Internal Einstein spaces and symmetry breaking

Volume 143B, number 4, 5, 6 P H Y S I C S L E T T E R S 16 Augus t 1984

INTERNAL EINSTEIN SPACES AND SYMMETRY BREAKING

R. COQUEREAUX

Centre de Physique Theorique, C N R S - L u m i n y , Case 907, F - 1 3 2 8 8 Marseille Cedex 9, France

Received 27 Februa ry 1984

We first define a general ised gauge invar ian t Y a n g - M i l l s lagrangian: the Ki l l ing metr ic - K=# on the group is replaced by a more general met r ic h,~# ( x ) ; the field h,~# ( x ) - a scalar f rom the s p a c e - t i m e poin t of view - is then covar ian t ly coupled to the gauge field A~, '~ and is also self-coupled via a na tura l scalar po ten t ia l (no parameters) . Non- t r iv ia l saddle poin ts of this scalar po ten t ia l cor respond to non- s t anda rd Eins te in metr ics on the group G. The associa ted shifts lead to an ent i re ly computab le mass spec t rum for the gauge field.

1. The lagrangian. When one considers the usual Yang-Mills lagrangian - K(F~,, F ~ ) and thinks about possible generalisations, a simple idea comes to the mind: replace the Killing metric - K ~ t ~ by a more general metric h,# i.e., replace a bi-invariant metric (G x G) on the internal space G by another metric which is less symmetric. It is almost im- mediately clear that h,B has to be at least G-invariant, in other words it has to be entirely char- acterized by its value at the identity of G, i.e. by its action on the elements of the Lie algebra (one can say also that, by an appropriate choice of basis, h~B is independent of the coordinates on the group). Therefore, the isometry group of G will be of the kind K x G (where K is a subgroup of G, possibly trivial). However if h ~ is nothing but a symmetric matrix of constant real numbers, it is easy to see that gauge invariance (with respect to G) is broken, unless in the case where h ~ is proportional to K,, a. To cure this disease, the obvious solution is to decide that h ~a will be a field h ~ a ( x ) and not a constant matrix; of course x is a point in space-time. From the intuitive point of view, we associate to each point x of space-time, a quantity h ~ a ( x ) measuring the shape of the group G at this point. Now of course, h ~ a ( x ) transforms according to some representa- tion of G (the bilinear symmetric) and we have to couple h ~ a ( x ) covariantly to the gauge field. It can

be checked that

- ¼F~,Ft '~h ,q~(x)

- ¼h"~(x )hVa(x )D~h ,~v (x )D~ 'h l~n(x )

+Dl, h , , ~ ( x ) O g h v s ( x ) ,

with

D~,h,, B ( x ) = 01,h,~#( x ) - Av~C~,,h~l ~ - A~C~vh~a

is indeed a gauge invariant quantity (we discard total divergences). Now, we have a Yang-Mills field and a kinematic term for the scalar field (h ,~p(x) is indeed a scalar field from the space-time point of view); however we would like to use a kind of potential term for the scalar field h, ,B(x ). Being a bilinear symmetric quantity, the most obvious choice for this potential is V ( h ) = - [sca lar curvature associated to h~#(x)]. Ex- plicitly, when G is unimodular (in particular when G is compact),

7 a 1 aa" 7 7" "r ° = - V [ h ] = h a a ' ( ½ C ~ , C ~ , v - z h hvv,C~#C~,#,),

C~aV being the structure constants associated to a basis ( X, }, orthonormal for the Killing form. The minus sign in front of V(h) is there for reasons of positivity.

