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Constraint effective potential and the σ-mass in the O(4) φ4-theory
Physics Letters B 273 ( 1991 ) 450-456 North-Holland
PHYSICS LETTERS B
Constraint effective potential and the o-mass in the O (4) 04-theory M. G 6 c k e l e r a b c d
a,b K.
J a n s e n c a n d T. N e u h a u s d
lnstitut J~r Theoretische Physik, R W T H Aachen, W-51 O0 Aachen, FRG H L R Z c/o KFA Jfilich. P.O. Box 1913, I4-5170 Jiilich, FRG Department of Physics 0319, University qfCalifornia at San Diego. 9500 Gihnan Drive, La Jolla, CA 92093-0319, USA Fakultiitj~rTheoretischePhysik, UniversitdtBielefeld, W-4800Bielefeld, FRG
Received 13 June 1991; revised manuscript received 23 August 1991
We present a numerical calculation of the o-mass from the constraint effective potential in the O (4) non-linear o-model in four dimensions. The o-mass is determined from the volume dependence of the susceptibility, which is calculated directly from the width of the effective potential. Furthermore, we compute the infinite volume value of the analogueof the pion decay constant F from the minimum of the effective potential. The extrapolation to infinite volume is based on chiral perturbation theory in combination with perturbation theory in the renormalized quartic coupling.We observe the divergenceof the susceptibilityof the length of the magnetization, which in four dimensions is proportional to the logarithm of the linear size of the lattice and is attributed to the pion cut of the theory. Our resulting high precision values of the renormalized quartic coupling agree very well with the work of Liischer and Weisz.
1. Introduction
Though in the electroweak theory the mass of the Higgs boson, or equivalently the quartic coupling is a free parameter, it is possible to give an upper b o u n d for its value. The reason is the triviality of the 0 4theory, which allows the standard model to be regarded as an effective theory, which has to break down when the cut-off is of the order of the Higgs mass. The upper b o u n d can be calculated by studying the scalar sector of the electroweak theory, the 0 ( 4 ) 0 4theory, alone [ 1 ], treating the gauge degrees of freedom perturbatively. Using non-perturbative analytical [2] and numerical [3,4] methods a value of m Max ~ 660 GeV was found. The challenge of numerical methods, which use of course finite lattices, is to give values of physical observables in the t h e r m o d y n a m i c limit. Therefore it is extremely important to control finite size effects. For the determination of the upper b o u n d the quantity R=
--
F
(1)
has to be calculated in the broken phase of the model.
M~ is the physical mass of the G-particle, defined by 450
the location of the real part of the pole in the scalar propagator, and F is the renormalized field expectation value or analogue of the pion decay constant. It is thus necessary to calculate M~ and F for infinite volume. For the renormalized field expectation value F it was demonstrated in ref. [ 5 ] that this can be done using results from chiral perturbation theory. Here we improve this analysis by adding higher-loop contributions of chiral perturbation theory. The determination of the renormalized o-mass is, however, more complicated. If the volume is large enough the Higgs field will couple to the massless Goldstone bosons, thus giving rise to pronounced finite size effects in propagators of the scalar field. Earlier determinations of the o-mass from the exponential decay of correlation functions relied on the assumption that the Goldstone modes couple weakly to the corresponding propagators. Here we present a more rigorous way to calculate Mo. The width of the distribution of the mean magnetization, i.e., the susceptibility, develops in four dimensions a singularity proportional to the logarithm of the linear size of the hypercubic lattice, caused by the pion cut of the theory. This volume dependence of the susceptibility may be derived from chiral perturbation theory. Us-
0370-2693/91/$ 03.50 © 1991 ElsevierSciencePublishers B.V. All rights reserved.
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ing in addition ordinary perturbation theory in the renormalized quartic coupling
M~ 2R-- 2F 2
(2)
we establish a relation between the susceptibility and M~, which allows a determination of the 6-mass. These theoretical aspects are described in the next section.
2. Theoretical aspects
2.1. The model
26 December 1991
Here three massless Goldstone bosons corresponding to the excitations in the 0 ( 4 ) directions a = i = 1, 2, 3 exist. Therefore, the correlation functions do not fall off exponentially at large distances, but with an inverse power of the distance. Specifically, the Goldstone boson two-point function ( ~0!,q~'~,) satisfies lim
4~21x-y12(~o;,4o~.) =Z(~ ij .
(7)
The quantity Z is the wave function renormalization constant and the renormalized fields are ~0~/x/Z. The constants Z, -Y"and F, which describe the low-energy behaviour of the Goldstone bosons, are related through
Z=FxflZ.
(8)
The lattice regularized 0 ( 4 ) 04-theory is defined by the action
2.2. Chiral perturbation theory
4
S=-2~c ~
Z q~q~+;,+2
.yeA It-- I
)-" ( q ) ~ q ~ - l )
2
xeA
+ Z q'~q'~-J .VE A
Z qso,
(3)
A'~ A
where x is the bare hopping parameter and 2 the bare quartic coupling. The scalar field is a four-component vector q~ ~ with a = 0 , 1, 2, 3. In (3) we have introduced an external source J which explicitly breaks the 0 ( 4 ) symmetry and is important for the application ofchiral perturbation theory. The field q) and the source J are related to their analogues in continuum normalization by
s=
J
We consider the model (3) in the limit of infinite bare quartic coupling, where 4 yeA
It--
1
x~A
4~',~4, ," = 1.
(5)
In this limit, the renormalized quartic coupling 2R reaches its maximal value for a given value of the 6mass. It is well known that in infinite volume the model is in the symmetry broken phase for ~c> x~(2) (for 2 = o e , ~c~(oe)-=~Cc=0.3045(7) [ 3 ] ) . In this phase we define ((pc~>=X,
(p',.>=0
(i=1,2,3).
