IC/8U/U

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

COLEMAH WEINBERG MODEL IK EINSTEIN SPACETIME

G, Denardo

and

E. Spallucci

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1984 Ml RAM ARE-TRIESTE

International Atomic Energy Agency

and

United nations Educational Scientific and Cultural Organization

IHTERHATIOHAL CENTRE FOR THEORETICAL PHYSICS

COLEMAH WEINBERG MODEL IN EINSTEIN SPACETIME *

G. Denardo

International Centre for Theoretical Physics, Trieste, Italy,Istituto di Fisica Teorioa dell'Universita1 di Trieste, Italy,

andIstituto Hasionale di Fisica Bucleare, Sesione di Trieste,

and

E. Spallucci

Istituto di Fisica Teorioa dell'Universita1 di Trieste, Italy.

ABSTRACT

We study radiative symmetry breaking, a la Coleman-Weinberg, in

the geometry of a static Einstein Universe. We prove that the symmetric

ground state <0t4> IO> = 0, in a non-minimally coupled massless >4"

theory, can become unstable at high curvature only if 0 £ "£ ~ X ^T 1.

As far as massless scalar (JED is concerned, we find that phase transitions

can already occur at low curvature. These conclusions are improved by means

of Renormalization Group techniques.

MIRAMARE - TRIESTE

February

* To be submitted for publication.

1. INTRODUCTION

In a fundamental paper Coleman and Weinberg proposed radiative

corrections as the driving mechanism For symmetry breaking in massless field

theories. In these models (from now on we simply refer to them as CW-models)

the classical scalar potential exhibits a single minimum in the origin of

the constant field configuration space. The addition of quantum effects leads

to the appearence of a new absolute minimum for a non vanishing vacuum value

of the field. More recently.radiative symmetry breaking has been advocated

by several authors in order to cure the deficiencies in the original

version of the Inflationary Cosmology . In this model the universe is

assumed to undergo one, or more, phase transitions between vacua with dif-

ferent symmetry properties.

Actually the generalization of the CW-model to the de Sitter space

time, which is involved in the Inflationary Cosmology,is not trivial at all.

For example, the effective potential does not seem to be physically meaningful

in this metric, because of non perturbative tunnelling phenomena which force(4)

the vacuum expectation value of any scalar field to be always aero

A further difficulty is due to the "thermal" character of the vacuum as a

consequence of the cosmological event horizon . For all these reasons we

consider it desirable to have a better understanding of the CW mechanism in a

simpler case before attacking the problem in a more realistic framework.

For the sake of simplicity we shall work in the static Einstein Universe

3with topology RxS and metric

— ^(1.1)

This background gravitational field has been fruitfully used for investigating

the effects of curvature and topology on the internal symmetry properties

of scalar field theories. Moreover (1.1) formally represents the geometry

of a closed Robertson-Walker Universe at a definite instant of time. Hence

assigning different values to the scale a, we can simulate the evolution of

an expanding cosmological spacetime. In such a framework it is natural to

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inquire how the couplings vary under the action of this scale transformation :

the answer comes from the Renormal izatton Group (ERG) equations. In flat

spacetime one improves the effective potential by taking into account

;he dependence of the effective coupling constants from the variable

t- -*w(̂ /p,,) where *-f is the classical field and '-Po the renormalization

point , The RG equations we shall use in this paper are siigbtJy

different. The basic idea is that in curved spacetime the short, or large,

distance behaviour of the Green functions is achieved by scaling the metric

(8)rather than coordinates or momenta . This procedure leads to a corresponding

variation of the curvature, therefore^ we shall use as variable

2 p 2 2

t = Xvn (R/6M. ) where R=6/a is the Ricci scalar for the metric (1.1) and

M. is the typical field scale.

The paper is planned as follows. In Sect.2 we consider a massless

non-minimally coupled A *P theory. We show that the symmetric vacuum

will even eventually became unstable beyond some large critical curva-(*)

ture only if we are close to the minimal coupling. In agreement with

previous results we recover that the model results to be Weyl invariant in

the infrared limit.

