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A Supersymmetry Primer Stephen P hep-ph/9709356 version 7, January 2016 A Supersymmetry Primer Stephen P. Martin Department of Physics, Northern Illinois University, DeKalb IL 60115 I provide a pedagogical introduction to supersymmetry. The level of discussion is aimed at readers who are familiar with the Standard Model and quantum field the- ory, but who have had little or no prior exposure to supersymmetry. Topics covered include: motivations for supersymmetry, the construction of supersymmetric La- grangians, superspace and superfields, soft supersymmetry-breaking interactions, the Minimal Supersymmetric Standard Model (MSSM), R-parity and its conse- quences, the origins of supersymmetry breaking, the mass spectrum of the MSSM, decays of supersymmetric particles, experimental signals for supersymmetry, and some extensions of the minimal framework. Contents 1 Introduction 3 2 Interlude: Notations and Conventions 14 3 Supersymmetric Lagrangians 19 3.1 The simplest supersymmetric model: a free chiral supermultiplet . ........ 20 3.2 Interactions of chiral supermultiplets . ..... 24 3.3 Lagrangians for gauge supermultiplets . .... 28 3.4 Supersymmetricgaugeinteractions . .... 29 3.5 Summary: How to build a supersymmetric model . ... 31 4 Superspace and superfields 34 4.1 Supercoordinates, general superfields, and superspace differentiation and integration 34 4.2 Supersymmetry transformations the superspace way . .......... 37 4.3 Chiral covariant derivatives . .. 39 4.4 Chiralsuperfields ................................... 41 4.5 Vectorsuperfields.................................. 43 4.6 HowtomakeaLagrangianinsuperspace . ... 44 4.7 Superspace Lagrangians for chiral supermultiplets . ........ 46 4.8 Superspace Lagrangians for Abelian gauge theory . ....... 48 4.9 Superspace Lagrangians for general gauge theories . .......... 51 4.10 Non-renormalizable supersymmetric Lagrangians . ........ 54 4.11 R symmetries ..................................... 57 5 Soft supersymmetry breaking interactions 58 1 6 The Minimal Supersymmetric Standard Model 60 6.1 The superpotential and supersymmetric interactions . ......... 60 6.2 R-parity (also known as matter parity) and its consequences . ...... 64 6.3 Soft supersymmetry breaking in the MSSM . .... 67 6.4 Hints of an Organizing Principle . 68 6.5 Renormalization Group equations for the MSSM . .... 73 7 Origins of supersymmetry breaking 80 7.1 General considerations for spontaneous supersymmetry breaking......... 80 7.2 Fayet-Iliopoulos (D-term)supersymmetrybreaking . 82 7.3 O’Raifeartaigh (F -term)supersymmetrybreaking . 83 7.4 The need for a separate supersymmetry-breaking sector . .......... 87 7.5 The goldstino and the gravitino . 89 7.6 Planck-scale-mediated supersymmetry breaking models . ........ 92 7.7 Gauge-mediated supersymmetry breaking models . ...... 97 7.8 Extra-dimensional and anomaly-mediated supersymmetry breaking ....... 102 8 The mass spectrum of the MSSM 105 8.1 Electroweak symmetry breaking and the Higgs bosons . ....... 105 8.2 Neutralinosandcharginos ............................. 113 8.3 Thegluino....................................... 117 8.4 Thesquarksandsleptons............................. 117 8.5 Summary: theMSSMsparticlespectrum . 122 9 Sparticle decays 127 9.1 Decaysofneutralinosandcharginos . .... 127 9.2 Sleptondecays .................................... 128 9.3 Squarkdecays..................................... 129 9.4 Gluinodecays..................................... 129 9.5 Decays to the gravitino/goldstino . 130 10 Experimental signals for supersymmetry 132 10.1 Signals at hadron colliders . 132 + 10.2 Signals at e e− colliders ............................... 139 10.3 Darkmatteranditsdetection . 144 11 Beyond minimal supersymmetry 148 11.1 R-parityviolation................................... 149 11.2 Extra vectorlike chiral supermultiplets . ..... 151 11.3 The next-to-minimal supersymmetric standard model . ......... 153 11.4 The µ-term from non-renormalizable Lagrangian terms . 155 11.5 Extra D-termcontributionstoscalarmasses . 158 12 Concluding remarks 160 Acknowledgments 161 References 161 2 “We are, I think, in the right Road of Improvement, for we are making Experiments.” –Benjamin Franklin 1 Introduction The Standard Model of high-energy physics, augmented by neutrino masses, provides a re- markably successful description of presently known phenomena. The experimental frontier has advanced into the TeV range with no unambiguous hints of additional structure. Still, it seems clear that the Standard Model is a work in progress and will have to be extended to describe physics at higher energies. Certainly, a new framework will be required at the reduced Planck 1/2 18 scale M = (8πG )− = 2.4 10 GeV, where quantum gravitational effects become P Newton × important. Based only on a proper respect for the power of Nature to surprise us, it seems nearly as obvious that new physics exists in the 16 orders of magnitude in energy between the presently explored territory near the electroweak scale, MW , and the Planck scale. The mere fact that the ratio MP/MW is so huge is already a powerful