tensor-helicity.studies.ipynb

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    "# Plots for the ``Tensor-Helicity''-project\n",
    "\n",
    "Below I code different amplitudes and compare $\\pi K$-invariant mass distribution.\n",
    "\n",
    "At the end, two cross-checks are performed."
   ]
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      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPackage Plots is already installed\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mMETADATA is out-of-date — you may not have the latest version of Plots\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mUse `Pkg.update()` to get the latest versions of your packages\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPackage QuadGK is already installed\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mMETADATA is out-of-date — you may not have the latest version of QuadGK\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mUse `Pkg.update()` to get the latest versions of your packages\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPackage GR is already installed\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mMETADATA is out-of-date — you may not have the latest version of GR\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mUse `Pkg.update()` to get the latest versions of your packages\n",
      "\u001b[39m"
     ]
    }
   ],
   "source": [
    "Pkg.add(\"Plots\")\n",
    "Pkg.add(\"QuadGK\")\n",
    "Pkg.add(\"GR\")"
   ]
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   "execution_count": 5,
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   "source": [
    "# masses\n",
    "mB = 5.279;\n",
    "mJpsi = 3.686;\n",
    "mπ = 0.139;\n",
    "mK = 0.485;\n",
    "mKs = 0.893;\n",
    "ΓKs = 0.05;\n",
    "mμ = 0.105;\n",
    "\n",
    "# physical region border for $m_{\\pi K}^2$-invariant\n",
    "s_th = (mπ+mK)^2+0.0001\n",
    "s_en = (mB-mJpsi)^2-0.0001;"
   ]
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  {
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   "execution_count": 6,
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      "text/plain": [
       "Wignerdhat (generic function with 1 method)"
      ]
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     "execution_count": 6,
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   "source": [
    "using GSL\n",
    "\n",
    "# Clebsches and d-function\n",
    "function ClebschGordon(j1,m1,j2,m2,j,m)\n",
    "    factor = (j1+j2-m)%2==1 ? -1 : 1\n",
    "    factor*sqrt(2*j+1)*sf_coupling_3j(2*j1,2*j2,2*j,2*m1,2*m2,-2*m)\n",
    "end\n",
    "\n",
    "function Wignerd(j,m1,m2,cosθ)\n",
    "#    print(\"WignerD(\",j,\",\",m1,\",\",m2,\",\",cosθ,\")\")\n",
    "    (m1 < 0) && print(\"Error: Something is wrong with Wignerdhat!\")\n",
    "    (m2 == 0) && sqrt(4*π/(2*j+1))*sf_legendre_sphPlm(j,m1,cosθ)\n",
    "end\n",
    "\n",
    "function Wignerdhat(j,m1,m2,cosθ)\n",
    "#    print(\"WignerD(\",j,\",\",m1,\",\",m2,\",\",cosθ,\")\")\n",
    "    (m1 < 0) && print(\"Error: Something is wrong with Wignerdhat!\")\n",
    "    (m2 == 0) && sqrt(4*π/(2*j+1))*sf_legendre_sphPlm(j,m1,cosθ)/sqrt(1-cosθ*cosθ)^(m1)\n",
    "end"
   ]
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   "source": [
    "## Example of the interest\n",
    "Consider two resonances:\n",
    " * $K^*(1410)$, $m = 1.414\\,$MeV, $\\Gamma = 232\\,$MeV\n",
    " * $K_2^*(1430)$, $m = 1.432\\,$MeV, $\\Gamma = 109\\,$MeV"
   ]
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   "execution_count": 7,
