Rotational and vibrational excitations of weak skyrmions

Volume 236, number 1 PHYSICS LETTERS B 8 February 1990

ROTATIONAL AND VIBRATIONAL EXCITATIONS OF WEAK SKYRMIONS

Yves BRIHAYE a, Burkhard KLEIHAUS b and Jutta KUNZ c,b a Physique Thborique et Mathdmatique, Universitb de l'Etat ~ Mons, B-7000 Mons, Belgium b FB Physik, Universitiit Oldenburg, Postfach 2503, D-2900 Oldenburg, FRG c NIKHEF-K, P.O. Box 41882, NL-1009 DB Amsterdam, The Netherlands

Received 11 November 1989

We perform a semiclassical quantization of weak skyrmions, classically stable solutions present in effective models of the weak interactions. In particular, we discuss the collective quantization of the zero modes associated with spatial and weak "vector" isospin rotations. We determine the collective rotational and vibrational excitation energies of weak skyrmions.

I. Introduction

By now perturbative calculations performed in the Weinberg-Salam model are in agreement with a con- siderable amount of experimental results. However, the interpretation of the Higgs sector remains prob- lematic, because the predicted Higgs boson has not yet been observed. It appears quite possible, that the Weinberg-Salam model only represents a low-energy effective model of the weak interactions, and at higher energies, say in the TeV region, a rich region of new weak interaction phenomena waits to be unravelled by future generations of accelerators. Here we focus on possible non-perturbative phenomena present in this energy range of the weak interactions, and consider in particular weak skyrmions.

The Skyrme model is an effective field theory of pions based on the non-linear a-model supplemented by a higher order stabilization term, the Skyrme term [ 1 ], which provides a rather satisfactory description of hadronic physics at low energies [2,3]. The theo- retical foundation for the Skyrme model is the 1 ~No

expansion [4], which suggests that baryons emerge as solitons in an effective field theory of mesons [ 5 ]. Similarly, the weak interactions mediated by the massive vector bosons W + and Z ° have been viewed as an effective "low energy" field theory, with a technicolour theory as the underlying non-abelian gauge field theory [6]. The 1/Nxc expansion then suggests, that also technibaryons arise as solitons of a

corresponding effective low energy field theory. Weak skyrmions or techni-skyrmions have previ-

ously been considered and interpreted as the technibaryons of an underlying technicolour theory [ 7-9 ]. Ambj~rn and Rubakov [8] have constructed the classical techni-skyrmion solution in a Skyrme model with chiral SU (2) L × SU (2) R symmetry and gauged SU(2)L. In this model the chiral matrix represents the techni-pion fields and the gauge bosons the massive vector fields W -+ and Z °. (The electromagnetic field has been neglected, implying degenerate masses of the three vector bosons.) Eilam, Klabucar and Stern [10,11] have studied weak skyrmions in the same effective model of the weak interactions, but obtained from another point of view. They have considered the Weinberg-Salam model in the limit of an infinitely heavy Higgs boson, where the model reduces to a gauged non-linear a-model, and added a gauged Skyrme term, representing higher order corrections arising in the quantum theory, much in the spirit of Gipson and Tze [ 12]. We have recently dis- covered a whole sequence of classical solutions of this effective model of the weak interactions [ 13-15 ]. The weak skyrmion is the energetically lowest solution of this sequence, and the higher solutions of this sequence constitute radial excitations of the weak skyrmion.

Eilam, Klabucar and Stern have investigated the dependence of the classical solution on the coupling strength of the higher order term and demonstrated

54 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )


that there exist two branches of classical solutions for the weak skyrmion, which merge and end in a cusp at a critical, maximal value of this coupling and of the mass of the weak skyrmion. Only the lower branch is classically stable [11,14]. Like the weak skyrmion also the higher radially excited solutions have two branches merging and ending in a cusp. Unfortu- nately, however, all higher solutions are classically unstable and therefore not eligible to the standard semi-classical quantization procedure.

