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Bounds on the renormalized couplings in an

arXiv:hep-lat/9303012v1 23 Mar 1993DESY 93-036MS-TPI-93-01SHEP-92/93-15Bounds on the renormalized couplings in anSU(2)L ⊗SU(2)R symmetric Yukawa modelL. Lin∗I.

Montvay†G. M¨unster∗M.

Plagge∗H. Wittig‡March 23, 1993AbstractThe vacuum stability lower bound on the mass of the Higgs boson is numericallyinvestigated in an SU(2)L ⊗SU(2)R symmetric Yukawa model, which describes twoheavy degenerate fermion doublets in the limit of vanishing gauge couplings.

Goodagreement with perturbation theory is found, although the couplings are strong.The upper bound on the fermion mass and renormalized Yukawa coupling is alsodetermined in the part of bare parameter space where reflection positivity has beenproven.1IntroductionCut-offdependent upper bounds on renormalized couplings arise in the Standard Model for non-asymptotically free couplings, if their continuum limit is trivial. Such “triviality bounds” showup in perturbation theory as an apparent inconsistency [1], and have been intensively studiedon the lattice by non-perturbative methods (for a recent review see [2]).

In Yukawa modelsthere is also a lower limit on the Higgs mass, which is called “vacuum stability bound” [3]. Onthe lattice this can be understood as due to quantum effects inducing a positive renormalizedquartic scalar coupling even if the bare quartic coupling takes its lowest possible value, namelyzero [4].In previous papers [5] the triviality upper bound on the Higgs mass has been investigatedby non-perturbative methods in a Yukawa model with chiral SU(2)L ⊗SU(2)R symmetry.

Thelattice formulation is based on the mirror fermion action [6] with exact decoupling of the mirrorfermions from the physical spectrum [7, 8]. (Another lattice formulation of the same continuum“target theory” is possible by using reduced staggered fermions [9].) Since the Hybrid MonteCarlo method [10] is used, the minimum number of fermion doublets which can be numericallysimulated, is two.

In this context it is important that we are working in the limit of vanishing∗Institut f¨ur Theoretische Physik I, Universit¨at M¨unster, Wilhelm-Klemm-Str. 9, D-4400 M¨unster, FRG†Deutsches Elektronen-Synchrotron DESY, Notkestr.

85, D-2000 Hamburg 52, FRG‡Department of Theoretical Physics, University of Southampton, UK1

gauge couplings, and hence by charge conjugation a mirror fermion doublet can be transformedto a fermion doublet.In fact, only the vanishing of the SU(3)colour ⊗U(1)hypercharge gaugecouplings is necessary. The chiral SU(2)L gauge coupling can be introduced, because SU(2) ispseudoreal [11, 12].In the present paper we continue the non-perturbative investigation of the same SU(2)L ⊗SU(2)R symmetric Yukawa model as in refs.

[13, 5].In particular, we concentrate on thevacuum stability lower bound and on the upper limit of the renormalized Yukawa coupling inthe region of positive scalar hopping parameter, where reflection positivity, implying unitarityin Minkowski space, can been proven [4]. Besides the physically relevant phase with brokensymmetry, the renormalized Yukawa coupling is also computed in the symmetric phase, in orderto check previous observations of very strong couplings there [14, 13].2Vacuum stability boundThe lattice action and the definition of different renormalized physical quantities closely followour previous papers on chiral Yukawa models with mirror fermions [14, 4, 13, 5].

The latticeaction is a sum of the O(4) (∼= SU(2)L ⊗SU(2)R) symmetric pure scalar part Sϕ and fermionicpart SΨ:S = Sϕ + SΨ . (1)ϕx is the 2 ⊗2 matrix scalar field, and Ψx ≡(ψx, χx) stands for the mirror pair of fermiondoublet fields (usually ψ is the fermion doublet and χ the mirror fermion doublet).

In the usualnormalization conventions for numerical simulations we haveSϕ =Xx12Tr (ϕ+x ϕx) + λ12Tr (ϕ+x ϕx) −12−κ4Xµ=1Tr (ϕ+x+ˆµϕx),SΨ =Xxnµψχh(χxψx) + (ψxχx)i−K±4Xµ=±1h(ψx+ˆµγµψx) + (χx+ˆµγµχx) + r(χx+ˆµψx) −(χxψx) + (ψx+ˆµχx) −(ψxχx)i+Gψh(ψRxϕ+x ψLx) + (ψLxϕxψRx)i+ Gχh(χRxϕxχLx) + (χLxϕ+x χRx)io. (2)Here K is the fermion hopping parameter, r the Wilson-parameter, which will be fixed to r = 1in the numerical simulations, and the indices L, R denote, as usual, the chiral components offermion fields.

