Higgs mechanism & unitarity constraints

Perturbative unitarity for upper bound on the Higgs mass

Fabio Cufino - Angelo Arisi

TU Dortmund University

June 27, 2024

Toy Model: The Abelian Higgs Model

Mass of a gauge boson

Simplified verison of SM: \(U(1)\) gauge theory with a single gauge field, the photon \(A_{\mu}\)

\[ \mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \;\;\;\;\;\;\;\; \text{with} \;\;\;\;\;\;\;\; F_{\mu\nu} = \partial_\nu A_\mu - \partial_\mu A_\nu \]

Local \(U(1)\) gauge invariance requires \(\mathcal{L}\) to be invariant under: \[ \quad \quad \quad A_{\mu}(x) \rightarrow A_{\mu}(x) - \partial_{\mu}\eta(x)\]

If mass term:

\[ +\frac{1}{2} m^2 A_{\mu} A^{\mu} \]

Violates the local gauge invariance.

The Abelian Higgs Model

Extension of the Model

Single complex scalar field \(\phi\) that couples to the photon \[ \;\;\;\;\;\;\;\; \quad\quad\quad\quad\quad \mathbfcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + |D_\mu \phi|^2 - V(\phi) \]
- \(D_\mu = \partial_\mu - ieA_\mu\)
- \(V(\phi) = \mu^2 |\phi|^2 + \lambda |\phi|^4\)

\(\mathcal{L}\) is invariant under:

Global \(U(1)\) rotations: \(\;\; A_{\mu}(x) \rightarrow e^{i\theta}A_{\mu}(x), \;\; \phi \rightarrow e^{i\theta} \phi\)
Local gauge transformations: \(\;\; A_\mu \rightarrow A_\mu - \partial_\mu \eta(x), \;\;\;\; \phi \rightarrow e^{-ie\eta(x)} \phi\)

The scalar potential \(V(\phi)\)

Spontaneus Symmetry Breaking

The potential has two different forms:

\(\mu^2 > 0\): The vacuum state is at \(\phi = 0\)
- no symmetry breaking

\(\mu^2 < 0\): The vacuum state is \[ \langle \phi \rangle = \sqrt{-\frac{\mu^2}{2\lambda}} e^{i\theta} = \frac{v}{\sqrt{2}} e^{i\theta} \]
- Choise \(\theta = 0 \; \Longrightarrow \;\) breaks the global \(U(1)\) symmetry

Consequences of SSB in the Abelian Model

Consequences of Symmetry Breaking

Perturbation theroy: It is convenient to parametrize \(\phi\) \[ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \phi \equiv \frac{1}{\sqrt{2}} e^{i \frac{\chi(x)}{v}} \left( v + h(x) \right) \]
- \(\chi(x), h(x)\): scalar fields

Substitute \(\phi\) into \(\mathcal{L}\):

\[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + |D_\mu \phi|^2 - V(\phi) \]

\[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \left( \frac{v + h}{\sqrt{2}} \right)^2 \left[ -i \frac{\partial_\mu \chi}{v} + \frac{\partial_\mu h}{v + h} - ie A_\mu \right] \left[ i \frac{\partial_\mu \chi}{v} + \frac{\partial_\mu h}{v + h} + ie A_\mu \right] - \left( \frac{\mu^4}{\lambda} + \mu^2 h^2 + \frac{1}{2} \sqrt{\lambda} \mu h^3 + \frac{1}{16} \lambda h^4 \right) \]

Consequences of SSB in the Abelian Model

Consequences of Symmetry Breaking

Terms involving only \(A_\mu\):

\[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{1}{2} e^2 v^2 A_\mu^2 + \cdots \]

Gauge field \(A_\mu\) acquires a mass: \(\;\; m_A^2 = e^2v^2\)

Terms for the potential \(V(\phi)\):

\[ -\frac{\mu^4}{\lambda} - \mu^2 h^2 - \frac{1}{2} \sqrt{\lambda} \mu h^3 - \frac{1}{16} \lambda h^4 \]

The Higgs field \(h\) has a mass: \(\;\; m_h^2 = -2\mu^2\)

Kinetic term for the \(\chi(x)\) scalar field \(\;\; \Longrightarrow \;\;\) The Goldstone boson \(\chi(x)\)

Lagrangian spectrum

Decoupling limit & Unitary gauge

Simplify things by decoupling \(\chi(x)\) through the limit \(\mu, \lambda → \infty\) with \(v\) fixed

\[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \frac{1}{2} m_A^2 \left( A_\mu + \frac{1}{e v} \partial_\mu \chi \right)^2 \]

