# Does the Higgs talk to itself?

Article: Indirect probes of the trilinear Higgs Coupling
Authors: Martin Gorbahn, Ulrich Haisch
Reference: 1607.03773

Hello particle munchers,

I’m back to further discuss our good friend the Higgs boson.

After its discovery in 2012, there are many properties of the Higgs boson which have since been established. In addition to its spin-0 nature and measuring its mass with high precision (125 GeV), the existence of its couplings to a number of Standard Model particles has also been established.

What has yet to be established is whether the Higgs can “talk to itself”. More precisely, does the Higgs have self interactions involving multiple Higgs bosons (see diagrams in Figure 1) and how strong are these self interactions? Measuring these interactions gives us direct information on the Higgs scalar potential, which is responsible for not only the electroweak symmetry breaking mechanism leading to the generation of mass for the Standard Model particles, but also has implications for the stability of our universe (see Footnote 1) .

The Higgs potential can be written in a very simple form (just a polynomial in the Higgs field H) as shown in Figure 1. The $H^2$ term represents the Higgs mass while the $H^3$ and $H^4$ terms represent the Higgs self interactions we are interested in. In the Standard Model $\lambda_3$ and $\lambda_4$ have a precise relation reflecting the underlying symmetries of the SM. By measuring interactions involving 3 and 4 Higgs bosons we can determine these parameters and therefore test directly this prediction of the SM. A deviation from from the Standard Model prediction would signal the presence of new physics which, needless to say, would send theorist into a drunk frenzy not seen since…oh wait, never mind :/.

Typically one attempts to measure $\lambda_3$ and $\lambda_4$ using $h \to hh$ and $h\to hhh$ decays respectively (see Footnote 2). The $h\to hhh$ rate is far too small at the LHC to be useful for extracting $\lambda_4$. Thus, most theoretical and experimental studies (see here and references therein) have focused on measuring $\lambda_3$ via $h\to hh$ decays.

Since the $h\to hh$ rate is also small, it requires producing many Higgs bosons so that after a long time and massive amounts of data, enough of them will decay off-shell into pairs of Higgs bosons to be seen at colliders. At LHC energies the production rates are not quite large enough to be able see this process if it conforms to the SM prediction. However, if new physics somehow makes the rate much larger than found in the SM, then perhaps it can be seen at the LHC.

The paper highlighted above is interesting because it proposes a new, but indirect, way to measure $\lambda_3$ at the LHC by looking at gluon fusion Higgs production and $h\to \gamma\gamma$ decays (where $\gamma$ is a photon), which i’ll focus on in todays post. If you recall my earlier post about how we see the Higgs boson in its decays to pairs of photons, you might remember that the interaction between the Higgs and photons is generated through loops of virtual charged particles. At leading one loop order, there can only be charged particles running in the loop and thus no Higgs bosons. This means that at leading order we are not sensitive to $\lambda_3$. However, if we go to next to leading order at two loops one can have Higgs bosons contribute as shown in Figure 2. In this case we see the Higgs boson can enter in the loops and in particular the $\lambda_3$ coupling at the vertex with 3 Higgs bosons indicated by the box.

Since these two loop processes are next to leading order, they are in general very small. However, the high precision with which $h\to \gamma\gamma$ will eventually be measured at the LHC allows for these tiny effects to be potentially probed. What is interesting is that the authors find that these indirect methods are competitive and complementary to $h\to hh$ decays at the LHC. They find it will be possible to eliminate regions of parameter space in beyond the Standard Model theories which can not be ruled out with the more conventional and direct $h\to hh$ decays.

Though the precision is still not great, due to the importance of establishing the precise form of the Higgs potential, it is crucial to have as many ways as possible of constraining its parameters. Since the LHC will just begin to probe the parameters of the Higgs potential before the end of running, a future collider which could produce far larger numbers of Higgs bosons will be crucial in this endeavor of determining if indeed the Higgs talks to itself.

Footnotes

1. I’ll discuss this interesting topic of the vacuum stability of the universe more in a future post, but see here for a great discussion or here for a famous paper about it. Or just ask Stephen Hawking.
2. You might ask how one Higgs boson can decay to multiple Higgs bosons when clearly one Higgs boson has less mass than two or more Higgs bosons. This of course is due to the subtleties of quantum mechanics which leads to the fact that particles can decay `off mass shell’ (more precisely with a 4-momentum squared which is not equal to its physical mass). This means that particles which naively are too light to decay to a particular more massive final state can still do so at the cost of a massive kinematic suppression.

1. Very similar study proposing additional indirect probes of $\lambda_3$