- Email: [email protected]
Biased discrete symmetry and domain wall problem
__ __ i!B
a
lSJune1995
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 3.52(1995) 214-219
Biased discrete symmetry and domain wall problem G. Dvali a,b*‘,Z. Tavartkiladze b, J. Nanobashvili b a Dipartimento di Fisica. Universita di Pisa and INFN. Sezione di Pisa, I-56126 Piss, Italy h Institute of Physics, Georgian Academy of Sciences, 380077 Tbilisi. Georgia
Received 28 November 1994; revised manuscript received 22 March 1995
Editor: L. Alvarez-Gaumt
Abstract We reconsider the cosmological evolution of domain walls produced by the spontaneous breaking of an approximate discrete symmetry. We show that domain walls may never collapse, even if the standard bound on the vacuum energy asymmetry is satisfied. Instead of disappearing, these defects may form stable “bound states”-double wall systems. Possible stability of such a wall is a dynamical question and consequently restricts the allowed range of parameters. In particular, in the two Higgs doublet standard model with an anomalous Z2 symmetry, the above restriction suggests the mass of the pseudoscalar Higgs (would be axion) being close to the mass of the scalar one.
1. Introduction Spontaneously broken discrete symmetries often play a fundamental role in particle physics models. Unfortunately, in a cosmological context such theories exhibit serious problems [ l] as they lead to the formation of the topologically stable domain walls 121. The enormous energy stored in these structures very soon comes to dominate the universe, unless the scale of discrete symmetry breaking is very small (< 10m2GeV or so) [ 1,2]. One famous solution for the domain wall problem is inflation 131. However, normally inflation deals with scales as high as say Mo N lOI GeV (the GUT scale). In this situation, it is clear that the topological defects formed below MC, and in particular those attributed to the electroweak phase transition, will sur-
’ E-mail: [email protected].
vive. Therefore, it seems that at least for the weak scale domain walls inflation may not be a good solution. Alternatively, cosmological troubles caused by the spontaneously broken discrete symmetry can be avoided if this symmetry is approximate, meaning that it is also explicitly broken by a relatively small amount. From the naturalness point of view this ‘solution’ only makes sense if explicit breaking results dynamically from some underlying physics, e.g. from the anomaly [4]. Another possibility [ 51 is that gravity may not respect ungauged discrete (or continuous) symmetries and explicit breaking can manifest itself through the higher dimensional Planck scale ( MP) suppressed operators in the scalar potential. Whatever the source of the explicit violation is, below a certain temperature it creates an energy difference between initialy degenerated vacua. Consequently one vacuum becomes energetically more favorable and the corresponding domains start to expand pushing the walls away, whereas unfavorable domains tend to collapse and disappear. The necessary condi-
0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDf 0370-2693 ( 95) 005 1 1-O
G. Dvali et al. /Physics
tion for the domain walls to disappear before dominating the universe has the following form [ 63
e>(g)*
(1)
P
where E is the vacuum energy density difference and (+ is the wall energy per unit surface. In the simplest case, when the domain walls are formed by a single component real Higgs field @ with vacuum expectation value (VEV) < Q, >= fV, the condition ( 1) can be enough to guarantee that walls collapse without any trouble for cosmology. In the case of a real scalar field the expectation value < @ > always vanishes across the wall, and two neighboring walls (bounding the collapsing domain from opposite sides) sooner or later will annihilate each other. However, in realistic theories the Higgs fields are usually represented by complex scalars and have several components. The elements of the discrete symmetry group can be mimicked by certain abelian (or even nonabelian) phase transformation (which of course is not a part of a continuous gauge group, since we are not interested in walls bounded by strings [ 71). This circumstance introduces a qualitatively new point in the wall structure implying that there is no topological reason for the absolute value of the Higgs field I@]to vanish inside the wall. In other words, whether this will happen becomes a matter of the choice of parameters in the scalar potential. For a wide range of these parameters the energetically most favorable path in the field space, connecting two degenerated vacua, never includes the point I@1 = 0. Thus, the Higgs field never vanishes inside the wall and we can define its “winding number” through the defect. In the present paper we show that in such cases domain walls may never collapse even if discrete symmetry is explicitly broken and ( 1) is satisfied. This has to do with the fact that the walls of the collapsing domain can annihilate each other only if the Higgs field winds oppositely through this walls. Two walls with the same winding can never annihilate each other and disappear. Instead, they form classically stable “bound states”-double wall systems. The expectation value of the Higgs field returns to its initial value after traversing such a composite wall. If the winding is parametrised by the phase arg@ = 8, we will have A6 = 27r through the wall. As far as we know, previously the 2~-walls have been considered only in the
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Letters B 352 (1995) 214-219
context of the walls bounded by cosmic strings [ 81, which often appear in axion cosmology. In such a case initially there are no walls, but rather axionic strings, formed due to a global U( l)pp-symmetry breaking. 27r-walls are formed later on when U( 1)~ gets explicitly broken (and only if there is no discrete subgroup left [ 91). Each 27r-wall is bounded by a string. Existence of the string network crucially determines the cosmological evolution of the walls and makes them decay without trace. In contrast 2r-walls discussed in the present paper can be cosmologically problematic, since there is no string network behind their existence. They are formed as bound states of would-be topologically stable domain walls when the latter collapse due to an explicit breaking of discrete symmetry. 2. General mechanism In this section we will study the mechanism of the 2?r-wall formation on a simple model including one SU( 3) -color triplet quark UL,Rtransforming under discrete Z2 symmetry: UL + uL.3 UR--+ -uR.
