Physics Letters B 272 (1991) 207-212 PHYSICS LETTERS B North-Holland

Flavoured cosmic strings and monopoles

J.L. Chkareu l i lnslitute qf Phsyics, Georgian Academy o/Sciences, SU-380 077 Thilisi, Georgia, US£7~

Received 11 August 1991

We analyze string and monopole solutions related with the spontaneously broken gauge family symmetry of quarks and leptons and consider their cosmological and astrophysical consequences. We sketch a new possible scenario for the dark matter in the Universe based on the evolution of the flavoured cosmic str ing-monopole networks. The scale of the broken family symmetry which originates such a string-dominated Universe is estimated to be I'H = 10 4 - l 0 m GeV, which meets the limitations following at present from the search for flavour-changing processes of quarks and leptons.

1. It is currently accepted that in the early Universe linear topological defects called cosmic strings could be formed as a result of a phase transition [ 1,2 ]. The mass per unit length of such strings is/~ ~ I ,'2, where I" is the vacuum expectation value (VEV) inducing the spontaneous breakdown of some starting sym- metry G of quarks and leptons. Superheavy strings usually related with grand unification such as SO (10) or E(6) (VGu= 10'5-10 ~6 GeV) can yield a cosmo- logically marked density fluctuation from which gal- axies evolve and can also produce a number of dis- tinctive observational effects [2 ]. Strings with V< 10 9 GeV would not be observed by their gravitational in- teractions. Even so, there could be some observa- tional phenomenology from strings related with their super-conductive charge-carrying ability [ 3 ].

Our interest in light cosmic strings is due to the highly attractive possibility that they might consti- tute the bulk of the dark matter in the Universe (re- quired to make ,Q= 1 according to the inflationary scenario [4] ) as was suggested by Vilenkin [5] and Kibble [6] a few years ago. Actually, if the above phase transition occurs at a typical grand-unification scale, this can certainly be ruled out. The Universe could be dominated by strings provided that it be- came so only recently after galaxy formation. Thus the existence of a second symmetry breaking scale at a value much lower than I)~tJ seems to be essential. In this letter we consider the horizontal (or flavour) gauge one, GH of quark-lepton families with the

minimal scale limitation 1~> 10 4 GeV, as such a symmetry, as it follows at present from the search for some flavour-changing neutral transitions of quarks and leptons [7]. We argue that the spontaneous breakdown of GH can naturally lead to the appear- ance of flavoured cosmic strings which could provide real physical grounds for scenarios of a string-domi- nated Universe (SDU).

2. While our discussion is suitable for any family symmetry GH which meets the well-known topologi- cal criteria, we consider below the gauge SU(3)H symmetry [ 8 ] with the chiral assignment of ordinary quarks and leptons (in the framework of the stan- dard e[ectroweak SU (2) X U ( 1 ) model )

U V c

f~,= [(u, d)R, eR] ~ . (1)

where the families are numbered by the S U ( 3 ) H in- dices o~= 1, 2, 3. This symmetry seems to be a most attractive one if we bear in mind the direct extension to the grand unified models such as SU (5) × SU (3) n or E ( 6 ) X S U ( 3 ) H as well as the natural description of quark masses and mixings [9-11 ].

The chiral nature ofSU (3)H requires the inclusion in the model of the complementary weak-isosinglet fermions (quarks and leptons) F~ and FR~, to be

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anomaly free. They receive large masses (of the order of VH) from the Yukawa couplings

~--k/~ ,i,, (2)

after the horizontal scalar fields rl~ n (the triplets rl~, c~/q and the sextets rlj, <~n' of SU (3) H, n stands for the number of scalars) develop their VEVs and SU (3)n breaks fully. At the same time the masses of the ordinary up and down quarks ( 1 ) and the charged leptons arise from the effective couplings of type

