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Coarse grid Yukawa interaction for staggered fermions
Nuclear Physics B349 (1991) 277-304 North-Holland
COARSF GRID YUKAWA INTERACTION FOR STAGGERED FERMIONS ifing B E N D E R
h~stitu!fiir Hochenergieph~ik~ Unirersit& Heiddberg~ Schr6derstr~e 90~ D6~O Heidelberg~ Germany Roger H O R S L E Y * and Werner W E T Z E L
hzstitut fiir Tileoretische Physik, Universit& Heidelberg, Phite~old~en~g 16~ D ~ Heiddberg Geema~'~ Fachbereich Physik, Unh'ersit& KaL~erslautern~ D6750 Kaiserslauter~ Germany
Received 22 March 1990 (Revised 23 July 1990)
We examine a coarse grid version of the Yukawa coupling between a staggered f e r r r ~ n field X and a scalar field ~ in an attempt to describe the case of four ferm~n flavors on t ~ lattice. Convincing evidence for the posithJ/ty of the fermion determinant is presented. phase diagram is analyzed by investigating the vacuum expectation value of 6 and the c~rM condensate of ~C by analytical and numerical tools. An expansion up to ~,o-loop order h performed to cover perturbative aspects and an analo~ r to the Ising mode| at large Yukawa coupling is exploited. In the numerical simulations we mainly rely on the Hybrid Monte ~ r [ o algorithm, although alternatives had to be used because the algorithm in its present form faih to be useful at large Yukawa couplings. Also the issue whether our model is renormai/zable at one loop order is discussed.
1. Introduction Recently first attempts have been made to study the properties of the Yukawa interaction by numerical simulation [1-4]. The main motivation comes from the role which the Yukawa interaction plays for the dynamical generation of fermion masses in the standard model of the electroweak interaction. Besides this and not unrelated, there is the basic question whether the Yukawa theory on its own exists as a quantum field theory, i.e. whether it shares the fate of the 4~4 model to be * Present address: HLRZ, c / o KFA, Postfach 1913, D5170 Jiilich, Germany and Institut fiir Theoretische Physik, RWTH Aachen, Germany. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
278
I. Bender et al. / Staggered fermions
trivial, or whether the combination of self-coupling and Yukawa coupling allows for a nontrivial fixpoint. There are several layers at which effects of the Yukawa interaction can be studied. On the most complex level one might consider a full fledged lattice model of the electroweak interaction, where all interactions including the gauge coupling are present (see for example refs. [4,5]). Qualitative studies at this level have become feasible through algorithmic improvements and the considerable increase of computing resources. However in this case the investigation of quantitative issues is still beyond present day reach. Lowering ambitions, one can restrict attention to the Yukawa sector alone. A case where this may be a reasonable starting point for extending the perturbative analysis of the standard model is the scenario of a heavy top quark, say with a mass of 300 GeV. Here the Yukawa coupling is of the order of 1 and nonperturbative me~hods are essential. To reduce the complexity even further one might ignore the weak isodoublet structure of the original theory altogether and investigate a Yukawa model for a fermion singlet coupled to a singlet scalar (pseudoscalar) field. In the numerical studies [2,3] the latter attitude was adopted and a scalar Yukawa coupling has been investigated. The fermions were treated as staggered. This choice is quite natural because the dynamical generation of the fermion mass is the central issue. The theory should therefore have some kind of 3'5 invariance preventing mass generation by radiative corrections. For staggered fermions enough chiral symmetry remains on the lattice, in contrast to the situation for Wilson fermions. Actually, some time ago it has been suggested [5, 6] in the context of the Yukawa interaction to modify the original Wilson prescription by letting the scalar field play the role of the constant but otherwise arbitrary r-parameter. In this way a Wilson-type single fermion model is obtained which also possesses a "shift" symmetry [7] preventing radiative mass generation as in the staggered fermion case. First results for this model have been presented in ref. [8]. Here we will not discuss this alternative further but stick to staggered fermions. Of course the usual price to pay for staggered fermions is that the number of fermion species is an integer multiple of four. It is also possible to give a staggered fermion formulation where the number of flavors is only a multiple of two [9]. It was shown in ref. [9] for gauge theories, that the fermion determinant is complex in the SU(N) case with N > 2 but real for S O ( N ) groups. When applied to the Yukawa case with a real scalar field this construction should also lead to a real determinant. Here, however, we will not consider this interesting idea further. Second, and more severely, there is the question of how precisely to write down the Yukawa coupling on a lattice. The difficulty has to do with the fact that spin and flavor degrees for staggered fermions are distributed over elementary cells. In fact, we have been motivated to look for an alternative to the coupling schemes used in refs. [1-3] because of the problems encountered when using these. The
L Bender et al. / Staggered]'ermions
279
investigations made so far were based on the following form of a lattice action:
s,= E [~(x)O x ( x )
m2
-½~,(x)oq,+--i-~(x
)2
'~ 4 +~,p(x) +y~,(x)x(x)~(x)
(i) with 4
O x ( x ) = E n,,( x )O,,x( x ) , p.=l
,7,,(x) = H ( - 1 ) * * v
O,,x(,:) = '~.( x ( x + ;,) - x ( x - ;,)) , 4
E l = Y. El /x
)
p.=l
O ~q~( x ) =q~( x + f~) + q~( x - ft) - 2 ~ ( x ) ,
(2)
and
,t,(x) =,p(x)
local coupling (model I a ) ,
4)(x) = ~ ~ q~(y)
hypercube coupling (model Ib).
(3)
yEx
Here X is the staggered fermion field, q~ is the scalar field with mass m and self-coupling A; y is the Yukawa coupling constant and the sum over x extends over all lattice sites. The two alternatives for the form of the Yukawa interaction are a strictly local coupling (model la) and a hypercube coupling (model lb) where ~b(x) is obtained by averaging ~p(y) over the hypercube associated with x. The chiral symmetry prohibiting a perturbative mass term is
x(x) --,~(x)x(x), ¢ ( x ) --, - ~ ( x ) ,
~ ( x ) --, - ~ ( x ) ~ ( x ) , E(x)
= H ( - 1) x" .
(4)
/.L
As it stands, the action in eq. (1) describing four fermion species cannot be used as a starting point for numerical simulations since the fermion determinant can become negative. The indefiniteness of the determinant is clearly visible for l~rge
_80
L Ben&r et aL / Staggered fennions
y or equivalently for a large background field 4~ and y = 0(1). In this limit the determinant for the fermion matrix
M,.,=D,.~,+y6(x)$,.,,
(5)
det M cx ~ 6 ( x ).
(6)
is given by A"
As a remedy to this problem previous investigations have resorted to a duplication of the fermion degrees of freedom. ~ adding to eq. (1) a second but otherwise identical copy of the staggered fermion field, the determinant becomes (det M ) 2, i.e. of definite sign. Although this doubling of fermion species from 4 to 8 leads to a model amenable to numerical simulation, it is not clear that by this procedure an action has been obtained which is well suited to approach continuum physics for 8 flavors. in the scaling limit the doubled model is expected to have an exact SU(8) flavor ss~nmetry. However, as it stands, only the SU(2) flavor symmetry introduced by the explicit doubling is manifest. For finite lattice spacing the eigenvalues of the fermion matrix M exhibit an SU(4) flavor symmetry only when the Yukawa coupling is zero; but this symmetry is broken already at first order in y. The use of a hypercube form for the coupling instead of a local one has been advocated as a step towards achieving a closer resemblance with the continuum action and therefore a reduction of lattice artefacts. As it turns out, model la and Ib have largely different phase diagrams at large Yukawa coupling. In the local coupling case a transition from the broken phase to an unbroken phase at large m 2 is always observed whereas for the hypercube case only the broken phase seems to exist for sufficiently large y. The argument put forward in ref. [2] to explain this observation is as follows: In the Ising limit, i.e. at large self-coupling the scalar field ~ is restricted to the values + 1. At large y the determinant of M is well approximated by eq. (6). For a local coupling we then have (det M) 2= 1 and we are back at the |sing model we started from at y = 0. For a hypercube coupling the fermion determinant will strongly suppress configurations where ~b is zcro and favor those configurations where the spins in the hypercube are aligned. This constraint could prevent the transition to a disordered phase as was indeed observed in the simulation. Because the behavior at large y is of particular interest, the absence of a clear transition line to the disordered phase appears as a quite unattractive feature of the hypercube form of coupling. Instead of following the strategy adopted by previous investigations we have looked for an alternative staggered fermion model for which the need of additional flavor doubling does not arise and where the spin and flavor symmetry of the Yukawa interaction are more transparent. Our candidate for such a 4-flavor action
I. Bender et al. / Staggered fermions
~
is presented in sect. 2. In sect. 3 the positivity of the corres~ndh~g fermion determinant is discussed. Sect. 4 gives the theoretical results ~or the p h a ~ diagram, obtained from a gap equation and an Ising ~ e l an~ ~, t ~ latter being valid at large Yukawa coupling. These results are then ~ p a r e d [n sect. 5 with the results obtained from numerical skmulations. There f o | | ~ s a d i ~ ~ in sect. 6 of whether our model is renormalizab|e on the I - l ~ level F~na|~ sect. 7 summarizes the results and draws some conclusions.
2. T h e ~ We propose, similarly to the Dirac-K[ihler approach [ 10|, to rep|ace q . (1) by a lattice action (model 11) formulated in terms of fields on a r ~ ~ ¢~~ lattice X: m2
s,,=
A
X ) O O ( X ) - ~. ( X ) o ~ ( X ) + --i-6( X ) 2 + ~ 6 ( X ) ~
+ 2y~(X)~(X)
(7)
with 4
D=~.E g.=l
[ r . a . - r 2 o . 1,
/',, --- y. ® I,
/'~ --- ys ® e s t .
(8)
where {Yu,Y~} = 2~$~,,~, (y5)2"-" 1 are ordinary 4-dimensional hermitian y matrices and --- means up to a unitary equivalence. In spite of its appearance, model II actually is very similar to model I. As far as the fermionic part of the kinetic energy is concerned, eq. (7) is directly obtained from eq. (1) by subdividing the x lattice into unit cells and identifying the degrees of freedom of X(x) within these cells with the 16 components making up the spin and flavor degrees of 0 ( X ) . In precise terms the connection is
x=2X-.~,
~ , ~ [0,1], Us
x(x)
= ~,/,,~(x),
= (£t - L ) .
(F~)
,
(9)
where the Kronecker 8 has to be understood as a condition modulo 2 between the components of .~, ~3 and the unit vector in the ~ direction.
282
I. Bender et al. / Staggered fermions
Having so far merely rewritten the kinetic energy of the fermions, the difference between eq. (7) and eq. (1) therefore only resides in the fact, that the scalar field ~k now lives on the coarse grid formed by the unit cells of the original lattice x and couples in an equal manner to all spinor and flavor degrees of $. Thus there are no other effects breaking spin and flavor symmetry than the one contained in the kinetic energy term. Due to the presence of a second derivative, the F5 term in eq. (7) is an irrelevant operator. The classical continuum limit of model II therefore represents a Yukawa model with an SU(4) flavor symmetry. In spite of the presence of both the F, and the Ff term, the free fermion model in fact also has an exact SU(4) invariance which is easily visible in the momentum space representation [11]. This "flavor" symmetry however is nonlocal in X space and is not respected by both the m o d e m and model-ll Yukawa couplings. The Ys t~q3e invariance of model I described in the introduction is also shared by model II. In the notation of eq. (7) it has the form: m
g, ~ ~ ,
n
g, ~ - g,Fs,
4~ -> - . ~
(10)
(with ~ -- Y5 ®5)- In this context we note that eq. (7) becomes a Wilson action if FA~ is replaced by 1. Indeed the F f term has the same function as the usual Wilson term; it removes the fermion zero modes at the boundary of the Brillouin zone. The above Y5 type invariance of course is lost under this replacement. The main reason why we have adopted model II as a starting point to investigate the Yukawa interaction is that it does not have the problems of model I discussed in sect. 1. The model is Ising-like at large y and A and is therefore expected to have two clearly separated phases like model Ia. Secondly the formula analogous to eq. (6) is now composed of factors where the scalar field values appear with a power of 16, making the determinant insensitive to the sign of the Higgs field. As discussed in sect. 2 all evidence indeed supports the conjecture that the fermion determinant of model II is nonnegative. The additional doubling of flavors required for model I can therefore be avoided here, i.e. we can investigate the case of four flavors.
