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Sphalerons in the singlet majoron model
Physics Letters B 287 (1992) 119-122 North-Holland
PHYSICS LETTERS B
Sphalerons in the singlet majoron model Kari Enqvist and Iiro Vilja
Nordita, Blegdamsvej17, DK-2100Copenhagen0, Denmark Received 29 May 1992 We study the sphaleron solutions in the singlet majoron model and evaluate the sphaleron energy E MM in an accurate approximation. We show that E MM is restricted to vary within the range of variation of the standard model sphaleron energy E TM(2). As the VEV of the singlet scalar responsible for the breaking of the global U ( 1) is increased, E MM--, ESM(~.eff) w i t h 2eft = ).(1 - y2/4fl~.), where fl and 7 are couplings in the Higgs-singlet potential.
The singlet majoron model [1] provides perhaps the simplest explanation for vanishingly small neutrino masses via spontaneous breaking of a global U ( 1 ) symmetry and the see-saw mechanism. The order parameter of the global symmetry breaking is an SU (2) × U ( 1 ) r singlet scalar field ~ , which couples to the right-handed neutrinos, giving them large masses via its vacuum expectation value f , as well as to the Higgs sector of the standard model. The introduction of the singlet scalar can thus be expected to modify the nature of the transitions between different phases in the electroweak theory. First order phase transitions in the singlet majoron model have recently been studied in a cosmological context in refs. [2,3]. Here we wish to consider sphaleron-mediated transitions between topologically distinct vacua of the electroweak sector [4]. We will study the sphaleron solutions in the majoron model to find out to what extent the sphaleron energy changes with respect to the standard model. Such considerations are motivated by the current discussion on the possibility of producing baryon number violation directly in colliders [ 5 ], where the actual production cross section depends directly on the energy of the sphaleron configuration. We should mention that sphalerons in an SU(2) model with an extra singlet scalar have recently been studied numerically by Kastening and Zhang [6]. Their singlet did not take a VEV, and hence is not directly relevant to the singlet majoron model, although their results are in a qualitative agreement with ours. As the simplest version of the see-saw mecha-
nism yields an electron neutrino mass ,-~ m~/f, current Majorana neutrino mass limits would imply that f >> 100 GeV. Hence the physically relevant case is the one with large f , and although we shall not commit ourselves to any particular value, in what follows we shall mostly focus on large f . (For a recent exposition of the experimental consequences of the singlet majoron model, see ref. [7].) A technical problem for considering the sphaleron solutions of the equations of motion in the majoron model is the large number of free parameters in the Higgs-singlet potential. In principle, for each parameter set it is straightforward to integrate the equations of motion numerically. In practice this is rather cumbersome. Instead we will here present an approximarive solution, the accuracy of which is sufficient for all practical purposes, and from which the parameter depence of the sphaleron energy can easily be deduced. In the majoron model the energy of a static sphaleron configuration with spherically symmetric action (setting 0w = 0 for simplicity) can, following ref. [4], be written in the form
EMM = 4n.f dr(~(F')2 + gS-~rE[(1-F)F]2 0
+lr2[(h,)2
+ (¢,)2 + t~E(zt)2] -t- [h(1 - F ) ]
+rZ[V(h,c~) -
0370-2693/92/$ 05.00 ~) 1992-Elsevier Science Publishers B.V. All rights reserved
V(f,f)]],
/
2
(1)
119
Volume 287, n u m b e r 1,2,3
PHYSICS LETTERS B
where V (h, 4)) is the scalar potential with global minimum at h = f = 247 GeV, and 4) = f . As in the standard model the fields F and h are related to the weak gauge field Wia and weak doublet H by wia•a dxi = ~ F ( r ) U
-1 d U ,
g
H=
'
(2)
x+iy) z
(3)
"
The fields 4) and Z are simply the modulus and the phase of a complex singlet field q), q) = ~ 1 e
•
ix .