(6)
In the symmetry broken phase of the ~4-theory the spectrum consists of the massive G-particle and three very light, on infinite volume massless, Goldstone bosons. Turning on the external source the Goldstone boson mass M~ becomes finite and is given in lowest order by X M~ = J ~52 •
(9/
If the Goldstone bosons are very light their Compton wave length exceeds the extension of the lattice. One may then expect that the finite size effects of the model are dominated by the Goldstone excitations. This scenario is described by chiral perturbation theory [6-8 ] in the so-called 4/-expansion [ 5 ]. It is an expansion in 1/F2L 2 with uo=ZjL 4 fixed, where L is the length of the box. It is valid when Co- M g 1 << L and M~L <~1. The partition function Y for the ~4-theory in the ¢/-expansion is given by
( zsv
~ = . I ' Y ( p , XjV) e x p p 2 k F 2 L 2 j
+ O ( ( F 2 L 2)
_3) , (10)
with Y=
2-11(u /,/
) ,
(11)
451
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3 fl, Pl = 1 + ~2 F
26 December 1991
while the susceptibility Z is related to the width of the distribution W(q5~),
L~
3 (fit~- 2fl2 -- ~ 8F-g~
ln(AML)),
(12)
Z= V(( ] ~ , 1 2 ) _ ( ]O~j ) 2 ) ,
(17)
and one finds
P2= ~(fl2 + 8--~51n(AzL)).
(13)
Here I~ is the modified Bessel function of first order, •V is a normalization factor, AM and As are scale parameters which determine the logarithmic dependence of M~ and X on j, respectively, fl~ and f12 are constants, which depend on the shape of the lattice only. For a hypercubic geometry, which will be considered in this paper, their values are fl~ = 0.140461 and f12= -0.020305. We want to emphasize that the parameters F, S, A~ and Az refer to the model at L = ~ and j = 0 .
2.3. Constraint effective potential The emphasis of the paper lies on the determination of the o-mass form the constraint effective potential. This is done for J = 0. The constraint effective potential per unit volume Uis defined by [9,10]
,2
Z = if7 2p2 + 0 -- 2 F 4
f12+ 8zt-~ln(AzL) +0
.
(18)
Hence Az can be determined from the volume dependence ofz.
2.4. Perturbation tkeotT In the scaling region of the O ( 4 ) model, which is reached if M~<0.5, the renormalized quartic coupling 2R is small [2]. It is then justified to apply renormalized perturbation theory in 2R. In this way it is possible to relate the scale parameters A±- and AM to M~. Exploiting the one-loop results given in appendix B of ref. [ 12 ] one finds
(14)
8zc2 lnA,~t=ln M R - - 7 + - - , gR
(19)
where c is a normalization constant and W is the distribution of the mean magnetization q~"= ( 1/ V) ×
8ZC2 -, In Az = In MR -- ~7 + -3gR
(20)
exp( -
VU) =cW,
f d 4 y (/)a ( x ) ,
w(~-)
= f DOexp[-S(~o)lc~(O'~-l f d4x~o'~(x)), (15) where J = 0. O f course, due to O(4)-symmetry, Was well as U depend only on the length I~"l of the mean magnetization. As is shown in ref. [ 11 ] the partition function of the ¢~4-model for J ¢ 0 determines W(~ ") and moments of the length of the magnetization can be calculated. Using the low-energy approximation of chiral perturbation theory for ~ , eq. (10), one finds for the average length of the magnetization the expression 3
([O'~L)=X(PI+~p2)+O(F-~), 452
(16)
where MR is the renormalized mass in the renormalization scheme used in ref. [ 12] and gR is the corresponding renormalized coupling, gR = M 2 / 2 F 2. The physical G-mass Mo as defined by the real part of the pole in the propagator is connected with MR by 2 2( I + M~=MR
gR ( 3 Z C X / ~ _ 1 3 ) + O ( g ~ ) ) .
(21)
Thus M . can be calculated using the known values of F, once the scale factor Az has been extracted from Monte Carlo data. Alternatively one may combine the results of chiral perturbation theory and perturbation theory in the renormalized coupling to arrive at the following expression for the second derivative of the constraint effective potential at its minimum:
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Z ( U" I m~,) - ' - - M~ [ 1 -t-gR (C 1 in L + c 2 ) ]
+O(F~ILz, g~),
(22)
where the constants c~ and c2 are given by 3
7
Cj -- 87~2, C'~=3fl2--_ --167r2 +
lnM R .
(23)
In this way, the renormalized coupling gR and the renormalized mass MR are connected with U" at the m i n i m u m [ 13,4]. We want to remark that the scale parameters As and A~/correspond to additional terms in the effective chiral lagrangian. Through those terms contributions of the scalar particle enter the finite size effects in this order of chiral perturbation theory. The form of the one-loop contribution of a scalar massive particle in one-loop renormalized perturbation theory added in ref. [ 14 ] was therefore not correct. The validity of formulae from chiral perturbation theory like (16) and (18) requires M Z ~ << L and therefore a finite distance in the hopping parameter ~c to the critical hopping parameter xc on given lattices with size L. If this condition is not met, higherorder terms have to be taken into account. Moreover, contributions due to the finiteness of M~ can no longer be neglected although they will eventually vanish like exp ( - M ~ , L ) and hence are invisible in chiral perturbation theory. Renormalized perturbation theory works only at small values of gR when Mo is sufficiently small, M~<0.5, say [2]. Otherwise at large values of M~, cutoff dependent scaling violations, e.g., in scattering amplitudes are to be expected and become sizeable in magnitude [ 15 ]. It is therefore nontrivial whether both expansions, chiral perturbation theory and renormalized perturbation theory, share an overlap region, where both expansions perform while their corresponding correction terms are small. In ref. [5] it was demonstrated that within the framework of chiral perturbation theory in the socalled ¢/-expansion region the theoretically expected functional behaviour of the field expectation value and the Goldstone propagator as a function of the external source j show very good agreement with the numerical data in the O ( 4 ) - m o d e l , if~c>0.31 on lattice sizes comparable to those used in this paper. In
26 December 1991
ref. [ 15 ] it was shown that cutoff effects amount to only a few percent in magnitude on scattering amplitudes and other quantities if M . < 1, thus indicating that deviations from scaling play no important role. It is this region of couplings K> 0.31 with F L > 2 and M~< 1 which we explore in this paper and where we faithfully assume validity of both expansions. This assumption can, however, only be justified further by the later observation of consistency of our final results with results from other sources.
