Goldstone bosons versus domain walls bounded by cosmic strings

Physics Letters B 265 ( 1991 ) 64-68 North-Holland

PHYSICS LETTERS B

Goldstone bosons versus domain walls bounded by cosmic strings G.R. Dvali Institute of Physics, Georgian Academy of Sciences. Tamarashvili street 6, SU-380 077 Tbilisi, Georgian Republic, USSR Received 1 I February 1991 ; revised manuscript received 30 May 1991

It is shown that two possible schemes of string bounded domain wall formation are closely related. Walls can be avoided if a certain type of couplings between scalars, responsible for string and wall formation, is excluded from the potential. This enlarges the continuous global symmetry of the model which, being broken, instead of domain walls gives rise to global strings and Goldstone bosons. A realistic example of spontaneously broken family symmetry is considered. It is shown that the existence of the axion in the model with local chiral flavour symmetry SU ( 3 ) H can solve the domain wall problem.

It is known that in various realistic models with spontaneously broken symmetries hybrid topological defects, such as domain walls bounded by cosmic strings, can be formed. This takes place if some discrete symmetry (e.g. Z2) appears as a separate factor at an intermediate stage of the hierarchical phase transition [ 1 ], or if the spontaneously broken continuous symmetry responsible for the string creation is approximate [2]. However, it can be seen that in fact these two possibilities are closely related, and work together in each case in which a hybrid structure is formed. For the beginning let us consider the hierarchical breaking pattern of a certain gauge symmetry G--, Z2®G'---,G' ( G ' = G ) . Choose a path in the manifold G parametrized by the angle 0 corresponding to the one parametric subgroup H ( 0 ) including Z z - { H ( 0 ) , H(2zt)}. Then the closed curve in the coordinate space with the configuration Z ( 0 ) = H ( 0 ) Z ( 0 ) of the scalar field Z (responsible for the breaking G ~ Z 2 Q G ' ) along it, cannot be contracted continuously to a point. This implies the existence of a cosmic string enclosed by the curve [ 3 ]. At the same time the vacuum expectation value (VEV) of the scalar 0 (breaking Z2) along the same closed curve 0(0) = H ( 0 ) 0 ( 0 ) must change discontinuously at a point, since 0(2n) :g 0(0) and the string becomes attached by the domain wall. The existence of such a wall with energy per unit area a ~ ( 0 ) 3 can have problematic cosmological consequences. However, 64

note that the energy density (as well as the thickness 8) of the wall and the scale of Z2 breaking are independent parameters. Therefore, in principle one can have a<< ( 0 ) 3 8>> ( 0 ) - t. Responsible for domain wall production are certain couplings in the potential depending on the relative orientation of Z and 0 in the H(0)-orbit space. Due to this, if such couplings are excluded from the potential, the degeneracy of the minimum energy state for 0 is not discrete but continuous under global H ( 0 ) transformations and, instead of domain walls, massless Goldstone states are created. The simplest example is the one discussed in ref. [4], with U( 1 ) symmetry spontaneously broken by means of two complex scalars Z and 0 with U ( 1 )charges l and noninteger Y respectively, and ;(>> 0. The vortex solution for Z in the cylindrical coordinates (0, p, z) takes the form [5] (for definiteness we consider the vortex with unit topological charge) X=r/(p) cxp(iO),

(I)

q(O)=O, r/(oo)=ll, where lz is the usual constant vacuum solution minimizing the potential energy. In the core of the string there is a U ( l )-magnetic flux corrcsponding to the gauge field Ao-, (igp) - ' as p--, oo (g is thc gauge coupling constant). Neglecting the back influence on solution ( 1 ), the kinetic term for the O's phase oJ away from the core is

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Assuming for a moment that I~1 is independent from 0, the vacuum solution for to is some function of 0 which minimizes the integral

f(0 )

2n

I=

-~-Y

2

dO,

(3)

0

subject to the condition to(2n) - t o ( 0 ) =2ztk,

(4)

with integer k. That function is to=kO+ c ,

(5)

with arbitrary constant c. The minimal energy solution is determined from the minimality of Ik - YI. This is the true solution if there are no explicitly phase dependent non-selfconjugate couplings in the potential between X and 0. In the above case no domain wall is produced. This is clear, since the global symmetry of the potential is U ( 1 ) x ® U ( I ) ~ which is broken down to U ( 1 ) o by Y. Being broken, U ( 1 ) o gives rise to global strings with different topological charges k and Goldstone bosons expressed in the arbitrariness c. Let us now consider the effects of possible phase dependent couplings. For Y= ~ they are m(x+(b2+h.c.) =2rn Ixl 1 0 1 2 c o s ( 0 - 2 t o ) ,