(4)
We shall study the most general scalar potential of an SU(2) doublet and a complex singlet field, which reads V = m 2 l H ] 2 + m~lq)] 2 + 2IHI 4
+7tH]2[~12 + p i l l 4 1. 2,.2
1~2~2
1. h2~.2
(5) whence the m i n i m u m of the potential, i.e., the true vacuum, is given by the relations f2 f2
=
- 2 7 m ~ + 4tim 2 7 2 - 42fl
>/0,
=
- 2 7 m 2 + 42mg 7.2 - 42fl
/> 0 ,
(6)
where one should require that 72 - 42fl < 0,
(7)
because otherwise there would not exist nondegenerate minima with non-zero f . In what follows we shall also assume that 2, fl > 0 to ensure the stability of the potential. Dynamics is now determined by the equations of motion, together with the boundary conditions. These 120
0 = r2F"-2F(1-F)(2F-
l) + ¼ h 2 ( 1 - F ) ,
0 = (r2h') ' - 2h(1 - F ) 2 - r 2 0 V Oh' 0 = (r2~b') ' -- r 2 0 V
where U E SU(2) is the matrix which carries the information about the winding number, 1( z 7 x-iy
are given by
0~'
1--~-h(r)U(
U=
6 August 1992
0 = (r2z') ',
(8)
where the prime stands for derivation with respect to the radial coordinate r. Obviously the solution of the last of these equations is Z = 0, which also minimizes the energy. To solve the remaining equations, one must specify the boundary conditions of the other fields F, h and 4). We require that the fields are (modulo gauge transformation) asymptotically at their true vacuums, which leads to the limiting values F ( v c ) = I,
h(ec) = f ,
4)(oc) = f .
(9)
As the gauge field and the doublet field must be analytical everywhere, and in particular at the origin, there exists a consistency condition for F and h, namely that for small r, F ( r ) ~., r z, h(r) ,.~ r, and thus F ( 0 ) = h (0) = 0. The crucial point here is that the value of 4) at the origin, ~b0,is not constrained by any consistency condition, but is a priori free and must be determined dynamically. By studying the equation of motion one can, however, say that for positive 7, ¢~0 > f and q~(r) is decreasing, whereas for negative 7, q~0 < f and ~(r) is increasing. A rough upper bound for the sphaleron energy E MM in the singlet majoron model is found by setting q~ = f , which is not a saddle-point configuration; one finds easily that EMM(~.,fl, y , f ) ~< ESM(2), where E TM is the sphaleron energy in the standard model. A lower bound can be obtained by disregarding the kinetic term ofq~ and then using its equation of motion, which yields an algebraic equation between h and 0. As the result we obtain
EMM(z, fl, 7, f ) > ESM(2eff),
(10)
where 2elf = 2
1- ~
>0.
(11)
Hence in the majoron model the sphaleron energy always lies in the range dictated by the sphaleron of
Volume 287, number 1,2,3
PHYSICS LETTERS B
the standard model. A similar conclusion can also be drawn from the numerical solutions of ref. [6], although there, in our notation, it was assumed that f = 0 and heft = 0. TO obtain a more accurate estimate for the sphaleron energy, we shall use the ansatz
6 August 1992
--
I000
-~",\
'
l
i,i
\k 0.950
c~] = fn 2 - -~flah 2,
E MM
(2,fl, 7, f)
~ EMM
4nf (2eff + (y2/4fl)(a-1) 2) -- g v'~B g2u 2
,
,
ii
,
~\\
(12)
where a is a parameter within the range 0 ~< a ~< 1, and rh 2 is determined by the correct asymptotic behaviour, f 2 = rh2_ ayf2/2fl. This ansatz reproduces in the case a = 0 the previous upper bound. Because h is always increasing, ~a is decreasing for positive 7 and increasing for negative y, as it should be according to the equation of motion. After scaling out the standard model VEV f , and neclegting momentarily a small (negative) contribution from the gauge-fieldHiggs interaction, we infer the upper bound
(o)
(c)
,.ooo
"~
0,995
}-
0.990
L -4
, "3
, -2
, -4
, 0
log (k/g)
-4
3
-2
log
-I
0
(X/g)
Fig. 1. The ratio EMM/E TM for various sets of parameters, as a function of 2 with f / f = 1 (dashed line), f / f = 10 (dotted line) and f / f = 100 (solid line): (a) (fl, 72) = (102,2fl,~); (b) (fl, 72) = (0.12,2fl2); (c) (fl,72) = (102,0.5fl2); (d) (fl, 72) = (0.12,0.5fl2). In (a) and (c) the curves for f / f = 10 and f / f = 100 coincide, indicating the saturation of the bound (10).
