Scattering in a Simple 2-d Lattice Model∗

arXiv:hep-lat/9210023v1 21 Oct 19921Scattering in a Simple 2-d Lattice Model∗C.R. Gattringera, I.

Hip and C.B. Lang baMax Planck Institut f¨ur Physik,F¨ohringer Ring 6, D-8000 Munich, GermanybInstitut f¨ur Theoretische Physik,Universit¨at Graz, A-8010 Graz, AustriaL¨uscher has suggested a method to determine phase shifts from the finite volume dependence of the two-particleenergy spectrum.

We apply this to two models in d=2: (a) the Ising model, (b) a system of two Ising fields withdifferent mass and coupled through a 3-point term, both considered in the symmetric phase. The Monte Carlosimulation makes use of the cluster updating and reduced variance operator techniques.

For the Ising system westudy in particular O(a2) effects in the phase shift of the 2-particle scattering process.1. INTRODUCTIONMost of the particle entries in the annual reviewof particle properties of the particle data group re-fer to resonances.

Usually they are created in two-particle collisions and observed in correspondingcross sections and scattering phase shifts. How-ever, in quantum field theory unstable states stillare one of the tantalizing problems.

In the latticeapproach, up to now efforts have concentrated onthe identification of stable particle states.Thedetermination of phase shifts and resonance pa-rameters has been reserved for a future of morepowerful computers and there are only very few(courageous) contributions [1]- [6].Recently, L¨uscher [2] has refined earlier ideas[1,3] on how to determine phase shifts in an elas-tic two-particle system from the energy spectrumin a finite volume and together with Wolff[4] thepower of the method was demonstrated in thed=2 O(3) model. Here we discuss some of our re-sults [6] where we use this method to determinethe phase shifts in a simple d=2 model with twoparticle types and resonating phase shifts.

Othercontributions to this conference present first re-sults for d=3 [7] and d=4 [8] (cf. also [9]) wherelife is harder and statistics scarcer.

The case d=2∗Presented by C.B. Lang.

Supported by Fonds zur F¨or-derung der Wissenschaftlichen Forschung in ¨Osterreich,project P7849.provides an excellent testbed to study some of thejuicy details of the approach.1.1. MethodLet us briefly review the idea.Consider thescattering of two identical particles of mass m ina box of finite spatial extension L. The time ex-tension is assumed to be sufficiently large, notto contribute to finite size effects.The size ofthe spatial volume L and the periodic b.c., how-ever, are responsible for the quantization of themomenta.

In the elastic regime 2m ≤W < 4m(or 3m, depending on the theory) the allowed mo-mentum values k are, in the two-dimensional case,related to the scattering phase shift via the quan-tization condition2δ(kn) + knL = 2nπ , n ∈N. (1)Assuming vanishing total momentum (CMS) thetotal energy of the 2-particle state is just twice theenergy of the back-to-back single particle statesWn = 2pm2 + k2n.

(2)Thus given m and a couple of low lying energylevels Wn one may obtain values of the (infinitevolume) phase shift at the corresponding valueskn. Varying L one may cover a whole range of mo-mentum values.

Relation (1) holds in that simpleform for d=2 but can be generalized to higherdimensions [2].

2One has to take care of the following restric-tions.• The interaction region and the single par-ticle correlation length ought to be smallerthan the spatial volume.• Polarization effects due to virtual particlesrunning around the torus should be undercontrol.• Lattice artifacts will turn up in O(a2) cor-rections.• For the determination of the energy spec-trum one should consider correlation func-tions of sufficiently many observables.In d=2 all these can be controlled.1.2. Model and SimulationWe choose a model, where two light particlesϕ couple to a heavier particle η giving rise to res-onating behaviour.

The action is given byS=−κϕXx∈Λ,µ=1,2ϕxϕx+ˆµ−κηXx∈Λ,µ=1,2ηxηx+ˆµ+g2Xx∈Λ,µ=1,2ηxϕx(ϕx−ˆµ + ϕx+ˆµ). (3)The values of the fields are restricted to {+1, −1}.The sums run over all sites (x0, x1) of the eu-clidean L × T lattice Λ with periodic boundaryconditions.

The 3-point term was introduced ina nonlocal but symmetric way, because ϕ2x ≡1.For g = 0 this is just a system of two indepen-dent Ising models, each describing in the scalingregion interacting bosonic fields with mass m(κ).The corresponding masses have been adjusted tomϕ ≃0.19 and mη ≃0.5 (mη defined by the reso-nance peak position) by calibrating the couplingsκϕ and κη.When kinematically allowed, the term propor-tional to g gives rise to transitions like η →ϕϕ rendering η a resonance in the ϕϕ channel.We study the model at g = 0, 0.02 and 0.04.Throughout this work we use T = 100; the spatialextension L varies between 12 and 60. For eachset of couplings and lattice size we performed typ-ically 2 × 105 measurements.

Our Monte Carlosimulation utilizes the cluster updating methodintroduced for the Ising model in [10]. The sta-tistical errors are estimated with the Jackknifemethod.

Details of the simulation technique andthe phase diagram can be found in [6].2. OBSERVABLES2.1.

Single particle stateThe operator of a ϕ state with momentump1,ν = 2πν/L, ν = −L/2 + 1, . .

. , L/2, is giventhrough1LXx1∈Λx0ϕx0,x1 exp (ix1p1,ν),(4)where Λx0 denotes a timeslice of Λ.

