Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

S T U D Y O F T H E P H A S E S T R U C T U R E O F A N S U ( 3 ) H I G G S M O D E L A T F I N I T E T E M P E R A T U R E ~

F. K A R S C H 1

*CERN, CH 1211 Geneva 23, Switzerland *
E. S E I L E R 2

*M a x Planck Institut fur Physik und Astrophysik - Werner Heisenberg Institut fur Physik - *
*D 8000 Munich 40, Fed. Rep. Germany *

a n d

I.O. S T A M A T E S C U

*Institut fur Theorie der Elementarteilchen, Freie Universiti~t Berlin, *
*D 1000 Berlin, Germany *

Received 18 February 1985

We analyse numerically an SU(3) Higgs model with complete symmetry breaking and radial degree of freedom on asymmetric, periodic lattices. The character of both the Higgs and deconfining transitions is found to depend on the Higgs self-coupling and on a parameter which may simulate the number of flavours. In particular, an increase in the latter leads to the disappearance of the deconfining transition for small Higgs masses.

*1. I n t r o d u c t i o n . * The phase structure o f gauge
models with Higgs fields that " b r e a k the s y m m e t r y
c o m p l e t e l y " in the traditional perturbative language
still remains insufficiently understood. The main
reason is the so-called duality [ 1 - 3 ] between con-
finement and the Higgs mechanism and - closely re-
lated to it - the nonexistence o f a simple order pa-
rameter characterizing the Higgs ( " b r o k e n s y m m e t r y " )
phase unambiguously.

On the other h a n d the p o p u l a r "inflationary scenarios" for the very early universe [4,5] rely in an essential way on the existence o f a first-order transi- tion between the " b r o k e n " and " u n b r o k e n " phases at

* Presented at the "Lattice co-ordination meeting", CERN, Geneva, December 1984.

1 Present address: University of Illinois, Urbana, IL 61801, USA.

2 Present address: Princeton University, Princeton, NJ 08544, USA.

60

least at high temperature. Earlier Monte Carlo searches for such a p h e n o m e n o n in SU(2) models b o t h with adjoint [6,7] and fundamental Higgs [8,9] do n o t always give unambiguous evidence o f its occurrence.

In this paper we go from SU(2) to SU(3).

F o r this group it is n o t quite as straightforward to " c o m p l e t e l y break the s y m m e t r y " as for SU(2);

a single fundamental SU(3) Higgs field would still leave an SU(2) subgroup unbroken. What would be needed is a " f l a v o u r " multiplet o f fundamental Higgs fields. On the other hand, in the case o f SU(2),

" c o m p l e t e b r e a k d o w n " is achieved b y a single fun- damental Higgs field which can also be described b y an SU(2) matrix, together with a r a d i a / d e g r e e o f free- dom if desired. This suggests to t r y to achieve "com- plete b r e a k d o w n " in SU(3) also b y choosing for the Higgs fields multiples o f SU(3) matrices. A t first we thought that this choice would not be in any essen- tial way different from a conventional model with a 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V.

V o l u m e 157B, n u m b e r 1 PHYSICS L E T T E R S 4 July 1985

multiplet of Higgs fields. The often surprising resuks we are describing in this article, however, make this belief rather doubtful. In particular the model shows a wealth of first-order transitions, some of the sort that that should be liked by inflationary cosmologists.

We are not sure, however, to what extent these phe- nomena are artifacts of the model. We are currently checking this by running a similar study of a model with a more conventional Higgs multiplet [10].

That our model might show some peculiar fea- tures could perhaps be expected because, unlike in the case of SU(2), the multiples of SU(3) matrices do not form a linear space, so a perturbative analysis is not easily possible. The smallest linear space in which our Higgs fields could be imbedded is the space of all complex 3 × 3 matrices [ 11 ] ; the restriction to the multiples of SU(3) matrices can be thought of as arising as a limiting case of a rather complicated po- tential.

