Complex scalar field

A complex scalar field has distinct particle and antiparticle excitations and a global phase symmetry.

This interaction preserves the global U(1) symmetry with relative charges $q_{\varphi }=1$, $q_{\Phi }=2$. The two conjugate interaction vertices have Feynman rule $-ig$, with either one charge-two arrow entering and two charge-one arrows leaving, or all arrows reversed. For $\varphi \varphi \to \varphi \varphi$ only the heavy $s$-channel exchange occurs at tree level. For $\varphi \bar{\varphi }\to \varphi \bar{\varphi }$, with outgoing particle labelled 3 and outgoing antiparticle labelled 4, only the heavy $u$-channel exchange occurs. For small momenta and heavy mediator mass, the amplitudes coincide but the identical-particle total scattering cross-section is smaller by the identical final-state symmetry factor $1/2$.

For the charge-two cubic complex-scalar interaction, integrating out a complex scalar field of mass $m$ generates the displayed local effective Lagrangian term when external energies and momenta are small compared with $m$. A constant $\Phi$ gives two real squared masses $m^2\pm g|\Phi |$. Expanding half the sum of their one-loop logarithms gives $V_1\supset -g^4|\Phi {|}^4{I}_4/4$, with the Euclidean massive loop integral $I_4=1/(96{\pi }^2{m}^4)$. Since ${\mathcal{L}}_{\mathrm{eff}}=-V_1$ for a constant field, $2!2!$ external attachments give the scattering amplitude $g^4/(96{\pi }^2{m}^4)$ for either heavy-heavy or heavy-antiparticle elastic scattering. This agrees with the unique charge-allowed light box diagram.

A global phase symmetry sends a complex field to $e^{i\alpha }\varphi$ with the same constant real $\alpha$ everywhere. It is distinct from allowing a spacetime-dependent phase together with a compensating gauge-field transformation. A global phase symmetry can produce a conserved Noether current and, in an appropriate ordered mean-field state, a continuum of degenerate phase choices.

In rescaled units, $V(\varphi )=-|\varphi {|}^2+e^{\lambda |\varphi {|}^2}$ is bounded below only for $\lambda >0$. With $s=|\varphi {|}^2$, its derivative is $-1+\lambda e^{\lambda s}$. For $\lambda \ge 1$ the unique minimum is $s=0$; for $0<\lambda <1$ the minima form a circle $s=-\ln \lambda /\lambda$. The latter phase has a massless angular Goldstone boson and canonical radial squared mass $-2\ln \lambda$. At $\lambda =1$ the quadratic curvature vanishes at the single symmetric minimum; that accidental critical masslessness is not a Goldstone mode.

For the connected tree scattering amplitude of one particle and one antiparticle of a complex scalar field, the Dyson series inserts $-i\lambda {\varphi }^{†2}{\varphi }^2/4$. There are $2!$ assignments of the particle and antiparticle external attachments to each pair of identical field factors, giving $2!2!=4$ and hence the displayed Feynman vertex. With relativistic normalization of a one-particle state, integrating the vertex position gives $i\mathcal{T}(2\pi {)}^4{\delta }^4(p_1+q_1-p_2-q_2)$ with $\mathcal{T}=-\lambda$. Vacuum diagrams and one-particle self-energy insertions are separate disconnected contributions; normal ordering eliminates the single-vertex vacuum and tadpole contractions.

For $\mathcal{L}=|\partial \varphi {|}^2-m^2|\varphi {|}^2-{\mu }^{ϵ}g|\varphi {|}^4/4$ in $d=4-ϵ$, the directed tadpole diagram has symmetry factor one. Its divergent Feynman contribution to $i{\Gamma }^{(2)}$ is $ig{m}^2/(8{\pi }^2ϵ)$. A mass insertion $-i{\delta }_{m^2}$ therefore gives the displayed minimal subtraction scheme counterterm. Differentiating at fixed bare mass and coupling gives the displayed ${\gamma }_m=\frac{1}{2}\mu d\log m^2/d\mu$. The equivalent two-real-field formulation has vertex coefficients $3g/2$ for a pure species and $g/2$ for two of each, giving total tadpole weight $(3g/2+g/2)/2=g$.

A nonlinear complex scalar field theory with a global phase symmetry and a curved target-space kinetic coefficient. In physical coordinates its classical density is displayed above; dimensionless coordinates measured in $M^{-1}$ put a common $M^2$ outside the reduced density. Its regular coordinate domain is ${\lambda }^2|\psi {|}^2<1$. Rotating localized solutions are charged complex sine-Gordon solitons. The local classical density does not by itself specify the treatment of the singular coordinate boundary or a global quantum completion.

The rest-frame field $\psi =\cos \alpha e^{i\sin \alpha t}/[\lambda \cosh (\cos \alpha x)]$ has mass $4M\cos \alpha /{\lambda }^2$ and charge $4[\mathrm{sgn}(\alpha )\pi /2-\alpha ]/{\lambda }^2$, for the generator $\delta \psi =i\psi$ and $\alpha \ne 0$. The conserved phase angle is a collective coordinate with momentum $Q$. Quantization of a periodic soliton coordinate gives integer $Q$ in units $\hslash =1$ and the displayed leading mass formula. On the regular branch $0<|Q|<2\pi /{\lambda }^2$, its concave increasing sine law prevents fragmentation into smaller like-charge states.

At zero rotation the field reaches ${\lambda }^2|\psi {|}^2=1$ at its center. Its energy remains finite, but its charge expression is singular and has the two displayed one-sided limits. The regular local branch excludes this endpoint. A global completion must say whether and how these two limiting charge labels represent a physical state.

