Accessing directly the properties of fundamental scalars in the confinement and Higgs phase

0.35 MB·English

by  Axel Maas

Accessing directly the properties of fundamental scalars in the confinement and Higgs
phase
Axel Maas∗†
Institute of Physics, Karl-Franzens-University Graz, Universita¨tsplatz 5, A-8010 Graz, Austria
(Dated: February 1, 2011)
Thepropertiesofelementaryparticlesareencodedintheirrespectivepropagatorsandinteraction
vertices. For a SU(2) gauge theory coupled to a doublet of fundamental complex scalars these
propagators are determined in both the Higgs phase and the confinement phase and compared to
theYang-Mills case, using lattice gauge theory. Since thepropagators are gauge-dependent, this is
donein theLandaulimit of the’t Hooft gauge, permitting toalso determinetheghost propagator.
It is found that neither the gauge boson nor the scalar differ qualitatively in the different cases.
In particular, the gauge boson acquires a screening mass, and the scalar’s screening mass is larger
1 than the renormalized mass. Only the ghost propagator shows a significant change. Furthermore,
1 indications are found that the consequences of theresidual non-perturbativegauge freedom dueto
0 Gribov copies could bedifferent in theconfinement and theHiggs phase.
2

n
a I. INTRODUCTION Aside from these questions, which concern the practi-
J cal viability of the weak isospin theory in its standard

1 model form, there is also a more conceptual question.
3 Although the theory of weak isospin can be very well It has been shown that the confinement phase and the
described with conventional perturbation theory after Higgs phase of the SU(2)-Higgs theory are not sepa-

] parametrizing the effective Higgs condensate [1], it still rated,ifthe theoryis investigatedwitha(lattice-)cutoff,
t
a poses quite a number of genuine non-perturbative ques- but are analytically connected [11]. Only in terms of
-l tions. The most obvious one is, what happens if the gauge-dependentquantitiescanaphasetransitionbe es-
p Higgs is very heavy. In this case, perturbative calcu- tablished, but the transition itself then turns out to be
e lations will break down eventually when reaching the gauge-dependent[12]. This would imply that there is no
h
[ energy domain of about 1 TeV [1]. But this is by far qualitative difference between a confining and a Higgs
not the only question. A possibly even more interest- phase in terms of gauge-invariant physics. Since a con-
3 ing question is, whether the theory in itself can provide fining phase is not perturbatively accessible, so would
v a dynamical mechanism for Higgs condensation, as in- neitherbe aHiggsphase. Thismustbe sortedouttoget
9
2 dicated by lattice simulations [2–7]. This would obliter- a full understanding of the field theoretical setup of the
7 atethenecessitytodriveelectroweaksymmetrybreaking weakisospinsectorofthe standardmodel. Furthermore,
0 entirely by physics beyond the standard model, though latest results indicate that the thermodynamic limit for
. not resolving the perceived hierarchy problem [8]. An- questionsconcerningthephasediagramisratherhardto
7
0 other serious obstacle is the possibility that the theory reach [6].
of weak isospin could indeed become trivial upon quan-
0 On the other hand, this also permits to study an en-
1 tization, very much like the ungauged φ4 theory [9] or tirely different set of problems in the same theory. In
: the ungauged Higgs-Yukawa theory [7, 10]. However,
v theconfinementphase,thecolorconfinementprocessap-
the possibility of a non-perturbative stabilization (sim-
i pears to be entirely the same as for quarks, but without
X ilar in concept, e. g., to asymptotic safety) has not been
thecomplicationsintroducedduetochiralsymmetryand
r finally ruled out, and deserves further investigation. Of fermions1. For an understanding of color confinement,
a
course, it could be that the top-Yukawa coupling will be
the weak isospin theory in the confinement phase there-
important in case of the standard model [9], but before
fore offers an ideal laboratory, as has already been ex-
requiring outside assistance, it should be better under-
ploited previously [14–18]. This is another incentive to
stood whether this is indeed necessary. Results concern-
study this theory in both phases.
ing this question are not yet fully satisfactory. In par-
None of the points above, with the possible exception
ticular, besides the phase diagram [3, 5, 6] usually only
of a heavy Higgs, imply that for precisionmeasurements
gauge-invariantboundstateshavebeeninvestigated[2,4]
inthe standardmodelnon-perturbativephysicsis neces-
forthisquestion,forwhichitmaybedifficulttoseparate
sary. Non-perturbativeeffectscouldeasilybesufficiently
genuine bound states from almost scattering states in a
suppressed over the whole accessible energy spectrum,
very weakly interacting theory.
evenatLHC,orpossiblyuptothePlanckmass,tomake
thesequestionspracticallyirrelevant. Andinfact,there-
∗Present address: Institute for Theoretical Physics, Friedrich-
Schiller-University Jena, Max-Wien-Platz 1, D-07743 Jena, Ger-
many 1 NotethattheoftencitedWilsoncriterionforconfinement[13]is
†[email protected] actuallyblindtothespinstructureoftheinvolvedparticles.
2
sults from LEP and Tevatron so far are in perfect agree- isospin model2 is given by [23]
ment with a domination of perturbative contributions.