Sect. 3 is devoted to the study of massless, scalar, (JED. Phase

transitions occur at low curvature, we evaluate the critical values of the

field and the curvature. The usual approximation X ~ £ is justified

through RG arguments.

2. MASSLESS A<P^ THEORY

This section is mainly devoted to introducing the formalism and to

recover already known results in a new, simple way. In flat spacetime a

single scalar massless field does not develop any physical non-vanishing

(*) The occurrence of an unphysical minimum at low curvature is discussed

in the appendix.

- 3 -

vacuum expectation value through radiative corrections. An apparent minimum

of the renormalized one loop effective potential stands outside the range

of validity of the one loop approximation. This result has been improved

by Renormalization Group (RG) analysis . In curved spacetime new

effects must be taken into account because of a possible nontrivial coupling

between the scalar field 0 and the background curvature R

(2.1)A,

D is the covariant D'Alembertian, A and T are both positive numbers

in order to avoid spontaneous symmetry breaking at the classical level.

We will show that only for " 5 ^ 0 radiative corrections can destabilize

the classical minimum inV-0, when the curvature becomes larger than a

critical value R . The one-loop effective potential for (2.1) formally

reads

22.(2-2)

</* is the classical field. 5"? and SX. are counterms, X i is a formal

expression for the spacetime volume. In curved spacetime there are two

useful methods to handle (2.2). One can use either the Heat Kernel

K(x,y;T) or the generalized "J -function 7(s)=ZA~ S

related to the operator - • * "SR * X*f'/2. 3 A

In the first case

{2.3)

while in the second

(*) We shall work as usual on the Rianannian section of the metric (1.1)

where the signature is ++ + +. Physical quantities are me asured in natural

units *n =c=l.

TffftfW»r'; *sS<

with ^'(sls d5/ ds . yi*t is an arbitrary regularization mass quite

analogous to the mass one introduces in dimensional regularization. <X»

is the typical scale of the A eigenvalues of A -In our problem a

is just the scale factor in (1.1} . 5(s) and K(x,x;f) are related by the

Mel1in transform

(2.5)

The Heat Kernel method is particularly useful for the study of ultraviolet

divergences. In fact, by inserting the asymptotic expansion of K(x,x; T)

(2.6)

in (2.3), the first three terms in the series exhibit all the ultraviolet

infinities of the theory. The B^ have the form Bm= ^-b m where b,,, is an

n-degree function of the curvature invariants. Actually (2.6) is a sort of

curvature powers expansions around flat spacetime. Therefore the first

divergent termsT after renormalization, provide a good approximation for

V(*f,R) as long as the curvature is small with respect to the typical field

scale. In order toimprove such art expression (for larger curvature) one can

(8)use RG techniques . In this second case one can start from (2.4)

neglecting the scaling variable AOL independent term ~5(Q), and writing

5(0) as

For a scalar field in

After this preliminary discussion let us consider the behaviour of

at low curvature. If the ultraviolet divergences exhibited by K(x,x; X )

are dimensionalLy regularized, one gets

Now we can either simply discard the poJe term l/£ f\t ( minimal subtraction

scheme) or explicitly renormalize A and ^

-<* (2.10a)

(2.10b )

The choice ^ -- fe £ 0 is dictated, as usual, by infrared divergences at

the origin whil^ R-0 requires some romrncnt. Any renormalization point is in

principle equally good to define *̂ and 5

'I'h... choice K-n is suggested both by consistency with our low Curvature

approximation ,-.;ni by the fact that in surh a way we will parametrize

V!1 :̂:!'! ir: tfrr-'a of I he physical coupliri^ •*•-• neasure in flat spacetime.

