Nuclear Physics B (Proe. Suppl.) 20 (1991) 617-620 617 North-Holland

CONSTRAINT EFFECTIVE POTENTIAL iN 0(4) ¢4 THEORY

Thomas Neuhaus Fakultft fLir Physik Universit3t Bielefeld D-4800 Bielefeld Federal Republic of Germany

We present a numerical determination of parameters of the constraint effective potential in the 0(4) non- linear a-model in 4 dimensions. We determine the infinite volume value of the analog of the pion decay constant F from the minimum of the effective potential, and the @-mass from the second derivative of the effective potential at the minimum. The extrapolation of these quantities to infinite volume is based on chiral perturbation theory in combination with perturbation theory in the renormalized quartic coupling. For the first time we observe the divergence of the suszeptibility of the length of the mean field proportional to the logarithm of the linear size of the lattice, which is caused by the two-pion cut of the theory, The form of the divergence determines the or-mass, or alternatively the renormalized quartic coupling, using one loop renormalized perturbation theory. The resulting values of the renormalized quartic coupling agree very well with the work of Liischer and Weisz. Using a nonlocal cluster algorithm we obtain high precision of our numerical data.

1. INTRODUCTION Though in the electroweak theory the mass of

the Higgs hoson, or equivalently the renormalized quartic coupling #R is a free parameter, triviality of the theory sets an absolute upper bound to the Higgs mass when cut-off is of the order of MH 1. The triv- iality bound was calculated during the last few years in the lattice regularized version of the theory em- ploying non-perturbative analytical 2 and numerical methods 3 and a value of - / ~ I ~ = ~ 660(40) GeV was established.

The challenge of numerical methods, which use of course finite lattices, is to give precise values of physical observables in the thermodynamic limit. The determination of the tree-level renor,~alized

~-~ requires the knowledge quartic coupling gR = 2~ of the infinite volume values of the analog of the pion decay constant F and the mass of the scalar particle. It is therefore extremeley important control the finite size effects of these quantities. For the pion decay constant F - or renormalized scalar field expectation value - it was demonstrated in Ref. 3 that this can be done successfully using the analytic results of chiral perturbation theory.

The scalar mass l~r~ is however more compli- cated to calculate. If the volume is large enough, the Higgs field will couple to a two-pion state of

massless Goldstone bosons, thus gMng rise to pro- nounced finite size effects in propagators of the scalar field. While in principle idears on the treatment of finite size effects for resonant states exist, no such attempt of a rigorous analysis has yet been made for the Higgs model.

Here we want to present a related way to cal- culate M~. The suszeptibility of the length of the mean field is equal to the inverse second derivative of the constraint effective potential at its minimum, if. the distribution is dose to gaussian. On general grounds one has the perturbative expansion

Vof? 1 l,~i- = ~-----~C: + gdcl ln(L) + e2]) (1.1)

on a L 4 lattice to O(gR) in renormalized perturba- tion theory, where Z is the wave function renormal- ization constant. The constants cl and c2 are cal- culable using chiral perturbation theory and renor- realized perturbation theory. One notices a two-pion contribution which in D~4 generates a divergence which is proportional to the logarithm of the linear size of the lattice. As already pointed out by Neu- berger 4, the strength of the divergence determines then the renormalized coupling 9R - or alternatively the renormalized massparameter MR, provided F is known from the minimum of the effective potential.

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618

The massparameter MR in turn is perturbativeley re- lated to the location of the real part of the pole of the scalar mass M~. Here we have assumed knowl- edge on the wave function renormalization constant Z, which is very close to unity in our model Z -~ 0.97 3

The work descibed here, which is closeley related to a forthcoming paper 5 was done in collaboration with M. GSckeler and K. Jansen.

1.1. THE MODEL AND SOME THEORY The lattice regularized 0(4) ¢4-theory is defined

by the action 4

zEA ~=I

where ~ is the bare hopping parameter and we

have set the bare quartic coupling to infinity. The

scalar field is a four component vector ~ with

a = 0 ,1 ,2 ,3 and is related to its analog in the con-

tinuum by the formula ~o ~ = V ~ ~. VVe study the model in the broken phase, where the infinite volume theory will posess a finite scalar field expec-

tation value E and a pion decay constant F = ~ where the quantity Z is the wave function renormal- ization constant. For a detailed discussion of the low energy behaviour of the theory and definitions of the constants ~, F and Z we refer to Ref. 3

We define the effective potential per unit of vol- ume Vcff through

exp - ( L 4 E I f ) = cW (1.3)

where c is a normalization constant and W is the probability distribution function of the mean field

1 7 , = ~ E ~ = , (1.4)

1 W(I 7 " l) = f OCexp ( -S(~))6(~ '~ - ~ Ecp,~).

(1.5) O(4)-invariance of the path integral ensures that V~ H is a function of the length of the mean field I ~-~ I alone. In the simulation on L 4 lattices we determine the expectation value