In this paper we determine the quantum spectrum of the classically stable weak skyrmion. We apply the standard method of semiclassical quantization. In particular, we consider the zero modes associated with spatial and weak "vector" isospin rotations. We in- troduce collective variables for these zero modes, and promote them to quantum operators. We proceed analogously to the quantization of skyrmions with vector mesons [ 3 ]. There is, however, an important difference to the skyrmions of the strong interactions: a residual gauge symmetry is present for the weak skyrmions. This gauge symmetry reflects itself also in the non-vanishing time-components of the vector fields. The gauge invariance was broken in the incomplete quantization procedure of ref. [ 11 ]. We define the model and discuss the classical solutions in section 2. In section 3 we perform the collective quantization of the rotational zero modes. We investigate the breathing mode of the weak skyrmion in section 4 and give our conclusions in section 5.

2. Classical solutions

We consider a non-linear o-model with S U ( 2 ) L

x S U ( 2 ) R symmetry and gauged S U ( 2 ) L , which is supplemented by a higher order Skyrme term [8- 11,14,15]

L = - ~ Tr(Fu~FU" ) + ½ (h ) 2 Tr(Du U*DuU)

+ 3~e2 Tr [ D, UU*, D,, UU* ] 2 (2.1)

with

D¢, U= (0,, -i½gAu) U, (2.2a)

and

F~ = 0,,A~ - a~A~ -i½g[&, A~I. (2.2b)

The chiral matrix U may be thought of either as representing the Higgs doublet with its length frozen at the vacuum expectation value ( h ) [ 10-12 ], or as representing the techni-pions [ 8,9 ]. In the latter case ( h ) should be replaced by F~/x/~, where F~ is the techni-pion decay constant.

To construct classical solutions of the model (2.1) we choose static ans~itze with generalized spherical symmetry

U=cos 0+i(•..f) sin 0, (2.3a)

-½gAi= a - ½ T~ + fl_ T~ + d T3, (2.3b) r r r

A o = 0 , (2.3c)

here 0, a , fl and 6 are functions of the radial coordi- nate r, the matrices T~ are defined as

T] -- (-~XT)i, (2.4a)

TZi = ri - ( ~'.r ),ri , (2.4b)

T 3 = (~'.~)~,, (2.4c)

where t h e , denote the Pauli matrices and 2,=x Jr. In terms of these functions the classical energy

functional reads

E=87r(h) idp[(ol '+2f la)2+(f l '_2ola)2 g o

2 +p2(O, +a)~+ Y (a2+f12_ ~)2

( ' )] +F l+ 3--2-~Sp2[F+4p2(O'+a)2 ] , (2.5)

where

F=2[(a-½+sin20)2+(f l+sinOcosO)2], (2.6a)

d r g2 a = - , p = g ( h ) ~ = - - (2.6b, c ,d )

p 2 ' e 2 '

and the prime denotes the derivative with respect to p.

An important property of this expression is the residual gauge symmetry present in the energy density. Indeed, it is invariant under the following transformation:

O--,O-x(p) (2.7a)

55


a ~ a + Z ' (p) (2.7b)

~o~o9+ 2Z(p) (2.7c)

ol= R cos oJ, fl=RsinoJ. (2.7d)

Along with refs. [ 10,1 1,14,15 ] we choose the gauge /~=ro=0.

Regularity of the solutions (see eqs. (2 .3) ) and fi- niteness of the energy require certain boundary conditions [ 8-1 1,14,15 ]. Here we consider only

0 ( 0 ) = ½ [ 1 + ( - - 1 )"];,r,

and

a(O)=a(oo)= ~,

0(oe )=~r , (2.8a)

(2.8b)

where n counts the number of zeros of the function 0'. The weak skyrmion has n = 1, while its radial excitations have n > 1 [ 14 ].

There are two branches of classical solutions for the weak skyrmion. The lower branch is classically stable, while the upper branch has one unstable mode [ 14 ]. In fig. 1 we present the classical solution for the weak skyrmion for e = 0.38, which is the critical value of e at the cusp. The energy of this classical solution is 13.80 TeV. Note that we use the vector boson mass Mw = 83 GeV, the coupling constant g = 0.67 and the relation Mw = g ( h )/xf22 (Mw = ~gF,~).

