- Email: [email protected]
Finite fermion density effects on the electroweak string
6 April 1995
PHYSICS LETTERS 6 PhysicsLetters B 348 (1995) 457-461
Finite fermion Qensityeffects on the electroweakstring G. Bimonte abbe*, G. Lozano a,2 a International Centre for Theoretical Physics, RO. Box 586. I-34100 Trieste, Italy b INFN, Sezione di Napoli, Napoli, Italy
Received 18 September1994 Editor: L. Alvarez-Gauti
Abstract We consider an electroweakstring in the backgroundof a uniform distribution of cold fermionic matter.As a consequence of the fermion number non-conservationin the Weinberg-Salammodel, the string producesa long-rangemagnetic field.
Since the pioneer&g work by Nielsen and Olesen [ 11, the study of vortex like solutions in gauge theorieshas attracteda constantinterest.This kind of objectsis by now believedto play a prominentrole in cosmology,for exampleby providing a mechanismto explain structureformation in the early universe [ 21. Until very recently,this type of configurationshave beenmainly studiedin theorieswith a nontrivial vacuum structure where the existenceof a topological conservationlaw ensures its stability. However, as recently stressedby Vachaspatiand Barrlola [3], the embedding of such configurations in larger gauge groups can be stable even in the absenceof such topological conservationlaws. This is what happens in the case of the &string [4], which results from embedding the U( 1) Nielsen-Olesenstring in the SU( 2) x U( 1) theoryof electroweakinteractions[ 51. In order to understandthe role of these configurations during the phase transition it is important to know how they are modified in typical thermodynamical situationssuch as non-zerotemperatureand finite matter density.In this letter we shall analyzethe case i E-mail addnzss:[email protected]. * E-mail address:[email protected].
in which the string is placed in a backgroundof cold neutralfermionic matter, a situation in which, due to the V-A characterof the electroweakinteractions,the fermion sectorof the model manifestsin a non trivial way. To illustrate the kind of phenomenaiha: occur, let us begin with the simpler U( 1) model and study the way in which a constantfermionic density affects the Nielsen-Olesenstring. We consider the system described by the Lagrangian density,
+ (U&4) (D”d)
- A(q52 - 7j2j2 + Lf .
(1)
Here the covariantdedvative for the scalar complex field 4 is defined as
a,+ = (4.lf W,)#
(2)
and 2, and Z,, are the vector potential and field strength. The interactionof the fermionswith the gaugefields is governedby the Lagrangiandensity
0370-2693/95/@9.50 @ 1995 Elsevier Science B.V. AI1 r&h@,reserved SSDIO370-2693(95)00211-l
G. Bimnnre, G. L.ozano/Physics Letters B 348 (1995) 457-461
458
where #; and J/L+are left-handedfermions with oppositechargeanda, are the Pauli matrices.Following Rubakov and Tavkhelidze161, we will considerthe case in which there is a cold (zero temperature)and neutral (n+ = n- = n/2) backgroundof fermions. In the weakly coupledregime,the bosonicfield can be treatedclassicallyand its contributionto the energy (in the static case) is simply given by .??b = I d3xE;Z; + lDj#12 + A(42 - 7j2)2] . J
(4)
As shown in Refs. [6,7] the presenceof a finite density manifeststhroughthe appearancein the effective energyof a Chern-Simonsterm [ 81 which resultsfrom the integrationof the fermionic determinant Ef-2
k
in recent years mainly in the context of high tcmperaturesuperconductivityand the quantumHall ?:ffeet [ lo,1 11. A generalfeature of these solutions ;s that as a result of the Chem-Simonsinteraction the vortex acquiresan electric chargeand consequentlya radia! electric field. The transformation(9) correspondsto the combined effect of an Euclidean rotation and a duality transformationin the xc -x3 plane.As a consequence, what in the 2 + 1 dimensionalcase correspondedto an electric chargewill now becomea flux of current along the string aud the radial electric field will be now convertedinto a tangentialmagnetic field. Making the ansatz 4 = f(r) exp(imd),
Z = -pZ@
(10)
+ Z3e3
the equationsof motion then read (rzy
- 2q’rfZ3 f 2kZi = 0
(11)
~3xEVkz. .z k 11
(5) I the coefficientk beingrelatedto the chemicalpotential P by k= m2 is’ The effective energy
(6)
subjectto the boundaryconditions Z3(00)
E = Eb + Ef
=o,
Ze(oQ)=-y,
f(w)
(7)
was originally usedin Refs. [ 6,9] whereit was shown that the trivial vacuumbecomesunstablefor chemical potentialslargerthan the critical value 32$ kit = -m, 8
(8)
where the massof the gaugeparticle is nrz = 2q2q2. We will insteadconsiderthecaseof small chemical,potentials p 5 petit in order to study how the fermionic backgroundmodifiesthe Nielsen-Olesenstring. Notice that by making in (7) the transformation A3 + iAo,
(12)
x3 -b -in’,
k -+ ik
(9)
we obtain the actian of the 2 + 1 dimensionalChemSimons-Higgssystem. Vortex like solutions in this and related models bave been object of much study
Z3(0>
< 00, Z@(O)=o,
f(0) =o.
