Extensions of the Standard Model

16 Pages·0.35 MB·

by  Fabio Zwirner

EXTENSIONS OF THE STANDARD MODEL
Fabio Zwirner
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
and
INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy
ThistalkbeginswithabriefgeneralintroductiontotheextensionsoftheStandardModel,reviewingthe
ideology of effective field theories and its practical implications. The central part deals with candidate
6 extensionsneartheFermiscale,focusingonsomephenomenologicalaspectsoftheMinimalSupersymmet-
9
ricStandardModel. Thefinalpartdiscusses somepossible low-energy implications offurtherextensions
9
near thePlanck scale, namely superstring theories.
1

n
a 1 Preamble (some facts and some ideology) theSMgaugefields. Thefirstlineofeq.(1)containstwo
J
operators carrying positive powers of Λ, a cosmological

9 It is quite obvious that the Standard Model (SM) must constanttermproportionaltoΛ4 andascalarmassterm
1 be extended. Among the ‘hard’ arguments supporting proportional to Λ2. Barring for the moment the discus-

thepreviousstatement,thestrongestoneisthefactthat sion of the cosmological constant term, which becomes
1
v the SM does not include a quantum theory of gravita- relevant only when the model is coupled to gravity, it
0 tional interactions. Immediately after, one can mention is important to observe that no quantum SM symme-
0 the fact that some of the SM couplings are not asymp- try is recovered by setting to zero the coefficient of the
3 totically free, making it almost surely inconsistent as a scalar mass term. On the contrary, the SM gauge in-
1
formalQuantumFieldTheory. Onecanaddtotheabove variance forbids fermion mass terms of the form ΛΨΨ.
0
6 the usual ‘soft’ argument that the SM has about 20 ar- The second line of eq. (1) contains operators with no
9 bitraryparameters,whichmayseemtoomanyforafun- power-likedependence onΛ,but only a milder,logarith-
/ damental theory. mic dependence, due to infrared renormalization effects.
h
p Whilstthisdoesnotgiveusdirectinformationonthe The operators of dimension d 4 exhibit two remark-
- formoftherequiredSMextensions,itbringsalonganim- able properties: all those allowe≤d by the symmetries are
p
portant conceptual implication: the SM should be seen actuallypresentintheSM;bothbaryonnumberandthe
e
h as an effective field theory, valid up to some physical individual lepton numbers are automatically conserved.
: cut-off scale Λ. The basic rule of the game1 is to write The third line of eq. (1) is indeed the starting point of
v
i downthemostgenerallocalLagrangiancompatiblewith anexpansionininversepowersofΛ,containinginfinitely
X theSMsymmetries[i.e. theSU(3) SU(2) U(1)gauge many terms. For energies and field VEVs much smaller
× ×
r symmetry and the Poincar´e symmetry], scaling all di- thanΛ,theeffectsoftheseoperatorsaresuppressed,and
a
mensionful couplings by appropriate powers of Λ. The thephysicallymostinterestingonesarethosethatviolate
resulting dimensionless coefficients are then to be inter- some accidental symmetries of the d 4 operators. For
preted as parameters, which can be either fitted to ex- example, a d = 5 operator of the form≤ ΨΨΦΦ can gen-
perimental data or (if one is able to do so) theoretically eratealepton-number-violatingMajorananeutrinomass
determined from the fundamental theory replacing the of order G−1/Λ (where G−1/2 300 GeV is the Fermi
SM at the scale Λ. Very schematically (and omitting all F F ≃
scale), as in the see-saw mechanism; some of the d = 6
coefficients and indices, as well as many theoretical sub-
four-fermion operators can be associated with flavour-
tleties, such as the problems in regularizing chiral gauge
changing neutral currents (FCNC) or with baryon- and
theories):
lepton-number-violating processes such as proton decay.
At this point, the question that naturally emerges is
= Λ4+Λ2Φ2
Leff the following: where is the cut-off scale Λ, at which the
+ (DΦ)2+ΨDΨ+F2+ΨΨΦ+Φ4 expansion of eq. (1) loses validity and the SM must be
6
ΨΨΦΦ ΨΨΨΨ replaced by a more fundamental theory? Two extreme
+ Λ + Λ2 +... , (1) but plausible answers can be given:
where Ψ stands for the generic quark or lepton field, Φ (I) Λ is not much below the Planck scale, MP
≡
for the SM Higgs field, and F for the field strength of G−1/2/√8π 2.4 1018 GeV, asroughly suggested
N ≃ ×
by the measured strength of the fundamental inter- m . Such constraint has a meaning which goes be-
H
actions, including the gravitational ones. yond perturbation theory, as suggested by the in-
frared structure of the SM renormalization group
(II) Λ is not much above the Fermi scale, as suggested equation for λ(Q) and confirmed by explicit lattice
bytheideathatnewphysicsmustbeassociatedwith computations4.
the mechanism of electroweak symmetry breaking.
Figure1includessomerecentrefinements5oftheoriginal
Intheabsenceofanexplicitrealizationatafundamental analysis, such as two-loop renormalization group equa-
level,eachoftheaboveanswerscanbeheavilycriticized. tions, optimal scale choice, finite corrections to the pole
The criticism of (I) has to do with the existence of the top and Higgs masses, etc. For very large cut-off scales,
‘quadraticallydivergent’scalarmassoperator,whichbe- Λ=1016–1019 GeV, the results are quite stable and can
comes more and more ‘unnatural’ as Λ increases above be summarized as follows: for a top quark mass close
the electroweak scale2. On general theoretical grounds, to 180 GeV, as measured at the Tevatron collider6, the
we would expect for such operator a coefficient of order only allowed range for the SM Higgs mass is 130 GeV
1, but experimentally we need a strongly suppressed co- <m < 200GeV. This means that, evenin the absence
H
efficient, of order G−1/Λ2. However, after taking into of a direct discovery of new physics beyond the SM, an-
F
accountquantumcorrections,thiscoefficientcanbecon- swer (I) could be falsified by LEP, the Tevatron and the
ceptually decomposed into the sum of two separate con- LHC in two possible ways: either by discovering a SM-
tributions, controlled by the physics below and above like Higgs boson lighter than 130 GeV, or by excluding
the cut-off scale, respectively. Answer (I) would then a SM-like Higgs boson in the 130–200GeV range!
require a subtle (malicious?) conspiracy between low-
energyandhigh-energyphysics,ensuringthedesiredfine-
tuning. The criticism of (II) has to do instead with the
d > 4 operators: in order to sufficiently suppress the
coefficients of the dangerous operators associated with
protondecay,FCNC,etc., the new physics at the cut-off
scale Λ must have quite non-trivial properties!
