Electroweak precision observables fit of the Higgs mass

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THE MINIMAL SUPERSYMMETRIC STANDARD MODEL

The supersymmetric solution

At the end of the last chapter, we pointed out that the Higgs boson mass suﬀers from quadratically divergent corrections. Thus, unless an unnatural cancellation between its tree-level value and its radiative corrections is assumed, the Higgs mass is pushed more than 16 orders of magnitude away from the value required for the electroweak interactions.
However, if one supposes that a complex scalar particle with mass ms exists and couples to the Higgs through a λs constant, then the left diagram in Figure 2.1 will give a contribution to mh equal to Δmh2 = λs ΛUV2 − 2ms2 ln ΛUV +… . (2.1) 16π2 ms
It is worth noting the relative minus sign between Equation 1.21 and 2.1. If each of the quarks and leptons of the Standard Model is accompanied by two complex scalar fields with λs = λ2f , then the quadratic divergences cancel. Actually, the presence of such bosonic partners can be the result of a new symmetry relating bosons and fermions, called supersymmetry (SUSY).
To prove that supersymmetry really provides the desired particle content, we shall examine the supersymmetry algebra. In quantum theory, a generator Q of supersym-metry must turn a bosonic state into a fermionic one and vice versa, thus they must carry spin angular momentum 1/2, as opposed to the Lorentz group or gauge group generators, all of which are bosonic. Since spinors are complex objects, the Hermitian conjugate of Q, Q†, is also a supersymmetry generator.
As Q and Q† are conserved, so is their anti-commutator. This implies the existence of a conserved quantity that does not transform trivially under the Lorentz transfor-mations, i.e. schematically:
{Q,Q†} = QQ† +Q†Q = P . (2.2)
However, the form of the conserved four-vector P is highly restricted by the Haag-Lopuszanski-Sohnius extension of the Coleman-Mandula no-go theorem [49, 50]. It states that in a relativistic quantum field theory with a conserved four-vector charge in addition to the energy-momentum there can be no scattering and so the theory is trivial. For instance, in two-body scattering, for fixed centre-of-mass energy, energy-momentum conservation leaves only two degrees of freedom, the two scattering angles. A second conserved four-vector would forbid almost all their possible values. Then P must be the energy-momentum generator of space-time translations.
In general, it is possible to have N ≥ 1 distinct copies of the supersymmetry generators Q, but phenomenological problems restrict the choice to N = 1, at least in four-dimensional field theories.
The particle states of supersymmetric theories, which are called supermultiplets, must be representations of the supersymmetry algebra. Their content in terms of bosonic and fermionic states can be derived from the algebra definition of Equation 2.2.
Consider the operator (−1)s, where s is the spin angular momentum. For any repre-sentation of the algebra
Tr[(−1)sP ] = Tr[(−1)sQQ† + (−1)sQ†Q] =
s† s † (2.3)
= Tr[(−1) QQ + Q(−1) Q ] =
= Tr[(−1)sQQ† − (−1)sQQ†] = 0.
The equality in the second line follows from the cyclic property of the trace, while in the third line we have used the fact that, since Q turns a boson into a fermion and vice versa, Q must anti-commute with (−1)s. For fixed momentum, Tr[(−1)sP ] is simply proportional to the number of bosonic degree of freedom minus the number of fermionic degree of freedom. Therefore each supermultiplet contains the same number of fermionic and bosonic degrees of freedom.
Furthermore, supermultiplets own other interesting properties. By definition, two states residing in the same supermultiplet are related by some combination of Q and Q†. Since Q and Q† do not carry Lorentz indices, they commute with P and, thereby, with P 2; as a consequence, all the states within the same supermultiplet have equal masses. The supersymmetry generators also commute with the generators of gauge transformation and so particles in the same supermultiplet must also have identical gauge quantum numbers.
With such particle content, supersymmetry guarantees the cancellation of quadratic divergences, not only at one loop, as showed previously, but also at all orders in pertur-bation theory. Actually under exact supersymmetry, the whole fermionic and bosonic contributions completely cancel, giving a vanishing total correction to scalar masses. At one-loop order, this can be easily verified by adding to Equation 2.1 the amplitude of the right diagram in Figure 2.1 and setting ms = mf .
Though, supersymmetry is not an exact symmetry of nature, otherwise a bosonic partner of the electron with mass equal to me ≃ 511 keV would have been discovered long time ago; we have no experimental evidence of the existence of any supersymmetric partner of the SM particles up-to-date.
Thus, supersymmetry must be broken.
However, if supersymmetry is still to solve the naturalness problem, the breaking terms in the Lagrangian must not reintroduce the quadratic divergences in radiative corrections to the scalar masses. This class of terms is referred to as soft supersymmetry breaking and leaves in the Δm2h only logarithmic terms of ΛUV: Δmh2 ∝ msoft2 ΛUV +…, (2.4)
ln msoft where msoft is the mass scale associated with the breaking terms, for example the mass splitting between SM particles and their supersymmetric partners.
From Equation 2.4, it must be clear that supersymmetric particles cannot be too heavy, but rather have masses of order 1 TeV at most, in order not to create again a fine tuning problem between the tree level Higgs mass and its correction, which is proportional to m2soft.
In the next sections, we review the supersymmetric extension of the Standard Model with the minimal possible addition of particles, the Minimal Supersymmetric Standard Model (MSSM).

