Why is SUSY Breaking so Difficult?! - Institute for Particle Physics ...

Outline1 Introduction and MotivationWhat is it?The Language of SupersymmetryProblems2 Breaking SupersymmetryThe MathsThe PhysicsThe Tools3 SummaryBooks(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 2 / 27

SupersymmetryThe IdeaSymmetry (with generator Q) which relates bosonic states |B 〉 tofermionic states |F 〉:Q |B 〉 = |F 〉 , Q |F 〉 = |B 〉The generators satisfy the following anticommutation relations:{ }Q α , Q ˙β= 2 σ µ α ˙β P µ ,[Q β, P µ]= 0 ,{Q α , Q β}= 0 ,[Q ˙α , P µ]= 0{ }Q ˙α , Q ˙β= 0(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 3 / 27

Why Supersymmetry?It’s personalColeman-Mandula Theorem: Unique extension of Poincaré algebra.For non-trivial scattering in > 1 + 1 dimensions, can onlyenlarge the known spacetime symmetries by spinorialgenerators.Mathematically, the extra symmetry – both exact and broken – is veryuseful for taming the behaviour of quantum field theories. This leadsto many interesting results. . .(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 5 / 27

The Language of SUSYSuperEverythingUseful notation: Introduce fermionic spacetime coordinates θ α , θ ˙αWrap up fields related by SUSY transformations into superfields —functions on the superspace (x µ , θ α , θ ˙α ):Chiral Superfield: Φ(x µ , θ α , θ ˙α ) = φ(y µ ) + √ 2 θ ψ(y µ ) + θ 2 F(y µ )where y µ = x µ − i θ σ µ θGauge Superfield: V a = θ σ µ θ A a µ + i θ 2 θλ a − i θ 2 θλ a + 1 2 θ2 θ 2 D a(in Wess-Zumino Gauge)Fermions ←→ sFermionsGauge bosons ←→ Gauginos(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 6 / 27

The Language of SUSYLagrangiansKinetic terms for Chiral superfields, and derivative interactions followfrom a real-valued function – the Kähler potential, K (·, ·):∫ () ∫L ⊃ d 2 θd 2 θ K Φ † , e gT a V a Φ = d 2 θd 2 θ Φ † e gV ΦTo restrict attention to renormalizable interactions, it is sufficient toconsider the canonical Kähler potential.Non-derivative interactions follow from a holomorphic function calledthe Superpotential, W (·). Don’t confuse it with the gauge fieldkinetic term!L ⊃∫d 2 θ W (Φ) + 1 4∫d 2 θ W a α W a α + h.c.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 7 / 27

The Language of SUSYLagrangiansTo break SUSY in a Lorentz invariant way, we’re going to want to knowthe scalar potential:V (φ i ) = F †iF i+ ∂W∂φ iF i + 1 2 Da D a − g φ † i T a φ iD aEliminating the auxiliary fields with their equations of motion we find:∣ V (φ i ) =∂W ∣∣∣2∣ + 1 )( )∂φ} {{ i 2 g2( φ † i T a φ iφ † k T a φ k} } {{ }F −termsD−terms(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 8 / 27

R-symmetryWhat is it?It’s a global U(1) symmetry which doesn’t commute with thesupercharges:[Q, R ] = Q , [Q, R ] = −QDifferent components of chiral superfields carry different R-charge:Φ = φ + ψ θ + F θ 2 =⇒ R[θ] = 1 , R[dθ] = −1s s s−1 s−2And hence so does the superpotential:[∫ ]R d 2 θ W (Φ) = 0 =⇒ R[W ] = 2(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 9 / 27

The Minimal Supersymmetric Standard ModelGetting rid of seriously unwanted stuffProblem:[Q β, P µ]= 0 =⇒[Q β, P µ P µ]= 0So particles in the same supermultiplet will have the same mass.Why have we not seen a single superpartner?Scalar electron?Fermionic photon?To keep susy as a useful organising principle for the high energydegrees of freedom, we need to carefully break it.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 12 / 27

Breaking SupersymmetryIt has to be doneFrom the susy algebra:Can find: H = P 0 = 1 4{ }Q α , Q ˙β= 2 σ µ α ˙β P µ)(Q 1 Q 1 + Q 1 Q 1 + Q 2 Q 2 + Q 2 Q 2So the vacuum energy: E = 〈 0 | H | 0 〉 0E > 0 ⇐⇒ Q α | 0 〉 ̸= 0As the scalar potential is also positive definite:V (φ) = F †iF i+ 1 2 Da D a 0Q α | 0 〉 = 0 ⇐⇒ F i = 0 & D a = 0(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 14 / 27

Dynamical SUSY BreakingHierarchy?Question: Does the need to break susy re-introduce a hierarchy?Answer: Not necessarily.Non-perturbative effects can can generate natural hierarchy of scale:( ) Λ b ( )( )= exp −8π 2=⇒ Λ = mµg 2 (µ)X exp −8π 2b g 2 (m X )Same way no-body worries why Λ QCD ≪ M Pl .e.g.Gaugino condensation:〈λλ〉∼ Λ3(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 15 / 27

Witten IndexSUSY vacua are easy to findConsider W = Tr (−1) F = ∑ B〈B|B〉 − ∑ F〈F|F〉As H is positive definite: H|B〉 = 0 =⇒ Q|B〉 = 0whereas for H|B〉 ̸= 0 we are guaranteed a state |F〉 = Q|B〉with the same energy but opposite statistics.◮ W counts the number of zero energy states ◭If W is non-zero, there must exist a susy vacuum.Independent of couplings — “topological”Non-zero for vector-like theories ⇒ complicates models(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 16 / 27

