- Email: [email protected]
The membrane abelian Higgs model
Volume 264, number 1,2
PHYSICS LETTERS B
25 July 1991
The membrane abelian Higgs model A n t o n i o A u r i l i a a,~, F r a n c o L e g o v i n i b,2 a n d E u r o Spallucci b,3 a Department of Physics, California State Polytechnic University, Pomona, CA 91768, USA b Dipartimento di Fisica Teorica, Universitd di Trieste, andlNFN, Sezione di Trieste, 1-34014 Trieste, Italy Received 7 February 1991
We develop the first and second quantized version of a model consisting of a closed bosonic relativistic membrane coupled to a three-index gauge potential Au~p(x). The model is reparametrization invariant as well as invariant under a generalized U ( 1)gauge transformation and is developed in analogy to scalar QED. We call it the membrane abelian Higgs model. For imaginary surface tension the absolute value of the membrane field has a non-vanishing vacuum expectation value and the gauge symmetry is spontaneously broken; as a consequence, the gauge field A~p(x) acquires a mass while the Goldstone bosons disappear from the physical spectrum. However, unlike the usual Higgs mechanism, the physical spectrum consists of spin-0 (pseudoscalar) particles.
It has long been known that relativistic extended objects possess a new kind o f U ( 1 ) gauge symmetry involving multilocal phase transformations o f the wave functionalwhich represents the extended object [ 1 ]. The new gauge functions and the associated compensating fields are represented by antisymmetric tensors o f higher rank. In particular, the three-index gauge potential A ~ p ( x ) represents the "gauge p a n ner" o f relativistic membranes in the sense that it mediates the interaction between surface elements according to the same general principle, i.e. local gauge invariance, dictating the coupling between charged point-like particles and vector gauge bosons. This kind o f generalized "electromagnetic potential" plays a relevant role both in panicle physics and cosmology [2 ]. Some remarkable examples o f physical effects driven by A , , p ( x ) are: the trapping mechanism in the quark-bag model [ 3 ]; dynamical generation and suppression o f the cosmological constant [ 4 ]; spontaneous generation o f inflationary domains [5 ]; bosonization o f Dirac fields in a R i e m a n n Cartan background geometry [ 6 ]; vacuum decay and bubble nucleation as a generalized Klein effect [ 7 ]. J Bitnet address: [email protected]. 2 Bitnetaddress: [email protected]. 3 Bitnetaddress: [email protected].
In this letter we wish to suggest a possible extension of the Higgs mechanism whereby the gauge field Au~p(x) acquires a mass as a consequence of spontaneous breakdown o f the extended U ( 1 ) gauge symmetry while the Goldstone bosons, represented here by axion fields in the K a l b - R a m o n d representation [ 8 ], disappear from the physical spectrum. A potential application of this idea was already considered in ref. [ 9 ] but here we consider the full functional field theory o f a bosonic membrane interacting in a gauge and reparametrization invariant way with the local field Au~p(x). This is a lagrangian theory where the dynamical variable qb(S) is a complex functional [ 1 ] o f the surface configuration S: X" (~ ~, ~2 ) modelling the spatially closed membrane in terms o f local coordinates ~J, ~:. For imaginary values o f the surface tension the functional gauge variance is spontaneously broken at the classical level. As a consequence, the absolute value o f qb(S) acquires a nonvanishing vacuum expectation value and the gauge field Au~p(x) becomes massive. This is what we call the m e m b r a n e Higgs mechanism. A similar phenomenon has been recently investigated for K a l b R a m o n d fields coupled to relativistic strings [ 10 ]. Our first objective is to derive a functional SchrSdinger-type equation from which we deduce the form o f the membrane hamiltonian operator as well
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
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Volume 264, number 1,2
PHYSICS LETTERSB
25 July 1991
as the membrane Green function. From here we proceed to construet an action functional for the membrane Higgs model which is invariant under the extended U ( 1 ) gauge transformation. Our starting point is the quantum mechanical amplitude from an initial membrane configuration IX, ) to a final configuration [272), in the presence of the Au~p(X) field, written as a path integral
where Po is the bare surface tension, g is the gauge coupling constant and
K(272,27,; g2)= S ~ [ X ( ~ ) ] ~ [ A ( x ) ]
Fu~,p~= 0luA~pol
×exp{--i(SM[X]+SF[A]+S~NT[A,X])}.
