see e.g. [33, 34]. Moreover, the Wick rota-tion to Minkowski space-times is hampered by the de-formation of the momentum integrals, which can lead toadditional poles and cuts in the integration contours.

The present renormalisation scheme achieves the re-quirement of analytic momentum integrals by using spec-tral representations of correlation functions. For thepropagators this is the Kallen-Lehmann spectral repre-sentation, similar spectral representations also exist forthe vertices, though getting increasingly difficult. We callthis renormalisation scheme spectral renormalisation: af-ter inserting the spectral representations in the loop inte-grals, the momentum integrands take an analytic form.This form is well-suited for using dimensional regulari-sation or related analytic computation techniques. Theloop-momentum integrations can be performed analyti-cally and we are left with spectral integrals. The wholenon-perturbative information is contained in the spec-tral functions of propagators and vertices. In most non-perturbative applications the respective spectral integralscan only be computed numerically.

FIG. 2. Functional DSE for the effective action of the scalartheory under investigation in this work. The notation is givenin Figure 1.

The scheme can be practically applied to any divergentdiagram that scales in the UV with loop momentum tosome natural power qm, m ∈ N (with m < nmax andnmax given by the renormalisability constraint). This isalways the case when using spectral representations forall correlation functions, but also works for classical ver-tices. This will be detailed in the present work within theexample of the Dyson-Schwinger approach introduced inthe next section, Section II A. The functional DSE forthe effective action, (4), is depicted in Figure 2, that forthe inverse propagator is depicted in Figure 3. For clas-sical vertices, all momentum integrals in functional ap-proaches, see e.g. Figure 3, are of the standard perturba-tive form, but with different spectral masses for all lines.Most of these integrals are known from perturbation the-ory results, e.g. [35]. Note that this re-parameterisationcomes at the cost of spectral integrals for each propaga-tor. Spectral representations of vertices lead to furtherspectral integrals as well as further classical propagatorswith spectral masses. In summary, in a spectral func-tional approach all momentum integrals are perturbative.Hence we can implement a symmetry-preserving regular-isation such as dimensional regularisation, leading to arenormalisation scheme with symmetry-consistent coun-terterms.

A relevant example for this important symmetry-preserving property are gauge theories. There, a mo-mentum cutoff or standard subtraction scheme requiresexplicitly or implicitly a mass counterterm for the gaugefield in order to keep the renormalised gauge field mass-less. In four dimensions this leads to a quadratic fine-tuning task instead of a logarithmic one, for a detaileddiscussion see [33, 34]. Spectral renormalisation withspectral regularisation removes the (explicit or implicit)necessity of a mass counterterm for the gauge field, andhence the quadratic fine-tuning task.

After briefly introducing Dyson-Schwinger equations,Section II A, and the Kallen-Lehman spectral represen-tation, Section II B, we set up spectral renormalisation inSection II C (spectral dimensional renormalisation), andSection II D (spectral BPHZ-renormalisation). Furtherexamples and the discussion of the fully non-perturbativesetup can be found in Section II E and Section II F. Theexplicit example used for demonstrating the propertiesand computational details of the spectral renormalisationscheme is the gap equation for scalar φ4- and φ3-theories.

3

FIG. 3. DSE of the two-point function for a general background field φ 6= 0. The vacuum polarisation and squint diagrams areproportional to S(3)[φ] ∝ φ. They vanish for φ = 0, where the standard form with tadpole and sunset diagrams is obtained.The notation is given in Figure 1.

A. Dyson-Schwinger equations

The central object in functional approaches such asDyson-Schwinger equations or functional renormalisationgroup equations is the quantum effective action Γ[φ]. Itis related to the generating functional Z[J ] of correlationfunctions including their disconnected parts via a Legen-dre transformation,

Γ[φ] = supJ

[∫ddxJ(x)φ(x)−W [J ]

]. (1)

Here, W [J ] = lnZ[J ] is the Schwinger functional thatgenerated connected correlation functions. The effectiveaction Γ[φ] is the generating functional for one-particle ir-reducible (1PI) correlation functions. They are obtainedfrom nth derivative of the effective action w.r.t. to thefields. In momentum space this reads,

Γ(n)[φ](p1, . . . , pn) =δnΓ[φ]

δφ(p1) . . . δφ(pn). (2)

A pivotal role in all functional approaches is played bythe full, field-dependent propagator G[φ]. It is simplythe inverse of the 1PI two-point function,

G[φ](p, q) =1

Γ(2)[φ](p, q) . (3)

With these prerequisites we straightforwardly arrive atthe master Dyson-Schwinger equation (DSE), the quan-tum equation of motion,

δΓ[φ]

δφ=δS

δϕ

[ϕ = G · δ

δφ+ φ

], (4)

where G · δ/δφ =∫ddq/(2π)dG(p, q)δ/δφ(q) in momen-

tum space, and S[φ] is the classical action. A more de-tailed derivation can be found e.g. in [36]. By takingfunctional derivatives of (4) with respect to φ, all higherorder 1PI correlation functions are generated. For exam-ple, the DSE for the two-point function is generated bytaking one derivative of (4), and is depicted in Figure 3.

For our explicit example of a φ4-theory, the classicalaction in (4) is given by,

S[ϕ] =

∫ddx

[1

2(∂µϕ)2 +

m2φ,0

2ϕ2 +

λφ,04!

ϕ4

]. (5)

It depends on two parameters or couplings, the bare four-point coupling λφ,0 and the bare mass parameter mφ,0.The third parameter required for renormalisation, thewave function renormalisation Zφ can be scaled out. Thisis conveniently done by setting it to unity at the renor-malisation scale.

We close this section with a few comments: To beginwith, the underlying Z2-symmetry of the theory, ϕ→ −ϕimplies the same symmetry for the effective action undertransformations of the mean field, φ → −φ. Accord-ingly, the odd vertices vanish at vanishing mean field:Γ(2n+1)[φ = 0] ≡ 0. Moreover, restricting ourselves toconstant background fields φc, we can formally expandthe three-point function in powers of the field,

Γ(3)[φc](p1, p2, p3) = φc

[Γ(4)(p1, p2, p3, 0)+O(φ2

c)

], (6)

due to the odd vertices vanishing at the expansion pointφ = 0. In (6) we have used the Fourier transform

φc(p) = (2π)dφcδ(p) of the constant field. We have alsointroduced Γ(n)(p1, p2, p3, 0, ..., 0), the n-point functionsat a vanishing background and n−3 vanishing momenta.The relation (6) gives rise to the polarisation and squintdiagram in Figure 3 for φc 6= 0, which are absent forφc = 0. In the broken phase the equation of motion(EoM) φ0 for constant fields is solved for a non-vanishingexpectation value of the field, i.e. φ0 6= 0. Accordingly,if evaluating the DSEs for correlation functions on theEoM, these diagrams are present.

B. Kallen-Lehmann spectral representation

Using the Kallen-Lehmann spectral representation [37,38], the propagator can be recast in terms of its spectralfunction ρ,

G(p0) =

∫ ∞0

dλ

π

λ ρ(λ, |~p|)p2

0 + λ2. (7)

For asymptotic states, the spectral function can be un-derstood as a probability density for the transition to anexcited state with energy λ. In this way, the spectralfunction acts as a linear response function of the two-point-correlator, encoding the energy spectrum of thetheory. The existence of a spectral representation im-poses tight restrictions on the analytic structure of the

4

propagator. In turn, the Euclidean propagator also con-strains the spectral function, for a rather non-trivial ex-ample for the latter constraints see [39].

From a complex analysis perspective, the spectral func-tion naturally arises as the set of non-analyticities of thepropagator, which are severely restricted in the complexplane by Cauchy’s Theorem. This results into the fol-lowing inverse relation between spectral function and theretarded propagator,

ρ(ω, |~p|) = 2 Im G(−i(ω + i0+), |~p|) , (8)

where ω is the real time zero momentum component.This formulation allows us to work only with the fre-quency argument and set the spatial momentum to zeroin practice, since the full phase-space can be restoredfrom Lorentz invariance. Hence, for the remainder ofthis work, |~p| will be dropped.

The existence of a spectral representation restricts allnon-analyticities of the propagator to lie on the real mo-mentum axis, as is manifest in (7) and (8). This crucialcondition as well as the generic structure in the com-plex plane already mentioned above allow us to recastthe spectral function into the form

ρ(λ) =π

λ

∑i

Zi δ(λ−mi) + ρ(λ) . (9)

The spectral function is split into a set of δ-functions anda continuous scattering part ρ, arising from branch cutsin the complex plane of the propagator. For a classicalpropagator the spectral function reduces to one mass-shell δ-function with Z1 = 1. There are no further poles,Zi>1 = 0, and the scattering part is absent, ρ = 0. Fur-ther details can be found e.g. in [40].

In the scalar φ4-theory, the spectral function of thescalar field is that of an asymptotic state, and henceis positive semi-definite and has the interpretation of aprobability density. Therefore it is convenient to nor-malise its integrated weight to unity within an appropri-ate renormalisation scheme. Within this scheme we havethe sum rule,∫

λ

λ ρ(λ) = 1 with

∫λ

≡∫ ∞

0

dλ

π, (10)

which implies Zi ≤ 1. The spectral weight is distributedbetween poles and cuts, and in the presence of scatteringstates the weight of the poles is less than one.

C. Spectral dimensional renormalisation

Next, we discuss the spectral renormalisation scheme,which is fully based on dimensional regularisation. Di-mensional regularisation renders the loop diagrams finiteand we can swap orders of the initially outer momen-tum and inner spectral integrals integrals (cf. upper partof Figure 4. This allows us to first perform the momen-tum integrals analytically. We are left with finite spec-tral integrals at finite ε > 0, that in general have to be

∫λ ∫qf(q, λ)

divergent

∫q ∫λ

Initial integral

F(p) := ∫q ∫λf(q, λ)

divergent

∫λ ∫qf(q, λ)

finite

momentum renormalization

divergent

∫λ ∫qf(q, λ)

spectral renormalization

finite

Performed integral

RENORMALIZATIONε → 0

ε → 0

ε → 0

dim. reg.

FIG. 4. Schematic illustration of the spectral renormalisationscheme. The function f is some arbitrary divergent integrand.The dependence on the external momentum p is suppressed.The upper two boxes have to be understood as finite by di-mensional regularisation, but divergent in the limit ε → 0 ind− ε dimensions. In a first step, the momentum integrals areanalytically evaluated via dimensional regularisation. Subse-quently, the spectral integrals are renormalised via spectralrenormalisation, either within the dimensional- (Section II C)or BPHZ-approach (Section II D).

done numerically since the spectral functions may onlybe known numerically.

