Renormalization of the energy-momentum tensor in three-dimensional scalar SU(N) theories using the Wilson flow
Luigi Del Debbio, Elizabeth Dobson, Andreas Jüttner, Ben Kitching-Morley, Joseph K. L. Lee, Valentin Nourry, Antonin Portelli, Henrique Bergallo Rocha, Kostas Skenderis
RRenormalisation of the energy-momentum tensor inthree-dimensional scalar SU(N) theories using the Wilson flow
Luigi Del Debbio, Elizabeth Dobson,
1, 2
Andreas Jüttner,
3, 4
Ben Kitching-Morley,
3, 4, 5
Joseph K. L. Lee, ∗ Valentin Nourry,
1, 4, 6
Antonin Portelli, Henrique Bergallo Rocha, and Kostas Skenderis
4, 5 (LatCos Collaboration) Higgs Centre for Theoretical Physics, School of Physics and Astronomy,The University of Edinburgh, Edinburgh EH9 3FD, UK Institute of Physics, The University of Graz,Universitätsplatz 5, A-8010 Graz, Austria 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 Université de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France
Abstract
A non-perturbative determination of the energy-momentum tensor is essential for understanding thephysics of strongly coupled systems. The ability of the Wilson Flow to eliminate divergent contact termsmakes it a practical method for renormalising the energy-momentum tensor on the lattice. In this paper,we utilise the Wilson Flow to define a procedure to renormalise the energy-momentum tensor for a three-dimensional massless scalar field in the adjoint of SU ( N ) with a ϕ interaction on the lattice. In this theorythe energy momentum tensor can mix with ϕ and we present numerical results for the mixing coefficientfor the N = 2 theory. ∗ [email protected] a r X i v : . [ h e p - l a t ] S e p ONTENTS
I. Introduction 2II. Generalities/Definitions 4A. Continuum and lattice SU ( N ) scalar action 4B. Energy-momentum tensor 5C. Wilson flow 7III. Lattice simulations 9A. Simulation setup 9B. Critical mass determination 10IV. Renormalisation of the EMT 11A. Numerical results 12V. Conclusion and outlook 15Acknowledgments 17A. Lattice perturbation theory calculations 171. Massless lattice integrals: V ( q ) and I µν ( q ) c C ( q ) and C µν ( q ) C ( t, q ) , C µν ( t, q ) I. INTRODUCTION
The energy-momentum tensor (EMT) plays a fundamental role in quantum field theories, byvirtue of being the collection of Noether currents related to space-time symmetries. It acts as thesource for space-time curvature in Einstein field equations, and its expectation value encodes theenergy and momentum carried by quantum excitations. One of the motivations for this study comesfrom the application of holography to cosmology [1]. In this holographic approach, cosmologicalobservables, such as the Cosmic Microwave Background (CMB) power spectra, can be described interms of correlators of the EMT of a dual three-dimensional quantum field theory (QFT) with no2ravity. The dual theories introduced in [1] comprise three-dimensional Yang-Mills theory, coupledto massless scalars ϕ in the adjoint of SU ( N ) with a ϕ interaction. Perturbative calculations ofthe correlators have been performed [2–5] and the prediction of holographic cosmology were testedfavourably against Planck data in [6]. The results in [6] however also implied that a non-perturbativeevaluation of the EMT is required in order to fully exploit the duality.Here we initiate the computation of non-perturbative effects by means of lattice QFT. A funda-mental limitation of the lattice framework is the fact that space-time symmetries, such as Poincaréinvariance, are explicitly broken at finite lattice spacing; these symmetries are restored only in thecontinuum limit. Consequently, the Ward identities associated to translations are violated, andthe EMT, which generates such transformations, has to be defined with care. On the lattice, theEMT has to be renormalized by tuning the coefficients of a linear combination of all operatorswith dimension not greater than the space-time dimension d , which are compatible with latticesymmetries. This ensures that the Ward identities are recovered in the continuum limit, up tocutoff effects. Perturbative analytic calculations using this method have been discussed extensivelyin [7, 8].Various strategies have been proposed to non-perturbatively renormalize the EMT on the lattice( cf. [9], and references therein) such as the shifted boundary condition [10–13], and the GradientFlow for probe operators [14–16], which is the strategy considered in this paper. The Wilson Floworiginated from [17], and the idea here is to construct probes from fields at some positive flow time,which are non-local in the elementary fields, that can eliminate the divergent contact terms presentin the correlators. The divergence properties and regularization of Ward identities of flowed gaugefields are discussed extensively in [18].In this paper we are particularly interested in renormalizing the EMT of a simpler version ofthe holographic dual theory, which is the class of d massless scalar QFTs with ϕ in the adjointof SU ( N ) and a ϕ interaction, regularized on a Euclidean space-time lattice [19]. This classof massless, super-renormalisable QFT, with the coupling g of mass dimension one, suffers fromsevere infrared (IR) divergences in perturbation theory. Perturbative calculations of correlationfunctions and renormalisation