IL NUOVO CIMENTO

VOL. 90 A, N. 3

1 Dicembre 1985

The Case of Dynamical Mass Hierarchies. H. SALLER

Max-Planck-Institut fiir Physik und Astrophysik Werner-Heisenberg-Institut ]i~r Physik - D-8000 Miinchen 40, B.1LD. (ricevuto il 3 Aprile 1985)

S u m m a r y . - Hierarchical mass ratios--of apparent importance in particle physics--can arise from logarithmic-type consistency equations related to a spontaneous symmetry breakdown. For ~ breakdown of an internal symmetry group G to a remaining subgroup H the induced unitary trasmutator fields connect G with the locally implemented subgroup H and establish, via their normalization, logarithmic consistency equations. The coupling constant of the breakdown enforcing potential can be determined by imposing local dilatation invarianee. As an example, the masses characteristic for a breakdown of SU4c>SU3®U1 and SU2c>U 1 are related by the logarithmic relation logm~(SU4)/m2(SU2)= 42--

23 .

PACS. ll.10. - Gauge field theories. PACS. 11.30. - Spontaneous symmetry breaking.

Introduction.

Particle physics abounds with mass hierarchies, i.e. mass ratios comprising several orders of magnitudes, e.g. re(electron) ~ 0.5 MeV, m(weak interaction) 170 GeV, perhaps re(unification) ~-, 1015 GeV, m(Planck) ~ 1019 GeV. These mass ratios are n o t understood, their u n d e r s t a n d i ~ g constitutes one of t h e basic problems in particle physics. One s t r a t e g y to solve this p r o b l e m starts from one (primordia~l?) mass and looks for g r o u p - t h e o r e t i c M l y - m o t i v a t e d mechanisms to establish particles at mass zero. A second step, so far c o m p l e t e l y unclear, has to give to the massless particles a v e r y small mass hierarchical with respect to the primordial one. 16 -

II Nuovo aimento A.

233

234

H. SALL]~R

A t t a c k i n g the p r o b l e m of masses a n d their hierarchies r e q u i r e s - - i n m y o p i n i o n - - a n analysis of t h e dilatation properties of the underlying dynamics. I n effective theories, e.g. the electroweak s t a n d a r d model, masses are simply i n t r o d u c e d via condensations implemented, e.g., with an effective ad hoe Higgs field potential. The conditions for a condensation violate not only the i n t e r n a l s y m m e t r y , e.g. SU2Q U1, b u t also the dilatation invariance: each symmetry breakdown contains also a dilatation breakdown. A s t u d y of the dilatation p r o p erties involved in a condensation m a y help to u n d e r s t a n d the problem of the mass hierarchies. F o r this purpose I consider in sect. 1 a classical f r a m e w o r k for the condensation of a n i n t e r n a l SUs s y m m e t r y . On this way also the arising effective gauge invariances related to the r e m a i n i n g symmetries, discussed earlier (i), are exhibited. Close analogies to the f r a m e w o r k of general r e l a t i v i t y (~,3) are m e n t i o n e d shortly. I n t h e condensation-induced r e a r r a n g e m e n t of t h e degrees of f r e e d o m care is t a k e n t h a t t h e dilatation degree of freedom is n o t <~lost ,~ b y some a p p r o x i m a t i o n . I n sect. 2 the dilatation properties are discussed for quantized fields. I t is shown t h a t t h e ad hoe p o t e n t i a l of t h e tiiggs fields responsible for a s y m m e t r y b r e a k d o w n can be used as a co u n t e r t e r m to m a k e their kinetic t e r m locally dilatation invariant. This is only possible since--because of the quantizationinduced singularities--the i n t e r a c t i o n has to be defined as the limit of a nonlocal expression, well k n o w n for p r o d u c t s of quantized fields. The local dilat a t i o n invariance determines the coupling constant of the condensation-inducing p o t e n t i a l (4). I n sect. 3 the r e a r r a n g e m e n t of the t h e o r y with a condensing S Us s y m m e t r y of sect. 1 is continued b y t a k i n g into account t h e local dilatation properties. I t is shown t h a t the condensation r e l a t e d local SU~ transformations (transm u t a t o r fields) are dimensionless ~lso in a quantized fl'amework. This leads, in sect. 4, to a consistency e q u a t i o n of a logarithmic t y p e for the mass m characteristic for the condensation (log m 2-~ const) (6). L o g a r i t h m i c - t y p e consistency equations, however, can easily induce hierarchical mass ratios, even s t a r t i n g f r o m (~o r d i n a r y ,~ numbers, as well k n o w n f r o m the ratio of the gap to t h e D e b y e f r e q u e n c y in superconductivity (*). The (~o r d i n a r y ,) numbers which are of i m p o r t a n c e here are t h e dimensionalities of the condensing groups, e.g. N ' - - I for SUN ( l o g m ' _ _ _ / ~ 1 ) .