(13) which, with the following values of the parameters:
where A = 2.15,
u
=
1 + \2ffl)
a
>t 1,
(14)
and B (2/g 2) is a smooth, increasing function given in ref. [4], which determines the sphaleron energy in the standard model. The correction to the gauge-field-Higgs interaction neglected in (13) reads oo
: 4.
1) f dr:(l -.)' < O.
(15)
o
Our estimate consists now of substituting the standard model sphaleron solutions for h and F into (15), and minimizing (13) and ( 15 ) with respect to a to obtain
EMM(2, fl, 7, f ) ~ EaMM +
AE a MM
rain'
(16)
which is still also an upper bound. For a numerical estimate it is sufficient to use a fit to the function B, which has been computed in ref. [4]. We shall write
B(A/g2) = A + B t a n h [ c i l n ( g ~ ) + c 2 ] ,
(17)
cl = 0.247,
B = 0.573, c2 = 0.0863,
(18)
has a (linear) correlation 0.99994. It is then a simple matter to minimize (16) with respect to a. The result is displayed in fig. 1, where we have drawn the energy E MM(2, fl, y, f ) in units of ESM (2) for different cases of parameter values. When the VEV of the singlet field f is close to f , the sphaleron energy is, for all practical purposes, equal to the standard model sphaleron. Increasing f will decrease the sphaleron energy until it saturates the bound (10), as is evident from the form (13), and as can be gathered also from fig. 1. The quantity which determines the degree of saturation is 7 f / f l f ; as it decreases, a in eq. (13) will be driven towards I. The largest ratio EMMIESM, or the largest modification of the standard model sphaleron, is obviously obtained in the limit 72 __, 4fl2. In realistic cases, however, the modifications are assured to remain minor. This can also be seen in fig. 2, where we have drawn the contours of fixed EMM/E TM in the (fl,;t)-plane for the case 72 = f12. To conclude, we have estimated the modification of the energy of the sphaleron configuration due to 121
Volume 287, number 1,2,3
PHYSICS LETTERS B
alone and, for a fixed Higgs self-coupling 2, will always be slightly less than in the standard model. As already pointed out in ref. [3], the singlet majoron model cannot offer radical changes for the washout o f the baryon number asymmetry [8] after the electroweak phase transition, and the same conclusion holds also for direct production of baryon number via sphalerons in high energy collisions.
/0
.,.< -I.5
-4.0
L
I
i
1.0
i
i
References
(b) 0.990
I< -I.5
).99~
o O. 999
-4.0 -4.(
-15
IO
Io9 Fig. 2. Contours for the fixed ratio EMM/ESM in the (/7,2)-plane for the case 7 2 = 2p. (a) f = 1 TeV; (b) f = 10 TeV.
effects o f the singlet scalar o f the majoron model, and found it to be small. The sphaleron energy will always stay between the range allowed by the standard model
122
6 August 1992
[ 1] Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. B 98 (1981) 265. [2] Y. Kondo, I. Umemura and K. Yamamoto, Phys. Lett. B 263 (1991) 93. [3] K. Enqvist, K. Kainulainen and I. Vilja, Nordita preprint 92/10 P. [4] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [5] A, Ringwald, Nucl. Phys. B 330 (1990) 1; O. Espinoza, Nucl. Phys. B 343 (1990) 310; L. McLerran, A. Vainshtein and M. Voloshin, Phys. Rev. D 42 (1990) 171. [6] B. Kastening and X. Zhang, University of California preprint UCLA/91/TEP/20. [7] G. Jungman and M.A. Luty, Nucl. Phys. B 361 (1991) 24. [8] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposnikov, Phys. Lett. B 155 (1985) 36.