Atthemoment,answer(I)isnotverypopularinthe
physics community, since we do not have the slightest
idea onhow the requiredconspiracycouldpossibly work
atthefundamentallevel. Conceptually,suchapossibility
canbetheoreticallytestedinanultraviolet-finiteTheory
ofEverything: asdaringasitmaysound,withtheadvent
and the continuing development of string theories, we
maynot be veryfar fromthe implementationofthe first
quantitative tests. More concretely, such a possibility Figure1: Boundsinthe(Mt,mH)plane,forvariouschoicesofΛ.
can be experimentally tested in the near future, via the
searchfortheHiggsbosonatLEP,attheTevatronandat Answer (II), instead, gives rise to a well-known con-
theLHC.Aclearpictureoftheimplicationsof(I)isgiven ceptual bifurcation:
infigure1,whichshows,forvariouspossiblechoicesofΛ
(IIa) Inthedescriptionofelectroweaksymmetrybreak-
in the SM, the values of the top quark and Higgs boson
ing, the elementary SM Higgs scalar is replaced
masses allowed by the following two requirements3:
by some fermion condensate, induced by a new
The SM effective potential should not develop, be- strong interaction near the Fermi scale. This in-
• sidesthe minimumcorrespondingtothe experimen- cludes old and more recent variants of the so-called
talvalueoftheelectroweakscale,otherminimawith technicolor models7 (‘extended’, ‘walking’, ‘non-
lower energy and much larger value of the Higgs commuting’, ...). The stringent phenomenological
field. In first approximation, this amounts to re- constraintson technicolor models coming fromelec-
quiring the SM effective Higgs self-coupling, λ(Q), troweak precision data will be mentioned later. On
not to become negative at any scale Q < Λ: for a the theoretical side, technicolor remains quite an
given value of the top quark mass M , this sets a appealing idea, still waiting for a satisfactory and
t
lower bound on the SM Higgs mass m . calculable model. The lack of substantial theoret-
H
ical progress in this field, however, may be due to
The SM effective Higgs self-coupling should not de- the technical difficulties of dealing with intrinsically
•
velop a Landau pole at scales smaller than Λ: for non-perturbative phenomena. This should not and
a given value of M , this sets an upper bound on certainly will not prevent the experimentalists from
t
keepinganopenmindwhenlookingforpossiblesig- where ϕ (i = H ,H ,Q,Uc,Dc,L,Ec) denotes the
i 1 2
nals of new physics. generic spin-0 field, and λ (A = 1,2,3) the generic
A
gaugino field. Observe that, since AU,AD and AE are
(IIb) The SM is embedded in a model with softly
matrices in generation space, contains in principle
soft
broken global supersymmetry, and supersymmetry- L
a huge number of free parameters. Moreover,for generic
breaking mass splittings between the SM particles
values of these parameters one encounters phenomeno-
and their superpartners are of the order of the elec-
logical problems with FCNC, CP violation, charge- and
troweak scale. This approach, generically denoted
colour-breaking vacua. All the above problems can be
as low-energy supersymmetry8, ensures the ab-
solved at once if one assumes that the running mass
sence of field-dependent quadratic divergences, and
parameters in , defined at the one-loop level and
soft
makesit‘technically’naturalthatthereexistsscalar L
in a mass-independent renormalization scheme, can be
masses much smaller than the cut-off scale. More-
parametrized, at a cut-off scale Λ close to M , by a
P
over, a minimal and calculable model is naturally
universal gaugino mass m , a universal scalar mass
1/2
singled out, the so-called Minimal Supersymmetric
m , and a universal trilinear scalar coupling A, whereas
0
Standard Model (MSSM). m2 Bµ remains in general an independent parame-
3 ≡ −
ter.
2 Extensions near the Fermi scale (mainly
MSSM phenomenology)
2.1 MSSM (and alternatives) vs. electroweak precision
data
This section reviews some phenomenological aspects of
SM extensions near the Fermi scale. Reflecting the con-
The theoretical interpretation of electroweak precision
tent ofthe parallelsessionsandthe personaltaste ofthe
data, in the framework of the SM and of its candidate
speaker,mostofitwilldealwiththeMSSManditsvari-
extensions (including the MSSM), has been the subject
ants. ofseveraltalksin the parallel9 andplenary10,11 sessions.
In order to set the framework for the following dis-
Universal effects, occurring via the vector-boson
cussion, it is useful to recall the defining assumptions of
self-energies, and parametrized in terms of convenient
the MSSM. The field content is organized in gauge and variables12 such as (S,T,U) or (ǫ ,ǫ ,ǫ ), have already
1 2 3
matter multiplets of N = 1 supersymmetry. The gauge
been discussed many times at this and previous confer-
groupisG=SU(3) SU(2) U(1) ,andthematter
C L Y ences, and the results can be summarized as follows:
× ×
content corresponds to three generations of quarks and
leptons, as in the SM, plus two complex Higgs doublets, The SM fits excellently all the data (with the value
•
onemorethanintheSM.Toenforcebaryon-andlepton- of the strong coupling constant extracted from the
number conservation in d = 4 operators, one imposes hadronic Z and τ branching ratios slightly higher
a discrete R-parity: R = +1 for all ordinary particles than, but still compatible with, the one extracted
(quarks, leptons, gauge and Higgs bosons), R = 1 for from deep-inelastic scattering).
−
their superpartners (spin-0 squarks and sleptons, spin-
1/2gauginosandhiggsinos). A globallysupersymmetric The MSSM gives at least as good a fit as the SM,
•
Lagrangian isthenfullydeterminedbythesuper- thanks to the fast decoupling properties of the vir-
SUSY
L
potential (in standard notation): tual effects of supersymmetric particles, as long as
their mass is increased above the m /2 threshold.