(2)
This quark gets a mass from the following Yukawa coupling &@‘aLaR+ h.c.
(3)
where @ = @)R+ i@~ = ]@le” is a complex Higgs scalar (0 -+ -@ under Zz), whose VEV breaks Z2 spontaneously. We choose a tree level scalar potential in the form:
u=-$$,*+$,@,~;(@*+h.c.) = -p,@,*
+ $I]4
- ~]@~*cos2&
(4)
For definiteness we assume all parameters m*, M*, h being real and positive. The possible quartic term Q4 which is allowed by all symmetries is excluded for the simplicity of the analysis. Note that the first two terms in (4) respect global U( 1)-symmetry Q -+ e% which is explicitly broken by the third one down to Z2 C U( 1) and m is the mass of the would-be Goldstone boson. The absolute value of the
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Letters B 352 (1995) 214-219
Higgs VEV is given by I@/ = V = ((M* + m*)/h)f. This vacuum is discretely degenerate, since the potential is minimized by any 0 = rrN (where N is integer), This degeneracy results in the formation of topologically stable domain walls. The topological constraint forces 8 to change by JABI= 7~across the wall. However, this defects is not truly stable, because the Z2-symmetry is anomalous [ 41. The important point about the wall structure is whether the absolute value of the scalar field vanishes inside the wall. Certainly, this is the case if m* > M*. In this situation the domain wall solution is given by the kink (antikink)
In such a case it costs a lot of energy (- M4 per volume) for I@1to vanish somewhere in the space and in particular in the wall vicinity. To approximate the wall structure it is useful to put I@[ = V = constant for a moment. The equation of motion for 19becomes then the well known sine-Gordon equation: 2@ -sin28=0 at2
(7)
(where 5 = mx and x is the coordinate transverse to the wall) which has a topologically stable solution (soliton) 0(t) =2stan-‘exp([>.
@R = fV tanh[xV( h/4) */*I
(3
(n is a coordinate perpendicular to the wall), whereas the pseudoscalar component @‘I is identically zero, since its effective [mass] * term I# = hai +m* - M*, which is spatially dependent trough the @R VEV, is positive everywhere. In such a case, the wall network will evolve along the lines discussed in [ 451. That is, we expect that after Z2 will be explicitly broken by the anomaly, walls can collapse without any cosmological harm (provided ( 1) is satisfied). A significant deviation from this scenario (to be discussed below) will occur if ]@I stays nonzero everywhere including the vicinity of the domain wall. In this case the energetically most favorable trajectory connecting two neighboring vacua 8 = TN and 8 = rr( N f l), goes through the saddle point 0’1 = k[ (M* - m*)/h] ‘I2 and @R = 0, whereas the point I@( = 0 is maximum. Of course, the condition M* > m* is not yet sufficient to conclude that the pseudoscalar will get a nonzero VEV inside the wall, since the gradient energy wants @I to stay constant everywhere. So one has to check the stability of the small perturbation about @I = 0 in the background of a kink. This can, for example, be done through the analysis of the linearized equation for the pseudoscalar mode with the potential M: on the existence of the bound state along the lines discussed by Witten [lo] for the bosonically superconducting string. Such a consideration shows that condensation of thepseudoscalar mode inside the wall occurs at least for m* < M2 N M* - m2.
(6)
(8)
This is a reasonably good approximation for our purposes, although in reality IQ,1cannot stay constant across the wall due to a back reaction. In turn this will alter the shape of e(t), but whatever the exact form of e(t) is, it should change by ]A@(= n across the soliton. At the QCD scale the color instanton effects become important and they explicitly break Z2-symmetry. The effective Z2-noninvariant term generated in the scalar potential by instanton vertex has the form Knst= -mu A3 cos
tI
(9)
where A is a typical scale factor of the order of the QCD scale AQCDN 200 MeV and m, is the quark mass. This term creates an energy density difference l = 2m,A3 between 8 = 2Nr and r3= (2N + 1)~ vacua. This energy difference creates the pressure difference which drives the domain walls. Assuming say m,A3 > 0, the preference is given to 0 = 2N7r domains which tend to expand, whereas those with 0 = (2N + 1>?r will tend to collapse and disappear. The dynamics of the collapsing walls is defined by several factors. At the early stages (when explicitly violating effects are negligible) these are mainly the wall tension and the frictional force. Crudely the time dependences of this two factors are proportional to N (aMi/t3)‘/* and N ( Mp/t)* respectively. At some point the system becomes dominated by the pressure difference (with corresponding force per unit area N l) which forces the false vacuum to collapse. For walls to disappear, the above has to happen before they dominate the universe and we are lead to the lower bound ( 1) for
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Letters B 352 (I 995) 214-219
the vacuum energy asymmetry (for more details see [2,4,5,61).
