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Not to be quoted or reproduced without permission. uesaeadSuperfields and Superspace 3 representations and algebra Supersymmetry 2 Motivation Physical 1 Contents . rbeso h tnadModel Standard the of Problems 1.5 symmetries of Importance symmetries 1.4 of Classes symmetry 1.3 principle: Basic QFT 1.2 theory: Basic 1.1 . hrlsuperfields Chiral 3.2 superspace about Basics 3.1 supersymmetry Extended 2.5 Poincaré group the 2.4 of Representations 2.3 algebra SUSY 2.2 spinors and Poincaré symmetry 2.1 .. ento n rnfraino h eea clrsu superfields scalar on general Remarks the of transformation and 3.1.4 Definition variables Grassmann of 3.1.3 Properties cosets and 3.1.2 Groups 3.1.1 of representations Massive of representations 2.5.3 Massless supersymmetry extended of 2.5.2 Algebra 2.5.1 Parity supermultiplet 2.4.4 Massive supermultiplet supermultiplet 2.4.3 Massless a in fermions and 2.4.2 Bosons 2.4.1 algebra Graded supersymmetry of 2.2.2 History 2.2.1 spinors Dirac spinors Weyl of 2.1.5 Products of 2.1.4 Generators of tensors invariant and 2.1.3 Representations group Lorentz the of 2.1.2 Properties 2.1.1 Model Standard the of Modifications 1.5.1 Model Standard The 1.4.1 N uesmer representations supersymmetry 1 = SL (2 , C ) – i – N N > > uesmer n P states BPS and supersymmetry 1 supersymmetry 1 SL (2 , C ) perfield 27 12 10 10 33 32 30 29 27 27 25 22 22 21 21 19 18 17 17 16 13 13 13 7 1 8 7 7 6 5 4 2 2 2 1 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ossetfaeokicroaigteetotere is theories two these incorporating framework consistent A ories. h hsclyosral lmnayprilswihaet are The which particles fields. elementary quantum observable are physically entities the fundamental the theory this irsoial ehave we QFT Microscopically theory: i Basic live we universe 1.1 the about facts relevant some review us Let Motivation Physical 1 (M model standard supersymmetric minimal the Introducing 6 breaking Supersymmetry 5 Lagrangians supersymmetric dimensional Four 4 4.3 superfields Vector 4.2 4.1 . rsadCn fteMSSM the of Cons and Pros problem 6.5 hierarchy MSSM The the in breaking 6.4 Supersymmetry 6.3 Interactions 6.2 Particles 6.1 Preliminaries 5.1 supersymmetry local about facts few A 4.5 theorems Non-renormalisation 4.4 .. raiglclsupersymmetry local Breaking 5.1.4 model 5.1.3 O’Raifertaigh 5.1.2 5.1.1 History 4.4.1 4.3.2 4.3.1 integral superspace a as Action Lagrangian superfield non-examinable vector 4.2.6 strength: Abelian field abelian - 4.2.5 Non superfield strength field 4.2.4 Abelian superfield gauge vector Zumino 4.2.3 Wess the of transformation and 4.2.2 Definition 4.2.1 Lagrangian superfield Chiral 4.1.1 N N 2 = supersymmetry global 1 = , D F N N lblsupersymmetry global 4 embreaking term embreaking term 4 = 2 = unu mechanics quantum – 1 – and pca relativity special ebsccnttet fmatter of constituents basic he unu edter (QFT) theory field quantum n. recttoscrepn to correspond excitations ir stofnaetlthe- fundamental two as SSM) In . 46 34 51 47 46 45 45 44 44 43 43 42 40 39 38 38 37 37 34 34 60 58 55 52 51 50 49 48 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. emos( fermions ymtyi motn o aiu reasons: various for important is symmetries Symmetry of Importance 1.4 hrce.Prilscnb lsie ntognrlclasse general two in There classified be interactions. can known Particles the all of character. mediators the as well as h andffrnei htfrin a esont aif th satisfy to shown be can fermions that is difference main The o lmnayprils ecndfietognrlcasso classes general two define can we particles, elementary symmetries For of Classes 1.3 s nature. science understand of better to history s role physical the important a to Throughout made be unchanged. can observables that transformation a is symmetry of A processes, concept elementary the study is to framework processes basic the is QFT symmetry If principle: Basic 1.2 differences. their all despite n h eitr falteohritrcin.Te r not are They interactions. see, el will other of we the mediator As all and principle. of light ferm mediators of are the (particle hadrons) and photon other the all include and hand neutrons protons, el make (including leptons (that the particles: atoms. matter elementary of All diversity vast occup cannot the fermions explaining identical therefore two that states which ple”, • • h ieetfilsi edtheory, field a in fields different the symmetries: Internal er sa is metry oa indices Roman xmlsaerttos rnltosad oegenerally, more and, translations transformations rotations, are Examples hoyatn xlctyo h pc-iecoordinates, space-time the on explicitly acting theory symmetries: Space-time tions scle a called is s htdefine that = n + oa symmetry local lblsymmetry global 1 2 ∈ ,b a, Z enn pca eaiiya elas well as relativity special defining eea relativity general + ae h orsodn ed.If fields. corresponding the label supersymmetry 2 1 symmetry .Bsn n emoshv eydffrn hsclbehaviou physical different very have fermions and Bosons ). hs r ymtista orsodt rnfrain of transformations to correspond that symmetries are These x µ hs ymtiscrepn otasomtoso field a on transformations to correspond symmetries These . ncs fsaetm dependent space-time of case in ; 7→ Φ . a x ( ′ x µ . ) ( x sasmer htuie oosadfermions and bosons unifies that symmetry a is – 2 – ν 7→ ) ,ν µ, , M a b Φ b ( 0 = crn n etio)adquarks and neutrinos) and ectrons x ) :bsn (spin bosons s: M h aeqatmsae and state, quantum same the y eea oriaetransforma- coordinate general . , oe ed aeaparticle-like a have fields fore, symmetries: f 1 a n olt er bu these about learn to tool one , mer a lydavery a played has ymmetry b 2 crmgei interaction), ectromagnetic os ooso h other the on Bosons ions. e , se evn h physical the leaving ystem scntn hntesym- the then constant is osrie ytePauli the by constrained 3 Pauli . oet-adPoincaré and Lorentz- M a b ( ecuinprinci- ”exclusion x s h symmetry the ) = n ∈ Z and ) r. 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Not to be quoted or reproduced without permission. h tnadMdli eldfie n urnl elconfirm well currently and well-defined is Model Standard Model The Standard The 1.4.1 • • gauged dimensions 4 Poincaré in symmetries: space-time h ymtyof symmetry the au (VEV) value hntemnmmi at is minimum the then ymtisi motn o tlattoreasons: two least at for important is symmetries neatos SU(2) interactions. ymtysneteoii sivrat oee ftepoten the if However invariant. is origin the since symmetry U(1) i)Teeitneo idnsmere mle httefun the that implies symmetries hidden of existence The (ii) i hsi aua a oitouea nrysaei h sy the in scale energy an introduce to way natural a is This (i) em ytennvnsigVV npriua,w ilseta o t for that see will we particular, In VEV. vanishing model non the by xlctyraie nntr.Tesadr oe stetyp further the are is dimensions model extra though standard of theories even The and symmetries supersymmetry nature. of ful in to the number realised resource of indefinite explicitly unlimited those an not essentially with and an theories in ca opens-up limite live we This a we symmetries vacuum observe manifest theory. the we only of that the symmetries fact because the the is despite This huge symmetry. be may nature of effect. this matter from the mass their and obtain bosons couplings, Yukawa gauge their electroweak the model, standard iue1 Figure h ag ed r pn1bsn,freapetephoton the example for bosons, spin-1 are fields gauge The . 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Not to be quoted or reproduced without permission. fcus en olwda h H.Teeprmn ilexpl will experiment The LHC. the at St followed the being generalise attain course that currently of principles the St pr underlying above making and III of energies particles Part way explore the traditional See to pr a the experiments are Model. of Experiments all Standard details. or the some more generalise solve to could order that in extensions find to wish We cl n e hsc eodteSadr oe.Ntc that scale Notice Planck Model. the Standard to the closer beyond physics new and scale uigteps 0yas oee,teeaesilsm unres some still are there However, ve experimentally years. carefully 20 past been a the has science during It all of century. cornerstones past the the of of one is Model Model Standard Standard the The of Problems 1.5 • • • • h omlgclcntn Λ rbe:poal h bigges the probably problem: (Λ) constant cosmological The htpril osiue h akmte bevdi h un the Model. in Standard observed the matter in dark tained the constitutes particle What usino h h irrh ssal ihrsett h qu the to respect supersymmetry. with scale stable is the hierarchy called the why of question h tnadMdlhsaon 0prmtr,wihms eme be must which parameters, 20 hand’. around ‘by has Model Standard The xetdt eo re ftehais cl nteter div theory the in scale heaviest the of order of be to expected 1? hsc.Λi h nrydniyo resaetm.Wyi (Λ is Why time. space free of density energy the is Λ physics. 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Not to be quoted or reproduced without permission. ewl otydsustoeΛcniuul once oide to connected continuously Λ those discuss mostly will We hcrnu group thochronous h ona´ ru orsod otebscsmere fs of symmetries basic coordinates the space-time to corresponds Poincaré group The spinors and Poincaré symmetry representations 2.1 and algebra Supersymmetry 2 iesoa arxrpeetto o the for representation matrix dimensional 4 A rttosadLrnzbot)and boosts) Lorentz and (rotations h eeaosof generators The n h oet ler rte ntrso ’ n ’ is K’s and J’s of terms in written algebra Lorentz the and hycnb eaae ewe hs htaecnetdt the to connected Λ are reversal parity that (i.e. those not between separated be can They enwso htlcly(..i em fteagba,w hav we algebra), the of terms in (i.e. locally group that show Lorentz now the We of Properties 2.1.1 oet rnfrain ev h erctensor metric the leave transformations Lorentz 1 hs oss ftesbru ftasomtoswihhave which transformations of subgroup the of consist These ie ute oiaint td supersymmetry. hap study so to motivation It ext further and QFT. a supersymmetry of gives requires framework theory basic string our consistency, beyond goes which theory h M [ h K M h µν i P K , µν µ M , 1 P , P , SO j = ] SO ρσ (1 σ ν i i i − (1 x , P ( )( 3) µ iǫ x , M J 0 = = = 3) µ diag(1 = ijk sfollows: as i ρσ J ↑ J eeaosfrtePicregopaetehermitian the are Poincaré group the for Generators . SO i = 7→ ) i i k µ frttosand rotations of , P M ν (1 2 1 µ x µσ , , ′ = η P ǫ µ 3) − ijk νσ [ η σ J Λ 1 νρ i Λ = , tasain)wt algebra with (translations) T − K , M = ∼ − − η i + jk – 7 – 1 = Λ Lorentz j P η , | SU = ] K , µσ M ν {z − µ η M ν )adtm eeslΛ reversal time and 1) } δ νρ (2) K µσ iǫ η ρ µν x µν ijk i η ν η ν × µσ fLrnzbot)cnb xrse as expressed be can boosts) Lorentz of is i K − diag(1 = + SU k − = , η eΛ=+ n Λ and +1 = detΛ translation (2) ρµ M |{z} M δ a . µρ σ µ [ , 0 J ν i η i − eilrltvt,i cson acts it relativity, pecial correspondence a e dniyadtoeta are that those and identity J , , νσ 1 . tt,ie the i.e. ntity, admnin lo This also. dimensions ra , j − = ] − T 1 0 0 M , diag( = iǫ ≥ − νσ ijk esta o its for that pens 1. )invariant: 1) η J µρ k . − rpror- proper 1 , 1 , M 1 , µν 1)). Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ecninterpret can We enwcntuttelinear the construct now We nain,weesteato of action the whereas invariant, ne parity Under oseti,tk vector 4 a take this, see To Transformations where hc satisfy which h a between map The evstedeterminant the serves .. ersnain n nain esr of tensors invariant and Representations 2.1.2 group. covering but h ai ersnain of representations basic The X 2 • NB nteohrhn,teei oemrhs nta isomorphi an (not homeomorphism a is there hand, other the On SL h udmna representation fundamental The h lmnso hsrepresentation this of elements The = σ h ǫ (2 123 A µ , i x ste4vco of vector 4 the is C 1= +1 = A , µ a h datg fbigsml once,so connected, simply being of advantage the has ) e P µ SU ˆ j ( , i σ ( = (2) X J J ǫ x ~ µ 123 SL = i 0 = 7→ A × 7→ . = i (2 7→ x A ~ iǫ Λ SU 0 x , X + ijk = C x , 0 ( ( 2 omtto relations commutation (2) J and ) B and ~ under A 2 i al matrices Pauli e ˜ det X 1 | 2 1 obntos(hc r ete emta o nihermiti anti nor hermitian neither are (which combinations SL k K , X stepyia spin. physical the as 1 0 0 1 x , n orsodn 2 corresponding a and , ~x ψ x | J 2 (2 SO SO α ′ 2 i ! →− 7→ SO SL , + x , = C = (1 , (1 = h i are: ) (2 (1 iK B 3 ~x

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Not to be quoted or reproduced without permission. h udmna n ojgt ersnain r h basi the are representations conjugate and fundamental The eaiiy atta ol emse yntraiigtec the realising not by missed be and could spinors that to fact importance the a relativity, then giving group, Lorentz the and oseti ewl osdrnwtedffrn ast as and raise to ways different the now consider will we this see To independent. • • • • • SL ohnl mixed handle To h otaain ersnain are representations contravariant The iia eain odfor hold relations Similar otecrettasomto ueis rule transformation correct the so ee¯ Here h ojgt representation conjugate The h erctensor metric The nents h arx˜ matrix the h nlg within analogy The ocnrvratrpeettosaentidpnetfrom independent not are representations contravariant so since hti why is That indices. (2 ψ , α C χ x .W ilsenx httecnrvratrpeettosa representations contravariant the that next see will We ). β ˙ µ = r called are hudlo h ae hte etasomtevector the transform we whether same, the look should x ǫ ǫ αβ = ( ǫ x sue orieadlwrindices lower and raise to used is αβ ψ x µ ψ β µ ǫ σ SO η αβ ′ σ ′ µ α , ih-addWy spinors Weyl right-handed = µν µ ) SL (¯ (1 α σ via α ( = = ˙ = , µ ǫ (2 ( ) and 3)- α ) ˙ σ χ ¯ αα β SL ˙ 7→ , η χ ˙ µ ¯ α ψ ˙ C µν N α ′ ) ˙ β α is ) (2 α = ) α ( = ˙ := − N ρ N = , 1 C N α SL − = − sivratunder invariant is ) β ǫ β 1 ǫ α ǫ N ) ˙ ( (2 σ αβ αβ β β x ˙ α N ∗ ǫ χ ˙ – 9 – ¯ , ν α ρσ ǫ β C β α ˙ ˙ σ α ˙ , χ nie,rcl httetasomdcompo- transformed the that recall indices, ) ¯ = β β ν ˙ β ) ˙ ⇒ ( ( = β σ σ , γ ˙ − µ ν N χ ) ) ¯ . ǫ ǫ β β ψ ′ α α ∗ αβ α ˙ ˙ γ β ˙ ˙ ˙ α β γ ˙ ˙ (Λ) α, ˙ · ǫ , ¯ = ( = SO det = β Λ = ˙ µ 12 1 = ν neto fteLrnzgroup Lorentz the of onnection (1 χ N ǫ +1 = N 1 ersnain of representations c stebscojcso special of objects basic the as β αβ , ˙ , , α )adi sdt raise/lower to used is and 3) µ ∗ ˙ ( 2 N ψ γ − ν = oe indices. lower ˙ oain ones. covariant β x ~σ ∗− ǫ , . , ν ) 21 1 ǫ ( . ) αβ σ β = ˙ µ α ) ˙ X χ . ¯ α . − α α ˙ ˙ 1 ehwvrnot however re via . , = SO ǫ SL α (1 ˙ β ˙ , χ ¯ (2 )or 3) β ˙ , C ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. nw as known en h rdc ftoWy pnr as spinors Weyl two of product spinors the Weyl Define of Products 2.1.4 of Generators 2.1.3 yhl.Arfrnebo lutaigmn ftecalculati the of many [ illustrating is book reference A half. by n nlgul o ¯ for analagously and hc aif h oet algebra Lorentz the satisfy which oeueu dniiscnenn the concerning identities useful Some o osdrtesiswt epc othe to respect with spins the consider Now e sdfietensors define us Let ne nt oet rnfrainwt parameters with transformation Lorentz finite a Under representation). niymti em ohave to seems antisymmetric o o,ltu utmninteidentities the mention just us let now, For 3 2 ǫ ]. 0123 = 1 = χ ψ ¯ α α ˙ efduality self ( : ( : h − σ ψ χ ¯ µν ǫ 0123 α α ˙ σ , 7→ 7→ σ λρ µν ,B A, ,B A, σ and µν i hytu omrpeettoso h oet ler (the algebra Lorentz the of representations form thus They . χ exp ¯ SL ¯ , β ˙ nisl duality self anti σ = = ) = ) exp µν (2 (¯ ( σ σ − χ χψ , 4 ¯ µν µν i satsmersdpout of products antisymmetrised as × C 2 ψ 2 ¯ i 3 0 ) ) ) 2 1 σ σ ¯ ω η − α β α β , ˙ ˙ , µν µν opnns u h efdaiycniin eue this reduces conditions duality self the but components, µρ µν := =¯ := 2 i 2 1 0 σ ω σ := := µν µν νλ = = σ χ χ σ ¯ µ α α ˙ + 4 4 µν = = α β i i hyaeipratbcuenaively because important are They . 0– 10 – 3 − and ψ ψ 2 ¯ 1 ⇒ ⇒ ψ SU i α α 2 ˙ η σ σ 1 ¯ β ǫ β α i νλ ˙ ˙ µ µ µνρσ 2ssandb the by spanned (2)s σ ǫ = = σ σ ¯ µνρσ σ J J µν ν ν i i µρ σ − − − − a efudo h xmlssheets. examples the on found be can ρσ = = σ ¯ − χ χ ¯ ρσ σ σ ¯ α α ˙ ν ν ω η ψ ψ 1 2 2 1 ¯ , µλ µν σ σ ¯ α α ˙ n o w opnn spinors component - two for ons σ σ µ µ pnr rnfr sfollows: as transform spinors , , σ i i α β β α (right-handed) νρ ˙ ˙ K , K , (left-handed) σ − matrices: η A νρ i i i and σ + = = µλ B − , i 2 2 : i i σ σ σ i i µν spinor being Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. hs xasosaeas eerdt as to referred also are expansions These rdcso h udmna ersnain( representation represent fundamental dimensional the higher of all products generate can we general In ones. undotted of contraction 2 emta ojgt falf rgt-addsio sarig a is spinor (right)-handed define left we a of conjugate Hermitian hoigthe Choosing tflosthat follows it e sgv w xmlsfrtnoigLrnzrepresentati Lorentz tensoring for examples two give us Let hc utfisthe justifies which hr nparticular in where ino esrpout ( products tensor of tion fteelementary the of ent hteq. that note We ψ 2 • • ψ 1 h onigo needn opnnsof components independent of counting The ieypoie h ih ubro he opnnsfrthe for components three of number right the provides cisely iial,teei nat-efda esr¯ tensor dual anti-self an is there Similarly, ( the ( h rc Tr trace the ( rvdstesmercones. symmetric the provides isioswt ieetciaiiscnb xaddi ter in expanded be can chiralities different with Bi-spinors h rdc fsioswt lk hrlte eopssin decomposes scalar chiralities a alike representations, with spinors of product The ec,tosio ere ffedmwt poiechiralit opposite with freedom of vector degrees spinor two Hence, lk isiosrqieadffrn e fmtie oexpan to matrices of set different a require bi-spinors Alike Thus . σ 1 1 2 2 µν , , 0) 0) σ ) α µ ⊗ ⊗ ψσ γ arcsfr opeeotoomlsto 2 of set orthonormal complete a form matrices ψ ǫ ( (0 ψ βγ α 1 2 µ α , ψ , χ ¯ h omrrpeet h nqeatsmerc2 antisymmetric unique the represents former The . { β )=(0 = 0) 1 2 . SU 2.1 ψψ obe to σ ( = ) ր = µ σ ¯ 2 ue( rule (2) mle that implies otato fipii otdidcsi otatt the to contrast in indices dotted implicit of contraction 1 2 ψ ν ǫ = ( } 2 r α αβ χψ ( niomtn rsmn numbers Grassmann anticommuting , 1 2 ψ 2 = χ , ( , 2 s α β ) ψψ 0) ψ ( = ) ) † 1 2 α η † ) ⊕ µν = ). ψ ψχ 2 j ¯ = ) α := (1 : 2 1 ⊗ , ψ , n efda niymti aktotensor two rank antisymmetric self-dual a and 1 2 A χ 0) 0) = α ( ǫ , ψ ψ 2 1 ¯ ¯ χ ⊗ ¯ αβ ↔ ( = ) α ˙ α r ˙ ǫ ⊗ , αβ Fierz ( B ψχ 1– 11 – (0 = j ψ ne emta ojgto.Teeoe the Therefore, conjugation. Hermitian under − , 2 + ) β ( 1 2 1 1 2 σ ψ ψ ψ , 1 2 ¯ ) ) identities. α ⊗ )adiscnuae(0 conjugate its and 0) α χ ⊕ ˙ ( µ s σ ¯ σ ψ 2 1 ( a erdcdt ucsieapplication successive to reduced be can χ µν ¯ := = σ j ) +1 2 µν ψ † µ ¯ σ χ (for ) ¯ µν ψ n(0 in ψ rmissl-ult rprypre- property self-duality its from = ) 2 β ∗ ǫ σ ψ T ( α µ toso h oet ru by group Lorentz the of ations 1 ons: σ , t(et-addsio.Thus spinor. (left)-handed ht σ χ α ˙ j αβ 1). 0 × , − ) 0). 6= β µ ψ arcswt epc to respect with matrices 2 α ( otoLrnzirreducible Lorentz two to ˙ 1 ψ ψ σ ψ so the of ms ¯ d, ψ iers oaLorentz a to rise give y 1 2 ψ µν ǫ × (1 αβ 2 = , χ arx h latter the matrix, 2 . , )representation. 0) ) 1 2 − n ( and .Tecomputa- The ). ψ σ 2 ψ α µ σ α 1 ˙ so , Actually, . µν ց ǫ σ ψ T implicit ) ψψ αβ µν := χ = . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ocnetteieso elsioswt h oestandard more the with spinors Weyl of ideas the connect To spinors Dirac 2.1.5 edefine We hnthese then where the define Finally, i.e. chirality, definite of part one eliminating h matrix The a aeeigenvalues have can uhta h cinof action the that such ecndfietefloigpoeto operators projection following the define can We C eoe the denotes ia spinors Dirac γ γ µ 5 aif the satisfy endas defined , ia conjugate Dirac P L P ± hrecnuainmatrix conjugation charge γ L 5 ciaiy.Tegnrtr fteLrnzgopare group Lorentz the of generators The (chirality). 1 γ Ψ := Ψ ob h ietsmo w elsioso poiechirality opposite of spinors Weyl two of sum direct the be to 5 lffr algebra Clifford D γ D sgvnas given is Σ 5 Ψ µν 2 1 = = D Ψ := 1 D C n =

