Flavour Physics: Now and in the LHC era

0.64 MB·English

by  Gino Isidori

Flavour Physics: Now and in the LHC era∗
Gino Isidori
Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
INFN, Laboratori Nazionali di Frascati, Via E.Fermi 40, I-00044 Frascati, Italy
Wepresentanoverviewofwhatwelearnedsofarfromlow-energyflavourobservables,concerning
physics beyond the Standard Model, and what we could still learn from further studies in flavour
8
physicsin thenext few years.
0
0
2
I. INTRODUCTION: THE MAIN explicitbreakingofleptonnumberatveryhighen-

n LESSONS OF FLAVOUR PHYSICS ergyscales,inagreementwiththe expectationsof
a Grand Unified Theories (GUT).
J
In the last few years there has been a great ex- If the SM is not a complete theory, it is nat-
9
perimental progress in quark and lepton flavour ural to expect new degrees of freedom around or
1
physics. In the quark sector, the validity of the slightly above the electroweak scale (the energy
] Standard Model (SM) has been strongly rein- domain that will be fully explored for the first
h forced by a series of challenging tests. As sum- time at the LHC). Indeed we cannot extend the
p marised by the plots shown in Fig. 1, all the validity of the SM above the TeV range without
-
p relevant SM parameters controlling quark-flavour a serious fine-tuning problem in the Higgs sector
e dynamics (the quark masses and the angles of (see e.g. Ref. [4]). In constructing a realistic SM
h theCabibbo-Kobayashi-Maskawamatrix[1])have extension we should then try to reconcile three
[ been determined with good accuracy. More im- apparently conflicting requirements:

1 portant, several suppressed observables (such as
i. new degrees of freedom around the elec-
v ∆M , ∆M , ACP , B → X γ, ǫ , ...) po-
9 tentiBadllysensBitsivetKoΨNewPhysicss(NPK)havebeen troweak scale,
3
measured with good accuracy, showing no devi-
0 ii. no significant deviations for the SM in the
ations from the SM. The situation is somehow
3 quarksector(aswellasnosignificanteffects
. similar to the flavour-conservingelectroweak pre- in EWPO);
1
cision observables (EWPO) after LEP: the SM
0
8 worksvery welland genuine one-loopelectroweak iii. non-standard flavour structures in the lep-
0 effects have been tested with relative accuracy in ton sector.
: the10%–30%range. SimilarlytotheEWPOcase,
v
alsointhequarkflavoursectorNPeffectscanonly The rest of this talk is devoted to discuss how
i
X appear as a small correction to the leading SM these three points can be reconciled, and why
contribution. they imply that a few specific measurements in
r
a The situation of the lepton sector is more un- theflavoursectorwillstillbeveryinterestingalso
certain but also more exciting. The discovery of in the LHC era.
neutrino oscillations has two very significant im-
plications: i) the SM is not complete; ii) there
exists new flavour structures in addition to the II. WHAT WE LEARNED SO FAR
three SM Yukawa couplings. We have not yet ABOUT NEW PHYSICS
enough information to unambiguously determine
howthe SMLagrangianshouldbe modifiedinor- We can follow three mainstrategies to describe
der to describe the phenomenonof neutrino oscil- and quantify what we learned so far about NP
lations. However, natural explanations point to- from quark-flavourobservables.
wardtheexistenceofnewdegreesoffreedomwith
I. Generic EFT approach.
As long as we are interested in processes occur-
ring well below the electroweak scale (such as B,
∗ ToappearinthetheproceedingsofLeptonPhoton2007 D and K decays), we can integrate out the new
(Daegu, Korea,Aug.13-182007). degreesoffreedomanddescribeNPeffects–infull
2
to pay is the loss of generality. This is quite a
high price given our limited knowledge about the
physics above the electroweak scale.
III. EFT with explicit flavour symmetries.
An interesting compromise between these two ex-
tremestrategiesisobtainedimplementingspecific
symmetryrestrictionsontheEFT.Theextracon-
straintsincreasethenumberofcorrelationsinlow-
energy observables. The experimental tests of
such correlations allows us to test/establish gen-
eralfeaturesoftheNPmodel(possiblyvalidboth
at low- and high-energies). In particular, B, D
andK decaysareextremelyusefulindetermining
the flavour-symmetrybreaking pattern of the NP
model. The EFT approaches based on the Mini-
mal Flavour Violation (MFV) hypothesis and its
variations(MFVatlargetanβ,n-MFV,...) have
exactly this goal.
