Nonperturbative infrared finiteness in super-renormalisable scalar quantum field theory
Guido Cossu, Luigi Del Debbio, Andreas Juttner, Ben Kitching-Morley, Joseph K. L. Lee, Antonin Portelli, Henrique Bergallo Rocha, Kostas Skenderis
NNonperturbative infrared finiteness in super-renormalisable scalar quantum fieldtheory
Guido Cossu,
1, 2
Luigi Del Debbio, Andreas J¨uttner,
3, 4, ∗ Ben Kitching-Morley,
3, 5, 4
Joseph K. L. Lee, Antonin Portelli, Henrique Bergallo Rocha, and Kostas Skenderis
5, 4 (LatCos Collaboration) Braid Technologies, Shibuya 2-24-12, Tokyo, Japan Higgs Centre for Theoretical Physics, School of Physics and Astronomy,The University of Edinburgh, Edinburgh EH9 3FD, UK School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK STAG Research Center, University of Southampton, Highfield, Southampton SO17 1BJ, UK Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK (Dated: October 1, 2020)We present a study of the IR behaviour of a three-dimensional super-renormalisable quantumfield theory (QFT) consisting of a scalar field in the adjoint of SU ( N ) with a ϕ interaction. A baremass is required for the theory to be massless at the quantum level. In perturbation theory thecritical mass is ambiguous due to infrared (IR) divergences and we indeed find that at two-loops inlattice perturbation theory the critical mass diverges logarithmically. It was conjectured long ago in[1, 2] that super-renormalisable theories are nonperturbatively IR finite, with the coupling constantplaying the role of an IR regulator. Using a combination of Markov-Chain-Monte-Carlo simulationsof the lattice-regularised theory, both frequentist and Bayesian data analysis, and considerations ofa corresponding effective theory we gather evidence that this is indeed the case. a. Introduction: Massless super-renormalisablequantum field theories suffer from severe infrared diver-gences in perturbation theory: the same power countingargument that implies good ultraviolet (UV) behavioralso implies bad IR behavior. Explicit perturbative com-putations (with an IR regulator) lead to IR logarithmswhich make the perturbative results ambiguous. Thefate of such IR singularities was discussed in [1, 2] whereit was argued that such theories are nonperturbativelyIR finite. In the examples analysed in [1, 2] the nonper-turbative answer, when expanded with a small couplingconstant, reduced to the perturbative result but withthe IR regulator replaced by the (dimensionful) couplingconstant.One motivation for the original studies was that in thehigh-temperature limit of four-dimensional Yang-Mills(YM) theory there is an effective dimensional reductionto three dimensions and the dimensionally reduced the-ory is super-renormalizable. Here our motivation comesfrom a new application of massless super-renormalisabletheories: such theories appear in holographic models forthe very early universe [3].The models introduced in [3] are based on three-dimensional SU ( N ) Yang-Mills theory coupled to mass-less scalars ϕ in the adjoint of SU ( N ) with a ϕ in-teraction. To compute the predictions of these modelsfor cosmological observables one needs a nonperturbativeevaluation of the relevant QFT correlators. This is thecase even in the regime where the effective expansion pa-rameter is small because of the IR singularities discussedabove. Moreover, understanding the IR behavior of this ∗ [email protected] QFT is important for another reason: in holographic cos-mology cosmic evolution corresponds to inverse RG flow,and the initial singularity in the bulk is mapped to theIR behavior of the dual QFT. Thus a mechanism for cur-ing the IR singularities would also provide a holographicresolution of the initial bulk singularity.In this Letter we initiate the study of such a theory us-ing lattice methods. We will focus on the simplest theorywithin this class: three-dimensional massless scalar QFTwith ϕ in the adjoint of SU ( N ) and a ϕ interaction reg-ularised on a Euclidean space-time lattice [4]. It turnsout this theory still provides an interesting holographicmodel. Irrespective of the holographic motivation we be-lieve that understanding the fate of IR singularities inthis QFT is an interesting problem in its own right andthis model provides the possibility to explicitly test thehypothesis in [1, 2].We address two central questions in this paper: Is thetheory nonperturbatively IR finite, and what is the criti-cal mass, i.e. what is the value of the bare mass suchthat the renormalised theory is massless? The latterquestion is crucial for future simulations at the mass-less limit where the holographic duality is defined [3].At two-loops the critical mass is both linearly UV di-vergent and logarithmically IR divergent. We proceedto a nonperturbative determination of the critical massin Markov-Chain-Monte-Carlo simulations of the discre-tised Euclidean path integral, where naively the inverseof the finite extent of the lattice L acts as the only IR reg-ulator. By studying the finite-size scaling (FSS) nonper-turbatively, within the effective theory and on the lattice,we find evidence for the absence of IR divergence beyondperturbation theory.The N = 2 model is equivalent to the O (3) vector a r X i v : . [ h e p - l a t ] S e p FIG. 1. One- and two-loop diagrams contributing to themass-renormalisation in double-line representation represent-ing matrix indices of the scalar propagator. model and the N = 3 model is in the same universal-ity class as the O (8) vector model [5], which have beenstudied widely in the literature [6], including studies oftheir critical mass [7]. For N > b. Lattice perturbation theory:
We consider thethree-dimensional Euclidean scalar SU ( N ) valued ϕ theory, S = (cid:90) d x Tr (cid:104) ( ∂ µ ϕ ( x )) + ( m − m c ) ϕ ( x ) + λϕ ( x ) (cid:105) , (1)with fields ϕ = ϕ a ( x ) T a where ϕ a ( x ) is real, and T a are the generators of SU ( N ) (Tr (cid:2) T a T b (cid:3) = δ ab ). In thefollowing we prefer to work with a rescaled version of theaction where the ’t Hooft scaling is explicit, S = Ng (cid:90) d x Tr (cid:104) ( ∂ µ φ ( x )) + ( m − m c ) φ ( x ) + φ ( x ) (cid:105) , (2)which we obtain from Eq. (1) by identifying φ = (cid:112) N/g ϕ and λ = g/N , where g is the ’t Hooft coupling whichshould be kept fixed in the large N limit. The field φ and coupling constant g have mass dimension one.To discretise the theory on a 3 d space-time lattice withlattice spacing a we replace partial derivatives by finitedifferences, ∂ µ φ ( x ) → δ µ φ ( x ) = ( φ ( x + ˆ µa ) − φ ( x )) /a ,and integrals by sums (cid:82) d x → a (cid:80) x ∈ Λ , where a is thelattice constant, ˆ µ a unit-vector in the µ direction andΛ the set of all lattice sites. We use periodic boundaryconditions.The diagrams that contribute to the critical mass, m c ,at the two-loop level are shown in Fig. 1. The IR-finite but linearly UV-divergent one-loop integral eval-uated with Mathematica is π/a (cid:90) − π/a d k (2 π ) k = Z a with Z = 0 . ... , (3)for lattice momenta ˆ k = a sin( ka/ p = 0 is D ( p ) = π/a (cid:90) − π/a d k (2 π ) d q (2 π ) k ˆ q ˆ r , (4) ag D (Λ) ( am c ) , N = 2 ( am c ) , N = 41-loop 2-loop 1-loop 2-loop0.1 0.05469(19) -0.03159 -0.03125 -0.04581 -0.045430.2 0.04953(13) -0.06318 -0.06194 -0.09161 -0.090240.3 0.04783(13) -0.09477 -0.09208 -0.13742 -0.134430.5 0.045311(92) -0.15796 -0.15088 -0.22904 -0.221160.6 0.044134(90) -0.18955 -0.17962 -0.27484 -0.26380TABLE I. Results for the two-loop integral D (Λ) and thecritical mass in lattice perturbation theory. where r = − k − q − p , and hatted quantities are definedas above. By naive dimensional counting, and confirmedby repeating the analysis of the IR-properties of this di-agram in [9] for d = 3, we find that the integral divergeslogarithmically in the IR: D ( p ) p → = D IR ( p ) = − log( | pa | )(4 π ) (5)(derivation in Sec. I of supplementary material).Following [1, 2], we introduce the IR regulator | p | = g/ (4 πN ) ≡ Λ. The two-loop expression for the criticalmass then evaluates to m c ( g ) = − g Z a (cid:18) − N (cid:19) + g D (Λ) N ( N ) , (6)where N ( N ) = 1 − /N + 18 /N . Representative valuesfor D (Λ) and m c ( g ) at one- and two-loops for N = 2 and4 are listed in table Tab. I. For the range of couplingspresented in the table the change from one- to two-loopcorresponds to a relative change in the range 1% to 6%. c. Finite-size scaling for m c : In this section we pro-vide details and results of our nonperturbative studies to-wards the determination of the critical mass. Our strat-egy is to compute it as a function of the IR cutoff givenin terms of the inverse lattice size 1 /L , by means of FSS.The observable we consider is the Binder cumulant, B = 1 − N (cid:104) Tr (cid:2) M (cid:3) (cid:105)(cid:104) Tr [ M ] (cid:105) , (7)where M is the magnetisation matrix defined below, and (cid:104)·(cid:105) indicates expectation value under the Euclidean pathintegral.For each choice of simulation parameters, we determinethe bare input mass, m ( ¯ B, g, L ), in the critical regionfor which the Binder cumulant takes some suitably cho-sen value ¯ B . The Binder cumulant in a finite volume of We do not know the correct proportionality factor accompanying g . The current choice is arbitrary and corresponds to defining ascheme. We evaluate the two-loop lattice momentum integral using theMarkov Chain Monte Carlo integration implemented in VE-GAS [10]. The error estimates we provide together with theresults are statistical only. extent L in the critical scaling region is described by ascaling function f ,¯ B = f (cid:16)(cid:0) m ( ¯ B, g, L ) − m c ( g ) (cid:1) /g x /ν (cid:17) , (8)where x = gL and ν is the critical exponent. Expanding f in the vicinity of the critical mass we find the leadingFSS behaviour m ( ¯ B, g, L ) = m c ( g ) + g x − /ν ¯ B − f (0) f (cid:48) (0) . (9) d. FSS in the Continuum Effective Theory: Beforeanalysing and interpreting simulation data for the FSSof the critical mass, we can gain further analytical un-derstanding of the critical behaviour. To this end weconsider the underlying effective field theory (EFT) ofthe zero-mode of the field φ , i.e. the magnetisation M = 1 L (cid:90) d x φ ( x ) , (10)and fluctuations χ around it, i.e. φ = M + χ . In thevicinity of the critical point long-distance contributionsdescribed by M dominate, motivating us to consider theleading-order effective action S eff = L Ng (cid:2) ( m − m c )Tr (cid:2) M (cid:3) + Tr (cid:2) M (cid:3)(cid:3) . (11)Following [11], we quantise the theory under the finite-volume path integral and find integral expressions for theBinder cumulant (for details see Sec II B of supplemen-tary material). Expanding again in the vicinity of thecritical point we recover Eq. (9) and compute the leading-order predictions ν | N =2 , = 2 / f (0) | N =2 ≈ . f (cid:48) (0) | N =2 ≈ − . f (0) N =4 ≈ . f (cid:48) (0) N =4 ≈ − . e. Lattice simulation: We implemented the modelin the GRID library [12, 13] with both the Hy-brid Monte Carlo [14] and a heat-bath over-relaxationalgorithm [15–18]. We generated ensembles of O (100k) field configurations for N = 2 ,
4, volumeswith
L/a = 8 , , , , , , ag =0 . , . , . , . , .