=7)
(14)
(15)
Although an exact solution to the equationscannot be obtained analytically, it is possible (in complete analogy to the 2 + 1 dimensionalcase [ 10,111) to expressthe first termsof an expansionin k in a closed r.,.rn involving the solutions of the Nielsen-Olesen vortex
z, = zTOz3 =
$$Zt”
+ ;) + O(k4)
-k(Z,N@ + f$ +Q(k3)
f = f”” + O(k4)
4
(16)
(17) (18)
wherethe superscriptNO refersto the Nielsen-Qlesen solutions, that is, solutions to Eqs. (1 l)-( 13) with k = 0. Thus, we see that the leading effect is the
G. Bimonte, G. Lozano / Physics L.&ten I) 348 (1995) 457-461
appearanceof a tangential componentof the quasimagneticfield Ci (we reservethe namemagneticfield for later when we introduce the unbroken CI(1) of electromagnetism) Ci = &QkZjk
(19)
CO= -rZj
(201
which can he associatedto a flux of current flowing along the string
J= d*xh, jl = iq(cbD#- d*LhCp) . I
(21)
459
and dependingon the ratio $! the behaviorat infinity will be determinti by the first or secondterm [ 121. We can now turn to the study of the electroweak case.The contributionof the bosonsto the static effective energyis & =
I
d3x[~(W$)2+$%$+]D@12+A(@2-~2)2] (29)
where the covariant derivativeof the Higgs doublet and the strengthsof the SU(2) x U( 1) gaugefie!ds are definedas
Using the boundaryconditions,it can be easily shown that J=2k@
(22)
where @ is the flux of the quasi-magneticfield
@= s
d2xC3 = -272
.
Ee-J;
5 --) -7
+ h.c.
(23)
(24)
$ (cfie-@’ + h.c.)
.(i) = .(if = n$Gu= ,Wo = ,(i)a = @a = n6; (33) CL &J UR R (where i = 1..t,fai a= 1, . .. . 3 denotethe family and color indices respectively) their contribution to the effective energyis given by [ 6,7] : EF = -4fP[Jks(W)
mz=2q2q2 .
- Ncs(%)1
(34)
&s(w) (25)
‘8 =552
where p=dz--ik,
(32)
Assuminga cold backgroundof fermionssatisfying
4 The asymptoticbehaviorfor the gaugefields at infinity canbederivedfrom(ll)-(13) z3 ---f
Bij = ai%j - aj%i .