C γ γ − iγ Ψ Ψ µ µ ψ := =( := − 0 0 D 0 4 α D i γ , γ 1 := γ ! := 5 γ and 1 1 0 C ν µν := γ χ o ! P , P , 2– 12 –

2 Ψ α

hreconjugate charge γ , D T ǫ = 3 2 =

αβ σ 0 ψ ¯ ¯ 0 ψ χ ¯ µ α ˙ ψ χ α α ¯ = = ˙ Ψ = ) ǫ

R R σ α α P ˙ α ! 0 0 ˙ µ Ψ η β L ! σ ˙ µν

! , ¯ 0

µν ! := D P = , ψ χ 1 − ¯ R , σ . 0 α α ˙ = 1 , 0 . µν D † 1 2 !

1 0 γ ! −

! 1 , 0 χ ¯ pnrΨ spinor ψ Dirac α . χ ˙ ¯ + α 0 , α ˙ ! ! γ 5 . . hoy define theory, D , C by , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Heuristically: ews otr oosit emos hsw edt introduc to need we thus fermions, into bosons turn to wish We eosatrn mltds nohrwrs hr sn non- th no for is except there scattering words, non-zero non-tri with other a symmetries in In internal this do amplitudes. cannot scattering one zero dimensions, 3+1 in that stated .. itr fsupersymmetry of History 2.2.1 algebra SUSY 2.2 ews to wish We algebra Graded 2.2.2 oagnrlDrcsio adiscag ojgt)cnb de be can conjugate) charge its (and spinor Dirac general a so aoaaspinors Majorana h ro nycniee “ considered loop-hole only a was proof there the theorems) no-go with usual (as However • • • • • • Ramond Wess Haag h culsatn on nsseai td fsupersymme of study field. systematic the on to point contributions starting other actual many the for way the opened They generators Golfand aor hspoeuewskona the as known was procedure This flavour. iesos(rmsrn theory). string (from dimensions ymtyo the of symmetry nafamous a In man nte16’,fo h td fsrn neatos ayha of many multiplets interactions, in strong organised of successfully were study and the covered from 1960’s, the In freape,i o tl eur hr ob interactions be to there require still you if example), (for ue onaeegopi ietpoutwt nenlsym internal have with theories product field direct quantum a in non-trivial Poincareé group only super the that show to Poincaré usin rueaotbge utpesicuigparti including multiplets bigger about arouse Questions . , and extend Lopuszanski , × Neveu-Schwarz Ψ Zumino and Q internal. D og theorem No-go Ψ α where , M h ona´ ler o-rval.The non-trivially. Poincaré algebra the Q Licktman Ψ = | aeproperty have boson S 17)woedw uesmercfil hoisi dimensi 4 in theories field supersymmetric down wrote (1974) arxi Poincaré is matrix - , bosonic M α Sohnius | ∝ i 1 Ψ 1 = M + 17)etne h ona´ ler oicuespinor include to Poincaré algebra the extended (1971) fermion , ( , oea,Mandula Coleman, 2. i Gervais generators”. Ψ = M ψ 17) eeaie h oea adl theorem Mandula Coleman the generalised (1975): α 2

= , i Q , 3– 13 – ψ ψ ¯ χ α α , ˙ α ! Sakita , Ψ × D Ψ = | nenl htcno i ieetspins different mix cannot that internal, ihfl way eightfold fermion C 17) eie uesmer n2 in supersymmetry devised (1971): Ψ = M C 97 adta h otgeneral most the that said 1967) | ∝ i ietproduct direct e , M eas fa mlctaxiom: implicit an of because 1 boson rva i fPoincaré and of mix trivial opsdas composed ilwyadsilhv non- have still and way vial oea adl theorem Mandula Coleman SU of emoi generator fermionic a e − metries. Gell-Mann (3) try. i Ψ i lso ieetspins. different of cles ymtygopof group symmetry a . rn aebe dis- been have drons M f hsi fe enas seen often is This the , 2 . f eern to referring and Nee- ons. Q . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o hs erqieagae algebra graded a require we this, For fa ler sc sagopgnrtr,agae ler is algebra graded a generator), group a as (such algebra an of where o uesmer,tebsncgnrtr r h Poincar´ the are generators bosonic the supersymmetry, For emoi generators fermionic UY ncase in SUSY, h ona´ ler,s ene ofind to need we so Poincaré algebra, the lo(o nenlsmer generators symmetry internal (for also rnfr ntesm a stecmuaosudraLrnzt Lorentz a under commutators for the relations as The way stance. same the in transform esalb sn h atta h ih adsdsms be must sides hand right the that fact the using be shall We conjugates. • eko h omtto eain [ relations commutation the know We ne natv rnfrain sa operator, an as transformation, active an Under Hence Similarly, opr hs w xrsin for expressions two these Compare hr esttergthn ieeulto equal side hand right the set we where (a) Since operator. an as and spinor σ η µν a Q : if 0 = α h Q Q − Q α α Q α ′ > N sasio,i rnfrsudrteepnnilo the of exponential the under transforms it spinor, a is M , O 2 i α ′ a = ω sa is µν exp = fa xeddSS.I hsscin ewl nydiscuss only will we section, this In SUSY. extended an of 1 µν Q U i ( α A ecnwr hsoeotb nwn how knowing by out one this work can we : σ ooi generator bosonic † , µν Q Q Q ¯ ) α (a) (c) α α A = ˙ U O ↔ − where , ⇒ β a n h 2 Q O i Q Q Q ≈ ¯ ω h β b Q α µν α ¯ − a hnb bandfo hs ytkn hermitian taking by these from obtained be then may α h M , ˙ Q , σ = Q ( M , eeaiaino i ler.If algebra. Lie of generalisation a - (e) µν − A 1 α T 1) β µν Q i 1 = M , + o and , h Q µν ) P α η 4– 14 – Q α i a α ′ µ , i η β α 2 , i b P , .,N ..., , − µν pt rtodrin order first to up Q O (b) (d) h T , ω (¯ = ψ η β b i ν µν O a 2 i ,[ ], | Q i h n a if 1 = i M ω ≈ ( = ncase In . Q α ′ P Q = σ µν . | α µν µ ψ µν α | ψ iC M , i P , , ) → i σ Q n where and , α 1 ab O ˙ e µν Q α ¯ Q ρσ β a O ˙ µ β − M ) ˙ α Q i n [ and ] generators e U α ¯ o e sa is , N β , µν β ˙ | 2 , ψ i 1 Q linear h ⇒ i eseko simple a of speak we 1 = ω emoi generator fermionic ω − β µν − M µν U M σ asomto,frin- for ransformation, , µν 2 i ψ µν exp = n htte must they that and µν Q SL M , | ω Q α µν P Q O α α (2 ρσ rnfrsa a as transforms | µ a α M ψ β , , led from already ] sa operator an is C h → i Q M − µν + generators ) β 2 i µν ω . N O µν ψ . n the and ( | 1. = . M ω U (2.3) 2 † µν ) Q α . U | ψ i , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. eabsncsaeand state bosonic a be oieta w ymtytransformations symmetry two that Notice • • • Q tk donsuig( using adjoints (take P (b) (d) (c) ossetcoc is choice consistent µ hr sn a ffixing of way no is There hstm,idxsrcueipisa ansatz an implies structure index time, This ic h ethn iecmue with commutes side hand left the Since u oidxsrcue htcmuao hudlo like look should commutator that structure, index to Due a nyhl o general for hold only can fteoperators: the of α , | µ F α , h i n P n Q hc slna in linear is which Q = 0 Q ν α = α α and P , = = = Q , , | Q B µ ¯ Q " − | | β i β i c c ˙ α o P o | | c : , 2 2 c µ ( ( · | ( σ | σ ( σ F , ν σ ν ν ) k i h µ ) n α Q σ ¯ α P ) ¯ ,i.e. 0, = emoi n,then one, fermionic a Q α Q ˙ α α µ ˙ β ν ˙ α h α Q ˙ α (¯ | P Q h σ Q , ) − B ¯ n ofi h constant the fix To . † Q Q , Q 6=0 {z t µ µ α i Q ˙ o ycneto,set convention, by so, , n ) = β α n σ , αβ steol a fwiigasnil emwt reindices free with term sensible a writing of way only the is α ˙ α Q if , Q µ β i P , = Q Q α ¯ ¯ o Q , # σ Q ¯ α α α ˙ ˙ ν c , β , i µ ) + | ,so 0, = n ( and α β Q 0 = i F ¯ Q | − ¯ + o β } β i 5– 15 – " ˙ β ˙ o = P Q o Q c c = , σ ν β α ( P | = 2 σ µ = Q , ⇒ ¯ ( 2 = h µ ( Q µ ¯ Q β σ k h ¯ ˙ ) n h ih adsd os’,teonly the doesn’t, side hand right the and ) α µ Q α α † aeteeeto rnlto.Let translation. a of effect the have ( ˙ t n α ) σ ˙ α ( α Q P , Q ( = h σ µν ¯ α c ˙ σ P , P α µ Q osdr[ consider , ˙ ¯ (¯ ) µ ) ν σ α µ Qσ , ) α : t i α ν β µ β , ˙ Q ) β ¯ ,dfiigtenormalisation the defining 2, = i ˙ | P M αβ µ Q B ˙ # β ¯ P ˙ 0 = ) µ o α →| 7→ i α µν µ ˙ ˙ Q + i . .TeJcb dniyfor identity Jacobi The ). β . 0 = " Q ¯ Q α ˙ α P , B , µ (translated) | h = ] P µ {z c P , 0 ∗ · (¯ ν σ i } µ # i ) αβ ˙ . | Q B β i Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ae aheeet ttsaelble like labelled are States element. each label lorcl h w aiisi h ona´ ru,oeo wh of Poincaré one vector the group, in Casimirs two the recall Also J omtswt l the all with commutes h aii operator Casimir The eain ihtegnrtr ftePoincaré group the of generators the with relations rmme htoenesacmlt omtn e fobserva of set group commuting lab rotation to complete the used a se are needs these to one since that check group, (remember must changed we the Poincaré of group, Casimirs the the changing are Poincaré we group the Since of Representations 2.3 ne oet rnfrain)adcmue with commutes and transformations) Lorentz under generator (where hr a oepestescn aii.Ee though Even Casimir. second the express to way short a h ona´ aiisaete ie by given then are Poincaré Casimirs The au of value ic the since ona´ utpesaelabelled Poincaré are multiplets 2 ihnteeirdcberpeettos the representations, irreducible these Within . • Let (e) ler sivratudrtesmlaeu change simultaneous the under invariant is algebra sal,ti omttrvnse u oteColeman-Mandu are the to due vanishes commutator this Usually, W ǫ 0123 C µ C U R h P 2 eciiggnrlsdspin generalised describing Q ttswti hs reuil ersnain ar th carry representations irreducible those within States . i 1 uoopim ftesprymtyagbakonas known algebra supersymmetry the of automorphisms (1) µ eaglobal a be omt ihalgenerators. all with commute = α salbl oieta tti ee h al jbnk vect Ljubanski Pauli the level this at that Notice label. a as T , − ǫ 0123 i i { J i +1). = Q ⇒ : J U α i i h 1 eeao,te,since then, generator, (1) n aesirdcberpeettosb eigenvalues by representations irreducible labels and C 1 = Q 1 7→ α , R , = 2 | , exp( ,ω m, W 3 h i } J µ P i satisfying µ iλ i J , = J where , = P ) 2 µ Q j i Q C , α 6– 16 – 1 2 = α | , ,j j, ǫ = µνρσ , m X i 3 =1 3 2 i iǫ 2 . J steegnau of eigenvalue the is Q P J ¯ ijk 3 h i α 2 ν ˙ Q = Q P eigenvalues ¯ J M M µ α α ˙ k 7→ µν ρσ i sivratudrtranslations), under invariant is (it 7→ R , W . snei rnfrsa vector a as transforms it (since µ e W exp( c novsthe involves ich i − W µ iRλ lirdcberepresentations irreducible el µ = a tnadcommutation standard has , − lst ae hm.Recall them). label to bles j Q 3 fayhn apn to happens anything if e iλ α − = aterm Exceptions theorem. la e ) iRλ eigenvalue e Q C Q ¯ − ¯ α 1 α ˙ j, ˙ , . and R . − rol provides only or j symmetry al Ljubanski Pauli 1 + ω steeigen- the is .,j ..., , j ( p j µ )of 1) + fthe of The . − 1 j , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. C o.Oecnhv atce fdffrn pnwti n multi one within spin different of particles have can One not. For 2.4 h omttr[ commutator the ofidmr aesw take we labels more find To algebra. closed a of generators oa oet nain unu edter sCTinvariant. CPT is theory field quantum invariant Lorentz local fermions, naysprymti utpe,tenumber the multiplet, supermultiplet supersymmetric a in any fermions In and Bosons 2.4.1 omt with commute opoeti,cnie the consider this, prove To hsnwoeao ( operator new This ( ˜ − 2 4 5 ) crepnigt uesi) edefine we superspin), to (corresponding • • e h tnadMdlPr I orefraruhpofo the of proof rough a for course III Part Model Standard the See e h atI rnilso unu ehnc course. Mechanics Quantum of Principles II Part the See F N Q asesprilshave particles Massless atcewt o-eoms sa reuil representat irreducible an is labels mass with non-zero with particle a Thus, asv ersnain r P self-conjugate. CPT are representations massive ttsaetu labelled thus are States atce n ymtiscus) Thus, course). Symmetries and Particles to asv particles, Massive ic hyleave they since λ N supersymmetry, 1 = α = | 1 = | p F µ 2 1 i , B C − qak,leptons), (quarks, uesmer representations supersymmetry 1 µ ( = λ P = i . µ := | P ,j m, hsdfie itegroups: little defines This . W − λ 2 ) − µ F iheigenvalue with utb nee rhl integer half or integer be must W , W ) ( ; p | F − p µ B µ µ ν ) p niomtswith anticommutes i nain.Fo h ento of definition the From invariant. j , F = ] µ − | 3 emo ubroperator number fermion = P B ( = C i . µ 1 4 i 1 | iǫ p 0 | W sgvnadlo o l lmnso h oet ru that group Lorentz the of elements all for look and given as µ Q = m, B , λ µνρσ ¯ 0; = 0 α i ˙ ( = P ( 1 = p (¯ C nain ne rot. under invariant σ µ ˜ µ = W 0 = | m 2 P µ λ , B | n ) p µ ρ 2 αβ i Q i ˙ B P | 0 | γ , := , ssilago Casimir, good a still is , , =: , α C σ 7– 17 – Q W , {z p 0 W | 2 = F mle htthe that implies β , and ) | = Q C p ± i C , 0 } µ µν ( α , − λ λ , n − = Z since F n P C = ) i i F B 0 ne CPT Under . 2 µν W . ,hv oain sterltl group, little their as rotations have ), , − j J | 5 F g · fbsn qastenumber the equals bosons of = µν 2 ( Q µ . p and ) − i λ iheigenvalue with α / eigenvalues 1) ( | 0 = − p := = − F W J m P hoe,wihsae that states which theorem, CPT | ) ( = stehelicity. the is F λ µ , W −| tflosthat follows it , B 1 2 | i lt ogtanwCasimir new a get To plet. F gaio) oethat Note (graviton). 2 = , − µ µ . F 1 o ftePoincaré group the of ion C i sb hmevsaenot are themselves by ’s ) P 4 ... , i F 2 hs ttstransform states those , ν . endvia defined , = λp = e.g. , ⇒ − µ W − B m setePr III Part the (see µ n ν W 2 λ ( P j − µ ( µ (Higgs), 0 = oee,is however, , j ) F ) hence 1), + Q , α n o F any of 0 = . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. nteohrhn,i a eeautdusing evaluated be can it hand, other the On et osdrtetae(nteoeao es,ie vrel over i.e. sense, operator the (in trace the consider Next, n h ento of definition the and ttso asesprilshave particles massless of States supermultiplet Massless 2.4.2 Tr where o eebrn that remembering So, C asesstate massless hc mle that implies which rmorpeiu omtto relation, commutation previous our From emyas n n element one find also may We 1 n = ( ⇒ Tr − P ) P ( [ F µ W n µ Tr ( o P − Q 0 h srpae yiseigenvalues its by replaced is µ ( , p skona h Wte index”. “Witten the as known is ) α W F µ Tr = 0 and ( Q ¯ λ , − , 0 n α | Q ˙ ) p Q ¯ Q | = ¯ ] F n µ | 2 ˙ [ C β p α ˙ W ˜ | λ , Q n p µ o 2 Q , µ λ , Q bosons 2 µ = λ , X i , 2 Q W α , ¯ ( 2 = n ntergthn side) hand right the on p i Q szr nterepresentation: the in zero is β C ¯ Q µ , ˙ 3 ( i ¯ o = nti representation this in , − α µν ˙ h 2 Q = = ˙ ) ¯ = ] B o ) − β C F ˙ | − | o p B σ 8 o i µν [ µ p Tr = W ) 2 1 µ ǫ − i λ , 0 | 03 ) ǫ p r eo osdrteagba(mlctyatn nour on acting (implicitly algebra the Consider zero. are 0 α µνρσ n,frmsls representations, massless for and, = µ jk , β ˙ λ , Tr = Tr = i Q ¯ P p P fermions 2 3 ˙ ( µ i P µ bosons + ] 0 = X X uhthat such ( ν [¯ eigenvalues - − [ σ 2 = M p n n Q ) j ¯ µ F σ , anticommute h − 2 8– 18 – h ˙ ρσ | ( B ⇔ F λp o h pcfi tt.Tecnlso is conclusion The state. specific the for − ( 2 Q | , k | E ) α ( F Q 0 {z σ ] ¯ F { − Q Q ¯ ( ¯ µ Q i α ˙ − Q Q ) 2 σ | ˙ ) = ] p F α | α ) α 0 α p 1 } ˙ µ , = β F | ˙ | µ λ , | Q p B + ¯ p P λ , − Q Q µ ¯ p ¯ µ β i µ ˙ λ , µ β σ λ , 1 2 n } β i ˙ ) ˙ i + 3 B ǫ ( = 2( = i ( = µνρσ + ( + i 0. = α = − mnso h multiplet) the of ements fermions ( 2 = β = λ ˙ Q | X E, σ P − − ylcperm. cyclic − α n µ 4 = Q ν ) F ( ) 1 2 F 2 1 (¯ − α 2 0 σ p {z ) σ | . β h Q , ˙ p ) ¯ p 3 ρσ F P µ W F µ ( 0 β ˙ ) E 0 σ µ | ) λ , Q α Q ¯ Q E , 0 ( β α , ¯ 3 ˙ ˙ β − | 2 ˙ ˙ Q Q

