Chapter III: Supersymmetry and searches at the LHC
3.3 SUSY in the SM
To extend the SM to be a supersymmetric theory, it is first necessary to place all of the known SM particles into supermultiplets. It is not possible for any of the known particles to be superpartners of one another [47], so this involves doubling the parti- cle content of the SM, introducing a new superpartner for each SM particle. The SM gauge bosons, having spin one, fall into vector supermultiplets and have spin-1/2 superpartners, called gauginos. The SM fermions fall into chiral supermultiplets and have scalar superpartners, calledsfermions.
The SM Higgs boson is a scalar and hence must be part of a chiral supermultiplet.
In fact, the structure of the SUSY Lagrangian requires that the theory contain at least twoHiggs chiral supermultiplets. One, denoted Hu, gives masses to the up- type quarks, and the other, denotedHd, gives masses to the down-type quarks and to the charged leptons [43]. After EWSB, there are a total of five scalar Higgs par- ticles: three neutral (two CP-even and one CP-odd) and two charged (one positive and one negative). One of the CP-even neutral scalars, denoted h0, has very sim- ilar interactions to those of the SM Higgs boson. While the masses of the other four Higgs scalars can be arbitrarily large, there is an upper bound on the mass of h0. At tree level it cannot exceed the mass of the Z boson, but higher-order loop corrections push the bound up to ~140 GeV, depending on the masses of the top superpartners [48]. The Higgs-like particle discovered by CMS and ATLAS in Run I of the LHC is therefore consistent withh0.
Placing the known SM particles into supermultiplets and adding one additional Higgs supermultiplet are the minimal changes required to formulate the SM in a SUSY framework. This Minimal Supersymmetric Standard Model (MSSM) forms the basis of much of the LHC SUSY search program.
SUSY breaking
Adding SUSY to the SM solves the hierarchy problem for the Higgs boson mass via the addition of two scalar degrees of freedom for every SM fermion. The couplings have the appropriate values such that the quadratic contribution from Eq. 3.14 is exactly cancelled by that from Eq. 3.15. Moreover, no MSSM scalar’s mass suffers from a quadratic divergence at any order in perturbation theory [43].
However, if nature were fully supersymmetric, all of the superpartners would have the same masses and couplings as their SM counterparts, and many of them would have been discovered already. We know therefore that if SUSY exists, it is inexact or spontaneously broken. Terms that break SUSY (i.e. that introduce differences between particles and their superpartners) can be introduced into the MSSM La- grangian without reintroducing the hierarchy problem into the theory. Such terms are calledsoftSUSY-breaking terms and are characterized by positive mass dimen- sion. They can take the following general forms [49, 50]:
• Bilinear and trilinear couplings for the scalar superpartners
• Masses for the scalar superpartners
• Masses for the gauginos
This set of soft SUSY-breaking terms is sufficient for all of the MSSM superpart- ners to have masses different from their SM counterparts. Introducing these terms adds a large number of parameters to the MSSM; the full Lagrangian with soft SUSY-breaking terms included has over 100 free parameters that are not present in the SM [43]. Simplifying assumptions are often made to reduce the number of parameters to a manageable level for interpretation of collider searches [51].
Electroweak symmetry breaking and the µparameter
Before SUSY is broken, the minimum of the Higgs potential is at 0 and there is no EWSB. The nature of EWSB in the MSSM depends on the SUSY-breaking param- eters in the theory. If electroweak symmetry is broken, the neutral components of HuandHdobtain VEVs, denotedvuandvdrespectively. We parameterize the ratio of the VEVs as
tanβ =vu/vd. (3.20)
The minimization condition for the Higgs potential in the MSSM is [52]:
1
2m2Z = m2H
d −m2H
utan2β
tan2 β−1 − µ2, (3.21)
wheremHD andmHu are soft SUSY-breaking parameters. The parameter µis part of the MSSM Lagrangian and is unrelated to the breaking of SUSY. To avoid large fine-tuning between the two terms of Eq. 3.21, µ should have a value near the SUSY-breaking scale. Variants of the MSSM, such as the NMSSM [53], dynam- ically generate µfrom a mechanism related to SUSY breaking, thus ensuring that these scales are close to one another.
R-parity
Unlike in the SM, the Lagrangian of the MSSM contains terms that explicitly vio- late the conservation of lepton and baryon number. Unless there is a mechanism to suppress these terms, the interactions allowed by the MSSM would not respect the observed limits on, e.g., the lifetime of the proton. One way to prevent this from happening is to introduce a quantity calledR-parity:
PR = (−1)3(B−L)+2s, (3.22)
and postulate that it is conserved in the MSSM. Here B and L are the baryon and lepton number of a particle andsis its spin. All of the particles in the SM have even R-parity, while all of their superpartners have oddR-parity.
