Skyrmion Phenomenology in the Standard Electroweak Model

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 production 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 isospin 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. HENRY TYE 33

TABLE I. Properties of the Skyrmionlike soliton as a func- in defining what is meant by "pinning the soliton and an- tion of e, or MH as defined in (2.4). The mass Mq is given by tisoliton down. " It is convenient to define three coordi- (2.8). The coupling constant g is analogous to g~~~ in hadron nate systems: one centered on the soliton and denoted by physics and is given by (2.14). The width is only an order-of- the subscript 5, one centered on the antisoliton and denot- magnitude estimate. ed by the subscript S, and a global coordinate system. MH Now consider a widely separated soliton-antisoliton pair. (TeV) I (width) Near the center of the soliton the field is T+J symmetric and may be written 20 0.34 1.5 0.8 200 GeV iF{r )x 15 0.42 1.9 1.9 60 GeV U=e 10 0.82 2.9 6.6 1 GeV 8 1.6 3.7 12.8 10 MeV The point at which U= —1 in global coordinates is de- 6 6.7 4.9 30 1 keV fined as the position af the soliton (or antisoliton), and the 5 28.5 5.8 53 1 eV orientation of the antisoliton with respect to the soliton is represented by a vector parametrizing the rotation of the antisoliton necessary to annihilate the soliton if the two were superposed. Then the configuration reached by slowly moving the soliton [constraint U(xi)= —1 and 4m fg U fixed) toward the antisoliton [constraint (2.16) Vs ~ 3 „0 U(xz)= —1 and V~U ~~ 0 fixed] along the line joining The numerical results are summarized in Table I. them until the separation is s = — can be said to ~ xi x2 ~ In the standard electroweak model, p is not fixed to be a soliton pinned at a distance s from an antisoliton and p(x) =f, in contrast with the Skyrme model. At very high will in general depend on the initial orientation. Of energies the Higgs mode is excited and in writing the ex- course, for sufficiently small s such a configuration would pression for the topalogical current it is necessary to ac- deviate considerably from a solution to the equation of count for excursions of the field away from the surface motion since the energy is na longer minimized by con- parametrized by the matrix U. In terms of the usual field centrating spatial dependence of the fields in two localized variables the topological current given in (2.6) is not con- regions. Assuming that this objection could be disregard- served when P fluctuates to zero. This means that soli- ed, a bound state would appear as a local minimum of the tons can decay; by using the standard semiclassical esti- energy as a function of separatian and orientation, and the mate for tunneling amplitudes the width may be estimat- low-lying excitations could be systematically treated in the ed. Here we shall simply borrow the method of Refs. 5 harmonic approximation. Recent numerical wark along and 9 to provide an order-of-magnitude estimate of the these lines has been published by other authors but width. The numerical values for the width shown in should be considered only a first step on the program Table I may not be reliable. But two of its key features described above, since in these papers the fields were not are expected to be correct: (1) the width is substantially allowed to relax in calculating the energy of a configura- smaller than that of an ordinary resonance, and (2) the tion. width decreases rapidly as the mass of the soliton in- In the second approach we proceed by using a simplify- creases. ing approximation which cannot be rigorously justified. The approximation to be taken is that one of the soliton- III. RESONANCES IN THE WS'CHANNEL antisoliton pair is treated as a point particle interacting with the field of the other soliton and described by a The large mass of the soliton (typically M 1 TeV) to- & quantum-mechanical wave function. Fortunately this gether with the fact that it carries topological charge, re- phenomenological approach can be tested by applying it quiring pair production, would make it a rare find even at to hadron physics, where the analagous states are the ri the SSC and puts it out of reach of lesser accelerators. As and mesons. The comparison of the calculated mass far as phenomenology is concerned, it is more useful to p and coupling to the known experimental values gives us consider low-energy manifestations of the strongly in- some encouragement as to the reliability of the method teracting Higgs sector. In several papers"" models adopted. with strongly coupled Higgs particles are analyzed using There is a choice as to how the soliton may be quan- techniques (low-energy theorems, analyticity and unitarity tized; the simplest case is to take it as a fermion isodoub- of the S matrix, etc.) adopted from studies of m-n scatter- let 4' which obeys the Dirac equation ing, with a general conclusion that there should be a prominent resonance, analogous to the q or the p meson [ a.B+gfp(a+—iysa"v )]+=E%, (3.2) in the strong interactions, in the 1—2-TeV range. In this discussion we advocate modeling such a resonance as a where sr+in"v= U with U the soliton field. The as- soliton-antisoliton bound state. sumption I+J=o for ql gives the form There are at least two approaches to the investigation of the posited bound state. One, which we will describe but not actually use, would involve pinning the soliton and (33) antisoliton down at some separation and calculating the static the configuration. Some care is necessary where acts on rows and isospin on columns of U energy of spin ~ ). 33 SKYRMION PHENOMENOLOGY IN THE STANDARD. . .

