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Volume 163B, number 1,2,3,4 PHYSICS LETTERS 21 November 1985

NAMBU-GOLDSTONE BOSONS IN CURVED SPACETIME

Takeo INAMI

Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan

and

Hirosi OOGURI

Department of Physics, Kyoto University, Kyoto 606, Japan

Received 26 July 1985

Spontaneous breakdown of continuous symmetry is investigated in field theories in maximally symmetric curved spacetime. It is shown that Coleman's theorem precluding the existence of Nambu-Goldstone bosons in two-dimensional flat spacetime gets modified in curved spacetime, depending in different manners on the sign of spacetime curvature.

1. lntroductfon. After the advent of the inflation- ary universe scenario based upon grand unified theo- ries, spontaneous symmetry breaking in curved space- time has attracted much attention [ 1 ]. Recent inves. tigations have revealed that symmetry breaking be- haviours in de Sitter (dS) space are different from those in flat spacetime. In this letter we will examine spontaneous symmetry breaking in anti-de Sitter (ADS) space. Besides being of mathematical interest, this problem is also of physical significance, because AdS space appears as a natural background geometry in extended supergravities.

It will be shown that, in contrast to the case in dS space, symmetry breaking properties in AdS space are similar to those in flat spacetime in dimensions greater than two. We encounter a surprise in two-dimensional theories. As proven by Coleman [2] long time ago, continuous bosonic symmetries cannot be broken spontaneously in two-dimensional flat spacetime, be- cause long-range fluctuations of a massless field sup- press long-range order. We shall f'md that in AdS space symmetries can be broken spontaneously even in two dimensions. The reason for different symme. try breaking behaviours in the three classes of curved spacetime will be clarified.

We will first consider two kinds of field theories

in two-dimensional AdS (AdS 2) space, which are sol. uble in the large-N limit, to demonstrate the above statement. A more general argument will then be giv. en concerning long-range correlations in curved space. time in two dimensions. By examining massless scalar propagators in AdS, fiat and dS spaces of arbitrary dimensions, we will be able to dwaw conclusions about whether symmetry can be broken spontaneous- ly in these spaces.

2. The large-N limits o f field theories in two dimen- sions. We first consider the theory of a set of N real scalar fields ~i(x) with quartic interactions in two- dimensional spacetime. The action of this model is

s = fd2x

1 2J i - i - ~o~ q~ - (g/SNXii)2l , (1)

where guy is the metric tensor of the background spacetime.

The O(N) symmetry of the action (1) is broken at the tree level i f# 2 < 0. In flat spacetime, the effects due to radiative corrections restore the symmetry [ 1 ], in accord with the general theorem [2]. To see wheth- er this is also the case with AdS 2 space, we evaluate

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Volume 163B, number 1,2,3,4 PHYSICS LETTERS 21 November 1985

the effective potential in this space incorporating quantum effects to the leading order of the 1IN ex- pansion.

For the purpose of studying symmetry breaking, it is slightly more convenient to rewrite the action (1) in the form including an auxiliary field. The two ac- tions describe the same dynamics.

s' = f d2x ~ [kg~Vau~ia~,(fl

- ~oq~i~ i - N(l~2/g)o + No2/2g] . (2)

The large-N effective potential for ~i and o is given by

(1/N)V((o i, o) = (1/2N)o()i() i + (I.12 /g)o -- (1/2g)o 2

--ki f G(x ,x ;m 2) dm 2 , (3) O

where G(x, y; m 2) is the propagator of a free scalar field of mass m in the given background metric.

We now proceed to the evaluation of the effective potenti~ in AdS 2 space. Since AdS space is maximal- ly symmetric, propagators can be evaluated exactly. The scalar propagator in the coincidence limit is given by [3]

G(x, x; m 2) = (i/47r)

X [-log(A2a 2) + ~(~ + x/m2a 2 + ~)1 , (4)

where A is an ultraviolet cutoff parameter, a is the radius of AdS space, and ~ is the polygamma func- tion. Propagators in arbitrary dimensions will be given later [eq. (11)]. The scalar field is assumed to obey the regular boundary conditions at spatial infinities [4]. The ultraviolet divergence appearing in (4) is the only divergence in the theory in consideration, and it can be cancelled by renormalization of/a 2. We define

g2 /g = la2 /g + (1/8rr)log(AZ/M2), (5)

where M is an arbitrary parameter with dimension of mass.

The vacuum configuration of the model is deter- mined by the following extremum conditions.

a V/a(fl = o~ i = 0, (6a)

0 V[ao = ~(~i()i _ N[(1/gXo - #2)

+ (1/41r)qJ(~ + ox/~a 2 + ~) - (1/81r) log(M2a2)]

= 0 . (6b)

The vacuum expectation value a must exceed -1 /4a 2 in order that the theory be stable against local fluctu- ations of the scalar fields [4].

