Table Of ContentPerturbative extension of the standard model
with
a 125 GeV Higgs boson and magnetic dark matter
Karin Dissauer,1,2,∗ Mads T. Frandsen,1,† Tuomas Hapola,1,‡ and Francesco Sannino1,§
1CP3-Origins & Danish Institute for Advanced Study DIAS,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
3 2InstituteofPhysics,Karl-FranzensUniversityofGraz,Universita¨tsplatz5,A-8010Graz,Austria
1
0
2 Abstract
n
a Weintroduceaperturbativeextensionofthestandardmodelfeaturinganewdarkmattersector
J
0 together with a 125 GeV Higgs. The new sector consists of a vectorlike heavy electron (E), a
3
complex scalar electron (S) and a standard model singlet Dirac fermion (χ). The interactions
]
h χ
among the dark matter candidate and the standard model particles occur via loop-induced
p
p- processes involving the operator SEχy, with y being the Yukawa-like coupling. The model is
e
h an explicit underlying realization of the light magnetic dark matter effective model introduced
[
earlier to alleviate the tension among several direct dark matter search experiments. We further
2
v
4 constraintheparametersoftheunderlyingtheoryusingresultsfromtheLargeHadronCollider.
4
1 The extension can accommodate the recently observed properties of the Higgs-like state and
5
. leads to interesting predictions. Finally we show that the model’s collider phenomenology and
1
1
2 constraintsnicelycomplementtheonescomingfromdarkmattersearches.
1
v: Preprint: CP3-Origins-2012-31,DIAS-2012-32.
i
X
r
a
∗
Electronicaddress: [email protected]
†
Electronicaddress: [email protected]
‡
Electronicaddress: [email protected]
§
Electronicaddress: [email protected]
1
I. ASEχyEXTENSIONOFTHESTANDARDMODEL
After the recent discovery [1, 2] of a new resonance with properties similar to the
standardmodel(SM)Higgsparticle,itisparticularlytimelytoinvestigateminimalexten-
sionsoftheSMfeaturingalsodarkmattersectors. Hereweputforwardarenormalizable
extension featuring, besides the Higgs at 125 GeV, also a dark matter sector. The new
sector consists of a vectorlike heavy electron (E), a complex scalar electron (S) and a SM
χ
singletDiracfermion( ). TheassociatedrenormalizableLagrangianis
L = L +χ ∂/χ− χχ+ / − − χ +
SEχy SM ¯i mχ¯ EiDE mEEE (SE y H.c.)
+ DµS†DµS−m2SS†S−λHSH†HS†S−λS(SS†)2 , (1)
where Dµ = ∂µ − ieswZµ + ieAµ, s and c represent the sine and cosine of the Weinberg
cw w w
,λ λ
angle. We assume the new couplings y and to be real and the bare mass squared
HS S
of the S field to be positive so that the electroweak symmetry breaks via the SM Higgs
χ
doublet (H). The interactions among , our potential dark matter candidate, and the SM
fields occur via loop-induced processes involving the SE¯χy operator in (1). If, however,
theHiggssectoroftheSMisnaturalanddescribed,forexample,bycompositedynamics
[3],onecanimaginethenewsector,and,inparticular,thescalarS,alsotobecomposite.
χ
Attheone-looplevel, developsthefollowingmagnetic-typeinteractions
λ
L = χχσ χ µν − sw λ χσ χ µν,
5 ¯ µν F χ¯ µν Z (2)
2 2c
w
µν µν λ
where F and Z are the photon and Z field strength tensors, and χ is related to the
electromagneticformfactorF (q2),
2
F (q2)
λ = 2 .
χ e (3)
2mχ
Theexplicitderivationoftheone-loop–inducedformfactorcanbefoundinAppendixA.
It is instructive to report the analytic form for a few interesting limits to learn about the
dependence upon couplings and masses. We start by considering the static limit F (0),
2
useful for dark matter direct detection experiments. To reduce the parameter space we
= =
take the masses of E and S to be degenerate, m m M, leading to the simplified
E S
expression
(cid:114) (cid:32) (cid:33)
y2 2 2+z z
F2(0) = 8π2 z 2−z tan−1 √4−z2 −1, (4)
2
wherez = mχ. Forsmallzwehave
M
(cid:32) (cid:33)
y2 z2 (cid:16) (cid:17)
F (0) = z+ +O z3 , (5)
2 16π2 3
which for z = 0 gives λχ = ey2/(32π2M). It is also interesting to consider a dark matter
candidate with mass of the order of the electroweak scale or slightly higher. For this
purpose a simple estimate for the electromagnetic form factor can be deduc√ed by setting
π−
3 3
z = 1(i.e. mχ = M),√andMaroundtheelectroweakscale,yieldingF2(0) = y2 24π2 and,
π−
3 3
therefore, λχ = ey2 48π2M . From the two limits it follows that the scale of the magnetic
momentiscontrolledbythecommonmassoftheheavystatesM.