(1)
Here, the classical action S = S M + S v + S I N T is defined as follows: we have a domain Y( in the space of the parameters ~ = (~o, ~ , ~2) which represents the lorentzian membrane manifold and we require that the theory be reparametrization invariant, i.e. invariant under an arbitrary coordinate transformation, or diffeomorphism, ~ , a in or(. In addition, we have an embedding g2 of ov¢into Minkowski space d/, that is, g2:~oug~12(~)=XU(~)e~# and we note, in particular, that the tangent space T ( ~ ) is mapped into the tangent space T(I2) in J/. Thus the tangent threevector in parameter space
is mapped into the tangent three-vector
j(u,,p= dXU dX" dX p d~O ^ ' ~ ^ d~2-
- ~.. X~,~X ~'"p ,
SF= -- 2"4! f daxF~,~p~F u~p~ , ,//
70
d3~.,~UUPAlavp,
(4)
(5)
represents the field strength of A~,~p. Under the assumption that the membrane is closed, the function integral ( 1 ) is computed over all the histories, xU= X u (~), of the three-volume ~2, interpolating between 2~j and 272 and having 271w 272 as the only boundary. Here g2 plays the same role as time in particle quantum mechanics and we have introduced it in the theory of membranes in analogy to the intrinsic time which in quantum cosmology is measured by the proper three-volume of the universe spatial sections [ 1 1 ]. Finally, we note that the functional measure [A (x) ] should include a sequence of suitable gauge fixing and ghost terms [ 12 ] which we have omitted in ( 1 ) since they do not affect the physical mass spectrum of the A~,,p(X) field. Next, we note that under the gauge transformation
"f
5S~= ~..
(6)
d X U ^ d X " AdXPO[uA~p]
- g f dXU^dX"A~,, 2
which represents the tangent element to the membrane itself at the point with coordinate ~ , ~2. Then, our definitions OfSM, SF, S~NTare
1
- g f 3!
the classical section S--- SM + SF + SXNTvaries according to
j ( ~ _ __OX" ^ __OX ~ 0~ ~ O~~
da~
d X ~ ^ d X ~ ^ d X p Au~, s9
6AA,p,=Ot,Ap, 1 , ~AXU(~)=0
O(X ~', X", X p) 0(~o,~,,~2)
at each point of the embedded submanifold x ~ = X" (~) which represents the world-history of the membrane in .¢/. For later reference, we also introduce the bivector
SM = --Po
SINT = ~.
(2)
(3)
(7)
where we have used Stokes' theorem to write ~SA as a surface integral over the membrane boundary worldsheet 8Q. Thus, if the membrane is closed, as we are assuming, and its world-tube is infinitely extended along the time-like direction, then 0s~ = 0 and the theory is gauge invariant ~ as well as reparametrization invariant. a~ A gauge invariant description of open membranesrequires the introduction of an extra Kalb-Ramond field [ 12].
Volume 264, number 1,2
PHYSICSLETTERSB
From the path integral ( 1 ) one can derive a functional SchrOdinger-type equation for the evolution kernel K as follows. If we take the initial configuration to be the vacuum state I-r~) = I0 ) , then K becomes a functional of/2 and Z2. By varying t2, without changing-re, we get the response of K to volume variations: 0 i ~ K(Z2, 0; 12) =poK(~z, 0; 12).
(8)
On the other hand, by varying the action S with respect to the embedding functions X(~), one obtains the classical equation of motion (Lorentz force equation )
OH#vp OXV OXP g FuvpaJ(vpa O~o A ~ A a~ z-31
(9)
which we have written in terms of the volume conju-
gate momentum #2 8LM Hu. p - Oku~p
k~.p+gA,~p
(10)
which satisfies the Klein-Gordon condition in the presence of an external field
25 July 1991
and (8) we finally arrive at the membrane "Schr6dinger equation ",
(0 , ipo~ - ~
)
~
,~ K(&, 0;co)=0.
(14)
with the boundary condition K(X2, 0; 0) =8(--rE). Note that eq. (14) is ambiguous as it stands since the second functional variation 82/5Xu (~) 0X ~(~') is an ill defined operator in the limit ~ ' . In the absence of a formal theory of renormalization for relativistic membranes coupled to an antisymmetric tensor field, we resolve the ambiguity on physical grounds by giving the membrane a finite thickness "a" which is identified with the inverse of a cutoff parameter in momentum space. This procedure, which is an extension of the regularization method of Seo and Sugamoto [ 13 ] for strings coupled to Kalb-Ramond fields, enables us to write a finite expression for the order parameter, eq. (27), which controls the membrane condensation into the vacuum and is responsible, ultimately, for generating a mass for the Au~p(x) field. However, the details of the renormalization procedure are not our immediate concern here since we use eq. (14) merely as a formal device to introduce a bare membrane Green function. This is obtained by summing over all possible values of the "time"/2 oo
1
- 3~" (Hu~p -gA,.p)2=p2o.
