Radiative Neutrino Mass with $Z_3$ Dark matter: From Relic Density to LHC Signatures

1.8 MB·

by  Ran Ding

Z
Radiative neutrino mass with Dark matter: From relic
3
density to LHC signatures
Ran Ding 1,∗ Zhi-Long Han 2,† Yi Liao 3,2,1,‡ and Wan-Peng Xie 1§
1 Center for High Energy Physics, Peking University, Beijing 100871, China
2 School of Physics, Nankai University, Tianjin 300071, China
3 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
6 Chinese Academy of Sciences, Beijing 100190, China
1
0 (Dated: January26,2016)
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n InthisworkwegiveacomprehensiveanalysisonthephenomenologyofaspecificZ3 darkmat-
a
J ter(DM)modelinwhichneutrinomassisinducedattwoloopsbyinteractionswithaDMparticle

4 thatcanbeacomplexscalaroraDiracfermion. BoththeDMpropertiesinrelicdensityanddirect
2
detectionandtheLHCsignaturesareexaminedingreatdetail,andindirectdetectionforgamma-ray

]
h excess from the Galactic Center is also discussed briefly. On the DM side, both semi-annihilation
p
- and co-annihilation processes play a crucial role in alleviating the tension of parameter space be-
p
e tweenrelicdensityanddirectdetection. Onthecolliderside,newdecaychannelsresultingfromZ
3
h
[ particlesleadtodistinctsignalsatLHC.Currentlythetrileptonsignalisexpectedtogivethemost

1 stringentboundforbothscalarandfermionDMcandidates, andthesignaturesoffermionDMare
v
5 verysimilartothoseofelectroweakinosinsimplifiedsupersymmetricmodels.
5
3
6
0
.
1
0
6
1
:
v
i
X
r
a
∗[email protected]
†[email protected]
‡[email protected]
§[email protected]
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I. INTRODUCTION
Neutrinomassandnonbaryonicdarkmatter(DM)offertwopiecesofrobustevidencefortheexistence
of physics beyond standard model (SM), but their origins remain mysterious. It would be appealing if
they could be understood in the same framework. At low energies neutrino mass can be accommodated
by a dimension-five operator in terms of the SM Higgs and lepton fields [1]. The operator can be realized
at tree level in three different manners [2] which correspond exactly to the three types of conventional
seesaws. Though simple enough, these models are difficult to test experimentally since they invoke very
highenergyscalesorveryweakcouplingstoSMparticlesinordertoinducetinyneutrinomasses. Oneway
to alleviate this problem is to push the neutrino mass to a radiative effect of new physics which provides
additionalsuppression. Forthispurpose, an(almost)exactdiscretesymmetryisusuallyrequiredtoforbid
thegenerationofneutrinomassatalowerorder. Suchasymmetrycanstabilizethelightestneutralmember
ofallparticlesthattransformnontriviallyunderthesymmetry,andmakesitanaturalDMcandidate.
Theabove ideaofDM-generatedneutrino masshasbeen extensivelyexploitedinthe literature[3–15].
ThesimplestdiscretesymmetryisaZ parity. However,ifitappearsasaremnantofabrokengaugegroup
2
(U(1) ),otherZ sarealsopossibleingeneral[16]. ThepossibilitywithN > 2hasbeeninvestigatedin
X N
Refs[4,17–22]. ComparedwiththeZ case,DMwithZ symmetryhasthefollowingdistinctfeatures:
2 N
• New DM annihilation processes such as semi-annihilation (SE-A) [17] become available that allow
for different numbers of DM particles to appear in the initial and final states. The processes can
changesignificantlytheevaluationoftheDMrelicdensity.
• DMparticleshavenewinterestingdecaymodesthatresultinricherphenomenologyanddistinguish-
ablesignaturesatcolliders[23,24].
• Multi-componentDMispossible. Inthiscase,annihilationprocessesbetweendifferentcomponents
leadtotheso-calledassistedfreeze-outmechanism[25],whichalsoinfluencestheDMrelicdensity.
In this paper, we focus on the first two features. We consider a specific Z DM model that induces
3
neutrino mass at two loops. The model was originally proposed in Ref. [4], in which DM can be either
a Dirac fermion or a complex scalar. Some phenomenological aspects of the model have been previously
studied in Ref. [26] with emphasis on the effect of SE-A processes on relic density and direct detection.
