Table Of ContentNeutral Higgs-pair production at Linear Colliders within the general 2HDM:
quantum effects and triple Higgs boson self-interactions
David Lo´pez-Val and Joan Sola`
∗ †
High Energy Physics Group, Dept. ECM, and Institut de Ci`encies del Cosmos
Univ. de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain
(Dated: January 8, 2010)
The pairwise production of neutral Higgs bosons (h0A0, H0A0) is analyzed in the context of the
future linear colliders, such as the ILC and CLIC, within the general Two-Higgs-Doublet Model
(2HDM). The corresponding cross-sections are computed at the one-loop level, including the full
set of contributions at order (α3 ) together with the leading (α4 ) terms, in full compliance
O ew O ew
withthecurrentphenomenological boundsandthestringenttheoretical constraintsinherenttothe
consistency of the model. We uncover regions across the 2HDM parameter space, mainly for low
0 tanβ 1andmoderate–andnegative–valuesoftherelevantλ parameter,whereintheradiative
5
1 correc∼tions to the Higgs-pair production cross section, σ(e+e− A0h0/A0H0), can comfortably
→
0 reach δσ/σ 50%. This behavior can be traced back to the enhancement capabilities of the
2 trilinea|r H|iggs∼self-interactions – a trademark feature of the 2HDM, with no counterpart in the
Minimal SupersymmetricStandardModel (MSSM).Thecorrections arestrongly dependenton the
n
a actualvalueofλ5 aswellasontheHiggsmassspectrum. Interestinglyenough,thequantumeffects
J arepositiveforenergiesaround √s 500GeV,therebyproducingasignificant enhancementin the
expected number of events precisely≃around the fiducial startup energy of the ILC. The Higgs-pair
8
productionratescanbesubstantial,typicallyafewtensoffemtobarn,thereforeamountingtoafew
] thousand events per 100fb−1 of integrated luminosity. In contrast, the corrections are negative in
h the highest energy range (viz. √s 1TeV and above). We conclude that a precise measurement
p of the 2H final states could carry∼unambiguous footprints of an extended (non-supersymmetric)
- Higgs sector. Finally, to better assess the scope of these effects, we compare the exclusive pairwise
p
production ofHiggs bosonswith theinclusive gauge boson fusion channelsleading to 2H+X final
e
h states, and also with the exclusive triple Higgs boson production. We find that these multiparticle
[ finalstates can behighly complementary in theoverall Higgs bosons search strategy.
2 PACSnumbers: 12.15.-x,12.60.Fr,12.15.Lk
v
8
9 I. INTRODUCTION least) one elementary spinless field – the so-called Higgs
8 boson. Itsrelevanceisparticularlyevidentifwetakeinto
2
account that the presence of one or more such fields in
. Itwasalreadyinthe60’swhenthepioneeringworksof
8 thestructureofthemodelisindispensablefortheunitar-
Higgs,Kibble,Englertandotherssuggestedtheexistence
0 ityofthe theory. Ifno Higgsbosonsexistbelowthe TeV
9 of a fundamental spinless building-block of Nature[1],
scale,weakinteractionswouldactuallybecomestrongat
:0 wplhaoinsetnhoens-pvoanntisahnienoguvsabcureuamkinexgpoefcttahteioSnUva(2lu)ecouUld(1e)x- that scale and e.g. the W+W− cross-section would vio-
v L ⊗ Y late the unitarity of the scattering matrix. Moreover, in
i gauge group of the Electroweak(EW) interactions down the absence of Higgs bosons the various particle masses
X to the U(1) abelian symmetry group of the Electro-
em could not be generated consistently (i.e. without spoil-
r magnetism. Indeed, this so-called Higgs mechanism is
a ingtheultravioletbehaviorofthetheoryathigherorders
the most fundamental longstanding issue that remains
of perturbation theory). In short, if no Higgs bosons are
experimentally unsettled in Particle Physics. For one
theretoprotectthe(presumed)perturbativestructureof
thingitistheonlyknownstrategycapableofbuildingup
the weak interactions, the latter would enter a perilous
a (perturbative) renormalizable quantum field theoreti-
runaway regime at high energies. It is, thus, essential
cal description of the Electroweak Symmetry Breaking
to confirm experimentally the existence of one or more
(EWSB) phenomenon. It is difficult to overemphasize
Higgs boson particles through their explicit production
that, to a great extent, the Higgs mechanism embodies
in the colliders.
thebackboneoftheSMstructureandthatinitsabsence
we would have to cope with an entirely different concep- Amazinglyenough,morethan40yearsaftertheHiggs
tionoftheinnerfunctioningoftheSMasaquantumfield mechanismwasnaturallyblendedintotheconventionally
theory (QFT) . acceptedSMlandscapeofthe strongandelectroweakin-
The central assumption here is the existence of (at teractions[2], it remains still thickly curtained and no
direct evidence has been found yet of its main testable
prediction: the existence of fundamental spinless parti-
cles. Notwithstanding the many efforts devoted at LEP
andatthe Tevatroninthe lastdecades,allexperimental
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] searchesforHiggsbosonsignatureshavecomeupempty-
2
handed, and one has to admit that no conclusive physi- can appear in the form of a non-supersymmetric frame-
cal signature for the existence of these elementary scalar work, the so-called general (unconstrained) Two-Higgs-
fields–noteventhesingleonepredictedbytheSM–has DoubletModel(2HDM),whichdoesalsoentailarichphe-
been found for the time being. Thus, we do not know nomenology [18]. Again, the same spectrum of -even
CP
whether the conventional Higgs boson exists at all, or if (h0,H0) and -odd (A0) scalar fields arise. However,
CP
therearenoHiggsparticlesofanysort. Onthecontrary, weshouldkeepinmindthatthemostdistinctive aspects
it could be that there are extensions of the SM span- of the general2HDM Higgs bosons could wellbe located
ningaricherandelusivespectrumofHiggsbosonswhich in sectors of the model quite different from the MSSM
have escaped all our dedicated experimental searches. case[19]. Indeed, eventhe verycouplingsinvolvedinthe
Be that as it may, this pressing and highly intriguing structure ofthe Higgs potential (viz. the so-calledtrilin-
enigma will hopefully be unveiled soon, specially with ear and quartic Higgs boson self-interactions) could be
the advent of the new generation of supercolliders, like the most significant ones as far as the phenomenologi-
thebrandnewLHC,thefutureInternationalLinearCol- cal implications are concerned, rather than just the en-
lider (ILC)[3] and, hopefully, the Compact Linear Col- hancedYukawacouplingswiththeheavyquarks. Insuch
lider (CLIC) too[4]. circumstance,weshouldbeabletodetectaverydifferent
Therefore, we have to (and we can) be well prepared kindof experimentalsignatures. Assuming,for instance,
to recognize all kind of hints from Higgs boson physics. thattheLHCwillfindevidenceofaneutralHiggsboson,
Letusrecallthatapartfromthesingleneutral( -even) a critical issue will be to discern whether the newcomer
Higgs boson H0 of the SM, other opportunitiesCParise in is compatible with the SM or any of its extensions and,
model building that could serve equally well the afore- in the latter case, to which of these extensions it most
mentionedpurposes. Perhapsthe mostparadigmaticex- likely belongs. In order to carry out this “finer” hunting
tension of the SM is the Minimal Supersymmetric Stan- of the Higgs boson(s), a TeV-range accessible linear col-
dardModel(MSSM)[5],whoseHiggssectorinvolvestwo lider machine (such as the aforesaid ILC/CLIC) will be
doublets of complex scalar fields. The physical spec- needed. In this work, we will show how useful it could
trumconsistsoftwochargedstates,H±,twoneutral - be such collider to discriminate neutral supersymmetric
even states h0,H0 (with masses conventionallychosenCPas Higgs bosons from generic (non-SUSY) 2HDM ones.
