Heavy quarks within electroweak multiplet

0.22 MB·

by  J. Besprosvany

Heavy quarks within electroweak multiplet
7
1 J. Besprosvany and R. Romero
0
2
Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,
n
a Apartado Postal 20-364, Ciudad de M´exico 01000, M´exico
J

5

]
h Abstract
p
-
p
e Standard-modelfieldsandtheirassociatedelectroweakLagrangianareequiv-
h
[
alently expressed in a shared spin basis. The scalar-vector terms are written

1
v
1 with scalar-operator components acting on quark-doublet elements, and shown
9
1
to be parametrization-invariant. Such terms, and the t- and b-quark Yukawa
1
0
. terms are linked by the identification of the common mass-generating Higgs
1
0
7 operating upon the other fields, after acquiring a vacuum expectation value v.
1
:
v Thus, the customary vector masses are related to the fermions’, fixing the t-
i
X
quark mass m with the relation m2+m2 = v2/2, either for maximal hierarchy,
r t t b
a
or given the b-quark mass m , implying m 173.9 GeV, for v = 246 GeV.
b t
≃
An interpretation follows that electroweak bosons and heavy quarks belong in
a multiplet.
Keywords: Top quark, mass, multiplet, Lagrangian, spin, electroweak
1
1 Introduction
A basis or representation choice can be useful, even essential, in the description of
a system and its dynamics. It may reveal otherwise-hidden connections between
its components, and provide a simpler framework to understand physical proper-
ties. Such a basis may describe effective degrees of freedom[1] accounting for col-
lective interactions, allowing for a simpler near free-particle description, in a first
approximation. For example, nucleon and associated boson interactive configurations
give a tractable account of nuclear-motion modes[2]. Within condensed matter and
low-temperature superconducting systems, a residual attractive interaction related
to phonons forms Cooper pairs[3], which propagate freely, and lead to frictionless
currents. In an application of this theory to quantum field theory and elementary
particles[4], afour-fermioninteraction produces fermionandcomposite-boson masses,
linking their values. The quark model[5] conceives mesons and baryons in terms of
constituent (dressed) quarks.
The standard model (SM) predicts mass values for the W and Z bosons[6] that
carry the short-range electroweak interaction, in terms of electroweak parameters,
through the Higgs mechanism[7, 8]. However, fermion masses have remained arbi-
trary, as they arise from independent Lagrangian terms. The similar order of magni-
tude of the measured masses[9] of the W, Z, the recently discovered scalar excitation,
associated with the Higgs[10], and the top quark (with the bottom quark’s the next
highest), suggests connections among them, and thus, a common energy scale. More-
over, fermions occupy the spin-1/2 and fundamental representations of the Lorentz
and scalar groups, respectively, as vector bosons belong to the adjoint representation
of each group,1,2 which implies bosons can be constructed in terms of fermions, and
hints at composite structures and/or a common origin.
A previously proposed SM extension[11], based on a shared extended spin space,
reproduces these SM-field features: SM fields are replicated, as the matrix structure
accommodates a fundamental-adjoint representation composite structure. This space
contains a (3+1)-dimensional subspace and one beyond 3+1, linked, respectively, to
Lorentz and scalar degrees of freedom[12]. At each dimension, a finite number of
Lorentz-invariant partitions are generated with specific symmetries and representa-
tions, reproducing particular SM features, where the cases with dimension 5+1[13],
7+1[14], and 9+1[15] were studied.
Complementarily to focusing on such an extension, in this paper, SM heavy-
fermion (F), vector (V), and scalar (S) fields are equivalently reexpressed in terms of
the obtained common basis[11] for both Lorentz and electroweak degrees of freedom,
1As the Higgs occupies the SU (2) fundamental representation.
L
2For the Abelian hypercharge group U(1) , gauge invariance ensures boson-fermion quantum-
Y
number additivity.
