Online optimization. Adaptive sampling design

One of the biggest discoveries in the history of science was that na-ture is not invariant under parity transformations. In layman’s terms this means that some experiments behave differently than their mir-rored analogue. The experiment that discovered the violation of parity symmetry was the Wu experiment. A full description of this experiment, although fascinating, strays from our current subject, so let’s just discuss the final result.

The Wu experiment discovered that the particles mediating the weak force (the W⁺, W⁻, Z bosons) only couple to left-chiral parti-cles. In other words: Only left-chiral particles interact via the weak-force and we will discuss at the end of this section why this means that parity is violated. All particles produced in weak interactions are left-chiral. Neutrinos interact exclusively via the weak force and therefore it is possible that there are no right-chiral neutrinos⁴¹. All

41We will see in a moment that massive, left-chiral particles always get a right-chiral component during propagation.

It is known from experiments that neutrinos have mass and therefore there should be a right-chiral neutrino component. Nevertheless, this right-chiral component does not participate in any known interaction.

other particles can be produced via other interactions and therefore can be right-chiral, too.

Up to this point, we used left-chiral and right-chiral as labels for objects transforming according to different representations of the Lorentz group. Although this seems like something very abstract, we can measure the chirality of particles, because there is a correlation to a more intuitive concept called helicity⁴².

42We will not discuss this here, because the details make no difference for the purpose of the text. The message to take away is that it can be done. A very nice discussion of these matters can be found in Alessandro Bettini. Intro-duction to Elementary Particle Physics.

Cambridge University Press, 2nd edi-tion, 4 2014. ISBN 9781107050402

In fact, most of the time particles do not have a specified chirality, which means they aren’t definitely left-chiral or right-chiral and the

corresponding Dirac spinorΨ has both components. Parity viola-tion was no predicviola-tion of the theory and a total surprise for every physicist. Until the present day, no one knows why nature behaves so strangely. Nevertheless, it’s easy to accommodate this discovery into our framework. We only need something that makes sure we always deal with left-chiral spinors when we describe weak interactions.

Recall that the symbolsχ and ξ denote Weyl spinors (two compo-nent objects),ψ Dirac spinors (four component objects, consisting of two Weyl spinors) andΨ doublets of Dirac spinors

Ψ=

ψ1

ψ2



. (7.99)

Then this "something" is the projection operator P_L:

P_Lψ=P_L

Such a projection operator can be constructed using the matrix⁴³

43Recall the definition of theγμ matri-ces in Eq. 6.13 and don’t let yourself get confused about the missingγ4matrix.

There is an alternative convention that usesγ4instead ofγ0and to avoid inter-ference between those conventions, the matrix here is commonly calledγ5.

γ5=iγ0γ1γ2γ3=

−1 0

0 1



. (7.101)

The matrixγ5is called the chirality operator, because states of pure chirality

are eigenstates ofγ5with eigenvalue−1 and+1 respectively.

The projection operator P_Lis then⁴⁴

44Maybe you wonder why we define PLas so complicated and do not start with the explicit matrix form right away. We do this, because it’s possible to work in a different basis where the matricesγμlook completely different.

(For more information about this have a look at Sec. 8.10). Take note that in the Lagrangian the Dirac spinors appear al-ways in combination with the matrices γμ. We can always add a 1=U⁻¹U, with some arbitrary invertible ma-trix U, between them. For example,

∂μΨγ¯ μΨ=∂μΨ U¯  ⁻¹U

course completely independent of such transformations, but we can use this to simplify computations. The basis we prefer to work with in this text is called Weyl Basis. In other bases the two com-ponents of a Dirac spinor are mixtures ofχLandξR. Nevertheless, the projec-tion operator defined as P_L = ¹⁻₂^γ5, always projects out the left-chiral component, because P^Weyl_L Ψ^Weyl = Ψ^Weyl_L ⇒ P_L^Ψ^ = ¹⁻₂^γ5 Ψ^ and we can define analogously

P_R= ¹+γ5

Now, in order to accommodate for the fact that only left-chiral particles interact via the weak force, we must simply include PL into all terms of the Lagrangian that describe the interaction of W_μ^± and Z_μwith different fields. The corresponding terms were derived in Sec. 7.2, and the final result was Eq. 7.58, which we recite here for convenience:

L =i ¯Ψγ_μ∂^μΨ+Ψγ^¯ _μσ_jW_j^μΨ−¹

4(W_μν)i(W^μν)i. (7.104)

The relevant term is ¯ΨγμσjW_j^μΨ and we simply add PL:

→L =^{i ¯}Ψγ_μ∂^μΨ+Ψγ^¯ _μσ_jW_j^μP_LΨ−¹

4(^Wμν)i(^Wμν)i. (7.105) Here P_Lacts on a doublet and is therefore

P_LΨ=

 P_L 0

0 P_L

 ψ₁ ψ2



=

(ψ₁)_L (ψ2)_L



. (7.106)

One PLis enough to project the left-chiral component out of both doublets ¯Ψ and Ψ. To see this we need three identities:

• (P_L)² = P_L, which is obvious from the explicit matrix form and because every projection operator must have this property⁴⁵.