( I 1)
i - - j dt'2 K(Z2, 0;/2) exp(ipol2) G(£2, O;po) = - Po 0
(15)
Then, introducing the gauge covariant derivative
~u-
a ~A.~flc ~;, ax~(¢------~+
(12)
and using eqs. ( 9 ) - ( 1 2 ) one can show that the response of K to the variation of the embedding functions X(~) takes the form of the eigenvalue equation
~u~UK(S.2,0;t2)=p2IIj(U"II2K(Z2,0;g2),
(13)
where IIXUql2= ½J(~J(o~. Combining now eqs. (13)
~2 The canonicalmomentumPu is related to Huu p by
and we assign a small imaginary part to Po in order to have a well defined integration, i.e. Po--*Po+ i~. Then G(2"2, 0; Po) is a solution of the inhomogeneous Schr6dinger equation
I(~
+ ~ Au,pJ("P)2-p~IIJ(UqI2] G(Z2, 0;po)
=-~(Z'2, 0).
(16)
For later convenience it is useful to write eq. (16) in terms of the membrane "hamiltonian" operator
Hz2G(-r2, 0; po) =t~(S2, O),
(17)
pu= 8S [( 1 " • '-'/2 " ] • ~X--'~" = ½Po -- "~.X°V'tXo'm) X, up+gA~upJXUP -~H, upX' u p . RI
71
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PHYSICSLETTERSB
1
× d2~ ~f/~ZT~. ~ U _ p g
S
N/ w,wl.Vab
,
(18)
where aab is the induced metric on 2;. This enables us to construct a gauge and reparametrization invariant membrane field theory based on the quantum mechanical model discussed above. The relationship between first and second quantized version of the system is established by the Green function G(X2, 0; Po). A suitable field action, which leads to G(S2, 0; Po) as a membrane tree-levelpropagator is SH[q~(2;), q~*(S), A]
=S,+SF,
(19)
s.-- I DX(~) ~*[SIH~¢[S] +Si,t [ q~* (Z)qo(27) ] ,
(20)
where qo[X] is a complex functional, and the gauge invariant term Si,t [ q~*(27) q~(X) ] describes self-interaction, i.e. branching and joining of membrane world-tracks. The total action (19) is modelled on scalar QED, and represents the basis of our formulation of the Higgs model for membranes. To bring out this analogy more explicitly, note that the field theory defined by (19) and (20) is invariant under the functional gauge transformation
25 July 1991
taneous symmetry breaking. On general grounds, selfenergy effects could renormalize p~ to a positive value, at least for some suitable value of the parameters of the model. This interesting effect amounts to a dynamical symmetry restoration and will be discussed in a forthcoming publication. To read off the physical spectrum of our system, it is convenient to parametrize the membrane field as in the usual Higgs model, q~[Z] = --~-e~O[Z]xp~[-
2~i fz dXU ^ dX~ Otu~l( x) ) (22)
and we assume that the spontaneously broken phase is characterized by a non-vanishing vacuum expectation value (01 (a[2~] [0) = 9o = constant. Eq. (22) shows that the "phason" field, or Goldstone boson, is associated with a Kalb-Ramond ranktwo gauge potential 0t~j. The appearance of this unusual representation of massless spin-0 fields [ 8 ] is a direct consequence of the transformation law (6) and represents a drastic departure from the usual Higgs mechanism as far as the physical spectrum is concerned. Indeed, with the field decomposition (22), the relevant part of the action ( 18 ) - (20) becomes 1 ~ d4 x FuupaF,uUp a 2.4--~
S= -
--!
q~[s] ~q~' is]
2 •
=exP(2~z=~aadXU ^
aab
-½ ~ D[ X] [pZ~o2+ g2 ~92
- gl OluOvp]) 2]
q~[Xl ,
(21) so long as the "compensating" field A~,,p transforms as in eq. (6). In analogy to the usual Higgs model, we expect gauge symmetry breaking as a consequence of closed membranes condensation into the vacuum. At the treeqevel this will occur for purely imaginary surface tension, i.e. when po2 <0. Note, however, that a negative value ofpg does not immediately imply spon72
1
S
dX"At,,l(X))q~[S]
(iy dX~'^dX"^dX p OtuA~ol(X))
=exp ~.
2
+aim.
(23)
Now, the Goldstone field 0 can be reabsorbed in A via a gauge transformation 1
Au,p =~4u~p+ g OtuO~pl, and thus, we find in the "unitary gauge" :
(24)
Volume 264, number 1,2
PHYSICS LETTERS B
1 f d4 x ffuupa~uupa 2.4!
S=-
_r-
g2v ~02~¢~,)+Si., - ½ ~ D [ X ] ( P ~ 0 2 + -~.