parameters, such as the critical mass or the EMT renormalisationcoefficients, contain IR logarithms, which makes the results dependent on the IR regulator. Thenon-perturbative IR finiteness of super-renormalisable theories, where the dimensionful couplingconstant acts as the IR regulator, has been conjectured and discussed in [20, 21], and has been con-firmed non-perturbatively for the theory under consideration in [22]. This allows us to renormalisethe theory non-perturbatively without IR ambiguity. In this paper we focus on the N = 2 theory;3heories with N > and the large N limit will be discussed in a later publication.This paper is organised as follows. In Sec. II we first introduce the scalar SU ( N ) theory in thecontinuum and on the lattice, and we define the EMT operator and correlators. We also define theWilson Flow, as well as the relevant correlators at finite flowtime. In Sec. III we list the parametersof the simulated ensembles for this study, and summarise the results of the critical mass determinednon-perturbatively in [22]. In Sec. IV we discuss the procedure to renormalise the EMT using flowedcorrelators, and finally present the numerical results for the N = 2 theory. We have also includeda number of appendices. In appendix A 1 we summarise the method to evaluate massless latticescalar integrals in d . In appendices A 2, A 3 and A 4, we present the lattice perturbation theorycalculations for the EMT operator mixing, correlators at vanishing flowtime, and correlators atfinite flowtime respectively. II. GENERALITIES/DEFINITIONSA. Continuum and lattice SU ( N ) scalar action The theory under consideration here is a three-dimensional Euclidean scalar su ( N ) valued ϕ theory, S [ ϕ ] = (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 ) , which arenormalised so that Tr (cid:2) T a T b (cid:3) = δ ab . Here λ is the ϕ coupling constant with mass dimensionone (which does not renormalise), m is the bare mass. Since the mass of the theory renormalisesadditively, we include the mass counterterm, or critical mass m c ( g ) , i.e. the value of the bare masssuch that the renormalised theory is massless. To make the ’t Hooft scaling explicit, hereafter thefollowing rescaled version of the action will be used, S [ φ ] = Ng (cid:90) d x Tr (cid:104) ( ∂ µ φ ( x )) + ( m − m c ) φ ( x ) + φ ( x ) (cid:105) , (2)which can be obtained by identifying φ = (cid:112) g/N ϕ and λ = g/N from Eq. (1).The theory is discretised on a three-dimensional Euclidean lattice by replacing the action with S [ φ ] = a Ng (cid:88) x ∈ Λ Tr (cid:34)(cid:88) µ ( δ µ φ ( x )) + ( m − m c ) φ ( x ) + φ ( x ) (cid:35) . (3)Here δ µ is the forward finite difference operator defined by, δ µ φ ( x ) = a − [ φ ( x + a ˆ µ ) − φ ( x )] , where ˆ µ is the unit vector in direction µ , Λ is a lattice with cubic geometry containing N L points (withperiodic boundary conditions), and a the lattice spacing.4 . Energy-momentum tensor In the continuum theory, the energy-momentum tensor (EMT) T µν is defined as the conservedcurrent of space-time symmetries. For our scalar SU ( N ) theory, it is given by [23] T µν = Ng Tr (cid:40) ∂ µ φ )( ∂ ν φ ) − δ µν (cid:34)(cid:88) ρ ( ∂ ρ φ ) + ( m − m c ) φ + φ (cid:35) + ξ (cid:32) δ µν (cid:88) ρ ( ∂ ρ φ ) − ( ∂ µ φ )( ∂ ν φ ) (cid:33)(cid:41) . (4)Here the term multiplying ξ is the improvement term. As will be calculated in the Appendix,this improvement term only contributes a finite part to the EMT correlation function that we areinterested in. As only the divergent parts are studied to obtain the divergent operator mixing(explained below), ξ will be set to 0 for the remainder of the text. In the continuum theory, due totranslational invariance, the EMT satisfies Ward-Takahashi identities (WI) of the form (cid:104) ∂ µ T µν ( x ) P ( y ) (cid:105) = − (cid:28) δP ( y ) δφ ( x ) ∂ ν φ ( x ) (cid:29) (5)where P ( y ) is any composite operator inserted at point y . If P is such that the RHS of Eq. (5) isfinite for separated points x (cid:54) = y , the LHS correlation function, which contains the divergence ofthe EMT, is finite up to contact terms. For this theory, it can be shown that the insertion of T µν does not introduce new UV divergences (as discussed in more detail in appendix A 2).On the lattice, the continuous translational symmetry is broken into the discrete subgroup oflattice translations; because of this a naïve discretisation of the EMT on the lattice, T µν = Ng Tr (cid:40) δ µ φ )( δ ν φ ) − δ µν (cid:34)(cid:88) ρ ( δ ρ φ ) + ( m − m c ) φ + φ (cid:35)(cid:41) , (6)which is obtained by replacing the partial derivatives ∂ µ φ ( x ) with the central finite difference δ µ φ ( x ) = a [ φ ( x + a ˆ µ ) − φ ( x − a ˆ µ )] (this is chosen in order to obtain a Hermitian EMT), does notsatisfy the WI Eq. (5). Now, the WI on the lattice includes an additional term [7], (cid:104) δ µ T µν ( x ) P ( y ) (cid:105) = − (cid:28) δP ( y ) δφ ( x ) δ ν φ ( x ) (cid:29) + (cid:104) X ν ( x ) P ( y ) (cid:105) , (7)where X ν is an operator proportional to a , which classically vanishes in the continuum limit.However, radiative corrections cause the expectation value (cid:104) X ν ( x ) P ( y ) (cid:105) to produce a linearly a − divergent contribution to the WI. Therefore, the naïvely discretised EMT will not