The Higgs Boson: what are the odds?

[little_monkey]

Higgs particle could be found by Christmas

http://www.bbc.co.uk/news/science-environment-14731690

The title of that article is slightly misleading. The hope is that by Christmas, the Higgs boson will either be ruled in or ruled out.

If it's ruled out, it will be like a tsunami, an earthquake and a hurricane all hitting your house at the same time. It would mean that most of what was done in the last 40 years is up for grabs. And many Physicists will see their reputation on the block, many among them Nobel prize winners. A lot is at stakes, more than what the general public is aware.

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Poop. I was kind of hoping for one last hurrah for the Tevatron before they shut it down....

Me too, but many working there will be out of a job by the end of the year. So that if the research is no longer done in favor of searching for a job, that is understandable.

[KichigaiNeko]

just want to subscribe to this thread..thanks

[lilphil1989]

I would say the odds are reasonably high for at least one Higgs' boson existing.

Otherwise, massive gauge bosons in a non-Abelian gauge theory make no sense at all. And they certainly exist in nature, the W's and Z bosons associated with the weak interaction have huge masses.

[leo-rcc]

...

*Nods head, pretending to understand, whilst slowly backing towards the door*

Darn, that leaves me alone with these eggheads.

[little_monkey]

I would say the odds are reasonably high for at least one Higgs' boson existing.

Otherwise, massive gauge bosons in a non-Abelian gauge theory make no sense at all. And they certainly exist in nature, the W's and Z bosons associated with the weak interaction have huge masses.

Indeed, just to give a general rundown of the theory, here:

(1) in QM: x → operator
But to satisfy Relativity, in which time is on an equal footing with space, in QFT: x → parameter, and Φ(x) → operator. Now Φ(x), a function of x, is called the “field”.

(2) L = T – V. The Lagrangian plays an important role. From Noether’s theorem, we know that if the Lagragian is invariant under a symmetry, this symmetry points to a conservation law.

Corresponding to L there is a Hamiltonian, H = T + V. The Hamiltonian is known to measure the energy of a system.

(3)In classical mechanics, let v = dx/dt, then L = (1/2) mv2 – V(x). The corresponding Hamiltonian is, H = (1/2) mv2 + V(x). Quantizing this, (ℏ =1),we get the Schroedinger equation: i∂Ψ(x)/∂t =( -(1/2m)∆ 2 + V(x)) Ψ(x).

(4) In Relativity, the energy equation is:

E2= p2c2 + m2c4.

Quantizing this, (c =1) yields the K-G equation:

(1/2) (∂μΦ)(∂μΦ) + (1/2) mΦ2 = 0.

From this, the Lagrangian can be deduced as: L = (1/2) (∂μΦ) 2 – (1/2) mΦ2.


(5) In QFT, the general Lagrangian is:

L = (1/2) (∂μΦ) 2 – V(Φ).

(6) Comparing (5) and (4), if V(Φ) contains any terms with Φ 2, its coefficient is taken to be the mass of the field quanta (particles).

Gauge theory:

From electromagnetism, it was known that Maxwell’s equations were gauge invariant. In QM, gauge invariance of the Lagragian involves three important steps:

(7) the wave function is redefined as Ψ → ΨeiqX
(8) the operator ∂μ → ∂μ + iqAμ
(9) the electromagnetic field Aμ → Aμ - ∂μX

(10) In QED, in equation (5), V(Φ) → - ¼ Fμν Fμν, where Fμν = ∂μAν - ∂νAμ

If you apply, 7,8,9,10 to equation (5), you get the invariance of the Lagrangian under gauge transformation, in which the photon mediates the electromagnetic force. Note that the photon has no mass.

In the weak force, the bosons involved have mass, and one had to figure out how to include a mass term, keeping the Lagrangian gauge invariant.

There is where number (6) comes into play under the notion of SPONTANEOUS SYMMETRY BREAKING.

In another post, I will draw the skeleton of the Higgs Mechanism.

EDIT: if there are corrections, let me know.

[Napoléon]

I have no idea wtf you guys are talking about but I sure would like to.

[little_monkey]

Higgs Mechanism:

Basically, I will only look at U(1) symmetry. Electroweak interactions need a U(1) x SU(2) symmetry, but SU(2) requires 2 by 2 matrices, and the software on this forum is inadequate to deal with matrices. But you can get the flavor just by doing U(1) symmetry and how mass is introduced in the Lagrangian of equation (5).

I will rewrite this equation as:

(11) L = ∂μΦ† ∂μΦ - ¼ Fμν Fμν – V(Φ†Φ).

(12) where V(Φ†Φ) = (m2)/(2φ2) [Φ†Φ - φ2 ] 2

Three important things to note:

(13) The field Φ is now a complex number, denoted by (Φ1, Φ2) or Φ = Φ1 + iΦ2 ( i being the imaginary number, square root of – 1), and Φ† = Φ1 – iΦ2.

(14) the minimum field energy is obtained when Φ†Φ = φ2.

(15) The number of possible vacuum states is infinite. We break this symmetry by requiring that Φ is real, we take the vacuum state to be (φ,0), and expand:

Φ = φ + (1/2½)h

Substituting 7,8,9, 12, and 15 into 11, we get

(17) L = {(∂μ - iqAμ)( φ + (½ ½)h)}{( ∂μ + iqAμ)( φ + (½ ½)h} - ¼ Fμν Fμν - (m2)/(2φ2) [2½φh + ½h2 ]2

After calculating the Lagrangian, we separate it into two parts:

(18) L = Lfree + Lint

where

(19) Lfree = ½∂μh∂μh - m2h2 - ¼ Fμν Fμν + q2φ2AμAμ

All the remaining terms are lumped into Lint, which offer no interest.

So, we can see that by breaking the symmetry, we end up with two massive particles. In equation 19, the second term refers to a scalar particle with mass equal to 2½m, associated with h (the higgs field) and the fourth term, a vector boson with mass 2½qφ, associated with Aμ( the electromagnetic field).

[popeyespappy]

Show off

[little_monkey]

Show off

Got me.

The thing with this stuff is that you can look at it as a stroke of genius or "creative cooking" theory. As long as there is justification -- read: its predictions are confirmed experimentally -- then it's the former. But should the higgs boson be ruled out, then for many detractors, this whole thing about the higgs mechanism giving mass to all the particles will be looked down as cooking theory. Now imagine with Supersymmetry Theory or String Theory, the amount of "creative cooking" in those two theories is just unbelievable, and those theories have extremely dim chances of ever being confirmed experimentally. Yet, many in the physics community go about as if these are the only game in town.

[lilphil1989]

I don't think accusations of "creative cookery" would be justified really. It's clear that a SU(2) x U(1) gauge theory perfectly describes the weak interaction and electromagnetism. Since experiment shows that the W and Z bosons are massive, and direct mass terms in the lagrangian would break the SU(2) invariance, the most parsimonious way to include the masses in the theory is the Higgs' mechanism.

Sure, it might not be the way that nature does things, but until that can be demonstrated experimentally, by Occam's razor the Higgs' mechanism is a better explanation than anything more exotic.