3. The fermion determinant
We here address the question of whether the fermion matrix M for a coarse grid Higgs field has a positive semidefinite determinant also for values of y away from y = 0 or y = oo. Although we cannot give a mathematical proof, there is a large body of evidence in support of this, which stems from investigating the spectrum of eigenvalues of M. Since M is a real matrix, complex eigenvalues of M will always come in conjugate pairs and thus give a strictly positive contribution to the determinant. A
I. Bender et al. / Staggered fermions
283
negative sign can therefore only arise if there is an odd number of eigenvalues present on the negative real axis. Our means to argue against suL:h a possibility for model II is to investigate how the behavior of the eigenvalues as a function ~ y changes when going from a fine grid Higgs field (model I) to a coarse grid (model II). We start by considering the spectrum of M for y = 0. For an x l a t t ~ with Nm x N 2 x N 3 x N4 sites and periodic boundary conditions the eigenva|ues ~" are given by ~ = +i~Y'.~ sin2(q~,/2),
N./2z.,
= 0, _+ l , . . . ,
+_
[ l 2
"
(ll)
where [ . . . ] stands for taking an appropriate integer part. This spectrum is eighffold degenerate due to the above mentioned nonlocal "'flavor" SU(4) ~ m e try and a corresponding SU(2) "spin" symmetry. For the zero mode (q--, 0) the two imaginary eigenvalues approach the real axis (at 0) leading to a doubling of the degeneracy. The presence of this zero mode in fact is the reason why periodic boundary conditions are considered here. Only in this case can small values of y suffice to reach a negative result for the determinant. We next examine how the zero mode is split when the Yukawa term is switched on. Since the other eigenvalues are away from the real axis they come into play for the positivity issue only when a complex conjugate pair coalesces on the real axis, and then splits into eigenvalues moving in opposite directions along the real axis. Such nonanalytic behavior has to do with a breakdown of perturbation theory arm is beyond the level of the present discussion. To first order in y we have to diagonalize the 16 x 16 matrix formed by the matrix elements of the Yukawa term in a basis for the unperturbed eigen~ates. It is easy to see that for a generic Higgs field the degeneracy is completely lifted for model I and that odd numbers of negative eigenvalues are not the exception if the Higgs field is sufficiently fluctuating. For model II, however, only a common shift along the real axis can take place. To second order in y the eigenvalues originating from the zero modes generally start to acquire imaginary parts because the free fermion matrix is antisymmetric whereas the Yukawa term is a symmetric matrix. A recombination to conjugate complex pairs, however, is only possible if the first order splittings are small enough to allow a substantial mixing of the corresponding states. Thus in the generic case we will remain at second order with an odd number of negative eigenvalues if there was an odd number of negative eigenvalues to order y. This settles the fate of the fermion determinant for models I.
284
L Bender et al. / Staggered fennions
To find out the magnitude of the imaginary parts for model II we have again to diagonalize a 16 × 16 matrix, this time formed by the matrix elements of the second order operator of Bloch's perturbation theory [12]. It turns out that the eigenvalues of this matrix still form an eightfold degenerate pair of eigenvalues on the imaginary, axis which can collapse to the real axis only in the trivial case where the Higgs field is constant, i.e. where it corresponds to a mass term. Thus at least to this order there is no mechanism visible which can lead to an odd number of negative eigenvalues for model II. It is interesting to note that this imaginary shift is generated solely by the F~5 term, i.e. the analogue of the Wilson term for staggered fermions. Let us now consider the situation at large y. For y = ~ the spectrum of M is given by the values of the H ~:,,,, , s field with a 16-fold degeneracy in the case of model II. To leading order in 1/y we have to consider the 16 × 16 matrix formed from the kinetic part D of M in the basis of the unperturbed states. Since these states are localized, it is evident from eq. (7) that only the Ff term contributes and that the resulting matrix is given by The eigenvalues of this matrix are _i. Thus similar to the situation at order O(y2), for model II the leading correction at large y moves the eigenvalues into the imaginary direction while preserving the degeneracy to the maximal possible degree. Having discussed the behavior at small and large y the intermediate values of y remain. Although highly improbable, it is nevertheless conceivable for model II that with increasing y several pairs of conjugate eigenvalues return to the real axis, separate such that an odd number of negative eigenvalues is present for some range of y and then recombine into pairs to move back into the complex plane. Examining the spectrum and the determinant numerically for a sizable number of random Higgs fields we have never observed such a behavior. Of course these studies were limited to moderate lattice sizes (maximally 6 4) due to storage requirements. Based on the evidence presented here it seems safe to make the conjecture that the determinant of M is strictly positive for model II with a possible zero occurring only in the trivial case of a vanishing Higgs field. As our discussion has shown, the FAt term not only removes extraneous fermion modes through the Wilson mechanism, but also has the function to shift real eigenvalues of the fermion matrix into the complex plane.
F.uFf/2.
4. The phase diagram - theoretical impressions
We shall now attempt to give a qualitative understanding of the phase diagram, as a function of m 2, h and y which we would expect from the action in eq. (7). A line of second order transitions should separate the unbroken phase ( ( t ~ ) - 0) from the broken phase ((4~) :g 0). We are particularly interested in the effects due to the fermion, and so we shall consider the whole y range: 0 < y < oo. The two
I. Bender el M. / Staggered fermions
~5
tools at our disposal are a loop expansion and an Ising analogy for large y. We shall now consider both of these in some detail. 4.1. THE LOOP EXPANSION
The Feynman rules corresponding to model II after shifting the sca|ar riced an amount ~bo are given in Appendix A. We nosy iml~3~sethat 6o ~ k l e s with t ~ vacuum expectation value to one loop order. Diagrammatically this reads a x
.......
+
....
+
....
$
~
=0
(I2)
%~.=t, 8
Upon evaluation, the following gap equation is obtained: ~ - 6 4 y ,- n ( ( 2 y 6 o ) 2 ) + ~m D ( m 2 + ~~ A 62~ ) = o , ,n z + g ~ 6 ~
where D is the lattice integral D(
iy/2
)
1
F
]k~
m 2
2E~(1
(I4)
cos
over the Brillouin zone B. Note that the coefficient 64 in front of the fermion loop contribution to eq. (13) includes as a factor the number of spin and flavor degrees of the fermion. The gap equation is understood as an implicit equation for 6~ as a function of the parameters m 2, ~ and y. For the critical line we set 4'o = 0 and obtain m 2 - 6 4 y 2 D ( 0 ) + ½ A D ( m 2) = 0 (D(0)=0.155 in 4 dimensions). Solving this equation gives either m 2 ( y , A ) o r y ( m 2 , A). Eq. (15) is certainly valid in the small-y, smalI-A region. As far as
intermediate values of y are concerned it may still give a reasonable description because it also is the correct answer in the large number of flavors limit. Taken at face value it predicts that the critical m 2 increases like y2 with a constant of proportionality of 64D(0)= 10. It is feasible to extend the above calculation to two loops. In terms of diagrams the gap equation for 4'o now reads . . . . . . .
X
+
+
....
-4 I
"
+
....
J
$S P ~ % i-
.3
+
-Q ....
qs
-!-
""|
-(D'
....
)
+
.
.
+ •
.
.
.
.0
0
(16)
2'S~
L ~nder et aL / Staggered fennions
Applying Feynman rules we again obtain an implicit equation for 60, this time representing the vacuum expectation value of 6 to 2-loop order. Like in the l-loop case this equation and the corresponding equation for the critical line can be so~ved by numerical methods. We shall present the numerical results later in sect. 5 when we will ~ able to confront them ~ t h corresponding results from a Monte Carlo simulation. A quand~' closely related to the vacuum eKpectadon value of 6 is the chiral conden~te ( $ $ ) . To 2-1~ps we have diagrammatically:
.... Q
+
.... O
=Y(~$)"
(17)
At l - t ~ p order this yields ( $ 6 ) --- 32y&oD((~y60)-), where 60 is to ~ taken from the corresponding gap equation, i.e. from eq. (13). For numerical results at both 1- and 2-loop order we again refer to sect. 5. 4.2. I S | N G A N A L L Y
We now examine our mode| at large y. For y = ~ the only effect of the fermion determinant (see eq, (5)) is to add to the scalar potential the term -81n(~ 2) giving rise to a potential barrier at 6 = 0 of infinite height. Thus also for positive m 2 the potential is always of the double well ~ and we can expect a close resemblance to a lsing-like situation with preassigned values for the magnitude of 6. The simplest way of exploiting the Ising analogy is to fix the magnitude of 62 to the l~ation 6 '~ of the minimum of the potential and then identify from the kinetic term in the action the corresponding strength of the nearest neighbour coupling/3. in this way we obtain
m
)
+7'
-m2 ,
/3=62 .
(19)
Eliminating 6o we end up with the simple result: 16 m 2 =
--
/3
A -/3
--
6
(20)
for the connection at y = o0 between the parameters m 2 and A and the nearest neighbor coupling /3 of the equivalent Ising model. Inserting the critical value t3 = 0.1495 of the 4-dimensional Ising model we obtain as a prediction that model
I. Bender et al. / Staggered fermions
II at y = 0c has a second order phase transition line given by m 2=
107 - 0.02.%t.
(2I)
To test the validity of this conjecture we have performed a n u m e k ~ I s ~ u l a ~ using a Metropo|is algorithm. It turns out that eq. (21) is i ~ e d we[[ satL~f~ (say with an uncertainty of ± 1) for a|| A va|ues. Encouraged by this success, we try to extend the Is[ng a n a ~ to the s i m a ~ where the leading 1/y correction to the y = = limit of ~ [ H ~ taken into accoum. Expanding the fermion determinant in l / y we obtain the f ~ ~ ~ correction term to the acdon:
2{4
Z l S - - - wy~ x
x)
+
6(x)6{x+u)
"
-"
In the spirit of the preceding consideration ~2 has to be treated here ~ a c o , t a m equal to 6~. The term _4S therefore effectively just re~rmaI/zes the nearest neighbour coupling of the y = = lhnit by the amount: 2
|
Solving for y2 we end up with 32A + (~A + 6m2)(m2 + ~/m4+ -~A ) y2 =
(24)
96~(m 2 - 16//$ + ( f l / 6 ) A )
'
which extends eq. (20) away from y = ~ into the domain of large values of y. The prediction following from eq. (24) for the critical line (/~ ---0.1495) at sma~ A is shown in fig. 1 together with the l-loop result discussed in the prcv/ous section. Contrary to our original expectation, the branch starting at y - - - ~ shows no tendency of being in any way connected with the branch originating from y - 0. Instead it bends back towards y - o0, reaching the boundary at m 2 - ~ with the asymptotic behavior m 2--- 8/3y 2 _ y a , i.e. with a constant of proportionality I0 times smaller than for the branch originating from y = 0. These characteristic properties of the y = cc branch also hold when A is no longer small. The asymptotic behavior is exactly the same. In addition, the minimum value which y reaches as a function of m 2 is independent of ~t and is given by ( ~ / / 3 -- 19, which seems large enough for an a posteriori justification of the underlying approximation scheme. Thus at larger A the only difference is the shift in m e of the intercept at y - co which as mentioned before has been thoroughly tested by a Metropolis simulation.
L Bemh'r et al. / Staggered fennions
2SS
1 oo
symmetric phase
10
broken phase
symmetric phase
o.1 1
10
1 O0
ml
1000
Fig. 1. The critical line fo~ small A in the ~y, m2Vplane as obtained from the l-loop calculation and fronl the Ising a n a l o g .
The above results lead to the conjecture that model I| has two domains where the symmetry is unbroken, namely a domain connected to y = 0 which can be accessed by perturbation theory, and a domain connected to y = = with an |sing-like behavior. In the next section we will add further support to this by presenting results from a numerical simulation where the full dynamics of the fermion is taken into account.
5. The phase diagram - numerical impressions
The results presented in the last section now have to be backed up by Monte Carlo calculations. We have simulated eq. (7) using the Hybrid Monte Carlo (HMC) method [13]. The fermion determinant is first rewritten as a functional integral in terms of a real pseudo-boson field r/ instead of the Grassmann field X (the positivity of det M is crucial here): det M = f [ d n ]
e -'"T{MMT~-'. .