6, and a number of bare mass param-eters in the vicinity of the perturbative prediction for m c ( g ) in Eq. (6). By using a wide range of couplings,a large range of lattice volumes was covered (0 . ≤ x ≤ .
8) while keeping simulation costs acceptable.Using multi-histogram reweighting [19] we obtained acontinuous representation of Eq. (7) as a function of thebare scalar mass. Example results for B ( N, g, L ) areshown in the top panel of Fig. 2 and the reweightingis illustrated in the bottom panel. The analysis was car-ried out under bootstrap resampling [20]. We determinedthe integrated autocorrelation time τ int , M , M and φ with the method of [21] with largest values being O(100).All data was binned into bins of size min(50 , τ int ). The − − . . . . . . . B ( N , g , L ) f (0) (EFT) B for am → ∞ (EFT) N = 2 L/a = 8
L/a = 16
L/a = 32
L/a = 48
L/a = 64
L/a = 96
L/a = 128 − . . . . . . . f (0) (EFT) B for am → ∞ (EFT) N = 4 ( m c − m ) /m c x /ν − . − . − . − . − . am ) . . . . . B − l oo p P T − l oo p P T − l oo p P T − l oo p P T − l oo p P T − l oo p P T − l oo p P T FIG. 2. Top: N = 2 (left) and N = 4 (right) results for theBinder cumulant, the EFT prediction for f (0) and the valueof the Binder cumulant in the large-mass limit (cf. Sec. d).The values on the x -axis have been rescaled using the val-ues of the critical exponent ν and the critical masses m c determined in Sec. f. Darker colour corresponds to largervalue of gL . Bottom: Data points from simulations, linesfrom reweighting with width corresponding to the statisticalerror. Intersects of N = 4, g = 0 . L/a = 128 , , , , , ,
16 with ¯ B = 0 .
50 indicatedwith y -error bars. The black vertical line indicates the 2-loopinfinite-volume value of the critical mass. reweighting allows for a model-independent determina-tion of m ( ¯ B, g, L ) by means of an iterative solution.Example results for m ( ¯ B, g, L ) are listed in Tab. II.We note the proximity of these finite-volume results tothe 2-loop infinite-volume prediction listed in Tab. I. f. Finite-size scaling analysis:
We now turn to thefitting of m ( ¯ B, g, L ). Guided by Eq. (9) we chose the fit
L/a ag am ) ( ¯ B = 0 . , g, L ) for N = 2. ansatz m ( ¯ B,g, L ) = m c ( g ) | − loop + g α + g (cid:32) x − /ν ¯ B − f f + βD IR (Λ IR ) N ( N ) (cid:33) , (12)The first term is the 1-loop expression for the criticalmass and it removes the linear UV divergence pertur-batively (cf. Eq. (6)). The second term parameterisesthe dependence on the IR cutoff for which we study, re-spectively, Λ IR = π gN and L . Potential residual schemedependence in the IR/UV regulator, e.g. normalisationfactors in the argument of D , are absorbed by α . Tobetter constrain the fit we simultaneously analyse datafrom various pairings of two ¯ B values in the vicinity of f (0) as predicted in the EFT (cf. Sec. d). For N = 4 weallowed one value of α per ¯ B value. For N = 2 excellentfit quality was achieved without this additional freedom.The central fits are for pairs ¯ B = { . , . }| N =2 and { . , . }| N =4 , respectively, for which we foundthe largest number of degrees of freedom describedsimultaneously. The ansatz in Eq. (12) provides anexcellent parameterisation ( p -values well above 5%) forthe simulation data over the entire range gL min (cid:38)
12 to gL max = 76 .
8. The case N = 2 is illustrated in Fig. 3for Λ IR = π gN . Tab. III summarises the fit results.The first error is statistical, and, where applicable,the second error is the maximum shift of the fit resultunder variation of gL min and the choice of ¯ B -pairs with¯ B ∈ { . , . , . , . , . , . , . , . , . }| N =2 and ¯ B ∈ { . , . , . , . , . , . }| N =4 , whilerequiring at least 15 degrees of freedom. Note thatthe result for β is compatible with the prediction fromperturbation theory, β = 1 (cf. Eqs. (4) and (12)).The result for ν for N = 2 agrees well with a previouslattice determination [22], ν = 0 . ν and f (0) agree at the few-per-centlevel (cf. Sec. d). Fits with Λ IR ∝ /L are not possiblefor similarly small values of gL min . For N = 2 the firstacceptable ( p ≥ .