Generally, the numerical integration of the spectral in-tegrals can be performed for finite ε, and in gauge theo-ries full manifest gauge-consistency of spectral renormal-isation requires that the limit ε → 0 is taken only afterperforming all integrals. However, it is convenient forthe numerical performance to do the spectral integrationat ε = 0. The same holds for the access to the analyticmomentum structure required for extracting Minkowskiproperties. In this case, for the limit ε → 0 we haveto set up a consistent renormalisation procedure beforeperforming the spectral integrals, which is worked outin detail in Section II D. In particular it is not sufficientto only remove the 1/ε-divergences that originate in themomentum integrations, as the spectral integrations leadto further 1/ε-terms. This relates to the swapping of theintegration order and performing the limit ε → 0 beforeevaluating all integrals.

This section is dedicated to the fully gauge-consistentrenormalisation scheme we call spectral dimensionalrenormalisation. In this scheme, the divergent parts ofthe spectral integrals are performed analytically beforetaking the limit ε→ 0. A simple example is the tadpolecontribution to the gap equation in Figure 3 in d = 3 di-mensions, which comes with a momentum-independentlinearly divergent term. This example is also relevant forour later computation in Section III and Section IV. Af-ter the momentum integration is performed, we arrive at

5

a finite result proportional to∫ ∞0

dλλµ2ελ1−2ε

(λ2 +m2)= −π

2

1

cos(πε2

) m( µ2

m2

)ε. (11)

In (11) we have used a trial spectral function that decaysfor large spectral values λ according to its momentum orspectral dimension,

ρtrial(λ,m) =1

λ2 +m2, (12)

with a positive mass m > 0. The trial spectral func-tion in (12) approximates the correct leading ultravio-let behaviour, if we neglect logarithmic corrections. Forlarge spectral values, the UV-asymptotic of the spectralfunction can be extracted from the leading momentumdependence of the respective propagator. In the presentexample, the latter is assumed to decay quadratically onits branch cut on the real momentum axis.

The finiteness of the result of the momentum integra-tion used on the left hand side of (11) also points at aspecific property of dimensional regularisation: in odddimensions, d = 2n + 1, it already removes all momen-tum divergences. In even dimensions, d = 2n, it removessubclasses of divergent ones, a prominent being the (one-loop) tadpole in a massless theory such as a gauge theory.

If we had put ε = 0 before the integration, (11) simplyis (linearly) divergent. This is the price to pay for theswapping of integration orders: the divergences are notfully covered by the momentum integrals any more. Theexample also entails that in odd dimensions includingour explicit computation in the φ4-theory in d = 3, alldivergences come via the spectral integrals.

In spectral dimensional renormalisation, spectral sin-gularities in even dimensions show up as 1/ε-terms. Indimensional regularisation in perturbation theory thesedivergences are typically removed recursively by intro-duction of appropriate counterterms. In spectral di-mensional renormalisation this procedure is applied too,while keeping a finite ε as well as isolating analyticallythe singular part of the spectral integrals with

ρ(λ) = ρIR(λ) + ρUV,an(λ) . (13)

In (13), the numerical ’infrared’ part ρIR(λ) decays suf-ficiently fast for large spectral values and renders the re-spective spectral integrals finite. In turn, the ultravioletpart ρUV,an(λ) carries the ultraviolet asymptotics ana-lytically. Therefore, the respective spectral integrals canbe treated analytically with dimensional regularisationas done in (11). With (12) we use the IR-UV–split

ρ(λ) = ρIR(λ, k) + ρtrial(λ, k) . (14)

We also emphasise that the theory does not depend onthe mass parameter k that regularises (in the infrared)the UV-part of the spectral function. In particular wehave ∂kρ(λ) ≡ 0, which entails the k-dependence of theinfrared part of the spectral function.

The (leading) ultraviolet behaviour of the spectralfunction at large spectral values is governed by ρUV,an =ρtrial and the infrared part decays with the fourth powerof the spectral value,

limλ→∞

ρIR(λ→∞) ∝ 1

λ4(15)

Inserting the split (14) into (11) leads us to the final finiteresult with ε = 0,∫ ∞

0

dλµ2ελ2−ερ(λ) =

∫ ∞0

dλλ2ρIR(λ)− π

2k , (16)

with a finite (in general numerical) integral over the in-frared part of the spectral function due to (15). Thenumerical convergence of this integral can be further im-proved systematically, if the UV-part ρUV(λ) also in-cludes sub-leading UV-terms of the full spectral function.

The finite result of (16) was obtained without the in-troduction of any possibly symmetry-breaking countert-erms. Since the systematics of this example is general, itcan be applied to all divergences of general diagrams. Asthe demonstrated spectral dimensional renormalisationprocedure is entirely based on dimensional regularisationand the use of spectral representation, it preserves allsymmetries of the theory at hand. Especially for the caseof gauge theories it is manifestly gauge-invariant or rathergauge-consistent. For example, it reflects the peculiarityof dimensional regularisation that integrals without anyexternal scale vanish identically. For k = 0 the integralover the UV-part of the spectral function vanishes andwe are left with the finite IR-part. Accordingly, no masscounterterms are needed in a massless theory such as agauge theory.

Figure 4 illustrates the general renormalisation work-flow, including spectral renormalisation. The schematicrepresentation holds for the case of spectral dimen-sional renormalisation as well as for the subtraction-based and more universal approach of spectral BPHZ-renormalisation, which is introduced in the following.

D. Spectral BPHZ-renormalisation

In fully symmetry-consistent spectral dimensionalrenormalisation as described in Section II C, one has toperform analytic spectral integrals with dimensional reg-ularisation on top of the momentum integrations. Al-ready the latter are more complicated than in standardperturbation theory at the same order, since the spec-tral representations lead to different masses for each line.While the momentum integration has to be necessar-ily analytic in order to access Euclidean and Minkowskispace-time, this is not necessary for the spectral integra-tion, whose non-perturbative infrared part has to be donenumerically in most cases anyway.

The need for additional analytic computations of spec-tral integrals can be circumvented by performing subtrac-tions on the spectral integrals which render the spectral

6

integrals finite. This can be done by subtracting a Taylorexpansion of the spectral integrand in momenta accord-ing to the BPHZ-scheme with Dyson’s formula. The pro-cedure is called spectral BPHZ-renormalisation. We shallsee, that its workflow is still described within Figure 4.It is the last ”spectral” step from the bottom right to thebottom left which is changed by moving from spectral di-mensional to the spectral BPHZ-procedure. We empha-sise that the underlying BPHZ-regularisation in generaldoes not preserve all symmetries of a given theory andin particular breaks gauge symmetry. This is not a con-ceptual problem, as the counterterms also break gaugeinvariance and the final result is gauge-consistent, seee.g. [41]. However, in numerical applications to gaugetheories the gauge-consistent spectral dimensional renor-malisation is arguably worth its price, in particular forinvestigations of the Gribov problem and the confinementmechanism.

In the present example of a scalar φ4-theory, theBPHZ-scheme is consistent with both space-time and theinternal Z2-symmetry. We will therefore utilise it for ex-plicit computation. We introduce and explain the setupwithin a specific example, the sunset graph in the gapequation of the scalar φ4-theory in d dimensions, see Fig-ure 3. This diagram also carries sub-divergences whilestill being relatively simple. In Section II E we addition-ally analyse an explicit one-loop example.

Due to the spectral representation of the sunset graph,the fully perturbative momentum integrals including thesubtraction can be solved analytically in both d = 3 andd = 4:

In d = 4 the sunset graph is superficially quadrati-cally divergent with logarithmic divergences in the sub-diagrams. The respective spectral power counting fol-lows from the momentum dimension of the spectral value,[λ] = 1, where [O] counts the momentum dimension of O.In particular, that of the spectral function is [ρ(λ)] = −2.This can be read off from the classical spectral functionof a field with mass m with ρcl(λ) = 2πδ(λ2 − m2).Trivial examples for such a power counting of spec-tral integrals are (11) and the propagator itself with[∫dλλρ(λ))/(ω2 + λ2)] = −2. We subtract the zeroth

and first order of the Taylor expansion in external mo-mentum about p2 = µ2 as well as the zeroth term in Tay-lor expansions about the external momenta of the subdia-grams. These counterterms contribute to the mass renor-malisation as well as the wave function renormalisationof the scalar field. The subtractions remove the leading

and also the next-to-leading order contributions in thespectral parameters λi in the integrand of the spectralintegral for λi →∞. Consequently, the integrand decaysfaster by two powers of λ2

i and the spectral integrals overλi with i = 1, 2, 3 are UV-finite.

In d = 3 there are no divergent subdiagrams and thesunset is superficially logarithmically divergent. We onlysubtract the zeroth order of the Taylor expansion, whichcontributes to the mass renormalisation.

We are left with finite spectral integrations for the sun-set graph in d = 3, 4 at ε = 0. The integrand dependsanalytically on the external momentum, the spectral val-ues λi with i = 1, 2, 3 of the three internal lines as wellthe respective spectral functions ρ(λi). In general, theremaining spectral integrals have to be performed nu-merically.

In summary, spectral BPHZ-renormalisation, as de-scribed in detail above, has the same workflow as in thelast section and is depicted in Figure 4: First we ap-ply dimensional regularisation to the momentum inte-grations and swap the order of momentum and spectralintegrals. Then we perform the momentum integrationanalytically, right bottom corner in Figure 4. Finally weapply the spectral BPHZ-step: the Taylor expansion inall momenta about the renormalisation scale µ, which al-lows us to take the limit ε → 0. This leaves us with thetask to perform the finite spectral integrals either ana-lytically or numerically, depending on the application.

E. One-loop example: φ3-theory in d = 6

For further illustration we now apply spectral BPHZ-renormalisation within the simple perturbative exampleof the one-loop DSE for the two-point function of the(renormalisable) φ3-theory in d = 6 dimensions with acoupling g/(3!)

∫xφ3. The advantage of this example is

that already at one-loop it requires both a mass renor-malisation and a wave function renormalisation. Hence,both can be discussed within a simple one-loop compu-tation. In contrast, in the φ4-theory the wave functionrenormalisation only arises at two-loop from the sunsetdiagram (even for φc 6= 0).