reproduce thecontinuum WI when the regulator is removed; T µν has to be renormalised by adjusting the coeffi-cients of a linear combination of lower-dimensional operators which satisfy the same symmetries.5n four dimensions, it has been shown in [7] that T µν potentially mixes with five lower-dimensionaloperators, which can generate such divergences. However, in three dimensions, dimensional countingindicates that divergent mixing can only occur with O = δ µν Ng Tr φ . The renormalised EMT onthe lattice can therefore be defined as an operator mixing, T Rµν = T µν − C δ µν Ng Tr φ . (8) C has to be tuned to satisfy the continuum WI up to discretisation effects when the regulator isremoved.At leading order (LO) O ( g ) ( i.e. one-loop) in lattice perturbation theory, C is shown to be C = ga c , (9)where c = (cid:18) − N (cid:19) (cid:18) W − (cid:19) , (10) Z = a (cid:90) πa − πa d k (2 π ) k = 0 . ..., (11)for lattice momentum ˆ k = a sin( ka/ , see appendix A 2. In the continuum limit, a → , the valueof C diverges. To account for this leading behaviour, we define C = ga c , (12)and by determining the value of c non-perturbatively, we are able to renormalise the EMT on thelattice. As mentioned in the introduction, the two-loop contribution diverges logarithmically withthe IR regulator.Before discussing the strategy to obtain the value of c non-perturbatively, we define an EMTcorrelator which will be useful in our analysis. Consider the momentum-space 2-point correlator, C µν ( q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T Rµν ( x ) Tr φ (0) (cid:105) . (13)Here q = πaN L n is the momentum where n is a vector with integer components. This particularcorrelator is chosen since Tr φ is the lowest dimension non-vanishing scalar operator in the theory.By inserting the definitions in Eqs. (8) and (12), we obtain C µν ( q ) = C µν ( q ) − ga c δ µν C ( q ) , (14)6here C µν ( q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T µν ( x ) Tr φ (0) (cid:105) , (15) C ( q ) = (cid:18) Ng (cid:19) a (cid:88) x ∈ Λ e − iq · x (cid:104) Tr φ ( x ) Tr φ (0) (cid:105) . (16)The superscript is used to distinguish the naïvely discretised EMT from the renormalised one.On the lattice, the correlator C µν ( q ) has a contact term which arises when the operators coincidein position space; in momentum space, this manifests as a constant (momentum-independent) con-tribution C µν (0) which needs to be subtracted before the proper continuum limit can be obtained, ˆ C µν ( q ) = C µν ( q ) − C µν (0) . (17)By dimensional counting, C µν (0) has a leading a − divergent contribution. We therefore define C µν (0) = κa δ µν . (18)Lattice perturbation theory at next-to-leading order (NLO) gives the following results for the variousexpressions from above (details can be found in appendix A 3): ˆ C µν ( q ) = N q (cid:18) − N (cid:19) π µν + O ( a ) , (19) ˆ C µν ( q ) = − N qg eff (cid:18) − N (cid:19) (cid:18) − N (cid:19) π µν + O ( a ) , (20) κ = − N (cid:18) − N (cid:19) (cid:18) W − (cid:19) , (21)where g eff = g | q | is the effective coupling , and π µν = δ µν − q µ q ν q the transverse projector . It canbe seen that ˆ C µν ( q ) has a leading N q behaviour; an overall q is expected from ˆ C µν ( q ) being adimension one correlator, where at LO ( i.e. one-loop) there is no coupling constant dependence,and at NLO ( i.e. two-loops) we encounter the first order expansion in the effective coupling g eff . Inboth terms, the planar diagram contribute to the leading N factor, whereas non-planar diagramscan be seen as N corrections to the leading planar diagram. The fact that the finite piece of ˆ C µν ( q ) is proportional to the transverse projector is a consequence of the WI. C. Wilson flow
From above, we see that the correlator C µν ( q ) contains divergent contributions in terms of ga c from the operator mixing , as well as κa due to the contact term. In order to non-perturbatively7enormalise the EMT operator, we need to isolate the contact term from the operator mixing, andwe will utilize the method of the Wilson Flow [17] to achieve this. For our scalar field φ ( x ) , definea flowed field ρ ( t, x ) governed by the flow equations, ∂ t ρ ( t, x ) = ∂ ρ ( t, x ) , ρ ( t, x ) | t =0 = φ ( x ) , (22)where t is the flowtime , a new parameter introduced into the theory. Solving by means of Fouriertransformation, one finds (cid:101) ρ ( t, k ) = e − k t (cid:101) φ ( k ) , (23)where (cid:101) ρ ( t, k ) is the Fourier transform of ρ ( t, x ) ; the flow effectively smears the field with radius √ t .The Wilson flow suppresses high-momentum modes exponentially, and thereby regulates thedivergent contact term present in the EMT correlator C µν ( q ) . We are therefore able to isolatethe divergent mixing c from the divergent contact term. There have been extensive discussionsof various implementations of the Wilson Flow for renormalising the EMT, which can be foundin [9, 12, 14–16, 18].In our case, we are interested in determining the flowed correlator C µν ( t, q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T Rµν ( x ) Tr ρ ( t, (cid:105) , (24)at finite flowtime. Here we replaced the operator Ng Tr φ ( x = 0) with the operator Ng Tr ρ ( t, x = 0) at finite flowtime t , and kept the renormalised EMT operator T Rµν ( x ) at flowtime t = 0 . Bydefinition, C µν (0 , q ) = C µν ( q ) . Since the operator mixing c is local to the EMT operator T µν ( x ) ,it is not affected by replacing the probe φ (0) with the one at finite flowtime ρ ( t, . On the otherhand, the divergent contact term C µν ( t, is suppressed. More explicitly we similarly define ˆ C µν ( t, q ) = C µν ( t, q ) − C µν ( t, , (25) C µν ( t,
