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Take $D_{\mu }={\partial }_{\mu }+i{K}_{\mu }$, with the Pauli matrices acting on the Higgs doublet. For electroweak hypercharge $Y=1/2$,This specifies every component of the gauge covariant derivative for the electroweak interaction. Expanding the gauge-covariant kinetic term makes its derivative, trilinear and quartic interactions explicit:where the Pauli matrix multiplication law givesThe antisymmetric Pauli contribution vanishes because $W_{\mu }^a{W}^{b\mu }$ is symmetric in $a,b$. Reversing the sign convention for $D_{\mu }$ reverses the linear gauge interactions consistently, without changing the masses.

A nonzero vacuum expectation value requires ${\mu }^2<0$. Minimizing the Higgs potential gives $v^2=-{\mu }^2/\lambda$. By an gauge transformation choosein unitary gauge. The electroweak doublet gauge-boson mass matrix follows by inserting the vacuum expectation value in the gauge-covariant kinetic term:Define the charged electroweak gauge bosons and the neutral rotation through the Weinberg angle bywith inverse $W_{\mu }^3=c_W{Z}_{\mu }+s_W{A}_{\mu }$ and $B_{\mu }=-s_W{Z}_{\mu }+c_W{A}_{\mu }$. ThenThe factors differ because $W^{+}$ and $W^{-}$ are conjugate fields whereas $Z$ is real. The massless photon corresponds to the unbroken Lie algebra generator $Q=T_3+Y$, which annihilates $⟨\varphi ⟩$. Thus three of the four real gauge bosons acquire mass, with $m_W=m_Z{c}_W$. The three would-be Goldstone bosons provide their longitudinal polarizations; the remaining scalar $h$ is the Higgs boson.

Introduce the left-handed lepton doublet $L=({\nu }_{eL},e_L{)}^T$, with $Y_L=-1/2$, and the right-handed singlet $e_R$, with $Y_R=-1$. Use the chiral projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ in the course's convention. The minimal Standard Model has no right-handed neutrino. The gauge-invariant fermion terms areThe doublet contraction in the Yukawa interaction is a singlet, and its total hypercharge is $+1/2+1/2-1=0$. A bare term ${\bar{e}}_L{e}_R$ would fail electroweak gauge invariance. The gauge covariant derivatives above already give all the requested fermion-gauge couplings. In terms of mass eigenstates, $e=g{s}_W=g^{\prime }{c}_W$ and they becomeThus the weak charged current is chiral and the neutrino has zero electric charge. The gauge-invariant electron Yukawa mass follows from the Yukawa interaction after electroweak symmetry breaking:Rephase $e_R$ to make $y_e$ real and positive. It givesThe electron Dirac mass is therefore compatible with the original gauge symmetry through the Higgs mechanism. The neutrino remains massless in this minimal renormalizable lepton sector.

Take a canonical Kähler potential $K={\Phi }^{†}\Phi$ and the Wess–Zumino model superpotentialThe F-term component givesEliminating the auxiliary field by $F=-\overline{W^{\prime }(\varphi )}$ gives the F-term scalar potential $V=|W^{\prime }(\varphi ){|}^2$. ConsequentlyThis is the supersymmetric relation between quartic and Yukawa couplings: there is one holomorphic parameter $g$, with the quartic coefficient $|g{|}^2$ and the Yukawa coupling $g$ in the displayed normalization. For real $g$ the quartic coefficient is the $g^2$ of the question. They cannot be varied independently while retaining this canonical supersymmetry.

The relation produces the supersymmetric cancellation of quadratic divergences. Bosonic and fermionic Feynman diagrams correcting a scalar mass have equal ultraviolet quadratic coefficients and opposite signs, so the mass is not driven directly to the ultraviolet cutoff scale. In a theory with soft supersymmetry breaking, residual corrections instead have the schematic sizewith coefficients and logarithms depending on the spectrum. Thus superpartner masses near the electroweak scale can stabilize the Higgs boson mass against a much larger cutoff, addressing the radiative part of the hierarchy problem. Heavy sparticles leave a corresponding residual sensitivity; the cancellation does not by itself explain every mass scale.

