BPS Solutions to a Generalized Maxwell-Higgs Model

0.19 MB·English

by  D. Bazeia

BPS Solutions to a Generalized Maxwell-Higgs Model
D. Bazeia1,2, E. da Hora1,3, C. dos Santos4 and R. Menezes2,5.
1Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil.
2Departamento de F´ısica, Universidade Federal da Campina Grande, 58109-970, Campina Grande, Para´ıba, Brazil.
3Department of Mathematical Sciences, Durham University,
DH1 3LE, Durham, County Durham, United Kingdom.
4Centro de F´ısica e Departamento de F´ısica e Astronomia,
Faculdade de Ciˆencias da Universidade do Porto, 4169-007, Porto, Portugal.
5Departamento de Ciˆencias Exatas, Universidade Federal da Para´ıba, 58297-000, Rio Tinto, Para´ıba, Brazil.
We look for topological BPS solutions of an Abelian-Maxwell-Higgs theory endowed by non-
standardkinetictermstobothgaugeandscalarfields. Here,thenon-usualdynamicsarecontrolled
by two positive functions, G(|φ|) and w(|φ|), which are related to the self-dual scalar potential
2 V (|φ|) of the model by a fundamental constraint. The numerical results we found present inter-
1 esting new features, and contribute to the development of the recent issue concerning the study of
0 generalized models and theirapplications.
2
n PACSnumbers: 11.10.Kk,11.10.Lm
a
J
I. INTRODUCTION Ingeneral,k-theorieshavebeenusedaseffective mod-
4
1 els mainly in the cosmological scenario. Here, many au-
In the context of classical field theories, topological thors have used the so-called k-essence models [6] to in-
] vestigatethepresentacceleratedinflationaryphaseofthe
h structures, such as kinks [1], vortices [2] and magnetic
universe [7]. Also, such models can be used to study
t monopoles [3], are described as static solutions to some
- strong gravitational waves [8], dark matter [9], tachyon
p nonlinear models. In particular, such models must al-
matter[10]andothers[11]. Therearealsoimportantmo-
e low for the spontaneous symmetry breaking mechanism,
h since topological solutions are formed during a symme- tivations concerning the strong interaction physics; see,
[ for instance, Ref. [12].
try breaking phase transition. In this sense, topologi-

1 cally non-trivial configurations are of great interesting In this context, the overallconclusion is that the exis-
v to physics[4], mainlyconcerningthe cosmologicalconse- tenceoftopologicalstructuresisquitesensibletotheuse
4 quencestheymayengender,sincesuchconfigurationscan of non-standard kinetic terms. Here, the more interest-
7 appear in a rather natural way, during phase transitions ing issue is that such structures can exist even as solu-
9 in the early universe. tions ofsome theorieswhichdo notallowfor the sponta-
2
In particular, vortices are described as rotationally neous symmetry breaking mechanism [13]. On the other
.
1 symmetric solutions of a planar Abelian-Maxwell-Higgs hand, a rather natural way to study such topologicalso-
0 model endowed by a fourth-order scalar potential for lutions is comparing them with their canonical counter-
2 the matter self-interaction, which also introduces sym- parts. Inthissense,someofushavealreadyinvestigated
1
metry breaking and nonlinearity. In the usual case, the topological solutions in the context of k-field models en-
:
v Maxwell term controls the dynamics of the gauge field, dowedby spontaneous symmetry breaking potentials for
Xi and the covariant derivative squared term controls the the scalar-matter self-interaction, and interesting results
dynamics of the scalar field. In such context, vortices can be found, for instance, in Ref. [14]. Other interest-
r
a are finite-energy solutions of a set of two coupled first- ing results concerning k-field models and their classical
orderdifferentialequations,namedBogomol’nyi-Prasad- solutions can be found, for instance, in Ref. [15].
Sommerfield (BPS) equations [5]. In this case, the BPS Another interesting issue concerning topological k-
vortices are the minimal-energy solutions of the model, solutions is that they can be either much largeror much
andtheyhaveinterestingapplications,mainlyconcerning smaller than their usual counterparts. In this sense, k-
the superconductivity phenomena [2] and the superfluid bosons can mediate either large-rangeor small-range in-
He4. teractions. Also, important physical quantities, such as
During the last years, beyond the standard configura- energy density and electric and magnetic fields, can ex-
tions previously cited, modified ones, also named topo- hibit proeminent variations on their tipical profiles, in-
logical k-solutions, have been intensively studied. These cluding on their maximum values; for a detailed treat-
solutions arise in a special class of theoretical field mod- ment of some of such features, see Ref. [16].