3. Numerical results
The numerical simulations have been performed with the reflection cluster algorithm [16]. This allowed us to sample very precise data within a reasonable computer time. We have used hypercubic lattices of volume V = L 4 with L ranging from L = 4 to L = 16. The accumulated statistics on each lattice was always of about 103 sweeps. The statistical errors have been determined by a blocking procedure: We have divided the data into blocks of 1000 configurations and treated the averages on each block as independent measurements. We discuss results at four values of the hopping parameter K=0.310, 0.320, 0.330 and 0.355. For the analysis of our data for the length and the susceptibility of the magnetization we adopt the following scheme. We fix the wave function renormalization constant Z to a value obtained in earlier studies [ 3,5 ]. Given the renormalized mass MR, eq. (16) with the help ofeqs. (12), (13), (19) and (20) becomes a function of F alone, making it possible to calculate from each measured data point separately the infinite volume value of the renormalized field expectation value F. The influence of MR on the field expectation value is small since it enters only at O ( ( F L ) - 4 ) . Hence an educated guess, e.g., on the basis of older results, allows one already to extract good estimates for F from the larger lattices. These can then be used to obtain a precise value of MR from the susceptibility, as will be explained in more detail below. If necessary, the whole procedure can be repeated with the new MR as input. In fig. 1 we present a finite size analysis for the length of the magnetization at x = 0 . 3 2 0 . Fixing the wave function renormalization constant Z to a con453
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PHYSICS LETTERS B
FL
Table 1 The renormalized field expectation value at infinite volume.
tc=O. 3 2 I
I
26 December 1991
I
.28
+
K
F
0.310 0.320 0.330 0.355
0.1574(2) 0.2465(1) 0.3050(1) 0.4068(1)
+
.26 ++ +
~c=O. 3 2
++ x
o
x
X
o
.24
o
I
I
0
.01
2
I
.02
.03 L - 2
Fig. I. Finite size analysis of the renormalized field expectation value F at to= 0.320 for lattices of sizes 64 to 164. The inequality FL> 2 is fulfilledfor lattices with L> 8. The upright crosses mark the measured data. The triangles have been obtained by subtracting the O ( 1/F2L 2) contribution of chiral perturbation theory. We remark here that for the actual calculation of this contribution we have replaced the continuum propagators of chiral perturbation theory by lattice propagators, thus accounting for some lattice effects. The X-symbols have then been obtained by subtracting furthermore the O( 1/F4L 4) contribution of chiral perturbation theory coming from the p~-term in eq. ( 16). Finally we arrive at the circles by subtraction of the contribution proportional top2 in eq. (16). stant value of Z = 0 . 9 7 , we introduce the quantity FL = ( I0 ~ I >Z - ~/2. The measured data points correspond to the upright crosses of fig. I. Taking for MR the value obtained below from the susceptibility we get for F the results shown as circles in fig. 1. They still show a pronounced finite size dependence, demonstrating that higher orders of chiral perturbation theory would be needed to account for the finite size effects on small lattices. Only for large values o f F L > 2 we obtain a precise description of the finite size effects, using chiral perturbation theory up to two-loop. This finding is confirmed by a subsequent analysis of the scaling behaviour of the effective potential itself [ 17 ]. Its scaling form was given in a recent work of G6ckeler and Leutwyler [ i 1 ]. The infinite volume values of the renormalized field expectation value are collected in table 1. We have measured the susceptibility of the magnetization directly in our simulation. In fig. 2 we show the results for the susceptibility as a function of the logarithm of L at x = 0 . 3 2 0 . The susceptibility exhibits a significant L-dependence. As the volume in454
. . . . .
i'
'
t . . . . . . . . .
s
I '
a ln(L)
Fig. 2. The susceptibility at x=0.320 as a function of the logarithm of L. creases a slow increase of Z, indicating the presence of the pion cut, is observed. According to eq. (18), s = ( Z / F 2) 3 / 1 6 n 2 is the theoretical prediction for the slope of the linear increase proportional to the logarithm of L. We draw in fig. 2 a straight line corresponding to this increase. We find significant deviations from a linear behaviour on the considered lattice sizes, even though the data asymptotically might show a linear behaviour in In L consistent with the predicted slope. We interpret these deviations as an indication that higher orders of the ~7/-expansion have to be taken into account. Since these have not been calculated analytically yet, we have tried to incorporate the next order by replacingp2 in ( 18 ) byp2 ( 1 + 7/ F 2 L 2 ) , where 7 is a free parameter which has to be determined by a fit. With this correction included the fit to Z presented by the curve in fig. 2 provides a good description of the data. The difference between the data and the straight line reflects the magnitude of the correction induced by the term oc 7. It can be seen that on the largest lattices this correction is small. Analogous fits have been performed for all values of ~c and for values of FL>~ 2. Fixing F to the values
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quoted in table 1 the fit determines the logarithm of the scale parameter As and y. The resulting values of In Az are collected in table 2. They are used to calculate MR by means ofeq. (20). Furthermore we present in table 2 values for the physical mass Mo calculated from eq. (21) and also for gROur analysis is based on formulae from chiral perturbation theory, which were derived in the continuum. One can, however, estimate the size o f possible lattice corrections by comparing the continuum propagators with the corresponding lattice propagators. In the cases of interest to us, the differences turn out to be small, e.g., the modification of the straight line in fig. 2 due to lattice effects amounts to at most 0.01 on smaller lattices and decreases rapidly as L gets larger. Note furthermore that for the calculation of the O ( ( F L ) -2) contribution in the analysis of the field expectation value the lattice propagators have been used. At x = 0 . 3 2 we have performed an alternative calculation of the G-mass as suggested and described in ref. [4 ]. There the G-mass is calculated by measuring the two-point function for different lattice m o m e n t a and plotting it against the free propagator. We have found similarly to ref. [4] a nice linear behaviour. From the intercept with the zero-momentum axis we determined the G-mass on the finite lattices and extrapolated them to infinite volume as suggested in ref. [ 4 ], assuming a 1/L 2-behaviour. The resulting value agrees with M~ as obtained above. However, the M~values do not agree with our earlier results at tc=0.355 and x = 0.330 [3] obtained from the exponential decay of a suitable propagator. We suspect that the discrepancy results from the fact that in the fit o f the propagator in coordinate space the effects of Goldstone modes have not been taken into account appropriately.