(6)

which together with the standard hermitian terms M21f~12+hlOI2lxI2+ht 1¢12+selfinteraction ofx

(7) can be included in the potential (the parameters m, M 2, h, h~ are real). The coupling (6) is a source of the domain wall. This can be directly seen from the equation for to, which in the approximation of 0-independent 101 has the well-known sine-Gordon form l-q~+3mlxIsin~=0,

~=0-2to,

(8)

with the domain wall solution [6 ] ~=4 tan - ' e x p ( 2 ~

x) ,

where x is the axis perpendicular to the wall. More strictly one has to account for the P-dependence of 101 as well as for the back reaction on solution ( 1 ). Then, in general, the phase of the vortex does not coincide with the azimuthal angle, but due to the topological charge it changes still by 2n after encircling the string. The existence of a domain wall can be

8 August 1991

understood easily. Indeed, the argument of the cosine in (6) has to change strictly by 2n (2k + l ) while going round the string and thus the cosine must go through all possible values (between + l and - l ) 2 k + 1 times. This implies the existence o f 2 k + l domain walls with energy density ~ m Izl I¢12 and energy per unit area tr~ 1012v/~ IXI. In the absence of the coupling (6), the potential for ~ is the usual Goldstone potential, with a continuously degenerate minimum along the circular vacuum valley and with a maximum in the center (at I~1 = 0 ) . As soon as the coupling ( 6 ) is switched on the degeneracy is spoiled and the valley is slightly lifted everywhere except at two points corresponding to the unbroken discrete symmetry ( w ~ to + rt). The energy density of the wall is determined by the height of the energetic barrier which is needed to cross for a transition from one vacuum (to) to another ( t o + n ) . For the region of parameters h l012> m IXI, the energetically most attractive way goes along the lifted vacuum curve, through the saddle point of height ~ m IXI I~12. At the same time such walls can be considered as a result of approximate U ( 1 ) symmetry breaking. Indeed, the coupling (6) explicitly breaks the additional global U ( 1 )~ symmetry. And, corresponding to this global symmetry, strings with topological charge k become attached by the 2k+ 1 domain walls. For m ~ 0 the U ( 1 )~ symmetry is restored and the walls disappear. The example considered demonstrates the fact that the two mechanisms of domain wall production in fact work simultaneously and cannot be separated. In any case, to obtain a discrete degeneracy of the potential minimum at some intermediate stage of phase transition one always needs to introduce (6)-type couplings which break the global H(0)-symmetry explicitly. In some realistic models such couplings can be excluded without spoiling the desired symmetry breaking pattern. This gives the possibility to avoid domain walls. What 1 have shown up to this stage is that domain walls might be substituted by Goldstone bosons and global strings. But now one has to worry about the cosmological consequences of these new objects. If they are dramatical, then the avoidance of walls paying such a price does not make any sense from the cosmological point of view. In general, the situation 65

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is different for different realistic cases and need to be investigated separately. In the present paper I consider two possibilities for which the above discussed mechanism of domain wall liquidation is cosmologically acceptable. The first is the case when the additional global U ( 1 ) is approximate (like Peccei-Quinn [12] symmetry). Then the corresponding global strings are unstable and might decay before dominating the universe [ 12]. Another is the case in which the additional U ( 1 ) is a local symmetry. In this situation the Goldstone boson is absorbed by the U ( I )gauge field and instead of global ones local strings are created. We consider as an example the model with a chiral local family (horizontal) symmetry SU (3) H [ 7 ]. The cosmic strings arising in this scheme were recently studied in ref. [8]. The three families of quarks and l e p t o n s f ~(u, d, e, v e ) , f 2 ( c , s, It, v ~ ) , f 3(t, b, 1:, v,) form the fundamental representation of SU (3) H, with left-handed fermion c o m p o n e n t s f ~ transforming as triplets and right-handed ones f ~ as antitriplets. The quark and lepton mass matrices are generated by means of the effective couplings (a, fl, 7 = 1, 2, 3 are SU (3)H-indices)

