Electroweak non-topological solitons and baryon number violation in the standard model

~FiiJVc~Li.~SA ~

Nuclear Physics B 378 (1992) 468—486 North-Holland

Electroweak non-topological solitons and baryon number violation in the standard model G.G. Petriashvili Institutefor Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, Russia and High Energy Physics Institute of Thilisi State University, University St. 9, Thilisi 380086, Georgia * Received 8 November 1991 Accepted for publication 31 January 1992

It is argued, that the presence of a heavy fermion doublet of fourth generation in the standard model would lead to the formation of non-topological solitonic states of two different types. Due to the complex vacuum structure of the model, the solitonic states are unstable with respect to the anomalous decay with fermion number non-conservation. If the mass of the fermion exceeds the critical value mcr = 10—18 TeV, then together with the possibility of the formation of the metastable non-topological soliton, there exists a possibility of the rapid anomalous decay of the bare fermion.

1. Introduction The non-perturbative effects of electroweak interactions are now of great interest. One of the such conjectured effects at TeV scale is the formation of non-topological solitons [1,21and their possible subsequent decay with baryon- and lepton-number non-conservation due to the complex vacuum structure [3,4] and triangle anomaly [51.At small energies this non-conservation is usually associated with instantons [61, which describe the semi-classical tunneling between the vacua with different topological numbers and the amplitude is suppressed by where a~= ct/sin2O~. exp(—217-/~~), However, the exponential suppression may be absent under the extreme conditions: at high temperatures [71or high densities [81,in high-energy collisions [9] and in decays of heavy particles [10—141.The characteristic energy scale, at which the anomalous processes become rapid, is set by the sphaleron mass, which determines the height of the energy barrier between the topologically distinct vacua, and is of order 2irm~/a~ 10 TeV [151. *

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In the standard electroweak model with heavy elementary fermions the study of the anomalous baryon- and lepton-number non-conservation in their decays is not straightforward because these fermions are likely to appear as non-topological solitons [1]. In the standard model with the heavy fermion doublet there may occur two different mechanisms of formation of non-topological solitons. The first mechanism is inherent to theories, where fermions acquire large masses due to strong Yukawa coupling to the Higgs field. It has been argued [161 within the semi-classical approximation, that if the Yukawa coupling (the mass of the bare fermion) is large enough, it might appear that the lowest energy state with unit fermion number is not a plane wave (bare fermion) but a bound state, where the inhomogeneous Higgs field expectation value produces the attractive potential for the fermion. The gauge fields are not essential in this mechanism; the size of the soliton is determined by the inverse mass of the bare fermion and its energy is bounded from above by the maximum value mmax 2 TeV. For brevity we call solitons of this type Higgs ones. Another possible mechanism of the formation of non-topological solitons is associated with the complex vacuum structure and level crossing [17]. This mechanism leads to the instability of a sufficiently heavy bare fermion with respect to the anomalous decay with fermion number violation. As opposed to the Higgs soliton the role of the gauge fields is crucial in formation of these solitons, which we call gauge ones. When the bosonic field configuration evolves from the trivial vacuum configuration to the topologically distinct one with unit topological number, the fermion energy level emerges from the positive continuum, crosses zero and disappears in the negative continuum. At small fermion masses, the state corresponding to the bare fermion is separated from the topologically non-trivial vacuum state by a finite energy barrier. The fermion can decay into the vacuum state due to semi-classical tunneling through this barrier, but the probability of such a process is exponentially small. If the mass of the fermion is comparable with the height of the energy barrier, it might appear energetically favorable for the system to produce a gauge and Higgs field condensate with non-zero Chern—Simons number and form a non-topological solitonic state, where the heavy fermion occupies the bound state level, emerged from the upper continuum. Increasing the Yukawa coupling, one increases the mass of this soliton, and at some critical value of the Yukawa coupling constant the solitonic state disappears and the bare fermion becomes classically unstable with respect to the anomalous decay. In this case the barrier between the bare fermion and the topologically non-trivial vacuum disappears, the decay, instead of tunneling, proceeds classically and the exponential suppression of the amplitude is absent. Anomalous fermion number non-conservation in decays of the system of elementary fermions has been investigated in ref. [141 in the framework of the two-dimensional abelian Higgs model. Within the semi-classical approximation it has been argued that at small energies the lowest energy state with given fermion

