numerics.cpp

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// ====================================================================
// This file is part of FlexibleSUSY.
//
// FlexibleSUSY is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published
// by the Free Software Foundation, either version 3 of the License,
// or (at your option) any later version.
//
// FlexibleSUSY is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with FlexibleSUSY.  If not, see
// <http://www.gnu.org/licenses/>.
// ====================================================================
 
// #define USE_LOOPTOOLS
 
#include "numerics.h"
#include "numerics2.hpp"
#ifdef USE_LOOPTOOLS
#include <clooptools.h>
#endif
#include <algorithm>
#include <cmath>
 
namespace softsusy {
 
namespace {
 
constexpr double EPSTOL = 1.0e-11; 
constexpr double TOL = 1e-4;
 
constexpr double dabs(double a) noexcept { return a >= 0. ? a : -a; }
constexpr double sqr(double a) noexcept { return a*a; }
constexpr double pow3(double a) noexcept { return a*a*a; }
constexpr double pow6(double a) noexcept { return sqr(pow3(a)); }
 
constexpr bool is_zero(double m, double tol) noexcept
{
   const double am = dabs(m);
   const double mtol = tol * am;
 
   if (mtol == 0.0 && am != 0.0 && tol != 0.0)
      return am <= tol;
 
   return am <= mtol;
}
 
constexpr bool is_close(double m1, double m2, double tol) noexcept
{
   const double mmax = std::max(dabs(m1), dabs(m2));
   const double mmin = std::min(dabs(m1), dabs(m2));
   const double max_tol = tol * mmax;
 
   if (max_tol == 0.0 && mmax != 0.0 && tol != 0.0)
      return mmax - mmin <= tol;
 
   return mmax - mmin <= max_tol;
}
 
double sign(double x) noexcept
{
   return x >= 0.0 ? 1.0 : -1.0;
}
 
// can be made constexpr in C++20
double fB(const std::complex<double>& x) noexcept
{
   using flexiblesusy::fast_log;
 
   const double re = std::real(x);
   const double im = std::imag(x);
 
   if ((std::abs(re) == 0.0 || std::abs(re) == 1.0) && im == 0.0) {
      return -1.0;
   }
 
   return std::real(-1.0 + fast_log(1.0 - x) - x*fast_log(1.0 - 1.0/x));
}
 
double fB(const std::complex<double>& xp, const std::complex<double>& xm) noexcept
{
   using flexiblesusy::fast_log;
 
   const double rep = std::real(xp);
   const double imp = std::imag(xp);
 
   if ((std::abs(rep) == 0.0 || std::abs(rep) == 1.0) && imp == 0.0) {
      return -1.0 + fB(xm);
   }
 
   const double rem = std::real(xm);
   const double imm = std::imag(xm);
 
   if ((std::abs(rem) == 0.0 || std::abs(rem) == 1.0) && imm == 0.0) {
      return -1.0 + fB(xp);
   }
 
   return std::real(-2.0 + fast_log((1.0 - xp)*(1.0 - xm))
      - xp*fast_log(1.0 - 1.0/xp) - xm*fast_log(1.0 - 1.0/xm));
}
 
} // anonymous namespace
 
/*
 * Real part of A0 1-loop function.
 *
 * @param m mass
 * @param q renormalization scale, q > 0
 * @return Re[A0(m, q)]
 */
double a0(double m, double q) noexcept
{
   return rea0(m*m, q*q);
}
 
/*
 * Real part of A0 1-loop function.
 *
 * Note: Returns correct result even for x < 0.
 *
 * @param x squared mass
 * @param q squared renormalization scale, q > 0
 * @return Re[A0(x, q)]
 */
double rea0(double x, double q) noexcept
{
   if (q <= 0) {
      return std::numeric_limits<double>::quiet_NaN();
   }
 
   if (x == 0) {
      return 0;
   }
 
   return x * (1 - 2*std::log(std::sqrt(std::abs(x/q))));
}
 
double ffn(double p, double m1, double m2, double q) noexcept {
   return a0(m1, q) - 2.0 * a0(m2, q) -
      (2.0 * sqr(p) + 2.0 * sqr(m1) - sqr(m2)) *
      b0(p, m1, m2, q);
}
 