26 December 1991
4. Summary and conclusion We have presented a determination of the infinite volume o-mass in the 0 ( 4 ) Ca-theory from the susceptibility of the length of the magnetization. This determination relies explicitly on the presence of the pion cut of the theory. The relevant formulae are derived from chiral perturbation theory in combination with one-loop renormalized perturbation theory in 2R. The necessary non-perturbative input is the second derivative of the constraint effective potential as determined by the numerical simulation. In general, chiral perturbation theory allows an expansion of finite size effects in terms of the variable 1/F2L 2. Here we find that at the considered values of FL> 2 chiral perturbation theory to order O( 1/ F4L4), in combination with one-loop renormalized perturbation theory, describes the finite size effects of the pion decay constant F to a precision of 1 in 1000. Fig. 3 contains a comparison of the quantity R=Mo/F as obtained in this calculation with the analogous quantity from the work of Lfischer and Weisz [2]. For our data we quote statistical errors while systematic errors due to higher-order corrections in 1/F2L: and 2R, though presumably small, are not precisely known. The agreement of both calculations is reinsuring if one considers the very different approaches. Finally we remark that this agree-
Ma/F I
I
I
3.,5
2.5
Table 2 Main results of the calculation. x
In Az
MR
M,,
gR
0.310 0.320 0.330 0.355
7.02(101) 5.45(13) 4.78(7) 4.09(5)
0.377(22) 0.675(7) 0.899(5) 1.322(8)
0.388(22) 0.701(7) 0.940(6) 1.396(8)
2.87(34) 3.750(85) 4.345(57) 5.287(67)
I
I
I
1
2
3
4
1/Mg
Fig. 3. Comparison of the results of Ltischer and Weisz (crosses) with our calculation (circles).
455
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m e n t s t r e n g t h e n s t h e case for o u r a s s u m p t i o n o f v a l i d i t y o f c h i r a l p e r t u r b a t i o n t h e o r y in c o m b i n a t i o n with renormalized perturbation theory at the considered values of couplings. W e c o u l d p r o b e o u r m e t h o d h e r e o n l y o n t h e edge o f t h e s c a l i n g region. It is a c h a l l e n g e to see, w h e t h e r t h e m e t h o d also w o r k s i n s i d e t h e scaling region, w h i c h we p l a n to s t u d y in t h e f u t u r e . T h i s will r e q u i r e s i m u l a t i o n s o n s u c h large l a t t i c e s as is d i c t a t e d b y t h e ine q u a l i t y F L > 2 . In light o f t h i s l i m i t a t i o n we m i g h t in t h e scaling r e g i o n o f c o u r s e try to d e s c r i b e t h e fin i t e size effects u s i n g o n l y r e n o r m a l i z e d p e r t u r b a t i o n theory.
Acknowledgement W e a p p r e c i a t e d i s c u s s i o n s w i t h J. Jersfik, H.A. K a s t r u p , H. L e u t w y l e r , H. N e u b e r g e r a n d Y. S h e n . T h e c o m p u t a t i o n s w e r e p e r f o r m e d o n t h e C R A Y YM P 8 / 3 2 at H L R Z Jfilich a n d t h e C o n v e x C 2 4 0 a t t h e U n i v e r s i t y o f Bielefeld.
References [1 ] R. Dashen and H. Neuberger, Phys. Rev. Left. 50 (1983) 189. [2] M. Liischer and P. Weisz, Phys. Lett. B 212 (1988) 472; Nucl. Phys. B 318 (1989) 705. [ 3 ] A. Hasenfratz, K. Jansen, C.B. Lang, T. Neuhaus and H. Yoneyama, Phys. Len. B 199 (1987) 531; A. Hasenfratz, K. Jansen, J. Jersfik, C.B. Lang, T. Neuhaus and H. Yoneyama, Nucl. Phys. B 317 (1989) 81; K. Jansen, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 422;
456
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C.B. Lang, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988 ) p. 158; T. Neuhaus, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 21. [4] J. Kuti, L. Lin and Y. Shen, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 397; Phys. Rev. Lett. 61 (1988) 678; in: Proc. Lattice Higgs Workshop (Tallahassee, FL) eds. B. Berg et al. (World Scientific, Singapore, 1988) p. 140; J. Kuti, L. Lin, Y. Shen and S. Meyer, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988) p. 216. [ 5 ] A. Hasenfratz, K. Jansen, J. Jers~ik, C.B. Lang, H. Leutwyler and T. Neuhaus, Z. Phys. C 46 (1990) 257; A. Hasenfratz, K. Jansen, J. Jers~ik, H.A. Kastrup, C.B. Lang, H. Leutwyler and T. Neuhaus, Nucl. Phys. B 356 (1991) 332. [6] J. Gasser and H. Leutwyler, Phys. Lett. B 184 (1987) 83; B 188 (1987) 477; Nucl. Phys. B 307 (1988) 763. [7] H. Leutwyler, Phys. Len. B 189 (1987) 197; Nucl. Phys. B (Proc. Suppl.) 4 (1988) 248. [ 8 ] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B 343 (1990) 241. [ 9 ] R. Fukuda and E. Kyriakopoulos, Nucl. Phys. B 85 ( 1975 ) 354. [ 10] L. O'Raifeartaigh, A. Wipf and H. Yoneyama, Nucl. Phys. B271 (1986)653. [ l 1 ] M. G6ckeler, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 347; M. G6ckeler and H. Leutwyler, Phys. Lett. B 253 ( 1991 ) 193; Nucl. Phys. B 350 ( 1991 ) 228. [ 12 ] J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142. [13] H. Neuberger, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988)p. 197. [ 14] T. Neuhaus, Nucl. Phys. B (Proc. Suppl.) 20 ( 1991 ) 617. [ 15 ] G. Bhanot, K. Bitar, U.M. Heller and H. Neuberger, Nucl. Phys. B 343 (1990) 467. [16] U. Wolff, Phys. Rev. Lett. 62 (1989) 361; C. Frick, K. Jansen and P. Seuferling, Phys. Rev. Lett. 63 (1989) 2613. [ 17 ] I. Dimitrovid, J. Nager, K. Jansen and T. Neuhaus, Phys. Len. B 268 (1991) 408.