(~) H. SALLE~: N~OVO(]4~e~tO A, 82, 299 (1984); I~PI-PAE/PTh 47/84; MPI-PAE/PTh

8~/84. (3) (a) (4) (5) (~e)

C.J. ISHAM, A. SAT.AM and J. STRATHD~E: A ~ . l~hys. (N. X.), 62, 214 (1971). H. SALLER: ~UOVO Gi~nento B, 82, 1 (1984); MPI-PAE/PTh 86/84. H. SALLE~: ~IPI-PAE]PTh 56]84. H. S~-LLE~: ~UOVO Cimento A, 79, 437 (1983). J. B~DEEN, L . N . COOt'~R and D. R. SCm~IEFFE~: Phys. l~ev., 108, 1175 (1957).

T H E CASE OF

D Y N A M I C A L MASS H I E R A R C H I E S

23~

A s h o r t a p p e n d i x shows how t h e c o m p o s i t e gauge fields, c o n s t r u c t e d for the g r o u n d - s t a t e i n v a r i a n c e subgroup H in sect. 1, h a v e to be n o r m a l i z e d in the q u a n t i z e d f r a m e w o r k .

1. - Gauge invariance f r o m s y m m e t r y breakdown.

A b r e a k d o w n ( b e t t e r condensation) of a non-Abelian s y m m e t r y gronp G gives rise to a new f o r m a t i o n of the degrees of f r e e d o m involved with respect to t h e c o n d e n s a t i o n - r e l a t e d g r o u n d - s t a t e properties. I n this r e a r r a n g e m e n t effective g a u g e invari,'mces are established which r e l a t e to t h e u n b r o k e n subgroup H c G a n d which reflect the free choice of t h e local H reference frames. The salient f e a t u r e s of such a s i t u a t i o n will be reviewed (1) for the case of a condensation for a n i n t e r n a l s y m m e t r y g r o u p G = SU~v with, occasionally, explicit illustrations for G/H = SU~/U~ a n d G/H = SU4/SU3@ U~ of possible r e l e v a n c e in p a r t i c l e physics (~). The concept to explain local invariances as a consequence of a condensation enforced choice of t h e g r o u n d s t a t e has b e e n proposed a n d studied (8) also for general r e l a t i v i t y with G/H = GL4,R/SO~,s. I n t h e following, s o m e t i m e s also this case will be alluded to. Although, in t h e followir~g, the l a n g u a g e of q u a n t u m physics m a y be used, the considerations in this section are classical. F o r the condensation, fields of an adjoint SU~- r e p r e s e n t a t i o n Oc(x) (c = 1, ..., N 2 - - 1 ) will be used, in a H e r m i t i a n N × N m a t r i x (*) 0~(x)

O(x) = rooo(x) , (1.1)

O(x) = I

[~o, ~ ] = i/o,~T~ ,

for SU~

(~12)O(x)

( (~o(~)/2)

Oo(x)

for S U , .

Since the m a t r i x 0~ is normal, i.e. [0(x), 0*(x)] = 0, it can be diagonalized at

(7) E.g., P. LANGACKER: Phys. Rep., 72, 185 (1981). (*) A concroto representation for S U4

/08 + ~Os + ~015

0~- io~ Os

- o~ + ~

~c(4) 0c =

015

+ v,-g

0~-- i05

Oo -- i01o t

06-- iO7

011- g01~.

20s

- ~

015

+ ~

01~- i014 _30,5

236

~t. s ~ L ~ l ~

each point with a u n i t a r y mutrix s~(x) e SU~, culled transmutator (1.2)

0~(x) = s~(x)r~(x)s*a(x),

r~(x) diagonal

0(x) will be assumed to be nonsingular (1.3)

det 0~(x) = det r~(x) ¢ 0

a n d continuous in its space-time dependence. I n this cnse the diagonal radial m a t r i x r~(x) determines at all space-time points the same subgroup / / c G commuting with r~(x). One obtains the transformation behavionr

(1.~)

O(x) v-~ gO(x)g* ,

g~eG,

s(x) ~ gs(x)h*(x) ,

h~(x) e ~ ,

[r(x), h(x)] = 0 ,

h(x) = h(g, 0(x)) .

Oae should realize t h a t the induced H transformations h(x) are space-time dedependent via their dependence on 0(x). I t is necessary to distinguish between the G-system (indices ~, fl, ...) and the H(loc)-system (indices i, ~, ...). For t h e simple case in which 3 / i s given b y (1.5)

H=~v~,,®~u~,®u~, .N~+.y~=~, .y,,~>o

(note SU1 = 1) one can extract a common dilatation factor r(x) from the radial m a t r i x r~(x) leaving a c o n s t a n t m a t r i x J5o

(1.01

+

0

l/

For simplicity the case (1.5) will be assumed. I t leads for the examples above to

(1.7)

L o -~-

I ~,/2

for Su, IU~ ,

[ ~.~,(4)12

for ~u,IsU~® u,.