05) fit is possible only after discardingall data with gL <
32 and for N = 4, gL <
24. The r.h.s.axis in Fig. 4 shows how the the p -value varies with thecut in gL . Generally, larger p -values for Λ IR ∝ g at agiven value of gL indicate that this ansatz provides abetter description of the data in terms of a χ -analysis.Inserting the the fit parameters in Tab. III into (12) and taking the limit x → ∞ we obtain predictionsfor the infinite-volume critical mass. For instance, for ag = 0 . am c ) = − . N = 2and ( am c ) = − . N = 4.We also address the question of the IR regulator withinthe framework of Bayesian inference with uniform priors α ∈ [ − . , . f (0) ∈ [0 , f (cid:48) (0) ∈ [ − , β ∈ [0 , ν ∈ [0 , N = 4,two values of α are used: α , ∈ [ − . , . IR ∝ g and Λ IR ∝ L ) werecalculated across a range of gL min cuts and pairings of ¯ B values. In Fig. 4 both the p -value, and the Bayes Factorof the central fit are shown across the range of gL min val-ues. In this plot, the graph is broken down into regionsaccording to the Jeffreys’ scale [27]. The Bayes Factor K is E E , where E and E are the marginal probabil-ities for model 1 (Λ IR ∝ g ) and model 2 (Λ IR ∝ /L )respectively. If log ( K ) is greater than 1 there is strongevidence for model 1 over model 2, and if it is greaterthan 2 it is decisive. The reverse is true for negative val-ues of log ( K ) in support of model 2. As the cut on gL min is reduced (more data is used) the evidence forΛ IR ∝ g increases, with there being decisive evidence un-der the Jeffreys’ scale for all gL min cuts for N = 2 andfor gL min ≤ . N = 4. The same pattern isseen for all ¯ B values.One can also obtain parameter estimates via. the pos-terior probability distribution, which we find to be in ex-cellent agreement with the results for the fit parametersfrom the χ analysis.In conclusion, Bayesian inference prefers the IR-finiteansatz over the IR-divergent one; complementary andconsistent with this, from χ fits we find the IR-finiteFSS ansatz (Λ IR ∝ g ) able to describe more degrees offreedom (i.e. larger range in gL ) with acceptable p -value. g. Conclusions and outlook: We present the firstnonperturbative study of the critical properties of athree-dimensional super-renormalisable scalar QFT with ϕ interaction and fields in the adjoint of SU ( N ) with N = 2 ,
4. When studied in lattice perturbation theorythe theory exhibits a logarithmic IR divergence for thecritical mass at 2-loop. The absence of this divergence inour numerical results from lattice simulations provides
N gL min gL max α i ν β f (0) f (cid:48) (0) p χ /N dof N dof χ fits to finite-size-scaling data. The first error is statistical and the second systematic as described inthe text. .
000 0 .
005 0 .
010 0 .
015 0 .
020 0 . / ( gL ) /ν − . − . − . − . − . a m ( g , L ) / g ag = ag = ag = ag = ag = L = ∞ FIG. 3. Central fit N = 2, ¯ B = 0 . , .
53. Dashed linescorrespond to the 2-loop prediction for the effective mass,solid lines to the fit result including error band. Value of ag increasing from bottom to top. At each coupling top set ofpoints corresponds to ¯ B = 0 .
52, bottom set to ¯ B = 0 . strong evidence for the IR-finiteness of the full theory.This constitutes one of the central results of this study.Further results are the nonperturbative determination ofthe critical mass. For the range of couplings consideredhere the critical mass agrees with 2-loop perturbationtheory at and below the percent level when employingthe dimensionful coupling constant g as IR regulator,confirming the expectation of [1, 2]. Our result for thecritical exponent is close to the leading-order effectivetheory prediction, where the effective fields correspondto the magnetisation of the full theory.Three-dimensional super-renormalisable QFT consist-ing of Yang-Mills theory coupled to adjoint scalar and/orfermionic matter are candidate theories for describing thephysics of the early Universe by means of holographic du-ality. Our determination of the critical point constitutesthe starting point towards the study of cosmology froma three-dimensional QFT. In view of the holographic du-ality cosmic evolution corresponds to inverse RG flowwhere the initial singularity is mapped to the IR be-haviour of the QFT. The absence of an IR singularityon the QFT side may thus be seen as the holographicresolution of the initial singularity in the bulk. Acknowledgements:
The authors would like towarmly thank Pavlos Vranas for his valuable supportduring the early stages of this project. We would like tothank Masanori Hanada for collaboration at early stages
FIG. 4. Top: N = 2, ¯ B = 0 .
52, ¯ B = 0 .
53 data, Bottom: N = 4, ¯ B = 0 .
42, ¯ B = 0 .
43 data. The p -value of the fit ofequation (12) with Λ IR ∝ g and Λ IR ∝ L (right y -axis) isshown by the orange squares and green triangles respectively.The black circles represent the log of the Bayes Factor, K = E E , where E and E are the marginal probabilities for fitswith (Λ IR ∝ g ) and (Λ IR ∝ /L ) respectively. [1] R. Jackiw and S. Templeton, How SuperrenormalizableInteractions Cure their Infrared Divergences, Phys. Rev. D23 , 2291 (1981).[2] T. Appelquist and R. D. Pisarski, High-TemperatureYang-Mills Theories and Three-Dimensional QuantumChromodynamics, Phys. Rev.