els, generically named k-theories, which are endowed by In the present context, some k-theories can also sup-
non-usualkineticterms. Asexpected,suchtermschange porttopologicalsolutionswithafinitewavelength. These
the dynamics of the overall model under investigation. solutions, generically named compactons [17], are quite
Moreover, it is important to reinforce that the idea of a differentfromthe standardtopologicalstructures,which
non-standarddynamics arisesina rathernaturalway,in interact even if separated by an infinite displacement,
the context of string theories. sincetheyhaveaninfinitewavelength: twoadjacentcom-
2
pactons will interact only if they come into close con- netic matter term. Both G(φ) and w(φ) are dimen-
| | | |
tact, due to their already explained finite wavelength. sionless functions to be given below, as functions of the
In this sense, compactons are most appropriated to de- amplitude of the scalar field.
scribe particle-like configurations than the usual non- Inthe caseofvortices,itis convenientto dealwithdi-
compactified topological structures. mensionlessvariables. So,forsimplicity,weintroducethe
So, in this workwe presentnew results concerningthe mass scale M, and use it to implement the scale trans-
topologicalsolutionsofanAbelian-Maxwell-Higgsmodel formations: xµ xµ/M, φ M1/2φ, Aµ M1/2Aµ,
endowed by non-usual kinetic terms to both gauge and e M1/2eand→υ M1/2υ,w→hereυ stands→forthesym-
→ →
scalar fields. Here, the non-standard dynamics are in- metry breaking parameter of the model. In this case,
troduced by two positive functions, G(φ) and w(φ), we get M3 , with being the dimensionless
G
| | | | L → L L
which couple with the Maxwell term and with the co- Lagrange density to be used from now on, which has
variantderivativesquaredterm,respectively. TheEuler- the same functional form of . Moreover, we omit
G
L
Lagrange equations of motion of such theory are hard the coupling constant related to the scalar matter self-
to solve, then we focus our attention only on the finite- interaction. Also, we take e and υ as real and positive
energy solutions of the BPS equations of the model. parameters.
These equations can be obtained by minimizing the en- To search for vortex solutions, the canonical proce-
ergy functional of the model, which can be done via dure is to deal with the Euler-Lagrangeequations of the
an important constraintbetween G(φ), w(φ) and the model. In the present case, these equations are
| | | |
scalarpotentialV (φ)forthematterself-interaction;see
| | G∂ Fµλ+Fµλ∂ G=Jλ , (2)
eqs.(15) and (16) below. In this case, rotationally sym- µ µ
metricBPSvorticesaredescribedasthe minimal-energy
and
solutions of the cited model, and they engender interest-
ing features, as explained below. w∂ ∂µ φ +∂ φ ∂µw e2A Aµ φ w
µ µ µ
| | | | − | |
Thispaperisoutlinedasfollows: inthenextSec.II,we 1 2 dw F2 dG 1 dV
introduce the model and develop the theoretical frame- = D φ . (3)
µ
2| | d φ − 8 d φ − 2d φ
work which allows us to get to its Bogomol’nyi-Prasad- | | | | | |
Sommerfield equations. In Sec.III, we prove the con- Here, we take F2 =F Fµν and
µν
sistence of the theoretical framework previously devel-
oped by using it to investigate the existence of new BPS Jµ = 2e2w φ2Aµ (4)
− | |
states. Here, we note that such states are constrained
to the choices made for G(φ) and w(φ): for any ac- as the modified 4-current vector.
ceptable pair of such functio|n|s, there is|a|corresponding TheGausslawfortime-independentfieldscanbewrit-
BPS configuration. Also in Sec.III, we show how to use ten as
the theoretical framework presented in this work to re- G∂ ∂kA0+∂ A0∂kG= 2e2w φ2A0 , (5)
covertherecovertheBPSresultsconcerningthestandard k k − | |
Maxwell-Higgsmodel. InSectionIV,weperformthenu- where E = −→A0. We note that the Eq. (5) is trivially
merical analysis concerning the new BPS states previ- verified by A−0∇=0. So, we fix this gauge and use it from
ously presented, we depict the corresponding minimal-
now on.