However,as long as it is not clearwhat happens atLHC S = β 1− 1 ℜtrU (x)
and beyond, the consequences of non-perturbative con- 2 µν
Xx (cid:16) µX<ν
tributions should be known.
+1φ+(x)φ(x)+λ φ(x)+φ(x)−1 2
2
(cid:0) (cid:1)
Tofindanswerstothequestionsabove,inthisworkthe
−κ φ(x)+U (x)φ(x+e )
possibility to access the properties of the (gauge-depen- µ µ
Xµ (cid:16)
dent) elementary degrees of freedom of the weak isospin
theory will be investigated. These are the gauge bosons, +φ(x+e )+U (x)+φ(x)
µ µ
whichwillbecalledtheW forsimplicity. Ofcourse,since (cid:17)(cid:17)
U (x) = U (x)U (x+e )U (x+e )+U (x)+
QED is neglected, the Z and W are degenerated here. µν µ ν µ µ ν ν
The second will be denoted Higgs. Their propagators 1
W = (U (x)−U (x)+)+O(a2)
will be determined in both the Higgs and confinement µ 2agi µ µ
phase, and will be compared to the quenched case, i. e. 4
withoutdynamicalHiggsparticles,andthusaYang-Mills β =
g2
theorywithgaugegroupSU(2). Forthenon-perturbative
(1−2λ)
determination a lattice implementation will be used. a2m2 = −8.
0 κ
Sincethepropagatorsaregauge-dependent,itisneces- In this expression a is the lattice spacing, Wµ the gauge
sary to fix a gauge. Given that non-perturbative gauge- boson field, φ the Higgs field, g the bare gauge coupling,
fixing is made complicated by the Gribov-Singer ambi- λ/κ2 is the bare self-interaction coupling of the Higgs,
guity [19, 20], a gauge is chosen in which these effects m0 the bare mass of the Higgs, the sums are over the
arecomparativelywellunderstood,whichistheLandau- lattice points x, and eµ are unit vectors on the lattice in
limit of the ’t Hooft gauge,correspondingto the Landau the direction µ.
gauge in Yang-Mills theory. Thus, as additional degrees For such a system the path integral, and thus the ex-
of freedom, the Faddeev-Popov ghosts, also all degener- pectation values of operators, can be evaluated using
ate,areavailable. AsaconsequenceoftheGribov-Singer MonteCarlosimulations. Themethodusedforthegauge
ambiguity,ona finite lattice, as employedhere,the Lan- partis describedin detailin[24]. For the Higgs part,for
dau gauge becomes a family of non-perturbative gauges each of the six sub-sweeps of the hybrid-overrelaxation
inYang-Millstheory[21]. Thiseffectwillalsobestudied cycle of the gauge field a local Metropolis update has
here. been performed, where the acceptance probability was
adjusted adaptively to be 50%. The results for observ-
ables like the action, the spatial average of the expecta-
This, rather exploratory,investigationis structuredas
tion value of the operator
follows: In section II the technical details of the simula-
tions,includinggauge-fixing,willbebrieflydiscussed. In η(x)=φ(x)+φ(x), (1)
sectionIIIthepropagatorsandtheirrenormalizationwill
be introduced and the central results presented. In sec-
the expectation value of the plaquette, and the lowest
tionIVtheconsequencesoftheGribov-Singerambiguity
masses of the isoscalar-scalar and isovector-vector exci-
willbeanalyzed. EverythingissummarizedinsectionV.
tations have been compared to literature values [2] to
Some technical details are deferred to an appendix.
check the code.
Inthe following,three different sets ofparameterswill
be investigated, all for a lattice size of N4 = 244. One
is the quenched case, i. e., the Higgs field is set to zero,
while the other two cases correspond to systems deep
inside the confinement phase and the Higgs phase3 [15].
2 Note that strictlyspeakingthe gauge groupisSU(2)/Z2 dueto
therequirementofanomalycancellationwhenincludingfermions
II. TECHNICALITIES [22]. Here,instead, theZ2 partismerelyexplicitlybroken, and
itisassumedthatthisdoesnotmakeadifferenceonthelevelof
correlationfunctions.
3 Thoughthesephasesarenotstrictlyseparatedonafinitelattice,
itispossibletodetermineaphasediagram. Beingsufficientlyfar
The method used here is essentially standard lattice
away fromthe cross-over region, the systems are then classified
gauge theory. The lattice version of the weak SU(2) by these names, though they are strictly speaking qualitatively
3
A certain problem is posed by translating the lattice invariant contribution in an expansion in terms of poly-
scale to a physical scale. In the Higgs phase it could be nomials in the fields. However,from the point of view of
suggestedtomatchwiththeHiggscondensate. However, the gauge-dependent elementary fields, it is a composite
this quantity is gauge-dependent, and in particular zero bound-state, very much like the σ-meson in QCD. It is
in the Landau gauge [12]. The possibility to instead use thus not directly associated with the usual perturbative
the spatially averaged expectation value of η is also not definition of the Higgs [1] in ’t Hooft gauges where the
useful: It is non-zero also in the confinement phase [2], Higgs is not an isoscalar. On the other hand, some dif-
and it is therefore dubious to associate it with the Higgs ferent relation may exist in the non-renormalizable uni-
condensate. The next possibility would be to match the tary gauge. Anyway, it should be noted that neither the
W boson pole mass, which is measured in experiment. polemassnorthe screeningmassofanelementaryHiggs
There are two reasons which make this approach rather is a renormalization-group invariant, even if it should
hard. The first is that the pole positions would have to be gauge-invariant. In contrast, the mass of the low-
bedeterminedbyanalyticalcontinuation,whichisrather est excitation in the η channel is renormalization-group-
unreliable with the number of lattice points available invariant. Thus a direct identification of both concepts
here. The second problem is that, at least in Yang-Mills is, at least, not obvious.