double gfn(double p, double m1, double m2, double q) noexcept {
   return (sqr(p) - sqr(m1) - sqr(m2)) * b0(p, m1, m2, q) - a0(m1, q)
      - a0(m2, q);
}
 
double hfn(double p, double m1, double m2, double q) noexcept {
   return 4.0 * b22(p, m1, m2, q) + gfn(p, m1, m2, q);
}
 
double b22bar(double p, double m1, double m2, double q) noexcept {
   return b22(p, m1, m2, q) - 0.25 * a0(m1, q) - 0.25 * a0(m2, q);
}
 
double b0(double p, double m1, double m2, double q) noexcept
{
#ifdef USE_LOOPTOOLS
   setmudim(q*q);
   double b0l = B0(p*p, m1*m1, m2*m2).real();
   //  return B0(p*p, m1*m1, m2*m2).real();
#endif
 
   m1 = std::abs(m1);
   m2 = std::abs(m2);
   p = std::abs(p);
   q = std::abs(q);
 
   if (m1 > m2) {
      std::swap(m1, m2);
   }
 
   // protect against infrared divergence
   if (is_zero(p, EPSTOL) && is_zero(m1, EPSTOL) && is_zero(m2, EPSTOL)) {
      return 0.0;
   }
 
   // p is not 0
   if (p > 1.0e-5 * m2) {
      const double m12 = sqr(m1), m22 = sqr(m2), p2 = sqr(p);
      const double s = p2 - m22 + m12;
      const std::complex<double> imin(m12, -EPSTOL);
      const std::complex<double> x = std::sqrt(sqr(s) - 4.0 * p2 * imin);
      const std::complex<double> xp  = (s + sign(s)*x) / (2*p2);
      const std::complex<double> xm = imin / (xp*p2);
 
      return -2.0*std::log(p/q) - fB(xp, xm);
   }
 
   if (is_close(m1, m2, EPSTOL)) {
      return -2.0*std::log(m1/q);
   }
 
   if (m1 < 1.0e-15) {
      return 1.0 - 2.0*std::log(m2/q);
   }
 
   const double m12 = sqr(m1), m22 = sqr(m2);
 
   return 1.0 - 2.0 * std::log(m2/q)
        + 2.0 * m12 * std::log(m2/m1) / (m12 - m22);
}
 
double db0(double /* s */, double x, double y) noexcept
{
   if ((std::abs(x) < 0.0001) != (std::abs(y) < 0.0001)) {
      return (x + y)/(2*sqr(x - y));
   } else if ((std::abs(x) < 0.0001) && (std::abs(y) < 0.0001)) {
      return 0;
   } else if (std::abs(y - x) < 0.001) {
      return (1./12*(1 - y/x) + 1./6)/x;
   }
   return ((x - y)*(x + y) + 2*x*y*std::log(y/x))/(2*pow3(x - y));
}
 
double b1(double p, double m1, double m2, double q) noexcept
{
#ifdef USE_LOOPTOOLS
   setmudim(q*q);
   double b1l = -B1(p*p, m1*m1, m2*m2).real();
   //    return b1l;
#endif
 
   // protect against infrared divergence
   if (is_zero(p, EPSTOL) && is_zero(m1, EPSTOL) && is_zero(m2, EPSTOL))
      return 0.0;
 
   const double p2 = sqr(p), m12 = sqr(m1), m22 = sqr(m2), q2 = sqr(q);
 
   const double pTolerance = 1.0e-4;
 
   if (p2 > pTolerance * std::max(m12, m22)) {
      return (a0(m2, q) - a0(m1, q) + (p2 + m12 - m22)
              * b0(p, m1, m2, q)) / (2.0 * p2);
   }
 
   if (std::abs(m1) > 1.0e-15 && std::abs(m2) > 1.0e-15) {
      const double m14 = sqr(m12), m24 = sqr(m22);
      const double m16 = m12*m14 , m26 = m22*m24;
      const double m18 = sqr(m14), m28 = sqr(m24);
      const double p4 = sqr(p2);
      const double diff = m12 - m22;
 