A dynamics which can implement for 0(x) the mathematical situation described above is given b y the Lagrangian

(1.s)

1

~2\s

~(o) = ~r (a,o)~- h~ ~l~r o~- T I ~ ,

237

THE CASE OF DYNAMICAL MASS ItlEtCAI~,CHIES

where the m i n i m u m of the potential is churucterized b y



(1.9)

=

--

<½r:(x))

--

m 2 2

fulfilluble with an ~ s y m m e t r i c ground state

(1.~o)


The p r o b a b l y more reMistic, b u t more complicuted ease of a dyn~micnl breakdown, e.g. vi~ a quantized SUN field # ( x ) (a----1, . . . , N ) , 1

will be t r e a t e d in ~ f o r t h c o m i n g p~per. I n the c~se of ge~lerM relutiv~ty t h e normM 4 × 4 m~trix g~,(x),the metricM tensor, is ~ssumed to be nonsingular (orientuble manifold), d e t g , ~ ¢ 0 und leads viu the di~gonMizution, g~,(x) ~- h~(x)~j~(x), to the t e t r a d h~(x) e GL~,~ as the GL~,~/SO~.~ tra, n s m u t a t o r . Also in this case ~ Dz -~ GI~,a/SI~,~ dilatntion fuetor r(x) = (-- det gu~)~ c~n be sep~rnted, h~(x) -~ r(x)h~(x), h~(x) e SL~,~. T h e g r o u n d - s t u t e L o r e n t z f o r m < g , ~ ( x ) ) - - - - M ~ is known from experiments with M s reluted to the Pl~nck m~ss. The s p u c e - t i m e - d e p e n d e n t H - t r a n s f o r m a t i o n behavionr of the t r a n s m u t n t o r s~( ) (1.5) induces composite H-gauge fields W~(x) constructed vi~ t h e i n t e r n a l d e r i w t i v e W~(x) of s(x)

i~s*= W~s*, --i~us=sW~, (1.11)

W.

=

(i3~s*)s

~

--

s * i ~ s ~-~ h W , h* ~- (i~h)h*,

w'; + wT/". W~(x) is the H - p r o j e c t e d p~rt of W,(x) a n d Mlows the construction of H - c o v a r i u n t derivatives [W~, L~] = 0 ,

(1.12)

(Lc) ~ generators oi H ,

~-

i~us*-- W~s* ~ W~ms* ,

-- iDus

=

-- i~us -- s W~ =

W~"

=

(iD~s*)s

iDus*

=

s W~/~ ,

--s*(iD~s),

~3 -~ T r s ~~3 i~us*

(1.13)

=

for SU~/Uz ,

L(4) a~1,...~8~15

for SUd/EU3Q U1.

238

H. S ~ L ~ R

For general relativity one obtains SOa,, gauge fields used, e.g., in the covariant derivatives of spin-½ fields (spin connection). The composite gauge fields W~ has nontrivial curvature since it is only the Hprojected p a r t of the complete internal derivative whose curvature vanishes:

w,,,= (1.14)

a,w,,--

~,,w,÷

i[w., w~] = o,

w-.. = ~ w -w- ~ %,' w." , , + , ~[

w.-] # o

The t r a n s m u t a t o r s(x) t r a n s m u t e s from linear G representations, e.g. ~,(x), ~(x) ~-.g~(x), g e G, to nonlinear G realizations (s,~), e.g. q d x ) ~ s*~(z)t~(x), q(x) ~-~ h(x)~(x), h e H, phenomenologicMly t a k e n as linear representations of the local group H~oo. The condensing field 0~(x) is rearranged according to (1.2) and gives the derivative

(1.15)

{

~,0 = s(~,r)s* d- is[W~, r]s* = s(IoS~r d- ir[W~/~, Lo]) s*,

Tr (8,0) ~= Tr {L~(~r) ~d- i( ~ r ~)[ W~,/~, Io] -- r~[ W~/~, Lo] [ W~/H,/,]}.

W i t h the relation

- ~o[ wy/", Lo] = ~o w~ ~" ,

(1.16) Tr/~o = 1~

a n d with the covariant derivative (1.12) one obtains the kinetic terms for radial field r(x) a n d t r a n s m u t a t o r s(x) in the e]]ective G ~ Hjo-invariant expression

(1.17)

Tr (~,9) ~ -~ ½(Ol,r) 2 -4- 2r 2 Tr Eo(D,s*)(D~s) .

The transition from the 9- to the (r, s)-formulation has its analogue in the transition from the g~, to the h~ formulation of gravity with the effective GLd,~Q S03,1 invariance there. The lst-order Goldstone field expansion for the condensing field

e(x),

e(z)=m~o÷O(z),

= o

(8) A. SALA• and J. STRATHD~]~: Phys. I~ev., 184, 1750 (1968). (9) H. SALLE~: ~UOVO Cimento A, 82, 259 (1984).

