Z. Phys. C - Particles and Fields 56, 231-235 (1992) Zeitschrifl P a r t i d e s for Physik C

and FL=t: �9 Springer-Verlag 1992

Weak Skyrmions and sphalerons

The case of finite Higgs mass Yves Brihaye 1, Jutta Kunz 2, Fanny Mousset 1

1 Department of Mathematical Physics, Mons University, Av. Maistriau, B-7000, Mons, Belgium 2 Institute for Theoretical Physics, University of Utrecht NL-3508 TA Utrecht, The Netherlands, and FB Physik, University of Oldenburg, P.O. Box 2503, W-2900 Oldenburg, Federal Republic of Germany

Received 15 January 1992

Abstract. When the gauged non-linear a-model is supple- mented by a Skyrme term stable as well as unstable classical solutions exist: weak Skyrmions and sphalerons. Without the Skyrme term there remains only a sequence of sphalerons, which can be continued to finite Higgs masses. Here we investigate a model containing both, a finite Higgs .mass and a Skyrme term. Stable weak Skyrmion solutions occur only above critical values of the Higgs mass and the Skyrme coupling constant.

Introduction

The idea that the bosonic sector of the electroweak inter- actions may allow for stable extended objects, related to the non-abelian character of the gauge group, was sug- gested a few years ago, when, in analogy to the strong interactions, the existence of weak Skyrmions was pre- dicted [,1-3]. Neglecting the gauge fields in a first approx- imation and taking the infinite Higgs mass limit, the Weinberg-Salam model becomes a non-linear a-model, for which stable solutions, Skyrmions, are known to exist, when the model is supplemented by a stabilizing higher order term, called a Skyrme term [-4]. Since the effective Lagrangian for the Weinberg-Salam model contains such a Skyrme term, when loop corrections are included, the existence of weak Skyrmions seemed possible [,1-3].

Weak Skyrmions were then constructed explicitly in the infinite Higgs mass limit, where the length of the Higgs field is frozen, for the Weinberg-Salam model supple- mented by a Skyrme term (but with vanishing mixing angle) [-5-7]. The presence of the SU(2) gauge fields, however, led to significant differences with respect to the Skyrmions of the strong interactions. Weak Skyrmions do not exist for any finite value of the Skyrme coupling constant, but only up to a critical value of this coupling, when a spike is encountered and a new branch of unstable solutions emerges [6]. Also, the 'topological number' as- sociated with the classical solutions is not conserved.

An attempt to obtain stable classical solutions for such an effective electroweak Lagrangian with a finite Higgs

mass was reported in [8]. The authors indeed constructed classical solutions, which should be stable on similar grounds as argued in [5-7]. H6wever, they chose a Sky- rme term, which is not a polynomial function of the Higgs field. More recently, therefore, a better Skyrme term for the finite Higgs mass effective model was proposed, but stable solutions were searched for without success [9].

In this letter, we briefly reconsider the effective model of [,8], but put our main emphasis on the better effective Lagrangian of [9]. We investigate for both cases the sets of classical solutions depending on two parameters, the Higgs mass and the Skyrme coupling constant, searching for stable weak Skyrmions. We show, that the infinite Higgs mass limit is reached in a continuous manner by the set of classical solutions for the effective Lagrangian [9]. Thus stable weak Skyrmions exist also for finite values of the Higgs mass in this model, provided the Higgs mass is sufficiently large ( ~ 100Mw). In addition further unstable solutions exist in this model, which are related to the sphaleron [10-11] and the 'deformed' sphalerons [12-14], unstable solutions of the pure Weinberg-Salam model (without Skyrme term).

In the following we first briefly recall the models [-8, 9], present the ansfitze for the fields, the resulting energy functionals and the boundary conditions. Then we discuss the classical solutions for the two models considered, and present our conclusions.

Classical equations

Let us consider the SU(2) | U(1) locally invariant Wein- berg-Salam model in the limit of vanishing mixing angle, Ow=O. In this limit the U(1) field decouples from the SU(2) fields and may consistently be set to zero. The model thus reduces to an SU(2) gauge theory coupled to a complex Higgs doublet ~b = (4)1, 4)2).

