Dynamical mass generation in S1 × R3

Nuclear Physics B169 (1980) 514--526 © North-Holland Publishing Company

D Y N A M I C A L M A S S G E N E R A T I O N IN S ~ x R 3 G. DENARDO and E. SPALLUCCI lstituto di Fisica Teorica dell'Universit~ Trieste Istituto Nazionale di Fisica Nucleate, Sezione th" Trieste, Italy

Received 19 December 1979

We study radiative corrections for a scalar field in fiat space-time with topology S I × R 3. Quantum effects, and this non-trivial topology, dynamically generate mass for the field quanta. This result is deduced by the study of the effective potential. For a model with a spontaneously broken symmetry it is possible to define a critical spatial length below which the symmetry is restored. The critical length is explicitly computed for the )~,4 theory and the O ( N ) model for large N.

1. Introduction Increasing attention has been devoted to the study of quantum field theory in space-time with non-trivial topology, i.e., not homeomorphic to R 4. A physical example of the deep connection between quantum phenomena and global space-time properties is the Casimir effect [1], that is, the vacuum energy modification between two perfectly conducting plates. Vanishing boundary conditions on the two surfaces allow only the normal modes with suitable wavelength. In some sense, it is the finite length of one space dimension which selects only certain quantum fluctuations, giving rise to the measured [2] attraction force acting on the plates. A very similar problem has been treated in a recent series of papers where quantum fields are studied in space-times where one or more spatial dimensions are compactified, i.e., they extend for a finite length and the extremes are made coincident. If one space axis, let us say x, is closed then, while the local geometry remains minkowskian, the topology is no longer R 4 but S t × R a. In this world it turns out that radiative corrections shift the mass pole in the propagators, also for massless fields, thus particles acquire mass dynamically [3, 4]. The amount of mass correction is inversely proportional to the length L of the compactified dimension, as it must be on dimensional grounds. This is not a unique situation where the mass of a particle is modified by the interaction with a periodic background structure: actually the electron in a crystal lattice is more massive than the free one. 514

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One can wonder about the consequences of compactifying the time coordinate. The answer is well-known because it physically means dealing with a finite temperature quantum field theory (FTQFT). In this work we want to present the close analogy between quantum field theory in a space-time with compactified time coordinate and that with compactified space coordinate, and discuss the physical implications of such a similarity. One can roughly guess this analogy by very simple dimensional arguments. In natural units temperature is the same as energy and energy is dimensionally an inverse length. So T o : I / L can be considered, at least on a purely formal ground, as the "typical temperature" of S ~ × R 3. More precisely, after Wick rotation t ~ - it has been performed, time or space axis are completely equivalent; hence we expect euclidean Green functions in S ~ × R 3 to exhibit the same properties as ordinary thermal Green functions. Finally, we recall the similar situation occurring in a curved space-time, with an event horizon, when it is analytically continued to complex values of some coordinate. Indeed the metric becomes periodic in the complexified coordinate, and this feature physically manifests itself as a thermal background radiation in the real space-time [5]. The temperature of this thermal bath turns out to be exactly the inverse period of the complexified coordinate. One of the most interesting results obtained in the framework of F T Q F T is the existence of a critical temperature [6, 7] above which a spontaneously broken symmetry can be restored. We will show that in a consistent way S ~ × R 3 exhibits a critical length, of the closed x axis, below which, again, a spontaneously broken symmetry can be restored. Another interesting question is related to the existence in S ~ × R 3 of both twisted and untwisted scalars and spinors [8]. Actually twisted (untwisted) fields satisfy antiperiodic (periodic) boundary conditions along x. So, the problem arises whether all such configurations can be considered as physically meaningful. In F T Q F T the scalar field naturally undergoes a periodicity condition in time while spinors are antiperiodic, and this obviously reflects their commutation and anticommutation properties [6]. Quite analogously in S I × R 3, on statistical grounds, one is led to choose the physical objects to be the untwisted scalar and the twisted spinor fields. This choice is consistent with the results obtained by lsham and Ford and avoids the problem of acausal propagation due to an improper assignment of boundary conditions on the fields. In more detail, the photon mass gained through the interaction with twisted spinors is real while untwisted spinors give rise to an imaginary mass [3]. Furthermore the regularized vacuum energy of a twisted spinor is lower than that of its untwisted configuration, showing up the correct feature of supplying the minimum energy for the ground state. The reverse takes place for scalars. Both pieces of information concerning the vacuum energy and the mass of a field are contained in the effective potential of the theory. We recall that, from the