Here we aim to implement a comprehensive analysis on DM properties and collider signatures. We will
show that both SE-A and co-annihilation (CO-A) processes have significant effects on the evaluation of
relicdensitywhileevadingstringentconstraintsfromdirectdetection. Moreover,thepresenceofmanynew
3
decaychannelsofZ particlesinducesaplentyofdistinctsignalsatLHCforbothscalarandfermionDM
3
candidateswhichwouldbeabsentforZ DM.
2
Therestofthispaperisorganizedasfollows. InSec. II,werecallthemodelanddiscusscurrentexperi-
mentalconstraintsonitsparameters. SectionsIIIandIVcontainthecorecontentofthiswork,inwhichwe
systematicallystudyDMpropertiesandLHCsignaturesforbothscalarandfermioncandidates. InSec. III,
weexplorethevastparameterspacethatsurvivestheconstraintsfromrelicdensityanddirectdetection;all
importantannihilationchannelswillbepresentedanddiscussedindetail. InthefollowingSec. IV,wefirst
exhaust all decay patterns according to the mass spectra of new particles, and then analyze various LHC
signaturesandcomparewiththerelevantLHClimits. Finally,Sec. Visdevotedtoconclusions.
II. MODELANDCONSTRAINTS
In the model under consideration [4] a global and exact Z symmetry is imposed to induce neutrino
3
masses at the two-loop level through interactions with new particles charged under Z . In one minimal
3
version of the model, one introduces two scalars χ (a = 1,2) and one Dirac fermion S, both of which
a
are neutral singlets of the SM gauge group, and one Dirac fermion doublet Ψ = (N,E) of hypercharge
Y = 1. Thefermionsareassumedtobevector-liketoavoidchiralanomalies. Thenewparticlestransform
−
underZ inthesamewayasχ χ ωwithω = exp(i2π/3),whileSMparticlesareneutral. TheYukawa
3 a a
→
and fermion mass terms involving new fields and the SM leptons F = (ν ,(cid:96) ), (cid:96) and Higgs boson
iL iL iL iR
Φ = (G+,φ0)are:
1 1
y(cid:48)ijF¯ Φ(cid:96) m S¯ S m Ψ¯ Ψ x(cid:48)aχ S¯CS x(cid:48)aχ S¯CS
L ⊃ − iL jR− S L R− Ψ L R− 2 L a L L− 2 R a R R
z(cid:48) S¯ Φ˜†Ψ z(cid:48) S¯ Φ˜†Ψ h(cid:48)aiχ†F¯ Ψ +H.c., (1)
L L R R R L a iL R
− − −
whereΦ˜ = iσ Φ∗. Andthescalarpotentialis
2
1 1
V = m2Φ†Φ+(M2)abχ†χ + λ (Φ†Φ)2+ λab;cd(χ†χ )(χ†χ ) (2)
− a b 2 1 2 2 a b c d
1
+λab(Φ†Φ)(χ†χ )+ (µabcχ χ χ +H.c.),
3 a b 6 a b c
whereM2 andλ areHermitian,µabc iscomplexandsymmetricinthethreeindices,andλab;cd = λcd;ab =
3 2 2
(λba;dc)∗ is complex as well. Some of the phases in the above couplings can be removed by field redefini-
2
tions but there are still many physical ones. To make the number of independent parameters under control
forourlaternumericalanalysis,wewillsimplyassumethatλab;cd = λ ,λab = λ ,µabc = µ,andthatthey
2 2 3 3
areallreal.