Irid'-ed, at '.he present ly available experimental energies in particle

ijhy;ics, s['v? st: mo ippcuTH quite fist. :;f,uvY.*r a n y other renor-

•!!--i.l i z . i ! i.-r. ; r •.••*:;•• l •:': :ii:ir:i: L:& p i y : . . _ l \ - •> ' - ( - p - r a b l e , i t s j r n p l y a m o u n t s

i , i •- • ••! ,;••• \: _ : ..iM ? M l '• . •• ' 'j-.^iiijpje, a s e ! [ - c t H i s i s t e n c y

: ! (*).• ; !•-. - ! <\ -v >; •*: • H-i-^v':., • ':-, ' . •'. a ^ n o r TFI a 1 x z a t i. o n p o i n t

par-ticular V- Lnd of seLf-

uu minimum oi' V!*fo;R) ,

jrmal \zf.fl

We will not explore this possibility here.

We recover SA and 5 \ from (2.10a) and (2.10b) respectively:

(2.11b)

By inserting (2«11) in (£,9) we cancel the poles and exchange^*! with ^

The one loop renormalized effective potential for small curvature reads

(2.12)

V("f. R) apparently exhibits a minimum for *f ̂ 0 (see the appendix arising)

from the balancing of the classical potential with the last term in (2.12).

But corresponding to this new minimum we have A ^ « R which is

inconsistent with our approximation, so we have to reject this result. We

know that in flat spacetime no physical minimum appears because of quantum

fluctuations at the one loop level, therefore it would be surprising to

destroy such a configuration only by adding a weak gravitational field.

Radiative symmetry breaking will eventually occur for large curvature. In

this second case we have except for finite renormalizations

(2.13)

We notice that frcm (2.13! It is easy to der the results of r<;£';.( ": :•.;

concerning the radiative instab11ity of the origin \ =0, We shall now

improve these perturbative results by means of RG techniques,

t /vYlfR/G/V)Let us define as scaling variable t = -/vYlfR/G/V), the one-loop

-function and the anomalous dimension tx then result to be

( 2 . 1 4 )

( 2 . 1 5 )

In flat spacetime t —»+ c» correspond to the ultraviolet and infrared

limit respectively while now t —•• + oo represent the large and small

curvature cases. So we shall obtain the behaviour of the coupling constant

as R varies with respect to the reference value of the "curvature" 6 H l

fixed by the typical scale of the Tfieid. The running coupling constants A

and "5 satisfy the equations

dt(2.16)

(2.17)

(2.16) and (2.17) give the following one-loop r(-.jlt •

X ;- (2.18)

] ' / 3 (2.L9!

32 FT1

We can say, borrowing the usual terminology, that A is infrared free and

rjaymptoticai ly-slave, indeed there j £ a pole at ]?- y^ e^f)!32H /3 Xft ),

anyway it occurs in a region of large X where our approximation is

untrustworthy.. Moreover for such an extreme curvature quantum grav; '̂y

effects too mu^t be taken into account. So we can safely use eq.(2.18) for

A onLy in the region O 4 X <*H . With this assurapLion in mii:d, we see

that Weyl invariance,i•e. ^ -1/6, is recovered in the infrared limit

supporting, in such a way, the stability of the minimum in the origin at

Low curvature. This feature also persists at high curvature if TJ >. 1/6.

On the other hand, for 0 <, YK K 1/6 we see that "̂ becomes

negative beyond the critical curvature

(2.20)

and the system^presumably , undergoes a second order phase transition. We

expect a new minimum ^j ^0 to appearwhen the curvature exceeds Re. This

result is consistent with our approximation only if /\ XJf\ (Rc/fc^A*) 4f- 1

which requires 0 ^ "£ **- A, 4C 1. Therefore one can infer from the

previous RG analysis that dynamical symmetry breaking, in a strong

gravitational field, can occur only when "J and A ^ are of the same order

of magnitude. We cannot say anything about the persistence of this minimum

for curvature much larger than R . This is a strong coupling region, i.e.

X)^l , where, presumably , non perturbative phenomena provide the most

relevant contributions.

