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For the MSSM superfield representations augmented by singlet right-handed neutrino chiral superfields $N_i$, the homogeneous cubic gauge-invariant superpotential contains the four Yukawa structures $Q{H}_{u}U$, $Q{H}_{d}D$, $L{H}_{d}E$, $L{H}_{u}N$ and the additional structures $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$. Weak-doublet pairs contract with the antisymmetric two-index tensor; $UDD$ contracts with the color three-index tensor. Consequently the $LLE$ coefficient is antisymmetric in its two $L$ indices, the $UDD$ coefficient in its two $D$ indices, and the $NNN$ coefficient symmetric. With $L(N)=-1$, the last five structures violate baryon number or lepton number, whereas the first four conserve both. Singlet fields must not be omitted when classifying a general cubic superpotential.

A right-handed neutrino is a singlet of the entire Standard Model gauge group. Its same-chirality bilinear with the field obtained by charge conjugation therefore admits the displayed renormalizable Majorana mass term, in addition to a Higgs-induced Dirac mass term. It violates lepton number by two units. Analogous terms for electrically charged Standard Model fermions carry nonzero electric charge; quarks also have nontrivial color charge. The presence of a right-handed field permits mass but does not force a nonzero mass parameter.

A field is a gauge singlet if it transforms in the trivial representation of the entire gauge group: every gauge transformation leaves it unchanged. A singlet of only one factor need not be a gauge singlet of the product group. For example a right-handed neutrino with representation $(1,1{)}_0$ is a Standard Model gauge singlet, whereas a color-singlet charged lepton is still acted on by the electroweak group. A singlet can have ordinary derivatives in its kinetic term and gauge-invariant masses, subject to Lorentz invariance and any additional global symmetries.

Each effectively massless active neutrino species contributes the same invisible Z boson partial decay width. Subtracting calculated or measured visible widths from the total width therefore measures the displayed number, assuming no additional invisible channels. The inference concerns particles lighter than $M_Z/2$ with ordinary weak couplings. It does not exclude heavier neutrinos or right-handed neutrinos that are gauge singlets. Early LEP resonance measurements were consistent with three light active species, with statistical uncertainty. See the OPAL report on its 1989 data for the original resonance-width comparison.

Matter parity assigns $-1$ to every matter chiral superfield, including a singlet $N$ describing a conjugate right-handed neutrino, and $+1$ to Higgs chiral superfields. Thus the quadratic superpotential term $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even, although it changes lepton number by two. The higher-degree operators $QQQL$ and $(L{H}_u{)}^2$ are also even and violate baryon number or lepton number. Exact R-parity therefore forbids the renormalizable trilinear violations $LLE$, $LQD$, $UDD$, $N{H}_d{H}_u$, $NNN$, but does not imply conservation of these continuous charges in every allowed interaction.

The gauge fields require their own Yang-Mills Lagrangian, $-W_{\mu \nu }^a{W}^{a\mu \nu }/4$. The non-Abelian field strength contains a term quadratic in the gauge potentials, so squaring it produces three-vector and four-vector self-interaction vertices, with derivative momentum factors in the cubic vertex and SU(2) structure constants in both. They exist in either scalar phase and must be added to the scalar-generated vertices. A gauge-fixed quantum calculation also has gauge-fixing and, where applicable, ghost terms; those are not extra physical boson species.

For ${\mu }^2>0$, the global phase transformation $\Phi \mapsto e^{i\alpha }\Phi$ leaves both the scalar potential and the covariant kinetic term unchanged while leaving $W_j$ fixed. Its Noether charge counts scalar particles minus antiparticles. Thus ${\varphi }_a{\varphi }_b\to {\varphi }_a{\varphi }_b$ and ${\varphi }_a{\varphi }_a^{\ast }\to W_3{W}_3$ are compatible with this charge, whereas ${\varphi }_a{\varphi }_b\to W_j{W}_k$ is forbidden: its initial charge is two and final charge zero. Charge conservation is necessary, not sufficient, for a particular amplitude to be nonzero; other quantum numbers and kinematics still apply.