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G. Denardo, E. Spallucci / Dynamical mass generation

physical point of view, it represents the sum of the "classical" potential energy and the vacuum energy density due to the zero-point fluctuations of the field. On the other hand, the second derivative of the effective potential is just the inverse, "dressed", propagator evaluated at zero external four-momentum, i.e., the radiatively corrected mass. All this motivated the study of the effective potential in S ~ × R 3 space-time. We are dealing with radiative correction problems; hence it is natural to wonder whether or not the topology change preserves the renormalizability of the theory. Ford and Birrel have shown, by explicitly computing Feynman diagrams, that A04 theory remains renormalizable in S t × R 3, up to the two-loop approximation [4]. In fact, summing all the graphs at this order, the L-dependent infinities, which would break renormalizability, cancel with one another. This result is actually valid at all orders, because in F T Q F T all the temperature-dependent divergent terms cancel order by order leaving alive only the usual zero-temperature divergences [9]. Hence, after the established analogy of space and time coordinates, if a theory is renormalizable in R 4 it remains so in S 1 × R 3. The paper is planned as follows: in sect. 2 we calculate the one-loop effective potential for h~,4 theory in the S t x R 3 background; in the limit of small L it turns out to be of the same form of the corresponding quantity in the high-temperature regime of FTQFT. In sect. 3 a "critical", spatial length is defined and its explicit form is recovered both for h~,4 and the O ( N ) model for large N; the physical meaning and the possible implications are discussed in sect. 4. Throughout the paper we shall use natural units: c=h=ka=

1.

2. One-loop effective potential In this section we shall compute the one-loop effective potential for a scalar field theory described by the lagrangian density

~[ , ( X ) ]

----½ 3 ~ ( X ) O a O ( x )

-- -~m2~b(x) 2 -- 4 ~ b ( x ) 4 .

(2.1)

The background geometry is the flat space-time but with topology S I × R 3, instead of R4: as 2--dt 2-dx 2-dx 2-dx 2 ;

(2.2)

here - oo < t, x 2 , x 3 < + oo, while 0 < x t < L and the points x t = 0 , x t = L are identified.

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According to the introductory discussion we assume ~(x) to be periodic along S ~. This implies the Wick rotated Feynman propagator to be of the form - i ( k ~ + ((2~r/L)n) 2 + k22 + k~ + m2) - t in other words, the four-momentum k~ has a discrete k I component: k I -- (27r/L)n, with n -- 0, - !, _+2. . . . . The local dynamics of the field is unchanged by the closure of the x I axis, hence the effective potential can be recovered by the usual shifting and truncation procedure. In the one-loop approximation one gets

+ v,(4,).

(2.3)

Vo(~) is the classical, L-independent, potential

Vo( )

'

a "'

(2.4)

where ~ is the constant field shift. The one-loop correction is

FlnF[ 2rrn]2 ] L~-Z- / + / ~ 2 + M 2 •

1

V(-~(2~r)32LSk

(2.5)

We have introduced the following notation into (2.5):

f,=---Y,,fd;

(2.6)

with

(2.7) M 2 ~ m 2 + ½~2. Let us study the "small L " case, L2M2<
1

M 2

L

V((~b) = (2vr~32L vf(¢~) + (2~r) 32L (v2 (O))M2L2.0 + O ( M 2 L 2 ) .