(25)
Configurations for r / c a n be obtained from gaussian distributed random fields p by applying the transformation r / = Mp. For each given r/ a molecular dynamics step
L Bender et al. / Staggered fermions
289
is now performed for the scalar field 4~ using a discretized leapfrog version of Hamiltons equations. The resulting d~ is then accepted or rejected in a Metropolis-like manner depending on the energy change 8 H accompan)ing tbe diseretization of the trajectory. This ensures that the generated Markov chain is exact, that is, in equilibrium we obtain a representative ensemble of configurat~ns for the underlying action. For the case of gauge theories, in part~ular QCD, the method has been well described and discussed in the literature, e.~ ref. [141. A potential problem with HMC is the possible low a c c e p t a ~ rate; by thks we mean that unless the step size is sufficient|y fine, many Metrol~|~ rejectm~ may ~ r before a new configuration is reached. We have ~ t | y m a ~ ~ on 8 ~ | a u ~ (i.e. coarse grid lattices of s~e 44) with a few test runs on 124 [att~es. P e ~ k : boundary conditions for the boson and spafia||y periodic but t e m ~ H y antipefiodic boundary conditions for the fermion have been used. For these | a t t ~ s~zes we have no acceptance prob|ems. The trajectory length has been set to ! w~ch seems sufficiently |arge to decorrelate successive configuratmns. For s m ~ y satisfactory results (acceptance rate ~ 70-80%) were obtained ~ i n g a ~ e p s/ze 8r = 0.1 i.e. 10 leapfrog steps to integrate the ~trajectory, wI~ie for larger y this dropped to 8 r = 0 . 0 1 . It should be noted that each | e a p f r ~ step requ~es a calculation of ~"= ( M M r ) - arl. We have used the conjugate gradient ~ t h o d for this task*. Typica| runs are over 1000-2000 trajectories per point in parameter space. The quantities measured were the vacuum expectation value (d~) and the chiral condensate (~6(X))- = ½E~ ~ x ( ~ ' X ( x ) ) . ~ where E ~ x denotes t ~ g the sum over positions inside a unit cell. A stochastic estimator for the ch~a[ condensate proved satisfactory [ 14]. We take
Y'~(~X)-(TrM-~)6=(~xp'(x)~'(x))
(26)
6.p"
x
where p' is a real Gaussian random field like p and g" for given d~ and p' is the solution of ~" = ( M M r ) - ~r f with 7/' Alp'. What should we expect? We are mainly searching for the critical line (surface) separating the domains of broken and unbroken chiral symmetry. The unbroken phase is characterized by d, fluctuating about 0. When approaching a critical surface these fluctuations should increase, turn into a 2-state signal with an initially high tunnel rate as we enter the broken phase and eventually settle at one preference of sign when far beyond criticality. Such a behavior is indeed observed in our data. In fig. 2 we show
* For H M C it can be shown that ( e x p ( - S H ) ) = 1 where 6 H is the change in the energy over a trajectory. This average is sensitive to the n u m b e r of conjugate iterations performed, which provides a check that the matrix inversion is sufficiently accurate.
L Bender et aL / Staggered fennions
290
O. -0.1
0
S~
IO00
IS00
2000
1
u. :L2~
-0,1
qej
1
:S
] -0,2~
.....................
0
l ........................
SO0
I ..............
,,,,~ . . . . . . . . . . . . . . . . . . . . . . . . . .
LO00
ISO0
1
2000
Fig. 2. (
transition point the gaussian distribution broadens and then splits into two gaussians. All the above mentioned effects can be seen. This allows a rough estimation of the critical value of y (note that we are not trying for a high precision result, but just trying to scan the phase diagram). It turns out that the H M C method reproduces well the predictions from weak coupling perturbation theory and also is an efficient means to proceed from weak coupling to the physics at larger y. What comes as a surprise is how closely the results continue to agree with those from say the 2-loop gap equation. For a
i. Bender et at. / Staggered fermim~s 250
o
,
~
,
200 150
,~..
-0,1!50 25(1
o
o
,
~
•
e
~
~ ~
~
,,~-
.
~
~f
~ i
~
~.,
-0.125
O. OQO
-O. 12~5
0.0O()
• Q. ,225
-0.125
0.0O0
0.125
,
~
-,
~,
,$,
.........
20Q 15(1 IQ0 5O QI
-0.250
O,, 250
25Qr'~r--"r~T 200 150 100 50 q
-C. 250
|
I
z
i
t'
0.250
Fig. 3. histogramscorresponding to the runs shown in fig. 2.
comparison we confront in figs. 4-11 numerical and gap equation results obtained for A = 6, 60 and m 2 = 2, 50. As evident from these plots, for the parameter range covered here there is not much room in the Monte Carlo data for phenomena beyond the reach of perturbation theory. The 2-loop correction helps quite a bit, in particular close to criticality. Qualitatively however already l-loop perturbation theory can reproduce the main features of the Monte Carlo data. For completeness, we mention that the results from the gap equation incorporate a finite size effect, i.e. the momentum integrals of the loops have been replaced by momentum sums appropriate to the size and the boundary conditions of the HMC lattice. As it turns out this finite size effect is numerically sma~l
L Bender e! aL / Staggered fennio~u~
9_
m2=2
~,= 6
1.5
<@>
.....
1-Loop
---
2-Loop
/
Id
0.5
0.4
0.6
Fig. 4. ( 6 } for m 2 = Z A = 6 as o b t a i n e d
y
0.8
by H M C
o n a n 84 lattice a n d 2 loops.
m==,50
Z= 6
0.6 .... <~>
0.5
1-Loop 2-Loop
0.4
0.3
0.2 /
0.1
...<
2
I
:I.,
,
,
I
3
,
,
,
,
y
I
4
Fig. 5. S a m e as fig. 4 but with m 2 = 50.
1
by p e r t u r b a t i o n
t h e o r y to I a n d
1. Bender et ul. / S~aggered fernzions
rn%2
293
,,sc
0.7
....
1-Loop
0.8
2-Loop
<0> 0.5
./
0.4 / /A
0.3 O.:P 0.1 O
6
0.5
0.6
0.7
y
O.S
0.9
Fig. 6. S a m e as fig. 4 b u t v.ith .~ = 6~.
m%,50
~SO
0.5
.... 0.4
1-Loop
~
2 Loop
i
.. ~ ~ , ~ ~ ~ -
<~)> 0.3
0,2
0,1
2,5
3
y
3.5
Fig. 7. Same as fig. 4 but with m 2 = 50, ~ = 60.
4
L B ~ d e r e+ aL / Sta:z.cered :6onnions
294
m2=2
~.=6
0.4
........ I-Loop 0.3
0,2
0.1
0.2 +
0.4
,78 for ,m:
0,4
'
+
"'
0.6
+
i!
+
2-Loop
0.3
0.8
1
Z A 6 as obtained from HMC on an 8 `+ lattice and by l-loop and 2-|oop perturbation theo~,.
+
•
+"
|
'+
"
/
1 -Loop
y
+'
~'
|
+
+
+
+
~
/
,"/° ///e
0.2
l
/
0.1
I
1
n
.. ~
l
2
,
m
.
.,
II
3
.
.
.
.
Y
Fig. 9. Same as fig. 8 but with m 2 = 50.
l
4
.
.
.
.
5
l~ Bender et d. / Sta~ered f e r ~
m2~
2~5
~eo
0.4
<~>
....
1-Loop 2-Loop
0.2
J /
s"
/ /
¢ol
0.5
O.S
0.7
y
O.S
0.9
I
Fig. [0. Same as fig. 8 ~zt w~th ~. = (~0o
m2=50 0.4
<~X >
•
.
....
u
~
~
~
b m
o
~ 60
~
B
~
~
~
~
~
~
1-Loop 2-Loop
0.3
0.2
I a
J J [ [ ! I
0.1
0
I
2
.|
3
Y
4
Fig. 11. S a m e as fig. 8 b u t with m 2 = 50, ,~ = 60.
5
2%
L Bender et aL / Staggered fi, nnions
I 00
~. = (6, 60)
Y
10
1
t
10
100
m2
1000
Fig. |2. H M C results for the location of the critical line in the m : - y plane for the values J~ = 6.0 (squares) and A = 60.0 (triangles), The error bars are roughly of the size of the data points. Also shown are corresponding predictions from l-loop perturbation theoD' (eq. (15)) and from the Ising analogy (eq, (24)).
already for an 8 4 lattice and so the results do not differ very. much from their infinite volume limit. Conversely this observation allows us to conclude that our simulation is not severely hampered by the use of a relatively small lattice (coarse grid of size 44). We now turn to the question of how the critical surface evolves when we come to truly large values of y. Fig. 12 represents a summary of the numerical results obtained by the H M C simulation. The plot shows, that the almost perfect agreement between HMC data and perturbation theory continues to hold true at large y. It also reveals a lack of Monte Carlo results for the Ising-like branch of the critical surface. The reason for this is, that H M C is bound to fad as a method for exploring this region. To simplify the discussion and also to bring about more clearly the essentials governing the critical surface at large y we consider the limit y -~ ~ keeping the ratio ~ = y / m fixed. The effective bosonic action then reduces to
S,, = E ( ½4~(X )2) _ Tr( In( M ) ) x
(27)
I. Bender et al. / Staggered ferm&ms
where ~ = m 6 . Thus we are left with a system of gaussian spins with spin-spin couplings induced by the presence of fermion loops and controlled by the paran~ter jS. This induced coupling is ferromagnetic in nature and h ~ the property being small both at small and large ~ (cfo subsect. 4.2). The leading effect at |arge :~ is the emergence of a double well potentia| with an infinite barrier at g = 0o Although this potential favors symmetry breaking, due to the lack of a s u ~ i e n t [ y large coupling between the spins, the system is essentially random, i.e. ~ i n f i i ~ occur frequently. It is precisely this feature that standard HMC cannot r e p r o d ~ e . Since energy is conserved along a trajectory, spin flips cannot occur in the absence of a coupling betwee~ the spins even for very large canonica| m ~ e n t a . Results obtained for (&) will therefore strongly depend on the initial configuration, rendering the method useless by this fact. From our theoretical analysis (loop expansion and |sing analogy) the s ~ e m described by the action in eq. (25) is expected to undergo a phase transition to a broken phase at :~--- 1 / ~ 6 4 D ( 0 ) ~ 0.3 and to return to the unbroken phase at ~ 1/8~--|. Hybrid Monte Carlo has no difficulties L,1 detecting the first transition. The critical coupling is 0.3, in close agreement with the weak coup[Lng prediction. This finding also provides some insight for the success of I- or 2-loop perturbation theory at intermediate and large y which first appeared as a s u r p ~ . For m 2 >> 16 the momentum dependence in the scalar propagator can be neglected, which sets the scale where eq. (27)can be used. Instead of y the relevant coupling then is ~5 and a value of the order of 0.3 is still sufficiently small for perturbation theory to be applicable. The HMC method ceases to function above ~ = 0.6. To expIore the model beyond this point we have switched to a Metropolis algorithm with an a priori distribution of the spins as obtained by evaluating the action in eq. (27) at constant ~. Compared to starting from a gauss|an distribution this has the big advantage that the double well nature of the effective potential at large ~5 is aheady incorporated into the selection of the spins. For the acceptance-rejection step each spin update of course requires a recalculation of the fermion determinant. For truly exploratory purposes on small lattices a direct evaluation by a matrix routine can be used. For larger lattices a much more efficient procedure probably is to represent the fermion determinant by a pseudo-boson integral as described at the beginning of this section and to perform in an alternating manner updates of the pseudo-boson field T/and the scalar field ~. Here a spin update only requires a calculation of (MMT) - ~T I which is much faster to perform. In a first attempt to detect the transition expected at ~5= 1.0 we have however renounced on following this route and have instead used the direct method on a 44 lattice. As it turns out even for such a small lattice a clear transition from unbroken to the broken phase and then back to the unbroken phase at larger )5 can be seen. The location of the second transition is found to be close to the expectation from the Ising analogy with an uncertainty of say + 0.2.