After adding the hypercharge field, the neutral-vector mass term becomesIts mass matrix has zero determinant. With $G=\sqrt{g^2+g^{\prime 2}}$, the orthonormal combinations areThe unbroken generator is electric charge, $Q=T_3+Y/2$, because the lower doublet component has $T_3=-1/2$ and $Y=1$. The charged combinations $W^{\pm }=(W_1\mp i{W}_2)/\sqrt{2}$ keep mass $gv/2$.

The observed charged weak interaction couples left-chiral fermions within SU(2) doublets. In the Standard Model, $Q_L=(u_L,d_L)$ and $L_L=({\nu }_L,e_L)$ are doublets, while $u_R,d_R,e_R$ are singlets. A vectorlike assignment would also give right-handed charged weak currents, contrary to this observed chiral structure. Once the measured electric charges are specified, $Y=2(Q-T_3)$ fixesAn optional right-handed neutrino is a singlet with $Y=0$. Conjugate left-chiral fields have the opposite hypercharges to these right-chiral fields. These assignments permit gauge-invariant Yukawa interactions; they are not obtained by assigning one common hypercharge to every chirality.

A bare Dirac mass couples left and right chirality, so the distinct electroweak representations prevent writing it as a gauge-invariant term before symmetry breaking. Instead the Yukawa interactions areWeak indices are contracted to singlets. Substituting the vacuum value gives mass matrices $M_f=Y_{f}v/\sqrt{2}$ and a physical Higgs coupling $-{\bar{f}}_L{M}_f{f}_R\sigma /v$ plus its conjugate. Biunitary diagonalization defines mass eigenstates; a mismatch between the up- and down-quark left rotations appears in the CKM matrix. Gauge-boson masses arise from the Higgs kinetic term, whereas these fermion masses arise from the Yukawa term.

With only the minimal field content there is no right-handed neutrino, so no renormalizable neutrino Yukawa term. Adding a gauge-singlet $N_R$ permits both a Dirac scale $m_D=y_{\nu }v/\sqrt{2}$ and a Majorana scale $M_R$. For one family the symmetric mass matrix in a left-chiral basis isIts signed eigenvalues are $(M_R\pm \sqrt{M_R^2+4m_D^2})/2$. Rephasing gives positive physical masses, and for $|M_R|\gg |m_D|$ the seesaw mechanism yieldsFor $m_D\sim 100\mathrm{GeV}$ and $M_R\sim 10^{14}\mathrm{GeV}$, the light scale is about $0.1\mathrm{eV}$. Integrating out the heavy state generates the dimension-five Weinberg operator, whose Higgs vacuum insertion gives the same suppression. Alternatively a Dirac neutrino can have a very small Yukawa coupling with lepton number conserved. A tiny observed neutrino mass does not by itself distinguish these mechanisms.

The SU(5) grand unified theory embeds the strong and electroweak groups in a single simple gauge group. Its 24 gauge generators contain the eight gluons, three weak generators and one hypercharge generator, plus twelve additional generators connecting color and weak directions. A common gauge coupling applies at the unification scale with the properly normalized hypercharge generator; low-energy couplings differ through their renormalization group evolution.

Using left-chiral fields throughout, one Standard Model generation fits intoThe dimensions are $3+2=5$ and $6+3+1=10$, so no right-handed neutrino is required in these multiplets. A singlet neutrino can be added. A suitable adjoint scalar vacuum breaks SU(5) to the Standard Model subgroup; a fundamental scalar contains an electroweak doublet and a colored triplet. The extra heavy vector bosons can mediate transitions between quarks and leptons, hence proton decay in the minimal model. This is a proposed unification, not an experimentally established identification of new particles.

The hierarchy problem concerns keeping an elementary weak-scale scalar light in the presence of much heavier physical scales. A heavy state coupled to it generically induces a threshold correction of order $\delta m_H^2\sim c{M}^2/(16{\pi }^2)$, with $c$ a product of couplings. Without protection, a weak-scale result then needs cancellation against a much larger parameter. In SU(5), this is sharpened by doublet-triplet splitting: the electroweak doublet must stay light while its colored partner must be sufficiently heavy to avoid rapid proton decay. Choosing a small tree-level doublet mass is not by itself a mechanism protecting it from subsequent heavy-particle corrections. Supersymmetry and other proposals change this protection problem, but the minimal SU(5) embedding alone does not solve it.