In the φ3-theory, the only diagrams in the gap equa-tion Figure 3 are the polarisation diagram and the squintdiagram. At one loop we only have to consider the polar-isation diagram. In d = 6, this diagram is quadraticallydivergent. The respective DSE for the (inverse) propa-gator reads schematically,

Γ(2)(p) = Zφ,0(µ)(p2 +m2

φ,0(µ))

+ g2

∫λ1,λ2

λ1λ2 ρ(λ1)ρ(λ2)F (p, µ;λ1, λ2) . (17)

The integrand F results from the momentum integration and depends on the renormalisation group scale µ due todimensional regularisation. Zφ,0 is the wave function renormalisation and m2

φ,0 is the bare mass squared. Both bare

parameters contain counterterms that remove the divergences in the diagrams within an expansion about p2 = µ2: the

7

! (p2 ! !2) [ "p2p2=!2

]p2=!2

spectral

renormalisation!spectral BPHZ

FIG. 5. Schematic spectral BPHZ-renormalisation procedure at the example of the one-loop scalar propagator DSE for theφ3-theory. The diagram is quadratically divergent in d = 6. First, the loop momentum divergences are discarded by the usualmomentum renormalisation part of dimensional regularisation (i.e. implicitly assumed on the LHS). By introducing mass andwave function counterterms, the diagram is subtracted by the first two terms of its own Taylor expansion around the RG scaleµ. This cancels the leading order quadratic and subleading logarithmic divergences of the spectral integrals.

constant quadratic divergence as well as the logarithmic divergence proportional to p2. This amounts to the choices

Zφ,0(µ) = 1− g2

∫λ1,λ2

λ1λ2 ρ(λ1)ρ(λ2)F (p, µ;λ1, λ2)

∂p2

∣∣∣∣p2=µ2

,

Zφ,0(µ)m2φ,0(µ) =m2

φ − g2

∫λ1,λ2

λ1λ2 ρ(λ1)ρ(λ2)

[F (p, µ;λ1, λ2)− µ2 F (p, µ;λ1, λ2)

∂p2

∣∣∣∣p2=µ2

], (18)

for wave function and mass renormalisation respectively. The counterterms propertional to g2 in (18) provide the firsttwo terms of the Taylor expansion about p2 = µ2 of the diagram. To see this, we insert (18) in the DSE (17),

Γ(2)(p) = p2 +m2φ + g2

∫λ1,λ2

λ1λ2 ρ(λ1)ρ(λ2)

[F (p, µ;λ1, λ2)− F (µ, µ;λ1, λ2)−

(p2 − µ2

) F (p, µ;λ1, λ2)

∂p2

∣∣∣∣p2=µ2

].

(19)

Eq. (19) is depicted in Figure 5. Accordingly, (18) imple-ments the standard renormalisation conditions, that thequantum corrections vanish at p2 = µ2,

Γ(2)(p2 = µ2) = Zφ(µ2 +m2φ)

∂p2Γ(2)(p2 = µ2) = Zφ , (20)

for the two-point function with Zφ = 1. In the presentφ3-example these two renormalisation conditions arecomplemented by that for the coupling g, which is alsologarithmically divergent in d = 6. As we have in-troduced this example only for illustration of spectralBPHZ-renormalisation, we refrain from discussing thisany further. Mode details on the spectral renormalisa-tion conditions can be found in Section III B.

Continuing with the discussion of spectral renormali-sation for the two-point function, the subtraction of Fin (19) by its own Taylor expansion at p2 = µ2 leadsto finite spectral integrals. We emphasise that this isnot achieved by simply subtracting the 1/ε-terms beforeperforming the spectral integration: Since F scales withλ2 for large λ with λ = λ1, λ2, the spectral integralsin (17) are quadratically divergent. After spectral BPHZ-renormalisation however, the subtracted scalar integrandin (19) scales as 1/λ2. The subtraction scheme cancelsthe leading and subleading contributions in λ to F andleads to finite spectral integrals.

F. Non-perturbative spectral renormalisation

The discussions of the last three sections, Section II Cto Section II E, entail that spectral renormalisation leadsto two different parts in the counterterms: the first partis related to the momentum divergences and has all theproperties of the counterterms in dimensional regular-isation. The second part comes from the spectral di-vergences. The counterterms in spectral dimensionalrenormalisation respects all symmetries including gaugesymmetries and is tantamount to dimensional regulari-sation and renormalisation. The counterterms in spec-tral BPHZ-renormalisation lack the full symmetries. Inparticular in gauge theories the BPHZ-counter terms arenecessarily not gauge-invariant, precisely for restoringgauge consistency of the full renormalised result.

Importantly, in both spectral renormalisation schemes,the spectral counterterms follow the same recursive rela-tions known from standard perturbation theory. Thismakes it a consistent renormalisation scheme to all or-ders of perturbation theory.

In non-perturbative applications of the present spec-tral approach, the non-perturbative information is solelypresent in the spectral functions of propagators and ver-tices. Moreover, within the DSE we only deal with one-and two-loop diagrams with n vertices derived from themaster DSE in (4), see also Figure 2. We have one classi-cal (bare) vertex and n − 1 full vertices as well as full

8

propagators. If recursively written in terms of loops,both, the finite full vertices and propagators, carry sub-tractions of the divergences in these diagrams that renderthe diagrams finite. The leftover subtractions from thesere-distribution renormalise the one or two explicit loopsin the DSE diagrams.

In summary, non-perturbative spectral renormalisa-tion only concerns the counterterms for the explicit loopsin the DSE, while the rest of the renormalisation is car-ried by the finite full vertices and propagators that haveto obey the renormalisation conditions. This is a consis-tent numerical non-perturbative renormalisation scheme.

III. φ4-THEORY IN 2+1 DIMENSIONS

The φ4-theory in 2 + 1 dimensions is super-renormalisable, and the initial two renormalisation condi-tions for the two-point function in Figure 3 reduce to thefirst one for the mass. Moreover, we do not need to renor-malise the coupling. This entails that in the DSE for thescalar two-point function spectral BPHZ-renormalisationas discussed in Section II D simply amounts to subtract-ing the zeroth order term in the Taylor expansion aboutp2 = µ2. After the momentum integrals are computedanalytically within dimensional regularisation, we are leftwith the finite spectral integrals.

After completing renormalisation, the iterative solu-tion procedure in the DSE is briefly described as follows:With a given input spectral function the renormalisedDSE is evaluated in Minkowski space-time. The inputspectral function is either the initial guess or the resultof the last iteration step. Then, an updated retarded two-point function is computed from the result. This allowsus to extract an updated spectral function, which is fedback as input into the next iteration step. In this sectionwe discuss the calculation sketched above in a detailedway, step by step.

A. Momentum integration and spectralrenormalisation

The DSE can be expressed as the sum of the bare two-point function and the loop diagrams Dj ,

Γ(2)(p) = p2 +m2φ,0 +

∑{j}

Dj(p) , (21)

with j = tad,pol,sun,squint. In (21) we have used thatthe φ4-theory in d = 3 is super-renormalisable and theonly divergent term is the mass term. From now on,we drop the µ-dependence of the spectral integrands fornotational simplicity. With the Kallen-Lehmann spec-tral representation for the full propagator (7), as well asmomentum-independent vertices, an arbitrary loop dia-

gram Dj in the DSE takes the form,

Dj(p) = gj

Nj∏i

(∫λi

λiρ(λi)

)Ij(p;λ1, ..., λNj ) . (22)

The prefactors gj are the products of the combinatorialprefactors in the DSE and the vertices of the correspond-ing diagram. In Table I we provide the prefactors for theMinkowski version of (22), see (25).Dj has Nj internal lines, each of them coming with

one spectral integral and a corresponding spectral func-tion. The Ij are nothing but a product of (momentum)loop integrals over Nj classical propagators with differentspectral masses λi,

Ij(p;λ1, . . . , λNj ) =

N loopsj∏k

∫d3qk(2π)3

Nj∏i

1

λ2i + l2i

. (23)

In (23) the momenta li are linear combinations of theloop-momenta qk and the external momentum p. The

number of loops is denoted by N loopsj in (23). The ana-

lytic solutions of these integrals in d = 3 are known fromperturbation theory, e.g. [35], and can be used here.

With the analytic expressions for Ij , we apply spec-tral BPHZ-renormalisation, c.f. Section II D. Please notethat only the logarithmically divergent sunset diagramDsun explicitly requires renormalisation, since the baretadpole simply can be absorbed in the definition of thebare mass squared. To cancel the logarithmic divergenceof this diagram, within the BPHZ-approach we subtractthe zeroth order term in the Taylor expansion of Dsun

about p2 = µ2. The renormalised diagram reads

Drensun(ω) = gsun

∫λ1,λ2

λ1λ2ρ(λ1)ρ(λ2)

×[Isun(ω;λ1, λ2)− Isun(µ;λ1, λ2)

]. (24)

B. Analytic continuation

The Ij(p) are evaluated in Minkowski space-timefor the retarded two-point function. This is done byparametrising the complex (Euclidean) frequency as p0 =−i(ω + iε) and taking the limit ε→ 0. In a slight abuseof notation we denote the continued expression as Ij(ω).They are given explicitly in Appendix C. The DSE inMinkowski space-time reads

Γ(2)(ω) = − ω2 +m2φ,0

+∑{j}

gj

Nj∏i

(∫λi

λiρ(λi)

)Ij(ω, µ;λ1, ..., λNj ) . (25)

The prefactors of the diagrams of the DSE (25) with theclassical vertex approximation and the skeleton expan-sion can be found in Table I. It can be deduced from the

9

1. Make initial guess ρ0

2. Calculate via DSEΓ(2)

3. Compute from propagatorρ

Iterate until convergence

FIG. 6. Iteration procedure for computing the spectral func-tion. With the initial guess ρ0 for the spectral function, thetwo-point function Γ(2) is computed via the DSE. The re-sulting spectral function is fed back into the DSE for thetwo-point function. This procedure is iterated until the con-vergence for the spectral function is reached.

analytic structure of the spectral integrands Ij(ω), thatthe support in ω of the imaginary part of the polarisationdiagram starts at ω = λ1 + λ2, as expected. Similarly,we find that the support of the sunset diagram starts atω = λ1 + λ2 + λ3. Hence, the support of the expressionsin the two-point function is given by multiples of the polemass. This shows how the Cutkosky cutting rules can beeasily extracted from the present spectral approach.

It is clear from (25) that the spectral renormalisationapproach allows for the implementation of physical on-shell renormalisation conditions. This is in contrast toEuclidean computations, where on-shell renormalisationcan only be implemented for massless modes. On-shellrenormalisation has the advantage that it minimises thequantum corrections in a study of the resonance spec-trum of a given theory. The renormalisation conditions(20) for a φ4-theory in d = 3 dimensions reduce to

Γ(2)[p2 = −m2pole] = 0 , (26)

since both the coupling and the kinetic term do not re-quire renormalisation. The triviality of the wave func-tion renormalisation or rather the vanishing anomalousdimension entail, that the leading large momentum be-haviour of the propagator is given by 1/p2. Accordingly,the canonical commutation relations of the scalar field areunchanged, as is the normalisation of dynamical states.The sum rule (10) is hence formally always satisfied in2+1 dimensions, and thus provides a non-trivially bench-mark test of our results.

C. Spectral integration and iteration

The remaining multi-dimensional integrals over thespectral parameters λi in (25) are solved numerically.Details on the numerical part of the calculation can befound in Appendix B. Subsequently, the spectral functionis extracted from the updated two-point function via (8)and fed back into the DSE. In this way, the DSE is solvediteratively by successively integrating the right hand side

of the DSE with the updated spectral function from thelast step until the solution converges, comp. Figure 6.Note that all the dynamical information is stored in thespectral functions, and the integrands Ij do not changewithin the iterations. Hence, each iteration only involvesthe numerical solution of the respective multidimensionalspectral integral of each diagram.