The reader may wonder why we choose such a complicated kinetic energy term for h~/~ and not

0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

403

Volume 143B, number 4, 5, 6 PHYSICS LETTERS 16 August 1984

something like (D,h,BD"hv8)8"v6a~; it should then be not iced that by making a l inear approx ima t ion : h ~B( x ) = 6oB + ~ ,a( x ), the first term in the expan- sion would indeed be of the above kind. The mot iva t ion for such a kinet ic energy term and for the scalar po ten t ia l comes f rom the following: when the above lagrangian is covar ian t ly coupled to Einstein gravity, the whole lagrangian can be re in te rpre ted as a theory of pure gravi ty (wi thout ma t t e r fields) in d imens ion 4 + d im(G) ; see ref. [1] where a more general s i tuat ion is investigated). Moreover , this kinet ic energy term is a k ind of genera l i sa t ion of the non- l inear o model : for a f ixed volume of G (det h = const.), the field h~a varies in the mani fo ld S L ( n ) / S O ( n ) , n = d im G, this is indeed the man i fo ld of G- invar i an t metr ics on G. The Q F T exper t will have also not iced that, by power count ing arguments , the above lagrangian looks b a d f rom the renormal i sa t ion po in t of view; however, if one remembers that it can be wr i t ten as a theory of pure gravi ty (in more d imen- sions), even if one expects s t rong divergences in the associa ted quan tum field theory, one also expects s t rong cancel la t ions; therefore the subject deserves a more de ta i led study.

2. Motioation for Einstein spaces and symmetry breaking. On one hand, we saw that the poten t ia l for scalar f ield is in te rpre ted (or def ined!) up to a sign as the scalar curvature of the internal space G at the po in t x, on the other hand, there is an old theorem (due to Hi lber t ) which says that saddle po in ts of the total scalar curvature - cons idered a funct ional on the space of metr ics - for f ixed volume, coincide with the Einstein metrics. Put t ing these two facts together and remember ing that the scalars h ~ ( x ) are covar ian t ly coupled to the gauge field, one m a y look for an analogue of the Higgs mechanism.

3. Einstein metrics on groups. The Kil l ing metr ic on a Lie group G is an Einste in metric, this is well known and is fo r tuna te in our context since this p rope r ty tells us that the usual Y a n g - M i l l s lagrangian is indeed associa ted to a saddle po in t of our more general lagrangian. There exist in general - bu t not a lways - m a n y other Einstein metr ics on a given Lie group G however we are only inter-

es ted here in those which are K x G invariant . As s ta ted previously, a very old theorem assures that for a given mani fo ld S, the total scalar curvature f~'Sd vol, cons idered like a funct ional on the space of metr ics with fixed volume admi ts saddle poin ts for all Einstein metrics. The cons t ra in t of fixed vo lume can easily be unders tood: consider for ins tance a usual two-sphere of radius R, its scalar curvature can be made a rb i t ra r i ly small or large by modi fy ing its rad ius (~- = 2 / R 2 ) ; this k ind of vari- a t ion is not interest ing, we have to fix the volume and s tudy the var ia t ion of the total scalar curvature. In the special case under study, the total scalar curvature is equal to the p roduc t ~- x vol(G) and we have only to look at var ia t ion of ~, ~- being cons idered as a funct ional on the space of G- in - var iant metr ics (of course, with such a restr ic t ion we only get a necessary condi t ion but, for all the cases t reated here, it can be shown to be sufficient). The basic s t ra tegy used to ob ta in non - s t a nda rd Einstein metr ics or groups is more or less always the same: one first chooses along K in G = U ~ c / K ( g K ). This amounts to consider G as a col lect ion of copies of G / K glued together and pa ramet r i sed by K; one then chooses some na tu ra l metr ic on g (for example the G × G invar iant Ki l l ing metr ic) and begins to "d i s to r t i t" in a way app rop r i a t e to the coset decompos i t ion ; in the ob ta ined family of new metrics, one looks for those where the Einstein condi t ion is satisfied, ei ther by comput ing direct ly the Ricci tensor or by looking at saddle po in ts of some functional . Let us do it expl ici t ly in the special case where G is a s imple compac t Lie group and where moreover K is a subgroup of G such that (G, K) is a symmetr ic pair . We now star t f rom the fol lowing b i - invar ian t (and Einstein) metr ic on G: g = - ( K i l l i n g form). W e can now compute the scalar curvature T G o f G in terms of the scalar curvatures ~.~/K, ~K of G / K and K associa ted to the cor respond ing res t r ic t ions of g. We f ind .re = ~.G/K + ~.K _ ¼rc~ ab~r'~b,, or

= k s + c k k - - k(1 - c ) .

