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* DOC: switch to inspire-HEP citation keys
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redeboer committed May 16, 2024
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118 changes: 50 additions & 68 deletions docs/bibliography.bib

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2 changes: 1 addition & 1 deletion docs/usage.ipynb
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"cell_type": "markdown",
"metadata": {},
"source": [
"In case of multiple decay topologies, AmpForm also takes care of {doc}`spin alignment </usage/helicity/spin-alignment>` with {cite}`marangottoHelicityAmplitudesGeneric2020`!"
"In case of multiple decay topologies, AmpForm also takes care of {doc}`spin alignment </usage/helicity/spin-alignment>` with {cite}`Marangotto:2019ucc`!"
]
},
{
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46 changes: 23 additions & 23 deletions docs/usage/dynamics/k-matrix.ipynb
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"cell_type": "markdown",
"metadata": {},
"source": [
"While {mod}`ampform` does not yet provide a generic way to formulate an amplitude model with $\\boldsymbol{K}$-matrix dynamics, the (experimental) {mod}`.kmatrix` module makes it fairly simple to produce a symbolic expression for a parameterized $\\boldsymbol{K}$-matrix with an arbitrary number of poles and channels and play around with it interactively. For more info on the $\\boldsymbol{K}$-matrix, see the classic paper by Chung {cite}`chungPartialWaveAnalysis1995`, {pdg-review}`2021; Resonances`, or this instructive presentation {cite}`meyerMatrixTutorial2008`.\n",
"While {mod}`ampform` does not yet provide a generic way to formulate an amplitude model with $\\boldsymbol{K}$-matrix dynamics, the (experimental) {mod}`.kmatrix` module makes it fairly simple to produce a symbolic expression for a parameterized $\\boldsymbol{K}$-matrix with an arbitrary number of poles and channels and play around with it interactively. For more info on the $\\boldsymbol{K}$-matrix, see the classic paper by Chung {cite}`Chung:1995dx`, {pdg-review}`2021; Resonances`, or this instructive presentation {cite}`meyerMatrixTutorial2008`.\n",
"\n",
"Section {ref}`usage/dynamics/k-matrix:Physics` summarizes {cite}`chungPartialWaveAnalysis1995`, so that the {mod}`.kmatrix` module can reference to the equations. It also points out some subtleties and deviations.\n",
"Section {ref}`usage/dynamics/k-matrix:Physics` summarizes {cite}`Chung:1995dx`, so that the {mod}`.kmatrix` module can reference to the equations. It also points out some subtleties and deviations.\n",
"\n",
":::{note}\n",
"\n",
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"In amplitude analysis, the main aim is to express the differential cross section $\\frac{d\\sigma}{d\\Omega}$, that is, the intensity distribution in each spherical direction $\\Omega=(\\phi,\\theta)$ as we can observe in experiments. This differential cross section can be expressed in terms of the **scattering amplitude** $A$:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (1)\n",
"{cite}`Chung:1995dx` Eq. (1)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -230,7 +230,7 @@
"We can now further express $A$ in terms of **partial wave amplitudes** by expanding it in terms of its angular momentum components:[^spin-formalisms]\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (2)\n",
"{cite}`Chung:1995dx` Eq. (2)\n",
"```\n",
"\n",
"$$\n",
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"The dynamical part $\\boldsymbol{T}$ is usually called the **transition operator**. It describes the interacting part of the more general **scattering operator** $\\boldsymbol{S}$, which describes the (complex) amplitude $\\langle f|\\boldsymbol{S}|i\\rangle$ of an initial state $|i\\rangle$ transitioning to a final state $|f\\rangle$. The scattering operator describes both the non-interacting amplitude and the transition amplitude, so it relates to the transition operator as:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (10)\n",
"{cite}`Chung:1995dx` Eq. (10)\n",
"```\n",
"\n",
"$$\n",
"\\boldsymbol{S} = \\boldsymbol{I} + 2i\\boldsymbol{T}\n",
"$$ (S in terms of T)\n",
"\n",
"with $\\boldsymbol{I}$ the identity operator. Just like in {cite}`chungPartialWaveAnalysis1995`, we use a factor 2, while other authors choose $\\boldsymbol{S} = \\boldsymbol{I} + i\\boldsymbol{T}$. In that case, one would have to multiply Eq. {eq}`partial-wave-expansion` by a factor $\\frac{1}{2}$."