More details on the convergence test can be found inAppendix B. There, the rapid convergence is illustratedat an exemplary case, see in particular Figure 15.

As a starting point for the iterative procedure, an ini-tial guess for the spectral function has to be made thatis close enough to the solution (in the attraction basinof the solution in terms of the iteration). In the presentcase with the on-shell renormalisation (26) the spectralfunction of the classical theory carries already the correctpole position by definition. In general this will improvethe convergence properties of the iteration. This is yetanother property that singles out on-shell renormalisa-tion. The classical spectral function is given by,

ρ0(ω) = δ(ω2 −m2pole)

=π

mpole

[δ(ω −mpole)− δ(ω +mpole)

]. (27)

The delta-function peaks are located at the physical polemass squared, and the delta-functions at ±mpole, are re-lated by anti-symmetry.

The branch cuts of the loop corrections generate acontinuous tail in the spectral function when insertingthe initial spectral function ρ0 on the right hand side ofthe DSE. This entails via the sum rule in (10), that theresidue of the mass pole decreases to Z < 1 due to thepositive weight of the tail. The mass pole residue of theupdated spectral function is obtained via the relation

Z = − 2mpole

∂ωΓ(2)(ω)

∣∣∣∣ω=mpole

. (28)

The counterterm for the mass is conveniently extractedfrom Re Γ(2)(ω = mpole) = 0.

The three-point function Γ(3)[φc] in the gap equationis evaluated at the constant field value φc = φ0, whichsolves the equation of motion, ∂φVeff[φ0] = 0, at eachstep of the iteration. For fields in the vicinity of φ0 wecan expand the effective potential in powers of φ2 − φ2

0

in the broken phase, leading to,

Veff[φ] =

∞∑n=2

vn2n!

(φ2 − φ2

0

)n. (29)

Accordingly, the two-, three- and four-point functions at

10

FIG. 7. DSE of the two-point function with the classical vertex approximation as used in Section IV A. The tadpole is notpresent because its contribution is absorbed into the mass renormalization in the bare inverse propagator, for more details canbe found in the main text. The two-loop terms constitute vertex corrections of the classical vertices in the one-loop diagrams.The DSE is not two-loop complete as further vertex corrections have been dropped due to the classical vertex approximation.The notation is given in Figure 1.

vanishing momentum are given by,

Γ(2)[φ0] =1

3v2 φ

20 ,

Γ(3)[φ0] = v2 φ0

(1 +

1

15

v3

v2φ2

0

),

Γ(4)[φ0] = v2

(1 +

1

5

v3

v2φ2

0 +1

105

v4

v2φ4

0

). (30)

Dropping the higher order terms O((φ2 − φ2

0)3)

in (29),the terms in parentheses of (30) all reduce to unity.We introduce the curvature mass m2

cur(mpole, λφ) as thevalue of the two-point function at vanishing momentum,i.e.,

m2cur(mpole, λφ) ≡ Γ(2)[φ0](p = 0) . (31)

Note that mcur is determined by the two free parame-ters of the theory, mpole and λφ. It is no new parameter.Using (31) in (30) and dropping the higher-order contri-butions, the three- and four-point functions at vanishingmomentum are given by,

Γ(3)[φ0] =√

3 v2mcur ,

Γ(4)[φ0] = v2 . (32)

v2 is nothing but the full four-vertex at vanishing mo-mentum, v2 = Γ(4)[φ0](p = 0). For a general, momentumdependent four-point function in the s-channel, we findfor the momentum-dependent three-point function,

Γ(3)(p) = Γ(4)(p)

√3

v2mcur , (33)

dropping the φ0-dependence of the correlators from nowon. Note that by choosing Γ(3) consistently with Γ(4) asdone above, the three-point function becomes dynami-cal and is hence updated through each iteration by itsdependence on the two- and four-point function.

IV. RESULTS

In this section we compute and discuss the solutionto the DSE for the propagator with two different ap-proximations for the vertices. At first we solve the DSE

with the classical four-point vertex and the related three-point vertex as derived in Section III C. Subsequently,the DSE is considered in a skeleton expansion. Both, thefull three- and four-point functions Γ(3),Γ(4), are basedon the bubble resummed s-channel approximation to thefour-point vertex derived from its Bethe-Salpeter equa-tion. We also use the DSE for Γ(4) to compute resultswithin a self-consistent version of this setup.

A. DSE with classical four-point vertex

In the present section we approximate the full ver-tices in the gap equation in Figure 3 with their classi-cal counterparts while keeping the two-loop terms. Thisis depicted in Figure 7. The two-loop terms constitutevertex corrections to the classical four-point function inthe tadpole diagram (sunset) and for the classical three-point function in the polarisation diagram (squint). Inthe latter case this is but half of the vertex correction, theother half has been dropped in the current approximationwhen approximating the full three-point function in thepolarisation diagram by its classical counterpart. Thisapproximation resums the propagator but is expected tofail in a regime where vertex corrections grow large.

For the four-point function the classical vertex approx-imation amounts to,

Γ(4) = S(4) = λφ . (34)

With (34) and (32), the three-point function is given by

Γ(3) = S(3)[φ0] =√

3λφmcur , (35)

see (32). The resulting prefactors gj of the diagrams arelisted in Table I. The tadpole diagram only contributes amomentum-independent term that shifts the mass, andis absorbed into the mass renormalisation.

All units will be given in the value of the pole mass.Put differently, we introduce dimensionless units

λφ →λφmpole

, ω → ω

mpole, p→ p

mpole, (36)

This also entails mpole = 1, and the massless limit istaken with λφ → ∞. The dimensionless units also em-

11

λϕ = �

λϕ = ��

λϕ = ��

� � � � � � �

����

����

����

��������� ω

����������������ρ(ω

)

2mpole

3mpole

λϕ = �

λϕ = ��

λϕ = ��

��-� ��-� ��� ��� ������

���

���

�������� �

�����������(�)

FIG. 8. Spectral function (left) and propagator (right) in the scalar theory for the coupling choices λφ = 5, 10, 20 from thefull DSE with classical vertices using on-shell renormalisation (26). All dimensionful quantities were rescaled in units of therespective mass pole result. The different height of the delta peaks encodes the magnitude of the residue relative to the otherspectral functions. The grey dashed lines mark the n-particle onsets. For large enough coupling, the three-particle onsetbecomes visible. The propagators were computed by the Kallen-Lehmann spectral representation.

phasise the well-known fact, that the physics of the φ4-theory in d = 2 + 1 is specified by one dimensionlessparameter, the ratio of coupling and (pole) mass.

We solve the DSE for three different values of the clas-sical coupling constant λφ = 5, 10, 20. The renormalisedmass is fixed by the on-shell RG-conditions (26) withmpole = 1, and the quantum corrections to the massvanish on-shell. For the smallest classical coupling usedhere, λφ = 5, we take the classical spectral function asinitial choice. For the further couplings we use as theinitial choice the full quantum spectral function of theclosest coupling value available. This stabilises the itera-tive procedure when successively moving further to largercouplings inducing larger quantum corrections.

The resulting spectral functions, as well as the corre-sponding propagators, are shown in Figure 8 for differentvalues of the classical four-point coupling, λφ = 5, 10, 20.The mass pole as well as the onset of the two-particlethreshold, φ→ φφ, at twice the pole mass are clearly visi-ble. The three-particle threshold, φ→ φφφ, at 3mpole be-comes visible for large enough coupling. Also, all highern-particle thresholds are present in the result. Since theyare suppressed by the inverse of the respective thresholdenergy squared, they are not visible in the plots. Themain effect of a stronger coupling (or small pole mass)can be understood intuitively very well. The residue ofthe mass pole becomes smaller, while the scattering cutgets larger contributions, i.e. scatterings get enhanceddue to the large coupling or the small pole mass. As men-tioned before, the only parameter present is λφ/mpole,and in the present units with mpole = 1 this simply is λφ,see (36). At the largest coupling value considered here,λφ = 20, also the higher scattering processes start con-tributing significantly: the threshold at 3mpole is clearlyvisible in the spectral function for this coupling, shownin the left panel of Figure 8. For large couplings also thehigher thresholds kick in. This scattering physics alsoleaves its traces in the Euclidean propagator, shown in

the right panel of Figure 8: while first the propagatordrops for small momenta with the increasing coupling(measured in the respective pole masses), it increasesagain for even larger ones due to the more pronouncedscattering physics present in the spectral function.

B. Fully non-perturbative DSE

The practical applicability of the spectral renormali-sation scheme has been shown in the last section withinthe DSE for the two-point function with classical ver-tices, see Figure 7. This approximation implements afull resummation of the propagator. The approximationalso includes some corrections in the higher order dia-grams in the DSE as already discussed in the last section.While these diagrams contribute to the higher-order scat-tering thresholds, this may not be sufficient in the limit ofasymptotically large couplings λφ/mpole →∞, also tan-tamount to small pole masses. In this regime the vertexcorrections should be taken into account consistently.

In the present section we discuss non-perturbative ex-pansion schemes of the DSE as well as resummations ofthe vertices. This allows us to study the strongly cor-related regime of the theory. A full quantitative studyis beyond the scope of the present contribution and isdeferred to future work.

A non-perturbative expansion scheme for the DSE isgiven by the skeleton expansion. In this expansion allvertices are full-dressed, and higher loop-order diagramswith dressed propagators and vertices have to be intro-duced successively. Instead of an expansion in classicalvertices it is an expansion in fully dressed ones. This ex-pansion is closely related to nPI-resummation schemes,in the φ4-theory it is related to a 4-PI scheme.

Here we consider the two-loop order of the skeleton ex-pansion in the broken phase. A first observation is, thatthe prefactor of the squint diagram vanishes: it is fully

12

FIG. 9. Truncated DSE of the two-point function as used in Section IV D. Notation is given in Figure 1. The full s-channelfour-vertex in the tadpole diagram enters via its spectral representation (39). The full vertices in polarisation and sunsetare approximated at zero frequency, see Section IV B 3. The different prefactor in front of the sunset diagram as comparedto Figure 7 is due to the tadpoles contribution to the the sunset topology, cf. Section IV B 2. Squint, kite and double-bubbletopology are dropped, as motivated in Section IV B.

contained in the polarisation diagram with two dressedthree-point functions Γ(3). At perturbative two-loop levelthe expansion involves a kite-diagram as well as a double-bubble diagram. Both topologies are only generated fromthe polarisation diagram in the DSE, more precisely fromvertex corrections of the dressed three-point function.These contributions have to be subtracted in terms ofexplicit kite and double-bubble diagram in the skeletonexpansion. For small field expectation values φ0 � 1, itis reasonable to neglect the kite diagram, since it scaleslike φ4

0 due to its four three-point functions. We willalso drop the double-bubble diagram which is of orderφ2

0. These approximations are discussed again later. Theremaining diagrams are the polarisation, sunset and tad-pole. The present approximation of the two-loop skeletonexpansion of the gap equation is depicted in depicted inFigure 9.