Particles and interactions

The MSSM [51, 52, 53, 54] is constructed by placing each of the known particles in supermultiplets and subsequently adding its associated supersymmetric partner (or simply superpartner).
As each supermultiplet contains the same number of fermionic and bosonic degrees of freedom, the simplest possibilities are supermultiplets comprising:
• a two-component Weyl spinor and a complex scalar field (called chiral or matter supermultiplet);
• a massless gauge boson, which has two helicity states, and a two-component Weyl spinor (called gauge supermultiplet); gauge bosons can eventually acquire masses after the gauge symmetry is spontaneously broken.
In renormalizable supersymmetric field theories with only one SUSY generator Q, all possible forms of supermultiplets are combinations of these two.
Since the SM fermions reside in diﬀerent representations of the gauge group than the gauge bosons, none of them can be identified with the superpartner of a gauge boson. Therefore we have to place them in chiral supermultiplets. Then, one gauge supermultiplet is needed per gauge boson.
The left-handed and right-handed parts of quarks and leptons are diﬀerent two-component Weyl spinors, with diﬀerent gauge properties, and each of them is accom-panied by a new complex scalar field. By convention, the names of scalar superpartners are obtained by adding the prefix s to the SM particle names, for example sleptons, squarks or, collectively, sfermions. Sfermions can be left- or right-handed, referring to the helicity state of their SM partner.
The supersymmetric copies of gauge bosons are Weyl fermions and are called gaug-inos. In particular, the names of the partners of the gluon, W , Z and photon are the gluino, wino, zino and photino. The composition of zino and photino in terms of the original (i.e. before gauge symmetry breaking) massless SU (2)L and U (1)Y gauginos is the same as of Z and photon in terms of the massless SU (2)L and U (1)Y gauge bosons. Equivalently, if supersymmetry were unbroken, the zino and photino masses would be exactly mZ and 0.
For reason that are postponed to Section 2.3, the one Higgs doublet model providing the SU (2)L × U (1)Y breaking does not work in supersymmetric extensions of the SM. The minimal choice for the Higgs sector is a pair of complex scalar SU (2)L doublets, with hypercharge Y = +1/2 and −1/2 respectively, that we will mark as
H+ H0
u and d . (2.5)
H0 H−
u d
These Higgs fields can only fit in chiral supermultiplets, together with four spin-1/2 higgsinos. After the scalar doublets acquire a non-zero ground state expectation value, three of the initial eight degrees of freedom become the longitudinal helicity states of the W ± and Z0 bosons. The remaining five turn into as many scalar fields: two neutral CP -even, the lightest called h, the heaviest H0, one neutral CP -odd, A0, and a positive and a negative one, H±.