(2.8)

We have indicated by v~ and v~ the following integrals:

v~(~)

fln[[2~rn] 2

-sk

cL(~k)-~

L

]

L~-L- / + £2],

I ((2~r/L)n)2 + f¢2 + M 2"

(2.9a) (2.9b)

G. Denardo , E. Spallucci / Dynamical mass generation

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Let us first compute v~(~). In order to regularize this divergent integral we adopt the dimensional continuation method. One introduces a D + 1 dimensional phase space and performs the substitution Y.n fd/¢--.Y,~ f d n f ¢ . In the notation of ref. [3] we can write (2.9b) as:

where

(2.10)

" v, L2 (*)IL~M=.0 = I ( D, 1,0, M 2)1L,M~.0 , D-,.3

I(D,I,0,M2)I L,M,.0 = ( , r D / 2 F ( I - i DI xi¢" )t ~2"/'/' ) "~D-2 F( 1

_lD,a,O))a.O

D--*3

D---*3

(2.11) For the meaning of F(I - i D,a,0) see the appendix. The final result is M2 M2 (2~r)3"---2L ( V L ( e P ) ) M 2 L 2 . 0 - 2 ~ 2 "

(2.12)

In order to evaluate v f ( ~ ) it is useful to consider first its derivative with respect to L:

:

2 OL

= - Z

--~nj

1

(2.13)

I~2 + ( ( 2 ~ r / L ) n ) 2 .

The integral in (2.13) is nothing but l u ( D , 1,0,0), whose explicit form, following ref. [3], is

lli( D, 1,0,0)

:- 0-a b-~-~ F ( - 1

½D,a,b

b-0'a'0 D---~3

(2.14) Finally, we get

1

0v~= 6 ~r2

(2~.)3 0L

L 4 90"

(2.15)

Now, by integrating (2.15) over L and multiplying by I / 2 L , we recover:

1 v~(~,)-(2~r)32L

': + c 90L 4 L

- -

(2.16)

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C is an L-independent integration constant. Dimensional arguments require C to be of the form c~M 3, with a a pure number. However, a M a / L is negligible, in the M2L2<
V((~)

---- - -

~r2 90L 4

1 2-~

- - - t -

( m 2 + ½~.~2)

(2.17)

The first term only depends on the global space-time structure through l / L 4. It is nothing but the Casimir, vacuum, energy density of the free field. This interpretation can be easily checked recalling the zero-point energy in S ~ × S ~ × S ~ × R Minkowski geometry [10]: 7r2 L°Z 90 L 3

E°---

(2.18)

One immediately sees that the energy density, in the limit L0--~ 0o is just - ~ , 2 / 9 0 L 4 . The periodicity boundary condition along x~ excludes all the vacuum oscillations whose wavelength does not fit into L. Roughly, it is the absence of these normal modes which generates the static negative energy distribution in the vacuum. The round bracket in (2.17) contains the vacuum energy shift ~,~2/48L2 due to the self-interaction; this term alters the mass of the field. Indeed, if we define the one-loop renormalized mass as 2 BV ~-0 ,

(2.19)

then we get as a consequence of the non-trivial space-time topology, that the physical mass is shifted by the finite quantity Am

=

X

24L 2

(2.20)

This result is in complete agreement with the calculation by Ford and Birrel of the "tadpole" correction to the free field propagator [4]. Now if one looks at the high-temperature form of the one-loop of effective potential in F T Q F T [6], one will recognize (2.17) once L is substituted by the inverse temperature ft. As we anticipated in the introduction, quantum field theory in S t × R a manifests the same features of F T Q F T . This analogy goes beyond the formal level and reflects the intrinsic nature of the vacuurm In both cases the ground state of the system is an "excited vacuum", either because it is heated, or because the boundary conditions, dictated by the non-trivial topology, select some particular zero-point oscillations.

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G. Denardo, E. Spallucci / Dynamical mass generation

3. The critical length At the beginning of sect. 2 we did not specify the sign of m 2. It is well-known that for m 2 < 0 and ~, > 0 the classical potential (2.4) assumes a symmetry breaking configuration with minima in ~ = +_(-6m2/~) I/2. When finite-temperature, quantum corrections are included, it can happen that the symmetry is restored above a critical temperature tic-i, defined by the condition [6] o r ' ° ~=o 7

,

2

=

(3.1) .