Due to the above arguments, in general one expects [ 41 the domain walls to disappear soon after the QCD phase transition, since for the weak scale walls (destabilized by QCD anomaly) condition (1) is satisfied (~(Mp/g)~ N 10” or so). A similar conclusion was made in [ 51 for the case of domain walls destabilized by “gravity induced” higher dimensional operators. However as it will be shown in the present paper, the resulting dynamics of the walls is more sensitive to the parameters than one would naively expect from ( 1). Namely for a wide range of parameters (forming a subrange of ( 1) ) the walls may not collapse completely. In particular this can happen for (6), meaning that the mass of the pseudoscalar mode (would-be axion) is more than twice as small as the scalar one (radial mode). In such a case only the walls having opposire signs of A0 can annihilate each other, whereas others can not. Instead the latter will be paired forming stable double wall systems across which 1601 = 27r. Such a wall pairing can never occur if J@J vanishes across the wall. In this case the vacua 0 = 27rN (or 0 = (2N + 1)~) corresponding to different N are identical. So in fact we have just two possible domains (say (+) and (-) respectively). This has to do with the fact that actually @does not “wind” through the wall, since it becomes ill defined at IQ,1= 0. The planar domain wall solution separating (-) and (+) ((+) and (-)) domainsis given by a kink (antikink) (5) rather than by a soliton (8). Now if we switch on the instanton effects the ( -) domains will start to collapse and finally disappear, since the two neighboring domain walls (kink and antikink) will annihilate each other. The situation will drastically change if a nonzero @ is energetically favored everywhere in space; that is, for example, if we are in the range of parameters given in (6). Now 8 is well defined for any spatial point x (including those in the vicinity of the domain wall) and as we show in the next section, this fact plays a crucial role in the subsequent evolution of the system.
3. Cosmological evolution of I@[ # 0 domain walls Now let us follow the history of the early universe in our model. At high temperature (T >> M, m) the Q-dependent part of the potential receives a correction (the Yukawa interaction is neglected) [ 111 AV(T)
= [@)2T2h/6
(10)
and we see that the mass difference between the scalar and pseudoscalar fields (m2) never vanishes. Therefore, at the moment of the phase transition, when (PR and @I pick up nonzero VEVs, they already “know” about the explicit breaking of the global would-be U( 1)-symmetry. At this stage, the probability of string-bounded wall formation is exponentially suppressed by the factor mR, where R is the size of the defect. Since the range of interest to us is m not much smaller (say within one order of magnitude) than M, the probability of large string-wall formation is negligible, and so we are left with a network of domain walls of very complicated geometry. Traversing each wall, 0 winds by A0 = f?r continuously interpolating between two values that differ by r. Differences among “winding numbers” of neighboring walls become very important after the explicit breaking of the discrete symmetry. The instanton induced term (9) in the scalar potential gives preference to certain domains (say to 6’ = 27rN) which tend to expand. The crucial point however is that not all 0 = (2N + l)?r domains will disappear. To see this, consider a collapsing domain 0 = 7r bounded by two infinite planar walls. It is clear that these boundaries can annihilate each other and disappear only if they correspond to opposite windings of 8. That is, the two neighboring domains (of which 0 = r is one) must have both 8 = 0 or both 0 = 2~ (and never one 0 and the other 27r). By contrast, if the rdomain were initially created between 0 and 27r ones, the corresponding walls could never annihilate each other since they would have the same A8. Instead of annihilation such walls will create stable bound states. The double wall solution can be approximated analytically if we assume I@]= constant. In such a case the equation for 0 is a double sine-Gordon equation 2d28/dc2
- sin 28 - a sin 8 = 0
(11)
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where a = &. The composite wall can be approximated by a stable double-soliton solution e=2arccos
a/2 - Z2 2Z ,
Letters B 352 (1995) 214-219
radius exceeds the critical value R, = (S + ,x2/a) 12. The tunneling rate is exponentially suppressed by a factor
Z2 = 2 + a/2 + 2~~tanh(2J). (12)
Note that the usual sine-Gordon equation (7) also admits a 27r-soliton solution which is however unstable and tends to decay into two solitons with A0 = 7r each. The last term in ( 11) opposes this decay and prevents two ?r-solitons from escaping each other, forming the bound state. The width of the boul,d state (typical distance between A0 = r walls) is 6 N t In t. Since A and V (N m) correspond to QCD and weak scales respectively, this width is by an order of magnitude larger than the width of the constituent r-solitons 6’ N A. The infinite planar 2r-wall is not topologically stable and can decay quantum mechanically. This decay goes through the tunneling process which creates a hole connecting 6 = 0 and 0 = 2~ vacua with each other. Imagine a closed path that starts in one of the domains (say in 8 = O), goes once through the hole and finally returns back to the base point by traversing the wall in some other place. The one that travels along the path, will find the phase 0 changing by A8 = 2~ at the end of the journey. Thus, I@1has to vanish at some point inside the region encircled by the path. By moving the path around continuously (without crossing the edge of the hole), the point I@ = 0 will follow a closed line. So, there is a loop of the cosmic string enclosing the hole somewhere. Since the space between two solitons is initially filled with the 6 = v phase, the formation of the hole is costly in the energy of its own edge. The latter includes the energy of the (cylindric) wall surface that bounds the hole and the energy of the string loop that is somewhere in this boundary. The hole will start to expand classically when this energy becomes overcomed by the energy of the punched out piece of the domain wall. The change in energy caused by a R-radius hole formation can be estimated to be: AE = vR12(fi2 + &J) - R(SE + 2g)l
(13)
where p2 % ~ITV~In E and u N 4mV2 are string and wall tensions respectively. The hole expands when its
so that even for Ifrn N 1 - 10 this wall can be stable for all practical purposes. Such a 2v-wall of horizon size (if formed) can be disastrous cosmologically. To destabilize this structure we should increase the parameter m2 and convert I@1= 0 in to the saddle point. In other words we have to increase the mass of the angular pseudoscalar mode (the would-beGoldstone boson) relative to scalar one (radial mode). The closer I@1= 0 is to the saddle point, the less costly the string bounded hole formation in the wall sheet is. In the limit when )@I= 0 becomes a saddle point, 2r-walls become classically unstable.