29g
io
et aL / Stag~red fen,nions
in the literature often the "statistical mechanic" notation for the action is used which reads =
s.
g{ (x)D fx)=z Ei
~,
2= l }2
g
=
]
The following relations to the parameters in the field theoretical language hold: 1
~(
1 = 2A') = 4 + ~m-,
2
(29)
=
We have replotted in fig. 13 the phase diagram (fig. 12) in order to allow a convenient comparison ~ith results of other authors. The ee~clusion to be drawn from the above results is, that the numerical sflnu|at~ons fully support the picture for the phase diagram obtained by analytical tools consisting of a loop e~ansion and an analogy with the lsing model.
10
y
l
~--(o, 60)
0
0
0
0
;~=60
0.1
. . . . . . . .
I 0 -4
I
0.001
,
. . . . . . .
I
0.01
,
,
. . . . . .
I
0.1
. . . .
,,,
1
Fig. 13. Fig. 12 redrawn in the (K, y')-plane defined by the "statistical mechanics" parametrization of the action in eq. (28). Note that only the lines referring to A = 0 are also lines of constant A', and that the line A = 6 would be indistinguishable from the line A -- 0 shown in this plot.
L Benderet aL / Staggeredfermions 6. C o n ~ u u m Hmit This section is devoted to the behavior of our ~ e l in the continuum [~m~L On the classicM level the flavor symmetry b r e ~ n g terms are p r ~ r t ~ to lattice spacing a and a 4-flavor symmetric theory [s r e v e r e d for a -~ 0° We ~ i [ now examine whether this also holds true at the 14oop level To simplify the ana|ysis we work in the unbroken p h i , Le° a p # y Feym~an ru|es corresponding to 4~o = O. The 5 ~ of diagrams o c c u m n g at I [ ~ ~ e shown in Appendix A. It is obvious that the pure|y ~ n ~ diagrams a, b ~ to be discussed any further. Their quadratic and ~gafitnmgc d ~ v e r g e ~ ~ be absorbed into a renormalgzafion of the scalar mass and the scalar s e l f - c o n i n g the usual way. We now prepare the stage to perform the sma[|-¢ expansion of the remamh~g nontrivial diagrams c to e. In general the Lmegra[s to be dealt w'ith at [ Mop are of the form:
u(~,a,d) =/p'~BH(p'+ #,m,a,d) where # represents a generic external momentum and p ' + # in H ~ meant to indicate that the internal momenta are linear combinations of externa| m o m e n ~ and the loop momentum p'. Variables are in physical units here, i.e. the B f i [ ~ m zone ranges from - ¢ r / a to + ¢r/a. For convenience the space-t/me dimension d has been introduced as a free parameter, i.e. we imagine the integration to be performed in d dimensions, putting d = 4 only at the end. Based on th~ representation a general formula can now be given for the coefficient of a ~ in the expa~ion of u at d = 4 (for a derivation see for instance ref. [15]) u( fi, a, 4)o~a,~ = d-,41im( fp.~ H(p' +~,m,a,d)[c~,~
+ a 4 - d £ ' ~ B H ( k' + a ~ ' a m ' a = l'd)[°~"÷'~) " ( 3 t ) Here k ' = a × p' is the dimensionless loop momentum with Brillouin zone B from - ~ - to +~r, • refers to integration from - ~ to +o0 and i 0 is the superficial degree of divergence of the diagram. The first term in eq. (31) represents the continuum integral obtained by expanding the integrand H to the given power of a putting the integration limits to infinity. The second term, a lattice integral obtained by expanding H around zero external momenta and zero mass, compensates for the neglect of the a dependence from the integration limits. Both terms are dimensionally continued away from d = 4 to regularize their ultraviolet and infrared behavior respectively. If a divergence is actually present, the correspond-
300
L Bender et aL / Staggered fermions
ing poles at d = 4 of the two terms will cancel each other with a left over piece propomonal to In(a). For i = 0 the coefficient in front of this logarithm is universal, i.e. it does not depend on the precise form of the lattice regularization as long as the lattice action has the correct classical continuum limit. Note, that eq. (31) can be used also as a starting point for a systematic analysis of lattice artefacts, i.e. positive powers i of a. In the present context, we are however only interested in the terms relevant to the continuum limit, i.e. the case i < 0. As a first application of eq. (31) we now consider the self-energy diagram c whose superficial degree of divergence is i o = 2. The corresponding integrand H c is given by
( s(p',p'+p,a) ) Hc(p'+p,p',a,d) =64y2 s(p',p',a)s(p' +p,p' +p,a) '
((°) (°)
s(r,t,a)=4Y'~ sin /.t
~r,
sin ~t,, cos
r~-t~,)
)
.
(32)
For i = - 2 we obtain as quadratic divergence from the lattice integral: uc( p, a, 4)1 ota--'~ = 64y2D(0)
(33)
with D(0) as given in eq. (14). For i = 0 (the linear divergence i -- - 1 vanishes) both types of integrals produce a result proportional to p2 with a pole at d = 4. As it should be, these poles cancel each other and we end up with 2 uc( p, a , 4 ) l o , a,,, = - ~ y 2 p 2 ( l n ( p 2 a 2 )
+
const.),
(34)
where the coefficient of the logarithm has the value expected for a 4-flavor symmetric Yukawa interaction. It is evident from these expressions that after mass and wavefunction renormalizafion, the lattice dependence completely disappears and that we reach the same result for the renormalized self-energy as would be obtained by any of the standard regularization techniques. For the vertex correction (diagram e) with superficial degree of divergence i 0 = 0 we obtain for i = 0 expression: Ue(/),a,4)lo(.,', = 4y 2 lim d--*4
(F.r.fp '~
(/) + P ' ) " ( / ) + P ' ) ~ 1 (/~p;)--~/~q-p')2 m2+(/~+p,)
_t_a4_d [
1
"k'eBs(k',k',a=l) 2
2
) " (35)
Here momenta on internal lines are again written in the schematic f o r m / ) + p'. As can be seen the lattice term is spin and flavor independent. The same holds true
!. Bender et .l. / Staggered fermions
30~
for the divergent piece of the continuum term since # and v become contracted. Renormalization of the Yukawa coupling then eliminates all dependence on the lattice regularization, i.e. the correct continuum limit is obtained. Finally we turn to the fermion self-energy diagram with i n = 1. For the c ~ f f i cient i = 0 in the small-a expansion of u a the situation is similar to the one found for the other diagrams, i.e. the same spin and flavor dependence as on the classical level is obtained. The linearly divergent piece however behaves differently. The result obtained by applying eq. (31) is =
(36)
-
Thus in contrast to the classical level where the flavor breaking F f term is proportional to a, on the l-loop level flavor breaking appears in a linear d~'ergence of the fermion self-energy. This clearly is an unwanted feature of our model, making its physical interpretation rather obscure. The reason that beyond the classical level our model seems unable to produce the expected continuum behavior probably has to be seen in the fact that not ~[ lattice symmetries of the free staggered fermion action are respected ~ the interaction. In the free case the following shift symmetry by one spacing of the flue grid lattice exists: x ( x ) --, x ) --,
x + x + g),
= 1-'I ( - 1 ) ='-
(37)
This directly follows from the property rt~(/z)= ~:~(v). It has been argued in refs. [ 11, 16] that the preservation of this symmetry guarantees the absence of nonrenorrealizable terms on the quantum level. A local coupling or a hypercube coupling as in models Ia, b conforms with this symmetry if & ( x ) ~ ~b(x + t~) is used. The rigid subdivision of the lattice into cells which is the basic element in the formulation of model II however does not. Our findings thus imply, that the preservation of the shift symmetry is not only a sufficient but also a necessary condition for perturbative renormalizability to hold true. The main motivation for a coarse grid description of the Yukawa coupling was to keep the number of flavors at four. The divergence encountered above indicates that this is not possible in the framework of staggered fermions [11, 16, 17]. 7. Conclusions
We have investigated an alternative to the models existing in the literature for describing a Yukawa interaction with dynamical mass generation on a lattice using staggered fermions. The model is characterized by the fact that the staggered
L Bender et al. / Staggered [ermions
302
fermion field is rewritten as a Dirac field on the coarse grid formed by elementary cells and then coupled in a spin and flavor independent way to a scalar field also living on the coarse grid lattice. The problem encountered in the other models with a nondefinite fermion determinant for the minimal case of four flavors can be circumvented here. We have investigated the phase structure of this model theoretically by performing an expansion to two loops and by exploiting the fact that the model becomes lsing-like at large Yukawa coupling. The theoretical analysis was then backed up by a series of numerical simulations using amongst others the, in generally very efficient, hybrid Monte Carlo algorithm. It was found that besides the broken and unbroken phases connected to the corresponding phases of the 4,4 theory another unbroken phases exists at large Yukawa coupling. In fact the latter phase and in particular the phase transition could not be investigated by the hybrid Monte Carlo method due to ergodicity problems. Instead we had to rely on Metropolis-like algorithms where spin flips of the scalar field are more direct!y build into the updating process. The theoretical results for the critical surface were fully supported by the numerical analysis. Finally the renormalizability issue of the model was investigated on the I-loop level. A problem with the behavior of the fermion self-ener~ was encountered which is probably related to the fact that the shift symmetry of the free staggered fermion action is not respected by our ansatz for the Yukawa interaction. This work was supported in part by the Deutsche Forschungsgemeinschaft. We thank I.O. Stamatescu for discussions at the initial stage of this project. We are grateful to the R H R K at Kaiserslautern and the RZ Karlsruhe for allocating computing time on their vector machines. R.H. thanks T. Pool, A. Kuhn, F. Speckert, M. Brug and M. Bootz for computational help. W.W. thanks the Fachbereich Physik of the University Kaiserslautern for the hospitality extended to him.
Appendix A I-LOOP DIAGRAMS, UNBROKEN PHASE j
~" "~ %
#
•
p ~,,, -,b % q~
(a)
#'
• •
(b)
(
q
(e)
Cd)
Ce)
L Bender et aL / Staggered [ennions
303
FEYNMAN RULES FOR MODEL I|
• ,i k I 2 ~ 4 ~,,,-(~. .) + ,,,'- + ~A,o
..........
#
....... "
! 2 = - ~ o ( m 2 + ~*~'o)
I J
.-...<
•
s
X• P J J
-----I • %
References [1] [2] [3] [4] [5] [6]
A. Hasenfratz and T. Neuhaus, Phys. Lett. B220 (1989) 435 J. Polonyi and J. Shigemitsu, Phys. Rev. D38 (1988) 3232 J. Kuti, Nuci. Phys. B (Proc. Suppl.) 9 (1989) 55 I.H. Lee and R.E. Shrock, Nucl. Phys. B305 (1988) 305 P.D.V. Swift, Phys. Lett. B!45 (1984) 256 J. Stair, Acta Phys. Pol. B17 (Zakopane 1985) 531; J. Stair, Nucl. Phys. B175 (1980) 307, L.H. Karsten, in Field theoretical methods in panicle physics, ed. W. Riih! (Plenum, New York, 1980) [7] M.F.L Golterman and D.N. Petcher, Phys. Lett. B225 (1989) 159 [8] W. Boek, A.K. De, K. Jansen, J. Jersfik, T. Neuhaus and J. Stair, Phys. l~tt. B232 (1989) 486 [9] H.S. Sharatchandra, H.J. Thun and P. Weisz, Nuel. Phys. B192 (1981) 205; C.P. van den Doel and J. Smit, Nucl. Phys. B228 (1983) 122 [1(3] P. Becher and H. Joos, Z. Phys. C15 (1982) 343; Lett. Nuovo Cimento 38 (1983) 293 [11] M.F.L Golterman and J. Smit, Nucl. Phys. B245 (1984) 61 [12] C. Bloch, Nucl. Phys. 6 (1958) 329 [13] S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195 (1987) 216
3{}4
L Bender et aL / Staggered fennions
[14] K. Bitar, A.D. Kennedy, R. Horsley, S. Meyer and P. Rossi, Nuci. Phys. B313 (1989) 348, 377 R. Gupta, G.W. KJlcup and S.R. Sharpe, Phys, Rev. D38 (1988) 1278, |288 [15] W. Wetzel, Nucl. Phys, B255 (1985) 659 {16] T. Jolicoeur, A. Morel and B. Petersson, Nucl, Phys, B274 (1986) 2Z'~ [17] P. MRtra and P. Weisz, Phys~ Letl, B|26 (1983) 355: O. Napoly, Phys. Letl, B132 (1983) 145
COARSF GRID YUKAWA INTERACTION FOR STAGGERED FERMIONS ifing B E N D E R
h~stitu!fiir Hochenergieph~ik~ Unirersit& Heiddberg~ Schr6derstr~e 90~ D6~O Heidelberg~ Germany Roger H O R S L E Y * and Werner W E T Z E L
hzstitut fiir Tileoretische Physik, Universit& Heidelberg, Phite~old~en~g 16~ D ~ Heiddberg Geema~'~ Fachbereich Physik, Unh'ersit& KaL~erslautern~ D6750 Kaiserslauter~ Germany
Received 22 March 1990 (Revised 23 July 1990)
We examine a coarse grid version of the Yukawa coupling between a staggered f e r r r ~ n field X and a scalar field ~ in an attempt to describe the case of four ferm~n flavors on t ~ lattice. Convincing evidence for the posithJ/ty of the fermion determinant is presented. phase diagram is analyzed by investigating the vacuum expectation value of 6 and the c~rM condensate of ~C by analytical and numerical tools. An expansion up to ~,o-loop order h performed to cover perturbative aspects and an analo~ r to the Ising mode| at large Yukawa coupling is exploited. In the numerical simulations we mainly rely on the Hybrid Monte ~ r [ o algorithm, although alternatives had to be used because the algorithm in its present form faih to be useful at large Yukawa couplings. Also the issue whether our model is renormai/zable at one loop order is discussed.