3.2

2 . 8

2 . 4

2.0

1.6

1.2

0.8

0.4

0.0 0.0

/

@

I

1 . 0

P 2 . 0

Fig. 1. The classical radial functions of lhe weak skyrmion are shown for the parameter ~=0.38.

3. Collective quantization

Classical solutions which represent local minima of the energy functional, apart from zero modes associated with continuous symmetries broken by the solution, may be quantized by standard techniques to obtain the corresponding physical particle spectrum. To quantize the classical weak skyrmion along the lower, stable branch we proceed analogously to the quantization of skyrmions in the strong interactions [ 1-3 ]. The zero modes are identified, and the variables associated with these zero modes are pro- moted to dynamical variables, the collective variables. Then the standard commutators are imposed for the collective coordinates and their canonically conjugate momenta.

The relevant zero modes of the weak skyrmion are associated with spatial rotations and with weak "vector" isospin rotations. Due to the special choice of the classical ans~itze, i.e. the generalized spherical symmetry, both are related. We parametrize the collective variables by the SU(2 ) matrix V(t) and ne- glect - consistent with the adiabatic approximation and the 1/NTc expansion - terms of higher order in time derivatives. This leads to the time dependent ans~itze

U(xi, t)= V(t)UC(xi)V(t) t , (3. la)

Ak(xi, t) = V( t)A~(&) V( t)* ( 3. lb)

(the superscript c denotes the classical ans~itze (2.3) ) and

-½gAo(xi, t)=dT~+(ff-1)T2+ST3. (3.1c)

with the matrices T a (see eq. (2 .4))

T" = K~ T ~, ( 3.2a )

and the isovector K

iKjrj= Vt(/. (3.2b)

Note that the time component of the gauge field is no longer zero. This is essential to insure invariance of the lagrangian density under the residual gauge transformation (2.7). Under this transformation, the induced new functions c~ and fi transform in the same way as c~ andfl, while the induced function 6 is invariant. Substituting eq. (3.1) into the total lagrangian we obtain

56


L = - ½a[ UC, Ay, Ao] Tr( V* f / )2 -E[ U% A~] ,

(3.3)

with the moment of inertia oo

a [ U c,A y,Ao]= 32z r ~ dp{ (pa ' + 2 f i 6 ) 2 3g3(h)

0

+ (pfl' - 2 a 6 ) 2 + ~ (pS ' )2+p2(2 W2+62 )

+ [ ~ ( ~ + 1 ) - }/7] 2 + [ /~ (5+ 1 ) + ½a] 2

+ [½(J+1)-I412+I~

+2~[ V262 + WZ(t~+pO ' )2

+ ( W. V)2+ ( lUla2 V)2I} (3.4)

and

W= ( a - s i n 20, f i - c o s 20) , (3.5a)

V = ( a - ½ cos 20, fl+ ½ sin 20) , (3.5b)

13 =Ott~-Fflff, I4=ol f f - - f la . (3.5c, d)

By minimizing the functional a[ U c, A~, Ao], eq. (3.4), with respect to the new induced functions cL ffand 5we obtain a set of linear differential equations for these functions. The equations are subject to the boundary conditions

a(0) =0, a ( ~ ) = 0 ,

fi' (0)=0, f i ( ~ ) = l ,

P ( 0 ) = 0 , 5 ( ~ ) = 0 , (3.6a)

and the functions ffand 5further satisfy the relation

f i ( 0 ) - 5 ( 0 ) = 1. (3.6b)

For the weak skyrmion we observe that for small values o f t the functions a , fland 5deviate very little from their trivial values corresponding to Ao=0, i.e. a = 0, fi= 1 and 5= 0. For larger values of e these de- viations become substantial as seen in fig. 2, where the induced functions are shown for the critical value ~=0.38 at the cusp.