Z
f =hUQUcH +hDQDcH +hELEcH +µH H . (2)
2 1 1 1 2
Naive versions of technicolor and extended techni-
To proceed towards a realistic model, one has to in- •
color models are ruled out (whereas some ‘walking’
troduce supersymmetry breaking. In the MSSM, super-
technicolor models may still work).
symmetrybreakingisparametrizedbyacollectionofsoft
terms, soft,whichpreservethegoodultravioletproper- Apointthathasattractedincreasingattentioninthe
L
ties of globalsupersymmetry. soft contains mass terms months before this Conference is the fact that, in some
L
for scalar fields and gauginos, as well as a restricted set extensions of the SM, non-universal effects on the Zbb
of scalar interaction terms vertex are also possible, which can modify appreciably
theSMpredictionforR Γ(Z bb)/Γ(Z hadrons).
1 b
−Lsoft = m˜2i|ϕi|2+ 2 MAλAλA IntheMSSM,onecanhav≡e13ext→rapositivec→ontributions
Xi XA toR fromloopsinvolvingstopsquarksandcharginosor
b
bottom quarks and neutral Higgs bosons, extra negative
+ hUAUQUcH +hDADQDcH
2 1 contributions to R from loops involving the top quark
b
(cid:0)
and the charged Higgs boson. In the technicolor frame-
+ hEAELEcH +m2H H +h.c. , (3) work, one has14 extra negative contributions to R in
1 3 1 2 b
(cid:1)
‘walking’ technicolor models, whereas contributions can figure2,thediscrepancywouldbelargeenoughthat,bar-
be of either sign in ‘non-commuting’ technicolor models. ring very special regions of the parameter space, which
The experimental data available before this may be already ruledout by indirect constraints or soon
Conference15 suggested16 thatanimprovedfittoα and ruled out by the forthcoming LEP run at √s = 130-
S
R could be obtained in the MSSM in the case of light 140GeV,theMSSMcanprovideonlyamodestimprove-
b
stops andcharginos(with generictanβ)and/orlightA0 ment in the quality of the fit.
(with tanβ m /m ). For the effect to be numerically
t b
∼
significant,thenon-standardparticlesintheloopsshould 2.2 MSSM and the decay b sγ
→
not be much heavier than m /2, otherwise fast decou-
Z As discussed in the parallel sessions18,19, the recent ex-
pling would take place and the effect rapidly vanish. A
quantitative estimate of this effect is given17 in figure 2, perimental observation of radiative B decays20 plays to-
which includes, besides the standard (t,W±) loop, also day a very important role in constraining many exten-
the (t,H±) loop and the (t˜,χ˜±) loops, in the simplified sions of the SM, and in particular the MSSM.
case of light t˜ and H˜±. Theexperimentalnumbermosteasilycomparedwith
R
After the new data presented at this Conference11, theory is the inclusive branching ratio
the picture appears more confused. Now both Rb and [BR(B →Xsγ)]exp =(2.32±0.67)×10−4. (6)
the analogousratio R havebeen measuredto better ac-
c
IntheSM,thisprocessisdescribed,atthe partoniclevel
curacy and with different methods. The theoretical SM
(b sγ) and at lowest order, by loop diagrams with
pαre(dmicti)o=n[0fo.1r2M5t =0.108007±an1d2Gα−eV1(,mmH) ==16258–.190000G0.e0V9,] inte→rnal top and W± lines. However, the theoretical de-
S Z Z
± ± termination of the inclusive branching ratio suffers from
and the averagedexperimental determinations are:
large uncertainties, mainly due to the QCD corrections,
which at the moment have been calculated only at lead-
EXP. TH.(SM) ing order21. A conservative estimate22 gives a total the-
oretical error of roughly 50%:
[BR(B X γ)]SM =(2.55 1.28) 10−4, (7)
Rb 0.2219 0.0017 0.2156 0.0005 → s th ± ×
± ±
whilst other less conservative estimates give theoretical
errors as low as 30%. The excellent agreement between
R 0.1540 0.0074 0.1724 0.0003 the two determinations (6) and (7) can be taken as an-
c
± ±
otherpieceofevidenceforSMradiativecorrections. This
isnotyetatthelevelofaprecisiontest,butalreadyrep-
We then have an excess in R at about the 3.5σ level
b
resents an important constraint on possible new physics
and a defect in R at about the 2.5σ level. Taking into
c
at the electroweak scale. For example, in the MSSM
account the measured value of the total hadronic width,
there are additional diagrams, corresponding to (t,H±)
Γ =(1744.8 3.0)MeV, (4) and(t˜,χ˜±)exchange,whichcangivequitelargecontribu-
h,exp
± tions to the rate23. For heavy supersymmetric particles,
which comfortably agrees with the SM prediction, thedatadisfavouralightchargedHiggs. Moregenerally,
a correlation is enforced between a light charged Higgs
Γh,SM =(1745.7 6.0)MeV, (5) and light stops and charginos, since one needs the right
±
amount of negative interference to fit the data. The sit-
onefinds the followingdiscrepancies: δΓb =11±3MeV, uationis illustrated infigure 3, whichdisplays17 contour
δΓ = 32 13MeV,δ(Γ +Γ )= 21 12MeV.Notice
c − ± b c − ± lines of Rγ BR(B Xsγ)MSSM/BR(B Xsγ)SM,
thatthediscrepancyinΓ +Γ ismuchlargerthantheer- ≡ → →
b c in the plane characterized by a common mass for the
roronΓh,andinsignandmagnitudecannotsupportany lightest stop and charginos (taken here to be t˜ and
R
longeranyintriguingconnectionbetweentheexperimen- H˜±) and by the charged Higgs mass, for the repre-
taleffectsonα andR . Also,fittingthedatawithinthe
S b sentative value tanβ = 1.5. The calculation of the
MSSMnowbecomesimpossible: forstops,charginosand
next-to-leadingQCDcorrections,announcedinaparallel
A0 all around 50 GeV, and tanβ ∼ mt/mb, the MSSM session18, would allow a significant reduction of the the-
couldmarginallyreproducetheobservedvalueofR ,but
b oretical error, and thus greatly enhance the constraints
the improvementin the fit to R with respect to the SM
c on the MSSM and on generic two-Higgs-doubletmodels.
would be negligible.