Take ${\tau }^a={\sigma }^a/2$, $Y_{\varphi }=1/2$, metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, and a Higgs potential with ${\mu }_h^2>0$, $\lambda >0$. The gauge-scalar electroweak interaction isHere $B_{\mu \nu }={\partial }_{\mu }{B}_{\nu }-{\partial }_{\nu }{B}_{\mu }$. With the printed plus-sign convention for $D_{\mu }$, define $[D_{\mu },D_{\nu }]=ig{W}_{\mu \nu }^a{\tau }^a+i{g}^{\prime }{Y}_{\varphi }{B}_{\mu \nu }$; thus $W_{\mu \nu }^a={\partial }_{\mu }{W}_{\nu }^a-{\partial }_{\nu }{W}_{\mu }^a-g{ϵ}^{abc}{W}_{\mu }^b{W}_{\nu }^c$. The minus sign in the nonlinear gauge field strength follows from this convention.

The minima have ${\varphi }^{†}\varphi =v^2/2$, with $v^2={\mu }_h^2/\lambda$. A gauge transformation rotates the vacuum to $(0,v{)}^T/\sqrt{2}$. In unitary gauge, the three angular Goldstone bosons are removed, leaving $\varphi =(0,v+H{)}^T/\sqrt{2}$ and one real Higgs boson. The generator $Q=T^3+Y$ annihilates the vacuum, because its lower component has $T^3=-1/2$ and hypercharge $1/2$. This identifies the unbroken electromagnetic gauge group.

The quadratic terms from the Higgs field kinetic term areDefine the Weinberg angle and physical fields byThe neutral combinations areThe neutral gauge-boson mass matrix is $(v^2/4)\left(\begin{array}{cc}{g}^2& -g{g}^{\prime }\\ -g{g}^{\prime }& g^{\prime 2}\end{array}\right)$. It has eigenvalues zero and $(g^2+g^{\prime 2})v^2/4$, with the zero eigenvector giving $A_{\mu }$. Expanding the Higgs potential about its minimum gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. Therefore the tree-level electroweak gauge-boson masses and scalar mass areAlso $m_W=m_Z\cos {\theta }_W$ and $e=g\sin {\theta }_W=g^{\prime }\cos {\theta }_W$. The three removed Goldstone bosons supply the longitudinal polarization vector degrees of freedom of the massive electroweak gauge bosons.

Replacing $v^2$ by $(v+H{)}^2$ in the neutral mass term gives all the Higgs boson couplings to Z bosons in unitary gauge:There are exactly the cubic $HZZ$ and quartic $HHZZ$ elementary tree-level Feynman diagrams. Differentiating with respect to the identical fields gives Feynman rules $2i{m}_Z^2{g}_{\mu \nu }/v$ and $2i{m}_Z^2{g}_{\mu \nu }/v^2$, respectively. There is no elementary $ZHH$ vertex for the real neutral radial field.

The authoritative PDF has $m_Z^2/v$, not the dimensionally incorrect $m_Z^2/v^2$ in the TeX conversion. The two identical Z bosons give the vertex factor two, so the scattering amplitude for the decay isup to an irrelevant overall phase. Apply the massive vector polarization sum, identifying the hint's $M_Z$ with $m_Z$:Since $k_1\cdot k_2=(m_H^2-2m_Z^2)/2$, putting $x=m_Z^2/m_H^2$ givesNo initial-spin average is needed for a Higgs boson. In its rest frame the two-body momentum is $|\mathbf{k}|=(m_H/2)\sqrt{1-4x}$. Integrating the energy and momentum delta functions in the Lorentz-invariant phase-space measure gives $d{\Phi }_2=\sqrt{1-4x}d\Omega /(32{\pi }^2)$, hence $\int d{\Phi }_2=\sqrt{1-4x}/(8\pi )$.

The printed generic width formula treats the daughters as labelled. Here the identical final-state symmetry factor is $1/2!$, without which the same physical configuration is counted twice. Thus the Higgs decay to two Z bosons hasEquivalently, $1/v^2=\sqrt{2}{G}_F$ gives the prefactor $G_F{m}_H^3/(16\sqrt{2}\pi )$. The threshold limit is zero and the large-mass limit is $m_H^3/(32\pi v^2)$. The additional factor for identical daughters is a required specialization of the PDF's generic phase-space formula, not an extra factor in the vertex.

Parameterize the scalar locally aswhere the five $X^{\alpha }$ are broken generators. A gauge transformation removes all ${\pi }^{\alpha }$ in unitary gauge. Their derivative terms combine with the corresponding gauge fields in $(D_{\mu }\Phi {)}^{†}{D}^{\mu }\Phi$, supplying the longitudinal polarizations of five massive vectors. This is the Higgs mechanism.

