Table Of ContentDESY14-248 HU-EP-14/67 SFB/CPP-14-105
Phase structure and Higgs boson mass in a
Higgs-Yukawa model with a dimension-6 operator
5
1 DavidY.-J.Chu
0
NationalChiaoTungUniversity,Hsinchu,Taiwan
2
E-mail: [email protected]
n
a Karl Jansen
J
NIC,DesyZeuthen,Germany
1
E-mail: [email protected]
]
t BastianKnippschild
a
l HISKP,Bonn,Germany
-
p E-mail: [email protected]
e
h
C.-J. DavidLin
[
NationalChiaoTungUniversity,Hsinchu,Taiwan
1 E-mail: [email protected]
v
6
Kei-IchiNagai
0
3 KMI,Nagoya,Japan
0 E-mail: [email protected]
0
. Attila Nagy∗
1
0 HumboldtUniversityBerlin;NIC,DesyZeuthen,Germany
5
E-mail: [email protected]
1
:
v
We investigatethe impactof a l j 6 term includedin a chirally invariantlattice Higgs-Yukawa
i 6
X
model.SuchatermcouldemergefromBSMphysicsatsomelargerenergyscale.Wemapoutthe
r
a phasestructureoftheHiggs-Yukawamodelwithpositivel 6andnegativequarticselfcouplingof
thescalarfields. Tothisend,weevaluatetheconstrainteffectivepotentialinlatticeperturbation
theoryandalsodeterminethemagnetizationofthemodelvianumericalsimulationswhichallow
usto reachalso non-perturbativevaluesofthe couplings. As a result, we find a complexphase
structurewithfirstandsecondorderphasetransitionsidentifiedthroughthemagnetization. Fur-
therweanalyzetheeffectofsuchaj 6termonthelowerHiggsbosonmassboundtosee,whether
thestandardmodellowermassboundcanbealtered.
The32ndInternationalSymposiumonLatticeFieldTheory,
23-28June,2014
ColumbiaUniversityNewYork,NY
∗Speaker.
©Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
1. Introduction
With the discovery of the Higgs boson all ingredients in the standard model (SM) of particle
physics havebeencompleted inasense, thatitindeed describes particle interactions intheregime
ofenergiesthatcanpresentlybeprobedexperimentally. Therearehoweverphenomenathatcannot
beexplained bythestandard model, likedarkmatter, darkenergy andtheincorporation ofgravity
in the framework of the other three known fundamental forces. Thus, the SM can account for
all phenomena and needs to be replaced at some yet unknown energy. In [1] this question has
been analyzed through the stability of the electro-weak vacuum and a lower bound for the Higgs
boson mass was found that is required to obtain a fully stable vacuum. This bound was found to
be at m >129.4±1.8 GeV. The main uncertainties originates from the top quark mass and the
H
strong coupling, determined atthemassoftheZ-bosona (m ). Themassbound wasobtained by
s Z
running allSMcouplings uptothePlanckscale and requiring that thequartic self coupling ofthe
scalar doublet l remains positive. A Higgs boson with a mass slightly below the bound derived
in[1]mayyieldametastablevacuum: Thesystemcanremaininalocalminimumofenergy with
a non-vanishing probability to tunnel into the global vacuum with lower energy. Depending on
the parameters, suchametastable state canhave ameanlifetimethat exceeds thelifetimeofthe
universe, sothatsuchascenario doesnothaveimmediateconsequences.
In this work we pursue an approach that investigates how the standard model Higgs boson
mass lowe bound can be altered by the inclusion of a higher dimensional operator in the electro-
weaksector. Explicitlyweaddal j 6termtotheaction. Withl >0,theactionremainsbounded
6 6
frombelowevenfornegativequarticselfcouplingofthescalarfields. Theinclusionofsuchaterm
mayappear naturally ifone considers the Higgs sector ofthe SMasalowenergy effective theory
obtainedbyintegratingoutdegreesoffreedomatsomehigherscalephysics. Forourinvestigations
we use a chirally invariant lattice formulation of the Higgs-Yukawa model as a limit of the SM
where only one family of quarks and the scalar doublet are considered. This model was already
successfully used to determine the cutoff dependence of the Higgs boson mass bounds for the
SM [2, 3], and to investigate the change on those bounds in case of a heavy fourth generation of
quarks [4].
