Boundary Conditions as Mass Generation
Mehanism for Real Salar Fields
Jose Alexandre Nogueiraand Pedro Leite Barbieri
Departamento deFsia,CentrodeCi^eniasExatas,
UniversidadeFederaldoEspiritoSanto,
29.060-900 -Vitoria-ES-Brazil,
E-mail:nogueirae.ufes.br
Reeivedon10Deember,2001. Revisedversionreeivedon7Marh,2002.
We onsider the eets of homogeneous Dirihlet's boundary onditions on two innite parallel
plane surfaesseparatedbysomesmalldistanea. Wendthatalthoughspontaneoussymmetry
breaking doesnot our for thetheory ofa massless, quartially self-interating realsalar eld,
thetheorybeomesatheoryofamassivesalareld.
I Introdution
The originof partile masshas beena puzzlefor
theoretial physiists. In the ontext of the standard
model, the Higgs mehanism is well aepted as the
mass generation mehanism. The underlying idea is
that the universe islled witha eld,the Higgs eld.
Thespin-zeroHiggseldisadoubletintheSU(2)spae
andarriesnon-zerohyperharge,anditisasinglet in
theSU(3)spaeofolor.
Bosoni gauge and fermioni matter elds aquire
mass from their interations with the Higgs eld. It
is of fundamental importane to this mehanism that
exited states (i. e., withone or more Higgs) are not
orthogonal tothe ground state(i. e., vauum). Sine
stateswithoneormoreHiggsarrynon-zeroSU(2)and
U(1)quantum numbersthentheyare non-zeroforthe
vauum as well. As a onsequene of this the SU(2)
and U(1) symmetries are spontaneously broken. The
Lagrangianissymmetriunder SU(2)andU(1)
trans-formationsbut thevauumis not[1℄. Tehniallythis
isahievedintroduingaHiggs potentialof imaginary
massandquartiinteration.
IntheColeman-Weinberg alternativeapproahthe
spontaneous symmetry breaking is indued by 1-loop
radiative orretions and the mass being vanishing in
thetreeapproximation[2℄. Inthemasslesssalar
ele-trodynamis,althoughthevauumexpetationvalueof
thelassialpotentialisunique(lassially<>=0),
initsrenormalizedformtheeetivepotentialin1-loop
beomes degenerate (< > is arbitrary). The mass
renormalization ondition xes a determined vauum
and breaks the symmetry of the theory. The
sponta-partiles to dynamially aquire mass whose value is
proportionaltothevauum expetationvalue. Thisis
only possible beause there aretwoparametersin the
theory: theeletromagneti ouplingonstant, e, and
theouplingonstantofthequartiself-interation,,
whihmustbeofordere 4
.
Inatheoryofmasslessrealsalareldtheexistene
ofjustoneparameterdoesnotallowanon-zerovauum
expetation valueandthetheoryremainsmassless.
Boundaryonditionsdonotaetthelassial
po-tential, but quantum orretions are aeted and the
eetive potential hanges. This may auseeets on
thequantum vauumexpetation value<>.
More-overboundaryonditions introdue anew parameter,
the length of the nite region. With these
onsidera-tionsweinvestigatetheeets ofboundaryonditions
onthevauumexpetationvalueh i=0andthemass
generation. Besidestheintrodutionofanew
parame-ter makesmoreattrativetherealsalareld theory.
Thestudy ofboundaryonditionseets in
Quan-tum Field Theory is not new. The Casimir eet [3℄,
whihisthehangeofthevauumenergydensitydueto
onstraintsonthequantumeld,induedbyboundary
onditions in spae-time was experimentally observed
in 1958 [4℄ and reently [5, 6℄, and a lot of
applia-tionshavebeenaomplished[7-10℄,suhas: gravity
models, blakholes,bag models, nonlinearmeson
the-ories desribing baryons as solitons, the osmologial
onstantproblem,ompatiationoftheextra
dimen-sions in Kaluza-Klein theories, quantum liquids [11℄,
ondensedmatter[12℄.
Inthis work we study the eets of the boundary
onditionson twoinnite parallelplane surfaes
sepa-rated bysomesmalldistane a. Thestudy ofthe 4
theoryisveryimportantinfaeofitsappliationsinthe
Weinberg-Salam model of weak interations, fermions
massesgeneration,insolidstatephysis[13℄,
ination-ary models[14℄,solitons[15℄andCasimireet[16℄.