THE

CASE

239

OF DYNA1KICAL MASS HIERARCHIES

is as useful and as restrictive as a 1st-order flat-space approach in gTavity, e.g. t h e N e w t o n i a n approximation. I n sect. 3 a f u r t h e r r e a r r a n g e m e n t of t h e L a g r a n g i a n (1.8) in t h e (r, s)formulation (1.18)

1

Lf(0) = ~ ( ~ r ) ~ + 2r ~ Tr

Eo(Dzs*)(D~s) -- -4h~(r2_ m~)~

is proposed which n e v e r destroys the effective G Q H~o0 invariance.

2. - L o c a l dilatation invariance and t h e strength o f t h e interaction.

I n the effective models for a condensation, t h e G ~ H s y m m e t r y breakdown is enforced b y an ad hoe p o t e n t i a l V, e.g. (1.10), i n t r o d u c e d e x a c t l y for this purpose. I n renormalizable theories, this interaction V is coupled with a dimensionless free c o n s t a n t h ~. Gauge interactions in gauge theories are d e t e r m i n e d (up to the gauge field normalization) b y the r e q u i r e m e n t of local invariance (covariant derivative). Also the ad hoe i n t e r a c t i o n p o t e n t i a l V in theories with condensation can be related to the kinetic t e r m via a local i n v a r i a n c e - - i n this case via the locaZ dilatation invarianee--which, in this way, determines the interaction coupling c o n s t a n t h 2. Such a connection between kinetic t e r m a n d interaction potential is only possible ]or quantized ]ietds. I t can also be characterized as a self-consistent d e t e r m i n a t i o n of the kinetic t e r m (4) taking into account a singularity avoiding definition of p r o d u c t s of quantized fields. I n classical theories, the local D1----GL4,R/SL4,R invariance is established via the metrical tensor, e.g. as shown for the d y n a m i c s of t h e SU~ adjoint field 0o(X) (2.1)

~f(Oo, g..) = ~ / Z ~ [g~, Tr (~Oo) (~,Oo) -- h~ (Tr O~-- ~ ]

j

with the D~ t r a n s f o r m a t i o n s defined b y dx.

(2.2)

~ exp [-- 2(x)]dx~,

g~,~(x) ~ exp [2)~(x) ]g~,~(x) ,

Oo(x) ~ Oo(x). 0o(x) has vanishing dimensions, whereas in quantized theories the m a t t e r fields,

e.g. O(x) (1.8), have nontrivial dimensions. Before carrying out t h e ideas above for the SUN condensing field 0~(x) (sect. 1) t h e y will be concretized in a simplified model with a o n e - c o m p o n e n t

240

H.

SALLER

complex field ¢(x) and the dynamics (2.3)

Ze(¢) = (~,¢*)(~¢) -- h~:(¢*¢)~:(x).

The small-distance singularity of the two-point function is not D~,~oo-invariant (*)

(2.4)

<¢(x+)¢*(x)> = i f ~ 1 = ¢ ~ * ( ~ ) - - (2~) 4 d p e x p [ - - ip ]p-~--

1 1 (2~) 2 ~ 2 ( x ± = x ~ 1 2 ) .

It has to be taken into account in the Laurent expansion of the operator produets, e.g. in the product of two fields

(2.5)

¢*(x_)¢(x+) -

1

~',¢. 8,,

1

(2~)~ p + ¢*¢(x) +

~-¢(x)

+

Under the (infinitesimal) D1 transformations (2.2) and (2.6)

~(x) ~ exp [Z(x)]¢(x) = (1 ÷ ~(x) ÷ ...)¢(x)

the kinetic term transforms inhomogeneously:

(2.7)

(a.¢.)(a,,¢)

~ (1 ÷ 4~t)(a,¢*)(8~) - - ( ¢ * ~ ) 82~. ÷ ....

The interaction in (2.3) has to be defined b y approaching the local limit via delocalized products, e.g. as given in (2.5). Delocalized products as (2.5), however, can produce inhomogeneous terms in their 1)1 behaviour, e.g.

(2.8)

@*(x_)¢(x+) ~ ,

[1 + ~(x_) + ~(x+) + ...]¢*(x )¢(x+) =

---- (1 + 22):¢*¢:(x)

(2z) 2 4 T

....

1 = (1 + 2 k ) : ¢ * ¢ : ( x ) - ~-~3-~ a~k + ..., where the space-time averaged limit lim ~,~',/~= ¼~', has been used.

(*) The analoguo of a not Ul,loo-invariant electron propagator in QED is we]l known.