*2. The model. *Our model has the following degrees
of freedom: The usual gauge fields (link variables

*{ U u ( x ) } , U u ( x ) E *SU(3)); Higgs fields living on sites,
consisting of radial and internal degrees of freedom
*( R ( x ) , *V(x)} (V(x) E SU(3)).

The action we choose has the form
2n ~ ] Re Tr *R x V x Ux, u V x + u R x + . *+
*S = - -3- x , . *

+ ~ ] R 2 + X ~ D ( R 2 - 1 ) 2 - ~ ~ ] R e T r U a p .

x x P l a q . (1)

This action contains the parameters K, X, 3 corre- sponding to the Higgs mass, coupling constant of the Higgs self-interaction and gauge coupling constant.

In order to fully specify the model we have to give a probability measure on the field configurations.

This will be given by

Z - l e - S 1-I *dUx, u r I d V x I-I R f x d R x . * (2)

*X , U * *X * *X *

*d U u, d V *denote the Haar measure on SU(3), dR the
Lebesgue measure on [o, co). It should be noted that
there is an additional free parameter f appearing in
this measure. Unlike the case of SU(2), where f = 3
would lead to the standard fundamental Higgs model,
here there is no canonical choice o f f . We will look at
various casesf = 3 , f = 8 , f = 17 and f = 50, and we

will see that the phase structure is rather sensitive to this p a r a m e t e r . f = 8 is suggested by the dimension of SU(3) and analogy with S U ( 2 ) , f = 17 by the di- mension of the space of complex 3 × 3 m a t r i c e s . f = 50 was chosen to have some comparison with QCD with fermions (see b e l o w ) . f = 3 is just a small value.

To understand, at least roughly the effect of vary-
ingf, consider that there were a natural choice for it,
*f = f o * say, leading to an effective potential without

logarithmic terms (e.g.f0 = 3 for SU(2)). Then using the measure eq. (2) w i t h f # : f 0 simulates a term

*- ( f - fO) l n R x * (3)

in the effective potential, producing additional struc-
ture at *R x *= 0. Due to quantum effects the loga-
rithmic singularity at zero will be smeared. F o r f > f 0
we thus get a peak at *R x *= 0, resembling a Higgs po-
tential, even if the original action eq. (1) has positive
mass squared (K < ~ (1 - 2X)). F o r f < f 0 on the
other hand, a Higgs-type potential (K > ~(1 - 2X))
will develop a second deep at the origin which can
lead to a first-order phase transition.

Similar effects have been observed for SU(2) with f = 0 , 3 [9].

*3. Results. *As remarked before our model has a
tendency to show first-order transition. As in QCD
with fermions [12-14] at finite temperature and low
K we find persistence of the first-order deconfinement
transition of the pure SU(3) gauge model. But we also
find first-order transitions by varying K at fixed/3,
corresponding to a transition between "broken" and

"unbroken" phases. Generally these transitions be- come weaker, sometimes become second order (or even disappear altogether) as we increase f and/or X.

The limiting case X = oo (corresponding to R = 1) seems to show an at most second-order transition.

On the other hand it is remarkable that at low f ( f =
3) and at ~, = 0 even the "spin model"/3 = co has a
first-order transition. This transition is not quite like
the usual magnetizing transition in spin models. The
main effect seems to be a transition between a phase
in which *R x *fluctuates around small values to a phase
in which *R x *fluctuates around a value of the order
of Rcl. Of course the phase with small *R x *values is
at the same time disordered because the link term in
the action is small for small *R x , *whereas in the other
phase we find spontaneous magnetization. So at the

61

V o l u m e 157B, n u m b e r 1 PHYSICS L E T T E R S 4 J u l y 1985

transition both

*<Rx> *

and the magnetization c ~ jump.
This way o f modelling ferromagnets with first- order transitions ( ' ~ h o u l e s s effect") m a y be o f some interest.

For X = 0 our models become unstable for K > ~.