energy modified solutions and comment on their main
We look for vortex solutions of the form
features. Finally, in Sec.V, we present our conclusions
and perspectives. φ(r,θ)=υg(r)einθ , (6)
From now on, we use standard conventions, including
natural units system, and a plus-minus signature for the
θ
planar Minkowski metric: diag(ηµν)=(+ ). A= (a(r) n) . (7)
−− −er −
b
Here, r and θ are polar coordinates, and n =
II. THE MODEL
1, 2, 3,... is the winding number (vorticity) of the
± ± ±
solution. In terms of (6) and (7), the Euler-Lagrange
Inthissection,weintroducethemodel. Itisdescribed equations (2) and (3) can be rewritten as
by the (2+1)-dimensional Lagrange density
d2a dG G da
2 2 2
G + =2e υ g aw , (8)
G = 1G(φ)FµνFµν+w(φ) Dµφ2 V (φ) . (1) dr2 (cid:18)dr − r(cid:19) dr
L −4 | | | | | | − | |
Here, Fµν = ∂µAν ∂νAµ is the usual Faraday field d2g 1dg a2g 1 1 da 2 dG
− w +
strength tensor, Dµφ = ∂µφ + ieAµφ is the covari- dr2 rdr − r2 − 4υ2 erdr dg
ant derivative and V (φ) is the spontaneous symmetry (cid:18) (cid:19) (cid:18) (cid:19)
breaking potential. A|lso|, G(φ) is the ”dieletric func- 1 dV 1 dg 2 g2a2 dw
tion”, and w(|φ|)|Dµφ|2 stand|s|for the non-standardki- = 2υ2 dg − 2 (cid:18)dr(cid:19) − r2 ! dg . (9)
3
Equations (8) and (9) are the Euler-Lagrange equations of such states is still possible, and it is closely related to
of motion to the profile functions a(r) and g(r), respec- an important constraint between G(g), w(g) and V (g):
tively.
To solve (8) and (9), we need to specify the model. d √GV =√2eυ2wg . (16)
In general, we can do it by choosing non-trivialforms to dg
G(g) and w(g). In this case, we have to keep in mind
that both these functions must be positive, in order to Asweclarifybelow,foranypositivechoicestoG(g)and
avoid problems with the energy of the model; see the w(g), there is acorrespondingsymmetrybreakingHiggs
expression for the energy density (15) below. Also, we potentialV (g)asasolutionof (16). Inthiscontext,gen-
need to choose a Higgs potential V (g) which allows for eralizedfirstorderequationscanbefoundbyminimizing
the spontaneous symmetry breaking mechanism. the energy functional (15).
Before that, we note that the limit w(g) 1 leads us To search for BPS vortex solutions in the modified
back to the model studied in [18], which is s→upported by model,weneedtoknowhowthefunctionsg(r)anda(r)
applications concerning the interaction between quarks behave,neartheoriginandasymptotically. Neartheori-
andgluons[19]. Inthiscase,thelimitG(g) 1leadsus gin, these functions must avoid singular fields. So, they
back to the usual Maxwell-Higgstheory. In→this context, have to behave according to
if we choose the symmetry breaking Higgs potential
g(r 0) 0 and a(r 0) n=1 , (17)
→ → → →
e2υ4 2 2
Vs(g)= 1 g , (10) wherewehavefixedn=1,forsimplicity. Also,the sym-
2 −
metry breaking vortex configurations must have a finite
(cid:0) (cid:1)
the Euler-Lagrange equations (8) and (9) can then be totalenergy. So,asaconditiontomaketheenergyfinite,
rewritten as the energy density (15) must vanish. Then, asymptoti-
cally, g(r) and a(r) have to obey
d2a 1da
2 2 2
=2e υ g a , (11)
dr2 − rdr g(r ) 1 and a(r ) 0 . (18)
→∞ → →∞ →
In the next Sec. III, we use the theoretical framework
d2g 1dg a2g 2 2 2 developed in this Section to investigate the existence of
+ =e υ g g 1 . (12)
dr2 rdr − r2 − BPS states in the generalized model. Also, we present
(cid:0) (cid:1) and comment the resulting numerical solutions. Finally,
Accordingtoourconventions,eqs.(11)and(12)arecom-
we show how to map the standard Maxwell-Higgs and
pletely solvable by the first order differential equations
Chern-Simons-Higgs first order equations; even in these
cases,Eq.(16)leadsustoaphysicallydifferentrotation-
dg ga
= , (13) ally symmetric solutions.
dr ± r
III. NEW BPS STATES
1da
2 2 2
= e υ g 1 . (14)
r dr ± −
(cid:0) (cid:1) We now pay due attention to the modified BPS states
Thesolutionsofeqs.(13)and(14)aretheBogomol’nyi- themselves. Here,we choose positive functional forms to
Prasad-Sommerfield (BPS) states of the standard G(g)andw(g). Then,wesolvetheconstraint(16)toget
Maxwell-Higgsmodel (10). These solutions are the well- totheconsistentHiggspotentialV (g),whichmustallow
known Abrikosov-Nielsen-Olesen (ANO) ones [2], which for the spontaneous symmetry breaking mechanism. A
solve the equations of motion(11) and (12) by minimaz- posteriori,weusetheseconventionstoobtainthefirstor-
ing the energy of the resulting vortex configurations. der differentialequations of the modified model, by min-
In general, for non-trivial choices to G(g) and w(g), imizing its energy functional (15). Then, in the next
the equations of motion (8) and (9) will be much more Sec.IV, we numerically solve these equations, according
sofisticated than the eqs.(11) and (12). So, in order to to the boundary conditions (17) and (18). Finally, in
get an useful insight about the non-trivial case, we con- Sec.V, we comment the main features of the resulting
sidertheexpressionfortheenergydensityofthemodified solutions.