(26)
(p&q s
w!?w; - E&qPWPW,E) Y 3 ‘J
Ncs(%) =& J d3Xeijk%ij%k3
(35) (36)
Comparingthis result with the 2 + 1 dimensionalcase, we seethat as a consequenceof the replacementk ---) ik, p has becamea complex number.Notice that as far as k* L rn: or equivalently ,u 5 pcrttr p has a positive real part. In addition, this real part is smaller comparedto the casek = 0 which clearly corresponds to the anti-screeningeffect of the Chern-Simonsterm. We then see that the value p = pcrtt correspondsto the point for which this effect completelyoverwhelms the screeningoriginating from the Higgs mechanism. As for the Higgs field, its asymptoticis
The effectsof the Chern-Simonsterm on the ground stateof the Weinberg-Salam modelin an externalmagnetic field and its influence on the phenomenonof W-condensation[ 141 has beenrecently analyzedby Poppitz [ 1.51.We will insteadconcentratehereon the influence of this term on the Z-string configuration and consequentlywe will set to zero the chargedW bosonsand the upper componentof the Higgs field. ¬ing as (b the lower componentof the H/ggs, the energytakesthe form
f-
E=
$t?
rni = 4A7*
+ $?-""+h.c.
(27) (28)
s
d3x[$(fij)* + :Zz + lD&\*
+ A(4* -Q*)* - (F”\(kl ZiFjk + $&zjk)]
(37)
G. Bimonte, G. Lazano /Physics Letters B 348 (1995) 4.57-W
460
where we have introducedthe electromagneticvector potential Ai and the neutralvector potential Zi by Ai= W~sin0+BicosB
138)
Zi= W~cose-
(39)
BisinB
tan6= g’ g and kt and k2 are de&d as 2
= &f,
h
(40)
I2
k2 = 9-P
f ’
(41)
Notice the absenceof Chern-Simonsterm assaciated to the field Ai, a fact which can be traced back to the vector characterof the electromagneticinteractions. Due to the presenceof the crossChern-Simons term, the field Zi will now act as a sourcefor Ai and it is not any more consistent(as in the purely bosonic case [4]) to set it to zero. Indeed,the equationsfor Ai are t?iF” = -k,&Qni I .
(42)
For ,X (i.e. k) small compared with the particle masses,we can replace the right-hand side of (42) by its value in the Nielsen-Olesen’case and then we obtain a tangentialmagneticfield 2:” * (43) r At, large distances,where Zt” goes to --m/q (q =
H = 2k l----e9
As we mentionedMore, all these considerations are valid for small chemical potentials and strictly speaking for small distancesfrom the cor5 of the string, A more complete analysis woald follow from the numericalstudy of the equationsof motion which result from the ansatz q5= f(r)exp(imC?), Z=Zee~+Z3q-j, A = Ae(r)ee + A3(rh,
(48)
(rZ3)’ -2$rf2Z3
(49)
I
f(rf9’
+2kzZ,‘+4kTrZs
-2qy3f2-2k2Z;+4k;F -
(m+4Z0)2 r2
=o.
f -q2Z?f
= 0 =0
(50)
-2Af(f2-q”)
(51)
Once the solutions are known, the electromagnetic potentials are calculated by direct integration (see Eq. (42) ) . At infinity, due to the long rangecharacter of electromagneti,m,we expecta substantialmodification of the Z-string profile. Indeed, the asymptotic behavior of the massivefields wil! turn from exponential to power like and additionally there will be a modification of the total,Z-flux. To the leadirg oroer in the chemicalpotential we obtain
ZO4
-;
- k;c,
(52)
7
z3 -i--
+r
.
-
c&z
I
which coincideswith the magneticfield generatedby an infinitely long wire carrying a current
(54) A3 --)
$kl
In(r)
Ae --f -k;k2c3-.