TH:E C A S E

OF DYNAMICAL

241

MASS I { I E R A I ~ , C t t I E S

Comparing the inhomogeneous D1 behaviour of b o t h the kinetic t e r m (2.7) and the limit of the delocMized p r o d u c t (2.8) one realizes t h a t - - w i t h an appropriate i n t e r a c t i o n strength h~--the dynamics (2.3) can be made D~,,o¢invariant. This c o n s t a n t h ~ is eusily d e t e r m i n e d as follows: (2.9)

[¢*(x_)~b(x+)]2 ~-~ [1 +- 22(x+) ~- 22(x_)] [¢*(x_)¢(x+)]~ ~

=(1 +4A + ~2~"O,~,A+ ...). .[2~,¢(~)?,~(~) + 4f,~(~} :¢,(~_)¢(~+): + :(~,(~_}¢(~+)),:] = 4 ~'~" (2~)~ 2~ 0,~,~:¢*¢:(x)

= (1 + 4~):(¢*¢)~:(x)

= (1 + ~ ) : ( ¢ * @ : ( x ) (2.~0)

1

2(2~)~

+

....

~ : ¢ * ¢ : ( x ) + ...,

h~/2(2~)~ = ~.

As m e n t i o n e d above, such a d e t e r m i n a t i o n of the coupling strength h ~ can also be characterized to be the self-consistent connection of the interaction and the kinetic t e r m via the singularity (2.4) ((( sel~-consistent quantiz,ootion ~)) (~). One obtains from the expansion of the delocMized interaction (2.11)

2

1

4

[¢*(x_)¢(x+)]~ = (2~),'(~)~

1

(2~)~ ~"

• :¢*¢'(x) + ~ , ¢ * ~ - ¢

+

¢*---'¢

+ ...

+:(¢*@:(x)

~fter subtraction of singularities for the space-time-averaged limit also kinetic terms

(2.12)

li ~ o { [ ¢ . ( x _ ) ¢ ( x + ) ] 2 _

2

1

4

1

}

(2~)," ( ~ ) ~ - - (2=)3 "~ :¢*¢ :(x) =

= :(¢*@:(x) + ~

1

(~,¢*)(~,¢) + total divergence.

The kinetic term (~,¢*)(~¢)(x) and the interaction :(¢*¢)~:(x) condition each other in a Dl.,oo-invariant theory leading to a well-determined coupling cons t a n t h 2 (2.10). h ~ measures the relative s t r e n g t h of the D1-bManced contributions o~ the two operators with dimension 4, i.e. (~,¢*)(~¢) and :(¢.¢)2;. F o r the Lagrangian (1.8) with the (N ~ - - J ) - c o m p o n e n t field @(x) in a quun-

242

H. s ~ a

tized t h e o r y ~ ( O ) - - - - T r ( ~ , O ) ' - - h ~ : ( T r O ~ - - 2 ) 2"

(2.13)

one obtains the two-point singularity

=

(2.14)

00(~) --)

1®1

1

2(2zt)* ~2"

Therefore, the interaction conditions the following kinetic terms: (2.15)

(Tr 0~)(Tr 0~)(x+, x_) = = 2 TrOO(D T r : 0 ( x + ) 0 ( x )" + 4 ~ r : O ( x + ) ~ ( ~ ) e ( x _ ) : 2N~ + 4 ~,

2(2z)'

2~ 2

Tr 0

0 + ... - - - -

-

2(4z) 2

+ ....

Tr (0,0) 3 + ....

The inverse strength o] the D~,~oo-indueed interaetion is proportiona~ to the number o] ]idds involved, here (2.16)

(4~) 2 _ l (N 2 _~ 2). hi

The requirement of local dilatation invariance (for the massless theory), t a k e n directly without probably necessary modifications, leads in the standard SU2(~ U~ GWS-model with a scalar field ~ ( x ) (~ ~ 1, 2) (2.17)

~(~) :

[iD~t 2 - h~(qJ*~ ~ m2) ~ ,

m ~ 170 G e V ,

to a determination of the interaction strength h~ h a n d the lowest-order mass of the massive neutral ttiggs mode ~0o,~~2 • ~o*--2m

h~- (4~)~ 3 '

m2(%) = 4 h ~ m 2 '

(2.1s) m~(¢o) _ 4(4~)~ ~ 2 1 0 m S

3. -

Effective

parametrization

3

'

for a dynamic

reCto) ~ 2.5 T o V --

with

"

condensation.

As described in sect. 1 the choice of a condensation-induced ground state leads to a d a p t e d field co-ordinates (r(x), s~(x)) with the rearranged effectively G(D H~oo-invariant Lagrangian (3.1)

Lf(0) = ½(O~r)2 -~ r ~ Tr Eo(D~s*)(D~s) -- V(0) .

TH:E CAS~ OF DYNAMICAL ]YIAS8 H I E R A R C H I E S

2~

Field variables are called e]/ective if t h e y arise in t h e liuearized p a r t of the t h e o r y , r(x) is a n effective field. To o b t a i n t h e effective kinetic t e r m for t h e t r a n s m u t a t o r s~(x) one usually chooses t h e l a r g e - d i s t a n c e e x p a n s i o n r 2 ( x ) = m 2 4- ..., which produces via ~f°(s)----m 2 T r E o ( ~ s * ) ( ~ s ) a t w o - p o i n t function for t h e t r a n s m u t a t o r (s(x+)s*(x_))

--

1 (2z) ~E;~@

llm~

~ .