PHYSICS LETTERS B
Sphalerons in the singlet majoron model Kari Enqvist and Iiro Vilja
Nordita, Blegdamsvej17, DK-2100Copenhagen0, Denmark Received 29 May 1992 We study the sphaleron solutions in the singlet majoron model and evaluate the sphaleron energy E MM in an accurate approximation. We show that E MM is restricted to vary within the range of variation of the standard model sphaleron energy E TM(2). As the VEV of the singlet scalar responsible for the breaking of the global U ( 1) is increased, E MM--, ESM(~.eff) w i t h 2eft = ).(1 - y2/4fl~.), where fl and 7 are couplings in the Higgs-singlet potential.
The singlet majoron model [1] provides perhaps the simplest explanation for vanishingly small neutrino masses via spontaneous breaking of a global U ( 1 ) symmetry and the see-saw mechanism. The order parameter of the global symmetry breaking is an SU (2) × U ( 1 ) r singlet scalar field ~ , which couples to the right-handed neutrinos, giving them large masses via its vacuum expectation value f , as well as to the Higgs sector of the standard model. The introduction of the singlet scalar can thus be expected to modify the nature of the transitions between different phases in the electroweak theory. First order phase transitions in the singlet majoron model have recently been studied in a cosmological context in refs. [2,3]. Here we wish to consider sphaleron-mediated transitions between topologically distinct vacua of the electroweak sector [4]. We will study the sphaleron solutions in the majoron model to find out to what extent the sphaleron energy changes with respect to the standard model. Such considerations are motivated by the current discussion on the possibility of producing baryon number violation directly in colliders [ 5 ], where the actual production cross section depends directly on the energy of the sphaleron configuration. We should mention that sphalerons in an SU(2) model with an extra singlet scalar have recently been studied numerically by Kastening and Zhang [6]. Their singlet did not take a VEV, and hence is not directly relevant to the singlet majoron model, although their results are in a qualitative agreement with ours. As the simplest version of the see-saw mecha-
nism yields an electron neutrino mass ,-~ m~/f, current Majorana neutrino mass limits would imply that f >> 100 GeV. Hence the physically relevant case is the one with large f , and although we shall not commit ourselves to any particular value, in what follows we shall mostly focus on large f . (For a recent exposition of the experimental consequences of the singlet majoron model, see ref. [7].) A technical problem for considering the sphaleron solutions of the equations of motion in the majoron model is the large number of free parameters in the Higgs-singlet potential. In principle, for each parameter set it is straightforward to integrate the equations of motion numerically. In practice this is rather cumbersome. Instead we will here present an approximarive solution, the accuracy of which is sufficient for all practical purposes, and from which the parameter depence of the sphaleron energy can easily be deduced. In the majoron model the energy of a static sphaleron configuration with spherically symmetric action (setting 0w = 0 for simplicity) can, following ref. [4], be written in the form
EMM = 4n.f dr(~(F')2 + gS-~rE[(1-F)F]2 0
+lr2[(h,)2
+ (¢,)2 + t~E(zt)2] -t- [h(1 - F ) ]
+rZ[V(h,c~) -
0370-2693/92/$ 05.00 ~) 1992-Elsevier Science Publishers B.V. All rights reserved
V(f,f)]],
/
2
(1)
119
Volume 287, n u m b e r 1,2,3
PHYSICS LETTERS B
where V (h, 4)) is the scalar potential with global minimum at h = f = 247 GeV, and 4) = f . As in the standard model the fields F and h are related to the weak gauge field Wia and weak doublet H by wia•a dxi = ~ F ( r ) U
-1 d U ,
g
H=
'
(2)
x+iy) z
(3)
"
The fields 4) and Z are simply the modulus and the phase of a complex singlet field q), q) = ~ 1 e
•
ix .