An additional integer-level completion identifies the charge labels modulo $k$ and has $k-1$ nonzero sectors. For even $k$, the two maximal labels describe one self-conjugate sector. This interpretation is specified in Dorey and Hollowood, section 2; it is additional to the local classical density.

A free complex scalar field has independent particle and antiparticle mode operators. With $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_p]$, the expansion $\varphi =\int d{\Pi }_p(a{e}^{-ipx}+b^{†}{e}^{ipx})$ and its adjoint reproduce equal-time canonical commutation relations when $[a(\mathbf{p}),a^{†}(\mathbf{q})]=[b(\mathbf{p}),b^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ and cross commutators vanish. The canonical momenta are $\pi ={\dot{\varphi }}^{†}$, ${\pi }^{†}=\dot{\varphi }$. An overall phase on $b$ can reverse the sign of the antiparticle term without changing the theory.

For positive kinetic coefficient $\zeta$ in ${\mathcal{L}}_0=\zeta ({\partial }_{\mu }{\varphi }^{\ast }{\partial }^{\mu }\varphi -m^2{\varphi }^{\ast }\varphi )$, the canonical momentum is ${\pi }_{\varphi }=\zeta {\dot{\varphi }}^{\ast }$. Positive-energy oscillator modes with unit oscillator commutators have amplitude $1/\sqrt{2\zeta E_{\mathbf{p}}}$. The mixed time-ordered propagator is $C={\zeta }^{-1}{D}_F$, with $D_F$ the ordinary scalar propagator. This also follows by inverting the quadratic form in the path integral. Rescaling $\varphi$ changes interaction coefficients as well as external correlation-function factors.

With the invariant measure $d{\Pi }_p=d^{3}p/[(2\pi {)}^{3}2E_p]$, a free complex scalar field has particle and antiparticle modes. At any fixed time the displayed inverse and $b^{†}(p)=\int d^{3}x{e}^{-ip\cdot x}(E_p\varphi -i\dot{\varphi })$ project onto its positive- and negative-frequency components. The equal-time canonical commutation relations then give $[a(p),a^{†}(q)]=[b(p),b^{†}(q)]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$; cross terms vanish because their coefficient contains $E_p-E_q$ on the momentum delta function.

For the global phase transformation $\delta \varphi =-i\alpha \varphi$, the Noether current is $j^{\mu }=i({\varphi }^{†}{\partial }^{\mu }\varphi -{\partial }^{\mu }{\varphi }^{†}\varphi )$. The Klein-Gordon equation gives ${\partial }_{\mu }{j}^{\mu }=0$. With zero vacuum charge, its normal-ordered charge is $Q=\int d{\Pi }_p(a^{†}a-b^{†}b)$. Thus $[Q,a^{†}]=a^{†}$, $[Q,b^{†}]=-b^{†}$ and $[Q,\varphi ]=-\varphi$. Particle and antiparticle creation carry opposite charges while both increase energy.

Substituting the free complex scalar field modes into $i\int ({\varphi }^{†}\dot{\varphi }-{\dot{\varphi }}^{†}\varphi )d^{3}x$ gives the formally ordered expression $Q_{\mathrm{bare}}=\int d{\Pi }_p(a^{†}a-b{b}^{†})$. Its divergent vacuum constant is removed by normal ordering, equivalently by choosing $Q|0⟩=0$. The resulting complex scalar charge operator has $[Q,a^{†}]=a^{†}$ and $[Q,b^{†}]=-b^{†}$. Subtracting the constant changes neither its action on fields nor current conservation, but it is essential when identifying the vacuum charge and the charges of one-particle states.

Spatial integration of the free Hamiltonian density cancels particle-antiparticle pair terms through $E_p^2-{\mathbf{p}}^2-m^2=0$. The raw Hamiltonian is $\int d{\Pi }_p{E}_p(a^{†}a+b{b}^{†})$. Normal ordering removes the field-independent vacuum energy and gives $H=\int d{\Pi }_p{E}_p(a^{†}a+b^{†}b)$. Both creation operators increase the energy by $E_p$, so a negative-frequency field mode represents a positive-energy antiparticle, not a negative-energy state.

For $\mathcal{L}={\partial }_{\mu }{\psi }^{\ast }{\partial }^{\mu }\psi -V(|\psi {|}^2)$, constant $\alpha$ preserves both terms. This is an internal symmetry of a classical field theory with circle group $U(1)$. A spacetime-dependent phase would introduce derivative terms, so the ordinary-derivative density has only the global symmetry. The associated current is the Noether charge of a complex scalar field construction; the orientation $e^{-i\alpha }$ fixes its overall sign.

In a phase-invariant vacuum, a correlator with $r$ insertions of $\varphi$ and $s$ of ${\varphi }^{\ast }$ gains the phase $e^{i\alpha (r-s)}$. Its invariance for all real $\alpha$ forces it to vanish unless $r=s$. Balanced local interactions and the absence of same-charge free contractions give the same rule order by order in Wick's theorem. A symmetry-breaking vacuum is an essential exception to the vacuum-invariance premise.

For the global phase symmetry of a complex scalar field oriented as $\delta \psi =-i\epsilon \psi$, the Noether current is $j^{\mu }=i({\psi }^{\ast }{\partial }^{\mu }\psi -\psi {\partial }^{\mu }{\psi }^{\ast })$. The Euler-Lagrange field equations $\square \psi +V^{\prime }(|\psi {|}^2)\psi =0$ and its complex conjugate give ${\partial }_{\mu }{j}^{\mu }=i({\psi }^{\ast }\square \psi -\psi \square {\psi }^{\ast })=0$ for a real differentiable potential. Integrating $j^0$ gives $Q$ when boundary flux vanishes. A charged scalar coupled to electromagnetism has electric charge $qQ$ with the chosen charge normalization; without that physical identification it is an internal charge.