M < M ) and one -odd state A0. We shall not The paper is organizedas follows. In the next section,
h0 H0 CP
dwell here on the whys and wherefores of supersymme- we remind the reader of some relevant Higgs boson pro-
try (SUSY) as a most sought-after realization of physics duction processes in the linear colliders. In Section III,
beyondtheSM.ItsufficestosaythatitprovidesaHiggs we describe the structure of the 2HDM and the relevant
sector which is stable (to all orders in perturbation the- phenomenological restrictions, including the properties
ory) under embedding of the low-energystructure into a ofthe trilinearHiggsself-interactionsandtheirinterplay
Grand Unified Theory, and in particular it provides the with the notion of unitarity and vacuum stability. In
most natural link with gravity[5]. SectionIV,wedevelopindetailtherenormalizationpro-
gramfortheHiggssectorofthegeneral2HDM.Thethe-
But, how to test the structure of the Higgs sector?
oretical setup for the one-loop computation of the Higgs
In the MSSM case, such structure is highly constrained
boson cross-sections is elaborated in Section V, leaving
by the underlying supersymmetry. A consequence of it,
SectionVItopresentacomprehensivenumericalanalysis
which is particularly relevant for our considerations, is
of our results. Finally, Section VII is devoted to a thor-
the fact that the self-interactions of the SUSY Higgs
oughgeneraldiscussionandtopresenttheconclusionsof
bosons turn out to be largely immaterial from the phe-
our work.
nomenological point of view, in the sense that they can-
not be enhanced (at the tree-level) as compared to the
ordinary gauge interactions and, therefore, cannot pro-
vide outstanding signatures of physics beyond the SM. II. HIGGS BOSON PRODUCTION AT LINEAR
The bulk of the enhancing capabilities of the MSSM COLLIDERS
Lagrangian is concentrated, instead, in the rich struc-
ture of Yukawa-like couplings between Higgs bosons and TheleadingHiggsbosonproductionmechanismsatthe
quarks or between Higgs boson and squarks,or even be- LHC have been studied thoroughly for the last twenty
tween squarksand chargino-neutralinos,as it was shown years, they are well under control both in the SM[20]
long ago in plentiful phenomenological scenarios involv- andintheMSSM[21,22]andtherearealsosomestudies
ingquantumeffectsonthe gaugebosonmasses[6,7]and inthe 2HDM[23]. Hopefully, they may soonhelp reveal-
gaugebosonandtopquarkwidths[8],aswellasinmany ing some clues on Higgs boson physics at the LHC —
other processes and observables (see e.g. [9–13]. For for a review see e.g.[17] and references therein. How-
the present state of the art, cf. [14, 15]; and for recent ever, it is not obvious that it will be possible to easily
comprehensive reviews, see [16, 17], for example. disentangle the nature ofthe potentially producedHiggs
The two-Higgs SU (2)-doublet structure of the Higgs boson(s). The next generationof TeV-classlinear collid-
L
sector in the MSSM is a trademark prediction of SUSY ers (based on both e+e and γγ collisions), such as the
−
invariance. Nonetheless,theverysamedoubletstructure ILC and the CLIC projects [3, 4], will be of paramount
3
importance in finally settling the experimental basis of 2HDM. For example, in Ref.[27] the tree-level produc-
the “Higgs issue” as the most fundamental theoretical tion of triple Higgs boson final states was considered in
construct of the SM of electroweak interactions. Thanks the ILC. There are three classesof processesof this kind
to its extremely clean environment (in contrast to the compatible with -conservation,namely
CP
large QCD background inherent to a hadronic machine
suchastheLHC),alinearcollider(linacforshort)should 1) e+e H+H h, 2) e+e hhA0,
− − −
→ →
allow for a precise measurement of the Higgs boson pa- 3) e+e h0H0A0, (h=h0,H0,A0) (1)
−
rameters,suchas: i) the Higgs bosonmass(or masses,if →
more than one); ii) the couplings of the Higgs bosons to where, in class 2), we assume that the two neutral Higgs
quarks, leptons and gauge bosons; iii) the Higgs boson bosons h must be the same, i.e. the allowed final states
self-couplings mentioned above, i.e. the trilinear (3H) are (hhA0) = (h0h0A0), (H0H0A0) and (A0A0A0). The
and quartic (4H) Higgs boson self-interactions. At the cross-sections in the 2HDM were shown to reach up to
endoftheday,alinacshouldallowustodigdeeperthan (0.1)pb[27], i.e. several orders of magnitude over the
everintothestructureoftheEWSBand,hopefully,even OcorrespondingMSSMpredictions[28]. Besidesthe exclu-
to reconstruct the Higgs potential itself. sive production of three Higgs bosons in the final state,
AdetailedroadmapofpredictionsforHiggs-bosonob- also the inclusive pairwise production of Higgs bosons
servablesatthelinearcollidersiscalledfor,withaspecial may be critically sensitive to the 3H self-interactions.
