Bulk vibrations and twists in global cosmic strings

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  • Volume 222, number 3,4 PHYSICS LETTERS B 25 May 1989


    Carl ROSENZWEIG and Ajit M. SRIVASTAVA Physics Department, Syracuse University, Syracuse, NY 13244-1130, USA

    Received 3 March 1989

    The Higgs fields surrounding loops of global cosmic strings contain the major component of the loop energy [E~R In(R/2) ]. These Higgs fields can undergo bulk vibrations and we demonstrate that these vibrations have a frequency which is larger, by a factor of ~/ln(R/2), than that of previously considered oscillations of strings. These bulk modes should dominate the processes of radiation emission. We also study configurations where a string threads a loop. Both strings are twisted. Twisted global strings

    RCUTOFF, usually have an energy which grows as 2 but in the above configuration we find the boundary conditions eliminate this extra energy. The final energy is just the sum of the energies of the untwisted loop and the untwisted string, with no interaction energy. The absence of interaction energy suggests that the twists may play an important role in the interaction of global strings, at least when one of the strings is a loop.

    I. Introduction 2. Bulk vibrations

    Cosmic strings are amongst the most interesting ideas that particle physics has contr ibuted to cosmol- ogy [ 1,2 ]. I f they ever existed they would have had profound influence on the evolution of the universe. Cosmic strings are formed in a phase transition caused by spontaneous breaking of a symmetry G~H, if the vacuum manifold G/H is mult iply connected. Re- suiting strings are called global ( local) strings i f the broken symmetry is global ( local). While much in- terest has focussed on local strings (as they seem to be most suitable for galaxy formation scenarios) [ 3,4 ] a good deal of attention has been paid to global strings. At this stage of our knowledge, with no evi- dence for the existence of any cosmic string, it is pre- mature to rule out one one or the other possibility. In this letter we wish to explore some hitherto ignored features of global strings. In section 2 we will show the existence of a new class of vibrational modes for global strings with a frequency larger than the usually considered oscillation. Section 3 discussed the pres- ence of twists in global strings. We discuss a configu- ration of strings where the presence of twists can change the usual expectations of interaction of strings.

    It is well known that the energy of an isolated seg- ment of a global string diverges logarithmically. However, for a loop of string, its size R provides a cutoff leading to finite energy ~ R In (R /2) . A string loop of radius R will oscillate with a typical fre- quency 09 ~ 1/R and these oscil lations lead to emis- sion of gravitational radiation as well as Goldstone bosons [2-5 ].

    The Higgs field configuration around a loop of global string has a significant energy density associ- ated with it and in that sense behaves like a lump of energy. This suggests that there are vibrational modes of this lump which should contribute to the emission of radiat ion from these strings. The conventional treatment of the oscillation of strings starts with the Nambu string action [ 2 ] and thus excludes these vi- brational modes (to study these modes one would need an action appropriate for three-dimensional ob- jects). We now turn to the examination of these modes.

    Consider a model with a single complex scalar field in which strings arise due to breaking of global U( 1 ) symmetry:

    Lp= O~tO~q~_ h (~pt~o_f 2) 2 ( 1 )

    368 0370-2693/89/$ 03.50 Elsevier Science Publishers B.V. ( North-Hol land Physics Publ ishing Div is ion)

  • Volume 222, number 3,4 PHYSICS LETTERS B 25 May 1989

    The energy associated with a given string configura- tion is

    E= I d3x [ I V~plz+ h(~*~o-f2) 2 ]

    =E .... + f d3x IV~ol =, (2) r~>2

    where 2 is the thickness of the string core and I ~ 12 =f 2 for r>~2.