1. Introduction Recently first attempts have been made to study the properties of the Yukawa interaction by numerical simulation [1-4]. The main motivation comes from the role which the Yukawa interaction plays for the dynamical generation of fermion masses in the standard model of the electroweak interaction. Besides this and not unrelated, there is the basic question whether the Yukawa theory on its own exists as a quantum field theory, i.e. whether it shares the fate of the 4~4 model to be * Present address: HLRZ, c / o KFA, Postfach 1913, D5170 Jiilich, Germany and Institut fiir Theoretische Physik, RWTH Aachen, Germany. 0550-3213/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)
278
I. Bender et al. / Staggered fermions
trivial, or whether the combination of self-coupling and Yukawa coupling allows for a nontrivial fixpoint. There are several layers at which effects of the Yukawa interaction can be studied. On the most complex level one might consider a full fledged lattice model of the electroweak interaction, where all interactions including the gauge coupling are present (see for example refs. [4,5]). Qualitative studies at this level have become feasible through algorithmic improvements and the considerable increase of computing resources. However in this case the investigation of quantitative issues is still beyond present day reach. Lowering ambitions, one can restrict attention to the Yukawa sector alone. A case where this may be a reasonable starting point for extending the perturbative analysis of the standard model is the scenario of a heavy top quark, say with a mass of 300 GeV. Here the Yukawa coupling is of the order of 1 and nonperturbative me~hods are essential. To reduce the complexity even further one might ignore the weak isodoublet structure of the original theory altogether and investigate a Yukawa model for a fermion singlet coupled to a singlet scalar (pseudoscalar) field. In the numerical studies [2,3] the latter attitude was adopted and a scalar Yukawa coupling has been investigated. The fermions were treated as staggered. This choice is quite natural because the dynamical generation of the fermion mass is the central issue. The theory should therefore have some kind of 3'5 invariance preventing mass generation by radiative corrections. For staggered fermions enough chiral symmetry remains on the lattice, in contrast to the situation for Wilson fermions. Actually, some time ago it has been suggested [5, 6] in the context of the Yukawa interaction to modify the original Wilson prescription by letting the scalar field play the role of the constant but otherwise arbitrary r-parameter. In this way a Wilson-type single fermion model is obtained which also possesses a "shift" symmetry [7] preventing radiative mass generation as in the staggered fermion case. First results for this model have been presented in ref. [8]. Here we will not discuss this alternative further but stick to staggered fermions. Of course the usual price to pay for staggered fermions is that the number of fermion species is an integer multiple of four. It is also possible to give a staggered fermion formulation where the number of flavors is only a multiple of two [9]. It was shown in ref. [9] for gauge theories, that the fermion determinant is complex in the SU(N) case with N > 2 but real for S O ( N ) groups. When applied to the Yukawa case with a real scalar field this construction should also lead to a real determinant. Here, however, we will not consider this interesting idea further. Second, and more severely, there is the question of how precisely to write down the Yukawa coupling on a lattice. The difficulty has to do with the fact that spin and flavor degrees for staggered fermions are distributed over elementary cells. In fact, we have been motivated to look for an alternative to the coupling schemes used in refs. [1-3] because of the problems encountered when using these. The
L Bender et al. / Staggered]'ermions
279
investigations made so far were based on the following form of a lattice action:
s,= E [~(x)O x ( x )
m2
-½~,(x)oq,+--i-~(x
)2
'~ 4 +~,p(x) +y~,(x)x(x)~(x)
(i) with 4
O x ( x ) = E n,,( x )O,,x( x ) , p.=l
,7,,(x) = H ( - 1 ) * * v
O,,x(,:) = '~.( x ( x + ;,) - x ( x - ;,)) , 4
E l = Y. El /x
)
p.=l
O ~q~( x ) =q~( x + f~) + q~( x - ft) - 2 ~ ( x ) ,
(2)
and
,t,(x) =,p(x)
local coupling (model I a ) ,
4)(x) = ~ ~ q~(y)
hypercube coupling (model Ib).
(3)
yEx
Here X is the staggered fermion field, q~ is the scalar field with mass m and self-coupling A; y is the Yukawa coupling constant and the sum over x extends over all lattice sites. The two alternatives for the form of the Yukawa interaction are a strictly local coupling (model la) and a hypercube coupling (model lb) where ~b(x) is obtained by averaging ~p(y) over the hypercube associated with x. The chiral symmetry prohibiting a perturbative mass term is
x(x) --,~(x)x(x), ¢ ( x ) --, - ~ ( x ) ,
~ ( x ) --, - ~ ( x ) ~ ( x ) , E(x)
= H ( - 1) x" .
(4)
/.L
As it stands, the action in eq. (1) describing four fermion species cannot be used as a starting point for numerical simulations since the fermion determinant can become negative. The indefiniteness of the determinant is clearly visible for l~rge
_80
L Ben&r et aL / Staggered fennions
y or equivalently for a large background field 4~ and y = 0(1). In this limit the determinant for the fermion matrix
M,.,=D,.~,+y6(x)$,.,,
(5)
det M cx ~ 6 ( x ).
(6)
is given by A"
As a remedy to this problem previous investigations have resorted to a duplication of the fermion degrees of freedom. ~ adding to eq. (1) a second but otherwise identical copy of the staggered fermion field, the determinant becomes (det M ) 2, i.e. of definite sign. Although this doubling of fermion species from 4 to 8 leads to a model amenable to numerical simulation, it is not clear that by this procedure an action has been obtained which is well suited to approach continuum physics for 8 flavors. in the scaling limit the doubled model is expected to have an exact SU(8) flavor ss~nmetry. However, as it stands, only the SU(2) flavor symmetry introduced by the explicit doubling is manifest. For finite lattice spacing the eigenvalues of the fermion matrix M exhibit an SU(4) flavor symmetry only when the Yukawa coupling is zero; but this symmetry is broken already at first order in y. The use of a hypercube form for the coupling instead of a local one has been advocated as a step towards achieving a closer resemblance with the continuum action and therefore a reduction of lattice artefacts. As it turns out, model la and Ib have largely different phase diagrams at large Yukawa coupling. In the local coupling case a transition from the broken phase to an unbroken phase at large m 2 is always observed whereas for the hypercube case only the broken phase seems to exist for sufficiently large y. The argument put forward in ref. [2] to explain this observation is as follows: In the Ising limit, i.e. at large self-coupling the scalar field ~ is restricted to the values + 1. At large y the determinant of M is well approximated by eq. (6). For a local coupling we then have (det M) 2= 1 and we are back at the |sing model we started from at y = 0. For a hypercube coupling the fermion determinant will strongly suppress configurations where ~b is zcro and favor those configurations where the spins in the hypercube are aligned. This constraint could prevent the transition to a disordered phase as was indeed observed in the simulation. Because the behavior at large y is of particular interest, the absence of a clear transition line to the disordered phase appears as a quite unattractive feature of the hypercube form of coupling. Instead of following the strategy adopted by previous investigations we have looked for an alternative staggered fermion model for which the need of additional flavor doubling does not arise and where the spin and flavor symmetry of the Yukawa interaction are more transparent. Our candidate for such a 4-flavor action
I. Bender et al. / Staggered fermions
~
is presented in sect. 2. In sect. 3 the positivity of the corres~ndh~g fermion determinant is discussed. Sect. 4 gives the theoretical results ~or the p h a ~ diagram, obtained from a gap equation and an Ising ~ e l an~ ~, t ~ latter being valid at large Yukawa coupling. These results are then ~ p a r e d [n sect. 5 with the results obtained from numerical skmulations. There f o | | ~ s a d i ~ ~ in sect. 6 of whether our model is renormalizab|e on the I - l ~ level F~na|~ sect. 7 summarizes the results and draws some conclusions.
2. T h e ~ We propose, similarly to the Dirac-K[ihler approach [ 10|, to rep|ace q . (1) by a lattice action (model 11) formulated in terms of fields on a r ~ ~ ¢~~ lattice X: m2
s,,=
A
X ) O O ( X ) - ~. ( X ) o ~ ( X ) + --i-6( X ) 2 + ~ 6 ( X ) ~
+ 2y~(X)~(X)
(7)
with 4
D=~.E g.=l
[ r . a . - r 2 o . 1,
/',, --- y. ® I,
/'~ --- ys ® e s t .
(8)
where {Yu,Y~} = 2~$~,,~, (y5)2"-" 1 are ordinary 4-dimensional hermitian y matrices and --- means up to a unitary equivalence. In spite of its appearance, model II actually is very similar to model I. As far as the fermionic part of the kinetic energy is concerned, eq. (7) is directly obtained from eq. (1) by subdividing the x lattice into unit cells and identifying the degrees of freedom of X(x) within these cells with the 16 components making up the spin and flavor degrees of 0 ( X ) . In precise terms the connection is
x=2X-.~,
~ , ~ [0,1], Us
x(x)
= ~,/,,~(x),
= (£t - L ) .
(F~)
,
(9)
where the Kronecker 8 has to be understood as a condition modulo 2 between the components of .~, ~3 and the unit vector in the ~ direction.
282
I. Bender et al. / Staggered fermions
Having so far merely rewritten the kinetic energy of the fermions, the difference between eq. (7) and eq. (1) therefore only resides in the fact, that the scalar field ~k now lives on the coarse grid formed by the unit cells of the original lattice x and couples in an equal manner to all spinor and flavor degrees of $. Thus there are no other effects breaking spin and flavor symmetry than the one contained in the kinetic energy term. Due to the presence of a second derivative, the F5 term in eq. (7) is an irrelevant operator. The classical continuum limit of model II therefore represents a Yukawa model with an SU(4) flavor symmetry. In spite of the presence of both the F, and the Ff term, the free fermion model in fact also has an exact SU(4) invariance which is easily visible in the momentum space representation [11]. This "flavor" symmetry however is nonlocal in X space and is not respected by both the m o d e m and model-ll Yukawa couplings. The Ys t~q3e invariance of model I described in the introduction is also shared by model II. In the notation of eq. (7) it has the form: m
g, ~ ~ ,
n
g, ~ - g,Fs,
4~ -> - . ~
(10)
(with ~ -- Y5 ®5)- In this context we note that eq. (7) becomes a Wilson action if FA~ is replaced by 1. Indeed the F f term has the same function as the usual Wilson term; it removes the fermion zero modes at the boundary of the Brillouin zone. The above Y5 type invariance of course is lost under this replacement. The main reason why we have adopted model II as a starting point to investigate the Yukawa interaction is that it does not have the problems of model I discussed in sect. 1. The model is Ising-like at large y and A and is therefore expected to have two clearly separated phases like model Ia. Secondly the formula analogous to eq. (6) is now composed of factors where the scalar field values appear with a power of 16, making the determinant insensitive to the sign of the Higgs field. As discussed in sect. 2 all evidence indeed supports the conjecture that the fermion determinant of model II is nonnegative. The additional doubling of flavors required for model I can therefore be avoided here, i.e. we can investigate the case of four flavors.