Earlier works showed that in the Higgs-Yukawa model with l = 0 the phase transition of
6
interest, i.e. the transition between a symmetric and a broken phase with small quartic couplings
and yukawa couplings generating quarks with a mass of the order of the top quark mass, is of
second order. This may change drastically with the inclusion of a l j 6 term due to the more
6
complex structure of the bosonic potential. As we will see below, depending on the choice of l
and l there appear lines of first order transition separating the symmetric and the broken phases
6
orevenfurthertransitions betweendifferentnon-zero magnetizations.
InthisworkwepresentresultsonthephasestructureandtheHiggsbosonmasshavingl 6=0.
6
We study both aspects perturbatively by means of Lattice perturbation theory using a constraint
effectivepotential (CEP)andnon-perturbatively vialatticesimulations.
2. The Higgs-Yukawamodel onthe lattice
The field content of the Higgs-Yukawa model in this work is given by a mass degenerate
fermion doublet y andthe complex scalar doublet j . Thecontinuum formulation oftheaction of
2
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
theHiggs-Yukawamodelisgivenby:
Scont[y¯,y ,j ]=Z d4x(cid:26)21(cid:0)¶ m j (cid:1)†(¶ m j )+21m20j †j +l (cid:16)j †j (cid:17)2+l 6(cid:16)j †j (cid:17)3(cid:27)
+Z d4x t¯¶/t+b¯¶/b+y(y¯Lj bR+y¯Lj˜tR)+h.c. , (2.1)
(cid:8) (cid:9)
withj˜ =it f ∗ andt beingthesecondPaulimatrix.
2 2
Foranydetailsonthenumericalimplementation ofthelattice simulations wereferto[5]. We
just mention that we use the polynomial Hybrid Monte Carlo algorithm as the basic simulation
tool. Forthediscretizationofthefermionsweusethechirallyinvariantoverlapoperator. Tosetthe
scaleweidentifythevacuumexpectationvalues(vev)ofthescalarfieldwiththephenomenological
valueofvev≈246GeV.
Theperturbative resultsareobtained byusingaconstraint effectivepotentialU(vˆ)[6,7]. The
basic idea is that the system is dominated by the zero mode of the scalar field, and therefore we
can integrate out allnon-zero modes. Thevevofthescalar fieldandthe Higgsboson masscanbe
obtained bytheglobalminimumandthecurvature attheminimumofthepotential, respectively:
dU(vˆ) d2U(vˆ)
0= , m2 = . (2.2)
dvˆ (cid:12)(cid:12) H dvˆ2 (cid:12)(cid:12)
(cid:12)vˆ=vev (cid:12)vˆ=vev
(cid:12) (cid:12)
Wepointout, thatweexplicitly keepthe(cid:12) latticeregularization(cid:12)intheperturbative approach. Thus,
we will work with only discrete momenta and we also use the overlap operator for the fermionic
kernel.
WecomparetwoexpressionsfortheCEPwhichdifferbythedecomposition oftheactioninto
aninteraction partthathastobeexpanded inpowersofthecouplings andaGaussianpartthatcan
beintegrated out. Thefirstexpression wasalready usedin[8,2]fortheHiggs-Yukawamodeland
isgiventofirstorderinl andl by:
6
m2
U1(vˆ)=Uf(vˆ)+ 20vˆ2+l vˆ4+l 6vˆ6
+l ·vˆ2·6(PH+PG)+l 6· vˆ2·(45PH2+54PGPH+45PG2)+vˆ4·(15PH+9PG) . (2.3)
(cid:16) (cid:17)
Inthepropagator sumsP = 1 (cid:229) 1 themassesareset“byhand”tozerofortheGold-
H/G V p6=0 pˆ2+m2
H/G
stone and to the Higgs boson mass obtained from eq. (2.2). U denotes the contribution from
f
integrating outthefermionsinthebackground ofaconstant field.
A second possibility to formulate the CEP is obtained, when taking zero-mode contributions
fromtheselfinteractionsofthescalarfieldintoaccount,byintegratingouttheGaussianpartinthe
non-zero modes ofthescalar field. Inthisapproach, logarithmic dependence onvˆappear. Further
thepropagator sumsandthefirstordercontribution intheselfcouplings change, yielding:
U2(vˆ)=Uf(vˆ)+m220vˆ2+l vˆ4+l 6vˆ6+21V p(cid:229)6=0log(cid:20)(cid:16)pˆ2+m20+12l vˆ2+30l 6vˆ4(cid:17)·(cid:16)pˆ2+m20+12l vˆ2+30l 6vˆ4(cid:17)3(cid:21)
+l 3P˜H2+6P˜HP˜G+15P˜G2 +l 6vˆ2 45P˜H2+54P˜HP˜G+45P˜G2 +l 6 15P˜H3+27P˜H2P˜G+45P˜HP˜G2+105P˜G3 ,
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(2.4)
with
P˜H=V1 p(cid:229)6=0pˆ2+m20+121vˆ2l +30vˆ4l 6, P˜G=V1 p(cid:229)6=0pˆ2+m20+41vˆ2l +6vˆ4l 6. (2.5)
This second method should be more precise but has a limited range of validity depending on the
parameters: Itfails,whentheargumentofthelogarithmsbecomenegativeleadinginthiscasetoa
complexpotential.