For T • fl~l the temperature dependent corrections become so large as to make the renormalized mass m~ • 0; as a consequence, the effective potential admits only the symmetric minimum in (b = 0. According to our point of view, it is natural to look for a "critical length" L c, below which a spontaneously broken symmetry is restored. L~ is defined by a condition equivalent to (3.1); that is, ~vL" I

1

2

2 .

(3.2)

On a length scale smaller than L~ zero-point oscillations are strong enough to shift the mass pole in the renormalized propagator from an imaginary to a real value. By inserting (2.17) into eq. (3.2), and solving for L¢ one gets 2 _

L¢ -

h 24m2

(3.3)

The result (3.3) is obtained by expanding the effective potential (2.5) up to the first order in L2M 2. It is interesting to check the validity of this approximation. This point can be clarified by the direct computation of L c from the definition (3.2) where V~ is given by the exact expression (2.5) and not by the approximated form (2.8). When radiative corrections a.re included the renormalized mass reads 1 2 8~P28Vi~-o+Sm2=--m2=m2+Sm2+-~ /"lk ((2¢r/L)n) 2+i~2+M2

, (3.4)

where 8m 2 is the mass counterterm, and eq. (3.2), i.e., the condition for restoring the symmetry, becomes = O.

(3.5)

521

G. Denardo, E. Spallucci / Dynamicalmassgeneration After evaluation of the integral in (3.4) the regularized expression for m 2 reads ~,¢rn/2 ( 2 ~ ' ) n-2

m2=m2+~m2+EL(2~r)D --~

r(I-½D)F(a,O;1-½D).

(3.6)

The function F(a,0; l - ~ D) is the sum of two contributions of which one is L-independent, divergent for D---,3, and cancels the 8m 2 term, the other is finite and L-dependent. The result is m~ -- m 2 + ~22f_ i/2(a,0),

(3.7)

f - i / 2 ( ° , O) ..~f°°du ( u2 - a2)l/2

(3.8)

where, since a = LM/2rt,

From (3.5) we get a general form for the critical length

2

x/

Lc=

(3.9)

m2 -I/2 (a'0)"

In the limit L2M2q~. 1, f_l/2(a,O)~.f_ i / 2 ( 0 , 0 ) - ~ and (3.3) is recovered. These results can be deduced from (A.3), (A.4). Up to this stage we have worked in the one-loop approximation, so it is natural to wonder what happens when higher-order corrections are included in the effective potential. In other words we are going to inquire whether a spontaneously broken symmetry will be restored for L < L c, in a framework non-perturbative in h, i.e., in the number of loops. In order to do this it is very useful, and widely used, to allow an O(N ) internal symmetry group for the field, because in this case a perturbative development in l / N is applicable, so that one is able by direct inspection to evaluate the complete contribution of a given order in 1/N, independently of the number of loops [6, 1 l]. In order to achieve this situation one uses a iagrangian: f_=½at~aO#tk °

,

o

I N

2

-~X~cho+'~ g X -

Nm2 g X,

(3.10)

where ~° is the N-component scalar field; X is a fictitious scalar field used as a device following ref. [12]; g/8N is the bare coupling constant. The one-loop effective potential from this lagrangian is [12]

N

i

V(~b,X)=--~gX+~X~

2 Nm2 N£ + g X + "~" ln(k 2 + X ) -

(3.11)

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G. Denardo, E. Spallucci / Dynamicalmassgeneration

The ground state is obviously defined by the stationarity conditions aV/Ox--O, ~V/O~ = O. The latter gives

2N 2g Nm2_ *2=Tx-

l

N fk k2_+ X

(3.12)

The symmetry is spontaneously broken whenever q~2[×, o :/: 0. The computation of the integral in (3.1 i) proceeds as for (3.4). The final result, after having absorbed the infinite part in the redefinition of the mass and the coupling constant, turns out to be, for L2m2<< I, 2N m2 g2

g,21x.0 =

N

12L2.

(3.13)

The symmetric ground state of the theory corresponds to th21x_0 = 0.

(3.14)

Solving this equation with respect to L, one easily obtains L 2 = - g / 2 4 m 2 which in the notation used for (2.1) reads L2 =

N~, 72m2 .