β p α ¯ ¯ } ˙ | i p p o ) µ o β ) µ ˙ F 0 0 0 1 µ .TeCasimirs The ). λ , α ˙ λ , Tr | 0 = | p F 0 = i µ ! n i i λ , = ( α . − β . ˙ λp i ) . . , F 0 o | p µ , (2.4) (2.5) λ , i , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. Thus, gi,teatcmuain-rlto for relation - anticommutation the Again, hr r,freape hrlmlilt with multiplets chiral example, for are, There n hr r oohrsae,sneEq. since states, other no are there and ag n gaugino) and gauge h ievle of eigenvalues The gates: ncs of case In supermultiplet Massive 2.4.3 n Casimirs and swl stegaio ihispartner: its with graviton the as well as where hs ehv w ttsi h uemlilt oo n f a and boson a supermultiplet: the in states two have we Thus, Q 6 1 oeta ehv sdntrluis therefore units, natural used have we that Note | p lt ahrta nthe in than rather plets usin h ow u atrﬁlsi the in ﬁelds matter put we do Why Question: µ Q Y ¯ λ , i 2 ˙ eoe superspin denotes = m − n − C ,i h eteo asfaeteeare there frame mass of centre the in 0, 6= 1 2 Q Q λ 1 i ¯ α 1 ˙ = scalar 0 = , = Y erae h eiiyb / unit a 1/2 by helicity the decreases Y √ i slepton Q squark ¯ 2 1 Higgs 4 β ˙ P = E o = µ Q Y P 1 J ( 2 = µ i Q i Higgsino lepton quark λ Y ¯ i 1 ˙ − = | = | are p p µ λ µ | 2 1 p σ 4 λ , , 1 µ = m y m fermion µ ± λ λ , ( ) λ i y 2 α = 2 Q 3 ¯ gravitino β i 1,s elblirdcberpeettosby representations irreducible label we so +1), = ˙ , fermion − { σ P ¯ , i 1 2 √ µ 2.6 , Q 1 2 , 1 4 9– 19 – i Q 1 E C 2 = ˜ = } ⇒ | ~ 2 p and λ ones? 1. = µ √ graviton λ Q 0 = , Q λ ¯ Q = ¯ ± 4 boson 2 = 1 m ˙ h 1 = ˙ E W 1 Q | Y ¯ p , , ( i λ | µ C ino 1 2 σ 1 2 p stekyt e h states: the get to key the is Q λ , Y , ¯ 6 etr rguemlilt ( multiplets gauge or vector- , − µ µν 0 photino fermion h omlsdsaei then is state normalised The . 1 ˙ ) λ , Z ,

P α gluino 2 1 j C − β i ˙ − µ i µν i ino 2 1 eigenvalues - Q . ¯ 2 = λ i = 1 ˙ and 0 = = 2 = Q gluon photon λ ,Z W, 1 iǫ { boson 1 = rin lsCTconju- CPT plus ermion, m 0 ijk | , p m 2 1

µ Y } λ , 4 k 1 0 0 1 Y supermulti- . p i i µ ! Y = i ( = α , √ , β ˙ 4 m, E | p λ 0 µ , | = λ , ,y m, 0 (2.6) , 2 1 i , 0) , i 1 . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ersnsadffrn spin different a represents Hence hs o h case the for Thus, ..for i.e. h eann state remaining The e snwcnie pcfi case, specific a consider now us Let y (a) rmtesprymtyagba Thus, algebra. supersymmetry the from pnb nti h hr ieto sePr IPicpe o Principles Eq. II Using Part unchanged. (see spin direction total third the the leaves but in unit 1 by spin emyoti h eto h uesmer utpe yderi by multiplet supersymmetry the of rest relations the obtain may We Let emyueEq. use may We usin o ow nwthat know we do How Question: 0 = a | 1 † Ω | | Ω Ω i h spin the , i i etelws egtsae niiae by annihilated state, weight lowest the be has a 1 † | j j 3 J i 2 1 = y := a 1 † Y [ ,w aestates have we 0, = j Q | 2.7 / i Ω √ a n superspin and n o a hc that check can you and 2 | α Q Ω i a 2 † ¯ 2 J , 1 † a = oderive to i 1 m ˙ , 2 1 † j i | | | | 3 4 = ] j = Ω Ω Ω object. | 3 a Ω i i i i 1 † [ J | ′ = i 2 1 Ω J | [ 2 = = = Ω J ( i := − i , | σ 3 | i j Q Ω ¯ a , i 3 ) y α a − i ˙ | | | − α β a y 1 † ,j m, ,j m, ,j m, = 2 † = ] Q .W define We 0. = a 2 † a r h ae ofrgiven for So same. the are 1 2 2 † | J β 1 † [ = ] i zero − , | | 4 3 {z a , 0– 20 – ,j m, = 0; = 0; = 4 Ω | Q 1 Ω m ¯ i | Ω } α 1 2 i J ˙ ; − Q ′ 2 − ¯ p p [ = 6 i = p j a , J a µ µ 2.9 3 µ σ = ¯ 2 † i ( j , j , 1 † j , y , 1 † | i σ = j J | ; a and , | − 3 3 i 3 3 Q zero Ω p | Q 3 ) ¯ 2 † − {z i β α a 0 = 0 = µ | 0 = ] = ˙ {z ˙ α i Ω | ˙ 0 := 1 † j , Q 1 Ω ? = ] J ¯ } Q i / a ± } ± i β 3 ˙ i i .Similarly, 2. 2 † =: | 1 i J √ Ω 1 2 , | := − 2 i Q Ω i ¯ = , i 2 Consequently, . =: j 2 1 2 i m ˙ := ( J unu ehnc notes) Mechanics Quantum f ( j 1 σ | J j 1)¯ + | i ± | i 3 m, ) Ω i | β α ˙ Ω ˙ iJ ′ ,y m, = Q i ¯ igtecommutation the ving 0 i a 2 β , 1 † ˙ | a , hc lowers/raises which , 0 j | 2 † Ω 3 i : | , Ω i + . i 1 2 = i . | m, 1 / 2 , (2.9) (2.7) (2.8) 1 / 2 i . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. na in h case The y (b) oieta nlbligtesae ehv h au of value the have we states the labelling in that Notice eas have also We hsyed iercmiaino w osbettlspins total possible two of combination linear a yields This ( utpe n h ausof values the and multiplet aiyitrhne ( interchanges Parity Parity 2.4.4 spin). different of states are there multiplet ersnain fteright-handed the of representation) ene h olwn rnfrainrlsfor rules transformation following the need we odncoefficients Gordan factor hsensures This ffc that effect iltr o otemr eea aeof case general more the to now turn will | aigdsusdteagbaadrpeettoso simple of representations and algebra the discussed Having supersymmetry Extended 2.5 hs ttsaecle clr( scalar called are states These | Ω Ω a 1 † | ↔ i ′ i a , Since : | | 2 ,y m, 2 † 0 6= η | · cigon acting ) Ω P ,j m, ′ a a (4 i i y uhthat such 1 † 2 † ogtlws egtsae ihadfie aiy ene lin need we parity, defined a with states weight lowest get To . y P Q utpe,wihi fcus neulnme fbsncadf and bosonic of number equal an course of is which multiplet, | | ˆ ¯ 2 states +2) rcessihl ieety h doublet The differently. slightly proceeds 0 6= Ω Ω 2 α ˙ = P P {z a i i Q ˆ | Ω 1 † α y | ′ j = = P ; i ˆ 3 µ p i ,whereas 0, = − | µ Ω k | P = 2 j , ˆ η k k ,B A, i i P − |±i = 3 1 3 | (recall j eae ietecmiaino w spins: two of combination the like behaves } 1 | i | | 3 ,j m, j m, − 1): = , ( = − ) Q j := ↔ α 2 1 P = = hnesaeb tt a spoe,snei supersymmetri a in since proper, is (as state by state change i oevr osdrtetwo the consider Moreover, . j Q P P ˆ ˆ 0 | 1 ( | and ⊗ y y + ,A B, Q | Q | · , ¯ Ω + + i 2 1 α α α ,j m, ˙ n suosaa ( pseudo-scalar and ) | ± i − | P P a ( = 2 2 1 1 Ω ˆ ˆ P ~ SU 2 † ; ; (2 ,ie ( i.e. ), − − i | p p seqeto nEape he ) n a the has and I). Sheet Examples on question (see ) j y 1 1 j = µ µ ,adsneprt swaps parity since and 0, = 3 2 states +2) 2 in (2) j , j , Ω i − {z y = = = ′ 1– 21 – 3 3 xeddsupersymmetry extended i 2 1 + − + , ) | 2 1 j η , ⊕ 2 1 − 3 P 2 1 2 1 SL ; i i 0) + η p ( ( P ∗ j Q σ µ + + P (2 1 2 ↔ j , 0 ˆ + (¯ α i ) σ , |±i nttl ehave we total, In . 3 α k k 0 C and 2 1 } (0 i β 4 2 ˙ ) )): ,a Eq. as ), , αβ Q , ˙ | | ¯ |−i ,j m, j m, β 1 2 ˙ = Q Q ¯ .Since ). m states. ) β α ˙ Q | 1 j = = ¯ j ( |±i ± and ne parity under α | · + N ˙ asv states massive 0 = y y ,j m, sadult(..si 1/2 spin (i.e. doublet a is 2.1.2 2 1 − − )sprymty we supersymmetry, 1) = y and { Q 2 2 1 1 > N . 2 1 (2 Q fixed ; ; = α y p p α and hw.Tedoublet The shows. states ) ↔ µ µ , {z y j j , j , Q 1. ¯ − − a combinations ear Q 3 3 β ¯ ˙ j rincstates. ermionic hogotthe throughout } − + 2 1 α 1 2 rmEq. from , P ˙ ˆ ; tas swaps also it , 2( = ihClebsch with p 1 1 2 2 wt phase (with µ i i j , . σ 3 | } µ i Ω ) , α i β ˙ and 2.8 P µ c . , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. T sa lmn of element an is hyfr naeinivratsbagbao nenlsymm internal of sub-algebra invariant abelian an form They sw i for did we As iC h H ihantisymmetric with label additional an get generators spinor supersymmetry the extended Now, of Algebra 2.5.1 .. asesrpeettosof representations Massless 2.5.2 gen straightforward a ex is of supersymmetry. algebra ingredient previous new the main of the derivation is charges central of existence ewl tr ihtemsls aewihi ipe n a v has and simpler is which case massless Let the with start will We aea for as same etestof set the be implies nodrt bantefl ersnain enwdefine now we representation, full the obtain to order In ecnimdaeysefo hsta h eta charges central the that this from see immediately can We operators Z a abc = AB ∈ p T G µ U P , c n ( = ( Let . Z satisfying Q N AB µ α A ,btwt oeegnausof eigenvalues some with but ), a E, i A , N | p † = Q G G N µ 0 ¯ xetfor except 1 = , λ , βB ˙ := ea nenlsmer ru,te en the define then group, symmetry internal an be lmnsta ontcmuewt h uesmer generat supersymmetry the with commute not do that elements ,w ilpoednwt ics asesadmsierepre massive and massless discuss to now proceed will we 1, = 0 h Z E , i H o rmteanticommutator the from 0 = | AB fteegnausof eigenvalues the If . eta charges central p n 2 µ ,te smlrto (similar then ), Q √ Q λ , M , 1 A E α A i Q , a , µν h 4 = i Q β B α A o = n T , A E Q = h Z α A a Z

:= i AB AB , ǫ 0 0 0 1 αβ N Q N = ¯ = Q , 2– 22 – 2 βB Z Z ˙ ! Q > 1). = √ ¯ − AB AB α S 1 A o ˙ α A E Z Z 1 β ˙ a i AB BA δ , 0, 6= A supersymmetry B A ( 2 = = B n n | = p omtn ihaltegenerators the all with commuting Q Q r l eo hnteRsmer is symmetry R the then zero, all are Q ¯ µ ⇒ ,B A, H λ , α A 1 A h α B ˙ Z Q , σ , ilb ugopof subgroup a be will AB i µ Q n 0 6= N ¯ ) 1 = Z 2 B a α rlsto fteoefor one the of eralisation β B Z , ˙ β A = o rain and creation- AB ˙ o eddsprymtis The supersymmetries. tended ⇒ P | , tis ealta [ that Recall etries. a , p CD 2 µ r motn implications. important ery µ aihsince vanish .,N ..., , = δ λ , B † A i o Q symmetry R B i ǫ = 2 A α = 0 = ˙ | h ler sthe is algebra The . β = p ˙ µ ( h Z λ , Z N δ † AB ) A ǫ i AB Q 12 niiain- annihilation B U 2 A T , Z 0 = | , ( sentations. H p AB N a µ T i λ , r,e.g. ors, .The ). ⊂ | a p T , N 0 = µ i G λ , b 1 = 0 = = ] to i . . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. l the all ogttefloigsae satn rmlws egtsta weight lowest from (starting states following the get to oeta h oa ubro ttsi ie by given is states of number total the that Note osdrtefloigexamples following the Consider • • • • N lsterCTconjugates. CPT their plus N N N qiaety tcnit fone of consists it Equivalently, unu edter st ernraial.W a e htt that see of can terms We in renormalisable. decomposed or be be vector-like, to can (i.e. is adjoint theory the ﬁeld in quantum be must which particle, oe ntrso two of terms in posed N hrlmultiplet. chiral hsi h single the is This a A aiu utpe ( multiplet - maximum 8 = etr-mlilt( multiplet - vector 4 = etrsprutpe lsaCTcnuaeadtwo and conjugate CPT a plus supermultiplet vector 2 = P efcnuaehpr-mlilt e Fig. see multiplet, - hyper self-conjugate CPT 2 = etrmlilt ssoni Fig. in shown as multiplet, vector 2 = ): states a a a | a Ω N A A A i † † † † . . . a | a a Ω B B X k ( N N i =0 † † | a − N Ω C

1) i † † | utpe ihsae with states with multiplet 4 = N N ...a Ω k i ! utpes n hrl n anti-chiral. one chiral, one multiplets: 1 = 1 λ † 0 | Ω = = i λ N 0 − helicity λ N λ λ λ X k = 0 0 0 1) 0 N =0 etradthree and vector 1 = λ + utpes one multiplets: 1 = + + . . . 1 + − 0 70 56 28 3– 23 – 8 1 1 6 4 1 4