3
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
3. Results onthe phasestructure
Forour analysis wechose two values for l , namely l =0.001 and l =0.1. Using the the
6 6 6
tree-level relation m =y·vev we further fix the Yukawa coupling to yield the physical top quark
t
mass. Weworkatseveralvaluesforl andperformscansinm2 or,equivalently, thescalarhopping
0
parameter in the lattice action k 1. For the here performed phase structure study we use the vev
as an order parameter. In infinite volume the vev is zero in the symmetric phase and non-zero in
the broken phase. In finite volume the vev never assumes an exactly zero value, so eventually an
extrapolationtoinfinitevolumemustbecarriedouttounambiguouslydeterminethephasestructure
ofthesystem.
1.6 λ=−0.36
1.4 0.8 λλ==−−00..3378 1
−1a 1.12 −1a 0.6 λ=−0.39 −1a
in 0.8 in in
vev 0.6 λ=−0.0080 vev 0.4 vev 112633××2342
0.4 λλ==−−00..00008858 0.2 0.1 220433××2408
0.2 λ=−0.0090 323×32
0 0
0.12276 0.12279κ0.12282 0.12285 0.116 0.117 κ 0.118 0.1227 0.12273 0κ.12276 0.12279
(a)l6=0.001 (b)l6=0.1 (c)l6=0.001,l =−0.0085
Figure1:Thefirsttwoplotsshowthevevasafunctionofk forvariousvaluesofl whilel 6iskeptfixedtol 6=0.001
(a)andl 6=0.1(b)obtainedon163×32lattices.Weshowacomparisonbetweendataobtainedfromlatticesimulations
(opensquares)withtheCEPU1(crosses)andU2(dots).In(c)thedependenceonthevolumeisillustratedforl 6=0.001
andl =−0.0085.
Asa first step we compare results obtained from the CEPand lattice simulations in figure 1a
and1bwhereweshowthevevagainstthehoppingparameterk ona163×32lattice. Weshowfor
both values of l the results from lattice simulations and the analytical results from both CEPsin
6
eqs.(2.3)and(2.4). Bothsetupsshowqualitatively thesamepicture: Withl >0andnegativebut
6
smalll ,thetransitionbetweenasymmetricphaseandabrokenphaseiscontinuouswhichsuggests
asecondordertransition, whileforsmallerl theorderparametershowsajumpatsometransition
value indicating a transition of first order. For l =0.001 both potentials describe the data very
6
well,althoughthequantitativeagreementisslightlybetterfortheCEPU ineq.(2.4). Thepotential
2
U in eq. (2.3) also fails to exactly reproduce the first order phase transition for l = −0.0088.
1
However,ifonedecreasesl further,alsothepotentialU showsafirstordertransition. Forl =0.1
1 6
theCEPU doesnotreproduce thebehavior ofthesimulation data. Thisisnotsurprising, sincein
2
this case analyzing the CEPU in the region of parameter space close to the the phase transitions
2
wemeetthedifficulty that theeffective potential becomes complex. QualitativelyU showsstill a
1
good agreement, but again the value of l where the transition turns first order is not reproduced
exactly. In1cweshow the volume dependence of thevevforfixedl =0.001 andl =−0.0085,
6
comparing results obtained from both potentials and simulation data. Wefind, that the qualitative
behavior ofthesimulationdataisverywellreproduced bybothperturbative approaches.
Since the constraint effective potential describes the results from the simulations reasonably
well, we willrely on the CEP to perform a more complete study of the phase structure with large
volumes. Furthermore, we restrict ourselves to the potential U . The exact location of the phase
1
transition in finite volume cannot easily be determined from the order parameter alone due to the
1Therelationbetweenk andm20isgivenby:m20= 1−8lkk 2−8k .
4
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
fact that the vev does not exactly go to zero. One possibility is to investigate the curvature of the
potentialatitsminimumwhichisidenticaltothesquaredHiggsbosonmassaccordingtoeq.(2.2).
The inverse curvature of the potential in the minimum is related to the susceptibility and the peak
position ofwhichcanthenbeusedforlocating ofthephasetransition point.