      if (std::abs(diff) < pTolerance * std::max(m12, m22)) {
         return
            - 0.5*std::log(m22/q2)
            + 1.0/12.0*p2/m22 + 1.0/120.0*p4/m24
            + diff*(-1.0/6.0/m22 - 1.0/30.0*p2/m24 - 1.0/140.0*p4/m26)
            + sqr(diff)*(1.0/24.0/m24 + 1.0/60.0*p2/m26 + 3.0/560.0*p4/m28);
      }
 
      const double l12 = std::log(m12/m22);
 
      return (3*m14 - 4*m12*m22 + m24 - 2*m14*l12)/(4.*sqr(diff))
         + (p2*(4*pow3(diff)*
                (2*m14 + 5*m12*m22 - m24) +
                (3*m18 + 44*m16*m22 - 36*m14*m24 - 12*m12*m26 + m28)*p2
                - 12*m14*m22*(2*sqr(diff) + (2*m12 + 3*m22)*p2)*l12))/
         (24.*pow6(diff)) - 0.5*std::log(m22/q2);
   }
 
   return (m12 > m22)
      ? -0.5*std::log(m12/q2) + 0.75
      : -0.5*std::log(m22/q2) + 0.25;
}
 
double b22(double p,  double m1, double m2, double q) noexcept
{
#ifdef USE_LOOPTOOLS
   setmudim(q*q);
   double b22l = B00(p*p, m1*m1, m2*m2).real();
#endif
 
   p = std::abs(p);
   m1 = std::abs(m1);
   m2 = std::abs(m2);
   q = std::abs(q);
 
   // protect against infrared divergence
   if (is_zero(p, EPSTOL) && is_zero(m1, EPSTOL) && is_zero(m2, EPSTOL))
      return 0.0;
 
   const double p2 = sqr(p), m12 = sqr(m1), m22 = sqr(m2);
   const double pTolerance = 1.0e-10;
 
   if (p2 < pTolerance * std::max(m12, m22)) {
      // m1 == m2 with good accuracy
      if (is_close(m1, m2, EPSTOL)) {
         return -m12 * std::log(m1/q) + m12 * 0.5;
      }
      // p == 0 limit
      if (m1 > EPSTOL && m2 > EPSTOL) {
         return 0.375 * (m12 + m22) - 0.5 *
            (sqr(m22) * std::log(m2/q) -
             sqr(m12) * std::log(m1/q)) / (m22 - m12);
      }
      return (m1 < EPSTOL)
         ? 0.375 * m22 - 0.5 * m22 * std::log(m2/q)
         : 0.375 * m12 - 0.5 * m12 * std::log(m1/q);
   }
 
   const double b0Save = b0(p, m1, m2, q);
   const double a01 = a0(m1, q);
   const double a02 = a0(m2, q);
 
   return 1.0 / 6.0 *
      (0.5 * (a01 + a02) + (m12 + m22 - 0.5 * p2)
       * b0Save + (m22 - m12) / (2.0 * p2) *
       (a02 - a01 - (m22 - m12) * b0Save) +
       m12 + m22 - p2 / 3.0);
}
 
double d0(double m1, double m2, double m3, double m4) noexcept
{
   const double m1sq = sqr(m1);
   const double m2sq = sqr(m2);
 
   if (is_close(m1, m2, EPSTOL)) {
      const double m3sq = sqr(m3);
      const double m4sq = sqr(m4);
 