M-'tp°{a~"~Xt,'}~' + b ~ " ~ i ' , , , ) ~ } f ~ , f ~ ,

(9a)

where a '''~, b ~''~ (n, m = 1, 2, ...) are coupling constants, Ey,a is the invariant antisymmetric tensor and M is a regulator mass (of the order of the horizontal interaction scale MH ) related with the masses of heavy intermediate scalars [9] or fermions [ I0]. ~o° is the neutral component of the usual electroweak doublet ~0= (to °, ~0- ). The large ( ~ M H ) VEVs of horizontal Higgs scalar sextets z,t~ ~ " ) and antitriplets (~,,,) during SU(3)H hierarchical breaking at the same time are fixing the hierarchical structure of the fermion mass matrices. This is the so called direct hierarchy scheme (DHS). In other realizations of the model a special choice of the set of intermediate heavy fermions gives the possibility to obtain an inverse relation between the quark mass hierarchy and the pattern of horizontal symmetry breaking (e.g. see ref. [ 11 ] ). This is the inverse hierarchy scheme (IHS) in which fermion masses are generated by the following effective operator: I ~ M - t ~ o ° ( a ~ " ' X ~ +b ~''~r;' ~ ( m ) -t)'afl]~ ,f Lf~ - -

where ( 66

) - J denotes the inverse matrix.

(9b)

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In order to obtain the correct quark and lepton family mass hierarchy, the biggest VEV must be developed on the Z33 component of the sextet Z,~ain the DHS and on the Zl~ component in the IHS case. For definiteness let us consider the DHS. Then, at this stage, continuous symmetry is broken down to SU (2) n and at the same time the quarks and leptons of the heaviest generation (t, b, T, v,) become ready to acquire the diagonal masses as soon as the electroweak symmetry is broken by the (~o°). Let us consider the string formed due to the residual discrete symmetry 7/4= {exp(imn/2 28); m = 0, 1, 2, 3} under a phase rotation of multiples of n/2 in the broken U ( I )H C S U ( 3 ) H / S U ( 2 ) H subgroup H(0)= exp (i028), corresponding to the generator 28 = diag ( 1, 1, - 2 ) . Note that the ~e2 ( m = 0 , 2) part of this 2~4 belongs to SU(2 )H. Further breaking of the horizontal symmetry by other components of sextets and triplets spoils this discrete symmetry, and the strings become attached by the domain walls. For example consider the VEV of the triplet ( ( " ) = (0, 0, r). Due to (9), this VEV generates 1-2 mixing in the fermion mass matrices. ( ( 3 ) breaks 7/4 (under which it changes sign) but leaves SU(2)H unbroken. Due to this, each minimal string is the boundary of only one domain wall. The sources of these walls are the following type of nonselfconjugate couplings:

m~ ('~X,~P+m2~,~,z~, /~,rZ""'ZPPZ"7" +h.c.

(10)

Their existence could be disastrous for cosmology. The above walls are absent if such cubic couplings are excluded from the potential. The absence of these couplings in the lagrangian seems to be natural, since they are not induced by any other (gauge or Yukava) interaction [ 11 ]. This implies the existence of an additional global U( 1 ) symmetry [11 ] of Peccei-Quinn [ 12 ] type. fL~exp(ir)fL,

fR~exp(-ix)fR,

(Z,()~exp(2ix)(g,().

(ll)

After a series of redefinitions in the phase transition chain SU ( 3 ) t j ® U ( l ) - , S U ( 2 ) H ® U ( I )'--,U ( l )po-" I, the global symmetry is broken and gives rise to a Goldstone boson ot of the invisible axion [ 13 ] type [ l 1 ]. It can be verified easily that the 7/a/Z2 factor does not appear separately after the first step of the

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phase transition, but enters as a subgroup in continuous U ( 1 )' symmetry. Really, the remaining global U ( 1 )' is a superposition of the original U ( 1 ) and the broken U ( 1 )H subgroup of the 28-rotations. This superposition is defined in such a way that the Z33 component (and automatically the third family fermions (t, b, ~, v~), which acquire masses at this stage) remain invariant under U ( 1 )' transformations. Thus, for the antitriplets the matrix of U ( 1 )' transformations has the form (-,exp[i(2S+4l)0](

(12)

