Chapter III: The hierarchy problem, naturalness, and supersymmetry . 11
3.3 Supersymmetry
Supersymmetry (SUSY) is a proposed symmetry of spacetime that intro- duces a bosonic (fermionic) partner for every fermion (boson) [9, 7, 8, 51, 52, 10, 53, 54, 6]. For many years, such a symmetry was thought to be im- possible since in 1967, Coleman and Mandula [55] published their no-go theorem that says internal symmetries, those that act on internal degrees of freedom (like spin) cannot be combined with spacetime symmetries in a nontrivial way. SUSY evades this theorem because it is based on asuper Lie
102 104 106 108 1010 1012 1014 1016 1018 1020 0.0
0.2 0.4 0.6 0.8 1.0
RGE scaleΜin GeV SMcouplings g1
g2
g3
yt
yb Λ
min TeV
6 8 10
0 50 100 150 200
0 50 100 150 200
Higgs pole massMhin GeV ToppolemassMtinGeV
LI=1045GeV678910 12141619 Instability
Non-perturbativity
Stability Meta
-stability
Figure 3.2: Renormalization of the SM gauge couplings g1 = √
5/3g0, g2, g3, of the top, bottom andτ couplings (yt, yb,yτ), of the Higgs quartic cou- plingλ, and of the Higgs mass parametermH(left) and SM phase diagram in terms of Higgs and top pole masses (right). The plane is divided into regions of absolute stability, metastability, instability of the SM vacuum, and nonperturbativity of the Higgs quartic coupling. The top Yukawa cou- pling becomes non-perturbative formt >230 GeV. The dotted contour-lines show the instability scaleΛIassumingα3(mZ) =0.1184 [47].
algebra, which may include fermionic symmetries and anticommutation re- lations as well as the usual bosonic symmetries and commutation relations.
Supersymmetric extensions of the SM are compelling mainly because they (a) yield a solution to the hierarchy problem, alleviating the fine-tuning
of fundamental parameters [19, 20, 21, 22, 23, 24], explained further in Sec.3.5.
(b) exhibit gauge coupling unification [13,14,15,16,17,18], and
(c) provide a weakly interacting particle candidate for dark matter [11,12].
SUSY can be thought of as an extension of the usual group of spacetime symmetries, known as the Poincar´e group. This group has generators re- lated to translation symmetry, Pm, and Lorentz symmetry, Mmn = −Mnm, which form a Lie algebra:
[Pm,Pn] = 0
[Pm,Mnp] = i(ηmnPp−ηmpPn)
[Mmm,Mpq] = i(ηmpMnp−ηnpMmq+ηnqMmp−ηmqMnp). (3.13)
As shown by Coleman and Mandula [55], the only way to extend this sym- metry with a new internal symmetry group G with bosonic generators Br
and Lie algebra,
[Br,Bs] = frst
Bt , (3.14)
with structure functions frst, is if the extended symmetry group is simply the direct product(Poincar´e)×Gwith a trivial Lie algebra,
[Br,Pm] = [Br,Mmn] =0 . (3.15) However, SUSY exploits a loophole in the Coleman-Mandula theorem, which only considers bosonic symmetry generators, by incorporating fermionic symmetry generatorsQthat generate SUSY transformations,
Q|bosoni =|fermioni, Q|fermioni=|bosoni. (3.16) Supersymmetries can be combined with the spacetime symmetries in a way that mixes the two symmetries, as exemplified by the super Lie algebra for N =1 SUSY,
{Qα,Qβ˙} =2σαmβ˙Pm
{Qα,Qβ} ={Qα˙,Qβ˙} =0
[Pm,Qα] = [Pm,Qα˙] = 0 . (3.17) In other words, two SUSY generators can combine to generate a spacetime translation.