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Baryon number violation

number is a non-topological soliton. The non-topological solitonic state is metastable, being separated from the non-trivial vacuum state by a finite energy barrier. However, if the energy of the fermionic system exceeds some critical value, the barrier disappears (the soliton becomes classically unstable) and the system freely rolls down, decaying into the vacuum state with zero fermion number. In this paper we consider the standard electroweak model with heavy fermion doublet. We assume for simplicity, that charged and neutral components of the doublet have equal masses and neglect the weak hypercharge interactions, taking the gauge group SU(2)L. In order to cancel the anomaly [181,we have to introduce some other fermion doublets, but we assume that these are light enough so that they play no essential role in the rest of our discussion. Throughout this paper we consider the classical approximation, i.e. we neglect the radiative corrections due to the boson loops and the contribution of the Dirac sea to the energy of the system. As we are forced to deal with strong Yukawa coupling, this approximation is far from being convincing, nevertheless we do hope that our results are qualitatively correct In the classical approximation the static energy is the sum of the bosonic classical energy Eh and the energy of the fermion occupying a certain energy level Ef ~.

ECl=Eb+Ef.

(1.1)

If the gauge and Higgs fields take their vacuum values, E~is the bare mass of the fermion. If the bosonic fields produce an attractive potential for fermions, E~is the energy of the lowest bound state level. We evaluate the lowest value of the static energy E~~(e~) under the condition that the energy of a fermion bound state level is equal to E~ mfef. The local minimum of E~i(E~)as a function of ~ corresponds to the non-topological soliton. We consider spherically symmetric bosonic field configurations, invariant under simultaneous space rotations and vector isospin transformations. First we consider the limit of zero gauge coupling, =

0, and discuss only the Higgs—fermion sector of the theory. We evaluate the dependence of the Yukawa coupling constant g on the Higgs self-coupling constant A, at which the heavy fermion forms the Higgs non-topological soliton. We investigate the classical mass of the soliton as a function of coupling constants g and A, and show that it is bounded from above by the maximum value mmax 1.2—2 TeV. Then we switch on gauge fields and show that at weak gauge coupling (a~~ 1) the Higgs non-topological soliton remains stable and again it represents the lowest energy state with unit fermion number. However, if the =

=

*

In fact, heavy fermions in the standard model cannot be included without violating either vacuum stability [191or applicability of the perturbation theory [20]. The first problem may be overcome by introducing extra bosonic fields, almost degenerate with fermions. It is beyond the scope of this paper to discuss these aspects of the model.

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gauge coupling is large enough (a~.>2irmw/mm~) the Higgs soliton becomes unstable with respect to the anomalous decay. In sect. 3 we estimate the critical value of gauge coupling constant and show that the Higgs soliton can decay classically only at a~ 1. In sect. 4 we discuss the possibility of rapid anomalous decay of a bare fermion. We show that at ~ << 1, mf 2~m~/a~ there appears the possibility of formation of a different (gauge) non-topological solitonic state. Within the spherically symmetric variational ansatz, we evaluate the dependence of the mass of the gauge soliton on the Yukawa coupling constant and estimate the critical value of the mass, at which the soliton decays classically with fermion number violation. Sect. 5 contains concluding remarks.