~ ff~ ~ n,,<~B, (3)

which are formed [ 10] as the consequence of the ex- change by some intermediate non-chiral weak-isosin- glet fermions %~ (WL~, WR~) with masses Me due to their Yukawa couplings

f fJ ~Y tIJ , I l l l-gltsR<-<'qO, qs~f(~'q,,o~/s, --L--at~ . . . . ( 4 )

where q~ is an ordinary Weinberg-Salam scalar whereas I{{ is a real scalar field - singleI and /o r octet of SU(3)H, (Mv){{vcI~. As follows from the Yu- kawa couplings (2) and (3) there is a global chiral symmetry U ( 1 ) n of F fermions, ordinary quarks and leptons and horizontal scalar fields

FL,R=:~exp(+-i09) FL,R, fL.R=c>exp(-T - i0)) fL.R,

G,,~exp (2io)) rl,, (5)

in addition to the local SU(3)H symmetry in the model.

For the full spontaneous breaking of SU (3) H and U ( 1 ) H only three basic scalar fields q~n are necessary which can be chosen as one sextet rl:o ~#I and two trip- lets GI~ n] . Their VEVs determine the mass spectrum of the horizontal gauge bosons of SU (3) H and simul- taneously the mass matrices of the quarks and lep- tons via the couplings ( 2 ) - ( 4 ). The simplest natural solution [9] for the VEVs of the scalars qg/~, q~,/~, q~/~ (in their Higgs potential),

(rl~ 'is ) =1126c~16/J2 , (6)

with the soft hierarchy rh =p-~lh =p3rl0 (p~ 0.2) leads to the concrete Fritzsch ansatz [ 12 ] for the mass ma- trices of the ordinary up and down quarks. We adopt the view that this ansatz reproduces mainly the ob- served picture of the quark mixings, and subse-

quently we have to do with the VEVs (6). As to the global chiral U ( 1 ) n symmetry ( 5 ) which

breaks through the same VEVs (6) it can be consid- ered as the starting Peccei-Quinn symmetry [ 13 ] of all quarks and leptons including the F fermions. On the other hand we could gauge it and could have ab initio the total local family symmetry S U ( 3 ) H × U ( 1 ) n with the horizontal hypercharge U ( 1 ) n [ 1 4 ]. In any case the presence of the additional U(1 )H proves to be essential for the topological features of our model if we start from the VEVs (6).

3. Let us consider now the string solutions related with the full breakdown of the family symmetry SU ( 3 ) n × U ( 1 ) H due to the VEVs (6). In accord with the general arguments the existence of the ordinary axially symmetrical strings means in our case the presence in the horizontal group space of the un- shrinkable loop of type

~ ( 0 ) =exp[ i ( a28 +b23 +cY)O] , (7)

where 28, 23 and Y are the diagonal generators of S U ( 3 ) H × U ( 1 )u. Applying U2(0) to the VEVs (6)

rl.(O)=f2(O)q,,(O), n = O , 1, 2, (8 )

We arrive at the conditions for the constants a, b and C

-4a+c=l , - a - b + c = m , qa+c=n, (9)

where by definition the topological indices l, m and n take the values +_ 1 and 0, that means the presence or the lack of the (basic) strings, respectively. For non-integer values of some of them there would be domain walls bounded by strings. Actually such walls are produced in the "pure" SU ( 3 ) H case (c = 0 ) as is easily seen from eqs. (9). The symmetry breaking looks as

Z 4 rll SU(3)H ~ SU(2)H X Z~ ~ 1. (10)

On the other hand, in the presence of the hypercharge U ( 1 )n a "hybrid" string is produced only in the final stage of the breaking:

S U ( 3 ) n X U ( 1 ) . n ° S U ( 2 ) n X U ( 1 ) H " 0 ( 1 ) H q2 ~ , (11)

after the generator Ysuccessively mixes with the gen-

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erators 2s and 23 of SU(3)H. For starting the global U( 1 )H symmetry the modified l~l( 1 )H in ( 1 1 ) coin- cides with the real Peccei-Quinn symmetry U ( 1 )pQ which relates now with the fermions of the first gen- eration only. Its spontaneous breakdown can lead to the appearance of the analogous (global) "hybrid" string which, subsequently, after an explicit violation o fU ( 1 )pQ by non-perturbative strong-interaction ef- fects, becomes the boundary of the cosmologically safe axion domain wall. Such global "hybrid" strings are well known (see e.g. ref. [ 15]; they were intensively discussed recently, particularly in relation to the fam- ily symmetry [ 16 ].