The effect of enforcing R-parity conservation in the MSSM is to remove the L- and B-violating terms from the Lagrangian. It also has a striking implication for collider phenomenology. If R-parity is conserved, then every interaction vertex in the theory contains an even number of particles withPR = −1. This implies that:
• Supersymmetric particles must be produced in pairs in collider events, and
• The lightest supersymmetric particle (abbreviated LSP) is absolutely stable, and every SUSY particle will eventually decay to a final state consisting of SM particles and LSPs.
The LSP also provides a particle candidate for the dark matter whose existence in the universe is suggested by astrophysical observations [54].
Theories that violate R-parity are sometimes considered in collider searches for SUSY, but in generalR-parity conservation is assumed to hold.
Summary of the MSSM superpartners
Here we briefly describe each of the classes of superpartner particles that exist in the MSSM, and discuss the interactions of each.
Neutralinos and charginos
The superpartners of the SM electroweak bosons (‘binos’ and ‘winos’), and those of the Higgs scalars (‘higgsinos’), mix with one another to form four neutral and two charged mass eigenstates. The neutral states (‘neutralinos’), denoted ˜χ01, ˜χ02, χ˜03, and ˜χ04, are mixtures of the superpartners of the SM BµandWµ0and the neutral Higgs scalars. The charged states (‘charginos’), denoted ˜χ±1 and ˜χ±2, are mixtures of the superpartners of theWµ1, theWµ2, and the charged Higgs scalars. The physical neutralino masses are obtained by diagonalizing the matrix [43]
MN˜ =
M1 0 −cβsWmZ sβsWmZ 0 M2 cβcWmZ −sβcWmZ
−cβsWmZ cβcWmZ 0 −µ sβsWmZ −sβcWmZ −µ 0
, (3.23)
where M1 and M2are SUSY breaking mass parameters,cW andsW denote cosθW and sinθW, andcβandsβdenote cos βand sin β. The chargino masses are obtained by diagonalizing the block matrix
MC˜ =
0 XT X 0
, X =
M2
√
2sβmW
√
2cβmW µ
. (3.24)
The decays of neutralinos and charginos most relevant for searches for hadronically produced SUSY at the LHC are illustrated in Figure 3.3. A neutralino can undergo a decay to a lighter neutralino and a neutral gauge or Higgs boson. A chargino can decay to a neutralino and a charged W or Higgs boson. Neutralinos and charginos can both decay to a fermion-sfermion pair.
Gluinos
The gluino is the color-octet superpartner of the SM gluon. Being charged under SU(3)c, it cannot mix with the other superpartners.
Gluinos decay via the squark-quark-gluino interaction vertex illustrated on the left side of Figure 3.4. If the gluino is heavier than at least one squark, it will undergo
the decay ˜g → qq. If it is lighter than the squarks, it will decay through o˜ ff-shell squarks, e.g. ˜g →qq¯χ˜01.
Being strongly interacting, gluinos may be produced with significant cross section at the LHC. Example diagrams illustrating gluino pair production are shown in the top row of Figure 3.5.
Figure 3.3: Two-body decays of MSSM neutralinos and charginos to gauge or Higgs bosons. Decays to a fermion-sfermion pair (not shown) are also possi- ble [43].
Figure 3.4: Feynman diagrams for MSSM three-point gluino (left), wino (center), and bino (right) couplings to fermions [43].
Figure 3.5: Diagrams illustrating gluino (top row) and squark (bottom row) produc- tion from two colliding gluons [43].
Squarks and sleptons
The scalar superpartners of the SM quarks are calledsquarks, and those of the SM leptons and neutrinos are calledsleptonsand sneutrinos. Their names are formed
by appending ‘s’ to the SM particle name:stop, sbottom, and so on. There are two squarks or sleptons for each SM quark and charged lepton, which correspond to the left- and right-handed field components; and one sneutrino for each SM neutrino.
In principle there can be arbitrary mixing of the fields in each sfermion family: the down-type squarks, the up-type squarks, the sleptons, and the sneutrinos. It is often assumed that the SUSY-breaking MSSM parameters that mediate this mixing are small or zero [43]. Mixing within the third generation of squarks and sleptons does arise, however, as a result of the large Yukawa couplings for the third generation.
Thus, e.g., the physical stops ˜t1and ˜t2may be mixtures of the left- and right-handed stop states ˜tL and ˜tR. The Yukawa couplings also affect the running of the sfermion masses under the renormalization group (RG) equations, tending to give the third- generation sfermions different (usually lower) masses than those of the first and second generation.
Strong production of squarks from colliding gluons is illustrated in the bottom row of Figure 3.5. The squarks interact with the gluino as shown in Figure 3.4; they can therefore decay to a quark and a gluino if this is kinematically allowed. A squark can also decay to a quark and a neutralino or chargino; this is the primary decay mode if the squark is lighter than the gluino.