(] 7 ~ t) (1.6)

which, if the distribution is gaussian, is the minimum of the effective potential - and the suszeptibility

x = L4((I 7 , 12) - (I 7 , I)~), (1.7)

T. Neuhaus / Constraint effective potential in 0(4) ~4 theory

which in the same approximation is equal to V~H . The spectrum of the broken phase contains very

light Goldstone bosons whose Compton wave length exceeds the extension of the lattice. In this situation one expects that finite size effects are dominated by the Go;dstone excitations. In chiral perturbation theory O one finds then an expansion of finite size effects of certain quantities in ~ I with L beeing the length of the box. The expansion is valid when M~ -I << L and M~L _~ 1. We quote the results 7

According to chiral perturbation theory to order 1 O ( ~ ) X is given by

3 Z 1 X = ~ ~-~[,6'2 + ~ In(ArL)]. (1.8)

For the length of the mean field we have

N - I <I 7 , I> = E(pl + L ~ - p ~ ) . (1.9)

The quantities pl and p2 are defined as follows :

3 I~i 8~L4{f l .~_2~2_~_~In(AML) l Pl = 1 + 2F-- ~ L~ - (1.10)

3 1 ,o2 = ~[/32 + ~ In (AsL)] . (1.11)

AM and As are scale parameters which determine the logarithmic dependence of the pion mass and

on a external source, f l l and /~2 are constants, which depend on the shape of the volume only. For a hypercubic geometry, which will be considered here their values are fl l -- 0.140461 and f12 = -0.020305. We emphasize that the parameters F, E, AM and As refer to the infinite volume theory.

In in the scaling region of the 0(4) model, which is reached if Mo ~ 0.5 the renormalized coupling gn is small 2. It is then justified to apply perturbation theory in gR- In this way it ;~ possible to relate the scale parameters As and AM to MR- Exploiting the one-loop results given in Api~.E of 8 one finds

7 8~r 2 In AM = In MR -- ~ + gR--n; (1.12)

7 81r 2 In A~ = In MR -- ~ + ~9n" (1.13)

Thus extracting As from Monte Carlo data it is pos- sible to calculate MR. In addition there exists a one

T. Neuhaus / Constraint effective potential in 0(4) ¢4 theory 619

L FL F~ 4 .3211(20) .2272(15) 5 .2943(10) 2409(8) 6 .2803(6) .2462(6) 7 .2701(4) .2464(4) 8 .2634(3) .2461(3) 9 .2600(3) .2469(3) 10 .2571(2) .2469(2) 11 .2552(2) .2469(2) 12 .2533(2) .2465(2) 14 )515(2) .2467(2) 15 .2502(1) .2466(1)

Table 1: Finite size analysis

X at n = 0.320.

x FL 0.672(17) 0.9 0.917(24) 1.2 1.147(32) 1.5 1.330(33) 1.7 1.457(34) 2.0 1.527(34) 2.2 1.617(37) 2.5 1.675(42) 2.7 1.757(53) 3.0 1.828(66) 3.5 1.903(26) 3.9

of F and raw data for

loop relation between MR and the physical mass M~ which is

M~ = MR(1 + 3 ~ 2 (3~rV~-- 13)). (1.14)

Finally we improve the finite size corrections to the minimum of the effective potential - or F - by adding the contribution of a massive scalar particle with mass MR. The one loop contribution is

6F 3 1 I (1.15) F 2

where the momentum sum on the finite lattice is regularized by subtraction of its continuum value.

1.2. RESULTS The numerical simulations have been performed

with a nonlocal cluster algorithm 9 This allowed us to sample very precise data within a reasonable com- puter time. We have used lattices ranging from 44 to 164 . The accumulated statistics was always about 100 ksweeps. We discuss results at four values of the hopping parameter ~ = 0.310, 0.320, 0.330, 0.355.

In Table 1 we present a finite size analysis for the length of the mean scalar field at a value of n = 0.320. F5 = (I ~ " ] ) /~/(Z) denotes the mea- sured data points, while Foo corresponds to the infi- nite volume estimate, which was extracted for each lattice size seperately, using the formulae (1.9) to (1.15). For values of F L >__ 1.5 - 2 we observe a very precise description of the finite size effects on F as predicted by chiral perturabtion theory in junction

F 0.310 0.1580(2) 0.320 0.2466(1) 0.330 0.3051(1) 0.355 0.4069(1)

Table 2: Infinite volume values of F.