0) = δ µν K ( t ) . (26)As recorded in Eqs. (18) and (21), at vanishing flowtime, K ( t = 0) = κa . However, as calculatedin Eq. (A33), at small finite flowtime, K ( t ) = ω √ t + O (cid:16) √ t (cid:17) , (27)where at LO in perturbation theory, ω = − N (cid:18) − N (cid:19) (cid:32) √ π / (cid:33) . (28)8e utilise this small t expansion to remove the contact term contribution in our correlationfunction in order to obtain the value of c . The strategy will be explained in further detail insection Sec. IV.In analogy to Eqs. (14)–(16) we have the relations C µν ( t, q ) = C µν ( t, q ) − ga c δ µν C ( t, q ) , (29)where C µν ( t, q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T µν ( x ) Tr ρ ( t, (cid:105) , (30) C ( t, q ) = (cid:18) Ng (cid:19) a (cid:88) x ∈ Λ e − iq · x (cid:104) Tr φ ( x ) Tr ρ ( t, (cid:105) . (31)Having defined the above correlation functions, we can now non-perturbatively renormalize theEMT on the lattice. The renormalisation scheme is defined by first imposing the Ward Identity q µ ˆ C µν ( t, q ) = 0 (32)on all lattice ensembles. Here q = a sin ( aq ) is the lattice momentum. This condition is imposed ona specific choice of momentum aq which is present in all ensembles, as constrained by the quantizedmomentum modes. This step gives a value of c for each choice of mass, volume and ’t Hooftcoupling. We then extrapolate the value c towards the massless and infinite volume limit to obtain c . This defines a massless renormalisation scheme, which is independent of the volume. We willalso investigate the dependence of c on the value of the ’t Hooft coupling. The implementation ofthe scheme and the numerical fits results will be explained in Sec. III. III. LATTICE SIMULATIONSA. Simulation setup
The theory is simulated using the Hybrid Monte Carlo algorithm [24], which was implimentedusing the GRID library [25, 26]. For this paper, we will focus on the N = 2 theory. The simulatedvolumes N L , ’t Hooft coupling in lattice unit ag (which is equivalent to the dimensionless lattice-spacing), and bare masses ( am ) are listed in Table I. For each of the three ’t Hooft couplings, twobare masses in the vicinity of the critical mass have been simulated (see Table II).The analysis is performed using bootstrap resampling [27], and only every 50th or 100th tra-jectory is sampled in order to reduce auto-correlation. The first 5000 trajectories are discarded9 g ( am ) N L trajectories sample frequency ag , two bare masses are simulated in three volumes to ensure the ensembles are thermalized. A representative example of the value of the observable M = Tr (cid:0) a (cid:80) x ∈ Λ φ ( x ) (cid:1) across one HMC simulation ( ag = 0 . , N L = 128 , ( am ) = − . ) isshown in Fig. 1. × × × × × × × M Trajectory ag = 0 . , N L = 128 , ( am ) = − . Figure 1. Example of observable M = Tr (cid:0) a (cid:80) x ∈ Λ φ ( x ) (cid:1) for ensemble with ag = 0 . , N L = 128 , ( am ) = − . B. Critical mass determination
To extrapolate to the massless point, the renormalised mass of the ensembles have to be de-termined, which requires the critical masses for each lattice spacing as input. These have beendetermined in [22] at two-loops in lattice perturbation theory, as well as non-perturbatively byanalysing the finite-size scaling of the Binder cumulant. The relevant masses are summarisedin Table II. 10 g ( am c ) in the infinite volume limit are calculated at NLO in lattice perturbationtheory, as well as non-perturbatively in [22], which are listed for each ’t Hooft coupling ag . These are usedin the later global fit to obtain c in the massless limit. IV. RENORMALISATION OF THE EMT
The renormalisation condition Eq. (32) implies that ˆ C µν ( t, q ) is purely transverse, i.e. , ˆ C µν ( t, q ) = F ( t, q ) π µν (33)where π µν = δ µν − q µ q ν q is the transverse projector with lattice momentum q . In other words, ˆ C µν vanishes in the direction with purely longitudinal momentum. For example, picking the momentumto be purely in the direction q l = ( q , q , q ) = (0 , , q ) , ˆ C ( t, q l ) = 0 . (34)Substituting the definition of ˆ C µν ( t, q l ) from Eqs. (25) and (29), we obtain ˆ C ( t, q l ) = C ( t, q l ) − C ( t,
0) = C ( t, q l ) − ga c C ( t, q l ) − K ( t ) = 0 (35) → c = ag C ( t, q l ) C ( t, q l ) − f g ( g √ t, q l ) , (36)where f g ( g √ t, q l ) = ag K ( t ) C ( t, q l ) . (37)Using the one-loop perturbative expressions for K ( t ) and C ( t, q ) from Eqs. (A27) and (A33), andexpanding in powers of σ = (cid:112) tq / , this gives f g ( g √ t, q l ) = ω (cid:48) ( q l ) g √ t + O (cid:16) q l √ t (cid:17) , (38)where ω (cid:48) ( q ) = √ aq )3 π / . (Details can be found in appendix A 4. The strategy to obtain the value of c is to first flow the correlators to a range of small finite flowtimes, picking a fixed momentum q l .Then, utilising Eq. (36), we fit the ratio on the left hand side of ag C ( t, q l ) C ( t, q l ) = c + f g ( g √ t, q l ) , (39)11s a function of the physical flowtime g √ t . We have tested a range of fit functions for f g , and havefound that the fit ansatz ag C ( t, q l ) C ( t, q l ) = c + Ω g √ t (40)provides a very good fit to the data. Here we keep the first term linear in the inverse physicalflowtime g √ t from Eq. (38), and leave Ω and c as fit parameters. From the fit we can extrapolate c from the y -intercept. A. Numerical results