"with $\\boldsymbol{I}$ the identity operator. Just like in {cite}`Chung:1995dx`, we use a factor 2, while other authors choose $\\boldsymbol{S} = \\boldsymbol{I} + i\\boldsymbol{T}$. In that case, one would have to multiply Eq. {eq}`partial-wave-expansion` by a factor $\\frac{1}{2}$."
]
},
{
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"Knowing the origin of the $\\boldsymbol{T}$-matrix, there is an important restriction that we need to comply with when we further formulate a {ref}`parametrization <usage/dynamics/k-matrix:Pole parametrization>`: **unitarity**. This means that $\\boldsymbol{S}$ should conserve probability, namely $\\boldsymbol{S}^\\dagger\\boldsymbol{S} = \\boldsymbol{I}$. Luckily, there is a trick that makes this easier. If we express $\\boldsymbol{S}$ in terms of an operator $\\boldsymbol{K}$ by applying a [Cayley transformation](https://en.wikipedia.org/wiki/Cayley_transform):\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (20)\n",
"{cite}`Chung:1995dx` Eq. (20)\n",
"```\n",
"\n",
"$$\n",
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"_unitarity is conserved if $\\boldsymbol{K}$ is real_. With some matrix jumbling, we can derive that the $\\boldsymbol{T}$-matrix can be expressed in terms of $\\boldsymbol{K}$ as follows:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (19);\n",
"{cite}`Chung:1995dx` Eq. (19);\n",
"compare with {eq}`T-hat-in-terms-of-K-hat`\n",
"```\n",
"\n",
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"The description so far did not take Lorentz-invariance into account. For this, we first need to define a **two-body phase space matrix** $\\boldsymbol{\\rho}$:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (36)\n",
"{cite}`Chung:1995dx` Eq. (36)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -338,7 +338,7 @@
"with $\\rho_i$ given by {eq}`PhaseSpaceFactor` in {class}`.PhaseSpaceFactor` for the final state masses $m_{a,i}, m_{b,i}$. The **Lorentz-invariant amplitude $\\boldsymbol{\\hat{T}}$** and corresponding Lorentz-invariant $\\boldsymbol{\\hat{K}}$-matrix can then be computed from $\\boldsymbol{T}$ and $\\boldsymbol{K}$ with:[^rho-dagger]\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (34) and (47)\n",
"{cite}`Chung:1995dx` Eqs. (34) and (47)\n",
"```\n",
"\n",
"$$\n",
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"With these definitions, we can deduce that:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (51);\n",
"{cite}`Chung:1995dx` Eq. (51);\n",
"compare with {eq}`T-in-terms-of-K`\n",
"```\n",
"\n",
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]
},
"source": [
"One approach by {cite}`aitchisonMatrixFormalismOverlapping1972` is to transform $\\boldsymbol{T}$ into $F$ (and its relativistic form $\\hat{F}$) through the **production amplitude $P$-vector**:\n",
"One approach by {cite}`Aitchison:1972ay` is to transform $\\boldsymbol{T}$ into $F$ (and its relativistic form $\\hat{F}$) through the **production amplitude $P$-vector**:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (114) and (115)\n",
"{cite}`Chung:1995dx` Eqs. (114) and (115)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -433,10 +433,10 @@
"\\hat{\\boldsymbol{K}} = \\sqrt{\\boldsymbol{\\rho}^{-1}} \\boldsymbol{K} \\sqrt{\\boldsymbol{\\rho}^{-1}}.\n",
"$$ (K-hat in terms of K)\n",
"\n",
"Another approach by {cite}`cahnMystery9801986` further approximates this by defining a **$Q$-vector**:\n",
"Another approach by {cite}`Cahn:1985wu` further approximates this by defining a **$Q$-vector**:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (124)\n",
"{cite}`Chung:1995dx` Eq. (124)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -447,7 +447,7 @@
"that _is taken to be constant_ (just some 'fitting' parameters). The $F$-vector can then be expressed as:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (125)\n",
"{cite}`Chung:1995dx` Eq. (125)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -459,7 +459,7 @@
"Note that for all these vectors, we have:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (116) and (124)\n",
"{cite}`Chung:1995dx` Eqs. (116) and (124)\n",
"```\n",
"\n",
"$$\n",
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"[^complex-conjugate-parametrization]: Eqs. (51) and (52) in {cite}`chungPrimerKmatrixFormalism1995` take a complex conjugate of one of the residue functions and one of the phase space factors.\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (73) and (74)\n",
"{cite}`Chung:1995dx` Eqs. (73) and (74)\n",
"```\n",
"\n",
"$$\n",
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"with $c_{ij}, \\hat{c}_{ij}$ some optional background characterization and $g_{R,i}$ the **residue functions**. The residue functions are often further expressed as:\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (75-78)\n",
"{cite}`Chung:1995dx` Eqs. (75-78)\n",
"```\n",
"\n",
"$$\n",
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"\n",
"with $\\gamma_{R,i}$ some _real_ constants and $\\Gamma^0_{R,i}$ the **partial width** of each pole. In the Lorentz-invariant form, the fixed width $\\Gamma^0$ is replaced by an \"energy dependent\" {class}`.EnergyDependentWidth` $\\Gamma(s)$.[^phase-space-factor-normalization] The **width** for each pole can be computed as $\\Gamma^0_R = \\sum_i\\Gamma^0_{R,i}$.\n",
"\n",
"[^phase-space-factor-normalization]: Unlike Eq. (77) in {cite}`chungPartialWaveAnalysis1995`, AmpForm defines {class}`.EnergyDependentWidth` as in {pdg-review}`2021; Resonances; p.6`, Eq. (50.28). The difference is that the phase space factor denoted by $\\rho_i$ in Eq. (77) in {cite}`chungPartialWaveAnalysis1995` is divided by the phase space factor at the pole position $m_R$. So in AmpForm, the choice is $\\rho_i \\to \\frac{\\rho_i(s)}{\\rho_i(m_R)}$."
