Search for an upper bound of the renormalized Yukawa coupling in a lattice fermion-Higgs model

NUCLEAR P H VS I C S B

Nuclear Physics B 371 (1992) 683—712 North-Holland

_______________

Search for an upper bound of the renormalized Yukawa coupling in a lattice fermion—Higgs model Wolfgang Bock 1,2 Asit K. De 1,2 Christoph Frick Karl Jansen ~and Thomas Trappenberg 1,2

*

2,1

1lnstitut für Theoretische Physik E, RWI’HAachen, D-5100Aachen, Germany 2HLP~c/o KFA Jülich, P.O. Box 1913, D-5l7OJülich, Germany ~ UCSD, Department of Physics B-019, La Jolla, CA 92093, USA Received 11 June 1991 Accepted for publication 7 October 1991

We study the scaling laws for the fermion mass and the scalar field expectation value in the weak coupling regionof the broken phase of a lattice regularized chiral-invariant SU(2)L® SU(2)R fermion—Higgs model with bare Yukawa coupling y and Wilson—Yukawa coupling w. In particular we concentrate on the region in the vicinity of the line A, which is the line of maximal values of y + 4w on the critical surface containing the gaussian fixed point. We have not found any indication for the existence of a nontrivial fixed point on that line or anywhere else in the weak coupling region. The renormalized Yukawa coupling YR as a function of the fermionic correlation length appears to be bounded from above. The upper bound obtained from the numerical data at w = 0 is compatible with the perturbative unitarity bound. Furthermore, in the weak coupling region, including the line A, it is not possible to choose w such that the unwanted fermion doublers would be removed from the physical particle spectrum.

1. Introduction In the last three years nonperturbative investigations of Yukawa models by means of the lattice regularization method have received a lot of attention [1—3]. At the beginning these studies were mainly motivated by the following two complexes of nonperturbative questions [4]:(i) At large values of the bare Yukawa coupling y a nontrivial fixed point might exist at which an interacting continuum limit can be constructed. The existence of such a nontrivial fixed point in a 4-dimensional lattice field theory would be of great interest since at present all the known examples seem to support the fact that any nonasymptotically free field *

Supported by the Deutsche Forschungsgemeinschaft

0550-3213/92/$05.00

© 1992

—

Elsevier Science Publishers B.V. All rights reserved

684

W. Bock etal.

/

Yukawa coupling

theory in 4 dimensions is trivial. (ii) If on the other hand the model turns out to be dominated by the gaussian fixed point even at large values of the bare Yukawa coupling the question arises whether there exists an upper bound to the Yukawacoupling generated fermion mass [5,6] and whether it can be determined by means of numerical simulations similar to the upper bound which was obtained some time ago for the mass of the u-particle in the P4-theory [7—9].Furthermore it is of importance to know how much this upper bound for the u-particle mass obtained in the framework of the ~4-theory is influenced if also the Yukawa coupling of the scalar field to heavy fermions is considered. In order to find answers to these nonperturbative questions first of all a consistent lattice regularization of the standard model of electroweak interactions would be very useful. The naive lattice regularization of a chiral gauge theory such as the standard model results in a vectorlike theory because of the axial charges of the fermion doublers. Over the years there have been a few proposals for a formulation of chiral gauge theories on the lattice [10—17].In a series of recent publications we have concentrated on the global symmetry limit of the model proposed by Smit and Swift [10]. In this approach a manifestly chiral invariant Wilson—Yukawa coupling term is added to the naive lattice regularized action of a chiral SU(2)L ® SU(2)R fermion—Higgs model in the hope to remove the unwanted doubler fermions from the physical spectrum by rendering them heavy dynamically. In our SU(2)L ® SU(2)R fermion—Higgs model both the members of the fermion doublet are coupled with the same Yukawa coupling to the Higgs field. The model has three coupling parameters: the hopping parameter K for the Higgs field, the Yukawa coupling y and the Wilson—Yukawa coupling w. The phase diagram of the model with dynamical fermions was presented in a previous publication [18]. In fig. 1 we display once more this phase diagram with naive fermions (w 0) including new results obtained recently on a larger (8~) lattice (see sect. 4). The phase diagram consists of several different phases and phase regions with ferromagnetic (FM), paramagnetic (PM), antiferromagnetic (AM) and ferrimagnetic (F!) properties. Throughout this paper we shall use the following notation for the various phases and phase regions: Symmetric phases: PMW: Paramagnetic phase with weak Yukawa couplings PMS: Paramagnetic phase with strong Yukawa couplings Broken symmetry phases and their regions: FM(W): Ferromagnetic phase (weak Yukawa coupling region) FM(S): Ferromagnetic phase (strong Yukawa coupling region) AM(W): Antiferromagnetic phase (weak Yukawa coupling region) AM(S): Antiferromagnetic phase (strong Yukawa coupling region) Fl: Ferrimagnetic phase. Appropriate order parameters which allow a distinction between the various phases are defined by means of the dimensionless scalar field ~I’~:the scalar field =

W. Bock et al.

Q5:

/

Yukawa coupling

FM(W)

685

FM(S)

IC 0.0

—

-

PMW PMS —0.5

-

-

w=O A —1.0

B D:6~

-

Fl 0

-

AM(S)

1

2

3

4~

Fig. 1. The phase diagram at w = 0. The phase transition points marked by the squares were obtained on a 6~lattice, the ones obtained on the 8~lattice are represented by the diamonds.

vacuum expectation value or magnetization K ~b) and the staggered magnetization <~P~~) where ~~st (1/V)E~e~cI~ with ~ ( 1)xl+x2+x3+x4 The symbols “(~)“ =

=

and “K~~~)” introduced here are meant to be dimensionless numbers, as will be defined in sect. 2 below. The relation of K P) to the physical vacuum expectation value v R will be given in subsect. 3.1. The determination of <1) on finite lattices is discussed in sect. 4. In the FM phase (cP) > 0 and K~S~) 0 whereas in the AM phase it is vice versa 0 and K~S~) > 0. In the paramagnetic phases PMW and PMS both order parameters vanish (K P 0) whereas they are both nonzero within the Fl phase (
=

=

=

686

W. Bock et al.

/

Yukawa coupling

extends down to K = — oo or whether it ends at a large negative value of K. The AM and F! phases do not seem easily to relate to a quantum field theory and will not be considered further in this paper. Within the precision of their localization in our numerical simulation, the various phase transition lines meet for w = 0 at the two quadruple points A and B. These points have the following coordinates (KA, yA)

(KB,

=(—0.78±0.05,1.000±0.005) and

yB)_(—~7S±O.O1,1.770±0.010).

(1.1)

We cannot rule out from our numerical data that the phase structure might be even more complex in the vicinity of these points, e.g. it is not excluded that the various phase transition lines at A actually meet at two triple points which lie very close to each other and which are connected by a very short FI—PMW phase transition line (indications for such a more complex phase structure were found in a U(1) model with naive fermions [23] and in a SU(2) 0 SU(2) model with staggered fermions [24]). If the latter case is true we denote by A the triple point at which the FM(W)—PMW phase transition line ends. The phase diagram at a nonzero value of w can be approximately obtained by shifting in fig. 1 the y = 0 axis by an amount of 4w in the positive y-direction [18]. For example at w = 0.5 the y = 0 sheet will now be approximately at the position y = 4w = 2 in fig. 1 and the phase diagram at y > 0 consists then of the FM(S), PMS and AM(S) phases only. In the 3-dimensional (K, y, w) coupling-parameter space the two points A and B become two lines which we shall again label by A and B. The shift of the phase diagram according to the rule described above was confirmed by various results of numerical simulations and analytical calculations as well [18]. In previous publications [20,22,25—27] we were able to show that keeping w = 0.5 fixed the unwanted doubler fermions can be removed completely from the physical spectrum in the scaling regions of the FM(S) and PM(S) phase by giving them a mass which is larger than the cut-off. Simultaneously, the mass amF (here a denotes the lattice spacing of the hypercubic lattice, which sets the cut-off A 1/a; thus amF is the mass in lattice units and mF in physical units) of the physical fermion and the mass am w of the W-boson (if the gauge fields were included and considered perturbatively) can be made arbitrarily small by tuning y to 0 and K to K~. The vanishing of the fermion mass (in lattice units) in the limit y “~0 is guaranteed by the Golterman—Petcher theorem [28]. Though the above results are quite promising for the formulation of a chiral gauge theory on the lattice, many questions remain unanswered, e.g. it is unclear whether the renormalized Yukawa coupling vanishes logarithmically or as a power of some energy scale as the scaling region is approached in the FM(S) phase. In the first case the model

