Volume 189, number 3 PHYSICS LETTERS B 7 May 1987

MINIMIZATION OF SO (3) INVARIANT LANDAU POTENTIALS FOR SPIN FOUR

Yves BRIHAYE and Jean NUYTS Umverstty of Mons, B-7000 Mons, Belgtum

Received 17 January 1987; revised manuscript received 19 February 1987

The absolute minima for the SO(3) mvanant Landau potential for the nine-dimensional (spin-four) wreduclble representation is discussed as a function of the couphng constants. The non-maximal D4 stablhty group appears This is stu&ed m relation to Michel's conjecture.

I. Introductton. Spontaneously broken symmetry is a concept that plays a central role in the construc- tion of many unification models. Practically, one introduces a Higgs-Landau potential in the theory. In most cases, the minimization of the potential and the research of the unbroken symmetry group is pla- gued by technical difficulties.

One reason for Michel's conjecture [ I ] was to simplify these problems. The conjecture states: If the representation of the symmetry group G of a Higgs-Landau potential II(~) on R n is irreducible (on the reals) its minima have stability groups maximal in K (the set of conjugation classes of stability groups on a n \ {0}).

Recently, some counter examples were con- structed *~ which generated much activity. As a con- tributlon to the literature about Michel's conjecture, we present a discussion of the absolute minima of the most general Hlggs-Landau potential for one irreducible representation of spin four of SO(3). A more detailed and techmcal version, including irre- ducible representations of SO (3) for spins lower than or equal to four can be found in ref. [ 4 ].

Let S be the field, a four-index completely sym- metric traceless tensor of SO(3), basis of the irre- ducible representation of spin s equal to four (or d equal to nine). The most general Landau potential

v = 2 1 t Q + q Q 2 + p P + k K (1.1)

:t See for example for finite symmetry groups ref. [2], and for continuous symmetry groups ref. [3].

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is constructed out of the quadratic lnvariant

Q = SabcdSabca , (1.2)

the unique cubic invariant

P = SabcdScdej~efab , ( 1.3 )

and the quartlc invarmnt

K = Sab,j SabkIScd, kScasl • ( 1.4)

Note that all other quartic invariants are linear com- binations of Q2 and K. There are four coupling con- stants /z, q, p and k. The asymptotic positiwty condition which implies that Vbe non-decreasing at infinity in every direction implies that

q>0, 1 5 q + k > 0 . (1.5a, b)

The extremality equations are

O V = 2 ( l z + Q q ) O Q + p O e + k O K = O , (1.6)

where the gradient 0 is taken in the nine-dimensional space of the field. Hence, eqs. (1.6) are nine equa- tions cubic in the fields. The absolute minimum is the extreme field which has the lowest value for V.

2. S tabth ty groups and ex t r eme soluttons. The solutions of (1.6) can be characterized by their sta- bility groups. On general grounds [ 5 ], this group can only be one of the following subgroups of SO(3): SO(3) itself, the symmetry group of the cube, Do~,

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Volume 189, number 3 PHYSICS LETTERS B 7 May 1987

D4, D3, D2, C2 and the group of the identity ~2. In order to solve (1.6) we have constructed explic-

itly special configurations of S with any of these subgroups as stability group. These special configu- rations were then introduced m (1.6). The solutions for the remaining parameters were obtained and the potentials were computed.

Using the fact that the potential can be rescaled by a positive factor and the field S by a real positive or negative factor, it is easy to show that if we define the renormalized potential W by

W = ( q / p 2 ) V (2.1)

(the case/z equal to zero can be recovered by a suit- able limiting procedure); the resulting potential W depends only on the two scale invariant quantines

y=k/q, x=qp/p 2 . (2.2a, b)

After some algebra one obtains the following results: (a) Neither the group restricted to the identity nor

the C2 group correspond to stability groups of extreme solutions.

(b) The trivml solution where all the fields are zero, as is the potential

Wso~3) = 0 , (2.3)

is SO(3) invarlant. (c) The group of the cube is the stability group of

an extreme solution provided that

A 2 = 9 - 1 2 8 × ( 1 5 + y ) > 0 . (2.4)

The value of the potential Wfor such a configuratmn is given by

15 ( 9 Wc~b~ = ( y + 15) 1 ( y + 15)X

34 A 3 ) + (y+15)Zx 2 + 2 ,1 (y+15)2x 2 , (2.5)

where d is the positive square root of (2.4). (The negative root leads to a higher value of W.)

(d) The group Doo is the stability of an extreme

:2 C, Is a group which consists of the n rotations of angles 2 k/n around one axis. D, is C, extended by one reflection (C2) around one axxs orthogonal to the ax~s of C,. The group of the cube ~s the subgroup of SO(3) which leaves a cube mvariant Co is SO(2). D~ is the extension of C~ by one reflexlon.

solution provided

272 =36 _27 .7 × (245 + 8y) > 0 , (2.6)

correspondingly the minimal potential takes the fol- lowing form:

( 315 245 1 - 25

WD~---- (245 + 8y) "7(245+ 8y)x

3tl +

21172(245 + 8y)2x 2

3 2"~'3 + 21172(24-~-~8y)2x2,] , (2.7)

where 27 is the positive square root of (2.6). (e) The group D4 is the stability group of an

extreme solutton provided that

32 (4y- 35) - 2 9 y Z x > 0 . (2.8)