4. Two doublet extention The above model is trivially extendable to the realistic example of [4]. This is the W(2) 61 U( 1) standard model with two Higgs doublets a,, @d and anomalous discrete symmetry Z2 needed for natural flavor conservation. Under 22: % + -%, QL --) QL,
@d 4 @d, UR
--+
-uR,
dR
+
(14)
dR
where QL = (i) is the left-handed quark doublet. Z2symmetry ensures that each Higgs doublet couples only to the fermions of a given charge. The scalar potential now takes the form
a -
Vphase
h(@:@d(* - h’(@~(~(@d(’ + bhase = -$D;@d)2
+
(15)
h.c.
(16)
where cy = u, d. If h > 0 the VEVs are pointing in the electrically neutral direction Qn = (‘$“‘). In this parameterization Vphase
= -~v,2v+0s(28),
8 = 8, -
6d.
(17)
G. Dvali et al./Physics
For A > 0, Vphaseis minimized by 0 = rk (k = 1,2,. . .) and there are topologically stable domain walls. If h + h’ is larger than A, nonzero expectation values Vu, Vd # 0 will be energetically flavored everywhere in the space and even in the vicinity of the domain wall. So that as before, the winding of 8 can be well defined. Explicit violation of Z2-symmetry by QCD instantons leads to the formation of “composite” 2r-walls much in the way discussed in Section 2. The source of the energy difference induced by the instanton vertex in the scalar potential has the form (for simplicity we assume just one family of quarks) : Vphase= -m,md~2~~~e
(18)
where as before A N A~co N 200 MeV and m,, md are masses of u and d-quarks respectively. The resulting double sine-Gordon equation for B has the form ( 11), where now a = mumdA2/V2m2. Once again, the resulting 27r-walls can be practically stable and cause trouble for cosmology, unless the parameter A is sufficiently large, implying that the mass of the would be axion is close to the Higgs mass. 5. Conclusions We have shown that for a certain range of parameters, compatible with the standard bound, the collapse of unstable domain walls may end up with the formation of classically (and practically) stable double wall systems. If formed, this structure can be problematic cosmologically and therefore their nonexistence requires additional restriction of the parameter space. Of course, the formation and subsequent evolution of this defects is a very complicated process and requires a much more detailed and perhaps numerical study. For example, collisions of the domain walls may enhance
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the rate of hole formation. At this point our analysis can be considered as a ‘first’ approximation, but nevertheless, it provides an interesting enough example of how cosmological considerations may restrict parameters in the particle physics models.
Acknowledgements
One of us (G.D.) would like to thank Goran Senjanovic and Vazha Berezhiani for discussions and INFN for support. References [ I] Ya.B. Zel’dovich, I.Yu. Kobzarev and L.B. Okun, JETP 40 (1975) 1. [2] For a review, see T.W.B. Kibble, J. Phys. A 9 (1976) 1387; Phys. Rep. 67 (1980) 183; A. Vilenkin, Phys. Rep. 121 (1985) 263. [3] A.H. Guth, Phys. Rev. D 23 (1981) 347; for a review, see, e.g. A.D. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Switzerland, 1990); K.A. Olive, Phys. Rep. 190 (1990) 307. [4] J. Preskill, SP Trivedi, E Wilczek and M.B. Wise, Nucl. Phys. B 363 (1991) 207. [5] B. Rai and G. Senjanovic, Phys. Rev. D 49 (1994) 2729; for earlier work on the possible role of gravity, see B. Holdom, Phys. Rev. D 28 (1983) 1419. [6] A. Vilenkin. Phys. Rev. D 23 (1981) 852. [7] T.W.B. Kibble, G. Lazarides and Q. Shah, Phys. Rev. D 26 (1982) 435. [ 81 A. Vilenkin and A.E. Everett, Phys. Rev. Lett. 48 ( 1982) 1867; for a review, see, e.g. J.E. Kim, Phys. Rep. 150 (1987) 1. [9] P Sikivie, Phys. Rev. Len 48 ( 1982) 1156. [lo] E. Witten, Nucl. Phys. B 249 (1985) 557. [ 111 S. Weinberg, Phys. Rev. D 9 (1974) 3357; L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320; A.D. Linde, Rep. Prog. Phys. 42 (1979) 389.