1. Bubble-resummed s-channel four-point function

It is left to specify the approximations for the three-point and four-point vertices. To begin with, we stilluse the relation (33) for the three-point function withthe assumption of small field values φ2

0. This leavesus with the four-point function, for which we resort toa bubble-resummed s-channel expansion, e.g. [40, 42],shown graphically in Figure 10. Algebraically, the mo-mentum dependence of the s-channel in the four-pointfunction can be expressed as

Γ(4)(p) =λφ

1 + λφΠfish(p). (37)

Here, Πfish is the one-loop part of the s-channel self-energy (apart from λ2

φ),

Πfish(p) =1

2

∫λ1,λ2

λ1λ2ρ(λ1)ρ(λ2) Ipol(p;λ1, λ2) . (38)

This is exactly the polarisation diagram also appearingin the DSE with prefactor gpol = 1, cf. Figure 5 and

(17). The analytically-continued expression for Γ(4)(ω) isobtained by simply replacing Ipol(p) in (38) with Ipol(ω).

The resulting Γ(4)(ω) depends on the full propagatorthrough the spectral functions in (38), and the equationsfor Γ(2) and Γ(4) are coupled. We also note that the s-channel resummation used here is obtained in the NLO-expansion of the 1/N -expansion of an O(N)-theory withN real scalar fields.

The iteration procedure does not change for such acoupled system, even though coupled systems genericallyshow worse convergence properties. For a given inputpair Γ(2) and Γ(4) we compute the next iteration fromthe right hand side of the DSE for Γ(2), Figure 9, and theresummed representation of Γ(4), (37). This is repeateduntil convergence is reached.

2. Tadpole contribution to the sunset topology

Before turning to the explicit approximation used inthe skeleton scheme, we emphasise again, that the fully-dressed tadpole diagram and the fully-dressed sunset di-agram are related. They both carry the s-channel ofthe four-point vertex. While the tadpole simply is pro-portional to the s-channel four-point vertex, the sunsetincludes the fish-diagram as a sub-diagram. Indeed, theperturbative two-loop sunset graph is a combination ofthe respective contributions, and the prefactor gsun of thesunset diagram in the skeleton expansion is such that theperturbative prefactor , c.f. Table I.

The tadpole diagram is proportional to the s-channelfour-point vertex and we use the full momentum-dependent four-point vertex obtained from the bubbleresummation (37). Inserting the diagrammatic vertexexpansion explicitly into the diagram, one sees that thedressed tadpole contributes to the sunset topology on theperturbative two-loop level. However, this contributiondoes not account for the full prefactor of the latter. Toarrive at the correct perturbative prefactor of the sun-set diagram, the prefactor of the fully-dressed sunset di-agram in the skeleton expansion needs to be adjustedaccordingly, see Figure 9 and Table I.

13

1

2

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FIG. 10. s-channel expansion of the momentum dependent four-point function Γ(4). With a bubble resummation one arrivesat (37). The notation is defined in Figure 1.

3. Vertex approximation in the skeleton expansion

In the sunset diagram, the two four-point vertices areaveraged due to the two loop momenta that run throughboth vertices. This averaging holds true for both the Eu-clidean branch as well as the Minkowski one. For this rea-son we approximate the full momentum-dependent four-point vertices by that at vanishing momentum, Γ(4)[φ0].

In our approximation with Γ(3)pol(p) = φ0Γ(4)(p) with

external momentum p, the vertices in the polarisationdiagram are, as in the tadpole, proportional to the s-channel four-point vertex. For the sake of simplicity we

also use Γ(3)pol = φ0Γ(4)(ω = 0). In any case, the spec-

tral integrands Ipol and Isun of polarisation and sunsetdiagram remain the same as in Section IV A. However,their prefactors gpol and gsun are modified by the skeletonexpansion, cf. Table I.

As outlined, our solution method requires an analyticsolution of the momentum integrals for all diagrams. Inthe current approximation this holds true for the polar-isation and sunset diagram. It is not the case for thetadpole diagram because the loop-momentum is prob-ing the non-trivial momentum structure of the resummedfour-point function (37). This problem can be resolvedif we can use a spectral representation for the resummedfour-point function, which is discussed in the followingsection.

C. Spectral representation for the four-pointfunction

While the existence and practical form of spectral rep-resentations for full four-point functions pose an intricateproblem, spectral representations for (approximations of)single exchange channels of the four-point function canbe derived. From a practical perspective we may treatsuch a channel similarly to a propagator. This is well-motivated by considering that the resonant channels of afour-point function correspond to particle exchange inter-actions. Technically, this can be made explicit by meansof an Hubbard-Stratonovich transformation. In analogyto a propagator we can make the same ansatz for a spec-tral representation for a single channel of the resummedvertex

Γ(4)(p) = λφ +

∫λ

λ ρ4(λ)

p2 + λ2, (39)

with

ρ4(ω) = 2 Im Γ(4)(− i(ω + i0+)

). (40)

In (39) the constant classical part λφ has to be separated.It has no spectral representation and does not need onefor the present purpose. Indeed, classical vertices havebeen already considered in Section IV A.

Our results confirm that the analytic structure of theresummed vertex is compatible with (39) and works well.This can be seen in Figure 11, the computational detailscan be found in the next section, Section IV D. The spec-tral function ρ4 displayed in the right panel exhibits acontinuous tail corresponding to the φφ → φφ scatter-ing continuum for ω ≥ 2mpole. The spectral representa-tion (39) of the four-point function is used in the tadpolediagram in complete analogy to that of propagators.

Importantly, the spectral representation of the vertexeffectively just leads to another classical propagator withspectral mass λ to the loop momentum integral. Themomentum flowing through the four-vertex is the sum ofthe loop and external momentum. Thus, the momentumintegral of the tadpole is identical to that of the polari-sation diagram, since the internal line of the four-vertexcarries p + q and the initial internal line just the loopmomentum q. The tadpole diagram can therefore be ex-pressed as

Dtad(ω) = gtad

∫λ1,λ2

λ1λ2ρ(λ1)ρ4(λ2)Ipol(ω;λ1, λ2) .

(41)The spectral integral is logarithmically divergent, sincethe vertex spectral function ρ4 drops off in the UV asλ−1 (as opposed to ρ ∼ λ−2 in the UV). Again, we em-ploy spectral BPHZ-renormalisation to subtract the ze-roth order term of the Taylor expansion of Ipol. Finally,the renormalised diagram reads

Drentad(ω) = gtad

∫λ1,λ2

λ1λ2ρ(λ1)ρ4(λ2)

×[Ipol(ω;λ1, λ2)− Ipol(µ;λ1, λ2)

]. (42)

D. Results for the coupled system of propagatorand vertices

The DSE in the skeleton expansion is solved for thecouplings also used in Section IV A, λφ = 5, 10, 20, mea-sured in the pole mass mpole = 1.

14

λϕ = �

λϕ = ��

λϕ = ��

������������ ���

��-� ��� ��� ����

��

��

�������� �

����-������Γ(�) (�)

λϕ = �

λϕ = ��

λϕ = ��

� � � � � ��

-��

-�

-�

�

��������� ω

����������������ρ�(ω

)

FIG. 11. Left: Comparison of the momentum dependent (Euclidean) four-vertex Γ(4)(p) in its initial form (37) with its spectralrepresentation (39) for the coupling choices λφ = {10,15,20} using a classical propagator, i.e. a delta pole spectral functionpeaked at mpole = 1. All dimensionful quantities were rescaled in units of the respective mass pole result. Right: Correspondingspectral functions of the four-point functions. The different height of the delta peaks encodes the magnitude of the residuerelative to the other spectral functions. Also here, all dimensionful quantities were rescaled in units of the respective mass poleresult.

For the first iteration, initial choices ρ0, ρ4,0 for thespectral function of the propagator and that the four-point vertex are required. For ρ0 we use the classicalspectral function (27), as already done in Section IV A.For the spectral function of the four-point function ρ4,0

compute it from the resummed representation of the four-point function, (37) with the initial choice of the spectralfunction of the propagator, ρ0. This results in

ρ4,0(ω) = 2 Imλφ

1 + λφΠfish,0(ω), (43)

with

Πfish,0 =1

2

∫λ1,λ2

λ1λ2ρ0(λ1)ρ0(λ2)Ipol(ω;λ1, λ2)

=1

2Ipol(ω;mpole,mpole) , (44)

with mpole = 1. While one could also simply take theclassical vertex, the convergence speed and potentiallyalso the convergence radius (coupling range) is increasedby the improved choice (43). For further couplings λφwe take as initial choices ρ0 and ρ4,0 the full solutions ρand ρ4 of the closest coupling value already computed.This procedure has already been used in Section IV A,and speeds up the convergence.

The spectral function and propagator obtained fromthe coupled system of resummed four-point function andDSE are displayed in the top panels of Figure 12. As forthe case of bare vertices, shown in Section IV A, we find adistinct one-particle mass pole as well as a scattering tail.The φ→ φφφ onset is not visible in the spectral functionfor any of the coupling configurations. It can be seenthat for increasing coupling λφ, the tail of the spectralfunction becomes more enhanced, since the higher scat-tering states are more accessible. In turn, the mass poleresidue decreases. The corresponding propagators in the

right panel show similar behaviour as the for the DSEwith bare vertices. The larger the coupling gets, the fur-ther the propagators deviate from the classical behaviourG(p)→ 1 for p→ 0.

In the bottom panels of Figure 12 we display the spec-tral function of the s-channel four-point function and theEuclidean four-point function itself. The consistency ofthe spectral representation has been discussed alreadyin the previous section, and is confirmed numerically,see Figure 11. Technically, the negativity of the spec-tral function displayed in Figure 12 can be understoodfrom the dominant quantum correction to the classicalvertex, which is negative. On the conceptual side, withinthe Hubbard-Stratonovich transformation ρ4 is related tominus the spectral function of the exchange particle.