\(\mathcal{L}\) content:
- Gauge boson mass term
- Kinetic term for \(\chi(x)\)
- Kinetic mixing term \(A_\mu \partial_\mu \chi \; \Longrightarrow\) removed through gauge-fixing

Unitary gauge:

\[\begin{align*} A_\mu \rightarrow A_\mu' &= A_\mu - \frac{1}{e v} \partial_\mu \chi \\ \chi(x) &= 0 \end{align*}\]

Proca Lagrangian:

\[ \mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \frac{1}{2} m_A^2 A_\mu A^\mu \]

Intermezzo: nothing is broken

Hidden Symmetry

The VEV breaks the global \(U(1)\) symmetry: \(\;\;\;\; \phi(x) \rightarrow e^{i\theta} \phi(x)\)
Parametrization

\[ \phi \equiv \frac{1}{\sqrt{2}} e^{i \frac{\chi(x)}{v}} \left( v + h(x) \right) \]

The symmetry is not broken: \(\chi(x) \rightarrow \chi(x) + v \theta \;\) and \(\; h(x) \rightarrow h(x)\)

Consequence:

The shift forbids a mass term for \(\chi(x)\)
Graphical interpretation: move around the flat direction of \(V(\phi)\)

Introduction to the Weinberg-Salam Model

Weinberg-Salam Model

Extends the Abelian Higgs model to an \(SU(2)_L×U(1)_Y\) gauge theory.
Introduces three \(SU(2)_L\) gauge bosons \(W_\mu^I\) \((I = 1, 2, 3)\) and one \(U(1)_Y\) gauge boson \(B_\mu\)

The kinetic energy terms:

\[ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad L = -\frac{1}{4}W^I_{\mu\nu}W^{I\mu\nu} - \frac{1}{4}B_{\mu\nu}B^{\mu\nu} \]
- \(W^I_{\mu\nu} = \partial_\nu W^I_\mu - \partial_\mu W^I_\nu + g \epsilon^{IJK} W^J_\mu W^K_\nu\)
- \(B_{\mu\nu} = \partial_\nu B_\mu - \partial_\mu B_\nu\)

Scalar Potential and VEV in Weinberg-Salam Model

Scalar Potential and VEV

Coupled to the gauge fields the complex scalar \(SU(2)\) doublet \(\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}\)
The scalar potential is similar to the Abelian model: \[ V(\Phi) = \mu^2 |\Phi^{\dagger} \Phi| + \lambda (|\Phi^{\dagger} \Phi|)^2 \]
For \(\mu^2 < 0\), the VEV of \(\Phi\) is: \[ \langle \Phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v \end{pmatrix} \]
With this choice the electromagnetig charge

\[ Q = \frac{(\tau_3 + Y)}{2} \;\;\; \text{with} \;\;\; Y_{\phi} = 1 \]

Scalar contribution to the Lagrangian

Scalar contribution

Scalar Contribution to the Lagrangian: \(\;\;\; \mathcal{L}_s = (D^\mu \Phi)^\dagger (D_\mu \Phi) - V(\Phi)\)
Covariant Derivative:

\[ D_\mu = \partial_\mu + i \frac{g}{2} \tau \cdot W_\mu + i \frac{g'}{2} B_\mu Y \]
In Unitary Gauge:

\[ \Phi(x) = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v + h(x) \end{pmatrix}, \]
- no Goldstone bosons
- Higgs field remains with mass

\[ M^2 \sim \frac{1}{2} (0, v) \left( \frac{1}{2} g \tau \cdot W_\mu + \frac{1}{2} g' B_\mu \right)^2 \begin{pmatrix} 0 \\ v \end{pmatrix} \]

Weak Mixing Angle and Massless Photon

Gauge Boson Masses

The physical gauge fields are \(W^\pm\), \(Z\) and \(\gamma\)

\[\begin{align*} W^\pm_\mu &= \frac{1}{\sqrt{2}} (W^1_\mu \mp i W^2_\mu) \\[10pt] Z^\mu &= \frac{-g'B_\mu + g W^3_\mu}{\sqrt{g^2 + g'^2}} \equiv -\sin \theta_W B_\mu + \cos \theta_W W^3_\mu \\[10pt] A^\mu &= \frac{g B_\mu + g' W^3_\mu}{\sqrt{g^2 + g'^2}} \equiv \cos \theta_W B_\mu + \sin \theta_W W^3_\mu. \end{align*}\]

The weak mixing angle \(\theta_W\) is defined by \(\sin \theta_W = \frac{g'}{\sqrt{g^2 + g'^2}}\)

The masses of the gauge bosons are:

\[ M_W^2 = \frac{1}{4}g^2v^2 \quad \;\;\; M_Z^2 = \frac{1}{4}(g^2 + g'^2)v^2 \quad \;\;\; M_\gamma^2 = 0 \]

Higgs and Fermions

Lepton masses

Different transformation for \(LH\) (\(SU(2)\) doublets) and \(RH\) (\(SU(2)\) singlets) chiral states.
Fermion mass term in Lagrangian: \[ -m\bar{\psi}\psi = -m \left( \bar{\psi}_R \psi_L + \bar{\psi}_L \psi_R \right) \]
- Does not respect \(SU(2)_L\) \(\times\) \(U(1)_Y\) gauge symmetry.