We now introduce the complex 2 • 2 matrix M for the Higgs field [-9]

•,--q•* r " (1)

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With this matrix the bosonic,part of the Lagrangian reads

1 | Tr (FuvFU~) + i Tr(DuM)t(DuM) 2g 2

_ 2(~ Tr(MtM)_ V2"~ 2 2J '

(2)

where the covariant derivative D u and the field strength tensor Fu~ are defined by

DuM=(Ou-iVu)M, (3)

Fuv=~uVv-~V,-i[Vu, V~]. (4)

The SU(2) symmetry is spontaneously broken, and the masses of the gauge and Higgs bosons are Mw = 9v/2 and MH = x / ~ v , respectively.

In this letter we consider the Lagrangian (2) supple- mented by a Skyrme term. The presence of such higher order terms in the effective Lagrangian, representing loop corrections, was largely motivated in [5, 6]. Here we study the sets of classical solutions for the Lagrangian (2), sup- plemented by two specific Skyrme terms.

The Skyrme term

1 ~ i = 3~ez Tr[DuUU *, DvUUt] 2, (5)

where the unitary matrix U constitutes the "angular" part of the polar decomposition of M,

h M = ~ U, h 2=Tr(M*M), (6)

was used in [5=7], where only the infinite Higgs mass limit was considered. However, this Skyrme term was also applied in [-8] together with the Lagrangian (2) for a finite value of the Higgs mass. Since in this case the Skyrme term (5) is not a polynomial in the Higgs field, this choice is not very appropriate for a finite Higgs mass.

A better choice for the Skyrme term is given by [9]

1 ~9ii= ~ T r ( ( D u M ) t ( D v M ) - ( # ~ v ) ) 2, (7)

which is a polynomial in M. Both Skyrme terms ~ and ~i , coincide in the infinite Higgs mass limit, when h~v.

To construct classical solutions of the equations of motion of the effective Lagrangians LP + LPI and ~ + ~it , we restrict ourselves to static, spherically symmetric con- figurations of the fields. The corresponding ansatz for the fields [8, 12]

v M = ~ L(r) (cos F(r) + i?aZ, sin F (r)), (8)

G(r) H(r) K(r) ~ ̂ Vii = ~-r giabFaT'b ~- ~-r (6ia- FiFa)'r'a -t- ~ riraT, a, (9)

Vo =0, (10)

involves the five radial functions L, F, G, H and K. The spherically symmetric enery functionals asso-

ciated with the Lagrangian terms (2), (5) and (7) read,

respectively,

I/2nMw'~j F((G-1)2+HZ-1)2+2(G, - H K ) 2

E:t -)J~ I +4xZL'2+2( H'+_] (G~I)K) 2+Lz(2P+(2xF'-K)2)

+ 2eX2 (L 2 -- 1)2] ' (11)

f2rtMw'~f ~ "~ ej P[P+2(2xF'-K) 2] L -jL jjo , (12 )

and

E,,= 02 )

+ 8L' 2), (13)

with

P = (G - 1 + cos 2F) 2 + (H -- sin 2F) 2, (14a)

the dimensionless coordinate

x = Mwr, (14b)

the prime means derivative with respect to x, and the parameters

42 I_{Mu~ 2 92 e=-gy=Z\Mwj , ~ = ~ . (15)

The energy functionals (11)-(13) contain a residual U(1) gauge symmetry, which can be used to eliminate one of the radial functions, except L(x), the length of the Higgs field. In the following we choose the gauge

F(x)=0. (16)

From the variational principle we then obtain a set of equations of motion for the dynamical degrees of freedom, L(x), G(x) and H(x). For the function K(x) we obtain only constraint equations, because K'(x) does not occur in the energy functionals.

Regular, finite energy solutions of the equations of motion require certain boundary conditions. Here two sets of boundary conditions are possible for the functions L(x), G(x) and H(x), independent of the parameters e and 3. The first set

G(0)=2, H(0)=0, L(0)=0, (17a)

G(oo)=2, H(oo)=0, L(oo)=l, (17b)

is consistent with the sphaleron solution [10, 11] of the Lagrangian 5 a, which exists for all values of the parameter E, representing the Higgs mass. The second set

G(0)=0, H(0)=0, L'(0)=0, (18a)

G(~)=0, H(oo)=0, L(oo)=l, (18b)

coincides with the boundary conditions of the 'deformed' sphaleron solutions of the Lagrangian LP, which were shown to exist only for MH> 12Mw[12-14].

