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The gauge-covariant kinetic term of the Higgs doublet gives $M_W^2(1+H/v{)}^2{W}_{\mu }^{+}{W}^{-\mu }+M_Z^2(1+H/v{)}^2{Z}_{\mu }{Z}^{\mu }/2$. Differentiation with respect to the external fields yields Feynman rules $2i{M}_V^2{g}_{\mu \nu }/v$ for one Higgs boson and two matching massive electroweak gauge bosons, and $2i{M}_V^2{g}_{\mu \nu }/v^2$ for two Higgs bosons and the same pair. Identical neutral fields supply combinatorial factors. The neutral radial Higgs boson has no corresponding elementary coupling to photons at tree level.

At tree level and energies much greater than a gauge-vector mass, an amplitude with longitudinal external gauge bosons equals the corresponding amplitude with their would-be Goldstone bosons, up to conventional external-state phases and relative $O(m_V/E)$ corrections for a nonzero leading amplitude. In a renormalizable gauge, the broken-symmetry identities relate $k_{\mu }{\mathcal{M}}^{\mu }$ to $m_V{\mathcal{M}}_{\chi }$; inserting ${\epsilon }_L^{\mu }=k^{\mu }/m_V+O(m_V/E)$ explains the result. Loop-level versions include field-normalization factors. This is different from an equivalence theorem for field redefinitions.

In unitary gauge, ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$, so this Higgs potential gives $M_H^2=2\kappa v^2$, a cubic coefficient $\kappa v$ and a quartic coefficient $\kappa /4$. The corresponding Feynman rules are $-3i{M}_H^2/v$ and $-3i{M}_H^2/v^2$. Writing the potential with an additional factor $1/2$ changes the relation between its named coupling and the Higgs boson mass; it does not change these mass-based rules.

For $V=\frac{1}{2}\lambda ({\varphi }^{†}\varphi -v^2/2{)}^2$ and the canonically normalized radial field, the expansion is $V=\frac{1}{2}\lambda v^2{H}^2+\frac{1}{2}\lambda v{H}^3+\frac{1}{8}\lambda H^4$. Comparing the quadratic term with $m_H^2{H}^2/2$ gives the displayed mass. Replacing the prefactor $\lambda /2$ by $\lambda$ changes the answer to $2\lambda v^2$, so the potential's normalization must be retained.

Higgsstrahlung produces a Higgs boson together with a massive electroweak gauge boson. At an electron-positron collider the process $e^{+}{e}^{-}\to Z^{\ast }\to ZH$ uses the Higgs boson coupling to Z bosons. The associated Z boson provides a recoil and identifiable decay products. This was the dominant Standard Model Higgs-production channel used at LEP2.

In a Heyting algebra, $a\to b$ is the greatest $x$ with $x\wedge a\le b$. Thus $x\le a\to b$ if and only if $x\wedge a\le b$. Every frame has this operation, obtained by taking the join of all such $x$. The open nucleus attached to $a$ is $x\mapsto a\to x$.

For $F\in C^2({\mathbb{R}}^n)$ with bounded Hessian matrix, Taylor's formula along the line from zero to $p$ gives $|F(p)|\le |F(0)|+|DF(0)||p|+\frac{1}{2}\Vert D^{2}F{\Vert }_{\infty }|p{|}^2$. Its gradient obeys $|DF(p)|\le |DF(0)|+\Vert D^{2}F{\Vert }_{\infty }|p|$. Consequently ${\int }_{\Omega }F(Du)$ is finite for $u\in H^1(\Omega )$ when $\Omega$ has finite measure, and $DF(Du)\in L^2$.

A nondegenerate Hermitian form over a quadratic finite field has an orthonormal basis. If all diagonal values vanished, evaluating $B(v+aw,v+aw)$ would give $\mathrm{Tr}(\bar{a}B(v,w))=0$ for every $a$, contradicting quadratic finite-field norm and trace surjectivity and nondegeneracy. Thus some $v$ has nonzero norm; rescaling using the field norm makes it one. Its orthogonal complement for a sesquilinear form is nondegenerate, so induction completes the basis.

In a two-dimensional nondegenerate Hermitian form over a quadratic finite field, choose an orthonormal basis $v,w$ and $a$ with $a\bar{a}=-1$. Then $e=v+aw$ is an isotropic vector. Choose $g$ with $B(e,g)=1$, and choose $c$ with $c+\bar{c}=B(g,g)$. The vector $f=g-ce$ satisfies $B(e,f)=1$ and $B(f,f)=0$. Thus $e,f$ give the displayed matrix, including in characteristic two. Iterating on nondegenerate orthogonal complement for a sesquilinear form leaves only a unit-norm line in odd dimension.

For $|t|<1$, this identity sums the Hermite function expansion into a Gaussian. One proof takes the Bargmann-Fock space inner product of the generating functions ${\pi }^{-1/4}\exp (-x^2/2+\sqrt{2}xz-z^2/2)$ at $\sqrt{t}z$. Writing $z=u+iv$, the two resulting real Gaussian integrals have quadratic coefficients $1+t$ and $1-t$ and linear coefficients $\sqrt{2t}(x+y)$ and $i\sqrt{2t}(x-y)$. Evaluation gives the formula. Setting $t=e^{-2s}$ and multiplying by $e^{-s}$ gives the harmonic oscillator transition kernel of $e^{-s(-d^2/d{x}^2+x^2)}$.