This t w o - p o i n t f u n c t i o n exhibits a n intrinsic dimension one for the dimensionless t r ~ n s m u t a t o r s(x) (mismatch o] dilatation properties) ~nd l e a d s - - c o n s e q u e n t l y - t o a not m ~ n i f e s t l y r e n o r m a l i z a b l e a p p r o a c h . T h e s a m e s i t u a t i o n occurs in t h e simple-minded quuntization of g r a v i t y b y t h e e x p a n s i o n g , ~ ( x ) = M ~ u ~ 4 - . . . leading to ~f'r'v(h) ~ % / - - g R ~_ M2(O~h~)(~qh~) .... A linearization for the kinetic t r a n s m u t a t o r t e r m , m o r e a p p r o p r i a t e with r e s p e c t to t h e dilatation properties, sucks up the f a c t o r r2(x) with dimension two a n d leads to a n e]leetive canonical pair o] transmutators (So(X), s2(x)) with dimensions (0, 2) So(X) = s(x) ~ gSo(X)h*(x), (3.2)

s2(x) = r~s(x) ~-~ e x p [2~(x)Jgs2(x)h*(x) ,

r(x) ~-~ exp [~(x)]r(x) . Now, a linearization of the kinetic t r a n s m u t a t o r t e r m proves possible w i t h o u t a dilatation mismatch (3.3)

~f(0) = ½(~r)2 4- ½ Tr Eo[(D~s*)(D~s~) 4- (D~s*)(D~so)] - - V(r, s) .

The t w o - p o i n t functions of t h e r e a r r a n g e d t h e o r y (3.3) give t h e singularities for a quantized t h e o r y 1 1 (r(x+)r(x_)) -~ - - (2~) 2 " ~ ,

(3.4) (so(x+)s*(x_D -~

2 1 @ E~ I (2~)~ ~

W i t h t h e decomposition of B into the effective variables (r, s) also t h e int e r a c t i o n has to be r e a r r a n g e d . Therefore~ one will e x p e c t also a n i n t e r a c t i o n for the t r a n s m u t a t o r s. The t r a n s m u t a t o r kinetic t e r m in the f o r m (3.1) is

244

m

SAL~

Dz,~o¢-invariant. Treating (So, s,) as distinct variables, however, the rearrangement (3.3) violates Dz,~o¢ invaria~_ce. In the spirit of sect. 2 a minimal Dz,~o° invariant (So, s2) theory is chosen by determining the coupling g~ in the effective parametrization

(~.5)

~(0) = ~1 (~.r)~--~h~ (r~-- m~)~ + 3- ½ TrEo[(D.s*)(D.s~) + --g~:

(3.6)

(D~,sz*)(Dt'so)]m ~)~ "= ~f(r)

( Tr~oS*S~+s'so

3- .Sf(s),

[Tr Eo(S*S~ + S*So)]~(x+, x ) = : 2 TrEo (s*_$2(~) 3- s*so(~))Tr :Eo(s*(x+)s~(x_) 3- ~(x+)so(x_)) : 3t.._A

+ 2 cr :Eo(S*(X+)S~Bo$*(~)s~(x_) + s*(x+) So~o;*(~)So(X_))" + (2=) ~ _

. . . .

2~

2(2~T~ + 1)

(4~) ~

~r Bo [(a,~)(a,s~) + (a,~,*)(a,~o)] +

. . .

with the result g~ (2_7f~ + 1) = 1 . (4=)~

(3.7)

In a quantized theory the normalization of the gauge fields W~ has to be reconsidered (appendix).

4. - Mass hierarchies

from logarithmic

mass ratios.

The G-symmetric condition for the unitary transmutator s~(x) in the classical framework

(~.1)

s(x)s*(x) =

1

,

s*(x)s(x) =

1

will be interpreted, in a quantized theory, as the ground-state part of the of the delocalized product (4.2)

limit

1 lira (So(X+)S*(X_)) ~- _~ 1 G 1 •

~--*o

The product So(X)S~*(x)=r2(x)l

c~Jn contain, in the quantized framework, a

245

T t I E CAS:E OF D Y N A M I C A L MASS H I E R A R C H I : E 8

G-symmetric ground-state contribution < :Tr.EoS*So : ( x ) ) z

(4.3)

m2,

1

< :SoS*:(x)) = m 2_~ 1 ® ~ d . To avoid double counting no nontriviM ground-state contributionof s~(x)s*(x) = ~-- (r2)~(x) m u s t be added:

<:s..s*:(~O) = o .