2. Fermion number violation in the standard model In this paper we consider the standard model with a heavy fermion doublet. The model is defined by the following lagrangian L =LAU+Lf, LAU

=

L~

=

~Tr

F~F~+ ~Tr(D~U)~D~U

i~JLy~D~I/JL + ~PRY~I’R

—

+A Tr(U~U—p2)2

g(~LU~1R+ ~IJRU~!JL),

—

where D1~~ 9~~ ~ + [A,~, Ar], A~is the SU(2) gauge field, U çf~+ i(T, ~) are 2 x 2 matrices, describing the Higgs field, ~I~L,R are left- and right-handed doublets of fermions, respectively. The model has global SU(2)R and local SU(2)L gauge symmetries, which are spontaneously broken. As a result of symmetry breaking, fermions and gauge bosons acquire masses due to the Yukawa and gauge interactions with the Higgs fields. At tree level the particle masses are =

=

—

=

m~=~g~v, mH=2%/~t~,

mf=gv,

where v 250 GeV is the vacuum expectation value of the Higgs field. The model has a non-trivial vacuum structure. In the gauge A0 0 the classical vacuum configurations, being defined by the minima of the bosonic part of static energy =

1~

Eb

=

fd3x{_ ~Tr

~JF

—

~Tr(D 1U)~D’U

+

~A Tr(U~U

—

v2)2J

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number violation

are represented by the pure gauge configurations A1=g~3~g1’,

U=g~v, where

g~(x)ESU(2)L.

Vacuum configurations with boundary conditions U v as x are related to the trivial vacuum configuration A, 0, U v by a time-independent gauge transformation, carrying an integer-valued topological number —~

=

n[gL]

=

~—~fd3x

—~ ~,

=

~

Any field configuration can be characterized by the Chern—Simons number eijk

r N~~=—~jdx Tr[~F,JAk—A1AJAk IT ~ is invariant under “small” gauge transformations, while under “large” gauge transformations it transforms as N~~[A~] =N~5[A} +n[g~]. In the case of the vacuum configuration, the Chern—Simons number is equal to the topological number of the vacuum. Neighbouring vacua with different topological numbers are separated by a finite energy barrier. The height of the barrier is determined by the sphaleron solution of the bosonic field equations [15]. This saddle-point solution has Chern—Simons number ~ and its energy is equal to the height of the barrier. In the theory with fermions, the transition between distinct vacua leads to fermion number violation. For small Yukawa coupling this non-conservation is associated with the following euclidean field configuration (instanton) [31: =

~,

2’

IT X

Al~1~~t = 0 x~+a~

A”°

iTX0[TXX] x~+a2 =

‘

u Inst

TX =

which in the gauge A 0 0 determines the path in the configuration space connecting the vacua with different topological numbers, X0 being the parameter along the path. When the bosonic fields evolve from one vacuum to another, the fermion energy levels move from the upper continuum to the lower one, or vice versa [17]. This level crossing leads to a change in the number of real fermions. If the system starts, e.g. from the trivial gauge vacuum (n 0) with filled Dirac sea plus one real fermion, and evolves to the non-trivial gauge vacuum with n 1, then the final state has filled Dirac sea and no real fermions, so that the fermion number is not conserved. At small fermion masses the initial state is metastable, being separated =

=

=

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Baryon number violation

473

from the vacuum state by a finite energy barrier. However, if the energy of the system is large enough, the barrier might disappear and the system might freely roll down, decaying into the vacuum state with zero fermion number. Comparing the rest energy of the fermion to the sphaleron mass, one obtains a crude estimate of the parameters of the model at which the anomalous decay may be expected to be unsuppressed mt>.msph,

i.e.

g>~c—.

(2.1)

In fact, the way of obtaining this estimate is extremely naive: if eq. (2.1) is valid, the lowest energy state with unit fermion number is not a free fermion even in the absence of the gauge interactions, but is a non-topological soliton whose mass is less then m~and whose expectation value of Higgs field is inhomogeneous and different from (~) v. These properties could in principle change drastically the estimate (2.1) and even make the unsuppressed anomalous decay impossible. We study this problem within the classical approximation in the rest of this paper. =