Standard quantization of the collective variables, parametrized by the matrix V(t), leads to the mass spectrum for the weak skyrmion, which takes the form of a rotator

2l ( /+ 1 ) Mr=E[ U c, A~] + . (3.7)

a[ U% A~,Ao]

1.0

0 .6

0 .2

- 0 . 2

- 0 . 6

- 1 . 0 i I ,

0.0 ~.0 2.0

p

Fig. 2. The induced radial functions of the weak skyrmion are shown for the parameter e = 0.38.

Table 1 This table contains data with respect to the collective quantization of the rotational zero modes of the weak skyrmion along its lower branch for several values of the parameter ~. Shown are the classical energy E ¢ in TeV, the moment of inertia a in TeV- t (see eq. (3.4)) and the rotational energy of the lowest excited mode E r°t ( l= 1 ) in TeV.

~,/4 E ¢ (TeV) a (TeV -t ) E ' ° ' ( I = 1 ) (TeV)

0.1 0.289 1.43× 10 -3 2792. 0.2 1.078 9.17 X 10 - 2 43.6 0.3 2.419 1.04× l0 ° 3.82 0.4 4.269 5.87× 10 ° 0.68 0.5 6.568 2.24X l0 t 0.18 0.6 9.196 6.73X l0 t 0.059 0.7 11.92 1.74X 102 0.023 0.785 13.98 4.20× 102 0.009

When the solitons are quantized as bosons [ 11 ], and - unlike the strong interactions - there is no anomalous term requiring the quantization of the solitons as fermions, the lowest state h a s / - - 0 and its energy equals the energy of the classical solution (up to an unknown constant shift due to zero point oscillations). The energy spacing within the collective rotational band is determined by the moment of inertia a[ UC, A~,Ao].

Several values of the classical energy EC=E[ U c, A~ ], the moment of inertia a=a[ U% A~, Ao] and the rotational energy Er°t(/= 1 ) for the weak skyrmion are shown in table 1. The values for the moment of

57


inertia (respectively the energy spacing) are of course lower (respectively higher) than those of ref. { 11 ], since those authors did not consider the induced time- components of the vector fields. In fact, the discrep- ancy is of the order of 30% for solutions close to the critical value of ~. For small values of e the results of ref. [ 11 ] almost coincide with our results, since there the vector field functions tend to their trivial values.

For intermediate values o f the parameter ~ the moment of inertia of the weak skyrmion a [ U c, A 7, Ao ] is very big. This leads to a very closely spaced rotational spectrum, where the excited quantum states of the weak skyrmion are almost degenerate with the ground state. In this region the collective quantiza- lion procedure is well justified. Close to the critical value of c the classical min imum becomes very shal- low and finally disappears at the cusp. Therefore the semi-classical quantization procedure becomes in- adequate close to the cusp. For small values of ~ on the other hand the classical energy of the weak skyrmion approaches zero like ¢~/2, while its moment o f inertia approaches zero like E 3/2, i.e. the rotational energy diverges like e -3/2. Thus for small values of E the quantum corrections to the energy exceed the classical energy by far, invalidating the collective quantization procedure for small values of e. Unlike Eiam and Stern [ 1 1 ], we therefore conclude that the model does not allow for arbitrarily light and small solitonic states.

4. Breathing modes

In the strong interactions there are various excited states of the nucleon. Besides the A-resonance, which corresponds to the first rotationally excited state in the Skyrme model, there is also the prominent Roper resonance, which has the same quantum numbers as the nucleon. In the Skyrme model the Roper resonance has been described as a vibrationally excited nucleon, performing monopole vibrations, in a so called breathing mode. Here we investigate the corresponding vibrational excitation of the weak skyrmion and determine the excitation energy.

For the monopole vibration of the weak solitons we consider the t ime-dependent scale factor 2( t ) [ 16 ]. The fields then scale according to

Uc (xi) ~ UC (2x, .) , (4. la)

A~( x,)-+AA~( 2& ) , (4.1b)

where the additional factor 2 in front of the gauge field leads to a homogeneous transformation of the co- variant derivative. In this case also, the time component of the gauge field acquires a non trivial value that we found convenient to parametrize as

,{ ½gAo(xi, t) = ~ {(2x/) ( ,~' t) . (4.2)

After inserting the scaled fields into the total lagrangian and making use of the virial theorem we obtain L(2, J.)