The model allows us to investigate the interplay between dark matter and ordinary
matter and the possibility that dark matter is either light with respect to the electroweak
scale or has the same mass scale. The model can be constrained by dark matter experi-
mentsandatthe(LHC).
The light dark matter limit allows us to explore models of long-range interactions
which can be useful to alleviate the tension between the experimental observations by
/
DAMA LIBRA [4] and the limits set by XENON100 [5, 6] and CDMS [7]. The light dark
ff
matter limit naturally maps in the e ective model of magnetic light dark matter put
forward in, e.g., Refs. [8–11], while allowing us to investigate its phenomenology at the
electroweakenergyscale.
Electroweakscaledarkmatterispotentiallyobservableincosmic-raydetectionexper-
iments. If the tentative observation of the 135 GeV Fermi line [12–15] is confirmed by
future data, the model studied here can be opportunely extended to explain it [16]. The
present model, albeit similar to the one of Ref. [16], was conceived independently in
ordertoprovideanUVcompletionoftheanalysispresentedinRef. [10]. Inadditionthe
ff
main di erences with the model presented in Ref. [16] are: Our model describes elastic
magnetic dark matter; we discuss all the relevant renormalizable interactions with the
SM, in particular, the coupling of the new sector to the SM fermions and the Higgs. The
coupling to the Higgs is quite important, allowing us to properly investigate the collider
phenomenologyofthemodel.
The paper is organized as follows. In Sec. II we allow for the heavy electron to decay
directly into SM fermions by adding mixing operators with the SM leptons. We will set
3
preliminary bounds on the mixing angles. In Sec. III we investigate collider constraints
χ
on the spectrum and couplings of the SE y sector. In particular, we will focus on the
γγ γ
invisible width of the Z and the Higgs alongside the decay of the latter into and Z.
ffi ff
Ifdarkmatterissu cientlylighttobeproducedattheLHC,itcana ectmissingenergy
signals[17]whichwillbealsoexploredforthismodel. InSec. IVwesummarizethelight
dark matter constraints set in Ref. [10] and translate them in terms of the fundamental
parametersofthemodel.
II. ALLOWINGETODECAYINTOSMLEPTONS
Heavy single charged leptons cannot be stable on cosmological scales because they
wouldappearasanomalouslyheavyisotopes. Theabundanceofsuchisotopesislimited
to be less than one in 1012 baryons [18] as discussed in, e.g., Refs. [19, 20]. Therefore one
needs to mix the heavy electron with the SM leptons. Here we assume, for simplicity,
τ χ
that the mixing is primarily with the lepton. In general, we can also mix with the
ffi
SM neutrinos provided that its decay time is su ciently long on cosmological scales to
χ
be a proper dark matter candidate. As a first approximation, we take to be stable and,
henceforth,suppressitsmixingwithSMneutrinos.
ThemixingLagrangian,aftertheHiggsfieldhasacquiredavacuumexpectationvalue
(cid:39)
v 246GeV,is
LEτ = L¯(cid:48) MR(cid:48) = (cid:18)τ¯(cid:48)L E¯(cid:48)L(cid:19)yτm√v2 ymE √v2Eτ(cid:48)R(cid:48). (6)
E R
Here the prime on the fields indicates gauge eigenstates. The existing SM mass term for
τ ff
the lepton is the first entry of the matrix. The remaining o -diagonal terms are added
to(1).
To achieve the mass eigenstates, we need to diagonalize M which is a nonsymmetric
realmatrix. Thediagonalizationproceduremakesuseoftwoindependentrotations,one
fortheleft-handedfieldsV andtheotherfortheright-handedfieldsU. Thediagonalized
matrixisthen
= (cid:62) ,
M V MU (7)
D
4
whereV andU arethefollowingrealorthogonalmatrices
cosθ sinθ cosα sinα
V = , U = . (8)
−sinθ cosθ −sinα cosα
(cid:62) (cid:62)
VandUdiagonalize,respectively,thesymmetricsquaredmassmatricesMM andM M.