vortex solutions: The model to be studied here has two unusual func-
tions, G(g)andw(g), whichwehaveto fix to determine
G 1da 2 dg 2 g2a2
2 how they affect the vortex solutions. The function G(g)
ε= +υ w + +V . (15)
2e2 (cid:18)rdr(cid:19) (cid:18)dr(cid:19) r2 ! standsfor a”dieletricfunction”, anditcontrolsthe non-
standard kinetic term to the gauge field. On the other
From Eq. (15), we note that the presence of non-trivial hand, the function w(g) controls the non-usual kinetic
G(g) and w(g) makes it hard to obtain the BPS states scalar matter term. Both these functions are dimension-
of the modified model. Even in this case, the existence less,andarefunctionsoftheamplitudeofthescalarfield.
4
Also, they must be positive, in order to avoid problems We note that the modified equations (24) and (25) can
with the energy of the model; see Eq. (15). notbeparametrizedintothestandardones,i.e.,eqs.(13)
A rather natural way to investigate the consistence of and (14). Even in this case, the energy of the modi-
the theoreticalframeworkdevelopedinthe previousSec. fied field solutions is bounded from below, and the Bo-
II is to study the BPS states of the standard Maxwell- gomol’nyi bound is
Higgs model, which is defined by G(g) = w(g) = 1. In
this case, Eq. (16) leads us to the potential (10). Then, E =3Es . (26)
the energy density (15) can be rewritten as
Here,asinthestandardcase,thetotalenergyoftheBPS
2 2
1 1 da dg ga states is quantized; see Eq. (20).
2 2 2
ε = eυ g 1 +υ
2 erdr ∓ − dr ∓ r In order to reinfoce the consistence of the theoretical
(cid:18) (cid:19) (cid:18) (cid:19)
υ2da υ2 d(cid:0) (cid:1) framework presented in this work, we introduce another
g2a . (19) modified model. It is defined by
∓ r dr ± r dr
The resulting total ener(cid:0)gy E(cid:1) is minimized by the first G(g)= g2+1 2 and w(g)=2g2 . (27)
orderequations(13)and(14). Inthiscontext,theenergy
of the BPS states is According to thes(cid:0)e conv(cid:1)entions, Eq. (16) leads us to the
standard Higgs potential (10). So, the modified model
2 2
E = ε(r)d r =2πυ n . (20) (27) has the same vacuum structure as the usual model.
s
| |
Z Eveninthiscontext,thenewmodelisnotaparametriza-
As usual, this is the lower bound of the energy func- tion of the standard one; see the BPS eqs.(29) and (30)
tional, i.e., the Bogomol’nyibound. Also, it is quantized below. The modified energy density can be written in
according to the winding number n. the form
Now, we introduce an interesting new model, which is
defined by g2+1 2 1 da eυ2 g2 1 2
ε = −
g2+3 2 (cid:0) 2 (cid:1) erdr ∓ (g(cid:0)2+1) (cid:1)!
2
G(g)= and w(g)=2 g +1 . (21) 2
g2 2 2 dg ga
(cid:0) (cid:1) +2υ g
(cid:0) (cid:1) dr ∓ r
According to these choices, we solve (16) to get to the (cid:18) (cid:19)
υ2 d
potential 2 2
a g 1 g +1 , (28)
± r dr −
2
V (g)=g V (g) , (22)
s (cid:0) (cid:0) (cid:1)(cid:0) (cid:1)(cid:1)
and the resulting energy functional is minimized by the
which allows for the spontaneous symmetry breaking first order equations
mechanism,asdesired. Wepointoutthatthisnewmodel
is not a parametrization of the standard Maxwell-Higgs dg ga
= , (29)
one, since the vacuum manifold of the two models are dr ± r
quite different: it is a circle for the usual model (10),
whileitisadotsurroundedbyacircleforthenewmodel
(22). 1da e2υ2 g2 1
= − . (30)
Using eqs.(21) and (22), the energy density (15) can rdr ± (g2+1)
(cid:0) (cid:1)
then be rewritten as
Then, the Bogolmol’nyibound is also givenby (20), and
g2+3 2 1 da eυ2g2 g2 1 2 we conclude that the modified solutions (29) and (30)
ε = −
(cid:0) 2g2 (cid:1) erdr ∓ (g2(cid:0)+3) (cid:1)! havethesameenergyofthestandard(13)and(14)ones.