    It will be most simple and convincing to choose a simple variational ansatz for ~o around a string and examine its properties. From eq. (2) the energy of a loop (excepting Ecore) is contained in the position dependent phase of the scalar field,

    ~0(r) =fexp [iO(r) ]. (3)

    We will take the string loop to be circular and in the xy plane. For our ansatz for ~o around this loop we look for a smooth series of surfaces of constant phase O(r) = 0. A convenient and tractable choice is a se- ries of paraboloids with z axis the axis of azimuthal symmetry (see fig. 1 ). We choose the scalar field phases such that 0=0 on the disc in the plane of the loop and 0= n on the outermost paraboloid which peaks at a height Zo above (and 0= -n at -Zo be- low) the center of the loop. Zo here is treated as a variational parameter. The phase 0 is fixed to vary linearly between 0 and n as one goes from the center with the loop to the outermost paraboloid along the z axis. Thus as one goes from + Zo to -Zo, phase 0


    (0,0.Z 0)

    " -~ @= rr paraboloid

    .- -...

    S I i \ < - @= 0 disc Y

    ~ string loop J

    Fig. 1. Paraboloids of constant Higgs field phase for a string loop.

    changes by 2n. With this ansatz the scalar field is given as (in cylindrical coordinates)

    (0(p, z) =fexp [iO(z, p) ]

    =fexp \ZOO/ ' (4) p being the distance from the z axis and R the radius of the loop. It is straightforward to calculate the en- ergy of the string loops,

    4 E, ooo=2n3f2Rln(~---~)(ffo+~-~). (5)

    E~oop is the total energy of the loop minus the energy of the core. Eloop, minimized at Zo= x/~ R, is

    167~ 3 2

    From eq. ( 5 ) one can see that there will be vibra- tions of surfaces of constant phase where Zo vibrates about its equilibrium value Zo=x/~R. The fre- quency for such vibrations is

    N/~ f fin(R/22)'] 1/2 09=4 ~\ V/~ J (7)

    This frequency is larger by a factor of [In(R/22) ] 1/2 (apart from coefficients of order unity) than the fre- quency of string oscillations previously considered [2].

    Many other bulk vibration modes are also possi- ble, e.g. lateral deformation of the surfaces of con- stant phase (where the peaks of the paraboloids in fig. 1 oscillate away from and towards the z axis). For convenience we have used a conical ansatz to study these modes (instead of the paraboloids of fig. 1 ). Here the surfaces of constant phase are cones, again with the axis of the cone being the z axis, and each cone passing through the string. Full 2~ variation in the phase is again achieved within two outermost cones with their tips at _+ Zo. We calculated the en- ergy of the loop while tilting the cones away from the z axis. The associated frequency of vibration was found to be the same as in eq. (7) (apart from coef- ficients of order unity). Zo vibrations in this ansatz also give the same frequency. This suggests that our results are not specific to the ansatz and are generally true.


  • Volume 222, number 3,4 PHYSICS LETTERS B 25 May 1989

    These vibrational modes have a significantly higher frequency than the ones previously considered for global strings. This will imply that these strings may radiate their energy much more rapidly and decay with much shorter lifetimes. For oscillations of global strings, it is known that the energy emitted in gravi- tational radiation is much less than that emitted in Goldstone bosons [ 3,4]. However, this result (and many other results concerning the radiation from the global strings) needs reevaluation in the light of these new bulk vibrational modes. It is important to realize that these bulk vibrations will be coupled to the os- cillations of the loop and a full treatment of this prob- lem will require the consideration of interaction be- tween the two kinds of modes. Our main purpose was to illustrate the existence of a new class of vibrational modes of global strings.

    3. Twists in cosmic strings

    Twists are another, potentially interesting feature of strings. They are not usually considered in the con- text of global strings because they introduce an addi- tional divergence into the string energy, a divergence which is stronger than the logarithmic divergence of an untwisted string. A twist in a global string is a change in the phase O of the Higgs field as we proceed parallel to the (z) axis of the string. This introduces a z gradient for ~ which when integrated out to a ra- dius R in eq. (2) gives a contribution proportional to R 2 [6].

    It turns out, however, that a very simple configu- ration of strings is able to completely remove this "twisted" energy contribution. Consider a twisted string surrounded by a loop of string. The boundary conditions require that the string forming the loop itself be twisted. (Conversely a twisted loop must be threaded by a twisted string [6 ].) The net effect of these twists is no new contribution to the energy. The total energy is just the sum of the energy of a free un- twisted string plus the energy of an untwisted loop. No interaction energy arises.

