The standard model of particle physics is a theory describing our knowledge of three out of four known fundamental forces: electromagnetic, weak and strong forces. It is formulated in terms of quantum field theory.

Since it was put together in the middle of last century, this theory has withstood a huge number of experimental tests, and has demonstrated prediction power to crazy accuracy.

Despite the enormous success, it is clear that the theory is not the final theory of everything. There are some items missing, where we don't have much clue about:

1. How does gravity fit into the picture?
2. What is the nature of dark matter?
3. Dark energy vs. expansion of the universe
4. Matter-antimatter asymmetry in the observable universe
5. Flavor problem
6. etc.

We hope to discover new physics that can help answer these questions! But for now let's start with the standard model.

There are different types of particles in the standard model: quarks, leptons and force carriers. It is nicely summarized by this picture:

The Higgs field is the only known fundamental spin-0 field.

Gauge invariance requires that everything is massless, but particles in our world has rest mass. Therefore the Higgs mechanism is proposed to reconcile them:

In the high-energy theory, all particles are massless. However the Higgs field has a unique "Mexican hat"-shaped potential, which causes it to spontaneously break symmetry and acquire a non-zero vacuum energy in the ground state. As a result of this symmetry breaking, fermions obtain mass through the Yukawa coupling of the Higgs.

In position space, while particles travel through the Higgs field, they attract some Higgs "cloud" around them and effectively makes the system resilient to external forces, which we call "having non-zero mass".

Range of Higgs cloud is order of 1-10 am, or 10^-17-10^-18 meters

Everyday mass has many origins, and only a small portion is attributed to the Higgs mechanism. For example in a proton, where there are three (valence) quarks and a sea of gluons and virtual quarks. Gluons are massless, and total rest mass of the valence quarks add up to less than 10 MeV/c². The rest (> 99%) is due to strong force that binds things together.

After the discovery of Higgs boson, many fundamental questions remain unanswered.

For example we still don't know the origin of fermion mass and why they take on the values in our world. The understanding of the origin of mass is delegated, from some mysterious mass terms that don't fit well into our gauge theories, to the Yukawa couplings of the Higgs field.

The importance of the discovery is that now we can write down a gauge-invariant self-consistent theory that nicely describe (most of) the world we see. It's a unprecedented triumph of 20^th century physics development.

This particle is one-of-a-kind in the standard model. By studying it closely we can perhaps find some hints or clues towards to answer of origin of mass, and other new types of physics we didn't know about.

The decay of the Higgs depends a lot on the Higgs mass, as can be seen in this figure. When the Higgs particle is heavy, the heavier decay products dominate (for example above threshold of 2 M_Z, 2 M_W and 2 M_t); whereas on the other hand, when the giants don't have a chance to go on-shell, decaying to b-quark pair dominates. Decays to gluons and photons are suppressed since they occur at loop level.

At Higgs mass of 125 GeV/c² things are especially interesting since it's right at the transition region where branching ratio to b-quark starts to drop, and di-bosons are rising. This helps a lot for the experiments, as there are a myriad of decay channels available to probe.

The "golden channel" refers to the decay of Higgs to 4 charged leptons through Z bosons and photons.

Branching ratio of Higgs to neutral di-bosons (Z, γ) is small, and it is further hit by the small branching fraction of Z and γ to charged leptons. We expect the cross section to be very small in this channel for M_H = 125 GeV/c².

Despite the small cross section, this channel is one of the best channels for Higgs studies. There are many reasons:

1. It's a closed final state, ie., all particles in the final state are visible. This is to be contrasted with the fully leptonic decay from W⁺W^- final state. Even though cross section is much higher, there are neutrinos escaping the detector. A closed final state allows us to actually see a sharp peak in the M_4l spectrum, something that is indisputable.
2. Experimentally leptons are easy to identify and has good resolution (for electrons and muons)
3. The only irreducible background is from the standard model ZZ/Zγ/γγ production, the background rate is manageable (it turned out to be roughly the same magnitude to the Higgs signal).
4. The rich final state of 4 leptons allows us to have a good handle on the properties of Higgs, or HVV vertex.

Due to the above reason this channel is usually called the "Golden Channel"

To understand what we will see in the golden channel, let's start with the diagrams. To the lowest order, there is one type of diagram involving Higgs, and two types of diagrams without Higgs.

We now know that the Higgs-like boson is roughly at 125 GeV/c² from the LHC experiments. At this mass the Higgs resonance is extremely narrow, roughly 6 MeV/c², or a relative width of 5x10^-5!

In standard model, during Higgs decay, among different pairings of vector bosons (ZZ, Zγ, γγ), we expect that contribution from ZZ to be the largest. This is mainly due to order of the diagrams. Photons don't have mass, and Higgs doesn't couple directly to photons. Any diagrams involving photons occur at at least one loop level.