3. Non-topological soliton Let us consider the lowest energy state with unit fermion number. In the classical approximation this state corresponds to the local minimum of the classical energy. As was pointed out above, for sufficiently large Yukawa coupling it might turn out that the classical energy has two different non-trivial minima, which correspond to non-topological solitonic states with different sizes. In order to evaluate these minima it is convenient to take the gauge and Higgs fields spherically symmetric and introduce the dimensionless variable p hv I x I. We shall set the parameter h to be equal to the Yukawa or the gauge coupling constant for Higgs and gauge solitons, respectively. The most general configuration, satisfying the spherical symmetry is given by =

~A0=

(n, T)a(p),

i

1 —f1(~o) f2(.°) 2p +(1-—n(n,T)) 2

-11--A=n(n,i-)b(p)+[nxr] U= (P(p) +i(n, i-)Q(p))v, where n

=

I x I.

(3.1)

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In the spherically symmetric external fields neither the angular momentum of the fermion, nor the weak isospin is conserved but only the sum K J + I is. The lowest K 0 energy state has the form =

=

+ ~L

=

=

{(UR(P) + UL(p))X + (UR(p) — UL(p))X +

i(n, i(n,

~)(wR(p) + WL(P))X]~ 1)(WR(p) — WL(P))X

)3/2

(3.2)

where ,y ~ja satisfies the condition K~ 0. For spherically symmetric configurations of gauge fields and the s-wave function of the fermion we have ~JJ~AQ~/1L 0, so there is no source for the zero component of the gauge field and we can take A0 0. Then the classical energy of the system =

=

=

=

take the form E~1 E1 + EA + EAU, =

2+

EA

=

EAU=

~hvfdP{(3~fi

~

—

+ 2bf2)

+f~

2bf 2 + 1)