a
lSJune1995
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 3.52(1995) 214-219
Biased discrete symmetry and domain wall problem G. Dvali a,b*‘,Z. Tavartkiladze b, J. Nanobashvili b a Dipartimento di Fisica. Universita di Pisa and INFN. Sezione di Pisa, I-56126 Piss, Italy h Institute of Physics, Georgian Academy of Sciences, 380077 Tbilisi. Georgia
Received 28 November 1994; revised manuscript received 22 March 1995
Editor: L. Alvarez-Gaumt
Abstract We reconsider the cosmological evolution of domain walls produced by the spontaneous breaking of an approximate discrete symmetry. We show that domain walls may never collapse, even if the standard bound on the vacuum energy asymmetry is satisfied. Instead of disappearing, these defects may form stable “bound states”-double wall systems. Possible stability of such a wall is a dynamical question and consequently restricts the allowed range of parameters. In particular, in the two Higgs doublet standard model with an anomalous Z2 symmetry, the above restriction suggests the mass of the pseudoscalar Higgs (would be axion) being close to the mass of the scalar one.
1. Introduction Spontaneously broken discrete symmetries often play a fundamental role in particle physics models. Unfortunately, in a cosmological context such theories exhibit serious problems [ l] as they lead to the formation of the topologically stable domain walls 121. The enormous energy stored in these structures very soon comes to dominate the universe, unless the scale of discrete symmetry breaking is very small (< 10m2GeV or so) [ 1,2]. One famous solution for the domain wall problem is inflation 131. However, normally inflation deals with scales as high as say Mo N lOI GeV (the GUT scale). In this situation, it is clear that the topological defects formed below MC, and in particular those attributed to the electroweak phase transition, will sur-
’ E-mail: [email protected].
vive. Therefore, it seems that at least for the weak scale domain walls inflation may not be a good solution. Alternatively, cosmological troubles caused by the spontaneously broken discrete symmetry can be avoided if this symmetry is approximate, meaning that it is also explicitly broken by a relatively small amount. From the naturalness point of view this ‘solution’ only makes sense if explicit breaking results dynamically from some underlying physics, e.g. from the anomaly [4]. Another possibility [ 51 is that gravity may not respect ungauged discrete (or continuous) symmetries and explicit breaking can manifest itself through the higher dimensional Planck scale ( MP) suppressed operators in the scalar potential. Whatever the source of the explicit violation is, below a certain temperature it creates an energy difference between initialy degenerated vacua. Consequently one vacuum becomes energetically more favorable and the corresponding domains start to expand pushing the walls away, whereas unfavorable domains tend to collapse and disappear. The necessary condi-
0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDf 0370-2693 ( 95) 005 1 1-O
G. Dvali et al. /Physics
tion for the domain walls to disappear before dominating the universe has the following form [ 63
e>(g)*
(1)
P
where E is the vacuum energy density difference and (+ is the wall energy per unit surface. In the simplest case, when the domain walls are formed by a single component real Higgs field @ with vacuum expectation value (VEV) < Q, >= fV, the condition ( 1) can be enough to guarantee that walls collapse without any trouble for cosmology. In the case of a real scalar field the expectation value < @ > always vanishes across the wall, and two neighboring walls (bounding the collapsing domain from opposite sides) sooner or later will annihilate each other. However, in realistic theories the Higgs fields are usually represented by complex scalars and have several components. The elements of the discrete symmetry group can be mimicked by certain abelian (or even nonabelian) phase transformation (which of course is not a part of a continuous gauge group, since we are not interested in walls bounded by strings [ 71). This circumstance introduces a qualitatively new point in the wall structure implying that there is no topological reason for the absolute value of the Higgs field I@]to vanish inside the wall. In other words, whether this will happen becomes a matter of the choice of parameters in the scalar potential. For a wide range of these parameters the energetically most favorable path in the field space, connecting two degenerated vacua, never includes the point I@1 = 0. Thus, the Higgs field never vanishes inside the wall and we can define its “winding number” through the defect. In the present paper we show that in such cases domain walls may never collapse even if discrete symmetry is explicitly broken and ( 1) is satisfied. This has to do with the fact that the walls of the collapsing domain can annihilate each other only if the Higgs field winds oppositely through this walls. Two walls with the same winding can never annihilate each other and disappear. Instead, they form classically stable “bound states”-double wall systems. The expectation value of the Higgs field returns to its initial value after traversing such a composite wall. If the winding is parametrised by the phase arg@ = 8, we will have A6 = 27r through the wall. As far as we know, previously the 2~-walls have been considered only in the
215
Letters B 352 (1995) 214-219
context of the walls bounded by cosmic strings [ 81, which often appear in axion cosmology. In such a case initially there are no walls, but rather axionic strings, formed due to a global U( l)pp-symmetry breaking. 27r-walls are formed later on when U( 1)~ gets explicitly broken (and only if there is no discrete subgroup left [ 91). Each 27r-wall is bounded by a string. Existence of the string network crucially determines the cosmological evolution of the walls and makes them decay without trace. In contrast 2r-walls discussed in the present paper can be cosmologically problematic, since there is no string network behind their existence. They are formed as bound states of would-be topologically stable domain walls when the latter collapse due to an explicit breaking of discrete symmetry. 2. General mechanism In this section we will study the mechanism of the 2?r-wall formation on a simple model including one SU( 3) -color triplet quark UL,Rtransforming under discrete Z2 symmetry: UL + uL.3 UR--+ -uR.