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     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 7,
     "metadata": {},
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    }
   ],
   "source": [
    "using Plots\n",
    "gr()"
   ]
  },
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   "cell_type": "code",
   "execution_count": 14,
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   "source": [
    "mKs = 1.414; ΓKs = 0.232;\n",
    "mKs2 = 1.432; ΓKs2 = 0.109;\n",
    "# Breit-Wigner denominator\n",
    "bw(s,msq,Γ)=1/(msq-s-1.0im*sqrt(msq)*Γ)\n",
    "# Breit-Wigner denominator\n",
    "λ(x,y,z) = x^2+y^2+z^2-2*x*y-2*y*z-2*z*x;\n",
    "function bwDalitz(s,msq,Γ)\n",
    "    phsp = sqrt(λ(s,mπ^2,mK^2)*λ(s,mJpsi^2,mB^2))/s\n",
    "    phsp*abs(bw(s,msq,Γ))^2\n",
    "end\n",
    "plot(s->bwDalitz(s,mKs^2,ΓKs), s_th, s_en, lab=\"K*(1410)\")\n",
    "plot!(s->bwDalitz(s,mKs2^2,ΓKs2), s_th, s_en, lab=\"K2*(1430)\", legend=:topleft)"
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    "## Compare three aproaches\n",
    "* Our amplitude, Eq. (B1)\n",
    "* LS-amplitude, Eq. (13)\n",
    "* Tensor amplitude, Eq. (19)\n",
    "\n",
    "Partial wave expansion for the amplitude\n",
    "$$\n",
    "A_\\lambda(s,\\cos\\theta) = \\frac{1}{4\\pi}\\sum (2j+1) A_\\lambda^j(s) d_{\\lambda 0}^j(\\theta)\n",
    "$$\n",
    "\n",
    "Integrating over muons is equivalent to summing over helicity\n",
    "$$\n",
    "\\frac{\\textrm{d}I}{\\textrm{d}s} = N\\big( |A_0^1(s)|^2 + 2|A_1(s)|^2 \\big) \\frac{\\lambda_{\\pi K}^{1/2}(s)\n",
    "\\lambda_{\\psi B}^{1/2}(s)}{s}\n",
    "$$\n",
    "Below, I demonstrate it numerically."
   ]
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       "ampsqBW (generic function with 1 method)"
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   "source": [
    "# phase space factor and intensity\n",
    "# the function depends on two helicity functions\n",
    "ph(s) = sqrt(λ(s,mπ^2,mK^2)*λ(s,mJpsi^2,mB^2))/s;\n",
    "function project_general(s,cS,cD,A10,A11)\n",
    "    ph(s)*(abs(A10(s,cS,cD))^2+2*abs(A11(s,cS,cD))^2)\n",
    "end\n",
    "\n",
    "# Blatt-Weisskopf functions\n",
    "BWs = [z->1./(1.+z), z->1./(9+3*z+z^2)]\n",
    "Rbw=5; # /GeV\n",
    "# s-channel break up momentu\n",
    "p(s) = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "q(s) = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "# Breit-Wigner pole squared and Blatt-Weisskopf numerator squared (sqrt(BWs)^2):\n",
    "ampsqBW(s) = abs(bw(s,mKs^2,ΓKs))^2*BWs[1]((Rbw*q(s))^2)*BWs[2]((Rbw*p(s))^2)"
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    "### Implementation of the helicity amplitudes\n",
    " * ```our``` := JPAC\n",
    " * ```LS``` := LS-scattering\n",
    " * ```ten``` := tensors in the decay kinematics\n",
    " * ```ten_sc``` := tensors in the scattering kinematics"
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    {
     "data": {
      "text/plain": [
       "A11_ten_sc (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
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    "# our amplitude\n",
    "function A10_our(s,cS,cD)\n",
    "    j = 1\n",
    "    lam=0\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    K00 = 2*mJpsi/sqrt(λ(s,mJpsi^2,mB^2))\n",
    "    K00*(p*q)^j*(s+mJpsi^2-mB^2)/(2*mJpsi^2)*(\n",
    "        sqrt((2*j-1)/(2*j+1))*ClebschGordon(j-1,0,1,0,j,0)*cS+\n",
    "        λ(s,mJpsi^2,mB^2)*sqrt((2*j+3)/(2*j+1))*ClebschGordon(j+1,0,1,0,j,0)*cD)\n",
