K\" All\'En-Lehmann Spectral Representation of the Scalar SU (2

$K\" All\'En-Lehmann Spectral Representation of the Scalar SU (2$

K¨all´en–LehmannSpectral Representation of the Scalar SU(2) Glueball

David Dudala,b,∗ Orlando Oliveirac,† Martin Roelfsa‡

a KU Leuven Campus Kortrijk–Kulak, Department of Physics, Etienne Sabbelaan 53 bus 7657, 8500 Kortrijk, Belgium b Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium c CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal

Abstract The estimation of the Källén-Lehmannspectral density from gauge invariant lattice QCD two point correlation functions is proposed, and explored via an appropriate inversion method. As proof of concept the SU(2) glueball spectrum for the quantum numbers J PC = 0++ is investigated for various values of the lattice spacing. The spectral density and the glueball spectrum are estimated using the published data of [1]. Our estimates for the ground state mass are in good agreement with the traditional approach published therein, which is based on the large time exponential behaviour of the correlation functions. Furthermore, the spectral density also contains hints of excites states in the spectrum and, despite using non-optimized gauge configurations, the method is able to estimate their mass values.

1 Introduction

A major effort to understand the dynamics of strong interactions is the computation of the hadronic spectra. In QCD, hadrons, namely mesons and baryons, are seen as bound states of quarks and gluons. Besides the conventional hadronic states, multiple-quark and pure glue bound states are also predicted by theory. Of these exotic states, so far only multi-quark states have been identified experimentally [2]. For glueballs there are a number of candidate states [3, 4, 5], but no established observation. Glueballs continue to attract a great deal of theoretical [6, 7, 8, 9, 10, 11, 12] and experimental attention, with an ongoing effort to identify these pure glue states unambiguously [13, 14, 15]. Additionally, glueball states have also been discussed as possible candidates for dark matter [1, 16].

arXiv:2103.11846v1 [hep-lat] 22 Mar 2021 A ﬁrst principles approach to access the hadron spectra, which has been advancing since the beginning of the eighties from last century [17], is lattice QCD. In a typical lattice QCD computation of the bound state mass, an appropriate two point correlation function is evaluated and, from its large time decay behaviour and the associated slope, the ground state mass is estimated [18]. It is common practise in the computation of bound masses to rely on techniques that improve the signal to noise ratio of the Monte Carlo simulation. These techniques allow access not only to the ground state mass with

∗[email protected] †[email protected] ‡[email protected]; Corresponding author

1 the chosen quantum numbers, but, sometimes, also allow the first excited state to be extracted. In practice however, the computation of the excited states masses has proven to be a difficult task. Herein, we aim to discuss an alternative way of accessing the particle masses: via the computation of the Källén-Lehmannspectral representation associated with the momentum space particle propagator [19]. A possible advantage of using spectral repre- sentations compared to a conventional lattice calculation is that it does not necessarily require the use of smearing, or other techniques, to improve the Monte Carlo signal to noise ratio, see e.g. [20, 21] and references therein. As seen below, the computation of the spectral function allows simultaneous extraction of the ground state and, to some extent, also of the 1st excited state. The interest in the Källén-Lehmannspectral representation goes beyond the determination of particle spectra. Besides accessing the spectra at zero and finite temperature, the spectral representation is linked with the analytical structure of the associated propagator [22], and it allows the computation of thermodynamical and transport properties [23]. Furthermore, in QCD, or any other confining theory, it can help to understand the confinement mechanism [24, 25, 26, 27, 28, 29, 30]. In the current work we focus on the SU(2) glueball states with quantum numbers J PC = 0++, i.e. the scalar glueball. However, in principle, the procedure can be extended to other quantum numbers and other gauge groups. As discussed below, the masses of the glueball 0++ obtained from the spectral function are in good agreement with the estimates of a more conventional mass calculation. Let us describe our procedure to access the Källén-Lehmannspectral representation ρ(ω) from a two point correlation function G(p2). The relation between these two functions reads Z Z 2 ∞ 2ωρ(ω) dω ∞ ρ(ω) dω G(p ) = 2 2 = . 0 ω + p ω ip −∞ − This definition assumes that the integrations are well defined [31] and, therefore, ρ(ω) approaches zero sufficiently fast as ω . Certain correlators contain polynomial terms that diverge for large p as happens| | → e.g. ∞ for the glueball operator, something which follows immediately from power counting and a dimensional analysis. However in these cases it is still possible to write down a sensible Källén–Lehmannspectral representation, if one first subtracts the polynomial part [31]. This corresponds to adding appropriate contact counterterms. So, if G(p) has a polynomial part, the propagator can be written as n 1 Z X− ∞ 2ωρ˜(ω) dω G(p2) = a (p2 p¯2)k + ( p2 +p ¯2)n (1) k − − ω2 + p2 k=0 0 n 1 Z X− ∞ ρ˜(ω) dω = a (p2 p¯2)k + ( p2 +p ¯2)n k − − ω ip k=0 −∞ − with

