Searching for Higgs in Excel

Simple linear expression

Before the discovery of the Higgs particle was comfirmed, as sumarised (NS, 07-Sep-2013, Instant Expert No.35), a much earlier issue of New Scientist (NS, 30-Mar-2002, p28) gave an excellent top-level summary of the justification for belief that the Higgs particle exists, and a later issue (NS, 03-Mar-2007, p8) gave a summary of the then present state of the hunt. The former explains how the photon, W+, W- and Z particles are all related, but that the Higgs particle interacts asymmetrically with them, so that photons are massless, but the other three are massive. Meanwhile, the latter article includes a graph, showing the contours of a three-dimensional plot of the expected mass of the Higgs boson, mh, against any pairing of masses of top quark, mt, and W boson, mw. Unfortunately, it only explicitly shows two contours (mh=114 GeV and mh=400 GeV).

Here is a somewhat simplistic expression for reading off other values of m_w, given any pair of values of m_t (in the vicinity of 173.34 GeV) and m_h (in the vicinity of 132 GeV, which is the mean of 115 and 150 (NS, 20-Feb-2010, p5)):

In this radiative correction expression for the W boson, K_a (a dimensionless constant of about 0.006) is the gradient of m_w against m_t for constant m_h; K_b (a dimensional constant of about -0.07 GeV) is the gradient of m_w against ln(m_h) for constant m_t.

The calculations on the rest of this page are based on data obtained from:

• Pour la Science, No.240, Oct-1997, p73
• A physorg.com web page, giving more detailed data than the New Scientist, 03-Mar-2007, p8 article
• A fnal.gov web page, giving further data, dated 23-Feb-2012

Specifying a reference point

The expression needs a reference point (K_to, K_ho, K_wo) to fix the position of the plane in its three-dimensional space. Conventionally, two of the coordinates are chosen to be zero, and the remaining one is set as the intersect on the corresponding axis.

Thus, setting K_to and ln(K_ho) to zero, and hence setting K_ho to one (namely, 1GeV), K_wo, as the intersect on the m_w axis, has a value of about 80.83 GeV.

The convenience of choosing one of the points of intersection with the axes as the reference is that two of the constants do not need to appear in the final equation (since subtracting zero, or dividing by unity, is an identity operation).

Although (0,1,80.83) (all in GeV) is the obvious choice of constants, it is not the only possibility. For example, using the reference point (0.0,1e-9,82.28) puts the m_h axis at 1eV rather than 1GeV, but begs the question "why eV, rather than J, or Planck-energy units, or some other unit?"

Using the intercept on the ln(m_h) axis instead would have been tempting, giving the reference point (0,K_ho,0). Unfortunately, K_ho then has a value of around 10⁵¹⁰, and Excel refuses to work with it directly.

Using the intercept on the m_t axis, namely the reference point (K_to,1,0), gives a value of K_to of -13109 GeV (and is little changed for the reference point (K_to,1e-9,0))

There is one further interesting point for this particular function: the one where K_t=K_h=K_w (since, although there are then three constants to handle in the equation, they all have the same value). This leads to the following expression:

An inverse function, to find m_h from given values of m_t and m_w, is therefore:

Relevance post Higgs

Now that the discovery of the Higgs particle is confirmed, the role of this expression can be reversed. The observed masses of the three particles can be used as the input (m_t,m_h,m_w) and the closest fit values can be found for the three constants (K_a,K_b,K_c).

For convenience, here is a short Excel spreadsheet that has been coded up to evaluate the two expressions given earlier.

It should be noted, though, that care should be taken when using the inverse function (m_h in terms of m_t and m_w) because the exponential function makes it very sensitive to small rounding errors in its argument.