The left- and right-chiral components of a charged Standard Model fermion transform in different electroweak group representations. A bare Dirac mass term $-m{\bar{f}}_L{f}_R+h.c.$ is therefore not invariant under the full electroweak gauge group. The Higgs doublet supplies the missing transformation properties. With $\tilde{H}=i{\sigma }_2{H}^{\ast }$, the gauge-invariant Yukawa interactions arewhere each $Y_f$ is a flavor-space Yukawa matrix. For example, hypercharges in the down-quark term add to $-1/6+1/2-1/3=0$, while those in the up-quark term add to $-1/6-1/2+2/3=0$. Thus the Higgs and conjugate Higgs are required in different terms.

After electroweak symmetry breaking, choose unitary gauge with $H=(0,(v+h)/\sqrt{2}{)}^T$. The fermion mass matrices becomeA singular value decomposition $U_{fL}^{†}{M}_f{U}_{fR}=\mathrm{diag}(m_{f,1},m_{f,2},m_{f,3})$ defines nonnegative physical tree-level masses. The mismatch $U_{uL}^{†}{U}_{dL}$ is the Cabibbo-Kobayashi-Maskawa matrix in the weak charged current. The corresponding physical Higgs boson couples diagonally as $-(m_f/v)h\bar{f}f$ in the minimal one-doublet theory. This relates its coupling strength to each mass, but the measured Yukawa couplings remain independent input parameters: the minimal theory does not predict their hierarchy.

The minimal renormalizable Standard Model contains no right-handed neutrino, so it has neither a neutrino Dirac Yukawa term nor a gauge-invariant renormalizable left-handed Majorana mass term. Its neutrinos are massless. Observed neutrino oscillations require going beyond that minimal field content or operator set: adding right-handed singlets permits Dirac masses, while the dimension-five Weinberg operator gives a Majorana neutrino mass of order $v^2/\Lambda$ after symmetry breaking. These alternatives should be distinguished from the charged-fermion Yukawa mechanism.

Chirality distinguishes the two inequivalent Weyl spinor representations of the proper Lorentz group. For a Dirac spinor, define ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the chiral projectorsThey satisfy $P_L{P}_R=0$ and $P_L+P_R=I$. Proper Lorentz transformations preserve each chiral representation, whereas parity exchanges them. Helicity instead measures the projection of spin angular momentum along momentum. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with chirality. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.

The Standard Model is a chiral gauge theory: left- and right-chiral fields have different gauge group representations. In the convention $Q=T_3+Y$, one generation containsThe minimal field content has no right-handed neutrino. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays weak isospin, hypercharge and color charge assignments, not assignments of massive-particle helicity.

The weak charged current contains $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ:Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The photon and gluon couplings of a charged massive fermion are vectorlike; this does not make the full electroweak interaction vectorlike.

A bare Dirac mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before electroweak symmetry breaking. A Yukawa interaction, such as $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. Up-type quarks use the conjugate doublet. Thus chirality explains the weak charge assignments, their parity violation, and why masses arise through the Higgs mechanism. The consistency of these chiral assignments also requires Standard Model anomaly cancellation.

Choose the first $SU(2)$ as $S{U}_w(2)$ and embed $S{U}_c(3)$ in the upper-left three-by-three block of $SU(4)$. The fundamental representations then restrict as $4\to 3\oplus 1$ and $\overline{4}\to \overline{3}\oplus 1$. Thus the representation in solution, forgetting hypercharge, restricts toThis contains every required $S{U}_w(2)\times S{U}_c(3)$ representation, with one extra singlet.

For the hypercharge embedding, let $X=\mathrm{diag}(1,1,1,-3)$ in the $SU(4)$ fundamental representation, and $T_{3R}=\mathrm{diag}(1/2,-1/2)$ in the second $SU(2)$. These commute with $S{U}_w(2)\times S{U}_c(3)$. The integer normalization in the paper isExplicitly, this circle group embeds in the covering group byBoth nontrivial blocks have determinant one, and the map is injective since the first three entries of the $SU(4)$ block recover $z$.

