B, D and K decays

6.1 MB·English

by  G. Buchalla

B, D and K decays
∗
Conveners: G.Buchalla,T.K.Komatsubara, F.Muheim,L.Silvestrini
Sectioncoordinators: Newphysicsscenarios: A.J.Buras,S.Heinemeyer, G.Isidori, Y.Okada,
F.Parodi,L.Silvestrini
WeakdecaysofhadronsandQCD:G.Buchalla
Radiativepenguin decays: P.Gambino,A.Golutvin
Electroweakpenguin decays: J.Berryhill, Th.Feldmann
8
Neutrinomodes: Y.Grossman,T.Iijima
0
0 Veryraredecays: U.Nierste, M.Smizanska
2 UTanglesfromtreedecays: M.Bona,A.Soni, K.Trabelsi,G.Wilkinson

n B-mesonmixing: V.Lubicz,J.vanHunen
a
Hadronicb sandb dtransitions: M.Ciuchini,F.Muheim
J
→ →
Kaondecays: A.J.Buras,T.K.Komatsubara
1
1 Charmphysics: D.M.Asner,S.Fajfer
Prospectsforfuturefacilities: T.Hurth
]
h Assessments: S.Heinemeyer, F.Parodi,L.Silvestrini
p
Contributingauthors:
-
p M.Artuso1,D.M.Asner2,P.Ball3,E.Baracchini4,G.Bell5,M.Beneke6,J.Berryhill7,A.Bevan8,
e I.I.Bigi9,M.Blanke10,Ch.Bobeth11,M.Bona12,F.Borzumati13,T.Browder14,T.Buanes15,
h
[ G.Buchalla16,O.Buchmu¨ller17,A.J.Buras18,S.Burdin19,D.G.Cassel20,R.Cavanaugh21,

M.Ciuchini22,P.Colangelo23,G.Crosetti24,A.Dedes3,F.DeFazio23,S.Descotes-Genon25,
1
v J.Dickens26,Z.Dolezˇal27,S.Du¨rr28,U.Egede29,C.Eggel30,G.Eigen15,S.Fajfer31,Th.Feldmann32,
3 R.Ferrandes23,P.Gambino33,T.Gershon34,V.Gibson26,M.Giorgi35,V.V.Gligorov36,B.Golob37,
3
A.Golutvin38,Y.Grossman39,D.Guadagnoli18,U.Haisch40,M.Hazumi41,S.Heinemeyer42,
8
1 G.Hiller11,D.Hitlin43,T.Huber6,T.Hurth44,T.Iijima45,A.Ishikawa46,G.Isidori47,S.Ja¨ger17,
1. A.Khodjamirian32,T.K.Komatsubara41,P.Koppenburg29,T.Lagouri27,U.Langenegger48,
0 C.Lazzeroni26,A.Lenz49,V.Lubicz22,W.Lucha50,H.Mahlke20,D.Melikhov51,F.Mescia52,
8
M.Misiak53,F.Muheim54,M.Nakao41,J.Napolitano55,N.Nikitin56 U.Nierste5,K.Oide41,
0
: Y.Okada41,P.Paradisi18,F.Parodi57,M.Patel17,A.A.Petrov58,T.N.Pham59,M.Pierini17,
v
S.Playfer54,G.Polesello60,A.Policicchio24 A.Poschenrieder18,P.Raimondi52,S.Recksiegel18,
i
X P.Rˇezn´ıcˇek27,A.Robert61,S.Robertson62,J.L.Rosner63,G.Ruggiero17,A.Sarti52,O.Schneider64,
ar F.Schwab65,L.Silvestrini4,S.Simula22,S.Sivoklokov56,P.Slavich66,C.Smith67,M.Smizanska68,
A.Soni69,T.Speer40,P.Spradlin36,M.Spranger18,A.Starodumov48,B.Stech70,A.Stocchi71,
S.Stone1,C.Tarantino22,F.Teubert17,S.T’Jampens12,K.Toms56,K.Trabelsi41,S.Trine5,S.Uhlig18,
V.Vagnoni72,J.J.vanHunen64,G.Weiglein3,A.Weiler20,G.Wilkinson36,Y.Xie54,M.Yamauchi41,
G.Zhu73,J.Zupan31,R.Zwicky3.
1 Syracuse University, Syracuse, NY,USA
2 CarletonUniversity, Ottawa,Canada
3 DurhamUniversity, IPPP,Durham,UK
4 Universita` diRomaLaSapienza andINFN,Rome,Italy
5 Universita¨t Karlsruhe, Germany
6 RWTHAachen,Aachen,Germany
7 FermiNationalAccelerator Laboratory, Batavia,IL,USA
8 QueenMary,UniversityofLondon,UnitedKingdom
9 University ofNotreDame,NotreDame,IN,USA
10 TechnischeUniversita¨t Mu¨nchen,GarchingandMax-Planck-Institut fu¨rPhysik,Mu¨nchen,Germany
∗Report of WorkingGroup 2of theCERNWorkshop“FlavourintheeraoftheLHC”,Geneva, Switzerland, November
2005–March2007.
1
11 Institut fu¨rPhysik,Universita¨t Dortmund, Germany
12 LAPP,Universite´ deSavoie, IN2P3-CNRS,Annecy-le-Vieux, France
13 ICTP,Trieste,ItalyandNationalCentralUniversity, Taiwan
14 University ofHawaiiatManoa,Honolulu, HI,USA
15 University ofBergen,Norway
16 Ludwig-Maximilians-Universita¨t Mu¨nchen, Mu¨nchen,Germany
17 CERN,Geneva,Switzerland
18 TechnischeUniversita¨t Mu¨nchen,Garching, Germany
19 TheUniversityofLiverpool, Liverpool, UnitedKingdom
20 CornellUniversity, Ithaca, NY,USA
21 University ofFlorida, Gainesville, FL,USA
22 Universita` diRomaTreandINFN,Rome,Italy
23 INFNBari,Italy
24 Universita` diCalabriaandINFNCosenza,Italy
25 LPT,CNRS/Universite´ deParis-Sud 11,Orsay,France
26 University ofCambridge, Cambridge, UnitedKingdom
27 IPNP,CharlesUniversity inPrague,CzechRepublic
28 NIC,FZJu¨lichandDESYZeuthen,Ju¨lich,Germany
29 ImperialCollege, London,UnitedKingdom
30 ETH,Zu¨richandPSI,Villigen, Switzerland
31 Ljubljana UniversityandJozefStefanInstitute, Ljubljana, Slovenia
32 Universita¨t Siegen, Siegen, Germany
33 Universita` diTorinoandINFN,Torino,Italy
34 University ofWarwick,Coventry, UnitedKingdom
35 Universita` diPisa,SNSandINFN,Pisa,Italy
36 University ofOxford,Oxford,UnitedKingdom
37 University ofLjubljana, Slovenia
38 CERN,Geneva,Switzerland andITEP,Moscow,Russia
39 Technion, Haifa,Israel
40 Universita¨t Zu¨rich,Zu¨rich,Switzerland
41 KEKandGraduateUniversityforAdvancedStudies (Sokendai), Tsukuba, Japan
42 IFCA,Santander, Spain
43 CalTech,Pasadena, CA,USA
44 CERN,Geneva,Switzerland andSLAC,Stanford, CA,USA
45 NagoyaUniversity, Nagoya, Japan
46 SagaUniversity, Saga, Japan
47 SNSandINFN,PisaandINFN,LNF,Frascati,Italy
48 ETH,Zu¨rich,Switzerland
49 Universita¨t Regensburg, Regensburg, Germany
50 Institut fu¨rHochenergiephysik, O¨sterreichische AkademiederWissenschaften, Wien,Austria
51 Institut fu¨rHochenergiephysik, O¨sterreichische AkademiederWissenschaften, Wien,Austriaand
NuclearPhysicsInstitute, MoscowStateUniversity, Moscow,Russia
52 INFN,LNF,Frascati,Italy
53 CERN,Geneva,Switzerland andWarsawUniversity, Warsaw,Poland
54 University ofEdinburgh, Edinburgh, UnitedKingdom
55 Rensselaer PolytechnicInstitute, Troy,NY,USA
56 Skobeltsin Institute ofNuclearPhysics,LomonosovMoscowStateUniversity, Russia
57 Universita` diGenovaandINFN,Genova,Italy
58 WayneStateUniversity, Detroit,MI,USA
59 EcolePolytechnique, CNRS,Palaiseau, France
2
60 Universita` diPaviaandINFN,Pavia,Italy
61 Universite´ deClermont-Ferrand, Clermont-Ferrand, France
62 McGillUniversityandIPP,Canada
63 EnricoFermiInstitute, University ofChicago, Chicago, IL,USA
64 EcolePolytechnique Fe´de´raledeLausanne (EPFL),Lausanne, Switzerland
65 Universitat AutonomadeBarcelona,IFAE,Barcelona, Spain
66 CERN,Geneva,Switzerland andLAPTH,Annecy-le-Vieux, France
67 Universita¨t Bern,Bern,Switzerland
68 Lancaster University, Lancaster, UnitedKingdom
69 Brookhaven NationalLaboratory, Upton,NY,USA
70 Universita¨t Heidelberg, Heidelberg, Germany
71 LAL,IN2P3-CNRSandUniversite´ deParis-Sud, Orsay,France
72 Universita` diBolognaandINFN,Bologna, Italy
73 Universita¨t Hamburg,Hamburg,Germany
Abstract
The present report documents the results of Working Group 2: B, D and K
decays,oftheworkshoponFlavourintheEraoftheLHC,heldatCERNfrom
November2005throughMarch2007.
With the advent of the LHC, we will be able to probe New Physics (NP) up
to energy scales almost one order of magnitude larger than it has been possi-
ble with present accelerator facilities. While direct detection of new particles
will be the main avenue to establish the presence of NP at the LHC, indirect
searches will provide precious complementary information, since most prob-
ably it will not be possible to measure the full spectrum of new particles and
their couplings through direct production. In particular, precision measure-
mentsandcomputations intherealmofflavourphysics areexpected toplaya
keyroleinconstraining theunknownparameters oftheLagrangian ofanyNP
modelemerging fromdirectsearches attheLHC.
The aim of Working Group 2 was twofold: on one hand, to provide a coher-
ent, up-to-date picture of the status of flavour physics before the start of the
LHC; on the other hand, to initiate activities on the path towards integrating
information onNPfromhigh-p andflavourdata.
T
This report is organized as follows. In Sec. 1, we give an overview of NP
models, focusing on a few examples that have been discussed in some detail
during the workshop, with a short description of the available computational
tools for flavour observables in NP models. Sec. 2 contains a concise dis-
cussion of the main theoretical problem in flavour physics: the evaluation of
the relevant hadronic matrix elements for weak decays. Sec. 3 contains a de-
taileddiscussionofNPeffectsinasetofflavourobservablesthatweidentified
as “benchmark channels” for NP searches. The experimental prospects for
flavour physics atfuture facilities are discussed in Sec.4. Finally, Sec. 5con-
tains some assessments on the work done at the workshop and the prospects
forfuturedevelopments.
4
Contents
1 Newphysicsscenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Model-independent approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 SUSYmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Non-supersymmetric extensions oftheStandardModel . . . . . . . . . . . . . . . . . . 25
1.5 Toolsforflavourphysicsandbeyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 WeakdecaysofhadronsandQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Charmlesstwo-bodyB decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Light-cone QCDsumrules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 LatticeQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Newphysicsinbenchmark channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Radiativepenguin decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Electroweakpenguin decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Neutrinomodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4 Veryraredecays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5 UTanglesfromtreedecays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6 B-mesonmixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.7 Hadronicb sandb dtransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
→ →
3.8 Kaondecays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.9 Charmphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4 Prospectsforfuturefacilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.1 OnthephysicscaseofaSuperFlavourFactory . . . . . . . . . . . . . . . . . . . . . . 186
4.2 SuperB proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.3 Accelerator designofSuperKEKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.4 LHCbupgrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5 Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.1 New-physicspatterns andcorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.2 Correlations betweenFCNCprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.3 Connection tohigh-energy physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.4 Discrimination betweennewphysicsscenarios . . . . . . . . . . . . . . . . . . . . . . . 240
5
1 New physics scenarios
1.1 Overview
TheStandardModel(SM)ofelectroweak andstronginteractions describes withanimpressiveaccuracy
all experimental data on particle physics up to energies of the order of the electroweak scale. On the
other hand, weknow that the SMshould beviewed asan effective theory valid uptoascale Λ M ,
W
∼
since,amongmanyotherthings, theSMdoesnotcontainasuitablecandidate ofdarkmatteranditdoes
notaccountforgravitational interactions. ViewingtheSMasaneffectivetheory,however,posesaseries
oftheoretical questions. Firstofall, thequadratic sensitivity oftheelectroweak scale onthecutoff calls
for a low value of Λ, in order to avoid excessive fine tuning. Second, several of the higher dimensional
operators which appear in the SM effective Lagrangian violate the accidental symmetries of the SM.
Therefore,theircoefficientsmustbehighlysuppressedinordernottoclashwiththeexperimentaldata,in
particularintheflavoursector. Unlessadditionalsuppressionmechanismsarepresentinthefundamental
theory,acutoffaroundtheelectroweakscaleisthusphenomenologicallynotacceptablesinceitgenerates
higherdimensional operators withlargecoefficients.
Wearefacing aformidable task: formulating anatural extension oftheSMwithacutoff close to
the electroweak scale and with avery strong suppression of additional sources of flavour and CPviola-
tion. While the simplest supersymmetric extensions of the SM with minimal flavour and CP violation,
such asMinimalSupergravity (MSUGRA)models, seem tobethephenomenologically mostviable NP
options,itisfairtosaythatafullyconsistentmodelofSUSYbreakinghasnotbeenputforwardyet. On
the other hand, alternative solutions of the hierarchy problem based on extra dimensions have recently
becomeverypopular,althoughtheyhavenotyetbeentestedatthesamelevelofaccuracyastheMinimal
SupersymmetricStandardModel(MSSM).WaitingfortheLHCtodiscovernewparticlesandshedsome
light on these fundamental problems, we should consider a range of NP models as wide as possible, in
ordertobereadytointerprettheNPsignalsthatwillshowupinthenearfuture.
In the following paragraphs, we discuss how flavour and CP violation beyond the SM can be
analyzedongeneralgroundsinamodel-independentway. Wethenspecializetoafewpopularextensions
oftheSM,suchasSUSYandlittleHiggsmodels,andpresent theirmostrelevantaspectsinviewofour
subsequent discussion ofNPeffectsinflavourphysics.
6
1.2 Model-independentapproaches
1.2.1 Generalconsiderations
Inmostextensions oftheStandard Model(SM),thenewdegrees offreedom that modifythe ultraviolet
behavior of the theory appear only around or above the electroweak scale (v 174 GeV). As long as
≈
weareinterested inprocesses occurringbelowthisscale(suchasB,DandK decays),wecanintegrate
outthenewdegreesoffreedomanddescribethenew-physicseffects–infullgenerality– bymeansofan
EffectiveFieldTheory(EFT)approach. TheSMLagrangianbecomestherenormalizable partofamore
general local Lagrangian which includes an infinite tower of higher-dimensional operators, constructed
intermsofSMfieldsandsuppressed byinversepowersofascaleΛ > v.
NP
Thisgeneralbottom-upapproachallowsustoanalyseallrealisticextensionsoftheSMintermsof
alimitednumberofparameters(thecoefficientsofthehigher-dimensional operators). Thedisadvantage
ofthisstrategyisthatitdoesnotallowustoestablishcorrelationsofNewPhysics(NP)effectsatlowand
highenergies(thescaleΛ definesthecut-offoftheEFT).Thenumberofcorrelationsamongdifferent
NP
low-energy observables is also very limited, unless some restrictive assumptions about the structure of
theEFTareemployed.
ThegenericEFTapproachissomehowtheoppositeofthestandardtop-downstrategytowardsNP,
whereagiventheory–andaspecificsetofparameters–areemployedtoevaluatepossibledeviationsfrom
theSM.Thetop-down approach usually allowsustoestablish severalcorrelations, both atlowenergies
andbetweenlow-andhigh-energy observables. However,thepricetopayisthelossofgenerality. This
isquiteahighpricegivenourlimitedknowledge aboutthephysicsabovetheelectroweak scale.
An interesting compromise between these two extreme strategies is obtained by implementing
specific symmetry restrictions on the EFT. The extra constraints increase the number of correlations in
low-energy observables. The experimental tests of such correlations allow us to test/establish general
features of the NP model (possibly valid both at low and high energies). In particular, B, D and K
decaysareextremelyusefulindeterminingtheflavour-symmetrybreakingpatternoftheNPmodel. The
EFTapproaches based on the Minimal Flavour Violation (MFV)hypothesis and its variations (MFV at
largetanβ,n-MFV,...) haveexactlythisgoal.
InSect.1.2.2weillustratesomeofthemainconclusions aboutNPeffectsintheflavoursectorde-
rivedsofarwithingeneralEFTapproaches. InSect.1.2.3weanalyseinmoredetailtheMFVhypothesis,
discussing: i) the general formulation and the general consequences of this hypothesis; ii) the possible
strategies to verify or falsify the MFV assumption from low-energy data; iii) the implementation of the
MFVhypothesisinmoreexplicitbeyond-the-SMframeworks,suchastheMinimalSupersymmetricSM
(MSSM)orGrandUnifiedTheories(GUTs).
1.2.2 GenericEFTapproaches andtheflavourproblem
TheNPcontributionstothehigher-dimensionaloperatorsoftheEFTshouldnaturallyinducelargeeffects
in processes which are not mediated by tree-level SM amplitudes, such as meson-antimeson mixing
(∆F = 2 amplitudes) or flavour-changing neutral-current (FCNC) rare decays. Up to now there is no
evidence of deviations from the SM in these processes and this implies severe bounds on the effective
scale of various dimension-six operators. For instance, the good agreement between SM expectations
and experimental determinations of K0–K¯0 mixing leads to bounds above 104 TeV for the effective
scale of ∆S = 2 operators, i.e. well above the few TeV range suggested by a natural stabilization of
the electroweak-symmetry breaking mechanism. Similar bounds are obtained for the scale of operators
contributing tolepton-flavour violating (LFV)transitions intheleptonsector, suchasµ eγ.
→
The apparent contradiction between these two determinations of Λ is a manifestation of what in
manyspecific frameworks(supersymmetry, technicolour, etc.) goesunder thenameofflavour problem:
if we insist on the theoretical prejudice that new physics has to emerge in the TeV region, we have to
conclude that the new theory possesses a highly non-generic flavour structure. Interestingly enough,
7
ηη γ
11
00..55
V
ub
V
00 cb
--00..55
--11
--11 --00..55 00 00..55 11
ρρ
Fig.1: Constraintsontheρ¯–η¯planeusingtree-levelobservablesonly,fromRef.[7](seealsoRef.[8]).
thisstructurehasnotbeenclearlyidentifiedyet,mainlybecausetheSM(thelow-energylimitofthenew
theory),doesn’tpossessanexactflavoursymmetry. Withinamodel-independentapproach,weshouldtry
todeduce thisstructure fromdata, usingtheexperimental information onFCNCtransitions toconstrain
itsform.
1.2.2.1 Boundson∆F = 2operators
In most realistic NP models we can safely neglect NP effects in all cases where the corresponding ef-
fective operator is generated at the tree-level within the SM. This general assumption implies that the
experimental determination of γ and V via tree-level processes (see Fig. 1) is free from the con-
ub
| |
tamination of NP contributions. The comparison of the experimental data on meson-antimeson mixing
amplitudes (both magnitudes and phases) with the theoretical SM expectations (obtained by means of
thetree-level determination oftheCKMmatrix) allowstoderive someofthe moststringent constraints
onNPmodels.
In a wide class of beyond-the-SM scenarios we expect sizable and uncorrelated deviations from
the SM in the various ∆F = 2 amplitudes.1 As discussed by several authors [2–6], in this case NP
effects can be parameterized in terms of the shift induced in the B –B¯ mixing frequencies (q = d,s)
q q
andinthecorresponding CPVphases,
B Hfull B¯
hBq|HeSffM|B¯qi = CBqe2iφBq = rq2e2iθq , (1)
h q| eff | qi
andsimilarlyfortheneutralkaonsystem. Thetwoequivalentparameterizations [(C ,φ )or(r ,θ )]
Bq Bq q q
havebeenshowntofacilitate theinterpretation oftheresultsoftheUTfit[7]andCKMfitter[8]collabo-
rationsfortheB case,showninFig.2.
d
Themainconclusionsthatcanbedrawnformthepresentanalysesofnew-physicseffectsin∆F =
2amplitudes canbesummarizedasfollows:
– Inallthethreeaccessible short-distance amplitudes (K0–K¯0,B –B¯ ,andB –B¯ )themagnitude
d d s s
ofthenew-physicsamplitudecannotexceed,insize,theSMshort-distancecontribution. Thelatter
1AsdiscussedforinstanceinRef.[1],thereisarathergenerallimitwhereNPeffectsin∆F =2amplitudesareexpected
tobethedominantdeviationsfromtheSMintheflavoursector. Thishappensunderthefollowingtwogeneralassumptions:
i)theeffectivescaleofNPissubstantiallyhigherthantheelectroweakscale;ii)thedimensionlesseffectivecouplingsruling
∆F =2transitionscanbeexpressedasthesquareofthecorresponding∆F =1coupling,withoutextrasuppressionfactors.
8
1-CL
]]
oo[[BBdd8800 11..55 CfK i tM t e r 0.9
φφ 6600 BEAUTY 2006
0.8
4400
0.7
11
2200
d)d) 0.6
--220000 θθ (ra2 (ra2dd 00..55 00..45
--4400 0.3
--6600 00 0.2
--8800 0.1
00 11 22 33 44 55 66
CC --00..5500 00..55 11 11..55 22 22..55 33 33..55 0
BBdd rr22
dd
Fig.2: ConstraintsontheeffectiveparametersencodingNPeffectsintheB –B¯ mixingamplitude(magnitude
d d
andphase)obtainedbytheUTfit[7](left)andCKMfitter[8](right)collaborations.
MM 33..55 MM 33..55
SSdd SSss
AA AA
PP// 33 PP// 33
NNdd NNss
AA AA
22..55 22..55
22 22
11..55 11..55
11 11
00..55 00..55
00 00
00 2200 4400 6600 8800 110000 112200 114400 116600 118800 00 2200 4400 6600 8800 110000 112200 114400 116600 118800
φφNNPP[[oo]] φφNNPP[[oo]]
dd ss
Fig.3:Constraintsontheabsolutevalueandphase(normalizedtotheSM)ofthenewphysicsamplitudeinB –B¯
d d
andB –B¯ mixingfromref.[9].
s s
is suppressed both by the GIM mechanism and by the hierarchical structure of the CKM matrix
(V):
G2M2
∆F=2 F W (V∗V )2 M¯ (Q¯i γµQj)2 M (2)
ASM ∼ 2π2 ti tj h | L L | i
Therefore, new-physics models with TeV-scale flavoured degrees of freedom and (1) flavour-
O
mixing couplings are essentially ruled out. To quantify this statement, we report here the results
oftherecentanalysis ofref.[9]. Writing
Ck
∆F=2 ij M¯ (Q¯iΓkQj)2 M ,
ANP ∼ Λ2 h | | i
(3)
9
whereΓk isagenericDiracandcolourstructure(seeref.[9]fordetails), onehas2
2 105 TeV C4 1/2
× ×| 12|
Λ> 2 103 TeV C4 1/2
 × ×| 13|
 3×102 TeV×|C243|1/2
– AsclearlyshowninFig.3,intheB –B¯casethereisstillroomforanew-physicscontribution up
d d
to the SM one. However, this is possible only if the new-physics contribution is aligned in phase
with respect to the SM amplitude (φNP close to zero). Similar, but thighter, constraints hold also
d
forthenewphysicscontribution totheK0–K¯0 amplitude.
– Contrary to B –B¯ and K0–K¯0 amplitudes, at present there is only a very loose bound on the
d d
CPVphase oftheB –B¯ mixing amplitude. Thisleaves open the possibility ofobserving alarge
s s
(B J/Ψφ)atLHCb,whichwouldbeaclearsignalofphysicsbeyondtheSM.
CP s
A →
Aswewilldiscussinthefollowing, thefirsttwoitemslistedabovefindanaturalexplanation withinthe
so-called hypothesis ofMinimalFlavourViolation.
1.2.3 MinimalFlavourViolation
A very reasonable, although quite pessimistic, solution to the flavour problem is the so-called Minimal
Flavour Violation (MFV) hypothesis. Under this assumption, which will be formalized in detail below,
flavour-violatinginteractionsarelinkedtotheknownstructureofYukawacouplingsalsobeyondtheSM.
Asaresult,non-standardcontributionsinFCNCtransitionsturnouttobesuppressedtoalevelconsistent
with experiments even for Λ few TeV. One of the most interesting aspects of the MFVhypothesis is
∼
that it can naturally be implemented within the EFT approach to NP [10]. The effective theories based
onthis symmetry principle allow ustoestablish unambiguous correlations among NPeffects invarious
raredecays. Thesefalsifiablepredictions arethekeyingredients toidentifyinamodel-independent way
whicharetheirreducible sourcesofflavoursymmetrybreaking.
1.2.3.1 TheMFVhypothesis
The pure gauge sector of the SM is invariant under a large symmetry group of flavour transformations:
= U(1)5,where
SM q ℓ
G G ⊗G ⊗
= SU(3) SU(3) SU(3) , = SU(3) SU(3) (4)
Gq QL ⊗ UR ⊗ DR Gℓ LL ⊗ ER
andthreeofthefiveU(1)chargescanbeidentifiedwithbaryonnumber,leptonnumberandhypercharge
[11]. Thislargegroupand,particularly theSU(3)subgroups controlling flavour-changing transitions, is
explicitly brokenbytheYukawainteraction
= Q¯ Y D H +Q¯ Y U H +L¯ Y E H + h.c. (5)
Y L D R L U R c L E R
L
Since isalreadybrokenwithintheSM,itwouldnotbeconsistenttoimposeitasanexactsymmetry
SM
G
beyondtheSM:evenifabsentathetree-level, thebreakingof wouldreappear atthequantumlevel
SM
G
becauseoftheYukawainteraction. Themostrestrictivehypothesiswecanmaketoprotectinaconsistent
wayflavourmixinginthequarksector, istoassumethatY andY aretheonlysourcesof breaking
D U q
G
alsobeyondtheSM.Toimplementandinterpretthishypothesis inaconsistent way,wecanassumethat
is indeed a good symmetry, promoting Y to be non-dynamical fields (spurions) with non-trivial
q U,D
G
transformation properties underthissymmetry
Y (3,¯3,1) , Y (3,1,¯3) . (6)
U ∼ Gq D ∼ Gq
2ThechoiceΓ4 = PL⊗PR givesthemoststringentconstraints. Constraintsfromotheroperatorsareuptooneorderof
magnitudeweaker.
10