theory, and likely also at least in the confinement phase, Partoftheinvestigationherewillthereforebetostudy
the W boson does not exhibit a pole [25–28]. Therefore, how these two concepts are related5. One possible out-
also this possibility is notuseful for the presentpurpose. come could, e. g., be a similar relation as for quarks and
Concerning the Higgs,similar considerationsapply, even mesons in QCD.
if its mass would have been determined in experiment. That completes the basic idea used here to determine
thescale. Thedetailsofthedeterminationofthescaleas
Hence a more ad-hoc procedure will be used here. For
implementedherearedeferredtoappendixA,wherealso
both the confinement and the Higgs phase, the mass m
η
systematic uncertainties are discussed. The resulting set
of the lowest lying state of the composite operator η(x),
ofparametersisthengivenintableI. Thebarecouplings
(1),willbedetermined. Thisstatewouldbeanisoscalar-
have been chosen such that the systems are sufficiently
scalar Higgsonium bound state (or scattering state). Its
far away from the phase transition such that no meta-
mass will be arbitrarily set to 250 GeV, motivated that
stable phases could occur while at the same time the
it may be twice as heavy as a single Higgs with its most
lattice spacing is not exceedingly small or large, based
possiblemassofabout125GeV[29]. Since,incontrastto
on the results in [2, 4, 15].
quarks in QCD, it is found below that the Higgs screen-
ing mass and its renormalized pole mass turn out to be Thisconcludesthegenerationofthe configurations. It
rather close, it could be expected on the basis of a sim- remainstogauge-fixthem. Asstatedintheintroduction,
pleconstituentmodelthatthisshouldgiveanacceptable this is potentially obstructed by the presence of Gribov
firstguess. Inthequenchedcase,this objectisratherte- copies. However, the presence of Gribov copies can also
dious to calculate. Instead, here an upscaled version of be turned into a virtue by using them to define different
thea(β)relationofYang-Millstheorywillbeused,toset non-perturbative gauges, depending on the selection of
ascalecompatiblewiththeoneintheconfinementphase. certain Gribov copies [21, 28, 30]. Still, all of these non-
That said and done, one should see the physical scale to perturbative gauges satisfy the associated perturbative
be rather of illustrative purpose, and it can always be gauge condition, which therefore has to be chosen first.
scaled out again to replace it with another scale. In the present case, this will be the Landau limit of
the ’t Hooft gauge. Thus, the gauge fields satisfy the
It should be noted4 that in the lattice literature the
condition
operator (1) is usually associated with the Higgs itself
rather than with a Higgs-Higgs-bound state [2, 4], and
∂ W =0 (2)
µ µ
thus its mass is denoted as the Higgs mass. The motiva-
tion for this is that it is the simplest operator available,
inthenowEuclideanspacetime. TodealwiththeGribov
and, provided no anomalous hierarchy is encountered,
copies,firstofalltheselectionofcopiesisrestrictedtothe
it will also be the lightest state in the isoscalar sector,
firstGribovhorizon,definedtobetheregionwithstrictly
and thus the lightest physical excitation of the theory.
positive semi-definite Faddeev-Popov operator [19].
In particular, it is the square of the radial mode of the
The Gribov-Singer ambiguity now leads to the fact
elementary Higgs field, as being its lowest order gauge-
thatpotentiallymorethanonegaugecopyofagivencon-
figurationsatisfiesbothconditions. Infact,inYang-Mills
theory appear to exist a large, possibly infinite, number
identical. Furthermore,inthesenseoftheWilsoncriterion,the
confinement phaseisnotconfiningduetoscreeningbypaircre-
ation[13], andintheHiggs phase thesymmetryis actuallynot
broken,butonlyhidden[1,12]. Therefore,thesetermsshouldbe 5 Similarconsiderationsapplytothelowestisovectorvectorchan-
taken only as a short-hand notice, and their true meaning kept nelinvestigatedinlatticecalculations[2,4],whichisacollective
inmindallthetime. excitation of elementary Higgs and W bosons, but the lowest
4 IamgratefultoChristianLangforadiscussionofthisissue. gauge-invariantvector-particlestate.
4
TABLE I: The three sets included for comparison. For setting the scale a, see the text and appendix A. The number of
configurations are given for the determination of the propagators, where 1080 thermalization sweeps and 108 decorrelation
sweeps have been performed in multiple independent runs. In the quenched case, the first number is for the W and ghost
propagator, thesecond forthequenchedHiggs propagator. m0 isthetree-levelmass oftheHiggs. Inthequenchedcasethisis
themass appearing in thecovariant Laplacian (4), see section III.