1 ∂nG(p2) a = , (2) n n! ∂(p2)n p2=¯p2 ρ(ω) ρ˜(ω) = , (3) (ω2 +p ¯2)n andp ¯2 is a reference momentum scale at which the subtraction is done. A derivation of the above relations can be found in appendix A. For the scalar glueball a dimensional analysis [10] shows that n = 3. The spectral function ρ(ω) can be obtained, if the

2 subtraction of the polynomial part can be performed. However, performing subtractions on numerical data is extremely sensitive to the choice ofp ¯ and results in relative rapid variation of ak with the subtraction point [32]. An elegant{ way} to perform the subtractions, without actually having to do these, is by considering the Fourier transform of G(p2) and look at the Schwinger function deﬁned as Z ∞ 2 ip4τ C(τ) = G(p) (τ) = G(p ) e− dp4 . (4) F{ } ~p=0,p =0 −∞ 46 Setting the subtraction reference momentump ¯ = 0, yields

C(τ) = G(p) (τ) (5) F{ } n 1 X− = ρ(ω) ( τ ) + 2π a ( 1)kδ(2k)(τ) (6) L{ } | | k − k=0 n Z n+1 X (2(n k)+2) ∞ 2k 3 + 4π( 1) δ − (τ) dω ω − ρ˜(ω), − k=2 0 see appendix A. The important observation however, is that C(τ) = ρ(ω) ( τ ) when τ = 0, and equal to a sum of (derivatives of) Dirac delta functionsL{ when} | |τ = 0. Therefore,6 ρ(ω) can be recovered by taking C(τ) for τ > 0, and inverting the Laplace transformation. Because the inverse Laplace transform is an ill-defined numerical problem, regularization is necessary in order to perform the inversion. We shall use Tikhonov regularization, similar to our previously published method [25, 33]. However, because glueballs are observable particles, their spectral density ρ(ω) is non-negative. Therefore we shall implement Tikhonov regularization using Non-Negative Least Squares (NNLS) [34], to ensure a positive spectral function ρ(ω) 0. Tikhonov regularization is not the only possible way to regularize the inversion, and≥ a number of different regularization strategies have been explored in literature to access the spectral function for particle correlators by various authors [35, 36, 37, 38, 39, 40, 30, 28, 41, 42]. An advantage of the Tikhonov regularization being that it keeps the optimization function a quadratic function, which translates into solving a modified linear system of equations. The most widely used approach is based on the maximum entropy method [35]. We will therefore use the same litmus test as in [35] to benchmark our approach, before turning to the actual glueball case.

2 The Numerical Method

1 In a lattice simulation the propagator G(pn) is computed on a ﬁnite set of evenly spaced momenta pn. Given a data set G(pn) := G(p0),...,G(pN 1) , the Schwinger function is computed using DFFT, resulting{ in} a data{ set C(τ ) ,− where} { k } N 1 X− i2πkn/N C(τk) = G(pn)e− . n=0 The Laplace transformation to access the spectral function

Z ∞ ωτk C(τk) = ρ(ω) (τk) = e− ρ(ω) dω , (7) L{ } 0 1Note the change in notation from G(p2) to G(p), where p now stands for the four component of the momentum. See Eq. (4) for the deﬁnition of the Schwinger function.