Are there any new string configurations in the fam- ily model with the VEVs (6)? This could be possible if there is some discrete symmetry Zx (N>~2) in the intermediate stage of SU (3)H symmetry breaking. It can be easily seen that in all cases when a scalar which breaks S U ( 3 ) u breaks also the horizontal hyper- charge U ( 1 ) n (local or global - this does not matter) we have the above "hybr id" strings only. Actually, the discrete symmetry Zx which can appear from the VEV of N-index symmetric scalars )(,~<~</J. e l (antisym- metric scalars - triplets o f SU (3) n - do not produce strings in this case) is always "absorbed" by the modified hypercharge symmetry 0 ( 1 ) n as in the above case with the 2-index scalar rl~ '~'~ .

Now let the N-index scalar )~ have no U ( 1 ) hyper- charge while its VEV develops one of the compo- nents, say, Z33..3. We have the chain of breakings

()~> Go>'ql > 112)

SU(3)H × U ( I )H ~ SU(2)H X ~ XU( 1 )n

llO 111 -~ SU (2)n - . "[]. (12)

We can see that a possible string solution related with Z must have a ZN configuration. Now, owing to all former scalars Go.~2 having hypercharge U( 1 )n and (Z:x /Z2) e U ( 1 ) H, a domain wall in the second stage in (12) is not produced. Instead of it there appears our old "hybrid" string related now with the scalar rio which forms the masses o f the third family of ordi- nary quarks and leptons as well as F fermions. More- over, the conditions of absence of the domain wall are found now to fix completely the order N of the allowed discrete symmetry Zx and the corresponding string configuration. Applying again the group trans-

formations (7) to the VEV (X33...3) we obtain in ad- dition to the conditions (9) the new one:

-2Na=k , k = + l . (13)

The conditions (9), (13) give rise to the final equa- tion between the topological indices in the model:

l - n = 3 k / N , (14)

which has only one non-trivial solution, N = 3, for ba- sic strings. The same solution appears if the scalar Z develops its VEV on any other component. Hence, the 3-index scalar )¢I~/~,', is singled out topologically.

Thus we are led to the Z3 string configurations with vertices where three strings join. An inner core of these strings is related with the scalar Z while the outer one is related with the scalar Go- They can form suit- able three-dimensional string networks in the course of evolution [2]. I f the SU(3)H symmetry breaks beforehand through the octet of scalars I~ ( S U ( 3 ) H - , U ( 1 ) s × U ( 1 ) 3 , this would actually be implied in (7 ) ) , flavoured monopoles are produced and then after string formation they turn out to be in the vertices of the networks. Subsequently we shall deal with such str ing-monopole networks.

The last point which concerns us is the scalar g',<~n~,l itself inducing the above Z3 strings. Adjusting the re- quired zero hypercharge assignment happens to be possible just for the 3-index scalars only. This follows directly from the crossing terms

z,<,,/<~.,,, q i ; 'sj~ • (h. n7 +h~ q~ ) ( 15 )

(h~, h2 are constants) in a general Higgs potential o f the horizontal scalar fields.

