Volume 2 15, number 2 PHYSICS LETTERS B 15 December 1988
GLOBAL COSMIC STRINGS, NAMBU-GOLDSTONE BOSONS
AND THEIR CONSTRAINTS ON UNIFICATION
R. HOLMAN Physics Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA
David B. REISS Department of Physics, University of Wisconsin, Madison, WI 53 706, USA
Received 15 September 1988
We show how the existence of a Nambu-Goldstone boson of the type recently argued for by Sikivie powerfully constrains possible models for grand unification. In particular, we show that the only models consistent with Sikivies arguments are those in which at least part of the electric charge operator lies outside the group containing color. This rules out most known grand unification schemes including many of those inspired by superstring theory. We discuss the implications of this fact and construct a toy omion model.
Recently, Sikivie [ 1 ] has claimed to deduce the existence of a Nambu-Goldstone boson (NGB), which he calls the omion, from cosmic ray and radio astronomy data. If this deduction is correct, it has far reaching implications for particle physics, astrophys- ics and cosmology.
In this letter, we wish to examine some of the ef- fects of the omion on grand unified model building. It turns out that the existence of the omion imposes severe constraints on viable GUTS, to the extent of ruling out almost all models constructed to date! The only viable models remaining are those in which some part of the electric charge operator lies outside the group containing SU ( 3 ),. Such models can and have been built (fortunately! ) [ 2 ] and we will indicate how the fermion charges of the global symmetry group G (whose spontaneous breaking gives rise to the om- ion) must be constrained. We will also discuss the feasibility of constructing phenomenologically rea- sonable grand unified omion models.
Let us first review Sikivies arguments for the ex- istence of the omion. When a global symmetry group G is spontaneously broken to a subgroup H, global cosmic strings can form if n, (G/H) is non-trivial [ 31.
Such strings will form in the early universe as the temperature drops past the critical temperature T, for this phase transition ( T, - v, where v is the vacuum expectation value of the Higgs field that breaks G to H). These strings may still be present today if no in- flation occurred after string formation. Harari and Sikivie [ 41 have shown that these global strings be- have quite differently from gauge strings. In particu- lar, bent strings relax by the radiation of NGBs whose spectrum behaves as k-.
Next, Sikivie assumes that the omions couple to photons via a term of the form
Here, @ denotes the omion field, N= Tr (QQ: ), where Q, is the electric charge operator and Q is the gener- ator of G of which $ is the NGB. The trace is taken over all fermions in the theory (assumed to all be left- handed). This coupling is required so that the omion can be converted to a photon in the presence of an external magnetic field (just as for the axion) [ 51. If the external magnetic field is assumed to be trans- verse to the omion momentum direction then one can
256 0370-2693/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 215, number 2 PHYSICS LETTERS B 15 December 1988
compute the differential y-ray flux that should be ob- served due to omions radiated from global strings being converted into photons [ 1 ]:
dpo~ 2.1 10-39(Jansky/ster) d~2 dk ~
oo .o (Gauss cm) 2 km/s Mpc] "
Here, L is the extent of the magnetic field in the di- rection of the omion momentum, l is the coherence length of the conversion process and the omion is as- sumed to be massles (i.e. a true Goldstone boson). An important feature of the photons emitted by the omion conversion process is that they are 100% po- larized in the direction of the external magnetic field.
Given these results, Sikivie argues that observa- tions of the flux of diffuse y-rays and polarization maps of extended radio sources imply that the omion must be present. The spectrum of y-rays in the 100 MeV- 106 GeV range seems to be proportional to k - 1 as in eq. ( 1 ), while features of the polarization maps that are hard to explain by use of standard emission mechanisms (such as synchrotron radiation) can be explained by a mixture of synchrotron emission and omion conversion processes. These data also indi- cate that the free parameter N must lie in the range 103-106. We shall see later that this large value gives rise to severe difficulties in constructing perturba- tively grand unified omion models.