System amη a−1 [GeV] β m20 [GeV2] κ λ Propagator
Quenched - 114(7) 2.2 (1232) - - 95/194
Confinement 3.2(2) 78(5) 2.0 -(2212) 0.25 0.5 623
Higgs 1.08+0.06 231+13 2.3 -(6542) 0.32 1.0 1227
−0.04 −9
ofsuchcopiesinthecontinuumandinfinite-volumelimit with µ = 80 GeV and m = 80 GeV. If the propaga-
W
[31, 32]. At leastin a finite volume,these canbe used to tor would be at tree-level, this would yield roughly the
design different gauges [21, 30]. In the next section, the experimentally observedpole mass of the W boson. The
so-called minimal Landau gauge [33] will be used, which resultingpropagatorZ Z(p2)/p2 andthe dressingfunc-
W
selects one random representative among the remaining tion Z Z(p) are shown in figure 1.
W
Gribov copies to represent a configuration. The conse-
The resultsshowthatthere is almostnodifference be-
quencesofalternativechoiceswillbediscussedinsection
tweenthecasewithoutHiggsandtheconfinementphase.
IV, a detailed description can be found in [21, 28, 30].
Inparticular,inbothcasesanon-zeroscreeningmassex-
The method used to fix this gaugeis describedin [24].
ists7, which is about 1.1m . For the Yang-Mills case, it
Since the gauge condition only involves the gauge fields, W
is known that, despite its simple appearance, the ana-
thesamemethodforgauge-fixingasinYang-Millstheory
lytic structure of the propagator is rather involved [25–
can be used. After obtaining the SU(2)-valued gauge-
28]. In particular, though a non-zero screening mass is
transformationg(x), the gauge-fixedHiggs field φ′ is ob-
present,thereappearsto benopole mass,andthe parti-
tained as
cle has no representation as a K¨allen-Lehmann state. A
φ′(x)=g(x)φ(x), first glimpse indicates that the corresponding Schwinger
function[27]intheconfinementphasehasessentiallythe
as required to make the action gauge-invariant.
sameformasinYang-Millstheory,suggestingthatthisis
not changing. Nonetheless, this requires further investi-
gation,inparticularforlargerlatticevolumesandbetter
III. PROPAGATORS
discretizations.
Since Landau gauge is weak-isospin symmetric, and In the Higgs phase, the propagator is quantitatively
thusthevacuumexpectationvalueoftheHiggsfieldvan- different from the confinement phase. In particular, it is
ishes [12], the isospin symmetry is fully conserved and much closer to a tree-level behavior. Nonetheless, there
manifest even in the Higgs phase, as a consequence of aresomedifferences,andinparticularits screeningmass
Elitzur’s theorem [34]. Thus, also the propagators are is larger than the mass at the renormalization point,
isospin symmetric. There are then three independent about 1.1mW. Therefore, there are sizable corrections
ones, the W propagator, the Higgs propagator, and the to its behavior. Still, these are essentially quantitative
Faddeev-Popov ghost [1]. effects8. Concerning the analytic structure, first indi-
The W propagator in Landau gauge is given by cations hint that at least quantitatively the Schwinger
function in the Higgs phase may be substantially differ-
p p Z Z(p2)
Dab =δab δ − µ ν W . ent. However,aqualitativedifferencetotheconfinement
µν (cid:18) µν p2 (cid:19) p2 phasecannotbediscernedunambiguouslywithoutlarger
The methods used to determine it and the ghost prop- and finer lattices. Therefore, this is not shown explicitly
agator below can be found in [24]. In Landau gauge, here.
one renormalizationconditionis requiredto fix the wave Instead of now directly investigating the Higgs propa-
function renormalization constant6 ZW. It will be con- gator, it is interesting to investigate the ghost first. Its
veniently chosen to satisfy
µ2
Z Z(µ2)= ,
W µ2+m2
W 7 Theactualvalueofthescreeningmassisverysensitivetolattice
artifacts[35–37],andthevalue,whichcanbereadfromtheplot,
shouldbetakenwithcare.
8 Note that the apparent absence of a maximum of the dressing
6 For simplicity, all wave-function renormalization constants are function in the Higgs phase is misleading. At sufficiently large
takentomultiplythepropagators,notalwaysinlinewithstan- momentapropagatorsinbothphaseswillcoincide,andbothrun
dard conventions. However, since only the combination ZWZ logarithmically to zero essentially perturbatively [1]. Hence, in
etc.playarolehere,thisisessentiallyirrelevant. allcases thereisamaximum,onlyitsheightandwidthchange.