FIG. 1: Fit of the CKM unitarity triangle within the
A. Generic EFT approaches and the flavour
SM [2] (see also [3]).
problem
TheNPcontributionstothehigher-dimensional
generality– by means of an Effective Field The- operators of the EFT should naturally induce
ory (EFT) approach. The SM Lagrangian be- large effects in processes which are not medi-
comes the renormalizable part of a more general ated by tree-level SM amplitudes, such as meson-
local Lagrangian which includes an infinite tower antimeson mixing (∆F = 2 amplitudes) or
of higher-dimensional operators, constructed in flavour-changingneutral-current(FCNC)rarede-
termsofSMfieldsandsuppressedbyinversepow- cays. On the other hand, it is usually a good
ers of a scale ΛNP > v = 174 GeV. This general approximation to neglect non-standard effects in
bottom-upapproachallowsustoanalyseallrealis- processeswhicharemediatedbytree-levelSMam-
ticextensionsoftheSMintermsofalimitednum- plitudes. A general analyses of ∆F = 2 observ-
ber of parameters (the coefficients of the higher- ables based on the latter assumption has recently
dimensional operators). The disadvantage of this been performed by the UTfit Collaboration [3]
strategy is that it does not allow us to establish (earlierstudiescanbefoundalsoinRef.[2]). The
correlations of New Physics (NP) effects at low results are summarised by the plots in Fig. 2.
and high energies (the scale ΛNP defines the cut- Firstofall,itisinterestingtonotethatpresent
offoftheEFT).Thenumberofcorrelationsamong data, in particular the determination of γ and
different low-energy observables is also very lim- |V |, allow a rather precise determination of the
ub
ited, unless some restrictive assumptions about CKMmatrixusingtree-levelprocessesonly(Fig.2
the structure of the EFT are employed. left). This allows a model-independent compari-
sonofthe experimentaldataonmeson-antimeson
II. Explicit NP models.
mixingwiththecorrespondingtheoreticalSMpre-
The generic EFT approach is somehow the op-
dictions. NP effects in ∆F = 2 amplitudes can
posite of the standard top-bottom strategy to-
simply be parametrizedin terms of a modulo and
ward NP, where a given theory –and a specific
a phase for each meson-antimeson amplitude,
set of parameters– are employed to evaluate pos-
sible deviations from the SM. The top-bottom hM|Hfull|M¯i
eff =C e2iφM (1)
approach usually allows to establish several cor- hM|HSM|M¯i M
relations, both a low-energies and between low- eff
and high-energy observables. However, the price such that the SM is recovered for C = 1 and
M
3
ηη γ oo] [] [ 8800
11 BBdd
φφ
6600
4400
00..55
2200
00
00
--2200
--4400
--00..55
--6600
--11 --8800
00 11 22 33 44 55 66
CC
--11 --00..55 00 00..55 11 BB
ρρ dd
FIG.2: Left: Constraintsontheρ¯–η¯planeusingtree-levelobservablesonly. Right: Constraintsontheeffective
parameters encoding NPeffects in theB –B¯ mixingamplitude (magnitude and phase) [3].
d d
φ = 0. The main conclusions which can be • AsclearlyshowninFig.2,intheB –B¯ case
M d d
drawn form the present analyses can be summa- there is still room for a new-physics contri-
rized as follows: butionupto∼50%oftheSMone(C can
Bd
be substantiallydifferentfromunity). How-
• In all the three accessible amplitudes (K0–
ever, this is possible only if the new-physics
K¯0, B –B¯ , and B –B¯ ) the magnitude of
d d s s contributionisalignedinphasewithrespect
the new-physics amplitude cannot exceed,
to the SM amplitude (φ close to zero). A
in size, the SM short-distance contribution. similar conclusionholdsBadlsofor theK0–K¯0
The latter is suppressed both by the GIM
amplitude.