In this normalization the fermion–mirror-fermion mixing mass is µψχ = 1−8rK.The fermionic part SΨ is given here for a single mirror pair of fermions. For the HybridMonte Carlo simulation the fermions have to be doubled by taking the adjoint of the fermionmatrix for the new species.

Taking the adjoint transforms fermions to mirror fermions andvice versa, but as noted before, without SU(3)colour ⊗U(1)hypercharge gauge couplings they areequivalent to each other.The numerical simulations were performed on 63·12 lattices at a bare scalar quartic couplingλ = 10−6. The small positive value of λ was chosen in order to be sure about the convergence ofthe path integral.

The Yukawa coupling Gχ was kept at zero, for exact decoupling of the mirrordoublets [8]. This allows to stay with the fermion hopping parameter K near its critical valueat K = 1/8, as described in ref.

[5]. In the broken (FM) phase at the fixed values Gψ = 0.25and Gψ = 0.30 the scalar hopping parameter was tuned to achieve a scalar mass of about2

Table 1:The main renormalized quantities and the bare magnetization ⟨σ⟩≡⟨|ϕ|⟩for severalbare couplings Gψ and κ-values. Points labelled by small letters are our data at λ = 10−6 whichwere all obtained on 63 · 12.

Points labelled by capital letters are the data for the upper boundat λ = ∞. For the points with double capital letters the lattice size was 83 · 16, whereas singlecapital letters denote data on 63 · 12.

All data are collected from typically 10000 trajectories,except for the points labelled a, i, A, C where only about 5000 trajectories were run. As Gχ = 0,the renormalized coupling GRχ was always zero within errors and is not included here.Gψκ⟨σ⟩vRmRσµRψgRGRψG(3)Rψa0.250.0950.251(7)0.196(11)0.79(7)0.271(8)42(11)1.38(4)1.42(20)b0.250.0990.393(7)0.241(10)0.59(9)0.407(9)18(3)1.69(4)1.56(16)c0.250.1010.525(6)0.304(9)0.57(3)0.59(3)10(2)1.68(10)1.53(40)d0.30.0900.438(6)0.27(1)0.75(7)0.55(6)23(3)2.04(10)e0.30.0950.754(4)0.417(11)0.76(7)0.91(6)10(1)2.19(11)f0.30.1001.260(5)0.67(2)0.84(6)1.503(12)4.8(8)2.24(7)g0.620.00.424(3)0.29(2)1.23(9)1.23(3)53(11)4.2(1.2)4.5(4.0)h0.630.00.469(3)0.351(11)1.65(13)1.25(16)63(16)3.5(4)i0.650.00.523(3)0.34(2)1.39(14)1.6(3)53(18)4.6(4)A0.30.300.439(1)0.400(13)1.17(7)0.55(2)26(7)1.36(6)BB0.30.270.270(2)0.25(1)0.77(3)0.342(2)31(4)1.35(6)1.36(12)C0.60.180.3524(18)0.36(2)1.36(10)0.86(8)36(6)2.4(3)DD0.60.180.3389(13)0.339(16)1.31(7)0.86(11)38(4)2.5(3)2.3(2.4)mRσ ≃0.6 −0.8 and a not too small fermion mass.

κ was always kept to be non-negative, inorder to be sure about reflection positivity, that is unitarity in Minkowski space [4]. The lastpoints in the positivity region were fixed at κ = 0, and then Gψ was tuned to obtain reasonablemasses.

The results are summarized in table 1, where also such points are included where themasses are not yet sufficiently tuned.Tuning the scalar and fermion mass is, of course, important in order to be close to thecritical line separating the broken (FM) and symmetric (PM) phases [15], and at the sametime avoid strong finite size effect. This prevents us from going on the 63 · 12 lattice to a verysmall Yukawa coupling Gψ ≪1, because then the fermion mass becomes too small.

At thestrongest Yukawa coupling (at κ = 0) the minimum of the scalar mass on our lattice is quitelarge (above 1), therefore we could not achieve the desired value mRσ ≃0.6 −0.8. This issimilar to the behaviour observed at λ = ∞[5].