N 2) 2 2 2 3 1 × × × × × × × × × × N k 3! λ λ 1 λ λ λ λ λ λ λ λ ! N + = = +1 = = = 2 = = = = = :s-aldbcuei otisavector a contains it because so-called a: ( 1 N − ± − k ± ± ± ± ± 1 0 1 2 2 1 1 2 0 1 − 3 1 2 2 N 2! 1 − N 1)( ubro states of number k ( N N te 2 = 2 − − N N .Aanti a edecom- be can this Again b. | | λ Ω N )= 1) )= 2) | etradone and vector 1 = = 1 = 1 hrlsupermultiplets chiral 1 = el ersnaini the if representation real) i N hc sanhltdby annihilated is which , < = N . 2 3 . . . hypermultiplets. 2 = tcnit fone of consists It . N N N N N N 1 0 2 3 his N multiplet 2 = N 1 = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. rmteerslsw a xrc eyipratgnrlcon general important very extract can we results these From N • • • nicia supermultiplet anti-chiral 1 = h aiu ubro uesmere is supersymmetries of number maximum The er stelretsprymtyfrrnraial ﬁeld renormalisable for supersymmetry renormalisable! largest the is metry against eomlsbetere have theories Renormalisable λ omsls atce fhelicity of particles massless no ealddsuso fti n h xeso fhsargumen his of extension the and this of discussion detailed neeymultiplet: every In opet oecnevdcret ( currents conserved some to couple a lob enfo h oea adl hoe) Also, theorem). Mandula Coleman the from seen be also can htteei oeta n rvtn e hpe 3i [ in 13 chapter See graviton. one than more is there that N = hrlsupermultiplet chiral 1 = ± rvt) u hr r ocnevdcret for currents conserved no are there But gravity). - 2 | λ | > stefc htmsls atce of particles massless that fact the is 2 iue2 Figure λ max − λ . λ λ 0 = N = min | etradhprmultiplets. hyper and vector 2 = tiplet supermul- Vector (a) tiplet supermul- hyper (b) λ 2 1 a a ≤ | a a | 2 † 1 † λ 2 † 1 † = λ | ∂ > λ implying 1 4– 24 – = µ λ λ j = N 2 xs s nyhave only (so exist 2 µ 1 = 0 = − in 0 = 2 1 1 2 a a a a 1 † 2 † 1 † 2 † λ N λ λ N 0 = = = .Teei togble that belief strong a is There 8. = N N ≤ ± 1 2 hrlsupermultiplet chiral 1 = .Therefore 4. etrsupermultiplet vector 1 = electromagnetism, - 1 | λ clusions: | > osprymtyi an in supersymmetry to t 4 | N hois rvt snot is Gravity theories. nsf htn o a for photons soft on ] λ 1 2 | ≤ > n o momentum low and > N ) n argument One 8). smtigthat (something 2 N supersym- 4 = ol imply would 8 ∂ µ T µν in Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o consider Now otayt h asescs,hr h eta hre a b can charges cases: central two the distinguish to here have case, we massless the to Contrary .. asv ersnain of representations Massive 2.5.3 • • hc snncia.Te the Then non-chiral. is which > N yemlilt have hypermultiplet, ontinclude not do t(hyaedult under doublets whereas interactions are chir weak (they the are feel it They not do leptons representations. and quarks fundamental complex on live ﬁnd: npriua h ii flwmmnu a ob assumed. be to had momentum low of limit the particular in edn o2 to leading afo h uesmer eeaosvns ( vanish generators supersymmetry the of half utpe,teecntb hrlt ihri hsmultiple this in either chirality be can’t there multiplet, ril by article ..a rdce 6=2 = 16 predicted as i.e. hrfr h nyhp o elsi uesmerctheo supersymmetric realistic a for b not hope have only particles These the particle. Therefore Model Standard each for agrta h asesoe ihol 2 only with ones massless the than larger ntesm ersnainadteeoeb o-hrl The non-chiral. be therefore N and representation same the in a ecia,frinstance for chiral, be can Z uesmer tlwenergies low at supersymmetry hr r 2 are There AB yemlilt o hs,tepeiu ruetdoesn argument previous the these, for - hypermultiplets 2 = 1 0 = uesmere r non-chiral are supersymmetries p µ Grisaru ( = N 2 N rain n niiainoperators annihilation and creation- m, tts aho hmwt ieso (2 dimension with them of each states, λ = 0 , and ± n a 0 α A tts u since but states, 1 Q ˙ λ , † a α A a ) so 0), a Pendleton = α A 4 β B α A , ˙ ˙ † † ttsi oa.Ntc htteemlilt r much are multiplets these that Notice total. in states ± Q N a ¯ a a := SU atce rnfrigi h don representation adjoint the in transforming particles 1 γ C β B βB α A ˙ ˙ ˙ ˙ λ with 1 = † † † E (2) = a o a a a δ D √ B β γ C α A ˙ ˙ ˙ N ˙ Q ≈ ± † † † † L 2 2 = α A | | | | | .All ). 5– 25 – m 1 2 > 10 Ω Ω Ω Ω Ω 17) oieti sntafl og theorem, no-go full a not is this Notice (1977). atce ihntemliltwudtransform would multiplet the within particles i i i i i eko htteSadr oe particles Model Standard the that know We . 2 1 a , 3 GeV. uesmer n P states BPS and supersymmetry λ m N × 0 2 1 > N = pn0 spin

tts u otefc hti htcase, that in that fact the to due states, 1 2 α A 4 4 1 0 0 1 Q 1 1 ˙ rdciga es n xr particle extra one least at predicting and - † × × × × 2 A utpes xetfrthe for except multiplets, 1 ! spin spin 0). = pn0 spin pn0 spin := , δ 3 λ A × y 1 1 2 2 B √ = ) nthe In 1). + Q pn1 spin ¯ o-aihn.Therefore non-vanishing. e .Teeoeonly Therefore t. 2 . − α A m ˙ 2 1 ethne nsd feel do ones left-handed nyecpin r the are exceptions only e bevd however. observed, een ttsaei h same the in are states yi:broken is: ry lsnergthanded right since al , twr eas they because work ’t N ae we case, 2 = N N N 1 = 1 = 2 = , 0 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ounadrwo zeroes. of row and column 7 • If N For o oeuiaymatrix unitary some for aigol 2 only having sasmo products of sum a As Q Prasad Z u oteplrdcmoiinterm ahmatrix each theorem, decomposition polar the to Due ttso minimal of States that if generally, More en h clrquantity scalar the Define hsi h P on o h mass the for bound - BPS the is This Z matrix ¯ AB α A > ˙ = − but 2 N HV 0 6= Γ ¯ ,w en h opnnso h antisymmetric the of components the define we 2, = V α A ˙ otemlilti hre smlrt h asescs in case massless the to (similar shorter is multiplet the so , H N and fapstv eidfiiehriinmatrix hermitian semi-definite positive a of ( = sod eoti Eq. obtain we odd, is V 8 = N Sommerfeld † H ) nta f2 of instead − Z N H 1 m m Choosing . AB =(¯ := Z > − N = AB 8 = AA (but 2 2 = U 1 N σ † , 0 H Tr = (satisfying Γ ) Tr 4 m 2 H .Te r hrceie yavnsigcombination vanishing a by characterised are They ). βα                 N α A ˙ n ob aan mlctysnwcigi bra/ket) a in sandwiching implicitly (again, be to m − N N 2.10 spstv eidfiie n h Γ the and semi-definite, positive is √ U −

states. n n := Z q Q vn emypromasmlrt transform similarity a perform may we even) H = − 0 0 0 0 0 0 0 0 0 0 1 ≥ . . . ihtebokmtie xedn to extending matrices block the with † α A 0 q o Z q V 1 6– 26 – 0 0 0 0 Tr 2 1 o . . . ǫ − , αβ q 2 m 0 UU − 8 = 1 r called are 1 N q : Γ ! n U 0 0 0 2 . . . α A U Z † AB Tr q = , 0 0 2 m . . . n = Q Q † 1 . ¯ ¯ √ ⇒ − N . .W derive We ). 0 γB βA ˙ . ˙ + Z · · · · · · · · · · · · P states BPS . † (¯ . Z Z U σ − . m 0 o Tr 4 − ) Z Γ ¯ γβ H ˙ † q βA ≥ o ˙ N a ewitna product a as written be can 2 0 n = o √ q ≥ Z H q N Z 2 2 0 1 deto (due AB ≥ † †                 . Z 0 n ntr phase unitary a and o α A obe to . , q 0 ( − N r endas defined are . ≥ which Bogomolnyi 1) / 2 0 n nextra an and . Q 7 2 a (2.10) 0) = such , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. uesmercfil hoydsrbn neatos Rec interactions. fields supermul describing by in theory states field particle supersymmetric 1 considered a just have we far, So Superfields and Superspace 3 ntesprymti ae ewn oda ihojcsΦ( objects with deal to want we case, supersymmetric the In eko hteeycniuu group continuous every that know cosets We and Groups superspace 3.1.1 about Basics 3.1 superspace? that is what But hr dim where • • • • hyaefntoso h coordinates the of functions are they r ucino coordinates of function are ϕ n 2 and h P odtoshl lc yblock: by block hold conditions BPS the rnfr ne h ue Poincaré group. super the under transform H e scnld hsscinaotnnvnsigcnrlch central non-vanishing about section this conclude us Let ii xrmlbakhls(hc r h n onso the of points end the are (which holes black Extremal (iii) i)BSsae r motn nudrtnigstrong-weak understanding in important are states BPS (iv) i)TeBSsae r tbesnete r h ihetcen lightest the are they since stable are states BPS The (ii) v nsrn hoyetne bet nw as known objects extended theory string In (v) i P ttsadbud aefrom came bounds and states BPS (i) rnfrsudrtePoincaré group the under transforms o ahbok If block. each for ϕ ( ed n tigtheory. string and field- ned h qiaec fms n hrermnsu fchar of us super reminds charge extended and for mass states of BPS equivalence the be Indeed, to happen stable) therefore oin h on eest neeg bound. energy an c to the refers of bound solutions The energy finite motion. localised are which systems, G 2( x µ − N dim = ihteproperties: the with ) : Λ k ) 0 states. M G < k < G →M −→ osdrfrexample: for Consider . k k k fthe of = = 0 = G N N X 2 2 q , i fsuperspace of r qa o2 to equal are = = G ⇒ ⇒ ⇒ n ensamanifold a defines 7– 27 – g x exp( = µ 2 2 2 soliton 2 N 2( nMnosispace-time Minkowski in N − N m tts lr hr multiplet short - ultra states, tts ogmultiplet long states, m ≥ k iα ) hr r 2 are there , mnpl- ouin of solutions (monopole-) a tts hr multiplets short states, 1 2 T max branes D a ) o i l htprilsaedescribed are particles that all ilt.Orga st rieat arrive to is goal Our tiplets. ( M q X −→ i ,snew ol en one define could we since ), G which ) Hawking − N r BPS. are re ihsm remarks: some with arges i t parameters its via rlycagdparticles. charged trally n α 2 opigdaiisin dualities coupling k ascleutosof equations lassical a o rainoperators creation e lc holes. black ged , rvt theories. gravity vprto and evaporation agMills Yang { α a } Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ealta h eea element general the that Recall edefine We hr rsmn parameters Grassmann where uaosbecause mutators ob oegnrl e’ en coset a define let’s general, more be To e.g. • • • • • • • { ikwk Poincaré = Minkowski oegenerally, More ec,teprmtrsaei h pee( sphere 2 the is space parameter the Hence, G/H G i.e. G for G oonly so G/H for G | aiodi h pee( circle) (a sphere - 1 the is manifold α a − h g | µ = = = n 2 β α = β x p = + ∗ Q , i N SL SU U = α α q U ∈ = | x 1 ∗ β exp = (1) 1 = α 1 ihelements with (1) µ implies (2 , 2 ihelements with (2) R | } 2 U SU 2 dniyn hsb a by this identifying , ue PoincaréSuper Q so , , ¯ 1 .Write 1. = hc a eietfidwt ikwk space. Minkowski with identified be can which C otiseetv information, effective contains × g α (1) superspace ˙ { (2) ihelements with ) o U Q M n exp = 2 × α /U P (1) i M ( 2 = , SL U θ ¯ k 4 ( (1) β SO 2 =1 α ˙ (2 (1) } ∋ 1 , C ( = x n α + ) = ∼ g σ k 2 ob h coset the be to +1) i /U = / { µ = ( θ ,so 1, = exp = β Q ω ) / α SO ¯ /SO oet = Lorentz α g g R ) 1 µν x , α g oet = Lorentz ˙ α 1,teietfiainis identification the (1), Q ˙ 3 1 θ = fsprPicregopi ie by given is Poincaré group super of θ , ¯ g (3) a P β M × + ( exp( = 1 ˙ n β = µ = eueatcmuainrltosfor relations anticommutation reduce ) µν } U ix S + i /SO G/H − M ( = e

3 0: = 1 lmn diag( element (1) β α α β 2 a . = ∗ α + 1 S SU x · and M Q ⇒ 8– 28 – α 2 2:Each (2): V n 2 iαQ ∗ { where Q 1 (2) . a − x , U ω µ + 2 1 V (1) β hr ope parameters complex where , µν ix P = ,then ), n h o α G/H = µ Q, θQ ∈ a , ω 3 = 2 S β g µν Q x SU x µ + 3 1 2 exp = S } ∈ 3 2 2 + a , − / ) 1 + + = θ g θ 2 and (2) G . ¯ { β x µ α , ω Q α ix 2 2 ix θ , 1 ¯ U ∈ sietfidwith identified is H Q e µν + i 2 4 iγ 3 α ∈ α (1). } SU α , ! = for e , i θ ¯ 2 = 2 + [0 ( ipie otetranslations the to simplifies α − ˙ U α a ) i.e. 1), = 2 a ewitnas written be can (2) o , iγ x 1 1 2 θ staeesadhermitian, and traceless is ¯ k (1) makes ) Q / π α ˙ ∈ θ ,s h corresponding the so ], 1 Q n α ¯ R ω α ∋ ( ˙ + σ ) hnteconstraint the then , µν µ h M α o ) , α α g 2 exp( = . β · SU Q ˙ Q h α ffcieyreal. effectively θ ¯ α 2 ∀ β (2) and ˙ ) , P h Q /U ¯ µ iβQ ∈ β . ˙ (1) = β H ocom- to satisfy 1 = .In ). ⊂ , g g S G = 2 . , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. si hr een onaytrs o nerl over integrals For terms. boundary no were there if as l t onsaeietfidi h oe.Ayrd(ak verti in (dark) neighbours red its Any with coset. identified the is in and identified are points its All iue3 Figure iesosteobto some of orbit the shows line o nerl,define integrals, For e sfis osdroesnl variable single one consider first us Let ok bu hssbetae[ are subject this about books uesaewsfis nrdcdi 94by 1974 in introduced variables first was Grassmann Superspace of Properties 3.1.2 ucinin function otems eea function general most the So xetfrtwo, for except . f lutaino h oe identity coset the of Illustration ( η η = ) sapwrsre,tefc that fact the series, power a as X k ∞ =0 f G k Z = η Z k d U η d 1 f 8 η η (1) = d d n [ and ] ( η f η × α slna.O ore t eiaiei ie by given is derivative its course, Of linear. is ) 1 f U =1= 1 := 0 direction. =0= 0 := 2 1 element (1) 9 + G/H ]. f η 9– 29 – 1 Salam hntyn oepn eei (analytic) generic a expand to trying When . = η η ⇒ qae ozero, to squares + ⇒ U g 1 (1) ne the under f and 2 δ a iecnan l h itntcstelements coset distinct the all contains line cal × Z ( |{z} η η U 0 d = ) 2 Strathdee 2 η η (1) edefine we , + H 0 = |{z} /U ... = 0 . η 1 U 1 = (1) η 1 2 1 ru hc sdvddout. divided is which group (1) , ,cnesalteterms the all cancels 0, = = U [ 6 2 f , 1:Tebu horizontal blue The (1): 0 7 .Recommendable ]. + f 1 . η d d f η = f 1 . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. .. ento n rnfraino h eea clrsuperfield scalar general the a of define transformation To and Definition 3.1.3 where h nerloe function a over integral The But oethat Note eiaie oki nlg oMnosicoordinates: Minkowski to analogy in work Derivatives hc utfistedefinition the justifies which or Treating et let Next, P rnfrainrlsEqs. rules transformation 1 µ − • • sarpeetto fteasrc operator abstract the of representation a is Z ϕ ϕ ia ucino pc-iecoordinates space-time of function rnfrainudrPoincaré under transformation ( ( { x µ d x ∂ µ 2 P µ α ϕ θ sas ibr etri oefnto space function some in vector Hilbert a also is ) θ ) ∂ , µ R α := sa prtr rnlto ihparameter with translation a operator, an as superfield , 1 7→ ϕ β θ ¯ dθ α } 2 1 ˙ + 1 = Z esioso rsmn ubr.Tersursaedfie by defined are squares Their numbers. Grassmann of spinors be α d exp( = ∂ 2 Z = { α ⇒ d θ ∂ ia Z θ ¯ θ R α d β ealpoete fsaa fields scalar of properties recall , 1 ˙ − µ , f η d d Z = 1 := P ia θ 2 ∂ θ ¯ d θ 1 µ β α d µ ˙ ( θ ¯ 3.1 ϕ Z ∂θ ∂θ } = η θ − θ P α ˙ θθ 2 = ) β .A o ut-iesoa integrals, multi-dimensional for As 0. = 4 1 α β f µ d .As,witni em of terms in written Also, 0. = = , 1 4 3.2 7→ ) ( θ ǫ d ⇒ η ϕ 2 = βα θ := ste qa oisderivative, its to equal then is ) = ( α θ ofis re in order first to 1 x Z dθ 2 exp( d Z µ − − θ θ =: ) α δ d θ 1 β 1 2 α dθ ia d 2 α − ǫ η θθ θ β ǫ µ θ αβ β = αβ ia α ( x d , P f 0– 30 – µ µ , 0 , µ θ, θθ = ϕ P 1 2 ⇒ + and 1 = ( ϕ µ Z x ) µ ⇔ 2 f θ ϕ ¯ d d θ ¯ 1 θ ¯ − a 2 ∂ θ ¯ i P = exp( η θ ¯ µ α 1 ˙ h θ = ) ¯ egttefloigrelationship: following the get we , θ a µ Z ¯ := a , ϕ α − ˙ β µ ˙ = θ ϕ cigon acting ¯ = ) ia 4 1 d Z := β ( ˙ θ ǫ x µ a θ α ¯ 2 µ ˙ d 1 4 F µ µ α β ∂ P ∂ ˙ f ˙ = 2 P θθ ): d ǫ θ P ⇒ θ θ 1 θ ¯ so , ¯ ¯ d ilcag tto it change will µ : µ θ α β ¯ α ˙ ˙ θ ˙ ) Z ¯ β i ˙ α ˙ = . 2 1 d 1 = d d θ ¯ = = ǫ F θ 2 α ¯ ˙ α ˙ θ β µ ¯ . ˙ oprn h two the Comparing . β d d ˙ − ǫ ( f δ η θ ¯ α θθ ˙ , α . θ ia ˙ = ¯ β ˙ . β ( ) ˙ µ . P θ ¯ − θ ¯ µ i∂ 1 = ) ϕ µ . = − . a (3.1) (3.2) µ ∂ µ ϕ. Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. of osdrn nifiieia transformation infinitesimal an Considering where s a on transformation super-) (but similar a perform shall We ti ovnett set to convenient is It ehv h rnfrainof transformation the have We representation: edoperator field ǫ eodya ibr vector Hilbert a as secondly h rnlto farguments of translation The ucin of functions Here, o h rnfre superfield transformed the for ) S S θ ( δS ( x α o eea clrsuperfield scalar general a For x µ , µ usin h steen term no there is Why Question: θ , ǫ θ ¯ c θ , α ˙ eoe parameter, a denotes α := a edtrie rmtecmuainrlto hc,o co of which, relation commutation the from determined be can α , ihafiienme fnneoterms: nonzero of number finite a with , θ ¯ θ ¯ α ˙ α θ ˙ ) δϕ = ) , S θ ¯ 7→ ( ( + and , ( x x + ) µ n exp θθ θ , Q ϕ i ) α c α ( c h θδψ θδ , ¯ x sacntn ob xdltr hc sivle ntetransl the in involved is which later, fixed be to constant a is i ( + ǫQ , S .Aan oprsno h w xrsin t rtorder first (to expressions two the of comparison a Again, 1. = θ , ¯ + ) ( α λ ¯ ˙ ǫ Q ) ( ¯ ( Q θθ x δx x α ˙ ( + ) + ) ¯ + o 7→ ) θψ Q x µ Q Q θ P ¯ ¯ µ ǫ ¯ + = S λ ¯ Q , ersnaino h pnra generators spinorial the of representation a ( α α µ ( 2 = ˙ ¯ S ( x ( θ exp ) θδ x − ǫ x ¯ + ) θ α ¯ S Q stekyt e t omtto eain with relations commutation its get to key the is ( + ) µ ¯ + = = = θ ¯ , χ S ¯ ( ic θ , i ) θ x ¯ ( ( θδρ − α x x α ( µ ˙ σ σ ǫ − − θ + ) ¯ θ , , µ i = µ implies, χ θ ¯ θ , ¯ ( i i i∂ θ ) ( ¯ 1– 31 – α α ǫQ ( µ α ˙ x θ ¯ ∂ ∂θ x α , ne h ue ona´ ru,fisl sa as firstly Poincaré group, super the under ) µ ( α ) ( + ) θ ˙ i ¯ S ∂ ∂ θ θσ + ) θ , ¯ ¯ θρ + ) θδM θθ P , α θ α α ¯ ¯ + ˙ ˙ ǫ → α ,oecnproma xaso npowers in expansion an perform can one ), µ µ ˙ Q ( = ) θ x ¯ + − ǫ S )( Q ( + ) θM θθ ¯ θθ ic ¯ + = θσ ) + ( c c ∗ x ( ) ⇒ ∗ ( ǫ ( S + ) δS ν σ θ S σ θ Q θ ¯ θ ( ¯ β ¯ µ θ θθ x ¯ ) exp ) ( x ) F 1+ (1 = + ) α µ σ Re S µ ( ) δD µν β ˙ θ µ ǫ ¯ ¯ + ) θ δN θ ) ¯ ¯ θ { ¯ ? i β δx β ( θ = . c ¯ ˙ ( uperfield. x α θ ˙ ) } ǫQ ¯ ∂x ) µ N θ ¯ D ∂x iǫ . ∂ θ , ( δS ∂ x ( 1 = µ Q ¯ + x µ ( + ) ( α (3.3) ) x + + ǫ ( + ) Q ¯ i ǫ ǫ ¯ ) re od nany in holds urse, α Q ¯ σ θ , ) , θ ¯ S σ θ α µ ˙ Q where , ¯ + θ ¯ µ α ) ǫ θ δV ¯ cigon acting α ˙ ) V µ . ( µ Q x ation ( (3.4) (3.6) (3.5) x ) α ) : in Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. usiuigfor Substituting so h eodeape he.Nt that Note sheet. examples second the on as of .. eak nsuperfields S on Remarks 3.1.4 other). each into transforming fermions and sasuperfield a is S δV • • • : µ lont htsprcvratdrvtvsstsyantico satisfy derivatives super-covariant that note Also hc satisfies which ∂ n therefore and ic [ since iercmiain fspred r uefilsaan(st again superfields are superfields of combinations Linear ntels tp eue h ebizpoet fthe of property Leibnitz the used we step, last the In If µ = S S 1 ǫσ δ n δρ saspredbut superfield a is λ ¯ and D n µ ∂ 2¯ = λ 2 = ¯ α D α D ǫ , + α , S α ǫD Q Q ǫD ⇔ , 2 ρσ D ¯ δ α δ ¯ + r uefilste oi h product the is so then superfields are Q β ( ˙ := , µ + ( tsatisfies it − o ∂ S β ǫ ¯ Q ǫ δ h α ¯ o Q 1 χ 2 + i D ¯ 2 ¯ S i α ˙ S = ] (¯ ∂ α = ) 2¯ = ( 2 2 and σ i .W edt en oain derivative, covariant a define to need We 0. 6= σ α = = ) ǫ , ν ( ν σ ∂ σ + − ǫN ¯ µ ν Q = = µ ∂ S n 2 ∂ ψσ ǫ ¯ ǫ α δS i egtepii em o h hne ntedffrn parts different the in changes the for terms explicit get we , i ) α D ) − ( ¯ + S S ( ( ∂ ∂ σ σ µ α δS 1 i S µ ( µ µ = σ snot: is δS ¯ µ ǫ ( ǫσ V 1 , V ) Q ǫ ν ) = ) ν ¯ α Q ν α ǫ i 2 Q δϕ µ i β i ( ¯ β + ˙ − ˙ + )( ǫ ( + ( β θ ¯ ˙ ∂ ¯ + Q ǫ i∂ o i i∂ β = Q ǫ i ¯ µ ˙ = 0 = (¯ ( σ ¯ δS ¯ + ∂ σ α ǫ µ σ 2– 32 – , ν ǫψ Q µ ¯ + µ [( ϕ ¯ µ = σ ǫ 1 ǫ ǫ ǫ ¯ ( ) , Q δD µ ) ) − ¯ ) Q ¯ + ǫ S ∂ ∂ ∂ Q ) S ¯ µ 2 ν V n S µ ǫ ¯ + n 1 δD M ) χ sattldrvtv.As,w aebosons have we Also, derivative. total a is N ,δψ χ, µ ¯ D ¯ Thus: . D ¯ S S ) ) D ⇒ ǫ D α ¯ 2 Q 2 ˙ α ¯ (3.7) ) α ˙ , ) , S + Q D ] := β 6= β α S δN δM Q o o S 1 i i S ( and ( − 2 = ǫ ǫ 2 = = = ¯ = Q = Q : mtto relations mmutation ∂ ssuperfield is ¯ α ǫρ ˙ 2 i ǫ Q ǫM ¯ + ¯ λ ¯ ¯ + agtowr proof). raightforward ∂ n n − + µ sdffrniloperators. differential as − D ǫ D ( ¯ ¯ ǫ ( + Q ǫσ 2 ¯ α iθ i Q α ˙ 2 i ˙ ¯ ) ǫσ ∂ µ , β ( ) , S σ λ µ ¯ ( µ Q 1 µ D ∂ ψσ ¯ σ ¯ ∂ − ǫ ¯ . α β β µ )( µ ˙ ˙ S S o o ) ρσ χ µ ¯ 2 i∂ β ) ǫ ¯ α ˙ µ µ 0 = ∂ 0 = ϕ ǫ ¯ µ ) + V . µ ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. fΦ=Φ( = Φ If ewn ofidtecmoet faspred satisfying Φ superfields a of components the find to want We superfields Chiral 3.2 convenience er ler.Teeaedffrn ye eedn pntec the upon depending types different are There algebra. metry opnnskeigi tl saspred ngnrlw can we general In superfield. a on as still it keeping components S efidfrtecag ncomponents in change the for find we n emoi (complex fermionic 4 and ne uesmer transformation supersymmetry a Under h hsclcmoet facia uefil r sfollows as are superfield chiral a of components physical The uiir field auxiliary hr h ethne uecvratdrvtv csas acts derivative supercovariant handed left the where n eed nyon only depends Φ and surs lpos Higgs), sleptons, (squarks, ehv ( have We ec h hrlspredcniinbecomes condition superfield chiral the hence is S • • • • • not edn osalrspred htaeirdcberepres irreducible are that superfields smaller to leading , etr(rra)superfield real) (or vector nicia superfield anti-chiral iersuperfield linear hrlspredΦsc that such Φ superfield chiral S o osatspinor constant For = nirdcberpeetto fsprymty ow a e can we so supersymmetry, of representation irreducible an ,θ, y, D f ¯ α ( Φ( ˙ x θ saspredol if only superfield a is ) θ α ¯ x nawyt edfie ae.O hl,teeae4bsnc(comp bosonic 4 are there shell, Off later. defined be to way a in ,te,since then, ), D n ( and 0 = ) ¯ µ α θ , ˙ ( = Φ α Φ( , θ ¯ α L y ˙ = ) D µ y ¯ uhthat such θ , α ψ ˙ and c D ¯ θ , uhthat such Φ ψ α α ¯ α α ˙ = ) S opnns efrigaTyo xaso fΦaround Φ of expansion Taylor a Performing components. ) D ) y ¯ some − µ θ = α ϕ ∂θ ˙ ∂ ncmoet,oefinds one components, In . ( = ) V ( δ y Φ sadffrniloperator, differential a is cθ √ x = Φ α µ i + ) =

