PHY646 - Quantum Theory and the

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 47

Wednesday, April 15, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Topic: The Weak .

The

The

The history of the weak can be traced back to the beta decay process. The energetics of decay indicates that there must be an unseen and electrically neutral particle. The conservation of in the neutron decay process requires that it must be a . In 1930 Pauli postulated the existence of this particle and in 1932 Fermi named it as the (“the little neutral one” in Italian). The neutron decay process is

− n → p + e + νe, (1) where νe is the anti-neutrino. The astonishingly long neutron decay time (about 10 minutes) demands that the mediating interaction be very, very weak. We now know that the indeed are an almost omnipresent indicator of any weak process. They are part of a family of known as . In Table. 1 we show the family: it contains the trio corresponding to the charged particles: the electron, the , and the particle, and their respective neutrino cousins. The anti-leptons have the same and opposite to that of their leptonic counterparts.

The Fermi Model of the Weak Interaction

All fermions participate in the weak interaction. But many weak interaction processes have a strong or electromagnetic pathway, which dwarfs and thus masks the weak interaction contribution. repel one another weakly, just as they do electrically, but unless the impact parameter is PHY646 - and the Standard Model Even Term 2020

− l νl ml (MeV) − e νe 0.511 − µ νµ 105.7 − τ ντ 1776.8

Table 1: The three generations (or flavors) of leptons, with the corresponding mass of the charged leptons. We do not yet know the of the neutrinos but the existence of mass (as well as their mass differences) is constrained by . on the scale of an atomic nucleus, the contribution of the weak interaction is effectively zero for any realistic system. However, processes involving neutrinos have no electromagnetic pathway, as is the case for neutron decay. While the ultimate description of the weak interaction will involve a mediator, as a first attempt, Fermi in 1933 proposed an interaction of the form

µ   LFermi int = 2GF ψpγ ψn ψeγµψν , (2) where the subscript for each spinor refers to the particle type, and GF is known as the Fermi constant. We know that this Lagrangian is incorrect in a number of ways. and are composite particles; a more accurate calculation would involve a down decaying into an . Another effect that is not incorporated in this Lagrangian is that the weak interaction violates . Also, Fermi theory does not incorporate a mediator particle. The value of the Fermi constant is approximately

−2 GF = (292.8 GeV) . (3)

(Note that this has a negative mass dimension and thus this theory is not renormalizable.) Against all these inadequacies, this model can compute the decay rate of a station- ary neutron to a moderate accuracy: It gives a neutron lifetime of about 1300 s, which is very close to the true neutron decay time of 882 s. Thus the 4-Fermi theory, which is the low- limit of a massive vector (we will see the details of this theory later), completely characterizes the most familiar effect of the weak : beta decay. At high energy, there is no weak force, per se, only an electroweak force, which is spontaneously broken down to the electric and weak at low energy. There are many aspects of electroweak physics that only become apparent at high energy, such as the existence of the weak force mediators: the W and Z .

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The Weak Force Mediators

There must be a mediator for the weak interaction. Indeed, there three: the W + and the W − (which are of one another) and the neutral Z0 (which, like a , is its own anti-particle). The masses of the weak mediators are huge. Experimentally measured masses are:

- MW = 80.385 ± 0.015 GeV,

- MZ = 91.1876 ± 0.0021 GeV.

For interactions significantly less energetic than the mediator masses, the virtual mediators have a very short range 1 ' 2.5 × 10−18 m, (4) MW well confined to the interior of an atomic nucleus. Explaining the mechanism for giving mass to these weak mediators turned out to be one of the major physics discoveries of our time. It resulted in the 2013 for Englert and Brout, and Higgs.

The Fermionic Doublet

Under weak interactions, particles can be sorted into doublets. Let us consider the under SU(2). We begin with a doublet of bi-spinors

! ψ Ψ = u . (5) ψd

Conceptually, these two components are as similar to one another as a spin-up electron is to a spin-down. Within the context of the weak interaction, the particle doublet might be ! ! ν u Ψ = e or , (6) e− d though we will focus on the lepton doublet for the time being. We may also define an adjoint

Ψ = (νe e). (7)

Just as we will use e to represent the bi-spinor of an electron, e will be the adjoint of an electron, not anti-particle. Under the assumption that all fermions are massless (which is not true), the free-field doublet Lagrangian may be written as µ L = iΨγ ∂µΨ. (8)