More generally, in the investigated range ofGψ the qualitative change of the renormalized quantities between 43 · 8 and 63 · 12 lattices isquite similar to the one at λ = ∞. This allows to choose the points with label c,d and g asoptimal in the three sets of points for the lower bound in table 1.

These are shown in figs.1 and 2 together with the one-loop perturbative estimates of the vacuum stability lower limitat the given cut-offs. In these figures also the data at λ = ∞are included, which show thebehaviour of the triviality upper bound for the renormalized quartic coupling.

For completenesswe display the results for λ = ∞in table 1 too, because more statistics has been collected forsome points since publication of [5].The agreement between numerical simulation data and the one-loop perturbative estimates3

is remarkably good. In particular, at the strongest couplings, within our errors, there is practi-cally no difference in the renormalized couplings between λ ≃0 and λ = ∞.

This means thatthe strong Yukawa coupling alone is able to induce the maximal possible renormalized quarticcoupling at the given cut-off.Table 1 also includes results for the renormalized Yukawa coupling G(3)Rψ, defined in terms ofa three-point function (see [5]). At tree level, and moreover in the one-loop approximation, itcoincides with GRψ.

The numerical results show that both couplings are consistent with eachother, even in those points where the errors are relatively large.3Yukawa couplingOne can see in table 1 that in the broken (FM) phase the renormalized Yukawa coupling GRψgrows with the bare coupling Gψ roughly linearly, up to quite strong values above the treeunitarity limit√2π ≃2.5. In previous numerical simulations in the symmetric (PM) phaseof the U(1)L ⊗U(1)R [14] and SU(2)L ⊗SU(2)R [13] symmetric Yukawa models very strongrenormalized Yukawa couplings were observed, as well.

We would like to see how strong therenormalized Yukawa couplings can be in the PM phase in the mirror-fermion decoupling limit.Since we do not want to go to the region with κ < 0 where reflection positivity could not beproven [4], the maximal possible value of GRψ should come from tuning Gψ along the κ = 0axis in the PM phase. We tune Gψ such that the physical scalar mass mφ is around 0.7 on the63 · 12 lattice to avoid large finite size effects.

We also run the same point on 83 · 12 and 83 · 16lattices to see how various quantities (especially the renormalized Yukawa couplings) changewith different volumes.The renormalized Yukawa coupling matrix GR ≡diag(GRψ, GRχ) is defined asGR ≡−m2R4√ZR˜ΓR(k) Z−1/2Ψ⟨ΦΨ¯Ψ⟩0 Z−1/2Ψ˜ΓR(k) ,(3)where the three-point Green’s function ⟨ΦΨ¯Ψ⟩0 and Z−1/2Ψare defined in [13]. ˜ΓR(k) is themomentum-space renormalized fermionic two-point vertex function defined at the smallest mo-mentum k = (⃗0, k4), k4 = π/T.

Near k = 0 the behaviour of ˜ΓR(k) is˜ΓR(k) ≃iγ4 ¯k4 + MR ,¯k4 ≡sink4 = sin( πT ) ,MR = 0µRµR0!,(4)where µR is the renormalized fermion mass. In previous work we did the leading-order ap-proximation by neglecting the term iγ4¯k4 in the measurement since T is large.

However, fromdata in [13], we suspect that it might cause some small but visible effects on GR.There-fore we decided to improve the measurement of GR by including this leading-order correction.Also, we remove a previous inconsistency in normalization in the definition of GR in the PMphase.Before, the convention was such that the full scalar propagator at zero momentum˜G(0) ≡Px,y⟨φSxφSy⟩/L3T, after renormalization, was normalized to 1/m2R. However, this doesnot correspond to the natural normalization convention we used in the broken phase [5].

Therethe convention corresponds to the one in the PM phase such that ˜G(0) is renormalized to 4/m2R.In order to have a fair comparison of the renormalized Yukawa couplings in both phases, wedecide to switch to this new convention of normalization for GR shown in (3). This impliesthat data on GR’s in [13] should be scaled down by a factor of two in this new normalization4

convention. (A similar inconsistency in the U(1)L⊗U(1)R symmetric model [14] can be removedby dividing the values of GR there by a factor of√2.

)One should notice that, as shown in (3), scalar quantities appearing in the definition of GRshould be the renormalized scalar mass mR and the wave function renormalization factor ZRdefined at vanishing momentum [16]. However, the scalar mass we actually use in the mea-surement is the physical scalar mass mφ obtained from a cosh fit of the scalar field correlationfunction along the time direction.

