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Numerical simulations of an SU(2)L ⊗ SU(2)R-symmetric Higgs-Yukawa model on the QUADRICS Q16
UCLEARPHYSICS O .1 PROCEEDINGS SUPPLEMENTS II
KI~SEVlKR
Numerical
Nuclear Physics B (Proc. Suppl.) 42 (1995) 639~541
Simulations
of a n
SU(2)c ,'£~ S U ( 2 ) R - s y m m e t r i c
Higgs-Yukawa
Model on the QUADRICS Q16 M. Plagge ~ * a n d D. Talkenberger b t ~Fakult~it fiir Physik, Universit/it Bielefeld, Universit~itsstr. 25, 33615 Bielefeld, G e r m a n y b l n s t i t u t fiir Theoretische Physik I, Universit/it M/luster, W i l h e l m - K l e m n > S t r . 9, 48149 Miinster, Germany We report on our work on the SU(2)L C! SU(2)R symmetric tliggs Yukawa Model with mirror fermion action. Our model describes a fermion Higgs system in the limit of vanishing gauge coupling. Setting the bare Yukawa coupling of the mirror fermions G x to zero, we want to determine the triviality bounds on the renormalized Yukawa coupling of the fermions G e e and the scalar self-coupling gn on 8a x 16 and 163 x 32 lattices.
1. I n t r o d u c t i o n F e r m i o n Higgs models with left right synm~etric action have been intensively studied over the last years (see [1] a n d references therein). T h e results o b t a i n e d so far are in good a g r e e m e n t with one loop p e r t u r b a t i o n theory b u t there are still strong finite size effects, especially at large values of the Yukawa coupling. W i t h increasing c o m p u t e r power c o m p u t a t i o n s on larger lattice sizes become feasible and we can get a b e t t e r control over the finite size efl>cts (beside the use of i m p r o v e d actions). T h e use of massivly parallel c o m p u t e r s boosts this development. T h e Q U A D R I C S Q16 is one of these which has a very a p p e a l i n g p r i c e / p e r f o r m a n c e ratio. In the following we report on our progress in i m p l e m e n t i n g the m o d e l which has been studied in [1] on the Q16. We present first, t i m i n g results of the p r o g r a m with d y n a m i c a l fermions and details of the i m p l e m e n t a t i o n .
2,~.;,2 m a t r i x 02 = ¢4 +i~r~4)~ k = 1,2, 3, where ak are the Pauli matrices and ¢~ are real variables. is the ferlnion pair of f e r m i o n ~ a n d m i r r o r ferInion k doublets, • = (t/,, 7~). Using c o n v e n t i o n a l n o r r n a l i z a t i o n s we have
{,
]2
+
x
;l=O
:r 4
-It ~
['7~,+,,7~ex+ V.+aT.x~
tt=--4
+,.,~ .¢,~
~ ,.
--
)] +
2. T h e M o d e l T h e lattice action of the m o d e l consists of a pure scalar p a r t a n d a m i x e d fermion scalar part S = S ~ , + S , . T h e scalar field c2 is represented as a *Supported by Deutsche Forschungsgenminschaft tamer grant PE 340/3-2 ISupported by Deutsche Forschungsgelneinschaft under grant Mu 757/4-3 0 9 2 0 - 5 6 3 2 / 9 5 / $ 0 9 . 5 0 © 1995 Elsevier Science B.V. SSDI 0920-5632(95)00336-3
All rights
reserved.