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To the fermionic doublet let us apply a global SU(2) gauge transformation of the form

a a Ψ → e−igα T Ψ, (9) a a Ψ → Ψeigα T , (10) where g is a parameter, which would serve as the weak coupling constant later, αa are parameters a 1 a and T = 2 σ are generators of the gauge transformation. We note that the above transformations must leave

ΨΨ = νeνe + ee (11) invariant. The statement that Lagrangians of Ψ are invariant under SU(2) transformations means that as far as the weak interaction is concerned, the two particles in the doublet are interchangeable. That is, turning all electrons to neutrinos and vice versa will not change the weak interaction calculation. The states 1 |eei; |νeνei; √ (|νeei + |eνei) (12) 2 should all be energetically identical under weak interaction. distinguishes very strongly between the two particles, as one (the electron) has charge and the other (the neutrino) does not. However, the strong force will ignore both neutrinos and electrons equally. We find that, in this notation for gauge invariant quantities, the L (specifically electron number Le here) is conserved. Anti-particles have a lepton number of −1. We have

- Electron, neutrino: Le = 1

- , anti-neutrino: Le = −1

The relation of νe to e is the same as the relation of spin up to spin down, and accordingly, we can introduce a quantity called the weak , normally labelled as T3. We have

1 T = + (for the upper component of doublet), (13) 3 2 1 T = − (for the lower component of doublet). (14) 3 2

Helicity

In 1956 Chien-Shiung Wu demonstrated that the weak interaction has handedness. In experimental weak decay reactions all neutrinos are created so that they are left handed, with anti-neutrinos produced as right handed. We say that “neutrinos are left handed” but in fact it is nearly impossible to directly measure the spin of a neutrino. We have to infer it from a neutrino’s partners in a reaction. Since lepton number is conserved, electrons and other charged leptons are often produced in weak interactions as well, and they are non-relativistic.

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For a spin-up fermionic particle propagating in the +z-direction we have

q m  E+p    0    u+(p) ∝ q  . (15)  E+p   m  0

In Weyl basis, the upper two components correspond to a left-handed helicity, while the bottom two correspond to a right-handed one. For massless particles, there is no distinction between the helicity (the dot product of the spin and the momentum) and the handedness. If a massless neutrino is left handed in one Lorentz frame, it will be left handed in all Lorentz frames. Let us look at the weak channel decay of a , which is a spinless composite particle made of a quark and an anti-quark. For π−, which is composed of ud, we have the decay channels

− − π → e + νe, (16) − − π → µ + νµ. (17)

Since are spin-0, the spins of the outgoing particles need to cancel each other. If the electron is spin-up, then the anti-neutrino must be spin-down, but with opposite momenta. Thus, we would expect that both should be left-handed or both right-handed. See Fig. 1.

Figure 1: Since pions are spin-0, the spins of the outgoing particles need to cancel each other. If the electron is spin-up, then the anti-neutrino must be spin-down, but with opposite momenta.

Due to the reasons we do not know yet, the theory of weak interaction presents itself as a chiral theory. That is, the the weak force must make the distinction between left- and right-handedness or particles, and thus parity by itself is not a symmetry of the weak force. While constructing a Lagrangian for the weak interaction, we need to redesign the interaction terms in the Lagrangian to accommodate the handedness of the weak interaction. Fortunately, we have a tool for distinguishing between left-handed and right-handed spinors: the γ5. We can project out the left- and right-handed components via 1 ψ = (1 ± γ5)ψ. (18) L,R 2 Thus, if a Lagrangian is to interact with only left-handed particles, we need to replace currents

5 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 appropriately. For instance, the left-handed current can take the form

1 J + = gν γµ(1 − γ5)e (19) µ 2 e with everything else remaining the same.

Thus the symmetry of the weak force is not SU(2), but is is SU(2)L. A symmetry acting only on left-handed particles and doing nothing to right-handed ones. Note that SU(2)L does not quite describe the weak interactions by itself, rather the combination SU(2)L and U(1) will describe the electromagnetic and the weak interactions. It is very interesting to know that our seems to distinguish between left and right.