    We can establish this result by considering the an- satz of fig. 1, but now with a string along the z axis. The paraboloids are no longer surfaces of constant phase. As we make a circuit around the central string staying always on one of the paraboloids of fig. 1 the

    phase must change by 2n. To effectuate this phase change we take as an ansatz

    O(z,p, O)=O+O(z,p), (8)

    where O(z, p) is as defined in eq. (4) and 0 is defined as follows.

    Consider a set of planes whose intersection is the central string, like the fins of a paddle wheel. Choose one plane to have the value zero and label the others continuously from 0 to 2n. Each plane will intersect each paraboloid along a curved line. Along this curve the phase O(z,p, O) will be given by eq. (8) where 0 is the fixed angular label of the plane. The energy of this configuration will be proportional to the square of the gradient of O(z, p, 0). There will be three terms: (1) IV0I 2, (2) VO.VO(z,p), (3) [V~9(z, p) ] 2. The first term leads to the energy of a free untwisted string, the third term is the energy of the untwisted loop cal- culated above (while considering bulk vibrations). The second term vanishes, since we have chosen 0 in such a way that VO lies entirely on the surface of the paraboloids and is thus perpendicular to VO. Thus our ansatz gives no interaction energy.

    The argument and ansatz are readily extended to non-symmetric orientations of the string through the loop. We continue to define the 0 dependence by the intersection of the paraboloids with the "paddle wheel planes" radiating from the string. Although analytic calculations with this coordinate system would be daunting, the geometrical nature of our argument will lead to the same result - no interaction energy. Of course we must restrict ourselves to configurations where the strings are never too close to each other.

    Although we have not shown that the ansatz we used was the one of lowest energy, this seems quite plausible to us. The ansatz respects the symmetries of both the string loop and the single string and some- what unexpectedly does not generate any interaction terms. Other natural choices for the configurations, such as using the physical string, even when dis- placed from the Z axis of fig. 1, as the axis where the surfaces of constant phase are peaked give rise to higher energies. One implication of our result is that the string-string interaction is somewhat more in- volved than has been hitherto appreciated. In this study [7], Shellard computed string-string interac- tions using segments of strings. He found the reason- able tendency for strings to align anti-parallel. Our


  • Volume 222, number 3,4 PHYSICS LETTERS B 25 May 1989

    calculation of the loop-string energy suggests that, if one of those strings is in a loop, the results can be drastically different. For example suppose the straight string piercing the loop is displaced from the center and slightly tilted in a direction perpendicular to its displacement. Shellard's calculations suggest that, depending on the orientation of the string, the string should be either attracted to or repelled from the cen- ter, since parallel strings repel and anti-parallel strings attract. Our result of no string-string interaction en- ergy contradicts this. Even if our ansatz does not give the lowest energy of the system it still rules out the possibility of a repulsive interaction in such situations.

    This result is not surprising since global strings have long range correlations and a free bit of string is clearly a very different object than a string in a loop. The different boundary conditions imply very different

    Higgs field configurations, even if the boundaries are quite distant with respect to the string-string dis- tances. Since loops are a common feature in cosmic string scenarios the question of intercommutivity of a string and a loop or of two string elements both in loops may be worthy of further study in the light of our results.


    [ 1 ] T.W.B. Kibble, Phys. Rep. 67 (1980) 183. [2] A. Vilenkin, Phys. Rep. 121 (1985) 263;

    N. Turok, Fermilab preprint. [ 3 ] R.L. Davis, Phys. Rev. D 32 ( 1985 ) 3172. [4] A. Vilenkin and T. Vachaspati, Phys. Rev. D 35 (1987) 1138. [5] T. Vachaspati, A.E. Everett and A. Vilenkin, Phys. Rev. D

    30 (1984) 2046. [6] R.L. Davis, Nucl. Phys. B 294 (1987) 867. [7] E.P.S. Shellard, Nucl. Phys. B 283 (1987) 624;

    see also R. Matzner, University of Texas preprint.



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