We find a continuous 2 → 2 scattering tail, startingat 2mpole for all coupling choices. The spectral functionis strongly enhanced with increasing coupling. Addition-ally, for larger coupling the spectral functions also clearlyshow the 1 → 3 scattering onsets starting at 3mpole,which was not visible in the propagator spectral func-tions (cf. top left panel of Figure 12). By simple dimen-sional analysis it becomes clear that the higher n-particlethresholds in the propagator spectral function are sup-pressed by their respective energy threshold squared.This is not the case for the vertex spectral function: It de-cays with λ−1, making the higher onsets less suppressed.In turn, the invisibility of four, five, and higher particleonsets is due to their decreasing amplitude, as every nexthigher onset comes with one additional loop. Further, wenote that the visible size of the 1 → 3 scattering onsethas its sole origin in the tadpole diagram. This diagramcontributes to the 1→ 3 scattering process due to the s-channel resummed four-point function, cf. Section IV B 2.The contribution of the sunset itself is very suppressed incomparison. This points towards a general feature of ourapproximation: In the large coupling (massless) limit, the

15

2mpole

3mpole

λϕ = �

λϕ = ��

λϕ = ��

� � � � � � �

����

����

����

����

��������� ω

����������������ρ(ω

)

λϕ = �

λϕ = ��

λϕ = ��

��-� ��-� ��� ��� ������

���

���

�������� �

�����������(�)

2mpole

3mpole

λϕ = �

λϕ = ��

λϕ = ��

� � �� �� ��

-��

-�

-�

��������� ω

����������������ρ�(ω

) λϕ = �

λϕ = ��

λϕ = ��

��-� ��� ��� ����

��

��

�������� �

������Γ(�) (�)

FIG. 12. Results from the skeleton expanded DSE (comp. Figure 9) with a bubble resummed s-channel expansion of the four-point function for coupling choices λφ = 5, 10, 20 using on-shell renormalisation (26). The curves were rescaled by the respectivemass poles. All vertices except for the tadpole one were approximated at ω = 0. TOP: Spectral function (left) and propagator(right). The weight of the continuous tail increases with coupling, the mass pole residue decreases. Higher n-particle onsets arenot visible in the spectral function. The different height of the delta peaks encodes the magnitude of the residue relative to theother spectral functions. The propagators were computed by the Kallen-Lehmann spectral representation. Increasing couplingmakes the propagators deviate more from the classical propagator, which approaches 1 at momentum p → 0. BOTTOM:Four-vertex spectral function (left) and four-vertex (right). Weight of the vertex spectral functions continuous tail increaseswith coupling. For the largest coupling choice, the three-particle onset is visible in the spectral function. The four-verticeswere computed by their spectral-like representation (comp. (39)). The quantum corrections in the IR increase with coupling.All vertices approach their classical value in the UV.

tadpole becomes the dominating diagram in the spectralfunction (for ω > 3mpole). We elaborate more on themassless limit in Appendix A.

While presenting the spectral and correlation functionsin units of the fixed mass pole mpole = 1 allows for a com-parison of the relative strength of the different contribu-tions, the approach to the massless limit is better studiedif the results are presented in units of a uniform interac-tion strength. This is achieved by measuring all resultsin units of the coupling λφ, see Appendix A. For the pa-rameters λφ/mpole = 5, 10, 20 studied here this entailsthat we consider theories with the coupling λφ = 1 withpole masses mpole = 1/5, 1/10, 1/20. Evidently, withinthese units the spectral functions pole position moves to-wards zero and the onset of the scattering states getsmore pronounced, see Figure 14.

E. Self-consistent skeleton expansion

Within the current approximation, we have explicitlydropped the kite and double-bubble diagrams. Implic-itly we have also dropped the squint diagram, that corre-sponds to vertex corrections to the three-point functionin the vacuum polarisation: we have used two dressedthree-point vertices that are derived from the bubble-resummation of the four-point function. Such an approx-imation of the three-point function does not contain thesquint topology.

An alternative approximation of the Dyson-Schwingerequation is derived as follows: We start from the initial,full propagator DSE (Figure 3) and consider it in a dia-grammatic expansion in orders of the constant field φ0.In our approximation, these higher orders come via thethree-vertices Γ(3)(p) = φ0Γ(4)(p)+O(φ3

0). Acting on theDSE with two derivatives w.r.t. the constant field φ0 andmultiplying by φ2

0/2 afterwards, one finds the schematic

16

2mpole

3mpole

λϕ = �

λϕ = ��

λϕ = ��

� � � � � � �

����

����

����

����

��������� ω

����������������ρ(ω

)

λϕ = �

λϕ = ��

λϕ = ��

��-� ��-� ��� ��� ������

���

���

�������� �

�����������(�)

2mpole

3mpole

λϕ = �

λϕ = ��

λϕ = ��

� � �� �� ��

-��

-�

-�

��������� ω

����������������ρ�(ω

) λϕ = �

λϕ = ��

λϕ = ��

��-� ��� ��� ����

��

��

�������� �

������Γ(�) (�)

FIG. 13. Results from the self-consistent skeleton expanded DSE (comp. Figure 9) with a bubble resummed s-channel expansionof the four-point function for coupling choices λφ = 5, 10, 20 using on-shell renormalisation (26). The polarisation diagram ishere expressed through the s-channel four-vertex, cf. (47). The curves were rescaled by the respective mass poles. All verticesexcept for the tadpole one were approximated at ω = 0. TOP: Spectral function (left) and propagator (right). The weight of thecontinuous tail increases with coupling, the mass pole residue decreases. Higher n-particle onsets are not visible. The differentheight of the delta peaks encodes the magnitude of the residue relative to the other spectral functions. The propagators werecomputed by the Kallen-Lehmann spectral representation. Increasing coupling makes the propagators deviate more from theclassical propagator, which approaches 1 at momentum p→ 0. BOTTOM: Four-vertex spectral function (left) and four-vertex(right). Weight of the vertex spectral functions continuous tail increases with coupling. For the largest coupling choice, thethree-particle onset is visible in the spectral function. The four-vertices were computed by their spectral-like representation(comp. (39)). The quantum corrections in the IR increase with coupling. All vertices approach their classical value in the UV.

relation

OΓ(2),diag

[φ2

0

](p) =

1

2φ2

0

[Γ(4)(p,−p, 0, 0)− λφ

], (45)

where OΓ(2),diag

[φ2

0

]represents all diagrams in the propa-

gator DSE with two external constant field legs, includingtheir initial prefactors. This includes the squint diagramand the vacuum polarisation.

On the right hand side of (45), we have the full four-point function subtracted by its classical value. We alsomade explicit the specific momentum dependence of thefour-point function. By differentiating twice w.r.t. to themomentum independent field φ0, Γ(4) only depends onone external momentum p.

This entails that we can re-express all diagrams ofOΓ(2),diag

[φ2

0

]through an s-channel four-point function.

What is missing is the classical part of the vertex DSE.It is included by adding and subtracting the classical ver-tex contribution multiplied by an appropriate prefactor

involving the constant field, 12φ

20λφ to the propagator

DSE. This leads us to (45). Evidently, the additionalconstant part − 1

2φ20λφ in (45) is absorbed in the mass

renormalisation.

The expectation value φ20 can be expressed in terms of

Γ(2)(0) = m2cur and Γ(4)[0]. This leads us to

1

2φ2

0 Γ(4)(p) =3

2m2

cur

Γ(4)(p)

Γ(4)(0), (46)

which has the scaling of a two-point function, and reducesto 3/2m2

cur at vanishing momentum. Making use of (45)and (46), we are led to the DSE for the propagator withMinkowski frequencies ω,

Γ(2)(ω) = − ω2 +m2pole +Dren

tad(ω) +Drensun(ω)

+3

2m2

cur

[Γ(4)(ω)− Γ(4)(mpole)

Γ(4)(0)

], (47)

17

and i = tad,sun. Note that the polarisation and squintdiagram have been absorbed into the last term of (47)proportional to m2

cur as a result of (45). Due to the on-shell renormalisation condition (26), all renormalised di-agrams vanish at ω = mpole, that is Dren

i (mpole) = 0. Insummary, (47) is exact up to higher orders of φ2

0, leavingus with a self-consistent systematic expansion scheme.The self-consistency refers to the fact that in the presentorder in φ2

0, the polarisation diagram is given exactly interms of the four-point vertex. Therefore, approxima-tions to the latter are transported to the former.

1. Vertex-approximation in the self-consistent skeletonexpansion

As discussed above, within the self-consistent DSEin (47), it suffices to specify the approximation for thefour-point function. Here we again resort to the bubble-resummed four-point function of (37), already used in theprevious section. This approximation of the four-pointfunction neglects in particular contributions in the DSEof the four-point function that originate in the squint di-agram. The self-consistency of (47) is reflected in thefact that the contribution of the polarisation diagram isgiven by its bubble-resummed 1/2φ2

0Γ(4),

1

2φ2

0 Γ(4)(p) =1

2φ2

0

[λφ − λφΠfishΓ(4)(p)

]. (48)

As for the four-point function, (48), lacks the contribu-tions from the squint diagram. Note also that these con-tributions are related to the u, t-channel. Hence, with(48) we consistently neglect the squint topology in theDSE if assuming dominance of the s-channel vertex cor-rections for the four-point vertex. We emphasise that theassumption of s-channel dominance is well-supported inthe large-N limit, but less so in the present N = 1 case.

Diagrammatically, the DSE is still represented by Fig-ure 9, with the polarisation diagram given by the lastterm in (47). The prefactors of tadpole and sunset di-agram are identical to that in the standard skeletonscheme used in Section IV D. They are listed in Table I.In the tadpole diagram, the four-point function again en-ters via its spectral representation (39).

2. Results

Numerical results for the self-consistent skeletonscheme are displayed in Figure 13. The propagator spec-tral function is similar in shape to that obtained in thestandard skeleton approximation, see Figure 12. How-ever, the magnitude of the scattering tail close to thethreshold is roughly a third for all coupling choices. Com-paring the two skeleton schemes one realises that the cor-rect momentum scaling behaviour of the standard skele-ton expansion came at the price of two dressed vertices.In turn, the self-consistent skeleton expansion has the

correct momentum scaling as well as the correct vertexstrength. Arguably, this property is particularly impor-tant in the vicinity of s-channel resonances or for asymp-totically large couplings.

Higher n-particle onsets are not visible in the prop-agator spectral function similarly to the results in thestandard skeleton expansion. We emphasise again, thatthey are present nevertheless, as well as easily accessi-ble in the present spectral approach. The three-particleonset can again be seen in the vertex spectral function(bottom left panel of Figure 13), although it is less pro-nounced as in Figure 12. The magnitude of the vertexspectral functions matches very well in the two differentschemes however.

The corresponding Euclidean correlation functions areshown in the top and bottom right panel. Both receiveslightly less quantum corrections as for the plain skeletonexpansion. For the propagator this is quite clear fromthe much smaller spectral functions in the self-consistentapproximation. For the four-point vertex, the differencesare less pronounced and we refrain from discussing them.For a more detailed discussion of the general features,see Section IV D.

3. Low-lying bound state close to phase transition

Lattice calculations [43–45] show an additional low-lying excitation in the spectrum of the scalar φ4-theoryin d = 2 + 1 close to the phase transition with a massof m ≈ 1.8mpole. This state has been interpreted asa bound state of the fundamental excitation in [45]. Itis also been observed within the recent functional RGstudy [46]. The approximation scheme underlying theEuclidean computation done in [46]. is close in spirit tothe s-channel approximation scheme used in the presentwork in Minkowski space-time In [46], the spectral func-tion of the propagator, numerically reconstructed fromEuclidean data, shows a bound state close to the phasetransition at a mass ratio consistent with that foundin the lattice studies. However, further away from thephase transition, the clear signal of the bound state islost in [46].

