Documents/LightInflaton/BrKpih.ipynb

BrKpih.ipynb

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      "source": "## Calculation of K->pi h decay\n\nThis is the formula (4.1) and (4.4) from the Light inflaton guide. No approximations were made, except for the one already present in (4.1). However, change from $m_K^2$ to $m_K^2-m_\\pi^2$ is 10% error, well beyond the precision of the calculation anyway\n$$\n\\begin{aligned}\n  A( K^+\\to \\pi^+\\chi)\n    & \\simeq\n      \\theta\\frac{M_K^2}{v}\\Bigg\\{\n        \\gamma_1\\frac{1-\\kappa}{2}\\left(1-\\frac{m_\\chi^2-M_\\pi^2}{M_K^2}\\right)\n  \\\\\\nonumber\n    & \\hphantom{\\simeq\n      \\theta\\frac{M_K^2}{v}\\Bigg\\{}\n        -\\gamma_2(1-\\kappa)\n        +\\frac{1}{2}\n        \\frac{3G_F\\sqrt{2}}{16\\pi^2} \n        \\sum_{i=c,t} V_{id}^* m_i^2 V_{is}\\; \n      \\Bigg\\}\n      \\end{aligned}\n$$\n$$\n\\begin{aligned}\n    \\text{Br}( K^+ \\to \\pi^+ \\chi)\n    &= \\frac{1}{\\Gamma_\\mathrm{total}(K^+)}\n    \\frac{|A(K^+\\to\\pi^+\\chi)|^2}{16\\pi M_K}\\frac{2|\\mathbf{p}_\\chi|}{M_K}\n  \\nonumber\\\\\n  \\label{BrKpih}\n    & \\approx\n      1.3\\times 10^{-3}\\cdot \\left( \\frac{2|\\mathbf{p}_\\chi|}{M_K} \\right)\\theta^2\n \\end{aligned}\n$$"
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      "cell_type": "code",
      "source": "from math import pi, sqrt",
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      "source": "mc=1.27\nmb=4.2\nmt=171.3\nVcd=-0.218\nVcs=0.94\nVcb=41.2e-3\nVtd=8e-3\nVts=-38.7e-3\nVtb=1\n\nmK=493.677e-3\nmpi=139.57018e-3\n\nGF=1.16637e-5\nv=1/sqrt(sqrt(2)*GF)\n\nGeVsec=6.58211899e-25\nGammaK=(1.2380e-8)**-1*GeVsec",
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      "source": "def BrKpih_coeff(mh):\n    nh=3\n    b=9\n    kappa=2*nh/3/b\n    ## Formfactors for kaon decay\n    gamma1=3.1e-7\n    x=0.5\n    FKslashFP=2.22\n    f=-0.11\n    gamma2=1/4*f*x*gamma1/(FKslashFP-1)\n\n    AKpihchiral=mK**2/v*( gamma1*(1-kappa)/2*(1-(mh**2-mpi**2)/mK**2)\n                          -gamma2*(1-kappa) )\n    AKpihquarks=mK**2/v*( 3*GF*sqrt(2)/(16*pi**2)/2*( Vcd.conjugate()*mc**2*Vcs\n                                                     +Vtd.conjugate()*mt**2*Vts ) )\n    AKpih=AKpihchiral+AKpihquarks\n\n    return abs(AKpih)**2/(16*pi*mK)/GammaK\n\ndef BrKpih(theta2, mh):\n    return theta2*BrKpih_coeff(mh)*(2*ph/mK)",
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      "source": "BrKpih_coeff(0.1)",
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