All new supersymmetric particles are represented by the symbols of the associated SM fields superposed by a tilde. Table 2.1 summarizes the new supersymmetric parti-cles and Higgs bosons predicted by the MSSM. However, the supersymmetric particles listed are not necessarily the mass eigenstates of the model. Indeed, after symmetry breaking, particles with the same set of quantum numbers in general mix. This is the case for the charged W , Hu , Hd and for the neutral Z, γ˜, Hu , Hd . The former com-bine to give two charginos χ˜±1, χ˜±2, while the latter mix-up into four neutralinos χ˜01, χ˜02, χ˜03, χ˜04. The subscripts indicate the order of the mass eigenvalues from the lightest to the heaviest. Also left- and right-handed sfermions in general mix. The corresponding lightest and heaviest mass eigenstates have subscripts 1 and 2 instead of L and R.
Now that we have drawn a picture of the particle content of the MSSM, we can outline the interactions between the diﬀerent constituents. A complete and accurate discussion of the full Lagrangian and the deriving Feynman rules can be found in [55].
After writing down the kinetic terms for the fields of the theory, the local gauge invariance requires the substitution of ordinary derivatives with covariant derivatives for scalar and fermions, analogously to Sections 1.1 and 1.2. From this simple proce-dure, the gauge interactions already present in the SM arise for fermions and scalars in chiral supermultiplets:
• gauge-fermion-fermion;
• gauge-scalar-scalar;
• gauge-gauge-scalar-scalar.
The invariance of the Lagrangian under supersymmetry transformations is ensured by the presence of the interactions:
• gaugino-scalar-fermion;
• (Higgs)4.
Their strengths are fixed to be gauge couplings by the requirements of supersymmetry, even though they are not gauge interactions from the point of view of an ordinary field theory. Of course, the existence of gauge couplings for a specific chiral supermultiplet depends on its gauge charges; for instance, neither a pair of neutrinos nor a pair of sneutrinos interact with the photon, since they are electrically neutral.
For non-abelian gauge groups, in addition to the usual gauge boson cubic and quartic self-interactions, a coupling between vector bosons and their superpartners emerges, since, in this case, also the covariant derivative of gauginos contains a part proportional to the gauge field. Thus, the following interactions exist:
• (gauge)3;
• (gauge)4;
• gauge-gaugino-gaugino.
The most general supersymmetric non-gauge interactions are described by the La-grangian: ∂W ∂W ∗ 1 ∂W − − ψiψj + c.c. , (2.6) ∂φi ∂φi 2 ∂φi∂φj where φi and ψi are the bosonic and fermionic components of the chiral supermultiplets and W is an analytic function of the scalar fields. Thus, once the gauge transformation properties of the fields are defined, the only missing input to build all the interactions of a supersymmetric theory is W , the superpotential. In the MSSM, it is:
∗i i,j ˜j ˜∗i i,j ˜j ˜∗i i,j ˜j Hd + H uHd. (2.7)
W = u˜R λu Q Hu − dR λd Q Hd − ℓR λℓ L
The matrices λu,d,ℓ are equivalent to those in Equation 1.13 and corresponds to the Standard Model Higgs mass parameter of Equation 1.8. The generated vertices are:
• (Higgs)2, (higgsino)2;
• (Higgs)2-(slepton)2, (Higgs)2-(squark)2, (slepton)2-(squark)2;
• Higgs-(slepton)2, Higgs-(squark)2;
• Higgs-lepton-lepton, Higgs-quark-quark;
• higgsino-slepton-lepton, higgsino-squark-quark.

