## Posts Tagged ‘LHC’

### The Higgs Particle For Semi-Dummies

May 16, 2010

With the LHC ramping up, the hype about the Higgs particle is at frenzy levels, but what exactly is it?  You may have heard some frivolous descriptions, for example “the God particle”, but at the very least you might have heard that it gives the elementary particles their mass.  But how does it do such a thing?  In this post I will try to explain what it’s all about in a way which you wont need a physics degree to understand.  Of course I’m not an expert so for any corrections or suggestions please comment!

The Standard  Model

The best description we have of all the elementary particles is called the Standard Model.  It’s very technical, and not intuitive at all (to me at least).  It’s pretty amazing that it describes the way Nature works with such amazing accuracy.  The model is a gauge theory, and before we get to the Higgs, we have to understand what that means

Wikipedia tells us:

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations.

Let’s define each of the important terms:

• field theory – A theory which assigns a value to every point in space and time.
• Lagrangian – Defines the complete dynamics of a physical system.  The langrangian may have terms which depend on mass, for example, a standard lagrangian for a classical object is: $\mathcal{L}= Kinetic Energy - Potential Energy = 1/2mv^2 - mgh$, where m is the mass of the object.
• invariant – independent of.  If Y is invariant under X, then the value of Y will not change if X changes (and all other inputs remain the same).
• continuous group – we’ve discussed Groups on this blog before because they’re so fundamental, but if you need a refresher it’s an algebraic object which has a set of elements and is closed under a product operation.  Each element has an inverse so that the product of an element with its inverse is the identity element.  A continuous group is one where the product is a continuous function: infinite and without any discontinuities.
• local transformations – these are the elements of the above group.  These types of groups are called Lie groups.  an example is rotations of the sphere.

Putting it all together, a gauge theory defines a system where the global state of the system is unchanged with specific local changes.

Symmetry Group

These “local changes” are the elements of symmetry groups.  An example of such a group is the Circle group, which is  the rotations of a circle about its axis, called U(1).  The rotations have a product  (adding angles together), an identity, (the rotation of 0 degrees), and an inverse, (a rotation in the opposite direction).  It’s called U(1) because this group of transformations is represented by the set of all unitary matrices of dimension 1.

The Standard Model has 3 such symmetry groups: U(1), SU(2), and SU(3).  They represent 3 of the fundamental forces of nature: electromagnetic, weak nuclear, and strong nuclear respectively.  SU stands for special unitary, and SU(n) is the group of special unitary matrices of n-dimensions.

Somewhat bizarrely, The Standard Model says that the generators of these symmetry groups actually represent particles!  For U(1) there is 1 generator, which is the photon.  SU(2) has 3 generators which are the Z, W+, and W- particles.  SU(3) has 8 generators which are the 8 different gluons.

Symmetry Breaking

The Lagrangians described by these symmetry groups describe the complete dynamics of these quantum systems, however, they do not allow for any of these particles to have mass.  While this is fine for the photon and the gluon, the Z, and W+- must have mass to explain their observed behavior: namely their very limited range.

Incredibly, one can add mass to the Lagrangian for the weak force, if the SU(2) symmetry is broken.  As a first example, let’s see what it means to break U(1) symmetry.  If someone hands you a perfect circle, it is impossible to differentiate any point on the circle from any other.  However, if one breaks the symmetry by marking a point, then all points can be differentiated by describing how far they are from the marked point.

Spontaneously broken symmetry with a Higgs Field

The Higg field breaks the symmetry of SU(2) and gives us mass terms in the Lagrangian.  This  field can be represented by a Mexican Hat Potential, at the very center of which is a local maxima which represents a meta-stable state.  The stable states are in a circle around this local maxima which are the minimums of the potential.

Fields in Quantum Field Theory have an associated particle, and the particle representing this Higgs field is our Higgs boson.

It is the only particle predicted by the Standard Model which has not been discovered, but the LHC hopes to change that.  What’s interesting is that there are other ways to get mass to appear in the lagrangians.  The Higgs field is the simplest and most intuitive, however, it is possible that there is no Higgs particle and that mass is derived in some other manner.  That would certainly shake up the field of physics!

UPDATE (7/22/2012): The LHC found the Higgs and it appears to be just as the Standard Model predicted!