2
in turn, recasting their Lagrangian components = + + , where the
FV SV SF
L L L L
aim is to derive SM connections: the identification of the scalar operator within the
and vertices links univocally its defining parameters, and hence the fields’
SF SV
L L
mass values. Indeed, such universal electroweakly-invariant terms lead, under the
Higgs mechanism, to a scalar whose lowest-energy condensate state pervades space,
and generates particle masses. Within the spin basis, this mechanism is similarly
represented; as these fields shape vectors on a matrix space with a single associated
scalar operator acting upon the others, their mass-generation property relates their
coefficients. We work in the classical framework afforded by the Lagrangian, and at
tree-level, but also rely on a quantum-mechanical interpretation.
The paper is organized thus: Section 2 chooses one among two vector bases within
. Section 3 deals with the equivalence for combinations of the scalars and
FV SV
L L
their conjugates, with universal couplings to vectors, shown explicitly in Supplement
2; similarly for their spin-base representation, in which these two scalar components
induce a projection onto a flavor doublet. A comparison follows to Yukawa terms
within , with a link to t and b masses, done in Section 4. In Section 5, we draw
SF
L
conclusions. In the Supplements, we also show the Lagrangian equivalence for all
terms in the two bases, with explicit expressions.
2 Fermion-vector Lagrangian: chiral basis in spin
space
Concentrating on the heaviest fermions, the SM two-quark3 electroweak interaction
Lagrangian[6] is4
1 1
= q¯ (x)[i∂ + gτaWa(x)+ g B (x)]γµq (x)+
LFV L µ 2 µ 6 ′ µ L
2 1
t¯ (x)[i∂ + g B (x)]γµt (x)+¯b (x)[i∂ g B (x)]γµb (x), (1)
R µ ′ µ R R µ ′ µ R
3 − 3
t (x)
where the spin-1/2 fields consist of q (x) = L , a left-handed hypercharge
L (cid:18) b (x) (cid:19)
L
Y = 1/3 SU(2) -doublet, and t (x), b (x), right-handed Y = 4/3, 2/3 singlets,
L R R
−
ψ1 (x)
respectively; each term contains two polarizations as, e. g., t (x) = tL ;
L (cid:18) ψ2 (x) (cid:19)
tL
ψα (x) are wave functions5 for quarks q = t,b, with spin components α = 1,2, and
qh
3A single generation is used, and CKM mixing is neglected; Eq. 1 describes the electroweak
interaction for one quark color, and a sum is assumed over each such term.
4
We use units with ¯h=c=1, and metric g =(1, 1, 1, 1)throughout.
µν
5For simplicity, spin and scalar representations are a−ssum−ed−that give the states’ form.
3
chirality h = L,R; Wa(x), a = 1,2,3, and B (x), are associated gauge-group weak
µ µ
and hypercharge vector bosons, with coupling constants, g, g , respectively; τa are
′
the Pauli matrices representing the SU(2) generators.
L
An extended (7+1)-dimensional [d] Clifford algebra comprises a sufficiently large
space to describe heavy SM particles[12, 14], with the 4-d Lorentz symmetry main-
tained, and spin-component generators 3Bσ , where σ = i[γ ,γ ], and µ,ν =
2 µν µν 2 µ ν
0,...,3; additional scalar generators use γ ,...,γ , producing the baryon-number op-
5 8
erator B = 1(1 iγ γ ), which conforms a spin-space projection partition, and gives
6 − 5 6
quarks 1/3 ( 1/3 for antiparticles,) and bosons 0.
−
In general, a Hermitian operator Op within this space characterizes a state Ψ with
theeigenvaluerule[Op,Ψ] = λΨ,forrealλ,and, underaunitarytransformation,Ψ
→
UΨU , given the ket-bra matrix structure[12], with the kets interpreted as conjugate
†
states. This defition is also consistent with the action of a derivative operator on a
Hilbert space: [−→∂ ,Ψ] = [−←∂−,Ψ] = [Ψ,←∂−]. The direct product trΨ†bΨa is consistent
associativity-wise with the operator rule, as tr[Op,Ψb]†Ψa = trΨ†b[Op†,Ψa]. Fields
are usually assumed to exist on a Cartesian basis; for example, a vector field has
components A (x) = g νA (x); alternatively, in the spin basis, they are expressed
µ µ ν
as A (x)γ γµ.