    "end\n",
    "function A11_our(s,cS,cD)\n",
    "    j = 1\n",
    "    lam=0\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    K10 = q\n",
    "    K10*(p*q)^(j-1)*(\n",
    "        sqrt((2*j-1)/(2*j+1))*ClebschGordon(j-1,0,1,1,j,1)*cS+\n",
    "        λ(s,mJpsi^2,mB^2)*sqrt((2*j+3)/(2*j+1))*ClebschGordon(j+1,0,1,1,j,1)*cD)\n",
    "end\n",
    "\n",
    "# LS amplitude, scattering\n",
    "function A1λ_LS(s,cS,cD,lam)\n",
    "    j = 1\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    p^(j-1)*q^j*(\n",
    "        sqrt((2*j-1)/(2*j+1))*ClebschGordon(j-1,0,1,lam,j,lam)*cS+\n",
    "        sqrt((2*j+3)/(2*j+1))*ClebschGordon(j+1,0,1,lam,j,lam)*p^2*cD)\n",
    "end\n",
    "A10_LS(s,cS,cD) = A1λ_LS(s,cS,cD,0)\n",
    "A11_LS(s,cS,cD) = A1λ_LS(s,cS,cD,1)\n",
    "\n",
    "# tensor amplitude, decay\n",
    "function A10_ten(s,cS,cD)\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    γ = (mB^2+s-mJpsi^2)/(2*mB*sqrt(s))\n",
    "    E1 = (s+mJpsi^2-mB^2)/(2*sqrt(s))\n",
    "    q*(E1/mJpsi*cS - γ*p^2*s/mB^2*(s-mJpsi^2-mB^2)/(2*mJpsi*mB)*cD)\n",
    "end\n",
    "function A11_ten(s,cS,cD)\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    q*(cS + p^2/2*s/mB^2*cD)\n",
    "end\n",
    "\n",
    "# tensor amplitude, scattering\n",
    "function A10_ten_sc(s,cS,cD)\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    E1 = (s+mJpsi^2-mB^2)/(2*sqrt(s))\n",
    "    E1/mJpsi*q*(cS - p^2*cD)\n",
    "end\n",
    "function A11_ten_sc(s,cS,cD)\n",
    "    p = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    q*(cS + p^2/2*cD)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plots\n",
    "Normalized differential width $1/\\Gamma\\, \\mathrm{d}\\Gamma/\\mathrm{d}m_{\\pi K}^2$ is plotted at every plot below\n",
    "for the different approaches and different dynamic models (pole/no pole; with/without BWs).\n",
    "\n",
    "First without resonance -- just energy-dependent factors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using QuadGK"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "funcs=[\n",
    "    s->project_general(s,0,1,A10_our,A11_our),\n",
    "    s->project_general(s,0,1,A10_LS,A11_LS),\n",
    "    s->project_general(s,0,1,A10_ten_sc,A11_ten_sc),\n",
    "    s->project_general(s,0,1,A10_ten,A11_ten)\n",
    "    ]\n",
    "norms = [quadgk(s->f(s),s_th,s_en)[1] for f in funcs]\n",
    "nfuncs = [s->funcs[i](s)/norms[i] for i=1:length(funcs)]\n",
    "plot(nfuncs, s_th:0.03:s_en,\n",
    "xlab=\"\\$M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{2})\\$\", ylab=\"Normalized differentian width\",\n",
    "title=\"Model-dependent factor (without barrier factors)\",\n",
    "    lab=[\"JPAC\",\"LS-scatt.\",\"CPT-scatt.\",\"CPT-decay\"],\n",
    "    ls=[:solid :solid :solid :dash],\n",
    "    c=[:red :blue :green :green],\n",
    "    grid=false)\n",
    "annotate!([(2.0, 0.8, text(\"\\$1/\\\\Gamma\\\\textrm{ d}\\\\Gamma/\\\\textrm{d}M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{-2})\\$\",12))])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "funcs=[\n",
    "    s->project_general(s,0,1,A10_our,A11_our)*ampsqBW(s),\n",
    "    s->project_general(s,0,1,A10_LS,A11_LS)*ampsqBW(s),\n",
    "    s->project_general(s,0,1,A10_ten_sc,A11_ten_sc)*ampsqBW(s),\n",
    "    ]\n",
    "norms = [quadgk(s->f(s),s_th,s_en)[1] for f in funcs]\n",
    "nfuncs = [s->funcs[i](s)/norms[i] for i=1:length(funcs)]\n",
    "plot(nfuncs, s_th:0.01:s_en,\n",
    "    lab=[\"JPAC\",\"LS\",\"CPT\"],\n",