In h~ MR Mo gR

.310 7.14(58) 0.376(14) 0.387(15) 2.83(23)

.320 5.37(8) [ 0.679(4) 0.707(4) 3.800(51)

.330 4.88(7) [ 0.891(5) 0.931(6) 4.269(54)

.355 I 4.09(3) t 1.322(4) 1.396(4) 5.285(33)

Table 3: Main result of the calculation.

with renormalized perturbation theory. The infinite volume values of the pion decay constant F at the various ~ values areshown in Table 2. As an e×_ample we show in Table 1. results for the suszeptibility X as a function of L at a value ~ = 0.320. We observe a

slow increase of X as the lattice size increases, nicely indicating the presence of the two pion cut. For val- ues of F L >_ 1.5 - 2 we have done a X2-fit to the measured x-data with the form

~P'(2Z 1 3" (1.16) x = + ~ - ~ ) ,

d~ering from (1.8) by the presence of a next or- der term in the ~ expansion. This term is made necessary by the insufficient quality of the f i t to the x-data using (1.8) alone, its inclusion resulted then in a good quality fit. The form of this correction is however not yet calculated in chiral perturbation theory, therefore we left 3' as a free parameter. We checked that in the subsequent f i t this correction was at most a few percent of the leading term. The fit to X - we fixed the values of Z and F (according to Table 2 ) - determines then the scale parameter In h ~ .

Table 3 contains the fitted values of the scale parameter In h~. Using the the perturbative rela- tions (1.13) and (1.14) we determine M R , M~ and gR- The results are collected in Table 3.

2. SUMMARY AND CONCLUSION We have presented a determination of the in-

finite volume a-mass in the 0(4) ~4-theory from

620 T. Neuhaus / Constraint effective potentiM in 0(4) ¢4 theory

M~ ~ R1 R2 0.70 3.24(1) 3.43(1) 1.12 2.92(2) 3.o5(2) 1.47 2.76(2) 2.87(2)

2.00 2.64(25) 2.50 2.51(20)

2.00 12.38(9) 2.45(9) - - 3.35 2.38(21)

Table 4: Comparison of our results (left rows) to results of Liischer and Weisz (rows to the right).

the logarithmic divergence of the suszeptibility of the length of the mean scalar field. This determi- nation relies explicitely on the presence of the two pion cut of the theory. To obtain this we have used results from chiral perturbation theory in combina- tion with one loop renormalized perturbation theory in gR. While the determination assumes validity of renormalized perturbation theory at the considered values of parameters of the theory, the determina- tion of Mo needs a nonperturbative input, which is the second derivative of the constraint effective po- tential as determined by the numerical simulation.

In general chiral perturbation theo~ allows an expansion of finite size effects in terms of the variable ~-1L2 of certain quantities. Here we find, that at the considered values of F L >_ 1.5-2 chiral perturbation theory to order O(~Z-~ ) , 1 in combination with one loop reuormalized perturbation theory, describes the finite size effects of the pion decay constant F to a precision of 1 in 1000. Such high precision could only be obtained by using a nonlocal cluster update.

The values of M~ obtained here are in complete agreement with the earlier methods, which used the momentum dependence of the 2-point function in combination with a finite size scaling ansatz moti- vated by perturbation theory.

Table 4 contains a comparison of the quantities R1 = - ~ and R2 = -~ as obtained in t,his calcula- tion with the analog quantity of the work of Lilscher and Weisz ~ 2. For our data we quote statistical

v

errors while systematic errors of higher order correc- tions in ~ and gR, thou&;, presumably small, are not preciseley known. The difference inbetween the quantities /~1 and R2 is however indicative for the

magnitude of possible systematic errors. The agree- ment of both calculations still is surprising, if one conside-s the different approaches.

We could probe our method here only on the edge of the scaling region. It is a challenge to see, whether the method also works inside the scaling region, which we plan to study in the future. This will require simulations on such as large lattices as is required by the inequality F L > 1.5 - 2. Having now at hand a better definition of M~, we also want to perform a study of the regularization dependence of the Higgs mass upper hound.

ACKNOWLEDGEMENTS We appreciate discussions with J. JersJk, H.A.

Kastrup, H. Leutwyler and H. Neuberger. The com- putations were performed on the CRAY Y-MP8/32 at HLRZ Jiilich and the Convex C240 at the Univer- sity of Bielefeld.

REFERENCES 1. R. Dashen and H. Neuberger, Phys. Rev. Lett.

50 (1983) 189.7.

2. M. LLischer and P. Weisz, Phys. Lett. B212 (1988) 472; Nucl. Phys. 8318 (1989) 705.

3. For further references see : K. Jansen, J. Jers~k, H. A. Kastrup, C.B. Lang, H. Leutwyler and T. Neuhaus, preprint HLRZ JLilich 90-9, PITHA 90-1 (June 1990), accepted for publication in Nucl. Phys. B.

4. H. Neuberger, in proceedings of the LATTICE HIGGS WORKSHOP in Tallahassee, Ed. B. Berg et. al., (1988) 197.

5. M. G6ckeler, K. Jansen and T. Neuhaus, Con- straint Effective Potential and the ~-mass in the O(4)-¢4-Theory, paper in preparation.

6. P. Hasenfratz and H. Leutwyler, Bern preprint BUTP-89/28 and references therein.

7. M. G6ckeler and H. Leutwyler, Jiilich preprint HLRZ 90-54 and Bern preprint BUTP-g0/28.

8. J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142.

9. U. Wolff, Phys. Rev. Lett. 62 (1989) 361; C. Frick, K. Jansen and P. SeuferJing, Phys. Rev. Lett. 63 (1989) 2613.