(2)
This quark gets a mass from the following Yukawa coupling &@‘aLaR+ h.c.
(3)
where @ = @)R+ i@~ = ]@le” is a complex Higgs scalar (0 -+ -@ under Zz), whose VEV breaks Z2 spontaneously. We choose a tree level scalar potential in the form:
u=-$$,*+$,@,~;(@*+h.c.) = -p,@,*
+ $I]4
- ~]@~*cos2&
(4)
For definiteness we assume all parameters m*, M*, h being real and positive. The possible quartic term Q4 which is allowed by all symmetries is excluded for the simplicity of the analysis. Note that the first two terms in (4) respect global U( 1)-symmetry Q -+ e% which is explicitly broken by the third one down to Z2 C U( 1) and m is the mass of the would-be Goldstone boson. The absolute value of the
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Letters B 352 (1995) 214-219
Higgs VEV is given by I@/ = V = ((M* + m*)/h)f. This vacuum is discretely degenerate, since the potential is minimized by any 0 = rrN (where N is integer), This degeneracy results in the formation of topologically stable domain walls. The topological constraint forces 8 to change by JABI= 7~across the wall. However, this defects is not truly stable, because the Z2-symmetry is anomalous [ 41. The important point about the wall structure is whether the absolute value of the scalar field vanishes inside the wall. Certainly, this is the case if m* > M*. In this situation the domain wall solution is given by the kink (antikink)
In such a case it costs a lot of energy (- M4 per volume) for I@1to vanish somewhere in the space and in particular in the wall vicinity. To approximate the wall structure it is useful to put I@[ = V = constant for a moment. The equation of motion for 19becomes then the well known sine-Gordon equation: 2@ -sin28=0 at2
(7)
(where 5 = mx and x is the coordinate transverse to the wall) which has a topologically stable solution (soliton) 0(t) =2stan-‘exp([>.
@R = fV tanh[xV( h/4) */*I
(3
(n is a coordinate perpendicular to the wall), whereas the pseudoscalar component @‘I is identically zero, since its effective [mass] * term I# = hai +m* - M*, which is spatially dependent trough the @R VEV, is positive everywhere. In such a case, the wall network will evolve along the lines discussed in [ 451. That is, we expect that after Z2 will be explicitly broken by the anomaly, walls can collapse without any cosmological harm (provided ( 1) is satisfied). A significant deviation from this scenario (to be discussed below) will occur if ]@I stays nonzero everywhere including the vicinity of the domain wall. In this case the energetically most favorable trajectory connecting two neighboring vacua 8 = TN and 8 = rr( N f l), goes through the saddle point 0’1 = k[ (M* - m*)/h] ‘I2 and @R = 0, whereas the point I@( = 0 is maximum. Of course, the condition M* > m* is not yet sufficient to conclude that the pseudoscalar will get a nonzero VEV inside the wall, since the gradient energy wants @I to stay constant everywhere. So one has to check the stability of the small perturbation about @I = 0 in the background of a kink. This can, for example, be done through the analysis of the linearized equation for the pseudoscalar mode with the potential M: on the existence of the bound state along the lines discussed by Witten [lo] for the bosonically superconducting string. Such a consideration shows that condensation of thepseudoscalar mode inside the wall occurs at least for m* < M2 N M* - m2.
(6)
(8)
This is a reasonably good approximation for our purposes, although in reality IQ,1cannot stay constant across the wall due to a back reaction. In turn this will alter the shape of e(t), but whatever the exact form of e(t) is, it should change by ]A@(= n across the soliton. At the QCD scale the color instanton effects become important and they explicitly break Z2-symmetry. The effective Z2-noninvariant term generated in the scalar potential by instanton vertex has the form Knst= -mu A3 cos
tI
(9)
where A is a typical scale factor of the order of the QCD scale AQCDN 200 MeV and m, is the quark mass. This term creates an energy density difference l = 2m,A3 between 8 = 2Nr and r3= (2N + 1)~ vacua. This energy difference creates the pressure difference which drives the domain walls. Assuming say m,A3 > 0, the preference is given to 0 = 2N7r domains which tend to expand, whereas those with 0 = (2N + 1>?r will tend to collapse and disappear. The dynamics of the collapsing walls is defined by several factors. At the early stages (when explicitly violating effects are negligible) these are mainly the wall tension and the frictional force. Crudely the time dependences of this two factors are proportional to N (aMi/t3)‘/* and N ( Mp/t)* respectively. At some point the system becomes dominated by the pressure difference (with corresponding force per unit area N l) which forces the false vacuum to collapse. For walls to disappear, the above has to happen before they dominate the universe and we are lead to the lower bound ( 1) for
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the vacuum energy asymmetry (for more details see [2,4,5,61).