    "    ls=[:solid :dash :dashdot],\n",
    "#    c=[:red :blue :blue :green :green],\n",
    "    xlab=\"\\$M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{2})\\$\", \n",
    "    ylab=\"Normalized differentian width\", # d\\\\Gamma / d M_{\\\\pi K}^2\n",
    "    title=\"with Blatt-Weisskopf factors\",\n",
    "    legend=:topleft,\n",
    "    grid=false\n",
    ")\n",
    "annotate!([(1, 0.75, text(\"\\$1/\\\\Gamma\\\\textrm{ d}\\\\Gamma/\\\\textrm{d}M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{-2})\\$\",12))])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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    "funcs=[\n",
    "    s->project_general(s,0,1,A10_our,A11_our)*abs(bw(s,mKs^2,ΓKs))^2,\n",
    "    s->project_general(s,0,1,A10_LS,A11_LS)*abs(bw(s,mKs^2,ΓKs))^2,\n",
    "    s->project_general(s,0,1,A10_ten_sc,A11_ten_sc)*abs(bw(s,mKs^2,ΓKs))^2\n",
    "    ]\n",
    "norms = [quadgk(s->f(s),s_th,s_en)[1] for f in funcs]\n",
    "nfuncs = [s->funcs[i](s)/norms[i] for i=1:length(funcs)]\n",
    "mainfunc = nfuncs\n",
    "plot(nfuncs, s_th:0.01:s_en, lab=[\"JPAC\",\"LS\",\"CPT\"],\n",
    "xlab=\"\\$M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{2})\\$\", ylab=\"Normalized differentian width\",\n",
    "title=\"without Blatt-Weisskopf factors\",\n",
    "    legend=:topright,\n",
    "    grid=false,\n",
    "    #    c=:black,\n",
    "    ls=[:solid :dash :dashdot])\n",
    "annotate!([(1.5, 0.2, text(\"\\$1/\\\\Gamma\\\\textrm{ d}\\\\Gamma/\\\\textrm{d}M_{\\\\pi K}^2\\\\textrm{ }(\\\\textrm{GeV}^{-2})\\$\",12))])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cross check I:\n",
    "Two methods to get the same plots:\n",
    " * Coding a parametrization of the helicity amplitudes from the paper.\n",
    " Analytic integration over lemptons turns to be is just a sum over helicities of $\\psi$.\n",
    " * Implementation of the full amplitude in terms of four vectors.\n",
    " Integration over leptions is a bit more tricky (details are below).\n",
    "\n",
    "### Analytic integraion over lepton current\n",
    " - Matrix element square is lorethz invariant\n",
    " - in s-channel CMS, amplitude is straightforward\n",
    " - integrals over phase space are analytical as long as amplitude depend on (s, sz)\n",
    " - avaraging over J/psi polarization is a little bit tricky (see comment in the cell below)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Zjl (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# tensor pre-factors\n",
    "function Cs(p1,p2,p3,p4)\n",
    "    s = mnorm(p3+p4)\n",
    "    p3-p4 - (mnorm(p3)-mnorm(p4))/s*(p3+p4)\n",
    "end\n",
    "Bs(p1,p2,p3,p4) = p3+p4\n",
    "function Ds(p1,p2,p3,p4)\n",
    "    # minus is removed due to the crossing\n",
    "    # i is removed to make amplitude real\n",
    "    [-sum((σ-ν)*(σ-ρ)*(σ-μ)*(ν-ρ)*(ν-μ)*(ρ-μ)*\n",
    "            p1[σ]*p2[ν]*p3[ρ]\n",
    "        for σ=1:4, ν=1:4, ρ=1:4)\n",
    "        for μ=1:4]\n",
    "end\n",
    "\n",
    "# kinematical functions\n",
    "function ζs(j,l,s,zs)\n",
    "    (l < j-1 || l > j+1) && return 0;\n",
    "    # three cases\n",
    "    (l==j) && return 0;\n",
    "    ps = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    qs = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    (l==j-1) && return (ps*qs)^(j-1)/(4*π)*sqrt((2*j+1)*(2*l+1)/2.0)*\n",
    "        ClebschGordon(j-1,0,1,1,j,1)*Wignerdhat(j,1,0,zs)\n",
    "    (l==j+1) && return λ(s,mJpsi^2,mB^2)*(ps*qs)^(j-1)/(4*π)*sqrt((2*j+1)*(2*l+1)/2.0)*\n",
