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Non-perturbative Yukawa coupling in scalar-fermion systems
92
Nuclear Physics B (Proc. Suppl.) 9 (1989) 92-95 North-Holland, Amsterdam
NON-PERTURBATIVE YUKAWA COUPLING IN SCALAR-FERMION SYSTEMS Anna HASENFRATZ Physics Department and Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306
We summarize our results of the 1-component real and complex scalar-fermion models. We show that the perturbative predictions are not valid for strong Yukawa coupling and investigate the possibility of a non-trivial tricritical fixed point.
1. INTRODUCTION In the standard model the masses of the fermions are generated by the spontaneously-broken scalar vacuum via
(1)
the Yukawa interaction. Perturbatively, one finds the tree
We simulated the quenched version of this model in the
level relation, m ] = y (~p), where (~) ~ 0 is the scalar field expectation value in the broken phase. The Yukawa
=
oo Ising limit. 3-5 The second model is a 1-component
complex system with 2 fermion flavors
coupling y is non-asymptotically free in the scalar-fermion sector of the standard model if one neglects gauge interactions. Its/~ function behaves the same way as the quartic scalar coupling in the perturbative (small y) region. In the
n
(2)
infinite cut-off limit both renormalized couplings are zero. The ratio ~r/y~ is fixed
(boundedfor
"~,/4
an effective theory)
which gives an upper bound for the quark masses.1 These
+
- 1) = + Z
arguments are from perturbation theory and are not necessarily valid for large y. We need non-perturbative methods t0investigate the strong Yukawa coupling region.
We simulated the full model 6,7 using the hybrid MC method. 8
In ~his paper we summarize our recent numerical results of the 1-component real and complex scalar fermion mod-
3. ANALYTICAL CONSIDERATIONS
els. We show that the perturbative predictions are not valid
For many respects the two models have similar prop-
for strong Yukawa coupling and investigate the possibility
erties. In the perturbative region the Yukawa coupling is
of a non-trivial tricritical fixed point. For further details
non-asymptotically free, its/3 function is positive, /3(y) _=
and new results the reader is referred to other contributed
dy/dt ~ y3. The renormalized coupling decreases as the
talks in this volume. 2
cutoff increases; it is zero in the infinite cutoff limit. The physical fermion mass is m f = y~ (~)r at tree level. When
2. THE MODELS We studied two different 1-component scalar fermion
the cut-off is taken to infinity, not only m / but the di-
models. In the first one real scalar fields are coupled to
mensionless ratio, mr~ (~)r (which can be used as the definition of the renormalized Yukawa coupling yr ), will
(naive) fermions with action
approach zero. A simple lowest-order strong coupling expansion shows that for large bare y the situation is different. 4 The tenor-
0920-5632/89/$03.50 © Elsevier Science Publishers B.V, (North-Holland Physics Publishing Division)
93
A. Hasenfratz / Non-perturbative Yukawa coupling malized fermion mass, and consequently yr, increases as
Renormalized quantities like the fermion mass should
the cutoff is removed. Therefore, there are two separate
clearly indicate the existence of the two regions. At finite
regions in the broken phase. While the phase structure of these systems can be very complicated, we can consider
correlation length the fermion mass will change rapidly or will be discontinuous at some finite y value. Local observ-
tWo basic and simplified scenarios (Figure 1.).
ables like ( ~ ¢ ) can be used to study the order of the phase
1) The change in the behaviour of the fermion mass is
transition(s).
caused by a (tri)critical point which lies on the sur-
If the system has a tricritical point we would like to
face separating the symmetric and broken phases. This
determine the two relevant critical exponents u 1 and u2,
phase transition surface can be second order every-
The change in ~ corresponding to a scale change in a scalar
where, or it can turn from second into first order at
mass m8 *~ rose t (or equivalently (~P)r -'-*
the tricritica] point. The tricritical point is ultraviolet-
,,(0 =
stable; by tuning the bare parameters around it, the renormalized parameters can take any value within the spontaneously broken phase. A continuum theory with
d/'%,(o)
(¢P)r e~) is
- ,~*) + ,,*
(s)
while the change in the renorraalized Yukawa coupling is
two relevant operators can be defined this way.
~(~) =
~-~/"~(~(o) - y*) + ~*
(4)
2) The symmetric and broken phases are separated by a first order phase transition.
In this case continuum
theory can be defined only at the perturbative fixed
where n* and y* are the critical couplings. Using Eq(4) we obtain
point. The change in the flow of the renormalized yr can still influence the value of the upper bound of the heavy quark mass in an effective model.