At tree level, there are two types of diagrams which are conventionally called the "s-channel" and the "t/u-channel" diagrams. There is no narrow Higgs resonance here so we expect to see a broad spectrum in the 4-lepton mass.

1. s-channel: we have a Z resonance. Width of Z is 2-3% so we expect to see a broader peak in the 4-lepton mass (compared to Higgs).

2. t/u-channel: two vector bosons couple directly to the quarks. Expect to see "turn-on" effects as a function of mass once it passes a new kinematic threshold.

At tree level, there are two types of diagrams which are conventionally called the "s-channel" and the "t/u-channel" diagrams. There is no narrow Higgs resonance here so we expect to see a broad spectrum in the 4-lepton mass.

1. s-channel: we have a Z resonance. Width of Z is 2-3% so we expect to see a broader peak in the 4-lepton mass (compared to Higgs).

2. t/u-channel: two vector bosons couple directly to the quarks. Expect to see "turn-on" effects as a function of mass once it passes a new kinematic threshold.

And so we arrive at the total M_4l spectrum. Can you identify all the features and relate to diagrams?

There are different ways new physics can enter the picture, assuming there is still a Higgs resonance and 4-lepton final state:

1. Inside a loop between Higgs and vector bosons. In this case we can integrate out the additional degrees of freedom, and treat the effect from new physics as an "effective" vertex of HVV.

2. Different kind of intermediate vector boson (for example Z')

3. As a heavy fermion with HFl coupling, where the heavy fermion radiates a Z/γ* which subsequently decays into two leptons.

We can parameterize the HVV vertex using a progressive approach: starting from the simplest term and include things progressively. There are three allowed Lorentz structures for each combination of intermediate vector bosons, corresponding to the following vertex:

Each coefficient can in principle be momentum dependent. As a start, we can treat them as constants, and later on when this first approximation is shown to be insufficient, we come back and add in the next term in the expansion.

This vertex can be mapped from (for example) the following Lagrangian:

Since Higgs has spin-0, other than the 4-momentum of Higgs there is no additional information carried. Therefore we can safely talk about the production side and the decay side to the lowest order.

Production side: dependent on empirical parton density functions. Related to total Higgs production cross section and QCD predictions. By nature there is relatively large uncertainty associated with physics observables from the production side.

Decay side: does not depend on PDFs. Starting from the center-of-mass frame of a spin-0 colorless particle allows us to precisely describe the decay distributions with very little uncertainty (it's the realm of QED!).

One of the big advantages of having 4 leptons in the final state is the wealth of observables at our disposal.

The figure shows one popular parameterization in the decay part.

1. Invariant mass of 4 leptons.
2. Invariant masses of lepton pairs.
3. Decay angles of the lepton pairs in the di-lepton rest frame.
4. Opening angle between two Vll plane.
5. Opening angle between one of the Vll plane and the ppVV plane.
6. The φ angle between one of the Vll plane and the ppVV plane.

The 8 decay observables, together with production side variables of Higgs 3-momentum and a φ angle between Higgs direction and one of the Vll planes, adds up to the full set of 12 observables. It matches the number of
degrees of freedom from the four leptons (3 each)

In standard model, Higgs couple primarily to two Z bosons, but couplings to virtual photons are also present. Decay properties are very different if we look at hypothetical models where only one only of coupling (ZZ, Zγ, γγ) is present. As an example di-lepton mass distributions are plotted here for Higgs with mass 125 GeV/c². M₁ is the larger of the two di-lepton masses, and M₂ is the smaller.

What causes the general shape of blue, orange and green?

Standard model Higgs boson has no spin. Z bosons and photons have spin 1. There are 3 ways to match up the spins with projections along the Z direction.

Under parity operation, directions flip sign while spin remains the same. The three states map to each other as

in-in ↔ out-out

zero-zero ↔ zero-zero

and we can form three parity eigenstates from these, which incidentally maps to the three terms we have in the Lagrangian:

(in-in - out-out) → A₃^VV

(in-in + out-out) & zero-zero → A₁^ZZ & A₂^VV

Interference is important! Since interference is multiplied by one order of each coupling involved, while the square terms are multiplied by two order of couplings, sometimes interference is the first thing we detect.

We also need interference to observe CP violation in this channel. Some diagrams (or combination of diagrams) may be CP-odd, but we have to square them to calculate the differential cross section, which makes it CP-even. Only the interference terms can preserve CP-oddness in the final differential cross section.

• (CP even diagram)² = CP even differential cross section → No CP violation
• (CP odd diagram)² = CP even differential cross section → No CP violation
• (CP even diagram)¹ x (CP odd diagram)¹ = CP odd differential cross section → CP violation

One of the first things we can learn about the property of Higgs is through hypothesis testing. In this approach a pair of hypotheses are considered, and we measure the ratio of "distance" of data to the hypotheses. This ratio tells us if data prefers one over another.