3 can be written as a matrix equation C = Kρ, with the elements of K deﬁned as

ω`τk Kk` := e− ∆ω. Since ρ needs to be obtained, and a direct solution is impossible due to the near zero singular values of K, the original problem is replaced by the minimisation of the Tikhonov regularizing functional 2 2 2 J = Kρ C + α ρ ρ∗ (8) α k − k2 k − k2 2 where α > 0 is the Tikhonov parameter and ρ∗ is a prior estimate for ρ. In order to impose the constraint ρ 0, define ≥ K C A = , b = , (9) α1 αρ∗ and rewrite Jα as J = Aρ b 2. (10) α k − k2 It is straightforward to show that (8) and (10) are equivalent. However, the formulation of the problem using Eq. (10) can be solved with a non-negative least squares (NNLS) solver such that ρ 0 is guaranteed [34]. ≥ Determination of α The regularization parameter α, in essence, provides a soft threshold to the singular values of K, such that the smallest singular values no longer cause numerical issues. Choosing α is a delicate affair, since setting it too small means the problem remains ill-defined, whereas setting it too large destroys a lot of the information contained in the data. Our preferred criterion for α2 relies on the Morozov discrepancy principle [33], which states that α2 should be chosen such that X Kρ C 2 = σ2, (11) k − k2 i i P 2 2 where i σi is the total variance in the data. The α obeying eq. (11) is guaranteed to be unique [43], and means that the quality of the reconstruction is identical to the quality of the data.

Construction of K The matrix K should perform the Laplace transform as truth- fully as possible, and therefore ω should range from [0, ωmax), where ωmax is suﬃciently large compared to any features that might appear in ρ(ω). In order to ensure this we 5 5 choose to sample ω evenly in logarithmic space from [10− , 10 ] GeV in Nω steps. How- ever, it should be remarked that the peak positions are very consistent, in both linear and logarithmic space, provided ωmax is large enough. For the problems under consid- eration, the peak positions do not change signiﬁcantly after ω > 100 GeV. A numerical study is provided in appendix B.

3 Results and Discussion

In this section we detail the results of applying the method of section 2 to various mock and real data. In section 3.1 the method is applied to a toy model based on a vector- meson spectral density to establish its reliability. Then, in section 3.2 the method is applied to recent lattice SU(2) propagator data for the 0++ glueball [1].

4 3.1 Meson toy-model

0.15 4 − 0.10 = 10 d 0.05

0.00

0.15 3 − 0.10 = 10 d 0.05

0.00

0.15 2 − 0.10 = 10 d 0.05

0.00 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 N = 5 N = 10 N = 15

Figure 1: Reconstruction the toy-model spectral density function for various d and N. The dashed black line is the original spectral function, while the blue full curve is the reconstructed spectral function as given by the Tikhonov regularized NNLS method.

In order to investigate the reliability of the method we consider a realistic toy-model, based on a vector-meson model decay into hadrons, as used before in [35]. This particular model needs a single subtraction, and therefore provides an excellent test of the method. To allow comparison of our results to those of [35], we shall use the exact same process to generate the mock data. The meson spectral density function is given by 2 2 Γρmρ 1 αs 1 ρ(ω) = Fρ + 1 + , (12) π (ω2 m2)2 + Γ2m2 8π π 1 + e(ω0 ω)/δ − ρ ρ ρ − with an energy-dependent width

g2 4m2 3/2 Γ (ω) = ρππ m 1 π θ(ω 2m ). (13) ρ 48π ρ − ω2 − π

The empirical values of the parameters are

mρ = 0.77 GeV mπ = 0.14 GeV mρ gρππ = 5.45 Fρ = gρππ

ω0 = 1.3 GeV δ = 0.2 GeV

2 As ω , this model behaves like ρ(ω ) = (1/4π )(1 + αs/π). Therefore the integral→ eq. ∞ (1) does not converge, and a single→ ∞ subtraction has to be performed, that means n = 1 in the notation used in eq. (2).

5 1 β Volume Conﬁgurations N max(C(τ))/ σC (τ) a√σ a[Λ− ] 3 h 4 i 2.1 10 12 1,000,000 7 3.28 10− 0.608(16) 0.356(27) 4 × × 4 2.2 12 9,999,990 7 1.04 10− 0.467(10) 0.273(20) 3 × 4 2.3 14 16 4.100,000 9 1.38 10− 0.3687(22) 0.216(15) 3 × × 4 2.4 16 24 2,030,000 13 1.55 10− 0.2660(21) 0.156(11) 3 × × 4 2.5 20 24 520,000 13 3.04 10− 0.1881(28) 0.110(8) × × Table 1: The glueball data sets of [1]. Selection of information taken from [1, Table I, Table II].