emphasis on those signatures which may be characteris- Sizable production rates, again in the range of 0.1 1pb
−
tic of the different extensions of the SM. For example, have recently been reported in Ref.[29], whose focus was
as indicated above, triple Higgs boson self-interactions on the inclusive Higgs boson-pair production at order
(3H) may play a cardinal role in this endeavor because, (α4 ) through the mechanism of gauge-boson fusion
O ew
in favorable circumstances, they could easily distinguish
between the MSSM and the general 2HDM Higgs sec- e+e− V∗V∗ hh+X
→ →
tors. Such 3H couplings can mediate a plethora of in- (V =W±,Z, h=h0,H0,A0,H±). (2)
terestingprocesses. Thekeypointhereisthe potentially
largeenhancementsthatthe3Hcouplingsmayaccommo- Similar effects have also been recently computed on 2H-
date inthe general2HDM case,incontrastto the super- strahlungprocessesof the guise e+e Z0hh [30]. In a
−
symmetric extensions. Actually, even for the SUSY case complementary way, 3H couplings can→also be probed in
there is a large number of works attempting to extract loop-induced processes. For instance, Ref. [31] presents
vestiges of non-standard dynamics in these couplings, a computation of the single Higgs bosonproduction rate
mainly through the radiative corrections that they can throughthescatteringprocessofi)tworealphotons,us-
undergo. For instance, the 3H-couplings have been in- ing the γγ mode of a linear collider, i.e. γγ h; and ii)
vestigated in [24–26], and in some cases considered for the more traditionalmechanism of virtual ph→oton fusion
possible phenomenological applications in TeV-class lin- in e+e colliders, e+e γ γ h+X. In either case,
− − ∗ ∗
ear colliders through the double-Higgs strahlung process theobtainedcrosssectio→nswithi→nthegeneral2HDM(up
e+e− HHZ or the double-Higgs WW-fusion mecha- to 1pb for the first mechanism, and 0.01pb for the lat-
nism e→+e−→H+H−νeνe. These processes,which include ter) are 1 to 2 orders of magnitude larger than the ex-
verticeslikeZZH,WWH,ZZHH,WWHHandHHH,are pected SM yields and also well above the MSSM results
possible both in the SM and its extensions, such as the (see [31] and references therein). Promising signatures
MSSM andthe general2HDM. Unfortunately, the cross- have also been reported within the general 2HDM us-
section turns out to be rather small both in the SM and ingloop-inducedproductionoftwoneutralHiggsbosons
in the MSSM, being of order 10−3 pb at most, i.e. equal through real γγ collisions [32], although in this case the
orlessthan1fb [25]. Evenworseisthesituationregard- cross-sectionsaresmallerthanintheprimarymechanism
ing the triple Higgs boson production in the MSSM, in γγ h mentioned before.
which – except in the case of some particular resonant →
A crucial observation concerning our aim here is the
configuration– the typical cross-sectionsjust border the
following: all the processes described above are directly
line of 0.01 fb or less [25]. In the latter reference, for
sensitive to the 3H self-couplings already at the leading
∼
instance, it has been shown that if the double and triple
order. In this paper, we continue exploiting the prop-
Higgs production cross-sections would yield sufficiently
erties of these couplings in the general 2HDM, but we
high signal rates, the system of couplings could in prin-
concentrate now on their impact at the level of indirect
ciple be solved for all trilinear Higgs self-couplings up
effectsthroughradiativecorrections. Thelattermaysig-
to discrete ambiguities using only these processes. But
nificantly affect processes that are kinematically more
thisisperhapsabittoooptimisticsinceinpracticethese
favored (e.g. because they have a smaller number of
cross-sections are manifestly too small to be all measur-
particles in the final state), but which are nevertheless
able in a comfortable way.
totally insensitive to the trilinear Higgs boson couplings
In stark contrast to this meager panorama within the at the lowest order. Specifically, we wish to concentrate
SM and the MSSM, 3H couplings have been tested for on identifying the largest quantum effects that the 3H
tree-level processes in the context of the unconstrained couplings can stamp on the cross-sections for two-body
4
neutral Higgs boson final states: point later)3. All in all, one arrives at the following ex-
pression for the tree-level potential:
e+e 2h (2h h0A0;H0A0). (3)
−→ ≡ v2 2 v2 2
Surprisingly enough, very little attention has been paid V(Φ1,Φ2)= λ1 Φ†1Φ1− 21 +λ2 Φ†2Φ2− 22
(cid:18) (cid:19) (cid:18) (cid:19)
to these basic production processes and to the calcula- v2 v2 2
tionofthecorrespondingradiativecorrectionswithinthe +λ3 Φ†1Φ1− 21 + Φ†2Φ2− 22
2HDM. In contrast, a lot of work has been invested on (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
them from the point of view of the MSSM[24, 25, 33– +λ4 (Φ†1Φ1)(Φ†2Φ2)−(Φ†1Φ2)(Φ†2Φ1)
38] – see also [39] and references therein1. To the best
h v v 2 i
of our knowledge, only Refs. [41–43] have addressed this +λ5 ℜe(Φ†1Φ2)− 12 2
topic in the general 2HDM, although they are restricted
h 2 i
ptoretsheentpproadpuerc,tiwoensohfacllhbaregceodnHceirgngesdpeaxicrsluHsi+veHly−.onIncotmhe- +λ6 ℑm(Φ†1Φ2) , (5)
h i
putingthequantumeffectsinvolvedontheneutralHiggs
where λ (i = 1,... 6) are dimensionless real parameters
boson channels (3) at order (α3 ) (and eventually also i
the leading (α4 ) terms).OAsewwe will see, very large and vi(i=1,2) stand for the non-vanishing VEV’s that
O ew the neutralcomponentofeachdoubletacquires,normal-
quantum effects (of order 50%) may arise on the cross- ized as follows: Φ0 =v /√2.
sections of the two-body Higgs boson channels (3) as a h ii i
The complex degrees of freedom encoded within each
result of the enhancement capabilities of the triple (and,
Higgs doublet in Eq.(4) are conveniently split into real
to a lesser extent, also the quartic) Higgs boson self-
ones in the following way:
interactions. These effects are completely unmatched
wteirtihstinictohfenMonS-SsMupearnsdymshmouetldricthHerigegfosrbeobseonhipghhylysicchs.arac- Φ1 = ΦΦ+10 = v1+φφ01+1+iχ01 ,
(cid:18) 1 (cid:19) √2 !