4. Thus we can have two types of strings in the fam- ily model: the single "hybrid" strings (global or local infinite strings as well as loops - from the scalar i"12, see ( 11 ) ) and the Z3 str ing-monopole networks with the double-core strings (from the scalars Z and Go re- spectively, see ( 12 ) ). According to Witten's general criterion [3] the transverse zero modes of the ordi- nary quarks and leptons would be trapped on these strings as well as the heavy (outside of strings) F and

fermions due to their Yukawa couplings ( 2 ) - ( 4 ) with the basic string-forming scalars G,, in the model. Therefore, generally we are led to superconducting strings and networks. The charge-carriers on them will be anomaly-free sets of fermions which are actually

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in this case (see couplings (2) and (4 ) ) the topolog- ically conjugated fermions (Fg, FR,) and (~L, , f~ ) moving on the strings in opposite directions. In ac- cordance with the structure of the VEVs of the scalars r h and 1"1o (6) only fermions of the first and second families (c~ = 1,2 ) are trapped on the "hybrid" strings whereas fermions of the third family (c~=3) are trapped on the Z3 string networks. Therefore, unlike the strings in the grand unification models our super- conducting strings are really flavoured.

5. The most interesting cosmological and astro- physical consequences of the discussed model follow mainly from the expectation that relatively light ( V. << Vc;u ) flavoured cosmic strings could lead to a SDU [5,6]. It is commonly assumed [ 1,2] for ordi- nary single strings that: (i) at the time of formation the strings have the shape of brownian trajectories with persistence length ~o < to (causality) which then scales in the course of its evolution with the horizon distance ~(t)~ct; (ii) intersecting strings intercom- mute (change partners ) and form closed loops which oscillate and disappear losing their energy by gravi- tational radiation.

These general assumptions which are strongly sup- ported by numerical simulation provide the basis of an attractive theory of galaxy formation [2]. With both of them any single string never dominates the Universe. On the other hand if there are some dy- namical reasons for revising [ 5,6 ] assumption (i) or (ii), our light "hybr id" strings could be the best can- didate for a SDU. While such strings with 1~ n < 10 9 GeV would not be observed gravitationally they could manifest themselves due to their charge-carrying ability [31.

However it seems more natural that a SDU could appear through topological reasons rather than dy- namical ones. Numerical simulation [ 17] of the sta- tistical properties and the evolution of Zx ( N = 3, 4, and one can expect this in the general case) string- monopole networks actually confirm that this could be the case. It was found [17] that: (iii) the system is strongly dominated by one infinite network wherein most of the string segments have lengths comparable to the typical distance between monopoles d; (iv) the system evolves in a self-similar fashion showing no tendency to relax to an equilibrium configuration or to decay into small nets, the intercommutings are

inessential inducing a difference in d of only ~ 10%; (v) the main energy-loss mechanism for the network is the radiation of gauge quanta by ultra-relativistic monopoles with deconfined SU (3)~ and /o r U( 1 )cm magnetic charge, just what prevents such networks in grand unification models to dominate the Universe,

Now one can make direct use of these results for our flavored Z3 str ing-monopole network and get all the same conclusions about its formation and evolu- lion but the last point (v). Instead of it we get that: (vi) all magnetic flux from the flavoured monopoles is confined in strings because of the full spontaneous breakdown of the family symmetry outside the strings (see ( 12 ) ). Therefore we are inevitably led to a SDU.

The typical scale d of our network as function of time can be estimated more or less accurately follow- ing to standard argumentation [ 1,2 ]. We adopt that the system approaches rapidly a quasi-equilibrium state which actually due to its basic features (iii), (iv) and (vi) is quite close to a true equilibrium and we can write the energy conservation law in an expand- ing Universe in the simple form

/)n = - 3 d (Pn + P . ) • (16) a

Here a(t) is the cosmic scale factor whereas Pn and Pn are the total energy density and the total pressure o f the system, respectively. For non-superheavy monopoles (the VEVs of scalars l)f and Z-I~:. are roughly of the same order) we merely arrive at the string ones Pn -~Ps ~ V n d - e and Pn -~ Ps = ?'Ps with 7 [17]

7 = ½ ( 2 < V s ) - l ) , -1~<7~< ½, (17)

where Vs is the transverse velocity of the strings and angular brackets indicate statistical averaging.

