V O L U M E 5 7 , N U M B E R S P H Y S I C A L R E V I E W L E T T E R S 4 A U G U S T 1986

Comment on "Monopoles on Strings"

Cosmic strings1 may be formed in symmetry break­ing of the form G — K ® Z2^H, where the group G has a simply connected manifold. All strings formed, including antistrings (with opposite direction of mag­netic flux and opposite circulation of the Higgs field), are topologically equivalent. However, Hindmarsh and Kibble2 have shown that, in a number of cases, one can distinguish locally opposite directions on Z2

strings, with oppositely directed regions joined by beadlike solitons. There are various ways in which Z 2

can be embedded in G. Here we show that in many realistic models the embedding with the lowest mass/unit length, ^ , gives strings in which the bead­like solutions of Ref. 2 do not occur.

A particular model of physical interest is3

SOCIO) — SU(5) ® Z2 . Along a circular path with az-imuthal coordinate 9 enclosing a string the Higgs field 4> is given by

<l>(9)~Qxp(iTpe/27r)<P^g(9)<P, (1)

where r can be any broken-symmetry generator of SO(10) satisfying exp(/r/?) = P, the nontrivial ele­ment of Z2 . We consider two possibilities: first, T = rjoc F, where Y is the operator whose matrix in the sixteen-dimensional representation of SO (10) is diago­nal with eigenvalues 1, —3, and 5 in the ten-, five-, and one-dimensional representations of SU(5), and second, T=*T2

:==:TRU where TR is the right-handed weak isospin; T\ and T2 are not related by a gauge rota­tion, as is clear since they have different eigenvalues.

Physically occurring strings will be those with the lowest value of fi. The symmetry breaking can be ac­complished by giving a vacuum expectation value to a Higgs field transforming like VL^L* where vL is the left-handed antineutrino; the discrete symmetry P is VjL—+ —vL. The Higgs field <f> is an eigenstate of r^ It is not an eigenstate of T2 , however, but of 7^3, and can be expanded in eigenstates of r2 as

4>-(<f>i+<l>-i)/2 + 4>o/y/2, (2)

where the subscripts indicate the eigenvalues of T2. This suggests that IL will be smaller for T = T2 , since $ 0

is invariant under rotations generated by r2 , so that, for r = T2, one has, loosely speaking, only half a string. We have verified this by computing the value of fx nu­merically by solving the coupled equations for <f> and the vector potential, analogous to the Nielsen-Olesen equations4; the radial integral of the energy density then yields LL. Details are given in Aryal and Everett.5

For T = TX we take the potential to have the form

K=x(4>W)2 , 0)

while for T == T2 we take

V-Xl^l + ̂ / l - ^ + k'i^-ti)2/!, (4)

with <f>i « 0_! . For X' > 0 this mimics the full SO(10) potential in having a minimum at 01==0O

==T7. For \ = 0.1 (the results are insensitive to X) and gauge coupling strength appropriate to the grand unification scale, we obtain /x = 3.2i72 for T = T\. For T = T2, 1.0 ^fx/r}2^ 2.85 for O ^ X ' ^ 0 0 , with a numerical accuracy = 5%. Thus, throughout the relevant range of X, T2 strings have a lower mass. Other choices of r are also possible, but are either gauge equivalent to r2, or can be argued to have higher mass.5

Strings and antistrings differ only in the path fol­lowed by g(9) between the two disconnected pieces of the manifold of // , and thus may be deformed into one another. The intermediate configurations will have the same mass, and strings and antistrings will be gauge equivalent, if there is a gauge transformation which takes T—+—T. For T = TU no such transforma­tion exists. Hence, T\ strings and antistrings are physi­cally distinct, and the ' 'beads" of Ref. 2 occur. How­ever, T2—* - T 2 under a 180° rotation generated by T/?2. Thus the stable r2 strings are gauge equivalent to antistrings and can be deformed into antistrings through equal-mass configurations. The string direc­tion thus cannot be defined gauge invariantly, and lo­calized beads connecting string segments of different directionality will not exist.

For the related symmetry-breaking pattern SO(10) —SU(5) ® U ( l ) r — SU(5) ® Z2 the Higgs field associated with the first stage of symmetry break­ing will force T = TU and beadlike solutions will occur.2

Similar conclusions should hold with other symmetry-breaking patterns; we expect beads with intermediate U(l) phases but not otherwise.

Allen E. Everett and Mukunda Aryal Department of Physics Tufts University Medford Massachusetts 02155

Received 21 March 1986 PACS numbers: 12.10.Dm, 11.15.Ex, 98.80.Cq

lA. Vilenkin, Phys. Rep. 121, 265 (1985). 2M. Hindmarsh and T. W. B. Kibble, Phys. Rev. Lett. 55,

2398 (1985). 3T. W. B. Kibble, G. Lazarides, and Q. Shafi, Phys. Lett.

113B, 237 (1982). 4H. Nielsen and P. Olesen, Nucl. Phys. B61, 45 (1973). 5M. Aryal and A. Everett, to be published.

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