Φ2 = ΦΦ+20 = v2+φφ02+2+iχ02 . (6)
III. THE TWO-HIGGS-DOUBLET MODEL: (cid:18) 2 (cid:19) √2 !
GENERAL SETTINGS AND RELEVANT
RESTRICTIONS Upon diagonalization of the Higgs potential (5) we may
obtain the physical Higgs eigenstates in terms of the
gauge (weak-eigenstate) basis:
The Two-Higgs-Doublet Model (2HDM) is defined
uponthecanonicalextensionoftheSMHiggssectorwith H0 cosα sinα φ0
a second SU(2)L doublet with weak hypercharge Y =1, h0 = sinα cosα φ10
so that it contains 4 complex scalar fields arranged as (cid:18) (cid:19) (cid:18)− (cid:19) (cid:18) 2(cid:19)
follows: G0 = cosβ sinβ χ01
A0 sinβ cosβ χ0
(cid:18) (cid:19) (cid:18)− (cid:19) (cid:18) 2(cid:19)
Φ1 =(cid:18)ΦΦ+101 (cid:19) (Y =+1), Φ2 =(cid:18)ΦΦ+202 (cid:19) (Y =+1()4). (cid:18)GH±±(cid:19)=(cid:18)−cosisnββ csoinsββ(cid:19) (cid:18)φφ±1±2 (cid:19) (7)
Withthehelpofthesetwoweak-isospindoublets,wecan The parameter tanβ is given by the ratio of the two
write down the most general structure of the Higgs po- VEV’s giving masses to the up- and down-like quarks:
tentialfulfillingtheconditionsof -conservation,gauge
CP Φ0 v
invariance and renormalizability. Moreover, a discrete tanβ h 2i = 2 . (8)
Z2 symmetry Φi → (−1)iΦi (i = 1,2) – which will ≡ hΦ01i v1
be exact up to soft-breaking terms of dimension 2 – is
Incidentally, let us note that this parameter could be
usually imposed as a sufficient condition to guarantee
ideally measured in e+e colliders[36] since the charged
the proper suppression of the Flavor-Changing Neutral- −
Current(FCNC)effectsthatwouldotherwisearisewithin Higgs event rates with mixed decays, e+e− H+H−
→ →
the quark Yukawa sector [44] 2 (we shall return to this tbτν¯τ,tbνττ¯, do not involve the mixing angle α. The
physical content of the 2HDM embodies one pair of -
even Higgs bosons (l, H0); a single -odd Higgs boCsPon
CP
1 For the analysis of the tree-level double Higgs production pro-
cesses in the MSSM, see e.g. the exhaustive overview[40]. For
adetailed‘anatomy’oftheMSSMHiggssectorandanupdated 3 This symmetry is automatic in the MSSM case, although it
accountofitsphenomenological consequences, seee.g. [17]. is again violated after introducing the dimension 2 soft-SUSY
2 Foralternativestrategiesofbuildinguprealizationsofthe2HDM breakingterms. TheseareessentialfortheEWSB,whichother-
withnoexplicitZ2symmetry,seee.g. [45]andreferencestherein. wisewouldnotoccur,seeEq.(15)below.
5
(A0); and a couple of charged Higgs bosons (H ). The [18] for an exhaustive treatment), since their contribu-
±
free parameters of the general 2HDM are usually chosen tion to the processes e+e h0A0/H0A0 under study
−
→
to be: the masses of the physical Higgs particles; the at the one-loop level is largely subdominant in the main
aforementioned parameter tanβ; the mixing angle α be- regions of parameter space. However, it is precisely the
tweenthetwo -evenstates;andfinallythecouplingλ , coupling pattern to fermions that motivates the distinc-
5
CP
whichcannotbe absorbedinthe previousquantities and tion between the different types of 2HDM’s, so let us
becomestiedtothestructureofthe Higgsself-couplings. brieflydescribethem. Assumingnaturalflavorconserva-
To summarize, the vector of free inputs reads tion[44],wemustimposethatatmostoneHiggsdoublet
can couple to any particular fermion type. As a result,
(Mh0,MH0,MA0,MH±,α,tanβ,λ5). (9) the coupling of the 2HDM Higgs bosons with fermions
(say the quarks) can be implemented in essentially two
Therefore, we are left with 7 free parameters, which in-
manners which insure the absence of potentially danger-
deed correspond to the original 6 couplings λ and the
i ous(tree-level)FCNC’s,towit(cf. TableIII)4: i)type-I
twoVEV’sv ,v –thelatterbeingsubmittedtothecon-
1 2
2HDM, wherein the Higgs doublet (Φ ) couples to all of
straint v2 v2+v2 =4M2 /g2, which is valid order by 2
≡ 1 2 W quarks, whereas the other one (Φ ) does not couple to
orderinperturbationtheory,whereM istheW mass 1
W ±
them at all; or ii) type-II 2HDM, in which the doublet
and g is the SU (2) gauge coupling constant. The di-
L
Φ (resp. Φ ) couples only to down-like (resp. up-like)
mension 2 term that softly breaks the Z symmetry can 1 2
2
right-handed quarks. In the latter case, an additional
be written as −m212Φ†1Φ2+h.c., with discretesymmetryinvolvingthechiralcomponentsofthe
m212 = 12λ5v2 sinβcosβ = G2√−F21 1+tatnanβ2β λ5, (10) fterreme-iloevnelseFcCtoNrCmpursotcebseseism,ep.ogs.eDdRiin→or−deDrRito, UbaRini→shUthRie
forthedownandupright-handedquarksinthethreefla-
the second equality being valid at the tree-level. Given vor families (i=1,2,3).
tanβ one may trade the parameter λ5 for m12 through Inturn,theHiggsself-couplingsλi intheHiggspoten-
this formula, if desired. tial can be rewritten in terms of tanβ and the physical
Since we are going to adopt the on-shell renormaliza- parametersoftheon-shellschemesuchasthemassesand
tion scheme[46–48] for the one-loop calculation of the the electromagnetic fine structure constant αem. At the
cross-sections (cf. section IV), it is convenient to intro- tree-level, we have
duce the electromagnetic fine structure constant αem = λ (1 tan2β) α π
e2/4π, which is one of the fundamental inputs in this λ = 5 − + em
scheme(togetherwiththephysicalZ0mass,M ,andthe 1 4 2MW2 s2W cos2β
Z
Higgs and fermion particle masses). The electron charge M2 cos2α+M2 sin2α
× H0 h0
e is connected to the original SUL(2) gauge coupling 1
−and weak mixing angle θW through the well-knownrela- −(cid:2)2(MH20 −Mh20) sin2α cotβ ,
tion e=gsinθ , which is also preserved order by order (cid:21)
W
λ (1 1/tan2β) α π
in perturbation theory. Then the W± mass can be re- λ = 5 − + em
latedto α andthe FermiconstantG inthe standard 2 4 2M2 s2 sin2β
em F W W
way in the on-shell scheme, namely M2 cos2α+M2 sin2α
× h0 H0
1
G√F2 = 2MW2παseinm2θW (1−∆r), (11) −(cid:2)2(MH20 −Mh20)sin2α tanβ(cid:21) ,
λ α π sin2α
where the parameter ∆r[48] vanishes at the tree-level, λ = 5 + em (M2 M2 ),
3 − 4 2M2 s2 sin2β H0 − h0
W W
but is affected by the radiative corrections to µ-decay
2α π
both from standard as well as from new physics, for in- λ4 = M2ems2 MH2±,
stance from the MSSM[8, 14] and also from the 2HDM W W
(aslseoe benetloewrs).imNploitciictelyththarto,uignhththeeabreolvaetiofnormsiunl2aθ,WMW= λ6 = M2α2ems2π MA20, (12)
1 M2 /M2, which is valid order by order in the on- W W
− W Z where s = sinθ , c = cosθ . From these La-
shell scheme. Since G can be accurately determined W W W W
F
grangian couplings in the Higgs potential, we can de-
from the µ-decay, and M has been measured with high
Z
precision at LEP, it is natural to take them both as ex-
perimentalinputs. Then,withthehelpofthenon-trivial
relation(11),theW masscanbeaccuratelypredictedin
±
a modified on-shell scheme where G , rather than M , 4 Strictly speaking, in the absence of CP-violation there are four
F W
manners to have natural flavor conservation, giving rise to four
enters as a physical input.