W e write a, a, h for indices in G, G / K , K, where n = d i m G , k = d i m K , s = n - k = d i m G / K and c, the index of K in G being def ined as follows: Kil l ing metr ic of G restr ic ted to K = gab = -C"aaCab~ , Kil l ing met r ic of K = gKab =

404


d - - C aeC Zd and calling c the coefficient gKaL = cgaT,.

The previous decomposi t ion of r G is a " K a l u z a - K l e i n reduct ion" (before K a l u z a - K l e i n ) where the "external" space is G / K , the " in t e rna l " space is K and the field strength is C",h. Notice that when (G, K) is a symmetric pair we have indeed c = 1 - s / 2 k but this would not be true in the general case. Fol lowing the general recipe, we now write L i e (G)= L i e ( K ) • P - orthogonal decomposi t ion for g - and consider the following family of metrics (t is a real parameter) on G: h = g / P + t 2 g / K .

These metrics, obta ined by a scaling of g in the direction K are no longer G × G invar iant but only G × K invar iant ; the scalar curvature of G is n o w

r ° = s / 2 + ( c k / 4 ) / t 2 - k(1 - c ) t 2 / 4 .

However, when t varies, the volume of G varies; in order to keep it fixed we jus t have to make a conformal rescaling and consider the family of metrics h = ( 1 / t 2 ) * / n h , then det h = ( 1 / t 2 ) k • ( t2) k = constant.

The associated scalar curvature reads

r 6 = ( , 2 ) * / " [ s / 2 + ( c k / 4 ) / t 2 - k(1 - c ) t 2 / 4 ] .

We now vary this expression with respect to t and find

d c G / d t = - ( s / 4 ) ( Z k + 2 ) / ( k + s ) . t 2 k / " - 3

× (t 2 - 1) [ t 2 - ( 2 k - s ) / ( Z k + s ) ] ,

where we used the property c = 1 - s / 2 k , valid for a symmetric pair. We find therefore two Einstein metrics: corresponding to the values t 2 = 1 and t 2 = (2k - s ) / ( 2 k + s) .

The first value corresponds of course to the b i - invar iant metric g used in the beginning, the other value corresponding to a non-s t andard G x K invar iant metric on G (for example, s tudying G = SU(3), if we choose K = SO(3), we obta in an S U ( 3 ) x SO(3) non-s t anda rd Einstein metric on SU(3) for the value t 2 = (2 x 3 - 5 ) / (2 x 3 + 5) = 1/11) . A general study based on the direct compu- ta t ion of the Ricci tensor is carried out in ref. [2]. M a n y comments and references on relative topics may be found in ref. [31.

4. S y m m e t r y breaking. In the Higgs setting, one looks for a non-zero local m i n i m u m V 0 of a suita- ble fourth degree G- invar ian t polynomial and per- form the shift qb(x) = V 0 + q~'(x). In our approach we write formally h , B ( x ) = h° ~ + h ' , a ( x ) , h°~B being a homogeneous Einstein metric for the group G.

Let us analyse the si tuat ion when G is un imod- ular (then C ~ b = 0) and when h°~a is a G × K invar iant Einstein metric obta ined by the above method (we even suppose that (G, K) is a symmetric pair, then the critical value of the scaling parameter is t 2 = (2k - s ) / ( 2 k + s) .

Using the defini t ion of D,, the term

L = - ¼h'~Bh ~a (D.h,~,8 D~'h.y~ + D~,h,,vD~'hCa )

can be expanded and we get

where M~a = J . ¢ + K~¢; K~a = C~yC~a being the Kil l ing form and J = h YY't~ t ~ t ~ '

aB " 8 8 " ~ a ' g ~ f l ~ ,'"

We now make formally the shift h ( x ) = h ° + h ' ( x ) . The calculat ion is straightforward, we find that

i A I A a A t ~ B _ 1 l lA A ~ B - 2.- . .¢ . . .1 . = 5 . . . . ¢1 . .~ + Rest,

where

= 0, = 0, & b = 1 ( t 2 + 1 / , 2) - 1.