PHYSICS LETTERS B
Constraint effective potential and the o-mass in the O (4) 04-theory M. G 6 c k e l e r a b c d
a,b K.
J a n s e n c a n d T. N e u h a u s d
lnstitut J~r Theoretische Physik, R W T H Aachen, W-51 O0 Aachen, FRG H L R Z c/o KFA Jfilich. P.O. Box 1913, I4-5170 Jiilich, FRG Department of Physics 0319, University qfCalifornia at San Diego. 9500 Gihnan Drive, La Jolla, CA 92093-0319, USA Fakultiitj~rTheoretischePhysik, UniversitdtBielefeld, W-4800Bielefeld, FRG
Received 13 June 1991; revised manuscript received 23 August 1991
We present a numerical calculation of the o-mass from the constraint effective potential in the O (4) non-linear o-model in four dimensions. The o-mass is determined from the volume dependence of the susceptibility, which is calculated directly from the width of the effective potential. Furthermore, we compute the infinite volume value of the analogueof the pion decay constant F from the minimum of the effective potential. The extrapolation to infinite volume is based on chiral perturbation theory in combination with perturbation theory in the renormalized quartic coupling.We observe the divergenceof the susceptibilityof the length of the magnetization, which in four dimensions is proportional to the logarithm of the linear size of the lattice and is attributed to the pion cut of the theory. Our resulting high precision values of the renormalized quartic coupling agree very well with the work of Liischer and Weisz.
1. Introduction
Though in the electroweak theory the mass of the Higgs boson, or equivalently the quartic coupling is a free parameter, it is possible to give an upper b o u n d for its value. The reason is the triviality of the 0 4theory, which allows the standard model to be regarded as an effective theory, which has to break down when the cut-off is of the order of the Higgs mass. The upper b o u n d can be calculated by studying the scalar sector of the electroweak theory, the 0 ( 4 ) 0 4theory, alone [ 1 ], treating the gauge degrees of freedom perturbatively. Using non-perturbative analytical [2] and numerical [3,4] methods a value of m Max ~ 660 GeV was found. The challenge of numerical methods, which use of course finite lattices, is to give values of physical observables in the t h e r m o d y n a m i c limit. Therefore it is extremely important to control finite size effects. For the determination of the upper b o u n d the quantity R=
--
F
(1)
has to be calculated in the broken phase of the model.
M~ is the physical mass of the G-particle, defined by 450
the location of the real part of the pole in the scalar propagator, and F is the renormalized field expectation value or analogue of the pion decay constant. It is thus necessary to calculate M~ and F for infinite volume. For the renormalized field expectation value F it was demonstrated in ref. [ 5 ] that this can be done using results from chiral perturbation theory. Here we improve this analysis by adding higher-loop contributions of chiral perturbation theory. The determination of the renormalized o-mass is, however, more complicated. If the volume is large enough the Higgs field will couple to the massless Goldstone bosons, thus giving rise to pronounced finite size effects in propagators of the scalar field. Earlier determinations of the o-mass from the exponential decay of correlation functions relied on the assumption that the Goldstone modes couple weakly to the corresponding propagators. Here we present a more rigorous way to calculate Mo. The width of the distribution of the mean magnetization, i.e., the susceptibility, develops in four dimensions a singularity proportional to the logarithm of the linear size of the hypercubic lattice, caused by the pion cut of the theory. This volume dependence of the susceptibility may be derived from chiral perturbation theory. Us-
0370-2693/91/$ 03.50 © 1991 ElsevierSciencePublishers B.V. All rights reserved.
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PHYSICS LETTERS B
ing in addition ordinary perturbation theory in the renormalized quartic coupling
M~ 2R-- 2F 2
(2)
we establish a relation between the susceptibility and M~, which allows a determination of the 6-mass. These theoretical aspects are described in the next section.
2. Theoretical aspects
2.1. The model
26 December 1991
Here three massless Goldstone bosons corresponding to the excitations in the 0 ( 4 ) directions a = i = 1, 2, 3 exist. Therefore, the correlation functions do not fall off exponentially at large distances, but with an inverse power of the distance. Specifically, the Goldstone boson two-point function ( ~0!,q~'~,) satisfies lim
4~21x-y12(~o;,4o~.) =Z(~ ij .
(7)
The quantity Z is the wave function renormalization constant and the renormalized fields are ~0~/x/Z. The constants Z, -Y"and F, which describe the low-energy behaviour of the Goldstone bosons, are related through
Z=FxflZ.
(8)
The lattice regularized 0 ( 4 ) 04-theory is defined by the action
2.2. Chiral perturbation theory
4
S=-2~c ~
Z q~q~+;,+2
.yeA It-- I
)-" ( q ) ~ q ~ - l )
2
xeA
+ Z q'~q'~-J .VE A
Z qso,
(3)
A'~ A
where x is the bare hopping parameter and 2 the bare quartic coupling. The scalar field is a four-component vector q~ ~ with a = 0 , 1, 2, 3. In (3) we have introduced an external source J which explicitly breaks the 0 ( 4 ) symmetry and is important for the application ofchiral perturbation theory. The field q) and the source J are related to their analogues in continuum normalization by
s=
J
We consider the model (3) in the limit of infinite bare quartic coupling, where 4 yeA
It--
1
x~A
4~',~4, ," = 1.
(5)
In this limit, the renormalized quartic coupling 2R reaches its maximal value for a given value of the 6mass. It is well known that in infinite volume the model is in the symmetry broken phase for ~c> x~(2) (for 2 = o e , ~c~(oe)-=~Cc=0.3045(7) [ 3 ] ) . In this phase we define ((pc~>=X,
(p',.>=0
(i=1,2,3).