different kindsofallowed2HDM’sthat donotleadtotree-level
Weshallnotdwellhereonthedetailedstructureofthe
FCNC’s,see[49],although wewillrestrictourselvesheretothe
Yukawa couplings of the Higgs boson to fermions (see twocanonicalones[18].
6
typeI typeII holomorphic function of the chiral superfields [5, 18]).
h0tt cosα/sinβ cosα/sinβ This implies that we cannot construct the MSSM with
h0bb cosα/sinβ sinα/cosβ two Y =+1 Higgs superfields. Then, in order to be able
−
H0tt sinα/sinβ sinα/sinβ to generate masses for both the top and bottom quarks
H0bb sinα/sinβ cosα/cosβ throughEWSB, the Φ doublet can be kept as it is with
2
A0tt cotβ cotβ Y =+1,althoughwemustreplaceΦ withtheconjugate
1
A0bb cotβ tanβ (Y = 1) SU (2) doublet
− − L
TABLEI:NeutralHiggsbosoncouplingstofermionsintype-I
and type-II 2HDM, using third family notation. For exam- H = H10 ǫΦ = Φ01∗ (Y = 1), (13)
ple, theSMHiggs boson coupling totop andbottom quarks, 1 (cid:18)H1−(cid:19)≡ ∗1 (cid:18)−Φ−1 (cid:19) −
gmf/2MW (f = t,b), multiplied by the corresponding fac-
t−orinthetableprovidesthe2HDMcouplingsoftheh0boson where ǫ = iσ , with σ the second Pauli matrix. Thus,
2 2
to these quarks. Worth noticing are the characteristic en-
hancement factors arising at large (and low) values of tanβ. for the first doublet the correspondence with the MSSM
case reads Φ = ǫH , and the second doublet (the one
1 − 1∗
withnochange)isjustrelabeledH . Moreover,thetree-
2
level relationships imposed by Supersymmetry between
rive the “physicalcouplings”,namely those affecting the
the λ couplings of the potential (5) are the following:
physical Higgs bosons in the mass-eigenstate basis. We i
call these couplings the triple (3H) and quartic (4H)
Higgs couplings. Their behavior and enhancement ca- λ = λ ,
1 2
pabilities are at the very core of our discussion. Indeed, παem
λ = λ ,
from our analysis it will become clear that they furnish 3 2s2 c2 − 1
W W
thedominantsourceofquantumcorrectionstotheHiggs-
2πα
em
pair production processes (3) within the general 2HDM. λ4 = 2λ1− c2 ,
The physical 3H and 4H couplings are not explicitly W
2πα
present in the 2HDM potential (5). They are derived λ = λ =2λ em . (14)
5 6 1− s2 c2
from it after spontaneous breaking of the EW symme- W W
tryandcorrespondingdiagonalizationoftheHiggsboson
Substituting (13) and (14) in (5) one obtains the usual
mass matrix using the rotationanglesα and β in (7). In
the particular case of the SM, the trilinear and quartic MSSM potential, which in practice must be supple-
Higgscouplingshavefixedvalues,whichdependuniquely mented with soft SUSY-breaking scalar mass terms
on the actual mass of the Higgs boson. In the MSSM, m2H2, including a bilinear mixing term m2 H H for
i i 12 1 2
however,and due to the SUSY invariance, the Higgs bo- the two Higgs doublets. This term is the analog of (10)
son self-couplings are of pure gauge nature, as we shall for the 2HDM, but in the soft-SUSY breaking context is
revisebrieflybelow. Thisisinfacttheprimaryreasonfor arbitrary. The result can be cast as follows:
thetinyproductionratesobtainedforthetripleHiggsbo-
sonprocesses(1)withintheframeworkoftheMSSM[25].
V(H ,H )= µ2+m2 H 2+ µ2+m2 H 2
In contrast, the general 2HDM accommodates trilinear 1 2 | | 1 | 1| | | 2 | 2|
HfuilglglisstcoisupdliisnpglsaywedithingTreaabtlepoIIt.ential enhancement. The +2πsα2ecm2 |H(cid:0)1|2−|H2|2(cid:1)2+ 2πsα2e(cid:0)m |H1†H2|2(cid:1)
W W W
As can be seen, the couplings in this table depend m2 ǫ H(cid:0)iHj +h.c. (cid:1), (15)
on the 7 free parameters (9). In the particular case − 12 ij 1 2
where λ =λ , this table reduces to Table 1 of Ref. [27]. (cid:16) (cid:17)
5 6
where µ is the higgsino mass term in the superpoten-
Theequalityofthesecouplingstakesplaceautomatically
e.g. in the MSSM, where in addition other simplifica- tial[5]. The obtained potential is one where all quartic
tions occur, as we discuss below. Let us note, for exam- couplings become proportional to α , which is tanta-
em
ple, that in the limit α = β π/2 the h0h0h0-trilinear mount to say that in the SUSY case these self-couplings
−
coupling in Table II reduces exactly to the SM form are purely gauge. Thus, after EWSB also the trilinear
iλSHMHH = −3igMH2/(2MW) – where we denote by H self-couplings will be purely gauge. As warned before,
the SMHiggsboson. Thissituationwouldcorrespondto this is the primary reasonfor their phenomenologicalin-
the so-called decoupling limit in the MSSM[50] since it conspicuousness. Under the supersymmetricconstraints,
is correlated with MA0 , although there is no such theentriesofTableIIboildowntotheMSSMformlisted
→ ∞
correlation in the general 2HDM. e.g. in [18]. Moreover, the five constraints (14) reduce
the number of free parameters from 7 to 2 in the su-
Let us recall that the MSSM Higgs sector is a type-II persymmetric context, typically chosen to be tanβ and
one of a very restricted sort: it is enforced to be SUSY M . Forthegeneral2HDMcase,however,weshallstick
A0
invariant. This is a very demanding requirement as, for totheformpresentedinTableII,wherethe7freeinputs
example, it requires that the superpotential has to be a are chosen as in (9).