2 DD D s ϕ ¯ y, sntaspreddeto due superﬁeld a not is f − α ( = ( := V θ ˙ ¯ y θθ 0 = Φ ∂ = ¯ ( + L † µ α 2 1 √ D ˙ ) + ) const − and 0 = i ∂ α atce qak,lpos igio and Higgsino) leptons, (quarks, particles x 2 D 3– 33 – ¯ µ 0 = Φ iθ µ ¯ θψ ǫ α ψ ˙ Q α y ( √ ( µ σ + tews,teewudb some be would there otherwise, , x x ∂ ) ∂ 2 α ρ ¯ + ) θ + ) ¯ Φ α β θψ ˙ σ ∂y ˙ ∂ L ∂ θσ iθ µ ǫ Φ .Tu hr sno is there Thus 0. = ρ Q µ ( θ = )( ¯ ¯ y

θF θθ µ µ θ, x D − L Φ + ) µ θ . θ ¯ ¯ † α + ( + . , 4 1 ( = iθσ x ( D + ) θF θθ ∂ ¯ θθ moecnitn constraints consistent impose α α δφ µ ˙ : +2 ( ) θ θ ¯ onstraint: ¯ nain ftesupersym- the of entations ϕ β = ) ˙ D ( = θ ¯ ¯ ) θσ iθ y i θ α ¯ ersnsasaa part scalar a represents ( ˙ µ ∂ ) ∂ ǫc σ .W en for deﬁne We 0. = Φ ) i θ ¯ ∂ Φ µ ( µ . β , µ θσ ˙ θ ¯ iiaesm fits of some liminate ∂ θ ¯

∂ ) y,θ α µ µ µ ϕ ) ∂y θ ¯ ϕ α . ∂ ˙ ( α ˙ µ − x ( x dependence - ) nΦ( on i ) ( δψ θσ lex µ ∝ y F ) µ α ˙ ϕ ǫ∂ θ , san is 0, = , µ x α F f µ ). ) . : Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. nodrt n nobject an find to order Lagrangian In superfield Chiral 4.1.1 Lagrangi superfields. build transforma chiral to supersymmetry only a prescription of under case a derivative) simplest need total we a to this (up For inclu which Model. superfields dard among couplings determine to want We Lagrangians 4.1 supersymmetric dimensional Four 4 rnfrain eepotthat: exploit we transformation, R n nert ozr.Fracia uefil = Φ superfield chiral a For zero. to integrate and So endt ewaee utpis( multiplies whatever be to defined Since arninfracia uefil ’ a ewitnas: written be can Φ’s superfield chiral a for Lagrangian d 4 δF • • • • • x Φ Φ fΦi hrl then chiral, is Φ If h rdc fcia uefilsi hrlspred sinc superfield, chiral a is superfields chiral of product The S o eea clrsuperfield scalar general a For uefil sacia superfield. chiral a is Φ superfield o hrlspredΦ= Φ superfield chiral a For N δF saohrttldrvtv em utlike just term, derivative total another is | 1 † F D n Φ and Φ ¯ 1 = and = α ˙ S R 2 lblsupersymmetry global d δD if 0 = 4 xF † L r elspred u ete hrlnranti-chiral. nor chiral neither but superfields real are Φ + r oa eiaie,te aen ffc nlclpyisin physics local on effect no have they derivatives, total are → = D ¯ R α ˙ Φ = Φ S ¯ d ¨ he potential Kähler - 4 i L x K | Φ uhthat such (Φ) .I eea,any general, In 0. = ( δF δψ δϕ F (Φ {z † δD + , santi-chiral. is ... Φ δF = = = † θθ ( + } ) S δF = = ) .Tu,freape ne UYtransformation, SUSY a under example, for Thus, ). i i √ = θθ √ √

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,the ), sattldrvtv ne supersymmetry a under derivative total a is √ ∂ θθ σ ǫ ue potential - super µ µ δD ¯ 2 )( ϕ . ψ sivrat hrfr,tems general most the Therefore, invariant. is µ ǫ θ ¯ holomorphic | W λ . . . ¯ σ ¯ + θ ¯ F nagnrlspred oethat: Note superﬁeld. general a in µ ) {z (Φ) − D ∂ ( + emtasom as: transforms term √ µ ( } ψ. 2 x σ ρ θθ ,the ), F ǫ µ )

F F ǫ ¯ in ewl tr ihthe with start will We tion. eteprilso h Stan- the of particles the de hste‘ the thus , + . e D function D h.c. n hc r invariant are which ans ¯ emtasom as: transforms term α ˙ ( ! S 1 . S 2 f F ( = ) Φ fachiral a of (Φ) − em Φ term’ D h action, the ¯ α ˙ S 1 ) S | F 2 + is Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. [ osbernraial emfor term renormalisable possible A foeaoso iesoaiysc htteLgaga has Lagrangian constr the to that need such we [ dimensionality theory, of renormalisable operators a of obtain to order itself). In superfield chiral a is fore hrfr eaela otefloiggnrlepesosf expressions general following the to lead are we Therefore no contain stesm sta fissaa opnn n h dimension the and component scalar its of that as mas same The the not. is are Φ dimensions mass negative with couplings but tnadfrin htis that fermion, standard nw sthe as known hsarayhnsthat hints already This h function The hs arnini nw as known is Lagrangian whose Where eas efr alrepninaon = Φ around expansion Taylor a perform also We rmteepninΦ= Φ expansion the From egtteepeso o Φ for expression the get We L ϕ = Φ weetesur rcessadfrdmninlt ft of dimensionality for stand brackets square the (where 1 = ] .Trso ieso ,sc as such 4, dimension of Terms 4. = ] | D ϕ eest the to refers + D − √ W K superpotential terms. 2 K Φ = (Φ) θψ skona the as known is Φ = + D θF θθ L F [ [ emo h orsodn uefil waee utpis( multiplies (whatever superfield corresponding the of term = W K Z W ϕ W † ⇒ sntasadr clrfil.I re ohv [ have to order In field. scalar standard a not is D Φ [ = [Φ] F Φ + ooopi ucino h hrlspredΦ(n there (and Φ superfield chiral the of function holomorphic a , † ( + ] ] Φ ϕ √ W , [

θ (Φ + ) [ Φ = θσ iθ esZmn model Zumino Wess D ≤ K ≤ 2 = ] θψ ] ysubstituting by K¨ahler potential µ in 4 in 4 + ... K ≤ † ∂ θ ¯ ϕ | Φ + 1 = ] θθF − sΦ is µ

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= µ + i , ∂W 2 tflosthat follows it [ ∂ϕ ϕ W [ W F ∗ ϕ ( ... λ elfnto fΦadΦ and Φ of function real a , ... , (Φ) θθ 2 = ] ] : + Φ (where m ( + [ ( + + ) ψ ≤

2 ∂ F = ] ϕϕ µ ... θθ σ ψ θθ | 1 2 + ( + 3 m ∗ 2 ( ) ) (Φ or . ∂W . µ and θψ ∂ϕ W h.c. θ Φ ¯ θ ¯ θ 3 2 {z ¯ ( ) ieso ftesuperfield the of dimension s iesoaiy4 eknow We 4. dimensionality F † K 2 − ) c arnini terms in Lagrangian a uct − = θψ ! rΦ Φ + ΦΦ or K g + and ϕ | + ) . D ∂W ϕ 4 1 of ∂ ) efil)adw want we and field) he Φ } 2 | g 3 ( 4 ... ψ θθ

r renormalisable, are , W Φ Φ= ∂ stesm sany as same the is ( ) 3 ∂ϕ : 2 ϕ , W θ ¯ ): L 2 θ † ¯ Φ eneed: we 4 = ] ) ∂ † µ † ic these since ∂ . µ W θ ϕ. ¯ θ ¯ Φ is (Φ) )( (4.2) (4.3) (4.1) θθ )). - Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. usiuigEqs. Substituting o rpgt.As,w a aiyeliminate easily can we Also, propagate. not oieta hsi udai n ihu n eiaie.T derivatives. any without and quadratic is this that Notice hsdfie h clrptnil rmisepeso ecan we expression potential its scalar From definite potential. scalar the defines This n usiuetersl akit h Lagrangian, the into back result the substitute and efiihtescinaottecia uefil arninw Lagrangian superfield chiral the about section the finish We • • L h ato h arnindpnigo h axlayfield’ ‘auxiliary the on depending Lagrangian the of part The Z W rmwec egttekntctrs e.g. terms, kinetic the get we whence from K qasteoefrtespinor the for one the equals ngnrl emyexpand may we general, In oeta spstv eidfiie( semi-definite positive is potential oevr h coefficient the Moreover, field The etrainter:dvrecsaermvdfo oeloo some from removed are divergences theory: perturbation efcoupling, self i ¯ j = samti nasaewihi ope ¨he aiodwit manifold Kähler - complex a is which space a in metric a is N ϕ ∂ K arnini atclrcs fstandard of case particular a is Lagrangian 1 = µ a tcnb edfo h udai term quadratic the from read be can it (as ϕ (Φ ∗ j † ∂ , 4.3 µ g ϕ Φ 2 , i − | 4.2 ) ϕ

D | i V 4 δ δ ψ ¯ L δF noEq. into hsi h oreo oe”iauos aclain nSU in cancellations ”miraculous” some of source the is This . ( S S δF = L F σ ¯ ( ( ( F ) F F ( µ ∗ . . . ( F ) ) ) ϕ ∂ ) µ ). + g = ψ = 0 = = 0 = →− 7→

fYkw coupling Yukawa of + 4.1 K ψ F F ∂ϕ F F (Φ eobtain we , ∂ a a era rmteterm the from read be can (as 2 i i

∂ϕ , ∗ K ∗ ∂W V Φ ∂ϕ ⇒ ⇒ + 6– 36 – + ¯ j ∗ j † ≥ and ) !