On $(2,1,4)$, $T_{3R}=0$, so the branching of a representation gives $(2,3{)}_1\oplus (2,1{)}_{-3}$. On $(1,2,\overline{4})$, the U(1) charges under $X$ are $-1$ on $\overline{3}$ and $3$ on $1$, while $6T_{3R}$ takes values $\pm 3$. HenceCombining the two summands givesThis is the Pati-Salam charge embedding, with $X=3(B-L)$. In terms of the common Standard Model convention $Q_{\mathrm{electric}}=T_{3L}+Y_{\mathrm{SM}}$, the paper's integer hypercharge is $Y=6Y_{\mathrm{SM}}$.

The extra neutral singlet can be interpreted as the left-handed charge conjugate ${\nu }_R^c$ of a right-handed neutrino. It is a sterile neutrino under the Standard Model gauge group, and completes the sixteen Weyl fermions of one family in this half-spin representation. Its neutrality permits a Majorana mass term without violating the remaining Standard Model gauge group.

One generation of left-handed Standard Model fermions fits into $\overline{5}\oplus 10$ of an SU(5) grand unified theory. With hypercharge defined by $Q=T_3+Y$, their restriction of a representation to $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$ isHere $Q$ in the multiplet list denotes the left-handed quark doublet, and the superscript $c$ labels left-handed charge-conjugate fields of the corresponding right-handed fermions.

The results of c therefore giveThus the required matter representations are present. The extra singlet has Standard Model representation $(1,1{)}_0$ and can describe the left-handed charge-conjugate field of a right-handed neutrino. Its gauge singlet character permits a Majorana mass term; a Yukawa interaction with a lepton doublet and a suitable Higgs field can then supply neutrino masses. The branching alone does not fix those interactions or their scales.

The extra $5$ hasIf it describes Weyl fermions, these are additional coloured and electroweak fermions in the conjugate Standard Model representations to the ordinary $\overline{5}$. With an additional $\overline{5}$, they can form a vectorlike gauge spectrum and acquire invariant masses. Pairing them with the only ordinary $\overline{5}$ would also remove that generation's $d^c,L$ fields, so a realistic heavy pairing needs extra matter. If the $5$ instead describes a scalar field, it has the quantum numbers of the usual $SU(5)$ Higgs field: a weak doublet and a coloured triplet. A single fermionic $15$ does not turn its extra fermions into a scalar Higgs field merely by branching.

There is a further consistency condition if these are the entire chiral matter content of an $SU(6)$ gauge theory. Normalize the cubic anomaly coefficient by $A(6)=1$. The dual representation has $A(\overline{6})=-1$, and the two-index antisymmetric representation has $A(15)=6-4=2$. Thus one copy of each has nonzero gauge anomaly, $A_{\mathrm{total}}=1$. Their inclusion of the desired $SU(5)$ multiplets is therefore not, by itself, a complete consistent chiral unified theory.

For example, anomaly cancellation holds for $15\oplus 2\overline{6}$, whose restriction of a representation isThis leaves one ordinary generation, a potentially massive vectorlike gauge spectrum, and two possible singlet neutrinos. This anomaly-free matter choice is also used in explicit $SU(6)$ model constructions.^[ref] These interpretations require additional choices of matter statistics, spontaneous symmetry breaking, and interactions; the weight systems determine the multiplet content.

Let $\epsilon =i{\sigma }_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, so ${\epsilon }^{-1}=-\epsilon$. Direct multiplication of the Pauli matrices gives $\epsilon {\sigma }_i^{\ast }\epsilon ={\sigma }_i$, equivalently $\epsilon {\sigma }_i^{\ast }{\epsilon }^{-1}=-{\sigma }_i$. For $U=\exp (i{\theta }_i{\sigma }_i/2)\in SU(2)$ this impliesTherefore ${\varphi }^c=\epsilon {\varphi }^{\ast }$ transforms as the same weak fundamental representation doublet as $\varphi$, while complex conjugation reverses its hypercharge to $-1/2$. This is the Higgs conjugate doublet. The Yukawa coupling product $\bar{L}{\varphi }^c{u}_R$ is an $SU(2)$ singlet, and its hypercharges add toIt is consequently gauge-invariant and can supply an up-quark mass.