5
p2(W propagator) W propagator
· 10-3
p) 2]
Z( -V Without Higgs
W Ge
Z 2 [p Confinement phase
p)/ Higgs phase
1 Z( 1
ZW p2+(80 GeV)2
0.1
0.5
0 0
0 100 200 300 400 0 100 200 300 400
p [GeV] p [GeV]
FIG. 1: The W propagator (right) and dressing function (left). Momenta are along the x-axis for low momenta and along
the xy-diagonal for high momenta, such that violations of rotational symmetry for the displayed points are negligible. For
larger momenta along the axes they become of the order of 10-20%. The quenched, confinement, and Higgs case are denoted
by triangles, circles, and squares, respectively. Statistical errors in this section contain contributions from both statistical
fluctuationsandthescaleuncertainty,thelatterdominating. Notethatthedressingfunctionisdimensionless, andistherefore
not affected by thescale uncertainty.
propagator is a scalar function given by associatedwiththe confinementprocess. Inmarkedcon-
trastistheghostpropagatorintheHiggsphase,whereit
Z G(p2)
Dab =−δab G . is almost a bare propagator,without any dressing. Indi-
G p2
rect evidence for this drastic difference has already been
In the Landau gauge, its tree-level mass is zero [1], and obtainedearlierfromcalculationsinCoulombgauge[44],
thus the renormalizationcondition andhasbeendiscussedasapossibilityinlinearcovariant
gauges [45–47].
Z G(µ2)=1,
G Hence, the most distinct difference between the con-
will be used, with µ = 80 GeV. The reason why this finement and Higgs phase so far is the rather different
gauge degree of freedom is interesting is that in Landau ghostpropagator. Thisalsofindsitsmanifestationinthe
gauge it is possible to combine it with the W propaga- effective coupling (3), as shown in figure 3. It is visible
tortoobtainanexpressionfortherenormalization-group thatthe couplinghasnoLandaupoleintheconfinement
invariant running gauge coupling as [38, 39] phase, resembling the situation in the Yang-Mills case
α(p2)=α(µ2)Z Z(p2)(Z G(p2))2, (3) [35–37]once more. Onthe otherhand, the couplingis in
W G
all cases infrared suppressed in this gauge, and starts to
wherethedependenceontherenormalizationscaleofthe follow the same qualitative behavior at large momenta.
propagatorshas been suppressed. This coupling is given Thus,fromthispointofviewtheonlydifferencebetween
intheso-calledminiMOMscheme,butcanbetranslated the Higgs and the confinement phase is in the size of the
into the MS scheme, and is known up to four loops per- coupling, rather than its low-momentum behavior.
turbatively [40]. The finalpropagatoris then the Higgs propagator. Its
Theresultfortheghostpropagatorisshowninfigure2. bare lattice version is obtained from the Fourier-trans-
It is immediately clear that there is a drastic difference formed Higgs field
between the confinement phase, again being essentially
identicaltoYang-Millstheory,andtheHiggsphase. The φ′(p)= ei2πNpxφ′(x),
significance of the infrared enhancement seen9 is likely
Xx
where p and x are lattice coordinatesand momenta,and
a symmetric lattice is assumed. The lattice Higgs prop-
9 Atverysmallmomentatheghostdressingactuallybecomesalso agator is then given by
finiteorlogarithmicallydivergent[41]. However,thesignificance
of this is currentlystill not fullyresolved, see, e. g. [28, 42, 43], κ
DLab = <φ′(p)a+φ′(p)b >,
fordetaileddiscussionsandinparticularforfurtherreferences. H V
6
p2(Ghost propagator) Ghost propagator
G(p) -2]ev Without Higgs
G G Confinement phase
Z 2 [
1.5 p Higgs phase
p)/
G(
G
Z
10-4
1
0.5
10-6
0 100 200 300 400 0 100 200 300 400
p [GeV] p [GeV]
FIG. 2: The ghost propagator (right) and dressing function (left). Momenta are along the x-axis for low momenta and along
the xy-diagonal for high momenta. The quenched,confinement, and Higgs case are denoted by triangles, circles, and squares,
respectively.
Running coupling term,
2) gτa 2
m( −D2 =− ∂ −i Wa +m2,
a2)/ (cid:18) µ 2 µ(cid:19) 0
p
(
a 1 with the generators of the gauge algebra τa, which on
the lattice is given by
Without Higgs
−D2 = − U (x)δ +U+(x−µ)δ −21δ
Confinement phase L Xµ (cid:16) µ y(x+eµ) µ y(x−eµ) xy(cid:17)
0.5 Higgs phase +m201δxy, (4)
where1isaunitmatrixinweakisospinspace. Sincethis
operatoris positive semi-definite, it canbe inverted. For
thispurpose,thesamemethodhasbeenusedasincaseof
the Faddeev-Popov operator in [24]. It should be noted
0 thatevenazeromassisnotaproblemforthismethod10.
0 100 200 300 400
p [GeV] With this, the bare scalar propagator is available for all
systems.
In contrast to the ghost and W propagator the
FIG.3: Theeffectivecoupling(3). Momentaarealongvarious
renormalization of the scalar propagator is somewhat
directions [24]. The quenched, confinement, and Higgs case
are denoted by triangles, circles, and squares, respectively. more complicated, since besides the multiplicative wave-
Notethattheoverallscaleisarbitraryafterdivisionbyα(µ). function renormalization also an additive mass renor-
malization is necessary. The renormalized propagator is
given by [1]
where the complex conjugationalsoinverts the direction
δab
of the momentum, as usual. This propagator can be as- Dab(p2)= ,
signedphysicalunitsinthesamewayasfortheW prop- H ZH(p2+m2)+ΠH(p2)+δm2
agator to obtain the continuum propagator [24].