The wave function renormalization factor Z3, which is de-fined through the residuum of the pole of the propagator, cannot be determined reliably fromour statistics. Therefore we define the wave function renormalization factor Zφ in terms of themass mφ and the susceptibility ˜G(0) by means ofZφ = m2φ ˜G(0) .

(5)In general, mR and ZR is different from mφ and Zφ. In a weakly interacting system (e.g.

purescalar λφ4 theory), they are very close to each other[16]. In our SU(2) mirror-fermion model,the bare Yukawa coupling Gψ is large, therefore there is no guarantee that they are still closeto each other.

On the other hand, the correct measurement of mR and ZR is crucial to themeasurement of GRψ and GRχ, we therefore also measure mR and ZR to see how they differfrom mφ and Zφ. Since we are on the finite lattice, we estimate ZR and mR fromZ−1R ≡[ ˜G(k)−1 −˜G(0)−1]/ˆk2 ,m2R = ZR˜G(0) ,(6)wherek = (⃗0, k4) ,k4 = 2πT ,ˆk2 = 4sin2(k42 ) .

(7)The results of the numerical simulations in the symmetric phase are collected in table 2.From the table one can clearly see that GRχ is now zero within error bars as expected fromthe shift symmetry at Gχ = 0 [8, 13]. This indicates that the term we used to neglect does,indeed, have some visible effect on the renormalized Yukawa couplings.

Meanwhile, the naturaldefinition of the renormalized couplings are taken at zero momentum since there is no infraredsingularity in the PM phase. But GR’s we measured according to (3) are actually defined atk = (0, 0, 0, π/T), which will approach zero in the infinite T limit.

If the inverse propagator hada non-negligible curvature near zero momentum, this would influence the determination of ZRand hence of other renormalized quantities. We therefore took measurements at two differentvalues of T, i.e.

: T = 12, 16, to see how we can get GR’s at zero momentum by extrapolation.Our data show that GRψ and GRχ basically stay unchanged as T goes from 12 to 16. Wetherefore believe that curvature effects in the propagator are not large and that the data wehave on GR’s are, to a good approximation, the renormalized Yukawa couplings defined at zeromomentum.The fluctuations of ZR and mR are quite large, as expected.

Within error bars, they agreewith Zφ and mφ. This shows that our measurement of GRψ and GRχ using mφ and Zφ can betaken as a good approximation to the GR’s defined in (3).As one can see from table 2, on our lattices the maximal value of GRψ in the region of non-negative scalar hopping parameter is more than twice the value of the tree unitarity bound.This supports our previous results in the symmetric phase on smaller lattices and less statistics[14, 13].

With the present correct normalization the difference between the maximal value ofGRψ in the symmetric and broken phase is not dramatic, although the values are still larger inthe symmetric phase (see tables 1–2).5

Table 2:The main renormalized quantities at λ = ∞, Gχ = 0, Gψ = 1.09, κ = 0 andK = 2/19 in the PM phase on various lattices. Data on 63 · 12 lattices are collected from 20000trajectories while on 83 · 12 and 83 · 16 we have about 6000 trajectories each.SizemφmRZφZRµRGRψGRχ63 · 120.69(3)0.60(17)1.52(5)1.11(38)0.664(5)5.84(16)-0.03(7)83 · 120.65(7)0.56(11)1.84(15)1.36(57)0.654(9)6.04(37)-0.12(17)83 · 160.55(3)0.51(11)1.71(33)1.44(73)0.63(1)5.87(54)0.07(9)41/N expansionThe results of numerical studies of the phase diagram of the model at both λ = ∞and λ = 10−6are reported in [5].

The particular interest in the phase structure at small λ arose after it wasreported [17] that models with naive fermions do not exhibit a ferrimagnetic (FI) phase andthat instead a first order phase transition is observed. Clearly the absence of any such transitionis of great importance for the study of bounds on the couplings within our model.Our Monte Carlo investigations show that at least the physically relevant phase transitionfrom the PM to the FM phase is second order.

In particular, the magnetization varies smoothlyacross the transition.The numerical analysis of the phase diagram was supplemented by a 1/Nf expansion inleading order, where Nf denotes the number of fermion–mirror-fermion doublet pairs.Our strategy was to calculate the one-loop effective potential to leading order in 1/Nf as afunction of the fluctuation fieldσx ≡φ4x −s −(−1)x1+x2+x3+x3 bs ,(8)where φ4x is the fourth real component of the scalar field ϕx, and s and bs are the positionsof the minimum of the effective potential at tree level with respect to ϕx and the staggeredscalar field bϕx, respectively. After performing a Fourier transformation to momentum space,we finally obtain the effective potential Veff[eσ(0), eσ(π)] as a function of the fluctuation field atboth the zero- and 4π-corners of the Brillouin zone.