—

2

dp[(0~P+bQ)2+(3~Q_bP)2+ ~(Pf

~~~fp2

+ 1)Q)

2—(f1 + Ef

=

1)~,

~(f~Q

+

2

(f

1

1)P)~+ A(~2 +

-

Q

-

i)~J~

gvc~,

where Cf is the lowest eigenvalue of the Dirac equation, which we write in terms of dimensionless variables 1+f —

0PUL

—

buL

f2

1 WL

—

—

p

p

bwL + 1—f1

p

~~UL +

UL

—

f2

~WL

p

2 ~WR

+

p

WR

p~(PuR+ QwR)

+ p(Pw~—

QuR)

+p~(Pu~QwL) —

3PuR~’i(1~L+QuL)

with the normalization condition 8ITfp2dp(u~+w~+u~+w~)= 1.

=

/iCfUL,

=

~LEfWL,

lLEtU~,

—I.LEfWR,

(3.3)

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Baryon number violation

Here ~ g/h. One can express the fermion energy by the eigenfunctions of eq. (3.3) in the following way =

E~ 8ITv~fp2 dP[_uL3PwL + WL3PUL +

uR(BPwR + _WR)

=

2f —WRaPUR

—

b(u~+ w~,)

1 —

~~ULWL

2~Q(uLwR

—

+

—

u~)+ 2~P(uLuR + wLwR)

URWL)j.

+

The spherically symmetric bosonic field configuration and the s-wave function of the fermion admit U(1) gauge transformations A~—sg~A~gj~’~

U —sg~U,

—

—*g~~1~,

~L

where g~(x) exp[i(n, 1)cr(x)]. Under these transformations the functions b, =

f2’

P,

f1

Q

f

1,

transform as

—~ficos(2a)

—f2

f2—sf2

sin(2a),

P—s P cos(a)

—

Q

sin(a),

cos(2cr)

Q

—~

while the transformation of the wave functions UL~UL

cos(a)

+WL

sin(a),

+f1

sin(2a),

b—sb+a

Q

cos(a) +P sin(a),

UL

and

WL~WL

WL

cos(a)

has the form sin(a).

~UL

In terms of spherically symmetric fields the Chern—Simons number reads N~~= __Jdp[b(fl2+f~_1)+~pflf2_3pf2(fl_1))I. In what follows we set the gauge b 0. In this gauge the bosonic field configuration with finite energy satisfies the boundary condition =

Q(0) Q(oc)

=

sin(s~),

=

0, P(c~) =

P(0)


cos(i3),

f1(0) f1(co)

=

=

1,

f2(0)

cos(2s~),

=

0,

f~(~)sin(2s~). =

(3.4) We first switch off the gauge fields, i.e. assume that gauge fields are frozen at their vacuum values f1 1, f2 0. In this case (g~ 0) the height of the barrier becomes infinite and the transition with fermion number violation is forbidden. The classical energy has only one non-trivial minimum, which corresponds to the =

=

—‘

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Baryon number violation

lowest energy state with unit fermion number. To study this state it is convenient to set the parameter h in eq. (3.1) equal to the Yukawa coupling constant g. Then the classical energy has the form (3.5)

E~iEt+u,

where

=

~

~,

dP[(3~P)2 +

~fp2

(~Q)2+

~(p2

+

Q2

—

1)21.

We have investigated numerically the existence of local non-trivial minima of the classical energy (3.5) at various values of the parameters g and A, making use of the minimization technique described in ref. [13]. Since the fermion energy functional is not positively definite the minimization of the classical energy is not straightforward. To find the minimum of the classical energy we evaluate the lowest value of c~(P,Q) under the condition Ef(P, Q) gvef, considering Cf as a parameter. Then the classical energy becomes a function of the parameter Ef, =

~

(3.6)

Ef+Eu(Ef)

Its minimum, if it occurs at Cf I <1, corresponds to the non-topological soliton. For small Yukawa coupling the classical energy has only the trivial minimum, which occurs at P 1, Q 0, Ef 1. This configuration describes the bare fermion at rest and the vacuum state of Higgs fields. However, if the Yukawa coupling is strong enough, the Higgs fields can produce an attractive potential for the fermion and form the bound state, which is a non-trivial local minimum of the classical energy and corresponds to the Higgs non-topological soliton (see fig. 1). We find that at g > g 0(A) (see fig. 2), the lowest energy state with unit fermion number is the Higgs soliton whose mass is less then the mass of the bare fermion. The mass of the soliton is related to the minimum of the function (3.5) as follows: =

=

=

m501(g, A) =gve~(g, A). In fig. 3 we plot the mass of the Higgs soliton as a function of Yukawa coupling g at various values of Higgs self-coupling A. As is clear from fig. 3 the mass of the Higgs soliton is bounded from above by a maximum value mm,,,, 1.2—2 TeV. When gauge fields are switched on (g~~ 0) the solitonic state is separated from the non-trivial vacuum state with IsI~~ 1 by a finite energy barrier. When the value of gauge coupling constant is close to its physical value g~ 0.6 (m~ 80 GeV), the height of the barrier is of order mSPh 10 TeV. On the other hand when gauge interactions are taken into account the mass of the Higgs soliton is shifted slightly downwards. Hence, at weak gauge coupling we have for all values =