mechanismandbythehierarchicalstructure
of the CKM matrix (|Vtd|,|Vts|≪1): • ContrarytoB –B¯ andK0–K¯0 amplitudes,
d d
G2M2 at present there is only a very loose bound
A∆F=2 ∼ F W (V∗V )2× on the CPV phase of the B –B¯ mixing
SM 2π2 ti tj s s
amplitude. This leaves open the possibil-
×hM¯|(Q¯i γµQj)2|Mi (i,j =d,s) (2)
L L ity of observing a large ACP(Bs → J/Ψφ)
at LHCb, which would be a clear signal of
Therefore, new-physics models with TeV-
physics beyond the SM.
scale flavored degrees of freedom and
O(1) flavour-mixing couplings are essen-
The strong bounds on Λ in models with generic
tiallyruledout: denotingbyC theflavour-
ij flavour structure (C ∼ 1) is a manifestation of
ij
mixing coupling in the NP model,
what in many specific frameworks (supersymme-
C try, technicolour, etc.) goes under the name of
A∆NPF=2 ∼ 2Λij2hM¯|(Q¯iLγµQjL)2|Mi (3) flavour problem: if we insist with the theoretical
prejudice that new physics has to emerge in the
the condition |A∆NPF=2|<|A∆SMF=2| implies TeVregion,wehavetoconcludethatthenewthe-
ory possesses a highly non-generic flavour struc-
9×103 TeV×|C |1/2
3.4 TeV sd ture. Interestingly enough, this structure has not
Λ< < 4×102 TeV×|C |1/2
|V∗V |/|C |1/2  bd been clearly identified yet, mainly because the
ti tj ij 7×101 TeV×|C |1/2
 bs SM, i.e. the low-energy limit of the new theory,

4
doesn’t possess an exact flavour symmetry. (in the down-quark mass-eigenstate basis). As a
Themostreasonable(butalsomostpessimistic) result, within this framework the coefficients of
solution to the flavour problem is the so-called the higher-dimensional operators have the same
Minimal Flavour Violation hypothesis [5, 6, 7, 8]. CKM suppression of the corresponding SM am-
Under this assumption, which will be discussed plitudes and the bounds on the new-physics scale
below, the first two items listed above find a nat- are in the few TeV range. This is already clear
ural explanation. from Eq.(3), once we set C = y2V∗V ; statisti-
ij t 3i 3j
cally well defined and updated bounds can be in
Ref. [3]. Moreover, the flavour structure of Y Y†
U U
B. Minimal Flavour Violation implies a well-defined link among possible devi-
ations from the SM in FCNC transitions of the
The mainidea ofMFV is that flavour-violating type s → d, b → d, and b → s (the only quark-
interactions are linked to the known structure of leveltransitionswhereobservabledeviationsfrom
YukawacouplingsalsobeyondtheSM.Asaresult, the SM are expected).
non-standard contributions in FCNC transitions
The idea that the CKM matrix rules the
turn out to suppressed to a level consistent with
strengthofFCNCtransitionsalsobeyondtheSM
experiments even for Λ ∼ few TeV. On the most
is a concept that has been implemented and dis-
interesting aspects of the MFV hypothesis is that
cussed in several works, especially after the first
it can easily be implemented within the general
results of the B factories (see e.g. Ref. [9]). How-
EFT approach to new physics [6, 7]. This allows
ever, it is worth stressing that the CKM matrix
us to establish general and unambiguous corre-
represents only one part of the problem: a key
lations among NP effects in various rare decays.
role in determining the structure of FCNCs is
These falsifiable predictions are a key ingredient
also played by quark masses (via the GIM mech-
toidentifyinamodel-independentwaytheflavour
anism), or by the Yukawa eigenvalues. In this re-
structure of the new-physics model.
spect,theaboveMFVcriterionprovidesthemax-
Inamore quantitativeway,the MFV construc-
imal protection of FCNCs (or the minimal vio-
tion consists in identifying the flavour symmetry
lation of flavour symmetry), since the full struc-
and symmetry-breaking structure of the SM and
ture of Yukawa matrices is preserved. Moreover,
enforceittheEFT.Inthequarksectorthisproce-
contrary to other approaches, the above MFV
dureisunambiguous: thelargestgroupofflavour-
criterion is based on a renormalization-group-
changing field transformations commuting with
invariant symmetry argument, which can easily
the gauge group is G = SU(3) ×SU(3) ×
q QL UR be extended to TeV-scale effective theories where
SU(3) , and this group is broken only by the
DR new degrees of freedoms, such as extra scalar
two 3ׯ3 structures of the Yukawa interaction:
fields (see e.g. [10]) or SUSY partners of the SM
Lquark = Q¯i (Y ) UjH + fields (see e.g. [11, 12]), are included. Finally,
Y L U ij R U
this symmetry and symmetry-breaking pattern
+Q¯i(Y ) DjH +h.c. (4)
L D ij R D can explicitly be implemented in well-motivated
UV completions of the SM valid up to very high
TheinvarianceoftheSMLagrangianunderG can
q
energy scales (see e.g. [13, 14]).