The Higgs sector

In Section 2.2, we anticipated that the Higgs sector of any supersymmetric extension of the SM model must contain at least two Higgs doublets. There are at least two reasons for this.
First, a single Higgs doublet would introduce a gauge anomaly, violating the local invariance of the model and causing the quantum theory to be inconsistent. Indeed, in the triangular Feynman diagram with one photon and two SU (2)L bosons at the vertices receive contributions from the loops of all the left-handed charged fermions. In the Standard Model the resulting total current vanishes and the gauge symmetry is preserved. The addition of a single left-handed charged higgsino would destroy this re-markable cancellation. Two Higgs doublet with opposite hypercharges, instead, would be accompanied by two left-handed higgsinos whose contributions to the anomalous current cancel each other.
The second motivation relies on the structure of supersymmetric theories and in particular to the superpotential. Indeed, since it must be analytic, it cannot contain both Hu and its complex conjugate. At the same time, the first term of the MSSM superpotential (Equation 2.7) needs a Higgs doublet with Y = +1/2 while the second and third require a Higgs doublet with Y = −1/2. If we leave out one of the Higgs multiplets, some quarks or leptons will be left massless.

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R-parity

The superpotential defined in Equation 2.7 does not contain all the renormalizable terms allowed by the gauge invariance of the MSSM. However, the inclusion of terms like
˜i ˜j ˜∗k , ˜i ˜j ˜∗k , ˜i Hu, ∗i ˜∗j ˜∗k (2.8)
LLℓR LQℓR L u˜R dR dR
would lead to the violation of leptonic and baryonic numbers, already discussed in Section 1.3. This is rather disturbing, since such processes are severely constrained by experiments.
To prevent any undesired eﬀect, we can postulate the conservation of the baryonic and leptonic numbers. However, this is clearly a step backward from the situation of the SM, where the preservation of these quantum numbers is an accidental consequence of the renormalizability. Even more important, baryonic and leptonic numbers are known to be violated by non-perturbative electroweak eﬀects negligible for all ordinary energy experiments, but important for models describing the early universe. Therefore in the MSSM, instead of the baryonic and leptonic number conservation, a new discrete symmetry, called R-parity, is required.
R-parity is a defined as
R = (−1)B−L+2s. (2.9)
Quarks and anti-quarks have, respectively, B = ±1 and L = 0, while leptons and anti-leptons have L = ±1 and B = 0. The letter s stands for the spin. This symmetry forbids the unwanted superpotential terms, without aﬀecting any of the others and without excluding the possibility of non-perturbative B and L violation.
It turns out that all the ordinary SM particles and the Higgs bosons have R-parity R = +1, while their supersymmetric partners have R = −1. Three important phe-nomenological implications follow:
• in collider experiments, sparticles are produced in even number;
• each sparticle can decay only into a state containing a odd number of sparticles;
• the lightest supersymmetric particle (LSP) is stable.
We can now have an insight into the MSSM phenomenology at colliders. After the production of a pair of sparticles, each of these decays into a SM and a SUSY particle. The two decay chains develop, resulting into a final state composed by two LSP’s and a number of SM particles equal to the number of disintegration processes. Because of gauge couplings, the SM particles associated to the SUSY chains are expected to be dominantly leptons at electron-positron colliders and quarks and gluons at hadron colliders.
The experimental signature of the passage of lightest supersymmetric particles at collider detectors is diﬀerent depending on the nature of the LSP.
A neutral and weakly interacting particle crosses the detector without leaving any trace or energy deposit, behaving as a neutrino. If the detector has a full coverage of the spheric angle around the collision point, then a non-zero energy balance can be reconstructed and the missing energy of the escaped particles calculated. A LSP with such characteristics, i.e. neutral, massive, stable and weakly interacting, exactly fits our description of a possible dark matter candidate given in Section 1.5.
If the LSP is a charged slepton, it will look like a muon. However, since the slepton is heavy, its time of flight through the detector diﬀers considerably from the muon one.
Another possibility is a coloured LSP, such as a squark or gluino. Because of colour confinement, the LSP hadronizes before crossing the detector, generating colour-singlet states called R-hadrons. The details of R-hadron interactions in matter are highly uncertain. However, the probability of an interaction between a squark or a gluino in the R-hadron and a quark in the target nucleon is low, since, according to perturbative QCD, the cross-section varies with the inverse square of the parton mass. Thus, stable R-hadrons escape the detector. But the light quarks bounded to the heavy parton may interact, causing small amount of energy losses and eventually charge flipping, providing additional discriminating signatures.

