Nambu-Goldstone bosons in curved spacetime

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


    Takeo INAMI

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


    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


  • 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



    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.


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



    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 scalar fields with mass m have been obtained previously both in dS [8] and AdS [3] spaces. In D-dimensional AdS space we have

    GAds(X, Xo;m2)_ --i 1 F(ct)F(~ +-~) 47rD/2 aD-2 r (2ot -D /2 + ~)

    X cosh-2a(o/a) (12)

    X F(ot, ct + ; 2or - D/2 + ~; cosh-2(o/a) - ie)

    c t=~(D- 1 + [ (D- 1) 2 +4m2a211/2}.


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

    The geodesic distance o between x and x 0 is given by

    cosh(o/a) = cos((t - to)/a ) cosh([x -XOI/a ) . (13)

    The propagator in dS space is

    - i 1 p(2~)r'(-2~ + O - 1) Gds(X'xo;m2)= (41r)D/2 aD_2 P(D/2)

    X F(2~,-2~ +D- 1;D/2; 1 -o2 /4a 2 + ie) (14)

    =~{D- 1+ [ (D- 1) 2 -4m2a 2]1/2}.

    The geodesic distance o in this space is

    0 2 = e( t+to) /a lx _ x012 -4a 2 sirth2((t - to)/2a ) .

    (15) We have chosen the Robertson-Walker type metric whose spacelike sections (t = const.) are flat.

    The (would-be) massless field O(x) has been defined by ~(x) = p(x)e i(x). If the fluctuation of the ~ field becomes strong enough, O(x) becomes ill-defined since it is determined up to an integral multiple of 21r. In fact, for any dimensionD, the massless limit (m 2 ~ 0) of the propagator Gds does not exist in dS space, be- cause of the infrared divergence. This divergence can- cels when the propagator is substituted into eq. (11) and the correlation function of ~ itself is well-defined.

    Substituting the large Ixl limits of eqs. (12) and (14) fo rd = 2 into eq. (10), we find

    (*(t, x)(t, Xo))

    ~-02(2Aa) -I/Nc exp [+(2/Nc) exp(-Ix - x 0 I/a)]

    (AdS space),

    ~- O2(1/AIx - x 0 I) lINe

    (flat space),

    ~- p2(1/AIx -x 0 I) lINe e-tlNca

    (dS space). (16)

    Here A is an ultraviolet cutoff parameter which is to be absorbed into the renormalization of P0"

    In the case of flat or dS space, the correlation of the order parameter field ~(x) vanishes at special infi- nities (Ixl ~ ~). Thus U(1) symmetry is restored at finite N. Symmetry breaking is an artifact of taking the limit N-~ ~.

    In AdS space, on the other hand, long-range order

    is achieved even at finite N. The gravitational potential of this space, which confines particles within a finite spacial interval, acts as an infrared cutoff. Therefore fluctuations of the Nambu-Goldstone mode decay exponentially near the spacial infinities.

    So far we have considered field theories in two. dimensional spacetime. A similar analysis can be per- formed for field theories in higher dimensional space- time by employing an effective lagrangian describing the ( massless field coupled minimally to the background metric. Examination of the large Ixl be- haviours of the propagators (12) and (14) in the mass. less limit will then tell whether symmetry breaking can occur in dS, flat and AdS spaces of arbitrary di- mensions. Concerning dS space, it can be shown that symmetry restoration takes place for arbitrary dimen. sions.

    4. Discussion. To summarize, we have considered the effects of spacetime curvature on symmetry break. ing behaviour in quantum field theories. In AdS space, symmetry can be broken for all dimensions D I> 2. Concerning fiat spacetime, the case ofD = 2 discussed above is exceptional, and symmetry can be broken for D > 2 as exemplified in many models.

    The situation is completely different for dS space [1]. In this space continuous symmetry is restored for arbitrary dimensions D provided the vacuum state is dS invariant. It is known [1] that the restoration of discrete symmetry also takes place for arbitrary di. mensions. These phenomena can be interpreted as fol. lows. D-dimensional dS space can be realized as a hy- perboloid embedded in (D + 1)-dimensional fiat space- time whose topology is R X S D- 1. For the dS invari- ant vacuum state, a propagator of a free field is analy- tic on this hyperboloid. Since the compactified space like section S D- 1 is immaterial to the problem of symmetry breaking, D-dimensional dS space is effecti- vely similar to one-dimensional space, in which even discrete symmetry is restored quantum mechanically.

    These results do not apply to fermionic symmetries like supersymmetry, since the infrared behaviour of the Nambu-Goldstone fermion is different from that of the Nambu-Goldstone bosom The authors have examined models in AdS 2 space which exhibit dynam- ical breakdown of supersymmetry, and the results will be published elsewhere [9]. It has been shown there that, in contrast to continuous bosonic symmetries


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

    discussed here, the patterns of supersymmetry break- ing in various models in AdS 2 space are qualitatively the same as those in flat space.

    We would like to thank H. Hata and T. Uematsu for reading the manuscript and for valuable comments. We are also grateful to A. Vilenkin for an illuminating discussion.


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    [2 ] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1933; S. Coleman, Commun. Math. Phys. 31 (1973) 259.

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    [4] P. Breitenlohner and D.Z. Freedman, Ann. Phys. 144 (1982) 249; L. Mezincescu and P.K. Townsend, University of Texas preprint, UTrG-8-84.

    [5] S. Coleman, R. Jackiw and H.D. Politzer, Phys. Rev. D10 (1974) 2497.

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