7
h0h0h0 3ie M2 (2cos(α+β)+sin2αsin(β α))
−2MWsin2βsW (cid:2) h0 −
cos(α+β)cos2(β α) 4λ5MW2 s2W
− − e2 i
h0h0H0 −2MieWcossi(nβ2−βαs)W h(cid:0)2Mh20 +MH20(cid:1)sin2α−(3sin 2α−sin2β) 2λ5MeW22 s2Wi
h0H0H0 2MieWsinsi(nβ2−βαs)W h(cid:0)Mh20 +2MH20(cid:1)sin2α−(3sin2α+sin2β)s2W 2λ5eM2W2 i
H0H0H0 3ie M2 (2sin(α+β) cos(β α)sin2α)
−2MWsin2βsW (cid:2) H0 − −
sin(α+β)sin2(β α)s2 4λ5MW2
− − W e2 i
h0A0A0 ie cos(α+β) 2M2 4λ5MW2 s2W sin(β α) M2 2M2
−2MWsW h sin2β (cid:16) h0 − e2 (cid:17)− − (cid:0) h0 − A0(cid:1)i
h0A0G0 ie M2 M2 cos(β α)
2MWsW (cid:0) A0− h0(cid:1) −
h0G0G0 ie M2 sin(β α)
−2MWsW h0 −
H0A0A0 ie sin(α+β) 2M2 4λ5MW2 s2W cos(β α) M2 2M2
−2MWsW h sin2β (cid:16) H0− e2 (cid:17)− − (cid:0) H0− A0(cid:1)i
H0A0G0 ie M2 M2 sin(β α)
−2MWsW (cid:0) A0 − H0(cid:1) −
H0G0G0 ie M2 cos(β α)
−2MWsW H0 −
h0H+H− −2MWiesW hcossi(nα2+ββ)(cid:16)2Mh20 − 4λ5Me2W2 s2W(cid:17)−(cid:0)Mh20 −2MH2−(cid:1)sin(β−α)i
H0H+H− −2MWiesW hsinsi(nα2+ββ)(cid:16)2MH20 − 4λ5Me2W2 s2W(cid:17)−cos(β−α)(cid:0)MH20 −2MH2−(cid:1)i
TABLE II: Trilinear Higgs boson self-interactions (iλ3H) in the Feynman gauge within the 2HDM. Here G0 is the neutral
Goldstone boson. These vertices are involved in theradiative corrections to theA0h0Z0(A0H0Z0) bare couplings.
In another vein, it is of paramount importance when weakened down to 92.8GeV for models with more
studying possible sources of unconventional physics to than one Higgs doublet[51]. In the 2HDM all the
make sure that the SM behavior is sufficiently well re- mass bounds can be easily satisfied upon a proper
produced up to the energies explored so far. Such a re- choiceofthe differentHiggsmasses–which,unlike
quirement translates into a number of constraints over the SUSY case, are not related with each other.
the parameter space of the given model. In particular,
these constraintsseverelylimit the a priorienhancement Second, an important requirement to be enforced
•
possibilities of the Higgs boson self-interactions in the is related to the (approximate) SUL+R(2) custo-
2HDM. dial symmetry satisfied by models with an arbi-
trary number of Higgs doublets [52]. In practice,
To begin with, we obviously need to keep track of this restriction is implemented in terms of the pa-
•
the exclusion bounds from direct searches at LEP rameter ρ, which defines the ratio of the neutral-
and the Tevatron. These amount to M &114GeV to-charged current Fermi constants. In general it
l
for a SM-like Higgs boson – although the bound is takes the form ρ = ρ +δρ, where ρ is the tree-
0 0
8
level value. In any model containing an arbitrary Ref.[57] for detailed analytical expressions of the
number of doublets (in particular the 2HDM), we contributions to (b sγ) from the 2HDM. Be-
B →
haveρ =M2 /M2c2 =1,andthenδρrepresents sides,datafromB lν maysupply additionalre-
0 W Z W → l
the deviations from 1 induced by pure quantum strictions onthe chargedHiggs mass at largetanβ
corrections. From the known SM contribution and for the type-II 2HDM [58]. For a very recent up-
the experimental constraints[51] we must demand datedanalysisofthedifferentB-physicsconstraints
that the additional quantum effects coming from on the 2HDM parameter space, see Ref. [45].
2HDMdynamicssatisfytheapproximatecondition
δρ . 10 3. It is thus important to stay in Furtherconstraintsapplytotanβ comingfromthe
| 2HDM| − • following two sources: i) The Z0 bb and B B
a region of parameter space where this bound is
→ −
mixing processes strongly disfavor tanβ below 1
respected. In our calculation we include the domi-
[56]; ii) The requirement that the Higgs couplings
nant part of these corrections, which involves one-
to heavy quarks remain perturbative translates
loop contributions to δρ mediated by the Higgs-
into an (approximate) allowed range of (0.1) <
boson and yields [53]
O
tanβ <60 [49].