ToendthisSection,weshowhowtousethetheoretical
2
2 2 dg ga framework developed in this work to map the standard
+2υ g +1
dr ∓ r Maxwell-andChern-Simons-Higgsfirstorderdifferential
(cid:18) (cid:19)
υ2 d(cid:0) (cid:1) equations. We do it by reviewing the model studied in
2 2
a g 1 g +3 . (23) [18], which is defined by w(g) = 1. In this case, as pre-
± r dr −
sentedinthatwork,aninterestingchoiceto the dieletric
(cid:0) (cid:0) (cid:1)(cid:0) (cid:1)(cid:1)
Then, the corresponding energy functional is minimized function is
by the equations
e2υ2 −1
2
dg ga G(g)=(1+λ) 1+2λ g . (31)
= , (24) k2
dr ± r (cid:18) (cid:19)
Here, λ is a real auxiliary parameter which controls the
1da e2υ2g2 g2 1 model, and k stands for the coupling constant corre-
= − . (25)
r dr ± (g2+3) sponding to the Chern-Simons term. In the limit λ 0,
(cid:0) (cid:1) →
5
Eq. (31) gives G(g) 1, and (16) reproduces the stan- In the present work, the numerical strategy to be em-
→
dardMaxwell-Higgssystem(10). Ontheotherhand,the ployed is the relaxation one. In this case, it is necessary
limit λ gives to input anapproximatedfield solution. Then, our algo-
→∞
rithmwill”relax”itintothecorrectone. Tostartthenu-
k2
G(g)= . (32) merical analysis, we consider a variation of the standard
2e2υ2g2 Maxwell-Higgsmodel,inwhichthepotentialfortheself-
interaction for the scalar field is not present. Then, we
We then solve Eq. (16) to get
useitssolutionstosolvetheself-dualMaxwell-Higgscase,
e4υ6 2 2 2 i.e., the equations (13) and (14), from which we get the
V (g)= g g 1 , (33)
k2 − well-understood Abrikosov-Nielsen-Olesen (ANO) con-
figurations. We then use these solutions to initialize the
(cid:0) (cid:1)
whichisexactlythesixth-ordersymmetrybreakingHiggs
numerical study of the modified theory.
potential one finds in the usual self-dual Chern-Simons-
Starting from such ANO configurations, we numeri-
Higgs model. Now, from eqs.(32) and (33), the energy
cally solve the modified BPS equations presented in the
density (15) can be written in the form
previousSection,i.e.,eqs.(24)and(25),andalsoeqs.(29)
k2 1 da 2e3υ4 2 and(30),fore=υ =n=1. Thesolutionsfortheprofile
ε = g2 g2 1 functions g(r) and a(r) are plotted in Figs. 1 and 2,
4e2υ2g2 erdr ∓ k2 −
(cid:18) (cid:19) respectively. Also, we depict the solutions for the corre-
2 (cid:0) (cid:1)
dg ga 1 d sponding energy densities; see Figure 3.
2 2 2
+υ υ a g 1 .(34)
dr ∓ r ± rdr −
(cid:18) (cid:19)
(cid:0) (cid:0) (cid:1)(cid:1) 1,0
Theresultingtotalenergyisminimizedbythedifferential
equations
dg ga
= , (35) 0,8
dr ± r
1da 2e4υ4
2 2
r dr =± k2 g g −1 , (36) 0,6
(cid:0) (cid:1)
and the Bogomol’nyi bound is E = E . The equations g(r)
s
(35) and (36) are just the BPS ones for the standard
Chern-Simons-Higgs system (33). In this sense, Eq. (31) 0,4
works as a unified way to map both the Maxwell- and
theChern-Simons-Higgsfirstorderdifferentialequations.
Finally, we point out that, despite the possibility of re-
0,2
producing suchChern-Simons equations,the unified pic-
ture(31)leadsustoaphysicallydifferentsituation,since
the modified model we consider here only supports non-
charged field solutions; see the Gauss law (5) above. In 0
this case, the modified nontopological field solutions are 0 2 4 6 8 10 12 14
unstable,andtheycandecayintotheelementarymesons r
of the model. This fact makes an important difference, FIG.1: NumericalBPSsolutions tog(r)forthemodels(21)
since the nontopological configurations presented by the (dash-dottedredline) and(27)(dashedblueline). Thestan-
self-dual Chern-Simons-Higgs theory are stable, and can dard solution is also depicted (solid black line), for compari-
not decay. son.