=

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/

477

Batyon number violation

\(~5)

(a,) 0.9 0.7 0.5

(a’).

0.3

(6) . (c,):

(c’)

2

,~2+ ~

Vy’ ~ d&’(.

8 /~
0.1 0

1 2

3

4 5 5

7 8 9 10

)~/3/47•5

p~m1/x/

Fig. 1. Field configurations for2the Higgs fermion non-topological soliton s,/t/i~i/i~ (g = 5, m~ = 50 0eV, energy Ef = 0.61 gv, + Q2,(b) wave function , (c) soliton density m5,,~ 8e~1 /ap = 0.9gv): measured (a) Higgs in units fieldofPthe bare fermion mass mf = gv. The distance is measured in units of the inverse bare fermion mass.

9

07y[ThV] i-ion- ~opoCoQLca~’

.s’o8’J~on 8

-2

7

/anewa~e

0~1 0,5 1 I

I

2

5 I

10 5000

mH[~eW Fig. 2. The dependence of the critical value of the Yukawa coupling constant g, above which the fermion can form the Higgs non-topological soliton for g~= 0, on the Higgs self-coupling A.

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G. G. Petriashvili

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Ba,yon number violation

m~g[TeV]

0.7S

/~ 3

I

4

5

5

I

7

8

.9

10

9

Fig. 3. The dependence of the mass of the bare fermion (dashed line) and the mass of the Higgs soliton (solid lines) on the Yukawa coupling g for g~= 0 and various values of mH.

of Yukawa coupling m501 g~(g)we found that the system from initial state, describing the bare fermion, freely rolls down to the non-trivial vacuum state with N~5 1. So in this case the heavy fermion decays classically with fermion number violation, instead of forming the non-topological soliton. The dependence of the critical value of the gauge coupling constant on the Yukawa coupling constant is shown in fig. 5 (solid line). This dependence is drastically different from that following from the naive estimate (2.1) (dashed line). =

=

4. Anomalous decay of base fermion In sect. 3 we argued within the classical approximation that it is energetically favorable for a heavy fermion (m~> g0(A)u 0.75—2 TeV) to form a bound state a Higgs non-topological soliton, which is the lowest energy state with unit fermion =

—

G. G. Petriashvili / Baryon number violation

~ /

7

/

V

479

/

~4OO~817~7m~

/ /

Fig. 4. The lowest value of the classical energy as a function of the fermion bound state energy for g = 7 and A 0: solid line — g~= 0, dashed line — g~= 7. The minimum of the solid line corresponds to the Higgs non-topological soliton.

number. However, if the fermion is sufficiently heavy, together with formation of the Higgs soliton, there exists a possibility of rapid anomalous decay of the bare fermion due to the complex vacuum structure and level crossing. If during changing the bosonic field configuration along the path, connecting the trivial vacuum configuration with the topologically distinct one with ~ 1, the fermionic energy level occupied by a heavy fermion emerges from upper continuum, crosses zero and disappears in the negative Dirac sea, and classical energy monotonically decreases (energy barrier is absent), one can expect that rapid anomalous decay of the base fermion might occur classically. When the gauge freedom is fixed, this =

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G.G. Petriashvili

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O(W

-~-

~ITITIT

4

5

6

7

8

/0

9

11

Fig. 5. The critical value of a~= g~/4ir, above which the Higgs soliton becomes classically unstable as a function of Yukawa coupling g (solid line). This dependence is drastically different from that following from the naive estimate (2.1) (dashed line).

path corresponds to the non-contractible vacuum loop [21]. In this section, making use of the simple spherically symmetric ansatz and a Rayleigh—Ritz type procedure, we determine approximately the minimum energy vacuum loop. This loop contains the sphaleron configuration [22] with N~5 for which there exists the =

-~,

normalizable solution of the Dirac equation with zero energy [23]. We show that if the fermion mass is large enough, the evolution of the system along the path determined by the minimum energy vacuum loop is not classically forbidden and can lead to rapid anomalous decay of the bare fermion with baryon and lepton number violation. Let us consider the following loop in the configuration space, given by the spherically symmetric ansatz: 2 ——