Leptons and

In the 1960s Glashow, Weinberg and Salam united electromagnetism and the weak interaction into a single gauge theory: the electroweak theory. This theory (also known as the GWS model) can predict a remarkable range of phenomena, including the existence of a fundamental scalar - the Higgs . The electroweak model is predicated on the idea that leptons and quarks are each group into doublets of the form ! ! ν u Ψ = e ;Ψ = . (20) Lepton e Quark d We have focused on leptons so far, but to get a sense of the power of the electroweak unification theory, we need to consider the similarities and differences between these two doublets. The main difference - and one outside of GWS theory - is that quarks participate in the , while leptons do not. 2 The electric charges on the particles also differ. The up quark has a charge of + 3 , while the neutrino is neutral. However, there is an important similarity, one which will drop out of the theory naturally, namely, the difference between the charges of the doublets are the same. That is,

Qu − Qd = Qν − Qe = +1. (21)

The Standard Model has three generations of SU(2) doublet pairs of quarks and leptons. Let us label them as ! ! ! ! ! ! ν ν ν u c t Li = eL , µL , τL ,Qi = L , L , L . (22) eL µL τL dL sL bL

Here, i = 1, 2, 3 runs over the generations. These all transform as left-handed Weyl spinors under Lorentz group.

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We index the right-handed fermions by the first-generation label

i eR = {eR, µR, τR} (23) i νR = {νeR, νµR, ντR} (24) i uR = {uR, cR, tR} (25) i dR = {dR, sR, bR} (26)

These all happen to be SU(2) singlets, so they are uncharged under the weak interaction. They transform as right-handed Weyl spinors under the Lorentz group. It is worth remarking that the right-handed neutrinos have not yet been observed in Nature, but we include them here in case they do exist.

The Group SU(2)L × U(1)Y

Electroweak unification is based on the symmetry breaking of the gauge group

SU(2)L × U(1)Y → U(1)EM . (27)

The high-energy U(1) symmetry is called the , denoted by U(1)Y . It is not to be confused with the low-energy U(1) associated with the electromagnetism, denoted by U(1)EM . As we will see, the massless particle we know as the photon is a linear combination of the hypercharge and one of the generators of SU(2). In the Standard Model, SU(2)L × 1 U(1)Y is broken by the vev of a complex doublet H with hypercharge Y = 2 called the Higgs multiplet. We can see that the quantity ΨΨ is invariant under both SU(2) and U(1). We suppose that only left-handed fermions participate in the SU(2) symmetry, and thus we want to focus on an

SU(2)L × U(1) symmetry transformation. As a result, we will write down a generalized form of possible transformations for left-handed and right-handed fields separately

0 0 a a ΨL → exp −ig YL α exp (−igα T )ΨL , (28)

0 where we have assumed the existence of two different “charges,” g and g , and YL corresponds to the hypercharge of the associated left-handed particles. Similarly, right-handed particles are invariant under

0 0 ΨR → exp −ig YR α ΨR , (29) only the U(1) transformation, and no SU(2). By convention, we define the of the left-handed lepton doublet as

1 Y = − . (30) L 2

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1 Also, we have the T3 = + 2 for the upper component (neutrinos, up quarks), and 1 T3 = − 2 for the lower (electrons, down quarks). The relation between (in electron units) and the weak isospin and hypercharge is

Q = Y + T3. (31)

Each generation of leptons and quarks has a left-handed doublet and two apparently independent right-handed singlets (with no weak isospin). The hypercharge for each of the singlets, however, is selected so that the electric charge is the same as for the left-handed particle. This step is rather straightforward, since right-handed fermions have a weak isospin of zero. If we demand that the electric charge of left- and right-handed fermions be identical, we quickly get

YR = −1 for right-handed electrons, (32)

YR = 0 for right-handed neutrinos. (33)

The right-handed electrons couple to the weak field, unlike right-handed neutrinos (which may not exist in our Universe). In Table 2 we show the handedness, weak isospin, electric charge of the first generation of particles in the Standard Model.

Particle Handedness T3 YQ 1 1 νL Left 2 − 2 0

νR Right 0 0 0 1 1 eL Left − 2 − 2 −1

eR Right 0 −1 −1 1 1 2 uL Left 2 6 3 2 2 uR Right 0 3 3 1 1 1 dL Left − 2 6 − 3 1 1 dR Right 0 − 3 − 3

Table 2: The first generation of particles in the Standard Model. We have Q = Y + T3. Note that the right-handed neutrino may not exist in our Universe.

Quarks also take part in weak interactions. The only real distinction between quarks and leptons within the context of the is that quarks have a different weak hypercharge: 1 YL = 6 for the left-handed doublet. It is a fact of nature, and not yet well understood, that the charges of quarks and electrons are simple ratios of one another. The strong interaction requires that quarks are grouped in three, and thus it is interesting that any group of three quarks will produce and integer charge. We do not have a fundamental basis for saying why some particles get the weak hypercharge that they do.

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References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

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