Technically the "distance" is usually measured with likelihoods. We find some observables to describe data, and evaluate the probability of certain value(s) appearing in data assuming the hypothesis is true.

Observables used in building these probabilities don't have to be fundamental like lepton momenta or direction; it can be as complex as needed. People usually form "discriminant" from theoretically motivated quantities that helps telling apart the two hypotheses. For example one can look at the generator-level differential cross section as the probability, and traditionally we see good sensitivity to tell models apart with this kind of discriminant.

We can take it one step further.

Observe that the total differential cross section (in some variables) is the sum of square terms multiplied by respective coupling squared, plus the interference shape multiplied by one order of each coupling. Therefore by generating the shape separately for different terms, we can very efficiently produce the distribution of discriminant from the combination of individual components.

This provides us a tool to directly measure relative size of one pair of couplings with good sensitivity, since the discriminant is designed to best tell apart the two extreme hypothesis.

However this poses a big problem when we want to measure multiple couplings at once. In order to maintain good sensitivity, dimensions of the discriminant grow quickly. Given that the discriminant are generally very complex objects, it is close to impossible to write down an analytic form of the shape, and therefore one need to generate events to fill a template. With large dimensions, this becomes intractable.

So we take it one step even further.

First we dismantle the discriminant, and return to the most basic variables to use: masses and angles.

This has a great advantage that generator-level distribution is calculated directly, and we can write down an analytic form of it.

The generator-level analytic expression is then convoluted with detector response through numerical methods to obtain likelihood that certain event configuration appears.

This is equivalent to generating infinitely many MC events and fill a big 8D template, and then read off the value in the bin of interest.

Since there are only 8 decay variables, we retain the full information of all decay kinematics, and there is no need to worry about optimality.

For the 8D full fits, the complication, of course, is that the calculation is not trivial to do.

Ideally since there are 12 degrees of freedom, we would want to do a 12 D convolution integral. However CMS has a very good resolution on lepton direction, a very good approximation is to assume that leptons do not change direction from generator level to detector level. This leaves us with 4 degrees of freedom to integrate over, which is computationally doable.

The integration is done via numerical integration methods that are not based on random sampling. The reason is that with random sampling uncertainty on the acquired value scales as 1/√N. If we want precision down to O(0.1%) for all possible terms (square terms and interference terms), things quickly become intractable. In the official CMS analysis, we have to evaluate 2000 numbers per event for parameter extraction.

One important feature of the interpretation of the event is that we have to make an objective choice. It is necessary to distinguish between different stages of reconstruction:

1. Generator level: an event is sampled from some parameterization of the truth distribution. There is a "true" value of all the parameters (for example masses and angles).
2. Detector level: the event is smeared according to lepton response functions. The value of parameters is somewhere close to the "true value" where this event is smeared from.
3. Interpreted level: in order to reconstruct the event, we choose a consistent way to obtain the observed values of parameters. Note that the observed values of parameters may not coincide with the "true" value this event is originated from. In fact, there are always multiple possible true values that give rise to the exact same event.

This is one of the many details to get right for the 8D full fit to work.

CMS is extremely good in reconstructing leptons at energy scales relevant to Higgs analyses. Resolution is reasonable and efficiency remains high throughout the detector. (Maybe give some benchmark numbers?)

Triggering, however, suffers a bit more. Since we didn't know what the mass of Higgs boson was, there is no dedicated trigger to target one specific mass. All we have is semi-general purpose triggers with ever-increasing threshold as beam intensity gets higher.

The tightest trigger we use for the analysis of 2012 data is the di-lepton triggers with thresholds of 18 GeV/c and 8 GeV/c.

Offline reconstruction cuts are then motivated by the trigger we use: at least 20 GeV/c for the hardest lepton, and 10 GeV/c for the second. Reject events with interpreted "first Z" mass below 40 GeV/c²

An additional cut on the second Z mass of 12 GeV/c² is also applied to allow us to go as low as possible while avoiding the Υ resonances.

The reconstruction interpretation from CMS is to look through all opposite-sign same-flavor lepton pairs, and mark the one closest to Z mass (~91.1876 GeV/c²) and call it the first Z.

There are in principle three types of pairings. Other than the one chosen with the above criteria ("CMS"), there is another one with opposite-charge pairing ("opposite"), and one with same-sign pairing ("same"). The plot shows percent of events selected from standard model H→ZZ→4l.

The current CMS result selects the majority events.

Here we show fit results as a function of dataset sizes. Black line indicate true value, where in this case it's zero. Each vertical strip is normalized to one. We see that at high number of events things converge to the right value.

Expected sensitivity is shown here, measured in terms of effective sigma.

To understand whether we have sensitivity to Hγγ couplings, first let's enumerate differences between the two:

1. Shape is very different, especially the M₁ and M₂ distributions.
2. Phase space for contributions involving photons is greatly boosted.
3. Different pairings of leptons can pick up different sensitivity.