One could then fix (somewhat arbitrarily) R to its
c 2.3 ‘Relaxed’ MSSM
SM value. In this case, the fit to the experimental data
would give R = 0.2205 0.0016, roughly 3σ in excess Some of the assumptions defining the MSSM are plau-
b
±
of the SM prediction. Still, as can be appreciated from sible but not really compulsory, even if they may find a
justification in some theoretical constructions going be- the magnitude of the top quark mass and the amount
yond the MSSM. When discussing the phenomenology of FCNC expected in the resulting, ‘relaxed’ version of
of low-energy supersymmetry, it is important to keep an the MSSM. In order to do so, one defines a SUSY GUT,
open mind and to study what happens when some of with universal soft mass terms, near the scale M , and
P
these assumptions are relaxed. follows the logarithmic renormalization group evolution
Two possibilities were discussed in the parallel ses- of the model parameters from M to M : the large top
P U
sions. The first one24 consists in writing down the most Yukawacouplingcontrolstheamountofnon-universality
general renormalizable superpotential compatible with generated at M . A possible limit to the predictivity
U
supersymmetryandtheSMgaugesymmetry,whichcon- of this analysis is, in my opinion, the assumption that
tains, besides the familiar MSSM terms of eq. (2), the the logarithmic RG evolution in the (M ,M ) interval,
U P
additional terms withβ-functionsascomputedinthespecificSUSYGUT
model, is a good approximation. However,this criticism
∆w =λQDcL+λ′LEcL+λ′′UcDcDc, (8)
doesnotspoiltheinterestofsuchananalysis: onecanin-
where λ, λ′ and λ′′ have to be interpreted as three- troduceageneralparametrizationfortheuniversalityvi-
index tensors in generation space. Novel analyses of olationsatMU,or,equivalently,attheelectroweakscale,
the phenomenological constraints on the R-parity vio- and study the bounds on these parameters coming from
lating couplings of eq. (8) were discussed in the parallel FCNC processes; these will have to be respected by any
sessions25,26. Newboundsfromnon-leptonicB-decays25, fundamental theory that claims to predict the soft mass
Z physics at LEP26 and D-decays26 were presented. It parameters of the MSSM.
wasalsoobserved25that,ifenoughthird-generationfields
are involved, some baryon- and lepton-number violating 2.4 How could the MSSM be falsified?
terms in (8) could coexist with couplings of order 10−1–
Alegitimatequestion,oftenaskedwhensearchesfornew
10−2.
particles29 aredescribed,isthefollowing: Howcouldthe
The second possibility consists in allowing non-
MSSM be falsified, in the absence of new experimental
universal soft supersymmetry-breaking terms. This hy-
discoveries?
pothesis is subject to very stringent constraints from
Apart from the generic ‘naturalness’ argument, re-
FCNC, as discussed in the parallel sessions27. An ex-
quiring the masses of supersymmetric particles to be
ample is the decay µ eγ, subject to the strong ex-
perimental bound BR→(µ eγ) < 5 10−11. Off- of the order of the Fermi scale, namely smaller than a
→ × few TeV, it is difficult to establish firmer theoretical up-
diagonalsleptonmasstermsingenerationspace,denoted
per bounds. Attempts to quantify an acceptable ‘mea-
here with the generic symbol δm2, would contribute to
sure of fine-tuning’ and use it to bound from above the
the above decay at the one-loop level, and the previous
supersymmetric particle masses30 are parametrization-
limit roughly translates into δm2/m2 < 10−3–10−5, if
˜l dependent,andshouldbetakenjustasindications,since
one assumes gaugino masses of the order of the aver-
they do not have a solid theoretical foundation.
age slepton mass m (a quite complicated parametriza-
˜l However, the Higgs sector of the MSSM is very
tion is needed to formulate the bound more precisely).
tightly constrained. At the classical level, the mass of
Similar constraints can be obtained by looking at the
the lightest CP-even neutral Higgs boson obeys the cel-
K0–K¯0, B0–B¯0 systems and at other flavour-changing
ebrated inequality m < m cos2β . This bound is
phenomena. It is important to recall that all these h Z| |
shifted by the radiative corrections31. For example, the
bounds are naturally respected by the strict MSSM,
leading one-loop correction, due to the exchange of the
where the only non-universality in the squark and slep-
top quark and of its scalar partners, involves a shift in
ton mass terms is the one induced by the renormaliza-
the ‘22’ diagonal entry of the CP-even mass matrix,
tion groupevolutionfrom the cut-off scale Λ to the elec-
troweakscale. However,thesameboundsrepresentquite (∆m2) = 3g2 m4t logmt˜1mt˜2 +... , (9)
non-trivial requirements on extensions of the MSSM, 22 8π2m2 sin2β m2
such as supersymmetric grand-unified theories (SUSY W t
GUTs) and string effective supergravities, since in gen- which clearly exhibits the relevant dependences on the
eral one expects non-universal contributions to the soft top and stop masses. Further refinements in the calcu-
supersymmetry-breaking masses. Various mechanisms lation of the radiative corrections to the MSSM Higgs
that could enforce the desired amount of universality, or masses, and in particular of the upper bound mmax on
h
a sufficient suppression of FCNC via approximate align- m , include the parametrization of mixing effects in the
h
ments of the fermion and sfermion mass matrices, have stop sector (which can give in some cases an extra pos-
been presented in the mini-review by Savoy27. Another itive shift in m ), the resummation of the leading log-
h
interestingrecentdevelopmentistheattempt28 toestab- arithms via the renormalization group (which in gen-
lish a link, in the framework of SUSY GUTs, between eraldecreasesthe upper bound on m ), the momentum-
h
dependenceoftheself-energiesandloopsofotherMSSM soft scalar masses at M . Many strong (and indeed un-
U
particles (which give in general small effects). The necessary)modeldependencesareintroduced! Beforego-
results of a state-of-the-art calculation5 are illustrated ing to this level of detail, one would need a believable
in figure 4, which displays contours of mmax in the theory at the scale M and, in my opinion, we have not
h U
(m ,tanβ) plane, for large average stop mass (m yet reached such a stage. Therefore, some of the inter-
t sq
≡
(m2 +m2 )/2) and negligible or maximal mixing ef- esting analyses presented in the parallel sessions19,35,36,
qfects,t˜1respect˜2tively. It should be stressed that mmax is technicallycorrectwithin theirassumptions,mustbe in-
h terpreted with a grain of salt!
the maximumpossible valueofm , essentiallysaturated
h
for m = 1 TeV, but not necessarily the theoretically
A
most probable value, since it is obtained by pushing the 3 ExtensionsnearthePlanckscale(superstrings
MSSM parameters to the limits of their plausible range and their possible low-energy implications)
of variation.