With $T^a={\lambda }^a/2$, the gauge bosons $A_{\mu }^{1,2,3}$ of the unbroken $SU(2)$ remain massless. The four bosons $A_{\mu }^{4,5,6,7}$ haveand $A_{\mu }^8$ hasThe five eaten Goldstone modes complete their third polarizations. The sixth real component of the complex triplet remains as a radial Higgs boson withThis realizes the degree-of-freedom count summarized by Fundamental-Higgs breaking of SU(3) to SU(2).

After electroweak symmetry breaking, the Higgs field has vacuum expectation value $⟨H⟩=(0,v/\sqrt{2}{)}^T$. A Yukawa interaction $-y_f{\bar{f}}_{L}H{f}_R+h.c.$ then becomes a fermion mass with $m_f=y_{f}v/\sqrt{2}$. The Higgs gauge-covariant kinetic term $|D_{\mu }H{|}^2$ giveswhile the radial fluctuation is the massive Higgs boson. The photon and all eight gluons correspond to unbroken gauge generators and remain massless. In the minimal renormalizable Standard Model, which has no right-handed neutrinos, the neutrinos also remain massless; adding right-handed neutrinos permits Higgs-generated Dirac masses.

The hierarchy problem is visible in the one-loop top-quark contribution to the Higgs boson mass parameter,In a supersymmetric theory the corresponding stop loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughlyThe two relevant one-loop Feynman diagrams are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the naturalness expectation $m_{\tilde{t}}$, and broadly other sparticle masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.

R-parity iswhere $B$ is baryon number, $L$ is lepton number and $s$ is spin angular momentum. Standard Model particles have $R_p=+1$, while their superpartners have $R_p=-1$. The renormalizable R-parity-violating superpotential of the Minimal supersymmetric Standard Model iswith ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$.

Combining ${\lambda }_{i1k}^{\prime }{L}_i{Q}_1{D}_k^c$ with ${\lambda }_{11k}^{\prime \prime }{U}_1^c{D}_1^c{D}_k^c$ produces proton decay through exchange of a virtual right-handed down-type squark. The tree diagram has $u$ and $d$ entering the baryon-number-violating vertex, an internal ${\tilde{d}}_{Rk}$ line, and the lepton-number-violating vertex emitting ${\bar{\nu }}_i$ and a light quark. Hadronization givesThe invisible particle is an antineutrino of any family $i=1,2,3$, and electric-charge conservation requires a positively charged pion. The antisymmetry of ${\lambda }_{11k}^{\prime \prime }$ requires $k=2$ or $3$.

Integrating out a squark of mass $m_{\tilde{q}}$ gives a dimension-six interaction with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. Ignoring order-one hadronic and phase-space factors,For $m_p\sim 1\mathrm{GeV}$ and $m_{\tilde{q}}\sim 10^4\mathrm{GeV}$, the factor before the couplings is about $10^{-18}$--$10^{-17}\mathrm{GeV}$. A lifetime above $10^{34}$ years is about $10^{65}{\mathrm{GeV}}^{-1}$, so ${\Gamma }_p\lesssim 10^{-65}\mathrm{GeV}$ andat order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.

The matter-parity factor $(-1{)}^{3(B-L)}$ is odd for every quark or lepton superfield and even for either Higgs superfield. Each term in $W_{\mathrm{RPV}}$ contains an odd number of matter superfields, so R-parity forbids all four types of term. Standard MSSM Yukawa interactions contain two matter superfields and remain allowed.

Finally, the requested four-point interaction contains a squark, an antisquark, a $W$ boson and a gluon. It comes from the squark gauge-covariant kinetic termThe cross term isbecause the colour and weak generators commute. With all fields incoming, its Feynman rule iswith the conventional $1/\sqrt{2}$ conversion when $W^I$ is replaced by $W^{\pm }$.

Use unitary gauge to setThenDefineThe propagating fields are $W_{\mu }^{\pm },Z_{\mu },A_{\mu }$, and the Higgs boson $h$. Expandingand reading the quadratic terms givesThe factors $(v+h{)}^2$ in the kinetic term also give the corresponding Higgs-vector interactions.

For a purely electromagnetic interaction continuum with five active quark flavors, the usual estimate is $N_{e^{+}{e}^{-}}/N_{\mathrm{had}}=3/11$. The unspecified particle resonance and the word “total” require qualifications before this can be treated as a unique physical prediction.