1 ,~2 1 ( 2 + l ) E c " L(2,'{)= 2 ( 9 + C 4 ) ~ - ~ (4.3)

\ --1 Note the similarity of the above expressions for the monopole vibration of the weak skyrmions with the corresponding expressions for the skyrmions in the strong interactions [ 161. The coefficients C2 and C4 are functionals depending on the new function { and on the classical functions a , /? and d:

C2 = 647r i g3(h ) dpp2(pO'+~)2, 0

C4~ 32a idpp212(o~'+2~'8]2 g 3 ( h ) o P /

(4.4a)

+ 2 ( f l ' - 2 ~ O ~ 2 + )2..}_ 2~(0' 7 , +S J -

(4.4b)

It is easy to check that the above expressions are indeed invariant under the gauge transformation (2.7) after one realizes that the new function ~ transforms exactly like the function d.

For small perturbations around the classical minimum at 2 = 1 the lagrangian eq. (4.3) reduces to the one of a harmonic oscillator. The vibrational excitation energy O)br, i.e. the energy of the breathing mode, is then given by

Oabr= • (4.5)

Minimizing the functional C2+C4 leads to a differential equation for the new function {. By minimiz-

58


Table 2 This table contains data with respect to the quantization of the breathing mode of the weak skyrmion along its lower branch for several values of the parameter ~. Shown are the classical energy E c in TeV, the constants 6"2 and C4 both in TeV -1 (see eq. (4.4)) and the vibrational energy of the breathing mode o~r in TeV (see eq. (4.5) ).

~/4 E c ('FEW) C2 (WeV -1 ) C4 (TeV -1 ) O~br (TeV)

0.1 0.289 0.21 × 10 -2 0.59× 10 -3 0.99× 101

0.2 1.078 0.13× 10 ° 0.40× 10 -~ 0.25)< 10 ~ 0.3 2.419 0.14)< 101 0.49)< 10 ° 0.11 )< 10 ~ 0.4 4.269 0.75 × 101 0.30× 10 ~ 0.63)< 10 ° 0.5 6.568 0.36)< 102 0.13)< 102 0.41 × 10 ° 0.6 9.196 0.71 × 102 0.43)< 102 0.28>( 10 ° 0.7 11.92 0.16>( 103 0.13× 103 0.20X 10 ° 0.785 13.98 0.29)< 103 0.39× 103 0.14× 10 °

ing this functional instead with respect to ~,=6-~, we have been able to show that the minimum is at- tained precisely for ~= 6.

Table 2 contains the excitation energy O)br of the breathing mode for the weak skyrmion for several values of ~ : g Z / e 2 along the lower branch.

5. Conclusions

The Weinberg-Salam model describes the low energy regime of the weak interactions well, but at higher energies modifications of the model may become necessary, and there may be an underlying non-abelian gauge theory, such as a technicolour theory, for which non-perturbative effects will play a central role. Starting from this reasoning and lacking the knowl- edge of the true effective weak interaction lagrangian, we have studied as a model lagrangian a Skyrme model with gauged SU(2)L, where the gauge fields describe the weak vector bosons. This effective lagrangian allows for soliton solutions, the energetically lowest solution describes the weak skyrmion or techniskyrmion. Analogy with the strong interactions then suggests (i) that even though being crude, the model predicts the properties of the weak soliton states at least with the right order of magnitude, (ii) that a rich set of quantum states, particles and reso- nances, resides in the TeV region.

To obtain the quantum spectrum of the weak skyrmions, we have semi-classically quantized the stable classical solutions, using collective coordinates. The zero modes associated with spatial and isospin rotations give rise to a rotational quantum spectrum.

Along with Eilam and Stern [ 11 ] we have quantized the solitons as bosons. (The consistency require- ments and implications of quantizing the solitons as fermions have been discussed by Hung and Tze [ 17 ]. ) The energy of the ground state, an l= 0 state then agrees with the classical energy (up to an unknown constant shift due to zero point oscillations, a higher order term in the semi-classical approximation). The excitation energy of the higher angular momentum states is determined by the moment of inertia a. The first excited state has an excitation energy of Er°t= 4/a.