~2 vnCy,(O - y*) = - I n C M O ) , )
+ c
(S)
where the constant depends on Yb" A similar equation can be obtained for Ul using Eq(3).
broken
K
\
4. MONTE-CARLO SIMULATIONS AND RESULTS We investigated both models at infinite scalar coupling using naive fermions. In order to justify the existence of the two regions it is sufficient to do a quenched calculation. We simulated action (1). on 8 3 x 10 lattices at several tc values corresponding to a scalar field expectation value (~) = 0.4 - 0.8. The renormalized Yukawa coupling is defined as Yr = mr~ (~)r, where in
(~)r
=
(~) Z1/2
we
approximated Z by 1. Figure 2 shows the dependence of Yr on (~)r for different y. For small y(_~ 1.0) yr is al-
A
Symm. Y
most constant in accordance with the marginal behavk~ur predicted by perturbation theory. For large y ( _ 2.0), yr increases in agreement with the strong coupling expansion.
For 1.0 < y < 2.0 we found that the conjugate gradient al-
FIGURE 1
gorithm converged very slowly. This behaviour can indicate
The sketch of the flow lines in the e~- y plane. The solid line represents the symmetry breaking phase transition. The direction of the arrows corresponds to the change in the coupling with increasing cut-off.
a phase transition, but a sudden change of the renormalized fermion mass, as described in Section 3, can make the fermion determinant small and cause a convergence problem as well. Consider the derivative of the fermionic
A. Hasenfratz / Non-perturbative Yukawa coupling
94
tricritical behaviour can be obtained. The critical exponent
MODEL I.
v2 might change as one includes dynamical fermions. 7%
15
The quenched calculation is consistent with a tricriti-
x: y = 4 . 0 ~: y=3.0 ~: y=2.5
~. "~ ~
cal point and there is no indication for a phase transition around y* in the broken phase.
+: y = 2 . 0
For action (2), the phase diagram was studied in
10
Ref. [7] for positive ~ values. This model has a very rich phase structure, especially if one considers negative ~ values where the interplay between ferromagnetic and antiferromagnetic interactions becomes important. 0 0.4
0.5
0.6
0.7
0,8
¢
We will
discuss the details of this calculation in a forthcoming
09
publication. 6
Here, we just show the dependence of yr
on (~) (Figure 3.). We see the same structure as in the
FIGURE 2
quenched simulation of model I. From the scaling of scalar Yr as the function of (~) for model I.
expectation values, the symmetry-breaking phase transition is probably of second-order for small y, while for large y it
correlation function at distance n with respect to y. If the fermion mass changes rapidly around some y, the derivative will have a strong peak there, approaching a 5 function like singularity in the continuum limit. The correlation function can be expressed as D o n / D
=- d e t ( M o n ) / d e t ( M )
where
M o n is the cofactor corresponding to the (0,n) element of
is more likely to be weakly first-order. The second critical exponent from Eq(5) is v 2 = 0.7 -I- 0.0S. It suggests a non-Gaussian tricritical point. These preliminary results were obtained on 64 lattices. Although we think the indications are strong, larger systems and better statistics are necessary to clarify the critical properties of the models.
the fermionic matrix. If we suppose that both D o n and D are smooth functions of y, the expression
d Don dy D
_
DlonD
-Don
DI
MODEL II. 12.5
~"~--~
~ ....
I .... ~F~'~-]-~
can be singular only if D = det(M) ~ 0. The quenched
+: y - 1 6 ~: y 1 2
100
\\
calculation predicts y ~ 1.4 for the change in the direction of the flow lines of the fermion mass. The critical value y* can be different from that as m f = Yr {~)r approaches
-
÷: y = 2 4 n: y 2.0
D2
~: y 1 0
75
50
zero at y* and will start to increase only at some larger value of the Yukawa coupling.
25
Because of the convergence problem, we could not accurately calculate (~b¢) for 1.2 < y < 1.6, but for other
O0
.... 02
1. . . . 03
I .... 04
I .... 05
¢
I .... 06
I,~ 07
0a
Yukawa coupling values it is a rather smooth function. FIGURE 3
The symmetry-breaking phase transition in the quenched simulation is identical to the scalar model phase transition which is second-order.
yr as the function of {~) for model II.