'
(25)
where Fu~p.= ~u~a.-= 0Lu~C.p.1 since the field strength is gauge invariant by definition. Finally, expanding (25) around the constant vacuum field configuration, we deduce the following lagrangian for d : L[~¢] = -
1 ~.paO_~.u~,po.- g2~2 ~.uuP ~'u)'p 2-4~-~ 2.3! ' (26)
¢ 2 = (01 fj DXx(~)
~2[s] - - I o > . a2fzd2~
aab
(27)
The details o f the above calculations will be given in a forthcoming publication, but we wish to underline now the physical meaning ofeq. (26): a non-vanishing vacuum expectation value for the membrane field develops a mass term for the ~-field, just as the usual Higgs field generates the mass of the gauge bosons. However, while in the usual Higgs mechanism the vectorial character, or spin content, of the gauge field is preserved in the transition from massless to massive particle, a new effect occurs when the gauge field is A..p (x): as a gauge field, AF,vp(x ) prossesses no dynamical degrees of freedom and represents a constant background field [3]; but the lagrangian (26) describes a spin-0, pseudoscalar particle o f mass g~o in a representation which is dual to the familiar Proca representation o f massive, spin-1 particles. Indeed, the lagrangian (26) leads to the constraint equation: 0~,~p = 0 or, in terms of the dual field, eu~po~"d*~= 0. This constraint eliminates the spin-1 component o f the (pseudo) vector field d *° but allows its time-like component to propagate as a free field by absorbing the Goldstone bosons which, in the present model, are represented by massless K a l b - R a m o n d fields [ 8 ]. Evidently the key factor which is responsible for this
25 July 1991
unusual phenomenon is the gauge transformation (6) which is dual to the usual gauge transformation of the electromagnetic field. In this sense the new Higgs mechanism for membranes is dual to the old Higgs mechanism for point particles. The possibility that such a dual Higgs mechanism might occur in the field theory o f extended objects was anticipated long ago [ 14] and in ref. [9] it was suggested that precisely such a mechanism might provide the solution o f the U ( l ) problem in QCD. In this connection, we wish to close this letter with a speculative idea which, we believe, is worth looking into: the "invisible" axions, originally required by the Peccei-Quinn mechanism [ 15 ], also play a major role in cosmology as a leading candidate for cold dark matter in the universe; an equally important cosmological role is played by the gauge field Au,p(x ) which provides, even in flat Minkowski space, the constant vacuum energy density necessary for the exponential expansion of the universe during the inflationary epoch [ 5 ] and may be responsible for the birth and evolution of the universe [ 6,7 ]. In the light o f the dual Higgs mechanism, it is tempting to identify the K a l b - R a m o n d Goldstone bosons with the axion fields and speculate that the reason why they are "invisible" is that they are "eaten up" by the gauge field Au~p(x). In this perspective, the Higgs mechanism for relativistic bubbles may well be regarded as the ultimate mechanism for generating mass out of the "bubbling" vacuum o f the early universe.
References [1 ] Y. Nambu, Phys. Rev. 23 (1976) 250. [2] For a review see for example A. Aurilia and E. SpaUucci, The role of extended objects in particle theory and cosmology,in: Proc. Trieste Conf. on Super-membranesand physics in 2+ 1 dimensions (Trieste, June 1989) (World Scientific, Sigapore). [ 3 ] A. Aurilia, Phys. Left. B 81 ( 1979 ) 203. [4] S.W. Hawking, Phys. Len. B 134 (1984) 403; J.D. Brown and C. Teitelboim, Nucl. Phys. B 297 (1988) 787; Phys. Lett B 195 (1987) 177; M.J. Duncan and L.G. Jensen, Nucl. Phys. B 336 (1990) 100; A new mechanism for neutralizing the cosmological constant, preprint UMN-TH-908/90, DOE/ER/40423-19. [ 5 ] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Len. B 147 (1984) 258; Nucl. Phys. B. 252 (1984) 523. 73
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PHYSICS LETTERS B
[6] A. Aurilia and E. Spallucci, Phys. Rev. D 40 (1990) 2511. [ 7 ] A. Aurilia and E. Spallucci, Phys. Lett. B 251 (1990) 39; A. Aurilia, R. Balbinot and E. Spallucci, Phys. Lett. B 262 (1991) 222. [8] M. Kalb and P. Ramond, Phys. Rev. D 9 (1974) 2273. [9] A. Aurilia, Y. Takahashi and P.K. Townsend, Phys. Lett. B 95 (1980) 265. [ 10] S.J. Rey, Phys. Rev. D 40 (1990) 3396. [ 11 ] B. DeWitt, Phys. Rev. 160 (1977) 1113.
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[ 12 ] A. Aurilia and Y. Takahashi, Progr. Theor. Phys. 66 ( 1981 ) 693. [ 13] K. Seo and A. Sugamoto, Phys. Rev. D 24 ( 1981 ) 1630. [ 14] A. Aurilia and F. Legovini, Phys. Lett. B 67 ( 1977 ) 299. [15 ] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.