(4.4)

If the conditions (4.2)-(4.4) are t a k e n into account in the Lagrangian ~ ( s ) (3.5) one obtains the large-distance r e a r r a n g e d form (4.5)

Z ( s ) = ~ Cr Eo[(-D.s*o)(D~s~) + (.D.s~)(9 S o ) ] - - ~

BoS~ s ~ - -

with oo°°indicating t h a t all extractions have been performed. After a trivial renormMization

(4.6)

s2(x)Eo:

r~2(x) '

~zg~

So(X) -~ ~o(x) ,

m 2-- too2,

the L a g r a n g i a n ~q~(s) (4.5) looks simpler:

(4.7) - m~(~o*q,2 + ~*,Po)} - g ~2o° ( ¢ r ($0, ~-. + $~*$o))~: •

The linearized p a r t of (4.7) leads to the Green's functions

(4.s)

The original two-point function in (3.4) is modified for large distances b y the mass m o. An additional soft-dipole p r o p a g a t o r arises for the dimensionless t r a n s m u t a t o r so(x) (<( anoma, lous Green's function ~>).

246

m

SALLE~

To have consistency of (&8) with the extracted pair condensation (4.2) the following relation has to hold:

(4.9)

--i

(4~)~ lira ~2j -- [dp exp [-- @$] ~ N -- ~--,o

1

( p 2 m~)~

There is no massless pole in the transmutator propagators (4.8). The longrange Goldstone ]orves arise ]rom bound states in the m a t r i x elements
(4.1o)

<~[SoLoS*(x)

l~> =

<~91r-~(x) O(x) lG>.

The analogue to the Goldstone forces in general relativity are the longrange interactions in h ~ ( x ) ~ hi(x ) ~ gt,,(x), i.e. gravity. For gravity, possibly also a quantized approach using a canonical t r a n s m u t a t o r pair (h~,(x), h~i,(x)) with a massive dipole at the Planck mass M ~ could prove appropriate. The t h e o r y (3.5) with the interactions defined as the local limits of delocalized expressions, e.g. (2.12), has to avoid the actual limit ~ --~ O. This is impossible with only one condensation involving only one mass. If, however, more than one mass and more than one condensation occurs, e.g. G ~ / / ~ (i--~ 1, 2), t h e G~, G~-invariant radial condensations violating dilatation invariance lead to ]inite logarithmic mass ratios, e.g.

(~.11)

(4~) ~ (4=) ~ _ 2g.~, - - 2g~,, ---- lim -- [dp exp [-- ip~] ~-,o = ~ j ip~--m?) ~

(p~±m:) ~

= log

.

Since the D1 behaviour effects equally all fields, it is allowed to subtract relations like (4.9) for different condensations. In a more sophisticated t h e o r y this a m o u n t s to the introduction of a universal mass scale where all D1 break downs a n d consequently all masses have to be measured in. Presumably this mass scale is related to the P l a n c k mass M ~, = M ~ , , reflecting a GL4,R/S08,1 condensation phenomenon. The 1/hTM dependence of the interaction strength g~ (3.7)

(4.12)

( 4 ~ ) ~ __ IV ~ q- 1

2g~

THE

CASE OF DYNAMICAL

247

M A S S HIEI~A~CHIES

suggests large condensation masses ]or > groups~ small masses ]or <(small >> groups caused by the number of components involved in the transmntator s~(x) ~ S U ~ , given by the dimension of the group ~ b 2 ----

(~.13)

log~

2

2

iv~- N~.

For the exnmples G~///~= SU4/SU3Q U~ G2/H~ = SU2/U~ one obtains the hierarchicul mass ratio

(4.14)

m2(S U,)

log m~(SU~ )

_

12,

m~(SU,)

m:(SU.~) ~_ 1.6.10 5 .

APPENDIX

Normalization of quantized composite gauge fields W~. The rearranged transmutator Lagrangiun (4.6)

(A.~)

~f(s) = Tr [(D,$*)(D'cp2) -[- ( n , cp*)(D'cpo)-

¢p*cp~--

- ~(¢P$cP2 + ¢p*~po)]

--

g~o(Tr '° ¢~o* ¢P, + ¢P*q~o)~

with the two-point functions

• , (x+)(~*, ~*)(x_) --

i fl - (2~),: p exp [-

ip~]1 ® I

p, lo,n~ (P~-1 m~)~1 I

p ' - - m~ ! contains the composite//-gauge field W~ as the//-projected part of the internal transmutator derivative

(A.3)

W~

--= --

s* ~ ~So --

N ¢p* 2 ~¢Po

classical.

248

H. SALLER

The H~oobehaviour of W~(x) (1.11) rests strongly on the classical normalization So(X)S*(x) ~ J which--in a quantized t h e o r y ~ c a n n o t be used because of the singularities in the two-point function <~o(x+)cp*(x_)} for ~-->0. Therefore, one has to reconsider the normalization Zw of the composite gauge field W~ which, in a quantized framework, will be defined by (A.4)

[

w,,(x) = - z . ~

]

q,*(x )~ ~,$o(X+) + . . . .