x,y
For the Hybrid M o n t e Carlo the n u m b e r offlayours has to be doubled so t h a t the last s u m becolues a sum over the lattice points and the two [tavours. T h e choice of the seven p a r a m e t e r s follows [1]. T h e W i l s o n p a r a m e t e r r is set to 1. In order
640
M. Plagge, D Talkenberger /Nuclear Physics B (Proc. SuppL) 42 (1995) 639-641
= 0.18
~c = 0.34
ith equal n on all b o a r d s
#0
board
# 15
F i g u r e 1. E x a m p l e r u n on the Q16
to s t u d y the t r i v i a l i t y b o u n d , we set the q u a r t i c coupling A to infinity. We choose p~,\ = 0.0 and G x = 0.0 to ensure the d e c o u p l i n g of the m i r r o r f e r m i o n s in the c o n t i n u u m limit. T h e fermionic h o p p i n g p a r a m e t e r K is fixed to K ~ = 0.125 [1]. T h i s leaves t h e f e r m i o n i c Yukawa coupling Gv, a n d the h o p p i n g p a r a m e t e r n as physical i n p u t p a r a m e t e r s . W e use ~ to t u n e the scalar m a s s and select the phase. 3. h n p l e m e n t a t i o n
on the QUADRICS
To s i m u l a t e the m o d e l we use the h y b r i d Monte C a r l o a l g o r i t h m with c o n j u g a t e g r a d i e n t for the m a t r i x inversion. For i n f o r m a t i o n on the p a r a n Jeter d e p e n d e n c e we refer to [1],[4] and [51. T h e t o p o l o g y of the Q16 is 2 × 2 x 32 which is n o t well s u i t e d for a l a t t i c e of size 8 a × 16. But it is p o s s i b l e to run t h e Q16 as 16 i n d e p e n d e n d Q1 b o a r d s w i t h 2 x 2 x 2 t o p o l o g y and p e r i o d i c bound a r y c o n d i t i o n s closing on each b o a r d . Therefore we decided to i m p l e m e n t the p r o g r a m in a sort of two step p a r a l l e l i s m . F i r s t we did a g e o m e t r i c p a r a l l e l i z a t i o n on the Q1 b o a r d s in d i v i d i n g the
l a t t i c e in 8 p a r t s a n d p u t t i n g each one on of the 8 C P U s . T h e second step is s i m p l y to run a different configuration on each of the 16 Q1 b o a r d s . T h e d r a w b a c k of this a p p r o a c h seems to be t h a t one has to e q u i l i b r a t e each configuration a n d so waste a lot of C P U time. F o r t u n a t e l y , this is not true in our case. Before we can s t a r t our m e a s u r e m e n t s , we have to t u n e the s c a l a r h o p p i n g p a r a m e t e r n in order to achieve a s c a l a r m a s s of 0.6 - 0.8 in l a t t i c e units a n d to choose the desired phase (here the broken, F M , phase; see [1]). Fig. 1 shows w h a t we do. We s t a r t on each Q1 w i t h a different value of ~;, t h e r m a l i z e these configurations and then d e t e r m i n e the scalar m a s s on the 16 b o a r d s . W h a t we get is shown in Fig. 2. F r o m this p i c t u r e we d e t e r m i n e the h o p p i n g p a r a m e ter value for the m e a s u r e m e n t s . W e pick o u t the configuration which lies closest to this values, duplicate it 16 t i m e s a n d r e s t a r t the p r o g r a m now with 16 equal c o n f i g u r a t i o n s a n d ~ values. If the new n value lies in between two old ~ values we do again a few t h e r m a l i z a t i o n sweeps, the n u m b e r of which is of the order 10 s m a l l e r t h a n on the original s t a r t configurations. A t the end we get 16 i n d e p e n d e n d results f r o m which d e t e r m i n e the q u a n t i t i e s of interest by the b o o t s t r a p m e t h o d . W i t h i n each c o n f i g u r a t i o n the errors are e s t i m a ted by a . j a c k k n i f e a p p r o a c h . T h e larger 163 × 32 l a t t i c e will be run on the D F G QH2 which will allow a m o r e flexible p a r t i tioning, so t h a t we can use the s a m e m e t h o d b u t with less configurations. A p r o b l e m which has to be faced qn the Q U A D RICS is its single precision f l o a t i n g p o i n t h a r d w a re. T h e r e are three p o i n t s in the p r o g r a m where global s u m s have to be c a l c u l a t e d a n d where the low precision can cause p r o b l e m s . In the m e t r o potis u p d a t e at the end of each t r a j e c t o r y s m a l l errors in the s u m m a t i o n p r o d u c e r a t h e r big effects on the u p d a t e p r o b a b i l i t y due to the e x p o n e n t i a l flmction involved. Errors in the scalar p r o d u c t s , which have to be e v a l u a t e d d u r i n g the conjugate g r a d i e n t i t e r a t i o n s , influence the convergence in an u n p r e d i c t a b l e m a n n e r . A n d , last n o t least, in the m e a s u r e m e n t r o u t i n e , where we e x p e c t the errors to play a less i m p o r t a n t role. W e have decided to use in all these cases a s u m m a t i o n procedure where the t e r m s are a d d e d tree like, so t h a t
M. Plagge, D. Talkenberger/Nuclear Physics B (Proc. Suppl.) 42 (1995) 639-641
2.5 IJ . . . . . . . . ' . . . . . .
,'~ /1l
' .........
.~,, , ~
' . . . . . . .