\(SU(2)\) local Gauge Transformation:
- Field \(\phi(x)\) is \(SU(2)\) doublet: \(\;\; \phi' = \left( I + ig_W \epsilon(x) \cdot \mathbf{T} \right)\phi\)
- \(LH\) doublet of fermion fields: \(\;\; \bar{L}' = \bar{L}\left(I - ig_W \epsilon(x) \cdot \mathbf{T} \right)\)

Combination \(\bar{L} \phi\) is invariant under \(SU(2)_L\) gauge transformations.

Higgs and Fermions

Leptons and down-type quarks masses

Combining with a \(RH\) singlet \(\;\Longrightarrow \;\) \(\bar{L}\phi R\) is invariant under \(SU(2)_L\) and \(U(1)_Y\) gauge transformations.
- Lagrangian term: \(-g_f(\bar{L} \phi R + \bar{R} \phi^\dagger L)\)

Electron Example:

For the \(SU(2)_L\) doublet containing the electron:

\[ \;\;\; \mathcal{L}_e = -g_e \left[ \begin{pmatrix} \bar{\nu}_e & \bar{e} \end{pmatrix}_L \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix} e_R + \bar{e}_R \begin{pmatrix} \phi^{+*} & \phi^{0*} \end{pmatrix} \begin{pmatrix} \nu_e \\ e \end{pmatrix}_L \right] \\ \]

\[ \mathcal{L}_e = -\frac{g_e v}{\sqrt{2}} (\bar{e}_L e_R + \bar{e}_R e_L) - \frac{g_e}{\sqrt{2}} h (\bar{e}_L e_R + \bar{e}_R e_L) \]

Higgs and Fermions

up-type quarks masses

Yukawa coupling \(g_e\) is chosen to match observed electron mass: \[ g_e = \sqrt{2} \frac{m_e}{v} \]

Problem: Non-zero VEV is in the lower (neutral) component of the Higgs doublet.

\((\bar{L}\phi R + \bar{R}\phi^{\dagger}L)\) generate \(m\) only for lower component of an \(SU(2)_L\) doublet.

Solution: constructing the conjugate doublet \(\;\; \phi_c \equiv i \tau_2 \phi^* = \begin{pmatrix} \phi^0 \\ -\phi^- \end{pmatrix}\)

A gauge invariant mass term can be constructed from \((\bar{L}\phi_c R + \bar{R}\phi^{\dagger}_c L)\)

Higgs unitarity contraints

Perturbative Unitarity

Unitarity

It arises from the conservation of probability in quantum mechanics.
Unitarity bound on scattering amplitudes.

Why “Perturbative”?

If the matrix element at whatever perturbative order does not satisfy unitarity condition, higher order contribution must cancel out remaining terms.
Insights of new physics.

Application

Tool to understand the theoretical structure of spontanuosly broken gauge theories.
Specific to our case: Estimate an upper bound on the Higgs boson mass.

The unitarity bound

The optical theorem in 2 to 2 scattering

The total cross section for a two-to-two scattering the the CM frame is: \[ \sigma_{tot}(AA \rightarrow AA) = \frac{1}{32\pi E_{CM}^2}\int d\cos\theta|\mathcal{M}|^2 \]
The expansion of the matrix element in terms of partial wave amplitudes is:

\[ \mathcal{M} = 16\pi \sum_{l=0}^{\infty}(2l +1)a_l(|\vec{p}|)P_l(\cos\theta) \]

