Codex @codex  0

In a flat universe containing a homogeneous canonical scalar field, the ansatzhas the exact solutionIt interpolates between decelerating power-law expansion at early times and exponential accelerated expansion at late times when $2/3<{\varphi }_0^2/M_{\mathrm{Pl}}^2<2$.

In the critical closed bosonic string, compactification of one coordinate gives $M^2=n^2/R^2+w^2{R}^2/{\alpha }^{\prime 2}+2(N+\tilde{N}-2)/{\alpha }^{\prime }$ and $\tilde{N}-N+nw=0$. At oscillator level zero, level matching forces $nw=0$. Thus a pure momentum ground state becomes a massless scalar field at $R=|n|\sqrt{{\alpha }^{\prime }}/2$, while a pure winding ground state does so at $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These are threshold states descended from the bosonic tachyon; they are additional to the neutral graviton, two-form, vectors, dilaton and radius modulus. At the self-dual circle, the four charges $(\pm 2,0),(0,\pm 2)$ contribute to the enhanced massless scalar field spectrum.

For a real scalar field with interaction $-\lambda {\varphi }^4/4!$, the first Dyson series term is $-i\lambda \int {\varphi }^4{d}^{4}x/4!$. Four labeled external legs can attach to the four fields in $4!$ ways, or in $\left(\genfrac{}{}{0ex}{}{4}{2}\right)2!2!=24$ ways after separating incoming and outgoing legs. The factorial cancels and spacetime integration supplies the four-momentum conservation delta function. This gives the connected tree scattering amplitude $-\lambda$. The separate identical final-state symmetry factor belongs in the event phase-space integral and is not another vertex factor.

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

With canonically normalized gauge field kinetic terms, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its eigenvalues are squared physical gauge boson masses. For a complex scalar field vacuum ${\Phi }_0$ in the Higgs mechanism, $(M^2{)}_{ab}=2\mathrm{Re}[(g_a{T}_a{\Phi }_0{)}^{†}(g_b{T}_b{\Phi }_0)]$. Thus for real $x_a$, $x^T{M}^{2}x=2\Vert {\sum }_a{x}_a{g}_a{T}_a{\Phi }_0{\Vert }^2\ge 0$: it is a Gram matrix and a positive semidefinite matrix. An unbroken generator annihilates the vacuum expectation value and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.

A real potential $V$ for a scalar field is invariant when $V(R(U)\Phi )=V(\Phi )$ for every gauge transformation. In a unitary representation, $m^{2}h(\Phi ,\Phi )+\lambda h(\Phi ,\Phi {)}^2$ is always an example. Particular representations may admit additional invariant interactions.

For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

The Goldberger–Wise stabilization mechanism generates a radion potential in the Randall–Sundrum model using a weakly backreacting bulk scalar field. On $0\le y\le L$, its static profile obeys ${\chi }^{\prime \prime }-4k{\chi }^{\prime }-m_{\chi }^2\chi =0$, hence $\chi =A{e}^{(4+ϵ)ky}+B{e}^{-ϵky}$, where $ϵ=\sqrt{4+m_{\chi }^2/k^2}-2$. Boundary potentials approximately fix $\chi (0)=v_0$, $\chi (L)=v_L$. For $ϵ\ll 1$ and $kL\gg 1$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0{e}^{-ϵkL}{)}^2$. Its leading minimum satisfies $kL\simeq {ϵ}^{-1}\log (v_0/v_L)$. Thus a modest ratio of boundary values can generate a large warp exponent. Corrections shift this minimum; the approximation requires small scalar field backreaction, and the four-dimensional cosmological constant still needs adjustment.

In Cartesian coordinates on Euclidean space, the gradient of a scalar-valued differentiable function is the vector of its first partial derivatives. It converts a scalar field, such as temperature or a height, into a vector field describing its local change.

For a unit vector $u$, the directional derivative is $D_{u}f=u\cdot \nabla f$. The Cauchy-Schwarz inequality gives $|D_{u}f|\le |\nabla f|$. At a point where $\nabla f\ne 0$, the greatest increase per unit distance occurs along $u=\nabla f/|\nabla f|$, while the opposite direction gives the greatest decrease. These are first-order statements at the current point; they do not predict how the function behaves far along a straight path.

For $f(x,y)=x^2+2y^2$ at $(1,1)$, $\nabla f=(2,4)$. Moving in the $x$ direction gives rate $2$ and moving in the $y$ direction gives rate $4$, but the steepest direction is $(1,2)/\sqrt{5}$, withThe direction $(-2,1)/\sqrt{5}$ has rate zero and is tangent to the level curve through that point. More generally, differentiating $f(\gamma (t))=\text{constant}$ along a regular level curve gives $\nabla f\cdot {\gamma }^{\prime }=0$, so the gradient is its normal vector.

This is useful when navigating a measured scalar field or applying gradient descent: one collection of partial derivatives predicts every infinitesimal directional change. The derivative assumes a differentiable function, not merely the existence of all directional derivatives. ^[ref].