W. Bock et aL

/ Yukawa coupling

687

in the FM(S) phase would lead to a “trivial” theory whereas in the latter one the model would have practically no interaction between the fermions and scalars even at a finite value of the cut-off. It is also an interesting question whether mirror fermion states might appear dynamically in the physical particle spectrum which could spoil the chiral couplings in the model in spite of the fact that the doubler

fermions are already completely removed. These questions are presently a subject of intense discussion [2,3]. In this paper we concentrate on the FM(W)—PMW phase transition, especially on the region of the FM(W) phase in the vicinity of the line A whose points apparently seem to be natural candidates for a possible nontrivial fixed point [23,29].In the region y + 4w 1, a perturbative tree-level calculation leads to the following relations for the masses of the physical fermion and the doubler fermions ‘~

(in lattice units) amF —y
amD

(y + 2nw)(~P>,

(1.2)

where n = 1, 2, 3, 4 denotes the number of the momentum components equal to ~ in the different edges of the Brillouin zone. As the FM(W)—PMW phase transition is approached within the FM phase ((‘1~’)~ 0) the masses of the doubler fermions stay at best of the order of the electroweak scale and will therefore remain in the physical particle spectrum. Our numerical data indicate that the whole FM(W)— PMW phase transition sheet is of second order. Therefore, the masses of all the fermions in lattice units are expected to vanish on the whole FM(W)—PMW phase transition sheet. The corresponding continuum limit leads to a theory with a spectrum which, because of the presence of the doubler fermions, differs substantially from that of the standard model. In spite of this fact a study of the SU(2)L 0 SU(2)R model in the FM(W) region, in particular in the region close to the point A, is of interest for the following reasons: (a) The study of the pure ~P4-theory has shown that the gaussian fixed point at A = 0 and K = 1/8 [cf. eq. (2.2)] turns out to be the only fixed point in this theory. Even at A = + ~ the c14-theory leads in the continuum limit to a trivial field theory. At A = + and small values of y + 4w the SU(2)L ® SU(2)R fermion—Higgs model is expected to be still influenced by the gaussian fixed point at A = 0, K = 1/8, y = 0 and w = 0 leading to a trivial theory with fermions. The question now arises whether this remains also true for larger y + 4w as the line A is approached. As mentioned already above, the points on the line A are possible candidates for a nontrivial fixed point. (b) If it turns out from the numerical simulations that no nontrivial fixed point exists on the line A and the whole FM(W)—PM(W) phase transition sheet including the line A is still influenced by the gaussian fixed point, an upper bound to the renormalized Yukawa coupling exists which probably can be calculated in a similar

688

W. Bock et al.

/

Yukawa coupling

way as the upper bound for the renormalized quartic coupling in the cP4-theory. (c) If on the other hand the line A is not influenced by the gaussian fixed point, close to that line the doubler fermions might be rendered substantially heavier than the electroweak scale or might even be decoupled completely from the physical particle spectrum by tuning the bare coupling parameters towards that line. Among further interesting questions, which we do not address in this paper (see ref. [39]), are the properties of the scalar boson masses, the physical meaning of the K <0 region of the FM(W) phase, etc. The outline of the paper is as follows: The model is introduced in sect. 2. Sect. 3 deals with analytic considerations. In this section we first define the renormalized Yukawa coupling on the lattice, we present formulas for the i-loop renormalization group /3-function for our model in the continuum, give the scaling laws for the field expectation value and the fermion mass and derive a perturbative unitarity bound for the Yukawa coupling. In sect. 4 we compare the numerical data for the scalar field expectation value (c1) and the fermion mass am F in the vicinity of the point A with the gaussian scaling laws. Sects. 5—6 deal with the determination of the renormalized Yukawa coupling at w = 0 using the definition ~ R = m F/v R = (am~/KI)) x ~ In sect. 5 we describe the numerical data for the ratio am ~I~)as a function of K and y at w = 0 and give an estimate of the Goldstone boson wave function renormalization constant Z~,..In sect. 6 we first compare the numerical data for YR obtained for small y with a perturbative i-loop calculation. Finally we present and discuss the numerical data for the renormalized Yukawa coupling in the whole FM(W) region at w = 0 as a function of the fermionic correlation length. In sect. 7 we address the question whether the doublers might be removed from the physical spectrum by tuning the bare coupling parameters at nonzero values of w towards the line A. We summarize our results and conclude in sect. 8.

2. The model The model on the euclidean lattice is defined by the action SSH+SF+Sy+Sw,

(2.1)

with

=

K~ ~

Tr{’D~~+ + 4

+

~

4 Tr{cP~~i~ +~

— 1)2)

(2.2)

W. Bock et al.

SF

=

SY

s~,~

~

Y~

—

~‘X(~T~XPR

+

/

Yukawa coupling

689

(2.3)

~+~2~,~I’x),

(2.4)

~XPL)~1’X,

+

+ ~X+

4(”X+~PR+~PL)wXj}.

(2.5)

Here SH denotes the action for the Higgs field, SF the kinetic term for the fermions, S~.the Yukawa term and S~the Wilson—Yukawa term which is added in the Smit—Swift approach to the naive lattice regularized action. The hopping parameter for the scalar field is labeled by K, A is the quartic coupling, y is the “normal” Yukawa coupling and w is the Wilson—Yukawa coupling. ~L and ~R are the left- and right-handed chiral projectors. In this paper we use only the fixed length scalar field Ii~which is a 2 X 2 SU(2) matrix, i.e. IXt~PJ~= 1. The length fixing, or freezing of the radial mode, corresponds to the choice of an 4-theory infinite quartic coupling A of the scalar field. The experience from the pure I~ suggests that such a model belongs to the same universality class as models with finite quartic coupling. The fermion fields ‘I’s and ~I’~are SU(2) doublets. Both fermions in the doublet couple with the same strength y to the Higgs field. Therefore the model has an additional SU(2)R flavour symmetry. The action is invariant under the global chiral SU(2)L 0 SU(2)R transformations ~Px—~, (aLPL+QRPR)~I’X, + ~

Q~PL),

(2.6)

(2.7) (2.8)

where 11L,R E SU(2)LR. When the gauge fields are included, the SU(2)L symmetry turns into the local gauge symmetry. We have performed the numerical simulations of the above model using the Hybrid Monte Carlo (HMC) algorithm which requires (a) that the fermion determinant is positive and (b) that the fermion matrix M decomposes for the heat bath update of the pseudofermion fields into a product M = TtT. (a) Because of the pseudoreality of the SU(2) group the eigenvalues of the fermion matrix appear in complex conjugate pairs [30] and the fermion determinant is real and positive. (b) Since in our case M is not hermitian and does not decompose into a product M = TtT we have to substitute M by MtM. This amounts to squaring the

690

W. Bock et aL

/

Yukawa coupling

fermion determinant (det(M))2 = det(MtM) which means to introduce implicitly two replicas of the fermionic fields. These identical flavours in the SU(2)L 0 SU(2)R model have the same chiral couplings and we thus avoid an explicit introduction of the mirror fermions, required, e.g. in the U(1) 0 U(1) model for the same purpose. The SU(2) model, however, is also not strictly chiral because of its pseudoreality [31]. The fermion spectrum at w = 0 contains 2 X 2 x 16 = 64 fermions with a degenerate mass. The three individual factors arise from the flavour doubling required for the HMC algorithm, the SU(2) degrees of freedom and the usual fermion doubling on the hypercubic lattice respectively. For a nonzero value of w the degeneracy of the masses due to the fermion doubling on the lattice will be removed and in the weak coupling region of the FM phase the masses of the doubler fermions will split according to the eq. (1.2). In this paper we will use the following notation for the Higgs field: 3

‘~P~=u~l ~

(2.9)

i= 1

where ~io~ + E~3. =2 1 and T~,j = 1,. 3 are the usual Pauli matrices. In the 1(~r~) FM phase the components are chosen such that the magnetization is given by . . ,

KcI.)

=

Ku).

(2.10)

With the condition (2.10), eq. (2.9) describes a decomposition of the scalar field into a longitudinal component u and three transversal components irt. In the FM phase the longitudinal component is associated with the massive u or Higgs particle whereas the transversal ones are associated with the three massless Goldstone bosons ir~ ~.2 and ir3.