The fit ranges for the physical flowtime g √ t required special attention. They must first besufficiently small to justify the small flowtime expansion of Eq. (38). This also ensures the smearingradius is sufficiently smaller than the length of the lattice ( gL = gaN L ) such that there will belittle finite volume contributions from the boundaries. The physical flowtime must also be largerthan the lattice spacing ( ag ) such that actual smearing occurs across lattice points. We thereforeimpose the range to be between ag < g √ t < . The momentum q l is chosen by the lowest discretemomentum from the smallest volume for each lattice spacing.The fits are shown in Fig. 2. The fit results including the choice of momentum and the valuesof c for each ensemble is summarised in Table III.In order to assess the mass, volume and lattice-spacing dependence of the value of c , we performa global fit c ( m R , gL, ag ) = c + p m R + p gL + p ( ag ) , (41)where m R = ( m − m c ) /g is the dimensionless renormalised mass (The values of m c has beensummarised in Table II), gL is the dimensionless length of the lattice, and ag the dimensionlesslattice spacing.From the fit data, it was observed that when two or more parameters are included in the globalfits, the fits are not improved since the parameters are already compatible with 0; we therefore onlyretain the models with no extra parameter dependence (i.e. a constant fit), and models with oneparameter. The fit values for c using different models, along with their corresponding definitions,are summarised in Table IV. The constant fit for model 1 is shown in Fig. 3, and the fits in thecorresponding free parameter for models 2 - 4, i.e. m R , gL , and ag , are shown in Fig. 4.In Fig. 5, the fit values of c for the different models from Table IV, along with the one-loop value c from Eq. (10) are plotted. The p -values for each of the fit models are acceptable ( p > . ),12 g N L ( am ) a | q l | dof χ /dof p -value c c using the form Eq. (40). The momentumchosen is the first discrete momentum from the smallest volume in each lattice spacing, and the flowtime fitrange is bounded by ag < g √ t < .model fit function c p p p dof χ / dof p -value1 c = c c = c + p m R c = c + p gL c = c + p ( ag ) c . For reference, the 1-loop perturbative value gives c ≈ . from Eq. (10). but the extra parameters are all compatible with zero; there is only very weak dependence on themass, volume and lattice spacing for our simulated ensembles. Again, it is worth pointing out thatthe finiteness of this value in the infinite volume limit is a non-peturbative feature of the theory.In perturbation theory, all terms of O (cid:0) g (cid:1) are IR divergent and depend on the IR regulator; but13 . − . − . . . .
06 0 2 4 6 8 10 c + f g ( g √ t ) g √ tag = 0 . ,N L = 64 , ( am ) = − . ag = 0 . ,N L = 64 , ( am ) = − . (a) ag = 0 . , N L = 64 − . − . − . − . − . . . .
06 0 2 4 6 8 10 c + f g ( g √ t ) g √ tag = 0 . ,N L = 128 , ( am ) = − . ag = 0 . ,N L = 128 , ( am ) = − . (b) ag = 0 . , N L = 128 − . − . − . − . − . − . − . − . . . .
06 0 2 4 6 8 10 c + f g ( g √ t ) g √ tag = 0 . ,N L = 256 , ( am ) = − . ag = 0 . ,N L = 256 , ( am ) = − . (c) ag = 0 . , N L = 256 − . − . . . . . . .
06 0 1 2 3 4 5 c + f g ( g √ t ) g √ tag = 0 . ,N L = 64 , ( am ) = − . ag = 0 . ,N L = 64 , ( am ) = − . (d) ag = 0 . , N L = 64 − . − . − . − . . . . . . .
06 0 1 2 3 4 5 c + f g ( g √ t ) g √ tag = 0 . ,N L = 128 , ( am ) = − . ag = 0 . ,N L = 128 , ( am ) = − . (e) ag = 0 . , N L = 128 − . − . − . . . . .
08 0 1 2 3 4 5 c + f g ( g √ t ) g √ tag = 0 . ,N L = 256 , ( am ) = − . ag = 0 . ,N L = 256 , ( am ) = − . (f) ag = 0 . , N L = 256 − . − . . . . . . .
06 0 0 . . . . c + f g ( g √ t ) g √ tag = 0 . ,N L = 64 , ( am ) = − . ag = 0 . ,N L = 64 , ( am ) = − . (g) ag = 0 . , N L = 64 − . − . − . . . . . . .
06 0 0 . . . . c + f g ( g √ t ) g √ tag = 0 . ,N L = 128 , ( am ) = − . ag = 0 . ,N L = 128 , ( am ) = − . (h) ag = 0 . , N L = 128 − . − . . . . . . . . .
08 0 0 . . . . c + f g ( g √ t ) g √ tag = 0 . ,N L = 256 , ( am ) = − . ag = 0 . ,N L = 256 , ( am ) = − . (i) ag = 0 . , N L = 256 Figure 2. Plots showing c against the inverse physical flowtime g √ t using Eq. (40) for three ’t Hooftcouplings and 3 volumes . The red and blue data points and fits are for the lighter and heavier masssimulations respectively. The value of c is the y -intercept on the fit. . . . . . . . . .
08 0 0 .
05 0 . .
15 0 . c /gL Model 1 constant fit versus 1 /gL
Figure 3. c global fit using Model 1 (constant fit). The value of c is plotted against gL − . . . . . . . . .
08 0 0 .
02 0 .
04 0 .
06 0 .
08 0 . c m R Model 2 fit versus m R (a) Model 2 (against m R ) − . . . . . . . . .
08 0 0 .
05 0 . .
15 0 . c /gL Model 3 fit versus 1 /gL (b) Model 3 (against gL ) − . . . . . . . . .
08 0 0 .
05 0 . .
15 0 . .
25 0 . .