In unitary gauge, ${\varphi }^c=((v+H)/\sqrt{2},0{)}^T$. Writing $y_u$ for the full arbitrary coefficient of the Yukawa coupling term, its physical form isA one-generation field rephasing makes the mass positive, and its Higgs boson coupling is $m_u/v$. The printed Yukawa expression has an additional trailing $m$ after $R^{+}$ which is never defined. If it denotes a constant, absorb it into $y_u=f^{+}m$; in the conventional dimensionless Yukawa notation without that factor, $y_u=f^{+}$. It does not affect the gauge-transformation argument.

For three families, use a singular value decomposition of the two mass matrices: $U_{uL}^{†}{M}_u{U}_{uR}=D_u$ and $U_{dL}^{†}{M}_d{U}_{dR}=D_d$, with positive diagonal entries. The original left-handed weak current then becomes a current between mass eigenstates withA complex $3\times 3$ matrix has eighteen real parameters, and $U^{†}U=I$ imposes nine independent real conditions, leaving nine. A generic unitary matrix can be built from three complex plane rotations and three diagonal phases: each plane rotation supplies one angle and one phase. Thus these are three rotation angles and six phases. Rephase each of the six Dirac quark mass eigenfields, applying the same phase to its two chiralities so the diagonal masses stay real. A common phase changes no entry of $V$, leaving five effective rephasings. Removing five of the six phases leaves three angles and one physical phase. This CKM parameter counting assumes generic nondegenerate masses; degeneracies give additional basis freedom.

Exactly massless neutrinos have an arbitrary unitary flavor rotation that changes no mass term. After diagonalizing the charged-lepton mass matrix, choose the neutrino basis with the same left-handed rotation. The apparent lepton charged-current mixing matrix becomes the identity. Thus mixing of exactly massless neutrinos is removable; it is not an analogue of physical CKM mixing.

For the requested CP symmetry calculation, write $j_{ij}^{\mu }={\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)d_j$. The supplied charge conjugation transformations, with the fermion-reordering sign, giveFor $\Gamma ={\gamma }^{\mu }(1-{\gamma }^5)$, use $C({\gamma }^5{)}^T{C}^{-1}={\gamma }^5$ and the given gamma matrix relation to obtainUnder parity, the current acquires the Lorentz parity matrix and $1+{\gamma }^5$ becomes $1-{\gamma }^5$. Under charge conjugation the charged vector also changes sign and becomes its conjugate. The two minus signs cancel, and the parity matrices cancel in the contracted operator. ConsequentlyWith $k=g/(2\sqrt{2})$, the interaction is $-k\sum (V_{ij}{O}_{ij}+V_{ij}^{\ast }{O}_{ij}^{†})$. Thereforewith the transformed density evaluated at $x_P$. This is the CP transformation of a charged quark current.

Physical noninvariance requires a phase that cannot be removed by quark rephasings. The invariant quartetis the Jarlskog invariant, where $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$ in the standard three-angle parametrization. If $J\ne 0$, no allowed row/column phases can make the matrix real, and the charged-current interaction violates CP symmetry. For example all three angles $\pi /4$ with $\delta =\pi /2$ give $J=1/(8\sqrt{2})\ne 0$.

The printed noninvariance claim is conditional on a nonzero physical phase. Arbitrary Yukawa matrices do not force it: $V=I$, or any real CKM matrix, gives a CP-invariant charged-current interaction. The correct statement is generic CP violation when the irreducible phase is nontrivial, as explained by quark rephasing and CP conservation.

Finally a right-handed neutrino has gauge representation $(1,1{)}_0$. Its Lorentz-invariant same-chirality bilinear with the field obtained by charge conjugation is therefore a gauge singlet and permits a Majorana mass term, conventionallyThe factor $1/2$ is the usual convention for a field paired with itself; it may be absorbed into the printed coefficient. A Higgs Yukawa interaction can independently give a Dirac mass term. The Majorana term violates lepton number by two units but preserves the gauge symmetries. Corresponding terms for the other charged Standard Model fermions carry nonzero electric charge, and quarks also carry nontrivial color charge. Thus this gauge-invariant right-handed neutrino Majorana mass is allowed while those charged-fermion analogues are forbidden. Merely introducing a right-handed neutrino permits mass; vanishing mass coefficients would still leave it massless.

Write $t=\log \mu$ and use ${\alpha }_i=g_i^2/(4\pi )$. The chain rule turns the renormalization-group beta function intoIntegrating gives the one-loop inverse gauge coupling evolutionThis expression applies over a range with unchanged active field content and in the perturbative regime. It uses $\log \mu$, not $\log {\mu }^2$.