3. Analytic considerations First we discuss the relation between the continuum and the lattice parametrization and define the renormalized Yukawa coupling. In the second subsection we give the i-loop renormalization group /3-function for the SU(2)L 0 SU(2)R fermion—Higgs model in the continuum and discuss the behaviour of the renormalized Yukawa coupling YR in the vicinity of the gaussian fixed point. In subsect. 3.3 we give some heuristic formulae for the scaling laws of the scalar field expectation value K P) and the fermion mass amF. In subsect. 3.4 we derive a perturbative unitarity bound for the renormalized Yukawa coupling in our model.

W. Bock et al.

/

Yukawa coupling

691

3.1. LATFICE PARAMETRIZATION AND THE DEFINITION OF THE RENORMALIZED YUKAWA COUPLING

The euclidean action of the target theory in the continuum we want to investigate nonperturbatively by the lattice action in eq. (2.1) is given by m2 0

1 S0= fd4x

+

~4Tr{(~~o)t(a~~Po)} + —~--4Tr{cP~~} + ~-~-4 Tr{(I~P~) } + YO~O(~OPR+

g0

~PL)~Q}.

2

(3.1)

The fields and the coupling parameters in eq. (3.1) are related to the corresponding lattice quantities in eq. (2.1) by the following transformations:

(3.2)

3”2

~o(x)=v~~I~/a,

~P0(x)=!1’~/a

and 1—2A K

6A —8,

y

g 0=

—i-,

y0=

-~==-.

(3.3)

The latter relation shows that the bare coupling y0 is zero on the y = 0 axis and becomes infinite on the K = 0 axis. Consequently as K is lowered from K = K~(y= 0) to K = 0 along the FM(W)—PMW phase transition the bare parameter y0 increases from 0 to + oo• From expression (3.3), formally y0 becomes imaginary for negative values of K. The continuum parametrization of the bare couplings covers only a subspace of the lattice bare parameter space. This, however, does not necessarily mean that the lattice model with K <0 is unphysical, since ultimately we are interested in the renormalized quantities which are supposed to obey universal scaling laws sufficiently close to the FM(W)—PMW phase transition. Until now the Osterwalder—Schrader reflection positivity [32] which guarantees unitarity of the theory in the continuum could not be proven for the model (2.1) neither for w> 0 nor at w = 0 for K <0 [33]. Especially the negative K region appears suspect because of the antiferromagnetic scalar field coupling having no analogy in the continuum. This, of course, might be interpreted as a warning. However, in our

numerical investigation we did not find any indication for the presence of states with negative norm in the spectrum. We shall therefore assume throughout this paper that the region K <0 of the FM(W) phase region has a quantum field theoretical interpretation. (See ref. [39] for the discussion of this issue.) In order to avoid confusion stemming from the peculiar properties of the relations between the bare quantities on the lattice and in the continuum it is recommended to turn to the renormalized quantities as soon as possible. We write

692

W. Bock et aL

/

Yukawa coupling

the relations of the bare to the renormalized quantities using lattice units for expressions in the lattice parametrization and physical units for the continuum parametrization denoted by the subscript “0”. The scalar field is renormalized as ~R,x

=

~

~O,R(x)

=

~0(x)/~I~,

with

~Rx

a~O,R(x). (3.4)

=

Here Z~.is the wave function renormalization constant of the Goldstone propagator of the scalar field D in the lattice parametrization. In the FM phase we define the renormalization of the scalar field by means of the ar-pole, since there the only asymptotic state in the bosonic particle spectrum is the Goldstone boson (the u-particle is unstable and decays into Goldstones or fermions). In particular, the Goldstone propagator in momentum space is

I G3~(ak)= ((i/3V)E

3

\

~

exp(iak(x—y)))

(3.5)

/

x,yj=1

and Z~is defined by G~,.(ak)Ik2..o=

z —i,

(3.6)

ak

since the Goldstone bosons are massless. The quantity

~2

=

2~(i— cos(ak)) is

the dimensionless lattice equivalent of the momentum squared in the continuum. Since periodic boundary conditions are used for the scalar fields the possible momenta ak,h are given by ak,~= 2nir/N,~with n = 0,. . . , N,~— 1. By N4 = NT we denote the time extension, by N123 = NL the space extension of the lattice and V = N~NT. The wave function renormalization constant Z0,,~of the field .10(x) is defined in analogy to its definition in the continuum. The renormalized scalar field expectation value is obtained from the magnetization (~) in the lattice parametrization by av~=

K~)/~j~,

(3.7)

yR 246 GeV in the scaling region. An appropriate definition of the renormalized Yukawa coupling in the broken phase is the ratio

in lattice units; the physical scalar field expectation value is then

YRT~~

~

(3.8)

The relations (3.7) and (3.8) do not involve the transformations (3.2) and (3.3) between the continuum and the lattice parametrizations (eqs. (3.1) and (2.1), respectively). Therefore we can use eq. (3.7) for the calculation of avR and eq.

W. Bock et al.

/

Yukawa coupling

693

(3.8) as a definition of the renormalized Yukawa coupling not only in the perturbative regime but also in the negative K region. An approximative relation between the renormalized Yukawa coupling and a bare coupling YB ~S obtained by integrating the renormalization group /3-function dj

[YB

~

/3(p)

=

1nd.i~t

(3.9)

~

Here j~denotes the running coupling constant at momentum scale ,u, A the cut-off and ~ a suitable scale at which YR is defined, YR = j(p. = i.~o).(Here and in the following we neglect cut-off effects in /3 for p. near A.) The bare Yukawa coupling may be defined as the value of 53 at the scale p. = A, YB—Y(/~L--A).

(3.10)

For small values of y0 and g0 the /3-function can be calculated up to a given order in perturbation theory and the bare Yukawa coupling YB will not differ very much from the coupling parameter y0 in the lagrangian (3.1). At larger values the parameter y0 does not have to be close to YB any more. Then and in particular in

the negative K region, which is not covered by the continuum (m~, y0) parametrization, the bare Yukawa coupling can be defined by eq. (3.10). This so defined bare coupling YB has in general no simple relation to the bare lattice parameters K and y. The situation for large y0 and in the negative K region is similar to that in the strong Yukawa coupling region where the bare Yukawa coupling y B is also not known a priori in terms of K and j~,as discussed in ref. [1]. 3.2. PERTURBATIVE ~3-FUNCIiON AND THE RENORMALIZED YUKAWA COUPLING IN THE CONTINUUM

For the action in eq. (3.1) a i-ioop calculation [34,35] for the /3-function in the continuum leads to /3(57)

=/3~573+

2. /30 =

The quantities 130,a’

/3o,a + /3ob + /3~ + /3o,d

/3o,b~/3O,c

=

(3.11)

ND/41r

and /3od are respectively the contributions from the

four i-loop diagrams shown in fig. 2, /3o,a

+2/16ir2,

$ob

+2/16~2,

/3~=

—4/16~r2, 130,d= +4ND/16~r2. (3.12)

694

W. Bock et at

a)

/

Yukawa coupling

b)

II

-.

I,

~

c)

~

d)~j~

I

4 ‘.

,

I

-.

,

Fig. 2. The four diagrams which have to be considered for the i-loop calculation of the renormalization group p-function. The dashed lines represent the scalar field and the solid lines the fermion fields.

The quantity ND denotes the number of fermion doublets which in our case is equal to 32. In a quenched simulation the internal fermion loops are neglected. Since the contributions of the diagrams a, b and c cancel among each other, the

1-loop renormalization group /3-function vanishes in the quenched approximation. Nonzero contributions then come from higher orders in the perturbative expansion. A relation between the renormalized Yukawa coupling YR and the bare

Yukawa coupling

YB

can be obtained from eq. (3.9). Using the 1-loop formula

(3.11) for the /3-function and identifying the bare Yukawa coupling parameter y0 in the lagrangian we obtain YR~ —

2/3~y~ ln(p.0/A)

YB

with the

(3.13)

,

where we neglect deviations coming from the regions p. = O(mF, m7) as well as p. = 0(A). In the continuum limit at the FM(W)—PMW phase transition in the vicinity of the gaussian fixed point the renormalized Yukawa coupling vanishes

logarithmically as a function of the energy scale p.0. In the scaling region of the FM(W) phase we can identify p.0 either with the fermion mass mF or with the u-particle mass

m~.In

the following we will make the choice p.0

=

mF.

<~)AND amF 4-theory the scalar-field expectation value obeys the following scaling lawIninthe the 1vicinity of the gaussian fixed point [8,36] 3.3. SCALING LAWS FOR

K~>(T) tt rv1Iln(r)I~’2,

(3.14)

W. Bock et al.

/

Yukawa coupling

695

where T is an appropriate scaling variable, e.g. r = (K K~)/K~.The exponent = 1/2 can be derived from a mean field calculation and the exponent ~2 = 1/4 is obtained from a renormalized perturbation analysis [36]. In the case of the SU(2)L 0 SU(2)R fermion—Higgs model the exponent s’~can be easily derived from the mean field equations given in ref. [18]. We find —

=

1/2. We have not determined the exponent of the logarithmic correction.