3. SCALAR QED

From the previous discussion it appears to be difficult to

destabilize perturbatively a symmetric minimum even in the presence of a

strong gravitational field. On the contrary f we expect nontrivial effects to

occur, already at low curvature, In quantum field theoric-s exhibiting a

symmetry breaking minimum in flat spacetime. Such a physical situation is

realized in massless scaLar QED

(3.1)

where T is a complex scalar field, "F^u =• "I)MA V — 3i/A^u. is the

Maxwell field strength, ol is the gauge parameter and & is the electric

chsrge. For the sake of simplicity we shall work in the Landau gauge oi — " 0

where the photon is purely transverse and the anomalous dimensions do not

explicitly depend on £&• .In this gauge the traces of the first coefficients

of the heat kernel expansion for the photon Green function are

C3.2)

(3.3)

The one-loop effective potential in terms of the order parameter reads

(3.4)

A lengthy calculation gives the following counterterms t

- 10 -

y(3.5)

(3.6)

Finally we get

V f ^ R ) = •£•!

1-0 (3.7)

Let us now assume \- CS. Then we can neglect scalar loops and retain in

(3.7) only the contribution of the gauge field. A further simplification is

achieved expressing <•£ in terms of the flat spacetime minimum

Slightly rearranging (3.7) we get

with

(3.8)

(3.9)

(3,10)

~ 11 -

4 * • 5 ••*:.

An additive constant Z^hiH /u^a^ has b.---<-:i : ;iL-cr :.--1 •• ••. < • • .

provide a vanishing vacuum energy derisi ty 5:o the fjymmo'.:-y broak i ;i^ :~ 1 n Ln.i.n.

in the absence of gravitation, i.e. Vf ̂ -\'\ R.TO ) --0 . lionet Vf *fj R •

represents a vacuum energy density measure^ wi th rf::-;pr.- i h... Mi r-iki ••.v̂ k1 I'fjr

Euclidean.) spacetime. We recal 1 that the chuice of \\\<: "•/.•-rro" in L N1' energy

scale is not arbi trary when gravi tat ion is present. I : wo -ons idor-

se lf-c consistently the cosmological constant /\ as a rpsu1t of the natter

and gauge fields zero points fluctuations th^r varuun. energy ionsiiy rust

be positive to be compatible with the Einstein net r ic and van i shing i i: i~h--

zero curvature limit.

We notice that the same form of V(tf-R) has bee?: :>btauipd jn re!'!!1

4far the Coleman-Weinberg potential in ds Sitter 5 spare; thiti is ••bvit.iua I v

due to the fact that both Einstein and rie Sitter metrics are mns^artt • wrvat

solutions of the gravitational field equations with a i^smo 3 ogi.-a i yns r,ai: t

However, the B quantity in ref.(l5) does not contain 2/J, This numt?ricaJ

discrepancy comes from the finite renormal izat LUIJ of ^ prv.diit -̂-.i by ( ?, 1 Ob .

Allen simply fixes y 1^ 1 in the correct way to r^produ'>: Lhe flat :-naretime

TLinimujfl in the limit R-»0« This procedure amounts t..o rrriurnsa 1 i ?• ir'•%

according to (2.10a) but does not correspond tu ._mr dei'initmn (2.L0b) ^f "£

Except for this point, everything proceeds L ike i. L ' i • •' 1 5 ), In part i culnr

one finds t when p ^ 2 , a first order phase transition for

^JL— 1

. | ta ;

(3.11b!