µ 0
Other scalar-symmetry generators include the hypercharge Y = 1 (1 iγ γ )[1+
o 6 − 5 6
i3(1+γ˜ )γ γ ], with γ˜ = iγ γ γ γ , the weak SU(2) terms
2 5 7 8 5 − 0 1 2 3 L
i
I1 = (1 γ˜ )(1 iγ γ )γ7,
5 5 6
8 − −
i
I2 = (1 γ˜ )(1 iγ γ )γ8, (2)
5 5 6
8 − −
i
I3 = (1 γ˜ )(1 iγ γ )γ γ ,
5 5 6 7 8
8 − −
and flavor generators[14]; as required, [Ii,Ij] = iǫ Ik, [Ii,Y ] = [B,Y ] = [B,Ii] =
ijk o o
[3Bσ ,Y ] = [3Bσ ,Ii] = 0.
µν o µν
The(7+1)-dspaceallowsforadescriptionofquarkfieldsΨ (x) = ψα (x)Tα+
qL α tL L
ψbαL(x)BLα, ΨtR(x) = αψtαR(x)TRα, ΨbR(x) = αψbαR(x)BRα, with hypPercharges 1/3,
4/3, 2/3, respectivePly, and spinor componentPs exemplified in Table 1, as the com-
−
plete list is found in Ref. [14]; the quantum numbers λ are obtained from the oper-
ator structure [Op,Ψ] = λΨ for the weak component I3, hypercharge Y (or charge
o
Q = I3 + 1Y ,) and spin-polarization 3iBγ1γ2 operators.
2 o 2
The SM Lagrangian in Eq. 1 can be equivalently written6 in this basis: as
FV
L
6The commutator is omitted as the operator acts trivially on one side.
4
derived in Ref. [12], and examined in Ref. [16]
1
LFV = tr{Ψ†qL(x)[i∂µ +gIaWµa(x)+ 2g′YoBµ(x)]γ0γµΨqL(x)+
1 1
Ψ†tR(x)[i∂µ + 2g′YoBµ(x)]γ0γµΨtR(x)+Ψ†bR(x)[i∂µ + 2g′YoBµ(x)]γ0γµΨbR(x)}Pf, (3)
while gauge and Lorentz symmetries can be checked with the above transformation
rule, or given the equivalence to the traditional formulation. A projection operator
P that connects the two expressions[16] can be omitted by finding phases for Ψ,
f
which translates into finding an adequate γ basis. The trace coefficient is usually 1,
µ
as the field normalization factor accounts for reducible representations. A complete
proof of the equivalence is given in Supplement 1.
hypercharge 1/3 left-handed doublet I3 Q 3iBγ1γ2
2
T1 1 (1 γ˜ )(γ5 iγ6)(γ7 +iγ8)(γ0 +γ3) 1/2 2/3 1/2
L = 16 − 5 −
(cid:18) B1 (cid:19) (cid:18) 1 (1 γ˜ )(γ5 iγ6)(1 iγ7γ8)(γ0 +γ3) (cid:19) 1/2 1/3 1/2
L 16 − 5 − − − −
(a)
I = 0 right-handed singlet Y Q 3iBγ1γ2
3 2
T1 = 1 (1+γ˜ )(γ5 iγ6)(γ7 +iγ8)γ0(γ0 +γ3) 4/3 2/3 1/2
R 16 5 −
B1 = 1 (1+γ˜ )(γ5 iγ6)(1 iγ7γ8)γ0(γ0 +γ3) 2/3 1/3 1/2
R 16 5 − − − −
(b)
Table 1: (a) Quantum numbers of massless left-handed quark weak isospin doublet,
and (b) right-handed singlets, with momentum along ˆz. The spin component along
±
zˆ, i3Bγ1γ2, is used.