clue to the character of physics beyond the Standard Model, because of the infamous “hierarchy problem” [1]. This is not really a difficulty with the Standard Model itself, but rather a disturbing sensitivity of the Higgs potential to new physics in almost any imaginable extension of the Standard Model. The electrically neutral part of the Standard Model Higgs field is a complex scalar H with a classical potential V = m2 H 2 + λ H 4 . (1.1) H | | | | The Standard Model requires a non-vanishing vacuum expectation value (VEV) for H at the minimum of the potential. This occurs if λ> 0 and m2 < 0, resulting in H = m2 /2λ. We H h i − H know experimentally that H is approximately 174 GeV from measurements ofq the properties h i of the weak interactions. The 2012 discovery [2]-[4] of the Higgs boson with a mass near 125 GeV implies that, assuming the Standard Model is correct as an effective field theory, λ =0.126 and m2 = (92.9 GeV)2. (These are running MS parameters evaluated at a renormalization H − scale equal to the top-quark mass, and include the effects of 2-loop corrections.) The problem 2 is that mH receives enormous quantum corrections from the virtual effects of every particle or other phenomenon that couples, directly or indirectly, to the Higgs field. 2 For example, in Figure 1.1a we have a correction to mH from a loop containing a Dirac fermion f with mass m . If the Higgs field couples to f with a term in the Lagrangian λ Hff, f − f f S H H (a) (b) 2 Figure 1.1: One-loop quantum corrections to the Higgs squared mass parameter mH , due to (a) a Dirac fermion f, and (b) a scalar S. 3 then the Feynman diagram in Figure 1.1a yields a correction λ 2 ∆m2 = | f | Λ2 + .... (1.2) H − 8π2 UV Here ΛUV is an ultraviolet momentum cutoff used to regulate the loop integral; it should be interpreted as at least the energy scale at which new physics enters to alter the high-energy 2 behavior of the theory. The ellipses represent terms proportional to mf , which grow at most logarithmically with ΛUV (and actually differ for the real and imaginary parts of H). Each of the leptons and quarks of the Standard Model can play the role of f; for quarks, eq. (1.2) should be multiplied by 3 to account for color. The largest correction comes when f is the top quark with λ 0.94. The problem is that if Λ is of order M , say, then this quantum correction to m2 f ≈ UV P H is some 30 orders of magnitude larger than the required value of m2 (92.9 GeV)2. This is H ≈ − only directly a problem for corrections to the Higgs scalar boson squared mass, because quantum corrections to fermion and gauge boson masses do not have the direct quadratic sensitivity to 0 ΛUV found in eq. (1.2). However, the quarks and leptons and the electroweak gauge bosons Z , W ± of the Standard Model all obtain masses from H , so that the entire mass spectrum of the h i Standard Model is directly or indirectly sensitive to the cutoff ΛUV. One could imagine that the solution is to simply pick a ΛUV that is not too large. But then one still must concoct some new physics at the scale ΛUV that not only alters the propagators in the loop, but actually cuts off the loop integral. This is not easy to do in a theory whose Lagrangian does not contain more than two derivatives, and higher-derivative theories generally suffer from a failure of either unitarity or causality [5]. In string theories, loop integrals are p2/Λ2 nevertheless cut off at high Euclidean momentum p by factors e− UV . However, then ΛUV is a string scale that is usually† thought to be not very far below MP. Furthermore, there are contributions similar to eq. (1.2) from the virtual effects of any heavy particles that might exist, and these involve the masses of the heavy particles (or other high physical mass scales), not just the cutoff. It cannot be overemphasized that merely choosing a regulator with no quadratic divergences does not address the hierarchy problem. The problem is not really the quadratic divergences, but rather the quadratic sensitivity to high mass scales. The latter are correlated with quadratic divergences for some, but not all, choices of ultraviolet regulator. The absence of quadratic divergences is a necessary, but not sufficient, criterion for avoiding the hierarchy problem. For example, suppose there exists a heavy complex scalar particle S with mass mS that couples to the Higgs with a Lagrangian term λ H 2 S 2. Then the Feynman diagram in − S| | | | Figure 1.1b gives a correction λ ∆m2 = S Λ2 2m2 ln(Λ /m )+ ... (1.3) H 16π2 UV − S UV S h i If one rejects the possibility of a physical interpretation of ΛUV and uses dimensional regular- 2 ization on the loop integral instead of a momentum cutoff, then there will be no ΛUV piece. 2 However, even then the term proportional to mS cannot be eliminated without the physically 2 unjustifiable tuning of a counter-term specifically for that purpose. This illustrates that mH is †Some attacks on the hierarchy problem, not reviewed here, are based on the proposition that the ultimate cutoff scale is actually close to the electroweak scale, rather than the apparent Planck scale.
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