#Measurements is 500~ Loffice size is 8516 '
~.5
)~
12 0.5
O.Ot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00
0.10
0.20
0.30
0.40
0.50
K"
Figure 2. r~ tuning at Gw = 0.3
641
to larger lattices. C o m p a r e d to the peak performance we achieved ,~, 45% in the conjugate gradient. These values are in good coincidence .with the results obtained with the performance analyzer. Lattices of size 43 x 8 can be run either on a single C P U or, geometrically parallelized, on a eight C P U Q1 board. We have measured a speed up factor of 6 for one hybrid trajectory. C o m p a r i n g the times for different lattice sizes on the Q1, an increase in the performance, measured in M F L O P S , with increasing lattice size can be seen. This is due to a better filling of the pipeline. To sum up, we can say t h a t the Q16 gives for our problem an increase in c o m p u t e r power of a factor 15 c o m p a r e d to a C R A Y YMP. This allows us to achieve high statistics in a reasonable time on lattice sizes up to 163 x 32. The main difficulties are the limitation on the code size and the single precision arithmetic.
REFERENCES at every step terms of equal size are added. For the r a n d o m n u m b e r s we use the internal r a n d l 6 library routine, so we have for each of the 128 C P U s an independend generator. A Q U A D R I C S feature which caused some problems is the limitation on the size of the executables. This limit lies at ~ 2.6MB, a size which can be reached very quickly by using compile time for loops even if the length of these loops is small (in our p r o g r a m typically 2-4).
4. T i m i n g results T h e m o s t time consuming part in the simulation is the conjugate gradient. It ntilizes almost 95% of the whole C P U time. We have compared the times for a single m a t r i x inversion with times from a C R A Y YMP. For a fixed n u m b e r of iterations we are a b o u t 10% faster on the Q I. If we use the residual as the stopping criteria, which is of course more realistic, we are about 10% slower. which results from an increase in the n u m b e r of' iterations. We think this increase is due to summ a t i o n errors in the scalar product, but we will have to clarify this point a bit more, before going
1.
2. 3.
,t. 5.
C. Frick, L. Lin, I. Montvay, G. Miinster, M. Plagge, T. Trappenberg, H.Wittig, Nucl. Phys. B 397 (1993) 431. I. Montvay, Phys. Lett. B 199 (1987) 89; Nucl. Plays. B (Proc. Suppl) 4_ (1988) 443. R. G u p t a , G. Gnralnik, G. Kilcup, A. Patel, S. Sharpe, T. Warnock, Plays. Rev. D 3_fi6 (1987) 2813. H. Wittig, Ph.D. thesis, University of Hamburg, 1992 M. Plagge, Ph.D. thesis, University of Miinster, 1993
KI~SEVlKR
Numerical
Nuclear Physics B (Proc. Suppl.) 42 (1995) 639~541
Simulations
of a n
SU(2)c ,'£~ S U ( 2 ) R - s y m m e t r i c
Higgs-Yukawa
Model on the QUADRICS Q16 M. Plagge ~ * a n d D. Talkenberger b t ~Fakult~it fiir Physik, Universit/it Bielefeld, Universit~itsstr. 25, 33615 Bielefeld, G e r m a n y b l n s t i t u t fiir Theoretische Physik I, Universit/it M/luster, W i l h e l m - K l e m n > S t r . 9, 48149 Miinster, Germany We report on our work on the SU(2)L C! SU(2)R symmetric tliggs Yukawa Model with mirror fermion action. Our model describes a fermion Higgs system in the limit of vanishing gauge coupling. Setting the bare Yukawa coupling of the mirror fermions G x to zero, we want to determine the triviality bounds on the renormalized Yukawa coupling of the fermions G e e and the scalar self-coupling gn on 8a x 16 and 163 x 32 lattices.
1. I n t r o d u c t i o n F e r m i o n Higgs models with left right synm~etric action have been intensively studied over the last years (see [1] a n d references therein). T h e results o b t a i n e d so far are in good a g r e e m e n t with one loop p e r t u r b a t i o n theory b u t there are still strong finite size effects, especially at large values of the Yukawa coupling. W i t h increasing c o m p u t e r power c o m p u t a t i o n s on larger lattice sizes become feasible and we can get a b e t t e r control over the finite size efl>cts (beside the use of i m p r o v e d actions). T h e use of massivly parallel c o m p u t e r s boosts this development. T h e Q U A D R I C S Q16 is one of these which has a very a p p e a l i n g p r i c e / p e r f o r m a n c e ratio. In the following we report on our progress in i m p l e m e n t i n g the m o d e l which has been studied in [1] on the Q16. We present first, t i m i n g results of the p r o g r a m with d y n a m i c a l fermions and details of the i m p l e m e n t a t i o n .
