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 ντ L
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 , ντ R : 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 T a − ig 0 YL B / a T a − ig 0 YQ B
L = iLi ∂/ − ig W / Li + iQi ∂/ − ig W / Qi
+ieiR ∂/ − ig 0 Ye B
/ eiR + iν iR ∂/ − ig 0 Yν B
i
/ νR
i
+iuiR ∂/ − ig 0 Yu B
/ uiR + idR ∂/ − ig 0 Yd B / diR .
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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,
Qi ∂/ Qi = Q†i γ 0 γ µ ∂µ PL Qi , (8)
with PL = 21 (1 − γ5 ) and
uiR ∂/ uiR = ui† 0 µ i
R γ γ PR uR , (9)
with PR = 21 (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
neutral gauge bosons, Wµ3 and Bµ , and then changing to the (Aµ , Zµ ) basis using the definitions we
encountered earlier
Zµ ≡ cos θW Wµ3 − sin θW Bµ ,
Aµ ≡ sin θW Wµ3 + cos θW Bµ ,
we get the following expression for the electron and neutrino couplings
1 3 0 1 3
L = eiL
− gW / + g YL B i
/ eL + ν L i / νLi
/ + gYL B
gW
2 2
/ i + g 0 Yν ν iR Bν
+ g 0 Ye eiR Be / R i
R
1 i / i 1 i / i i / i i / i
= e − + YL eL AeL + + YL ν L AνL + Ye eR AeR + Yν ν R AνR + Z terms. (10)
2 2
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
− + YL = Q = −1 =⇒ YL = − . (11)
2 2
It also gives
Q = Ye = −1. (12)
Now plugging in YL = − 21 in the term 21 + YL ν iL Aν
/ Li 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.
Similarly, using that the up quark has electric charge + 32 , and the down quark has electric charge
− 31 we need YQ = 16 , Yu = 23 , and Yd = − 31 . (See Table. 1.) It turns out that given the particle
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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) − − − −
U (1)Y − 12 −1 0 1
6
2
3 − 13 1
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 + eR iD
L = L 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
Dµ = ∂µ − i √ Wµ+ T + + Wµ− T − − i Zµ T 3 − sin2 θW Q − ieAµ Q.
(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−
+g Wµ+ JW + Wµ− JW + Zµ0 JZµ + eAµ JEM
µ
, (16)
where the charged currents are
µ+ 1
JW = √ (ν L γ µ eL + uL γ µ dL ) , (17)
2
µ− 1
√ eL γ µ νL + dL γ µ uL ,
JW = (18)
2
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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
and the neutral currents are
1 h µ 1 1
JZµ νL + eL γ − + sin θW eL + eR γ µ sin2 θW eR
µ 2
= ν Lγ
cos θW 2 2
µ 1 2 2 µ 2 2
+ uL γ − sin θW uL + uR γ − sin θW uR
2 3 3
i
µ 1 1 2 µ 1 2
+ dL γ − + sin θW dL + dR γ sin θW dR ,
2 3 3
µ µ µ 2 µ 1
JEM = eγ (−1) e + uγ + u + dγ − d. (19)
3 3
In the above we have used
1
T ± = (σ 1 ± iσ 2 ) = 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 ψψ = me ψ L ψR + me ψ R ψL . (21)
However, in electroweak theory, a mass term like eL eR 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 (ν L eL ) eR + h.c.
√v
2
v
= −λf √ eL eR + h.c., (22)
2
where λf is the fermion coupling. After the Higgs field Φ gets a vacuum expectation value, a mass
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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
term will be generated: −me (eL eR + eR eL ), with
λe
me = √ 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 Lσ2 Φ∗ . To see
that Lσ2 Φ∗ is SU (2) invariant, we note that, since Φ and L are fundamentals under SU (2), we have
the infinitesimal transformations
1
δΦ = iθk σk Φ, (24)
2
1
δL = iθk σk L, (25)
2
giving
1 1
δ(Lσ2 Φ∗ ) = − iθk Lσ2 σk∗ Φ∗ − iθk Lσk† σ2 Φ∗ = 0. (26)
2 2
To get Eq. (26) we have used
! !
T 0 −i ψ1
ψR σ2 ψ R = (ψ1 ψ2 ) = −i(ψ1 ψ2 − ψ2 ψ1 ), (27)
i 0 ψ2
σjT σ2 + σ2 σj = 0, (28)
and σ2∗ = −σ2 . Thus we define
e ≡ iσ2 Φ∗ ,
Φ (29)
which transforms in the fundamental representation of SU (2) and has hypercharge − 12 . Then we
can write −λf L Φν
e R 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
λν . 10−12 . (30)
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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
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 Lepton, 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 top quark 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
m2h
λ= , (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 weak hypercharge 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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