The qualitative features of the transitionsfrom PM to FM (PM to AFM [15]) can now be studied via the dependence of Veffon the fieldeσ(0) (the field eσ(π)).A first observation is that to leading order in 1/Nf the effective potential is a quadraticfunction of eσ(0) and eσ(π), and therefore we do not expect a first order phase transition whichwould rather require a quartic term, resulting in a double-well structure of the potential.For λ = 0 we obtain estimates for the critical value of the scalar hopping parameter κ fromthe two gap equations−16 κcr + 2 −8Nf G2ψZqq2(q2 + µ2q)2 + G2ψ s2 q2=0(for PM ↔FM),(9)16 κcr + 2=0(for PM ↔AFM),(10)where qµ = sin(qµ), and the integral is taken over the Brillouin zone. It is therefore only thetransition PM to FM which is affected by fermionic contributions in leading order.

Its transitionline bends down for increasing Yukawa couplings and finally intersects the straight transition6

line from PM to AFM at κcr = −1/8. Obviously, in leading order of the 1/Nf expansion thequalitative behaviour of the phase transition lines is similar to all our numerical studies [5, 15].According to the 1/Nf expansion the FI phase exists in leading order, since solutions tothe minimum of Veffat non-zero values of both eσ(0) and eσ(π) are found to exist.

Furthermore,the position of the minimum of Veffvaries smoothly across all phase transition lines, thereforesuggesting the existence of second order phase transitions only.The findings from the 1/Nf expansion together with the Monte Carlo results are strongevidence that the allowed region of renormalized couplings can safely be studied at λ ≃0 aswell.5ConclusionsThe main conclusion of the numerical simulations of heavy fermions in the SU(2)L ⊗SU(2)Rsymmetric Yukawa model is that the estimates of the upper and lower bounds on renormalizedcouplings obtained in one-loop perturbation theory work well. All qualitative features of theone-loop β-functions are supported, including the fixed point in the ratio of the fermion toscalar mass µRψ/mRσ (see fig.

2). The upper limit of the renormalized couplings (at the fixedratio of gR/G2Rψ) is provided by our requirement of reflection positivity.

If this would not beimposed, the line towards the upper right corner of fig. 1 would be continued, but probablynot much beyond our values.

The reason is the appearance of the ferrimagnetic (FI) phasetransition in the region of negative scalar hopping parameters, somewhat beyond the maximalbare Yukawa coupling we consider. The existence of the FI phase at small and large valuesof the bare quartic coupling λ is implied by both numerical simulations and 1/N expansion.These conclusions are also supported in a recent numerical simulation of the same continuum“target theory” as ours, by using reduced staggered fermions [18].Although the observed qualitative behaviour is certainly consistent with the one-loop per-turbative scenario implying the triviality of the continuum limit, one has to keep in mind thatpresent simulations are done at relatively low cut-offs.

In particular, the evolution of the cou-plings towards smaller values at decreasing cut-offs should be investigated in the future. Atpresent the renormalized Yukawa coupling can have quite large values (see tables 1–2).

In thesymmetric phase it reaches more than twice the tree level unitarity bound with both scalar andfermion masses about equal to 0.5 in lattice units.AcknowledgementsWe thank Christoph Frick for useful discussions. The Monte Carlo calculations for this workhave been performed on the CRAY Y-MP/832 of HLRZ J¨ulich.7

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Figure captionsFig. 1.The data for the upper and lower bounds on gR as a function of G2Rψ together withthe perturbative estimates for scale ratios Λ/mRσ = 3 (solid curve) and Λ/mRσ = 4 (dottedcurve).

Open points denote the data for the lower bound (points c, d, g in table 1), whereas fullsymbols are data for the upper bound. The 63 · 12-lattice is represented by triangles, whereaspoints on 83 · 16 are denoted by squares.Fig.

2.The mass ratio µRψ/mRσ as a function of GRψ in comparison with one-loopperturbative estimates for mRσ = 0.75 (dotted curve), mRσ = 1 (full curve) and mRσ = 1.25(dashed curve). The solid horizontal line represents the fixed point at infinite cut-off.

Theexplanation of symbols is as in Fig. 1.9

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