PHYSICS LETTERS B
25 July 1991
The membrane abelian Higgs model A n t o n i o A u r i l i a a,~, F r a n c o L e g o v i n i b,2 a n d E u r o Spallucci b,3 a Department of Physics, California State Polytechnic University, Pomona, CA 91768, USA b Dipartimento di Fisica Teorica, Universitd di Trieste, andlNFN, Sezione di Trieste, 1-34014 Trieste, Italy Received 7 February 1991
We develop the first and second quantized version of a model consisting of a closed bosonic relativistic membrane coupled to a three-index gauge potential Au~p(x). The model is reparametrization invariant as well as invariant under a generalized U ( 1)gauge transformation and is developed in analogy to scalar QED. We call it the membrane abelian Higgs model. For imaginary surface tension the absolute value of the membrane field has a non-vanishing vacuum expectation value and the gauge symmetry is spontaneously broken; as a consequence, the gauge field A~p(x) acquires a mass while the Goldstone bosons disappear from the physical spectrum. However, unlike the usual Higgs mechanism, the physical spectrum consists of spin-0 (pseudoscalar) particles.
It has long been known that relativistic extended objects possess a new kind o f U ( 1 ) gauge symmetry involving multilocal phase transformations o f the wave functionalwhich represents the extended object [ 1 ]. The new gauge functions and the associated compensating fields are represented by antisymmetric tensors o f higher rank. In particular, the three-index gauge potential A ~ p ( x ) represents the "gauge p a n ner" o f relativistic membranes in the sense that it mediates the interaction between surface elements according to the same general principle, i.e. local gauge invariance, dictating the coupling between charged point-like particles and vector gauge bosons. This kind o f generalized "electromagnetic potential" plays a relevant role both in panicle physics and cosmology [2 ]. Some remarkable examples o f physical effects driven by A , , p ( x ) are: the trapping mechanism in the quark-bag model [ 3 ]; dynamical generation and suppression o f the cosmological constant [ 4 ]; spontaneous generation o f inflationary domains [5 ]; bosonization o f Dirac fields in a R i e m a n n Cartan background geometry [ 6 ]; vacuum decay and bubble nucleation as a generalized Klein effect [ 7 ]. J Bitnet address: [email protected]. 2 Bitnetaddress: [email protected]. 3 Bitnetaddress: [email protected].
In this letter we wish to suggest a possible extension of the Higgs mechanism whereby the gauge field Au~p(x) acquires a mass as a consequence of spontaneous breakdown o f the extended U ( 1 ) gauge symmetry while the Goldstone bosons, represented here by axion fields in the K a l b - R a m o n d representation [ 8 ], disappear from the physical spectrum. A potential application of this idea was already considered in ref. [ 9 ] but here we consider the full functional field theory o f a bosonic membrane interacting in a gauge and reparametrization invariant way with the local field Au~p(x). This is a lagrangian theory where the dynamical variable qb(S) is a complex functional [ 1 ] o f the surface configuration S: X" (~ ~, ~2 ) modelling the spatially closed membrane in terms o f local coordinates ~J, ~:. For imaginary values o f the surface tension the functional gauge variance is spontaneously broken at the classical level. As a consequence, the absolute value o f qb(S) acquires a nonvanishing vacuum expectation value and the gauge field Au~p(x) becomes massive. This is what we call the m e m b r a n e Higgs mechanism. A similar phenomenon has been recently investigated for K a l b R a m o n d fields coupled to relativistic strings [ 10 ]. Our first objective is to derive a functional SchrSdinger-type equation from which we deduce the form o f the membrane hamiltonian operator as well
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
69
Volume 264, number 1,2
PHYSICS LETTERSB
25 July 1991
as the membrane Green function. From here we proceed to construet an action functional for the membrane Higgs model which is invariant under the extended U ( 1 ) gauge transformation. Our starting point is the quantum mechanical amplitude from an initial membrane configuration IX, ) to a final configuration [272), in the presence of the Au~p(X) field, written as a path integral
where Po is the bare surface tension, g is the gauge coupling constant and
K(272,27,; g2)= S ~ [ X ( ~ ) ] ~ [ A ( x ) ]
Fu~,p~= 0luA~pol
×exp{--i(SM[X]+SF[A]+S~NT[A,X])}.