In natural units, neglecting final masses, single-photon annihilation givesHere $\alpha$ is the fine-structure constant, $Q_f$ the quark electric charge in elementary-charge units, and the factor three counts orthogonal colour final states. Consequently the inverse hadronic R ratio isWith $u,d,s,c,b$ accessible, the denominator is $3[2(2/3{)}^2+3(1/3{)}^2]=11/3$. The corresponding values for $u,d,s$ and $u,d,s,c$ are $1/2$ and $3/10$. Inclusive Quantum chromodynamics corrections increase the hadronic denominator, approximately by $1+{\alpha }_s/\pi$ away from thresholds and particle resonances. These are annihilation ratios before selection and detection efficiency corrections, not ratios of uncorrected detector counts.

A future muon-collider Higgs resonance is a plausible reading of part (c), since its narrow decay width makes a detuning of a few $\mathrm{MeV}$ relevant. Even then, two further assumptions are necessary. First, being a few $\mathrm{MeV}$ below the Higgs boson peak does not eliminate it: for detuning $\Delta =\sqrt{s}-m_H$, the ideal narrow Breit-Wigner resonance cross section retains the relative factorFor ${\Gamma }_H=4\mathrm{MeV}$ and $\Delta =-4\mathrm{MeV}$ this is $1/5$, not zero. Resonant hadronic production can still change the requested denominator substantially.

Second, at $\sqrt{s}\simeq 125\mathrm{GeV}$ the Z boson continuum is not negligible. A more complete annihilation estimate including both tree-level Feynman diagrams usesHere $s_W^2$ is the squared sine of the weak mixing angle, $T_3^f$ is weak isospin, and $N_c^f$ is one for a lepton and three for a quark. Thus the continuum ratio is $F_e/[3(2F_u+3F_d)]$, rather than necessarily $3/11$. For example, $s_W^2=0.231$, $m_Z=91.1876\mathrm{GeV}$ and ${\Gamma }_Z=2.4952\mathrm{GeV}$ give approximately $0.102$ at $125\mathrm{GeV}$ before Quantum chromodynamics corrections.

If the unspecified particle resonance instead means the Z boson, detuning by a few $\mathrm{MeV}$ is tiny compared with its $2.5\mathrm{GeV}$ decay width. The resonant ratio remains approximately $\Gamma (Z\to e^{+}{e}^{-})/\Gamma (Z\to \mathrm{hadrons})=1/20.767\simeq 0.0482$.^[ref] This is a concrete counterexample to treating any slight downward detuning as permission to discard the particle resonance. The source supplies neither its identity nor the detuning and selection needed for a unique all-process ratio. The boxed continuum formula is the charge-counting answer under its stated approximation; a literal total must also include remaining resonant and other accepted hadronic channels.

A bare gauge boson mass would spoil the electroweak gauge structure. In the Standard Model a complex Higgs doublet instead has a symmetry-breaking potential. Choose the conventionThe Higgs mechanism rearranges three scalar components into the longitudinal polarizations of the massive W bosons and Z bosons; the unbroken electromagnetic gauge symmetry leaves the photon massless. One physical scalar excitation, the Higgs boson $h$, remains.^[ref] Using tree-level Feynman diagrams,These equations use natural units. Yukawa interactions generate charged-fermion masses and Higgs couplings proportional to those masses; the minimal model does not account for observed neutrino masses. The gauge kinetic term supplies Higgs couplings to W and Z pairs. The physical Higgs exchange also cancels the otherwise excessive high-energy growth of longitudinal vector-boson scattering. The mechanism does not fix the Yukawa couplings or the Higgs self-coupling, and most proton mass arises from strong-interaction dynamics rather than directly from the light-quark Higgs masses.

The mass-generation mechanism is a statement about the field and its vacuum expectation value; the observable boson is the excitation about that vacuum.

The entrance Yukawa coupling is proportional to the incident lepton mass, so the partial Higgs width into that lepton pair is proportional to its mass squared. With the question's masses,The same factor enhances direct s-channel Higgs boson formation when all other factors are held fixed. The much larger muon Yukawa coupling motivates a muon-collider Higgs resonance. Heavier muons also radiate less strongly in a curved orbit, but their decay lifetime makes a collider technically demanding. The resonance remains narrow, so a sufficiently small beam-energy spread is essential to retain a large fraction of the ideal peak.

Production and identification. At a hadron collider, initial partons carry fractions of the proton momenta; observable production rates combine parton-level amplitudes with parton distribution functions.

Why the three masses form a test. At tree level, electroweak symmetry breaking gives $m_W=gv/2$ and $m_Z=\sqrt{g^2+g^{\prime 2}}v/2$. The Higgs mechanism relates masses to the vacuum field, but it does not by itself predict the top Yukawa coupling or the Higgs self-coupling, so the three masses are not fixed independently of measured parameters.