Soft SUSY breaking

Masses are added to supersymmetric particles by an explicitly soft supersymmetry breaking Lagrangian, that contains mass term for gauginos, such as mg˜g˜ and for scalar fields, mi,j φi∗φj , with i and j running on the family indices.
These are the only possible mass terms, since explicit vector boson and matter fermion masses are prohibited by gauge symmetry. This simple observation clarifies why none of the supersymmetric partners has been discovered yet. Actually, all the known particles acquire mass only through the Higgs mechanism, so they must have m ∼ v/ 2 ≃ 174 GeV, while their superpartners have an explicit mass, that can be in principle as high as 1 TeV.
Moreover, the scalar interactions
∗i i,j ˜j ˜∗i i,j ˜j ˜∗i i,j ˜j Hd + c.c. (2.10)
u˜R au Q Hu − dR ad Q Hd − ℓR aℓ L
are soft supersymmetry breaking terms and should be taken into account.
Before this section, the parameter in Equation 2.7 was the only newly introduced parameter. The soft SUSY breaking Lagrangian, instead, needs around a hundred new parameters: the gaugino masses, the Higgs boson masses, the 3 × 3 mass matrices for sleptons and squarks, the 3 × 3 au,d,ℓ complex matrices.
However, experimental evidences, especially from the flavour physics sector, already strongly constrain such parameters. For example, a non-diagonal mass matrix of slep-tons would violate the stringent limits on the muon decay rate to an odd number of electrons, quoted in Equation 1.18. Equivalent arguments subsist for the mass ma-trix of squarks, which is restricted by limits on the flavour changing neutral currents. Under these assumptions, often called universality relations, the number of required parameters amounts to about twenty.

Spontaneous SUSY breaking

The relatively simple form of the SUSY breaking matrices deriving from experimental evidences is presumed to be the result of the existence of an underlying principle gov-erning supersymmetry breaking. Indeed, suppose the diagonality conditions of sparticle mass matrices are exact at some very high energy scale. Then, the matrices must be evolved using the renormalization group equations (RGE) to the electroweak energy scale to perform predictions for the observables. Even though the diagonality relations are no longer exact, the flavour violating eﬀects are enough suppressed to be compat-ible with experiments. Thus, the universality relations should be interpreted as high energy boundary conditions to the renormalization group equations.
Models that explain the origin of the soft supersymmetry breaking terms should address also the existence of such high energy conditions.
An interesting theoretical reason to believe in some simpler high energy principle is the unification of gauge couplings in the MSSM. The three coupling constants as-sociated to the SU (3)C × SU (2)L × U (1)Y gauge groups can be evolved toward high
energies by solving the renormalization group equations. While in the Standard Model the three coupling constants fail to meet, the particle content of the Minimal Super-symmetric Standard Model give the possibility to unify the gauge couplings at the scale ∼ 1016 GeV. Figure 2.2 shows the graph of the evolution of the inverse squared gauge couplings in the SM and in the MSSM, including two loop eﬀects [52].
The theoretical challenge is to explain the soft breaking parameters with a model for spontaneous supersymmetry breaking.
In principle, we could include in the MSSM a field whose vacuum expectation value leads to supersymmetry breaking, just as we insert a Higgs field to break the electroweak gauge symmetry. However, it has been shown that this cannot lead to phenomenologically viable models [56].
The solution is to introduce also a hidden sector which consists of some fields that do not have any direct coupling to the visible sector containing the MSSM. Supersymmetry is spontaneously broken in this hidden sector. A weak interaction, called mediator and coupling the two sectors, then induces a supersymmetry breaking for the Standard Model particles and their superpartners. If the mediating interaction is independent of the flavour of the particles, the resulting soft supersymmetry breaking term will satisfy universality relations like those of Section 2.5.
Among the diﬀerent proposals for the mediators, the most competitive two are based on gravity and on gauge interactions.
These models also oﬀer predictive frameworks useful for phenomenological analyses of supersymmetry, since they describe all the MSSM masses and interactions in terms of few new parameters. For example, Minimal Supergravity (mSUGRA) [57, 58, 59, 60, 61, 62] is based on gravity mediation and is determined by only five parameters: three are defined at the unification energy scale, the universal mass of scalars (M0) and fermions (M1/2) and the strength of the cubic scalar coupling (A), while the remaining two parameters fix the Higgs sector at the electroweak scale, the ratio of the vacuum expectation values of the two Higgs doublets (tan β) and the sign of the Higgs mass parameter in the superpotential ( ). Among the Gauge Mediated Supersymmetry Breaking (GMSB) models [63, 64], the most popular is defined by six parameters: Λ, the eﬀective SUSY mass scale, N , the number of mediator generations, M , the mediator mass scale, Cg , the intrinsic SUSY breaking to messenger scale, and again tan β and the sign of .