' In terms of the scaled variables y =gfr, e=E/gf, (3.2}be- lowing Hung and Thacker" and Brown and Goble, we comes will apply the same arguments to longitudinal gauge- boson scattering to obtain a crude estimate of the width of 1 —t} — —s the bound state. y H H J (3.4) Using (2.3) the tree-level scattering amplitude for t} — —s —C J, H~'is y Tkl, ij TChiral + TSkyrme + TAnti with s=sin[F(y)], c=cos[F(y)], It =H/y, and j =J/y. The antisoliton profile Fis controlled by the Hamiltonian T~„., =,(s5'15"'+t5"5'+u5"5"), 2$ (3.6) H=2tr~ y dy F'2+ O y2 g f Tskyrme=——— [(2s' —t2 —u 2)5"5"' 8 2f4 2 4s2 Ss 3F' + F' + +(2t2 2 2)5ik5it 2e +(2u 2 s2 t2)5il5l'k] y c H — —2sHJ (3.5) . TAnti [(t2+u2)5i15kl+(s2+u2)5ik5it 8 2f4 The equation of motion derived from this Hamiltonian and the Dirac equation were solved simultaneously via + (s 2+ t 2 )5il5ik] power series for small y and numerically for larger y us- This amplitude may be written as a sum over partial ing the multiple shooting method. Results are given in Table II for several values of e(Mtt). In this table Hs waves and H~;,„refer to the energy of the scalar field alone. —16m ttq(s, 1)PJ(cos8)Pkt ', (3.7) Tkl,z g t)(2J+ i, Also, we follow Ref. 7 and use a value f =65 MeV I,J which fits the Skyrmion mass to the proton mass, rather where P' ' projects onto isospin I. Here we analyze in de- than taking =93 MeV as determined experimentally. f tail just the I=J=1 component, so If we choose f =93 MeV, the resonance mass is around 700 MeV. Note that a coupling g =M „, /f was used p i t (5ik5il 5ily k)' (3.8) rather than the value calculated from (2.13}which is considerably larger for large MH. The reason for this is that and the estimate (2.13) depends only on the long-distance tail k (3.9) of the soliton while from (2.6) and (2.12) it is apparent 12irf 2 that the mass is primarily dependent on the short-distance with k the center-of-mass-frame momentum. Note that structure of the soliton. Since the fermion (representing the contributions to tii from Tsk~~e and TA„„cancel. the soliton) spends much of its time in a region where the For elastic unitarity to hold field from the antisoliton is large, it is clearly preferable ' to have its coupling to this field representative of the Imttg ——, (3.10) g ~ ttq ~ dominant contribution to the energy of the real soliton- antisoliton system. which gives a parametrization for ttz in terms of (real) The quantum numbers of this bound state are indeter- phase shifts 5tJ. minate, but we think it is reasonable to assume that there ttg =28 sln5tg . (3.11) is a spin-0 isosinglet and a spin-1 isotriplet corresponding to the rl and p in the strong interactions. It has been Reassured by the success of relations based on elastic uni- shown that for the p, unitarization of current-algebra re- tarity at describing mm scattering phenomenology above sults for n nscattering g-ive a relation between the mass the inelastic threshold, we assume this extended domain and the width in good agreement with experiment. ' Fol- of validity holds also for longitudinal gauge-boson scatter-

TABLE II. Properties of the antisoliton-particle resonance. Here we treat the soliton as a fermion f=250 GeV. For the strong-interaction case, we choose f =65 MeV, corresponding to that of Ref. 7. Mass (TeV) %'idth (GeV)

1 9.2 9.0 —0.13 0.99 1.95 390 5 6.3 13.2 —0.45 0.87 1.36 135 10 5.7 14.6 —0.52 0.84 1.15 81 20 5.2 16.0 —0.57 0.81 0.95 46 Strong interactions (f =65 MeV) g M~ If e Hn;, „Igf Mass (MeV) Vridth (MeV~ —0.63 570 1428 MINOT G. CLEMENTS AND S.-H. HENRY TYE 33

ing. To find a unitarized version of (3.9), note that (3.10} physics. On the other hand, in the electroweak interac- and (3.11}imply tions the ultimate reality behind the Higgs sector is still a

—1 mystery. It is certainly possible that the standard model, Imtu (3.12) in which the scalar fields are elementary, is literally correct. If this is the case, and the scalar fields are strongly coupled, the application of Skyrme dynamics l rIJ +gIJ($) (3.13) should be more suitable as well as more accurate for the 2 electroweak model than for hadron physics. Here the masses of the co-like and the p-like resonances are essen- with gIJ meromorphic. This is related to the low-energy In the version, the ui-like reso- expression (3.9) by matching at threshold, which fixes the tially degenerate. Skyrme nance, if it exists, is expected to be heavier than the p-like scattering length in the effective range expression of g resonance. 1 Another possibility is that there is some technicolor-like g11 — + ~11+ ''' (3.14) 2 2 theory operating behind the scene and the Higgs scalars play the role of pions in QCD. In this case one should ex- yielding the unitarized amplitude pect a richer spectrum, with light technipions and reso- k nances corresponding to the ~, etc., in hadronic physics. ' (3.15) Given that the Higgs sector is strongly coupled, once suf- 12m + —,riik —(i/2)k f ficient energy is attained to start seeing resonances it If r» (0 there is a resonance at should be easy for experiment to tell the difference between the minimal electroweak model and versions with — 96n mg —— f (3.16) composite Higgs scalars. 11 There are arguments, '4 based on the supposed triviality ' which has a width of A4} and lim„„O(n} theories, that the Higgs particle must be light in the standard electroweak model. Howev- 3 PlR er, we have seen that if the Higgs boson is heavy the (3.17) 96m Higgs sector supports nonperturbative structure such as f solitons and resonances. It is unclear how valid quantita- For bound-state masses of the order of those displayed in tively the triviality arguments are in the face of this struc- Table I, the width is between 50 and 400 GeV. Similar ture and we feel that they should not provide an excuse analysis for the isoscalar-scalar resonance gives for ignoring the possibility of strongly coupled scalars and the interesting physics they would provide. Also, a PPl —— I 00 (3.18) strongly interacting Higgs sector in a supersymmetric ver- 32m f sion of the electroweak model is a distinct and interesting possibility. IV. GENERAL REMARKS ACKNOWLEDGMENT It is generally accepted that the strong interactions are completely and correctly described by QCD. The Skyrme This research was supported by the National Science model is just a low-energy effective theory for hadron Foundation.

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