2
The W-fermion vertex in , Eq. 3, contains the matrix element F Wi F ,
LFV h ′| oµ| i
where the W contribution
Wi = gγ γ Ii (4)
oµ 0 µ
describes the SU(2) inherently chiral action on fermion states F , F , as it carries
L ′
| i | i
the projection L = 1(1 γ˜ ), predicted by the spin basis[14]; it is thus the natural
5 2 − 5
choice. For example, this property is absent for W i = gγ γ I i, where I i are the
o′µ 0 µ ′ ′
SU(2) generators without L ; although an equivalent interaction term results within
L 5
this space, it requires the inclusion of L within the vertex; worse, [Y ,I 1,2] = 0.
5 op ′
6
5
3 Scalar-vector Lagrangian: extended charge-conjugate
symmetry
In the SM, the Higgs particle is present[6] in the SU(2) U(1) gauge-invariant in-
L Y
×
teracting Lagrangian-density component = H (x)F (x)F(x)H(x), with F (x) =
SV † † µ
L
i∂µ + 21gτ · W¯(x) + 21g′Bµ(x), W¯(x) = (Wµ1(x),Wµ2(x),Wµ3(x)), and the Y = 1
η (x)+iη (x)
complex-doublet scalar H(x) = 1 1 2 , composed of two charged
√2 (cid:18) η (x)+iη (x) (cid:19)
3 4
(upper), and two neutral (lower) fields. can be equivalently written (with
SV
B (x) B (x)) in terms of the orthogonaLl Y = 1 combination H˜(x) = iσ H (x),
µ µ 2 ∗
→ − −
which uses an antiunitary transformation expressing charge-conjugation invari-
C
ance (in addition to the CP symmetry in the electroweak sector, and approximate
SU(2) SU(2) symmetry[17]; a Hilbert space is assumed;) this is also a consequence
L R
×
of the SU(2) property that the conjugate representation is obtained from a similar-
ity transformation, which ensures independence of the doublet choice. Supplement 2
shows that = tr[F H¯ (x)] FµH¯ (x), where H¯ (x) = (χ H(x),χ H˜(x))
is a 4 4 mLatSrVix, χ , χ′µarχetχcbompl†ex′, anχdtχbχ 2+ χ 2= 1,χtw,χibth F H¯ t (x) =b(i∂ +
21gτ·Wׯ(x))H¯χtχb(xt)+gb′H¯χtχb(x)Bµ(x)τ3,w| ht|ich|isbd|iagonalinH(′xµ),Hχ˜tχ(bx),andheµnce
does not mix them. Moreover is a sum of weighted positive-definite terms, mean-
SV
L
ingonlythecombination χ 2+ χ 2 results. Thisgeneralizestheexpression[18,19]for
t b
SV in terms of H¯ 1 1 (x|). |Wit|h t|he U(1) overall phase, a three-parameter subspace
L √2√2
ofthenorm-conserving constraint χ 2+ χ 2= 1isgenerated. Weassociatethisisom-
t b
etry with the invariance unde|r |: σ| |F H¯ (x)σ = F H¯ (x), with
LSV C 2K ′µ χtχb 2K ′µ −χ∗tχ∗b K
the complex conjugate operator; is also invariant under the transformation
SV 3
defined as H¯ (x) H¯ (x)τ L, together with the combination≪τ .
