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Home , Hypercharge, Weak hypercharge

PHY646 - and the

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 49

Monday, April 20, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Topic: Assignments of Fields. Charged and Neutral Currents. Mass Terms.

Hypercharge Assignments of Fields

Let us denote the hypercharges of left-handed fields by YL and YQ. They are the same for each generation. Let us also denote the hypercharges of the right-handed fields by Ye, Yν, Yu and Yd. They are also the same for each generation. That is, for left-handed fields ! ! ! νeL νµL ντL L1 = ,L2 = ,L3 = : with same hypercharge YL, (1) eL µL τL ! ! ! uL cL tL Q1 = ,Q2 = ,Q3 = : with same hypercharge YQ, (2) dL sL bL and for the right-handed fields

eR, µR, τR : with same hypercharge Ye, (3)

νeR, νµR, ντR : with same hypercharge Yν, (4)

uR, cR, tR : with same hypercharge Yu, (5)

dR, sR, bR : with same hypercharge Yd. (6)

Using the index i = 1, 2, 3 to denote the generations we can write the Lagrangian for the gauge interactions in the following form

a a 0  a a 0  L = iLi ∂/ − igW/ T − ig YLB/ Li + iQi ∂/ − igW/ T − ig YQB/ Qi i 0  i i 0  i +ieR ∂/ − ig YeB/ eR + iνR ∂/ − ig YνB/ νR i 0  i i 0  i +iuR ∂/ − ig YuB/ uR + idR ∂/ − ig YdB/ dR. (7) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

We note that the also have charges under SU(3)QCD, and they are not shown in the above expression. Since we will almost always be performing computations in the broken phase, where left- and right-handed spinors combine into a single Dirac representation, it is generally easier to use the Dirac-spinor notation from the start, where L and R indicate implicit projectors. That is,

† 0 µ Qi∂Q/ i = Qi γ γ ∂µPLQi, (8)

1 with PL = 2 (1 − γ5) and i i i† 0 µ i uR∂u/ R = uRγ γ PRuR, (9)

1 with PR = 2 (1 + γ5). Let us find out the values for the hypercharges for various fields. Since the hypercharge is associated with the U(1) group, they could be arbitrary real numbers. To find out what the actual hypercharges are in the Standard Model, we can use the known electric charges. First isolating the 3 neutral gauge , Wµ and Bµ, and then changing to the (Aµ,Zµ) basis using the definitions we encountered earlier

3 Zµ ≡ cos θW Wµ − sin θW Bµ, 3 Aµ ≡ sin θW Wµ + cos θW Bµ, we get the following expression for the and couplings

 1  1  L = ei − gW/ 3 + g0Y B/ ei + νi gW/ 3 + gY B/ νi L 2 L L L 2 L L 0 i i 0 i i + g YeeRBe/ R + g YννRBν/ R  1  1   = e − + Y ei Ae/ i + + Y νi Aν/ i + Y ei Ae/ i + Y νi Aν/ i + Z terms. (10) 2 L L L 2 L L L e R R ν R R

Since the electric charges are the coefficients of the coupling to the , we can read off from this equation the relationship between hypercharges and electric charges. Using the convention that the electron is defined to have electric Q = −1, we see that

 1  1 − + Y = Q = −1 =⇒ Y = − . (11) 2 L L 2

It also gives

Q = Ye = −1. (12)

1 1  i / i Now plugging in YL = − 2 in the term 2 + YL νLAνL in the above Lagrangian we see that νL must be neutral, which is in agreement with Nature. For νR to be neutral, we also need Yν = 0. 2 Similarly, using that the up has + 3 , and the has electric charge 1 1 2 1 − 3 we need YQ = 6 , Yu = 3 , and Yd = − 3 . (See Table. 1.) It turns out that given the particle

2 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 content of the Standard Model, the hypercharges must satisfy certain constraints. In particular the constraint

YL + 3YQ = 0. (13)

This forces the electric charge of the electron to be exactly three times the electric charge of the down quark and exactly opposite to the charge of the .

    νL uL Field L = eR νR Q = uR dR H eL dL

SU(3) − − −    − SU(2)  − −  − −  1 1 2 1 1 U(1)Y − 2 −1 0 6 3 − 3 2

Table 1: Hypercharges and group representations of the Standard Model fields.  indicates that the field transforms in the fundamental representation, and − indicates that a field is unchanged.

Charged and Neutral Currents

To work out the physical consequences of the fermion-vector couplings, we should write the Lagrangian (suppressing the generation index i)

     L = L iD/ L + eR iD/ eR + QL iD/ QL + uR iD/ uR + dR iD/ dR, (14) in terms of the vector-boson mass eigenstates, using the form of the covariant derivative

g + + − − g 3 2  Dµ = ∂µ − i√ Wµ T + Wµ T − i Zµ T − sin θW Q − ieAµQ. (15) 2 cos θW

Then Eq. (14) takes the form

     L = L i∂/ L + eR i∂/ eR + QL i∂/ QL + uR i∂/ uR + dR i∂/ dR  + µ+ − µ− 0 µ µ +g Wµ JW + Wµ JW + ZµJZ + eAµJEM, (16) where the charged currents are

µ+ 1 µ µ J = √ (νLγ eL + uLγ dL) , (17) W 2 µ− 1 µ µ  J = √ eLγ νL + dLγ uL , (18) W 2

3 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 and the neutral currents are     µ 1 h µ 1 µ 1 2 µ 2  JZ = νLγ νL + eLγ − + sin θW eL + eRγ sin θW eR cos θW 2 2 1 2   2  + u γµ − sin2 θ u + u γµ − sin2 θ u L 2 3 W L R 3 W R  1 1  1  i + d γµ − + sin2 θ d + d γµ sin2 θ d , L 2 3 W L R 3 W R  2  1 J µ = eγµ (−1) e + uγµ + u + dγµ − d. (19) EM 3 3

In the above we have used 1 T ± = (σ1 ± iσ2) = σ± (20) 2 µ to simplify the W boson currents. Notice that JEM associated with the photon field is indeed the standard electromagnetic current.