2 Assuming αs = 0.3, the value (1/4π )(1 + αs/π) = 0.0277 can be used as the prior, but identical to [35] we shall use the slightly smaller value ρprior = 0.0257. In order to generate mock data, we compute Corig(τk) as the Laplace transform 1 of ρ(ω), on N points τk spaced by ∆τ = 0.085 fm = 0.43078 GeV− . The standard deviation of the noise is chosen as τ σ(τ ) = d C (τ ) k , k orig k ∆τ where d is a parameter which controls the noise level, identical to that of [35]. The mock data set is then generated as C(τ ) = (µ = C (τ ), σ2 = σ(τ )2). k N orig k k These mock data sets were then inverted, ignoring C(τ0), for various values of N and d, using the method of section 2, to test the robustness of the method. No inversions without positivity constraints were performed, as signiﬁcant positivity violations were observed in initial trials. Here we used Nω = 1000 in the construction of K. The results are shown in ﬁg. 1. Both more data-points, or less noise, are found to improve the quality of the reconstruction. Interestingly, more data-points do lead to a greater variance in the reconstructions, though their quality is improved with the number of data-points. A direct comparison with [35, Figure 4] is complicated by the absence of uncertainties on that Figure, but the performance of the methods seems comparable using the naked eye.

3.2 SU(2) Glueball data Yamanaka et al. [1], see also [44], have provided us with the Schwinger functions for the SU(2) pure Yang-Mills glueballs, using lattice simulations for β = 2.1, 2.2, 2.3, 2.4, 2.5. The Schwinger functions were computed using the raw data, that is, without any smearing applied. The lattice volumes, the number of configurations, the number of Schwinger function time slices N, and the uncertainties in each data set, are shown in table 1. The simulations of Yamanaka et al. rely on large ensembles of configurations, resulting in data sets with very low statistical uncertainties. This makes the results of the Tikhonov regularized inversion reliable, as seen in section 3.1. Additionally, the number of Schwinger function time slices N for these data sets is 7, 9, or 13, meaning the variance in the inversion will be small, as was observed in section 3.1. For further details about the data sets, we refer the reader to [1]. The Schwinger functions for the various simulations can be seen in fig. 2, whereas fig. 3 shows the spectral density functions as obtained without any constraints on ρ(ω), using the ip-method published in [33]. Here we used Nω = 10000 in the construction of K to create high quality reconstructions. However, this is unnecessarily large, and was done purely to improve the

6 1.0 β = 2.1,N = 7 β = 2.1 β = 2.2,N = 7 1.00 β = 2.2 β = 2.3,N = 9 β = 2.3 0.8 0.75 β = 2.4,N = 13 β = 2.4

) β = 2.5,N = 13 β = 2.5 1

τ 0.6 0.50 ( ) ω /C ( )

ρ 0.25 τ

( 0.4 C 0.00 0.2 0.25 − 0.0 0.50 − 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 1 τ /GeV− ω /GeV

Figure 2: C(τ)/C(τ1) for all data sets. Figure 3: ρ(ω) for all data sets, using the ip-method. The y-axis has been normalized to give the last peak an intensity of 1. Clearly positivity constraints should be imposed.

aesthetics of fig. 4. See appendix B for the dependence of the ground state mass on Nω. As fig. 3 illustrates, there are significant positivity violations and rapid oscillations when positivity is not enforced, especially for small ω, a tell-tale sign of overfitting. On the other hand, when positivity is imposed, the resulting spectral functions in fig. 4 all dis- play a clear mass gap that correspond to a ground state mass of 1.4 GeV. Moreover, the infrared oscillations are gone. ∼ The exact mass values are given in table 3, which lists the 0++ ground state mass as obtained from the spectral density function, as well as those calculated by Yamanaka et al. [1].