χtχb → χtχb 3 C 3
Further extension can be made for the scalars in the spin basis by attaching the
γ˜ operator, which leads to four doublet components that emerge naturally in the
5
7+1 spin basis, with the Higgs potential not altered under different definitions (chiral
ones or not.) Thus Table 2 presents two of these scalar elements (with two additional
as their conjugates), with coordinate dependence
1 1
φ (x) = [η (x)+iη (x)]φ+ + [η (x)+iη (x)]φ0
1 √2 1 2 1 √2 3 4 1
1 1
φ (x) = [η (x)+iη (x)]φ+ + [η (x)+iη (x)]φ0, (5)
2 √2 1 2 2 √2 3 4 2
and whose quantum numbers associate them to the Higgs doublet. These are unique
within the (7+1)-d space[14]. Although new scalar fields are introduced in principle,
here we concentrate on the SM-equivalent projections. Given the SM Higgs conjugate
representation H˜(x) the scalar components are interpreted through the assignments
6
0 baryon-number scalar I Y Q 3iBγ1γ2
3 2
φ+ 1 (1 iγ γ )(γ7 +iγ8)γ0 1/2 1
1 = 8 − 5 6 1 0
(cid:18) φ0 (cid:19) (cid:18) 1 (1 iγ γ )(1+iγ γ γ˜ )γ0 (cid:19) 1/2 0
1 8 − 5 6 7 8 5 −
φ+ 1 (1 iγ γ )(γ7 +iγ8)γ˜ γ0 1/2 1
2 = 8 − 5 6 5 1 0
(cid:18) φ0 (cid:19) (cid:18) i (1 iγ γ )(1+iγ γ γ˜ )γ7γ8γ0 (cid:19) 1/2 0
2 8 − 5 6 7 8 5 −
Table 2: Scalar Higgs-like pairs
(see Table 2),
H(x) φ (x) φ (x)
→ 1 − 2
H˜ (x) φ (x)+φ (x). (6)
† → 1 2
This leads to the equivalent expressions
SV = tr [F′′(x),Haf(x)]† [F′′(x),Haf(x)] sym (7)
L { ± ±}
1
= 2tr{[F′′(x),Haf(x)+2[I3,H†af(x)]]†±[F′′(x),Haf(x)+2[I3,H†af(x)]]±}sym,
(8)
where we introduced H (x) = aφ (x) + fφ (x), and F (x) = [i∂ + gWi(x)Ii +
af 1 2 ′′ µ µ
1g B (x)Y ]γ γµ; thesubindexsymmeansonlysymmetricγ γ componentsaretaken,
2 ′ µ o 0 µ ν
to avoid the Pauli components; and the index means the commutator and the anti-
±
commutator should be used for the temporal and spatial γ components, respectively.
µ
I3 adds the phase needed for H†af(x) to SU(2)L-transform as Haf(x). The equality
for implies that it accommodates SM parity-conserving representations in terms
SV
L
of chiral scalars. The complex parameters a, f, are constrained by the normalization
rule a 2+ f 2= 1. These properties for are shown explicitly in Supplement 2.
SV
| | | | L
The spin representation can be connected with that of H¯ (x) with the expres-
χtχb
sion H (x) = 1 (χ H (x) + χ H (x)), where H (x) = φ (x) + φ (x), H (x) =
af √2 t t b b t 1 2 b
φ (x) φ (x), with φ defined in Eq. S26, and this parameterization applies the
1 − 2 i
unitary transformation χ = 1 (a+f), χ = 1 (a f).
