PHYSICAL REVIE%' D VOLUME 33, NUMBER 5 1 MARCH 1986 Skyi=itiion phenomenology in the standard electroweak model Minot G. Clements' and S.-H. Henry Tye Newman Laboratory ofNuclear Studies, Cornell University, Ithaca, New York 14853 {Received 21 October 1985} If the Higgs-boson mass in the standard electroweak model is large, the Higgs sector has a Skyr- mionlike soliton solution. Its properties are studied here. Also resonances in the 8'8' channel are studied in a phenomenological approach. I. INTRODUCTION calculate its mass. To check the approximation in our phenomenological analysis we apply this method to the Most aspects of the standard electroweak model seem Skyrme model for the strong interactions, in which the to be well understood both theoretically and experimental- corresponding state is expected to be the g and/or the p ly, with the notable exception of the Higgs sector. Even meson. The width of this resonance is estimated using in the simplest version with one Higgs doublet, the mass unitarity arguments. We briefly comment, in Sec. IV, on MH can vary over orders of magnitude without affecting the difference between the strongly interacting Higgs the agreement of the model with experiment in the "low- model and the technicolor model and on the limits of va- energy" regime explored up until now. If the Higgs-boson lidity of our approach. mass is sufficiently large, i.e., MH & 1 TeV, the standard model acquires a strongly interacting scalar sector and II. PROPERTIES OF SOLITONS nonperturbative methods are required to elucidate the ' consequences. lt has been observed that in the large-MH Symmetry breaking in the minimal Glashow- limit the standard model is equivalent to a gauged non- Weinberg-Salam model occurs through the intervention of linear o model augmented by higher-derivative terms with a complex scalar isodoublet of fields P which it is con- coefficients proportional to ln(MH) (Ref. 2). Thus the venient to parametrize as Higgs sector is similar to the Skyrme modeli scaled up by a factor of 2500. This naturally suggests the existence of p(x)U(x) =P +is P, (2.1) a heavy quasistable soliton analogous to the Skyrmion, as where U is an SU(2) matrix field and p is a scalar field, in proposed by Gipson and Tze. The properties of such a terms of which the scalar part of the Lagrangian is soliton have bo.m discussed in the literature. ~ Its mass 2 is estimated to be a few TeV, well within the range of the —— L =(tempt)~+ Tr(P'Utt)„U) —A, (p — ) + Superconducting Super Collider (SSC). f Here we study the phenomenological properties of the (2 2) Skyrmion in electroweak interactions. To derive the pro- duction rate and detection signals for these solitons, it is Note that L is invariant under SU(2)LXSU(2)a transfor- necessary to estimate their coupling to the "pion" (longi- mations. When p is shifted by its vacuum expectation f it tudinal gauge boson) field. The method for doing so was betximes the physical Higgs particle and, through the in- developed by Adkins, Nappi, and Witten, who applied it teraction terms which are not shown explicitly here, the to the study of Skyrmions as baryons. In the electroweak Wand Z gauge bosons are given masses. Although in the model there is no a priori reason why the soliton should unitary gauge the kinetic term for U combines with the be a fermion rather than a boson. For the case that it is a mass term for the gauge bosons, in this paper we use the boson, we extend their method for calculating the cou- Landau gauge so that the unphysical Higgs or longitudi- pling. The calculation of couplings, masses, and widths nal gauge-boson modes described by U do not mix with for the soliton sector is presented in Sec. II. All results the transverse gauge bosons and may be regarded as in- are expressed as a function of Mtt, or equivalently, the dependently propagating particles. parameter e introduced by Skyrme, and the weak- In the limit of large 1(, it is well known that for energies interaction scale f=250 GeV. E &&2fi/k, the theory behaves like a nonlinear o model A more readily acemsible signal may be resonances in (NLSM). As quantum field theories NLSM's are non- the W+ W, Z Z, and 8E channels, which we consid- renormalizable; loop corrections induce interactions with er in Sec. III. Our approach is to regard these resonances four or more derivatives and these new terms have diver- as soliton-antisoliton bound