For gauge coupling unification, define the normalized inverse couplings $A_1=3/(5{\alpha }_1)$, $A_2=1/{\alpha }_2$, $A_3=1/{\alpha }_3$, and slopes $b_1=(3/5){\beta }_1$, $b_2={\beta }_2$, $b_3={\beta }_3$. At the common scale each inverse coupling is $A_U$. With $L=\log (M_{\mathrm{GUT}}/M_Z)$,Subtract the equations for factors 1 and 2 to solve $L=2\pi (A_1-A_2)/(b_1-b_2)$, then insert this into the difference of factors 3 and 2:This one-loop normalized hypercharge unification relation requires $b_1\ne b_2$. A scale above $M_Z$ also requires the inferred $L$ to be positive. If those two slopes coincide, unification instead requires $A_1=A_2$ and their difference cannot fix the scale.

In the convention $Q=T_3+Y$, one matter generation and one Higgs doublet have the following Standard Model representations. Spin is in units of $\hslash$; L/R describe chirality, not the spin projection. The $U(1{)}_Y$ entry is its hypercharge eigenvalue, defining the one-dimensional representation $e^{iY\omega }$.

There is no right-handed neutrino in the minimal field content being counted. Right-handed matter can alternatively be written as left-handed charge-conjugate Weyl fermions, in conjugate colour representations with opposite hypercharges. The quadratic indices and squared charges in the beta coefficients are unchanged.

For $SU(3)$, one generation supplies two colour fundamentals from $Q_L$ and one each from $u_R,d_R$. With three generations there are $n_f=3(2+1+1)=12$ Weyl fermions counted as fundamental copies, and no coloured scalar. ThusFor $SU(2)$, the three colours of $Q_L$ give three weak-doublet copies and $L_L$ gives one. There are $n_f=3(3+1)=12$ doublets across three generations and $n_s=1$ complex Higgs doublet, givingThese are spectator multiplicities in gauge beta functions: the colour index multiplies the weak contribution, while the weak index multiplies the colour contribution. Each two-component fermion is counted once; it must not be counted again as a separate antiparticle.

For $U(1{)}_Y$, sum squared hypercharges over all colour and weak components. One generation givesThe two complex Higgs components give ${\sum }_s{Y}_s^2=2(1/2{)}^2=1/2$. There is no Abelian gauge self-interaction contribution. ThereforeThe Higgs is counted only once for the whole model, whereas the fermionic generation is repeated three times. These are the Standard Model one-loop gauge coefficients in the unnormalized hypercharge convention. The grand-unified normalization instead has $b_1=41/10$. The ratio of slopes in the unification relation is $({\beta }_3-{\beta }_2)/(b_1-{\beta }_2)=-115/218$.

Write $t=\log \mu$ and use ${\alpha }_i=g_i^2/(4\pi )$. The renormalization-group beta function impliesIntegrating the one-loop running coupling equation givesHere the sign of ${\beta }_i$ is the sign convention in the question: a negative coefficient makes the inverse coupling increase towards high energy.

For gauge coupling unification, let $L=\log (M_{\mathrm{GUT}}/M_Z)/(2\pi )$ and let $a_G$ be the common inverse coupling at the unification scale. The normalized hypercharge coupling is $(5/3){\alpha }_1$, soSubtract the second relation from the first to determine $L$, then subtract it from the third. Provided $3{\beta }_1/5-{\beta }_2\ne 0$, this provesThe factors $3/5$ belong to the inverse normalized coupling; dropping them changes the prediction.

Use the Standard Model gauge group $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ and convention $Q=T_3+Y$. One fermion family and one complex Higgs doublet have the following Standard Model representations. The left-handed quark and lepton doublets contain two Weyl spinors each; the right-handed fields are single Weyl species.The row $u_R$ denotes the right-handed up quark. Equivalently an all-left-handed table replaces the three right-handed fields by $u_R^c:(\overline{3},1{)}_{-2/3}$, $d_R^c:(\overline{3},1{)}_{1/3}$, and $e_R^c:(1,1{)}_1$; these are alternative descriptions, not extra fermions. The minimal model has no right-handed neutrino. For completeness, the family-independent gauge fields are the spin-one gluons $(8,1{)}_0$, weak gauge bosons $(1,3{)}_0$, and hypercharge gauge boson $(1,1{)}_0$; they are not counted once per family.