G M2 M2
δρ2HDM = 8√2Fπ2nMH2±(cid:20)1− MH2± −A0MA20 ln MHA2±0 (cid:21) • Btioesniadlessetthoefraevqauiliarbemleeenxtspeernismueenstfarlomdatthae, tahneoardedtii--
+ cos2(β α)M2 MA20 lnMA20 calconsistencyofthemodel. Inparticular,weshall
− h0 M2 M2 M2 introduce the following set of conditions ensuring
(cid:20) A0 − h0 h0
M2 M2 the stability of the vacuum [59–61]:
H± ln H±
− M2 M2 M2
H± − h0 h0 (cid:21) λ1+λ3 >0; λ2+λ3 >0
M2 M2
+ sin2(β α)M2 A0 ln A0 2 (λ1+λ3)(λ2+λ3)+2λ3+λ4
− H0 M2 M2 M2
(cid:20) A0 − H0 H0 +pmin[0,λ5 λ4,λ6 λ4]>0. (17)
M2 M2 − −
H± ln H±
− M2 M2 M2 The unitarity bounds deserve a separate discussion. In-
H± − H0 H0 (cid:21)o deed,theyturnouttobeafundamentalingredientofour
(16)
computation. As we have already been mentioning, the
Fromthisexpression,itisclearthatarbitrarymass trademark behavior that we aim to explore shall depend
splittings between the Higgs bosons could easily critically on the enhancement capabilities of the Higgs-
overshoot the limits on δρ. However, we note that boson self-interactions which, in turn, will be sharply
ifMA0 MH± thenδρ2HDM 0,andhence ifthe constrained by unitarity. The basic idea here is that,
→ →
masssplitting betweenMA0 andMH± is notsigni- within perturbative QFT, the scattering amplitudes are
ficant δρ2HDM can be kept under control 5. Let us “asymptoticallyflat”,meaningthattheycannotgrowin-
also recall in passing that the δρ correction trans- definitelywiththeenergy. Thisistantamounttosaythat
lates into a contribution to the parameter ∆r of the unitarity of the S-matrix must be guaranteed at the
(11) given by (c2 /s2 )δρ. Therefore, the tight perturbative level. In a pioneering work by Lee, Quigg
− W W
bounds on ∆r, together with the direct bounds on and Thacker (LQT)[62], the above arguments were first
the ratio of chargedand neutral weak interactions, appliedtotheSM.Asaveryrelevantoutcome,anupper
restrict this contribution as indicated above. bound for the Higgs boson was obtained:
Also remarkable are the restrictions over the 8π
• charged Higgs masses coming from FCNC radia- MH2 <8πv2 = √2G ≡ML2QT ≃(1.2TeV)2. (18)
tiveB-mesondecays,whosebranchingratio (b F
sγ) 3 10−4[51] is measured with suffiBcie→nt The basic idea beneath the LQT argument is to com-
≃ ×
precision to be sensitive to new physics. Current pute the scattering amplitudes in a variety of processes
lowerbounds renderMH± &295GeV fortanβ &1 (namely scalar-scalar, scalar-vector and vector-vector
[55, 56]. It must be recalled that these bounds ap- scattering; most particularly, the longitudinal compo-
ply for both Type-I and Type-II models in the re- nentsofthevectorbosons)anddemandallofthemtobe
gion of tanβ < 1. In contrast, they are only rel- inaccordancewith the generic tree-levelunitarity condi-
evant for Type-I in the large tanβ regime, since tion. The latter is a pure quantum-mechanical require-
for them the charged Higgs couplings to fermions ment,whichcanbeexpressedintermsofthes-wavescat-
are proportional to cotβ and hence the loop con- tering lenght in the high-energy limit:
tributionsarehighlysuppressedatlargetanβ –cf.
a 1/2. (19)
0
| |≤
Takingintoaccounttheequivalencebetweenthelongitu-
5 Formorerefinedanalysesonδρconstraints, seee.g. [54]. dinalcomponentsofthegaugeandtheGoldstonebosons
9
(whichisexactto order ( M2 /s), s beingthe center- 1 λ5 λ6
O W f1 = f2 = 2λ3+ + ,
of-mass energy squared), we can convince ourselves that 16π 2 2
only the scalar-scalar procpesses will be relevant to this 1 n λ5+λo6
p = 2(λ +λ ) .
concern. Moreover,itcanalsobe shownthatthe bulk of 1 16π 3 4 − 2
the contribution is brought about by the quartic scalar n o (20)
self-couplings – the triple ones are suppressed as 1/s.
Thereby the quartic Higgs self-couplings (4H) turn out
to be the most strongly constrained by the bounds of IV. RENORMALIZATION OF THE 2HDM
Eq. (19). HIGGS SECTOR
Several authors have exported the above ideas to the In this Section, we discuss in detail the renormaliza-
2HDM[63–76]andalsotoitscomplex( -violating)ex- tion of the Higgs sector. The renormalization of the SM
tensions [77]. The strategy here is toCcPompute the S- fields and parameters is performed in the conventional
matrix elements S in a set of 2 2 scattering pro- on-shell scheme in the Feynman gauge, see e.g. Ref.
ij
cesses involving Higgs and Goldsto→ne bosons. In this [46–48, 78, 79]. At present, highly automatized proce-
case there are more scalar-scalar channels than in the duresareavailableforloopcalculations,especiallyinthe
SM,whicharecoupledamongthemselves. Atthis point, MSSM, see e.g. [16] and[80–83]. However, in our case
we could perform a unitary transformation U relating the calculation of the cross-sections for the processes (3)
the S-matrix expressed in the original weak-interaction is performed within the general (non-supersymmetric)
eigenstate basis (S ) with the S-matrix written in the 2HDMandwemustdealwiththe renormalizationofthe
w
physicalmass-eigenstate basis (S ). Since, however,the Higgs sector in this class of generic models. To this end,
m
formerismuchsimplerthanthelatterandbothcarryex- weattachamultiplicativewave-function(WF)renormal-
actly the same information, we may generalize the origi- izationconstantto eachofthe SUL(2) Higgs doublets in
nal LQT procedure[62] by focusing on S and requiring the 2HDM,
w
that its eigenvalues satisfy the tree-levelunitarity condi-
tion αi <1 ( i). In practice, inspired by the scattering Φ+1 Z1/2 Φ+1 , Φ+2 Z1/2 Φ+2 ,
lengt|h a|nalysi∀s (19), we require αi < 1/2 ( i). This (cid:18) Φ01 (cid:19)→ Φ1 (cid:18)Φ01 (cid:19) (cid:18) Φ02 (cid:19)→ Φ2 (cid:18) Φ02 (cid:19)
leads to a set of conditions over s|eve|ral linear∀combina- (21)
tions of the quartic couplings in the original Higgs po- The renormalized fields are those on the r.h.s. of these
tential [68–76], as can be seen from the explicit form of expressions. At one-loop we decompose ZΦ1,2 = 1 +