The present direct computation of spectral functionsis not performed close to the phase transition, that isfor mpole → 0. This regime will be studied elsewherewithin the self-consistent skeleton expansion developedin the present section. Our results for mpole 6= 0 indicatethat the bound state may indeed only exist close to thephase transition. However, the present s-channel approx-imation does not allow for a fully conclusive statement,as the latter requires a multi-channel analysis. Yet, thes-channel resummation typically captures the dominantresonances and is trustworthy as next-to-leading orderin a 1/N -expansion in the large-N limit. The spectralproperties of O(N) models as well as the phase transi-tion regime will be studied elsewhere.

18

V. CONCLUSION

In this work, we developed a spectral functional ap-proach for the direct non-perturbative computation ofreal-time (time-like) correlation functions. This approachis based on a novel renormalisation scheme called spectralrenormalisation: This renormalisation scheme is basedon the use of spectral representations and dimensionalregularisation. The spectral representation allows to per-form the momentum integrals within dimensional reg-ularisation. This leaves us with the spectral integrals,whose spectral divergences can be renormalised withindimensional regularisation as well, see Section II C. Thisscheme is called spectral dimensional renormalisation,and respects all symmetries of the theory at hand thatare maintained within dimensional regularisation. Thelatter set also includes gauge symmetries, and hence spec-tral dimensional renormalisation as developed here is amanifestly gauge-invariant renormalisation scheme.

The renormalisation step can also be done with a stan-dard subtraction procedure within a Taylor expansion inmomenta as done in BPHZ-renormalisation This schemeis called spectral BPHZ-renormalisation. It maintainsless symmetries than spectral dimensional renormalisa-tion and in particular requires counterterms in gaugetheories that break gauge symmetry (or rather BRST-symmetry). The appeal of spectral BPHZ renormalisa-tion lies in its relative numerical simplicity.

In summary, spectral renormalisation allows for directaccess to the real momentum axis by analytic continua-tion as a result of the fully analytic solution of all mo-mentum integrals, while maintaining all symmetries ofthe theory at hand.

We performed explicit, non-perturbative computationswithin the spectral Dyson-Schwinger approach to thescalar φ4-theory in 2+1 space-time dimensions. Our re-sults include the spectral function of the scalar propa-gator and that of the four-point function (s-channel),and are obtained by solving the spectral DSE recursively.We have first considered the approximation with classicalvertices and full propagators, but including the two-loopdiagrams in the DSE. The resulting spectral functionsshow a distinct one-particle pole and a clear scatteringtail with onset at twice the pole mass. The spectral func-tion contains all higher scattering thresholds, which areeasily accessible due to the analytic nature of the momen-tum integrations. For increasing couplings, the 1 → 3scattering onset becomes more pronounced.

We also considered an approximation with non-perturbative vertices, based on a skeleton expansionscheme. The respective four-point function was givenby an s-channel bubble-resummation. The propagatorspectral function again shows a distinct one-particle poleand a continuous scattering tail. The spectral functionof the resummed four-vertex features a scattering tailas well and additionally shows a distinct onset for the1 → 3 scattering process. For larger couplings, the fullynon-perturbative nature of the approximation leads to

large, though only quantitative differences compared tothe classical vertex computation.

In the last part of this work, Section IV E, we have de-veloped a self-consistent skeleton expansion scheme. Theself-consistency was obtained by relating a class of dia-grams with three-point functions to the s-channel four-point function. This entails that the approximation usedin the computation of the four-point function is also usedwithin the three-point function diagrams. The resultsare qualitatively similar to that of the standard skele-ton scheme. The magnitude of the scattering tail of thepropagator spectral function turns out much smaller inthe upgraded scheme, and indicates an overestimation ofthe polarisation diagram before. We expect that the useof such a self-consistent scheme is important close to thephase transition of the theory or, more generally, in thepresence of resonant s-channel interactions.

In conclusion, we have developed and put to work afully non-perturbative spectral functional approach tothe computation of real-time correlation functions. Theapproach was successfully tested within the scalar φ4-theory in 2+1 dimensions. It is specifically attractivein gauge theories, as spectral dimensional renormalisa-tion preserves gauge invariance. We hope to report onrespective results soon.

VI. ACKNOWLEDGMENTS

We thank G. Eichmann, C. Fischer, M. Hasenbusch,C. Jung, P. Lowdon, N. Stamatescu, M. Salmhoferand L. von Smekal for discussions. This work is sup-ported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excel-lence Strategy EXC 2181/1 - 390900948 (the HeidelbergSTRUCTURES Excellence Cluster) and under the Col-laborative Research Centre SFB 1225 (ISOQUANT) andthe BMBF grant 05P18VHFCA.

Appendix A: Massless limit

The scalar φ4-theory in 2+1 dimensions depends onone dimensionless parameter, λφ/mpole. In the presentwork we have shown all results in terms of the respec-tive pole mass. Hence, the above parameter simply re-lates to different couplings λφ, measured in units of thepole mass mpole = 1. If interested in the massless limitof the theory, it is more convenient to keep the cou-pling fixed λφ = 1, and to depict all results for differ-ent pole masses. For example, in the work we have usedλφ = 5, 10, 20 with mpole = 1, which can be read asλφ = 1 and mpole = 1/5, 1/10, 1/20. This leads us to thespectral functions for propagator and four-vertex as wellas the respective Euclidean correlators itself of the scalarfield, depicted or rather redrawn in Figure 14. Rescalingthe results in this way it gets clear the massless limit isreadily investigated through the limit λφ → ∞. For a

19

����� =��

����� =���

����� =���

��� ��� ��� ��� ���

�

�

�

�

��������� ω

����������������ρ(ω

)

����� =��

����� =���

����� =���

��-� ��-� ����

���

���

���

�������� �

�����������(�)

����� =��

����� =���

����� =���

��� ��� ��� ���

-���

-���

-���

��������� ω

����������������ρ�(ω

)

����� =��

����� =���

����� =���

��-� ��� ��� �������

����

����

����

�������� �

����-������Γ(�) (�)

FIG. 14. Results from the skeleton expanded DSE (comp. Figure 9) with a bubble resummed s-channel expansion of thefour-point function for coupling choices λφ = 5, 10, 20 using on-shell renormalisation (26). All vertices except for the tadpoleone were approximated at ω = 0. The curves were rescaled by their respective coupling parameters, i.e. mpole,i = 1

λφ,iand

consequently λφ =λφ,iλφ,i

= 1 for all curves. TOP: Spectral function (left) and propagator (right). The weight of the continuous

tail decreases with larger mass pole, the mass pole residues increases. The different height of the delta peaks encodes themagnitude of the residue relative to the other spectral functions. The propagators were computed by the Kallen-Lehmannspectral representation. The pole contributions are dominant, since for smaller mass pole the propagator is strongly enhanced.

consistent treatment of the DSE, the vertices need to bewell defined and have the appropriate scaling propertiesin this limit. This is given for the resummed s-channelfour-point function that was introduced in Section IV Band used in calculation of Section IV D and Section IV E.

Note that in the skeleton expansion as approximatedin this work, the only other appearing vertex, which isthe three-point function, is obtained directly from Γ(4)

and hence has the same property. The resummation ishence suitable for studying the massless case in our skele-ton expansion. In all diagrams except for the tadpole, allvertices are approximated at frequency zero. In conse-quence, they do not carry loop momentum and merely en-ter as multiplicative factor, which makes them well undercontrol in the large coupling limit. For the tadpole how-ever, this is not the case. Here, the four-point functionenters via its spectral representation (39). The additiveclassical contribution drops out by renormalisation, as itcontributes momentum-independently. What is left is thevertex spectral function contributing to the tadpole loopintegral, cf. (42). With increasing coupling, this spectralfunction gets larger, see Figure 12 or Figure 13. Since

the multidimensional spectral integrals need to be eval-uated numerically, a UV cutoff for the integrals needs tobe chosen that minimises the error caused by doing so.This results in increasingly long-range integrals for thelarge coupling limit. The appropriate treatment of theseintegrals hence results in a technical obstacle, which willbe the subject of a follow-up project. Apart from thetechnical aspect, this suggests that the tadpole diagramis the dominant contribution to the DSE and thus to thespectral function for ω > 3mpole in the s-channel approx-imation. As a result, one is left solely with polarisationand tadpole diagram, further simplifying the setup.

Appendix B: Numerics

In order to compute the spectral integrals in (25), theintegrands Ij are discretised on suitable, evenly spacedmomentum grids of usually around 100 points. The gridsare chosen differently for each diagram such that peakedor discontinuous structures like the onset jumps in theimaginary parts are ideally resolved. The spectral in-

20

����� = �

����� = �

����� = �

����� = �

����� = ��

��� ��� ��� ��� ��� ���

����

����

����

��������� ω

����������������ρ(ω

)

Convergence

boundary

��� ��� ���

����

����

����

�/�����

�����������������

FIG. 15. Example of a convergent iteration of the scalar DSE with classical vertices. The curves were not rescaled by therespective mass poles to demonstrate convergence of the iteration also for the pole mass. Instead, renormalisation was doneat µ = 0. Units thus given by the input parameters. The spectral functions are alternating, approaching the final orangecurve (left). 20 iterations have been performed. The curves from iterations 10 to 19 were left out as they were graphicallyindistinguishable from the final curve. An iterative behaviour as displayed is taken taken to be convergent, i.e. signalling asolution to the DSE where left and right hand side of the equation coincide. The corresponding pole positions also convergevery quickly (right). Left of the convergence boundary, assuming a (relative) precision of 10−4, all points are identical.

tegrations are performed numerically in Mathematicawith standard global adaptive integration strategies us-ing a relative precision goal of 10−3. All diagrams is in-terpolated separately in real and imaginary part in orderto treat the sharp onset of the imaginary parts properly.All interpolations are performed using B-splines up toorder 2 as all interpolants are real due to the split ofreal and imaginary part. The spectral function is thencomputed from the interpolated diagrams.

For a given set of parameters, convergence was usuallyreached within less than 10 iterations. Figure 15 demon-strates convergent behaviour. We estimate the relativeprecision of our routine to be ≥ 10−4. Based on that,all mass poles shown in the right panel of Figure 15 left

Classical vertices Skeleton

gpol − 32Γ(2)(ω = 0)λφ − 3

2Γ(2)(ω = 0)Γ(4)(ω = 0)

gsunset − 16λ2φ

112

Γ(4)(ω = 0)2

gsquint32λ2φΓ(2)(ω = 0) 0

gtad 0 12

TABLE I. Prefactors of the propagator DSE diagrams in thedifferent approximation schemes. The prefactors are obtainedby the standard DSE prefactors and the loop-momentum andspectral parameter independent parts of the vertices. Thetadpole factor in the approximation with classical vertices isset to zero, as the tadpole is absorbed completely in the massrenormalisation. The sunset prefactor in the skeleton expan-sion compensates the two-loop contributions of the tadpolewith full four-vertex. In the self-consistent approximationof Section IV E, the skeleton prefactors apply with exceptionof the polarisation diagram, which is given by (48).

of the convergence boundary are indistinguishable. Nor-malising the scattering onsets of all spectral functionsin the left panel to be identical, the continuous tail alsoconverges pointwise beyond the convergence boundary,based on above precision estimate.