Requiring (i) that the classical solution is a pronounced energetical minimum (up to zero modes) and (ii) that the quantum excitation energies are at least smaller than if not small compared with the classical energy, restricts the allowed range for the coupling strength ~ of the higher order term and therefore also restricts the predicted energy range for the weak skyrmions and their rotational excitations. From the work of Ambjorn and Rubakov [ 8 ] it fol- lows that a pronounced minimum develops only for

< 0.2, while a comparison of the classical energy and the rotational excitation energy yields the bound

> 0.01. We therefore expect the ground state energy of the weak quantum soliton roughly between 3 and 10 TeV. In most of this range the excitation energy of the first rotationally excited state is then small, rang- ing between several hundred and several tens of GeV, resulting in a closely spaced rotational band. The excitation energy of the collective vibrational state, the breathing mode, shows a weaker dependence on the coupling e. A typical frequency (in the allowed range of ~ ) is on the order of a few hundred GeV.

59


Besides the quantum spectrum also the decay modes and the electromagnetic properties of the weak solitons are of great physical relevance. Using pertur- bation theory for the weak mixing angle 0w, Eilam and Stern [ 11 ] have determined the electric charges of the lowest states: the ground state l= 0 is neutral, while the isotriplet 1= 1 has charges 0 and _+ 1. We plan to determine the magnetic moments of the weak skyrmion and of its excited states. Remember that a perturbative calculation by Klinkhamer and Manton [ 18 ] of the magnetic moment of another non-perturbative weak excitation, the well-known sphaleron of the Weinberg-Salam model, has revealed a very big classical magnetic dipole moment of this solution, It ~ 8 0 e / M w .

Acknowledgement

We gratefully acknowledge discussions with E.R. Hilt', A. Lande and P.J. Mulders.

References

[ 1 ] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 127; Nucl. Phys. 31 (1962) 556.

[2] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552; G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825.

[3] U.-G. MeiBner, Phys. Rep. 161 (1988) 213; B. Schwesinger, H. Weigel, G. Holzwarth and A. Hayashi, Phys. Rep. 173 (1989) 173.

[ 4 ] G. 't Hooft, Nucl. Phys. B 72 ( 1974 ) 461 ; Nucl. Phys. B 75 (1974) 461.

[5] E. Witten, Nucl. Phys. B 160 (1979) 57. [6] S. Weinberg, Phys. Rev. D 13 (1976) 974;

L. Susskind, Phys. Rev. D 2 0 (1979) 2619; E. Farhi and L. Susskind, Phys. Rep. 74 (1981) 277.

[ 7 ] E. d 'Hoker and E. Farhi, Phys. Lett. B 134 ( 1984 ) 86; Nucl. Phys. B 241 (1984) 109.

[8] J. Ambjorn and V.A. Rubakov, Nucl. Phys. B 256 (1985) 434.

[9] V.A. Rubakov, Nucl. Phys. B 256 (1985) 509. [10] G. Eilam, D. Klabucar and A. Stern, Phys. Rev. Len. 56

(1986) 1331. [ 11 ] G. Eilam and A. Stern, Nucl. Phys. B 294 ( 1987 ) 775. [ 12] J.M. Gipson and H.C. Tze, Nucl. Phys. B 183 ( 1981 ) 524;

J.M. Gipson, Nucl. Phys. B 231 (1984) 365. [ 13] J. Kunz and Y. Brihaye, Phys. Lett. B 216 (1989) 353. [ 14] Y. Brihaye and J. Kunz, Mod. Phys. Lett. A, in press. [ 15] Y. Brihaye andJ . Kunz, Z. Phys. C 41 (1989) 663. [16] C. Hajduk and B. Schwesinger, Phys. Lett. B 140 (1984)

172. [ 17 ] P.Q. Hung and C.-H. Tze, Z. Phys. C 39 (1989) 371. [ 18 ] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 ( 1984 )

2212.

60

Documents

Rotational and vibrational excitations of weak skyrmions