For the Standard Model one-loop gauge coefficients, include three fermion families and one Higgs doublet. With $T(\mathbf{N})=1/2$, the colour Dynkin index sum in one family isThe first factor two counts the two members of $Q_L$; the colour trace is already included in the Dynkin index. There is no coloured scalar. HenceFor $SU(2{)}_L$, the three colours of $Q_L$ and the lepton doublet give $3(1/2)+1/2=2$ per family. The Higgs contributes $T_2(\varphi )=1/2$. ThusFor $U(1{)}_Y$, sum squared hypercharges over every colour and doublet component. One family contributesThe two complex Higgs components contribute $2(1/2{)}^2=1/2$. There is no Abelian gauge self-interaction term, soThis ${\beta }_1$ is for the unnormalized coupling $g_Y$ used in the question. For $g_1^{\mathrm{GUT}}=\sqrt{5/3}{g}_Y$, the corresponding coefficient is $(3/5){\beta }_1=41/10$. This distinction is required for a consistent unification calculation.

Take $T_i={\tau }_i/2$ and the hypercharge generator $Y=I_2/2$ on the scalar. The two commuting factors of the electroweak interaction have independent invariant gauge kinetic terms and therefore independent coupling constants. Non-Abelian gauge invariance requires the same $g$ for all three $SU(2)$ generators, while the $U(1)$ factor has its own $g^{\prime }$; no symmetry of this product group requires $g=g^{\prime }$.

A nonzero Higgs doublet can be rotated by $SU(2)$ toThe fundamental representation acts transitively on normalized complex doublets, so this fixes an internal direction without changing the vacuum norm. A general infinitesimal generator annihilates this vacuum only ifFor real parameters this means ${\alpha }_1={\alpha }_2=0$ and $\beta ={\alpha }_3$. Thus the connected stabilizer subgroup is one-dimensional:This proves the claimed electroweak symmetry breaking and identifies the unbroken electric charge generator.

The gauge covariant derivative isWriting the connection as ${\mathcal{A}}_{\mu }=g{A}_{\mu }^i{T}_i+g^{\prime }{B}_{\mu }Y$, its local transformation is ${\mathcal{A}}_{\mu }^{\prime }=U{\mathcal{A}}_{\mu }{U}^{-1}-i({\partial }_{\mu }U)U^{-1}$. Direct substitution gives $D_{\mu }^{\prime }(U\varphi )=U{D}_{\mu }\varphi$. Unitarity then provesso the scalar kinetic term is gauge-invariant.

At the constant vacuum the gauge covariant derivative becomesConsequentlywhere $\mathcal{V}=(A^1,A^2,A^3,B{)}^T$ andThe factor $1/2$ is the canonical real-vector mass-term normalization; the displayed electroweak doublet gauge-boson mass matrix contains squared masses when the gauge kinetic terms are canonical.

The electric charge generator acts on gauge fields through the Adjoint representation, not through the scalar's two-by-two matrix. With $U=e^{i\alpha Q}$, the brackets $[T_3,T_1]=i{T}_2$ and $[T_3,T_2]=-i{T}_1$ give $\delta A^1=\alpha A^2$, $\delta A^2=-\alpha A^1$, and neutral $A^3,B$. Hence the electromagnetic charge matrix of electroweak gauge fields in the requested basis isChoose positive gauge couplings by the gauge-field sign convention. Set $s_W=g^{\prime }/\sqrt{g^2+g^{\prime 2}}$ and $c_W=g/\sqrt{g^2+g^{\prime 2}}$. The Weinberg angle rotation diagonalizes the neutral block, while $W^{\pm }=(A^1\mp i{A}^2)/\sqrt{2}$ obey $\delta W^{\pm }=\pm i\alpha W^{\pm }$. The physical electroweak gauge bosons areThe charges are in units of $e=g{s}_W=g^{\prime }{c}_W$. Thus the tree-level electroweak gauge-boson masses satisfy $m_W=m_Z{c}_W$, and the unbroken photon is massless.