For the fermion mass we expect a similar scaling law amF(’r) cx r”~Iln(r)I”~.

The perturbative relation

am F

(3.15)

y K ck) suggests a mean field exponent v

3 = 1/2 also for the fermion mass. If we substitute eqs. (3.14) and (3.15) into the relation (3.8) we find YR cx I ln(r) 2~4) for sufficiently small T (assuming a non-vanishing value of Z~. at the phase 2. transition). This relation is compatible with eq. (3.13) forInr—~0if v2—v4—1/ our numerical simulation we are only able to determine the mean field exponents reliably. The values for the exponents v 2 and i.’~ of the logarithmic corrections cannot be extracted from the numerical data for K ‘TI> and amF since the available range for the correlation lengths is restricted by the limited extension of the lattices we are using. 3.4. UNITARITY BOUND

In this subsection we derive the perturbative unitarity bound for the Yukawa coupling in our model from the unitarity condition for the J = 0 partial wave amplitude in the tree approximation. The considerations presented in this subsection closely follow ref. [37].The calculation provides an independent estimate of the value of the Yukawa coupling at which it ceases to remain weak. Our purpose is to compare the numerically obtained renormalized Yukawa coupling in the scaling regions of the lattice model with the perturbative unitarity bound in order to judge whether the renormalized couplings are weak or strong. The lagrangian density of the Yukawa interaction part in the continuum Minkowski space can be written as =

—Yo,R{~t’o,Ruo,R~Po,R + ~‘O,R1Y5’~TO,RT~’0,R}.

(3.16)

As before the subscripts “0” denote fields and couplings in the continuum. We have written the renormalized coupling YR instead of the bare coupling Yo since

our calculations are limited to the tree level. In the following (F1 F2) represents one of the ND = 32 degenerate SU(2) doublets in our model at w = 0. In our model F1 and F2 have the same mass amF. The corresponding antifermions are labeled by F1 and F2.

696

W. Bock eta!.

/

Yukawa coupling

For the determination of the unitarity bound we will consider the processes + F~(p

2) —+ ~A’(p3) +

(3.17)

F.A’(p4)

where i, j = 1, 2. The momenta of the respective particles are given by Pk’ where k = 1,. . . 4. The helicities of the involved fermions and antifermions in the initial and final state are defined by A, A and A’, A’, respectively. Only the combinations (A, A;A’, ~‘)=(+, + ; +, +),(—, ; —, —),(+, + ; —, —),(—, — ; +, +)can contribute to the J = 0 partial wave amplitude, which is given by ,

—

1d(cosO)T.

(3.18)

a0= (1/32~r)J~ The matrix

is related to the S-matrix by the formula S = 1

T

+

iT. The variable 0

is the angle between the 3-momenta p 1 and p3 in the center-of-mass frame. The upper bound for the renormalized Yukawa coupling is obtained from the relation [38] IRe(Amax(ao)) I

~

4,

(3.19)

where Amax(ao) is the eigenvalue of the matrix a0 with the largest absolute value. Let us first consider the simple elastic process +

F~(p2)—‘ ~j~(p3)

+

F1~(p4).

(3.20)

In this case the matrices T and a0 reduce to single numbers. There are contribu3-exchanges. tions to the process (3.20) only from the s-channel u and s-channel ~ Here and in the following we drop the subscripts “0” on u and ir’s. The amplitudes for these two processes are given by T.~±;++(u) =

where p

=

Ip 1

E

=

I p~I

=

I

=

I p2 I

=

I p~I = I p3°I

=

—

s4p2y2 — m~ and

I p3 I = I p4 I

)

T+,+;±±(~r =

—

s4E2y2 — m~’

(3.21)

is the magnitude of the 3-momentum and

I p4° I the energy of each particle in the center-of-mass

frame in which we have p1 +p2 = 0 and +p4 by = 0. the center-of-mass frame 2 isp3given theInsquare of the total energy the Mandelstam variable s = (p1 + p2)(FM) m~.= 0 and in the symmetric phases 2E. In the broken symmetry phase rn,,.

=

m 0. #

0. In the relativistic limit, however, the amplitudes do not depend on

the phase. In the relativistic limit s —‘ + ~ each of the amplitudes simplifies to —y~and the J = 0 partial wave amplitude for the process (3.20) is given by a0

=

—y~/8ir.

(3.22)

W. Bock et at

/

Yukawa coupling

697

The unitarity condition (3.19) then gives YR ~ 2v’~ 3.54.

(3.23)

Next we consider the processes (3.17) for the case of one fermion doublet, i.e. ND = 1. The J =0 coupled channel matrix a0 is now a 4 X 4 matrix corresponding to the 4 relevant states Fj~F~,F~F~,Fj Fj and F~F~.First we consider the processes which are diagonal in the flavour indices, e.g. i = j. In the following we present a list of the relevant diagrams, the corresponding T-matrix elements and the resulting expressions in the relativistic limit: (i) the u-exchange in the s-channel T;A,~~(u) =

AA’~A~A~x~ ~

—

—y~AA’&A~ôAx~, (3.24)

3-exchange in the s-channel (ii) the ~r S

—

=

~

~

~

(3.25)

(iii) the u-’exchange in the t-channel 4y2 =

—

—

{E sin(0/2)A3A

‘~

A’ —

mF

cos(0/2)ô~,A~}

x {E sin(0/2)~~,_~ — mF cos(0/2)8~~~) S~+OC

~

2

—

(3.26)

YR~A,—A’~,--~’,

(iv) the 7r3-exchange in the t-channel sin2(0/2)ÔA,_A,6X,_X, s~±~

T~A~X’(~3)=

~

(3.27)

The Mandelstam variable t is given by t = (p 2 = —4p2 sin2(0/2). 1 the —p3)flavour indices, e.g. i ~ j, come The contributions which are off-diagonal in from (i) the u-exchange in the s-channel

=

—

‘~AA’ôAx~A~x,~

—y~AA’6Ax6A~~~, (3.28)

698

W. Bock et at

/

Yukawa coupling

(ii) the ir3-exchange in the s-channel 4E2

~‘L~A’,~’( ~r3)

=

2

—

~

-~-~-~

(3.29)

~

(iii) the ~r1-and ir2-exchanges in the t-channel

or

~2)

=

~sin2(0/2)8A,_A,~,_~,

s~+T

(3.30)

From the above expressions (3.24)—(3.30) one can easily conclude that the elements which are off-diagonal in the flavour and/or helicity indices cancel. The coupled channel matrix a 0 for one fermion doublet (ND = 1) is then a diagonal matrix with each of the four elements on the diagonal equal to —Y~/(87r)which is the same as in eq. (3.22) leading to the same unitarity bound YR ~ 3.54 as obtained from considering the simple process (3.20). In Xthe case of can ND(* degenerate doublets the coupled a0 matrix is 4ND 4N~ and be 1) written as a direct product of the 4channel X 4 matrix for one doublet and a ND X ND matrix with all elements equal to one: The largest eigenvalue of this ND X ND matrix is ND. Consequently, the unitarity bound with ND = 32, as is the case for our model at w = 0, is given by YR~2\//ND =~/~7~~i0.63.

(3.31)

In sect. 6 we will compare this bound with the numerical data obtained in the scaling region of the FM(W) region, especially in the vicinity of the point A at w =0.

4. Numerical investigation of the scaling laws at

w

=

0

In this section we discuss our numerical results for the scalar field expectation am F in different parts of the FM(W) phase at w =0. Since there is no spontaneous symmetry breaking on a finite lattice we rotate each configuration of the scalar field I~ obtained during the Monte Carlo

value K ‘P) and the fermion mass

simulation in the 0(4)

SU(2)L 0 SU(2)R/Z(2) symmetry space so that (1/V) ~

~XLotu 1 =

(4.1)

W. Bock eta!.

/

Yukawa coupling

699

is proportional to a unit matrix. Then we define the scalar field expectation value K c1> by means of the rotated configuration, i.e. K~)1=((1/V)~c~X).

(4.2)

[9], compensates for the This procedure, used in simulations of the pure ~I~4-theory drift of the system on finite lattices through the set of degenerate ground states in a phase with broken symmetry. This drift would cause the vanishing of an observable which is not invariant with respect to the symmetry transformations

even if its expectation value in the thermodynamic limit is nonzero. It has been demonstrated that the rotation provides a very good approximation to the infinite volume quantities [42]. We have determined the fermion mass from the propagators K lPL~1~L>and <~1~’R~’R> using

the usual method which is described in greater detail in ref. [20].