If Sikivie's analysis is correct, we can infer a great deal about a grand unified model building just from the existence of the omion. We first claim that the omion must be distinct from the axion. There are a couple of ways to see this. One way is to note that the omion must be massless if it is to give rise to the re- quired spectrum [ 5 ]. The point is that if the omion were the axion, the k - ~ spectrum would turn into a spectrum of the form
1 (k2_m2)l/2 3)
where ma is the axion mass. For the omion hypothesis to be correct Sikivie
states that the k - 1 spectrum must extend down to 10 8 Hz. This corresponds to an axion mass of 10-7 eV
which would correspond to an axion decay constant of 1014 GeV. This is ruled out by the cosmological bounds onf~ [ 6 ].
The other way (which is somewhat model depen- dent) uses constraints on axion models coming from the existence of axion domain walls . QCD in- stanton effects give leave intact a Z(N' ) subgroup of the Peccei-Quinn U ( 1 ) that is then broken sponta- neously giving rise to domain walls. These walls are cosmologically unacceptable [ 3 ] and must be gotten rid of before they dominate the energy density of the universe. There are essentially three ways in which this can be done. The first is to use inflation to push the walls outside of our horizon. The others are to use the Lazarides-Shafi [ 8 ] or the Georgi-Wise [9 ] mechanisms. If inflation were to be used, the strings (at the intersections of the domain walls) which ra- diate the omions would be eliminated and hence there would be no evidence of them today. The Georgi- Wise mechanism requires that the parameter N' = (2n/To)Nbe equal to 1 or 2. Here To is the pe- riodicity of the 0 parameter of QCD and depends on the second index of the various fermion representa- tions . The point is that in all known models To=2n(O( 1 ) ) which for large Nimplies N' must be quite large. In the Lazarides-Shafi mechanism, the action of Z(N') is arranged to have the same effect as that of the center of the gauge group. The centers of simple gauge groups are small except for SU (n) groups of large dimension. Since N must be quite large in order to be compatible with the data described above it would be very hard, if not impossible to im- plent the afore-mentioned mechanisms. Thus we conclude that the omion and the axion must be dif- ferent particles.
We may rephrase this conclusion in group theoret- ical terms. Since the Peccei-Quinn U ( 1 ) PO must have a color anomaly, and since this is what gives rise to the axion mass (through color instanton effects), we must demand that the charge Q, of which the omion is the NGB, be anomaly free with respect to SU (3)c. More generally, if SU(3)c is contained in a larger simple group, then we must demand that Tr(QT]) vanish for all generators T~ of this group.
However, if the omion is to have the required cou- pling to two photons [eq. ( 1 ) ] then Q must have a QED anomaly, i.e. N=Tr(QQ~) must not vanish. If U ( 1 ) EM were unified with SU (3) c in a simple group
Volume 215, number 2 PHYSICS LETTERS B 15 December 1988
then these conditions would be incompatible since Q~ would be a linear combination of generators of the simple group. Hence we conclude that the only uni- fication schemes compatible with the existence of the omion are those in which at least some part of the electric charge operator lies outside the group con- taining SU (3)c. This then means that all of the stan- dard grand unified models are incompatible with the existence of the omion! In particular, superstring models based on compactification on Calabi-Yau spaces [ 11 ] are ruled out if the omion exists.
One interesting consequence of this requirement on omion models is that the magnetic monopole prob- lem of most grand unified models [ 12 ] does not exist in omion models. This is for the same reason mag- netic monopoles do not appear when the standard model undergoes spontaneous symmetry breakdown i.e. the U( 1 )'s containing EM are just reshuffled after symmetry breaking.
Can grand unified models consistent with the ex- istence of the omion be built? Our conditions require that a consistent grand unified model have a sym- metry group of the form: (G~ G2 )~oca~ Gglobal, where SU(3)c c G~ and the generator of U( 1 )EM is a linear combination Q~,=aT~ +bTz, where T~, T2 are in the Cartan subalgebras of G~, G2 respectively and b is non-zero. Furthermore, we must have that N= Tr(QQ~)=b2Tr(QT2)#O[Tr(QT~)=O as men- tioned above ].