(6)
In the symmetry broken phase of the ~4-theory the spectrum consists of the massive G-particle and three very light, on infinite volume massless, Goldstone bosons. Turning on the external source the Goldstone boson mass M~ becomes finite and is given in lowest order by X M~ = J ~52 •
(9/
If the Goldstone bosons are very light their Compton wave length exceeds the extension of the lattice. One may then expect that the finite size effects of the model are dominated by the Goldstone excitations. This scenario is described by chiral perturbation theory [6-8 ] in the so-called 4/-expansion [ 5 ]. It is an expansion in 1/F2L 2 with uo=ZjL 4 fixed, where L is the length of the box. It is valid when Co- M g 1 << L and M~L <~1. The partition function Y for the ~4-theory in the ¢/-expansion is given by
( zsv
~ = . I ' Y ( p , XjV) e x p p 2 k F 2 L 2 j
+ O ( ( F 2 L 2)
_3) , (10)
with Y=
2-11(u /,/
) ,
(11)
451
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3 fl, Pl = 1 + ~2 F
26 December 1991
while the susceptibility Z is related to the width of the distribution W(q5~),
L~
3 (fit~- 2fl2 -- ~ 8F-g~
ln(AML)),
(12)
Z= V(( ] ~ , 1 2 ) _ ( ]O~j ) 2 ) ,
(17)
and one finds
P2= ~(fl2 + 8--~51n(AzL)).
(13)
Here I~ is the modified Bessel function of first order, •V is a normalization factor, AM and As are scale parameters which determine the logarithmic dependence of M~ and X on j, respectively, fl~ and f12 are constants, which depend on the shape of the lattice only. For a hypercubic geometry, which will be considered in this paper, their values are fl~ = 0.140461 and f12= -0.020305. We want to emphasize that the parameters F, S, A~ and Az refer to the model at L = ~ and j = 0 .
2.3. Constraint effective potential The emphasis of the paper lies on the determination of the o-mass form the constraint effective potential. This is done for J = 0. The constraint effective potential per unit volume Uis defined by [9,10]
,2
Z = if7 2p2 + 0 -- 2 F 4
f12+ 8zt-~ln(AzL) +0
.
(18)
Hence Az can be determined from the volume dependence ofz.
2.4. Perturbation tkeotT In the scaling region of the O ( 4 ) model, which is reached if M~<0.5, the renormalized quartic coupling 2R is small [2]. It is then justified to apply renormalized perturbation theory in 2R. In this way it is possible to relate the scale parameters A±- and AM to M~. Exploiting the one-loop results given in appendix B of ref. [ 12 ] one finds
(14)
8zc2 lnA,~t=ln M R - - 7 + - - , gR
(19)
where c is a normalization constant and W is the distribution of the mean magnetization q~"= ( 1/ V) ×
8ZC2 -, In Az = In MR -- ~7 + -3gR
(20)
exp( -
VU) =cW,
f d 4 y (/)a ( x ) ,
w(~-)
= f DOexp[-S(~o)lc~(O'~-l f d4x~o'~(x)), (15) where J = 0. O f course, due to O(4)-symmetry, Was well as U depend only on the length I~"l of the mean magnetization. As is shown in ref. [ 11 ] the partition function of the ¢~4-model for J ¢ 0 determines W(~ ") and moments of the length of the magnetization can be calculated. Using the low-energy approximation of chiral perturbation theory for ~ , eq. (10), one finds for the average length of the magnetization the expression 3
([O'~L)=X(PI+~p2)+O(F-~), 452
(16)
where MR is the renormalized mass in the renormalization scheme used in ref. [ 12] and gR is the corresponding renormalized coupling, gR = M 2 / 2 F 2. The physical G-mass Mo as defined by the real part of the pole in the propagator is connected with MR by 2 2( I + M~=MR
gR ( 3 Z C X / ~ _ 1 3 ) + O ( g ~ ) ) .
(21)
Thus M . can be calculated using the known values of F, once the scale factor Az has been extracted from Monte Carlo data. Alternatively one may combine the results of chiral perturbation theory and perturbation theory in the renormalized coupling to arrive at the following expression for the second derivative of the constraint effective potential at its minimum:
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Z ( U" I m~,) - ' - - M~ [ 1 -t-gR (C 1 in L + c 2 ) ]
+O(F~ILz, g~),
(22)
where the constants c~ and c2 are given by 3
7
Cj -- 87~2, C'~=3fl2--_ --167r2 +
lnM R .
(23)
In this way, the renormalized coupling gR and the renormalized mass MR are connected with U" at the m i n i m u m [ 13,4]. We want to remark that the scale parameters As and A~/correspond to additional terms in the effective chiral lagrangian. Through those terms contributions of the scalar particle enter the finite size effects in this order of chiral perturbation theory. The form of the one-loop contribution of a scalar massive particle in one-loop renormalized perturbation theory added in ref. [ 14 ] was therefore not correct. The validity of formulae from chiral perturbation theory like (16) and (18) requires M Z ~ << L and therefore a finite distance in the hopping parameter ~c to the critical hopping parameter xc on given lattices with size L. If this condition is not met, higherorder terms have to be taken into account. Moreover, contributions due to the finiteness of M~ can no longer be neglected although they will eventually vanish like exp ( - M ~ , L ) and hence are invisible in chiral perturbation theory. Renormalized perturbation theory works only at small values of gR when Mo is sufficiently small, M~<0.5, say [2]. Otherwise at large values of M~, cutoff dependent scaling violations, e.g., in scattering amplitudes are to be expected and become sizeable in magnitude [ 15 ]. It is therefore nontrivial whether both expansions, chiral perturbation theory and renormalized perturbation theory, share an overlap region, where both expansions perform while their corresponding correction terms are small. In ref. [5] it was demonstrated that within the framework of chiral perturbation theory in the socalled ¢/-expansion region the theoretically expected functional behaviour of the field expectation value and the Goldstone propagator as a function of the external source j show very good agreement with the numerical data in the O ( 4 ) - m o d e l , if~c>0.31 on lattice sizes comparable to those used in this paper. In
26 December 1991
ref. [ 15 ] it was shown that cutoff effects amount to only a few percent in magnitude on scattering amplitudes and other quantities if M . < 1, thus indicating that deviations from scaling play no important role. It is this region of couplings K> 0.31 with F L > 2 and M~< 1 which we explore in this paper and where we faithfully assume validity of both expansions. This assumption can, however, only be justified further by the later observation of consistency of our final results with results from other sources.