Table of contents :

1 The Standard Model of particle physics
1.1 The minimal gauge Lagrangian
1.2 The Higgs mechanism
1.3 Precision tests
1.3.1 Lepton magnetic moments
1.3.2 Flavour physics
1.3.3 W boson mass
1.4 Higgs mass limits
1.4.1 Experimental limit on the Higgs boson mass
1.4.2 Electroweak precision observables fit of the Higgs mass
1.5 What is missing?
2 The Minimal Supersymmetric Standard Model
2.1 The supersymmetric solution
2.2 Particles and interactions
2.3 The Higgs sector
2.4 R-parity
2.5 Soft SUSY breaking
2.6 Spontaneous SUSY breaking
3 The ATLAS experiment at the LHC
3.1 The Large Hadron Collider
3.2 Detector overview
3.3 Inner tracking system
3.4 Calorimetry
3.4.1 Electromagnetic calorimetry
3.4.2 Hadronic calorimetry
3.4.3 Forward calorimetry
3.5 Muon spectrometer
3.6 Trigger and Data Acquisition systems
3.7 Computational aspects
4 Expected ATLAS performance
4.1 Particle reconstruction
4.1.1 Photons and electrons
4.1.2 Muons
4.1.3 Jets
4.1.4 Hadronic τ decays
4.1.5 Tagging b-flavoured jets
4.1.6 Missing transverse energy
4.2 Trigger performance
4.3 SM Higgs boson discovery potential
4.4 MSSM Higgs discovery potential
4.5 SUSY discovery potential
5 Missing transverse energy studies and monitoring
5.1 Introduction
5.2 Measurement of Emiss
T direction
5.3 Fake Emiss
T correlation with jet direction
5.4 Monitoring of Emiss
T quality
5.5 Sensitivity tests of the Emiss
T monitoring system
6 Higgs searches in cascade decays of SUSY particles
6.1 Motivations and phenomenology
6.2 Event generation and detector simulation
6.3 Scan of Minimal SUGRA parameter space
6.4 Searches for h → b¯b signature
6.5 More complex signatures involving b pairs
6.6 Di-leptonic signatures
6.7 Searches for h → γγ signature
6.8 Full simulation studies
6.8.1 Detector simulation comparison
6.8.2 Reconstruction of h → b¯b signal
6.8.3 Trigger issues
7 Extrapolation of SUSY parameters
7.1 Global fits of LHC measurements
7.2 The SFitter tool
7.3 Statistical approach
7.3.1 Likelihood function
7.3.2 Confidence intervals
7.3.3 Probing new physics
7.4 Markov-chain Monte Carlo methods
7.5 Results for benchmark Point 1
7.6 Results for benchmark Point 2
8 Conclusions
A Calculation of cascade kinematics
Acknowledgments
Résumé
Bibliography