Use the unitary Fourier transform with phase $e^{-i\lambda x}$. It commutes with $-d^2/d{x}^2+x^2$, because differentiation transforms to multiplication by $i\lambda$ and multiplication by $x$ transforms to $id/d\lambda$. The Hermite differential equation has a unique polynomial solution of each degree up to scale, and multiplying by the Gaussian gives a Hermite function. Transforming a polynomial times this Gaussian again gives a polynomial times the Gaussian of the same degree; comparison of leading coefficients gives the eigenvalue $(-i{)}^n$.

For a hereditary hypergraph property of $r$-uniform hypergraphs, ${\mathcal{P}}_n$ is the family on the labelled vertex set $[n]$. For $n\ge r$, its normalized speed lies between zero and two and is nonincreasing. Choose a uniformly random member on $n+1$ vertices and record its edge indicators. Each edge occurs in $n+1-r$ vertex-deletion restrictions. The Shearer inequality bounds its information entropy by the sum of the restriction entropies divided by $n+1-r$. Each restriction has at most $|{\mathcal{P}}_n|$ possible values. Division by $\left(\genfrac{}{}{0ex}{}{n+1}{r}\right)$ gives $\log d_{n+1}\le \log d_n$. Empty families are handled directly by heredity.

Every infinite-dimensional Banach space contains either an unconditional basic sequence or a hereditarily indecomposable Banach space. For a space with a basis and no unconditional basic sequence, apply the block Ramsey theorem to finite sequences whose odd and even spans contain nearby unit vectors. A diagonal sequence of successively sharper games gives a subspace in which every pair of infinite-dimensional subspaces has angle zero.

A set has countable-subset closure when every countable subset of it is one of its elements, including the empty subset. In ZFC, the hereditarily countable sets form the least such set. First realize them as the separated subset of $V_{{\omega }_1}$ with countable transitive closure. Countable unions show closure; well-founded induction proves leastness because each of these sets is a countable collection of earlier members.

A countable well-founded relation that is an extensional relation codes a transitive set through the Mostowski collapse theorem. Adding a distinguished point selects one decoded set. Every hereditarily countable set has such a code by enumerating its transitive membership ancestry. There are only continuum many codes. In ZFC, selecting least codes in a well-order gives an injection of the hereditarily countable sets into a set of size $2^{{\aleph }_0}$.

The transitive set of hereditarily countable sets satisfies every ZFC axiom except the Axiom of power set. For internal definable functions on a countable domain, ambient Axiom schema of replacement gives a countable range of hereditarily countable sets, which belongs to the structure by countable-subset closure. The same closure proves axiom schema of collection and retains countable choice-function graphs. All subsets of $\omega$ are elements, but their uncountable full power set is not an element, so Power Set fails.

For a finite module acted on by a cyclic group, write $D=\sigma -1$ and $N=1+\sigma +\cdots +{\sigma }^{n-1}$. The two Tate cohomology group sizes are $|\ker D|/|NM|$ and $|\ker N|/|DM|$. The equalities $|NM|=|M|/|\ker N|$ and $|DM|=|M|/|\ker D|$ show that these sizes are equal. The exact-sequence multiplication rule for the Herbrand quotient consequently makes it unchanged by finite-index modifications of lattices.

For a symmetric reentrant notch opening towards negative $x_1$, let its semi-angle be $0\le \gamma \le \pi /2$. On the upper free face, tensile tangential stress selects $p=-k$, $\varphi =\pi /4-\gamma$. The adjacent centred fan has radial $\beta$ lines and circular $\alpha$ lines: $\varphi =\vartheta -\pi /2$, $p=-k+2k[\vartheta -(3\pi /4-\gamma )]$, for $\pi /4\le \vartheta \le 3\pi /4-\gamma$. It joins a constant stress region containing the forward axis, where $\varphi =-\pi /4$ and $p=-k(1+\pi -2\gamma )$. Therefore ${\sigma }_{22}=k(2+\pi -2\gamma )$ in that fully plastic notch-tip region. This is a local stress field, not a claim that an arbitrary specimen is plastically yielded everywhere.

In a uniform isotropic material, these scalar P-SV displacement potentials separate longitudinal P waves from vertically polarized SV-waves in an $x,z$ plane. They satisfy ${\varphi }_{tt}={\alpha }^2\Delta \varphi$ and ${\psi }_{tt}={\beta }^2\Delta \psi$, with compressional and shear speeds $\alpha ,\beta$. Reversing the sign convention for $\psi$ changes its polarization column but not the physical P-SV directional impedance matrix or the separate modal energy fluxes.

The displayed space curve, $s>0$, lies on a cone and is parametrized by arc length. It has a constant tangent angle with its axis, while the turn spacing shrinks geometrically toward the limiting apex. Its curvature is $1/(\sqrt{2}s)$: differentiate its unit tangent with respect to $s$. Infinitely many turns accumulate at the apex with finite arc length; the apex itself is not a regular point of the curve.