      if (is_zero(m2, EPSTOL)) {
         // d0 is undefined for m1 == m2 == 0
         return 0.;
      } else if (is_zero(m3, EPSTOL)) {
         return (-m2sq + m4sq - m2sq * std::log(m4sq/m2sq))/
            sqr(m2 * m2sq - m2 * m4sq);
      } else if (is_zero(m4, EPSTOL)) {
         return (-m2sq + m3sq - m2sq * std::log(m3sq/m2sq))/
            sqr(m2 * m2sq - m2 * m3sq);
      } else if (is_close(m2, m3, EPSTOL) && is_close(m2, m4, EPSTOL)) {
         return 1.0 / (6.0 * sqr(m2sq));
      } else if (is_close(m2, m3, EPSTOL)) {
         return (sqr(m2sq) - sqr(m4sq) + 2.0 * m4sq * m2sq * std::log(m4sq / m2sq)) /
            (2.0 * m2sq * sqr(m2sq - m4sq) * (m2sq - m4sq));
      } else if (is_close(m2, m4, EPSTOL)) {
         return (sqr(m2sq) - sqr(m3sq) + 2.0 * m3sq * m2sq * std::log(m3sq / m2sq)) /
            (2.0 * m2sq * sqr(m2sq - m3sq) * (m2sq - m3sq));
      } else if (is_close(m3, m4, EPSTOL)) {
         return -1.0 / sqr(m2sq - m3sq) *
            ((m2sq + m3sq) / (m2sq - m3sq) * std::log(m3sq / m2sq) + 2.0);
      }
 
      return
         (m4sq / sqr(m2sq - m4sq) * std::log(m4sq / m2sq) +
          m4sq / (m2sq * (m2sq - m4sq)) -
          m3sq / sqr(m2sq - m3sq) * std::log(m3sq / m2sq) -
          m3sq / (m2sq * (m2sq - m3sq))) / (m3sq - m4sq);
   }
   return (c0(m1, m3, m4) - c0(m2, m3, m4)) / (m1sq - m2sq);
}
 
double d27(double m1, double m2, double m3, double m4) noexcept
{
   if (is_close(m1, m2, EPSTOL))
      m1 += TOL * 0.01;
 
   const double m12 = sqr(m1), m22 = sqr(m2);
 
   return (m12 * c0(m1, m3, m4) - m22 * c0(m2, m3, m4))
      / (4.0 * (m12 - m22));
}
 
double c0(double m1, double m2, double m3) noexcept
{
#ifdef USE_LOOPTOOLS
   double q = 100.;
   setmudim(q*q);
   double psq = 0.;
   double c0l = C0(psq, psq, psq, m1*m1, m2*m2, m3*m3).real();
#endif
 
   const double m12 = sqr(m1), m22 = sqr(m2), m32 = sqr(m3);
   const bool m1_is_zero = is_zero(m1, EPSTOL);
   const bool m2_is_zero = is_zero(m2, EPSTOL);
   const bool m3_is_zero = is_zero(m3, EPSTOL);
 
   if ((m1_is_zero && m2_is_zero && m3_is_zero) ||
       (m2_is_zero && m3_is_zero) ||
       (m1_is_zero && m3_is_zero) ||
       (m1_is_zero && m2_is_zero)) {
      return 0.;
   }
 
   if (m1_is_zero) {
      if (is_close(m2,m3,EPSTOL))
         return -1./m22;
      return std::log(m32/m22)/(m22 - m32);
   }
 
   if (m2_is_zero) {
      if (is_close(m1,m3,EPSTOL))
         return -1./m12;
      return std::log(m32/m12)/(m12 - m32);
   }
 
   if (m3_is_zero) {
      if (is_close(m1,m2,EPSTOL))
         return -1./m12;
      return std::log(m22/m12)/(m12 - m22);
   }
 
   if (is_close(m2, m3, EPSTOL)) {
      if (is_close(m1, m2, EPSTOL))
         return - 0.5 / m22;
      return m12 / sqr(m12-m22) * std::log(m22/m12) + 1.0 / (m12 - m22);
   }
 
   if (is_close(m1, m2, EPSTOL)) {
      return - (1.0 + m32 / (m22-m32) * std::log(m32/m22)) / (m22-m32);
   }
 
   if (is_close(m1, m3, EPSTOL)) {
      return - (1.0 + m22 / (m32-m22) * std::log(m22/m32)) / (m32-m22);
   }
 
   return (m22 / (m12 - m22) * std::log(m22 / m12) -
           m32 / (m12 - m32) * std::log(m32 / m12)) / (m22 - m32);
}
 