beformallyrecoveredelevatingtheYukawamatri-
ces to spurionfields with appropriatetransforma- As shown in Fig. 3, the MFV hypothesis pro-
tionpropertiesunderGq. The hypothesisofMFV vides a natural (a posteriori) justification of why
states that these are the only spurions breaking no NP effects have been observed in the quark
Gq also beyond the SM. Within the effective the- sector: by construction,most of the clean observ-
ory formulation, this implies that all the higher ables measured at B factories are insensitive to
dimensional operators constructed from SM and NP effects in this framework. However, it should
Yukawa fields must be (formally) invariant under be stressed that we are still very far from having
Gq. proved the validity of this hypothesis from data.
It is then easy to realize that, similarly to A proof of the MFV hypothesis can be achieved
the pure SM case, the leading coupling ruling all only with a positive evidence of physics beyond
FCNCtransitionswithexternaldown-typequarks the SM exhibiting the flavour pattern (link be-
is (Y Y†) ≈ y2V∗V , with y = m /v ≈ 1 tween s→d, b→d, and b→s) predicted by the
U U ij t 3i 3j t t
5
ηη γ ηη γ
11 11
00..55 β ∆∆mmsd ∆md 00..55 β ∆∆mmds
ε V V
K ub ub
00 Vcb 00 Vcb
sin(2β+γ)
--00..55 α --00..55 α
--11 --11
--11 --00..55 00 00..55 11 --11 --00..55 00 00..55 11
ρρ ρρ
FIG. 3: Fit of the CKM unitarity triangle within the SM (left) and in generic extensions of the SM satisfying
theMFV hypothesis(right) [3].
MFV assumption[6]. So far we haveonly bounds istence ofnew sourcesofflavoursymmetry break-
on NP effects in the flavour sector, and it could ing. A typical example is the MSSM with generic
wellbe thatthe new theoryincludes non-minimal flavour structures [18]: here each flavour observ-
sources of flavour symmetry breaking with spe- ableisusedtosetalimitonaspecificcombination
cific flavour structures, such as those discussed in of non-diagonal entries of the sfermion mass ma-
Ref.[15]. Itisalsoconceivablethatthereisnotan trices (see e.g. Ref. [19] for a recent discussion).
underlyingflavoursymmetry,andthesuppression
Theimportanceofflavourobservablesislessob-
of FCNCs is of dynamical origin. This happens
vious in constrained models, such as MSSM sce-
forinstance inscenarioswithhierarchicalfermion
narioswith MFV.The situationhereturns outto
wave functions [16], which are well motivated by
be even more interesting than in generic models:
models with warped extra-dimensions [17].
the number of free parameters is substantially re-
Last but not least, it is worth to stress that ducedandagivenobservableputconstrainswhich
even within the pessimistic MFV framework the are relevant for several other processes (even be-
lepton sectorcouldstill be veryexciting. The im- yond the flavour sector). As a result, the consis-
plementationoftheMFVhypothesisinthelepton tency of the model is probed to a deep level.
sector is not as straightforward as for the quark
An illustrationof this factin the contextof the
sector [7]. But if the breaking of lepton flavour
mSUGRAscenariohasbeenpresentedinRef.[20]:
and totallepton number are decoupled, rareLFV
the information derived by B →X γ poses a sig-
decays such as µ → eγ could be within the reach s
nificant constraint on the model, which is com-
of the next generation of experiments even in a
patiblewiththosederivedfromflavourconserving
MFV framework [7].
processes. In particular, the heavy stop mass re-
quiredbyB →X γ isoneofthemainingredients
s
which pushes the mass of the light Higgs boson
C. Flavour constraints in explicit models above the LEP bound [21].
As recently shown in Ref. [22], there are also
In all explicit NP scenarios the constraints of specific supersymmetric MFV frameworks which
flavour physics play a very important role. This are essentially ruled out by the recent results of
is obvious in cases where the model allow the ex- flavour physics. In particular, the present con-
6
straints from B(B → τν), B(B → µ+µ−) and helicity-suppressed amplitudes. These are con-
s
B(B → X γ), puts in serious difficulties the finedtotheB-mesonsystem(becauseofthelarge
s
SO(10)GUTmodelofDermisekandRaby[23],a b-quark Yukawa coupling), with the notable ex-
specificexampleofMFVscenariowithlargetanβ. ception of K → ℓν decays. We can divide the
most interesting observables in three classes: the
charged-current processes B(K) → ℓν, the rare
III. FLAVOUR PHYSICS IN THE LHC decays B → ℓ+ℓ−, and the FCNC transition
s,d
ERA B →X γ.
s
It is worth to stress that, beside the theoreti-
If new particles or,more generally,new degrees cal interest, the large tanβ regime of the MSSM
of freedom, are present in the TeV energy range, could also provide a natural explanation of the
there are good chances that part of them will be a =(g−2) /2 anomaly, which is now a solid 3σ
µ µ
discovered at the LHC. This does not mean that effect: ∆a =aexp−aSM ≈(2.9±0.9)×10−9[24].
µ µ µ
the complete structure of the new model can eas- The size of this discrepancy is large compared
ilybedeterminedattheLHC:thedirectdiscovery to the electroweak SM contribution (∆ae.w. ≈
µ
ofnewparticlesisonlyoneofthe ingredientsnec- 1.5 × 10−9). This large discrepancy can easily
essary to achieve this goal. As already discussed be explained by the fact that a is a(flavour-
µ
in the previous section, some of the parameters conserving) helicity suppressed observable, whose
of the model (in particular its flavour structure) non-standard contribution can be enhanced com-
can only be determined with improved measure- pared to the SM one by increasing the value of
ments in the flavour sector. A brief survey of the tanβ:
mostinterestinglow-energyflavourobservablesin
2
thisperspective,focusingonMSSMscenarioswith M
MFV (or approximate MFV), is presented in the ∆aMµSSM ≈tanβ×∆aµe.w.× W (6)
following. Mslept!
For values of tanβ ∼> 10 the MW/fMslept suppres-
sion can easily be compensated for sleptons well
A. Helicity-suppressed observables and the
large tanβ scenario above the W mass, in prefect agreefment with the
constraints of electroweak precision tests.
The Higgs sector of the MSSM consists of two
SU(2) scalardoublets,coupledseparatelytoup-
L 1. B(K)→ℓν
and down-type quarks
Ltree = Q¯ Y U H +Q¯ Y D H + The charged-current processes P → ℓν are the
H L U R U L D R D
simplest flavour-violating helicity suppressed ob-
+L¯ Y E H +V(H ,H )+h.c. (5)
L E R D U D servables. HerebothSMandHiggs-mediatedcon-
tributions (sensitive to tanβ) appear already at
A key parameter of this sector is the ratio of the
the tree level. The H± contribution is propor-
two Higgs vevs: tanβ = hH i/hH i. Varying
U D tionalto the Yukawa couplingsofquarksandlep-
tanβ leads to modify the overallnormalizationof
tons, but it can compete with the W± exchange
the twoYukawacouplingsand,fortanβ ∼40–50,
thanks to the helicity suppressionof P →ℓν [25].
we can achieve the interesting unification of top
Takinginto accountthe resummationofthe lead-
and bottom Yukawa couplings.
ing tanβ corrections to all orders,the H± contri-
The variation of tanβ do not change the mis-
butions to the P → ℓν amplitude within a MFV
alignmentinflavourspaceofthetwoYukawacou-
supersymmetric framework leads to the following
plings. This implies that flavour-changing ob-
ratio [26, 27]:
servables not suppressed by powers of down-type
quark masses (i.e. most of the experimentally ac- B(Pℓν)
cessibleobservables)arenotsensitivetothe value RPℓν = BSM(Pℓν)
of tanβ. If the model has a MFV structure,
the phenomenological consequences of tanβ ≫ 1 SU=SY 1− m2P tan2β 2(7)
show up only in the few observables sensitive to m2 (1+ǫ tanβ)
(cid:20) (cid:18) H±(cid:19) 0 (cid:21)
7
2. B →ℓ+ℓ−
β)
n(
a
t 80 The important role of B(Bs,d → ℓ+ℓ−) in the
large tanβ regime of the MSSM has been widely
discussed in the literature (see e.g. Ref. [26, 33,
lavi
netKaon WG 34, 35] for a recent discussion). Similarly to
60
P → ℓν decays, the leading non-SM contribution
in B →ℓ+ℓ− decaysis generatedby a single tree-
level type amplitude: the neutral Higgs exchange
B → A,H → ℓ+ℓ−. Since the effective FCNC
40
couplingoftheneutralHiggsbosonsappearsonly
at the quantum level, in this case the amplitude
95% CL from K→µν/π→µν
has a strong dependence on other MSSM param-
20 95% CL from B→τν eters in addition to M and tanβ. In particu-
H
lar, a key role is played by µ and the up-type tri-
linear soft-breaking term (A ), which control the
100 200 300 400 500 U
charged Higgs mass (GeV/c2) strengthofthenon-holomorphicterms. Thelead-
ing parametric dependence of the scalar FCNC
amplitude from these parameters is given by
fFrIoGm.B4:(BP→resτenνt) acnondsBtr(aKint→s iµnνt)h[e32M].H–tanβ plane AHiggs(B →ℓ+ℓ−)∝ mMbm2ℓµMA2U tan3β×floop
A q˜
For tanβ ∼50 and M ∼0.5 TeV the neutral-
A
Higgs contribution to B(B → ℓ+ℓ−) can easily
s,d
lead to an O(100) enhancement over the SM ex-
where ǫ denotes the effective coupling which pectation. This possibility is already excluded by
0
parametrizes the non-holomorphic corrections to experiments: the upper bound B(Bs →µ+µ−)<
the down-type Yukawa interaction [28, 29]. For a 5.8×10−8 [36] is only about 15 times higher that
natural choice of the MSSM parameters, Eq. (7) the SM prediction of 3.5×10−9 [37]. This limit
impliesasuppressionwithrespecttotheSMinB poses interesting constraints on the MSSM pa-
decays of few×10% (but an enhancement is also rameter space, especially for light MH and large
possible for very light MH±) and an effect 100 values of tanβ (see e.g. Fig. 5). However, given
times smaller in K decays (where the branching the specific dependence onAU and µ, the present
ratio is necessarily smaller than BSM). B(Bs →µ+µ−) bound does not exclude the large
tanβ effects in (g − 2) and P → ℓν already
µ
discussed. The only clear phenomenological con-
In the B case only the τ modes has been ob-
clusion which can be drawn for the present (im-
served: B(B →τν)exp =(1.41±0.43)×10−4[30].
proved) limit on B(B → µ+µ−) is the fact that
s
In the Kaon system the precision of B(K → µν)
the neutral-Higgs contribution to ∆M [38] is
is around0.3%[31]. In the limit of negligiblethe- Bs
negligible.
oreticalerrors,we should therefore expect similar
bounds inthe M –tanβ plane fromB andK de-
H
cays. This limit is far from being realistic, due to
the sizable errorson f (determined from Lattice 3. B→Xsγ
P
QCD) and V (which must be determined with-
uq
out using the information on P → ℓν decays). The radiative decay B → X γ is one of the
s
But again the present level of precision is such observables most sensitive to non-standard con-
that the B and K decays set competitive bounds tributions, not only in the large tanβ regime of
intheM –tanβ plane(seeFig.4). Bothchannels the MSSM. Contrary to pure leptonic decays dis-
H
have interesting possibility of improvementin the cussed before, B → X γ does not receive effec-
s
near future. tivetree-levelcontributionsfromtheHiggssector.
8
FIG. 5: B(Bs → µ+µ−) as a function of tanβ in the
mSUGRAscenario [34].
FIG. 6: Combined bounds from low-energy ob-
The one-loop charged-Higgs amplitude, which in- servables in the tanβ–MH plane assuming heavy
creasesthe rate comparedto the SM expectation, squarks and dark-matter constraints in the A-funnel
can be partially compensated by the chargino- region [41] (Mq˜ = 1.5 TeV, AU = −1 TeV, µ =
squark amplitude, giving rise to delicate cancel- 0.5 TeV, Mℓ˜ = 0.4 TeV, 1.01 < RBsγ < 1.24; the
lations. As a result, the extraction of bound in light-blue area is excluded by the dark-matter condi-
the M –tanβ plane from B(B → X γ) (within tions).
H s
the MSSM) is a non trivial task.
Despite the complicated interplay of various
a supersymmetric contribution to a of O(10−9)
non-standard contributions, B →X γ is particu- µ
s
larlyinterestinggiventhegoodtheoreticalcontrol is both compatible with the present constraints
from B(B →X γ) and it implies a suppressionof
of the SM prediction and the small experimental s
B(B →τν)withrespecttoitsSMpredictionofat
error. According to the recent NNLO analysis of
least 10% [41]. A more precise determination of
Ref. [39], the SM prediction is
B(B →τν) is therefore a key element to test this
B(B →X γ)SM =(3.15±0.23)×10−4 scenario.
s Eγ>1.6 GeV
to be compared with the experimental aver-
age [40]:
B. Rare K decays
B(B →X γ)exp =(3.55±0.24)×10−4
s Eγ>1.6 GeV) Among the many rare K and B decays, the
Theseresultsallowasmallbutnonnegligiblepos- K+ → π+νν¯ and K → π0νν¯ modes are unique
L
itive non-standard contribution to B(B → X γ) since their SM branching ratios can be computed
s
(asexpectedifthecharged-Higgsamplitudewould to an exceptionally high degree of precision, not
dominate over the chargino-squark one), which matched by any other FCNC processes involv-
represents one of the most significant constraint ing quarks. It is then not surprising that K →
in the MSSM parameter space. πνν¯ decays continue to raise a strong theoreti-
Anillustrationofthe typicalcorrelationsofthe cal interest, both within and beyond the SM (see
low-energyflavourconstraintsintheM –tanβ,in e.g. Ref. [42]).
H
a generic scenario with heavy squarks and dark- Because of the strong suppression of the s→d
matter conditions satisfied in the A-funnel re- short-distance amplitude in the SM [V V∗ =
td ts
gion, is shown in Fig. 6. One of the most in- O(10−4)], rare K decays are the most sensitive
teresting aspects of this scenario is the fact that probesofpossibledeviationsfromthestrictMFV
9
FIG. 7: Predictions of different NP models for B(K+ → π+νν¯) and B(K+ → π+νν¯) [courtesy of F. Mescia].
The 95% C.L. exluded areas of B(K+ →π+νν¯) refer to theresult of the BNL-E787/949 experiment [50].
ansatz. Several recent NP analyses confirm the NP [46, 47, 48, 49].
high discovery potential of these channels (see
Fig. 7 and Ref. [42]). The latter has also im-
proved thanks three significant improvements on
C. Lepton Flavour Violation and LF
the SM predictions of K → πνν¯ rates: i) the non-universality
NNLO calculation of the dimension-six charm-
quark contribution to K+ → π+νν¯ [43]; ii) the
LFV couplings naturally appear in the MSSM
first complete analysis of dimension-eight and
once we extend it to accommodate the non-
long-distance(up-quark)contributionsrelevantto
vanishing neutrino masses and mixing angles
K+ → π+νν¯ [44]; iii) a new comprehensive anal-
by means of a supersymmetric seesaw mecha-
ysis of matrix-elements and isospin-breaking ef-
nism [51]. In particular, the renormalization-
fects, relevant to both channels [45]. Thanks to
group-induced LFV entries appearing in the left-
these recentworks,the irreducible theoreticalun-
handed slepton mass matrices have the following
certaintiesonbothbranchingratiosareatthefew
form [51]: δij = c (Y†Y ) , where Y are the
% level. LL ν ν ν ij ν
neutrino Yukawa couplings and c is a numeri-
ν
It is worth stressing that if a deviation from cal coefficient of O(0.1–1). The information from
the SM is seen in one of the two K →πνν¯ chan- neutrino masses is not sufficient to determine in
nels,akeyindependentinformationaboutthena- a model-independent way all the seesaw parame-
ture of NP can be obtained also from the two ters relevant to LFV rates and, in particular, the
K → π0ℓ+ℓ− (ℓ = e,µ) modes. The latter are neutrino Yukawa couplings. To reduce the num-
L
not as clean as the neutrino modes, but are still beroffreeparametersspecificSUSY-GUTmodels
dominated by SD dynamics and very sensitive to and/or flavour symmetries need to be employed.
10
Twomainroadsareoftenconsideredinthelitera-
ture: thecasewherethecharged-leptonLFVcou-
plings are linked to the CKM matrix (the quark
mixing matrix) and the case where they are con-
nected to the PMNS matrix (the neutrino mixing
matrix) [52].
Once non-vanishing LFV entries in the slep-
ton mass matrices are generated, LFV rare de-
cays are naturally induced by one-loop diagrams
with the exchange of gauginos and sleptons. For
largevaluesoftanβ theradiativedecaysℓ →ℓ γ,
i j
mediated by dipole operators, are linearly en-
hanced,incloseanalogytothetanβ-enhancement
of ∆a = (g −gSM)/2. A strong link between
µ µ µ
these two observable naturally emerges [53]. We
can indeed write
B(ℓ →ℓ γ) 48π3α ∆a 2
i j µ
= ×
B(ℓ →ℓ ν ν¯ ) G2 m2
i j ℓi ℓj F (cid:20) µ (cid:21)
2
f M2/M2,µ2/M2
× 2c 2 ℓ˜ ℓ˜ δij 2 (8)
g2c(cid:16)M22/Mℓ˜2,µ2/Mℓ˜2(cid:17) (cid:12) LL(cid:12) FIG. 8: B(µ → eγ) vs. ∆aµ = (gµ −gµSM)/2 in the
 (cid:16) (cid:17) (cid:12)(cid:12) (cid:12)(cid:12) MSSM assuming |δL12L|=10−4 [26].
where f and g are O(1) loop functions. In the
2c 2c
limitofadegenerateSUSYspectrum,thisimplies
∆a 2 104 in the case of a CKM-type hierarchy. In the
B(ℓ →ℓ γ) ≈ µ × latter case B(τ → µγ) can exceed 10−9 and be
i j 20×10−10
(cid:20) (cid:21) within the reach of a super-B factory. The en-
1×10−4 δ12 2 [µ→e] hancement of B(τ → µγ) can be even larger in
×(2×10−5 (cid:12)δLL23LL(cid:12)2 [τ →µ] (9) nreocne-nstulpyedrsisycmusmseedtriinc fRraemf.e[5w5o]r.ks, such as the one
(cid:12) (cid:12)
The strong correlati(cid:12)on b(cid:12)etween ∆a and the An independent and potentially large class of
(cid:12) (cid:12) µ
rate of the LFV transitions ℓ → ℓ γ holds well LFV contributions to rare decays in the large
i j
beyond the simplified assumptions used to derive tanβ regime of the MSSM comes from Higgs-
these equations (see Fig. 8). The normalization mediated amplitudes. Similarly to the quark sec-
|δ12 | = 10−4 used in Fig. 8 for B(µ → eγ) corre- tor, non-holomorphic couplings can induce an ef-
LL
spondstotheMFVhypothesisintheleptonsector fective FCNC Higgs coupling also in the lepton
with M > 1012 GeV [7]. As canbe seen,for such sector [56]. Gauge- and Higgs-mediated ampli-
ν ∼
natural choice of δ the µ→eγ branching ratio tudes leads to very different correlations among
LL
is in the 10−12 range, i.e. well within the reach of LFV processes [52, 57, 58] and their combined
the MEG experiment [54]. study can reveal the underlying mechanism of
Ratios of similar LFV decay rates, such as LFV.
B(τ → µγ)/B(µ → eγ), are much more easy to Finally, as recently pointed out in Ref. [59],
be predicted, being free from the overall normal- Higgs-mediatedLFVeffectsatlargetanβ canalso
ization uncertainty. These predictions depend es- induce visible deviations of lepton-flavour univer-
sentially only onthe flavourstructure of the LFV sality in charged-currentprocesses. If the slepton
couplings. The search for B(τ → µγ) is thus a sector contains sizable (non-minimal) sources of
key element in trying to determine the structure LFV, we could hope to observe deviations from
of flavour symmetry breaking in the lepton sec- the SM predictions in the B(P → ℓν)/B(P →
tor. In particular, B(τ → µγ)/B(µ → eγ) ranges ℓ′ν) ratios. The deviations can be O(1%) in
from to 102 in the case of a PMNS hierarchy, to B(K → eν)/B(K → µν) [59], and can reach