t √2 b √2 −
Under the Higgs mechanism, the SM scalars acquire[7, 8] a vacuum expectation
0
value v, and only the neutral field η (x) survives: η (x) = v, H(x) = v ,
3 h 3 i h i √2 (cid:18) 1 (cid:19)
while the charged and imaginary components are absorbed into vector bosons, as
seen explicitly in the unitary gauge. Idem in the spin basis, as can be proved by the
Lagrangian equivalence or directly; then,
v
H (x) = H = (χ H0 +χ H0), (9)
h af i n 2 t t b b
7
where the normalized Higgs operator H is defined, with the same 0,+ component
n
conventions as for the φi, implying, as trH0†iH0j = 2δij, i,j = t,b,
hH†af(x)Haf(x)i = (|a|2+|f|2)v2/2 = (|χt|2+|χb|2)v2/2 = v2/2. (10)
The vector-Higgs vertex in determines the vector-boson masses, and within
SV
L
the spin basis, the trace is taken consistently with H . Thus, the mass component
n
= tr([H ,gWm(x)Im+g′Y B (x)] [H ,gWm(x)Im+g′Y B (x)]+ H ,(gWk(x)Ik+
LSVm n 0 2 o 0 † n 0 2 o 0 { n i
g′Y B (x))γ γi H ,(gWl(x)Il+g′Y B (x))γ γj ) is produced. Fortheneutral mas-
2 o i 0 }†{ n j 2 o j 0 }
sivevectorboson,onederivesthenormalizedZ (x) = ( gW3(x)+g B (x))/ g2+g 2,
µ − µ ′ µ ′
and massless photon A (x) = (g W3(x) + gB (x))/ g2 +g 2, giving, e. pg., the 0-
µ ′ µ µ ′
component p
1 † 1
= tr H ,W3(x)gI3 +B (x) g Y H ,W3(x)gI3 +B (x) g Y (11)
LSZm0 (cid:20) n 0 0 2 ′ o(cid:21) (cid:20) n 0 0 2 ′ o(cid:21)
= Z2(x) 1 tr H ,g2I3 1g 2Y † H ,g2I3 1g 2Y = 1Z2(x)m2,
0 g2 +g 2 (cid:20) n − 2 ′ o(cid:21) (cid:20) n − 2 ′ o(cid:21) 2 0 Z
′
where m = v g2 +g 2/2, m = 0. Similarly, for , the Wi basis in Eq. 4
Z ′ A LSWm oµ
emerges, and definpes the masses of the charged boson fields W (x) = 1 (W1(x)
µ± √2 µ ∓
iW2(x)). Thus, thecharged-vectorbosoncomponent = Wi(x)Wj(x)tr[H ,Wi ]
µ LSWm0 0 0 n o0 †
[H ,Wj ] = m2 W+(x)W (x), i,j = 1,2 contains m2 = tr[H ,W+] [H ,W+] =
n o0 W 0 0− W n o0 † n o0
v2g2/4, with W = 1 gγ γ I , I = I1 iI2. This assignment is unique as this is
o±µ √2 0 µ ± ± ±
the only way to maintain not only the vertex condition (gauge invariance,) but also
normalization (above.) When written in terms of H = H + H , interpreted as a
n n†
fermion Hamiltonian, m2 = tr[H,W+] [H,W+] and the other part is not affected,
W o0 † o0
as [H ,W+] = 0,
n† oµ
4 Scalar-fermion Lagrangian: heavy-quark doublet’s
mass constraint
The Yukawa fermion-scalar interaction can be similarly parameterized in the Clifford
basis
√2
−LSF = tr v [mtΨ†tR(x)H†t(x)ΨqL(x)+mbΨ†qL(x)H†b(x)ΨbR(x)]+{hc}.(12)
8
where m and m are the top and bottom masses, respectively. We note that the
t b
Higgs scalar components have the correct chiral action over fermions: under the
projection operatorsL , andR = 1(1+γ˜ ), e. g., R H (x)L = H (x), L H (x)R =
5 5 2 5 5 t 5 t 5 b 5
H (x), L H (x)R = 0, R H (x)L = 0. For Eq. 12, the underlying mass operator
b 5 t 5 5 b 5
is H (x) = √2(m H (x) + m H (x)), giving, under the Higgs mechanism, H =
m v t t b b m
H (x) = m H + m H . Examples of quark massive basis states are summarized
m t t b b
h i
on Table 3, for both u and d-type quarks, with their quantum numbers. Only one
polarization and one flavor are shown, as a more thorough treatment of the fermion-
flavor states are given elsewhere[14].
This results in, e. g.,
HhT1 = m T1 , HhTc1 = m Tc1,
m M t M m M − t M
HhB1 = m B1 , HhBc1 = m Bc1, (13)
m M b M m M − b M
where Hh = H + H , and tc1, bc1 correspond to negative-energy solution states
m m m† M M
(and similarly for opposite spin components) and Eq. 13 justifies the m and m
t b
massive quarks Hh Q 3iBγ1γ2
m 2
T1 = 1 (T1 +T1) m 2/3 1/2
M √2 L R t
B1 = 1 (B1 B1) m 1/3 1/2
M √2 L − R b −
Tc1 = 1 (T1 T1) m 2/3 1/2
M √2 L − R − t
Bc1 = 1 (B1 +B1) m 1/3 1/2
M √2 L R − b −
Table 3: Massive quark eigenstates of Hh
m
mass interpretation.