(3.15)

This quantity is not perturbative in h in the sense that both one- and two-loop graphs have been simultaneously taken into account. The role that the critical length could play in the theory of extended hadrons will be shortly discussed at the end of sect. 4. 4. Discussion

In this last part of the paper we shall comment about some features of the previous results. In sect. 2 we studied the effective potential in the LEM2<< l limit. When the opposite case, L2M2>> l, occurs then k I becomes a continuum variable and (2.5) goes smoothly into the usual form: dak ln(k 2 + M2). (2~r) 4

(4.1)

We remark that for L2M2<< i it is enough to expand V ( ( ~ ) up to the first order in L2M 2 in order to obtain all the physically relevant contributions in the one-loop approximation. Higher-order terms in the expansion produce corrections of the

G. Denardo, E. Spallucci / Dynamicalmass generation

523

form M 4 and M41nM2L 2 which are either negligible or complex, in a certain range of the parameters. The formal identity between (2.17) and the high-temperature effective potential in F T Q F T , allows us to fix C = - M 3 / 1 2 w in (2.16). If one does not like this a posteriori reasoning, one can straightforwardly evaluate C by recognizing in the C / L piece the n = 0 contribution from y.n. Indeed, it is d/~

O)= I f

M3 12~rL "

ln(/~2 + M 2 ) = _ _ _

(4.2)

Finally, let us briefly consider the effective potential for spinor fields. We add the following fermionic part to the (2. l) lagrangian: (4.3) tk(x) is a twisted spinor field, i.e., it is antiperiodic along x~. The one-loop correction to the classical effective potential turns out to be [6]:

U~(~)=-2£1n[(n+½)2(21r)

2 + ic2 + g

2] ,

(4.4)

where M = m + f ~ . It is very easy to compute (5.4) once (2.17) is known, in fact we can use the following property:

~.. n I -

l(2n+l;L)= oo

~.

I(n;L)-

n~--oo

~_, I ( n ; 2 L ) , nl

(4.5)

--oo

where I stands for an integral of the kind l = f d ° l ~ [ l ~ 2 + 2 p l ~ + H ] -° comparing (4.4) with (2.5) and exploiting (4.5) one recovers: U#( ~, ) = - 4(2V,2£(~,) - V # ( ~ ) ) .

By

(4.6)

A trivial calculation gives V((%) =

m7~2 180L 4

+ - - M2 12L 2 '

(4.7)

which again agrees with the corresponding quantity in F T Q F T . The physical meaning of the quantities in (4.7) is quite similar to those of the corresponding terms in (2.17). Consistently with our comments on (2.17) we want to stress the meaning of vacuum energy which is attributed to the first term there and in (4.7).

G. Denardo , E. Spallucci / Dynamical mass generation

524

Explicitly, the zero-point energy for a scalar untwisted field is

(4.8) in which we remark that ] ~ 2 - - k 2 + k 2 regularized form is

l

l

'°

and d / ~ = d k 2 d k 3. Its dimensionally

+...]"'

2 L (2~r) 2

( , , 'too). (4.9) Discarding the piece divergent for D --2, we get the finite quantity

3L 4

2~r

! "

(4.10)

I In the limit L2m2<( I, f_ 3/2(Lm/21r, O)~f_3/2(O,O)=F~ and the first term of (2.17) is recovered. In this paper we tried to clarify the physical mechanism through which the global structure of space-time dynamically influences the mass of the elementary particles. The answer seems to be in the modification induced in the form of the effective potential by the presence of a compactified spatial dimension. In our opinion the study of the effective potential is a promising approach to the understanding of quantum effects in space-time with non-trivial topology. On the groundwork, if one tries to reproduce the mass of a physical particle entirely via this kind of mechanism, it is clear that L must be chosen comparable with the Compton wavelength of the particle itself. This is very reminiscent of what happens for field theories in 4 + D dimensional flat space-time with D compactified extra dimensions. After the technique of dimensional reduction has been applied, the D extra dimensions manifest themselves in the physical world only in the mass spectrum of the field [13]. In this context, our S~x R 3 space-time appears as a simplified useful "laboratory" for investigating the way through which space compactification generates masses. Another set of models, where, presumably, all these effects could fit in, contains theories like the strong gravity [14] and bag [15] models, which describe hadrons as micro-universes, with one or more compact dimensions, embedded in the Minkowski space-time [16].