It is possible to fit the data for
y >_ 2.0 with the formula Eq(5). One gets the most consistent result with v 2 = 0.55 + 0.04 and y* = 1.0 4- 0.5,
5. CONCLUSIONS
but the fit is not too good. In the small y region marginal-
We studied 1-component real and complex scalar
ity sets in so fast that no information about the possible
model coupled to (naive) fermions with Yukawa interaction.
A. Itasenfratz / Non-perturbative Yukawa coupling
Both quenched and unquenched MC calculations showed the existence of two different regions in the broken phase. We argue that this behaviour can be the consequence of an ultraviolet-stable tricritical fixed point. Our results are indicative of the existence of such a tricritical point but further studies are needed to firmly establish the existence of a non-trivial fixed point. ACKNOWLEDGMENTS The work reported here was done in collaboration with W. Liu and T. Neuhaus. This work was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the U.S. Department of Energy through Contract No. DE-FC0585ER250000 and by the Florida State University Computing Center with allocation of computer time on the ETA10. REFERENCES
1) N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B158 (1979) 295 M. Lindner Z.Phys. C31 (1986) 295
95
M. B. Einhorn and G. J. Goldberg, Phys.Rev.Lett. 57 (1986) 2115. 2) For a recent summary, see J. Kuti, plenary talk in the "Lattice '88" conference at Fermilab, 1988. and contributions by I. Montvay, Y. Shen and J. Shigemitsu, present volume. 3) J. Shigemitsu, Phys. Lett. 189B (1987) 164; OSU preprint DOE/ER/01545-397. I. Montvay, preprint DESY-87-077. 4) A. Hasenfratz, preprint SCRI-88-63, to appear in the proceedings "Lattice Higgs Workshop", Tallahasse 1988 and A. Hasenfratz, T. Neuhaus preprint SCRI-88-125 (1988). 5) J. Polonyi and J. Shigemitsu, OSU preprint DOE-ER01545-403. J. Shigemitsu, OSU preprint DE/ACO2-76ER01545-40, to appear in the proceedings "Lattice Higgs Workshop," Tallahasse 1988. 6) A. Hasenfratz, W. Liu and T. Neuhaus, in preparation. 7) D. Stephenson and A. Thornton, Edinburgh preprint 88/436 (1988). A. Thornton, Edinburgh preprint 88/444 (1988). 8) S. Duane, A. D. Kennedy,B.J. Pendleton and D. Roweth, Phys. Lett. 195B (1987) 216.
Nuclear Physics B (Proc. Suppl.) 9 (1989) 92-95 North-Holland, Amsterdam
NON-PERTURBATIVE YUKAWA COUPLING IN SCALAR-FERMION SYSTEMS Anna HASENFRATZ Physics Department and Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306
We summarize our results of the 1-component real and complex scalar-fermion models. We show that the perturbative predictions are not valid for strong Yukawa coupling and investigate the possibility of a non-trivial tricritical fixed point.
1. INTRODUCTION In the standard model the masses of the fermions are generated by the spontaneously-broken scalar vacuum via
(1)
the Yukawa interaction. Perturbatively, one finds the tree
We simulated the quenched version of this model in the
level relation, m ] = y (~p), where (~) ~ 0 is the scalar field expectation value in the broken phase. The Yukawa
=
oo Ising limit. 3-5 The second model is a 1-component
complex system with 2 fermion flavors
coupling y is non-asymptotically free in the scalar-fermion sector of the standard model if one neglects gauge interactions. Its/~ function behaves the same way as the quartic scalar coupling in the perturbative (small y) region. In the
n
(2)
infinite cut-off limit both renormalized couplings are zero. The ratio ~r/y~ is fixed
(boundedfor
"~,/4
an effective theory)
which gives an upper bound for the quark masses.1 These
+
- 1) = + Z
arguments are from perturbation theory and are not necessarily valid for large y. We need non-perturbative methods t0investigate the strong Yukawa coupling region.