The eets of the boundary onditions on a 4
theory wererstly onsidered,untilwhere weare
on-erned,byFord[17℄,FordandYoshimura[18℄andTom
[19℄. Theyfoundthatamasslesssalareldatthetree
level ould develop a mass at the one-loop level as a
onsequene of both the 4
self-interation and the
boundaryonditions. Inthisworkweshowthat ifthe
length of the nite region is small enoughthe theory
will beame one massiveat theorder ~ 0
. That is
be-ause and the lengthof the region nite, a, are
in-dependentparametersandtheseondderivativeofthe
eetivepotentialbeomeoftheorder~ 0
,eventhough
itisaone-loopresult,thenitannotbesubtratedout
by themassountertermof theorder ~. So thereis a
physialmass(measure)andweannotset the
renor-malized massto zero. Hene boundaryonditions are
apossiblemassgenerationmehanismwhenthelength
of nite region is small enough. We understand a is
smallenoughwhenthereisatypiallengthsalewhere
a 2
isoftheorder. Thereforethattypiallengthsale
determinesthemasssaleoftheory.
Itisimportanttostressthatwedonotusethe
imag-inarymasstermoftheHiggspotentialinordertoavoid
thatitinduesspontaneoussymmetrybreaking.
There-forewith thepurpose to evaluate theatual eets of
boundary onditions we onsider the real salar eld,
with aseond parameter introdued by theboundary
onditions.
As will be seen subsequently the boundary
ondi-tions do not indue a degenerate vauum state,
al-thoughthereisatypiallengthsaleoftheniteregion
wherethemasslesssalareld aquiresmass.
Theoutline ofthe paperis asfollows: Setion 2is
abrief review of how to alulate the eetive
poten-tial at 1-loop. In setion 3 we alulate the eetive
potentialformasslessrealsalareld,andevaluatethe
vauum stateand the renormalizedmass. In ontrast
to[19℄,wederivetheeetivepotentialattheone-loop
levelasasumofmodiedBesselfuntions,thenwetake
thelengthofthenite region,a,verysmall toexpand
eah term of thesum in powerof a and nally throw
awayhigher-orderterms. Insetion4wepointoutour
onlusionsandsomespeulations.
II Eetive Potential
In the funtional method approah of quantum
eld theory, the eetive potential is found as a loop
expansion(orequivalentlyin powersof ~), that is, its
lassialamountplusquantumorretions[20-26℄.
Let(x)beasinglerealsalareldinaMinkowski
spae-time,subjetedtothepotentialV(). The
ee-tivepotentialtotherstorderintheloopexpansionis
givenby
V
ef (
)=V
l (
)+
1
2 ~
lndet
Æ 2
S[
℄
Æ( x)Æ( y)
=V
l (
)+V
(1)
ef (
); (1)
d
where thelassialeld
is the vauum expetation
valueinthepreseneofanexternalsoureJ(x) ,taken
as a onstant value
= , therefore, in the limit
J ! 0, S[℄ is the lassialation and is the four
dimensional spae-timevolume.
Performing the analyti ontinuation to the
Eu-lidean spae-time [21 - 23℄, the lassial ation an
bewrittenas
S[℄= Z
d 4
x
1
2
+V
l ()
; (2)
wheretheEulideansummationonventionisassumed
for repeated indexes. From Eq.(2) we get the matrix
m(x;y)ofthequadrativariationoftheationS[℄
m(x;y) Æ
2
S[℄
Æ(x)Æ(y) =Æ
4
(x y)[ Æ
+V
00
l
DuetotheEulideananalytiontinuationthe
op-eratorm isareal,elliptialandself-adjoint. For
oper-atorswiththesepropertieswedeneageneralizedzeta
funtion[7,8,23and27℄. Iff
i
garetheeigenvaluesof
theoperatorm(x;y),thenthegeneralizedzetafuntion
assoiatedtoM(x;y)(m!M = m
2
)isdened by
M (s)=
X
i
i
2
s
; (4)
where we have introdued an unknown sale
parame-ter , with dimensions of (length) 1
ormass in order
to keepthe zetafuntion dimensionless for alls. The
introdutionofthesaleparameter,anbebest
un-derstood when we observe that a hidden splitting of
thedivergentintegralthereisintheproeedingofzeta
funtion regularization,that is, aseparationof the
di-vergent and nite parts of the V
ef (
) ( on page 208
[23℄andonpage88[24℄). Itiswell-knowntherelation
[21℄
lndetM = d
M (0)
ds
: (5)
Now,theeetivepotentialintheone-loop
approx-imationanbewrittenas
V
ef (
)=V
l (
)
1
2 ~
d
M (0)
ds
: (6)
Due to theregularityof the generalizedzeta
fun-tion at s = 0 [27℄, the evaluation of the eetive
po-tential (Eq.(6)) gives a nite result with no need of
substrationofanypole,oradditionofinnite
ounter-terms. Evidently, the ttingof thetheory parameters
taking into aount the observed results, leads to the
renormalizationonditions
d 2
V
ef
d 2
=hi
=m 2
R
; (7)
d 4
V
ef
d 4
=hi =
R
; (8)
where m
R
is the renormalized mass,
R
is the
renor-malizedoupling onstantand < >istheminimum
of the eetive potential (subtration or
renormaliza-tionpoint)[25℄. Sine
takesonvalue<>in the
groundstate, then<>is alled thevauum
expe-tationvalueof,<>=<0jj0>.