double d1_b0(double /* p2 */, double m2a, double m2b) noexcept
{
   m2a = std::abs(m2a);
   m2b = std::abs(m2b);
 
   if ((m2a < 0.0001) != (m2b < 0.0001)) {
      return (m2a - m2b) * (m2a + m2b) / (2 * pow3(m2a - m2b));
   } else if (m2a < 0.0001 && m2b < 0.0001) {
      return 0.;
   } else if (std::abs(m2b - m2a) < 0.001) {
      return 1. / (6 * m2a) + (m2a - m2b) / (12 * sqr(m2a));
   }
 
   return ((m2a - m2b) * (m2a + m2b) + 2 * m2a * m2b * std::log(m2b / m2a)) /
          (2 * pow3(m2a - m2b));
}
 
double c00(double m1, double m2, double m3, double q) noexcept
{
  // taken from Package X
  const double m12 = sqr(m1), m22 = sqr(m2), m32 = sqr(m3), q2 = sqr(q);
 
  double ans = 0.;
 
  if (is_close(m1, 0., EPSTOL) && is_close(m2, 0., EPSTOL)
      && is_close(m3, 0., EPSTOL)) {
      // IR singularity
     ans = 0.;
  } else if (is_close(m2, 0., EPSTOL) && is_close(m3, 0., EPSTOL)) {
     ans = 3./8. + 1./4*std::log(q2/m12);
  } else if (is_close(m1, 0., EPSTOL) && is_close(m3, 0., EPSTOL)) {
     ans = 3./8. + 1./4*std::log(q2/m22);
  } else if (is_close(m1, 0., EPSTOL) && is_close(m2, 0., EPSTOL)) {
     ans = 3./8. + 1./4*std::log(q2/m32);
  } else if (is_close(m1, 0., EPSTOL)) {
     if (is_close(m2, m3, EPSTOL)) {
        ans = 1./8 + 1./4*std::log(q2/m22);
     } else {
        ans = 3./8 - m22*std::log(m22/m32)/(4*(m22-m32)) + 1./4*std::log(q2/m32);
     }
  } else if (is_close(m2, 0., EPSTOL)) {
     if (is_close(m1, m3, EPSTOL)) {
        ans = 1./8 + 1./4*std::log(q2/m12);
     } else {
        ans = 3./8 - m12*std::log(m12/m32)/(4*(m12-m32)) + 1./4*std::log(q2/m32);
     }
  } else if (is_close(m3, 0., EPSTOL)) {
     if (is_close(m1, m2, EPSTOL)) {
        ans = 1./8 + 1./4*std::log(q2/m12);
     } else {
        ans = 3./8 - m22*std::log(m12/m22)/(4*(m12-m22)) + 1./4*std::log(q2/m12);
     }
  } else if (is_close(m2, m3, EPSTOL)) {
    if (is_close(m1, m2, EPSTOL)) {
        ans = 1./4*std::log(q2/m12);
    } else {
        ans = (3*m12-m22)/(8*(m12-m22))
            - 1./4*(sqr(m12)*std::log(m12/m22))/sqr(m12-m22) + 1./4*std::log(q2/m22);
    }
  } else if (is_close(m1, m2, EPSTOL)) {
     ans = (3*m32-m12)/(8*(m32-m12)) - 1./4*(sqr(m32)*std::log(m32/m12))/sqr(m32-m12)
         + 1./4*std::log(q2/m12);
  } else if (is_close(m1, m3, EPSTOL)) {
     ans = (3*m22-m12)/(8*(m22-m12)) - 1./4*(sqr(m22)*std::log(m22/m12))/sqr(m22-m12)
         + 1./4*std::log(q2/m12);
  } else {
     ans = 3./8 - 1./4*sqr(m12)*std::log(m12/m32)/((m12-m22)*(m12-m32))
         - 1./4*sqr(m22)*std::log(m22/m32)/((m22-m12)*(m22-m32)) + 1./4*std::log(q2/m32);
  }
 
  return ans;
}
 
} // namespace softsusy