Under the assumption of a single mass-producingfield operator, we match a repa-
rameterized H in Eq. 11 that gives the Z (x) mass, to the fermion-mass term H , in
n µ m
Eq. 13, resulting in √2H =H ; a multiplet structure is suggested. In other words,
n m
the operator identification derives from their mass eigenvalues, expressed schemat-
ically as Z √2H Z 2= m2 and t H + H t = m , and the proportionality
|h | n| i| Z h | m m† | i t
constant is derived accordingly. In this association, the simple real-field Z (x) nature
µ
justifies its use (similarly for each Wi(x)), as opposed to the complex W (x). Thus,
µ µ±
the vacuum expectation value reproduces the parameterization in Eq. 9, and iden-
tifies χ , χ as Yukawa parameters: χ = m / v , χ = m / v . The same argument
t b t t √2 b b √2
can be made using the second scalar form in Eq. 8, with the appropiate redefinition
of the Yukawa constants. This results in
−LSF = tr√2[Ψ†tR(x)H†af(x)ΨqL(x)+Ψ†qL(x)HafΨbR(x)]+{hc}. (14)
9
Using Eq. 10, we obtain the relation for the t, b quark masses
( a 2+ f 2)v2/2 = m 2+ m 2= v2/2. (15)
t b
| | | | | | | |
The commutator arrangement in Eq. 11 is used in the above comparison; as it
is set on the demand of a normalized scalar, the argument strengthens on the use
of the same Z operator acting on fermions in Eq. 3. The coefficient matching in
derives from the underlying freedom of choice in , and, in turn, from the
SF SV
L L
underlying three-parameter τ - symmetry that can be equally implemented in the
3
C
spin basis. Looking at the matrix structure, the γ operator within H makes it a
0 m
rank-2 reducible-representation operator, as expressed in Eq. 9; indeed, H connects
m
two fermion spin polarizations, but hits a single W state’s components twice as H
m
duplicates the scalar representations, requiring the 1 normalization factor. In yet
√2
another interpretation, this relation is obtained from the normalization restriction
in the Yukawa term in Eq. 10, dividing out the energy scale set by the vacuum
expectation. To the extent that these arguments rely on a common metric vector
space, they are geometric.
Equation15assumestheparity-conservingcondition, constrainingthequarkmasses.7
For maximal hierarchy[12], with a, f dependence on one comparable large scale
O(a) O(b), (m m ,) we get 1 v 173.95, for v = 246 GeV, m = 0; alterna-
≃ b ≪ t √2 ≃ b
tively, the quark-b mass input predicts the top quark-mass as m = v2/2 m2
t − b ≃
173.90 GeV, for[9] m = 4 GeV (while renormalization effects give[2p0] m (m ) 2
b b t
∼
GeV.) These two calculations are consistent with the measured top pole mass[9]
m¯ = 173.21 0.51 0.71 GeV, where systematic and statistic errors are quoted,
t
± ±
respectively. Future precision improvements will test the limits of this tree-level cal-
culation, with view of the bottom-quark influence.
5 Conclusions and Outlook
In summary, the formalism used places fields on a basis that simultaneously contains
SM bosons and fermions. SV and SF terms are linked through the mass render-
ing of the scalar operator within them, and using independence of its components
acting on different fermion-doublet elements, within the electroweak SV vertex, im-
plicitly expected, but which we now expose. Supporting a SM prediction of a unique
scalar, normalization input from the scalar-vector vertex, and the mass-parameter
interpretation in the SF vertex relates v and m . The same relation can be argued
t
by considering the scalar operator’s matrix rank, or assuming normalized Yukawa
components.
7We neglect t-b mixing as the CKM matrix is nearly diagonal[9], confirming this method can be
applied here.
10