Sphalerons and 'deformed' sphalerons are unstable classical solutions of the Lagrangian LP. The 'deformed'

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sphalerons have a lower energy than the sphaleron (for the same value of Mn). For M , < 12Mw the sphaleron pos- sesses a single direction of instability, and the 'deformed' sphaleron does not exist. For Mn> 12Mw the first 'de- formed' sphaleron possesses a single direction of instabil- ity. The sphaleron then has two directions of instability, with the number of directions of instability increasing by one at each successive critical value of the Higgs mass, where a new 'deformed' sphaleron solution emerges.

Numerical results

Case I

Finite energy, classical solutions of the energy functional E+E~r exist only for the set of boundary conditions (18), when ~ > 0. In [8], one branch of solutions was shown to exist for all values of Mn and for Skyrme coupling con- stants on the interval 4 �9 [0.0, 0.3]. The solutions of this branch were identified with weak Skyrmions and con- sidered to be stable.

We extended this analysis to higher values of 4 and constructed a new branch of solutions, which merges with the sphaleron or the first 'deformed' sphaleron in the limit 4 = 0. We found, that solutions always exist in pairs for all values of Mn on finite intervals of the Skyrme coupling constant 4,

4 �9 [0, 4 , ( M , ) ] . (19)

When fixing Mn and varying 4, the energies of these solutions form two branches, which merge in a spike at 4=~c~. This is illustrated in Fig. 1 for M , = ~ and Mn = Mw. The lower energy branches are those obtained in [8], representing the weak Skyrmions. The critical value ~cr increases monotonically as M , decreases, e.g.

~r (Ms =~)~0 .38 , 4~r(Mn=2OMw)"~0.41,

~r = Mw) ~ 0.66. (20)

No solutions were found for 4 > 4 , , in complete analogy

16.00

12.00 ~ ~

z 0.00 - - - ~ - ~ - - - - . . . . - - - f J / - - - . . . . . . . . / I

/ 4.00

/ / /

0.00 /, . . . . . . . . i . . . . . . . . . I . . . . . . . . . i . . . . . . . . . i . . . . . . . . . J . . . . . . . . . i . . . . . . . . .

0. O0 0. i0 0.20 0.30 0.40 0.50 0.60 0.70

Fig. 1. The energy (in TeV) is shown as a function of ~ for the classical solutions of the functional E+Et for the Higgs masses M n =oo (solid) and M n = M w (dashed)

with the limiting case M , = ~ , where the critical pheno- menon was first observed [6].

There is an interesting point concerning the behaviour of the upper branch solutions, when the limit 4 ~ 0 is approached. For M n > 12Mw, the solutions (functions, energy, etc.) converge to the first 'deformed' sphaleron. In contrast, for Mn< 12Mw, the solutions converge to the sphaleron.

Let us now turn to the stability of the classical solu- tions. An argument based on catastrophe theory states that "as a function of external parameters (here e, 4), solutions will bifurcate at values, where eigenvalues of the matrix of small fluctuations pass through zero" [16]. This argument was invoked in [6, 9] for the weak Skyrmions, and it was shown to be true indeed for an analogous set of solutions in the strong interactions [17]. (The analysis [17] can be carried over to the case of weak Skyrmions in the infinite Higgs mass limit, since on the one hand the additional fluctuations in the pion field decouple from those in the vector field, and since on the other hand neglecting the fluctuations in the Higgs field function F does not change the critical points, where zero modes are encountered [14, 15].) Relying on this argument in the present case, we conclude that the Lagrangian ~ + s possesses stable non-trivial solutions for all values of the Higgs mass and for 4 �9 [0, ~ , ( M , ) ] . Like the sphaleron for Mn< 12Mw and the first 'deformed' sphaleron for Mn>12Mw, into which the upper branch solutions merge for 4~0 , these solutions possess a single negative mode, whose eigenvalue increases as r increases (for Mn fixed). At the spike where 4--4or, this eigenvalue crosses zero, and then the eigenvalue turns positive along the lower branch, which therefore should represent stable weak Skyrmions.

