.. title: Typesetting the CKM unitarity relations
.. slug: typesetting-the-ckm-unitarity-relations
.. date: 2008-06-01 12:00:00
.. type: text
.. category: blog

You wouldn't think that typesetting the Cabibbo-Kobayashi-Maskawa (CKM) matrix's
unitarity and normalisation relations would be hard, would you? Well, depending
on how anal retentive you are it's either just a case of whacking in the right
subscript indices or it's a minor voyage of discovery into how LaTeX doesn't
always know best. When it comes to math typesetting I'm quite picky, so you can
safely assume that I consider this to be a nice example of the latter!


A first try
-----------

First, lets try it the naive way. Put this in your LaTeX source, using
:code:`\usepackage{amsmath}` in the document preamble:

.. code-block:: latex

    \begin{subequations}
      \label{eq:CKMUnitarityRelations}
      \begin{equation}
        V_{ud}V_{ub}^{*} + V_{cd}V_{cb}^{*} + V_{td}V_{tb}^{*} &= 0 \qquad (db) \label{eq:CKMRelnDB} \\
        V_{us}V_{ub}^{*} + V_{cs}V_{cb}^{*} + V_{ts}V_{tb}^{*} &= 0 \qquad (ds) \label{eq:CKMRelnDS} \\
        V_{ud}V_{us}^{*} + V_{cd}V_{cs}^{*} + V_{td}V_{ts}^{*} &= 0 \qquad (sb) \label{eq:CKMRelnSB}
      \end{equation}

      \begin{equation}
        V_{ud}V_{td}^{*} + V_{us}V_{ts}^{*} + V_{ub}V_{tb}^{*} &= 0 \qquad (ut) \label{eq:CKMRelnUT} \\
        V_{cd}V_{td}^{*} + V_{cs}V_{ts}^{*} + V_{cb}V_{tb}^{*} &= 0 \qquad (ct) \label{eq:CKMRelnCT} \\
        V_{ud}V_{cd}^{*} + V_{us}V_{cs}^{*} + V_{ub}V_{cb}^{*} &= 0 \qquad (uc) \label{eq:CKMRelnUC}
      \end{equation}
    \end{subequations}

Here I've divided the relations first into those between rows and then between
columns, using the amsmath package to do the subequation labelling, alignment
around the equals sign and so on. Here's the output:

.. image:: /images/ckm1.png

Now I don't know about you, but the unequal (math mode) spacing of the flavour
indices, the variations in subscript height between conjugated and unconjugated
V's and the different widths of the terms look pretty scrappy to me. Let's try
to fix these little visual quirks.


Improved method
---------------

Now a more careful approach, using the maybemath, amsmath and hepnicenames packages. This time, we need to put a bunch
of definitions in the preamble or in a personal style/package/class:

.. code-block:: latex

    \usepackage{amsmath,maybemath,hepnicenames}
    \DeclareRobustCommand{\mymath}[1]{\ensuremath{\maybebmsf{#1}}}
    \DeclareRobustCommand{\CkmElementConj}[2]{\mymath{V_{{#1}{#2}}^{*}}\xspace}
    \DeclareRobustCommand{\CkmElement}[2]{\mymath{V_{{#1}{#2}}^{\phantom{*}}}\xspace}
    \DeclareRobustCommand{\Vud}{\CkmElement{\Pup}{\Pdown}} \DeclareRobustCommand{\Vus}{\CkmElement{\Pup}{\Pstrange}}
    \DeclareRobustCommand{\Vub}{\CkmElement{\Pup}{\Pbottom}} \DeclareRobustCommand{\Vcd}{\CkmElement{\Pcharm}{\Pdown}}
    \DeclareRobustCommand{\Vcs}{\CkmElement{\Pcharm}{\Pstrange}} \DeclareRobustCommand{\Vcb}{\CkmElement{\Pcharm}{\Pbottom}}
    \DeclareRobustCommand{\Vtd}{\CkmElement{\Ptop}{\Pdown}} \DeclareRobustCommand{\Vts}{\CkmElement{\Ptop}{\Pstrange}}
    \DeclareRobustCommand{\Vtb}{\CkmElement{\Ptop}{\Pbottom}} \DeclareRobustCommand{\VudConj}{\CkmElementConj{\Pup}{\Pdown}}
    \DeclareRobustCommand{\VusConj}{\CkmElementConj{\Pup}{\Pstrange}}
    \DeclareRobustCommand{\VubConj}{\CkmElementConj{\Pup}{\Pbottom}}
    \DeclareRobustCommand{\VcdConj}{\CkmElementConj{\Pcharm}{\Pdown}}
    \DeclareRobustCommand{\VcsConj}{\CkmElementConj{\Pcharm}{\Pstrange}}
    \DeclareRobustCommand{\VcbConj}{\CkmElementConj{\Pcharm}{\Pbottom}}
    \DeclareRobustCommand{\VtdConj}{\CkmElementConj{\Ptop}{\Pdown}}
    \DeclareRobustCommand{\VtsConj}{\CkmElementConj{\Ptop}{\Pstrange}}
    \DeclareRobustCommand{\VtbConj}{\CkmElementConj{\Ptop}{\Pbottom}} %% CKM element pairing for unitarity relations
    \newlength{\CKMPairWidth} \settowidth{\CKMPairWidth}{\Vtd\VtbConj}
    \DeclareRobustCommand{\@Vbox}[1]{\makebox[\CKMPairWidth]{#1}}
    \DeclareRobustCommand{\VCkmPair}[2]{\ensuremath{\@Vbox{{#1}{#2}}}}

These definitions are mostly for compactness. The definitions of the CKM
elements and conjugates include a superscript phantom asterisk to make the two
take up the same vertical space, and the :code:`\VCkmPair` macro fixes each term to be
the same width. The flavour indices are implemented using the quark macros from
my hepnicenames package (in the `hepnames` set) to do the particle symbols via
commands like :code:`\Pcharm`.

Now the equations themselves are written as follows,

.. code-block:: latex

    \begin{subequations}
      \label{eq:CKMUnitarityRelations}
      \begin{align}
        \VCkmPair{\Vud}{\VubConj} + \VCkmPair{\Vcd}{\VcbConj} + \VCkmPair{\Vtd}{\VtbConj} &= 0 \qquad (\Pdown\Pbottom) \label{eq:CKMRelnDB} \\
        \VCkmPair{\Vus}{\VubConj} + \VCkmPair{\Vcs}{\VcbConj} + \VCkmPair{\Vts}{\VtbConj} &= 0 \qquad (\Pdown\Pstrange) \label{eq:CKMRelnDS} \\
        \VCkmPair{\Vud}{\VusConj} + \VCkmPair{\Vcd}{\VcsConj} + \VCkmPair{\Vtd}{\VtsConj} &= 0 \qquad (\Pstrange\Pbottom) \label{eq:CKMRelnSB}
      \end{align} %

      \begin{align}
        \VCkmPair{\Vud}{\VtdConj} + \VCkmPair{\Vus}{\VtsConj} + \VCkmPair{\Vub}{\VtbConj} &= 0 \qquad (\Pup\Ptop) \label{eq:CKMRelnUT} \\
        \VCkmPair{\Vcd}{\VtdConj} + \VCkmPair{\Vcs}{\VtsConj} + \VCkmPair{\Vcb}{\VtbConj} &= 0 \qquad (\Pcharm\Ptop) \label{eq:CKMRelnCT} \\
        \VCkmPair{\Vud}{\VcdConj} + \VCkmPair{\Vus}{\VcsConj} + \VCkmPair{\Vub}{\VcbConj} &= 0 \qquad (\Pup\Pcharm) \label{eq:CKMRelnUC}
      \end{align}
    \end{subequations}

Here's the output:

.. image:: /images/ckm2.png

Not bad, eh? And that, as they say, is that.

----------------------

**Postscript:** you may now find that the `subdepth
<https://www.ctan.org/pkg/subdepth>`_ package makes a lot of this redundant by
automatically fixing the subscript depth. That package was written long after
these instructions, and since I don't need to typeset CKM relations anymore I've
not tried it out. It is used by the new version of `hepparticles
<https://www.ctan.org/pkg/hepparticles>`_, though, and works very nicely indeed.