(1)
Here, the classical action S = S M + S v + S I N T is defined as follows: we have a domain Y( in the space of the parameters ~ = (~o, ~ , ~2) which represents the lorentzian membrane manifold and we require that the theory be reparametrization invariant, i.e. invariant under an arbitrary coordinate transformation, or diffeomorphism, ~ , a in or(. In addition, we have an embedding g2 of ov¢into Minkowski space d/, that is, g2:~oug~12(~)=XU(~)e~# and we note, in particular, that the tangent space T ( ~ ) is mapped into the tangent space T(I2) in J/. Thus the tangent threevector in parameter space
is mapped into the tangent three-vector
j(u,,p= dXU dX" dX p d~O ^ ' ~ ^ d~2-
- ~.. X~,~X ~'"p ,
SF= -- 2"4! f daxF~,~p~F u~p~ , ,//
70
d3~.,~UUPAlavp,
(4)
(5)
represents the field strength of A~,~p. Under the assumption that the membrane is closed, the function integral ( 1 ) is computed over all the histories, xU= X u (~), of the three-volume ~2, interpolating between 2~j and 272 and having 271w 272 as the only boundary. Here g2 plays the same role as time in particle quantum mechanics and we have introduced it in the theory of membranes in analogy to the intrinsic time which in quantum cosmology is measured by the proper three-volume of the universe spatial sections [ 1 1 ]. Finally, we note that the functional measure [A (x) ] should include a sequence of suitable gauge fixing and ghost terms [ 12 ] which we have omitted in ( 1 ) since they do not affect the physical mass spectrum of the A~,,p(X) field. Next, we note that under the gauge transformation
"f
5S~= ~..
(6)
d X U ^ d X " AdXPO[uA~p]
- g f dXU^dX"A~,, 2
which represents the tangent element to the membrane itself at the point with coordinate ~ , ~2. Then, our definitions OfSM, SF, S~NTare
1
- g f 3!
the classical section S--- SM + SF + SXNTvaries according to
j ( ~ _ __OX" ^ __OX ~ 0~ ~ O~~
da~
d X ~ ^ d X ~ ^ d X p Au~, s9
6AA,p,=Ot,Ap, 1 , ~AXU(~)=0
O(X ~', X", X p) 0(~o,~,,~2)
at each point of the embedded submanifold x ~ = X" (~) which represents the world-history of the membrane in .¢/. For later reference, we also introduce the bivector
SM = --Po
SINT = ~.
(2)
(3)
(7)
where we have used Stokes' theorem to write ~SA as a surface integral over the membrane boundary worldsheet 8Q. Thus, if the membrane is closed, as we are assuming, and its world-tube is infinitely extended along the time-like direction, then 0s~ = 0 and the theory is gauge invariant ~ as well as reparametrization invariant. a~ A gauge invariant description of open membranesrequires the introduction of an extra Kalb-Ramond field [ 12].
Volume 264, number 1,2
PHYSICSLETTERSB
From the path integral ( 1 ) one can derive a functional SchrOdinger-type equation for the evolution kernel K as follows. If we take the initial configuration to be the vacuum state I-r~) = I0 ) , then K becomes a functional of/2 and Z2. By varying t2, without changing-re, we get the response of K to volume variations: 0 i ~ K(Z2, 0; 12) =poK(~z, 0; 12).
(8)
On the other hand, by varying the action S with respect to the embedding functions X(~), one obtains the classical equation of motion (Lorentz force equation )
OH#vp OXV OXP g FuvpaJ(vpa O~o A ~ A a~ z-31
(9)
which we have written in terms of the volume conju-
gate momentum #2 8LM Hu. p - Oku~p
k~.p+gA,~p
(10)
which satisfies the Klein-Gordon condition in the presence of an external field
25 July 1991
and (8) we finally arrive at the membrane "Schr6dinger equation ",
(0 , ipo~ - ~
)
~
,~ K(&, 0;co)=0.
(14)
with the boundary condition K(X2, 0; 0) =8(--rE). Note that eq. (14) is ambiguous as it stands since the second functional variation 82/5Xu (~) 0X ~(~') is an ill defined operator in the limit ~ ' . In the absence of a formal theory of renormalization for relativistic membranes coupled to an antisymmetric tensor field, we resolve the ambiguity on physical grounds by giving the membrane a finite thickness "a" which is identified with the inverse of a cutoff parameter in momentum space. This procedure, which is an extension of the regularization method of Seo and Sugamoto [ 13 ] for strings coupled to Kalb-Ramond fields, enables us to write a finite expression for the order parameter, eq. (27), which controls the membrane condensation into the vacuum and is responsible, ultimately, for generating a mass for the Au~p(x) field. However, the details of the renormalization procedure are not our immediate concern here since we use eq. (14) merely as a formal device to introduce a bare membrane Green function. This is obtained by summing over all possible values of the "time"/2 oo
1
- 3~" (Hu~p -gA,.p)2=p2o.