    "        ClebschGordon(j+1,0,1,1,j,1)*Wignerdhat(j,1,0,zs)\n",
    "end\n",
    "function βs(j,l,s,zs)\n",
    "    (l < j-1 || l > j+1) && return 0;\n",
    "    # three cases\n",
    "    (l == j) && return 0;\n",
    "    ps = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    qs = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    (l == j-1) && return (ps*qs)^(j)/λ(s,mJpsi^2,mB^2)*\n",
    "        (s+mJpsi^2-mB^2)/sqrt(2)*\n",
    "        4/(4*π)*sqrt((2*j+1)*(2*l+1)/2.0)*(\n",
    "            ClebschGordon(j-1,0,1,0,j,0)*Wignerdhat(j,0,0,zs)/sqrt(2)+\n",
    "            ClebschGordon(j-1,0,1,1,j,1)*Wignerdhat(j,1,0,zs)*zs\n",
    "        )\n",
    "    (l == j+1) && return (ps*qs)^(j)*\n",
    "        (s+mJpsi^2-mB^2)*\n",
    "        4/(4*π)*sqrt((2*j+1)*(2*j+3)/2.0)*(\n",
    "            ClebschGordon(j+1,0,1,0,j,0)*Wignerdhat(j,0,0,zs)/sqrt(2)+\n",
    "            ClebschGordon(j+1,0,1,1,j,1)*Wignerdhat(j,1,0,zs)*zs\n",
    "        )\n",
    "end\n",
    "function δs(j,l,s,zs)\n",
    "    (l < j-1 || l > j+1) && return 0;\n",
    "    # three cases\n",
    "    (l == j-1 || l == j+1) && return 0;\n",
    "    ps = sqrt(λ(s,mJpsi^2,mB^2)/(4*s))\n",
    "    qs = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    (l == j) && return (ps*qs)^(j-1)*sqrt(2.0)/(4*π)*(2*j+1)*Wignerdhat(j,1,0,zs)\n",
    "end\n",
    "\n",
    "function Zjl(j,l,p1,p2,p3,p4)\n",
    "    s = mnorm(p3+p4)\n",
    "    t = mnorm(p1+p3)\n",
    "    u = mB^2+mJpsi^2+mπ^2+mK^2-s-t\n",
    "    zs = (s*(t-u)+(mπ^2-mK^2)*(mJpsi^2-mB^2))/sqrt(λ(s,mB^2,mJpsi^2)*λ(s,mπ^2,mK^2))\n",
    "#    print(\"zs = \",zs,\"\\n\")\n",
    "    Cs(p1,p2,p3,p4)*ζs(j,l,s,zs)+\n",
    "        Bs(p1,p2,p3,p4)*βs(j,l,s,zs)+\n",
    "        Ds(p1,p2,p3,p4)*δs(j,l,s,zs)\n",
    "end"
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    {
     "data": {
      "text/plain": [
       "amplsq (generic function with 1 method)"
      ]
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   "source": [
    "# definition of the amplitude and square of matrix element\n",
    "# i.e. contracted with lepton current\n",
    "function ampl(p1,p2,p3,p4,pars)\n",
    "    amp = [0,0,0,0]\n",
    "    for j in 1:length(pars)\n",
    "        ps = pars[j]\n",
    "        (length(ps) != 3) && continue\n",
    "        amp += (ps[1] == 0) ? 0 : ps[1]*Zjl(j,j-1,p1,p2,p3,p4);\n",
    "        amp += (ps[2] == 0) ? 0 : ps[2]*Zjl(j,j,  p1,p2,p3,p4);\n",
    "        amp += (ps[3] == 0) ? 0 : ps[3]*Zjl(j,j+1,p1,p2,p3,p4);\n",
    "    end\n",
    "    amp\n",
    "end\n",
    "\n",
    "function amplsq(p1,p2,p3,p4,q1,q2,pars)    \n",
    "    amp = ampl(p1,p2,p3,p4,pars)\n",
    "    ampc = conj(amp);\n",
    "    real(mprod(amp,q1)*mprod(ampc,q2)+mprod(amp,q2)*mprod(ampc,q1)-mprod(amp,ampc)*mJpsi^2/2)\n",
    "#    real(amp[2:4]⋅ampc[2:4])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
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    {
     "data": {
      "text/plain": [
       "amplsp_analytic (generic function with 1 method)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# scalar products\n",
    "mprod(P1,P2) = P1[1]*P2[1]-P1[2:4]⋅P2[2:4]\n",
    "mnorm(P1) = mprod(P1,P1);\n",
    "# Integration\n",
    "function amplsp_analytic(s,zs,pars)\n",
    "    # states along z axis\n",
    "    p = sqrt(λ(s,mB^2,mJpsi^2)/(4*s))\n",
    "    E1 = sqrt(p^2+mJpsi^2)\n",
    "    E2 = sqrt(p^2+mB^2)\n",
    "    p1 = [E1, 0,0,p]\n",
    "    p2 = [E2, 0,0,p]\n",