But there is a clear signal o f a phase transition at/3 =

**0% K = K c < ** ~-, ( a t f = 17). This transition also occurs
for/3 < ~ and seems to become actually stronger
there.

Even more remarkable is the following phenome- non: For f = 8, X = 0.1 we have a clear second-order transition in the spin model (/3 = oo). This transition, however, seems to become first order as we turn on the gauge coupling (fl < oo) and then disappear again for very strong gauge coupling (the transition is defi- nitely gone by/3 = 0.5 but we did not try to localize the end point).

We summarize our results in four figures. The cal- culations have been done on the Cray-1 computers o f IPP Garching and o f WRB Berlin. We used vectorized programs as have been described in ref. [15]. The simulations have been done on lattices o f size NtNs 3 with N t = 2, N s = 4, 6, 8. We take typically some thousands o f sweeps per point.

Fig. la shows a rough phase diagram f o r f = 8, X = 0.1. We use the values o f R = <R 2> and o f the Polyakov loop expectation P = (Re Tr II t

*U(x,t), 0 *

to distinguish
three regions:
R , P large : the Higgs region, R small, P large : the deconfined region, R, P small : the conffmement region.

A l t h o u g h R , P are not order parameters, the various transitions are accompanied b y jumps or strong varia- tion in these expectations. Figs. l b - l d show typical behaviour o f various observables across the transition lines.

Here and in the following:

A = ~/3(Re Tr(Uap)),

*H *

^{= }*~K(Re TrRxV x + Ux,#Vx+~Rx+#), *

e G = A (time-like plaquette) - A (space-like plaquette),

t h ( p / 1 0 )

. 5 1.

**1.°° ** **' ** **' ** **' ** **' ** **' ** **' ** **' ** **' ** **' ** **/ 1 **

R, P: l a r g e

. 5 , E

.4 "" A

Q .3

F . 5 t h l 2 ~ e ) a e

. 2

| R : s m a l l . 1 2 5 R , P : s m a l l K I P : l a r g e

.1

0 , , , R , . , , 0

1 2 3 Z, 5 6 10

I. I. oo o ° o I . - ] z~ n Az~ o o

### , ta°°° oo

.5 & 5 7 o o o 8D o o

0 , i i i f i 0 ' , i T ,

1. L J B v vv

I v ~vvv

**ooe **•
.5

.35 .is 0 4 . 9 ~ ; , : , : /..95 . 2 7 " ' " " .3

ae ff ae

F ~ . ]. General picture for f = 8, ~ = 0.1. (a) Phase structure a n d transitions: Higgs and finite t e m p e r a t u r e . T h e capital letters (A, B, E, ...) denote r u n s along ~ = const, or r = const.

lines, b y w h i c h t h e transitions were localized: first-order tran- sitions (A, Q, B, R); second-order or weaker (F, K, L); no transition (E, M). (b) T h e behaviour o f s o m e YM-observabtes across the transitions A, L, B: o: P; o: A X 1/2 (X 1/4 for B);

~x: eG × 4. (c) Same for s o m e Higgs observables: o: R X 1/10;

X : H X 1 / 1 0 ; V : e H X 4.

### 1 °°

° o o o oOO o °

b ) .5 ~ s ~ , , ,,

oeO

o 8 ~ , ~ ,

**eoll **

^{• }

c ) .~ " " 7 " ;'

0

e H = H (time-like link) - H (space-like link),

where N V is the lattice volume.

Next we illustrate the fact o f the Higgs transition as function o f f , b o t h for the spin model (fl = oo; fig.

2) and for the Higgs model (/3 < ~ ; fig. 3). As noticed above, the transition o f the spin model, which for f = 8 is second order, becomes first order f o r f = 3 while getting weaker for f = 17. For/~ < oo the transition is

**Volume 157B, number **1 P H Y S I C S L E T T E R S 4 J u l y 1985

**it:i ** ^{1.: }

• o l

. 5 x x

**0 ** x x x

**.:h ** **.~3 **

. 5 **s **

**o o • **

0 0

### vV%

.3-1

.2-1 .1-1

**0 **

**• ** **• o e e " **

x x
**x **
x x > F

**.~2' ' '.:~6 ** **.1'7 /8 **

a e

1.