(4)
We shall study the most general scalar potential of an SU(2) doublet and a complex singlet field, which reads V = m 2 l H ] 2 + m~lq)] 2 + 2IHI 4
+7tH]2[~12 + p i l l 4 1. 2,.2
1~2~2
1. h2~.2
(5) whence the m i n i m u m of the potential, i.e., the true vacuum, is given by the relations f2 f2
=
- 2 7 m ~ + 4tim 2 7 2 - 42fl
>/0,
=
- 2 7 m 2 + 42mg 7.2 - 42fl
/> 0 ,
(6)
where one should require that 72 - 42fl < 0,
(7)
because otherwise there would not exist nondegenerate minima with non-zero f . In what follows we shall also assume that 2, fl > 0 to ensure the stability of the potential. Dynamics is now determined by the equations of motion, together with the boundary conditions. These 120
0 = r2F"-2F(1-F)(2F-
l) + ¼ h 2 ( 1 - F ) ,
0 = (r2h') ' - 2h(1 - F ) 2 - r 2 0 V Oh' 0 = (r2~b') ' -- r 2 0 V
where U E SU(2) is the matrix which carries the information about the winding number, 1( z 7 x-iy
are given by
0~'
1--~-h(r)U(
U=
6 August 1992
0 = (r2z') ',
(8)
where the prime stands for derivation with respect to the radial coordinate r. Obviously the solution of the last of these equations is Z = 0, which also minimizes the energy. To solve the remaining equations, one must specify the boundary conditions of the other fields F, h and 4). We require that the fields are (modulo gauge transformation) asymptotically at their true vacuums, which leads to the limiting values F ( v c ) = I,
h(ec) = f ,
4)(oc) = f .
(9)
As the gauge field and the doublet field must be analytical everywhere, and in particular at the origin, there exists a consistency condition for F and h, namely that for small r, F ( r ) ~., r z, h(r) ,.~ r, and thus F ( 0 ) = h (0) = 0. The crucial point here is that the value of 4) at the origin, ~b0,is not constrained by any consistency condition, but is a priori free and must be determined dynamically. By studying the equation of motion one can, however, say that for positive 7, ¢~0 > f and q~(r) is decreasing, whereas for negative 7, q~0 < f and ~(r) is increasing. A rough upper bound for the sphaleron energy E MM in the singlet majoron model is found by setting q~ = f , which is not a saddle-point configuration; one finds easily that EMM(~.,fl, y , f ) ~< ESM(2), where E TM is the sphaleron energy in the standard model. A lower bound can be obtained by disregarding the kinetic term ofq~ and then using its equation of motion, which yields an algebraic equation between h and 0. As the result we obtain
EMM(z, fl, 7, f ) > ESM(2eff),
(10)
where 2elf = 2
1- ~
>0.
(11)
Hence in the majoron model the sphaleron energy always lies in the range dictated by the sphaleron of
Volume 287, number 1,2,3
PHYSICS LETTERS B
the standard model. A similar conclusion can also be drawn from the numerical solutions of ref. [6], although there, in our notation, it was assumed that f = 0 and heft = 0. TO obtain a more accurate estimate for the sphaleron energy, we shall use the ansatz
6 August 1992
--
I000
-~",\
'
l
i,i
\k 0.950
c~] = fn 2 - -~flah 2,
E MM
(2,fl, 7, f)
~ EMM
4nf (2eff + (y2/4fl)(a-1) 2) -- g v'~B g2u 2
,
,
ii
,
~\\
(12)
where a is a parameter within the range 0 ~< a ~< 1, and rh 2 is determined by the correct asymptotic behaviour, f 2 = rh2_ ayf2/2fl. This ansatz reproduces in the case a = 0 the previous upper bound. Because h is always increasing, ~a is decreasing for positive 7 and increasing for negative y, as it should be according to the equation of motion. After scaling out the standard model VEV f , and neclegting momentarily a small (negative) contribution from the gauge-fieldHiggs interaction, we infer the upper bound
(o)
(c)
,.ooo
"~
0,995
}-
0.990
L -4
, "3
, -2
, -4
, 0
log (k/g)
-4
3
-2
log
-I
0
(X/g)
Fig. 1. The ratio EMM/E TM for various sets of parameters, as a function of 2 with f / f = 1 (dashed line), f / f = 10 (dotted line) and f / f = 100 (solid line): (a) (fl, 72) = (102,2fl,~); (b) (fl, 72) = (0.12,2fl2); (c) (fl,72) = (102,0.5fl2); (d) (fl, 72) = (0.12,0.5fl2). In (a) and (c) the curves for f / f = 10 and f / f = 100 coincide, indicating the saturation of the bound (10).