In the region

K>

0 we have mostly fixed Y at several values and varied

K

in the

scaling region of the FM(W)—PMW phase transition. The y-values we have chosen for the numerical simulations are: = 0.2, 0.3, 0.4, 0.5 and 0.6. In addition we have ~‘

fixed ic at K = 0.2, 0.1 and 0.05 varying Y. For K si 0 we have performed scans at the following fixed K values: K = 0, —0.5, —0.65, —0.72 and —0.75, varying now

the value of Y in the scaling region of the FM(W)—PMW phase transition. The scans at K = —0.72 and —0.75 fall very close to the point A whose coordinates are given in eq. (1.1). For the scans at K ~ 0 a more appropriate definition of the scaling variable T in eq. (3.14) is given by ‘r = (y Y~)/Y~since at negative K the FM(W)—PMW phase transition line is almost perpendicular to the K = 0 axis. For each point we have typically performed 300—400 HMC trajectories. The fermion propagators are measured at every fifth scalar field configuration. In order to —

estimate the influence of statistics on our data we have increased the number of HMC trajectories at several points up to 1200—3000. For all data the scaling behaviour of K P) and am F is found to be consistent with eqs. (3.14) and (3.15). The mean field exponents determined from the numerical data for K 1) and am F are within the error bars consistent with the

predicted value of 1/2. In fig. 3 we display the numerical data for the fermion mass and the scalar field expectation value for the three K values which are closest to the point A. For direct comparison the results for an exemplary positive K = 0.1 are also included. Fig. 3 clearly shows that (arnF)2 and K J)2 depend linearly on Y confirming the gaussian nature of the exponents ~ and p 3~Going from positive to approaching point A we find only a change in the slopes, i.e. in the factors of proportionality in eqs. (3.14) and (3.15): for (am F) the slope increases, whereas it decreases for K 1~>2 It looks quite unlikely that the point A is a nontrivial fixed point, unless the domain of the nontrivial fixed point at A is

negative

K

unnaturally narrow so as to escape its detection even at

K

=

—

0.75 or the

700

W. Bock et al.

a)

/

Yukawa coupling

0.1__X:K=O.1 Li 0.08

-

0.06

-

(0)2

: ,c=—0.65 ,c=—0.72

~

S

: ic=—0.75

0.95 1 1.05 y 2 and (b) for @I)2 as a function of ~vfor the three values of K which are Fig. 3. Data (a) for (amF) closest to the point A (squares, diamonds and circles) and for one exemplary positive K (crosses). The data at positive K are shifted by an amount of 4 y = 0.44 in the positive y direction. For the fits only the data represented by the solid symbols were taken into account. The fits are given at negative K by the solid lines and at positive K by the dashed lines.

exponents of the logarithmic corrections are different from those at the gaussian fixed point. Some results for the fermion mass am F in the FM phase for the unquenched model were already presented in a previous publication [20]. They indicated that the lines of constant fermion mass might flow into the point A. Had this scenario been true, the point A would be probably a nontrivial fixed point. If this scenario remained valid also at a nonzero value of w it would open up the possibility to decouple the fermion doublers completely within the FM(W) region by tuning towards the line A from the weak coupling side. To settle this question we have translated all our results at w = 0 for (amF)2 (as they are shown for example in fig. 3 for three K values) by a simple interpolation into a plot of the lines of constant (amF)2 in the (K, y) coupling parameter space. These lines are drawn in fig. 4, which displays the weak coupling region of the phase diagram shown in fig. 1. From fig. 4 it can be clearly concluded that at least for w = 0 the lines of constant (amF)2 do not flow into the point A. The analysis of the scaling laws allows a very accurate determination of the phase transition points by fitting the numerical data for K P) and arnF with the Ansätze (3.14) and (3.15) for sufficiently small values of T. The obtained positions for the phase transitions on the 8~lattice are marked in fig. 1 by the diamonds. We have also investigated the scaling properties of the magnetization K 1> approaching the FM(S)—PMS phase transition within the strong coupling side of the FM phase and of the staggered magnetization K cP~>tuning K within the F! phase towards the

/

W Bock et aL 0.2

—

~.

~..

IC

B

Yukawa coupling

‘~ ~

~ B

0

0

701

It 0

~

w=0

2 at w = 0. From left to right (am F) changes from 0.1 to 0.8 in steps of 0.1.

Fig. 4. The lines of constant (amF)

Fl—FM phase transition. In both cases the critical exponents are coming out to be also consistent with 1/2. Furthermore, it is noticeable that the width of the scaling regions is quite different for Kc1) and amF. As can be seen from fig. 3, our data at the fixed K values —0.65, —0.72 and —0.75 show that the straight line behaviour of (am F) continues up to Y 1.20 which lies already beyond the Y interval in fig. 3. On the contrary, the scaling region of K CP) extends only up to Y 1.05.

5. The ratio

K 1> and the Goldstone wave function renormalization constant Z.

am F /

Before we present in sect. 6 the numerical data for the renormalized Yukawa coupling YR using the definition (3.8), we will first discuss in this section the numerical results for the ratio am F/K ~P) and the wave function renormalization constant Z~. of the Goldstone propagator. Our studies show that the ratio amF/ K can be determined quite accurately even if the statistics and the lattice volumes are moderate. On the other hand, the preliminary analysis of the Goldstone propagator presented in this paper reveals that a precise determination of the wave function renormalization constant Z,. requires a substantial increase of statistics and lattice size. In particular in the region near the point A we can give only a rough estimate of this quantity using mostly an 8~lattice and performing runs with typically 300—400 HMC trajectories per point. In order to get an impression of the finite-size effects we have performed runs at some selected points in the FM(W) phase also on 6~,6~i2,8~10,8~12,iO~and 10~12lattices. To di)

702

W. Bock et a!.

Yukawa coupling

________________________

amF (4i)

/

I••’’I’••’I’’’’

3

—

-

w=0

O:j~=—O.72

D:,~=—0.65 X:~O

::~::°

2-

1

-

-

~

1*~*4— -+++

0

0

•...I,~H 2

4

-+-— ~

6

—

•: y=O.98

j-ET1 w291 498 m296 498 lSBT

y=0.63 •: y=0.65

— +

U:

8

10 EF

Fig. 5. The ratio amF /(cP) as a function of the fermionic correlation length fF = i/(amF) for the scans at the various fixed values of K and the scan at y = 0.4. The data on the larger latticevolumes are represented by the solid symbols. From left to right these symbols correspond to following lattice volumes: (K, y) = (0, 0.63) (squares): 6~,8~1Q, io~and 8~,(K, y) = (0, 0.65) (diamonds): 6~12, 8~12, 8~10,8~,10~12and io~,(K, y) = (—0.65, 0.98) (circles): 6~12,8~12,8~and 10~12.For clarity, only the error bars larger than the size of the symbols are shown in this and further figures. The dashed lines connect the data points on the 8~lattice.

estimate the effect of the statistics we increased the number of HMC trajectories at several points in the FM(W) phase up to 1200—3000. 5.i. THE RATIO amF/(ck)

In fig. 5 we display the ratio amF/K CP) as a function of the fermionic correlation length eF = 1/(arnF) for a set of values of constant K ~ 0 and also for constant Y = 0.4. All the open symbols represent results obtained on an 8~lattice. The results represented by the solid symbols (obtained on various lattices) are discussed later in this subsection. For each scan the ratio amF/ K d’) decreases rapidly as ~ grows in the interval 0 ~ 3. For 3 the ratio arnF/Kd’) is within the error bars almost independent of ~ In this range arnF and K d) are simultaneously within their scaling regions and obey approximately the scaling laws (3.14) and (3.15). According to these relations the ratio is supposed to decrease logarithmically if the theory turns out to be trivial. Since on the lattices we have used for the simulations the available interval for the correlation length is very narrow and the error bars at the larger values of ~ are quite large the logarithmic decrease is not visible in the numerical data. For the three points (K, Y) = (0, 0.63), (0, 0.65) and (—0.65, 0.98) in particular, we have included results for various lattice volumes. These data are represented in fig. 5 by the solid symbols. If one follows the data in fig. 5 from left to right these solid symbols correspond respectively to the lattice volumes listed below:


e~

W. Bock eta!.

/

Yukawa coupling

703

amp (

4~)IYR

• amp. 2.0

—32

-~jy- Er—

-

-

.

w=0 •:yR 1.5

-

8~

——

+ 0.0

-

..-~

-

0

—0.2

—0.4

~ —0.6

‘C

—0.8

Fig. 6. The ratio amF / ( P> and the renormalized Yukawa coupling YR as a function of K for the fixed ferinionic correlation length ~F = 3.2. The data were obtained by a simple interpolation from the data shown in figs. 5 and 8.