Nelson & SeibergR-symmetry causing problemsThe argument of Nelson & SeibergDynamical SUSY breaking in a generic, calculable model requires aspontaneously broken R-symmetry.To find a SUSY vacuum we want to solve: ∂ i W eff (φ j ) = 0No symmetries: n equations in n unknowns — soluble.Global U(1) non-R symmetry: W eff carries no charge, so mustbe a holomorphic function of the n − 1 variables:X a =φ a 〈φ n〉 q a/qn(a = 1, . . . , n − 1) (if 〈 φ n〉≠ 0)One vacuum equation trivially satisfied.n − 1 equations in n − 1 unknowns — soluble.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 17 / 27

Nelson & SeibergR-symmetry causing problemsThe argument of Nelson & SeibergDynamical SUSY breaking in a generic, calculable model requires aspontaneously broken R-symmetry.Global U(1) R symmetry: W eff carries charge 2, so it must be possibleto write it in terms of a new holomorphic function:W eff= φ 2/qnn f (X a )Vacuum equations: ∂ a f (X b ) = 0f (X b ) = 0 (if 〈 φ n〉≠ 0)n equations in n − 1 unknowns — insoluble.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 17 / 27

Mediating SUSY BreakingBreaking the MSSMCan’t engineer tree-level spontaneous breaking like the Electroweaksector of the Standard Model:Each supermultiplet still obeys Sum Rules:m 2 e d+ m 2 es + m 2 e b= 2. ( md 2 + m s 2 + mb2 )Right-hand side ≈ 2. (5 GeV ) 2 =⇒ All sQuarks lighter than7 GeV . RULED OUT.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 18 / 27

Mediating SUSY BreakingBreaking the MSSMSeparate out the supersymmetry breaking sector from the MSSMsector, and couple them with messenger fields:The phenomenology, i.e. the pattern of soft terms, depends quitehighly on the form of these messengers.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 19 / 27

Mediating SUSY BreakingCommon Messenger SectorsGravity mediation — Planck suppressed operators◮ FCNC◮ Universality assumptions ⇒ mSUGRAGauge mediation — charged under SM gauge fields◮ flavour blind◮ µ problemAnomaly mediation — supergravityGaugino mediation — extra dimensions. . .(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 20 / 27

Radiative Electroweak Symmetry BreakingAnother nice featureSoft susy breaking terms are generated at the mediation scale M X .Although mH 2 u> 0 at this scale, RG flow tolower energies drives mH 2 unegative.〈 〉=⇒ Hu ≠ 0=⇒ Electroweak symmetry breakingWriting tan β = 〈 H u〉/〈Hd〉, the Higgs vacuum conditions become:µ 2 = m 2 H d+(tan 2 β − 1)m 2 Hu1 − tan 2 β− 1 2 m 2 ZBµ =sin 2β2(m 2 H u+ m 2 H d+ 2µ 2 )Usually used to constrain the parameters of ✘✘ ✘ SUSY models.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 21 / 27

SoftSusy2.0A useful toolTo calculate the particle spectrum of a model, consistent with EWSB:Run Standard Model to M ZM Z — Radiative corrections to g i and Y jM X — Impose soft susy breaking boundary conditions(not µ or Bµ)√Q SUSY ≡ m e tm1e t— Use EWSB conditions to2iteratively determine µ and BµM Z — Repeat until results convergeCalculate the Higgs and sparticle masses.(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 22 / 27

Other Useful ToolsThe Model-building ToolkitSpectrum calculators:SoftSusy — C++SPheno — F90SuSpect — FORTRANISAJET — F77Monte Carlo Event Generators:HERWIG++SherpaPYTHIAOther stuff:CPsuperH — MSSM Higgs Pheno with CP violationmicrOMEGAs — SUSY DM Relic Density calculatorFor a comprehensive list, see:www.ippp.dur.ac.uk/montecarlo/bsm(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 23 / 27

SummaryWhy Supersymmetric Model Building is DifficultIn general SUSY doesn’t like being broken◮ Witten Index◮ Nelson–Seiberg ArgumentMSSM breaking has 105 new parameters!Very little is actually known about the relation between SUSY andthe real world (same for all BSM physics). Hopefully this willchange soon!There are tools to help make this connection(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 24 / 27

Suggested ReadingNiceModern Supersymmetry — J. TerningThe Quantum Theory of Fields, Vol III — S. WeinbergA Supersymmetry Primer — S. P. Martin◮ hep-ph/9709356Supersymmetry in Elementary Particle Physics — M. E. Peskin◮ arXiv:0801.1928Lectures on Global Supersymmetry — P. Argyres◮ www.physics.uc.edu/%7Eargyres/661/index.html(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 25 / 27

Reasons to like SUSYIt makes life easierThe extra structure imposed by supersymmetry gives us more controlover non-perturbative aspects of field theory. Many of the followingresults can be traced back to the holomorphy of the superpotential:Perturbative Non-renormalisation Theorem◮ Superpotential is tree-level exact◮ Gauge kinetic function only runs at 1-loopExact beta functionAffleck–Dine–Seiberg SuperpotentialGaugino CondensationSeiberg Duality(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 26 / 27

Other NicetiesWhat has SUSY ever done for us?Q α & Q ˙α ←→ N = 1 SUSYQ K α & Q K˙α ←→ Extended SUSY: N = 2, 4 (8)K = 1, . . . , NSeiberg–WittenAmplitude calculationsIntegrabilityAdS/CFT Correspondence(I.P.P.P. — Durham University) SUSY Breaking Heidelberg 27 / 27