Intheaseoftheoriesofnullmass,thesubtration
point for the renormalization ondition (8) annot be
taken at hi = 0 due to the logarithmi singularity.
Even so in that ase there is no intrinsi mass sale;
therefore allthe renormalizationpoints areequivalent
andtheondition(8)isreplaedby
d 4
V
ef
d 4
=M
=
R
; (9)
where M is a arbitraryoating renormalizationpoint
[23℄.
III Mass Generation
Now,letusonsiderthetheoryofamassless,
quar-tially self-interating real salar eld (x) satisfying
homogeneous Dirihlet's boundary onditions on two
inniteparallelplanesurfaesseparatedbysomesmall
distanea.
TheLagrangiandensityforthistheoryis
L= 1
2
4!
4
: (10)
TheLagrangianaboveisevenin,soitisinvariant
un-derdisreetsymmetry(G-parity)denedbythe
trans-formation! .
Inthisasethezetafuntion,dened by Eq.(4),is
givenby[7,8,24,28-33℄
m ( s)=
1
X
N=1 Z
+1
1
( 2) 3
a d
3
k
k 2
+
2
N 2
a 2
+
2
2
s
:
(11)
Usingtheintegral[28℄
Z
+1
1
k 2
+A 2
s
d m
k=
m
2
s m
2
(s) A
2
m
2 s
;
(12)
andusingtheformula[34℄
1
X
N=1
N 2
+B 2
p
= 1
2 B
2p
+
1
2
2B 2p 1
(p) "
p 1
2
+4 1
X
N=1 K
p 1
2
( 2NB)
(NB) 1
2 p
#
; (13)
whereK
(x) aremodiedBesselfuntions,weget
m (s)=
3
2 s
3
2
(s) (
)
3 2s
+
16 2
1
( s 1)(s 2) ( ) 4 2s + + 4 2 ( ) 4 2s (s) 1 X N=1 K s 2 (2Na ) (Na ) 2 s ; (14)
where wedene 2
=
2
forthesakeofsimpliity.
Inorder toalulatetheone-loopeetivepotential,weompute
m
( 0)and 0
m
(0)fromEq.(14)andusethem
in Eq.(6) 1 V ef ( )= 2 2 4! 4 + ~ 24a 3 3 + ~ 64 2 4 4 ln 2 2 2 3 2 + ~ 8 2 4 4 1 X N=1 K 2 (2Na ) (Na ) 2 ; (15)
realling that K
(x)=K
(x). Wenotie that,aswehaveto takeabsolutevalue of
in theformula (13),the
seond termontheright-handsideofEq.(15)doesnotyieldsymmetrybreaking.
Letm0beanintegernumber,suhthat,2ma
<1and2(m+1)a
1. Then,thesumin Eq.(15)an
bewriteas
~ 8 2 4 4 1 X N=1 K 2 (2Na )
( Na
) 2 = = ~ 8 2 4 4 m X N=1 K 2 (2Na ) ( Na ) 2 + ~ 8 2 4 4 1 X N=m+1 K 2 (2Na ) (Na ) 2 : (16) Sine 2Na
<1,foranyN m, weexpandthe rsttermontheright-handside ofEq.(16)using therelation
[35 -39℄
x 2 K (x)= 1 2 1 X a=0 ( 1) a x 2 2a ( a)
( a+1) + + 1 X a=0 ( 1) x 2 2+2a
(a+1) (+a+1) h
(a+1)+ (+a+1) 2ln
x
2 i
; (17)
for >0,toget
~ 8 2 4 4 1 X N=1 K 2 ( 2Na ) (Na ) 2 = ~ 16 2 a 4 m X N=1 1 N 4 ~ 16 2 a 2 2 2 m X N=1 1 N 2 + ~ 16 2 4 4 m X N=1
[ln(Na
)+ 3=4℄+ ~ 8 2 4 4 m X N=1
O(Na
) 2 + + ~ 8 2 4 4 1 X N=m+1 K 2 ( 2Na ) (Na ) 2 ; (18)
where istheEulernumber.