    "    # scattered states\n",
    "    q = sqrt(λ(s,mπ^2,mK^2)/(4*s))\n",
    "    E3 = sqrt(q^2+mπ^2)\n",
    "    E4 = sqrt(q^2+mK^2)\n",
    "    sinθ = sqrt(1-zs^2)\n",
    "    p3 = [E3,-q*sinθ,0,-q*zs]\n",
    "    p4 = [E4, q*sinθ,0,q*zs]\n",
    "    # amplitude\n",
    "    amp = ampl(p1,p2,p3,p4,pars)\n",
    "    ampc = conj(amp)\n",
    "    # avaraging over muon angles is easy in J/psi rest frame\n",
    "    # the answer there is \\vec A \\cdot \\vec A (some numerical factor)\n",
    "    # therefore, I have to find \\vec A^2 in Jpsi rest frame\n",
    "    # I do it using scalar product of A to p1\n",
    "    vecTildeAsq = mprod(amp,p1)*mprod(p1,ampc)/mJpsi^2 - mprod(amp,ampc)\n",
    "    # attach muon current avaraged over muons angles\n",
    "    ksq = λ(mJpsi,mμ^2,mμ^2)/(4*mJpsi^2)\n",
    "    return 2*vecTildeAsq*(mJpsi^2/4-ksq/3)\n",
    "end"
   ]
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   "cell_type": "code",
   "execution_count": 32,
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   "source": [
    "# first, our amplitude from tensor form\n",
    "Msq_our(s,zs,cS,cD) = 3*(abs(A10_our(s,cS,cD)*Wignerd(1,0,0,zs))^2+2*abs(A11_our(s,cS,cD)*Wignerd(1,1,0,zs))^2)\n",
    "function project(s)\n",
    "    ph(s)*quadgk(zs->Msq_our(s,zs,0,1),-0.999,0.999)[1]\n",
    "end\n",
    "norm1 = quadgk(s->project(s),s_th,s_en)[1]\n",
    "plot(s->project(s)/norm1,s_th:0.03:s_en, lab=\"sum of helicities\", title=\"JPAC model\")\n",
    "\n",
    "# second, numerical projection, helicity\n",
    "function project(s)\n",
    "    ph(s)*quadgk(zs->amplsp_analytic(s,zs,[[0,0,1]]),-0.999,0.999)[1]\n",
    "end\n",
    "norm1 = quadgk(s->project(s),s_th,s_en)[1]\n",
    "plot!(s->project(s)/norm1,s_th:0.03:s_en, lab=\"full model with tensors\")\n",
    "\n",
    "# analytic integrals\n",
    "norm1 = quadgk(s->project_general(s,0,1,A10_our,A11_our),s_th,s_en)[1]\n",
    "plot!(s->project_general(s,0,1,A10_our,A11_our)/norm1,s_th:0.03:s_en, lab=\"analyt. cosT integr\")"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cross-check II\n",
    "Below is comparison to Adam's Szczepaniak implementation of the ``JPAC``-equations.\n",
    "Evenytially, Adam's distribution mathces well my tensor inplementation.\n",
    "It means there is a factor $\\sqrt{s}$ differentce in codes!\n",
    "\n",
    "-- to be fixed!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "adam_data = readdlm(\"fort.13\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
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     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# plot adam function normalized to 1\n",
    "normB = sum(adam_data[:,2])\n",
    "dx = adam_data[2,1]-adam_data[1,1]\n",
    "bar(adam_data[:,1],adam_data[:,2]/normB/dx, lab=\"Adam\")\n",
    "# plot my function\n",
    "plot!(mainfunc[1], s_th:0.01:s_en, lab=\"Misha\", c=:black,\n",
    "xlab=\"MpiK(GeV)\", ylab=\"Normalized differentian width\",\n",
    "title=\"just pole without Blatt-Weisskopf factors\",\n",
    "    legend=:topleft)\n",
    "# plot a my function / sqrt(s)\n",
    "# that is guess of possible mismatch in our codes\n",
    "# see function ``mainfunc`` defined above\n",
    "renorm3 = quadgk(s->mainfunc[1](s)*sqrt(s),s_th,s_en)[1]\n",
    "corr_mainfunc = s->mainfunc[1](s)*sqrt(s)/renorm3\n",
    "plot!(corr_mainfunc, s_th:0.01:s_en, lab=\"mKpi*Misha\", c=:red)"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": []
  }
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