the various eigenvalues αi at the tree-level: δZΦ1,2 +O(αew). These WF renormalization constants
intheweak-eigenstatebasiscanbeusedtoconstructthe
WF renormalizationconstants Z =1+δZ in the
hihj hihj
mass-eigenstatebasisbymeansofthe setoflineartrans-
formations(7). We areinterestedonly in the production
of neutral Higgs bosons in this paper, and therefore we
shallonlyquotetherelationsreferringtothem. Thecor-
1
a± = 16π 3(λ1+λ2+2λ3) nreostpaotniodninδgZδZhihj δinZth)eanreeutdreatleHrmigignsedseacstofrol(luowsins:g the
n hihi ≡ hi
λ λ
±(r9(λ1−λ2)2+(4λ3+λ4+ 25 + 26)2 , δZh0 = sin2αδZΦ1 +cos2αδZΦ2
1 o δZ = cos2αδZ +sin2αδZ
b = λ1+λ2+2λ3 H0 Φ1 Φ2
± 16π δZ = sin2βδZ +cos2βδZ
n ( 2λ +λ +λ )2 A0 Φ1 Φ2
(λ λ )2+ − 4 5 6 , δZ = cos2βδZ +sin2βδZ
± 1− 2 4 G0 Φ1 Φ2
r
c = d = 1 λ +λ +2λ o δZh0H0 = sinα cosα(δZΦ2 −δZΦ1)
± ± 16π 1 2 3 δZA0G0 = sinβ cosβ(δZΦ2 −δZΦ1), (22)
n
(λ λ )2
(λ λ )2+ 5− 6 , where in the last two equations we have also included
1 2
±r − 4 the mixing of the two -even Higgs bosons and that of
1 λ5 λ6 o the -odd state withCPthe neutral Goldstone boson in
e = 2λ λ +5 ,
1 3 4 CP
16π − − 2 2 the Feynman gauge. The different counterterms δZ
1 n λ5 λ6 o must be anchoredby specifying a set of subtractionchoihnj-
e = 2λ +λ + ,
2 16π 3 4− 2 2 ditions on a finite number of 2-point functions. To that
1 n λ λo purpose,weneedtospecifytherenormalizedself-energies
5 6
f = 2λ λ +5 ,
+ 16π 3− 4 2 − 2 correspondingto the gaugebosons andeachofthe phys-
1 n λ λ o ical Higgs-boson fields in the model. First of all let us
5 6
f = 2λ3+λ4+ , define the relation between the polarization tensor of a
− 16π 2 − 2
n o
10
gauge boson and the transverse self-energies. We define we define Σµ (q) ( iqµ)(iΣ (q2)) = qµΣ (q2),
φV ≡ ∓ φV ± φV
it as follows, where q is the external momentum of φ flowing into
±
(outof) the blob. As with the tensor iΣµν(q), we equate
ΣµVνV′(q)= −gµν + qµq2qν ΣVV′(q2)+..., (23) the vector ΣµφV(q) (in this case, with no i-factor) to the
(cid:18) (cid:19) corresponding blob with external lines φ and V 6. With
where the dots indicate that we neglect the longitudinal theseconventions,wemaywritethevariousscalar-scalar
part (proportional to qµqν) because our calculations for self-energies needed for this calculation involving both
alinearcolliderinvolveextremelylightexternalfermions HiggsbosonsandGoldstonebosonsindiagonalormixing
(electronsandpositrons)andasaresultthe longitudinal form. For the -even sector:
CP
contributions are suppressed as m2/M2. The same ap-
plies to the qµqν terms in the transeversVe part, of course. ΣˆH0(q2)=ΣH0(q2)+δZH0(q2−m2H0)−δm2H0
cWiaeteiddetnotiftyhe+sitΣruµcνt(uqr)ewoifththtehepoFleayrinzmataionndtiaengrsaomr. aAsssoa- Σˆh0H0(q2)=Σh0H0(q2)+ 21δZh0H0(q2−m2H0)
result, the unrenormalized self-energy part is defined in
1
such a way that −iΣVV′(q2) (notice the minus sign) is +2δZh0H0(q2−m2h0)−δm2h0H0
equated to the blob of externallines V and V . For sim-
plicity, we define ΣV(q2) ΣVV(q2) and sim′ ilarly for Σˆh0(q2)=Σh0(q2)+δZh0(q2−m2h0)−δm2h0. (27)
≡
other fields. Thus e.g. the free propagator for the gauge
boson V becomes “dressed” by the quantum effects as For the -odd sector:
CP
follows:
Σˆ (q2)=Σ (q2)+δZ (q2 m2 ) δm2
igµν igµν A0 A0 A0 − A0 − A0
q2−−MV(0)2 → q2−MiV−g(0µ)ν2Z+VΣV(q2) ΣΣˆˆAG00G(q02()q2=)=ΣGΣ0A(q02G)0+(q2δ)ZG0q2−δm2G0
= − , (24)
q2−MV2 +ΣˆV(q2) +δZA0G0 q2− M2A0 −δm2A0G0. (28)
(0) (cid:18) (cid:19)
M being the bare mass of V and where in the sec-
V
ond equality we have introduced the renormalized form The structure of the Higgs sector beyond the leading
of the propagator. Similarly, for self-energies involving order is somewhat more involved. Similarly as in the
two scalars (cf. Fig.1) we identify +iΣφiφj(q2) with the SM, scalar-vector mixing terms arise in the Lagrangian
Feynmanblobwithexternallegsφi andφj. Forinstance, of the general 2HDM. Such contributions originate from
the dressed propagator of the scalar φ with bare mass the gauged kinetic terms of the Higgs doublets,
m(0) is given by
φ
=(D Φ ) (DµΦ )+(D Φ ) (DµΦ ), (29)
K µ 1 † 1 µ 2 † 2
i iZ L
φ
= . (25)
q2−m(φ0)2+Σφ(q2) q2−m2φ+Σˆφ(q2) wdehreivrea,tiivneorueradcosn(vuesnintigonpsr,etthtyesStUan(d2a)Lrd×nUo(t1a)tYioncso)v,ariant
The last equality refers to the corresponding renormal-
ized scalar field propagator, mφ being the renormal- D =∂ +i e Waτ +i e Y B , (30)
ized mass and Σˆφ the renormalizedself-energy. Without µ µ 2sW µ a 2cW µ
wantingtoappeartoopedagogical,weremindthereader
with e being the electron charge. Upon spontaneous
that the renormalization transformation at one-loop is −
EWsymmetrybreaking,the followingscalar-vectormix-
performed by canonically introducing renormalization
ing terms are generated:
constants relating every bare field φ(0) to the renormal-
i
ized ones as follows φ(0) = Z1/2φ = (1 + 1δZ )φ – ie
which in this case takies on thie pairticular fo2rm i(21)i – LS0V = 2s (v1∂µφ−1 +v2∂µφ−2)Wµ++h.c.
W
and at the same time decomposing the bare masses and e
(v ∂µχ0+v ∂µχ0)Z0. (31)
couplingsasthe sumofthe renormalizedquantity plus a −2s c 1 1 2 2 µ
W W
counterterm (m(0) = m +δm ; g(0) = g +δg ). With
i i i i i i
thesedefinitions,wecancheckthattherelations(24)and
(25)areconsistentatone-loopifwedefine the renormal-
ized self-energy of V and φ as 6 These conventions are slightly different from previous compre-
hensive calculations of Higgs boson physics presented by some
Σˆ (q2)=Σ (q2)+δZ (q2 M2) δM2 ofus inthepast[9](basicallythey differonthe signoftheself-
V V V − V − V energies)andtendtobemoreinagreementwiththeconventions
Σˆφ(q2)=Σφ(q2)+δZφ(q2−m2φ)−δm2φ. (26) ofthestandardpackagesFormCalc andFeynHiggs[80,81]. But
therearesomesmalldifferences;forinstance,thestandardfunc-
Finally, for blobs involving a vector (V) and a scalar tion SelfEnergy[S→V,MS] of FormCalc corresponds precisely
(φ) field (cf. Fig.1), there is a free vector index µ and toour−iΣφV.