Therst sum onthe right-hand side of Eq.(16) exists, provideda( a =a)be of order n
(n 0), sine ( =
)mustbeoforder 0
. Hene,uptohigher-orderterms,theeetivepotentialbeomes
V ef ( )= 2 2 4! 4 + ~ 24a 3 3 ~ 16 2 a 4 m X N=1 1 N 4 + ~ 16 2 a 2 2 2 m X N=1 1 N 2 : (19)
Forlargem,wemayapproximatethetwosumsin Eq.(19)toRiemannzetafuntionsandtoobtain
Theminimumoursat
=hi ,where
dV
ef
d
=hi
=0: (21)
Dierentiating Eq.(20),wehave
hi
2
3 hi
2
+ ~
3
8a hi+
~ 2
48a 2
=0: (22)
Therewillbenon-trivialsolutionofEq.(22)ifthesum
ofthetermsbetweenbraketsvanishes. Howeverthere
isnota(withrespettotheordersof)whihsatises
Eq.(22). ThereforetheminimumofV
ef
issatisedfor
h i=0: (23)
Eq.(23)showsthat the vauum is non-degenerate,
and therefore spontaneous symmetry breaking annot
our. Atually,theboundaryonditionsallowjustone
onstant vauum solution, hi = 0. If the result had
beenhi=onstant6=0,theeetivepotentialwould
nothave been used anda solutionhi =
0
(x) would
have been expeted beause the boundary onditions
breakthetranslationalinvariane.
AlthoughEq.(20)is nite,that isnotanal result
beausetheouplingonstantin itisanarbitrary
pa-rameter. Thereforewe mustt it to therenormalized
oupling onstant using therenormalization ondition
Eq.(8)to get=
R .
The renormalized mass of the salar eld is given
by
d 2
V
ef
d 2
l
=hi=0 =
~
R
96a 2
=m 2
R
; (24)
thatunlikewhat weexpeteditanbenon-zero.
Eq.(24)showsifa 2
is oforderthenthemasswill
beoforder 0
. Thisresultisinagreementwithour
ini-tialassumption. Ifa 2
isoflowerorderthanthenthe
masswill liefaroutsidetheexpetedrangeof validity
of our approximation. It followsthat there is a
typi-allengthsalegivenbytheparameteraofthetheory.
Withinthislengthsale,themasslesssalareldtheory
withself-interationbeomesmassiveduetoboundary
onditionseets.
Although topologial mass has already been
ob-tained[17 -19℄ toorder foratheory whih is
mass-lessatthetree-graphlevel,westressthefat thatifa
issmall enoughthetheory willbeomeonemassivein
order (~ 0
). This is beause and a are independent
parameters, so the mass term found an be of order
zero-loop,eventhoughitisaone-loopresult(itisfrom
radiativeorretions). OurresultofEq.(24)isin
agree-mentwiththatobtainedbyDavidJ.Toms[19℄.
IV Conlusion
Wehavestudied thetheory of amassless,
quarti-ally self-interating real salar eld satisfying
homo-parallel plane surfaes separated by some small
dis-tanea. Asaresult,weinfer:
i) For small a( a / n
, n> 0) there is notanyorder
ofawith regardto orderofwhih leavestheground
state(vauum) degenerate,i. e., spontaneous
symme-trybreakingdoesnotour. So,theeetivepotential
anbeusedtoevaluatetherenormalizedmassandthe
Casimirenergy.
ii) There is a typial length sale of the nite region
where massless salar eld aquires mass, i. e., if
a/
n
, with n1,the theorybeomesatheoryof a
massive real salar eld. Therefore, when a beomes
small enough the theory undergoes a transition from
one massless to one massive. That typial length of
niteregiondeterminesthemasssaleofthetheory.
iii)Sinespontaneoussymmetrybreakingdoesnot
o-ur, the mass generation is only due to the boundary
ondition.
Finally,wespeulatethatboundaryonditionsmay
be amehanismof mass generation, i. e., that
mass-lesstheories dened in niteregions ofspae-time
be-ome massivetheorieswhenthe lengthof niteregion
issmallenough. Thereforeweonjeturethat
onne-mentmaybeaandidateforanalternativemehanism
forthemassgeneration ofquarks.
It is lear we do not laim that these are atual
boundaryonditions whih produe the masses of the
partiles. Weonlyonsiderboundaryonditionmaybe
analternativemehanismforpartilemassgeneration.
Aknowledgments
J. A. Nogueira is espeially grateful to Professor
OlivierPiguetwhohashelpedto larifymanyobsure
points. We would like to express our thanks to Dr.
Manoelito Martins de Souza and to Dr. Franisode
Assis RibasBoso. It is apleasure to thankDr. Luiz
C. Albuquerquefor helpful omments. This work was
supportedinpartbytheNationalAgenyforResearh
(CAPES)(Brazil).
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