2,~.;,2 m a t r i x 02 = ¢4 +i~r~4)~ k = 1,2, 3, where ak are the Pauli matrices and ¢~ are real variables. is the ferlnion pair of f e r m i o n ~ a n d m i r r o r ferInion k doublets, • = (t/,, 7~). Using c o n v e n t i o n a l n o r r n a l i z a t i o n s we have
{,
]2
+
x
;l=O
:r 4
-It ~
['7~,+,,7~ex+ V.+aT.x~
tt=--4
+,.,~ .¢,~
~ ,.
--
)] +
2. T h e M o d e l T h e lattice action of the m o d e l consists of a pure scalar p a r t a n d a m i x e d fermion scalar part S = S ~ , + S , . T h e scalar field c2 is represented as a *Supported by Deutsche Forschungsgenminschaft tamer grant PE 340/3-2 ISupported by Deutsche Forschungsgelneinschaft under grant Mu 757/4-3 0 9 2 0 - 5 6 3 2 / 9 5 / $ 0 9 . 5 0 © 1995 Elsevier Science B.V. SSDI 0920-5632(95)00336-3
All rights
reserved.
x,y
For the Hybrid M o n t e Carlo the n u m b e r offlayours has to be doubled so t h a t the last s u m becolues a sum over the lattice points and the two [tavours. T h e choice of the seven p a r a m e t e r s follows [1]. T h e W i l s o n p a r a m e t e r r is set to 1. In order
640
M. Plagge, D Talkenberger /Nuclear Physics B (Proc. SuppL) 42 (1995) 639-641
= 0.18
~c = 0.34
ith equal n on all b o a r d s
#0
board
# 15
F i g u r e 1. E x a m p l e r u n on the Q16
to s t u d y the t r i v i a l i t y b o u n d , we set the q u a r t i c coupling A to infinity. We choose p~,\ = 0.0 and G x = 0.0 to ensure the d e c o u p l i n g of the m i r r o r f e r m i o n s in the c o n t i n u u m limit. T h e fermionic h o p p i n g p a r a m e t e r K is fixed to K ~ = 0.125 [1]. T h i s leaves t h e f e r m i o n i c Yukawa coupling Gv, a n d the h o p p i n g p a r a m e t e r n as physical i n p u t p a r a m e t e r s . W e use ~ to t u n e the scalar m a s s and select the phase. 3. h n p l e m e n t a t i o n
on the QUADRICS
To s i m u l a t e the m o d e l we use the h y b r i d Monte C a r l o a l g o r i t h m with c o n j u g a t e g r a d i e n t for the m a t r i x inversion. For i n f o r m a t i o n on the p a r a n Jeter d e p e n d e n c e we refer to [1],[4] and [51. T h e t o p o l o g y of the Q16 is 2 × 2 x 32 which is n o t well s u i t e d for a l a t t i c e of size 8 a × 16. But it is p o s s i b l e to run t h e Q16 as 16 i n d e p e n d e n d Q1 b o a r d s w i t h 2 x 2 x 2 t o p o l o g y and p e r i o d i c bound a r y c o n d i t i o n s closing on each b o a r d . Therefore we decided to i m p l e m e n t the p r o g r a m in a sort of two step p a r a l l e l i s m . F i r s t we did a g e o m e t r i c p a r a l l e l i z a t i o n on the Q1 b o a r d s in d i v i d i n g the
l a t t i c e in 8 p a r t s a n d p u t t i n g each one on of the 8 C P U s . T h e second step is s i m p l y to run a different configuration on each of the 16 Q1 b o a r d s . T h e d r a w b a c k of this a p p r o a c h seems to be t h a t one has to e q u i l i b r a t e each configuration a n d so waste a lot of C P U time. F o r t u n a t e l y , this is not true in our case. Before we can s t a r t our m e a s u r e m e n t s , we have to t u n e the s c a l a r h o p p i n g p a r a m e t e r n in order to achieve a s c a l a r m a s s of 0.6 - 0.8 in l a t t i c e units a n d to choose the desired phase (here the broken, F M , phase; see [1]). Fig. 1 shows w h a t we do. We s t a r t on each Q1 w i t h a different value of ~;, t h e r m a l i z e these configurations and then d e t e r m i n e the scalar m a s s on the 16 b o a r d s . W h a t we get is shown in Fig. 2. F r o m this p i c t u r e we d e t e r m i n e the h o p p i n g p a r a m e ter value for the m e a s u r e m e n t s . W e pick o u t the configuration which lies closest to this values, duplicate it 16 t i m e s a n d r e s t a r t the p r o g r a m now with 16 equal c o n f i g u r a t i o n s a n d ~ values. If the new n value lies in between two old ~ values we do again a few t h e r m a l i z a t i o n sweeps, the n u m b e r of which is of the order 10 s m a l l e r t h a n on the original s t a r t configurations. A t the end we get 16 i n d e p e n d e n d results f r o m which d e t e r m i n e the q u a n t i t i e s of interest by the b o o