f1—1

2(1 C

_2

)

p2+a2

___

,

f2—

——

2E

1

e

,

__________

%/2

P

1

b—0, —

p

,

Q=_V1_e2

(4.1)

1/p2+a2

where ~ g/g~(in this section it is convenient to set h g~, in eq. (3.1)], a is the variation parameter and C !S the parameter of the loop. At e ±1 the configuration given by eq. (4.1) describes the classical vacuum of bosonic fields, while at 0 it coincides with the instanton configuration ~ U Inst with euclidean time equal to X11 0. Configuration (4.1) satisfies the boundary condition (3.4), where arccos(c), and its Chern—Simons number (calculated in the gauge, when =

=

=

=

=

=

—

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Baryon number violation

481

fields vanish at spatial infinity) is Ncs

=

‘arccos(E)

—Vi

—

—

e2 [1

+

~(1

—

In what follows, we assume ~ ~ 1, a~ 1. Then the fermion energy and its wave function are found by inserting eq. (4.1) into the Dirac equation (3.3) and neglecting the terms of order O(1/~).The solution has the form uR=[p+a]

exp[_~V1_e2~/p2+a2j,

1 1+

UL

1 ViE

UR, V/p2+a2

WR

WL

=

=

0,

where the fermion energy is independent of the parameter a and is equal to gce. It is worth pointing out that at C 0 this solution coincides with the exact solution found in ref. [11]. In order to determine the minimum energy vacuum loop we substitute ansatz (4.1) into the expression for the classical energy =

=

m

d

IT

3 1(e) a +d2(e)a+d3()a

a~4 E~1=2—~ _(1_2)

+w+O

1_f1

—

where d

2 1(c)

=

d3(C)

=

2

+

(1—c2),

2(C)

w

2),

For given

C

d

e

+

~2(1

=

+

2(1

—

=a~~ ~g~g/4IT.

—C

the classical energy is minimal at a2()=~_[~/d 22+12did3 _d2}.

(4.2)

Substituting eq. (4.2) into eq. (4.1) we evaluate the vacuum ioop with minimum energy. This loop contains the configuration with N~5 ~, which is a reasonable approximation to the sphaleron configuration at A/g~~ 1. The dependence of the classical energy on the parameter C along the minimum energy vacuum ioop is shown in fig. 6 for several values of the parameter w. =

G. G. Petriashvili / Baryon number violation

482

o(w

3 —

/

/, I,

./.// -o-~’--o.~/4~ -0.2

-7’

,. ,.

—

—

—~‘=2.55

_______

0

~2

0.4

0,6

0,8

f

£

IIII

/

I I I ‘i /I

-1

mw=mH

/

-2

/ /

-3

Fig. 6. The classical energy as a function of the parameter e along the minimum energy vacuum loop for m~= mH and various values of the parameter w = g~g/4~r:dashed-dotted line — w wc~. At m~= m~we have w 0 = 1.9, o,~= 2.6. The minimum of the solid line corresponds to the gauge non-topological soliton.

At w 1.9 (see fig. 6 dashed-dotted line) the state corresponding to the bare fermion is separated from the topologically non-trivial vacuum state by a finite energy barrier. The height of the barrier is determined by the solution corresponding to the maximum of the classical energy and is of order of the sphaleron mass. The fermionic state is metastable, its decay proceeds via tunneling and the decay amplitude is exponentially suppressed. At w0 <~
G.G. Petriashvili

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/

Baryon number violation

483

(b,)

128

0.7 (a’). P2÷Q2veesus

0

5

/0

15

20

m~/x/

II,’,

25

30

35

40

Fig. 7. Field configurations for the gauge non-topological soliton (g

45 =

50

49, mH

=

50 GeV,

m,,,

1

2 +=Q2 4.51, as m~/a~ a function= 11.8 of m~ TeV, I x ,Ef(b)= gauge 3.1 m~/a~ field —= f~ 8.2+TeV, f~asm~ a function = gv = 12.25 of m~ TeV): I x , (a) (c)Higgs fermion field wave — Pfunction —

V~1+1I,/v3as a function

of mf I x

pn[TeVJ (ci 12

(~)

(a)

7.1 (a).’ (L,)

(c)

10

I

41

42

I

43

I

41

I

I

4~ 45

47

.

mpeane

lv~I-e

I

I

I

48

49

50

51

9

Fig. 8. The dependence of the mass of the gauge soliton (a), height of the energy barrier (b) and the mass of the bare fermion (c) on the Yukawa coupling g for mH = 50 0eV.

G. G. Petriashvili / Baryon number violation

484