F 2 ∂W ∂ϕ ) lotemass the Also 0). ∂W ∂ ∂ϕ F F µ sn h edequations field the using =: ∗ ϕ W + F i F + ∂ (Φ µ − + + ∂W ∂ϕ ϕ ∂W ∂ϕ g V i ¯ h.c. j ∗ rudΦ around ) ( ( ∂W ∗ F ϕψψ ∂ϕ ∗ ) m = ( i en httefield the that means his N ∗ ϕ ∗ 0 = 0 = 2 aiyseta ti positive a is it that see easily − ) lodtrie h scalar the determines also ) ϕ F t w remarks, two ith . . . arnin:tescalar the Lagrangians: 0 = , † ∗ 1 2 ϕ m orcin,al Fig. la a corrections, p F + ntesaa potential) scalar the in i o h ope scalar complex the for ae h ipeform: simple the takes K ∂ ∂ϕ = 1 2 2 i W ¯ j ∂ ∂ϕ 2 ϕ 2 ∂ W i coordinates h µ 2 ψψ ϕ ncomponents, in ψψ i ∂ + µ = ϕ h.c. ¯ j ∗ m/ . F 2 ψψ . does SY ϕ 4 ). i . . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ecndfieagnrlsdguetasomtoso etrfi vector of transformations gauge generalised a define can We hc nue tnadguetasomto o h vecto the for transformation gauge standard a induces which opigt h uaacouplings Yukawa the to coupling iue4 Figure hr ehv hfe oefils(notably fields some shifted have we where components oga ehv osrce arninta sivratun invariant is that Lagrangian a constructed transformation. have we as long .. ento n rnfraino h etrsuperfield vector superfield vector the general of most transformation The and Definition 4.2.1 superfields Vector 4.2 divergence. leading the to contribute hnw a choose can we Then htteei nte novn h rlna iga coming diagram trilinear the involving another is there that V fΛi hrlspred then superfield, chiral a is Λ If ( ,θ, x, usin a o eieteerltosb usiuigi for in substituting by relations these derive you Can Question: θ ¯ = ) . n opdarm hc il orcin otesaa assqu mass scalar the to corrections yield which diagrams loop One C , C M ( + + ( x , σ θ + ) N θ ¯ θ ¯ V ϕ , µ ) µ , D θ V θ ¯ iθχ ψ , 7→ µ , V λ ( µ F ( ( 1 2 x φ x x ( + ) n emoi ns( ones fermionic 4 + 4 and ( + ) ( ) V ihnΛt ag wysm ftecmoet of components the of some away gauge to Λ within ψ M V i µ ψ λ (Λ − ¯ n hrfr nue aclaino h edn diverge leading the of cancellation ensures therefore and ) − + 1 2 − i 7→ V = θθ θ σ ¯ ∂ χ ( Λ ¯ µ iN ) ,θ, x, µ ( ∂ V † θ ¯ x D C V savco uefil.I a components: has It superfield. vector a is ) µ χ µ = ) + ) χ 2Re( ¯ 7– 37 – λ ¯ ( θ ¯ + D x = 0 = = = ( = ) x ) i and ) ϕ 2 i ) − F i √ Λ V − θθ + ∂ 2 † ϕ λ ( − µ 2 1 ψ rmptnilterms potential from ,θ, x, o ovnec.Teeae8bosonic 8 are There convenience. for ) 2 1 =: σ M ¯ − ϕ µ Λ ( ( ∂ θθ † x + ϕ µ θ ¯ V χ + ) a h form the has ) χ ( ) † µ , α ( ϕ λ , θ x ¯ † + θ ¯ ) opnn of component r iN ) lsvia elds α ∂ ). e h eeaie gauge generalised the der ( µ D x . α ) ( rd UYrltsthe relates SUSY ared. x ) mgϕϕ − − 2 i 1 2 † θ ¯ ∂ ϕ θ ¯ µ † V u tdoesn’t it but , ∂ M µ Λ C ? ( c.Note nce. ( x x ) ) V − as , iN , φ 4 ( x ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ne uesmer,teecnet eeaiet hrls chiral to generalise fields concepts these supersymmetry, Under oain derivative covariant ne local under h hsclcmoet favco uefil are: superfield vector a of components physical The above: Λ of components the choose can gauge We Zumino Wess 4.2.2 esmer.Hwvr ne obnto fsupersymmetr of combination a transformation under However, persymmetry. oeta h esZmn ag sntsprymti,since supersymmetric, not is gauge Zumino Wess the that Note eoesprymty edefined we supersymmetry, Before ee stecia uefil enn h eeaie gaug generalised the defining superfield chiral exp( the is Λ Here, sa bla ed-srnt.Tesprymti nlg is analogy supersymmetric The strength. - field abelian an as hc schiral. is which rnfrainproperties: transformation ealta o-uesmerccmlxsaa field scalar complex non-supersymmetric a superfield that strength Recall field Abelian 4.2.3 N etrspredi esZmn ag eue oteform the to reduces gauge Zumino Wess in superfield vector ( oesof Powers γ V , .Ti ensthe defines This 0. = WZ W − usin o osoeko that know one does How Question: V ± 2 ( ,θ, x, iq , ocntutagueivratqatt u fΦand Φ of out quantity invariant gauge a construct To . V Φ Z ) sas hrli is. Φ if chiral also is Λ)Φ V lo) the gluon), , 7→ 7→ WZ θ ¯ U ( = ) ϕ 1 ihcharge with (1) V exp( ( r ie by given are x V ¨ V ) WZ + ′ ¨ WZ − 2 7→ i 2 D 7→ σ θ iq µ Λ = )Φ Λ) λ V µ = exp esZmn W)gauge (WZ) Zumino Wess WZ θ − ′′ ¯ and ) ∂ 2 1 V µ Λ ecnedu ihavco edi esZmn gauge. Zumino Wess in field vector a with up end can we iqα µ − ( † q λ ¯ ( θθ x iqV n oa parameter local and oguio and gauginos to ) ( ( + ) W ( ) F x ) µν α θ µ ¯ θ ¯ ⇒ ϕ rnfrslike transforms ) := ( V θθ = x µ ) ) V Φ 8– 38 – V , − ∂ θ µ ¯ † µ λ ¯ 1 4 V , exp(2 V ,ψ F ψ, ϕ, ( x ν ( W D ) ¯ D − µ D α ¯ ( qV sa uiir ed ob endlater. defined be to field, auxiliary an is ( + n esalueti rmnwo.A on. now from this use shall we and , x ) schiral? is WZ ∂ ) 2+ α D nsc a oset to way a such in ν Φ ) V ( V α n θ x ¯ µ 7→ V µ θ ϕ ). ¯ ) orsodn ogueparticles gauge to corresponding ⊂ 0 = ope oaguefield gauge a to coupled θλ pred n etrsuper- vector and Φ uperfields K V rnfrain.Nt that Note transformations. e µ ( ( x V x ∀ sgueinvariant. gauge is ) + ) eips h following the impose we , n eeaie gauge generalised and y n V + WZ ∈ ∂ N 1 2 µ 7→ α . ( θθ ( x C V ( ) ) ¨ WZ ′ = ¨ θ ¯ θ ¯ χ ) ne su- under D = ( V x M µ ) via . = Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. to alta h edsrnt tensor strength field the that call h edsrnt superfield strength field The where of invariance gauge generalised see we Hence, obtain To eiaie ipiyto simplify derivatives h omttrtrsaedet h ae apelHudr f Hausdorff Campbell Baker the to due ponentials are terms commutator The iθσ nWs uiogue h uesmercfil teghcnb can strength field supersymmetric the gauge, Zumino Wess In quantity. covariant gauge a obtain and .. o bla edsrnt:non-examinable supersymmetric strength: section field this abelian In - Non 4.2.4 utlk nteaeincs,w att epΦ keep to want we case, abelian the in like Just omto Φ formation rus h ag ere ffedmte aevle ntea the in values take then freedom of generators hermitian degrees by gauge spanned The groups. rnfrainlaw transformation UF µ D θ ¯ α µν aibe(where variable W = U α − W ∂ W ( 1 α ,θ y, α a α 7→ ne ntr rnfrain.Smlry edefine we Similarly, transformations. unitary under 2 + Λ = Λ ( exp(2 ,θ y, ncmoet,i sms ovnett rewrite to convenient most is it components, in exp( = ) e ⇒ i iq ( = ) σ Λ X qV V µ ,bttenncmuaientr fΛand Λ of nature non-commutative the but Φ, W = θ ¯ exp( ) V λ ) 7→ ′ θσ a α α exp( = ) D α ′ T ∂ ( λ − F V µ µ y µ a α a = λ := θV ¯ µν ¯ + ) a 1 4 ′ and Y ( : a V , y W µ ( exp = ) + ) D ( V ¯ := := α x − D θ D U ¯ = ) ¯ F lonessm oicto nnnaeintere.Re- theories. non-abelian in modification some needs also α 8 − ) α T 1 1 ag hoisaegnrlsdt o-bla gauge non-abelian to generalised are theories gauge (1) ˙ µν q D iq D θ a ∂ ∂ = α : θσ Λ ( ( α 2 µ µ fnnsprymti agMlstere transforms theories Mills Yang non-supersymmetric of i = y D D λ V − ¯ ¯ † ( + ) (Λ exp(2 ) µ a ν V D a a ∂ ¯ ¯ θV ¯ ( α a ) V ˙ y + X 9– 39 – − − ( : ( + ) a µ ,θ, y, exp( ( T V q Λ y σ ∂ + a ) µν qV W † ν ) − θ V µ , ¯ − b Y α + ) θ σ µ exp( ) λ a ¯ † − : 2 ) 2 µν i e c α qV θ 2 λ + + f qV 2 F , θ iq bc 2 h θ ¯ ) V i ) µν D T 2 nain ne h ag trans- gauge the under invariant Φ f q α 2 1 a D − ∂ h a ( F , V µ and h α b y iq a ( T , V µν Y , X ) a ,θ, y, exp(2 Λ) bc µ ( − ( y Λ + Λ V b F y i ) µ µν b ) hntesupercovariant the then )), θ V ¯ i V ) ( − vlae as evaluated e i qV θθ = V ν r l ag invariant! gauge all are nteshifted the in c + V c ) ( ) i † ( ( ruafrmti ex- matrix for ormula scae i algebra Lie ssociated i θθ ,θ, y, if σ ... nocsanonlinear a enforces µ + ( ) abc ) α θ σ ¯ ... β T . ˙ ) µ ∂ c f ) µ α a λ β ¯ ˙ bc β D y ˙ ( µ y µ ) λ ¯ = a β x ˙ ( µ y ) + Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o hrlΛ u rcdr ocntutasial Lagrangi suitable slight a look construct expressions to the procedure (although Our case supersymmetric Λ. chiral for eoeatcigvco uefil arnin,ltu first of us variance let Lagrangians, superfield Lagrangian vector attacking superfield Before vector Abelian 4.2.5 ihSS,teKähler potential the SUSY, With case. • • • • • loolsterm Iliopoulos e nrdeto uesmerctere sta nextra an that is theories supersymmetric L of ingredient new A ag iei function kinetic gauge ℜ d iei emfor term kinetic Add Introduce hc srnraial if renormalisable is which d iei emfor term kinetic a Add i.e. n ert iei emas term kinetic rewrite and scagls) hra o o-bla ag hoyteg the theory gauge non-abelian a corresponding for whereas chargeless), is ob obde.Ti sterao h Itr nyeit for exists only term FI the reason the is This forbidden. be to hoybcuetecrepnigguefil sntcagdu charged not is field gauge corresponding the because theory h parameter The nrdc oain derivative covariant Introduce ti loSS/ag nain (for invariant SUSY/gauge also is It . ( τ 1 = ) K ∂ µ sivratudrorgnrlsdguetransformation. gauge generalised our under invariant is Φ ϕ∂ /g V µ 7→ 2 ϕ o general For . L uhthat such ∗ K D ne oa transformations local under : exp( D ξ = µ sacntn.Ntc htteF emi ag nain o a for invariant gauge is term FI the that Notice constant. a is em)wudtasomudrteguegopadteeoeh therefore and group gauge the under transform would terms) Φ = ϕ − ... 2 V . := iq A + † ihcoupling with )Φ Λ) L µ exp(2 f kin f ∂ to 4 Φ,hwvr ti o-eomlsbe ewl call will We non-renormalisable. is it however, (Φ), Φ saconstant a is (Φ) µ 1 g L K ϕ 2 , L L I F D qV = F Φ = − µ µν = Φ ) = eedn nguepotential gauge on depending qA iq f F Φ † 0– 40 – Φ ( (Φ) sntivratunder invariant not is Φ † µν D V , Φ V ξ τ µ µ U F , A , ϕ ϕ W 1 ag hois n nw sthe as known and theories) gauge (1)

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ϕ D → . − ϕ∂ iϕ µ ϕ ∗ Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. o-bla eeaiaino the of generalisation non-abelian invariant: ﬁeld ealta h Iterm FI the that Recall potential scalar a get we u ftesaa oeta rdcsannzr VEV non-zero a predicts VEV potential The scalar VEVs. the their if around but ﬁelds expands always one that Note oa oeta sgvnby given is potential total h ulsrcueof structure full the hs ehv enta ngnrltefunctions the general in that seen have we Thus, where sa) fw nwterepesosw nwalteinteract the all know we expressions their know we If usual). ihu UY h eainhpbtenteaction the between relationship integral the superspace SUSY, a Without as Action L and 4.2.6 Bailin See well. as transformations gauge generalised =Λ := Λ owiedw iia xrsinfrSS cin,recall actions, - SUSY for expression similar a down write To hspoie lgn aso expressing of ways elegant provides This eedu ihtems eea cinivligsvrlchir several involving action general most the with up end We L ntennaeinextension, non-abelian the In φ S a,oewrites: one say, , h a a K = = T sa don ru ae,and label, group adjoint an is Tr V a Φ , n ( K Z i † ϕ V W ,

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) 2 ξV Φ ξV F ( θ ( = V ϕ i ( ( + ) + D W , θθ a nyapa nabelian in appear only can Tr v ) + ( 1 = ) + n h.c. ϕ e Φ .Tgte with Together ). V h iq i ( 0 W Z ξ D Λ + f , S ) W α → ( + d ϕ W iA 2 α Φ , W θ = ) ,j i, = and 0 α 0 W i K W f ) 2– 42 – emrqie netataet epi gauge it keep to trace extra an requires term / ξ , α

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Not to be quoted or reproduced without permission. hr the where N Φ. superfield ooopi function holomorphic edcmoethe decompose diagrams We Feynman than QFT monopoles. in and things more obviously are There where osdrthe Consider su of number 4.3.1 the larger The t arise. actions and constrained. on functions constraints more these be to will constraints actions put will supersymmetry uesmercatosadwl hrfr edtrie by determined be therefore will and actions supersymmetric the supersymmetry, global in that know We on? depend actions For 4.3 t..I h aewy cin o xeddsupersymmetrie extended for actions way, same the r are In couplings etc.). the of some which (in actions supersymmetric hr eoe oectff hs ttmnsapyt h Wil the to apply statements These that: cutoff. Note some denotes Λ where Seiberg h ulprubtv cinde o oti n correctio any contain not does action perturbative full The UYenforces SUSY 2 = • • N exp etraiepoessuulyivleseries involve usually processes Perturbative N F UY ehda action an had we SUSY, 1 = N non-pert 2 = and 2 = − f A , Φ = (Φ) µ g 4 N c 2 Witten and lblsupersymmetry global o ntnecudb the be could instance for etrmultiplet vector 2 = sannprubtv xml n xaso npwr of powers in expansion (no example non-perturbative a is N λ F ∂ prepotential 2 = ∂ r ecie yavco superfield vector a by described are W civdsc nepninin expansion an such achieved 2 Φ Φ aldthe called (Φ) F 2 nteaction. the in 0 = F K , F = Φ = (Φ) ( (Φ S Φ Φ , eedn on depending prepotential Φ 2 2 F † ln ntno expansion instanton ψ λ = ) F 3– 43 – as 1loop A ϕ Φ Λ µ K 2 2 N and 2 + 1 cin r atclrcsso non- of cases particular are actions 1 = 1loop) (1 : te level) (tree : i : P F K f n non-pert Φ N , a a ewitni em fasingle a of terms in written be can † n W exp(2 g UY[ SUSY 2 = n , sfrmr hn1loop, 1 than more for ns ltd h oeta spositive, is potential the elated, r atclrcssof cases particular are s f ihsalcoupling small with V P and K V eeoetecorresponding the herefore n the and ) k , esmere h more the persymmetries a W ∂ ∂ ξ a el ..instantons e.g. tell, can k oineetv action. effective sonian F Φ htwl the will What . , exp 11 f − ]. and ϕ − h.c. , g g ψ c 2 ξ possible). k h extra The . yachiral a by n1994, In . g ≪ N ≥ N 1. 1 = 2 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. san As 4.3.2 In orsodnei d n5dmnin ult ofra fiel conformal a to dual possess to dimensions well happens 5 as that in theories AdS field do is to in applies correspondence theories benefit string) same (the (and gravity understood describe to one allowing where ob nain ne ofra rnfrain.Ti sth is le This one in transformations. gravity) conformal (without under invariant theories be field to to th certain ”dual” that are realisation the space to led have theories field and N cl,teeoew have we therefore scale, uesae,ta n unu orcin naSS hoyc theory SUSY a in corrections quantum generalisation any (a that theory superspace), perturbation supergraph using by hr h function the where hr r oeipratpoete of properties important some are There theorems Non-renormalisation 4.4 ns ml that: imply ones) h o-eomlsto ftesproeta soeo the of of behaviour one simple The is theory. superpotential field the supersymmetric of non-renormalisation The lohv neetn consequences. interesting have also ne modular under safiieter,mroe its moreover theory, finite a is 4 = N • • • ,teeaen refntosa l,btw aeafe param free a have we but all, at functions free no are there 4, = h neatosin interactions The f W a N Φ nyrcie n op-corrections - loop one receives only (Φ) , N Φ and (Φ) b xml,cnie h etrmultiplet, vector the consider example, 4 = , 4 = c , d oma form S ξ duality are g contains SL not N , ofra invariance conformal K (2 eomlsdi etraintheory perturbation in renormalised supersymmetry. 4 = , r orce re yodri etraintheory perturbation in order by order corrected are |    Z f arx ial,a naie ao eeomnsi string in developments major aside, an as Finally, matrix. ) no N ψ λ 2vector =2 = A ϕ δ {z 1 µ ucin of functions β τ τ 1 ucinvnse.Culnsrmi osata any at constant remain Couplings vanishes. function    = 7→ } K 4– 44 – + , F aτ cτ µν |{z} i W |    2 hr r ietasomto properties transformation nice are There . Θ F π ˜ θ , + + ψ N µν f or 3 2hyper =2 d b ϕ ϕ + and {z θ ¯ , 3 2 hsrsl adsm te similar other some (and result This . F f ψ µν |{z} ξ 2 4 g n h o-eomlsto of non-renormalisation the and π    oiso rvt nAt eSitter de Anti in gravity of eories F 2 in } µν an hr hyaentwell not are they where mains fteuulFymnrlsto rules Feynman usual the of .Tepieeapeo this of example prime The ). . N e sdmnin hthappen that dimension, ss otipratrslsin results important most d/F correspondence AdS/CFT UY twsshown was It SUSY. 1 = nb rte as written be an hoyi dimensions 4 in theory d eter: R d 4 g θ ξ , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. orcin in corrections n1993, In UYt oa ymty h nwri e,adtecorrespon the and yes, is answer The symmetry. local a to SUSY potential h potential the n ooopiiyagmnst sals hs eut in results these establish to arguments holomorphicity and − n1977 In History 4.4.1 o oedtis e e.[ Ref. see details, more For ehv enta uefil rnfrsudrsupersymmet under transforms Φ superfield supersymmetry a local that seen about have facts We few A 4.5 h orsodn charge corresponding the opigt current a to coupling o i eda ihlocal with deal we did How oeta nthe in that Note h usinaie fw a make can we if arises question The t.Temetric The dependent ity. space-time Poincaré parameters the make we When o osdrlclSS.Tegnrlsdguefil stesp the is field gauge generalised The SUSY. local consider Now ∂ associated µ 3 T | W µν h clrptnilo lblSUSY global of potential scalar The | 2 mle osatttlmomentum total constant implies 0 = oe rmteaxlayfilso h rvt multiplet. gravity the of fields auxiliary the from comes Grisaru Seiberg V F supercurrent ( = V f F unu orcin nycm nteform the in come only corrections quantum , g K M µν V bv sn ogrpstv.Teeta(eaie atrpro factor (negative) extra The positive. longer no is above , − F pl bsdo tigter ruet by arguments theory string on (based Siegel saguefil ope ote“current” the to couples field gauge a as 1 ) ∞ → i J exp = ¯ j ∂ µ J i ∂ W α i nitrcinterm interaction an via q µ ii,gaiyi eope n h lblsprymti s supersymmetric global the and decoupled is gravity limit, , 5 D sconstant is n UYcharge SUSY and scin27.6). (section ] α Rocek P ¯ j i ( W W