t s t r a p m e t h o d . W i t h i n each c o n f i g u r a t i o n the errors are e s t i m a ted by a . j a c k k n i f e a p p r o a c h . T h e larger 163 × 32 l a t t i c e will be run on the D F G QH2 which will allow a m o r e flexible p a r t i tioning, so t h a t we can use the s a m e m e t h o d b u t with less configurations. A p r o b l e m which has to be faced qn the Q U A D RICS is its single precision f l o a t i n g p o i n t h a r d w a re. T h e r e are three p o i n t s in the p r o g r a m where global s u m s have to be c a l c u l a t e d a n d where the low precision can cause p r o b l e m s . In the m e t r o potis u p d a t e at the end of each t r a j e c t o r y s m a l l errors in the s u m m a t i o n p r o d u c e r a t h e r big effects on the u p d a t e p r o b a b i l i t y due to the e x p o n e n t i a l flmction involved. Errors in the scalar p r o d u c t s , which have to be e v a l u a t e d d u r i n g the conjugate g r a d i e n t i t e r a t i o n s , influence the convergence in an u n p r e d i c t a b l e m a n n e r . A n d , last n o t least, in the m e a s u r e m e n t r o u t i n e , where we e x p e c t the errors to play a less i m p o r t a n t role. W e have decided to use in all these cases a s u m m a t i o n procedure where the t e r m s are a d d e d tree like, so t h a t
M. Plagge, D. Talkenberger/Nuclear Physics B (Proc. Suppl.) 42 (1995) 639-641
2.5 IJ . . . . . . . . ' . . . . . .
,'~ /1l
' .........
.~,, , ~
' . . . . . . .
#Measurements is 500~ Loffice size is 8516 '
~.5
)~
12 0.5
O.Ot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.00
0.10
0.20
0.30
0.40
0.50
K"
Figure 2. r~ tuning at Gw = 0.3
641
to larger lattices. C o m p a r e d to the peak performance we achieved ,~, 45% in the conjugate gradient. These values are in good coincidence .with the results obtained with the performance analyzer. Lattices of size 43 x 8 can be run either on a single C P U or, geometrically parallelized, on a eight C P U Q1 board. We have measured a speed up factor of 6 for one hybrid trajectory. C o m p a r i n g the times for different lattice sizes on the Q1, an increase in the performance, measured in M F L O P S , with increasing lattice size can be seen. This is due to a better filling of the pipeline. To sum up, we can say t h a t the Q16 gives for our problem an increase in c o m p u t e r power of a factor 15 c o m p a r e d to a C R A Y YMP. This allows us to achieve high statistics in a reasonable time on lattice sizes up to 163 x 32. The main difficulties are the limitation on the code size and the single precision arithmetic.
REFERENCES at every step terms of equal size are added. For the r a n d o m n u m b e r s we use the internal r a n d l 6 library routine, so we have for each of the 128 C P U s an independend generator. A Q U A D R I C S feature which caused some problems is the limitation on the size of the executables. This limit lies at ~ 2.6MB, a size which can be reached very quickly by using compile time for loops even if the length of these loops is small (in our p r o g r a m typically 2-4).
4. T i m i n g results T h e m o s t time consuming part in the simulation is the conjugate gradient. It ntilizes almost 95% of the whole C P U time. We have compared the times for a single m a t r i x inversion with times from a C R A Y YMP. For a fixed n u m b e r of iterations we are a b o u t 10% faster on the Q I. If we use the residual as the stopping criteria, which is of course more realistic, we are about 10% slower. which results from an increase in the n u m b e r of' iterations. We think this increase is due to summ a t i o n errors in the scalar product, but we will have to clarify this point a bit more, before going
1.
2. 3.
,t. 5.
C. Frick, L. Lin, I. Montvay, G. Miinster, M. Plagge, T. Trappenberg, H.Wittig, Nucl. Phys. B 397 (1993) 431. I. Montvay, Phys. Lett. B 199 (1987) 89; Nucl. Plays. B (Proc. Suppl) 4_ (1988) 443. R. G u p t a , G. Gnralnik, G. Kilcup, A. Patel, S. Sharpe, T. Warnock, Plays. Rev. D 3_fi6 (1987) 2813. H. Wittig, Ph.D. thesis, University of Hamburg, 1992 M. Plagge, Ph.D. thesis, University of Miinster, 1993