Themixinganglesaregivenbytherelations
√
+
α = − 2v(myτ mEyE) ,
tan2 (9)
v2(y2τ + y2)−m2 −m2
2 E E
tan2θ = − 2(yτyEv22 +mmE) . (10)
m2 −m2 + v2(y2τ − y2)
E 2 E
τ
The physical mass of the lepton fixes one of the eigenvalues. Constraints on the
θ
left-handed mixing angle come from the Z decay and from the electroweak precision
measurements as discussed in Ref. [21]. These constraints lead to an upper limit, which
atthe95%confidencelevelis
sin2θ < 0.0018. (11)
Depending on the values of the (yet unknown) physical parameters, the right-handed
α θ
angle can be larger or smaller than . In the future, an interesting way to further
ττ
constraintheright-handedanglewouldbebymeasuringthedecayoftheHiggsto .
To simplify the analysis here, we demand E to be stable on collider detection time
scales. Thisrequiresthemixinganglestobeoftheorderof
sin2θ ∼ sin2α ∼ 10−14, (12)
renderingthemixingeffectsnegligible1.
However, in this limit at least one of the new charged particles is stable and will hit,
onceproduced,thewallsoftheLHCdetectors. TheCMSCollaborationexcludedmasses
1However, if the mixing is large, it will lead to interesting phenomenological consequences which we
plantoinvestigateelsewherewhilewementionhereafew. Forexample,theSEχycouplingleadstothe
followingmixingoperatorsbetweenagenericSMlepton,χ,andthescalar,S:
S(E¯ cosθ+(cid:96)¯ sinθ)χy+S(E¯ cosα+(cid:96)¯ sinα)χy+H.c., (13)
L L R R
inducingacorrectiontothe g−2ofthegivenSMlepton(cid:96)aswellasamodificationofitselectricdipole
moment.
5
below 427 GeV for particles with one e charge [22]. The limit is imposed for a lepton
with SM-like neutral-current interactions. If we interpret the CMS results in the context
of our model, we find that E and S have to be more massive than 393 GeV and 243 GeV,
respectively.
III. CONSTRAINTSFROMTHELARGEHADRONCOLLIDER
Having at our disposal the explicit Lagrangian for this particular extension of the
SM, we can now systematically analyze the various phenomenological constraints and
potentialsignalsattheLHC.
A. InvisiblewidthoftheHiggs
χ
Via loop processes the Higgs can decay into two fermions giving a contribution on
the invisible width of the Higgs. The explicit computation and relevant formulas are
χ
derived in Appendix B. Of course, for this process to occur, has to have a mass less
thanoraroundhalfofthemassoftheHiggs. Weassumethatthedominantcontribution
χ
comesfromthedecayoftheHiggsintotwo s,and,therefore,theinvisiblewidthis
(cid:32) (cid:33)3
m (cid:16) (cid:17) 4m2χ 2
Γ [H] ≈ Γ[H → 2χ] = H 4v2λ2 y4|A|2 1− . (14)
inv 32π HS m2
H
The loop function A is given in Appendix B. From the results of Ref. [23], one deduces
σ
therough1 estimate
Γ → χ
[H 2 ]
≤ . .
015 (15)
Γ +Γ → χ
[H 2 ]
SM
In Fig. 1 we present the associated disfavored parameter space in the (mχ, M) plane
assumingverylargevaluesofλ = y = 4πtomaximizetheconstraints2.
HS
2Theselargevaluesofthecouplingsstretchthevalidityofperturbationtheoryandaretakenforillustrative
reasons. Itwouldbeinterestingtoextendtheperturbativeanalysistohigherorders.
6
300
250
]
V
e
G
[200
M
150
10 20 30 40 50 60
0.150
m [GeV]
χ
λ = = π
FIG.1: Theshadedregionoftheparameterspace(mχ,M)assuming HS y 4 is
σ
excludedatthe1 level.
B. InvisiblewidthoftheZ
χ
If hasamassbeloworaroundhalfoftheZmass,thenonecanusetheinvisiblewidth
oftheZasafurtherconstraint. ThemeasuredinvisiblewidthoftheZis[24]
ΓZ = 499.0±1.5MeV, (16)
inv
yieldinga95%confidencelevelupperlimit[25]
Γnew < 2.0MeV (17)
inv
on the invisible width contribution from new physics which does not interfere with the
neutrinopairproduction.
→ χχ
TheZ ¯ decaywidthformulais
(cid:115)
ΓZ→χχ¯ = e2s2w 1− 4m2χ
96c2wm2χmZπ m2Z (18)
(cid:16) (cid:16) (cid:17) (cid:17)
24m2m2F (m2)F (m2)+8F2(m2)m2 2m2 +m2 +F2(m2)m2(8m2 +m2) .
χ Z 1 Z 2 Z 1 Z χ χ Z 2 Z Z χ Z
χχ
It is clear from Fig. 2 that the invisible constraint on the Z boson decay width into ¯ is
= π
weakevenforlargevaluesofthecoupling y 4 .