D23 , 2305 (1981).[3] P. McFadden and K. Skenderis, Holography for Cosmol-ogy, Phys. Rev.
D81 , 021301 (2010), arXiv:0907.5542[hep-th].[4] J. K. L. Lee, L. Del Debbio, A. J¨uttner, A. Portelli, andK. Skenderis, Towards a holographic description of cos-mology: Renormalisation of the energy-momentum ten-sor of the dual QFT, in (2019) arXiv:1909.13867 [hep-lat].[5] F. Delfino, A. Pelissetto, and E. Vicari, Three-dimensional antiferromagnetic CP(N-1) models, Phys.Rev.
E91 , 052109 (2015), arXiv:1502.07599 [cond-mat.stat-mech].[6] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi,and E. Vicari, Critical exponents and equation ofstate of the three-dimensional Heisenberg universalityclass, Phys. Rev. B , 144520 (2002), arXiv:cond-mat/0110336.[7] A. Pelissetto and E. Vicari, Critical mass renormalizationin renormalized φ theories in two and three dimensions,Phys. Lett. B751 , 532 (2015), arXiv:1508.00989 [hep-th].[8] A. Pelissetto and E. Vicari, Three-dimensional ferromag-netic CP(N-1) models, Phys. Rev. E , 022122 (2019),arXiv:1905.03307 [cond-mat.stat-mech].[9] M. L¨uscher and P. Weisz, Coordinate space methodsfor the evaluation of Feynman diagrams in lattice fieldtheories, Nucl. Phys.
B445 , 429 (1995), arXiv:hep-lat/9502017 [hep-lat].[10] P. Lepage, VEGAS, https://github.com/gplepage/vegas.[11] J. Zinn-Justin,
Quantum Field Theory and Critical Phe-nomena; 4th ed. , Internat. Ser. Mono. Phys. (ClarendonPress, Oxford, 2002).[12] P. Boyle, A. Yamaguchi, G. Cossu, and A. Portelli,Grid: A next generation data parallel C++ QCD library,(2015), arXiv:1512.03487 [hep-lat].[13] P. A. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli,Grid: A next generation data parallel C++ QCD library,
Proceedings, 33rd International Symposium on LatticeField Theory (Lattice 2015): Kobe, Japan, July 14-18,2015 , PoS
LATTICE2015 , 023 (2016).[14] S. Duane, A. D. Kennedy, B. J. Pendleton, andD. Roweth, Hybrid Monte Carlo, Phys. Lett.
B195 , 216(1987). [15] F. R. Brown and T. J. Woch, Overrelaxed heat-bathand Metropolis algorithms for accelerating pure gaugeMonte Carlo calculations, Physical Review Letters ,2394 (1987).[16] S. L. Adler, Overrelaxation algorithms for lattice fieldtheories, Physical Review D , 458 (1988).[17] Z. Fodor and K. Jansen, Overrelaxation algorithm forcoupled gauge-Higgs systems, Physics Letters B , 119(1994).[18] B. Bunk, Monte-Carlo methods and results for theelectro-weak phase transition, Nuclear Physics B (Pro-ceedings Supplements) , 566 (1995).[19] A. M. Ferrenberg and R. H. Swendsen, New Monte CarloTechnique for Studying Phase Transitions, Phys. Rev.Lett. , 2635 (1988).[20] B. Efron, Bootstrap Methods: Another Look at the Jack-knife, The Annals of Statistics , 1 (1979).[21] U. Wolff (ALPHA), Monte Carlo errors with lesserrors, Comput. Phys. Commun. , 143 (2004),[Erratum: Comput. Phys. Commun.176,383(2007)],arXiv:hep-lat/0306017 [hep-lat].[22] M. Hasenbusch, Eliminating leading corrections to scal-ing in the three-dimensional O ( N ) symmetric φ model: N = 3 and N = 4, J. Phys. A , 8221 (2001),arXiv:cond-mat/0010463.[23] J. Buchner, A. Georgakakis, K. Nandra, L. Hsu,C. Rangel, M. Brightman, A. Merloni, M. Salvato,J. Donley, and D. Kocevski, X-ray spectral modelling ofthe AGN obscuring region in the CDFS: Bayesian modelselection and catalogue, Astron. Astrophys. , A125(2014), arXiv:1402.0004 [astro-ph.HE].[24] F. Feroz and M. P. Hobson, Multimodal nested sam-pling: an efficient and robust alternative to markov chainmonte carlo methods for astronomical data analyses,Monthly Notices of the Royal Astronomical Society ,449 (2008).[25] F. Feroz, M. Hobson, and M. Bridges, MultiNest: anefficient and robust Bayesian inference tool for cosmologyand particle physics, Mon. Not. Roy. Astron. Soc. ,1601 (2009), arXiv:0809.3437 [astro-ph].[26] F. Feroz, M. P. Hobson, E. Cameron, and A. N. Pettitt,Importance nested sampling and the multinest algorithm,arXiv preprint arXiv:1306.2144 (2013).[27] H. Jeffreys, The theory of probability (OUP Oxford,1998).[28] C. Vohwinkel, Unplublished.[29] A. Lewis, GetDist: a Python package for analysing MonteCarlo samples, (2019), arXiv:1910.13970 [astro-ph.IM].