(the mass of the bare fermion) increases the mass of the gauge soliton increases and at the critical value the mass of the gauge soliton becomes equal to the height of energy barrier (see fig. 8); the local minimum disappears and the soliton becomes classically unstable. At w ~ w~~(A/g,~,)(see fig. 6, dashed line) the classical energy monotonically decreases along the minimum energy vacuum loop. Hence the evolution of the system along this path is not classically forbidden. The initial state, corresponding to c 1, describes the bare fermion and trivial vacuum of bosonic fields. At 0 bosonic fields form a sphaleron-like configuration with ~ 4, while the fermion energy level crosses zero and disappears in the negative energy sea at 1. The final state, corresponding to 1, describes the topologically non-trivial vacuum, containing no real fermions. Thus, if the mass of the bare fermion exceeds the critical value mcr (Ocr2mw/aw 10—18 TeV, its anomalous decay might proceed without tunneling (classically) and its lifetime might be small. This value of the critical mass of the fermion is in good agreement with the naive estimate (2.1). =

=

=

=

=

—

—

=

5. Conclusions In this paper within the classical approximation we argue that the standard electroweak model with a heavy fermion doublet might contain non-topological solitonic states, at least of two different types. These solitons have different masses and sizes and differ from each other first of all by the dependence of their lifetime on the mass of the heavy fermion. At a~<< 1 and mf > g0(A)v 0.75—2 TeV due =

to the strong Yukawa coupling the fermion forms the Higgs non-topological soliton, which is the lowest energy state with unit fermion number. The mass of the Higgs soliton is bounded from above by the maximum value mmax
=

G. G. Petriashvili

/

Baryon number violation

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again remains a possibility of formation of the Higgs non-topological soliton. Both these possibilities might lead to exotic events in experiments at future accelerators. In particular, in the anomalous decay of a heavy fermion the fermion number of every SU(2)L doublet would be changed by unity (if there exist any new non-singlet light fermions, all of them would also show up). This means that in such a process one might expect multi-jet and multi-lepton events. On the other hand the non-topological solitons are long lived, but not absolutely stable. The mixing between the fourth and third generation forces the heavy fermion, that forms the soliton, to decay. The decay of the bound fermion inside the soliton, leads to instability of the empty bosonic bag with respect to the production of gauge and Higgs bosons. We have estimated the number of produced particles and their spectrum in the decays of solitonic bags of Higgs and gauge solitons approximating the bag configurations by a coherent state of free fields. The coherent state was chosen in such a way that the field operators averaged over it give the soliton configuration. In this way we obtain that the bosonic (namely Higgs) bag of Higgs non-topological soliton is formed by a small (even less then one) number of hard Higgs bosons. This fact implies poor applicability of the classical approximation, but it indicates that the number of the produced Higgs bosons is indeed of order one. For the gauge non-topological soliton we have found that in the decay of the bosonic bag a large number (of order of a;1) of gauge and Higgs bosons, with average momenta of order of their masses, is produced, as it should be for the decay of an extended field configuration. The number of different species of gauge bosons (W ~, Z) are equal to each other (this is the consequence of the spherical and isotopic symmetry of the ansatz (3.1)), while the number of Higgs bosons is substantially smaller. Note that the decay of the non-topological soliton is a superposition of the decay of the solitonic bag and the decay of the bound fermion. We note also that the fermion inside the bag is in a localized state that may affect its decay. It is of interest to study whether it is possible to distinguish the decays of non-topological solitons from decays of conventional heavy fermions with the same masses. We hope to turn to this problem in future. The author is deeply indebted to V.A. Rubakov for numerous discussions and comments, to N.S. Amaglobeli, V.A. Matveev, A.N. Tavkhelidze, for interest and encouragement and to Institute for Nuclear Research of the Russian Academy of Sciences, where this work was done, for warm hospitality. References [1] R. Friedberg and T.D. Lee, Phys. Rev. D16 (1976) 1096; D18 (1978) 2623 [2] Y. Nambu, NucI. Phys. B130 (1977) 505; J.M. Gipson and H.Ch. Tze, NucI. Phys. B183 (1981) 524;

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