(0, 0.63) (squares): 6~,8~10,iO~and 8~, (0, 0.65) (diamonds): 6~12,8~i2,8~i0,8~,i0~12and iOn, (K, y) = (—0.65, 0.98) (circles): 6~i2,8~i2,8~and i0~12. All these points fall nicely on the dashed lines connecting the data points for the scans at K = 0 and K = —0.65 on the 8~lattice. This shows that the finite-size effects are quite small for the quantity amF/ K d’). On the contrary, the individual quantities am F and K P) depend strongly on the lattice size. It can be seen from fig. 5 that the data points for amF/K d) on the larger lattices are spread over a large range of ~F values while simultaneously the value of the ratio amF/KcI~) remains practically unchanged. In the upper part of fig. 6 we have translated the 8~data of fig. 5 by a simple interpolation into a plot of the ratio am F/K P) (represented by the solid squares) against K for the fixed fermionic correlation length ~ = 3.2 at which the fermion mass and the scalar field expectation value are simultaneously2 within their = 0.1 in fig. scaling 4. The regions. This value of ~F corresponds to the line with (amF) plot shows that the ratio amF/Kd~) increases approximately linearly as the hopping parameter K is decreased along a line of constant fermionic correlation length. For fixed correlation lengths ~F> 3.2 we find similar results. The results represented in fig. 6 by the diamonds will be discussed in sect. 6. (K, Y) (K, y)

= =

5.2. THE GOLDSTONE WAVE FUNCTION RENORMALIZATION CONSTANT Z,,.

The definition of the Goldstone wave function renormalization constant Z.,,. is given in eq. (3.6) by the limit k2 —~ 0 of the Goldstone propagator in momentum

704

W. Bock et at

/

Yukawa coupling

space G,,.(ak) (3.5). In the pure scalar theory it turns out [40], that the inverse propagator G; 1(ak) is proportional to ak2 for all lattice momenta. Therefore Z~. can be determined numerically by simply fitting the data for G; 1(ak) to ak2/(Z 1,.).2 With increasing Yukawa coupling Y we observe deviations from the linear behaviour of the inverse propagator for larger momenta [39]. In this caseak a straightforward way to determine Z~,. is to take only the smallest momentum into account, e.g. Z~ 7.

G,,.(ak = (0, 0, 0, 2’~/NT))[2(1

—

cos(2ir/NT)J

(5.1)

for lattices with NT ~ NL [29]. Recently we could show that the momentum dependence of the scalar propagators can be described very accurately by the 1-fermion-loop to the self 1(ak) to k2 —~contribution 0 for the evaluation of energy, allowing a better extrapolation of G~ Z 77.. This will be described in more detail in a subsequent publication [39]. However, we find that the straightforward method using eq. (5.1) leads to values for Z~. which are consistent with the results of this more sophisticated analysis. Since the latter method requires additional information, as e.g. an estimation of the fermion mass in the infinite volume, we will use in the present paper exclusively the definition (5.1). Our numerical results for Z.,,. in4F~ theThis FM(W) regionhas show rapid monotonic behaviour beena confirmed also by decrease decreasing K at fixed the more with sophisticated analysis method mentioned above. To illustrate it we list in table 1 some values of Z~. which are obtained by a simple interpolation at the fixed fermionic correlation length ~F = 3.2. Furthermore, in the scans at fixed K ~ 0 we find the values of Z~. to decrease slightly as the FM(W)—PMW phase transition is approached. All in all our results indicate that the value of Z 1. keeps on decreasing as the point A is approached more and more. This brings about the interesting possibility of a vanishing Z~,. exactly at the point A. Our data at K> 0, K = 0 and K = —0.5 show clearly that Zfl. has a nonvanishing value at the FM(W)—PMW phase transition. Because of the universality we expect this to be true for the whole FM(W)—PMW phase transition line except possibly at point A. The strong decrease of Z,,. compensates the increase of the ratio am F/K P) shown in fig. 6. As can be seen from this figure the resulting values of the renormalized Yukawa coupling YR determined by means of eq. (3.8) are almost TABLE 1 The wave function renormalization constant Z,,. at several values of K for the fixed fermionic correlation length fF = 3.2. The values were determined by an interpolation ofdata which were calculated on an 8~lattice

K

Z,,.

0 0.73(6)

—0.5 0.32(6)

—0.65 0.25(7)

—0.72 0.23(6)

—0.75 0.19(11)

W. Bock eta!.

/

Yukawa coupling

705

constant as a function of K for fixed ~ We will discuss this in more detail in sect. 6. In order to get an impression of the volume dependence of Z.,,. defined by means of relation (5.1) we varied at several points in the coupling parameter space the size of the lattice from 6~to 10~12.We find that the numerical values of Z~,. decrease when the largest size of the lattice (usually NT ~ NL) is increased, i.e. when the value of the smallest lattice momentum (usually akT = 2’lr/NT) is decreased, whereas they remain almost constant when NL is varied for NT ~ NL fixed. For example when the lattice size is increased from 8~to 8~i2the value of Z0. decreases approximately by a factor 1.5 at (K, 3’) = (0, 0.65) and by a factor 2 at (—0.65, 0.98). This indicates that the estimation of Z,. with the definition (5.1) on smaller lattices, especially on 8~mainly used in the present paper, leads to (K, Y)

=

values which lie systematically above the values in the thermodynamic limit. Smaller values of the lattice momenta (i.e. larger lattices) are required to observe the asymptotic behaviour of eq. (3.6). We stress that the volume dependence of Z~. makes itself less felt in the renormalized Yukawa coupling because of the square root appearing in eq. (3.8). We find that the measured Goldstone propagator at large negative K in the FM(W) region differs substantially from a free propagator (3.6) for large momenta ak11 >> 2~/N~ [41] so that the use of eq. (3.6) requires really small momenta. This feature of the scalar propagators will also be elaborated in a future publication [39]. 6. The renormalized Yukawa coupling We present in this section the results for the renormalized Yukawa coupling in the FM(W) region at w = 0 calculated numerically by means of relation (3.8). First we compare our numerical data for positive K and small Y with the perturbative i-loop formula given in eq. (3.13) with p.0 = mF and investigate whether deviations appear. For comparison with the perturbative relation (3.13) we have plotted in fig. 7a the quantity 1/Y~obtained for K ~ 0 against 1/Y~ 2/3d ln(amF). If our results are well described by the i-loop formula in the continuum they should fall in this plot on a straight line with slope 1 and some intercept on the abscissa which depends on the special choice of the energy scale Po~The dashed line in fig. 7a has a slope of 1. Fig. 7a shows clearly that within the error bars the numerical results agree quite well with the i-loop formula where for the special choice p.0 = mF the intercept comes out to be very small. Remarkably, even the results corresponding to Yo = + m, i.e. at K = 0 (represented in fig. 7a by the circles) are in a excellent agreement with the expression (3.13). In fig. 7b we display the data for YR which were calculated in the quenched approximation at K = 0.3 1—0.4 and which were already presented in a previous —

706

W. Bock et a!.

1/yR2 10.0 7.5

Yukawa coupling

I.’’’I’’’.