In the ansatz (A.4) for W~(x) one encounters a problem well known from usual gauge theories: point splitting is not gauge invariant. Oriented at gauge theories for scalar fields, I have proposed an approach (1) using only 1st-order derivatives in the Lagrangian ~(s) (A.1) which will be described in the following. The 2nd-order equations for the linearized theory of ~e(s) (A.1) (A.5)

(-- a ~ - m~)qn = q~,,

(-- ~ ' - - m ~ ) q , , = o

are replaced b y 1st-order equations (*)

(A.6)

{

~ o 2°¢0 = ¢1,

ire ~°¢~ = ¢3,

(.o) (.2)

with formal 4-component Dirac fields (commuting!)

(A.7)

1 ¢o=~

~0 ~o

q~o

1

,

¢,=~

?~ ~'2

.

q~

Equations (A.6) can be derived from the reordered linearized part of .~(s) (A.:)

(h.8)

where ¢1 and ¢3 serve as Lagrangian multipliers. Starting from (A.8), the two-point functions (A.2) are supplemented b y the

(') This approach is very close to the Duffin, Kemmer, Petiau formalism.

THE

CASE

OF

DYNA~IICAL

I~IASS I I I E H A R C H I : E S

2~

1st derivative propagators

(A.9)

(~+)(~, ~, ~, ~o)(~_)

1 2 2

p~ --

(p~--

m~) ~

( p ~ - - m~) ~

(p~ - ~n~)~

( p ~ - m~) ~

p~ - - m~

p~ - - m~

m~

p

p~

P

mo

- (2i), f 4 p exp [ - @$]1 (:~1

p 2 __ m~

m~

~0 2 - -

0

0

0

0

m 0

p~ --

m~

p~ - - m~

For the g~uge field W~(x) (A.4) one has to consider the p r o d u c t ~o~.¢~ ~+ ~1Y~¢o which contains l s t - o r d e r intern~I derivatives of ¢~o

= ~.*~o + ~o~¢0,

~r~= ~-

i~.

Therefore, the guuge field (A.4) is defined viu the local limit

(A.I~)

~-~o

2

"

The small-distance singulurity in the products involved in (A.11) (A.12)

i /dp exp [-- ip~] (p2_p m~)2 ~ (4zl)~" 2i ~~21 ® 1

(¢o(X+)~(x_)> -- (2~) 4

produces uu inhomogeneous t e r m (~o) for the transformation beh~viour of the deloc~lized p r o d u c t (A.11) ¢o,1(x) ~-~ g¢o,~(x)h*(x),

(A.13) (A.14)

¢~(x )7~¢o(X+) ~-~ h(x_)¢l(x_)7,¢o(x+)h*(x+)

=

= [Lh(X) __ 2$~h(x)][:~7~¢°:(x)~ (4~) ~N_~...][h.(x)_~h.(x_)]_~_2 (4~) ~ (~,h)h* (lo) H . P . Di~H~ and N. J. WINTEH: NUOVOCimento A, 70, 467 (1970). 17 - 1l N u o v o Cimento A.

÷ ....

250

H. SALLER

To obtain the required gauge behaviour

(A.~5)

w,,~ aW,,h* + (i~,h)h*

t h e g a u g e field n o r m a l i z a t i o n Z ~ h a s t o b e c h o s e n t o b e

(A.~6)

•

RIASSU~TO

zw= (~)~ 2N

"

(*)

I rapporti di massa gerarehiei - - di palese importanza nella fisica delle particelle - possono derivare da equazioni di consistenza di ripe logaritmieo legato ad u n a r o t t u r a di simmetria spontanea. Per u n a rottura del gruppo di simmetria i n t e r n a G nel sottogruppo rimanente H i eampi del trasmutatore unitario indotto connettono G con fl sottogruppo implementato loealmente e stabiliseono, mediante la lore normalizzazione, equazioni eli eonsistenza logaritmiehe. La costante di aeeoppiamento del potenziale ehe induce la r o t t u r a si pub determinaro imponendo invarianza locale di dilatazione. Come esempio, si eorrelano Ie masse caratteristiehe di u n a rottura di S U 4 ~ - S U s ® U 1 e S U z ~ U x mediante la relazione logaritmiea l o g m s ( S U 4 ) [ m 2 ( S U ~ ) = 4 3 - 2 ~. (*)

T r a d u z i o n e a c/ara della l~edazione.

C.~T~afi zSmaMHqec~mx MaccoBblX . e p a p x ~ . Pe3mMe (*). - - Hepapxn~ecmae MaCCOBLIe OTHOrrteHn~ MOryT BO3HIIKaTb H3 ypaBHeHn'l~ aenpOTnBOpe~nBOCTnnorapa~MmIecI SUB ® 171 a SU2 ~ Ux; CB~3aHBI norapnc}M~ecIcRM OTHOIIIettIIeMlog m~(SUc)[m~(SU2) = 4 ~ - - 2%

(*) IIepesec)eNope~)ag~uea.