.5

**x ** x
x
x x x

0

### .~ ' l h

v

x
**i **

**x ** **x X X ~ **

.07 .09

**a) ** **b) ** **c) ** **d) ** **e} **

Fig. 2. **The spin model transition **(/3 = *~). (a) f = 3, k = 0.1 ; (b) f = 8, h =0.1 ; (c) f = 17, X = 0.1 ; (d) R = 1 ; (e) f = 17, h = 0.

**Here: **o: R X 1/10 (R × 1 / 2 0 for ( e ) ) ; X : H × 1 / 5 ; V : e H X 4 ; o : Q ~ .

**steeper becoming first order for f = 8 while apparent- **
**ly remaining second order for f = 17. Finally, going to **
**X= 0 seems also to strengthen the transition, while **
**fixingR x = 1 weakens it. **

**Finally we discuss the fate of the deconfining tran- **
**sition. For small K it is first order, weakening as ex- **
**pected with increasing K (and also with increasingf **

**1.5- **

. 5

**o o ~ **

**, ** **r **

v v ,7 v ~ T v

**• ** **! • **

**e e ** **x x **

**• ** **x **

× x X

**.~s ** **.5 **

a e

**b) **

1 5 #, ~ ~ a 1 1.5

z~

. 5 , ~r , , 0 5

**ooooou **
**ooo ooo **

**, ** **, ** **f ** ,.

I v v v v 1. v v v V ~ V v

~7

.5 • • " ' " ~ .5 :.,

x x

Ot " * * o

**.is ** **22 ** **'1'1'112' **

**c) ** **d) **

1 5 tszx zx

. 5 ~

**1-i ** v v

**e e **
.5"

O' **x X **

1 0 5 .115

a e

**e) **

Fig. 3. **The Higgs transition at/3 **< o0. (b) f = 8, h = *0 . 1 , / 3 = 5 . 6 *
( Q ) ; ( c ) f = 17, k = 0 . 1 , / 3 = 5 . 6 ; ( d ) R = 1,/3 = 6; ( e ) f = 17,
h = 0,/3 = 5.2. **Here o: **P ; u: A = 1/4; a: eG X 4 **(right scale); **

e : R × 1 / 1 0 ( R × 1 / 2 0 f o r ( e ) ) ; × : H X 1 / 5 ; V : e H × 4.

**and/or k) see fig. 4. In fig. 4 we present the situation **
**for f = 50, ~ = 0. The reason for using such a high **
**value for f i s the following. To the lowest order in **
**the effect o f the Higgs fields can be obtained from **
**an effective action with a Polyakov loop term (we **
**consider lattices with N t < 4) **

### (2K)Nt(1R'2)Ntl/TrNs\ t---1 *U(x't)'o+h'c') " * (4)

**Here **

**R-- ~ _ f dr r/+ **

**R-- ~ _ f dr r/+**

**2 exp( - r 2) _ f + 1**

**f dr r£ **

**f dr r£**

**e x p ( - r 2)**

**2**

**(5)**

**To the same order in n QCD with N f fermions pro- **
**duces a similar term [12] **

**N t **

**(2K)Nt2/Vf(Trt[I 1= **

**U(x,t),6+h.c.), **

**U(x,t),6+h.c.),**

**(6)**

**Thus to the lowest order in K the two models coin-**

**cide if we take**

f = 1 2 ( 6 N f ) 1 *] N t _ 1 , * ( 7 )

**leading t o f ~ 50 f o r N f = 3 , N t = 2. Notice that we **
**could not obtain the same effect by choosing a larger **
**g because this latter is the expansion parameter. What **
**we need is to introduce many more flavours and this **
**we can simulate economically by choosing a large f. **

**The results in fig. 4 show the weakening o f the **

### 63

Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

0 th (13/10) .5

**• ** **- ** **-t- ****" q .****.****.****.** **X **

i I - .