states. Unfortunately, such gent coefficients. Since the corresponding linear o model states are very difficult to analyze and some simplifying (LSM) is renormalizable the divergent terms in the NLSM approximations are required to make any progress. The correspond to finite quantum corrections in the LSM major approximation made here is to treat one of the two which diverge as the Higgs-boson mass approaches infini- bodies in the bound state, say, the soliton, as an elementa- ty; this has been explicitly verified to one loop by Appel- ' ry particle. Then we can adopt the method of Ref. 9 to quist and Bernard. Thus the Higgs-boson mass can be 33 1424 SKYRMION FHENOMENOLOGY IN THE STANDARD. 1425 regarded as a cutoff whose presence makes itself known to the low-energy physics through subtraction pieces in the M=2m'~ x dx F' + J x renormalized Lagrangian. For the semiclassical analysis undertaken here our starting point will be the one-loop regularized Lagrangian (2.8) L = Tr((FU B„U) with x=efr, s=sinF(r), and F'=dF/dx. Numerical minimization of (2.8) gives M=117f/e to be compared + ', (T.(a„UUt, a„UUt) with M=73f/e in the case where Lq is absent. This is 32e fairly close to the theoretical lower bound 3Tr—I a„UU', a„UU'I') M&3n ~10~=93.6 (2.9) determined Gipson and Tze. The isospin-rotation col- Lo+Ls+L (2.3) by lective coordinates can be quantized as in Ref. 7 to give a tower of states with I=J. These states have masses where terms proportional to gauge couplings are not relevant for the analysis and have been The neglected. M-MI '"+" e (2.10) second term in (2.3) is just that of Skyrme's model and the ™+2 standard parametrization of the multiplying coefficient where has been used. Here the parameter f is the weak- 4s interaction scale f=250 GeV, and the parameter e is re- x'dxs' 1+F'+, =804. (2.11) lated to the Higgs-boson mass through f0 1/2 If M» 1 TeV then e & 10 and the difference between the (2.4) mass of the static Skyrmion (I=J=O} and the mass of the Skyrmion quantized as a fermion (I=J= —,') is less than 5%; in what follows we will ignore this difference. In the low-energy limit, calculation of the coupling of the soliton to longitudinal gauge bosons is straightfor- In (2.4) f is meant to represent some scale typical of the ward. %'e consider two eases: the soliton quantized as an weak interactions, although for definiteness we shall take isoscalar scalar or as an isodoublet fermion. Details of it as 250 GeV. Hence (2.4) is reliable only for large the electroweak model will determine which quantization ab- MH/f. Note that the third term L„ in (2.3), which is is appropriate; i.e., the soliton quantization must give sent in the Skyrme model, can destroy the positive defin- zero for the Written global SU(2) index for the electroweak iteness of the Hamiltonian if the field derivatives are model. If the soliton is a fermion, the simplest chirally large. However in this case, higher-derivative corrections invariant interaction is to (2.3) would become important and should restore the positive definiteness. (This requires the inclusion of Li =gf(%'I, U%'it +'pit U pL ) . (2.12) higher-order loop contributions. ) The approximate L in ' Writing U=e ' ' and the nonrelativistic form (2.3) should be sufficient for low-energy phenomenology, using for it is easy to find the m field sourced the static where low energy means on the order of a few TeV or less. 4 by fermion. with To obtain the static solution which has maximal sym- Comparison the soliton field, which at large distances from the soliton center has the form metry we follow the standard ansatz by requiring the iso- spin and spin vectors to satisfy T+J=O so that U may be . 8 8 U = 1+i—r.x— (2.13) written r2 2r' U iF(r)x v' (2.5) gives In terms the g= MfB . (2.14) of low-energy degrees of freedom 3 parametrized U, the topological current is by Here the numerical solution for U determines B=6.7/e f . For the soliton pair production cross sec- Tr(a„UU'a UU'a. UU') (2.6} &4~' tion, 2 cr(Wr Wr ~SS)= g1vlng M 1 The chirally invariant interaction of lowest dimension for = —— — o" (2.7) Q (F(r) sin[2F(r)] ~ I the scalar soliton is Lr —a4 4Tr(e)„U cPU) (2.15) for the ansatz (2.5). Here we just consider the case with F(0)=ir and F( ao ) =0. From (2.3) the mass of the static with 4 the field operator for the soliton, and a procedure soliton is similar to that used in the previous case gives MINOT G. CLEMENTS AND S.-H.