InthenextSection,wepresentthemodifiedfirstorder
numericalsolutionsfortheprofilefuntionsg(r)anda(r). In Figure 1, we present the solutions for the profile
Also,wecommentonthemainfeaturesofsuchsolutions. functiong(r),andweseethatboththemodifiedprofiles
reachtheir vacuumvalues more slowlythantheir canon-
icalcounterpart. Inthissense,suchsolutionshaveacore
IV. NUMERICAL SOLUTIONS
whichisgreaterthantheusualMaxwell-Higgsone. Here,
the conclusion is that, in general, the introduction of a
Let us focus attention on the modified numerical so- non-canonical dynamics allows for the existence of self-
lutions themselves. The equations to be solved are the dual field solutions g(r) with a increased characteristic
first order ones (24) and (25), and (29) and (30). Also, length. Also, we note that the solution for (24) and (25)
we solve the standard equations (13) and (14), for com- goes to its vacuum configuration more slowly than that
parison. In all cases, the functions g(r) and a(r) must for (29) and (30). So, beyond the fact of increasing the
obey the boundary conditions (17) and (18). core of the solutions, we note that (21) increases it more
6
than the (27). We believe that this fact is related with decrescent for all values of the independent variable. In
the corresponding self-dual Higgs potential, which is of this context, this difference reinforces our previous con-
sixth-order for the model (21), and of fourth-order for clusion,since suchmodified behaviourmimics the one of
the (27) one; see equations (22) and (10) above. the energy density of the usual self-dual Chern-Simons
theory.
1,0
2,5
0,8
2,0
0,6
1,5
a(r)
e(r)
0,4
1,0
0,2
0,5
0
0 2 4 6 8 10 12 14 0,0
r 1 2 3 4 5 6 7 8 9 10
r
FIG. 2: Numerical BPS solutions to a(r). Conventions as in
FIG. 1. FIG. 3: Plots of the energy density ε(r). Conventions as in
FIG. 1.
In Fig. 2, we depict the numerical solutions for the
The solution for the BPS energy density of the modi-
function a(r). Here, as in the Fig. 1, we note that the
fied model (27) is also depicted in Figure 3, and we see
modifiedsolutionsgototheirvacuumstatesmoreslowly
that its behaviour is qualitatively similar to that of the
thanthestandardMaxwell-Higgsprofile,andsosuchso-
Maxwell-Higgs model: it reaches its maximum value as
lutions have a increased characteristic length. This be-
r goes to 0, and it is monotonically decrescent for all r.
haviour reinforces the previous conclusion, which states
Here, however, there are two important differences: the
that a non-standard dynamics leads to a self-dual field
firstone is onthe maximumvalue ofthe modifiedprofile
solutions with a increased core. We also see that, be-
itself,whichissmallerthantheusualone,andthesecond
yond the fact of increasing the core of a(r), the model
oneisonthecharacteristiclengthofthenon-standardso-
(21) enlarges it more than the (27). Finally, we point
lution, which is greater than its canonical counterpart.
out the existence of a proeminent plateau in the profile
Animportantissueconcerningthestudyoftopological
related to (21), near the origin: such structure is also
structuresisthecomputationoftheirtopologicalcharges,
present in the self-dual Chern-Simons-Higgs case, which
which must be conserved. In the present case, i.e., for
is governed by the potential (33). In this context, it is
electricallynon-chargedfieldsolutionsoftheform(6)and
interestingto note thatthe existence ofsuchproeminent
(7),this chargeis relatedtothe magneticfieldgenerated
plateau seems to be closely related to the vacuum mani-
by such solutions themselves. To investigate this issue,
fold of a sixth-order symmetry breaking potential, since
we introduce the topological current
it is also present in the modified model (21), which is
governedby (22). Jµ =ǫµνλ∂ A , (37)
ν λ
The Fig. 3 encloses the solutions for the energy densi- which is clearly conserved. Here, ǫ012 = 1. The 0th-
ties of the new BPS states; see equations (23) and (28). −
componentofsuchcurrent,thatis,thetopologicalcharge
Weseethattheprofilefortheenergydensitycorrespond-
density, can be written as
ingtothemodifiedmodel(21)isquitedifferentfromthat
of the standard Maxwell-Higgs theory: in the canonical J0 =∂ A ∂ A =B , (38)
x y y x
−
case, the energy density reaches its maximum value in
andweseethatitisdirectlyproportionaltothemagnetic
the limit r 0, and it is monotonically decrescent for
→ field B. In this case, the topological charge is given by
all r. On the other hand, in the modified case (21), the
energy density reaches its maximum value at some finite
2
Q = Bd r Φ . (39)
distance R from the origin, and it is not monotonically T ≡ B
Z
7
Here, Φ stands for the flux of the magnetic field. that of the usualMaxwell-Higgs case,since it reachesits
B
According(7),(17)and(18),themagneticfluxΦ can maximum value as r 0, and it is monotonically de-
B
→
be rewritten in the form crescent ever. Here, we point out the behaviour of the
modified solution, which assures the conservation of the
2πn
Φ = , (40) topological charge; see eqs.(39) and (40).