There are some theoretical considerations that can be used to set a bound on the parameters in the
scalar potential, such as perturbativity, unitarity [27], and Z not to be spontaneously broken [21]. These
3
4
constraints are easily respected in our numerical analysis. Since Z is exact, new particles do not mix
3
with SM particles but can mix among themselves. We assume that χ are diagonalized by an angle α
1,2
to χ of masses M M . The electrically neutral fermions S and N also mix due to the Yukawa
L,H χL ≤ χH
couplings z(cid:48) by an angle β to N of masses M . Our convention is that N (N ) is dominantly a
L,R 1,2 N1,2 1 2
singlet S (doublet N) for small β but either mass order is possible. In terms of the mass-eigenstate fields
the couplings will involve the mixing angles. For the Yukawa couplings, we simply replace the primed
couplingsbyunprimedones,e.g.,xa andhai. Forthescalarself-couplings,theangleαentersexplicitly;
L,R
e.g., the χ χ χ coupling (now a, b, c = L, H) is proportional to µg , in which an index L (H) is
a b c abc
associatedwithafactorofα = cosα sinα(α = cosα+sinα),forinstance,
− +
−
g = α3, g = α2α , etc. (3)
HHH + LHH + −
ThereforewecantakeM ,m ,α,β,hai,xa ,λ ,λ ,andµasourinputparameters.
χL,H N1,2 L,R 2 3
0 0
φ φ
S S
N χc N
ν χ χ ν
i a b j
FIG.1. Feynmandiagramforneutrinomass.
Theaboveinteractionsinduceneutrinomassesatthetwo-looplevel[4]asshowninFig. 1:
µsin22β (cid:88)(cid:88) (cid:16) (cid:17)
(m )ij = haihbjg ( 1)k+l xcIabc +xc Iabc , (4)
ν 4(4π)4 abc − L Lkl R Rkl
a,b,c k,l
wherea,b,c = L,H refertoscalarsχ andk,l = 1,2tofermionsN . Theloopfunctionsare
L,H 1,2
M (cid:90) 1 δ(x+y+z 1) (cid:20) ξalnξa ξbclnξbc (cid:21)
Iabc = l dxdydz − k k + kl kl ,
Lkl M z(1 z) (1 ξa)(ξa ξbc) (1 ξbc)(ξbc ξa)
k 0 − − k k − kl − kl kl − k
(cid:90) 1 δ(x+y+z 1) (cid:20) (ξa)2lnξa (ξbc)2lnξbc (cid:21)
Iabc = dxdydz − k k + kl kl , (5)
Rkl (1 z) (1 ξa)(ξa ξbc) (1 ξbc)(ξbc ξa)
0 − − k k − kl − kl kl − l
with
M2 xM2+yM2+zM2
ξa = a, ξbc = l b c . (6)
k M2 kl z(1 z)M2
k − k
OurloopfunctionsagreewithRef. [15]whichsharesthesametopologyofFeynmandiagramsinadifferent
model, but the relative sign of the two terms differs from that in Ref. [26] which computes neutrino mass
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fromaFeynmandiagramofsametopologyinanotherscenariooftheZ model. Theinduced3 3neutrino
3
×
massmatrixhasadegeneratestructureimplyingamasslessneutrinoineithernormalorinvertedhierarchy.
Withatwo-loopsuppressionfactorof(4π)−4 10−5,itiseasytoaccommodateamassoforder0.1eVfor
∼
theothertwoneutrinosbyassumingreasonablevaluesofnewparameters. Forheavymassesofsameorder,
the loop functions are of order 0.1. As will be shown below, the constraints from lepton flavor violating
(LFV)transitionscanbetriviallyfulfilledwithhai 0.01. Thisthenrequiresµxa sin2(2β) 0.1GeV.
∼ L,R ∼
γ
E
−
ℓj χa ℓi
FIG.2. FeynmandiagramforLFVprocess(cid:96) (cid:96) γ.