3. Numerical results
The numerical simulations have been performed with the reflection cluster algorithm [16]. This allowed us to sample very precise data within a reasonable computer time. We have used hypercubic lattices of volume V = L 4 with L ranging from L = 4 to L = 16. The accumulated statistics on each lattice was always of about 103 sweeps. The statistical errors have been determined by a blocking procedure: We have divided the data into blocks of 1000 configurations and treated the averages on each block as independent measurements. We discuss results at four values of the hopping parameter K=0.310, 0.320, 0.330 and 0.355. For the analysis of our data for the length and the susceptibility of the magnetization we adopt the following scheme. We fix the wave function renormalization constant Z to a value obtained in earlier studies [ 3,5 ]. Given the renormalized mass MR, eq. (16) with the help ofeqs. (12), (13), (19) and (20) becomes a function of F alone, making it possible to calculate from each measured data point separately the infinite volume value of the renormalized field expectation value F. The influence of MR on the field expectation value is small since it enters only at O ( ( F L ) - 4 ) . Hence an educated guess, e.g., on the basis of older results, allows one already to extract good estimates for F from the larger lattices. These can then be used to obtain a precise value of MR from the susceptibility, as will be explained in more detail below. If necessary, the whole procedure can be repeated with the new MR as input. In fig. 1 we present a finite size analysis for the length of the magnetization at x = 0 . 3 2 0 . Fixing the wave function renormalization constant Z to a con453
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PHYSICS LETTERS B
FL
Table 1 The renormalized field expectation value at infinite volume.
tc=O. 3 2 I
I
26 December 1991
I
.28
+
K
F
0.310 0.320 0.330 0.355
0.1574(2) 0.2465(1) 0.3050(1) 0.4068(1)
+
.26 ++ +
~c=O. 3 2
++ x
o
x
X
o
.24
o
I
I
0
.01
2
I
.02
.03 L - 2
Fig. I. Finite size analysis of the renormalized field expectation value F at to= 0.320 for lattices of sizes 64 to 164. The inequality FL> 2 is fulfilledfor lattices with L> 8. The upright crosses mark the measured data. The triangles have been obtained by subtracting the O ( 1/F2L 2) contribution of chiral perturbation theory. We remark here that for the actual calculation of this contribution we have replaced the continuum propagators of chiral perturbation theory by lattice propagators, thus accounting for some lattice effects. The X-symbols have then been obtained by subtracting furthermore the O( 1/F4L 4) contribution of chiral perturbation theory coming from the p~-term in eq. ( 16). Finally we arrive at the circles by subtraction of the contribution proportional top2 in eq. (16). stant value of Z = 0 . 9 7 , we introduce the quantity FL = ( I0 ~ I >Z - ~/2. The measured data points correspond to the upright crosses of fig. I. Taking for MR the value obtained below from the susceptibility we get for F the results shown as circles in fig. 1. They still show a pronounced finite size dependence, demonstrating that higher orders of chiral perturbation theory would be needed to account for the finite size effects on small lattices. Only for large values o f F L > 2 we obtain a precise description of the finite size effects, using chiral perturbation theory up to two-loop. This finding is confirmed by a subsequent analysis of the scaling behaviour of the effective potential itself [ 17 ]. Its scaling form was given in a recent work of G6ckeler and Leutwyler [ i 1 ]. The infinite volume values of the renormalized field expectation value are collected in table 1. We have measured the susceptibility of the magnetization directly in our simulation. In fig. 2 we show the results for the susceptibility as a function of the logarithm of L at x = 0 . 3 2 0 . The susceptibility exhibits a significant L-dependence. As the volume in454
. . . . .
i'
'
t . . . . . . . . .
s
I '
a ln(L)
Fig. 2. The susceptibility at x=0.320 as a function of the logarithm of L. creases a slow increase of Z, indicating the presence of the pion cut, is observed. According to eq. (18), s = ( Z / F 2) 3 / 1 6 n 2 is the theoretical prediction for the slope of the linear increase proportional to the logarithm of L. We draw in fig. 2 a straight line corresponding to this increase. We find significant deviations from a linear behaviour on the considered lattice sizes, even though the data asymptotically might show a linear behaviour in In L consistent with the predicted slope. We interpret these deviations as an indication that higher orders of the ~7/-expansion have to be taken into account. Since these have not been calculated analytically yet, we have tried to incorporate the next order by replacingp2 in ( 18 ) byp2 ( 1 + 7/ F 2 L 2 ) , where 7 is a free parameter which has to be determined by a fit. With this correction included the fit to Z presented by the curve in fig. 2 provides a good description of the data. The difference between the data and the straight line reflects the magnitude of the correction induced by the term oc 7. It can be seen that on the largest lattices this correction is small. Analogous fits have been performed for all values of ~c and for values of FL>~ 2. Fixing F to the values
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quoted in table 1 the fit determines the logarithm of the scale parameter As and y. The resulting values of In Az are collected in table 2. They are used to calculate MR by means ofeq. (20). Furthermore we present in table 2 values for the physical mass Mo calculated from eq. (21) and also for gROur analysis is based on formulae from chiral perturbation theory, which were derived in the continuum. One can, however, estimate the size o f possible lattice corrections by comparing the continuum propagators with the corresponding lattice propagators. In the cases of interest to us, the differences turn out to be small, e.g., the modification of the straight line in fig. 2 due to lattice effects amounts to at most 0.01 on smaller lattices and decreases rapidly as L gets larger. Note furthermore that for the calculation of the O ( ( F L ) -2) contribution in the analysis of the field expectation value the lattice propagators have been used. At x = 0 . 3 2 we have performed an alternative calculation of the G-mass as suggested and described in ref. [4 ]. There the G-mass is calculated by measuring the two-point function for different lattice m o m e n t a and plotting it against the free propagator. We have found similarly to ref. [4] a nice linear behaviour. From the intercept with the zero-momentum axis we determined the G-mass on the finite lattices and extrapolated them to infinite volume as suggested in ref. [ 4 ], assuming a 1/L 2-behaviour. The resulting value agrees with M~ as obtained above. However, the M~values do not agree with our earlier results at tc=0.355 and x = 0.330 [3] obtained from the exponential decay of a suitable propagator. We suspect that the discrepancy results from the fact that in the fit o f the propagator in coordinate space the effects of Goldstone modes have not been taken into account appropriately.