Fermion Mass Terms

Let us discuss how fermion mass terms are generated in the electroweak theory. As anticipated, the plays a crucial role in giving mass to . Before introducing the Higgs boson, we do not really have a left- and a right-handed electron, but rather two separate unrelated fields that happen to have the same electric charge. That is, fields eL and eR with electric charge Q = −1. In QED, left- and right-handed fermions are connected by a Dirac mass term:

meψψ = meψLψR + meψRψL. (21)

However, in electroweak theory, a mass term like eLeR explicitly breaks the SU(2) invariance, and thus forbidden. This is the place where the Higgs boson comes to the rescue. To write down the electron mass terms, we can use the Higgs doublet; then the masses appear only after electroweak breaking. Let us look at the term

YYukawa = −λf LΦeR + h.c. ! 0 = −λf (νL eL) eR + h.c. √v 2 v = −λf √ eLeR + h.c., (22) 2 where λf is the fermion coupling. After the Higgs field Φ gets a , a mass

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term will be generated: −me(eLeR + eReL), with

λe me = √ v. (23) 2

Following the similar path, we see that the charged and the down-type quarks (d, s, b) will get masses, and no additional breaking of SU(2) is required. Since λe is a renormalizable coupling, it must be treated as an input to the theory. Thus the GWS theory allows the electron to be very light, but it cannot explain why the electron is so light compared to the charged force mediators, ± Wµ . ∗ To give masses to the remaining fermions, we can use the SU(2) invariant term Lσ2Φ . To see ∗ that Lσ2Φ is SU(2) invariant, we note that, since Φ and L are fundamentals under SU(2), we have the infinitesimal transformations

1 δΦ = iθ σ Φ, (24) 2 k k 1 δL = iθ σ L, (25) 2 k k giving 1 1 δ(Lσ Φ∗) = − iθ Lσ σ∗Φ∗ − iθ Lσ†σ Φ∗ = 0. (26) 2 2 k 2 k 2 k k 2 To get Eq. (26) we have used

! ! T 0 −i ψ1 ψR σ2 ψR = (ψ1 ψ2) = −i(ψ1ψ2 − ψ2ψ1), (27) i 0 ψ2

T σj σ2 + σ2σj = 0, (28)

∗ and σ2 = −σ2. Thus we define ∗ Φe ≡ iσ2Φ , (29)

1 which transforms in the fundamental representation of SU(2) and has hypercharge − 2 . Then we can write −λf LΦeνR as a term that gives a mass to the neutrino (or the up-type quarks).

Neutrino Masses in the Standard Model

It is sometimes said that the Standard Model does not allow for the neutrino to have mass. This is not really true. As seen above, if the up-type quark can acquire mass through the , then can as well. However, given that the observational constraints on the upper limit on the sum of neutrino masses is less than an electron-volt, this puts an upper limit on any of the neutrino coupling constants of −12 λν . 10 . (30)

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The Yukawa coupling terms are given in Table. 2. The key point is that the coupling for neutrinos is six orders of magnitude smaller than any other.

Generation Charged , l− Up-Type Quark Down-Type Quark 1 3 × 10−6 1 × 10−5 3 × 10−5 2 6 × 10−4 7 × 10−3 5 × 10−4 3 1 × 10−2 1 2 × 10−2

Table 2: Approximate value of the coupling constant λf for each of the charged fermions. (See PDG for the latest values.)

Stability of Electroweak Vacuum

We see that all the coupling constants are small, with the tantalizing exception of the top, which is, within experimental errors, 1. The theory of renormalization tells us that the effective coupling terms in the Lagrangian (including values of λf ) are generally energy dependent. Just as the Higgs field gives the mass, the top quark (and others) produce corrections to the Higgs potential. At very high top quark mass, the Higgs vacuum becomes unstable, eventually decaying to a lower vacuum state. As the situation stands, the Higgs potential appears to be metastable, but this suggests that there are no hidden Standard Model particles at yet-higher masses just waiting to be discovered. Indeed, it is well known that top quark quantum corrections tend to drive the quartic Higgs coupling λ, which in the Standard Model is related to the Higgs mass by the tree-level expression m2 λ = h , (31) 2v where v is the Higgs field vacuum expectation value, to negative values which render the electroweak vacuum unstable [3].

We also note that the electroweak bosons have YW = 0. So unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.

References

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

[2] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).

[3] S. Alekhin, A. Djouadi and S. Moch, “The top quark and Higgs boson masses and the stability of the electroweak vacuum,” Phys. Lett. B 716, 214-219 (2012) doi:10.1016/j.physletb.2012.08.024 [arXiv:1207.0980 [hep-ph]].

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