q µ x nitra ymtis eitoue ag field gauge a introduced We symmetries? internal in ) M ∗ δ K = = Φ etrd oieta o nt auso h lnkmass, Planck the of values finite for that Notice restored. = pl := 2 Q Z ǫ ( ! α hwduig”uegah”ta,ecp o loop 1 for except that, ”supergraphs” using showed ucino pc-iecoordinates space-time of function a Z Z d 4 ∂ = x V d i d ( W i 3 K F 3 Z 5– 45 – J x ( T x ǫ smdfidi uegaiyt (where to supergravity in modified is − Z Q d + 1 4 ) 0 µ d θ i ¯ j ¯ + 0 3 M n D x = 1 A ... pl ǫ 2 i = J D W Q µ o ¯ α J ( 0 os . const Φ ) ∂ µ . os . const . i h current The . K ¯ j W . ) Witten . W ∗ ipeadeeatwy[ way elegant and simple a T − µν eoti hoyo grav- of theory a obtain we , igter is theory ding via 3 n3/2 in yas ry | M W 95 sdsymmetry used 1985) g µν pl 2 | J 2 µ T ) gravitino µν scnevdand conserved is ǫ ( Conservation . x ,ie extend i.e. ), supergravity otoa to portional ∂ i Ψ = α µ calar ∂ϕ with ∂ 10 A i ): ]. µ . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. eko htfields that know We Preliminaries 5.1 breaking Supersymmetry 5 ϕ ne nt n nntsmlgopeeet.B Goldstone’ By elements. broken group infinitesimal and finite under ehv a have we infinitesimally ig ehns fthe if mechanism Higgs θ (¯ npriua h ( the particular in e scnie h niomtto relation anticommutation the consider us Let hshstovr motn implications: important very two has This iial,w pa fboe UYi h aumstate vacuum the if SUSY broken of speak we Similarly, orcin o dnmcly) n a od tnon-pertur super also it do break it to to then has order level, one in ‘dynamically’), tree that (or at means corrections unbroken This is theory. SUSY perturbation global If result: Since σ vac orsod otemsls odtn oo ti setnb eaten is (this boson Goldstone massless the to corresponds ν ) • • βα ˙ stevleta h field the that value the is E nboe SUSY, broken In nrydniyi titypositive, strictly is density energy W , 8 ≥ ftevcu tt ( state vacuum the if (¯ sntrnraie oalodr nprubto hoy we theory, perturbation in orders all to renormalised not is σ (¯ o n tt,since state, any for 0 0 σ U ) ν βα ˙ 1 ymty n let and symmetry, (1) ) βα ˙ n ϕ Q n ν i α Q )cmoetuig¯ using component 0) = δϕ ϕ α , 7→ i Q , ¯ Q fguetere rnfr as transform theories gauge of U β Q = ˙ ¯ α o 1 saguesymmetry). gauge a is (1) β ˙ exp( | vac o αϕ iα = ϕ 6 i vac (¯ 2 = iα Q ϕ ( α X α a ) α ,so 0, = =1 i 2 T ae hni iiie h potential the minimises it when takes a ( = T rnfrsi o-rva a,i.e. way, non-trivial a in transforms Q a Q ⇒ ) a ϕ σ Q α α ) ν ) i α i | † ) > E j = vac j βα ( ( + ˙ ϕ 6– 46 – ( Q δρ ϕ σ j ρ h ( 6 i α 0 vac vac σ Q δϕ , exp( 0 ) = † µ α 0 = ) 0 = ) | ) j α ( + [ 1 { † Q β iϑ Q Q ˙ : α P 0 6= α α ncmlxplrcodnts then coordinates, polar complex in ) ( Q µ , . δϑ , Q i spstv semi-definite positive is Q α ¯ α ) β 4 = ) = ˙ † } † . | Q vac ( + batively. 2( = α iα hoe,guesmer is symmetry gauge theorem, s i η Q a µν satisfies = α ymtyi h quantum the in symmetry 4 = ( h ag oo i the via boson gauge the y nrknt l resin orders all to unbroken σ ) T P † µ a Q µ ) . α ) α α i β P j ] ˙ 4 = | P ϕ vac 0 aea important an have µ j i otatdwith contracted 4 = V P > ( ν ϕ ,hnethe hence 0, .Suppose ). , E Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ste ohtasomnntiilyudrteLrnzgrou Lorentz the under to non-trivially simplifies transform condition both they as eneed we foeof one If iue5 Figure energy osdrtetasomto asudrSS o component for SUSY under laws - transformation the Consider 5.1.1 ozr edconfiguration field nonzero 8 e pnaeu ymtybekn oe nteSadr Model Standard the in notes breaking symmetry spontaneous See V ( F ϕ . h min δϕ embreaking term snneo ag ymtyi rknweee h potential’ the whenever broken is symmetry Gauge nonzero. is ) aiu ymtybekn cnro:SS sboe,wheneve broken, is SUSY scenarios: breaking symmetry Various i , h δψ i , h δF (c) a intact (a) ϕ min 6 i SUSY ,te UYi rkn u opeev oet invariance, Lorentz preserve to But broken. is SUSY then 0, = faguenon-singlet. gauge a of 0 6= δF δψ δϕ SUSY h = = = ψ i √ √ i √ = ⇒h ⇐⇒ 2 2 ¯ 2 F ǫ ǫψ 7– 47 – ǫ h σ ¯ ∂ µ µ + ∂ ϕ µ F i i . ψ √ 6 i 0 = 2 0 = σ µ ∂ ǫ ¯ (d) µ . (b) ϕ SUSY course. gauge .S u UYbreaking SUSY our So p. facia uefil Φ, superfield chiral a of s gauge iiu satie ta at attained is minimum s h iiu potential minimum the r Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. suul eepn h ed rudthe around fields the expand we usual, As If nytefrincpr fΦwl change, will Φ of part fermionic the Only ecno have cannot We The model O’Raifertaigh 5.1.2 and osdrtefrinms term mass fermion the Consider hs ene omnms h clrpotential: dete we scalar breaking, the SUSY minimise the to of need effects we some this, see to order In SUSY. etk h xml of example the take we ¨ he oeta n ueptnilaegvnby given are superpotential and Kähler potential ocall so rmthe From oeGlsoebsn.Rmme htthe that Remember boson). Goldstone some sgvnby given is m V 2 F ’afrag model O’Raifertaigh < − K = ψ h emSS raigi qiaett oiievcu expect vacuum positive a to equivalent is breaking SUSY term M 2 ϕ g F a 2 2 2 Φ = i hntemnmmo h oeta sat is potential the of minimum the then , odtn fermion Goldstone edequations, field − ∂W ∂ϕ = 2 1 i ψ i † F h i δϕ Φ h i ∗ ϕ i o all for 0 = i 3 ∂ϕ W , ∂W ∂ϕ i ∂ 2 = i j F ∂ϕ W − − − 0 = h novsatilto hrlspred Φ superfields chiral of triplet a involves F F F ϕ − ∗ j h 2 3 1 1 δF ∗ ∗ ∗ term i = , n opt h pcrmo emosadscalars. and fermions of spectrum the compute and 0 = ψ i = i = = = V rthe or j ( 1 = F 0 = g ) SUSY g 2 ∂ϕ ∂ϕ ∂ϕ ∂W ∂W ∂W = h , Φ

ϕ 2 ϕ = 1 3 2 1 1 , goldstino 3 2 i − iutnosy oti omof form this so simultaneously, 3 (Φ , aumepcainvalues expectation vacuum rirr = arbitrary − 2 1 K 2 = = = 8– 48 – 3 2 i − ⇒h ⇐⇒ ¯ j m − ψ 1 h ϕ M g F 2 δψ 1 ∂W ∂ϕ ϕ g

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2 Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. iue6 Figure ic hyhv ieetmasses: different have they since egnrlygthairadlgtrspratessneth since superpartners lighter and heavier get generally We ofiuain hr:frany for (here: configurations umrsn,w aetesetu:so nFig. in show spectrum: the have we Summarising, nteLgaga,wihyed the yields which Lagrangian, the in ϕ wihtet ooi n emoi at ieety vani differently) parts fermionic and bosonic treats (which 5.1.3 SUSY. where ψ osdravco superfield vector a Consider λ xmlsset3 hr o r se owr u oedetails some out work to asked are you where 3, sheet examples V eeaieteqartctrsin terms quadratic the examine we ϕ 1 quad 1 3 saglsio(hc,aan snttefrincprnrof partner fermionic the not is again, (which, goldstino a is ilb massless. be will un u ob h odtn deto (due goldstino the be to out turns sacmlxfil,wihw utsltit t eladimagin and real its into split must we which field, complex a is j = D ersnste’pn fteprils hsi eei o t for generic is This particles. the of ’spin’ the represents . − embreaking term xml fafltdrcin fteptniltksisminimum its takes potential the If direction: flat a of Example m 2 g 2 m ( δλ a 2 ϕ STr 3 2 = + ∝ m ϕ ϕ M ψ M 3 ∗ 2 1 2 D ǫ ∈ 2 + ) V 2

R 0 = − ,te ti adt aeafltdrcin sarsl,tescal the result, a As direction. flat a have to said is it then ), ( = := M = 2 ,V λ, 2 h ⇒ g V m , ψ | 2 ϕ X : i m j 3 µ masses h | 2 2 D , δψ ( D − m , + 9– 49 – 1 1) 6 i ψ h ∝ i ), M 2 2 j = 0 = +1 2 = | F ϕ (2 b 2 1 2 7 6 i | j m . 2 = ) odtrietesaa masses, scalar the determine To 0). = 1) + ψ 3 ⇒ = M ⇒ m = shes: e 2 j 2 supertrace 2 + m . M n odtn oo) See boson). Goldstone any 0 = ϕ SUSY . r parts ary o otnosrneo field of range continuous a for 1 e ee ietybroken directly level ree g 2 0 = m , 2 . of . m , M ϕ 3 ..STr i.e. = √ ϕ 1 2 2 ( a = rfield ar + M ib M 2 ),

Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. iue7 Figure .. raiglclsupersymmetry local Breaking 5.1.4 model. • • • h clrpotential scalar The hti h both why is That fwytevcu nrydniytdyi lotzero, almost is today density t energy is vacuum (which the why problem of constant cosmological the for important egaiy hra rknSS ntegoa aerequired case global the in SUSY broken whereas pergravity, The ag nrysaeadsilt eptevcu nryzr.T zero. energy vacuum the keep to still and supersymmetr scale break energy to large wit possible potential is the it in however, appears supergravity squared scale breaking SUSY the h uegaiymliltcnan e uiir ﬁelds - auxiliary new contains multiplet supergravity The otiuint h omlgclcntn hti a o la too far is that constant cosmological the to t contribution need we order supersymmetry, of global heavy, In minimum. expec its vacuum at the potential to corresponds essentially density energy SUSY. omlgclcntn rbe,though. problem, constant cosmological . F assltigo h el n mgnr ato h hr sc third the of part imaginary and real- the of splitting Mass emi rprinlto proportional is term - V ( F ) ∼ exp = 0 e rhair hs lblSS ol aual iea give naturally would SUSY global Thus, heavier. or GeV 100 h V V F i ( F and 0 = ) ∝ a eaiegaiainlterm, gravitational negative a has

M K DW pl 2 ( ! h V 0– 50 – 6 i = ( K r osbeatrSS raigi su- in breaking SUSY after possible are 0 = − ∂W ∂ϕ 1 ) i ¯ j D + i D W M 1 pl 2 ¯ j W ∂K ∂ϕ O ∼ ∗ F rge, − g . W ta miial viable empirically an at y (10 ainvleo h scalar the of value tation elc funderstanding of lack he alar with ongtv em.In terms. negative no h 3 aesuper-particles make o − h O ∼ i osnot does his V | 3 M ϕ W eV) i h 3 F pl 2 | 10GeV) (100 > nteO’Raifertaigh the in 2 g ) 6 i 4 .Ti svery is This 0. .Tevacuum The ). o broken for 0 = . solve 4 since , the Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. ic hrlspred nycnanlf-addfermions left-handed anti contain only Standard superfields containing chiral Since superfields chiral are there Secondly, nrdcn h iia uesmercsadr oe (M on model based standard is MSSM supersymmetric The minimal the Introducing 6 xeietlydsoee edsae into states field discovered experimentally is fal ehv etrspred otiigteStand the ( containing under superfields representations vector their have write we all, of First Particles 6.1 • • • • • • • ih addfrincfils(hc r culyleft-hand actually are (which fields fermionic handed right h ue ig effect: Higgs super The ig bosons/higgsinos: higgs (s)leptons oaodagueaoay n ogv asst ontp quar down-type to masses give to and anomaly, gauge a avoid to h eodo hc sanwHgsdultntpeeti h St the in present not a doublet is Higgs MSSM new the a is which of second the sqak:lpo number lepton (s)quarks: gluons/gluinos y”aig h odtn.Tegaio ean ases h massless, remains graviton The goldstino. the ”eating” by W B ag edt e as iia hnmnnhpesi supe in happens fiel phenomenon gravitino similar massless N originally A the joins mass. field a ”eaten” goldstino is get boson to Goldstone field corresponding gauge the which in anism hc otisteguebsnof boson gauge the contains which B = bosons/gauginos uegaiy n ie tams,i hssnetegravit the sense this in mass, a it gives and supergravity) 1 = bosons/winos − 1 / | Q o nanti-quark. an for 3 i L left-handed o lepton, a for 1 = = w ig obe model doublet Higgs two {z 3 SU , H | L 2 2 i (3) , 6 1 pnaeul rknguetere els h ig mech Higgs the realise theories gauge broken Spontaneously = B C = } left-handed u , × L = {z SU SU 1 ,weesbro number baryon whereas 0, = L 1 L , , 0. = | (2) (3) 2 = 2 i c , , − L U C − 2 1 B W G 1– 51 – = o nanti-lepton. an for 1 SU , × N (1) 2 1 (1 = } (8 = (1 = H , U h xr ig obe snee norder in needed is doublet Higgs extra The . supermultiplets. 1 = Y e , (1) . 3 ¯ (2) , , , , 1 Y 1 3 1 L , , , , | 0) × U , 0) ni(right-handed) anti 0) 1 i c − ni(right-handed) anti , N 2 3 (1) = (1 = UY ems talo the of all fit must We SUSY. 1 = epaecag ojgtd i.e. conjugated, charge place we , Y d , {z oe atradHgsfields. Higgs and matter Model {z s(r-ig mechanism): (pre-Higgs as ) 1 r oe ag oos We bosons. gauge Model ard wihi h ag edof field gauge the is (which d , , d,dntdby denoted ed), 2 1 B , , i c B 0. = owever. − +1) = 1 = 2 1 ytecorresponding the by nadMdl Thus, Model. andard } n eevsismass its receives ino sadleptons. and ks / o (s)quark, a for 3 3 ¯ smer.The rsymmetry. , 1 , 3 1 c SSM) } - Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. . Interactions 6.2 he-yecag ag boson gauge three-hypercharge a ymtya h unu ee.Hprcagdcia fermio chiral Hyper-charged level. quantum the at symmetry iue8 Figure th (see symmetry is electroweak generator of charge electric breaking the the after that Note nSS,w d h igiodoublet Higgsino the add we SUSY, In utb acle yaohrHgsn obe ihopposite with doublet Higgsino another by cancelled be must ymtyta speeti h relvlLgaga sbr is Lagrangian is tree-level symmetry the the in Here, present is that symmetry a oaihi iegnepootoa to proportional divergence logarithmic a utrnraietedarmaa yadn a adding by away diagram the renormalise must utpidb oeknmtcfco hc stesm o eac for same the is which factor kinematic some by multiplied hrlfrin a eeaean generate may fermions the Chiral in symmetries perturbative perturbatively. discrete multiplicative to u hsbreaks this But unu ee.Fortunately, level. quantum • • • K ag opig r eomlsd hc nsu iigthem giving up ends which dependence renormalised, are couplings Gauge usin a o hwthat show you Can Question: T f a a Φ = = nmlu rp rprinlt Tr to proportional graph Anomalous . en h elMn arcsand matrices Gell-Mann the being τ a i † hr Re where exp(2 U hc ace nodpnec pnteeeg cl twhic at scale energy the upon dependence onto matches which , (1) V )Φ Y U enn that meaning , { i τ (1) srnraial,where renormalisable, is a } Y = A l hrlfrin nteter rvli h op n yie and loop, the in travel theory the in fermions chiral all : V 4 B g o ahfrinfml nteSadr Model. Standard the in family fermion each for 0 = Q π a 2 A := vertex. anomaly = eemnstegueculn constants. coupling gauge the determines := g T 3 3 SU T Hf LH H U X ˜ a (2) A G (1) 1 hc ilsannzr otiuinto contribution non-zero a yields which , 2– 52 – i a nteter,a hw yFig. by shown as theory, the in L 0 = Y Y + + σ i 3 { i ol o eacnitn ymtya the at symmetry consistent a be not would Y/ g Y − en h al matrices. Pauli the being 2 naSadr oe family? Model Standard a in 3 1 2 B .Bro n etnnme correspond number lepton and Baryon 2. } Hf RH σ X µ hc utvns for vanish must which i B W ν i B i Y ns + i ρ 3 one emi h Lagrangian. the in term counter g f Y emo.If fermion. h rvli h opcnrbtn to contributing loop the in travel M n r hsconserved, thus are and SM, knb unu corrections. quantum by oken Y 2 Y B, tnadMdlcourse), Model Standard e : H ˜ 2 . eomlsto scale renormalisation U A (1) 8 snnzr,one non-zero, is hsi where is This . Y ob valid a be to n is one h A This . ld Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. SU iue10 Figure 9 Figure instance. G ˜ op.Teeaeohrcnrbtn igas oeinvolvi some diagrams, contributing other are There loops. (3) eas fadtoa otiuin rmsursadgluin and squarks from contributions additional of because o etec te tal e Fig. see all: at other each meet not otiuin nteMS,tegueculnsalme tar a at meet all couplings gauge the E MSSM, the in contributions Eq. eesr odto o rn nfidTer,wihol ha only which Theory, Unified Grand (above a for condition necessary otntknt be to taken (often o riayQDi is it QCD ordinary For where rbn them: probing C SU , ≈ otiuint h n opQDbt function beta QCD loop one the to Contribution . 6.1 eomlsto ftesrcueconstants structure the of Renormalisation . 2 (2) β × M a sue oetaoaegueculnsmaue tsm ener some at measured couplings gauge extrapolate to used is L sacntn eemndb hc atce rvli h loo the in travel particles which by determined constant a is GUT 10 and g 16 3 ≈ U G e,weeswt utteSadr oe otiuin,th contributions, Model Standard the just with whereas GeV, µ (1) 2 dg g M × Y 3 dµ a Z 10 groups. ( µ rmLPcntans osm te scale other some to constraints) LEP from , 16 ) β 3 = GeV). = β a − g 7 a 3 / ( µ (16 ) g , 3 ⇒ π 3– 53 – 2 + 10 hra o h SM tis it MSSM, the for whereas ) g a − h etn ftegueculnsi a is couplings gauge the of meeting The . 2 ( µ = ) g a − α 2 gloso ursadsurs for squarks, and quarks of loops ng β ( a µ g 3 3 0 := rmgluon from ) G − ˜ st h op,a nFig. in as loops, the to os g 2 g a 2 β 3 / a 4 π ln n ag coupling gauge one s nraiainscale enormalisation soitdwt the with associated µ µ µ G ihteSUSY the With . 0 β op n gluino and loops ntetheory. the in p 3 = yscale gy − 3 g / 3 (16 ydo ey (6.1) π µ 2 9 0 ) . Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. • • • • aiywudhv motn hsclimplications: physical important have would parity R netenurlHgscmoetdvlp aumexpectat vacuum a develops component Higgs neutral the Once h fundamental a in asst pqak,dw ursadlpos sw hl see. shall we as leptons, and quarks down quarks, up to masses h rttretrsin terms three first The ewiedw ueptnilcnann l em hc ar which terms all containing superpotential a down write We eta hgsn,poio io.I ste odcandida good a then matter. is dark e It acting be zino). must LSP photino, stable (higgsino, a neutral that say then constraints Cosmological stable. is (LSP) superpartner lightest The n oor(eas n eeae o-eovcu expecta vacuum non-zero instance). a for generates one (because colour and oesubset. some hr ehv upesdgueindices. gauge suppressed have we where o h Itr:w uthave must we term: FI the For n ymtyta ok sadsrt utpiaiesymmet multiplicative discrete a is works that symmetry One > naaoosmne.Tefut emi astr o h two the for in term terms mass the a of is all term If fourth The manner. analogous an rtndecay proton tfrisalo h em in terms the of all forbids It ossetwt u ymtis foede hs n obtain one this, does one If W symmetries. our with consistent m number 2 0 u ) 10 W / = =( := √ W 34 R hr oterm no there usin hc em break terms Which Question: W p ,tefis embcms( becomes term first the 2, W PV RP er.I re ofri rtndcya xr ymtyshou symmetry extra an decay proton forbid to order In years. Y becomes R L R U R p hra hs in those whereas , p ) + ij v = ( = =( := W p 2 / SU → PV RP √ ( λ Y Y o h pqak.Tedw ur n etnmse rce i proceed masses lepton and quark down The quarks. up the for 2 e ijk 3 index, (3) − U U + h em in terms The . ) 1) W ) ij + L ij λ 3( Q PV RP i k ′′′ π Q L B i xa 0 H i j − W ihnscns hra xeiet a hti hudbe should it that say experiments whereas seconds, within H H 1 e L H r rsn,teitrcinsoni Fig. in shown interaction the present, are k c R )+2 2 2 W b ,b a, W p u 1 u + e PV RP jx c j c S orsodt tnadYkw opig n give and couplings Yukawa standard to correspond PV RP k c Y ξ ǫ 1 = ( + λ U in ab ,ohrietesaa oeta rascharge breaks potential scalar the otherwise 0, = = ijk ′ ) u hr xs te xmlswihol ban only which examples other exist there but , ij ra either break ( = 4– 54 – W , W Y v L sfundamental as 2 2 R ( D PV RP i / Y p Q ) √ 1:Sadr oe particles Model − Standard : +1 U ij l osrebro number baryon conserve all L j ) 2 superpartners : 1 Q ? ij d u k c n hc break which and [ i Li x u H L x + u 1 H jx c B d λ 2 0 + j c u ijk ′′ or . . . jx c + u L iligaDrcms matrix mass Dirac a yielding , − i c Y SU : d E d j c L x L 2 nie,tefis term first the indices, (2) d H efrcl ekyinter- weakly cold for te k c i invlefrasquark, a for value tion H 2 w lse fterms, of classes two s + ry o value, ion etial n colour- and lectrically + u eomlsbeand renormalisable e 1 Writing jx c e parity R B κ j c ] i . ig(n)fields. Higgs(ino) h is Why ? + L 11 i db imposed. be ld B H H µ x ol allow would H 2 n lepton and 1 = , endas defined . 2 0 1 H =( := , 2 2 , v as 3 2 + n Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. L λ iue11 Figure W lssprymti ois where copies, supersymmetric plus fcomponents: of ecno ra uesmer ietyi h SM ic it since MSSM, the in directly supersymmetry MSSM break the cannot in We breaking Supersymmetry 6.3 transpose. hths’ enosre.Apyn tt pqak:2 quarks: up to it Applying observed. been hasn’t that say: photon, the to this Applying a ihMS ed)aditrcin,ada additional an and interactions, and fields) MSSM with qakms be must squark 1 ′′ hidden j iltn em utb rsn o h rtnt ea.Tem The decay. to proton the for present be must terms violating 9 1 = • • hsi aiirsrcuefrpol xedn h Standar the extending people for structure familiar a is This ∗ oeta h em in terms the that Note × λ n nsu ihLP tteedo h eas hs ontinte not do These decays. the of end the at LSPs with up ends One nclies h nta tt is state initial the colliders, In ar.We uepril eas tms ea oanother to particles. decay model must standard it some decays, plus superparticle a When pairs. eetr n ec pera naacdo msig momen ‘missing’ or unbalanced as appear hence and detector, 112 ′′ λ 11 ′ etr hc rasSS n a t w ed wihd o d not do (which fields own its has and SUSY breaks which sector, u rtndcydet ayn n etnnme iltn int violating number lepton and baryon- to due decay Proton . j 1 c ∗

d . 1 c etr MSSM sector, d 2 c observable lighter etk the take we : L hnteu ur,aanti antbe bevd eintrod We observed. been hasn’t this again quark, up the than = 2 1 W ! λ PV RP F 112 ′′ − ←→ − u ˜ em suuli re oﬁdteLgaga nterms in Lagrangian the ﬁnd to order in usual as terms a edt aoaafrinstructure fermion Majorana to lead can 1 ∗ 3 R d m C 1 p R † γ 2 1 mligta ueprilsaepoue in produced are superparticles that implying +1, = Cd +2 stecag ojgto arxand matrix conjugation charge the is