Case H

We now consider the classical solutions of the energy functional E + E n , whose structure is more involved in this case. Let us first summarize the results of [9]. For any M , one branch of solutions, emerging from the sphaleron in the limit 4 = 0 and obeying the boundary conditions (17), can be constructed. These branches have been con- tinued to large values of 4, and they do not seem to terminate at any finite value of 4. The number of unstable modes along these branches is for small 4 equal to the number of unstable modes of the sphaleron, and changes by one, whenever a bifurcation is encountered.

By a simple argument of continuity, one expects that branches of new solutions exist as well, which start from the 'deformed' sphaleron, whenever it exists, i.e. when Mn> 12Mw, obeying the boundary conditions (18). The interesting questions then are: Do these branches form a spike, as in case I and in the infinite Higgs mass limit? Are there stable branches, whose solutions could be iden- tified with weak Skyrmions?

We describe our results by means of Figs. 2-4, where we show the energies and the 'topological' numbers [5, 6] for the various classical solutions as functions of 4. Figure 2 presents the energies of the classical solutions for Mn = 20Mw. Indeed, a branch of solutions starts from the

234

16.00

15 NO"

14.00 '

13.00

12. O0

ii.00 ......... i ......... , ......... , ......... r ......... i ......... 0.00 0.20 0.40 0.60 0.80 1.00 1.20

Fig. 2. The energy (in TeV) is shown as a function of ~ for the classical solutions of the functional E + E u for the Higgs mass M n = 20M w. The dashed line represents the branch emerging from the sphaleron, the solid line the one emerging from the 'deformed' sphaleron. The solid line merges into the dashed one at the critical value ~r m0.95

0.60

0.40 " ~ . . . .

0.20 �9

0.00 ......... i ......... ~ ......... , ......... , .........

0.00 1.00 2.00 3.00 4.00 5.00

Fig. 4. The 'topological' number Q is shown as a function of ~ for the classical solutions of the functional E + E u, emerging from the 'deformed' sphaleron, for the Higgs masses M n = 1 5 M w, M n = 2 0 M w and M n = 136M w. The solutions on the branch coming from the sphaleron have Q=0.5, independent of M n and ~. The curve for M , = 15M w reaches the value Q=0.5 at r the one for M n = 2 0 M w at ~0 .95 and the one for M n = 136M w at ~ 4 . 4

20,00

18.00

16.00

1 4 . 0 0

12.00

i 0 . 0 0

/ / / / / / / / ~ / / ~

/ / / / / / / / / / / / / / / / / / / / / / / / /

Fig. 3. The energy (in TeV) is shown as a function of ~ for the classical solutions of the functional E + E u for the Higgs mass M n = 136M w. The dashed line represents the branch emerging from the sphaleron, the solid line the set of three branches emerging from the 'deformed' sphaleron. The solid line merges into the dashed one at the critical value r The points A, B, C and D have coordinates (0, 12.43), (0, 11.78), (0.40, 14.11) and (0.28, 12.31)

'deformed' sphaleron (solid). This branch merges with the branch coming from the sphaleron (dashed) at a critical value of 4, 4 , ---0.95. This critical value 4or is read off more easily from Fig. 4, which shows the 'topological ' number Q as a function of 4. At the critical value 4 , the value of the sphaleron branch Q=0.5 is attained.