As in the abeliancase,therewill be a tangentialcomponentof the quasi-magneticfield C= -rz+*
= -kp(
Zf’)‘ee
:
(46)
We can estimatethe maximum,value of the magnetic field to be kmm H mix -J-. 9
(53)
l-4
(47)
1 r2
(55) (56)
Summarizing,we have shown that the presenceof cold neutralmatterconsiderablymodifies the original Z-stringprofile leadingto the appearance of long range magneticfields associatedwith currentsflowing along the string. Clearly,an importantissueto be addressedconcerns the stability of theseconfigurations.As it was established in Ref. [ 131,the bare electroweakstring is un-
G. Bimonte. G. Lazano /Physics Letters B 348 (1995) 457-461
461
stable for realistic values of the coupling constants. The questionis then to study whetherthe fermionscan improve the stability of the Z-string. Although the answer of this issuein the full electroweakmodel seems rathercomplicate4 an encouragingresult comesfrom the analysis of the semilocal string. In fact, for this case,stability is improved.This canbe shownby looking at the potential term in the Schriidinger-likeequation satisfiedby the perturbationsand noticing that it is less negativethan in the barecase.It would be interesting to explore in the electroweaktheory the region of stability as a function of the coupling constantsand the fermion density (and eventuallyinclude temperature corrections) in order to establishthe role of these configurationsin a cosmologicalsetting.
[3] T. Vachaspatiand M. Baniola, Phys. Rev. Len. 69 ( 1992) 1867. [4] T. Vachaspati,Phys. Rev. Let. 68 (1992) 1977. [5] Y. Nambu, Nucl. Phys. B 130 (1977) 505; M.B. &horn and A. Savit, Phys. Lett. B 77 (1978) 295; N.S. Manton, Phys. Rev. D 28 (1983) 2019. [El V.A. Rubakov and A.N. Tavkhelidze, Phys. Lett. B !65 (1985) 109. [7] A.N. Redlich and L.C.R..Wijewardhaoa,Phys. Rev. Lett. 54 (1985) 970; K. Tsokos, Phys. Lett. B 157 (1985) 413. [S] R. Jackiw and S. Templeton,Phys. Rev. D 23 (1981) 2291; J. Schonfeld, Nucl. Phys. B 185 (1981) 157. [Y] V.A. Rubakov, Prog. Theor. Phys. 75 (1986) 366; V.A. Matveev, VA. Rubakov, A.N. Tavkhelidze and V.F. Takamv, Nucl. Phys. B 282 ( 1987) 7000. [lo] S.K. Paul and A. Khare, Phys. Lea. B 174 (1986) 420; H. de Vega and EA. Schaposnik,Phys. Rev. Lett. 56 (1986)
We would like to thank Prof. Abdus Salam,the InternationalAtomic EnergyAgency and UNESCO for hospitality at the InternationalCentre for Theoretical Physics.
[ 111 V.I. lnozemtsev,Europhys. Lett. 5 (1988) 113; G. Lozano, M.V. Maniaq and EA. Schaposnik,Phys. Rev. D 38 (1988) 601. [ 121 L. Periv&ropoulos, Phys. Rev. D 48 (1993) 596. [ 131 T. Vachrspati,Nucl. Phys. B 397 (1993) 648; M. James,L. Perivolaropoulosand T Vachaspati,Nucl. Phys. B 395 (1993) 534. [ 141 J. Ambjam and l? Olesen, Nucl. Phys. B 315 (1989) 606; B 330 (1990) 193; Int. J. >Mod.Phys.A 5 (1993) 4525. [15] E.R. Poppitz, F~ys. Lett. B 309 (1993) 114.
.
I
2565.
References [ 1] H.B. Nielsen and P. Olesen, Nut!. Phys. B 61 (1973) 45. 121 For a recentreview see,R. Brandenberger,lnt. J. Mod. Phys. A 9 (1994) 2117.
PHYSICS LETTERS 6 PhysicsLetters B 348 (1995) 457-461
Finite fermion Qensityeffects on the electroweakstring G. Bimonte abbe*, G. Lozano a,2 a International Centre for Theoretical Physics, RO. Box 586. I-34100 Trieste, Italy b INFN, Sezione di Napoli, Napoli, Italy
Received 18 September1994 Editor: L. Alvarez-Gauti
Abstract We consider an electroweakstring in the backgroundof a uniform distribution of cold fermionic matter.As a consequence of the fermion number non-conservationin the Weinberg-Salammodel, the string producesa long-rangemagnetic field.