1.2 β = 2.1 β ω ± HWHM /GeV 1.0 β = 2.2 2.1 1.382 + 0.049 − 0.041 β = 2.3 3.712 + 1.048 − 0.486 β = 2.4 0.8 β = 2.5 2.2 1.444 ± 0.003

) 3.354 + 0.023 − 0.031 ω

( 0.6 ρ 2.3 1.401 + 0.241 − 0.138 0.4 3.941 + 1.929 − 1.106 2.4 1.505 + 0.085 − 0.064 0.2 2.670 + 0.431 − 0.307 6.118 + 2.644 − 1.520 0.0 0 1 2 3 4 5 6 7 8 9 10 2.5 1.392 + 0.103 − 0.079 ω /GeV 3.798 + 1.833 − 0.950

Figure 4: ρ(ω) for all datasets, subject to Table 2: Maxima of ﬁg. 4 in ρ(ω) 0. The y-axis has been normalized order of increasing ω/GeV, to give≥ the maximum an intensity of 1. with corresponding left and right Half Width at Half Maximum (HWHM).

1 The values of a√σ and a[Λ− ] are given in table 1. A string tension √σ of 0.44 GeV was assumed to convert to physical units. Since the toy model study from section 3.1 suggests that a small number of data points, combined with high quality data, leads to a low variance in the inversion, we 1 assume that the uncertainties in a√σ or a[Λ− ] are the dominant sources of uncertainty in the calculated masses. The uncertainties in the glueball masses as given in table 3

7 Traditional [1] Spectral representation β amφ mφ[Λ] mφ /GeV amφ mφ[Λ] mφ /GeV 2.1 1.853(13) 5.21(39) 1.34(56) 1.910(52) 5.364(15) 1.382(38) 2.2 1.517(10) 5.55(41) 1.43(34) 1.532(32) 5.613(12) 1.444(30) 2.3 1.241(6) 5.75(54) 1.48(54) 1.174(8) 5.436(38) 1.401(10) 2.4 0.924(8) 5.93(43) 1.53(16) 0.910(8) 5.832(53) 1.505(14) 2.5 0.696(6) 6.32(46) 1.63(23) 0.595(10) 5.408(88) 1.392(23)

Table 3: The 0++ glueball masses as presented in Table 3 of [1] compared with the spectral representation method. The physical units were calculated assuming √σ = 0.44 GeV.

1 were therefore calculated by propagating the uncertainties in a√σ or a[Λ− ] respectively. The scalar glueball masses as obtained by [1], and those extracted using the spectral density method, agree within one standard deviation. This is actually quite remarkable, since we based ourselves on the unsmeared data, whilst the mass esimates from [1] were obtained after smearing, which usually improves the ground state signal. In addition to the ground state, the spectra presented in fig. 4 also show a clear indication of an excited state, though its position shows much greater variance with β than that of the ground state. Table 2 lists the ω/GeV values of all the local maxima in fig. 4, and the left and right Half Width at Half Maximum (HWHM) values of each local maximum. Except for β = 2.4, all conditions indicate a single excited state at 3.3 3.9 GeV. The only exception is β = 2.4, which has a second maximum at 2.670(25)− GeV and a third one at 6.118(57) GeV. The relatively wide peaks for the excited states indicate that we might need a better signal, as there are in principle no decay channels open for these 1st excited states in pure gluodynamics, not even to the 0++ ground state. We do not have a clear-cut understanding of why the β = 2.2 (very sharp 1st excited state) and the β = 2.4 (2nd excited state and relative dislocation of the 1st excited state peak) are a bit different, but we dare to speculate it has either to do with the much larger configuration number for β = 2.2, or with the β = 2.4 data set encompassing a considerably larger lattice volume, combined with still a quite large statistical ensemble, relative to the other β-values considered. We should also keep in mind that we are aiming at reliable continuum estimates, which implies β should be sufficiently large, so probably values of β < 2.4 are suboptimal from this viewpoint. The size of the (inverse) lattice spacing also determines the maximally accessible momentum scale, forcing our spectral function estimates to vanish in the UV, in contradistinction with the continuum tail going as ω4, as follows from power counting or explicit computation [45]. Interestingly, re-expressing the β = 2.4 groundstate and 1st excited state mass in units of string tension, we get 3.420(32)√σ and 6.069(56)√σ respectively, values that compare quite well with an independent SU(2) estimate giving 3.781(23)√σ for the ground state and 6.126(38)√σ for the 1st excited state2.