However, the toughest constraint to satisfy is that N~ 103-106. In the normalization of eq. ( 1 ) the Q charges must be integers with no common factor. In most models these charges are of order unity, so that if the non-abelian gauge group fermion representa- tions have relatively small dimension, then N will be too small to agree with observations. One potential solution is to use a large (reducible or irreducible) fermion representation which contributes signifi- cantly to N and acquires a large mass so as to decou- pie from low energy physics. There are some potential problems with this approach, however. Since the sca- lars giving these fermions their large mass will break the global charge Q on acquiring a vacuum value, constraints on the energy density of omions require that this vacuum value be less than 1017 GeV [ 1 ]. Other constraints may make this bound even smaller .
Let us see how far one can go in constructing a more
or less reasonable grand unified omion model (though we do not claim it will be aesthetically pleasing! ). In order not to have outlandishly large values of the global charges carried by the fermions, we will take the gauge group to be of the form (GtXG2X U(1 )ehoca~U(1)gJobab where the Gi are non-abe- lian. This will allow us to be up large values of N due to the dimensionality of the fermion representations. To be specific, we will take both of the Gi's to be SO (10). We will decouple the heavy and light sectors from each other so as to make life a bit easier. The standard model is taken to be contained in one of the SO ( 10 )'s only. The light fermions will be placed in Nf copies of ( 16, 1; 0, 0), where the last two entries are the local and global U ( 1 ) charges respectively. The fermions contributing to Nare (210, 210; 1, 1 ), (210, 210; -1 , 1), (210, 210; 0, -2 ) . It can be checked that all local anomalies are canceled by this assignment of charges. Furthermore, it is also easy to check that the global U( 1 ) has no QCD anomaly [in fact it has no anomaly with respect to either SO (10) ]. In order to calculate N we must specify the electric charge operator Qy. Since the standard fermion fam- ilies are neutral with respect to U(1)O taking the electric charge to be
oy=a~ +aQ, (4) where Ql is the standard electric charge operator con- tained in SO (10), and a is an arbitrary real number, will have no effect on low energy physics. We may now compute N and find that it is given by N= 2a z (210) 2 ~ 8a2 X 10 4. I f a z ~ 2, N can be as large as 105. An appropriate Higgs sector can be constructed that gives the new fermions a large mass, and breaks the required symmetries. Note that the other SO ( I 0) need not be broken at all and may give rise to confin- ing forces acting only on the new fermions.
We see then that the large number of fermions re- quired in omion models can be repackaged in a rea- sonable way. There is, however, a problem with this model, and in fact, all omion models that achieve large N via the introduction of many new fermions. These models cannot be perturbatively unified! The problem is that at some point we will have (in the above model) 630 fermions contributing to the fl functions involved! This will generally overwhelm the gauge contribution making it so that the Landau sin- gularity is reached before unification. A possible way
Volume 215, number 2 PHYSICS LETTERS B 15 December 1988
out of this problem may be provided by the use of the non-perturbative unification framework of Maiani, Parisi, and Petronzio . In this formalism the boundary conditions on the coupling constants are that they become strong at some scale. Then the low energy couplings are near the infra-red stable fixed point, and this knowledge allows some statements to be made about the various mass scales of the problem.
We mentioned above that there exist superstring models with the feature that electromagnetism and color are not contained in the same simple group. Can these models have the required global symmetry? Re- cent work by Banks and Dixon [ 14 ] suggests that the answer to this question is no. They have arguments showing that there are no continuous global symme- tries in the effective field theories for the classical su- perstring vacua. There are some exceptions to these arguments, however, and we are not sure at present whether the omion symmetry falls into one of these exceptions.
In conclusion, we see that the existence of the om- ion has serious consequences on grand unified model building. All of the standard models based on simple groups must be discarded in order to be consistent with the properties of the omion. Furthermore, the constraints on the value of N required from observa- tions certainly allow even less leeway than usual in model building. Although we have constructed a toy omion model it is certainly not aesthetically pleasing and also suffers from problems with perturbative unification. It remains to be seen whether any fully consistent grand unified model can be constructed.
We would like to thank Professor Tom Kephart for a useful conversation. The work of R.H. was sup- ported in part by the DOE while that of D.B.R. was supported by the US Department of Energy under Contract No. DE-AC02-76ER00881,
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