Since the pioneer&g work by Nielsen and Olesen [ 11, the study of vortex like solutions in gauge theorieshas attracteda constantinterest.This kind of objectsis by now believedto play a prominentrole in cosmology,for exampleby providing a mechanismto explain structureformation in the early universe [ 21. Until very recently,this type of configurationshave beenmainly studiedin theorieswith a nontrivial vacuum structure where the existenceof a topological conservationlaw ensures its stability. However, as recently stressedby Vachaspatiand Barrlola [3], the embedding of such configurations in larger gauge groups can be stable even in the absenceof such topological conservationlaws. This is what happens in the case of the &string [4], which results from embedding the U( 1) Nielsen-Olesenstring in the SU( 2) x U( 1) theoryof electroweakinteractions[ 51. In order to understandthe role of these configurations during the phase transition it is important to know how they are modified in typical thermodynamical situationssuch as non-zerotemperatureand finite matter density.In this letter we shall analyzethe case i E-mail addnzss:[email protected]. * E-mail address:[email protected].
in which the string is placed in a backgroundof cold neutralfermionic matter, a situation in which, due to the V-A characterof the electroweakinteractions,the fermion sectorof the model manifestsin a non trivial way. To illustrate the kind of phenomenaiha: occur, let us begin with the simpler U( 1) model and study the way in which a constantfermionic density affects the Nielsen-Olesenstring. We consider the system described by the Lagrangian density,
+ (U&4) (D”d)
- A(q52 - 7j2j2 + Lf .
(1)
Here the covariantdedvative for the scalar complex field 4 is defined as
a,+ = (4.lf W,)#
(2)
and 2, and Z,, are the vector potential and field strength. The interactionof the fermionswith the gaugefields is governedby the Lagrangiandensity
0370-2693/95/@9.50 @ 1995 Elsevier Science B.V. AI1 r&h@,reserved SSDIO370-2693(95)00211-l
G. Bimnnre, G. L.ozano/Physics Letters B 348 (1995) 457-461
458
where #; and J/L+are left-handedfermions with oppositechargeanda, are the Pauli matrices.Following Rubakov and Tavkhelidze161, we will considerthe case in which there is a cold (zero temperature)and neutral (n+ = n- = n/2) backgroundof fermions. In the weakly coupledregime,the bosonicfield can be treatedclassicallyand its contributionto the energy (in the static case) is simply given by .??b = I d3xE;Z; + lDj#12 + A(42 - 7j2)2] . J
(4)
As shown in Refs. [6,7] the presenceof a finite density manifeststhroughthe appearancein the effective energyof a Chern-Simonsterm [ 81 which resultsfrom the integrationof the fermionic determinant Ef-2
k
in recent years mainly in the context of high tcmperaturesuperconductivityand the quantumHall ?:ffeet [ lo,1 11. A generalfeature of these solutions ;s that as a result of the Chem-Simonsinteraction the vortex acquiresan electric chargeand consequentlya radia! electric field. The transformation(9) correspondsto the combined effect of an Euclidean rotation and a duality transformationin the xc -x3 plane.As a consequence, what in the 2 + 1 dimensionalcase correspondedto an electric chargewill now becomea flux of current along the string aud the radial electric field will be now convertedinto a tangentialmagnetic field. Making the ansatz 4 = f(r) exp(imd),
Z = -pZ@
(10)
+ Z3e3
the equationsof motion then read (rzy
- 2q’rfZ3 f 2kZi = 0
(11)
~3xEVkz. .z k 11
(5) I the coefficientk beingrelatedto the chemicalpotential P by k= m2 is’ The effective energy
(6)
subjectto the boundaryconditions Z3(00)
E = Eb + Ef
=o,
Ze(oQ)=-y,
f(w)
(7)
was originally usedin Refs. [ 6,9] whereit was shown that the trivial vacuumbecomesunstablefor chemical potentialslargerthan the critical value 32$ kit = -m, 8
(8)
where the massof the gaugeparticle is nrz = 2q2q2. We will insteadconsiderthecaseof small chemical,potentials p 5 petit in order to study how the fermionic backgroundmodifiesthe Nielsen-Olesenstring. Notice that by making in (7) the transformation A3 + iAo,
(12)
x3 -b -in’,
k -+ ik
(9)
we obtain the actian of the 2 + 1 dimensionalChemSimons-Higgssystem. Vortex like solutions in this and related models bave been object of much study
Z3(0>
< 00, Z@(O)=o,
f(0) =o.