4 Summary and Conclusion

In this work we discuss the computation of the Källén-Lehmannspectral density function ρ from two point correlation functions G(p2) that can have a (divergent) polynomial part. The extraction of the finite part of G(p2) that is associated with ρ requires the subtraction of that polynomial part. To avoid doing the subtraction directly on the

2We thank A. Athenodorou and M. Teper for sharing these yet unpublished results with us.

8 numerical data, an operation that can be cumbersome and a source of large uncertainties, we consider the Schwinger function C(τ) instead of G(p2). Once the Schwinger function has been calculated from G(p2), subtractions are performed simply by removing the data point C(0). The spectral density is then obtained by taking the inverse Laplace transform of C(τ) for τ > 0. Further, the inversion method as described in [33], that does not constrain ρ, is compared with the reformulation of the problem imposing a positive ρ, that relies on the non-negative least square method to perform the inversion. The numerical method was first tested on mock data from a given spectral density. The test followed the same rules as used in the original maximum entropy method inversion paper [35], with similar results. The inversion method was then applied to extract the Källén-Lehmannspectral density function from the lattice SU(2) glueball propagator data with quantum numbers J PC = 0++, based on the unsmeared data of [1]. We found that the inversion method is robust and allows estimation of the ground state and excited states masses, albeit that the latter must be considered with some more care. The comparison of the mass estimates between a conventional lattice approach [1] and the method described herein shows that the latter achieves a better statistically accurate result for the ground state, despite the lack of smearing. Further, the spectral method is able to give information in the 0++ glueball spectra not appearing from the conventional approach invoked in [1]. The new spectral method is sufficiently general to be applied to any spectroscopic calculation. In future research, we will scrutinize the usage of smeared data to boost the contribution of the lowest lying states, which should allow us to access at least the 1st excited state with a better accuracy. Afterwards, the analysis performed in the current manuscript might also be extended to the analysis of the spectra of other glueball states with different quantum numbers.

5 Acknowledgements

The authors are indebted to N. Yamanaka, H. Iida, A. Nakamura and M. Wakayama for sharing the Schwinger function estimates based on their data of [1]. We also thank A. Athenodorou and M. Teper for useful communication. O.O. was partly supported by the FCT – Funda¸c˜aopara a Ciˆenciae a Tecnologia, I.P., under project numbers UIDB/04564/2020 and UIDP/04564/2020, while the work of D.D. and M.R. was supported by KU Leuven IF project C14/16/067.

A Subtracted correlator

A correlator G(p2) can be expressed in terms of a Källén–Lehmannspectral representation as Z 2 ∞ 2ωρ(ω) dω G(p ) = 2 2 , (14) 0 ω + p if this integral converges. However, if the correlator contains divergences polynomial in p2, it is still possible to find a Källén–Lehmannspectral representation, if one first subtracts the polynomial part. In order to do this, consider the Taylor expansion of G(p) aroundp ¯:

n 1 X− ∂kG(p2) G(p2) = (p2 p¯2)k + G˜ (p2). (15) − ∂(p2)k n k=0 p2=¯p2

9 2 where Gñ(p ) is the remainder after the first n terms have been isolated. At zeroth order, we find

2 2 2 G˜1(p ) = G(p ) G(¯p ) − Z 2 2 ∞ 2ωρ(ω) dω = ( p +p ¯ ) 2 2 2 2 (16) − 0 (ω +p ¯ )(ω + p ) Through induction, we obtain Z ˜ 2 2 2 n ∞ 2ωρ(ω) dω Gn(p ) = ( p +p ¯ ) 2 2 n 2 2 (17) − 0 (ω +p ¯ ) (ω + p ) This allows us to state that n 1 Z X− ∞ 2ωρ˜(ω) dω G(p2) = a (p2 p¯2)k + ( p2 +p ¯2)n k − − ω2 + p2 k=0 0 n 1 Z X− ∞ ρ˜(ω) dω = a (p2 p¯2)k + ( p2 +p ¯2)n (18) k − − ω ip k=0 −∞ − with