35 0 . c ag Model 4 fit versus ag (c) Model 4 (against ag ) Figure 4. c global fits model 2, 3, 4, as defined in Table IV. Each plot is plotted against the respective freefitting parameter for each model, i.e. m R , gL , and ag ; the value for c is the y -intercept of the fit line. as shown in [22] the theory is in fact non-perturbatively IR finite, where the dimensionful couplingeffectively acts as the IR regulator in the infinite volume limit. Comparing the non-perturbativeresult for c with the one-loop perturbative value, the non-perturbative value is approximately smaller than the one-loop result. This is qualitatively expected, as the higher order terms inperturbation theory (with the IR regulator replaced by the coupling) changes sign at every order,and so the two-loop result is a correction of the opposite sign to the one-loop value. V. CONCLUSION AND OUTLOOK
We have presented a procedure to non-perturbatively renormalise the EMT on the lattice for athree-dimensional scalar QFT with ϕ interaction and fields in the adjoint of SU ( N ) . We have alsopresented numerical results of the EMT operator mixing for the theory with N = 2 . The methodutilises the Wilson Flow to define a probe at positive flowtime, which can eliminate the divergentcontact term present in EMT correlator. This allows us to determine the mixing coefficient withthe lower-dimensional operator δ µν Ng Tr φ . This ensures that the Ward Identity can be restored in15 .
03 0 .
035 0 .
04 0 .
045 0 .
05 0 .
055 0 .
06 0 .
065 0 .
07 0 . c ) p -value: 0.12Model 2: ( c , p ) p -value: 0.09Model 3: ( c , p ) p -value: 0.12Model 4: ( c , p ) p -value: 0.09c3global fit summary Figure 5. The value of c from each of the model in Table IV is represented with a blue point, and the redline shows the 1-loop perturbation theory value from Eq. (10). the continuum limit, up to cut-off effects.The context of our investigation is to predict the CMB power spectrum for holographic cosmo-logical models, and to test them against observational data. The next step of the investigation isto determine the renormalised EMT two-point function, C µνρσ ( q ) = (cid:104) T µν ( q ) T ρσ ( − q ) (cid:105) , for this classof scalar theories. This two-point function can be used to compute the primordial CMB powerspectra in the holographic cosmology framework. On the lattice, this correlator contains a largecontact term of order O ( a ) . This large contact term presents significant statistical noise to thesignal of the renormalised two-point function. We are currently exploring using the Wilson Flow toeliminate the presence of such contact term, which will allow us to make a fully non-perturbativeprediction for the CMB power spectra with the SU ( N ) scalar theory as the dual theory.We are also working towards simulating and performing the renormalisation of the EMT forthree-dimensional QFTs with adjoint SU ( N ) scalars coupled to gauge fields. This is the class oftheories preferred by the fit of the perturbative predictions to Planck data [6]. Much work havebeen performed to study the EMT on the lattice for gauge theories [28] and gauge theories withfermions [8]. The implementation of the Wilson Flow for renormalising the EMT has also beenstudied for gauge theories [15, 18]. We are exploring related methods to perform renormalisationof the EMT for theories with scalar fields coupled to gauge fields. This will take us closer to fullytesting the viability of holographic cosmological models as a description of the very early Universe.16 CKNOWLEDGMENTS
Appendix A: Lattice perturbation theory calculations
In this appendix we present the details of the lattice perturbation theory (LPT) calculationsin Sec. II. We will first evaluate two lattice scalar integrals in appendix A 1, which are necessary tocalculate the EMT c coefficient mixing in appendix A 2, the correlators C ( q ) , C µν ( q ) at vanishingflowtime in appendix A 3, and the correlators C ( t, q ) , C µν ( t, q ) at finite flowtime in appendix A 4.
1. Massless lattice integrals: V ( q ) and I µν ( q ) To evaluate the relavent massless lattice integrals, we generalise the method used in [29] to threedimensions. Using a set of recursion relations, any massless, one-loop lattice scalar integrals inthree dimensions of the form B ξ ( p ; n ) = lim δ → (cid:90) π/a − π/a d k (2 π ) ˆ k n ˆ k n ˆ k n (ˆ k + ξ ) p + δ with ξ (cid:28) , p ∈ Z , n ∈ Z , (A1)17an be reduced to a linear combination of two constants, Z = B (1; { , , } ) ≈ . and Z = B (1; { , , } )3 ≈ . . (A2)Here, ˆ k = a sin( ka/ is the lattice momentum. These two constants have been determined to highprecision using the Lüscher–Weisz coordinate-space method [30].The two momentum-dependent scalar lattice integrals required for the following LPT calculationsare V ( q ) = (cid:90) π/a − π/a d k (2 π ) k + m )( (cid:91) q − k + m ) , (A3) I µν ( q ) = (cid:90) π/a − π/a d k (2 π ) k µ ( q − k ) ν (ˆ k + m )( (cid:91) q − k + m ) , (A4)where k = a sin( ka ) . By expanding the expressions in powers of the external momenta [31, 32] andusing the recursion relations from before, in the massless limit, these evaluate to lim m → V ( q ) = 18 q + a (cid:18) Z + 9 Z − (cid:19) + a q (cid:18) Z − Z + 427648 (cid:19) + O (cid:0) a q (cid:1) (A5) lim m → I µν ( q ) = δ µν a (cid:18) − Z (cid:19) + q
64 (2 δ µν − π µν )+ a (cid:20) Z δ µν q µ + q (2 π µν − δ µν ) (cid:18) Z − Z + 41152 (cid:19)(cid:21) + a (cid:104) (cid:0) δ µν ( q + q + q ) + 6 q q µ δ µν + 4 q µ q ν ( q µ + q ν ) (cid:1) (52 − Z − Z )+ 360 δ µν q µ (6 Z + 9 Z −