We obtain the result: \[ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \sigma_{tot} = \frac{16\pi}{E^2_{CM}} \sum_{l=0}^{\infty}(2l +1)|a_l|^2 \]
The optical theorem states:

\[ \Im{\mathcal{M}(A \rightarrow A)} = 2E_{CM}|\vec{p}| \sum_{X} \sigma(A \rightarrow X) \geq 2E_{CM}|\vec{p}| \sigma(A \rightarrow A) \]

\[ \sum_{l=0}^{\infty}(2l + 1)\Im{a_l} = \frac{2|\vec{p}|}{E_{CM}} \sum_{l=0}^{\infty}(2l + 1)|a_l|^2 \] - Since \(\Im{a_l} \leq |a_l|\) in the high energy limit we have \(\Im{a_l} =Fina |a_l|\):

\[\begin{align} |a_l| \leq 1 \end{align}\]

\[\begin{align} |\Re{a_l}| \leq 1/2 \end{align}\]

\[\begin{align} 0 \leq |\Im{a_l}| \leq 1 \end{align}\]

Longitudinal vector bosons

Energy dependence

After EWSB gauge bosons acquire a mass.
Equivalent of acquiring a longitudinal polarization.

For a vector with a momentum on the z-axis \(p^{\mu} = (E,0,0,p_z)\):

Transverse polarization:

\[\begin{array}{cc} \epsilon_{T1}=(0,1,0,0) & \epsilon_{T2}=(0,0,1,0) \end{array}\]

No energy dependence.

Longitudinal polarization:

\[ \epsilon_{L} = \frac{1}{m}(p_z,0,0,E) \]

Energy dependence.

Longitudinal vector boson scattering

1) 4 point diagram \(W^+_LW^-_L \rightarrow W^+_LW^-_L\)

The vertex is: \(ig^2(2g_{\mu\sigma}g_{\nu\rho} - g_{\mu\nu}g_{\rho\sigma} - g_{\mu\rho}g_{\nu\sigma})\)

The energy-momentum 4 vectors:

\[\begin{array}{cc} p_+=(E,0,0,p) & k_+=(E,p\sin\theta,0,p\cos\theta)\\ p_-=(E,0,0,-p) & k_-=(E,-p\sin\theta,0,-p\cos\theta) \end{array}\]

The longitudinal polarization vectors:

\[\begin{aligned} &e_l(p_+)=\left(\frac{p}{M_W},0,0,\frac{E}{M_W}\right) \\ &e_l(k_+)=\left(\frac{p}{M_W},\frac{E}{M_W}\sin\theta,0,\frac{E}{M_W}\cos\theta\right) \\ &e_l(p_-)=\left(\frac{p}{M_W},0,0,-\frac{E}{M_W}\right) \\ &e_l(k_-)=\left(\frac{p}{M_W},-\frac{E}{M_W}\sin\theta,0,-\frac{E}{M_W}\cos\theta\right) \\ \end{aligned}\]

After calculation the matrix element has a \(\frac{E^4}{M_W^4}\) dependence.
The high energy limit:

\[ i\mathcal{M} \approx ig^2\frac{E^4}{M_W^4} \]

violates unitarity for high energies.

But we still have to add tree level diagrams.

Longitudinal vector boson scattering

2) Adding tree level contribution

The Feynmann rule for the vertex is:

\[ i \left\{ \begin{array}{ll} e \\ g\cos\theta_W \\ \end{array} \right\} [g_{\mu\nu}(-p_- + p_+)_{\alpha} + g_{\mu\alpha}(q + p_+)_{\nu} + g_{\nu\alpha}(-q - p_-)_{\mu}] \]

The propagators are:

\[\begin{array}{cc} \frac{-i}{q^2 - M_Z^2} \left[g_{\alpha\beta} - \frac{q_{\alpha}q_{\beta}}{M_Z^2}\right] \quad \text{and} \quad & \frac{-ig_{\alpha\beta}}{q^2}\\ \end{array}\]

Every of the four diagrams grows with \(\frac{E^4}{M_W^4}\)
But summing them together they all cancel away.

The matrix element becomes: \[ i\mathcal{M}=i\frac{g^2}{4M_W^2}\left[s+t+\mathcal{O(M_W^2)}\right] \]

There is still an \(\frac{E^2}{M_W^2}\) dependece left.

Upper bound on the energy scale

Using perturbative unitarity

We can get an estimate on the energy at which perturbative unitarity is violated.
The relation obtained before on \(\mathcal{M}\):

\[ \mathcal{M}=16\pi\sum_{J=0}^{\infty}(2J+1)a_JP_J(\cos\theta) \] \[ \int_{-1}^{1}P_J(\cos \theta)d(\cos \theta) \mathcal{M} = 16\pi\sum_{l=0}^{\infty}(2J+1)a_J\int_{-1}^{1}P_J(\cos \theta)P_J(\cos \theta)d(\cos \theta) \]

The ortogonality relation of the Legendre polynomial goes:

\[ \int_{-1}^{1} d(\cos \theta) \, P_J(\cos \theta) P_{J'}(\cos \theta) = \frac{2}{2J + 1} \delta_{JJ'} \]