( I 1)
i - - j dt'2 K(Z2, 0;/2) exp(ipol2) G(£2, O;po) = - Po 0
(15)
Then, introducing the gauge covariant derivative
~u-
a ~A.~flc ~;, ax~(¢------~+
(12)
and using eqs. ( 9 ) - ( 1 2 ) one can show that the response of K to the variation of the embedding functions X(~) takes the form of the eigenvalue equation
~u~UK(S.2,0;t2)=p2IIj(U"II2K(Z2,0;g2),
(13)
where IIXUql2= ½J(~J(o~. Combining now eqs. (13)
~2 The canonicalmomentumPu is related to Huu p by
and we assign a small imaginary part to Po in order to have a well defined integration, i.e. Po--*Po+ i~. Then G(2"2, 0; Po) is a solution of the inhomogeneous Schr6dinger equation
I(~
+ ~ Au,pJ("P)2-p~IIJ(UqI2] G(Z2, 0;po)
=-~(Z'2, 0).
(16)
For later convenience it is useful to write eq. (16) in terms of the membrane "hamiltonian" operator
Hz2G(-r2, 0; po) =t~(S2, O),
(17)
pu= 8S [( 1 " • '-'/2 " ] • ~X--'~" = ½Po -- "~.X°V'tXo'm) X, up+gA~upJXUP -~H, upX' u p . RI
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1
× d2~ ~f/~ZT~. ~ U _ p g
S
N/ w,wl.Vab
,
(18)
where aab is the induced metric on 2;. This enables us to construct a gauge and reparametrization invariant membrane field theory based on the quantum mechanical model discussed above. The relationship between first and second quantized version of the system is established by the Green function G(X2, 0; Po). A suitable field action, which leads to G(S2, 0; Po) as a membrane tree-levelpropagator is SH[q~(2;), q~*(S), A]
=S,+SF,
(19)
s.-- I DX(~) ~*[SIH~¢[S] +Si,t [ q~* (Z)qo(27) ] ,
(20)
where qo[X] is a complex functional, and the gauge invariant term Si,t [ q~*(27) q~(X) ] describes self-interaction, i.e. branching and joining of membrane world-tracks. The total action (19) is modelled on scalar QED, and represents the basis of our formulation of the Higgs model for membranes. To bring out this analogy more explicitly, note that the field theory defined by (19) and (20) is invariant under the functional gauge transformation
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taneous symmetry breaking. On general grounds, selfenergy effects could renormalize p~ to a positive value, at least for some suitable value of the parameters of the model. This interesting effect amounts to a dynamical symmetry restoration and will be discussed in a forthcoming publication. To read off the physical spectrum of our system, it is convenient to parametrize the membrane field as in the usual Higgs model, q~[Z] = --~-e~O[Z]xp~[-
2~i fz dXU ^ dX~ Otu~l( x) ) (22)
and we assume that the spontaneously broken phase is characterized by a non-vanishing vacuum expectation value (01 (a[2~] [0) = 9o = constant. Eq. (22) shows that the "phason" field, or Goldstone boson, is associated with a Kalb-Ramond ranktwo gauge potential 0t~j. The appearance of this unusual representation of massless spin-0 fields [ 8 ] is a direct consequence of the transformation law (6) and represents a drastic departure from the usual Higgs mechanism as far as the physical spectrum is concerned. Indeed, with the field decomposition (22), the relevant part of the action ( 18 ) - (20) becomes 1 ~ d4 x FuupaF,uUp a 2.4--~
S= -
--!
q~[s] ~q~' is]
2 •
=exP(2~z=~aadXU ^
aab
-½ ~ D[ X] [pZ~o2+ g2 ~92
- gl OluOvp]) 2]
q~[Xl ,
(21) so long as the "compensating" field A~,,p transforms as in eq. (6). In analogy to the usual Higgs model, we expect gauge symmetry breaking as a consequence of closed membranes condensation into the vacuum. At the treeqevel this will occur for purely imaginary surface tension, i.e. when po2 <0. Note, however, that a negative value ofpg does not immediately imply spon72
1
S
dX"At,,l(X))q~[S]
(iy dX~'^dX"^dX p OtuA~ol(X))
=exp ~.
2
+aim.
(23)
Now, the Goldstone field 0 can be reabsorbed in A via a gauge transformation 1
Au,p =~4u~p+ g OtuO~pl, and thus, we find in the "unitary gauge" :
(24)
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1 f d4 x ffuupa~uupa 2.4!
S=-
_r-
g2v ~02~¢~,)+Si., - ½ ~ D [ X ] ( P ~ 0 2 + -~.