In the quenched case, the elementary Higgs field φ is
notavailable. Thequenchedpropagatoristhenobtained,
10 IncontrasttotheFaddeev-Popovoperator,thisoperatorhasno
in analogy to the quenched quark propagator,by the in-
trivialzeromodes,andthusaninversionevenatzeromomentum
version of an operator. For a fundamental Higgs, this
ispossible. However,sinceconstantmodesaffecttheresultona
is just the fundamental covariantLaplacian with a mass finitelattice,thisisnotdonehere.
7
Higgs propagator phase, little is visible from the non-perturbative interac-
· 10-3 tions.
Without Higgs It should be noted that the propagators in the Higgs
phase all exhibit the perturbatively expected behavior.
Confinement phase
In particular, a non-zero screening mass is found for the
0.05 Higgs phase W andthe negativemasssquaredofthe Higgsbecame a
1 positive effective mass. Thus, despite the vanishing vac-
-2]V p2+(125 GeV)2 uumexpectationvalueandthusthenotexplicitlyhidden
Ge symmetry, the dynamics seem to be Higgs-like,and thus
p) [ to be dynamically generated. However, one should be
(H wary about this statement for two reasons: First, this is
D
a resultona single lattice, andlattice artifactsmay play
a significant role. Secondly, this is not qualitatively dif-
ferent from the confinement phase, again reemphasizing
that the non-perturbative distinction of the Higgs and
the confinement phase is still rather obscure.
00 100 200 300 400 Finally, both the renormalizedmass andthe screening
p [GeV] mass of the elementary Higgs agree with a very simple-
mindedconstituentHiggsmodelfortheHiggsoniumstate
FIG.4: TheHiggspropagator. Momentaarealongthex-axis (1). This therefore supports the view of the elementary
forlowmomentaandalongthexy-diagonalforhighmomenta. Higgs building up the gauge-invariantstates in the form
The quenched, confinement, and Higgs case are denoted by of bound-states.
triangles, circles, and squares, respectively.
IV. NON-PERTURBATIVE
where Π is its self-energy, and Z and δm2 are the
H H GAUGE-DEPENDENCE
wave-function and mass renormalization constants, re-
spectively. The two renormalization conditions imple-
As noted, the Gribov-Singerambiguityimplies that in
mented here are
principlethereexistsmorethanonesolutiontothegauge
δab condition(2). The gaugeusedsofarreducesthenumber
Dab(µ2) = of Gribov copies by fixing to the first Gribov region in
H µ2+m2
H Landau gauge, and then choosing randomly a represen-
∂DHab(µ2) = − 2µδab , tative from the residual gauge orbit left. This implies
∂p (µ2+m2 )2 that in this minimal Landau gauge the obtained gauge-
H
dependent correlation functions are effectively averaged
where µ = 125 GeV and the would-be pole mass m is with a certain weight overthe residual gauge orbits [21].
H
chosen to be 125 GeV. Using a derivative with respect To the best of our current knowledge, this weight func-
to the momentum instead of the momentum squared tionis flat, andthus the distribution onthe gaugeorbits
is more appropriate for the lattice with its more-or-less is faithfully reproduced.
evenlyspacedmomentaatthemostreliableintermediate This is only one admissible way to treat the residual
scales than the more conventional momentum squared. gauge freedom. Many alternatives have been investi-
Of course, this is also sufficient to determine the two gated, see e. g. [21, 30, 48–50] . In Yang-Mills theory, at
unknown renormalization constants, though the result leastatfinite volumesanddiscretizationsthe correlation
is therefore a bit different from the usually employed functionsdependonthisnon-perturbativechoice,ingen-
schemes. eral up to the typical scale Λ . Though of course this
YM
The results for the renormalizedHiggs propagatorare dependency is irrelevant for the determination of gauge-
showninfigure4. Itisimmediatelyvisiblethattheprop- invariant physical observables like cross-sections, it can
agator in all cases deviates only slightly from an almost be useful to choose an alternate gauge for technical rea-
bare propagator with the renormalized mass. The only sons. E.g.,thereexistshintsthatsomenon-perturbative
significantdeviationsfromthebarepropagatorarefound gaugesin Yang-Millstheory may be more amendable for
at small momenta, and the screening mass D (0)−1/2 is the explicit construction of the Hilbert space, while oth-
H
in all cases larger than the renormalized mass. In the ers,duetoalackofinfraredsingularities,aremoreuseful
Higgs phase, its value is only slightly larger, about 130 in various practical calculations. It is therefore worth-
GeV,whileintheconfinementphaseitisabout145GeV. while to study the dependence of correlation functions
Thus, there remain some non-trivial infrared modifica- on this gauge choice also in the case with matter fields.
tions of the Higgs propagation. Other than that, the For that purpose, the methods described in [21, 49]
Higgs propagator is essentially unaffected compared to will be adopted here. In particular, Gribov copies are
its tree-level behavior. Thus, even in the confinement thensearchedforbyamulti-startalgorithminthegauge-
8
fixing procedure. In the present case, five random starts other hand, the W propagator shows no change within
are performed for each configuration, since only an in- the statistical error, though when including more Gri-
dicative result is desired here. It is inherent to this bov copies this is known to change for pure Yang-Mills
method that only a lower limit to the gauge dependence theory at fixed lattice parameters [48]. Nonetheless, the
of the propagatorscan be achieved. running coupling, derived from the gauge propagators,
The first finding is that no Gribov copies are found is consequently significantly dependent on the gauge. In
in the Higgs phase. Besides the interesting option that contrast, the Higgs propagator is not gauge-dependent
indeed in the Higgs phase only configurations contribute within the statistical errors. This is not surprising, as
to the path integral significantly which have no Gribov thegaugeconditionsincludeonlytheghostandW fields.