21

Appendix C: Analytic expressions of the diagrams

Analytic expressions for the integrands in (25) before and after analytic continuation:

Polarization:

Ipol(p;λ1, λ2) =1

4πparctan

(p

λ1 + λ2

),

Ipol(ω;λ1, λ2) =1

4πω

[artanh

(ω

λ1 + λ2

)+ i arg

(1− ω

λ1 + λ2

)] (C1)

Sunset:

Isun(p;λ1, λ2, λ3) =1

(4π)2

(1

2log

(1

(λ1 + λ2 + λ3)2 + p2

)− λ1 + λ2 + λ3

parctan

(p

λ1 + λ2 + λ3

)),

Isun(ω;λ1, λ2, λ3) =1

(4π)2

(1

2log

(1

(λ1 + λ2 + λ3)2 − ω2

)

− λ1 + λ2 + λ3

ω

[artanh

(ω

λ1 + λ2 + λ3

)+ i arg

(1− ω

λ1 + λ2 + λ3

)])(C2)

Squint:

Isquint(p;λ1, λ2, λ3, λ4) =1

(8π)2ωλ4

(2 log

(λ2 + λ3 + λ4

λ2 + λ3 − λ4

)artanh

(ω

λ1 + λ4

)+ i

[Li2

(ip− λ1 − λ4

λ2 + λ3 − λ4

)

− Li2

(ip− λ1 + λ4

λ2 + λ3 + λ4

)+ Li2

(−ip− λ1 + λ4

λ2 + λ3 + λ4

)− Li2

(−ip− λ1 − λ4

λ2 + λ3 − λ4

)]),

Isquint(ω;λ1, λ2, λ3, λ4) =1

(8π)2ωλ4

(Re

[2 log

(λ2 + λ3 + λ4

λ2 + λ3 − λ4

)artanh

(ω

λ1 + λ4

)− Li2

(ω − λ1 − λ4

λ2 + λ3 − λ4

)+ Li2

(−ω − λ1 − λ4

λ2 + λ3 − λ4

)+ Li2

(ω − λ1 + λ4

λ2 + λ3 + λ4

)− Li2

(−ω − λ1 + λ4

λ2 + λ3 + λ4

)]− i θ

(ω − |λ1 + λ4|

)Im

[2 log

(λ2 + λ3 + λ4

λ2 + λ3 − λ4

)artanh

(ω

λ1 + λ4

)− Li2

(ω − λ1 − λ4

λ2 + λ3 − λ4

)+ Li2

(−ω − λ1 − λ4

λ2 + λ3 − λ4

)+ Li2

(ω − λ1 + λ4

λ2 + λ3 + λ4

)− Li2

(−ω − λ1 + λ4

λ2 + λ3 + λφ

)])(C3)

[1] E. E. Salpeter, Phys. Rev. 84, 1226 (1951).[2] E. Salpeter and H. Bethe, Phys.Rev. 84, 1232 (1951).[3] G. Eichmann, H. Sanchis-Alepuz, R. Williams,

R. Alkofer, and C. S. Fischer, Prog. Part. Nucl.Phys. 91, 1 (2016), arXiv:1606.09602 [hep-ph].

[4] S. Aoki et al., Eur. Phys. J. C77, 112 (2017),arXiv:1607.00299 [hep-lat].

[5] G. Krein, M. Nielsen, R. D. Puff, and L. Wilets, Phys.Rev. C47, 2485 (1993), arXiv:nucl-th/9303010 [nucl-th].

[6] S. Floerchinger, JHEP 05, 021 (2012), arXiv:1112.4374[hep-th].

[7] S. Strauss, C. S. Fischer, and C. Kellermann, Phys. Rev.

Lett. 109, 252001 (2012), arXiv:1208.6239 [hep-ph].[8] S. M. Dorkin, L. P. Kaptari, T. Hilger, and B. Kampfer,

Phys. Rev. C89, 034005 (2014), arXiv:1312.2721 [hep-ph].

[9] R.-A. Tripolt, N. Strodthoff, L. von Smekal, andJ. Wambach, Phys. Rev. D89, 034010 (2014),arXiv:1311.0630 [hep-ph].

[10] S. M. Dorkin, L. P. Kaptari, and B. Kampfer, Phys. Rev.C91, 055201 (2015), arXiv:1412.3345 [hep-ph].

[11] J. M. Pawlowski and N. Strodthoff, Phys. Rev. D92,094009 (2015), arXiv:1508.01160 [hep-ph].

[12] S. Dorkin, M. Viebach, L. Kaptari, and B. Kampfer, J.

22

Mod. Phys. 7, 2071 (2016), arXiv:1512.06596 [nucl-th].[13] T. Yokota, T. Kunihiro, and K. Morita, PTEP 2016,

073D01 (2016), arXiv:1603.02147 [hep-ph].[14] K. Kamikado, T. Kanazawa, and S. Uchino, Phys. Rev.

A95, 013612 (2017), arXiv:1606.03721 [cond-mat.quant-gas].

[15] C. Jung, F. Rennecke, R.-A. Tripolt, L. von Smekal,and J. Wambach, Phys. Rev. D95, 036020 (2017),arXiv:1610.08754 [hep-ph].

[16] J. M. Pawlowski, N. Strodthoff, and N. Wink, Phys.Rev. D98, 074008 (2018), arXiv:1711.07444 [hep-th].

[17] S. Jia and M. R. Pennington, Phys. Rev. D96, 036021(2017), arXiv:1705.04523 [nucl-th].

[18] T. Yokota, T. Kunihiro, and K. Morita, Phys. Rev. D96,074028 (2017), arXiv:1707.05520 [hep-ph].

[19] Z. Wang and P. Zhuang, Phys. Rev. D96, 014006 (2017),arXiv:1703.01035 [hep-ph].

[20] M. Bluhm, Y. Jiang, M. Nahrgang, J. M. Pawlowski,F. Rennecke, and N. Wink, Proceedings, 27th In-ternational Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018): Venice, Italy,May 14-19, 2018, Nucl. Phys. A982, 871 (2019),arXiv:1808.01377 [hep-ph].

[21] R.-A. Tripolt, C. Jung, N. Tanji, L. von Smekal, andJ. Wambach, Proceedings, 27th International Conferenceon Ultrarelativistic Nucleus-Nucleus Collisions (QuarkMatter 2018): Venice, Italy, May 14-19, 2018, Nucl.Phys. A982, 775 (2019), arXiv:1807.04952 [hep-ph].

[22] R.-A. Tripolt, J. Weyrich, L. von Smekal, andJ. Wambach, Phys. Rev. D98, 094002 (2018),arXiv:1807.11708 [hep-ph].

[23] Z. Wang and L. He, Phys. Rev. D98, 094031 (2018),arXiv:1808.08535 [hep-ph].

[24] R. Alkofer, A. Maas, W. A. Mian, M. Mitter, J. Parıs-Lopez, J. M. Pawlowski, and N. Wink, Phys. Rev. D 99,054029 (2019), arXiv:1810.07955 [hep-ph].

[25] G. Eichmann, P. Duarte, M. T. Pena, and A. Stadler,Phys. Rev. D100, 094001 (2019), arXiv:1907.05402 [hep-ph].

[26] E. L. Solis, C. S. R. Costa, V. V. Luiz, and G. Krein, FewBody Syst. 60, 49 (2019), arXiv:1905.08710 [hep-ph].

[27] G. Aarts and J. Berges, Phys. Rev. D 64, 105010 (2001),arXiv:hep-ph/0103049.

[28] Yu. B. Ivanov, F. Riek, H. van Hees, and J. Knoll, Phys.Rev. D72, 036008 (2005), arXiv:hep-ph/0506157 [hep-

ph].[29] D. Roder, J. Ruppert, and D. H. Rischke, Nuclear

Physics A 775, 127 (2006).[30] G. Marko, U. Reinosa, and Z. Szep, Phys. Rev. D92,

125035 (2015), arXiv:1510.04932 [hep-ph].[31] L. Shen and J. Berges, Phys. Rev. D 101, 056009 (2020),

arXiv:1912.07565 [hep-ph].[32] L. Shen, J. Berges, J. Pawlowski, and A. Rothkopf,

(2020), arXiv:2003.03270 [hep-ph].[33] A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski,

and N. Strodthoff, Phys. Rev. D94, 054005 (2016),arXiv:1605.01856 [hep-ph].

[34] A. K. Cyrol, M. Mitter, J. M. Pawlowski, andN. Strodthoff, Phys. Rev. D 97, 054006 (2018),arXiv:1706.06326 [hep-ph].

[35] A. K. Rajantie, Nucl. Phys. B480, 729 (1996), [Erratum:Nucl. Phys.B513,761(1998)], arXiv:hep-ph/9606216 [hep-ph].

[36] R. Alkofer and L. von Smekal, Phys. Rept. 353, 281(2001), arXiv:hep-ph/0007355.

[37] G. Kallen, Helv. Phys. Acta 25, 417 (1952).[38] H. Lehmann, Nuovo Cim. 11, 342 (1954).[39] A. K. Cyrol, J. M. Pawlowski, A. Rothkopf, and

N. Wink, SciPost Phys. 5, 065 (2018), arXiv:1804.00945[hep-ph].

[40] M. E. Peskin and D. V. Schroeder, An Introduction toquantum field theory (Addison-Wesley, Reading, 1995) p.842.

[41] E. Kraus, Annals Phys. 262, 155 (1998), arXiv:hep-th/9709154.

[42] K. Fukushima and J. M. Pawlowski, Phys. Rev. D 86,076013 (2012), arXiv:1203.4330 [hep-ph].

[43] V. Agostini, G. Carlino, M. Caselle, and M. Hasenbusch,Nucl. Phys. B484, 331 (1997), arXiv:hep-lat/9607029[hep-lat].

[44] M. Caselle, M. Hasenbusch, and P. Provero, Nucl. Phys.B556, 575 (1999), arXiv:hep-lat/9903011 [hep-lat].

[45] M. Caselle, M. Hasenbusch, P. Provero, and K. Zarembo,Nucl. Phys. B623, 474 (2002), arXiv:hep-th/0103130[hep-th].

[46] F. Rose, F. Benitez, F. Leonard, and B. Delamotte,Phys. Rev. D93, 125018 (2016), arXiv:1604.05285 [cond-mat.stat-mech].