The stronger test comes from radiative corrections. With accurately measured $\alpha$, $G_F$ and $m_Z$, the electroweak precision test with the W, top and Higgs masses usesThe top-bottom loop contributes approximately $\Delta \rho =3G_F{m}_t^2/(8\sqrt{2}{\pi }^2)$, while the Higgs contribution has a characteristic logarithmic mass dependence. Hence direct measurements of $m_t$ and $m_H$ turn the precision calculation into a prediction for $m_W$. Comparing that prediction with the direct W-mass measurement overconstrains the Standard Model; disagreement beyond experimental and theoretical uncertainties could indicate additional particles or interactions in the loops. The test concerns the relation among independently measured quantities, rather than merely the existence of the three particles.

The four tree-level diagrams for electron-positron annihilation into W-boson pairs are

The first two diagrams contain the electromagnetic interaction and weak interaction neutral electron currents and the gauge boson self-interaction vertices $\gamma W^{+}{W}^{-}$ and $Z{W}^{+}{W}^{-}$. The third has two weak charged current vertices joined by an electron neutrino propagator. In the fourth, the incoming electron-Positron pair couples to the Higgs boson, which then couples to $W^{+}{W}^{-}$.

The small factor is at the initial Higgs-electron vertex. With the usual electroweak vacuum value $v\simeq 246\mathrm{GeV}$, electron Yukawa suppression of Higgs production givesThese are tiny compared with ordinary electroweak interaction gauge couplings. The decay amplitude for the Higgs boson diagram carries this small coefficient, and its separate squared contribution carries its square. Thus the Higgs diagram is negligible because the electron Yukawa coupling is proportional to the electron mass. It vanishes in the massless-electron approximation, even though the Higgs boson-W boson coupling itself is not small. The remaining three gauge diagrams must be added coherently, including their interference.

In gluon-fusion Higgs production, two gluons couple to a closed top quark loop, and the Higgs boson couples to that massive quark through its Yukawa interaction. The right panel of the diagram in c shows one explicit colour-flow component: internal $t_r\to t_b$ at one gluon insertion, then $t_b\to t_r$ at the other; the colour-neutral Higgs boson insertion preserves the internal color charge. Include both loop orderings. The color charge factor isso the two incoming gluons are projected onto a colour singlet. The Yukawa coupling is $y_t=\sqrt{2}{m}_t/v$, and the $h\bar{t}t$ vertex has strength $m_t/v$. No direct tree-level $hgg$ mass coupling is needed. Integrating out a sufficiently heavy top quark generates an effective interaction proportional to $(h/v)G_{\mu \nu }^a{G}^{a\mu \nu }$, which is allowed by gauge invariance even though the gluon is massless.

There is also no kinematic obstacle: two massless particles with nonparallel momenta can have nonzero total invariant mass. In particular, head-on gluons have $\hat{s}=4E_1{E}_2$, providing the energy needed for the Higgs boson at the LHC.

Using c and omitting ${\partial }_{\mu }\Phi$, the covariant derivative isThe two components are orthogonal, so their squared magnitudes add; $(W_{\mu }^{+}{)}^{†}=W_{\mu }^{-}$. Thus the Higgs field kinetic contribution to the Lagrangian density isThe two $v^2$ terms are gauge boson mass terms. Comparing with $m_W^2{W}^{+}{W}^{-}+\frac{1}{2}{m}_Z^2{Z}^2$ gives $\boxed{m_W=gv/2,m_Z=Gv/2}$ and $m_W/m_Z=\cos {\theta }_W$. The factor $1/2$ for the real Z boson field is essential.

The two $2vh$ terms are the cubic Higgs boson coupling to W bosons and Higgs boson coupling to Z bosons. They produce the three-leg $h{W}^{+}{W}^{-}$ and $hZZ$ Feynman vertices, allowing Higgs decay to two W bosons, Higgs decay to two Z bosons when kinematically available, and the corresponding production mechanisms with virtual gauge bosons. The two $h^2$ terms are quartic $hh{W}^{+}{W}^{-}$ and $hhZZ$ interactions, including vector-boson scattering into a Higgs boson pair. Written in terms of the masses,The corresponding Feynman vertex factors are $2i{m}_V^2{g}_{\mu \nu }/v$ for one $h$ and $2i{m}_V^2{g}_{\mu \nu }/v^2$ for two $h$, for $V=W,Z$, after differentiating with respect to identical fields. These six contributions contain no photon mass or direct $hAA$ interaction because $Q\Phi =0$.