j i
→
As is well known, precise measurements of LFV transitions set strong constraints on relevant interac-
tions. The diagram in Fig. 2 for the lepton radiative decay (cid:96) (cid:96) γ yields the branching ratio [28, 29]:
j i
→
BR((cid:96)j → (cid:96)iγ) = BR((cid:96)j → (cid:96)iν¯iνj)16πG3α2M4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) hai∗hajF (cid:32)MMχ22a(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2, (7)
F E (cid:12)a=L,H E (cid:12)
wheretheloopfunctionF(x)isgivenby
1
F(x) = [1 6x+3x2+2x3 6x2lnx]. (8)
−12(1 x)4 − −
−
Currently, the most stringent limit comes from BR(µ eγ) < 5.7 10−13(90% C.L.) [30], while the
→ ×
limits on τ decays are less stringent, BR(τ eγ) < 3.3 10−8(90% C.L.) [31] and BR(τ eγ) <
→ × →
4.4 10−8(90%C.L.)[31]. ForZ particlesofsimilarmasses,M M ,wehaveF(x) 1/24;then
× 3 χL,H ∼ E ∼
theaboveboundstranslateintotheconstraintsontheYukawacouplingshai:
(cid:18) M (cid:19)2
hae∗haµ (cid:46) 5.1 10−5 E ,
| | × 100GeV
(cid:18) M (cid:19)2
hae∗haτ (cid:46) 3.0 10−2 E ,
| | × 100GeV
(cid:18) M (cid:19)2
haµ∗haτ (cid:46) 3.4 10−2 E . (9)
| | × 100GeV
6
For instance, when M 200 GeV, the above become hae∗haµ (cid:46) 0.0002, hae∗haτ (cid:46) 0.12, and
E
∼ | | | |
haµ∗haτ (cid:46) 0.14. Thus, without requiring a special flavor structure we can assume safely a universal
| |
bound hai (cid:46) 0.01. 1
| |
ThemasssplittingbetweenthechargedfermionE andneutralonesN willcontributetothecustodial
1,2
symmetry breaking measured by the parameter ∆T. Using the formulas in, e.g., Refs. [33, 34] and the
fitting result ∆T < 0.2 [35], we can set a bound on the mass splitting. In the most stringent case for a
| |
large mixing sinβ 0.66, one gets M M < 250 GeV. In the opposite case of a small mixing
∼ | N2 − N1|
sinβ < 0.1, M M is restricted to be less than a few GeV [36]. In our numerical analysis we will
| N2 − E|
alwaysworkwithasmallβ andassumeN , E aredegenerate. Furthermore,foralightDMparticleχ or
2 L
N therearecolliderconstraintsoninvisibleHiggsdecays,whichwillbeexaminedinthenextsection.
1
III. DARKMATTERPHENOMENOLOGY
In this section, we investigate DM phenomenology of the Z model. For this purpose, we generate the
3
CalcHEP [37] file by using the FeynRules [38] package, which is used by micrOMEGAs4.1 [39] to
calculatetheDMrelicdensityandDM-nucleonscatteringcrosssection. Weimplementrandomscansovera
vastparameterspace(withatotalof3 105samples),fortherangesorvaluesoftheinputparametersshown
×
in Table I. The constraints from relic density and direct detection are then imposed on each sample. For
relic density, we use the combined Planck+WP+highL+BAO 2σ range, 0.1153 < Ω h2 < 0.1221 [40].
DM
Asfordirectdetection,weadoptthecurrentlymostrestrictivespin-independentlimitprovidedbytheLUX
experiment[41]2. Wemakesomecommentsbeforepresentingournumericalresults:
• There are four new neutral particles in this model, χ and N , resulting in a rich annihilation
L,H 1,2
pattern. In addition to the standard annihilation (ST-A) processes, the SE-A and CO-A processes
playacrucialroleintheDMrelicdensity. Thelattertwoprocessesaresensitivetothemassrelations
of new particles and should be thoroughly examined. We will discuss this issue in great detail in
Sec.IIIA.
• For DM-nucleon scattering, it is sufficient to include tree-level contributions since one-loop terms
are subleading. In the case of scalar DM, χ interacts with quarks through the Higgs-portal λ
L 3
term, so that direct detection can set a stringent constraint on χ χ† b¯b annihilation channel.
L L →
ThisisacommonfeatureofHiggs-portalmodelsaswewilldiscussinSec.IIIB.ForfermionDM,
1WerecallthatinthecaseoffermionDMwithaZ symmetryitishardtoprovidecorrectrelicdensitywithsuchsmallYukawa
2
couplings[32].
2Since the exclusion limit varies with the DM particle mass M , we interpolate the LUX data to obtain the corresponding
DM
exclusionlimitforeachrandomlygeneratedM .
DM
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N can scatter with quarks via the t-channel Z exchange due to the singlet-doublet mixing. Direct
1
detectionthenimposesaverystringentboundonthemixingangleβ. Theadditionalt-channelHiggs
exchangealsocontributesviathez YukawatermsbutissubleadingtotheZ exchange. Theangle
L,R
β is therefore varied in a much narrower interval than α. The constraints from the Higgs and Z
invisibledecaysandelectroweakprecisionmeasurementsalsopreferasmallβ.
