Hypercharge Assignments of Fields

PHY646 - Quantum Field Theory and the Standard Model 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: Hypercharge Assignments of Fields. Charged and Neutral Currents. Fermion 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 quarks 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 chirality projectors. That is, y 0 µ Qi@Q= i = Qi γ γ @µPLQi; (8) 1 with PL = 2 (1 − γ5) and i i iy 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 bosons, 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 electron and neutrino 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 photon, 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 charge 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 quark has electric charge + 3 , and the down quark 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 proton. 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 boson 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 − ip 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 = p (νLγ eL + uLγ dL) ; (17) W 2 µ− 1 µ µ J = p 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 Higgs boson plays a crucial role in giving mass to fermions. 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: me = me L R + me R L: (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 symmetry breaking. Let us look at the term YYukawa = −λf LΦeR + h:c: ! 0 = −λf (νL eL) eR + h:c: pv 2 v = −λf p eLeR + h:c:; (22) 2 where λf is the fermion coupling. After the Higgs field Φ gets a vacuum expectation value, a mass 4 / 6 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 term will be generated: −me(eLeR + eReL), with λe me = p v: (23) 2 Following the similar path, we see that the charged leptons 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 Lσyσ Φ∗ = 0: (26) 2 2 k 2 k 2 k k 2 To get Eq. (26) we have used ! ! T 0 −i 1 R σ2 R = ( 1 2) = −i( 1 2 − 2 1); (27) i 0 2 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 Higgs mechanism, then neutrinos 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 .

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