copies inside the first Gribov region, the second pos- However, if a ’t Hooft gauge outside the Landau limit is
sibility is that the strongly volume and discretization- used, this may change. Still, the gauge-dependence in
dependent [21] number of Gribov copies for the se- the casewithHiggsis rathersimilartothatinthe Yang-
lected parameters is just so small that none have been Millscaseforallcorrelationfunctions,againemphasizing
found. Given the experience with results on small two- the similarity between both cases.
dimensionalvolumesinYang-Millstheory[21,30],where Intotal,thenon-perturbativegauge-dependenceinthe
alsoveryfewcopiesarepresent,andtheappreciablevari- confinementphaseis sizable,andshouldbekeptinmind
ation of the propagators in the investigated momentum if, e. g., results should be transferred between different
window here, either possibility would indicate that the calculationsorevenmethods. Ontheotherhand,within
residualgaugeorbitintheHiggsphasehasasignificantly the limited scope of this investigation, no effect in the
different structure than in the confinement phase. Higgs phase is found.
Analternativewouldbethattheemployedmulti-start
algorithm is just not successful in finding copies. Again,
given its successes for the Yang-Mills theory [21, 49, 50], V. SUMMARY
this wouldimply a significantlydifferentstructure ofthe
residual gauge orbit in the Higgs phase.
Summarizing, the present study shows the feasibility
Irrespectiveoftheprecisereasontheresidualgaugede- ofdeterminingthegauge-dependentcorrelationfunctions
pendence of the correlationfunctions in the Higgs phase inYang-Mills-Higgssystemsbeyondperturbationtheory,
is thus not existent, and in the following only the con- in both the confinement and the Higgs phase, even for a
finementphasewillbeinvestigated,andcomparedtothe very heavy Higgs. It therefore provides a route to unify
Yang-Mills case. In that case, the average number of the non-perturbative gauge-invariantdescription usually
GribovcopiesfoundalmostsaturatesthenumberofGri- employedinlatticecalculationswiththeapproachofper-
bov copies checked, i. e., there have been found 4.38(3) turbation theory, which operates directly on the gauge-
copies in the confinement phase per configurationin five dependent elementary degrees of freedom. In addition,
attempts. Inthe Yang-Mills case,this number is slightly first hints how to provide such a relation have been ob-
larger, 4.55(6). tained.
Toprovideanestimateofthevariabilityofthecorrela- This exploratory investigation has furthermore shown
tion functions, these will be determined for the absolute that the difference between the would-be confinement
Landau gauge and the max-B gauge, see [21] for their phase and the would-be Higgs phase is essentially man-
definition. These gauges have been found so far to limit ifest in the gauge-fixing sector outside the physical do-
the variability of the correlation functions in Yang-Mills main,inagreementwith previousstudies andarguments
theory at a fixed number of Gribov copies. The first of [44–46]. It has also been shown that, within the limited
these gaugesattempts to minimize the W propagatorby range of gauges investigated here, there is little quali-
the choiceofthe Gribovcopy,while the secondattempts tative difference between both phases otherwise. How-
to maximize the ghost propagatorin the infrared. ever, the behavior found in the would-be Higgs phase
The result for the three correlation functions and the is in agreement with the expectations for a dynamically
running coupling are shown in figure 5. The dependence induced Higgs effect. On the other hand, the confine-
of the quenched Higgs propagator is left out intention- ment phase showed little difference compared to Yang-
ally, since it does not represent a dynamical variable, Mills theory. This is similar to the case when including
and therefore its gauge dependence is irrelevant for the dynamical fundamental quarks instead of scalars, where
present purpose. the difference haven been found to be also rather small
Forthe ghostpropagatorthe gaugevariabilityexceeds (see, e. g., [26, 52]).
the statistical error. Thus, it is significantly gauge-de- In the present case, it is impossible to assess what
pendent, even given the small numbers of Gribov copies the continuum and infinite-volume limit of these results
used. In fact, its dependence is expected to increase sig- will be, if the theory should be non-trivial. Therefore, it
nificantly even further when enlarging the search space mustnowbe the primaryaimto further developthis ap-
[50, 51], as is the case in Yang-Mills theory, if the ra- proach, and push in both phases towards the continuum
tio of search space to average number of Gribov copies field theory, or at least to a useful cutoff-theory in case
found is so close to one as in the present case. On the of triviality. This will not be simple in this approach,
9
W propagator - Gauge dependence p2(Ghost propagator) - Gauge dependence
· 10-3
p)
G(
G Without Higgs
Z
0.1
1.5 Confinement phase
2]
-V
e
G
2 [p
p)/
Z( 1
W
Z
0.5
0
0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160
p [GeV] p [GeV]
Higgs propagator - Gauge dependence Running coupling - Gauge dependence
· 10-3
2] )
-V m(
Ge a2)/
p) [ (p
(H a
D
0.04
0.5
0.02
0 0
0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160
p [GeV] p [GeV]
FIG. 5: Thegauge dependenceof thecorrelation functions in comparison between theYang-Mills theory and theconfinement
phase. The top left panel shows the W propagator, the top right panel the ghost dressing function, the bottom-left panel
the Higgs propagator (only the confinement phase is shown, see text), and the bottom right panel shows the running gauge
coupling. The data points are in the minimal Landau gauge, and the hatched area represents the variability of the average
valuewith thegauge. Diagonally hatchedareas are for theconfinementphase, and vertically hatchedareas are for thecase of
Yang-Mills theory. For clarity, scale uncertainties have been suppressed in this figure, as they are not relevant since the ratio
of scales of both cases is fixed.