• Since the quartic coupling λ is only related to the DM self-interaction and has no further phe-
2
nomenologicaleffect,weassumeafixedvalueforitinthescan.
• AccordingtoourdiscussioninSec.II,weassumeauniversal hai 0.01toavoiddangerousLFV
| | (cid:39)
processes,thougharelativelylarge haτ canbeaccommodatedbyaspecificflavorstructure.
| |
M M M M M M α β λ λ µ xa hai
χL N1 χH − χL N2 − N1 2 3 L,R | |
[1,1000] [1,1000] [1,500] [1,500] [0,π] [0.001,0.2] 0.1 [0.001,1] 10 [0.1,1] 0.01
TABLEI.Therangesorvaluesoftheinputparametersusedinourscan. AllmassesandµareintheunitsofGeV.
TheSMHiggshasthemassM =125GeV.
h
A. Analysisofparameterspace
Wepresentournumericalscansinthissubsection. Figure3displaysthedistributionsofsurvivedsam-
ples in the (M , M ) and [log(M /M ), log(M /M )] plane. From the figure, we learn some
N1 χL N2 N1 χH χL
importantfeatures:
• It is clear from the left panel that both scalar χ and fermion N DM samples can survive in the
L 1
whole mass regions that we explored, but the scalar one has a much more number. This can be
explained as follows. Since both CO-A and SE-A processes depend on specific mass relations, the
survived samples resulting from them do not distinguish much between χ and N . The difference
L 1
originates instead from ST-A processes. While χ annihilates into gauge boson pairs and produces
L
the correct relic density in a vast mass region, N mostly annihilates into light and b quark pairs
1
through the Higgs exchange, which can only give the correct relic density in a relatively low mass
region.
• The right panel shows that survived samples tend to cluster in regions of small mass splitting for
RD+LUX points, where CO-A and SE-A processes are generally dominant. Since ST-A channels
are not sensitive to mass splitting and tend to cause a more scattered distribution in the plane, the
8
clustering indicates that direct detection imposes a more severe constraint on ST-A processes. We
willillustratethisfeatureexplicitlyinsectionIIIB.
N1:RD ΧL:RD N1:RD+LUX ΧL:RD+LUX
N1:RD ΧL:RD N1:RD+LUX ΧL:RD+LUX
2.0
1000
800
1.5
L
ΧL
LV 600 (cid:144)M
Ge MΧH 1.0
HMΧL Hg10
400 o
L
0.5
200
0 0.0
0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
MN1HGeVL Log10HMN2(cid:144)MN1L
FIG.3. Distributionsofsurvivedsamplesinthe(M , M )(leftpanel)and[log(M /M ), log(M /M )]
N1 χL N2 N1 χH χL
(right)plane. Theblue(green)pointsonlypasstherelicdensity(RD)constraintinthecaseofN (χ )DM,while
1 L
thered(orange)pointssatisfybothRDandLUXconstraints.
Inordertoinvestigatetheparameterspacemorecomprehensively,thedistributionsofsurvivedsamples
inthe[M ,λ ]forχ DMareshowninFig.4. Hereλ = λ α2 isthehχ χ† coupling. Similarly,
χL hχL L hχL 3 − L L
thedistributionsforN DMsamplesaredisplayedinthe[M , β]and[M M , M M ]plane
1 N1 N2− N1 | N2− E|
inFig.5;herebothM M andβ enterthechargedfermionmassM . Wesummarizetheproperties
N2 − N1 E
seeninthefigures:
• AsshowninFig.4forχ DM,survivedsamplesaredistributedinanarrowbandinthe[M ,λ ]
L χL hχL
plane,andmostRD+LUXsamplespreferaheavyDM.
• Direct detection indeed imposes a stringent limit on the mixing angle β in the case of N DM. For
1
samplespassingtheLUXbound,β cannotexceed2◦. Inaddition,thechargedfermionmassM is
E
dominatedbyM ,withamasssplittingdeterminedbyM M andsinβ. Themaximalsplit-
N2 N2 − N1
tingreaches17GeVwhenM M 500GeV.However,uponimposingtheLUXconstraint,
N2 − N1 (cid:39)
M alwaysstaysnearlydegeneratewithM .