The local minimum off the origin becomes meta-stable fur R>RL and turns

into an inflexion point at

- 12 -

(3.12b)

. Relations (3.11) and (3.12) become more transparent if we consider

a particular case. For the sake of simplici ty, let us suppose 5 rv^ VQ and

A'-J 6. . Then V(*?j R) can be approximated with

(3.13)

The model described by (3.13) formally appears to be equivalent to the

usual scalar QED with a small* positive, "mass squared" m =Y^* ^n the

"massless" limit E»0 the original Coleman-Weinberg theory is recovered and

2 2V( *? t fl«O ) exhibits a minimum in *-? =H . For R> 0 the origin

turns from a local maximum to a local minimum. The critical values (3.11)

read

(3.14a)

and the inflexion point becomes

(3.14b)

(3.15a)

(3.15b)

Tt is also clear that these results are perturbatively reliable, in fact

fi, 13,, i o »

e*^i - 2 e1 ^ 32 n̂

To invest igate th-^ K^F-I! ing behaviour of !.he e'Tect ive -•;

let us consider a^air. an approximation like (? . 1 '^ )

— 1 "'. —

$/#

(3.16)

The dots stand for ̂ independent, geometrical terms. If we neglect the

scalar field contributions (3.16) takes the same form as the effective

potential in the de Sitter spacetime, in the region S H * * ^

The fi.- functions and the anomalous dimensions Ji,-j are solutions of the

RG equations

(-it+ P*&

(3.17a,

(3.17c)

(3.17d)

where

7, and H ;ji-̂ in:.' usual sc^Jar ['i

T'cii'jrin.iL ;7.ir;.jr. roristarts *xc*-^

arm nhc.ton, one loop, wave function

"he dependence on the new scaling

variable t B XH (R/6/) rather than the usual t2= JU, fl

(3.20)

(3.21)

Equations (3.l?a),(3.17c), (3.17d) have the same forra as in flat spaoetime,

therefore it is immediate to obtain

e -- eV«,«rt*

.*- -Y.e - e*/

The new anomalous dimension I-| is determined by (3.17b)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

The running coupling constants can be derived from

u t

1 f

(3.27)

(3.28)

- 15 -

Equation (3.27) is the easiest to solve, or:rj si:oLy finds

(3.30)

The behaviour of © is quite similar tu triaf

purely scalar theory, we shall not repeat ii'.r:'

(2.18).

The solution of (3.28) requires a mo

In terms of the variable A / S we obtain

X in (2.18) for the

a i ̂ r-ussiori ful lowing

nvolved ralculrit L^n ,

e io 10(3.31)

The integration constant

(3.32)

is fixed by the initial condition XCo">/ (S.CO")

Putting together (3.31) and (3.32) we find

e(fe).IO

io{3.33)

Proceeding in the same way we can solve equations [3.29) too:

\oor T [M*1

c j

1 a - - M

( 3.3

t i " f o r m o f I1

. ' . ' ; ; ( • • . ' • • . ) •••ral. d i H f u s t i L on ; I t h e

•"•an s u e s s t h . i i t } e v e n

for small € , A runs over all possible values. This feature justifies

the choice A.*-£ in (3.17) and supports the discussion about symmetry

breaking. Finally we notice that for "f^ > A /(, also "Eft) i l/( so that

the origin will be at least a local minimum. 'Sft)can eventually become

negative only if ^ ^ ^ u& leading, presumably, to a second order phase

transition.

- 17 -

APPENDIX

It is a useful exercise to prove that, despite the appearance, the

effective potential (2.12) does not exhibit any genuine minimum except for the

origin. We Know, from the flat spacetime calculation , that the A term

cannot compensate the classical potential at physically acceptable values

of A J W T / T ' O , so we may drop it.

It appears that the last term can in principle balance the other ones and

produce a minimum in *™f ** ^ *r 0 . The new minimum become meta-

stable when the system satisfies the critical conditions

^ = 0 (A. 2)

R«Re,

turns into an inflexion point if

(A .3)

From (£.2),{A.3) we get the critical values characterizing this first order

phase transition

(A.4)

-5)

(A..6)

- 18 -

(3, * 2H.*.7)

Unfortunately it results to be

(A.8)

so, even if we take -£ =0 the ratios (A .8) are very large being A « l

We must conclude that {A ,41 (A .5) , (A.6) and (A.7)are inconsistent with the low

curvature approximation which was the basic assumption in order to recover

- 19 -

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