(where I is a unit matrix). For the values of the parameter O=mn/2 ( m = 0 , l, 2, 3) this gives the discrete Z4 c U ( l )' symmetry group. Due to this there are no local 2~4-strings and domain walls. The breaking of global U ( t )pQ at the last step of the phase transition gives rise to global strings. They become attached by the axion domain walls, since U( 1 )Po is explicitly broken by instantons. Due to the results of ref. [ 2 ], these structures decay before they can dominate the universe. However, in this case dangerous axion walls without boundaries might also be produced. This requires a special fermion content of the theory with the U ( 1 ) po color anomaly A¢ = 1 [ 11 ]. In the above model the scale of PQ symmetry VpQ is in the same time the lowest possible scale of the horizontal interaction VH. On the other hand, the hierarchy of phase transition (11) reflects the observed hierarchy in the quark and lepton mass matrices. Therefore, astrophysical constraints on VpQ ( Vr~) give respective constraints on the horizontal symmetry scale. As was shown in ref. [ 14 ], the analysis of the radiative decay ofaxion strings gives a stronger upper bound, VpQ< 10 ~° GeV, then the accepted [ 15 ] bound, VpQ< 1012 GeV. Since the current lower limit coming from supernova 1987A is ~ VpQ> l0 ~° GeV [ 16 ], this in fact (up to uncertainties on the 10 I° GeV scale) closes the axion window. And the standard axion appears to be consistent only with inflationary scenarios. However, the axion appearing in our model differs significantly from the usual invisible axion discussed in refs. [ t4-16]. This particle has both flavour diagonal and flavour nondiagonai couplings with ordinary quarks and leptons and thus is simultaneously a familon and a majoran [ I 1 ]. Its coupling is differently suppressed with fermions of different genera-

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tions. This is obvious since e.g. in the direct hierarchy model U ( 1 )PO is in fact a symmetry of the first family fermions (u, d, e, re), while in the inverse hierarchy scheme under U ( 1 ) p o only the quarks and leptons of the heaviest generation (t, b, x, v~) are transforming. Due to this, e.g. in the second case, the couplings of a with ordinary matter (u, d, e, re) is strongly suppressed. This removes the lower bound from the supernova up to V p Q ( V H ) > 8 × I 0 s GeV [ 11 ]. Stronger restrictions Vpo(VH) > 108 follow from the familon decays such as x--, pct, la--,ect, K--,Tttt .... So there is a wide window 101°> VN> 108 GeV. I refer the reader for more details to ref. [ 11 ] where the physical and astrophysical consequences of the axion = familon = majoron arising in SU (3) H® U ( I )H theory are investigated. The above example can be considered as a strong argument in favor of the existence ofaxions in models with spontaneously broken family symmetry. The scheme considered can be interpreted as something like the inverse problem of the one discussed in ref. [ 17 ]. In this reference it is shown that cosmologically dangerous Z~axion domain walls (more than one wall on each cosmic string) can be avoided if U ( I ) p Q is broken by the Higgs scalar transforming under an exact local symmetry including 71~,.Note that in our case U ( l )pQ is broken on the last stage of the phase transition, when the local SU (3)r~ symmetry is already destroyed. Another possibility, as I have mentioned above, is to consider local U( l ) symmetry. In this case one needs a special set of intermediate hypothetical fermions for the cancellation of U( l ) anomalies. Then, Goldstones are absent and breaking of U ( 1 ) at the last stage of the phase transition gives rise to a local vortex.

I would very much like to thank Z. Berezhiani, T. Bibilashvili, J. Chkareuli, E. Gurvich, O. Kancheli and S.M. Mahajan for very useful discussions.

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[3] T.W. Kibble, G. Lazarides and Q. Shaft, Phys. Lett. B 113 (1982) 237. [4] A. Everett, T. Vachaspati and A. Vilenkin, Phys. Rev. D 31 (1985) 1925. [5] H. Nielsen and P. Olesen, Nucl. Phys. B 61 (1973) 45. [ 6 ] S. Coleman, in: New phenomena in subnuclear physics, ed. A. Zichichi (Plenum, New York. 1977). [ 7 ] J.L. Chkareuli, Pis'ma Zh. Eksp. Teor.Fiz. 32 (1980) 684. [8 ] T. Bibilashvili and G. Dvali, Phys. Lett. B 248 (1990) 259. [91Z.G. Berezhiani and J.L. Chkareuli, Yad. Fiz. 37 (1983) 1043. [10] Z.G. Berezhiani, Phys. Lett. B 129 (1983) 99. [ I I ] Z . G . Berezhiani and M.Yu. Khlopov. Proc. XXIVth Rencontre de Moriond Series, Moriond particle physics Meetings (1989) p. 179; Z. Phys. C 49 (1991) 73.

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