1283×256 0.1232 0.135
−1′′Uvev()1 110034 963164263333××××11639242280.51vev κ κ 000000......111111222222222223024680 ssbyyrmmokmbmernbeoertotkrkrie
ein
n
rosso2v1nesdrt κ 00000.....111111122305050 symmbetrroik
en
rosso2v1nesdrt
102 -0.008 -0.01 -0.388 -0.389
0.1218 0.105
0.12273 0.12276 0.12279 0 -0.003 -0.006 -0.009 -0.012 -0.015 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6
κ λ λ
(a)l6=0.001,l =−0.0085 (b)l6=0.001 (c)l6=0.1
Figure2: Thefirstplotshowsthevev(inlet)andtheinversecurvatureattheminimumofthepotentialasafunctionof
k forvariousvolumes. Thefirstpeakisagoodindicatorforasecondordertransition,whilethesecondpeakwithout
volume dependence may refer to across over. The other figures show the result of a morecomplete phase structure
scan for l 6=0.001 (middle) and l =0.1 (right) obtained from theCEPU1 (2.3). Therearetwo phases - abroken
andasymmetricone-separatedbylinesoffirstandsecondorderphasetransitions. Furtherthereisasmallregionin
parameterspace,wherethereisalsoafirstordertransitionbetweentwobrokenphases(forl 6=0.001andl 6=0.1).
Thelinesconnectingthedatapointsaretoguidetheeye.
Infigure2aanexampleplotisshownthatillustratesthebehavioroftheorderparameterandthe
inverse curvature ofthepotential atitsminimum asameasure ofthesusceptibility forl =0.001
6
and l =−0.0085. The inverse curvature shows two peaks. The first peak is getting higher with
increasing volume, a typical behavior of the peak of the susceptibility which is diverging at the
critical coupling whenthevolumeisincreased toinfinity. Thesecondmaximumdoesnotincrease
withincreasing volumewhichsuggests thatthisisacrossover transition.
Performing such scans systematically, we obtain the phase diagrams in the k −l -plane for
both fixed l =0.001 in figure 2b and l =0.1 in figure 2c. Qualitatively both diagrams show
6 6
the same behavior: for l = 0 there is a single phase transition of second order separating the
symmetric and the broken phases. If l is decreased, the transition point moves to smaller k . A
some point, a line of crossover transition appears in the broken phase which turns into a line of
first order transition. Atsome critical point the line of second order transition joins the first order
transition line. Beyond that, only a first order transition separates the symmetric and the broken
phases. For l = 0.1 the region in the parameter space where the first order and the crossover
6
transitions appear in the broken phase is very narrow. Given the fact that the agreement between
theCEPandthesimulation dataforavalueofl =0.1isnotcompletely satisfactory, atestofthe
6
phasestructure fromnumerical simulations wouldbeverydesirable.
4. Results onthe Higgsbosonmass
In this section, the influence of the l j 6 term on the lower Higgs boson mass bound is dis-
6
cussed. Especially we look at the cutoff dependence of the lower Higgs boson mass. In figure. 3
weshowtheperturbativeresultsobtainedwiththepotentialU forbothvaluesofl used. Wealso
1 6
show the result of the lower Higgs boson mass bound for the case of vanishing j 6 term. All data
with non-zero l and l show a similar behavior. For small cutoff values the mass increases sub-
6
stantially, a phenomenon which is not present for the case of vanishing couplings. This behavior
is related to the mass shift from positive l values, which contribute significantly at small cutoff
6
5
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
scales. The Higgs boson mass then has a minimum at some given cutoff (for the data that show
a second order transition) and increases as similar to the SM bound. Further we can see that the
Higgsboson massgetssmaller, whenl isdecreased. Asacriterion, howlowthequartic coupling
can be driven, we use a value of l where the transition between the broken and the symmetric
phaseturnsfromsecondtofirstorder.
100
GeV 6800 GeV 111024000 λλλλS====M−−−−b0000o....u3333n8888d9850 GeV 6800 λλSλλM====b−−−−o0000u....n00000000d88888705((21nstdoorrddeerr))
in in 80 in
mH 40 λ=SM−0b.o0u0n8d0 mH 60 mH 40
λ=−0.0085 40
20 λ=−0.0088 20
λ=−0.0089 20
0 0 0
0 500 1000 1500 2000 0 2000 4000 6000 0 400 800 1200 1600
Λ Λ Λ
inGeV inGeV inGeV
(a)l6=0.001 (b)l6=0.1 (c)l6=0.001
Figure 3: Shown is the cutoff dependence of the Higgs boson mass obtained from the CEP according to eq. (2.2)
forl =0.001ona643×128-lattice(left)andl =0.1ona1923×384(middle)andpreliminaryresultsfromlattice
simulationsforl 6=0.001.Inallplotswealsoshowthestandardmodellowermassbound(l 6=l =0).