"[^phase-space-factor-normalization]: Unlike Eq. (77) in {cite}`Chung:1995dx`, AmpForm defines {class}`.EnergyDependentWidth` as in {pdg-review}`2021; Resonances; p.6`, Eq. (50.28). The difference is that the phase space factor denoted by $\\rho_i$ in Eq. (77) in {cite}`Chung:1995dx` is divided by the phase space factor at the pole position $m_R$. So in AmpForm, the choice is $\\rho_i \\to \\frac{\\rho_i(s)}{\\rho_i(m_R)}$."
]
},
{
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"The production vector $P$ is commonly parameterized as:[^damping-factor-P-parametrization]\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eqs. (118-119) and (122)\n",
"{cite}`Chung:1995dx` Eqs. (118-119) and (122)\n",
"```\n",
"\n",
"$$\n",
Expand All @@ -541,7 +541,7 @@
"[^damping-factor-P-parametrization]: Just as with [^phase-space-factor-normalization], we have smuggled a bit in the last equation in order to be able to reproduce Equation (50.23) in {pdg-review}`2021; Resonances; p.9` in the case $n=1,n_R=1$, on which {func}`.relativistic_breit_wigner_with_ff` is based.\n",
"\n",
"```{margin}\n",
"{cite}`chungPartialWaveAnalysis1995` Eq. (121)\n",
"{cite}`Chung:1995dx` Eq. (121)\n",
"```\n",
"\n",
"$$\n",
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"cell_type": "markdown",
"metadata": {},
"source": [
"[^pole-vs-resonance]: See {pdg-review}`2021; Resonances`, Section 50.1, for a discussion about what poles and resonances are. See also the intro to Section 5 in {cite}`chungPartialWaveAnalysis1995`."
"[^pole-vs-resonance]: See {pdg-review}`2021; Resonances`, Section 50.1, for a discussion about what poles and resonances are. See also the intro to Section 5 in {cite}`Chung:1995dx`."