26 December 1991
4. Summary and conclusion We have presented a determination of the infinite volume o-mass in the 0 ( 4 ) Ca-theory from the susceptibility of the length of the magnetization. This determination relies explicitly on the presence of the pion cut of the theory. The relevant formulae are derived from chiral perturbation theory in combination with one-loop renormalized perturbation theory in 2R. The necessary non-perturbative input is the second derivative of the constraint effective potential as determined by the numerical simulation. In general, chiral perturbation theory allows an expansion of finite size effects in terms of the variable 1/F2L 2. Here we find that at the considered values of FL> 2 chiral perturbation theory to order O( 1/ F4L4), in combination with one-loop renormalized perturbation theory, describes the finite size effects of the pion decay constant F to a precision of 1 in 1000. Fig. 3 contains a comparison of the quantity R=Mo/F as obtained in this calculation with the analogous quantity from the work of Lfischer and Weisz [2]. For our data we quote statistical errors while systematic errors due to higher-order corrections in 1/F2L: and 2R, though presumably small, are not precisely known. The agreement of both calculations is reinsuring if one considers the very different approaches. Finally we remark that this agree-
Ma/F I
I
I
3.,5
2.5
Table 2 Main results of the calculation. x
In Az
MR
M,,
gR
0.310 0.320 0.330 0.355
7.02(101) 5.45(13) 4.78(7) 4.09(5)
0.377(22) 0.675(7) 0.899(5) 1.322(8)
0.388(22) 0.701(7) 0.940(6) 1.396(8)
2.87(34) 3.750(85) 4.345(57) 5.287(67)
I
I
I
1
2
3
4
1/Mg
Fig. 3. Comparison of the results of Ltischer and Weisz (crosses) with our calculation (circles).
455
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m e n t s t r e n g t h e n s t h e case for o u r a s s u m p t i o n o f v a l i d i t y o f c h i r a l p e r t u r b a t i o n t h e o r y in c o m b i n a t i o n with renormalized perturbation theory at the considered values of couplings. W e c o u l d p r o b e o u r m e t h o d h e r e o n l y o n t h e edge o f t h e s c a l i n g region. It is a c h a l l e n g e to see, w h e t h e r t h e m e t h o d also w o r k s i n s i d e t h e scaling region, w h i c h we p l a n to s t u d y in t h e f u t u r e . T h i s will r e q u i r e s i m u l a t i o n s o n s u c h large l a t t i c e s as is d i c t a t e d b y t h e ine q u a l i t y F L > 2 . In light o f t h i s l i m i t a t i o n we m i g h t in t h e scaling r e g i o n o f c o u r s e try to d e s c r i b e t h e fin i t e size effects u s i n g o n l y r e n o r m a l i z e d p e r t u r b a t i o n theory.
Acknowledgement W e a p p r e c i a t e d i s c u s s i o n s w i t h J. Jersfik, H.A. K a s t r u p , H. L e u t w y l e r , H. N e u b e r g e r a n d Y. S h e n . T h e c o m p u t a t i o n s w e r e p e r f o r m e d o n t h e C R A Y YM P 8 / 3 2 at H L R Z Jfilich a n d t h e C o n v e x C 2 4 0 a t t h e U n i v e r s i t y o f Bielefeld.
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C.B. Lang, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988 ) p. 158; T. Neuhaus, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 21. [4] J. Kuti, L. Lin and Y. Shen, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 397; Phys. Rev. Lett. 61 (1988) 678; in: Proc. Lattice Higgs Workshop (Tallahassee, FL) eds. B. Berg et al. (World Scientific, Singapore, 1988) p. 140; J. Kuti, L. Lin, Y. Shen and S. Meyer, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988) p. 216. [ 5 ] A. Hasenfratz, K. Jansen, J. Jers~ik, C.B. Lang, H. Leutwyler and T. Neuhaus, Z. Phys. C 46 (1990) 257; A. Hasenfratz, K. Jansen, J. Jers~ik, H.A. Kastrup, C.B. Lang, H. Leutwyler and T. Neuhaus, Nucl. Phys. B 356 (1991) 332. [6] J. Gasser and H. Leutwyler, Phys. Lett. B 184 (1987) 83; B 188 (1987) 477; Nucl. Phys. B 307 (1988) 763. [7] H. Leutwyler, Phys. Len. B 189 (1987) 197; Nucl. Phys. B (Proc. Suppl.) 4 (1988) 248. [ 8 ] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B 343 (1990) 241. [ 9 ] R. Fukuda and E. Kyriakopoulos, Nucl. Phys. B 85 ( 1975 ) 354. [ 10] L. O'Raifeartaigh, A. Wipf and H. Yoneyama, Nucl. Phys. B271 (1986)653. [ l 1 ] M. G6ckeler, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 347; M. G6ckeler and H. Leutwyler, Phys. Lett. B 253 ( 1991 ) 193; Nucl. Phys. B 350 ( 1991 ) 228. [ 12 ] J. Gasser and H. Leutwyler, Ann. Phys. 158 (1984) 142. [13] H. Neuberger, in: Proc. Lattice Higgs Workshop (Tallahassee, FL), eds. B. Berg et al. (World Scientific, Singapore, 1988)p. 197. [ 14] T. Neuhaus, Nucl. Phys. B (Proc. Suppl.) 20 ( 1991 ) 617. [ 15 ] G. Bhanot, K. Bitar, U.M. Heller and H. Neuberger, Nucl. Phys. B 343 (1990) 467. [16] U. Wolff, Phys. Rev. Lett. 62 (1989) 361; C. Frick, K. Jansen and P. Seuferling, Phys. Rev. Lett. 63 (1989) 2613. [ 17 ] I. Dimitrovid, J. Nager, K. Jansen and T. Neuhaus, Phys. Len. B 268 (1991) 408.