We simulated the full model 6,7 using the hybrid MC method. 8
In ~his paper we summarize our recent numerical results of the 1-component real and complex scalar fermion mod-
3. ANALYTICAL CONSIDERATIONS
els. We show that the perturbative predictions are not valid
For many respects the two models have similar prop-
for strong Yukawa coupling and investigate the possibility
erties. In the perturbative region the Yukawa coupling is
of a non-trivial tricritical fixed point. For further details
non-asymptotically free, its/3 function is positive, /3(y) _=
and new results the reader is referred to other contributed
dy/dt ~ y3. The renormalized coupling decreases as the
talks in this volume. 2
cutoff increases; it is zero in the infinite cutoff limit. The physical fermion mass is m f = y~ (~)r at tree level. When
2. THE MODELS We studied two different 1-component scalar fermion
the cut-off is taken to infinity, not only m / but the di-
models. In the first one real scalar fields are coupled to
mensionless ratio, mr~ (~)r (which can be used as the definition of the renormalized Yukawa coupling yr ), will
(naive) fermions with action
approach zero. A simple lowest-order strong coupling expansion shows that for large bare y the situation is different. 4 The tenor-
0920-5632/89/$03.50 © Elsevier Science Publishers B.V, (North-Holland Physics Publishing Division)
93
A. Hasenfratz / Non-perturbative Yukawa coupling malized fermion mass, and consequently yr, increases as
Renormalized quantities like the fermion mass should
the cutoff is removed. Therefore, there are two separate
clearly indicate the existence of the two regions. At finite
regions in the broken phase. While the phase structure of these systems can be very complicated, we can consider
correlation length the fermion mass will change rapidly or will be discontinuous at some finite y value. Local observ-
tWo basic and simplified scenarios (Figure 1.).
ables like ( ~ ¢ ) can be used to study the order of the phase
1) The change in the behaviour of the fermion mass is
transition(s).
caused by a (tri)critical point which lies on the sur-
If the system has a tricritical point we would like to
face separating the symmetric and broken phases. This
determine the two relevant critical exponents u 1 and u2,
phase transition surface can be second order every-
The change in ~ corresponding to a scale change in a scalar
where, or it can turn from second into first order at
mass m8 *~ rose t (or equivalently (~P)r -'-*
the tricritica] point. The tricritical point is ultraviolet-
,,(0 =
stable; by tuning the bare parameters around it, the renormalized parameters can take any value within the spontaneously broken phase. A continuum theory with
d/'%,(o)
(¢P)r e~) is
- ,~*) + ,,*
(s)
while the change in the renorraalized Yukawa coupling is
two relevant operators can be defined this way.
~(~) =
~-~/"~(~(o) - y*) + ~*
(4)
2) The symmetric and broken phases are separated by a first order phase transition.
In this case continuum
theory can be defined only at the perturbative fixed
where n* and y* are the critical couplings. Using Eq(4) we obtain
point. The change in the flow of the renormalized yr can still influence the value of the upper bound of the heavy quark mass in an effective model.
~2 vnCy,(O - y*) = - I n C M O ) , )
+ c
(S)
where the constant depends on Yb" A similar equation can be obtained for Ul using Eq(3).
broken
K
\
4. MONTE-CARLO SIMULATIONS AND RESULTS We investigated both models at infinite scalar coupling using naive fermions. In order to justify the existence of the two regions it is sufficient to do a quenched calculation. We simulated action (1). on 8 3 x 10 lattices at several tc values corresponding to a scalar field expectation value (~) = 0.4 - 0.8. The renormalized Yukawa coupling is defined as Yr = mr~ (~)r, where in
(~)r
=
(~) Z1/2
we
approximated Z by 1. Figure 2 shows the dependence of Yr on (~)r for different y. For small y(_~ 1.0) yr is al-
A
Symm. Y
most constant in accordance with the marginal behavk~ur predicted by perturbation theory. For large y ( _ 2.0), yr increases in agreement with the strong coupling expansion.
For 1.0 < y < 2.0 we found that the conjugate gradient al-
FIGURE 1
gorithm converged very slowly. This behaviour can indicate
The sketch of the flow lines in the e~- y plane. The solid line represents the symmetry breaking phase transition. The direction of the arrows corresponds to the change in the coupling with increasing cut-off.
a phase transition, but a sudden change of the renormalized fermion mass, as described in Section 3, can make the fermion determinant small and cause a convergence problem as well. Consider the derivative of the fermionic
A. Hasenfratz / Non-perturbative Yukawa coupling
94
tricritical behaviour can be obtained. The critical exponent
MODEL I.
v2 might change as one includes dynamical fermions. 7%
15
The quenched calculation is consistent with a tricriti-
x: y = 4 . 0 ~: y=3.0 ~: y=2.5
~. "~ ~
cal point and there is no indication for a phase transition around y* in the broken phase.