Due to the above arguments, in general one expects [ 41 the domain walls to disappear soon after the QCD phase transition, since for the weak scale walls (destabilized by QCD anomaly) condition (1) is satisfied (~(Mp/g)~ N 10” or so). A similar conclusion was made in [ 51 for the case of domain walls destabilized by “gravity induced” higher dimensional operators. However as it will be shown in the present paper, the resulting dynamics of the walls is more sensitive to the parameters than one would naively expect from ( 1). Namely for a wide range of parameters (forming a subrange of ( 1) ) the walls may not collapse completely. In particular this can happen for (6), meaning that the mass of the pseudoscalar mode (would-be axion) is more than twice as small as the scalar one (radial mode). In such a case only the walls having opposire signs of A0 can annihilate each other, whereas others can not. Instead the latter will be paired forming stable double wall systems across which 1601 = 27r. Such a wall pairing can never occur if J@J vanishes across the wall. In this case the vacua 0 = 27rN (or 0 = (2N + 1)~) corresponding to different N are identical. So in fact we have just two possible domains (say (+) and (-) respectively). This has to do with the fact that actually @does not “wind” through the wall, since it becomes ill defined at IQ,1= 0. The planar domain wall solution separating (-) and (+) ((+) and (-)) domainsis given by a kink (antikink) (5) rather than by a soliton (8). Now if we switch on the instanton effects the ( -) domains will start to collapse and finally disappear, since the two neighboring domain walls (kink and antikink) will annihilate each other. The situation will drastically change if a nonzero @ is energetically favored everywhere in space; that is, for example, if we are in the range of parameters given in (6). Now 8 is well defined for any spatial point x (including those in the vicinity of the domain wall) and as we show in the next section, this fact plays a crucial role in the subsequent evolution of the system.
3. Cosmological evolution of I@[ # 0 domain walls Now let us follow the history of the early universe in our model. At high temperature (T >> M, m) the Q-dependent part of the potential receives a correction (the Yukawa interaction is neglected) [ 111 AV(T)
= [@)2T2h/6
(10)
and we see that the mass difference between the scalar and pseudoscalar fields (m2) never vanishes. Therefore, at the moment of the phase transition, when (PR and @I pick up nonzero VEVs, they already “know” about the explicit breaking of the global would-be U( 1)-symmetry. At this stage, the probability of string-bounded wall formation is exponentially suppressed by the factor mR, where R is the size of the defect. Since the range of interest to us is m not much smaller (say within one order of magnitude) than M, the probability of large string-wall formation is negligible, and so we are left with a network of domain walls of very complicated geometry. Traversing each wall, 0 winds by A0 = f?r continuously interpolating between two values that differ by r. Differences among “winding numbers” of neighboring walls become very important after the explicit breaking of the discrete symmetry. The instanton induced term (9) in the scalar potential gives preference to certain domains (say to 6’ = 27rN) which tend to expand. The crucial point however is that not all 0 = (2N + l)?r domains will disappear. To see this, consider a collapsing domain 0 = 7r bounded by two infinite planar walls. It is clear that these boundaries can annihilate each other and disappear only if they correspond to opposite windings of 8. That is, the two neighboring domains (of which 0 = r is one) must have both 8 = 0 or both 0 = 2~ (and never one 0 and the other 27r). By contrast, if the rdomain were initially created between 0 and 27r ones, the corresponding walls could never annihilate each other since they would have the same A8. Instead of annihilation such walls will create stable bound states. The double wall solution can be approximated analytically if we assume I@]= constant. In such a case the equation for 0 is a double sine-Gordon equation 2d28/dc2
- sin 28 - a sin 8 = 0
(11)
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where a = &. The composite wall can be approximated by a stable double-soliton solution e=2arccos
a/2 - Z2 2Z ,
Letters B 352 (1995) 214-219
radius exceeds the critical value R, = (S + ,x2/a) 12. The tunneling rate is exponentially suppressed by a factor
Z2 = 2 + a/2 + 2~~tanh(2J). (12)
Note that the usual sine-Gordon equation (7) also admits a 27r-soliton solution which is however unstable and tends to decay into two solitons with A0 = 7r each. The last term in ( 11) opposes this decay and prevents two ?r-solitons from escaping each other, forming the bound state. The width of the boul,d state (typical distance between A0 = r walls) is 6 N t In t. Since A and V (N m) correspond to QCD and weak scales respectively, this width is by an order of magnitude larger than the width of the constituent r-solitons 6’ N A. The infinite planar 2r-wall is not topologically stable and can decay quantum mechanically. This decay goes through the tunneling process which creates a hole connecting 6 = 0 and 0 = 2~ vacua with each other. Imagine a closed path that starts in one of the domains (say in 8 = O), goes once through the hole and finally returns back to the base point by traversing the wall in some other place. The one that travels along the path, will find the phase 0 changing by A8 = 2~ at the end of the journey. Thus, I@1has to vanish at some point inside the region encircled by the path. By moving the path around continuously (without crossing the edge of the hole), the point I@ = 0 will follow a closed line. So, there is a loop of the cosmic string enclosing the hole somewhere. Since the space between two solitons is initially filled with the 6 = v phase, the formation of the hole is costly in the energy of its own edge. The latter includes the energy of the (cylindric) wall surface that bounds the hole and the energy of the string loop that is somewhere in this boundary. The hole will start to expand classically when this energy becomes overcomed by the energy of the punched out piece of the domain wall. The change in energy caused by a R-radius hole formation can be estimated to be: AE = vR12(fi2 + &J) - R(SE + 2g)l
(13)
where p2 % ~ITV~In E and u N 4mV2 are string and wall tensions respectively. The hole expands when its
so that even for Ifrn N 1 - 10 this wall can be stable for all practical purposes. Such a 2v-wall of horizon size (if formed) can be disastrous cosmologically. To destabilize this structure we should increase the parameter m2 and convert I@1= 0 in to the saddle point. In other words we have to increase the mass of the angular pseudoscalar mode (the would-beGoldstone boson) relative to scalar one (radial mode). The closer I@1= 0 is to the saddle point, the less costly the string bounded hole formation in the wall sheet is. In the limit when )@I= 0 becomes a saddle point, 2r-walls become classically unstable.