Supplemental Materials: Nonperturbative IR finiteness in super-renormalisable scalarQFT
I. IR BEHAVIOUR OF THE TWO-LOOP CORRECTION
The finite-size-scaling (FSS) analysis in this paper centers around the fit ansatz in Eq. (8), which uses the analyticalexpression for the IR-behaviour of the theory in Eq. (5). We derive it by studying the IR behaviour of the 3 d lattice-regularised 2-loop integral, D ( p ) = π/a (cid:90) − π/a d k (2 π ) d q (2 π ) k ˆ q ˆ r , (S1)where ˆ k = a sin( ka/ r = − k − q − p . The derivation is done conveniently in coordinate space, where D ( p ) = (cid:88) x ∈ Λ e − ipx G ( x ) , (S2)with the coordinate-space scalar propagator G ( x ). Below we retrace the steps in d = 3 taken by L¨uscher and Weisz [S9]for d = 4, to derive the large- x expansion of the free scalar lattice propagator.The long-distance behaviour of the propagator should be independent of the discretisation. Following [S9], wetherefore rewrite G ( x ) in terms of the continuum scalar propagator and a smooth momentum cutoff, G ( x ) = π/a (cid:90) − π/a d p (2 π ) e ipx p p → ∼ ∞ (cid:90) −∞ d p (2 π ) e ipx e − ( ap ) p , = a (cid:90) ∞ dt ∞ (cid:90) −∞ d p (2 π ) e ipx e − t ( ap ) = 14 π √ x Erf (cid:34) √ x a (cid:35) . (S3)For large x we therefore expect G ( x ) x →∞ ∼ π √ x . (S4)We now introduce the auxiliary function, H ( x ) = π/a (cid:90) − π/a d p (2 π ) e ipx ln (cid:0) ( a ˆ p ) (cid:1) , (S5)and the Vohwinkel relation [S28], (cid:0) δ ∗ µ + δ µ (cid:1) G ( x ) = x µ H ( x ) , (S6)with the lattice derivatives δ ∗ f ( x ) = a ( f ( x ) − f ( x − ˆ µa )) and δf ( x ) = a ( f ( x + ˆ µa ) − f ( x )).Eq. (S6) can be shown as follows: Consider the symmetrised lattice derivative of the coordinate space propagator, (cid:0) δ ∗ µ + δ µ (cid:1) G ( x ) = 1 a π/a (cid:90) − π/a d p (2 π ) i sin( ap µ ) e ipx ˆ p . (S7)Observing that 1 a ap µ (ˆ p ) = ∂∂p µ ln (cid:32)(cid:88) ν ( a ˆ p ν ) (cid:33) , (S8)and using integration by parts we find (cid:0) δ ∗ µ + δ µ (cid:1) G ( x ) = 1 a π/a (cid:90) − π/a d p (2 π ) i sin( ap µ ) e ipx ˆ p = i π/a (cid:90) − π/a d p (2 π ) (cid:16) ∂ p µ ln (( a ˆ p )) (cid:17) e ipx = x µ π/a (cid:90) − π/a d p (2 π ) ln (( a ˆ p )) e ipx = x µ H ( x ) . (S9)Using (S6) for large x we obtain H ( x ) x →∞ = 1 x µ ∂ µ G ( x ) = − π ( x ) / . (S10)Comparing with Eq. (S4) we identify G ( x ) x →∞ = − π ) H ( x ) , (S11)which, by inverse Fourier transformation allows us to conclude that D ( p ) p → = − ln(( ap ) )(4 π ) . (S12) II. FINITE-SIZE SCALING EFFECTIVE FIELD THEORY
In this section we consider the continuum action (1) S [ φ ] = Ng (cid:90) d x tr (cid:40)(cid:88) µ [ ∂ µ φ ( x )] + ( m + m c ) φ ( x ) + φ ( x ) (cid:41) , (S13)expressed as a function of the renormalised parameters m and g . A. Effective theory
The theory is expected to undergo a phase transition when the renormalised mass becomes close to 0 ( i.e. thecorrelation length diverges). In a finite cubic volume, no transition can occur because no length in the system canexceed the spatial extent, L . However, at the massless point, various statistical quantities will scale non-trivially with L according to the critical exponents of the theory. Moreover, to analyse close-to-critical lattice simulation results, itis important to understand the behaviour of finite-size effects.In a periodic and cubic volume T , a momentum vector k is quantised as πL n , where n is a vector with integercomponents. In massless perturbation theory, loop integrals become sums, such as I = 1 L (cid:88) k k , (S14)for the tadpole diagram. Even with a UV regulator, such a sum is undefined because of the explicit term it contains.This problem arises from a sickness of the finite-volume free theory which is defined by a Gaussian integral with anon-invertible covariance matrix. More explicitly, this matrix is given by the Laplacian operator, which has an isolatedzero-mode in the finite-volume massless theory. However, in the full theory, the exponential in the path integral issystematically damped by the quartic term tr[ φ ( x ) ] in the action. This indicates that in the massless theory, thecontribution from the field zero-mode has to be treated non-perturbatively.The magnetisation M defined in (10) is the zero-momentum component of the field φ mentioned above, and wedefine the decomposition φ = χ + M , (S15)where χ has a vanishing zero-mode. Close to the critical regime, the theory will be dominated by the long-distancecontributions from M . Therefore one can try to investigate finite-volume effects by using an effective theory wherethe higher-frequency modes χ are integrated out. We build this effective theory followiing the procedure describedin [S11, Sec. 37.3]. The effective action S eff [ M ] is defined byexp( − S eff [ M ]) = 1 C (cid:90) D χ exp( − S [ φ ]) , (S16)where C is a normalisation factor defined by S eff [0] = 0. The effective action has to be invariant under the Z symmetry M (cid:55)→ − M and the gauge symmetry M (cid:55)→ Ω † M Ω for any Ω in SU( N ). This means that the only terms S eff [ M ] can contain have the form tr( M k ) l where kl is an even integer. B. Leading-order effective action