E N2
9
ΧL:RD ΧL:RD+LUX
1.
0.5
0.2
ΧL
h
Λ
0.1
0.05
0.02
0 200 400 600 800 1000
M HGeVL
ΧL
FIG.4. Distributionsofsurvivedsamplesforχ DMinthe[M , λ ]plane. Thesymbolsofpointsarethesame
L χL hχL
asinFig.3.
N1:RD N1:RD+LUX N1:RD N1:RD+LUX
20
12°
10°
15
8°
ÈE
M
Β 6° -N2 10
M
È
4°
5
2°
0 0
0 200 400 600 800 1000 0 100 200 300 400 500
M HGeVL M -M
N1 N2 N1
FIG.5. DistributionsofsurvivedsamplesforN DMinthe[M , β](leftpanel)and[M M , M M ]
1 N1 N2 − N1 | N2 − E|
(right)plane. ThesymbolsofpointsarethesameasinFig.3.
10
ThedirectsearchesforinvisibleHiggsdecayshavebeencarriedoutbyATLASandCMSintheweakbo-
sonfusion(WBF)[42,43]andZhassociatedproductionchannels[43,44],withthe95%CLupperbounds
onBR(h invisible)of28%(ATLAS),65%(CMS)intheWBFchanneland75%(ATLAS),83%(CMS)in
→
theZhchannel,respectively. Alternatively,invisibleHiggsdecaysalsogetconstrainedbyfittingtovisible
Higgsdecays[45–47]. Theupperboundthusfoundisstronger,BR(h invisible) < 25%(95%CL)[47],
→
whichwillbeemployedinourfollowingdiscussion. ThedecaywidthtoscalarorfermionDMreads
(cid:115)
λ2 v2 M2
Γ(h χ χ†) = hχL 1 4 χL,
→ L L 16πM − M2
h h
(cid:32) (cid:33)3
M M2 2
Γ(h N N¯ ) = h (M M )2sin42β 1 4 N1 , (10)
→ 1 1 32πv2 N2 − N1 − M2
h
withv 246GeV. WetakeΓ = 4.07MeVforthevisibledecaywidthatM = 125GeVandEq.10for
vis h
≈
theinvisibleoneΓ . ThescatterplotoftheinvisibledecaybranchingfractionBR = Γ /(Γ +Γ )
inv inv inv vis inv
ispresentedinFig.6asafunctionofM forRDandRD+LUXsurvivedsamples,wheretheshadedarea
DM
indicatestheregionexcludedbytheupperboundfromRef.[47]. Wefoundthatforχ DM,sampleswith
L
M < 55 GeV are totally excluded while for N DM the corresponding bound can be relaxed to about
χL 1
28GeV.
OnecanalsoconverttheupperboundoninvisibleHiggsdecaysintoexcludedregionsinthe[M ,λ ]
χL hχL
([M , β])planeforχ (N )DM.AsshowninFig.7,thecorrelationsamongparametersmanifestthem-
N1 L 1
selves more explicitly. In this manner, we obtain the most stringent bound λ (cid:46) 0.01 with M <
hχL χL
55GeV forχ DM,orβ (cid:46) 4◦ withM M = 500GeV forN DMinthemoststringentcase. We
L N2 − N1 1
noticethattheseconstraintsarelessstringentthanfromdirectdetectioninthesamemassregions,suchthat
all of RD+LUX samples easily survive for either χ or N DM. Finally, N DM also contributes to the
L 1 1
invisibleZ decayifkinematicallyallowed,
(cid:32) (cid:33)(cid:115)
M3 sin4β M2 M2
Γ(Z N N¯ ) = Z 1+2 N1 1 4 N1. (11)
→ 1 1 12πv2 M2 − M2
Z Z
The LEP 95% CL upper bound of 3 MeV [48] translates to β (cid:46) 17◦, which is weaker than from invisible
Higgs decays. For light N , the decays h N N¯ /N N¯ , N N¯ and Z N N¯ /N N¯ , N N¯ may
2 1 2 2 1 2 2 1 2 2 1 2 2
→ →
also be allowed. These decay modes could provide more severe constraints [36], but are still weaker than
theLUXconstraint. Thereforewewillnotconsiderinvisibledecaysasanindividualconstraintinourlater
analysis.