Solving eq. ( 16 ) with ?, ( 17 ) which does not change with time (self-similarity ( iv) ) leads to the typical length scale d( t ) of a network:

(.(,) d ( t ) = \ ~ / d(to)

/ \ 1 + ; ' / \ 3 ( 1 + 7 ) / 4 / \

= ~ \/~q/ \ t o /

where we have used for the cosmic scale factor a(t) (according to the evolution equation)

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a( t <te , ) x t ~/2 ,

a ( t < ts) ~ t 2 / 3 ,

a ( t > ts)Jet 2/3~1 +;') (19)

Here t o ~ M p V ~ 2 is the time of formation of the strings (Mp is the Planck mass), tcq ~ 10 ~1 s is the time when the densities of matter and radiation are equal, and ts is the time when the strings begin to dominate the Universe ( ts> [eq) . Requiring now the causality condition to be satisfied in each era (19) separately, we get a more stringent constraint on y ( 17):

- ~ . < 7 . < 0 . (20)

Finally, for the ratio of the string mass densityps and the mass density of ordinary matter p ~ l / G t 2 we obtain

v

p \ d o J k ip ,] \ le ,] '

where e H = ( V H / M p ) 2, tp is the Planck time and

do=d( to) <to. Now, if string dominance (Ps /P~ 1 ) actually starts

at the present time ts~ 10 ~7 s, eq. (21) for 7 in (20) gives constraints on the scale VH (for ts--tcq they are practically the same):

I,~ ~ 104-101° G e V , (22)

which meet the limitations following from the labo- ratory experiments, Vu> 104 GeV [7]. One can eas- ily see from eq. (21) that if the flavoured Z3 strings are formed in a second-order phase transition (do ~ 1/VH) they dominate the Universe very soon after formation. Thus we have to assume a first-order phase transition with sufficiently large bubbles (of the or- der of to).

In this SDU scenario the Universe at present is built up mainly from a three-dimensional flavoured Z3 string-monopole network with the practically straight strings having relativistic velocities. The typical dis- tance between the strings (monopoles) in the net- work according to eq. (18) is d=10~3-10 ~s cm for the intervals (20) and (22). Thus there should be a lot of strings of the network passing our galaxy and even (for low scale Vu) through the Sun. For the Earth the typical time between two successive en- counters with the strings is proved to be about one year if VH ~ 104 GeV.

As to observational manifestations of our network, they are related with its superconducting charge-car- rying ability [3] rather than with the gravitational interactions. As was argued in section 4 the zero modes of the third families o f the f, F and ~u fermions (quarks and leptons) are trapped on the outer core of the strings (due to the scalar qo) in the network. The heavy (outside of the strings) F and qJ fermions can acquire the Fermi momentum p ~ 10 m GeV as the string crosses a galaxy [3], and then leave the string creating mainly top- and beauty-flavoured bar- yons and mesons (as well as z-type leptons) with en- ergy E - ~ at a rate [3] of 10 ~2 particles/cm of string/s. In a low-scale SDU such phenomena could be expected from time to time in the form of very extended (of the order of Earth dimensions) air showers in cosmic rays.

6. To conclude, it has been found that in the family unified SU (3)~ model there is a quite distinctive in- terplay between quark-mixing and flavoured topo- logical defects. Among them the Z3 str ing-monopole configurations with double core strings are unambig- uously singled out by the requirement of the lack of domain walls in the model. They form an infinite network which inevitably comes to dominate the Universe in the course of its evolution. For the first time there appears in the SDU a strict upper limit on the family unification scale VH~< 10 m GeV, which meets the limitation following from laboratory ex- periments on flavour-changing processes. The charge- carrying ability of the network predicts occasional flavoured signals in high-energy cosmic rays.

I acknowledge gratefully the stimulating conversa- tions with R. Barbieri, Z. Berezhiani, G. Dvali, G. Fiorentini, O. Kancheli, A. Masiero, E. Paschos and A. Vilenkin.

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