The potential is then

5 . 3 4 - ( 3 2 - 2 7 y x ) 2 y WD4= X22 14y 3 (2.9)

(f) The group D 3 is the stabihty group of an extreme field provided that

(23 .5 -7+ 13y) + 4y2x< 0. (2.10)

The potential is then

W D ~ = - 1 6 ( 1 - 9 ( y + 16) 4yx

) 16y3x 2 ( y + 2 0 ) . (2.11)

(g) For the group D2, let us define

X~ = a + B + y= 3p/2k, (2.12a)

272 =a2 +f12 +y2

= [-71 . t+27~(2k-7q)] / (3k+49q) , (2.12b)

273 =O~3 +f13 +73

=XI (47X 2 - 39X2)/56. (2.12c)

Provided

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Volume 189, number 3 PHYSICS LETTERS B 7 May 1987

D Y~/\ so(31

Fig. 1 Phase dmgram for the absolute mlmmum m the s=4 case as a function of the eouphng constants and more precisely as a function of the scale mvarmnt quantmes x and y (2.2). Useful coordinates are. for the triple points T = (1/560, 20), B = (0, m ), C= ( - ~ , 0); for the asymptotic point A= (~, -15) , and for the crossing points wath the coordlonate axis a = (1/240, 0), b = (0, 5(72 -310/3)/(37/3-8)), c= (0, 255)

- 18X32 + 36XI X 2 S 3 - 8X13,~3 --~ 3X 3 ,

- 2 1 X ~ X 2, +9X4X2 - X 6 > 0 , (2 .13)

i.e. p rov ided there are three real (o~, fl, y) solut ions of (2.12 ), the potent ia l cor responding to an ext reme field with stabil i ty group D2 is

Wo2 = [ 1 6 ( 3 y + 4 9 ) ] - '

X [ _ 2 4 . 7 2 + 2 3 . 3 2 . 1 7 y - J x 1

+ 3 4 x - 2 y - a ( 8 0 - y ) ] . (2 .14)

Once the algebraic expressions (2 .3) , (2 .5) , (2 .7) , (2 .9) , (2 .11) and (2.14) are obtained, it is easy to compute their absolute m i n i m u m as a function of the scale invar iant quant i t ies x and y in their respective allowed domains: (2.4), (2.6), (2.8), (2.10), (2.13). Note also that by the asymptot ic pos i t iv i ty condi t ion ( l . 5 b ) , one has to restrict the analysis to the half- plane

y > - 1 5 . (2 .15)

We have noted that nei ther D2 nor D 3 cor respond to an absolute min imum. On the other hand, there are regions in coupling constants space where SO (3) , Doo, the group o f the cube and D4 are the stabil i ty group o f the absolute min imum.

The object o f fig. 1 is to draw the phase d iagram of the absolute m i n i m a as a funct ion o f x and y in the upper half-plane ( y > - 15) m a p p e d onto a rectan- gle. The figure exhibi ts the occurrence of three t r iple points T, B, C; o ther useful points are given in the

capt ion o f the figure, namely the intersect ions with the coordinate axis (a, b, c) and the asymptotic point A. The curve from T to A has the form

x = l / 1 6 ( y + 1 5 ) , (2 .16)

and from T to B the form

x = 3 4 / 2 4 . 7 ( 8 y + 245) . (2.17)

The other curves have much more complicated forms and we only give their asymptot ic behav iour

T - C : y 2 x - ~ - 0 . 7 2 3 f o r y - - . 0 ,

c - C : y 2 x _ ~ - 12.18 f o r y - * 0 ,

c -B :yx "~ -c~-g for y - ~ ,

as the exact expressions are po lynomia ls o f high degree in y and x.

3. Conclusions. Contrary to one 's expectation based on a vers ion o f Michel ' s conjecture we obta in a very large doma in o f absolute m i n i m u m for the non-max- imal discrete group D4 while the maximal cont inu- ous D ~ group is restr icted to a small band in the middle o f the phase diagram.

The authors would like to thank Dr. H. Caprasse for in t roducing them to the use o f R E D U C E which was essential in obta in ing some o f the analyt ical results. One of the authors (J .N.) is grateful to Pro- fessor H. Ruegg and the Univers i ty o f Geneva for a visi t which p rompted this investigation.

References

[1] L Michel and L Ra&catl, Ann Phys 66 (1971) 758, Ann Inst Henn Pomcar6 A18 (1973) 185, L Michel, m. Regards sur la Physique Contemporame (CNRS, Pans, 1980) p 157, prepnnt CERN-TH 2716 (1979).

[2] D. Mukamel and M Jarlc, Phys. Rev. B29 (1984) 1465, M. Janc, Phys Rev Left 51 (1983) 2073

[ 3 ] M Abud, G. Anastaze, P Eckert and H Ruegg, Phys Lett. B 142 (1984) 371,Ann. Phys. 162 (1985) 155; J. Burzlaff, T. Murphy and L O'Ralfeartalgh, Phys. Lett B 154 (1985) 159

[4] Y. Bnhaye and J Nuyts, Mmlmlzatxon of Landau potentials mvanant under SO(3), Mons preprlnt (December 1986),

[5] L. Mlchgl, Rev Mod Phys. 52 (1980) 638

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