'
(25)
where Fu~p.= ~u~a.-= 0Lu~C.p.1 since the field strength is gauge invariant by definition. Finally, expanding (25) around the constant vacuum field configuration, we deduce the following lagrangian for d : L[~¢] = -
1 ~.paO_~.u~,po.- g2~2 ~.uuP ~'u)'p 2-4~-~ 2.3! ' (26)
¢ 2 = (01 fj DXx(~)
~2[s] - - I o > . a2fzd2~
aab
(27)
The details o f the above calculations will be given in a forthcoming publication, but we wish to underline now the physical meaning ofeq. (26): a non-vanishing vacuum expectation value for the membrane field develops a mass term for the ~-field, just as the usual Higgs field generates the mass of the gauge bosons. However, while in the usual Higgs mechanism the vectorial character, or spin content, of the gauge field is preserved in the transition from massless to massive particle, a new effect occurs when the gauge field is A..p (x): as a gauge field, AF,vp(x ) prossesses no dynamical degrees of freedom and represents a constant background field [3]; but the lagrangian (26) describes a spin-0, pseudoscalar particle o f mass g~o in a representation which is dual to the familiar Proca representation o f massive, spin-1 particles. Indeed, the lagrangian (26) leads to the constraint equation: 0~,~p = 0 or, in terms of the dual field, eu~po~"d*~= 0. This constraint eliminates the spin-1 component o f the (pseudo) vector field d *° but allows its time-like component to propagate as a free field by absorbing the Goldstone bosons which, in the present model, are represented by massless K a l b - R a m o n d fields [ 8 ]. Evidently the key factor which is responsible for this
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unusual phenomenon is the gauge transformation (6) which is dual to the usual gauge transformation of the electromagnetic field. In this sense the new Higgs mechanism for membranes is dual to the old Higgs mechanism for point particles. The possibility that such a dual Higgs mechanism might occur in the field theory o f extended objects was anticipated long ago [ 14] and in ref. [9] it was suggested that precisely such a mechanism might provide the solution o f the U ( l ) problem in QCD. In this connection, we wish to close this letter with a speculative idea which, we believe, is worth looking into: the "invisible" axions, originally required by the Peccei-Quinn mechanism [ 15 ], also play a major role in cosmology as a leading candidate for cold dark matter in the universe; an equally important cosmological role is played by the gauge field Au,p(x ) which provides, even in flat Minkowski space, the constant vacuum energy density necessary for the exponential expansion of the universe during the inflationary epoch [ 5 ] and may be responsible for the birth and evolution of the universe [ 6,7 ]. In the light o f the dual Higgs mechanism, it is tempting to identify the K a l b - R a m o n d Goldstone bosons with the axion fields and speculate that the reason why they are "invisible" is that they are "eaten up" by the gauge field Au~p(x). In this perspective, the Higgs mechanism for relativistic bubbles may well be regarded as the ultimate mechanism for generating mass out of the "bubbling" vacuum o f the early universe.
References [1 ] Y. Nambu, Phys. Rev. 23 (1976) 250. [2] For a review see for example A. Aurilia and E. SpaUucci, The role of extended objects in particle theory and cosmology,in: Proc. Trieste Conf. on Super-membranesand physics in 2+ 1 dimensions (Trieste, June 1989) (World Scientific, Sigapore). [ 3 ] A. Aurilia, Phys. Left. B 81 ( 1979 ) 203. [4] S.W. Hawking, Phys. Len. B 134 (1984) 403; J.D. Brown and C. Teitelboim, Nucl. Phys. B 297 (1988) 787; Phys. Lett B 195 (1987) 177; M.J. Duncan and L.G. Jensen, Nucl. Phys. B 336 (1990) 100; A new mechanism for neutralizing the cosmological constant, preprint UMN-TH-908/90, DOE/ER/40423-19. [ 5 ] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Len. B 147 (1984) 258; Nucl. Phys. B. 252 (1984) 523. 73
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[6] A. Aurilia and E. Spallucci, Phys. Rev. D 40 (1990) 2511. [ 7 ] A. Aurilia and E. Spallucci, Phys. Lett. B 251 (1990) 39; A. Aurilia, R. Balbinot and E. Spallucci, Phys. Lett. B 262 (1991) 222. [8] M. Kalb and P. Ramond, Phys. Rev. D 9 (1974) 2273. [9] A. Aurilia, Y. Takahashi and P.K. Townsend, Phys. Lett. B 95 (1980) 265. [ 10] S.J. Rey, Phys. Rev. D 40 (1990) 3396. [ 11 ] B. DeWitt, Phys. Rev. 160 (1977) 1113.
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[ 12 ] A. Aurilia and Y. Takahashi, Progr. Theor. Phys. 66 ( 1981 ) 693. [ 13] K. Seo and A. Sugamoto, Phys. Rev. D 24 ( 1981 ) 1630. [ 14] A. Aurilia and F. Legovini, Phys. Lett. B 67 ( 1977 ) 299. [15 ] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.