+: y = 2 . 0
For action (2), the phase diagram was studied in
10
Ref. [7] for positive ~ values. This model has a very rich phase structure, especially if one considers negative ~ values where the interplay between ferromagnetic and antiferromagnetic interactions becomes important. 0 0.4
0.5
0.6
0.7
0,8
¢
We will
discuss the details of this calculation in a forthcoming
09
publication. 6
Here, we just show the dependence of yr
on (~) (Figure 3.). We see the same structure as in the
FIGURE 2
quenched simulation of model I. From the scaling of scalar Yr as the function of (~) for model I.
expectation values, the symmetry-breaking phase transition is probably of second-order for small y, while for large y it
correlation function at distance n with respect to y. If the fermion mass changes rapidly around some y, the derivative will have a strong peak there, approaching a 5 function like singularity in the continuum limit. The correlation function can be expressed as D o n / D
=- d e t ( M o n ) / d e t ( M )
where
M o n is the cofactor corresponding to the (0,n) element of
is more likely to be weakly first-order. The second critical exponent from Eq(5) is v 2 = 0.7 -I- 0.0S. It suggests a non-Gaussian tricritical point. These preliminary results were obtained on 64 lattices. Although we think the indications are strong, larger systems and better statistics are necessary to clarify the critical properties of the models.
the fermionic matrix. If we suppose that both D o n and D are smooth functions of y, the expression
d Don dy D
_
DlonD
-Don
DI
MODEL II. 12.5
~"~--~
~ ....
I .... ~F~'~-]-~
can be singular only if D = det(M) ~ 0. The quenched
+: y - 1 6 ~: y 1 2
100
\\
calculation predicts y ~ 1.4 for the change in the direction of the flow lines of the fermion mass. The critical value y* can be different from that as m f = Yr {~)r approaches
-
÷: y = 2 4 n: y 2.0
D2
~: y 1 0
75
50
zero at y* and will start to increase only at some larger value of the Yukawa coupling.
25
Because of the convergence problem, we could not accurately calculate (~b¢) for 1.2 < y < 1.6, but for other
O0
.... 02
1. . . . 03
I .... 04
I .... 05
¢
I .... 06
I,~ 07
0a
Yukawa coupling values it is a rather smooth function. FIGURE 3
The symmetry-breaking phase transition in the quenched simulation is identical to the scalar model phase transition which is second-order.
yr as the function of {~) for model II.
It is possible to fit the data for
y >_ 2.0 with the formula Eq(5). One gets the most consistent result with v 2 = 0.55 + 0.04 and y* = 1.0 4- 0.5,
5. CONCLUSIONS
but the fit is not too good. In the small y region marginal-
We studied 1-component real and complex scalar
ity sets in so fast that no information about the possible
model coupled to (naive) fermions with Yukawa interaction.
A. Itasenfratz / Non-perturbative Yukawa coupling
Both quenched and unquenched MC calculations showed the existence of two different regions in the broken phase. We argue that this behaviour can be the consequence of an ultraviolet-stable tricritical fixed point. Our results are indicative of the existence of such a tricritical point but further studies are needed to firmly establish the existence of a non-trivial fixed point. ACKNOWLEDGMENTS The work reported here was done in collaboration with W. Liu and T. Neuhaus. This work was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the U.S. Department of Energy through Contract No. DE-FC0585ER250000 and by the Florida State University Computing Center with allocation of computer time on the ETA10. REFERENCES
1) N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B158 (1979) 295 M. Lindner Z.Phys. C31 (1986) 295
95
M. B. Einhorn and G. J. Goldberg, Phys.Rev.Lett. 57 (1986) 2115. 2) For a recent summary, see J. Kuti, plenary talk in the "Lattice '88" conference at Fermilab, 1988. and contributions by I. Montvay, Y. Shen and J. Shigemitsu, present volume. 3) J. Shigemitsu, Phys. Lett. 189B (1987) 164; OSU preprint DOE/ER/01545-397. I. Montvay, preprint DESY-87-077. 4) A. Hasenfratz, preprint SCRI-88-63, to appear in the proceedings "Lattice Higgs Workshop", Tallahasse 1988 and A. Hasenfratz, T. Neuhaus preprint SCRI-88-125 (1988). 5) J. Polonyi and J. Shigemitsu, OSU preprint DOE-ER01545-403. J. Shigemitsu, OSU preprint DE/ACO2-76ER01545-40, to appear in the proceedings "Lattice Higgs Workshop," Tallahasse 1988. 6) A. Hasenfratz, W. Liu and T. Neuhaus, in preparation. 7) D. Stephenson and A. Thornton, Edinburgh preprint 88/436 (1988). A. Thornton, Edinburgh preprint 88/444 (1988). 8) S. Duane, A. D. Kennedy,B.J. Pendleton and D. Roweth, Phys. Lett. 195B (1987) 216.