3. The fermion determinant
We here address the question of whether the fermion matrix M for a coarse grid Higgs field has a positive semidefinite determinant also for values of y away from y = 0 or y = oo. Although we cannot give a mathematical proof, there is a large body of evidence in support of this, which stems from investigating the spectrum of eigenvalues of M. Since M is a real matrix, complex eigenvalues of M will always come in conjugate pairs and thus give a strictly positive contribution to the determinant. A
I. Bender et al. / Staggered fermions
283
negative sign can therefore only arise if there is an odd number of eigenvalues present on the negative real axis. Our means to argue against suL:h a possibility for model II is to investigate how the behavior of the eigenvalues as a function ~ y changes when going from a fine grid Higgs field (model I) to a coarse grid (model II). We start by considering the spectrum of M for y = 0. For an x l a t t ~ with Nm x N 2 x N 3 x N4 sites and periodic boundary conditions the eigenva|ues ~" are given by ~ = +i~Y'.~ sin2(q~,/2),
N./2z.,
= 0, _+ l , . . . ,
+_
[ l 2
"
(ll)
where [ . . . ] stands for taking an appropriate integer part. This spectrum is eighffold degenerate due to the above mentioned nonlocal "'flavor" SU(4) ~ m e try and a corresponding SU(2) "spin" symmetry. For the zero mode (q--, 0) the two imaginary eigenvalues approach the real axis (at 0) leading to a doubling of the degeneracy. The presence of this zero mode in fact is the reason why periodic boundary conditions are considered here. Only in this case can small values of y suffice to reach a negative result for the determinant. We next examine how the zero mode is split when the Yukawa term is switched on. Since the other eigenvalues are away from the real axis they come into play for the positivity issue only when a complex conjugate pair coalesces on the real axis, and then splits into eigenvalues moving in opposite directions along the real axis. Such nonanalytic behavior has to do with a breakdown of perturbation theory arm is beyond the level of the present discussion. To first order in y we have to diagonalize the 16 x 16 matrix formed by the matrix elements of the Yukawa term in a basis for the unperturbed eigen~ates. It is easy to see that for a generic Higgs field the degeneracy is completely lifted for model I and that odd numbers of negative eigenvalues are not the exception if the Higgs field is sufficiently fluctuating. For model II, however, only a common shift along the real axis can take place. To second order in y the eigenvalues originating from the zero modes generally start to acquire imaginary parts because the free fermion matrix is antisymmetric whereas the Yukawa term is a symmetric matrix. A recombination to conjugate complex pairs, however, is only possible if the first order splittings are small enough to allow a substantial mixing of the corresponding states. Thus in the generic case we will remain at second order with an odd number of negative eigenvalues if there was an odd number of negative eigenvalues to order y. This settles the fate of the fermion determinant for models I.
284
L Bender et al. / Staggered fennions
To find out the magnitude of the imaginary parts for model II we have again to diagonalize a 16 × 16 matrix, this time formed by the matrix elements of the second order operator of Bloch's perturbation theory [12]. It turns out that the eigenvalues of this matrix still form an eightfold degenerate pair of eigenvalues on the imaginary, axis which can collapse to the real axis only in the trivial case where the Higgs field is constant, i.e. where it corresponds to a mass term. Thus at least to this order there is no mechanism visible which can lead to an odd number of negative eigenvalues for model II. It is interesting to note that this imaginary shift is generated solely by the F~5 term, i.e. the analogue of the Wilson term for staggered fermions. Let us now consider the situation at large y. For y = ~ the spectrum of M is given by the values of the H ~:,,,, , s field with a 16-fold degeneracy in the case of model II. To leading order in 1/y we have to consider the 16 × 16 matrix formed from the kinetic part D of M in the basis of the unperturbed states. Since these states are localized, it is evident from eq. (7) that only the Ff term contributes and that the resulting matrix is given by The eigenvalues of this matrix are _i. Thus similar to the situation at order O(y2), for model II the leading correction at large y moves the eigenvalues into the imaginary direction while preserving the degeneracy to the maximal possible degree. Having discussed the behavior at small and large y the intermediate values of y remain. Although highly improbable, it is nevertheless conceivable for model II that with increasing y several pairs of conjugate eigenvalues return to the real axis, separate such that an odd number of negative eigenvalues is present for some range of y and then recombine into pairs to move back into the complex plane. Examining the spectrum and the determinant numerically for a sizable number of random Higgs fields we have never observed such a behavior. Of course these studies were limited to moderate lattice sizes (maximally 6 4) due to storage requirements. Based on the evidence presented here it seems safe to make the conjecture that the determinant of M is strictly positive for model II with a possible zero occurring only in the trivial case of a vanishing Higgs field. As our discussion has shown, the FAt term not only removes extraneous fermion modes through the Wilson mechanism, but also has the function to shift real eigenvalues of the fermion matrix into the complex plane.
F.uFf/2.
4. The phase diagram - theoretical impressions
We shall now attempt to give a qualitative understanding of the phase diagram, as a function of m 2, h and y which we would expect from the action in eq. (7). A line of second order transitions should separate the unbroken phase ( ( t ~ ) - 0) from the broken phase ((4~) :g 0). We are particularly interested in the effects due to the fermion, and so we shall consider the whole y range: 0 < y < oo. The two
I. Bender el M. / Staggered fermions
~5
tools at our disposal are a loop expansion and an Ising analogy for large y. We shall now consider both of these in some detail. 4.1. THE LOOP EXPANSION
The Feynman rules corresponding to model II after shifting the sca|ar riced an amount ~bo are given in Appendix A. We nosy iml~3~sethat 6o ~ k l e s with t ~ vacuum expectation value to one loop order. Diagrammatically this reads a x
.......
+
....
+
....
$
~
=0
(I2)
%~.=t, 8
Upon evaluation, the following gap equation is obtained: ~ - 6 4 y ,- n ( ( 2 y 6 o ) 2 ) + ~m D ( m 2 + ~~ A 62~ ) = o , ,n z + g ~ 6 ~
where D is the lattice integral D(
iy/2
)
1
F
]k~
m 2
2E~(1
(I4)
cos
over the Brillouin zone B. Note that the coefficient 64 in front of the fermion loop contribution to eq. (13) includes as a factor the number of spin and flavor degrees of the fermion. The gap equation is understood as an implicit equation for 6~ as a function of the parameters m 2, ~ and y. For the critical line we set 4'o = 0 and obtain m 2 - 6 4 y 2 D ( 0 ) + ½ A D ( m 2) = 0 (D(0)=0.155 in 4 dimensions). Solving this equation gives either m 2 ( y , A ) o r y ( m 2 , A). Eq. (15) is certainly valid in the small-y, smalI-A region. As far as
intermediate values of y are concerned it may still give a reasonable description because it also is the correct answer in the large number of flavors limit. Taken at face value it predicts that the critical m 2 increases like y2 with a constant of proportionality of 64D(0)= 10. It is feasible to extend the above calculation to two loops. In terms of diagrams the gap equation for 4'o now reads . . . . . . .
X
+
+
....
-4 I
"
+
....
J
$S P ~ % i-
.3
+
-Q ....
qs
-!-
""|
-(D'
....
)
+
.
.
+ •
.
.
.
.0
0
(16)
2'S~
L ~nder et aL / Staggered fennions
Applying Feynman rules we again obtain an implicit equation for 60, this time representing the vacuum expectation value of 6 to 2-loop order. Like in the l-loop case this equation and the corresponding equation for the critical line can be so~ved by numerical methods. We shall present the numerical results later in sect. 5 when we will ~ able to confront them ~ t h corresponding results from a Monte Carlo simulation. A quand~' closely related to the vacuum eKpectadon value of 6 is the chiral conden~te ( $ $ ) . To 2-1~ps we have diagrammatically:
.... Q
+
.... O
=Y(~$)"
(17)
At l - t ~ p order this yields ( $ 6 ) --- 32y&oD((~y60)-), where 60 is to ~ taken from the corresponding gap equation, i.e. from eq. (13). For numerical results at both 1- and 2-loop order we again refer to sect. 5. 4.2. I S | N G A N A L L Y
We now examine our mode| at large y. For y = ~ the only effect of the fermion determinant (see eq, (5)) is to add to the scalar potential the term -81n(~ 2) giving rise to a potential barrier at 6 = 0 of infinite height. Thus also for positive m 2 the potential is always of the double well ~ and we can expect a close resemblance to a lsing-like situation with preassigned values for the magnitude of 6. The simplest way of exploiting the Ising analogy is to fix the magnitude of 62 to the l~ation 6 '~ of the minimum of the potential and then identify from the kinetic term in the action the corresponding strength of the nearest neighbour coupling/3. in this way we obtain
m
)
+7'
-m2 ,
/3=62 .
(19)
Eliminating 6o we end up with the simple result: 16 m 2 =
--
/3
A -/3
--
6
(20)
for the connection at y = o0 between the parameters m 2 and A and the nearest neighbor coupling /3 of the equivalent Ising model. Inserting the critical value t3 = 0.1495 of the 4-dimensional Ising model we obtain as a prediction that model
I. Bender et al. / Staggered fermions
II at y = 0c has a second order phase transition line given by m 2=
107 - 0.02.%t.
(2I)
To test the validity of this conjecture we have performed a n u m e k ~ I s ~ u l a ~ using a Metropo|is algorithm. It turns out that eq. (21) is i ~ e d we[[ satL~f~ (say with an uncertainty of ± 1) for a|| A va|ues. Encouraged by this success, we try to extend the Is[ng a n a ~ to the s i m a ~ where the leading 1/y correction to the y = = limit of ~ [ H ~ taken into accoum. Expanding the fermion determinant in l / y we obtain the f ~ ~ ~ correction term to the acdon:
2{4
Z l S - - - wy~ x
x)
+
6(x)6{x+u)
"
-"
In the spirit of the preceding consideration ~2 has to be treated here ~ a c o , t a m equal to 6~. The term _4S therefore effectively just re~rmaI/zes the nearest neighbour coupling of the y = = lhnit by the amount: 2
|
Solving for y2 we end up with 32A + (~A + 6m2)(m2 + ~/m4+ -~A ) y2 =
(24)
96~(m 2 - 16//$ + ( f l / 6 ) A )
'
which extends eq. (20) away from y = ~ into the domain of large values of y. The prediction following from eq. (24) for the critical line (/~ ---0.1495) at sma~ A is shown in fig. 1 together with the l-loop result discussed in the prcv/ous section. Contrary to our original expectation, the branch starting at y - - - ~ shows no tendency of being in any way connected with the branch originating from y - 0. Instead it bends back towards y - o0, reaching the boundary at m 2 - ~ with the asymptotic behavior m 2--- 8/3y 2 _ y a , i.e. with a constant of proportionality I0 times smaller than for the branch originating from y = 0. These characteristic properties of the y = cc branch also hold when A is no longer small. The asymptotic behavior is exactly the same. In addition, the minimum value which y reaches as a function of m 2 is independent of ~t and is given by ( ~ / / 3 -- 19, which seems large enough for an a posteriori justification of the underlying approximation scheme. Thus at larger A the only difference is the shift in m e of the intercept at y - co which as mentioned before has been thoroughly tested by a Metropolis simulation.
L Bemh'r et al. / Staggered fennions
2SS
1 oo
symmetric phase
10
broken phase
symmetric phase
o.1 1
10
1 O0
ml
1000
Fig. 1. The critical line fo~ small A in the ~y, m2Vplane as obtained from the l-loop calculation and fronl the Ising a n a l o g .
The above results lead to the conjecture that model I| has two domains where the symmetry is unbroken, namely a domain connected to y = 0 which can be accessed by perturbation theory, and a domain connected to y = = with an |sing-like behavior. In the next section we will add further support to this by presenting results from a numerical simulation where the full dynamics of the fermion is taken into account.
5. The phase diagram - numerical impressions
The results presented in the last section now have to be backed up by Monte Carlo calculations. We have simulated eq. (7) using the Hybrid Monte Carlo (HMC) method [13]. The fermion determinant is first rewritten as a functional integral in terms of a real pseudo-boson field r/ instead of the Grassmann field X (the positivity of det M is crucial here): det M = f [ d n ]
e -'"T{MMT~-'. .
(25)
Configurations for r / c a n be obtained from gaussian distributed random fields p by applying the transformation r / = Mp. For each given r/ a molecular dynamics step
L Bender et al. / Staggered fermions
289
is now performed for the scalar field 4~ using a discretized leapfrog version of Hamiltons equations. The resulting d~ is then accepted or rejected in a Metropolis-like manner depending on the energy change 8 H accompan)ing tbe diseretization of the trajectory. This ensures that the generated Markov chain is exact, that is, in equilibrium we obtain a representative ensemble of configurat~ns for the underlying action. For the case of gauge theories, in part~ular QCD, the method has been well described and discussed in the literature, e.~ ref. [141. A potential problem with HMC is the possible low a c c e p t a ~ rate; by thks we mean that unless the step size is sufficient|y fine, many Metrol~|~ rejectm~ may ~ r before a new configuration is reached. We have ~ t | y m a ~ ~ on 8 ~ | a u ~ (i.e. coarse grid lattices of s~e 44) with a few test runs on 124 [att~es. P e ~ k : boundary conditions for the boson and spafia||y periodic but t e m ~ H y antipefiodic boundary conditions for the fermion have been used. For these | a t t ~ s~zes we have no acceptance prob|ems. The trajectory length has been set to ! w~ch seems sufficiently |arge to decorrelate successive configuratmns. For s m ~ y satisfactory results (acceptance rate ~ 70-80%) were obtained ~ i n g a ~ e p s/ze 8r = 0.1 i.e. 10 leapfrog steps to integrate the ~trajectory, wI~ie for larger y this dropped to 8 r = 0 . 0 1 . It should be noted that each | e a p f r ~ step requ~es a calculation of ~"= ( M M r ) - arl. We have used the conjugate gradient ~ t h o d for this task*. Typica| runs are over 1000-2000 trajectories per point in parameter space. The quantities measured were the vacuum expectation value (d~) and the chiral condensate (~6(X))- = ½E~ ~ x ( ~ ' X ( x ) ) . ~ where E ~ x denotes t ~ g the sum over positions inside a unit cell. A stochastic estimator for the ch~a[ condensate proved satisfactory [ 14]. We take
Y'~(~X)-(TrM-~)6=(~xp'(x)~'(x))
(26)
6.p"
x
where p' is a real Gaussian random field like p and g" for given d~ and p' is the solution of ~" = ( M M r ) - ~r f with 7/' Alp'. What should we expect? We are mainly searching for the critical line (surface) separating the domains of broken and unbroken chiral symmetry. The unbroken phase is characterized by d, fluctuating about 0. When approaching a critical surface these fluctuations should increase, turn into a 2-state signal with an initially high tunnel rate as we enter the broken phase and eventually settle at one preference of sign when far beyond criticality. Such a behavior is indeed observed in our data. In fig. 2 we show
* For H M C it can be shown that ( e x p ( - S H ) ) = 1 where 6 H is the change in the energy over a trajectory. This average is sensitive to the n u m b e r of conjugate iterations performed, which provides a check that the matrix inversion is sufficiently accurate.
L Bender et aL / Staggered fennions
290
O. -0.1
0
S~
IO00
IS00
2000
1
u. :L2~
-0,1
qej
1
:S
] -0,2~
.....................
0
l ........................
SO0
I ..............
,,,,~ . . . . . . . . . . . . . . . . . . . . . . . . . .
LO00
ISO0
1
2000
Fig. 2. (
transition point the gaussian distribution broadens and then splits into two gaussians. All the above mentioned effects can be seen. This allows a rough estimation of the critical value of y (note that we are not trying for a high precision result, but just trying to scan the phase diagram). It turns out that the H M C method reproduces well the predictions from weak coupling perturbation theory and also is an efficient means to proceed from weak coupling to the physics at larger y. What comes as a surprise is how closely the results continue to agree with those from say the 2-loop gap equation. For a
i. Bender et at. / Staggered fermim~s 250
o
,
~
,
200 150
,~..
-0,1!50 25(1
o
o
,
~
•
e
~
~ ~
~
,,~-
.
~
~f
~ i
~
~.,
-0.125
O. OQO
-O. 12~5
0.0O()
• Q. ,225
-0.125
0.0O0
0.125
,
~
-,
~,
,$,
.........
20Q 15(1 IQ0 5O QI
-0.250
O,, 250
25Qr'~r--"r~T 200 150 100 50 q
-C. 250
|
I
z
i
t'
0.250
Fig. 3. histogramscorresponding to the runs shown in fig. 2.
comparison we confront in figs. 4-11 numerical and gap equation results obtained for A = 6, 60 and m 2 = 2, 50. As evident from these plots, for the parameter range covered here there is not much room in the Monte Carlo data for phenomena beyond the reach of perturbation theory. The 2-loop correction helps quite a bit, in particular close to criticality. Qualitatively however already l-loop perturbation theory can reproduce the main features of the Monte Carlo data. For completeness, we mention that the results from the gap equation incorporate a finite size effect, i.e. the momentum integrals of the loops have been replaced by momentum sums appropriate to the size and the boundary conditions of the HMC lattice. As it turns out this finite size effect is numerically sma~l
L Bender e! aL / Staggered fennio~u~
9_
m2=2
~,= 6
1.5
<@>
.....
1-Loop
---
2-Loop
/
Id
0.5
0.4
0.6
Fig. 4. ( 6 } for m 2 = Z A = 6 as o b t a i n e d
y
0.8
by H M C
o n a n 84 lattice a n d 2 loops.
m==,50
Z= 6
0.6 .... <~>
0.5
1-Loop 2-Loop
0.4
0.3
0.2 /
0.1
...<
2
I
:I.,
,
,
I
3
,
,
,
,
y
I
4
Fig. 5. S a m e as fig. 4 but with m 2 = 50.
1
by p e r t u r b a t i o n
t h e o r y to I a n d
1. Bender et ul. / S~aggered fernzions
rn%2
293
,,sc
0.7
....
1-Loop
0.8
2-Loop
<0> 0.5
./
0.4 / /A
0.3 O.:P 0.1 O
6
0.5
0.6
0.7
y
O.S
0.9
Fig. 6. S a m e as fig. 4 b u t v.ith .~ = 6~.
m%,50
~SO
0.5
.... 0.4
1-Loop
~
2 Loop
i
.. ~ ~ , ~ ~ ~ -
<~)> 0.3
0,2
0,1
2,5
3
y
3.5
Fig. 7. Same as fig. 4 but with m 2 = 50, ~ = 60.
4
L B ~ d e r e+ aL / Sta:z.cered :6onnions
294
m2=2
~.=6
0.4
........ I-Loop 0.3
0,2
0.1
0.2 +
0.4
,78 for ,m:
0,4
'
+
"'
0.6
+
i!
+
2-Loop
0.3
0.8
1
Z A 6 as obtained from HMC on an 8 `+ lattice and by l-loop and 2-|oop perturbation theo~,.
+
•
+"
|
'+
"
/
1 -Loop
y
+'
~'
|
+
+
+
+
~
/
,"/° ///e
0.2
l
/
0.1
I
1
n
.. ~
l
2
,
m
.
.,
II
3
.
.
.
.
Y
Fig. 9. Same as fig. 8 but with m 2 = 50.
l
4
.
.
.
.
5
l~ Bender et d. / Sta~ered f e r ~
m2~
2~5
~eo
0.4
<~>
....
1-Loop 2-Loop
0.2
J /
s"
/ /
¢ol
0.5
O.S
0.7
y
O.S
0.9
I
Fig. [0. Same as fig. 8 ~zt w~th ~. = (~0o
m2=50 0.4
<~X >
•
.
....
u
~
~
~
b m
o
~ 60
~
B
~
~
~
~
~
~
1-Loop 2-Loop
0.3
0.2
I a
J J [ [ ! I
0.1
0
I
2
.|
3
Y
4
Fig. 11. S a m e as fig. 8 b u t with m 2 = 50, ,~ = 60.
5
2%
L Bender et aL / Staggered fi, nnions
I 00
~. = (6, 60)
Y
10
1
t
10
100
m2
1000
Fig. |2. H M C results for the location of the critical line in the m : - y plane for the values J~ = 6.0 (squares) and A = 60.0 (triangles), The error bars are roughly of the size of the data points. Also shown are corresponding predictions from l-loop perturbation theoD' (eq. (15)) and from the Ising analogy (eq, (24)).
already for an 8 4 lattice and so the results do not differ very. much from their infinite volume limit. Conversely this observation allows us to conclude that our simulation is not severely hampered by the use of a relatively small lattice (coarse grid of size 44). We now turn to the question of how the critical surface evolves when we come to truly large values of y. Fig. 12 represents a summary of the numerical results obtained by the H M C simulation. The plot shows, that the almost perfect agreement between HMC data and perturbation theory continues to hold true at large y. It also reveals a lack of Monte Carlo results for the Ising-like branch of the critical surface. The reason for this is, that H M C is bound to fad as a method for exploring this region. To simplify the discussion and also to bring about more clearly the essentials governing the critical surface at large y we consider the limit y -~ ~ keeping the ratio ~ = y / m fixed. The effective bosonic action then reduces to
S,, = E ( ½4~(X )2) _ Tr( In( M ) ) x
(27)
I. Bender et al. / Staggered ferm&ms
where ~ = m 6 . Thus we are left with a system of gaussian spins with spin-spin couplings induced by the presence of fermion loops and controlled by the paran~ter jS. This induced coupling is ferromagnetic in nature and h ~ the property being small both at small and large ~ (cfo subsect. 4.2). The leading effect at |arge :~ is the emergence of a double well potentia| with an infinite barrier at g = 0o Although this potential favors symmetry breaking, due to the lack of a s u ~ i e n t [ y large coupling between the spins, the system is essentially random, i.e. ~ i n f i i ~ occur frequently. It is precisely this feature that standard HMC cannot r e p r o d ~ e . Since energy is conserved along a trajectory, spin flips cannot occur in the absence of a coupling betwee~ the spins even for very large canonica| m ~ e n t a . Results obtained for (&) will therefore strongly depend on the initial configuration, rendering the method useless by this fact. From our theoretical analysis (loop expansion and |sing analogy) the s ~ e m described by the action in eq. (25) is expected to undergo a phase transition to a broken phase at :~--- 1 / ~ 6 4 D ( 0 ) ~ 0.3 and to return to the unbroken phase at ~ 1/8~--|. Hybrid Monte Carlo has no difficulties L,1 detecting the first transition. The critical coupling is 0.3, in close agreement with the weak coup[Lng prediction. This finding also provides some insight for the success of I- or 2-loop perturbation theory at intermediate and large y which first appeared as a s u r p ~ . For m 2 >> 16 the momentum dependence in the scalar propagator can be neglected, which sets the scale where eq. (27)can be used. Instead of y the relevant coupling then is ~5 and a value of the order of 0.3 is still sufficiently small for perturbation theory to be applicable. The HMC method ceases to function above ~ = 0.6. To expIore the model beyond this point we have switched to a Metropolis algorithm with an a priori distribution of the spins as obtained by evaluating the action in eq. (27) at constant ~. Compared to starting from a gauss|an distribution this has the big advantage that the double well nature of the effective potential at large ~5 is aheady incorporated into the selection of the spins. For the acceptance-rejection step each spin update of course requires a recalculation of the fermion determinant. For truly exploratory purposes on small lattices a direct evaluation by a matrix routine can be used. For larger lattices a much more efficient procedure probably is to represent the fermion determinant by a pseudo-boson integral as described at the beginning of this section and to perform in an alternating manner updates of the pseudo-boson field T/and the scalar field ~. Here a spin update only requires a calculation of (MMT) - ~T I which is much faster to perform. In a first attempt to detect the transition expected at ~5= 1.0 we have however renounced on following this route and have instead used the direct method on a 44 lattice. As it turns out even for such a small lattice a clear transition from unbroken to the broken phase and then back to the unbroken phase at larger )5 can be seen. The location of the second transition is found to be close to the expectation from the Ising analogy with an uncertainty of say + 0.2.