since the Higgs propagator itself will not be sufficient Acknowledgments
to differentiate between an only very weakly interacting
theory and a trivial one. This requires determination of
I am grateful to C. B. Lang and O. Rosten for help-
the vertices, a technique well-developed for Yang-Mills
ful discussions and to R. Alkofer, J. Greensite, and J.
theory [24, 53], and straightforwardly applicable to the
M. Pawlowski for comments on the manuscript. This
present setting. Also, results in the confinement phase
work was supported by the FWF under grant number
are hopefully shedding light on how the confinement of
M1099-N16. The ROOT framework [54] has been used
fundamental charges in dynamical theories is manifest
in this project. Computing time was provided by the
in low-order correlation functions [17], and how it is
HPC center at the Karl-Franzens-University Graz and
different from screening due to the Higgs effect.
the Slovak Grant Agency for Science, Grant VEGA No.
2/6068/2006.
10
Effective h mass into account. A convenient way is to fit the correlator
with the ansatz [55]
u.) 4
a.
1)) ( Cf(t)=acosh(m(t−N/2))+bcosh(n(t−N/2)), (A1)
+
C(t 3
C(t)/ where all quantities are in lattice units. The result, to-
n( Confinement phase getherwiththerawdata,isshowninfigure7. Errorsare
l
determined by fitting besides the central value the one-
2 σ intervals, and as fit intervals always the time interval
Higgs phase
is used for which the correlation function is a positive,
monotonouslydecayingfunctionof time, andis adjusted
for both upper and lower one-σ bounds.
1
Despite the large number of configurations included,
see table II, it is in particular for the confinement case
not possible to unambiguously identify an excited state.
0
0 1 2 3 4 5 6 7 Usage of an enlarged base of operators may here be use-
t/a ful in future calculations [55]. Therefore, only a single
state fit is possible. The resulting fit parameters, to-
FIG. 6: The effectivemass [55] as a function of time. gether with the number of configurations, are given in
table II. The procedure in the main text is then used to
h correlator fit associate a scale with the lower mass in the Higgs phase
andthe only mass in the confinementphase. The result-
C(t) (a.u.)105 CFHioitgngfsin pehmaesnet phase ittnhiogenvsstaialnutiesseticsictgaioilvneernIrIoIinr..HtIatobwilseepvIe.rorT,pthahgeeaetdrerrdoerstshoinrnogtuhfguehnscactlailolcenaisls,cujaunlasdt-
104 thus also the running coupling depending only on these,
Fit
are dimensionless quantities, and therefore only the de-
termination of the momentum in these cases receive an
103 additional error. Furthermore, since the quenched case
has a uniquely assigned scale based on the scale in the
confinement case, the scale erroris irrelevantwhen com-
102
paring only the confinement and the quenched case, and
can therefore be dropped in section IV.
10 Unfortunately, with the available lattice techniques it
cannever be guaranteedthat there is not a further state
still present, which is still lighter than the lightest one
1
0 1 2 3 4 5 6 7 found, in particular if having a very small pre-factor.
t/a This is a source of systematic uncertainty in the scale
determination. To illustrate this uncertainty, it will now
FIG. 7: The quality of determination of the scale. Shown is be assumed that in the confinement phase there exists
theraw data together with a fit of the type(A1). a lighter state with the same mass as the lighter state
in the Higgs phase. For this hypothetical situation the
error on this scale can be ignored. The results for the
Appendix A: Determination of the scale and W sector in this case are shown in figure 8 and for the
systematic uncertainty Higgs sector in figure 9. In the gauge sector nothing
dramatic changes. Only the W propagator becomes less
As noted in section II, the ground-state energy of the tree-level-like,whiletherunningcouplingbecomessome-
state (1) is used to set the scale. This energy can be what more similar between both cases. Still, the ghost
extracted from the exponential decay of its correlation is the one most different between both cases, being a
function C(t) at asymptotically large times [55], in par- strongly momentum-dependent function in the confine-
ticular from a plateau in ln(C(t + 1)/C(t)). However, ment phase, but almost momentum-independent in the
the correlator may also contain contributions from ex- Higgs case. In the Higgs case, the changes are essen-
cited states at finite times. As is visible in figure 6, this tially negligible. Thus, the statements made in the main
isindeedthecase,atleastintheHiggsphase. Thus,itis text are likely only (weakly) quantitatively affected by
necessarytotakethe presenceofexcitedstatesexplicitly the systematic uncertainty in the scale determination.