If one ignores entirely the corrections coming from the non-zero frequencies χ , then it is clear from the originalaction (S13) that the effective action is given by S eff [ M ] = L Ng [ m tr( M ) + tr( M )] , (S17)where the mass counter-term is absent because no dynamics from χ is included. For an observable O [ M ], the tree-levelexpression is given by (cid:104) O [ M ] (cid:105) = 1 Z eff (cid:90) su ( N ) d M O [ M ] exp( − S eff [ M ]) , (S18)where Z eff is defined by (cid:104) (cid:105) = 1. In other words, the effective theory is a random matrix theory on the space oftraceless hermitian matrices. For an SU( N )-invariant function f on su ( N ), the Weyl integration formula reducesintegrating f ( M ) over su ( N ) to the integral over its N − (cid:90) su ( N ) d M f ( M ) = π N ( N − (cid:81) Nj =1 Γ( j ) (cid:90) d N − λ V (¯ λ ) f [diag(¯ λ )] (S19)Here diag( ξ ) is the diagonal matrix where the diagonal elements are the components of the vector ξ , “barred” vectors¯ λ are defined by ¯ λ = ( λ , . . . , λ N − , − (cid:80) N − j =1 λ j ) , (S20)and V ( ξ ) is the Vandermonde determinant V ( ξ ) = (cid:89) j At small z , the function Ψ kl ( z ) is a linear expansion of the formΨ kl ( z ) = Ψ kl (0) − z Ψ (cid:48) kl (0) + O ( z ) , (S28)where Ψ (cid:48) kl (0) = − (cid:90) d N − µ V (¯ µ ) (¯ µ k ) l ¯ µ exp (cid:0) − ¯ µ (cid:1) . (S29)Using this expansion, one can expand the Binder cumulant function f ( z ) = f (0) + zf (cid:48) (0) + O ( z ) , (S30)where f (cid:48) (0) is a combination of Ψ kl (0) and Ψ (cid:48) kl (0) for ( k, l ) ∈ [(4 , , (2 , , (0 , m = m − m c , where the m c is defined in the infinite-volume theory. Considering anarbitrary number ¯ B close to f (0), we define the finite-volume critical mass m ( L ) to be the bare mass such that theBinder cumulant is equal to ¯ B f (cid:40) √ Ng (cid:2) m ( L ) − m c (cid:3) ( gL ) (cid:41) = ¯ B . (S31)Then we use the expansion (S30) to obtain m ( L ) = m c + g √ N ¯ B − f (0) f (cid:48) (0) 1( gL ) + O (cid:18) L (cid:19) . (S32)The EFT predictions for f (0) and f (cid:48) (0) can be evaluated numerically using standard integration methods applied tothe integrals (S25) and (S29). The values we found for N = 2 , , D. Exact formulas for N = 2 For N = 2, (S25) is a one-dimensional integral that can be computed explicitlyΨ kl ( z ) = − kl (cid:104) Γ (cid:0) kl +34 (cid:1) F (cid:16) kl +34 ; ; z (cid:17) − √ z Γ (cid:0) kl +54 (cid:1) F (cid:16) kl +54 ; ; z (cid:17)(cid:105) if z ≤ − kl +14 Γ (cid:0) kl +32 (cid:1) U (cid:16) kl +34 , , z (cid:17) if z > F and U are the hypergeometric functions F ( a ; b ; z ) = Γ( b )Γ( a )Γ( b − a ) (cid:90) d t t a − (1 − t ) b − a − e zt , (S34) U ( a, b, z ) = 1Γ( a ) (cid:90) + ∞ d t t a − (1 + t ) b − a − e − zt . (S35) III. BAYESIAN ANALYSIS The primary quantity used here to determine the favoured form of the infrared regulator, Λ IR , is the Bayes Factor, K . This quantity is equal to the ratio of the marginalised probabilities of one model over the other, given the data.We can derive its expression through application of Bayes Theorem. p ( M | data) = (cid:90) dα p ( M ( α ) | data) p ( α | M ) , (S36)= (cid:90) dα p (data | M ( α )) p ( M ( α )) p (data) p ( α | M ) , = p ( M ) p (data) (cid:90) dα L ( M ( α )) p ( α | M ) , where M is a model of the data, α is the parameters of that model. The first line of (S36) simply separates theprobability of a model given the data into the contributions from all possible parameter values, weighted by the priorof those parameter values, p ( α | M ). In the second line Bayes theorem is applied. In the final line the definition of thelikelihood function, L is used. Taking the ratio of (S36) between two competing models yields the Bayes Factor, K (data) = p ( M | data) p ( M | data) = (cid:82) dα L ( M ( α )) p ( α | M ) (cid:82) dα L ( M ( α )) p ( α | M ) p ( M ) p ( M ) . (S37)The fraction p ( M ) p ( M ) expresses any prior belief of the likelihood of one model over the other. In this Letter, as is oftenthe case, this has been set to 1. In this analysis, M and M are both of the form given in equation (12), where theydiffer only by the expression of Λ IR , with model 1 using Λ IR = π gN and model 2 using Λ IR = L . The models thereforeshare the same model parameters, and the same uniform priors. Under the statistical bootstrap [S20], the probabilitydensity of the data points is assumed to follow a multivariate Gaussian distribution, with covariance matrix, Cov,estimated from the bootstrap samples. Writing the data as a vector of inputs x , and output masses m , we have thefamiliar Gaussian distribution for L ( M ( α )): L ( M ( α )) = 1 (cid:112) (2 π ) k | Cov | exp (cid:18) − 12 ( m − M ( α ))( x )) T · Cov − · ( m − M ( α ))( x )) (cid:19) , (S38)where we have denoted the dimension of the data vector m by k .Since the covariance matrix is defined through the bootstrap independently of the parameters α , the pre-factor tothe exponential may be brought outside of the exponential, where it cancels between the numerator and denominatorof K . Taking a logarithm of K gives the log of the Bayes factor, which one can interpret using the Jeffreys’ scale [S27].The integrand of the Bayesian evidence integral, L ( M ( α )) p ( α | M ), is significant as it is the posterior probabilitydistribution of the model parameters given the data. As an example, the posterior distribution of the model parametersfor the central fits of N = 2 and N = 4 are shown in figures S1 and S2 respectively. FIG. S1. Posterior probability density obtained using [S23–S26] and plotted with [S29], for N = 2 data with ¯ B = 0 . 52 and¯ B = 0 . 53 and a gL min cut of 12 . 8. The red (‘x’) points and red-solid lines are the predictions of the EFT. The black (‘+’)points and the black-dashed lines show the parameters of the maximum likelihood estimate. The yellow dot-dashed line showsthe value of ν found in [S22].[S1] R. Jackiw and S. Templeton, How Superrenormalizable Interactions Cure their Infrared Divergences, Phys. Rev. D23 ,2291 (1981).[S2] T. Appelquist and R. D. Pisarski, High-Temperature Yang-Mills Theories and Three-Dimensional Quantum Chromody-namics, Phys. Rev. D23 , 2305 (1981).[S3] P. McFadden and K. Skenderis, Holography for Cosmology, Phys. Rev. D81 , 021301 (2010), arXiv:0907.5542 [hep-th].[S4] J. K. L. Lee, L. Del Debbio, A. J¨uttner, A. Portelli, and K. Skenderis, Towards a holographic description of cosmology:Renormalisation of the energy-momentum tensor of the dual QFT, in (2019) arXiv:1909.13867 [hep-lat].[S5] F. Delfino, A. Pelissetto, and E. Vicari, Three-dimensional antiferromagnetic CP(N-1) models, Phys. Rev. E91 , 052109(2015), arXiv:1502.07599 [cond-mat.stat-mech].[S6] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Critical exponents and equation of state of thethree-dimensional Heisenberg universality class, Phys. Rev. B , 144520 (2002), arXiv:cond-mat/0110336.[S7] A. Pelissetto and E. Vicari, Critical mass renormalization in renormalized φ theories in two and three dimensions, Phys.Lett. B751 , 532 (2015), arXiv:1508.00989 [hep-th].[S8] A. Pelissetto and E. Vicari, Three-dimensional ferromagnetic CP(N-1) models, Phys. Rev. E , 022122 (2019),arXiv:1905.03307 [cond-mat.stat-mech].[S9] M. L¨uscher and P. Weisz, Coordinate space methods for the evaluation of Feynman diagrams in lattice field theories,Nucl. Phys. B445 , 429 (1995), arXiv:hep-lat/9502017 [hep-lat].[S10] P. Lepage, VEGAS, https://github.com/gplepage/vegas.[S11] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena; 4th ed. , Internat. Ser. Mono. Phys. (Clarendon Press,Oxford, 2002). FIG. S2. Posterior probability density obtained using [S23–S26] and plotted with [S29], for N = 4 data with ¯ B = 0 . 42 and¯ B = 0 . 43 and a gL min cut of 12 . 8. The red (‘x’) points and red-solid lines are the predictions of the EFT. The black (‘+’)points and the black-dashed lines show the parameters of the maximum likelihood estimate.[S12] P. Boyle, A. Yamaguchi, G. Cossu, and A. Portelli, Grid: A next generation data parallel C++ QCD library, (2015),arXiv:1512.03487 [hep-lat].[S13] P. A. Boyle, G. Cossu, A. Yamaguchi, and A. Portelli, Grid: A next generation data parallel C++ QCD library, Pro-ceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015): Kobe, Japan, July 14-18, 2015 , PoS LATTICE2015 , 023 (2016).[S14] S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett.