The hierarchy problem

(On the origin of the Higgs potential) Electroweak symmetry breaking (EWSB) in the SM is triggered by the Higgs VEV:

1 1 V (h)= µ2h2 + λh4 −2 4

λ µ2 = λv2 = 4M 2 104 GeV2 << M 2 1038 GeV2 g2 W ∼ P ∼

Why so different? Even worse, at the quantum level, scalar masses are extremely sensitive to heavy states

1 h h M 2 ∼ 16π2

mass of the particle in the loop Not the same situation for fermions or gauge bosons ➠ gauge symmetries can protect them No symmetry in the SM protects the Higgs mass

In general: mΨ¯ Ψ vs µ2 H 2 L R | | { { Not a singlet if Always a singlet ΨR transform: under phase transformations Ψ eiθΨ R → R (chiral symmetry)

2 Expected: µ heavier scale ~ MGUT , M P , M string ∼ ² ² ² ² ➥ This is the hierarchy problem Let me emphasize that is not a problem of consistency but of naturalness

Example:

Fine-tune system 4

R( Ω ) 0.14 0.12 0.1 0.08 0.06 -5 Ω 0.04 10 0.02

4.1 4.2 4.3 4.4 T(K)

FIG. 1 Data from Onnes’ pioneering works. The plot shows the electric resistance of the mercury vs. temperature.

therefore, due to the Maxwell equation 1 ∂B E = , (1.1) ∇ ∧ − c ∂t Analogythe magnetic with field isSuperconductivity frozen, whereas it is expelled. This implies that superconductivity will be destroyed by a critical magnetic field Hc such that H2(T ) f (T )+ c = f (T ) , (1.2) EWSB ⇔ Breaking of U(1)EM s 8π n

where fs,n(T ) are the densities of free energy in the the superconducting phase at zero magnetic field and the density Higgs Modelof free ⇔ energy GL in the Model normal phase. The behaviorh = of thee− criticale− magnetic field with temperature was found empirically to be parabolic (see Fig. 2) ￿ ￿ ￿ ￿

T 2 H (T ) H (0) 1 . (1.3) B c ≈ c − T Give the GL Model a ￿ ￿ c ￿ ￿ good description H (T) ______c Bc1 N phase of superconductors? H (0) c h =0 0.8 ￿ ￿ 0.6

0.4 SC phase 0.2 h =0

￿ ￿￿ ___T T T 0.2 0.4 0.6 0.8 T1 c 1.2 c

FIG. 2 The critical field vs. temperature. 4

4 R( Ω ) 0.14 R( Ω ) 0.12 0.14 0.1 0.12 0.08 0.1 0.06 0.08 -5 Ω 0.04 10 0.06 -5 Ω 0.020.04 10 0.02 4.1 4.2 4.3 4.4 T(K) 4.1 4.2 4.3 4.4 T(K) FIG. 1 Data from Onnes’ pioneering works. The plot shows the electric resistance of the mercury vs. temperature.

FIG. 1 Data from Onnes’ pioneering works. The plot shows the electric resistance of the mercury vs. temperature. therefore, due to the Maxwell equation therefore, due to the Maxwell equation 1 ∂B E = , (1.1) ∇ ∧ −1c∂B∂t E = , (1.1) Analogythe magnetic with field isSuperconductivity frozen, whereas it is expelled.∇ This∧ implies− c ∂t that superconductivity will be destroyed by a critical magneticthe magnetic field H fieldc such is frozen, that whereas it is expelled. This implies that superconductivity will be destroyed by a critical magnetic field H such that c H2(T ) f (T )+ c = f (T ) , (1.2) ⇔ EM s H28(πT ) n EWSB Breaking of U(1) f (T )+ c = f (T ) , (1.2) s 8π n where⇔fs,n(T ) are the densities of free energy in the the superconducting phase at zero magnetic field and the density Higgs Modelofwhere free energyfs,n GL(T in) are the Model the normal densities phase. of free The energy behaviorh = in the of thee− superconductingcriticale− magnetic phase field with at zero temperature magnetic field was and found the empirically density toof be free parabolic energy in (see the Fig. normal 2) phase. The￿ behavior￿ of￿ the critical￿ magnetic field with temperature was found empirically to be parabolic (see Fig. 2) T 2 H (T ) H (0) 1 2 . (1.3) B c ≈ c − TT Give the GL Model a Hc(T ) Hc(0) ￿1 ￿ c ￿ ￿. (1.3) ≈ ￿ − Tc ￿ good description ￿ ￿

H______c (T) H______c (T)Bc1 N phase of superconductors? H (0) 1 Hc (0) c h =0 0.80.8 ￿ ￿ NO, it only works close 0.60.6 to the critical line 0.40.4 SC phase

0.20.2 h =0 only there h is small and it ￿ ￿￿ makes sense￿ to￿ Taylor-expand ___T___T 0.2 0.40.4 0.60.6 0.80.8 1 1 1.21.2TcTc T the potential: Tc 2 2 4 VFIG.(FIG.h 2)= 2TheThe critical criticalm field fieldh vs. vs. temperature. temperature.+ λ h + | | | | ··· Possibilities that theorists envisage to tackle the Hierarchy Problem:

1) Supersymmetry: Protecting the Higgs mass by a symmetry 2) Composite Higgs: The Higgs is not elementary: As in superconductivity: h ~ ee or QCD: pions ~ qq- 3) Large extra dimensions: Gravity strong at the EW-scale: Λ ~ Mstring ~ TeV

➥ In all cases New Physics at ~TeV Strong motivation for the LHC ! Supersymmetry

Following notation and formulae of “A Supersymmetry Primer”, Stephen P. Martin (hep-ph/9709356) We want a symmetry to protect the Higgs mass:

Idea: Scalar Fermion symmetry trans. since fermion masses protected by chiral It exists, it is a Supersymmetry: symmetry Simplest case: 2 1 Ψ = Majorana fermion = ∂µΦ + i Ψ¯∂/Ψ L | | 2 Φ = Complex scalar Invariant under:

Φ Φ + δΦ δΦ ξ¯(1 γ5)Ψ → → − µ Ψ Ψ + δΨ δΨ i(1 γ5)γ ξ∂µΦ → → − Parameter of the trans. The scalar must be massless!! being a Majorana fermion 2 The systematic cancellation of the dangerous contributionsto∆mH can only be brought about by the type of conspiracy that is better known to physicists as a symmetry. Comparing eqs. (1.2) and (1.3) strongly suggests that the new symmetry ought to relatefermionsandbosons,becauseofthe 2 relative minus sign between fermion loop and boson loop contributions to ∆mH .(NotethatλS must be positive if the scalar potential is to be bounded from below.) If each of the quarks and leptons of the Standard Model is accompanied by two complex scalars with λ = λ 2,thentheΛ2 contributions of S | f | UV Figures 1.1a and 1.1b will neatly cancel [3]. Clearly, more restrictions on the theory will be necessary to ensure that this success persists to higher orders, so that, for example, the contributions in Figure 1.2 2 and eq. (1.4)The from systematic a very cancellation heavy fermion of the are dangerous canceled contribution by the twosto-loop∆mH ecanffects only of be some brought very about heavy by bosons. Fortunately,the type of conspiracy the cancellation that is better of all known such contribu to physiciststions as to a scalarsymmetry. masses Comparing is not only eqs. (1.2)possible, and (1.3) strongly suggests that the new symmetry ought to relatefermionsandbosons,becauseofthe but is actually unavoidable, once we merely assume that thereexistsasymmetryrelatingfermionsand relative minus sign between fermion loop and boson loop contributions to ∆m2 .(Notethatλ must bosons, called a supersymmetry. H S be positive if the scalar potential is to be bounded from below.) If each of the quarks and leptons of the A supersymmetryStandard Model transformation is accompanied turns by two a complex bosonic scalars state into with aλ fermionic= λ 2,thenthe state, andΛ2 vicecontributions versa. The of S | f | UV operator FiguresQ that 1.1a generates and 1.1b such will transformations neatly cancel [3]. mustClearly, be more an anticommuti restrictions onng the spinor, theory with will be necessary to ensure that this success persists to higher orders, so that, for example, the contributions in Figure 1.2 Q Boson = Fermion ,QFermion = Boson . (1.5) and eq. (1.4) from| a very! | heavy fermion! are canceled| by the! two| -loop! effects of some very heavy bosons. Fortunately, the cancellation of all such contributions to scalar masses is not only possible, Spinors arebut intrinsically is actually unavoidable, complex objects, onceSupersymmetry we merely so Q† assume(the hermitian that therAlgebraeexistsasymmetryrelatingfermionsand conjugate of Q)isalsoasymmetry generator.bosons, Because calledQ aandsupersymmetryQ† are fermionic. operators, they carry spin angular momentum 1/2, so it is clear that supersymmetryA supersymmetry must(Maximal transformation be a spacetime extension turns symmetry. a bosonic of Poincare state The into p inossible a afermionic QFT) forms state, for such and vicesymmetries versa. The in an interactingoperator quantumQ thatMinimal generates field theory SUSY such are transformations ( highlyN=1): restricted One must extra by be the angenerator anticommuti Haag-Lopuszanski-Sohnius Qng spinor, with extension of the Coleman-Mandula theorem [4]. For realistic theories that, like the Standard Model, have chiral Q Boson = Fermion ,QFermion = Boson . (1.5) fermions (i.e., fermions whose left-| and! right-handed| ! pieces transform| ! diff|erently! under the gauge group) and thus the possibility of parity-violating interactions,thistheoremimpliesthatthegeneratorsQ and Spinors are intrinsically complex objects, so Q† (the hermitian conjugate of Q)isalsoasymmetry Q† must satisfy an algebraSchematic of anticommutation form: and commutation relations with the schematic form generator. Because Q and Q† are fermionic[Q, Mµν operators,]=Q they carry spin angular momentum 1/2, so it is clear that supersymmetry must be a spacetime symmetry.µ The possible forms for such symmetries in Q, Q† = P , (1.6) an interacting quantum field theory{ are highly} restricted by the Haag-Lopuszanski-Sohnius extension of the Coleman-Mandula theorem [4].Q, For Q realistic= Q†,Q theories† =0that,, like the Standard Model, have chiral(1.7) fermions (i.e., fermions whose left- and{ µ right-handed} { µ piec}es transform differently under the gauge group) [P ,Q]=[P ,Q†]=0, (1.8) and thus the possibility of parity-violating interactions,thistheoremimpliesthatthegeneratorsQ and where P µQ†ismust the satisfy four-momentum an algebra of generator anticommutation of spacetime and commutation translations.relationsHere with we have the schematic ruthlessly form sup- pressed the spinor indices on Q and Q†;afterdevelopingsomenotationwewill,insection3.1,derµ ive Q commutes with P² andQ, Q any† =generatorP , of the gauge symmetries: (1.6) the precise version of eqs. (1.6)-(1.8) with indices{ restor} ed. In the meantime, we simply note that the µ Q, Q = Q†,Q† =0, (1.7) appearance of P on the right-handThe Fermion side ofand eq. Boson{ (1.6)µ } is have unsurprising,{ µ equal} masses since and it transforms charges under Lorentz [P ,Q]=[P ,Q†]=0, (1.8) boosts and rotations as a spin-1 object while Q and Q† on the left-hand side each transform as spin-1/2 objects. where P µ is the four-momentum generator of spacetime translations. Here we have ruthlessly sup- The single-particlepressed the spinor states indices of a on supersymmetricQ and Q†;afterdevelopingsomenotationwewill,insection3.1,der theory fall into irreducible representations of theive supersymmetrythe precise algebra, version called of eqs.supermultiplets (1.6)-(1.8) with.Eachsupermultipletcontainsbothfermionandboson indices restored. In the meantime, we simply note that the states, whichappearance are commonly of P µ on the known right-hand as superpartners side of eq. (1.6)of each is unsurprising, other. By since definition, it transforms if Ω underand Ω Lorentzare | ! | !! membersboosts of the and same rotations supermultiplet, as a spin-1 object then whileΩ Qisand proportionalQ† on the left-hand to some side combination each transform of Q as spin-1/2and Q | !! † operatorsobjects. acting on Ω ,uptoaspacetimetranslationorrotation.Thesquared-massoperator P 2 The single-particle| ! states of a supersymmetric theory fall into irreducible representations of− the commutes with the operators Q, Q†,andwithallspacetimerotationandtranslationoperators,so supersymmetry algebra, called supermultiplets.Eachsupermultipletcontainsbothfermionandboson it follows immediately that particles inhabiting the same irreducible supermultiplet must have equal states, which are commonly known as superpartners of each other. By definition, if Ω and Ω are 2 | ! | !! eigenvaluesmembers of P of,andthereforeequalmasses. the same supermultiplet, then Ω is proportional to some combination of Q and Q − | !! † The supersymmetryoperators acting generators on Ω ,uptoaspacetimetranslationorrotation.Thesquared-masQ, Q† also commute with the generators of gauge transformations.soperator P 2 | ! − Thereforecommutes particles with in the the same operators supermultipletQ, Q†,andwithallspacetimerotationandtranslationoperators must also be in the same representation of the gauge,so group, andit follows so must immediately have the same that particles electric charges, inhabiting weak the same isospin, irreducible and color supermultiplet degrees of freedom. must have equal Eacheigenvalues supermultiplet of P contains2,andthereforeequalmasses. an equal number of fermion and boson degrees of freedom. To prove − 2s this, considerThe the supersymmetry operator ( generators1) whereQ,s Qis† also the commute spin angular with the momentum. generators of By gauge the transformations. spin-statistics − theorem,Therefore this operator particles has in eigenvalue the same supermultiplet +1 acting on must a bosonic also bestatein the and same eigenvalue representation1actingona of the gauge group, and so must have the same electric charges, weak isospin, and color degrees of− freedom. Each supermultiplet contains an equal number of fermion and boson degrees of freedom. To prove this, consider the operator ( 1)2s where s is5 the spin angular momentum. By the spin-statistics − theorem, this operator has eigenvalue +1 acting on a bosonic state and eigenvalue 1actingona −

5 Minimal Supersymmetric SM (MSSM)

Imposing supersymmetry to the SM ➡ MSSM The spectrum is doubled: SM fermion ➡ New scalar (s-”...”) SM boson ➡ New majorana fermion (“ ...“-ino) ... but not yet realistic: The model has a quantum anomaly (due to the Higgsino) and the down-quarks and leptons are massless Extra Higgs needed ➡ Two Higgs doublets:

Hu :(1, 2, 1) ➞ give mass to the up quarks

Hd :(1, 2, 1) ➞ give mass to the down quarks − and leptons + two Higgsino doublets:

Hu :(1, 2, 1) H :(1, 2, 1) ￿d − ￿ MSSM Spectrum Names spin 0 spin 1/2 SU(3)C ,SU(2)L,U(1)Y 1 squarks, quarks Q (uL dL) (uL dL) ( 3, 2 , 6 ) 2 ( 3families) u u∗ u† ( 3, 1, ) × Squarks ! R! R − 3 1 d d∗ d† ( 3, 1, ) !R R 3 sleptons, leptons L (ν e ) (ν e ) ( 1, 2 , 1 ) Sleptons ! L L − 2 ( 3families) e e∗ e† ( 1, 1, 1) × ! R! R + 0 + 0 1 Higgs, higgsinos Hu (H H ) (H H ) ( 1, 2 , + ) u! u u u 2 0 0 Higgsinos 1 NamesH (HspinH 1/2−) (HspinH 1 −) SU(3)C ,SU( 1,(2)2L,,U(1)) Y d d d !d !d − 2 gluino, gluon g g ( 8, 1 , 0) Table 1.1: Chiral supermultiplets in the Minimal Supersymm! ! etric Standard Model. The spin-0 fields winos,Gauginos W bosons W W 0 W W 0 ( 1, 3 , 0) are complex scalars, and the spin-1/2 fields±! are left-handed± two-component Weyl fermions. 0 0 bino, B boson " B " B ( 1, 1 , 0) completely different reason: because of the structure of supersymmetric theories, only a Y =1/2Higgs chiral supermultipletTableparticles: 1.2: can Gauge have R-parity the supermultiplets Yukawa = 1 couplings! in the Minimal necess1) Superpart.ary Supersymme to give interact massestric Standard in to pairs charge Model. +2/3 up-type quarks (up, charm,superpartners: top), and only R-parity a Y = =1 -1/2HiggscanhavetheYukawacouplingsnecessarytogive2) Lightest superpart. stable − masses to charge 1/3 down-type quarks (down, strange, bottom) and to the chargedleptons.We by taking a− neutrino and a Higgs scalar to be superpartners, instead of putting them in separate will callsupermultiplets. the SU(2)L-doublet This would complex amount scalar to the fields proposal with thatY =1 the /Higgs2and bosonY = and1/ a2bythenames sneutrino should beHu theand − Hd,respectively.same particle.† The This weak attempt isospin played components a key role in ofsomeHu ofwith the firTst3 attempts=(1/2, to1 connect/2) have supersymmetry electric charges to + 0 − 1, 0 respectively,phenomenology and [5], are but denoted it is now (H knownu , Hu to). not Similarly, work. Even the ignorSU(2)ingL the-doublet anomaly complex cancellation scalar problemHd has mentioned above, many insoluble0 phenomenological problemswouldresult,includinglepton-number T3 =(1/2, 1/2) components (Hd , Hd−). The neutral scalar that corresponds to the physical Standard non-conservation− and a mass for at least one of the0 neutrinos0 in gross violation of experimental bounds. Model Higgs boson is in a linear combination of Hu and Hd ;wewilldiscussthisfurtherinsection7.1. The genericTherefore, nomenclature all of the superpartners for a spin-1/2 of superpartner Standard Model is partito appendcles are “-ino” really new to the particles, name of and the cannot Standard be identified with some other Standard Model state. Model particle, so the fermionic partners of the Higgs scalars are called higgsinos. They are denoted The vector bosons of the Standard Model clearly must reside ingaugesupermultiplets.Their+ by Hu,fermionicHd for the superpartnersSU(2)L-doublet are generically left-handed referred Weyl to spinor as gauginos. fields, The withSU weak(3) isospincolor gauge components interactionsHu , 0 0 C Hu andofH QCDd , H ared−. mediated by the gluon, whose spin-1/2 color-octet supersymmetric partner is the gluino. As ! ! ! Weusual, have now a tilde found is used all to of denote the chiral the supersymmetric supermultiplets partner of a minim ofaStandardModelstate,sothesymbolsal phenomenologically viable exten- ! ! ! sion offor the the Standard gluon and Model. gluino They are g and areg summarizedrespectively. in The Table electroweak 1.1, classified gauge symmetry accordingSU to(2) theirL U transfor-(1)Y is + 0 0 +× 0 mationassociated properties with under spin-1 the gauge Standard bosons ModelW ,W gauge,W− groupand BSU,withspin-1/2superpartners(3)C SU(2)L U(1)Y ,whichcombinesW , W , W − 0 × ×0 0 uL,dL and Bν,e,calledL degreeswinos ofand freedombino.Afterelectroweaksymmetrybreaking,the into! SU(2)L doublets. Here we follow aW standard, B gauge convention, eigenstates that all chiralmix supermultiplets to give mass eigenstates are definedZ0 and inγ terms.Thecorrespondinggauginomixturesof of left-handed Weyl spinors, so thatW 0 and the"Bconjugates0 are" called" of ! 0 the right-handedzino (Z )andphotino( quarks and leptonsγ); if supersymmetry (and their superpartne were unbroken,rs) appear they would in Table be mass 1.1. eigens This protocoltates with for " ! definingmasses chiralm supermultipletsZ and 0. Table turns 1.2 summarizes out to be the very gauge useful supermultiplets for constructing of a supersymmetric minimal supersymmetric Lagrangi- extension! of the Standard! Model. ans, as we will see in section 3. It is also useful to have a symbol for each of the chiral supermultiplets The chiral and gauge supermultiplets in Tables 1.1 and 1.2 make up the particle content of the as a whole;Minimal these Supersymmetric are indicated Standard in the second Model (MSSM). column of The Table most 1 obv.1.ious Thus, and for interesting example, featureQ stands of this for the SUtheory(2)L-doublet is that chiralnone of supermultiplet the superpartners containing of the StandarduL,uL (with Modelparticleshasbeendiscoveredasof weak isospin component T3 =1/2), and dL,dthisL writing.(with T3 If= supersymmetry1/2), while wereu stands unbroken, for the thenSU there(2)L wo-singletuld have supermultiplet to be selectrons containinge and e uwith∗ ,u† . − L R R R There aremasses three exactly families equal for to eachme of=0 the.511 quark... MeV. and A lepton similar! supe statementrmultiplets, applies Table to each 1.1 of lists the other the first-family sleptons ! representatives.and squarks, A familyand there index wouldi =1 also, 2 have, 3canbea to be a masslessffixed to gluino the chiraland photino. supermultiplet These particles names! would (Q!i,!u havei,...) when needed,been extraordinarily for example easy (e1, e to2, detecte3)=( longe, µ, ago.τ). The Clearly, bar ther on efore,u, d, esupersymmetryfields is part ofis a the broken name, symmetry and does not denotein the any vacuum kind state of conjugation. chosen by Nature. An important clue as to the nature of supersymmetry0 breaking0 can be obtained by returning The Higgs chiral supermultiplet Hd (containing H , H−, H , H−)hasexactlythesameStandard to the motivation provided by the hierarchy problem.d d Supersd ymmetryd forced us to introduce two Model gauge quantum numbers as the left-handed sleptons and leptons Li,forexample(ν, eL, ν, complex scalar fields for each Standard Model Dirac fermion,!which! is just what is needed to enable a eL). Naively, one might therefore suppose that we2 could have been more economical in our assignment cancellation of the quadratically divergent (ΛUV)piecesofeqs.(1.2)and(1.3).Thissortofcancellation ! ! 2 † also requires that the associated dimensionless couplings should be related (for example λS = λ ). Other notations in the literature have H1,H2 or H,H instead of Hu,Hd.Thenotationusedherehasthevirtueof| f | making itThe easy necessary to remember relationships which Higgs between VEVs gives couplings masses indeed to which occu typerinunbrokensupersymmetry,aswewill of quarks. see in section 3. In fact, unbroken supersymmetry guaranteesthatthequadraticdivergencesinscalar squared masses must vanish to all orders in perturbation theory.‡ Now, if broken supersymmetry is still to provide a solution to the hierarchy problem even in the presence of supersymmetry breaking, then 8 ‡Asimplewaytounderstandthisistorecallthatunbrokensupersymmetry requires the degeneracy of scalar and fermion masses. Radiative corrections to fermion masses areknowntodivergeatmostlogarithmicallyinanyrenormal- izable field theory, so the same must be true for scalar masses in unbroken supersymmetry.

9

in the second line of eq. (3.72)]. One can think of this as the “

Figure 3.3g we have the coupling of gaugino a to chiral a fermi

in Feynman a diagram istraditionally drawn as solid a fermio

part of the Lagrangian. Figure 3.3c shows the coupling of gaa

form of the covariant derivative; they come from eqs. (3.59)

between gauge bosons and fermion and scalar fields that must o

are necessary only for consistent loop amplitudes.) Figure

gauge boson vertices of the Standard Model. (We do not show th

in eq. (3.57). In the MSSM these are exactly the same as the wel

MSSM. Figures 3.3a and 3.3b are the interactions of gauge bos

when the gauge group isnon-Abelian, for example for

the scalar squared-mass term as an insertion in the propagat

for scalara propagator in theorya with exact supersymmetry Figure 3.3 shows the gauge interactions supersymmetric in a

fermion propagator; note the directions of the arrows in Fig

M

in the theory are constructed in the usual way using the fermi

ik

∗

as indicated by the second and third terms in eq. (3.50). The p

M

which are entirely determined by the superpotential mass pa

with coupling dimensions of [mass] and [mass]

kj

.Thefermionmassterms

corrections to scalar masses, as discussed in the Introduct

tion of Figure 3.1c is exactly of the special type needed to ca Figure 3.2 shows the only interactions corresponding to ren

in the second line of eq. (3.72)]. One can think of this as the “ 28

i i Figure 3.3g we have the coupling of gauginoa to chirala fermi in Feynman a diagram is traditionally drawn as solid a fermio

i i part of the Lagrangian. Figure 3.3c shows the coupling of aga

form of the covariant derivative; they come from eqs. (3.59)

between gauge bosons and fermion and scalar fields that must o i are necessary only for consistent loopj amplitudes.) Figurei j j i

i j kj ij kj j gauge boson vertices of the Standard Model.i (We do not show th

j j in eq. (3.57). In the MSSM these are exactly the same as the wel k k MSSM. Figures 3.3a and 3.3b are the interactions of gauge bos

when the gauge group isnon-Abelian, for example for

M (a) (b) (c) (d) (e) (f)

Figure 3.3: Supersymmetric gauge interaction vertices. supersymmetrization” of Figure 3.3e or M

the scalar squared-mass term as an insertion in the propagat

(a) (a) (b) (c) (d) (e)

ij

the conjugate interaction

on and complexa scalar [the first term ij

∗ for scalara propagator in theorya with exact supersymmetry

Figure 3.3 shows the gauge interactions supersymmetricin a

nl i n es u p e r i m p o3 s e do naw a v yl i n e .I n jkn and Figure 3.2: Supersymmetric dimensionful couplings: (a) (s

,a n d( e )s c a l a rs q u a r e d - m a s st e r m

ugino to gaugea boson; the gaugino line

Figure 3.2: Supersymmetricfermion dimensionful propagator; note the directions of the couplings: arrows in Fig (a) (scalar) interaction vertex M y and (b)

3 s3 . 3 c , d , e ,jkn fa r ej u s tt h es t a n d a r di n t e r a c t i o n s in∗

and (3.65)-(3.67) inserted in the kinetic

Figure 3.2: Supersymmetric dimensionful couplings: (a) (scalar) interactionM in vertex Min∗ y and (b) ij

in the theory are constructed in the usual way using the fermi

i i ik

the conjugate interaction ∗ M y∗ ,(c)fermionmasstermM and (d) conjugate fermion mass term in M ij

jkn ccur in any gauge theory because of the

as indicated by the second and third terms in eq. (3.50). The p SU j

the conjugate interaction M yjkn∗ ,(c)fermionmasstermM and (d)M conjugate fermion mass term ij which are entirely determined by the superpotential mass pa kj

TypeM ∗ ,and(e)scalarsquared-masstermof interactions M ∗ M .

(a) kj

ij with coupling dimensions of [mass] and [mass] kj ik (3)

M ∗ ,and(e)scalarsquared-masstermM ∗ M . each lead chirality-changing toa insertion inthe ij ik .Thefermionmassterms

i i j i i j j i

C

ei n t e r a c t i o n so fg h o s tfi e l d s ,w h i c h 2 corrections to scalar masses, as discussed in the Introduct

l-known QCD gluon and electroweak .F i r s t ,t h e r ea r e( s c a l a r )

j k j k ons, which derive from the first term tion of Figure 3.1c is exactly of the special type needed to ca Figure 3.2 shows the only interactions corresponding to ren

color and

Getting them from “supersymmetrization”:or, the arrow direction ispreserved. i

(g) ure 3.2c,d. There is no such arrow-reversal

j ;a sd e p i c t e dij nF ij g u r e3 . 2 e ,i fo n et r e a t s

(b) i i i

(a) (b) (c) (d) (e) theory. Figures 3.3a,b,c occur only

k j k0 j Hk0 0 0 0 0 H u H H H H 28 i i i u i u u u u

Figure 3.2: Supersymmetric dimensionful couplings:(a) (a) (scalar)3 interaction(b) vertex M(c) yjkn and (b) (d) (e)

∗ on mass

rametersin

SU

M ! ! ! ! in ij ropagators of the fermions and scalars

the conjugate interaction M y ,(c)fermionmasstermM and (d) conjugateion [compare Figure 1.1, fermion and eq. (1.11)]. mass term3 jkn j ∗

jkn Figure 3.2: Supersymmetric dimensionful couplings: (a) (scalar) interaction vertex Min∗ y and (b) in (2)

kj tL tR† tL tL ormalizabletR† j and supersymmetrictR† t verticesL tLj tR† j tR∗ j tL j tR∗ji y i in i i ij i i (b) ncel the quadratic divergences in quantum

M ,and(e)scalarsquared-masstermjkn M Mthe. conjugate interaction M y ,(c)fermionmasstermM and (d) conjugate fermion mass term

ij∗ ∗ ∗ jkn∗ (a) (b) (c) (d) (e) j jik j (a) j (b) (c) (d) L (e)

k k k k kj

M

M ,and(e)scalarsquared-masstermM M . weak isospin inthe

,(c)fermionmassterm ∗ ∗

ij (a) !(a) ik(b) M ! (b) (c) ! (c) !

ij

(h)

M i ij (c)

(a) (a) (b) (b) (c) (c) (d) (d) (e)M (e)

supersymmetrization” of Figure 3.3e or

Figure 5.1: The top-quarkFigure 5.1: Yukawa The couplingtop-quark (a) Yukawa and its coupling “supersymmetrizations” (a)and scalar squared and mass its “super (b),symmetrizations” (c), all of (b), (c), all of

and Yukawa couplings ik

∗ gauginos 3

on and complexa scalar [the first term (f)

Figure 3.3: Supersymmetric gauge interaction vertices.

ij k

nl i n es u p e r i m p o s e do naw a v yl i n e .I n

couplings in Figure 3.2a,b,

M strength y .

t M strength yt. 3 3 jkn jkn

(a) ugino to agauge boson; the gaugino line

and

Figure 3.2: SupersymmetricFigure 3.2: Supersymmetric dimensionful couplings: dimensionful (a) (s couplings:calar) interaction (a) (scalar) vertexinteractionM y vertexand (b)M ∗ y and (b) s3 . 3 c , d , e , fa r ej u s tt h es t a n d a r di n t e r a c t i o n s the conjugate interaction ∗ ij

in in and (3.65)-(3.67) inserted in the kinetic ∗ kj

Figure 3.2: Supersymmetric dimensionful couplings: (a) (s

,a nin d( e )s c a l a rs q uin a r e d - m a s st e rij m ij .

the conjugate interactionthe conjugateM y interaction∗ ,(c)fermionmasstermM y∗ ,(c)fermionmasstermM and (d) conjugateM and fermion (d) conjugate mass term fermion mass term ccur in any gauge theory because of the jkn M jkn SU

space. All of the gaugespace. [SU All(3) ofC thecolor gauge and SU [SU(2)(3)LCweakcolor isospin] and SU and(2)L familyweak indices isospin] in and eq. family (5.1) are indices in eq. (5.1) are

kj kj ij

i M ,and(e)scalarsquared-masstermM M . αβ (3)

Mij∗ ,and(e)scalarsquared-masstermsuppressed.j ij∗ The “µ term”, as itM isik∗ traditionallyM . called,ik∗ can be written out as µ(H ) (H ) ! ,where αβ

suppressed.(a) The “µ(b)term”, as it is traditionally(c)slepton, squark called,(d) can beu writtenα d β out(e) as µslepton,(Hu)eachα( leadH chirality-changing to a dsquark)β! ,where insertion inthe

(a) (c) (b) (c) (d) (e)

(f) (g) (h) (i) ei n t e r a c t i o n so fg h o s tfi e l d s ,w h i c h

(f) (g) (h) C (i) (a)

αβ αβ l-known QCD gluon and electroweak

2

! is used to tie together SU(2) weak isospin indices α, β =1, 2inagauge-invariantway.Likewise, ons, which derive from the first term

! is used toL tie together SU(2)L weak isospin indices α, β =1, 2inagauge-invariantway.Likewise,.F i r s t ,t h e r ea r e( s c a l a r )

color and

ia j iaαβ j αβ or, the arrow direction is preserved.

the term uyuQH (d)can be written out as u (yu) Q (H ) ! ,wherei =1, 2, 3 is a family index,

u the term uy QH can be writteni jα outa asu βu (y ) Q (H ) ! ,wherei =1, 2, 3 is a familyure 3.2c,d. There index, is no such arrow-reversal

Figureu 3.3:u Supersymmetric(i) gauge interactionu i jαa vertices.u β

Figure 3.3: Supersymmetric gauge interaction;a sd e p i c tvertices. e di nF i g u r e3 . 2 e ,i fo n et r e a t s

(g)

theory. Figures 3.3a,b,c occur only

M i

and a =1, 2, 3isacolorindexwhichislowered(raised)intheand a =1, 2, 3isacolorindexwhichislowered(raised)inthe(b) 3 (3)representationof3 (3SU)representationof(3)C . SU(3)C . calar)

y

k

ij The µ term in eq. (5.1)The µ isterm the supersymmetric in eq. (5.1) is the version supersymmetric of the Higgs version boson mass of the in Higgs the Standard boson mass in the Standard

ijk on mass and (d) conjugate fermion mass term

j SU

tionModel. of ItFigure is unique, 3.1c because is exactly terms of theH H specialor H typeH are needed forbidden to ca inncel the superpotential,the quadratic divergenceswhich must be inrameters quantum

∗ u ∗ ,

Model. It is unique,u becausedtion termsd ofH Figureu∗Hu or 3.1cHd∗H isd are exactly forbidden of the in special the superpotential, type neededropagators which to of the fermions ca mustncel and scalars be the quadratic divergences in quantum

ion [compare Figure 1.1, and eq. (1.11)].