In the search for a more fundamental theory going be-
Similarly, only slightly weaker bounds can be estab-
yond the MSSM, and allowing us to predict some of its
lished within supersymmetric models with non-minimal
Higgssectors5. Therefore,excluding the predicted Higgs manyparameters,we havetoday agreatadvantagewith
respect to the early eighties, since we can make use of
sectors stands out as the most promising option for fal-
the impressive progress of string theories over the last
sifying the MSSM and its non-minimal variants at fu-
ture accelerators32. Positive evidence for supersymme- decade.
Superstrings37 (perhapstobereplaced,someday,by
try, however, can only come from the discovery of some
the conjectured ‘M-theory’, of which the various string
(R-odd) supersymmetric particle.
theories may be different perturbative expansions) are
the only known candidate for a consistent, ultraviolet-
2.5 ‘Constrained’ MSSM
finite quantum theory of gravity, unifying all fundamen-
A remarkable fact, extensively advertisedin the last few tal interactions. There are perturbatively stable four-
years, is the following: combining the extracted values dimensional solutions of the heterotic string with nice
of the effective gauge couplings at the weak scale and phenomenological properties such as N = 1 supersym-
the leading logarithmic evolution of the latter33 in the metryinflatfour-dimensionalspace-time,agaugegroup
MSSM (with no new thresholds), one gets a consistent GcontainingtheSMgaugegroupSU(3) SU(2) U(1),
× ×
pictureofapproximateunificationofthegaugecouplings threechiralfamilies(andpossiblyextrastuff),andmore.
at a scale M 2 1016 GeV. Incidentally, the fact that supersymmetry seems to play
U
∼ ×
This stunning success, however, does not allow us averyimportantroleforthequantumstabilityofsuper-
to single out a unique SUSY GUT replacing the MSSM stringvacuamaybetakenasanadditionalmotivationto
at the scale M ! In constructing such a theory, there favourlow-energysupersymmetryovertechnicolor: how-
U
is freedom to choose the unified gauge group, the repre- ever, it should be kept in mind that so far superstrings
sentations in the Higgs sector, the parameters of the su- have not been able to give us any definite insight about
perpotential couplings (including, in general, a number the scale of supersymmetry breaking.
of explicit mass terms), the structure of the soft terms The generalfeature to be stressedis that string the-
afterspontaneoussupersymmetrybreaking. Evenchoos- ories contain one explicit mass scale, the string scale,
ing the simplest and most famous SUSY GUT, minimal whichfixes amassunitandactsasaphysicalultraviolet
SUSY SU(5)34, predictivity is limited by the freedom to cut-off. All the other physical scales (M ,M ,m ,...,
P U Z
choosethemassesofsomeofthe heavyHiggsmultiplets, in realistic models), and all the dimensionless couplings
and by the likely existence of corrections to the SUSY- ofthelow-energyeffectivetheory(probablysomeversion
GUT Lagrangian, in the form of little-suppressed non- oftheMSSM),arecontrolledbytheVEVsofsomescalar
renormalizableoperators,induced by physics at possible fields, called moduli, corresponding, in the effective su-
nearbyscales(compactificationscale,stringscale,Planck pergravity theories, to perturbatively flat directions of
scale). Moreover, minimal SUSY SU(5) must certainly the scalar potential. The inclusion of non-perturbative
be modifiedto incorporatearealisticfermionmassspec- quantum effects is expected to spontaneously break su-
trum and to solve the doublet–triplet splitting problem. persymmetry and to remove the degeneracyin the mod-
The moral of the story is that, when performing uli space, thus selecting the correct vacuum.
phenomenological analyses, it may be dangerous to put The special duality properties of string theories38
bounds on the MSSM mass spectrum by imposing ad- (someofwhichhavetheircounterpartatthe field-theory
ditional constraints such as ‘strict’ gauge coupling unifi- level, as discussed at this Conference by E. Verlinde39)
cation, ‘strict’ bottom–tau Yukawa coupling unification, can play a crucial role in controlling these phenom-
proton decay as described by minimal SUSY SU(5), or ena. The best-known string dualities are the so-called
radiative electroweak symmetry breaking with universal T-dualities, of which the simplest example is the equiv-
alence between a string compactified on a circle of ra- a realistic and consistent framework, gravitational in-
dius R and the same string compactified on a circle of teractions cannot be neglected. One is then led to
radius 1/R (in appropriate string units). These duali- N = 1, d = 4 supergravity, seen as an effective theory
ties areperturbative,inthe sensethat the duality trans- below the Planck scale, within which tree-level calcula-
formations do not act on the dilaton field, whose VEV tions can be performed and some qualitative features of
controlsthe couplingconstantassociatedwith the string the ultraviolet-divergent one-loop quantum corrections
loop expansion, so they can be consistently defined in be studied. Of course, infrared renormalization effects
the weak-coupling limit. The dualities at the origin of can be studied, but they are plagued by the ambiguities
a lot of recent excitement are however the so-called S- duetothecountertermsfortherenormalizableoperators.
dualities, which interchange weak and strong coupling, To proceed further, one must go to N =1, d=4 super-
and are therefore inherently non-perturbative. As ex- strings,seenasrealizationsofafundamentalultraviolet-
plainedby Verlinde, a prototypeofS-duality is the well- finite theory, within which quantum corrections to the
known electric–magnetic duality of QED. In supersym- low-energyeffectiveactioncanbeconsistentlytakeninto
metric theories, electric–magnetic duality is expected to account, with no ambiguities due to the presence of ar-
be part of a larger set of transformations, acting both bitrary counterterms.