29g
io
et aL / Stag~red fen,nions
in the literature often the "statistical mechanic" notation for the action is used which reads =
s.
g{ (x)D fx)=z Ei
~,
2= l }2
g
=
]
The following relations to the parameters in the field theoretical language hold: 1
~(
1 = 2A') = 4 + ~m-,
2
(29)
=
We have replotted in fig. 13 the phase diagram (fig. 12) in order to allow a convenient comparison ~ith results of other authors. The ee~clusion to be drawn from the above results is, that the numerical sflnu|at~ons fully support the picture for the phase diagram obtained by analytical tools consisting of a loop e~ansion and an analogy with the lsing model.
10
y
l
~--(o, 60)
0
0
0
0
;~=60
0.1
. . . . . . . .
I 0 -4
I
0.001
,
. . . . . . .
I
0.01
,
,
. . . . . .
I
0.1
. . . .
,,,
1
Fig. 13. Fig. 12 redrawn in the (K, y')-plane defined by the "statistical mechanics" parametrization of the action in eq. (28). Note that only the lines referring to A = 0 are also lines of constant A', and that the line A = 6 would be indistinguishable from the line A -- 0 shown in this plot.
L Benderet aL / Staggeredfermions 6. C o n ~ u u m Hmit This section is devoted to the behavior of our ~ e l in the continuum [~m~L On the classicM level the flavor symmetry b r e ~ n g terms are p r ~ r t ~ to lattice spacing a and a 4-flavor symmetric theory [s r e v e r e d for a -~ 0° We ~ i [ now examine whether this also holds true at the 14oop level To simplify the ana|ysis we work in the unbroken p h i , Le° a p # y Feym~an ru|es corresponding to 4~o = O. The 5 ~ of diagrams o c c u m n g at I [ ~ ~ e shown in Appendix A. It is obvious that the pure|y ~ n ~ diagrams a, b ~ to be discussed any further. Their quadratic and ~gafitnmgc d ~ v e r g e ~ ~ be absorbed into a renormalgzafion of the scalar mass and the scalar s e l f - c o n i n g the usual way. We now prepare the stage to perform the sma[|-¢ expansion of the remamh~g nontrivial diagrams c to e. In general the Lmegra[s to be dealt w'ith at [ Mop are of the form:
u(~,a,d) =/p'~BH(p'+ #,m,a,d) where # represents a generic external momentum and p ' + # in H ~ meant to indicate that the internal momenta are linear combinations of externa| m o m e n ~ and the loop momentum p'. Variables are in physical units here, i.e. the B f i [ ~ m zone ranges from - ¢ r / a to + ¢r/a. For convenience the space-t/me dimension d has been introduced as a free parameter, i.e. we imagine the integration to be performed in d dimensions, putting d = 4 only at the end. Based on th~ representation a general formula can now be given for the coefficient of a ~ in the expa~ion of u at d = 4 (for a derivation see for instance ref. [15]) u( fi, a, 4)o~a,~ = d-,41im( fp.~ H(p' +~,m,a,d)[c~,~
+ a 4 - d £ ' ~ B H ( k' + a ~ ' a m ' a = l'd)[°~"÷'~) " ( 3 t ) Here k ' = a × p' is the dimensionless loop momentum with Brillouin zone B from - ~ - to +~r, • refers to integration from - ~ to +o0 and i 0 is the superficial degree of divergence of the diagram. The first term in eq. (31) represents the continuum integral obtained by expanding the integrand H to the given power of a putting the integration limits to infinity. The second term, a lattice integral obtained by expanding H around zero external momenta and zero mass, compensates for the neglect of the a dependence from the integration limits. Both terms are dimensionally continued away from d = 4 to regularize their ultraviolet and infrared behavior respectively. If a divergence is actually present, the correspond-
300
L Bender et aL / Staggered fermions
ing poles at d = 4 of the two terms will cancel each other with a left over piece propomonal to In(a). For i = 0 the coefficient in front of this logarithm is universal, i.e. it does not depend on the precise form of the lattice regularization as long as the lattice action has the correct classical continuum limit. Note, that eq. (31) can be used also as a starting point for a systematic analysis of lattice artefacts, i.e. positive powers i of a. In the present context, we are however only interested in the terms relevant to the continuum limit, i.e. the case i < 0. As a first application of eq. (31) we now consider the self-energy diagram c whose superficial degree of divergence is i o = 2. The corresponding integrand H c is given by
( s(p',p'+p,a) ) Hc(p'+p,p',a,d) =64y2 s(p',p',a)s(p' +p,p' +p,a) '
((°) (°)
s(r,t,a)=4Y'~ sin /.t
~r,
sin ~t,, cos
r~-t~,)
)
.
(32)
For i = - 2 we obtain as quadratic divergence from the lattice integral: uc( p, a, 4)1 ota--'~ = 64y2D(0)
(33)
with D(0) as given in eq. (14). For i = 0 (the linear divergence i -- - 1 vanishes) both types of integrals produce a result proportional to p2 with a pole at d = 4. As it should be, these poles cancel each other and we end up with 2 uc( p, a , 4 ) l o , a,,, = - ~ y 2 p 2 ( l n ( p 2 a 2 )
+
const.),
(34)
where the coefficient of the logarithm has the value expected for a 4-flavor symmetric Yukawa interaction. It is evident from these expressions that after mass and wavefunction renormalizafion, the lattice dependence completely disappears and that we reach the same result for the renormalized self-energy as would be obtained by any of the standard regularization techniques. For the vertex correction (diagram e) with superficial degree of divergence i 0 = 0 we obtain for i = 0 expression: Ue(/),a,4)lo(.,', = 4y 2 lim d--*4
(F.r.fp '~
(/) + P ' ) " ( / ) + P ' ) ~ 1 (/~p;)--~/~q-p')2 m2+(/~+p,)
_t_a4_d [
1
"k'eBs(k',k',a=l) 2
2
) " (35)
Here momenta on internal lines are again written in the schematic f o r m / ) + p'. As can be seen the lattice term is spin and flavor independent. The same holds true
!. Bender et .l. / Staggered fermions
30~
for the divergent piece of the continuum term since # and v become contracted. Renormalization of the Yukawa coupling then eliminates all dependence on the lattice regularization, i.e. the correct continuum limit is obtained. Finally we turn to the fermion self-energy diagram with i n = 1. For the c ~ f f i cient i = 0 in the small-a expansion of u a the situation is similar to the one found for the other diagrams, i.e. the same spin and flavor dependence as on the classical level is obtained. The linearly divergent piece however behaves differently. The result obtained by applying eq. (31) is =
(36)
-
Thus in contrast to the classical level where the flavor breaking F f term is proportional to a, on the l-loop level flavor breaking appears in a linear d~'ergence of the fermion self-energy. This clearly is an unwanted feature of our model, making its physical interpretation rather obscure. The reason that beyond the classical level our model seems unable to produce the expected continuum behavior probably has to be seen in the fact that not ~[ lattice symmetries of the free staggered fermion action are respected ~ the interaction. In the free case the following shift symmetry by one spacing of the flue grid lattice exists: x ( x ) --, x ) --,
x + x + g),
= 1-'I ( - 1 ) ='-
(37)
This directly follows from the property rt~(/z)= ~:~(v). It has been argued in refs. [ 11, 16] that the preservation of this symmetry guarantees the absence of nonrenorrealizable terms on the quantum level. A local coupling or a hypercube coupling as in models Ia, b conforms with this symmetry if & ( x ) ~ ~b(x + t~) is used. The rigid subdivision of the lattice into cells which is the basic element in the formulation of model II however does not. Our findings thus imply, that the preservation of the shift symmetry is not only a sufficient but also a necessary condition for perturbative renormalizability to hold true. The main motivation for a coarse grid description of the Yukawa coupling was to keep the number of flavors at four. The divergence encountered above indicates that this is not possible in the framework of staggered fermions [11, 16, 17]. 7. Conclusions
We have investigated an alternative to the models existing in the literature for describing a Yukawa interaction with dynamical mass generation on a lattice using staggered fermions. The model is characterized by the fact that the staggered
L Bender et al. / Staggered [ermions
302
fermion field is rewritten as a Dirac field on the coarse grid formed by elementary cells and then coupled in a spin and flavor independent way to a scalar field also living on the coarse grid lattice. The problem encountered in the other models with a nondefinite fermion determinant for the minimal case of four flavors can be circumvented here. We have investigated the phase structure of this model theoretically by performing an expansion to two loops and by exploiting the fact that the model becomes lsing-like at large Yukawa coupling. The theoretical analysis was then backed up by a series of numerical simulations using amongst others the, in generally very efficient, hybrid Monte Carlo algorithm. It was found that besides the broken and unbroken phases connected to the corresponding phases of the 4,4 theory another unbroken phases exists at large Yukawa coupling. In fact the latter phase and in particular the phase transition could not be investigated by the hybrid Monte Carlo method due to ergodicity problems. Instead we had to rely on Metropolis-like algorithms where spin flips of the scalar field are more direct!y build into the updating process. The theoretical results for the critical surface were fully supported by the numerical analysis. Finally the renormalizability issue of the model was investigated on the I-loop level. A problem with the behavior of the fermion self-ener~ was encountered which is probably related to the fact that the shift symmetry of the free staggered fermion action is not respected by our ansatz for the Yukawa interaction. This work was supported in part by the Deutsche Forschungsgemeinschaft. We thank I.O. Stamatescu for discussions at the initial stage of this project. We are grateful to the R H R K at Kaiserslautern and the RZ Karlsruhe for allocating computing time on their vector machines. R.H. thanks T. Pool, A. Kuhn, F. Speckert, M. Brug and M. Bootz for computational help. W.W. thanks the Fachbereich Physik of the University Kaiserslautern for the hospitality extended to him.
Appendix A I-LOOP DIAGRAMS, UNBROKEN PHASE j
~" "~ %
#
•
p ~,,, -,b % q~
(a)
#'
• •
(b)
(
q
(e)
Cd)
Ce)
L Bender et aL / Staggered [ennions
303
FEYNMAN RULES FOR MODEL I|
• ,i k I 2 ~ 4 ~,,,-(~. .) + ,,,'- + ~A,o
..........
#
....... "
! 2 = - ~ o ( m 2 + ~*~'o)
I J
.-...<
•
s
X• P J J
-----I • %
References [1] [2] [3] [4] [5] [6]
A. Hasenfratz and T. Neuhaus, Phys. Lett. B220 (1989) 435 J. Polonyi and J. Shigemitsu, Phys. Rev. D38 (1988) 3232 J. Kuti, Nuci. Phys. B (Proc. Suppl.) 9 (1989) 55 I.H. Lee and R.E. Shrock, Nucl. Phys. B305 (1988) 305 P.D.V. Swift, Phys. Lett. B!45 (1984) 256 J. Stair, Acta Phys. Pol. B17 (Zakopane 1985) 531; J. Stair, Nucl. Phys. B175 (1980) 307, L.H. Karsten, in Field theoretical methods in panicle physics, ed. W. Riih! (Plenum, New York, 1980) [7] M.F.L Golterman and D.N. Petcher, Phys. Lett. B225 (1989) 159 [8] W. Boek, A.K. De, K. Jansen, J. Jersfik, T. Neuhaus and J. Stair, Phys. l~tt. B232 (1989) 486 [9] H.S. Sharatchandra, H.J. Thun and P. Weisz, Nuel. Phys. B192 (1981) 205; C.P. van den Doel and J. Smit, Nucl. Phys. B228 (1983) 122 [1(3] P. Becher and H. Joos, Z. Phys. C15 (1982) 343; Lett. Nuovo Cimento 38 (1983) 293 [11] M.F.L Golterman and J. Smit, Nucl. Phys. B245 (1984) 61 [12] C. Bloch, Nucl. Phys. 6 (1958) 329 [13] S. Duane, A.D. Kennedy, B.J. Pendleton and D. Roweth, Phys. Lett. B195 (1987) 216
3{}4
L Bender et aL / Staggered fennions
[14] K. Bitar, A.D. Kennedy, R. Horsley, S. Meyer and P. Rossi, Nuci. Phys. B313 (1989) 348, 377 R. Gupta, G.W. KJlcup and S.R. Sharpe, Phys, Rev. D38 (1988) 1278, |288 [15] W. Wetzel, Nucl. Phys, B255 (1985) 659 {16] T. Jolicoeur, A. Morel and B. Petersson, Nucl, Phys, B274 (1986) 2Z'~ [17] P. MRtra and P. Weisz, Phys~ Letl, B|26 (1983) 355: O. Napoly, Phys. Letl, B132 (1983) 145