3

M (2)

interaction vertexcorrectionsanalytic in the to chiral scalar superfields masses, as (or discussed equivalently in the in the Introduct scalar fields)ion [compare treated as Figure complex 1.1, variables, and eq. as (1.11)].

analytic in(f) the chiral superfields(g)corrections (or equivalently to scalar(h) in masses, the scalar as fields) discussed(i) treated inormalizable as the complex and Introduct supersymmetric variables, verticesion [compare as Figure 1.1, and eq. (1.11)].

j

L ncel the quadratic divergences in quantum

shown in section 3.2. We canin also see from the form of eq. (5.1)gauginoswhy both H and H are needed in order i

Figure 3.2 showsshown the in onlyy section interactions 3.2. We can corresponding also see from theto ren formormalizable of eq.u (5.1)d andwhy supersymmetric both Hu and Hd are vertices needed in order (b) jkn

weak isospin inthe

(a) (b) (c) (d) (e) M

(f) (a) (g) (b) (h)∗ Figure 3.3: Supersymmetric(c) (i)Figure gauge 3.2 shows interaction(d) the vertices. only interactions(e) corresponding to renormalizable and supersymmetric vertices

to give Yukawa couplings, and thus masses, to all of2 the quarksandleptons.Sincethesuperpotential3 M

(d) with coupling dimensionsto give Yukawa of [mass] couplings, and [mass] and thus.First,thereare(scalar) masses, to all of the quarksandleptons.Sincethesuperpotentialcouplings in Figure2 3.2a,b, 3 ij

,(c)fermionmassterm

(e) with coupling dimensions of [mass] and [mass] .First,thereare(scalar) couplings in Figure 3.2a,b,

ij

(h)

must be analytic, the uQHu Yukawa terms cannot be replaced by somethingHiggs likeij uQH∗.Similarly, ijk and scalar squared mass

must be analytic, the uQH Yukawa terms(c) cannot be replaced byd something like uQH .Similarly, Figure 3.3: Supersymmetric gauge interaction vertices. u and∗ Yukawa couplings

which are entirely determined byM the superpotential mass parameters M and Yukawa couplings yd , ij ijk

3

i tion of Figure 3.1cHiggs is exactly of the specialwhich type needed are entirely to cancel determined the quadratic by divergences the superpotential in quantum mass parameters M and Yukawa couplings y , couplings in Figure 3.2a,b, the dQH and eLH terms cannot beik replaced by something like dQH and eLH .Theanalogous

as indicatedd by thethed seconddQHd andand thirdeLH∗ d termsterms cannotin eq. (3.50). be replaced The by propagators somethingu∗ of likeu∗ thedQH fermionsu∗ and eLH andu∗.Theanalogous scalars

M correctionsk to scalar masses, as discussedas in indicated the Introduct byion the [compare second Figure and 1.1, third and terms eq. (1.11)]. in eq. (3.50). The propagators of the fermions and scalars Yukawa couplings wouldYukawa be allowed couplings in woulda general be non-supersymallowed in agauginos generalmetric non-supersym two Higgs doubletmetricHiggsinoij model, two Higgs but are doublet model, but are

tion of Figure 3.1c is exactly of the specialin the type theory neededFigure are 3.2constructed to shows cancel the the only in quadratic the interactionskj usual divergences way corresponding using the in to quantum ren fermiormalizableon mass andM supersymmetricand scalar vertices squared mass ij forbidden by the structure of supersymmetry.. So we need both H and H ,evenwithoutinvokingthe kj forbidden by the structureij ofin supersymmetry. the2 theory areu So constructed we needd 3 both inHu theand usualHd,evenwithoutinvokingthe way using the fermion mass M and scalar squared mass j

corrections to scalar masses, as discussedMargumentMik∗ M in.Thefermionmassterms thewith based Introduct coupling on anomaly dimensionsion [compare cancellation of [mass] FigureM mentioned andand 1.1, [mass]M and inij.First,thereare(scalar)kj theeach eq. Int (1.11)].roduction. lead to a chirality-changingcouplings in Figureij insertion 3.2a,b, in the in argument based on anomalyM cancellationM .Thefermionmassterms mentioned in theij Introduction.M and Mijk each lead to a chirality-changing insertion in the ∗ ∗ ij

fermion propagator;which are entirelyi note determined the directions by the of superpotential theik arrows mass in Fig paurerameters 3.2c,d.M Thereand Yukawa is no couplings such arrow-reversaly ,

Figure 3.2 shows the only interactionsThey corresponding Yukawa matrices toThe ren determine Yukawaormalizable the matrices current and determine supersymmetric masses and the CKM current vertices mix massesing angles and ofCKM the ordinarymixing angles quarks of the ordinary quarks

(c) fermion propagator; note the directions of the arrows in Figure 3.2c,d. There is no such arrow-reversal jkn as indicated by the second and third terms in eq. (3.50). The propagators of the fermions and scalars j (f) (g) (h) (i)

forand a leptons, scalar(f)2 after propagator theand neutral leptons, in a scalar theory after(g) components the with3 neutral exact of scalarH supersymmetryu and components(h)Hd get VEVs.;asdepictedinFigure3.2e,ifonetreats of H Sinceand H theij(i)get top VEVs. quark, Sincebottom the top quark, bottom

with coupling dimensions of [mass] and [mass] .First,thereare(scalar) couplings in Figure 3.2a,b,(d) u d in the theory are constructed in the usualfor way a scalar using the propagator fermion mass inM a theoryand scalar with(i) squared exact mass supersymmetry;asdepictedinFigure3.2e,ifonetreats and (b)

which are entirely determined by thethequark superpotential(e) scalar andM tau squared-massM leptonkj.Thefermionmassterms massquark are pa the and termrameters heaviest tau as lepton anM fermions insertionij areandM theij Yukawa inand heaviestin the theM Standar couplings propagateach fermions leaddModel,itisoftenusefultomakean toor,y inijk a the chirality-changing, Standar arrow directiondModel,itisoftenusefultomakean insertion is preserved. in the

ik∗ M ij

Figure 3.3: Supersymmetricthe scalar squared-mass gauge interaction term as an vertices. insertion in the propagator, they arrow direction is preserved.

approximationFigure 3.3Figure that shows only 3.3: the the Supersymmetric gauge (3, 3) family interactionscalar) components gauge in a of interaction supersymmetric each of yu, vertices.yd andtheory.ye are Figures important: 3.3a,b,c occur only

fermion propagator;approximation note the that directions only the of (3 the, 3) arrows family in components Figure 3.2c,d. of There each is of noyu such, yd arrow-reversaland ye are important: ijk as indicated by the second and third terms in eq. (3.50). The propagators ofij the fermions and scalars

Figure 3.3 shows the gauge interactions in a supersymmetric theory., Figures 3.3a,b,c occur only

for a scalar propagator in a theory withij and (d)exact conjugate fermion supersymmetry mass term ;asdepictedinFigure3.2e,ifonetreats when the gauge group is non-Abelian,j for example for SU(3)C color and SU(2)L weak isospin in the

in the theory are constructed in the usual way using the00 fermi 0on mass M00and00 0 scalar 0 squared00 mass00 0 0 00 0 3 when the gauge group is non-Abelian, for example for SU(3) color and SU(2) weak isospin in the

kj ij the scalar squared-mass term as an insertion in the propagator, the arrow direction is preserved. C L MSSM.tion Figures ofy Figure 3.3a00 3.1c and is 0 3.3b exactly, arey theof the interactionsinteraction00 special vertex 0 type of, gauge neededy bos to00ons, cancel which 0 the. derive quadratic from divergences(5.2) the first term in quantum

Mik∗ M .Thefermionmasstermstion of FigureM 3.1cand is exactlyMFigureiju each of 3.3 the lead shows special to the ay gauge chirality-changingu typed interactions00 needed 0 to in, caa insertion supersymmetricncelyd thee 00 in quadratic thetheory. 0 , divergences Figuresye 3.3a,b,c00 in occur quantum 0 only. (5.2)

i ≈   ≈  ≈  MSSM.  Figures≈ ≈ 3.3a and 3.3b are≈ the interactions of gauge bosons, which derive from the first term in eq.corrections (3.57). In the to00 scalar MSSMyt masses, thesei are as exactly discussed00 theyb in same the as Introduct the wel00l-knownion [compareyτ QCD gluon Figure and 1.1, electroweak and eq. (1.11)].

fermion propagator; notecorrections the directions to scalar of thewhen masses, arrows the gauge in as Fig discussed groupure 3.2c,d. is non-Abelian, in the There00 Introduct is fory not example suchion arrow-reversal for [compareSU(3)00C color Figureyb and SU 1.1,(2) andL weak eq.00 isospin (1.11)].y inτ the   (d) in eq. (3.57). In the MSSM these are exactly the same as the well-known QCD gluon and electroweak

for a scalar propagator in a theory withgauge exact bosonFigureMSSM. supersymmetry vertices 3.2Figures shows of 3.3a the;asdepictedinFigure3.2e,ifonetreats the and Standard only 3.3b are interactions Model. the interactions (We corresponding do of not gauge show bosons, to th reneinteractionsofghostfields,which whichormalizable(e) derive from and the first supersymmetric term vertices Figure 3.2In shows this limit, the only only theIn interactions this third limit, family only corresponding and the Higgs thirdgauge fields family to contribu boson ren andormalizable Higgs verticeste to fieldsthe of MSSM and contribu the supersymmetric Standard superpotential.te to the Model. MSSM It vertices is (We superpotential. do not show It is theinteractionsofghostfields,which the scalar squared-mass term as anare insertion necessarywithin in coupling eq. the only (3.57). propagat for dimensions In consistent theor, MSSM the arrow theseof loop [mass] are amplitudes.) direction exactly and the [mass] is same preserved. Figure2.First,thereare(scalar) as thes3.3c,d,e,farejustthestandardinteractions well-known QCD gluon and3 couplings electroweak in Figure 3.2a,b, with couplinginstructive dimensions to write ofinstructive [mass] the superpotential and to [mass] write in2 the.First,thereare(scalar) terms superpotential of the sepa inrate termsSU(2) of3L thecouplingsweak sepa isospinrate inSU components Figure(2) weak 3.2a,b, isospin components between gaugegauge boson bosons vertices and fermion of the+ Standard0 and scalar Model.are0 fields necessary (We that do not must only show o for thccureinteractionsofghostfields,which consistent in any gauge loop theoryij amplitudes.)L because Figure of the s3.3c,d,e,farejustthestandardinteractionsijk Figure 3.3 shows the gauge interactions[Q3 =(whichtb), inL a are3 =( supersymmetric entirelyντ τ), Hu determined=(Hutheory.Hu), H byd Figures=( theH superpotentiald H 3.3a,b,cd−+), u03 = occurt, d3 mass= onlyb0, ije3 pa=rametersτ], so: M and Yukawaijk couplings y , which are entirely determinedare necessary[Q by3 only=( thetb for), superpotential consistentL3 =(ντ loopτ), H amplitudes.)ubetween mass=(Hu paHu gauge Figurerameters), Hds3.3c,d,e,farejustthestandardinteractions=( bosonsHMd Hd and−and), u fermion3 Yukawa= t, d3 = and couplingsb, e scalar3 = τ], fields so:y , that must occur in any gauge theory because of the

when the gauge group is non-Abelian,form foras of example indicated the covariant for bySU the derivative;(3) secondcolor andthey and thirdSU come(2) terms fromweak eqs. in isospin eq. (3.59) (3.50). inand the The (3.65)-(3.67) propagators inserted of the in fermions the kinetic and scalars

C L M between gauge bosons and0 fermionj+ and scalar fields that0 must occur in any gauge0 theory because of the

as indicated by the secondWMSSM and thirdyt( termsttH intbH eq.) (3.50).yformb(btH− of The0 thebbH propagators covariant+) yτ (τντ derivative;H of− theττH0 fermions) they come and scalars from0 eqs. (3.59) and (3.65)-(3.67) inserted in the kinetic part of the Lagrangian. Figureu 3.3c showsu the couplingd in ofd a gauginod to a gauged boson;ij the gaugino line

MSSM. Figures 3.3a and 3.3b are the interactionsin theform theory of of the gauge covariant are≈ constructed bos derivative;ons,W− whichMSSM in they the derive− comeyt usual(ttH from fromu −∗ way eqs.tbH the (3.59) usingu first)− yand termb the(btH (3.65)-(3.67) fermid− −bbHond mass) insertedyτ (Mτν inτ H theandd− kineticττ scalarHd ) squared mass + 0≈ 0 −y − ij− − −

in the theoryin are a Feynman constructedkj diagram in the is+ traditionallyµ usual(H H way− H using drawnH part). the asij of a+ fermi the solid Lagrangian.on fermio0 mass0 nlinesuperimposedonawavyline.InM Figureand scalar 3.3c shows squared the(5.3) coupling mass of a gaugino to a gauge boson; the gaugino line u d u d jkn

in eq. (3.57). In the MSSM these are exactlyM Mpart the of same.Thefermionmassterms the Lagrangian. as the well-known Figure 3.3c QCD shows+M gluon theµ(H couplinguand andHd− M electroweak ofH auHeach gadugino). lead to a gauge to a boson; chirality-changing the gaugino line insertion in(5.3) the − j