on the gauge coupling g, controlling the F2 term in the In recent years,two approachesto the problem have
Lagrangian, and on the vacuum angle θ, controlling the been followed. On the one hand, four-dimensional tree-
associated FF˜ term, combined into a single chiral su- level string solutions, in which N = 1 local supersym-
perfield S. There is mounting evidence that S-duality metry is spontaneously broken via orbifold compacti-
is indeed a symmetry of the ten-dimensional heterotic fications, have been constructed40: none of the exist-
string compactified on a six-torus, as well as of globally ing examples is fully realistic, however they represent a
supersymmetric N = 4 Yang-Mills theories. Even more useful laboratory to perform explicit and unambiguous
interestingly, examples are being found of dual pairs of string calculations. On the other hand, many studies
stringtheories,in whichone string theoryatstrong cou- have been performed within string effective supergrav-
pling is equivalent to another string theory at weak cou- ity theories, assuming that supersymmetry breaking is
pling. Most of the evidence collected so far concerns induced by non-perturbative phenomena such as gaug-
string theories that would have unbroken N > 1 super- ino condensation41: the loss in predictivity is compen-
symmetry in d = 4, but the physically most important sated by the possibility of a more general parametriza-
goalisclearlytounderstandthetheorieswithN =1and tion,includingnon-perturbativeeffectsthatarestillhard
N = 0 supersymmetries in four dimensions: it would be to handle at the string theory level. With the advent of
great if one could study non-perturbative phenomena in string–stringdualities,itisevenconceivablethatthetwo
realistic string models just by going to the dual, weakly- approaches may be related (in an interesting paper that
coupled theory! Important conceptual developments are appearedafter this Conference42, itisarguedthatstring
rapidlytakingplacealsointhisrespect. Waitingforsolid tree-level breaking in a type II string solution may be
results, applicable to realistic cases, we are already wit- dual to non-perturbative breaking in a heterotic coun-
nessing a change of perspective in the approach to some terpart).
phenomenological problems. In the rest of this talk, I Before proceeding with the discussion, it may be
would like to mention some of them, not because they useful to recall some basic facts of N = 1, d = 4
areparticularlyimportant,butbecausetheyaretheones supergravity43. Thetheorycanbeformulatedwiththree
in which I have recently been involved. types of supermultiplets: in addition to the chiral and
vector supermultiplets, already present in global super-
symmetry, we need to introduce the gravitational su-
3.1 Supersymmetry breaking
permultiplet, whose physical degrees of freedom are the
At the level of dimensionless couplings, the MSSM is spin-2gravitonanditssupersymmetricpartner,thespin-
morepredictivethantheSM,sinceitsquarticscalarcou- 3/2 gravitino. Up to higher-derivative terms, the the-
plingsarerelatedbysupersymmetrytothegaugeandthe ory is completely determined by two functions of the
Yukawa couplings. The large amount of arbitrariness in chiral superfields: one is the K¨ahler function (z,z) =
G
the MSSM phenomenology is strictly related to its ex- K(z,z) + log w(z)2, which controls the kinetic terms
| |
plicitmassparameters,thesoftsupersymmetry-breaking andtheinteractionsofthechiralmultiplets;thisfunction
masses and the superpotential Higgs mass. Such arbi- is conventionally decomposed into a K¨ahler potential K
trariness cannot be removed within theories with softly and a superpotential w. The other is the gauge kinetic
broken global supersymmetry, such as SUSY GUTs: functionf (z),whichcontrolsthe kinetic termsandthe
ab
to make progress, spontaneous supersymmetry breaking interactions of the vector supermultiplets. It is custom-
must be introduced. ary to work in the natural supergravity units, where all
To discuss spontaneous supersymmetry breaking in masses are expressed in units of the Planck mass, i.e.
M = 1 by convention. An important difference with It is possible to formulate supergravity models
P
•
global supersymmetry is that the scalar potential is no where the classical potential is manifestly positive-
longer positive-semidefinite, but takes the form semidefinite, with a continuum of minima corre-
sponding to broken supersymmetry and vanishing
V0 = Da 2+ Fi 2 3eG, (10) vacuumenergy,andthegravitinomassslidingalong
| | | | −
a flat direction 44,45. A recent development is the
where the first two terms are positive-semidefinite, in
construction of models of this type where gauge
analogy with the usual F- and D-term contributions of
andsupersymmetry breakingaresimultaneously re-
globalsupersymmetry,whereas the last term, associated
alized, with goldstino components along gauge-non-
with the auxiliary field of the gravitational supermul- singlet directions46.
tiplet, is negative-definite. The novel structure of the
potential in supergravity theories permits the breaking This special class of supergravity models emerges
•
of supersymmetry with vanishing vacuum energy, if the naturally, as a plausible low-energy approximation,
last term in eq. (10) cancels exactly the remaining ones from four-dimensional string models, irrespectively
attheminimum: theorderparameterforthebreakingof of the specific dynamical mechanism that triggers
local supersymmetry in flat space is the gravitino mass, supersymmetry breaking. Due to the special geo-
m2 = eG, which fixes the scale of all supersymmetry- metricalpropertiesofstringeffective supergravities,
3/2
breakingmasssplittings,andthereforeoftheMSSMsoft the coefficient ofthe one-loopquadratic divergences
mass terms in the low-energy limit. intheeffectivetheory, Str 2,canbewrittenas47
M
The generic problems to be solved by a satisfactory
Str 2(z,z)=2Qm2 (z,z), (11)
mechanismforspontaneoussupersymmetrybreakingcan M 3/2
be succinctly summarized as follows:
where Q is a field-independent coefficient, calcula-
ble from the modular weights of the different fields
Classicalvacuumenergy. ThepotentialofN =1
• belonging to the effective low-energy theory, i.e.
supergravitydoesnothaveadefinitesignandscales
asm2 M 2: alreadyattheclassicallevel,onemust the integer numbers specifying their transformation
3/2 P properties under the relevant duality. The non-
arrange for the vacuum energy to be vanishingly
trivial result is that the only field-dependence of
small with respect to its natural scale.