The most economical supersymmetric extension of the standard model is the Minimal Supersymmetric Standard Model (MSSM). For supersymmet- ric theories, the Lagrangian can be written in terms of vector superfields V(x,θ,θ) and chiral superfields Φl(x,θ,θ) that are functions ofsuperspace, an extension of spacetime that includes anticommuting Grassmanian vari- ables θ and θ. Tab3.1 shows the particles of the MSSM, described by chi- ral superfields Qi,Uic,Li,Eci,Hu,Hd, and the representations in which they transform under the MSSM gauge group.
The MSSM Lagrangian can be split into three terms, the K¨ahler potential K(Φl,Φ†l,V), which describes the kinetic and gauge-covariant terms, a su- perpotentialW(Φl), which describes the mass and interaction terms, and a
SU(3)C SU(2)L U(1)Y
Q 3 2 1/6
Uc 3 1 −2/3
Dc 3 1 1/3
L 1 2 −1/2
Ec 1 1 1
Hu 1 2 1/2
Hd 1 2 −1/2
Table 3.1: Table summarizing the representations in which the chiral super- fields transform under the MSSM gauge group. See Tab.2.1for a description of the columns.
gauge kinetic termG(V), LMSSM =
Z d4θK(Φl,Φ†l,V) + Z
d2θW(Φl) +h.c.
+
Z
d2θG(V) +h.c.
=
Z d4θ
∑
V
Φ†legVΦl+ Z
d2θW(Φl) +h.c.
+
Z d2θ1
4WαWα+h.c.
, (3.18) whereWαis the vector superfield strength2and the indexlruns over all the matter superfields Φl = Qi,Uic,Li,Eci,Hu,Hd. The superpotential can be split into two components based on R-parity [56], a discrete Z2 symmetry often assumed in SUSY model building, defined for each particle as
PR = (−1)3(B−L)+2s, (3.19) where B is the baryon number, L is the lepton number, and s is the spin of the particle. With this assignment, all SM particles have even R-parity (PR = +1), while the superpartners have odd R-parity (PR = −1). The R-parity conserving terms in the superpotential are
WRPC=−UcyuQHu+DcydQHd+EcyeLHd+µHuHd , (3.20) and theR-parity violating terms are
WRPV= 1
2λijkLiLjEck+λ0ijkLiQjDck+µ0iHuLi+1
2λ00ijkUicDcjDkc. (3.21) If R-parity is conserved, there are three important phenomenological con- sequences:
2The vector superfield strength isWα=−14D2(e−VDαeV).
• The lightest superparter, called the “lightest supersymmetric particle”
or LSP, must be stable.
• Each superpartner other than the LSP must eventually decay into a state that contains an odd number of LSPs (usually just one).
• In collider experiments, superpartners can only be produced in even numbers (usually two).
In particular, at a proton-proton collider like the LHC (described in Ch. 4), the SUSY processes with the largest cross sections are the pair production of gluinos and squarks through gluon-gluon and gluon-quark fusion, which are among the reactions of QCD-level strength [45]:
gg→egeg,eqieqj, (3.22)
gq→egeqi , (3.23)
qq→egeg,eqieqj, (3.24)
qq→eqieqj. (3.25)
Some of the diagrams corresponding to the gluon-gluon fusion reactions in Eqn. 3.22 are depicted in Fig. 3.3 [45]. Assuming all SUSY particles other than those being pair-produced are decoupled from the theory (as is done in the simplified model approach, see Sec. 3.8 [57, 58, 59, 60, 61, 62]), the SUSY pair-production cross sections for each SUSY particle may be com- puted at next-to-leading-order (NLO) plus next-to-leading-logarithm (NLL) accuracy [63, 64, 65, 66, 67, 68, 69]. For the √
s = 8 TeV and 13 TeV LHC, these SUSY production cross sections are plotted as a function of the super- partner mass in Fig.3.4. Due to the hierarchy of SUSY cross sections, there is a corresponding hierarchy in terms of discovery potential at the LHC, with the largest mass reach for gluinos then squarks then top and bottom squarks then electroweak SUSY partners (see Sec.3.4) then, finally, sleptons.