M M kj.Thefermionmasstermsik∗ M ij and M each lead to− ij a chirality-changing insertion in the

ik∗ Figure 3.3gin a we Feynman have the diagram coupling is traditionally of a gauginoij drawnin a to Feynmanas a a chiral solidand (b) fermio fermidiagramnlinesuperimposedonawavyline.Inon and is traditionally a complexαβ scalar drawn [the as first a solid term fermionlinesuperimposedonawavyline.In gauge boson vertices of the Standard Model. (We do not show theinteractionsofghostfields,which(e) The minusfermion signs propagator; insideThe the minus parentheses note signs the inside appear directions the because parentheses of the of the arrows appearantisymmetry because in Figure of of the the 3.2c,d.antisymmetry! symbol There used is of no the to such!αβ symbol arrow-reversal used to are necessary only for consistentfermion propagator; loopin amplitudes.) the second noteFigure line the 3.3gFigure of directions we eq.s3.3c,d,e,farejustthestandardinteractions have (3.72)]. the couplingof the One arrows ofcan a gaugino thinkFigure in Fig of to 3.3gthis aure chiral as3.2c,d. we fermi the have “onsupersymmetrization” There the and a coupling complex is no such scalar of a arrow-reversal gaugino[the of first Figure0 term to a 3.3e chiral or fermion and a complex scalar [the first term tie upfor the aSU scalar(2)L propagatorindices.tie up the TheSU other in(2) a theory minusindices. signs with The in exact other eq. (5.1) minus supersymmetry were signs chosen in eq. so;asdepictedinFigure3.2e,ifonetreats (5.1) thattheterms were chosenytttH so thau, tthetermsy ttH0, for a scalar propagator0 in the in second a theory0 line of with eq. (3.72)]. exactL One supersymmetry canin think the of second this;asdepictedinFigure3.2e,ifonetreats as theline “supersymmetrization” of eq. (3.72)].0 One of0 can Figure think 3.3e of or this as thet “supersymmetrization”u of Figure 3.3e or between gauge bosons and fermion andybbbH scalard ,and fieldsyτ ττ thatHd ,whichwillbecomethetop,bottomandtaumasseswhen must0 occur in any0 gauge theory because of the Hu and Hd get VEVs, 0 0 the scalar squared-massybbbH ,andyτ termττH as,whichwillbecomethetop,bottomandtaumasseswhen an insertion in the propagator, the arrow directionHu isand preserved.H get VEVs, form of the covariant derivative;the scalar they squared-masseach come have from overall term eqs. positive (3.59)as and signs insertionand in (3.65)-(3.67) eq. (5.3).d in the propagat insertedor, in the the kinetic arrow direction is preserved. d Figure 3.3each shows have overall the gaugeijk positive interactions signs in eq. in (5.3). a supersymmetric theory. Figures 3.3a,b,c occur only

part of the Lagrangian. FigureFigure 3.3c 3.3 shows showsSince the the the coupling Yukawa gauge of interactions interactions a gauginoy to ainin gauge a general supersymmetric boson; supersymmetrici 28 theijk gauginotheory. theory line Figures must be 3.3a,b,c completely occur sym- only when the gaugeSince group the is Yukawa non-Abelian, interactions fory example28 in a general for SU supersymmetric(3)C color and theorySU(2) mustL weak be completely isospin in sym- the metric under interchange of i, j, k,weknowthatyu, yd and ye imply not only Higgs-quark-quark and in a Feynman diagramwhen is traditionally the gauge drawn group as is a non-Abelian, solidmetric fermio undernlinesuperimposedonawavyline.In for interchange example of fori, j,SU k,weknowthat(3)C color andyu, ySUd and(2)yLe weakimply not isospin only Higgs-quark-quark in the28 and Higgs-lepton-leptonMSSM. Figures couplings 3.3a as and in the3.3b Standard are the Model, interactions butalsosquark-Higgsino-quarkandslepton- of gauge bosons, which derive from the first term Figure 3.3g we have theMSSM. coupling Figures of a gaugino 3.3a and to a 3.3b chiralHiggs-lepton-lepton are fermi theon interactions and a couplings complex of gauge asscalar in the [the bos Standardons, first which term Model, derive butalsosquark-Higgsino-quarkandslepton- from the first term Higgsino-leptonin eq. (3.57). interactions. In the ToMSSM illustrate these this, are Figure exactlys5.1a,b,cshowsomeoftheinteractionsinvolving the same as the well-known QCD gluon and electroweak in the second line of eq. (3.72)]. One can think of thisHiggsino-lepton as the “supersymmetrization” interactions. To illustrate of Figure this, 3.3e Figure or s5.1a,b,cshowsomeoftheinteractionsinvolving in eq. (3.57).the In top-quark the MSSM Yukawa these coupling are exactlyy .Figure5.1aistheStandardModel-likecouplingofthetopq the same as the well-known QCD gluon and electroweakuark gauge bosonthe vertices top-quark oft the Yukawa Standard coupling Model.y .Figure5.1aistheStandardModel-likecouplingofthetopq (We do not show theinteractionsofghostfields,whichuark gauge bosonto vertices the neutral of the complex Standard scalar Higgs Model. boson, (We which do notfollowst show fromthefirsttermineq.(5.3).Forvariety, theinteractionsofghostfields,which are necessaryto only the neutral for consistent complex loopscalar amplitudes.) Higgs boson, which Figure followss3.3c,d,e,farejustthestandardinteractions fromthefirsttermineq.(5.3).Forvariety, are necessary onlybetween for consistent gauge bosons loop amplitudes.) and fermion and Figure scalars3.3c,d,e,farejustthestandardinteractions fields that must occur in any gauge theory because of the 28 31 between gauge bosonsform of and the fermion covariant and derivative; scalar fields they that come must from occur eqs. in any (3.59)31 gaugeand theory (3.65)-(3.67) because inserted of the in the kinetic form of the covariantpart of derivative; the Lagrangian. they come Figure from 3.3c eqs. shows (3.59) theand coupling (3.65)-(3.67) of a gaugino inserted to a gauge in the boson; kinetic the gaugino line part of the Lagrangian.in a Feynman Figure diagram 3.3c shows is traditionally the coupling drawn of a gaugino as a solid to a fermio gaugenlinesuperimposedonawavyline.In boson; the gaugino line in a Feynman diagramFigure is 3.3g traditionally we have the drawn coupling as a of solid a gaugino fermionlinesuperimposedonawavyline.In to a chiral fermion and a complex scalar [the first term Figure 3.3g we havein the the second coupling line of of a eq. gaugino (3.72)]. to One a chiral can think fermion of this and as a complex the “supersymmetrization” scalar [the first term of Figure 3.3e or in the second line of eq. (3.72)]. One can think of this as the “supersymmetrization” of Figure 3.3e or

28 28 i i

i j i j j i j k j k (a) (b) (c) (d) (e)

3 jkn Figure 3.2: Supersymmetric dimensionful couplings: (a) (scalar) interaction vertex Min∗ y and (b) in ij the conjugate interaction M yjkn∗ ,(c)fermionmasstermM and (d) conjugate fermion mass term kj Mij∗ ,and(e)scalarsquared-masstermMik∗ M .

Up to scalar trilinear and quartics: (a) (b) (c) (d) (e)

i i j i j j k (a) (b) (f) (c) (g) (d) (h) (i)

Figure 4.1: Soft supersymmetry-breaking terms: (a) GauginFigureomass 3.3:M Supersymmetrica;(b)non-analyticscalarsquared gauge interaction vertices. 2 i ij ijk mass (m )j;(c)analyticscalarsquaredmassb ;and(d)scalarcubiccouplinga . tion of Figure 3.1c is exactly of the special type needed to cancel the quadratic divergences in quantum that are singlets or in the adjointcorrections representation to scalar of a masses, simplefactorofthegaugegroup,thenthereare as discussed in the Introduction [compare Figure 1.1, and eq. (1.11)]. also possible soft supersymmetry-breakingFigure 3.2 Dirac shows mass the term onlysbetweenthecorrespondingfermions interactions corresponding to renormalizableψa and supersymmetric vertices and the gauginos [54]-[59]: with coupling dimensions of [mass] and [mass]2.First,thereare(scalar)3 couplings in Figure 3.2a,b, which are entirely determined by the superpotential mass parameters M ij and Yukawa couplings yijk, = M a λaψ +c.c. (4.3) as indicatedL − byDirac the seconda and third terms in eq. (3.50). The propagators of the fermions and scalars This is not relevant for the MSSMin the with theory minimal are field constructed content, in which the does usual not way have using adjoint the represen- fermion mass M ij and scalar squared mass kj ij tation chiral supermultiplets. Therefore,Mik∗ M .Thefermionmassterms equation (4.1) is usually takenM to beand theM generalij each form lead of to the a chirality-changing insertion in the soft supersymmetry-breaking Lagrangian.fermion propagator; For some interes note theting directions exceptions, of see the refs. arrows [54]-[65]. in Figure 3.2c,d. There is no such arrow-reversal The terms in soft clearly dofor break a scalar supersymmetry, propagator because in a theory they with involve exact only supersymmetry scalars and gauginos;asdepictedinFigure3.2e,ifonetreats L and not their respective superpartners. In fact, the soft terms in are capable of giving masses to all the scalar squared-mass term asLsoft an insertion in the propagator, the arrow direction is preserved. of the scalars and gauginos in a theory,Figure even 3.3 if the shows gauge the boso gaugens and interactions fermions in in chiral a supersymmetric supermultipletstheory. Figures 3.3a,b,c occur only are massless (or relatively light).when The the gaugino gauge masses groupM isa are non-Abelian, always allowed for example by gauge for symmetry.SU(3) Thecolor and SU(2) weak isospin in the 2 i j C L (m )j terms are allowed for i, jMSSM.such that Figuresφi, φ 3.3a∗ transform and 3.3b in are complex the interactions conjugate representations of gauge bosons, of which derive from the first term each other under all gauge symmetries; in particular this is true of course when i = j,soeveryscalar in eq. (3.57). In the MSSM these are exactly the same as the well-known QCD gluon and electroweak is eligible to get a mass in this way if supersymmetry is broken. The remaining soft terms may or may gauge boson vertices of the Standard Model. (We do not show theinteractionsofghostfields,which not be allowed by the symmetries. The aijk, bij,andti terms have the same form as the yijk, M ij, are necessary only for consistent loop amplitudes.) Figures3.3c,d,e,farejustthestandardinteractions and Li terms in the superpotential [compare eq. (4.1) to eq. (3.47) or eq. (3.77)], so they will each be allowed by gauge invariance if andbetween only if gauge a corresponding bosons and su fermionperpotential and scalar term is fields allowed. that must occur in any gauge theory because of the The Feynman diagram interactionsform of corresponding the covariant to derivative; the allowed they soft come terms from in eq. eqs. (4.1) (3.59) are shownand (3.65)-(3.67) inserted in the kinetic in Figure 4.1. For each of the interactionspart of the in Lagrangian. Figures 4.1a,c Figure,d there 3.3c is anothershows the with coupling all arrows of a reversed, gaugino to a gauge boson; the gaugino line corresponding to the complex conjugatein a Feynman term diagram in the Lagrangi is traditionallyan. We will drawn apply as these a solid general fermio resultsnlinesuperimposedonawavyline.In to the specific case of the MSSMFigure in the 3.3g next we section. have the coupling of a gaugino to a chiral fermion and a complex scalar [the first term in the second line of eq. (3.72)]. One can think of this as the “supersymmetrization” of Figure 3.3e or 5TheMinimalSupersymmetricStandardModel

In sections 3 and 4, we have found a general recipe for constructing Lagrangians28 for softly broken supersymmetric theories. We are now ready to apply these general results to the MSSM. The particle content for the MSSM was described in the Introduction. In this section we will complete the model by specifying the superpotential and the soft supersymmetry-breaking terms.

5.1 The superpotential and supersymmetric interactions The superpotential for the MSSM is

W = uyuQH dydQH eyeLH + µH H . (5.1) MSSM u − d − d u d

The objects Hu, Hd, Q, L, u, d, e appearing here are chiral superfields corresponding to the chiral supermultiplets in Table 1.1. (Alternatively, they can be just thought of as the corresponding scalar fields, as was done in section 3, but we prefer not to put the tildes on Q, L, u, d, e in order to reduce clutter.) The dimensionless Yukawa coupling parameters yu, yd, ye are 3 3matricesinfamily ×

30 60 40 60 40 [GeV]

d 20 H [GeV]

d 20

0 H 0 50 0 100 150 200 250 300 0 50H [GeV]100 150 200 250 300 u H [GeV] Figure 7.1: A contour map of the Higgs potential, for a typicalcasewithtanu β cot α 10. Figure 7.1: A contour map of the Higgs potential, for a typicalcasewithtan≈− ≈ β cot α 10. The minimum of the potential is marked by +, and the contours are equally spaced equipotentials.≈− ≈ The minimum of the potential0 0 is marked by +, and the contours are equally0 spaced equipotentials. Oscillations along the shallow direction, with Hu/Hd How10, correspond supersymmetry to the mass works? eigenstate h ,while Oscillations along the shallow direction,≈ with H0/H00 10, correspond to the mass eigenstate h0,while the orthogonal steeper direction corresponds to the mass eigenstateu H d.≈ the orthogonal steeper direction corresponds to the mass eigenstate H0. t t˜ t t˜ t˜ 0 0 t˜ h 0 h 0 0 2 h h h 0 ∆(mh0 )= 2 + + h ∆(mh0 )= + + +

Figure 7.2: ContributionsFigure to the 7.2: MSSM Contributions lightestFermion to Higgs the MSSM mass loop lightestfrom top-quark Higgs massBoson and from top-squark loop top-quark one-loop and top-squark one-loop diagrams. Incompletediagrams. cancellation, Incomplete due to softcancellation, supersymme due totry soft breaking, supersymme leadstry to breaking, a large positive leads to a large positive 2 2 0 correction to m 0 in the limit of heavy top squarks. correction to mh in the limit of heavyh top squarks.μ² = + A μ² = - A and is traditionally chosenand isto traditionally be negative; chosen it follows to be that negative;π/2 it< followsα < 0 that (providedπ/2 m0 (provided). ThemA0 >mZ ). The − − A Z Feynman rules for couplingsFeynman of the rules mass for couplings eigenstate of Higgs the mass sca eigenstatelars to the Higgs Standard scalars Model to the quarks Standard and Model quarks and leptons and the electroweak vector bosons, as well as{ to the various sparticles, have been worked out leptons and the electroweak vector bosons, as well as to the variousμ²total sparticles, = 0 have been worked out in detail in ref. [182, 183]. in detail in ref. [182, 183]. The masses of A0, H0 and H can in principle be arbitrarily large since they all grow with b/ sin(2β). The masses of A0, H0 and H can in principle be arbitrarily± large since they all grow with b/ sin(2β). In contrast,± the mass of h0 is bounded above. From eq. (7.20), one finds at tree-level [184]: In contrast, the mass of h0 is bounded above. From eq. (7.20), one finds at tree-level [184]:

m 0

69 Its not the first time that symmetries force doubling the known spectrum:

Relativistic quantum field theories: Particle ➞ Antiparticles

Made the electron-mass corrections not linearly divergent:

e− γ e+

γ e− e−

e−

d4k i γµi(p k + m )γ iδm + . . . = ( ie)2 − ￿ − ￿ e µ − e − (2π)4 k2 (p k)2 m2 ∆m m￿ e ￿ ￿ e ∝ e − − • The second diagram is the correct interpretation of the contributions from Ep Ek < 0 − • Note this represents an interaction with virtual e+e- pairs in the vacuum. • The net result is only logarithmically divergent:

δme 3 = α log (mere) = 18% for re = 1/MPlanck me −2π

11 Model particles and the Higgs bosons have even R-parity (PR =+1),whileallofthesquarks,sleptons, gauginos, and higgsinos have odd R-parity (P = 1). R − The R-parity odd particles are known as “supersymmetric particles” or “sparticles” for short, and they are distinguished by a tilde (see Tables 1.1 and 1.2). If R-parity is exactly conserved, then there can be no mixing between the sparticles and the PR =+1particles.Furthermore,everyinteractionvertex in the theory contains an even number of P = 1 sparticles. This has three extremely important R − phenomenological consequences: The lightest sparticle with P = 1, called the “lightest supersymmetric particle” or LSP, must • R − be absolutely stable. If the LSP is electrically neutral, it interacts only weakly with ordinary matter, and so can make an attractive candidate [73] for the non-baryonic dark matter that seems to be required by cosmology. Each sparticle other than the LSP must eventually decay into astatethatcontainsanoddnumber • of LSPs (usually just one). In collider experiments, sparticles can only be produced in even numbers (usually two-at-a-time). • We define the MSSM to conserve R-parity or equivalently matter parity. While this decision seems to be well-motivated phenomenologically by proton decay constraints and the hope that the LSP will provide a good dark matter candidate, it might appear somewhat artificial from a theoretical point of view. After all, the MSSM would not suffer any internal inconsistency if we did not impose matter parity conservation. Furthermore, it is fair to ask why matter parity should be exactly conserved, given that the discrete symmetries in the Standard Model (ordinary parity P ,chargeconjugationC, time reversal T ,etc.)areallknowntobeinexactsymmetries.Fortunately,it is sensible to formulate matter parity as a discrete symmetry that is exactly conserved. In general, exactly conserved, or “gauged” discrete symmetries [74] can exist provided that they satisfy certain anomaly cancellation conditions [75] (much like continuous gauged symmetries). One particularly attractive way this could occur is if B L is a continuous gauge symmetry that is spontaneously brokenatsomeveryhighenergy − scale. A continuous U(1)B L forbids the renormalizable terms that violate B and L [76, 77], but this − gauge symmetry must be spontaneously broken, since there is no corresponding massless vector boson. However, if gauged U(1)B L is only broken by scalar VEVs (or other order parameters) thatcarry − even integer values of 3(B L), then P will automatically survive as an exactly conserved discrete − M remnant subgroup [77]. A variety of extensions of the MSSM in which exact R-parity conservation is guaranteed in just this way have been proposed (see for example [77, 78]). It may also be possible to have gauged discrete symmetries that do not owe their exact conservation to an underlying continuous gauged symmetry, but rather to some other structure such as can occur in string theory. It is alsoBut possible if supersymmetry that R-parity is broken, is or isexact: replaced by some alternative discrete symmetry. We will briefly consider these as variations on the MSSM in section 10.1. MF = MB ➡ e.g. Me = Me˜ 5.3 Soft supersymmetryIt must be breakingbroken to in give the MSSMmasses to the superpartners To complete the description of the MSSM, we need to specify thesoftsupersymmetrybreakingterms. In section 4, we learnedSupersymmetry how to write down breaking the most must general afford set of such“soft terms terms”: in any supersymmetric theory. Applying this recipe(terms to that the do MSSM, not spoil we the have: good UV properties of the Susy) 1 MSSM = M gg + M W W + M BB +c.c. Lsoft −2 3 2 1 ! $ u au QH d#ad#QH e"a"e LH +c.c. − "" u − d − d ! $ 2 " 2 2 2 † 2 Q"† m "Q L† m L" u m" u†" d m d e m e† − Q − L − u − d − e 2 2 m H∗Hu m H∗H (bHuH +c.c.) . (5.12) − " Hu u" −" Hd " d "d − " "d " " " ˜ ˜ +µHuHd 36 for 3 families, more than100 terms are possible!! 60 40 60 40 [GeV]

d 20 H [GeV]

d 20

0 H 0 50 0 100 150 200 250 300 0 50H [GeV]100 150 200 250 300 u H [GeV] Figure 7.1: A contour map of the Higgs potential, for a typicalcasewithtanu β cot α 10. Figure 7.1: A contour map of the Higgs potential, for a typicalcasewithtan≈− ≈ β cot α 10. The minimum of the potential is marked by +, and the contours are equally spaced equipotentials.≈− ≈ The minimum of the potential0 0 is marked by +, and the contours are equally0 spaced equipotentials. Oscillations along the shallow direction, with Hu/Hd 10,How correspond supersymmetry to the mass eigenstate works?h ,while Oscillations along the shallow direction,≈ with H0/H00 10, correspond to the mass eigenstate h0,while the orthogonal steeper direction corresponds to the mass eigenstateu H d.≈ the orthogonal steeper direction corresponds(including to the mass soft-masses) eigenstate H0. t t˜ t t˜ t˜ 0 0 t˜ h 0 h 0 0 2 h h h 0 ∆(mh0 )= 2 + + h ∆(mh0 )= + + +

Figure 7.2: ContributionsFigure to the 7.2: MSSM Contributions lightestFermion to Higgs the MSSM mass loop lightestfrom top-quark Higgs massBoson and from top-squark loop top-quark one-loop and top-squark one-loop diagrams. Incompletediagrams. cancellation, Incomplete due to softcancellation, supersymme due totry soft breaking, supersymme leadstry to breaking, a large positive leads to a large positive 2 2 0 correction to m 0 in the limit of heavy top squarks. correction to mh in the limit of heavyh top squarks.μ² = + A μ² = - A + mstop² B and is traditionally chosenand isto traditionally be negative; chosen it follows to be that negative;π/2 it< followsα < 0 that (providedπ/2 m0 (provided). ThemA0 >mZ ). The − − A Z Feynman rules for couplingsFeynman of the rules mass for couplings eigenstate of Higgs the mass sca eigenstatelars to the Higgs{ Standard scaSuperpartnerslars Model to the quarks Standard expected and Model quarks and leptons and the electroweak vector bosons,μ² astotal well ~ m asstop to² the various sparticles, have been worked out leptons and the electroweak vector bosons, as well as to the various sparticles, havearound been v worked ~ 100 outGeV in detail in ref. [182, 183]. in detail in ref. [182, 183]. The masses of A0, H0 and H can in principle be arbitrarily large since they all grow with b/ sin(2β). The masses of A0, H0 and H can in principle be arbitrarily± large since they all grow with b/ sin(2β). In contrast,± the mass of h0 is bounded above. From eq. (7.20), one finds at tree-level [184]: In contrast, the mass of h0 is bounded above. From eq. (7.20), one finds at tree-level [184]:

m 0

69 Constraints on superpartner masses from flavor physics:

γ γ γ W − Breaking of lepton µR eR µL eR " symmetry: µ− ! B ! γe− µ− νµ νe eγ− µ− ! B ! e− γ W − µR eR µL eR (a)! ! (b) ! (c)! " µFigure− 5.6:! SomeB 11! of the diagramse− thatµ− contributeνµ toνe the processe−µ µ−e γ in! modelsB with! leptone− Exp: BR(µ eγ) < 10− ➡ me˜ mµ˜ − → − → flavor-violating soft supersymmetry breaking￿ parameters (indicated by ). Diagrams (a), (b), and (c) (a)! ! (b) ! 2 2 × (c)! contribute to constraints on the off-diagonal elements of me, mL,andae,respectively. Figure 5.6: Some of˜ the diagramsup¯ to that 1% contribute - 0.1%˜ to the¯ process µ− e−γ ˜in models¯ with lepton s¯ s˜R∗ dR∗ d s¯ s˜R∗ dR∗ d s¯ →s˜R∗ dL∗ d flavor-violating soft supersymmetry breaking parameters (indicated by ). Diagrams (a), (b), and (c) 2 2 × contribute tog constraintsg on the off-diagonalg elementsg of me, mL,andgae,respectively.g ˜ s˜ ˜ s˜ ˜ s˜ d s˜∗dR d˜ R ¯s d ds˜L∗ d˜L s¯ d dLs˜∗ Rd˜ s ¯ s¯ ! R R∗ ! d s¯ ! R R∗ ! d s¯ ! R L∗! d Large contributions_ (a) (b) (c) g g g g 0 g g to K-K mixing: Figure 5.7: Some of the diagrams that contribute to K0 K mixing in models with strangeness- ˜ s˜ ˜ s˜ ↔ ˜ s˜ violatingd softdR supersymmetryR s breakingd parametersdL L (indicateds by d). ThesedL diagramsR contributes to ! ! ! ! × ! ! constraints on the off-diagonal elements of (a) m2 ,(b)thecombinationofm2 and m2 ,and(c)a . (a) (b)d d (c) Q d 0 Figure 5.7: Some of the diagrams that contribute to K0 K mixing in models with strangeness- ↔ ➡ ms˜ violatingmthed˜ bino softBupis supersymmetry nearly to a0.1% mass eigenstate. breaking- 0.001% This parameters result is to (indica be comparedted by to). the These present diagrams experimental contribute upper to 11 ￿ limit Br(µ eγ)exp < 1.2 10− from [106]. So,2 if the right-handed× slepton2 squared-mass2 matrix constraints on→ the off-diagonal× elements of (a) m ,(b)thecombinationofm and mQ,and(c)ad. m2 were! “random”, with all entries of comparabled size, then the prediction ford Br(µ eγ)wouldbe e → too large even if the sleptons and bino masses were at 1 TeV. Forlightersuperpartners,theconstraint theon binoµ ˜R,Be˜Rissquared-mass nearly a mass mixing eigenstate. becomes This correspondingly result is to be more compared severe. toThere the present are also experimental contributions upper to 11 2 limitµ Br(eγµthate dependγ) < on1 the.2 off10-diagonalfrom elements [106]. So, of the if the left-handed right-handed slepton slepton squared-mass squared-mass matrix m maL,trix → → exp × − m2comingwere! “random”, from the diagram with all shown entries in fig. of comparable 5.6b involving size, the then charg theed winoprediction and the for sneutrinos, Br(µ e asγ)wouldbe well as e 0 → toodiagrams large even just if like the fig. sleptons 5.6a but and with bino left-handed masses were sleptons at 1 TeV. andeither Forlightersuperpartners,theconstraintB or W exchanged. Therefore, the slepton squared-mass matrices must not have significant mixings for eL, µL either. onµ ˜R, e˜R squared-mass mixing becomes correspondingly more severe.! There" are also contributions to Furthermore, after the Higgs scalars get VEVs, the ae matrix could imply squared-mass terms that 2 µ eγ that depend on the off-diagonal elements of the left-handed slepton squared-massMSSM matrix mL, →mix left-handed and right-handed sleptons with different lepton flavors. For! example,! soft contains coming from the diagram shown in fig. 5.6b0 involving the charg0 ed wino and the sneutrinos,L as well as eaeLH +c.c. which implies terms H (ae) e µ H (ae) µ e +c.c. These also contribute diagramsd just like fig. 5.6a but with−% left-handedd & 12 R∗ sleptonsL −% and & deither21 R∗ BL or W 0 exchanged. Therefore, to µ eγ,asillustratedinfig.5.6c.Sothemagnitudesof(ae) and (ae) are also constrained → 12 21 the!by slepton experiment! squared-mass to be small, matrices but in must a way not that have is! more significant! stronglymixings dependent! ! for e onL, otherµL either. model parameters ! " Furthermore, after the Higgs scalars get VEVs, the ae matrix could imply squared-mass terms that [85]. Similarly, (ae) , (ae) and (ae) , (ae) are constrained, although more weakly [86], by the 13 31 23 32 MSSM mixexperimental left-handed limits and right-handed on Br(τ eγ)andBr( sleptons withτ µ diγff).erent lepton flavors. For! example,! soft contains → 0 → 0 L eaeLHThered +c. arec. which also important implies terms experimentalHd (ae constraints)12eR∗ µL onHd the(ae sq)21uarkµR∗ e squared-massL +c.c. These matrices. also contribute The −% & −% & 0 tostrongestµ eγ,asillustratedinfig.5.6c.Sothemagnitudesof( of these come from the neutral kaon system. The effectiveae) Hamiltonian12 and (ae)21 forareK0 alsoK constrainedmixing ! → ↔ !bygets experiment contributions to be from small, the but diagrams in a way in Figure that is 5.7,! more! among strongly others, dependent if! MSSM! oncontains other model terms that parameters mix Lsoft [85].down Similarly, squarks and (ae)13 strange, (ae)31 squarks.and (ae The)23, ( gluino-squark-quarkae)32 are constrained,vertices although in Figure more 5.7 weakly are all [86], fixed by by the experimentalsupersymmetry limits to onbe of Br( QCDτ interactioneγ)andBr( strength.τ µγ). (There are similar diagrams in which the bino and → → winosThere are are exchanged, also important which can experimental be important constraints depending on on the therelative squark sizes squared-mass of the gaugino matrices. masses.) The 0 m20 strongestFor example, of these suppose come that from there the is neutral a non-zero kaon right-handed system. The down-squark effective Hamiltonian squared-mass for mixingK ( Kd)21mixingin the basis corresponding to the quark mass eigenstates. Assuming that theMSSM supersymmetric↔ correction gets contributions from the diagrams in Figure 5.7, among others, if soft contains terms that mix to ∆m m m following from fig. 5.7a and others does notL exceed, in absolute value, the down squarksK ≡ andKL − strangeKS squarks. The gluino-squark-quark vertices in Figure 5.7 are all fixed by supersymmetry to be of QCD interaction strength. (There are similar diagrams in which the bino and winos are exchanged, which can be important depending on the relative sizes of the gaugino masses.) m2 For example, suppose that there is a non-zero right-handed down-squark squared-mass mixing ( d)21 in the basis corresponding to the quark mass eigenstates. Assuming that the supersymmetric correction to ∆m m m following from fig. 5.7a and others does not exceed, in absolute value, the K ≡ KL − KS 38

38 Soft terms must be generated in a clever way

Most interesting possibilities:

1) Gauge mediation 2) Gravity/Moduli/Extra-dim mediation The famous scenario “minimal sugra” not a model, just an Ansatz:

All gaugino masses equal = M1/2

At Q=MGUT All scalar masses equal = M0

All trilinear equal = A0 ➥ Why? At Q=MZ 600{

500 H I don’t know, d 2 2 1/2 (µ +m ) 0 but experimentalists Hu 400 like it a lot!

300 M3 M 2 m Mass [GeV] 1/2 M 200 1 squarks 100 sleptons m0

0 2 4 6 8 10 12 14 16 18 Log10(Q/1 GeV) Figure 7.4: RG evolution of scalar and gaugino mass parameters in the MSSM with typical minimal supergravity-inspired boundary conditions imposed at Q =2.5 1016 GeV. The parameter µ2 + m2 0 × Hu runs negative, provoking electroweak symmetry breaking.

a reasonable approximation, the entire mass spectrum in minimal supergravity models is determined 2 by only five unknown parameters: m0, m1/2, A0,tanβ,andArg(µ), while in the simplest gauge- mediated supersymmetry breaking models one can pick parameters Λ, M , N , F ,tanβ,and mess 5 " # Arg(µ). Both frameworks are highly predictive. Of course, it is easy to imagine that the essential physics of supersymmetry breaking is not captured by either of these two scenarios in their minimal forms. For example, the anomaly mediated contributions could play a role, perhaps in concert with the gauge-mediation or Planck-scale mediation mechanisms. Figure 7.4 shows the RG running of scalar and gaugino masses inatypicalmodelbasedonthe minimal supergravity boundary conditions imposed at Q =2.5 1016 GeV. [The parameter values 0 × used for this illustration were m =80GeV,m =250GeV,A = 500 GeV, tan β =10,and 0 1/2 0 − sign(µ)= +.] The running gaugino masses are solid lines labeled by M1, M2,andM3.Thedot-dashed lines labeled H and H are the running values of the quantities (µ2 + m2 )1/2 and (µ2 + m2 )1/2, u d Hu Hd which appear in the Higgs potential. The other lines are the running squark and slepton masses, with dashed lines for the square roots of the third family parameters m2 , m2 , m2 , m2 ,andm2 d3 Q3 u3 L3 e3 2 2 (from top to bottom), and solid lines for the first and second family sfermions. Note that µ + mHu runs negative because of the effects of the large top Yukawa coupling as discussed above, providing for electroweak symmetry breaking. At the electroweak scale, the values of the Lagrangian soft parameters can be used to extract the physical masses, cross-sections, and decay widths of the particles, and other observables such as dark matter abundances and rare process rates. There are a variety of publicly available programs that do these tasks, including radiativecorrections;seeforexample[204]-[213],[194]. Figure 7.5 shows deliberately qualitative sketches of sample MSSM mass spectrum obtained from three different types of models assumptions. The first is the output from a minimal supergravity- 2 2 inspired model with relatively low m0 compared to m1/2 (in fact the same model parameters as used for fig. 7.4). This model features a near-decoupling limit fortheHiggssector,andabino-likeN1 LSP, nearly degenerate wino-like N2, C1,andhiggsino-likeN3, N4, C2.Thegluinoistheheaviest ! ! ! ! ! ! 80 1) Gauge mediation