=7)
(14)
(15)
Although an exact solution to the equationscannot be obtained analytically, it is possible (in complete analogy to the 2 + 1 dimensionalcase [ 10,111) to expressthe first termsof an expansionin k in a closed r.,.rn involving the solutions of the Nielsen-Olesen vortex
z, = zTOz3 =
$$Zt”
+ ;) + O(k4)
-k(Z,N@ + f$ +Q(k3)
f = f”” + O(k4)
4
(16)
(17) (18)
wherethe superscriptNO refersto the Nielsen-Qlesen solutions, that is, solutions to Eqs. (1 l)-( 13) with k = 0. Thus, we see that the leading effect is the
G. Bimonte, G. Lozano / Physics L.&ten I) 348 (1995) 457-461
appearanceof a tangential componentof the quasimagneticfield Ci (we reservethe namemagneticfield for later when we introduce the unbroken CI(1) of electromagnetism) Ci = &QkZjk
(19)
CO= -rZj
(201
which can he associatedto a flux of current flowing along the string
J= d*xh, jl = iq(cbD#- d*LhCp) . I
(21)
459
and dependingon the ratio $! the behaviorat infinity will be determinti by the first or secondterm [ 121. We can now turn to the study of the electroweak case.The contributionof the bosonsto the static effective energyis & =
I
d3x[~(W$)2+$%$+]D@12+A(@2-~2)2] (29)
where the covariant derivativeof the Higgs doublet and the strengthsof the SU(2) x U( 1) gaugefie!ds are definedas
Using the boundaryconditions,it can be easily shown that J=2k@
(22)
where @ is the flux of the quasi-magneticfield
@= s
d2xC3 = -272
.
Ee-J;
5 --) -7
+ h.c.
(23)
(24)
$ (cfie-@’ + h.c.)
.(i) = .(if = n$Gu= ,Wo = ,(i)a = @a = n6; (33) CL &J UR R (where i = 1..t,fai a= 1, . .. . 3 denotethe family and color indices respectively) their contribution to the effective energyis given by [ 6,7] : EF = -4fP[Jks(W)
mz=2q2q2 .
- Ncs(%)1
(34)
&s(w) (25)
‘8 =552
where p=dz--ik,
(32)
Assuminga cold backgroundof fermionssatisfying
4 The asymptoticbehaviorfor the gaugefields at infinity canbederivedfrom(ll)-(13) z3 ---f
Bij = ai%j - aj%i .
(26)
(p&q s
w!?w; - E&qPWPW,E) Y 3 ‘J
Ncs(%) =& J d3Xeijk%ij%k3
(35) (36)
Comparingthis result with the 2 + 1 dimensionalcase, we seethat as a consequenceof the replacementk ---) ik, p has becamea complex number.Notice that as far as k* L rn: or equivalently ,u 5 pcrttr p has a positive real part. In addition, this real part is smaller comparedto the casek = 0 which clearly corresponds to the anti-screeningeffect of the Chern-Simonsterm. We then see that the value p = pcrtt correspondsto the point for which this effect completelyoverwhelms the screeningoriginating from the Higgs mechanism. As for the Higgs field, its asymptoticis
The effectsof the Chern-Simonsterm on the ground stateof the Weinberg-Salam modelin an externalmagnetic field and its influence on the phenomenonof W-condensation[ 141 has beenrecently analyzedby Poppitz [ 1.51.We will insteadconcentratehereon the influence of this term on the Z-string configuration and consequentlywe will set to zero the chargedW bosonsand the upper componentof the Higgs field. ¬ing as (b the lower componentof the H/ggs, the energytakesthe form
f-
E=
$t?
rni = 4A7*
+ $?-""+h.c.