4. Two doublet extention The above model is trivially extendable to the realistic example of [4]. This is the W(2) 61 U( 1) standard model with two Higgs doublets a,, @d and anomalous discrete symmetry Z2 needed for natural flavor conservation. Under 22: % + -%, QL --) QL,
@d 4 @d, UR
--+
-uR,
dR
+
(14)
dR
where QL = (i) is the left-handed quark doublet. Z2symmetry ensures that each Higgs doublet couples only to the fermions of a given charge. The scalar potential now takes the form
a -
Vphase
h(@:@d(* - h’(@~(~(@d(’ + bhase = -$D;@d)2
+
(15)
h.c.
(16)
where cy = u, d. If h > 0 the VEVs are pointing in the electrically neutral direction Qn = (‘$“‘). In this parameterization Vphase
= -~v,2v+0s(28),
8 = 8, -
6d.
(17)
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For A > 0, Vphaseis minimized by 0 = rk (k = 1,2,. . .) and there are topologically stable domain walls. If h + h’ is larger than A, nonzero expectation values Vu, Vd # 0 will be energetically flavored everywhere in the space and even in the vicinity of the domain wall. So that as before, the winding of 8 can be well defined. Explicit violation of Z2-symmetry by QCD instantons leads to the formation of “composite” 2r-walls much in the way discussed in Section 2. The source of the energy difference induced by the instanton vertex in the scalar potential has the form (for simplicity we assume just one family of quarks) : Vphase= -m,md~2~~~e
(18)
where as before A N A~co N 200 MeV and m,, md are masses of u and d-quarks respectively. The resulting double sine-Gordon equation for B has the form ( 11), where now a = mumdA2/V2m2. Once again, the resulting 27r-walls can be practically stable and cause trouble for cosmology, unless the parameter A is sufficiently large, implying that the mass of the would be axion is close to the Higgs mass. 5. Conclusions We have shown that for a certain range of parameters, compatible with the standard bound, the collapse of unstable domain walls may end up with the formation of classically (and practically) stable double wall systems. If formed, this structure can be problematic cosmologically and therefore their nonexistence requires additional restriction of the parameter space. Of course, the formation and subsequent evolution of this defects is a very complicated process and requires a much more detailed and perhaps numerical study. For example, collisions of the domain walls may enhance
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219
the rate of hole formation. At this point our analysis can be considered as a ‘first’ approximation, but nevertheless, it provides an interesting enough example of how cosmological considerations may restrict parameters in the particle physics models.
Acknowledgements
One of us (G.D.) would like to thank Goran Senjanovic and Vazha Berezhiani for discussions and INFN for support. References [ I] Ya.B. Zel’dovich, I.Yu. Kobzarev and L.B. Okun, JETP 40 (1975) 1. [2] For a review, see T.W.B. Kibble, J. Phys. A 9 (1976) 1387; Phys. Rep. 67 (1980) 183; A. Vilenkin, Phys. Rep. 121 (1985) 263. [3] A.H. Guth, Phys. Rev. D 23 (1981) 347; for a review, see, e.g. A.D. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Switzerland, 1990); K.A. Olive, Phys. Rep. 190 (1990) 307. [4] J. Preskill, SP Trivedi, E Wilczek and M.B. Wise, Nucl. Phys. B 363 (1991) 207. [5] B. Rai and G. Senjanovic, Phys. Rev. D 49 (1994) 2729; for earlier work on the possible role of gravity, see B. Holdom, Phys. Rev. D 28 (1983) 1419. [6] A. Vilenkin. Phys. Rev. D 23 (1981) 852. [7] T.W.B. Kibble, G. Lazarides and Q. Shah, Phys. Rev. D 26 (1982) 435. [ 81 A. Vilenkin and A.E. Everett, Phys. Rev. Lett. 48 ( 1982) 1867; for a review, see, e.g. J.E. Kim, Phys. Rep. 150 (1987) 1. [9] P Sikivie, Phys. Rev. Len 48 ( 1982) 1156. [lo] E. Witten, Nucl. Phys. B 249 (1985) 557. [ 111 S. Weinberg, Phys. Rev. D 9 (1974) 3357; L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320; A.D. Linde, Rep. Prog. Phys. 42 (1979) 389.