Its connectedcorrelation function over temporal distance t de-cays exponentially ∝exp (−p0,νt) defining p0,ν;in particular p0,ν=0 = mϕ.For the determination of the energy spectrum aprecise knowledge of the single particle mass andrelated finite size effects is important. We find [6],that our results for p0,ν follow with high precisionthe spectral relation for the lattice propagator ofa Gaussian particle with mass m,p0,ν = cosh−1(1 −cos p1,ν + cosh m).

(5)This expression deviates from the continuum dis-persion relation (d.r.) p0,ν = (m2 + p21,ν)1/2 by aleading correction O((ap)2).The observed mass, as compared to the “real”mass at vanishing lattice spacing and infinite vol-ume, is also affected by polarization due to selfinteraction around the torus.We confirm thisbehaviour and find good agreement with the ex-pected exponential decrease [11].We also de-termined the wave function renormalization con-stants for the fields [6].2.2.

Scattering sectorWe consider operators with total zero momen-tum and quantum numbers of the η,N1(x0)=1LXx1∈Λx0ηx1,x0,(6)

3Figure 1. Phase shifts for the Ising model for twovalues of the mass, determined with the contin-uum d.r.

(2) ; full line: δ = −π/2, dashed line:2-particle threshold.Nj(x0)=1L2Xx1,y1∈Λx0eipj(x1−y1)ϕx1,x0ϕy1,x0,with pj = 2π(j −2)L, j = 2, 3 . .

. (7)and measure all (connected) cross-correlationsMnm(t) = ⟨Nn(t)Nm(0)⟩c (t denotes the sepa-ration of the time slices).The operators Nj>1describe two ϕ-particles in the CM system withrelative momentum 2pj.

Because of the interac-tion they do not correspond to eigenstates of ourmodel. Indeed they are eigenstates of the Gaus-sian model of free bosons.However, if the setis complete, a diagonalization of the correlationmatrix provides the necessary information on theenergy spectrum of this channel.The transfermatrix formalism yields the spectral decomposi-Figure 2.

Like fig. 1, but now using the latticed.r.

(9)tionMnm(t) =∞Xl=1v(l)n∗v(l)m e−tWl,(8)where v(l)n= ⟨l|Nn0⟩are the projections of thestates |Nn0⟩(generated by the operators Nn outof the vacuum) on the energy eigenstates ⟨l| ofthe scattering problem.The number of operators considered should bechosen larger than the number of states in theelastic regime 2mϕ ≤W < 4mϕ and not largerthan L/2 to be linearly independent. A larger setprovides a better representation of the eigenstatesbut enhances the numerical noise.We work with between 4 and 6 operators de-pending on L and distances t = 1 .

. .

8. Detailson the determination of the eigenspectrum in thescattering sector and on the representation of theeigenstates are discussed in [6].

43. RESULTS FOR THE PHASE SHIFTS3.1.

ResonanceIn [6] we present our results for the observed en-ergy levels and the resulting phase shifts. Sincethe pure Ising model has an S-matrix equal to −1[12], the phase shift starts with −π/2 and thenshows, for non-vanishing g, a clear resonance be-haviour manifesting itself in a fast increase by π.It can be nicely approximated by a standard ef-fective range resonance formula [2,6].3.2.

O(a2) correctionsLet us consider the Ising case (g = 0) in moredetail.Here we increased the statistics signifi-cantly (1.5 million measurements) and repeatedthe calculation for m = 0.19 and a higher valueof the mass m = 0.5 (fig.1). As mentioned weexpect a phase shift of −π/2 and the results arein agreement with this.

At small k the energy isclose to the 2-particle threshold and the statisti-cal error of the energy transforms via (2) into arelatively larger error of k and thus δ(k). Higherk, on the other hand, stem from large values ofthe energy with intrinsically larger statistical fluc-tuations.

However, due to the enhanced statisticswe do identify a systematic deviation from −π/2increasing with k and m. We attribute this be-haviour to O(a2) corrections.As mentioned earlier, the d.r. (2) gives thetotal energy of the asymptotic 2-particle statewhich (under the assumption of localized inter-action region) is just twice the energy of the out-going particles.

Now the O(a2) corrections of thesingle particle d.r. have been nicely described byreplacing the continuum d.r.

by the lattice rela-tion (5). We therefore replace (2) by the corre-sponding lattice expression,Wn = 2 cosh−1(1 −cos kn + cosh m).

(9)Our data for Wn now produce slightly differentvalues of kn and δ exhibited in fig.2, in betteragreement with a constant value of −π/2.Weconclude that the leading O(a2) corrections canbe expressed by replacing the continuum d.r. bythe lattice relation, at least in the 2 dimensionalIsing Model.

In general we suspect that latticeartifacts in the phase shift can be diminished bystudying carefully the dispersion relation of singleparticle states.4. CONCLUSIONWe have determined phase shifts in the Isingmodel and resonating phase shifts in a model withtwo types of particles and a three-point coupling.We find that L¨uscher’s suggestion for determin-ing these phase shifts is indeed a very reliablemethod, at least in d=2.

The leading O(a2) ef-fects in the 1- and 2-particle channel may be ex-plained by the differences between lattice vs. con-tinuum dispersion relations.REFERENCES1.M. L¨uscher,Commun.

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