New sector Susy breaking sector

gauge bosons

MSSM

Gauge interactions are “flavor blind”: Universal masses for squarks/sleptons with equal charges ~ ! = 74 TeV N = 1 8 dL tan" = 3 µ < 0 ~ uR Very predictive (in the minimal case).~ t1 ~ Just calculate loops:6 g 1 ~ F ! S" t2 ~ m / M 4 Figure 6.3: Contributions to the MSSM gaugino masses B,W,g˜ in gauge-mediated supersymmetry breaking models come ~e from one-loop graphs involving virtual messenger parti- ! " L cles. S ! " Figure 6.4: MSSM2 scalar squared masses in gauge-mediated supersymmetry breaking models arise in Replacing S and FS by their VEVs, one finds quadratic mass terms in the potential for the messenger leading order from these two-loop Feynman graphs. The heavy dashed lines are messenger scalars, the scalar leptons: gaugino masses ~ solid lines are messenger fermions,scalar the wavy lines masses are ordinaryeR Standard Model gauge bosons, and the V = y S 2( ! 2 + ! 2)+ y S 2( q 2 + q 2) solid lines with wavy lines superimposed are the MSSM gauginos. | 2! "| | | | | | 3! "| | | | | y2 FS !! + y3 FS qq +c.c. − ! " ! " order Mmess yI S for I =2, 3. The running mass parameters can then be RG-evolved down to the # $ ∼ " # +quarticterms. (6.49) electroweak scale to predict the physical masses to be measured by future experiments. Depends on 3 parameters:The scalars of the MSSM do not get any radiative~ corrections totheirmassesatone-looporder.! = 74 TeV N = 1 The first line in eq. (6.49) represents supersymmetric mass terms that go along with eq. (6.44), while The leading contribution to their masses comes from the two-loop graphs shown in! Figure = 15 6.4, TeV with N = 5 88 dL tan" = 3 µ < 0 the second line consists of soft supersymmetry-breaking masses. The complex scalar messengers !, ! the messenger fermions (heavy solid lines) and messenger scalars (heavy dashed lines)tan" and = ordinary 3 µ < 0 thus obtain a squared-mass matrix1) equal μ to:-term: Higgsino mass gauge bosons and gauginos running~ around the loops. By computing these graphs, one finds that each uR 2 MSSM scalar φi gets a squared mass given by: y2 S y2∗ FS∗ ~ | ! "| − ! "2 (6.50) 2) Susy-breakingy2 FS y2 S scale: F α 2 α 2 t1α 2 % − ! " | ! "| & m2 =2Λ2 3 C (~i)+ 2 C (i)+ 1 C (i) , (6.55) φi ~ 4π 3 4π 2 4π 1 2 ~ g!" # t1 " # " # $ with squared mass eigenvalues y2 S y2 FS .Injustthesameway,thescalarsq,q get squared 66 | ! "| ±| ! "| g masses y S 2 y F . 3) Scale where the soft-terms | 3! "| ±| 3! S"| with the quadratic Casimir invariants Ca(i)asineqs.(5.27)-(5.30).Thesquaredmassesineq.(6.55) So far, we have found that the effect of supersymmetry breakingistospliteachmessengersuper-

are positive (fortunately!).1 1 multiplet pair apart: are induced: M The terms au, ad, ae arise first at two-loop~ order, and are suppressed by an extra~ factor of αa/4π compared to the gaugino masses. So, tou aR very good approximation one has, at~ the messenger scale, 2 2 2 2 dL !, ! : mfermions = y2 S ,mscalars = y2 S y2 FS , (6.51) t2 ~ ~ m / M | ! "| | ! "| ±| ! "| m / M 2 2 2 2 au = ad = ae =0, (6.56) q,q : m = y3 S ,mscalars = y3 S y3 FS . (6.52) 44 ~ fermions | ! "| | ! "| ±| ! "| t a significantly stronger condition than eq. (5.19).~ 2 Again, eqs. (6.55) and (6.56) should be applied at The supersymmetry violation apparent in this messenger spectrum for FS =0iscommunicatedto e ! "$ an RG scale equal to the average mass of the messengerL fields running in the loops. However, evolving the MSSM sparticles through radiative corrections. The MSSMgauginosobtainmassesfromthe1-loop the RG equations down to the electroweak scale generates non-zero a , a ,anda proportional to the Feynman diagram shown in Figure 6.3. The scalar and fermion lines in the loop are messenger fields. u d e corresponding Yukawa matrices and the non-zero gaugino masses, as indicated in section 5.5. These Recall that the interaction vertices in Figure 6.3 are of gauge coupling strength even though they do not ~ will only be large22 for the third-family squarks and sleptons,intheapproximationofeq.(5.2).TheeL involve gauge bosons; compare Figure 3.3g. In this way, gauge-mediation provides that q,q messenger parameter b may also be taken to vanish near the messenger scale, but this is quite model-dependent, loops give masses to the gluino and the bino, and !, ! messenger loops give masses to the wino and and in any case b will be non-zero when it is RG-evolved to the~ electroweak scale. In practice, b can be bino fields. Computing the 1-loop diagrams, one finds [159] that the resulting MSSM gaugino masses fixed in terms of the other parameters by the requirement ofe coRrrect electroweak symmetry breaking, are given by as discussed below in section 7.1. ~ eR α Because the gaugino masses arise at one-loop order and the scalar squared-mass contributions M = a Λ, (a =1, 2, 3), (6.53) a 4π appear at two-loop0 order, both eq. (6.53) and (6.55) correspond to the estimate eq. (6.27) for msoft,with M y S .Equations(6.53)and(6.55)holdinthelimitofsmall F /y S 2,correspondingto mess ∼ I" # 106 109 " S# I " # 1012 1015 in the normalization for αa discussed in section 5.4, where we have introduced a massGiudice,Rattazzi parameter mass splittings97 within each messenger supermultiplet that are small compared to the overall messenger mass scale. The sub-leading corrections in an expansion in F /y S 2 turn out [160]! to = be 15 quite TeV small N = 5 8 " S# I "M# (GeV) Λ FS / S . (6.54) unless there are very large messenger mass splittings. ≡! " ! " tan" = 3 µ < 0 The model we have described so far is often called the minimal model of gauge-mediated supersym- (Note that if F were 0, then Λ =0andthemessengerscalarswouldbedegeneratewiththeir ! S" metry breaking. Let us now generalize it to a more complicatedmessengersector.Supposethatq,q fermionic superpartners and there would be no contribution to the MSSM gaugino masses.) In contrast, Figure 3: Different supersymmetric particle masses in units of the B-ino mass M1,asafunc- the corresponding MSSM gauge bosons cannot get a corresponding mass shift, since they are protected ~ ~ by gauge invariance. So supersymmetry breaking has been successfully communicated to the MSSM g t1 60 (“visible sector”). To a good approximation, eq. (6.53) holds for the running gaugino massestion at ofan RG the messenger6 mass M.Thechoiceofparametersisindicated,andinbothcasesit scale Q0 corresponding to the average characteristic mass of the heavy messenger particles, roughly of

corresponds to a B-ino1 mass of 100 GeV. ~u ~ 59 R dL ~ m / M 4 ~ t2

~ 2 eL

~ eR 0 106 109 21 1012 1015 M (GeV)

Figure 3: Different supersymmetric particle masses in units of the B-ino mass M1,asafunc- tion of the messenger mass M.Thechoiceofparametersisindicated,andinbothcasesit corresponds to a B-ino mass of 100 GeV.

21 the condition of vanishing cosmological constant, is given by [272, 81]

F0 m3/2 = . (3.1) √3MP Here M =(8πG )−1/2 =2.4 1018 GeV is the reduced Planck mass. We denote by F the total P N × 0 contribution of the supersymmetry-breaking VEV of the auxiliary fields, normalized in such a 2 way that the vacuum energy of the globally supersymmetric theory is V = F0 .ThusF0 does not coincide with the definition of F ,whichappearsinthesparticlemassesthroughΛ = F/M.

While F0 is the fundamental scale of supersymmetry breaking, F is the scale of supersymmetry breaking felt by the messenger particles, i.e. the mass splitting inside their supermultiplets. The ratio k F/F depends on how supersymmetry breaking is communicated to themessengers. ≡ 0 If the communication occurs via a direct interaction, this ratio is just given by a coupling constant, like the parameter λ in the case described by eq. (2.5). It can be argued that this couplingPredicts should be smallera very than light 1, by gravitino requiring perturbativ = Massity up tosuppressed the GUT scale by [16]. M IfP the: communication occurs radiatively, then k is given➥ partner by some loop of factor, the andgraviton therefore it is much smaller than 1. We thus rewrite the gravitino mass as

2 F 1 √F m3/2 = = 2.4eV, (3.2) k√3MP k !100 TeV " where the model-dependent coefficient k is such that k

If R parity is conserved, all supersymmetric particles follow decay chains that lead to grav- itinos. In order to compute the decay rate we need to know the interaction Lagrangian at lowest order in the gravitino field. Since, for √F M ,thedominantgravitinointeractionscome $ P from its spin-1/2 component, the interaction Lagrangian canbecomputedinthelimitofglobal µ supersymmetry. In the presence of spontaneous supersymmetry breaking, the supercurrent JQ satisfies the equation ∂ J µ = F γµ∂ G,˜ (3.3) µ Q − 0 µ 33 Quantum corrections in AdS5 2) Gravity/Moduli/Extra-dim mediation: ...to be discussed later gauge or gravity

Susy-breakingNo-Susy Sector MSSM Matter H AdS5 R

Spectrum at high-energies (Q~1/R)Gp=AdSpropagators model dependent 4 1 + m2 (H) GB V(H)= d p ln B p (2π)4 2 F An example: Scalar" 1 + mF (massesH) Gp # = 0 (at tree-level) ! Gaugino masses = M1/2 ≠0 Lightest superpartner: NeutralinoFINITE (From (mixture non-local of gaugino effects) and Higgsino)

e.g. from gauge bosons 2 0.1 2 mH ≈ L $ %

One-loop below the composite scale 1/L This implies that M M M ,sothelightestneutralinoisactuallymostlywino,witha | 2| ! | 1| ! | 3| lightest chargino that is only of order 200 MeV heavier, depending on the values of µ and tan β.The decay C± N π produces a very soft pion, implying unique and difficult signatures in colliders 1 → 1 ± [173]-[177]. Another! large! general class of models breaks supersymmetry using the geometric or topological properties of the extra dimensions. In the Scherk-Schwarz mechanism [178], the symmetry is broken by assuming different boundary conditions for the fermion andbosonfieldsonthecompactifiedspace. In supersymmetric models where the size of the extra dimension is parameterized by a modulus (a massless or nearly massless excitation) called a radion, the F -term component of the radion chiral supermultiplet can obtain a VEV, which becomes a source for supersymmetry breaking in the MSSM. These two ideas turn out to be often related. Some of the variety of models proposed along these lines can be found in [179]. These mechanisms can also be combined with gaugino-mediation and AMSB. It seems likely that the possibilities are not yet fully explored.

7ThemassspectrumoftheMSSM

7.1 Electroweak symmetry breaking and the Higgs bosons In the MSSM, the description of electroweak symmetry breaking is slightly complicated by the fact + 0 0 that there are two complex HiggsHiggs doublets sectorHu =(Hu ,Hu)andHd =(Hd ,Hd−)ratherthanjustone in the ordinary Standard Model. The classical scalar potential for the Higgs scalar fields in the MSSM is givenOnly by 3 parameters:

2 2 0 2 + 2 2 2 0 2 2 V =(µ + m )( H + H )+(µ + m )( H + H− ) | | Hu | u| | u | | | Hd | d | | d | + 0 0 +[b (H H− H H )+c.c.] u d − u d 1 2 2 0 2 + 2 0 2 2 2 1 2 + 0 0 2 + (g + g" )( H + H H H− ) + g H H ∗ + H H−∗ . (7.1) 8 | u| | u | − | d | − | d | 2 | u d u d | The terms proportional to µ 2 come from F -terms [see eq. (5.5)]. The terms proportional to g2 and 2 | | g" are the D-term contributions, obtained from the general formula eq. (3.75) after some rearranging. Finally, the terms proportional to m2 , m2 and b are just a rewriting of the last three terms of Hu Hd eq. (5.12). The full scalar potentialquartic of coupling the theory also includes many terms involving the squark and slepton fields that we can ignore here, since they do not get VEVs because they have large positive squared masses. related to gauge-couplings We now have to demand that the minimum of this potential shouldbreakelectroweaksymmetry down to electromagnetism SU(2) U(1) U(1) ,inaccordwithexperiment.Wecanusethe Spectrum: L × Y → EM freedom to make gauge transformations to simplify this analysis. First, the freedom to make SU(2)L gauge transformations2 x allows4 = 8 us scalars to rotate = away 3 Goldstones a possible VEVforoneoftheweakisospincomponents (eaten by W, Z) + of one of the scalar fields, so without loss +3 of generalityneutral Higgs we cantake = h, HH,u =0attheminimumofthe A + potential. Then one can check that a minimum + Charged of the Higgs potential = satisfyingH⁺, H⁻ ∂V/∂Hu =0mustalso have Hd− =0.Thisisgood,becauseitmeansthatattheminimumofthepotential electromagnetism is necessarily unbroken, since the charged components of theHiggsscalarscannotgetVEVs.After + setting Hu = Hd− =0,wearelefttoconsiderthescalarpotential

V =(µ 2 + m2 ) H0 2 +(µ 2 + m2 ) H0 2 (bH0H0 +c.c.) | | Hu | u| | | Hd | d | − u d 1 2 2 0 2 0 2 2 + (g + g" )( H H ) . (7.2) 8 | u| − | d | The only term in this potential that depends on the phases of the fields is the b-term. Therefore, a redefinition of the phase of Hu or Hd can absorb any phase in b,sowecantakeb to be real and positive.