Pro-45Another defining condition of any projection operator is P_LPR=PRPL=^0, which is here fulfilled as you can check by using the explicit form of P_L,P_Rand γ5.

jecting twice must be the same as projecting one time.

• {γ5,γ_μ} =γ5γ_μ+γ_μγ5 =0, which you can check by brute force computation⁴⁶.

46Or using another identity{γμ,γν} = γμγν+γνγμ= ¹₂ημν, whereημνis the Minkowski metric and the definition of γ5=iγ0γ1γ2γ3.

• (P_L)^† = P_L, becauseγ5is real, as can be seen from the explicit matrix form: γ5=

−1 0

0 1



The second identity simply tells us thatγ₅γ_μ = −γ_μγ₅, i.e. that we can switch the position ofγ5and anyγ_μmatrix, as long as we include a minus sign. This tells us

γ_μP_L =γ_μ1−γ5

2 = ¹+γ5

2 γ_μ=P_Rγ_μ. (7.107) We can now rewrite the relevant term of Eq. 7.105:

Ψγ¯ μσ_jW_j^μP_LΨ=Ψγ^¯ μσ_jW_j^μ(P_L)²Ψ

= Ψ^¯

Ψ^†γ0

γ_μP_L

 

PRγμ

σ_jW_j^μP _LΨ

ΨL

=Ψ^†γ  ₀P_R

PLγ0

γ_μσ_jW_j^μΨ_L

 =

Using P_L^†=PLand(AB)^†=((AB)^T)^=(B^TA^T)^=B^†A^†

(P_LΨ)^†γ₀γ_μσ_jW_j^μΨ_L

= (P _LΨ

=ΨL

)^†γ0γμσ_jW_j^μΨL

=Ψ^¯Lγ_μσ_jW_j^μΨL  (7.108) Now we know how we can describe mathematically that only left-chiral fields interact via the weak force, but why does this mean that parity is violated? To understand this we need to parity

transform this term, because if it isn’t invariant, the physical system

in question is different from its mirror image⁴⁷. Here we need the ⁴⁷This means an experiment, whose outcome depends on this term of the Lagrangian, will find a different outcome if everything in the experiment is arranged mirrored.

parity operator for spinors⁴⁸P_spinorand vectors⁴⁹Pvector. The

trans-48The parity operator for spinors was derived in Sec. 3.7.9. Using the γμmatrices, we can write the parity operator derived there as P_ =γ0 =

49The parity operator for vectors is

simply Pvector= as already mentioned in Eq. 3.132.

formation yields by looking at the explicit form of the matrices. Furthermore, we have P_vectorW⁰=W⁰and PvectorWⁱ= −Wⁱ, which follows from the explicit form of Pvector. We conclude these two minus signs cancel each other and the parity transformed term of the Lagrangian reads:

(P_spinorΨ)^†γ₀γ_μσ_j(PvectorW^μ)jP_L(P_spinorΨ) =Ψγ^¯ _μσ_jW_j^μP_RΨ=Ψγ^¯ _μσ_jW_j^μP_LΨ (7.110) Therefore this term isn’t invariant and parity is violated.

Parity violation has another important implication. Recall that we always write things below each other between two big brackets

if they can transform into each other⁵⁰. For example, we use four- ⁵⁰This is explained in appendix A.

vectors, because their components can transform into each other through rotations or boosts. In this section we learned that only left-chiral particles interact via the weak force and the correct term in the Lagrangian is ¯Ψγ_μσ_jW_j^μP_LΨ = Ψ^¯_Lγ_μσ_jW_j^μΨ_L. In physical terms this term means that the components of the left-chiral doublets, which means the two spin ¹₂ fields(ψ₁)L,(ψ₂)L, can transform into each other through weak-interactions. Right-chiral fields do not interact via the weak force and therefore(ψ₁)R,(ψ2)Raren’t transformed into each other. Therefore writing them below each other between two big brackets makes no sense. In mathematical terms this means that right-chiral fields form SU(2)singlets, i.e. are objects transforming according to the 1 dimensional representation of SU(2), which do not change at all, as explained in Sec. 3.6.3. So let’s summarize:

• Left-chiral fields are written as SU(2)doublets: Ψ_L =

(ψ₁)L

(ψ₂)L

 , because they interact via the weak force and therefore can trans-form into each other. They transtrans-form under the two-dimensional representation of SU(2):

Ψ_L →Ψ^_L=e^iaσ2Ψ_L (7.111)

• Right-chiral fields are described by SU(2)singlets:(ψ1)R,(ψ2)R, because they do not interact via the weak force and therefore can’t transform into each other. Therefore they transform under the one-dimensional representation of SU(2):

(ψ₁)_R→ (ψ₁)^_R=e⁰(ψ₁)_R= (ψ₁)_R

(ψ₂)R→ (ψ₂)^_R=e⁰(ψ₂)R= (ψ₂)R (7.112) Now we move on and try to understand how mass terms for spin

12 particles can be added in the Lagrangian without spoiling any symmetry.