Forthecaseofl =0.001thelowerHiggsbosonmassboundissignificantlydecreasedasalso
6
foundin[9]. Thus,forsuchsmallvaluesofl thehereconsideredextensionofthestandardmodel
6
withaj 6-termisfullycompatiblewiththe126GeVHiggsbosonmass. Forl =0.1thesituation
6
changes since here the lower bound meets the 126 GeV Higgs boson mass at a cutoff of around
800GeV,whichisstillinsidethescalingregion. Thus,largevaluesofl cannotbeadmittedasan
6
extension ofthestandard model. Forboth values ofl and atlarge cutoffs theinfluence ofthej 6
6
becomes negligible and we find the SM-like behavior of the lower Higgs boson mass bound as a
function ofthecutoff.
Preliminary results on the cutoff dependence of the lower Higgs boson mass obtained from
non-perturbative lattice simulations can be found in fig. 3c for a fixed value of l =0.001. We
6
show data for values of l corresponding to ones used in figure 3a. Note that for some of the
intermediate l values the order of the phase transition is still not clear. Although the data for
l &−0.0087, where there is still a second order phasetransition, still have large error bars, it is
evident, thatalsonon-perturbatively thestandard modellowerbound caneasily bedecreased well
below126GeVwhilekeeping m /L wellbelow1/2andhencestaying inthescalingregion.
H
5. Summary andconclusion
We studied the phase structure of a chirally invariant lattice Higgs-Yukawa model – allow-
ing non-perturbative computations – with the addition of a l j 6 term as a simple model for an
6
extension of the standard model. Having l > 0 allows to set the quartic coupling l < 0. We
6
found good agreement between a lattice perturbation theory approach using the CEP and Monte
Carlosimulationsforthebehaviorofthevacuumexpectationvalueasfunctionofthebaremass. A
systematic studyofphasetransitions usingthevevandthesusceptibility ledtothephasediagrams
showninfigures2band2c. Wefoundtransitionsoffirstandsecondorderseparatingthesymmetric
and the broken phase as well as first order transitions separating two broken vacua indicating the
possibilityofmetastablestates. Theappearanceoffirstorderphasetransitionsinthepresenceofa
j 6-term canbeveryinteresting forthecase ofanon-zero temperature. Itmight leadtoascenario
whereasimpleadditionofaj 6-termcanprovideastrongenoughfirstorderphasetransitionstobe
6
PhasestructureandHiggsbosonmassinaHiggs-Yukawamodelwithadimension-6operator AttilaNagy
compatible withelectro-weak baryogenesis intheearlyuniverse[10]. Anotheraspectofthephase
diagram in figure 2 is that at fixed bare Higgs boson mass one can move from a symmetric to a
broken phasebyonlychanging thevalueofthequarticcoupling.
Further we investigated the influence of the dimension-6 operator on the Higgs boson mass
bound with respect to the question whether the addition of this operator can be compatible with a
126 GeV Higgs boson mass and whether the lower bound can be altered compared to the Higgs-
Yukawa limit of the standard model (i.e. l =0). We found that for the values of l considered
6 6
here,atlargevaluesofthecutoffthelowerboundcanbesignificantlydecreasedbeforeitbecomes
compatible withthecase ofl =0for increasing cutoffs. However, foralarger value ofl =0.1
6 6
the lower Higgs boson mass bound meets the 126GeV Higgs boson mass already at a cutoff of
about800GeV.Thus,suchalargevalueofl isexcluded. Weplantodetermineacriticalvalueof
6
l from which on an extension of the standard model with a j 6-term is not compatible anymore
6
with the 126 GeV Higgs boson mass. This can in turn provide bounds on models beyond the SM
thatgenerateeffectively suchaterm.
6. Acknowledgements
Thesimulations havebeenperformedatthethePAXclusteratDESY-Zeuthen,HPCfacilities
at National Chiao-Tung University and the cluster system j at KMI in Nagoya University. This
work is supported by Taiwanese NSC via grants 100-2745- M-002-002-ASP (Academic Summit
Grant), 102-2112-M-009-MY3, and by the DFG through the DFG-project Mu932/4-4, and the
JSPSGrant-in-Aid forScientificResearch(S)number22224003.
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