(27) (28)
s
d3x[$(fij)* + :Zz + lD&\*
+ A(4* -Q*)* - (F”\(kl ZiFjk + $&zjk)]
(37)
G. Bimonte, G. Lazano /Physics Letters B 348 (1995) 4.57-W
460
where we have introducedthe electromagneticvector potential Ai and the neutralvector potential Zi by Ai= W~sin0+BicosB
138)
Zi= W~cose-
(39)
BisinB
tan6= g’ g and kt and k2 are de&d as 2
= &f,
h
(40)
I2
k2 = 9-P
f ’
(41)
Notice the absenceof Chern-Simonsterm assaciated to the field Ai, a fact which can be traced back to the vector characterof the electromagneticinteractions. Due to the presenceof the crossChern-Simons term, the field Zi will now act as a sourcefor Ai and it is not any more consistent(as in the purely bosonic case [4]) to set it to zero. Indeed,the equationsfor Ai are t?iF” = -k,&Qni I .
(42)
For ,X (i.e. k) small compared with the particle masses,we can replace the right-hand side of (42) by its value in the Nielsen-Olesen’case and then we obtain a tangentialmagneticfield 2:” * (43) r At, large distances,where Zt” goes to --m/q (q =
H = 2k l----e9
As we mentionedMore, all these considerations are valid for small chemical potentials and strictly speaking for small distancesfrom the cor5 of the string, A more complete analysis woald follow from the numericalstudy of the equationsof motion which result from the ansatz q5= f(r)exp(imC?), Z=Zee~+Z3q-j, A = Ae(r)ee + A3(rh,
(48)
(rZ3)’ -2$rf2Z3
(49)
I
f(rf9’
+2kzZ,‘+4kTrZs
-2qy3f2-2k2Z;+4k;F -
(m+4Z0)2 r2
=o.
f -q2Z?f
= 0 =0
(50)
-2Af(f2-q”)
(51)
Once the solutions are known, the electromagnetic potentials are calculated by direct integration (see Eq. (42) ) . At infinity, due to the long rangecharacter of electromagneti,m,we expecta substantialmodification of the Z-string profile. Indeed, the asymptotic behavior of the massivefields wil! turn from exponential to power like and additionally there will be a modification of the total,Z-flux. To the leadirg oroer in the chemicalpotential we obtain
ZO4
-;
- k;c,
(52)
7
z3 -i--
+r
.
-
c&z
I
which coincideswith the magneticfield generatedby an infinitely long wire carrying a current
(54) A3 --)
$kl
In(r)
Ae --f -k;k2c3-.
As in the abeliancase,therewill be a tangentialcomponentof the quasi-magneticfield C= -rz+*
= -kp(
Zf’)‘ee
:
(46)
We can estimatethe maximum,value of the magnetic field to be kmm H mix -J-. 9
(53)
l-4
(47)
1 r2
(55) (56)
Summarizing,we have shown that the presenceof cold neutralmatterconsiderablymodifies the original Z-stringprofile leadingto the appearance of long range magneticfields associatedwith currentsflowing along the string. Clearly,an importantissueto be addressedconcerns the stability of theseconfigurations.As it was established in Ref. [ 131,the bare electroweakstring is un-
G. Bimonte. G. Lazano /Physics Letters B 348 (1995) 457-461
461
stable for realistic values of the coupling constants. The questionis then to study whetherthe fermionscan improve the stability of the Z-string. Although the answer of this issuein the full electroweakmodel seems rathercomplicate4 an encouragingresult comesfrom the analysis of the semilocal string. In fact, for this case,stability is improved.This canbe shownby looking at the potential term in the Schriidinger-likeequation satisfiedby the perturbationsand noticing that it is less negativethan in the barecase.It would be interesting to explore in the electroweaktheory the region of stability as a function of the coupling constantsand the fermion density (and eventuallyinclude temperature corrections) in order to establishthe role of these configurationsin a cosmologicalsetting.
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We would like to thank Prof. Abdus Salam,the InternationalAtomic EnergyAgency and UNESCO for hospitality at the InternationalCentre for Theoretical Physics.
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