In Fig. 3 the energies of the classical solutions for M u = 136M~ are shown, and we observe a more subtle situation. For a certain interval of the parameter 4 there are four different branches of solutions. The solutions (solid) emerging from the 'deformed' sphaleron (point B) do not directly merge with the solutions (dashed) coming from the sphaleron (point A). Instead, the branch coming from the 'deformed' sphaleron terminates at a critical

value 4~1~ ~ ~ 0.40 (point C) in a spike. There another branch originates, which is present over the finite interval 4 e [0.28, 0.40], i.e. up to a second critical value 4~ 2) ..~0.28 (point D). At the second critical point a third branch emerges, which connects these solutions with those coming from the sphaleron at the third critical point 4(3~ ~ a a Of all these finite energy solutions the energeti- cr ~ . . . . cally lowest solutions are those in the interval FY <z~ y(t)] L'~er , "~er A, along the branch C - D . Again Fig. 4 show the correspond- ing 'topological ' numbers for the classical solutions, and the critical values 4~i] obviously correspond to the turning points, where the slopes d Q / d 4 are infinite.

Invoking the argument from catastrophe theory again to discuss the stability of the solutions, we conclude, that the branch C - D corresponds to a set of stable solutions, representing weak Skyrmions. We reach this conclusion as follows: The branch B - C smoothly emerges from the 'deformed' sphaleron, which has a single unstable mode. At the critical point 4~1r ) a zero mode is encountered, and the negative mode turns into a positive mode. Along the branch C - D there is then no negative mode. At the next critical point 4 (2) another zero mode is encountered, and the here emerging branch has again a single negative mode. This branch then merges at the third critical point 4~ 3) with the sphaleron. Its negative mode merges with the lower negative mode of the sphaleron, while the upper negative mode of the sphaleron crosses zero.

Repeating the numerical analysis for different values of M n , we find that the domain of existence of the lowest, stable branch C - D (i.e. the interval [0.28, 0.40] for M , = 136Mw) increases, when M , increases, and that it approaches the interval [0, 0.38] in the infinite Higgs mass limit. On the other hand, when M n is decreased, the domain of existence of this stable branch diminishes, approaching the value 0.41, and it disappears for M n "~ 100Mw.

235

Conclusions

In this letter our main concern has been to establish, that weak Skyrmions can exist also for finite values of the Higgs mass in effective models of the weak interactions. For this purpose, we have investigated the classical solu- tions of an effective Lagrangian [9], regular in the Higgs field. The model [9], consisting of the Weinberg-Salam Lagrangian with a Skyrme term added, indeed allows for weak Skyrmions, when the Higgs mass is chosen large enough, and in the infinite Higgs mass limit the classical solutions appoach the previously known set of solutions [6, 13]. Invoking the arguments of Appelquist and Be- rnard [18], however, assuming that the proper effective Lagrangian should be the l- loop renormalized elec- troweak Lagrangian, we cannot attribute much physical significance to the precise numerical values, we obtained in our analysis for the range of existence of the weak Skyrmions (in terms of the parameters MH and 9/e) since this Lagrangian [18] would contain other terms in addi- tion to a Skyrme term. In this respect an extension of our analysis is definitely called for. But we expect, that the general picture obtained, will remain valid.

The set of classical solutions of the effective Lagran- gian [9] exhibits a plethora of branches and bifurcations for large values of the Higgs mass. The set of solutions of the pure Weinberg-Salam model, the sphaleron and the 'deformed' sphalerons, can be smoothly extended to finite values of the Skyrme coupling constant. But all branches emerging from 'deformed' sphalerons end in bifurcations, which either represent the onset of new branches or the mergence into the sphaleron branch. Since the nth 'de- formed' sphaleron has n unstable modes, one expects that only the 1st 'deformed' sphaleron, when continued to finite values of the Skyrme coupling constant, will allow for a stable branch of solutions and thus weak Skyrmions. While the 1st 'deformed' sphaleron exists for Mn > 12Mw, a stable weak Skyrmion branch appears in the model only for MH > 100Mw. To really prove stability of this branch

a normal mode analysis would be necessary. Here we have only relied on arguments from catastrophe theory, known to be correct in the infinite Higgs mass limit [17] and for the pure Weinberg-Salam model [14, 15]. With these arguments, though, we can establish a Morse equality for the model [9]

•= ~ ( - 1)qNq =0, (21) q

where the Morse index q is the number of negative modes of a solution and Nq is the number of solutions with Morse index q. For finite values of the Higgs mass Z is conserved, irrespective of bifurcations.

Acknowledoements. We gratefully acknowledge discussions with J. Baacke, H. Lange and A. Stern.

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