Str 2 occurs via the gravitino mass. Since all
M
(m /M ) hierarchy. In a theory where the only supersymmetry-breaking mass splittings, including
3/2 P
•
explicit mass scale is the reference scale M (or the those of the massive string states not contained in
P
string scale), one must find a convincing explana- theeffectivetheory,areproportionaltothegravitino
tion of why it is m3/2 ∼< 10−15MP (as required by a mass, this sets the stage for a natural cancellation
natural solution to the hierarchy problem), and not of the O(m23/2MP2) one-loop contributions to the
m M . vacuum energy. Indeed, there are explicit string ex-
3/2 P
∼
amplesthatexhibitthisfeature. Ifthispropertycan
Stability of the classical vacuum. Even assum- persist at higher loops (an assumption so far), then
•
ingthataclassicalvacuumwiththeaboveproperties the hierarchy m M can be induced by the
3/2 P
canbearranged,theleadingquantumcorrectionsto logarithmiccorrectio≪nsdueto light-particleloops45.
the effective potential of N = 1 supergravity scale
again as m2 M 2, too severe a destabilization of In this special class of supergravity models one nat-
3/2 P •
the classical vacuum to allow for a predictive low- urally obtains, in the low-energy limit where only
energy effective theory. renormalizable interactions are kept, very simple
mass terms for the MSSM states (m ,m ,µ,A,B
0 1/2
Universality of squark/slepton mass terms. in the standard notation), calculable via simple al-
•
Such a condition (or alternative but equally strin- gebraic formulae from the modular weights of the
gent ones) is phenomenologically necessary to ade- correspondingfields andeasilyreconcilablewiththe
quatelysuppressFCNC,butisnotguaranteedinthe phenomenologicaluniversality requirements47. This
presence of general field-dependent kinetic terms. last result can indeed be obtained also in a slightly
less restrictive framework48.
From the above list, it should already be clear that the
generic properties of N = 1 supergravity are not suf- Just to give the flavour of the argument, we present
ficient for a satisfactory supersymmetry-breaking mech- here an ultra-simplified example, which retains the rele-
anism. Indeed, no fully satisfactory mechanism exists, vant qualitative features of the general case, without its
but interesting possibilities arise within string effective full technical complexity.
supergravities. Thebestresultsobtainedsofararelisted Consider a supergravity theory containing as chiral
below: superfields a gauge-singletT (to be thought of as one of
the superstring moduli fields), and a number of charged supersymmetry is broken and the gravitino mass,
fields Cα (to be thought of as the matter fields of the m2 = k2/(T +T)3 = 0 if one takes for simplicity
3/2 6
MSSM and possibly others), with K¨ahler potential Cα = 0, is classically undetermined. The modulus
field T corresponds to a flat direction, as in the no-
K = 3log(T +T)+ Cα 2(T +T)λα +... , (12) scalemodels44,anditsfermionicpartnerT˜playsthe
− | |
Xα role of the goldstino in the super-Higgs mechanism.
and superpotential
Str 2 can be put in the form of eq. (11), with
w =d CαCβCγ. (13) • M
SUSY αβγ
Q= 2+ (1+λ ), (16)
α
The model exhibits a classical invariance under the fol- −
Xα
lowing set of transformations, parametrizing the contin-
where the first addendum is the contribution of the
uous group SL(2,R):
massive gravitino and the second one the contribu-
aT ib
T − , Cα (icT +d)λαCα, (ab cd=1). tion of the matter fields.
→ icT +d → −
(14) In MSSM notation, the following very simple mass
•
The above symmetry can be interpreted as an approxi- terms are generated:
matelow-energyremnantofaT-dualityinvarianceunder
(m2)
thediscretegroupSL(2,Z),correspondingtotherestric- 0 α =1+λ , (17)
tion of the transformations (14) to the case of integer m23/2 α
(a,b,c,d) coefficients, and generated by the two trans-
(A)
formations T 1/T and T T +i. One can think αβγ =3+λ +λ +λ , (18)
of this SL(2,Z→) as an exact q→uantum symmetry of the m3/2 α β γ
underlying string model. In the language of supergrav- (µ) λ +λ
αβ α β
ity, the K¨ahler potential transforms as K K +φ+φ, =1+ , (19)
→ m3/2 2
where φ is an analytic function, and the superpotential
as w wexp( φ), so that the full K¨ahler function (B)αβ λα+λβ
→ − G =2+ . (20)
remains invariant. m3/2 2
Without specifying the dynamics which induces the
The above example can be easily generalized to include
spontaneous breaking of local supersymmetry, one can
gaugeinteractions,withanon-trivialmodulidependence
try to parametrizethe latter with a superpotentialmod-
of the gauge kinetic function: non-vanishing gaugino
ification of the form
masses can then be generated, proportional to the grav-
w =w +∆w, ∆w =k=0, (15) itino mass, and eq. (16) can be modified accordingly. It
SUSY
6
is important to stress that, in this framework, the phe-
wherekisaconstant,independentofthemodulusfieldT,
nomenologically desirable universality properties of the
whichcanbethoughtofasthelarge-T limitofamodular
soft mass terms can naturally arise as a consequence of
formofSL(2,Z). Inthecaseinwhichothermodulifields
T-duality. Furthermore, a non-vanishing µ-term can be
are present, such as the dilaton–axion field S associated
generated for the MSSM, proportional to m , even if
withthegaugecouplingconstant,onecanreplacek with 3/2
the supergravitysuperpotentialdoesnotcontainanyex-
a suitable function of S, with the correcttransformation
plicit Higgs mass term.
properties under a possible S-duality. Notice that the
The weakest point of the above construction is the
superpotentialmodificationintroducedabovebreaksthe
absence of a string calculation showing that, if there is
invariance under T 1/T, but preserves the shift sym-
metryT T+iα. →Alow-energystructureequivalentto cancellation of the O(m23/2MP2) contributions to the ef-
→ fectivepotentialatoneloop,thiscancellationcanpersist
the one introduced here has been found in explicit con-
athigherloops. Sinceintheeffectivetheoryonecaniden-
structions ofstring orbifoldmodels with string tree-level
tifysomequadraticallydivergenttwo-loopgraphs49,such
breaking40,buttheseresultscouldhavemoregeneralva-
an assumption is far from obvious. However, there are
lidity,andapplyalso,withtheappropriatemodifications,
hints47 thatthenumericalcoefficientofeq.(16)mightbe
to the case of non-perturbative breaking.