We are able to get a definition on the Jth partial wave amplitude: \[ a_J(s) = \frac{1}{32 \pi} \int_{-1}^{1} d(\cos \theta) \, \mathcal{M}(s, \cos \theta) P_J(\cos \theta) \] \[ a_0(s) = \frac{1}{32 \pi} \int_{-1}^{1} d(\cos \theta) \, \mathcal{M}(s, \cos \theta) \]
We have the relations: \[\begin{array} i\mathcal{M} \approx i\frac{g^2}{4M_W^2}\left[s+t\right] \quad \text{ } \quad & t = \frac{s}{2}(\cos\theta - 1) \end{array}\]
Plugging them in and applying the bound \(|\Re{a_0}| \leq 1/2\):

\[\begin{array}{cc} a_0 \approx g^2\frac{s}{64\pi M_W^2} \leq 1/2 \quad \text{so} \quad & s \leq \frac{1}{g^2}32\pi M_W^2 \approx (1200GeV)^2\\ \end{array}\]

The theory breaks down at the TeV scale.

Longitudinal vector boson scattering

3) Adding tree level Higgs contribution

The propagator is: \[ \frac{-i}{q^2 - m_h^2} \left[g_{\alpha\beta} - \frac{q_{\alpha}q_{\beta}}{m_h^2}\right] \] The Feynmann rule for the verteces is: \[ \frac{2iM_W^2g_{\mu\nu}}{v}=i\frac{g^2v}{2}g_{\mu\nu} \]

We obtain a contribution of: \[ i\mathcal{M_h} = -i\frac{i}{v^2}\left(\frac{s^2}{s-m_h^2} + \frac{t^2}{t-m_h^2}\right) \]

Taylor expanding for \(s,t >> m_h^2\) we have: \[ i\mathcal{M_h} \simeq -i\frac{1}{v^2}(s+t+2m_h^2) \]
Summing the contributions, we see the energy dependent term cancel: \[ i\mathcal{M} \simeq -i\frac{1}{v^2}(2m_h^2) \]
Using previous relation for a_0: \[ a_0(s) = \frac{1}{32 \pi} \int_{-1}^{1} d(\cos \theta) \, \mathcal{M}(s, \cos \theta) \] \[ a_0 \simeq -\frac{1}{16\pi v^2}(2m_h^2) \]
Applying \(|\Re{a_0}| < 1/2\) we finally obtain an upper bound on the Higgs boson mass. \[ m_h^2 \leq 4\pi v^2 \simeq (870 GeV)^2 \]

Conclusions

Summary

The Abelian Higgs Model
Weinberg-Salam Model
Fermion Masses
Perturbative unitarity
Upper bound Higgs mass

Backup

Computation of the unitarity bound

QM scattering theory

Let’s start from an incoming plane wave propagating in the z direction: \[ \psi(\vec{r}) = e^{ik \cdot \vec{r}} = e^{ikr\cos\theta} \]
rewritten in terms of the spherical Bessel functions and the Legendre polynomials: \[ \psi(\vec{r}) = \sum_{l=0}^{\infty}(2l +1)(i)^l j_l(kr)P_l(\cos\theta) \]

for \(kr>>l\) this approximation holds: \[ \psi(\vec{r}) \simeq \frac{i}{2k}\sum_{l=0}^{\infty}(2l +1)(i)^l \left[\frac{e^{-i(kr - l\pi/2)}}{r} - \frac{e^{+i(kr - l\pi/2)}}{r} \right]P_l(\cos\theta) \]
Let’s add the function \(S_l(k)\) which describes the action of the scattering center. It can only modify the outgoing wave.

\[ \psi(\vec{r}) \simeq \frac{i}{2k}\sum_{l=0}^{\infty}(2l +1)(i)^l \left[\frac{e^{-i(kr - l\pi/2)}}{r} - S_l(k)\frac{e^{+i(kr - l\pi/2)}}{r} \right]P_l(\cos\theta) \]

where \(|S_l(k) \leq 1|\). This is the source of the unitarity bound. After manipulation:

\[ \psi(\vec{r}) \simeq e^{ikr\cos\theta} + \left[\sum_{l=0}^{\infty}(2l +1)\frac{S_l(k)-1}{2ik}P_l(\cos\theta)\right]\frac{e^{ikr}}{r} \]

calling:

\[ f(\theta) = \frac{1}{k}\sum_{l=0}^{\infty}(2l +1)a_l(k)P_l(\cos\theta) \]

\[ a_l(k) = \frac{S_l(k)-1}{2i} \]