65 2 2 g + g 2 V = m2 H0 2 + m2 H0 2 m2 H0H0 + h.c. + 2 Y H0 2 H0 2 1| 1 | 2| 2 | − 3 1 2 8 | 1 | − | 2 | ￿ ￿ ￿ ￿ v1 0 2 2 H1 = H2 = v1 + v2 vF = 174 GeV ￿ ￿ 0 ￿ ￿ v2 ≡ ￿ ￿ ￿ ￿ ￿

0 + H1 H2 8 real scalars - 3 ‘eaten’ by W ±, Z H1 = , H2 = 0 H− H ￿ 1 ￿ ￿ 2 ￿ = 5 physical scalars 2 CP-even real neutral: h, H 1 CP-odd real neutral: A 1 complex charged: H+ 2 unknown parameters (since v 2 = H 2 + H 2 ): ￿ u￿ ￿ d￿ v parameters: 2 2 2 1 H m1, m2, m3 vF , tan β = , mA u −→ v2 1) tan β = ￿ ￿ mA H 2) ￿ d￿ At tree-level: 2 m2 m2 tan2 β 2m2 m2 = 1 − 2 sin2β = 3 Z 2 2 mh < mA < mH 2 ￿ tan2 β 21 ￿ mA mH2+ =2mA 2+ m−W 2 2 mA=m1 + m2 mH+ = mA + mW 1 mh < mZ m2 = m2 + m2 (m2 m2 )2 + 4sin2 2βm2 m2 h,H 2 A Z ± A − Z A Z { ￿ ￿ ￿ m m h ≤ Z Was a great prediction for Higgs hunters at LEP! The radiatively corrected masses of the neutral CP–even and the charged Higgs bosons are displayed in Fig. 1.7 as a function of MA for the two values tan β =3and30.Thefullset of radiative... correctionsbut quantum has been included effects and the (mostly “no–mixing” loops scenario withof top/stop)Xt =0(left) and “maximal mixing” scenario with Xt = √6MS (right) have been assumed. The SUSY scale has been set to M =2TeVandtheotherSUSYparametersexceptfort t˜ A to 1 TeV; t˜ S t t˜ the SM input parameters are fixed to mt =178GeV,mb =4.88 GeV and αs(MZ )=0.1172. The program HDECAY [129] which incorporates the routine FeynHiggsFast1.2 [130] for the calculation of the radiative corrections in the MSSM Higgs sector, has been used. modify Figurethe 7.3:bound: Integrating out the top quark and top squarks yields large positive contributions to the quartic Higgs coupling in the low-energy effective theory, especially from these one-loop diagrams.

An alternative way to understand the size of the radiative correction to the h0 mass is to consider an effective theory in which the heavy top squarks and top quarkhavebeenintegratedout.Thequartic Higgs couplings in the low-energy effective theory get large positive contributions from the the one-loop diagrams of fig. 7.3. This increases the steepness of the Higgspotential,andcanbeusedtoobtainthe same result for the enhanced h0 mass. An interesting case, often referred to as the “decoupling limit”, occurs when mA0 mZ.Then 2 2 2 ! mh0 can saturate the upper bounds just mentioned, with mh0 mZ cos (2β)+ loop corrections. The Upper bound 0 0 ≈ particles A , H ,andH± will be much heavier and nearly degenerate, forming an isospin doublet that ~130decouples GeV from sufficiently low-energy experiments. The angle α is very nearly β π/2, and h0 has the − same couplings to quarks and leptons and electroweak gauge bosons as would the physical Higgs boson of the ordinary Standard Model without supersymmetry. Indeed, model-building experiences have shown that it is not uncommon for h0 to behave in a way nearly indistinguishable from a Standard Model-like Higgs boson, even if mA0 is not too huge. However, it should be kept in mind that the couplings of h0 might turn out to deviate significantly from those of a Standard Model Higgs boson. Top-squark mixing (to be discussed in section 7.4) can resultinafurtherlargepositivecontribution 2 to m 0 .Atone-looporder,andworkinginthedecouplinglimitforsimplicity, eq. (7.24) generalizes to: h Djouadi 05 Figure 1.7: The masses of the2 MSSM2 Higgs2 bosons3 as2 a function2 2 of MA for2 two2 values2 2 2 2 2 m 0 = m cos (2β)+ sin β y m ln m˜ m˜ /m + c s (m m )ln(m /m ) h Z 4π2 t t t1 t2 t t˜ t˜ t˜2 t˜1 t˜2 t˜1 tan β =3and 30, in the no mixing (left) and maximal mixing! (right) scenarios with MS =2 − " 1# TeV and all the other SUSY parameters set to 1 TeV. The+c4 fulls4 ( setm2 of radiativem2 )2 corrections(m4 m4 )ln(m2 /m2 ) /m2 . (7.25) is included with m =178GeV, m =4.88 GeV and α (M )=0t˜ t˜ .1172t˜2 −. t˜1 − 2 t˜2 − t˜1 t˜2 t˜1 t t b s Z $ % & Here c and s are the cosine and sine of a top squark mixing angle θ ,definedmorespecificallybelow As can be seen, a maximalt˜ valuet˜ for the lighter Higgs mass, M 135 GeV, is obtainedt˜ following eq. (7.71). For fixed top-squark masses,h ∼ the maximum possible h0 mass occurs for rather for large MA values in thelarge maximal top squark mixing mixing, scenarioc2s2 with= m tan2/[mβ2=30[themassvalueisalmost+ m2 2(m2 m2 )/ln(m2 /m2 )] or 1/4, whichever is less. t˜ t˜ t t˜2 t˜1 − t˜2 − t˜1 t˜2 t˜1 constant if tan β is increased].It follows For that no the stop quantity mixing, in or square when brackets tan β is in small, eq. (7.25) tan β is< always3, the less than m2[ln(m2 /m2)+3]. t t˜2 t upper bound on the h Theboson LEP mass constraints is smaller on by the more MSSM than Higgs 10 sector GeV makein each the case case∼ and of large the top-squark mixing noteworthy. Including these and other important correctionsmax [185]-[194], one can obtain only a weaker, but still combined choice tan β =3andXt =0,leadstoamaximalvalueMh 110 GeV. Also for very interesting, bound ∼ large MA values, the A, H and H± bosons [the mass of the latter being almost independent of the stop mixing and on the value of tan β]becomedegenerateinmass.Intheoppositemh0 < 135 GeV (7.26) max ∼ case, i.e. for a light pseudoscalar Higgs boson, MA < M ,itisMh which is very close to h 2 in the MSSM. This assumes that all of the sparticles that can contribute to m 0 in loops have masses M ,andthemassdifference is particularly small for∼ large tan β values. h A that do not exceed 1 TeV. By adding extra supermultiplets to the MSSM, this bound can be made even weaker. However, assuming that none of the MSSM sparticles have masses exceeding 1 TeV and that all of the couplings in53 the theory remain perturbative up to the unification scale, one still has [195]

mh0 < 150 GeV. (7.27) ∼ This bound is also weakened if, for example, the top squarks are heavier than 1 TeV, but the upper bound rises only logarithmically with the soft masses, as canbeseenfromeq.(7.24).Thusitisafairly robust prediction of supersymmetry at the electroweak scalethatatleastoneoftheHiggsscalarbosons must be light. (However, if one is willing to extend the MSSM inacompletelygeneralwayabovethe

70 particles do not couple to gauge boson pairs; the CP–odd and charged Higgs boson couplings to respectively, hZ and hW ,arealsoproportionaltothisfactorandvanishinthedecoupling 2 limit. In addition, in contrast to the SM where the self–couplings are proportional to MHSM , the trilinear and quartic Higgs couplings in the MSSM are all proportional to the gauge couplings and never become large; in fact, they all tend to either zero or 1whenexpressed ± in units of M 2 /v,asseenin 1.3.3. Nevertheless, since these particles are the remnantsof Z § the electroweak symmetry breaking which occurs at the Fermi scale, they are expected to

have masses not too far from this scale, i.e. MH,A,H± < (1 TeV). ∼ O 1.4.2 Constraints from direct Higgs searches The neutral Higgs bosons

The search for the Higgs bosons was the main motivation for extending the LEP2 energy up to √s 209 GeV [159]. At these energies, there are two main processesfortheproduction # of the neutral Higgs bosons of the MSSM: the Higgs–strahlung process [158,160–162] already discussed in the SM Higgs case [see I.4.2], and the associated production of CP–even and § CP–odd Higgs bosons [163,164]; Fig. 1.14. In the case of the lighter h particle, denoting the

SM Higgs cross section by σSM,theproductioncrosssectionsaregivenby

+ 2 + σ(e e− hZ)=ghZZ σSM(e e− hZ) → → 3 + 2 + λAh σ(e e− hA)=ghAZ σSM(e e− hZ) 2 2 (1.160) → → × λZh(λZh +12MZ /s) where λ =(1 M 2/s M 2/s)2 4M 2M 2/s2 is the two–body phase space function; the ij − i − i − i j additional factor for the last process accounts for the fact that two spin–zero particles are produced and the cross section should be proportional to λ3 as discussed in I.4.2.2. LEP searches: ij § e+ Z e+ A Z∗ Z∗ • •

e− h e− h Figure 1.14: Diagrams for MSSM neutral Higgs production at LEP2 energies.

2 2 with2 decays2 to taus and+ bottoms Since gAhZ =cos(β α)whileghZZ =sin(β α), the processes e e− hZ and + − 11 − +→ e e− hA are complementary .Inthedecouplinglimit,MA MZ , σ(e e− hA) → 2 + ' 2 → vanishes since g 0whileσ(e e− hZ)approachestheSMlimitsinceg 1. In hAZ ∼ → hZZ ∼ ! ! 11As will be discussed in the next section, this remark can be extended to the heavierm CP–even-max Higgs boson and the complementarity is doubled in this case: there is one between the processes e+e−h° HZ and tan m -max tan h° e+e− HA as for the h boson, but there is also a complementarity between the production of→ the h and H bosons.→ The radiative corrections to these processes will also be discussed in 4.1. 10 10 § 69 Excluded by LEP

Excluded 1 1 by LEP Theoretically Inaccessible 0 20 40 60 80 100 120 140 0 200 400 2 2 mh° (GeV/c ) mA° (GeV/c ) ! ! No Mixing

tan No Mixing tan

10 10

Excluded by LEP

Excluded Theoretically by LEP 1 Inaccessible 1

0 20 40 60 80 100 120 140 0 200 400 2 2 mh° (GeV/c ) mA° (GeV/c )

Figure 1.17: 95% CL contours in the tan β–Mh (left) and tan β–MA (right) planes excluded by the negative searches of MSSM neutral Higgs bosons at LEP2,fromRef.[167].They are displayed in the maximal mixing (top figures) and no–mixing (lower figures) scenarios with MS =1TeV and mt =179.3 GeV. The dashed lines indicate the boundaries that are excluded on the basis of Monte–Carlo simulations in the absence of a signal. [171, 172] but the obtained bounds are not yet competitive with those discussed above.

The charged Higgs boson

+ In e e− collisions, the production of a pair of charged Higgs bosons [163, 173] proceeds through virtual photon and Z boson exchange; Fig. 1.18a. The cross section depends only

73 After LEP, a heavy stop is essential to keep the MSSM alive Higgs bound: 3m4 m2 =< m2 + t ln(m /m )+ h Z 2π2v2 stop t ···

Needed to be large to be above the experimental bound

Higgs searches rules out a big chunk of the parameter space of the MSSM! In minimal Sugra:

2 2 M0 /µ

2 2 Only the thin M1/2/µ “withe spike” is left! Giudice, Rattazzi MSSM Higgs hunting at the LHC

Bad news: h too light to decay to WW/ZZ A, H⁺ have very small couplings to WW/ZZ H small regions with sizable couplings to WW/ZZ Good news: Regions where the decays of H, A, H⁺ to leptons are enhanced (Large Tanβ region) Due to: m = Y H τ τ ￿ d￿

can be larger than in the SM, if H is smaller than v ￿ d￿ Neutral MSSM Higgs CMS NOTE-2010/008 ! Associated production with b-jets Near future: ! combining three "" channels " with at least one leptonic decay: !had!#, !had!e, !e!# 5# discovery ! cross-sections scaled from 14 TeV to 7 TeV 95% exclusion • Large range covered, down to tan$ ~ 15 at low mA

Expected sensitivity to the MSSM Higgs boson More interestingly:in the MSSM!! channel could be ruled out F. Ronga (ETH Zurich) – Planck 2010 – June 3, 2010if a Higgs ➞ WW/ZZ with mass ~16021 GeV is discovered in the first LHC run 3. MSSM Higgses: detection at the LHC In the long run....

50 ! ATLAS-1 40 ATLAS - 300 fb

tan maximal mixing 30

0 0 0 -+ 20 h H AH

0 -+ 0 0 0 h H h H A 10 9 8 7 6 0 h only 5

4 LEP 2000 0 0 3 h H LEP excluded

2 + 0 0 0 -+ 0 - h H AH h H

1 50 100 150 200 250 300 350 400 450 500 m (GeV) A

LaThuile, 06/03/2009 Theory aspects of Higgs searches – A. Djouadi –p.18/24 Superpartners at Hadron Colliders Superpartners at Hadron Colliders

Neutral gaugino + Higgsino mix: Mass-eigenstate = χ 0 neutralino

Charged gaugino + Higgsino mix: Mass-eigenstate = χ + chargino Main consideration: Due to R-parity superpartners are produced in pairs, and decay, in cascade, down to the lightest one (neutral) that, being stable, goes awayPhysics at the ILC fromSUSY, LHC+IL theC detector Typical SUSY Event at LHC

A classic:

J¨orgen Samson (DESY) Physics at the ILC 22. January 2007 28 / 38 Physics at the ILC SUSY, LHC+ILC Strategy: Detect leptons or jets + Missing ET Typical SUSY Event at LHC

J¨orgen Samson (DESY) Physics at the ILC 22. January 2007 28 / 38 Final states with same-charge dilepton due to the Majorana nature of the gluino Tevatron

From P. Wittich at “Physics at the LHC 2010” conference susy in jets + met:generic squark/gluino production

• Large production cross section, bkgnds from multi-jet, Z→νν, top • Optimize searches as a function of (Missing ET, njet) • No excess seen so far • Limits for 2 (2.1)/fb of data for CDF (D0) • interpret results in mSUGRA-like SUSY scenario

!"#

20 Trileptons: Chargino-Neutralino Search, 3.2/fb ~ • Very clean signature: 0 • Missing ET due to undetected ν, χ 1 • 3 isolated leptons, lower momentum~

• 3 identified leptons (e,μ) • 2 identified leptons + track (l) • “Tight” and “loose” e,μ categories

• Rejection using kinematic selections on: ml+l-, njets, Missing ET, Δφ between leptons...