givenatopologicalinterpretation,sosuchanassumption
In the supergravity theory defined above, by apply-
is not completely arbitrary.
ing the standard formalism one can easily verify the fol-
Under the assumption that no terms (m2 M 2)
lowing results: O 3/2 P
are generated by string quantum corrections to the ef-
Thanks to the identity F 2 3eG, the scalar fective potential, the possibility arises of treating the
T
• | | ≡
potential of eq. (10) is automatically positive- gravitino mass m as a dynamical variable of the low-
3/2
semidefinite. At any minimum of the potential energy theory valid near the electroweak scale, namely
the MSSM. Then the actual magnitude of the gravitino planationforthenumericalvaluesofsomeofthefermion
mass could be determined by the logarithmic quantum masses,forexamplethoseofthethird-generationquarks.
corrections45,ascomputedintheMSSM.Theminimiza- To review the logic ofthe argument,we beginby re-
tionconditionoftheone-loopeffectivepotentialV ,with callingthat,intheMSSM,theRGEsforthetopandbot-
1
respect to m , would take the form50: tom Yukawa couplings admit an effective infrared fixed
3/2
curve,analogoustotheeffectiveinfraredfixedpointthat
m23/2∂∂mV21 =2V1+ S6tr4πM2 4 =0. (21) otonezeorob.taNinesg5l7ecbtiyngseftotrinsgimtphleicibtyottthoemτYYuukkaawwaa ccoouupplliinngg
3/2
andtheelectroweakgaugecouplings,anapproximatean-
The aboveequationcanbe interpretedasdefining anin- alytical equation for the infrared fixed curve is58,55
frared fixed point for the vacuum energy, with the two
2πE 4α α
tsecramlinsginantdhethseecsocanldinmgevmioblaetriornepbryesqeunatnintgumthceocrraencotnioicnasl, αt+αb ∼< 3F ·f(cid:20)(αt+tαbb)2(cid:21) , (22)
respectively. Onecanshowthat,forreasonablevaluesof
where to a good approximation (2πE/3F) (8/9)α
S
the boundary conditions on the dimensionless parame- ≃
and f is a hypergeometric function bounded by 1
ters, an exponentially suppressed hierarchy m M ≤
3/2 ≪ P f 12/7. This infrared behaviour is illustrated55 in
can be generated. ≤
figure 5. The previous considerations are only sufficient
Of course, the reason why m can be treated as
3/2 to set an upper bound, of order 200 GeV, on the com-
a dynamical variable in the effective low-energy theory bination m2/sin2β+m2/cos2β, but cannot predict the
is the existence of a very flat direction for the mod- t b
values of the top and bottom quark masses.
ulus on which it depends monotonically. This means
However,ifaftersupersymmetrybreakingsomevery
that, after the inclusion of the (m4 ) quantum cor-
O 3/2 flat directions are left over in moduli space, and the
rections,therewillbesomeverylightgauge-singletspin-
Yukawa couplings have some functional dependence on
0 fields, with ‘axion-like’ or ‘dilaton-like’ couplings and
the corresponding fields, then also the top and bottom
masses (m2 /M ), i.e. in the 10−3–10−4 eV range
O 3/2 P Yukawa couplings can be treated as dynamical variables
if m23/2 ∼ G−F1, with interesting astrophysical and cos- of the low-energy theory. A point stressed in a parallel
mological implications, including a number of potential session is that, in this two-variable problem, the details
phenomenological problems51. of the moduli-dependence of the two Yukawa couplings
may or may not impose some constraints on the mini-
3.2 Infrared moduli physics and the flavour problem mization problem to be solved. Irrespectively of these
details, it can be shown that minimization of the vac-
Oncethetaboohasbeenbroken,byconsideringaparam-
uumenergyalmostinvariablybringsthelow-energycou-
eter of the MSSM (in the previous example, the overall
plings very close to the infrared fixed curve of fig. 5b.
scale of its mass terms) as a dynamical variable at the
Unconstrained minimization, however, favours the so-
electroweak scale, and some partial success obtained (a lution h = hmax h = 0. Constrained minimiza-
possibleexplanationforthem M hierarchy,atthe t t ≫ b
3/2 P tion, instead, can produce the phenomenologically de-
≪
price of one important assumption and some unsolved sired solution h hmax h = 0, provided that some
cosmologicalproblems), it is not a big step to generalize t ≃ t ≫ b 6
constraint in the moduli space forbids the configuration
the game to other MSSM parameters,to see if there is a
h (M ) = 0. Barring these model-dependent details,
b U
chance that other problems can be solved.
which could be worked out only in a specific string con-
For example, in the case of non-universal soft mass
struction, one is still left with a definite prediction
terms one has in general a very severe problem with
FCNC27, unless one can find a good reason to justify m2 m2 12
the alignment of the quark and squark mass matrices. (MtIR)2 ∼< sin2tβ + cos2bβ ∼< 7 (MtIR)2, (23)
Ithasrecentlybeenproposed52 thatalsothe relativean-
glesbetweenthequarkandsquarkmassmatricesmaybe where MtIR ≃ (4/3) α3/(α2+α′)mZ ≃ 195 GeV. It
considered as dynamical variables: again, minimization seems difficult to gopbeyond the above result without
of the vacuum energy can induce at least partial align- a more detailed knowledge of the moduli-dependence of
ment. Analogous considerations have also been made53 Yukawacouplingsinstringtheoriesandofthemechanism
byconsideringapossibledynamicsjustabovetheSUSY- for spontaneous supersymmetry breaking.
GUT scale M .
U
Along similar lines, two groups50,54,55 have consid- 4 Conclusions
ered the possibility of treating some of the Yukawa cou-
plings of the MSSM as dynamical variables, as reviewed Amongthe SMextensionsatthe Fermiscale,theMSSM
inaparallelsession56. Thegoalistofindadynamicalex- stands out as the theoretically most motivated and the