B
e
andwenotethatthetopologicalchargeQ isconserved,
T
and it is quantized according to the winding number n. V. ENDING COMMENTS
In this sense, to better specify the field configurations
studied in this work, a rather natural way is to consider
In the present letter, we have considered the existence
their corresponding magnetic fields. In the present con-
of rotationally symmetric BPS solutions in a (2+1)-
text, this field is given by
dimensional space-time. We have investigated such so-
lutions in a modified self-dual Maxwell-Higgs model en-
1da
B = . (41) dowed by a non-standard dynamics. Here, the modifica-
−rdr
tionwasintroducedintermsoftwonon-trivialfunctions,
The modified numerical profiles for the magnetic field G(φ) and w(φ), which must be positive, in order to
| | | |
(41) areplottedinFigure4. Also forsuchfield, the non- avoidproblemswiththeenergydensity(15)ofthemodel.
usualdynamicsintroducedearlierleadstoBPSsolutions So,whileG(φ)coupleswiththeMaxwelltermand,asa
| |
with a increased core, since the modified profiles have a consequence,changesthedynamicsofthegaugefieldina
characteristic length which is greater than the standard non-usualway,w(φ)coupleswiththesquaredcovariant
| |
one. In this case, as for the energy densities previously derivativeofthenon-chargedscalarfield,thenleadingto
depicted, the magnetic field associated to (21) is quite a non-standard dynamics to such field. In this context,
different from the canonical self-dual one, which reaches consistent first order equations were obtained since the
its maximum value for r 0, and is monotonically de- non-trivial functions G(φ) and w(φ) are constrained
→ | | | |
crescentever: thenon-standardsolutionreachesitsmax- to the symmetry breaking Higgs potential V (φ) of the
| |
imumvalueatsomefinitedistancefromtheorigin,andit modified model; see Eq. (16).
isnotmonotonicallydecrescentforallr. Suchbehaviour WehaveintegratedtheBPSequationsbymeansofthe
is similar to that of the self-dual magnetic field related relaxation method, and the numerical results we found
to the canonical Chern-Simons-Higgs model. aredepictedinFigs. 1and2,forsomeinterestingchoices
tothefunctionsG(φ)andw(φ);seeeqs.(21)and(27).
| | | |
1,0 According these solutions, we conclude that both the
profile functions g(r) and a(r) are quite sensible to the
choices made for G(φ) and w(φ). In particular, it is
| | | |
important to reinforce that each of the models (21) and
0,8 (27) is related to a very specific symmetry breaking po-
tentialV (φ)and,asaconsequence,suchmodelspresent
| |
distinct vacuum manifolds; for details, see eqs.(10) and
(22).
0,6
Also, using the previous results we found for g(r) and
B(r) a(r), we have integrated both the energy density ε(r)
andthemagneticfieldB(r)forthemodifiedmodels(21)
0,4 and (27), and these solutions are depicted in Figs. 3
and 4, respectively; see eqs.(23), (28) and (41). The nu-
mericalanalysis revealsthat the solutions corresponding
the model (21) behave as those predicted by the Chern-
0,2
Simons-Higgs theory, which reach their maximum val-
ues at some finite distance from the origin, and are not
monotonically decrescent for all r. On the other hand,
0 thesolutionscorrespondingthemodel(27)behaveasthe
0 2 4 6 8 10 Maxwell-Higgsones,sincetheyreachtheirmaximumval-
r uesasr 0,andaremonotonicallydecrescentforallval-
→
FIG. 4: Plots of the magnetic field B(r). Conventions as in uesoftheindependentvariable. So,asaconsequence,we
FIG. 1. conclude that also the energy density ε(r) and the mag-
neticfieldB(r)arebothsensibletothechoicesmadefor
To end this Section, we discuss the solution for the G(φ) and w(φ).
| | | |
magnetic field (41) related to the model (27). This solu- We hope that this work may stimulate subsequent
tion is also depicted in Fig. 4, from which we note that analysis in the field, concerning mainly the features that
the behaviour of such magnetic field is quite similar to the modified solutionsengender. Also,we pointoutthat
8
the variationonthe characteristiclength ofthe modified models to mimic the very same solutions engendered by
solutions presented here is closely related to the effects the standard Maxwell-Higgs theory. In particular, such
of anisotropy, which is a feature tipically present in the issue is now under consideration, and we hope to report
effective field models used to describe the behaviour en- new results in a near future.
gendered by low-energy condensed matter systems. In The authors would like to thank CAPES, CNPq
this sense, we point out that such effective models are (Brazil)andFCTProjectCERN/FP/116358/2010(Por-
usually based on the Lorentz-breaking ideia, since it in- tugal) for partial financial support. Also, we are grate-
troduces the issue of anisotropy explicity. In the context ful to C. Adam, F. Correia and D. Rubiera-Garcia for
of Lorentz-violating models, such effects where already useful discussions. E. da Hora thanks the Department
studied by some of us; see, for instance, Ref. [20]. of Mathematical Sciences of Durham University (United
A very interesting issue concerns the use of modified Kingdom),foralltheirhospitalitywhiledoingthis work.