G. Denardo , E. Spallucci / Dynamical mass generation

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For instance, we recall that in the "bag" theory the vacuum is assumed to exist in two distinct phases: inside the bag it supports free and massless partons 0aadron phase), while outside (normal vacuum) it forbids constituents free propagation. The simplest way to achieve such a confinement is to attribute a very high mass to the quarks outside the hadronic bubbles. If masses are generated through the Higgs mechanism, then symmetry must be spontaneously broken in the normal vacuum and restored inside the hadron. Now, by requiring for the bag a spatial extension close to the critical length L¢ all the previous features follow naturally. This is obviously a toy-model; anyway, it incorporates the appealing idea that quark confinement is a consequence of the hadron non-trivial topology. Particularly interesting in this framework should be the study of gauge fields, which we plan to present in a forthcoming paper. We thank Dr. E. Gava and Dr. R. Percacci for discussions.

Appendix For the sake of the reader's convenience, we report from ref. [3] those formulae which are more widely used throughout the paper:

F(X;a,b)=

Z

[ ( n + b ) 2 + a 2 ] -x,

(A.I)

F( )~; a,b ) = ¢r' /2a' - 2x g()~ - ½) ~ 4sinTO,.A(a,b), f~(a,b) - - ~ d u ( u

2 - a 2 ) - ~ R e [ e 2"("÷'~) - ! ] - ' ,

fx(0,0) = ( 2 ~ r l 2 X - ' r ( ! - 2h)~'(I - 2 A ) ,

Reh < 0.

(a.2) (A.3)

(A.4)

From (A.4) it is easy to prove that f_ 1/2(0,0)--gI and f_3/2(0,0)= ± 240"

References [I] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wentschap. 51 (1940) 793 [2] D. Tabor and R.H.S. Winterton, Proc. Roy. Soc. A312 (1969) 435 131 L.H. Ford, Vacuum polarization in a non-simply connected space-time, King's College preprint [4] N.D. Birrell and L.H. Ford, Renormalization of self-interacting scalar field theories in a non-simply connected space-time, King's College preprint [51 G.W. Gibbons and M.J. Perry, Proc. Roy. Soc. A358 (1978) 467 [61 L. Dolan and R. Jackiw, Phys. Rev. I)9 (1974) 3320 [71 S. Weinberg, Phys. Rev. 139 (1974) 3357 [81 S.J. Avis and C.S. lsham, Proc. Roy. Soc. A363 (1978) 581; C.J. lsham, Proc. Roy. Soc. A362 (1978) 383

526 [9] [10] [11] [12] [13] [14]

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M.B. Kislinger and P.D. Morley, Phys. Rev. DI3 (1976) 2771 L.H. Ford, Casimir effect for a self interacting scalar field, King's College preprint R. Jackiw, Phys. Rev. D9 (1974) 1686 S. Coleman, R. Jackiw and H.D. Politzer, Phys. Rev. DI0 (1974) 2491 E. Cremmer and J. Scherk, Nucl. Phys. BI03 (1976) 399 C.J. lsham, A. Salam and J. Strathdee, Phys. Rev. D3 (1971) 867; 8 (1973) 2600; B. Zumino, in Lectures on elementary particles and quantum field theory, ed. S. Deser, M. Grisaru, H. Pendleton (MIT Press, Cambridge, Mass., 1970) [15] A. Chodos, R.L. Jaffe, K. Johnson and V.F. Weisskopf, Phys. Rev. D9 (1974) 9471; W.A. Bardeen, N.S. Chanowitz, S.D. Drell, M. Weinstein and T.M. Yan, Phys. Rev. DII (1975) 1094 [16] A. Salam and J. Strathdee, Phys. Rev. DiS (1978) 4596