(13) which, with the following values of the parameters:
where A = 2.15,
u
=
1 + \2ffl)
a
>t 1,
(14)
and B (2/g 2) is a smooth, increasing function given in ref. [4], which determines the sphaleron energy in the standard model. The correction to the gauge-field-Higgs interaction neglected in (13) reads oo
: 4.
1) f dr:(l -.)' < O.
(15)
o
Our estimate consists now of substituting the standard model sphaleron solutions for h and F into (15), and minimizing (13) and ( 15 ) with respect to a to obtain
EMM(2, fl, 7, f ) ~ EaMM +
AE a MM
rain'
(16)
which is still also an upper bound. For a numerical estimate it is sufficient to use a fit to the function B, which has been computed in ref. [4]. We shall write
B(A/g2) = A + B t a n h [ c i l n ( g ~ ) + c 2 ] ,
(17)
cl = 0.247,
B = 0.573, c2 = 0.0863,
(18)
has a (linear) correlation 0.99994. It is then a simple matter to minimize (16) with respect to a. The result is displayed in fig. 1, where we have drawn the energy E MM(2, fl, y, f ) in units of ESM (2) for different cases of parameter values. When the VEV of the singlet field f is close to f , the sphaleron energy is, for all practical purposes, equal to the standard model sphaleron. Increasing f will decrease the sphaleron energy until it saturates the bound (10), as is evident from the form (13), and as can be gathered also from fig. 1. The quantity which determines the degree of saturation is 7 f / f l f ; as it decreases, a in eq. (13) will be driven towards I. The largest ratio EMMIESM, or the largest modification of the standard model sphaleron, is obviously obtained in the limit 72 __, 4fl2. In realistic cases, however, the modifications are assured to remain minor. This can also be seen in fig. 2, where we have drawn the contours of fixed EMM/E TM in the (fl,;t)-plane for the case 72 = f12. To conclude, we have estimated the modification of the energy of the sphaleron configuration due to 121
Volume 287, number 1,2,3
PHYSICS LETTERS B
alone and, for a fixed Higgs self-coupling 2, will always be slightly less than in the standard model. As already pointed out in ref. [3], the singlet majoron model cannot offer radical changes for the washout o f the baryon number asymmetry [8] after the electroweak phase transition, and the same conclusion holds also for direct production of baryon number via sphalerons in high energy collisions.
/0
.,.< -I.5
-4.0
L
I
i
1.0
i
i
References
(b) 0.990
I< -I.5
).99~
o O. 999
-4.0 -4.(
-15
IO
Io9 Fig. 2. Contours for the fixed ratio EMM/ESM in the (/7,2)-plane for the case 7 2 = 2p. (a) f = 1 TeV; (b) f = 10 TeV.
effects o f the singlet scalar o f the majoron model, and found it to be small. The sphaleron energy will always stay between the range allowed by the standard model
122
6 August 1992
[ 1] Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. B 98 (1981) 265. [2] Y. Kondo, I. Umemura and K. Yamamoto, Phys. Lett. B 263 (1991) 93. [3] K. Enqvist, K. Kainulainen and I. Vilja, Nordita preprint 92/10 P. [4] F.R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212. [5] A, Ringwald, Nucl. Phys. B 330 (1990) 1; O. Espinoza, Nucl. Phys. B 343 (1990) 310; L. McLerran, A. Vainshtein and M. Voloshin, Phys. Rev. D 42 (1990) 171. [6] B. Kastening and X. Zhang, University of California preprint UCLA/91/TEP/20. [7] G. Jungman and M.A. Luty, Nucl. Phys. B 361 (1991) 24. [8] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposnikov, Phys. Lett. B 155 (1985) 36.