7
0.14
0.12
0.10
]
V
e
M
0.08
[
)
χ
χ
→0.06
Z
(
Γ
0.04
0.02
0.00
0 10 20 30 40
m [GeV]
χ
χ = π χ
FIG.2: Zdecaywidthtoa pairasafunctionofmχ taking y 4 fortheSE y
coupling.
C. Monojets
Monojet signatures, associated to missing energy, can emerge in this model from dia-
ff
gramsofthekindreportedinFig. 3. Analyticexpressionsforthecorrespondingdi eren-
g q g q
χ χ
E¯ S¯
γ,Z γ,Z
S E
χ¯ χ¯
q¯ E q¯ S
FIG.3: Twodiagramsgivingamonojetsignatureatone-looplevelthrough
quark-gluoninitialstates. Similardiagramsarisealsoviaquark-quarkinitialstates.
tial cross sections in the center-of-mass frame are given in Appendix C. We have imple-
mented the momentum-dependent loop functions given in Appendix A into MadGraph5
8
[26]. This allows us to compute the monojet cross section and compare it with the ex-
perimental results. We take here the common mass for the new charged particles to be
= π
M 300 GeV and the coupling y to be 4 . In Fig. 4 we have plotted the cross section for
a monojet signal as a function of mχ and the 95% confidence level upper limit set by the
CMS Collaboration [17]. From the plot it is clear that the coupling y needs to be at least
10-1
CMS limit, pp@ 7 TeV
b] 10-2 SEχy monojetp
p
[ SEχy monojet, static limit
)
j
χ
χ
→
p
p
σ( 10-3
10-4
0 50 100 150 200 250 300
m [GeV]
χ
FIG.4: Thelowerline: monojetcrosssectionasafunctionofmχ usingparameter
= = π
valuesM 300GeVand y 4 . Theupperline: CMScollaboration95%confidence
levelupperlimitonthemonojetsignalcrosssection.
π
3 times larger than 4 to reach the current experimental sensitivity. Thus, the monojet
searchesdonotconstrainyetthemodelparameters.
Monojets arising through a magnetic moment interaction have been studied in Refs.
λ
[27,28]. Inthesepapersthemagneticmoment χistakentobeamomentum-independent
≈
constant, and limits on its value are derived. In the static limit and for mχ 10 GeV, we
≈ π ≈
recovertheirresultswhentaking y 4 andM 10GeV.
D. DirectLHCprobeoftheSsector
The scalar S has properties reminiscent of a selectron except that the heavy electron is
vectorlike, and, therefore, only the scalar S feels the Higgs directly. This is true provided
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wedonotmixthenewelectronwiththeSMleptonsviageneralizedYukawainteractions.
Due to this property and the requirement of the renormalizability of the theory, the S
sectorcanbeprobeddirectlyusingprocessesinvolvingtheHiggs. Herewewillconsider
theHiggstotwoneutralgaugebosonsprocesses.
1. h → γγ
The measured signal strength in the Higgs to a two-photon decay channel is slightly
biggerthantheexpectationfromtheSM[29,30]. Anicefeatureofthismodelisthatitcan
explain the enhancement via the S contribution to this process. The enhancement fixes
λ
thesignofthecoupling oncetheSMHiggstotopcouplingisemployed.
HS
TheproductionoftheHiggsisnotalteredcomparedtotheSM.Thisallowsustowrite
thesignalstrengthrelativetotheSMoneas
Γh→γγ
µ = σ(pp → h → γγ) = ΓSM SEχy ,
(19)
σ (pp → h → γγ) Γ −Γh→γγ +Γh→γγ +Γh→χχΓh→γγ
SM
SM SM SEχy SEχy SM
Γh→γγ Γ
where denotes a partial width and the total width. The partial width of the
X SM
χ χ
Higgsintotwo sdependsonthephysicalvalueofthe mass,whichherewetaketobe
about 10 GeV. The needed formulas to calculate the Higgs decay width to two photons
aregiveninAppendixD.ThetotalwidthoftheHiggsiscalculatedwiththeaidofHDECAY
[31].
Thebest-fitvalueforthesignalstrengthreportedbytheATLASCollaboration[29]is
µ = . ± . ,
19 05 (20)
ATLAS
√ √
achieved by combining the s = 7 TeV and s = 8 TeV data samples of size 4.8 fb−1
and 5.9 fb−1, respectively. In the context of our model, we can test which pair of values
,λ µ
of (M ) gives within the quoted error. In Fig. 5 the shaded areas represent
HS ATLAS
excludedparametercombinationsbasedonthisATLASmeasurement.
2. h → γZ
→ γ
The model can be tested further when the process h Z will be measured. A
ff → γ
thorough analysis of how a singlet scalar, or other non-SM particles, a ects the h Z
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