4) + q (4 π µν − δ µν )(236 − Z − Z ) (cid:105) + O (cid:0) a q (cid:1) . (A6)
2. EMT operator mixing: c Here we calculate the perturbative renormalisation of T µν on the lattice. The naïve discretisationof the EMT is T µν = Ng Tr (cid:40) δ µ φ )( δ ν φ ) − δ µν (cid:34)(cid:88) ρ ( δ ρ φ ) + ( m − m c ) φ + φ (cid:35) + ξ (cid:16) δ µν δ − δ µ δ ν (cid:17) φ (cid:41) (A7)Here the term multiplying ξ is the improvment term, which has been taken to be 0 in the maintext. The calculations here retraces the steps taken for the d case in [7].By considering operators which have a lower dimension than T µν , the only operator capable ofproducing divergent mixing is δ µν Tr φ . We therefore defined the renormalised EMT in Eq. (8) as T Rµν = T µν − C δ µν Ng Tr φ , (A8)18ith C being the divergent mixing coefficient. To calculate the mixing coefficient perturbatively,consider the insertion of T µν in the two point correlator, i.e. (cid:104) φ a φ b T µν (cid:105) . The one-loop diagramsare shown in Fig. 6. Both in the continuum and on the lattice, diagram (a) in Fig. 6 is finite,and contributes to the WI. However, for diagram (b), as a result of the breaking of translationalinvariance, the LPT result diverges, even though in the continuum the PT result is finite (thiscould be calculated by replacing the lattice momenta ˆ q with the continuum momenta q , and theintegration limit by (cid:82) ∞−∞ ). φ a φ b T µν diagram (a) φ a φ b T µν diagram (b) Figure 6. The insertion of T µν in a two point correlator, i.e. (cid:104) φ a φ b T µν (cid:105) , up to one-loop. The black dotrepresents the insertion of T µν Using LPT, diagram (b) in Fig. 6 evaluates to B µν ( q ) = − δ ab (cid:18) N − N (cid:19) (cid:34) − I µν ( q ) + δ µν (cid:32)(cid:88) α I αα ( q ) + ( m − ξ ( q µ q ν − δ µν p )) V ( q ) (cid:33)(cid:35) . (A9)Using [31], the divergent term of B µν ( q ) can be isolated with B µν (0) , while the remaining termsare finite or proportional to the lattice spacing. This evaluates to B µν (0) = δ µν a (cid:18) N − N (cid:19) (cid:18) Z − (cid:19) = C Ng δ µν . (A10)Using the definition from Eq. (9), C = ga c , (A11)to absorb the leading a behaviour, we obtain c = (cid:18) − N (cid:19) (cid:18) Z − (cid:19) . (A12)This gives the result in Eq. (10). 19 . Correlators at vanishing flowtime: C ( q ) and C µν ( q ) The first two-point correlation function to calculate is defined in Eq. (16): C ( q ) = (cid:18) Ng (cid:19) a (cid:88) x ∈ Λ e − iq · x (cid:104) Tr φ ( x ) Tr φ (0) (cid:105) . (A13)The one- and two-loop diagrams are shown in Fig. 7(a) and (b) respectively. Ng Tr φ Ng Tr φ (a) 1 loop Ng Tr φ Ng Tr φ (b) 2 loop Figure 7. Perturbative expansion of C ( q ) at 1- and 2-loops. Note that the two-loop diagram is simply the square of the one-loop diagram up to an overallcolour factor. These diagrams evaluate to C ( q ) = tr( T a T b ) tr( T c T d )( δ ac δ bd + δ ad δ bc ) V ( q ) = N (cid:18) − N (cid:19) V ( q ) , (A14) C ( q ) = − (cid:16) gN (cid:17) tr( T a T b ) tr( T c T d T e T f ) tr( T g T h ) ( δ ac (16 δ bd δ eg + 8 δ be δ dg ) δ fh ) V ( q ) = − N g (cid:18) − N + 3 N (cid:19) V ( q ) . (A15)In the massless limit, using Eq. (A5), these yield C ( q ) = N g (cid:18) − N (cid:19) (cid:20) ( g/q ) + ( ag ) 14 Z + 9 Z −
412 + ( ag ) ( q/g ) Z − Z + 43456 + O (cid:0) ( ag ) (cid:1)(cid:21) , (A16) C ( q ) = − N g (cid:18) − N + 3 N (cid:19) (cid:104) ( g/q ) + ( ag )( g/q ) (cid:18) Z + 9 Z − (cid:19) + ( ag ) (14 Z + 9 Z −
144 + ( ag ) ( q/g ) (cid:18) Z − Z + 41728 (cid:19) + O (cid:0) ( ag ) (cid:1) (cid:105) . (A17)Now we evaluate the correlation function in Eq. (13): C µν ( q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T Rµν ( x ) Tr φ (0) (cid:105) = C µν ( q ) − gc a δ µν C ( q ) , (A18)where C µν ( q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T µν ( x ) Tr φ (0) (cid:105) , (A19) C ( q ) = (cid:18) Ng (cid:19) a (cid:88) x ∈ Λ e − iq · x (cid:104) Tr φ ( x ) Tr φ (0) (cid:105) (A20)20 µν Ng Tr φ (a) 1 loop T µν Ng Tr φ (b) 2 loop Figure 8. Perturbative expansion of C µν ( q ) at 1- and 2-loops. The relevant one- and two-loop diagrams for the correlator C µν ( q ) are shown in Fig. 8(a) and(b) respectively, and they evaluate to C µν ( q ) = N (cid:18) − N (cid:19) ( δ µν (cid:88) α I αα ( q ) − I µν ( q )) + ( ξ ( q µ q ν − δ µν p ) − δ µν m ) C ( q ) (A21) C µν ( q ) = − N g (cid:18) − N − N (cid:19) ( δ µν (cid:88) α I αα ( q ) − I µν ( q )) V ( q )+ (cid:0) ξ ( q µ q ν − δ µν q ) − δ µν m (cid:1) C ( q ) . (A22)At one-loop, Eq. (A21), C µν contains only the tree-level EMT, so C µν = C µν . There is nocontribution coming from the operator mixing c , which comes with another order O ( g ) . However,the term δ µν (cid:80) α I αα ( q ) − I µν ( q ) presents a divergent contact term at C µν (0) , C µν (0) = − N a (cid:18) − N (cid:19) (cid:18) Z − (cid:19) δ µν = κa δ µν . (A23)The integral producing this contact term is similar to that in c in Eq. (A10), with the onlydifference being the colour factor. This contact term has to be subtracted before the continuumlimit of the correlator is taken.For the two-loop expression, it can be shown that after subtracting the correlator gc a δ µν C ( q ) to renormalise the EMT from Eq. (A18), the correlator is UV finite; no extra divergences otherthan the one coming from the operator expansion appear.