5– 55 – esne - messenger 2 m R ∗ γ 2 ˜ sector − ,wihwudpeitamsls photino massless a predict would which 0, = ( λ 112 ′′ ) ∗ ! u oe oicuenurn masses. neutrino include to Model d ˜ m esne sector messenger 1 d u 2 1 − ←→ R T C ti lmn spootoa to proportional is element atrix m ∗ u 2 ˜ d L 2 rsre STr preserves R −

tum. lgtr superparticle (lighter) m hidden sector rcin.Both eractions. u 2 ˜ R ,tu n up one thus 0, = 9 rcl interact irectly o instance, For . ! atwt the with ract . T M denotes 2

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G hsgt rudtesprrc ue hr stpclya o an typically is There rule. supertrace the around gets This raigsco nteMS otx sta of coupling that gauge is free context MSSM totically the in sector breaking eoti = Λ obtain we omo UYbekn em nteMS,icuig(oehere (note including MSSM, w the in fields, terms sector breaking breaki SUSY messenger SUSY of the the form for carry possibilities and several fields, MSSM are the with couplings have ∼ raigi h idnsco,wees∆ whereas sector, hidden the in breaking n h hoybcmsnnprubtv,oecnobtain can one non-perturbative, becomes theory the and SMfields): MSSM dynamically SM 10 5 • • • ee ˜ Here, × ehv led ensvrleape fSS raigtheor breaking SUSY of examples several seen already have We h UYbekn ed aeculnswt h esne se messenger the with couplings have fields breaking SUSY The . nml mediated anomaly ag mediated gauge h rvtn esamass a gets gravitino The h relvlcnrbto ssprse o oereason. some for T suppressed factors. is loop contribution by tree-level vacuu suppressed the a but get present, supergravity always of are fields effects auxiliary the case, this In ewn ∆ want We h UYbekn assltigbtenMS atce and particles MSSM between splitting mass breaking SUSY The esne ed r hre ne both under charged are fields Messenger becomes ..TV nta ae h rvtn mass gravitino the case, that In TeV. i.e. UYbekn oteMS ed.Tu,∆ Thus, fields. MSSM the to breaking SUSY rvt mediated gravity opig r upesdb h nes lnkmass Planck inverse t the strength by gravitational suppressed are with couplings couples field mediating the If 10 17 g 6.1 steguioo h idnsco ag ru,and group, gauge sector hidden the of gaugino the is e.I ses oarnefrΛ for arrange to easy is It GeV. 10 ihsm eafnto offiin.Sligteeuto,wi equation, the Solving coefficient. function beta some with . SU M m (3) exp[ ≈ SUSY e n know and TeV 1 × g SUSY − SUSY 2 SU ( M M (2) SUSY g ) /β eoe ag tsm nrysaeΛ. scale energy some at large becomes m ]. × 3 2 M = U f∆ of (1) ∆ ol esm ag cl uha h tigscale, string the as such scale large some be could G M m p m SUSY pl 6– 56 – ∆ ≪ m × re e rmtesprHgsmechanism. Higgs super the from TeV order ≈ = m M steSS raigta nsu nthe in up ends that breaking SUSY the is G 10 · n h idnsco ed r igesof singlets are fields sector hidden the and m M hntegueculn eoe large, becomes coupling gauge the When . SUSY agn condensation gaugino G M 18 SM 2 3 pl M 2 SUSY e,so GeV, ∼ m pl ≈ and M β =: ∼ M 2 stehde ag ru eafunction beta group gauge hidden the is SUSY . ihmydtrieteexplicit the determine may hich 10 M gives pl G h G pl 11 g ˜ SUSY ealguegroup gauge verall g SM ˜ h aua cl fgravity. of scale natural the , ∼ O ∼ i GeV M h tnadmdl the model, standard the o xetto au.The value. expectation m that Vadi h LSP. the is and eV ag op transmit loops Gauge . × SUSY e a edmnn if dominant be may hey e.OepplrSUSY- popular One ies. goe ote.There them. to over ng . (Λ G ee oeasymp- some here, : M ueprils ∆ superparticles, tr hc nturn in which ctor, / SUSY 3 (16 g ,bekn SUSY breaking ), SUSY ilrenormalise will th π , 2 ) g steSUSY the is − O ∼ 2 (Λ) (∆ → m m 0, ), , Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. uesmer sepce ob rkni h bevbesec searching. observable is the LHC in the determi which broken They for be superparticles to terms. expected breaking is SUSY supersymmetry soft different break the SUSY of between forms Particular theory. the into divergences M o enrdc h irrh rbe.Te r ftefr ( form the of are symmetr They renormalisable problem. all hierarchy down the write reintroduce we not term, second the In hi rbesi natv edo eerh nalscenario all In research. of contributions has field sector active servable an an advantages is phenomenological problems has their scenarios these of Each Sometimes, fields): L indices: gauge suppressing MSSM, hs ayprmtr.Frisac,teCntandMS ( MSSM Constrained the instance, For parameters. many these L R SUSY λ p SUSY m , xlcty eprmtiealo h em htsfl brea softly that terms the of all parametrise we Explicitly, ′ ij 2 ( = m , iue12 Figure Q ( = ˜ A m ij 2 Li ∗ m U 3 2 A , ( H ) 3 2 m | m ij switnas written is 1 ijk ij 2 Q Q 2 H ˜ ˜ ϕ ) Li 2 ij r called are agn odnainadsprrvt eitdSS brea SUSY mediated supergravity and condensation Gaugino . i ∗ + H Q ϕ ˜ 2 Lj j h.c. g u ˜ clrmasses scalar + Rj ∗ + + ) m ( + L ˜ {z ∆ µB ′ ij 2 i ∗ otSS raigterms breaking SUSY soft m m ( ( A m ϕ fe,seichg cl oespoierltosbetwee relations provide models scale high specific Often, . 1 2 D L i L 2 | ˜ ϕ H ) ) ij j ij 1 2 Q = + L | ˜ ˜ + Li j h.c. ˜ + H m L Λ u 1 ~ SUSY 2 2 } 7– 57 – ) d | M Ri ˜ H Rj ∗ + SUSY ( 2 m | ( + 2     + U 2 ˜ + agn masses gaugino ) A ij L | 2 1 1 2 E u ˜ SUSY M M ) Rj ∗ ij {z 3 hyd o enrdc quadratic reintroduce not do they : λ M L g ˜ ˜ + λλ g ˜ Li . } d + ˜ H n eito a ierelations give can mediation ing Ri ,teLgaga o h ob- the for Lagrangian the s, log scale 1 2 1 ( UYi the in SUSY k iavnae n solving and disadvantages d e ˜ + where m M hc a oefo some from come may which Rj ∗ nain em hc do which terms invariant y o,adtemse fthe of masses the and tor, D 2 2 ˜ rlna couplings trilinear eteaon ywhich by amount the ne W + ) | A ˜ ij ijk W d i ˜ ˜ Rj ∗ and ϕ {z + i ˜ + ϕ 1 2 j j e M ϕ Ri ae different label R k } king 1 ( p B m ˜ preserving B. + E 2 ˜ ˜ ) h.c. ij e ˜ Rj ∗     + . n Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. iue13 Figure h U cl,adrcielreqatmcretos sFig as corrections, quantum large receive and scale, GUT the U cl.Tesaas(. h ih-addelectron right-handed gaugino the the (e.g that scalars implies The and scale. symmetry GUT GUT has which assumed, h lnkmass Planck problem The hierarchy The 6.4 lcrwa scale electroweak ieeti antd.Atal h rbe a eformulat be involves can problem problem hierarchy the Actually The magnitude. Model. in different Standard the of scale parameters, where ffe UYbekn aaeesdw to down parameters breaking SUSY free of tigter rohrfil hoy ie h constraints the gives theory) field other or theory string breaking. eaaey hnw r cteiga energies at though scattering scale are GUT we the Below When separately. scale. GUT the at mass-degenerate asseautda htrnraiainscale renormalisation that at evaluated masses 11 n hudral nld tan include really should One I 3 ste3b dniymti.Tu nte‘MS’ erdc th reduce we ‘CMSSM’, the in Thus matrix. identity 3 by 3 the is neapeo eomlsto nteMS.Apriua hig particular A MSSM. the in renormalisation of example An . µ 2 + M Hu 2 M M ew eoe eaiea h lcrwa cl,tigrn ele triggering scale, electroweak the at negative becomes , pl -200 ≈ 600 200 GeV 400 ≈ m M A m 0 10 10 Q 2 U ˜ 1 2 1 2 2 β 19 = = = = m e sa nrysaeascae ihsmer breaking symmetry with associated scale energy an is GeV = ( Ql µ A e sa nrysaeascae ihgaiyadthe and gravity with associated scale energy an is GeV M m m v 2 2 4 0 +m 2 2 L 2 ˜ 2 /v Y = = = U 1 Hu A , swl,tertoo h w ig aumepcainvalues. expectation vacuum Higgs two the of ratio the well, as m M m 6 2 0 2 ) U 2 3 ˜ D 1/2 log 8– 58 – =: = 11 = µ ∼ 8 10 m ≈ M 3: O A ( M D 2 µ 1 e,i sago prxmto ouethe use to approximation good a is it GeV, (1) ˜ E 1 0 10 SOFTSUSY3.0.5 M / Y /GeV) 2 eseta n fteHgsms squared mass Higgs the of one that see We . 2 = M ( D Er µ 1 / A , 3 2 m M 2 +m 12 , n h ethne squarks left-handed the and E 2 1 ˜ m m E Hd := Er 0 = 14 2 SPS1a and m ) h asssltadrenormalise and split masses the , A 1/2 r l asdgnrt tthe at degenerate mass all are s 0 2 0 . I di w parts: two in ed Y A 3 16 13 E hs w clsbigso being scales two these 0 hs eain odat hold relations These . shows. toeksymmetry ctroweak nryter is theory energy h ag number large e Ql r also are ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. way. e snwtiksm oeaottetcnclheacyprob hierarchy technical that: the know about we more some think now us Let oe nsm way. some in model nSS,bsn aetesm assa h emos ic qu the Since that implies fermions. SUSY the invariance, as too. gauge protected masses of same because the massless have bosons SUSY, In i)Oc ehv ovd() h sti irrh tbeund stable hierarchy this is why (i), solved have we Once (ii) 12 13 i h is Why (i) • • • oieta with that Notice hsde eyo unu rvt iliga ffcieqatmfie quantum effective an yielding gravity quantum on rely does This clymc agr h ags r xetdt be to expected lo are the largest since the unnatural larger: be much to ically considered is Model Standard The etrbsn r asesdet ag naine htmea that invariance, particles gauge gauge the to for due massless are bosons Vector h ig steol clrpril nteSadr oe.T Model. 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Model Standard the in 9– 59 – 2 op iecretoso re Λ order of corrections give loops , ψ after 13 me O ∼ H R e R esannzr value non-zero a gets (10 ? e.I h tnadModel Standard the In lem. dter htat nteusual the in acts that theory ld 17 qak n lposare sleptons and squarks pcnrbtost some to contributions op rqatmcorrections? quantum er lada xlnto of explanation an and al e.Teeoeee if even Therefore GeV. ) rlrectffsaeΛis Λ scale cutoff large er oajs r“n tune” “fine or adjust to cl,lo corrections loop scale, k obe to d / kw opig othe to couplings ukawa s ietms term mass direct a ns, pcretosaetyp- are corrections op (4 ol eur cancel- require would rsadlposare leptons and arks eei osymmetry no is here n ouin,aside solutions, any π ual maintained turally .Ti ol ruin would This ). osadfermions and sons m A µ H → ≈ getthe ugment A 2 GeV. 126 12 2 µ / ). (16 + lem. ∂ π µ 2 α ) Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. eody UYipista nteepii optto fl of computation explicit the in that implies SUSY Secondly, hs etr oe r evl ae nRf [ Ref. on based heavily are notes lecture These Acknowledgements hrfr fsprymtywr xc,frin n bosons and fermions exact, M were supersymmetry loop. if the Therefore in particle some of mass h aeculn.Ee hnSS ssfl rkn hs lea of these term broken, a softly is only SUSY with vertice when us the Even leaving relates SUSY coupling. defining same couplings the the the against that cancel fact loops the bosonic the of divergences leading hr sa3hu xmnto o hscus.Yuwl easke expect. pa be can some you will through questions You work of should course. kind you the this ever, for As examination questions. hour possible 3 a is There Exam III Part the Appendix: esatwt ito ntrciefaue fteMSSM: the of features unattractive of MSSM list a the with of start Cons We and Pros 6.5 t close be LHC. to the masses at accessible superparticle therefore the expect we Thus, heavy. hs UYpolm a esle ihfrhrmdlbuildin model further with solved be can problems SUSY These SUSY • • • • • • • ecoewt nodrdls fwa-cl UYssuccesses SUSY’s weak-scale of list ordered an with close We The lcrwa ymtybek radiatively. breaks symmetry Electroweak hr are There h SMcnan ibedr atrcniae if candidate, matter dark viable a contains MSSM The works. unification Gauge problem. hierarchy technical the solves SUSY m tde o ra n Msmere?(oetog htoc it correcti once quantum that large from though it (Note protects symmetries? SUSY SM tree-level, any at break not does it hr smr tutr oteMS aaee pc,tog ( though space, parameter i MSSM clue the upo this to that current, structure be more neutral could is changing It there flavour bounds. cou a experimental mixing in strong this very mix then quarks general, the in ph make flavour mix from heavily out ruled could is particles space parameter this of all Nearly opiae aaee space. parameter complicated H scoet h lcrwa cl hni ilpoetteHig the protect will it then scale electroweak the to close is h hudti e hni rnil ecudpti obe to it put could we principle in when be, this should Why . µ problem. ∼ 0 xr reprmtr nteSS raigsco,makin sector, breaking SUSY the in parameters free extra 100 µ in W R p O utbe must ( 4 1 π M SUSY < 0– 60 – O nΛ,where Λ), ln 1 e,snei otiue tte-ee to tree-level at contributes it since TeV, (1) 1 ]. M emoi op.Ti sdeto due is This loops. fermionic o igas(e Fig. (see diagrams oop SUSY nec iga oinvolve to diagram each in s R h lcrwa cl,and scale, electroweak the o ol edgnrt,btif but degenerate, be would tppr ogta dafor idea an get to papers st p g. sconserved. is ons). ocmlt u f4 of out 3 complete to d : igdvrecscancel, divergences ding sc osrit:SUSY constraints: ysics eeytligu that us telling merely s dapa nlosand loops in appear ld steSS breaking SUSY the is sfo eoigtoo becoming from gs iei h CMSSM). the in like O ∼ hc hr are there which n sstt esmall be to set is ( M l P ,because ), o a for g 4 ,the ), Copyright © 2016 University of Cambridge. Not to be quoted or reproduced without permission. References 1]N ebr n .Witten, E. and Seiberg N. [11] 1]N Seiberg, N. [10] 2 ...MulrKrtn .Wiedemann, Müller-Kirsten, A. H.J.W. [2] 3 ..Bcbne n ..Kuzenko, S.M. and Buchbinder I.L. [3] Schlotterer, O. and Krippendorf S. Quevedo, F. [1] 4 .Weinberg, S. [4] 5 .Weinberg, S. [5] 9 .d Witt, de B. Leites D. [9] Kirillov, A.A. Berezin, F.A. [8] Strathdee, A. J. and Salam A. [7] Strathdee, A. J. and Salam A. [6] et B Lett. ocpuladcluainldetails calculational and conceptual Dimensions Extra and Supersymmetry uegaiy r aktruhsuperspace through walk A or, supergravity, 19)45 [arXiv:hep-th/9407087]. 485] (1994) Press University D 477. Press University uesmercYang-Mills, Supersymmetric 11 17)1521. (1975) 318 auans essSprymti o-eomlzto T Non-renormalization Supersymmetric Versus Naturalness Supermanifolds, 19)49[arXiv:hep-ph/9309335]. 469 (1993) h unu hoyo ed,Vlm Foundations I Volume fields, of theory quantum The h unu hoyo ed,Vlm I:Supersymmetry III: Volume fields, of theory quantum The oooeCnesto,AdCnnmn In Confinement And Condensation, Monopole U (1992). CUP ul hs B Phys. Nucl. nSpred n em-oeSymmetry, Fermi-Bose And Superfields On Transformations, Supergauge 1– 61 – ol Scientific World , da n ehd fsprymtyand supersymmetry of methods and Ideas nrdcint superanalysis, to Introduction ri:0119 [hep-th]. arXiv:1011.1491 , uesmer,a nrdcinwith introduction an Supersymmetry, 426 R Press CRC , 19)1 Ertmii.B [Erratum-ibid. 19 (1994) abig etrson Lectures Cambridge ul hs B Phys. Nucl. Cambridge , edl(1987). Reidel Cambridge , heorems, N hs Rev. Phys. 76 430 2 = (1974) Phys.