[1] D.Finkelstein, J. Math. Phys. 7, 1218 (1966). ski, J. Math. Phys. 47, 052302 (2006).
[2] A.A.Abrikosov,Sov.Phys.JETP5,1174 (1957). H.B. [13] T. H.R. Skyrme,Proc. R.Soc. A 262, 237 (1961).
Nielsen and P. Olesen, Nucl. Phys.B 61, 45 (1973). [14] D. Bazeia, E. da Hora, C. dos Santos and R. Menezes,
[3] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974). A. M. Phys. Rev.D 81, 125014 (2010). D. Bazeia, E. daHora,
Polyakov,JETP Lett. 20, 194 (1974). R. Menezes, H. P. de Oliveira and C. dos Santos, Phys.
[4] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Rev.D81,125016(2010).C.dosSantosandE.daHora,
Other Topological Defects (Cambridge University Press, Eur.Phys.J.C70,1145(2010);Eur.Phys.J.C71,1519
Cambridge, England, 1994). N.Manton and P.Sutcliffe, (2011). C. dos Santos, Phys. Rev.D 82, 125009 (2010).
Topological Solitons (Cambridge University Press, Cam- [15] C. Adam, N. Grandi, J. Sanchez-Guillen and A.
bridge, England, 2004). Wereszczynski, J. Phys. A 41, 212004 (2008). Erratum-
[5] E. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976). M. ibid. A 42, 159801 (2009). C. Adam, J. Sanchez-Guillen
Prasad and C. Sommerfield, Phys. Rev. Lett. 35, 760 and A. Wereszczynski, J. Phys. A 40, 13625 (2007).
(1975). Erratum-ibid.A42,089801(2009).C.Adam,N.Grandi,
[6] E.Babichev, V.Mukhanovand A.Vikman,J. High En- P. Klimas, J. Sanchez-Guillen and A. Wereszczynski, J.
ergy Phys. 02, 101 (2008). Phys. A 41, 375401 (2008). C. Adam, P. Klimas, J.
[7] C. Armendariz-Picon, T. Damour and V. Mukhanov, Sanchez-Guillen and A. Wereszczynski, J. Phys. A 42,
Phys.Lett. B 458, 209 (1999). 135401 (2009).
[8] V. Mukhanov and A. Vikman, J. Cosmol. Astropart. [16] E.Babichev,Phys.Rev.D74,085004(2006).D.Bazeia,
Phys.02, 004 (2005). L. Losano, R. Menezes and J. C. R. E. Oliveira, Eur.
[9] C. Armendariz-Picon and E. A. Lim, J. Cosmol. As- Phys. J. C 51, 953 (2007). X. Jin, X. Li. and D. Liu,
tropart. Phys.08, 007 (2005). Classical QuantumGravity 24, 2773 (2007).
[10] A.Sen, J. High Energy Phys. 07, 065 (2002). [17] P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564
[11] J. Garriga and V. Mukhanov, Phys. Lett. B 458, 219 (1993). P. Rousenau and E. Kashdan, Phys. Rev. Lett.
(1999). R. J. Scherrer, Phys. Rev. Lett. 93, 011301 104, 034101 (2010).
(2004). A. D. Rendall, Classical Quantum Gravity 23, [18] D. Bazeia, Phys.Rev.D 46, 1879 (1992).
1557 (2006). [19] R. Friedberg and T. D. Lee, Phys. Rev. D 15, 1694
[12] T. H. R. Skyrme, Proc. R. Soc. A 260, 127 (1961). L. (1977); Phys.Rev.D 16, 1096 (1977); Phys.Rev.D 18,
Faddeev and A. J. Niemi, Phys. Rev. Lett. 82, 1624 2623 (1978).
(1999).D.A.Nicole,J.Phys.G4,1363(1978).S.Deser, [20] M. N. Barreto, D. Bazeia and R. Menezes, Phys. Rev.
M.J.DuandC.J.Isham,Nucl.Phys.B114,29(1976). D 73, 065015 (2006). D. Bazeia, M. M. Ferreira Jr., A.
H. Aratyn, L. A. Ferreira and A. H. Zimerman, Phys. R. Gomes and R. Menezes, Physica D 239, 942 (2010).
Rev.Lett.83,1723(1999).D.Bazeia,J.MenezesandR. R. Casana, M. M. Ferreira Jr., E. da Hora and C. M.
Menezes, Phys. Rev. Lett. 91, 241601 (2003). C. Adam, Cantanhede, in preparation.
J.Sanchez-Guillen,R.A.VazquezandJ.A.Wereszczyn-