I

-

unquenched

—

O:y~=oo

-

2.5

/

——

-

8

-

a) ~

b)

-

(K=0)

-

10.0

-

7.5

-

5.0

—

2.5

—

0.0

~

1/y 2—2~ 0 0ln(am~)

quenched

..,5 — ——

-

8~i6 w=o ~

Fig. 7. For comparison with perturbation theory the numerical data for the renormalized Yukawa coupling are plotted in (a) at K ~ 0 against 1/y~— 2f3~In(ainp) for the unquenched calculation and in (b) at K = 0.31—0.4 against 1/y~ for the quenched calculation. In both graphs the dashed lines

represent a slope of 1.

publication [19]. Since /3~= 0 in the quenched approximation, we have plotted these data in fig. 7b as a function of i/Y~.In this plot the dashed line represents again a slope of 1. Also here the data lies quite well along a straight line with the predicted slope. The remainder of this section will deal with the renormalized Yukawa coupling in the FM(W) region at negative K. For the fixed fermionic correlation length = 3.2 we display in fig. 6 also the data for the renormalized Yukawa coupling as a function of K. These data were obtained from an interpolation of the YR values displayed in fig. 8. The figure shows that the obtained values of YR are within the error bars practically independent of K since the increase of the ratio amF/K di> is just compensated by the decrease of the Goldstone wave function renormalization constant as described above. In fig. 8 we display our numerically obtained results for the renormalized Yukawa coupling YR in the FM(W) region as a function of the fermionic correlation length ~F = i/(amF) for various values of K we have chosen for the scans in the funnel. The results for the scan at Y = 0.4 are also included in this figure. To visualize the effect of the multiplication by ~ the reader should compare this plot with fig. 5 where we have plotted the ratio am F/K ‘P) as a function of ~ for the same scans as in fig. 8. It is clearly demonstrated by fig. 8 that for negative K the renormalized Yukawa coupling is almost independent of K. As the fermionic correlation length grows within the interval 0 <~F ~ 3 the renormalized Yukawa coupling decreases rapidly. As pointed out in sect. 5 only in the region ~F 3 the

W. Bock et a!.

YR

/

Yukawa coupling

707

_______________________________

2.0

~

1

f

O:~=—O.72 D:~=—0.65

w=0

-

-

O:,c=—0.50

X:~=O +:y= 0.40 2o~098

•: y=O.63 •: y=0.65 0.0

0

Ill 2

4

6

.1.. 8

10

Fig. 8. The renormalized Yukawa coupling YR as a function of the fermionic correlation length = 1/(amF) for the scans at the various values of K and the scan at y = 0.4. The data on the larger lattice volumes are represented by the solid symbols. From left to right these symbols correspond to following lattice volumes: (K, y) = (0, 0.63) (squares): 6~,8~10,io~and 8~,(K, 3’) = (0, 0.65) (diamonds): 6~12,8~12,8~10,8~,10~12and io~,(K, y)=(—O.65, 0.98) (circles): 6~12,8~12,8~and 10~12.The dotted curve represents the envelope (6.2) for the 1-loop formula given in eq. (3.13).

scalar field expectation K d) and the fermion mass am F are simultaneously within their scaling regions. As ~F is increased above about 3 the renormalized Yukawa coupling 3’R flattens out and becomes within the error bars approximately a constant as a function of ~ Of course, the expected logarithmic decrease of YR cannot be seen in our results because of the large error bars and the smallness of the available interval for the fermionic correlation length. It can be seen clearly from our data that in the FM(W) phase the renormalized Yukawa coupling is bounded from above. An upper bound for the renormalized

Yukawa coupling can be obtained from the condition that the mass of the heaviest particle (the u-particle here) enters its scaling region. Unfortunately the mass am,~ of the u-particle could not be determined reliably with the accumulated statistics. We estimate am, to be considerably larger than arnF and avR and hence the corresponding scaling region to be much narrower. Here we apply only the weaker condition that the scalar field vacuum expectation value is at the edge of its scaling region, that is at ~F 3 (see fig. 3). The obtained upper bound is thus presumably overestimated since in accordance with triviality the renormalized Yukawa coupling seems to decrease monotonically as the FM(W)—PMW phase transition is approached further and the scaling region for am,, is entered. From the data shown in fig. 8 we find that this upper bound for the renormalized Yukawa coupling is roughly given by YR

~ 0.80(30).

(6.1)

708

W. Bock eta!.

/

Yukawa coupling

All our results for YR for different values of fixed K and Y form roughly a universal curve as a function of ~ Interestingly, the dotted curve 1 YR

I ln(amF) I

(6.2)

which is obtained from the perturbative i-loop formula (3.13) by performing formally the limit Y~—p + matches this universal behaviour of YR quite well. The discussion of the finite-size effects of am F/K CP> and Z,,. in sect. 5 indicates that the values of YR are overestimated on small lattices. The data which were obtained at the points (K, 3’) = (0, 0.63), (0, 0.65) and (—0.65, 0.98) on various lattice volumes are represented in fig. 8 by the solid symbols. One recognizes that the data on the larger lattices lie mostly below the data on the smaller lattices. However, with the present data we are not able to perform an extrapolation to the infinite volume; this is attempted in a subsequent publication [39]. The value for the upper bound in (6.1) may be compared with the perturbative unitarity bound given by eq. (3.31). In fig. 8 the unitarity bound is represented by the horizontal dashed line. The value in eq. (6.1) is approximately 30% larger than the unitarity bound in eq. (3.31). However, both values are still compatible within the error bars given in eq. (6.1). This means that the Yukawa couplings in our model are not exceeding the perturbative unitarity limit. Taken the agreement with 1-ioop perturbation theory as illustrated in fig. 7, they might be even lower.

Furthermore it is remarkable that in fig. 8 the numerical data obtained on the larger lattice volumes lie already mostly below the perturbative unitarity bound. These results differ from the findings in ref. [29] where the numerical data for the renormalized Yukawa coupling in the symmetric phase of a model with mirror fermions were substantially larger than the perturbative unitarity bound. Note that also in ~P4-theory the upper bound for the renormalized quartic coupling determined by means of analytical and numerical calculations is significantly smaller than the perturbative unitarity bound [8,9]. The scaling laws obtained for the vacuum expectation value of the scalar field and for the fermion mass and now the results for the renormalized Yukawa coupling are completely consistent with the triviality of the theory for the whole FM(W) scaling region near the phase transition to the PMW phase. At K = —0.75 we are within our resolution practically at point A. Our results therefore demonstrate that the likelihood of point A being a nontrivial fixed point is very low.

7. The doubler fermion masses in the vicinity of the line A In this section we address the question whether in the FM(W) region the doubler fermions might be rendered sufficiently heavy by tuning the bare couplings

W. Bock eta!.

/

Yukawa coupling

‘‘‘‘I~’’’I’~’I’’’.

mD

709

I

T 1.00

O:~=—O.72

6~12, 8~

-

~:I1~I

w

>

D:,c=—0.65

-

0

0.75

-

-

0.50

-

-

x

1

0.250.00

0

2

4

6

8

1/amD Fig. 9. The ratio mD/yR as a function of the ratio l/(am~)for several values of

K.

at nonzero values of w towards the line A. In order to find an answer to this question we have calculated the mass amD of the lightest doubler fermion state at nonzero values of the Wilson—Yukawa coupling w by means of a numerical simulation. We have performed two extensive scans in the FM(W) phase using 6~12and 8~lattices. For the first scan we have kept Y = 0.1 fixed and varied w for several fixed K values: K = 0, —0.2, —0.5, —0.65 and —0.72, whereas for the second one w = 0.15 was kept fixed and now the parameter y was changed for several fixed K~K = 0, —0.5 and —0.65. The value K = —0.72 is very close to the line A. In fig. 9 we have plotted the ratio mD/vR=(amD)/Kd) X .~/2 as a function of 1/(amD). The wave function renormalization constant Z,. was again determined from the Goldstone propagator in the momentum space by means of eq. (5.1). The value of the ratio mD/DR determines how heavy the doubler fermion mass can be in comparison with the electroweak scale v R 246 GeV. We have used l/(amD) instead of 1/(amF) since for the scan at = 0.1 the values for 1/(amF) are much larger than the lattice extension in the spatial directions (amF has to vanish at ‘ = 0 [28]) and therefore may be unreliable. We have found that amD scales to zero with mean field exponents as the FM(W)—PMW phase transition sheet is approached and naturally the correlation length 1/(amD) can be taken as a reasonable measure for the distance from the phase transition. The figure shows clearly that the ratio mD/DR cannot be rendered larger than 1 independent of the value of K and the proximity to the phase transition. This means that the mass of the lightest doubler fermion state cannot be made larger than the electroweak scale. As 1/(amD) grows and the FM(W)—PMW phase transition is approached the ratio mD/DR decreases and the doubler fermion mass becomes even smaller in units of the electroweak scale. Of particular physical interest is the value of mD/DR in the region where all the particles in the physical ~‘

710

W. Bock eta!.

/

Yukawa coupling

spectrum scale, including the heaviest particle, i.e. the u-particle in our model. As we mentioned above, am,, could not be determined accurately within our statistics yet, but we estimate it to be larger than arnD. So the values of mD/DR for the largest values of i/amD, mD/DR 0.25 (see fig. 9), are to be taken as most relevant.