0 1 2 3 Z. 5 6

**a) **

i i = ~ i , i

Z

o

**"t o **

.1 $ { D

### oo ,{,,,

0 .05

### b)

z x

. 5 o z x

e °

0 " , , J

4.91~5.1

### c) ..""

-.10 O I I 0 0

**j ** **o **

### 22 O°oOO

### oO **] 2o°. **

o x xxx~

° ~ N'e'~ 0 . . . .

.03 .05 .07 .09 0 1 2 3

ae **r~ **

### d)

### i1/°

^{. 0 7 }

^{.1 }

7 '

0

o

### J

.67 ' ' .~

**el **

:.1

Fig. 4. The deconfining transition for f = 50, h = 0. (a) The
transition line. (b) The jump in some observables as function
of K (points R, C, D); o: P; a: eG. (c) The behaviour across
the transition at C: o: P; **n : **A × 1/2; a: e G X 4. (d) The gen-
eral behaviour along the lines X, Z. o: P; X : H × 1/20 (both
left-hand scale); o : A × 1/2, o: R X 1/50 (both right-hand
scale). (e) Cross-over region below # = 2.4. o: P; o: A; z~: eG;

V: e H.

( f o r small ~ still first o r d e r ) d e c o n f i n i n g t r a n s i t i o n up t o its c o m p l e t e disappearance: t h e t r a n s i t i o n does n o t e x t e n d o u t s i d e t h e r e g i o n / 3 / > 2.4, K ~< 0.1 (we did n o t t r y to localize precisely t h e e n d p o i n t ) . H o w -

ever a k i n d o f cross-over region m a y b e o b s e r v e d f o r /3 ~< 2.4. T h e e f f e c t o f b o s o n i c m a t t e r on t h e d e c o n - fining t r a n s i t i o n for small masses c o n f i r m s t h u s t h e q u a l i t a t i v e b e h a v i o u r o b t a i n e d already in t h e K pa- r a m e t e r e x p a n s i o n at larger masses [12]. This has, o f course, n o direct i m p l i c a t i o n for light f e r m i o n s effects.

O n e o f t h e authors (IOS) wants t o t h a n k P.

H a s e n f r a t z , C. Lang and V. L i n k e f o r discussions and remarks.

*References *

[ 1 ] G. 't Hoof t, Recent developments in gauge theory (Carg~se, 1979), ed. G. 't Hooft (Plenum, New York, 1980).

[2] K. Osterwalder and E. Seiler, Ann. Phys. (NY) 110 (1978) 440.

[3] E. Fradkin and D. Shenker, Phys. Rev. D19 (1979) 3682.

[4] A. Guth, Phys. Rev. D23 (1981) 347.

[5] A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220.

[6] F. Karsch, E. Seiler and I.O. Stamatescu, Phys. Lett.

131B (1983) 138.

[7] T. Munehisa and Y. Munehisa, Nucl. Phys. B215 [FS7]

(1983) 508.

[8] H. Kuhnelt, C.B. Lang and G. Vones, Nucl. Phys. B230 [FS10] (1984) 31.

[9] V.P. Gerdt, A.S. Ilchev, V.K. Mitryushkin, I.K. Sobolev and A.M. Zadorozhny, preprint Dubna E2-84-313 (1984).

[10] C. Hege, V. Linke, E. Seller and I.O. Stamatescu, work in progress.

[11] G. Mack, Commun. Math. Phys. 65 (1979) 91.

[12] P. Hasenfratz, F. Karsch and I.O. Stamatescu, Phys.

Lett. 133B (1983) 221.

[13] J. Polonyi, H.W. Wyld, J.B. Kogut, J. Shigemitsu and D.K. Sinclair, Phys. Rev. Lett. 53 (1984) 644.

[14] T. ~elik, J. Engels and H. Satz, Phys. Lett. 133B (1984) 427.

[ 15 ] I.O. Stamatescu, preprint MPI, MPI-PAE-PTh 92/83 (1983).