In other words, when t 2 = 1 (i.e., we expand a round the bi - invar iant metric of G), the gauge field stays massless; however, when t 2 = (2k - s ) / ( 2 k + s )

(i.e. we expand a round a non-s tandard Einstein metric on G, G × K invariant , with the nota t ions of section 6.2.4) [3] then, the components of the gauge field taking their value in Lie(K) stay massless bu t the components lying in the subspace P ( L I E ( G ) = L i e ( K ) + P ) acquire a mass m 2 = 2 s 2 / (4k 2 _ s 2) in dimensionless units.

Let us assume for example that our in ternal space is G = SU(4) and that the volume is fixed, then if we expand the scalar fields h ~a a round the b i - invar iant metric, we get 15 massless gauge fields bu t we can also expand h~t ~ a round the following

405


-TGI

Fig.

I I ) t 2

G x K invar iant Einstein metrics: (i) K = S(U(2) x U(2)) = SU(2) x SU(2) × U(1)

for t 2 = 3 /11 . (ii) K = USp(4) for t 2 = 3 /5 . (iii) K = SO(4) for t 2 = 1 /7 .

In case (i) we get k = 7 massless gauge fields and s = 8 massive fields of mass m 2 = 32 /33 . In case (ii) we get k = 10 massless gauge fields and s = 5 mass ive fields of mass m 2 = 1 /15 . In case (iii) we get k = 6 massless gauge fields and s = 9 massive fields of mass m 2 = 18 /7 .

In the previous examples , the pa i r (SU(4), K) is symmet r ic bu t there exist o ther saddle poin ts (other Einste in metrics) involving non-symmet r i c pairs.

There is a diff icul ty which is be t te r expla ined by looking at fig. 1 which is the graph of the po ten t ia l V(H) for the one -pa ramete r fami ly of metr ics a l ready discussed in sect ion 3; using typi- cal values of the pa rame te r we get the fol lowing curve"

The po in t A cor responds to the s t andard bi- in- var iant metr ic on G and B to a non - s t anda rd Einste in metric. This " p o t e n t i a l " is therefore not b o u n d e d f rom below - even with the f ixed volume res t r ic t ion - moreover the non - s t anda rd Einste in met r ic of this fami ly cor responds to a local maxi- m u m of the curve; in a more general s i tuat ion,

Einste in metr ics co r respond to saddle poin ts which are nei ther min ima nor max ima of the total scalar curvature function. The " phys i c a l " in te rpre ta t ion of the above results is therefore unclear : we expla ined what happens if we expand a non trivial saddle point , but

(1) are we al lowed to " e x p a n d " a round them? (2) even if we can, why do it and are these

saddle po in ts impor t an t in a qua n tum mechanica l perspect ive?

If we want to s tudy global aspects of symmet ry break ing and use f iber bund le techniques, there is ano ther diff icul ty which arises when we try to def ine global ly the shifted field h~B; however the same diff icul ty usual ly over looked a l ready exist in the Higgs set t ing when we try to def ine the shif ted field dp'; one solut ion is to work in a trivial bund le (as people do usual ly when discussing Higgs mech- an i sm - see however ref. [4]).

C o m p a r e d with the usual Higgs approach , the above ideas have a clear geometr ical and intui t ive in te rpre ta t ion , they also lead to calcula t ions wi thout a rb i t r a ry parameters . They still suffer f rom a lack of in te rp re ta t ion bu t show a new di rec t ion in the s tudy of " s p o n t a n e o u s symmet ry breaking" .

This let ter covers the last par t of a set of lectures given at Szczyrk (Poland) in Sep tember 1983. More detai ls m a y be found in the set of lecture notes [5].

References

[1] R. Coquereaux and A. Jadczyk, Commun. Math. Phys. 90 (1983) 79.

[2] J.E. D'Atri and W. Ziller, Mem. Amer. Math. Soc. 18 (1979) 215.

[3] R. Coquereaux, CERN TH 3639, unpublished. [4] D. Bleecker, Department of Mathematics (1983). [5] R. Coquereaux, 1983 Szczyrk Lectures, Marseille CPT-83/P.

1556, to be published in Acta Phys. Polonica.

406

Documents

Internal Einstein spaces and symmetry breaking