4. Correlators at finite flowtime: C ( t, q ) , C µν ( t, q ) At finite flowtime, the lattice integrals are regulated by the flowtime t . In perturbation theory,the kernel for each propagator has an extra exponential factor, e − tq , where q is the momentum ofthe propagator. We first evaluate the correlator C ( t, q ) = (cid:18) Ng (cid:19) a (cid:88) x ∈ Λ e − iq · x (cid:104) Tr φ ( x ) Tr ρ ( t, (cid:105) (A24)21t finite flowtime. This correlator is obtained by replacing Tr φ ( x ) with Tr ρ ( t, in C ( q ) . Sincethe regulated correlators are finite, we look at the continuum limit ( a → ) of the correlator inperturbation theory. At one-loop, this evaluates to C ( t, q ) = N (cid:18) − N (cid:19) (cid:90) d k (2 π ) e − tq e − t ( q − k ) ( k + m )(( q − k ) + m ) . (A25)In the massless limit, C ( t, q ) = N g (cid:18) − N (cid:19) (cid:34) − π (cid:90) σ d s e − s Erfi( s ) s (cid:35) (cid:18) gq (cid:19) , (A26)where σ = (cid:112) tq / , and Erfi( z ) = − i Erf( iz ) is the imaginary error function , which has the series Erfi( z ) = π − / (cid:0) z + z + · · · (cid:1) about z = 0 . Expanding in σ , this evaluates to C ( t, q ) ≈ N g (cid:18) − N (cid:19) (cid:34) (cid:18) q tπ (cid:19) / (cid:18) q t − (cid:19)(cid:35) (cid:18) gq (cid:19) + O (cid:0) σ (cid:1) . (A27)Similarly, we look at the continuum limit of C µν ( t, q ) = Ng a (cid:88) x ∈ Λ e − iq · x (cid:104) T µν ( x ) Tr ρ ( t, (cid:105) (A28)at finite flowtime. In the continuum limit, the EMT does not require renormalisation, we cantherefore drop the superscript. At one-loop, C µν ( t, q ) = N (cid:18) − N (cid:19) (cid:90) d k (2 π ) δ µν k · ( q − k ) − k µ ( q − k ) ν + ξ ( q µ q ν − δ µν q )( k + m )(( q − k ) + m ) e − tq e − t ( q − k ) (A29)In the massless limit, this evaluates to C µν ( t, q ) = − N (cid:18) − N (cid:19) q π / (cid:34) √ π Erfi( σ )( π µν (3 + 2 σ ) − δ µν ) σ − e − σ + 8 √ π (1 − ξ ) π µν (cid:90) σ e − s Erfi( s ) s d s − − ξ ) π / π µν + e − σ (4 δ µν − π µν ) σ − (cid:35) (A30)where π µν = δ µν − q µ q ν q is the transverse projector. To obtain the “flowed contact term” K ( t ) from Eq. (25), we utilise the fact that the contact term is the longitudinal part of the correlator C µν ( t, q ) . We separate the above expression for C µν ( t, q ) into a transverse part, C µν ( t, q ) transverse (which is proportional to the π µν ), and the remaining longitudinal part C µν ( t, q ) longitudinal . The22ransverse part C µν ( t, q ) transverse = − π µν N (cid:18) − N (cid:19) q π / (cid:34) √ π Erfi( σ )(3 + 2 σ ) σ − e − σ + 8 √ π (1 − ξ ) (cid:90) σ e − s Erfi( s ) s d s − − ξ ) π / + 6 e − σ σ − (cid:35) (A31)is finite, as ensured by the WI. The remaining longitudinal part gives C µν ( t, q ) longitudinal = − δ µν N (cid:18) − N (cid:19) q π / (cid:34) √ π Erfi( σ )( − σ − e − σ + 4 e − σ σ − (cid:35) . (A32)When expanded about σ = 0 , the leading order term contributing to the contact term C µν ( t, q ) longitudinal is C µν ( t, q ) longitudinal ≈ − δ µν N (cid:18) − N (cid:19) q π / σ + O ( σ )= − δ µν N (cid:18) − N (cid:19) √ π / √ t + O ( σ )= δ µν K ( t ) , (A33)which gives us the result in Eq. (27).Using Eqs. (A27) and (A32), the perturbative expression for the ratio in Eq. (37) can be calcu-lated, f g ( g √ t, q l ) = ag K ( t ) C ( t, q l )= − √ π / aqg √ t + O ( σ ) , (A34)giving the result in Eq. (38). 23
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