Channel SM expected Data Good agreement between data and SM prediction → set limit Trilepton 1.5 ± 0.2 1 Lepton+trk 9.4± 1.2 6 !"# Peter Wittich 16 Chargino-neutralino results PRL 101, 251801 (2008) •interpret null result in mSugra SUSY scenario as excluded region in mSUGRA m -m space a convenient/conventional benchmark 0 !

m0 = 60 GeV, tan β=3, A0=0, μ>0

CDF @ 2 fb-1 ~ ~ ~ ~

Limits depend on relative χ0 -l masses ± 2 2 Excludes mχ 1 < 164 (154 Exp.) GeV/c ! mχ2 > ml ~ increases BR to e/μ ! mχ2 ! ml reduces acceptance

D0 limit in 2.3/fb: Phys. Lett. B 680, 34 (2009) !"#

Peter Wittich 17 Miss 2 b jets + ET - ~q and LQ ZH νν¯b¯b → • Final state familiar from Higgs searches • missing ET and b quarks pp¯ ˜b ¯˜b bχ˜0¯bχ˜0 • Also good signal for leptoquarks and SUSY → 1 1 → 1 1 • event selection: LQ3 ντ b • b tagging (D0: neural-net algo) → miss jet1 jet2 • two b-tagged jets, ET , Sign., ΣET 6 p + p miss T T • optimize pT, ET , HT, Xjj for SUSY/LQ3 Xjj = to the final uncertainty. The total systematic uncertaintysignals SM CDF RunII DATA (L=2.65 fb-1) HT on the SM predictions varies between 19% and 18% as Total Syst. Uncertainty SM + MSSM ∆M increases. In the case of the MSSM signal, various 102 low !M analysis102 2 ~ Mb = 123 GeV/c sources of uncertainty on the predicted cross sections at 1 2 M 0 = 90 GeV/c prospino2 10 "# 10 NLO, as determined using , are considered: 1 low δM(LSP, b squark) the uncertainty due to PDFs is computed using the Hes-pre-tagtagged 1 1 sian method [27] and translates into a 12% uncertainty on the absolute predictions; variations of the renormal- 100 200high ! 300M analysis 200 400 600 2 ~ Mb = 193 GeV/c ization and factorization scale by a factor of two change 10 1 10 2 M#0 = 70 GeV/c events per bin " the theoretical cross sections by about 26%. Uncertain- 1 ties on the amount of initial- and final-state gluon radi- 1 1 hi δM(LSP, b squark) ation in the MSSM Monte Carlo generated samples in- 6 troduce a 10% uncertainty on the signal yields. The 3% 100 200 300 200 400 600 uncertainty on the absolute jet energy scale translates E/ T [GeV] HT + E/ T [GeV] tointo the a 9% final to uncertainty. 14% uncertainty The on total the MSSM systematic predictions. uncertainty SM CDF RunII DATA (L=2.65 fb-1) !"# Total Syst. Uncertainty SM + MSSM onOther the sources SM predictions of uncertainty varies include: between a 4% 19% uncertainty and 18%FIG. as 1: Measured E/T andPeterH WittichT +E /T distributions (black dots) 21 for low-∆M (top)2 and high-∆M analyses (bottom),2 compared ∆dueM toincreases. the determination In the case of the ofb-tagging the MSSM efficiency, signal, and various 10 low !M analysis10 to the SM predictions (solid lines) and the2 SM+MSSM pre- ~ a 2% to 1% uncertainty due to the uncertainty on the Mb = 123 GeV/c sources of uncertainty on the predicted cross sections at 1 dictions (dashed lines). The shaded bands2 show the total M 0 = 90 GeV/c trigger efficiency. The total systematicprospino2 uncertainty on 10 "# 10 NLO, as determined using , are considered:systematic uncertainty on the SM1 predictions. thethe MSSMuncertainty signal due yields to varies PDFs between is computed 30% and using 32% theas Hes- ∆M increases. Finally, an additional 6% uncertainty on 1 1 sianthe quoted method total [27] integrated and translates luminosity into is also a 12% taken uncertainty into 100 200 300 200 400 600 onaccount the absolute in both SM predictions; background variations and SUSY of signal the pre- renormal-background predictions are takenhigh !M into analysis account. For each 2 M~ = 193 GeV/c MSSM point considered, observedb and expected limits izationdictions. and factorization scale by a factor of two change 10 1 10 2 M#0 = 70 GeV/c are computedevents per bin separately for both" low- and high-∆M theFigure theoretical 1 shows cross the measured sections byE/T aboutand HT 26%.+ E/T Uncertain-dis- 1 analyses, and the one with the best expected limit is tiestributions on the compared amount to of the initial- SM predictions and final-state after all gluon final radi- 1 1 selection criteria are applied. For illustrative purposes, adopted as the nominal result. Cross sections above 0.1 ation in the MSSM Monte Carlo generated samplespb in- are excluded at 95% C.L. for the range of sbottom troducethe figure a indicates 10% uncertainty the impact on of thetwo signalgiven MSSM yields. sce- The 3% 100 200 300 200 400 600 narios. The data are in agreement with the SM predic- masses considered. Similarly, the observed numbers of E/ T [GeV] HT + E/ T [GeV] uncertaintytions within uncertainties on the absolute for each jet of the energy two analyses scale translates at events in data are translated into 95% C.L. upper limits intolow and a 9% high to∆ 14%M. uncertainty In Table 2, the on observed the MSSM number predictions. of for sbottom and neutralino masses, for which the uncer- Otherevents and sources the SM of predictions uncertainty are include: presented a in 4% each uncertainty case. taintiesFIG. on the 1: theoretical Measured crossE/T sectionsand HT + areE/T alsodistributions included (black dots) dueA global to theχ2 determinationtest applied to all of data the pointsb-tagging in Fig. effi 1,ciency, and andin the limitfor low-calculation,∆M (top) and and where high- both∆M analysesanalyses are (bottom), com- compared aincluding 2% to correlations1% uncertainty between due systematic to the uncertainty uncertainties, on thebined into the the same SM way predictions as for the (solidcross section lines) limits. and the Fig- SM+MSSM pre- ure 2 showsdictions the expected (dashed and lines). observed The exclusion shaded bands regions show the total triggergives a 30% efficiency. probability The for data total to systematic be consistent uncertainty with the on SM. in the sbottom-neutralinosystematic uncertainty mass on plane. the SM For predictions. the MSSM the MSSM signal yields varies between 30% and 32%scenario as considered, sbottom masses up to 230 GeV/c2 ∆M increases.(2.65 Finally, fb−1) anlow additional∆M high ∆ 6%M uncertaintyare on excluded at 95% C.L. for neutralino masses below mistags 51.4 ± 18.218.5 ± 5.5 2 the quoted total integrated± luminosity± is also taken into70 GeV/c . This analysis extends the previous CDF lim- QCD jets 7.6 1.91.6 0.2 background predictions are taken into account.2 For each account in bothtop SM background 21.2 ± 3.47 and.8 ± 1 SUSY.3 signal pre-its [4] on the sbottom mass by more than 40 GeV/c . dictions. Z → νν¯+jets 27.7 ± 8.810.9 ± 3.5 In summary,MSSM we point report considered, results on a searchobserved for sbottom and expected limits Z/γ∗ → l+l−+jets 0.5 ± 0.20.11 ± 0.04 are computed separately for both low- and high-∆M Figure 1 shows→ the measured± E/ ±and HT + E/ dis-pair production in pp collisions at √s =1.96 TeV, based W lν+jets 22.3 7.37T.3 2.4 T −1 tributionsWW,WZ,ZZ compared to the3.1 ± SM0.51 predictions.4 ± 0.2 after all finalon 2.65analyses, fb of CDF and Run the II one data. with The the events best are expected se- limit is ± ± selection criteriaSM prediction are applied. 133.8 26 For.447 illustrative.6 8.8 purposes,lected withadopted large E as/T theand nominal two energetic result. jets Cross in the sections final above 0.1 Events in data 139 38 state, and at least one jet is required to originate from a the figure indicates the impact of two given MSSM sce- pb are excluded at 95% C.L. for the range of sbottom b quark.masses The measurements considered. are Similarly, in agreement the with observed SM numbers of narios. The data are in agreement with the SM predic-predictions for backgrounds. The results are translated TABLE II: Number of events in data for the two analyses events in data are translated into 95% C.L. upper limits tionscompared within to SM uncertainties predictions, including for each statistical of the two and analyses sys- into at 95% C.L. upper limits on production cross sections lowtematic and uncertainties high ∆M summed. In Table in quadrature. 2, the observed numberand of sbottomfor sbottom and neutralino and neutralino masses in a masses,given MSSM for sce- which the uncer- tainties on the theoretical cross0 sections are also included events and the SM predictions are presented in each case.nario for which the exclusive decay b1 bχ1 is assumed, ! → ! AThe global resultsχ2 test are applied translated to into all data 95% pointsconfidence in Fig. level 1, andand significantlyin the limit extend calculation, previous CDF and where results. both analyses are com- including(C.L.) upper correlations limits on the between cross section systematic for sbottom uncertainties, pair We thankbined the in Fermilab the same sta wayff and as for the the technical cross section staffs limits. Fig- givesproduction a 30% at probability given sbottom for and data neutralino to be consistent masses, us- with theof the participatingure 2 shows institutions the expected for their and vital observed contribu- exclusion regions SM.ing a Bayesian approach [28] and including statistical and tions. Thisin the work sbottom-neutralino was supported by the mass U.S. Department plane. For the MSSM systematic uncertainties. For the latter, correlations be- of Energyscenario and National considered, Science sbottom Foundation; masses the Italian up to 230 GeV/c2 tween systematic(2.65 uncertainties fb−1) low on signal∆M effihighciencies∆M and Istitutoare Nazionale excluded di Fisica at 95% Nucleare; C.L. for the neutralino Ministry of masses below mistags 51.4 ± 18.218.5 ± 5.5 2 ± ± 70 GeV/c . This analysis extends the previous CDF lim- QCD jets 7.6 1.91.6 0.2 2 top 21.2 ± 3.47.8 ± 1.3 its [4] on the sbottom mass by more than 40 GeV/c . Z → νν¯+jets 27.7 ± 8.810.9 ± 3.5 In summary, we report results on a search for sbottom Z/γ∗ → l+l−+jets 0.5 ± 0.20.11 ± 0.04 → ± ± pair production in pp collisions at √s =1.96 TeV, based W lν+jets 22.3 7.37.3 2.4 −1 WW,WZ,ZZ 3.1 ± 0.51.4 ± 0.2 on 2.65 fb of CDF Run II data. The events are se- ± ± SM prediction 133.8 26.447.6 8.8 lected with large E/T and two energetic jets in the final Events in data 139 38 state, and at least one jet is required to originate from a b quark. The measurements are in agreement with SM TABLE II: Number of events in data for the two analyses predictions for backgrounds. The results are translated compared to SM predictions, including statistical and sys- into 95% C.L. upper limits on production cross sections tematic uncertainties summed in quadrature. and sbottom and neutralino masses in a given MSSM sce- 0 nario for which the exclusive decay b1 bχ1 is assumed, ! → ! The results are translated into 95% confidence level and significantly extend previous CDF results. (C.L.) upper limits on the cross section for sbottom pair We thank the Fermilab staff and the technical staffs production at given sbottom and neutralino masses, us- of the participating institutions for their vital contribu- ing a Bayesian approach [28] and including statistical and tions. This work was supported by the U.S. Department systematic uncertainties. For the latter, correlations be- of Energy and National Science Foundation; the Italian tween systematic uncertainties on signal efficiencies and Istituto Nazionale di Fisica Nucleare; the Ministry of Supersymmetric top in the e+μ+bb+MET, 3.1/fb • 3rd generation again - special role in SUSY Miss ¯ • Look for decay mode in e μ final state with ET >18 GeV pp¯ t˜1t˜1 • Low SM backgrounds (Z→ττ,ttbar) → Miss • Reject with δΦ(lepton, ET ) cuts (t˜1 ν˜b￿) = 100% • no explicit b tag required B → R parity conserving • Consider small and large δm(stop, sneutrino) • drives kinematics of accepted events

• Bin events in two kinematic variables D’ Preliminary Result 140 -1 e!+ee Obs (1.1 220

60 4 LEP II excluded

240 LEP I excluded 060 50 80 100 100 150 120 200 140 250160 180 300 200 350 H [GeV] T Stop mass (GeV) D0 Conference Note 5937-CONF Blue: this result Peter Wittich 23 LHC

From P. Jenni at “Physics at the LHC 2010” conference !"#$%&%'%()$*+,$-.&&%&/$0%)) ()-#(12$3('4"$53(26#$#74##18 #&1$9:;:$'"#$!#<('-=&$-#(4"$

>$'2?%4()$#7(3?)#@$&='#$'"('$'"#$3%AA%&/ '-(&A<#-A#$ #&#-/2$?#-B=-3(&4# $#&'#-A 1%-#4')2$'"#$CDBB#4'%<#$E(AAFG$1#'#4'=-A$3.A'$ 6#$0#))$.&1#-A'==1$B=-$'"#A#$3#(A.-#3#&'A$ !"#$%&'&()Ͳ'%*+,-%&'& 267-/08-,.9:;+889/<9,=>+.:??@ )' !-.-/*-,,01$2345 !"#$%&#'()$*+,-'./(.'&+0(1,.(2!23(4&.#$+"'*(&#(#0'(567 8$9)$+&#$-'(4",#*:(%,)'";)'4'9)'9#<= !"#$%$&'

>0'(%&** ( *+&"'( 4.,?')( 1,. *@A&.B* &9)(C"A$9,* D$""(?'

!"#$%&'&()Ͳ'%*+,-%&'& !($)(*+,' #/4$+&""/(EFG(>'H ?/(EIJK )% !-.-/*-,,01$2345 267-/08-,.9:;+889/<9,=>+.:??@ 2 The systematic cancellation of the dangerous contributionsto∆mH can only be brought about by the type of conspiracy that is better known to physicists as a symmetry. Comparing eqs. (1.2) and (1.3) strongly suggests that the new symmetry ought to relatefermionsandbosons,becauseofthe 2 relative minus sign between fermion loop and boson loop contributions to ∆mH .(NotethatλS must be positive if the scalar potential is to be bounded from below.) If each of the quarks and leptons of the Standard Model is accompanied by two complex scalars with λ = λ 2,thentheΛ2 contributions of S | f | UV Figures 1.1a and 1.1b will neatly cancel [3]. Clearly, more restrictions on the theory will be necessary to ensure that this success persists to higher orders, so that, for example, the contributions in Figure 1.2 and eq. (1.4) from a very heavy fermion are canceled by the two-loop effects of some very heavy bosons. Fortunately, the cancellation of all such contributions to scalar masses is not only possible, but is actually unavoidable, once we merely assume that thereexistsasymmetryrelatingfermionsand bosons, called a supersymmetry. A supersymmetry transformation turns a bosonic state into a fermionic state, and vice versa. The operator Q that generates such transformations must be an anticommuting spinor, with

Q Boson = Fermion ,QFermion = Boson . (1.5) | ! | ! | ! | !

Spinors are intrinsicallyOther complex MSSM objects, so goodies:Q† (the hermitian conjugate of Q)isalsoasymmetry generator. Because Q and Q† are fermionic operators, they carry spin angular momentum 1/2, so it is clear that supersymmetry must be a spacetime symmetry. The possible forms for such symmetries in an interacting quantum field● Gauge theory coupling are highly unification restricted by the Haag-Lopuszanski-Sohnius extension of the Coleman-Mandula● theorem The lightest [4]. For supersymmetric realistic theories particlethat, like (LSP) the Standard Model, have chiral fermions (i.e., fermions whose left- and right-handed pieces transform differently under the gauge group) can be Dark matter and thus the possibility of parity-violating interactions,thistheoremimpliesthatthegeneratorsQ and Q† must satisfy an algebra● ofLocal anticommutation supersymmetry and must commutation incorporaterelations gravity: with the schematic form

µ Q, Q† = P , (1.6) { } Q, Q = Q†,Q† =0, (1.7) ● Fits well EWPT{ µ from} {LEP/Tevatronµ } [P ,Q]=[P ,Q†]=0, (1.8) where P µ is the four-momentum generator of spacetime translations. Here we have ruthlessly sup- ➥ pressed the spinor indices on Q and Q It†;afterdevelopingsomenotationwewill,insection3.1,der has allowed us to write more than 20,000 papers ive the precise version of eqs. (1.6)-(1.8) with indices restored. In the meantime, we simply note that the appearance of P µ on the right-hand side of eq. (1.6) is unsurprising, since it transforms under Lorentz boosts and rotations as a spin-1 object while Q and Q† on the left-hand side each transform as spin-1/2 objects. The single-particle states of a supersymmetric theory fall into irreducible representations of the supersymmetry algebra, called supermultiplets.Eachsupermultipletcontainsbothfermionandboson states, which are commonly known as superpartners of each other. By definition, if Ω and Ω are | ! | !! members of the same supermultiplet, then Ω is proportional to some combination of Q and Q | !! † operators acting on Ω ,uptoaspacetimetranslationorrotation.Thesquared-massoperator P 2 | ! − commutes with the operators Q, Q†,andwithallspacetimerotationandtranslationoperators,so it follows immediately that particles inhabiting the same irreducible supermultiplet must have equal eigenvalues of P 2,andthereforeequalmasses. − The supersymmetry generators Q, Q† also commute with the generators of gauge transformations. Therefore particles in the same supermultiplet must also be in the same representation of the gauge group, and so must have the same electric charges, weak isospin, and color degrees of freedom. Each supermultiplet contains an equal number of fermion and boson degrees of freedom. To prove this, consider the operator ( 1)2s where s is the spin angular momentum. By the spin-statistics − theorem, this operator has eigenvalue +1 acting on a bosonic state and eigenvalue 1actingona −

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