Eq. (6) admits two possible types of solutions, de- pending on whether o = 0 or q~i = 0 is chosen as the solution of eq. (6a). The former corresponds to broken O(N) symmetry, while the latter to unbroken O(N). The critical value of the mass parameter is found to be

Mc(a ) = a -1 exp(-4~rg2/g- "),). (7)

Here 7 is Euler's constant. WhenM ~Mc(a ). In the fiat spacetime lim- it (a -> +oo), Mc(a ) converges to zero and the phase with broken O(N) symmetry disappears, reproducing the result of Coleman, Jackiw and Politzer [5].

To summarize, we have demonstrated in the O(N) symmetric model in AdS 2 space that there exists a region of the coupling parameter M in which the sym- metry of the vacuum is broken spontaneously from O(N) to O(N - 1). The phase diagram of this model is depicted in fig. 1. It is not difficult to check that this vacuum of O(N - 1) symmetry is a global mini- mum.

Another interesting two-dimensional model is the Gross-Neveu model [6], in which the issue is a possible breakdown of chiral U(1) symmetry. The analysis of

1/o

>M

Fig. 1. The phase diagram of the O(N) symmetric model. The inverse radius of AdS 2 space is measured upward on the verti- cal axis, and the horizontal axis represents the flat spacetime limit.

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Volume 163B, number 1,2,3,4 PHYSICS LETTERS 21 November 1985

1/a

3~t~

Fig. 2. The phase diagram of the Gross-Neveu model. The dashed horizontal axis indicates that the theory has no smooth flat spacetime limit.

this model in the large-N limit goes on the same lines as the previous case, and here we only state the result.

We introduce the mass parameter 3~t deffmed renor- malization-group-invariantly in terms of the bare four- Fermi coupling constant go as

g20 = 21r/log(A 2/M 2), (8)

where A is an ultraviolet cutoff parameter. The critical value of/~/is found to be

/14c(a) = a - 1 e- ~. (9)

The chiral U(1) symmetry is broken for/1~/> ~l~c(a), and it is restored for/l~r < 3~tc(a ). Fig. 2 shows the phase diagram of this model.

3. Long-range correlations. We have discussed spon- taneous breakdown of continuous bosonic symme- tries in AdS 2 space relying on the 1IN expansion. In the case of the O(N) symmetric model, we claim that the feature observed in the large-N limit also persists at ffmite N. Since the number of the (would-be) brok- en generators is (N - 1), the ratio between indepen- dent fields i of the model and the broken generators remains finite even in the limit of infinite N. Thus the symmetry breaking behaviour at finite N is expected to be qualitatively the same as that in the large-N lim- it.

In the Gross-Neveu model, on the other hand, there are N chiral fermions against only one generator of the chiral U(1) symmetry, and one might suspect that this U(1) symmetry is restored at finite N. This is actually the case in flat spacetime [7]. To answer the question whether similar restoration of symmetry at finite N occurs or not in the case of AdS 2 space, we shall examine the long-range correlation of the order parameter field.

We consider for simplicity breakdown of U(1) sym- metry. Extension of our argument to other continu- ous symmetry will be straightforward. Suppose that (x) is the order parameter field of U(1) symmetry and ~* acquires a non-vanishing value at the mini- mum point of the large-N effective potential. We esti- mate the long-distance behaviour of the two-point function (q~*(x)(0)) to the next leading order of the 1IN expansion by making use of the large-N effective action P[( )].

When q~(x) is parametrized as p(x)e i(x), P[ = pe i ] is invariant under O(x) ~ O(x) + a with constant or, and the mode corresponding to p(x) has mass gap. Therefore, the part of I ~ which is relevant in the in- frared region is the following:

ri R = ~Nc f d2x x/-S~gU~,3uO3~,O . (10)

Here c is some dimensionless constant. Other terms containing 0 involve higher derivatives and are irrele- vant. Therefore the two-point function is given by

X exp[(O(x)O(XO)) - ~((02(x)) + (02(x0)))] (11)

as x goes to spatial infinities, where P0 is the large-N vacuum expectation value of p(x). The RHS of this equation is ultraviolet divergent, and this divergence must be cancelled by the renormalization of PO.

From this equation, one can read off the interrela- tion between the long-range correlation of U(1) sym- metry and fluctuations of the (would-be) Nambu- Goldstone mode O(x). If the latter is amplified with- out bound near spatial infinities, the former is sup- pressed and symmetry is restored. This is indeed the case with two-dimensional flat spacetime and dS space of arbitrary dimensions, as shown below.

The propagator of scala