Furthermore it can be seen from fig. 9 that within the error bars the ratio is practically independent of K. A similar behaviour was observed in sect. 6 for the renormalized Yukawa coupling YR at w = 0. Our results make it very unlikely that the doubler fermions can be made sufficiently heavy so as to be regarded decoupled from the physical spectrum by tuning the bare coupling parameters towards the line A. Consequently, in the weak coupling region FM(W) the Smit—Swift proposal is not appropriate for the formulation of a chiral theory on the lattice. However, our previous analysis of the Smit—Swift model has shown that the doubler fermions can be removed completely from the physical particle spectrum within the strong coupling region [20,22,25].

mD/DR

8. Conclusion Analyzing the scaling behaviour of the fermion mass and the scalar field expectation value in the weak coupling region of the broken phase at w = 0, approaching the phase transition between FM(W) and PM(W) we could not find any indication for a nontrivial fixed point. Especially, this includes the point A: (i) no deviation from the gaussian scaling behaviour was observed even very close to the point A, (ii) the lines of constant (amF)2 do not flow into the point A as we suspected

before [20],

(iii) the renormalized Yukawa coupling YR is bounded from above. The renormalized Yukawa coupling from our nonperturbative calculation agrees nicely with i-loop perturbation theory. We show that in the FM(W) phase YR as a function of the fermionic correlation length ~F = i/(amF) is bounded from above. The numerical upper bound YR = 0.80(30) is compatible with the unitarity bound which we have derived for the special case of our fermion—Higgs model. Interestingly, the upper bound is in principle already saturated at K = 0 and does not change significantly if the value of K is decreased further and the point A is approached. The reason is that the increase of the ratio amF/K~) is just compensated by a dramatic decrease of the Goldstone wave function renormalization constant Z,,.. From the trend of our results with decreasing K values one may even speculate that the Goldstone wave function renormalization constant Z,. might just vanish at the point A. In the framework of the Smit—Swift approach, i.e. with w > 0, we could show that in the weak coupling region of the broken phase the doubler fermions cannot

W. Bock eta!.

/

Yukawa coupling

711

be made substantially heavier than the electroweak scale even if the bare parameters are tuned towards the line A. Consequently the unwanted doubler fermions will remain as additional fennionic particles in the spectrum in the continuum limit. However, our previous results have shown that the doubler fermions can be removed completely from the physical particle spectrum in the strong coupling region [20,25]. We thank J. Jersák and J. Smit for many helpful comments and for reading the manuscript. We have also benefitted from discussions with I. Montvay and J. Vink. The continuous support of H.A. Kastrup is gratefully acknowledged. The numerical simulations were performed on the CRAY Y-MP/832 at HLRZ Jülich and on the VP200-EX at RWTH Aachen.

References [1] [2] [3] [4]

J. Smit, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 3 J. Shigemitsu, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 515 M.F.L. Golterman, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 528 I. Montvay, in Heavy flavours and high-energy collisions in the 1—100 TeV range, ed. A. Ali and L. Cifarelli (Plenum, New York, 1989) p. 469; J. Jersák, in Higgs particle(s) — Physics issues and searches in high energy collisions, Erice, 1989, ed. A. Mi (Plenum, New York, 1990) p. 39; J. Kuti, NucI. Phys. B (Proc. Suppl.) 9 (1989) 55; A. Hasenfratz, in Proc. 1989 Theoretical Advanced Study Institute in Elementary particle physics, ed. T. DeGrand and D. Toussaint (World Scientific, Singapore, 1989); H. Neuberger, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 17 [5] M.B. Einhorn and G.J. Goldberg, Phys. Rev. Lett. 57 (1986) 2115 [6] J. Shigemitsu, Phys. Lett. B226 (1989) 364; 1.-H. Lee, J. Shigemitsu and R.E. Shiock, Nucl. Phys. B330 (1990) 225 [7] P. Hasenfratz, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 3 [8] M. Lüscher and P. Weisz, NucI. Phys. B318 (1989) 705; Phys. Lett. B212 (1988) 472 [9] A. Hasenfratz, K. Jansen, C.B. Lang, T. Neuhaus and H. Yoneyama, Phys. Lett. B199 (1987) 531; A. Hasenfratz, K. Jansen, J. Jersák, C.B. Lang, T. Neuhaus and H. Yoneyama, Nucl. Phys. B317 (1989) 81; J. Kuti, L. Lin and Y. Shen, Phys. Rev. Lett. 61(1988) 678; G. Bhanot, K. Bitar, U.M. Heller and H. Neuberger, NucI. Phys. B343 (1990) 467 [10] J. Smit, Nucl. Phys. B175 (1980) 307; J. Smit, Acta. Physica Polonica B17 (1986) 531; P.D.V. Swift, Phys. Lett. B145 (1984) 256 [11] J. Smit, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 451 [12] S. Aoki, Phys. Rev. Lett. 60 (1988) 2109; Phys. Rev. D38 (1988) 618; D40 (1989) 2729 [13] K. Funakubo and T. Kashiwa, Phys. Rev. Lett. 60 (1988) 2113; Phys. Rev. D38 (1988) 2602 [14] S. Aoki, I-H. Lee, J. Shigemitsu and R.E. Shrock, Phys. Lett. B243 (1990) 403; S. Aoki, I-H. Lee and R.E. Shrock, Nucl. Phys. B355 (1991) 383 [15] I. Montvay, Phys. Lett. B199 (1987) 89; B205 (1988) 315 [16] A. Borrelli, L. Maiani, G.C. Rossi, R. Sisto and M. Testa, Nucl. Phys. B333 (1990) 335 [17] G.T. Bodwin and E. Kovacs, Nucl. Phys. B (Proc. SuppL) 20 (1991) 546 [18] W. Bock, A.K. De, K. Jansen, J. Jersák, T. Neuhaus and J. Smit, Nuct. Phys. B344 (1990) 207 [19] W. Bock, A.K. De, K. Jansen, J. Jersák and T. Neuhaus, Phys. Lett. B231 (1989) 283

712

W. Bock eta!.

/

Yukawa coupling

[20] W. Bock and AK. Dc, Phys. Lett. B245 (1990) 207 [21] W. Bock, A.K. De and J. Smit, preprint HLRZ-91-81 [22] A.K. De, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 572 [23] A. Hasenfratz, W. Liu and T. Neuhaus, Phys. Lett. B236 (1990) 339 [24] J. Berlin, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 597 [25] W. Bock, A.K. De, K. Jansen, J. Jersák, T. Neuhaus and J. Smit, Phys. L.ett. B232 (1989) 486 [26] A.K. De, NucI. Phys. B (Proc. Suppl.) 17 (1990) 484 [27] W. Bock, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 559 [28] M.F.L. Golterman and D.N. Petcher, Phys. Lett. B225 (1989) 159; NucI. Phys. B (Proc. Suppl.) 17 (1990) 470 [29] K. Farakos, G. Koutsoumbas, L. Lin, J.P. Ma, I. Montvay and G. Münster, Nucl. Phys. B350 (1991) 474; L. Lin, I. Montvay, G. Münster and H. Wittig, DESY 90-105; NucI. Phys. B (Proc. Suppl.) 20 (1991) 601 [30] I.M. Barbour, W. Bock, C.T.H. Davies, AK. De, D. Henty, J. Smit and T. Trappenberg, NucI. Phys. B368 (1992) 390 [31] I-H. Lee and R.E. Shrock, NucI. Phys. B305 [FS23] (1988) 305 [32] K. Osterwalder and R. Schrader, Commun. Math. Phys. 31(1973) 83; 42 (1975) 281 [33] J. Smit, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 542 [34] N. Cabbibo, L. Maiani, G. Parisi and R. Petronzio, NucI. Phys. B158 (1979) 295 [35] J. Smit, unpublished notes [36] E. Brézin, J.C. Le Guillou and J. Zinn-Justin, in Phase transitions and critical phenomena, Vol. VI, ed. C. Domb and MS. Green (Academic Press, New York, 1976) [37] MS. Chanowitz, MA. Furman and I. Hinchliffe, Nuci. Phys. B153 (1979) 402 [38] W. Marciano, G. Valencia and S. Willenbrock, Phys. Rev. D40 (1989) 1725 [39] W. Bock, A.K. De, C. Frick, J. Jersák and T. Trappenberg, preprint HLRZ-91-82 [40] J. Kuti and Y. Shen, Phys. Rev. Lett. 60 (1988) 85; L. Un, J. Kuti and Y. Shen, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 26 [41] K. Jansen, NucI. Phys. B (Proc. Suppl.) 20 (1991) 564 [42] A. Hasenfratz, K. Jansen, J. Jersák, C.B. Lang, H. Leutwyler and T. Neuhaus, Z. Phys. C46 (1990) 257; A. Hasenfratz, K. Jansen, J. Jersák, H.A. Kastrup, C.B. Lang, H. Leutwyler and T. Neuhaus, Nucl. Phys. B356 (1991) 332