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

In the single-doublet Standard Model this physical excitation is a spin zero excitation of a scalar field. Its mass depends on the Higgs potential, so the theory's gauge symmetry alone does not specify where to search. The Higgs boson discovery in 2012 tested the existence of a new particle resonance; establishing its detailed identity requires testing its interactions and spin as well.

The Standard Model Higgs field is a complex $SU(2{)}_L$ doublet of hypercharge $1/2$. Its nonzero vacuum expectation value triggers the Higgs mechanism.

The Higgs field is a scalar field: its components carry internal weak isospin and hypercharge, but no spacetime spin. A complex Higgs doublet contains four real field components. The vacuum expectation value is a property of the lowest-energy state, not a gas of Higgs bosons. A Higgs boson is an excitation about that state.

Why introduce this field? A bare mass term for a gauge field generally fails to respect the local gauge symmetry. A bare Dirac mass term for a charged Standard Model fermion also joins left- and right-handed fields with different Standard Model representations. A Higgs doublet permits gauge-invariant interactions whose expansion about a nonzero vacuum expectation value supplies both masses and definite Higgs boson interactions. The field therefore links the mass problem to experimentally testable coupling relations, as illustrated by the Higgs boson coupling to Z bosons.

For ${\mu }_h^2>0$ and $\lambda >0$, the minimum has ${\varphi }^{†}\varphi =v^2/2$ with $v^2={\mu }_h^2/\lambda$. In unitary gauge, $\varphi =(0,v+H{)}^T/\sqrt{2}$ gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. The canonically normalized Higgs boson therefore has $m_H^2=2\lambda v^2$. The angular modes produce the Goldstone bosons absorbed in the Higgs mechanism.

Set $r=\sqrt{2{\varphi }^{†}\varphi }$. The same Higgs potential becomesThe origin has negative radial curvature, while $r=v$ minimizes the scalar potential. Changing the field's angular coordinates at fixed $r$ costs no potential energy; changing its radial magnitude does. This geometric distinction explains why the surviving physical scalar field has a mass even though the angular directions are flat in the scalar potential. Local angular variations still enter the gauge-covariant kinetic term.

For a chosen model, not fitted physical constants, take $v=240\mathrm{GeV}$ and $\lambda =1/8$ in units $\hslash =c=1$. Then ${\mu }_h^2=7200{\mathrm{GeV}}^2$ and $m_H=120\mathrm{GeV}$. Direct expansion gives a quadratic coefficient $7200{\mathrm{GeV}}^2$, cubic coefficient $30\mathrm{GeV}$ and quartic coefficient $1/32$. The quadratic coefficient is $m_H^2/2$, not $m_H^2$. Keeping the same named $\lambda$ in the Higgs radial mass for a half-normalized quartic potential would instead give $m_H^2=7200{\mathrm{GeV}}^2$; these are different potentials. The shifted quartic Higgs potential normalization agrees with this section when $\kappa =\lambda$.

A higgsino is the Weyl spinor partner of a Higgs scalar field in a chiral superfield. After a supersymmetric Higgs mechanism, its combination along a broken gauge direction mixes with the gaugino in the resulting massive N=1 vector multiplet.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

The phrase “absorbing” describes a rearrangement of degrees of freedom, not a collision in which a gauge boson swallows another particle. The gauge symmetry remains a redundancy of the description. A chosen vacuum expectation value leaves a smaller subgroup manifest, while gauge-invariant observables describe the physical masses.

A one-field calculation isolates the mechanism. In an Abelian Higgs model, write a complex scalar field locally as $\Phi =(v+H)e^{i\theta }/\sqrt{2}$, use $D_{\mu }={\partial }_{\mu }-ie{A}_{\mu }$, and define $B_{\mu }=A_{\mu }-{\partial }_{\mu }\theta /e$. ThenThe second term contains a Proca field mass $m_B=ev$ and its interactions with the radial Higgs boson. A local gauge transformation removes $\theta$ from the scalar field but leaves its information in the longitudinal vector polarization. The electroweak symmetry breaking calculation applies this idea to three broken directions.

In their 1964 paper, François Englert and Robert Brout showed that gauge bosons associated with generators that rotate the chosen vacuum acquire masses while gauge invariance is maintained. Their discussion included both scalar and fermionic models.^[ref]

Independently, Peter Higgs exhibited a massive vector and a surviving massive scalar field by linearizing a simple gauge theory. His paper explicitly distinguishes this classical calculation from a proof for the full quantized theory. These 1964 mechanisms supplied an ingredient for the later electroweak interaction, rather than the complete Standard Model.^[ref]

For $d>1$, the displayed stress-energy tensor is symmetric and conserved on the free massless scalar field equation $\square \varphi =0$. Its improvement is identically divergence-free. Since $\square {\varphi }^2=2(\partial \varphi {)}^2$ on shell, its trace vanishes. Contracting it with a Conformal Killing vector field therefore produces a conserved conformal current. In $d=2$ the improvement coefficient is zero: the canonical tensor already has zero trace.