]
},
{
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2 changes: 1 addition & 1 deletion docs/usage/helicity/formalism.ipynb
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"\n",
":::{tip}\n",
"\n",
"For more information about the helicity formalism, see {cite}`chungSpinFormalismsUpdated2014`, {cite}`richmanExperimenterGuideHelicity1984`, and {cite}`kutschkeAngularDistributionCookbook1996`.\n",
"For more information about the helicity formalism, see {cite}`chungSpinFormalismsUpdated2014`, {cite}`Richman:1984gh`, and {cite}`kutschkeAngularDistributionCookbook1996`.\n",
"\n",
":::\n",
"\n",
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4 changes: 2 additions & 2 deletions docs/usage/helicity/spin-alignment.ipynb
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"cell_type": "markdown",
"metadata": {},
"source": [
"One way of aligning the spins of each sub-system, is Dalitz-Plot Decomposition (DPD) {cite}`mikhasenkoDalitzplotDecompositionThreebody2020`. DPD **can only be used for three-body decays**, but results in a quite condense amplitude model expression.\n",
"One way of aligning the spins of each sub-system, is Dalitz-Plot Decomposition (DPD) {cite}`Marangotto:2019ucc`. DPD **can only be used for three-body decays**, but results in a quite condense amplitude model expression.\n",
"\n",
"We can select DPD alignment as follows:\n",
"\n",
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"cell_type": "markdown",
"metadata": {},
"source": [
"The second spin alignment method is the 'axis-angle method' {cite}`marangottoHelicityAmplitudesGeneric2020`. This method results in much larger expressions and is therefore much less efficient, but theoretically it **can handle $n$-body final states**. It can be selected as follows:"
"The second spin alignment method is the 'axis-angle method' {cite}`Marangotto:2019ucc`. This method results in much larger expressions and is therefore much less efficient, but theoretically it **can handle $n$-body final states**. It can be selected as follows:"
]
},
{
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4 changes: 2 additions & 2 deletions docs/usage/kinematics.ipynb
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"Kinematics for a three-body decay $0 \\to 123$ can be fully described by two **Mandelstam variables** $\\sigma_1, \\sigma_2$, because the third variable $\\sigma_3$ can be expressed in terms $\\sigma_1, \\sigma_2$, the mass $m_0$ of the initial state, and the masses $m_1, m_2, m_3$ of the final state. As can be seen, the roles of $\\sigma_1, \\sigma_2, \\sigma_3$ are interchangeable.\n",
"\n",
"```{margin}\n",
"See Eq. (1.2) in {cite}`bycklingParticleKinematics1973`\n",
"See Eq. (1.2) in {cite}`Byckling:1971vca`\n",
"```"
]
},
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"\n",
"\n",
"```{margin}\n",
"See §V.2 in {cite}`bycklingParticleKinematics1973`\n",
"See §V.2 in {cite}`Byckling:1971vca`\n",
"```"
]
},
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9 changes: 4 additions & 5 deletions src/ampform/dynamics/__init__.py
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Expand Up @@ -45,9 +45,8 @@ class BlattWeisskopfSquared(sp.Expr):
Each of these cases for :math:`L` has been taken from
:cite:`pychyGekoppeltePartialwellenanalyseAnnihilationen2016`, p.59,
:cite:`chungPartialWaveAnalysis1995`, p.415, and
:cite:`chungFormulasAngularMomentumBarrier2015`. For a good overview of where to use
these Blatt-Weisskopf functions, see :cite:`asnerDalitzPlotAnalysis2006`.
:cite:`Chung:1995dx`, p.415, and :cite:`Chung:1995dx`. For a good overview of where
to use these Blatt-Weisskopf functions, see :cite:`ParticleDataGroup:2020ssz`.
See also :ref:`usage/dynamics:Form factor`.
"""
Expand Down Expand Up @@ -139,7 +138,7 @@ class EnergyDependentWidth(sp.Expr):
r"""Mass-dependent width, coupled to the pole position of the resonance.
See Equation (50.28) in :pdg-review:`2021; Resonances; p.9` and
:cite:`asnerDalitzPlotAnalysis2006`, equation (6). Default value for
:cite:`ParticleDataGroup:2020ssz`, equation (6). Default value for
:code:`phsp_factor` is `.PhaseSpaceFactor`.
Note that the `.BlattWeisskopfSquared` of AmpForm is normalized in the sense that
Expand Down Expand Up @@ -189,7 +188,7 @@ def relativistic_breit_wigner(s, mass0, gamma0) -> sp.Expr:
"""Relativistic Breit-Wigner lineshape.
See :ref:`usage/dynamics:_Without_ form factor` and
:cite:`asnerDalitzPlotAnalysis2006`.
:cite:`ParticleDataGroup:2020ssz`.
"""
return gamma0 * mass0 / (mass0**2 - s - gamma0 * mass0 * sp.I)

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5 changes: 2 additions & 3 deletions src/ampform/helicity/__init__.py
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Expand Up @@ -884,9 +884,8 @@ def formulate_isobar_wigner_d(transition: StateTransition, node_id: int) -> sp.E
Wigner-:math:`D` functions in a *sequential* two-body decay. Note that this source
chose :math:`\Omega=(\phi,\theta,-\phi)` as argument to the (conjugated)
Wigner-:math:`D` function, just like the original paper by Jacob & Wick
:cite:`jacobGeneralTheoryCollisions1959`, Eq. (24). See p.119-120 and p.199 in
:cite:`martinElementaryParticleTheory1970` for the two conventions, :math:`\gamma=0`
versus :math:`\gamma=-\phi`.
:cite:`Jacob:1959at`, Eq. (24). See p.119-120 and p.199 in :cite:`Martin:1970hmp`
for the two conventions, :math:`\gamma=0` versus :math:`\gamma=-\phi`.
Example
-------
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