1 ∂nG(p2) a = , (19) n n! ∂(p2)n p2=¯p2 ρ(ω) ρ˜(ω) = . (20) (ω2 +p ¯2)n

Because ρ is an odd (real) function for bosonic degrees of freedom [35], so isρ ˜. We will now calculate the Fourier transform of the subtracted correlator, in order to obtain C(τ).

n 1 Z X− ∞ ρ˜(ω) dω G(p) = a (p2)k + ( p2)n (21) F{ } kF F − ω ip k=0 −∞ − | {z } | {z } A B where we have setp ¯ = 0. For the ﬁrst term A we obtain

n 1 n 1 X− X− A = a (p2)k = 2π a ( 1)kδ(2k)(τ) (22) kF k − k=0 k=0 and for the second Z ∞ ρ˜(ω) dω B = ( p2)n (23) F − ω ip −∞ − 2 n Z ∞ 2ω( p ) = dω ρ˜(ω) 2− 2 (24) 0 F ω + p Z " n # n ∞ X 2k 3 (2(n k)+2) 2n τ ω = 2π( 1) dω ρ˜(ω) 2 ω − δ − (τ) + ω e−| | − − 0 k=2 n Z n+1 X (2(n k)+2) ∞ 2k 3 2n = 4π( 1) δ − (τ) dω ω − ρ˜(ω) + ρ˜(ω)ω ( τ ) − L | | k=2 0

10 As a result the total transform is given by

n 1 X− G(p) = 2π a ( 1)kδ(2k)(τ) + ρ(ω) ( τ ) (25) F{ } k − L{ } | | k=0 n Z n+1 X (2(n k)+2) ∞ 2k 3 + 4π( 1) δ − (τ) dω ω − ρ˜(ω) − k=2 0

An important consequence of this is that all the subtractions are turned into (derivatives of) delta functions, which end up at τ = 0. Therefore, taking only τ > 0, we can say that

C(τ) = ρ(ω) (τ). L{ }

11 B Numerical dependence on Nω To produce the glueball spectra given in section 3.2, ω was sampled evenly in logarithmic 5 5 space between [10− , 10 ] GeV, in Nω = 10000 steps, when constructing the Laplace kernel K. However, this number of steps Nω is unnecessarily large, and was chosen merely to generate aesthetically pleasing graphs. To demonstrate the convergence of the maxima of the spectral functions, fig. 5 shows the ω coordinate of the first and second maximum in the β = 2.1 dataset for increasing Nω. In both the linearly spaced and logarithmically evenly spaced scenarios, the ground state converges rapidly, and the second maximum soon follows suit. In addition, fig. 6 shows the full spectrum in both scenarios, normalized such that the first maximum has ρ(ω) = 1. There is excellent agreement between the two methods for the determination of the ground state mass, but the convergence of the second maximum is better when using logarithmically evenly spaced samples.

10 10 1st 1st 9 2st 9 2st

8 8

7 7

6 6

5 5

4 4

/GeV of local maximum 3 /GeV of local maximum 3 ω ω

2 2

1 1

0 0 26 27 28 29 210 211 26 27 28 29 210 211 Nω Nω (a) Results for logarithmic sample of ω in (b) Results for linear sample of ω in [10−5, 105] GeV. [0, 100] GeV.

Figure 5: The position of the ﬁrst and second maxima for the β = 2.1 data set, as a function of the number of samples Nω in the Laplace kernel K.

103 103

102 102

101 101

6 6 Nω = 2 Nω = 2 0 0 10 7 10 7 Nω = 2 Nω = 2 8 8 Nω = 2 Nω = 2 9 9 , normalized ﬁrst maximum 1 Nω = 2 , normalized ﬁrst maximum 1 Nω = 2 ) 10− ) 10− 10 10 ω N = 2 ω N = 2

( ω ( ω

ρ 11 ρ 11 Nω = 2 Nω = 2 2 2 10− 10− 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 ω /GeV ω /GeV (a) ω sampled uniformly in logarithmic (b) ω sampled uniformly in linear space be- space between [10−5, 105] GeV. tween [0, 100] GeV.

Figure 6: ρ(ω) for the β = 2.1 data set, as a function of the number of samples Nω in the Laplace kernel K. ρ(ω) was normalized such that the ﬁrst maximum has ρ(ω) = 1.

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